PHYSICAL REVIEW 8 VOLUME 43, NUMBER 1 1 JANUARY 1991

Positron annihilation from nearly localized Fermi liquids: A probe of pairing

Bulbul ChakrabortyDepartment of Physics, Brandeis University, P. 0. Box 9110, Waltham, Massachusetts 02254-9110

(Received 1 June 1990)

Based on a theory of positron annihilation from systems with strongly correlated electrons, it is

predicted that positron-annihilation spectroscopy (PAS) can be an extremely useful probe of pairingand superconductivity in such systems. Specifically, it is shown that if the nearly localized electronsform pairs which are small, then the annihilation characteristics are changed significantly fromthose in the unpaired state, and the changes reAect both the amplitude and the symmetry of the or-der parameter. This has obvious consequences for the high-temperature superconductors whichhave small coherence lengths and where, it has been argued, the Hubbard-model description may beappropriate.

Positron-annihilation spectroscopy (PAS) has longbeen recognized as a powerful tool for studying the elec-tronic properties of metals. ' In recent years, PAS hasbeen applied to the study of the Cu02-based superconduc-tors. ' These are systems which are believed to fallwithin the general category of nearly localized Fermiliquids. '

PAS studies of the new superconductors yielded a fewsurprises. It was observed that, unlike in earlier super-conductors, the positron lifetime changes on goingthrough the superconducting transition. The nature ofthis change was found to depend on the purity of thesamples, but the lifetime in the normal and superconduct-ing states were always found to differ significantly.Theoretical studies of positron annihilation have beenmostly focused on an electron gas. One of the predic-tions of these theories is the insensitivity of PAS to theBCS (Ref. 10) superconducting transition. " This couldlead one to speculate that the origin of effects observed inthe oxide superconductors lie in the intrinsically differentresponse of positrons to pairing in a system of nearly lo-calized fermions as compared to a weakly interactingFermi liquid.

It is not difficult to envisage a simple physical systemin which positrons may respond to pairing. One couldimagine a solid made up of two different types of siteswith a small concentration of one type ("hole" sites) lessattractive to the positron than the other, and distributedrandomly among the others. The electrons that the posi-tron annihilate with are imagined to be tightly bound tothese two types of sites. Because the hole sites are less at-tractive to the positron, the overlap with the electronsaround "hole" sites is less than that for the filled sites.As one increases the concentration of holes, therefore,the total electron-positron overlap decreases. Keepingthe hole concentration fixed, one could change the natureof the distribution from a random distribution of singleholes to pairs of holes. The electron™positron overlap isstill affected, because the positron would now be kept outof a larger region containing more tightly bound elec-trons. This is due to the difference between the repulsive

potential of a pair of holes as compared to two isolatedholes. ' The situation is very similar to the problem of adivacancy versus a vacancy. Further, the pair can havedifferent shapes and the overlap would reQect the shapeof this pair. The electron-positron overlap would then bea sensitive probe of the symmetry and amplitude of thepairing of these holes.

Approximately, this qualitative picture emerges from amodel of "strongly correlated" electrons, and, in thiswork, the positron-annihilation characteristics of thismodel are calculated and shown to have essentially thesame features as discussed above.

The most important prediction of the present study isthat PAS would provide detailed information about thenature of pairing in the high-temperature superconduc-tors (cf. Fig. 1) if indeed, they fall within the category ofstrongly correlated systems; if not, then PAS should notexhibit any dramatic changes on going through the su-perconducting transition. Current experiments ' indi-cate that changes do occur but more detailed experimen-tal studies have to be carried out before PAS can be usedto make any definite statements regarding the nature ofthe oxide superconductors.

A positron interacts strongly with the electrons of thesystem being studied. It is, therefore, crucial to take intoaccount the effects of electron-positron correlation on theannihilation characteristics. There is an extensive litera-ture on the subject of electron-positron interaction in ahomogeneous electron gas and its effects on PAS.These studies have also been extended to inhomogeneoussystems within a local-density-functional (LDF) formal-ism. ' ' In an electron gas, one is concerned with the in-teraction of a delgcalized positron with a delocahzed elec-tron. In contrast, in the nearly localized electron sys-tems, one is closer to the atomic limit and the electron-positron interaction is better described as a short-rangedreal-space interaction. The treatment of this problem isbeyond the realm of applicability of the LDF formalism.

