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Effect of localization on the hall conductivity in the two-dimensional system in strong magnetic fields
~)
Solid State Communications, Vol. 88, Nos. i 1/12, pp. 951-954, 1993. Printed in Great Britain.
0038-1098/9356.00+.00 Pergamon Press Ltd
EFFECT OF LOCALIZATION ON THE HALL CONDUCTIVITY IN THE TWO-DIMENSIONAL SYSTEM IN STRONG MAGNETIC FIELDS H. Aoki Cavendish Laboratory, Madingiey Road, Cambridge CB3 0HE, England and T. Ando Institute of Applied Physics, University of Tsukuba, Sakura, Ibaraki 305, Japan
(Received 16January 1981 by Y. Toyozawa) Some exact properties of the Hall conductivity in the two-dimensional electron system in strong magnetic fields are investigated in the presence of immobile carriers. It is shown that oxy should be exactly the integral multiple of--e2/h when the Fermi energy lies in the localized regime. RECENTLY yon Klitzing etal. [1] and Kawaji and Wakabayashi [2] experimentally verified that the plateau value of the Hall conductivity, am, in the two. dimensional electron system in MOS inversion layers in strong magnetic fields (H ~ 100 kO¢) is equal to the integral multiple o f - - e2/h, potentially providing a high precision method of measuring the universal quantity e2]h. The plateau region of a m is shown to arise from immobile carriers around the tails of the Landau subband in which region the d.c. conductivity a ~ vanishes. The cartier localization in this system has been explained by the localization due to randomness [3, 4], electronelectron interaction [5, 6] or combined effect of b o t h [7]. However, the result that o ~ = --Ne2/h (AT = integer) despite the existence of localization is particulaxly intriguin 8 because one might expect, say, axy= -eCnmobae/H with nmoba e being the number of mobile carriers. In the present paper we clarify this point theoretically, and elucidate some exact properties of the Hall conductivity. The Hamiltonian of a two-dimensional system in a magnetic field H with random potential V(r) is given by
~c = " 2 + v(r), 2m
(1)
o~
nec = -- ~ - + Zxaxy ,
(2)
with
(3)
~o.. = ± e ~ (<>-<~?~>>), 2
where the double brackets represent the correlation functionwith canonical average and configuration averages over randomness. In the right.hand.s.ide of equation (2) we have employed the relation e <(~rl)) = -- nec/H, where n is the total number of states below the Fermi energy, EF, per unit area. The Hall conductivity in twodimensional systems has been formulated by Ando et al. [9], and we start from the general expression
Aoxy = e2h ins f d E f ( E ) <
TrX
x I ~ ' I m E _ ~f+i~l ] _ [ ~ , ~ . ] >
ReE_3f+i ,
(4)
where S is the area of the system and y is a positive irtfmitesimal. If we denote the eigenstates of the Hamiltertian by la) . . . . with eigenenergy Ea . . . . . equation (4) can be written as
where w = p + (e/c)A with A being a vector potential. e2h If 1 a (Ea --Ea + i6)z The electronic structure cons~ts of Landau subbands [8]. The total number of states in a subband for a spin x [(alkll~)(~lf'la)-(alI?lO)(131~'la)l~ (5) direcUon (and valley for Si MOS) per unit area is equal w i t h f ( E ) being the Fermi distribution function. If la) is to l/2~rl 2, where 1 = (ch/eH) ~/2 is the cyclotron radius. a localized state, we have, for any l/I), As is shown by Kubo etaL [10], we can express the coordinate of an electron as a sum of relative and center (al~'l~) -- (ih) -t (otlXI/~) (E~, --Ea), (6) coordinates of cyclotron motion by x = ~ + X and y = so that the contribution from the ath state to Aoxy in n + Ywith ~ = (/2/h)lry and n = -- (12/h)Tr,. The Hall conductivity is expressed as equation (5) reduces to Previously published in: Solid State Commun. Vol. 38, Nos. 7-12, pp. 1079-1082 (1981)
9521
Ao~,
= f(E,~)ec/H,
(7)
