BCS Theory: Ground State and Gap Equation

Hello! Let's dive into the full microscopic theory of superconductivity, building directly on our last discussion.

In the previous lesson, we solved the Cooper problem and discovered that the normal metallic state is unstable against the formation of a bound pair of electrons (a Cooper pair) for any arbitrarily weak attractive interaction. This was a crucial first step, but it only involved two electrons. The challenge now is to describe the ground state of the entire system, where a macroscopic number of electrons participate in this pairing.

Today, we will construct the celebrated Bardeen-Cooper-Schrieffer (BCS) ground state wavefunction. This state is not an eigenstate of particle number but rather a coherent superposition of states with different numbers of Cooper pairs. By applying a variational principle to this wavefunction, we will derive the famous BCS self-consistency equation for the superconducting energy gap, $\Delta$, which is the central quantity of the theory and a directly measurable experimental parameter.

1. From Cooper Pairs to a Many-Body Condensate

The Cooper problem hints that the true ground state is not a simple filled Fermi sea, but one where electrons near the Fermi surface are bound into pairs. The insight of Bardeen, Cooper, and Schrieffer was to propose a many-body wavefunction that captures this collective pairing.

The BCS Hamiltonian

We begin with the model Hamiltonian that isolates the essential physics. In second quantization, it consists of the kinetic energy of the electrons (measured relative to the chemical potential, ${\xi }_{\mathbf{k}}={ϵ}_{\mathbf{k}}-\mu$) and an effective pairing interaction that scatters a pair of electrons with opposite momentum and spin from $({\mathbf{k}}^{\prime },\uparrow ),(-{\mathbf{k}}^{\prime },\downarrow )$ to $(\mathbf{k},\uparrow ),(-\mathbf{k},\downarrow )$.

$${\mathcal{H}}_{BCS}=\sum _{\mathbf{k},\sigma }{\xi }_{\mathbf{k}}{c}_{\mathbf{k}\sigma }^{†}{c}_{\mathbf{k}\sigma }+\sum _{\mathbf{k},{\mathbf{k}}^{\prime }}{V}_{\mathbf{k}{\mathbf{k}}^{\prime }}{c}_{\mathbf{k}\uparrow }^{†}{c}_{-\mathbf{k}\downarrow }^{†}{c}_{-{\mathbf{k}}^{\prime }\downarrow }{c}_{{\mathbf{k}}^{\prime }\uparrow }$$

Here, $c_{\mathbf{k}\sigma }^{†}$ and $c_{\mathbf{k}\sigma }$ are the standard fermionic creation and annihilation operators. The interaction term $V_{\mathbf{k}{\mathbf{k}}^{\prime }}$ is assumed to be attractive ($V_{\mathbf{k}{\mathbf{k}}^{\prime }}<0$) for electrons within a shell $\hslash {\omega }_D$ around the Fermi surface.

The BCS Ground State Wavefunction

Schrieffer's brilliant ansatz for the ground state is a product over all momentum states $\mathbf{k}$, where each state is a superposition of being empty or being occupied by a Cooper pair:

$$|{\Psi }_{BCS}⟩=\prod _{\mathbf{k}}\left(u_{\mathbf{k}}+v_{\mathbf{k}}{c}_{\mathbf{k}\uparrow }^{†}{c}_{-\mathbf{k}\downarrow }^{†}\right)|0⟩$$

Here, $|0⟩$ is the vacuum state. The coefficients $u_{\mathbf{k}}$ and $v_{\mathbf{k}}$ are complex variational parameters that we need to determine. They represent probability amplitudes:

• $|v_{\mathbf{k}}{|}^2$ is the probability that the pair state $(\mathbf{k}\uparrow ,-\mathbf{k}\downarrow )$ is occupied.
• $|u_{\mathbf{k}}{|}^2$ is the probability that this pair state is empty.

For the state to be normalized, we require $|u_{\mathbf{k}}{|}^2+|v_{\mathbf{k}}{|}^2=1$. Notice that this state is a superposition of states with different total particle numbers, which breaks the U(1) gauge symmetry associated with particle number conservation. This broken symmetry is the defining feature of the superconducting state.

