- Email: [email protected]
Theory of resonant tunneling through quantum dots in the presence of electron-phonon interaction
Superlattices
and Microstructures,
THEORY
403
Vol. 10, No. 4, 1991
OF RESONANT PRESENCE
TUNNELING THKOllGH OF ELECTRON-PHONON
QIIANTUM DOTS 1NTERACTIC)N
IN THE
P. Selbmann and 0. Ktin Department of Physics. Humboldt Universltv Invalidenstrasse 110. O- 1040 Berlin. FR( i (Received
Structures
where
the
electrons
15 October
of a two-dimensional
1990)
electron
gas are confined
to discon-
nected regions can be fabricated by the use of appropriate gate geometries. The transpolr between these electrostatically defmed quantum dors takes place by tunneling. lising the tunneling Hamiltonian approach we present a theoretical model of the system including electron-phonon interaction. The relevant coupling constants are determmed from realistic wave functions for the expected confinement potentials. The phonon part of the Hamilton nian is diagonalized using a canonical transformation. Starting from the determination of the transmission matrix for the interacting system we caiculare the current-voltage charac teristics for different temperatures and phonon coupling strengths
1. INTRODUCTION
Resonant tunneling phenomena and their poapplications (fast switches, device tential etc.) have been demonstrated for transistors, structures planar barrier various 111. it has become possible to produce Recently, quasi-one-dimensional (QlD) and quasi-zerodimensional (QOD) electron systems. quantum and dots, by nanolithowires respectively, Transport gra.phg measurements for single quantum dot resonant tunneling structures so far have been reported for two different configurations. Reed et al. 121 use deep etched quantum dots, based on modulation doped GaAs heterostructures, .AIGaAs’InGaAs -multilayer relatively level spacing of with a large about 25 meV. In comparison. electrostatically defined quantum dots in patterned gate structures [3,4] have the advantage of a tunable confinement potential in the plane of the two-dimensional electron gas. But. since is in these devices the dot level spacing typically of t.he order of only some meV their operation is restricted to low temperatures and small voltages. Generally, properties of the tunneling such systems are determlned by their specific takes elertronic structure. The tunneling contacts via place between QlD-states in the localized QOD-levels in the dot: only the is conserved in such a electron total energy process. The obserl,ability of resulting resonance peaks and negative dIPferentia1 conductivitbin the I-V-characteristics - and thus the practical applicability of the devices -
0749-6036/91/080403+04$02.00/0
is mainly controlled by temperature effects the resonant states to and coupling of lattice \,ibrations. In the present paper we use the tunneling Hamiltonian approach (51 to resonant tunneling to develop a sequential theoretical for realistic simple model quantum dot structures which takes these into effects account. Numerical results are presented which correspond to the case of a split gate quantum dot device. In the concluding discussion we give some suggeslions for further work.
2. ELECTRONIC
STATES
An accurate model for any quantum dot structure would require the self-consistent solution of Schroedinger’s equation and Poisson’s equation. Due to the complicated boundar! conditions and the difflcultles of problem in low-dimensional the screening this is an extensive task which is systems, beyond the scope of the present paper inwe use a slmpllfled description which stead, the quanthe electrostatic and decouples tum-mechanical parts of the problem. For the two-dlmenslonal quantization in the z-direction we assume the elec$ric quantum limit: is considered. only the lowest subband, E 0, The electrons of the two-dimensional electron gas (in the x-y plane) are confined by potential self-consistent the superimposed V(x.y). It has been shown by numerical 161 that \‘(x.y) and analytical 171 calculations
0 1991 Academic
Press
Limited
Superlattices
404
can be approximated for not too high carrier so
*2
2
V(x,y)
X
quency
x
x
2
(r=x,y)).
The
eigenfunctions,
E
of
-
2DEG.
n.m=0.1.2
ground
wave
harmonic
is
N
are
easily
found.
(2’)
state
H,(yrr)
function are
F
the
oscillator
of
the
normalized stat,es
subdivided into full Hamiltonlan,
regions is H.
