arXiv:1103.4742v1 [cond-mat.mes-hall] 24 Mar 2011
Giant Spin-Hall Effect induced by Zeeman Interaction in Graphene
D. A. Abanin,1R. V. Gorbachev,2K. S. Novoselov,2A. K. Geim,2and L. S. Levitov3
1Princeton Center for Theoretical Science and Department of Physics, Princeton University, Princeton, NJ 08544
2Manchester Centre for Mesoscience and Nanotechnology,
University of Manchester, Manchester M13 9PL, UK
3Department of Physics, Massachusetts Institute of Technology, Cambridge, MA 02139
We propose a new approach to generate and detect spin currents in graphene, based on a large
spin-Hall response arising near the neutrality point in the presence of external magnetic field. Spin
currents result from the imbalance of the Hall resistivity for the spin-up and spin-down carriers
induced by Zeeman interaction, and do not involve spin-orbit interaction. Large values of the spin-
Hall response achievable in moderate magnetic fields produced by on-chip sources, and up to room
temperature, make the effect viable for spintronics applications.
The spin-Hall effect (SHE) is a transport phenomenon
resulting from coupling of spin and charge currents: an
electrical current induces a transverse spin current and
vice versa[1, 2]. The SHE offers tools for electrical ma-
nipulation of electron spins via striking phenomena such
as current-induced spatial segregation of opposite spins
and accumulation of spin at the boundary of current-
carrying sample [3, 4]. All SHE mechanisms known to
date rely on spin-orbit interaction. The two main va-
rieties of SHE — intrinsic SHE and extrinsic SHE —
arise due spin-orbit terms in the band Hamiltonian and
spin-dependent scattering on impurities, respectively.
Single layer graphene has emerged recently as an at-
tractive material for spintronics that features long spin
diffusion lengths, gate tunable spin transport[6, 7], and
high-efficiency spin injection. However, to realize the
full potential of graphene, several issues must be ad-
dressed. First, the measured spin lifetimes are orders of
magnitude shorter than theoretical predictions[6–11] call-
ing for identifying and controlling extrinsic mechanisms
of spin scattering[10–14]. Second, the low intrinsic spin-
orbit coupling values[8, 15] render the conventional SHE
mechanisms ineffective, depriving graphene spintronics of
a crucial control knob for spin transport.
Here we outline a new approach to generate and probe
spin currents in graphene, based on a SHE response in the
presence of magnetic field that does not rely on spin-orbit
interaction. Spin currents are generated by the combined
effect of spin and orbital coupling to magnetic field. The
Zeeman splitting lifts the up/down spin degeneracy and
imbalances the Hall resistivities of the two spin species
(see Fig.1 inset). This leads to a net transverse spin
current in response to an applied charge current. The re-
sulting SHE response, called below ZSHE for brevity, is
an essentially classical effect that offers a robust and effi-
cient way to generate spin currents electrically in a wide
range of temperatures and magnetic fields. The ZSHE
response is sharply enhanced near the Dirac point (DP).
This makes the effect viable for spintronics applications.
The enhancement at the DP, which results from special
transport properties of the Dirac fermions, is illustrated
in Fig.1. Transport is unipolar at high doping from the
Carrier density n [1011 cm−2]
Spin−Hall coefficient θSH
FIG. 1: Spin-Hall response induced by an external magnetic
field in graphene in the absence of spin-orbit coupling. The
SHE coefficient θSH, Eq.(3), peaks at the Dirac point (DP).
Spin currents at the DP originate from the the imbalance of
Hall resistivities for spin up and down due to Zeeman split-
ting EZ (inset, red and blue curves). Steep behavior of ρxy
leads to large imbalance in the spin-up and spin-down Hall
response at chemical potentials |µ|<
can be reached already at moderate field strengths and high
temperatures, Eq.(14). Parameters used: B = 1T, disorder
broadening γ = 100K, electron-hole drag coefficient η = 2.3?.
∼∆µ. Large values θSH
DP, dominated by carriers of one type, with ρxyfollowing
the standard quasiclassical expression,
ρxy(n) = −B
Transport near the DP is bipolar, which produces smear-
ing of the 1/n singularity in ρxy by the effects of two-
particle scattering as well as disorder. This leads to a
steep linear dependence in ρxy(n) at the DP (Fig. 1 in-
set), which is also seen in experiment (Fig. 3). The large
values of ∂ρxy/∂n, despite the smallness of the Zeeman
splitting, can yield giant ZSHE response.
