VOLUME 39, NUMBER 9 P H Y S I C A L R E V I E W L E T T E R S 29 AUGUST 1977

the d i s o r d e r e d S i ( l l l ) l x l su r face . Then the ob­se rva t ion of SiH4, a s tab le spec ies in the gas phase , on the (311) sur face should be taken as evidence of the exis tence of the SiH3 su r face phase .

Another impor tan t observat ion is that the ob ­tained energy dis t r ibut ion in the field evaporat ion of Si is wide, a lmos t 35 eV (half-width at half-maximum), compared to an energy sp read of l e ss than 10 eV in the case of a me ta l in the dc evap­orat ion mode.17*20 This s t r ik ing difference may be a r e su l t of the fact that field penet ra t ion at the semiconductor su r face is ve ry l a rge c o m ­pared to that at a me ta l sur face where the s c r e e n ­ing length is an angs t rom or so . 2 4 The energy dis t r ibut ion in field evaporat ion is of g rea t i n t e r ­est and is the bas ic quantity for developing a good f ie ld-evaporat ion theory .

In conclusion, we have demons t ra ted the unique capabil i ty of a magne t i c - s ec to r a tom-p robe FIM to give impor tan t r e s u l t s in a semiconduc to r -sur face study. Using Si e m i t t e r s we have de tec t ­ed and identified s i l icon Si+ , s i l icon monohydride SiH+ , s i l icon dihydride SiH2

+, and s i lane SiH4 +

as f ie ld-evaporat ion products f rom var ious s u r ­face r eg ions . These observat ions can be taken as evidence for the format ion of s i l icon dihydride and t r ihydr ide sur face phases upon the i r i n t e r a c ­tion with hydrogen as well as the monohydride sur face phase .

We a r e grateful to R. H. Good, J r . , J . E. Rowe, H. D. Hags t rum, E. I. Givargizov, and T. T. Tsong for helpful d i scuss ions and comments on this study. This work was supported by the R e ­s e a r c h Init iation Grant of The Pennsylvania State Univers i ty and by the National Science Foundation.

* ̂ Deceased.

One of the most puzzl ing fea tu res of the sp in-g l a s s e s (dilute al loys such a s CuMn and AuFe) i s t he i r "dua l " na ture : Measu remen t s of the s t a t -

!G. E. Becker and G. W. Gabeli, J . Chem. Phys. 38, 2942 (1963).

2H. Ibach and J. E. Rowe, Surf. Sci. 43, 481 (1974). 3J. A. Appelbaum and D. R. Hamann, Phys. Rev. Lett.

34, 806 (1975). ~~~*T. Sakurai and H. D. Hagstrum, Phys. Rev. B :12, 5349 (1975).

5K. C. Pandey, T. Sakurai, and H. D. Hagstrum, Phys. Rev. Lett. 35, 1728 (1975).

6T. Sakurai and H. D. Hagstrum, J. Vac. Sci. Tech. 13, 807 (1976).

7T. Sakurai, M. J. Cardillo, and H. D. Hagstrum, J. Vac. Sci. Tech. 14, 397 (1976).

8T. Sakurai and H. D. Hagstrum, Phys. Rev. B 14, 1593 (1976).

9T. Sakurai and H. D. Hagstrum, to be published. 10J. A. Appelbaum, G. A. Baraff, D. R. Hamann, H. D.

Hagstrum, and T. Sakurai, to be published. n J . E. Rowe and H. Ibach, Phys. Rev. Lett. 32, 421

(1974). 12F. Jona, H. D. Shih, A. Ignaliev, D. W. Jepsen, and

P . M. Marcus, to be published. 13M. B. Webb, Bull. Am. Phys. Soc. 21, 936 (1976). 14E. W. Miiller, J . A. Panitz, and S. B. McLane, Rev.

Sci. Instrum. J39, 83 (1968). 15A. J. Helmed and R. J. Stein, Surf. Sci. 49, 645

(1975). 16A. J. Melmed and J . A. Panitz, private communica­

tion. 17E. W. Miiller and T. Sakurai, J . Vac. Sci. Tech. 11,

878 (1974). 18T. Sakurai, E. W. Miiller, R. J . Culbertson, and

T. T. Tsong, to be published. 19H. Ibach and J . E. Rowe, Phys. Rev. B 10, 710. (1974). 20A. R. Waugh and M. J . Southon, J . Phys. D 6, 1017

(1976). 21E. W. Miiller, S. V. Krishnaswamy, and S. B. Mc­

Lane, Phys. Rev. Lett. 31, 1282 (1973). 22T. Sakurai, T. T. Tsong, and E. W. Miiller, Phys.

