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What do we know for sure about the cuprate superconduc-
tors?
1. Macroscopic EM properties
[Ref.: Tinkham (1996 edition), Ch. 9] In this and the next lecture, I shall address the question: How much, if anything, can we infer about the general nature of the normal and/or superconducting states of the cuprates without recourse to a specific microscopic model?^1 Probably the single most important piece of experimental information we have on cuprate superconductivity concerns flux quantization and the Josephson effect. These experiments were done on YBCO at an early stage, and gave the results that are standard for classic superconductors, i.e. the unit of flux quantization is h/ 2 e (not e.g. h/e or h/ 4 e) and the Josephson frequency-voltage relation is ω = 2eV /ℏ. However, there is one subtle point that is often overlooked: the circuits used in the experiments were without exception such that the “paths” with respect to which the flux is quantized (etc.) lie entirely in the ab-plane. It is theoretically conceivable (though to my mind improbable, in view ofthe considerations below) that a direct experiment using an “all c-axis” circuit, should it be possible, would give a different result. The significance of these results is that, according to the argument of Part I, lecture 14, they provide very strong evidence that the superconducting state of the cuprates possesses long-range order in the two-particle correlation function (and does not have it in the one-particle one), which is, crudely speaking, equivalent to the statement that the “topology” of the wave function corresponds to formation of Cooper pairs just as in the classic superconductors. If we assume, as it almost universally is done, that this result holds for the c-axis as well as for the ab-plane, then this knowledge is sufficient for us to set up a Ginzburg-Landau description in terms of an order parameter which, just as in the classic superconductors, will have the physical significance of the center-of-mass wave function of the Cooper pairs. However, in distinction to the case of a classic (isotropic) superconductor the parameters of the theory will evidently distinguish between ab-plane and c-axis. At this point, anticipating the conclusions to be obtained in the next lecture, we might ask whether the fact that the internal state of the Cooper pairs will turn out, almost certainly, to be “exotic”, that is to have a symmetry lower than that of the lattice, will effect the validity of the GL description? The answer is no, at least so long as it corresponds to a single nondegenerate irreducible representation of the crystal symmetry group (see next lecture), but the reason is quite subtle: Although the OP does in a sense possess an “orientation,” that orientation is not free to adjust itself arbitrarily, but is pinned to the original crystal lattice, and therefore does not constitute a real “degree of freedom” which needs to be explicitly taken into account. Were the orientation free to
(^1) For the purpose of this discussion, I will generally assume, unless there is a good reason not to, that experimental results which may have been obtained on a single cuprate are representative of the properties of the cuprates as a whole.
adjust, as for example in the case of the l-vector in superfluid 3 He-A, it would have to be incorporated explicitly and the description would become more complicated. Thus, we proceed just as in the classic case but with allowance for the anisotropy: we treat the coordinate r for the moment as a continuous variable and define, just as there, a complex scalar order parameter Ψ(r) and write the usual terms proportional to |Ψ(r)|^2 and |Ψ(r)|^4 in the free energy. The gradient term, however, must now be treated as a tensor quantity γab with eigenvalues γ‖, γ⊥ corresponding to in-plane^2 and c-axis variation. Thus, the relevant form of the free energy density is
F {Ψ(r)} = −α(T )|Ψ(r)|^2 +
β(T )|Ψ(r)|^4 +
αβ
γαβ (T )
∇α + 2ieAα(r)
Ψ∗(r)
∇β − 2 ieAβ (r)
Ψ(r)
where as usual in the limit T → Tc we assume the temperature + magnetic field depen- dence β(T ) ∼ const ≡ β, γαβ (T ) ∼ const ≡ γαβ , α(T ) = α 0 (1 − T /Tc) (α 0 = const). It is worth taking a moment to discuss the limits of validity of eqn. (1). Strictly speaking, it is valid only in the limits T → Tc and infinitely slow spatial variation. A generalization to arbitrary T can (as in the classic case) be simply achieved by replacing the first two terms in F by a more general function Floc{|Ψ(r)|^2 , T }, and usually does not change things qualitatively. The question of the spatial variation, however, is more tricky. We recall that for a given eigenvalue γ of γαβ , the GL healing length ξ(T ) is given by ξ(T ) ≡ (γ(T )/α(T ))^1 /^2 = ξ 0 (1 − T /Tc)−^1 /^2 where ξ 0 ≡ (γ 0 /α 0 )^1 /^2. Crudely speaking, ξ(T ) is the distance over which the order parameter has to bend appreciably either in amplitude or in phase before the bending energy exceeds the original condensation energy; thus, the