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Spin Transport and Accumulation in a 2D Weyl Fermion System
T. Tzen Ong1, 2 and Naoto Nagaosa1, 2
1RIKEN Center for Emergent Matter Science (CEMS), Saitama 351-0198, Japan 2Department of Applied Physics, University of Tokyo, Tokyo 113-8656, Japan (Dated: October 19, 2017)
In this work, we study the spin Hall effect and Rashba-Edelstein effect of a 2D Weyl fermion
system in the clean limit using the Kubo formalism. Spin transport is solely due to the spin-torque
current in this strongly spin-orbit coupled (SOC) system, and chiral spin-flip scattering off non-SOC
scalar impurities, with potential strength V and size a, gives rise to a skew-scattering mechanism
for the spin Hall effect. The key result is that the resultant spin-Hall angle has a fixed sign, with
SH O
V2 vF2 /a2
(kF
a)4
being a strongly-dependent function of kF a, with kF and vF being the
Fermi wave-vector and Fermi velocity respectively. This, therefore, allows for the possibility of
tuning the SHE by adjusting the Fermi energy or impurity size.
arXiv:1701.00074v3 [cond-mat.mes-hall] 18 Oct 2017
The spin Hall effect (SHE) has a long and rich history,
starting with the initial proposal of asymmetric Mott
scattering by Dyakonov and Perel [1, 2]. This extrin-
sic mechanism was re-introduced in 1999[3, 4], while an
intrinsic SHE was first proposed in 2003[5, 6]. The pro-
posal of a two-dimensional (2D) Z2-protected Quantum Spin Hall (QSH) state[7], and its successful prediction
in HgTe/CdTe quantum well [8] quickly followed; thus
giving rise to a new field of topological materials[9, 10],
which now include 2D QSH states [11], 3D topological in-
sulators (TI)[12, 13], topological Kondo insulators[14, 15]
and Weyl semi-metals[16].
One of the most striking characteristic of 3D TI ma-
terials is the existence of spin-momentum locked chiral
Weyl fermions on the surfaces, which are expected to
provide highly efficient spin-charge conversion[17, 18], via
the spin Hall effect or spin accumulation in the Rashba-
Edelsten effect[19]. Hence, there is a strong interest
in spintronic TI heterostructures, with many theoret-
ical works[20<32>25], discussing a plethora of spin-charge
phenomena, including magnetoresistance effects, inverse
spin-galvanic effect, and spin-transfer torque, which have
stimulated a flurry of experimental efforts[18, 26<32>30].
In heavy-metal/ ferromagnet systems, e.g. FePt/Au,
a giant spin Hall angle (SHA) of 0.1 has been
reported[31], which has been interpreted as resonant
skew-scattering off the Fe impurities[32]. However, re-
cent experiments on TI heterostructures[26, 29] have reported values of tan SH > 100%, with combined sur-
face and bulk contributions. In order to disentangle the
surface Weyl fermion contribution from the bulk bands,
a Cu-layer inserted TI/Cu/ferromagnet heterostructure has recently been engineered, with tan SH 50% [30].
Similar to the anomalous Hall effect, there are both
intrinsic Berry curvature and extrinsic scattering contri-
butions to the SHE. For systems with weak spin-orbit
coupling (SOC), it has been shown[33] that the extrinsic
skew scattering mechanism dominates in the clean limit;
hence, the spin Hall conductivity xzy scales with the lon-
gitudinal conductivity yy, and the SHA, SH
=
xzy yy
is
a well-defined measure of the SHE. The Rasha-Edelstein
effect is a closely related transport-driven spin accumula-
tion phenomena, which also scales with yy in the clean limit; the spin accumulation Si = i E is proportional to the applied electric field E (along -direction) with a coefficient i . For the strongly SOC-coupled Weyl system considered here, the main results are that due
to spin-momentum locking, chiral spin-flip scattering off
non-magnetic
impurities
drives
an
O(
1 ni
)
skew-scattering
mechanism,
and
that
Rashba-Edelstein
is
an
O(
1 t
)
ef-
fect; here, ni is the impurity concentration and t is the
transport scattering rate.
We adopt the Kubo formula framework for
calculating yy, xzy and yi , given by the retarded current-current correlation functions, yy =
-lim0
lim
k0
Im
yy (k,)
,
xzy
=
-lim0
lim
k0
I
m
xzy (k,)
,
and yi
=
-lim0
lim
k0
I
m
yi (k,)
;
where, yy(k, ),
xzy(k, ), and yi (k, ) are the current-current, spin current-current and spin accumulation-current correla-
tion functions respectively.
In spin-orbit coupled systems, the proper definition
of the spin current is more subtle as spin is not a
conserved quantity. Ref. [34] presented a bulk con-
served spin current that satisfies a continuity equation,
dSz dt
+
<EFBFBD>
(Js
+
P )
=
0,
with
an
additional
spin-torque
density term, <20> Pi = i [Si, H0], as well as the conven-
tional
spin
current
jsz
=
1 2
{v,
S
z
}.
Hence,
the
trans-
port spin current is the sum of a spin-polarized and a
spin-torque current, Jsi = jsi + Pi, succintly expressed as
the
time-derivative
of
a
spin-dipole
operator,
J^s
=
d(r^S^) dt
.
As pointed out by several groups[35<33>37], there is no fi-
nite conventional spin current for Weyl systems; hence,
spin transport for Weyl fermions is solely due to the spin-
torque density P coming from quantum-mechanical evolution of the electron spin.
We consider elastic scattering near the Fermi energy, EF , of 2D Weyl fermions (Dresselhaus-type vF k <20> sys-
2
(a)
ky
kF
(b) |k,
T (k, k)
T (k, k)
kx
|k,
x
x
x
=
+
+
+...
=
FIG. 1: Fig. (a) shows a colour density plot of the FS contribution to the Rashba-Edelstein effect yy (Eq. 15c). When the FS is shifted by ky = eEyt due to an external electric field Ey, the non-equilibrium distribution gives rise to a net Sy . Fig. (b) illus-
trates spin-dependent skew scattering, T , (k, k) and
T ,(k, k) having positive () and negative (-) chirality respectively, with the helical Weyl fermions defining positive () chirality.
tem) from a dilute (ni 1) random distribution of non-
magnetic impurities, with scattering off each impurity
given by Himp =
r
c (r)V
e-
|r|2 a2
c (r),
with
impurity
size a. Note that the results can be easily translated
into the Rashba-type vF z^ <20> k <20> case via rotation of the momentum by 90. Choosing the chemical potential <20>
to lie in the upper helical band, we obtain the following
Hamiltonian as,
H = H0 + Himp
(1)
H0 =
ck,vF k <20> , ck, - <20> ck,ck,
k,,
H imp =
c
k,
Vk
,k
ck
,
(2)
k,k
Here, Vk,k =
n Vnein(k-k ),
and
Vn
V a2 2
(kF 2n(
a)n n+1
2
)
,
while vF and i [1, ] are the Fermi velocity and spin
Pauli matrices, and kF a determines Vn, which will be
shown to control the skew scattering strength. Since the
impurity is non-magnetic, the system is invariant under time-reversal symmetry, T = Ki2, H = T HT -1. All
the scattering events from an impurity are summed up in
the T -matrix, and the spin-dependent skew scattering is captured by the <20> terms, illustrated in Fig. 1. The fol-
lowing Dyson equations, in operator formalism, give the effective Green's function, G^ eff = G^ 0 + G^ 0T^ G^ 0, and T matrix, T^ = V^ + V^ G^ 0T^ , with G^ 0 being the bare Green's
function, and Fig. 2 shows the Feynman diagram for the
effective Green's function.
FIG. 2: Feynman diagram for Geff (k, k, , ) that sums up the infinite set of scattering events from a sin-
gle impurity. This is captured by the T -matrix, which is represented by the diamond symbol in the second line above.
G0(k, in)
=
in
+
<EFBFBD>
1 -
vF k
<EFBFBD>
(3a)
= g00(k, in)1 + g0a(k, in)(cos x + sin y)
T (k, k, in) = Tnim(|k|, |k|, in)eink e-imk i (3b)
nm
Rotational symmetry of the Hamiltonian allows us
to carry out a multipole expansion of G0(k, in) and
the T -matrix, where g00(k, in)
=
, in +<2B>
(in+<2B>)2-vF2 k2
and
g0a(k, in)
=
. vF k
(in+<2B>)2-vF2 k2
We assume the T -matrix
varies slowly near EF , i.e. absence of resonances, thereby
simplifying the radial integral and reducing the Dyson
equation to a set of coupled algebraic recurrence equa-
tions for the retarded T -matrix coefficients, Tnim(|k| = |k| = kF , = EF ).
Tnzm<EFBFBD> = n,m Vn 1 - Vn<56>1 g00(EF )
1 - Vn g00(EF )
<EFBFBD> 1 - Vn<56>1 g00(EF )
- VnVn<56>1
g01(EF )
2
-1
(4a)
Tn<EFBFBD>m
=
n1,m 2
VnVn1 g01(EF )
1 - Vn1 g00(EF )
<EFBFBD> 1 - Vn g00(EF ) - VnVn1 g01(EF ) 2 -1 (4b)
The T -coefficients reduce to two set of coupled equa-
tions for T z<> = Tn0m <20> Tn3m and T <20> = Tn1m <20> iTn2m,
given in terms of Vn and the momentum-averaged re-
tarded Green's functions, g0i,(R)() =
dk 2
kg0i,(R)(k,
)
(refer to SOM for calculation details). The arguments
of the T -matrix coefficients are dropped, understanding that they are evaluated at kF and EF . Defining the symmetric and asymmetric parts of the spin-flip scattering as T S/A = T1+0 <20> T--10, T03 T030, and T13 T131, we can now
write down the s and p-wave channels of the T -matrix.
