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Non-BCS Superconductivity: Diagnostics
The “classic” superconductors—that is, those whose behavior is well accounted for by the BCS theory, which essentially means all those known prior to 1986 with the exception of some “heavy-fermion” superconductors—have, or are believed to have, at least four general properties in common:
(1) The transition temperature Tc never exceeds 25K.
(2) The normal state appears to be well described by textbook (Fermi liquid) theory.
(3) The principal mechanism of formation of Cooper pairs is a phonon-induced attrac- tion.
(4) The symmetry of the Cooper pairs is s-wave (or more precisely, the “simplest” representation of the appropriate crystal symmetry group).
In addition, most although not all of the classic superconductors have three other prop- erties in common:
(5) The structure, although possibly anisotropic, is essentially 3-dimensional.
(6) The system is not particularly close to other types of ordering transition (e.g. magnetic ones).
(7) Superconductivity is not particularly sensitive to chemical stoichiometry (in the case of alloys).
All of the “exotic” classes of superconductors to be discussed in this part of the course (are believed to) fail to satisfy at least one of conditions (1)–(7): The cuprates fail all of them. Clearly, the properties (1), (5), (6) and (7) can be read directly from experiment. What about (3) and (4)? (We will consider (2) in a later lecture.)
Property ←− Class −→ Classic BKBO MgB
Heavy- fermions
Organics Ruthenates Fullerenes Ferro- pnictides
Cuprates
Tc < 25 K ( √ ) × √ √ √ × × × FL normal state √ √ × × × √ × No neighboring phase trans. √ √ × √ √ √ × OP s-wave √^ √^?? × × × Phonon mechanism √^ √^ ×?? √^ × Crystal structure simple √^ × √^ × × × × × Stoichiometry-insensitive √^ √^ √^ √^ √^ × ×
Diagnostics of non-phonon mechanism
A. (Absence of ) isotope effect
We first recall the McMillan expression for Tc (bearing in mind that the specific form is derived for a particular choice of the phonon DOS, namely that characteristic of Nb):
Tc =
θD
- 45 exp −
1 .04(1 + λ) λ − μ∗(1 + 0. 63 λ)
Here θD is the phonon Debye frequency, λ the effective electron-phonon coupling constant defined by
λ ≡ 2
∫ (^) ωD
0
α^2 (ω)F (ω)dω ω
where F (ω) is the phonon DOS and α^2 (ω) is a suitably averaged electron-phonon cou- pling constant, and finally μ∗^ is the effective (renormalized) Coulomb coupling constant, given by μ∗^ = N (0)〈Vc〉(1 + N (0)〈Vc〉 ln
F/θD)
It is clear that the only way in which μ∗^ depends on the isotopic mass M of the ions is through the cutoff θD in the denominator. What is also crucially important is that the quantity λ is independent of M ; this is most easily seen by noting that apart from an M -independent constant λ is simply the “local” compressibility of the lattice, a static quantity which, at least within the harmonic approximation, cannot depend on the ionic mass. Suppose then that we calculate the dependence of Tc on isotopic mass M. If we were to neglect the dependence through μ∗, we would get Tc ∝ θD ∝ M −^1 /^2 , or in terms of the conventionally defined isotope shift parameter α ≡ −∂(ln Tc)/∂(ln M ), α = 1/2; this is the original BCS result. If we take into account also the dependence via μ∗, we find
α =
[
1 .04(1 + λ(1 + 0. 62 λ)μ∗^2 [λ − μ∗(1 + 0. 62 λ)]^2
]
Note that even if μ∗^ is small, the correction to the BCS value α = 1/2 may be appreciable if λ is also small. Although many of the classic superconductors do show a value of α close to 1/2, values right down to zero and occasionally even negative do occur: note that such negative values are not incompatible with eqn. (4). However, values of α > 1 /2 are not found. In attempting to compare eqn. (4) with the experimental data on (possibly) exotic superconductors, one must bear in mind that isotopic substitution could conceivably affect Tc in other ways, e.g. by changing the lattice structure (this is particularly true for substitution of H (^1 H) by D (^2 H)). However, crudely speaking, the occurrence of an isotope effect with α close to 1/2 is prima facie evidence for a phonon mechanism, and conversely the absence of an isotope effect is at least some evidence for a non-phonon one.
