E LECTRON TRANSPORT THROUGH SINGLE
MAGNETIC MOLECULES
E LECTRON TRANSPORT THROUGH SINGLE
MAGNETIC MOLECULES
Proefschrift
ter verkrijging van de graad van doctor
aan de Technische Universiteit Delft,
op gezag van de Rector Magnificus prof. ir. K. C. A. M. Luyben,
voorzitter van het College voor Promoties,
in het openbaar te verdedigen op woensdag 28 maart 2012 om 10:00 uur
door
Alexander Sergeyevich Z YAZIN
Physicist, M.V. Lomonosov Moscow State University, Moskou, Rusland
geboren te Obninsk, Sovjet-Unie.
Dit proefschrift is goedgekeurd door de promotor:
Prof. dr. ir. H. S. J. van der Zant
Samenstelling van de promotiecommissie:
Rector Magnificus
Prof. dr. ir. H. S. J. van der Zant
Prof. dr. K. Park
Prof. dr. A. Cornia
Prof. dr. ir. M. R. Wegewijs
Prof. dr. ir. T. M. Klapwijk
Prof. dr. ir. L. M. K. Vandersypen
Dr. J. M. Thijssen
voorzitter
Technische Universiteit Delft, promotor
Virginia Tech, Verenigde Staten
University of Modena and Reggio Emilia,
Italië
RWTH Aachen, Duitsland
Technische Universiteit Delft
Technische Universiteit Delft
Technische Universiteit Delft
Keywords: single-molecule electronics, three-terminal transport, molecular spintronics, magnetic anisotropy, spin excitations.
Printed by: Gildeprint Drukkerijen - Enschede
Cover: Photograph of the experimental setup emphasizing rotating sample stage
used in experiments described in chapter 6.
Cover design and photo: A. S. Zyazin
Copyright © 2012 by A. S. Zyazin
Casimir PhD Series, Delft-Leiden, 2012-8
ISBN 978-90-8593-119-5
An electronic version of this dissertation is available at
.
Памяти моей мамы
C ONTENTS
1 Introduction
1.1 Motivation . . . . . . .
1.2 Molecular electronics
1.3 Molecular spintronics
1.4 Thesis outline . . . . .
References . . . . . . . . . .
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1
2
2
7
8
9
2 Theoretical background
2.1 Three-terminal transport through molecular junctions . . . . . . . . .
2.1.1 Single molecule as a quantum dot . . . . . . . . . . . . . . . . .
2.1.2 Weak coupling: Coulomb blockade and sequential tunneling .
2.1.3 Intermediate coupling: elastic and inelastic cotunneling and
Kondo effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.4 The role of spin . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Single-molecule magnets . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.1 Superparamagnetism . . . . . . . . . . . . . . . . . . . . . . . .
2.2.2 Spin Hamiltonian: zero-field splitting and quantum tunneling
of magnetization . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.3 Beyond the spin Hamiltonian: single-ion levels and interaction
2.2.4 An example of SMM: Fe4 . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17
18
18
20
3 Device fabrication and measurement setup
3.1 Experimental setup . . . . . . . . . . . . . . . . . .
3.1.1 Low-temperature insert . . . . . . . . . . .
3.1.2 Sample rotator . . . . . . . . . . . . . . . .
3.1.3 Measurement electronics . . . . . . . . . .
3.2 Device fabrication . . . . . . . . . . . . . . . . . .
3.2.1 Electron-beam lithography . . . . . . . . .
3.2.2 Molecule deposition and electromigration
3.3 Measurements technique . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . .
49
50
50
51
53
54
54
55
56
57
vii
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24
29
32
33
34
38
40
42
viii
C ONTENTS
4 Magnetic anisotropy in multiple charge states of an Fe4 molecule
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Experimental details . . . . . . . . . . . . . . . . . . . . . . . . .
4.3 Fitting to a model . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.5 Supplementary information . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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59
60
61
65
66
67
79
5 High-spin Kondo effect and spin transitions
5.1 Higher-energy excitations . . . . . . . . .
5.2 Spin transition calculations . . . . . . . .
5.3 High-spin Kondo effect . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . .
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83
84
87
93
97
6 Gate-voltage spectroscopy
6.1 Measurement and analysis technique
6.2 Results and discussion . . . . . . . . .
6.3 Numerical calculations . . . . . . . . .
6.4 Conclusion . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . .
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99
100
101
105
108
108
7 Spin blockade in a single-molecule junction
7.1 Introduction . . . . . . . . . . . . . . . . .
7.2 Measurements . . . . . . . . . . . . . . . .
7.3 Model calculations . . . . . . . . . . . . .
7.4 Discussion . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . .
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111
112
112
117
121
122
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Summary
125
Samenvatting
129
Curriculum Vitae
133
List of Publications
135
Acknowledgements
137
1
I NTRODUCTION
1
2
1. I NTRODUCTION
1.1 M OTIVATION
T
he idea of using single organic molecules as building blocks for functional electronic devices was first proposed in 1974 by Aviram and Ratner [1]. They suggested a molecular system consisting of a π-conjugated acceptor (tetracyanoquinodimethane, known in literature as TCNQ) and a π-conjugated donor (tetrathiofulvalene or TTF) separated by a σ-bonded (triple methylene) tunneling bridge. In
this molecule the TCNQ subunit is electron poor, TTF is electron rich, and the purpose of methylene is to prevent the π-levels of TTF and TCNQ to interact strongly.
This structure thus roughly resembles the structure of a conventional semiconductor p-n diode [2] with TTF corresponding to the n-doped region, TCNQ to the
p-doped region and methylene to the depletion layer. According to Aviram and
Ratner’s calculations the response of such a molecule to an applied voltage shows
rectifier properties.
It took, however, a long time before the technology development allowed this
proposal to be verified experimentally. The obvious challenge is to establish robust
electrical contacts to a single molecule, which is a nanometer-sized object. It must
be said that although significant progress in this area has been made during the
last two or three decades, this is still an open problem. A variety of techniques described in the next section have been suggested and tested in the research laboratories. Some of them are indeed viable for studying properties of single molecules
on chip, but industry standards of yield and reproducibility are still out of reach;
commercial single-molecule devices will therefore not appear in the market in the
near future. Nevertheless, several fundamental questions of molecular electronics
can already be posited. How do the properties of materials change when single
molecules are isolated on a surface and attached to the electrodes? How can mechanical, optical or magnetic properties of single molecules affect electron transport and how can electron transport affect these properties? What functionality
can these effects offer? Can we, for instance, store and process information in the
single molecules? The work presented in this thesis aims to address some of these
questions, in particular relating to the magnetic molecular properties.
1.2 M OLECULAR ELECTRONICS
To study electron transport through a single molecule, one first has to fabricate
electrical contacts to it. Given the sizes of the molecules which are typically several
nanometers, it is necessary to produce electrodes with nanometer-scale separation. This is far beyond the capabilities of conventional mass-production technologies. As of late 2011 the state of the art deep ultraviolet lithography, used in semiconductor industry, results in a 22 nm node [3]. Further reducing critical dimensions requires a transition to extreme ultraviolet lithography [4], which involves
1.2. M OLECULAR ELECTRONICS
3
major technological challenges, requires extremely expensive equipment and still
cannot achieve a resolution below 10 nm. Conventional electron-beam lithography, widely used in the research labs for nanostructure fabrication, is also limited
by 10-15 nm due to electron scattering and proximity effects [5]. Hence, it is not
surprising that the history of molecular electronics is to a large extent the history
of contact fabrication techniques.
The idea of contacting single molecules became feasible after the invention of
the scanning tunneling microscope (STM) in the beginning of 1980’s [6]. The STM
is based on a tunneling contact between an atomically sharp probe and a conducting sample. It can work in two regimes. In order to image the sample surface
a constant tunneling current between the tip and the sample is maintained by a
feedback loop during the scanning of the surface. In the other regime the feedback
loop is switched off and a current-voltage characteristics is recorded at a given spot
of the sample surface. If the metallic substrate is covered by a submonolayer of
organic molecules, the tip can be placed on top of a molecule and electrical measurements of the current from the tip through the molecule to the surface can be
performed [7]. By scanning the surface with a feedback control on, an image of the
same molecule can be acquired; this combined inspection is the main advantage
of this technique. Apart from conductance measurements the STM can be used
as a tool for inelastic tunneling spectroscopy (IETS) [8], where features of d I /dV curves can be assigned to molecular excitations of vibrational [9] or electronic [10]
origin.
The junctions obtained using this approach are inherently asymmetric: the
coupling of the molecule to a metallic surface is generally much higher than to the
tip. Furthermore, the strong coupling to the surface leads to a broadening and distortion of the molecular orbitals which modifies both imaging and electrical measurements. To reduce this effect a thin insulating layer, such as sodium chloride can
be introduced between the molecules and the conductive substrate [11]. This technique allows imaging of individual orbitals and improves the resolution of IETS.
Spin excitations can then be also resolved in IETS as the insulating layer protects
the d -electron states from coupling to the metallic substrate [12].
Another application of a scanning probe technique, the STM break junction,
first used by Xu and Tao [13], involves the repeated breaking and formation of a
metallic contact between tip and substrate in the presence of molecules in solution.
The molecules, with a certain probability, can form a conducting bridge between
the metallic surface and the tip when the tip is retracted. The current is recorded
at a fixed bias voltage during the retraction and stable conductance values can be
attributed to the molecule. This approach is particularly interesting because the
molecule is attached to both electrodes (the tip and the surface). Using this technique the conductance as a function of the length of the polymer molecular wire
4
1. I NTRODUCTION
has been investigated [14]. The current can also be measured as a function of the
bias voltage at a fixed length of the junction.
The combination of imaging capabilities and electrical measurements together
with the possibility of in situ deposition of the molecules in ultra high vacuum
(UHV) makes the STM a powerful tool with a very high degree of control on the
measured structures. Remarkably, it can be used even for experiments with single
atoms [15–17]. However, the junctions formed by traditional STM approach are inevitably asymmetric, the stability of the STM break junctions is quite limited and
overall, STM-based devices are not scalable: it is difficult to form more than one
operational molecular junction on a sample using STM. From this point of view
lithography-based approaches, where junctions form a planar structure on a chip
are more favourable.
Several approaches have been followed: mechanically controllable break junctions (MCBJ) [18–21], electromigration [22–27], electrochemical approaches [28–
30], shadow evaporation [31–34], ’molecular lithography’, using self-assembled
monolayers of organic molecules as a mask [35, 36] and ’self-aligned lithography’,
using a thick chromium-oxide shadow mask [37–39], to name a few. All of them
have their advantages and drawbacks. Here, we will only discuss MCBJ, shadow
evaporation and electromigration approaches, which proved to be most suitable
for fabrication of single-molecule junctions.
The MCBJ technique is built on a similar principle as the STM break junction
discussed earlier. It stems from experiments with notched metallic wires glued on
a flexible substrate, bending of which is translated to the stretching of the wire. In
their 1992 experiment Muller et al. [40] showed that breaking of the wire caused by
such a stretching can lead to a reversible transition from a metallic conductance
to a tunneling junction. In 1997 Reed et al. [18] performed a similar experiment,
but in the presence of a 1 mM solution of benzene-1,4-dithiol molecules. After
evaporation of the solution in an inert atmosphere, they found a gap of 0.7 V in
the I (V ) curves, which was interpreted as either a Coulomb blockade signature or
a mismatch between the contact Fermi level and the lowest unoccupied molecular
orbital (LUMO) of the molecule. The key factor determining the stability of the contact or of the tunneling gap is the large reduction factor between the elongation of
the piezo element pushing the substrate and electrode separation. In Reed’s experiment the spacing between two parts of the wire, set by the piezo voltage, was 8 Å,
while the length of the molecule is 8.46 Å. In lithographically defined MCBJ devices,
where the two metal electrodes are suspended over a micron-sized trench, the ratio of the electrode separation to the piezo elongation can be as low as 10−5 [41],
leading to a subAngstrom control of the gap size.
If the junctions are broken in a cryogenic vacuum, stable atomically clean metal
point contacts result. This makes MCBJ devices an ideal testbed for experiments,
1.2. M OLECULAR ELECTRONICS
5
where simple organic molecules (typically benzene derivatives) are functionalized
with different anchoring groups. The results of these tests are very important as
the molecules of interest must form strong and electronically transparent chemical bonds to the electrodes in order to obtain stable electronic devices. Thiol
(SH) [21, 42], amine (NH2 ) [21] and fullerene (C60 ) [43] groups have been employed
among others. Although similar experiments can also be done with STM break
junctions [44, 45], the results can differ for two techniques. For instance, it was
found that while the thiol groups resulted in a spread of molecular conductance
in the STM setup [45], this effect was negligible in MCBJ experiments [21, 42]. The
spread results from the deformation of electrodes caused by the fact that the sulfurgold bond is stronger than the gold-gold bond [46]. In contrast to STM setup such
a deformation is negligible in the MCBJ setup because the speed of electrodes retraction is typically two orders of magnitude smaller in MCBJ.
STM and MCBJ meausurements are typically two-terminal, in which a
molecule bridges a gap between two electrodes (source and drain). Introduction of
a third electrode, a gate, allows studies in which the molecule can be oxidized and
reduced: molecular levels can be brought into and out of resonance with Fermi levels of the electrodes and transport through excited and different charge states can
be probed [47]. Although a side gate was introduced into an STM setup and signatures of Coulomb blockade were measured in transport through single carboran
clusters as early as in 1996 [48], these attempts did not receive further development. A gate electrode has also been introduced in the MCBJ setup [20, 49]. However, these devices suffer from a low gate coupling rendering gating inefficient in a
reasonable voltage range. Moreover, it was found that the gate voltage can induce
mechanical movement of the source and drain, complicating interpretation of the
data [50].
One of the techniques allowing efficient gating of molecular transport is
shadow evaporation [31–34]. Gold source and drain electrtodes are evaporated in
UHV conditions onto a tilted substrate (which can be a surface of oxidized gate)
through a suspended mask. If the tilt angle is high, there is no overlap between
source and drain. Reducing the angle decreases the size of the gap. The separation
is controlled through an in situ measurement of the tunneling conductance. After
the desired gap is obtained, molecules of interest can be deposited into the gap
by quench condensation without breaking vacuum. This approach results in clean
devices with an efficient gate coupling. However, quench condensation limits the
choice of molecules to those with high vapor pressures, like oligophenylenevynilene (OPV) derivatives [33, 34].
To date most three-terminal measurements are performed using the electromigration approach [22–27]. In this approach a thin metallic, typically gold, wire,
lithographically defined on top of an oxidized gate, is broken by a current sent
6
1. I NTRODUCTION
through it [22]. The mechanism, responsible for the breaking is the momentum
transfer between conducting electrons and diffusing metal atoms. Electromigration can be done in the solution containing the molecules, so that the self-assembly
takes place during the breaking. To reduce the risk of violent, abrupt breaking,
feedback schemes, monitoring the resistance of the wire in situ, have been introduced [24]. Electromigration has several advantages: an efficient gate can be easily
fabricated, it is fairly simple compared to the shadow evaporation method, it does
not limit the choice of molecules, and the number of junctions that can be fabricated on one sample is limited mainly by its surface area. Due to these advantages,
the electromigration method has been widely used, however it was found that it
can result in the formation of gold nanoparticles in the vicintity of the source-drain
gap. These nanoparticles can exhibit Coulomb blockade and the Kondo effect [51],
mimicking molecular signatures. Data analysis in principle allows discrimination
between samples containing molecules and gold nanoparticles: the latter typically
reside on the oxide surface, thus having much stronger gate coupling compared
to the ones with a molecule, because the molecule is attached to the source and
drain, which partially screen the gate electric field [52]. It was also discovered that
electromigration, stopped before the transition to tunneling transport (i.e. a wire
with a resistance lower than R 0 = 1/G 0 = h/2e 2 = 12.9 kOhm), induces further selfbreaking of the wire [26]. Such a self-breaking prevents nanoparticle formation.
Despite the extensive studies to improve its performance, electromigration is
still plagued by several drawbacks. First, it is impossible to control the geometry
and separation of the electrodes on the atomic scale, while these are key factors determining electronic coupling and thus the conductance of the junction [53]. Second, the yield of electromigration is low: only a few percent of the junctions show
spectroscopic molecular signatures. Third, the stability of gold electromigrated
junctions is limited: due to the high mobility of gold atoms (which eases electromigration), junctions go through irreversible changes at temperatures higher
than about 200 K. The high mobility at high temperatures makes it impossible to
perform room temperature measurements or repeated measurements of the same
sample after the sample is warmed up. Recently, however, it was found that the use
of platinum instead of gold increases stability significantly both at low and room
temperatures [27].
Nevertheless, electromigration remains the technique of choice for fabrication
of three-terminal single-molecule devices, and it is used, for example, for the fabrication of all samples discussed in this thesis. Very recently, a similar principle
(breaking with a large current) has been applied to fabrication of graphene electrodes [54], which is a promising direction in molecular electronics. The difference between the two techniques is that in case of graphene, the mechanism of the
breaking is Joule heating in combination with burning, as the presence of oxygen
1.3. M OLECULAR SPINTRONICS
7
is crucial for formation of the gap.
1.3 M OLECULAR SPINTRONICS
The field of spintronics, studying active control and manipulation of spin degrees
of freedom in solid-state systems, is an established fundamental discipline that
found its way to commercial applications in an amazingly short time and holds
great expectations for the future [55, 56]. The field of molecular spintronics [57, 58]
emerged recently when the development of molecular electronics made possible
creation of devices containing single molecules with magnetic properties [59, 60].
Possible applications of these devices include switching [61, 62], high-density data
storage [63] and quantum information processing [64, 65]. They can also be used
for exploration of a reach fundamental physics of the magnetic molecules: effects
like quantum tunneling of magnetization (QTM) [66] are expected to be observed
in the electron transport [67, 68]. The variety of molecules that can be employed
in these devices include single transition metal atom complexes, single-molecule
magnets (SMMs), spin crossover compounds and polyoxometalates(POMs). In this
section we briefly review major experimental results in this field.
The first experiments on electron transport through magnetic molecules were
reported in 2006 by groups from Delft [59] and from Harvard and Cornell [60]. In
these experiments three-terminal devices, containing an individual Mn12 SMM,
were fabricated using electromigration. The results however looked different.
In Harvard/Cornell experiment a characteristic zero-field splitting (ZFS) of the
ground spin multiplet, caused by the magnetic anisotropy, was observed in four
samples as excitations in single-electron tunneling and inelastic cotunneling. The
origin of these excitations was confirmed by the measurements in the magnetic
field. The value of ZFS varied from 0.25 to 1.34 meV for different samples. Each
sample showed two charge states, but ZFS was present only in one of them. In
Delft experiment no ZFS was found, however the data exhibited striking features as
negative differential conductance and complete current suppression. These effects
were explained by the model, taking into account mixing of the two lowest-lying
spin multiplets (S = 10 and S = 9 for neutral Mn12 molecule). Two years after the
publication of these data it was found that Mn12 degrades upon deposition onto a
gold surface [69]. The results of Heersche et al. and Jo et al., however, should not be
disproved by this finding as they showed magnetic nature of the observed effects,
and although the molecule could undergo some structural changes, the signatures
of non-zero spin and magnetic anisotropy were clear.
The next milestone result was published by Cornell group in 2008 [70]. They
fabricated three-terminal junctions with endofullerene molecule (N@C60 ) using
platinum electrodes. These molecules are highly symmetric and thus are not ex-
8
1. I NTRODUCTION
pected to possess magnetic anisotropy. Two charge states with several SET excitations were observed. Magnetic field data showed that there was a change of
one of the ground states, caused by level crossing at a field of about 6 T. This observation was used to determine the charge and the spin S values for each of the
states, starting with an assumption that neutral endofullerene molecule preserves
its spin S = 3/2 when deposited on a surface. They found that at zero gate voltage
the molecule was twicely reduced (carried two extra electrons) and one electron
could be removed by applying a negative voltage to the gate. In the twicely reduced
state the spin was S 2− = 3/2 and in another charge state at zero field the ground
state spin was S 1− = 1. At the field of ∼ 6 T the level crossing occured between
S 1− = 1 and S 1− = 2 charge states. Thus the data showed for the first time signatures
of spin-state transition in a single electron transport. Recently these observations
were confirmed by Roch et al. [71], who observed the same transitions in inelastic
cotunneling regime.
In 2010 a purely electrically-controlled transition from high-spin (S = 5/2) to
low-spin (S = 1/2) in a single Mn2+ coordination complex was reported [72]. In
this molecule a Mn2+ ion is coordinated by two terpydine ligands. The gate voltage
induced a reduction of a terpydine moiety which enhanced the ligand field on the
Mn ion. An SET transport between these two charge states was suppressed by a
spin blockade [73]. Interestingly, that sample also showed strongly gate-dependent
singlet-triplet splitting at a low-spin side. The gate-dependence was explained by a
difference in the gate coupling for two terpydine moieties, arising from asymmetric
nature of an electromigrated junction.
A different kind of device architecture was reported very recently by Urdampilleta et al. [74]. They presented a spin-valve device in which a gated single-walled
carbon nanotube contacted with palladium electrodes, is laterally coupled through
supramolecular interactions to TbPc2 SMMs, deposited on a surface. The localized
magnetic moments of the SMMs lead to a magnetic field dependence of the electron transport through the nanotube with the measured magnetoresistance values
reaching 300% at low bias voltage. The conductance is strongly anisotropic with
respect to magnetic field, reflecting anisotropy of TbPc2 molecules. This result is
particularly interesting because the spin valve operation is achieved without use
of magnetic electrodes, the fabrication of which proved to be extremely challenging [75–77].
1.4 T HESIS OUTLINE
The content of this thesis is outlined below:
Chapter 2 lays the theoretical foundation for the thesis. It starts with discussion
of the three-terminal transport through single molecules in two different regimes:
1.4. T HESIS OUTLINE
9
weak and intermediate coupling to the leads. Next, we discuss the interplay between spin properties and electron transport. The following section introduces
the physics of single-molecule magnets in the spin Hamiltonian approximation
and beyond this model when exchange interaction between constituent ions are
taken into account. The chapter finishes with the discussion of properties of an
Fe4 molecule in the bulk phase, so that these properties can be compared to the
properties of individual molecules studied in the following chapters.
Chapter 3 describes the experimental setup and device fabrication procedures.
Electromigration as the technique of establishing single-molecule contacts is discussed in detail. Furthermore, the chapter introduces the measurement methods
used in the experiments.
In chapter 4 the effect of electrical gating on the magnetic anisotropy of single
Fe4 molecules is studied. Characteristic zero-field splitting is observed in multiple charge states and anisotropy parameters are derived. It is found that reduction
and oxidation by the gate voltage lead to an enhancement of the anisotropy. Moreover, nonlinear Zeeman effect was observed originating from mismatch between
anisotropy axis and magnetic field direction. Orientation of the molecule with respect to the magnetic field is derived from the fitting to theoretical models.
Chapter 5 discusses higher energy excitations of the samples from the previous
chapter. Although the energy scale of these excitations corresponds to the energy
of spin transition in the bulk phase, their magnetic field behaviour differs from
the trivial Zeeman effect. Calculations of a transport through a model exchangecoupled anisotropic system shows that spin excitations appear in the transport
characteristics as a band consisting of several transitions in contrast to the zerofield splitting which appears as a single line. The magnetic field evolution of such
a band is found consistent with experimental observations. The chapter finishes
with a discussion of two different samples, in which a high-spin Kondo effect was
observed.
Chapter 6 presents experiments in which a sample is rotated in the magnetic
field. Gate-voltage spectroscopy is introduced as analysis method for determination of anisotropy parameters. This technique based on ground-state-to-groundstate transition observation is found useful when the zero-field splitting cannot be
observed. Using this analysis a change of the anisotropy axes orientation was found
after the sample rotation.
Chapter 7 concludes the thesis with a discussion of a spin blockade of a current through an Fe4 junction. The blockade in these measurements is slightly violated and the negative differential conductance (NDC) is observed. This situation
is modelled with a five-levels system. A microscopic mechanism of such a blockade
is suggested.
10
1. I NTRODUCTION
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2
T HEORETICAL BACKGROUND
17
18
2. T HEORETICAL BACKGROUND
2.1 T HREE - TERMINAL TRANSPORT THROUGH MOLECU LAR JUNCTIONS
2.1.1 S INGLE MOLECULE AS A QUANTUM DOT
S
ingle molecules, objects with nanometer scale dimensions, are essentially
quantum-mechanical systems. To get a feel of the energy scales we view them
for the time being as small spheres, containing a finite number of electrons (some
molecules, such as C60 have indeed a spherical shape). In classical electrostatics
charging an isolated metallic sphere with one electron requires an energy
EC = e 2 /2C = e 2 /8π²0 r,
(2.1)
where e is an electron charge, C is a capacitance of the sphere, ²0 is a dielectric
constant and r is a radius of the sphere. For a diameter 2r = 1 nm, this expression
gives a value EC ≈ 1.4 eV. Already at room temperature (T ∼ 300 K) this energy is
much larger than the energy of thermal fluctuations k B T ≈ 25 meV, so the number
of electrons, residing at a molecule is well defined, and its charge is quantized.
A spatial confinement of an electron is a well-known in quantum mechanics
problem of a particle-in-a-box [1]. The solution yields a discrete set of energy levels
that are available for electrons:
E nx n y nz =
2
2
π2 ħ2 n x2 n y n y
( 2 + 2 + 2 ),
2m L x L y L y
(2.2)
where L x,y,z are lateral dimensions of a system, n x,y,z are quantum numbers. For
2 2
π ħ
L x = L y = L z = 1 nm this gives a level separation ∆ ∼ 2mL
2 ≈ 0.4 eV. This energy quantization is a generic phenomenon: it has been also observed in metallic
nanoparticles and semiconductor heterostructures. In single molecules this spectrum is modified by interactions of electrons with nuclei that lead to a formation of
molecular orbitals with a level spacing specific for a particular kind of molecules.
Furthermore, the spectrum of the molecule trapped in between two metallic electrodes is modified compared to a free isolated molecule.
Source and drain electrodes are usually described as bulk metallic electron
reservoirs. The number of electrons in these reservoirs is variable as they are connected to a bias voltage source. The density of states in the electrodes follows the
Fermi-Dirac distribution:
1
f (E , µ) =
.
(2.3)
exp[(E − µ)/k B T ] + 1
Here, µ is the chemical potential of the electrode. At zero temperature and in the
absence of a bias voltage the chemical potential by definition equals the Fermi energy: µ = E F . The Fermi-Dirac distribution is then just a step function changing
2.1. T HREE - TERMINAL TRANSPORT THROUGH MOLECULAR JUNCTIONS
19
from 1 at energies E < µ to 0 at energies E > µ; all electronic states below Fermi
energy are filled and all states above this energy are empty.
Chemical potentials of the electrodes can be shifted by applying a bias voltage
Vb , according to
µS − µD = eVb ,
(2.4)
where µS (D) is the chemical potential of the source (drain) electrode. In all the
following discussion we assume that the source is grounded as is usually the case
in an experiment, i.e.,
VS = 0.
(2.5)
In this case µD = E F +eVb and µS = E F . The inequality between chemical potentials
leads to a difference in the population of the two electrodes and thus to a flow of
the electrons (electric current) from source to drain. This flow occurs via electron
tunneling. If a small island like a molecule or a nanoparticle is present in a gap betweeen two contacts, two tunneling events are needed: first, the electron tunnels
from the source to the molecule and second, it is transferred from the molecule to
the drain. The tunneling processes are fast and in most cases they can be considered elastic: the energy of electron is conserved during each single tunneling event.
The complete transfer of the electron from the source to the drain, however, can be
inelastic, with the energy partially absorbed by the molecule, as will be discussed
in the following sections.
The current depends both quantitavely and qualitatively on the electronic couplings of the molecule to the leads ΓS,D . This parameter characterizes the overlap
of the molecular wavefunction with that of the density of states in the contacts. It
can also be introduced as a tunneling rate:
ΓS,D =
ħ
.
τS,D
(2.6)
Here τS,D is an average time within which a single electron tunnels from the source
to the molecule/from the molecule to the drain. The total coupling is the sum of
couplings to the source and the drain: Γ = ΓS + ΓD . The coupling leads to a broadening of the molecular energy levels. The levels can also shift compared to an isolated molecule due to image charge formation in the contacts and electronic polarization [2–4]. These two effects have a strong influence on transport properties.
Three different regimes are possible: weak (Γ << EC , ∆), intermediate (Γ ∼ EC , ∆)
and strong (Γ >> EC , ∆) coupling. These regimes are discussed in detail in the following two sections.
20
2. T HEORETICAL BACKGROUND
2.1.2 W EAK COUPLING : C OULOMB BLOCKADE AND SEQUENTIAL
TUNNELING
When the levels broadening due to interaction with the leads is small (Γ << EC , ∆)
one is in the weak coupling limit. In this regime the transport properties are similar
to those of a single-electron transistor [5–7]. In these devices a tiny metallic or
superconducting island is separated from two electrodes by tunneling barriers of
a low transparency. The island is capacitevely coupled to a third electrode, gate,
that controls a potential of the island. In molecular devices the island is reduced to
a single molecule. The difference in sizes is reflected in the difference in spectra:
whereas level separation in the metallic single-electron transistors is typically very
small, in the molecules it is not and the separate levels can be distinguished in the
transport measurements at relatively high temperatures.
