Van Hove SINGULARITIES IN BCS THEORY Van Hove SINGULARITIES IN BCS THEORY
By Armando Ga.ma Goicochea, B.Sc.
A Thesis
Submitted to the Faculty of Graduate Studies
in Partial Fulfilment of the Requirements
for the Degree
Master of Science
McMaster University
August 1992 MASTER OF SCIENCE{1992) McMASTER UNIVERSITY (Physics) Hamilton, Ontario TITLE: Van Hove Singularities in BCS Theory AUTHOR: Armando Gama Goicochea, B.Sc. (Universidad Nacional Autonoma de Mexico) SUPERVISOR: Dr. Jules P. Carbotte
NUMBER OF PAGES: vii, 115
ii ABSTRACT
The influence of a logarithmically dependent (van Hove singularity) electronic density of states is studied in the weak-coupling limit. Through analytic and numerical analysis it is found that the model can give rise to temperatures in the 100 K range, and that universal BCS ratios such as 2tfa./k8 Tc and tfa.C/'rTc do not change essentially from their constant BCS values. The consequences of this model on the calculation of the isotope effect and specific heat are discussed in detail and compared to recent experimental results.
iii To my parents,
Eva Goicochea and Armando Gama ACKNOWLEDGEMENTS
I wish to thank my supervisor, Dr. Jules P. Carbotte for his advice and encouragement. His devotion for his work makes of him an excellent role model. This is perhaps the most trascendentallesson one can learn from him.
I am also indebted to Drs. David Goodings and Yuki Nogami for their careful reading of this thesis, and for the very useful comments they had.
For the multiple enlightening discussions we had, his interest in this work, and his willingness to offer help, I would like to thank Mohamed Man sor. I have also benefitted from discussions with Phil Fischer.
My gratitude goes to all those who have made these two years a very enjoyable learning experience, in particular Peter Arberg, Anthony Boey,
Alex Bunker, Charles Curry, Mohamed Mansor, Peter Mason, R.achid Ouyed, Harald Schwichow, and Mark Walker. For sharing their enormous talent and
friendship with me I thank them all. It is an honour to count them as friends.
The generous hospitality of Harald Schwichow is also acknowledged.
The staff of the Department of Physics and Astronomy has always
been very helpful. Their efficiency and friendliness are exceptional. They all
certainly contribute to enhance the reputation of the Department.
For financial assistance, thanks are due to the Department of Physics and Astronomy of McMaster University. Finally, I would like to thank the Universidad Nacional Aut6noma
de Mexico for taking care of my graduate education through a D.G.A.P.A.
scholarship.
v Table of Contents
Chapter 1 Introduction ...... 1
Chapter 2 The BCS Theory ...... 5
2.1 Historical Introduction ...... 5
2.2 The Ground State ...... 8 2.3 Zero-Temperature Energy Gap ...... 10 2.4 Temperature Dependent Energy Gap ...... 12 2.5 The Critical Temperature...... 13
2.6 The Isotope Effect ...... 15
2.7 Thermodynamic Properties ...... 16
2.8 Concluding Remarks...... 18
Chapter 3 IDgh Tc Superconductivity ...... 21
3.1 Historical Introduction ...... 21 3.2 Structural Properties ...... 23 3.3 Physical Properties ...... 24
3.4 Concluding Remarks...... 28
Chapter 4 Van Hove Singularities in BCS Theory ...... 31
vi Table of Contents
4.1 Introduction ...... 31
4.2 Energy Dependence in the DOS...... 32
4.3 The Model ...... 34
4.4 The Critical Temperature...... 36
4.5 Zero Temperature Energy Gap...... 47
4.6 The Isotope Effect ...... 56
Chapter 5 Thermodynamic Properties ...... 43
5.1 Zero Temperature Condensation Energy ...... 43
5.2 :Free Energy ...... 66
5.3 Specific Heat ...... 76
5.4 Concluding Remarks...... 94
Chapter 6 Conclusions ...... 97
Appendices...... 101
A The Density of States...... 101
B A Useful Integral ...... 105
C Weak-Coupling Constant ...... 107 D The Energy Gap near Tc ...... 109
References ...... 112
vii Chapter 1
Introduction
A few years ago, the idea of superconductivity occurring at temper atures around 100 K would probably have seemed futile, or hopeless. And there was reason to believe so, since for decades scientists worked hard to raise the critical temperature, Tc (temperature at which a material undergoes the superconducting transition), only a few degrees above 20 K. This ideal became reality in 1987, when superconductivity was found to occur in copper-based oxides at temperatures close to 100 K. Through ingenious chemical substitutions, the highest Tc was later pushed up to 125
K, in 1988, and remains the record Tc known to date. These events constitute a major breakthrough that has profoundly revolutionized not only physics, but science in general. It has also produced a flood of multidisciplinary work hardly seen before.
Soon after the discovery of high Tc oxides it became clear that the con ventional microscopic theory of superconductivity, formulated by J. Bardeen,
1 2 1 Introduction
L. N. Cooper, and J. R. Schrieffer (BCS) in 1957 did not provide an ade quate description of these new materials. Experimental results such as a very small or near zero isotope effect, and unusual temperature dependence in the normal state resistivity led some authors to formulate models that were not based on a Fermi liquid description, and did not invoke a phonon-mediated pairing mechanism. By contrast, others believe that the Fermi liquid ap proach is valid, and that once the appropriate pairing mechanism is found, the BCS formalism will provide a good description of the oxide supercon ductors. A common feature found among these new materials is the presence of Cu. and 0 atoms occupying a nearly square planar arrangement. These
Cu.02 planes, as they are usually referred to, are probably the most funda mental pieces, for they are believed to be responsible for superconductivity in the layered compounds. This is the reason why superconductivity in these compounds is generally considered quasi two-dimensional. On the other hand, it has been known for a long time that the density of states (DOS) for electrons in a two-dimensional periodic potential exhibits logarithmic van Hove singularities (vHs), which arise from saddle points in the energy dispersion relation. This motivated some theorists recently to introduce the concept of vHs in the DOS of the new Cu.-based superconductors, in an attempt to explain the strikingly high Tc's that they show. It has also been claimed that the small isotope effect can be understood with the help of this energy dependent DOS. 1 Introduction 3
We believe that this conclusion is of considerable importance. Moti vated by this interesting suggestion, we have introduced a vHs in the elec tronic DOS near the Fermi energy, within the framework of the weak-coupling
BCS theory, and some properties have been calculated and compared to their constant-DOS counterparts. Our aim is to determine whether or not this model is able to account for some of the properties of the oxides, and to ob tain simple approximate analytic expressions for important superconducting parameters such as the critical temperature, and the specific heat jump.
In Chapter 2 we outline briefly some of the important ideas involved in the BCS theory, as well as a few basic equations. No attempt is made to include all the relevant concepts of the theory, nor to provide a detailed derivation of the equations, for there is plenty of excellent literature on the subject accesible to those interested.
Chapter 3 is a summary of some relevant characteristics of the high Tc cuprate superconductors. Because this is a rapidly evolving field, one cannot hope more than to merely describe a few common structural and physical properties, especially those that are important in the development of our model.
The consequences of a vHs on the critical temperature are studied in
Chapter 4, along with some other calculations regarding the zero temperature energy gap, and the isotope effect coefficient. The influence of the vHs in the thermodynamical behavior of weak coupling superconductors is the subject of Chapter 5. The free energy, tem perature dependent gap, and specific heat are investigated for this model and compared to the BCS results. 4 1 Introduction
Fina.lly, the most important conclusions are gathered in Chapter 6. Additiona.lly, Appendices A, B, C, and D are included to provide some com pleteness to the work. Chapter 2
The BCS Theory
2.1 Historical introduction
The phenomenon of superconductivity was discovered in 1911 by
Kamerlingh Onnes1, who named it based on the striking electrical properties of this state, i.e. infinite conductivity below the transition temperature Te.
He also discovered that superconductivity is destroyed if a strong enough magnetic field is applied , which is now called the critical field He .
A superconductor is more than a perfect conductor, it is also a per fect diamagnet. This means that the magnetic field vanishes in the interior of a bulk specimen when cooled below its transition temperature Te. This phenomenon, known as the Meissner effect, was discovered in 1933 by W.
Meissner and R. Ochsenfeld2•
5 6 2 The BCS Theory
In 1934, C. J. Gorter and H. B. Casimir formulated3 a phenomeno logical theory that accounted for the thermodynamic properties of supercon ductors, based on a two fluid model. The following year, F. and H. London proposed4 an electromagnetic theory of superconductivity that was able to account for the infinite conductivity and the Meissner effect. However it was not until1957 that a proper theory of the supercon ducting state based on first principles could be formulated. This was due to a number of reasons, but certainly one among the most important ones was the extremely small energies involved in the superconducting transition, which
amount typically to l0-8 eV per electron, while the Coulomb repulsion is of order 1 eV per electron. Therefore it was very important to recognize the relevant correlations that a model should have to give rise to superconduc
tivity, leaving the rest of the larger effects aside, since presumably they do not change between the normal and superconducting phases. A major step toward the formulation of a microscopic theory came with the discovery in 1950 by Maxwell and Reynolds and collaborators5 of the so-called isotope effect, because it indicated that superconductivity arises from the interaction between the quantised motion of the ions (phonons) and the electrons.
This experimental clue motivated some theoretical work which un fortunately ran quickly into mathematical difficulties. For example, some ap
proaches tried to apply perturbation theory, unsuccessfully. It is now known
that perturbation theory cannot account for superconductivity to any order in the expansions.
It was Fritz London himself who conceptualised some of the basic ideas about the nature of the superconducting state that turned out to be 2 The BCS Theory 7 ratified later on by the correct microscopic theory, e.g. the pairing of elec trons, and the energy gap, among others. The following step toward a microscopic theory was taken by Leon
Cooper, who showed6 that if there is an effective attractive interaction be tween a pair of quasiparticles above the Fermi sea, they will form a bound state, no matter how weak the interaction. Put in other words, the Fermi sea is unstable against the formation of the so-called Cooper pairs. A powerful method of approach in low temperature physics consists of studying the nature of the ground state of a system, and then the elementary thermal excitations. And here the Landau theory of the Fermi liquid7 seemed most appropriate. According to Landau, as long as the interactions between particles do not lead to discontinuos changes in the microscopic properties of the system, a quasiparticle picture is valid, i.e. the thermal excitations of an interacting phase are in a one-to-one correspondence to the low-lying excitations of a non-interacting Fermi gas. However, Cooper's problem is a two-body problem, and in a real superconductor there are about 106 pairs within the volume occupied by the bound state of a given pair. The average radius of a Cooper pair can be identified with a coherence length eo, whose order of magnitude can be estimated from the uncertainty principle:
(2.1) which is quite large, considering that superconductivity is a quantum me chanical effect. Here, kp is the Fermi wave vector. The coherence length is a fundamental quantity in BCS theory, since it indicates the spatial spread of the wave function of a superconductor, and in this sense it sets a minimum 8 2 The BCS Theory to the size that a specimen should have for the theory to be aplicable. On the other hand, the strong overlap of the pairs as well as the exclusion principle and required anti-symmetry of the wave functions invalidate a Bose-Einstein condensation picture for superconductivity.
There is another relevant length for superconductivity, known as the London penetration depth, or A£. However its physical meaning is entirely different from that of eo. The length AL measures the depth of penetration of a magnetic field applied to a bulk specimen. Both lengths depend on the mean free path of the electrons in the normal state. The ratio"= AL/eo sets the difference between type I and type II superconductors, i.e., "'< 1 implies a type I superconductor, whereas"> 1 indicates type IJ9.
2.2 The Ground State
The fundamental and now classic work of J. Bardeen, L. Cooper, and J. Schrieffer8 (from now on, BCS) took Cooper's treatment of the two-body problem as a starting point. They considered quasiparticle states occupied by pairs (k1 j,k2 !), such that each pair had opposite spin and the same total momentum k1 + k2=q, common to all pairs. If q =/; 0 there is a net current flow; for q = 0 the energy is minimum and for every state k i occupied there is also another state -k! occupied. The problem was then finding the ground state for the reduced Hamil- tonian
Bred = L fkDks - L v k'k'4t ~ (2.2) k,s k,k' 2 The BCS Theory 9 where the first term gives the energy of the unperturbed quasiparticles, and the second is the pairing interaction, in which quasiparticles from the state
(k j ,-k ! ) are scattered to (k' j ,-k' ! ). The operator f4 = ~r c~k! creates a pair with momentum and spin (k j,-k !). They obey the commutation relations
[btc, '41 = 1 - ( ntcr + ntc!), [btc, btc1 = [f4, '41 = o (2.3) with nu = blcbJc the number operator. To solve (2.2) BCS used a variational approach, with the following, now known as the BCS-wave function:
IG >= II
< GlnuiG >=N (2.5) with N being the fixed number of particles. This introduces a Lagrange Mul- tiplier, p., which turns out to be the chemical potential. After minimizing the quantity
< GIHIG > -p. < GlnuiG > (2.6) one finds9
(2.7) and
(2.8) 10 2 The BCS Theory
(2.9) where Ek = Ek- p. is the single-particle energy measured with respect to p.; .6.k is the gap parameter that satisfies the gap equation ( 2.8), 0 is a unit volume and Ek is the quasiparticle energy.
