Theory of simultaneous excitonic-superconductivity condensation II

Physica C 158 (1989) 15-31 North-Holland, Amsterdam

THEORY OF SIMULTANEOUS E X C I T O N I C - S U P E R C O N D U C T I V I T Y C O N D E N S A T I O N I I Experimental evidences and stoichiometric interpretations K.W. W O N G Department of Physics and Astronomy, University of Kansas, Lawrence, Kansas 66045, USA W.Y. C H I N G Department of Physics, University of Missouri-Kansas City, Kansas City, Missouri 64110, USA Received 20 December 1988

We discuss a variety of experimental observations which are consistent with theory of the excitonic-enhancement model (EEM) presented earlier. The experimental works discussed are: (1) isotope substitution; (2) fluorinated YBa2Cu3OT_,; (3) infrared optical spectra; (4) specific heat and tunneling gap; (5) Hall effect and nuclear spin relaxation; (6) positron annihilation; (7) ultrasound velocity and sound attenuation; (8) Meissner effect and critical current; (9) antiferromagnetism and oxygen deficiency; ( 10 ) flux quantization; and ( 11 ) photoemission. A simple stoichiometric interpretation on the existing high temperature superconducting oxides based on the specific stacking of chemical subsystems is also presented. It is argued that according to EEM theory, a superconducting oxide must contain two stable oxides, one having excitonic levels such as Cu20; the other having intrinsic hole population at the top of the valence band such as CuP. A systematic search for other potential high Tc compounds is also suggested.

1. Introduction In an earlier p a p e r referred as I [ 1 ], we have discussed the excitonic e n h a n c e m e n t m o d e l o f superconductivity ( E E M ) . This theory o f simultaneous excitonic-superconductivity c o n d e n s a t i o n was presented in two different approaches: ( a ) a t w o - b a n d model H a m i l t o n i a n with emphasis on the existence o f an intrinsic hole population; ( b ) the off-diagonallong-range-ordering ( O D L R O ) o f positively charged excitonic quasi-particles. O u r purpose, obviously, is to suggest the EEM as a possible m e c h a n i s m for high Tc superconductivity. Eventually, a correct high Tc theory must be able to satisfactorily explain all comm o n features o f different high Tc materials, similar to the BCS theory [2] which is capable o f explaining all c o m m o n features in the low Tc metallic superconductors. Therefore, it is p r u d e n t that in this follow-up paper we a t t e m p t to explain some o f the observed experimental results within the context o f the EEM model, or at least to show that these observations are consistent with EEM. A successful theory should also have some predictive power. In partic-

ular, one would like to conjecture the existence o f other possible superconducting compounds. Based on the EEM concept, we find that the c o m p o s i t i o n a n d structure o f all the currently discovered high Tc superconducting oxides can be simply interpreted in terms o f stoichiometry a n d stacking o f two basic chemical subsystems: on that is excitonic such as Cu20; the other possesses intrinsic hole population, or equivalently is a p-type c o n d u c t o r such as C u P . A step-by-step recipe is then p r o p o s e d to search systematically for other high Tc superconducting compounds. Before we discuss the experimental evidences in support o f EEM a n d the stoichiometric interpretation o f superconducting oxides, we shall surmise the basic points o f EEM presented in p a p e r I. ( 1 ) A high Tc material must have a semiconductor-like or a semi-metal-like b a n d structure. Intrinsic hole p o p u l a t i o n must exist at the top o f the valence b a n d ( V B ) which is separated from the conduction b a n d ( C B ) by a gap. The normal conductivity is then carried by holes with its F e r m i surface in VB. We have calculated the electronic structures o f nearly all

0 9 2 1 - 4 5 3 4 / 8 9 / $ 0 3 . 5 0 © Elsevier Science Publishers B.V. ( N o r t h - H o l l a n d Physics Publishing Division )

16

K. W. Wong, W. Y. Ching / Theory of simultaneous excitonic-superconductivity condensation H

the newly discovered superconducting oxides [3-7] and they all have either semiconductor-like or semimetal-like band structures. (2) Electrons excited to the CB can form excitons with the holes in the VB via strong Coulomb attraction. These excitons are not charge neutral because of the condensed long-range-ordering and the presence of intrinsic hole states in the VB. The condition for the excitonic bonding depends on the size of the band gap G (or band overlap for semi-metal), the static dielectric constant % and the reduced effective mass tensor of the electron and holes. For G of the order of 1 eV, eo is expected to be of order l0 for a semiconductor. However, for a material with large intrinsic hole population, eo can be even smaller because of the larger separation between the Fermi surface and the CB minimum. First-principles interband optical calculations [ 4,8,9 ] indicates that eo for the high Tc oxides are of the order 12-16 range without taking into consideration the difference due to the presence of the hole states. More realistic calculations may reduce % even further and therefore are consistent with EEM. (3) Below the critical temperature To, a simultaneous excitonic and superconductivity condensation of positively charged excitons is realized. This condensation results in the renormalization of the normal state Fermi surface. In contrast to the normal BCS metallic superconductor in which the Fermi surface is measured from the bottom of CB, the Fermi surface of this new condensation of positively charged quasi-particles must be measured from the top of the VB. This can also be viewed as the off-diagonal-longrange ordering [ODLRO] of charged fields which are linear superpositions of electron and hole states. The total gap amplitude AT satisfies a simple Pythagorean sum rule of two orthogonal gap amplitudes: ']BCS, the usual phonon-mediated BCS amplitude and Acx the diagonal long range ordering excitonic amplitude: A~ =dacs -2 +dcx -2 •

( 1.1 )

While ']BCS is Debye frequency limited, Aex is not. A sufficiently large 71ex component can enhance z~T significantly resulting in high temperature superconductivity. The temperature dependence for dT should also be consistent with the BCS weak coupling limit. (4) In EEM theory, the positively charged quasi-

particles, i.e. charged excitons, are l~ormed by linear combination of CB electron and VB holes. Such a linear combination breaks charge symmetry and thus gives rise to fractional charge. The "Cooper pairs" formed by these fractionally charged quasi-particles must then give non-integer flux quantization, and slightly reduced Meissner effect. The specific heat on the other hand remains similar to that given by BCS weak coupling theory; except that a shift of the Fermi surface would also accompany this simultaneous breaking of the Cooper and excitonic pairs. Thus a double specific heat jump near Tc is expected. In the following section (section 2), we will discuss various experimental evidences in support of the EEM theory. In section 3, we present a stoichiometric interpretation of superconducting and related non-superconducting phases of various classes of perovskite oxides. A plausible procedure to hunt for other potentially high Tc compounds is then suggested. The paper ends with a brief conclusion in the last section.

