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Magnetic impurity and other ‘pair-breaking’ effects
References: de Gennes Chapter 8; AJL QL section 5.
One of the most striking experimental facts about (classical) superconductivity is that while it is rather insensitive to nonmagnetic impurities, even a rather small concentration of magnetic impurities (ie, those corresponding to a finite local moment) can lead to a drastic suppression of Tc or even the complete vanishing of superconductivity. The basic reason for this and related affects is that such impurities destroy the invariance of the (conduction-electron) Hamiltonian under time reversal. In the following, we suppose until further notice that the effect of magnetic impurities is to add to the conduction- electron Hamiltonian a term of the form
∆K =
m
JmSm · σ(rm) (1)
where σ(r) ≡ σαβ ψ† α(r)ψβ (r) is the conduction-electron spin density at the point r. The spins Sm are taken to be classical and random both in orientation and in position. In lecture 9 we considered the case of nonmagnetic impurities, and showed that by pairing time-reversed eigenstates (|n ↑〉, |¯n ↓〉) of the single-particle Hamiltonian, we would obtain almost as large an (average) value of the quantity F (r, r) as in the ‘pure’ case. To recapitulate the argument, we write in that case
ΨBCS =
n
(un + vnan↑an¯↓)|vac〉 (2)
and the quantity F (r, r) takes the value
F (r, r) ≡ 〈ψ↑†(r)ψ†↓(r)〉 =
n
unvnφn(r)φ¯n(r) (3)
but since φ¯n(r) ≡ φ∗ n(r) this becomes simply
F (r, r) =
n
unvn|φn(r)|^2 (4)
and with an appropriate choice of the parameter unvn (= ∆n/ 2 En) this can be made approximately as large (or larger) as its value for the pure case. Thus, the pair term in the potential energy, which for the simple contact potential considered is simply
〈V 〉pair = V 0
|F (r)|^2 dr (5)
is also just as large as in the pure case. For a system lacking time reversal invariance we cannot repeat this argument, because in general the eigenstates of the single-particle Hamiltonian no longer occur in pairs related by time reversal. We therefore have two obvious choices: (a) pair in eigenstates
of if Hˆ 0 (which are not in general time-reversed) (b) pair in time reversed states (which are not in general eigenstates of Hˆ 0 ). Of course, intermediate choices are also possible. Before embarking on a quantitative calculation, let’s try to consider the pros and cons of choices (a) and (b) qualitatively. If we make choice (a), then from the kind of general arguments developed in lecture 5 we would expect that the best choice is to pair eigenstates of Hˆ 0 with nearly degenerate energies n. (These will not of course in general be eigenstates of spin). Such a choice will lead to a depression of the quantity F (r, r), which is effectively equivalent to a suppression of the constant V 0 ; since Tc depends exponentially on V 0 , we expect it to be strongly depressed (except in very special circumstances, cf. below). So let’s consider alternative (b)(which is (something like) what the system actually does, cf. below). We then lose little or nothing on the pairing contribution to 〈V 〉, but the price is that we increase the kinetic energy; crudely speaking, we have to start our pairing from a “pseudo-Fermi sea” that is the Fermi sea that would describe the normal system subject to a Hamiltonian without the magnetic terms. What does it cost us to create this “pseudo-Fermi sea” from the true normal- groundstate (i.e., the groundstate of Hˆ 0 including the