BCS Superconductivity: Quasiparticles & Thermodynamics

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Hello!

In our last lesson, we derived the BCS variational ground state, a coherent condensate of Cooper pairs. By minimizing the system's energy, we found the self-consistent gap equation for the order parameter $Δ$ and calculated the condensation energy, proving that the superconducting state is energetically favored over the normal metallic state. Along the way, we encountered the expression $E_{k} = ξ_{k}^{2} + ∣ Δ_{k} ∣^{2}$ , but its physical role remained to be seen.

Today, we will uncover the profound physical meaning of this quantity. Our focus will be on the elementary excitations that exist above the superconducting ground state. You will learn to:

Analyze the spectrum of these excitations, known as Bogoliubov quasiparticles, revealing the true nature of the energy gap.
Understand how this gapped spectrum dictates the key thermodynamic properties of a superconductor, such as its electronic specific heat.

This lesson will bridge the gap between the abstract ground state wavefunction and the tangible, measurable properties that define a superconductor.

1. Diagonalizing the Hamiltonian: The Emergence of Quasiparticles

In the previous lesson, we used a variational approach to find the ground state. A more formal and powerful method to solve the BCS Hamiltonian is the Bogoliubov-Valatin transformation. This is a canonical transformation that re-expresses the Hamiltonian in terms of new fermionic operators, with the goal of making it diagonal. This technique is a cornerstone of many-body physics, with analogues for both fermionic and bosonic systems.

The core idea is to define a new set of fermionic quasiparticle operators, $γ_{k σ}$ , which are linear superpositions of the original electron creation and annihilation operators:

γ_{k ↑} γ_{- k ↓}^{†} = u_{k} c_{k ↑} - v_{k} c_{- k ↓}^{†} = v_{k}^{*} c_{k ↑} + u_{k}^{*} c_{- k ↓}^{†}

Here, $u_{k}$ and $v_{k}$ are the same coefficients from our variational wavefunction, satisfying $∣ u_{k} ∣^{2} + ∣ v_{k} ∣^{2} = 1$ . This transformation "mixes" particles and holes. The new operator $γ_{k ↑}$ annihilates a quasiparticle, which is a coherent mixture of an electron with momentum $k$ and a hole with momentum $- k$ .

The following video segment provides an excellent conceptual overview of this procedure, showing how it diagonalizes the Hamiltonian and reveals the quasiparticle picture.

20. Fermi gases, BEC-BCS crossover

In this clip, Prof. Ketterle explains how the BCS Hamiltonian, which contains complicated four-operator terms, can be simplified using a mean-field approximation and then diagonalized via the Bogoliubov transformation. This reveals a simple picture of a gas of non-interacting quasiparticles.

Please watch from 43:27 to 54:52. Focus on how the interacting Hamiltonian is first reduced to a bilinear form using a mean-field approximation for the pairing term. Then, observe how the Bogoliubov transformation converts this into a diagonal Hamiltonian describing independent quasiparticle excitations. The final form of the Hamiltonian is the key result.

As you saw, applying this transformation to the mean-field BCS Hamiltonian, and choosing the coefficients $u_{k}$ and $v_{k}$ to eliminate the off-diagonal "cross terms," results in a beautifully simple diagonal form:

H_{BCS} = E_{GS} + k, σ \sum E_{k} γ_{k σ}^{†} γ_{k σ}

This form tells us everything we need to know:

Ground State: The term $E_{GS}$ is the energy of the BCS ground state, which is the vacuum for these new quasiparticles (i.e., $γ_{k σ} ∣ Ψ_{GS} ⟩ = 0$ for all $k, σ$ ).
Excitations: The second term describes a gas of independent fermionic excitations. $γ_{k σ}^{†}$ creates a quasiparticle (or "Bogoliubon") with momentum $k$ and spin $σ$ .
Excitation Energy: The energy required to create one such quasiparticle is precisely $E_{k} = ξ_{k}^{2} + Δ^{2}$ . This finally reveals the physical meaning of this expression: it is the quasiparticle dispersion relation.

