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Width effect on the magnetoconductivity of 2D electron systems
Solid State Communications, Printed in Great Britain.
Vol. 60, No. 5, pp. 4699472,
0038-1098/86 $3.00 + .OO Pergamon Journals Ltd.
1986.
WIDTH EFFECT ON THE MAGNETOCONDUCTIVITY
OF 2D ELECTRON SYSTEMS
A. Isihara and L. SmrEka* Statistical Physics Laboratory,
Department
of Physics, State University of New York at Buffalo, Buffalo, NY, 14260, USA
(Received 2 1 April 1986 by il4.F. Collins)
Based on a parabolic well model, it is shown that the magnetoconductivity of narrow 2D electron systems is characterized by a modified cyclotron frequency and a crossover parameter which is a ratio of the cyclotron frequency to the curvature of the parabolic potential. Explicit results are derived for systems with finite width under the assumption that quantum oscillations are small.
THERE HAS BEEN a surge of interest in the past several years in narrow two-dimensional electron systems in Si inversion layers. These systems show very unique properties ranging from two dimensions to one [l] . Especially, irregular but reproducible spiked structures have been observed in ultra narrow systems. While in such cases hopping conduction dominates, we are intereseted in the present article in intermediate cases in which width effects on the Shubnikov-de Haas oscillations can be expected. In fact, Skocpol er al. [l] reported that when the width of Si MOSFETs is reduced from 140 to 1lOnm the oscillations are markedly changed. Moreover, for a given width, the conductivity depends on the magnetic field as well as on the gate voltage. In what follows, we shall present a simple but workable model which can describe width effects on the SdH oscillations. This model may be applied only to two-dimensional systems with finite width so that the basic energy levels and the degree of freedom are preserved. We assume that quantum oscillations are small. Our theory is based on a parabolic-well model in which the electrons are subject to a potential [2] : mR2 v(y) = 2y2,
;
(i f
2
= m/(1-02);cr
cj2 = ~3 + a2 ;r .
1
(2”n!;lrT)l,2 expi -Y$ekJ)>
(4)
where
112
=
where x represents the extended direction of the narrow channel, B is the magnetic field, and the last term represents the parabolic potential centered at y = 0. One can write down Jl(x, y) = e ik”u(y) and the equation for u(y). From thus obtained equation, we find that the eigenvalues and eigenfunctions are given by
(1) QJY) =
where m is the effective mass, y is the coordinate in the lateral direction, and a is a parameter to represent the “width” w through W(E)
in this model is energy dependent. The curvature increases as the width and electron energy Increase so that the electrons are more strongly “confined” in the parabolic well. This model gives the advantage of being able to accomodate a perpendicular magnetic field without causing mathematical difficulties. It is simple but workable if the width is fmite. The Schorodinger equation of an electron in such a parabolic potential well and in a magnetic field can be expressed by [3]
(2)
That is, s1 is the angular frequency which measures the curvature of the parabolic well. Note that the “width” *Present address: Institute of Physics, Czechoslovak Academy of Sciences, 18040 Praha 8, Na Slovance 2, Czechoslovakia. 469
= w,/G = (fi/ijm)1’2 = 1dt.1.