2.3 Zero Temperature Energy Gap
In order to solve the non-linear equation (2.8), BCS introduced the following simplifying model for Vkk':
(2.10) with V a constant and wn the Debye frequency. Using (2.10), (2.8) is trans formed into
(2.11) which does not depend on k, thus
(2.12) where N (E) represents the density of single-spin states. A further approxima tion is to assume N{t) can be considered constant within the rim ltl ~ 1iwn, centered at the Fermi level, i.e. N{£) ~ N(O). This provides the energy gap in the ground state, according to BCS theory:
.6.{0) = 21iwve -·, (2.13) in the weak-coupling limit, N(O)V < 1. 2 The BCS Theory 11
The condensation energy per unit volume, defined as the difference between the energies of the superconducting and normal states, is given by
1 """ _ Ek Ek 1 """ ~F = - L..J Ek[---] -- L..J-~~ 0 k lekl Ek 0 k 2Ek 1iwD E2 ~2 (2.14) = 2N(O) dE[E- -- -] 1o E 2E ~ _.!_N(0)~(0) 2 2
because liwn >~.Equation (2.14) shows that the superconducting state is a lower energy state and also that the energy involved in the transition is very small, of the order of ~(0) 2 ,..,. 10-6eV. These were some of the first indications of the essential correctness of the theory.
k
Figure 2.1 Quasiparticle energy dispersion for the superconducting state.
Note the modification with respect to the free electron picture, near kF. 12 2 The BCS Theory
The energies of the excited states are given by (2.9). Figure 2.1 shows the quasiparticle energy dispersion, where the similarity with a free electron dispersion curve, except for energies close to kp, is seen because of the pres- ence of the gap. Note that even when Ek = 0 there is still a minimum energy
Ek = ~k of these excitations. Hence one says there is an energy gap, ~' be tween the ground state and the lowest single-particle excitations. One can also think of these excitations as quasiparticles which behave almost like
normal electrons, i.e., they are uncorrelated. These excitations exist in a background of still correlated electrons, which conserve their superfluidity and are hard to excite. This picture has similarities to the two-fluid models proposed earlier in the development of the theory of superconductivity.
2.4 Temperature Dependent Energy Gap
At finite temperature, the gap equation is
where f(E) is the Fermi function:
1 f(E) = ef3E + 1 {2.16)
1 {2.17) /3 = kBT
with kB being the Boltzmann constant. The solution to (2.15) gives the en-
ergy gap as a function of temperature. In general, ~ depends on k, but unless
stated otherwise it will be considered independent of k. As shown in Figure
2.2, the energy gap becomes zero at the critical temperature, Tc, when the 2 The BCS Theory 13 material undergoes a transition to the normal state and all the electrons have been excited beyond fl., breaking up the coherence of the superconducting state. Also note that at low temperatures (<:OAT fTc) the gap remains es- sentially constant. This is because as the temperature is lowered it becomes harder and harder to excite electrons above the gap, given the macroscopic coherence of the ground state wave function. Therefore the number of elec trons that can be thermally excited is reduced; they fill empty places in the quiescent Fermi sea, and so become "inert", i.e. they have negligible in:Huence on the gap, and so it remains constant.
2.5 The Critical Temperature
As mentioned before, when fl.= 0, T = Tc, so from (15) (to simplify notation, € = ~:), 1 = v L: d~: ~~~i) [1- 2/(1~:1)] (2.18) and again, if N(~:) is considered constant around the Fermi energy EF, and for 1iwn ~ kBTc (typically, 1iwn"' leV, kBTc"' 10-3eV), the solution to (2.18) is
1 kBTc = 1.131iwne -NriiiV (2.19) in the weak-coupling limit.
One can note immediately that a law of corresponding itates can be obtained when (2.13) and (2.19) are combined in what is usually called the gap-to-Te ratio: 2 tl.(O) = 3.53 (2.20) kBTc 14 2 The BCS Theory
o Tin 33.5 Me • Tin 54 Me - BCS theory
Figure 2.2 Temperature-dependent behavior of the energy gap, from equa tion (2.15). The points correspond to experimental results10•
This universal ratio is the result of a highly oversimplified model, and in reality, experimental data on superconductors show certain scatter about that value. In particular, some superconducting elements, like lead or mercury, show rather large values of such ratio ("' 4 or bigger). Such materials are usually called strong-coupling superconductors and have been the subject of numerous and very interesting studies11• We, however, shall not discuss them in this work. An interesting feature of ( 2.19) is that, once again, it shows that the energy involved in the normal-superconducting transition, "'kBTc, is very small. On the other hand, Tc is proportional to nw D, in agreement with the isotope effect, which is discussed next. 2 The BCS Theory
2.6 The Isotope Effect
As mentioned before, it is observed experimentally that there is a correlation between the critical temperature and the isotopic mass of a su perconductor. Mathematically,
MfJTc = constant, (2.21) where f3 is called the isotope effect coefficient, which in BCS theory equals
~, and can be written as
(2.22) because wn in (2.19) is interpreted as a phonon frequency that varies as M-112.
In general, if the superconducting material is a compound, equation (2.22) describes only a partial isotope effect, f3i, for an isotopic mass M,. The total isotope coefficient is just
(2.23)
It has been mentioned before that the isotope effect was of paramount im- portance in the development of the microscopic theory of superconductivity because it revealed that the electron-phonon interactions play a basic role as the pairing mechanism between electrons. It is, nevertheless, not conclu- sive if the mechanisms involved in the pairing are not phononic, and in any case, the total isotope coefficient (2.23) should be considered when making an assessment, rather than the partial coefficient (2.22), because even if the 16 2 The BCS Theory contribution from one element to (2.21) is negligible this does not prevent other elements in a compound from showing significant values of {3.
2. 7 Thermodynamic Properties
The next objective would be to know the thermodynamic properties of the system. In order to do so, the entropy for a system of fermions is
S = -2kB L {[1 - /(Ek)]ln(1- /(Ek)) + /(Ek)ln/(Ek)} (2.24) k with /(Ek) given by (2.16). The specific heat at constant volume is
Cv = T(8S) = -13(88) (2.25) OT v 8{3 v with {3 as in (2.17). Inserting (2.24) into (2.25), the electronic specific heat in the superconducting state is
8 C5 = 2kB/3 L f3Ekef3~ [Ek + _/!_ ~,] (2.26) v k (1 + ef3E~o ) 2 2Ek 8{3 and to get the normal state it suffices to set ~k = 0, and so Ek = ltkl,
R 2 /3Ek c: = 2kBf3L I-Itke (2.27) k (1 + e/3Ek )2 which is equal to "(T, with 'Y = j1r2 N(O)k~ the Sommerfeld constant. When
T = Tc, the specific heat difference (~Cv:: C~- c:) is
/31Ekl) !lA2 ~C (T.) = -k T. R2 ~ Eke _u~_k (2.28) v c B w ~ ( (1 + e/31E~o1)2 Pc OT Tc
BCS predicts another universal law when the size of the specific heat jump
is normalised to the normal- state value, at Tc:
(2.29) 2 The BCS Theory 17 where (2.20) has also been used. For temperatures below, but close to Tc it is possible to show that
(2.30)
The theory predicts also that the superconducting transition is a second order phase transition, because the entropy is continuous at Tc and there is no latent heat involved. In addition, at very low temperatures equation
(2.23) shows exponential behavior, i.e. C8 (T) ,..., e-d/ksT, just as it is found experimentally.
At finite temperature, the free energy per unit volume for a super- conductor is:
2 F 8 (T) = 2kBT ~In ( 1-J(Ek) ) 1 ~ [ Ek- Ek AE: ( 1- 2/(Ek) )] (2.31) ----g- + 0 + 2 and for the normal state Ek = ltkl, Ak = 0, as before. The critical magnetic field is given by
pN (T) - pS (T) = 0 H'; (T) (2.32) 87r but the left-hand side of the equation above reduces to (2.14) at T = 0. So, we find that
(2.33)
This results gives rise to another famous universal ratio in BCS theory:
0Hc(0)2"'T! = 0.168. (2.34)
From (2.29) one can derive approximate analytical solutions for Hc(T). For temperatures close to zero,
2 Hc(T) = Hc(O) [1-1.057(~) ] (2.35a) 18 2 The BCS Theory and just below Tc,
Hc(T) = 1.74Hc(O} [ 1- (~)]. (2.35b}
It is frequently found more useful to work with the deviation function D(TfTc) rather than He, since D measures the deviation He has from the empirical law 2 Hc(T) = Hc(O) [1- (~) ] (2.36) which could also be obtained from the two-fluid models3• D is defined as
2 D(T) = Hc(T) _ [ ]. (2.37) Tc Hc(O} 1- (T)Tc
2.8 Concluding Remarks
Before closing this brief review of some of the results of the BCS theory, it should be mentioned that the theory was successfully applied to a wide variety of properties such as electromagnetic properties, thermal con ductivity, ultrasonic attenuation, nuclear spin relaxation, electron tunneling, critical fields and currents, etc. A fairly complete review of the theory of superconductivity is beyond the scope of this work, but for the reader in terested in this matter there are excellent sources in the literature (see, for example, refs. 8-10). The emphasis here is rather on the exposition of some of the ideas involved in the BCS theory that provide a theoretical background for the forthcoming calculations.
The BCS theory, although greeted with skepticism at first, became widely accepted later on. Especially from the mid-60's, when it was definitely settled, until the late 80's the" tyranny" of the BCS theory was absolute. This 2 The BCS Theory 19 was due to its ability to provide a. qualitative (and, in some instances, even quantitative) understanding of superconductivity; and also due to its power to predict new phenomena., like the Josephson effect. Nonetheless, since its formulation it was clear that further refinement was needed in order to ac count, for example, for the deviations that actual superconductors show from the calculated values of ratios such as {2.20), {2.26) and {2.31). The feeling was that the model BCS proposed for the electron-phonon interaction was rather crude and that a. more comprehensive treatment of it would naturally lead to better agreement with the experiments. With the introduction of field theoretic techniques, the understand ing of the electron-phonon interaction was greatly enhanced. This was done first by Migdal13, for the normal state, and extended later by Nambu, and
Eliashberg14, to superconductors. These theories are accurate to terms of or der (m/M)112 (Migdal's theorem), where m and M are the electron and ion masses, respectively; so these treatments are highly accurate.
In particular, Eliashberg derived two coupled, non-linear integral equations for a. renorma.liza.tion factor, Z(w), and a. ga.p, ~(w), which involve the electron-phonon interaction in the function a(w)2F(w). The complexity of these equations does not allow for exact analytic solutions. However,when solved numerica.lly, remarkable quantitative agreement is found with experi ment. Where BCS fails Eliashberg theory succeeds in providing an excellent understanding of superconductors where the pairing is phononic in origin. Moreover, the Eliashberg equations reduce to the BCS theory in the weak coupling limit, i.e. when the frequencies are large compared to kBTc so that
~(w) can be regarded as a. constant independent of frequency within the en ergies of order kBTc. Also, in the weak-coupling limit one may neglect the 20 2 The BCS Theory difference between the renormalization function in the normal and super conducting states. Eliashberg theory is particularly successful in describing properties of strong-coupling superconductors11, and remains a research area of great current interest.
In most cases the agreement between theory and experiment is re markable. BCS were surprised themselves of the accurateness of their theory. Their primary goal was just trying to explain the behavior of superconduc tors such as lead and tin at low temperatures, and they found the frame work for the understanding of a rich range of phenomena. As John Bardeen put it once, " ... everything fitted together neatly like the pieces of a jigsaw
puzz1 e... "12 .
Nevertheless, this success brought some negative consequences. If ex perimentalists obtained results that contradicted the theory, people would think they did the experiment wrong. As time passed the activity in the field decreased and the belief that there was not very much challenge left in it
and that Tc 's beyond the 30 K range would probably never be acomplished started to spread. Then, in 1986 the discovery by J. G. Bednorz and K. A. Miiller of a copper-oxide based superconductor with critical temperature around 36 K
revolutionized and gave new life to the field. But this will be the subject of the next chapter. Chapter 3
High Tc Superconductivity
3.1 Historical Introduction
The search for high temperature superconductivity is surely not a recent issue. It has been motivated for a long time by the potential tech nological applications, which at temperatures easily (and cheaply) achieved are almost unlimited. However, for many years this search faced multiple fruitless ends. Most superconductors known in the 1960's had Tc's well be low 20 K. It is not surprising that for the majority of workers in the field
(Bardeen himself included) the possibility of superconductivity ocurring at high temperatures appeared unlikely.
21 22 3 High Tc Superconductivity
This view seemed justified by the microscopic theory that accounted so well for most known superconducting materials, assuming the electron phonon pairing mechanism. By 1964, Little15, and later Ginzburg16 had pro posed exitonic mechanisms that raised hopes for the discovery of supercon ductivity at high Tc 's. These models did not enjoy too much popularity, partly because they did not seem to represent any real material.
On the experimental side, the research in high Tc superconductivity was almost entirely concentrated in the so-called A15 compounds, whose member, NbaGe, held the record Tc=23.2 K for more than twenty years17 . But superconductivity was not only sought for in metals, and the first oxide superconductor was discovered in 1964. By 1973 a few of them were known.
But even the best oxide superconductor known at that time, BaBil-xPbxOa
(Tc""' 13 K) could not compete with the A15 compounds. Nevertheless, the work in the oxides proved useful when in 1986 J. G. Bednorz and K. A. Miiller reported the ground breaking discovery of super conductivity in the 30 K-range in the La-Ba-Cu-0 system18• A few months later, Chu and co-workers19 were able to raise Tc up to 90 K by substitution
of Y for La. Reports of Tc ""' 100 K appeared in China almost simultaneously. A frenetic period of research started then that has produced about 18000
publications over the last four years. This is a unique phenomenon in the history of science.
After the discovery of Y1Ba2Cua07-.s and related compounds, a num ber of copper-oxide superconductors have been developed. These include
the so-called electron-type superconductors, such as N d2-zCexCu04, and the
highest Tc (""' 125 K) material known to date, Tl2Ba2Ca2Cua010. 3 High Tc Superconductivity 23
This new era of superconductivity has also motivated the review of old theories and the proposal of new models, and has challenged in many ways our present understanding of solid state physics.
