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MOBILITY GAP AND ANOMALOUS DISPERSION
D. Mattis, F. Yonezawa
To cite this version:
D. Mattis, F. Yonezawa. MOBILITY GAP AND ANOMALOUS DISPERSION. Journal de Physique Colloques, 1974, 35 (C4), pp.C4-123-C4-125. �10.1051/jphyscol:1974421�. �jpa-00215612�
JOURNAL DE PHYSIQUE Colloque C4, suppldment au no 5, Tome 35, M a i 1974, page C4-123
MOBILITY GAP AND ANOMALOUS DISPERSION (*)
D. C. MATTIS and F. YONEZAWA (**) Belfer Graduate School of Science, Yeshiva University
New York, New York 10033, U S A
Rksum6. - L'alliage binaire est Btudit dans l'approximation du potentiel cohtrent (CPA) et l'on obtient que les relations de dispersion ~k(w) d'une quasi-particule ont parfois une anomalie dans le sens que dsk(w)/dw est negatif. I1 est dkmontre que dans la region d'anomalie les ttats ne se propagent pas, et que cette region correspond a un gap de mobilitk.
Abstract. - A binary alloy is treated by the coherent potential approximation (CPA) and the dispersion relations cr(w) for a quasi-particle are found to be anomalous in some region, in the sense that dcs(w)/dw is negative. It is proved that the states in the region of anomalous dispersion correspond to a mobility gap.
1. Introduction. - We consider the question of the existence of a mobility gap, of whether in a disordered medium there is a range of allowed energy levels which are all non propagating states. This concept, first conjectured by Mott [I], Cohen et al. [2] and others [3], is of great physical interest in the electronic band structure of liquids, amorphous semiconductors and disordered alloys. As a prototype of disordered systems, the random binary alloy has been successfully and accurately studied by the coherent potential approximation (CPA) [4, 51. This method allows one to study the effects of varying the concentrations x A and xB = 1 - x , of the constituents and has recently been extended to include the effect of long range order [6].
In the present work, we study the CPA dispersion relations for a propagating particle and discover that under certain conditions there is a region of anomalous dispersion. We find that inclusion into propagating wavepackets of states within this region leads to a clear violation of causality. Contours must then be chosen such that these states are excluded.
One identifies the region of anomalous dispersion as, in fact, the mobility gap, terminating in well defined mobility edges where we find m* = co and the mobility to be zero. The criterion we find for the mobility gap, while entirely self-consistent within CPA, does not agree in detail with an earlier criterion based upon
probabilistic considerations of electron diffusion, and we suggest this topic as worthy of further study.
2. Dispersion relations. - The one-electron Hamil- tonian in the tight-binding representation is :
where the atomic level Ei is either EA or EB. Off- diagonal matrix elements tij are simply assumed to be independent of the types of atoms at i and j. The ensemble-averaged local Green function is given by the CPA in the form [5]
and
in which 5 is defined as t = G-I + [. All energy variables are scaled by U = I EB - EA I and the Green function G by U - l so that & A = EA/U, cB = EB/U and (' is the dimensionless self-energy measured by U.
The band energy ek is related to t i j by a Fourier trans- form
where N is the total number of atoms in the alloy.
(*) This work is sponsored by the Air Force Office of Scien- 1, order to make an analytic study possible, we
tific Research Air Force Systems Command, USAF, under
grant No AFOSR-72-2153B. employ a special model for the density-of-states
(**) Partly supported by grants AFOSR 73-2430 and AFOSR function
72-2153B. On leave from Department of Applied Physics, po(&) = N-' B(E - ck)
Tokyo Institute of Technology, Moguroku, Tokyo 152, Japan. k (2.4)
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1974421
C4- 1 24 C . MATTIS AND F. YONEZAWA
which has the simple semicircular form
and zero otherwise. 6
-
d / U is the dimensionless bandwidth. It is easy to show, making use of eq. (2.4) and (2.5) that the Green function is related to 4and ( in the following way [5] :
and therefore the CPA equation is written as
which yields a cubic equation for t. We confine oursel- ves to the most tractable case x, = x, = 3. Essential arguments and conclusions are not altered even when x , # +. It is worth noting that this special example also corresponds to the alloy analogy approxi- mation of the Hubbard model [8] for a half-filled band, and all the equations remain the same mutatis mutandis. Our results can therefore also be used to discuss the metal-insulator transition in the Hubbard model [9]. The zero of energy is chosen at mid-band such that = - E~ = $, and thus the band is symme- tric about E = 0. For a half-filled band, the Fermi energy is E , = 0.
It has been shown previously [8, 101 that the quasi- particle spectrum consists of two separate bands when the ratio 6 = A/U is small enough while it consists of a single continuous band when 6 is large enough. The critical value 6,, at which a density-of- states gap appears, is calculated by examining the cubic equation for < and is found to be 6, = 2.
We next construct an equation for I using eq. (2.6) and (2.7) ;
To derive the dispersion relations for a quasi-particle, we first locate the poles by means of w - R - E, = 0, R being Re I. We write Im I = - T . If we associate G(w) with e- '"', causality requiresr(o) % Ofor all values of o.