characteristic (and there is no argument, e.g. from symmetry, that the relevant electron- phonon matrix element should vanish), that is a strong prima facie argument that the formation of the gap has nothing to do with phonons. Finally, we should remark that just as the electron behavior in tunneling reflects the effects of phonons, the converse should also be true: the phonon spectrum should show some effects of the onset of superconductivity in the electron system. These effects should be small, of the order of the dimensionless attenuation due to phonon-electron collisions in the normal phase, i.e. cs/vF ∼ 10 −^2 where cs is the phonon (sound) velocity, however they should be outside the experimental error. In particular, one would expect an anomalous contribution to the ultrasound attenuation when the frequency ω is close to the gap edge ω = 2∆), c.f. lecture 8 of part I. While the existence of such anomalous attention does not unambiguously establish a phonon mechanism for superconductivity, its absence would tend to indicate that the coupling of phonons to electrons is very weak and thus to cast doubt on such a mechanism.
Diagnostics of non-s-wave pairing.
In BCS theory as described in part I of the course, it is assumed that the “pseudomolec- ular” wave function ϕ(r 1 r 1 σ 1 σ 2 ) which enters the many-body wave function
Ψ(r 1 σ 1 r 2 σ 2.. .) = N A {(ϕ(r 1 r 1 σ 1 σ 2 )ϕ(r 3 r 4 σ 3 σ 4 ).. .} (7)
can be factorized into a spin part which is a singlet and an orbital part:
ϕ(r 1 r 1 σ 1 σ 2 ) = 2−^1 /^2 (↑ 1 ↓ 2 − ↓ 1 ↑ 2 ) · ϕ˜(r 1 r 1 ) (8)
It then follows that the “pair wave function” F has the same structure, i.e. 〈ψ↑(r 1 )ψ↓(r 2 )〉 = −〈ψ↓(r 1 )ψ↑(r 2 )〉 ≡ F (r 1 r 2 ), 〈ψα(r 1 )ψα(r 2 )〉 ≡ 0. Furthermore, F (r 1 r 2 ) is assumed, in the free-space case, to correspond to center of mass at rest (in thermal equilibrium) so that F (r 1 r 2 ) = F (r 1 − r 2 ), and finally to correspond to isotropic (s-wave) internal structure of the pair, i.e. F (r 1 − r 2 ) = F (|r 1 − r 2 |) with no dependence on the direction of r 1 − r 2. In the presence of a crystalline lattice we have to modify the last couple of statements slightly: although F (r 1 r 2 ) is not in general now simply a function of |r 1 −r 2 |, it is invariant under all operations of the crystal symmetry group (crystal translations and point-group operations).^2 We cannot assume that this simple state of affairs will hold for all possible systems in which Cooper pairs form (in fact, as early as the 70’s it was established not to hold in superfluid 3 He). Let us define a generalized “pair wave function” by
Fαβ (r 1 , r 2 ) ≡ 〈ψα(r 1 )ψβ (r 2 )〉. (9) (^2) In the simplest case, we can use a Bloch-wave basis and assume that only one band intersects the Fermi surface. Then we can introduce the quantity Fk ≡ 〈a† k↑ a†−k↓〉, where the a† k are Bloch-wave creation operators) and the statement is that Fk is invariant under transformations k → k′^ induced by the point group of the crystal.
