Download Coulomb Energy Savings in High-Tc Superconductors: ILT Model vs. Incoherent Hopping - Prof and more Exams Physics in PDF only on Docsity!
Where is the energy saved?
A rather generic and unsatisfying feature of all the theoretical scenarios discussed in the last lecture is that, while purporting to explain part or (occasionally) all of the existing theoretical data, they make few if any quantitative predictions about phenomena as yet unobserved. (In a few cases a theory may seem to do so, but on close examination it usually turns out that the ‘prediction’ is in effect a simple extrapolation of trends already observed and is thus not terribly surprising.) In this lecture I review a couple of theoretical scenarios which, by contrast, have attempted to make quantitative predictions about experimental quantities which at the time of the prediction were not measured, and examine the extent to which the predictions agree with subsequent experiment. The first is the ‘inter-layer tunneling’ scenario of P.W. Anderson and collaborators, the second my own ‘midinfrared’ scenario. Both focus primarily on the ‘mechanism’ of the superconducting transition (in fact, neither as such claims to explain the N state behavior in detail∗), and both emphasize, in rather different ways, the question ‘where is the energy saved?’ One more feature which the two scenarios have in common (and in which they differ from the vast majority of other theoretical proposals in the literature) is that both take very seriously the observed dependence, in homologous series, of Tc(n) on the layer multiplicity n. So it is appropriate to start by saying a word about this: To recapitulate the results quoted in lecture 6, in the Ca-spaced homologous series (Bi, Tl, Hg) Tc(n) increases with n as far as n = 3, thereafter apparently decreasing (at least in the cases, such as the Tl-2 series, where one can be reasonably sure that the material is single-phase). In all the cases quoted, the Tc’s for n ≤ 3 appear to fit rather well the formula T (^) c(3) − T (^) c(2) T (^) c(2) − T (^) c(1)
However, we should strongly caution that this apparently impressive coincidence among four different series (Bi, Tl-1, Tl-2, Hg) is almost certainly less significant than it looks, since the ratio α of T (^) c(2) − T (^) c(1) to the single-layer transition temperature T (^) c(1) is quite different for the Bi and Tl-1 series (α ∼ 2) to that for the Tl-2 and Hg series (α ∼ 0 .5). My own belief is that one should attach significance to the relation (1) only in these
cases (Tl-2, Hg, possibly Ca) where T (^) c(1) is already of order 100K. The above statements refer exclusively to Ca-spaced homologous series: when the spacer is (entirely) Sr or Ba, by contrast, the 2- and 3-layer members are almost invariably nonsuperconducting,† probably though not certainly because of the effect of ‘intruder’ oxygens. Clearly, there are two major classes of explanation for the observed behavior of Tc as a function of n.
(1) The mechanism of superconductivity is entirely (or overwhelmingly) confined to the single CuO 2 planes, and the reason that Tc(n) increases with n in the Ca- ∗In work which is logically closely related to the ILT scenario, Anderson has claimed to explain the N-state properties. However, no direct quantitative test has been suggested for this part of the scenario. †But cf. Jim Eckstein’s recent result in Bi 2 Sr 3 Cu 2 O 8.
spaced series is that the properties of the individual planes are ‘improved’ in some way in multilayer materials.
(2) The increase of Tc is a result of some kind of interaction between the different CuO 2 planes.
Let’s first discuss some possible class-1 explanations. In the early days, it was often postulated that the effect of adding more planes was simply to increase (or otherwise improve) the number of carriers per plane. Certainly, NMR measurements in Hg-2223‡ do seem to indicate that the number of carriers in the central plane is different (probably less) than that in the outer ones of the trilayer, and this is not entirely unexpected from the point of view of the electrostatics. However, a major argument against attributing the increase in Tc to this effect is simply that it is by now very well established that in all superconducting cuprates, including the single-layer ones like Tl-2201, Tc has a maximum as a function of doping; it then immediately follows that no change in the doping can by itself increase Tc above this maximum. A similar argument refutes the suggestion that adding extra planes changes the buckling; all the evidence is that the ‘optimal’ buckling is zero, and this is already attained in the one-layer Tl and Hg compounds. A third class-1 proposal is that as a result of adding extra planes the dielectric screening within the individual planes is changed; this is certainly likely to be true, since e.g. for a 3-layer material the rather polarizable oxide layers are replaced, for the middle plane, by very unpolarizable Ca++^ ions. The hypothesis that it is this effect which is responsible for the increase in Tc is more difficult to refute, and in fact has the advantage that it might go some way towards explaining why the introduction of (highly polarizable) intruder oxygens appears to depress Tc so severely: however, it would not seem to explain why it is depressed way below the one-plane value, nor why in the Ca-spaced compounds Tc decreases for n ≥ 4. So it seems at least possible that the true explanation of the variation of Tc with n is of class 2; both the principal scenarios to be discussed in this lecture are of this class.
