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5.2 From Fermi’s Golden Rule to the Totel Scattering Rate

Fermi’s Golden Rule is a method to calculate the transition rate from an initial state |α,k∥⟩ to a final state |α′ , k ∥ ′ ⟩ due to a perturbation. As described in the section 4.1.1.2, the general form of the transition probability is given by

α′ ′ 2π-∑ ′ ′ ′ ˆ 2 ′ ′ Sα(k∥,k∥) = ℏ |⟨α ,k∥|⟨nq |Hint|nq ⟩|α,k ∥⟩| δ(Eα(k∥) - Eα (k∥)± ℏω ) q
(5.2)

The energy exchange between the electrons and the lattice occurs via phonons and the Delta distribution ensures energy conservation. Here, ℏω is the energy of the absorbed or emitted phonons. For the phonon interaction, the perturbation potential can be written in the following form

i(q⋅x-ω t) Hˆint = ˆUqe q
(5.3)

Inserting the wave function (3.1) into the matrix element appearing in Fermi’s Golden Rule, gives

′ ′ ∫ 1 ⋆ ′ i(k -k′)⋅x |⟨α,k ∥|H ˆint|α,k∥⟩| = AΦ α′(z)|⟨nq|H ˆint|nq⟩|Φα(z)e ∥ ∥ dx V ∫ ∫ 1 i(k - k′+q )⋅x = dz dx ∥Φ ⋆α′(z)Φ α(z)eiqzz|Uq |-e ∥ ∥ ∥ α A = |Uq |Fα′(qz)δk∥+q∥,k′∥ (5.4)
where Uq = ⟨nq′|Ûq|nq⟩ and the form factor is given by
∫ α ⋆ iqzz Fα′(qz) = Φα′(z)Φ α(z)e dz
(5.5)

In equation (5.2), the summation over q can be split into a sum over qz and another sum over q∥. Due to the momentum conservation, the sum over q∥ gives only one term for q∥ = k∥′- k∥. The remaining summation in the z direction can be tranformed to an integration over qz according to

∑ L ∫ → -z- dqz qz 2π

Thus, Fermi’s Golden Rule can be written as

α′ ′ Lz ∫ 2 α 2 ′ Sα (k ∥,k ∥) =-ℏ- |Uk ′∥-k∥,qz||Fα′(qz)| δ(Eα(k∥) - Eα′(k∥)± ℏω )dqz
(5.6)

Finally, the total scattering rate can be calculated as follows

1 A ∫ ′ α′ = ----2 S αα (k∥,k ′∥)dk ′∥ τα(k∥) (2π ) ∫ ∫ = ---V-- dk ′ dq |U ′ |2|F α(q )|2δ(E (k )- E ′(k ′) ± ℏω) (2π )2ℏ ∥ z k∥-k∥,qz α′ z α ∥ α ∥ (5.7)

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