2. Some experimental evidences In this section, we shall discuss a variety of recent experiments on high Tc superconductors and show how these results are consistent with the EEM theory. These experiments are selected for the purpose of analyzing EEM and are by no means exhaustive or even highly representative. We shall also bear in mind that any experimental data may not be consistent with each other and some of the experimental claims remain controversial. Better and more refined experiments in the near future will definitely change and modify some of the data quoted. Therefore, our discussions, even though with quantitative numbers involved, are meant to be qualitative and general. Because numerous experimental results are available, we shall subdivide our discussion into eleven subsections: 2.1: Isotope substitution of ~80; 2.2: Fluorine-substitution; 2.3: Infrared optical spectra; 2.4: Specific heat and tunneling gap; 2.5: Hall effect and nuclear spin relaxation; 2.6: Positron annihilation; 2.7: Ultrasound velocity and sound attenuation;

K. W. Wong, W.Y. Ching/ Theory of simultaneous excitonic-superconductivity condensation H 2.8: Meissner effect and critical current; 2.9: Antiferromagnetism and oxygen deficiency; 2.10: Flux quantization; 2.11: Photoemission. 2. l.Isotope substitution of ~80 In the previous paper [ 1 ], we have obtained the total excitation gap/t T in terms of a Pythagorean sum of A~x and Aacs: -2 -'~ /i~"~ =Ae~ +A~cs •

(2.1)

Furthermore, we have also found that for systems with large intrinsic hole populations, such as YBa2Cu307 _ ,- [ 1 - 2 - 3 ] and La2_xSr.,-CuO4 [ 2 - 1-4 ] compounds, the effective excitonic gap A¢x can be well approximated by: Ae~ =/i~x/2~v •

(2.2)

Where ~v is the renormalized Fermi-surface energy measured from the middle o f the gap and/ie~ and Jlex are the excitonic binding gap before and after the renormalization. From the calculated electronic structure [3] of YBaECU307 ..... we estimate that /2v = 2.5 eV. It is possible to deduce ,]~ and/ie~ from the oxygen isotope effect measurements [ 10-12 ], if we assume the total gap /IT is related to T~ by 2/IT=3.5kT~ as derived in paper I. Only the ~lBcs component should be sensitive to the isotope mass effect [ 1 ]. Let us assume a perfect inverse squareroot dependence of ABcs on isotope mass and use the superscript i to denote the isotope-substituted case. We can write:

_, 2 [

(2.3)

L/i.csJ where ~mi=(mi-m)x

(2.4)

and x is the fraction of isotope substitution. Now, since Te is proportional to AT, we have:

AT-

T~ --I

Tc '

(2.5)

where 5T~ is the shift in Tc due to isotope substitution. By eliminating ~I¢~from the gap equations (2. I ) and (2.3) for T~ and (2.5) for 8T~, we obtain:

aT

L\am'/

17

Tc / J

" (2.6)

Using the experimental values of sample I of ref. [ I 0 ] , x=0.9, 8Tc=0.5 K, Tc=92 K, we have ABCS//IT =0.33 and Aex/AT=0.94 for YBa2Cu307 ..... We note that ~IBCS=0.33A T no longer exceeds the phonon mediated limit give by the BCS theory. The same analysis can be applied to other superconducting oxides on which careful isotope substitution experiments have been done. These include the earlier measurement on Lal.85Sro.~sCuO4 [ l i, 12 ] and the more recent copper-free cubic superconductors Bao6Ko.4BiO3 and BaPbo.vsBio.2sO3 [13]. The results are summarized in table I together with that for [ I-2-3 ] compound. We obtained ABCS//IT=0.66, 0.65, 0.63 respectively for the above three superconductors with Tc < 40 K. The corresponding ratios for Aex/AT are 0.75, 0.76 and 0.78. To illustrate more clearly the relative contribution of ~IBcs and ~lex to To, we plot in fig. I an illustrative diagram for various high Tc oxides. The horizontal axis is 2zlBcs/3.5k, and the vertical axis is 2~lex/3.5k both in units of K. The length of the arrow corresponds to Tc for each material according to eq. (2. I ). The pure BCS superconductor Nb3Ge with Tc=25 K is shown as a horizontal line. The vertical arrow indicates the possible BCS limit ABcs because of the dependence on Debye frequency. The dashed arrow indicates the Tc o f 125 K for Tl2Ca2Ba2Cu3Oio under the assumption that ilBcs in TleCa2Ba2Cu301o is of the same order as YBa2Cu3OT. Assume 35 K is the limit of Tlacs/3.5k then for a Tc of 300 K, one must seek a material with ~3e~/3.5k~ 298 K. Therefore, to achieve room temperature superconductivity Aex should completely dominate over ABcs. From fig. 1, we note that the effective BCS gap 71Bcs for YBa2Cu3OT_x and Lal.85Sro.15CuO4 are quite close. This is resonable because in both materials the Debye frequency is controlled by the mass of 160 in a C u - O plane. Furthermore, their respective Fermi energies are nearly equal, ~F=/t+ G/2 = 1.5 eV (the negative sign is for YBA2Cu307_x and the positive sign is for LaLssSro isCuO4).

18

K. W.. Wong, IV..Y. Ching / Theory of simultaneous excitonic-superconductivity condensation H

Table I Analysis of isotope-substitution data base in the two-component gap theory of EEM.

T~ (K) % of ~80 ATe (K) ZJBCS/ZJT

dcx/AT 2Aacs/3.5k (K) 2A~/3.5k (K)

YBa2Cu307_x

La:_xSr~CuO4

Bao.6Ko.4BiO3

BaPbo.75Bio.T503

92 0.90 0.5 0.33 0.94 30.4 86.5

35 0.68 0.6 0.66 0.75 23.1 26.3

28 0.65 0.45 0.65 0.76 18.2 21.3

10.5 0.85 0.20 0.63 0.78 6.62 6.5

Energy bands of YBazCu307

5 (fit) B~lab.~Bi.~O,

12C -I

IOC

~e)

(b) s~,.,K,./aioa

~ a

(e) u,,_,s~c,,o,

~ 2

(d) ~,,,c,,~,_.

~ ~-

( . ) ~-2~z~

80 ~ k

(f) 60

4"

-1

Nb,.e

F

¢.(~)~~

X

S

Y

F

Energy bands of YBaaCusO6F !

5¸

I<

4-

4c

~~

20

i

~

BC,SLimit !

>,

i

)

0

20 40 60 ZAncs / 3 . 5 k Fig. I. Decompositionof Jscs, 3e~ components using isotope shift data and the Pythagoras sum rule: dT2 =ABCS -2 +Ae~ -2 for oxide superconductors. The length of the hypotenuse corresponds to the magnitude of To. Detailed explanation is in the text. 2.2. Fluorine-substitution.

There are numerous experimental reports [ 14-16 ] which indicate that Tc for the YBazCu307 system can be raised by partial replacement of O with F. We have calculated and reported the electronic structures o f some F-substituted YBazCu307 compounds [ 5 ]. For the simplicity o f argument, let us concentrate on the substitution of one O, site. We see from fig. 2 that

F

X

S

Y

F

Fig. 2. Comparison of normal state band structures near Ef of: (a) YBa2Cu307; (b) YBa2Cu306Ft (F substitute on Ot site). the general character o f the bands remains relatively similar to the unsubstituted case with the following differences: (a) The heavy-hole band at Y is lowered; (b) the indirect band gap is narrowed to 0.86 eV; (c) the Fermi level is 1.72 eV below the VB m a x i m u m at S. Substituting the result o f '~ex obtained from isotope analyses for YBa2Cu3OT_x into eq. (2.2), we get Aex = [ 2 ~ F ~ e x ] ,/2

= [0.94×3.SkTcflv] '/2=0.25 eV.

(2.7)

K.W. Wong, W.Y. Ching/ Theoryofsimultaneous excitonic-superconductivitycondensationH Now, referring to eq. (2.17 ) in paper I which gives the condition for excitonic condensation and the solution of Aex, we obtain the exciton binding: 0.5 IEBI =G/2+A2x/G=0.56 eV.