The interlayer tunneling scenario¶
The ILT scenario (or, as its proponents would call it, theory) for HTS dates from Septem- ber 1987, and was vigorously maintained by Anderson and a few collaborators for about ten years thereafter. Although it is by now largely discredited, I believe it is well worth reviewing in some detail, since it seems to me to have many of the generic characteris- tics that one would hope for in any eventually successful theory, in particular in that it makes very specific and quantitative predictions, and hence is eminently ‘falsifiable’ in the sence of K.R. Popper. The ILT scenario rests on two major hypotheses: ‡Michalak et al., Physica C 235 , 1673 (1994). ¶Ref.: P.W. Anderson, The Theory of Superconductivity in the High-Tc Cuprates, Princeton Univer- sity Press, 1997, ch. 7.
compose a bilayer than for a pair of neighboring unit cells). It is supposed that it is this energy saving which is all, or at least a large part (let us say η) of the superconducting condensation energy Econd. The remarkable thing about this idea is that it is subject to more or less direct experimental test, since in a single-plane material the coefficient J is directly proportional to the (3D) c-axis superfluid density∗^ ρs⊥. In fact, directly from the considerations of lecture 9 we have the relation
∆T⊥ =
2 md
ρs⊥ (3)
where ∆T⊥ (∝ J) ≡ ηEcond is the energy saving per unit volume and d is the interplane spacing. Using the relation between ρs⊥ and the c-axis penetration depth λ⊥ and as- suming the local magnetic permeability μ to be 1, we obtain a prediction for λ⊥ which is conveniently written in the form†
λ⊥ = η−^1 /^2 λILT, λILT ≡
mc^2 Econd
a 0 A 16 πd
where Econd is now the condensation energy per formula unit (i.e. per CuO 2 ) at T = 0, a 0 is the Bohr radius and A the area per formula unit. Note that the formula for λILT does not involve the poorly known c-axis dielectric constant ⊥. It is interesting to compare the prediction (4) with that which would follow from the application to each inter-plane ‘junction’ of the Ambegaoker-Baratoff formula (part I, lecture 13)
IcRn = π∆ 2 e
with ∆ taken to have the BCS value 1. 76 kBTc. If we call the value of λ⊥ predicted in this way λρ we have according to the results of lecture 7
λρ =
ℏc^2 0 Rn π∆
2 d
We saw in lecture 7 that according to the Basov plot,‡^ most of the measured values of λ⊥ whether for single- or multilayer cuprates,§^ seem to fall close to λρ. It is clear that if the predicted value of λILT for a given single-plane cuprate is of the same order as that of λρ, an examination of the experimentally measured λ⊥ will not
∗The case of a multilayer cuprate is more complicated, since in this case Econd comes predominantly from the ‘strongest’ tunneling links whereas ρs⊥ is determined primarily by the ‘weakest’ ones: so there is no one-one correspondence between the two quantities. †AJL, Science 274 , 587 (1996). Note that I have incorporated, here, an extra factor of 2 into the definition of λILT. ‡D.N. Basov et al., Phys. Rev. B 50 , 3511 (1994). §In the multilayer case we would expect both λ⊥ and ρ⊥ to be diminished by the ‘easiest’ links, so this result is expected
necessarily be very informative. This is the case for La 2 −xSrxCuO 4 at various dopings since both λILT and λρ lie in the range 3 − 15 μ (cf. Anderson, Science 279 ,1197 (1998)). However, the situation is quite different for Tl-2201: in this case λρ is around 20μ, whereas from the measured value of the condensation energy we have λILT ∼ 0. 9 μ (note this corresponds to a discrepancy of a factor ∼ 500 in the more physically meaningful quantity ρs⊥!). For Hg-1201 the value of λρ is ∼ 10 μ; the prediction for λILT is less accurate than for Tl-2201 because the condensation energy is less reliably measured, but λILT should again be of order 1μ. This is a crucial test of the ILT model, at any rate in the simple form embodied in eqn. (1), should be a measurement of λ⊥ for these two materials.∗^ This has been accomplished by Moler, Kirtley and co-workers by a very direct technique (imaging of ‘c-axis vortices”: note that while the resolution in these experiments is insufficient to measure λab, it is more than adequate for the measurement of λ⊥). The results are unambiguous:‡‡^ λ⊥ ≈ 17 − 21 μ for Tl-2201 and ≈ 8 for Hg- 1201, in clear agreement with the predicted λρ and order-of-magnitude disagreement with λILT. Thus the ILT scenario, at least in its simplest and most discussed version, seems to be unambiguously refuted.[ Two cautions: (1) It may be just possible to rescue the scenario if one is prepred to go beyond the simple ansatz (1), but only at the cose of postulating rather wild fluctuations in the c-axis tunneling properties for which there seems to be no obvious independent evidence (see AJL, ref. cit.)
