2.2. Quantum Transport Equation for Ferromagnetic Conductors
2.2.2. Keldysh Technique
The standard way to proceed in non-equilibrium many-particle physics is to set up the kinetic equation for the Keldysh Greens function. For the remaining part of this chapter, we set~= 1 and only restore it in the final results.
The further treatment is done in the Wigner representation which is obtained from usual spatio-temporal representation via the transformation
G(k, ,ˇ r, t) = Z
d3z Z
dτ e−ikz+iτG(rˇ + z 2, t+τ
2;r−z 2, t−τ
2) , (2.6) where now, operator multiplication has to be carried out by applying the Moyal product∗. Explicitly, this product is defined as
C(k, ,r, t) =A(k, ,r, t)∗B(k, ,r, t)≡A(k, ,r, t) ei2(←−∂r−→∂k−←−∂t−→∂−←−∂k−→∂r+←−∂−→∂t) B(k, ,r, t) , (2.7) where ←−
∂ and −→
∂ denote derivatives acting only to the left and right, respectively. The trans-formation (2.6) introduces center of mass coordinates r, t and Fourier transformed relative coordinates that are, respectively, k and . For a short summary of useful properties of the Wigner transformation, see Appendix A.
The Greens function ˇG is defined as expectation value of the field operators, time ordered along the Keldysh contour [50]. The ordering along back and forward time Keldysh contour gives rise to an additional 2×2 matrix structure (denoted by ˇ) which in an appropriate basis takes the convenient form [51]
Gˇ =
GˆR Gˆ<
0 GˆA
. (2.8)
GˆR and ˆGA are the retarded and advanced Greens functions, well known from equilibrium theory and carry information about the spectrum of the system, in particular, one obtains the spectral density simply from
Aˆ=i( ˆGR−GˆA) =−2Im ˆGR , (2.9)
which generally obeys the normalization condition Z ∞
−∞
d
2π A(k, ,ˆ r, t) = ˆ1.
The lesser component ˆG<(k, ,r, t) describes the occupation of states of the many particle system and is given in terms of electron field operators by
Gˆ<α,β(r,r0,t,t0) =iD
ψβ†(r0,t0)ψα(r,t)E
, (2.10)
where the grand canonical average is taken and the indicesα, βare one of↑,↓. In equilibrium, it takes the form
Gˆ<(k, ,r) =iA(k, ,ˆ r)fD() (2.11) with the Fermi-Dirac distribution function
fD() = 1
eβ(−EF)+ 1 , where β= 1/kBT denotes the inverse temperature.
With the knowledge of ˆG<at hand, one can easily obtain various physical quantities of interest such as the quasi-particle spin-charge density
Nˆ(r, t) =
Z d3k (2π)3
Z ∞
−∞
d
2πi Gˆ<(k, ,r, t) , (2.12) and spin-charge current density
Jˆ(r, t) =
Z d3k (2π)3
Z ∞
−∞
d
2πi vkGˆ<(k, ,r, t) , (2.13) with the quasi-particle group velocity given byvk=∂kk=k/m.
The spin-charge density ˆN is a 2×2 matrix in spin space and its trace yields the charge-densityNc=−eTr ˆN, while the spin-density is given byS = ~2Tr(ˆ~σNˆ). In a similar manner, we obtain the charge currentjc=−eTrJˆand the spin-kcurrent jk= ~2Tr(ˆσkJˆ).
In order to find ˇGfor our specific physical system, we need an equation of motion, called the Dyson equation
h((+eϕ(r))ˆ1−H) ˇˆ 1−Σˇi
∗Gˇ = ˇ1 , Gˇ∗h
((+eϕ(r))ˆ1−H) ˇˆ 1−Σˇi
= ˇ1 , (2.14)
where ˇ1 is the unit matrix in spin and Keldysh space.