The total charge Q of the molecule is given by
Q = C S (V − VS ) +C D (V − VD ) +CG (V − VG ),
(2.7)
where V is the potential of the molecule and VS,D are the potentials of the source
and the drain, C S is a capacitance to the source, C D to the drain and CG to the gate.
A total capacitance C of the molecule in the three-terminal junction is
C = C S +C D +CG .
(2.8)
After substitution (2.8) to Eq. (2.7) we find a potential of the molecule
V=
Q
Cs
CD
CG
+ VS
+ VD
+ VG
.
C
C
C
C
(2.9)
The total charge is restricted to integer multiples of the electron charge: Q = Ne.
The energy of the system can then be written as a sum of the electrostatic work
of adding N electrons to the molecule and the single-particle energies E n of all
occupied molecular levels:
−Ne
Z
U (N ) =
V dq +
0
N
X
n=1
En =
N
(Ne)2
VS C S + VD C D + VG CG X
− Ne
+
En .
2C
C
n=1
(2.10)
The chemical potential is then by definition
µ(N ) ≡ U (N ) −U (N − 1) = (N − 1/2)
e2
VS C S + VD C D + VG CG
−e
+ EN .
C
C
(2.11)
This derivation is performed within so-called constant interaction model [8]: capacitance values and single-particle levels E n are assumed independent of the
2.1. T HREE - TERMINAL TRANSPORT THROUGH MOLECULAR JUNCTIONS
a
b
μ(N+1)
N-1
Vb
21
N
μ
N+1
μ(N)
μ(N)
Eadd/βe
S
Eadd
D
μ(N-1)
S
Eadd
D
μ(N-1)
VG
F IGURE 2.1: Single-electron transport in the weak coupling regime. (a) Typical d I /dVb versus Vb and
VG map (stability diagram or conductance map) in the absence of excitations. White lines represent
peaks in d I /dV due to levels brought into the bias window, in the grey areas d I /dVb = 0. (b) A sketch
of SET transport. Source and drain electrodes are represented with their densities of states. Tunneling
under the barriers between the electrodes and the molecule is only possible when there is a molecular
level present between the Fermi levels of the electrodes.
charge, which is not always a valid assumption for the molecular devices. However, the basic features of transport in the weak coupling regime are described by
this model very well and it can always be used as a starting point when analyzing
the measurements results.
The mechanism of transport is sketched in the Figure 2.1b. At low bias voltage
there is not enough energy to charge the molecule and the current is suppressed:
the molecule is in Coulomb blockade. The Fermi levels of the contacts (whose position is controlled by the bias voltage) determine the bias window. Transport is
only possible when there is a resonant level of the molecule present in this window. Indeed, the density of states in the source must be nonzero for the energy
of the level, otherwise there are no electrons in the source with that energy. The
density of states in the drain, however, must be zero to allow tunneling of an electron in, according to Pauli exclusion principle. By application of a gate voltage the
molecular levels can be shifted with respect to the contact Fermi levels and thus
be brought into the bias window. When the chemical potential of an empty charge
state is within the bias window the Coulomb blockade is lifted and current can be
measured. At zero bias this condition is
µN +1 (VG ) = µS = µD .
(2.12)
The point at the gate voltage where (2.12) is satisfied is called the degeneracy-point.
Figure 2.1a shows a typical transport measurements result represented as a
map that plots differential conductance (d I /dVb ) versus bias (Vb ) and gate (VG )
22
2. T HEORETICAL BACKGROUND
a
b
μ(N+1)
Vb
N-1
N
μ(N+1)
Eadd
Eexc/e
Eadd/βe
Eadd
Eexc
μ(N)
S
Eexc
μ(N)
D
μ(N-1)
S
D
μ(N-1)
VG
F IGURE 2.2: (a)Typical stability diagram with SET excitations present as slanted lines inside SET areas
parallel to the diamond edges. (b)A sketch of SET transport through an excited level. Source and drain
electrodes are represented with their densities of states. Tunneling through an excited state is only
possible when both excited and ground state levels are between the Fermi levels of the electrodes.
voltages (so-called stability diagram). This pattern represents a classical signature
of Coulomb blockade.Slanted lines enclose diamond-shaped regions with a zero
conductance (Coulomb diamonds). The number of electrons inside the Coulomb
diamonds is fixed and every diamond corresponds to a well-defined redox state
of the molecule. Going from a negative to a positive gate voltage, one lowers the
molecular levels, leading to a consecutive addition of electrons (reduction of the
molecule). Going from a positive to a negative gate voltage, the levels are shifted
up, electrons are subtracted and the molecule is oxidized. It must be noted, however, that it is not an easy task to tell the exact number of electrons residing on
the molecule for each single diamond. The zero gate voltage does not necessarily correspond to a neutral state: the molecule can be charged by the electrostatic
environment (for instance, defects in the substrate or gate oxide or from another
molecule residing in the vicinity of the junction) or due to the interaction with the
leads.
The Coulomb diamonds are separated by the regions of single-electron, or sequential, tunneling (SET), where the current is non-zero. The differential conductance is, however, zero (the current does not vary with the bias voltage) except at
the Coulomb diamond edges. The diamond edges mark the onset of SET when
the chemical potential of the next charge state matches the the potential of the
source (for positively inclined lines) or the drain (for negatively inclined lines):
µ(N )(VG ) = µS(D) (Vb ). At this point the electron can tunnel into the molecule or off
the molecule (see Figure 2.1b), switching the current on. Using (2.11) and (2.5) one
can find a general condition for the onset of SET. For µ(N ) = µS (diamond edges
2.1. T HREE - TERMINAL TRANSPORT THROUGH MOLECULAR JUNCTIONS
23
with positive slope):
VD =
1
(2N − 1)e E N C
(CG VG −
−
).
CG +C D
2
e
(2.13)
For µ(N ) = µD (diamond edges with negative slope):
VD = −
1
(2N − 1)e E N C
+
).
(CG VG +
CS
2
e
(2.14)
The slopes of the edges on the conductance map can be used to extract quantitative information about capacitances. As can be seen from the Eq. (2.13) and
(2.14) the positive slope of the edge is α+ = CG /(CG + C D ) and the negative slope
is α− = −CG /C S . The gate coupling parameter β ≡ CG /C = (1/α+ + |1/α− |) gives
the ratio between the applied gate voltage and the shift of the molecular levels
∆E : ∆E = βeVG . This expression can be used to determine the addition energy
E ad d ≡ µN + 1 − µN , required to add an electron to the molecule: E ad d = βe∆VG ,
where ∆VG is the distance between two consequtive degeneracy-points. Alternatively E ad d can be found as the distance from the zero-bias axis to the crossing
point of two consequtive diamond, multiplied by an electron charge (see also Figure 2.1a).
At the stability diagram shown in Figure 2.2a there are also lines of non-zero
differential conductance, parallel to the diamond edges and terminating on them.
These lines mark the onset at which molecular excited states start to contribute to
the transport. Here the chemical potential of the excitation matches the potential of the source (for positively inclined lines) or the drain (for negatively inclined
lines): µexc (VG ) = µS(D) (Vb ), and at this point the state becomes accesible for tunneling. This situation is shown in Figure 2.2b. The opening of a new tunneling
channel leads to an abrupt rise of the current and a peak in d I /dVb . Note that for
excitation to occur it is necessary to have also the ground state in the bias window.
That is why the excitation lines are terminating at the diamond edges.
For the SET region separating charge states with N and N + 1 electrons lines
terminating at the left edge of the region correspond to the excitations of the charge
state N and lines terminating at the right edge to the excitations of the state N + 1.
The energy of the excitation E exc can be read off as the distance from the zero-bias
axis to the crossing point of the ecitation line with a diamond edge (see Figure 2.2a).
When the energy spectrum of the molecule in the junction is given, a theoretical
description of the transport can be obtained by solving a master equation (also
called rate equation) for probabilities of each state’s occupation [9, 10]:
d P/d t = WP,
(2.15)
24
2. T HEORETICAL BACKGROUND
where P is the vector of the occupation probabilities of n levels P i , i = 1..n and W
is a n × n matrix, whose elements Wi j are rates for transitions j → i for i 6= j , and
P
Wi i = − W j i . These rates can be written as
j 6=i
1 S S
D
(Γ f + ΓD
i j f i j ),
ħ ij ij
(2.16)
1 S
D
(Γ (1 − f iSj ) + ΓD
i j (1 − f i j )).
ħ ij
(2.17)
Wji =
Wi j =
Here, ΓS(D)
are the bare rates (determined by the thicknesses of the tunneling barriij
ers) for electron tunneling from the molecule in state j into the source (drain), leaving the molecule in state i , µi j are electrochemical potentials of the corresponding
transitions and f iS(D)
are the Fermi functions:
j
f iSj =
1
,
1 + exp((µi j + V /2)/k B T )
(2.18)
1
.
(2.19)
1 + exp((µi j − V /2)/k B T )
P
The steady-state current can then be found as I = e Wi j P i . In the simplest
f iDj =
i,j
case all the bare rates are equal to electronic couplings to the leads: ΓS,D
= ΓS,D .
ij
The current through one level then reads
I=
e ΓS ΓD
.
ħ ΓS + ΓD
(2.20)
One consequence of the last equation is that Γ can be estimated as the full width
at half maximum (FWHM) of the conductance peak at the diamond edge when
plotted as d I /dVb versus Vb .
2.1.3 I NTERMEDIATE COUPLING : ELASTIC AND INELASTIC COTUN NELING AND KONDO EFFECT
The resonant sequential tunnelling, described in the previous section, is a firstorder process in Γ. When the electronic coupling is comparable to the levels spacing or the temperature (Γ ∼ ∆, k B T ), not only the SET current rises as can be expected from Eq. (2.20), but also higher-order tunneling processes become important. In particular, they lead to a detectable current inside the Coulomb blockade
regions. In this section we will discuss three different higher-order processes: elastic cotunneling, inelastic cotunneling and the Kondo effect.
2.1. T HREE - TERMINAL TRANSPORT THROUGH MOLECULAR JUNCTIONS
S
D
S
25
D
F IGURE 2.3: Elastic cotunneling.
Figure 2.3 shows schematically an elastic cotunneling process. There are no
states availanble in the bias window and transport is forbidden in the first order.
However, Heisenberg’s energy-time uncertainty principle allows an electron to tunnel on the molecule from the source, as long as it tunnels off to the drain or back
to the source within a very short time period ∆t . ħ/(EC + ∆). This virtual process
conserves an energy and the molecule is in the same state at the end of the process
as it was in the beginning. Since this process is elastic, it can occur at arbitrarily
low bias voltage and leads to a constant nonzero background conductance in the
Coulomb blockade regions.
To estimate the rate of elastic cotunneling Γel note first that tunneling from the
molecule to the drain is energetically favourable. The chance for it to occur within
the time ħ/(EC + ∆) is given by ΓD ħ/(EC + ∆). Taking into account the tunneling
rate from source to molecule ΓS we get
Γel ' ΓS ΓD ħ/(EC + ∆).
(2.21)
For Γ << EC + ∆, which is the requirement of the weak coupling regime, Γel << Γ
and elastic cotunneling is negligible. However, it gives a substantial contribution to
the current when this condition is not fulfilled. Note that Eq. (2.21) is only a rough
estimation of the elastic cotunneling rate. A rigorous derivation can be found, for
instance, in Ref. [11].
Inelastic cotunneling is a virtual tunneling proccess leaving a molecule in an excited state. It is schematically shown in Figure 2.4. An electron from the molecule
tunnels to the drain, being immediately replaced by another electron from the
source, which occupies the excited state. Relaxation to the ground state takes place
subsequently. In contrast to elastic cotunneling this process is suppressed at low
bias, since energy is transferred from the electron to the molecule. The onset of
inelastic cotunneling occurs at Vb ' ∆/e and in stability diagram it appears as a
horizontal line parallel to the zero-bias axis and terminating at the Coulomb dia-
26
2. T HEORETICAL BACKGROUND
b
Vb
a
S
D
S
D
VG
F IGURE 2.4: Single-electron transport in an intermediate coupling regime. (a) Typical stability diagram.
The horizontal inelastic cotunneling lines inside the Coulomb blockade region connect to slanted SET
lines. (b) A sketch of inelastic cotunneling transport via a virtual state.
mond edges. At these points the horizontal line should meet the corresponding
SET excitation lines. The SET lines, however, can be indistinguishable due to level
broadening if the coupling is too strong. In this case inelastic cotunneling can provide valuable spectroscopic information. The technique basing on the same principle, called inelastic electron tunneling spectroscopy (IETS) has been widely used
in a surface science [12, 13], being recently adopted for STM research of nanoscale
objects [14, 15].
The line shape of the cotunneling trace (d I /dVb trace at fixed gate voltage)
can vary from a step [16] to a peak [17](see Figure 2.5). Neglecting the intrinsic
linewidth the step shape can be approximated by the following formula [18]:
· µ
¶
µ
¶¸
dI
∆ − eVb
∆ + eVb
= A e + A i (F
+F
,
dVb
kB T
kB T
(2.22)
where F (x) = [1 + (x − 1)e x ] / [e x − 1]2 , A i is a constant prefactor and A e is the contribution from the elastic cotunneling. This expression gives two steps of width
5.4k B T centered at Vb = ±∆/e.
More frequently in experiments the cotunneling threshold is seen as a conductance peak. This happens due to the fact that cotunneling current is different for
the ground state and the excitated state that becomes available above the threshold: I exc 6= I g r . The total inelastic cotunneling current reads I = P exc (I exc + R) +
P g r (I g r + Γi nel ), where P g r (exc) are the occupations of the ground (excited) states,
Γi nel is the rate of inelastic cotunneling and R is the relaxation rate from the excited to the ground state [11]. This relaxation is not due to the interaction with
environment: it is another cotunneling process with a rate comparable to Γi nel .
2.1. T HREE - TERMINAL TRANSPORT THROUGH MOLECULAR JUNCTIONS
27
Immediately above the threshold P exc rises, thus lowering P g r , which inhibits the
current and results in a decrease of d I /dV . It should be noted here that because
of the relaxation even at zero temperature cotunneling steps would have a width of
ħR.
Another important higher-order process is the Kondo effect. It was first observed in 1930’s in bulk metals where the inclusion of ferromagnetic impurities resulted in a conductance minimum at low temperatures [19]. Only thirty years later
it was shown by Jun Kondo, that in second order perturbation theory the antiferromagnetic interaction of conductance electrons with localized spins leads to a divergence of the scattering probability for electrons close to the Fermi energy E F [20].
Although such a divergence means inapplicability of the perturbation theory, it can
be used to explain the conductance behaviour, and the effect is named after Kondo
since then. In 1988 it was shown theoretically that a similar effect in localized states
connected to reservoirs by tunnel barriers causes a maximum of the conductance
at zero bias due to the increased density of states around E F [21]. Experimentally
the Kondo effect can be observed in semiconductor quantum dots [22, 23] and in
single-molecule devices [24, 25] in the intermediate coupling regime.
The Kondo effect is a rich and complicated many-body phenomenon; its theoretical description requires the use of numerical techniques like perturbative poorman scaling [26] or non-perturbative numerical renormalization group (NRG) [27].
A review of these methods can be found, for instance, in Ref. [28, 29]. We will restrict the discussion to the simplified picture that is sketched in Figure 2.6b. When
the number of electrons residing on the molecule is odd, the molecule has a nonzero spin, and its ground state is spin-degenerate. At zero bias elastic cotunneling
can bring an unpaired electron to the drain and replace it with an electron from
the source with an opposite spin orientation. Such a spin-flip process is a conduction mechanism responsible for the spin-1/2 Kondo effect. The stability diagram
is shown in Figure 2.6a. The zero-bias conductance peak is only present in the
diamonds with an odd occupation of the molecule: the ground states for evenoccupied states are non-degenerate, all electrons are then paired and there is no
a
b
dI/dVb
dI/dVb
Ai
Vb
Ae
Δ/e
Vb
Δ/e
F IGURE 2.5: d I /dVb (Vb ) trace for inelastic cotunneling. (a) Steps of d I /dV smeared with the temperature mark an onset of inelastic cotunneling. The shape of the curve is described by 4.2 (b) A d I /dVb
peak due to higher-order processes.
28
2. T HEORETICAL BACKGROUND
Vb
a
b
even
odd
even
S
D S
D
VG
F IGURE 2.6: Kondo effect. (a) A stability diagram. Kondo resonance is observed a zero-bias d I /dV peak
in Coulomb diamonds with an odd population. (b) A sketch of Kondo spin-flip.
Kondo effect. Note, however, that this odd-even asymmetry can be violated in case
of nontrivial orbital filling, where a high-spin Kondo effect can arise [30]. Moreover, the Kondo effect can also occur in case of other types (not spin) of degenarate
states like, for example, orbital degeneracy in carbon nanotubes [31].
The energy scale for formation of the Kondo state is given by the Kondo temperature TK , that depends on the charging energy and the coupling. For singlemolecule devices the coupling is typically stronger than the one that can be
reached in semiconductor quantum dots. Consequently, the Kondo temperature is
typically higher: TK ∼ 10 − 50 K for molecules and TK . 1 K for quantum dots. The
height of the Kondo peak (the zero-bias conductance) has a characteristic temperature dependence (Figure 2.7b) [32]:
G(T ) = G el +G 0
4ΓS ΓD
[1 + (21/s − 1(T /TK )2 ]−s ,
ΓS + ΓD
(2.23)
where G el is the constant offset due to elastic cotunneling, G 0 = e 2 /ħ is the conductance quantum and s is a parameter, depending on the spin S of the dot [33]:
s = 0.22 for S = 1/2 and it is less for higher spin values. The maximum conductance
is predicted to increase logarithmically with decreasing temperatures and to saturate at a value G el + 2e 2 /ħ in case of symmetric lead-dot coupling (unitary limit).
This model is empirical and does not always work well for high spins. There are different models proposed [34], but up to now it is not clear, which model should be
used for S > 1/2-Kondo effect. All of them, however, predict the same trend of logarithmic scaling and saturation at low temperatures T << TK with less steep curves
and slower saturation for higher spins.
The shape of the conductance peak at zero magnetic field (Figure 2.7a)is
2.1. T HREE - TERMINAL TRANSPORT THROUGH MOLECULAR JUNCTIONS
a
b
dI/dVb
29
c
dI/dVb
FWHM
log T
Vb
T
0
F IGURE 2.7: Characteristic dependencies of the Kondo effect. (a) The zero bias Kondo resonance.
Dashed line shows the splitting at high magnetic field. (b) Temperature dependence of the maximum
of the zero-bias conductance. (c) Temperature dependence of the full width at half maximum (FWHM).
Lorentzian with a temperature-dependent width (Figure 2.7c) [35]:
FW HM =
2
e
q
(πk B T )2 + (2k B TK )2 .
(2.24)
At low temperatures F W H M ' 2k B TK /e. Note that close to the diamond edges the
molecule is in the so-called mixed-valence regime [32], and the Kondo resonance
is broader than in the middle of the diamond.
In a high magnetic field B the Zeeman effect splits the Kondo peak in two components, separated by twice the Zeeman splitting: ∆Vb = 2g µB B , where g is the
Landé factor, and µB is the Bohr magneton [36]. This splitting is only visible above
some critical field B c ∼ k B TK /2g µB [37]. Such a Zeeman splitting and the logarithmic temperature dependence are hallmarks of the Kondo effect.
Finally, when the molecule is in the strong coupling regime (Γ >> EC , ∆, k B T ),
the states in the leads and in the molecule are strongly hybridized and transport
occurs via elastic coherent tunneling from the source to the drain without stopping
on the molecule. The Coulomb blockade signatures are washed out and the current
is independent of the gate voltage.
2.1.4 T HE ROLE OF SPIN
Molecular excitations mentioned in the previous sections can be of different nature: vibrational [38], electronic [39] or magnetic [40]. The latter ones are of special
importance for this thesis. As was already discussed, spin is crucial for the emergence of the Kondo effect, however its role in quantum dot devices is much broader.
Probably the simplest example of spin excitations is a singlet-triplet transition
(Figure 2.8), observed in GaAs heterostructures [41], carbon nanotubes [42] and
single molecules [40]. In quantum dots with an even occupation the ground state
spin configuration is determined by the exchange interaction with characteristic
30
a
2. T HEORETICAL BACKGROUND
S=0
b
Energy
,
S=1
1
m=+
S=1, m=0
S=1
, m=
-1
S=0
S=1
Magnetic field
c
Vb
S=0
B
F IGURE 2.8: Singlet-triplet transition. (a) Possible electronic configurations. Triplet components have
same energy at zero magnetic field. Excited singlet state may have the same energy when coupling
is antiferromagnetic or is higher for ferromagnetic coulping. (b) Energy spectrum in the magnetic
field. (c) d I /dVb versus magnetic field B measurement in inelastic cotunneling regime (for a constant gate voltage). The kink corresponds to a change of the ground state. At this point transition
|S = 0〉 → |S = 1, m = 0〉 (horizontal line), switches to a transition |S = 1, m = −1〉 → |S = 1, m = 0〉, whose
energy depends on the field.
energy J . For ferromagnetic exchange coupling (J < 0) two electrons residing in the
highest occupied level may either form a singlet (S = 0) state or promote one electron to a next level to form a triplet (S = 1), depending on the relative magnitude of
the level splitting ∆ and the exchange energy J [42]. For |J | > ∆ the triplet state is
energetically preferred, and a zero-bias S = 1 Kondo peak is observed. For ∆ > |J |
the singlet state has a lower energy and the Kondo effect is not expected. In this
case singlet-triplet transition can be induced by the bias voltage and can be observed as an inelastic cotunneling line at voltage Vb = ±(∆ − |J |)/e. The triplet state
should not be confused with an excited singlet state involving two levels separated
by excitation energy ∆. Applying a magnetic field causes the triplet to split up as
shown in Figure 2.8b. The energy of each component is then ∆ − |J | + g µB m, where
2.1. T HREE - TERMINAL TRANSPORT THROUGH MOLECULAR JUNCTIONS
31
m = −1, 0, +1. Interestingly, at the magnetic field B = (∆ − |J |)/g µB B the singlet
state and the m = −1 component of the triplet are degenerate. At higher fields this
triplet component becomes the ground state and the magnetic field dependence
has a kink at this point (Figure 2.8c) as the bias voltage probes excitation energies
relative to the ground state and the energy of the ground state starts to decrease
from this point. For antiferromagnetic coupling (J > 0) antiparallel configuration
of spins (total spin S = 0) is preferred energetically [40]. Triplet state can also be
excited by the bias voltage, however the excitation energy at zero field is just J in
this case.
The singlet-triplet transition is a special case of spin excitations. Spin excitations can be also measured in a weak coupling regime as SET resonances, showing
up as lines in the conductance maps that run parallel to the Coulomb diamond
edges [43]. In this case the transition occurs between two different charge states.
If the ground state occupied with N electrons has spin S and the next charge state
(N + 1 electrons) has spin S + 1/2 an SET line can be observed for the transition
|N , S〉 → |N + 1, S − 1/2〉. The state |N + 1, S − 1/2〉 is an excited state reducing the
spin of the molecule. This line is also subject to the Zeeman effect in the magnetic
field and a change in the ground state can occur.
In general, any molecular excitations, including those probed by electron transport, should comply to selection rules, resulting from the symmetry of the system [44]. These rules only allow the excitations with a restricted set of quantum
numbers. For all the other transitions corresponding matrix elements are equal to
zero. In particular, spin excitations can only be observed in SET when
∆S = ±1/2; ∆m = ±1/2,
(2.25)
or in inelastic cotunneling when
∆S = ±1, ±2, ...; ∆m = 0, ±1, ±2, ...
(2.26)
Here ∆S, ∆m are the differences in spins and spin projections for two states involved in the transition. Note that in the rule (2.26) the maximum allowed ∆S and
∆m depend on the order of the process. For second order inelastic cotunneling
(which is the most relevant case for experiments) ∆S = ±1, ∆m = 0, ±1. The selection rules 2.25 and 2.26 can be understood as a spin conservation law: in the
SET excitation process the molecule is exchanging angular momentum with a single electron, travelling from the source to the drain. Since the spin of the single
electron is 1/2, it is the only possible change in the molecular spin. Second order
inelastic cotunneling involves two electrons, thus |∆S| can be either 0 or 1. The former case implies that the spin state of the molecule is not changed and is relevant,
for example, for vibrational excitations.
2. T HEORETICAL BACKGROUND
Vb
32
S1
S2
Δ
VG
F IGURE 2.9: Spin blockade. For a molecule with ∆S > 1/2 the current at low bias is zero. The dashed
lines indicate where the Coulomb diamond edges would be without spin blockade. The transport is
only possible through an excitation of the left charge state with the spin S 1 = S 2 ± 1/2, which has an
energy ∆.
The rule (2.25) has an interesting consequence for the SET conductance. Consider a molecule that for two adjacent charge states has spins differing by more
than 1/2. Selection rules imply that transport between the two ground states is impossible. At low bias the conductance is zero and the current sets on only when the
transition involving excited states with ∆S = 1/2 is brought into the bias window.
In the stability diagram a gap appears at low bias separating the SET regions (see
Figure 2.9). This situation is called spin blockade [45, 46].
2.2 S INGLE - MOLECULE MAGNETS
Single-molecule magnets (SMMs) are a class of metalorganic compounds that
show intramolecular ferromagnetic behaviour at low temperature [47]. This was
first observed in Mn12 compounds [48], whose low-temperature magnetization
data indicated an S = 10 ground state, strong magnetic anisotropy of Ising type
and slow magnetic relaxation below 10 K [49]. A polycrystalline powder of these
molecules showed a magnetic hysteresis of molecular origin [50], i.e. not associated with domain-wall formation like in classical ferromagnets. Subsequently
similar effects were observed in different compounds, including iron, nickel,
chromium clusters. Moreover, quantum magnetic effects like quantum tunnelling
of magnetization (QTM) [51] and quantum phase interference [52] were demonstrated for the first time in the SMMs. In this section a brief general introduction to
the physics of SMMs is given; a more broad review of molecular magnetism can be
found, for example, in [47, 53].
33
KV
Energy
2.2. S INGLE - MOLECULE MAGNETS
Magnetization orientation
F IGURE 2.10: Anisotropy energy barrier for a classical superparamagnet.
2.2.1 S UPERPARAMAGNETISM
The main characteristics of SMMs, granting them a magnetic behaviour, are their
large spin and magnetic anisotropy that forces the spin to align at a certain direction. In classical physics the magnetic anisotropy of a particle, A, is proportional to
its volume V :
A = K V.
(2.27)
For Ising anisotropy the particle has the lowest energy when its magnetic moment
is parallel to an easy axis. The energy of the system as a function of the orientation
of the magnetic moment is shown in Figure 2.10. The bottom of the left well corresponds to magnetization down, the bottom of the right well to magnetization up
and the top to magnetization perpendicular to the easy axis.
At low temperatures the system is blocked in one of the wells and its magnetization is stable. When the height of the barrier is comparable to the thermal energy,
the particle can reverse its magnetization. For an ensemble of particles half of them
will be in the left well and half in the right. The system will no longer be magnetized at zero field. An external magnetic field lowers the energy of one well and
increases the energy of another one, so each particle is magnetized along the field
as a paramagnet. However, the total response of the ensemble is now large, and
its magnetic susceptibility is much larger than that of a paramagnetic system. This
form of magnetism is called superparamagnetism.
The characteristic time for the magnetization reversal due to thermal activation, τ, follows Arrhenius behaviour:
τ = τ0 exp
KV
.
kB T
(2.28)
The temperature at which the relaxation time equals the time of the measurement
is called the blocking temperature and depends on the size of the particle. This
dependence is crucial for applications. For instance, it sets a limit on the storage
34
2. T HEORETICAL BACKGROUND
density of hard disk drives (superparamagnetic limit): the minimum size of singledomain particles that can be used is such that their blocking temperature equals
room temperature.
Single-molecule magnets exhibit essentially superparamagnetic behaviour,
which is, however, mated with their quantum nature. Quantum mechanics invokes
two important corrections to the picture drawn above. First, the energy of the system is quantized, and second, its magnetization can be reversed not only by the
thermal activation but also by the tunneling through the barrier. These effects will
be discussed in the next section.
2.2.2 S PIN H AMILTONIAN : ZERO - FIELD SPLITTING AND QUANTUM
TUNNELING OF MAGNETIZATION
SMMs are complex objects, consisting of several interacting magnetic centers,
which are subject to a ligand field. The complete description of their electronic
structure requires tedious calculations, however a few approaches have been developed for the interpretation of experimental data without involving fundamental
theories. One of them, used throughout this thesis, is the spin Hamiltonian approach [47]. It is a general phenomenological model applicable not only to molecular magnetism, but also to artificially engineered magnetic nanostructures like
single transition metal ions on anisotropic substrates [54, 55] or single magnetic
impurities in semiconductors [56]. The spin Hamiltonian is suitable for a description of the ground spin multiplet, provided that all the excited spin multiplets have
much higher energy. A central assumption behind the spin Hamiltonian model is
that the orbital moment is quenched.
