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Phil. Trans. R. Soc. A (2011) 369, 3554–3574
doi:10.1098/rsta.2011.0137
REVIEW
Introduction to spin-polarized ballistic hot
electron injection and detection in silicon
B Y I AN A PPELBAUM*
Department of Physics, Center for Nanophysics and Advanced Materials,
University of Maryland, College Park, MD 20742, USA
Ballistic hot electron transport overcomes the well-known problems of conductivity and
spin lifetime mismatch that plague spin injection attempts in semiconductors using
ferromagnetic ohmic contacts. Through the spin dependence of the mean free path in
ferromagnetic thin films, it also provides a means for spin detection after transport.
Experimental results using these techniques (consisting of spin precession and spin-valve
measurements) with silicon-based devices reveals the exceptionally long spin lifetime
and high spin coherence induced by drift-dominated transport in the semiconductor.
An appropriate quantitative model that accurately simulates the device characteristics
for both undoped and doped spin transport channels is described; it can be used to
recover the transit-time distribution from precession measurements and determine the
spin current velocity, diffusion constant and spin lifetime, constituting a spin ‘Haynes–
Shockley’ experiment without time-of-flight techniques. A perspective on the future of
these methods is offered as a summary.
Keywords: spin-polarized electrons; ballistic hot electron transport; spin injection and
detection; semiconductor spintronics; spin precession
1. Background
The problems of spin-polarized electron injection and detection are central to
the field of semiconductor spintronics. Since ferromagnetic metals have a large
asymmetry between the spin-up and spin-down density of states and therefore
high spin polarization at the Fermi level, it is natural to assume that ohmic
transport of these electrons into a semiconductor will provide a robust injection
mechanism for spin-polarized currents. However, in the diffusive regime required
by ohmic transport (where the electron mean free path (mfp) is smaller than all
other length scales, and an electron temperature and chemical potential are well
defined), injection and detection efficiencies are suppressed by the large differences
in conductivity and spin lifetimes between metals and semiconductors, resulting
in the so-called ‘fundamental obstacle to spin injection’ [1–4].
*[email protected]
One contribution of 10 to a Discussion Meeting Issue ‘The spin on electronics’.
3554
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Review. Spin transport in silicon
3555
l
energy
semiconductor
Dm
FM metal
m
m
L
Figure 1. Spin-dependent band diagram showing the electrochemical potentials near the interface of
a metallic FM (high conductivity and short spin lifetime) with a semiconductor (low conductivity
and long spin lifetime).
The problem can be simply viewed in the following way: since the spin-up
and spin-down conductivities in the (non-magnetic) semiconductor are identical,
a spin-polarized current must be driven by an asymmetric spin-dependent
electrochemical potential (m↑/↓ ) drop across the semiconductor. This necessarily
requires an electrochemical splitting Dm = m↑ − m↓ = 0 at the interface between
the ferromagnet (FM) and semiconductor. In terms of the semiconductor bulk
conductivity sS , the semiconductor spin transport length scale L, cross-sectional
area A, spin polarization P and total current I , Ohm’s law gives
Dm = 2P · IR = 2PI
L
.
sS A
(1.1)
On the other hand, in the metallic FM, the spin current flowing towards the
FM–semiconductor interface is caused instead by a spin-dependent conductivity
s↑/↓ . Therefore, the interfacial electrochemical splitting can also be written as
l
l
Dm = I (R↓ − R↑ ) = I
−
,
(1.2)
s↓ A s↑ A
where l is the length scale over which the non-equilibrium electrochemical
splitting in the FM decays away from the interface via spin relaxation. Since
diffusive√transport dominates in the metal, this length scale is the ‘spin diffusion
length’ Dt, where t is the spin lifetime in the FM metal and D is the diffusion
coefficient. Unfortunately, this time scale is typically very short, so the length
scale l is very small. The spin-dependent electrochemical potentials near the
FM–semiconductor interface are schematically illustrated in figure 1.
With the definition of the FM bulk polarization as b = (s↑ − s↓ )/(s↑ + s↓ ), we
can rewrite the electrochemical splitting on the FM side (equation (1.2)) as
Dm =
Il
4b
,
sFM A 1 − b2
(1.3)
where sFM is the average value of conductivity (s↑ + s↓ )/2. At an ideal
interface where the absence of interfacial spin relaxation preserves continuity
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3556
I. Appelbaum
of the electrochemical potentials [5,6], the expressions given for the interfacial
electrochemical splitting in equations (1.1) and (1.3) must be equal. This
then allows us to calculate the polarization of the injected current in the
semiconductor as
2b
sS l
,
(1.4)
P≈
sFM L 1 − b2
which is valid in the low-b limit (typical values for elemental FMs Ni, Co and
Fe are b 0.4 [7]). Note that the conductivity of FM metals is several orders of
magnitude larger than any semiconductor, even in the degenerately doped regime.