T (k, k ) = T 01 + T03z + T13 ei(k-k ) - e-i(k-k ) z
+ T S + T A eik - + T S - T A e-ik +
2
2
+ T S + T A e-ik + + T S - T A eik -
(5)
2
2
3
with detailed expressions for the T -matrix coefficients
shown in the SOM. Charge-transport is dominated by the largest term, |T 0| V0, while spin-flip scatterings are captured by the T S/A<> terms. Upon projection
into the upper helical band, we obtain a chiral spin-flip scattering term, T S sin(k - k ), which comes from 3rd and higher orders in perturbation; T S V0V12N0(EF )2, in agreement with previous work [37]. Hence, the skew
scattering strength can be tuned by varying kF a, i.e. either the Fermi level or the impurity size a.
It is now straightforward to calculate the effective
Green's function in the dilute impurity limit (ni
(R)
(R)
-1
1)[38], G (k, ) = - vF k <20> - (k, ) ,
(R)
where the retarded self-energy is (k, ) =
(R)
(R)
ni k1 V (k, k1)Geff (k1, )T (k1, k, ). The appear-
(R)
(R)
ance of Geff (k, ) instead of G0 (k, ) reflects the pres-
ence of multiple impurities. We assume an average
quasi-particle scattering rate near the Fermi surface, i.e.
(R)
Im[ (kF , EF )], and take vF and EF to be exper-
imentally determined parameters, thereby dropping the
real part of the self-energy.
= 01 - a (cos x + sin y) - b(sin x - cos y) + i3 z
0 = niNe(f0)f (EF ) |T 0|2 + |T03|2 -2 |T13|2 + |T A|2 - |T S|2
a = 4niNe(f1)f (EF ) |T S |2 - |T A|2
(6a)
(6b) (6c)
We have carried out a multipole expansion of , and
the main quasi-particle scattering channels relevant to
transport are the s and p-wave 0 and a terms (re-
fer to SOM for complete expressions of all ). As we
shall show later, the transport scattering rate, t, will
be given in terms of 0 and a. The angular momentum resolved density of states (DOS) is defined as Ne(fi)f () =
kdk 2
I
m
geiff (k, ) , and Ne(f0)f (EF ) and Ne(f1)f (EF ) cor-
respond to the s and p-wave components respectively.
Since scattering events that result in a change of angular
momentum, i.e involving the l = 1 component Ne(f1)f (EF ), will also cause a spin-flip due to spin-orbit coupling, we
see that 0 and a are due to spin-independent and dependent scattering respectively.
The effective Green's function is therefore given by,
(R)
-1
Geff (k, ) = + <20> - vF k <20> - i(k, )
(7)
= ge0ff (k, )1 + geaff (k, ) (cos x + sin y) + gebff (k, ) (sin x - cos y) + ge3ff (k, )z
where,
ge0ff (k, )
=
((k)
+ i(k))( + <20> 2(k) + 2(k)
-
i0)
(8a)
geaff (k, )
=
((k)
+ i(k))(vF |k| + ia) 2(k) + 2(k)
(8b)
with (k) = ( + <20>)2 - vF2 |k|2 - 02 + a2 + b2 - 32, and (k) = 2 ( + <20>)0 + vF |k|a .
(R)
A similar multipole expansion of Geff (k, ) has been done, and we show here only the main s and p-wave
terms, ge0ff (k, ) and geaff (k, ), with complete expressions for the scattering-induced gebff (k, ) and ge3ff (k, ) terms relegated to the SOM for brevity. From Eqs. (8a)
& (8b), it is clear that Weyl fermions in the s and p-wave
channels pick up a 0 and a scattering rate respectively,
and we shall show later that it is chiral scattering between
the s and p-wave electrons that drive the SHE.
(R)
(R)
Geff (k, ) and (k, ) are determined self-
(R)
consistently by solving Eqns. 6a & 7, i.e. (k, ) is
calculated using the disorder-averaged density of states,
Ne(fi)f () =
kdk 2
I
m
geiff (k, )
.
However, in the dilute
impurity
limit,
Ne(f0)f/(1)(EF )
=
N0
(EF 2
)
(1
+
O( ))
[38];
allowing us to drop the O(ni) corrections.
As stated earlier, the DC longitudinal charge conduc-
tivity, spin-Hall conductivity and spin accumulation are
given by analytic continuation of the corresponding Mat-
subara correlation functions,
yy(k, in) = - d e-in T jy(k, )jy(k, 0) (9a)
0
yi (k, in) = - d e-in T i(k, )jy(k, 0) (9b)
0
xzy(k, in) = - d e-in T Pxz(k, )jy(k, 0) (9c)
0
Note that yy and yi are equal up to a factor of evF for Weyl fermions due to spin-momentum locking, i.e. j^y = evF ^y. The spin torque current, Pxz, arises from the intrinsic quantum-mechanical evolution of the elec-
tron spin, and the z-component of the spin-torque cur-
rent along x^ is,
Pxz (k)
=
i kx
dS^z (k) dt
= 2vF ikx
cp,
p
p
+
k 2
x -
y
p
+
k 2
(10)
y cp+k,
x
The Feynman diagrams for these correlation functions are shown in Fig. 3, with chiral spin-flip scattering starting to contribute at third-order in perturbation theory. Fig. 3 shows the infinite subset of Feynman ladder diagrams summed up in the Bethe Salpeter equation for the scattering vertex,
4
y
(k
+
p,
p,
im
+
in,
in)
=
y
+
T (k + p, k + q, im + in)Geff (k + q, im + in)
q
y
<EFBFBD> (k + q, q, im + in, in)Geff (q, in)T (q, p, in)
(11)
x
+
x
+
x
=
+
+
=
FIG. 3: Feynman diagram for the effective scattering
y
vertex, (p, ), is shown in the second line. This includes an infinite subset of scattering events from the dilute concentration of impurities. The first line shows all the scattering events from a single impurity, and the second and third diagrams in the first line are the leading-order contributions to skew scattering.
Here, k and im are the external momentum and fre-
quency, and the uniform DC limit of the conductivities is
obtained by analytic continuation of im + i, and
taking the limit k 0 followed by 0. Hence, we
only need to calculate the on-shell component of the scat-
y
y
tering vertex (p, ) = (p, - i, + i). The Bethe-
y
Salpeter equation for (p, ) is solved self-consistently
y
by expanding (p, ) =
n ineini
in
y
multipole
terms, assuming that the T -matrix and (p, ) vary
slowly near EF (see SOM for details). Keeping only the
y
s- and p-wave channels, and evaluating (|p| = kF , =
EF ) at the Fermi surface, we obtain,
y
(kF
,
EF
)
=
(0px
cos
+
i0py
sin
)
1
(12)
+10(EF )x + 20(EF ) y +(3px (EF ) cos + i3py (EF ) sin ) z
where,
20
=
0 , t
3px
=
-
s t
.
(13)
After analytic continuation of the current-current correlation functions in Eq. (9a) - (9c), we find that the main contributions come from the 20 charge-transport and 3px spin-transport scattering vertices (refer to SOM for all
the -coefficients). We can therefore define a transport and chiral spin-flip scattering rate respectively as,
t
=
(
1 2
0
+
a),
s
=
niN0(EF 2
)
|T
0
||T
S
|.
(14)
The main results of this paper are the charge and spin
conductivities, and the Rashba-Edelstein coefficient,
yy
=
(evF )2
N0(EF 2
)
1 t
xzy
=-
evF2
N0(EF 2
)
1 t
0
s +
a
yy =
evF
N0(EF 2
)
1 t
(15a) (15b) (15c)
Our
key
finding
is
Eq.
(15b),
which
shows
an
O(
1 ni
)
skew scattering contribution to the SHE. Explicitly writ-
ing out the spin and angular-momentum scattering chan-
nels for xzy = evF2 Re[3px(a0(EF ) - 0a(EF ))], where
ij() =
dp 2
p2
gei(fRf) (p,) p
gejf(Af )
(p,
),
we
see
that
chiral
spin-flip scattering between the s and p-wave electrons
is the cause of the skew-scattering mechanism, and the
strength of which is measured via the spin-Hall angle,
SH
=
-e
s 0 + a
(16)
Here, e < 0 is the electron charge, and power counting of
t 0 niV02N0(EF ) and s niV02V12N0(EF )3, gives
SH O
V2 vF2 /a2
(kF
a)4
. This is our key result: SH has
a fixed positive sign, and is a strongly-dependent function
of kF a; hence, the SHE can be tuned by EF .
Finally, we briefly discuss the effects of band bending in
Weyl
systems.
The
leading
O(
1 m
)
correction
comes
from
including
a
conventional
spin
current,
jsz
=
1 2
{v,
S
z},
with v =
k m
.
However, it has been pointed out[35<33>37]
that jsz y for Rashba-type systems; hence, up to
O(
1 m
),
band
bending
does
not
give
rise
to
a
spin
current
for Weyl fermion systems.
In conclusion, we have analysed both the spin Hall
and Rashba-Edelstein effects in a 2D Weyl electron sys-
tem. Our results show that strong spin-orbit coupling in the band-structure is sufficient to cause chiral spin-
flip scattering of the helical electrons off non-SOC scalar
impurities, resulting in a skew-scattering contribution to
the SHE. The strength of this mechanism is measured by
the
SHA,
SH
=
-e
s 0 +a
-e
O
V2 vF2 /a2
(kF
a)4
, and
we highlight the fact that the skew scattering strength
can be tuned by varying kF a, thereby providing an
experimentally-accessible parameter for controlling the
SHE.
