arXiv:1109.5969v1 [cond-mat.mes-hall] 27 Sep 2011
Preprint v5 (with supplement). Dated: 28 September 2011
Non-linear spin Seebeck effect due to spin-charge interaction in graphene
I. J. Vera-Marun,a)V. Ranjan, and B. J. van Wees
Physics of Nanodevices, Zernike Institute for Advanced Materials, University of Groningen,
The abilities to inject and detect spin carriers
are fundamental for research on transport and
manipulation of spin information1,2.
tronic spin currents have been recently studied
in nanoscale electronic devices using a non-local
lateral geometry, both in metallic systems3and
in semiconductors4. To unlock the full potential
of spintronics we must understand the interac-
tions of spin with other degrees of freedom, going
beyond the prototypical electrical spin injection
and detection using magnetic contacts. Such in-
teractions have been explored recently, for exam-
ple, by using spin Hall5–7or spin thermoelectric
effects6,8,9. Here we present the detection of non-
local spin signals using non-magnetic detectors,
via an as yet unexplored non-linear interaction
between spin and charge. In analogy to the See-
beck effect10, where a heat current generates a
charge potential, we demonstrate that a spin cur-
rent in a paramagnet leads to a charge potential,
if the conductivity is energy dependent. We use
graphene11as a model system to study this ef-
fect, as recently proposed12. The physical concept
demonstrated here is generally valid, opening new
possibilities for spintronics.
Previous reports on detection of spin signals using non-
magnetic contacts have made use of spin-orbit interac-
tion via the (inverse) spin Hall effect5,6. Recently, large
non-local signals in graphene have been attributed to an
effect with similar phenomenology, given by the differ-
ence in Hall resistance between two (spin) channels in-
duced by an applied perpendicular magnetic field7. Both
effects produce a charge potential transversal to the di-
rection of the spin current and are valid in the linear
regime. In the present work we deal with a different con-
cept based on a non-linear interaction between spin and
charge which results in charge potentials longitudinal to
the spin current12. This effect is solely based on the en-
ergy dependence of the conductivity σ(ǫ), not requiring
spin-orbit interaction nor external magnetic fields.
To explain the concept of detection of spin signals used
here it is useful to make an analogy with thermoelectrics.
As shown in Fig. 1a, a temperature gradient sets up
a heat current. Under open circuit conditions, this re-
sults in a built up voltage V = −S(T2− T1), with S
the Seebeck coefficient of the conducting system. For
the case of diffusive spin transport13(see Fig. 1b) the
electrochemical potential of each spin channel can be
s = 1/2
s = -1/2
s = ±1/2
FIG. 1. Analogy between spin and heat transport illus-
trated by electronic distributions f(ǫ). a, A temperature
gradient (T2 > T1) sets up a heat current, with high-energy
electrons moving towards the cold region and low-energy elec-
trons moving towards the hot region. When the conductivity
is energy dependent (here ∂σ/∂ǫ > 0) a Seebeck voltage is
built up under open circuit conditions, to compensate for the
different conductivities of high- and low-energy electrons. b,
A gradient in spin accumulation ∆µ sets up a spin current,
with majority spin electrons moving towards the region with
lower ∆µ and minority spin electrons moving in the opposite
direction. Similar to thermoelectricity, a voltage is built up
if the conductivity is energy dependent due to the different
conductivities of the two electron spin species.
described as µ± = µavg± ∆µ, with ∆µ the spin ac-
cumulation (created by electrical spin injection) in the
conductor, which decays with a characteristic spin re-
laxation length λ. The gradient in spin accumulation
sets up a spin current which results in a built up voltage
V = −(β/e)(∆µ2− ∆µ1), with β the conductivity spin
polarization of the system13. While S is a general prop-
erty of conductors, β is in general zero except for fer-
romagnetic or ferrimagnetic materials. Therefore, pure
spin currents are not expected to generate charge voltages
in a paramagnetlike graphene. The latter is not true if we
consider spin transport away from the Fermi level. When
a sizable ∆µ is present, each spin channel experiences a
different conductivity even in a paramagnet, as long as
the conductivity is energy dependent. So we consider
a spin polarization of the conductivity induced by ∆µ,
which can be approximated as12β = −∆µσ−1∂σ/∂ǫ.