2. Deriving the Gap Equation: A Variational Approach

To find the ground state, we must find the values of $u_{\mathbf{k}}$ and $v_{\mathbf{k}}$ that minimize the expectation value of the energy $⟨{\Psi }_{BCS}|{\mathcal{H}}_{BCS}|{\Psi }_{BCS}⟩$. Since the wavefunction does not have a definite particle number, it's more convenient to work in the grand canonical ensemble and minimize the quantity $⟨\mathcal{H}-\mu N⟩$, where $\mu$ is the chemical potential.

The expectation value of the energy can be shown to be:

$$⟨{\mathcal{H}}_{BCS}⟩=\sum _{\mathbf{k}}2{\xi }_{\mathbf{k}}|v_{\mathbf{k}}{|}^2+\sum _{\mathbf{k},{\mathbf{k}}^{\prime }}{V}_{\mathbf{k}{\mathbf{k}}^{\prime }}{u}_{\mathbf{k}}^{\ast }{v}_{\mathbf{k}}{u}_{{\mathbf{k}}^{\prime }}{v}_{{\mathbf{k}}^{\prime }}^{\ast }$$

The first term represents the kinetic energy of the paired electrons, and the second term is the pairing interaction energy. The minimization procedure, subject to the constraint $|u_{\mathbf{k}}{|}^2+|v_{\mathbf{k}}{|}^2=1$, is a standard problem in calculus of variations.

The key step is to define the superconducting gap parameter, ${\Delta }_{\mathbf{k}}$:

$${\Delta }_{\mathbf{k}}=-\sum _{{\mathbf{k}}^{\prime }}{V}_{\mathbf{k}{\mathbf{k}}^{\prime }}⟨c_{-{\mathbf{k}}^{\prime }\downarrow }{c}_{{\mathbf{k}}^{\prime }\uparrow }⟩=-\sum _{{\mathbf{k}}^{\prime }}{V}_{\mathbf{k}{\mathbf{k}}^{\prime }}{u}_{{\mathbf{k}}^{\prime }}{v}_{{\mathbf{k}}^{\prime }}^{\ast }$$

This parameter acts as an order parameter for the superconducting state. It is a measure of the pairing amplitude and is proportional to the expectation value of the Cooper pair annihilation operator.

The minimization procedure yields the following key results:

1. The optimal coherence factors are given by:

   $$|u_{\mathbf{k}}{|}^2=\frac{1}{2}\left(1+\frac{{\xi }_{\mathbf{k}}}{E_{\mathbf{k}}}\right)\text{and}|v_{\mathbf{k}}{|}^2=\frac{1}{2}\left(1-\frac{{\xi }_{\mathbf{k}}}{E_{\mathbf{k}}}\right)$$

2. The energy $E_{\mathbf{k}}$ is the energy of a quasiparticle excitation (more on this later) and is given by:

   $$E_{\mathbf{k}}=\sqrt{{\xi }_{\mathbf{k}}^2+|{\Delta }_{\mathbf{k}}{|}^2}$$

   This shows that the excitation spectrum has a minimum energy of $|{\Delta }_{\mathbf{k}}|$ at the Fermi surface (${\xi }_{\mathbf{k}}=0$). This is the famous superconducting gap.

3. Plugging the expression for $u_{\mathbf{k}}{v}_{\mathbf{k}}^{\ast }={\Delta }_{\mathbf{k}}/(2E_{\mathbf{k}})$ back into the definition of ${\Delta }_{\mathbf{k}}$ gives the BCS self-consistency equation:

   $${\Delta }_{\mathbf{k}}=-\sum _{{\mathbf{k}}^{\prime }}{V}_{\mathbf{k}{\mathbf{k}}^{\prime }}\frac{{\Delta }_{{\mathbf{k}}^{\prime }}}{2E_{{\mathbf{k}}^{\prime }}}=-\sum _{{\mathbf{k}}^{\prime }}{V}_{\mathbf{k}{\mathbf{k}}^{\prime }}\frac{{\Delta }_{{\mathbf{k}}^{\prime }}}{2\sqrt{{\xi }_{{\mathbf{k}}^{\prime }}^2+|{\Delta }_{{\mathbf{k}}^{\prime }}{|}^2}}$$

This equation is called "self-consistent" because the gap $\Delta$ appears on both sides. It's an integral equation that must be solved to find the value of the gap.