In
addition,
we
have
also
found
an
O(
1 t
)
Rashba-
Edelstein effect due to spin-momentum locking of the
Weyl fermions. We gratefully acknowledge I. Mertig, K.
Kondou and Y. Tokura for helpful discussions, and this
work was supported by CREST, Japan Science and Tech-
nology Agency (JST).
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Supplementary Online Material: Spin Hall Effect on Topological Insulator Surface
arXiv:1701.00074v3 [cond-mat.mes-hall] 18 Oct 2017
CONTENTS
I. 2D Weyl Fermion and Chiral Skew Scattering from Non-magnetic Impurity
1
II. Effective Greens Function and Quasi-particle Scattering Rate
5
III. SHE & Rashba Edelstein Effect Correlation Functions
7
IV. Vertex Correction
11
V. Longitudinal Charge Transport and SHE DC Conductivities
15
References
23
I. 2D WEYL FERMION AND CHIRAL SKEW SCATTERING FROM NONMAGNETIC IMPURITY
We consider elastic scattering near EF of 2D Weyl fermions (Dresselhaus-type vF k <20>
system) from a dilute (ni 1) random distribution of non-magnetic impurities, at positions
Ri, with impurity scattering Himp =
r,Ri
V
e-
|r -Ri |2 a2
c(r)1
c
(r),
and
the
impurity
size
a determines the strength of skew scattering. Note that the results can be easily translated
into the Rashba-type vF z^ <20> k <20> case by rotating the momentum by 90. The chemical
potential <20> is chosen to lie in the upper helical band, with the upper/ lower helical Weyl
fermions
being
<EFBFBD>,k
=
1 2
(<28>
ck,
+
eik ck,),
and
the
Hamiltonian
is,
H = H0 + Himp
(1)
H0 =
ck,vF k
<EFBFBD>
, ck,
-
<EFBFBD>
c k,
ck,
k,,
H imp =
c k,
Vk,k,
ck
,
(2)
k,k
The non-magnetic impurity is modelled with a scattering potential V and a Gaussian
profile,
V
e . -
r2 a2
Hence
the
scattering
matrix
element
of
2D
Weyl
fermions
off
this
impurity
is,
Vk,k, =
k,
V e-
r2 a2
k,
1
=
Vnein(k-k )1
(3)
n
where
Vn
=
e k a V a2 8
-
1 8
kF2
a2
F
I
(
n-1 2
,
kF2 a2 8
)
-
I
(
n+1 2
,
kF2 a2 8
)
. V a2 (kF a)n
2
2n
(
n+1 2
)
We
have
assumed
that transport involves mainly the quasi-particles near EF , i.e. |k| = |k| kF , and have used
the result
0
r
drJn(kF
r)e-
r2 a2
=
k a e a2
8F
-
1 8
kF2
a2
I
(
n-1 2
,
) kF2 a2 8
-
I
(
n+1 2
,
) kF2 a2 8
, with J(n, z)
and I(n, z) being the Bessel and modified Bessel functions of the first kind respectively, and
(n) is the Gamma function
All the scattering events from a single impurity are captured in the T -matrix, given by the Dyson equation T^ = V^ + V^ G^ 0 T^ . Making use of the rotational symmetry of the system,
we express the Greens function and T -matrix in a multipole-expansion,
G0(k, in)
=
in
+
1 <20> - vF k <20>
(4)
= g00(k, in)1 + g01(k, in)(cos k x + sin k y)
g00(k, in)
=
(in
in + <20> + <20>)2 - vF2 k2
g01(k, in)
=
(in
vF k + <20>)2 -
vF2 k2
, where
T (k, k)
Tnimeink e-imk i
nm
= V (k, k) +
n1n2n3
dk1 2
k1dk1 2
Vn1
ein1
(k
-k1
)
<EFBFBD> g00(k1, in)1 + g01(k1, in)(cos k1 x + sin k1 y)
<EFBFBD>Tnj2n3 (k1, k)ein2k1 e-in3k j
(5)
The Pauli matrices are defined as i [1, ]. As discussed in the main paper, we shall
assume that there are no resonances, so the T -matrix varies slowly as a function of k near
EF . Approximating the T -matrix as a constant near kF , the dk1-integral is carried out
only over the Green's function. This is the momentum-averaged retarded Green's function,
g0i,(R,A)(in)
kdk 2
gi,(R,A)(k,
in),
and
the
results
are,
g00,(R,A)(EF )
=
i 2
N0(EF
)sgn(EF
)
g01,(R,A)(EF )
=
<EFBFBD>
i 2
N0(EF
)sgn(EF
)
(6a) (6b)
Here,
N0(EF )
=
EF 2vF2
is the bare density of states,
and in terms of the momentum-
averaged retarded Greens functions, the retarded T -matrix is now given by,
2
T (k, k) =
Vnein(k-k )1nm + Vneink e-imk g00(EF ) Tn0m1 + Tn1mx + Tn2my + Tn3mz (7)
nm
+ g01(EF ) Tn--1m1 + Tn--1mz + Tnz-+1m- + g01(EF ) Tn++1m1 + Tn++1mz + Tnz+-1m+
The
coefficients
of
the
T -matrix
are
Tnzm<EFBFBD>
Tn0m
<EFBFBD> Tn3m,
Tn<EFBFBD>m
1 2
(Tn1m
<EFBFBD> iTn2m),
and
are
now defined by the following set of coupled recurrence equations,
Tnzm+ = Vnnm + Vn g0(EF ) Tnzm+ + 2Vn g1(EF ) Tn++1m
Tn+m
= Vn
g0(EF )
Tn+m +
1 2
Vn
g1(EF )
Tnz-+1m
Tnzm- = Vnnm + Vn g0(EF ) Tnzm- + 2Vn g1(EF ) Tn--1m
Tn-m
= Vn
g0(EF )
Tn-m +
1 2
Vn
g1(EF )
Tnz+-1m
(8)
The T -coefficients reduce to two set of coupled equations for T z<> = Tn0m <20> Tn3m and T <20> = Tn1m <20> iTn2m, given in terms of Vn and the momentum-averaged retarded Green's functions, gi,(R)(EF ) . The arguments of the T -matrix coefficients are dropped, understanding that they are evaluated at kF and EF . Some straightforward, albeit tedious, algebra allows us to solve Eq. 8.
Tnzm+
=
(1 - Vn
Vn (1 - Vn+1 g0(EF )) nm g0(EF ) ) (1 - Vn+1 g0(EF ) ) - VnVn+1
g1(EF )
2
Tn+m
=
1 2 (1 - Vn-1
VnVn-1 g1(EF )n-1m g0(EF ) ) (1 - Vn g0(EF ) ) - VnVn-1
g1(EF )
2
Tnzm-
=
(1 - Vn
Vn (1 - Vn-1 g0(EF )) nm g0(EF ) ) (1 - Vn-1 g0(EF ) ) - VnVn-1
g1(EF )
2
Tn-m
=
1 2 (1 - Vn+1
VnVn+1 g1(EF )n+1m g0(EF ) ) (1 - Vn g0(EF ) ) - VnVn+1
g1(EF )
2
(9)
Therefore, the T -matrix coefficients are,
Tn0m
=
1 2 (1 - Vn
Vn (1 - Vn+1 g0(EF )) nm g0(EF ) ) (1 - Vn+1 g0(EF ) ) - VnVn+1
g1(EF )
2
+
1 2 (1 - Vn
Vn (1 - Vn-1 g0(EF )) nm g0(EF ) ) (1 - Vn-1 g0(EF ) ) - VnVn-1
g1(EF )
2
Tn3m
=
1 2 (1 - Vn
Vn (1 - Vn+1 g0(EF )) nm g0(EF ) ) (1 - Vn+1 g0(EF ) ) - VnVn+1
g1(EF )
2
-
1 2 (1 - Vn
Vn (1 - Vn-1 g0(EF )) nm g0(EF ) ) (1 - Vn-1 g0(EF ) ) - VnVn-1
g1(EF )
2
Tn1m
=
1 2 (1 - Vn-1
VnVn-1 g1(EF )n-1m g0(EF ) ) (1 - Vn g0(EF ) ) - VnVn-1
g1(EF )
2
3
+
1 2 (1 - Vn+1
VnVn+1 g1(EF )n+1m g0(EF ) ) (1 - Vn g0(EF ) ) - VnVn+1
g1(EF )
2
Tn2m
=
-
i 2
(1
-
Vn-1
VnVn-1 g1(EF )n-1m g0(EF ) ) (1 - Vn g0(EF ) ) - VnVn-1
g1(EF )
2
+
i 2 (1 - Vn+1
VnVn+1 g1(EF )n+1m g0(EF ) ) (1 - Vn g0(EF ) ) - VnVn+1
g1(EF )
2
(10)
We calculate the T -matrix up to order O(V0V12), at which skew scattering appears, and keep only the l = 0 and l = 1 channels. Defining the symmetric and asymmetric parts of the spin-flip scattering as T S/A = T1+0 <20> T--10, we can now write down the s and p-wave channels of the T -matrix.