In addition, the spin diffusion length in the FM metal is many orders of magnitude
shorter than the spin transport length in the semiconductor owing primarily
to the short (long) lifetime in the metal (semiconductor). The polarization of
the electron current flowing through the semiconductor is then dependent on
the product of two (very) small quantities, sS /sFM and l/L. Unless the bulk
conductivity of the FM is close to 1 (i.e. a 100% polarized ferromagnetic halfmetal—still not possible in practice), ohmic injection of spin-polarized electrons
is therefore hopelessly inefficient.
Figure 1 by itself illustrates the problem quite clearly. The interfacial splitting
creates an electrochemical gradient in the FM metal that opposes the flow of
the spin-up electrons one hopes to inject across the interface. Owing to current
conservation, the splitting is therefore necessarily negligibly small.
To circumvent this problem, the constraints of ohmic injection must be lifted.
For instance, the requirement that the electrochemical splittings be equal on
the FM and semiconductor side can be relaxed by inserting a resistive tunnel
barrier [1,4]. However, direct tunnel injection requires that the applied voltage
drop mostly across the barrier so high-resistance low-doped and nearly intrinsic
semiconductors cannot be used; work with this approach has employed essentially
only material doped beyond the metal–insulator transition. Alternatively, the
high conductivity of the FM can be addressed by using a ferromagnetic
semiconductor, which typically has a much lower carrier concentration than a
metal [8,9].
Here, we consider a mechanism for spin injection that differs fundamentally
from ohmic injection in that
— the mfp is not the smallest length scale in the device and
— electrochemical potentials cannot be uniquely defined in thermal
equilibrium.
The above-described obstacle to spin injection therefore does not apply. With our
techniques, the physical length scale of the metallic electron injection contacts
is shorter than the mfp, and conduction occurs through states far above the
Fermi level (when compared with the thermal energy kB T ), far out of thermal
equilibrium. Because this transport mode uses electrons with high kinetic energy
that do not suffer inelastic scattering, it is known as ‘ballistic hot electron
transport’. We will find that this transport mode is useful, not only for spinpolarized electron injection into semiconductors, but also (and perhaps more
importantly) for spin detection after transport through them.
Phil. Trans. R. Soc. A (2011)
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Review. Spin transport in silicon
3557
2. Hot electron generation and collection
Tunnel junctions (TJs, which consist of two metallic conductors separated by
a thin insulator) [10–15] are the ideal hot electron source. The electrostatic
potential energy qV provided by voltage bias V tunes the energy of hot electrons
emitted from the cathode, and the exponential energy dependence of quantummechanical tunnelling assures a narrow distribution. Because of its robust
insulating native oxide, aluminium (Al) is often employed during TJ fabrication;
although in principle any insulator can be used, it has been empirically found that
the best are Alx O, MgO and AlN [16,17]. Some of the hot electrons thus created
can travel without inelastic scattering (i.e. ballistically) through the TJ anode
and couple with conduction states in the semiconductor to achieve injection.
Metals have a very high density of electrons at and below the Fermi energy,
EF . Therefore, sensitive hot electron collection by the semiconductor conduction
band relies on an electrically rectifying barrier to eliminate transport of these
thermalized electrons across the metal–semiconductor interface, which would
dilute the injected hot electron current. This ‘Schottky’ barrier is ideally created
by the difference in work functions of the metal and the electron affinity of the
semiconductor [18–21], but in reality its height qf is determined more by the
details of surface states which lie deep in the bandgap of the semiconductor that
pin the Fermi level [22]. Hot electrons created by a TJ can be collected by a
Schottky barrier if qV > qf.
The first proposal for a TJ hot electron injector was by Mead, who suggested
another TJ or the vacuum level as a collector [23–25]. Very soon thereafter, Spratt
et al. [26] showed that a semiconductor collector could be used to realize this
device with an Au–Alx O–Al TJ. A schematic band diagram of this device is
shown in figure 2.
There is of course always a leakage current owing to thermionic emission over
this Schottky barrier at non-zero temperature; because typical barrier heights
are in the range 0.6–0.8 eV [27] for the common semiconductors Si and GaAs,
hot electron collection with Schottky barriers is often performed at temperatures
below ambient conditions to reduce current leakage to negligible levels provided
by the super-exponential suppression with lower temperature.
