Physcia C 158 (1989) 326336 NorthHolland, Amsterdam
BONDCHARGE R E P U L S I O N AND H O L E SUPERCONDUCTIVITY J.E. HIRSCH Department o f Physics, B019, University o f California, San Diego, La Jolla, CA 92093, USA
Received 23 February 1989 Revised manuscript received22 March 1989
Is is proposed that a bondcharge repulsion term that necessarily arises in deriving a tightbinding Hamiltonian from first principles playsa fundamental role in superconductivity.This term is repulsive for bondingstates and attractive for antibonding states. Togetherwith the reduction of the local Coulombrepulsion due to atomic polarization this effect can explain the occurrence of superconductivityin solids without involvingthe electronphononinteraction.
1. Introduction The trend favoring superconductivity in elements towards the right side of the periodic table is well known [ 1 ]. We have recently proposed [ 2 ] that high Tc superconductivity in oxides is simple the extreme manifestation of this trend: highest Tc will occur when conduction occurs through holes in anions with filled shells, and the high Tc oxides are simply a realization of metallic oxygen. In this paper we analyze a simple fact that supports this point of view that has heretofore been unrecognized: a bondcharge repulsion term that arises in the derivation of a Hubbardlike tight binding model from first principles [ 3,4 ] turns into a bondcharge attraction when hole rather than electron states are at the Fermi surface. Such a term was also found recently through overlap matrix elements of the anion outer shell cloud in a model for conduction of holes through oxygen [5,6]. In ref. [6] the superconducting state that results from such a term was examined, and it was proposed that it contains the essential features of the superconducting state in high Tc oxides. Here we propose that this term together with the reduction of Coulomb repulsion by local atomic polarizability constitute the essential mechanism giving rise to superconductivity in all solids, rather than the electronphonon interaction. In section 2 we discuss the origin of the parameters that enter in a Hubbardlike tight binding Ham
iltonian. Section 3 discusses the properties of the model Hamiltonian obtained, in particular the different properties for electrons and holes. Section 4 discusses in a qualitative way the reasons that can lead to superconductivity for holes in this model Hamiltonian. In section 5 we derive and analyze the equation for the critical temperature within BCS theory, and we conclude in section 6 with a discussion of the implications of these results.
2. Parameters in thightbinding Hamiltonian Consider the derivation of a manybody tight binding Hamiltonian for a single band, as discussed by Hubbard [ 3 ]. The Hamiltonian has the form:
H= ~ Tij(c+cjo+h.c.)+½~',(ijl kl) C ia+Cjop + ClotCka ( ij ) a
ijkt o~7'
(1) with
q ~(rRj) To=fd3rq~.(r_R~)[  ~m h2 V2+ U~(r)_I (2) with and
09214534/89/$03.50 © Elsevier Science Publishers B.V. ( NorthHolland Physics Publishing Division )
Us(r) the single particle ionic pseudopotential,
J.E. Hirsch ~Bondcharge repulsion and hole superconductivity
(ij l kl)= f dard3r' ~o*(rRi)~o(rRk) e2
× ,T~_,,~o*(r'Rj)~o(r'Rt). it¢
j
(3)
In Hubbard's derivation ~o(rRi) is a Wannier state centered at site i, and c + creates an electron in that Wannier state. In that interpretation Tij is obtained from the Fourier transform of the band energy Ek. Here, however, w~ will assume that ~o(rRi) is an atomic orbital at site i, or a linear combination of local atomic orbitals. Although this introduces some complications related to nonorthogonality of atomic orbitals at different sites it makes the physics of the problem more transparent. We should also add that our notation in eq. (1) differs from Hubbard's in that T~j does not include the effect of HartreeFock terms. It should also be pointed out that the Coulomb interaction eq. (3) will be screened in the solid both by atomic electrons not included in our single band Hamiltonian and by conduction electrons themselves. We will return to these points later. Let us assume for simplicity that only nearestneighbor overlap terms are important and longer range overlaps can be neglected. We will also assume that the phase of the atomic orbitals can be chosen so that the sign of~0(rR~) and ~o(rRj) are the same in the region between i and j. This then implies that Tij in eq. (2) is necessarily negative as both the kinetic energy and the single particle potential contribute with the same sign. For two sites, the bonding state ((a~+~02) will then have lower energy than the antibonding state (~0t ~02). The physics is of course that the bonding state has higher charge density in the region between the atoms than the antibonding state and is independent of our choice of phases. The interaction terms eq. (3) involving at most two atoms are:
327
portant. The largest neglected terms will be of the form (4b) for more distant sites but they will be exponentially suppressed by metallic screening. Terms involving three sites, even if nearestneighbors, will be suppressed by at least one additional overlap factor as compared to the ones in eq. (4) and thus can also be neglected. The term we will focus on here is the hopping interaction that arises from the interaction eq. (4d):
Vh=At ~ (c+cj~+h.c.) (ni,_~+nj_o).