In the present work, the t-J model ' is adopted fordescribing the dynamics of the strongly correlatedvalence electrons. These electrons make up only a small

1991 The American Physical Society

43 POSITRON ANNIHILATION FROM NEARLY LOCALIZED. . . 379

~ I I I I ftf;+g bt b; =1 .

Or 95

0;9C4

Ox95C4

0.85

0.80.9

0 2 4 6 8 10p(1, 1)

0 2 4 6 8 10p(I.o)

FIG. 1. Plots of R (p) |coming from all the core electrons (cf.text) which dominate the electron-positron overlap] in the su-

perconducting state (s- or d-wave pairing) for a square lattice.The parameters entering the calculation are chosen to be typicalof the CuO, -based superconductors; 5=0.10, pF (Fermimomentum of holes)=~/Sa, where a is the lattice constant,6 =Oe 10. (a) depicts R (p) along the (1,1) direction and p is inunits of pF. (b) R (p) along (1,0). R&(p) is taken to be Gaussianwith half-width equal to 2/a.

fraction of the total number of electrons is systems likethe CuOz-based superconductors. The remaining elec-trons are relevant for positron annihilation since theydominate the total electron-positron overlap. In thepresent model, it is assumed that the relevant time scaleof motion of these electrons is much longer than the posi-tron or the valence electrons, and are treated as frozencore-like electrons. The interaction of the positron withthese core electrons can be incorporated into a periodicpotential. The interaction of the positron with the Hub-bard electrons is modeled by a short-ranged potential.The nature of the pairing interaction between the elec-trons is left largely unspecified in this model. Variousmechanisms for pairing within the context of a Hubbardmodel have been suggested' and different scenarios willbe considered.

There have been a number of studies of various aspectsof the t-J model' and only the features relevant to thestudy of PAS will be noted here. The t-J Hamiltonianacts on a subspace of states with no doubly occupiedsites. Close to the half-filled limit, the t-J model de-scribes a few holes or empty sites propagating in the pres-ence of a lattice of spins. The description of holes orempty sites as particles is a valid concept only in thenearly localized regime where the no-double-occupancyconstraint is enforced. It does not carry over to a nonin-teracting gas of electrons. In the slave-fermion represen-tations, ' the spins and holes are represented by bo-sons and fermions, respectively, via a forrnal transforma-tion of the electron operators: c; =f; b; . Here ft keepstracks of the holes, and b,- keeps track of the spins, andthere is a no-double-occupancy constraint enforcedthrough

In the half-filled limit, there are no holes and the posi-tron moves in a periodic potential. The effects of the in-teraction of the positron with the "core" electrons is in-corporated into this potential. The residual interactionto be taken into account away from half-filling is that be-tween the positron and the holes or empty sites. This is arepulsiue interaction which has the effect of keeping thepositrons away from the empty sites and leads to an "an-ticorrelation" between the holes and the positron. Itshould be stressed that this is a transformation, dictatedby convenience, of the original electron-positron interac-tion problem to one that is more natural in the nearly lo-calized regime.

In considering positron annihilation, the spins can beignored except for their role in renormalizing the holemotion. In mean-field treatments, ' ' the effective hop-ping amplitude t is found to be proportional to the holeconcentration for small doping. A model Hamiltoniandescribing the holes and the positron can be written as

H=t g f f~+g Tjd; dj+ V QNn; . (1)&Ij& i j

Here n, =ftf, and N, =d,. d, are the hole and positrondensity, respectively. Wannier orbitals have been used asthe single-particle basis set for holes and positrons, withthe caveat that the positron Wannier orbitals are cen-tered, not at an atomic site, but at an interstitial site. Thelabel i can therefore be thought of as labeling a unit cellcentered on an atomic site. The last term in Eq. (1),describing the "on-site" repulsion between the holes andthe positrons, is the interaction term that can give rise tointeresting effects in PAS.

