FroelichCooper

Introduction

Crucial experimental evidences:

• Below {$T_c$} a gap opens in electronic excitations. This is shown by many experiments. Historically the key evidence came from the specific heat, dominated in the normal metal by the small Sommerfeld linear term {$c=\gamma T$}. It jumps to a larger value at {$T_c$} predicted by Landau theory as {$c = (a^2/b + \gamma)T$}. The jump itself is a subtle effect, also predicted by BCS. At {$T\ll T_c$} {$c(T)$}, contrary to Landau's model, {$c(T)$} falls exponentially below Sommerfeld {$\gamma T$} line, as per a finite gap.
• Flux is quantized. In Deaver and Fairbanks experiment on a superconducting tube, magnetic field flux penetrates by integer numbers of the flux quantum {$\Phi_0 = h/q = 2\, 10^{-15}$} Wb, so that the charge {$q$} of the supercarriers turns out to be {$|q|= h/\Phi_0 = 3.2 \, 10^{-19}$} C, i.e an electron pair.
• The critical temperature of isotopically pure samples (e.g. {$^{113}$}Sn and {$^{123}$}Sn scales with the square root of isotope mass ratio. This suggests that transition temperatures are proportional to Debye temperatures, hence that phonons play an important role.

The BCS model captures the mechanism of superconductivity in its simplest form: it supposes a weak attraction among electrons, that gives rise to the formation of weakly bound pairs of electrons. When thermal energy is below some scale {$k_B T_c$}, proportional to the binding energy, the Fermi gas (or the Fermi liquid) is unstable. A state with electrons coupled in pairs has lower energy, as demonstrated by a variational approach, with respect to the Fermi ground state. The page discusses the origin of the tiny attractive interaction and the proof produced by Leon Cooper that two electrons can bind however small the attractive coupling. This is surprising in view of the fact that a point-like Coulomb potential does bind an electron in vacuum, but notably an external positive point charge in the Fermi gas fails to produce bound states. A clever feature of the Cooper model is the key to successfully justify a pair binding and these notes go through a simplified variational Hamiltonian and its trial wave function that shows this.

A final introductory remark is on BCS model place in the universe of superconductivity. The actual model describes a simplified weak coupling, and a specific one, due to phonons, and sketched in Fig. 1. Electron-electron attraction mediated by the electron-phonon interaction is actually responsible for all early discovered superconductors, until the cuprates, and for some of the most recent as well (fullerides, MgB₂, H₂S under high pressure). The scheme, however, requires an electron attractive (or pairing) mechanism of any strength (e.g strong, instead of weak coupling) and it is valid also when considering other mediating interactions, e.g. spin coupling, or other neglected terms in the electron-electron interaction as well.

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Fig. 1 Cartoon description of electron-phonon interaction producing a net attraction between two electrons: an electron at time zero sets off nearby positive ions in a motion towards its position; the motion results in a tiny distortion at later times (retarded), equivalent to a higher positive charge density; a second electron is attracted towards this enhanced positive charge.[Figure from Bristol Univ. Superconductivity ]

Electron-phonon coupling

The electron-phonon coupling provides an example of interaction that can result in an attractive effective electron electron potential, in second order perturbation. The process is the following: imagine an electron Bloch state {$k$} scattered to {$k^\prime$} in an inelastic first order perturbation process, with creation of a phonon of wave vector {$q$}. The Fermi golden rule rate is {$$ \frac{|V_{k,k^\prime-q}|^2}{\epsilon_k-\epsilon_{k^\prime}-\omega_q}$$} The interaction potential {$V$} between electrons and phonons is so tiny that it is totally neglected when calculating both electron and phonon eigenstates. It originates from the non-adiabatic term, ignored in the Born-Oppenheimer approximation. The phonon emission process calculated above does not necessarily conserve crystal momentum and energy, but the second order process where the phonon is emitted and then re-absorbed by the {$k^\prime$} electron does. Notice that {$\epsilon_k-\epsilon_{k^\prime}= \hbar\omega$} must be of order {$k_BT$} since it goes from an occupied state to an empty state at the surface of a Fermi sea. The Fermi golden rule rate of the re-absorption is like that of the emission, with the same matrix element square, call it {$g = |V_{k,k^\prime-q}| $} for simplicity, but the denominator has {$+\omega_q$} in place of {$-\omega_q$}. The rate of the second order process is then

{$$ \frac {g^2}{\hbar^2(\omega^2-\omega_q^2)}\approx - \frac {g^2}{\hbar^2\omega_D^2}$$}

The approximation is justified because in this process, at the low temperatures where superconductivity takes place, there is a well defined hierarchy of energies, from tiny to very large: {$g\ll k_B T \ll \omega_q\approx \omega_D\ll E_F$}, where {$\omega\ll\omega_D$} is the Debye frequency, and we may neglect {$\omega$} in the denominator. The rate is equivalent to a negative zero order elastic matrix element {$\langle k|V_{\mathrm {eff}}|k\rangle$}, i.e. and effective attractive potential, {$-V_0$} {$$-V_0=-\frac {g^2}{\hbar^2\omega_D^2}$$} representing an effective attractive interaction between two electrons.

