BCS Density of States and STS

Hello! Let's continue our exploration of BCS theory.

In our last lesson, we successfully constructed the BCS ground state and used the Bogoliubov transformation to find the true elementary excitations of the system: gapped quasiparticles with a dispersion relation $E_{\mathbf{k}}=\sqrt{{\xi }_{\mathbf{k}}^2+|{\Delta }_{\mathbf{k}}{|}^2}$. This gapped spectrum is the single most important consequence of the theory, responsible for virtually all of the hallmark properties of a superconductor.

Today, we will investigate the direct consequences of this gapped spectrum on an observable quantity: the electronic density of states (DOS). We will calculate the new DOS for the superconducting state and then see how it is directly and precisely measured using Scanning Tunneling Spectroscopy (STS). This will provide a powerful and concrete link between the abstract many-body theory we've developed and real-world experiments.

1. Calculating the BCS Density of States

The density of states, $N(E)$, tells us how many electronic states are available per unit energy. In a normal metal, we often approximate the DOS near the Fermi energy as a constant, $N_N(E)\approx N(0)$. The formation of the superconducting gap radically alters this picture.

To find the superconducting DOS, $N_S(E)$, we can use a powerful argument based on the conservation of states. The Bogoliubov transformation is a unitary transformation, which means it reshuffles the states but does not create or destroy them. Therefore, the number of states in a small energy interval $dE$ for the quasiparticles must be the same as the number of states in the corresponding interval $d\xi$ for the original electrons.

We can write this as:

$$N_S(E)dE=N_N(\xi )d\xi$$

Assuming the normal state DOS is constant, $N_N(\xi )=N(0)$, and that we are dealing with an isotropic s-wave gap, $\Delta$, we have:

$$N_S(E)=N(0)\left|\frac{d\xi }{dE}\right|$$

From the quasiparticle energy $E=\sqrt{{\xi }^2+{\Delta }^2}$, we can solve for $\xi$ to get $\xi =\pm \sqrt{E^2-{\Delta }^2}$. This expression only makes sense for $|E|\ge \Delta$, confirming there are no states within the gap. Differentiating with respect to $E$ gives:

$$\frac{d\xi }{dE}=\frac{E}{\sqrt{E^2-{\Delta }^2}}$$

Plugging this back, and considering that both positive and negative $\xi$ contribute to the same energy $E$, we arrive at the famous BCS density of states:

$$N_S(E)=\{\begin{array}{ll}N(0)\frac{|E|}{\sqrt{E^2-{\Delta }^2}}& \text{for }|E|\ge \Delta \\ 0& \text{for }|E|<\Delta \end{array}$$

This result is remarkable. Let's break down its features:

• The Superconducting Gap: There is a complete absence of single-particle states in the energy range from $-\Delta$ to $+\Delta$. This is the energy cost to break a Cooper pair and create two quasiparticle excitations.
• Coherence Peaks: The DOS diverges at the gap edges, $E=\pm \Delta$. These are known as BCS coherence peaks. They arise because all the states that were originally inside the gap in the normal state are "piled up" just outside of it.

For a formal derivation and discussion, the following resource is excellent.

2. Probing the DOS: Normal-Insulator-Superconductor (NIS) Tunneling

How can we be sure this exotic density of states is real? In 1960, Ivar Giaever performed a groundbreaking experiment that measured it directly using the phenomenon of quantum tunneling.

The setup, now a cornerstone of condensed matter experiments known as Scanning Tunneling Spectroscopy (STS), involves a junction between a normal metal (N) and a superconductor (S), separated by a very thin insulating layer (I). This is an NIS junction.

As the video explains, by applying a voltage $V$ across the junction, we shift the chemical potential of the normal metal relative to the superconductor by an amount $eV$. At zero temperature, electrons from the metal can only tunnel into unoccupied states in the superconductor. The tunneling current $I(V)$ is therefore proportional to the integral of the available states in the superconductor:

$$I(V)\propto {\int }_{\Delta }^{eV}{N}_S(E^{\prime })d{E}^{\prime }$$

The key insight is that if we measure the differential conductance, $dI/dV$, we get a quantity that is directly proportional to the density of states at the energy corresponding to the bias voltage:

$$\frac{dI}{dV}\propto N_S(eV)$$

This gives us a direct experimental tool to "see" the BCS density of states. When the bias voltage $|eV|$ is less than the gap $\Delta$, there are no states to tunnel into, so the conductance is zero. As $|eV|$ reaches $\Delta$, tunneling becomes possible, and the conductance shoots up, tracing the coherence peak.

