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Indirect exchange mechanism of high-Tc superconductivity Pressure gradients of Tc in hole-doped and electron-doped superconductors
PhysicaC 173 (1991) 409-413 North-Holland
Indirect exchange mechanism of high-T, superconductivity Pressure
gradients
of
T,in
hole-doped
and
electron-doped
superconductors
Laurens Jansen a, Leena Chandran b and Ruud Block ’ a c/o Theoretische Physik, ETH-Hiinggerberg, CH-8093 Ziirich, Switzerland b Institut ftir Quantenelekronik, ETH-Hiinggerberg, CH-8093 Ziirich, Switzerland ’ J.H. van? HoffInstitute, University ofAmsterdam, 1OI8 WVAmsterdam, The Netherlands
Received 6 November 1990 Revised manuscript received I2 December 1990
Pressure gradients of T, in both hole-doped and electrondoped high-T, superconductors are given a unified interpretation on the basis of an indirect-exchange mechanism of pairing. Delocalised electrons created by doping, in new narrow-band states within the charge transfer gap, pair via closed-shell oxygen anions. A recently proposed “antisymmetric” relationship between pressure gradients in hole-doped (dTJdP> 0) and electron-doped (dTJdP< 0) superconductors is readily understood on this basis. It is not related to a difference in the sign of carrier charges.
1. Introduction The effect of pressure on the transition temperature T, of high-Tc superconductors has received much attention since the early experiments by Chu et al. [ 11. It was found that dT,/dP is always positive for hole-doped superconductors, with a characteristic difference between relatively low and high T,-values. The quantity dln TJdP ranges [2] from 1 x lo-’ kbar- ’ for La,.,,Sr,,,CuO, (T,=39 K) to 0.15x10-* kbar-’ for BizCazSrzCu~Olo (T,=108 K) thus decreasing markedly with increasing T, ( YBa2Cu306.8, T, = 92 K, has an exceptionally low value of 0.08 x 10m2 kbar- ’ ) . Extensive recent measurements by Markert et al. [ 3 ] of pressure gradients in electron-doped superconductors Lnz_,M,CuO.,_, (Ln=Nd, Pr, Sm, Eu; M =Ce, Th) showed, surprisingly, negative values, with Ndl.ssCeo.L5C~04_-y (dln T,/dP=O?O.l x lo-’ kbar-‘) on the borderline. The same characteristic difference between lower and higher T, values (between z 4 K and z 24 K) is seen again but with opposite sign from the holedoped compounds: dln TJdP becomes markedly less negative with increasing T,. The magnitudes of dln TJdP and dln T,/dln V= -B dlnT,/P (volume V and bulk modulus B= - VdP/dP) are of the same
order for the two categories. This remarkable sign difference led the authors to postulate an “antisymmetric” behaviour of electron- versus hole-doped superconductors. A negative sign of dln TJdP has also been measured by Uwe et al. [ 41 for the classical superconducting oxide BaPb, _XBiX03; its value being at maximum T,= 12 K -0.24x 10e2 kbar-’ (x=0.25). It is not clear whether this compound formally belongs to the “hole”- or the “electron”doped family. In this paper we demonstrate that this “symmetry difference” finds a ready explanation on the basis of a recently proposed [ 6,7 ] indirect-exchange pairing mechanism for high-T, superconductivity. The analysis does not support the generally held belief that holes (hole pairs ) in hole-doped and electrons (electron pairs) in electron-doped superconductors are the charge carriers. The possibility of indirect-exchange interactions between conduction electrons giving rise to an attractive interaction (unretarded) and hence superconductivity, was first investigated [ 5 ] in the context of simple metals. It was shown that pairing via closed-shell cation cores, in a BCS swave channel, can take place under certain conditions [ 5 ] (e.g. in metallic cesium via Cs’+ cores). In the case of high-T, superconductors [6,7] cou-
0921-4534/91/$03.50 0 1991 - Elsevier Science Publishers B.V. (North-Holland)
410
L. Jansen et al. / Pressuregradients
of T, in hole- and electron-doped superconductors
pling occurs via the diamagnetic oxygen anions in direct analogy with indirect exchange (superexchange) in insulating transition-metal oxides, e.g. NiO. Conduction electrons in these superconductors are created by the doping process. They occupy delocalised states in a new, correlated, band (experimental width a few hundred meV) within the charge transfer gap of the undoped material. Strong experimental support for the validity of this approach to high-T, superconductivity has recently been provided by Allen et al. [ 81 from a resonant photoemission study of Nd1.85Ce0.,5Cu04_-y as compared with an earlier [9] similar analysis of La1.85Sr0.15Cu04. The authors conclude (quote) “for the metallic of electron-doped Nd,_,Ce,CuO,_,, EF lies in new states created in the insulator gap above the x=0 valence band. Exactly the same is true for the hole-doped material La,_,Sr,CuO,_,. Further, EF has essentially the same position in the gap for both hole and electron doping. These experimental facts contrast sharply with the expectation in many models that for hole or electron doping, respectively, EF would lie in the insulator’s valence-band or conduction-band states, which are separated by a gap of 2 eV”. Similar conclusions were drawn by Namatame et al. [lo] from a photoemission study of the same systems, and very recently by Takahashi et al. [ Ill, comparing photoelectron spectra of superconducting BiZSr2CaCu208 and nonsuperconducting Bi2Sr,Ca0.4Y,,6CuzOS. Other experimental support is reviewed in ref. [ 71 as well as in an analysis of doping phenomena in hole- and electron-doped superconductors [ 121. Quantitative agreement with most recent experiments on various properties has been illustrated in an earlier paper [ 13 1.