3. An Alternative View: The Bogoliubov Transformation

Given your background in QFT, you'll appreciate a more elegant method to arrive at the same result: the Bogoliubov-Valatin transformation. This technique diagonalizes the BCS Hamiltonian by defining new fermionic operators, which represent the true elementary excitations of the system—the quasiparticles.

We start with the mean-field version of the BCS Hamiltonian, where we replace the four-fermion interaction term with its average effect:

$${\mathcal{H}}_{MFT}=\sum _{\mathbf{k}\sigma }{\xi }_{\mathbf{k}}{c}_{\mathbf{k}\sigma }^{†}{c}_{\mathbf{k}\sigma }-\sum _{\mathbf{k}}({\Delta }_{\mathbf{k}}^{\ast }{c}_{-\mathbf{k}\downarrow }{c}_{\mathbf{k}\uparrow }+{\Delta }_{\mathbf{k}}{c}_{\mathbf{k}\uparrow }^{†}{c}_{-\mathbf{k}\downarrow }^{†})+\frac{|{\Delta }_{\mathbf{k}}{|}^2}{V}$$

This Hamiltonian is quadratic, but not diagonal, as it contains terms like $c^{†}{c}^{†}$ and $cc$ that create and destroy pairs. The Bogoliubov transformation defines new quasiparticle operators $\gamma$ as a linear combination of the original electron operators:

$$\begin{array}{rl}{c}_{\mathbf{k}\uparrow }& =u_{\mathbf{k}}^{\ast }{\gamma }_{\mathbf{k}\uparrow }+v_{\mathbf{k}}{\gamma }_{-\mathbf{k}\downarrow }^{†}\\ c_{-\mathbf{k}\downarrow }^{†}& =-v_{\mathbf{k}}^{\ast }{\gamma }_{\mathbf{k}\uparrow }+u_{\mathbf{k}}{\gamma }_{-\mathbf{k}\downarrow }^{†}\end{array}$$

Choosing $u_{\mathbf{k}}$ and $v_{\mathbf{k}}$ correctly (the same coefficients we found variationally!), this transformation diagonalizes the Hamiltonian into a simple form:

$${\mathcal{H}}_{MFT}=E_{GS}+\sum _{\mathbf{k}}{E}_{\mathbf{k}}({\gamma }_{\mathbf{k}\uparrow }^{†}{\gamma }_{\mathbf{k}\uparrow }+{\gamma }_{-\mathbf{k}\downarrow }^{†}{\gamma }_{-\mathbf{k}\downarrow })$$

This describes a gas of non-interacting fermionic quasiparticles with a gapped energy spectrum $E_{\mathbf{k}}=\sqrt{{\xi }_{\mathbf{k}}^2+|{\Delta }_{\mathbf{k}}{|}^2}$. The conditions required to make the Hamiltonian diagonal precisely reproduce the BCS gap equation.

Test your understanding!

A Bogoliubov quasiparticle is a superposition of an electron and a hole. At the Fermi momentum ($\mathbf{k}={\mathbf{k}}_F$, so ${\xi }_{\mathbf{k}}=0$), what are the values of $|u_{\mathbf{k}}{|}^2$ and $|v_{\mathbf{k}}{|}^2$? What does this tell you about the character of a quasiparticle at the Fermi surface?

Show answer

At the Fermi surface, ${\xi }_{\mathbf{k}}=0$, so $E_{\mathbf{k}}=|{\Delta }_{\mathbf{k}}|$. The probabilities become:

$$|u_{\mathbf{k}}{|}^2=\frac{1}{2}(1+0)=\frac{1}{2}\text{and}|v_{\mathbf{k}}{|}^2=\frac{1}{2}(1-0)=\frac{1}{2}$$

This means the quasiparticle is an exactly 50/50 mixture of an electron and a hole. Far above the Fermi surface (${\xi }_{\mathbf{k}}\to \infty$), $|u_{\mathbf{k}}{|}^2\to 1$ and $|v_{\mathbf{k}}{|}^2\to 0$, so the quasiparticle becomes purely electron-like. Far below (${\xi }_{\mathbf{k}}\to -\infty$), $|u_{\mathbf{k}}{|}^2\to 0$ and $|v_{\mathbf{k}}{|}^2\to 1$, so it becomes purely hole-like.

