- Email: [email protected]
Correlation functions and quantized noise in mesoscopic
Superlattices
and Microstructures,
CORRELATION
Vol. 11, NO. 2, 1992
FUNCTIONS Tilmann
205
AND
QUANTIZED
NOISE
Kuhn,
Lino Reggiani,
IN MESOSCOPIC
SYSTEMS
and Luca Varani
Dipartimento di Fisica, Universitd di Modena Via Campi 213/A, 41100 Modena, Italy (Received
19 May 1991)
We present a theoretical calculation of the current correlation functions and the associated noise spectra of a quasi one-dimensional system terminated by ideal contacts. The shape of the correlation functions is directly related to the microscopic carrier dynamics. When applying a voltage we find: In the classical limit the low-frequency noise spectrum recovers the well-known shot noise formula at high voltages. In the degenerate limit it is suppressed due to quantum-mechanical correlations. The extension of the theory to a multi-subband system shows that in the presence of an applied voltage the step height of the low-frequency spectral density is reduced to one half of its equilibrium value.
1. Introduction
with
Electronic noise in mesoscopic systems in the absence of dissipation is of physical interest due to its direct relation to fundamental constants. In particular, noise spectra at low-frequencies have been investigated recently in several theoretical [l-3] and experimental (4-71 papers. The aim of this contribution is to present a theoretical analysis of the noise spectrum under equilibrium and nonequilibrium conditions in the full frequency range, or equivalently, to analyze the full time dependence of the current correlation function. As a matter of fact, the shape of the correlation function can be directly related to characteristic times of the carrier transport [3]. Therefore, a study of the full time dependence of fluctuations in mesoscopic systems can provide additional insight into the microscopic dynamics.
We consider a two-terminal device of length L, terminated by ideal contacts. The cross-section is assumed to be so small that the electronic states in the transverse directions are quantized and can be described by a quantum number 0. In the stationary state, the current noise in this system is determined by the correlation function CI(t) of current fluctuations under the condition of a constant voltage applied at the contacts:
0749-6036/92/020205
+ I(t
+ 05 $02.00/O
+m S1(w)= J 2
- (I)2
(1)
dl eiYtCr(t)
(3)
The current operator jzTaTz, t) can be expressed in terms of the Wigner operator gp in subband r~ which, by definition, is given by g&z,l)
2. Theory
Cr(t) = +(O)I(t)
where j&z, t) is the current operator in subband a and brackets indicate averages with respect to the appropriate statistical operator. Equation (2) follows from a quantum generalization of the Ramo-Shockley theorem (81. The current spectral density is then obtained from the Wiener-Khintchine theorem according to
=
-J&
Jdrfeikzt !tj&(%
+ ;,
t)*,,,(z
- fJ)
(4)
. according
to j,?,
=
$Idkkg,(k,z,t).
Here,
Sz,,
(9, ,) denote the field creation (annihilation) operators’in subband a with spin s and m is a scalar effective mass. With these definitions, the current correlation function can be expressed in terms of the “Wigner correlation function” 6fPa as
Cl(t) = (-$)2
c
pkdk’
a,0 __-
0 1992 Academic Press Limited
206
Superlattices
X
Ii
dz dz’k k’6f,p(k, z, t; k’, t’, 0)
(5)
-L/2
and Microstructures,
Vol. 11, No. 2, 1992
contact
with 6f+G,r,t;k’,z’,O)
= ~(g,(k,z,t)SS(k’,z’,O) + gp(k’, t’, OMk,
z,t))
- (gp(k> 2, t))(gp(k’,
z’, 0)) (6)
In the absence of scattering and under the assumption of a constant electric field E = U/L inside the system, where U is the voltage applied at the contacts, the Wigner correlation function satisfies the equation of motion [$
+ 5;
+ ~~]af.s(k,*,t;k’,=f,O)
= 0
(7) Fig.