To complete the analogy, we define α = σ−1∂σ/∂ǫ and
FIG. 2. Sample geometry and non-local measurement
configuration. a, Coloured SEM image of the device (after
measurement and sample failure). Tunnel contacts have elec-
trodes made of 5/25-nm-thick Ti/Au (Contacts 1, 4 and 5) or
30-nm-thick Co (Contacts 2 and 3). Top inset: optical image
before measurement. Bottom inset: AFM image of graphene
before contact deposition.
b, Configuration for measuring
linear spin resistance using a magnetic detector. c, Configu-
ration for measuring non-linear resistance using non-magnetic
the Lorentz number L0= (π2/3)(k2
expressions for V are only valid when the coefficients S
and β are independent of the driving forces T and ∆µ,
respectively. In reality, the Seebeck coefficient is given
by the Mott formula10,14S = −L0eαT.
T ≈ T2− T1, the Seebeck voltage depends quadratically
on the driving force as V ∝ L0eα(T)2. Similarly, for spin
transport in a paramagnet, the induced spin polarization
mentioned above (β = −α∆µ) also results in a quadratic
dependence on the driving force as V ∝ (α/e)(∆µ)2. Due
to the common factor α the effect described here has sim-
ilar behavior as the Seebeck coefficient, showing oppo-
site polarity for electron and hole regimes. Furthermore,
because ∆µ is (to lowest order) linear on the injection
current3,15, the result is a second order signal V ∝ I2.
For our proof of concept we use graphene which, apart
from being a two-dimensional platform for relativistic
quantum mechanics11, has proven to be an excellent sys-
tem for spin transport where large values of ∆µ ≈ 1 meV
can be obtained15. The sample is shown in Fig. 2a. It
B/e2). The previous
In the limit
-400 -2000 200 400
-40 -200 20
FIG. 3. Linear spin detection using a magnetic de-
tector. a, Spin-valve effect in non-local linear resistance R1
by sweeping an in-plane magnetic field at Vg = 0 V. Two
well-defined values correspond to parallel (RPa
∆R1 = RPa
versus Vg. The dashed line is the square
resistance Rsq of graphene between Contacts 2 and 3 with
VD ≈ −20 V. c, Hanle spin precession curve by sweeping a
perpendicular magnetic field at Vg = 10 V. The solid line is
a fit with the one-dimensional Bloch equation. The obtained
parameters are D = 0.025 m2/s and τ = 71 ps, with con-
tact spin polarization P = 9 %. d, Spin relaxation length
λ extracted from Hanle curves (as that shown in c) taken at
several values of Vg. The solid line is a fit with a Gaussian
function for parameterization purposes.
1 ) and anti-
1) alignment of Co contacts. b, Spin resistance
1 − RAp
consists of a lateral graphene field-effect transistor cov-
ered with a thin aluminum oxide barrier that yields high-
resistance contacts for efficient spin transport16and elec-
trostatic gating via the Si/SiO2substrate, as previously
reported15,17. Besides using magnetic Co contacts for
electrical spin injection and detection (see Fig. 2b), we
also include Ti/Au contacts. These non-magnetic con-
tacts are used to electrically detect spins in the config-
uration shown in Fig. 2c. There are two key aspects to
such a measurement configuration. First, the use of non-
magnetic detectors simplifies the analysis of the non-local
signal, because for the case of using magnetic detectors
both linear and non-linear signals are expected due to
direct detection of ∆µ12,18. Second, the use of two mag-
netic contacts as source and drain for charge current al-
lows us to measure the non-local signal for both parallel
VPaand anti-parallel VApalignment of their magnetiza-
tions and thereby to focus on the difference between the
two states ∆V . This way we can exclude background sig-
nals which do not depend on ∆µ and can be present in
non-local measurements18. We use a lock-in technique to
determine the linear V1and second order V2components
-100 -80-60-40-200 20 40
FIG. 4. Non-linear spin detection using non-magnetic
detectors. a, Linear non-local signal versus in-plane mag-
netic field, no spin-valve effect is observed. b, Second order
signal (same measurement as in a) showing spin-valve effect.
Two well-defined values correspond to parallel (VPa
) alignment of the Co contacts. Curves in a and
b correspond to (from top to bottom) Vg = 30,0,−40,−80
and −90 V and are offset vertically for clarity. Each curve
is the average of 10 measurements. All data for a root mean
square current of 5 µA (configuration as in Fig. 2c). c, Non-
linear spin resistance ∆R2 = RAp
data point the average value of V2 for (anti)parallel configura-
tion, and their standard deviation, was extracted from curves
as those shown in b. The solid line is a result from numerical
2 ) and anti-
versus Vg. For each
of the resulting root mean square signal. From them we
extract the non-local resistances Ri contributing to the
total signal V = R1I + R2I2. All measurements are at
room temperature, unless otherwise noted.
