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CONDUCTION NEAR A MOBILITY EDGE
N. Mott
To cite this version:
N. Mott. CONDUCTION NEAR A MOBILITY EDGE. Journal de Physique Colloques, 1981, 42
(C4), pp.C4-27-C4-31. �10.1051/jphyscol:1981403�. �jpa-00220707�
CoZZoque supp26ment au n O I O , Tome 42, octobre 1981
CONDUCTION NEAR A M O B I L I T Y
EDGE
N.F. Mott
Cavendish Laboratory, Cambridge, U. K.
Abstract.- The e x i s t e n c e o r o t h e r w i s e of a minimum m e t a l l i c c o n d u c t i v i t y amin i s examined. It i s found t h a t t h e formulae given p r e v i o u s l y f o r t h i s q u a n t i t y a r e c o r r e c t i n t h e l i m i t of high temperatures. A t low t e m p e r a t u r e s , following t h e c o n s i d e r a t i o n s of Mottl and S t e i n and ~ r e ~ 2 , we f i n d t h a t . i t s e x i s t e n c e depends on t h e index s i n t h e behaviour of t h e l o c a l i z a t i o n l e n g t h l / a , where a % (Ec
-
E l S . I f s Z 213, amin e x i s t s ; i f n o t , a behaves l i k e arnin(E-
E , ) ~ where t = 213-
s and omin i s t h e v a l u e o b t a i n e d p r e v i o u s l y . I f s i s 0.6, a s some c a l c u l a t i o n s s u g g e s t , t i s 0.067 and a very r a p i d r i s e w i t h E i s p r e d i c t e d . This i s i n f a i r agreement w i t h t h e o b s e r v a t i o n s of Rosenbaum e t a 1 3 on Si:P. The s i t u a t i o n i n two dimensions i s reviewed, and some of t h e r e s u l t s of t h e s c a l i n g t h e o r y of Abraham e t a14 i n 3d a r e c r i t i c i s e d .The p r e s e n t a u t h o r ( ~ o t t ~ ) f i r s t suggested t h a t , f o r a degenerate e l e c t r o n gas i n a d i s o r d e r e d medium, t h e r e e x i s t s a q u a n t i t y c a l l e d t h e minimum m e t a l l i c conduc- t i v i t y , and denoted by amin. This can b e d e f i n e d i n two ways. I f t h e Fermi energy EF l i e s below t h e m o b i l i t y edge E c , t h e c o n d u c t i v i t y a t high temperatures w i l l b e given by
A t low temperatures conduction w i l l be by v a r i a b l e range hopping. We b e i i e v e ( 1 ) t o be e s s e n t i a l l y c o r r e c t . The p o i n t t o b e d i s c u s s e d i n t h i s paper i s w i e t h e r , i n t h e l i m i t of low T, t h e r e e x i s t s a range of AE (= EF - E,) f o r which a tends t o a f i n i t e v a l u e l e s s than t h e c a l c u l a t e d v a l u e of omin a s T + 0. There i s evidence t h a t t h i s i s s o f o r Si:P o v e r a narrow range of c o n c e n t r a t i o n (3.2 If 0.03 x 1018 ~ m - ~ ) , o b t a i n e d by Rosenbaum e t a13.
The q u a n t i t y omin was c a l c u l a t e d i n t h e f o l l o w i n g way from t h e model used by
~ n d e r s o n 6 i n 1958. For e n e r g i e s above b u t n e a r t o t h e m o b i l i t y edge, t h e one- e l e c t r o n wave f u n c t i o n was assumed t o be of t h e form ( c f . ~ o t t ~ f o r a r e c e n t d i s c u s s i o n )
where t h e J, a r e atomic wave f u n c t i o n s and t h e $ n a r e ran'dom phases. The conduc- t i v i t y was ?hen c a l c u l a t e d from t h e Kubo-Greenwood formula (Mott and ~ a v i s . 8 ) ~ b e i n g p r o p o r t i o n a l t o
where n , n+l r e f e r t o a d j a c e n t s i t e s . amin i s o b t a i n e d when EF l i e s a t a m o b i l i t y edge. (3) i s c o r r e c t i f f l u c t u a t i o n s i n t h e cn a t e n e g l e c t e d . The v a l u e s o b t a i n e d
,
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1981403
JOURNAL DE PHYSIQUE
are in substantial agreement with experiment in the high T limit, and seemed to justify the neglect of fluctuations in cn.