1:
with the general solution 6fmp(k,t,t;k’,z’,0)
=
S~~e(k-~,I-~+~,O;k’,=l,O)
Trajectories in the phase space of the carriers at the quasi Fermi levels for different values of the applied voltage. Solid line: eU/EF = 0, dashed line: eU/EF = 0.5, dotted line: eU/&F = 1.01, dash-dotted line: eU/Ep = 2.0.
(8)
depending on the initial value of 6fclp. In order to obtain this initial value, the properties of the contacts must be specified. We assume that, by arriving at a contact a carrier is thermalized immediately, thus losing memory of its previous state. Therefore, no correlations can exist between the system and the contacts. Furthermore, we assume that the contacts always remain at thermal equilibrium. Hence, the carriers emitted by a contact are described by a Fermi-Dirac distribution. The correlation function is then given by
PI: +LI2 cr(t)
=
J
;(-$)‘C
u
-L/2
g(+$-*-$$,
d.z
J
dk
~(_4_p_*)
t/r Fig.
where fLl(k,z) is the Fermi-Dirac distribution for the subband a with the quasi Fermi level determined by the emitting contact. 3.
Results
for a Single-Subband
System
Equation (8) for the Wigner correlation function can easily be interpreted by looking at the classical trajectories of the carriers in the k-z phase space of the system. Due to the assumption of ideal contacts, there can be correlations only for times shorter than the transit time of the carriers. In the classical limit where fp[l - fol] x f=, all carriers contribute to the noise. In the degenerate limit, however, due to the fac-
2:
1
Normalized correlation functions of current fluctuations in the degenerate limit for difThe normalization ferent applied voltages. constants are Crs = 4e2kBT/hrT and TT = 1;(2EF/m)-‘12.
tor fcl[l -fa] only the carriers close the the quasi Fermi levels of the contacts contribute to the noise. Thus, the correlation function wilI be zero for times longer than the transit time at the Fermi level. In this latter case, the correlation function is effectively given by an integral along the trajectories at the quasi Fermi levels and can be calculated analytically. In Fig. 1 the trajectories are plotted for differint values of the applied voltage and Fig. 2 shows the corresponding current
Superlattices
and Microstructures,
correlation functions voltage (solid lines),
Vol. 7 7, NO. 2, 1992
Cr. In the absence of an applied the trajectories are straight lines
207
This shape is similar to the case of a vacuum tube with a constant velocity emitter, the physical origin, however, is quite different. While in a vacuum tube it is due to the fact that all carriers are emitted with the
same velocity, here it is the Pauli principle which is responsible for the fact that only carriers with a single velocity contribute to the noise. Comparing Eq. (10) with the Nyquist formula .91(O) = 4kBTG, we recover for the static conductance G the fundamental value 2e2 /h. With increasing voltage the spectrum becomes less regular because of the two different, in general incommensurable, transit times. For eU/cF > 1 the negative part of the correlation function results in a maximum of the spectrum at nonzero frequency, a behaviour which is known from other nonequilibrium phenomena, where correlations in the carrier dynamics become important [lo]. When the degree of degeneracy EF/kBT decreases, carriers in an increasing energy range around the quasi Fermi levels contribute to the noise and the correlation function has to be calculated numerically. The qualitative features remain the same, but the abrupt changes in the correlation functions as well as in the spectra become smeared out. In Fig. 4 we report the low-frequency value of the spectral density, normalized to its zero voltage value, as a function of the applied voltage for different degrees of degeneracy. In going from low to high voltages its asymptotic values are always reduced by a factor of two. The reason is that at high voltages only carriers moving in the direction of the field reach the other contact and contribute to the low-frequency noise while at equilibrium both directions give a contribution. It is interesting to compare the spectral density at high voltages with the expected value for shot noise. Indeed, in the classical limit we recover SI = 2eI because all carriers contribute to the current as well as to the noise. In the degenerate limit, however, this value is reduced
Fig. 3: Current spectral densities corresponding to the correlation functions of Fig. 2. The normalization constants are S’ro = 8e’kBTlh and TT = L(2eF/m)-‘J2.