maximum gradient of the OP that is physically realistic is of order ξ−^1 (T ). The GL description will therefore be a generally valid description, at given T , if ξ(T ) exceeds by an appreciable margin any “microscopic” lengths in the problem (since correction terms, e.g. of the form |∇Ψ|^4 may be expected to become appreciable when the bending is over such a microscopic length). We recall for orientation that in the standard BCS case the longest such microscopic length is the (nearly temperature- independent) pair radius ξp, which in BCS theory is of the same order as the prefactor ξ 0 in ξ(T ); thus, for t = 1 − T /Tc 1 the GL description is generally valid. In the case of a layered system like the cuprates, this argument goes over unchanged as regards the ab- plane behavior. However, in the case of c-axis bending it may turn out that the prefactor ξ 0 (c )of ξ(c)(T ) is only of the order of a few ˚A or even less, and thus, in particular, smaller than the characteristic microscopic scale of the lattice structure (i.e. the (effective) c-axis cell dimension, ∼ 6 − 15˚A). In this case, eqn. (1) will still be a valid description in the limit T → Tc, but for T appreciably away from Tc it may need to be replaced by a more microscopic description: cf. below. Before we do so, however, let us derive from eqn. (1) some simple consequences that should be valid, at least, in the limit T → Tc. We first define the GL healing lengths
(^2) We implicitly assume isotropy within the ab-plane; where this is not present (as in YBCO) the appropriate generalization is obvious.
Note that if η 1, this means that for all but the smallest values of θ the critical field is determined by the condition Hc(θ) sin θ = H c⊥ 2 , i.e. the c-axis component of the field is equal to the critical field in this direction. The physical reason for this result is that almost all the energy of the vortices is associated with currents flowing in the ab-plane, very little with those flowing along the c-axis. If one looks at the detailed expressions for ξ and λ, one sees that the f (T ) which occurs in eqn. (2) is given by^4
f (T ) =
2 e^2
|α(T )||Ψ(T )|^2
2 e^2
∆F (T )
where ∆F (T ) is the condensation energy of the superfluid phase relative to that of the normal phase at the same temperature. Since this quantity can be measured directly in specific-heat experiments, the product ξi(T )λi(T ) can in principle be found for any given
cuprate. If one could also measure H c⊥ 2 and H c‖ 2 accurately, one would be able to combine this result with eqns. (3, 4) and obtain accurate values for all four quantities ξab, ξc, λab and λc, In practice, it is difficult to implement this program because the large fluctuations that occur in the EM behavior of the high-temperature superconductors mean that the whole concept of an “upper critical field” is not very well defined experimentally, see below. One might think that an alternative procedure would be to measure the (eigenvalues of the) lower critical field H c‖ 1 , H c⊥ 1 and relate them to λab(T ) and λc(T ) by the standard argument, but in practice reproducible measurements of Hc 1 turn out to be notoriously difficult in the cuprates. Probably the most reliable technique for estimating the eigenvalues of ξ and λ in the cuprates is to combine eqn. (7) with direct (e.g. microwave or μSR) measurements of λ. Where this is done, one finds that the prefactor ξab of the (1 − T /Tc)−^1 /^2 in ξab(T ) is comparable for YBCO (a-axis) and Bi- 2212, and of order 15 − 25˚A. The c-axis values are however very different: for (optimally doped) YBCO the experimental value^5 of λc(0) is ∼ 1. 1 μ; since λa(0) ∼1600˚A, this gives by (2) a ξc of the order of 2 − 3˚A (already smaller than the inter-bilayer spacing). For BSCO-2212 λc(0) has the enormous value 100μ, so ξc would be of the order of 0.05 ˚A, making the range near Tc where the 3D GL theory is applicable very small.^6 We now turn to the question of how to proceed when the 3D GL description fails because ξc(T ) becomes. the inter-multilayer spacing. So far we have avoided the ques- tion of “where” in the unit cell the superconductivity is located (nothing has depended on this, since we have used a continuum description), but we must now face up to it. So we raise the question: do we know for sure that superconductivity in the cuprates is primarily associated with the CuO 2 planes? Certainly this seems the overwhelmingly
(^4) This relation is often written (cf. Tinkham eqn. 9.6) in terms of the (isotropic) thermodynamic critical field Hc(T ), but the latter is not independently measurable for a type-II superconductor. (^5) Bonn et al., in G V. Note that what is quoted is the zero-temperature penetration depth λ(0) rather than the prefactor λ 0 of (1 − T /Tc)−^1 /^2 in λ(T ); the anisotropy of the two quantities is only slightly different, see Bonn et al., Fig. 18. (^6) Tinkham (p. 321) quotes values of γ(≡ our η) of 7 for YBCO and 150 for Bi-2212. The former is consistent with the above estimate but the latter smaller by a factor ∼ 3.
