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Negative-U superconductivity in the two-band model with the pair-transfer interaction
Physica C 177 (1991) 303-309 North-Holland
Negative-U superconductivity in the two-band model with the pairtransfer interaction K. Y a m a j i Electrotechnical Laboratory, 1-1-4 Umezono, Tsukuba 305, Japan Received 16 April 1991
The superconducting transition is studied in a two-band model which has an interband pair-transfer interaction between the conducting band and a hidden band not crossing the Fermi level. The interband ladder-diagram process enhances the effect of the pair-transfer interaction, remarkably reducing the effective on-site Coulomb energy and leading to high-Tosuperconductivity with a negative U. The isotope effect is shown to weaken with increasing To, if in addition we take account of the BCS interaction. The tendency of increase of Tc with decrease of the Madelung potential difference AVMbetween Cu and O sites in cuprates is understandable in the present model.
1. Introduction In oxide superconductors there are several b a n d s within a few eV from the F e r m i level eF according to b a n d calculations. Since the two-band model has been known to enhance the superconducting transition temperature Tc o f BCS-type superconductivity [ 1,2 ], the search for the secret o f the high Tc in the multiple-band feature is an indispensable a p p r o a c h in the extensive efforts to clarify the mechanism o f highTc superconductivity. W h e n we construct the seco n d - q u a n t i z e d C o u l o m b interaction between electrons which are in two tight-binding b a n d s consisting o f different-type atomic orbitals, 0~ ( r ) a n d 02(r), on identical a t o m i c sites, there arises an off-diagonal C o u l o m b integral on one a t o m i c site, called the exchange-like integral, which is defined by
K- f dr f dr'OT(r)OT(r') x(e2/lr-r'l)O2(r')O2(r).
(1)
Due to the finite value o f K which is r e d u c e d to the Slater integrals and is o f the o r d e r o f 0.5 ~ 1 eV for both Cu 3d a n d O 2p orbitals [ 3 ], there a p p e a r s an i n t e r b a n d pair-transfer interaction through which BCS-like pairs consisting o f electrons with up a n d down spin in one b a n d are transferred between the two bands. This process p r o m o t e s BCS-type super-
c o n d u c t i v i t y [2,4]. The effect o f this interaction was f o u n d to be r e m a r k a b l y e n h a n c e d by a process expressed by i n t e r b a n d l a d d e r - d i a g r a m s [ 5 ]. This enh a n c e m e n t particularly can easily occur when both b a n d s cross the F e r m i level a n d their F e r m i surfaces have the p r o p e r t y o f being nesting [ 5,3 ]. In this paper, however, we report that the e n h a n c e m e n t can be as i m p o r t a n t and cause large effects in the case where the second b a n d does not cross the F e r m i level. A simplified m o d e l in such a situation is treated both analytically a n d numerically. We find that m o d e r a t e sets o f p a r a m e t e r values lead to high-Tc superconductivity. The expression for Tc obtained reveals that the pair-transfer interaction reduces the effective onsite C o u l o m b energy in the c o n d u c t i o n band. This reduction is enormously magnified by the ladderd i a g r a m process, resulting in a negative-U situation where high-To s u p e r c o n d u c t i v i t y appears. This effect m a y be relevant in oxide, organic and transitionmetal superconductors. This result allows one to und e r s t a n d the observation on cuprates that Tc increases with decrease o f the M a d e l u n g potential difference A VM between the in-plane Cu a n d O sites, if we assume ihat the two bands reside mainly on the in-plane oxygen atoms. W h e n we take account o f the BCS-type interaction in addition, the isotope effect is seen to weaken quickly with increase o f To.
0921-4534/91/$03.50 © 1991 - Elsevier Science Publishers B.V. (North-Holland)
K. Yamaji / Negative-U superconductivity in the two-band model
304
t,-k,t
A part of this work was reported in ref. [6].
1,k,l
\
2. Model and interband ladder process The Hamiltonian of our two-band model is defined by 2 j=l
kt7
"~ ~ Uj E E ~kl+k2,*3+k4C}k'~C~*2°'Cjk3~'Cjk4° j= 1 2 N klk2k3k4 ~0'
+
Z
klk2k3k4
6,,+,~,,,+,, ~
(cI,,,c~,~c~,,,c~,,,
+h.c.)
+
~ CTk,~,C~kw'C2kso'ClJ.,.
,
(2)
where ej, is the energy of t h e j t h band in reference o f the Fermi level eF; cj,,,(ctj,~ ) is the annihilation (creation) operator o f the electron in the jth band with wave vector k and spin ~, N is the n u m b e r o f the electronic sites; ~ is the on-site Coulomb energy for the jth band; U' is the interband on-site Coulomb energy between the two bands; we call the interaction with the exchange-like integral K as the coupling constant by this name (or "interband pair-transfer interaction" ), since it causes the transfer of a pair o f electrons with up and down spin between the two bands. We use the convention kB = h = 1. As the irreducible vertex would transfer a BCS pair from the second band to the first, there appear interband ladder diagrams as shown in fig. 1 with the advent o f the pair-transfer interaction in the diagrammatic perturbation expansion in powers o f the interaction terms in eq. (2) [5 ]. The ladder diagrams are s u m m e d up to Fl 2 ( - k , k + q ) = K / R e { [ 1 - U' + K ) H l 2 ( q , (.or) ]
× [ 1 - ( U ' - K ) H T 2 ( q , toy) ]},
(3)
where Re{ } denotes the real part; k and q stand for (k, ton) and (q, toy), respectively; o g n = n T ( 2 n + 1) and my = 2nTu where T is the temperature and n and v are integers; this irreducible part links a BCS state consisting of (2, k + q , ?) and (2, - k - q , $) to another pair state o f (1, - k , 1") and (l, k, J,). Hl2(q,
iiill 2,k+q,t
2,-k-q,~
Fig. 1. Interband ladder diagram. Lines with an arrow are for electron Green's functions specified by a band index, a symbol for wave vector and frequency, and a spin direction. Horizontal dashed lines denote both the interband pair-transfer and Coulomb interactions. ogv) is the interband polarization function given by f(Elk) --f(~2k+q) /-/12(q, f-t/v) = ( 1/N) ~ ia). + e2,+q -Elk '
(4)
w h e r e f ( e ) is the electron distribution function. This is positive at least for 09~= 0 where the absolute value o f this function is maximized. Therefore, eq. (3) means that the effective matrix element F~2 for the pair transfer between the two bands is strongly enhanced compared to the bare value K by the denominator, especially when the factors in the square bracket approach zero. Even in the case where the second band does not cross the Fermi level, this enhancement is possible and can lead to important results such as a high T¢. To demonstrate this, we treat a simplified model with two bands defined in the next section.