The Hamiltonian consists of several terms. The first one is the Zeeman term,
responsible for interaction with the external magnetic field B:
H Z = µB B · g · Ŝ.
(2.29)
Here µB is the Bohr magneton, Ŝ is the spin operator of the system and g is the
Landé tensor. The Zeeman term results in a splitting of the spin multiplet in a
magnetic field. However, due to anisotropy the multiplet is split already at zero
field. This splitting can be accounted for by introducing the following term:
H Z F S = Ŝ · D · Ŝ,
(2.30)
where D is a real, symmetric tensor. It has three orthogonal eigenvectors and if the
coordinate axes x, y, z are chosen parallel to these eigenvectors, D is diagonal and
(2.30) takes the form
H Z F S = D xx Ŝ 2x + D y y Ŝ 2y + D zz Ŝ 2z ,
(2.31)
35
U
Energy
2.2. S INGLE - MOLECULE MAGNETS
-S
0
S
Sz
F IGURE 2.11: Anisotropy energy barrier for a SMM.
where Ŝ x , Ŝ y , Ŝ z are the spin operators. Subtracting the constant 21 (D xx +D y y )S(S +
1) from (2.31), one obtains
H Z F S = D Ŝ 2z + E (Ŝ 2x − Ŝ 2y ),
(2.32)
where D = D zz − 21 (D xx + D y y ) and E = 12 (D xx − D y y ). Let |m〉 be the eigenvectors
of Ŝ z : Ŝ z |m〉 = m|m〉, where m = −S, −(S − 1), ..., S. In the basis of |m〉 the first term
of (2.32) is diagonal, while the second term has no diagonal elements, i.e. for any
m 〈m|Ŝ 2x − Ŝ 2y |m〉 = 0. Equation (2.32) can then be rewritten in terms of spin raising
and lowering operators Ŝ ± = Ŝ x ± i Ŝ y as
2
2
H Z F S = D Ŝ 2z + (E /2)(Ŝ +
+ Ŝ −
).
(2.33)
The zero-field splitting depends on the symmetry of the system. For cubic symmetry D xx = D y y = D zz and D = E = 0, so that there is no splitting. In case of the
axial symmetry D xx = D y y , so E = 0 and
H Z F S = D Ŝ 2z .
(2.34)
The eigenvectors of the system are the eigenvectors |m〉 of S z and the eigenvalues
are W (m) = Dm 2 − S(S + 1)/3, so the spin multiplet is split at zero field into S + 1
components (S + 1/2 in case of half-integer spin). The value of the zero-field splitting (ZFS) between the states |m = S〉 and |m = S − 1〉 is
∆ = (2S − 1)D.
(2.35)
Note that the states with m 6= 0 remain doubly-degenerate: W (m) = W (−m).
The D term can be positive or negative. In the former case the levels with lowest |m| are the most stable and the ground state of the system has the spin in the
plane perpendicular to the symmetry axis. This is called easy-plane or hard-axis
anysotropy. For negative D the highest |m| states have the lowest energy and the
36
a
2. T HEORETICAL BACKGROUND
b
QTM
QTM
F IGURE 2.12: Quantum tunneling of magnetization. (a) Magnetic susceptibility measurements for a
Mn12 crystal (adapted from [51]). Arrows mark steep parts of hysteresis curve, where QTM happens. (b)
Level diagram for zero (left) and finite field (right), allowing QTM.
spin is aligned with the symmetry axis. This is easy-axis (also called Ising type)
anisotropy. The anisotropy creates a parabolic barrier for spin reversal (see Figure (2.11) and the system behaves as a quantum superparamagnet. The height of
the barrier is
U = −DS 2
(2.36)
for integer spin and
U = −D(S 2 − 1/4)
(2.37)
for half-integer spin.
If the symmetry is lower than axial, E 6= 0. For |E | ¿ |D| the second term in
Hamiltonian (2.32) can be treated as perturbation. In fact by definition |E /D| ≤ 1/3:
it is easy to see that a variation of |E /D| beyond this range is equivalent to a renaming of the axes. When E 6= 0 the eigenstates are not the pure eigenstates of
S z anymore and mixing of the |m〉 levels increases with increasing E . From equation (2.33) it is evident that the E -term only mixes states that have ∆m = 2. In case
of integer spin a non-zero E thus removes the degeneracy of |m〉 and | − m〉 states.
For half-integer spin this degeneracy is preserved by time reversal symmetry [57]
and the |m〉 and | − m〉 state remain degenerate Kramers doublets for any value of
E at zero magnetic field.
For integer spin the lowest energy m = ±S〉 states are simultaneously occupied
at zero field. The spin tunnels under a barrier with a frequency ω = ∆T /2ħ, where
∆T is the splitting, induced by the E -term, called tunnel splitting. This quantum
tunneling of magnetization (QTM) can also be observed at finite magnetic field if
the Zeeman interaction brings two states coupled by the E -term close to degeneracy as illustrated in Figure 2.12b (at avoided crossing points). In the measurements
of the magnetization as a function of field steps appear in the hysteresis loop at
these values of B [51](see Figure 2.12a). Recently QTM has been observed in self-
2.2. S INGLE - MOLECULE MAGNETS
37
assembled monolayers of SMMs [58]. It has also been predicted that QTM has an
impact on the electron transport through a single SMM [59, 60], although these effects have not yet been observed experimentally. Note that QTM is only possible
for states with the same parity and does thus not occur at zero field in half-integer
spin systems.
QTM can be produced not only by the E -term, but by any term in the Hamiltonian that does not commute with Ŝ z , for instance, by a transverse (to an easy
axis) magnetic field. The Hamiltonian in the presence of both longitudinal (B z )
and transverse (B x ) components of the field reads
H = D Ŝ 2z + g µB B z Ŝ z + g µB B x Ŝ x .
(2.38)
When B x = B z = 0 the eigenvectors |m〉 and | − m〉 are degenerate. When B x 6= 0
however, they are not. This can be easily seen for the case of S = 1/2. Let Z be a
direction of the total field (B x , 0, B z ). The Hamiltonian (2.38) then equals
H = g µB
q
B x2 + B z2 S Z − |D|/4.
(2.39)
It has two eigenvalues
E ± = ±(g µb /2)
q
B x2 + B z2 .
(2.40)
These eigenvalues are never degenerate at B x 6= 0 as the curves E + (B z ) and E − (B z )
avoid crossing at B z = 0. Their difference has a minimum there and QTM occurs.
This effect can be generalized for any spin value: for a given ansitropy Hamiltonian, the eigenvalues of the Hamiltonian are non-degenerate except for a final set
of values B x , B y , B z .
Strictly speaking, the quadratic Hamiltonian (2.32) is not complete: the terms
of higher even orders (up to 2S order) must be included to reproduce the energy
k
spectrum. It is usually done by using Stevens operators ÔN :
X k k
HZ F S =
B N ÔN ,
(2.41)
N ,k
k
where B N
are parameters with N = 2, 4, ..., 2S and the integer k satisfies −N ≤ k ≤
+N . The explicit form of the Stevens operators can be found in [47]. It is, however, generally a good approximation to consider higher order terms significantly
smaller than low order terms. This approximation is used everywhere in this thesis.
To complete the spin Hamiltonian, terms responsible for hyperfine H H F and
spin-orbit coupling HSO must be included. They are given by:
H H F = Î · A · Ŝ; HSO = (ζ/2S)L̂ · Ŝ.
(2.42)
38
2. T HEORETICAL BACKGROUND
a
b
L
L
L
2
eg
Δ0
M
t2g
L
2
x -y
2
z
xy
xz, yz
2
2
x -y
2
z
xz, yz
xy
L
L
F IGURE 2.13: (a)A sketch of octahedrally coordinated metal ion. (b) Ligand field splitting for octahedral
coordination (left), elongated (center) and compressed (right) tetragonal distortion
Here, A is the hyperfine splitting tensor, Î is the nuclear magnetic moment, ζ is
spin-orbit coupling constant and L̂ is the total orbital operator. These two terms
are not taken into account in this thesis since both the nuclear magnetic moment
and orbital moment of the molecule studied in this work are negligible. In general,
the complete spin Hamiltonian is the sum of the terms described above:
HS = H Z F S + H H F + HSO + H Z .
(2.43)
2.2.3 B EYOND THE SPIN H AMILTONIAN : SINGLE - ION LEVELS AND IN TERACTION
The spin Hamiltonian approach described in the previous section proved very useful for the interpretation of the low-energy intra- ground spin multiplet spectrum
of SMMs. However, it cannot be used for calculations of intermultiplet spin transitions, that can be probed in the experiments, nor does it give insight in the origin
of the anisotropy. Microscopic models taking into account every single transition
metal ion must then be employed.
Each magnetic ion in the molecule is interacting with the ligands, and in the
ligand field theory [61] this interaction is assumed to be purely electrostatic. The
electrons of the metal ions try to avoid regions where the negatively charged ligands
are. In octahedrally coordinated complexes six ligand atoms or groups of atoms
are symmetrically arranged around a metal ion (see Figure 2.13a). Its d -orbitals
are not degenerate as in the free atoms: the orbitals d x 2 − d y 2 and d z 2 pointing
to the ligands have higher energies than the d x y, d xz and d y z orbitals. In terms
of group theory the latter three orbitals span the irreducible representation t 2g of
a full octahedral group O h , while the former two span the e g representation. They
2.2. S INGLE - MOLECULE MAGNETS
eg
39
d
4
d
5
d
6
d
7
t2g
eg
t2g
F IGURE 2.14: Ligand field splitting and possible high and low spin configurations.
are therefore called t 2g and e g orbitals respectively. The ligand field splitting ∆0
between these two sets of orbitals depends on the ligands.
The ground states for octahedral transition metal ions can easily be determined. For the d 1 , d 2 , d 3 , d 8 , d 9 configurations there is only one way of populating
two energy levels with electrons. For the d 4 , d 5 , d 6 , d 7 configurations the electrons
may prefer to pair in the t 2g level, giving rise to a low-spin configuration, or they can
occupy both levels with parallel spins, giving rise to a high-spin configuration (see
Figure (2.14)). The low-spin configuration is favourable when the ligand field splitting ∆0 is higher than the exchange energy and high-spin results when the situation
is opposite. In some complexes the two energies are comparable and the high-spin
state can be thermally populated. These are called spin-crossover compounds [62].
The transition between the two spin configurations can be triggered by temperature, pressure or light irradiation [63]. Recently it was also demonstrated that spin
crossover can be induced by a gate voltage in a single-molecule junction [64].
Ground states for d 3 , d 8 , high spin d 5 and low spin d 6 configurations are orbitally non-degenerate. All the other configurations have either doubly or triply
degenerate ground states. Orbitally degenerate states are unstable and degeneracy
is removed by the Jahn-Teller effect for the doubly degenerate states or by spinorbit interaction for the triply degenerate states. Without going into details, we
note that this leads to a tetragonal distortion of octahedral symmetry and that the
levels can be split further in the two ways shown in Figure 2.13b.
The ions in the compound molecule are coupled to each other by exchange
interaction. For two ions it is described by the Heisenberg Hamiltonian:
12
HH
= Ŝ1 · J12 · Ŝ2 ,
(2.44)
where Ŝ1 , Ŝ2 are the spin operators for ions 1 and 2, and J12 is a matrix, which in the
general case is asymmetric. For systems consisting of a large number of magnetic
40
2. T HEORETICAL BACKGROUND
ions, the Heisenberg Hamiltonian is a sum of all the individual Hamiltonians (2.44)
describing the interaction between all ion pairs:
HH =
X
i 6= j
ij
HH .
(2.45)
In case of isotropic exchange interaction the Heisenberg Hamiltonian becomes
12
HH
= J 12 Ŝ1 · Ŝ2 .
(2.46)
Here, J 12 = −(1/3)TrJ 12 is the exchange coupling constant. When J 12 is positive the
interaction is antiferromagnetic and antiparallel alignment of the spins is preferred
energetically. For negative J 12 the interaction is ferromagnetic and parallel spin
configurantion is favourable.
The spectrum of the molecule can be obtained by diagonalizing the Hamiltonian which is the sum of single-ion spin Hamiltonians (2.43) and the Heisenberg
Hamiltonian (2.45):
X
H = HSi + H H .
(2.47)
i
When the coupling is isotropic and the exchange coupling constant is much larger
than the single-ion anisotropy the system is in the strong exchange limit. The net
spin and anisotropy parameters of the molecule can be found by the projection of
those of the individual ions. These parameters can then be used in the molecular spin Hamiltonian (giant spin model [65]). The strong exchange limit is a valid
approximation for most of the SMMs, including those studied in this thesis.
2.2.4 A N EXAMPLE OF SMM: F E4
Mn12 is by now the most extensively researched single-molecule magnet. Not only
numerous magnetometry, EPR, NMR, X-ray and neutron diffraction experiments
were carried on, but also electron transport through single Mn12 clusters was investigated [66, 67]. Nevertheless, for this work another class of molecules, the tetranuclear Fe(III) compound, was chosen. It is referred to as Fe4 in the following.
There are several reasons for this choice. The most important one is that in
the time span between the first transport experiments with SMMs and the start
of this project it was shown that Mn12 degrades upon monolayer deposition onto
a gold surface [68]. Partial reduction of Mn ions to Mn(II) was observed, accompanied by a significant decrease in the magnetic polarization and by an increased
instability of the monolayers. Similar experiments with Fe4 monolayers, however,
showed that no alteration of electronic and magnetic properties took place during the deposition process [69]. Furthermore, the Fe4 compound can be thermally
evaporated in ultrahigh vacuum (UHV) conditions [70]. Although this deposition
2.2. S INGLE - MOLECULE MAGNETS
41
F IGURE 2.15: Structure of the Fe4 molecules (color code: iron=purple, oxygen=red, carbon=grey, sulfur=yellow). Left: Magnetic core with four S = 5/2 iron (III) ions antiferromagnetically coupled to give a
molecular spin S = 5. Center: Fe4 C9 SAc derivative. Right: Fe4 Ph derivative.
technique was not used in this work it can be advantageous for the fabrication of
clean single-molecule junctions.
Molecules studied in this work have the formula [Fe4 L2 (dpm)6 ], where Hdpm =
2,2,6,6-tetramethyl-heptan-3,5-dione. Two derivatives, Fe4 Ph and Fe4 C9 SAc, were
synthesized by functionalizing the ligand H3 L = R-C(CH2 OH)3 with R = phenyl and
R = 9-(acetylsulfanyl)nonyl, respectively [71, 72] (see Figure 2.15). The difference
in ligands might be crucial for electron transport. In Fe4 Ph there are no chemical
anchoring groups, and the molecules are kept in the junctions only by the van der
Waals forces, whereas sulfur atoms in Fe4 C9 SAc presumably create strong covalent
bonds with the gold atoms of the electrodes, which can lead to an improved stability of the devices. On the other hand, the long alkyl chains of Fe4 C9 SAc are not
π-conjugated and therefore are not conducting. One can expect thus a stronger
electronic coupling Γ in the Fe4 Ph devices [73].
Four iron(III) ions are arranged in the molecules as a triangle with one ion in the
center and three in the vertices. Each ion is in a high spin d 5 electronic configuration, having spin S i = 5/2. The central ion is coupled antiferromagnetically to other
ions with a coupling constant much higher than the coupling of peripheral ions to
each other: J 1 À J 2 . The magnetic anisotropy and exchange coupling parameters
for bulk crystals of the two derivatives are presented in Table 2.1. The strong ex-
Fe4 Ph
Fe4 C9 SAc
S
5
5
g
2.0
2.0
D
-0.052
-0.051
E
0.0029
7.4×10−4
ZFS
0.47
0.46
U = DS 2
1.30
1.28
J1
2.03
1.98
J2
0.038
0.036
TABLE 2.1: Magnetic anisotropy and exchange coupling parameters for Fe4 Ph and Fe4 C9 SAc. The units
are meV for all the parameters except of S and g . The table is based on [71] for Fe4 Ph and on [72] for
Fe4 C9 SAc.
42
2. T HEORETICAL BACKGROUND
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an energy ∼ 6 meV, much higher than the anisotropy barrier [74]. Hence the spin
Hamiltonian fully describes the low-energy spectrum of magnetic excitations. This
is not the case in the Mn12 compounds, where the excited S = 9 multiplet overlaps
with the ground state S = 10 multiplet (the energy of the former is 4 meV and the
anisotropy barrier is 5.7 meV).
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Mechanics - Non-relativistic theory (Pergamon Press, 1991), pp. 223–226, 3rd
ed.
[58] M. Mannini, F. Pineider, C. Danieli, F. Totti, L. Sorace, P. Sainctavit, M.-A. Arrio,
E. Otero, L. Joly, J. C. Cezar, et al., Quantum tunneling of the magnetization in
a monolayer of oriented single-molceule magnets, Nature 468, 417 (2010).
[59] C. Romeike, M. R. Wegewijs, W. Hofstetter, and H. Schoeller, QuantumTunneling-Induced Kondo Effect in Single Molecular Magnets, Phys. Rev. Lett.
96, 196601 (2006).
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[60] C. Romeike, M. R. Wegewijs, and H. Schoeller, Spin Quantum Tunneling in
Single Molecular Magnets: Fingerprints in Transport Spectroscopy of Current
and Noise, Phys. Rev. Lett. 96, 196805 (2006).
[61] A. B. P. Lever and E. I. Solomon, Inorganic electronic structure and spectroscopy
(Wiley Interscience, New York, 1999), pp. 1–91.
[62] P. Gütlich and H. A. Goodwin, Spin Crossover in Transition Metal Compounds
I (Springer, Berlin, 2004).
[63] O. Kahn and C. J. Martinez, Spin-Transition Polymers: From Molecular Materials Toward Memory Devices, Science 279, 44 (1998).
[64] V. Meded, A. Bagrets, K. Fink, R. Chandrasekhar, M. Ruben, F. Evers,
A. Bernand-Mantel, J. S. Seldenthuis, A. Beukman, and H. S. J. van der Zant,
Electrical control over the Fe(II) spin crossover in a single molecule: Theory and
experiment, Phys. Rev. B 83, 245415 (2011).
[65] S. Hill, S. Datta, J. Liu, R. Inglis, C. J. Milios, P. L. Feng, J. J. Henderson, E. del
Barco, E. K. Brechin, and D. N. Hendrickson, Magnetic quantum tunneling:
insights from simple molecule-based magnets, Dalton Trans. 39, 4693 (2010).
[66] H. B. Heersche, Z. de Groot, J. A. Folk, H. S. J. van der Zant, C. Romeike, M. R.
Wegewijs, L. Zobbi, D. Barreca, E. Tondello, and A. Cornia, Electron Transport
through Single Mn12 Molecular Magnets, Phys. Rev. Lett. 96, 206801 (2006).
[67] M.-H. Jo, J. E. Grose, K. Baheti, M. M. Deshmukh, J. J. Sokol, E. M. Rumberger,
D. N. Hendrickson, J. R. Long, H. Park, and D. C. Ralph, Signatures of Molecular Magnetism in Single-Molecule Transport Spectroscopy, Nano Lett. 6, 2014
(2006).
[68] M. Mannini, P. Sainctavit, R. Sessoli, C. Cartier dit Moulin, F. Pineider, M.-A. Arrio, A. Cornia, and D. Gatteschi, XAS and XMCD Investigation of Mn12 Monolayers on Gold, Chem. Eur. J. 14, 7530 (2008).
[69] M. Mannini, F. Pineider, P. Sainctavit, C. Danieli, E. Otero, C. Sciancalepore,
A. M. Talarico, M.-A. Arrio, A. Cornia, D. Gatteschi, et al., Magnetic memory of
a single-molecule quantum magnet wired to a gold surface, Nature Mater. 8,
194 (2009).
[70] L. Margheriti, M. Mannini, L. Sorace, L. Gorini, D. Gatteschi, A. Caneschi,
D. Chiappe, R. Moroni, F. B. de Mongeot, A. Cornia, et al., Thermal Deposition
of Intact Tetrairon(III) Single-Molecule Magnets in High-Vacuum Conditions,
Small 5, 1460 (2009).
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2. T HEORETICAL BACKGROUND
[71] S. Accorsi, A.-L. Barra, A. Caneschi, G. Chastanet, A. Cornia, A. C. Fabretti,
D. Gatteschi, C. Mortaló, E. Olivieri, F. Parenti, et al., Tuning Anisotropy Barriers in a Family of Tetrairon(III) Single-Molecule Magnets with an S = 5 Ground
State, J. Am. Chem. Soc. 128, 4742 (2006).
[72] L. Gregoli, C. Danieli, A.-L. Barra, P. Neugebauer, G. Pellegrino, G. Poneti,
R. Sessoli, and A. Cornia, Magnetostructural Correlations in Tetrairon(III)
Single-Molecule Magnets, Chem. Eur. J. 15, 6456 (2009).
[73] A. Danilov, S. Kubatkin, S. Kafanov, P. Hedegard, N. Stuhr-Hansen, K. MothPoulsen, and T. Bjornholm, Electronic Transport in Single Molecule Junctions:
Control of the Molecule-Electrode Coupling through Intramolecular Tunneling
Barriers, Nano Lett. 8, 1 (2008).
[74] S. Carretta, P. Santini, G. Amoretti, T. Guidi, R. Caciuffo, A. Candini, A. Cornia,
D. Gatteschi, M. Plazanet, and J. A. Stride, Intra- and inter-multiplet magnetic
excitations in a tetrairon(III) molecular cluster, Phys. Rev. B 70, 214403 (2004).
3
D EVICE FABRICATION AND
MEASUREMENT SETUP
Parts of this chapter have been published in Synthetic Metals 161, 591 (2011).
49
50
3. D EVICE FABRICATION AND MEASUREMENT SETUP
magnet
power supply
matrix box
IVVI rack
1K pot
insert
voltmeter
4He
cryostat
sample
stage
piezo
controller
lock-in
Adwin
computer
current source
for a Hall bar
sample
temperature
controller
magnet
F IGURE 3.1: Schematic of the measurement setup.
3.1 E XPERIMENTAL SETUP
T
he measurement setup is designed for low-temperature high-vacuum highmagnetic field electrical characterization of the samples. It also supports room
temperature preparation for the measurements including molecule deposition and
electromigration. This arrangement protects the samples sensitive to electrostatic
discharge from the shocks that can be induced during transfer from one setup to
another. A schematic overview of the setup is present in Figure 3.1.
3.1.1 L OW- TEMPERATURE INSERT
The sample is mounted on the sample stage in the inner vacuum chamber (IVC)
of the 1K pot insert from Janis Research Company. The overall length of the insert
is ∼ 1400 mm; its diameter is 48 mm. The volume of the IVC is approximately 1.9
l. When the system is immersed in liquid helium, the vacuum in the IVC can be
maintained below 10−6 mbar using an external turbomolecular pump. The insert
is equipped with a 1K pot with the volume of 145 ml. The 1K pot is connected
through a neeedle valve to a helium cryostat. If this valve is slightly open, continous pumping of the helium through the 1 K pot with a rotary pump results in a
further cooling down to a temperature as low as 1.6 K. A temperature of 1.8 − 1.9 K
is routinely obtained. The 1K pot is supplied with a calibrated Cernox thermometer
and a 25 Ohm heater for operation above 4.2 K. The heater and the thermometer
are connected to a PID temperature controller Cryo-con® 32 through a hermetically sealed 10-pin electrical feedthrough at the top of the insert. Since the IVC is
3.1. E XPERIMENTAL SETUP
51
maintained under vacuum during low-temperature operation, the sample stage is
thermally connected to the 1K pot via a removable copper cold finger. The length
of the cold finger is chosen such that the sample is mounted in a maximum field
position of the magnet.
Electrical wiring consists of six wires for the thermometer and the heater mentioned above and 48 0.00500 phosphor bronze wires, connected from two 24-pin
Fischer feedthroughs at the room temperature flange on top of the insert through
the IVC evacuation pumping tube to the cold connectors installed on top of the
1K pot. These wires carry electrical signals to and from the sample. All wires are
thermally anchored onto copper bobbins installed at the bottom of the IVC flange
(4.2 K).
The sample stage consists of a plastic sample holder, which is closed with a
home-made brass liquid cell with a removable cap. The volume of the cell is 0.2 ml.
Prior to electromigration the sample is mounted in the holder, covered with the cell
and the cell is filled with a solution of the molecules. After the electromigration but
before closing the IVC, the cap is removed in order to pump out the solution during the IVC evacuation. This prevents crystallization of the solution during cooling
down. To prevent the solution from leaking, a Viton® seal is introduced between
the sample and the cell. Viton® was chosen because of its chemical and thermomechanical properties: it does not react with toluene and is mechanically stable
during cooling down and at low temperatures.
3.1.2 S AMPLE ROTATOR
In the experiments described in chapter 5 the sample holder is connected to a
piezoelectric rotary stepper positioner, ANRv51/LT/HV, manufactured by attocube
systems AG. This positioner (called rotator in the following text) allows 360◦ rotation in both directions around the horizontal axis at a broad temperature range
(10 mK - 373 K), high vacuum (10−8 mbar) and high magnetic fields (up to 31 T)
with a maximal step size below 250 at 4 K. The exact step size depends on the load
and the wire configuration and differs for different experiments.
The rotator has two PZT ceramic actuators. The stepping mechanism is a socalled inertial drive; it is illustrated in Figure 3.2. A series of triangular voltage
pulses is applied to a piezo actuator. During a slow expansion of the piezo the
inertial load does not move with rescpect to the piezo due to the friction force.
After having reached its final elongation the piezo is rapidly contracted. This sudden contraction induces an inertial force on the load, exceeding the friction force,
which leads to its net displacement.
The voltage is supplied from the piezo step controller ANC150 through a lowohmic (3.1 Ohm at room temperature) wire. The normal operational conditions
require peak voltage of 20 V at helium temperatures, however, sometimes a higher
52
3. D EVICE FABRICATION AND MEASUREMENT SETUP
Voltage
A
Load
Piezo
Voltage
Time
B
Load
Piezo
Voltage
Time
C
Load
Piezo
Time
F IGURE 3.2: Inertial drive principle. At a time moment A the actuator is in the starting position. Time
moment B is after a slow expansion of the piezo. Time moment C is after a rapid contraction of the
piezo.
initial voltage (up to 70 V) is required to start rotator motion. To prevent sample
damage that such a high signal can cause, the wire supllying the piezo voltage is
fed through a separate 3 mm diameter capillary tube inside the insert.
The frequency of the voltage pulses determines rotation speed. To achieve the
highest precision a low frequency (20-60 Hz) should be used. For achieving a fast
rotation (for instance, to rotate a sample 90 degrees) frequencies up to 3 kHz are
practical. High frequencies as well as high voltages, however, lead to a substantial
heat dissipation and care should be taken during such an operation.
The wiring of the sample, consisting of 32 copper wires linking the back side of
the sample holder through the connectors on the 1K pot to the phosphore bronze
wiring of the insert, can cause a drag to the movement and even complete blocking
of the rotator. To reduce this drag very thin (0.06 mm) wires are used. Different
arrangements of the wires were tried: the wires bound together to a straight bundle, helical wire bundles, bundles attached to the cold finger at different heights
and loose wires. No strategy absolutely preventing blocking was found, but the
loose wires configuration proved to cause less blocking than any other configuration used. Care should be taken though to assure that no single wire is blocked by
the cold finger or stuck in between the rotating and fixed parts of the sample stage.
When performing temperature dependent measurements it should be taken
3.1. E XPERIMENTAL SETUP
53
into account that the rotator is made of titanium and has thus a poor thermal conductivity. The temperature of the sample, attached to the rotator and having no
other heat connections apart from electrical wiring, can differ by several Kelvins
from the temperature of the 1K pot, which is set and indicated by the temperature
controller. It is therefore necessary to wait several minutes to allow for a thermal
equilibrium in the system when the temperature of the system is changed by temperature controller. It is also necessary to keep the pressure in the IVC as low as
possible to exclude non-radiative heat exchange between the sample and the liquid helium bath through the walls of the IVC. The latter is especially important at
temperatures above 4.2 K when any residual helium in the IVC that could be liquified during cooling down is evaporated to form an exchange gas and reduce the
temperature inside the IVC.
To sense the orientation of the sample we integrated a home-made lithographically fabricated InAs Hall bar to the sample holder. It is connected to the standalone current source and the Hall voltage is read out by a digital multimeter Agilent. These connections are made using four of the 32 wires, connecting the sample
holder to the measurement electronics. This method requires a non-zero magnetic
field for determination of the sample orientation. The Hall voltage across the bar is
V H = R H I B sinα/d ,
(3.1)
where I is the current sent through the bar, B is the magnetic field, d is the bar
width, R H is the Hall coefficient and α is the angle between the current direction
and the magnetic field. The orientation is then determined as follows. First, the
current I is set (usually to the full scale of the current source, which is 17 mA) and
the Hall voltage V H at a finite magnetic field is measured. Then the sample is rotated such that V H goes through its maximum value V H ⊥ and this value is noted.
