Aug 15, 1993 - ... Stevens Institute of Technology, Hoboken,New Jersey 07080. N. J. M. ... Four transverse subbands are includedin the numerical calculations.
PHYSICAL REVIEW B
15 AUGUST 1993-II
VOLUME 48, NUMBER 8
One-dimensional
confinement
effects on miniband superlattice
transport
in a semiconductor
X. L. Lei Shanghai Institute of Metallurgy, Chinese Academy of Sciences, 865 Changning Road, Shanghai, 200050, China and Department of Physics and Engineering Physics, Stevens Institute of Technology, Hoboken, New Jersey 07080
Department
N. J. M. Horing, and H. L. Cui of Physics and Engineering Physics, Stevens Institute of Technology, Hoboken, New Jersey 07080
K. K. Thornber I.ndependence Way, Princeton, (Received 3 December 1992)
NEC Research Institute, Inc
, g
New
Jersey 085/0
We analyze steady-state miniband conduction for a superlattice subject to transverse confinement in one direction. Force and energy balance equations are presented for these asymmetric, nonparabolic band systems, taking account of impurity, acoustic-phonon, and polar-optic phonon scatterings. Four transverse subbands are included in the numerical calculations. Nonlinear drift velocity and electron temperature are examined as functions of the applied electric field at lattice temperatures between 45 and 300 K, for a series of confined superlattices with miniband widths ranging from 50 to 900 K. Negative differential velocity is manifested in all cases investigated. We find that as lateral extension increases, the linear mobility rises, the critical field decreases, and the nonlinear velocity-field curves become steeper. When the lateral extension d varies from 5 to 45 nm, almost all transport properties monotonically approach the corresponding three-dimensional values, except at low temperature (T = 45 K), where the peak drift velocity exhibits moderate oscillations. The velocity-field behavior of the A = 900 K and d = 45-nm system appears to be particularly well suited for use in high-frequency oscillators.
4
I. INTRODUCTION
electric-field-induced progressive localization, ' have not only stimulated systematic experimental and theoretical but also stud. ies of the physics of miniband transport, brought into focus the prospect of superlattice-based microwave oscillators. In the conventional transferred-electron (Gunn) oscillator, the formation, transit, and decay times of the charged carrier packet place fundamental restrictions on the frequency attainable in a given structure. A nonin which resonant device geometry has been proposed the electrons are confined to drift along a thin layer and the image charge of the drifting carrier packet resides on a planar electrode placed parallel to the layer, so that nearly all of the space-charge field is normal to the drift direction, and the above-mentioned restrictions are removed. This suggests the use of a twosuperlatdimensional, rather than a three-dimensional, tice system to make microwave oscillators in which the
carriers are transversely confined in one direction by a potential wall due to the presence of a higher band-gap material. It has been shown that complete lateral confinement (two directions) of the carriers has a tremendous inhuence on longitudinal miniband transport. The investigation of miniband conduction in superlattices subject to one-dimensional (1D) confinement [which we refer to as two-dimensional (2D) superlattices] is not only of great importance in achieving a comprehensive understanding of carrier confinement eKects, but it also ofFers crucial information relating to the potential use of the superlattice system in nonresonant devices. The purpose of this paper is to examine linear and nonlinear miniband transport in a two-dimensional superlatextice using the Lei-Ting balance-equation approach tended to an arbitrary energy band. Numerical calculations are carried out here for several series of GaAs-based 1D-confined superlattices (2D superlattices) with various transverse extensions and diKering miniband widths. Comparing the results for extremely 1D-confined superlattices with those for corresponding three-dimensional (3D) ones, we find that scatterings in confined systems are more effective in inducing frictional forces. This results in a much higher linear resistivity and a significant shift of the critical field E, (at which the drift velocity reaches its peak value) towards higher field strength for 1D-confined superlattices. However, as the transverse extension d increases, E decreases monotonically and is
0163-1829/93/48(8)/5366(8)/$06. 00
5366
Years ago, Esaki and Tsu introduced the concept of the superlattice as a new type of man-made material, with the expectation of negative difFerential conductivity (NDC), in its vertical transport, which may lead to high-frequency devices. The recent experimenof such Bragg-difFraction-induced tal demonstration Esaki-Tsu conduction and the recognition that NDC in semiconductor superlattices is also a manifestation of
48
1993
The American Physical Society
ONE-DIMENSIONAL
48
not very far from that of a 3D superlattice d
in the case of
= 45nm.