Rev. 181, 530 (1969). 23D. C. Gupta, Solid State Tech. 14, 33 (1971). 24T. T. Tsong and E. W. Miiller, Phys. Rev. 181, 530

(1969).

ic low-field suscept ibi l i ty x , 1 Mossbauer sp l i t ­t ing,2 and muon-p reces s ion r a t e 3 suggest that these alloys exhibit a sha rp phase t r ans i t ion at

Cluster Mean-Field Theory of Spin-Glasses

C. M. Soukoulis and K. Levin The Department of Physics and The James Franck Institute, The University of Chicago, Chicago, Illinois 60637

(Received 18 April 1977)

hi view of the large amount of experimental evidence in support of a cluster descrip­tion of the spin-glasses, we propose and solve a mean-field model of these systems in which dynamical clusters are the basic entity. The model can explain an important and unresolved puzzle % why the susceptibility has a cusp at a freezing temperature Tc9

while the specific heat has a rounded maximum at a significantly higher temperature0

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a freezing temperature Tc. Thermoelectric pow­er,4 resistivity,5 and magnetic specific-heat6 Cm

measurements do not reveal any anomalies at Tc. Moreover, Cm generally has a broad maximum at a temperature Tc about 20-40% above Tc. Theo­retical approaches to a study of the spin-glasses have been equally divided and only partially suc­cessful. Mean-field theories7 '8 lead to sharp anomalies in both \ and Cm. Also inconsistent with experimental observations are the cluster theories9 '10 based on the notion of thermal block­ing of rigid clusters of spins. While these can explain some hysteresis effects9—the magnitude of the zero-temperature resonant magnetization10

and field-sensitivity experiments10—they predict9

a rounded peak in x. The cluster percolation the­ory of Smith11 yields a cusp in x, but the behavior of Cm near Tc is indeterminate.

The purpose of this Letter is to propose a sim­ple model for the spin-glasses which incorporates both the mean-field-theoretic and cluster notions. In this way, it is shown how a sharp cusp in x can appear at Tc9 while at the same time the spe­cific heat exhibits a broad maximum at a higher temperature. In contrast to previous rigid-clus­ter theories,9"n we emphasize here that the in­ternal dynamics of the clusters are important and play an essential role in determining the behavior of Cm.

The cluster model Hamiltonian is given by

* = - EJvx* v i<j

where J{j° is the intracluster exchange constant and JvXthe randomly distributed intercluster ex­change constant. In order for the cluster con­cept to make sense, the former interactions are assumed stronger than the latter. Greek indices refer to a particular cluster and Roman indices

to a given spin within that cluster. Here § v

=J^iSivis the total spin of a cluster. Thus the first term in the Hamiltonian describes interac­tions between clusters while the second refers to intracluster interactions0 We assume that J vX is independent of the spin indices i and j . This is justified provided the clusters are sufficiently far apart so that the intercluster exchange inter­actions are insensitive to the location of the spins within a cluster. For simplicity we take the clus­ters to be identical in size and shape. These then represent the average cluster in the alloy. The effects of including a distribution of cluster sizes within mean-field theory (in which all but one cluster is averaged over) are quantitative rather than qualitative. These will be discussed below. The model may be justified from a phe-nomenological point of view. A rather complete list of experimental support for it is given in Ref. 10. Microscopically, one may view the clusters as units of spins which are spatially in­terconnected and therefore strongly correlated. While these will occur in any statistically ran­dom arrangement of impurities, chemical clus­tering, which is usually present, will enhance their effects.12 As in the theory of Edwards and Anderson7'8 (EA), the intercluster exchange in­teractions which are given by a near-neighbor Gaussian distribution,

P(JvX) = (2ir) ' ^ J ^ e x p t - J , , 72J 2 ) , (2)

are treated within a random-mean-field theory. The intracluster interactions, however, are treat treated exactly.

The free energy is calculated with use of the replica method, which is believed to be reliable for temperatures at and above Tc. It is this tem­perature region which we will focus on here. We find that

• kTlimn'ifudJ^Tr^M^ £ Jv $va^£- ^EHciav)TlP(JvX) - l ] , n ~^0 a v<\ a v

(3)

where the intracluster Hamiltonian is

H„ (4)

and where the trace is taken over n replicas labeled a of the actual system. To evaluate the integral in (3), a mean-field approximation of the intercluster interaction is used so

that commutators like7 [Sy-Sx , S^-S^.J, etc., with z^A, are set equal to zero. Our mean-field decoup­ling of the resultant expression for F introduces two variational parameters: q- {'§v

a *'§ve>) for a^/3,

and M- (§lJa '^v

a). The former is analogous to the usual EA spin-glass order parameter. It differs only in that Sv denotes the cluster spin rather than the spin of one impurity atom. Here M is the total spin of each cluster; in the EA theory the analogous parameter is a (nonvariational) constant. Follow -

