VOLUME 44, NUMBER 17 P H Y S I C A L R E V I E W L E T T E R S 28 APRIL 1980

Tunneling States in Metallic Glasses: A Structural Model

M. Banville Groupe de Recherches en Semiconducteurs et Dielectriques, Departement de Physique,

Universite de Sherbrooke, Sherbrooke, Quebec J1K2R1, Canada

and a(a) R. Harris^'

Physics Department, McGill University, Montreal, Quebec H3A2T8, Canada (Received 20 December 1979)

We analyze the metas table states for individual atoms within a computer-generated structural model of a metallic glass, and discover that their probability of occurrence is a strong function of the degree of relaxation of the model. We propose that the relaxa­tion process simulates closely the annealing of metallic glasses below the glass transi­tion temperature, so that the annealing process should reduce the magnitudes of the vari­ous low-temperature anomalies seen in these materials.

PACS numbers: 61.40.Df

The structure of metallic glasses has received considerable recent attention in the literature: A review is given by Car gill.1 In general, some variant of the dense random packing of atomic spheres gives a good account of the structure, reproducing, for example, the characteristic double peak at the next-nearest-neighbor distance in the radial distribution function (RDF) for glass­es with only one type of metallic atom.1 In more particular amplications, however, it has been found that the simplest, unconstrained packings of hard spheres differ significantly from the structures of real glasses, and improvements have been achieved through two approaches. In the first of these2 it has been shown that "chemi­cal" constraints can be very significant in those glasses which contain "glass formers" such as P, C, or Si. In the second case,3 '4 more re le­vant to the present paper, it has been shown that a structural relaxation to equilibrium under the influence of suitable interatomic forces can also materially improve the agreement with experi­mentally determined RDF's.

More recently, a variety of experimental r e ­sults have posed more detailed questions about the structure of metallic glasses. Recent low-temperature measurements of the specific heat,5

the thermal conductivity,5'6 and the dispersion6"8

and the attenuation8*9 of ultrasonic waves as well as the low-temperature resistivity anomaly10

have all received explanations in terms of "two-level systems" (TLS's) or "tunneling states" as ­sociated in some way with the structure. These explanations have all been motivated by the mod­el proposed by Anderson, Halperin, and Varma11

and by Phillips12 to explain analogous phenomena in insulating glasses. In what follows, we pro­

pose a mechanism for this model which is par­ticular to metallic glasses, and which leads us to a new assessment of existing data.

In the original papers11 '12 the TLS's were mod­eled by double-well oscillators, having energy barr iers V between the two wells and an energy difference AE between the two minima. The atoms or groups of atoms represented by this moedl were thus able to tunnel from one well to the other. In order to explain the experimental data, it is necessary to postulate that there be a finite probability of finding both V and AE close to zero. Measurable quantities such as the spe­cific heat, thermal conductivity, and ultrasonic response are then related to the probability den­sity of TLS's having AE equal to zero, a quantity which has a value around 5x l0" 2 e V 1 atom"1 i r ­respective of the material studies. For the pur­pose of subsequent discussion, we note also that this value includes only those TLS's with barr i ­e r s low enough, V<VmaK> that tunneling can oc­cur during the time of the experimental measure­ment.11

In the insulating glasses, the TLS's have been associated with structural features specific to the glassy state,13 but there is no universal agree­ment on their precise nature. In the absence of direct experimental evidence, a computer simu­lation of the structure of a covalently bonded glass was carried out by Smith14 in an attempt to identify possible TLS's. His results were en­couraging, although with disappointing statistics in the physically interesting range of parameters. In the metallic glasses, however, there has until now been no attempt, either experimental or theo­retical, to associate any particular structural features with the TLS's presumed to be present.

1136 © 1980 The American Physical Society

V O L U M E 44, N U M B E R 17 PHYSICAL REVIEW LETTERS 28 A P R I L 1980

In what follows, we report on our investigations of a computer-generated model which we relaxed to varying degrees suing a standard algorithm which allows each atom in turn to move down the potential energy gradient. Our chosen model comprised the central 1170 atoms of the 8000-atom structure of hard spheres of one size con­structed by Bletry,15 and we used a Lennard-Jones potential V(r) =4e[(or/r)12 - (cr/r)6] to relax this structure from its original density (packing frac­tion) of 0.54.

