Table Of ContentStationary states and screening equations in the spin-Hall effect
J.-E. Wegrowe1∗
1 Ecole Polytechnique, LSI, CNRS and CEA/DSM/IRAMIS, Palaiseau F-91128, France
(Dated: February 24, 2017)
7 Abstract
1
0
2 Thecharacterizationofthestationarystatesinthespin-Halleffectisdiscussedwithintheframe-
b
work of the phenomenological two spin-channel model. It is shown that two different definitions of
e
F
the stationary states can be applied in the spin-Hall effect, leading to two different types of state in
3
2 the bulk: zero transverse spin-current or non-zero pure spin-current. This difference is due to the
]
l treatmentoftheregionneartheedges, inwhichelectricchargeaccumulationoccurs. Thescreening
l
a
h equations that describe the accumulation of electric charges due to spin-orbit coupling are derived.
-
s
The spin-accumulation associated to spin-flip scattering and the spin-Hall accumulation due to
e
m
spin-orbit coupling are two independent effects if we assume that the screening length is small with
.
t
a
respect to the spin-diffusion length. The corresponding transport equations are discussed in terms
m
- of the Dyakonov-Perel equations.
d
n
o
c PACS numbers:
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2
v
7
1
0
6
0
.
1
0
7
1
:
v
i
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r
a
∗Electronic address: [email protected]
1
I. INTRODUCTION
The spin-Hall effect (SHE) is one of the most important effect in the new field of spin-
orbitronics. However, in spite of a large number of excellent reports about SHE (see [1–9]
and references therein), a fundamental ambiguity seems to persist in the definition of the
stationary regime, leading to different predictions as to the presence of pure spin-current on
the stationary state [10].
The SHE describes the spin-accumulation generated at the edges of a paramagnet with
strong spin-orbit coupling (SOC), while injecting an electric current [11]. The phenomeno-
logical description of the SHE is based on the Dyakonov-Perel transport equations [1, 2],
which are a generalization of Ohm’s law for the spin-dependent electric carriers in the pres-
ence of SOC. As shown in Fig.1, the two spin-channel model can be used in order to describe
the SHE: it suffices to assume that the spin-orbit scattering plays the same role as an effec-
(cid:126)
tive magnetic field H in a two-channel version of the Hall effect. The main property of this
so(cid:108)
(cid:126) (cid:126)
effective magnetic field is expressed by the reciprocity relation H = ±H : the effective
so(cid:108) so
magnetic field is acting as an external static magnetic field applied along the direction of
(cid:126)
the effective field H for the electric carriers belonging to the ↑ channel, while it is acting
so↑
in the opposite direction for the electric carriers belonging to the ↓ channel (see Fig.1).
Consequently, the system can be viewed as a simple superimposition of two standard Hall
devices with two populations of electric charges experiencing opposite magnetic fields. More
(cid:126)
precisely, the magnetic field along the H direction generates a charge accumulation δn
so↑ ↑
(cid:126)
at the edge, while the magnetic field along the H direction generates a charge accumula-
so↓
tion δn = −δn at the same edge. The superimposition of the two sub-systems leads to a
↓ ↑
vanishing electric charge accumulation δn = δn +δn = 0 and non-zero spin-accumulation
↑ ↓
∆n = δn −δn (cid:54)= 0.
↑ ↓
The usual theoretical approach to the SHE consists in calculating the stationary state
after summing over the total system, i.e. for zero charge accumulation [4–9]. The result is
a generation of a transverse pure spin current J = −J (cid:54)= 0.
