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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[5] and

spin-dependent scattering on impurities[1], respectively.

Single layer graphene has emerged recently as an at-

tractive material for spintronics that features long spin

diffusion lengths[6], gate tunable spin transport[6, 7], and

high-efficiency spin injection[9]. 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

−4 −2024

0

0.04

0.08

0.12

Carrier density n [1011 cm−2]

Spin−Hall coefficient θSH

0

0

µ

∆µ

Ez

ρxy

↑(↓)

100K

200K

300K

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

nec.

(1)

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

Page 2

2

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

∇

?

φ +n↑

eν↑

?

=ρ↑

xy

ρ↑

xx

E,∇

?

φ +n↓

eν↓

?

=ρ↓

xy

ρ↓

xx

E,(2)

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

θSH=ρ↑

ρ↑

ns=

θSHweE

ν−1

↑

+ ν−1

↓

,

xy

xx

−ρ↓

xy

ρ↓

xx

≈ EZ

∂

∂µ

ρxy

ρxx, (3)

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

SHE coefficient.

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 [4]. 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[18]), and using E = 10V/µm (which corre-

sponds to maximum current density in graphene [19]), 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 [4], 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 [20] (fields up to

1.4T are achievable using widely available Neodymium

Boron magnets). State-of-the-art microelectromagnets

have similar characteristics [21]. 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

qe(h)

?

E +v

c× B

?∂fe(h)(p)

∂p

= St[fe(p),fh(p)],(5)

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

Page 3

3

this route, we follow Ref.[26] 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

fe(h)(p) =

1

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. [25]). 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):

qi

?

E +Vi

c

× B

?

= −Pi

τdis

i

− η

?

i′

ni′(Vi− Vi′),(7)

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

specified below.

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):

i

, the

mi=

1

v0

?d2ppx∇axfi(p)

?d2ppx

p∇axfi(p)=

1

v0

?d2pp2

?d2pp2

xgi(p)

x

pgi(p)

,(8)

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

i

and carrier densities ni in (7) are ex-

pressed through the distribution function (6) with ai= 0:

2ζ(2)kBT/v2

0≈ 3.29kBT/v2

0.

1

τdis

i

=

2

ni

?

d2p

(2π)2

fi(p)

τdis

i

(εp),ni= 2

?

d2p

(2π)2fi(p), (9)

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 [13]. In both cases the scattering time has an

approximately linear dependence on particle energy,

τdis(ε)|ε|>

∼γ= ?|ε|/γ2,γ = v0

?

e?/µ∗

(10)

where the disorder strength parameter γ is expressed

through mobility. The value µ∗= 6 · 104cm2/V · s mea-

sured in graphene on BN [27] yields γ ≈ 120K. Similar

values are obtained from the ρxx-based DP width. Tak-

ing ∆n ≈ 1010cm−2[18], 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.

ρ↑(↓)

xx + iρ↑(↓)

xy

=

1

e2

˜ γe˜ γh+ ηne

mh˜ γe+ ηnh

mh˜ γe+ η(ne−nh)2

me˜ γh

ne

me˜ γh+nh

memh

. (11)

Here ˜ γi=

frequency.

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],

1

τdis

i

− iΩi, with Ωi= qiB/mic the cyclotron

ρ↑(↓)

xx =

mT

2nTe2τ

?1 + τ2Ω2?+η

e2,nT=

π

12

k2

?2v2

BT2

0

, (12)

and ρ↑(↓)

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 [27], 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.[18]. 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<

∼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.

xy

= 0. Here nT is the density of thermally ac-

∼T∗.

2

+π

2

e2

?v0≈ 6, which ac-

∼γ. The peak

∼T∗and

i (µ,T) = mi(µ,T)v2

0?/γ2which links

Page 4

4

−4−2024

−60

−30

0

30

60

Carrier density n [1011 cm−2]

Hall resistivity ρxy [kΩ]

−202

−40

−20

0

20

40

n [1011 cm−2]

ρxy [kΩ]

1T

2T

4T

8T

12T

2T

1T

0.5T

0.25T

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

regimes T>

∼T∗and T<

interpolation formula as

∼T∗, can be described by a single

∂ρxy

∂n

????

n=0

=

?2v2

∗,πT2/3)

0

min(T2

B

nTec,

(13)

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

π

?2v2

kBT

0, is

θSH|n=0=λE2

0EZ

2γ2kBT,

E0= v0

?

