In this section we describe the weak localization corrections of disordered electron systems in qua-siclassical treatment of Eq. (3.21). We start with the Dyson equation, which is a summary of the Feynman-Dyson theory in a particularly compact form. In that way, the exact Green function consists of two terms, the unperturbed Green function G0 and all connected terms with the potential described by the so-called self energy Σ. The corresponding analytical expression of the Dyson equation is given by
Gαβ(x1, x2) = G0αβ(x1, x2) + Z
dxdx0GoαΓ(x1, x)ΣΓµ(x, x0)Gµβ(x0, x2), (3.56) where the Green function G is determined by the self energy Σ, and also Σ determined by G. According to the standard model of disorder potential as illustrated in Fig. 3.2, the impurity potential V is taken as the Gaussian random variable with zero mean and the variance given by
hV (r)i = 0, hV (r)V (r0)i = niv02δ(r − r0) (3.57)
U(x) U(x’)
iε x x’
Figure 3.2: Lowest order self-energy in the Born approximation for before (top left) and after (top right) averaging over the impurity distributions. The dashed line denotes the impurity average. The bottom panel is illustrating the sequence of the rainbow diagrams selected by the self-consistent solution for the Green function [18].
where ni and vi are the impurity density and scattering amplitude. Since the interaction is invariant under transformation and the system is spatially uniform, we can find an algebraic expression of the Green function in the momentum-energy space
G(p, ) = 1
ω − (p) − Σ(p, ω), (3.58)
The Green functions, Go and G, are all diagonal in the matrix indices in the absence of the SOC. The simple model potential given in Eq. (3.57) shows that only an even number of cross insertions are different from zero. Hence, the lowest order impurity-average, the so-called Born approximation represented by Fig. 3.2, plays an important role in the effective self-energy in the impurity technique. Let us consider the retarded self-energy shown in Fig. 3.2
ΣR1(p, ω) = niv02X
p0
G0(p0, ω), (3.59)
To evaluate the above equation we have to pass from momentum to energy variable X
p0
(· · · ) → N0
Z ∞
−µ
d(p0)(· · · ) → N0
Z ∞
−∞
d(p0)(· · · ) (3.60)
where N0 represents the density of states in the absence of perturbation. In the above equation, we assumed that the biggest energy scale is the Fermi energy. In the large values of p0, the real part of the integral divergence over p0, its values do not depend on ω and p. In fact, this is the consequences of the simple model taken from the scattering potential. A more realistic model cures this problem by introducing a cutoff frequency for the scattering process with a large momentum. In the following, we will consider the imaginary part of the self-energy, since the real part has been absorbed in just a shift of the chemical potential. By performing the integral, the imaginary part of the self-energy in the limit of the Born approximation reads
ΣR1(p, ω) ≡ − i
2τ0, (3.61)
where τ0= 2πN0niv20represents the elastic quasiparticle relaxation time. To proceed in the perturbative expansion, one should consider the above result into the Green function and compute it again for the
next iteration for Σ2. This leads to the replacement of the imaginary part of the integral G0. But as the integral does not depend on its modulus, this yields a self-consistent solution of Dyson equation for the Green function, as illustrated in Fig. 3.2. Hence, we can consider the Σ1 as a total self-energy Σ.
To evaluate the integral over the momentum, we assumed that the variation of the integrand, set up by the position of the pole, is much smaller than the lower limit of the integral −µ. Here we confirm this expression of the small expansion parameter
1
τ << µ ≈ F → pFl >> 1, (3.62) where l = vFτ is the mean free path. As shown in Eq. (3.58) for the Dyson equation, the disorder-dressed Green functions becomes
GR/A(p, ω) = 1
ω − (p) ± i/2τ0. (3.63)
However, these results are obtained in the lowest order Born approximation, but we can go beyond that order. At the end of the chapter, we will take care of the side-jump and skew scattering corrections, when the extrinsic SOC is also present. What we have learned up to now is that the disorder effects can be taken into account via the inclusion of the self-energies. Now we will obtain an explicit expression for the collision integral describing the scattering from impurities. We recall the Boltzmann collision integral in Eq. (3.52),
I ≡ Z d
2πiIK = − Z d
2π
Σ, ˇˇ Gp
K
, (3.64)
Now we have to transform it to the locally covariant formalism according to the transformation of Eq. (3.44). This procedure is the same with the transformation of the kinetic equation obtained in the previous section. In particular, the covariant transformation of the Keldysh collision integral gives us
UΓ(x, x1)Σ(xˇ 1, x3)⊗, ˇG(x3, x2) UΓ(x2, x) =hΣ(xˇ˜ 1, x3)⊗,G(xˇ˜ 3, x2)i
(3.65) after using the unitarity of the Wilson line by inserting
UΓ(x3, x)UΓ(x, x3) = 1
between the self-energy and the Green function. The locally covariant self-energy according to the shift of Eq. (3.44) and Eq. (3.45) yields
ˇ˜
Σ = niv20X
p0
Gˇ˜p0+1
2{Aµ(∂p0,µ− ∂p,µ),Gˇ˜p0}
= niv20X
p0
ˇ˜
Gp0, (3.66)
where the derivative with respect to , cancels in the two terms. The derivative with respect to p, vanishes and another one with respect to p0 can be neglected because it is constant after integrating.