3.2 Structural Properties
We shall review now very briefly some common structural character istics of the high Tc oxides.
One of the most important features that these materials share is, no
doubt, the fact that they all possess one or more Cu02 planes. The copper atoms occupy the vertices of a nearly square arrangement, with oxygen atoms in between. These planes are crucial, since they have been recognized as the
essential ingredient for the ocurrence of high Tc superconductivity. Figure 3.1 shows the layered structure of the most extensively studied
compound, Y1Ba2Cua01-6· The structure contains two Cu02 planes separated by Y atom;:;:, and intercalated by two planes containing Ba, Cu, and 0. These planes act as charge reservoirs for the Cu02 layers. They are the best candi
dates for chemical substitution, because doping modifies the charge carrier
concentration in the Cu02 planes.
It was early believed that the Cu-0 chains in Y1Ba2Cua07_6 were indispensable for superconductivity in the cuprates, but the discovery of the
Bi-based compounds, whose structures have no copper chains, invalidated this idea and the theoretical models based on it. These superconducting structures are relatively easy to reproduce in polycrystaJ.line form. Good single crystals and high quality samples involve 24 3 High Tc Superconductivity more sophisticated techniques. Microscopic observations indicate that the oxides are plagued with structural defects, that are believed by some to play an important role in the superconducting state. These defects, as well as anisotropy, instability, mechanical properties, etc. have been extensively studied.
3.3 Physical Properties
The cuprates possess a wide variety of physical properties, which are perhaps best described by their generic phase diagram (Fig. 3.2). For very low doping concentrations the system is an insulating antiferromagnet (Mott insulator). They have large Neel temperature (the temperature at which the spins order), which drops rapidly with increasing hole content. As doping is increased the antiferromagnetic state disappears and the system shows metallic behavior, and superconductivity. The transport properties of the layered oxides are highly anisotropic, as may be expected from their crystal structure. It is known, for example, that the electrical resistivity perpendicular to the Cu.02 planes can be 105 times larger than within the planes. Moreover, the resistivity along the Cu.02 planes is found to depend linearly on the temperature.
The coherence length ~, and the penetration depth .A are also quite different for the cuprates, and appear rather extreme when compared to the conventional materials. Different experiments reveal that for Y1Ba2Cu.a01-6,
~ab=14±2A, ~c=l.5-3A, .A=1400A, and Ac "'7000A. By contrast, the data for 3 High Tc Superconductivity 25
Nb are: .A=350A, ~=400A. Therefore. the cuprates are" clean:' superconduc tors, i.e., >.j~ >>1. Because of the small ~'s, the overlapping of the Cooper pairs is considerably smaller than in the low Tc materials, and this may lead to fluctuation effects near Tc. It also makes tunneling difficult, and this influ ences the data obtained by means of this, otherwise powerful, experimental technique.
0(2)
0(31 Ba -0(4) 0(1) c~ a
20 Figure 3.1 Crystal structure for Y1Ba2Cu307-.s, after Jorgensen et al. .
Note the two different crystallographic positions that Cu occupies. 26 3 High Tc Superconductivity
T
MET
AFM SUP
X
Figure 3.2 Schematic phase diagram for the high Tc superconductors, as
function of doping. AFM is the antiferromagnetic insulator, SUP the super
conducting, and MET metallic phases. 3 High Tc Superconductivity 27
Another important characteristic of the oxides is their extremely high zero-temperature critica.l magnetic field, Hc2(0). It is estimated that Hc2(0) is
80-320 T for YiBa2Cua01-.s, whereas the strongest magnetic field achievable to date is about 50 T. Consequently, quenching of the superconducting state through a.n applied magnetic field is not possible yet.
The isotope effect for the cupra.tes is found to be genera.lly sma.ll or near zero. This is often used as a.n argument against a.n electron-phonon mechanism in these new ma.teria.ls. However, the question is not whether there is a.n electron-phonon interaction, but if it is strong enough to produce the observed Tc 's. The strength of the interaction remains a. debatable issue, with some results favoring strong-coupling (e.g. high values of2!1/kBTc), and other indicating weak-coupling.
There are features of the high Tc ma.teria.ls that are not found in conventiona.l superconductors. The energy gap varies with the method of measurement, their norma.l state is not well understood, etc. Despite of aJ.l this, they still share severa.l similarities, such as the existence of Cooper pairs, a. discontinuity in the specific heat a.t Tc, qua.lita.tively similar behavior under the influence of a. magnetic field (Type II), etc. This makes the formulation of a. successful theory a. formidable task.
From the theoretica.l point of view, severa.l models for high Tc su perconductivity have been formulated, none of them have been genera.lly accepted. The close proximity of the superconducting and a.ntiferroma.gnetic states in the phase diagram is usua.lly taken as the starting point of the novel theories. These theories ( anyon theory, resonating valence bond, etc. ) con stitute a. funda.menta.lly different approach and are not developed far enough to a.llow detailed comparison with experiments. Therefore it becomes very 28 3 High Tc Superconductivity difficult to ratify, or discard such theories.lt is believed by many that the BCS formalism (or more fundamentally, the Eliashberg formalism) underlies the physics of the high Tc's, with possibly a modified pairing mechanism.
3.4 Concluding Remarks
The discovery of superconductivity at temperatures above that of liquid nitrogen (77 K) produced widespread euphoria and raised high ex pectations on the applications of the newly found materials. Most of the applications depend on the ability of superconductors to be used in the gen eration of very high magnetic fields, and to carry large currents with no loss. It soon became clear that these expectations were rather naive and de spite important developments over the last five years, the most ambitious applications for the cuprates perhaps will have to wait a few more years. The problem of high temperature superconductivity is extremely complex due to several reasons, such as sample quality, and other mate rial related characteristics, which strongly influence most experiments. And even though no theory can be considered satisfactory at this point, some facts are now well established. Before agreement is found between different key experiments, and as new theoretical techniques to deal with these systems are developed, no result from any model can be considered conclusive. And as our conceptual base of condensed matter theory is changed, nothing is cast in stone.
Needless to say that this is a vast and rapidly evolving field, and it would be pointless to even pretend to summarize such an extensive subject. 3 High Tc Superconductivity 29
The objective has been solely to review some important properties of high Tc, copper-based superconductors, that will later become relevant, in the context of the work that is hereby reported. Chapter 4
Van Hove Singularity in BCS Theory
4.1 Introduction
The preceding chapter provided a brief overview of the status of high
Tc superconductivity in the cuprates, and there is no reason to repeat it here. It is important, though, to bear in mind that our understanding of the high Tc copper-oxide superconductors is by no means complete, and that perhaps some authentically new ideas are needed at this point to further our knowledge of the nature not only of the superconducting, but also the normal state in these new materials. Even though the cuprates have presented a serious challenge to the predictions of the BCS theory, this is still the most popular and perhaps fruitful approach we have. The present chapter attempts to explore some of the consequences of incorporating a van Hove singularity (vHs) in the
31 3:1 4 Van Hove Singularity in BCS Theory electronic density of states (DOS), in the weak-coupling limit. Therefore the apparatus of the BCS theory is used throughout this work.
4.2 Energy Dependence in the DOS
Most of the results presented in Chapter 2 were obtained under the assumption that the density of electronic states at the Fermi energy f.F was essentially constant, for energies le- f.FI :::; nwv. For most conventional su perconductors such assumption seems to be justified. However, there have been several suggestions in the past that incorporate energy dependence in the DOS in the energy scale of nwD (see for example, ref. 22, and references therein). The motivation to do so was basically the high Tc's of some of the so-called A15 compounds (NbaSn, for example). It is no surprise that the discovery of new materials with even higher Tc's revived this idea. The particular functional dependences of the DOS on energy sug gested over the past are diverse, but the most common examples are sharp peaks (-lel-t), Lorentzians, and van Hove singularities. A van Hove singu larity in the DOS arises when the group velocity v9 = 8e(k)/8k vanishes, e.g., at the saddle points in the energy dispersion relation e(k) vs k (see Appendix A). These are integrable singularities (in 3D) and their behaviour depends on the dimensionality of the system. It was first shown some forty years ago23 that the DOS for a system of non-interacting electrons in a two-dimensional
(2D) periodic potential always exhibits logarithmic singularities. These have attracted considerable attention recently, in the context of the high Tc oxides, because it is believed they may lead to unusually high Tc 's (-100 K). 4 Van Hove Singularity in BCS Theory 33
It is well known that the origin of superconductivity in the high
Tc oxides resides in the Cu02 planes, which are only weakly coupled along the c-axis (for most high Tc materials the resistivity along the c-axis is four orders of magnitude bigger than that in the a - b plane, ref. 21). Thus, their electronic transport properties can be presumably understood by two dimensional models§. This was one of the original ideas that motivated the incorporation of a vHs into BCS theory, in an attempt to account for the unusually high
Tc 's that the cuprates display. As mentioned before, the model had been used in the past as a Tc enhancement mechanism in the A15 compounds25 , and other systems with reduced dimensionality26• More recently, it has been applied to the high Tc cuprates27-30.
Despite the fact that high Tc 's seem only possible on the weak coupling limit if a vHs-model is applied and not with standard BCS theory, the suggestion has been severely criticized on several grounds, mainly be cause the disorder present in the crystal structures of the oxides, and the small but measurable c-axis dispersion can smear out any structure in the
DOS on the scale of nw D. Also, there is no direct experimental evidence for a sharp peak in the DOS near the tp. On the other hand, a simple 2D band structure model with nearest neighbor interactions leads to a square Fermi surface at half-fillingt; when this is the case, nesting of the Fermi surface
(i.e., the 2D Fermi surface is a square) can occurU.28, which in turn usu ally leads to instabilities such as charge- and spin-density waves, inimical to superconductivity32.
§ It should be pointed out, though, that a strictly 2D system cannot show long-range superconducting order, see ref. 24. t One spin per lattice site. 34 4 Van Hove Singularity in BCS Theory
In spite of these arguments againts a vHs in the DOS, most of which are still controversial, the model has attracted a great deal of attention from the scientific community working in superconductivity , and it does pose a very interesting challenge that experiments, eventually, will resolve.
4.3 The Model
For the sake of simplicity, the following form for a 2D-vHs in the electronic density of states will be assumed:
(4.1) where NoC is a constant background, 6 is the filling factor, which plays the role of a chemical potential, for it indicates the highest occupied level relative to the singularity. The energy E is referred to the Fermi energy, EF·
We shall be concerned here primarily with small values of 6, since it is presumably close to EF where the vHs is most effective as a Tc enhancement mechanism.
It is possible to show33 that (4.1) follows from a tight-binding calcu lation including nearest- and next-nearest neighbor interactions on a square lattice, simulating the Cu02 planes. Figure 4.1 shows the energy dependence of N(E) for different choices of the filling factor. The model (4.1) will be used throughout this thesis to calculate different properties using the BCS for malism. Observe that the DOS is sharply peaked about the value of -6. For energies away from such a value the DOS is essentially constant. One expects that in that range of energies the behavior of a superconductor should be as predicted by the standard BCS model, i.e., N(E) "' constant, for lEI ~ 1iwv. 4 Van Hove Singularity in BCS Theory 315
Consequently, we should be able to account for the BCS results as the vHs is displaced from the Fermi energy.
30.0 .s-o I I I I .S-'0 I ------I I I .S•-'0 I --- I I I I I I 25.0 I I I I I I I I t t t t .....-.. t 0 t ....., t t t :zt I t t 20.0 t .....-.. t t w I '....., t :zt t ' ' ..' ..••- •• 15.0 ... . ' ·.