3. Simple two-band model The band structure o f our model is illustrated in fig. 2. Band 1 defined by elk--E, is quadratic in k, i.e., ek= ( k 2 - k 2 ) / 2 m with electron band mass m and is nearly half filled. We choose the zero level o f energy at the Fermi energy Ev, as shown in fig. 2. Band 2 is assumed to be completely flat lying at E2 > 0. This model mimics the oxide superconductors but the electron-hole reversal is made for simplicity. Whether band 2 is full or empty does not matter. For simplicity we assume that the system is two-dimen-
K, Yamaji / Negative-Usuperconductivity
in the two-band model
305
A( lk, 2k’)= k
-(T/N)T,~(k,k’)G(2k’)(2,
-k’)
>
(7b)
1, -k’)
,
(7~)
A(2k, lk’)= - (T/N)T:,(k’,
&,=O
k)G( lk’)G(
A(2k, 2k’)= -(T/N)U2G(2k’)G(2,
Fig. 2. Band structure model with band 1 and 2 defined by clr=(kZ-k:)/2m and elk=&, respectively. The Fermi level cF is set to zero. D denotes the band width and B the width of the occupied part.
sional so that the state density N(E) of band 1 is constant; the results do not critically depend on the dimensionality of the system. Then, we get at the absolute zero temperature
-k’),
(7d)
where 1k, e.g., denotes (k, w,) of a band-l electron; G(lk)=l/(iw,-e,); G(2k)=I/(iw,-E,). In eq. (7a) we take account of the enhancement of U, due to the intraband ladder-diagrams; the q- and o,-dependences are neglected, since they do not seem to be important in this part of the calculations. Such an enhancement does not occur for U, since band 2 is off tF. We have neglected the interband bubble process which was shown to grow as well, P approaching unity, since it is overwhelmed by the ladder process [51.
where D= 2n/ma2 is the band width defined by the approximate radius 2&/a of the Brillouin zone with a being the lattice constant of the assumed square lattice; we get N( 0) = 1/D, B is the occupied energy range of band 1. When the factor P defined by P= [(u’
+K)/D]
ln[ (E2+B)/E2]
(6)
approaches unity, r,2 in eq. (3) is enhanced so that superconductivity is expected to occur. We investigate the parameter dependence of the superconducting transition temperature T, in the next section.
4.2. Approximate expression for T, In the eigenvalue C A(ik,jk’) Jk’
equation
for the matrix A
u(jk')=h(ik)
,
u(2k) has no k-dependence and a weak w,-dependence. Approximately equating u( 2k) = constant, we obtain T, as T,= l.l3,/mexp(
l/p)
A(lk,
T=K/
lk’)=-(T/N) x[ui/(l-Ui/D)lG(lk’)G(l,
-k’),
(7a)
(9)
(10)
-r2/(2E2+U2)l,
T, is determined as the temperature at which the full vertex of the Cooper channel starts to diverge in the normal state side. As is shown in a previous paper[ 31, this condition is reduced to the condition that unity is reached by the largest eigenvalue of the matrix A defined by the following matrix elements:
,
~=NO)[U,/(l-U,/D)
4. Determination of T, 4.1. Eigenvalue problem determining T,
(81
1--
U’+KlnE2+B ~ D E,
>I .
(11)
This result for T, was obtained in ref. j2] with the assumption of UI =O but the effective enhancement of K to r was not noticed and r in eq. ( 10) was set to K. It should be noted that this enhancement brings about a surprisingly large effect. When E2 is small enough for the P factor defined
306
K. Yamaji / Negative-U superconductivity in the two-band model
by eq. (6) to approach unity, which can easily occur for moderate sets of parameter values, F is divergently magnified in eq. ( 11 ). Then,/t in eq. (10) is so much reduced that it becomes a large negative number, leading to a large value of T~ in eq. (9). Negative/~ means a negative-U situation in band 1. This effective reduction of U1 is considered to be caused by the virtual excitation of the electron pair from band 1 to band 2 through the pair-transfer interaction. Further, this reduction is enormously magnified by the effective enhancement of K to F due to the interband ladder process. When in addition we taken account of the BCS interaction in band 1 with its dimensionless coupling constant 2 and cut-off energy tOo, Tc is given by Tc = 1.13 tOo exp[ - 1/ ( 2 - / t * ) ] , /z*=p/[ I +/t ln(x/B(D-B)/too)
(12)
].
(13)
With the increase of the role of negative ~t in driving superconductivity, the isotope effect weakens, as is seen from a _ 0 In Tc/O In tOD = 1 -- [/t*/(2--/t*) ]2.
(14)
When T¢ is high, I/t* I >> 2. In cuprate superconductors the value for this derivative is reported to be 0.32 and 0.08 with T~ values 33 and 92 K for L a l . 8 5 S r o . 1 5 C u O 4 [ 7 ] and YBCO [ 8 ], respectively. In fig. 3 these values are plotted in the 2or versus Tc plane and fitted by a curve defined by eqs. ( 1 2 ) (14). Data for BPBO [9] and BKB [10,11] are added. The parameter values 2=0.14 and o90= 110 kB are chosen as common parameters as a crude approximation. Although these values may be Slightly too small, the acceptable quality of fitting is. encouraging.
