M2-ICFP 2017/2018 the BCS Theory of Superconductivity (Bardeen – Cooper – Schrieffer, 1957) This Chapter Content: 1. Idea Of

M2-ICFP 2017/2018 The BCS theory of superconductivity (Bardeen – Cooper – Schrieffer, 1957)

This chapter content: 1. Idea of attractive electron-electron interaction. 2. Cooper bound state. 3. Schrieffer’s many-fermion wavefunction. Off-diagonal long range order. 4. The mean-field BCS Hamiltonian Occupation factors. Gap function. 5. Bogolubov transformation* (* - treated as tutorial). 6. Ground state energy. Density of elementary excitations. Coherence length. 7. Predictions of the BCS theory.

1. Idea of attractive electron-electron interaction.

The first idea in the BCS theory is the existence of an effective attractive interaction between electrons near Fermis surface (Frölich, 1950). Where it comes from?

Bare electron gaz – Coulomb repulsion:

In the solid, there is a screening due to ions and other electrons. The electrons are not “bare” anymore, but “quasiparticles” (excitations consisting of a moving electron together with interactions (for example, exchange-correlation hole – as other electrons avoid the same place – by both, Coulomb and Pauli)).Due to the screening the quasiparticles interact weakly (Thomas-Fermi):

In good metals the Thomas-Fermi screening length 푟푇퐹 < 1푛푚. It means that the repulsive interaction between quasiparticles (we will call them abusively “electrons”, as they carry charge “e”) vanishes for |푟 − 푟′| > 푟푇퐹.

Now, the electrons interact with crystal lattice, they absorb and emit phonons. How does electron-phonon interaction arise? Usually the phonons are modelled by the effective Hamiltonian – a set of quantum harmonic oscillators, one for each vector 풒 and phonon mode (reminder: there are 3푁푎 phonon modes in a crystal with 푁푎-atoms per unit cell):

+ where 푎풒휆and 푎풒휆 are creation and annihilation operators of the phonon mode 휆.

Now imagine an electron of a state │푘1⟩ to scatter to a state │푘′1⟩. One of the channels to do so is to emit/absorb a phonon of momentum 풒 / −풒: ′ 풌1 = 풌1 + 풒

Microscopically, the electron scattering │푘1⟩ → │푘′1⟩ may be seen as a creation of an oscillating charge density (due to the beats between the initial and final state), with a 휀 − 휀 frequency 휔 = 푘1 푘′1 , and the spatial periodicity 2휋/푞. This charge density excites the ℏ lattice of ions. For instance, in locations when an excess electron density appears, the ions will tend to approach (via attractive Coulomb term). This results in a periodic (in space and in time) oscillation of ions. Now, the frequency of this forced oscillation is 휔, but its phase depend on 휔0. Indeed, instead of considering a full Hamiltonian of phonons, let us model a lattice as a simple classical oscillator with the self-frequency 휔0. The extra electron density due to scattered electrons can be seen as an external force 푒푖휔푡 . This results in a simple equation of motion for an ion: 푓 푥 + 휔2푥 = 푒푖휔푡 0 푀 푖휔푡 The solution is 푥 = 푥0푒 , with: 푓 1 푥0 = 2 2 푀 휔0 − 휔

For 휔 < 휔0 the motion of the atoms will be in phase with 푓. This means that higher concentration of ions will appear where higher electronic density was.

N.B. Notice that the ions being much heavier than the electrons, the lattice oscillations may continue even if the initial “force” due to electron charge density already disappeared.

The process of the electron scattering and consequent lattice distortion is the phonon emission. It may be represented by a vertex of a Feynman diagram: 풌1 풌′1 = 풌1 − 풒

풒

The locally created higher ion density may attract a second electron, let say with the initial momentum 풌2. However, this attraction is not very efficient as in the real space the wavelength of the electron is 2휋/풌2 while that of the lattice oscillation - 2휋/푞 . However, if the second electron scatters in the location of the lattice distortion and acquires a new momentum 풌′2 = 풌2 − 풒, the attraction will be stronger since the spatial extend of the electronic wave beat matches the lattice distortion period. This process may be represented as a vertex of a Feynman diagram: 풒

풌2 풌′2 = 풌2

Putting the two vertexes together we get the Feynman diagram for electron-electron interaction via the phonon exchange: 풌1 풌′1 = 풌1 − 풒 푔풒휆

풒

풌2 풌′2 = 풌2 + 풒 푔풒휆

N.B. In this quantum mechanical process there is no dissipation, as a part of the energy of the first electron is transmitted as a quantum (phonon) to the second electron.

N.B. The phonon-assisted electron-electron interaction is “retarded” as it involves the motion of heavy (푀) atoms. Also, the characteristic time of this interaction is phonon times (nanoseconds) and not electron times (femtoseconds).

N.B. The electron-phonon vertex 푔풒휆 is considered the same for direct and inverse processes.

