Physica C 161 (1989) 185-194 North-Holland, Amsterdam
HOLE SUPERCONDUCTIVITY: THE STRONG COUPLING LIMIT J.E. H I R S C H Department of Physics, Universityof California, San Diego, La Jolla, CA 92093, USA Received 24 July 1989 Revised manuscript received 15 September 1989
A Hamiltonian describing hole superconductivity is studied in the limit where the single particle hopping amplitude t goes to zero. It is shown that superconductivity can occur in this regime with purely repulsive Coulomb interactions, similarly to what occurs in the weak coupling case. The critical temperature remains finite as t-,0. No bound pairs exist above Tc except for extremely low density of holes, and above T¢holes are localized.
Superconductivity occurs in solids when electrons form pairs, loosely bound Cooper pairs [ 1 ] in the weak coupling limit or tightly bound "Ogg pairs"  or "Shafroth-Blatt-Butler" [ 3 ] pairs in the strong coupling limit. Until recently the only known ways to achieve the attractive interaction between electrons needed for pairing were second order processes by which a boson excitation (phonon, exciton, plasmon or m a g n o n ) is created and reabsorbed, leading to a (usually low frequency) attractive interaction between two fermions. Although the strong coupling limit (in the sense of tightly b o u n d pairs) has been considered in the literature [ 4 ], such a limit is not realistic for a pairing interaction originating from these second order processes, as it is hard to see how such a process could overcome the short-range Coulomb repulsion between electrons to lead to a tightly bound pair. However, a new type o f interaction has recently been proposed as the origin o f superconductivity in solids, that results from a first rather than a second order process [ 5-8 ]. This "hopping interaction":
V h = - - A t ~ (c+cjo+h.c.)(n,_o+nj._o)
originates in an off-diagonal matrix element of the Coulomb interaction between electrons and gives rise to repulsion or attraction depending on the phases of
the wave functions involved; in particular, very general arguments show [ 7 ] that it is repulsive for states at the bottom of an electronic energy band (electrons) and attractive for states at the top of the band (holes). In the weak coupling limit it has been shown that this interaction can give rise to superconductivity when the net total Coulomb interaction is repulsive [ 8 ], as it should be, without the necessity of invoking second order processes. In this paper, we show that the same is true in the strong coupling limit. In addition, we argue that this limit is not an unphysical regime of the model but that it can arise for realistic parameters, particularly in cases where conduction occurs through holes in closed shell anions as in the oxide superconductors [ 5 ]. As pointed out by Legget [ 9 ], the BCS formalism [ 10 ] can describe the superconducting ground state over the entire coupling range encompassing weakly bound Cooper pairs as well as strongly bound "diatomic molecules". As the temperature is raised the possibility arises that in the strong coupling regime the transition to the non-superconducting state occurs at the Bose condensation transition temperature, and that bound pairs exist above To. That is, the transition could be driven by center of mass excitations o f the pairs, occurring when the center of mass m o m e n t u m distribution of the pairs "spreads out" and no longer macroscopically occupies the q = 0 state. This is the scenario that has usually been used to describe the superconducting transition in the
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•I.E. Hirsch / H o l e superconductivity." the strong coupling limit
strong coupling limit, and is not described by BCS theory [ 11 ]. However, as we will see in what follows, in our case the Bose condensation transition temperature is found to be substantially higher than the BCS transition temperature describing pair dissociation in the physical parameter range. Thus when the temperature is raised from zero, pairs dissociate well before their center of mass excitations start to play a role, and no bound pairs exist above To. Hence we have a remarkably simple situation where BCS theory can describe both the ground state as well as the finite temperature transition over the entire physical coupling range, with the coherence length ranging all the way from infinity down to a single lattice spacing. This paper is organized as follows. In section 2 we define the model, explain where we expect the strong coupling limit to be applicable and solve it in this limit in perturbation theory. We obtain the binding energy of the pairs and the condition on the parameters to give rise to pairing. In section 3 we discuss some bounds on the interaction parameters that arise from the constraint that the Coulomb interaction is repulsive. In section 4 we discuss the strong coupling limit of BCS theory and show that it correctly describes the binding energy of a pair in that limit; we also compare the transition temperature for pair dissociation and for Bose condensation, and show that in the parameter range determined by the constraints of section 3 the Bose condensation temperature is much larger than the temperature for pair dissociation. We conclude in section 5 with a discussion of the results.