We chose to estimate the range of appropriate values for € by comparing the activation energy for the diffusion of a three-dimensional t ransi­tion metal atom in a body-centered cubic lattice (typically 0.5 to 5.0 eV) with the corresponding value calculated with our potential. In this way we assign € values in the range from 0.01 to 0.1 eV. Of course, strictly speaking, neither a Len­nard-Jones interatomic potential nor the similar Morse potential gives a good description of metal­lic binding, although for computational reasons they have often been used in other simulation cal­culations.3 Nevertheless, the highest density achieved in our simulation (0.6320) compares favorably with that measured for a simple MgZn glass.4

Apart from the density, an essential feature of our model structure is the presence of voids, which are somewhat analogous to vacancies in a crystalline material, but which are scarcely large enough to accommodate an extra atom. Al­though we can make no quantitative comparison between the characteristics of the voids in our low-density model and those in a typical metallic glass, with density around 0.69,1 we believe that there is a qualitative relationship, since the measurement of atomic diffusion in metal-metal­loid glasses16 '17 has been interpreted in terms of the "porous" nature of the structure.

The significance of the voids became apparent as , at various stages during the relaxation proc­ess , we searched for the simplest TLS's, corre­sponding to the motion of single atoms. We ex­amined the environments of a limited number of atoms in the center of the model structure, using a detailed analysis of the numbers of common nearest neighbors possessed by pairs of atoms in the structure,18 which enabled us to identify those situations most likely to give rise to TLS's. In particular, we chose as prime candidates those atoms which were the only common neigh­bor of a given pair of atoms. In each case we then searched for alternate metastable minimum-

energy positions for the single atom in the force field of its neighbors, and, when successful, tab­ulated the relevant parameters. Detailed analy­sis revealed that all our TLS's were associated with the voids in the structure, with the tunnel­ing motion consisting of a single atom moving between two or more alternate equilibrium posi­tions within the void. Although we did not simul­taneously take account of possible cooperative r e ­arrangements of the neighbors around the chosen single atom, we verified that such rearrange­ments were not significant in any interesting cas­es , in accord with the work of Gibbs19 who ex­amined the stability of vacant sites in a relaxed structure.

Typical results of our analysis are given in Fig. 1, which shows the distribution of TLS pa­rameters AE and V, as defined in the inset, after 325 relaxation steps per atoms, when the density had reached 0.6065. The axes are labeled in units of €, and also in units with c =0.1 eV, approxi­mately representative of a real glass, as dis­cussed earlier.

To estimate the probability density correspond­ing to the specific heat, we first evaluate the pa­rameter V ^ following Ref. 11. Using a value of 100 a.u. for the mass of a tunneling atom we ob­tain Fmax^O.OS eV, a value which depends only weakly on the energy of zero-point motion. By projecting the calculated distribution of Fig. 1 onto the AE axis, using only points with V< Vmax,

20

V /units\ Vote )

12

1.0 cV

0 4 AE 8

(units of €)

1.0cV 12

FIG. 1. Probabi l i ty d is t r ibut ion of T I S ' s for the cen t ra l 300 atoms of the model s t r u c t u r e re laxed to densi ty 0.6065. The axes co r respond to the b a r r i e r height V and the energy difference AE, as shown in the inset , m e a s u r e d in units of the LJ e . The p r o b a ­bil i ty densi ty cor responding to the specific heat is a project ion onto the A E axis of points with V< V m a x

= 0.05.

1137

VOLUME 44, NUMBER 17 PHYSICAL REVIEW LETTERS 28 APRIL 1980

we obtain a histogram for the probability density, from which we can extract the value for AE =0 as required for the specific heat. The appropri­ate values are -0.02 states eV"1 atom"1 with £ = 0.1 eV and -1 .0 states eV"1 atom"1 with € =0.01 eV, so that, given the large statistical e r ro r s , there is qualitative agreement with the experi­mental data.

However, a much more significant feature of our results concerns our analysis of the struc­ture when it is relaxed by a further 1520 steps per atom, to a density of 0.6320. At first sight, the changes in the structure, as monitored by the density or the radial distribution function (RDF), are only minor. However, as is clearly shown in Fig. 2, the probability density of TLS's in the interesting range of energies drops by almost an order of magnitude, a result which is consistent with our analyses of other "fully" relaxed struc­tures,20 and which also explains the null result of a search for TLS's by Gibbs.19 This substantial reduction in the probability density might be r e ­garded as an artifact of our relaxation procedure were it not for considerable evidence that sug­gests a comparison of the structures of densities 0.61 and 0.63 with those of real metallic glasses before and after annealing below their glass tran­sition temperatures.