T (k, k ) = T 01 + T03z + T13
e - e i(k-k )
-i(k-k )
z
+
TS
+ 2
T A eik -
+
TS
- 2
T A e-ik +
+ T S + T A e-ik + + T S - T A eik -
2
2
and the coefficients are defined as,
T0
=
1 2
V0 1 - V1 g0(EF ) (1 - V0 g0(EF ) ) (1 - V1 g0(EF ) ) - V0V1 g1(EF ) 2
+
1 2
V0 1 - V-1 g0(EF ) (1 - V0 g0(EF ) ) (1 - V-1 g0(EF ) ) - V0V-1 g1(EF ) 2
=
V0
2
1 - V0 g0(EF )
T03
=
1 2
V0 1 - V1 g0(EF ) (1 - V0 g0(EF ) ) (1 - V1 g0(EF ) ) - V0V1 g1(EF ) 2
-
1 2
V0 1 - V-1 g0(EF ) (1 - V0 g0(EF ) ) (1 - V1 g0(EF ) ) - V0V-1 g1(EF ) 2
=
V02V1 g1(EF ) 2
2
1 - V0 g0(EF )
T13
=
1 2
V1 1 - V2 g0(EF ) (1 - V1 g0(EF ) ) (1 - V2 g0(EF ) ) - V1V2 g1(EF ) 2
-
1 2
V1 1 - V0 g0(EF ) (1 - V1 g0(EF ) ) (1 - V0 g0(EF ) ) - V1V0 g1(EF ) 2
4
(11) (12a) (12b)
=
-
1 2
V0V12 g1(EF ) 2
2
1 - V1 g0(EF )
TS
=
1 2 (1 - V0
g0(EF )
V0V1 g1(EF ) )(1 - V1 g0(EF )
) - V0V1
g1(EF )
2
+
1 2
(1
-
V0
g0(EF )
V0V-1 )(1 - V-1
g1(EF ) g0(EF )
) - V0V-1
g1(EF )
2
= V0V12 g0(EF )
g1(EF )
2
1 - V0 g0(EF )
TA
=
1 2 (1 - V0
g0(EF )
V0V1 g1(EF ) )(1 - V1 g0(EF )
) - V0V1
g1(EF )
2
-
1 2
(1
-
V0
g0(EF )
V0V-1 )(1 - V-1
g1(EF ) g0(EF )
) - V0V-1
g1(EF )
2
=
V0V1 g1(EF )
2
1 - V0 g0(EF )
(12c) (12d) (12e)
We point out that upon projecting into the upper helical band, i.e. calculating the matrix elements k, + T S(eik - + e-ik +) k, + = 2 T S (cos(k - k ) - i sin(k - k )), we find that the spin-flip scattering gives rise to a skew-scattering term 2iT S sin(k - k ) in the chiral band basis, which will drive the SHE.
II. EFFECTIVE GREENS FUNCTION AND QUASI-PARTICLE SCATTERING RATE
The retarded T -matrix calculated in Eq. 5 includes only scattering from a single impurity, and in the dilute impurity limit, the T -matrix for scattering from all impurities can be calculated in the non-crossing approximation NCA) [1]) by including scattering events from other impurities in the bare Greens function leg, i.e. replacing G0 by Geff , in the calculation of the T -matrix. Hence, this forms an implicit self-consistent solution for the retarded and advanced Geff function and T -matrix.
T
(R)
(k,
k)
=
niV
(k,
k)
+
ni
V
(k,
(R)
k1 )Gef f
(k1,
)T
(R)
(k1,
k,
)
k1
T
(A)
(k,
k)
=
niV
(k,
k)
+
ni
V
(k,
(A)
k1 )Gef f
(k1,
)T
(A)
(k1
,
k,
)
k1
(13)
5
(R)
In the non-crossing approximation, the retarded self-energy (k, ) and quasi-particle
(R)
scattering rate (k, ) = Im[ (k, )] are given by,
(R)
(k, ) = ni
V
(k,
(R)
k1 )Gef f
(k1
,
)T
(R)
(k1,
k,
)
k1
(R)
(A)
(R)
(k, ) = Im[ (k, )] = T (k, k1, )Aeff (k1, )T (k1, k, )
(14)
k1
(R)
The spin-dependent spectral weight is given by Aeff (k, ) = 2Im[Geff (k, )]. Similar to
the calculation of the T -matrix, the dk-integral for the self-energy is done using the ap-
proximation that the T -matrix varies slowly near kF , leaving only the dk-integral of the spin-dependent spectral weight, which is none other than the density of states,
Ne(f0)f () =
kdk 2
I
m[ge0f
f
(k,
)]
Ne(f1)f () =
kdk 2
I
m[ge1f
f
(k,
)]
(15)
As
pointed
out
in
the
main
paper,
Ne(f0)f/(1)(EF )
=
N0 (EF 2
) (1
+
O())
in
the
dilute
limit;
hence, we will approximate Ne(f0)f/(1)(EF )
N0(EF ) 2
=
|EF2 | 4vF2
and Ne(f1)f (EF )
N0 (EF 2
)
sgn(EF
).
This finally gives the result for the quasi-particle lifetime near the Fermi surface, i.e. =
(R)
(kF , EF ) = Im[ (kF , EF )], which is shown below. The real part of the self-energy
that renormalizes vF and <20> are ignored here, as vF and <20> are taken to be experimentally
determined parameters.
= 01 + a (cos x + sin y)
- b(sin x - cos y) + i3 z
0 = niNe(f0)f (EF ) |T 0|2 - 2 |T 3|2 + |T S|2 - |T A|2 a = -4niNe(f1)f (EF ) |T S|2 + |T A|2 b = 2niNe(f0)f (EF ) |T 0||T A| + |T 3||T S| 3 = 4niNe(f1)f (EF ) |T 0||T S| + |T 3||T A|
(16a) (16b)
The effective Greens function in the dilute impurity limit is now given by,
(R)
-1
Geff (k, ) = + <20> - vF k <20> - i(k, )
= ge0ff (k, )1 + geaff (k, ) (cos x + sin y)
+ gebff (k, ) (sin x - cos y) + ge3ff (k, )z
ge0ff (k, )
=
((k)
+ i(k))( + <20> - i0) 2(k) + 2(k)
(17a)
6
geaff (k, )
=
((k)
+ i(k))(vF |k| 2(k) + 2(k)
+
ia)
gebff (k, )
=
ib((k) + i(k)) 2(k) + 2(k)
ge3ff (k,
)
=
-
3((k) + i(k)) 2(k) + 2(k)
(17b)
where the denominator terms are (k) = ( + <20>)2 - vF2 |k|2 - 02 + a2 + b2 - 32, (k) = 2 ( + <20>)0 + vF |k|a .
III. SHE & RASHBA EDELSTEIN EFFECT CORRELATION FUNCTIONS
Within the Kubo formalism, the longitudinal charge conductivity and spin-Hall con-
ductivity, yy and xzy, are given by the retarded current-current and spin current-current correlation functions respectively,
y(Ry )(k, ) = -i
dt eit(t) [jy(k, t), jy(k, 0])
-
xzy,(R)(k, ) = -i
dt eit(t) [Jxz(k, t), jy(k, 0])
-
(18a) (18b)
Similarly, it is straightforward to derive a Kubo formula for the spin-accumulation due
longitudinal charge transport, i.e. the Rashba-Edelstein effect.
S
=
lim
0
lim
k0
E ei(k <20> r-t)
dt(t) [S(k, t), j(k, 0)]
-
i,(R)(k, ) = -i
dteit [Si(k, t), j(k, 0)]
-
(19a) (19b)
The spin current Jxz has two components, one is the conventional spin current jxz due
to band-bending effects, and the other is the spin-torque current Pxz, which are defined as
follow,
jxz(k, ) =
c ( k1,
)
(k
+ k1)x m
z
ck+k1, (
)
k1
Pxz(k, )
=
2ivF kx
c ( ) k1,
(k1
+
k 2
)x
y
-
(k1
+
k 2
)y
x
ck+k1, ( )
k1
(20a) (20b)
We will now separate the SHE into two contributions, xzy(1) and xz(y2), coming from the
conventional spin current and the spin torque current respectively. All the Matsubara cor-
relation functions, yy(k, in), yi (k, in), xz(y1)(k, in) and xzy(2)(k, in), are given below,
7
and analytic continuation (in + i) will give the corresponding retarded correlation
functions.
yy(k, in) = - d e-in T U (, 0)jy(k, )jy(k, 0)
0
yi (k, in) = - d e-in T U (, 0)Si(k, )jy(k, 0) 0
xzy,(1)(k, in) = - d e-in T U (, 0)jxz(k, )jy(k, 0) 0
xz,y(2)(k, in) = - d e-in T U (, 0)Pxz(k, )jy(k, 0) 0
(21a) (21b) (21c) (21d)
The correlation functions are written in the interaction representation, and U(, 0) is the
S-matrix, which can be formally expanded as an infinite series of interacting terms involving
Hint. Hence, the correlation functions are evaluated by expanding the S-matrix, and we show the expansion for xzy,(1)(k, ) below.
xz,y(1)(k, ) = -
(-1)n n!
n=0
d1 . . .