4. Solving the Gap Equation and the Role of Temperature

For the simple model where $V_{\mathbf{k}{\mathbf{k}}^{\prime }}=-V_0$ for $|{\xi }_{\mathbf{k}}|,|{\xi }_{{\mathbf{k}}^{\prime }}|<\hslash {\omega }_D$, the gap $\Delta$ becomes independent of $\mathbf{k}$. At zero temperature ($T=0$), the self-consistency equation becomes:

$$1=\frac{V_0}{2}\sum _{|{\xi }_{\mathbf{k}}|<\hslash {\omega }_D}\frac{1}{\sqrt{{\xi }_{\mathbf{k}}^2+{\Delta }^2}}$$

Converting the sum to an integral over energy with a constant density of states $N(0)$ near the Fermi level, we can solve for the gap $\Delta (0)$:

$$\Delta (0)=2\hslash {\omega }_D{e}^{-1/N(0)V_0}$$

Note the similarity to the Cooper pair binding energy from our last lesson, but the factor of 2 in the exponent is now gone! This is because we are now considering a collective, many-body condensate, not just an isolated pair.

At finite temperature, quasiparticles can be thermally excited, which affects the pairing. The thermal average of the occupation number of quasiparticles follows a Fermi-Dirac distribution, $⟨{\gamma }_{\mathbf{k}}^{†}{\gamma }_{\mathbf{k}}⟩=f(E_{\mathbf{k}})$. This modifies the gap equation to:

$${\Delta }_{\mathbf{k}}(T)=-\sum _{{\mathbf{k}}^{\prime }}{V}_{\mathbf{k}{\mathbf{k}}^{\prime }}\frac{{\Delta }_{{\mathbf{k}}^{\prime }}(T)}{2E_{{\mathbf{k}}^{\prime }}}\tanh \left(\frac{E_{{\mathbf{k}}^{\prime }}}{2k_{B}T}\right)$$

Solving this equation shows that the gap $\Delta (T)$ decreases with increasing temperature, eventually vanishing at a critical temperature $T_c$.

Conclusion

Today we have constructed the foundation of the microscopic theory of superconductivity. By postulating the BCS wavefunction, we have moved from a two-electron problem to a full many-body description of the superconducting condensate.

Key Takeaways:

• The BCS ground state is a coherent superposition of Cooper pair states, described by the product wavefunction $|{\Psi }_{BCS}⟩={\prod }_{\mathbf{k}}(u_{\mathbf{k}}+v_{\mathbf{k}}{c}_{\mathbf{k}\uparrow }^{†}{c}_{-\mathbf{k}\downarrow }^{†})|0⟩$.
• Using a variational method or the Bogoliubov transformation, we derived the BCS self-consistency equation for the superconducting gap $\Delta$: ${\Delta }_{\mathbf{k}}=-{\sum }_{{\mathbf{k}}^{\prime }}{V}_{\mathbf{k}{\mathbf{k}}^{\prime }}\frac{{\Delta }_{{\mathbf{k}}^{\prime }}}{2E_{{\mathbf{k}}^{\prime }}}$.
• The elementary excitations above the ground state are gapped fermionic quasiparticles with energy $E_{\mathbf{k}}=\sqrt{{\xi }_{\mathbf{k}}^2+|{\Delta }_{\mathbf{k}}{|}^2}$.
• At $T=0$, the gap has a non-perturbative form $\Delta (0)=2\hslash {\omega }_D{e}^{-1/N(0)V_0}$, and it vanishes at the critical temperature $T_c$.

Preview of the Next Lesson

The most profound consequence of the pairing is the opening of the energy gap $\Delta$ in the quasiparticle excitation spectrum. This gap drastically alters the density of states (DOS) near the Fermi energy, creating a unique signature of the superconducting state. In the next lesson, we will calculate this new DOS and see how it can be directly measured in experiments, particularly Scanning Tunneling Spectroscopy (STS), providing one of the most direct and powerful confirmations of BCS theory.