y↑ y↓
However,thisgenerationofapurespincurrentisnotsatisfyingfromthethermodynamical
point of view. Indeed, the second law of thermodynamics imposes that the stationary state
should correspond to the minimum power dissipation. As shown in a recent report based
on this variational principle [10], the minimum power dissipation in the spin-Hall system
2
a) b)
G G
H
δn <0 y δn >0 y
– – – – – – – + + + + + + + + +
x x
+ + + + + + + + + – – – – – – –
-H
c)
G
δn=0 y
+ + + + + + + + +
– – – – – – –
H
SO x
– – – – – – –
+ + + + + + + + +
FIG. 1: : Schematic representation of the spin-Hall effect with the electrostatic charge accumula-
tion δn at the boundaries: (a) usual Hall effect with a magnetic field H(cid:126) in the up direction (b)
(cid:108) ↑
usual Hall effect with a magnetic field H(cid:126) in the down direction, (c ) the addition of configuration
↓
(a) and (b) leads to the modelisation of the spin-orbit scattering assuming an effective magnetic
field H(cid:126) acting on the two different electric carriers, defining the spin up and spin down channels.
so
is reached for J = J = 0 in the bulk, except for Corbino configurations [10]. The
y↑ y↓
corresponding system is then less constrained than the previous one.
The aim of this work is to study these two results in the framework of the usual definition
dn /dt = 0 of the stationary state, and to understand the apparent contradiction on the
(cid:108)
basis of the screening equations. It is shown that the last result (J = J = 0) is obtained
y↑ y↓
if the condition of stationary state is applied before summing over the two channels (with
charge accumulation δn (cid:54)= 0) [3], while the former result (J = −J (cid:54)= 0) is obtained if the
(cid:108) y↑ y↓
stationarity condition is apply after summing over the two channels (charge accumulation
δn = 0).
The paper is organized as follow. The first section treats the standard Hall effect in
the formalism of mesoscopic non-equilibrium thermodynamics. The screening equation is
3
derived and the usual approximations related to constant conductivity and small Debye-
Fermi length are analyzed. The second section is devoted to the two spin-channel model
for the SHE without spin-flip scattering, with the derivation of the screening equations.
The third section presents the derivation of the screening equation that includes spin-flip
scattering. The section four analyzes the corresponding transport equations in terms of the
Dyakonov-Perel equations.
II. HALL EFFECT
The goal of this section is to characterize the stationary state of the standard Hall device
onthebasisofthelawdn/dt = 0(wherenisthedensityofelectriccarriers),andinvestigating
the approximations that consists in assuming constant conductivity and small screening
length.
Ohm’s law reads:
(cid:126) (cid:126)
J = −ηˆn∇µ, (1)
where n is the density of electric carriers of charge q, µ is the chemical potential, and
the mobility tensor ηˆ is related to the conductivity tensor σˆ by the relation ηˆ = σˆ/(qn).
If the sample under consideration is an isotropic planar layer, to which a magnetic field
(cid:126)
H is applied along the direction (cid:126)e perpendicular to the layer, the mobility tensor in the
z
orthonormal basis {(cid:126)e ,(cid:126)e ,(cid:126)e } is expressed by the matrix:
x y z
η η
H
ηˆ= , (2)
−η η
H
where the coefficient η is the Hall mobility (which is an odd function of H). According
H
to the Drude relation we have σ = q2nτ/m∗ (where τ is the relaxation time and m∗ is the
effective mass) so that η = qτ/m∗ does not depend on n.
(cid:126)
Introducing a unit vector p(cid:126) in the direction of the applied field H, the vector form of
Eq.(1) reads:
(cid:126) (cid:126) (cid:126)
J = −ηn∇µ+p(cid:126)×nη ∇µ. (3)
H
On the other hand, the chemical potential is defined by :
(cid:126)
µ = kTln(n/n )+V +V(H)+µ , (4)
0 G 0
4
where T is either the Fermi temperature (T = T ) in the case of a metal or the temperature
F
of the thermostat in the case of a non-degenerated semi-conductor, k is the Boltzmann
constant, n is the constant density of the electric carriers that corresponds to the electric
0
neutrality of the material, and µ describes the chemical potential related to relaxation
0
processes, which is also constant in the bulk material. The electric potential V is produced
G
by the electric generator and imposes the constant electric field qE0 = −∂V /∂x along the
x G
(cid:126)
x axis. On the other hand, the electric potential V(H) takes into account the effect of
(cid:126)
the magnetic field H. Due to the Lorentz force, the consequence of the application of the
magnetic field is a redistribution of the electric carriers in the direction y, which results in a
charge accumulation δn = n−n at the edges. The charge accumulation generates, in turn,
0
(cid:126)
an electric field qE = −∂V(H)/∂y defined by Gauss’s law:
y
∂E qδn
y
= (5)
∂y (cid:15)
(cid:126) (cid:126)
For convenience, the two contributions ∇V and ∇V(H) of the gradient of the chemical
G
potential can be combined in a single electric field qE(cid:126) = −∇(cid:126)V −∇(cid:126)V(H(cid:126)) = qE0(cid:126)e +qE (cid:126)e .