2. Strong coupling limit
U and V, all positive, are the Coulomb matrix elements (ii[ 1/rlij), (ii[ 1/r[ii) and (ijl 1/r[/j), respectively, with
( ~il l /rlkl) C2
= f d3rd3r'O*(r)OT(r')-(~Ol(r')~k(r)
where ¢i is an atomic or Wannier function at site i, and the phase convention is chosen so that ~ and q~j have the same sign in the region between i andj. The hopping for holes, t, is given by: t = to - 2At
where to> 0 is the bare hopping for electrons between nearest neighbors:
-to= f d3rq)*(r)[-~mV2+~U(r)lOj(r)
( C iaCja--k h . c . ) (05
- A T ~ (c+cjo+h.c.)(n,._a+nz_~) + U ~[ n~rn. + V Z ninj i
to = ~ d3rd3r'O*(r)q)j(r) ] ~ - I x i ( r
nearest neighbor sites. The parameters At,
with Xi the ionic wave function and Z the "effective" ionic charge. On the other hand, the hopping interaction is given by:
At=~ d3rd3r'O*(r)Oj(r) Ir-r'[ e2 I¢i(r' ) 12
with V(r) the ionic potential. We assume the hopping for holes, t, remains of the same sign as to, i.e. positive. In that case, the hopping interactioh between two holes in the filled band eq. ( 1 ) is attractive, because the hole wavefunction has crystal momentum k ~ 0 . However, eq. (4) shows that t can be much smaller than to and in particular t could become much smaller than all other interactions in the problem. Thus it is a relevant physical question to discuss the properties of the Hamiltonian eq. (2) in the limit t ~ 0 , with the other interactions remaining finite. We denote by H ( t = 0) the Hamiltonian eq. (2) in that limit. This situation is likely to be relevr.m for the case of the oxide superconductors. Recall that in the tight binding limit the bare hopping to can be written as : Ze 2
The tight-binding Hamiltonian for holes is given by :
and it does not depend explicitly on the ionic charge. These forms suggest that the single electron hopping amplitude to will become smaller compared to At as
J.E. Hirsch /Hole superconductivity: the strong coupling limit
the effective ionic charge Z decreases, so that the hopping for holes eq. (4) will become smaller when the ions are more anion-like. To be more specific, the binding energy for an electron in O ° is only 1.45 eV, indicating that the effective ionic charge is very small. Thus, for conduction by holes through 0 2 anions one may have a situation where the hopping t [eq. (4) ] at the top o f the band is close to zero. As holes are added to the filled band the effective single particle hopping increases, since within a Hartree decoupling of the hopping interaction
t(nh) =to --2At+nhAt
for the system with nh holes. Hence we conclude that the strong coupling limit is likely to be applicable for a dilute system o f holes in anions with filled shells. Consider a single pair o f holes o f opposite spin. If they do not occupy nearest neighbor sites they cannot deloealize if t = 0 , and the state I~t) = I * ) , , I + ) ,
which is of course not an eigenstate o f H ( t = 0). Under application of the Hamiltonian we obtain: H(t=0) [s)ij = V l s ) s j - x / ~ A t ( I ?~); 1 0 ) j + 10)~ I ?~ )j)
and another application of H ( t = 0) yields back the original singlet state plus all other singlet states at bonds that have one site in c o m m o n with the origin one. Figure 1 shows the accessible bonds in a twodimensional geometry. For At~U<< 1, which is the realistic regime, one can eliminate the intermediate states with doubly occupied sites and in second order perturbation theory obtains the effective Hamiltonian:
where the operator b + creates a hard-core boson (a
I I I
I I I
a I I I I
Fig. 1. The singlet pair indicated by the solid line and arrows can hop with equal amplitude to any of the six dashed bonds in this two-dimensional geometry, through an intermediate state where one of the arrows moves on top of the other. singlet pair) at a bond labeled by /t. (Formally,
b-~ = (cir+cj+ + +ci+c/~)/v/2, w i t h / t the bond connecting i andj. ) The sum ( / z u ) runs over nearest neighbor bonds, meaning two bonds that share one site in common. The parameters are given by: tefr-
