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AN ELECTRONIC MODEL FOR AMORPHOUS SYSTEMS

J. SCHREIBER Joint institute for Nuclear Research, Head Post office, P.O. Box 79, 101000 Moscow, USSR

Linearly coupled diagonal and off-diagonal randomness are considered using a "cluster-effective-lattice" method where the potential parameters in the cluster are assumed to fluctuate according to a Lorentzian distribution. The density of states, the magnetic properties within a Stoner-like theory, and the localization of electrons are studied.

In amorphous or liquid systems diagonal disorder and, first of all, off-diagonal randomness (ODR) appear [1,2] whereas in general both kinds of disorder are coupled. Hence we use the following tight binding model for a system of N identical atoms forming an amorphous structure. Instead of the total Hamiltonian we only consider the projection onto the sub- space spanned by the atomic orbitals IP~). Using a second quantization representation the ap- proximated hamiltonian may be written as [2]

i jo io

(1)

Because of the non-orthogonality of ]p~) we have [ai~, ci~+~.] = (S-~)~i, where S is the matrix of overlap integrals. The second term in eq. (1) represents only the Hubbard-like part of electron correlation. We restrict our discussion to cases where (i) the atomic orbitals are well enough localized and (ii) the amorphous structure has a well established short range order. Therefore we can suppose that Vii and S o are different from zero only for i = j and for nearest-neighbours (NN). Considering s-like states IPi), e~ = Vii, Vii, and S o are functions of the atomic distances. Therefore the structure fluctuations of these quantities are coupled so that really only one random variable exists. In a first approximation, expanding ei, V~j, and Sij linearly in the variations of the distances l i - Jl and neglecting three centre integrals, the following simple relations can be obtained:

ei = A ~ (Vii - Vo) + B,

Sir = C(V~i - Vo) + D, (2)

where V0 belongs to the averaged atomic posi- tions.

The density of states is given by the relation (cf. [2])

p ( E ) = - 1 / w N ~ S,j Im Go(E + i0 +) i,j

= I l N ~ pi(E), i

(3)

where Gij is the Zubarev-Green function. We find that Gij obeys the equation

( E S i t - Vit)Gij = 8ij. (4) I

The structure averaging for p ( E ) is performed by a self-consistent "cluster-effective-lattice" method. Thereby it is required that the averaged local density of states pi(E) for an atom sur- rounded by its Z NN and embedded in an effective m e d i u m - characterized by an effective lattice (the Be lattice or regular one with equivalent short range order) and a coherent hopping integral Vc- i s equal to the cor- responding quantity for the effective medium. Numerical computations become easy if we assume that in the cluster the Vii fluctuate statis- tically independent according to a Lorentzian distribution with mean value V0 and width F (cf. [2]). Results are shown for a Ni-like effective structure (fig. 1) [3, 4]. Switching on ODR, an asymmetrical change of p ( E ) is obtained, where the sign of the coupling between e~ and V~i influences the results qualitatively. We note that the case A V o < 0 is the physically realistic one since the ei-level will be lowered if Iv01 becomes greater. The consideration of random overlap integrals ( C a 0) leads also to an asymmetrical change of p(E) .

Now we investigate the amorphous magnetism within the Hart ree-Fock approximation (HF) for the model (1) (cf. [3, 4]). Assuming that U~ is only determined by the sort of atoms, and incorporating the results for p(E), the Stoner-

Physica 86-88B (1977) 752-754 © North-Holland

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<Tt~

~ r

Fig. 1. Ni - l ike d e n s i t y of s t a t e s p(E) for d i f fe ren t k ind of

d i so rde r . (a) F = 0 ( c o r r e s p o n d i n g to c rys ta l ) ; (b) F ~ 0 ,

IAIF= 13 /12×0 .06 ; (c) F = 0 . 0 6 , A = 13/12; (d) F = 0.06, A =

- 1 3 / 1 2 . V , , < 0 , B=C=D=O. E ~ = F e r m i leve l for e lec- t ron n u m b e r pe r a t o m n = 1.88.