where we have employed the relation IX, Y] = il 2 . From this result we can derive important general proper. ties of the Hall conductivity at T = 0. (i) As long as the Fermi energy lies in the localized regime of the energy spectrum, oxy remains constant independent of n, the total number of electrons below EF. (ii) If all the states are localized below EF, we have oxy = 0 because Aor# exactly cancels -- nec/H in (2). Let us now study the limit of strong magnetic fields, in which the energy spectrum comprises separate Landau subbands. In this case there is a situation in which Landau subbands are just filled up to the N t h subband when EF lies in the gap between the N t h and (N + 1)th bands with n = N/27rl:. Now we can show that (iii) Aoxy = 0, i.e. ox~ = --Ne2/h, when we have electrons Idled up to the N t h Landau subband. This is readily seen from equation (5), if we note that we can replace f(Eo,) b y f ( E a ) [1 --f(E#)] in that equation. It follows that, in the limit of strong magnetic fields, (iv) o ~ assumes the constant value --Ne2/h when E r lies in the localized regime between the N t h and (N + 1)th subbands according to the properties (i) and (iii), as is shown in Fig. 1, and (v) all the states cannot be localized, since we have the properties (ii) and (fii). We can further show that the conclusion (iii) is still valid even if we take the electron-electron interaction into account. This can be proved from a symmetry, called the electron-hole symmetry, inherent in the Hamiltonian for the present system. In the quantum limit in which we can consider a single Landau subband, the Hamiltonian is given by ~T = ~ + ~'2 with ~
Vol. 88, Nos. 11/12
HALL CONDUCTIVITY IN THE TWO-DIMENSIONAL SYSTEM
-- f dr ~+(r)V(r)~(r)
(8)
!
X
e2/h . . . . !. . . . . . . . . 1
:
!
,
i I
, i
4
t
where v(r) is the electron-electron interaction and ~ (r) is a field operator for the Landau subband (assumed to be N). It can be expanded as
x
/x": ,
I/nee
,
,
/
!
:: s t
,l
1
2T~
|2n
Fig. 1. The density of states (D), d.c. conductivity (oxx), and the Hall conductivity (or#) are schematically shown as a function ofthe electron concentration (n) for a Landau subband. Shaded regions in the density of states correspond to localized regime. Coulomb and exchange terms arising from the rearrangment of ~ 3 ~ 4 into ~ 1 ~2 can be absorbed in the choice o f energy origin. Similarly center-coordinate operators X and Y are changed into -- X and -- Y, repe.ct.ively. Since Aor# is expressed as e 2 (((]:'X)> -(CYY)>) and is an odd function of H, we can conclude that the Hall conductivity possesses a property as Aor#(V;H;v) = Aor#(-- V;--H; 1 --o)
I--v),
(I I)
dr1 dr2 ~+(rt)~+(r2)v(r2 --r2)0(r2)O(rl), where v is the occupation ratio of the subband defined (9)
=
, '
"
0
= --Aor# (-- V;H; ~2 = ~-
t
,i i i i t i i i t
',
', .....
, I
i
¢Nx(r)aNx,
(10)
where ¢b~x(r) is the usual wave function of the N t h Landau level with the center coordinate X being diagnnalized and aNx is the annihilation operator. Let us consider a transformation U, which converts the quantifies as aNx -~ a ~ x and a ~ x "+ a~x. Then it can be shown that the Hamiltonian becomes that of a system in which V(r) and Hare replaced by -- V(r) and --H, respectively, apart from an additive constant. The
by v = 2~rl2 nN with n jr being the electron concentration of the subband. By this formula, we can relate or# in the lower edge of a Landau subband to that in the upper edge of the band. In particular,it is seen that the property (iii) that or# = --Ne2/h when the Landau subbands are filled and consequently (iv) are still valid in the presence of many.body effect, because the vanishing or# in the band bottom just corresponds to or# = --e2/h in the band top (Fig. 1). Thus, if we plot or# for the whole Landau subbands, it takes the value of integral multiple o f - - e2/h each time E~- falls in the localized regime between adjacent subbands (Fig. 2). In this figure a localized region is depicted as getting narrower for a higher subband as has been suggested theoretically [3, 4].