(5)
<6> a
Note that At is positive with the phase convention we have chosen for atomic orbitals. Hubbard [ 3] gives the following rough estimates for these parameters in 3d transition metals: U ~ 20 eV, V~6 eV, At~0.5 eV, X ~ 1/40 eV, and further argues that metallic screening will reduce particularly Vto 2 to 3 eV. It is clear that this reduction will be less for At. U will mainly be reduced by local atomic polarization, to perhaps 5 to 10 eV. As a different example [4] for benzene these parameters are estimated at U= 17 eV, V=9 eV, At=3.3 eV, X = 0 . 9 eV [7]. It is physically clear that the ordering U> V> At> Xwill always occur. We will also neglect X [eq. (4c)] in what follows as we expect it to be always substantially smaller than the others because it involves an additional overlap factor.
3. The model Hamiltonian The Hamiltonian that results from the considerations in the previous section has the form:
H=to ~, (c+cj~+h.c.)+U~ n,,n, +V Z ninj <6>
"
+At ~, (c+cj~+h.c.)(ni,_~+nj,_~),
(6)
<6> a
U= (ii] 1/r[ ii),
(4a)
6), X = (0[ 1/r[ji) = (ii[ 1/rJjj), At= ( ii[ 1/r[ ~i) ,
(4b)
V= (6[ 1/r]
(4c) (4d)
and we are omitting site indices on the left because we will assume only nearestneighbor terms are im
with t o =  T o for i, j nearest neighbors and ni= ni~+ ni_~. Recall also that the parameters to, At, U, V are all positive. Equation (6) is unfortunately not quite correct if atomic orbitals at different sites are not orthogonal. The normalized eigenstates of the single particle Hamiltonian with nearest neighbor hoppings only defined by eq. (2) are:
J.E. Hirsch ~Bondcharge repulsion and hole superconductivity
328
1 1 ~ (r)  x/lv~ ( 1 + S 2 a e ik'a) 1/2 E eik'RqPi(r) 1
(7)
with d a vector connecting nearest neighbor sites, and S = (~0o, (o~) the nearest neighbor overlap integral (positive with our phase convention). The single particle energies are to ~6 eik'd Ek=  1 qS~6e it''~
(8)
which defines a kdependent hopping matrix element. When we transform back to real space the hopping Hamiltonian will involve further than nearestneightbor hoppings. This is of course what we would have obtained starting from a Wannier representation at the outset. An approximate way out to keep the simple physics of atomic orbitals is to use fillingdependent parameters. In particular, the effective nearestneighbor hopping at the Fermi surface is: t0
to(n) = 1+ S ( E6ei*'6),~
(9)
where we have averaged in the denominator over the Fermi surface. Similarly, the interaction parameters at the Fermi surface will be renormalized: U
U~ U(n) = ( 1 + S ( Eaeit"a),v)z
(10)
and the other interaction parameters in the same way. Note that the interactions becomes stronger as the band filling increases, as S > 0 . Equation (9) could suggest that the hopping also becomes larger, i.e. the effective mass decreases. However, the hopping is modified by the Hartree terms that arise from the last two terms in eq. (6). The effective single particle hopping is:
t(n) =
to + nb V ( n ) / 4  nAt(n) 1 +S( E6eik"~),~
(11)
where n is the average site charge density
n=(~nio) and nb is the average bond charge density:
(12)
nb= (<~) (C~Cj~+h.c.)l "
(13)
The contribution from the hopping interaction tends to decrease t(n) as the band is filled, and n increases steadily from 0 to 2. The bond charge nb first increases from 0, goes through a maximum and then decreases to 0 as the band becomes full. Assuming the effect of At is larger than the effect of the denominator, eq. ( 11 ) suggests then that the effective mass for holes at the top of the band can be substantially larger than for electrons at the bottom of the band. This is indeed often observed [8,9]. Indeed, it can even happen that the At correction in eq. ( 11 ) causes a change in sign o f t ( n ) for n approaching 2. In that case, the single panicle band will develop a maximum and then drop as we approach the edge of the Brillouin zone. This is also often observed [ 8 ]. Consider now the sign of the hopping interaction eq. ( 5 ). The essential point is that it depends on the phases of the wave function. Near the bottom of the band ( k ~ 0) it is repulsive, as the wave function has the same sign on nearest neighbor sites. Near the top of the band, however, the wave function has opposite sign on nearestneighbor sites and the interaction is attractive. This is seen clearly if we make a particlehole transformation together with a sublattice rotation to interchange k = 0 and k = m c i+~_ _ (   l ) cii av
