One Dimensional Metals
Sascha Ehrhardt
Introduction
• One dimensional metals are characterized by a high anisotropy concerning some
of their physical properties. The most obvious physical property is the electrical
resistance, where the resistance in the direction with the highest conductivity (zdirection) is smaller, up to a factor 100000 or more, then in the direction with the
lowest conductivity (x-direction).
• These one dimensional metals can be realized via organic or inorganic compounds
like:
– Conjugated polymers
– Carbon nanotubes
– Metallic nanowire
– Organic crystals
Charge transfer salts
• A special group of organic crystal are the charge transfer complexes like e.g. TTFTCNQ where TTF (Tetrathiavulvalene) and TCNQ (Tetracyanoquinodimethane)
are planar molecules, which are stacked in the z-direction as shown in figure 1.
• TTF and TCNQ form a charge transfer complex, where m donors (TCNQ with
m = 1) transfer a charge δ to the n acceptors (TTF with n = 1) via the reaction
[Dm ] + [Xn ] *
) [Dm ]+δ [Xn ]−δ ,
with δ = 0.59 for TTF-TCNQ.
• An important group of those crystals are the dimer charge transfer salts with m = 2
and n = 1 where the transferred charge is δ = 1.00.
• This charge transfer leads to charge holes in the π-bonds and therefore the donor
molecules can overlap under the formation of a band structures with an metallic
character.
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• This overlap is most pronounced in the z-direction, medial in the y-direction and
weak in the x-direction as shown in figure 2.
• The acceptors are just anions, which do not form a band structure.
Figure 1: Charge-transfer salt TTF-TCNQ
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Figure 2: Resistivity of (T M T SF )2 P F6 in xyz-directions
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Fermi Gas, Fermi Liquid and Luttinger Liquid
• The Fermi-gas and Fermi-liquid models describes electron gases in three dimensions. The Fermi-gas model describes non interacting electrons and the Fermiliquid describes interacting electrons. In the one dimensional case the interactions
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http://iopscience.iop.org/0295-5075/74/1/110/fulltext/epl9262.html
http://www.pi1.uni-stuttgart.de/forschung/organic/one dimen/Bechgaard salts/
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between the electrons are that strong, that they can not be described by the Fermiliquid model any more, and therefore this 1D electron gas is described via a model
called Luttinger-liquid.
• When regarding the distribution function of a Fermi-gas, as shown in figure 3, one
can see that the occupation probability of the wave vectors k is one up to the Fermi
wave vector kF and drops down to zero above kF . In the case of a Fermi-liquid
we have unoccupied states below kF and occupied states above kF and therefore
a smudged Fermi-surface. But there is still a discontinuous jump at kF , where
the interactions are the stronger the smaller this jump is. When we finally regard
the Luttinger-liquid, one can see that this jump vanishes because of the strong
electron-phonon interactions.
• When we look now at the low energy excitations in the near of the Fermi-surface,
this will lead to a particle-hole state with a discrete momentum and energy in the
case of a Fermi-gas. Because in a Fermi-liquid this excitation also effects other
electrons we will get distributed momentum and energy, which is now called a
fermionic quasi-particle. In the case of the Luttinger-liquid the movement of one
electrons effects all other electrons and therefore this excitation is an collective
excitation called bosonic excitation. Further this excitation splits in a spinon with
spin 1/2 and no charge, and a spinon with charge e and no spin.
Figure 3: Distribution functions (a) Fermi-gas (b) Fermi-liquid (c) Luttinger-liquid
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Charge density waves
• We start with a crystal, having a lattice constant a in the z-direction and one electron per lattice point. At high temperature we get a metallic dispersion function
with a half filled band.
• By lowering the temperature, below the so called Peierls-temperature, small sinusoidally lattice distortions creates a superstructure with a new lattice constant 2a.
Now we have a 2a-periodic potential, which leads to the formation of a band-gap
∆CDW and therefore we now have an insulator.
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David Saez de Jauregui, Transportuntersuchungen an quasi-eindimensionalen organischen Leitern
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• This sinusoidally lattice distortions leads to a corresponding sinusoidally charge
density with the same periodicity.
• Having more or less then one electron on each lattice point results in other periodicity of the lattice distortion respectively charge density wave.
Electronic correlations
• Electronic correlations are Coulomb interaction between movable electrons, which
are described by the Hubbard-model:
X X
X
X
+
(c+
H = −t
ni↑ ni↓ + V
ni nj
iσ cjσ + cjσ ciσ ) + U
<i,j> σ
|
i
{z
}
Kinetic energy
|
<i,j>
{z
}
On-site repusion
|
{z
}
Inter-site repusion
• When the kinetic term dominates the system this will lead to a metallic state, with
a three quarter filled band in the case of dimer charge-transfer-salts D2 X1 .
• When the on-site Coulomb repulsion term dominates the system this will lead to
a constant charge density. Lowering the temperature below the transition temperature TDM , a phase transition to the Dimer-Mott-state occurs, where each two
molecules approach, which doubles to lattice constant a to 2a and again we have
a 2a-periodic potential, which leads to the formation of a band-gap and therefore
we have an insulator.
• When finally the inter-site Coulomb repulsion term dominates the system, this
will lead to a alternating charge density. When lowering the temperature at TCO
a phase transition to the charge-ordered-state occurs with different charge on each
two molecules. Here also a band gap occurs, but the reason here is the on-site
Coulomb repulsion and not a structural effect because of periodic lattice potentials.
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Literature and Papers
• Siegmar Roth, David Carroll, One-Dimensional Metals, Wiley-VCH
• Naoki Toyota, Low-Dimensional Molecular Metals, Springer
• David Saez de Jauregui, Transportuntersuchungen an quasi-eindimensionalen organischen Leitern
• M. Dressel, Ordering phenomena in quasi-one-dimensional organic conductors
• B. Dardel, Unusual Photoemission Spectral Function of Quasi-One-Dimensional
Metals
• B.J. Kim, Distinct spinon and holon dispersions in photoemission spectral functions from one-dim. SrCuO2
• A.0. Patil, A.J. Heeger, Optical Properties of Conducting Polymers
• Johannes Voit, One-dimensional Fermi liquids
• G. Grüner, The dynamics of charge-density waves
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