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Microscopic Properties of BCS Superconductors (cont.)
References: Tinkham, ch. 3, sections 7–
Notations: In last lecture, examined inter alia responses of system to probes that couple respectively to total S and total (transverse) current J. Both S and J have special property that they are diagonal in Bogoliubov quasiparticle operators, e.g.∑ S = σ σα
† pσαpσ. If we consider probe which does not have this property, life becomes more complicated.
1. Tunnelling
Two metals (N or S) separated by thin oxide-layer
oxide-layer barier
metal 1 metal 2
s.p. states k s.p. states q
barrier, voltage drop ∆V applied. What current I flows from 2 to 1? Usual description of tunnelling: Bardeen-Josephson Hamiltonian:
HˆT =
kqσ
(Tkqσa† kσaqσ + H.C.) (no spin flip) (1)
where k denotes a state in 1 and q are in 2. Usual assumption: no special symmetries, etc., in Tkqσ, also doesn’t depend appreciably on energy. [probably OK if all relevant energies barrier height.]
(a) Suppose both metals are normal:
2nd order perturbation theory: (neglect any dependence of Tkq on σ).
Pq→k = 2 π ℏ
|Tkq|^2 nqσ(1 − nkσ)δ(k − q)
Pk→q = 2 π ℏ
|Tkq|^2 nkσ(1 − nqσ)δ(k − q) (2)
⇒ I 2 → 1 = − 2 πe ℏ
kqσ
|Tkq|^2 (nqσ − nkσ)δ(k − q)
Suppose average over ˆk, qˆ, σ is |T |^2 and this is not a function of or ′, then summing over σ,
I = − 2 πe ℏ
2 |T |^2
dd′^ ρ 1 ()ρ 2 (′)(n 2 () − n 1 (′))δ( − ′) (3)
where ρ 1 , 2 () = single spin DOS at energy , and n 1 , 2 () = thermal (or other) occupation factor. Usually, ρ 1 , 2 () ∼ N 1 , 2 (0). Then
I = −
2 πe ℏ
|T |^22 N 1 (0)N 2 (0)
d[n 2 () − n 1 ()] (4)
Now in thermal equilibrium, n 1 , 2 = f ( − μ 1 , 2 ), f = Fermi function. Also, to a very good approximation, μ 1 − μ 2 = eV [note sign!]. Thus^1
I = −
2 πe ℏ
|T |^22 N 1 (0)N 2 (0)
−∞
d[f ( + eV ) − f ()] (5)
It is clear that independently of temperature the
is simply −eV. Hence Gnn ≡ I 2 → 1 /V 21 is given by
Gnn =
2 πe^2 ℏ
2 N 1 (0)N 2 (0)|T |^2 (6)
The matrix element |T |^2 is of course very sensitive to the details of the junction, but crucial point is that it is not expected to change (much) when one or both metals become superconductors.
(b) Now suppose one metal, for definiteness 1, is S with energy gap ∆, the metal (2) remaining N.
Assume for the moment zero temperature and let μ 2 be > μ 1. The ME Tkqa† kσaqσ must now be expressed in terms of Bogoliubov operators for metal 1:
Tkqa† kσaqσ = Tkq(ukα† kσ + σvkα−k,−σ)aqσ (7)
The term in vk does not contribute to Pq→k at T = 0, since impossible to destroy a Bogoliubov quasiparticle, so
Pq→k = 2 π ℏ
|Tkq|^2 u^2 kδ(Ek − q)nq (8)
The return current, Pk→q, is similarly given by the Hermitian conjugate of (7) (read vk = v−k):
Pk→q = 2 π ℏ |Tkq|^2 v^2 kδ(Ek + q)(1 − nq) (9)
since e−^ in 2 created. If we choose eV for definiteness to be positive, then the return current is zero (at T = 0) and the total current 2→ 1 is given by
I 2 → 1 = −
2 πe^2 ℏ
kqσ
|Tkq|^2 u^2 kδ(Ek − q)nq (10)
where nq = θ(eV − q): note that for V = 0 the current vanishes as it should. Now u^2 k = 12 (1 − k/Ek), and provided |Tkq|^2 doesn’t depend appreciable on Ek (usually true), the contribution of the k-term vanishes, because states with ±k have the same Ek: thus we can simply set the u^2 k = 1/2.
(^1) Provided eV, kBT bandwith, it is clear that only states close to Fermi surface contribute.