(2) The Moler-Kirtley experiments cannot exclude a much watered-down version of the model, in which most of the condensation energy (in fact all in single-plane materials) comes from (unspecified) in-plane effects but the ‘boost’ in multilayer materials derives from c-axis tunneling. See S. Chakravarty et al., PRL 82 , 2366 (1999); Phys. Rev. B 67 100504 (2003)
]
In the rest of this lecture I want to discuss a rather different approach to the problem of energy saving.†^ For simplicity I will start with the case of a single-plane material, and subsequently generalize the discussion to the multiplane case. Consider, then, a single- plane cuprate such as Tl-2201 or Hg-1201. I start from four fundamental assumptions:
(1) The usual separation of the electrons into ‘core’ and ‘conduction’ electrons is le- gitimate, so that we can write down an effective Hamiltonian for the conduction electrons alone, with the effects of the core electrons completely embodied in an external potential and the screening of the inter-conduction electron Coulomb in- teraction. ∗The first indication that λ⊥ was λILT came from the c-axis optical experiments of van der Marel and coworkers, who established the absence of a c-axis plasmon for ν > 100cm−^1. However, the conversion from νρ to λ⊥ involves the c-axis dielectric constant, which has been the subject of some controversy. ‡‡Moler et al., Science 279 , 1193 (1998); PRL 81 , 2140 (1998) †Ref.: AJL, Proc. Natl. Acad. Sci. 96 , 8365 (1999).
transition), i.e. we assume it to find the best ‘normal’ ground state.¶^ If we now relax the constraint, we know from experiment that the system will form pairs, and the reason it does so, at T = 0, is, trivially, to reduce its total energy. It follows that at least one of Tˆ‖, Uˆ and Vˆ must decrease, possibly though not necessarily at the expense of an increase in the other two. Let us suppose, for the sake of definitions, that 〈 Vˆ 〉 decreases (without any assumption about 〈 Tˆ‖〉 and 〈 Uˆ 〉; nothing actually depends on this. Now, as we shall see in a moment, the value of 〈 Vˆ 〉 in an arbitrary state of the system can be expressed as a sum rule, i.e. as a sum of contributions from different regions of wave vector q and frequency ω: so a natural question is: In what regime(s) of q and ω does the saving∗∗ of Coulomb energy occur? Oddly enough, until recently this question seems hardly to have been asked in the literature. Let’s try to be a bit more quantitative. An exact expression for the Coulomb energy is
〈 Vˆ 〉 =
q
Vq〈ρˆq ρˆ−q〉 =
2 π
q
0
dω Im χ(q, ω),
(Vq ≡ e^2 / 2 0 sc|q|)
where χ(q, ω) is the complete (true) density-density correlation function of the system. Note that this expression is exact, independently of the effects of lattice structure. How- ever, the latter now leads to a complication: we would like to express χ(q, ω) in terms of Vq and the ‘bare’ correlation function χ 0 (q, ω), by which we mean the quantity defined diagrammatically by omitting all those graphs in χ(q, ω) which can be cut into two by cutting a single Coulomb line of momentum q. The problem is that χ 0 is actually a matrix in the reinforced space, i.e. it is specified by two arguments, q and q + K, and this leads to rather a messy form for Vˆ (see appendix B of AJL, ref. cit.). For pedagogic simplicity I will therefore assume at this point that the matrix can be approximated by its diagonal terms: while in the general case this approximation may introduce some error, it can be shown (ref. cit.) that it does not affect appreciably the arguments I shall give concerning the long-wavelength limit. With the above approximation we then have
χ(q, ω) =
χ 0 (q, ω) 1 + Vqχ 0 (q, ω)
and inserting this into the expression for 〈 Vˆ 〉
〈 Vˆ 〉 = −
2 π
q
dω Im
1 + Vqχ 0 (q, ω)
Note that formula (10) is actually valid in any number of dimensions, provided that Vq has the appropriate form. In particular, in the bulk 3D case we have Vq = e^2 / 0 scq^2 ,
¶The nearest approximation to this in real life is to cool in a strong magnetic field, as in the experi- ments of Boebinger. ∗∗Or increase, if 〈 Vˆ 〉 indeed increases.