4.3. Numerical solution The value of T~ was also derived by numerically calculating the largest eigenvalue of matrix,4 defined by eq. (7). In the eigenvalue equation (8), there is neither dependence on k of band 2 nor dependence on the direction of k of band I. Choosing discrete points of k for band l and restricting the employed range ofto~ as in the previous paper [ 12 ], the matrix was reduced to the size of ~ 300 X 300. A typical ex-
_ Q
C
0
Tc (K)
I00
Fig. 3. Curve for 01n Tc/01n O~DVS. Tc defined by eqs. (12) and ( 14 ) with 2 = 0.14 and mD= l l 0 ks- The closed circle denotes the observed values for YBCO [ 8 ]; closed square for La 1.85S1"0.15CUO4 [7]; cross for BPBO [ 9]; open and closed triangles for BKB [ 10,11 ], respectively.
ample of the results is shown in table 1 for the case where U , = U 2 = U ' = - U = I eV, K = 0 . 5 eV and D / 2 = B = 1 eV. As the P factor approaches unity with decrease of E2, Tc becomes finite and grows enormously large. T¢ is in units ofeV ( ~ 10 000 K), since all relevant quantities are of the order of 1 eV. T~ appr~ is the value of T¢ given by eq. (9). It is in fair agreement with the numerical results in the above mentioned features. The eigenvector u ( l k ) has a broad bump around Ik l = k F and it quickly diminishes with the increase of Ito, IIn fig. 4 the critical value E~ cr~ of E2 below which superconductivity appears is plotted as a function of K for five values of U = - U1 = U2= U'. With the increase of K, E~ Cr~ increases. In a wide range of K, E~ ~ increases with increasing U. This must be due to a large enhancement ofF12 in eq. (3) for a larger U'. Figure 5 shows E~ cr~ determined from/~=0 with eqs. (10) and ( 11 ). Although this approximate E~ "~ is slightly larger than the numerical one, it reproduces the qualitative features. Especially the limiting value ofE~ cr~ for K ~ 0 is in good agreement. It is determined by the divergence of F i n eq. ( 11 ) and given by E ~ cr)
=n/[exp(V/U'
)- 1] .
(15)
In the case of U= 0, there is a qualitative difference. E~ c° is infinite from eqs. (10) and ( 11 ) but it is actually finite according to numerical calculation as
K. Yamaji / Negative-U superconductivity in the two-band model
307
Table 1 The values of Tc numerically obtained tabulated as a function of P defined by eq. (6). The value of E2 is given by P from eq. ( 6 ) . / t is defined by eq. (10). T c(appr) is given by eqs. ( 9 ) - ( 1 1 ) . Parameter values employed are K=0.5 eV, Ul = U2= U'-= U = 1 eV and D~ 2 = B = 1 eV P
0.5 0.6 0.7 0.75 0.8 0.82 0.84 0.86 0.88 0.9
1.5
I
U = 1.8 e
E2 (eV)
/t
1.06 0.82 0.65 0.58 0.52 0.50 0.48 0.47 0.45 0.43
0.77 0.54 --0.03 -0.64 - 1.84 -2.64 -3.78 - 5.49 - 8.17 - 12.70
T c(appr)
Tc
(eV)
(eV)
1 × 10 -15 0.24 0.66 0.77 0.87 0.94 1.00 1.04
0.01 0.10 0.20 0.31 0.45
2.0
I
V
~
I
I
~
A
1.5
1.0
1.0
" ' 0.5
U = 1.8 eV 1.5
ILl
0.5
1.0
1.5
0.5 ~0.5
K(eV) I
Fig. 4. The upper bound E~~) of Ez for the appearance of super-
conductivity is plotted against the exchange-like integral K for five values of U---U~= U2= U'. E~ or> was obtained by numerically solving the largest eigenvalue of the matrix A defined by eq. (7).
shown in fig. 4. For superconductivity actually to occur, it seems necessary that p defined by eq. (10) is smaller than a finite negative value.
5. Discussion
5.1. Aspects of the theory to be pursued We have neglected the effect o f F|2 to the self-energy and vertex correction, as is usually done in many theories of non-conventional mechanism of superconductivity, since we want to show the possibility of high Tc due to the enhanced FI~ in a simple and
0
0.5
1.0
1.5
K (eV) Fig. 5. E~Cr) vs. Kfor four values of U-~ Uz = U2= U' determined by solving # = 0 with eqs. (10) and ( 11 ). For U=0, E~ or) is infinite; p < 0 for any value of E2.