(2.8)

Ignoring the possible change in IEB I due to the substitution of F, we can easily calculate the new Tc to be 198 K f r o m eq. (2.1), (2.2) and (2.8), not too far from the reported Tc of about 155 K in the F-substituted samples [ 14 ]. The assumption that IEB I remains unchanged however, might not be valid, because in a two dimensional hydrogen problem, the ground state is given by lEa[ =4R(I~*M¢)/E~ where R is the Rydberg constant. Even if % is not affected by F substitution, ~t* is clearly reduced as the heavy hole band in the unsubstituted [ 1-2-3 ] now drops significantly below the VB maximum. Thus this effect can reduce 3~x more than the decrease in G and /zF can increase 3ex. Apart from this comment, we also need to view A2~/G as an electrostatic energy density. This new electrostatic energy density is doubled from 0.06 to 0.13 eV. This translates to a two body excitonic binding of A~x= 0.33 eV, approaching G/2; while in the non-substituted case Aex= 0.25 eV, only ½ of G/2. This much larger excitonic binding could easily cause lattice instability. Thus, in our model, the F-substituted [ 1 - 2 - 3 ] compounds are necessarily unstable. This is consistent with many experimental reports that fluorinated- [ 1-2-3 ] compound is somewhat difficult to synthesize. We have seen from the above analysis that a change in gap G influences 3~x and 2~x as given by eq. (2.7) and (2.8). Thus, an applied pressure large enough to alter the band gap of these high T~ oxides will result in a change in T~. If G decreases under pressure as to be expected for a semiconductor-like material, then T~ will increase under pressure. This was observed experimentally for the [ 1-2-3 ] compound where a semiconductor-like band gap exists [ 17 ].

2.3. Infrared optical spectra Many optical absorption measurements in the infrared region have been performed especially on the [ 1-2-3 ] compound [ 18-27 ]. Results on the earlier measurements are less reliable because of poor sample quality. Generally speaking, excitonic-related features in the absorption spectra have been tenta-

19

tively identified. The result appears to be consistent with the idea of a two-component gap [ 18 ]. The absorption edge at 190 to 200 c m - ~can be consistently interpreted as due to 3e~, while the so called plasma edge absorption at 60 c m - 1, if interpreted as due to 3Bcs, gives 3Bcs/AT =0.28. Smaller than the value we deduced from the oxygen isotope data [10]. Therefore, it is possible that the isotope shift in Tc for [ 1-2-3 ] reported in ref. [ 10] may be slightly too large. The presence of single excitonic binding in high Tc materials should be reflected in the far infrared absorption spectra. The simultaneous excitation of a quasi-hole and quasi-electron pair in the a-b plane requires 2dex energy. In YBa2fu307_x, 2Aex----0.5 eV as given by eq. (2.7), which represents no angular averaging over the anisotropic excitonic gap Aex. Therefore, in a polycrystal [ 1-2-3 ] sample, we expect an infrared absorption at 0.5/v/~ eV~0.37 eV after angle-averaging which has been observed [22,23]. But in the epitaxial film, we should have 2Aex-0.5 eV as reported by Geserich et al. [25]. It has been suggested [28] that, if excitonic excitations exist in the high Tc oxides, one should be able to identify a series of transition lines analogous to the Rydberg series of the hydrogen atom which are common features in many excitonic insulators or semiconductors irrespective of whether these excitonic condensed. The Rydberg series is fitted by Ralston to Kamaras et al. [22] reflectance data, based on: hvnm=R( 1/n 2- 1/rn2). However, such a series, if it exists, will be mixed with strong phonon lines difficult to isolate. Furthermore, the frequency of transition between excitonic energy levels may not follow a H-atom Rydberg series because of the strong anisotropic nature of the excitonic states. Recently, we have solved the anisotropic hydrogen problem [ 29 ]; the result still agrees well with Ralston's analysis [ 28 ], except we have h u , m = R [ ( n - ½ ) - 2 - ( m - ½ ) -2] and for the Lymann series which in our case reproduces the broad small bump from 10 000 to 12 000 cm-~ in the reflectance curve observed by Kamaras et al. [22 ].

2.4. Specific heat and tunneling gap In both specific heat [30-33 ] and tunneling experiments [ 33 ], a typical BCS-like excitation gap ex-

20

K.W. Wong, W.Y. Ching / Theory of simultaneous excitonic-superconductivitycondensation H

plains rather well the observed experimental data, although there remains some disagreement as to whether A C / y T c gives the strong-coupling or weakcoupling limit of the BCS-like superconductivity [30,31]. The difficulties associated with the estimation on AC/;,Tc arise from the subtraction for the lattice contribution to C, as well as the proper estimation of the non-superconducting fraction in the experimental sample. Furthermore, in the [ 1 - 2 - 3 ] compound, the strong anisotropic nature of the crystal could lead to a sizeable deviation from the T 3 dependence of the lattice specific heat [ 34 ] obtained from an isotropic Debye oscillator model. In fact, the T dependence of the specific heat could be due to both the anisotropy of the crystal and may be the existence of an antiferromagnetic ordering in these materials. Overall, speci.fic heat measurements support the existence of a BCS-like excitation spectrum with a finite energy gap which agrees with tunneling experiments [35] and A C / y T c lies close to the BCS limit of 1.43. In a recent and more careful experiment on the YBa2Cu307_x single crystal, two specific heat gaps were reported near Tc [36]. The specific heat jump at Tc was fitted to the BCS calculated result with a clear indication that BCS theory provides a good agreement when thermal critical fluctuation of the Gaussian type in three dimensions is included. The second jump was not satisfactorily explained in that paper. According to EEM however, this jump can be explained by the Fermi-surface shift when excitons are condensed, causing not only a specific heat jump from normal to superconducting state but also a jump of the normal component of the specific heat itself due to a change of the Fermi momentum pf. We have calculated the second jump, A C , = - ( z ) (PfkTc) (½,]ex/er) which gives a value of 0.0064Tc [ m J / g K ] that agrees well quantitatively with the experimentally observed value. Details of this analysis have been reported earlier [ 37 ]. -

2.5. Hall effect and nuclear spin relaxation

It was pointed out in the introduction of paper I that the presence of electrons in the CB just above Tc is important to EEM. Without it, it is difficult for the excitonic phase to form, particularly since the CB is quite far from the Fermi surface of the unfilled VB.

In Lal.85Sro.15CuO4,the bottom of CB is only I eV above the Fermi surface, thus thermal excitation may provide electrons in CB at room temperature. However, in YBa2Cu307_x, the CB is almost 3 eV above the Fermi surface. Normally, it would not be possible to thermally excite electrons into CB unless some donor states are available below CB. From the one-electron normal state calculation [ 3 ], we notice that the bottom of CB is mainly derived from the 3z 2 - r 2 orbitals of Y 4d state. Such a state is highly polarized in the z direction of the crystal; therefore, we expect the presence of any n-type carriers in the normal phase be easily detected along the z direction. Recent Hall mobility experiment on single crystal of YBa2Cu307 confirms the presence of such carriers [38], thus giving strong evidence for the presence of electrons in the CB despite the 3 eV gap above the Fermi surface. Nuclear spin relaxation experiment can directly measure the electronic-pair binding energy. There is some indication from such experiments [39] that the electronic pair binding is a two-component one, unlike conventional BCS theory, which give a onecomponent binding energy gap. EEM theory is a twocomponent binding gap theory which satisfies the Pythagorean sum rule. Since the nuclear spin relaxation mechanism coupled to the excitonic state and the BCS state could be quite different, we would expect such an experiment to be able to distinguish these two components. I f a measurement of the excitonic density below Tc can be carried out, we should be able to investigate the temperature dependence of the excitonic component of the gap to verify clearly that it is a simultaneous condensation. Positron annihilation experiment can provide one of such indirect measurements. 2.6. Position annihilation