Although it has been evident for some time3

that the relaxation procedure qualitatively im­proves agreement with experimentally deter­mined radial distribution functions (RDF's), it was only recently that energy-dispersive x - r ay -diffraction (EDXD) techniques21 have explicitly demonstrated the changes in the RDF's during an-

/unitsN Uf e ;

12 i~

1.0<2V \~

c ; i : L i L ^

O 4 A ,_ 8 1.0«2V 12 AE

(units of e)

FIG. 2. Probability density of TLS's for the central 300 atoms of the model structure relaxed to density 0.6320. Notation is otherwise the same as in Fig. 1.

1138

nealing. Quite explicitly, the effect of annealing is to enhance the structures seen in the RDF's, or to cause a "transformation towards a more stable amorphous state." This, of course, is precisely the function of any relaxation procedure, and, indeed, our calculations of the RDF are in qualitative agreement with the EDXD data. If we further assume that the annealing is associated with the disappearance of the voids in the struc­ture, an assumption which is consistent with the reduction of the diffusivity by two orders of mag­nitude upon annealing,17 then we can estimate the corresponding activation energy as £ 20e, which, with e taken as 0.1 eV, is in qualitative agree­ment with the data.21 '22

Further evidence that the effects of our relaxa­tion actually correspond to the annealing of real glasses is also to be found in a published analy­sis of low-temperature thermal-conductivity da­ta.6 In this paper it was shown that "mild heat treatment" actually increased the magnitude of the conductivity by some 45%: On the basis of the usual TLS model,11 '12 this was related to a reduction in the probability density of TLS's. Similar evidence relating to the low-temperature anomaly in the electrical resistivity10 can also be found,23 but there have been no published anal­yses of relevant specific-heat or ultrasonic data.

To summarize, we propose that the TLS's which exist in metallic glasses correspond to the motion of individual atoms within voids in the in­completely relaxed structure. We further pro­pose that the effect of annealing is to eliminate the TLS's by the suppression of the voids, and thus to eliminate the characteristic anomalous properties of these materials which appear at low temperatures. A straightforward test of our ideas would thus be the reevaluation of the effect of annealing on such properties: We suggest that measurement of the dispersion of ultrasound be­fore and after annealing would be very worth­while. K correct, our ideas offer the prospect of a clearer understanding of the structure of metallic glasses and perhaps, of a resolution of some of the current theoretical problems.10

One of us (M.B.) is grateful to J. P. Gaspard for his hospitality during an extended visit to the University of Liege, for suggesting the configura-tional analysis,18 and for providing the coordi­nates of the partially relaxed structure of Bletry (Ref. 15). Thanks are also due to our colleagues for useful discussions.

This work was supported in part by the Natural Sciences and Engineering Research Council of

VOLUME 44, NUMBER 17 PHYSICAL REVIEW LETTERS 28 APRIL 1980

Canada (NSERC) and by the Formation de Cher-cheurs et Action Concertee (FCAC) program of the government of Quebec.

( aTo whom all correspondence should be addressed. 1G. S. Cargill, III, in Solid State Physics, edited by

H. Ehrenreich, F. Seitz, and D. Turnbull (Academic, New York, 1975), Vol. 30, p. 227.

2J. F. Sadoc, J. Dixmier, and A. Guinier, J. Non-Cryst. Solids JL2, 46 (1973).

3L. von Heimendahl, J. Phys. F 5, L147 (1975), and 9t 161 (1979); J. A. Barker, J. L. Finney, and M. R. Hoare, Nature (London) 257, 120 (1975); A. Rahman, M. J. Mandell, and J. P. McTague, J. Chem. Phys. 64, 1564 (1976).

4J. Hafner and L. von Heimendahl, Phys. Rev. Lett. 42, 386 (1979).

5J. E. Graebner, B. Golding, R. J. Schutz, F. S. L. Hsu, and H. S. Chen, Phys. Rev. Lett. £9, 1480 (1977).

