Spin-polaron pairing and high-temperature superconductivity

Spin-polaron pairing and high-temperature superconductivity

• _ Solid State Communications, Vol. 67, No. 4, pp. 363-367, 1988. Printed in Great Britain. i • : ? 0038-1098/88 $3.00 + .00 Pergamon Press pl...

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Solid State Communications, Vol. 67, No. 4, pp. 363-367, 1988. Printed in Great Britain.




0038-1098/88 $3.00 + .00 Pergamon Press plc

SPIN-POLARON PAIRING AND HIGH-TEMPERATURE SUPERCONDUCTIVITY Hiroshi Kamimura, Shunichi Matsuno and Riichiro Saito Department of Physics, University of Tokyo, Bunkyo-ku, Tokyo, 113 Japan (Recieved 11 May 1988 by T. Tsuzuki) We show that in superconducting materials with CuO~ layers spin-polarons associated with itinerant holes in the % band (Cu dz 2 - Op~ band) are created by strong intra-atomic exchange interaction (Hund's coupling) with localized dx~-y2 holes. Then the interplay of Hund's coupling and superexchange between Cu2+(dx2-y2) spins gives rise to an attractive exchange interaction between spinpolarons, which leads to the formation of spin-singlet spin-polarons pairs. This contributes to high temperature superconductivity. By solving an gap equation it is shown that the s-wave pairing reproduces the observed x dependence of T as well as the right magnitudes of T in (Lal.fir)2Cu04 satisfactorily.

We first note that the ground state of a Cu 2+ ion (3d9), placed in a cubic ligand field is 2E, orbitally doubly degenerate, s and then that 2E state is split into two states with dx~-y2 and dz 2 symmetry by the Jahn-Teller interaction and surrounding lower symmetry field, with dx2-y2 state being the lowest electronic state for a 3d hole in a CuO. octahedron elongated along the z axis. These dx2*-y2 and dz 2 orbitals are hybridized with oxygen p orbitals, forming eu and % Bloch orbitals, respectively. Then we note that in pure La2CuO4, every hole in the eu band is localized at Cu 2+ site by the electron correlation, that is a hole at Cu 2+ site has dx2-y2 character and thus pure La2CuO * has a threedimensional(SD) antiferromagnetic ground state ordering of the Cu 2+ spins by superexchange through intervening O2- ions. As the experimental facts the N~el temperature T N of pure La2CuO 4 is 200~ 250K~'1°. Above T N the spins are ordered over large distances two-dimensionally, and the antlferromagnetic exchange interaction between Cu 2+ spins in the CuO 2 layers (J) is of order of 1000K 1°. The ground state of this spin system may be a resonating valence bond (RVB) type state, as Anderson2 proposed. When divalent ions such as Sr are substituted for trivalent ions La, electrons (or extra holes) are removed from (or created in) the % band (Cu dz2-O p band) but not from (or in) the eu band (Cu dx2-yt O p band), because of strong intra-atomic exchange interaction (Hund's coupling). This is the key point of our mechanism. From the ligand field theory 8 the intra-atomic exchange interaction in Ni2+(3d8) which is an isoelectronie ion with Cu ~+ is estimated to be of order of 1.5 to 2 eV (~20000K). Thus it is much stronger, compared with superexchange interaction J. All theoretical models proposed so far have not taken

THE DISCOVERY, OF HIGH-TEMPERATURE superconductivity* has led to intensive searches for new mechanisms of superconductivity. Anderson~ first pointed out the possibility of superconductivity in strongly correlated electron systems near the metalnonmetal transition. Since then a number of theoretical studies have been made on Hubbard-type models. All these models are based essentially on a single C u d - O p band, although some models stress the dx2-y2 character S while others the oxygen p character for extra holes~'4. Kamimura 5 first pointed out that the involvement of two bands of dx2-y2 and dz 2 types interacted by the Hund's coupling as well as the electron correlation play important roles in giving rise to high temperature superconductivity. In this paper, as the extension of previous work 5'6'7, it is shown that the interplay of Hund's coupling and superexchange interactions between holes in the dx2-y2 band gives rise to an effective attractive interaction between spin-polarons created in the dz 2 band by doping. Based on this interaction T for (Lal.fir)2CuO 4 systems is calculated as a function of x from a gap equation. It is shown that a calculated value of T as well as its x dependence reproduce the experimental results satisfactorily when we use the result of real band structure calculations for the one-electron energy dispersion in solving a gap equation. The present mechanism is based on a CuO 2 layer system so that we believe that high temperature superconductivity in La--(Ba, Sr)--Cu--O, Y--Ba--Cu--O, Bi--Sr--Ca--Cu--O and T1--Sr--Ca--Cu--O compounds all of which have a CuO 2 layer system can be explained by the present mechanism. Here we consider a mechanism in La--(Ba, Sr)--Cu--O systems for simplicity. 363