In our classical model the interaction remains attractive only for 휔 < 휔0, and becomes repulsive for 휔 > 휔0 . In a real solid not one but 3푁푎-modes exist, indexed as 휔풒휆(wave vector 풒 and phonon mode 휆). The quantum mechanical effective interaction between the electrons due to exchange of a virtual phonon {풒, 휆} turns out to be of the form similar to our classical considerations above:

i.e. it is attractive for 휔 < 휔풒휆. The electron-phonon vertex 푔풒휆 comes in square, as two vertexes enter the process. An important result due to Migdal is that:

where 푚 is the effective mass of the electron and 푀 is the mass of the ion. We note, that 1 our classical model nicely captures 푉 ~ behavior (see the above expression for 푥 ) ! 푒푓푓 푀 0

N.B. Since 푚/푀 is very small, the electron-electron attraction is supposed very weak.

It is clear that considering all existing {풒, 휆} phonons would be very complicated (incompatible with analytic calculations!). That is why BCS strongly simplified the 푉푒푓푓. They considered, that since in a solid the phonon frequencies are limited by Debye frequency, 휔퐷 (typically, ℏ휔퐷~20푚푒푉, 푇퐷 = ℏ휔퐷/푘퐵 ∼ 200퐾), one can replace 푔풒휆 by 푔풆풇풇 : 2 2 휔퐷 푉푒푓푓 = |푔푒푓푓| 2 2 휔 − 휔퐷 i.e. the interaction is attractive for all frequencies up to 휔퐷. Moreover, since the superconductivity takes place at temperatures much lower than the Debye energy, 푘퐵푇 ≪ ℏ휔퐷, BCS yet simplified 푉푒푓푓 taking it frequency independent:

The corresponding Hamiltonian for the effective electron-electron attraction is:

(mistake in the formula 휎1(2) as no spin flip is expected). the sum is done for all pairs of 휀 − 휀 electrons │푘 ⟩, │푘 ⟩ for which 휔 = 풌1+풒 풌1 < 휔 . The effective Hamiltonian kills two 1 2 ℏ 퐷 states │풌1⟩, │풌2⟩ and creates (scatters them to) two new states │풌1 + 풒⟩, │풌2 − 풒⟩. To be effective, the initial states │풌1⟩, │풌2⟩ should be occupied, and the final states │풌1 + 풒⟩,

│풌2 − 풒⟩ – empty. At low temperature this is impossible for the electronic states deep below 휖퐹 as all such states are occupied. This is also impossible for the unoccupied states far above 휖퐹, as all of them are empty. Therefore, the effective electron-electron attraction is only possible for the electron energies 휖푘푖 within the energy belt ±ℏ휔퐷of the Fermi surface

|휖푘푖 − 휖퐹| < ℏ휔퐷:

2 2 |푔풒휆| 1 N.B. Isotope effect. The effective interaction should be 푉푒푓푓 = −|푔푒푓푓| ~ 2 ~ 2 . But 휔퐷 푀휔퐷 2 휔퐷~푘/푀 (푘 is the effective lattice spring constant). Thus, in the BCS model 푉푒푓푓 does not −훼 depend on ion mass! What about the experimentally observed isotope effect, 푇푐 ∝ 푀 ? As we will see later, the isotope effect originates from the energy width ~2ℏ휔퐷 around the Fermi energy over which the electrons form Cooper pairs. This energy width varies with the ion mass as ∝ 푀−1/2.

2. Cooper bound state (Cooper pairs). The second step in the BCS theory was done by Cooper who found that the effective attractive interaction between two electrons above the Fermi sea, 휖푘푖 > 휖퐹, leads to a bound state of a pair (Cooper pair) with an energy per electron < 휖퐹!!

Assume two electrons above the Fermi sea in a thin belt ℏ휔퐷, i.e. (휖푘푖 − 휖퐹 < ℏ휔퐷). Consider = 0퐾 , i.e. all states with 푘 < 푘퐹 are occupied:

The two-particle wave function of these extra electrons is:

where 푹푐푚 = (풓1 + 풓2)/2 is the position of the centre of mass, and 풌푐푚 is the total momentum of the pair, and 휑(풓1 − 풓2) describes the mutual motion of two electrons in the centre of mass coordinates. Cooper considered that the minimum energy of the pair occurs when 풌푐푚 = 0, i.e. the pair does not move.

N.B. There are two reasons for why it is OK. 1 – any centre-of-mass motion leads to an additional kinetic term in the energy. 2 – In order to be efficient, we need a lot of electrons to participate in the effective Hamiltonian (6.11). Let us consider the two states 풌1and 풌2 in a free electron gas (Sommerfeld) which scatter to the two states 풌′1and 풌′2. Since the total momentum is conserved, ℏ풌1 + ℏ풌2 = ℏ풌′1 + ℏ풌′2 = ℏ풌푐푚, the number of free electronic states participating in the scattering process (6.11) depends on 풌푐푚:

It is clear, that at 풌푐푚 = 0 all electrons of the belt participate in the process (6.11), while their number is rapidly reduced with rising 풌푐푚.