clear way. N o w that this possibility is established here, these are the problems in the next step. Numerical calculation of correlation functions of the superconducting order parameter in finite-size systems seems to be a more straight-forward way to check the validity of the present result. The remarkable denominator o f F~2 in eq. (3) is confirmed by Entin-Wohlman and Imry by an ordinary RPA treatment, in which it is obtained as due to screening (actually anti-screening) of the external field acting on the spin density [ 13 ]. The values of Tc obtained for moderate sets of parameter values are tremendously large. In actual systems the appreciable width of band 2 and also a large
308
K. Yamaji / Negative-U superconductivity in the two-band model
value of U2 must moderate the increase of To. Further, T~ may be much lowered by pair breaking effects [ 14 ] which may be inherent in oxides but are not taken into account in the present treatment. Another possibility is that the large effective negativeU situation may bring about a metal-insulator transition which hinders the realization of tremendously high T~. Other consequences resulting from the effective negative-U situation arising in the present model are also problems to be pursued. The present results are based on the assumption of weak coupling. We presume, however, that their applicability region is extended to the intermediate coupling region, since such is often the case in other problems. 5.2. Applicability to oxides
We expect that the present results are relevant to oxide, organic and transition-metal superconductors. Concerning oxides, it has been established that the stable observed high-To superconductivity in cuprates occurs in the CuO2 plane. It is becoming more and more clear that at least one Fermi-liquid-like band crosses the Fermi level [15 ]. We expect it to be composed of Cu 3dx2_y2 and O 2p~ orbitals. As for the second band, there are results suggesting the presence near eF of bands mainly composed of other O 2porbitals such as in-plane p* [ 16,17] and p~(z) [ 18 ] and also of the band composed ofCu 3dz2 [ 19 ]. All the bands near ~F contribute additively in enhancing Tc in the present scheme. However, we are tempted to consider that an O 2p~ band gives a predominant contribution in cuprate superconductors due to the reason below. For the Cu 3d orbitals the values of the on-site Coulomb energy may be too large. The observed value of Tc has a tendency to increase with decrease of the difference AVM of the Madelung potentials between the in-plane Cu and O sites, i.e., with increase of the electron Madelung potential VM(O) for O in reference of that of Cu [20]. This is in accordance with the observation that with an increase of T~, the hole number of the O site increases [21,22]. Since with the increase of VM(O) in reference to VM(Cu), the center of the O p~ bands is considered to be shifted upwards, resulting in a smaller value of E2, our model can explain the ob-
servation if we identify one or two of the O p~ bands as the second band. This picture may be opposed with the vanishing of Tc in the overdoped LaSrCuO systems. The latter effect may be due to the lowering of the O p~ bands pressed down by the doped holes through the interband on-site Coulomb energy or due to the spill of holes to the apical oxygen, which causes a wider distribution of the Bloch function amplitude in the unit cell and consequently diminishes the effective values of K and U' as coupling constants. The assumption of U~ = U2 = U' ( = U) is for presenting the essence of the results in a simple form but it is considered to be a fair approximation. In the present model U~ cannot exceed band width D, since then U~/( 1 - U t / D ) diverges in eq. (7a), causing a ferromagnetic instability. With D ~ 2 eV, the employable value of U is smaller than the values employed for the O 2p orbitals in models treating the insulator state, e.g., ~ 4 eV [23 ]. We suppose, however, that the values of U~ and U', are renormalized in the metallic state from the bare values due to the effect of, e.g., bubble diagrams which are not considered explicitly here. The renormalized values must surely be smaller than D and of the order of D / 2 ~ 1 eV. Since the present mechanism drives the s-wave superconductivity, the electron-phonon interaction must work in enhancing Tc in concert and inscribe its features in the gap parameters etc. But it is the question if the latter interaction alone can drive the high Tc of the oxides or not. The present results suggest that/t* might take an appreciable negative value. Theoretical investigation on the isotope effect was animated by the recent work of Tsuei et al. [24], which showed that the isotope exponent ot sensitively depends on the fine structure of the state density near the chemical potential and consequently on the doping level in L a - S r - C u - O . Carbotte et al. showed that ot may be increased with a decrease of Tc due to the effect of pair breaking [25]. These works, however, give no explanation for the tendency of decrease of ot with increase of the highest Tc achieved in each group such as L a - S r - C u - O or Y - B a - C u - O . The present work provides one.
K. Yamaji / Negative-U superconductivity in the two-band model
6. Summary We have studied T¢ of a two-band model with a c o n d u c t i o n b a n d a n d a dispersionless empty b a n d off the Fermi level. We took account of the exchange-like or interband pair-transfer interaction a n d all kinds of on-site C o u l o m b energies. The i n t e r b a n d ladder process is pointed out to largely enhance the i n t e r b a n d pair-transfer process in the present situation as well as for moderate sets of parameter values. F r o m the c o n d i t i o n of the divergence of the Cooper channel vertices, we have derived an approximate expression of Tc. Its semiquantitative validity was assured by the value of Tc numerically calculated. The expression for Tc suggests,that the superconductivity results from the reduction of the on-site C o u l o m b energy due to the virtual excitation of two carriers to the second band, which the i n t e r b a n d ladder process enhances so much that a negative-U situation is realized. W h e n we consider the BCS-type attractive interaction in addition, the isotope index was found to quickly decrease with increase of T¢. The observation of increase of Tc with increase of the Madelung potential difference A VM may be in accordance with the presence model. It looks possible that the present model provides the m e c h a n i s m of the high T¢ in oxide superconductors.
Acknowledgements The author is heartily grateful to Prof. J. K o n d o for encouragement. He is also thankful to Dr. Y. Asai a n d Dr. T. Yanagisawa for discussions.