This step is necessary because the sample can be already slightly rotated during
cooling down. The orientation at the V H ⊥ corresponds to the sample perpendicular to the field. The angle between the sample and the field after consecutive rotations is determined as
α = arcsinV H /V H ⊥ .
(3.2)
Note however that this angle α in general is not the same as the angle θ between
an easy axis of a molecule and the magnetic field as the molecule can be oriented
arbitrarily with respect to the sample (see discussion in chapter 6).
3.1.3 M EASUREMENT ELECTRONICS
The measurement electronics is controlled and read out by the ADwin-Gold (Jäger
Computergesteuerte Messtechnik GmbH) unit, containing digital signal processor
54
3. D EVICE FABRICATION AND MEASUREMENT SETUP
(DSP), with a maximum clock frequency of 400 MHz. The maximum data acquisition rate using 16 bit ADCs is about 200 kHz, which allows real-time control and
analysis of the signals. This fast control is especially important for the electromigration procedure. The output is provided by two 16 bit DACs. The ADwin is connected via a USB interface to a personal computer, which runs LabVIEW software
for controlling the experiments.
The home-built electronics is hosted in a shielded IVVI rack. It consists of several interchangeable modules with current and voltage sources, current-voltage
converters, voltage amplifiers and calibrated resistors. The rack is powered by two
12 V batteries, equipped with low-pass filters and optically coupled isolation amplifiers at its inputs and outputs to reduce interference from external sources. The
ADwin can be connected directly to the IVVI rack for DC measurements or a lockin amplifier can be used for AC measurements and a direct measurements of the
dI/dV. The sample can be accessed via a matrix box, whose inputs can be grounded
to prevent voltage spikes on the devices. The matrix box is connected to the insert
wiring using two 24-pin Fischer cables.
3.2 D EVICE FABRICATION
3.2.1 E LECTRON - BEAM LITHOGRAPHY
For the fabrication of the nanoelectrodes we use electron-beam lithography followed by electromigration [1]. Four lithography steps are needed. The first step
defines contact pads and alignment markers by evaporation of 50 nm of gold on
top of a 3 nm of titanium sticking layer on an oxidized silicon substrate. This step is
done on an oxidized 76-mm silicon wafer, which is diced after evaporation into 12
19 × 19 mm2 chips. Secondly, we fabricate the gate electrode by evaporating 75 nm
of aluminium. Subsequent oxidation in an O2 chamber at a pressure of 50 mTorr
forms a thin (2-4 nm) aluminium oxide layer. During the next step we evaporate
on top of the oxidized gate a thin (12 nm) and narrow (100 nm) gold wire, which
will be subject to electromigration. To promote electromigration we do not use a
sticking layer at this stage. The final step connects the thin gold wire to the contact pads with thick (110 nm) gold or AuPd electrodes (with a sticking layer). Each
19 × 19 mm2 chip is diced after this step into 16 identical 3 × 3 mm2 samples. Each
3 × 3 mm2 sample consists of 32 devices with two shared gates and four shared
sources.
Prior to measurements the samples are glued to 32-pin leadless ceramic chip
carriers using two-component epoxy UHU Plus®. Epoxy is cured under ambient
conditions for 16-24 hours, after which the samples are rinsed with acetone, blown
dry with nitrogen, cleaned in a UV-ozone cleaner for 10 minutes at the room temperature and wire-bonded.
3.2. D EVICE FABRICATION
55
3.2.2 M OLECULE DEPOSITION AND ELECTROMIGRATION
Molecule deposition is done by placing the samples into a home-made liquid cell,
filled with a dilute toluene solution of molecules, which is embedded inside the
4He probe (see section 3.1). The solution is prepared inside a glovebox in a nitrogen atmosphere with a concentration of oxygen less than 3 ppm. High-purity 99.7%
toluene over molecular sieve, containing ≤ 0.005% water, manufactured by Fluka®,
is used. The best yield of single-molecule devices is achieved with a ∼ 1 mM concentration.
To create single-molecule junctions we use electromigration of gold wires with
an active feedback. During this process the voltage is increased above the electromigration threshold (of order 100 mV) while measuring the current through the
device. As soon as there is a significant change (typically around 10%) in the wire
resistance, the voltage is reduced to a starting value and the process is repeated
until the wire resistance reaches ∼ 5 kOhm. This scheme protects the devices from
a sudden breaking and junctions with a desired resistance are formed with a reproducibility about 80%. In other cases, though, abrupt breaking occurs before the
resistance reaches ∼ 5 kOhm or electromigration does not start even when higher
voltage (up to 2 V) is applied.
Due to the high mobility of gold atoms the process of breaking continues at
room temperature even without applied voltage [2]. We use this self-breaking as a
final step in the nanogap preparation. When the resistances reach 1-1000 MOhm
the sample space is subsequently evacuated and the devices are cooled down to
1.6-1.7 K. The entire electromigration and self-breaking process is performed in a
liquid environment at room temperature, assuring that the molecules are exposed
to nanogaps between the electrodes already at the stage of the creation of the gaps,
minimizing the risk of contamination. Furthermore, in contrast to having a selfassembled monolayer of molecules prior to electromigration, the molecules in our
scheme are not subject to excessive local heating due to the high currents needed
for electromigration to occur. Another important issue is that self-breaking avoids
the formation of gold nanoparticles, which can occur when the junctions are broken abruptly; transport through these nanoparticles can mimic transport through
molecules [3]. Based on the characterization of thousands of samples we do not
find any evidence of gold nanoparticles in the gap when self-breaking is used.
A drawback of the electromigration technique is that one cannot control the
junction geometry on the atomic scale, while it is known that even the smallest
changes in the atomic positions can result in significant variations in the transport
characteristics. In particular, the electronic coupling of the molecule to the leads
Γ cannot be controlled. This parameter is of crucial importance as it determines
the width of spectroscopic features and hence the resolution of the measurements.
Moreover, when the device is in the intermediate coupling regime (Γ ∼ ∆, EC ) a
56
3. D EVICE FABRICATION AND MEASUREMENT SETUP
a
b
c
4
4
x 10
x 10
600
9
8
Current (pA)
400
8.5
200
6
0
8
-200
4
-400
7.5
2
-600
-2
0
Vg (V)
2
-2
0
Vg (V)
2
-2
0
Vg (V)
2
F IGURE 3.3: Gate traces taken at V = 50 mV for three different devices. (a) Typicval gate leakage signature. (b) A trace showing a weak monotonic gate dependence. (c) Coulomb blockade signature.
broad zero-bias Kondo resonance can result, masking low-energy spectroscopic
features. However, as it is shown in chapter 5, even in this regime information on
the spin state of the molecule can be extracted [4].
3.3 M EASUREMENTS TECHNIQUE
The first step in the low-temperature characterization of the samples is a quick
measurement of the current as a function of the gate voltage VG (in the range of
−3... + 3 V) at a fixed bias voltage V (typically 50 mV.) At gate voltages beyond this
range gate leakage usually sets in and the devices can be damaged by high current. This measurement allows to preselect devices that can potentially contain a
molecule for further characterization and discard all the others. Typically 2-4 devices per sample are selected at this stage. Three examples of gate traces are given
in Figure 3.3.
The trace shown in Figure 3.3a is a typical signature of a gate leakage. Most
of our devices show such traces. Figure 3.3b shows an example of a monotonic
linear rise of the current with the gate voltage. There are also samples showing a
similar decline of the current. The change in current does not exceed a few percent,
the gating is thus not efficient and such devices are discarded. Finally, Figure 3.3c
shows nonlinear nonmonotonic behaviour of the current. The peak at this Figure
is a signature of a Coulomb blockade; such devices thus can contain a molecule
or a gold nanoparticle. Only the devices showing the third type of gate traces are
taken for further examination.
The next step is a rough measurement of d I /dV as a function of gate and bias
voltage. The gate voltage is typically changed between −3 and 3 V with a step
of 50 mV. For each gate voltage the IV curve is measured in the range of −100
to 100 mV. This is a DC measurement and the d I /dV is obtained by a numerical
derivation of the measured current. Two examples of these rough d I /dV maps are
3.3. M EASUREMENTS TECHNIQUE
57
a
b
dI/dV (mS)
dI/dV (nS)
100
100
0.2
600
50
400
0
V(mV)
V(mV)
50
0.1
0
200
0
-50
-50
0
-100
-2
0
Vg (V)
2
-100
-2
0
Vg (V)
2
-0.1
F IGURE 3.4: Rough stability diagrams. (a) Device with molecular signatures. (b) Device with gold grains
shown in Figure 3.4. Figure 3.4a shows the stability diagram with only two charge
states in the full gate range. The island is weakly coupled to the gate (gate coupling
β ∼ 0.02). These are the signatures of conductance through a molecule. On the
other hand, Figure 3.4b shows the device with five charge states accessible within
the gate range with addition energies ∼ 100 meV and higher gate coupling. Such a
device is regarded as containing gold nanoparticles.
For molecular devices, further characterization is necessary. It is done using
a lock-in amplifier to directly measure the differential conductance. The lock-in
amplifier reduces the noise and gives more flexibility in terms of the sensitivity, so
that the regions of the stability diagram with a different (sometimes by a few orders
of magnitude) d I /dV like SET and inelastic cotunneling can be measured. We are
mostly interested in low-bias (below 20 mV) features as at these energy values the
magnetic excitations of the Fe4 molecule are expected to be present. Besides, at low
bias the spectroscopic features are usually narrower and can be better identified.
When the features of potential interest are found, magnetic field measurements
can be performed. The magnetic field can be swept from zero to a finite value with
a particular rate (typically from 0 to 8 T within 2 hours) and a series of the d I /dV
traces can be measured at a certain fixed gate voltage during this sweep. Alternatively, the bias voltage can be fixed at zero or a finite value and gate traces can be
recorded when sweeping the magnetic field. This gate-voltage spectroscopy is especially useful when looking for signatures of magnetic anisotropy or for a magnetic
hysteresis. It is discussed in more detail in chapter 6. Another kind of measurement, involving the magnet, is measurement of a stability diagram at a non-zero
field.
All the measurements described above can be performed at elevated temperatures using the temperature controller (see section 3.1). Such measurements are
standard for instance in the characterization of samples exhibiting the Kondo effect [5–7].
58
3. D EVICE FABRICATION AND MEASUREMENT SETUP
R EFERENCES
[1] E. A. Osorio, T. Bjørnholm, J.-M. Lehn, M. Ruben, and H. S. J. van der Zant,
Single-molecule transport in three-terminal devices, J. Phys.: Condens. Matter.
20, 374121 (2008).
[2] K. O’Neill, E. A. Osorio, and H. S. J. van der Zant, Self-breaking in planar fewatom Au constrictions for nanometer-spaced electrodes, Appl. Phys. Lett. 90,
133109 (2007).
[3] H. S. J. van der Zant, Y.-V. Kervennic, M. Poot, K. O’Neill, Z. de Groot, J. M.
Thijssen, H. B. Heersche, N. Stuhr-Hansen, T. Bjørnholm, D. Vanmaekelbergh,
et al., Molecular three-terminal devices: fabrication and measurements, Faraday
Discuss. 131, 347 (2006).
[4] A. S. Zyazin, H. S. J. van der Zant, M. R. Wegewijs, and A. Cornia, High-spin
and magnetic anisotropy signatures in three-terminal transport through a single
molecule, Synthetic Met. 161, 591 (2011).
[5] D. Goldhaber-Gordon, H. Strikman, D. Mahalu, D. Abusch-Magder, U. Meirav,
and M. A. Kastner, Kondo effect in a single-electron transistor, Nature 391, 156
(1998).
[6] E. A. Osorio, K. O’Neill, M. Wegewijs, N. Stuhr-Hansen, J. Paaske, T. Bjørnholm,
and H. S. J. van der Zant, Electronic Excitations of a Single Molecule Contacted
in a Three-Terminal Configuration, Nano Lett. 7, 3336 (2007).
[7] A. F. Otte, M. Ternes, K. von Bergmann, S. Loth, H. Brune, C. Lutz, C. Hirjibehedin, and A. Heinrich, The role of magnetic anisotropy in the Kondo effect,
Nature Phys. 4, 847 (2008).
4
M AGNETIC ANISOTROPY IN
MULTIPLE CHARGE STATES OF AN
F E4 MOLECULE
We have measured quantum transport through an individual Fe4 single-molecule
magnet embedded in a three-terminal device geometry. The characteristic zero-field
splittings of adjacent charge states and their magnetic field evolution are observed
in inelastic tunneling spectroscopy. We demonstrate that the molecule retains its
magnetic properties, and moreover, that the magnetic anisotropy is significantly enhanced by reversible electron addition / subtraction controlled with the gate voltage.
Single-molecule magnetism can thus be electrically controlled.
This chapter has been published in Nano Lett. 10, 3307 (2010).
59
60
4. M AGNETIC ANISOTROPY IN MULTIPLE CHARGE STATES OF AN F E4 MOLECULE
F IGURE 4.1: Fe4 molecule. (a) Structure of the Fe4 molecules (color code: iron=purple, oxygen=red,
carbon=grey, sulfur=yellow). Left: Magnetic core with four S = 5/2 iron (III) ions antiferromagnetically
coupled to give a molecular spin S = 5. Center: Fe4 C9 SAc derivative. Right: Fe4 Ph derivative. (b) Energy diagram of the ground spin multiplet at zero field. The S z 6= 0 levels corresponding to different
orientations of the spin vector along the easy axis of the molecule are doubly-degenerate. The S z = +5
and S z = −5 states are separated by a parabolic anisotropy barrier U . An important property of the Fe4
molecule is the large exchange gap to the next S = 4 high-spin multiplet [9] in the neutral state, which
is 4.80 meV and 4.65 meV, for Fe4 Ph and Fe4 C9 SAc, respectively [10, 11]. Transport below a bias voltage
of a few mV therefore only probes magnetic excitations of the ground high-spin multiplet, in contrast
to the Mn12 derivatives [12]. (c) Drawing of a three-terminal junction with a single Fe4 Ph molecule
bridging two gold electrodes (yellow) on top of an oxidized aluminum gate (grey).
4.1 I NTRODUCTION
R
ationally designed magnetic molecules [1, 2] can be used as building blocks
in future nanoelectronic devices for molecular spintronics [3], classical [4] and
quantum information processing [5, 6]. They usually have long spin coherence and
spin relaxation times due to a weak spin-orbit and hyperfine [7] coupling to the environment. Of crucial importance for applications is the ability to adjust the magnetic properties by external stimuli. In bulk samples tuning magnetic properties by
light has already been demonstrated [8], but on the single-molecule level it has not
been achieved. In addition, to tune magnetic properties on a single-molecule level,
the use of local electric fields is preferred as it allows for a direct and fast spin-state
control.
Addressing individual magnetic molecules on chip [13] has proven to be extremely challenging. Attempts to incorporate archetypal single-molecule magnets
4.2. E XPERIMENTAL DETAILS
61
(SMMs) Mn12 into a three-terminal device [12, 14] have been followed by observations that these complexes undergo electronic alterations when self-assembled
on gold surfaces [15]. In this letter we demonstrate electric-field control over the
magnetic properties of an individual Fe4 molecule connected in a planar threeterminal junction. In the neutral state the bulk properties of this SMM are well
documented [10, 11] and more importantly they are retained upon deposition on
gold [16]. From transport measurements we find that a Fe4 molecule in a junction
can still behave as a nanoscale magnet with the anisotropy barrier close to the bulk
value. In addition, upon reduction or oxidation induced by the gate voltage the
barrier height increases; i.e., by charging, the molecule becomes a better magnet.
Characteristic of a SMM is its magnetic anisotropy, creating an energy barrier
U which opposes spin reversal; i.e., the high spin of the molecule points along a
preferred easy axis, making it a nanoscale magnet. The anisotropy lifts the degeneracy of the ground high-spin multiplet, even in the absence of a magnetic field.
The splitting of the lowest two levels is known as the zero-field splitting (ZFS); see
Figure 4.1b. The magnetic anisotropy is described by the parameter D, which for
the Fe4 molecule in the bulk phase equals D ∼
= 0.051 − 0.056 meV; the ground state
spin for this class of molecules S = 5, the anisotropy barrier U = DS 2 ∼
= 1.3−1.4 meV
and the ZFS is (2S − 1)D ∼
= 0.46 − 0.50 meV [10, 11].
4.2 E XPERIMENTAL DETAILS
We use Fe4 molecules with formula [Fe4 L2 (dpm)6 ] (Hdpm = 2,2,6,6-tetramethylheptan-3,5-dione) (see Figure 4.1). Two derivatives, Fe4 Ph and Fe4 C9 SAc, were
synthesized by functionalizing the ligand H3 L = R-C(CH2 OH)3 with R = phenyl
and R = 9-(acetylsulfanyl)nonyl, respectively, and were prepared as described elsewhere [10, 11]. Their stability in a dry toluene solution at a 8 mM concentration was checked using spectroscopic techniques (see Supporting Information).
Nanometer-spaced electrodes were fabricated using self-breaking [17] of an electromigrated gold wire in a toluene solution of the molecules at room temperature [18]. Three-terminal transport measurements were performed at a temperature T = 1.6 K. By varying the gate voltage Vg the molecular levels are shifted,
thereby providing access to magnetic properties in adjacent charge states of the
SMM. Currently this is not possible with other techniques and therefore little is
known about the magnetic properties of charged SMMs.
To quantify the anisotropy of an individual molecule we have performed
transport-spectroscopy measurements in the Coulomb blockade regime. In total
we have measured 648 devices, of which 48 showed Coulomb blockade signatures
and two-dimensional conductance maps [18] (d I /dV versus gate and bias voltage)
were measured on these. The observation of molecule-related features (e.g. exci-
62
4. M AGNETIC ANISOTROPY IN MULTIPLE CHARGE STATES OF AN F E4 MOLECULE
tations) depends strongly on the dominant transport mechanism and electronic
coupling to the leads. Typically, the strength of the electronic coupling in molecular junctions is significant so that the resolution of single-electron tunneling spectroscopy is limited by the tunnel-broadening. In this case sharper higher-order
inelastic cotunneling peaks [19] may resolve the ZFS, but if their broadening due
to Kondo correlations exceeds the ZFS, only a single, broad zero-bias Kondo peak
remains, masking the low-bias ZFS features. We identified 9 stable devices with
indications of transport through a high-spin molecule: four devices showed broad
Kondo peaks in two adjacent charge states, indicating that at least one of the charge
states is a high-spin state. These Kondo features will be the subject of a subsequent
paper. Two samples showed a magnetic field-dependent transition at an energy
scale of 1.5 meV and another one showed a low-bias current suppression indicative
of spin blockade [20], involving a high-spin ground state. The microscopic origins
of these observations are not clear at the moment; it should be noted, however, that
the 1.5 meV value is close to the anisotropy barrier U . Two devices showed transitions in the inelastic cotunneling spectra at an energy scale below 1 meV, close to
the bulk value of the ZFS. In this chapter we focus on these two devices.
In Figure 4.2a and 4.3a,b we present the measurements as differential conductance maps at zero magnetic field. Sample A (Figure 4.2) features the Fe4 Ph derivative and sample B (Figure 4.3) the Fe4 C9 SAc derivative. The plots show conducting
regions characteristic of single-electron tunneling (SET). Outside the SET-regime,
the molecule is in a well-defined charge state with N electrons and the current is
suppressed (Coulomb blockade). Figure 4.2a and 4.3a,b clearly show a sizable conductance in the blockade regime due to tunnel processes of second (“cotunneling”)
or higher order. Here, inelastic cotunneling d I /dV steps or peaks appear at a bias
voltage Vb = ±∆/|e|, and their observation allows for the determination of molecular excitation energies ∆ in a particular charge state [19]. The charge state can
be changed by the application of a gate voltage when crossing a highly conductive SET region. By making the gate voltage more positive, the electron number
increases and the molecule is reduced; by making it more negative, the electron
number decreases and the molecule is oxidized.
In Figure 4.2a (sample A) three inelastic cotunneling lines are visible in the left
charge state. For large negative gate voltages these occur at V ∼
= ±0.6, ±4.6 and
±6.7 mV. For the right charge state there are lines at Vb ∼
= ±0.9 mV and ±5 mV. For
sample B cotunneling excitations appear at V ∼
= ±0.9 mV and ±5 mV (Figure 4.3a)
in the left charge state and at V ∼
= ±0.6 mV and ±7 mV (Figure 4.3b) in the right one.
Additional information on the nature of the low-energy excitations can be obtained by measuring their evolution in a magnetic field. In Figure 4.2b,c and Figure 4.3c,d d I /dV is plotted as a function of V for different magnetic field values at
two adjacent charge states for each sample. The energy of the lowest excitation in-
4.2. E XPERIMENTAL DETAILS
63
F IGURE 4.2: Characteristics of sample A. (a) Color plot of d I /dV versus V and VG at T = 1.6 K and
B = 0 T. In the black regions in the middle of the plot single-electron transport (SET) is allowed. The
lock-in amplifier saturates in these high conductance regions. (b) d I /dV as a function of V for VG =
-1.5 V and various magnetic field values. Successive curves are offset by 250 nS. (c) Same as (b) for VG =
2 V and an offset of 400 nS. (d) and (e) Excitation energy as a function of magnetic field for the same VG
values as in (b) and (c). Red lines are fits with D = 0.06 meV, g = 2.1, S = 5 in (d) and D = 0.09 meV, g =
1.8, S = 11/2 in (e) and θ = 0◦ for both cases.
64
4. M AGNETIC ANISOTROPY IN MULTIPLE CHARGE STATES OF AN F E4 MOLECULE
F IGURE 4.3: Characteristics of sample B. (a) and (b) Color plots of d I /dV versus V and VG at T = 1.6 K
and B = 0 T highlighting the behavior in two adjacent charge states. (c) d I /dV as a function of V at VG
= 0 V for successive magnetic fields B . Curves are offset by 125 nS. Red lines are fits to a Lambe-Jaklevic
formula (see Supporting Information). (d) Same as (c) for VG = 1.6 V with curve offsets 50, 80, 140 and
220 nS. (e) Energy of the first excitation as a function of B at VG = 0.1 V. The red line is a fit to model (1)
for S = 9/2, D = 0.09 meV, g = 2.0 and θ = 60◦ . (f) Same as (e) for VG = 1.1 V. The red line is a fit for S = 5,
D = 0.06 meV, g = 2.1 and θ = 71◦ .
4.3. F ITTING TO A MODEL
65
creases with magnetic field without splitting and is symmetric upon field reversal
(see Supporting Information). Such behavior is a hallmark of ZFS described by the
spin Hamiltonian [2, 21–23]
ˆ
~ ·~
Ĥ = −D Ŝ 2z + g µB B
S,
(4.1)
where the first term is the uni-axial magnetic anisotropy with the easy axis z. The
~ , where g
Sˆ with magnetic field B
second term is the Zeeman interaction of spin ~
is the Landé factor and µB the Bohr magneton. Importantly, model (1) predicts a
non-linear dependence of the excitation energy on the magnetic field if the angle
between the easy axis of the molecule and the magnetic field is substantial (see
Supporting Information). This non-linearity is caused by the mixing of |S z 〉 states
due to the presence of a transverse component of the field.
4.3 F ITTING TO A MODEL
To quantitatively compare the data with the model, the excitation energy has been
determined from individual d I /dV curves. For sample A (Figure 4.2b,c), we have
taken the peak positions as the ZFS excitation energy and this energy is plotted versus magnetic field in Figure 4.2d,e. For sample B, the inelastic cotunneling excitations have the form of conductance steps, reflecting a weaker molecule-electrode
tunneling Γ than for sample A. We have fitted the measured d I /dV to a LambeJaklevic formula [24, 25] and plotted the excitation energies in Figure 4.3e,f.
We first compare the data of sample A with the model. For the left charge state
(Figure 4.2b) the energy of the first excitation at B = 0 T is close to the ZFS value
(0.5 meV) in the bulk [10]. We assume that this is the neutral state, and since the
Fe4 molecule maintains its ground state spin S = 5 when deposited on gold [16],
we estimate D ∼
= 0.06 meV, so that U ∼
= 1.5 meV. The increase of the excitation en~ and the easy-axis below 45◦ .
ergy with B is linear, implying an angle θ between B
It should be noted that the angle cannot be controlled in our junctions and is expected to vary for different samples. Using model 4.5) we have calculated the energy difference between the ground and the first excited state originating from the
same side of the anisotropy barrier as a function of applied magnetic field. Using
the angle θ as an additional parameter, the best fit of the data to the model, shown
in Figure 4.2, yields g = 2.1 and θ = 0◦ .
Reduction and oxidation of the molecule inevitably change its magnetic properties, although it is not a priori clear in which way. Three-terminal spectroscopic
measurements can provide an answer since the ZFS and the change in the total
spin upon charging can be obtained independently. First, the difference in spin values of adjacent charge states can be determined from the shift of the degeneracypoint in a magnetic field (see Supporting Information). We infer an increased
66
4. M AGNETIC ANISOTROPY IN MULTIPLE CHARGE STATES OF AN F E4 MOLECULE
S = 11/2 in the reduced charge state of sample A. From the measured ZFS we
then find an enhanced D ∼
= 0.09 meV and U ∼
= 2.7 meV. From the magnetic field
dependence, we infer g = 1.8 and θ = 0◦ (Figure 4.2e) when fitting the data to
the model 4.5. For sample B we observe the bulk ZFS value in the right charge
state (Figure 4.3d,f ) and therefore identify this as the neutral state with S = 5 and
D∼
= 0.06 meV. A clear non-linear Zeeman effect is now observed, indicating that
the field is at a substantial angle with the easy-axis. We have fitted the data to the
model obtaining g = 2.1 and θ = 71◦ . Also for the left charge state the Zeeman effect
is non-linear (Figure 4.3e) and a three-parameter fit yields D = 0.09 meV, g = 2.0
and θ = 60◦ . Here, we have used S = 9/2 for the oxidized state (see Supporting Information) and estimate U ∼
= 1.8 meV. Note that for small magnetic fields the data
show a deviation from the model, which is not yet understood.
4.4 D ISCUSSION
F IGURE 4.4: Calculated d I /dV as a function of bias voltage and magnetuc field. (a) KE result for θ=0◦
using the parameters estimated for sample A with electron temperature T = 1.6 K and the gold conduction bandwidth W = 8.1 eV. Shown is the field evolution of the inelastic cotunneling step for the S = 5
state at gate voltage VG = -1.5 V. The conductance is scaled to its maximum value. (b) Same as (a) but
using the parameters estimated for sample B with gate voltage VG = 1.6 V and θ = 71◦ . (c) d I /dV traces
taken from (b) corresponding to sample B. (d) NRG result for θ=0◦ for the Kondo model using the same
parameters as in (a) with zero temperature and an exchange tunneling constant J = 0.1/ρ (ρ is density
of states). (e) Same as (d), but now for θ = 71◦ . (f) d I /dV traces taken from (d) corresponding to sample
A.
In view of the rich SMM excitation spectrum the question arises why only the
4.5. S UPPLEMENTARY INFORMATION
67
ZFS excitation is observed up to a bias of several mV. We have performed extensive
calculations which indicate that this indeed should be the case (see also Supporting Information). In Figure 4.4a,b we show d I /dV maps calculated using quantum
kinetic equations (KE), accounting for tunnel processes to first and second order in
Γ [26]. We model the low-energy spectrum of two successive charge states including the charge-dependent ZFS. Figure 4.4 shows that for the experimental temperature the ZFS d I /dV -step is dominant, showing a linear Zeeman effect for θ = 0◦
(Figure 4.4a) and a non-linear behavior for θ = 71◦ (Figure 4.4b). Numerical renormalization group (NRG) calculations using a Kondo model [27] cover the regime
where the tunnel coupling dominates over the thermal energy (in contrast to the
KE approach). Figure 4.4d,e show that the ZFS now appears as a peak due to the
exchange scattering through the SMM indicating significant Kondo correlations.
In summary, we have demonstrated electric-field control over the anisotropy of
a single magnetic molecule in a three-terminal junction. We found a stronger magnetic anisotropy upon both reduction and oxidation induced by the gate voltage.
This enhancement may be related to the alteration of single-ion anisotropy, which
should be substantial when changing the redox state, but more studies including
quantum chemistry calculations have to be performed to confirm this. Our findings open the route to new approaches in tuning magnetism on a molecular scale
and the possibility of manipulating individual magnetic molecules for future use
in nanoelectronic applications.