Now, let us examine the remaining parts of the wave function. In principle, the spin part of the wave function can be either spin singlet:

or triplet:

Cooper considered only singlet pairing.

N.B. While several hundreds of superconducting compounds were already discovered, 99.9% are the singlet ones, only a few are probably (!) triplet superconductors. The search for triplet pairing is one of the most important topics in superconductivity, as using it one can have superconductivity to survive in strong magnetic fields, and also create unusual (Majorana fermion-like) excitations.

Fermion antisymmetry implies that

Since the spin singlet is an odd function of 휎1 and 휎2, the wavefunction 휑(풓1 − 풓2) should be even, 휑(풓1 − 풓2) = +휑(풓2 − 풓1). We can expand 휑(풓1 − 풓2) in terms of planar waves:

where 휑풌 are expansion coefficients to be found; in general, 휑풌 = 휑−풌 because 휑(풓) is even. The total singlet wavefunction can therefore be presented as a sum of Slater determinants:

푖풌풓 where 휓풌(풓) = 푒 . Because the Fermi sea is fully occupied for < 푘퐹 , the summation is for 푘 > 푘퐹 .

N.B. This Slater determinant only couples the states with opposite momenta and spins!

We can also write in terms of vectors:

with:

Putting │Ψ⟩ in the Schrödinger equation 퐻│Ψ⟩ = (퐸 + 2휖퐹)│Ψ⟩ , and multiplying on the left by⟨Ψ풌│ one picks out the terms for a given 풌. It gives:

(I think in this expression we have a factor of 2 in the second term, since we had a double sum there). where 퐸 is the total energy of the two-particle state (with respect to 2휖퐹). Notice that in the Schrödinger equation the energy 휖풌 of the state 풌 is measured with respect to the Fermi energy 휖퐹, 휖풌 → 휖풌 − 휖퐹. Also, 휖풌 = 휖−풌 . N.B. The sum is taken for 푘′ > 푘퐹 but only until a wave vector corresponding to the upper energy limit, 휖퐹 + ℏ휔퐷, as above the effective BCS interaction is zero.

The energy 퐸 can be found from self-consistency. Let us define 퐶 = ∑푘 휑풌 (such a sum appears in the right part of (6.18) ). Evidently 퐶 = ∑푘′ 휑풌′. Putting 퐶 into (6.18) we get:

Taking a sum ∑풌 over (6.23) gives:

and finally:

Converting the sum ∑풌 훾풌(휖) to an integral over the density of states ∫ 훾푘(휖) 푔(휖)푑휖 we obtain:

Finally, we obtain:

2 , where 휆 = |푔푒푓푓| 푔(휖퐹) ≪ 1 is the electron-phonon coupling parameter.

− ퟐ⁄ − 1⁄ −퐸 = 2ℏ휔퐷푒 휆 , and not −퐸 = 2ℏ휔퐷푒 휆 (6.27*)

N.B. In these calculations we simplified 푔(휖) = 푔(휖퐹) for the whole integration window |ℏ휔퐷| ≪ 휖퐹. This is well justified for good metals, as 휖퐹 ≫ ℏ휔퐷. In other conductors (semimetals, doped semiconductors, correlated electronic systems, etc.) such a simplification is not valid.

This is a very remarkable result: 1. We started with two electrons with initial total energy > 2휖퐹 , added a weak attractive interaction, and we ended up with a bound state with the total energy 퐸 < 2휖퐹! This is always true even if the electron-phonon interaction is extremely small.

2. This means that each time the system “creates” two electrons above the Fermi sea – it lowers its total energy with respect to the initial Fermi-Dirac one-electron distribution, even at 푇 = 0!! Indeed, if the system “takes” two electrons from the

occupied levels (휖 < 0, 𝑖. 푒. < 휖퐹) and “sends” them to the two states 풌 and – 풌 above the Fermi sea, the kinetic energy of such a system is increased by at least 2휖푘 - the kinetic energy of two non-interacting (free) electrons above the Fermi level. However, due to the pairing interaction, these two electrons form a bound state with 2 the negative (potential, binding) energy 퐸 = −2ℏ휔 exp (− ), i.e. with respect to 퐷 휆 their initial state 풌 and – 풌 the energy of these two electrons has lowered by 2 2휖 + 2ℏ휔 exp (− ). The balance is therefore negative – i.e. the paired state is TD 푘 퐷 휆 more stable than the initial Fermi sea with non-interacting electrons. Therefore, the system with a weak attractive interaction will “pump” pairs of electrons above the Fermi level and transform them in pairs. It immediately tells us that the usual Fermi- Dirac distribution of electron energies (a step function at 푇 = 0) is not the optimal one! Specifically, even at T=0 the system will enlarge the distribution around the Fermi level in order to lower its total energy by forming Cooper pairs of electrons. The picture below represents the electron distribution in 풌-space after creating two electron and two hole pairs:

In principle, at 푇 = 0 the system will try to convert all available electrons near the Fermi level into pairs.