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[2 ] J. Kondo, Prog. Theor. Phys. 29 (1963) 1. [ 3 ] K. Yamaji, J. Phys. Soc. Jpn. 59 (1990) 677. [4] K. Yamaji and S. Abe, J. Phys. Soc. Jpn. 56 (1987) 4237. [ 5 ] K. Yamaji, Solid State Commun. 64 ( 1988) 1157. [6] K. Yamaji and T. Kinoshita, in: Proc. of Int. Symp. Superconductivity (Nov. 1990, Sendai), to be published. [7] B. Batlogg, G. Kourouklis, W. Weber, R.J. Cava, A. Jayaraman, A.E. White, K.T. Short, L.W. Rupp and E.A. Rietman, Phys. Rev. Lett. 59 (1987) 912. [8] K.J. Leary, H.C. zur Loye, S.W. Keller, T.A. Faltens and W.K. Ham, Phys. Rev. Lett. 59 (1987) 1236. [9 ] B. Batlogg, R.J. Cava and M. Stavola, Phys. Rev. Lett. 60 (1988) 754. [10] D.G. Hinks, D.R. Richards, B. Dabrowski, D.T. Marx and A.W. Mitchell, Nature 335 (1988) 419. [1 I]B. Batlogg, R.J. Cava, L.W. Rupp Jr., A.M. Mujsce, J.J. Krajewski, J.P. Remeika, W.P. Peck Jr., A.S. Cooper and G.P. Espinosa, Phys. Rev. Lett. 61 (1988) 1670. [ 12] In ref. [3], eq. (3.9) must be corrected by multiplying the summand by N(kjvJkjlkyj,Jkj,r )1/2. [ 13]0. Entin-Wohlman and Y. Imry, Phys. Rev. B 40 (1989) 6731. [ 14] P.A. Lee and N. Read, Phys. Rev. Lett. 56 (1987) 2691. [ 15 ] T. Takahashi, H. Matsuyama, H. Katayama-Yoshida, Y. Okabe, S. Hosoya, K. Seki, H. Fujimoto, M. Sato and H. Inokuchi, Phys. Rev. B 39 (1989) 6636. [ 16] Y. Guo, J.M. Langlois and W.A. Goddard III, Science 239 (1988) 896. [ 17] Y. Asai, J. Phys. Soc. Jpn. 58 (1989) 3264. [ 18 ] M. Hirao, T. Uda and Y. Murayama, 2nd ISSP Int Symp. on Phys. Chem. of Oxide Superconductors (Jan. 1991, Tokyo ); M. Hirao, private communication. [ 19] e.g., K. Takegahara, H. Harima and A. Yanase,Jpn. J. Appl. Phys. 26 (1987) L352. [20] J.B. Torranceand R.M. Metzger, Phys. Rev. Lett. 63 (1989) 1515. [21 ] D.M. de Leeuw, W.A. Groen, L.F. Feiner and E.E. Havinga, Physica C 166 (1990) 133. [22] K. Asayama, K. Kitaoka, K. Fujiwara, K. Ishida, S. Ohsugi and G. Zheng, Abstract of the Spring Meeting of Phys. Soc. Jpn. Part 3 (1991) 173. [23 ] e.g., Y. Ohta, T. Tohyama and S. Maekawa, Phys. Rev. Lett. 66 (1991) 1228. [24] C.C. Tsuei, D.M. Newns, C.C. Chi and P.C. Pattnaik, Phys. Rev. Lett. 65 (1990) 2724. [25] J.P. Carbotte, M. Greeson and A. Perez-Gonzalez, Phys. Rev. Lett. 66 (1991) 1789.
Negative-U superconductivity in the two-band model with the pairtransfer interaction K. Y a m a j i Electrotechnical Laboratory, 1-1-4 Umezono, Tsukuba 305, Japan Received 16 April 1991
The superconducting transition is studied in a two-band model which has an interband pair-transfer interaction between the conducting band and a hidden band not crossing the Fermi level. The interband ladder-diagram process enhances the effect of the pair-transfer interaction, remarkably reducing the effective on-site Coulomb energy and leading to high-Tosuperconductivity with a negative U. The isotope effect is shown to weaken with increasing To, if in addition we take account of the BCS interaction. The tendency of increase of Tc with decrease of the Madelung potential difference AVMbetween Cu and O sites in cuprates is understandable in the present model.
1. Introduction In oxide superconductors there are several b a n d s within a few eV from the F e r m i level eF according to b a n d calculations. Since the two-band model has been known to enhance the superconducting transition temperature Tc o f BCS-type superconductivity [ 1,2 ], the search for the secret o f the high Tc in the multiple-band feature is an indispensable a p p r o a c h in the extensive efforts to clarify the mechanism o f highTc superconductivity. W h e n we construct the seco n d - q u a n t i z e d C o u l o m b interaction between electrons which are in two tight-binding b a n d s consisting o f different-type atomic orbitals, 0~ ( r ) a n d 02(r), on identical a t o m i c sites, there arises an off-diagonal C o u l o m b integral on one a t o m i c site, called the exchange-like integral, which is defined by
K- f dr f dr'OT(r)OT(r') x(e2/lr-r'l)O2(r')O2(r).
(1)
Due to the finite value o f K which is r e d u c e d to the Slater integrals and is o f the o r d e r o f 0.5 ~ 1 eV for both Cu 3d a n d O 2p orbitals [ 3 ], there a p p e a r s an i n t e r b a n d pair-transfer interaction through which BCS-like pairs consisting o f electrons with up a n d down spin in one b a n d are transferred between the two bands. This process p r o m o t e s BCS-type super-
c o n d u c t i v i t y [2,4]. The effect o f this interaction was f o u n d to be r e m a r k a b l y e n h a n c e d by a process expressed by i n t e r b a n d l a d d e r - d i a g r a m s [ 5 ]. This enh a n c e m e n t particularly can easily occur when both b a n d s cross the F e r m i level a n d their F e r m i surfaces have the p r o p e r t y o f being nesting [ 5,3 ]. In this paper, however, we report that the e n h a n c e m e n t can be as i m p o r t a n t and cause large effects in the case where the second b a n d does not cross the F e r m i level. A simplified m o d e l in such a situation is treated both analytically a n d numerically. We find that m o d e r a t e sets o f p a r a m e t e r values lead to high-Tc superconductivity. The expression for Tc obtained reveals that the pair-transfer interaction reduces the effective onsite C o u l o m b energy in the c o n d u c t i o n band. This reduction is enormously magnified by the ladderd i a g r a m process, resulting in a negative-U situation where high-To s u p e r c o n d u c t i v i t y appears. This effect m a y be relevant in oxide, organic and transitionmetal superconductors. This result allows one to und e r s t a n d the observation on cuprates that Tc increases with decrease o f the M a d e l u n g potential difference A VM between the in-plane Cu a n d O sites, if we assume ihat the two bands reside mainly on the in-plane oxygen atoms. W h e n we take account o f the BCS-type interaction in addition, the isotope effect is seen to weaken quickly with increase o f To.
0921-4534/91/$03.50 © 1991 - Elsevier Science Publishers B.V. (North-Holland)
K. Yamaji / Negative-U superconductivity in the two-band model
304
t,-k,t
A part of this work was reported in ref. [6].
1,k,l
\
2. Model and interband ladder process The Hamiltonian of our two-band model is defined by 2 j=l
kt7
"~ ~ Uj E E ~kl+k2,*3+k4C}k'~C~*2°'Cjk3~'Cjk4° j= 1 2 N klk2k3k4 ~0'
+
Z
klk2k3k4
6,,+,~,,,+,, ~
(cI,,,c~,~c~,,,c~,,,
+h.c.)