Positron annihilation technique has been applied to both [ 2 - 1 - 4 ] and [ 1-2-3 ] superconductors [ 4046], and very recently to the Tl-related superconductors [47]. The major aim is to detect the presence of a Fermi surface and to see if there is an electronic structure change as the temperature is varied above and below To. The main conclusion from these experiments is that there exists an electronic structure change at To, supporting the general concept of

K. IV. Wong, W. Y. Ching / Theory of simultaneous excitonic-superconductivity condensation II

EEM which suggests a simultaneous excitonic as well as superconductivity condensation at To. The observed difference in the positron lifetime above and below To, which was not observed in the low T~ metallic superconductors, can be qualitatively explained as follows: Other than the impurity related annihilation, the positrons annihilate mainly with free electrons in the CB, not the electrons below the Fermi surface which are in general very heavy. According to EEM, these electrons in the CB can be collectively bounded into positively charged excitons, which in turn form ODLRO Cooper pairs below To. Therefore, as temperature is further reduced, there will be less and less electrons in the CB that are not in an excitonic-Cooper pair state. Thus we expect an increase in the positron lifetime. Quantitatively, we can analyze as follows: If the Hamiltonian of the interaction between the CB electrons and the positron fields is H, then the number of transition processes per unit time is: 2n//~l f q/~H~u~,dzl2p,

(2.9)

where p is the number of CB electrons available and ~uis the positron wave function. The total number of CB electrons should be given by p~= 3N(Aef/er), if we assume a spherical Fermi distribution of intrinsic holes, where N is the intrinsic number of hole and AEr is the shift of the Fermi surface due to condensation of excitons. Thus assuming that A¢~( T ) = J~x (0) ( 1 - T~ Tc) ~/2 as given by a second order phase transition and A~foCA~x(0), we get poc 3(N/~f),]~x(0) [ 1 - ( 1 - T / T c ) W 2 ] . Since the positron lifetime is inversely proportional to the transition number, we see clearly that the positron lifetime increases as T decreases. In fact if we assume N/~r to be a constant for all the high Tc materials found to date, we can deduce the increase of lifetime from z~ to ro by the following equation: 1

1

- - - - - =A 71~x(0). Zc ro

(2.10)

For cases of very high Tc, the BCS component hecomes negligible and we have A~(0)--~AT(0), the right hand side ofeq. (2.10) then being proportional to T~. In fig. 3, we plot ( l / r e - - 1/Zo) versus T~ using the re, ~o and T~ values from the positron experiments

21

/ / / M.I

f

o:

CO

o

/

,~.,6

1 / / f / /

I

/ o

/

7 / / / i

0

=

i

i

i

50

,

,

i

i

t

100

i

i

i

i

i

~ T

150-

c

Fig. 3. Plot of T¢ vs ( 1/To- 1/To) for different high Tc superconductors. []: Lal.ssSro.15CuO4 [44]; *: YBa2Cu307 [45]; o: TI2Ca2Ba2Cu3010.3+ a [47 ]. Note the data for LaLssSr0.~sCuO4 are expected to be above the dashed line because of the relatively low To. For detailed explanation, see text.

for Y B a 2 C u 3 0 7 ( T = 8 4 K) [45], Lal.85Sro.15CuO4 ( T = 3 3 K) [44] and T12.2Ca2Ba2Cu3Olo.3+ad ( T = 124 K) [47]. It can be seen that the approximate linear relationship of eq. (2.10) can be established. The deviation from the straight line for the Lal ssSro.15CuO4 data is due to the relatively important Aacs component in this compound which was neglected in replacing Aexby AT in eq. (2.10 ). If more experimental data on well-characterized samples become available and they all follow the same straight line, then we might have found a way to verify EEM. On the other hand, in a defective polycrystalline sample or an O deficient sample, positrons will naturally perpetuate at the defects. Electron tunneling through defects tends to break up Cooper pairing, thus the number of free electrons available in a very defective polycrystal should simply be proportional to 71ex(T), making the positron lifetime decreasing at lower T, reversing the trend of the perfect single crystalline case. This seems to explain that in the earlier measurement [40] on powdered sample of YBa2Cu3OT_x, the positron lifetime actually decreases as T is decreased below To.

2. 7. Ultrasound velocity and sound attenuation The longitudinal sound velocity of a fluid is given b y B/pm, where B is the bulk modulus and Pm is the

mass density. In a usual BCS superconductor, neither B or Pm should change across Tc, because the su-

22

K. W. Wong, W. Y. Ching / Theory of simultaneous excitonic-superconductivity condensation H

perconducting transition is not accompanied by a lattice transition or a change in carrier density. Thus we do not expect a significant longitudinal sound velocity change across To A careful analysis based on conventional BCS theory gives a slight softening for the sound velocity in the range of 10 ppm below Tc [48 ]. Surprisingly, ultrasound measurements [ 49 ] for the high Tc oxides indicate a hardening of longitudinal sound velocity to 103 ppm. Such a significant variation can only come about if either the bulk modulus B is hardened, or the mass density Pm of the carrier is changed, or both. EEM theory indicates that both of these changes can happen when an excitonic phase is present. From conservation of net charge, we see that the Fermi energy must expand to account for the electrons in the CB as given by eq. (2.24) of paper I:

~r=[z- G/2 + A~x/2fi.

(2.11 )

The term A~x/2[t can be viewed both as an increase of holes and as a change to the band gap G, AG=-dZ~x/[z. For YBazCu307_x, AG is estimated using Aex and/2 values from isotropic analysis to be about - 0.024 eV. Such a change in AG corresponds to a very large internal pressure change on the lattice and should result in a hardening of the bulk modulus which could be accompanied by lattice parameter anomalies if the lattice is easily deformed. In fact lattice anomaly of this kind has been observed in some ~crystals experimentally [ 50 ]. Due to the very nature of lattice deformability, we should expect lattice parameter anomalies to be highly sample dependent. Samples with more twinning are more likely to show such lattice anomalies near T~. This shift in the Fermi energy Ef also implies a corresponding total energy and mass density change of the carriers if the single particle mass is constant. Such a sound velocity anomaly should occur only near T~. For lower temperatures, EEM theory is also similar to BCS theory in that it has a finite energy gap in the excitation spectrum, thus we expect the sound attenuation to behave quite similar to that given by BCS theory, namely an exponential type T dependence below T~ [48].

2.8. Meissner effect and critical current One of the necessary criteria for superconductivity is the presence of the Meisser effect. Following the

EEM theory [ 1 ] and using a self-consistent field approach [ 51 ], it is not difficult to derive the total current under the influence of an external magnetic field provided by a vector potential A: Jtotal ( O ) =Jdia -'l'-Jpara

-

2e 2 A(O) ~ [2Bcs/E~]tU~-V~,]2X~r~, mc

(2.12) where the matrix elements Us, Vg Xk, Yk are the coefficients of two successive canonical transformations in the two-band model EEM theory discussed in paper I. Rewriting/totaj(0) in terms o f ~k and Ek which are given by [ 1 ]:

~,,,= eb-~.___~2 + 21~ j ( k _ q ) [ U e

V~,l

(2.13)

and E k = (~a"~-~b) "~ (~2"~I-A2x)I/2 2

(2.14)

we have 2e 2 ~,ota, ( 0 ) = - - - A ( O ) mc

E (2~cs/Ek) k

X ( E 2 +Aacs) -2 -1/2 ~,(~,-2 +Zl~x) -1/2

(2.15)

This current,/,ota~ (0) can again be expressed in terms of the BCS form: Jto,al (0) =./Bcs (0) + A,/(0),

(2.16)

where /acs(0)=-

2e2A(0) ~ (JBcs/E,) -2 ( E k2+ d a2c s ) -1/2 mc

k

(2.17) and

2e2A ( O) Z, ['J2cs/E*]

Ag(O) = -~c

X [E~ + j 2 c s ] l / 2 [ 1 - (1 +A 2)'/2] , where

2k + /fex/~k .