II. BALANCE
EQUATIONS FOR 2D
S UP ERLAT TICKS We consider a two-dimensional superlat tice system formed by periodically arranged potential wells and barriers of finite height along the z direction (see Fig. 1 for a sketch of the structure). In the transverse plane, electrons are free to move along the y direction but are confined in a narrow well of width d in the x direction by inGnitely high potential walls. Thus the single-electron state of the system can be described by an x-direction quantum number n, a y-direction wave vector k„, a longitudinal miniband index l, and a longitudinal wave vector k ( — vr/d k, vr/d, where d is the period of the sup erlat tice) . We assume that the energy gap between the lowest and second longitudinal minibands is large enough so that only the lowest miniband need be taken into account. The single-electron state concerned can be described by the wave function
.
(n7r
l (
1
4
T:
dvg
= eE/m*, + A;
dt
dh,
dt
@g
l'
(z),
= 1, 2, 3, .. . and L„ is the sample size in the y direction. The corresponding energy dispersion, within the tight-binding approximation along the z direction, is where n
given by
v„= —
with e(ky)
= kz/2m,
e„= eg e, (k
+ ei(k, )
and
(n
)
—1),
=
—(1 —cosk 2
7r2
2md
d),
—W.
)
f
v(k ) (e„(ky, k ), T, ) ik
as the center-of-mass velocity, i.e. , the average drift velocity of the carriers, and 6, is the average electron energy per carrier. Furthermore,
'
)
m.* e„(ky, k, ) = e„ + e(ky)
', = eEvg
+. A„,
In this [v(k ) = dei(k )/dk, is the velocity function in the z direction], we have
nunky
x exp(ikey)
5367
being the miniband width and m the electron band mass of the background material. Here we have chosen the ground subband bottom as the zero-energy reference level. When a uniform electric Geld E is applied parallel to the superlattice axis, the carriers are accelerated by the Geld and scattered by impurities and by phonons, resulting in an overall drift motion and heating of the carrier system. Such a transport state of the system is described in the balance-equation theory by the center-ofmass (c.m. ) momentum Pg = Npg (N is the total number of carriers) and the relative electron temperature these are determined by effective force and energy balance equations of the form
( (
(2)'~'
..
CONFINEMENT EFFECTS ON MINIBAND.
' f (e„(ky, k, ), T, )
(8)
n)ky)kz
introinverse-effective-mass, duced to describe the response of the miniband electron system to an external Geld. In the above equations is an ensemble-averaged
f (e„(k„,k, ), T ) = (exp[(e„(k„,k
)
—p)/T, ] + 1)
is the Fermi distribution function at the electron temperature p is the chemical potential determined by the condition that the total number of the electrons equals
T,
N
=2
)
f(e„(ky, k, ), T, ),
(1O)
n, ky, k,
:
e„(ky, k, )—e„+ e(k„) + e, (k, —pg) is the relative electron energy of subband n in the x direction. The energy dispersion (4) enables us to write the average drift velocity and the inverse effective mass as vg
= v~ cry' 1
FIG. 1. 2D superlattice (1D-confined superlattice).
sin(p&gd) cry
cos(pgd),
(12)
X. L. LEI, N. J. M. HORING, H. L. CUI, AND K. K. THORNBER
= Ad/2,
with v
nT
=
= Ad
1/M*
—)
Our purpose is to investigate the eKect of transverse confinement of the carriers on their longitudinal miniIn this, the details of the band transport properties. elastic-scattering mechanisms (impurity site distribution or surface roughness, etc. ) and the fine structure of the phonon modes are not critical. To facilitate a comparison of Bloch miniband conduction features among superlattices having diÃering degrees of x extensions in terms of a simple treatment, we assume that impurities are randomly distributed in the background and that the phonons are three-dimensional bulk modes. Thus we can write the impurity- and phonon-induced frictional accelerations A, and A~, and the energy-transfer rate from the electron system to the phonon system W in the following form:
/2, and
cos(k d) f (e (k„, k, ), T, ).