(14)
assuming we have a bipartite lattice structure (sc or bcc, for example). The interaction eq. (5) becomes: V~°le~=At
~ (c'~*~c~o+h.c.)(n~_o+n~_~)
(15)
while the hopping remains of the same sign, but n in eq. ( 11 ) is replaced by 2  nh, with nh the number of holes. The interaction eq. (15) is most attractive at the bottom of the hole band, i.e. for few number of holes. The reader might be surprised at the lack of symmetry in these considerations. The fact that we have obtained an attractive interaction for holes starting with a repulsive Coulomb interaction for electrons may appear rather mysterious. Shouldn't one pay a price for this? In fact one does, in the increase in the other Coulomb repulsions due to the correction eq.
J.E. Hirsch~Bondchargerepulsionand holesuperconductivity (10). To further clarify this issue, let us consider the interaction parameters for a simple diatomic molecule. Figure 1 shows schematically the bonding and antibonding orbitals for a diatomic molecule: ~b
~, (r) +~02(r) x/2(l+S) ,
(16a)
~o~(r)  ~02(r) ~bCa= x / / ~ l _ S ) '
(16b)
with S = ( ~ , ~02) the overlap integral. The Coulomb repulsion for two electrons in the bonding state is: [~b(r)q/b(r') ~
~b(r)~b(r')]
U+ V+ 2X 4At 2(1+S) z +2(1+S) 2

(17a)
and in the antibonding state:
[~(r)qla(r') ~ U+ V+ 2X 2(1_S) 2

¢ta(r)qla(r')1 4At 2 ( 1 _ S ) z.
(17b)
That is, in the antibonding state At gives a negative contribution but the other terms give a larger posititive contribution than in the bonding state, because S > 0. This is, of course, obvious from fig. (1): the charge density at the sites is larger in the antibonding than in the bonding state, giving rise to larger site Coulomb repulsions. Superficially, it may appear that what we gain in an attractive hopping interaction we lose in the larger onsite and nearestneighbor repulsion. That this is, however, not quite so is the topic of the next section.
4. How can holes do it?
The Hamiltonian for holes in a single band that emergies from the previous considerations has the form: Z (c,~cj~+h.c.)+U + Zi ni~ni,+V ~ n, nj
M=t
a
At
"
(cj+cj~+h.c.)(ni_o+nj_o) (ij} cr
(18)
329
where we are now using c + to denote a hole creation operator. Strictly speaking, the parameters in eq. (18) depend somewhat on hole density, as discussed in the previous section. We will only take into account qualitatively in what follows. There are several reasons why the Hamiltonian eq. (18) can give to superconductivity for realistic parameters, which are discussed in what follows. 4. I. Reduction of U by local atomic polarization As discussed in the previous section, Uis larger for holes than for electrons because of the larger onsite charge density in the antibonding state. However, this increase can be offset to a large extent by the larger interaction of the hole with the atomic shell that reduces the effective U on the site. If ot denotes the interaction of the hole with the electronic shell in the atom, the bare U is reduced by an energy of order  tx2/to by the relaxation of the electronic cloud, with 09 a typical excitation energy of the cloud. In the antibonding state both U and ot2 are enhanced by a factor (1 + S ) 2 / ( 1  S ) 2 with respect to the bonding state, so that the effective U on the atom is not necessarily larger in the antibonding state compared to the bonding state, despite the increased site charge density. For atoms with many electrons in the outer shell (anions) this reduction of U will be substantially more effective than for those with few electrons in the outer shell (cations). Thus, the enhancement of the local U in the antibonding state will difficult to overcome in the left region of the periodic table and easier in the right region. Similarly, this local screening will become more effective for atoms with more electrons, i.e. moving down in the periodic table. Note that this screening by localized charge can be expected to be substantially more effective for onsite interactions than for interactions involving interstitial charge density. 4.2. Phase space factor Consider the interaction that enters in a reduced Hamiltonian for Cooper pairs: 1
(19)
330
J.E. Hirsch ~Bondcharge repulsion and hole superconductivity
4.4. Momentum dependence of the interaction
From eq. (18) we obtain: Vkk = 2At ~ (eik'~+e i*''6) + U+ V ~ e i(kk')'6
(2o) with 6 the vectors connecting nearest neighbors. For zero momenta k and k' this yields: Vo0 =  4zAt+ U + z V .