(2.18)

For the YBa2Cu307_x, we have previously obtained from the discussion on isotope effect in section 2.1, Aex=0.25 eV and from the band result, G / 2 = 0 . 5 eV.

K. W. Wong, IV..Y. Ching / Theory of simultaneous excitonic-superconductivity condensation H

The upper limit of AJ ( 0 )/JBcs (0) can therefore be deduced by taking the limit 27k= ( V / ( 2 n ) 3 ) fd3k and setting ~k= G/2 in the integrand of eq. (2.18), we get: AJ ( 0 ) = --0-1JBCS ( 0 ) .

(2.19)

It should be noted that the mass m in JBcs which gives the diamagnetic current as given by eq. (2.16) is anisotropic. Therefore/Bcs and the Meissner effect is also anisotropic. Hence for YBa2Cu307_x, we have a reduction in Meissner effect of approximately 10%. For La2_,SrxCuO4, the Meissner effect reduction will be smaller. The critical current Jc (0) under an imposed electric field can also be obtained from the time-dependent self-consistent field method. The result is more complicated [52]. However, we can qualitatively conclude from the flux discussion in paper I, (and of course from the self-consistent field result) that the calculated critical current is only reduced by a factor proportional to the charge reduction tf of the Cooper pairs of quasi-holes. Since r/2~ 10-3, the critical current is not significantly reduced for both La2_ ,SrxCuO4 and YBa2Cu3Ov_x systems. The above analysis on both Meissner effect and critical current agrees quite well with experiments [32,53-55], as was seen by the fact that Jc approaches 106 A / c m 2 at T = 0 for good epitaxial films.

2. 9. Antiferromagnetism and oxygen deficiency An antiferromagnetic phase has been observed in both L a - X - C u - O and Y - B a - C u - O systems [ 56,57 ]. ¢.1 This phase is present in both superconducting and non-superconducting crystallographic phases. It is generally believed that Cu in these systems is responsible for the antiferromagnetic phase. This physical feature has led to many model theories [ 58 ] based principally on the connection between antiferromagnetism and superconductivity. Recent experiments on oxygen deficiency in the Y - B a - C u - O system, clearly demonstrate the role played by oxygen deficiency on the onset of the antiferromagnetic phase [ 57 ]. It has also been established that the most likely site of O vacancy is the O~ along the C u - O chain [59]. The removal of O1 creates a pure Cu plane, thus favors a two-dimensional antiferromagnetic ordering of Cu magnetic moments. Indeed when x = 0 . 5 in YBa2CuaO7_x, superconductivity is

23

quenched and we have an onset of the antiferr0magnetic phase [58]. This interpretation is reinforced by the similar observation in the case of Fe-substitution for Cu in the C u - O chain [60]. Again, both the quenching of superconductivity and the onset of antiferromagnetism shift towards lower x values. It was also reported [61 ] that the total replacement of Y by Gd in the [ 1-2-3 ] compound also leads to an antiferromagnetic phase below 2 K. However, that phase coexists with superconductivity with no evidence in specific heat anomaly. The interrelationship between superconductivity and antiferromagnetism is not a part of the EEM theory; rather, it seems to suggest that antiferromagnetism, when present in the C u - O plane, quenches superconductivity. One may be able to explain this feature by studying the spin-spin interaction between the Cu ions and the excitonic states. Since the excitonic states are plane polarized and since it is degenerate between the S state and P state, an antiferromagnetic order in the C u - O plane will lower the energy of the P excitonic state leading to a parallel spin phonon induced coupling which thus might not be favorable to superconductivity. To see this more closely, let us recall [ 1 ] that the excitonic equation is given by;

I - ~-~, h V2+G_Vc(r)]q/(r, sl s2) =E~J(r, sl, s2 ) ,

(2.20) (2.20)

where s I and s 2 are the spins of the electron in CB and hole in VB respectively. Ignoring spin-spin and spin-orbit interactions, we see that E is degenerate in spin s=sl +s2 variable. However, since these excitons are totally non-localized in a simultaneous excitonic-superconductivity condensed phase, the coupling of the total spin to the ionic core spin 8i cannot be neglected, particularly if the ionic core spin ordered in a ferro- or antiferromagnetic phase. Thus it is important to add to the two-band Hamiltonian [ 1 ] a Jss's~ term. Whether Js is positive or negative, we have a reduction of energy for the s = l state. Hence the leading term in the effective phonon-induced quasi-hole-quasi-hole interaction is not the BCS spinopposite pair, but rather we get the parallel spin pair term.

24 Hph . . . . --

K. W. Wong, W. Y. Ching / Theory of sirnultaneous excitonic-superconductivity condensation H --

~ J(q, k, k'

+

+

) OLk + qs OLk , -- q s OLk, s OLk s •

(2.21) I f J is symmetric, then Hp%ono. is exactly zero. However, i f J is antisymmetric, then H~h. . . . is similar to the heavy fermion superconductor, which would substantially reduce the phonon mediated gap component. Recent discovery of superconductivity near 30 K in the cubic [62] Bao.6Ko.4BiO4 which contains no magnetic atoms seems to indicate that magnetism plays no or little role in high Tc superconductivity in perovskite oxides. On the other hand isotope effect in Bao.6Ko.4BiO4 remains too small [ 13 ] to argue for the conyentional BCS mechanism as was clearly explained in section 2.1. 2. I0. Flux quantization

One consequence of EEM theory that has not yet been fully investigated experimentally is the charge reduction in flux quantization as discussed in section 3 of paper I. There, we have obtained a charge reduction o f S e / 2 e = 2q 2. This quantity, in the meanfield approximation, is given by q=A~x/4flv. Using the value ofd~x = 0.25 eV, and 9F = 2.50 eV obtained earlier for YBa2Cu307_x, and A~x=0.084 eV, ~'~F = 1 eV for La2_ ,-Sr,.CuO4, we obtain 5 e / 2 e = 0.135% and 0.086% for YBa2Cu3OT_~- and La2_xSrxCuO4 respectively in the isotropic limit. This charge reduction is quite different from that arising from the Chern-Simon gauge in a 2-dimensional quantum field problem [ 63 ]. In that case, which is also known as the "anion" theory, it will give fractional charge even in the normal phase [64 ]. In the EEM theory, this charge reduction should vanish at T~ since ACx vanishes above T~. A more detailed discussion of charge reduction predicted by these two different theories is given elsewhere [ 63 ]. In an earlier AC Josephson measurement [67], a charge quantization of 95% of 2e with an error of + 5% was indicated. While another experiment on quantized flux pattern [68] gave a strong bias towards a charge reduction, larger than that normally accountable due to large magnetic field. However, in a recent experiment on a large-bridge Josephson device, it was reported [65] that a small charge reduction of 1.5 + 0.5% over ten regular-spaced Shapiro steps in a [ 1-2-3] sample was detected, This result gives a net reduction exceeding the experi-

mental error though it is larger than our estimation. However, the difference is within error caused by the anisotropy of A¢x which we have neglected in estimating Be. An even more careful flux quantization measurement also using a large bridge device has recently been reported [66]. In that work, an average over different samples under different temperatures and fields was performed, giving a total average reduction of only 10-5 and the author concluded that there is no charge reduction. Nonetheless, this small statistically averaged reduction is not incompatible with our prediction, since some samples' conditions were under high field values and the temperatures are too close to Tc. A more careful AC Josephson experiment at low temperature and low field with accuracy of at least 10 -4 is certainly called for at this time. 2.11. Photoemission