3. Spin-polarized hot electron transport
The mfp of hot electrons in ferromagnetic metal films is spin dependent: majority
(‘spin-up’) electrons have a longer mfp than minority (‘spin-down’) electrons.
Therefore, the ballistic component of an initially unpolarized hot electron current
will become spin polarized by spin-selective scattering during transport through
a ferromagnetic metal film, where the polarization is given by
e−l/lmaj − e−l/lmin
,
(3.1)
e−l/lmaj + e−l/lmin
where l is the FM film thickness, lmaj is the majority spin mfp and lmin is the
minority spin mfp. This ‘ballistic spin-filtering’ effect can be used not only for
spin polarization at injection, but also for spin analysis of a hot electron current
at detection, much as an optical polarizing filter can be used both for electric field
P=
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3558
energy
insulator
I. Appelbaum
metal
emitter
cathode
qV
qf
semiconductor
collector
metal
base
anode
Figure 2. Schematic of a TJ used as a source for ballistic hot electron injection into a semiconductor
conduction band. The emitter cathode electrostatic potential energy (qV ) must exceed the Schottky
barrier height qf. (Online version in colour.)
polarization and analysis of photons by changing the relative orientation of the
optical axis. The realization of this important spin analysis concept has enabled
the spin injection, transport and detection in Si described in this text.
4. Spins in silicon
Silicon is a relatively simple spin system [28]; Si could be considered as the
‘hydrogen atom’ of semiconductor spintronics.
— A low atomic number (Z = 14) leads to a reduced spin–orbit interaction
(which scales as Z 4 in atomic systems).
— Spatial inversion symmetry of the diamond lattice leads to a spindegenerate conduction band, eliminating D’yakonov–Perel [29] spin
scattering in bulk Si.
— The most abundant isotope of Si (≈ 92%) is 28 Si, which has no nuclear
spin, and this abundance can be fortified with isotopic purification to very
high levels [30]. Therefore, the spin-nuclear (hyperfine) coupling is weak,
when compared with other semiconductors where no such nuclear-spin-free
isotope exists.
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Review. Spin transport in silicon
3559
Owing to its apparent advantages over other semiconductors, many groups
tried to demonstrate phenomena attributed to spin transport in Si [31–39],
but this was typically done with ohmic FM–Si contacts and two-terminal
magnetoresistance measurements or in transistor-type devices [40,41], which
are bound to fail owing to the ‘fundamental obstacle’ for ohmic spin injection
mentioned in §1 [1–4]. Although weak spin-valve effects are often presented, no
evidence of spin precession is available so the signals measured are ambiguous
at best [42,43]. Indeed, although magnetic exchange coupling across ultra-thin
tunnelling layers of Si was seen, not even any spin-valve magnetoresistance was
observed, [44] except with SiGe [45].
These failures were addressed by pointing out that only in a narrow window of
FM–Si resistance-area (RA) product was a large spin polarization and hence large
magnetoresistance expected [46]. Subsequently, several efforts to tune the FM–
Si interface resistance were made [47–51]. However, despite the ability to tune
the RA product by over eight orders of magnitude and even into the anticipated
high-magnetoresistance window (for instance, by using a low-work-function Gd
layer), no evidence that this approach has been fruitful for Si can be found, and
the theory has yet been confirmed only for the case of low-temperature-grown
5 nm thick GaAs [52,53].
Subsequent to the first demonstration of spin transport in Si by Appelbaum
et al. [54], optical detection (circular electroluminescence analysis) was shown
to indicate spin injection from an FM metal and transport through several tens
of nanometres of Si, using first an Alx O tunnel barrier [55] and then tunnelling
through the Schottky barrier [56], despite the indirect bandgap and relative lack of
spin–orbit interaction that couple the photon angular momentum to the electron
spin. These methods required a large perpendicular magnetic field to overcome
the large in-plane shape anisotropy of the FM contact, but later a perpendicular
anisotropic magnetic multilayer was shown to allow spin injection into Si at zero
external magnetic field [57]. While control samples with non-magnetic injectors
do show negligible spin polarization, again no evidence of spin precession was
presented, although it is permitted by the geometry [58].