(i4)
n, ky, k
The effective force and energy balance equations (5) and (6) can be written as d
/vg) = eEdnT
«Ev )
I
' dt gory
I
cos(pgd)
+
2
(A,
+ A„),
2W
= eEdc T sin(pgd)—
(i6)
+ q. ) —v(k-)]
l~(q) I'I J-,-(q*) I'Ig(q. ) I' [v(k. m)n, k„)kz )q
))
IM(q,
A)
&(
(k
I
I
J
„(q )
I
Ig(q, )
I
[v(k,
[f (em(kyl
kz) I Te)
le(q, e (ky,
+ q, ) —v(k
f (en (ky
&
kz
+ q. ), Te)]
k, ) —e„(ky, k, + q, ))
I
)]
m, n, ky, k, , q, A
k
k )
(k
)
&
)
[f(em(ky) kz))Te) f(en(ky) kz + qz)yTe)] I.(q, e (k„, k, ) —e„(ky, k, + q. ) ) I'
(e (k„, k, ) —e„(ky, k, + q, ) l
(O~ g 5
ET& and
k, q,
m, n, k„,
&) I'I
IM(q,
J-,-(q*) I'lg(q-) I'~a, ~
A
(ky, k, ), T, ) (q, e (k„, k, )
[f (e fe
/A~p)
ET)
—e„(ky, k, +q. )b
(ky, k, )
f (e„(ky—k + q, ), T, )] ,
—e„(k„,k, +q, ))l'
Here u(q) is the 3D Fourier transform of the impurity potential, n; is the impurity concentration, and M(q, A) is the electron-phonon matrix element in the electron 3D plane-wave representation for phonons of wave vector q in branch A, having frequency Qz p, n(x) = (e —1) is the Bose function, and e(q, w) is the carrier dielectric function In these equations in the random-phase approximation. (q )I is a form factor due to the confinement of the x-direction subband wave functions: I
2
J-,-(q*) = d Notethat
J
(q )
=J
2=2
(q
),
J
(q )
[( -d*)' —(
J
dx exp(iq x) sin
=J
(
frn7r I
x
i I
. /n~ x .
siil
I
I
q)*, and—
4' 2 mnq
d
2
— )' '][( *d-)' —( + )' ']
Finally, Ig(q, )l is a form factor determined by the wave function of the superlattice miniband. tight-binding limit for the envelope function it is simply the form factor of a single quantum d dzlg(z)l exp(iq z), P(z) being the single-well function.
f
(21) In the extreme well: g(q, )
CONFINEMENT EFFECTS ON MINIBAND. . .
ONE-DIMENSIONAL
48
III. FERMI ENERGY
AND
exp
5369
FUNCTION
In terms of the electron sheet density in the y-z plane N2 or the electron line density along the y direction for each superlattice period, Xi —X2d, Eq. (10) for the determination of the Fermi energy can be written as 1
)f
dzp
2T.
(1 —cos z) ~, r
(22)
where m~
0
N21
8 m is the zero-temperature Fermi energy of a one-dimensional E 1 is the function de6ned by
free-electron system with line density Nq and mass m, and
2
1
F
x &dx exp(x —y) + 1'
i(y) = 7r
(24)
The az function defined by Eq. (14) can be written as
dzcoszI'
i ~
At zero temperature, n7
i
we have
—
=1 =
27r
and the zero-temperature
1= —) where
s„b, (p——— — (1 —cosz) l . 2T~ T~ ) g T~
8(x) = 1 for x
dz cos z
Fermi level
dz
0
(1 —cos z) p —s —— 2
8
s&
= p(0)
is determined
(1 —cos z) p(0) —r„——
) 0, and 8(x) =
2 7r 2
&
p, (0)
—s„——(1 —cos z)
T
(s&)
0
(s~)
1
(27)
limit, the Fermi level is given by
1 ) ' 'exp(A/2T, ) ) Ip(A/2T, ) A(sd /T, )
fs~&
1 2
by
In the nondegenerate
0 otherwise.