The only generic requirement on F is that it respects the Fermi statistics, i.e. that
Fαβ (r 1 , r 2 ) = −Fβα(r 2 , r 1 ) (10)
but this still leaves many possibilities. It is convenient to restrict ourselves, as above, to the case where the Fermi surface is intersected only by a single band, and to assume that the center of mass is at rest. Then we have a more compact description in terms of the Fourier-transformed pair wave function in the Bloch-wave basis:
Fαβ (k) ≡ 〈a† kαa†−kβ 〉 = −Fβα(k) (11)
We can classify possible forms^3 of Fαβ (k) by their parity, i.e. by whether Fαβ (k) = + or −Fαβ (−k) [note spin indices are in the same order!], corresponding to even or odd parity respectively. It follows at once^4 from (10) (or (10)) that even-parity solutions must be spin singlet (i.e. odd in α β and odd-parity ones spin triplet (even in α β). In the singlet case Fαβ is manifestly just a product of spin and orbital functions:
Fαβ (k) = Fαβ (−k) = (iσy)αβ F (k) (12)
while for the triplet case Fαβ (k) may in general be a superposition of three different functions associated with the three Zeeman substates Sz = 1, 0 , −1. In the latter case it is convenient to characterize Fαβ (k) by a vector d(k) (in general complex) defined by
d(k) ≡ (iσyσ)αβ Fβα(k) (13)
In the (common) case where d(k) is a real vector, its physical significance is that there exists for any given k a direction along which the spin state of the relevant pair is S = 1, Sz = 0 ( √^12 (↑ 1 ↓ 2 + ↓ 1 ↑ 2 )), and the direction of d is just this direction (while its magnitude is a measure of the amplitude of the pair wave function, just as in the singlet case). In this case (real d) if we choose any axis in the plane perpendicular to d, the pairs appear to be formed in a linear superposition of Sz = 1 (↑ 1 ↑ 2 ) and Sz = −1 (↓ 1 ↓ 2 ) states, with a relative phase that depends on the specific choice.^5
Diagnostics of spin triplet (odd-parity) states.
- Knight shift
The most obvious difference between spin, triplet and spin singlet Cooper pairs is that the former, unlike the latter, can co-exist with a substantial spin polarization even in the
(^3) Superpositions of even- and odd-parity states can in principle occur, but the conditions for such a solution to be stable are extremely stringent and there is no evidence for them in any known (pure) Cooper-paired system. (^4) (11) itself follows strictly only because the operators a† kαa†−kβ are evaluated at the same time. It is
in principle possible to consider a scenario in which 〈a† kα(t)a†−kβ (t′)〉 is odd in the time variable t − t′ and thus vanishes for t = t′: then the “spin-parity connection” can be broken. (^5) For further details on the d-vector notation, see e.g. AJL, RMP 47 , 331 (1975).
averaged over the direction of k: cf. below. So if the DOS retains a singularity, we still expect an HS peak.
Diagnostics of non-s-wave orbital states^6
Let’s start with the simple case that the spin configuration of the pairs is a singlet, then the orbital configuration is characterized by a single scalar order parameter Fk ≡ 〈a† k↓a†−k↑〉, which is in general complex and must have even parity (Fk = +F−k). Equiv- alently, we can work in terms of the “gap parameter” ∆k, which is related to Fk by
Fk = ∆k/ 2 Ek, Ek ≡
^2 k + |∆k|^2
In the following I shall assume that ∆k is not appreciably a function of the magnitude of k, at least over a range k ∼ kBTc. Thus, it is the variation of ∆k, with direction on the Fermi surface that is of interest in the following. The quantity which is of primary interest for diagnostics of the orbital pairing state is the single-particle DOS in the superconducting state at T = 0,
Ns(E) ≡
k
δ(E − Ek). (15)
It is clear that this quantity is sensitive to the variation of ∆k over the Fermi surface; in particular, if ∆k is everywhere bounded below by some minimum value ∆min, then Ns(E) ≡ 0 for E < ∆min. If on the contrary ∆k has one or more nodes, i.e. tends to zero at one or more point or line on the Fermi surface, then Ns(E) will be finite for finite E. Let us make this a little more quantitative: The total number of states Ns(< E) with energies less than E is the number for which (^2 k + |∆(ˆn)|^2 ) ≤ E^2 , and hence it is (almost) intuitively obvious that the density of states dNs(< E)/dE is proportional to the area of that part of the Fermi surface that has |∆(ˆn)| < E. Formally, we can see this by writing
Ns(E) ≡
k
δ(E − Ek) = N (0)
dΩ 4 π
dE δ
E − E(, nˆ)
≡ N (0)
dΩ 4 π
d(E′, ˆn) dE′^
dE′^ δ(E − E′) ≡ N (0)
dΩ 4 π
d(E′, nˆ) dE′
Since (E, nˆ) = (E^2 − |∆(ˆn)|^2 )^1 /^2 , this becomes
Ns(E) = N (0)
∆(ˆn)≤E
dΩ 4 π
E
E^2 − |∆(ˆn)|^2
and thus, apart from a numerical constant that depends on the form of the node, is indeed proportional to the area for which |∆(ˆn)| < E. Thus we have, for point and line nodes in 3D and point nodes (the only possibility) in 2D, the results
(^6) Ref. Kuramoto & Kitaoka, Dynamics of Heavy Electrons, Sections 5.1–3.