2. Indirect exchange and high-T, superconductivity Apart from the source of the attractive interaction the indirect-exchange formalism follows the BCS approach. It can be summarized as follows [ 5-7 1. Consider 2 conduction electrons and n closed-shell cores (cations in metals and oxygen anions in high-T= superconductors), each core represented by two, spinpaired, electrons in an “effective” orbital w(v) to be determined. The scattering matrix elements { Vqk}are evaluated in a carrier space whose basis vectors {&}
are fully antisymmetrized products of spin-space vectors vkt ( 1 ), v__kl(2) and the (antisymmetrized) core ground-state function &( 3, 4, ... , 2n+ 2). Formally, the vectors v/k and &, are solutions of Schroedinger equations for, respectively, one conduction electron and 2n core electrons in the average fields of all the other charges. Vqk is obtained to be proportional to [ 5 ] (r is the distance between the two conduction electrons) e2<% -q]l/r]k,
-k)+n<@qi~~j@k),
(1)
where V is a difference operator containing coulomb interactions between the two conduction electrons as well as between conduction and core electrons. The density of cores is n and P, is a linear combination of permutations exchanging conduction electrons with those of the cores. The first term of ( 1) thus refers to direct electron-electron interactions and the second to indirect exchange via core electrons. If the sum of these two terms is negative for Iq 1, 1k 1zz IkF 1, then superconductivity can occur. The corresponding gap do is evaluated using an iteration procedure [ 5 1. In first approximation &=2]q]exp(-l/]W])for
W-CO,
(2)
in terms of 1WI, the indirect-exchange coupling strength. The transition temperature is then of the form (3) where (AU) is a characteristic temperature and I WI formally replaces N(0) V in the BCS expression. To obtain numerical results for I WI, we replace the formal solutions @,, and wk by simple approximations. First, the (Wannier) function w(r) at the site of an oxygen anion is assumed to be of Gaussian form W(r) r (a/n)3/2exp(
.
-cxr2/2)
(4)
The “size” parameter cx is set to be 0.20 au -2, a value taken over from an earlier [ 141 calculation of superexchange in transition-metal oxides. For a conduction electron we assume a tight-binding, theta, function [ 15 ] of the form ykk(r)=(8/2~)3’2(1/n)
c i
exp{-@(lr
-Rj
l*)}*exp(ik.r)
L. Jansen et al. /Pressure gradients of T, in hole- and electron-doped superconductors
= 7
exp{ -&
(Gf)}-exp(i(G,+k)*r)
411
(5)
with core-lattice vectors {R,} or reciprocal vectors {G,}. Values for the Gaussian parameters /3 are left to be determined. We treat the conduction band in an effective-mass approximation. Since only the density of cores occurs in the expression ( 1) for the coupling, we distribute the cores in a bee (or hcp) arrangement for computational convenience. Transport properties of high-T, superconductors are anisotropic largely due to the layered arrangement of oxygen anions. The coupling strength 1WI is a sensitive function of cr, /? and 1k, I. For fixed (Yit can be “chemically” varied in three ways. The first, changing j? at constant IkF 1, is realised in the cuprate families by varying the number of CuOz layers per formula unit. The value of /I depends on the oxygen density as pzi3. This follows from dimensional analysis and the fact that for pox= 0, p= 0 (i.e. no localisation). The second, changing I kF I while leaving p constant is achieved, for e.g. in Laz_XSrXCu04, by varying the dopant concentration x. As a third possibility, p and IkFl are varied simultaneously through a change in oxygen stoichiometry, e.g. varying 6 in YBa2Cu307_-6. Figure 1 reproduced from ref. [ 71 shows W for one CuOz layer plotted as a function of j?, with cr = 0.20 aue2. The W values apply [ 5 ] for an atomic density of p,,=8.6x102’ cme3 and a lkFl value of 0.341 au-’ based on metallic Cs. For the high-T, superconductors considered, a conversion factor (p,,/ pcS)1’3 for one CuOz layer [ 71 must be applied, together with the corresponding ratio of IkF I values. Since the latter is not known with precision, we absorb it in the effective-mass factor m*/m,. The markings in the figure refer to members of the three families (nominal compositions) and Tl2(Bi~)Ca,-iBa~(Sr~)Ct@~~+~ Tl, Ca,_ , Ba2Cuf12N+ 3, calculated on the basis of one gauge value: /3=0.15 au-* for the N= 1 member of the T12 series. It is found that relative transition temperatures are accurately reproduced for m*/m,= 4.55, a value well within the range reported in literature [ 16,171. Details of the calculations and a discussion of the results are given in ref. [ 71. It is important to remark that for fixed p and low doping, i.e. small carrier density p, I WI varies lin-
Fig. 1. Indirect exchange coupling strength W(cu, b, 1kF( ) for one Cu02 layer in the cuprates, as a function of the tight-binding (Gaussian) parameter /I (au?) for conduction electrons coupled via oxygen anions with (Gaussian) size parameter LY= 0.20 au-‘, and for a value of 1kF) = 0.34 1 au- ’ (metallic Cs, see text). ( 0 ) (exptl. compositions) and ( 0 ) (nominal compositions) refer to the high-T, family Tl,CaN_ IBa2Cu,+GZN+4, (A ) (nominal) to the Bi,-series (Bi, Sr instead of Tl, Ba), and + (exptl. compositions) to the Tl, family. The number N of CuOz layers increases from left to right, starting with N= 1. Data taken from ref. [7].