We start by characterizing spin transport in the linear
regime. The results in Fig. 3 show a non-local spin-valve
effect, demonstrating spin transport between Contacts 2
and 3, for a center-to-center separation of L = 1.0 µm
and width of graphene w ≈ 1.1 µm. The obtained spin
resistance ∆R1≈ 4 Ω is nearly constant versus the gate
voltage Vg applied to the substrate.
charge carrier density ngin graphene as ng= γ(Vg−VD),
with VDthe condition for charge neutrality (Dirac point)
and γ = 7.2 × 1014m−2V−1. On the other hand, the
graphene square resistance Rsq depends on Vg, chang-
ing by a factor of 5.The observed ∆R1 vs Rsq be-
havior can be understood by the standard relation15
∆R1 = (P2Rsqλ/w)exp(−L/λ), with P the spin po-
larization of the magnetic contacts, as being due to the
charge carrier density dependence of λ with a minimum
at the Dirac point. The latter is attributed in graphene
to the Elliot-Yafet mechanism for spin relaxation19–21
Vg controls the
Au Co CoAu
FIG. 5. Role of charge density distribution. a, Resis-
tance of graphene between Contacts 4 and 5 (Au detectors).
The red solid line is a fit using a phenomenological description
including two Dirac curves (solid lines labeled 1 and 2) and
a constant 0.35 kΩ for the low resistance Contact 5 (dashed
line). b, Extracted parameter α = σ−1∂σ/∂ǫ for each of the
two Dirac curves mentioned in a. c, Schematic representa-
tion of charge density distribution within graphene. Different
properties are considered for graphene under and next to the
contacts (1) and for the region away from the contacts (2).
(Supplementary A). Furthermore, the previous relation
is valid only for contact resistances Rc≫ Rsqλ/w, where
the contacts do not affect the spin transport16,17,22. We
take into account both considerations in our modelling
below, by parameterizing λ (see Fig. 3d) and by including
the finite resistance of the contacts used for spin injection
Next, we demonstrate non-linear detection of spins by
using non-magnetic contacts. In Fig. 4b is shown a clear
spin-valve effect in the second order component V2 of
the non-local signal. The transitions in V2occur at the
switching fields of the magnetic contacts used for current
injection. We observe at zero gate voltage that VAp
2 , consistent with the presence of a larger ∆µ for the
anti-parallel magnetic configuration15and a positive sign
of the parameter α for electron transport12. Therefore,
we expect that the sign of the non-linear spin resistance
∆R2should follow that of α and change sign when going
from transport in the electron (Vg > VD) to the hole
(Vg< VD) regime. The latter is confirmed by observing
that ∆R2 < 0 for gate voltages Vg ≪ VD. We remark
that the measured spin-valve signal cannot be explained
by spurious detection of potentials in the current carrying
part of the sample, due to the absence of spin-valve signal
in the first order response (Fig. 4a).
The gate voltage dependence of the non-linear spin re-
sistance is presented in Fig. 4c. The ∆R2 vs Vg curve
shows a maximum of ≈ 5 kΩ/A for electron transport.
We did not observe a clear sign change when crossing
the Dirac point, whereas for hole transport there was
a minimum value of only ≈ −2 kΩ/A. To understand
this electron-hole asymmetry we looked into the charge
transport properties of the detector circuit, between Con-
tacts 4 and 5. The Dirac curve in Fig. 5a shows that,
while there is a reasonable symmetry for Vg close to
VD = −9 V, this is not the case for larger Vg, as evi-
denced by the kink visible at Vg = −55 V. Such kinks
in the Dirac curve have been described as arising due to
electron doping from metal contacts having a thin oxide
layer that prevents charge density pinning23. Our con-
tacts are deposited onto a thin oxide barrier. Therefore
we interpret the Dirac curve of the detector circuit as
being composed of two contributions, the graphene un-
der (and next to) the contacts and that away from the
contacts (curves 1 and 2 in Fig. 5a, Supplementary B).
