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Physics 216: Special topics in many-body physics, Spring 2003
Lecture III
Most of this lecture will be devoted to answering, at a less computational and more physical level, some questions about the BCS superconducting state developed in the last lecture. The calculations are mostly in Schrieffer chapter 2. We claimed before that the BCS state was qualitatively different from the Fermi liquid in its “elementary excitations”, the low-lying eigenstates that carry current, heat, etc. (the superflow described by the Ginzburg-Landau equation carries current, but no heat).
First, let us show that Ek =
√ (k − μ)^2 + ∆k^2 can be interpreted as an excitation energy. Suppose that we insist that an electron be at p ↑ and not at −p ↓, thereby “breaking a Cooper pair.” The point of doing this is to create an excited state of well-defined momentum p and spin up: all the Cooper pairs at momentum k 6 = p still have momentum 0 and zero spin, so clearly the overall state has the desired momentum p and spin up. The change in energy is, from the kinetic term (p − μ)(1 − 2 vp^2 ), (1)
and from the pair interaction term (the new state has no interaction between the pair at p and any other pair, while the old state did have such interactions)
− 2
∑
k
Vpkukvkupvp = 2∆pupvp. (2)
Note that we are continuing to assume throughout this lecture that the uk and vk have been chosen real. Now we can simplify these using our previous expressions
vk^2 =
( 1 − k − μ Ek
) , ukvk = ∆k 2 Ek
The total excitation energy is
Wp↑ − W 0 =
(p − μ)^2 Ep
∆p^2 Ep = Ep. (4)
Suppose we try to force an electron into state p ↑, starting from the BCS state.
|ΨBCS 〉 =
∏
k
(uk + vkb† k)| 0 〉. (5)
We get c† p↑|ΨBCS 〉 = up
∏
k 6 =p
(uk + vkb† k)| 0 〉 = up|ψp↑ 〉. (6)
Note that, just as above, we obtain a state with one broken Cooper pair; now, however, the state is not normalized because of the factor up. What if we try to subtract an electron from state −p ↓? We get c−p↓|ΨBCS 〉 = −vp
∏
k 6 =p
(uk + vkb† k)| 0 〉 = −vp|ψp↑ 〉. (7)
Here the minus sign is forced by our anticommutation relation for the fermion operators. So these two operations create the same state, but with different coefficients. This is only possible because the BCS state has uncertain particle number: if it had a well-defined particle number N , then
c−p↓|ΨBCS 〉 would have number N − 1 and c† p↑|ΨBCS 〉 would have number N + 1. We will say a bit more about the physical significance of this in the next lecture.
Recalling that |uk|^2 + |vk|^2 = 1, define a new operator
γ p†↑ = upc† p↑ − vpc−p↓. (8)
This operator adds momentum p and spin-up. The orthogonal combination does the same:
γ−p↓ = vpc† p↑ + upc−p↓. (9)
Note that γ−p↓|ΨBCS 〉 = 0, so γ has the right property to be an annihilation operator. The
Hermitian conjugages of the above define the operators γp↑ and γ†−p↓.
In words, these operators behave like creation and annihilation operators above a vacuum which is not the state of zero particles, but the BCS ground state. They satisfy the same anticommutation relations as ordinary fermion operators c, by construction. For example,
{γp 1 s 1 , γ p† 2 s 2 } = δp 1 p 2 δs 1 s 2 , (10)
and applying two annihilation or two creation operators annihilates any state. This is an impor- tant picture in modern condensed matter and high-energy physics: complex ground states are new “vacua” which support novel excitations. This transformation to the γ operators was invented by Bogoliubov, and the particles created by this transformation are known as Bogoliubov quasi- particles, or (jokingly) “bogolons”. Note that the charge of a Bogoliubov quasiparticle is not defined.
Second, let’s solve the gap equation for the simplest case, since we claimed before that the gap equation would show interesting nonanalytic behavior. Assume that the pairing potential is given by
Vkk′ =
{ −V < 0 if |k − μ| and |k′ − μ| < ωc 0 otherwise
Now the gap equation clearly gives zero gap for |k − μ| ≥ ωc. The gap is constant and nonzero for |k − μ| < ωc: with N (0) the density of states at the Fermi level (the density of states always appears when we convert sums over k to integrals over energy)
∆ = N (0)V
∫ (^) ωc
−ωc
dE
E^2 + ∆^2
This implies
(N (0)V )−^1 =
∫ (^) ωc/∆
−ωc/∆
dE
E^2 + 1
= sinh−^1 (ωc/∆). (13)
So, we find finally that for this simplified potential
∆ =
ωc sinh( (^) N (0)^1 V )
≈ 2 ωce−^1 /(N^ (0)V^ ). (14)
In the last formula we have assumed weak coupling (V small).
As you might expect, this precise form for the gap should not be trusted for a more realistic potential. However, the prediction of BCS that there is a new energy scale for ∆, much smaller than the obvious energy scales like V , has a number of testable experimental consequences. Basically
impurity effects was developed by Abrikosov and Gorkov using the Green’s function techniques to be developed in the next few weeks.