4.5 S UPPLEMENTARY INFORMATION
S1. M ATERIALS AND M ETHODS
Synthesis and Solution Studies
Pure crystalline samples of Fe4 Ph.Et2 O and Fe4 C9 SAc were prepared as described
elsewhere [10, 11]. Their stability in toluene-d8 solution was checked by 1 H-NMR
spectroscopy using a Bruker FTDPX200 NMR spectrometer at 200 MHz and 30◦ C
(chemical shifts are referred to external TMS). The spectra recorded on freshlyprepared and aged samples (71 and 163 hours) show no significant differences,
pointing to a complete stability of the tetrairon(III) complexes over a time frame
much longer than required for the fabrication of three-terminal devices. The spectra, shown in Figure 4.5, are dominated by the very broad, paramagnetically-shifted
signal of tBu protons around 10.5 ppm (the NMR properties of Fe4 complexes are
discussed in [10]). Total stability of Fe4 C9 SAc in toluene was confirmed also by
electronic absorption spectra, which remained unchanged over a 1-week period
(6.2·10−4 M solution). When deposited in the transport junction, the molecule may
not be in the neutral state at zero applied gate and bias voltage due to the junctionspecific electrostatic environment. The neutral state, however, can be stabilized by
68
4. M AGNETIC ANISOTROPY IN MULTIPLE CHARGE STATES OF AN F E4 MOLECULE
F IGURE 4.5: (a) 1 H-NMR spectrum of Fe4 Ph recorded at 30◦ C in toluene-d8 (8.2 · 10−3 M solution). The
narrow peaks below 8 ppm are due to protio impurities in the solvent, lattice Et2 O and traces of the free
Hdpm ligand. (b) 1 H-NMR spectrum of Fe4 C9 SAc recorded at 30◦ C in toluene-d8 (7.5·10−3 M solution).
The narrow peaks below 8 ppm are due to protio impurities in the solvent, traces of free Hdpm ligand
and 1,2-dimethoxyethane (solvent used for crystallization). The spectra are offset for clarity.
adjusting the gate voltage. This ability distinguishes three-terminal measurements
from present STM and mechanically controllable break-junction experiments.
Measurement setup
The differential conductance d I /dV was measured using lock-in detection of the
tunnel current I by adding a 120-500 µV AC modulation at ∼300 Hz to the DC bias
voltage V . V was applied to one of the electrodes and the other one was grounded.
The gate voltage is defined with respect to the ground.
S2. T RANSPORT M EASUREMENTS IN A M AGNETIC F IELD
Magnetic Field Data and Line Shape Fitting
To obtain the magnetic field dependence of the transition energy shown in Figure 4.2b,c and 4.3c,d we repeatedly swept the bias voltage V at fixed gate voltage
VG while continuously increasing B from 0 to 8 T at a rate of 2-4 T/hour. Typically
∼200 d I /dV traces are recorded per magnetic field sweep giving ∼200 points for
the energy transitions shown in Figure 4.2d,e and 4.3e,f. The change in magnetic
field during the time needed to take a single d I /dV trace is ∼ 8/200 = 0.04 T. For
samples A and B two examples of d I /dV maps in the varying magnetic field are
shown in Figure 4.6a,b.
For sample B the line shapes of the d I /dV traces around the inelastic excitation
voltages V = ±∆/e were fitted to a Lambe-Jacklevic type expression [24]
µ
¶
µ
¶
−e(V + V0 ) + ∆
e(V + V0 ) + ∆
d I /dV = A e + A i + F
+ Ai −F
,
(4.2)
kB T
kB T
4.5. S UPPLEMENTARY INFORMATION
69
F IGURE 4.6: Magnetic field dependence of the ZFS inelastic cotunneling excitation. (a) and (b) d I /dV
as a function of B and V for samples A (VG = −1.5 V) and B (VG = 0.1 V) respectively. (c) and (d) Conductance map (d I /dV as a function of VG and V ) at B = 8 T for samples A and B. (e) and (f) ZFS evolution
in the magnetic field ranging from -8 to 8 T for the N charge states of samples A and B. The red lines are
the fitting curves identical for both magnetic field polarities, using the same parameters as the fitting
curves in Figure 4.2d and 4.3f.
70
4. M AGNETIC ANISOTROPY IN MULTIPLE CHARGE STATES OF AN F E4 MOLECULE
where F (x) = [1 + (x − 1)e x ] / [e x − 1]2 and e > 0 is the magnitude of the electron
charge. To account for the slight bias asymmetry of the measured data (independent of the magnetic field), a bias offset V0 was included and the constants of the
forward- and reverse-bias inelastic parts, A i + , A i − , were allowed to differ. This
merely serves the purpose of more accurately fitting the resonance position, the
quantity of main interest, as given by the inflection point of the d I /dV step. In the
χ2 -fitting procedure all parameters were allowed to vary, except the experimental
temperature T = 1.6 K.
In addition to the d I /dV maps as function of V and VG shown in Figures 4.2 and
4.3 we have also measured the corresponding conductance maps at high magnetic
field B = 8 T shown in Figure 4.6c,d. These clearly show that the lowest inelastic
cotunneling excitation increases for all gate voltages in both samples.
We have also performed measurements in a reversed magnetic field to verify
~loc
that the excitation at zero field does not originate from a local magnetic field B
instead of from magnetic anisotropy. We find that in all measurements the peak
position is symmetric upon field reversal as expected for ZFS. Two examples are
shown in Figure 4.6e,f. The sharp kink at zero field in Figure 4.6e cannot be ex~ +B
~loc . We conplained by assuming an isotropic spin in a total magnetic field B
~ were parallel to B
~loc , a nonsider two limiting cases. If the applied external field B
symmetric field-dependence and an excitation at zero energy for B = −B loc would
result, in contrast to all our reverse-field data which are symmetric around B = 0.
~loc the peak position would
If, on the other hand, the field were perpendicular to B
remain symmetric showing a zero-field energy gap of order g µB loc . However, since
this gap results from an anticrossing it is rounded on a magnetic field energy scale
of the same order. In contrast, our measurements, in particular in Figure 4.6e, show
a large gap and a sharp kink.
Fitting the ZFS Peak Positions
Since inelastic cotunneling probes energy excitations within a particular charge
state (controlled by the gate voltage), the magnetic parameters can be determined
for this state. Thus with a gate voltage the magnetic parameters of oxidized / reduced states of the SMM can be accessed independently. To compare the energy
dependence of the ZFS excitation to model (1) in the main text, one needs the values for the spin S, the D parameter, the g -factor and the angle θ between easy-axis
and magnetic field. After assigning the spin state as described in the main text, the
D parameter directly follows from the measured gap at zero field. These are listed
in Table 4.1. With the values of S and D fixed, we calculated the energy spectrum
as a function of magnetic field. From this we extracted the energy of the excitation from the ground state into the lowest state originating from the same side of
the anisotropy barrier. This is denoted by “ZFS-excitation” in the following. In Fig-
4.5. S UPPLEMENTARY INFORMATION
71
ure 4.7a the magnetic field evolution of the calculated ZFS excitation energy is plotted for different angles θ up to about 70◦ for which such states can be distinguished.
The results of Figure 4.7a can thus be used to compare with the measured peak positions. The non-linear behavior of the ZFS excitation results from the increasing
transverse field component (higher angles θ) and is referred to as non-linear Zeeman effect [28].
We note further that for higher angles spin states from different sides of the barrier start to be mixed and cannot be distinguished anymore. Due to field induced
quantum tunneling of the spin through the barrier, M is no longer a good quantum
number.
In Figure 4.7b we show the field evolution of the two lowest excitations for a
larger angle θ = 80◦ and g = 2.0. In this case the magnetic field mixes the two states,
originating from opposite sides of the anisotropy barrier, resulting in the anticrossing at a finite magnetic field value on the order of a few Tesla. For the parameters
considered here, in particular the thermal smearing, such anticrossing gaps cannot
be resolved in the transport (we have checked this with detailed model calculations
of the type reported in Section S3).
For sample A a large range of θ and g parameters can reproduce the linear magnetic field dependence of the ZFS excitation observed in both charge sectors. This
originates from the nearly linear dependence predicted by the model for a wide
range of angles in Figure 4.7a: deviations from linear behavior occur only for θ
larger than approximately ∼ 50◦ . A fit for θ and g is determined by minimizing the
RMS deviation between the calculated and measured energy positions (Table 4.1).
For example, assuming θ = 0, the best fit for the left (neutral) charge state is obtained for g = 2.1 close to the bulk value g = 2.0 [10, 11]. Higher angles with higher
g -factors also give a fit to the linear behavior, albeit with slightly higher errors. For
the right charge state g = 1.8 gives the best fit when fixing θ = 0◦ . Alternatively,
fixing g = 2.0, one obtains θ = 26◦ for this charge state.
For sample B, a distinct non-linear Zeeman effect is observed in Figure 4.3e,f.
For the right N -electron charge state the non-linear behavior can be well-fitted
with g = 2.1 and θ = 71◦ . For the left (N − 1)-electron charge state we obtain a
reasonable agreement for B > 2 T with D = 0.09 meV, g = 2.0, and θ = 60◦ . The
small deviation for low fields is not understood. It may also be present in the N
charge state, but the scatter in the data point prohibits a definite conclusion about
this.
From our transport measurements we thus conclude that Fe4 SMMs contacted
in these two molecular junctions retain their magnetic anisotropy and that this
anisotropy is enhanced upon both oxidation and reduction. Furthermore, the data
also indicates that the g -value and the orientation of the easy-axis does not change
substantially upon charging. The observed 50% enhancement of the uni-axial
72
4. M AGNETIC ANISOTROPY IN MULTIPLE CHARGE STATES OF AN F E4 MOLECULE
F IGURE 4.7: Excitation energies calculated by diagonalization of the Hamiltonian (1) of the main text.
(a) Magnetic field evolution of the ZFS excitation for different angles θ with the easy-axis using the parameters estimated for sample B in the N -electron state (Table 4.1). (b) Magnetic field evolution of
the two lowest excitations from the ground state for θ = 80◦ with the easy-axis using the parameters of
(a). The cotunneling resonance position follows the state with the largest amplitude of the spin wave
function on the same side of the anisotropy barrier as the ground state. For small fields it the follows
the upper blue line, but beyond the anticrossing at the field 5.3D/g µB = 2.6 T the observed resonance
follows the lower lying red excitation branch.
F IGURE 4.8: Zeeman shift of the degeneracy point (maximum of the conductance) for sample A (a) and
B (b) (lock-in amplifier measurement). The shift of the degeneracy-point from B = 0 T (black curves)
for sample A is ∆VG = −0.05 V for B = 8 T (red curve), and for sample B ∆VG = −0.07 V for B = 7 T (red
curve).
4.5. S UPPLEMENTARY INFORMATION
73
anisotropy may be related to an alteration of single-ion anisotropy, which should
be substantial when changing the charge state. Controlled oxidation/reduction of
SMMs in bulk crystals is difficult to achieve and consequently there are no experiments to compare our data with. Quantum chemistry calculations would therefore
be helpful in elucidating the properties of oxidized and reduced SMMs [29].
Zeeman Shift of the Degeneracy Point
~ ) and ²N +1 (B
~ ) of the two neighIn a magnetic field the ground state energies ²N (B
boring charge sectors experience a different Zeeman shift. Therefore the chemical
~ ) = ²N +1 (B
~ ) − ² N (B
~ ) changes and a shift in the gate voltage is required
potential µ(B
to maintain the single-electron resonance in linear response. This shift thus only
contains information about the difference of the magnetic parameters of two adjacent charge states. In contrast, above we used the inelastic cotunneling to estimate
the properties of the individual charge states.
For sample A, the shift of the degeneracy point is determined by
¡
¢ ¡
¢
~ ) − µ(~0) = ²N +1 (B
~ ) − ²N +1 (~0) − ²N (B
~ ) − ²N (~0) ,
−eβ∆VG = µ(B
(4.3)
where β = C g /C is the gate coupling [18], C g and C are the effective gate- and total
capacitance of the molecule. (For sample B replace N → N − 1.) Note that due to
~ relative
the magnetic anisotropy, the shift depends on the direction of the vector B
to the easy axes of the SMM in the two different charge states. We calculate the
change in the ground state energy for both charge states in 4.3 due to the magnetic
~ using the parameters in Table 4.1 estimated from the field evolution of the
field B
inelastic peak position. Figure 4.8a,b shows low-bias d I /dV gate traces at high
and zero magnetic field for samples A and B respectively. For sample A the shift
of the degeneracy point is ∆VG = −0.05 V for B = 8 T, while for sample B ∆VG =
−0.07 V for B = 7 T. With an estimated gate coupling β = 0.011 for sample A and β =
0.017 for sample B we obtain chemical potential shifts of 0.55 meV and 1.19 meV
respectively.
We first consider sample A for which we found that the angles with the field are
small. Fixing both angles to be equal to zero, the shift can be given analytically:
¡
¢
eβN +1,N ∆VG = µB g N +1 S N +1 − g N S N .
(4.4)
The measured negative gate-voltage shift directly indicates that the spin S increases when adding an extra electron (N → N + 1). When calculating the shift for
sample A with the fitted g -factors at fixed angle θ = 0, we obtain 0.28 meV which
is smaller than the measured value 0.55 meV. This may be explained by relaxing
the assumption of exactly equal orientation of the easy-axis in both charge states.
Small angle differences may already account for this. Fixing g = 2.0 in both charge
74
4. M AGNETIC ANISOTROPY IN MULTIPLE CHARGE STATES OF AN F E4 MOLECULE
Sample
A
B
g
θ (◦ )
Electron
number
N
N +1
S
D (meV)
5
11/2
0.06
0.09
2.1
1.8
2.0*
0.0
0.0*
26
N −1
N
9/2
5
0.09
0.06
2.0
2.1
60
71
TABLE 4.1: Parameters for the various spin ground states estimated from the inelastic cotunneling ZFS
peak position and its magnetic field evolution. For the N + 1 state of sample A two alternative fits are
given, where ∗ indicates the parameter that was held fixed in the fitting procedure.
states, the experimental value 0.55 meV is reproduced for θ N = 18◦ and θ N +1 = 0.
The reason for the increase is as follows: due to the larger angle with the field, the
Zeeman shift of the N electron state becomes suppressed (non-linear Zeeman effect) and compensates less of the linear Zeeman shift of the N + 1 electron state
(cf.( 4.4)), resulting in a larger chemical potential shift.
For sample B, both Zeeman shifts are in the non-linear regime due to large
~ . Taking the parameters from the main text,
angles between the easy axis and B
S N −1 = 9/2, g N −1 = 2.0, g N = 2.1, θ N −1 = 60◦ and θ N = 71◦ we obtain a shift of
0.42 meV, which is smaller than the experimental value of 1.19 meV. We note however that small changes in the parameters yield good fits of model (1) to the data in
Figure 4.3e as well. The Zeeman shift depends sensitively on these parameters and
in the absence of further information (e.g. the expected values of g or θ) no definite
conclusion can be drawn.
S3. T RANSPORT C ALCULATIONS
In this section we address a central question posed by the experimental data: why
only a single magnetic excitation, the zero field splitting characteristic of a SMM,
is observed despite the multitude of other excitations within the lowest spin multiplet. This may be contrasted to the standard magnetic measurements where hysteresis loops show several steps corresponding to different transitions [30]. We introduce the detailed transport models and the results of complementary calculations, extending previous theoretical works [27, 31–35]. We first present a perturbative picture and calculations which apply mainly to sample B, and then consider
renormalization effects which are more important for sample A which exhibits a
stronger tunnel coupling.
4.5. S UPPLEMENTARY INFORMATION
75
M ODERATE COUPLING , N ON - EQUILIBRIUM - R EAL -T IME P ERTURBA TION T HEORY (S AMPLE B)
We start with a perturbative argument for the dominant role of the ZFS excitation
using the sketch in Figure 4.1b. The ZFS excitation presents a bottleneck to all other
transitions if the spin-projection onto the easy-axis is conserved, i.e., when there is
no magnetic field component transverse to this axis. For the single electron tunneling (SET) regime this was discussed in Refs. [14, 31]. For cotunneling, involving
an electron-hole excitation [19], the change in the magnetic quantum number M
of the SMM is at most 1 for non-magnetic tunnel barriers. Therefore the excitation
above the ground state M = −S is the state with M = −S + 1, and the excitation energy is given by the ZFS (2S − 1)D. The transitions between subsequent magnetic
states M → M + 1 with 0 > M > −S have excitation energies (2|M | − 1)D. These
decrease linearly with M and are all smaller than the ZFS (2S −1)D. The ZFS transition therefore presents a bottleneck: the initial states for the lower lying transitions
M → M + 1 for M = −S + 1, −S + 2, ... are not occupied before the highest transition M = −S → M = −S + 1 has taken place. Therefore there are no resonant steplike features in the conductance for V < (2S − 1)D/e: at the ZFS excitation voltage
V = (2S − 1)D/e all allowed transport processes are activated at once. Clearly, the
above argument is weakened if transverse perturbations are taken into account,
as they break the selection rule |∆M | = 1 for inelastic tunneling. This violation can
arise from both a transverse magnetic field component as well as an intrinsic transverse anisotropy. However, we now show that under the experimental conditions
both effects are weak: low temperatures and small spin (but still S > 1/2) are required to see excitations other than the ZFS.
To include intrinsic transverse anisotropy we extend the Hamiltonian (1) in the
main text to the transport model first discussed in [31]. The Hamiltonian accounts
for the lowest spin multiplet of the two subsequent charge states of the SMM. For
charge state eN we have
£
¤
~ ·~
HˆN = −D N (S N )2z + 21 E N (S N )2+ + (S N )2− + g N µB B
SN
(4.5)
and correspondingly for state e(N − 1). Here (S N )± are the spin raising and lowering operators in charge state N . We take the parameters of sample B estimated in
section S2. For simplicity the magnetic field is taken to be at an angle of 71◦ with
the easy axis for both charge states. We consider the maximal effect of transverse
anisotropy terms by taking the ratio E N /D N = 0.2 which is larger than any of the ratios reported for neutral Fe4 derivatives in the bulk [10, 11]: E N /D N < 0.07. For the
charged state we assume the same ratio: E N −1 = 0.2D N −1 . The electron tunneling
P
is accounted for by a standard tunnel Hamiltonian HT = l r kσ t rl σ d l†σ c r kσ + h.c.
Here c r kσ is the electron operator of the electrode r = L, R, with density of states
76
4. M AGNETIC ANISOTROPY IN MULTIPLE CHARGE STATES OF AN F E4 MOLECULE
ρ and bandwidth W , and d l†σ (d l σ ) creates (annihilates) an electron with spinprojection σ in a single-particle state l of the SMM. As shown in [31] all the occurring tunnel couplings between the lowest spin multiplets can be fixed by specifying
only one for each electrode and making use of the Wigner-Eckart theorem. We take
the coupling to the leads to be equal, ΓL = ΓR = Γ, to identify the importance of
non-equilibrium effects. The transport properties are calculated using the quantum kinetic equation with rates calculated in perturbation theory using real-time
diagrams. In contrast to [31], the calculation includes all terms of both first and
second order in the tunneling coupling Γ (i.e., up to fourth order in HT ) and is valid
for Γ ¿ k B T . We also account for non-diagonal density matrix elements between
non-degenerate states, which we have shown to be of crucial importance if selection rules are violated [26], as is the case here. The calculations were done neglecting relaxation of the magnetic states due to processes other than the tunneling.
Including these will only further weaken the importance of transitions involving
states originating from the opposite side of the barrier.
In Figure 4.9a-d we show results of transport calculations for the Coulomb
blockade regime of the charged sector N with S = 5. Figure 4.9a shows the energy spectrum ²iN , where i labels the eigenstates, and the inset shows all lowenergy excitations ²iN − ²N
, i 6= j between states close to the ground state. In
j
Figures 4.9b,d the evolution of d I /dV and the occupations with bias voltage are
shown. Figure 4.9c shows the field evolution of the sub-ZFS conductance features corresponding to the inset of Figure 4.9a for the lowest temperature trace of
Figure 4.9b. Clearly, the ZFS is the dominant transition at the experimental temperature. Only when lowering the temperature k B T well below the ZFS energy
(2S − 1)D ∼
= 0.54 meV do new features develop below the ZFS threshold. These
are associated with quantum spin-tunneling through the uni-axial barrier [27, 36].
Such processes are enhanced when the transverse anisotropy or magnetic field
become stronger and the molecular spin S becomes smaller. Under the experimental conditions the above calculations indicate that the transverse anisotropy
effects can not be detected and therefore can be neglected when explaining the
experimental data. We note that at the lowest temperatures a peak develops on
top of the ZFS step since we account for non-equilibrium populations which the
Lambe-Jaklevic formula (4.2) neglects. At the experimental temperature this however cannot account for the enhancement seen in the other sample (A) which is
due to renormalization effects (see next section).
Strong Coupling, Equilibrium - Numerical Renormalization Group (Sample A)
The above calculations are perturbative in the tunnel coupling Γ to the electrodes
and it remains to be shown that the dominant role of the ZFS excitation in the differential conductance persists even for stronger coupling of the SMM to the elec-
4.5. S UPPLEMENTARY INFORMATION
77
a
b
c
d
F IGURE 4.9: Transport characteristics for sample B calculated from kinetic equations with parameters
as indicated in the text. (a) Energy spectrum for S N = 5. Inset: Lowest energy excitations between the
M∼
= ∓S and the M ∼
= ∓S ± 1 states. (b) Conductance for B = 0.375 T and at a gate voltage VG = 1.6 V
corresponding to Figure 4.4b. The two curves for temperatures T = 0.05, 0.3 K below the experimental
value of 1.6 K illustrate the existence of features below the ZFS gap due to quantum spin-tunneling. (c)
Map of d I /dV vs. V and B for the same parameters as in (b) for T = 0.05 K. The arrows mark the B =
0.375 T cross-section at which the conductance and occupations are shown in (b) and (d), respectively.
The ZFS excitation appears as the boundary to the white region since we have adjusted the color scale to
emphasize the small conductance features at lower bias. (d) Occupation probabilities for the four lowest
energy states of the SMM corresponding to the T = 0.05 K conductance curve in b. At low bias V ∼
=
0.12 mV the state with opposite maximum spin M ∼
= S N becomes accessible from the M ∼
= −S N ground
state without noticeably changing the conductance. This spin-forbidden process is only weakly allowed
by the transverse anisotropy and the transverse magnetic field. Next, at V ∼
= 0.39 mV and 0.47 mV the
two transitions between excited states M ∼
= SN → M ∼
= −S N + 1 and M ∼
= SN → M ∼
= S N − 1 lead to an
enhancement of the M ∼
= −S N ground state occupation and a change in the conductance. When the
spin-allowed ZFS bottleneck excitation M ∼
= −S N → M ∼
= −S N + 1 is reached at V ∼
= 0.56 mV all excited
states become increasingly occupied with bias V and the full inelastic current starts to flow.
78
4. M AGNETIC ANISOTROPY IN MULTIPLE CHARGE STATES OF AN F E4 MOLECULE
trodes, i.e., on the order of the experimental temperature or even larger Γ > k B T .
To this end we have performed numerical renormalization group (NRG) [37, 38]
calculations at zero temperature. To focus on the renormalization of inelastic cotunneling processes, we consider the Coulomb blockade regime where transport
through the SMM can be modeled by the following Kondo Hamiltonian
H = H N + J~
S N ·~
s + Helectrodes
(4.6)
as discussed in [31].
We assume that the dimensionless exchange coupling to the leads, ρ J , is effectively antiferromagnetic in sign, where ρ is the electronic density of states of the
electrodes. Our NRG calculations provide insight into the effect of order higher
spin-flip processes on the inelastic tunneling peak shape. In order to access the
differential conductance d I /dV we use the exact transport formula in [39], which
in the limit of low temperature can be evaluated using the non-equilibrium spindependent molecular spectral function A σ (ħω) as follows, assuming symmetric
tunnel coupling ΓL = ΓR = Γ:
µ µ
¶
µ
¶¶
e2 Γ X
eV
eV
dI
(V ) =
Aσ
+ Aσ −
.
dV
ħ 2 σ
2
2
(4.7)
Within NRG we are able to calculate non-perturbatively the equilibrium spectral
function [40]. Approximating A σ (ħω) in (4.7) by its equilibrium value, our conductance results are thus exact in the linear response regime (at zero temperature) and
provide a good approximation for the case of non-linear response at small temperature. We compute the spectral function in the Kondo model from the imaginary
part of the conduction band T-matrix, generalizing an approach presented in [41]
to spins S N > 1/2. As we are only interested in qualitative comparison to the experiment, we normalize the resulting conductance plots to the maximum of d I /dV .
The NRG results, shown in Figure 4.4f, indicate that strong conductance resonances develop due to spin-flip processes at strong coupling. We therefore infer that the peaks observed for sample A are related to the latter. The observed
split peak at zero field in Figure 4.6b can then be interpreted as a high-spin Kondo
peak which is split by the magnetic anisotropy, so that only ZFS (M = −S N ↔ M =
−S N + 1) spin-flip scattering processes prevail. Since the observed peak occurs at
finite bias it is further suppressed by non-equilibrium dephasing [42].
We finally note that we have chosen to omit the transverse anisotropy term (i.e.
E N = 0) which gives rise to a different low-temperature Kondo effect at zero bias for
a half integer spin SMM [27]. This Kondo effect involves only the doubly degenerate
ground state of such an isolated SMM [27]. By setting the temperature to zero in
the NRG calculations this different Kondo effect, however weak, will fully develop.
R EFERENCES
79
Extensive calculations show that this leads to a very narrow additional zero bias
Kondo resonance without affecting the inelastic peaks of interest here. For the Fe4
parameters of sample A, assuming again a transverse anisotropy of E = 0.2D (cf. the
perturbation theory above), the width of the zero-bias peak is approximately 103
times smaller than the ZFS. This shows that already for small finite temperatures,
well below the experimental value, this Kondo effect is suppressed. The transverse
anisotropy term can thus be neglected in explaining the experimental data.
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5
H IGH - SPIN KONDO EFFECT AND
SPIN TRANSITIONS
Parts of this chapter have been published in Synthetic Metals 161, 591 (2011).
83
84
5. H IGH - SPIN KONDO EFFECT AND SPIN TRANSITIONS
5.1 H IGHER - ENERGY EXCITATIONS
W
e now focus on higher bias voltages excitations in the tunneling spectra of
the devices discussed in the previous chapter [1]. Their stability diagrams
are shown in Figure 5.1. In the neutral state of sample A the energies of these transitions are (3.6 ± 0.4) and (6.1 ± 0.5) meV. For the neutral state of sample B they are
(3.9 ± 0.5) and (7.2 ± 0.5) meV. In bulk crystals spin transitions S = 5 → S = 4 have
been observed in the same energy range (5.7 − 8.3 meV) using inelastic neutron
scattering [2]. There is only one excitation in each of the adjacent charge states:
the energies are (5.0±0.4) meV for the reduced state (sample A) and (5.6±0.4) meV
for the oxidized state (sample B).
In the neutral state of sample A (Figure 5.2a,c) the 6.1 meV transition slightly decreases its energy to (5.5 ± 0.7) meV when the magnetic field strength is increased
to 8 T. The 3.6 meV transition does not change its energy. We must note, however,
that the precise determination of excitations energies is a hard task since transitions tend to broaden and may merge when the field is increasing. In the reduced
state of sample A (Figure 5.2b,d) the exitation energy is field independent within
experimental error.
In the oxidized state of sample B (Figure 5.3a,c) the excitation energy is (6.3 ±
0.4) meV at 8 T. Note the nonlinear increase for negative bias voltage: up to magnetic fields B ∼ 2 T the energy stays the same and then goes up. This behaviour
is similar to the magnetic field evolution of a ZFS when the angle θ is substantial
(see discussion of this sample in the previous chapter). It can be expected that spin
transitions follow the same pattern under that condition. This nonlinearity is also
present at the neutral state of sample B (Figure 5.3b,d): at B = 8 T the energy of
the highest-transition has increased to (7.8 ± 0.4) meV. The energy of the 3.9 meV
transition stays constant within the error when the magnetic field is swept to 8 T.
Attributing these transitions to spin or vibrational excitations is not straightforward. One would naturally expect vibrational excitations to remain constant under
magnetic field variation. Using this criterion a spin origin can be assigned to the
6.1 meV transition in the neutral state of sample A, to the 7.2 meV transition in the
neutral state of sample B and to the 5.6 meV transition in the oxidized state of sample B. The two transitions close to 4 meV would then be assigned to vibrational excitations of the molecule. However, spin origin cannot be ruled out completely for
these transitions since strong anisotropy can cause a significant deviation from the
familiar linear Zeeman evolution. To support this statement we have performed
transport calculations on a simple model system, details of which are discussed in
the following section.
5.1. H IGHER- ENERGY EXCITATIONS
(a)
85
(b)
dI/dV(μS)
10
dI/dV(μS)
20
30
0
15
10
N
-5
N+1
10
1.0
V (mV)
20
V (mV)
1.5
25
5
0
0.5
-10
5
N-1
N
0.0
0
-10
-2
-1
0
Vg (V)
1
-20
2
0.2
0.4
0.6
Vg (V)
-0.5
0.8
F IGURE 5.1: Stability diagrams of sample A (a) and B (b)
(a)
dI/dV(μS)
10
N
dI/dV(μS)
10
1.2
2.0
V (mV)
1.4
2.5
5
1.6
0
3.0
N+1
1.8
5
V (mV)
(b)
2.0
0
1.5
0.1
-5
-5
1.0
0.5
0.8
-10
0
2
4
B (T)
6
-10
0
8
(c)
10
6
8
2
4
B (T)
6
8
0.0
10
5
V (mV)
V (mV)
4
(d)
5
0
-5
-10
2
0
-5
0
2
4
B (T)
6
8
-10
0
F IGURE 5.2: Magnetic field evolution of sample A. (a) d I /dV as a function of magnetic field B and bias
voltage V for a fixed gate voltage Vg = −1.5 V (neutral state). (b) The same as (a) but for Vg = 2 V (reduced
state). (c) Higher-energy transitions as a function of magnetic field for the neutral state, extracted from
(a). ZFS is not shown for clarity. (d) The same for the reduced state, extracted from (b).