p = ln — T
(1 —cos z) p —s —— 2
(2g)
with
)
—
4 T. )
~
el~' —'&«- &
Ii (b, /2T, ) Ip(A/2T, )
~
'
(29)
(So)
Ip(x) and Ii(x) being the modified Bessel functions.
IV. LINEAR MOBILITIES The steady-state balance equations, obtained from Eqs. (5) and (6), determine the dc drift velocity and the electron temperature under the inhuence of a constant electric field along the z direction. In the weak electric-Beld limit these equations require T, = T, and the inverse linear mobility 1/p equals the sum of the impurity and phonon contributions 1/V
The impurity-limited
inverse mobility
= 1/S'+1/V&.
1/p, ; can be expressed in the form
X. L. LEI, N. J. M. HORING, H. L. CUI, AND K. K. THORNBER
5370
n'
dk,
eN2d2~~~T
27'
1 p,
,
2"; ) .Iu-, -(q~
q-)I'lg(q.
'
)l'lqwl
m $A
+ q, )d —sink,
x lsin(k
(e"-,- + 1) (1+ e —"-,-)
d]
in which Vm, ~
=
1+
8mT
cos(k, d) —cos(k,
q2
+ q, )d+
"
(1 —cosz)
+
+
(q„, q, ) is an effective two-dimensional impurity potential obtained from the statically screened and formfactor weighted impurity potential lu(q)l in the 3D plane-wave representation and lu
I
lu-, -(q. q. )l' = The inverse linear mobility induced by optic (Bz g simplified
q* 2vr
I
I
(q)I'
e(q, 0)
z
IJ
)I.
(
(34)
I
~ O„dispersionless)
phonons can also be expressed in a similar
form 2m,
vapo
&) '
eN, d2o, ~~T
)
dk 27r
where n'(x)
= dn(x)/dx, qy
SmT
IM
„(q„,q„o)l
1+ mA q2
cos(k, d) —cos(k,
I
„(q„,q, o)f
+ q, )d —sin(k,
d)]
and
+ q )d+
and IM „(q„,q, , o) is an efFective, screened dimensional electron-phonon matrix element,
IM
lg(q, )l lsin(k,
7A
two-
IM(q, o) I'
dq
V. NUMERICAL CALCULATIONS FOR NONLINEAR TRANSPORT To examine the eKects of transverse confinement on longitudinal miniband transport for electrons subject to one-dimensional confinement in the superlattice, we have numerically solved the steady-state limit of balance equations (5) and (6) for a series of GaAs-based 2D superlattices with miniband widths ranging from 50K to 900K, x extensions d = 5, 10, 30, and 45nm, at lattice temperatures T from 45 to 300 K. Depending on as many as four subbands in the x direction are included. Intrasubband and intersubband scatterings due to impurities, acoustic phonons (deformation potential and piezoelectric couplings with electrons), and polaroptic phonons are taken into account in the calculation. Transverse forin factors (q ) are evaluated exactly as given by Eq. (21), while the longitudinal form factor g(q, ) is estimated by assuming the single-well wave function P(z) = d i~2 for 0 z & d, and P(z) = 0 otherwise. A Thomas-Fermi-type static screening is used for charged impurity and polar-optic scatterings. The screening form
4
d,
J
(
2(e —e' )
+
20
+
(1 —cosz)
+
(36)
is not critical to the impurity scattering, since its role is subsumed in a total linear low-temperature mobility p(0), which is subject to experimental determination. This static screening approximation may slightly underestimate the contribution of the polar-optic phonons, but it turns out to have little eKect on the results presented in this paper. All the material parameters used in the calculations are taken to have typical values for GaAs, the same as those used in Ref. 18. Figures 2(a) and 2(b) show our calculated results for the drift velocity vp and the electron temperature T at lattice temperature T = 45K as functions of electric field E for a series of extremely 1D-confined (d superlattices with differing mini5nm) quantum-weil band widths L = 50, 150, 220, 300, and 450K, but having the same period d = 15nm, carrier sheet density %2 — 1.5 x 10ii/cm, and linear mobility p(0) 1.0m2/&s at 4.2K. The normalized drift velocity vg/v„ is plotted as a function of normalized electric field E/E (E, is the critical field at which the drift velocity reaches its peak value v„) in Fig. 2(c) for all the curves plotted in Fig. 2(a), together with that predicted by the Esaki-Tsu and Boltzmann theories,
2E/E
1+ (E/E. )' This figure should be compared with a similar figure for
DIMENSIONA L CONFINEME
48
T EFFECTS 0 MINIBAND
-
~
.