Fig. 4: Low-frequency value of the current spectral density, normalized to the zero voltage value, as a function of the applied voltage for different degrees of degeneracy.
and a symmetry exists between carriers moving in opposite directions. The correlation function has a triangular shape and vanishes at times longer than the transit time at the Fermi level TT = L(2s~/rn)-~/~. If we apply a small voltage (dashed lines), the trajectories become parabolic and the transit times of carriers moving in opposite directions are different. Cr(1) now clearly evidences the two contributions of the trajectories with different transit times. It vanishes at a time later than that in equilibrium because the transit time of the carriers moving against the field now is longer. With increasing voltage the Cr(l) changes continuously until eU reaches the Fermi energy. Above this value the carriers moving against the electric field cannot anymore reach the other contact: As a consequence, their transit time changes abruptly and Cl(l) now exhibits a negative part due to the change in their direction during the fight (dotted lines). If the voltage increases further, their transit time decreases and the negative part becomes smaller (dash-dotted lines). Figure 3 shows the frequency dependence of the current spectral densities corresponding to the correlation functions of Fig. 2. In the absence of an applied voltage, SI is given by 8e’kBT Sr(w) = h
sin(iWrT) iWTT
2
1
00)
208
Superlattices
-_
eU/Vp . . . . .._ __
2:
&/V>
0.50
= 0.75
I
/
/ i
I ---1
I
’ ~ _I
eL,/Vs = 1 .OO
-
1 I ’
I
----~
J !
I
and Microstructures,
Vol. I 1, No. 2, 7992
ever, are different, leading to a more complicated shape of the correlation function. For simplicity, we shall take a constant energy splitting V, between the subbands. Figure 5 shows the low-frequency value of the current spectral density as a function of the position of the Fermi level for different values of the applied voltage for the case (a) V,/kBT = 1000 and (b) V*,lkBT = 10. For a typical splitting energy of 2 meV case (a) would correspond to a temperature T = 0.023 K and (b) to T = 2.3 K. In the absence of voltage the spectral density exhibits plateau values at v8e2kBT/h, where Y takes integer values. This corresponds to the fact that there are always,equal contributions from carriers moving in opposite directions. In the presence of an applied voltage, this symmetry is broken and we find also half integer values of Y. If the voltage coincides with the level splitting, the integer steps are suppressed. As an interesting result we observe that already at a temperature which is one order of magnitude below the level splitting (case (b)), the steps are clearly detectable only in certain ranges of the applied voltage. If the voltage is close to the center between two subbands (dotted line in Fig. 5 (b)), the spectral density exhibits only a weak modulation with respect to the position of the Fermi level.
5. Conclusions
t
0.5
0.
0.5
1.0
1.5
2.0
2.5
:3 0
Fig. 5: Low-frequency
value of the current spectral density for a multi-subband system with equidistant subband splitting as a function of the position of the Fermi level (a) for a strongly and (b) for a weakly degenerate system. The normalization constant is Sro = 8e2kBT/h.
by a factor kBT/cF due to the correlations introduced by the Pauli principle. Again, all carriers contribute to the current, but only those in an energy range I~BT around the Fermi edge give a contribution to the noise. 4. Results
for a Multi-Subband
System
As can be seen from the general formulain Eq. (Q), the correlation function of a multi-subband system is given by the sum over the contributions of the various subbands. Due to the different confinement energies the transit times and the degrees of degeneracy, how-
We have presented a detailed theoretical analysis of the noise in two-terminal quasi one-dimensional structures under ballistic conditions. In the degenerate limit it is found that only carriers at the Fermi energy contribute to the noise. In this case, the correlation function directly reflects the microscopic dynamics showing e.g. the asymmetry in the presence of an applied voltage. At increasing voltages, the lowfrequency spectral density is reduced by a factor of two due to the fact that carriers moving against the field direction cannot anymore reach the opposite contact. Then, in the classical case the well-known shot noise value is recovered, while in the degenerate case the noise is suppressed due to the correlations induced by the Pauli principle. In a multi-subband system, the low-frequency spectral density in the absence of an applied voltage exhibits steps at multiples of 8e2knT/h. Under an applied voltage also steps at half integer multiples of this value appear. This is due to the fact that each subband can either contribute to the current in both directions or only in one direction, depending on the position of the Fermi level in this subband. Acknowledgments - This work has been supported by the finalized project “Materiali e Dispositivi per 1’Elettronica a State Solid0 (MADE%)” of the Consiglio Nazionale delle Ricerche (CNR).