natural assumption, since at first sight these planes are the only element the various superconducting cuprates have in common. J. Dow has challenged this view, pointing out that another element that may be common to all superconducting cuprates is the “charge reservoir” layers.^7 It is of course clear that these layers, as their name implies, play an essential role in high-temperature superconductivity, at least as donors of holes (or electrons); but if the claim is that they are conducting and the Cooper pairs (?) form primarily in them rather than in the CuO 2 planes, it would seem to stretch credulity that the chemically and structurally very different charge reservoir layers found in (e.g.) YBCO, Tl-2201, Bi-22l2 and Hg-1201 nevertheless give rise to such similar Tc’s and (at least qualitatively) similar behavior in the superconducting state.^8 This is not to deny that there are important questions concerning these layers, not the least of which is the extent to which they are metallic; it is entirely possible that if they are, Cooper pairing is induced in them by a proximity-type effect from the CuO 2 planes. If so, this may affect the quantitative details of the description, of the c-axis properties, but I shall assume it does not affect the general scheme to be described. Thus, I shall take it as a given from now on that the principal seat of superconductivity is indeed the CuO 2 planes. Let’s start by considering a “single-plane” material such as Tl-2201. Then it is natural to define a discrete order parameter Ψn(r‖) for the n-th CuO 2 plane, where r‖ is the in-plane (xy-) coordinate. As a function of this order parameter the GL free energy would be expected to have the usual “bulk” terms, and the terms corresponding to inplane bending should also be essentially identical to those for the 3D case. However, we should expect that for “bending” along the c-axis the continuous gradient terms would be replaced by an expression proportional to the square of the difference of the discrete quantities Ψn(r‖) and Ψn+1(r‖). The lowest-order expression that is compatible with gauge invariance, etc., is
F (^) bend(n) (r‖) = |Ψn+1(r‖) − Ψn(r‖)|^2 ≡ |Ψn+1(r‖)|^2 + |Ψn(r‖)|^2 − 2 |Ψn(r‖) ‖ Ψn+1(r)| cos ∆ϕn(r‖)
where ∆ϕn(r‖) = ϕn+1(r‖) − ϕn(r‖), ϕn being the phase of the complex quantity Ψn. Thus, this term has the characteristic form of a Josephson coupling between neighboring CuO 2 planes. The total free energy obtained in this way has the form
F =
n
dr‖Fn{Ψn(r‖)} (9)
where
Fn{Ψn(r‖)} ≡ −α(T )|Ψn(r‖)|^2 +
β(T )|Ψn(r‖)|^4
+γ‖(T )|(∇⊥ − 2 ie A‖(r)/ℏ)Ψn(r‖)|^2 (10) +K|Ψn+1(r‖) − Ψn(r‖)|^2 (^7) At first sight the “infinite-layer” system Ca 1 −xSrxCuO 2 has no charge reservoir layers, but the question is complicated by uncertainties as to whether this material is single-phase. (^8) cf. also the existence of the “infinite-layer” compound, which has no charge-reservoir component.