In transforming Ω into Bogoliubov quasiparticle operators, we must remember thatˆ the transformation involves σ: (cf. Lecture 6).
a† kσ = ukα† kσ + σvkα−k,−σ (15)
etc. Carrying out the transformation explicitly and using the ACR’s:
Ω =ˆ
kk′σσ′
Vkk′σσ′ (t)
uku′ kα† kσαk′σ − vkv k′σσ′α†−k′,−σ′ α−k,−σ +
σ′ukvk′ α† kσα†−k′,−σ′ + σuk′ vkα−k,−σαk′σ
It is convenient to redefine the variables of summation and introduce the notation θσσ′^ ≡ σσ′^ = +1 if σ = σ′, −1 if σ 6 = σ′. We further assume (note order of indices) that
V−k′,−k,−σ′,−σ = ηθσσ′ Vkk′σσ′ (17)
where η = ±1, i.e. V is even (type-I) or odd (type-II) under time reversal. Thus the expression for Ω becomesˆ
Ω =ˆ
kk′σσ′
Vkk′σσ′^ (ukuk′^ − ηvkvk′^ )(α† kσαk′σ + ηθσσ′^ α†−k′,−σ′ α−k,−σ) + (18)
(ukvk′^ + ηvkuk′^ )(α† kσα†−k′,−σ′ + ηθσσ′^ α−k,−σαk′σ′^ )
As emphasized by Tinkham, the factor ηθσσ′ is unimportant because it relates pro- cesses which are mutually incoherent, but the factors of η in the overall coefficients are crucial. What it means is that the effective matrix for scattering of the Bogoliubov quasiparticle is multiplied, relative to that for N-state particles, by factor
(ukuk′ − ηvkv′ k) (19)
Thus the transition probabilities are multiplied by this factor squared:
(ukuk′ − ηvkvk′ )^2 =
kk′ EkEk′
− η
∆^2
EkEk′
Provided that V is not strongly dependent on k (usually true) then when we sum over k and k′^ this expression multiplied by δ(Ek − Ek′ − ℏω), the kk′ terms cancel, and we are left with a factor
Rη(Ek, Ek′^ ) =
1 − η
∆^2
EkEk′
In the limit k, k′ → 0 (Ek, Ek′ → ∆) this tends to 0 for η = +1 and 1 for η = −1. In the same way, the factor multiplying the two-quasiparticle creation operator term α† k,σα†−k′,−σ′ , namely (ukvk′ + ηvkuk′ )^2 , becomes after the same summation over ±k,
R˜η =^1 2
1 + η
∆^2
EkEk′
which has the opposite behavior to R. Let’s now consider an expression of the general form
J(ω) ≡
m
Z−^1 e−βm
n
|〈n|
kk′σσ′
Vkk′σσ′ a† kσak′σ′ | 0 〉|^2 δ
ω − (m − n)
− (kσ k′σ′)
(23) where the states m are energy eigenstates of the many-electron system with (many- electron) energies m, and the matrix element Vkk′σσ′ is even or odd under time reversal as discussed above. Moreover, assume that the effect of averaging over the directions of k and k′^ doesn’t introduce any special energy-dependence, so that the replacement
∑
kk′σσ′
|Vkk′σσ′ |^2 → (dn/d)^2
d
d′^ |V |^2 ( ≡ k) (24)
is legitimate (depending on the actual structure of Vkk′ , the quantity |V |^2 may itself involve factors of dn/d). As we shall see, the expressions for the ultrasonic attenuation, nuclear spin relaxation and electromagnetic absorption are all of this type. In the normal phase, we get, allowing for both “forward” and “backward” process,
Jn(ω) = |V |^2 (dn/d)^2
dd′^
f () − f (′)
δ
ω − (′^ − )
= |V |^2 (dn/d)^2
d
f () − f ( + ℏω)
= |V |^2 (dn/d)^2 ℏω
Now consider the S phase, and, suppose for the moment that ω < 2∆(T ) so that only “scattering” terms contribute.∫ The difference, now, is that all ’s (except for the d’s) are replaced by the corresponding E’s, and moreover the matrix element squared contains a factor Rη(E, E′).^3
Js(ω) = |V |^2 (dn/d)^2
dd′^
f (E) − f (E′)
Rη(E, E′)δ
ω − (E′^ − E)
= |V |^2 (dn/d)^2 · 4
∆
dE
E
E′
f (E) − f (E′)
Rη(E, E′)
where E′^ ≡ E + ℏω, ′^ ≡ ′(E′), and as above Rη(E, E′) ≡ 12 (1 − η ∆ 2 EE′^ ) Suppose now that we have not only ω < 2∆(T ) but ω ∆(T ). Then we can expand in ω and neglect terms proportional to ω or higher except in the difference f (E) − f (E′), which we approximate as −ℏω(∂f /∂E):
Js(ω) = |V |^2 (dn/d)^2
∆
dE (E^2 /^2 )
1 − η
∆^2
E^2
(−∂f /∂E) × ℏω (27)