early with IkF I ccp “3, since as long as IkF I < aLI variation of the theta function across an oxygen anion may be neglected. As a result, the indirect-exchange coupling for each pair remains the same, and I WI is simply proportional to the density of states at EF At high doping the condition IkF I -ze aLr2 is no longer fulfilled; the overlap between theta function and oxygen valence shell diminishes, resulting in a rapid decrease of I WI. Thus, for fixed cr and 8, 1WI (and hence T,) as a function of IkFI (varying x) first increases, reaches a maximum and then decreases. The indirect-exchange mechanism therefore accounts [ 121 for the observed remarkable behaviour of T, as a function of doping in e.g. La2_,SrXCu04
[1813. Pressure dependence of transition temperatures The pressure dependence
of T, arises through the
412
L. Jansen et al. /Pressure gradients of T, in hole- and electron-doped superconductors
characteristic temperature (Ao) taken proportional to the Fermi energy EF [ 5 1, the density of states, as well as through the volume-dependent parameter /3. From eq. (3) we have, with dEF/dP=2EF/3B, dln TJdP=2/3B+ x(dl
W/W-)1.
(11 W2) [ 1WI /3B+
(2/?/3B)
(6)
The first term inside the brackets comes from the density of states while the second is the contribution to dl WI /dP from p. An important consequence of eq. (6) is that dln TJdP changes sign from positive to negative with increasingp, at dl WI /d/3= - [ I WI / j?] ( I WI + l/2), i.e. for a value of /.I to the right of the maximum in I WI. In fig. 1 this point lies between /I= 0.20 and 0.2 1 aue2. This implies that for p> 0.2 1 au2, i.e. for N> 3 in the T12- and N> 4 in the Tli- series, the theory predicts a negative pressure gradient. A numerical application of eq. (6) for the superconductor La,,ssBao.,SC~04 was presented recently [ 13 1, yielding good agreement with experiment. We note from eq. (6) that for p values to the left of the maximum in I WI, dln TJdP is always positive, its value being larger for smaller I WI, i.e. lower T,, in striking agreement with experiments. Another aspect of the effect of pressure on transition temperatures namely, the behaviour of dln TJ dP with dopant concentration x, as found e.g. by Tanahashi et al. [ 191 in La2_$.GXCu04, can directly be inferred from fig. 1. T, itself first increases with x, saturates between x=0.14 and 0.24, after which it decreases. Beyond x= 0.32, superconductivity has disappeared, although the metallic conductivity keeps increasing [ 181. On the other hand the derivative dln TJdP, positive except for anomalous behaviour around x= 0.12, first decreases, reaches a minimum, and then increases. To understand this effect we note that at low doping I WI (constant 8) is proportional to l&l, increasing with x. Thus, the W-curve becomes deeper, and 1 / I WJ more shallow and smaller, resulting in less positive values for both 1WI -dependent terms in eq. (6), i.e. a smaller value for dln TJ dP. At higher doping, 1WI reaches a maximum and then decreases. Accordingly, dln TJdP, as a function of x, reaches a (positive) minimum, and then increases again, as is found experimental [ 193. This behaviour of dln TJdP with x, but with opposite sign, applies also to superconducting oxides