537i
120
dx=sn 10am
40 80
30
~ ~
60
~
y ~ ~ ~
e
45IUII
~ ~ ~ + ~
~
~ ~ ~ ~ ~
~
~
~
20 150K
10
20
4=
SOK I
)
)
I
30
20
0
0
4p
1
6
7
10
8
].1
12
E (kV/cm)
E (kV/™) 4pp
. (b)T
45K
35o —
300
-
3d
220 K
J
15
xiii
250
~
2PP
150 100
0 0
3
30
20
5
6
8
9
1
E (kV/gm)
I
10
)
'4
40
F (kV/& 1.0
(c)
p. 8—
0.8— 'a p. 6
04 0.2
~~ ~~
~ ~~~
—~
-
tI =15 ~m
g —5 ~m
~
0.0::
FT-B
~~~ ~ ii
I
)
I
I
I
I
—~
p2
'
T-45 K 220 K
p. o::
. FT-B
~y~
~~~ ~~
2
E/Ec
I
2
F/E
p l
d = 5nm, bl i y
p4
d =10Ilm 0~ ~
= 1.0m
s
00 ) an curve represe re resents t }1e p mann theories q.
d
tron temp erature
T
3. Drift velocIty 5g a and elecctron t emperature norma as func tlons o the el ecctric-field, as we as t a functi on o f the norma ].jzed c d quan turn-wel l erlattices w and linear mo 2o 3p and 4 resp ectiv
13.5 x10
cm.
tice Ic having the same density N,,
= =4.5x
s P
and wI t}1
5372
X. L. LEI, N. J. M. HORING, H. L. CUI, AND K. K. THORNBER
3D superlattices. Figures 3(a) and 3(b) show the calculated results for T = 45 K as funcvg and T at lattice temperature tions of electric field E for a series of 1D-con6. ned quantum-well superlattices having diKering x extensions, d = 5, 10, 20, 30, and 45nm, and respective 2D densities N2 —1.5, 3.0, 6.0, 9.0, and 13.5 x 10 i/crn2, but having the same superlattice period d = 15 nm, miniband width A = 220K, and linear mobility p, (0) = 1.0m2/Vs at 4.2K. The corresponding calculated curve for a 3Dsuperlattice having the same d, A, and p, (0) and with carrier sheet density N, = 4. 5 x 10 /cm (per sheet) is also shown. In order to clarify the variation of transport properties with variation of d and to provide a comparison between 2D and 3D, the carrier sheet density for each lattice has been chosen such that the 3D carrier density is kept at the same value for all the 2D and 3D superlattices. The normalized curves of vg/v„vs E/E, for all the above systems are plotted in Fig. 3(c). One can see from Fig. 3(a) —3(c) that the critical field E, decreases monotonically and the electron temperature curve changes monotonically with increasing x extension d and in passing from 2D to 3D superlattices. The peak drift velocity v„and the vg/v~ vs E/E, curves, however, oscillate in passing from an extremely confined system to a 3D one. These oscillations are typical subband effects due to the relative positions of the Fermi energy with respect to the subband bottoms. The d values at which maxima or minima in v„appear depend on carrier density and miniband width. These subband eKects are gradually smeared with increasing lattice temperature. In Figs. 4(a) and 4(b) we plot the calculated results for vd and T at lattice temperature T = 300K as functions of electric field E for a series of 1D-con6ned quantum-well superlattices with diÃering x-extensions d = 10, 20, 30, and 45 nm and respective 2D densities N2 —2.0, 4.0, 6.0, and 9.0 x 10i /cm, but having the same values for superlattice period d = 10 nm, miniband width L = 300 K, and linear mobility p(0) = 1.0m2/Vs at 4.2 K. Also shown are the calculated results for a 3D superlattice having the same d, 4, and p(0), and with carrier sheet density N, = 2.0 x 10ii/cm2 (per sheet). The