(5)
That is, the system becomes equivalent to an ordinary three-dimensional electron system in a magnetic field provided the modified cyclotron frequency 15 and magnetic length 1 are used. The effective mass r% refers to the motion in the x direction which is extended. Note that G is generally larger than m. Although cr varies between 0 and 1, the limiting case of Q = 1 must be avoided because k diverges. cwis a convenient parameter
470
WIDTH EFFECT ON THE MAGNETOCONDUCTIVITY
to represent changes in confinement. In what follows, we assume that 6 is fmitie. The magnetoconductivity can now be obtained in terms of the Green’s function (See Appendix). As usual, we assume that there is a maximum wave number given - %3(n + f))] 1’2/h where eiM is the by kM = [Z(E~ band maximum. When averaged over the impurity configurations, the average Green’s function is characterized by the self-energy ZZ= A - ir. We assume that those electrons with energy less than the band maximum are scattered by randomly distributed impurities with concentration c in a unit cell V. The strength of the impurity potential will be denoted by 6. If 6 > 0, the potential is repulsive and if 6 < 0, it is attractive. For small fi, we may approximately evaluate the self-energy through the Soven equation [4] :
= (1 -aZ)ao
u,,
+
ix2uo 1+ G2r20
where =
00
Vol. 60, No. 5
1+
l+Scj’r; 2(1 + 37;)
-Ag g,
1’ (12)
Noe2r0/m,
(13)
is the zero field conductivity, and Ag/go is the ratio of the oscillating part of the density of states to the density of states in the zero filed case and is given by
Ag -= go
exp (- 2rrsro/hi3)
(14)
c6
lZ=
(6)
1 - 6K(e + i0)’
where K(z) = +
1 n,k
--!-
(7)
Z--~’
We remark that for finite temperatures a factor X&h X, should be introduced to the s-sum, where X, = 2n2skT/f6. Otherwise, the same expression holds. Note also that g = go + Ag, and that equation (10) states EM
The summation in K(z) is subject to the condition that E,
(8)
Furthermore, for z = E + i0, the non-oscillating be expressed as Ko(z) = R(E) + iF(e).
part can (9)
Note that the imaginary part F is related to the densityof-states. Hence, we impose the normalization condition Q4 -1
1 Im Ke(e+iO)de R. 0
= 1.
(10)
Because of the decomposition expressed by equation (8), it turns out that the zero field real and imaginary parts R. and F. rather than R and F characterize the magnetoconductivity and other quantities. For the same reason, the real and imaginary parts of the self-energy are also characterized dominantly by these zero field quantities. In fact, for a short and strong impurity potential, we arrive at ro
=
-c
Fo(E)
[RO(E)12P
A0
=
-c&.
(11)
Note that A, represents the energy shift and PO is associated with the relaxation time through r. = li/2ro. The magnetoconductivity is given to first order in the ratio AK/K0 (or in Ag/go listed below) by
i’
go(e)de
= 1.
-ca The magnetoconductivity given by equation (12) consists of two terms. The first term is due to size quantization and the second to field quantization. Since these two terms are proportional to Nero, the overall averaged behavior of the conductivity will be determined by those of No and ro. These quantities are given explicitly by the real and imaginary parts of the auxiliary function K. in accordance with equation (9). We have found that No increases almost linearly and r. decreases with gate voltage. Therefore, when plotted against gate voltage, uxx is expected to show a maximum. The experimental data of Skocpol ef al. show at least a saturation at a certain gate voltage and, in the case of 120 nm, an indication of a maximum around gate voltage of 18 V. The magnitude of the second non-oscillating term in our magnetoconductivity formula is reduced directly by the magnetic field. This is due to our approach in which field effects are more directly taken into consideration than size effects. Unless this is the case, we could not use the modified Landau level variables such as & and T. Since the non-oscillating part is dominant in general in accordance with our treatement, the oscillations in the magnetoconductivity will apear in superposition to the curve representing the case of zero magnetic field. A similar behavior has been observed in ordinary Si accumulation layers [5] . The oscillations are determined by those in the density-of-states, which in
turn are dependent on width and magnetic field. Since r0 = b/2F0, the oscillating terms in equation (14) decrease very fast if 0~7~ < 1 so that only the first term can be contributing. Our approximation based on small oscillating terms is particularly good then. Note that the two oscillating terms in equation (12) differ in phase. In the absence of magnetic field, the second term in equation (12) vanishes, while the amplitude of the first term becomes proportional to a. Hence, we expect larger oscillations for narrower strips. While this behavior is understandable and appears to agree with the experiment of Skocpol et al., the oscillations become irregular when the width becomes narrower. Finally, for ultra narrow channels the spiked structure appears. The present theory is not applicable to such quasi onedimensional cases. Acknowledgement - This work was supported ONR under Contract NO001 4-84-Ka387.
by the
%&F,T)
3. 4. 5. 6.