_ ! ( I~>,l*>j- I~>,lt>j) Is>~j-x/5
I I I I i
with rn, n non-nearest-neighbor sites is an eigenstate of the Hamiltonian with energy Et= 0. If the two holes are on nearest neighbor sites, we consider the single state:
U-V V- 4 ( A t ) 2 U- V "
This effective Hamiltonian neglects interactions between neighbor pairs, but this should be irrelevant in the dilute limit. Note that the connectivity o f the lattice on which the Hamiltonian eq. ( 11 ) is defined is higher than the original one. The number of nearest neighbors to a given bond is: g=2(z-1) with z the number of nearest neighbors to a site. (For illustration purposes only we mention that for a twodimensional square lattice the connectivity that results is the same as the one that exists for oxygen holes in a two-dimensional CuO2 lattice if hopping can occur through an intermediate Cu site only. ) The wave function of a single pair of holes is now delocalized and given by: 1
I~,ox>=-~= Y~ Is>,,j VlVb
with i, j nearest neighbor sites and Nb the number of bonds. Its energy is (Co-gtefr), or
,I.E. Hirsch /Hole superconductivity: the strongcoupling limit
188 E¢~ = V - 4 z ( At'2 ~ U-V
We define the parameters: (15a)
and the binding energy of a pair Eb as the difference between the single pair energy and the energy of two isolated holes, so that E b = - E x . Equations (14) and ( 15 ) yield K2
z( U - W / z )
This simple perturbative treatment yields the binding energy of a pair in the regime A t / U < 1. A more general expression for the binding energy of a single pair can also be easily obtained, even if the condition At/U<< 1 is not satisfied. Application o f the Hamiltonian to the wavefunction eq. ( 13 ) yields: H I ~'e, > = VI ~%x > = 2At,,/~t ~ , >
with I ~up > = - - ~
the condition for binding and delocalization is:
UV (At)Z> 4z
so that in particular it occurs for V-*0 or as z--,oo. It is clear that a ground state with such delocalized pairs is a superconducting state. Thus, eq. (21 ) is the condition on the parameters to have a superconducting ground state in the strong coupling limit. The coherence length for the pairs in such a state is a single lattice spacing. We recall that in the weak coupling limit solution o f the BCS equation gave rise to the condition for superconductivity [ 6 ]: EK> x/( 1 + E U ) (1 + g W ) - 1
with g the density o f states at the Fermi energy. For g--,oo eq. (23) reduces t o e q . (21). Fig. 2 shows the phase boundaries eq. (23) for increasing values of g as well as the limiting phase boundary eq. (21 ).
Y, I t t > ,
I/ . . . .
1.0 the wave function for a pair of holes at the same site, delocalized over the entire lattice. Application of H ( t = 0 ) to I~p) yields again l~U~x), so that we obtain the 2 × 2 Hamiltonian:
I . . . . . - a / . .t. . l
At (eV) 0.6
H = (_2AV tx/~
0.4 in the basis of I~U~x) and I~%). Its lowest energy E o = --Eb yields the pair binding energy:
Eb = --
(20) which reduces to eq. (16) on expanding in At/ ( U - V ) . The wave function for the pair is simply obtained by diagonalizing eq. ( 19 ) and it consists of a delocalized singlet eq. ( 13 ) with a small admixture of doubly occupied sites, eq. (18) for large U. Thus, two holes will bind in such an itinerant pair or remain separate and localized depending on whether E b > 0 or E b < 0 . From eq. (20) we see that
I .... 0.5
I .... i
I .... 1.5
] .... 2
I .... 2.5
v (ev) Fig. 2. Phase boundaries for the occurrence of superconductivity, eq. (23). The five lines shown correspond to g=0 (straight line), 0.0625, 0.25, 1 and oo in units of states/eV/spin. All curves are tangent at U= W= K, which correspondsto V= 1.25 eV, At= 0.625 eV for U= 10 eV, z=8. The line labeled g=0 coincides with the boundary eq. ( 31 ) above which the model becomes unphysical. For given g, superconductivityoccurs for parameters between the curve eq. (21) for that g and the line labeled g=0. The dashed line is the boundary eq. (30a), and the boundary eq. (30b) is outside the range of the figure.