The central quantity in PAS is the electron-positronpair momentum distribution function, R (p):

( P)=p(f dr I dr'exp[ —ip (r —r')]

Xdr, (r')der(r')d&, (r)@r(r)),where N, and 42 are the electron and positron fieldoperators, respectively. The function R (p) measures theprobability of finding an electron-positron pair with amomentum p. The integral of this over all momentagives the total overlap, the inverse of which determinesthe positron lifetime. ' '

The overlap with the core electrons, i.e., all the elec-trons except the ones in the t-J model, dominates the to-tal overlap since these are the majority of electrons.From the physical picture presented earlier, this is ex-pected to be significantly affected by the interaction withthe holes, and it is, therefore, sufficient to consider onlythis part of the overlap in analyzing the effects of pairingon R (p). The calculations presented below willa posteriori justify this assumption.

The repulsive term in the Hamiltonian has the effect ofkeeping a hole and a positron from occupying the samesite. This does not lead to any localization effects becausethe concentration of holes is small, and there is only onepositron in the system at a time. Taking a cue from the

380 BULBUL CHAKRABORTY 43

variational treatments of the Hubbard model, the calcula-tion of R (p) will be based on a Gutzwiller-type wavefunction for the hole-positron ground state:

q[=II;(1 g—n;N; )%0 . (3)

The state %0 is the ground state in the absence of anypositron-hole interaction. The correlation factor g is thevariational parameter, and serves to reduce the weights ofthose configurations where the hole and the positron arein the same unit cell. The value of the variational param-eter g lies between 0 and 1. The effects due to the pairingof holes can be investigated by allowing for a nonzeropairing order parameter in %'o. The evaluation of thecore contribution to R (p) [cf. Eq. (2)] involves calculat-ing this function for a giUen configuration of holes, and

then averaging over all possible hole configurations. '

This quenched average is difficult to calculate in general.However, because of the short-ranged nature of the in-teraction, the possible hole configurations can be dividedinto two categories, one with no holes occupying thesame unity cell as the positron, and the other with a holeand positron in the same unit cell. Then R (p) can beevaluated by considering the two classes separately andadding up their contributions. The projection into thetwo classes can be conveniently carried out by using theoperator f;tf; since

f, f,

++blab;

=1.

This leads to

R(p)=z( f f +xb; b; f~f +xb bj d;d, x J dr 1 dr exp[ —ip '[r —r')]g;, (r p;([r p'&[(r')p," r')[.I,J a

(4)

Here [)[[; and 1'; are the e+ Wannier state and the one-electron core states, respectively. It has been assumedthat only electron and positron wave functions centeredin the same unit cell overlap. In the absence of any in-teraction, the expectation value reduces to ( d;"d ) .

To calculate R (p), the Gutzwiller-type variationalwave function [Eq. (3)] is used. This task is furthersimplified by noting that there are two small parametersin the problem, the hole concentration 6 and the positrondensity no. Keeping leading terms of the expansion in 6and no, the function

where the pairs are large compared to the interparticleseparation.

In the normal state„+o contains no condensate frac-tion, and therefore there are no anomalous expectationvalues like (f; f )[[. Keeping only the leading terms inan expansion in 5 and no, II(q) can be evaluated to be

II (q)II(q) =I+gII (q)

It is the approximation of small 5 and no that makes itpossible to write down the closed-form expression forII(q) and is not possible in general. 22

The superconducting ground state is characterized by anonzero expectation value ( c; c. )o. In the slave-fermion representation, this would imply nonzero valuesfor both ( f;fi ) and (b; b~ ~ ). The latter is nonzero aslong as the ground state has some singlet correlations,and the interesting question to ask is what effect, if any,the pairing of the holes has on the electron-positron pairmomentum distribution. Allowing for the appearance ofthe anomalous expectation values in Eq. (7), and retainingonly leading terms in 5, no, and (f,f ), leads to

F(q)=+exp[ —iq (R; —R )]

X, , +

can be rewritten as

F(q)=G (q) —gy(q')[ll (q+q') —II(q+q')]q'

=6 (q) —5(II (q+po) —II(q+po)) . (6)

The functions appearing in Eq. (6), G, y II, and II, arethe discrete Fourier transforms of (d; dj )o, (f; f& )o, .