7. 8. 9. 10. 11.
di2e2 af * --y-G
J
-H)v$(e
--H))de,
(A.1)
where eF is the Fermi energy, f(e) is the Fermi distribution, v, is the component of the electron velocity in the extended direction, and (. . .) denotes averaging with respect to impurity configurations. The Green’s function G(z) = (z - H)-’ can be used in the integrand because of the relationship 6 (e - H) = - Im G(e + iO)/n. In carrying out the impurity averaging, we adopt the familiar method to replace the average (Gv,G) by (G )v, (G) where the averaged Green’s function is given by (G) = (z -H-z)-’ , with Z for the self-energy of the electron. Following Soven [4] we determine the selfenergy for a delta-function type short range impurity potential based on the self-consistent equation:
REFERENCES
2.
= -
< Tr{v,&
z = cs+ 1.
471
WIDTH EFFECT ON THE MAGNETOCONDUCTIVITY
Vol. 60, No. 5
c( 1 - c)62K(z - Z) 1+ [Z.(c-1)6]K(z-Z)’
(A.2)
A.B. Fowler, A. Hartstein 8t R.A. Webb, Phys. Rev. Lett. 48,196 (1982), Physica 117/118B, 661 (1983); A. Hartstein, R.A. Webb, A.B. Fowler & J.J. Wainer,Surf. Sci. 142, 1 (1984); R.G. Wheeler, K.K. Choi, A. Goel, R. Wisnieff & D.E. Prober, Phys. Rev. Lett. 49, 1674 (1982); R.G. Wheeler, K.K. Choi & R. Wisnieff, Surf. Sci. 142, 19 (1984); W.J. Skocpol et al., Surf Sci. 142, 14 (1984). G.F. Giuliani, J.J. Quinn & S.C. Ying, Phys. Rev. B28, 2969 (1983). M.J. Harrison, Phys. Rev. A29, 2272 (1984). L. SmrEka, J. Phys. C: Solid State Phys., to be published. P. Soven,Phys. Rev. 156,809 (1967). F.F. Fang, A.B. Fowler & A. Hartstein, Phys. Rev. B16,4446 (1977). R. Kubo, S. Miyake & N. Hashitsume in Solid State Physics Vol. 6, (Edited by F. Seitz & D. Tumbull) Academic Press, (1965). P. StFeda & L. SmrEka, Phys. Status Solidi (b) 70,537 (1975), A. Bastin, C. Lewiner, 0. Betbeder-Matibet & P. Nozieres, J. Phys. Chem. Solids 32,181l (1971). T. Ando &Y. Uemura, J. Phys. Sot. Japan 36,959 (1974), T. Ando, J. Phys. Sot. Japan 37, 1233 (1974). B. Belicky, S. Kirkpatrick & H. Ehrenreich, Phys. Rev. 175,747 (1968). K. Ohta, Japan J. Appl. Phys. 10, 850 (1971), J. Phys. Sot. Japan 31,1627 (1971). L. Smrcka & A. Isihara, Solid State Commun. 57, 259 (1986). P.N. Argyres, J. Phys. Chem. SoZids 4, 19 (1958). E.M. Lifshitz & A. Kosevich, Zh. Eksp. Teo. Fiz. 29,730 (1955).