2. The Quasiparticle Spectrum and Density of States

The dispersion relation $E_{k}$ is the most important feature of the BCS theory. Let's analyze it.

Recall that $ξ_{k} = ε_{k} - μ$ , so $ξ_{k} = 0$ at the Fermi surface.
At the Fermi surface, the excitation energy is $E_{k_{F}} = 0^{2} + Δ^{2} = Δ$ .
Far from the Fermi surface, where $∣ ξ_{k} ∣ ≫ Δ$ , the energy is $E_{k} \approx ∣ ξ_{k} ∣$ , which is just the normal electron/hole excitation energy.

This means there is a minimum energy $Δ$ required to create any excitation in the system. This is the famous superconducting energy gap. Unlike a normal metal where you can create electron-hole pairs with infinitesimally small energy, a superconductor has a robust gap in its excitation spectrum.

This figure compares the excitation spectrum of a normal metal (left) and a BCS superconductor (right). In the normal metal, electron-hole pair excitations can have arbitrarily small energy. In the superconductor, the formation of Cooper pairs opens a gap of size $2Δ$ in the spectrum, and the elementary excitations are gapped quasiparticles with a minimum energy of $Δ$ .

Quasiparticle Density of States (DOS)

This gapped spectrum drastically alters the density of states. We can find the new DOS, $N_{S} (E)$ , from the normal state DOS, $N_{N} (E)$ , by conserving the number of states: $N_{S} (E) d E = N_{N} (ξ) d ξ$ . Assuming the normal state DOS is roughly constant near the Fermi energy, $N_{N} (ξ) \approx N (0)$ , we get:

N_{S} (E) = N (0) \frac{d ξ}{d E} = N (0) \frac{E}{E ^{2} - Δ ^{2}} for E > Δ

For $E < Δ$ , $N_{S} (E) = 0$ .

The quasiparticle DOS is zero inside the gap and diverges at the gap edge, $E = Δ$ . This "coherence peak" is a hallmark of BCS theory and is directly observable in experiments like electron tunneling.

Test your understanding!

Imagine a scanning tunneling microscopy (STM) experiment where you measure the differential conductance ( $d I / d V$ ) between a metallic tip and a BCS superconductor. The $d I / d V$ is proportional to the local density of states of the sample. What would you expect the $d I / d V$ vs. voltage $V$ curve to look like at low temperatures?

Show answer

At zero voltage, there would be zero conductance because there are no states available within the gap $- Δ < e V < Δ$ . As the voltage increases to $∣ V ∣ = Δ/ e$ , the conductance would abruptly turn on and show sharp peaks, corresponding to the singular peaks in the BCS density of states at the gap edges. For $∣ V ∣ > Δ/ e$ , the conductance would approach the value for a normal metal. This "U-shaped" gap with sharp peaks is a classic signature of BCS superconductivity.

3. Thermodynamic Properties at Finite Temperature

The quasiparticle picture makes it straightforward to understand the thermodynamic properties. At any finite temperature $T > 0$ , a thermal population of quasiparticles will be excited according to the Fermi-Dirac distribution, $f (E_{k}) = (e^{E_{k} / k_{B} T} + 1)^{- 1}$ .

For a detailed derivation of the finite-temperature properties, the following notes are an excellent resource.

PHYS 798C Spring 2024 Lecture 11 Summary

These lecture notes from the University of Maryland provide a concise yet thorough derivation of the key thermodynamic properties of a BCS superconductor.

Please study the following sections: Section B: Finite-Temperature Self-Consistent Gap Equation: Understand how the presence of thermally excited quasiparticles modifies the gap equation. This leads to the temperature dependence of the gap, \Delta(T). Section C: Temperature Dependent Gap: Pay attention to the two asymptotic limits: the exponential flatness at low T and the mean-field behavior near T_c. Section D: Thermodynamic Quantities: Focus on the derivation and physical interpretation of the electronic entropy and heat capacity. Note the exponential suppression at low T and the characteristic jump at T_c.