(21)
It can be seen that the hopping interaction acquires a large multiplicative factor due to "phase space" as compared to the onsite repulsion U. Thus, the onsite repulsion becomes increasingly unimportant as the number of nearest neighbors increases. This fact was discussed in ref. [6]. The nearest neighbor repulsion increases also with the number of neighbors but the hopping interaction has an extra factor of 4. It can be seen that the very rough estimates given in section 2 are not too far from satisfying the condition Voo< 0: If we take U= 10 eV, V=2 eV, At=0.85 eV, for example, Voo is attractive for z > 7.
4.3. Enhancement of At from overlap factors The interaction eq. ( 15 ) implies that a larger amplitude for hopping of holes exists if other holes are present in the atom one is hopping to or from. Such a modulated hopping interaction can arise from overlap (FrankCondon) factors if interaction with other atomic degees of freedom is included. Zawadowski [10] has recently discussed such an "occupationinfluenced hopping rate" for a twoband model. In refs. [5] and [6] an interaction of the form eq. ( 15 ) was derived for an oxygen hole metal from the larger overlap of the atomic outer shell wave function with one and two holes as compared to the one with zero and one hole. This is likely to be a general phenomenon for holes, although to a lesser degree in atoms other than anions. As the outer shell of the atom becomes emptier when holes are added, the disruption caused by the second hole will be smaller than the one caused by the first hole because the electrons of the outer shell become increasingly bound to the atom as holes are added. This will then yield an enhanced ability for a hole to hop to or from an atom where another hole of opposite spin is present giving an additional contribution to the hopping interaction term eq. (15).
It was shown in ref. [6 ] that the momentum dependence of the attractive part of the interaction in eq. (20) allows the possibility of superconductivity even for Vkk'> 0 over the entire Brillouin zone. The reason is the same as for electronphonon interactions: the BCS gap function can change sign at high energy where the interaction is repulsive to take advantage of it, or equivalently reduce the repulsion in the low energy region. In the case discussed here, in the presence of nearest neighbor repulsion, this effect is somewhat offset by the fact that the nearest neighbor repulsion has a similar momentum dependence as the attraction, but we still find that superconductivity can occur even for parameters where the total interaction is always repulsive.
4.5. Large effective mass As discussed in section 3, the holes can acquire a large effective mass near the top of the band from the Hartree term in the hopping interaction [eq. ( 11 ) ]. This will give a large density of states at the Fermi energy and favor superconductivity, particularly if the density of states is lower away from the Fermi energy, as is shown in the next section.
5. The Tc equation We solve the BCS equation for Tc 1 1  2f(Ek,   / 0 Ak=  N ~k Vkk'Z~k' 2(~ k,
(22)
with the interaction eq. (20). The form of the attractive interaction implies a gap function of the form [6]:
Ak=Am(l~e'k'n+C)
(23,
and yields the coupled equations:
1 =K(I1 +clo)  W(I2 +cI~ ) ,
(24a)
c=K(I2 +cI, )  U(I1 +clo) ,
(24b)
with
331
J.E. Hirsch ~Bondcharge repulsion and hole superconductivity
K = 2zAt,
(25a)
W=zV,
(25b)
and 1 (1 Z i*'6"\'l2f(E*//) I'=~r~k z , e ) " 2~k~) "
(26)
From eq. (24) on eliminating c we obtain a single equation for To:
1=2KI1 WI2 UIo+ (K 2  WU)(IoI2I21).
