Question

In this paper we have the following:

The corresponding quasiparticle density is given by the equation
nqp=4N0∫[infinity]ΔdEEE2−Δ2−−−−−−−√f(E),where N0 is the single-spin density of electron states at the Fermi energy.

Why is there a factor of 4 here, rather than a factor of 2? Given that each quasiparticle has two spin states, and the normalized density of states is rho(E)=EE2−Δ2√ shouldn't we have 2N0?

Reference 21 to the paper, in Appendix A, in fact says:

We define the quasiparticle density Nqp asNqp≡∫+[infinity]−[infinity]dϵ2N(0)f(E),
where E=(ϵ2+Δ2)1/2. N(0) is the normal-state single-spin density of states at the Fermi surface. The factor of 2 takes into account the spin degree of freedom for the electrons.

And in Appendix B:

Here the excess quasiparticle density δNqp is defined as δNqp=∫[infinity]ΔdE4N(0)EE2−Δ2−−−−−−−√[f(E)−f(E,T)].
We note that this is not the quasiparticle density difference between the driven state and the equilibrium state, because we have used the steady-state energy gap in Eq. (B4) [above equation] to calculate the quasiparticle density.

Answer 1

The factor of 4 in the quasiparticle density equation accounts for spin states and the contributions of electron-like and hole-like quasiparticles in a superconducting state. It relates to the Pauli exclusion principle and the statistics governing electron distributions in energy states, especially at the Fermi level.

Step-by-step explanation:

The question seems to arise from a discrepancy in the factors in the expressions for the quasiparticle density in a superconductor. In physics, particularly in the context of superconductivity and electron behavior in metals, understanding the quasiparticle density, density of states, and the Fermi factor is crucial. The factor of 4 in the given expression for nqp can be related to both the spin degeneracy and the electron-like and hole-like excitations that quasiparticles represent in a superconducting state.

The derivation of the density of states, which is fundamental to modeling electronic properties, involves solving Schrödinger's equation and considering the Pauli exclusion principle, as only two electrons with opposite spins can occupy a given energy level. At absolute zero (0 K), all electrons are situated in energy states up to the Fermi energy, filling the lowest available states first due to Pauli's principle, with the Fermi factor indicating the probability of an energy state being filled.

The mathematical expressions and physical principles in the student's question are tied to the underlying Quantum Mechanics governing electronic properties in materials. In general, the density of states increases with energy, which indicates that higher energy levels have more available states for electrons. To comprehend the number of particles within a narrow energy interval, one must consider the product of the density of states and the Fermi factor. Hence, the factor of 4 may account for both spin states and the additional factor due to electron-like and hole-like quasiparticle contributions.