3D, point: Ns(E) ∝ E^2 3D, line: Ns(E) ∝ E 2D, point: Ns(E) ∝ E. So far we discussed explicitly the spin singlet case. The triplet case is a bit more complicated, but simplifies considerably in the so-called unitary case in which the vector d(ˆn) describing the OP is real. In that case, for any particular direction of ˆn (and its opposite −ˆn) we can always choose our spin axis so that the spin state is pure S = 1, Sz = 0
√ 2 (↑^1 ,^ ↓^2 +^ ↓^1 ↑^2 )
and the pairing can thus be described by a single
complex number Fk (or ∆k), the only difference with the singlet case being that Fk (or ∆k) must now have odd parity (Fk = −F−k). The above formulae for the density of states, expressed in terms of the quantity |∆(ˆn)| (which is of course invariant under the choice of axes) are then unchanged. In the nonunitary case we have in general two different gaps for a given ˆn, and must then sum the two densities of states resulting from them to get a total DOS. It is clear that the occurrence of gap nodes on the Fermi surface will have a profound effect on the behavior at low temperatures of those properties that involve the normal component: crudely speaking, the density of the latter vanishes exponentially as T → 0 for a gap that is everywhere finite, but only as a power of T if there are nodes. Specifically, if the DOS vanishes as En^ for E → 0, then for the asymptotic behavior of various physical quantities simple scaling arguments predict the following^7 :
cv ∝ T n+ λ(T ) − λ(0) ∝ T n T 1 − 1 ∝ T 2 n+ Ks ∝ T n^ (spin singlet case).
I now turn more briefly to the way in which the symmetry of the order parameter affects the presence or not of gap nodes and hence the low-energy DOS. In the 3D freespace case things are relatively straightforward: for spin singlet pairing, barring pathologies^8 , the form of the OP must correspond to either a single spherical harmonic Y (^) ml or to a combination of Y (^) ml for the same l, where l = even because of the necessity for even parity. The case l = 0 (s-wave) is the simple BCS case and has a gap that is constant over the Fermi surface. For l ≥ 2 it is impossible to form a state that does not have, at least, point nodes, so the low-energy DOS is power law. For spin triplet pairing, again the only possibility is that the vector d(ˆn) has components each of which is some combination of spherical harmonics Y (^) ml with the same (odd) l. For l = 1 this allows states in which the total gap magnitude |∆(ˆn)| is finite for all ˆn (e.g. the “Balian- Werthamer” state d(ˆn) = const ˆn); for l ≥ 3 no such state is possible and again one must have at least point nodes. The situation is different in 2D, where nodeless states are possible for any l (e.g. ∆(ˆn) ∝ exp ilϕ).
(^7) Ks is the Knight shift (relative to its T = 0 value). (^8) Mixtures of spherical harmonics with different l are in principle possible but require very stringent conditions on the coupling constants.