like theory gives: (i) The Stoner criterion Upp(EF)> 1. The

index p denotes the paramagnetic phase. For diagonal disorder and for A V0>0 ferromag- netism (FM) is monotonously weakened increasing F. In contrast to that for the "physical" case (AV0<0) FM is stabilized in a certain range of F. The reason is a shift of the Fermi level to the top of the peak (fig. 1). (ii) The Curie temperature Tc and the magnetization m(T = O) (fig. 2). For sufficiently great values of F both quantities are monotonously weakened by fluctuations of V~i in all cases. Important is the relation between the lowering of Tc and m(T = 0). Experimental data show that in amorphous Ni-samples m(T = 0) is decreased more than Tc

compared with the crystalline phase [1]. This tendency is reproduced in the case A Vo<O. Other kinds of coupling and only diagonal disorder yield the inverse tendency. (iii) The characteristic flattening of the m(T[Tc)/m(O) curve is not found within our model, since the Stoner theory is not appropriate for describing the temperature dependence of m(T).

Recently we have derived some expressions for the localization function L(E) within the Economou and Cohen theory for the model of amorphous systems described above (cf. [2]). Here we discuss only the most successful approximation which was proposed by Licardello and Economou for the Anderson model. Per- forming the structure averages in the same way as for p(E) and choosing a Bethe lattice for the effective structure, we have found the following results (U~ = 0). Due to the coupling of ~ and V~ i an asymmetrical shift of the mobility edges Ec is caused by increasing potential fluctuations. This is expected from the results for p(E). For larger F it is observable that the fluctuations of V 0 lead to a delocalization effect. Fluctuating overlap integrals yield also asymmetrical delocalization. Consequently, because of the essential effect of ODR in amorphous systems, ODR has to be included when properties of electronic conductivity in these systems are discussed. Regarding amorphous transition metals the electron correlation may have an effect in producing localized states. Within the alloy analogy for the model (1), which yields an effective cellular disorder, the above mentioned localization function can also be calculated [2]. As the density of states for F ~ 0 is smeared out in the gap a conductivity transition is now related to the mobility bands only (fig. 3). Let us assume that

"For ~.%

m,. t',,

i i

I I I I I 0 O.0 5 0 4

I^ll-

Fi 8. 2. D e p e n d e n c e of the Cur i e t e m p e r a t u r e T . = T~(F)/T~(O) and m a g n e t i z a t i o n m, = m(F)lm(O) at T = 0 u p o n

lair. - - A = 13n2; - - - r-~0, Ial-~ o~; . . . . . A=- 13/~2. v,,< 0, UllVol = 15.0, B = c = o =0.

r / , ' / / / I,-,

/ / / , I '~

[ / .~ I .~ ~": / / " q I " ! :

/ t . . . . . . . " i ! j i / , u / , ,., i /, '.D .......... '

l - " ~ = ' : " :

o ~ Iz 15

• E.¢

Fig. 3. Mobi l i ty b a n d s for the H u b b a r d mode l in the p a r a m a g n e t i c case (n T = n $ = ½). - - U = 3; - - - U = 6 ;

- . - U = 9 ; . . . . U = 1 2 . V o > 0 , A = - I , Z = 6 , B = C = D = 0 .

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U[Vo and F will be l o w e r e d if p r e s s u r e is ap- p l ied to a m o r p h o u s non -me ta l l i c s y s t e m s . Then we m e e t the pos s ib i l i t y of r ea l i z a t i on of a non- m e t a l - m e t a l t r ans i t ion . This t r ans i t i on is a m i x e d one of the M o t t and A n d e r s o n t ype w h e r e the d o m i n a n t m e c h a n i s m is due to e lec- t ron co r re l a t ion .

The a u t h o r is ve ry g ra te fu l to Dr. W. John and Mr. J. R ich t e r fo r va luab le d i s cus s ions .

References

[1] G.W. Wright, Amorphous Transition Metal Films, 7th Int. Colloquium on Magnetic Thin Films, Regensburg (1975).

[2] J. Schreiber, in: Proc. of VII-th Autumn School on Magnetism, Gaussig (1975).

[3] J. Kanemori, J. de Physique C4 (Suppl.) (1974) 131. [4] J. Richter, J. Schreiber and K. Handrich, Phys. Status

Solidi (b) 74 (1976) K125.

Download - An electronic model for amorphous systems

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