The conventional SHE is described by the spin-Hall
conductivity which relates transverse spin current and
the electric field [1, 2]. To identify the relevant quantity
FIG. 2: (a) Schematic for spin accumulation in the SHE
regime. An electric current in a graphene strip drives trans-
verse spin current, resulting in spin density build-up across
the strip, Eq.(3). (b) Generation and detection of spin cur-
rent in the H-geometry. Electric current passed through the
region of local magnetic field drives spin current along the
strip. Voltage generated via inverse SHE is detected using
probes 3, 4. Hanle-type oscillation due to spin precession can
be induced by external magnetic field applied in-plane.
for ZSHE, we consider spin accumulation in a simplified
situation when the two spin species are independent, each
described by its own conductivity tensor. For a strip of
width w carrying uniform current, shown in Fig.2(a), the
transverse gradients of electrochemical potential for each
spin projection are
with the up/down spin concentrations n↑(↓)and the den-
sities of states ν↑(↓). Ignoring spin relaxation, we esti-
mate spin density at the edge ns= n↑− n↓as
with EZthe Zeeman splitting (for full treatment see Ap-
pendix A). Here we used the smallness of EZ compared
to the DP smearing ∆µ (see Fig.1) to express θSHas a
derivative with respect to µ. Our analysis shows that
the quantity θSHplays a role identical to the ratio of the
spin-Hall and ohmic conductivities ξSH = 2σSH/σxx in
the conventional SHE. We will thus refer to θSH as the
For realistic parameter values, Eq.(3) yields large θSH
at the peak (see Fig.1). For B = 1T, using disorder
strength estimated from mobility in graphene on a BN
substrate, γ ≈ 100K (see Eq.(10)), we find θSH = 0.1.
This is more than two orders of magnitude greater than
the SHE values in typical spintronics materials with spin-
orbit SHE mechanism. Say, we estimate ξSH≈ 5 · 10−4
from the spin and charge resistance measured in InGaAs
system . The ’giant’ values θSHare in fact to be ex-
pected, since the ZSHE can be viewed as a classical coun-
terpart of the SHE at kBT < EZdiscussed in Refs.[16, 17]
characterized by quantized σSH= 2e2/h.
Large θSH values result in ‘giant’ spin accumulation.
From Eq.(3), taking θSH= 0.1 and the density of states at
disorder-broadened DP ν↑(↓)≈√∆n/π?v0(with density
inhomogeneity ∆n ≈ 1010cm−2typical for graphene on
BN substrate), and using E = 10V/µm (which corre-
sponds to maximum current density in graphene ), we
estimate nsat the edges of a 2µm-wide graphene strip:
ns≈ 3 · 1010cm−2.(4)
Such large densities, which exceed the DP width ∆n, can
be easily detected by spin-dependent tunneling. The esti-
mate (4) is also five orders of magnitude greater than the
spin accumulation per atomic layer observed in a three-
dimensional GaAs , ns≈ 5 · 105cm−2.
Another attractive feature of the ZSHE is that it can
enable local generation and detection of spin currents.
Permanent micromagnets can generate fields up to 1T
concentrated to regions of size ∼ 0.5µm  (fields up to
1.4T are achievable using widely available Neodymium
Boron magnets). State-of-the-art microelectromagnets
have similar characteristics . In an H-geometry, pic-
tured in Fig. 2(b), spin currents can be generated on
one end of graphene strip and detected on the opposite
end. External B field, applied in-plane or at an angle to
the graphene sheet, can be used to induce spin precession
which will manifest itself in Hanle-type oscillations of the
voltage measured between probes 3, 4. This setup can
serve as an all-electric probe of spin currents [18, 22, 23].
To model the dependence of θSH on B, T and disor-
der, we employ the quantum kinetic equation approach
of Refs.[24, 25]. For a spatially uniform system, we have
where fe(h)(p) is the distribution function for electrons
and holes, and qe= −qh= e. To describe transport near
the DP, it is crucial to account for the contributions of
both electrons and holes. The collision integral describes
momentum relaxation due to two-particle collisions and
scattering on disorder [24, 25].