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Cooper pair

The simplest model capturing the essence of the BCS mechanism is given by the following Ansatz. Consider two electron states with k and -k just above the Fermi energy {$E_F$}, for which {${\cal H}_0|\pm k\rangle = \epsilon_k|\pm k\rangle$}, such that they form an antisymmetric total wave function, e.g. {$$ |\psi_k\rangle = |k\rangle |-k\rangle\frac{|\uparrow\downarrow\rangle-|\downarrow\uparrow\rangle}{\sqrt2}.$$} In the following the spin wave function is ignored since the Hamiltonian of the system {${\cal H}={\cal H}_0+V$} is spin independent. Here all effective periodic potentials and the mean field electron repulsion are included in {${\cal H}_0$}, but the weak second order attractive, phonon-mediated interaction {$V$} is not.

Now consider the trial state {$|\Psi\rangle =\sum_k \phi_k |\psi_k\rangle$}. The matrix elements of the Hamiltonian between {$\langle \psi_k|$} and {$|\Psi\rangle$} obey the following equation {$$2\epsilon_k\phi_k+\sum_{k'}V_{kk'}\phi_{k'}=E\phi_k$$} where {${\cal H}_0|k\rangle =\epsilon_k|k\rangle$} and {$E$} is the variational minimum energy for {$|\Psi\rangle$}. We wish to show that these two states make a bound state, i.e. that {$E$} is less than the starting value {$>2E_F$}

Rearrange the matrix equation to give

{$$\sum_{k'}V_{kk'}\phi_{k'}=(E-2\epsilon_k)\phi_k$$}

and assume the simple form of the effective potential described above, {$-V_0$} for {$0\le\epsilon_k\le\hbar\omega_D$}, zero otherwise. The matrix equation reduces to {$$\frac {\phi_k} {V_0} = \frac {\sum_{k'}\phi_{k'}}{E-2\epsilon_k}$$} which can be summed over {$k$} to yield, after simplifying the two sums over {$\phi_k$}

{$$ \frac 1 {V_0} = \sum_k\frac 1 {2\epsilon_k-E}$$}

This is a simplified version of the BCS equation. The right hand side may be calculated converting the sums over {$k$} into an integral over energies {$\epsilon$} containing the density of states. Remember that we expect a very weakly bound state, with {$\hbar\omega_D\gg E_F-E/2>0$}

{$$\int_{E_F}^{E_F+\hbar\omega_D} \frac {g(\epsilon)d\epsilon}{2\epsilon-E}=g(E_F)\ln \frac{2E_F-E+2\hbar\omega_D}{2E_F-E}\approx g(E_F)\ln\frac{\hbar\omega_D}{E_F-E/2}$$}

Taking the exponential of both members and defining a dimensionless pseudopotential {$\lambda = g(E_F)V_0$}, a rearrangement provides a gap

{$$ \Delta = 2E_F - E = 2\hbar\omega_D \mathrm{e}^{-\frac 2 \lambda}$$}

The crucial point here is that the right hand side is positive defined, hence the energy of the pair, {$E$}, is lower than {$2E_F$}. At low temperature pairs are more stable than free electrons very close to the Fermi surface.

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Consequences of Cooper pairs

The gap equation above (a mock one, see here for the true BCS gap equation) shows that, however small the effective potential {$V_0$}, a solution exists with energy smaller than the unperturbed pair energy in the metal, {$2E_F$}, by a binding energy, the gap {$\Delta$}. This is a very small interval of forbidden energy values for normal electrons that opens at the Fermi surface. It would not exist for two electrons in free space, where week potentials are not bonding, and it is made possible by two simultaneous key ingredients:

1. the presence of the Fermi sea, and
2. the attractive, phonon-mediated, effective potential

Its onset below {$T_c$} produces a condition that is reminiscent of a small gap semiconductor, in that electronic excitations decay exponentially for {$T<\Delta/k_B$}. However, its ground state is composed of completely filled (condensed) Cooper pairs, almost like composite bosons. Therefore it displays extraordinary properties (totally unknown in normal bands, be them insulators, small gap semiconductors or half-filled metal bands), such as the zero electrical resistance, the Meissner effect, the jump in heat capacity at {$T_c$}, etc.

For instance zero resistance is the consequence of the gap: the pair wave function is coherent until it breaks by scattering, that however must produce an excitation (two normal unbound electrons) of finite energy cost {$\Delta$}. an event exponentially less probable as {$T$} lowers. Zero resistivity implies perfect conduction, but the Cooper pair wave function obeys also the second Ginsburg-Landau equation, with flux quantization. This constraint between the phase of the complex wave function and the vector potential {$\boldsymbol A$} implies gauge symmetry breaking, i.e. the magnetic flux quantization and the Meissner effect.

One important fact to keep in mind is that, the more the metal is a simple one, the more the {$|k\rangle|-k\rangle$} terms in the Ansatz state are delocalized waves. That is way the radius of the bound pair state, roughly equal at {$T=0$} to the {$\xi$} G-L parameter, may be very large. This justifies the cartoon of Fig. 1: an electron passing, attracting a very slow ion, the next electron passing long after to still see the ion displacement. This is called a retarded interaction, and it is the reason why a state formed of Cooper pairs cannot really be called a composite boson.

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