For a more rigorous derivation of this relationship, you can refer to the following notes. Your background in second quantization and Fermi's golden rule will make the argument straightforward to follow.

3. Experimental Confirmation of the BCS DOS

The agreement between the theoretical BCS density of states and the measured $dI/dV$ curves from tunneling spectroscopy is one of the most stunning confirmations of the theory.

This figure beautifully illustrates our main points:

• The measured conductance curves in panel (a) show exactly the predicted features: a flat region (the gap) around zero bias, flanked by sharp peaks (the coherence peaks).
• Comparing panel (a) with the theoretical DOS in panel (b) shows a near-perfect match. The slight broadening of the experimental peaks is due to finite temperature and lifetime effects, which are not included in our ideal formula.
• As the temperature increases, the gap shrinks and the coherence peaks broaden and become less sharp, exactly as predicted by the finite-temperature BCS theory we touched on in the last lesson. Panel (c) shows this temperature dependence of the gap, which closes at the critical temperature $T_c$.

Test your understanding!

You perform an STS measurement on an unknown conventional (s-wave) superconductor at very low temperature. The $dI/dV$ vs. $V$ plot shows two very sharp peaks located at $V=-1.5$ mV and $V=+1.5$ mV, with almost zero conductance between them.

1. What is the value of the superconducting gap $\Delta$ for this material?
2. Based on the BCS prediction $2\Delta (0)\approx 3.52k_B{T}_c$, what is your estimate for the critical temperature $T_c$ of this superconductor? (Use $k_B\approx 8.617\times 10^{-5}$ eV/K).

Show answer

1. The conductance peaks occur at $eV=\pm \Delta$. Therefore, the superconducting gap is $\Delta =1.5$ meV.
2. We have $\Delta =1.5\times 10^{-3}$ eV. The BCS relation is $T_c\approx \frac{2\Delta }{3.52k_B}$. $$T_c\approx \frac{2\times (1.5\times 10^{-3}\text{ eV})}{3.52\times (8.617\times 10^{-5}\text{ eV/K})}\approx \frac{3\times 10^{-3}}{3.03\times 10^{-4}}\text{ K}\approx 9.9\text{ K}$$ So, the estimated critical temperature is approximately 9.9 Kelvin.

Conclusion

Today, we have bridged the gap between the microscopic BCS theory and a key experimental observable. The creation of Cooper pairs leads to a dramatic rearrangement of the single-particle states, which is the direct cause of superconductivity.

Key Takeaways:

• The BCS ground state has a gapped quasiparticle excitation spectrum, which results in a distinct superconducting density of states (DOS).
• The BCS DOS is characterized by a gap of width $2\Delta$ centered at the Fermi energy and divergent coherence peaks at the gap edges $|E|=\Delta$. The functional form is $N_S(E)\propto |E|/\sqrt{E^2-{\Delta }^2}$.
• Scanning Tunneling Spectroscopy (STS) on a Normal-Insulator-Superconductor (NIS) junction provides a direct measurement of this DOS, because the differential conductance is proportional to the DOS: $dI/dV\propto N_S(eV)$.
• The outstanding agreement between STS measurements and the theoretical BCS DOS provides powerful evidence for the validity of the theory.

Preview of the Next Lesson

We have seen that single-particle tunneling into a superconductor is forbidden for energies inside the gap. However, this is not the whole story. A different, two-particle process can still occur at a Normal-Superconductor (NS) interface without an insulating barrier. In the next lesson, we will explore this fascinating phenomenon: Andreev reflection. We will see how an incoming electron can be reflected as a hole, transferring a Cooper pair into the superconductor, and what novel transport signatures this process creates.