which have /I in the region of negative dln TJdP, a class which includes the electron-doped materials. The effect has been observed by Uwe et al. [4] in BaPb, _XBi,Os while varying x and by Market-t et al. [3] in the electron-doped superconductors Ln,.ssMo.lSCu04_-y, at fixed x but with, successively, Ln=Pr, Nd, Pm, Sm, Eu (M=Ce and Th). The transition temperature decreases following the Lnseries given, both with Ce and Th. The same substitution in the Ln-series also results in dln TJdP assuming progressively more negative values, as predicted. In addition, in the Sm,_,Ce,CuO,_, system, these authors find [3] values of dln TJ dP= - 1.6x lop2 kbar-’ and -0.13~ lop2 kbar-’ forxz0.12 (T,=3.9 K) and 0.18 (T,=14.7 K) respectively, again following the predicted pattern. A particularly interesting example of enhanced ]dln TJdPI outside of optimal doping was found with the superconductor YBa2Cu408 (two sets of Cu0 chains separating the Cu02 planes) by Bucher et al. [20]. For a measured T, of 80 K, dTJdPz0.55 K/kbar i.e., dln TC/dP=0.7X 10e2 kbar-’ which is very large for a 80 K transition temperature. However, Miyatake et al. [ 2 1 ] established that T, increases to 90 K (at ambient pressure) by a 1O”/osubstitution of Ca for Y, corresponding to the limit of solubility of Ca, thus not yet the maximum T, from substitution. At the same time, dT,/dP decreases from 0.51 K/kbar for x=0 (in Y,_,Ca,; T,=82 K) to 0.26 K/kbar for x=0.1 (T,=89.8 K), i.e. to one half of the undoped value. When one further takes into account the fact the bulk modulus B for this compound is abnormally low, namely 1120 kbar at 300 K and 1200 kbar at 30 K [ 22 1, compared to zz 1500 kbar for other high-Tc cuprates, and that the pressure gradient is inversely proportional to B (see eq. (6) ), then the above two enhancing factors may, without conspicuous deviation from the other cuprates, amply explain the observed high value of 0.5 5 K/kbar at 80 K. It is physically unfounded, and conceptually misleading (albeit standard practice in recent literature) to invoke so-called “valence sum rules” [23] in conjunction with an observed decrease in the oxygen (chain, or apical)-to-copper (plane) distance in the cuprates under pressure, in order to conclude that the Cu “effective valence” in the planes increases, implying hole doping of these planes. This increased hole density is then supposed
L. Jansen et al. /Pressure gradients of T, in hole- and electron-doped superconductors
to be the cause of the increasing T, with pressure. The validity of such procedures for the interpretation of high-pressure data has recently been questioned by Pei et al. [24], and their applicability to high-T, cuprates severely criticised by Kwei et al. [ 25 1. We will return to these concepts in a separate paper.
4. Concluding remarks In summary, we find no difference of principle in the behaviour of pressure coefficients between holedoped and electron-doped superconductors. The fact that in the first category dln TJdP has been found to be positive and in the second category negative is “accidental” in that this difference is determined by whether the corresponding /3 values lie to the left or to the right of the maximum in the appropriate 1W(jl) 1 curve. This in turn depends on the detailed electronic effects of the doping procedure, which are different for the two types of superconductors. As mentioned earlier negative pressure gradients are predicted to occur also with “hole-doped” high-T, compounds for which j? lies to the right of the maximum in I W(p) 1, i.e. for N> 3 in the Tlz- and for N> 4 in the Tl,-series. Data to test this prediction on the known compounds Tlz (N= 4) and Tl, (NC 5, 6) is, to our knowledge, not yet available.
References [ I] C.W. Chu, P.H. Hor, R.L. Meng, L. Gao, Z.J. Huang and Y.Q. Wang, Phys. Rev. Lett. 58 (1987) 405. [2] J.E. Schirber, E.L. Venturini, B. Morosin and D.S. Ginley, Physica C 162-164 (1989) 745. [ 3 ] J.T. Markert, J. Beille, J.J. Neumeier, E.A. Early, C.L. Seaman, T. Moran and M.B. Maple, Phys. Rev. Lett. 64 (1990) 80.