+
~ CTk,~,C~kw'C2kso'ClJ.,.
,
(2)
where ej, is the energy of t h e j t h band in reference o f the Fermi level eF; cj,,,(ctj,~ ) is the annihilation (creation) operator o f the electron in the jth band with wave vector k and spin ~, N is the n u m b e r o f the electronic sites; ~ is the on-site Coulomb energy for the jth band; U' is the interband on-site Coulomb energy between the two bands; we call the interaction with the exchange-like integral K as the coupling constant by this name (or "interband pair-transfer interaction" ), since it causes the transfer of a pair o f electrons with up and down spin between the two bands. We use the convention kB = h = 1. As the irreducible vertex would transfer a BCS pair from the second band to the first, there appear interband ladder diagrams as shown in fig. 1 with the advent o f the pair-transfer interaction in the diagrammatic perturbation expansion in powers o f the interaction terms in eq. (2) [5 ]. The ladder diagrams are s u m m e d up to Fl 2 ( - k , k + q ) = K / R e { [ 1 - U' + K ) H l 2 ( q , (.or) ]
× [ 1 - ( U ' - K ) H T 2 ( q , toy) ]},
(3)
where Re{ } denotes the real part; k and q stand for (k, ton) and (q, toy), respectively; o g n = n T ( 2 n + 1) and my = 2nTu where T is the temperature and n and v are integers; this irreducible part links a BCS state consisting of (2, k + q , ?) and (2, - k - q , $) to another pair state o f (1, - k , 1") and (l, k, J,). Hl2(q,
iiill 2,k+q,t
2,-k-q,~
Fig. 1. Interband ladder diagram. Lines with an arrow are for electron Green's functions specified by a band index, a symbol for wave vector and frequency, and a spin direction. Horizontal dashed lines denote both the interband pair-transfer and Coulomb interactions. ogv) is the interband polarization function given by f(Elk) --f(~2k+q) /-/12(q, f-t/v) = ( 1/N) ~ ia). + e2,+q -Elk '
(4)
w h e r e f ( e ) is the electron distribution function. This is positive at least for 09~= 0 where the absolute value o f this function is maximized. Therefore, eq. (3) means that the effective matrix element F~2 for the pair transfer between the two bands is strongly enhanced compared to the bare value K by the denominator, especially when the factors in the square bracket approach zero. Even in the case where the second band does not cross the Fermi level, this enhancement is possible and can lead to important results such as a high T¢. To demonstrate this, we treat a simplified model with two bands defined in the next section.
3. Simple two-band model The band structure o f our model is illustrated in fig. 2. Band 1 defined by elk--E, is quadratic in k, i.e., ek= ( k 2 - k 2 ) / 2 m with electron band mass m and is nearly half filled. We choose the zero level o f energy at the Fermi energy Ev, as shown in fig. 2. Band 2 is assumed to be completely flat lying at E2 > 0. This model mimics the oxide superconductors but the electron-hole reversal is made for simplicity. Whether band 2 is full or empty does not matter. For simplicity we assume that the system is two-dimen-
K, Yamaji / Negative-Usuperconductivity
in the two-band model
305
A( lk, 2k’)= k
-(T/N)T,~(k,k’)G(2k’)(2,
-k’)
>
(7b)
1, -k’)
,
(7~)
A(2k, lk’)= - (T/N)T:,(k’,
&,=O
k)G( lk’)G(
A(2k, 2k’)= -(T/N)U2G(2k’)G(2,
Fig. 2. Band structure model with band 1 and 2 defined by clr=(kZ-k:)/2m and elk=&, respectively. The Fermi level cF is set to zero. D denotes the band width and B the width of the occupied part.
sional so that the state density N(E) of band 1 is constant; the results do not critically depend on the dimensionality of the system. Then, we get at the absolute zero temperature
-k’),
(7d)
where 1k, e.g., denotes (k, w,) of a band-l electron; G(lk)=l/(iw,-e,); G(2k)=I/(iw,-E,). In eq. (7a) we take account of the enhancement of U, due to the intraband ladder-diagrams; the q- and o,-dependences are neglected, since they do not seem to be important in this part of the calculations. Such an enhancement does not occur for U, since band 2 is off tF. We have neglected the interband bubble process which was shown to grow as well, P approaching unity, since it is overwhelmed by the ladder process [51.
where D= 2n/ma2 is the band width defined by the approximate radius 2&/a of the Brillouin zone with a being the lattice constant of the assumed square lattice; we get N( 0) = 1/D, B is the occupied energy range of band 1. When the factor P defined by P= [(u’
+K)/D]
ln[ (E2+B)/E2]
(6)
approaches unity, r,2 in eq. (3) is enhanced so that superconductivity is expected to occur. We investigate the parameter dependence of the superconducting transition temperature T, in the next section.
4.2. Approximate expression for T, In the eigenvalue C A(ik,jk’) Jk’
equation
for the matrix A
u(jk')=h(ik)
,
u(2k) has no k-dependence and a weak w,-dependence. Approximately equating u( 2k) = constant, we obtain T, as T,= l.l3,/mexp(
l/p)
A(lk,
T=K/
lk’)=-(T/N) x[ui/(l-Ui/D)lG(lk’)G(l,
-k’),
(7a)
(9)
(10)
-r2/(2E2+U2)l,
T, is determined as the temperature at which the full vertex of the Cooper channel starts to diverge in the normal state side. As is shown in a previous paper[ 31, this condition is reduced to the condition that unity is reached by the largest eigenvalue of the matrix A defined by the following matrix elements:
,
~=NO)[U,/(l-U,/D)
4. Determination of T, 4.1. Eigenvalue problem determining T,
(81
1--
U’+KlnE2+B ~ D E,
>I .
