Impurity Scattering

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 levelE_F 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-levelE_F. 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(∆, E_F) = ∆

2EF+ q

4E_F² −∆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 ∆→2E_F, 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. E_F 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.

2.2.3. Impurity Scattering

The self-energy ˆΣ incorporates scattering by magnetic and non-magnetic impurities, com-monly calculated in the self-consistent Born approximation which truncates the series of irreducible diagrams due to multiple scattering after the first one:

Σ =ˇ =hVˆGˇViˆ ^imp_.

In physical terms, this means that all kinds of interference effects like weak localization are dropped from the theory. In this approximation, the self-energy for spin-independent impurity scattering takes the form

Σˇ_i = χ⁽ⁱ⁾

Z d³k

(2π)³ G ,ˇ (2.25)

and likewise for magnetic impurity scattering, Σˇ_mag = The tensorial structure of χ^(m) accounts for situations in which scattering is non-isotropic in spin space. We note that the impurity concentration has been incorporated into χ⁽ⁱ⁾ and χ^(m), which then simply scales linearly with the density of impurities. We can rewrite the magnetic self-energy using simple algebra and the properties of the Pauli matrices

3 where Tr takes the trace over the spin degrees of freedom, while trχ^(m)=P₃

i=1χ^(m)_ii .

The explicit structure of χ^(m) to be used throughout this chapter is motivated in the fol-lowing. If there exists a ferromagnetic exchange coupling between the internal impurity spin and the ferromagnetic order parameter, the spin is preferably aligned along this direction.

Consequently, the impurity will scatter the electron with a different magnitude depending on its spin. In the case of uniaxial symmetry (the symmetry axis is denoted by~n),

χ^(m)_ij =χ_⊥^(m)(δij −ninj) +χk^(m)ninj .

For the above example of ferromagnetic interaction between impurity spins and order parame-ter, the unit vector~nactually corresponds to the local magnetization direction,m~ =~n. In the Born approximation and restricting ourselves to this special case of uniaxial symmetry, the self-energy for both spin-isotropic and magnetic impurity scattering is written in the compact notation

Σ = ˇˇ Σ_i+ ˇΣ_mag =

Z d³k

(2π)³ χˆG ,ˇ (2.27)

where the operation ˆχ on a general spin-matrix ˆX is defined as ˆ

Our model of impurities introduces various types of relaxation which are summarized in the table below along with the explicit expressions in terms of the parameters of our model.

Momentum relaxation in Spin-↑,↓channel 1/τ↑,↓ = 2πχ⁽ⁱ⁾ν±+ 2π(χ_k^(m)ν±+ 2χ_⊥^(m)ν∓) Transverse momentum relaxation 1/τ = 2πν₀(χ⁽ⁱ⁾+ trχ^(m))

Longitudinal, spin-flip 1/T₁ = 4πν₀ 2χ_⊥^(m)

Transverse, spin-dephasing 1/T₂ = 4πν₀(χk^(m)+χ_⊥^(m))

Transverse spin excitations are also subject to momentum relaxation, albeit with the elastic mean-free timeτ which is different from the elastic mean-free time τ↑,↓ in the spin-↑,↓ con-duction channels. The longitudinal, or spin-flip relaxation timeT₁ describes the time it takes for a non-equilibrium magnetization in the direction of m~ to relax to its equilibrium value.

The transverse, or spin-dephasing timeT₂ describes the decay of transverse spin-excitations.

Mathematically, this behavior is already evident from the specific form of the self energy which is essentially given by (2.28). We will see this more explicitly later, when we derive the extended Bloch equation (2.121) along with the scattering term (2.129).

Our specific model of impurity scattering has three independent parameters χ⁽ⁱ⁾, χ_⊥^(m) and χ_k^(m), therefore, we take τ, T₁ and T₂ as independent parameters of our system. Then the momentum relaxation time, or the elastic mean-free time for the spin-up and down conduction channels can be expressed as

1 τ↑,↓

= 1±P γ

τ , (2.29)

where we defined the scattering asymmetry γ ≡(1− _T^τ₁) and we see that _τ² = _τ¹

↑ + _τ¹

↓, even though γ is not an independent parameter in our model.

Before continuing, let us inspect the range of possible values for our parameters. First, τ is dominated byχ⁽ⁱ⁾when magnetic scattering is weak, and essentially can be chosen arbitrarily.

T₁ and T₂ are of course not affected by non-magnetic impurities, and from their explicit expression, we see that the spin-flip time T₁ is only affected by the transverse fluctuating field χk^(m), while for the spin-dephasing time T₂, both components χk^(m) and χ_⊥^(m) contribute equally. For the spin-isotropic situation χk^(m) = χ_⊥^(m), both times are equal T₁ = T₂, and for a vanishing perpendicular component, we have of course T₁ = ∞. On the other side, a vanishing longitudinal fluctuating field χk^(m) = 0 leads to T1 =T2/2, so that in general, one has the condition

2T₁ ≥T₂ , (2.30)

which is a well known inequality in the field of nuclear spin dynamics [52]. In addition, non-magnetic impurities always contribute to momentum relaxation, so one is not free in the choice of τ with respect to T₂ und usually, non-magnetic scatterers are dominant, so that τ < T2. Even in the best case scenarioχ⁽ⁱ⁾= 0, the momentum relaxation timeτ ranges from T₂ to 2T₂, so in order to cover all situations, we keep τ ≤T₂, T₁.