More recently, four-terminal non-local measurements on Si devices have been
made, for instance, with Alx O [59] or MgO [60–63] tunnel barriers, or Schottky
contacts using a ferromagnetic silicide injector and detector [64]. Only earlier
papers [61–63] present convincing evidence of spin precession, although van’t
Erve et al. [59] do show a Hanle measurement that unfortunately has low
signal-to-noise, obscuring geometrical effects of precession expected from the
oblique magnetic field configuration used [65]. Recent claims of ‘electrical creation
of spin polarization in silicon at room temperature’ [66], supported by Hanle
measurements using a three-terminal geometry (without any actual evidence of
transport through Si), remain ambiguous in part owing to the likely influence of
interface states [67]. This has been recently rectified by convincing measurements
of spin precession at room temperature in the less ambiguous non-local fourterminal geometry [63]. However, all the devices used heavily doped and therefore
metallic (and not semiconducting) Si.
As mentioned in §1, another way to overcome the conductivity–lifetime
mismatch for spin injection is to use a carrier-mediated ferromagnetic
semiconductor heterointerface. The interfacial quality may have a strong effect
on the injection efficiency so epitaxial growth will be necessary. Materials such
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3560
I. Appelbaum
as dilute magnetic semiconductors Mn-doped Si [68], Mn-doped chalcopyrites
[69–73] or the ‘pure’ ferromagnetic semiconductor EuO [74] have all been
suggested, but none as yet have been demonstrated as spin injectors for Si. It
should be noted that while their intrinsic compounds are indeed semiconductors,
owing to the carrier-mediated nature of the ferromagnetism, it is seen only in
highly (i.e. degenerately) doped and essentially metallic samples in all instances.
5. Ballistic hot electron injection and detection devices
Two types of TJs have been employed to inject spin-polarized ballistic hot
electrons into Si. The first used ballistic spin filtering of initially unpolarized
electrons from a non-magnetic Al cathode by a ferromagnetic anode base layer
in direct contact to undoped, 10 mm thick single-crystal Si(100) [54]. Despite
spin-valve transistor (SVT) measurements suggesting the possibility of 90 per
cent spin polarization in the metal, [75,76] only 1 per cent polarization was
found after injection into and transport through the Si. It was discovered
later that a non-magnetic Cu interlayer spacer could be used to increase the
polarization to over 50 per cent [77,78], probably owing to Si’s tendency to
readily form spin-scattering ‘magnetically dead’ alloys (silicides) at interfaces with
ferromagnetic metals.
Despite the possibilities for high spin polarization with these ballistic spin
filtering injector designs, the short hot electron mfps in FM thin film anodes
cause a very small injected charge current of the order of 100 nA with an
emitter electrostatic potential energy approximately 1 eV above the Schottky
barrier and a contact area approximately 100 × 100 mm2 . Because the injected
spin density and spin current are dependent on the product of spin polarization
and charge current, this technique is not ideal for transport measurements.
Therefore, an alternative injector using an FM TJ cathode and non-magnetic
anode (which has a larger mfp) was used for approximately 10 times greater
charge injection and hence larger spin signals, despite somewhat smaller potential
spin polarization of approximately 15 per cent [79,80]. These injectors can be
thought of as one-half of a magnetic TJ, [81] with a spin polarization proportional
to the Fermi-level density of states spin asymmetry, rather than exponentially
dependent on the spin-asymmetric mfp as is the case with the ballistic spin
filtering described above.
Although the injection is due to ballistic transport in the metallic contact,
the mfp in the Si is only of the order of 10 nm [82], so the vast majority of
the subsequent transport to the detector over a length scale of tens [54,77,79],
hundreds [65,80] or thousands [83] of micrometres occurs at the conduction
band edge following momentum relaxation. Typically, relatively large accelerating
voltages are used so that the dominant transport mode is carrier drift; the
presence of rectifying Schottky barriers on either side of the transport region
assures that the resulting electric field does nothing other than determine the
drift velocity of spin-polarized electrons and hence the transit time [84,85]—there
are no spurious (unpolarized) currents induced to flow. Furthermore, undoped
Si transit layers are primarily used; otherwise, band bending would create a
confining potential and increase the transit time, potentially leading to excessive
depolarization (see §6a) [86].
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3561
Review. Spin transport in silicon
injector
single-crystal
undoped Si
350 μm
VC1
VE
IC1
n-Si
NiFe
Cu
Al
Cu
Al
CoFe
Al2O3
energy
detector
IC2
10 nm
8 nm
distance
Figure 3. Schematic band diagram of a four-terminal (two for TJ injection and two for FM
semiconductor–metal–semiconductor detection) ballistic hot electron injection and detection device
with a 350 mm thick Si transport layer. (Online version in colour.)