0
dn T jxz(k, )Hint(1) . . . Hint(n)jy(k, 0) (22)
0
The n = 0 term in Eq. 22 is just the bare bubble diagram, and the n = 2 term will give
the first correction to the scattering vertex.
xz,y(1,n=2)(k, in) = -
d
0
d1
0
d2e-in
0
evF c
T
c ( k1,
)
(k
+ k1)x m
z
ck+k1,
(
)
k1 ,k2
<EFBFBD>H int(1)H int(2)ck2,(0)y ck+k2,
= - evF mc
1
z G<>1 (p + k, i1 + in)V<>1<EFBFBD>2 (p + k, p + q)
p,q
i1
<EFBFBD>G<EFBFBD>2(p + q, i1 + in)y G<>3(p + q - k, i1)
<EFBFBD>V<EFBFBD>3<EFBFBD>4(p + q - k, p)G<>4(p, i1)(k1 + k)x
(23)
This corresponds to the Feynman diagram for the vertex correction from a single scattering event. Notice that only elastic scattering is considered here, as each scattering event does not change the energy of the electron; hence, all the Green's functions on the upper (and lower) legs of the bubble diagram have the same energy, e.g. in Eq. 23, G<>1(p + k, i1 + in) and G<>2(p + q, i1 + in) undergo a change of momentum and spin upon scattering off V<>1<EFBFBD>2(p + k, p + q), but do not exchange energy with the impurity.
Since energy is conserved in the upper and lower legs of the bubble diagram, we can now include the effect of all the scattering events from a single impurity on the vertex correction
8
by replacing the scattering potential V<>1<EFBFBD>2(k, k) by the full T -matrix to obtain,
xzy,(1,T )(k,
in)
=
- evF c
1
(p + k)x Tr m
zG(p + k, i1 + in)T (p + k, p + q)
p,q
i1
<EFBFBD>G(p + q, i1 + in)yG(p + q - k, i1)T (p + q - k, p)G(p, i1) (24)
Finally, scattering events from all the impurities can be included by defining a scattering
y
vertex (p+k, k, i1 +in, in), whereby an infinite subset of scattering events are included
in the Bethe-Salpeter equation,
y
(p
+
k,
k,
i1
+
in,
in)
=
y
+
T (p + k, p + q, i1 + in)Geff (p + q, i1 + in)
q y
<EFBFBD> (p + q, q, i1 + in, in)Geff (q, in)T (q, k, in) (25)
and the full correlation function is therefore,
xzy,(1)(k,
in)
=
-
evF c
1
(p + k)x m
p
i1
<EFBFBD> Tr
G(p,
i1)zG(p
+
k,
i1
+
y
in) (p
+
k,
p,
i1
+
in,
i1)
(26)
This infinite subset of ladder diagrams includes all the scattering corrections to the vertex
from all the impurities, but does not include diagrams where scattering events from different
impurities cross each other, i.e. this is the non-crossing approximation, which is reasonable
in the dilute impurity limit.
Now let us evaluate the uniform limit of the Matsubara correlation function, lim xz,y(1)(k, in),
k0
by first doing the sum over the i1 frequencies using the standard method of integrating
over the poles of nF (z) = (ez + 1)-1 in the complex z-plane. The poles of nF (z) are at
z
=
i
2(n+1)
,
with
residue
of
-
1
,
and
the
sum
i1 is replaced by an integration over the
complex plane,
xzy,(1)(k
=
0,
in)
=
- evF mc
dz 2i
P (z,
z
+
in)nF
(z)
P(z, z + in) =
px Tr
G(p,
z )z G(p,
z
+
y
in) (p,
p,
z,
z
+
in)
(27)
p
The integral over the complex z-plane will also pick up the branch cuts of the Green's function G(p, z) and G(p, z + in), which leads to branch cuts at z = vF |p| - <20> = (p) and z + in = vF |p| - <20> = (p), and the upper ( + i) and lower ( - i) paths along the branch cuts will give the following retarded and advanced contributions to the correlation function.
9
xzy,(1)(k = 0, in) = -
d 2i
nF
()
P( + i, + in) - P( - i, + in)
+P( - in, + i) - P( - in, - i)
(28)
Therefore, the retarded correlation function is obtained by analytic continuation in
+ i,
xz,y(1)(k
=
0,
)
=
- evF mc
d 2i
(nF
()
-
nF
(
+
))P (
-
i,
+
+
i)
-nF ()P( + i, + + i) + nF ( + )P( - i, + - i) (29)
Following the standard discussion in [2], the most singular contribution comes from P( -
i, + + i).
Since
the
SHE
conductivity
is
given
by
xz y (
=
0)
=
- lim I 0
m[
xzy
(k=0,)
],
hence we will calculate the following contribution to the retarded SHE correlation function.
xzy,(1)(k
=
0,
)
=
- evF mc
d 2i
(nF
()
-
nF
(
+
))P (
-
i,
+
+
i)
xzy,(1)(k = 0, = 0) = -Im
evF mc
d 2i
dnF () d
P
(
-
i,
+
i)
P( - i, + i) =
px Tr
(A)
G (p,
)
z
(R)
G (p,
y
)
(p,
p,
-
i,
+
i
)
(30)
p
The other correlation functions for the spin-torque current contribution to the SHE
(xz,y(2)(k, )), the Rashba-Edelstein effect (yi (k, )), and the charge current conductivity
(yy(k, )) are derived in a similar manner, and we obtain,
yy(k = 0, ) = lim
k0
evF c
2
d - 2i
d2p (2)2
(nF
()
-
nF
(
+
))
<EFBFBD> Tr
(A)
G (p,
)y
(R)
G (p
+
k,
y
) (p,
k1
+
k,
)
(31)
xzy,(2)(k = 0, ) = xzy,(2a)(k = 0, ) + xz,y(2b)(k = 0, )
(32)
xz,y(2a)(k
=
0,
)
=
lim 2ievF2 k0 c
d - 2i
d2p py + (2)2 kx
ky 2
(nF () - nF ( + ))
<EFBFBD> Tr
(A)
G (p,
)xG(R)
(p
+
k,
y
)
(p,
p
+
k
,
)
xz,y(2b)(k
=
0,
)
=
lim 2ievF2 k0 c
d - 2i
d2p (2)2
px + kx
kx 2
(nF (
+
)
-
nF ())
<EFBFBD> Tr
(A)
G (p,
)y
(R)
G (p
+
k,
y
)
(p,
p
+
k,
)
yi (k
=
0,
)
=
lim evF k0 c
d - 2i
d2p (2)2
(nF
(
+
)
-
nF
())
<EFBFBD> Tr
(A)
G (p,
)iG(R)(p
+
k,
y
)
(p,
p
+
k,
)
(33)
10
IV. VERTEX CORRECTION
For four fermion correlation functions, like the current-current and spin current-current
correlation functions, we have to consider the effects of impurity scattering on the scattering
vertex[2], in addition to the quasi-particle self-energy corrections. This arises from an infinite
subset of Feynman ladder diagrams shown in the main paper, and is summed up in the Bethe
y
Salpeter equation for the scattering vertex (k + p, p, i1 + in, in) (Eq. 34).
y
(k
+
p,
p,
i1
+
in,
in)
=
y
+
T (k + p, k + q, i1 + in)Geff (k + q, i1 + in)
q y
<EFBFBD> (k + q, q, i1 + in, in)Geff (q, in)T (q, p, in) (34)
Here, k and i1 are the external momentum and frequency, and the DC uniform limit of
the conductivities are obtained by analytic continuation of i1 + i, setting the limit
k 0, and then setting 0, i.e. lim lim. Hence, we only need to calculate the on-shell
0 y
k0
y
component of the scattering vertex (p, ) = (p, - i, + i), which is defined by,
y
(p,
)
=
y
+
T (p, q, + i)Geff (q, + i)
q
y
<EFBFBD> (q, )Geff (q, - i)T (q, p, - i)
= y +
(R)
(R)
y
(A)
(A)
T (p, q, )Geff (q, ) (q, )Geff (q, )T (q, p, )
(35)
q
(R)
Note that both the advanced and retarded Green's function and T -matrices, Geff (p, ),
(A)
(R)
(A)
Geff (p, ), T (p, q, ) and T (p, q, ) enter into the Bethe-Salpeter equation due to
the branch cut in the complex plane, when the integral over the complex plane is car-
ried out. Similar to the assumption for the T -matrix, the scattering vertex is assumed
to be momentum-independent near EF , and we will do a similar multipole expansion of
y
(|p| = kF , , = EF ) =
n ineini, keeping only the l = 0 and l = 1 scattering chan-
nels.
y
(|p|
=
kF
,
,
=
EF
)
=
i0i
+
0px cos + i0py sin
1+
ipx cos + iipy sin
i
(36)
Hence, the Bethe-Salpeter equation is reduced to,
y
(p,
)
=
y
+
dq 2
T
(R)
(|p|
=
|p
+
q|
=
kF
,
p,
p+q ,
)
(37)
<EFBFBD>
qdq 2
(R)
G (p
+
q,
y
)
(p
+
q,
(A)
)G (p
+
q
,
)
11
(A)
<EFBFBD>T (|p + q| = |q| = kF , p+q, q, )
ineini = y +
n
n1 ...n7
dq 2
T i1 n1n2
ei(n1k
-n2
k+q
)
T i5 n6
n7
ei(n6
k
-n7
k+q
)
i1
i2
i3
i4
i5
<EFBFBD>
qdq 2
gi2,(R) n3
(|p
+
q|,
)e-in3p+q
i3 n4
e-in4p+q
gi4,(A) n5
(|p
+
q|,
)e-in5p+q
Since the in coefficients are assumed to be invariant near kF , the dq-integral is carried out over all the spin and angular momentum resolved Green's function components, gmi,(R)(|p + q|, ) gnj,(A)(|p + q|, ). As the Weyl fermions are spin-momentum locked; hence, the spin i and momentum m indices are related, i.e. m = 0 for i = [0, 3], and m = <20>1 for
i [1, 2]. We can now define,
ij() = kdk gi,(R)(|k|, )gj,(A)(|k|, )
(38)
2
We have carried out a change of variable from + <20> here, thereby absorbing the factors
of <20> that appear in the Green's function into , which is now the energy measured from EF .