The approach based on Eq.(5) is valid in the quasi-
classical regime, when particle mean free paths are long
compared to wavelength. This is true when the colli-
sion rate is small compared to typical particle energy,
which requires weak disorder γ ≪ kBT, where γ is de-
fined in Eq.(10), and weak effective fine structure con-
stant α = e2/?v0κ ≪ 1 (κ is the dielectric constant).
The kinetic equation (5) can be solved analytically
in the limit of small α [24, 25]. Rather than pursuing
this route, we follow Ref. to obtain transport coeffi-
cients from the balance of the net momentum for differ-
ent groups of carriers, electrons and holes, taken to be
moving independently. We use a simple ansatz
e(εp−pae(h)−µe(h))/kBT+ 1,εp= v0|p|, (6)
where µe= −µhare the chemical potentials of electrons
and holes. The quantities aeand ah, which have the di-
mension of velocity, are introduced to describe a current-
carrying state.This ansatz corresponds to a uniform
motion of the electron and hole subsystems, such that
the collision integral for the e-e and h-h processes van-
ishes (as follows from the explicit form of the collision
integral given in Ref. ). Thus only the e-h collisions
contribute to momentum relaxation, resulting in mutual
drag between the e and h subsystems.
Eq.(5) yields coupled equations for ensemble-averaged
velocities and momenta of different groups of carriers (6):
where i, i′label the e and h subsystems with different
spins. The ensemble-averaged scattering times τdis
carrier densities ni, and the electron-hole drag coefficient
η, describing collisions between electrons and holes, are
The quantities Vi, Piare proportional to each other,
Pi = miVi. An explicit expression for mi as a func-
tion of T, µ can be found by expanding the distribution
functions (6) to lowest non-vanishing order in ae(h):
where gi(p) = fi(p)(1 − fi(p)). The integrals over p,
evaluated numerically, give the effective mass as a func-
tion of T and µ. At charge neutrality, setting µe(h)= 0,
we find mT =9ζ(3)
The times τdis
and carrier densities ni in (7) are ex-
pressed through the distribution function (6) with ai= 0:
where τdis(ε) is the transport scattering time, Eq.(10),
and the factor of two accounts for valley degeneracy.
We pick the model for disorder scattering to account
for the experimentally observed linear dependence of con-
ductivity vs. doping, σ = µ∗|n|, where µ∗is the mobility
away from the DP. This is the case for Coulomb impuri-
ties or strong point-like defects, such as adatoms or va-
cancies . In both cases the scattering time has an
approximately linear dependence on particle energy,
∼γ= ?|ε|/γ2,γ = v0
where the disorder strength parameter γ is expressed
through mobility. The value µ∗= 6 · 104cm2/V · s mea-
sured in graphene on BN  yields γ ≈ 120K. Similar
values are obtained from the ρxx-based DP width. Tak-
ing ∆n ≈ 1010cm−2, we find γ ∼ ?v0
To obtain ρ↑(↓)
xy , we solve Eq.(7), accounting only for
the drag between electrons and holes of the same spin. It
can be shown that including the drag between species of
opposite spin does not change the overall behavior of the
transport coefficients and SHE (see Appendix C). Eq.(7)
can be conveniently analyzed using complex-valued quan-
tities Px+ iPy, Vx+ iVy, giving complex resistivity
√∆n ≈ 100K.
xx + iρ↑(↓)
˜ γe˜ γh+ ηne
mh˜ γe+ ηnh
mh˜ γe+ η(ne−nh)2
Here ˜ γi=
As a sanity check, we consider the behavior at charge
neutrality. Setting ne = nh, me = mh, etc., gives ρxx
which is a sum of the Drude-Lorentz resistivity and the
electron-hole drag contribution analyzed in Refs.[24, 25],
− iΩi, with Ωi= qiB/mic the cyclotron
?1 + τ2Ω2?+η
tivated electrons (holes) at the DP, having fixed spin
projection.Disorder scattering (first term) dominates
at low temperatures T<
∼T∗= γ??/η (at B = 0), while
electron-hole drag (last term) dominates at T>
The value for the electron-hole drag coefficient η can
be obtained by matching the last term in Eq.(12)), di-
vided by 2 to account for spin, to the analytic result
ρxx≈ 8.4?α2/e2[24, 25]. We evaluate α using the effec-
tive dielectric constant κ =ε0+1
counts for screening by substrate and for intrinsic screen-
ing in the RPA approximation. Taking ε0 ≈ 4 for BN
substrate , yields α ≈ 0.37, giving η ≈ 2.3?.