,I.E. Hirsch /Hole superconductivity." the strong coupling limit
a¢1 (r) - 02 (r) gtp(r)_ (l+aZ)l/2
The parameter range where superconductivity occurs increases somewhat as one approaches the strong coupling regime g - ~ . It is remarkable, however, how little the phase boundary eq. (23) changes in going from very weak to infinitely strong coupling.
with 0 < a < 1. The condition Vc> 0 translates into
V'c=aZU-I (l+a4~) V-2a(l+aZ)At>O 2
3. Bounds on interaction parameters
Thus, we have here a particularly simple situation where superconductivity can arise from Coulomb interactions between electrons. With the usual density-density interactions, pairs do not form in the strong coupling limit unless the net interaction between two particles is attractive, and the hopping processes weaken the binding of the pair. In our case instead, the hopping processes are the origin of the attractive interaction. However, the parameters in the model need to be such that the net Coulomb interaction between holes (or electrons) is repulsive for the model to be physical. Let us examine the Coulomb interaction between two electrons in the single particle states gt~ and ~p involving two nearest neighbor sites:
gc=~ d3rd3r'l~'~(r) 12 Ir-r t qJp(r') .
By definition Vc is positive, and this imposes constraints on the allowed values of U, V and At. If qJ~ and ~,p are orbitals at the same site, V~= Uso that we require U> 0. If ~ and ~B are orbitals at nearest neighbor sites Vc= V which requires V> 0. If ~,~ and ~p are antibonding orbitals: gt~(r) = ~uB(r)-
O,(r) - O z ( r ) x/~
the condition V~> 0 translates into U+ V - 4 A t > 0
q~l(r)-aOz(r) ~,,~(r)- (l+a2),/2
and it can be seen, for example, that if V = 0 this condition will be violated for arbitrarily small At for sufficiently small a, while eq. (26) is satisfied. To obtain a sufficient condition we set At= V in eq.(28) and obtain
V,=aE(U_3V) + (1 2a)_____~4V
which implies that a sufficient condition for the Coulomb interaction to be repulsive for states involving two sites is At< V
which is physically reasonable. The condition eq. (22) is still satisfied under those constraints for reasonable values of the parameters, for example U= 10 eV, V=2 eV, At=0.8 eV, z = 8 , as considered in ref. [81. For states involving many sites we consider the Coulomb interaction between Bloch states ~u,= ~Uk, g/p= g/k', with 1
where we have included the possibility of a finite overlap integral S=(01, 02) between orbitals on nearest neighbor sites connected by the lattice vector 6. The Coulomb interaction is:
and the corresponding interaction for the electrons in bonding states is larger, U+ V+ 4At, so that it is positive if eq. (26) holds. Unfortunately, eq. (26) is not a sufficient condition for V~ to be positive for arbitrary two-site states. The most general condition is obtained by choosing
U+ zV+ 2At Za(eik'n+ e ik''n)
and a sufficient condition for Vc> 0 is obtained by taking k- 6 = k'- 6 = n: Voo= U+zV-4zAt>O
which is also the reduced interaction between Cooper pairs with momentum ka, k ' 6 = ~ [ 8 ]. Note that even
J.E. Hirsch /Hole superconductivity." the strong coupling limit
if Voo is small the actual Coulomb repulsion Vc can be large because of the denominators in eq. (32), which also become smallest for k6=k'6= re. We have shown in various cases that superconductivity occurs under the constraint eq. (33) . Unfortunately, condition (33) does not guarantee that the Coulomb repulsion between electrons in arbitrary non-translationally invariant states is positive in our model. However, the electron states that enter in defining the Cooper instability are Bloch states of the form (31 ) so that it is reasonable to limit the model to the parameter range where the Coulomb interaction between those states is repulsive, as given by condition eq. (33). Note that condition (33) can be written as: K<
The pairing interaction in our case is given by:
Vkk" = U+ V ~ ei(k-k')~--2At ~ (eika+eik'~)
with 6 vectors connecting a site to its nearest neighbors. From these equations, using an ansatz for the gap: Ak = z J , , ( - - ~ 2 2 +C )
( D = bandwidth) one obtains the equation [ 6 ]:
1 =2KI~- WI2- UIo+ (K 2 - WU) (IoI2-IZ~)
eik.[~ z 1 - 2f(Ek)
which is the reverse of condition (23) for g~0. Thus, for given g the allowed parameter range that will give rise to superconductivity lies between the curve for that g in fig. 2 and the line g = 0. In addition, condition (30) puts additional constraints, so that in fig. 2 the allowed parameter range is to the right of the dashed line. It can be seen that the parameter range that gives rise to superconductivity under these constraints is reasonable.