account of the effect of such strong intra-atomic exchange interaction. In this context, when extra holes are created in the % band, the spin of a hole in lower dx2-y 2 orbital tends to be polarized parallel with the spin of % hole by the strong intra-atomic exchange interaction K at Cu sites at which % holes exist. Since the ground state of pure La2CuO , is the RVB type spin singlet, the RVB type state is partially destroyed by the above spin polarization. This situation is schematically shown in Fig.1. We call the spinpolarized state introduced by extra d holes in the % band a spin-polaron. Radius of a spin-polaron r 0 is determined by the competition between the energy gain due to spin polarizations caused by the intraatomic exchange and the energy loss due to the destruction of RVB type state. Here we assume that r 0 is of order of the nearest neighbour distance between Cu sites in a CuO 2 layer. The simplest model Hamiltonian describing the above mechanism consists of three parts; the transfer interaction of an extra % (~dz 2) hole (Htr), the superexchange interaction between Cu ~+ spins of dx=-y2 holes (HAF) and the intra-atomic exchange between spins of % and dx2-y ~ holes at the same Cu site (Hints=);

H = Htr -J- HAF "3L Hintr ` = E


t=~(a=a+a,~=+h.c.) + JES=. Sm - - K E s , . S , n,m


where tnm represents the transfer of an % hole between Cu n and m sites through p orbitals of intervening oxygen sites, a a + (a a) the creation (annihilation) operator of an % hole at C u n site, J the superexchange coupling between the spins S. and

Hund's e v band



(Cudx=+OP}~ f~

dxZ . zz -e-









Fig. 1. Schematic illustration of the formation of a spin-polaron. In pure La~CuO. in which there are no holes in ev band the holes i n ' t h e lower eu band are localized in Cu dx~-y~ orbital by the strong electron correlation and their spins form the spin-singlet RVB type state. These spins which fluctuate quantummechanically are denoted by circles in the figure. Behaviour of spin polarization by introducing an extra hole in the upper % band at Cu 0 site is shown by arrows.

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Tc J




Fig. 2. The calculated result of T for the s-wave pairing of spin-polarons in (La. Srx)2CuO4 as a function of x for three values of J:tt, (a) 1000K, (b) 1500K and (c / 2000K. Sm of dx2-y 2 holes at the nearest neighbour C u n and m sites (J>0, antiferromagnetic) and K the intraatomic exchange integral between the spin of the %(dz 2) hole s= and that of dx2-y2 hole So at Cu n-site (K>0, ferromagnetic). Assuming that the double occupancy in dx2-y 2 and dz 2 orbltals are prohibited by the strong electron correlation and K is the strongest interaction among K, J and trim, the above model Hamiltonian (1) can be deduced from a general Hubbard type Hamiltonian for two bands introduced by Aoki and Kamimura,~ in which the Wannier orbitals are taken as a basis set and as a result the oxygen p orbitals are apparently eliminated. When a number of spin-polarons are created, they form spin-singlet pairs. This is easily understood from Fig.1. Suppose there is a spin-polaron with its spin directed upwards. The magnitude of the total spin of a spin-polaron lies in between 1/2 and 1. We note from Fig.1 that the spins at the neighbouring Cu sites a,b, etc. around a Cu 0 site at which two holes exist in % and eu bands are antiparallel with the spins of the two holes with paralell spin direction at Cu 0 site. When another extra d hole is created in the ev band somewhere, a spin-polaron is again produced in the sacrifice of loosing the ground state energy of the RVB type state. However, if the extra d hole is accommodated at one of the neighbouring Cu sites a,b,- • • around Cu 0 site by making its spin antlparallel, the energy loss due to the destruction of the RVB type state is small, compared with that in case where two spin-polarons are produced far from each other. This is clearly understood from Fig.1. That means that there always exists an attractive exchange interaction between spin-polarons so that all the spin-potarons created by doping form spin-singlet pairs. Now we discuss how a spin-polaron and a spinpolaron pair can move in a crystal. According to the