86
5. H IGH - SPIN KONDO EFFECT AND SPIN TRANSITIONS
(a)
dI/dV (nS)
10
(b)
N-1
N
0
400
-5
-10
(c)
V (mV)
V (mV)
5
600
200
0
2
4
B (T)
6
0
200
-5
100
0
2
-10
0
2
(d)
4
B (T)
6
8
10
5
V (mV)
V (mV)
300
0
5
0
0
-5
-5
-10
0
5
-10
8
10
dI/dV (nS)
400
10
800
2
4
B (T)
6
8
4
B (T)
6
8
F IGURE 5.3: Magnetic field evolution of a sample B. (a) d I /dV as a function of magnetic field B and bias
voltage V for a fixed gate voltage Vg = 0 V (oxidized state). (b) The same as (a) but for Vg = 1.6 V (neutral
state). (c) Higher-energies transitions dependence on a magnetic field for the oxidized state, extracted
from (a). ZFS is not shown for clarity. (d) The same for the neutral state, extracted from (b).
5.2. S PIN TRANSITION CALCULATIONS
a
87
b
0.02
S2
J
0.01
Energy (eV)
S1
0
-0.01
-0.02
-0.03
0
2
4
B (T)
6
8
F IGURE 5.4: (a) Schematic representation of the model magnetic dimer. (b) Energy spectrum obtained
by a diagonalization of the Hamiltonian (5.1) using S 1 = S 2 = 5/2, J = −2 meV, D i = −0.3 meV and
magnetic field collinear to the single-ions easy axes.
5.2 S PIN TRANSITION CALCULATIONS
Our model system consists of two magnetic ions with spins S~ˆ1 and S~ˆ2 , coupled
by an exchange interaction J (Figure 5.4a). Each ion is anisotropic with uniaxial
anisotropy D i . The system is subject to a Zeeman interaction with an external mag~ . The full Hamiltonian of the system reads
netic field B
~ , S~ˆ1 + S~ˆ2 ).
H Z F S = D i Sˆ12z + D i Sˆ22z + J (S~ˆ1 , S~ˆ2 ) − g µB (B
(5.1)
Ferromagnetic coupling J < 0 favors a high-spin configuration of the dimer.
When |J | À |D i |, the system is in the strong exchange limit: the total spin multiplets are separated from each other with large energy gaps and the energy spectrum looks like that of single-molecule magnets.
To calculate the current through the system we use a rate equation approach [3,
4] (see also chapter 2). We assume that the single-electron transport is occuring
through one of the ions and we define two charge states of the dimer. We fix the
ions spins in the charge state I as S 1 = 5/2 and S 2 = 2 and in the charge state II
(with one extra electron compared to the state I) as S 1 = S 2 = 5/2. We also assume
for simplicity that D i parameters and the exchange coupling J remain unchanged
upon charging although it is not generally the case as shown in chapter 4 [1].
First, we find the eigenstates of the Hamiltonian (5.1) for both charge states. To
do that we merge two Hilbert spaces of single ions using tensor multiplication [5, 6].
88
5. H IGH - SPIN KONDO EFFECT AND SPIN TRANSITIONS
Each ion is described by the spin-state vector:




a1
b1
|ψ1 〉 =  ...  , |ψ2 〉 =  ...  .
a 2S 1 +1
b 2S 2 +1
(5.2)
The states of the dimer are represented by a direct product of these two vectors:

a1 b1


a1 b2
.
|ψ1 ψ2 〉 = |ψ1 〉 ⊗ |ψ2 〉 = 


...
a 2S 1 +1 b 2S 2 +1

(5.3)
If an operator has to act only on one spin, it has to be tensor-multiplied by the
identity matrix: for example, if we act with Sˆz on the first ion, then
(Sˆz |ψ1 〉) ⊗ |ψ2 〉 = (Sˆz ⊗ I)|ψ1 ψ2 〉.
(5.4)
If we act on the second ion then
|ψ1 〉 ⊗ (Sˆz |ψ2 〉) = (I ⊗ Sˆz )|ψ1 ψ2 〉.
(5.5)
For S 1 = S 2 = 5/2 (charge state II), J = −2 meV, D i = −0.3 meV the spectrum
evolution in the magnetic field is shown in Figure 5.4b. The field is chosen collinear
to the easy axes of the ions. At the charge state II there are 6 multiplets with a total
spin S = 0, 1, ..., 5. Each multiplet S consists of 2S + 1 states giving rise to 36 levels in
total. At the charge state I (not shown) there are 5 multiplets with a total number of
30 levels.
The ground spin multiplet S = 5 is split by the anisotropy. We find the zero-field
splitting between the lowest two doubly-degenerate states (m = ±5 and m = ±4)
∆ Z F S ≈ 1.2 meV. Using this value we obtain the dimer’s uniaxial anisotropy parameter D (not to be confused with single ion’s anisotropy D i ): D = ∆ Z F S /(2S − 1) ≈
0.13 meV. The anisotropy barrier is U = DS 2 ≈ 3.3 meV. The energy of the next
(S = 4) spin multiplet is found to be ∆S ≈ 11.2 meV determined as the energy difference between the |m = ±4, S = 4〉 and |m = ±5, S = 5〉 states at zero field. The energy
gap between two spin multiplets is then ∆S −U ≈ 7.9 meV.
The current can be calculated after obtaining the steady-state (left part is zero)
solution of the equation
d P/d t = WP.
(5.6)
Here, P is the vector of the energy levels occupation probabilities including levels
in both charge states P i , i = 1, ..., 66 and W is a 66 × 66 matrix, whose elements Wi j
5.2. S PIN TRANSITION CALCULATIONS
89
are rates for transitions j → i for i 6= j , and Wi i = −
P
j 6=i
W j i are rates of depopulat-
ing states i . Note that the rates for transitions between the levels within the same
charge state (Wi j , where i 6= j and both i ≤ 30 and j ≤ 30, or both i > 30 and j > 30)
are zero.
Non-zero rates are the product of two terms: one related to the electrochemical
potential of the transition relative to the Fermi levels of the leads (that are tuned
by the bias voltage) and one related to Clebsch-Gordan coefficients. The first term
is given by equation (2.16) or (2.17). To compute the second term we must first
rewrite the dimer’s states, that are given now in an uncoupled basis produced by a
tensor multiplication of individual ions spin eigenvectors |m 1 m 2 〉 = |m 1 〉 ⊗ |m 2 〉, in
terms of a total spin states [7]
|Sm〉 =
X
m 1 ,m 2
C SSm
|m 1 m 2 〉.
1 S 2 m1 m2
(5.7)
Here for charge state I m 1 = −5/2, ..., 5/2, m 2 = −2, ..., 2, S = 1/2, ..., 9/2. For charge
state II m 1 = −5/2, ..., 5/2, m 2 = −5/2, ..., 5/2, S = 0, ..., 5. m = −S, ..., S for both charge
states and C SSmS m m are the Clebsch-Gordan coefficients. The dimensionality of
1 2 1 2
the resulting vectors is different for different spin multiplets (2S+1 for spin S), however they still describe of course the stationary states of the system.
Due to the selection rules single-electron tunneling is only possible between
the states whose total spin is different by 1/2; all the other transition rates are zero.
At low bias it is thus only possible to observe excitations from the ground multiplet of the charge state I (S = 9/2) to the ground or the first excited multiplet of
the charge state II (S = 5 or S = 4). S = 9/2 components are described by the 10dimensional vectors, S = 5 components are 11-dimensional and S = 4 components
are 9-dimensional:

 5, j 
 4, j 
Ψ1
Ψ1
Ψ9/2,i
1
 9/2,i 
 5, j 
 4, j 
Ψ

Ψ 
Ψ 
|Ψ9/2,i 〉 =  2  , |Ψ5, j 〉 =  2  , |Ψ4, j 〉 =  2  .
 ... 
 ... 
 ... 
5, j
4, j
Ψ9/2,i
Ψ
Ψ9
10
11

(5.8)
The rate of transition between |Ψ9/2,i 〉 and |Ψ5, j 〉 is proportional to
10
10
1 X
1 X
5, j
5, j
i,j
5,5−k+1
5,4−k+1
Ψ9/2,i
Ψk )2 + ( C 9/2,1/2,9/2−k+1,−1/2
Ψ9/2,i
Ψk+1 )2 .
RCG = ( C 9/2,1/2,9/2−k+1,1/2
k
k
2 k=1
2 k=1
(5.9)
The first term in (5.9) is the rate associated with tunneling of a "spin up" electron
while the second term is the rate associated with a "spin down" electron [8]. For
90
5. H IGH - SPIN KONDO EFFECT AND SPIN TRANSITIONS
transitions between |Ψ9/2,i 〉 and |Ψ4, j 〉 the rate is proportional to
9
9
1 X
1 X
i,j
4, j 2
4, j
4,4−k+1
RCG = ( C 9/2,1/2,9/2−k+1,−1/2
Ψ9/2,i
Ψ
)
+
C 4,4−k+1
Ψ9/2,i Ψ )2 .
(
k
k
2 k=1
2 k=1 9/2,1/2,9/2−k+2,1/2 k+1 k
(5.10)
Finally the tunneling matrix elements are given by
1 i,j S S
D
R (Γ f + ΓD
i j fi j )
ħ CG i j i j
(5.11)
1 i,j S
D
R (Γ (1 − f iSj ) + ΓD
i j (1 − f i j )).
ħ CG i j
(5.12)
Wji =
and
Wi j =
The stability diagrams and d I /dV traces obtained by solving the rate equation (5.6) at magnetic fields B = 0 T and B = 8 T and at a temperature T = 2 K (close
to experimental conditions) are shown in Figure 5.5. There are two SET lines: a thin
line at low bias and a broad line at higher bias. The energy of the first excitation is
∼ 1.2 meV, which is the energy of the zero-field splitting ∆ Z F S of the charge state
II. The second line is a band with an energy ∼ 9 − 11 meV, which is the energy of
intermultiplet excitations, changing the spin of the dimer from S = 5 to S = 4.
To resolve the fine structure of the higher energy excitations the calculations
were also performed with the same set of parameters except the temperature has
been set to a lower value of T = 0.5 K. The d I /dV curves at B = 0 T and B = 8 T are
shown in Figure 5.5d. Analysis of the energies allows us to assign the transitions as
shown in Table 5.1.
Note that there is only one transition between the ground state multiplets:
|9/2, 9/2〉 → |4, 4〉. This is consistent with the previous measurements and calculations performed within the giant spin approximation [1, 8, 9]. The reason for the
absence of all the other excitations in the low-energy spectrum is that in the case
of easy-axis anisotropy the level spacing between consecutive |m〉 states decreases
with increasing energy of the states (decreasing m). The largest spacing is thus
between the ground state and the first excited state (|5, 5〉 and |5, 4〉 in this case).
Therefore, when the bias voltage becomes large enough to induce a ZFS transition
|9/2, 9/2〉 → |5, 4〉 and occupy the |5, 4〉 state, the bias is already large enough to induce all the other transitions between the ground state multiplets S = 9/2 and S = 5
and thus to occupy all the other states.
In contrast, for the transitions from the ground state multiplet S = 9/2 to the excited state multiplet S = 4, there are more components visible. This is because the
lowest energy, required to induce such a transition, is higher than the anisotropy
energy barrier U . Therefore, when the bias voltage becomes sufficient, the state
5.2. S PIN TRANSITION CALCULATIONS
91
a
b
30
10
30
10
20
8
20
8
10
6
0
4
6
10
V (m V )
V (m V )
4
0
2
2
-10
-10
0
-20
-30
c
-2
-30
0
0.01
0.02
Vg (V)
0.03
0.01
0.02
Vg (V)
0.03
e
d
5
2
4
dI/dV (a.u.)
dI/dV (a.u.)
0
d
6
3
2
1.5
c
a b
1
0.5
1
0
-30
0
-20
-2
-20
-10
0
Vb (mV)
10
20
30
0
24
26
V (mV)
28
30
F IGURE 5.5: (a) and (b) Calculated d I /dV maps for temperature T = 2 K. Only the excitations of the
charge state II are shown. (a) B = 0 T. (b) B = 8 T. (c) d I /dV trace at fixed Vg = −3 mV. Solid line is at
zero field, dashed line is at B = 8 T. (d) Fine structure of the high-energy band, calculated at T = 1 K for
B = 0 T (solid line) and B = 8 T (dashed line). Labelling corresponds to Table 5.1
label
a
b
c
d
e
Energy at B = 0 T
9.8 meV
10.1 meV
10.4 meV
10.7 meV
11.2 meV
transition
|9/2, 1/2〉 → |4, 0〉
|9/2, 3/2〉 → |4, 1〉
|9/2, 5/2〉 → |4, 2〉
|9/2, 7/2〉 → |4, 3〉
|9/2, 9/2〉 → |4, 4〉
TABLE 5.1: Fine structure of the spin transition. Energies are deduced from intersections of the excitation lines with a diamond edge at T = 0.5 K.
92
5. H IGH - SPIN KONDO EFFECT AND SPIN TRANSITIONS
30
8
7
Vb (mV)
28
6
5
26
4
3
2
24
1
0
2
4
B (T)
6
8
0
F IGURE 5.6: Calculated magnetic field evolution of SET excitations at a fixed gate voltage Vg = −1 mV.
contributing to this transition is occupied, and the transition takes place. With increasing bias more transitions become energetically available resulting in multiple
peaks in d I /dV .
Another difference between ground-to-ground and ground-to-excited spin
multiplet transitions is their magnetic field behavior. As can be seen from Figure 5.5d the components of the latter split in two lines, whereas only one ZFS
component is seen in the magnetic field. This behaviour can be explained using
the same argumentation: for the states with a negative m energy spacing between
the levels is smaller than for the states with the same |m| from another side of the
well. In the ground spin multiplet state |5, −5〉 has a higher energy at a finite magnetic field than state |5, 5〉 and is not occupied at low bias. Due to the selection rule
|∆m| = 1/2 it can only become occupied when it becomes possible to overcome
an anisotropy barrier, however the energy of transition |5, −5〉 → |5, −4〉 is smaller
and this transition is not seen. For the S = 9/2 → S = 4 transitions all the states
contributing to any transitions are available when these transitions become energetically allowed.
We have also performed calculations of the magnetic field evolution of SET
lines at a fixed bias voltage. The results for T = 2K and the angle between the magnetic field and the easy axis of the molecule θ = 0 are shown in Figure 5.6. Note
that for stronger electronic coupling Γ the splitting of the lines can appear as their
broadening, like in the measured data for sample A.
5.3. H IGH - SPIN KONDO EFFECT
93
The model discussed in this section is of course a much simpler system than the
Fe4 molecule. Also the calculations were performed for SET transport whereas the
measurements revealed inelastic cotunneling excitations. However, the main conclusion of the calculations is general: in contrast to the ZFS excitations spin transitions appear in the electron transport not as a single line, but as a broad band, consisting of multiple transitions between different components of the spin multiplets
involved. Moreover, the amplitude of the transition connecting the lowest-energies
components (like those with |m| = S at zero field, marked as "e" in Figure 5.5d) is
not much higher than the amplitude of other components. Consequently, magnetic field behaviour of the spin multiplets is quite complicated even for magnetic
field direction matching the easy axis of the molecule.
5.3 H IGH - SPIN KONDO EFFECT
The Kondo effect was observed previously in various molecules [10–14], and it was
shown theoretically that quantum tunneling of magnetization (QTM) can induce a
pseudospin 1/2 Kondo effect in SMMs [15, 16]. In a single Fe atom residing on a surface, Kondo resonances have been probed by scanning tunneling microscopy [17].
No experimental observation of a Kondo effect of any type in a SMM has been reported.
In general, the Kondo effect is probably the most frequently observed feature
in our measurements. Virtually every sample (consisting of 16-24 electromigrated
junctions in total) contains at least one device demonstrating a zero-bias conductance resonance. However, care should be taken when interpreting such data, as it
is known that even bare gold electromigrated junctions without molecules can produce Kondo effect [18]. Only devices with additional signatures of single molecules
are measured in detail.
We have observed five devices in which zero-bias conductance resonance was
present in two adjacent charge states. Here we focus on two of them, for which
the stability diagrams are presented in Figure 5.7a,b. An essential requirement
for the observation of the Kondo effect is a non-zero spin. Since the difference in
spin values for adjacent charge states cannot be lower than the free electron spin
1/2, at least one of the states in such devices necessarily must have a high spin in
the ground state, i.e., S ≥ 1. This argumentation implies that the common evenodd electron orbitals filling order, where even number of electrons residing on the
molecules results in a molecular spin S = 0 and odd number results in S = 1/2, is
violated. This situation is unlikely for gold nanoparticles, that are in general diamagnetic, whereas it is natural for high-spin anisotropic single-molecule magnets.
Figure 5.7a,b demonstrate conductance maps of two samples (sample C with
Fe4 Ph derivative, sample D with Fe4 C9 SAc) exhibiting zero-bias resonances in ad-
94
5. H IGH - SPIN KONDO EFFECT AND SPIN TRANSITIONS
(a)
dI/dV (μS)
(b)
dI/dV (nS)
300
50
4
15
3
5
0
2
-5
-10
200
V (mV)
V (mV)
10
0
100
1
0
-15
0
0.5
1
Vg (V)
-50
-1.5
1.5
(c)
-1
-0.5
Vg (V)
0
0.5
(d)
2
dI/dV (nS)
dI/dV (μS)
35
1.5
1
15
0.5
-5
0
V (mV)
(e)
5
-5
(f)
dI/dV (nS)
2
dI/dV (μS)
25
1.5
0
V (mV)
5
30
20
1
10
0.5
-5
0
V (mV)
5
-5
0
V (mV)
5
F IGURE 5.7: Kondo effect. (a) d I /dV map of sample C. (b) d I /dV map of sample D. (c)-(f) d I /dV traces taken at B = 0 T (blue curves) and B = 8 T (red curves). (c) Sample C: Vg = 0 V. (d) Sample D:
Vg = −0.65 V. (e) Sample C: Vg = 1.7 V. (f) Sample D: Vg = −0.2 V.
5.3. H IGH - SPIN KONDO EFFECT
95
b
dI/dV (uS)
20
2K
8K
20K
35K
15
10
2K
8K
12K
20K
50
dI/dV (nS)
a
40
30
20
5
-10
0
10
V (mV)
c
-10
-5
0
5
10
V (mV)
d
2K
8K
20K
35K
15
10
2K
8K
12K
20K
40
dI/dV (nS)
dI/dV (uS)
20
30
20
5
-10
0
10
-10
-5
V (mV)
0
5
10
Vb (mV)
F IGURE 5.8: dI/dV traces taken at different temperatures. (a) Sample C, Vg = −0.6 V. (b) Sample D,
Vg = −0.65 V. The curves are offset for clarity. (c) Sample C, Vg = 2.4 V. (d) Sample D, Vg = −0.2 V.
a
b
c
20
45
20
dI/dV (uS)
40
15
35
15
30
10
25
10
1
10
T (K)
1
10
T (K)
1
10
T (K)
F IGURE 5.9: Temperature dependences of the zero-bias conductance. Solid lines are fits to the S = 1/2
theoretical dependence (eq. 5.13) (a) Sample C: Vg = −0.6 V. (b) Sample C: Vg = 2.4 V. (c) Sample D:
Vg = −0.2 V.
96
5. H IGH - SPIN KONDO EFFECT AND SPIN TRANSITIONS
jacent charge states. For both samples the resonance width increases when approaching the degeneracy-points (Vg ≈ 1.0 V for sample C and Vg ≈ −0.5 V for
sample D). At these points states with electrons number different by one are degenerate, giving rise to a large single-electron tunneling current. The observed
gate-dependence of the intensity of the zero-bias resonance implies that the Kondo
effect and Coulomb blockade result from the same molecule.
Figure 5.7c-f demonstrate dI/dV curves at different charge states of the two
devices taken at approximately the same gate voltage offset from the degeneracypoints. While for a sample C (Figure 5.7c,e) the zero-bias resonances have almost
the same amplitude in the two charge states, there is a significant difference in the
peak height and width for a sample D (Figure 5.7d,f). This difference can be caused
by a change of the Kondo temperature TK in the two charge states due to a stronger
spin screening at the right state [19]. At a magnetic field B = 8 T the peaks show the
expected splitting of ∆ = 2g µB B with g ≈ 2.
We performed differential conductance measurements at different temperatures. The results are shown in Figure 5.8. Consistently with the magnetic field data
the temperature evolution of d I /dV is similar for two charge states of the sample C
(Figure 5.8a,c), but looks very different for two charge states of the sample D (Figure 5.8b,d). In one of the states of this sample maximum value of d I /dV shows
almost no temperature dependence (Figure 5.8b).
The zero-bias conductance as a function of temperature is shown in Figure 5.9
for both charge states of the sample C and for the right-hand charge state of the
sample D. We find that the amplitudes of the conductance peaks follow a logarithmic behaviour. We fit the experimental curves to an empirical S = 1/2 Kondo
temperature dependence [20]
G(T ) = G el +G 0
4ΓS ΓD
[1 + (21/s − 1(T /TK )2 ]−s ,
ΓS + ΓD
(5.13)
using G el ,G 0 and TK as fit parameters and fixing s = 0.22. For the sample C the
fits yield Kondo temperatures of (13 ± 1) K at Vg = −0.6V (left-hand charge state)
and (10 ± 4) K at Vg = 2.4V (right-hand charge state). For the sample D Kondo
temperature derived from the fit at Vg = −0.2V is (14 ± 3) K. Similar analysis for the
left-hand charge state of the sample D is impossible as the amplitude of the zerobias peak changes very slowly in the available temperature range. In combination
with a small value of the amplitude compared to the background conductance this
indicates that the Kondo temperature is lower than the lowest temperature reached
in these measurements.
We point out that the Kondo effect in these samples does not involve QTM,
since the level broadening due to the hybridization with the electrodes is too
strong. The molecule-lead coupling is Γ ≈ 6 meV for the sample C and Γ ≈ 37 meV
R EFERENCES
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for the sample D, as estimated from the width of Coulomb peaks [21]. These values
are much larger than the height of the anisotropy barrier in the ground spin multiplet (S = 5) of the neutral Fe4 molecule (U = 1.4 meV). Such a high coupling reintroduces the degeneracy of the spin multiplet so that the molecular spin is effectively
screened. Observation of QTM-induced Kondo effect remains thus an intriguing
challenge that requires lower temperature and electronic coupling.
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Effect in Electromigrated Gold Break Junctions, Nano Lett. 5, 1685 (2005).
[19] N. Roch, S. Florens, T. A. Costi, W. Wernsdorfer, and F. Balestro, Observation of
the Underscreened Kondo Effect in a Molecular Transistor, Phys. Rev. Lett. 103,
197202 (2009).
[20] D. Goldhaber-Gordon, J. Göres, M. A. Kastner, H. Strikman, D. Mahalu, and
U. Meirav, From the Kondo Regime to the Mixed-Valence Regime in a SingleElectron Transistor, Phys. Rev. Lett. 81, 5225 (1998).
[21] E. A. Osorio, T. Bjørnholm, J.-M. Lehn, M. Ruben, and H. S. J. van der Zant,
Single-molecule transport in three-terminal devices, J. Phys.: Condens. Matter.
20, 374121 (2008).
6
G ATE - VOLTAGE SPECTROSCOPY
99
100
6. G ATE - VOLTAGE SPECTROSCOPY
6.1 M EASUREMENT AND ANALYSIS TECHNIQUE
I
n three-terminal single-molecule junctions the gate voltage can be used to add
or subtract electrons one by one to the molecule (reproducible reduction or oxidation of the molecule respectively). A finite bias voltage on the other hand can
probe transitions to the excited states of the molecule. If no bias voltage is applied
the molecule stays in its ground state. When the gate voltage is swept so that one
electron is added and the bias voltage is fixed at zero the molecule undergoes a
transition from the neutral ground state (N ) to the oxidized ground state (N +1). In
a stability diagram this transition corresponds to a crossing point of the Coulomb
diamond edges and to a Coulomb peak at the d I /dV versus Vg trace at V = 0.
If a magnetic field is applied the energies of the adjacent ground states E N and
E N +1 will change. In the simplest case the spin of molecule changes by ∆S = 1/2
upon addition of the electron. If the molecule is isotropic the chemical potential
changes linearly with the magnetic field B :
∆E (B ) = (E N +1 (B ) − E N (B )) − (E N +1 (0) − E N (0)) = −g µB B ∆S,
(6.1)
where g is the Landé factor and µB is the Bohr magneton. This change of the chemical potential corresponds to a shift of the d I /dV peak
∆Vg (B ) = −∆E (B )/β = −g µB B ∆S/β,
(6.2)
where β is the gate coupling [1]. In case ∆S = +1/2, i.e. if the spin increases upon
oxidation the shift is towards negative gate voltages; for ∆S = −1/2 the shift is towards positive gate voltages. If the molecule is anisotropic, however, its spectrum
depends on the mutual orientation of the magnetic field and anisotropy axes [2]
and in general case, the energy evolution ∆E (B ) is nonlinear.
In this work we measure ∆VG (B ) to quantify the anisotropy of an individual single-molecule magnet. We call this technique a gate-voltage spectroscopy
since it does not involve a bias voltage in contrast to conventional transport spectroscopy. We study the Fe4 Ph derivative [3] from a tetrairon (III) family [4]. The
molecules are deposited from solution into gold electromigrated three-terminal
junctions [5]. Details of the fabrication process and junction preparation are reported elsewhere [2, 6].
Measurements are performed at a temperature of 1.8-1.9 K in a cryostat
equipped with a superconducting magnet. To directly assess the anisotropic properties the sample is mounted on a piezoelectric rotating stage with a rotation axis
perpendicular to the magnetic field direction. The orientation of the sample with
respect to the field is determined by measuring the Hall voltage across a homemade InAs Hall bar integrated into the chip carrier (see chapter 3 for more details
on the experimental setup).
6.2. R ESULTS AND DISCUSSION
101
a
b
dI/dV (nS)
20
2000
1500
0
1000
500
-10
-20
dI/dV (nS)
V (mV)
10
0
0.6
0.7
0.8
Vg (V)
0T
3T
6T
2000
0.9
1500
1000
500
0.68
0.70
0.69
Vg (V)
0.71
F IGURE 6.1: (a) d I /dV map as a function of gate Vg and bias V voltage at a magnetic field B = 0 T for
sample A. (b) d I /dV as a function of gate voltage Vg at V = 0 for three different magnetic field values.
6.2 R ESULTS AND DISCUSSION
Figure 6.1a shows a stability diagram of a sample in the intermediate coupling
regime at zero magnetic field. We estimate the electronic coupling Γ from the full
width at half maximum (FWHM) of the differential conductance peak along the diamond edge [1]. From such a d I /dV versus V trace we find a value of ∼ 3 meV.
This value is higher than the ZFS of Fe4 molecules (∼ 0.6 − 0.9 meV) [2] and SET
excitations at this energy scale cannot therefore be resolved. Moreover, there is a
broad zero-bias resonance present in both charge states indicative of a Kondo effect. The fact that a Kondo resonance is visible in adjacent charge states points at
high-spin ground states as discussed in chapter 5. In the remainder of this chapter
we will show that the single-molecule magnetic anisotropy can be characterized
using gate-voltage spectroscopy.
We measured d I /dV versus Vg curves using a lock-in modulation of 0.1 mV
RMS bias voltage with no DC signal applied between source and drain, while
sweeping the magnetic field perpendicular to the sample plane (α = 90◦ ). Figure 6.1b shows Coulomb peaks obtained with this measurements at three different magnetic field values. The measurement was repeated after the sample had
been rotated such that its plane was parallel to the field (α = 0◦ ). To obtain the
gate voltage values corresponding to the maxima of d I /dV we fitted the measured
Coulomb peaks to a Gaussian function. The extracted maximum gate voltages
are plotted against the magnetic field in Figure 6.2a. The peak position shows a
clear nonlinear dependence indicating magnetic anisotropy. Moreover, the lowfield behaviour differs significantly for these two angles, which is a direct evidence
102
6. G ATE - VOLTAGE SPECTROSCOPY
a
b
-893
sample A
sample B
Vg (mV)
Vg (mV)
700
698
-894
696
-6
-4
-2
0
B (T)
2
4
-895
-6
6
-4
-2
0
B (T)
2
4
F IGURE 6.2: Coulomb peak position as a function of magnetic field for the sample plane perpendicular
to the field (circles) and parallel to the field (crosses). (a) Sample A. (b) Sample B.
of anisotropy.
To obtain a quantitative estimate of the anisotropy parameters we performed
similar measurements at several angles α and compared the results of these measurements with numerically calculated Coulomb peak positions (Figure 6.3). The
calculation procedure is as follows: we first find the ground states energies E N +1
and E N for each magnetic field value by diagonalizing the spin Hamiltonian
~ ·~
Ĥ N (N +1) = D N (N +1) Ŝ 2zN (N +1) + g N (N +1) µB B
SˆN (N +1) .