Vol. 88, Nos. 11/12
HALL CONDUCTIVITY IN THE TWO-DIMENSIONAL SYSTEM Bo
_d
//nec H
,.C:
X
/-
2
ss S
I
/
s¢ ° =,~"
2 21'T,[2r3 Fig. 2. The Hall conductivity ax~ is schematically shown versus 2~12n together with a straight line representing -
-
nee/H.
We have shown that exy assumes the value --Ne2/h despite the presence of immobile carriers. The physical meaning of this property can be explained as follows. In the case o f strong magnetic field, the .matrix element of the current/x reduces to that o f - - e X . The equation of motion for X is given [10] as + E,
.
(12)
Since exy is a nondissipative quantity we have
where the summation for a' is over the occupied states and we have Ca'(r) = ~" C~x ¢h~x (r).
(14)
X
Note that or' contains effects of the infinitesimal external electric field Ey. If all'the states associated with the Landau subband are occupied, we have
Y*
,. I2 = 27tl
x
f dr aV(O ay =
O,
(15)
where we have used ~. iCh~x(r)12 = 1 x 2lr12 "
953
For a localized state, the effective field of the random potential modified by a slight deformation of the wave function cancels the applied electric field, and the localized state does not contribute to the Hall conductivity. On the other hand, electrons in extended states feel an effective field in the direction of the external field and move faster because of the severe constraint (l 5) and because their wave functions should be orthogonal to those of localized states. The fast motion of electrons in the extended states exactly cancel the decrease of current due to localized electrons. The properties (iii)-(v) described above have been derived in the limit of strong magnetic field. There might be corrections due to mixing of different Landau subbands. From equation (5) such corrections can be shown to vanish to the lowest order in (l-'/hwe) 2 , where F is the broadening of a Landau subband and hw c is the cyclotron energy. Higher order corrections have not been investigated and are left for a future study. We should remark that, within the single-site approximation, we can show [12] that Aaxy = 0 for an arb~itrary magnetic field when EF lies in a gap region where the density of states vanishes. A similar conclusion was recently reached by Prange [ 13], who studied the problem of a single impurity with a delta potential. We have also shown that oxy exactly vanishes if all the states are localized. We have a finite osy both experimentally and theoretically, which shows that all the states are not localized in this system, indicating that the theory of Abrahams et aL [14] does not apply, at least in its single-parameter scaling scheme, to two-dimensional systems in strong magnetic fields. As for the symmetry in Aoxy, in a special case in which the random potential consists of equal amount of attractive and repulsive scatterers, we end up with Aox~ (v) = -- Aox~ (1 -- v). Although this might not exactly be the case in real MOS systems, it is highly likely [ 15 ] that there are nearly equal amount of positive and negative impurity charges in the interface region of the inversion layer. In summary, we have shown some distinct properties of the Hall conductivity in the two-dimensional system in strong magnetic fields, which exhibits a characteristic behavior involving a universal quantity e2/h.
Acknowledgements - The authors are indebted to Prof. S. Kawaji and Dr J. Wakabayashi for discussion and for showing them experimental results prior to publication. They are also grateful to Prof. Y. Uemura and Dr M. Pepper for valuable discussions. One of the authors (H.A.) acknowledges the Science Research Council for financial support.
(16)
REFERENCES .
K. yon Klitzing, G. Dorda & M. Pepper, Phy~ Rev. Lett. 45,494 (1980).
954 2.
3.
4. 5. 6. 7. 8. 9.