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n e u t r o n diffraction experiments 1° the Cu 2+ spins fluctuate quantum-mechanically even in a superconducting state. This is due to the strong twodimensional n a t u r e of the Cu 2+ spin system in CuO 2 layers. This fact is also supported by computor slmultation work on a 2D Heisenberg antiferromagnetic system by Q u a n t u m Monte Carlo method u. Because of this q u a n t u m fluctuation of spins, the spin state of a d hole at each Cu ~+ site is a mixture of up- and down-states. Then the combined action of the spin polarization by the strong intra-atomic exchange interaction and a spin fluctuation can make a spinpolaron pair as well as a spin-polaron tunnel resonantly in a crystal by the transfer interaction of an e hole. This situation m a y be considered as the double exchange mechanism in a 2D Heisenberg spin systems. In this context a spin-polaron associated with a doped % hole can hop from a CuS+-like site to neighbouring Cu 2+ sites even though their spin states are different. Here we have used the word of a Cu a+like site because an e~ hole is a mixture of Cu dz 2 and oxygen p orbitals, especially the Pz character of oxygen sites above and below a Cu site along the z axis l='la, and thus the valency of a Cu ion in the e v hole state has a value between 2 and 3. Then the transfer of a spin-polaron and the attractive antiferomagnetic exchange interaction between spin-polarons J ff in a CuO 2 layer can be described by the following effective transfer Hamiltonian Herf --




T h.c.) -t- Jeff n~mSn"Sin' (2)

where ~ m is the effective transfer integral of a spinpolaron between Cu n and m sites in a crystal, and C ~+ and C ~ represent creation and annihilation operators of a spin-polaron at C u n site, respectively. F u r t h e r we have expressed the spin of a spin-polaron at n site by s with s = l / 2 , although the magnitude of s is expected to be a value in between 1/2 and 1. The above attractive interaction with coupling constant J~ff is responsible for the formation of spinpolaron pairs associated with the doped e holes, and it contributes to the occurrence of superconductivity. This spin-polaron pair is a boson-like particle constructed from fermions, and the Bose-Einstein condensation occurs below a condensation temperature. Using the band mass of an e hole m* which is obtained from a band structure calculation ha3 and found to be of order of a free electron mass and the three-dimensional formula for a Bose-Einstein condensation temperature T o = 3.31t~2n2/3(2m*kn) -1 , T O is estimated to be 1500 K for doping concentration n = 102~ cm -3, which is much higher than a superconducting critical temperature T . Thus the superconducting state of the spinpolaron pairs is just like the BCS state. 14'2 Then we can obtain a BCS type solution for I-Iamiltonian (2). Here we should remark that our Cooper pair is formed by the real-space pairing and that a correlation length of our Cooper pair is of order of the extension of a spin-polaron pair which is of order of 6 to 8 A along a CuO 2 layer. In our mechanism a



pair is formed for spin-polarons lying in a CuO 2 layer, but such pairs can move in a three-dimensional crystal, because the e b a n d has a considerable dispersion along the z axis. 12a3 In the m e a n field approximation the gap equation corresponding to Hamiltonian (2) is obtained in the wave vector representation as follows; A (k,T)=(I/2N)~, V(k,k')A (k',T)


× tanh(E(k')/2kBT)/n(k' ) where

V(k,k') = J . ( - - ~,(k-k')+ ~,(k+k')),


E(k) = (~ (k) + A2(k))'/~,


7 (k) = c o s k a + coskya


with u being the chemical potential, a the lattice constant between Cu 2+ ions and e(k) the energy dispersion of an e hole. Since the doping concentration of divalent ions is very low, that is of order of 1021 cm -a, ~ for e holes is not so large. Therefore, the Fermi level is located near the top of the e band. Because of a small wave vectors k corresponding to a small Fermi surface for e holes, we can approximate the order p a r a m e t e r for spin-polaron pairing A(k) as

As(k)= a(cosk~a+ coska)


for s-wave pairing, and Ad(k)---- h(cosk~a--coska)


for d-wave pairing. T h e n T c is easily calculated for both types of spin-polaron pairing as a function of doping concentration x in (Lal. Srx)2CuO ,. In order to calculate T¢ of (Lal.xSr~)2CuO ~ as a function of x (that is ~), we need information on (k) and Jeff" Suppose the transfer integral of a spinpolaron ~ m in equation (2) is the same as that of an e v hole tnm in equation (1); "~nm = t~m T h e n we can use the result of a real b a n d structure calculation for the e v band in (Lal.xSrx)2CuO412'13 for e(k) in equation (4b). The numerical result of b a n d structure calculation for the e v b a n d by Shiraishi et al is is fitted by the following analytical expression (k) = A(cosk~a + coskya) + Bcoskza coskya + C(cos2k=a + cos2kya) + Dcosk a coskya



with A = 2080, B = --820, C = --470, and D -- --260 in the energy unit of kelvin K. In this case a remaining p a r a m e t e r in the gap equation is Jeff" Jeff is of order of superexchange interaction J (~1000 K) 1°'15 but a little enhanced by the intra-atomic exchange K. Thus it is reasonable to take a value of 1000~1500 K for Jeff" In our calculation of T we vary Jeff as 1000 K, 1500 K, and