5
0
+ Hra
+ HT
(1)
where
(51
H 0 = HL + Ha + H u is the Hamiltonian the ticles in (wires), and the They are assumed from each other, term. Each contact chemical potential. equals the applied
for non-interacting parcontacts right left and dot region, respectivelyto be completely independent i.e. they commute term by is characterised by its the dif$ereace of which voltage. Er-Er=eU
contains the free phonons PH interaction, electron-phonon which stricted to the dominant coupling lized dot states in our model: H
and
rhe
is
reloca-
to
2 (3)
e a
nnlo’
+ hOy(rn+i)
-rfr’ = S,
Y
(2)
.). The
one-dimensional
where
CURRENT
Schroedinger’s
@0(z)
F,(Y)
= E’ + hw (n+i) x 0
(@0(z)
fre-
wr -characteristic
“In0 for this potential dot we have = F”(x)
H = H
wire
mass,
I nmo(~,y,~)
F,(r)
OF THE TUNNELING
The system is The (see flg.1). written as
dot
*y
2
eigenvalues,
equation For the
“In0
+
2
3. CALCULATION
Vol. IO, No, 4, 1997
(1)
cm*-effectice
E
*2 !JJ (3
least
(at
= *2 mcr 2
and
x
2
by parabolas densities).
and Microstructures,
normalization
y =(m*w poly, and H , is the Hermite r r :h) nomial (r=x,y). In the case of the wire the free motion in the y-dlrection is described by the corresponding plane wave expressions (with the ID wave vector, k) In (2). exp(iky)
(hzk2~2m*), respectively. In the following section we Hamiltonian tunneling method electronic states.
-
(bfr,b
constant,
’ l/2
operayors
operators,
dot
states
c+c
1 i
i.
-
i=n,m.
electron M,,(q)
coupling matrix). This contribution pan be diagonalized exactly by a polaron transe.g. formation (see The remaining 1811. transfer part
and
wlll apply the these using
phonon for
Hr=
c
T
Ir”,rn”’
k,n %“’
a
+
+ T r.,~n.d:ncmn,+ k “Clnn’ c k.n In,“’
h.c
(7)
(a:”(dt,)
-
creation
operator
for
electron
state with wave vector h in subband n in the left (rigth) contact) describes the tunneling between wire and dot states. The transition matrix elements T, for tunneling from region
I to region T
1J
=
!f* 2m
j is ca;iulated
according
/dS
‘4, VY’:)
(‘4: V Y Y J
to
151 (8)
where the integral is taken on a surface in the separating both barrier regions. The application of the transfer matrix method 191, which guarantees continuity of the proyields bability current, with the wave functions given by (2) Flgure I: Idealized energy band model for coupled quantum dot structure under bias direction).
the (Y-
(9)
Superlattices
and Microstructures,
where
-2wax 16k
ray
quires equals
e
”
40.5
Vol. 10, No. 4, 1991
1
Her; v is r=(2m (Us-E),h the
wave
) in
length.
derivative
region The
with
barrier and dot the plane y=y
electron velocity. decay- constant of
1s the
function
normalization the
tJi$,aclassical
II
prime
(IV). on
respect
to
F
LY is
.
I to region
region
;d& oorl(m.n’I.E’I
III this
6(E+%
J2n
(13)
IIII-”
This condition is used occupation number S(E) for thus the total current
a
WI yry.
The in III
(III and IV) standard The current is calculated by perturbation theory to lowest order with respect to HT 181. For the tunneling current from
I-111
redot
4. RESULTS
to the
determine dot levels
the and
AND DISCUSSION
denotes
wave functions are matched between the regions II and
0
For a stationary system contlnulty that the outgolng current from the the Incoming current
yields
-E’Ilnr(E)-E(E’II
calculated numerically the I-V We have characteristics of quantum dot structures with typical parameters for a split gate device. In figure 2 It is shown how different wlre subbands contribute to the full characteristics. Each channel carries current only as long as the voltage dependent position of the resonance levels in the dot is above the corresponding subband edge. The steep descent of the curves is due to the fact that the tinewidth becomes off-resonant. The Fermi energy In the leads determines the heights and the positions of the peaks. In figure 3 we have plotted the I-V curves for two different temperatures. Any structure in the curves is washed out when the temperature becomes comparable to the dot level spacing,
(10) 4
(n
F -
number) tained spectral
Fermi
distribution,
5
-
dot
occupation
in
All the essential physics are conthe spectral densities, SB. The wire density, dy,a~[(k.n).E], is that of
an ID free particle of energy vector k in subband n. The combined effect of both electron-phonon Interaction on represented tegral [ IO]
a oo,l(m.nl,El=
in
form
J
E
wave 10
type
L I I___& I
I
0
0
in-
‘\
I
c
20
tunneling and looT can be
of a convolution
rmn(E)
with
0
/‘\
-.
iQ 30
,’ --------. 5
A
‘!