Superlattices
and Microstructures,
Vol. 7 1, No. 2, 1992
References 1. G. B. Lesovik, JETP Letters 49, 592 (1989). 2. M. Biittiker, Physical Review Letters 65, 2901 (1990). 3. L. Reggiani and T. Kuhn, Proc. of the NATO Advanced Study Institute “Granular Nanoelectronits", edited by D. K. Ferry, J. R. Barker, and C. Jacoboni (Plenum, New York, 1991). 4. Y. P. Li, D. C. Tsui, J. J. Heremans, J. .4. Simmons, and G. W. Weimann, Applied Physics Letters 57, 774 (1990).
209 5. A. J. Kil, R. J. J. Zijlstra, M. F. H. Schuurmans, and J. P. Andre, Physical Review B41,5169 (1990). 6. Y. P. Li, A. Zaslavsky, D. C. Tsui, M. Santos, and M. Shayegan, Physical Review B41,8388 (1990). 7. C. Dekker, A. J. Scholten, F. Liefrink, R. Eppenga, H. van Houten, and C. T. Foxon, Physical Review Letters 66, 2148 (1991). 8. B. Pehegrini, Physical Review B34, 5921 (1986). 9. T. Kuhn and L. Reggiani, to be published. 10. T. Kuhn, L. Reggiani, and L. Varani, Physical Review B42, 11133 (1990).
and Microstructures,
CORRELATION
Vol. 11, NO. 2, 1992
FUNCTIONS Tilmann
205
AND
QUANTIZED
NOISE
Kuhn,
Lino Reggiani,
IN MESOSCOPIC
SYSTEMS
and Luca Varani
Dipartimento di Fisica, Universitd di Modena Via Campi 213/A, 41100 Modena, Italy (Received
19 May 1991)
We present a theoretical calculation of the current correlation functions and the associated noise spectra of a quasi one-dimensional system terminated by ideal contacts. The shape of the correlation functions is directly related to the microscopic carrier dynamics. When applying a voltage we find: In the classical limit the low-frequency noise spectrum recovers the well-known shot noise formula at high voltages. In the degenerate limit it is suppressed due to quantum-mechanical correlations. The extension of the theory to a multi-subband system shows that in the presence of an applied voltage the step height of the low-frequency spectral density is reduced to one half of its equilibrium value.
1. Introduction
with
Electronic noise in mesoscopic systems in the absence of dissipation is of physical interest due to its direct relation to fundamental constants. In particular, noise spectra at low-frequencies have been investigated recently in several theoretical [l-3] and experimental (4-71 papers. The aim of this contribution is to present a theoretical analysis of the noise spectrum under equilibrium and nonequilibrium conditions in the full frequency range, or equivalently, to analyze the full time dependence of the current correlation function. As a matter of fact, the shape of the correlation function can be directly related to characteristic times of the carrier transport [3]. Therefore, a study of the full time dependence of fluctuations in mesoscopic systems can provide additional insight into the microscopic dynamics.