HALL CONDUCTIVITY IN THE TWO-DIMENSIONAL SYSTEM S. Kawaji & J. Wakabayashi, Physics in High Magnetic Fields (Edited by S. Chikazumi & N. Miura), p. 284. Springer-Vedag, Berlin Heidelberg (1981). H. Aoki & H. Kamimura, Solid State Commun~ 21, 45 (1977). H. Aoki, J. Phys. ClO, 2583(1977);J. Phys. CII, 3823 (1978);So//d State Commun. 31,999 (1979). M. Tsukada,J. Phys, Soc. Japan 42,391 (1977). H. Fukuyama, Solid State Commun~ 19,551 (1976); H. Fukuyama, P.M. Platzman & P.W. Anderson,Phys. Rev. BI9, 5211 (1979). H. Aoki, Surf. 3ci. 73,281 (1978);J. Phys. C12, 633 (1979). T. Ando & Y. Uemura,J. Phys. Soc. Japan 36,959 (1974); T. Ando,J. Phys. Soc. Japan 36, 1521 (1974); 37,622 (1974); 37, 1233 (1974). T. Ando, Y. Matsumoto & Y. Uemura, J. Phys.
Vol. 88, lqos. 11/12
Soc. Japan 39,279 (1975). It has already been noted in this paper that oxy= --Ne2/h despite the presence of impurity states in the single-site approximation. 10. R. Kubo, S.J. Miyake & N. Hashitsume, Solid State Phys. (Edited by F. Seitz & D. Turnbull), Vol. 17, p. 269, Academic Press, London (1965). 11. R. Kubo,J. Phys Soc. Japan 12,570 (1957). 12. T. Ando (unpublished). 13. R.E. Prange, Phys~ Rev. Lett (to be published). 14. E. Abrahams, P.W. Anderson, D.C. Licciardello & T.V. Ramakrishnan, Phys. Rev. Lett. 41,673 (1979). 15. See, for example, M. Pepper, The Physics of Si02 and Its Interfaces (Edited by S.T. Pantelides), p. 407. Pergamon Press, Oxford (1978).
Solid State Communications, Vol. 88, Nos. i 1/12, pp. 951-954, 1993. Printed in Great Britain.
0038-1098/9356.00+.00 Pergamon Press Ltd
EFFECT OF LOCALIZATION ON THE HALL CONDUCTIVITY IN THE TWO-DIMENSIONAL SYSTEM IN STRONG MAGNETIC FIELDS H. Aoki Cavendish Laboratory, Madingiey Road, Cambridge CB3 0HE, England and T. Ando Institute of Applied Physics, University of Tsukuba, Sakura, Ibaraki 305, Japan
(Received 16January 1981 by Y. Toyozawa) Some exact properties of the Hall conductivity in the two-dimensional electron system in strong magnetic fields are investigated in the presence of immobile carriers. It is shown that oxy should be exactly the integral multiple of--e2/h when the Fermi energy lies in the localized regime. RECENTLY yon Klitzing etal. [1] and Kawaji and Wakabayashi [2] experimentally verified that the plateau value of the Hall conductivity, am, in the two. dimensional electron system in MOS inversion layers in strong magnetic fields (H ~ 100 kO¢) is equal to the integral multiple o f - - e2/h, potentially providing a high precision method of measuring the universal quantity e2]h. The plateau region of a m is shown to arise from immobile carriers around the tails of the Landau subband in which region the d.c. conductivity a ~ vanishes. The cartier localization in this system has been explained by the localization due to randomness [3, 4], electronelectron interaction [5, 6] or combined effect of b o t h [7]. However, the result that o ~ = --Ne2/h (AT = integer) despite the existence of localization is particulaxly intriguin 8 because one might expect, say, axy= -eCnmobae/H with nmoba e being the number of mobile carriers. In the present paper we clarify this point theoretically, and elucidate some exact properties of the Hall conductivity. The Hamiltonian of a two-dimensional system in a magnetic field H with random potential V(r) is given by
~c = " 2 + v(r), 2m
(1)
o~
nec = -- ~ - + Zxaxy ,
(2)
with
(3)
~o.. = ± e ~ (<>-<~?~>>), 2