Knowing
that
G(R)(k, )G(A)(k, )
=
A(k,) I m[(k,)]
A(k,)
,
this
means
that
ij ()
is
basically
the spin-resolved density of states divided by the quasi-particle scattering rate. The domi-
nant terms are the s-wave, p-wave and s-p spin-flip DOS, 00(), aa() and 0a() = (a0())
respectively, which are calculated to be,
00()
=
1 2vF2
+
02
2(0 + a) 2(0 + a)
aa() 0a()
= =
1 21vF2 2vF2
2(0 +
a)
+
a2 2(0 + a)
2(0 +
a) (
-
i0)(1
-
ia
)
(39)
The above set of coupled equations for the -coefficients are then solved analytically, and
the finite terms are shown below; and the other terms 00, 30, 1px, 1py , 2px and 2py are equal to zero.
10(EF ) = 2ni(|T13||T A| + |T 0||T03| - 2i|T A||T S|)(00 + aa)
<EFBFBD> 1 - ni |T 0|2 + |T03|2 + 2|T S|2 - 2|T A|2 - 2|T13|2 (00 + aa)
-1
-2ni |T13|2 - |T03|2 (0a + a0)
= asym,1 + 30 - iasym,3 + O( )
t
EF
(40a)
12
20(EF ) = 1 - ni |T 0|2 + |T03|2 - 2|T13|2 00 - ni |T S|2 + |T A|2 aa + i ni|T13||T S|(0a + a0)
<EFBFBD> 1 - ni |T 0|2 + |T03|2 + 2|T S|2 - 2|T A|2 - 2|T13|2 (00 + aa)
-1
-2ni |T13|2 - |T03|2 (0a + a0)
= 0 + a + i 31,s + O( )
t
t
EF
(40b)
0px(EF ) = ni |T A|2| + |T S|2 (0a + a0) + 2|T 0||T13|(0a - a0) <20> 2ni(|T13||T A| + |T 0||T03| - 2i|T A||T S|)(00 + aa)
<EFBFBD> 1 - ni |T 0|2 + |T03|2 + 2|T S|2 - 2|T A|2 - 2|T13|2 (00 + aa)
-1
-2ni |T13|2 - |T03|2 (0a + a0)
+ni 2|T A||T13|(00 + aa) - 2i|T A||T S|(0a + a0)
<EFBFBD> 1 - ni |T 0|2 + |T03|2 - 2|T13|2 00 - ni |T S|2 + |T A|2 aa + i ni|T13||T S|(0a + a0)
<EFBFBD> 1 - ni |T 0|2 + |T03|2 + 2|T S|2 - 2|T A|2 - 2|T13|2 (00 + aa)
-1
-2ni |T13|2 - |T03|2 (0a + a0)
=
asym,1
- iasym,3 t
-
a(30
+ asym,1 - iasym,3) 4t(0 + a)
+ O(
EF
)
(40c)
0py (EF ) = 2i ni |T13||T A|(00 + aa) + |T A||T S|(0a + a0) <20> 2ni(|T13||T A| + |T 0||T03| - 2i|T A||T S|)(00 + aa)
<EFBFBD> 1 - ni |T 0|2 + |T03|2 + 2|T S|2 - 2|T A|2 - 2|T13|2 (00 + aa)
-1
-2ni |T13|2 - |T03|2 (0a + a0)
-i ni |T S|2 + |T A|2 (0a + a0)
<EFBFBD> 1 - ni |T 0|2 + |T03|2 - 2|T13|2 00 - ni |T S|2 + |T A|2 aa + i ni|T13||T S|(0a + a0)
13
<0C> 1 - ni |T 0|2 + |T03|2 + 2|T S|2 - 2|T A|2 - 2|T13|2 (00 + aa)
-1
-2ni |T13|2 - |T03|2 (0a + a0)
=
i 4
a t
+
i 2
(asym,1
-
iasym,3)(asym,1 - t(0 + a)
iasym,3
+
30)
+
O( EF
)
(40d)
3px(EF ) = 2ni |T03||T S| - i|T 0||T A| 00 <20> 2ni(|T13||T A| + |T 0||T03| - 2i|T A||T S|)(00 + aa)
<EFBFBD> 1 - ni |T 0|2 + |T03|2 + 2|T S|2 - 2|T A|2 - 2|T13|2 (00 + aa)
-1
-2ni |T13|2 - |T03|2 (0a + a0)
-2ni (|T 0||T S| + i|T A||T03|)00 - i|T 0||T13|(0a + a0)
<EFBFBD> 1 - ni |T 0|2 + |T03|2 - 2|T13|2 00 - ni |T S|2 + |T A|2 aa + i ni|T13||T S|(0a + a0)
<EFBFBD> 1 - ni |T 0|2 + |T03|2 + 2|T S|2 - 2|T A|2 - 2|T13|2 (00 + aa)
-1
-2ni |T13|2 - |T03|2 (0a + a0)
=
- s t
-
i 31
+ asym,2 t
+
3sasym,1 2t(0 + a)
(40e)
3py (EF ) = ni |T 0||T A| + i|T03||T S| 00 <20> 1 - ni |T 0|2 + |T03|2 - 2|T13|2 00 - ni |T S|2 + |T A|2 aa + i ni|T13||T S|(0a + a0)
<EFBFBD> 1 - ni |T 0|2 + |T03|2 + 2|T S|2 - 2|T A|2 - 2|T13|2 (00 + aa)
-1
-2ni |T13|2 - |T03|2 (0a + a0)
-ni (|T03||T A| + i|T 0||T S|)00 + |T 0||T13|(0a + a0) <20> 2ni(|T13||T A| + |T 0||T03| - 2i|T A||T S|)(00 + aa)
<EFBFBD> 1 - ni |T 0|2 + |T03|2 + 2|T S|2 - 2|T A|2 - 2|T13|2 (00 + aa)
14
-1
-2ni |T13|2 - |T03|2 (0a + a0)
=
-
3 4t
-
i
3s 2t
+
1 2
(30
+
asym,1)(31 + asym,2) t(0 + a)
-
sasym,3
-
i 2
s(30
+
asym,1) + asym,3(31 t(0 + a)
+
asym,2)
Hence, using the results of ij(EF ) listed above, the scattering vertex is,
y
(|k|
=
kF
,
,
EF
)
=
10(EF
)1
+
20(EF
)
y
+
(0px (EF
)1
+
3px (EF
)
z)
cos
+i 0py (EF )1 + 3py (EF ) z sin
(40f ) (41)
Since 20 is the scattering vertex channel for longitudinal electrical conductivity, we have
defined
a
transport
scattering
rate
t
=
(
1 2
0
+
a
-
2t
),
in
terms
of
0,
a,
and
an
additional
transport contribution, t = 2niN0(EF )(|T13|2 - |T03|2). Since t V04V12N0(EF )5, it is
much weaker than 0 V02N0(EF ) and a V02V12N0(EF )3, and we do not display t in
the main paper, but instead, display it here for completeness.
In addition, there are spin flip scattering rates arising from |T A| and |T S|, s =
niN0 2
(EF
)
|T
0||T
S
|,
asym,1
=
2niN0(EF )|T13||T A|,
asym,2
=
niN0 2
(EF
)
|T03||T
A|,
asym,3
=
niN0 2
(EF
)
|T
S
||T
A|,
30
=
ni
N0(EF 2
)
|T03
||T
0|,
31
=
niN0 2
(EF
)
|T13
||T
0|,
3s
=
niN0 2
(EF
)
|T03||T
S
|
and
31,s
=
ni
N0(EF 2
)
|T13||T
S
|,
which
are
proportional
to
TS
and
T A,
the
symmetric
and
asymmetric component of the T -matrix, as well as the z components of the T -matrix, T03
and T13.
V. LONGITUDINAL CHARGE TRANSPORT AND SHE DC CONDUCTIVITIES
We calculate the longitudinal charge conductivity, the Rashba-Edelstein effect, and the
spin torque contribution to the SHE here. The retarded correlation functions for the spin-
torque current contribution to the SHE (xzy,(2)(k, )), the Rashba-Edelstein effect (yi (k, )),
and the charge current conductivity (yy(k, )) are shown below, and the DC conductivities
are all given by first taking the limit of lim k 0, then taking the DC limit of lim 0,
(DC)
=
-lim lim 0k0
I
m[
(k,)
].
yy(k = 0, ) = lim
k0
evF c
2
d - 2i
d2p (2)2
Tr
(A)
G (p,
)y
(R)
G (p
+
k,
y
)
(p,
p
+
k,
)
15
<0C> (nF () - nF ( + ))
(42)
xz,y(2)(k = 0, ) = xzy,(2a)(k = 0, ) + xz,y(2b)(k = 0, )
(43)
xzy,(2a)(k
=
0,
)
=
lim 2ievF2 k0 c
d - 2i
d2p (2)2
Tr
(A)
G (p,
)xG(R)(p
+
k,
y
)
(p,
p
+
k,
)
<EFBFBD>
py + px
ky 2
(nF () - nF ( + ))
xz,y(2b)(k
=
0,
)
=
-lim 2ievF2 k0 c
d - 2i
d2p (2)2
Tr
(A)
G
(p,
)y
(R)
G
(p
+
k,
y
)
(p,
p
+
k,
)
<EFBFBD>
px + px
kx 2
(nF ()
-
nF ( +
))
yi (k
=
0,
)
=
lim evF k0 c
d - 2i
d2p (2)2
Tr
(A)
G
(p,
)i
(R)
G
(p
+
k,
y
) (p,
p
+
k,
)
<EFBFBD> (nF ( + ) - nF ())
(44)
We have specialized to the case of a charge current along y^ in the expression for the Rashba-
Edelstein effect. For the SHE Kubo formula, we have to Taylor expand the Green's function
(R)
G (p
+
k,
)
=
(R)
G (p,
)
+
(R)
ki
dG
(p,) dpi
,
which
is
shown
in
detail
below.