G x x y y
Inserting the expression of the chemical potential Eq.(4) into Omh’s law Eq.(3) we obtain:
(cid:16) (cid:17)
(cid:126) (cid:126) (cid:126) (cid:126) (cid:126)
J = qnηE −D∇n−p(cid:126)× qnηE −D ∇n , (6)
H
where the diffusion constants are defined as D = ηkT and D = η kT.
H H
The divergence of Eq.(6) reads:
div(J(cid:126)) = qnηdiv(E(cid:126))+qη∇(cid:126)n.E(cid:126) −D∇2n, (7)
(cid:126) (cid:126) (cid:126) (cid:126) (cid:126)
where we have used the vector relations div(nE) = E.∇n + ndiv(E) and div(p(cid:126) × ∇µ) =
∇(cid:126)µ.r(cid:126)ot(p(cid:126)) − p(cid:126).r(cid:126)ot(∇(cid:126)µ) = 0. The consequence of the last relation is that the Hall term
in the right hand side of Eq.(6) disappears from the definition of the stationary states.
Nevertheless, the Hall effect is still present as it is responsible for the charge accumulation
δn, i.e. as it is responsible for the derivative of the electric field ∂E /∂y. The stationarity
y
(cid:126)
condition writes ∂n/∂t = −div(J) = 0. In the case of a recombination between electrons
˙
and holes, or for other relaxation mechanisms, we should add the relaxation parameter R
(cid:126) ˙ ˙
such that div(J) = −R (an explicit expression of R can be found in [5]). The expression
the stationarity condition leads to the following screening equation for the electric charge
accumulation δn:
∂2δn δn ∂δn q2 (cid:90) R˙
− = δn(y(cid:48))dy(cid:48) + (8)
∂y2 λ2 ∂y (cid:15)kT D
D y
5
(cid:113)
where λ = (cid:15)kT is the screening length. This is an exact result for the Hall bar assuming
D q2n
translation invariance along x. In the case n (cid:29) |δn|, the screening length is the Debye
(cid:113)
length λ ≈ λ¯ ≡ (cid:15)kT .
D D q2n0
A. Constant conductivity approximation
When |δn| (cid:28) n , the conductivity σ = qη(n + δn) ≈ qηn is constant. If we further
0 0 0
˙
assumethatR ≈ 0, thenEq.(8)reducestothewell-knownscreeningequationatequilibrium:
δn
∇2δn− ≈ 0. (9)
λ2
D
¯
The electrostatic charge accumulation decays exponentially to zero with a typical length λ .
D
(cid:126) (cid:126)
Consequently, the last term qη∇n.E in the left hand side of Eq.(7) accounts for the variation
of the conductivity σ due to the electrostatic charge accumulation over a distance λ from
D
the border of the Hall device. The approximation |δn| (cid:28) n as been well established as
0
correct for conventional systems.
B. Small Debye-Fermi length and bulk approximation
If we assume λ small, we have in second order:
D
(cid:18) (cid:90) (cid:19) (cid:18) (cid:90) (cid:19)
∂δn 1 ∂δn 1
−δn = δn(y(cid:48))dy(cid:48) ≈ δn(y(cid:48))dy(cid:48) , (10)
∂y n ∂y n
y 0 y
˙
A moderate non-conservation of the electric charges R (cid:54)= 0 has no consequence in the frame-
work of this approximation. The approximation of small Debye-Fermi length B together
with that of constant conductivity A (the right-hand side of Eq.(10) is vanishing) results in
the bulk approximation δn ≈ 0.