Impurity scattering due to ˆVimp is described in terms of the self-energy ˆΣ, and is to be discussed in a moment. In Keldysh space, ˆΣ possesses the same structure as the Greens function ˇG,
Σ =ˇ
ΣˆR Σˆ<
0 ΣˆA
. (2.15)
We assume throughout the rest of this article that scattering is weak such that
ΣEF , (2.16)
whereEF is the Fermi-level for the conduction electrons, and which will allow us later to take advantage of the quasi-particle approximation.
Spectral Density
First, we have a look at equilibrium properties of the system by calculating the retarded Greens function, from which we readily obtain the spectral density ˆA using Eq. (2.9). Essen-tially, we do this by first solving for the retarded Greens function and taking the imaginary part thereafter. The starting point is the Dyson equation for the Greens function and taking the retarded component of (2.14), we explicitly obtain
[−k+δµ(r, t)] ˆ1 +∆−δ∆
2 m(r, t)ˆ~ ~σ
∗GˆR/A= ˆ1 . (2.17) GˆR/Aonly differs by the boundary condition which can be simply incorporated by substituting → ±i0+ in Eq. (2.17). Furthermore, since Σ EF, we will not include self-energy corrections to the spectrum, i.e. we neglect the line broadening and assume a delta-peaked spectrum, commonly referred to as quasi-particle approximation. Also, the external electric field affects the spectrum only in second order of the field [51].
To proceed, we expand the Moyal product
∗ ≈1 + i 2(←−
∂r−→
∂k−←−
∂t−→
∂−←−
∂k−→
∂r+←−
∂−→
∂t) +. . . , (2.18) commonly referred to as gradient expansion, and iteratively determine ˆGR/A in orders of gradients, whereas, for our purpose, we need only up to order ∂r2. The zeroth order term of the spectral density ˆA= ˆA0+ ˆA1+. . . is readily found to be
Aˆ0 = 2πh
Pˆ↑δ(−↑k) + ˆP↓δ(−↓k)i
, (2.19)
and for higher order terms we refer to Eq. (D.3) in the Appendix. Here, we defined the projectors in spin-space ˆP↑,↓ = 12(ˆ1±m~~σ) that project on the local spin up/down direction,ˆ and↑,↓k ≡k∓∆2 is the dispersion relation for majority and minority spin bands (see Fig. 2.1).
Thek-integration of the spectral function yields the density of states, here defined as number of states per unit energy and volume,
Z d3k
(2π)3 A(k, ,ˆ r, t) = 2πν(,ˆ r, t). (2.20) The matrix density of states in zeroth order correction or without magnetization gradient present is due to ˆA0,
ˆ
ν0 = ˆP↑ν↑+ ˆP↓ν↓=ν0(ˆ1 +P ~m(r, t)ˆ~σ) , (2.21)
k ↑k
↓k
EF
∆
Figure 2.1.: The parabolic band structure for the spin-up and down bands is split by the ferromagnetic exchange field ∆. The Fermi levelEF is measured with respect to the mid-point of the two band-edges.
where we have introduced the polarization of the Fermi surface P = ν↑−ν↓
ν↑+ν↓
, (2.22)
andν0 = 12(ν↑+ν↓) denotes the average density of states. The density of states for majority and minority spin bands are
ν↑,↓() =
Z d3k
(2π)3 δ(−↑,↓k ). (2.23)
ν↑,↓ without energy argument denotes the density of states at the Fermi-levelEF. For smooth
~
m(r, t), the density of states adiabatically follows the variations of the magnetization, and higher order terms ˆA1, ˆA2,. . . constitute corrections due to non-adiabaticities.
The polarization parameter at the Fermi level explicitly reads P(∆, EF) = ∆
2EF+ q
4EF2 −∆2
, (2.24)
and, in place of ∆ will be used as a fundamental parameter in the subsequent investigations.
For ∆→0, the polarization vanishes, and in the opposite limiting case when ∆→2EF, the Fermi-level reaches the band edge of the spin-down band and the system is fully polarized, i.e.
the half-metallic regime withP = 1. EF is measured with respect to the mid-point between the two band-edges (zero on the energy axis). The situation is depicted in Figure 2.1.