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VOLUME 39, NUMBER 9 P H Y S I C A L R E V I E W L E T T E R S 29 AUGUST 1977

ing Refs . 7 and 8, we find that the free energy p e r c lus t e r i s

where

•eff . -$jv+J(q/3)l/27-§v (J2/6kT)(M^q)^v^v,

(5)

(6) i<j

Here J = zl/2J with 2 the number of c lus t e r nea r -ne ighbor s of a given c lus t e r . In der iving (5) we have used the fact that al l t h r e e t e r m s in the effective c lus te r Hamiltonian Hef{ commute with one another . The physical in te rpre ta t ions of the va r ious contr ibutions to HeU a r e as follows: The f i r s t t e r m is the i n t r a c l u s t e r exchange; the second r e p r e s e n t s the in te rac t ions of the random anisotropy field with the spins in the c lus t e r . This field a r i s e s because of i n t e r c lu s t e r in te rac t ions and i t s s t rength is given in t e r m s of the o r d e r p a r a m e t e r q. The th i rd t e r m i s an effective fe r romagne t ic i n t r a c l u s t e r exchange t e r m which a l so der ived f rom in t e r c lu s t e r in te rac t ions . In the usua l mean-f ield t heo r i e s , th is i s a c -number and i s combined with the f i r s t t e r m in Eq. (5).

The o r d e r p a r a m e t e r s q and the c lus t e r moment M a r e der ived var ia t ional ly by us ing [dF(q, M)/ dq]M = 0 and [dF(q,M)/dM]g = 0. This yields the coupled equations, with Z ^ T r e x p ( - )3#e f f):

M=(2irr**fd3re'r2*Tr($v'$vem*^n)/Z

and

M-q = (2Tr)-3/2fd3rre-r2/2(3/q)l/2(kT/J)Tr(7-1§ve-

When a dis t r ibut ion of c lus te r s i ze s i s included 1 we find that these equations a r e eas i ly modified.13 ' Because our numer i ca l r e su l t s (see below) show the s a m e qual i ta t ive behavior for al l c lu s t e r s i z e s we will cons ider , for s impl ic i ty , the dis t r ibut ion to be sharply peaked about some value N0 Both the s ta t ic suscept ibi l i ty x and the specific heat Cm p e r c lus t e r may be calculated in t e r m s of M and q,

X=g2liB2(M-q)/3kT, (9)

where gjiB i s the magnet ic moment of an i m p u r i ­ty a tom. The specific heat i s given by a sum of two t e r m s , Le 0 ,

C=C„ '+C„ (10)

The f i r s t contribution a r i s e s p r i m a r i l y f rom in­t e r c l u s t e r contr ibut ions and is equal to

C mi n t e r = ^ [ ( J 7 6 f e T ) ( ( f - M 2 ) ] ,

while the i n t r a c l u s t e r contr ibution is

dT f* r e . 2 / 2 Tr( f f c l e ' S//( reff,

(11)

(12)

Because it re f lec ts the dynamics of a finite num­b e r of sp ins , C m

i n t r a has a rounded maximum as a function of t e m p e r a t u r e which occu r s at the c h a r a c t e r i s t i c i n t r a c l u s t e r exchange- in te rac t ion energy. On the other hand, the f i r s t express ion on the r ight-hand side of Eq. (10) yields a c u s p ­like contribution to Cm l ike that found by EA.

^ e f f f ) / z .

(7)

(8)

The magnitude of C ml n t r a grows as the number of

spins N in the c lus t e r i nc r ea se s , 1 4 w h e r e a s for l a rge iVthe magnitude of C m

i n t e r i s independent of iVo7'8 It is because t he r e a r e two dis t inct en ­ergy sca l e s in the model, cor responding to the s t rength of the i n t r ac lu s t e r and i n t e r c l u s t e r ex ­change in te rac t ions , that the i n t r a c l u s t e r m a x i ­mum occurs at a t e m p e r a t u r e T0 which i s differ­ent f rom—in fact h igher than—Tc0 At low T, the in t r ac lu s t e r contribution v a r i e s as e"x/ T while C m

i n t e r is l inear in T as found exper imenta l ly . 6

For c lu s t e r s in which M is not s t rongly t e m p e r ­a tu re dependent, we find f rom Eq. (9) that , as in previous mean-f ie ld t h e o r i e s , 7 ' 8 the suscep t ib i l i ­ty x has a sha rp cusp at Tc. In this ca se , the cha rac t e r i s t i c in terac t ion which d e t e r m i n e s the behavior of x is the in t e r c lu s t e r exchange.