3. Noticeable, by “pumping” pairs of electrons from the occupied states to the empty states in order to create Cooper pairs, the electron system creates pairs of holes below the Fermi sea. These pairs may also have appropriate momenta; taking into account the electron-hole symmetry, the holes can also form Cooper pairs!

4. Interestingly, the Cooper bound state exists only because there is the filled Fermi sea below the two considered electrons! In general, in 3D an attractive interaction does not necessarily leads to a bound state. The presence of the Fermi sea is therefore an essential gradient of BCS theory.

3. Schrieffer’s many-fermion wavefunction. Off-diagonal long range order (please read Chapter 5 of Annett’s book before working this out) On the basis of Cooper’s result, BCS understood that the Fermi sea is unstable with respect to Cooper pair formation. The main problem was how to build a many-body wavefunction out of paired fermions. A straightforward way would be to write an antisymmetric 푁 particle function out of pair functions:

where the two electron wavefunction Ψ(풓1, 휎1, 풓2휎2) is (see also the expression (6.12) above):

, and the sum over 푃 denotes the sum over all the 푁! permutations of the 푁 particle 푃 labels 풓1, 휎1, 풓2휎2, etc. The sign (−1) is positive for an even permutation and —1 for odd permutations. This alternating sign is necessary so that the many-body wave function has the correct fermion antisymmetry

The problem with such a wavefunction is that it contains a fixed (푁) number of particles. It dos not allow for any ∆푁, and therefore, it cannot have a definite overall phase, since ∆휃∆푁~1.

The third step in building the BCS theory was done by Schrieffer who suggested to write the wave function in a form of a coherent state (see Chapter 5, sec.5.2, 5.7), similarly to a coherent state made of harmonic oscillator wavefunctions 휓푖(푥):

, with

In fact, to solve the latter problem, one creates annihilation and creation operators:

These have remarkable properties:

It is therefore possible to write any wavefunction 휓푛(푥) by applying 푛-times the creation (ladder) operator to the ground state 휓0(푥):

It was shown (first in optics of lasers) that is possible to create a many-body coherent state │α⟩ as:

The remarkable property of this state is made of an infinite number of eigenstates but the number of particles 푁 is not fixed. Consequently, such a state is phase coherent (see Chapt. 5 of Annett’ book). Since the number of particle is huge, this is a macroscopic coherent state.

The constant 퐶 is found from:

Using the ladder relation (5.12) it is possible to rewrite this state as:

, or even in a yet shorter form:

+ By analogy, Schrieffer used fermion creation/annihilation operators 푐̂ 푘↑(↓) and 푐푘̂ ↑(↓) which change the occupation of a given single electron state:

which (because of Pauli exclusion principle) have several properties:

Starting from Cooper problem, Schrieffer introduced a pair creation operator:

which creates, out of vacuum, a pair of electron states with zero total momentum and zero total spin. Then, by analogy with the coherent state in the laser system, eq. (5.16), Schieffer suggested a coherent trial function made of pairs of electrons

where the sum is made over all 풌 which give effective attraction (6.10), and 훼풌 are complex parameters to be found by minimization of the total energy of the system.

Unlike ladder operators, the fermion operators have different properties that decide the + properties of 푃̂ 풌, and 푃̂풌. Unlike Bosonic operators, they not obey Bose rules:

, yet they do commute:

+ 2 and, importantly, (푃̂ 풌) = 0 :

+ + since 푐̂ 푘↑푐̂ 푘↑ = 0 (Pauli exclusion).

Using these properties, the Schrieffer’s trial wavefunction

is simplified to:

The normalization constant can be found from 1 = ⟨Ψ퐵퐶푆|Ψ퐵퐶푆⟩: ∗ ̂ ̂+ 2 1 = ∏⟨0풌│(1 + 훼풌푃풌) (1 + 훼풌푃 풌)│0⟩ = ∏(1 + |훼풌| ) 풌 풌

We can finally rewrite the normalized BCS state in a form:

where

∗ 1 푢풌 = 2 1/2 (6.39) (1+|훼풌| )

∗ 푎풌 푣풌 = 2 1/2 (6.39) (1+|훼풌| )

(Attention: incorrect (6.39) and (6.40) in the early editions of the book!) and where

N.B. This is not the unique notation for 푢풌 and 푣풌, but the most widely used in the modern literature. ∗ N.B. Notice that in this notation 푢풌 is a fully real number, 푢풌 = 푢풌. N.B. The correct parameters 푢풌 and 푣풌 are found by minimizing the total energy of the superconducting state.