(f,~f )0(d;td )o, and (ftf d; d. ), respectively.In writing down the last step of Eq. (6), it has been as-

sumed that the Fermi momentum of the holes, whichdetermines the length scale of y(q), is small compared tothe scale over which II(q) —II (q) varies significantly.Since we are interested in studying the effects of pairingon II —II, this implies that we are restricting ourselvesto pair sizes which are small compared to the interholeseparation. This is the opposite limit of Cooper pairs

[1+gll (q)]II(q)=II (q) —gS(q)+O(g ),S (q) =& I

a(q') I'I6'(q —q') I'

Here b, (q) is the discrete Fourier transform of (f,f ), . .

the pairing wave function on a lattice. ' This modifiesthe electron-positron pair momentum distribution to

43 POSITRON ANNIHILATION FROM NEARLY LOCALIZED. . . 381

R„„(p)= g f drexp( —ip r)g (r)P(r) (10)

where the forms f; =P (r —R, ) and P, =P(r —R;), havebeen used, and P(r) is the wave function of the orbitalmaking up the lowest-lying positron band.

It is clear from the above equation that the pairmomentum distribution coming from the overlap of thepositron with the tightly bound electrons, which is thedominant part of the overlap, is modified by the real-space pairing of holes, and that it reAects the symmetry ofthe pairing. Since this is the main conclusion of thiswork, it is worth reviewing the assumptions which wentinto deriving it. The main assumption was that the pairsare small. This implies that ~b, (q)~ has to vary overlength scales large compared to the Fermi momentum ofthe holes, since, otherwise, the convolution in Eq. (6)would smear out the effect. One therefore needs both (a)strong correlations under which the electron-positron in-teraction can be transformed to a hole position interac-tion, and (b) short-ranged pairs. The extreme short-ranged pairing limit is nearest-neighbor pairing on thelattice, and results for R (p), calculated for two differentsymmetries of pairing, are presented in Fig.. 1. A squarelattice is assumed, which would apply to Cu02 systems ifR (p) along the C axis is a constant. The results havebeen derived within a Gutzwiller approximation butcould have been derived using other perturbative tech-niques. The actual expansion parameter in Eq. (8) is g5and the terms left out are of order g 6 5 or smaller.Even for g —1, these terms are small compared to theleading ones.

As shown in Fig. 1, a nearest-neighbor pairing with ex-tended s-wave symmetry on a square lattice with a latticeconstant a, which leads to

b(p) =b.[cos(p a)+cos(p~a)],

manifests itself in angular correlation experiments asmodulating the isotropic distribution R„„(p). Figurel(a) shows R (p) along the (1,1) direction; along this samedirection, there would be no effect due to pairing if thesymmetry of the pairing was extended d wave[(cos(p a) —cos(p~a)]. Along the (1,0) direction [Fig.1(b)] both symmetries would modify R(p). This clearlydemonstrates that monitoring the change in the angularcorrelation function as one goes through the supercon-ducting transition temperature would lead to a directmeasurement of the symmetry of the order parameter.

g~(p+ po)Rs(p)=R&(p) —5 R„„(p) .

1+gn'(p+p, )

Here R& is the distribution in the normal state, Rz in thesuperconducting state, and

The effects occur over large regions in momentum(large compared to pF) and are not expected to be washedout by resolution effects, since typical resolutions arecomparable to pz. It should be remarked that anyeffect expected in a BCS superconductor would occurnear p p

The positron lifetime measures the inverse of the totaloverlap and, integrating R (p), it can be easily shown thatthe temperature dependence of the lifetime measures

b, ( T). However, the valence contribution has been total-ly neglected in this analysis and has to be added to thecore contribution before comparing to experiments. Thetemperature dependence of the lifetime had been ob-tained previously using qualitative arguments. ' Adifferent approach has been used in Ref. 25 to understandthe origin of this temperature dependence.

It should be stressed that throughout this paper it hasbeen assumed that the positron samples the Cu02 planes.If this is not true, then the application of the presentanalysis is dubious, since the core electrons probably donot dominate the overlap when the positron is in thechain. There are indications that the positron preferen-tially samples the chains in YBa2Cu3O7 but samples theplanes in La2Cu04-based superconductors. The latterwould therefore be the ideal system to investigate.