Many photoemission experiments have been performed on the [ 2 - 1 - 4 ] and [ 1-2-3 ] compounds. The major motivation for such experiment is to see if a sharp Fermi surface can be detected and if the measured photoemission spectra are consistent with the one-electron band structure calculation on these materials. The photoemission results are usually less conclusive because the technique is very sensitive to the surface condition of the sample which is turn may be related to the oxygen content. Most measurements were on polycrystalline samples but singlecrystal data began to appear. Early measurement on sintered sample of the [ 12-3 ] compound using angle-integrated photoemission spectroscopy [70-83] produced data in disagreement with the band structure calculation and it was concluded that electrons in the high Tc oxides are strongly correlated and the one-electron band picture breaks down for such a system. This has led to several theoretical models based mainly on magnetic interactions [ 58 ]. Recently however, more accurate angle-resolved photoemission experiments [ 74,75 ] on epitaxially-grown single crystal thin film of the [ 1-2-3 ] compound has been carried out. Not only a clear Fermi-edge has been observed, but also the dispersive nature of the VB states is detected in strong support of the band picture for the high Tc oxides. The EEM theory starts with a two-band description based on the validity of the one-electron

K. W. Wong, W. Y. Ching / Theory of simultaneous excitonic-superconductivity condensation H

band picture for the normal state in these superconducting oxides. Therefore the validity of the single particle description is of paramount importance to the EEM theory. Although there are still questions about the measured peak positions relative to the Fermi energy being closer (by 1-2 eV) than those obtained by the local density calculation, this should not be regarded as the evidence for the failure of the band picture. As pointed out by the authors of ref. [75], such discrepancies have also been observed in transition metals and even in simple metals. Although EEM theory predicts a condensation of charge non-neutral excitons which will result in a small shift of Fermi level away from the top of VB, this shift of the order of Jcx~0.012 eV in the [ 1 - 2 - 3 ] compound and may be too small to be detected by photoemission experiment. Although the most current results from photoemission experiment appear to be compatible with EEM theory, more accurate measurements on well characterized samples are clearly needed.

3. Stoichiometric interpretations of high Tc oxides In the previous section, we have shown that many experimental observations are consistent with EEM theory. The basic requirements for the EEM theory to be operational are: (a) the material must be able to form excitons, and (b) there must be intrinsic holes in the VB. It is well known that CuO is a p-type conductor [76 ] and Cu20 is a typical semiconductor with well established excitonic states [ 77 ]. These facts are confirmed by our recent first-principles calculations of the electronic structures of Cu20 and CuO crystals [78]. Thus it would appear that these two oxides must play a major role in the stoichiometry of the Cu-containing high Tc materials. In this section we show that the superconducting or the nonsuperconducting nature of the ceramic oxides can be analyzed in terms of its chemical subsystem of stable oxides. Amazingly, it turns out that our simple stoichiometric interpretation indicates that for Cu-based superconducting oxides, both CuO and Cu20 phases should be present, while for the non-superconducting ones, either one will be missing. We shall apply our simple stoichiometric interpretation to the following superconducting systems and the related non-

25

superconducting phases: (i) YBa2Cu307_x; (ii) La2_xSrxCuO4, (iii) Bi2Can_lSr2CunO2n+4 based system; (iv) T12Can_iBa2CunO2n+4 and TI~Ca,_ ~Ba2Cu,O2,+3 systems; and (v) non-copper containing systems. According to EEM, the basic requirement for a material to be potentially superconducting at high temperature is the containment of two stable components, one is excitonic and the other a p-type conductor. As such, the materials are not necessarily limited to oxides since there are whole classes of stable sulfides, halides and nitrides, etc. which may be excitonic or having intrinsic hole population in their normal state band structures. We shall return to this point at the end of this section. 3. I. Y - B a - C u - O system

The general structure of YBa2Cu307 compound is considered as being composed of the three stacking cubes, with the centers of the top and bottom cube being occupied by a Ba atom, and the middle cube occupied by a Y atom [79 ]. It is also known that both Y and Ba can be replaced by other rare earth elements and the superconducting properties are only marginally changed [77]. Therefore, it is safe to conclude that the rare-earth elements probably do not play an important role, except on the formation of the orthorhombic structure. Y and Ba both form stable oxides. For Y, we only have one stable oxide in Y203. Therefore, if such a subsystem exists in the crystal, it must be ½ of a Y203 in the middle cube. We note that inside the middle cube, there are eight corner Cu atoms and four O in between the Cu on both the upper and lower planes accounting for a total of two O and one Cu. Since Y takes away 1 . 5 0 to form ½Y203, this implies that what remains in the middle cube is ½ of a Cu20. Following a similar argument, let us turn to either the top or bottom cube. Ba can form two stable oxides, namely BaO and BaO2. There are a total of 2.5 O in each cube with eight corner Cu atoms. Since it is known that the [ 1-2-3] system has p-type carriers, it is reasonable to expect that one CuO, which has intrinsic holes [78] is in the top and bottom cubes. The remaining chemical subsystem is easily deduced to be ½ [ BaO + BaO2 ]. The following stoichiometric interpretation for YBa2fu307 is then established:

26

K. W. Wong, W. Y. Ching / Theory of simultaneous excitonic-superconductivity condensation H

YBa2Cu307 -- - ½[BaO+BaO2] + C u O upper cube d- ½[Y203 + C u 2 0 ] middle cube

q_ ½[BaO+BaO2] + C u O lower Cube

presence of CuO and Cu20 is a necessary but not a sufficient condition. A final electronic band calculation is needed to determine if excitonic condensation is possible. 3.2. Z a 2 _ a S r x C U O 4 system

Such a stoichiometric interpretation implies that we have two p-type conductors CuO sandwiching a semiconductor Cu20 and can also explain the anisotropy in the conductivity of the crystal in the normal phase [38]. Furthermore, if oxygen is randomly removed from the top and bottom cubes, we will destroy the CuO subsystem, thus reducing the intrinsic holes in the band. This may lead to two effects: (a) In the normal phase, the planar conductivity will be reduced and the system will behave more like a semiconductor than a metal [59,80]. (b) In the superconducting phase, Jex will be increased, due to the reduction of fry. However, Aacs will be reduced for the same reason. The end results are: ( 1 ) a broader transition; (2) there may be a change in To; (3) certainly a reduced Meissner effect; and (4) if a quantized flux measurement is made, a larger reduction on the charge. It is equally revealing to carry out simple stoichiometric interpretations for two of the well-known non-superconducting Y-Ba-Cu-O phases, YBa2Cu306 and YBaCuOs, the semiconducting green phase. They can be decomposed as follows: YBa2Cu306 = ½Y203 + B a O + B a O 2 + 3Cu20 and YBa2 CuOs

=/Y203

+BaO2 + ½Cu20.