III. SPIN-HALL EFFECT IN THE TWO SPIN CHANNEL MODEL
Thegoalofthissectionistointroducethetwospinchannelslabeledbytheindex(cid:108), andto
(cid:126) ˙
characterize the stationary states defined by the conservation laws dn /dt = −divJ −R = 0
(cid:108) (cid:108)
(in the absence of spin-flip scattering).
The spin-dependent conductivity σ of the spin-channels can be introduced through the
(cid:108)
Drude formula: σ = q2n τ/m∗ where the relaxation τ is not spin-dependent, and m∗ is
(cid:108) (cid:108)
6
the effective mass. Hence, the mobility η = σ /(qn ) does not depend on n and is spin
(cid:108) (cid:108) (cid:108)
independent. Consequently, the diffusion constant is also spin-independent D = σ kT/n =
(cid:108) (cid:108)
ηkT. In contrast, the spin-orbit diffusion is spin-dependent D = η kT = ±η kT (where
so(cid:108) so(cid:108) so
the sign (+) corresponds to the spin ↑ and the sign (−) corresponds to the spin ↓), with the
relation η = −η = η . Ohm’s law reads:
so↑ so↓ so
(cid:126) (cid:126)
J = −ηˆn ∇µ , (11)
(cid:108) (cid:108) (cid:108)
where the mobility tensor is given by:
η η 0 0
so
−η η 0 0
ηˆ = so . (12)
(cid:108)
0 0 η −η
so
0 0 η η
so
In a vector form we have:
(cid:126) (cid:126) (cid:126)
J = −ηn ∇µ ±p(cid:126)×n η ∇µ (13)
(cid:108) (cid:108) (cid:108) (cid:108) so (cid:108)
where the sign (+) corresponds to the spin ↑ and the sign (−) corresponds to the spin ↓.
As shown in the last Section, Eq.(13) is a generalization of the Dyakonov-Perel equations.
According to the model depicted in Fig.1, the chemical potential of the charge carriers
in each spin channel is given by Eq.(4):
µ = kTln(n /n )+V +V(H )+µ , (14)
(cid:108) (cid:108) 0 G so(cid:108) 0(cid:108)
where V is the contribution of the electric generator which imposes a constant electric
G
field along the x direction: qE0 = −∂V /∂x. The contribution V(H ) takes into account
x G so(cid:108)
the effect of SOC through the effective magnetic field H , and µ accounts for the spin-
so(cid:108) 0(cid:108)
dependent properties of the electric carriers.
As described in Fig.1 and in the introduction, the effect of the effective magnetic field
H = ±H is equivalent to that of the usual Hall effect for each spin-channel. It leads to
so(cid:108) so
a redistribution of the electric charges δn such that δn = −δn . This distribution defines
(cid:108) ↑ ↓
a spin-dependent electric field E along the y-direction through Gauss’s law ∂E /∂y =
y(cid:108) y(cid:108)
qδn /(cid:15). The two contributions V and V(H ) in Eq.(14) can be combined into a single
(cid:108) G so(cid:108)
electric field qE(cid:126) = −∇(cid:126)V = −∇(cid:126)V −∇(cid:126)V(H ) = qE0(cid:126)e +qE (cid:126)e . Inserting the expression
(cid:108) (cid:108) G so(cid:108) x x y(cid:108) y
7
of the chemical potential Eq.(14) into the Omh’s law Eq.(13) yields:
(cid:16) (cid:17)
(cid:126) (cid:126) (cid:126) (cid:126) (cid:126)
J = qn ηE −D∇n ∓p(cid:126)× qn ηE −D ∇n , (15)
(cid:108) (cid:108) (cid:108) (cid:108) (cid:108) H (cid:108)
(cid:126) (cid:126)
It is important to note here that the choice of V instead of V ( or E instead of E) in
(cid:108) (cid:108)
the expression of the chemical potential Eq.(14) corresponds to the model depicted in Fig.1
where each spin channel is treated separately. If the two sub-systems of Fig.1, are combined
before applying the stationarity condition, the symmetry property δn = −δn related to
↑ ↓
the two sub-systems reduces to a property of the total system δn = δn + δn = 0. The
↑ ↓
corresponding system is then analogous to the Corbino configuration [10] in which charge
accumulation is forbidden (due to the absence of edge), and this configuration imposes a
Hall current for both subsystems in the stationary state: we have J = −J , instead of
y↑ y↓
J = J = 0. In agreement with the Dyakonov-Perel equations, most publications about
y↑ y↓
the SHE [8] assumed implicitly the more constrained situation δn = 0 and the generation of
a pure spin-current. The case of the less constrained system δn = −δn is developed below,
↑ ↓
(cid:126) (cid:126)
keeping in minde that the other situation is recovered with replacing E instead of E .