-100.0 0.0 100.0 £(meV)
Figure 4.1 Logarithmic van Hove singularity to be used in this work. The
different curves correspond to different values of the filling factor 6. The
background constant in the DOS, C, is equal to 10. From equation (4.1). 36 4 Van Hove Singularity in BCS Theory
4.4 The Critical Temperature
One of the first properties one would want to calculate for any model of superconductivity is the critical temperature. From the linearized BCS equation (2.18) we have, using (4.1):
_2 -~liwD [ml...!!._l +C]tanh[-E] dE (4.2) NoV- -liwv E + 6 2k8 Tc E •
As such, ( 4.2) cannot be solved exactly. However, some approximations are useful: tanh [ ••:T.] "' n>., ~ 1•1 S 2k.T, (4.3) { tanh[ k ~ 1, if lEI> 2k8 Tc 28 Te] In order to remove the absolute value in the logarithm in (1) the limits of integral (2) must be split, giving rise to four integrals, from -1i.wv to -2k8 Tc; from -2k8 Tc to -6; from -6 to 2k8 Tc, and finally from 2k8 Tc to 1i.wv. Two more approximations will prove useful:
valid for 6 < 2k8 Tc, and
In (a± bx) 1 2 a a2 3 J x dx ~ 2ln (bx) =F bx + 4(bx)2 + O[(ax) ], (4.5) which follows from using the series representation ofln(1+x) and integrating term by term (see Appendix B). Using (4.3)-(4.5) in (4.2), and after some laborious but straightforward algebra, we arrive at
O = 1n2 (2k8 Tc) _ 2(1 + C)In(2k8 Tc) __2_ EF EF NoV (4.6) EF ) ] 2 2 1 ( 6 ) 2 62 - [In ( 1i.w + C + (1 +C) - 2 2k T. - 21i.w2 + 1. D B c D 4 Van Hove Singularity in BCS Theory 37
This is a quadratic equation, when the term 8/k8 Tc is considered independent of k 8 Tc being 8 small, in ln(2k8 Tc/t:F), that admits two solutions, but we consider here only the negative sign in the square root, for that is the one with physical significance *:
2 2 [ ( fF ) ] 2 1 [( 8 ) 2 62] ln( ~~Tc) = [1+C]- lt.T v + In n.w + c + 2 2k T. + n.w2 - 1; .no D B C D (4.7) solving for k 8 Tc, we get the final expression for the critical temperature for a vHs in the DOS when the displacement of the singularity from the t:F is small:
k 8 Tc = 1.36t:F 2 x exp{c- ~ v + [~n(~ ) +cr + ~ r. + (~ 1 0 D [(2/B C f D f]- }· (4.8) Equation (4.8) has the advantage that it reduces simply to
(4.9)
2 when the vHs lies right at the Fermi level, with -ffov ~ ln ( Jf!;;), and C=O, a result first derived by Hirsch and Scalapino26 for an attractive Hubbard model. An estimate of Tc using equations (2.19) for BCS, and (4.8) for vHs gives a T:98 at least three orders of magnitude bigger than TcBcs, for typical values of n.wD and NoV. This enhancement in the critical temperature is due primarily to two factors: first, the energy scale is different because t:F is larger than nwD; and second, the argument of the exponential in (4.9) varies as
-1/../'Avas, with AvHs =NoV, whereas in the BCS model it goes as -1/'ABcs,
* A simple argument shows the +ve sign is not admissible for C = 0: being 2k8 Tc < fF, thenln(2k8 Tc/t:F) < O,i.e., ln(2k8 Tc/t:F)-1 < -1. Butfrom(4.7) this is J2/NoV + ... < -1!!. 38 4 Van Hove Singularity in BCS Theory where ABCS = N(EF)V. The latter one makes the exponential (and thus, Tc) smaller. Notice that No in Ayn, does not correspond to N(O) = N(EF) in ABcs, though. The point of our argument, nevertheless, holds because as shown in
Appendix C, one can define )."H" as Aa" = Na"V, where Na" is an averaged DOS, smaller than N(EF)· A very important implication of (4.9) is the absence of the isotope effect, for there is no phonon-frequency dependence on Tc. It must be recalled, though, that (4.9) is an oversimplified equation whose only purpose is a quick estimate of the critical temperatures attainable within a simple vHs model. The frequency wn is not the thermodynamic Debye frequency, at least for the high Tc oxides, but rather an upper cutoff in the phonon spectrum.
Equation ( 4.2) can be solved numerically to get more accurate esti mates of Tc. This has been done and the results are compared in Fig. 4.2 to the approximate solution (4.8) for a typical choice of the parameters C, EF, nw v, and N0V. The approximate solution (dotted line in Fig. 4.2) is good, overall. It provides the correct order of magnitude and shows the same general trend as that of the exact solution (solid line in same Figure), as a function of the filling factor. However, it seems to overestimate Tc; this is presumably due to terms neglected in the expansions (4.3)-(4.5), which, if included, would reduce Tc, especially for large values of 6. As expected, the Tc's pea.k about
6 = 0, decrease with increasing 6, and become very small when the offset of the vHs is of the order of 'f&wv, which is where most of the electrons involved in the superconducting transition lie. It seems encouraging that for relatively small values of NoV one can get Tc's of the order of 100 K or larger. However, caution must be observed and one has to be careful not to give too much numerical significance to these calculated values of Tc when comparing to 4 Van Hove Siugularity in BCS Theory 39
40
30
20
10
o~~~~~~~~~~~~~~~~~~~ -2 -1 0 1 2
Figure 4.2 Comparison between the exact numerical solution (solid line) to
(4.2), a.nd its approximation (dotted line) (4.8), for the parameters NoV =
0.084, 1i.w0 = 65 meV, C = 0, a.nd EF = 500 meV. The Tc using the BCS result (2.19) for the same set of parameters is 0.00578 K.
experiments, since the values of the parameters used cannot be measured
directly a.nd only rough estimates are available. Besides, it is well known 40 4 Van Hove Singularity in BCS Theory that BCS is generally incorrect when it comes to the evaluation of Tc even for conventional superconductors because it does not provide precise knowl- edge of the phonon structure. For this it is necessary to go to the Eliashberg formalism.
N0V=0.06 --- 0.0~ ------0.08----. 0.09--- 40 0.10 ------
20
0.5 1 1.5
Figure 4.3 Critical temperature for increasing values of the coupling con stant NoV, obtained by solving numerically eq. {4.2). The parameters used for all curves are: C=O, 1i.wv = 65 meV, and f.F = 500 meV.
The Tc values depend strongly on the choice of the coupling constant
NoV, as shown in Fig. 4.3, because that is the dominant term, as one can 4 Van Hove Singularity in BCS Theory 41 see from eq. (4.8), although the curves in Fig. 4.3 correspond to numerically solving eq. (4.2). It is found that the disagreement between the exact a.nd approximate solutions to ( 4.2) becomes larger a.s the coupling constant NoV is increased. Again, this is expected, since one of the assumptions made when deriving (4.8) was that NoV remained very small(< 1).
60--~------~~~-T~--r-~-r~~ 4==0.0 ---...: 10.0 ------· 20.0---- 30.0--- 40.0 ------50.0----- 40 ,- --- ., ., -- ., ., / / --- __.. __.. // ,- / ~ / / / / / .,...... ----- 20 / / ---- // / .,------1 / _., .,- -- 1 . --- / __... -----· 1 / _,- ·""' __... ------/ __..,. ------1 / . / --- '/ ,./ / . ------· I _,. __.., . ..---·· / 1. 'L---~------o~~~~~~~~~~----~~~_.~----._~ 0 50 100 150 ~MD (meV)
Figure 4.4 The critical temperature a.s function of 1i.w D, from ( 4.2). For all
curves: C=O, 1i.wD = 65 meV, a.nd EF = 500 meV.
Very interesting behavior is observed when ( 4.2) is solved a.s a func
tion of the frequency hwD, which is shown in Fig. 4.4. One notes that the 42 4 Van Hove Singularity in BCS Theory critical temperatures are increased as the vHs is moved toward EF, and "-wD is increased. For small values of "-w D there appears to be a linear relation between Tc and 1iwD, which changes roughly when the frequencies are bigger than the value of 6, due to a competition between both quantities, as we can see in eq. (4.8). This Figure also displays saturation in Tc when the values of
1iw D and 6 are large. Moreover, for such values Tc is approximately propor tional to 1iwD, in agreement with eq. (2.19). The solution (4.8) fails also for very small values of nwD, which does not surprise, for the foundation for (4.3) is to assume 1iwD/k8Tc- oo. In practice this presents no difficulty because "-wD (when interpreted as a phonon frequency) extends up to 100 meV for the oxides, and even for Tc's as high as 120 K, the ratio remains big, of the order of 10 at least.
The constant background in the DOS, C, is introduced to make our model more realistic, however its inftuence on Tc appears to be minor, as
depicted in Fig. 4.5. It does not change the qualitative behavior of Tc as a
function of 6, with the critical temperatures increasing for increasing C's, by
roughly a factor of exp(C) (see eq. (4.8)) for small 6. This is obviously so
because increasing C means increasing the overall DOS, and in particular
the height of the peak (see Fig. 4.1), therefore making Tc bigger.
Equation (4.8) has appeared in the literature recently30•33• A com
parison with the BCS equation (2.18) is not shown in the Figures presented here because the critical temperature obtained is extremely small, and almost never surpasses the 0.1 K range with the parameters used in our Figures. 4 Van Hove Singulariiy in BCS Theory 43
80 C=0.00--- "" 0.10 ...... " \ 0.30----- \ 0.50--- \ 0.70 ------1.00 ---·- 40 " "' " ,, ,, .. , ,, 20 ,,
0.5 1 1.5
Figure 4.5 Influence of the constant background C on Tc. For the numerical
solution of (4.2) we used nw 0 = 65 meV, EF = 500 meV, and NoV= 0.084. 4 Van Hove Singularity in BCS Theory
One could also incorporate a. joint mechanism into the model as suming a. general electronic mechanism through a. Coulomb pseudopotential
1-'* plus the phononic already considered. This option is particularly appeal ing when applied to the cupra.tes because it is believed that a. pure phononic mechanism is unlikely to be responsible for their behaviour, a.nd a. joint mech anism seems to be able to account for some of their properties, a.t least a.t the qualitative level34• Following Carbotte et al. , the critical temperature is calculated for a. joint mechanism a.nd the results are shown in Fig. 4.6. Start ing from the Eliashberg equations it is possible to show35 that the pairing parameter NoV is approximately given by
A- 1-'* NoV= l+A (4.10) where A is a.n effective mass renormaliza.tion parameter, related to the elec tron -phonon interaction, a.nd 1-'* is the Coulomb pseudopotential which must be negative if it is to represent a.n electronic pairing mechanism. Introducing
(4.10) in ( 4.2) does not bring a.ny qualitative modification to Tc. However, the a.dva.nta.ge of doing so is that one ca.n consider a. purely phononic (J.L* = 0), electronic (A= o), or joint pairing.
In order to obtain the Tc's shown in Fig. 4.6, we have assumed a. delta. function of height A, centered a.t a. frequency wE for the electron-phonon spectral density, i.e., a 2 F(w) = A6(w-wE), a.nd a. negative 1-'*. The solid line in
Fig. 4.6 shows the value of Tc for increasing 6, for a. joint, mainly electronic mechanism. Note that even if no phonon contribution is assumed (dotted line) the Tc's ca.n be quite high (- 100 K). But if the electronic mechanism is removed (J.L* = 0), the Tc's are so sma.ll ( ...... 0.23 K) that it is not possible to show them in the scale of Fig. 4.6. One ca.n interpret this as a.n indication 4 Van Hove Singularity in BCS Theory 45
(not a conclusive one, however) that the role that phonons play in the high
Tc cuprates does not appear to be as important for superconductivity as it is in the conventional, low Tc materials.
100
·····... JJ-·=-0.13, A=0.65
80 ··.,·. ···------J.lr·=-0.13, A=O.D •. .... •. .. •• 60 •• .. .. ••. .. . •• . . 40 ·.. ·. ···· ... . ·. ' .. ,, 20 ···· ...... -.
0 ~~~--~~--._~~--~~--~~~--~~--- 0 50 100 150 cJ(meV)
Figure 4.6 Introduction of an electronic mechanism, through (4.10), in ( 4.2).
For both curves, C = 0, WE= 65 meV, and t.F = 500 meV.
Equation (4.8) is valid only for 6 < 2k8 Tc. When the vHs is far away
from t.F the DOS can be approximated in a Taylor series expansion about t.F 46 4 Van Hove Singularity in BCS Theory
(the constant background C is set equal to zero):
(4.11) with 1 (4.12) C,. = nln(!f) for E < 6. Substitution of this form of the DOS in eq. ( 4.2} yields:
(4.13)
Now we apply the approximations (4.3} and proceed as before. The first integral in eq. (4.13}, after simple algebra, is equal to
(4.14a) while the second is
2L,.,--~ C2m (nwD)2m . (4.14b) m=l 2m 6 ' where n =2m. Substituting these results in eq. ( 4.13) gives
1 ~ C2m (nwD)2m} ( 4.15) kaTe= 1.36nwDexp { - N(EF)V + ~ 2m - 6- .
This equation provides a Te significantly lower than that from eq. (4.8). The energy scale is now proportional to nw D, in agreement with the BCS result, eq. (2.19}. For 6 > nwD, where the approximation of N(E) as a series works well, eq. (4.15} predicts a reduction in Te, because the coefficients
C2m are always positive and decreasing with increasing 6. These features, roughly shown in Fig. 4.2, are expected, because if the vHs is far from EF the carriers, which lie within kaTe of the Fermi energy, are not aware of the peak in the DOS. Therefore the material would behave like a conventional superconductor, with a somewhat enhanced DOS. 4 Van Hove Singularity in BCS Theory 47
There are important features that have been left out for simplicity, although they may be in some instances very interesting. In particular, eq.
(4.2) will overestimate Tc because it neglects inelastic scattering which acts as a pair-breaking mechanism36, and therefore reduces Tc. Nevertheless, even if such an effect is taken into account it is still possible to get critical tem peratures in the 100 K range, within the vHs model37• This closes our section on the calculation of the critical temperature. Much work remains to be done in this area, since, as we just mention, there are effects that can be incorporated into the model to make it more feasible, particularly for the high Tc oxides such as anisotropy, interlayer coupling, etc. , but they are beyond the limited scope of this work. Besides, one must be aware that any calculation of Tc using BCS theory is necessarily incomplete.
4.5 Zero-Temperature Energy Gap
In Chapter 2 we discussed the existence of an energy gap in the quasiparticle energy spectrum, which is temperature dependent. We also showed that BCS theory provides an approximate expression for the zero temperature gap parameter, _6.(0), that is found tO be proportional to nwD, and is exponentially dependent on the coupling constant NoV, just as the critical temperature (see eq. (2.13)). This led to one of the most important universal ratios in the theory, namely 2.6.(0)/k8 Tc = 3.53. We now calculate ..1.(0} for a vHs in the DOS, and compare it later to the BCS result. Once this has been done we shall proceed to calculate the ratio ( 2.20). 4 Van Hove Singularity in BCS Theory
The zero-temperature gap equation in BCS is given by (2.11). Insert ing the model (4.1) for N(e),
_2 =jnwD [lnl~l+c] de ( 4.16) NoV -liwD e + 6 yfe2 + A(0)2.