By solving eq. (2.8), ~ ( o ) = o - R ( o ) is obtained as a function of o and shown in figure 1.
It can be proved that, for 6 > 2, & ( a ) passes through the origin. The derivative ds(o)/dw at the origin can be either positive or negative for a given value of 6.
This is seen by solving eq. (2.8) for I o 1 -4 1. We obtain for 6 > 2 :
which is negative in the range 2 < 6 < 2 45. Figure l b corresponds to the negative slope case, while l c shows the more usual positive slope. It can be shown ana- lytically that there is no negative slope region for 6 < 2.
FIG. 1. - Dispersion relations ; &(a) u s o for (a) 6 = 1.6,
(b) 6 = 2.4 and ( c ) 6 = 3.0. Solid lines correspond to real
solutions of the cubic equation while dotted lines represent the real parts of the complex conjugate solutions for (. The thick line on the &(a) axis shows the allowed region over which the variable ex is defined. At the bottom of each figure is shown the imaginary part T(a) of 5. (a) corresponds to a separated band case with a finite density-of-states gap while (b) and (c) are
exan~ples with no density-of-states gaps.
3. Anomalous dispersion. - We now argue that quasi-particle states in the region of anomalous dispersion do not propagate and must therefore be considered localized. Consider a wave packet centered at w, :
where f ( a - a,) is an envelope function. Assuming f ( o - a,) is appreciable only around o = o,,
we expand about oo as follows :
which defines both m* and E,,. Note that m* is usually positive, although it is negative in the region of ano- malous dispersion. Assuming spherical symmetry in k, we have :
sin kr F(r, t ) = eCioot J d&k PO(&,) r x
MOBILITY GAP AND ANOMALOUS DISPERSION C4-125
where 52 = w - coo. We also expand k about ko :
which serves to define Vo = delak 1 , and zk = E, - eke.
Taking slowly varying factors outside the k integral, we have
where :
The contour for I, is on the real z-axis, and closed by an infinite semicircle in the upper-half complex z-plane. The contour for I, is closed by an infinite semi-circle in the lower-half-plane, where there are no poles. Thus only I, contributes and we have
for t 2 m* r/Vo ; it is necessary to introduce a sink at r = 0 to take care of the disappearing particles.
This sink can be interpreted as the localized states.
We therefore conclude that only states where m* > 0 contribute to electronic propagation.
4. Mobility edges. - It follows from the arguments in the preceding section that the states indicated by an arrowed portion on the w-axis in figure l b are loca- lized and that the mobility of quasi-particles in these states is zero. The mobility edges at o, are determined by the vanishing of de(o)/do. The mobility gap cha- racterized by o, thus determined is shown in figure 2 by cross-hatching.
FIG. 2. - The dependence of the quasi-particle spectrum and the mobility gaps upon 6 = A/U and w. The density of states is non-vanishing in dotted regions. Regions of localized states within our mobility gap are crosshatched, those within the
Economou-Cohen gap [7] are single-hatched.
d52f (SZ) e-iRu. For m* > 0 this
Note added in proof. - From (3.7) we see that is an outgoing spherical wavepacket, decaying as it mean free path is I = V o / r and group velocity moves out (exp [- r r / v o ] ) because of incoherent v = V0/m*. Thus z = 1/V = mx/T, and the mobility scattering. For m* < 0 this is an incoming wave p = e2 ~ / m * = e 2 / r is a constant or slowly varying which grows as it approaches the origin, owing to just outside the mobility edge. We thus find p is the selfsame exponential factor. In addition, there discontinuous at the mobility edge, the agreement is a serious problem with the wavepacket description with a well-known conjecture by Mott.
References
[I] MOT-, N. F., Adv. Phys. 16 (1967) 49. VELICKY, B., KIRKPATRICK, S. and EHRENREICX, H., Phys.
[2] COHEN, M. H., FRITZSCHE, H. and OVSHINSKY, S., Phys. Rev. 175 (1968) 747.
Rev. Lett. 22 (1969) 1065. [5] YONEZAWA, F., Phys. Rev. B 7 (1973) 5170.
161 PLISCHKE, M. and MATTIS, D., Phys. Rev. B 7 (1973) 2430.
[31 Two recent reviews on the subject, by EDWARDS, S. F. and [71 ECONO~OU, E. N. and coHEN, M, H., phys. R ~ ~ . B 5 (1972) by E c o n o ~ o u , E. N., appear in cr New Developments 293.
in Seiniconducfors >), WALLACE, P. R., HARRIS, R., [8] HUBBARD, J., pWC. R. sue. A 281 (1964) 401.
ZUCKERMANN, M. (Eds., Nordhoff International Publ., [g] YONEZAWA, F. and wATABE, M,, phyS. R ~ ~ . 8 (1973)
Leyden, Holland) 1973. 4540.
[4] SOVEN, P., Phys. Rev. 156 (1967) 809; [lo] ONODERA, Y. and TOYOZAWA, Y., J. Phys. Soc. Japan YONEZAWA, F., Prog. Theor. Phys. 40 (1968) 734 ; 24 (1968) 341.