where c is the impurity concentration and 6 is the strength of the impurity potential. This self-consistent equation is reduced to the type used by Andoand Uemura [7] if the constant shift, the-c” term and the denominator are neglected. The neglect of the denominator which comes from the T matrix amounts to ignoring multiple scatterings. The above form modifies what reported by Velicky et al. [8], and for the model it is exact. Ohta [9] used a similar self-consistent equation with a simplified denominator for an impurity potential which depends on distance. This dependence is represented by an integration as in Ohta’s case. The auxiliary function K(z) can be obtained in accordance with its definition of equation (7). As in the cases of Ohta, and also Ando [7], the n sum in that equation must be cutoff at a maximum n which corresponds to Ed. While these procedures are used frequently, the experimental data of Skocpol et al for width of around 140-l 10 nm indicate that quantum oscillations can be considered to be small. The overall variation of the magnetoconductance is similar to the case of zero field. For this reason, we split the function K into non-oscillating and oscillating parts. Since the density of states is given by
APPENDIX:
C = A-iI’,
DERIVATION OF MAGNETOCONDUCTIVITY
The conductivity obtained from [6]
per
unit
sample
length
L is
g(e) = - i Im K(e - X(E)),
(A.3)
it is also split into two parts in which the non-oscillating part is considered dominant. From equation (A.2) we can express the self-energy also in a similar way. As usual, it can be written as (A.4)
where A respresnts energy shifting, and F, which is related to the mean life time by r = h/2I’, gives rise to broadening. We then arrive at equation (11) for a strong
472
WIDTH EFFECT ON THE MAGNETOCONDUCTIVITY
short range impurity potential, and from equation (A.l) and equation (A.3) the final conductivity result in equation (14). Note that in the limit (Y= 0, m = m" and our results reproduce the correct threedimensional expressions [6, 11 ‘J as we expect from the expression for the energy levels. On the other hand, the opposite limit of cr = 1 must be treated separately because m” diverges. While this divergence reproduces the correct limiting energy, the degree of freedom in the k direction which is
Vol. 60, No. S
used in the evaluation of the function K(z) can no longer be removed after integration over k. In fact, in this limit K(z) diverges because it is proportional to m”I”. That is, the dimensional change associated with the limit can not be accomodated because our calculations have been based on the finiteness of m”. Like the case of symmetry changes, it is difficult to treat dimensional changes in a continuous way. Thus, the case of (Y= 1 must be treated separately.
Vol. 60, No. 5, pp. 4699472,
0038-1098/86 $3.00 + .OO Pergamon Journals Ltd.
1986.
WIDTH EFFECT ON THE MAGNETOCONDUCTIVITY
OF 2D ELECTRON SYSTEMS
A. Isihara and L. SmrEka* Statistical Physics Laboratory,
Department
of Physics, State University of New York at Buffalo, Buffalo, NY, 14260, USA
(Received 2 1 April 1986 by il4.F. Collins)
Based on a parabolic well model, it is shown that the magnetoconductivity of narrow 2D electron systems is characterized by a modified cyclotron frequency and a crossover parameter which is a ratio of the cyclotron frequency to the curvature of the parabolic potential. Explicit results are derived for systems with finite width under the assumption that quantum oscillations are small.
THERE HAS BEEN a surge of interest in the past several years in narrow two-dimensional electron systems in Si inversion layers. These systems show very unique properties ranging from two dimensions to one [l] . Especially, irregular but reproducible spiked structures have been observed in ultra narrow systems. While in such cases hopping conduction dominates, we are intereseted in the present article in intermediate cases in which width effects on the Shubnikov-de Haas oscillations can be expected. In fact, Skocpol er al. [l] reported that when the width of Si MOSFETs is reduced from 140 to 1lOnm the oscillations are markedly changed. Moreover, for a given width, the conductivity depends on the magnetic field as well as on the gate voltage. In what follows, we shall present a simple but workable model which can describe width effects on the SdH oscillations. This model may be applied only to two-dimensional systems with finite width so that the basic energy levels and the degree of freedom are preserved. We assume that quantum oscillations are small. Our theory is based on a parabolic-well model in which the electrons are subject to a potential [2] : mR2 v(y) = 2y2,
;
(i f
2
= m/(1-02);cr
cj2 = ~3 + a2 ;r .