6J. R. Matey and A. C. Anderson, J. Non-Cryst. Solids 23, 129 (1977), and Phys. Rev. B !L6, 3406 (1977).

7G. Bellessa, P. Doussineau, and A. Levelut, J. Phys. (Paris), Lett. 38, L65 (1977); G. Bellessa, J . Phys. C10, L285 (1977); G. Bellessa and O. Bethoux, Phys. Lett. 62A, 125 (1977).

8B. Golding, J. E. Graebner, A. B. Kane, and J. B. Black, Phys. Rev. Lett. 41, 1487 (1978).

9P. Doussineau, A. Levelut, G. Bellessa, and O. Bethoux, J. Phys. (Paris), Lett. 38, L483 (1977).

10R. W. Cochrane, R. Harris , J . O. Strom-Olsen, and M. J. Zuckermann, Phys. Rev. Lett. 215, 676 (1975).

n P . W. Anderson, B. I. Halperin, and C. M. Varma,

In this Letter we report the first observation of low-energy positron (e*) diffraction (LEPD) from a solid surface, Cu(l l l ) . 1 LEPD offers the possibility of becoming a quantitative tool for the study of surfaces to complement the well-established technique of low-energy electron dif­fraction (LEED)c The change in the sign of the

Philos. Mag. _25, 1 (1972). 12W. A. Phillips, J. Low Temp. Phys. 1, 351 (1972). 13S. Hunklinger and W. Arnold, in Physical Acoustics,

edited by R. N. Thurston and W. P. Mason (Academic, New York, 1976), Vol. 12, p. 155.

14D. A. Smith, Phys. Rev. Lett. 42, 729 (1979). 15J. Bletry, Z. Naturforch 329, 445 (1977). 16D. Gupta, K. N. Tu, and K. W. Asai, Phys. Rev.

Lett. £5, 706 (1975); H. S. Gill and J. H. Judy, J. Appl. Phys. 50, 1648 (1979).

17H. S. Chen, L. C. Kimerling, J. M. Poate, and W. L. Brown, Appl. Phys. Lett. £2, 461 (1978).

18M. Banville and J. P. Gaspard, to be published. 19S. Gibbs, M.Sc. thesis, McGill University, 1977

(unpublished). 20Our analysis (M. Banville and R. Harris , to be

published) has also been applied to the model structures built by Rahman et al. and by von Heimendahl (see Ref. 3). For the purposes of comparison we note that in terms of the effective time defined by Rahman et al. their simulation ran for 100 units, whereas our final structure had been relaxed for 127 units.

21T. Egami, J. Appl. Phys. J50, 1564 (1979), and J. Mat. Sci. 13, 2587 (1978).

22H. Fujimori, S. Ohta, T. Masumoto, and K. Naka-moto, in Rapidly Quenched Metals Til, edited by B. Cantor (Metals Society, London, 1978), Vol. 2, p. 232; R. Hasegawa and R. C. O'Handley, J. Appl. Phys. 50, 1551 (1979).

23S. Schmid-Marcic and J. A. Mydosh, Solid State Commun. 17, 795 (1975); S. B. Dierker, H. Gudmunds-son, and A. C. Anderson, Solid State Commun. 29, 767(1979).

charge from e~ to e+, the absence of an exchange term in the scattering Hamiltonian, and differ­ences in correlation effects make the interactions of positrons with a surface significantly different from those of electrons. As there is no readily available means for producing large quantities of low-energy positrons, the development of a

Low-Energy Positron Diffraction from a Cu(l l l ) Surface I. J. Rosenberg, A. H. Weiss, and K. F. Canter

Department of Physics, Brandeis University, Waltham, Massachusetts 02254 (Received 25 February 1980)

The first observation of low-energy positron diffraction from a solid surface is r e ­ported. Slow (20-400-eV) monochromatic positron beams were focused onto a Cu(lll) surface and their elastically scattered distributions detected with a channel electron multiplier. Measurements of the scattered intensity versus angle as a function of inci­dent energy show peaks at the predicted (01) and (02) diffraction angles. Profiles of in­tensity versus energy at fixed angles exhibit maxima corresponding to the primary Bragg peaks.

PACS numbers: 61.14.Fe, 78.70.Bj, 71.60.+ Z

© 1980 The American Physical Society 1139