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ON THE INTERACTION BETWEEN ELECTRONS
AND TUNNELLING LEVELS IN METALLIC
GLASSES
J. Black, B. Gyorffy
To cite this version:
JOURNAL DE PHYSIQUE
Colloque C6, supplément au n" 8, Tome 39, août 1978, page C6-941
ON THE INTERACTION BETWEEN ELECTRONS AND TUNNELLING LEVELS IN METALLIC GLASSES*
J.L. Black and B.L.GyorffyDept, of Physios, Brookhaven National Laboratory, Upton, New York 11973, U.S.A.
Résumé.- Nous considérons un modèle simple dans lequel les électrons d'un verre métallique sont soumis à un potentiel local dépendant du temps, dû à des doubles puits à deux niveaux.Nous mon-trons que ce modèle possède des divergences intéressantes qui sont toutefois très différentes des prédictions du modèle "s-d Kondo" de Cochrane et al./9/.
Abstract.- We consider a simple model in which the condution electrons of metallic glass experien-ce a local time-dependent potential due to two-level tunnelling states. We show that the model exhibits interesting divergent behavior which is, nevertheless, quite different from that predic-ted by the "s-d Kondo" model of a Cochrane et al./9/.
Recently there has been considerable experi-mental evidence for the existence of tunnelling le-vels in metallic glasses /l,5/.Such tunnelling sta-tes, presumably arising from local atomic rearrange-ments as proposed by Anderson et al./6/ and Philips
111, explain a variety of low-temperature anomalies in insulating glasses /8/. In the case of metallic glasses, an interesting new question arises. How do they interact with the degenerate Fermi system of the conduction electrons ? Here we report on a preli-minary study of a simple model which treats the tun-nelling states as localized scattering centers with an internal degreee of freedom. We show that the ins-tability of the electron gas to such perturbations leads to interesting behavior which nevertheless differs from that encountered in the Kondo and x-ray problems.
Considering the same problem, Cochrane et al./9/ have proposed a model which is in one-to-one correspondence with thes-d KondoHamiltonian /10/ and have consequently predicted a logarithmic contribu-tion to the resistivity. However, this analogy relies upon the questionable assumption that the current-carrying condution electrons can be labeled by an index which corresponds to the spin of the electron in the s-d model. As an alternative we consider a model in which an electron undergoes ordinary
poten-tial scattering dependent upon the state of the tun-nelling level but not upon an internal electronic degree of freedom corresponding to the spin.
Describing the tunnelling level by an s = /\
pseudospin, the Hamiltonian is
where e, is the single-particle electron energy, A is the tunnelling-level asymmetry /8,11/ and Ao is the quantum-mechanical tunnelling frequency /8,11/. The coupling constant K..accounts for the change in potential seen by the electrons when the tunnelling system changes states. This Hamiltonian is similar to a model for crystal-field-split levels conside-red by Fulde and Peschel /12/ and is essentially the same as one recently studied by Kondo /13/.Clear-ly Eq.(l) neglects many important scattering mecha-nisms as well as all k-dependence of K,,.Furthermo-re Eq.(l) may not accurately K,,.Furthermo-reflects the situation when the electronic mean free path is smaller than the extent of the tunnelling system (which is not known).Nevertheless, we feel that this model is use-ful for.a qualitative understanding of the electron-tunnelling-level interactions.Rotating the pseudo-spin basis states in Eq.(l) diagonalizes the non-interacting part of the Hamiltonian :
where (2S0)2=A2 + A2,V,, = K,.(A/280) and V =-K,, (Ao/260).
Using Abrikosov's pseudofermion representation for S and S /14/ and finite-temperature perturbation theory, the electronic-self-energy 2_(E ) is given
Permanent Address / H.H.Wills Physics Laboratory, Univ. of Bristol, U.K.
-
1(iEn
-
5)
,
= f 6,s p is the density of states at EF, and D is the bandwidth. The vertex part yaal is given by a perturbation expansion in terms of the bare coupling constant matrix2Vffa1 = VN (sgnAa)
gas,
+ V1(
1-
6aff ).
(4) The first corrections, yg), to Eq. (4) are the particle-particle and particle-hole bubble dia- gram, which are separately logarithmically divergent As in the Kondo /l4/ and x-ray edge 1151 problems, these divergences arise from the instability of the electron gas to a sudden localized change in the external potential. However, the algebra of the a summation results in qualitatively different beha- vior of the vertex part.For example, a11 logarithmic corrections cancel if the "magnetic field" 6, is zero 1131. Furthermore, even when 8,
)
0, there is no diver- gent contribution tor(3)
(E=O), as would be the case in the Kondo problem /lo/. In fourth order, however, we find to find to leading logarithmic ac- curacy.B60
Im z'4)(0) = 411 p3 tanh'
-T
V'v2
en2
-z-!I/
j-
2kBT ' ( 5 )In order to gain further insight into the nature of this divergence we have studied the lowest order "parquet" equations 1141 for the vertex part Yml within this approximation. We find that the diagonal component ym is not renormalized but the off-diagonal part deverges as a power law at the energies E + En = f is,. Namely, for electron ener- gies E < 16,1 at T = 0
y (+ i6,,
-
is,,+
is0 + E) = V1
1
ei2p(0)v/~n.
(6)While reminiscent of both the Kondo and the x-ray edge singularity behavior, the above result differs from both.
This conclusion is further supported by a separate analysis of Eq. (1) in which we rely on the work of Schotte 1161 and Blume et al. 1171 and use AoSx as a perturbation. This procedure is entirely analogous to the reformulation of the Kondo problem as a sequence of x-ray edge problems 1151 by Ander- son, et al. 1181. In their language, our model differs from the Kondo problem because "x-ray edge" is replaced by "x-ray photoemission" 1191, in which the creation of a deep core hole (i.e., spin flip) is
not
accompanied by the injection of an electron at the Fermi surface.Research supported by the Div. of Basic Energy Sciences, U.S. Dept. of Energy under Contract NO.
EY-76-C-02-0016 ; and NSF Grant DMR 76-82946.
Thus the relationship between our model and the usual Kondo problem, in terms of the respective free energies F MG and FK is
FMG = FK(€ = 2
-
J l = Ail, H = 8 ) + constant. (7) The Kondo free energy, FK,is&fined in the presence..
of a magnetic field, H, and depends upon the usual coupling constants
1181
pJl and pJ/, :L - .
Thus the weak-coupling (pK,/<< 1) version of the me-
, ,
tallic glass free energy corresponds to a particular anisotropic strongroupling-Kondo problem (1 < pJfl
, , >> p~l). This relationship does not, of course, im- ply that the response functions in our model are simply related to those in the Kondo problem. Never- theless, it does suggest that the metallic glass di- vergences can be studied by renormalization group methods 1181. Work is in progress on this approach and on the question of response functions.
References
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(1977) 1480. /2/ Lasjaunias, J., Ravex, A., and Thoulouze, D.,(1977) preprint.
/3/ Matey, J. and Anderson, A., J. Non-Cryst. Solid 23 (1977) 129.
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(1 977) L-65.151 Golding, B., Graebner, J., and Haemmerle, W., Proc. Int. Conf. on Lattice Dynamics (Sept. 1977 Paris).
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