and since the (longitudinal) dielectric constant ‖(q, ω) is conventionally defined as∗ 1 + (e^2 / 0 scq^2 )χ 0 (q, ω), eqn. (10) takes the simple form
〈 Vˆ 〉 = −
2 π
q
dω Im
‖(q, ω)
where the integrand – Im (^) ‖(^1 q,ω) is usually called the loss function and is directly mea- sured in transmission EELS experiments. In 2D one has to be a little more careful, since no true analogy to ‖ exists: it is most convenient to write expression (10) in the form
〈 Vˆ 〉 = −
2 π
q
0
dω Im (1 + qK(q, ω)/sc)−^1 (12)
where the quantity K(q, ω), which has the dimensions of length, is related to the 3D dielectric constant by
K(q, ω) =
2 π d¯(‖(q, ω) − b) (13)
where d¯ is the interplane separation and b is the ‘background’ (non-conduction-electron) contribution†^ to ‖(q, ω). Let’s now consider the contribution to the fundamental expression (10) for 〈 Vˆ 〉 from different regions of q (in the N phase). To do this, it is convenient to note the Kramers- Kronig relation for χ 0 (q, ω):
1 π
Im χ 0 (q, ω) ω
dω = χ 0 (q) (14)
where χ 0 (q) is the ‘bare’ static susceptibility, which we expect to be not too strongly varying with q and of the order of magnitude of χ 0 (0) ∼ dn/d. From this relation we see that in general‡^ is we would expect Im χ 0 (q, ω) to be of the general order of magnitude of dn/d (and then, by the KK relation applied to general ω,the real part should be of the same order of magnitude). This then leads us to define a 2D ‘Thomas-Fermi’ wave vector qtf by the relation
Vqtf
dn d
For a noninteracting-band model with effective mass m∗, we have dn/d = m∗/ℏ^2 and so qtf = 2(m∗/m) − rmsc^1 a− 0 1 (16)
(a 0 = Bohr radius). With m∗/m ∼ sc ∼ 4 this gives qtf ∼ 4˚A−^1 , irrespective of the density of carriers. Note incidentally that in 3D we can apply the same general arguments, and obtain the standard result
qtf =
[
e^2 0 sc
dn d
)] 1 / 2
[
π
m m∗
sca 0
] 1 / 2
kF^1 /^2 (17)
∗Modulo some rather messy questions concerning the treatment of the factor sc. †The equality b = sc holds only in specific models. ‡These arguments need some modification in the limit q → 0, cf. below.
An advantage of the ‘small-q’ assumption is that it permits us to explain the be- havior of Tc(n) in multilayer cuprates in terms of the effects of the inter-plane Coulomb interaction (which must be there!). See AJL, PRL 83 , 392 (1999), where it is shown inter alia that this hypothesis leads to the formula, valid for ‘not too large’ n,
Tc(n) − Tc(1) = const (1 − 1 /n) (20)
The ideal test of the MIR scenario would he differential transmission EELS experiments, but these turn out to he technically very difficult. The recent optical experiments of van der Marel and co-workers show that if one can extrapolate the measured changes in (q, ω) (actually ⊥(q, ω)) at the N-S transition from qξ 1 to qξ 1, then the magnitude of the change in (10) is roughly adequate to account for the condensation energy, but the sign is wrong!! Thus at the moment the whole question is very much up in the air.