(R)
(R)
(R)
dG (p, ) = G (p, ) p + G (p, )
dpx
p px
px
(45a)
(R)
G (p, ) p = p px
dg0 dp 1
+
dg3 dp
z
+
dga (cos
dp
px
+
sin
p y )
+
dgb (sin
dp
px
-
cos
p y )
cos p
(R)
G (p, ) = px
ga(- sin px + cos py) + gb(cos px + sin py)
- sin p p
(R)
(R)
(R)
dG (p, ) = G (p, ) p + G (p, )
(45b)
dpy
p py
py
(R)
G (p, ) p = p py
dg0 dp 1
+
dg3 dp
z
+
dga (cos dp
px
+
sin
p y )
+
dgb (sin dp
px
-
cos
p y )
sin p
(R)
G (p, ) = py
ga(- sin px + cos py) + gb(cos px + sin py)
cos p p
Following the same approximation of an average -matrix near EF , the spin current-current
correlation function is then given in terms of the -coefficients, and the spin-resolved density
of
states
ij(EF ),
as
well
as
the
quantity
involving
the
integral
of
(A)
G
(k,
)
(R)
dG
(k,)
,
dk
which
we term ij(),
ij()
-
dp 2
p2
dgi,(R)(p, dp
) gejf,(fA)(p,
)
(46a)
16
00() =
dp 2
vF
p2
2(-vF p + ia)( - i0) (p)2 + (p)2
+
4(vF
p(p)
-
a(p))((p) + i(p))( ((p)2 + (p)2)2
-
i0)
((p) - i(p))( + i0) (p)2 + (p)2
=
1 2vF2
i2 8(0 + a)2
-
16(0 +
a)
+
i0(02 + a2) 4(02 - a2)2
+
i(202 - 16(0
0a + + a)2
a2)
-
1 8
+
O(
)
(46b)
aa() =
dp 2
vF
p2
2(-vF p + ia)(vF p + ia) (p)2 + (p)2
+
4(vF
p(p)
-
a(p))((p) + i(p))(vF ((p)2 + (p)2)2
p
+
ia
)
((p) - i(p))(vF p - ia) (p)2 + (p)2
=
1 2vF2
i2 8(0 + a)2
-
16(0 +
a)
+
i0(02 + a2) 4(02 - a2)2
-
i(0 - 3a)a 16(0 + a)2
-
04 + 602a2 + a4 8(02 - a2)2
+
O( )
(46c)
aa()
-
00()
=
1 2vF2
-
02a2 (02 - a2)2
-
i
(0 8(0
- a) + a)
(46d)
0a() =
dp 2
vF
p2
2(-vF p + ia)( - i0) (p)2 + (p)2
+
4(vF
p(p)
-
a(p))((p) + i(p))( ((p)2 + (p)2)2
-
i0)
((p) - i(p))(vF p - ia) (p)2 + (p)2
=
1 2vF2
2 i 8(0 + a)2
+
16(0 +
a)
+
i
a(02 + a2) 4(02 - a2)2
-i
(02 - 0a + 2a2) 16 (0 + a)2
+
03a 2(02 - a2)2
+
O(
)
(46e)
a0() =
dp 2
vF
p2
2(-vF p + ia)(vF p + ia) (p)2 + (p)2
+
4(vF
p(p)
-
a(p))((p) + i(p))(vF ((p)2 + (p)2)2
p
+
ia
)
((p) - i(p))( + i0) (p)2 + (p)2
=
1 2vF2
2 i 8(0 + a)2
+
0a2 2(02 - a2)2
-
3 16(0 +
a)
+
ia(502 + a2) 4(02 - a2)2
-
i0(0 + 5a) 16 (0 + a)2
-
303a + 20a3) 2(02 - a2)2
+
O(
)
(46f )
0a() - a0()
=
1 2vF2
4(0 +
a)
-
i02a (02 - a2)2
+
0a(202 + a2) (02 - a2)2
+
O(
)
(46g)
17
Note that = + <20> is the energy measured from EF ; hence, the DC conductivities
will depend on ij(EF ). We now re-write the SHE correlation function as a sum of several
terms, xzy,(2)(k, ) = xzy,(2a)(k, ) + xz,y(2b)(k, ), where xz,y(2a)(k, ) and xz,y(2b)(k, ) are the
kyx and kxy terms respectively.
(R)
It is then necessary to Taylor expand G (p + k, ) =
(R)
G (p,
)
+
ki
(R)
dG
(p,) dpi
,
and
z,(2a1)(k, )
is
the
zeroth-order
term,
while
z,(2a2)(k, )
and
(R)
(R)
z,(2a3)(k,
)
are
the
kx dG
(p,) dpx
and
ky
dG
(p,) dpy
terms
respectively;
thus,
giving
xz,y(2a)(k,
)
=
xzy,(2a1)(k = 0, ) + xz,y(2a2)(k = 0, ) + xzy,(2a3)(k = 0, ) and xz,y(2b)(k, ) = xz,y(2b1)(k =
(R)
0, ) + xz,y(2b2)(k
=
0, ) + xz,y(2b3)(k
=
0, ).
Finally,
we
make
use
of
the
chain
rule
dG
(p,) dpi
=
(R)
(R)
dG
(p,) p dp pi
+
dG
(p,)
pi
,
which
give
xz,y(2a1)(k
=
0, )
=
xzy,(2a1P 1)(k, ) +
xz,y(2a1P 2)(k, )
(R)
respectively,
with
xzy,(2a1P 1)(k, )
and
xzy,(2a1P 2)(k, )
being
proportional
to
the
dG
(p,) p dp pi
(R)
and
dG (p,) pi
terms respectively.
A similar procedure is carried out for the other terms,
and we have symmetrized the expressions for xzy,(2a)(k, ) and xz,y(2b)(k, ) by doing a shift
of
variable
py +
ky 2
py
and
px +
kx 2
px
respectively.
The
results
are
shown
below.
xzy,(2a)(k,
)
=
lim
k0
2ievF2 c
d 2i
p
(nF
()
-
nF
(
+
))
py kx
<EFBFBD> Tr
(A)
G (p
-
k 2,
)
x
R)
G (p
+
k 2,
)
(y)
(p,
)
= xzy,(2a1)(k = 0, ) + xzy,(2a2)(k = 0, ) + xzy,(2a3)(k = 0, )
xzy,(2a1)(k
=
0,
)
=
lim 2ievF2 k0 c
1 kx
d 2i
(nF () - nF ( + ))
p
<EFBFBD> Tr
(A)
G
(p,
)xGR)(p,
(y)
) (p,
)
p sin
=0
(47a) (47b)
xzy,(2a2)(k
=
0,
)
=
lim 2ievF2 k0 c
kx kx
d 2i
(nF
()
-
nF
(
+
))
p
sin 2
p
(R)
(A)
<EFBFBD>
Tr
(A)
G
(p,
)x
G
(p,
)
(y)
(p,
)
- Tr
G
(p,
)
xG(R)
(p,
(y)
)
(p,
)
px
px
= lim 2ievF2 kx k0 c kx
d 2i
(nF
()
-
nF
(
+
))
p
sin 2
p
(R)
(R)
<EFBFBD>
Tr
(A)
G
(p,
)x
G
(p, ) p p px
+
G
(p,
)
px
(y)
(p,
)
18
(A)
(A)
- Tr
G
(p, ) p p px
+
G
(p,
)
px
xG(R)
(p,
(y)
) (p,
)
= xzy,(2a2P 1)(k = 0, ) + xzy,(2a2P 2)(k = 0, )
(47c)
xzy,(2a2P 1)(k
=
0,
)
=
lim 2ievF2 k0 c
kx kx
d 2i
(nF
()
-
nF
(
+
))
p
sin 2
p
(R)
<EFBFBD>
Tr
(A)
G
(p,
)x
G
(p, p
)
p px
(y)
(p,
)
(A)
- Tr
G
(p, p
)
p px
xG(R)(p,
(y)
) (p,
)
= lim 2ievF2 kx k0 c kx
d 2i
p
(nF
()
-
nF
(
+
))
p
sin 2
p px
(R)
(A)
<EFBFBD>
Tr
(A)
G
(p,
)x
G
(p,
)
(y)
(p,
)
- Tr
G
(p,
)
xG(R)
(p,
(y)
)
(p,
)
p
p
= 2ievF2 c
d 2i
(nF
()
-
nF
(
+
))
1 2
<EFBFBD>
1 4
2s() (2aa() - 2(aa()))
+0py () a0() + 0a() - (a0()) - (0a())
+i3px() a0() - 0a() + (a0()) - (0a())
+ O( ) EF
(47d)
xzy,(2a2P 2)(k
=
0,
)
=
lim 2ievF2 p0 c
kx kx
d 2i
p
(nF
()
-
nF
(
+
))
p
sin 2
px
(R)
(A)
<EFBFBD>
Tr
(A)
G
(p,
)x
G
(p,
)
(y)
(p,
)
- Tr
G
(p,
)
xG(R)
(p,
(y)
)
(p,
)
= 2ievF2 c
d 2i
(nF
()
-
nF
(
+
))
1 2
<EFBFBD>
1 4
30py
+3px ()
() i0a
0a ()