where the double brackets represent the correlation functionwith canonical average and configuration averages over randomness. In the right.hand.s.ide of equation (2) we have employed the relation e <(~rl)) = -- nec/H, where n is the total number of states below the Fermi energy, EF, per unit area. The Hall conductivity in twodimensional systems has been formulated by Ando et al. [9], and we start from the general expression
Aoxy = e2h ins f d E f ( E ) <
TrX
x I ~ ' I m E _ ~f+i~l ] _ [ ~ , ~ . ] >
ReE_3f+i ,
(4)
where S is the area of the system and y is a positive irtfmitesimal. If we denote the eigenstates of the Hamiltertian by la) . . . . with eigenenergy Ea . . . . . equation (4) can be written as
where w = p + (e/c)A with A being a vector potential. e2h If 1 a (Ea --Ea + i6)z The electronic structure cons~ts of Landau subbands [8]. The total number of states in a subband for a spin x [(alkll~)(~lf'la)-(alI?lO)(131~'la)l~ (5) direcUon (and valley for Si MOS) per unit area is equal w i t h f ( E ) being the Fermi distribution function. If la) is to l/2~rl 2, where 1 = (ch/eH) ~/2 is the cyclotron radius. a localized state, we have, for any l/I), As is shown by Kubo etaL [10], we can express the coordinate of an electron as a sum of relative and center (al~'l~) -- (ih) -t (otlXI/~) (E~, --Ea), (6) coordinates of cyclotron motion by x = ~ + X and y = so that the contribution from the ath state to Aoxy in n + Ywith ~ = (/2/h)lry and n = -- (12/h)Tr,. The Hall conductivity is expressed as equation (5) reduces to Previously published in: Solid State Commun. Vol. 38, Nos. 7-12, pp. 1079-1082 (1981)
9521
Ao~,
= f(E,~)ec/H,
(7)
where we have employed the relation IX, Y] = il 2 . From this result we can derive important general proper. ties of the Hall conductivity at T = 0. (i) As long as the Fermi energy lies in the localized regime of the energy spectrum, oxy remains constant independent of n, the total number of electrons below EF. (ii) If all the states are localized below EF, we have oxy = 0 because Aor# exactly cancels -- nec/H in (2). Let us now study the limit of strong magnetic fields, in which the energy spectrum comprises separate Landau subbands. In this case there is a situation in which Landau subbands are just filled up to the N t h subband when EF lies in the gap between the N t h and (N + 1)th bands with n = N/27rl:. Now we can show that (iii) Aoxy = 0, i.e. ox~ = --Ne2/h, when we have electrons Idled up to the N t h Landau subband. This is readily seen from equation (5), if we note that we can replace f(Eo,) b y f ( E a ) [1 --f(E#)] in that equation. It follows that, in the limit of strong magnetic fields, (iv) o ~ assumes the constant value --Ne2/h when E r lies in the localized regime between the N t h and (N + 1)th subbands according to the properties (i) and (iii), as is shown in Fig. 1, and (v) all the states cannot be localized, since we have the properties (ii) and (fii). We can further show that the conclusion (iii) is still valid even if we take the electron-electron interaction into account. This can be proved from a symmetry, called the electron-hole symmetry, inherent in the Hamiltonian for the present system. In the quantum limit in which we can consider a single Landau subband, the Hamiltonian is given by ~T = ~ + ~'2 with ~
Vol. 88, Nos. 11/12
HALL CONDUCTIVITY IN THE TWO-DIMENSIONAL SYSTEM
-- f dr ~+(r)V(r)~(r)
(8)
!
X
e2/h . . . . !. . . . . . . . . 1
:
!
,
i I
, i
4
t
where v(r) is the electron-electron interaction and ~ (r) is a field operator for the Landau subband (assumed to be N). It can be expanded as
x
/x": ,
I/nee
,
,
/
!