1
(2”n!;lrT)l,2 expi -Y$ekJ)>
(4)
where
112
=
where x represents the extended direction of the narrow channel, B is the magnetic field, and the last term represents the parabolic potential centered at y = 0. One can write down Jl(x, y) = e ik”u(y) and the equation for u(y). From thus obtained equation, we find that the eigenvalues and eigenfunctions are given by
(1) QJY) =
where m is the effective mass, y is the coordinate in the lateral direction, and a is a parameter to represent the “width” w through W(E)
in this model is energy dependent. The curvature increases as the width and electron energy Increase so that the electrons are more strongly “confined” in the parabolic well. This model gives the advantage of being able to accomodate a perpendicular magnetic field without causing mathematical difficulties. It is simple but workable if the width is fmite. The Schorodinger equation of an electron in such a parabolic potential well and in a magnetic field can be expressed by [3]
(2)
That is, s1 is the angular frequency which measures the curvature of the parabolic well. Note that the “width” *Present address: Institute of Physics, Czechoslovak Academy of Sciences, 18040 Praha 8, Na Slovance 2, Czechoslovakia. 469
= w,/G = (fi/ijm)1’2 = 1dt.1.
(5)
That is, the system becomes equivalent to an ordinary three-dimensional electron system in a magnetic field provided the modified cyclotron frequency 15 and magnetic length 1 are used. The effective mass r% refers to the motion in the x direction which is extended. Note that G is generally larger than m. Although cr varies between 0 and 1, the limiting case of Q = 1 must be avoided because k diverges. cwis a convenient parameter
470
WIDTH EFFECT ON THE MAGNETOCONDUCTIVITY
to represent changes in confinement. In what follows, we assume that 6 is fmitie. The magnetoconductivity can now be obtained in terms of the Green’s function (See Appendix). As usual, we assume that there is a maximum wave number given - %3(n + f))] 1’2/h where eiM is the by kM = [Z(E~ band maximum. When averaged over the impurity configurations, the average Green’s function is characterized by the self-energy ZZ= A - ir. We assume that those electrons with energy less than the band maximum are scattered by randomly distributed impurities with concentration c in a unit cell V. The strength of the impurity potential will be denoted by 6. If 6 > 0, the potential is repulsive and if 6 < 0, it is attractive. For small fi, we may approximately evaluate the self-energy through the Soven equation [4] :
= (1 -aZ)ao
u,,
+
ix2uo 1+ G2r20
where =
00
Vol. 60, No. 5
1+
l+Scj’r; 2(1 + 37;)
-Ag g,
1’ (12)
Noe2r0/m,
(13)
is the zero field conductivity, and Ag/go is the ratio of the oscillating part of the density of states to the density of states in the zero filed case and is given by
Ag -= go
exp (- 2rrsro/hi3)
(14)
c6
lZ=
(6)
1 - 6K(e + i0)’
where K(z) = +
1 n,k
--!-
(7)
Z--~’
We remark that for finite temperatures a factor X&h X, should be introduced to the s-sum, where X, = 2n2skT/f6. Otherwise, the same expression holds. Note also that g = go + Ag, and that equation (10) states EM
The summation in K(z) is subject to the condition that E,
(8)
Furthermore, for z = E + i0, the non-oscillating be expressed as Ko(z) = R(E) + iF(e).
part can (9)
Note that the imaginary part F is related to the densityof-states. Hence, we impose the normalization condition Q4 -1
1 Im Ke(e+iO)de R. 0
= 1.
(10)
Because of the decomposition expressed by equation (8), it turns out that the zero field real and imaginary parts R. and F. rather than R and F characterize the magnetoconductivity and other quantities. For the same reason, the real and imaginary parts of the self-energy are also characterized dominantly by these zero field quantities. In fact, for a short and strong impurity potential, we arrive at ro
=
-c
Fo(E)
[RO(E)12P
A0
=
-c&.