Following the procedure used to solve (4.2) in section 4.4, the interval of integration is split, so that the equation above becomes
2 {liwD (e-o) de -NoV= }6 In ey Je2 + A(0)2 (4.17) + 11iwD 1n(e + 6) de _ cjliwD de -6 eF yfe2 + A(0)2 -liwD Je2 + A(0)2
If it is assumed that 161 > A(O), and the square root in the denominator of the integrals above is approximated by e because the gap is very small, the first integral in the right hand side of (4.17) is
where (4.5) has been used. For convenience, the limits of integration in the second integral in (4.17) are separated, from -6 to o, and from o to nw D. Using the same technique as in (4.18), the integral from o to nwD is easily found to be
Then, in the integration from -o to o the interval from -A(O) to A(O) is separated, to be solved afterwards, and the remaining range of energies is integrated in a straitforward manner, following the examples above. The result is
(4.20) 4 Van Hove Singularity in BCS Theory 49
In order to solve the remaining integral from -d(O) to d(O), we approximate the square root by its average value over that interval:
J.6.(0) ..jE2 + d(Q)2dE [y'E2 + d(Q)2] = -.6-(0) = 1.15d(0); (4.21) av J.6.(0) dE -.6.(0) this leads to
1 1.6.(0) ln (E+6)-- dEO:t-2 [ ln (d(O))-- +-1(-- 8 )2 -1l (4.22) 1.15d(0) -.6.(0) EF - 1.15 EF 2 d(O)
where the approximation ln(1 +x) ~ x-; was used, because we are interested in small values of 6, such that 161 < d(O); for 161 > d(O) the other integrals ( 4.18-4.20) apply.
Finally, the term proportional to C in ( 4.17) can be readily evaluated,
considering nw0 > d(O). It is conveniently written as
(4.23)
Now we substitute the results (4.18)-(4.20), (4.22), and (4.23) into (4.17) to get, after some algebra, d(O) = 2.39EF x•%J>{c- N!v+ [m(:J+cr +*~} +(Ll~o>)'] -1}' (4.24)
which has the same dependence on NoV, nw0 , EF, and 6 as the equation for Tc, (4.8). In particular, Fig. 4.7 shows the comparison between the numerical solution to (4.16) (solid line), and the approximation (4.24) (dotted line), for the same set of parameters used in Fig. 4.2. The approximation appears to be
somewhat better overall than that for Tc. For the given choice of parameters,
do for a constant DOS, i.e., the BCS model, is equal to 0.0008864 meV by solving (2.12) numerically, whereas for a vHs can be four orders of magnitude 50 4 Van Hove Singularity in BCS Theory bigger. This increase is obviously due to the larger energy scale of (4.24), compared to (2.13), and to the square root in the exponential term.
One could produce for 6.o figures like those we have presented for Tc, but instead the ratio R = 26.o/kBTc will be studied. This ratio is of great interest in the theory of superconductivity because even if the critical temperature and (or) the gap depend on the details of the materials, their ratio should not, because it is a universal law. Using ( 4.8) and ( 4.24) this ratio is -.IA+l(..L)2 e V 2 .o.o ( R = 3.53 2 4.25) e -JA+t(,,;TJ
2 with A= N~V +[In(~)+ cr -1 + H~) • This ratio goes to the BCS value, 3.53, in the limit when the offset of the vHs is very large and when the singularity lies at the Fermi level.
In Fig. 4.8 we have calculated R numerically, using (4.16), and (4.2)
for different values of the coupling constant, N0V. It is interesting to see that the introduction of a vHs in the DOS does not change the value of R more than 3 or 4 % unless unphysical values of the parameters are used. The curves in Fig. 4.8 show saturation to the BCS value as the displacement of the
singularity is increased. With acceptable values of the coupling constant (,....,
0.2) one can get R's of the order of four. As the pairing constant is increased, the maximum and minimum in R are displaced, contrary to what happens when the constant background is increased (see Fig. 4.9). 4 Van Hove Singularity in BCS Theory 51
•• •• 8.0 •• '\ ' '• •• '• '' '' '' ~ '' 4.0 '' s •' . '' 0 •• '' - •• '.
0. 0 b:;;:::::S:::~.&.--l--J...-L.-..L..-.1.-L-L-.L-..J.--&...... L.L.-L::::!::~ -2.0 -1.0 0.0 1.0 2.0 d/ruo
Figure 4.7 Comparison between the numerical solution to (4.16), solid
line, a.nd its approximation ( 4.24 ), dotted line. The parameters used are
NoV=0.084, tF=500 meV, 1i.wn=65 meV, and C=O. The corresponding re
sult for a constant density of states is 0.0008864 meV. 52 4 Van Hove Singularity in BCS Theory
'\ • \ • \ 1.03 \\ .\ \ .q \'\ \\ \\ \\ \ \\ \ \ ~ \ \ NoV::0.08 ----- \ \ 0.07 -··-················ ~ \ \ .\ ' I 0.08------\ \ :: \ \ 0.09---- : \ : \ \ .\ 0.10 ·-·-·-·-·-·- 1.01 • • : \ \ \ \ \ \ \ \ \ \ \ \ \ \. \ \ \ \. 1 ~ \ ~' \ ' ...... --...... _.,. ,_,...... - ~...... --· • 'lo.·· - .....-- --- '...... ').r· .ac.··-""< '~ ~--.,-· - _.,- ---· 0 0.2 0.4 0.8 0.8 1
Figure 4.8 The ratio R as a function of 6 for increasing values of the coupling constant. The parameters involved are the same used for Fig. 4.7. 4 Van Hove Singularity in BCS Theory 53
1.03
1.02 C=O.O 0 -e-;. 0.1 ------~ 0.3 ------C"J -lO 0.5 ---- C"J 0.7 ------:r 1.01 1.0 ------· N
1 .. ------
0 0.2 0.4 0.6 0.8 1
Figure 4.9 Influence of a constant background added to the logarithmically dependent density of states on the gap to Tc ratio, after solving ( 4.16) for
the parameters used in Fig. 4. 7. 4 Van Hove Singularity in BCS Theory
It appears that the only significant contribution of C to R is to change the slope of the curves, but the maximum remains roughly the same. This can be understood by examining (4.25}, which remains essentially constant for very small6's regardless of the value of C. Note its primary contribution, ec, from 6.o cancels with that of k 8 Tc. The influence of the frequencies in the ratio R is examined in Fig. 4.10, for different positions of the vHs with respect to the Fermi energy.
Notice that as the singularity is placed increasingly far away from Ep the ratio becomes more independent of nwm and closer to the BCS limit. Except when the singularity is right at the Fermi level, the variation of R with respect to the canonical value 3.53 is never bigger than 1 %, as function of
1iw0 • However, for frequencies up to about 50 meV, and for vHs's very close to Ep (0~ 6 ~ 10 meV), the biggest deviations of R from 3.53 come from the low frequency range. One may ask how these results compare to real superconductors, par ticularly the cuprates. This comparison presents serious difficulties because of some intrinsic experimental problems in the determination of 6.o, for the high
Tc oxides. Just to mention a few examples, the small coherence length and the difficulty to get good single crystals make tunneling measurements (one of the most accurate techniques to determine the energy gap in conventional
superconductors) uncertain. Also, given the anisotropic transport properties
of the oxides one can expect the gap to be anisotropic as well. Under such
circumstances it is no surprise to find in the literature a wide spectrum of
values of 26.o/kBTc for the high Tc cuprates, ranging from about 1, to up to 10, or even higher (see, for example, ref. 35). Therefore our calculations 4 Van Hove Singularity in BCS Theory 55 for a vHs in the weak-coupling, isotropic BCS theory do not seem to sup port extreme values of the ratio R. And neither does the use of other energy
38 40 dependent DOS, nor the inclusion of anisotropy, in the BCS formalism - •
1.038 1.034 1.032 ...... ····· 1.004 .·· .· .· .· .··
0 ... ····· -E-;_1.002 . ~ ',. ..·· \ / t'l ~ ,• -10 '. .,•' t'l ~ ~ . ~ - ~ )1.000 =-·...:., .· ,• N 't ' <::::•--=-...:.::::- --,L-- -- ', ~ ...... ____ ,• ______' ' ' ' -----.£-:-:...... _._ ...... ------.______---- __ ·... ' ' .· .· -- --... - 0.998 ...... · --...-- '...... ·----,•*'' ' ' ---..... -...... ---._ ' ...... 0.998 ------0.0 50.0 100.0 150.0 ~ (meV)
Figure 4.10 The ratio R as function of nw 0 (in meV) for different 8's. For label on the curves see Fig. 4.4. 4 Van Hove Singularity in BCS Theory
4.6 The Isotope Effect
In Chapter 2 we discussed the crucial role that the discovery of the isotope effect played in the understanding of the mechanism responsible for superconductivity in the conventional (electron-phonon) materials. With the discovery of superconductivity at high temperatures several groups quickly realized the importance of performing isotopic substitution experiments in the oxides.
Substitution of 0 18 for 0 16 in La2-zSrzCtt.04-y revealed that the (par
41 42 tial) isotope effect coefficient f3oz was between 0.14 and 0.35 - • These re sults were frequently quoted to argue that those phonon modes involved in the isotopic substitution of oxygen were not responsible for superconductiv ity in this compound. Moreover, it was believed that {3 was negligible for most high Tc compounds. Recent experiments seem to point in another direction. Measure
43 44 45 ments of f3 in La2-zSrzCu.04 , Yi-zPrzBa2Cu.a01-.S , andY Ba2-zLazCu.aOz show strong dependence of {3 with doping x. It is found that {3 can be even larger than the canonical BCS value of 0.5.
Figure 4.11 shows Crawford et al. 43 results for {3 and Tc as function of Sr doping (x), in La2-zSrzCu04. This has motivated work for BCS-like, phonon-mediated pairing mechanisms30•33• It is the purpose of this section to investigate whether or not a vHs in the DOS can help interpret the results shown in this Figure. 4 Van Hove Singularity in BCS Theory 57
0.80
40 ..•. •. .• 0.60 : . . 30 •• • ..• ••. .• . •. . •. •• .. •• ~ • •. •. 0.40 • •• 20 .• •• •• •• •• ••. •. 0.20 10 •. •. •.' ___ .. ~ _. .. ---- ~---- ...... ------
0 L-..L--L-..L.....l.~....L.....JL....-L....I.-.L-L....L.-J-.L.-.1-..L....L--L.....L...JL.....L...J...... L-L.....I 0.00 0.05 0.1 0.15 0.2 0.25 0.3
Figure 4.11 Experimental results for Tc and /3 for La2-xSrxCu04 against doping, after Crawford et al. 43 The solid hexagons represent Tc measure ments, the triangles refer to /3. 58 4 Van Hove Singularity in BCS Theory
In order to do so let us recall that the isotope effect exponent f3 is defined as
{4.26) which leads to
{3 = oln(Tc) (4.27) ain(M). where M is the isotopic mass, which is related to the frequency as
(4.28) for an elastic constant "'· Using eq. (4.8) for the critical temperature, and after simple algebra, f3 = * 8;~, and finally, - 1 In(e;) +C + H~)2 (4.29) f3-2Jn(ll~F)+C-1( 6 )2' B c 2 ~ valid for 6 < 2k8 Tc. This equation, which was derived by Tsuei et al. first, and then by Xing et al., shows that /3 is minimum at 6 = 0, i.e., when k8 Tc is maximum. This is roughly what is found by Crawford et al. (see Fig. 4.11). We have obtained /3 numerically using (4.27) and (4.2), and the results are shown in Fig. 4.12 for different values of the constant background C. One interesting feature of this Figure is that as the value of C is increased, the minimum of f3 increases as well. For example, for C = 5, /3 = 0.42. Also, there appears to be singular behavior at 161 = hwn (/3 is symmetrical with respect to 6, which is the basic symmetry of the density of states). A more detailed calculation, differentiating (4.2) with respect to 1iwn (instead of using directly
( 4.8)) and then performing the integration over the energies, yields
2 _ 1 [ 2C + ln(E}/I1iw~- 6 1) l 2 2 {3- 2 2C + ln(e E}/I{2k 8 Tc) - 621) + (6/2k 8 Tc)lnl(6- 2k8 Tc)/(6 + 2k8 Tc)l (4.30) 4 Van Hove Singularity in BCS Theory 59
which clearly shows a logarithmic singularity at 151 = hw0 , independent of the magnitude of C. That explains why the peak in Fig. 4.12 remains the same for all the values of C.
1
0.8
~ 0.6
0.4
0.2
o~._._._~~----~--~~------~---- 0 0.5 1 1.5 2 d/rMJJ
Figure 4.12 Isotope effect exponent /3 for increasing values of constant C.
The parameters used in the calculation are: eF=500 meV, hw 0 =65 meV, NoV=0.084. For labels in curves see Fig. 4.5. 60 4 Van Hove Singularity in BCS Theory
The model can predict absence of the isotope effect, as we discussed in section 4.4, when eq. ( 4.9) is considered. Therefore a null isotope effect coefficient does not necessarily discard a phononic mechanism. When the vHs is far away from the Fermi level, and using the expansion for the DOS
(4.11), eq. (15) for k 8 Tc, and keeping C -=F 0, the isotope coefficient is
00 1 [ (1iw )2m] (4.31) {J = 2 1 + l; C2m T •
Notice that (4.31) is independent of C, and goes to the BCS value in the limit of very large offset of the vHs. This can also be observed in Fig. 4.12.