(cid:108)
(cid:126) ˙
From Eq.(13), we derived the expression of the stationarity condition div(J ) = −R/2 :
(cid:108)
−D∇2n +qηn divE(cid:126) +qη∇(cid:126)n .E(cid:126) = −R˙/2. (16)
(cid:108) (cid:108) (cid:108) (cid:108) (cid:108)
Equation (16) can be put into the form:
∂2δn δn q R˙
(cid:108) (cid:108) (cid:126) (cid:126)
− − ∇δn .E = , (17)
∂y2 λ2 kT (cid:108) (cid:108) 2D
D(cid:108)
(cid:113)
where the screening length λ = (cid:15)kT is approximatively equal to the Debye length
D(cid:108) q2n
(cid:108)
(cid:113)
λ ≈ (cid:15)kT .
D q2n0
Using ∂E /∂y = qδn /(cid:15), equation (17) reduces to :
(cid:108) (cid:108)
(cid:32) (cid:33)
∂2δn R˙ ∂δn 1 (cid:90)
λ2 (cid:108) − −δn = (cid:108) δn dy(cid:48) (18)
D(cid:108) ∂y2 2D (cid:108) ∂y n (cid:108)
(cid:108) y
For each spin channel, Eq.(18) is equivalent to that of the simple Hall effect Eq.(8) because
thetwospin-dependentequationsaredecoupled. Thisisalsoequivalenttoasimplescreening
equation of a transport process without Hall effect, in which the charge accumulation δn is
(cid:108)
imposed at the interfaces by other means.
8
In the approximation in which we assumed constant conductivity (|δn| (cid:28) n) and
˙
R ≈ 0 we obtain:
∂2δn δn
(cid:108) (cid:108)
− ≈ 0, (19)
∂y2 λ2
D
Theeffectoftheelectrostaticchargeaccumulationdecreasesexponentiallyoverthescreening
length λ :δn ∼ δn e−y/λD where δn is the charge accumulation at the edges produced
D (cid:108) 0(cid:108) 0(cid:108)
(cid:126)
by the effective field H . The spin accumulation takes place ∆n = δn −δn = 2δn (cid:54)= 0
so(cid:108) ↑ ↓ ↑
with zero total charge accumulation δn = 0 [2, 8].
On the other hand, in the approximation of small Debye length but without assuming
constant conductivity, entails the relation:
∂δn 1 (cid:90)
−δn = (cid:108) δn dy(cid:48). (20)
(cid:108) (cid:108)
∂y n
(cid:108) y
The approximation of small Debye-Fermi length B together with that of constant conduc-
tivity A (the right-hand side of Eq.(10) is vanishing) results in the bulk approximation
(cid:126) (cid:126)
δn ≈ 0 and ∆n ≈ 0. This result is also valid in the case of spin-independent field E = E
(cid:108) (cid:108)
(or V(H ) = V in Eq.(14)).