:: s t
,l
1
2T~
|2n
Fig. 1. The density of states (D), d.c. conductivity (oxx), and the Hall conductivity (or#) are schematically shown as a function ofthe electron concentration (n) for a Landau subband. Shaded regions in the density of states correspond to localized regime. Coulomb and exchange terms arising from the rearrangment of ~ 3 ~ 4 into ~ 1 ~2 can be absorbed in the choice o f energy origin. Similarly center-coordinate operators X and Y are changed into -- X and -- Y, repe.ct.ively. Since Aor# is expressed as e 2 (((]:'X)> -(CYY)>) and is an odd function of H, we can conclude that the Hall conductivity possesses a property as Aor#(V;H;v) = Aor#(-- V;--H; 1 --o)
I--v),
(I I)
dr1 dr2 ~+(rt)~+(r2)v(r2 --r2)0(r2)O(rl), where v is the occupation ratio of the subband defined (9)
=
, '
"
0
= --Aor# (-- V;H; ~2 = ~-
t
,i i i i t i i i t
',
', .....
, I
i
¢Nx(r)aNx,
(10)
where ¢b~x(r) is the usual wave function of the N t h Landau level with the center coordinate X being diagnnalized and aNx is the annihilation operator. Let us consider a transformation U, which converts the quantifies as aNx -~ a ~ x and a ~ x "+ a~x. Then it can be shown that the Hamiltonian becomes that of a system in which V(r) and Hare replaced by -- V(r) and --H, respectively, apart from an additive constant. The
by v = 2~rl2 nN with n jr being the electron concentration of the subband. By this formula, we can relate or# in the lower edge of a Landau subband to that in the upper edge of the band. In particular,it is seen that the property (iii) that or# = --Ne2/h when the Landau subbands are filled and consequently (iv) are still valid in the presence of many.body effect, because the vanishing or# in the band bottom just corresponds to or# = --e2/h in the band top (Fig. 1). Thus, if we plot or# for the whole Landau subbands, it takes the value of integral multiple o f - - e2/h each time E~- falls in the localized regime between adjacent subbands (Fig. 2). In this figure a localized region is depicted as getting narrower for a higher subband as has been suggested theoretically [3, 4].
Vol. 88, Nos. 11/12
HALL CONDUCTIVITY IN THE TWO-DIMENSIONAL SYSTEM Bo
_d
//nec H
,.C:
X
/-
2
ss S
I
/
s¢ ° =,~"
2 21'T,[2r3 Fig. 2. The Hall conductivity ax~ is schematically shown versus 2~12n together with a straight line representing -
-
nee/H.
We have shown that exy assumes the value --Ne2/h despite the presence of immobile carriers. The physical meaning of this property can be explained as follows. In the case o f strong magnetic field, the .matrix element of the current/x reduces to that o f - - e X . The equation of motion for X is given [10] as + E,
.
(12)
Since exy is a nondissipative quantity we have
where the summation for a' is over the occupied states and we have Ca'(r) = ~" C~x ¢h~x (r).