We have per fo rmed numer i ca l calculat ions to study sys temat ica l ly the r e s u l t s of our model for a wide range of p a r a m e t e r s . We cons idered c l u s ­t e r s containing N^ 6 spins coupled by e i the r n e a r -neighbor fe r romagne t i c o r an t i fe r romagnet ic ex ­change interactionso Because the i n t r ac lu s t e r contribution to the specific heat i s s m a l l e r for the fe r romagne t ic than for the an t i fe r romagnet ic case , 1 4 the mos t favorable r e s u l t s and those which a r e i l lus t ra ted h e r e a r e for an t i fe r romagnet ic coupling. We a lso chose two va lues of the p a r a m ­e te r N. For the case N = 3 the ra t io of the ex ­change consta tns \J/J0\ =0.55, and for N=6 th i s was given by =0.32. While the p a r a m e t e r s w e r e chosen to yield reasonably good ag reement with

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V O L U M E 39, N U M B E R 9 P H Y S I C A L R E V I E W L E T T E R S 29 AUGUST 1977

FIG0 1„ T e m p e r a t u r e dependences of the (a) s ta t ic suscept ibi l i ty and (b) specific hea t p e r c lus t e r for an t i fe r romagnet ic i n t r ac lu s t e r exchange in te rac t ions with N = 3 (solid curve) and N = 6 (dashed curve) . In the inse t s a r e plotted the r e s u l t s obtained within the usua l mean-f ie ld theory ,

experiment, the results obtained were not atypi­cal.

In Fig. 1(a) is shown the temperature depen­dence of the normalized susceptibility x(T)/\(Tc). The solid (dashed) curve is for N = 3 (N=6). The behavior of x is similar to what is found in the quantum-mechanical generalization of the EA theory,7 '8 which is plotted in the inset. Because of the T dependence of M, we find that x deviates slightly from a Curie-law behavior above Tc.

Finally, in Fig0 1(b) is shown the temperature dependence of Cm. The solid and dashed curves correspond to N = 3 and N= 6, respectively. A broad rounded maximum occurs at a temperature TQ~1.1T. There is also a small cusp at T .

Fluctuation effects15 may make this small feature undetectable experimentally. For comparison purposes, the quantum-mechanical EA-model r e ­sults are plotted in the inset. It is evident that the model represents an improvement over this previous one which has a sharp cusp at Tc and de­creases monotonically above this temperature in disagreement with experiment.

This work was supported by the National Science Foundation and by the National Science Founda­tion-Materials Research Laboratory. One of us (K.L.) acknowledges a fellowship from the Alfred P. Sloan Foundation.

1V. Cannella and J . A. Mydosh, P h y s . Rev. B j>, 4220 (1972).

2 B. Window, in Magnetism and Magnetic Materials — 1974, ATP Conference Proceed ings No. 24, edited by Co D. Graham, J r . , G. H. Lander , and J . J0 Rhyne (American Insti tute of P h y s i c s , New York, 1975), Vol. 1, p . 167.

3D. E. Murnick, A. T . F io ry , and W. J . K o s s l e r , P h y s . Rev. Let t . 36 , 100 (1976).

4 P. J . Ford etal., Solid State Commun. 13 , 857 (1973). 5P0 J . Fo rd and J . A. Mydosh, P h y s . Rev. B_14, 2057

(1976); and J . A. Mydosh etal., P h y s . Rev. B ^ O , 2845 (1974).

6 L. E. Wenger and P . H. Keesom, P h y s . Rev. B JUL, 3497 (1975), and 13, 4053 (1976).

7S. F . Edwards and P„ W, Anderson, J . P h y s . F £ , 965 (1975); K. H. F i s c h e r , P h y s . Rev. Let t . 34, 1438 (1975).

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10K. Binder , Z. P h y s . B £ 6 , 339 (1977); see a l so K. Binder and K. Schroeder , P h y s . Rev0 B^U, 2142 (1976).

n D 0 A. Smith, J . P h y s . F 4, L226 (1974), and 5, 2148 (1975).

12In a " c o r r e l a t e d " - c l u s t e r (Ref< 10) r a t h e r than spa ­t i a l - c l u s t e r descr ip t ion , the average c lus t e r s i z e s should be taken to be T dependent. We have cons idered these effects and find that in the region of t e m p e r a t u r e s of i n t e re s t (near Tc and above) our r e s u l t s a r e not quali tat ively affected.

13The essen t ia l modification is to int roduce a sum over the c lus t e r s ize JV t i m e s the probabi l i ty d i s t r i ­bution P(N) on the r ight -hand side of these equat ions. That they a r e so l i t t le affected comes about physical ly because in mean-f ield theory a d is t r ibut ion in the s i ze of ave raged-ove r spins is not of pa r t i cu l a r significance.

14J. C. Bonner and M. E. F i s h e r , P h y s . Rev0 135, A640 (1964).

t5A. B. H a r r i s , T„ C. Lubensky, and J . - H . Chen, P h y s . Rev. Let t . 3j6, 415 (1976); Ce J ayap rakash , J . Chalupa, and M. Wor t i s , Phys . Rev. B JJ5, 1495 (1977).

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