magnetic impurities)? A back of envelope argument goes as follows: consider a spin (or more generally time-reversal) eigenstate made up out of eigenstates of Hˆ 0 close to the Fermi energy. Such a state will have a width Γ (≡ ℏ/τK , see below) which tends to a constant near F , i.e., it is made up of a packet of eigenstates of Hˆ 0 which have spread ∼ Γ/2. Thus, to reconstitute a “pseudo-Fermi sea” out of such spin eigenstates we need to supply an extra energy ∼ Γ/ 2 to the number of states involved in the rearrangement, which is ∼ (dn/d)Γ/2. The total energy required is thus ∼ 14 (dn/d)Γ^2 ≡ 12 N (0)Γ^2. On the other hand, the condensation energy of the superconducting state relative to the “pseudo-Fermi sea” normal (pseudo-) groundstate is 12 N (0))∆^2 where ∆ is the energy gap in the material without magnetic impurities. Thus, we expect superconductivity to become energetically unfavorable even at T = 0 (i.e. Tc → 0) when Γ ∼ ∆. It is amusing that while the numerical factors in the above argument are clearly rather arbitrary, the exact criterion for the disappearance of superconductivity does in fact turn out to be Γ = ∆ (see below). If for a pure material (in the absence of nonmagnetic impurities), we introduce the relaxation time τK against T -violating effects, so that Γ ≡ ℏ/τK , and the corresponding mean free path lK ≡ vF τK , and recall the definition of the Pippard coherence length ξ 0 , this criterion can be rewritten lK = ξ 0 /π. Now let us turn to a more exact calculation. In the general case, the presence of terms like JiSix σx(ri), in the Hamiltonian means that the normal-state eigenfunctions are not in general eigenstates of spin, and the formulae then become rather messy: cf. de Gennes section 8.1. Let us therefore specialize to a rather artificial case which gives the essentials, namely one in which the impurity spins are constrained to lie along the z-axis, with however a random sign (and, still at random positions). Then the normal- state eigenfunctions can (indeed must) be chosen to be eigenstates of σz : we label them φn↑(r), φn¯↓(r). Since we wish the pair wave function F (r, r) to be as large as possible,
and similarly for the second equation, multiplying by φm↓(r) and integrating,
(|n↑| + m↑)dnm =
∆(r)φ∗ m↓(r)u^0 n(r)dr (12)
Now, the self-consistency equation for ∆ has the form
∆(r) = V 0
n
un(r)v∗ n(r) tanh βEn/ 2 (13)
Where we substitute the form (10) of un(r) and vn(r) in the RHS, the zeroth-order term gives zero since we automatically have u^0 n(r)v^0 n(r) = 0. The terms linear in ∆ give an expression of the form
K(r, r′)∆(r′)dr′, where K(r, r′) is given after a little algebra by the expression
K(r, r′) = V 0
nm
{ (^) u^0 n(r)u^0 n∗ (r′)φ∗ m↓φm↓(r)(r
′) tanh β|n↑| 2 |n↓| + m↑
v^0 n∗ (r)v n^0 (r′)φ∗ m↑(r′)φm↑(r) tanh β|n↓|/ 2 |n↓| − m↑
which is a generalization of de Gennes equation (7.8). Using the fact that the second term only exists for n↓ < 0 and so tanh β|n↓|/ 2 ≡ − tanh βn↓/2, and interchanging n and m in the second term, we can write this in a more compact form:
K(r, r′) = V 0 / 2
nm
{ (^) tanh(βn↑) + tanh(βm↓) n↑ + m↓
× φn↑(r)φm↓(r)φ∗ m↓(r′)φ∗ n↑(r′)
K(r, r′) is the kernel of the linearized gap equation
∆(r) =
K(r, r′)∆(r′)dr′^ (16)