corresponding normalized curves of vg/v„vs E/E, are plotted in Fig. 4(c). At this lattice temperature, subband effects on miniband transport disappear and with an increase of the x extension d from 10 to 45nm, the miniband conduction behavior of 1D-con6ned superlattices goes monotonically towards that of the 3D limit. at lattice temperaWe have also obtained results ture T = 300K for a series of superlattices having a wider miniband width A = 900K, and linear mobility p(0) = 1.0m /Vs at 4.2K. The 1D-confined superlattices have x extensions d = 10, 20, 30, and 45 nrn and 2D density N2 —1.5, 3.0, 4. 5, and 6.75 x 10 /crn, respectively. The 3D superlattice has a carrier sheet density N, = 1.5 x 10ii/cm2. A significant feature is that the wider miniband case exhibits not only higher values of peak drift velocities but also steeper negativedifFerential-velocity (vg vs E and vd/v„vs E/E ) curves. Obviously, such a wide miniband width is desirable for a high-frequency oscillator.
30
(a)
.
".
45nn
3d
25
20
CO
15
10
0
10
0
30
20
40
E (kV/cm)
~ 500— +
400300
200
I
I
I
I
I
I
10
0
I
I
I
I
20
I
30
40
E (kV/cm) (c) 1.0
0.8
0~~
0.4
ET-B
5=300 K 0.2
d=10 nm
0.0 '. 0
E/E, FIG. 4. Drift velocity vz (a) and electron temperature T, (b) as functions of the electric field, as well as the normalized drift velocity vz/v~ (c) as a function of the normalized electric field E/E, for a series of 1D-confined quantum-well superlattices with diKering x extensions. Here T = 300 K, A = 300 K, d = 10 nm, and linear mobility p(0) = 1.0 m /Vs at 4.2 K. Curves correspond to d = 10, 20, 30, and 45 nm, and respective 2D densities N2 ——2.0, 4.0, 6.0, and 9.0 x 10 /cm . Also shown are results for a 3D superlattice having the same d, A, and p, (0), and with a carrier sheet density N, = 2.0 x 10 /cm per period.
ONE-DIMENSIONAL
48
CONFINEMENT EFFECTS ON MINIBAND.
VI. DISCUSSION
..
5373
we investigated. The only subband effect is the moderate oscillation of the peak drift velocity at T = 45K, as shown in Fig. 3(a). With the increase of the lattice temperature to around 200K, this v„oscillation disappears. This stands in contrast to twodimensionally confined superlattices (1D superlattices), where strong oscillatory linear mobility, electron temperature, critical Geld, and peak drift velocity appear at lattice temperatures well beyond 300K. The weakness of subband e8'ects in a 2D superlattice is also related to the gapless electron energy spectrum. The confined superlattice structures discussed here, particularly the system with A = 900K and d = 45 nm, seem quite suitable for the development of useful devices. For example, one could proceed employing the resistivestructure originally proposed gate Geld-eft'ect-transistor which has exhibited NDC by Cooper and Thornber, and microwave (Gunn) oscillations. While these studies had the resistive gate separated from the semiconductor region by a thin ( 100 A) insulating layer, our present considerations would replace the semiconductor by a superlattice having layers perpendicular to the gate, with the bias field produced by the resistive gate (instead of the usual diode-type two-terminal arrangement).