2000 K. In Fig. 2 we show the calculated result of T c in (Lal. Sr,)2CuO 4 as a function of x for three values of J tf in the case of s-wave pairing. As seen in this figure, T first increases with increasing x, then takes a ma xim u m value around x=0.06 and then decreases with further increasing x. This trend is consistent with observed oneJ e Further, if we adopt 1500 K for Jar, the calculated T reproduces well the experimented results not only qualitatively but also quantitatively. The reason why our mechanism has reproduced an experimental value of T satisfactorily is that (1) the two-dimensional superexchange interaction between the spins of dx2-y 2 holes in a CuO~ layer J which plays an essential role in the formation of spinpolaron pairing is large, such as of order of 1000 K, and (2) the density of states associated with the motion of spin-polaron pairs which is represented by those of extra holes created in the e band is very high at the Fermi level. As regards the latter we show the density of states of the % band near its top fitted to by Shiraishi et al.'s energy band 1~ by curve a in the upper right corner of Fig.3. As seen in this figure,the density of states of the e band is very high near the top of the band. This contributes to a high value of T in a small doping concentration, as shown by curve a in the left part of Fig.3, where the caluclated value of T for Jar = 1500 K is again presented. The sharp decrease of T c with increasing x is due to the sharp decrease of the density of states. If we adopt a schematic cosine two-dimensional energy band for s (k) such as (k)= t0(coskxa + coskya),


and take the band width 4t 0 to be the same as the calculated band width of the ev band, the calculated T c varies as a function of x for Jeff = 1500 K, as shown by curve b in the left part of Fig.3, where the density of states corresponding to the energy band (7) over the whole band width is also shown by curve b in the lower right corner. By comparing the results of T c calculated for real 3D and schematic 2D bands shown in Fig.3 we conclude that the high density of states at the Fermi energy contributes crucially to the occurrence of higher T c together with a larger value of J. It is also seen that a three-dimensional dispersion of e(k), that is a 3D motion of spinpolaron pairs is important in reproducing the correct behaviour in the x dependence of T . We have also calculated T c for d-wave pairing as a function of x, and found that T c is nearly zero for x<0.3 and then increases for larger values of x. Thus the d-wave pairing does not contribute to the superconductivity in (Lal.fir)2CuO * systems. The existence of the % band near the Fermi level which is a mixture of Cu dz 2 orbital, p~ orbitals of oxygen sites above and below Cu along the z axis and pa orbitals of oxygen sites in a CuO 2 layer, that is the existence of holes in oxygen p z orbitals as well as in copper dz 2 orbital has been recently supported


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' OJ,2 ' ev




Fig. 3 Curve a: The calculated results of T for J r ~ l S 0 0 K (the same as Fig. 2) (left) and the denmty of states of the e band near the top of the band which has been ca~iculated by fitting to the e band obtained numerically by Shiraishi et alJ 2'1~ [right). Curve b: T for Ja,= I S 0 0 K calculated by using the density of states for a two dimensional cosine band shown in the lower right corner. Note that the energy scale for the density of states is different for curves a and b; that is near the top of the band for curve a while the whole band width for curve b. ¢


by the experiment of polarized X-ray absorption spectra 17, and also by the crystal field calculation is. In our theory the two kinds of valence of copper ions such as Cu 2+ and Cu 2+a with 0 < 6 < 1 . 0 coexist in a superconducting state. This coexistence of mixed valence of copper ions have been also supported by experiments of Cu K-edge XANES 19'e°'21, Cu Auger spectroscopy experiments 2~, and computer modelling studiesJ 3 In our theory the holes created in the ev band contribute to transport properties in a normal state. It is clear that the Hall coefficient and the Seebeck coefficient must have the sign of holes and that the number of carriers should be equal to the doping concentration of extra holes in the e band. These are all consistent with experimental results e*'2s. Observed linear temperature behaviour of the resistivity along the layers above Tc e6-2s can be also explained by the existence of a very small nearly cylindrical Fermi surface for the % holes. This will be discussed in later publication. Acknowledgements: We would like to thank Nobuyuki Shima and Kenji Shiraishi for providing information on the results of their band structure calculations and for their helpful discussions. This work was supported by a Grant-in-Aid fronl the Ministry of Education, Science and Culture.

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