Having described both spin and charge transport in
the linear regime, we now construct a minimal one-
dimensional model that allows quantitative comparison
with experiment. As mentioned above, we model ∆µ
by considering the induced conductivity spin polariza-
tion β, finite resistance contacts, and gate voltage de-
pendency of λ. We use a fixed P = 9 % for the magnetic
contacts (extracted in the regime Rc≥ 5Rsqλ/w) and a
width profile for graphene as extracted from atomic force
microscopy (AFM). Furthermore, we describe the Dirac
curve for each graphene region using the approximate
relation24σ = νe(n2
and nia background carrier density due to the presence
of electron-hole puddles and thermally generated carri-
ers. We then extract the parameter α for each region12
(see Fig. 5b) likewise to the extraction of the Seebeck
coefficient of graphene from the Dirac curve14,24.
The model, schematically shown in Fig. 5c, has the
extension of the graphene region 1 beyond the contact
edge as the only free parameter. Scanning photocurrent
work25has shown that the doping in graphene decays
gradually from the contact edge extending up to a dis-
tance of ≈ 0.3 µm. For simplicity we consider a con-
stant doping up to a distance of 0.15 µm. The modelled
∆R2 vs Vg curve (solid line in Fig. 4c) successfully re-
produces both the trend and the magnitude of the data.
The agreement yields certainty to our interpretation of
the measured ∆R2signal as arising due to the non-linear
interaction between spin and charge.
The magnitude of ∆R2is only slightly limited by the
finite resistance of our contacts (Rc≈ 7 kΩ). Assuming
infinite contact resistance, we predict only up to a 2 times
increase in ∆R2. On the other hand, the use of high-
quality tunnel contacts22with P = 30% would yield a
10 times increase. And since αmax∝ 1/√ni, similar to
the Seebeck coefficient14, decreasing niby two orders of
magnitude by using a boron nitride substrate26would
yield a further 10 times increase. Therefore the herewith
demonstrated effect is a real candidate for spin detection.
It can be regarded as a step in the logical progression
from linear interactions between spin and charge towards
interactions between spin and heat, as studied in the field
of spin caloritronics27.
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Graphene is obtained from HOPG graphite by mechan-
ical exfoliation and deposited on a highly n-doped Si sub-
strate covered with a thermal oxide layer 300 nm thick.
The Si substrate is used as electrostatic gate. Graphene
is first covered by 0.8 nm Al followed by natural oxidation
to obtain a thin aluminum oxide layer. Electron beam
lithography (EBL), deposition by evaporation, and lift-
off were performed twice, first for the Ti/Au contacts and
then for Co. We observed in previous samples a lower re-
sistance of the Ti/Au contacts, which forbade us to use
them for spin detection16. This problem was possibly
related to the required baking of PMMA for the second
EBL step, which may cause diffusion of Ti/Au through
the oxide barrier. To solve it we increased the thickness
of the oxide barrier only for the Ti/Au contacts by de-
position and oxidation of extra 0.3 nm Al. This yielded
non-invasive Au contacts with similarly high resistances
as those of the Co contacts.
Characterization took place at room temperature and
at 77 K (Supplementary A) in a cryostat with a base
pressure ≈ 1×10−7mbar. The sample was first annealed
under vacuum at 130◦C for ≈ 24 hr for removal of ph-
ysisorbed water, resulting in low hysteresis in the Dirac
curve (∆VD< 10 V). Measurements were performed us-
ing an a.c. current source 1–5 µA and recording simul-
taneously the first and second harmonic responses using
two lock-in systems. All current and voltage signals re-
ported are root mean squared values. Therefore the resis-
tances were extracted as R1= V1/I and R2=√2V2/I2.
Excitation frequency was kept ≤ 3 Hz to prevent signals
due to capacitive coupling, as determined by frequency
scans. Contribution from higher harmonics was found to
We developed a one-dimensional finite-element code
using the program MATLAB to find numerical solutions
for the two-channel spin diffusion equations13
with the inclusion of an element-specific conductivity
spin polarization12β, as described in the main text. Ele-
ment length was kept ≤ 10 nm. I±and ∆µ were set to be
continuous across element boundaries. We consider point
contacts located at the center of the fabricated electrodes,
which could either inject a spin current PI or detect the
electrochemical potential µavg+ P∆µ. Spin relaxation
under contacts with finite resistance (4.9, 9.2, 7.3 and
< 1 kΩ for Contacts 2, 3, 4 and 5, respectively) was
implemented as in ref. 17, with the extra consideration
of contact spin polarization P. The extracted parame-
ters for the curve Dirac 2 (graphene between contacts)
were ν = 3900 cm2/Vs and ni = 3.5 × 1015m−2, con-
sistent with previous experiments on SiO2substrates24,
whereas for Dirac 1 we obtained ν = 800 cm2/Vs and
ni = 2 × 1016m−2. To obtain ∆R2 we calculated the
difference in potential V = −µavg/e between the Au de-
tectors, for both parallel and anti-parallel configurations
and d.c. currents I = ±5 µA. The result was then fit-
ted with ∆V = ∆R1I + ∆R2I2. The odd contribution
|∆R1| < 3 mΩ was found to be negligible.
