Vortices and chemical potential in topological superconductors

Question

I am trying to read up some review articles about Majorana physics in topological material, but I am not really familiar with the condensed matter terminology (with condensed matter in general I should say) since I come from a high-energy background, so I come across quite a lot of vocabulary and visualization issues.

From the little that I know from semi-conductor physics, a gap is an energy difference between two bands such that a particle cannot go from one band to the other without being given at least that amount of energy. With that picture of band structure in mind, I don't really know what a gap potential is, nor how it can have vortices in it. For instance, in a $p_x + i p_y$ superconductor in 2D, it is said that Majorana fermions appear in vortices in the superconducting pairing potential, or when the gap is closed by variations in the chemical potential. I am wondering if there is an intuitive picture of what a "pairing potential vortex" is, without getting into solving the BdG equations for the superconductor, and on how Majorana fermions actually appear in them?

Moreover, another question that pops into mind is related to the use of chemical potential in the Hamiltonians describing superconductors. Statistical mechanics tells us that the chemical potential is the energy necessary to add one particle to a system from a reservoir, and also conveniently describes the energy costs related to diffusion processes in solutions. How does one interpret the chemical potential in a superconductor then? I have read somewhere that a non-homogeneous chemical potential $\mu(x)$ is a sign of an electric field $E(x) \sim \mu'(x)$, so it seems there would be a relation between $\mu$ and the electrostatic potential, but I don't find anything information that explains the relations between all these quantities in superconductors.

Answer 1

It seems to me that what you need is much more an introduction to superconductivity than to Majorana modes physics. I suggest you to open any book called superconductivity to have more details than the ones I give below.

A standard description of a superconductor consists in saying it is a perfect metal with an electron-electron attractive potential. A perfect metal is just a gas of free electrons, characterised by a Fermi level. In condensed matter it is usually called a chemical potential, since it is the energy you need to give to an extra particle to go into the metal, and so it has the same meaning as in statistical physics. The attractive potential destabilises the Fermi sea at low temperatures, a mechanism called the Cooper instability. Then the Fermi sea is no more a good description of a superconductor, and one should prefer a kind of semi-conductor terminology, since a gap appears at the (previously called) Fermi level. The new energy to add a particle to a superconductor is thus the chemical potential plus the gap energy. The gap is nevertheless (4 to 5) orders of magnitude smaller than the Fermi level (in BCS superconductors), so you can continue to discuss everything in terms of the chemical potential alone.

What you call a pairing potential is usually call the superconducting gap, noted $\Delta\left(x\right)$ and which can effectively be position dependent. When $\Delta\rightarrow 0$ locally, the superconductor hosts a vortex. The superconducting gap corresponds to the mean-field decoupling of the electron pair correlator $F\sim \left<\hat{c}\hat{c}\right>$.

A good way to vary the chemical potential is indeed to apply a voltage drop to the system. This can be done in space dependent fashion. Doping is an other possibility in semi-conductors, but not for metal, by definition of a metal. More precisely, the doping should not change the metallic nature of the system (that's only true for conventional / BCS superconductor, or to simplify to mono-atomic metals).

To understand the role of the chemical potential for the Majorana modes, you need the formula $$E_{0}=\left|B-\sqrt{\Delta^{2}+\mu^{2}}\right|$$ which tells you that the effective gap of a Zeeman ($B$-field) plus spin-orbit plus superconducting wire of initial (without Zeeman and spin-orbit say) gap $\Delta$ depend on the chemical potential. An edge modes (call them topological mode if you wish) will appear for a gap inversion. People are usually saying you need to close and reopen the gap. So you can do that in practise by a space-dependent chemical potential.

More details about the above formula:

Oreg, Y., Refael, G. & von Oppen, F. Helical liquids and Majorana bound states in quantum wires. Phys. Rev. Lett. 105, 177002 (2010) or arXiv:1003.1145.

The detailed calculation of the edge mode due a gap inversion in semi-conductor is done in:

Volkov, B. A. & Pankratov, O. A. Two-dimensional massless electrons in an inverted contact. JETP 42, 178 (1985).

More details about the space dependency of the chemical potential:

Alicea, J., Oreg, Y., Refael, G., von Oppen, F. & Fisher, M. P. A. Non-Abelian statistics and topological quantum information processing in 1D wire networks. Nat. Phys. 7, 412–417 (2011) or arXiv:1006.4395.