() + a0() + ia0()
+ +
i3b()
O( EF
+ )
i
b3
()
(47e)
xzy,(2a3)(k
=
0,
)
=
lim 2ievF2 k0 c
ky kx
d 2i
(nF
()
-
nF
(
+
))
p
sin 2
p
(R)
(A)
<EFBFBD>
Tr
(A)
G
(p,
)x
G
(p,
)
(y)
(p,
)
- Tr
G
(p,
)
xG(R)
(p,
(y)
)
(p,
)
py
py
19
= lim 2ievF2 ky k0 c kx
d 2i
(nF
()
-
nF
(
+
))
p
sin 2
p
(R)
(R)
<EFBFBD>
Tr
(A)
G
(p,
)x
G
(p, ) p p py
+
G
(p,
)
py
(y)
(p,
)
(A)
(A)
- Tr
G
(p, ) p p py
+
G
(p,
)
py
xG(R)
(p,
(y)
)
(p,
)
= xzy,(2a3P 1)(k = 0, ) + xzy,(2a3P 2)(k = 0, )
(47f )
xzy,(2a3P 1)(k
=
0,
)
=
lim 2ievF2 p0 c
ky kx
d 2i
p
(nF
()
-
nF
(
+
))
p
sin 2
p py
(R)
(A)
<EFBFBD>
Tr
(A)
G
(p,
)x
G
(p,
)
(y)
(p,
)
- Tr
G
(p,
)
xG(R)
(p,
(y)
)
(p,
)
p
p
= 2ievF2 c
d 2i
(nF
()
-
nF
(
+
))
1 2
<EFBFBD>
1 4
1s ()
400() - 4(00()) - 2aa() + 2(aa())
+0px() a0() + 0a() - (a0()) + (0a())
+3py () -3a0() + 30a() - 3(a0()) + 3(0a())
+ O( ) EF
(47g)
xzy,(2a3P 2)(k
=
0,
)
=
lim 2ievF2 p0 c
ky kx
d 2i
p
(nF
()
-
nF
(
+
))
p
sin 2
py
(R)
(A)
<EFBFBD>
Tr
(A)
G
(p,
)x
G
(p,
)
(y)
(p,
)
- Tr
G
(p,
)
xG(R)
(p,
(y)
)
(p,
)
= 2ievF2 c
d 2i
(nF
()
-
nF
(
+
))
1 2
<EFBFBD>
1 4
0px ()
0a() - a0()
- 3py ()
0a() + a0()
+ O( )
EF
(47h)
xz,y(2b)(k,
)
=
lim
k0
-
2ievF2 c
d 2i
p
(nF
()
-
nF
(
+
))
px kx
<EFBFBD> Tr
(A)
G (p
-
k 2
,
)
y
R)
G (p
+
k 2
,
)
(y)
(p,
)
= xz,y(2b1)(k = 0, ) + xz,y(2b2)(k = 0, ) + xz,y(2b3)(k = 0, )
(47i)
xz,y(2b1)(k
=
0, )
=
lim
k0
-
2ievF2 c
1 kx
d 2i
(nF () - nF ( + ))
p
<EFBFBD> Tr
(A)
G
(p,
)y
R)
G
(p,
(y)
)
(p,
)
p cos
20
=0
(47j)
xz,y(2b2)(k
=
0, )
=
lim
k0
-
2ievF2 c
kx kx
d 2i
(nF
()
-
nF
(
+
))
p
cos 2
p
(R)
(A)
<EFBFBD>
Tr
(A)
G
(p,
)y
G
(p,
)
(y)
(p,
)
- Tr
G
(p,
)
y
(R)
G (p,
(y)
) (p,
)
px
px
= lim - 2ievF2 kx
k0
c kx
d 2i
(nF
()
-
nF
(
+
))
p
cos 2
p
(R)
(R)
<EFBFBD>
Tr
(A)
G
(p,
)y
G
(p, ) p p px
+
G
(p,
)
px
(y)
(p,
)
(A)
(A)
- Tr
G
(p, ) p p px
+
G
(p,
)
px
y
(R)
G (p,
(y)
) (p,
)
= xz,y(2b2P 1)(k = 0, ) + xz,y(2b2P 2)(k = 0, )
(47k)
xz,y(2b2P 1)(k
=
0, )
=
lim
p0
-
2ievF2 c
kx kx
d 2i
p
(nF
()
-
nF
(
+
))
p
cos 2
p px
(R)
(A)
<EFBFBD>
Tr
(A)
G
(p,
)y
G
(p,
)
(y)
(p,
)
- Tr
G
(p,
)
y
(R)
G (p,
(y)
) (p,
)
p
p
= - 2ievF2 c
d 2i
(nF
()
-
nF
(
+
))
1 2
<EFBFBD>
1 4
2s ()
400() - 4(00()) - 2aa() + 2(aa())
+0py () a0() + 0a() - (a0()) - (0a())
+i3px() 30a() - 3a0() + 3(0a()) - 3(a0())
+ O( ) EF
(47l)
xz,y(2b2P 2)(k
=
0, )
=
lim
p0
-
2ievF2 c
kx kx
d 2i
p
(nF
()
-
nF
(
+
))
p
cos 2
px
(R)
(A)
<EFBFBD>
Tr
(A)
G
(p,
)y
G
(p,
)
(y)
(p,
)
- Tr
G
(p,
)
y
(R)
G (p,
(y)
) (p,
)
= - 2ievF2 c
d 2i
(nF
()
-
nF
(
+
))
1 2
<EFBFBD>
1 4
0py ()
0a() - a0()
- 3px()
i0a() + ia0()
+ O( ) EF
(47m)
xz,y(2b3)(k = 0, ) = xz,y(2b3P 1)(k = 0, ) + xz,y(2b3P 2)(k = 0, )
(47n)
21
xz,y(2b3P 1)(k
=
0, )
=
lim
k0
-
2ievF2 c
ky kx
d 2i
p
(nF
()
-
nF
(
+
))
p
cos 2
p py
(R)
(A)
<EFBFBD>
Tr
(A)
G
(p,
)y
G
(p,
)
(y)
(p,
)
- Tr
G
(p,
)
y
(R)
G (p,
(y)
) (p,
)
p
p
= - 2ievF2 c
d 2i
(nF
()
-
nF
(
+
))
1 2
<EFBFBD>
1 4
1s() (2aa() - 2(aa()))
+0px() a0() + 0a() - (a0()) - (0a())
+3py () a0() - 0a() + (a0()) - (0a())
+ O( ) EF
(47o)
xz,y(2b3P 2)(k
=
0, )
=
lim
k0
-
2ievF2 c
ky kx
d 2i
p
(nF
()
-
nF
(
+
))
p
sin 2
py
(R)
(A)
<EFBFBD>
Tr
(A)
G
(p,
)x
G
(p,
)
(y)
(p,
)
- Tr
G
(p,
)
xG(R)
(p,
(y)
)
(p,
)
= - 2ievF2 c
d 2i
(nF
()
-
nF
(
+
))
1 2
<EFBFBD>
1 4
30px ()
a0() - 0a()
- 3py ()
0a() + a0()
+ O( )
EF
(47p)
Therefore, summing up all the different contributions, we finally obtain the SHE correlation function,
z,(2)(p = 0, ) = 2ievF2 c
d 2i
(nF
()
-
nF
(
+
))
<EFBFBD> 21 0px() 0a() - a0() + 0py () a0() - 0a()
1s() 00() - aa() - (00()) + (aa()) +2s() aa() - 00() - (aa()) + (00()) +3px() ia0() - i0a() + i(a0()) - i(0a())
+3py () 0a() - a0() + (0a()) - (a0())
+
O(
)
(48)
Using
the
results
for
ij ( )
and
ij ( )
from
above,
where
0a() - a0()
=
-
i 2vF2
,
Im[aa() - 00()]
=
-1 2vF2
, (0-a)
8(0 +a )
and
Re[0a() - a0()]
=
1 2vF2
4(0 +a )
=
, N0()
4(0 +a )
we
see
that
the
main
O(
1
)
contributions
come
from
the
3px ( )
scattering
channel.
The uniform DC longitudinal charge and spin-Hall conductivity are given by yy =
-lim0
lim
k0
Im
y y (k ,)
,
xzy
=
-lim0
lim
k0
I
m
xz y (k,)
,
and keeping
only
the
O(
1
)
terms,
they
22
are,
yy
=
1 2
(evF )2 Re
220(EF ) 00(EF )
=
(evF )2
N0(EF ) 2t
+
O
EF
(49)
xzy,(2)
=
h<EFBFBD>evF2 Im
i3px(EF )
Re[0a(EF ) - a0(EF )]
=
-h<>evF2
N0(EF ) 2t 0
s +
a
+
O
EF
(50)
yy
=
h<EFBFBD>evF 2
Re
22s(EF )00(EF )
=
h<EFBFBD>evF
N0(EF 2t
)
+
O
EF
(51)
Hence, we see that the SHE is driven by scattering between the s and p-wave electrons
due to the symmetric spin-flip T S term, which occurs at 3rd-order in perturbation. Eq. 40e,
3px (EF )
=
- s t
-
i 31+asym,2 t
+
, 3s asy m,1
2t (0 +a )
shows
that
the
asymmetric
spin-flip
term
TA
also
contributes but as a sub-leading term, .
[1] J. Rammer, Quantum Transport Theory, Frontiers in Physics (Book 99) (Westview Press, 2004). [2] G. D. Mahan, Many-Particle Physics, 3rd ed., Physics of Solids and Liquids (Springer US,
2000).
23