We consider a two-terminal device of length L, terminated by ideal contacts. The cross-section is assumed to be so small that the electronic states in the transverse directions are quantized and can be described by a quantum number 0. In the stationary state, the current noise in this system is determined by the correlation function CI(t) of current fluctuations under the condition of a constant voltage applied at the contacts:
0749-6036/92/020205
+ I(t
+ 05 $02.00/O
+m S1(w)= J 2
- (I)2
(1)
dl eiYtCr(t)
(3)
The current operator jzTaTz, t) can be expressed in terms of the Wigner operator gp in subband r~ which, by definition, is given by g&z,l)
2. Theory
Cr(t) = +(O)I(t)
where j&z, t) is the current operator in subband a and brackets indicate averages with respect to the appropriate statistical operator. Equation (2) follows from a quantum generalization of the Ramo-Shockley theorem (81. The current spectral density is then obtained from the Wiener-Khintchine theorem according to
=
-J&
Jdrfeikzt !tj&(%
+ ;,
t)*,,,(z
- fJ)
(4)
. according
to j,?,
=
$Idkkg,(k,z,t).
Here,
Sz,,
(9, ,) denote the field creation (annihilation) operators’in subband a with spin s and m is a scalar effective mass. With these definitions, the current correlation function can be expressed in terms of the “Wigner correlation function” 6fPa as
Cl(t) = (-$)2
c
pkdk’
a,0 __-
0 1992 Academic Press Limited
206
Superlattices
X
Ii
dz dz’k k’6f,p(k, z, t; k’, t’, 0)
(5)
-L/2
and Microstructures,
Vol. 11, No. 2, 1992
contact
with 6f+G,r,t;k’,z’,O)
= ~(g,(k,z,t)SS(k’,z’,O) + gp(k’, t’, OMk,
z,t))
- (gp(k> 2, t))(gp(k’,
z’, 0)) (6)
In the absence of scattering and under the assumption of a constant electric field E = U/L inside the system, where U is the voltage applied at the contacts, the Wigner correlation function satisfies the equation of motion [$
+ 5;
+ ~~]af.s(k,*,t;k’,=f,O)
= 0
(7) Fig.
1:
with the general solution 6fmp(k,t,t;k’,z’,0)
=
S~~e(k-~,I-~+~,O;k’,=l,O)
Trajectories in the phase space of the carriers at the quasi Fermi levels for different values of the applied voltage. Solid line: eU/EF = 0, dashed line: eU/EF = 0.5, dotted line: eU/&F = 1.01, dash-dotted line: eU/Ep = 2.0.
(8)
depending on the initial value of 6fclp. In order to obtain this initial value, the properties of the contacts must be specified. We assume that, by arriving at a contact a carrier is thermalized immediately, thus losing memory of its previous state. Therefore, no correlations can exist between the system and the contacts. Furthermore, we assume that the contacts always remain at thermal equilibrium. Hence, the carriers emitted by a contact are described by a Fermi-Dirac distribution. The correlation function is then given by
PI: +LI2 cr(t)
=
J
;(-$)‘C
u
-L/2
g(+$-*-$$,
d.z
J
dk
~(_4_p_*)
t/r Fig.
where fLl(k,z) is the Fermi-Dirac distribution for the subband a with the quasi Fermi level determined by the emitting contact. 3.
Results
for a Single-Subband
System
Equation (8) for the Wigner correlation function can easily be interpreted by looking at the classical trajectories of the carriers in the k-z phase space of the system. Due to the assumption of ideal contacts, there can be correlations only for times shorter than the transit time of the carriers. In the classical limit where fp[l - fol] x f=, all carriers contribute to the noise. In the degenerate limit, however, due to the fac-
2:
1
Normalized correlation functions of current fluctuations in the degenerate limit for difThe normalization ferent applied voltages. constants are Crs = 4e2kBT/hrT and TT = 1;(2EF/m)-‘12.