The dependence of transport coefficients on T, B and
carrier density n, predicted from Eq.(11), can be directly
compared to experiment. Fig. 3 shows ρxy(n) measured
in graphene on BN, on samples similar to those described
in Ref.. The modeled ρxy(n) captures the main fea-
tures of the data: the 1/n dependence at large n and
a steep linear dependence near the DP. The linear re-
gion broadens with temperature at T>
in ρxx(n) features similar thermal broadening (see Ap-
pendix C). The SHE coefficient, found from Eq.(3), is
plotted in Fig.1.
We now explore the behavior of transport coefficients
near the DP, making estimates separately for T>
∼T∗. This can be conveniently done using an interpo-
lation formula τdis
the ensemble-averaged scattering time (9) and the effec-
tive mass (8) in the entire range of T and µ of interest.
= 0. Here nT is the density of thermally ac-
?v0≈ 6, which ac-
∼γ. The peak
i (µ,T) = mi(µ,T)v2
Carrier density n [1011 cm−2]
Hall resistivity ρxy [kΩ]
n [1011 cm−2]
FIG. 3: Measured ρxy(n) for a high-mobility graphene sample
on BN substrate at T = 250K. The dependence follows the
quasiclassical formula (1) away from the DP, and is linear with
a steep slope near the DP. Inset: Results for ρxy(n) obtained
from the two-carrier model, Eqs.(7),(11), for disorder strength
γ = 180K found by fitting the min/max distance in measured
ρxy for B = 1T. Other parameters: η = 2.3?, T = 250K.
We find the slope of ρxy at the DP by expanding
Eq.(11) in small n = ne− nh (see Appendix D for full
treatment). The result, which simplifies in each of the
interpolation formula as
∼T∗, can be described by a single
where only terms first-order in B have been retained.
The SHE coefficient, Eq.(3), found by combining the
results (13) and (12), and using thermally broadened
density of states at the DP derived in Appendix B,
∂n/∂µ =2 ln2
where E0 is the cyclotron energy. The functional form
is the same in both regimes, θSH∝ B2/T, with different
The 1/T growth of θSHsaturates at kBT ≈ γ, reaching
maximum value θSH,max≈1
We expect suspended graphene [28, 29] to feature an
even stronger SHE than graphene on BN. Using typical
mobility µ∗= 2 · 105cm2/Vs , we estimate γ ∼ 65K,
whereas the temperature dependence of the conductivity
at the DP  yields γ ∼ 10K. For either value of γ,
Eq.(14) predicts very large values θSHat the DP.
Based on these estimates, we expect strong SHE re-
sponse already at moderate fields B<
accumulation and locally tunable SHE response, which
was discussed above, SHE can also manifest itself in a
non-zero Hall voltage in response to spin-polarized cur-
rents injected from magnetic contacts.
∼T∗= 24ln2/π2and λT<
∼1T. Besides spin
Since our SHE mechanism does not rely on the rela-
tivistic dispersion of excitations, it can also be realized in
other zero-gap semiconductors, in particular graphene bi-
layer. It also applies, with suitable modifications, to the
valley degrees of freedom in graphene. It was predicted
that a (non-quantizing) magnetic field can lift valley de-
generacy and produce a Zeeman-like valley splitting .
This will imbalance the Hall resistivities and result in a
valley-Hall effect of a magnitude similar to the SHE.