(14)
X
Note that or' contains effects of the infinitesimal external electric field Ey. If all'the states associated with the Landau subband are occupied, we have
Y*
,. I2 = 27tl
x
f dr aV(O ay =
O,
(15)
where we have used ~. iCh~x(r)12 = 1 x 2lr12 "
953
For a localized state, the effective field of the random potential modified by a slight deformation of the wave function cancels the applied electric field, and the localized state does not contribute to the Hall conductivity. On the other hand, electrons in extended states feel an effective field in the direction of the external field and move faster because of the severe constraint (l 5) and because their wave functions should be orthogonal to those of localized states. The fast motion of electrons in the extended states exactly cancel the decrease of current due to localized electrons. The properties (iii)-(v) described above have been derived in the limit of strong magnetic field. There might be corrections due to mixing of different Landau subbands. From equation (5) such corrections can be shown to vanish to the lowest order in (l-'/hwe) 2 , where F is the broadening of a Landau subband and hw c is the cyclotron energy. Higher order corrections have not been investigated and are left for a future study. We should remark that, within the single-site approximation, we can show [12] that Aaxy = 0 for an arb~itrary magnetic field when EF lies in a gap region where the density of states vanishes. A similar conclusion was recently reached by Prange [ 13], who studied the problem of a single impurity with a delta potential. We have also shown that oxy exactly vanishes if all the states are localized. We have a finite osy both experimentally and theoretically, which shows that all the states are not localized in this system, indicating that the theory of Abrahams et aL [14] does not apply, at least in its single-parameter scaling scheme, to two-dimensional systems in strong magnetic fields. As for the symmetry in Aoxy, in a special case in which the random potential consists of equal amount of attractive and repulsive scatterers, we end up with Aox~ (v) = -- Aox~ (1 -- v). Although this might not exactly be the case in real MOS systems, it is highly likely [ 15 ] that there are nearly equal amount of positive and negative impurity charges in the interface region of the inversion layer. In summary, we have shown some distinct properties of the Hall conductivity in the two-dimensional system in strong magnetic fields, which exhibits a characteristic behavior involving a universal quantity e2/h.
Acknowledgements - The authors are indebted to Prof. S. Kawaji and Dr J. Wakabayashi for discussion and for showing them experimental results prior to publication. They are also grateful to Prof. Y. Uemura and Dr M. Pepper for valuable discussions. One of the authors (H.A.) acknowledges the Science Research Council for financial support.
(16)
REFERENCES .
K. yon Klitzing, G. Dorda & M. Pepper, Phy~ Rev. Lett. 45,494 (1980).
954 2.
3.
4. 5. 6. 7. 8. 9.
HALL CONDUCTIVITY IN THE TWO-DIMENSIONAL SYSTEM S. Kawaji & J. Wakabayashi, Physics in High Magnetic Fields (Edited by S. Chikazumi & N. Miura), p. 284. Springer-Vedag, Berlin Heidelberg (1981). H. Aoki & H. Kamimura, Solid State Commun~ 21, 45 (1977). H. Aoki, J. Phys. ClO, 2583(1977);J. Phys. CII, 3823 (1978);So//d State Commun. 31,999 (1979). M. Tsukada,J. Phys, Soc. Japan 42,391 (1977). H. Fukuyama, Solid State Commun~ 19,551 (1976); H. Fukuyama, P.M. Platzman & P.W. Anderson,Phys. Rev. BI9, 5211 (1979). H. Aoki, Surf. 3ci. 73,281 (1978);J. Phys. C12, 633 (1979). T. Ando & Y. Uemura,J. Phys. Soc. Japan 36,959 (1974); T. Ando,J. Phys. Soc. Japan 36, 1521 (1974); 37,622 (1974); 37, 1233 (1974). T. Ando, Y. Matsumoto & Y. Uemura, J. Phys.
Vol. 88, lqos. 11/12
Soc. Japan 39,279 (1975). It has already been noted in this paper that oxy= --Ne2/h despite the presence of impurity states in the single-site approximation. 10. R. Kubo, S.J. Miyake & N. Hashitsume, Solid State Phys. (Edited by F. Seitz & D. Turnbull), Vol. 17, p. 269, Academic Press, London (1965). 11. R. Kubo,J. Phys Soc. Japan 12,570 (1957). 12. T. Ando (unpublished). 13. R.E. Prange, Phys~ Rev. Lett (to be published). 14. E. Abrahams, P.W. Anderson, D.C. Licciardello & T.V. Ramakrishnan, Phys. Rev. Lett. 41,673 (1979). 15. See, for example, M. Pepper, The Physics of Si02 and Its Interfaces (Edited by S.T. Pantelides), p. 407. Pergamon Press, Oxford (1978).