We can see clearly that both of these non-superconducting oxides might contain no p-type CuO and become semiconductor-like. It may be argued that YBa2Cu306 can be equally well decomposed as: YBa2Cu306 = ½Y203 + 2BaO2 + ½ C u O + C u 2 0 . However, they still have to satisfy the conditions for simultaneous excitonic-superconductivity condensation, that is, the formation of excitonic states can lower the ground state energy. In YBa2Cu306, we do find unfilled hole states in VB, but the heavy hole band is no longer present. This results in the reduction of excitonic effective mass, reducing the condensation possibility. Therefore, we may say that the

A similar analysis can be applied to the [ 2 - 1 - 4 ] compound which has a composition of L a / S r - O planes sandwiches by C u - O planes. Hence LaECuO4 can be interpreted as La203+CuO which does not contain Cu20. Thus La2CuO4 itself without Sr/Ba "dopant" should not be a high Tc superconductor. On the other hand, we can create a stacking pattern similar to [ 1-2-3 ], with a LaSrCuO4 subunit cell sandwiched by two La2CuO4 subunit cells, making a supercell LasSrCu3Ol2, if the center subunit cell LaSrCuO4 is decomposed into ½[ La203 + 2SRO2+ Cu20 ]. Dividing this supercell by 3, we get La2_xSrxCuO4 with x = ] . Since we can sandwich LaSrCuO4 with more La2CuO4 layers, x = 0 . 3 3 presents an upper limit for La2CuO4, a fact quite consistent with experimental observation [27]. Again let us point out that the presence of Cu20 in the LaSrCuO4 cell is due to the choice of SrO2 rather than ½[SrO+SrO2] as in the YBa2Cu307 case considered earlier. Of course these different choices cannot be easily fixed without actually considering the crystal geometry involved. Thus the stoichiometric assignment is not quite unique in many cases. However, it is still highly useful and instructive as we will see from analyzing some non-superconducting Labased oxides. Recently, other related L a - S r - C u - O systems were synthesized and their structures identified [ 81 ]. It was found that La2SrCu206, LasSrCu6Ol5 and LaaBaCusO~3 are not superconducting. We find the following possible compositional interpretations: (a) La2SrCu206: Three possible interpretations: (i) [La203+SrO+2CuO]; (ii) [La203 + SrO2 + C u 2 0 ] ; (iii) [La203+ ½ ( S r O + S r O 2 + 2 C u O + C u 2 0 ) ]. Only (iii) contains both CuO and Cu20 and is therefore potentially superconducting. A sharp resistivity-drop signature was actually observed [ 81 ]. (b) LasSrfu6015: Two possible interpretations are possible: (i) ½[ 5La203 + SrO + SrO2 + 12CuO]; (ii)

K. W. Wong, W. Y. Ching / Theory of simultaneous excitonic-superconductivity condensation H

½[ 5La203 + 2SRO2 + 10CuO + Cu20 ] which does not have the SrO and SrO2 mixing. Only (ii) has a change of being superconducting. If O deficiency is present, it could contain more Cu20. Again, a resistivity-drop signature was observed in this compound [ 81 ]. (c) La4SrCusOt 3: only one unique composition can be identified: (i) [2La203+SrO2+5CuO] which cannot have high Tc superconductivity because Cu20 is always missing. No resistivity drop signature was observed in this phase. Our conclusion in this regard is that both La2SrCu206 and LasSrCu6Ol5 might have some hidden high Tc phases. However, both LasSrCu6Ol5 and La2SrCu206 are oxygen poor per copper-wise as compared to La2CuO4, thus these systems may favor antiferromagnetism instead of superconductivity as discussed in section 2.9. 3. 3. B i - C a - S r - C u - O

system

Recently, a new family of B i - C a - S r - C u - O based high Tc oxide has been reported [82-85]. Its crystal structure has been determined [86-88]. Let us assume the structure and composition as determined by Sunshine et al. [89]. Which gives a stoichiometric composition of Bi2+,Sr2Cat _xCu2Os. The Ca/Bi centered-cube has a close resemblance to the Y-centered cube in the [ 1-2-3] system. In our stoichiometric interpretation for [ 1-2-3 ] earlier, the Y centered cube is considered to be ½ [ Y 2 0 3 + C u 2 0 ] . The presence of Cu20 was emphasized as causing the occurrence of excitonic states. In this new oxide, we note that Ca only forms one stable oxide, namely CaO. Thus for the case of x = 0 , this cube appears to be composed of CaO + CuO, i.e. Cu20 is absent. But since the structure of this new oxide is so similar to the [ 1-2-3 ] it would be hard to believe that y3+ can be replaced by Ca 2+ alone. On the other hand ifBi 3÷ is present as a "dopant" for Ca 2+, then it is possible to interpret this Ca/Bi cube as ( 1 - x ) C a O + 0.5xBi203 + ( 1 - x ) C u O + 0.5xCu20, for x < 1, the presence of Cu20 is analogous to ½ [Y203+Cu2O] in the Y cube. This interpretation gives the Bi "dopant" an important physical meaning, namely, if x = 0 , no Cu20 exists, and therefore no excitonic states. Furthermore when x=0.5, this cube becomes ½[BiCaCu204] and sandwiching ½[SrO+SrO2] in the middle could form the Bi-

27

CaSrCu2Os+x, the 1112 phase. The rest of the Bi2+xCa~ _xSrzCu208 system could be interpreted as Sr-O planes sandwiching two Bi-O planes in between the C u - O planes. The chemical composition is then [ B i 2 0 3 + 2 S r O + C u O ] . Hence,

Bi2+xSr2Cal_xCu208 -

( 1 - x ) C a O + 0.5xBi203 + ( 1 - x ) C u O + 0.5xCu20 Bi-doped Ca cube 4

[Bi203 + 2 S r O + C u O ] Bi/Sr cube

Again, we have the CuO composition for the hole states in VB. The 2223 phase only involves stacking another central Ca/Bi cube on the 2122 phase. Thus the amount of Cu20 in the unit cell is doubled. Other structures in the Bi based superconductors are similar to the Tl based superconductors which we will analyse below. 3.4. Tl2Can_tBa2CunO2(n+ 2) a n d Tl l Can_ lBa2CunO2n + 3 series

For the recently discovered Tl-related series [ 90,91 ], let us first consider Tl2Can_ iBa2Cu~O2~n+ 2~. We note that for n = 2, this compound is similar to the Bi2CaSr2Cu208 that has been discussed above. The importance of Ca/T1 mixing is vital, and is in fact observed experimentally [92]. For n = 1, we see that the perfect system T12Ba2CuO6 cannot be decomposed into both CuO and Cu20 subsystems since T12Ba2CuO6=T1203+2BaO+CuO unless we have some O or Ba deficiency. In order to satisfy the stoichiometric rules established earlier and be potentially superconducting, the proper composition should be T12Ba~.sCuO6 or T12Ba2CuOs.7s. Again this is borne out experimentally and the decomposition is T12Bal.sCuO6 = T1203 + 3 [BaO+BaO2] + ½[ C u O + ½Cu20], T12Ba2CuO6_0.25 = T1203 + 2 B a O + ½[ C u O + ½Cu20] . But for n = 3 , this series needs to dopant or deficiency to be superconducting. The composition is