The ballistic hot electron spin detector comprises a semiconductor–FM–
semiconductor structure (both Schottky interfaces), fabricated using ultra-high
vacuum metal-film wafer bonding (a spontaneous cohesion of ultra-clean metal
film surfaces that occurs at room temperature and nominal force in ultra-high
vacuum) [87,88]. After transport through the Si, spin-polarized electrons are
ejected from the conduction band over the Schottky barrier and into hot electron
states far above the Fermi energy. Again, because the mfp in FMs is larger
for majority-spin (i.e. parallel to magnetization) hot electrons, the number of
electrons coupling with conduction band states in an n-Si collector on the other
side (which has a smaller Schottky barrier height owing to contact with Cu) is
dependent on the final spin polarization and the angle between spin and detector
magnetization. Quantitatively, we expect the contribution to our signal from each
electron with spin angle q to be proportional to
cos2 2q e−l/lmaj + sin2 2q e−l/lmin
= 12 (e−l/lmaj − e−l/lmin ) cos q + (e−l/lmaj + e−l/lmin ) .
(5.1)
Because the exponential terms are constants, this has the simple form ∝ cos q +
const; in the following, we disregard the constant term as it is spin independent.
The spin transport signal is thus the (reverse) current flowing across the
n-Si collector Schottky interface. In essence, this device (whose band diagram
is schematically illustrated in figure 3) can be thought of as a split-base tunnelemitter SVT with several hundred to thousands of micrometres of Si between the
FM layers.
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3562
spin transport signal (IC2) (pA)
I. Appelbaum
120
115
110
sweep
direction
105
100
T = 150 K
–200
–100
0
100
200
in-plane magnetic field (×1000/4p A m–1)
Figure 4. In-plane magnetic field measurements show the ‘spin-valve’ effect and can be used to
calculate the spin polarization after transport. (Online version in colour.)
Two types of measurements are typically made: ‘spin valve’ in a magnetic field
parallel to the plane of magnetization, and spin precession in a magnetic field
perpendicular to the plane of magnetization. The former allows the measurement
of the difference in signals between parallel (P) and anti-parallel (AP) injector–
detector magnetization and hence is a straightforward way of determining the
conduction electron spin polarization,
P=
IP − IAP
.
IP + IAP
(5.2)
Typical spin-valve measurement data, indicating ≈ 8% spin polarization after
transport through 350 mm undoped Si, are shown in figure 4.
Measurements in perpendicular magnetic fields reveal the average spin
orientation after transit time t through the Si, owing to precession at frequency
u = gmB B/h̄, where B is magnetic field. If the transit time is determined only by
drift, i.e. t = L/mE, we expect our spin transport signal to behave ∝ cos gmB Bt/h̄.
However, owing to transit time uncertainty Dt caused by random diffusion, there
is likewise an uncertainty in spin precession angle Dq = uDt, which increases as
the magnetic field (and hence u) increases. When this uncertainty approaches 2p
radians, the spin signal is fully suppressed by a cancellation of contributions
from anti-parallel spins, a phenomenon called spin ‘dephasing’ or the Hanle
effect [89].
On a historical note, our device is essentially a solid-state analogue of
experiments performed in the 1950s that were used to determine the g-factor
of the free electron in vacuum using Mott scattering as the spin polarizer and
analyser and spin precession in a solenoid [90]. In our case, we already know
the g-factor (from e.g. electron spin resonance lines), so our experiments in
strong drift electric fields where spin dephasing is weak can be used to measure
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Review. Spin transport in silicon
3563
(a) 120
T = 150 K
spin signal (pA)
115
110
105
sweep direction
100
–200
–100
0
100
200
perpendicular magnetic field (×1000/4p A m–1)
FFT-normalized
(b) 1.0
0.8
0.6
0.4
Δt
0.2
0
5
10
time (ns)
15
Figure 5. (a) A typical spin precession measurement shows the coherent oscillations owing to drift
and the suppression of signal amplitude (‘dephasing’) as the precession frequency rises. Our model
simulates this behaviour well. Black line, experiment; grey line, model. (b) The real part of the fast
Fourier transform (FFT) of the precession data in (a) reveals the spin current arrival distribution.