413
[ 41 H. Uwe, T. Osada, A. Iyo, K. Murata and T. Sakudo, Physica C 162-164 (1989) 743. [ 51W.W. Schmidt, R. Block and L. Jansen, Phys. Rev. B 26 (1982) 3656. [6] L. Jansen, Physica C 156 (1988) 501. [7] L. Jansen and R. Block, Physica A 161 (1989) 385. [8] J.W. Allen, C.G. Olson, M.B. Maple, J.-S. Kang, L.Z. Liu, J.-H. Park, R.O. Anderson, W.P. Ellis, J.T. Marker& Y. Dalichaouch and R. Liu, Phys. Rev. Lett. 64 (1990) 595. [9] Z.-X. Shen, J.W. Allen, J.J. Yeh, J.-S. Kang, W. Ellis, W. Speiser, I. Lindau, M.B. Maple, Y.D. Dalichaouch, M.S. Torichachvili, J.Z. Sun and T.H. Geballe, Phys. Rev. B 36 (1987) 8414. 110‘I H. Namatame, A. Fujimori, Y. Tokura, M. Nakamura, K. Yamaguchi, A. Misu, H. Matsubara, S. Suga, T. Ito, H. Takagi and S. Uchida, Phys. Rev. B41 (1990) 7205. K. 11’ 1T. Takahashi, H. Matsuyama, H. Katayama-Yoshida, Seki, K. Kamiya and H. Inokuchi, Physica C 170 ( 1990) 416. 112 L. Chandran and L. Jansen, preprint. 113 L. Jansen and L. Chandran, Europhys. Lett. 13 ( 1990) 543. R. Block and L. Jansen, Phys. Rev. B 29 ]14 i P. Straatman, (1984) 1415. 115 H.W. Myron, M.H. Boon and F.M. Mueller, Phys. Rev. B 18 (1978) 3810. 116 L.P. Gor’kov and N.B. Kopnin, Sov. Phys. Usp. 3 1 ( 1988) 850. 162-164 (1989) 8. 117 G. Gruener, PhysicaC ]18 J.B. Torrance, Y. Tokura, A.I. Nazzal, A. Bezinge, T.C. Huang and S.S.P. Parkin, Phys. Rev. Lett. 61 (1988) 1127; J.B. Torrance, A. Bezinge, A.I. Nazzal and S.S.P. Parkin, Physica C 162-164 (1989) 291. 119 ‘I N. Tanahashi, Y. Iye, T. Tamegai, C. Murayama, N. Mori, S. Yomo, N. Okazaki and K. Kitazawa, Jpn. J. Appl. Phys. 28 ( 1989) L762. [ 201 B. Bucher, J. Karpinski, E. Kaldis and P. Wachter, Physica C 157 (1989) 478. [ 2 1 ] T. Miyatake, M. Kosuge, N. Koshizuka, H. Takahashi, N. Mori and S. Tanaka, Physica C 167 ( 1990) 297. 1221H.A. Ludwig, W.H. Fietz, M.R. Dietrich, H. Wiihl, J. Karpinski, E. Kaldis and S. Rusiecki, Physica C 167 ( 1990) 335. 123 See e.g. I.D. Brown, J. Solid State Chem. 82 (1989) 122. D.G. Hinks, B. Dabrowski, P. ~24 S. Pei, J.D. Jorgensen, Lightfoot and D.R. Richards, Physica C 169 (1990) 179. 125 G.H. Kwei, A.C. Larson, W.L. Hults and J.L. Smith, Physica C 169 (1990) 217.
Indirect exchange mechanism of high-T, superconductivity Pressure
gradients
of
T,in
hole-doped
and
electron-doped
superconductors
Laurens Jansen a, Leena Chandran b and Ruud Block ’ a c/o Theoretische Physik, ETH-Hiinggerberg, CH-8093 Ziirich, Switzerland b Institut ftir Quantenelekronik, ETH-Hiinggerberg, CH-8093 Ziirich, Switzerland ’ J.H. van? HoffInstitute, University ofAmsterdam, 1OI8 WVAmsterdam, The Netherlands
Received 6 November 1990 Revised manuscript received I2 December 1990
Pressure gradients of T, in both hole-doped and electrondoped high-T, superconductors are given a unified interpretation on the basis of an indirect-exchange mechanism of pairing. Delocalised electrons created by doping, in new narrow-band states within the charge transfer gap, pair via closed-shell oxygen anions. A recently proposed “antisymmetric” relationship between pressure gradients in hole-doped (dTJdP> 0) and electron-doped (dTJdP< 0) superconductors is readily understood on this basis. It is not related to a difference in the sign of carrier charges.
1. Introduction The effect of pressure on the transition temperature T, of high-Tc superconductors has received much attention since the early experiments by Chu et al. [ 11. It was found that dT,/dP is always positive for hole-doped superconductors, with a characteristic difference between relatively low and high T,-values. The quantity dln TJdP ranges [2] from 1 x lo-’ kbar- ’ for La,.,,Sr,,,CuO, (T,=39 K) to 0.15x10-* kbar-’ for BizCazSrzCu~Olo (T,=108 K) thus decreasing markedly with increasing T, ( YBa2Cu306.8, T, = 92 K, has an exceptionally low value of 0.08 x 10m2 kbar- ’ ) . Extensive recent measurements by Markert et al. [ 3 ] of pressure gradients in electron-doped superconductors Lnz_,M,CuO.,_, (Ln=Nd, Pr, Sm, Eu; M =Ce, Th) showed, surprisingly, negative values, with Ndl.ssCeo.L5C~04_-y (dln T,/dP=O?O.l x lo-’ kbar-‘) on the borderline. The same characteristic difference between lower and higher T, values (between z 4 K and z 24 K) is seen again but with opposite sign from the holedoped compounds: dln TJdP becomes markedly less negative with increasing T,. The magnitudes of dln TJdP and dln T,/dln V= -B dlnT,/P (volume V and bulk modulus B= - VdP/dP) are of the same