10
UlmV-
+
,J,‘mV
-.
r4,,nl(m.nl,E’l (111
dE’ (E-E’)‘+P~(E)
where
I,,,(E)
= n ~\T_j’
is the tunneling.
intrinsic den
6(E-Ern,)
is
level the
broadening dot spectral
(12)
due to density
interactlon electron-phonon intludlng only simplest on the Here we concentrate II II case of a single Einstein model of phonons with energy ho=36 meV for GaAs. It should be mentioned, broadening for
that the phonon induced 1s negligible compared to
a11 temperatures
of interest
level I,,(E)
Figure 2: Resonant tunneling current via three lowest dot levels. Contributions of wire subbands n=O (a). n=l (b). n=2 (cl. the total current cd). Quantum numbers of dot levels are m=l (solid), m=2 (dashed). m=3 (dashed-dotted); (wb=180nm,wa=25nm, Us=22meV, wire:
Er=5meV;
1.5meV.
T=3K)
level
spacing
dot
the the and the and
2.5meV.
406
Superlattices
for different Figure 3: Tunneling current temperatures (a) T=2.5K, (b) T=5K (parameters see fig. 2a, the fourth dot level is included here).
and Microstructures,
Vol. IO, No. 4, 199 7
strength, g=O.O5, the polaron shift is large enough to bring the lowest dot level out of resonance; the corresponding feature at low still seen curve disvoltages, in (a), appears. These few examples show that the proposed model can be used to calculate the I-V characteristics for multi-channel quantum dot resonant tunneling structures with electroncoupling at phonon finite temperatures. It can be combined with a selfconsistent deterthe mination of the relative position of energy levels [21 to simulate real devices Further extensions could be made by lnclusion of level mixing effects [12l due to the real confinement potential profile and coupling to dispersive (acoustic) phonons.
REFERENCES
Ill
i
10
121
: 05
[31
I41 Figure 4: Resonant tunneling current via the two lowest dot levels wlth coupling to LO phonons. Coupling strength: (a) g=O.O25. (b) g=O.O5 (wo=170nm, ws=lOnm. Us=53meV, Er=4meV. level
spacing
dot:
4meV.
wire:
lmeV,
I51
T=6K) 161
The effect of the electronas expected. phonon interaction is demonstrated for two coupling strengths in figure 4. different Although the phonon Induced broadening is still small compared to Pmn(E) we observe the occurence
of
phonon
sidebands
’ in
the
levels.
For
the
higher
181 191
the
transmission. The corresponding current Is about one order of magnitude smaller than the contribution. In addition, the zero phonon curves are shifted by a voltage which equals the polaron shift ghw,, (g-coupling strength) of
171
coupling
1101 1111 1121
Y.0 Cho. F.Capasso. K.Mohammed, and IEEE Journal of Quantum Electronics QE-22,1853,(1986). J.N.Randall. J.H.Luscombe. M.A.Reed, R.J.Matyi. W.R.Frensley, R.J.Aggarwal, T.M.Moore, and A.E.Wetsel. Advances in Solid State Physics 29.267,(1989). B.J.van Wees, H.van Houten. C.W.J.Beenakker, J.W.WillIamson,L.P Kouwenhoven. D.van Marel. and C.T.Foxon. Physical Review Letters 60,848,(1988). H.Ahmed.J.E.Frost, C.G.Smith, M.Pepper, D A.Ritchie. D.G.Hasko, D.C.Peacock, and G.A.C.Jones, Superlattices and Microstructures 6,599,(1989). C.B.Duke, Solid State Physics 10,edited by F.Seitz, D.Turnbull, and H.Ehrenreich, Academic Press, London and New York 1969. SurS.E.Laux. D.J.Frank, and F.Stern, face Science 191,101.(1988). J.H.Davles, Semiconductor Science and Technology 3,995,(1988). G.D.Mahan, “4lany Particle Ph~x~c’s’. flenum Press, New York and London, 1981 E.O.Kane in “Tunneling Phenomena in Soljds”. edited by E.Burstein and 1) Lundquist. Plenum Press, New York. 1969 C.O.Almbladh and P.Minnhagen, Physical Review B 17,929,(1978) C.B.Duke and G.D.Mahan. Physical Keview 139,A1965,(1965). C.W.Bryant. Physical Review B 39. 3145. (1989).