(11)
Note that A, represents the energy shift and PO is associated with the relaxation time through r. = li/2ro. The magnetoconductivity is given to first order in the ratio AK/K0 (or in Ag/go listed below) by
i’
go(e)de
= 1.
-ca The magnetoconductivity given by equation (12) consists of two terms. The first term is due to size quantization and the second to field quantization. Since these two terms are proportional to Nero, the overall averaged behavior of the conductivity will be determined by those of No and ro. These quantities are given explicitly by the real and imaginary parts of the auxiliary function K. in accordance with equation (9). We have found that No increases almost linearly and r. decreases with gate voltage. Therefore, when plotted against gate voltage, uxx is expected to show a maximum. The experimental data of Skocpol ef al. show at least a saturation at a certain gate voltage and, in the case of 120 nm, an indication of a maximum around gate voltage of 18 V. The magnitude of the second non-oscillating term in our magnetoconductivity formula is reduced directly by the magnetic field. This is due to our approach in which field effects are more directly taken into consideration than size effects. Unless this is the case, we could not use the modified Landau level variables such as & and T. Since the non-oscillating part is dominant in general in accordance with our treatement, the oscillations in the magnetoconductivity will apear in superposition to the curve representing the case of zero magnetic field. A similar behavior has been observed in ordinary Si accumulation layers [5] . The oscillations are determined by those in the density-of-states, which in
turn are dependent on width and magnetic field. Since r0 = b/2F0, the oscillating terms in equation (14) decrease very fast if 0~7~ < 1 so that only the first term can be contributing. Our approximation based on small oscillating terms is particularly good then. Note that the two oscillating terms in equation (12) differ in phase. In the absence of magnetic field, the second term in equation (12) vanishes, while the amplitude of the first term becomes proportional to a. Hence, we expect larger oscillations for narrower strips. While this behavior is understandable and appears to agree with the experiment of Skocpol et al., the oscillations become irregular when the width becomes narrower. Finally, for ultra narrow channels the spiked structure appears. The present theory is not applicable to such quasi onedimensional cases. Acknowledgement - This work was supported ONR under Contract NO001 4-84-Ka387.
by the
%&F,T)
3. 4. 5. 6.
7. 8. 9. 10. 11.
di2e2 af * --y-G
J
-H)v$(e
--H))de,
(A.1)
where eF is the Fermi energy, f(e) is the Fermi distribution, v, is the component of the electron velocity in the extended direction, and (. . .) denotes averaging with respect to impurity configurations. The Green’s function G(z) = (z - H)-’ can be used in the integrand because of the relationship 6 (e - H) = - Im G(e + iO)/n. In carrying out the impurity averaging, we adopt the familiar method to replace the average (Gv,G) by (G )v, (G) where the averaged Green’s function is given by (G) = (z -H-z)-’ , with Z for the self-energy of the electron. Following Soven [4] we determine the selfenergy for a delta-function type short range impurity potential based on the self-consistent equation:
REFERENCES
2.
= -
< Tr{v,&
z = cs+ 1.