28

K. 14: Wong. W. Y. Ching / Theory of simultaneous excitonic-superconductivity condensation H

Tl2CaeBa2Cu3Oio = T1203 + 2CaO + BaO + BaO2 + CuO + C u 2 0 . Therefore, the [ 2 - 2 - 2 - 3 ] compound should also be the most stable one in the series. From these analysis, we see that a general decomposition for this series is not applicable. The other newly discovered series [93] TI~Ca,,_ iBa2Cu,,O2,~+3is much simpler to interpret and we do not have to invoke the presence of dopant or deficiency. It is easily decomposed into Tll Ca,,_ i Ba2Cu,,O2,,+ 3 = ½(T1203) + ( n - 1 ) C a O + BaO+ BaO2 + (n- l)CuO+½Cu20. We see clearly the CuO conducting planes are present in the ( n - 1 )Ca cubes and the Ti/Ba cubes provide the Cu20 excitons. This clearly implies T1BazCuO5 is nonsuperconducting, since CuO will be missing. Lastly there is also a cubic superconducting compound Ba i _ ,T1 ~.CuO2+~.reported recently [ 94 ]. This material should decompose into Bal_, TI,-CuO2 +,. = ½( 1 - x ) [BaO+ BaO2 + 2 C u O ] + ½x [T1203 + Cu20] provided y = ½( 1 - x ) . The presence of dopant and excess of oxygen should make such a structure inherently unstable. 3.5. Non-copper-containing oxide systems

The non-copper high Tc material Baj_xK,.BiO3_a is quite amazing [62]. We notice that like copper, potassium also has two stable oxides; namely the common excitonic ionic oxide K20 and the less common low temperature p-type conducting oxide KO> Similar to our previous treatment on copperbased oxides, we can decompose this potassium oxide as follows: Ba~_ ,K.,-BiO3_ ,. = ½( 1 - x ) [BaO + BaO2 ] +0.5Bi203 + ½x] ½K20+ KO2] which leads to y = x / 4 . Hence, for the reported Bao.6Ko.4BiO3 compound to O deficiency is predicted to be about 0. I. This cubic system needs both

dopant and deficiency to be superconducting and is very likely not stable despite being cubic. This also means this superconductor is difficult to synthesize because either a loss or gain of O will destroy superconductivity by eliminating either K20 or KO2. The lead oxide compound BaPbo.75Bio.2503 with a Tc of about 12 K was discovered [95] as early as 1975. It can be decomposed nicely into: BaPbo.Ts Bio.2sO3 = [ Bi2 O3 ] + 0.5 [ BaO + BaO2 ] + ] [ PbO2 + PbO ] . Here again, we note the presence of both the excitonic ionic oxide PbO and the conducting oxide PbO2. This cubic system has no need for oxygen deficiency and should be more stable than the Bal_ ,-K,-BiO3_v compound. Our analysis above which is based on the structure properties of CuO and CuO2 is over simplified. Obviously, there are many other oxides and chalcogenides that have either a p-type conductivity character [96,97] or exhibit excitonic states [98]. Thus we should extend our analysis to imply in a much more general sense that what is required in EEM stoichiometry is the simultaneous presence of two stable compounds with similar electronic characteristics to CuO and Cu20. The (Ba~_xK~)BiO3 system discussed in section 3.5 which does not contain Cu is an example that high Tc superconductivity can be present in systems beyond those containing both Cu and O. The apparent successful stoichiometric interpretation of the existing superconducting oxides based on the EEM theory leads us to believe that a systematic search for other non-conventional superconductors may he possible. We suggest below a simple step to step recipe for such a systematic search which will at least narrow down the potential candidates for high Tc superconductors. Step 1: We search for metallic oxides, (also halides, sulfides, nitrides, chalcogenides, etc.) that contain two stable compounds in the same category of which one is excitonic and the other is p-type conducting. Step 2: The general crystal structures of these two compounds in the same category should be reasonably compatible in the crystalline cell dimensions. Step 3: We search for a perovskite with base crys-

K. W. Wong, IV. Y. Ching / Theory of simultaneous excitonic-superconductivity condensation H

tal parameters compatible with the pair of compounds tentatively identified in the previous steps. Step 4: We now try to combine these compounds by mixing with certain rare earth oxides or CaO, etc) that can increase the possibility of stacking them together in planar geometry. Step 5: We apply our compositional analysis to obtain the optimal proportionality mixing before heating. We should always heat under a rich oxygen (or chlorine, fluorine, etc. ) flow. Step 6: Band calculations on hypothetical structures o f potentially superconducting compounds should be carried out to check if the one-electron band structures are favorable to the EEM mechanism. Under these conjectures, we suspect that the conducting oxide VO and the ionic oxide VO2 together with BaO, BaO2 and CaO might be a possible mixture for a new superconductor family. Let us consider the stacking o f two types of cubes: ( 1 ) A cube with Ca at the center, 8 corner V atoms with O~ between the V's on the top and bottom planes. This cube has the composition C a O + V O . (2) A cube with Ba at the center, 8 corner V atoms with O, between all the V's. This cube has the composition ½[ BaO + BaO2 + VO + VO2 ] or BaO + VO2 or BaO2 + VO. The last choice of composition will make this c o m p o u n d non-superconducting. The general formula for our suggested possible new superconducting family is CanBamV,+mO2n+3,,.

29

must have an intrinsic hole state near the top of the VB. Based on these ideas, systematic search for other high Tc compounds similar to our suggestion on Vbased series is facilitated. Many elements posses multivalency characteristic in compounds such a Ba or Sr in these high Tc oxides. Thus unique identification of a certain subsystem-decomposition is not possible. Our stoichiometric decomposition can only be used as a quick check to see if there is possibility of the containment in the subsystem of the needed excitonic a n d / o r conducting oxides. Sometimes, it could be slightly modified with a dopant or oxygen deficiency to create the needed containment. Thus we must caution the readers that our naive stoichiometric interpretation on EEM is a useful guide, but experimental "trial and error" on material mixing remains the most essential element of making new high Tc superconducting compounds.

Acknowledgement The authors have benefited from many discussions with their colleagues, especially Professors Y.C. Jean, Y.H. Kao, F.T. Chan, and J.P. Ralston. This work is supported by U.S. Department of Energy Grant No. DE-FG02-84ER45170.

References 4. Conclusion The basic theory of EEM was presented in paper I. In this follow-up paper, we have discussed rather extensively the experimental evidences in support of EEM theory. To our knowledge, none o f the other currently proposed theories can explain such a diversified array of experimental measurements. More accurate experiments on better prepared samples and on other newly discovered systems will certainly provide additional information to check out the validity of the EEM theory. It is also rather amazing that the complicated crystal structure and composition of the high Tc compounds can be naively interpreted stoichiometrically as the stacking of chemical subsystems. EEM theory contends that one of the subsystems must be an excitonic insulator and the other

[1] K.W. Wong and W.Y. Ching, this issue, Physica C 158 (1989) 1. [ 2 ] J. Bardeen, L.N. Cooper and J.R. Schrieffer, Phys. Rev. 108 (1957) 1175. For a detailed discussion on the relationship between BCS theory and high To see W.A. Little, Science 242 (1988) 1390. [3] W.Y. Ching, Y. Xu, G.-L. Zhao, K.W. Wong and F. Zandiehnadem, Phys. Rev. Lett. 59 (1987) 1333. [4] G.-L. Zhao, Y.N. Xu, W.Y. Ching and K.W. Wong, Phys. Rev. B 36 (1987) 7203. [5] Y.-N. Xu, W.Y. Ching and K.W. Wong, Phys. Rev. B 37 (1988) 9773; W.Y. Ching, Y.-N. Xu and K.W. Wong, Symposium AA, High Temperature Superconductivity, MRS Fall Meeting, Boston, No. 30 ( 1987). [6] W.Y. Ching, G.-L. Zhao, Y.-N. Xu and K.W. Wong, to appear in J. Mod. Phys. B. [7] W.Y. Ching, S. Weng and K.W. Wong, to be published.

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