(6.3)
We do not take into account an electrostatic charging energy since it does not
change in the magnetic field and thus does not contribute to a peak shift. The
chemical potential value E N +1 − E N is then divided by β = 0.08 (the gate coupling
obtained from the slopes of Coulomb diamond edges from Figure 6.1a [1]) and
the resulting gate voltage is offset such that it matches the experimental data at
B = 0 T. We assume that the left-hand charge state in Figure 6.1a is the neutral one
α
90◦
60◦
30◦
0◦
θN
63◦
74◦
83◦
87◦
θN +1
62◦
73◦
82◦
85◦
TABLE 6.1: Angles θi between the easy-axes of two charge states N and N + 1 and magnetic field, estimated from a comparison of the measurements of the sample A with the model calculations. The four
values of α correspond to four different positions of the sample rotator.
6
6.2. R ESULTS AND DISCUSSION
103
a
b
700
α = 90
o
700
B
α = 60
o
φ
698
Vg (mV)
Vg (mV)
θ
e.a.
γ
696
696
-6
-4
-2
0
B (T)
2
4
6
c
-6
-4
-2
0
B (T)
2
4
-2
0
B (T)
2
4
6
d
α = 30
o
700
Vg (mV)
700
Vg (mV)
698
698
α=0
o
698
696
696
-6
-4
-2
0
B (T)
2
4
6
-6
-4
6
F IGURE 6.3: Comparison of the Coulomb peak position as a function of a magnetic field with the model
(sample A). Solid lines are the best matching calculated curves. For all the curves S N = 5, S N +1 = 9/2,
D N = −0.056 meV, D N +1 = −0.068 meV. The angles θN and θN +1 are given in Table 6.1 . (a) Sample
perpendicular to the field (α = 90◦ ). Inset: setup geometry. Easy axis (e.a.) makes an angle θ with the
magnetic field B and an angle γ with the rotator plane. φ is the angle between the projection of the
easy-axis on the rotator plane and B (b) Sample at an angle α = 60◦ with the field. (c) Sample at an angle
α = 30◦ with the field. (d) Sample parallel to the field (α = 0◦ ).
and that its spin and anisotropy correspond to those of the Fe4 Ph bulk phase [3]:
S N = 5, D N +1 = −0.056 meV. We also assume that the g -factors are equal in both
charge states: g N = g N +1 = g . Next, taking into account that at high magnetic
~ ·~
fields the Zeeman interaction g µB B
Sˆ dominates over the anisotropy D Ŝ 2z and that
the energy gap between the two charge states increases with B in this region i.e.,
the Coulomb peak shifts to a higher gate voltage while B increases, we conclude
that the spin of the reduced charge state is lower than the one of the neutral state:
S N +1 < S N . Since no spin blockade signatures are observed, the spin difference
should be 1/2, so S N +1 = 9/2. There are four fitting parameters: g , D N +1 and the
angles θN and θN +1 between the easy-axes and the magnetic field. The impact
104
6. G ATE - VOLTAGE SPECTROSCOPY
of the different parameters is discussed in detail in the next section, here we only
point out that it is natural to keep the same values of g and D N +1 for all the angles
α and change only θN and θN +1 . With these assumptions we found that the peak
positions are well reproduced when using g = 2.0 and D N +1 = −0.068 meV. The fits
are shown in Figure 6.3 as the solid red lines. The values of θN and θN +1 are given
in Table 6.1.
Using these values one can determine the angle γ between the rotator plane
and the easy axis. This angle should not change when the sample is rotated and its
preservation would be an indication for the consistency of the fitting procedure. It
can be shown that θ is bound to γ by the following relationship:
cosθ = cos γcos φ.
(6.4)
Here, φ is the angle between the projection of the easy-axis on the rotator plane
and the magnetic field (see inset in Figure 6.3a). The change of this angle ∆φ upon
rotation is equal to the rotation angle: ∆φ = ∆α.
We write equation (6.4) for two angles α1 = 90◦ and α2 = 60◦ :
cosθ1 = cos γ cos φ1 ,
(6.5)
cosθ2 = cos γ cos φ2 .
(6.6)
Taking into account that φ2 −φ1 = α2 −α1 = −30 and dividing (6.5) by (6.6) we find
◦
cos φ1
cos θ1
=
cos θ2 cos(φ1 − 30◦ )
(6.7)
Substituting θ1 = 63◦ and θ2 = 74◦ (the angles found from the fits for the charge
state N (Figure 6.3a,b)) and after simple trigonometry we obtain cos φ1 = 0.90, from
which it follows that γ = 60◦ (equation (6.5)).
We have repeated this procedure for different pairs of angles α1,2 . The results
are summarized in Table 6.2. We find that apart from the cases of α1 = 30◦ , α2 = 0◦
and α1 = 60◦ , α2 = 0◦ the values of γ are close to each other, indicating consistency
of our analysis. The two above mentioned cases correspond to a very small value of
cos θ2 (the easy-axis almost perpendicular to the field) and in this region γ cannot
be determined reliably using this method.
We have also performed gate-voltage spectroscopy on another sample; the results are shown in Figure 6.2b. Note that in this case a 90◦ rotation of the sample
stage almost did not change the shape of the magnetic field dependence. This indicates that the easy-axes in both charge states are almost matching with the rotation
axis (perpendicular to the rotator plane); i.e. the θi ’s are close to 90◦ . This is consistent with the shape of the curves: our analysis shows that a local maximum around
6.3. N UMERICAL CALCULATIONS
α2 = 90
α2 = 60◦
α2 = 30◦
α2 = 0◦
◦
α1 = 90◦
x
60◦
62◦
63◦
105
α1 = 60◦
60◦
x
68◦
73◦
α1 = 30◦
62◦
68◦
x
81◦
α1 = 0◦
63◦
73◦
81◦
x
TABLE 6.2: Angles γ found by substitution of angles θN , obtained from the fits for angles α1,2 into
equation (6.7).
zero magnetic field is characteristic for devices in which one of the angles θi is close
to 90◦ (see next section).
6.3 N UMERICAL CALCULATIONS
As was already discussed, the description of gate-voltage spectroscopy of
anisotropic molecules involves a large number of parameters. Even the simplest
case of easy-axis anisotropy involves already eight parameters: the spins S i , two g factors, the two D i parameters and the angles θi between the easy-axes and the
magnetic field for both charge states. Introduction of transverse anisotropy requires four additional variables: the values of E i and the angles ψi between the
hard-axes and the magnetic field. We performed a series of calculations with different sets of parameters to determine the influence of each on the magnetic field
evolution of the peak position. The results are presented in Figure 6.4.
We start by fixing S N = 5 and S N +1 = 9/2, g =2, D N = D N +1 = −0.056 meV and
varying the angle θN = θN +1 (Figure 6.4a). Interchanging the spins would lead to
a change of direction of the curve from increasing with the field to decreasing. We
find that the Zeeman evolution becomes nonlinear when the angles θ are about
60◦ and it is linear below that value. Note the characteristic shape of the curve for
θ = 90◦ : the curve has a local maximum at zero field.
We further investigate the case when both angles θ are close to 90◦ . In Figure 6.4b we see that the local maximum at zero field disappear quickly as the D N +1
exceeds D N . Nevertheless, there are two distinct regions in all these curves: at low
field (below B c =∼ 3 T ≈ 3D N /µB ) the spectrum is dominated by anisotropy and at
higher field the evolution is the linear Zeeman dependence. The same two regions
are seen if D N =D N +1 and one of the angles is different from 90◦ (Figure 6.4c,d). The
maximum at zero is more pronounced and sharp when θN +1 deviates further from
90◦ . This can be easily understood: the energy of the ground state N + 1 is changing faster in the field when θN is smaller while the energy of the state N with the
easy axis perpendicular to the field stays almost constant until the magnetic field
dominates over anisotropy.
106
6. G ATE - VOLTAGE SPECTROSCOPY
a
b
0.8
0.6
Energy (meV)
Energy (meV)
0.3
0.2
0.1
0.4
0.2
0
0
-6
-4
-2
0
B (T)
2
4
-0.2
-6
6
c
-4
-2
0
B (T)
2
4
6
-4
-2
0
B (T)
2
4
6
d
0.2
Energy (meV)
Energy (meV)
0
-0.5
0.1
0
-0.1
-1
-6
-4
-2
0
B (T)
2
4
6
-0.2
-6
F IGURE 6.4: Calculated energy of the ground state of the charge state N + 1 as a function of magnetic
field. For all the diagrams S N = 5, S N +1 = 9/2. (a) D N = D N +1 = −0.056 meV; θN = θN +1 = 40◦ (black),
60◦ (red), 70◦ (blue), 80◦ (green), 90◦ (purple). (b) θN = θN +1 = 90◦ , D N = −0.056 meV; D N +1 = −0.056
(black), −0.06 (red), −0.07 (blue), −0.08 (green), −0.09 (puple) meV. (c) D N = D N +1 = −0.056 meV, θN =
90◦ ; θN +1 = 0◦ (black), 30◦ (red), 45◦ (blue), 60◦ (green), 75◦ (purple), 90◦ (cyan). (d) D N = D N +1 =
−0.056 meV, θN = 90◦ ; θN +1 = 90◦ (black), 88◦ (red), 86◦ (blue), 84◦ (green), 82◦ (purple).
6.3. N UMERICAL CALCULATIONS
107
a
b
700
Vg (mV)
Vg (mV)
700
698
698
696
696
-6
-4
-2
0
B (T)
2
4
6
c
-6
-4
-2
0
B (T)
2
4
6
-4
-2
0
B (T)
2
4
6
d
700
Vg (mV)
Vg (mV)
700
698
696
696
-6
698
-4
-2
0
B (T)
2
4
6
-6
F IGURE 6.5: Comparison of the Coulomb peak position as a function of a magnetic field with the model
(sample A). Solid lines are best matching calculated curves. For all the curves S N = 5, S N +1 = 9/2,
g N = 2.0, g N +1 = 1.7, D N = −0.056 meV, D N +1 = −0.039 meV, angles θN and θN +1 are given in Table 6.3
. (a) Sample perpendicular to the field (α = 90◦ ). (b) Sample at the angle α = 60◦ with the field. (c)
Sample at the angle α = 30◦ with the field. (d) Sample parallel to the field (α = 0◦ ).
108
6. G ATE - VOLTAGE SPECTROSCOPY
α
90◦
60◦
30◦
0◦
θN
74◦
76◦
79◦
82◦
θN +1
73◦
75◦
78◦
81◦
TABLE 6.3: Angles θi between the easy-axes of two charge states N and N + 1 and magnetic field, estimated from comparison of the measurements of the sample A with the model calculations (Figure 6.5).
The four values of α correspond to four different positions of the sample rotator.
The previous calculations were done under the assumption that the g -factors
are the same in both charge states. However, if that is not the case, the shape of
the gate-voltage spectroscopy curves changes drastically. It can even change from
increasing with a field to decreasing: for S N = 5, S N = 9/2, g N = 2 this happens
when g N +1 exceeds ∼ 2.2. This value of g N +1 is independent on the anisotropy
parameters.
Relaxing the assumption that the g -factors shoud be the same in both charge
states, we made an alternative fitting of the data from sample A, which is shown in
Figure 6.5. Here, g N +1 = 1.7, leaving g N , S N , S N +1 , D N the same as in the previous
fits and changing D N +1 , θN and θN +1 . Note that the resulting anisotropy in the reduced state D N +1 is now lower than D N in contrast to the previous fits. The angles
θN and θN +1 , yielded from the fits, are listed in Table 6.3. The angle γ, obtained by
the analysis similar to the one described in the previous section, is estimated to be
∼ 72 − 74◦ .
6.4 C ONCLUSION
Gate-voltage spectroscopy, a technique probing electron transport between the
ground states, which allows a single-molecule magnetic anisotropy determination
when the zero-field splitting cannot be resolved, is introduced. Using this method
we have observed a variation in the electron transport data upon the rotation of
the sample containing single-molecule Fe4 Ph devices in the magnetic field, which
is a direct evidence of a magnetic anisotropy. Moreover, our data indicate that the
easy axes of the adjacent charge states in these devices match or almost match.
The detailed quantitative information about anisotropy parameters, however, cannot be obtained as no unique fit of the measurements results to the model can be
produced due to a large number of parameters.
R EFERENCES
109
R EFERENCES
[1] E. A. Osorio, T. Bjørnholm, J.-M. Lehn, M. Ruben, and H. S. J. van der Zant,
Single-molecule transport in three-terminal devices, J. Phys.: Condens. Matter.
20, 374121 (2008).
[2] A. S. Zyazin, J. W. G. van den Berg, E. A. Osorio, H. S. J. van der Zant, N. P. Konstantinidis, M. Leijnse, M. R. Wegewijs, F. May, W. Hofstetter, C. Danieli, et al.,
Electric Field Controlled Magnetic Anisotropy in a Single Molecule, Nano Lett.
10, 3307 (2010).
[3] S. Accorsi, A.-L. Barra, A. Caneschi, G. Chastanet, A. Cornia, A. C. Fabretti,
D. Gatteschi, C. Mortaló, E. Olivieri, F. Parenti, et al., Tuning Anisotropy Barriers in a Family of Tetrairon(III) Single-Molecule Magnets with an S = 5 Ground
State, J. Am. Chem. Soc. 128, 4742 (2006).
[4] A. L. Barra, A. Caneschi, A. Cornia, F. Fabrizi de Biani, D. Gatteschi, C. Sangregorio, R. Sessoli, and L. Sorace, Single-Molecule Magnet Behavior of a Tetranuclear
Iron(III) Complex. The Origin of Slow Magnetic Relaxation in Iron(III) Clusters,
J. Am. Chem. Soc. 121, 5302 (1999).
[5] K. O’Neill, E. A. Osorio, and H. S. J. van der Zant, Self-breaking in planar fewatom Au constrictions for nanometer-spaced electrodes, Appl. Phys. Lett. 90,
133109 (2007).
[6] A. S. Zyazin, H. S. J. van der Zant, M. R. Wegewijs, and A. Cornia, High-spin
and magnetic anisotropy signatures in three-terminal transport through a single
molecule, Synthetic Met. 161, 591 (2011).
7
S PIN BLOCKADE IN A
SINGLE - MOLECULE JUNCTION
We study three-terminal transport through a single-molecule magnet Fe4 . Current
suppression at low bias voltage is observed in the single-electron tunneling regime.
Magnetic field measurements show that this suppression is due to a spin blockade,
resulting from a spin difference between adjacent charge states higher than 1/2. The
blockade is weakly violated and is accompanied by a negative differential conductance line with a temperature-dependent position. We also present rate equation
calculations modelling experimental data.
111
112
7. S PIN BLOCKADE IN A SINGLE - MOLECULE JUNCTION
7.1 I NTRODUCTION
E
lectron transport in single-molecule junctions has been studied extensively
during the last decade. Three-terminal conductance measurements can give
information on vibrational, electronic and spin properties of different redox states
of molecules. Among these, spin properties are of special interest as they can
be used to build various nanoelectronic devices for data storage and processing.
Consequently, the topic of molecular spintronics [1] has attracted a lot of attention from theorists and synthetic chemists. However, only a few reports on experimental investigation of electron transport through single magnetic molecules have
been published so far [2–9]. Spin can have a dramatic effect on an electron transport in nanostructures: under certain conditions current can be completely suppressed due to spin selection rules. Spin blockade of single-electron transport has
been studied in detail in semiconductor quantum dots. Two different types of this
phenomenon are known: Pauli spin blockade resulting from the exclusion principle [10] and the blockade resulting from spin conservation [11, 12]. The latter occurs when the difference in ground-state spin values of two adjacent charge states
∆S > 1/2. Single-electron transport between such states is prohibited by the selection rule |∆S| = 1/2. The blockade is lifted when the bias voltage is large enough to
induce spin transitions within each charge state so that excited states with ∆S = 1/2
become populated. Only two experimental observations of current suppression
due to spin effects in single molecules are known [2, 5]. Here, we present data on
spin blockade of the second type in a different class of magnetic molecules, the
single-molecule magnet (SMM) Fe4 [13, 14]. A distinct peculiarity of these data is a
weak violation of the blockade with a temperature-dependent negative differential
conductance (NDC) feature.
7.2 M EASUREMENTS
Our devices are made by electromigration of a gold wire in a solution of the
molecules with an active feedback [15] followed by a self-breaking [16]. Details
of the fabrication procedures are described in Ref. [8]. The device layout is shown
in Figure 7.1a. We measure differential conductance d I /dV as a function of gate
Vg and bias V voltage at low temperature T = 1.7 K. In this chapter we focus on a
particular sample with intermediate coupling to source and drain electrodes featuring both Coulomb blockade (CB) and the Kondo effect. The molecule used
in this device is a Fe4 single molecule-magnet (SMM) functionalized with two 9(acetylsulfanyl)nonyl chains [13, 14, 17].
The transport measurements are presented in Figure 7.2a. There are two regions of single-electron tunneling (SET) separating three charge states indicated by
I, II and III. A Kondo resonance is clearly visible in the rightmost charge state (III);
7.2. M EASUREMENTS
113
a
b
eg
t2g
D
S
Gate
V
100 MΩ
Vg
F IGURE 7.1: (a) A sketch of a three-terminal molecular device. An Fe4 core functionalized with two alkyl
chains bridges a gap between source (S) and drain (D) electrodes. A gate changes electrostatic potential
on the molecule enabling access to different charge states. (b)Electronic configuration of Fe4 molecule
in the ground state. Five d -electrons fill both t 2g and e g orbitals, resulting in a spin S i = 5/2 for each
Fe(III) ion. Antiferromagnetic coupling of the central ion to the outer ions leads to a total spin S = 5.
a weak Kondo peak is present in charge state I as shown by a d I /dV trace taken at
this region (Figure 7.3a). The most striking feature is the suppression of the current
in the left SET region (Vg ≈ −0.1 V) for bias voltages smaller than ∼ 5 mV. The behaviour of the right edges of the middle Coulomb diamond (charge state II) is also
peculiar: at bias voltages ±1.4 mV the diamond edges connect to a negative differential conductance feature and disappear at higher voltages. In Figure 7.2c dashed
lines indicate the position of the edges where they should have been. We will argue
that the current suppression at low bias is a signature of a spin blockade.
In the suppressed current region faint lines with increased conductance can
still be distinguished in the low-bias regime, indicating a weak violation of blockade. A line indicated by the red arrow in Figure 7.2a runs parallel to the diamond
edges and crosses zero bias at a gate voltage Vg = −0.045 V. At V = 1.2 mV it connects to a curved low conductance line (indicated by the green arrow) and disappears. This line connects to a crossing point of visible diamond edges. A similar
effect is observed at negative bias. The conductance in the curved lines is positive,
but lower than the value inside the CB regions. After subtraction of the background
signal, the conductance is negative and we therefore refer to it as NDC in the following. Strikingly, the position of the NDC line linearly depends on the temperature (see Figure 7.4). A similar temperature shift of an electron tunneling resonance
has been observed in semiconductor heterostructures by Deshpande et al. [18] We
discuss possible mechanism of the NDC and its temperature dependence later in
the text.
114
7. S PIN BLOCKADE IN A SINGLE - MOLECULE JUNCTION
F IGURE 7.2: d I /dV maps as a function of Vg and V . (a) B = 0 T. (b) B = 8 T. (c) The same map as (a)
but using different color scales emphasizing SET excitations. Yellow dotted lines are missing Coulomb
diamond edges. They are reconstructed by a translation of SET lines through the local maxima of d I /dV
at Vg = 0 (see Figure 7.5). White dotted lines show interpolation of the SET excitations to the points
where they cross missing diamond edges. These points determine the energies of excitations. (d) Zoomin at the left degeneracy-point emphasizing weak low-bias features.
7.2. M EASUREMENTS
115
F IGURE 7.3: d I /dV traces for Vg = −0.065 V. At this gate voltage all peaks in d I /dV correspond to
excitations of the state I. (a) Magnetic field B = 0 T. Small peak at V = 0 mV is a Kondo resonance. (b)
B = 8 T. Arrows indicate the splitting of the first excitation. The second excitation is broadened, however
its splitting cannot yet be resolved.
Several excitations are clearly seen in the left SET region. In order to determine
their energies we extend the suppressed diamond edges and extrapolate the excitations to the points where they cross the suppressed edges (Figure 7.2c). From
this, we infer the energies of 3.2, 5.1 and 6.3 meV for the first three excitations of
the state I and 3.7, 6.1, 8.9 and 10.9 meV for the first four excitations of the state II;
the 3.2 and 3.7 meV excitations lift the blockade, as above these lines the current
is substantially larger. Note that excitations lines terminate at the points where
they cross excitations lines of the other charge state rather than that they extend to
the diamond edges. This is a signature of an excited-state to excited-state transition [4]. For the right SET region, involving states II and III, excitations are found
at an energy of 1.4 meV for both state II and state III. It is important to note that
the spectrum for transitions between states II and III is different from the transition spectrum between II and I. The slopes of the Coulomb diamonds edges are
also considerably different. The slopes are determined by the capacitances of the
molecule to the gate and leads [19]. Although the capacitances may be different for
different orbitals [20], we believe that in this case the two crossing points belong to
different objects (quantum dots) in parallel. The main argument supporting this
scenario is the lower II-III diamond edge with positive slope: it runs through the
SET region of I-II transition as if they are indeed due to separate tunneling processes.
To obtain more information about the spin states we performed magnetic field
measurements. Figure 7.2b feature the d I /dV map (stability diagram) at B = 8 T. It
shows a zero-bias dip in region I, resulting from the splitting of the very weak zerobias resonance observed at a zero field. The splitting corresponds to the energy
116
7. S PIN BLOCKADE IN A SINGLE - MOLECULE JUNCTION
F IGURE 7.4: d I /dV traces at Vg = −0.027 V and B = 0 T taken at different temperatures. Low positive
bias NDC voltage decreases with the temperature. Conductance is normalized by a subtraction of a
zero-bias value. (b) Temperature dependence of an NDC position (marked with arrow in (a)). Red line
is a fit to V = V0 + bT , where V0 = 4.14 mV and b = −0.263 mV/K.
expected from a Kondo resonance 2g µB B , with g = 2. The Kondo resonance at
region III is also split with the same energy. In a similar way as described before, we
have determined the energies of the SET excitations; the result is given in Table 7.1.
All observed excitations of charge state I increase their energy in a magnetic field;
for charge state II they decrease in energy. From this one can conclude that the
spectrum of I corresponds to a high-spin ground state and low-spin excitations,
while for II the ground state is low spin and the excitations have a higher spin.
Further information about ground state spins can be obtained from the evolution of the degeneracy-points in a magnetic field. As can be seen from Figure 7.5a
the left degeneracy-point shifts by ∆Vg = 47 mV when the magnetic field increases
to B = 8 T. We can estimate the difference in the spin ground states I and II from this
shift by S I − S I I = ∆Vg β/g µB B , where β is the gate coupling, g = 2 is Landé factor
and µB is Bohr magneton. This equation is just the change in ground-state energy
between two states with spin S I and S I I corrected for the gate coupling parameter. It does not account for magnetic anisotropy of the molecule nor for a mixing
of the states; it is thus a lower bound for the spin difference. From the slopes of the
CB edges we estimate β = 0.03 [19], so that S I − S I I ≥ 3/2, thereby confirming the
spin origin of the transport blockade. The other crossing point shifts by −19 mV
(Figure 7.5b). Using β = 0.02 we obtain S I I I − S I I = 1/2, consistent with the observation that the degeneracy-point is not suppressed.
We now focus on the low-lying excitations in the I-II region lifting the blockade.
The first excited state of II is split in two components (see Figure 7.3b). They decrease in energy by ∼ 2.1 and ∼ 1.2 mV at B = 8 T. The lowest spin multiplet that
7.3. M ODEL CALCULATIONS
I
II
117
B =0T
3.2
5.1
6.3
B =8T
4.2
5.8
7.0
∆E
+1.0
+0.7
+0.7
3.7
1.6
2.3
4.2
5.4
7.1
-2.1
-1.2
-1.9
-0.7
-1.8
6.1
8.9
TABLE 7.1: Excitations energies (in meV) for charge states I and II at two different magnetic fields. For
charge state II first two excitations are split in two lines; both energies are given. ∆E is the difference
between the value at 8 T and the zero field value.
can exhibit such a behaviour is S = 2 and the two components are M S = +2 and
M S = +1. The ground-state spin should then be S I I = 0. Taking into account the
selection rules for SET transitions |∆S| = 1/2, |∆M S | = 0, 1/2 we infer that the first
excitation of state I has spin S = 3/2 or S = 5/2 (see Figure 7.6). It is easy to see that
there is no possible configuration that can satisfy the condition S I − S I I = 3/2, so
that the ground state spin in I, S I , should be 5/2 or 7/2, indicating that mixing or
anisotropy does play a role.
We now briefly discuss the other SET region separating II and III. Similar behaviour (NDC in a connection with suppressed CB edges at higher voltagea) was
previously computed for the SMM Mn12 (see Figure 4b in [2]). It can result from
a low-lying state in the spectrum of II with a small tunnel matrix element to the
ground state of III. If this state can be occupied via a cascade of transitions similar
to shown on Figure 7.6c the transport to the ground state III will be blocked after
this cascade becomes energetically allowed. We will not focus on this NDC in this
chapter.
7.3 M ODEL CALCULATIONS
To gain a better insight in the mechanism of the NDC and its temperature dependence we have performed model calculations. We use a rate equation approach [21, 22] to calculate d I /dV maps as a function of the gate Vg and bias V
voltages. The procedure is as follows: we first numerically find a stationary solution of the equation
d P/d t = WP,
(7.1)
where P is the vector of the occupation probabilities of n levels P i , i = 1..n and W
is a n × n matrix, whose elements Wi j are rates for transitions j → i for i 6= j , and
118
7. S PIN BLOCKADE IN A SINGLE - MOLECULE JUNCTION
F IGURE 7.5: Gate voltage traces for V = 0 mV at magnetic field B = 0 T (solid lines) and B = 8 T
(dashed lines). Peak positions correspond to degeneracy-points. (a) Left degeneracy-point. (b) Right
degeneracy-point.
a
b
c
S=3/2, M=+3/2
S=3/2
S=2, M=+1
S=2
S=2, M=+2
S=5/2
S=5/2
S=0
I
II
S=0
I
II
II
III
F IGURE 7.6: (a) and (b) Possible state diagram for charge states I and II at (a) B = 0 T and (b) B = 8 T.
Color of arrows labelling transitions corresponds to a color of arrows on Figure 7.2a. (c) Possible scheme
of transitions between states II and III. Transition labelled with a red dotted line has a very small rate
compared to other processes. The nature of states is not important for the appearance of NDC and
missing diamond edge.
Wi i = −
P
j 6=i
W j i . These rates can be written as
1 S S
D
(Γ f + ΓD
i j f i j ),
ħ ij ij
(7.2)
1 S
D
(Γ (1 − f iSj ) + ΓD
i j (1 − f i j )).
ħ ij
(7.3)
Wji =
Wi j =
Here ΓS(D)
are the bare rates (determined by thicknesses of tunneling barriers)
ij
for electron tunneling from a molecule in a state j to a source (drain), leaving a
7.3. M ODEL CALCULATIONS
119
are the Fermi functions:
molecule in a state i , and f iS(D)
j
f iSj =
1
,
1 + exp((µi j + V /2)/k B T )
(7.4)
f iDj =
1
.
1 + exp((µi j − V /2)/k B T )
(7.5)
µi j are electrochemical potentials of the corresponding transitions. The current
P
can then be found as I = e Wi j P i .
i,j
We have calculated conductance maps with 4, 5 and 6 levels using a wide range
of parameters. We found that at least 5 levels need to be taken into account to
show blockade violation resembling the one observed in the measurements. The
parameters in the calculation are the energies of the excited levels E 2 , E 4 and E 5
and the bare tunneling rates Γi j , i = 1, 2, j = 3, 4, 5, between all the levels within
different charge states. For simplicity we have assumed Γi j = Γ j i and that the electronic couplings to the source and drain electrodes are equal for both charge states.
The latter assumption affects only the relative heights of symmetric d I /dV peaks
at positive and negative bias.
In the simulations a violation of spin blockade can be obtained in two ways. In
the first one the rate of transition between the ground states, Γ13 is assumed to be
much smaller than all the other rates. This accounts for violations due to transverse anisotropy. In the other approach Γ13 = 0 and a mixing between the levels
within the same charge state is introduced. This accounts for relaxation due to a
transfer of the angular momentum to the leads. We have used both approaches,
and we found that calculations with Γ13 = 0 and mixing better reproduce measurements data than calculations with small Γ13 > 0. Therefore we enforced Γ13 = 0 and
introduced a mixing within each charge state by adding decay rates Γ12 , Γ34 , Γ35 .
−(E i −E j )
Corresponding excitation rates are Γi j = Γ j i exp k T , for i > j .