nf2
deg(e)
Note that we are defining the origin of energy at the center of the band. We obtain the integral Io within the usual BCS approximation, valid in the limit (D/2 +//)/kaT c > > 1, (D/2//)/kBTc>> 1, as: D 2
~/2
Io=g(//)ln2~71t flc[(~) _ / / 2 ]
(31)
with 2y/n = 1.13. Within the same approximation, 11 and 12 yield:
Ii = ~/2~Alg(//),
(32a)
//2
with: Al =   ~
//
+~A,,
1[1+ A2= 2
//
(33a) z
8AI and gA2 vanish if the density of states is constant throughout the band. Otherwise they are given by: D/2
5At = ~
de sgn(E#)8(E),
(34a)
D/2 D/2
8A2 = ~5
dE l e  / / 1 8 ( e ) ,
(34b)
D/2
with 6 ( e ) = gg(e) ~ 1.
(35)
(28) The Tc equation then becomes:
and D=2zt is the bandwidth. becomes: I,=
(30)
(27)
Equation (27) displays some interesting features. The first three terms on one hand and the last term on the other hand show two different ways in which attractive and repulsive interactions compete. In the first 3 terms they simply add up (with their respective sign), each multiplied by a differently weighted integral L because of their different momentum dependence. One can see from eq. (26) that quite generally we have Io > 11 > 12, as the weighting factor is always less than 1. Thus, the onsite repulsion with no momentum dependence gives the largest effect, and the nearestneighbor repulsion the smallest. The last term in eq. (27) is more subtle and embodies the "pseudopotential effect" mentioned in the last section. The attractive interaction comes in squared, and the nearestneighbor repulsion because of its momentum dependence couples with the onsite repulsion to yield an "inverse pseudopotential effect" that competes with the attractive piece. Equation (27) will clearly have a solution for sufficiently large K, but whether this occurs for realistic parameters remains to be examined. Because the interactions have the same momentum dependence as the hopping energy we can rewrite everything in terms of energy. The kinetic energy is: Ek= t ~ e i*'6 6
zJ(~) =Am (   ~ / 2 FC) .
(
Equation
~ )ll2f(,#)  ~~ 2(e//)
O/2
with g(~) the density of states, and eq. (23):
(26)
re= (29)
2yr {D'~2
2]'/2
Lk5)  / / J
(36a)
with 2# a = 1 +2kAl +  ~ AI w+A2w+ (k2wu)A 2 ' (36b)
33 2
3.E. Hirsch ~Bondcharge repulsion and hole superconductivity ~2
b= D/2#2k+ (k2wu)A2w
(D/2)2  u ,
(36c)
and
k=Kg(#), w=Wg(#), u=Ug(u).
(37a) (37b) (37c)
It is instructive to rewrite eq. (36) in terms of the hole density. We define an effective density t/from
Equation (41 ) predicts superconductivity first occurring for small values of t/. Tc first increases with t/due to the prefactor and then decreases as the denominator in the exponent goes to zero. This behavior, found in ref. [6 ], persists here in the presence of nearest neighbor repulsion. The condition for existence of superconductivity is b > 0 [eq. (41c)]. As the most favorable situation for this is t/= 0, this condition, for constant density of states ( f b = 0 ) , becomes:
k~>~/(l+w)(11u)l. # D/2
(ln),
(38)
t/is the true hole density n if the density of states is constant throughout the band, g = 1/D. For low hole density, g ( E ) ~ g ( # ) for e~
(43)
Figure 2 shows this phase boundary for density of states (per spin) g o t ) = 1 eV/atom and z = 8 in var
# r,
j deg(E).~2g(~)(#+D/2)
n=2
(39)
D/2 v
so that n
if= g(#)~ .
v
(40)
The Tc equation becomes: Tc=
Z D ~ e
a/b ,
(41a) w
a= [l+k(1~)]2
w(13t/+ ~3 t/2) wu( l  t / ) 2 +
fa,
(41b)
u+fb,
(41c)
Fig. 1. Bonding (b) and antibonding (ab) charge densities for a diatomic molecule (schematic). The signs of the wave functions are indicated.
k2t/2
b=k(k+2)(1t/)+2   W ( 1   t ] ) 2   WU
lt/+
~IO
At where fa and fib are corrections for nonconstant density of states, given by:
fa=2kfAl 2( 6b=
1 a ) f A l
(eV) 0.5
w+ 8A2w
+ (kZwu)[2(l t/)fAl + (fAl)2], (k2wu)6A2.