We would like to thank J. G. Holstein, B. Wolfs, and M.
de Roosz for technical assistance. This work was financed
by the Zernike Institute for Advanced Materials and by
EU FP7 ICT Grant No. 251759 MACALO.
SUPPLEMENTARY INFORMATION A
Here we present spin transport data acquired at liquid-
nitrogen temperature (77 K). First, we show linear spin
transport at 77 K and compare it with that at room
temperature. Then we present partial data on non-linear
spin detection using non-magnetic detectors at 77 K.
-400 -2000 200400
-40 -200 20
FIG. 6. Linear spin detection using a magnetic de-
tector at 77 K. a, Two curves showing the spin-valve ef-
fect in non-local linear resistance R1 by sweeping an in-plane
magnetic field at Vg = 0 V. Two well-defined values corre-
spond to parallel (RPa
Co contacts. b, Spin resistance ∆R1 = RPa
Vg. The dashed line is the square resistance Rsq of graphene
between Contacts 2 and 3 with VD ≈ −20 V. c, Hanle spin
precession curve by sweeping a perpendicular magnetic field
at Vg = 10 V. The solid line is a fit with the one-dimensional
Bloch equation. The obtained parameters are D = 0.053 m2/s
and τ = 86 ps, with contact spin polarization P = 7 %. d,
Spin relaxation length λ extracted from Hanle curves (as that
shown in c) taken at several values of Vg.
1 ) and anti-parallel (RAp
1) alignment of
We start by characterizing spin transport in the lin-
ear regime at 77 K. The results in Fig. 6 show a non-
local spin-valve effect, again demonstrating spin trans-
port between Contacts 2 and 3. The spin resistance is
∆R1 ≈ 5 Ω and shows a minimum close to the Dirac
point. This result for ∆R1is similar to that at room tem-
perature (shown in the main text) but ≈ 20 % larger. A
larger ∆R1at 77 K can be understood from the analysis
of Hanle spin precession curves (see Fig. 6c) from where
we extract spin relaxation lengths ≈ 60 % larger than at
room temperature. The effect of largervalues of λ at 77 K
is slightly compensated in our sample by a lower contact
spin polarization P = 7 %. We remark that the gate volt-
age dependence of the spin relaxation length λ =
at 77 K (see Fig. 6d) shows a minimum close to the Dirac
FIG. 7. Non-linear spin detection using non-magnetic
detectors at 77 K. a, Second order signal showing spin-
valve effect at 77 K. Two well-defined values correspond to
contacts. The curves correspond to (from top to bottom)
Vg = 20,0,−10,−20 and −40 V and are offset vertically for
clarity. Each curve is the average of 26 measurements. All
data for a root mean square current of 5 µA. b, Non-linear
spin resistance ∆R2 = RAp
black squares). For each data point the average value of V2for
(anti)parallel configuration, and their standard deviation, was
extracted from curves as those shown in a. The data for room
temperature (open red circles) is also shown for comparison.
2 ) and anti-parallel (VAp
) alignment of the Co
versus Vg at 77 K (closed
point, similar to the data at room temperature. The lat-
ter is is indicative of the Elliot-Yafet mechanism of spin
relaxation in single-layer graphene19–21being dominant
both at room temperature and at 77 K.
Finally, in Fig. 7 we demonstrate non-linear detection
of spins by using non-magnetic contacts at 77 K. Al-
though our data at low temperature is limited, it shows
a similar behavior of ∆R2as that at room temperature.