and Microstructures,
THEORY
403
Vol. 10, No. 4, 1991
OF RESONANT PRESENCE
TUNNELING THKOllGH OF ELECTRON-PHONON
QIIANTUM DOTS 1NTERACTIC)N
IN THE
P. Selbmann and 0. Ktin Department of Physics. Humboldt Universltv Invalidenstrasse 110. O- 1040 Berlin. FR( i (Received
Structures
where
the
electrons
15 October
of a two-dimensional
1990)
electron
gas are confined
to discon-
nected regions can be fabricated by the use of appropriate gate geometries. The transpolr between these electrostatically defmed quantum dors takes place by tunneling. lising the tunneling Hamiltonian approach we present a theoretical model of the system including electron-phonon interaction. The relevant coupling constants are determmed from realistic wave functions for the expected confinement potentials. The phonon part of the Hamilton nian is diagonalized using a canonical transformation. Starting from the determination of the transmission matrix for the interacting system we caiculare the current-voltage charac teristics for different temperatures and phonon coupling strengths
1. INTRODUCTION
Resonant tunneling phenomena and their poapplications (fast switches, device tential etc.) have been demonstrated for transistors, structures planar barrier various 111. it has become possible to produce Recently, quasi-one-dimensional (QlD) and quasi-zerodimensional (QOD) electron systems. quantum and dots, by nanolithowires respectively, Transport gra.phg measurements for single quantum dot resonant tunneling structures so far have been reported for two different configurations. Reed et al. 121 use deep etched quantum dots, based on modulation doped GaAs heterostructures, .AIGaAs’InGaAs -multilayer relatively level spacing of with a large about 25 meV. In comparison. electrostatically defined quantum dots in patterned gate structures [3,4] have the advantage of a tunable confinement potential in the plane of the two-dimensional electron gas. But. since is in these devices the dot level spacing typically of t.he order of only some meV their operation is restricted to low temperatures and small voltages. Generally, properties of the tunneling such systems are determlned by their specific takes elertronic structure. The tunneling contacts via place between QlD-states in the localized QOD-levels in the dot: only the is conserved in such a electron total energy process. The obserl,ability of resulting resonance peaks and negative dIPferentia1 conductivitbin the I-V-characteristics - and thus the practical applicability of the devices -
0749-6036/91/080403+04$02.00/0
is mainly controlled by temperature effects the resonant states to and coupling of lattice \,ibrations. In the present paper we use the tunneling Hamiltonian approach (51 to resonant tunneling to develop a sequential theoretical for realistic simple model quantum dot structures which takes these into effects account. Numerical results are presented which correspond to the case of a split gate quantum dot device. In the concluding discussion we give some suggeslions for further work.
2. ELECTRONIC
STATES
An accurate model for any quantum dot structure would require the self-consistent solution of Schroedinger’s equation and Poisson’s equation. Due to the complicated boundar! conditions and the difflcultles of problem in low-dimensional the screening this is an extensive task which is systems, beyond the scope of the present paper inwe use a slmpllfled description which stead, the quanthe electrostatic and decouples tum-mechanical parts of the problem. For the two-dlmenslonal quantization in the z-direction we assume the elec$ric quantum limit: is considered. only the lowest subband, E 0, The electrons of the two-dimensional electron gas (in the x-y plane) are confined by potential self-consistent the superimposed V(x.y). It has been shown by numerical 161 that \‘(x.y) and analytical 171 calculations
0 1991 Academic
Press
Limited
Superlattices
404
can be approximated for not too high carrier so
*2
2
V(x,y)
X
quency
x
x
2
(r=x,y)).
The
eigenfunctions,
E
of
-
2DEG.
n.m=0.1.2
ground
wave
harmonic
is
N
are
easily
found.
(2’)
state
H,(yrr)
function are
F
the
oscillator
of
the
normalized stat,es
subdivided into full Hamiltonlan,
regions is H.
