The Cooper instability
Piers Coleman - Introduction to Many-Body Physics
14.2 The Cooper instability
Cooper imagined adding a pair of electrons above the Fermi surface in a state with no net momentum, described by the wavefunction $$|\Psi\rangle = \Lambda^\dagger| FS\rangle,$$ where $$\Lambda^\dagger = \int d^3xd^3x'\ \phi(\mathbf{x}-\mathbf{x}')\psi^\dagger_\downarrow(\mathbf{x})\psi^\dagger_\uparrow(\mathbf{x}')$$ creates a pair of electrons, while $|FS\rangle = \prod_{k<k_F}c^\dagger_{\mathbf{k}\uparrow}c^\dagger_{-\mathbf{k}\downarrow}|0\rangle$ defines the filled sea. If we Fourier transform the fields, writing $\psi^\dagger_\sigma(\mathbf{x})=\frac{1}{N}\sum_\mathbf{k}c^\dagger_{\mathbf{k}\sigma}e^{-i\mathbf{k}\cdot \mathbf{x}}$, then the pair creation operator can be recast as a sum over pairs in momentum space: $$\Lambda^\dagger=\sum_\mathbf{k}\phi_\mathbf{k}c^\dagger_{\mathbf{k}\downarrow}c^\dagger_{-\mathbf{k}\uparrow},$$ where $$\phi_\mathbf{k}=\int d^3x e^{-i\mathbf{k}\cdot\mathbf{x}}\phi(\mathbf{x})$$ is the Fourier transform of the spatial pair wavefunction.
When an electron pair is created, electrons can only be added above the Fermi surface, so that $$|\Psi\rangle= \Delta^\dagger |FS\rangle= \sum_{|\mathbf{k}|>k_F}\phi_\mathbf{k}|\mathbf{k}_P\rangle,$$ where $|\mathbf{k}_P\rangle \equiv |\mathbf{k}\uparrow,-\mathbf{k}\downarrow\rangle = c^\dagger_{\mathbf{k}\uparrow}c^\dagger_{-\mathbf{k}\downarrow}|FS\rangle$. Now suppose that the Hamiltonian has the form $$H=\sum_\mathbf{k} \epsilon_\mathbf{k}c^\dagger_{\mathbf{k}\sigma}c_{\mathbf{k}\sigma}+\hat{V},$$ where $\hat{V}$ contains the details of the electron-electron interaction; if $|\Psi\rangle$ is an eigenstate with energy $E$, then $$H|\Psi\rangle=\sum_{|\mathbf{k}|>k_F}2\epsilon_\mathbf{k}\phi_\mathbf{k}|\mathbf{k}_P\rangle + \sum_{|\mathbf{k}|,|\mathbf{k}'|>k_F}|\mathbf{k}_P\rangle\langle \mathbf{k}_P|\hat{V}|\mathbf{k}'_P\rangle \phi_{\mathbf{k}'}.$$
Identifying this with $E|\Psi\rangle = E\sum_\mathbf{k}\phi_\mathbf{k}|\mathbf{k}_P\rangle$, so comparing the amplitudes to be in the state $|\mathbf{k}_P\rangle$, $$E\phi_\mathbf{k}=2\epsilon_\mathbf{k}\phi_\mathbf{k}+\sum_{|\mathbf{k}'|>k_F} \langle \mathbf{k}_P|\hat{V}|\mathbf{k}'_P\rangle \phi_{\mathbf{k}'}.$$
The beauty of this equation is that the details of the electron interactions are entirely contained in the pair scattering matrix element $V_{\mathbf{k},\mathbf{k}'}=\langle \mathbf{k}_P|\hat{V}|\mathbf{k}_P'\rangle$. Microscopically, this scattering is produced by the exchange of virtual phonons, and the scattering matrix element is determined by the electron-phonon propagator $$V_{\mathbf{k},\mathbf{k}'}=g^2_{\mathbf{k}-\mathbf{k}'}D(\mathbf{k}'-\mathbf{k}, \epsilon_\mathbf{k}-\epsilon_{\mathbf{k}'}).$$
Cooper noted that this matrix element is not strongly momentum-dependent, only becoming attractive within an energy $\omega_D$ of the Fermi surface, and this motivated a simplified model interaction in which $$V_{\mathbf{k},\mathbf{k}'}=\begin{cases}-g_0/V& (|\epsilon_\mathbf{k}|,\ |\epsilon_{\mathbf{k}'}|<\omega_D)\\ 0& (\text{otherwise}).\end{cases}$$
We can simplify equation: $$(E-2\epsilon_\mathbf{k})\phi_\mathbf{k}=-\frac{g_0}{V}\sum_{0<\epsilon_{\mathbf{k}'}<\omega_D}\phi_{\mathbf{k}'},$$ so that by solving for $\phi_\mathbf{k}$, $$\phi_\mathbf{k}=-\frac{g_0/V}{E-2\epsilon_\mathbf{k}}\sum_{0<\epsilon_{\mathbf{k}'}<\omega_D}\phi_{\mathbf{k}'},$$ then summing both sides over $\mathbf{k}$ and factoring out $\sum_\mathbf{k}\phi_\mathbf{k}$, we obtain the self-consistent equation $$1=-\frac{1}{V}\sum_{0<\epsilon_\mathbf{k}<\omega_D}\frac{g_0}{E-2\epsilon_\mathbf{k}}.$$
Replacing the summation by an integral over energy, $\frac{1}{V}\sum_{0<\epsilon_\mathbf{k}<\omega_D}\rightarrow N(0)\int^{\omega_D}_0$, where $N(0)$ is the density of states per spin unit volume at the Fermi energy, the resulting equation gives $$1=g_0N(0)\int^{\omega_D}_0\frac{d\epsilon}{2\epsilon-E}=-\frac{1}{2}g_0N(0)\ln \left[ \frac{2\omega_D-E}{-E}\right] \approx -\frac{1}{2}g_0N(0)\ln \left[ \frac{2\omega_D}{-E}\right],$$ where, anticipating the smallness of $|E|\ll \omega_D$, we have approximated $2\omega_D-E\approx 2\omega_D$. In other words, the energy of the Cooper pair is given by $$E=-2\omega_De^{-\frac{2}{g_0N(0)}}.$$