The amplitude of the effect on R (p) depends on thehole concentration, the amplitude of the pairing wavefunction 5, and the variational parameter g. In the limitof small 5 and large V, and t 5, g =-1—T5/V. Here Tis the nearest-neighbor hopping amplitude for the posi-tron. Making the assumption that T and V are compara-ble would lead to g =1—6, which is not a small number.It is expected that the electron-hole repulsion has to becomparable to the positron bandwidth before any in-teresting effects occur. The amplitude of the pairingwave function is roughly given by the ratio of the energygap to the Fermi energy. ' Since T, is large and the Fer-mi energy is small in the oxide superconductors, this ismuch larger than in conventional superconductors, and isof the order of 0.1. This is the value used in Fig. 1.

In conclusion, it has been shown that, in a model ofstrongly correlated systems, the positron can be a sensi-tive probe of electron pairing. The model was construct-ed to be on the opposite end of the spectrum from a delo-calized electron gas to stress the difference between astrongly correlated Hubbard-like system and a system ofweakly interacting delocalized electrons. One of the mostinteresting unresolved issues in the high-temperature su-perconductors is the nature of pairing and condensation.The present work opens up the possibility of using PASas probe of the nature of pairing in these systems.

The author would like to thank S. Berko and E. Jensenfor many useful discussions.

See articles in Positron Annihilation, edited by L. Doriken-VanPratt, M. Doriken, and D. Segers (World Scientific,Singapore, 1988); R. N. West, Adv. Phys. 22, 263 (1974).

~For reviews see, Positron Solid State Physics, edited by W.Brandt and A. Dupasquier (North-Holland, Amsterdam,

1983)~

E. C. Von Stetten et al. , Phys. Rev. Lett. 60, 2198 (1988); D. R.Harshman et al. , Phys. Rev. 8 38, 848 (1988); Y. C. Jeanet al. , J. Phys. Condensed Mat. 1, 2989 (1989).

4L. C. Smedskjaer et al. , Physica C {Amsterdam) 156, 269

382 BULBUL CHAKRABORTY 43

(1988).5C. Gross, R. Joynt, and T. M. Rice, Phys. Rev. B 36, 381

(1987).P. W. Anderson, Science 235, 1196 (1987).

7J. P. Carbotte and S. Kahana, Phys. Rev. 139, A213 (1965); J.P. Carbotte, Positron Solid State Physics, Ref. 2.

8S. Kahana, Phys. Rev. 129, 1622 (1963).9J. Arponen and E. Pajanne, Ann. Phys. (N.Y.) 121, 343 (1979).'OJ. R. Schrieffer, Theory of Superconductivity iBenjamin, Read-

ing, MA, 1964).R. Benedek and H.-B. Schuttler, Phys. Rev. B 41, 1789 (1990).Bulbul Chakraborty, Phys. Rev. B 39, 215 (1989).Bulbul Chakraborty, Phys. Rev. B 24, 7423 (1982).

~4E. Boronski and R. M. Nieminen, Phys. Rev. B 34, 3820(1986).

~5See, for example, P. A. Lee, Phys. Rev. Lett. 63, 680 (1989).See, for example, G. Baskaran, Z. Zou, and P. W. Anderson,Solid State Commun. 63, 973 (1987).C. L. Kane, P. A. Lee, T. K. Ng, B. Chakraborty, and N.Read, Phys. Rev. B 41, 2653 (1990).

' D. Yoshioka, J. Phys. Soc. Jpn. 5S, 32 (1989).

K. Flensberg, P. Hedegk. rd, and M. B. Pedersen, Phys. Rev. B40, 850 (1989).C. Jayaprakash, H. R. Krishnamurthy, and S. Sarkar, Phys.Rev. B 40, 2610 (1989).In most treatments of R (p), the order of averaging and in-tegration is not important and the averaging is done beforethe integration. In the present case the order has to bepreserved since there can be significant differences betweenthe time scales associated with the dynamics of the holes andthe positron.W. Metzner and D. Vollhardt, Phys. Rev. Lett. 59, 121 (1987).One might wonder about the fate of the t -J-model correlationand the pairing correlation in the state O'. It can be arguedthat these remain essentially the same as in %0, provided thatthe positron kinetic energy T is small compared to V and theHubbard U. If T becomes the largest energy scale in the sys-tern then the whole model breaks down.

24Typical experimental angular resolution is -0.5 mradX0. 5mrad. The pF chosen corresponds to 0.6 mrad.T. McMullen, Phys. Rev. B 41, 877 (1990).