transit time with t = h/gmB B2p , where B2p is the magnetic field period of the
observed precession oscillations, despite the fact that we make DC measurements,
not time-of-flight [84,85]. Typical spin-precession data, indicating a transit time
of approximately 12 ns to cross 350 mm undoped Si in an electric field of ≈
580 V cm−1 , are shown in figure 5a.
One important application of this transit time information from spin
precession is to correlate it to the spin polarization determined from spinvalve measurements and equation (5.2) to extract spin lifetime. By varying the
internal electric field, we change the drift velocity and hence average transit
time. A reduction of polarization is seen with an increase in average transit
time (as in figure 6a) that we can fit well to first order using an exponentialdecay model P ∝ e−t/t , and extract the time scale, t [80]. In this way, we have
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3564
I. Appelbaum
(a)
spin polarization (%)
15
10
5
0
50
100
150
200
transit time (ns)
250
(b)
spin lifetime (ns)
1000
500
0
60
90
120
temperature (K)
150
Figure 6. (a) Fitting the normalized spin signal from in-plane spin-valve measurements to an
exponential decay model using transit times derived from spin precession measurements at variable
internal electric field yields measurement of spin lifetimes in undoped bulk Si (filled squares, 60 K;
open circles, 58 K; filled diamonds, 100 K; open diamonds, 125 K; filled triangles, 150 K). (b) The
experimental spin lifetime values obtained as a function of temperature are compared with Yafet’s
[91] (black solid line) T −5/2 power law for indirect-bandgap semiconductors and Cheng et al.’s [92]
(grey solid line) T −3 derived from a full band structure theory; Huang & Appelbaum [93] (filled
circles). (Online version in colour.)
observed spin lifetimes of approximately 1 ms at 60 K in 350 mm thick transport
devices [94].1 The temperature dependence of spin lifetime is compared with
the T −5/2 power law predicted by Yafet [91] and the more recent (and more
complete) theory of Cheng et al. [92] giving T −3 in figure 6b. However, the fully
1 In the study of Huang et al. [80], a more conservative estimate of the spin lifetime (e.g. 520 ns
at 60 K) was obtained by fitting to the transit time dependence of an alternative quantity expected
to be proportional to the spin polarization, rather than using equation (5.2) directly.
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Review. Spin transport in silicon
3565
analytic recent theory of Li & Dery [95] shows that characterization of spin
lifetime across wide temperature scales with a single power law is a potentially
problematic oversimplification given the dominance of different electron–phonon
spin scattering mechanisms in different temperature regimes.
If we can adequately model the transport and the expected signals, we can make
a spin-polarized electron version of the Haynes–Shockley experiment (originally
used to measure the diffusion coefficient, mobility and lifetime of minority charge
carriers) [96]. This experiment was very useful in the design of bipolar electronics
devices such as junction transistors. By measuring the spin transport analogues
of these parameters, we imagine enabling the design of useful semiconductor
spintronic devices [97].
6. Spin transport modelling
Modelling the spin transport signal involves summing all the detector signal
contributions of spins that begin at the injector at the same time and with the
same spin orientation (i.e. initial conditions on spin density s(x, t = 0) = d(x)),
where s(x, t) is determined by the drift–diffusion–relaxation equation
v2 s
v(vs) s
vs
=D 2 −
− ,
vt
vx
vx
t
(6.1)
where D is the diffusion coefficient and v is the drift velocity. The solution to this
partial differential equation with the d(x) Dirac-delta initial condition is called
‘Green’s function’, which can be used to construct the response to arbitrary spin
injection conditions, including the DC currents used so far in experiments.
Unlike open-circuit voltage spin detection [43,98,99], our ballistic hot electron
mechanism employs current sensing. This must be accounted for in the modelling,
and since electrons crossing the metal–semiconductor boundary do not return
via diffusion, it imposes an absorbing boundary condition on the spin transport
[100]. Whereas voltage detection is sensitive to the spin density s(x = L, t) and
the boundaries imposed on the drift-diffusion Green function are at infinity, here
our signal is sensitive to the spin current Js (x = L, t) = −D(ds/dx)|x=L , where we
must impose s(x = L, t) = 0.
(a) Undoped silicon
Assuming v is a constant independent of position x (as it is in undoped
semiconductors), Green’s function of equation (6.1) satisfying this boundary
condition can be found straightforwardly using the method of images,
1
2
2
(6.2)
s(x, t) = √
e−(x−vt) /4Dt − eLv/D e−(x−2L−vt) /4Dt e−t/t .
2 pDt
The corresponding spin current at the detector is
1
L −(L−vt)2 /4Dt −t/t
e
e
.
Js (x = L, t) = √
2 pDt t
Phil. Trans. R. Soc. A (2011)
(6.3)
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3566
spin current
I. Appelbaum
1 ns
0
3 ns
5 ns
5
10
time (ns)
7 ns
15
9 ns
11 ns
13 ns
0
20
40
60
distance (µm)
80
100
Figure 7. Evolution of a Delta function spin distribution (Green’s function) injected on the left side
at the time increments shown, with drift velocity 106 cm s−1 , diffusion coefficient 100 cm2 s−1 and
absorbing boundary conditions at the detector (x = 100 mm). Inset: simulated spin current, given
by the gradient of Green’s function at the detector, as in equation (6.3). (Online version in colour.)