We thank D. Goldhaber-Gordon, L. M. K. Vander-
sypen and M. Soljacic for useful discussions and acknowl-
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APPENDIX A: SPIN ACCUMULATION
Here we analyze spin accumulation in the ZSHE
regime. Spin accumulation in two-dimensional electron
gasses, arising due to weak SHE induced by spin-orbit
interaction, was considered in several papers, in particu-
lar: S. Zhang, Phys. Rev. Lett. 85, 393 (2000). There
are several new aspects in our problem that warrant spe-
cial treatment. Specifically: (i) In the presence of exter-
nal magnetic field charge transport is described by ρxx
and ρxy separetly for each spin projection; (ii) Unlike
the semiconductor case, the expected SHE response is
not necessarily small, θSH∼ 1.
We consider spin accumulation in the strip geome-
try, −w/2 < y < w/2, with current driven along the
strip, such that external electric field E is applied along
x direction. To solve the transport problem, we intro-
duce electrochemical potentials for each spin projection,
ϕ↑(↓)= φ + n↑(↓)/ν↑(↓), where φ is the electric poten-
tial, n↑(↓)is the deviation of the local density for spin up
(down) from its equilibrium value, and ν↑(↓)is the den-
sity of states for the two spin projections. The equations
for the current density are given by,
j↑(r) = ˆ σ↑∇ϕ↑(r),
where ↑ and ↓ label carriers with up-spin and down-spin.
Here ˆ σ↑and ˆ σ↓are 2 × 2 matrices describing the longi-
tudinal and Hall conductivity. The continuity equation
for each spin projection can be written as
j↓(r) = ˆ σ↓∇ϕ↓(r),(15)
∇j↑(r) = −γs(n↑(r)−n↓(r)),∇j↓(r) = −γs(n↓(r)−n↑(r)),
where γsis the rate of spin relaxation. The terms with
time derivative are omitted, as appropriate for the DC
The transport equations should be supplemented by
the condition of electro-neutrality, n↑(r) = −n↓(r). We
solve transport equations in a strip geometry, where cur-
rent is driven by an electric field parallel to the strip,
with the boundary conditions of zero normal current at
the strip edges, jy(y = ±1
density ns= n↑− n↓profile across the strip
2w) = 0. We obtain the spin
ns(y) = eE¯ νℓsθSH
where ¯ ν = 2/(ν−1
for the two spin projections, E is the electric field, ℓsis
the spin relaxation length
↓) is the average density of states
For a narrow strip, w ≪ ℓs, spin relaxation can be ig-
nored. In this case, the expression for ns(y) agrees with
the estimate (3) in the main text.
To estimate the numerical value of the spin density
at the DP, where it is maximum, we make several as-
sumptions. First, we assume that the density of states
is disorder-broadened at the Dirac point, ¯ ν = ν↑= ν↓≈
in Eq.(10) of the main text. For an estimate we use the
value γ ∼ 100K, which corresponds to density inhomo-
geneity ∆n ≈ 2ν(0)γ ≈ 1010cm−2typical for graphene
on BN substrate. Second, we will assume w = 2µm, and
for the SHE coefficient, we will assume θSH= 0.1, as es-
timated in the main text. Taking E = 10V/µm, as in
Ref. , for the spin density at the edge we find
0, where γ is the disorder strength parameter defined
ns(y = ±w/2) ≈ 3 · 1010cm−2.
This is at least five orders of magnitude the spin density
per atomic layer observed in GaAs , ns≈ 5×105cm−2.
Such giant spin accumulation is due to the larger value
of θSH in graphene, and due to the fact that graphene
can sustain large current densities and electric fields.
APPENDIX B: THE CARRIER DENSITY AND
THE DENSITY OF STATES
We start with deriving a closed form expression for
the carrier density, evaluated separately for the elec-
trons and holes with a fixed spin projection: ni(µ,T) =
defined by Eq.(6) of the main text with ai= 0. Integrat-
ing over p, we find the dependence of ne(h)on tempera-
ture and chemical potential as power series in g = eµ/kBT
(2π)2fi(p,µ,T), where the distribution function fiis
ln(1 + g′)
Although the series converge only for −1 < g ≤ 1, the
final expression is valid for both g < 1 and g > 1. At the
DP, setting µ = 0, we obtain
ne(h)(µ = 0) =ζ(2)
This quantity, denoted as nT in Eq.(12) of the main text
and elsewhere, gives the density of thermally excited car-
riers (electrons or holes) with a fixed spin projection.