order for the two categories. This remarkable sign difference led the authors to postulate an “antisymmetric” behaviour of electron- versus hole-doped superconductors. A negative sign of dln TJdP has also been measured by Uwe et al. [ 41 for the classical superconducting oxide BaPb, _XBiX03; its value being at maximum T,= 12 K -0.24x 10e2 kbar-’ (x=0.25). It is not clear whether this compound formally belongs to the “hole”- or the “electron”doped family. In this paper we demonstrate that this “symmetry difference” finds a ready explanation on the basis of a recently proposed [ 6,7 ] indirect-exchange pairing mechanism for high-T, superconductivity. The analysis does not support the generally held belief that holes (hole pairs ) in hole-doped and electrons (electron pairs) in electron-doped superconductors are the charge carriers. The possibility of indirect-exchange interactions between conduction electrons giving rise to an attractive interaction (unretarded) and hence superconductivity, was first investigated [ 5 ] in the context of simple metals. It was shown that pairing via closed-shell cation cores, in a BCS swave channel, can take place under certain conditions [ 5 ] (e.g. in metallic cesium via Cs’+ cores). In the case of high-T, superconductors [6,7] cou-
0921-4534/91/$03.50 0 1991 - Elsevier Science Publishers B.V. (North-Holland)
410
L. Jansen et al. / Pressuregradients
of T, in hole- and electron-doped superconductors
pling occurs via the diamagnetic oxygen anions in direct analogy with indirect exchange (superexchange) in insulating transition-metal oxides, e.g. NiO. Conduction electrons in these superconductors are created by the doping process. They occupy delocalised states in a new, correlated, band (experimental width a few hundred meV) within the charge transfer gap of the undoped material. Strong experimental support for the validity of this approach to high-T, superconductivity has recently been provided by Allen et al. [ 81 from a resonant photoemission study of Nd1.85Ce0.,5Cu04_-y as compared with an earlier [9] similar analysis of La1.85Sr0.15Cu04. The authors conclude (quote) “for the metallic of electron-doped Nd,_,Ce,CuO,_,, EF lies in new states created in the insulator gap above the x=0 valence band. Exactly the same is true for the hole-doped material La,_,Sr,CuO,_,. Further, EF has essentially the same position in the gap for both hole and electron doping. These experimental facts contrast sharply with the expectation in many models that for hole or electron doping, respectively, EF would lie in the insulator’s valence-band or conduction-band states, which are separated by a gap of 2 eV”. Similar conclusions were drawn by Namatame et al. [lo] from a photoemission study of the same systems, and very recently by Takahashi et al. [ Ill, comparing photoelectron spectra of superconducting BiZSr2CaCu208 and nonsuperconducting Bi2Sr,Ca0.4Y,,6CuzOS. Other experimental support is reviewed in ref. [ 71 as well as in an analysis of doping phenomena in hole- and electron-doped superconductors [ 121. Quantitative agreement with most recent experiments on various properties has been illustrated in an earlier paper [ 13 1.
2. Indirect exchange and high-T, superconductivity Apart from the source of the attractive interaction the indirect-exchange formalism follows the BCS approach. It can be summarized as follows [ 5-7 1. Consider 2 conduction electrons and n closed-shell cores (cations in metals and oxygen anions in high-T= superconductors), each core represented by two, spinpaired, electrons in an “effective” orbital w(v) to be determined. The scattering matrix elements { Vqk}are evaluated in a carrier space whose basis vectors {&}
are fully antisymmetrized products of spin-space vectors vkt ( 1 ), v__kl(2) and the (antisymmetrized) core ground-state function &( 3, 4, ... , 2n+ 2). Formally, the vectors v/k and &, are solutions of Schroedinger equations for, respectively, one conduction electron and 2n core electrons in the average fields of all the other charges. Vqk is obtained to be proportional to [ 5 ] (r is the distance between the two conduction electrons) e2<% -q]l/r]k,
-k)+n<@qi~~j@k),
(1)
where V is a difference operator containing coulomb interactions between the two conduction electrons as well as between conduction and core electrons. The density of cores is n and P, is a linear combination of permutations exchanging conduction electrons with those of the cores. The first term of ( 1) thus refers to direct electron-electron interactions and the second to indirect exchange via core electrons. If the sum of these two terms is negative for Iq 1, 1k 1zz IkF 1, then superconductivity can occur. The corresponding gap do is evaluated using an iteration procedure [ 5 1. In first approximation &=2]q]exp(-l/]W])for
W-CO,
(2)
in terms of 1WI, the indirect-exchange coupling strength. The transition temperature is then of the form (3) where (AU) is a characteristic temperature and I WI formally replaces N(0) V in the BCS expression. To obtain numerical results for I WI, we replace the formal solutions @,, and wk by simple approximations. First, the (Wannier) function w(r) at the site of an oxygen anion is assumed to be of Gaussian form W(r) r (a/n)3/2exp(
.