so(cid:108)
IV. SPIN-HALL EFFECT WITH SPIN-FLIP RELAXATION
It is easy to take into account phenomenologically the spin-flip relaxation in the frame-
work of the two channel model. As a consequence, a well-known interface effect due to
non-local giant magnetoresistance [13, 14] will be superimposed to the spin-Hall effect de-
scribedabove. Indeed, ifthesurfaceisreplacedbyaninterfacewithaferromagneticmaterial
[4] the boundary conditions related to the spin system cannot be left free, and the situa-
tion described in the previous section should be revisited. In particular, it is necessary
(cid:126)
to take into account the gradient ∇µ of the spin-dependent chemical potentials µ near
0(cid:108) 0(cid:108)
the interface. We must then consider four forces in the gradient of the chemical poten-
tial ∇(cid:126)µ = kT∇(cid:126)ln(n ) − qE0(cid:126)e − qE (cid:126)e + ∇(cid:126)µ . The pumping force ∇(cid:126)(∆µ ) (cid:54)= 0 where
(cid:108) (cid:108) x x y(cid:108) y 0(cid:108) 0
∆µ = µ −µ is responsible for the out-of-equilibrium state of the two spin populations
0 0↑ 0↓
[12]. This force also leads to the well-known giant magnetoresistance (GMR) effect [15–18].
In the framework of the two channel model, the condition for the spin conservation
becomes:
˙
R
(cid:126)
div(J ) = − ∓L∆µ (21)
(cid:108) 0
2
9
where the spin-flip Onsager coefficient L is related to the spin diffusion length (see below)
measured in GMR experiments [12, 15–18]. Note that L is inversely proportional the spin-
relaxation time τ . Equation (17) becomes:
sf
∂2δn δn ∂δn 1 (cid:90) 1 (cid:18) ∂2µ ∂µ ∂δn (cid:19) 1 (cid:16) (cid:17)
(cid:108)− (cid:108) − (cid:108) δn dy(cid:48)+ n 0(cid:108) + 0(cid:108) (cid:108) = R˙ ±2L∆µ .
∂y2 λ2 ∂y λ2 n (cid:108) kT (cid:108) ∂y2 ∂y ∂y 2ηkT 0
D(cid:108) D(cid:108) (cid:108) y
(22)
We will not try to solve this system of two coupled equations, but the analysis performed in
the previous sections remains applicable.
Let us focus on the bulk approximation δn ≈ 0, which is also valid in the region
(cid:108)
λ (cid:28) y (cid:28) l , in which ∆µ is not zero. Assuming also, for the sake of simplicity tha
D sf 0
˙
R ≈ 0, Eq. (22) then reduces to:
∂2µ L
0(cid:108)
= ± ∆µ . (23)
∂y2 n η 0
0
We recognize the spin-accumulation equation for ∆µ which is formally equivalent to that
0
of the spin-accumulation described in the context of GMR:
∂2∆µ ∆µ
0 0
+ = 0, (24)
∂y2 l2
sf
(cid:112)
where l = σ /(qL) with σ = ηqn . Independently, we can look at the approximation of
sf 0 0 0
small λ , without assuming constant conductivity. Multiplying Eq.(22) by λ2 yields:
D D
(cid:32) (cid:33)
∂2δn R˙ 1 ∂δn (cid:90) λ2 (cid:18) ∂2µ ∂µ ∂δn (cid:19) λ2
λ2 (cid:108) − −δn − (cid:108) δn dy(cid:48) + D n 0(cid:108) + 0(cid:108) (cid:108) = ± D∆µ,
D ∂y2 2D (cid:108) n ∂y (cid:108) kT (cid:108) ∂y2 ∂y ∂y l2
(cid:108) y sf
(25)
In a metal, the Debye length λ is of the order of a nanometer while the spin-flip relax-
D
ation length l is few tens of nanometers. To second order in λ , Eq.(25) reduces to:
sf D
1 ∂δn (cid:90)
−δn − (cid:108) δndy(cid:48) ≈ 0 (26)
(cid:108)
n ∂y
(cid:108) y
Equation Eq.(26) which describes the SHE with spin-flip scattering is the same as Eq.(20)
which describe SHE without spin-flip scattering.
To conclude this section, we can state that despite the existence of the GMR-like spin-
accumulation at the interface over a distance l , there is no qualitative change introduced
sf
by the spin-flip relaxation in the bulk SHE, as long as the spin-diffusion length is much
10