(42a) (42b)
This completes the derivation of the T¢ equation. We now examine some of its consequences.
I
I
I
I
~I
2
3
4
V(eV) Fig. 2. Phase boundaries from eq. (43) for three values of U. g (#) = l eV/atom, z = 8. Superconductivity will occur in the region labeled SC.
J.E. Hirsch I Bondcharge repulsion and hole superconductivity
333
ious cases. It can be seen that the parameters required for superconductivity are quite realistic. The phase boundaries are found to be almost independent of g(/z), but the critical temperature depends on it. The behavior of Tc versus ~q from eq. (41 ) is qualitatively as discussed in ref. [ 6 ], going through a maximum and then decreasing to zero. An example is shown below. Tc is larger for larger g(/~) if the other parameters are kept constant. It is interesting to examine the effect of nonconstant density of states on T~. We take a simple parameterization of the density of states variation eq.
of the (electron) band (at < 0), Tc increases for small but goes to zero more rapidly, while in the opposite case Tc is depressed for low hole density but extends to higher occupation. We will discuss in more detail elsewhere the dependence of Tc on the various parameters. Here we will conclude by discussing some of the implications of these results in the next section.
( 3 5 ) as:
In this paper we have pointed out that a term that arises from Coulomb interactions in the derivation of a manybody tight binding Hamiltonian is attractive for electrons in antibonding states, or equivalently for holes. This term describes a hopping process modulated by the presence of other particles on the sites. We have given various arguments to support the possibility that this term could overwhelm the direct Coulomb repulsion and lead to superconductivity. Finally, we have derived an equation for the critical temperature within BCS theory and shown that it predicts superconductivity in parameter ranges that are not unrealistic. Let us now imagine the following farfetched possibility: superconductivity in all materials is caused by this modulated hopping interaction rather than by electronphonon interactions. Can we find clear evidence to either rule out or support this possibility? The first major objection is the isotope effect, that suggested the possible importance of electronphonon interactions in the first place. However, note that our hopping interaction eq. (15) will be very sensitive to phonons modulating the size of At. In the presence of coupling to phonons we will have for the term involving sites i and j:
al
6(e)=~(eU)
(44)
so that at is the total density of states variation, a~ = ~ ( D / 2 )  t ~ (  D / 2 ) , and we take it to be a fraction of the density of states at the Fermi energy g ( # ) . The quantities 8At and 8A2 become: at 8.4, = ~[ l + ( 1  J q ) 2] ,
(45a)
al 8/12 = 7(1  a ) [ 3 + ( l  n ) 2 ]
(45b)
I
o
Figure 3 shows the effect on Tc. If the density of states is larger around the Fermi level than at the bottom I
I
I
3
I
I
Ig(E)
,/,,
/
,
2
',
lf"~ I ~
• /
,
X\
IF/'
2
.
I
'1
\
///
6. Discussion
At= At(°)[ 1 fl(qgqj) ] \\ I
I
I
0.1
0.2
0.3
\',~l ~''~"
0.4
.
I
0.5
Fig. 3. Tc (arbitrary units) versus hole occupation showing the effect of nonconstant density of states. The curves labeled 1, 2, 3 correspond to at=0.5, 0 and  0 . 5 , respectively [eq. (44)]. The inset shows schematically the variation o f (hole) density of states in the three cases.