471
WIDTH EFFECT ON THE MAGNETOCONDUCTIVITY
Vol. 60, No. 5
c( 1 - c)62K(z - Z) 1+ [Z.(c-1)6]K(z-Z)’
(A.2)
A.B. Fowler, A. Hartstein 8t R.A. Webb, Phys. Rev. Lett. 48,196 (1982), Physica 117/118B, 661 (1983); A. Hartstein, R.A. Webb, A.B. Fowler & J.J. Wainer,Surf. Sci. 142, 1 (1984); R.G. Wheeler, K.K. Choi, A. Goel, R. Wisnieff & D.E. Prober, Phys. Rev. Lett. 49, 1674 (1982); R.G. Wheeler, K.K. Choi & R. Wisnieff, Surf. Sci. 142, 19 (1984); W.J. Skocpol et al., Surf Sci. 142, 14 (1984). G.F. Giuliani, J.J. Quinn & S.C. Ying, Phys. Rev. B28, 2969 (1983). M.J. Harrison, Phys. Rev. A29, 2272 (1984). L. SmrEka, J. Phys. C: Solid State Phys., to be published. P. Soven,Phys. Rev. 156,809 (1967). F.F. Fang, A.B. Fowler & A. Hartstein, Phys. Rev. B16,4446 (1977). R. Kubo, S. Miyake & N. Hashitsume in Solid State Physics Vol. 6, (Edited by F. Seitz & D. Tumbull) Academic Press, (1965). P. StFeda & L. SmrEka, Phys. Status Solidi (b) 70,537 (1975), A. Bastin, C. Lewiner, 0. Betbeder-Matibet & P. Nozieres, J. Phys. Chem. Solids 32,181l (1971). T. Ando &Y. Uemura, J. Phys. Sot. Japan 36,959 (1974), T. Ando, J. Phys. Sot. Japan 37, 1233 (1974). B. Belicky, S. Kirkpatrick & H. Ehrenreich, Phys. Rev. 175,747 (1968). K. Ohta, Japan J. Appl. Phys. 10, 850 (1971), J. Phys. Sot. Japan 31,1627 (1971). L. Smrcka & A. Isihara, Solid State Commun. 57, 259 (1986). P.N. Argyres, J. Phys. Chem. SoZids 4, 19 (1958). E.M. Lifshitz & A. Kosevich, Zh. Eksp. Teo. Fiz. 29,730 (1955).
where c is the impurity concentration and 6 is the strength of the impurity potential. This self-consistent equation is reduced to the type used by Andoand Uemura [7] if the constant shift, the-c” term and the denominator are neglected. The neglect of the denominator which comes from the T matrix amounts to ignoring multiple scatterings. The above form modifies what reported by Velicky et al. [8], and for the model it is exact. Ohta [9] used a similar self-consistent equation with a simplified denominator for an impurity potential which depends on distance. This dependence is represented by an integration as in Ohta’s case. The auxiliary function K(z) can be obtained in accordance with its definition of equation (7). As in the cases of Ohta, and also Ando [7], the n sum in that equation must be cutoff at a maximum n which corresponds to Ed. While these procedures are used frequently, the experimental data of Skocpol et al for width of around 140-l 10 nm indicate that quantum oscillations can be considered to be small. The overall variation of the magnetoconductance is similar to the case of zero field. For this reason, we split the function K into non-oscillating and oscillating parts. Since the density of states is given by
APPENDIX:
C = A-iI’,
DERIVATION OF MAGNETOCONDUCTIVITY
The conductivity obtained from [6]
per
unit
sample
length
L is
g(e) = - i Im K(e - X(E)),
(A.3)
it is also split into two parts in which the non-oscillating part is considered dominant. From equation (A.2) we can express the self-energy also in a similar way. As usual, it can be written as (A.4)
where A respresnts energy shifting, and F, which is related to the mean life time by r = h/2I’, gives rise to broadening. We then arrive at equation (11) for a strong
472
WIDTH EFFECT ON THE MAGNETOCONDUCTIVITY
short range impurity potential, and from equation (A.l) and equation (A.3) the final conductivity result in equation (14). Note that in the limit (Y= 0, m = m" and our results reproduce the correct threedimensional expressions [6, 11 ‘J as we expect from the expression for the energy levels. On the other hand, the opposite limit of cr = 1 must be treated separately because m” diverges. While this divergence reproduces the correct limiting energy, the degree of freedom in the k direction which is
Vol. 60, No. S
used in the evaluation of the function K(z) can no longer be removed after integration over k. In fact, in this limit K(z) diverges because it is proportional to m”I”. That is, the dimensional change associated with the limit can not be accomodated because our calculations have been based on the finiteness of m”. Like the case of symmetry changes, it is difficult to treat dimensional changes in a continuous way. Thus, the case of (Y= 1 must be treated separately.