Next, we analyze the density of states ν = ∂n/∂µ. The
density of states per one spin projection is represented by
a sum of an electron and a hole contribution
where the minus sign reflects the fact that electrons and
holes have opposite chemical potentials, µe= −µh= µ.
We use the above expression for ne(h)(µ) to find
?ln(1 + g) + ln(1 + g−1)? kBT
= ln 2cosh
π (?v0)2. (24)
This expression describes temperature-broadened density
with a crossover temperature kBT = |µ|/(2ln2).
APPENDIX C: THE EFFECT OF DRAG
BETWEEN CARRIERS OF OPPOSITE SPIN
Our analysis of transport coefficients in the main text
was based on a simplified model which neglected drag be-
tween carriers of opposite spin. To evaluate the accuracy
of this approach, here we discuss a more general model
which accounts for drag between carriers of either spin.
To emphasize this difference, we will use notation ˜ η for
the drag coefficient instead of η used in the main text.
As we will see, the more general approach predicts an es-
sentially identical behavior of transport coefficients, and
a qualitatively similar behavior of the SHE coefficient.
We consider coupled transport of four carrier species.
For simplicity, we will take the drag coefficient values to
be the same for all carrier species, a reasonable approx-
imation at high kBT. Ensemble-averaged velocities and
momenta in the presence of electric and magnetic fields
are governed by Eq.(7) of the main text, which we dupli-
cate here for reader’s convenience,
− ˜ η
ni′(Vi− Vi′). (26)
The quantities ni, mi, τdis
text, with the chemical potential µ replaced by µ±EZ/2
for spin up (down) electrons. The value of the drag co-
efficient is found by matching analytic results[24, 25].
Because Eqs.(26) account for drag between all carrier
species, not just those with parallel spins, the value ˜ η
found below is different from the one obtained in the
The transport coefficients are determined as follows.
First, for given values µ and T, we obtain ni, mi by
numerically evaluating integrals in the relation
are defined as in the main
ni(µ,T) = 2
Carrier density n [1011 cm−2]
Longitudinal resistivity ρxx [kΩ]
FIG. 4: The longitudinal resistivity, obtained from the model
(26) which takes into account drag between carriers of either
polarity and spin, shown for several temperatures. Param-
eters used: B = 0T, γ = 100K, ˜ η = 1.15?. The peak in
ρxx gets broadened as the temperature increases, as ρxx(0)
approaches the high-temperature limiting value, Eq.(31).
and in Eq.(8).
tering time (9) and the effective mass (8) in the entire
range of T and µ of interest.
After that, we solve the four coupled linear equations
(26) and find the currents of the spin-up and spin-down
For ensemble-averaged scattering time
i , we use the interpolation formula τdis
0?/γ2which links the ensemble-averaged scat-
i (µ,T) =
j↑= ne↑eVe↑− nh↑eVh↑,
j↓= ne↓eVe↓− nh↓eVh↓.
Then the charge and spin currents are expressed through
j↑and j↓as follows,
jc= j↑+ j↓,
js= j↑− j↓, (29)
These expressions can be used to calculate the charge and
spin conductivity tensors. We obtain the dimensionless
SHE coefficient by evaluating the ratio of the transverse
spin current and the longitudinal charge current,
where the transverse and longitudinal components are
taken with respect to the electric field E.
For charge transport, this model leads to the behav-
ior of transport coefficients which is essentially identical
to that obtained from the simplified model used in the
main text, Eq.(11), albeit with a doubled value of the
drag coefficient, η → 2˜ η. To see this, we note that the
same for the up-spin and down-spin carriers. Further,
the quantities ni, mi, τdis
coincide for spin up and down
when Ez= 0. Therefore, in the limit EZ→ 0, the rela-
tions (26) can be satisfied by v↑
ing this into account and eliminating variables for one
cv × B?
that drive transport are the
Carrier density n[1011 cm−2]
Spin−Hall coefficient ξSH
FIG. 5: The SHE coefficient ξSH, Eq.(30), obtained from the
model (26) which takes into account drag between carriers of
either polarity with parallel an opposite spins. Parameters
used: B = 1T, γ = 100K, ˜ η = 1.15?. Similarly to θSH, illus-
trated in Fig. 1, ξSH exhibits a sharp peak at the DP, which
gets more pronounced as T is lowered. The quantities ξSH
and θSH are identical at low T, when transport is dominated
by scattering on disorder. At high T, when transport is dom-
inated by the effects of electron-hole drag, the peak in ξSH is
significantly broader than the peak in θSH.
spin projection we obtain equations for the other pro-
jection which are identical to the equations in the main
text up to a replacement η → 2˜ η. Small but finite EZ
changes the transport coefficients, however the difference
between the spin-up and spin-down remains small as long
as EZ≪ kBT,γ.