5
0
+ Hra
+ HT
(1)
where
(51
H 0 = HL + Ha + H u is the Hamiltonian the ticles in (wires), and the They are assumed from each other, term. Each contact chemical potential. equals the applied
for non-interacting parcontacts right left and dot region, respectivelyto be completely independent i.e. they commute term by is characterised by its the dif$ereace of which voltage. Er-Er=eU
contains the free phonons PH interaction, electron-phonon which stricted to the dominant coupling lized dot states in our model: H
and
rhe
is
reloca-
to
2 (3)
e a
nnlo’
+ hOy(rn+i)
-rfr’ = S,
Y
(2)
.). The
one-dimensional
where
CURRENT
Schroedinger’s
@0(z)
F,(Y)
= E’ + hw (n+i) x 0
(@0(z)
fre-
wr -characteristic
“In0 for this potential dot we have = F”(x)
H = H
wire
mass,
I nmo(~,y,~)
F,(r)
OF THE TUNNELING
The system is The (see flg.1). written as
dot
*y
2
eigenvalues,
equation For the
“In0
+
2
3. CALCULATION
Vol. IO, No, 4, 1997
(1)
cm*-effectice
E
*2 !JJ (3
least
(at
= *2 mcr 2
and
x
2
by parabolas densities).
and Microstructures,
normalization
y =(m*w poly, and H , is the Hermite r r :h) nomial (r=x,y). In the case of the wire the free motion in the y-dlrection is described by the corresponding plane wave expressions (with the ID wave vector, k) In (2). exp(iky)
(hzk2~2m*), respectively. In the following section we Hamiltonian tunneling method electronic states.
-
(bfr,b
constant,
’ l/2
operayors
operators,
dot
states
c+c
1 i
i.
-
i=n,m.
electron M,,(q)
coupling matrix). This contribution pan be diagonalized exactly by a polaron transe.g. formation (see The remaining 1811. transfer part
and
wlll apply the these using
phonon for
Hr=
c
T
Ir”,rn”’
k,n %“’
a
+
+ T r.,~n.d:ncmn,+ k “Clnn’ c k.n In,“’
h.c
(7)
(a:”(dt,)
-
creation
operator
for
electron
state with wave vector h in subband n in the left (rigth) contact) describes the tunneling between wire and dot states. The transition matrix elements T, for tunneling from region
I to region T
1J
=
!f* 2m
j is ca;iulated
according
/dS
‘4, VY’:)
(‘4: V Y Y J
to
151 (8)
where the integral is taken on a surface in the separating both barrier regions. The application of the transfer matrix method 191, which guarantees continuity of the proyields bability current, with the wave functions given by (2) Flgure I: Idealized energy band model for coupled quantum dot structure under bias direction).
the (Y-
(9)
Superlattices
and Microstructures,
where
-2wax 16k
ray
quires equals
e
”
40.5
Vol. 10, No. 4, 1991
1
Her; v is r=(2m (Us-E),h the
wave
) in
length.
derivative
region The
with
barrier and dot the plane y=y
electron velocity. decay- constant of
1s the
function
normalization the
tJi$,aclassical
II
prime
(IV). on
respect
to
F
LY is
.
I to region
region
;d& oorl(m.n’I.E’I
III this
6(E+%
J2n
(13)
IIII-”
This condition is used occupation number S(E) for thus the total current
a
WI yry.
The in III
(III and IV) standard The current is calculated by perturbation theory to lowest order with respect to HT 181. For the tunneling current from
I-111
redot
4. RESULTS
to the
determine dot levels
the and
AND DISCUSSION
denotes
wave functions are matched between the regions II and
0
For a stationary system contlnulty that the outgolng current from the the Incoming current
yields
-E’Ilnr(E)-E(E’II
calculated numerically the I-V We have characteristics of quantum dot structures with typical parameters for a split gate device. In figure 2 It is shown how different wlre subbands contribute to the full characteristics. Each channel carries current only as long as the voltage dependent position of the resonance levels in the dot is above the corresponding subband edge. The steep descent of the curves is due to the fact that the tinewidth becomes off-resonant. The Fermi energy In the leads determines the heights and the positions of the peaks. In figure 3 we have plotted the I-V curves for two different temperatures. Any structure in the curves is washed out when the temperature becomes comparable to the dot level spacing,
(10) 4
(n
F -
number) tained spectral
Fermi
distribution,
5
-
dot
occupation
in
All the essential physics are conthe spectral densities, SB. The wire density, dy,a~[(k.n).E], is that of
an ID free particle of energy vector k in subband n. The combined effect of both electron-phonon Interaction on represented tegral [ IO]
a oo,l(m.nl,El=
in
form
J
E
wave 10
type
L I I___& I
I
0
0
in-
‘\
I
c
20
tunneling and looT can be
of a convolution
rmn(E)
with
0
/‘\
-.
iQ 30
,’ --------. 5
A
‘!