(46)
with q,, qj the ionic coordinates. That is, the interaction becomes more or less attractive with the ions moving closer and further apart, respectively. Note, however, that the interaction At enters squared in several terms in the exponent in eq. (41 ). We have on the average: (At) 2 = (At (°))
2fl262
(47)
with c~2= ((qiqj)2). AS t~20C 1 / M 1/2, with M the
334
J.E. Hirsch ~Bondchargerepulsion and hole superconductivity
ion mass, the k 2 terms in eq. (41) become larger for a lighter isotope. The dominant effect will be an increase in the first term in eq. (41 c ), leading to a larger Tc for a smaller ionic mass. If the isotope effect originates in a change in the exponent, we have, for Tcoc e  1/,~. To(2+ 52) =e5~/~2
(48)
T~(,l) indicating that the effect becomes smaller in strong coupling (large 2). This is consistent with the absence of isotope effect in the highest T~ materials. It is not impossible that the observed isotope effect of 0.5 in some materials is just a coincidence. Certainly in many materials this value is not observed [ 11 ]. Another serious objection to our proposal may be the argument that structure in the tunneling characteristics of strong coupling superconductors has been observed that correlates with the structure in the phonon spectrum [ 12 ]. However, it is clear from eq. (46) that the lattice vibrations should have a noticeable effect on superconductivity through their modulation of the attractive interaction eq. (15). Thus, it would be surprising if the phonon structure would not show up in the gap. This is, of course, completely different from the electronphonon interaction causing superconductivity through modulation of the single particle hopping term. It should be possible to calculate the effect of phonons coupling through the interaction eqs. (46) and ( 15 ) on the tunneling characteristics of our model to confirm or rule out this possibility. Another objection to our model might be that eq. (41 ) for Tc can yield wildly different values of T~ by slightly changing the interaction parameters. This is not very satisfying, as one would like to understand, for example, why Ts's of 1000 K do not occur. On the other hand, similar objections can be raised about the electronphonon mechanism, where a slight adjustment o f / t * can make superconductivity disappear. We believe there is a deep reason why the parameters in our model are what they are in nature to give rise to the small Tc'S observed, but do not have a quantitative understanding. A qualitative argument will be given below. Let us now enumerate some arguments in favor of our picture. First, it provides a unified picture of superconductivity in high Tc oxides and conventional
materials. There is something very distressing about "drawing the line" between electronphonon and another mechanism at 23 K (Nb3Ge), 30 K (Bal _xKxBiO3 ) or 40 K (La2CuO4_y), particularly because there is not a discontinuous change in properties observed. The model discussed here provides a natural link between superconductivity in the conventional materials, the high Tc oxides and the "transient superconductors" like CdS [ 13,2 ]. Another argument in favor of our model arises from examining the band structures [ 8 ] of the bcc superconducting elements V, Nb, and Ta. They all show a small holelike pocket centered at the F point that is an ideal situation for superconductivity in the model discussed here. As is well known, Nb is the element with higher Tc and both Nb and V form the AI5 compounds with the highest Tc's among conventional superconductors. For example, observation of a small "hole box" centered at the F point in the A15 compound V3Sihas been reported [ 14] from positron annihilation studies, as well as hole pockets near the edge of the Brillouin zone [ 15 ]. The model discussed here suggests that superconductivity in these compounds is driven by those parts of the Fermi surface. Note that the Fermi energy in Nb and the A 15 compounds is located near a peak in the density of states, and becomes substantially lower below it. In our model, this feature is consistent with high critical temperature (fig. 3). Other superconducting elements like Ti and Zr also show holelike bands near the F point [ 8] but here the Fermi surface encloses a larger number of holes. Consistent with our model, Tc in these elements is substantially lower than in V and Nb. Other superconducting elements do not show this feature. However, they also do not form a crystal structure that is a bipartite lattice like bcc. In Pb, for example [ 16], that has an fcc structure, the antibonding band crosses the Fermi surface quite far from the 1"point. Note, however, that this lattice structure is "frustrated": there is no way to accommodate a purely antibonding state where all nearest neighbors to a given site have opposite phases to it. The simplest analogy is a twodimensional x y triangular antiferromagnet. How does such a system order? By orienting its nearest neighbor spins at a 60 ° angle to each other. We speculate that part of the Fermi surface in Pb and other "frustrated superconductors"