Hence, the locally covariant self-energy has the same functional form as the original self-energy. The Keldysh component of the collision integral has the form
I˜K = −ihΣ,ˇ˜ Gˇ˜iK
(3.67)
= −iniv20X
p0
( ˜GRp0− ˜GAp0) ˜GKp − ( ˜GRp− ˜GAp) ˜GKp0
By using Eqs. (3.47-3.48) for the locally covariant Green functions, the Boltzmann collision integral for the impurity scattering process yields
I[f ] = −2πniv20X
p0
δ(p− p0)(f (p, r, t) − f (p0, r, t)). (3.68)
Now we are able to obtain the solution of the Boltzmann Eq. (3.52) in the diffusive approximation.
Notice that, by taking integration over the momentum p, the collision integral vanishes and reproduces the continuity equation derived in Eq. (3.55) with the density and current defined in Eq. (3.54). In diffusive approximation, the distribution function can be expanded in the spherical harmonic as
f (p, r, t) ≡ hf i + 2ˆp · f , (3.69)
where the terms kept up to the order of p-wave symmetry and h· · · i denote the integration over the momentum direction. Hence, the collision integral becomes
I(f ) = −1
τ2ˆp · f . (3.70)
We multiply both side of Eq. (3.52) by ˆp = (cos(φ), sin(φ)), and then take the integration over angle φ.
We get
−1 τf = p
2m
∇˜rhf i − e
2h{ˆpE · ∇p, hf i}i − e
2mh{ˆp(p × B · ∇p), 2ˆp · f }i, (3.71) where the U (1) × SU (2) fields are given by
E = −∂tA − ∇rφ + ie[φ, A], (3.72)
Bi = 1
2ijkFjk. (3.73)
The first term, according to Eq. (3.54) for the current term, is the diffusive contribution including the covariant derivative with respect to the SU(2) gauge fields. Under uniform circumstance, this term differs from zero due to the covariant nature of derivatives. The second term is the usual drift contribution due to the external electric field, whereas the third one yields a Hall contribution. Then we get
f = −τ p 2m
∇˜rhf i + eτ
4 {E, ∂phf i}i + eτ
2m{B×, f }i (3.74)
where the gradient with respect to the momentum is replaced by ∇p= ˆp∂p− ˆφ∂φ/p with ˆφ = (−sinφ, cosφ).
As shown in the definitions of density and current in Eq. (3.54), we can write the expression for the number and spin components
n = Tr[ρ], J0= Tr[σaρ] (3.75)
Sa= 1
2Tr[σaρ], Ja= 1
2Tr[σaJ]. (3.76)
Let us start with the drift term as Jdrif t=X
p
p m
eτ
4 {E, ∂phf i} = eN0
Z
dpD(p)1
2{∂phf i, E} = −e
2{σ(µ), E}, (3.77) where the diffusion coefficient is given by
D(p) = τ
mp, with p= p2
2m (3.78)
and µ = ρ/N0is the spin-dependent chemical potential, and σ(µ) = N0D(µ). In equilibrium, the density current is defined by ρeq = N0F + N0Ψ. By expanding D(µ) around F, we have D0 ≈ D(F) and Da≈ τ Sa/(N0m), and therefor
σ(µ) = N0D(F)σ0+ τ
mSaσa. (3.79)
Hence, the particle and spin currents of drift term reads J0drif t= −eN0D0E0−e
2N0DaEa, Jadrif t= −e
4N0D0Ea−e
2N0DaE0. (3.80) By taking the integration over momentum, the diffusion term becomes
Jdif f = −N0
Z
dp∇˜rhf i = −1
2{D(µ), ˜∇rρ}. (3.81)
Then we get
J0dif f = −D0∇rn − 2Da[ ˜∇rS]a, Jadif f = −1
2Da∇rn − D0[ ˜∇rS]a. (3.82) By using the same procedure, we obtain the Hall terms as
J0Hall=eτ
mB0× J0+eτ
mBa× Ja, JaHall= eτ
mB0× Ja+ eτ
4mBa× J0. (3.83) Hence, the particle and spin currents in general, may be written as
J = −eN0D0E0−e
2N0DaEa− D0∇rn − 2Da[ ˜∇rS]a+eτ
mB0× J0+eτ
mBa× Ja (3.84) and
Ja= −e
4N0D0Ea−e
2N0DaE0−1
2Da∇rn − D0[ ˜∇rS]a+eτ
mB0× Ja+ eτ
4mBa× J0. (3.85) In Chapter 6, we will use the above equations, together with the continuity-like equation derived in Eq. (3.55) to analyze some aspects of the current-induced spin polarizations in a disordered Rashba-Dresselhaus model, where we will consider the effect of extrinsic SOC as well. Although the SU(2) gauge field is a powerful method, it is useful to show how the same results can be obtained in a different way. In the next section we will introduce the diagrammatic Kubo formula for various spin transport coefficients.