B
The d I /dV map calculated for a particular set of parameters (E 2 = 2.5 meV, E 4 =
1 meV, E 5 = 4 meV; Γ15 = Γ23 = Γ25 = 1 meV, Γ14 = 0.1 meV, Γ24 = 1 × 10−4 meV) is
shown in Figure 7.7b. Mixing is only allowed for the right-hand charge state (Γ12 =
0, Γ34 = Γ35 = 1×10−5 meV). At the left-hand side of the crossing point the molecule
is in the state 1 at zero bias. At the right hand side the occupation probability of
state 3 is 0.34 and for state 4 it is 0.67. The lowest-energy transition is between states
4 and 1. NDC occurs when the transition 1 → 5 becomes energetically allowed: this
transition triggers a cascade 4 → 1 → 5 → 2 → 3. The state 3 acts as a blocking state,
as the transition 3 ↔ 1 is forbidden and the transition 3 → 2 is not accessible yet.
The population of this state therefore leads to a decrease in the current and thus an
NDC. Interestingly, the NDC threshold bias voltage is lower than µ51 /e −βVg , which
120
7. S PIN BLOCKADE IN A SINGLE - MOLECULE JUNCTION
F IGURE 7.7: (a) Five levels model mimicking experimental data. Energies of the levels: E 2 = 2.5 meV,
E 4 = 1 meV, E 5 = 4 meV. The transition between states 1 and 3 is forbidden, the rates (in meV) for
the transitions 1↔4 and 2↔3 are shown in the figure. The rates for the transitions 1↔5 and 2↔3 (not
shown for clarity) are 1 meV. The mixing rates are W12 = 0, W34 = W35 = 1 × 10−5 meV. (b) d I /dV map
for the model calculated using the temperature T = 1.7 K. (c) d I /dV traces calculated for V g = 0.06 V
at different temperatures. (d) Temperature dependence of the positive bias dip position at Vg = 0.06 V.
The curve can be fitted to a straight line V = V0 + bT , where V0 = 5.5 mV and b = −1.4 mV/K. At higher
temperatures dependence start to saturate due to approaching zero bias.
7.4. D ISCUSSION
121
corresponds to the energy required to access the 1 → 5 transition [22]. The reason
is that state 4, mainly contributing to transport at low bias, becomes depopulated
when the rate of transition 1 → 5, W51 equals W14 . At finite temperature W51 = W14
already at an energy below µ51 due to a difference in the bare rates Γ. In contrast to
a regular SET excitations this energy is temperature-dependent (see Figure 7.7d).
The dependence is weaker for stronger mixing. The NDC line approaches then
the tip of the SET region, where it connects to the 1 → 5 SET excitation at zero
temperature.
It is important to note that maximum current at zero bias does not occur at the
degeneracy point separating two charge states. It occurs at the point where the
ground state 1 has the same energy with the excited state 4. The degeneracy-point
for two ground states 1 and 3 is located at a more negative gate voltage. In this case
interpolation of SET lines to crossing with the suppressed edges as in Figure 7.2c
would lead not to the energy values with respect to the ground state but to the
energy differences with the state 4. The spin difference determined from the shift of
levels in the magnetic field would then also be with respect to the spin of the state 4.
When analyzing the measurements data we however assumed that the blockade is
violated by GS-GS transitions and derived all the energies under that assumption.
The values of the energies and the spins for the excited states can thus be higher
than our estimations, if the transport through the real ground state is completely
blocked as in our model.
7.4 D ISCUSSION
Our model does not take into account several aspects: for instance, inelastic cotunneling and Kondo correlations are neglected as well as the broadening of the levels
due to the interaction with the leads. Moreover, there are more than 5 levels in the
measured spectrum. We also did not include in the model a second dot that gives
rise to the right SET region. All these can have a noticeable effect on a transport,
however, the model presented already captures the main features of the data including the weak violation of blockade, followed by a curved NDC line, the features
also visible in the measurements data.
Spin blockade is not the only possible mechanism of transport suppresion at
low bias. For instance, current can be inhibited by the vibrational modes, an effect known as Franck-Condon blockade [23]. However, as can be seen from the
magnetic field dependence, the excitations lifting blockade reduce spin difference
between states I and II. The Frank-Condon scenario is then unlikely.
The microscopic origin of spin blockade, however, requires further investigation. A possible scenario is a low spin - high spin transition of one of the iron ions
constituting the Fe4 core upon (dis)charging, similar to transitions occuring in spin
122
7. S PIN BLOCKADE IN A SINGLE - MOLECULE JUNCTION
crossover compounds [24]. In the neutral state of the molecule each Fe(III) ion is
octahedrally coordinated, and the ligand field splitting is smaller than the exchange
energy. Five d -electrons then occupy both t 2g and e g orbitals, giving rise to a high
spin (S i = 5/2) single ion state. Antiferromagnetic coupling of a central ion to peripheral ions leads to a total spin S = 5 for the Fe4 molecule (see Figure 7.1b). Upon
addition of an extra electron to one of the ions the ligand field might overcome the
exchange energy. This would lead to a situation where all six electrons are paired
in the lower orbitals giving a zero spin to that ion. The total spin of the molecule is
then changed either to S = 15/2 if the central ion is charged or to S = 5/2 if the electron is added to one of the peripheral ions. The difference between spin values in
the two charge states |∆S| = 5/2 > 1/2, fulfilling the condition for a spin blockade.
As discussed in Sec. II, the spin in the state with an extra electron (II) is lower than
the spin in the neutral state (I), implying that the charge is localized on one of the
outer ions.
It is an open question why other experiments with Fe4 molecules [7, 8] did not
show similar effects. The interaction with the electrodes can play a decisive role
in the performance of molecular junctions. This interaction changes from sample to sample leading naturally to a spread in device characteristics. It is known,
for instance, that image charges in the leads can stabilize additional electrons at
the parts of the molecule that are closer to contacts [20, 25]. This can also explain
localization of an electron on a peripheral iron ion, the situation that otherwise
is implausible due to symmetry considerations. The second dot in parallel that
gives rise to a right hand side SET region can also influence a transport through the
molecule.
In conclusion, we observed a spin blockade of a single-electron transport
through individual Fe4 SMM embedded into a three-terminal device. This blockade is weakly violated and accompanied with a temperature-dependent NDC resonance. Further theoretical investigation would be helpful to reveal the origin of
this blockade.
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S UMMARY
Recent advances in molecular electronics and in particular developments in the
single-molecule junction formation techniques make it possible to build functional devices, in which not only the conductance but also other specific properties
of single molecules can be studied. A particularly interesting direction is molecular spintronics whose subject is the interplay between magnetic and electronic
properties of the molecules. This thesis describes three-terminal electron transport measurements through single magnetic molecules. In the introduction to this
thesis we briefly discuss the progress in the field of experimental molecular spintronics achieved so far as well as the progress in molecular electronics.
Theoretical background needed to understand the results of this thesis is given
in the chapter 2. The theory concerns some concepts from mesoscopic physics
related to electron transport in nanostructures and from ligand field theory related to the molecular magnetism. We show that a molecule trapped in a threeterminal junction behaves as a quantum dot, where electrons can be added and
subtracted one by one by passing a current. Different transport processes are discussed in detail: single-electron tunneling, inelastic cotunneling and the Kondo
effect. These processes can be used to obtain detailed spectroscopic information
about the molecule, and in particular, information about its magnetic properties.
The phenomenological description of these properties using a spin Hamiltonian
approach is given in the next section: magnetic anisotropy and quantum tunneling of magnetization (QTM) are introduced. Although there is no direct evidence of
QTM in the experimental results of this thesis, this phenomena is important for understanding properties of single-molecule magnets. Subsequently single-molecule
magnets are discussed as systems containing several transition metal ions coupled by an exchange interaction. This discussion is important for understanding
spin transitions and for understanding the change in magnetic properties when
the molecule is charged. At the end of the chapter 2 the molecule studied in all the
experiments described in the thesis, Fe4 is introduced.
In chapter 3 we describe in detail the experimental setup used in the measurements: the low-temperature insert, measurements electronics and the sample rotator, incorporated in the insert to directly observe magnetic anisotropy. This rotator was only used for experiments reported in the chapter 6. The chapter continues with a discussion of the fabrication technique used throughout this thesis,
125
126
S UMMARY
electromigration assisted by a self-breaking, and concludes with a discussion of the
measurements methods.
The discussion of experimental results starts in chapter 4 with the measurements that showed that the magnetism of a single Fe4 SMM preserves when the
molecule is isolated on a surface and contacted in a three-terminal junction. The
observation of a zero-field splitting (ZFS) of a ground spin multiplet indicates the
presence of magnetic anisotropy. This ZFS appears as an inelastic cotunneling excitation at low bias (< 1 mV) changing its energy when an external magnetic field is
applied. In one sample this change is linear with the field and in another sample it
is nonlinear. The nonlinear case corresponds to a situation in which the anisotropy
easy axis exhibites a large angle with the magnetic field direction. We also observed
that the energy of ZFS and thus the anisotropy parameters are different in different
charge states of the molecule. An important conclusion is that single-molecule
magnetism can thus be controlled electrically by applying voltage to the gate electrode.
Chapter 5 starts with a discussion of the higher-energy excitations observed in
the samples discussed in the previous chapter. It is shown that some of them may
originate from transitions between the ground and excited spin multiplets. To support this claim calculations of a single-electron transport through a simple model
system are performed. This model consists of two magnetic ions with high spins
coupled by an exchange interaction. It is shown that in contrast to an intramultiplet excitation, which appears in transport as a single ZFS line, an intermultiplet
spin excitation shows a band of transitions between different components of the
multiplets with a complex behaviour in an applied magnetic field. The chapter
concludes with a discussion of different two samples, in which a high-spin Kondo
effect has been observed.
Although the analysis of the measurements in chapter 4 showed strong evidence of the anisotropy, it was not a direct observation. A variation of the response
to the magnetic field upon changing the sample orientation was not available in
those measurements. A sample rotator was incorporated into the setup to make
the direct observation of the anisotropy possible. Unfortunately, no devices, that
showed unambiguously the ZFS, were measured after the implementation of the
rotator. However, we were able to observe the anisotropy directly by measuring
the evolution of Coulomb peaks separating adjacent charge states in the magnetic
field. For several samples we found that this evolution was nonlinear and different
for different sample orientations. The analysis of the data, outlined in the chapter
6, showed that in these cases the easy axes of the adjacent charge states were close
to parallel.
The last chapter of the thesis discusses the measurements of a sample showing a transport blockade. The current is supressed at low bias voltage and the
S UMMARY
127
degeneracy-point is missing. This behaviour is expected when the spins in adjacent charge states differ by more than 1/2 because of the spin selection rules. The
blockade is slightly violated and a negative differential conductance is observed in
the suppressed current region. We model this situation by solving a rate equation
for a five-level system with arbitrarily chosen level structure. A mechanism that can
lead to the observed spin blockade in Fe4 is suggested. It is, however, not clear why
the blockade is not observed more frequently in the measurements.
Alexander Sergeyevich Zyazin
January, 2012
S AMENVATTING
Recente ontwikkelingen in de moleculaire elektronica en met name de ontwikkelingen in de fabricage van moleculaire juncties maken het bestuderen van nieuwe
functionele bouwstenen mogelijk, waarmee niet alleen de geleiding, maar ook andere specifieke eigenschappen van enkele moleculen kunnen worden onderzocht.
Een bijzonder interessante richting is de moleculaire spintronica met als onderwerp de wisselwerking tussen magnetische en elektronische eigenschappen van
moleculen. Dit proefschrift beschrijft de metingen aan drie-terminal elektronentransport door enkele magnetische molecuulen. In de inleiding van dit proefschrift
bespreken we kort de vooruitgang op het gebied van de experimentele moleculaire
spintronica alsmede de vooruitgang in de moleculaire elektronica.
De theoretische achtergrond nodig om de resultaten van dit onderzoek te begrijpen wordt in hoofdstuk 2 beschreven. Een aantal begrippen uit de mesoscopische fysica wordt besproken die betrekking heeft op elektronentransport in nanostructuren; verder wordt de ligandveldtheorie en de verbinding met moleculaire
magnetisme besproken. We laten zien dat een molecuul gevangen in een drieterminal junctie zich gedraagt als een „quantum dot”, waarin de elektronen één
voor één toegevoegd en afgevoerd kunnen worden. Verschillende transportprocessen worden in detail besproken: enkel -elektron tunneling, inelastisch cotunneling
en het Kondo effect. Deze processen worden gebruikt om gedetailleerde spectroscopische gegevens te verkrijgen over het molecuul, en in het bijzonder over de
magnetische eigenschappen. Vervolgens wordt de fenomenologische beschrijving
van deze eigenschappen met behulp van een spin Hamiltoniaan verder uitgediept:
magnetische anisotropie en kwantum tunneling van de magnetisatie (QTM) worden geïntroduceerd. Hoewel er geen direct bewijs van QTM in de experimentele
resultaten van dit proefschrift zijn gevonden, is dit fenomeenvan belang voor het
begrijpen van de eigenschappen van enkel-molecuul magneten. Tenslotte worden
enkel-molecuul magneten besproken die bestaan uit overgangsmetaal ionen gekoppeld door een „exchange” interactie. Deze bespreking is belangrijk voor het
begrijpen van spin-overgangen en voor het begrijpen van de verandering in magnetische eigenschappen wanneer het molecuul geladen is. Als voorbeeld wordt het
molecuul Fe4 uitvoerig beschreven: het molecuul dat uitvoerig bestudeerd is in de
experimenten beschreven in dit proefschrift.
In hoofdstuk 3 beschrijven we in detail de experimentele opstelling gebruikt:
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S AMENVATTING
de lage-temperatuur insert, de meetelektronica en de samplerotator, die in de insert is ingebouwd om de magnetische anisotropie direct te kunnen observeren. De
rotator is alleen gebruikt voor de experimenten beschreven in hoofdstuk 6. Het
hoofdstuk gaat verder met een bespreking van de fabricagetechniek die gebruikt is
voor alle samples in dit proefschrift. De techniek bestaat uit een speciale vorm van
elektromigratie waarin het breken van gouden draden zonder aangebrachte krachten gebeurt („self-breaking”). Het hoofdstuk eindigt met een beschrijving van de
meetmethoden.
De bespreking van de experimentele resultaten begint in hoofdstuk 4 met de
metingen waaruit bleek dat het magnetisme van een enkel Fe4 molecuul behouden blijft wanneer het wordt geïsoleerd op een oppervlak en gecontacteerd in een
drie-terminal junctie. De observatie van een „zero-field splitting” (ZFS) van het
spinmultiplet van de grondtoestand duidt op de aanwezigheid van magnetische
anisotropie. De ZFS laat zich zien als een inelastische cotunneling excitatie bij
lage-bias voltages (< 1mV) waarvan de energie verandert als een extern magnetisch veld wordt aangelegd. In één sample is de verandering lineair in het aangelegde veld; in een ander sample is de verandering niet lineair. Dit niet-lineaire
geval komt overeen met een situatie waarin de voorkeursrichting van de spin een
grote hoek vertoont met de magnetische veld richting. Ook waargenomen is dat de
energie van de ZFS en daarmee de anisotropie parameters verschillend zijn in verschillende ladingtoestanden van het molecuul. Een belangrijke conclusie is dat het
enkel-molecuul magnetisme elektrisch kan worden gemodificeerd door een spanning aan te leggen op de gate elektrode.
Hoofdstuk 5 begint met een beschriving van excitaties waargenomen bij hogere
energies. De metingen zijn verricht op dezelfde samples als besproken in het vorige hoofdstuk. Er wordt aangetoond dat een aantal van die excitaities afkomstig
zijn van overgangen tussen spinmultiplets van de grond- en geexciteerde toestanden. Ter ondersteuning van deze claim zijn enkel-elektron transport berekeningen
uitgevoerd gebaseerd op een een simpel model systeem. Dit model bestaat uit twee
magnetische ionen met een hoge spin gekoppeld door een „exchange” interactie.
Het blijkt dat in tegenstelling tot een intramultiplet excitatie, die in het transport als
een enkele ZFS lijn zichtbaar wordt, een intermultiplet spin-excitatie een band van
overgangen toont tussen de verschillende onderdelen van de multipletten met een
complex gedrag in een magnetisch veld. Het hoofdstuk eindigt met een bespreking
van twee ander samples, waarin een hoge-spin Kondo effect is waargenomen.
Hoewel de analyse van de metingen in hoofdstuk 4 een sterk bewijs van magnetische anisotropie laat zien, is het geen directe waarneming ervan. Een variatie van
de transporteigenschappen optredend bij het wijzigen van de oriëntatie van het
sample in een aangelegd magneetveld is het directe bewijs van anisotropie. Een
samplerotator werd daarom in de opstelling gebouwd. Samples met een duide-
S AMENVATTING
131
lijke ZFS werden niet gevonden na installatie van de rotator. Echter, direct bewijs
voor de magnetische anisotropie is gevonden door een nieuwe spectroscopische
meetmethode gebaseerd op het meten van de positie van Coulomb pieken in een
magnetische veld. Voor verschillende samples vonden we dat de positie een nietlineaire verband vertoonde en dat het verband veranderde voor verschillende sample oriëntaties. De analyse van de gegevens toont aan dat in deze gevallen de voorkeursrichting van de hoge spin ongeveer loodrecht op het aangelegde magneetveld
staat en nagenoeg onveranderd blijft wannner het molecuul geladen is.
Het laatste hoofdstuk van dit proefschrift bespreekt metingen aan een sample
waarin elektronentransport geblokkerd is. De stroom wordt onderdrukt bij een
lage bias spanning en het „degeneracy point” ontbreekt. Dit gedrag kan verklaard
worden met een model waarin de spins in aangrenzende ladingstoestanden met
meer dan een 1/2 verschillen. Blokkade treedt dan op als gevolg van de spin selectieregels. De metingen laten zien dat de blokkade niet volledig is en een negatieve differentiële geleiding is zichtbaar in het gebied waar de stroom onderdrukt is.
We modelleren deze situatie door het oplossen van de de transportvergelijkingen
voor een vijf-level systeem met een willekeurig gekozen level struktuur. Een mechanisme dat leidt tot de waargenomen spin blokkade in Fe4 is voorgesteld. Het is
echter niet duidelijk waarom de blokkade niet vaker waargenomen is in de metingen.
Alexander Sergeyevich Zyazin
Januari, 2012
C URRICULUM V ITAE
Alexander Sergeyevich Z YAZIN
01-07-1984
Born in Obninsk, USSR
1994-1999
Gymnasium, Obninsk
1999-2001
Physics and Technics School, Obninsk
2001-2007
Diploma in Physics
M.V. Lomonosov Moscow State University
Thesis title: Nanogaps fabrication using platinum electroplating
Supervisor: Dr. E.S. Soldatov
2007-2011
Ph.D. in Physics
Delft University of Technology
Thesis title: Electron transport through single magnetic molecules
Supervisor: Prof. Dr. Ir. H.S.J. van der Zant
2011-present
R&D Scientist
Teledyne DALSA, Eindhoven
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L IST OF P UBLICATIONS
4. E. Burzuri, A. S. Zyazin, A. Cornia, H. S. J. van der Zant, Direct observation of magnetic
anisotropy in an individual Fe4 single-molecule magnet, in preparation.
3. A. S. Zyazin, J. .W. G. van den Berg, H. S. J. van der Zant, Spin blockade in a singlemolecule junction, in preparation.
2. A. S. Zyazin, H. S. J. van der Zant, M. R. Wegewijs, A. Cornia, High-spin and magnetic
anisotropy signatures in three-terminal transport through a single molecule, Synthetic
Met. 161, 591 (2011).
1. A. S. Zyazin, J. W. G. van den Berg, E. A. Osorio, H. S. J. van der Zant, N. P. Konstantinidis, M. Leijnse, M. R. Wegewijs, F. May, W. Hofstetter, C. Danieli, A. Cornia, Electric
Field Controlled Magnetic Anisotropy in a Single Molecule, Nano Lett., 10, 3307 (2010).
135
A CKNOWLEDGEMENTS
As any other Ph.D. thesis this one shows only one name on the cover. But as
with any other Ph.D. thesis many more people have contributed to it in one
way or another. Now it’s time to thank them all.
The first word of gratitude is addressed of course to my supervisor. Herre,
it’s been already more than 5 years since I wrote you an e-mail where I asked
about Ph.D. opportunities in your group. When I came for an interview I
was amazed with your energy, enthusiasm, and most of all with your informal working style. I didn’t really have to think a lot when you offered me a
position. When I moved to Delft and started to work I realized that you’re
open-minded, easy person, who was always willing to help and invest some
time when things are going (or can start going) not really good. Supervising
my Ph.D. was perhaps not the easiest venture , but I definitely learnt a lot
of invaluable lessons from you, and not only how to write papers and make
presentations. Besides, it was always fun to listen to your stories at the coffeetable and at TPKV. I wish your h-factor to grow, your football team to win
and your professor inauguration speech to be finally given. Thank you for
all.
The work on this thesis could not even start without good collaborations.
Molecules, that I studied, were synthesized and characterized by Chiara
Danieli and Prof. Andrea Cornia from Modena. Thank you for all this work
and for all the chemistry advice you gave and physics discussions we had
during our meetings and by e-mail. Theoretical support from Prof. Maarten
Wegewijs and his group in Aachen and Jülich was absolutely indispensable.
The main contributors there were Maarten himself and Nikos Konstantinidis. If I count all e-mails received from them, I think it will easily approach
four-digit numbers. Thank you for your calculations and for conveying the
fact that voltage should be measured in kelvins. I also thank Martin Leijnse
(now in Copenhagen) for discussions and Falk May (used to be in Frankfurt,
now in Mainz) for transport calculations in the strong coupling regime. Discussions with Prof. Kyungwha Park from Virginia Tech, whom I met at the
APS meeting in Dallas, were also very fruitful. I thank Kyungwha, Maarten
and Andrea for their time and I’m looking forward to seeing them in Delft in
my defense committee.
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ACKNOWLEDGEMENTS
Herre can definitely be proud of his Molecular Electronics and Devices
group. I really enjoyed being a part of it for 4.5 years. Being surrounded
by smart, skillful and enthusiastic colleagues was always motivating and encouraging. I should especially mention Edgar Osorio, who was my (and not
only) “babysitter”, who taught me everything about fabrication and measurements, sometimes staying in the lab with me after midnight. Thank you
for that and maybe even more important, thank you for creating an unforgettable MED atmosphere, for your endless jokes about Herre and for serious discussions about Hugo Chavez. Also thank you for the sense of realism
you always brought into speculations about data.
I shared an office with Christian Martin and Samir Etaki and these guys
were perfect officemates. Maybe partially because Samir was not so frequent
guest in our office, preferring a company of his dilfridge. But occasionally it
was writing time or Fortran time and then it was also The White Stripes time
in the room. And of course there was a March meeting in Dallas with Deep
Ellum bars, evening train rides and a diner in a place called ‘Dick’s Last Resort’. I wish you success with your Ph.D. defense, which will happen a month
after mine, and with your further academic career. No doubts that you as a
person combining excellent experimental skills with an ability to do complex
theory will find your way there. And Christian was just a perfect officemate.
The quality standards you set for your own research could be an example for
all of us, if only it was possible to fulfill them in molecular electronics, without involving mechanical breakjunctions. I hope your plants feel wet enough
in London and a German flag is in its place. Soon after Christian left his desk
was occupied by Enrique Burzuri, who became my ally in the fight with magnetic molecules and a sample rotator. I hope that despite all the frustration
we will manage to make a nice story of this fight and I wish you all the best
with your further research. Studying Fe4 was never easy and studying even
more sophisticated POMs doesn’t promise to be easy either, but in the end
that’s what makes it interesting.
Going outside of my office, I should mention Ferry Prins, the only Ph.D.
I know, who is capable of working on ten projects in parallel, bringing them
all to success. We started more or less at the same time and I still remember
how you used to leave at 5, but I think nobody from the younger MED generations will believe me. Anyway, your eyes always sparkled when you talked
about science and I think it was not only due to the lenses. Success with your
postdoc in MIT and thanks for all the nice music I got to know because you
had it on in the lab. Apart from me and Ferry the second MED generation included Jos Seldenthuis, a theorist in an experimental group, whom I should
thank, in particular, for the TEXtemplate for this thesis, and of course for the-
ACKNOWLEDGEMENTS
139
oretical discussions both in the corridor and at the molecule meetings. But
there were more theorists at the molecule meetings, although not officially
members of MED: Chris Verzijl, Fatemeh Mirjani and their boss Jos Thijssen.
Thank you for dissolving our experimental discussions with non-equilibrium
Green’s functions formalism and density functional theory. Jos, thank you
also for participating in my defense.
Coming back to MED, I should thank Anne Bernand-Mantel for the work
she started with the sample rotator, after which I only had to change a few
things to finally set it up. And thank you for all discussions, it was very helpful to have another person next door working on molecular magnetism at the
early stage of my Ph.D. Sorry that I didn’t come for a postdoc to your group in
Grenoble (: I should also mention Jeppe Fock, who stayed in our group two
times for a few months and with whom we shared a room at a very nice conference in Emmetten, Switzerland. Your ability to work on three setups in the
same time is impressive, but I’m still wating for your cruciform paper. All the
best to Luna and Nanna. Diana Dulic as a person, who was giving a sound
foundation for molecular electronics research in Delft during last couple of
years, cannot be skipped. It’s a pity that we haven’t put Fe4 into a mechanical breakjunction. After Diana has left to Hasselt, all MCBJ activities is led
by Mickael Perrin, probably the only person, whom I don’t have to wish good
luck, since he already has an impressive amount of papers and yet unpublished results after his M.Sc. thesis and the first Ph.D. year (and a coauthor in
Nature Publishing Group staff, but that won’t make it any easier).
I had a pleasure to supervise a Master project, and I think that together
with Johan van den Berg we did a very good job and this opinion is not only
based on the final grade. I believe we both learnt a lot and this project was
a crucial part of a very nice paper. Another M.Sc. student I worked with,
although not being officially a supervisor, was Arjan Beukman. I’m happy to
see that both Johan and Arjan decided to go for a Ph.D. and I wish them all
the success.
Unfortunately, space limitations do not allow me to write about every
MED Ph.D. student and postdoc, so I just list them all in one sentence:
Benoit, Menno, Andreas, Bo, Jay, Gijs, Saverio, Monica, Anna Molinari, Helena, Jeroen, Hangxing, Ignacio, Abdulaziz, Warner, Hidde, Harold, Ben,
Venkatesh, Michele, Carlos, Andres, Anna Spinelli and Ricardo. I thank you
all and I wish those of you who is still on his or her way to a Ph.D. best of luck.
And also best of luck to Gary Steele and Sander Otte with their tenure tracks.
I am very grateful to all the technicians who works to make our work easier: Bram van der Enden, Remco Roeleveld, the cleanroom team and especially Mascha van Oossanen for solving all the problems with pumps, wire
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ACKNOWLEDGEMENTS
bonders and inserts stuck in cryostats. I am also very grateful to the administative staff who did all the necessary paperwork and helped me all the way
from a visa application 5 years ago to taking care of thesis delivery, which is
only going to happen: Maria Roodenburg, Irma Peterse, Monique Vernhout
and Annette Bor from the FOM office.
Outside TU my life would be paler, if not the mailing list "Ура, товарищи!". I’m already missing our BBQ’s at Delftse Hout (на нашем месте),
birthday parties, poker nights, carting races and of course a Euro 2008 quarterfinal game Russia - Netherlands. I thank Vadim and Oxana, who were
the main organizers of the mailing list and of the majority of events. I wish
your son Maxim to grow healthy and strong. Anton (Доктор Г.), I will never
forget a crazy weekend after Saverio’s defense. Too bad you can’t come for
mine, I hope it will be at least a sunny day in Stavanger. Zhenya, sorry that
I didn’t come for yours, but when are you finally gonna move to Eindhoven?
Masha, thank you for agreeing to be my paranymph in this busy time between your trips and your own Ph.D. defense. Slava, my another paranymph,
thank you for all the funny moments and for being a person, who never says
"no", whether it’s about help with relocation or about drinking one more.
Please don’t forget my backpack. И побольше лимона. It’s amazing, given
the size of our community, that four cute girls were born in the time span
of two months. It’s a pity that leaving Delft we also had to leave this club of
young parents. Alimzhan and Raushan, Sasha and Yulya, Denis and Lena, I
wish you a lot of patience and a lot of joy with Sarbinaz, Emilia and Taya.
Olga and Sanjay, thank you for your hospitality. Oleg, Stas, Petya, Viktor,
Tanya, Marat and Sasha, Maxim, Dima, Daniela, Vlada, Milan, Marko and
all the other Russian, Ukrainian, Belorussian, Kazakh and Serbian friends,
thank you all.
Отдельно хочу поблагодарить свою семью: Лену, Франка, Алешу и
Елизавету Петровну. Без вашей поддержки многое было бы невозможно, а многое другое было бы гораздо труднее. Особое спасибо папе.
Словами невозможно выразить мою признательность за все, что вы с
мамой для меня сделали. Я уверен, она была бы сейчас счастлива. Наконец, я хочу за все поблагодарить свою жену Лину: за любовь, за поддержку, за заботу и конечно же за нашу очаровательную дочку Анюту.