Let's summarize the key thermodynamic signatures that arise from the gapped quasiparticle spectrum.

Temperature-Dependent Gap $Δ (T)$

The existence of quasiparticles reduces the number of electrons participating in the condensate, which in turn reduces the pairing strength. This means the gap itself is temperature-dependent, shrinking as $T$ increases. The self-consistent gap equation becomes:

1 = N (0) V \int_{0}^{ℏ ω_{D}} \frac{d ξ}{ξ ^{2} + Δ ( T ) ^{2}} tanh (\frac{ξ ^{2} + Δ ( T ) ^{2}}{2 k _{B} T})

Solving this equation gives $Δ (T)$ , which decreases from its zero-temperature value $Δ (0)$ and vanishes at the critical temperature $T_{c}$ , where the system reverts to the normal state. Near $T_{c}$ , the gap closes as $Δ (T) \propto 1 - T / T_{c}$ , the classic behavior for a second-order phase transition in mean-field theory.

Electronic Specific Heat $C_{e} (T)$

The most dramatic thermodynamic consequence is on the electronic specific heat.

Low Temperature ( $T ≪ T_{c}$ ): In a normal metal, $C_{e} \propto T$ . In a superconductor, the number of thermally available quasiparticles is proportional to $exp (- Δ/ k_{B} T)$ due to the gap. This leads to an exponentially suppressed specific heat:
$C_{e} (T) \propto exp (- \frac{Δ ( 0 )}{k _{B} T})$
This exponential behavior was one of the earliest and strongest pieces of evidence for an energy gap in superconductors.
Near the Critical Temperature ( $T \to T_{c}$ ): As the system transitions from the superconducting to the normal state, the specific heat exhibits a discontinuous jump. BCS theory predicts a universal value for this jump in the weak-coupling limit:
$\frac{C _{e} ( T _{c} ) - C _{n} ( T _{c} )}{C _{n} ( T _{c} )} = 1.43$
where $C_{n} (T_{c}) = γ T_{c}$ is the normal state electronic specific heat. The observation of this jump is considered definitive proof of a bulk superconducting transition.

Typical behavior of the electronic specific heat $C_{e}$ for a superconductor (solid line) compared to a normal metal (dashed line). Note the exponential decay at low T and the sharp, discontinuous jump at the critical temperature $T_{c}$ .

The same gapped quasiparticle spectrum also explains the behavior of other properties, like NMR relaxation rates (which show a characteristic "Hebel-Slichter peak" just below $T_{c}$ ) and ultrasonic attenuation, providing a unified and powerful framework for understanding superconductivity.

Conclusion

In this lesson, we have moved from the abstract ground state to the physically observable properties of a superconductor by analyzing its elementary excitations.

Key Takeaways:

Bogoliubov Quasiparticles: The elementary excitations in a BCS superconductor are gapped quasiparticles, which are superpositions of electrons and holes.
Quasiparticle Spectrum: The energy to create a quasiparticle is $E_{k} = ξ_{k}^{2} + Δ^{2}$ . This creates a minimum energy gap $Δ$ in the excitation spectrum.
Density of States: The gapped spectrum leads to a singular peak in the quasiparticle density of states at the gap edge, which is a key experimental signature.
Thermodynamic Signatures: The energy gap is responsible for the characteristic thermodynamic properties of superconductors, most notably the exponential suppression of the electronic specific heat at low temperatures and the specific heat jump at $T_{c}$ .

Preview of the Next Lesson:

We have now seen how a condensate of fermionic pairs gives rise to collective phenomena. In our next lesson, we will pivot to study a different kind of quantum fluid: a weakly interacting Bose gas. We will derive the Gross-Pitaevskii equation, a mean-field theory that describes the ground state and dynamics of a Bose-Einstein condensate. You will see that despite the different particle statistics, conceptual tools like mean-field theory and the idea of a macroscopic wavefunction are central to understanding both systems.