Thus, for example, for Bi-2212 where λc at low T has been measured at low temperatures to be ∼ 100 μ, we infer^11 Jc ∼ 2kA/cm^2. It should be emphasized that the relation (14) between Jc and γ⊥ (and hence also eqn. (16)) applies only in the “true LD limit” where ξc(T ) d; in the opposite case, the d in eqn. (14) is replaced, up to a factor ∼ 1, by ξc(T ). Thus for example, we expect the critical current to vary as (1 − T /Tc) for T not too close to Tc but to crossover eventually to a (1 − T /Tc)^3 /^2 behavior as T approaches Tc and we get into the “3D GL” regime. It seems that for YBCO this regime is already reached at 1 − T /Tc ∼ 0 .1, while for the much more anisotropic BSSCO-2212 compound it occurs only at 1 − T /Tc ∼ 10 −^3 and hence is barely visible: BSCCO is almost always in the “true LD” limit. One may ask whether, apart from its a priori plausibility, there is any direct experi- mental evidence for the picture of the CuO 2 planes as a series of Josephson junctions in series? If this view is correct, then one would expect that under appropriate conditions the nonlinear current-voltage characteristics would show the typical Josephson features, and indeed this seems to be the case^12 in Bi-2212. Note that in the true LD limit, in strong distinction to the GL case, the critical current can be exceeded without heating the sample into the normal phase. So far, we assumed we are dealing with a single-plane material. What about multi- plane materials such as Bi-2212? The most obvious assumption (which I have implicitly used a couple of times above, when referring to experimental data on this compound) is that the CuO 2 planes within a single multilayer are coupled together as strongly that it is legitimate in the present context to treat each multilayer as a single plane; then the above analysis goes though unchanged. However, it is not in fact quite certain that the coupling within a given multilayer is much stronger than that between different multilay- ers, and there are even a few pieces of evidence (e.g. the fact that the λc of Hg-1223, as inferred by Panagopoulos et al. from powder magnetization measurements,^13 is a factor of 5 larger than that of Hg-1201) which might suggest the opposite conclusion. I would regard this question as currently open: if indeed the “unexpected” result is correct, a quantitatively correct account of the bilayer cuprates would require the appropriate (and obvious) generalization of the LD model (the 3D GL model is, of course, insensitive to this complication). A great deal of experimental work on the static magnetic properties of the cuprates has been done and interpreted in terms of the LD theory (or its limiting form, the 3D GL theory). A particularly interesting situation occurs when the external magnetic field is neither parallel nor perpendicular to the ab-plane. Under these conditions one expects to produce vortices that are on average parallel to the field. However, it is easily verified that to produce a given current in the c-direction costs an energy γ‖/γ⊥ times that necessary to produce the same current in the ab-plane (E ∼ J^2 /ρs ∼ J^2 /γ!), and thus the currents much prefer to flow in the planes. The result is a set of so-called “pancake” vortices that are staggered from one plane to the next, and the magnetization is not (^11) Assuming that a bilayer can be treated in this context as equivalent to a single plane, cf. below. (^12) Kleiner and Milller, Phys. Rev. B 49 , 1327 (1994). (^13) However, note the caveats in lecture 7 on this technique.
parallel to the field but oriented more along the c-axis: see Tinkham, Section 9.3.
Let’s now very briefly turn to the question of the resistivity of the cuprate supercon- ductors in a magnetic field. This could itself easily be the subject of a whole course: here I have space only for the barest essentials. The two basic qualitative points to appreciate are (1) that “superconductivity” in the sense of zero resistivity cannot be maintained in the presence of vortices, unless these are pinned, and (2) that because of the very different orders of magnitude of the relevant parameters, in particular temperature, it is far more difficult to pin vortices than in a classic superconductor. As a result, the question “are cuprate superconductors in a magnetic field really superconducting?” does not have a trivial answer. To take point (1) first, a vortex of circulation ˆn
vs · dl = κ (ˆn = direction of axis) placed in a flow field such that the flow velocity at ∞ is v will find a so-called Magnus force of magnitude
FM = ρv × κ (17)
where ρ is the density of the fluid forming the vortex. The Magnus force has nothing to do with quantum mechanics (it was originally discovered in classical fluids); for a neutral system it is straightforward to obtain it by considering a tube of finite width and calculating the total kinetic energy as a function of vortex position (for a charged system where the vortex is effectively of finite extent, ∼ λ, this argument in its simple form does not work, but more sophisticated arguments give the same result (cf. Tinkham, Section 5.2)). In the case of a superconductor, κ is equal to ˆn (h/ 2 m), and if we assume that the bulk velocity u is associated with the same “density” ρ as appears in (16) (i.e. the superfluid density ρs) then we can rewrite (16) in terms of the electric current density J(r): FM = (J × nˆ)Φ 0 (Φ 0 ≡ h/ 2 e) (18)
Note that this relation is independent of the value of ρs and hence of T. If the Magnus force FM is not balanced by some “pinning” force that tends to keep the vortex close to a given impurity (etc.), then its effect will be to accelerate the vortex transverse to the current J; eventually its effect will be balanced by some frictional force, and the vortex will reach a terminal (steady-state) velocity u, which is the simplest case would be expected to be proportional to v. Now, consider the total phase difference ∆ϕ 12 ≡
1 ∇ϕ^ ·^ dl^ between two points in the system separated along the direction of current flow: for definiteness we choose a straight contour to connect them. Whenever a vortex moves across the contour, the integral decreases by an amount 2π. But according to the Josephson relation, we expect the voltage difference V 12 between the points 1 and 2 to be proportional to the rate of charge of ∆ϕ 12 : d dt
∆ϕ 12 = 2 e ℏ
V 12 (19)
Consequently, the average voltage V 12 is
V 12 =
2 e
d dt
∆ϕ 12 =
2 e
2 π nvus 12 ≡ Φ 0 nvus 12 (20)