As we discussed before, the quasiparticles – holes can also form pairs. Therefore, the full BCS wavefunction should be

4. The mean-field BCS Hamiltonian. Occupation factors. Gap function. The mean-field BCS Hamiltonian. The trial wavefunction should be put in the Schrödinger equation with the many-body BCS Hamiltonian:

which, considering only the most efficient 풌 ↑, −풌 ↓ pairs becomes:

The total energy, which depends on parameters 푢풌 and 푣풌, is then:

The total energy of the TD stable state is the minimum E over all possible parameters 푢풌 and 푣풌, considering the total mean number of particle 〈푁〉 fixed.

Occupation factors Let us give a physical meaning to parameters 푢풌 and 푣풌. For instance, the average occupation of Bloch state 풌 ↑ is

2 , and also 〈푛̂푘↓〉 = |푣풌| . By applying the commutation rules one finds:

(factor of 2 – two possible spins per state 풌). Similarly, the kinetic energy is

The expectation value of the interaction part of the BCS Hamiltonian can be calculated using

The total energy expressed in terms of 푢풌 and 푣풌 is therefore

Now, we should find sets of parameters 푢풌 and 푣풌 which minimize (6.50) while keeping 〈푁〉 constant. In Annett’s book (Sec.6.5, p.p.137-139) the method of Lagrange multipliers is used, ∗ ∗ differentiating over 푢풌 and 푣풌 as independent variables. The result is that the total energy is minimized when

where 휇 is the chemical potential. Here we introduced new parameters:

and the BCS gap parameter ∆

Notice, that the BCS gap does not depend individually on 풌 or on 푢풌 and 푣풌. It is a value in energy units.

Here a similar result can be found from (6.50) taking into account that 푢풌 is fully real number ∗ (6.39), 푢풌 = 푢풌. We shell follow Shmidt and simply and differentiate (6.50) with respect to a 2 particular |푣풌| term. At the TD equilibrium: 휕퐸 2 = 0 휕(|푣풌| )

From (6.50) we write: ∗ 2 휕(푣풌푢풌) ∗ 0 = 2휖풌 − 2|푔푒푓푓| 2 ∑ 푣풌′푢풌′ 휕(|푣풌| ) 푘′ ∗ 휕(푣풌푢풌) ∗ 2 Let us evaluate the term 2 . Since 푢풌 is real, 푣풌푢풌 = 푣풌푢풌 = 푣풌√1 − |푣풌| = 휕(|푣풌| ) 푖휃풌 2 2 푒 √|푣풌| √1 − |푣풌| . Here 휃풌 is the complex angle of 푣풌 which is a priori independent on 2 |푣풌| . The derivation results in 휕(푣 푢∗ ) 푒푖휃풌 1−2|푣 |2 1 1−2|푣 |2 1 1−2|푣 |2 풌 풌 = 풌 = 풌 = 풌 . Putting this into the equation gives 2 2 2 −푖휃풌 ∗ 휕(|푣풌| ) 2 √|푣풌| √1−|푣풌| 2 푒 |푣풌|푢풌 2 푣풌푢풌 ∗ 2 푣풌푢풌 |푔푒푓푓| ∗ 2 = ∑ 푣풌′푢풌′ 1 − 2|푣풌| 2휖풌 푘′ or, using the BCS gap definition (6.55), ∗ 푣풌푢풌 ∆ 2 = 1 − 2|푣풌| 2휖풌

Now, let us multiply the left and the right part of this equation by its conjugate (usually ∆ is real, but we can keep for generality, ∆∗∆= ∆2): ∗ 2 2 푣풌푣풌푢풌 ∆ 2 2 = 2 (1 − 2|푣풌| ) 4휖풌 2 Using (6.41) and substituting 푢풌 we obtain a quadratic equation for |푣풌| : ∆2 (| |2)2 | |2 푣풌 − 푣풌 + 2 2 = 0 4(휖풌 + ∆ )

2 2 We can also define 퐸풌 = √휖풌 + ∆ (equivalent to (6.58) in Annett,

, with 휇 = 0 in our case, and write

2 2 2 2 ∆ (|푣풌| ) − |푣풌| + 2 = 0 4퐸풌

The result is 2 1 휖풌 |푣풌| = (1 − ) 2 퐸풌 and consequently, 2 1 휖풌 |푢풌| = (1 + ) 2 퐸풌

2 2 The occupation numbers |푢풌| and |푣풌| .at 푇 = 0 are shown in the figure below. For the 2 distribution of the electrons with energy (or 푘) |푣풌| evolve around the chemical potential on a scale of ~2∆. It is therefore smeared out with respect to the step-like Fermi-Dirac distribution of non-interacting electrons (red dashed line).

Gap function. The BCS gap can be evaluated from the known self-consistency equation and BCS gap equation. From our result ∗ 푣풌푢풌 ∆ 2 = 1 − 2|푣풌| 2휖풌 and 2 1 휖풌 |푣풌| = (1 − ) 2 퐸풌 we immediately get:

Putting this in the expression (6.55) gives

This is very important equation in the BCS theory as it allows one to estimate the gap ∆. Exactly as in Cooper problem we can take attention to sum for 푘′ > 푘퐹 but only until a wave vector corresponding to the upper energy limit, 휖퐹 + ℏ휔퐷 . Then we replace

and obtain

The expression for the BCS gap energy has a similar form with respect to the binding energy of a single Cooper pair (6.27); there is however an important factor of 2 in the exponent, i.e. the BCS gap |∆| is much larger than the binding energy of a single Cooper pair. In fact, the BCS gap IS NOT the binding energy of a single Cooper pair!