Following the same reasoning as in the main text, we
fix the drag coefficient value at the half the drag coeffi-
cient used in the simplified model: 2˜ η = 2.3?. We obtain
the density dependence ρxy(n) which is very similar to
that shown in Fig. 3 of the main text. Behavior of the
longitudinal resistivity ρxx(n) at B = 0 is illustrated in
Fig. 4. The peak in ρxx at the DP becomes lower and
broader as the temperature is increased. The resistiv-
ity at the DP decreases, at high temperatures kBT ≫ γ
saturating at the value
xx = ˜ η/e2≈ 5kΩ,(31)
where the factor 1/2 is introduced to convert the resis-
tivity for one spin projection, given by Eq.(12) of the
main text, to net resistivity. The results in Fig. 4 show
that ρxx(0) is very close to this limiting value already at
T = 300K, indicating that transport at these tempera-
tures is dominated by electron-hole collisions.
The behavior of the SHE coefficient ξSH, Eq.(30),
shown in Fig.5, is overall similar although not identical
to the behavior of θSHfound in the main text. Near the
DP, ξSHreaches large values, similar to those of θSH(see
Fig. 1 in the main text). The peak in ξSH is, however,
significantly broader. Furthermore, ξSHexhibits a mono-
tonic decay away from the DP; in contrast, θSHexhibits
a sign change. We believe the sign change to be pecu-
liar for the two-component model which ignores the drag
between spin-up and spin-down carriers.
The SHE coefficient ξSH determines physical observ-
ables, such as spin accumulation density, which is given
by Eq.(3) of the main text with a substitution θSH→ ξSH.
The two models therefore predict the same values of ns
at the DP.
APPENDIX D: ANALYTIC ESTIMATES FOR
THE SHE COEFFICIENT
Here we derive the expression for ∂ρxy/∂n, Eq.(13),
and an estimate for the SHE coefficient, Eq.(14) of the
main text. First, we consider the high-temperature
regime, T ≫ T∗, where transport coefficients are dom-
inated by the effects of electron-hole drag. In this limit,
the first term in the numerator of Eq.(11), which is small
compared to the other two terms, can be neglected. Fur-
thermore, the last term in the denominator, which is
quadratic in density, cannot affect the slope of ρxy at
the DP, and we can drop it as well. Then Eq.(11) is
mh˜ γe. (32)
Using the relations ˜ γi=
− iΩiand the interpolation
0?/γ2, after simple algebra we obtain = miv2
0? − ieBn/c
0? + ieBn/c, (33)
where n = ne− nh is the carrier density. Then, using
the fact that ne+ nh= 2nT+ O(n2), and expanding to
linear order in B, we obtain
The second relation gives the slope of the Hall resistivity
at the DP, Eq.(13) of the main text (case T > T∗).
In the low-temperature regime, T ≪ T∗, we can neglect
the second and third terms in the numerator of Eq.(11),as
well as the third term in the denominator, which gives
˜ γe˜ γh
Once again using the relations ˜ γi =
interpolation formula τdis
above expression as
− iΩi and the
0?/γ2, we rewrite the
0+ ieBn/c, (36)
where we have neglected the term proportional to ΩeΩh
in the numerator, which is of the order B2. Expanding
to linear order in n, we obtain
Using the relation nT=
, we obtain the slope
of ρxyat the DP, Eq.(13) of the main text (case T < T∗).
Finally, we obtain the SHE coefficient at the DP, de-
fined in Eq.(3) of the main text. This is done by combin-
ing the results for ∂ρxy/∂n, which we have just derived,
with the result for thermally broadened density of states
at the DP, Eq.(25), and the above results for ρxxat the
DP. This gives Eq.(14) of the main text.