.I.E. Hirsch ~Bondcharge repulsion and hole superconductivity
with high Tc's is in a region that yields close to the m a x i m u m possible antibonding character to the states on it. Note that because z = 12 in this case it can tolerate some degree of frustration and still yield Tc's comparable to the unfrustrated z = 8 cases like Nb. Note also that the Hall coefficient o f Pb is positive, indicating the existence o f hole states at the Fermi surface. As pointed out in ref. [2], a correlation between positive values o f the Hall coefficient and high superconducting transition temperatures was noted long ago [ 17 ]. As a Fermi surface generally will have both hole and electronlike states, as well as open and closed orbits, one would not expect a very direct correlation between Hall coefficient and Tc'S but rather a statistical one, as observed. The model discussed here implies that it is those pieces o f the Fermi surface that lie on holelike bands with m a x i m u m antibonding character that drive the systems superconducting. A valid model should also be capable of explaining why some materials do not become superconducting. For example, why don't pdoped Si and Ge become superconductors? The answer within our model is that the n u m b e r o f nearest neighbors z = 4 is not sufficient to overcome the Coulomb repulsion. As z increases superconductivity becomes increasingly favored in our model as the effect o f the onsite repulsion becomes smaller, as discussed in section 4. As another example, why d o n ' t simple or noble metals become superconductors? Within our model it is because they do not have holelike Fermi surfaces and thus the interaction discussed here is not attractive. A more detailed discussion o f the band structure features o f superconducting elements and compounds is outside the scope o f this paper. We believe, however, that the model discussed here has the possibility o f providing a framework to correlate specific features of the band structures with the occurrence of superconductivity. Finally, a valid model should have a prediction, even if difficult to verify. In ref. [ 6 ] an asymmetry was found in the calculated N  I  S tunneling characteristics due to the energy dependence of the gap eq. (30). If the model proposed here is correct, such asymmetry should be found in all superconductors. The sign o f the asymmetry, a slightly larger peak in
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the 0I/0 V curve when the tip is positive with respect to the sample, will reflect the fundamental asymmetry between electrons and holes that leads to superconductivity. To conclude, we discuss the reason why it is difficult to achieve high superconducting transition temperatures. Qualitatively, it goes as follows: the ideal situation for superconductivity in our model is having all bonding orbital filled and almost all antibonding orbitals filled. However, that situation will be highly unstable: after all, it is the bonding electrons that give stability to the solid and the antibonding ones oppose it. This is for example, why the diatomic molecule, He2 does not form. Thus, the best one can do is to achieve this ideal situation for superconductivity on parts o f the Fermi surface, and pushing it too far will render the system unstable. The lattice instabilities that one observes when Tc gets too high are the successful attempts of the solid to get rid o f its antibonding electrons. We note in closing that Varma and Dynes [ 18 ] long ago pointed out the importance of antibonding states at the Fermi surface in transition metal and A15 superconductors. Their interpretation o f this phenomena was, however, based on the electronphonon mechanism.
Acknowledgements This work was supported by NSF Grant ~DMR8517756. I have benefitted from interactions with F. Marsiglio.
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[6] J.E. Hirsch and F. Marsiglio, UCSD preprint, December 1988 and UCSD preprint, February 1989. [7] R.G. Parr et al., J. Chem. Phys. 18 (1950) 1561. [8]V.L. Moruzzi, J.F. Janak and A.R. Williams, Calculated Electronic Properties of Metals (Pergamon, New York, 1978). [ 9 ] A.P. Cracknell, Adv. Phys. 18 ( 1969 ) 681 ; G. Grimvall, Phys. Scr. 14 (1976) 63. [ 10] F. Zawadowski, Phys. Rev. B 39 (to be published). [ 11 ] P.B. Allen, Nature 335 (1988) 396. [12]D.J. Scalapino, in Superconductivity, ed. R.D. Parks (Dekker, New York, 1969 ) p. 449.
[ 13] G.C.W. Homan et al., Physica B + C 107 (1982) 9. [ 14] A.A. Manuel et al., Solid State Comm. 31 (1979) 955. [ 15] S. Berko and M. Weger, Phys. Rev. Lett. 24 (1970) 55. [ 16 ] F.R. McFeely et al., Solid State Comm. 17 ( 1975 ) 1415. [ 17 ] I.M. Chapnik, Sov. Phys. Doklady 6 (1962) 988; Phys. Lett. A72 (1979) 255. [ 18 ] C.M. Varma and R.C. Dynes, Superconductivity in d and fband metals, ed. D.M. Douglass (Plenum, New York, 1976) p. 507.