tor fcl[l -fa] only the carriers close the the quasi Fermi levels of the contacts contribute to the noise. Thus, the correlation function wilI be zero for times longer than the transit time at the Fermi level. In this latter case, the correlation function is effectively given by an integral along the trajectories at the quasi Fermi levels and can be calculated analytically. In Fig. 1 the trajectories are plotted for differint values of the applied voltage and Fig. 2 shows the corresponding current
Superlattices
and Microstructures,
correlation functions voltage (solid lines),
Vol. 7 7, NO. 2, 1992
Cr. In the absence of an applied the trajectories are straight lines
207
This shape is similar to the case of a vacuum tube with a constant velocity emitter, the physical origin, however, is quite different. While in a vacuum tube it is due to the fact that all carriers are emitted with the
same velocity, here it is the Pauli principle which is responsible for the fact that only carriers with a single velocity contribute to the noise. Comparing Eq. (10) with the Nyquist formula .91(O) = 4kBTG, we recover for the static conductance G the fundamental value 2e2 /h. With increasing voltage the spectrum becomes less regular because of the two different, in general incommensurable, transit times. For eU/cF > 1 the negative part of the correlation function results in a maximum of the spectrum at nonzero frequency, a behaviour which is known from other nonequilibrium phenomena, where correlations in the carrier dynamics become important [lo]. When the degree of degeneracy EF/kBT decreases, carriers in an increasing energy range around the quasi Fermi levels contribute to the noise and the correlation function has to be calculated numerically. The qualitative features remain the same, but the abrupt changes in the correlation functions as well as in the spectra become smeared out. In Fig. 4 we report the low-frequency value of the spectral density, normalized to its zero voltage value, as a function of the applied voltage for different degrees of degeneracy. In going from low to high voltages its asymptotic values are always reduced by a factor of two. The reason is that at high voltages only carriers moving in the direction of the field reach the other contact and contribute to the low-frequency noise while at equilibrium both directions give a contribution. It is interesting to compare the spectral density at high voltages with the expected value for shot noise. Indeed, in the classical limit we recover SI = 2eI because all carriers contribute to the current as well as to the noise. In the degenerate limit, however, this value is reduced
Fig. 3: Current spectral densities corresponding to the correlation functions of Fig. 2. The normalization constants are S’ro = 8e’kBTlh and TT = L(2eF/m)-‘J2.
Fig. 4: Low-frequency value of the current spectral density, normalized to the zero voltage value, as a function of the applied voltage for different degrees of degeneracy.
and a symmetry exists between carriers moving in opposite directions. The correlation function has a triangular shape and vanishes at times longer than the transit time at the Fermi level TT = L(2s~/rn)-~/~. If we apply a small voltage (dashed lines), the trajectories become parabolic and the transit times of carriers moving in opposite directions are different. Cr(1) now clearly evidences the two contributions of the trajectories with different transit times. It vanishes at a time later than that in equilibrium because the transit time of the carriers moving against the field now is longer. With increasing voltage the Cr(l) changes continuously until eU reaches the Fermi energy. Above this value the carriers moving against the electric field cannot anymore reach the other contact: As a consequence, their transit time changes abruptly and Cl(l) now exhibits a negative part due to the change in their direction during the fight (dotted lines). If the voltage increases further, their transit time decreases and the negative part becomes smaller (dash-dotted lines). Figure 3 shows the frequency dependence of the current spectral densities corresponding to the correlation functions of Fig. 2. In the absence of an applied voltage, SI is given by 8e’kBT Sr(w) = h
sin(iWrT) iWTT
2
1
00)
208
Superlattices
-_
eU/Vp . . . . .._ __
2:
&/V>
0.50
= 0.75
I
/
/ i
I ---1
I
’ ~ _I
eL,/Vs = 1 .OO
-
1 I ’
I
----~
J !