(11)
This result for T, was obtained in ref. j2] with the assumption of UI =O but the effective enhancement of K to r was not noticed and r in eq. ( 10) was set to K. It should be noted that this enhancement brings about a surprisingly large effect. When E2 is small enough for the P factor defined
306
K. Yamaji / Negative-U superconductivity in the two-band model
by eq. (6) to approach unity, which can easily occur for moderate sets of parameter values, F is divergently magnified in eq. ( 11 ). Then,/t in eq. (10) is so much reduced that it becomes a large negative number, leading to a large value of T~ in eq. (9). Negative/~ means a negative-U situation in band 1. This effective reduction of U1 is considered to be caused by the virtual excitation of the electron pair from band 1 to band 2 through the pair-transfer interaction. Further, this reduction is enormously magnified by the effective enhancement of K to F due to the interband ladder process. When in addition we taken account of the BCS interaction in band 1 with its dimensionless coupling constant 2 and cut-off energy tOo, Tc is given by Tc = 1.13 tOo exp[ - 1/ ( 2 - / t * ) ] , /z*=p/[ I +/t ln(x/B(D-B)/too)
(12)
].
(13)
With the increase of the role of negative ~t in driving superconductivity, the isotope effect weakens, as is seen from a _ 0 In Tc/O In tOD = 1 -- [/t*/(2--/t*) ]2.
(14)
When T¢ is high, I/t* I >> 2. In cuprate superconductors the value for this derivative is reported to be 0.32 and 0.08 with T~ values 33 and 92 K for L a l . 8 5 S r o . 1 5 C u O 4 [ 7 ] and YBCO [ 8 ], respectively. In fig. 3 these values are plotted in the 2or versus Tc plane and fitted by a curve defined by eqs. ( 1 2 ) (14). Data for BPBO [9] and BKB [10,11] are added. The parameter values 2=0.14 and o90= 110 kB are chosen as common parameters as a crude approximation. Although these values may be Slightly too small, the acceptable quality of fitting is. encouraging.
4.3. Numerical solution The value of T~ was also derived by numerically calculating the largest eigenvalue of matrix,4 defined by eq. (7). In the eigenvalue equation (8), there is neither dependence on k of band 2 nor dependence on the direction of k of band I. Choosing discrete points of k for band l and restricting the employed range ofto~ as in the previous paper [ 12 ], the matrix was reduced to the size of ~ 300 X 300. A typical ex-
_ Q
C
0
Tc (K)
I00
Fig. 3. Curve for 01n Tc/01n O~DVS. Tc defined by eqs. (12) and ( 14 ) with 2 = 0.14 and mD= l l 0 ks- The closed circle denotes the observed values for YBCO [ 8 ]; closed square for La 1.85S1"0.15CUO4 [7]; cross for BPBO [ 9]; open and closed triangles for BKB [ 10,11 ], respectively.
ample of the results is shown in table 1 for the case where U , = U 2 = U ' = - U = I eV, K = 0 . 5 eV and D / 2 = B = 1 eV. As the P factor approaches unity with decrease of E2, Tc becomes finite and grows enormously large. T¢ is in units ofeV ( ~ 10 000 K), since all relevant quantities are of the order of 1 eV. T~ appr~ is the value of T¢ given by eq. (9). It is in fair agreement with the numerical results in the above mentioned features. The eigenvector u ( l k ) has a broad bump around Ik l = k F and it quickly diminishes with the increase of Ito, IIn fig. 4 the critical value E~ cr~ of E2 below which superconductivity appears is plotted as a function of K for five values of U = - U1 = U2= U'. With the increase of K, E~ Cr~ increases. In a wide range of K, E~ ~ increases with increasing U. This must be due to a large enhancement ofF12 in eq. (3) for a larger U'. Figure 5 shows E~ cr~ determined from/~=0 with eqs. (10) and ( 11 ). Although this approximate E~ "~ is slightly larger than the numerical one, it reproduces the qualitative features. Especially the limiting value ofE~ cr~ for K ~ 0 is in good agreement. It is determined by the divergence of F i n eq. ( 11 ) and given by E ~ cr)
=n/[exp(V/U'
)- 1] .
(15)
In the case of U= 0, there is a qualitative difference. E~ c° is infinite from eqs. (10) and ( 11 ) but it is actually finite according to numerical calculation as
K. Yamaji / Negative-U superconductivity in the two-band model
307
Table 1 The values of Tc numerically obtained tabulated as a function of P defined by eq. (6). The value of E2 is given by P from eq. ( 6 ) . / t is defined by eq. (10). T c(appr) is given by eqs. ( 9 ) - ( 1 1 ) . Parameter values employed are K=0.5 eV, Ul = U2= U'-= U = 1 eV and D~ 2 = B = 1 eV P
0.5 0.6 0.7 0.75 0.8 0.82 0.84 0.86 0.88 0.9
1.5
I
U = 1.8 e
E2 (eV)
/t
1.06 0.82 0.65 0.58 0.52 0.50 0.48 0.47 0.45 0.43
0.77 0.54 --0.03 -0.64 - 1.84 -2.64 -3.78 - 5.49 - 8.17 - 12.70
T c(appr)
Tc
(eV)
(eV)
1 × 10 -15 0.24 0.66 0.77 0.87 0.94 1.00 1.04
0.01 0.10 0.20 0.31 0.45
2.0
I
V
~
I
I
~
A
1.5
1.0
1.0
" ' 0.5
U = 1.8 eV 1.5
ILl
0.5
1.0
1.5
0.5 ~0.5
K(eV) I
Fig. 4. The upper bound E~~) of Ez for the appearance of super-
conductivity is plotted against the exchange-like integral K for five values of U---U~= U2= U'. E~ or> was obtained by numerically solving the largest eigenvalue of the matrix A defined by eq. (7).
shown in fig. 4. For superconductivity actually to occur, it seems necessary that p defined by eq. (10) is smaller than a finite negative value.
