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[20­25], 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, 26­30].

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

+

·

(Js

+

P )

=

0,

with

an

additional

spin-torque

density term,  · 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[35­37], 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 ·  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^ × k ·  case via rotation of the momentum by 90. Choosing the chemical potential µ

to lie in the upper helical band, we obtain the following

Hamiltonian as,

H = H0 + Himp

(1)

H0 =

ck,vF k · , ck, - µ 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 ± 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

+

µ

1 -

vF k

·



(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 +µ
(in+µ)2-vF2 k2

and

g0a(k, in)

=

. vF k
(in+µ)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± = n,m Vn 1 - Vn±1 g00(EF )

1 - Vn g00(EF )

× 1 - Vn±1 g00(EF )

- VnVn±1

g01(EF )

2

-1
(4a)

Tn±m

=

n1,m 2

VnVn1 g01(EF )

1 - Vn1 g00(EF )

× 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 ± Tn3m and T ± = Tn1m ± 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 ± 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 ·  -  (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, ) =  + µ - vF k ·  - 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))( + µ 2(k) + 2(k)

-

i0)

(8a)

geaff (k, )

=

((k)

+ i(k))(vF |k| + ia) 2(k) + 2(k)

(8b)

with (k) = ( + µ)2 - vF2 |k|2 - 02 + a2 + b2 - 32, and (k) = 2 ( + µ)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

× (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[35­37]

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).

5

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[35] J.-i. Inoue, G. E. W. Bauer, and L. W. Molenkamp, Phys. Rev. B 67, 033104 (2003).

[36] E. G. Mishchenko, A. V. Shytov, and B. I. Halperin, Phys. Rev. Lett. 93, 226602 (2004).

[37] N. Sugimoto, S. Onoda, S. Murakami, and N. Nagaosa, Phys. Rev. B 73, 113305 (2006).
[38] J. Rammer, Quantum Transport Theory, Frontiers in Physics (Book 99) (Westview Press, 2004).

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 · 

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^ × k ·  case by rotating the momentum by 90. The chemical

potential µ is chosen to lie in the upper helical band, with the upper/ lower helical Weyl

fermions

being

±,k

=

1 2

(±

ck,

+

eik ck,),

and

the

Hamiltonian

is,

H = H0 + Himp

(1)

H0 =

ck,vF k

·

, ck,

-

µ

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 µ - vF k · 

(4)

= g00(k, in)1 + g01(k, in)(cos k x + sin k y)

g00(k, in)

=

(in

in + µ + µ)2 - vF2 k2

g01(k, in)

=

(in

vF k + µ)2 -

vF2 k2

, where

T (k, k) 

Tnimeink e-imk i

nm

= V (k, k) +
n1n2n3

dk1 2

k1dk1 2

Vn1

ein1

(k

-k1

)

× g00(k1, in)1 + g01(k1, in)(cos k1 x + sin k1 y)

×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 )

=

±

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±



Tn0m

± Tn3m,

Tn±m



1 2

(Tn1m

± 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 ± Tn3m and T ± = Tn1m ± 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 ± 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 µ are ignored here, as vF and µ 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, ) =  + µ - vF k ·  - 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))( + µ - 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) = ( + µ)2 - vF2 |k|2 - 02 + a2 + b2 - 32, (k) = 2 ( + µ)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 · 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

×H int(1)H int(2)ck2,(0)y ck+k2,

= - evF mc

1 

z Gµ1 (p + k, i1 + in)Vµ1µ2 (p + k, p + q)

p,q

i1

×Gµ2(p + q, i1 + in)y Gµ3(p + q - k, i1)

×Vµ3µ4(p + q - k, p)Gµ4(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, Gµ1(p + k, i1 + in) and Gµ2(p + q, i1 + in) undergo a change of momentum and spin upon scattering off Vµ1µ2(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 Vµ1µ2(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

×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
× (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

× 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| - µ = (p) and z + in = vF |p| - µ = (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

(

+

))

× 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 ( + ))

× 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 ())

× 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

())

× 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
× (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
× (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)

×

qdq 2

(R)
G (p

+

q,

y
)

(p

+

q,

(A)
)G (p

+

q

,

)

11

(A)
×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

×

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 = ±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  + µ   here, thereby absorbing the factors

of µ 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)

× 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)

× 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) × 2ni(|T13||T A| + |T 0||T03| - 2i|T A||T S|)(00 + aa)
× 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)
× 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)

× 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) × 2ni(|T13||T A| + |T 0||T03| - 2i|T A||T S|)(00 + aa)
× 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)
× 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

× 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 × 2ni(|T13||T A| + |T 0||T03| - 2i|T A||T S|)(00 + aa)
× 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)
× 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)

× 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 × 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)

× 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) × 2ni(|T13||T A| + |T 0||T03| - 2i|T A||T S|)(00 + aa)
× 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

× (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,

)

×

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,

)

×

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,

)

× (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  =  + µ 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

× 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

× 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)

×

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)



×

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)



×

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)

×

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

×

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)

×

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

×

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)

×

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)



×

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)

×

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

×

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)

×

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

×

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

× 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

× 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)

×

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)



×

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)

×

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

×

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)

×

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

×

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)

×

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

×

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)

×

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

×

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

(

+

))

× 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¯evF2 Im 

i3px(EF )

Re[0a(EF ) - a0(EF )]

=

-h¯evF2

N0(EF ) 2t 0

s +

a

+

O

 EF

(50)

yy

=

h¯evF 2

Re

22s(EF )00(EF )

=

h¯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