The magnitude of ∆R2at both temperatures is similar
within the experimental uncertainty, except for an almost
2 times higher value at Vg = −10 V close to the Dirac
point. The observation of similar results at room temper-
ature and at 77 K are a confirmation of our interpretation
of the non-linear spin-valve signal as solely arising from
an interaction between spin and charge, which does not
directly involve heat as in the case of spin thermoelectric
SUPPLEMENTARY INFORMATION B
Here we discuss on the identification of contributions to
the Dirac curve from graphene regions under and around
the contacts and those away from the contacts. We also
discuss on the nature of the contacts and their possible
contributions to the non-linear spin signal.
In the main text we showed how the Dirac curve for
graphene between the two Au detectors is composed of
two distinct contributions. The main contribution corre-
sponds to regions of the graphene channel located away
from the contacts, with a Dirac point VD= −9 V. A mi-
nor contribution, visible as a kink in the hole regime23,
7 Download full-text
doping at room temperature. a, Dirac curve of graphene
between Contacts 2 and 3 (Co-Co, green), graphene between
Contacts 3 and 4 (Co-Au, red) and graphene between Con-
tacts 4 and 5 (Au-Au, black). Vertical dashed lines indicate
the location of the Dirac point VD for different graphene re-
gions. b, Non-linear spin resistance ∆R2 = RAp
Vg. The red solid line is for the model described in the main
text, considering graphene regions under both Au and Co con-
tacts to behave the same with VD = −55 V. The dashed green
line is for considering graphene under the Co contacts to have
a Dirac point VD = −20 V. The data for room temperature
(black circles) is also shown for comparison.
Effect of Co and Au contacts on graphene
corresponds to regions of graphene located under (and
next to) the Au contacts with VD= −55 V (due to con-
tact doping). We also observed similar kinks for the Dirac
curves for graphene between the adjacent Co injector and
Au detector, and for graphene between the two Co con-
tacts used for spin injection (see Fig. 8a). The kinks in
the Dirac curves indicate that the Co contacts also dope
the graphene channel but with a Dirac point close to
VD= −20 V, different than for graphene around the Au
contacts (VD= −55 V).
The resulting ∆R2 for the model presented in the
main text corresponds to the simple case of assumption
that all contacts have the same effect on graphene, with
VD= −55 V. In Fig. 8b we also show the result of incor-
porating in the model a different contribution from the
Co contacts, with VD = −20 V. Notice this considera-
tion does not have a significant effect on the modelled
∆R2. There are two reasons for this observation. First,
the graphene regions modified by the presence of the
Co contacts are not within the detector circuit. There-
fore, charge potentials generated due to their α param-
eter have no influence on the signal detected between
the Au contacts. Second, though the graphene regions
under the Co contacts do have an influence on the ∆µ
profile via their resistivity (Dirac curve), this influence is
small because these regions are narrow compared to the
full extent of graphene over which ∆µ decays. So the
consideration of doping effects under the Co contacts is
not critical for understanding the non-linear spin signal
measured via the Au contacts.
A fundamental question is whether the contacts them-
selves contribute to the measured non-linear spin signal.
-40 -200 20 40
FIG. 9. Contact characterization at 77 K. a, Gate volt-
age dependence of resistances of Contacts 2, 3 and 4. Data
for a root mean square current of 2 µA. b, Differential resis-
tances of Contacts 2, 3 and 4 versus d.c. current bias, for a
root mean square modulation of 0.1 µA.
This signal, generated via the non-linear interaction be-
tween spin and charge, relies on achieving a large enough
∆µ and having a sizable α parameter. Owing to the
large conductivity of metals, the achieved spin accumu-
lation within the Au and Co metals (≈ 1 µeV)3is much
lower than in graphene. So we do not expect a sizable
signal coming from the bulk of the metallic contacts.
The discussion above leaves us with the final possi-
bility that the graphene-metal interface could produce a
sizable signal. Spin thermoelectric effects have been ob-
served in high-quality tunnel contacts9, as expected from
the strong energy dependence of electron transmission
through a tunnel barrier. To address this issue we have
characterized the charge density and bias dependence of
contact resistances in our device. From the results in
Fig. 9 we observe that the contacts only change up to
20 % with gate voltage, and have linear I −V character-
istics (constant dV/dI within 10 % for the explored bi-
asing currents). These contact characteristics have been
previously observed on similar samples and were ascribed
to transport dominated by relatively transparent regions
in the oxide barrier17. In this case we do not expect that
the interface would exhibit a sizable α parameter and
its contribution to the non-linear spin signal would be
negligible. We conclude the latter is applicable to our
device, as we did not require to include this effect in our
model in order to achieve a satisfactory description of the