temperature we found
Comparing the linear mobilities and the nonlinear vd E curves of extremely 1D-conGned superlattices with those of corresponding 3D counterparts, we Gnd that scatterings in confined systems are more effective in inducing frictional forces. This gives rise to a much larger linear resistivity and a significant shift of the critical field E towards a higher Geld strength in 1D-confined systems. This is unique to 2D superlattices with an almost dispersionless polar-optic phonon scattering dominant. As in a 3D superlattice, the electron energy spectrum of a 2D superlattice still fills the whole half axis of energy greater than the miniband bottom, and phonon scatterings are always allowed energetically. In 3D systems, energy conservation limits the available phonon wave vectors to a spherical surface in 3D space. In the case of a 2D system, energy conservation imposes a restriction only on phonon wave vectors q„and q to form a circle in 2D space, while q is limited only by the form factor ~g(q ) due to x confinement. For the case of small d this gives a much larger available phase space for q . The situation is different in two-dimensionally conGned systems (1D superlattices), where, due to energy gaps existing in the electron energy spectrum, not all of the phonon scatterings are energetically allowed. This may result in a reduced frictional force, in contrast to 2D sysvs
~
tems. confined superlat tices, neither In one-dimensionally the linear mobility nor the critical field E exhibit any oscillation with varying x extension, even at the lowest
J. Res. Dev. 14, 61 (1970). A. Sibille, 3.F. Palmier, H. Wang, and F. Mollot, Phys. Rev. Lett. 64, 52 (1990). F. Beltram, F. Capasso, D.L. Sivco, A. L. Hutchinson, and S.N. G. Chu, Phys. Rev. Lett. 64, 3167 (1990). H. T. Grahn, K. von Klitzing, K. Ploog, and G.H. Dohler, Phys. Rev. B 4$, 12 094 (1991). R. Tsu and L. Esaki, Phys. Rev. B 48, 5204 (1991). X.L. Lei, N. J. Horing, and H. L. Cui, Phys. Rev. Lett. 66, 3277 (1991). A.A. Ignatov, E.P. Dodin, and V.I. Shaskin, Mod. Phys. Lett. B 5, 1087 (1991). X.L. Lei and I.C. da Cunha Lima, J. Appl. Phys. 71, 5517 L. Esaki and R. Tsu, IBM
(1992). X.L. Lei, N. 3.M. Horing, and H. L. Cui, 3. Phys. Condens. Matter 4, 9375 (1992). A. Sibille (unpublished).
X.L. Lei (unpublished).
ACKNOWLEDC MENT This work was partially supported by the National Science Foundation (Grant No. ECS-9113993).
M. Hadjazi, A. Sibille, P. 3. Palmier, and Mollot, Electron. Lett. 27, 1101 (1992). J.A. Cooper and K.K. Thornber, IEEE Electron. Device Lett. EDL-6, 50 (1985). X.L. Lei and X.F. Wang, J. Appl. Phys. (to be published). X.L. Lei and C.S. Ting, Phys. Rev. B 32, 1112 (1985). X.L. Lei, Phys. Status Solidi B 170, 519 (1992). X.L. Lei, J. Phys. C 18, L593 (1985). X.L. Lei, J.L. Birman, and C.S. Ting, J. Appl. Phys. 58, 2270 (1985). X.L. Lei, N. J.M. Horing, H. L. Cui, and K.K. Thornber, Solid State Commun. 86, 231 (1993). Y. Yin, J.A. Cooper, P. G. Neudeck, M. L. Balzan, and A. E. Geissberger, Appl. Phys. Lett. 54, 1884 (1989). J.A. Cooper, Y. Yin, M. L. Balzan, and A. E. Geissberger, IEEE Electron. Device Lett. EDL-10, 493 (1989). Y. Yin, H. Fu, J.A. Cooper, M. L. Balzan, and A. E. Geissberger, IEEE Electron. Device Lett. EDL-ll, 538 (1990).
