Effective Interaction for Electrons Intermediated by Phonons

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The Effective Interaction for Electrons
Intermediated by Phonons

The effective interaction of electrons which is intermediated by a phonon is:

2hω_q|M_q|²
^{______________________}
(E_k-E_k+q)² - (hω_q)²

Derivation:

One electron interacts with the lattice of ions, perturbing it and creating a phonon. The phonon produces a fluctuation in the position of of the ions perturbing the fields that another electron is reacting to. The perturbed fields then alters the momentum of the other electron.
The total Hamiltonian for the electrons, phonon and their interaction is of the form
H = H_e + H_ph + H'
where H_e, H_ph, H' are the Hamiltonians for the electrons, phonons and the electron-phonon interaction, respectively, and are given in the second quantization form by:
H_e = Σω_q a⁺_qa_q
H_ph = Σε_q c⁺_qc_q
H' = iDΣc⁺_k+qc_q (a_q - a⁺_-q)

where a⁺ and a are the fermion (electon) operators, and c⁺ and c are the boson (phonon) operators.
If the Hamiltonian H = H₀+λH' is transformed using a canonical transformation of the form e^-SHe^S where S is chosen so as make λH' + [H₀,S] = 0 the effective electron-electron coupling is given by the expected value over the wave functions of the electrons parameterized by the number of phonons before and after the interaction (n,m):
<n|S|m> =
<n|H'|m>/(E_m - E_n)
when E_m is not equal to E_n.
The expected value of H' is then determined from the difference of:

<1|S|0> =

-iDΣc⁺_k-qc_k

^{__________________}

(ε_k-ε_k-q -hω_q)

<0|S|1> =

iDΣc⁺_k'+qc_k'

^{__________________}

(ε_k'-ε_k'+q +hω_q)

Since ω_-q = ω_q the expression for the expected value of H' contains the term

1
______________
(ε_k-ε_k+q-hω_q)

-

1
______________
(ε_k'-ε_k'-q+hω_q)

which reduces to:
hω_q

________________

(ε_k-ε_k+q)²-(hω_q)²

This accounts for most of the functional elements of the effective interaction.
Reference:
Charles Kittel, Quantum Theory of Solids, p.148 and pp. 151-153).

2hωq|Mq|2 ______________________(Ek-Ek+q)2 - (hωq)2

H = He + Hph + H'

He = Σωq a+qaq Hph = Σεq c+qcq H' = iDΣc+k+qcq (aq - a+-q)

<n|S|m> = <n|H'|m>/(Em - En)

<1|S|0> = -iDΣc+k-qck __________________ (εk-εk-q -hωq)

<0|S|1> = iDΣc+k'+qck' __________________ (εk'-εk'+q +hωq)

1______________ (εk-εk+q-hωq) - 1______________ (εk'-εk'-q+hωq)

hωq ________________ (εk-εk+q)2-(hωq)2

2hω_q|M_q|²
^{______________________}
(E_k-E_k+q)² - (hω_q)²

H = H_e + H_ph + H'

H_e = Σω_q a⁺_qa_q
H_ph = Σε_q c⁺_qc_q
H' = iDΣc⁺_k+qc_q (a_q - a⁺_-q)

<n|S|m> =
<n|H'|m>/(E_m - E_n)

<1|S|0> =

-iDΣc⁺_k-qc_k

^{__________________}

(ε_k-ε_k-q -hω_q)

<0|S|1> =

iDΣc⁺_k'+qc_k'

^{__________________}

(ε_k'-ε_k'+q +hω_q)

1
______________
(ε_k-ε_k+q-hω_q)

-

1
______________
(ε_k'-ε_k'-q+hω_q)

hω_q

________________

(ε_k-ε_k+q)²-(hω_q)²