Anderson pointed out that, as long as time-reversal symmetry is preserved, every state at a given energy has a time-reversed state which is degenerate in energy. Hence, even in a nonmagnetic impurity potential, we can form Cooper pairs from pairs of states related by time-reversal. These pairs will have total spin zero. With magnetic impurities or other time-reversal-breaking conditions, it is no longer clear how to pair up noninteracting eigenstates to form Cooper pairs. An experimental prediction of this argument is that magnetic impurities should destroy superconductivity much more rapidly than nonmagnetic impurities (potential scattering). We will talk much more about magnetic impurities later in this course, in case you are wondering how this microscopic classification of impurities arises.
What are some spectacular consequences of superconductivity? Here we will discuss only one, the Josephson effect, which can be understood to some degree using just Ginzburg- Landau. We will do that, although certainly no one would believe the counterintuitive result that we’ll find without a justification in terms of a microscopic theory like BCS. Even with a detailed argument in the BCS picture, the theoretical predictions of Josephson were quite controversial until their spectacular confirmation by experiments of Rowell, Shapiro, and others.
The system we want to understand consists of two identical superconducting regions separated by a potential barrier that allows Cooper pairs to pass through with some small probability. Suppose that one region has phase Φ 1 and the other has Φ 2. Just as the current density is related to the derivative of the phase in the LG equation, we expect the current through the barrier to be a function of the phase difference: j(Φ 1 − Φ 2 ). The function j should be periodic with period 2π and also odd, and with a bit more insight we can find its exact form.
We give an argument from Landau and Lifshitz; a slightly different one is in Schrieffer. The key to understanding the Josephson effect is the requirement of gauge invariance under the nonrel- ativistic gauge transformation
V → V −
c
∂χ ∂t
, A → A + ∇χ, Φ → Φ + 2 eχ ¯hc
If the tunnel barrier were infinite, the condition on the Ginzburg-Landau fields ψ 1 , ψ 2 at the two sides of the barrier, assuming the barrier is perpendicular to xˆ, is
(∂x −
ie∗Ax ¯hc )ψ 1 = (∂x −
ie∗Ax ¯hc )ψ 2 = 0. (19)
This is gauge-invariant and forces the supercurrent given below equal to zero. If the barrier is not infinitely high, then we expect that instead of 0, there will be some function of ψ, which can be expanded in a power-series under the usual assumption in the LG equation that ψ is small, and replacing e∗^ = 2e:
(∂x − i 2 eAx ¯hc
)ψ 1 = −ψ 2 λ−^1 , (∂x − i 2 eAx ¯hc
)ψ 2 = ψ 1 λ−^1. (20)
Here λ is some unknown length scale related to the barrier thickness and height. Now we can plug one of these equations into the expression for the supercurrent:
j = − ie¯h 2 m
( ψ∗ 1 ∂ψ ∂x
− ψ 1 ∂ψ∗ ∂x
) − 2 e^2 mc
Axψ 1 ∗ ψ 1. (21)
If this form looks unfamiliar, note that the first part is just the ordinary particle current in quantum mechanics, and the second part is forced by gauge invariance. We are left with the simple form
j =
ie¯h 2 mλ (ψ∗ 1 ψ 2 − ψ 1 ψ∗ 2 ). (22)
This predicts that j ∝ sin(Φ 1 −Φ 2 ) since, for identical superconductors on both sides, the magnitude of ψ is the same.
So, even at no applied voltage, a supercurrent flows if there is a phase difference between the two superconductors. What happens when a voltage is applied is quite amazing. The phase Φ = Φ 1 −Φ 2 does not evolve at zero bias voltage: ∂ ∂tΦ = 0. When a voltage is applied, we would like to find a gauge transformation that mostly eliminates the voltage but creates a phase difference, since we know how to handle phase differences. The system should be gauge invariant under transformations of the form (we will use this sort of nonrelativistic gauge transformation several times in this course)
V → V −
c
∂χ ∂t , A → A + ∇χ, Φ → Φ +
2 eχ ¯hc
The gauge transformation we need is
V → V −
c
∂χ(t) ∂t
2 e ¯hc
χ(t). (24)
This leaves the vector potential constant since χ has no spatial dependence. Then a gauge-invariant relation between Φ and V is ∂Φ t
2 e ¯h
V = 0. (25)
This can be checked: at V = 0, this predicts constant phase difference, as expected. Then for nonzero V , we have
Φ = Φ(0) − 2 e ¯h
V t ⇒ j = jm sin(Φ(0) − 2 e ¯h
V t). (26)
So our simple theory predicts that, when a voltage is applied, there should be an oscillating current at frequency ωJ = 2|eV |/¯h. (27)
The Josephson frequency constant in useful units is 2e/¯h = 483.6 MHz/μV. Actually it is has been measured to at least eight decimal places. This is the first of several “perfect numbers” that we’ll come across. The preciseness of its quantization (one part in 10^8 or more) is a consequence of its simplicity: it depends only on the assumptions of gauge invariance and quantum coherence.