it is expressed entirely in terms of the normal-state eigenfunctions φmσ(r). In the “pure BCS” case, where the eigenstates are mutually orthogonal plane waves, we may take ∆(r) = const, and it may be straightforwardly verified that the only contribution to the integral over r′^ in equation (16) comes from m ≡ k, n ≡ −k; we thus recover the BCS equation for Tc. A slight generalization of this argument applies to the case of nonmagnetic disorder. We now ask: is it possible to relate K(r, r′), and hence the gap equation, more generally to some experimentally measureable properties of the normal phase? Actually, this is a question of rather general interest, for example in cases where the physical conditions, and hence ∆(r), vary substantially in space: in fact, in such cases analysis of equation (15) recovers for us the GL equation, with an identification of the parameters entering it: see de Gennes sections 7.1–2. However, we are interested in the case where the impurity distribution, while microscopically random, is on average homogenous. To
analyze this, and some other similar cases, it is useful to transform the expression for K(r, r′) somewhat. Step 1 is to note that because of the identity
tanh β/2 = 2kBT
n
− iℏωn
where ω ≡ (n + 1/2)2πkBT and the sum runs over all positions or negative integers, we can write the expression involving the ’s as
2 kBT
ωm
n↑ − iℏωm
m↓ + iℏωm
We can then introduce continuous variables of integration , ′^ such that
n →^
dN (0), and a quantity
Q(, ′^ : r, r′) ≡
n
m
δ(n↑ − )δ(m↓ − ′)φn↑(r)φm↓(r)φ∗ m↓(r′)φ∗ n↑(r′) (19)
Then
K(r, r′) = V 0 kBT
ω
d
d′^ Q(′^ : rr′) ( − iℏω)(′^ + iℏω)
Step 2. (this is the trickiest): We now argue that since the states near the Fermi energy in the normal phase are in no way “special,” the quantity Q(′^ : rr′) is very insensitive to the “center of mass” variable ( + ′)/2, and thus we can replace^3 the quantity
n
m δ(n↑^ −)δ(m↓^ − ′) by (−′)− 1 ∑ mn(1−fn↑)fm↓^ δ
n↑ −m↓ −(−′)
where the f ’s are Fermi functions. Thus
Q(, ′^ : r, r′) = Q( − ′^ : r, r) =
nm
fm↓(1 − fn↑)δ
n↑ − m↓ − ( − ′)
×φm↓(r)φ∗ m↓(r′)φ∗ n↑(r′)φn↑(r) (21)
Equation (21) looks very reminiscent of the imaginary part of a correlation function. In fact, let us for orientation consider the case where Hˆ 0 is real and thus the φm↑,↓(r) can be chosen real. Then we can, trivially, add and remove complex conjugation, and thus rewrite Q in the form
Q(−′^ : r, r′) =
mn
fm↓(1−fn↑)φ∗ n↑(r)φ∗ m↓(r)φm↓(r′)φn↑(r′)δ
n↑ −m↓ −(−′)
But the expression on the RHS is simply the imaginary part of the normal-state corre- lation function 〈〈S+(r)S−(r′)〉〉(ω) for ω = − ′! The great advantage of this result is that we can now use our phenomenological knowledge of the behavior of such correlation functions in the normal phase to calculate Q and hence the kernel K(r, r′).
(^3) This follows because in the latter comparison in view of its antisymmetry with respect to − ′, can replace (1 − fn↑)fm↓ by 12 {(1 − fn)fm − (m n)} = 12 (fn − fm) and R (^) ∞ 0 f^ (E^ +^ ω)^ −^ f^ (E^ −^ ω) = 2ω
In the literature, it is conventional to subtract from equation (27) the τs = ∞ gap equation (with transition temperature Tc 0 ) and express the result in the form
ln(Tc 0 /Tc) = ψ
4 πτskBTc
− ψ(1/2) (29)
where ψ(1/2) is the so called ”digamma function” defined by (Γ(z) ≡ Euler Γ-function)
ψ(z) ≡ Γ′(z)/Γ(z) = − 0 .577 +
∑^ ∞
v=
1 /v − 1 /(v + z)
− z−^1 (30)
Note that the equation for the relative reduction of Tc is expressed entirely in terms of the single dimensionless parameter α ≡ ℏ/(2πτskB Tc). A particularly simple result holds for the value of ℏ/τs which completely destroys superconductivity. This is most easily obtained by evaluating the RHS of expression (28) for β → ∞ and comparing with the τs = ∞, T = 0 gap equation to obtain
Re