U+ W 2
As the single particle hopping amplitude t-~0 we obtain, in the dilute limit (n--*0) at zero temperature:
Ii = 0
and eq. (40) becomes:
I+(u+W~I K2-WU Fo \ --~1[° -----ff~ - O
4. BCS theory in the strong coupling limit The superconducting state within the mean field approximation (BCS theory) is defined by the equations
A k = - - l ~ vkk,Zik, 1-2f(Ek')
1 U+_Z'~ 4 K2- WU Io = - 2 \
and since the binding energy of a pairs is given by:
for the gap, and l Eg-/t[ l _2f(Ek) ] n = 1--N~k Ek
for the fermion density (or the chemical potential a), with
Ek = X/(~k --/~) 2+ LJ2
~ = --t ~ e ik'~ .
we recover the same result for the binding energy obtained from the strong coupling analysis in section 2 (eq. (20)). Thus, solution of the BCS equation correctly yields the binding energy of the pairs in the dilute strong coupling limit. The critical temperature within BCS theory is defined by the temperature where ,J~--.0. Eq. (41) yields
J.E. Hirsch I Hole superconductivity: the strong coupling limit
tgh ( - lt / ka T~ ) -2/t
kBT~ = 2nh2
11 = 0
with p the number of bosons per unit volume and m their effective mass. For our model we obtain:
and eq. (36) determines the chemical potential: tgh(-/.t/2kBT~)
so that l
I°=kaT ~ 21n[ ( 2 - n ) /n]
and hence the critical temperature is given by: linE/
kaTe =2In[ ( 2 - n ) /n] Eb
with Eb given by eq. (20). The "gap ratio" EblkBTc is shown in fig. 3 versus the density of holes. It is always larger than the weak coupling value 3.53 and can become very large at low hole densities. The critical temperature within BCS theory gives the temperature at which pairs dissociate. However, as mentioned earlier, in strong coupling the transition could be driven by center of mass excitations of the pairs, not described by BCS theory. This Bose condensation transition temperature is given by [ 12 ]:
4n kB T~ - 2.6122•3 n 2/3/ef f
( 51 )
for a bcc lattice, with teff given by eq. (12a) [for a simple cubic lattice n gets replaced by n/2 in eq. (51 )]. This expression neglects the hard core constraint as well as interactions between nearest neighbor pairs, but these effects should be irrelevant in the dilute limit. We now compare the two critical temperatures, eqs. (49) and (51). It is clear that as n--,0 the Bose transition temperature is lower, and so it will determine the critical point. However, this only occurs in general for extremely low densities. As a typical example, in fig. 4 we show the transition temperature for z = 8 , U = 1 0 eV, V=0.5 eV and the maximum allowed value of At for these parameters [from eq. (33) ], At=0.4375 eV. The Bose condensation transition temperature is larger than the BCS transition temperature for all n > 0.006. For the same parameters but smaller At the crossing occurs at even smaller n. 1500
. . . .
. . . .
. . . .
, . . . . .
U: tO eV V--O.5eV
I I I
/ I I
250 2 ~l - -
. . . . . . . . . . . . . . . . . .
I .... I .... [ .... 0.2 0.3 0.4 0.5 n Fig. 4. Bose condensation transition temperature eq. (51 ) (dashed line labeled B) and BCS transition temperature eq. (49) (full line labeled F ) for the parameters given in the figure and the m a x i m u m allowed At for this case, At = 0.4375 eV. The two curves cross at n = 0.0064. 0
I .... 0.2
[ .... 0.3
I .... 0.4
Fig. 3. Ratio of pair binding energy Eb to BCS transition temperature versus density of holes in the strong coupling limit. The dashed line indicates the weak coupling value 2A/kaTe= 3.53.