In 1976 the present author I examined the effect of fluctuations in the c I"
These were found to depend on the index s when the radius of localization, Ila, 1s expressed as
If in 3d s 2 213 the fluctuations are unimportant and a minimum metallic conduc- tivity exists; but, if s < 213, o(E) drops to zero at Ec. The same result is obtained by Stein and ~ r e ~ 2 . The ar ument is as follows:- If in the Anderson model we consider a volume containing N potential wells, we expect a fluctuation
9
AV/V = N - ~ / ~ in the potential V taken over a large number of such volumes. For an electron with energy Ec, half these volumes will have their mean potential greater than V by an amount 6V, so that a wave function there will behave like exp(+$r) with
because we expect Ec to vary linearly with V. For our block with side Na (5) is then proportional to N - ~ s I ~ , and in the expression exp(+Br), Br = BNa % N I - ~ s / ~ . Fluctuations in '4 diverge with N if s < 213 (or < 1 in 2d) and the assumption that fluctuations in cn have a negligible effect is incorrect. The present author9 has also calculated what happens if E lies above Ec, if s < 213. As before, in blocks of size ( ~ a ) ~ , there are fluctuations AV/V % 1/N3l2. Writing E
-
Ec = BE,then if these fluctuations are small compared with AE, they should not affect o . We therefore calculate o in blocks of side Na for which AV = AE, so that N = (E/AE)~/~.
Our characteristic length is thus a(E/~E)~/~and within blocks of this size fluctua- tions of Y will occur of magnitude
where X is a constant of order unity. The matrix elements in the Kubo-Greenwood formula will thus, in one of these blocks, behave like
where n is the phase difference between the wave functions. An integration over q suggests that the mean of < Y ~ ( ~ / ~ x ) Y ~ > ~ should behave like (AE/E)~, with t = 213
-
sSo we estimate, if s < 213, that
a % 0 I(E
-
E ~ I I E ~ I ~min o < o
min o > o
min '
Here omin is the author's calculated value (4) and the second formula is valid in the Ioffe-Regel regime (L % a), that is up to a = 113 e2/Iia. Thereafter the Boltzmann treatment is applicable.
Calculations of s have given 0.6 (Abram and Edwards lo, Anderson I) and 213 (Freed l2). If the former value is correct, (6) gives a very rapid rise in o, and leads to qualitative agreement with the results of Rosenbaum et a1 for Si:P. The prediction that for o > om. the conductivity varies as {N(E)]~ can be checked from the observations of ~asakijg on the specific heat. A rapid rise of comparable amount is observed in both.
where D is the diffusion coefficient and T the lifetime; for electron-electron collisions Ili .:1/T. The behaviour (6) depends on long-range fluctuations in the cn, so that with increasing T, a rises to omin. The present author9 finds
giving a very rapid rise towards omin for increasing T, again in qualitative
agreement with the observations14 on Si:P. That is why the author's previous values (2) are, we believe, correct in eqn (I) at high T, but if s = 0.6 they are not at low T.
Turning now to two dimensions, we believe that a sharp transition exists at Ec between energies for which localization is exponential, which gives variable range hopping, and power-law localization which does not. For E above Ec, the conduc- tivity behaves (for k~9.. > 1) as
where L is either 2; or the size of the specimen (Mott and Kaveh15, sarma16). For moderate values of L we have
The logarithmic term with L = !ti I/T, which has been observed in many experiments, is normally only of order 0.1, but in principle a + 0 as L -t m . The theoretical value of the conductivity, including the logarithmic term, for E near a mobility edge (kFR % 1) has not yet been obtained; the derivation of (10) was first given by Abraham et a14 and alternative proofs by Gorkov et all7 and Kaveh and T4ott18, and Mott and Kaveh13 for eqn (7).