2?eB/c, (14)

where E0 is the cyclotron energy. The functional form

is the same in both regimes, θSH∝ B2/T, with different

prefactors λT>

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 [29], we estimate γ ∼ 65K,

whereas the temperature dependence of the conductivity

at the DP [29] 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<

∼T∗= 12ln2/π2.

2λE2

0EZ/γ3.

∼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 [30].

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-

edge support from Naval Research Grant N00014-09-1-

0724 (LL).

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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)),

(16)

where γsis the rate of spin relaxation. The terms with

time derivative are omitted, as appropriate for the DC

transport.

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

sinhy

ℓs

cosh

2ℓs

2w) = 0. We obtain the spin

ns(y) = eE¯ νℓsθSH

w

,θSH=σ↑

xy

σ↑

xx

−σ↓

xy

σ↓

xx

,(17)

where ¯ ν = 2/(ν−1

for the two spin projections, E is the electric field, ℓsis

the spin relaxation length

↑

+ν−1

↓) is the average density of states

ℓ2

s=

¯ σxx

2¯ νγs,

1

¯ σxx

=1

2

?

1

σ↑

xx

+

1

σ↓

xx

?

.(18)

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↓≈

γ

π?2v2

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. [19], 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 [4], 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.

(19)

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) =

2?

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

d2p

(2π)2fi(p,µ,T), where the distribution function fiis

ne(h)(µ,T) =

1

π

?kBT

?v0

?2?

?2?g

g −g2

ln(1 + g′)

22+g3

32−g4

42+ ...

?

=

1

π

?kBT

?v0

0

g′

dg′.(20)

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)

2π

?kBT

?v0

?2

=

π

12

?kBT

?v0

?2

.(21)

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

ν↑(↓)=∂ne

∂µ−∂nh

∂µ,

(22)

Page 6

6

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

ν↑(↓) =

?ln(1 + g) + ln(1 + g−1)? kBT

?

2kBT

π?2v2

0

(23)

= ln 2cosh

µ

?

2kBT

π (?v0)2. (24)

This expression describes temperature-broadened density

of states:

ν↑(↓)

|µ|≫kBT=

|µ|

π (?v0)2,ν↑(↓)

|µ|≪kBT=2ln2

π

kBT

(?v0)2

(25)

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,

qi

?

E +Vi

c

× B

?

= −Pi

τdis

i

− ˜ η

?

i′

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

main text.

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

i

are defined as in the main

ni(µ,T) = 2

?

d2p

(2π)2fi(p,µ,T), (27)

−4 −2024

0

2

4

6

8

Carrier density n [1011 cm−2]

Longitudinal resistivity ρxx [kΩ]

100 K

200 K

300 K

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).

τdis

mi(µ,T)v2

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

electrons,

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↓.

(28)

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,

ξSH= 2js,⊥/jc,?

(30)

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

forces qe(h)

?E +1

same for the up-spin and down-spin carriers. Further,

the quantities ni, mi, τdis

i

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

e= v↓

h, v↑

h= v↓

e. Tak-

Page 7

7

−4−2

Carrier density n[1011 cm−2]

024

0.04

0.08

0.12

Spin−Hall coefficient ξSH

100K

200K

300K

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

ρxx=1

2ρ↑(↓)

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

simplified as

(ρxx+ iρxy)↑(↓)≈η

e2

ne

mh˜ γe+nh

ne

me˜ γh+nh

me˜ γh

mh˜ γe. (32)

Using the relations ˜ γi=

formula τdis

i

1

τdis

i

− iΩiand the interpolation

0?/γ2, after simple algebra we obtain = miv2

(ρxx+ iρxy)↑(↓)≈η

e2

γ2(ne+ nh)/v2

γ2(ne+ nh)/v2

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

ρxx=

η

e2,ρxy=?2v2

0

T2

∗

B

nTecn.

(34)

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

(ρxx+ iρxy)↑(↓)≈

1

e2

˜ γe˜ γh

ne

me˜ γh+nh

mh˜ γe.(35)

Once again using the relations ˜ γi =

interpolation formula τdis

above expression as

1

τdis

i

− iΩi and the

i

= miv2

0?/γ2, we rewrite the

(ρxx+ iρxy)↑(↓)≈

1

e2

γ4/?2v4

0

γ2(ne+ nh)/?v2

0+ ieBn/c, (36)

Page 8

8

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

ρxx=

γ2

2nTe2?v2

0

,ρxy=

B

nTec

n

4nT. (37)

Using the relation nT=

π

12

?

kBT

?v0

?2

, 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.