From (4.31), {J increases with decreasing 6, and so does k 8 Tc (see eq. (4.15)). These features resemble the trend of the data shown in Fig. 4.11, for x < 0.12. The logarithmic energy dependence of the DOS was believed to ex plain qualitatively the results shown in Fig. 11, because according to ( 4.29), or (4.30), {J can be significantly smaller than the BCS value, and it has a minimum when Tc peaks (at 6 = 0). Also, {J can reach values much larger than 0.5. Nevertheless, it must be pointed out that, although the model presented here seems to account for the general behavoir of some experiments on the isotope effect, a more careful observation shows that the minimum values of {J obtained for given choices of the parameters for a vHs generally exceed those found experimentally, unless the constant background is neglected. Figure
4.13 shows calculated values of {J as function of Tc in comparison with exper iments. The data are taken from Frank et al. (pentagons), and Bornemann et al. (solid hexagons). It is not our purpose here to get the best fit but to show that only when C = 0 can one get relatively good agreement with experi ment. This assumption, though, may not be justified, as band tight-binding calculations for two dimensional lattices clearly indicate the addition of the 4 Van Hove Singularity in BCS Theory 61 constant term C in the model for the vHs, eq. (4.1) (Xing et al. ). When this term is included the apparent agreement with experiment disappears, as shown in Fig. 4.13. The solid line corresponds to C = 0, and the dotted line, to C = 2.0. The lack of agreement for this last curve is evident, even given the relatively small value of the background. In view of these facts, it seems unlikely that a vHs can account for the isotope effect in the cuprates.
1 •Y1_,Pr;Ba,pu:!J., t;YBa._;..,cu:P.,. 0.8 lo..' lo lo I 0 I\ .I t' . '' , ... '. ~ 0.6 ...... -·········· '•,, ·. '• -... -... ··. 0.4 .. ·· ...
0.2 0 • 0 • 0 o~._~~~--~._~~_.~~._~~~~~ 0 50 100 150 T,/.K) Figure 4.13 The isotope effect coefficient f3 as function of Tc. The solid line corresponds to C = 0, and the dotted line is for C = 2. The parameters are
the same as in Fig. 4.12, except NoV= 0.11 in both cases. Experimental data.
were taken from Franck et al. (pentagons), and Bornemann (solid hexagons). 62 4 Van Hove Singularity in BCS Theory
A number of objections have been raised against this model recently.
First, it is known that any energy dependent DOS leads to departure from the canonical value46• On the other hand, if a more realistic version of the vHs is used, say by introducing effects such as interlayer hopping, this causes {3 to shift towards the BCS value, thus smearing out the vHs {Xing et al. ). And even if a different type of energy dependent DOS is used (a Lorentzian, for instance) the small values of {3 remain very difficult to explain using a purely phononic mechanism34• Under these circumstances, the recent experimental measurements of the isotope effect coefficient remain an open problem, and their explanation does not appear to be possible within the context of a model involving DOS effects alone, even with the inclusion of the vHs. Chapter 5
Ther111odynarnic Properties
5.1 Zero Temperature Condensation Energy
The condensation energy was defined in Chapter 2 as the difference between the energies of the superconducting and normal phases. It was found to be proportional to the square of the zero temperature gap parameter, if the DOS at the Fermi energy is assumed constant. In this section we shall be devoted to the calculation of the condensation energy when a logarithmic singularity is introduced in the DOS.
In order to simplify the calculations, we are going to consider the case when the vHs lies exactly at EF, i.e., 6 = 0. From eq. {2.14), and neglecting temporarily a normalising unit volume 0,
63 5 Thermodynamic Properties
We now split the logarithm into two terms, and group together the constant terms, ln(tF) and C, leaving the In(ltl) aside for the moment. The integral above for the constant terms becomes (after using the change of variables
E = ~osinh(9)),
(5.2) where the approximation i!' > 1 has been used. For the term proportional to In(ltl), through a similar procedure, and neglecting terms proportional to
2 (~ ) , the result is D (5.3)
Adding eqs. (5.2) and (5.3),
~F = - 21 No~o2 [ In (2EF)~0 +Cl (5.4) which is valid for 6 = 0, and ~ « 1. D
When 6 > 0, and to first order in the approximation of In( E + 6) as a
Taylor series about E = 0, we get a constant term only, ln(o), and consequently,
(5.5)
Equation (2.32) provides us with the critical magnetic field He, once the condensation energy is known:
H;(o) =41rNo~~[~ne;) +c], for 6 > 0 (5.6) = 41r No~~ [1n( ~:) + c], 6=0.
Equation (5.1) has been solved numerically and combined with (2.29) in order to obtain the exact critical magnetic field at zero temperature. The results
are shown in Fig. 5.1. For convenience we have plotted the dimensionless
0 quantity A = He , where 0 is a unit volume. For a constant DOS A is 2ao 1rNo 10 5 Thermodynamic Properties 65 equal to 1. however the presence of a vHs in the DOS modifies this picture substantially, as Fig. 5.1 shows. For 6 = 0. A is maximum, and decreases with increasing 6. The maximum value of A is not dependent on the frequency, as long as this is not very low, but it rather depends on the zero temperature gap parameter, ~o, as (5.6) suggests.
2.2 ' '~',. \\ ;..... 2 \\' .\. ' \'\. ' .. \\. ' \ '\. ' '•·· ... 1.8 ·,'' ~ ' ' ..... ' ., ' "' ' ' ·· ...... ·, ' ' ·-.. " ' ' -.... ··-... ·, .. ' ' ...... 1.8 ' ..... _ ·,... ' ~ ...... ··-...... '· "- ...... '· ...... '·...... ~ ...... -- ...... -- 1.4 -...... _---..... __
1.2 '---1--'-.....lo..-'-...... -'--.L..-..L--L-.L..~...... _.L.....I~--L..-L-...J...... L.....J 0 0.5 1 1.5 2 d/G>u
Figure 5.1 Zero temperature critical magnetic field to zero gap ratio, as a
function of the offset parameter 6 (see text for definition of A). The param-
eters used are:eF=500 meV. C = 0, and ~0 =10 meV. 66 15 Thermodynamic Properties
5.2 Free Energy
We now turn our attention to the temperature dependent free energy.
From eq. (2.31), and substracting the normal state energy, the free energy difference is (6-F = pN- F 5 ):
where (4.1) must be subtituted for N(e) in the equation above. The com plexity of this equation makes it difficult to provide an accurate analytical solution. However, it has been solved numerically for a number of choices of
6, as shown in Fig. 5.2. In order to make the analysis easier the resulting free energy difference has been normalised to its value at zero temperature when the singularity is at the Fermi energy. Shifting the singularity away from ep produces a depression on the free energy difference, with respect to its value
at 6 = 0. Figure 5.3 compares the normalised free energy difference for a vHs
at the Fermi level, and at 50 meV away from it, to the BCS result. To obtain this last curve one must substitute the vHs in the DOS for a constant, i.e., No.
Note that, although the trend is basically the same, as the vHs is displaced the normalised free energy difference is shifted toward the BCS value, in agreement with most calculations presented so far. The conclusion here is, again, that the vHs is most effective only when located at or near the Fermi energy. 5 Thermodynamic Properties 61
1 d= 0 meV 10 ------20 ---- 0.8 30 --· 40 ------50 ----
0.6 ~ '?t ~ b rz..
0.2 -- ...... ------0.2 0.4 0.8 0.8 1 T/Tc
Figure 5.2 Free energy difference normalised to its value at T = 0 for 6 = 0, from eq. (5.7), as a function of the reduced temperature. The parameters
used are: ep= 500 meV, nwD= 65 meV, NoV=0.084, and C=O. 68 5 Thermodynamic Properties
c5= 0 meV 50 BCS 0.8
0.8
0.4
0.2
0.2 0.4 0.6 0.8 1 T/Tc
Figure 5.3 Normalised free energy difference vs the reduced temperature for
two displacements of the vHs. The BCS result~ obtained through numerical
integration of (5.7) with N(e) =No for e~s about f.F, is included for compari- son. The numerical parameters involved are the same used in the calculations
of Fig. 5.2. with the exception of NoV= 0.20. 5 Thermodynamic Properties 69
The influence of the constant background C in the DOS is schemati cally represented in Fig. 5.4. where the BCS curve is also included to facilitate the comparison. It is found that the inclusion of C makes the superconduct ing deviate even more from the BCS model. Both of the vHs curves were obtained for 6 = 0.
C=O c = 2 ------· BCS ----
0 ~~._._~~--~~----_.~~--~~~._._ ___ 0 0.2 0.4 0.6 0.8 1 T/Tc
Figure 5.4 Modification of the free energy when a finite, constant back
ground is introduced in the DOS. The BCS free energy is included for com
parison. Same parameters as in the previous Figure. 10 5 Thermodynamic Properties
tS= 0 meV 10 20 0.8 30 40 50 -0 0.6 ...... == -E-t -== 0.4 ------' ' ---- ' ' 0.2 ' ' ------------...... -...... ' ' -----.--- -- '
0.2 0.4 0.6 0.8 1 T/Tc
Figure 5.5 Thermodynamic critical magnetic field He, nomalised to its value
at T = 0, and 6 = 0, as a function of the reduced temperature T fTc. From (5.7), and (2.29), with the parameters used for Fig. 5.2. 5 Thermodynamic Properties 71
In Figure 5.5 we plot the temperature dependent thermodynamic critical field He as a function of the reduced temperature, for six choices of the offset parameter 6 in the vHs. For the sake of convenience and clarity we have normalised the critical field to the zero temperature value, when o = 0, just as was done for the free energy. Based on the results shown in Fig. 5.2 it is not surprising to see that He gets reduced as the vHs is displaced away from the origin, for the critical field has been obtained from the free energy, making use of eq. {2.32). Despite the fact that the maximum critical field is obtained when the singularity is placed at the Fermi energy, one finds that the influence of the temperature on He diminishes as o increases, especially at low tem peratures. This is due to the fact that the further away the vHs is from EF, the more constant the DOS appears to the electrons with energies lEI~ 1i.wv, and therefore the fastest it is for the normalised He to saturate when the temperature is reduced.
A comparison between the BCS result, and the vHs model for two choices of the background constant C, similar to that shown in Fig 5.4 is presented in Fig. 5.6 for the critical field. For T ""' Tc, and for very low temperatures the vHs field is well approximated by the BCS field. However for intermediate temperatures a small but significant discrepancy appears being larger for the largest C. 72 5 Thermodynamic Properties
c = 0 ...... c = 2 --- BCS 0.8
-0 0.8 ...... -=E-4 -::C 0.4
0.2
0 ~~~~~~~~~~~~~~~~~~~~ 0 0.2 0.4 0.8 0.8 1 T/Tc
Figure 5.6 Critical field for a vHs with a constant background in the DOS (dashed line), and with no background (dotted line). The constant DOS,
BCS result (solid line) is included for comparison. Same parameters as in
Fig. 5.4. 5 Thermodynamic Properties 73
It is customary to calculate the so called deviation function of the critical magnetic field, defined as
D(t)=---Hc(t) [ 1- (T)2]- (5.8) Hc(O) Tc which is an measure of the difference between the normalised field and the result obtained from the two fluid thermodynamic model. In Figure 5. 7 we plot D against the square of the reduced temperature for the same values of 6 shown in Fig. 5.5. The maximum deviation occurs for a vHs located at the Fermi energy, and as this is moved away from EF by increasing 6, the deviation function becomes almost independent of it. As the vHs is displaced from the origin the function D approaches the BCS curve until they become almost indistinguisable (see Fig. 5.8). The general features of these curves do not depart meaningfully from the BCS deviation function, and have the same sign. Note that the difference between the vHs- or BCS- Hc(T)/Hc(O), and the empirical formula proportional to the square of (T/Tc) is, for our choice of parameters, never larger than 7 %. Therefore the empirical law remains a good first approximation. An interesting feature shown in Fig. 5.8 is the increasing departure from the empirical t2 law when the background
DOS (C) is increased. This is simply a consequence of the differences that we noticed in Fig. 5.4, although they are more clearly seen in Fig. 5.8. 74 5 Thermodynamic Properties
0
•• •• •• •• •• •• •• •• -0.02 •. .. . ' .. ,,· .. ····· ...... ______
-0.04
Figure 5.7 Deviation function, as defined by (5.8), vs. the square of the
reduced temperature. For labels on curves and parameters see Fig. 5.5. 5 Thermodynamic Properties 15
BCS 0.02 c5=0.0 C-0.0 ·----·----·--·---- c5=100.0 c-o.~---- d=O.OC B.O ----
0 .1 .. J -.g--o.o2 .:i ./) ...... ; -0.04 ·.. ..····I \ ·. ··...... ··· .· I \ ...... ,,' / "' ··· ...... •••• ·· -0.06 "' ----/ / 0 0.2 0.4 0.6 0.8 1
2 Figure 5.8 Variations of the deviation function as a function of t , induced
by changes in the location of the vHs (8}, and the background constant C.
For the calculation of the curves the following data have been used: NoV
=0.20, 1iwn=65.0 meV, and tF=500 meV. '76 5 Thermodynamic Properties
5.3 Specific Heat
One of the purposes of this section will be the calculation of the specific heat difference normalised to its value at Tc for a vHs, among some other quantities. In order to do so, one may use ( 2.28) which is given in terms of the derivative with respect to the temperature of the square of the energy gap. It is reasonable, therefore, to find out first which are the essential differences that one finds when the gap function ~(T) is calculated for a vHs, and for a constant DOS.