I .... 0.1
J.E. Hirsch ~Hole superconductivity: the strong coupling limit
To visualize the behavior for other values of V, we plot in the inset of fig. 5 the n-dependence of the transition temperatures:
f ( n ) = n 2/3
15°°°b l ,,'"" to.21-
/ / //
I .... 0.6
v (eV) Fig. 6. Hole density at which the Bose transition temperature eq. (51) becomes equal to the BCS transition temperature eq. (49) versus V. U= 10 eV, z = 8 and At is the maximum value allowed by eq. (33). In the region labeled B the transition is driven by center of mass excitations of the pairs, and pairs still exist above To. In the region labeled F, the transition occurs through pair dissociation and above Tc there are localized single holes. Note that for smaller values of At the region labeled F would become larger, and that for smaller U the curve would approach zero at smaller values of V.
,,, f . / ¢ - - - 1
o'U,. , ,' , .J 10000 --
for the Fermi (BCS) case, and in fig. 5 we show To~ f ( n ) for both cases, with At chosen at the maximum allowed value from eq. (33) (line labeled g = 0 in fig. 2). The critical temperature is obtained by multiplying the value of T J f ( n ) for given V with the value o f f ( n ) in the inset for given n. The values of the hole density where the Bose transition temperature equals the BCS transition temperature as function of V are shown in fig. 6. It can be seen that the Bose Tc becomes smaller than the BCS T~ for nonnegligible values of n only for very small V. However, this regime is unphysical as it does not satisfy the constraint eq. (30). The case shown in fig. 4 is about the smallest value of V allowed, and for larger
21n[ ( 2 - n ) / n ]
for the Bose case, and
v (eV) Fig. 5. Tc divided b y f ( n ) [eq. (52) ] versus V. U= 10 eV, z=8 and At is the maximum allowed value from eq. (33). For the Bose case (dashed line labeled B), Tdf(n) =4~t~f/2.6122/3,with teff given in eq. (12a). For the Fermi (BCS) case (full line labeled F), T¢/f(n ) = Eb, with Eb given in eq. (20). The inset shows f(n) for the Bose case (dashed line) and the BCS case (full line), and Tc is obtained by multiplying the numbers given in the figure and in the inset for given V and n. Note that the binding energy Eb and hence Te go to zero at V= 1.25 eV, the point where the curves for g = o o and g = 0 in fig. 2 are tangent.
Vthe crossing occurs for even smaller values of n as can be seen in fig. 6. Hence, we conclude that in our model in the parameter range of interest the transition temperature in strong coupling is determined by pair dissociation as described by BCS theory rather than by center of mass excitations of the pairs, as given by the Bose condensation transition temperature, and consequently that no bound pairs exist above To. The ground state and the transition are therefore properly described by BCS theory in both weak and strong coupling. The pair wave function goes smoothly over from an extended state with somewhat larger amplitude on nearest neighbor sites, as found in ref. [ 8 ], to a tightly bound state with amplitudes only at nearest neighbor sites (large) and at the same site (small) as given by the solution of the Hamiltonian eq. (19). The transition temperature increases smoothly from the weak coupling regime towards the strong coupling limit eq. (49) as the density of states increases. As an example, we show in fig. 7 Tc versus bandwidth D = 1/g obtained from the numerical solution
. . . .
40o/ 2oo / 0
. . . .
Hirsch ~Hole superconductivity: the strong coupling limit I
. . . .
Fig. 7. Critical temperature versus bandwidth from the BCS equation (assuming constant density of states). U = 10 eV, z = 8 and the values of V and the bandwidth D are given in eV. At is chosen as the m a x i m u m allowed value from the constraint eq. (33). At= 0.9375 eV, At=0.8125 eV and At=0.75 eV for V=2.5 eV, 2 eV and 1.75 eV, respectively. The dashed lines give the analytic approximation to Tc valid for To~D<< 1 (eq. (41) of ref. ( 6 ) ) , and the dotted lines the strong coupling limit of Tc (eqs. (49) and (20) with z replaced by 3). In that limit, Tc=861 K, 343 K and 160 K for V=2.5 eV, 2 eV and 1.75 eV, respectively.
of the BCS equation for n = 0.1 and three values of the interactions, for the case of a constant density of states. The strong coupling limit of Tc for that case is obtained from eqs. (49) and (30), except that z in eq. (20) is replaced by 3 for a constant density of states. It is indicated by the dotted lines in fig. 7. The dashed lines give the analytic approximation to the critical temperature in the weak coupling limit (eq. (31) ofref. ( 6 ) ) .