-cxr2/2)
(4)
The “size” parameter cx is set to be 0.20 au -2, a value taken over from an earlier [ 141 calculation of superexchange in transition-metal oxides. For a conduction electron we assume a tight-binding, theta, function [ 15 ] of the form ykk(r)=(8/2~)3’2(1/n)
c i
exp{-@(lr
-Rj
l*)}*exp(ik.r)
L. Jansen et al. /Pressure gradients of T, in hole- and electron-doped superconductors
= 7
exp{ -&
(Gf)}-exp(i(G,+k)*r)
411
(5)
with core-lattice vectors {R,} or reciprocal vectors {G,}. Values for the Gaussian parameters /3 are left to be determined. We treat the conduction band in an effective-mass approximation. Since only the density of cores occurs in the expression ( 1) for the coupling, we distribute the cores in a bee (or hcp) arrangement for computational convenience. Transport properties of high-T, superconductors are anisotropic largely due to the layered arrangement of oxygen anions. The coupling strength 1WI is a sensitive function of cr, /? and 1k, I. For fixed (Yit can be “chemically” varied in three ways. The first, changing j? at constant IkF 1, is realised in the cuprate families by varying the number of CuOz layers per formula unit. The value of /I depends on the oxygen density as pzi3. This follows from dimensional analysis and the fact that for pox= 0, p= 0 (i.e. no localisation). The second, changing I kF I while leaving p constant is achieved, for e.g. in Laz_XSrXCu04, by varying the dopant concentration x. As a third possibility, p and IkFl are varied simultaneously through a change in oxygen stoichiometry, e.g. varying 6 in YBa2Cu307_-6. Figure 1 reproduced from ref. [ 71 shows W for one CuOz layer plotted as a function of j?, with cr = 0.20 aue2. The W values apply [ 5 ] for an atomic density of p,,=8.6x102’ cme3 and a lkFl value of 0.341 au-’ based on metallic Cs. For the high-T, superconductors considered, a conversion factor (p,,/ pcS)1’3 for one CuOz layer [ 71 must be applied, together with the corresponding ratio of IkF I values. Since the latter is not known with precision, we absorb it in the effective-mass factor m*/m,. The markings in the figure refer to members of the three families (nominal compositions) and Tl2(Bi~)Ca,-iBa~(Sr~)Ct@~~+~ Tl, Ca,_ , Ba2Cuf12N+ 3, calculated on the basis of one gauge value: /3=0.15 au-* for the N= 1 member of the T12 series. It is found that relative transition temperatures are accurately reproduced for m*/m,= 4.55, a value well within the range reported in literature [ 16,171. Details of the calculations and a discussion of the results are given in ref. [ 71. It is important to remark that for fixed p and low doping, i.e. small carrier density p, I WI varies lin-
Fig. 1. Indirect exchange coupling strength W(cu, b, 1kF( ) for one Cu02 layer in the cuprates, as a function of the tight-binding (Gaussian) parameter /I (au?) for conduction electrons coupled via oxygen anions with (Gaussian) size parameter LY= 0.20 au-‘, and for a value of 1kF) = 0.34 1 au- ’ (metallic Cs, see text). ( 0 ) (exptl. compositions) and ( 0 ) (nominal compositions) refer to the high-T, family Tl,CaN_ IBa2Cu,+GZN+4, (A ) (nominal) to the Bi,-series (Bi, Sr instead of Tl, Ba), and + (exptl. compositions) to the Tl, family. The number N of CuOz layers increases from left to right, starting with N= 1. Data taken from ref. [7].
early with IkF I ccp “3, since as long as IkF I < aLI variation of the theta function across an oxygen anion may be neglected. As a result, the indirect-exchange coupling for each pair remains the same, and I WI is simply proportional to the density of states at EF At high doping the condition IkF I -ze aLr2 is no longer fulfilled; the overlap between theta function and oxygen valence shell diminishes, resulting in a rapid decrease of I WI. Thus, for fixed cr and 8, 1WI (and hence T,) as a function of IkFI (varying x) first increases, reaches a maximum and then decreases. The indirect-exchange mechanism therefore accounts [ 121 for the observed remarkable behaviour of T, as a function of doping in e.g. La2_,SrXCu04
[1813. Pressure dependence of transition temperatures The pressure dependence
of T, arises through the
412
L. Jansen et al. /Pressure gradients of T, in hole- and electron-doped superconductors
characteristic temperature (Ao) taken proportional to the Fermi energy EF [ 5 1, the density of states, as well as through the volume-dependent parameter /3. From eq. (3) we have, with dEF/dP=2EF/3B, dln TJdP=2/3B+ x(dl
W/W-)1.