This has been done by solving numerically the gap equation (2.15) for a constant DOS, and for a vHs model (4.1). Figures 5.9 and 5.10 show that incorporating a vHs in the DOS has no substantial influence in the temperature dependence of the energy gap. In other words, the gap behaves just as if the DOS was a constant, as BCS assumed. This result, which may appear somewhat surprising at first, could have been foreseen because (2.15) shows that the temperature dependence of the gap is determined by the Fermi function f (see (2.16)). The DOS, being constant or not, does not have an explicit temperature dependence, and therefore cannot change anything
but the magnitude of the gap. However, notice that the plots in Figs. 5.9
and 5.10 show results for the normalised gap, hence its absolute magnitude
has no influence on the behaviour of ~ when the temperature is varied. The argument above allows us to simplify some calculations concern
ing the specific heat. Assuming that the DOS is constant, i.e. N(e) = N0, it 5 Thermodynamic Properties is possible to show (see Appendix D) that to the lowest order in the approx imations, the energy gap is given by
{5.9) with t = T /Tc, the reduced temperature, and Ao the zero-temperature gap, as usual. Now, Figs. 4.8-4.10 show that, unless unphysical parameters are used, the departure of ratio (2.20) from its canonical, 3.53 value is just a few percent. Thus it can be safely assumed to be constant. When (2.20) is substituted in (5.9) one obtains:
{5.10)
The specific heat difference at Tc is, after the change of variables x = f3e in (2.28), and using (5.10),
{5.11)
Now, for the sake of simplicity, let us assume that the singularity is at the origin, and also oo, :W¥.B c -
EF ) roo ez roo ln{x) } AC(Tc) = 2(9.38)k~TcNo { [In ( ksTc ] Jo dx (1 + ez)2 + Jo dx ez + e-z + 2 · (5.12)
To solve the first integral, note that (l_;;.,)2 = -U, and so this integral is equal to! when evaluated. On the other hand, the second integral in (5.12) can be approximated by t .. : {5.13) 78 5 Thermodynamic Propertie• 0.8 0.8 -0 -~ ~ 0.2 0 ~~~._._._._._._._._._._~~~~~~~ 0 0.2 0.4 0.6 0.8 1 T/Tc Figure 5.9 Temperature dependence oi the energy gap 6. normalized to its zero temperature value, for a vHs with increasing 6. The parameters used are those also used for Fig. 5.5. Note the curves are almost indistinguishable. 5 Thermodynamic Properties 79 0.8 0.8 -0 -<2 -'E-4 -<2 0.4 0.2 o~~~~~~~~~~~~~~~~~~~ 0 0.2 0.4 0.6 0.8 1 T/Tc Figure 5.10 The BCS gap for a constant DOS (solid curve) is compared to a vHs at fJ= 50 meV (dotted line), and to one vHs at fJ= 100 meV (dashed line). The parameters involved in the calculation are those of Fig. 5.8. 80 5 Thermodynamic Properties With these results the specific heat difference at Tc for 6 = 0 is: l>C(T,) = 9.38N0 k~T,{ In [ k:.;;M] + C} (5.14) with M ~ 0.8813. For 6 positive and far from EF one can expand the logarithm in the DOS, about e = 0, to obtain In(~+ 6) ~ ln(6) + ;6- ~(;6f + ~(;6)3 + ... (5.15a) for 6 > 0, and x > -6/k8 Tc; similarly 2 3 (5.15b) In(-=-+{3 6) -In(6)-- ~-{36 !(~)2 {36 - !(~)3 {36 - ... for x < -{36. Taking -f3c6 -+ -oo, and using (5.15), 00 2 (X ) ez 1 (k 8 Tc)21r 1-oo dxln fi + 6 (1 + ez)2 ~ In(6)- 2 -6- 3 (5.16) which gives the specific heat difference at Tc as 2 8 ~C(Tc) (9.38)NokBTc2 [ In (EF) 1r (k -Tc)2 C ] · (5.17) = T + 6 - 6 + This equation is valid for 6 > 0, and larger than k8 Tc. Equations (5.14), and (5.17) were developed with the purpose of calculating the magnitude of the specific heat jump at Tc, which we shall attempt now. The specific heat in the normal state is (from (2.27)): /iwD ef3c£ CN(Tc) =2kBf3~ deN(e)~ ( f3 )2 1liw 1 + e c£ 2 -~~D [ EF ] 2 ez (5.18) =2NokBTc -zD dx lnlx/{3 + 61 + C x (1 + ez)2 where x is defined as in (5.11). Following the calculation of .AC(Tc), let us assume first that 6 = 0. After some simple algebra, and taking the limit XD-+ oo, 5 Thermodynamic Properties 81 The integral in (5.19) is, after using Gradshteyn and Ryzhik' Tables of Integrals47, formula (4.354), no. 3 (using v =2), pp. 576; and formula ( 4.354), no. 1 (for v =2), pp. 576, (5.20) with .,P(v) the digamma function, .,P(v) = dlnJlJ''). The normal specific heat at Tc becomes after some manipulation, at 6=0, for &ln/k8 Tc > 1, with 1 = 2Nok~ ~, i.e., (5.21) When the singularity is away from EF, for 6 > 0 one can use the approximation (5.15) for the DOS. The normal specific heat at Tc is then (5.22) where the contribution of the second integral in (5.22) to eN is negligible because 6/3 > 1; and (1_;;.,)2 is strongly peaked about 6 = 0. The first integral in (5.22) is 7r2 1 (ksTc)2 Joe 4 ex 3ln(6)- 2 -6- _ x (1 + e:r:)2dx + ··· (5.23) 00 The contribution of odd powers of x to ( 5.23) is zero because the function (1_;;.,)2 is even. The integral in (5.23) is found to be, after straightforward 4 calculations, equal to l51r • When these results are gathered eN is 82 5 Thermodynamic Properties (5.24) which was obtained assuming 6 > 0, 6f3c > 1. One can redefine the coefficient 1, so that CN(Tc) is equal to 'Y*Tc, with 1* as given by the coefficient of Tc in (5.21), or (5.24). The maximum 1* corresponds to the vHs at 6 = 0, and decreases with increasing 6. Now, the size of the specific heat jump at the critical temperature is found to be: 2 (k T. )2 ~C(Tc) =1.43[ln(tF/6)+T ::a,r- 2+C], (5.25) 'YTc ln(tF/6)- Zfir( kBlc) + C which is valid naturally for 6 > 0, while the result for a singularity at the origin deviates very little from 1.43. It is clear from Fig. 5.11 that the con stant C does play an important role in the size of the specific heat jump at Tc. By increasing C the difference between the vHs-, and BCS- specific heat jump values is reduced. In particular, the discrepancy is of about 16% for C=4. Nevertheless, a value of 4N0, when added as a background to the vHs would raise noticeably the DOS beyond what is found through band structure calculations, or some experiments35, and thus it is perhaps unre alistic. The constant No is always assumed to be equal to 1 ev-1spin-1 so that the DOS is properly normalized. On the other hand, the DOS with no background induces a maximum deviation of almost 40 % with respect to the BCS value, at 6=0. For all cases the maximum deviations occur at the origin, and saturation towards the BCS limit appears when 6 is increased suficiently. All of these features are found, qualitatively at least, in (5.25), except the ocurrence of the maximum size of the jump at the origin, since our formula 5 Thermodynamic Properties 83 gives 1.43, just like BCS. Nonetheless. it does show that increasing C should reduce this ratio, and also goes to the proper limit when 6 is increased. 1.3 C=O 1 ------· 2 ----- 3 --- (J 4 ------· ~ 1.2 ~ ~. ...-4 ...... (J -E-t u 1.1 -<1 1 0.5 1 1.5 2 8/GJrJ Figure 5.11 The specific heat jump at Tc, for different values of the constant C. For the calculation, the following parameters were chosen: 1iw D =65 me V, €F=500 meV, Tc=92.0 K. 84 5 Thermodynamic Properties Equation (5.25) also helps in the interpretation of Fig. 5.12, which shows enhanced differences from the BCS case. The coefficient of Tc in the denominator of ( 5.25) is larger than the corresponding coefficient in the nu- merator. When the critical temperature is raised, the denominator is reduced faster than the numerator is increased, giving a net increment in ~~gc) with 1.4 T.=55.0 K 50.0 () 80.0 ~ 1.2 92.0 ~. 125.0 ..-4 -'e-.() -u 1 0 0.5 1 1.5 2 d/Glo Figure 5.12 Specific heat jump normalized to 1.43 (BCS) for some typical Tc. For all curves, C=O. The rest of the parameters are the same as in the previous Figure. 5 Thermodynamic Properties 85 The physical origin of the trends shown in Figs. 5.11, and 5.12 can be traced back to the gap-to-Te ratio for different o's and to the thermally averaged DOS. The experimental determination of the ratio ~~gc) for the high Tc superconductors is unfortunately not an easy task. One of the major dif ficulties is the accurate measurement of the Sommerfeld "Y, because at the scale of these critical temperatures the normal state electronic specific heat is dominated by the phonons. Phillips et al. 48 have estimated 'Y and ~~gc) for Y1Ba2Cua07_,s, and report a value of about 4.8, which is clearly inconsis tent with weak-coupling. Junod et al. 49 find a similar estimate. This could indicate that strong coupling is needed in order to account for the specific heat behavior in the cuprates. However, it has been shown50 that the dimen sionless ratio of the slope to the size of the jump in the specific heat obtained from experiments exceeds considerably the maximum strong-coupling value from Eliashberg theory. This has been atributed to different phenomena, such as :H.uctuations51 , indicating that we may be missing some important effects that lie deep into the microscopic understanding of the oxides, and remains a puzzle. From the discussion above, and from comparison of our results with some other groups' it is evident that a vHs in the DOS, within the weak coupling limit in BCS theory is not capable of explaning satisfactorily some specific heat measurements carried out for the cuprates. Although Tsuei et al. have presented recently some new evidence in favor of a vHs scenario, based precisely on the specific heat52• We have also calculated the specific heat difference between the su perconducting and normal states, as a function of the reduced temperature, 86 5 Thermodynamic Properties as seen in Figs. 5.13-5.15. In Fig. 5.13 6.C is calculated for different positions of the vHs with respect to the Fermi level. It is found that the change of sign occurs at slightly higher reduced temperatures when 6 is increased. This is due to the faster decay of the superconducting contribution to 6.C(T), which is a consequence of the decay of ll. with increasing 6. To see how the vHs results compare to the standard BCS theory, ~BCS is compared in Fig. 5.14 with a vHs at the origin, and another at 6=100 meV. This Figure shows how the introduction of the vHs shifts the change of sign towards lower temperatures, especially when it is located at the Fermi energy. Also note that for T close to Tc the slope of ll.C(T) is reduced by the vHs. The absolute minimum in fg( Junod et al. have measured ll.C(T), among some other quantities, for Y1Ba2(Cu1-~Fe~)a01-6, and compared their measurements to the weak coupling BCS behavior49• They find that the peak at T = Tc is very sharp, the change of sign occurs at considerably higher reduced temperatures than the BCS curve, and the minimum value of 6.C is unexpectedly lower. With the addition of a vHs we have found the opposite results, which indicate that the assumption of a logarithmic singularity in the DOS is not supported by specific heat experiments, and therefore cannot be considered an adequate model for high Tc superconductivity. 5 Thermodynamic Properiies 87 1 tJ= 0 meV 10.0 ------0.8 20.0 ----- 30.0 --- 40.0 ------0.6 50.0 ------e.: -u ~ 0.4 E-t -u - 0.2 0.4 0.6 0.8 1 T/T. Figure 5.13 Specific heat difference Cs- CN, normalized to its value at Tc, as a function of the reduced temperature, for different positions of the vHs. The parameters chosen are: EF=500 meV, NoV=0.084, 1iwD=65 meV, and C=O. 88 5 Thermodynamic Properties 6== 0 meV 100.0 0.8 BCS 0.6 -~ t) -~ ...... 0.4 E-4 -t) -~ 0.2 -0.2 L...... I...... _..--L.-.L.~..;J;:...... J-..L-.L..-.L..-1~--L~--L-...J-..L-.L..-.L....J 0 0.2 0.4 0.6 0.8 1 T/T. Figure 5.14 The specific heat difference, normalized at Tc, as a function of the reduced temperature. The solid and dotted curves correspond to vHs at 8=0 meV, and 100 meV, respectively. The BCS curve (constant DOS) is also included for reference. Para.meters:N0V=0.2; the rest as in Fig. 5.13. 5 Thermodynamic Properties 89 C=O.O ··············· 0.8 2.0 ----- ,.://. I,.'' 1/ I/ 0.6 ,,,./,: -E-to // /:' -u ,,: ~ 0.4 1:' -E-t ,,:, u 1/ - -0.2 ~....__.__... _ _...__.._""""--.....__...__'---'--"'---'---I..--'--L--'--~'---' 0 0.2 0.4 0.6 0.8 1 T/T. Figure 5.15 The influence of constant Con ~Cis shown, for the same set of parameters as in Fig. 5.14. 90 15 Thermodynamic Propertiea Another commonly used dimensionless ratio is Jf BCS value) is plotted in Figs. 5.16, and 5.17, as a function of 6. Once again, (5.26} provides a good qualitative description of the exact results. Note how an increment in 6 produces a reduction on the value of the ratio, and in the limit of large values of 6, the ratio goes to the BCS limit. On the other hand, when the background constant Cis increased (5.26) predicts a reduction in the ratio, in agreement with the exact solution (see Fig. 5.16}. Similarly, (5.26} predicts an enhancement of the ratio when Tc is increased, which can be observed in Fig. 5.17. The curves in the last two Figures appear to have "inverted" be havior, compared to most of the curves that have been shown. The reason for this behavior relies on the variation of the gap to Tc ratio, (2.20), which appears inverted in (5.26). See for example, Fig. 4.9. Before ending this Chapter we would want to analyze the low tem perature behavior of the electronic specific heat. This is important not only because it provides thermodynamic information about the system, but also for historical reasons, since the finding of an exponential temperature depen dence in the superconducting specific heat was one of the most important indications of the existence of an energy gap, and an excellent confirmation for the BCS theory. 5 Thermodynamic Properties 91 1 ~ C=O ~ 1 .....~ ------· 2 ----- ~ 3 --- ;:: 4 ------0.8 0.6 l....-..ll...-..lo---J...... J..--1....-L...-l...... J.-..I...-L...... J--L--L.-l--L..-1...