One may ask whether the strong coupling limit Hamiltonian eq. (11 ) could describe also a dilute system of electrons rather than holes. After all, the parameters in that Hamiltonian are independent of the sign of the hopping interaction, which is attractive for holes and repulsive for electrons. In other words, the Hamiltonian eq. (2) with opposite sign for the hopping interaction and t = 0 is also equivalent to the Hamiltonian eq. ( 1 1 ). However, such a Hamiltonian does not make physical sense. As dis-
cussed, the strong coupling limit for holes, t-,0, is reasonable because of the hopping renormalization eq. (4). For electrons at the bottom of the band there is no such renormalization and hence the strong coupling description is not applicable. In summary, we have shown that the model of hole superconductivity has a well-defined strong coupling limit that describes nearest neighbor singlet pairs in a superconducting state at low temperatures. This occurs for parameters in the model that correspond to repulsive Coulomb interactions, i.e. are physical. The parameter range for the interactions that gives rise to superconductivity is similar to the one obtained in the weak coupling regime. Tc remains finite in the strong coupling limit and is larger than in the weak coupling regime, and so is the "gap ratio" Eb/ 2kBTc. The transition is described by BCS theory over the entire coupling range, and no bound pairs exist above T¢. Thus, the model eq. (2) with the constraints eqs. (30) and (33 ) provides a remarkably simple case where the superconducting transition can be rigorously studied with the coherence length ranging from infinity down to a single lattice spacing. Of course, the strong coupling results only become exact for very low density of holes; at higher densities interaction between pairs, not described by either BCS theory or the non-interacting Bose gas treatment, should become important. By remarkable good fortune, it is only in the dilute regime that we expect the strong coupling limit to be achieved, as discussed in section 2. For the oxide superconductors the strong coupling limit should be applicable in the regime of low hole doping. The BCS transition discussed in this paper describes a transition from a superconducting state (with short coherence length) to a state where pairs unbind and the single particles become localized, as described by the wavefunction eq. (8). In this connection we recall that for temperatures above the superconducting transition and low hole doping the oxide superconductors exhibit precisely this localization feature, as indicated, for example, in fig. 4 of ref. [ 13 ]. For larger hole doping the single hole hopping amplitude eq. (7) increases, the normal state should become more metallic and the superconducting state should cross over towards the weak coupling regime. This cross-over from strong to weak
J.E. Hirsch / Hole superconductivity: the strong coupling firnit
coupling as a function of hole doping will be discussed in detail elsewhere.
Acknowledgements This work was supported by NSF-DMR-84-51899: I am grateful to X.G. Wen for a stimulating conversation and to F. Marsiglio for stimulating discussions.
References [ 1 ] L.N. Cooper, Phys. Rev. 104 (1956) 1189.  R.A. Ogg, Phys. Rev. 69 (1946) 243. [ 3 ] M.R. Schafroth, J.M. Blan and S.T. Butler, Helv. Phys. Acta 30 (1957) 93.
[ 4 ] See, for example, A.S. Alexandrov and J. Ranninger, Phys. Rev. B24 (1981) 1164.  J.E. Hirsch, Phys. Lett. A 134 (1989) 451.  J.E. Hirsch, Physica C 158 (1989) 326.  J.E. Hirsch, Phys. Lett. A 138 (1989) 83. [ 8 ] J.E. Hirsch and F. Marsiglio, Phys. Rev. B 39 (1989) 11515; in: Proc. Int. Conf. on Materials and Mechanisms of Superconductivity (Stanford, CA, 1989 ) to be published in Physica C 162-164; Phys. Lett. A, to be published. 19 ] A.J. Legget, in: Modem Trends in the Theory of Condensed Matter, eds. A. Pekalski and J. Przystawa (Springer, Berlin, 1980) p. 14. [ 10 ] J. Bardeen, L.N. Cooper and J.R. Schrieffer, Phys. Rev. 108 (1957) 1175. [ 11 ] P. Nozieres and S. Schmitt-Rink, J. Low-Temp. Phys. 59 (1985) 195. [ 12 ] K. Huang, Statistical Mechanics (McGraw-Hill, New York, 1956) ch. 12. [ 13] J. Torrance et al., Phys. Rev. Lett. 61 (1988) 1127.