I
and Microstructures,
Vol. I 1, No. 2, 7992
ever, are different, leading to a more complicated shape of the correlation function. For simplicity, we shall take a constant energy splitting V, between the subbands. Figure 5 shows the low-frequency value of the current spectral density as a function of the position of the Fermi level for different values of the applied voltage for the case (a) V,/kBT = 1000 and (b) V*,lkBT = 10. For a typical splitting energy of 2 meV case (a) would correspond to a temperature T = 0.023 K and (b) to T = 2.3 K. In the absence of voltage the spectral density exhibits plateau values at v8e2kBT/h, where Y takes integer values. This corresponds to the fact that there are always,equal contributions from carriers moving in opposite directions. In the presence of an applied voltage, this symmetry is broken and we find also half integer values of Y. If the voltage coincides with the level splitting, the integer steps are suppressed. As an interesting result we observe that already at a temperature which is one order of magnitude below the level splitting (case (b)), the steps are clearly detectable only in certain ranges of the applied voltage. If the voltage is close to the center between two subbands (dotted line in Fig. 5 (b)), the spectral density exhibits only a weak modulation with respect to the position of the Fermi level.
5. Conclusions
t
0.5
0.
0.5
1.0
1.5
2.0
2.5
:3 0
Fig. 5: Low-frequency
value of the current spectral density for a multi-subband system with equidistant subband splitting as a function of the position of the Fermi level (a) for a strongly and (b) for a weakly degenerate system. The normalization constant is Sro = 8e2kBT/h.
by a factor kBT/cF due to the correlations introduced by the Pauli principle. Again, all carriers contribute to the current, but only those in an energy range I~BT around the Fermi edge give a contribution to the noise. 4. Results
for a Multi-Subband
System
As can be seen from the general formulain Eq. (Q), the correlation function of a multi-subband system is given by the sum over the contributions of the various subbands. Due to the different confinement energies the transit times and the degrees of degeneracy, how-
We have presented a detailed theoretical analysis of the noise in two-terminal quasi one-dimensional structures under ballistic conditions. In the degenerate limit it is found that only carriers at the Fermi energy contribute to the noise. In this case, the correlation function directly reflects the microscopic dynamics showing e.g. the asymmetry in the presence of an applied voltage. At increasing voltages, the lowfrequency spectral density is reduced by a factor of two due to the fact that carriers moving against the field direction cannot anymore reach the opposite contact. Then, in the classical case the well-known shot noise value is recovered, while in the degenerate case the noise is suppressed due to the correlations induced by the Pauli principle. In a multi-subband system, the low-frequency spectral density in the absence of an applied voltage exhibits steps at multiples of 8e2knT/h. Under an applied voltage also steps at half integer multiples of this value appear. This is due to the fact that each subband can either contribute to the current in both directions or only in one direction, depending on the position of the Fermi level in this subband. Acknowledgments - This work has been supported by the finalized project “Materiali e Dispositivi per 1’Elettronica a State Solid0 (MADE%)” of the Consiglio Nazionale delle Ricerche (CNR).
Superlattices
and Microstructures,
Vol. 7 1, No. 2, 1992
References 1. G. B. Lesovik, JETP Letters 49, 592 (1989). 2. M. Biittiker, Physical Review Letters 65, 2901 (1990). 3. L. Reggiani and T. Kuhn, Proc. of the NATO Advanced Study Institute “Granular Nanoelectronits", edited by D. K. Ferry, J. R. Barker, and C. Jacoboni (Plenum, New York, 1991). 4. Y. P. Li, D. C. Tsui, J. J. Heremans, J. .4. Simmons, and G. W. Weimann, Applied Physics Letters 57, 774 (1990).
209 5. A. J. Kil, R. J. J. Zijlstra, M. F. H. Schuurmans, and J. P. Andre, Physical Review B41,5169 (1990). 6. Y. P. Li, A. Zaslavsky, D. C. Tsui, M. Santos, and M. Shayegan, Physical Review B41,8388 (1990). 7. C. Dekker, A. J. Scholten, F. Liefrink, R. Eppenga, H. van Houten, and C. T. Foxon, Physical Review Letters 66, 2148 (1991). 8. B. Pehegrini, Physical Review B34, 5921 (1986). 9. T. Kuhn and L. Reggiani, to be published. 10. T. Kuhn, L. Reggiani, and L. Varani, Physical Review B42, 11133 (1990).