Green’s function in equation (6.2) and the spin current derived from it in
equation (6.3) (using typical values for v, D and L at various times t) are shown
in figure 7.
Note that the expression in equation (6.3) is simply the spin density in the
absence of the boundary condition [98,99] multiplied by the spin velocity L/t.
Because our measurements are under conditions of strong drift fields, we have
approximated this term with a constant (e.g. the average drift velocity v) in
previous work [80,93,101]. To first order, this approximation involves a rigid shift
of the signal peak to lower arrival times of ≈ 2D/v 2 and a subsequent error in the
measured velocity of 2D/L. There is no first-order error in the dephasing. Because
typical values for these variables are of the order of v > 106 cm s−1 , L > 10−2 cm,
and 102 < D < 103 cm2 s−1 , the relative error associated with this approximation
(2D/Lv in both cases) is then just a few per cent.
We can use this Green’s function in the kernel of an integral expression to
model, for instance, the expected precession signal as the weighted sum of all the
cosine contributions from electrons with different arrival times (0 < t < ∞),
∞
Js (x = L, t) cos ut dt.
(6.4)
0
Note that because Js is causal (i.e. Js (t < 0) = 0), we can extend the lower
integration bound to −∞ so that this expression is equivalent to the real
part of the Fourier transform of Js (x = L, t). We can therefore use precession
Phil. Trans. R. Soc. A (2011)
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Review. Spin transport in silicon
3567
measurements as a direct probe of the spin current transit time distribution
by making use of the Fourier transform, as shown in figure 5b. By fitting the
data to the simulation, we can extract the transport variables v and D; along
with the spin lifetime measurement, this comprises a spin Haynes–Shockley
experiment.
Equation (6.4) is valid only for the case of a purely perpendicular magnetic
field B = Bz ẑ [102]. In the most general case, if there is also an in-plane
field component By ŷ (owing to misalignment or a static field) and/or if the
perpendicular component of the field Bz is strong enough to overcome the in-plane
magnetic anisotropy of the film, the cos ut term must be replaced with
(Bz cos q1 + By sin q1 )(Bz cos q2 + By sin q2 )
cos ut
By2 + Bz2
+
(Bz sin q1 − By cos q1 )(Bz sin q2 − By cos q2 )
,
By2 + Bz2
(6.5)
where in the simplest case, the magnetization rotation angles q1,2 =
arcsin(tanh(Bz /h1,2 )), and h1,2 are the demagnetization fields of injector and
detector ferromagnetic layers (typically several tesla) [86]. From equation (6.5),
we can show that small misalignments from the out-of-plane direction during
single-axis precession measurements cause only in-plane magnetization switching
of injector and detector and a negligible correction to the amplitude of precession
oscillations to first order [65]. Furthermore, the expression is now composed of a
coherent term proportional to cos ut and an incoherent term independent of t.
In the case of extremely strong dephasing (e.g. caused by geometrically induced
transit length uncertainty), the first term contributes a negligible amount to the
integral expression in equation (6.4) and the signal is dominated by the field and
magnetization geometry dictated in the second term [83].
(b) Doped silicon
In the above, we have assumed that the drift velocity v in equation (6.1) was
independent of x. This assumption is not valid in doped semiconductors owing to
the presence of inhomogeneous electric fields and band bending caused by ionized
dopants. Equation (6.1) then no longer has constant coefficients, so in general its
Green function cannot be solved analytically, and a computational approach must
be taken.
Direct simulation of an ensemble of electrons can be used to assemble a
histogram of transit times, which in the limit of large numbers of electrons
yields the transit-time distribution function. This ‘Monte Carlo’ method, which
incorporates the proper boundary conditions for our current-based spin detection
technique automatically, was used to model spin transport through doped Si [86].