We can extend the above calculation to finite temperature, 푇 > 0. For this we should modify the formula (6.50). The elementary excitations are fermions; they occupy the energy states respecting the Fermi-Dirac distribution 1 푓 = 풌 퐸 exp ( 풌 ) + 1 푘퐵푇

N.B. in this expression we use 퐸풌 (and not 휖풌) since the fermions are elementary excitations and not the paired electrons.

In (6.50) the first term generates pairs of electrons (풌; −풌). Let us estimate the occupation factor of this state. The probability that the electron state 풌 is occupied is 푓풌. The probability that one of the two necessary pair states, i.e. 풌 or – 풌 is occupied is simply 2푓풌. With this probability the states 풌 cannot contribute to “receive” a pair from the vacuum. Thus, the value 1 − 2푓풌 is then the probability that the pair of states (풌; −풌) is empty, i.e. these 2 states are ready to “receive” the electrons “generated” by |푣풌| from “vacuum”. In the second term the factor 1 − 2푓풌 should appear 2 times, once for the pair of states (풌; −풌), and second time for the pair of states (풌′; −풌′). We can then rewrite (6.50) in the form 2 2 ∗ ∗ 퐸 = ∑ 2휖풌|푣풌| (1 − 2푓풌) − 2|푔푒푓푓| ∑ 푣풌푢풌 (1 − 2푓풌) ∑ 푣풌′푢풌′(1 − 2푓풌′ ) 풌 풌 풌′ 2 Now we minimize the energy with respect to |푣풌|

휕퐸 2 = 0 휕(|푣풌| )

We obtain, similarly to 푇 = 0 case ∗ 푣풌푢풌 ∆(푇) 2 = 1 − 2|푣풌| 2휖풌 where the temperature gap function ∆(푇) is defined as

2 ∗ ∆(푇) = |푔푒푓푓| ∑ 푢풌푣풌 (1 − 2푓풌) 풌

∗ Using (6.61) and the expressions for 푢풌, 푣풌 and 푓풌 we obtain the integral equation for ∆(푇)

2 ∆(푇) 2 ∆(푇) = |푔푒푓푓| ∑ (1 − ) 2퐸풌 퐸풌 풌 exp ( ) + 1 푘퐵푇

By replacing (as usual) the sum over 풌 by the integral over 푔(휀)푑휀 we obtain

1 ℏ휔퐷 푑휀 √휀2 + ∆(푇)2 = ∫ 푡ℎ ( ) 2 2 2 2푘 푇 푔(휖퐹)|푔푒푓푓| 0 √휀 + ∆(푇) 퐵

In this formula ∆(푇) is the integral parameter. Unfortunately, this integral cannot be taken analytically). Graphically, the dependence is

Near critical temperature 푇퐶, the gap evolves as ∆(푇) ∝ √푇퐶 − 푇, and vanishes vertically at 푇퐶. Since ∆(푇 = 푇퐶) = 0 the integral becomes

1 ℏ휔퐷 푑휀 휀 = ∫ 푡ℎ ( ) 2 휀 2푘 푇 푔(휖퐹)|푔푒푓푓| 0 퐵 퐶

After integration we obtain −1/휆 푘퐵푇퐶 ≅ 1.14 ℏ휔퐷푒 2 −1/휆 with 휆 = 푔(휖퐹)|푔푒푓푓| . Since we already know from (6.67) that ∆(푇 = 0) = 2 ℏ휔퐷푒 , we finally obtain the famous ratio between the zero-temperature BCS gap and the critical temperature of superconductivity 2∆(0) ≅ 3.52 푘퐵푇퐶

5. Bogolubov transformation* (*- treated as tutorial).

To prepare this tutorial it is recommended that you read the Sec. 6.6 of Annett’s book.

6. The ground state energy. Density of elementary excitations. Coherence length. The ground state energy. We will estimate now the ground state energy (at 푇 = 0). Instead of calculating 퐸 (6.50), we will evaluate the energy difference between the SC and normal states: 푊 = 퐸 − 퐸푁 The energy of the normal state is simply

퐸푁 = ∑ 2휖푘

푘<푘퐹 , here the factor 2 comes from the fact that the sum is done by pairs of (푘; −푘) non- interacting electronic states. We have 2 2 2 ∗ ∗ 푊 = ∑ 2휖풌 (푣푘 − 1) + ∑ 2휖풌푣풌 − |푔푒푓푓| ∑ 푣풌푢풌푣풌′푢풌′ ′ 풌<푘퐹 풌>푘퐹 풌풌 where in the last term the sum is made only over the energy window ±ℏ휔퐷. This term is 2 2 ∗ ∗ ∆ −|푔푒푓푓| ∑ 푣풌푢풌푣풌′푢풌′ = − 2 풌풌′ |푔푒푓푓|