-..J.-.J..-.L.-J 0 0.5 1 1.5 2 8/GJrJ Figure 5.16 Ratio 'YT'! f H;(o) normalized to the BCS value, for different C's. The following values were used: hw 0 =65 meV, fF=500 meV, Tc=92 K. 92 5 Thermodynamic Properties , ..... -...... -~ ~ .,_ •••"1 I ' ...... _ ~--. /... ,I. ' ' ~~ -- ...... _ ... ~...... _ I ·-...... _ ------...... 1 1/ / -...... ______-- -- ...... ~-:__:_- ...... __,------==- 1 : I I ------·--·---- 0' ~ ! 1/ / T0 =35 K---- : 'I I 50 ....~ i I i f 'I ; 80 ~ : I j 92 : IJ I ~ : 1/ I 125 0.8 f I/ j : 11 I j II j : 1/ I ;1 l l I • I • I 0.5 1 1.5 2 4/rMD Figure 5.17 The same ratio plotted in the preceding Figure is calculated here, but this time for different choices of the critical temperature, when C=O. Same parameters as in Fig. 5.16. 5 Thermodynamic Properties 93 At low temperatures, the energy gap ~(T) can be approximated by ~(0) (e.g., see Figs. 5.9, and 5.10}. Thus, its derivative with respect to the temperature is null, and the electronic specific heat in the superconducting state will be given by 2 '"'-'D ePE E2 Cs = 2kBf3 -""'-' dtN(t) (1 + ePE)2 (5.27) 1 D 1 2 from (2.23}, withE~ y't2 + ~~=~o[1 + (t;-)] 1 • For k!T < 1, ePE -PE (1 + ePE)2 ~ e · (5.28) This exponential, when substituted in ( 5.27} is very sharp and only signifi cantly different from zero about t=O. It is then permissible to Taylor expand E about t=O. Keeping only the first two terms, and changing variables to X= t;, (5.27} is 2:D ~,o. .,2 2 0 Cs = 2kBf3 ~3e-f3.6.o 1-zD dxN(x~o)e- 2 • (5.29) Given the rapid convergence of the integrand, and since W.UD/~o < 1, xv can be extended to infinity with no appreciable error. Now let us consider a vHs located at the origin. After some algebra one gets 0 Cs(T) = 2v. rn= .t.1fkB~No ( -~ )3/2[ ln (tF)- + C + -ln1 (2~o-e'Y )]-~ e a (5.30) kBT ~o 2 kBT with 'Y ~ 0.577 ... , Euler's constant. Here the exponential behavior appears immediately, which is the dominant temperature dependent term. For a vHs away from the origin we proceed as we have done in previ ous sections, by expanding the logarithm in the DOS in a Taylor series about t = 0. After straightforward integration, term by term of the first four terms in the expansion, the specific heat is 2 Cs(T), = 2.;'2;kBa.No(k~)" [m(';) +C+ ks2~a,. ]·-•7r (5.31) IS Thermodynamic Properties which is also exponentially dependent on the temperature. The appearance of the gap in the exponential term in (5.30), and {5.31) is indicative of the activation energy behavior. These equations also show that Cs -+ 0 when T-+ 0, as is found experimentally. The coefficient of exp[-Ao/ks11 between brackets in (5.30), and (5.31) is obviously introduced by the vHs53• 5.4 Concluding Remarks The results presented in this Chapter, particularly those regarding the electronic specific heat, along with those shown in Chapter 4 reafirm our conviction that the vHs scenario is not capable of explaning the behavior of the high Tc superconductors as a whole, at least within the weak-coupling, BCS-theory framework. Moreover, it has been shown that even when the strong-coupling approach is used, the vHs model does not provide agreement with some key experimental results in the cuprates34, and the behavior of quantities such as Tc, /3, AC, etc. is qualitatively the same we have shown in this work. It appears to us that the single apparent success of the vHs model is that with it high Tc 's of the order of those found in the oxides are easily attainable. On the other hand, the model fails to explain the isotope effect measurements, and is inconsistent when accounting for the electronic specific heat, as some experiments have shown. There have been several criticisms of the vHs model and its role in the oxides. Effects of correlations, disorder, interlayer coupling on the model, and the lack of unquestionable experimental evidence for it, are some examples. While most of these criticisms have been rebutted by some authors, more 5 Thermodynamic Properties 95 and more assumptions seem to be needed (e.g., interlayer coupling must be smaller than k8 Tc, any structural disorder lies outside the Cu02 sheets, etc. } in order to make the vHs picture more credible54- 57• Until some compelling experimental evidence is available, the inclusion of too many particular as sumptions may reduce the reliability and utility of the model. Chapter 6 Conclusions Recent reports on the isotope effect in high Tc superconductors have resurrected the idea of a vHs in the electronic DOS. The suggestion has attracted attention because it combines the very appealing features of high transition temperatures with a varying isotope effect coefficient, that can even be larger than 0.5. Roughly speaking these are characteristics found experimentally. We believe that this is an attempt that deserves careful consider ation. Thus, we have calculated a number of properties for weakly coupled superconductors when a logarithmically dependent (vHs) DOS is introduced, in BCS theory. More particularly, it is found that the critical temperature can be approximated by the relation (6.1) 97 98 6 Conclusions when the vHs lies at the Fermi energy EF, and the coupling constant NoV is small. Using (6.1) with modest values for the parameters, one finds that Tc's in the 100 K range can easily be reached. Displacing the vHs away from EF reduces Tc towards the constant-DOS limit. The influence of other parameters, such as 1i.w 0 and C, on Tc was also investigated. Our conclusion here is that the filling factor 6 plays the most important role, once NoV has been fixed. The zero temperature energy gap Ao was also calculated and found to be proportional to Tc, in agreement with conventional BCS theory. Ac cordingly, the BCS gap to Tc ratio is not essentially modified by the vHs. Our results for the isotope effect coefficient {3 indicate that it is a minimum when Tc is maximimum, i.e., at EF· It grows as the filling factor is increased, reaching values that can be well over the BCS prediction. Quali tatively, this is supported by recent experimental observations. Nonetheless, our results indicate that the minimum values of {3 cannot be explained by means of a vHs. Moreover, using a quantitative strong coupling approach only makes the disagreement larger-57• Among the thermodynamic properties that were dealt with in this thesis, the specific heat is especially important. It was found that the specific heat difference between the normal and superconducting states at Tc is a function of the filling factor, as opposed to the constant BCS value. Also, the specific heat jump at Tc is dependent upon the position of 6 and the constant background in the DOS. However, the size of the specific heat jump found in some experiments generally exceeds the estimations we have presented here using the vHs approach. The calculations for the specific heat difference ~C(T), normalized at Tc also show considerable disagreement with Junod's 6 Conclusions 99 49 group's experiments for Yi.Ba2Cua07_6· We found that the change of sign for !1C(T)/!1C(Tc) occurs at lower reduced temperatures than those from BCS, and the slope of !1C(T) near Tc is also reduced from the BCS result when the vHs is invoked, in contradiction to Junod et al.' results. The calculation of universal ratios such as 2!1o/k8 Tc, !1C(Tc)hTc, and "(T; / Hc(0)2 for a vHs present small deviations from their corresponding values for a constant DOS, as long as the singularity is not too sharp. Comparison with values obtained from experiments is perhaps premature at this stage because the scatter in the data is still too large to be ignored, and sample quality is expected to improve some measurements. However, with the ex perimental data available at present for the isotope effect coefficient and the specific heat difference, the assumption of a logarithmic singularity in the DOS does not seem tenable. Any reliable calculation should include many-body effects, Coulomb repulsion, anisotropy, and perhaps an electronic component to the pairing61 , if accurate agreement is sought for with measured properties of the high Tc oxides. Eliashberg theory would be a good candidate to carry out this task. However, one may question the validity of the vHs approach when the results apparently fail to reproduce, even qualitatively, the experiments. One reason for this apparent failure is perhaps the simplicity of the model that has been used here; therefore more work is required. Properties such as the susceptibility, penetration depth, and transport properties deserve attention. Even if the vHs does not prove eventually to be an appropriate model for superconductivity at high temperature one hopes that this idea is important elsewhere. Appendix A The Density of States When the energy dispersion relation, E(k), is known for then-band, the number of states per unit energy is given by: (A.l) where L is the side of the crystal, and the integration is carried over the volume bounded by the constant energy surfaces, with energies E, and E +dE. An infinitesimal element of volume will be given by a base dS, and a height dk 1., so that 1dk = j dSdk1. (A.2) 101 102 Appendix A where dk.1. is measured normal to the surface. But note that Vkt: is also normal to the surface, and, moreover, de dk.1. = IVkel (A.3) which enables us to write the density of states from (A.l) as V dS (A.4) N(e) = (21r) 3 JIVkel· The integral is performed over the area of the surface of constant energy e, in k-space. One must keep in mind that each state acomodates two electrons with opposite spins. The result (A.4) refers to a single branch n, of the energy dispersion relation. A very important contribution comes from points at which IVkel = 0. These critical points produce the so-called van Hove singularities ( vHs ). In general there are four types of such critical points: maxima, minima, and two types of saddle points. In a one-dimensional lattice, a vHs makes the DOS diverge as 1/../i., withe being the energy measured from the vHs. For a two dimensional lattice, the DOS diverges logarithmically. In three dimensions these singularities are integrable. A case of particular interest in the context of this work is, naturally, the two-dimensional case. With the purpose of illustrating how a logarithmic singularity can arise from a two-dimensional lattice we proceed as follows: assume the simplest saddle point in the energy dispersion relation, so that (A.5) the DOS per spin at such saddle point is, per volume V, Appendix A 103 (A.6) After simple integration, one finds N(e) = Nolnl e: I (A.7) where EF is the Fermi energy, and No is a constant. A more detailed discussion of the DOS and vHs can be found in most textbooks in solid state physics, see for example refs. 58-60. Appendix B A useful integral The integral j ln(a: bx) dx (B.1) cannot be expressed as a finite combination of elementary functions (see ref. 47). An approximate solution can be obtained using the series representation of ln(1 + x). Now, In( a+ bx) = ln(bx) +In(1 + b:) (B.2) for bx :F 0. H -1 < ajbx ~ 1, then 1(a)2 In (1 +bxa) ~bx-2a bx + ... (B.3) 105 106 Appendix B so, using the results from (B.2), and (B.3), the integral above (B.l) is 2 ln(a+bx)dx=!ln2 (bx)-~+!(~) +... (B.4) J x 2 bx 4bx which holds for bx =/: 0, and -1 < afbx ~ 1. Appendix C Weak-coupling constant One can define an average DOS as follows: 1iw f_~D N(e)de (C.l) Na, = 1iw • I -nwDD de Now the model (4.1) for N(e) is introduced in the equation above: N •• = 2~. [L::ml <~51 de+ 2/iwDC] . (C.2) Introducing x = e + 6, and y = e- 6, (C.3) and, after rearrangement, 107 108 Appendix C 1 I €~ I 6 ln.wD-61 ] (C.4) Nav = No [ 2ln 1iw~ _ f1l + 21iwD In 1iwD + 6 + 1 + NoC. A case of special interest arises when the vHs is located at the Fermi energy, i.e., 6 = 0: N •• = No[m(,:;J + l+Cl (C.5) so that a coupling constant Aav can be defined by (C.6) where Vis the constant pairing potential. The weak-coupling regime will be given by the values for which Aav < 1. Appendix D The Energy Gap near Tc For a constant DOS the temperature dependent energy gap is given by (2.15): 1 = N(O)V {1iwo dt: [1- 2/( .je2 + ~2 (T))] (D.1) lo Je2 + ~2(T) where N(O) is the constant DOS at the Fermi nergy, and f is the Fermi function, (2.16). It is easily shown that 2 1 - 2/ ( J ,2 + d2(T)) = tanh ( J ' (D.2) 2:;(T)), and now the series representation for tanh(x/2) is used, from ref. 47, pp. 36: 109 110 Appendix D (D.3) This yields 1iwD 00 1 1 = 4N(O)VkBT dE I: 2 2 .6.2(T) (D.4) 1o k=l wk + E + where wk = 7rkBT(2k -1) are the so-called Matsubara frequencies. Expansion of the sum yields (D.5) valid for small .6-(T), which is guaranteed if T is close to Tc. If only terms up to second order in .6-(T) are kept, 1 = 4N(O)VkBTf: [ {1iwo 2dE 2- .6_2(T) {1iwo ( 2 dE 2)2]· (D.6) k=l Jo wk + E Jo wk + E When .6. = 0 we should get T = Tc, i.e., 1 = 4N(O)VkBTc f: {1iwo dE (D.7) 2 2 k=l Jo wk + E but from the BCS equation for Tc, (2.19), 1 = N(O)Vln[1.13:;';J (D.8) thus, by equating (D.7) to (D.S) one can observe that the equality formally holds for any temperature T, i.e., ~ 11iwD dE [ nw ] 4kBT L.... 2 2 = 1n 1.13k TD . (D.9) k=l 0 Wk +E B Appendix D 111 On the other hand, in the term proportional to ~ 2 (T), nwD can be extended to infinity because the integrand converges rapidly. Therefore (D.10) and also (D.ll) therefore, 00 roo dt. 1r 1 7 (D.12) Ek 2= 4 3 EOO k=l o (w~+t.2) (1rksT) 8 where E is the Riemann zeta-function47• By substituting these results into (D.6) one obtains: _1_ = In[1.131iwv] - ~2(T) 7 E(3) (D.13) NoV ksT (1rksT) 2 8 but the left hand side of the equation above is equal to equation (D.S). 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