The technique involves discretely stepping through time a duration dt,
modelling
√ drift with a spatial translation v(x)dt, and diffusion with a translation
of ± 2Ddt (where the sign is randomly chosen to model the stochastic
nature of the process), until x > L. The drift velocity v(x) can be constructed
from a depletion-approximation model of the transport layer to determine
the electric field profile E(x) and a mobility model to get v(E) [85]. Our
Phil. Trans. R. Soc. A (2011)
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3568
I. Appelbaum
1
potential energy (eV)
detector
1000
0
500
–1
–2
0
1000 2000 3000 4000
3V
–3
–4
4.46 V
1000
–5
6V
injector
500
–6
–7
0
0
0.5
0.02 0.04 0.06 0.08 0.10
1.0
1.5
2.0
distance (μm)
2.5
3.0
Figure 8. Depletion-approximation conduction-band diagrams of a 3.3 mm thick n-type (7.2 ×
1014 cm−3 ) doped Si spin-transport layer with an injector–detector voltage drop of 3 V (where
the transport is dominated by diffusion against the electric field at the detector side and results
in the exponential-like arrival distribution with a typical time scale of 1000 ns shown in the top
inset); 4.46 V (where the bias is enough to eliminate the electrostatically neutral region and fully
deplete the transport layer); and 6 V (where the potential well has been eliminated and transport is
dominated by drift in the unipolar electric field, resulting in the Gaussian-like arrival distribution
with a typical time scale of 0.050 ns shown in the bottom inset).
experimental studies with n-type (≈ 1015 cm−3 phosphorus) 3.3 mm thick Si
devices indicate that despite highly ‘on-ohmic’ spin transport where the transit
time can be varied by several orders of magnitude resulting from a confining
electrostatic potential, Monte Carlo device modelling shows that spin lifetime
is not measurably impacted from its value in intrinsic Si [86]. Figure 8 shows
example band diagrams under different bias conditions (calculated in the
depletion approximation) and corresponding arrival time distributions from the
Monte Carlo simulation. Other computational approaches employ numerical finite
differences such as direct simulation of the dynamics in the drift–diffusion–
relaxation–precession equation using the Crank–Nicolson technique [103,104], or
if no time-dependent dynamics are required, a simple matrix inversion scheme
(which is more computationally efficient, if less extensible, than the Monte Carlo
described here).
Recently, we have shown that under the right voltage and temperature
conditions, measurements on similar n-type phosphorus-doped Si devices
can exhibit precession features with both large magnetic fields in the
106 /4p A m−1 range, and substantial (but incomplete) dephasing with much
smaller characteristic widths of ≈50 − 100 × 1000/4p A m−1 [105]. The possibility
that this phenomenon originates from transit-time delays associated with
capture/re-emission in shallow impurity traps, including sensitivity to the excitedstate spectrum, was suggested along with a quantitative model.
Phil. Trans. R. Soc. A (2011)
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Review. Spin transport in silicon
3569
7. Discussion
There has been significant progress in using ballistic hot electron spin injection
and detection techniques for spin transport studies in Si. However, there
are limitations of these methods. For example, the small ballistic transport
transfer ratio is typically no better than 10−3 to 10−2 ; the low injection
currents and detection signals obtained will result in sub-unity gain and limit
direct applications of these devices. In addition, our reliance on the ability
of Schottky barriers to serve as hot electron filters presently limits device
operation temperatures to below room temperature (although materials with
higher Schottky barrier heights could extend this closer to room temperature;
recent results with reverse Schottky-asymmetry detectors indicate evidence of
spin transport at 260 K [78]), and also limits application to only non-degenerately
doped semiconductors. Carrier freeze-out in the n-Si spin detection collector at
approximately 20 K introduces a fundamental low-temperature limit as well [93],
although undoped collectors are promising if lower temperatures are desired.
Despite these shortcomings, there are also unique capabilities afforded by
these methods, such as independent control over internal electric field and
injection current, and spectroscopic control over the injection energy level [106].
Furthermore, because injection is driven by electron kinetic energy and not
potential energy, it is also the only method to study spin transport in truly
semiconducting (low doped, highly resistive) materials. Unlike, for instance,
optical techniques, other semiconductor materials should be equally well suited
to study with these methods. The purpose is to use these devices as tools to
understand spin transport properties for the design of spintronic devices just as
the Haynes–Shockley experiment enabled the design of electronic minority-carrier
devices [97].
There is still much physics to be done with ballistic hot electron spin injection
and detection techniques applied to Si. Recently, we have adapted our fabrication
techniques to assemble lateral spin transport devices, where in particular very
long transit lengths [83] and the effects of an electrostatic gate to control the
proximity to an Si–SiO2 interface can be investigated [107,108]. Hopefully, more
experimental groups will develop the technology necessary to compete in this
wide-open field. Theorists, too, are eagerly invited to address topics such as
whether this injection technique fully circumvents the ‘fundamental obstacle’
because of a remaining Sharvin-like effective resistance [109], or whether it
introduces anomalous spin dephasing [93,110].
Biqin Huang, Hyuk-Jae Jang, Jing Li and Jing Xu provided invaluable fabrication, measurement
and programming efforts, which have made much of this work possible. This work has been
financially supported by the Office of Naval Research, DARPA/MTO and the National Science
Foundation.
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