We see that there is a crucial difference between type-I (η = +1) and type-II (η = −1) cases. For type-I the factor 1 − η∆^2 /E^2 ≡ ^2 /E^2 just cancels the E^2 /^2 , and we get the
(^3) Note the overall factor of 4, coming from R^ ∞ −∞ d^ →^2
R (^) ∞ ∆ dE^ E ||
(b) Nuclear spin relaxation
(treatment in some texts rather confusing!) In the original Hebel-Slichter ex- periments, system was allowed to come to equilibrium (I ∼ H) in a finite field H > Hc(T ) (so sample is N). Next field was turned off (system → S state), so equilibrium value of nuclear spin I is zero, allowed to relax for a time t, then field switched on again and nuclear spin measured, hence obtain its decay, fitted to I(t) = I 0 exp(−t/T 1 ). Thus experiment measures T 1 − 1 (T ). In normal states, T 1 − 1 ∝ const T (Korringa law). The process which relaxes I involves a nuclear spin flipping down accompanied by an electron spin flipping up, hence is proportional to the matrix element of S+(r) =
kk′^ Jkk′^ σˆ+a
† kσak′σ′^ =^
kk′^ Jkk′^ a
† k↑ak′↓^ where^ J−k′,−k^ =^ Jk,k′^. This is type-II (note role of θσσ′ !), so the relation between T 1 −s^1 (T ) and the corresponding state in the normal rate at that T is
T 1 −s^1 (T )/T 1 −n^1 (T ) =
0
dE(−∂f /∂E)
E^2 + ∆^2
E^2 − ∆^2
- a rise just below Tc (the famous Hebel-Slichter peak), followed by a drop to 0 as T → 0. For comparison with experiment^5 see e.g. Rickayzen Fig. 3-4.
(c) EM absorption. At first sight this should be similar to ultrasound absorption (since both involve the absorption of a photon (phonon) with scattering of an electron), the principal difference being that since the ME is proportional to J, a T -odd operator, it is type-II rather than type-I. Actually this analogy obscures an important difference: we have cs vF but c vF and correspondingly, in the Sommerfeld (free-Fermi- gas) model of the N state, the US absorption is finite but the EM absorption zero! Thus to get any N-state EM absorption at all we need to take into account impurity scattering, and in general the relation between σ 1 S (ω) and σ 1 n(ω) is complicated, see e.g. Tinhham section 3.10.5. Simple formulas only result for σ 1 n(ω) ∼ const over ∼ ∆; this is so if τn ∆ 1 or l ξ. The operator corresponding to absorption of EM radiation of wave vector q and frequency ω is the Fourier transform of the electric current j(r), i.e.
jq ≡
eℏ m
kσ
ka† k+q/ 2 ,σ ak−q/ 2 ,σ (34)
Since the EM wave is transverse, we have to project jq i.e., k, on the direction ⊥ q · (jq · Aq). It is clear that this perturbation is type-II. However, unlike the case of nuclear spin relaxation, the relevant frequency ω can easily be > ∆(T ), so we cannot neglect 2-quasiparticle creation. At zero T , this is the only process, and (^5) In original experiment HS observed rise only by a factor ∼ 2.
we get for the real part of the A.C. conductivity at T = 0∗
σ 1 s/σ 1 n =
dE/||
dE′/|′|{ R˜−(E, E′)δ(ℏω − (E + E′))} (35)
∆
dE ·
EE′
∆^2
EE′
E′^ ≡ E + ℏω
It is clear that this expression vanishes for ℏω < 2∆(0): for ℏω > 2∆(0), it can be expressed as an elliptic integral (see e.g. Tinkham section 3.9.3). At finite T , the 2-quasiparticle creation term is multiplied by a factor
(1 − f )(1 − f ′) − f f ′^ = 1 − f (E) − f (E′) (36)
[note that since f (E) < 1 /2, this is positive!] In addition, we get a “quasiparticle scattering term” even for ω < 2∆(T ): this has a structure qualitatively similar to that of the nuclear spin relaxation.
Further discussion of the static EM properties is given in Lecture 9.
∗I suspect that the apparent difference of (35) from Tinkham’s (3.94) is due to his use of a different sign convention for E, E′.