Altshuler et all9 first showed that a proper treatment of the effect of
electron-electron interactions on the density of states in two dimensions also leads to a correction to u of the form (10). Both effects are present, but respond differently to a magnetic field, as has been shown experimentally by Uren et alZ0 in observations on inversion layers. A similar analysis was earlier applied to three dime~sional systems by Altshuler and ~ r o n o v ~ l, giving an addition to the resistivity as T2, in contrast to the Baber term in T ~ . The former appears when 2 is compara- tively small, and has been observed in cold worked bismuth and Si:P.
Finally we comment on the scaling theory of Abrahams et a14, which first gave correctly the logarithmic term (7) in two dimensions and which has also been used, in 3d, to disprove the existence of a minimum metallic conductivity. We do not think the latter conclusion is correct, for the following reason. These authors introduce a dimensionless conductance g defined by
for a block of size L. g is a function of L and of the energy E. They then intro- duce a "universal scaling function" defined by
JOURNAL DE PHYSIQUE
and suppose that 6 is a function of g only. For large g, that is for extended state conduction, this is so; it can be shown that
where A and B are numbers independent of E or L. Integration gives in 2d
so the dependence on L is obtained correctly but nothing can be said about the constant term. In three dimensions, integration gives
where l/Lo is a constant of integration. The term AIL is correct, giving a varfa- tion as T when L is replaced by R;, but is not observed because of the larger TZ term given by the interaction theory.
For small g we believe that
(in contrast to Abrahams et a1 who give g
*
e-aL). In 2d, both assumptions are the same and sowhich is also independent of L and E; thus it is maintained that an extrapolation can be made over the whole range of g. Pichard and Sarma16 also remark on this property of 2d systems. In three dimensions on the other hand
whence
6 may still be a function of g only, and may therefore exist in the region where they make use of it to obtain o(Ec) = 0. However, since U is obtained by integrating (10) and involves an unknown constant of integration, it seems impossible that it could give the value of o(E) under any conditions, or ~redict that o(E) = 0.
References
1 MOTT N.F., Comun. Physics
1
(1976) 203.2 STEIN J. and KREY W., Z. Phys. B
2
(1979) 287; ibid 37 (1980) 13.3 ROSENBAUM T.F., ANDRES K., THOMAS G.A. and BHATT R X . , Phys. Rev. Lett.
43 (1980) 1723.
4 BRAHAMS E., ANDERSON P.W., LICCIARDELLO D.C. and RAMAKRISNAN T.V., Phys. Rev.
Lett.
42
(1979) 673.5 MOTT N.F., Adv. in Phys.
5
(1967) 47; Phil. Mag. (1972) 1015.6 ANDERSON P.W., Phys. Rev.
109
(1958) 1492.7 MOTT N.F., Phil. Mag. (in press, 1981).
8 MOTT N.F. and DAVIS E.A., Electronic Processes in Non-Crystalline Materials, 2nd ed. (1979) Oxford.
9 MOTT N.F., Phil. Mag. (in press).
10 ABRAM R.A. and EDWARDS S.F., J. Phys. C
2
(1972) 1183.SASAKI W . , J . P h y s . Soc. ( J a p a n ) 4 9 , S u p p l . A (1980) 3 1 .
ROSENBAUM T.F., ANDRES K. a n d T H O ~ S G.A., S o l i d S t . Commun.
2
(1980) 663.MOTT N.F. a n d KAVEH M., J . P h y s . C ( i n p r e s s ) . SARMA G. ( t h i s volume).
GORKOV L.P., LARKIN A.I. a n d KHMELNITZKII D., JETP L e t t .
30
(1979) 229.KAVEH M. a n d MOTT N.F., J . Phys. C ( i n p r e s s ) .
ALTSHULER B.L., KHMELNITZKII D., LARKIN A . I . a n d LEE P.A., Phys. Rev. B
2
(1980) 5142.
UREN M . J . , DAVIES R.A. a n d PEPPER M . , J. Phys. C
12
(1980) L985.ALTSHULER B.L. a n d ARONOV A.G., S o l i d S t a t e C o m u n .