5. Discussion
5.1. Aspects of the theory to be pursued We have neglected the effect o f F|2 to the self-energy and vertex correction, as is usually done in many theories of non-conventional mechanism of superconductivity, since we want to show the possibility of high Tc due to the enhanced FI~ in a simple and
0
0.5
1.0
1.5
K (eV) Fig. 5. E~Cr) vs. Kfor four values of U-~ Uz = U2= U' determined by solving # = 0 with eqs. (10) and ( 11 ). For U=0, E~ or) is infinite; p < 0 for any value of E2.
clear way. N o w that this possibility is established here, these are the problems in the next step. Numerical calculation of correlation functions of the superconducting order parameter in finite-size systems seems to be a more straight-forward way to check the validity of the present result. The remarkable denominator o f F~2 in eq. (3) is confirmed by Entin-Wohlman and Imry by an ordinary RPA treatment, in which it is obtained as due to screening (actually anti-screening) of the external field acting on the spin density [ 13 ]. The values of Tc obtained for moderate sets of parameter values are tremendously large. In actual systems the appreciable width of band 2 and also a large
308
K. Yamaji / Negative-U superconductivity in the two-band model
value of U2 must moderate the increase of To. Further, T~ may be much lowered by pair breaking effects [ 14 ] which may be inherent in oxides but are not taken into account in the present treatment. Another possibility is that the large effective negativeU situation may bring about a metal-insulator transition which hinders the realization of tremendously high T~. Other consequences resulting from the effective negative-U situation arising in the present model are also problems to be pursued. The present results are based on the assumption of weak coupling. We presume, however, that their applicability region is extended to the intermediate coupling region, since such is often the case in other problems. 5.2. Applicability to oxides
We expect that the present results are relevant to oxide, organic and transition-metal superconductors. Concerning oxides, it has been established that the stable observed high-To superconductivity in cuprates occurs in the CuO2 plane. It is becoming more and more clear that at least one Fermi-liquid-like band crosses the Fermi level [15 ]. We expect it to be composed of Cu 3dx2_y2 and O 2p~ orbitals. As for the second band, there are results suggesting the presence near eF of bands mainly composed of other O 2porbitals such as in-plane p* [ 16,17] and p~(z) [ 18 ] and also of the band composed ofCu 3dz2 [ 19 ]. All the bands near ~F contribute additively in enhancing Tc in the present scheme. However, we are tempted to consider that an O 2p~ band gives a predominant contribution in cuprate superconductors due to the reason below. For the Cu 3d orbitals the values of the on-site Coulomb energy may be too large. The observed value of Tc has a tendency to increase with decrease of the difference AVM of the Madelung potentials between the in-plane Cu and O sites, i.e., with increase of the electron Madelung potential VM(O) for O in reference of that of Cu [20]. This is in accordance with the observation that with an increase of T~, the hole number of the O site increases [21,22]. Since with the increase of VM(O) in reference to VM(Cu), the center of the O p~ bands is considered to be shifted upwards, resulting in a smaller value of E2, our model can explain the ob-
servation if we identify one or two of the O p~ bands as the second band. This picture may be opposed with the vanishing of Tc in the overdoped LaSrCuO systems. The latter effect may be due to the lowering of the O p~ bands pressed down by the doped holes through the interband on-site Coulomb energy or due to the spill of holes to the apical oxygen, which causes a wider distribution of the Bloch function amplitude in the unit cell and consequently diminishes the effective values of K and U' as coupling constants. The assumption of U~ = U2 = U' ( = U) is for presenting the essence of the results in a simple form but it is considered to be a fair approximation. In the present model U~ cannot exceed band width D, since then U~/( 1 - U t / D ) diverges in eq. (7a), causing a ferromagnetic instability. With D ~ 2 eV, the employable value of U is smaller than the values employed for the O 2p orbitals in models treating the insulator state, e.g., ~ 4 eV [23 ]. We suppose, however, that the values of U~ and U', are renormalized in the metallic state from the bare values due to the effect of, e.g., bubble diagrams which are not considered explicitly here. The renormalized values must surely be smaller than D and of the order of D / 2 ~ 1 eV. Since the present mechanism drives the s-wave superconductivity, the electron-phonon interaction must work in enhancing Tc in concert and inscribe its features in the gap parameters etc. But it is the question if the latter interaction alone can drive the high Tc of the oxides or not. The present results suggest that/t* might take an appreciable negative value. Theoretical investigation on the isotope effect was animated by the recent work of Tsuei et al. [24], which showed that the isotope exponent ot sensitively depends on the fine structure of the state density near the chemical potential and consequently on the doping level in L a - S r - C u - O . Carbotte et al. showed that ot may be increased with a decrease of Tc due to the effect of pair breaking [25]. These works, however, give no explanation for the tendency of decrease of ot with increase of the highest Tc achieved in each group such as L a - S r - C u - O or Y - B a - C u - O . The present work provides one.
K. Yamaji / Negative-U superconductivity in the two-band model
6. Summary We have studied T¢ of a two-band model with a c o n d u c t i o n b a n d a n d a dispersionless empty b a n d off the Fermi level. We took account of the exchange-like or interband pair-transfer interaction a n d all kinds of on-site C o u l o m b energies. The i n t e r b a n d ladder process is pointed out to largely enhance the i n t e r b a n d pair-transfer process in the present situation as well as for moderate sets of parameter values. F r o m the c o n d i t i o n of the divergence of the Cooper channel vertices, we have derived an approximate expression of Tc. Its semiquantitative validity was assured by the value of Tc numerically calculated. The expression for Tc suggests,that the superconductivity results from the reduction of the on-site C o u l o m b energy due to the virtual excitation of two carriers to the second band, which the i n t e r b a n d ladder process enhances so much that a negative-U situation is realized. W h e n we consider the BCS-type attractive interaction in addition, the isotope index was found to quickly decrease with increase of T¢. The observation of increase of Tc with increase of the Madelung potential difference A VM may be in accordance with the presence model. It looks possible that the present model provides the m e c h a n i s m of the high T¢ in oxide superconductors.
Acknowledgements The author is heartily grateful to Prof. J. K o n d o for encouragement. He is also thankful to Dr. Y. Asai a n d Dr. T. Yanagisawa for discussions.
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