∫ (^) c
0
d/( + i/ 2 τs) =
∫ (^) c
0
d/
^2 + ∆^2 (31)
which for large c yields 1/ 2 τs = ∆(0). The general behavior of Tc as a function of ℏ/τK is
N
gapped S
Tc
Tc 0 as shown. For small impurity concentration the slope is given approximately by kB(Tc 0 − Tc) ∼= πℏ/ 2 τK. The region just below the N-S transition temperature which is shaded in the figure is very interesting. As shown by Abrikosov and Gor’kov in their original paper, in this regime that superconductor is gapless, that is, there exist Bogoliubov quasiparticles of arbitrarily low energy. I follow the discussion of de Gennes (Section 8.2): Consider for definiteness the case T = 0, but with a concentration of impurities close to critical; then we may reasonably assume that the “gap” ∆ is small, and work as above to lowest order in it. To order ∆ we have for the energy eigenvalues
En = |n| + |∆|^2 P
m
|〈n|K|m〉|^2 |n| + |m| , P = principal part (32)
If the one-electron part of the Hamiltonian is invariant under time reversal, then the only state m occuring in the sum is degenerate with n, so that
En = |n| + |∆|^2 /(2|n|) (33)
This is the beginning of an expansion in ∆/|n|: it clearly works for |n| → ∞ but fails for |n| → 0. If the system is not invariant under K, then the second term is not
singular as |n| → 0 and the perturbation theory may work. Suppose in particular K relaxes exponentially to zero with time constant τK , then
En = |n| + |∆|^2 P
d′^ Im χK (n − ′)/(n + ′) (34)
= |n| + |∆|^2 P
d′^
τK 1 + (n − ′|)^2 τ (^) K^2
n + ′
= |n| + 2 |∆|^2 |n| (2n)^2 + (ℏ/τK )^2
If now we take |n| ℏ/τK , this tends to
En = |n|(1 + 2(∆τK /ℏ)^2 ) (35)
which can be arbitrarily small. The density of states is
Ns() = N (0)d/dE ∼=
1 + 2(∆/ℏ)^2
(2)^2 − (ℏ/τK )^2 (2)^2 + (ℏ/τK )^2
so for < ℏ/τK is less than the N-state value but for > ℏ/τK greater. [cf. de Gennes Fig. 8.5.] The above considerations work for most kinds of pair-breaking effects. However, we should always bear in mind that solutions we have obtained are at most variational ansatz, and we cannot exclude that there may exist other solutions which as it were differ by a finite amount from the simple perturbation-theoretic ones. As an example, consider the case of a constant finite Zeeman field (assumed to act only on the spins and not on the orbital degrees of freedom). We could follow through the above calculation, but now the spectrum of S+ is a δ-function at − ′^ = 2μBH, the energy necessary to flip a spin. Correspondingly, the zero-T linearized gap equation becomes
(N (0)V 0 )−^1 =
∫ (^) c
0
d + μBH
∼= ln(c/μBH) (37)
(where in the last equality we assume c is large). The zero-field T = 0 gap ∆ satisfies the relation
(N (0)V 0 )−^1 =
∫ (^) c
0
d √ ^2 + ∆^2
∼= ln(2c/∆) (38)
and thus the critical field at T = 0 should apparently be given by μBH = ∆/2. However, this conclusion is not correct. To see this, let us compare the energies of the normal state in field H, and the paired state obtained by refusing to let the particles polarize in the field and then proceeding as if in field 0. Relative the normal state in zero field, the first has energy −(1/2)μ^2 B H^2 (dn/d) = −μ^2 B H^2 N (0), while the second has energy (cf. Lecture 6) −(1/2)∆^2 N (0). Thus the second is stable for μBH < ∆/
2, i.e. beyond the limit given by the perturbation calculation. The latter is actually the limit of metastability of the N phases, i.e. the “supercooling” field. cf. Maki and Tsuneto, Prog. Theor. Phys. 21, 945 (1964). (A further complication: FFLO state).