Now let us evaluate the first and the second terms. In the first sum all terms are positive, 2 since 휖풌(푘 < 푘퐹) < 0 and 푣푘 < 1. Then, by changing the direction of energy 휖풌 → −휖풌, the sum over 풌 < 푘퐹 can be replaced by the sum over 풌 > 푘퐹 2 2 휖풌 ∑ 2휖풌 (푣푘 − 1) = − ∑ 2휖풌 푢푘 = (휖풌 → −휖풌) = ∑ 휖풌(1 − ) 퐸풌 풌<푘퐹 풌<푘퐹 풌>푘퐹 i.e. it is exactly equal to the second term: 2 2 휖풌 ∑ 2휖풌 (푣푘 − 1) + ∑ 2휖풌푣풌 = 2 ∑ 휖풌(1 − ) 퐸풌 풌<푘퐹 풌>푘퐹 풌>푘퐹

We finally get: 2 휖풌 ∆ 푊 = 2 ∑ 휖풌(1 − ) − 2 퐸풌 |푔 | 풌>푘퐹 푒푓푓 By replacing the sum over 풌 by integration over 퐸 we get: ℏ휔퐷 휀 ∆2 푊 = 2푔(퐸 ) ∫ 휀 푑휀 (1 − ) − 퐹 2 2 2 0 √휀 + ∆ |푔푒푓푓| After the integration we obtain 1 2 2 2 2 ℏ휔퐷 ℏ휔퐷 ℏ휔퐷 ℏ휔퐷 ∆ 푊 = 푔(퐸 )∆2 {( ) − [1 + ( ) ] + 퐴푟푠ℎ( )} − 퐹 |∆| |∆| |∆| |∆| 2 |푔푒푓푓| ℏ휔퐷 1 Now, using ℏ휔퐷 ≫ |∆| and 퐴푟푠ℎ ( ) = 2 , the last two terms cancel, and we |∆| |푔푒푓푓| 푔(휖퐹) obtain 1 ℏ휔 2 ℏ휔 ℏ휔 2 2 푊 = 푔(퐸 )∆2 {( 퐷) − 퐷 [1 + ( 퐷) ] } 퐹 |∆| |∆| |∆|

1 2 | | 2 2 2 ℏ휔퐷 ∆ = 푔(퐸퐹)∆ ( ) {1 − [1 + ( ) ] } |∆| ℏ휔퐷

2 | | 2 2 ℏ휔퐷 1 ∆ 1 2 ≈ 푔(퐸퐹)∆ ( ) {− ( ) } = − 푔(퐸퐹)∆ |∆| 2 ℏ휔퐷 2 i.e. indeed, the SC ground state has a lower energy than the normal one. It is therefore TD stable. N.B. The formula for W is often interpreted as the result of integration of characteristic SC energy ∆ 푝푒푟 푝푎𝑖푟 with the DOS over an energy window ~∆. While this gives a correct result, it has no particular physical meaning.

Density of elementary excitations. Experimentally, the energy of the ground state can be measured through the transition to the normal state in the magnetic field. We previously obtained: 퐻2 푊 = −휇 퐶. Now we can connect the TD critical field to the microscopic characteristic of 푇퐷 0 2 the superconductor 1 퐻2 푔(퐸 )∆2= 휇 퐶 2 퐹 0 2 or

퐻퐶 = |∆|√푔(퐸퐹)/휇0

(put typical numbers for 푔(퐸퐹), use |∆|~1푚푒푉, and obtain typical values for 퐻퐶. Compare with the experiment).

Density of elementary excitations. We will calculate now one of the most important results of the BCS model – the existence of an energy gap around the Fermi level in the excitation spectrum of a BCS superconductor. Typically we would like to add/remove a single electron/hole to the superconductor, and see how this modifies its total energy.

Let us first consider a unique Cooper pair (풒; −풒) inside the superconductor. What is the typical (negative) energy contribution 푤풒 of this Cooper pair to the total energy 퐸 of the superconductor? From (6.50) and the following calculations we can isolate a single term 풒 2 2 ∗ ∗ 푤풒 = 2휖풒|푣풒| − 2|푔푒푓푓| 푣풒푢풒 ∑ 푣풌′ 푢풌′ 푘′ The total energy is indeed 퐸 = ∑풒 푤풒. If we put already known expressions for |푣풒|, ∆, and ∗ 푣풌푢풌 ∆ the expression 2 = that we obtained above, we get 1−2|푣풌| 2휖풌 휖풒 2 ( ) 휖 1 − 2|푣 | 휖2 퐸 휖2 + ∆2 1 풒 풒 풒 2 풒 풒 푤풒 = 2휖풒 (1 − ) − 2∆ ∆= 휖풒 − − 2∆ = 휖풒 − = 휖풒 − 퐸풒 2 퐸풒 2휖풒 퐸풒 2휖풒 퐸풒