10
UlmV-
+
,J,‘mV
-.
r4,,nl(m.nl,E’l (111
dE’ (E-E’)‘+P~(E)
where
I,,,(E)
= n ~\T_j’
is the tunneling.
intrinsic den
6(E-Ern,)
is
level the
broadening dot spectral
(12)
due to density
interactlon electron-phonon intludlng only simplest on the Here we concentrate II II case of a single Einstein model of phonons with energy ho=36 meV for GaAs. It should be mentioned, broadening for
that the phonon induced 1s negligible compared to
a11 temperatures
of interest
level I,,(E)
Figure 2: Resonant tunneling current via three lowest dot levels. Contributions of wire subbands n=O (a). n=l (b). n=2 (cl. the total current cd). Quantum numbers of dot levels are m=l (solid), m=2 (dashed). m=3 (dashed-dotted); (wb=180nm,wa=25nm, Us=22meV, wire:
Er=5meV;
1.5meV.
T=3K)
level
spacing
dot
the the and the and
2.5meV.
406
Superlattices
for different Figure 3: Tunneling current temperatures (a) T=2.5K, (b) T=5K (parameters see fig. 2a, the fourth dot level is included here).
and Microstructures,
Vol. IO, No. 4, 199 7
strength, g=O.O5, the polaron shift is large enough to bring the lowest dot level out of resonance; the corresponding feature at low still seen curve disvoltages, in (a), appears. These few examples show that the proposed model can be used to calculate the I-V characteristics for multi-channel quantum dot resonant tunneling structures with electroncoupling at phonon finite temperatures. It can be combined with a selfconsistent deterthe mination of the relative position of energy levels [21 to simulate real devices Further extensions could be made by lnclusion of level mixing effects [12l due to the real confinement potential profile and coupling to dispersive (acoustic) phonons.
REFERENCES
Ill
i
10
121
: 05
[31
I41 Figure 4: Resonant tunneling current via the two lowest dot levels wlth coupling to LO phonons. Coupling strength: (a) g=O.O25. (b) g=O.O5 (wo=170nm, ws=lOnm. Us=53meV, Er=4meV. level
spacing
dot:
4meV.
wire:
lmeV,
I51
T=6K) 161
The effect of the electronas expected. phonon interaction is demonstrated for two coupling strengths in figure 4. different Although the phonon Induced broadening is still small compared to Pmn(E) we observe the occurence
of
phonon
sidebands
’ in
the
levels.
For
the
higher
181 191
the
transmission. The corresponding current Is about one order of magnitude smaller than the contribution. In addition, the zero phonon curves are shifted by a voltage which equals the polaron shift ghw,, (g-coupling strength) of
171
coupling
1101 1111 1121
Y.0 Cho. F.Capasso. K.Mohammed, and IEEE Journal of Quantum Electronics QE-22,1853,(1986). J.N.Randall. J.H.Luscombe. M.A.Reed, R.J.Matyi. W.R.Frensley, R.J.Aggarwal, T.M.Moore, and A.E.Wetsel. Advances in Solid State Physics 29.267,(1989). B.J.van Wees, H.van Houten. C.W.J.Beenakker, J.W.WillIamson,L.P Kouwenhoven. D.van Marel. and C.T.Foxon. Physical Review Letters 60,848,(1988). H.Ahmed.J.E.Frost, C.G.Smith, M.Pepper, D A.Ritchie. D.G.Hasko, D.C.Peacock, and G.A.C.Jones, Superlattices and Microstructures 6,599,(1989). C.B.Duke, Solid State Physics 10,edited by F.Seitz, D.Turnbull, and H.Ehrenreich, Academic Press, London and New York 1969. SurS.E.Laux. D.J.Frank, and F.Stern, face Science 191,101.(1988). J.H.Davles, Semiconductor Science and Technology 3,995,(1988). G.D.Mahan, “4lany Particle Ph~x~c’s’. flenum Press, New York and London, 1981 E.O.Kane in “Tunneling Phenomena in Soljds”. edited by E.Burstein and 1) Lundquist. Plenum Press, New York. 1969 C.O.Almbladh and P.Minnhagen, Physical Review B 17,929,(1978) C.B.Duke and G.D.Mahan. Physical Keview 139,A1965,(1965). C.W.Bryant. Physical Review B 39. 3145. (1989).