(11 W2) [ 1WI /3B+
(2/?/3B)
(6)
The first term inside the brackets comes from the density of states while the second is the contribution to dl WI /dP from p. An important consequence of eq. (6) is that dln TJdP changes sign from positive to negative with increasingp, at dl WI /d/3= - [ I WI / j?] ( I WI + l/2), i.e. for a value of /.I to the right of the maximum in I WI. In fig. 1 this point lies between /I= 0.20 and 0.2 1 aue2. This implies that for p> 0.2 1 au2, i.e. for N> 3 in the T12- and N> 4 in the Tli- series, the theory predicts a negative pressure gradient. A numerical application of eq. (6) for the superconductor La,,ssBao.,SC~04 was presented recently [ 13 1, yielding good agreement with experiment. We note from eq. (6) that for p values to the left of the maximum in I WI, dln TJdP is always positive, its value being larger for smaller I WI, i.e. lower T,, in striking agreement with experiments. Another aspect of the effect of pressure on transition temperatures namely, the behaviour of dln TJ dP with dopant concentration x, as found e.g. by Tanahashi et al. [ 191 in La2_$.GXCu04, can directly be inferred from fig. 1. T, itself first increases with x, saturates between x=0.14 and 0.24, after which it decreases. Beyond x= 0.32, superconductivity has disappeared, although the metallic conductivity keeps increasing [ 181. On the other hand the derivative dln TJdP, positive except for anomalous behaviour around x= 0.12, first decreases, reaches a minimum, and then increases. To understand this effect we note that at low doping I WI (constant 8) is proportional to l&l, increasing with x. Thus, the W-curve becomes deeper, and 1 / I WJ more shallow and smaller, resulting in less positive values for both 1WI -dependent terms in eq. (6), i.e. a smaller value for dln TJ dP. At higher doping, 1WI reaches a maximum and then decreases. Accordingly, dln TJdP, as a function of x, reaches a (positive) minimum, and then increases again, as is found experimental [ 193. This behaviour of dln TJdP with x, but with opposite sign, applies also to superconducting oxides
which have /I in the region of negative dln TJdP, a class which includes the electron-doped materials. The effect has been observed by Uwe et al. [4] in BaPb, _XBi,Os while varying x and by Market-t et al. [3] in the electron-doped superconductors Ln,.ssMo.lSCu04_-y, at fixed x but with, successively, Ln=Pr, Nd, Pm, Sm, Eu (M=Ce and Th). The transition temperature decreases following the Lnseries given, both with Ce and Th. The same substitution in the Ln-series also results in dln TJdP assuming progressively more negative values, as predicted. In addition, in the Sm,_,Ce,CuO,_, system, these authors find [3] values of dln TJ dP= - 1.6x lop2 kbar-’ and -0.13~ lop2 kbar-’ forxz0.12 (T,=3.9 K) and 0.18 (T,=14.7 K) respectively, again following the predicted pattern. A particularly interesting example of enhanced ]dln TJdPI outside of optimal doping was found with the superconductor YBa2Cu408 (two sets of Cu0 chains separating the Cu02 planes) by Bucher et al. [20]. For a measured T, of 80 K, dTJdPz0.55 K/kbar i.e., dln TC/dP=0.7X 10e2 kbar-’ which is very large for a 80 K transition temperature. However, Miyatake et al. [ 2 1 ] established that T, increases to 90 K (at ambient pressure) by a 1O”/osubstitution of Ca for Y, corresponding to the limit of solubility of Ca, thus not yet the maximum T, from substitution. At the same time, dT,/dP decreases from 0.51 K/kbar for x=0 (in Y,_,Ca,; T,=82 K) to 0.26 K/kbar for x=0.1 (T,=89.8 K), i.e. to one half of the undoped value. When one further takes into account the fact the bulk modulus B for this compound is abnormally low, namely 1120 kbar at 300 K and 1200 kbar at 30 K [ 22 1, compared to zz 1500 kbar for other high-Tc cuprates, and that the pressure gradient is inversely proportional to B (see eq. (6) ), then the above two enhancing factors may, without conspicuous deviation from the other cuprates, amply explain the observed high value of 0.5 5 K/kbar at 80 K. It is physically unfounded, and conceptually misleading (albeit standard practice in recent literature) to invoke so-called “valence sum rules” [23] in conjunction with an observed decrease in the oxygen (chain, or apical)-to-copper (plane) distance in the cuprates under pressure, in order to conclude that the Cu “effective valence” in the planes increases, implying hole doping of these planes. This increased hole density is then supposed
L. Jansen et al. /Pressure gradients of T, in hole- and electron-doped superconductors
to be the cause of the increasing T, with pressure. The validity of such procedures for the interpretation of high-pressure data has recently been questioned by Pei et al. [24], and their applicability to high-T, cuprates severely criticised by Kwei et al. [ 25 1. We will return to these concepts in a separate paper.
4. Concluding remarks In summary, we find no difference of principle in the behaviour of pressure coefficients between holedoped and electron-doped superconductors. The fact that in the first category dln TJdP has been found to be positive and in the second category negative is “accidental” in that this difference is determined by whether the corresponding /3 values lie to the left or to the right of the maximum in the appropriate 1W(jl) 1 curve. This in turn depends on the detailed electronic effects of the doping procedure, which are different for the two types of superconductors. As mentioned earlier negative pressure gradients are predicted to occur also with “hole-doped” high-T, compounds for which j? lies to the right of the maximum in I W(p) 1, i.e. for N> 3 in the Tlz- and for N> 4 in the Tl,-series. Data to test this prediction on the known compounds Tlz (N= 4) and Tl, (NC 5, 6) is, to our knowledge, not yet available.
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