2 2 , where 퐸풒 = √휖풒 + ∆ . Note that 푤풒 is indeed negative, as expected. Now, let us try to add a single electron into the state 풒 (once put, its energy will be simply 휖풒 ). The first step to do is to insure that there is no Cooper pair (풒; −풒) in the superconductor, i.e. the pair of states (풒; −풒) is surely empty (by Pauli exclusion principle, our extra electron cannot occupy already occupied state). The energy of the superconductor with one missing Cooper pair (풒; −풒) is 푊− (풒;−풒) = 푊 − 푤풒 Now, we can put our extra electron in 풒. Putting it there will rise the total energy of the system 푊− (풒;−풒) by 휖풒 푊+푞 = 푊− (풒;−풒)+ 휖풒 = 푊 − 푤풒 + 휖풒

Finally, putting the expression for 푤풒 we obtain 푊+푞 = 푊 + 퐸풒 i.e. in order to put one single electron to the state 풒 (energy 휖풒) one needs to rise the

2 2 superconductor energy by 퐸풒 = √휖풒 + ∆ . This is an energy to be furnished to the superconductor in order to make our extra electron to “be accepted”. This energy has a minimum value |∆|, which corresponds to the energy to furnish in order to place one electron at the Fermi level (휖풒 = 0). To occupy higher energy states, an energy > |∆| should be added to the superconductor. It means, if we take a free electron (outside superconductor) with a kinetic energy less than |∆|, such an electron will be “rejected” from the superconductor! In other words, the spectrum of elementary excitations of a superconductor (to add an electron) has a forbidden gap equal to the BCS gap energy |∆|.

If we consider taking a single electron out of superconductor, this is equivalent of putting a hole at 휖풒 < 0. The process is fully reverse of the previous one 푊−푞 = 푊 − 퐸풒 at is the energy of the superconductor will be lowered by at least |∆|. This means that within the energy window ퟐ|∆| around the Fermi level no excitations can be added or removed from the superconductor. On this “elementary excitation language” the superconductor appears as… an insulator!

Now it is straightforward to calculate the density of such elementary excitations. From the figure above we already anticipate that this excitation density will be higher near 퐸 = ∆ where there are more available excitations per 푑퐸. By definition, the DOS is 푔(퐸푒푥푐) = 푑푛/푑퐸풒 where 푑푛 is the number of excitations in the energy window 푑퐸풒. We can write 푑푛 푑푛 푑휖풒 푔(퐸푒푥푐) = = 푑퐸풒 푑휖풒 푑퐸풒 푑푛 Here is simply the DOS in the normal state of the same material. We can approximate it, 푑휖풒 as previously, 푑푛 푔(휖) = ≈ 푔(휖퐹) 푑휖풒 Now, 푑휖풒 퐸 = 푑퐸풒 √퐸2 − ∆2 (we dropped index 풒). We finally get 퐸 푔(퐸푒푥푐) = 푔(휖퐹) √퐸2 − ∆2 This is a very important result. It comes directly from the BCS theory and predicts the existence of a gap 2|∆| at the Fermi energy. This DOS also has two singularities at the gap edges, at 퐸 = ±|∆|. These are often called “coherence peaks”.

The importance of this prediction is that the very peculiar shape of this DOS can be verified in a direct tunneling experiment. In fact, at very low temperature 푇 ≪ 푇퐶 the tunneling conductance as a function of voltage applied between the two tunneling electrodes (one is normal and other (sample) is superconducting) is directly proportional to 푔(퐸푒푥푐). The picture below shows a series of experimental tunneling conductance spectra measured in a Au (normal)-Nb(superconductor) junction at different temperatures. The spectrum at very low temperature (dots) fits exactly the BCS density of states (solid line):

N.B. The pioneering tunneling experiment was done by Ivar Giaever. For this discovery, he received Nobel Prix in 1973.

Coherence length. The BCS theory allows one to estimate the superconducting coherence length (which appears in the GL model due to the gradient term in the free energy). We can use the uncertainty relation ∆푘∆푥~1

2 2 Now the ∆푘 can be estimated from the region were the occupation factors |푣풌| and |푢풌| vary significantly. It takes place near zero, at the energy scale of about ~2∆.

7. Predictions of the BCS theory. Now, let us summarize the predictive strength of the BCS theory. 1. Electron pairs. 2. Isotope effect. 3. Gap in the excitation spectrum. 4. Gap vs 푇퐶 (not discussed above). 5. Gor’kov’s derivation of the GL model from the BCS theory.

Problems with the BCS theory. 1. Simplified electron-phonon interaction – Unable to predict superconductivity in real materials 2. Only valid for good metals in which 휖퐹 ≫ ℏ휔퐷 3. Only valid in a case ofml 4. a very weak electron-phonon coupling 5. Only valid for materials in which 풌 is a good quantum number.