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7/24/2019 Calculation of Transmission Tunneling Current Across Arbitrary
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Calculation of transmission tunneling current across arbitrary potential barriers
Yuji Andoand Tomohiro Itoh
Citation: Journal of Applied Physics 61, 1497 (1987); doi: 10.1063/1.338082
View online: http://dx.doi.org/10.1063/1.338082
View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/61/4?ver=pdfcov
Published by theAIP Publishing
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7/24/2019 Calculation of Transmission Tunneling Current Across Arbitrary
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Calculation
of
transmission tunneling current
across
arbitrary potential
barriers
Yuji Ando and Tomohiro itoh
Microelectronics Research Laboratories
NEC
Corporation 4-1-1.
114iyazaki
Miyamae-ku Kawasaki
213,
Japan
(Received 11 September 1986; accepted for publication 5 November 1986)
This pape r presents a simple method for accurately calculating quantum mechanical
transmission probability and current across arbitrary potential barriers by using the multistep
potential approximation. Th is method is applicable to various potential balTiers and wells,
including continuous variations of potential energy and electron effective mass. Various
potential barrier structures and a hot-electron transistor are analyzed to show the feasibility
of
this method.
I
INTRODUCTION
Recently, from the viewpoint of high-speed and new
functional device application,I-3 there has been an increas
ing interest in resonant tunneling in quantum-wen
and
su
periattice structures. The WKB (Wentzel-Kramers-BriI
louin) approximation,
the
conventional method used to
calculate the transmission coefficient across potential bar
riers, fails to explain the resonance phenomena. Further-
more, the WKB method is inaccurate in regions where the
potential profile varies abruptly,4 Abruptly varying poten
tial functions are, however, frequently encountered at the
interface between two different materials in heterojunction
structures.
Another method for calculating the transmiss ion coeffi
cient
is
to solve Schrodinger s equations
through
potential
barriers. Chandra and Eastman
5
calculated the transmission
coefficient for a triangular barr ier by solving SchrOdinger s
equations via the numerical method.
On the other
hand,
Gundlach
6
calculated
the
tunneling
current
for a trapezoidal
barrier
by
connecting the Airy functions, exact solutions for
Schrodinger s equations,
at
two interfaceso
The
same proce
dure has been applied
to
triangular barriers
by
Christodou
lides
et al.
7
Lui and Fukuma
8
showed this calculation to be
applicable to use with arbitrary piecewise linear potential
barriers. These calculations are, however, unsuitable for de
signing quantum-well and superlattice structures, because of
the complicated treatment involved. Furthermore, varia
tions
of
electron effective masses have never been taken into
account in these analyses,
This paper presents a simpler method for accurately cal
culating the t ransmission coefficient
and current
across arbi
trary potential barriers.
In
the present method, we approxi
mate variations of potential energy and electron effective
mass by multistep functions (multistep potential approxi
mation).
The
transmission coefficient is calculated by con
necting
momentum
eigenfunctions.
9
The details of the cal
culation procedures are described in Sec. II. As mentioned
above, various potential barriers, including continuous var
iations of potential and effective mass,
can
be analyzed easily
by using the present method. For example, rectangular and
parabolic potential barriers, fabricated with
GaAs/
AIGaAs
heterostructures, are analyzed in Sec.
III.
Section
IV
de-
scribes
the
analysis for hot-electron transistors
HETs)
an application
to quantum
size effect devices.
II. CALCULATION PROCEDURE
A Transmission probability across arbitrary potentia
barriers
In the present calculation, instead of dealing with co
tinuous variations
of
potential energy, we split the potenti
barrier
up into
segments, in
which
potential energy
can
b
regarded as a constant. In the limit as the divisions becom
finer and finer, a continuous variation will be recovered.
Let
us assume
the
potential barrier
to
be a sequence of
small segments.
An
example, in
the
case where N
=
10,
shown in Fig. 1 where
the
potential
barrier
U x),
the
effe
tive mass m* x) ,
and the
permittivity
E X)
are approxima
ed by
the
multistep functions
U x)
= =
U[(x
j
._ +x j ) /2 ]
,
m* x)
=mj=m*[ x j
.
J
+Xj)/2]
,
x) =j
[ X
i
-
1
+ x
j
) /2J,
for x
j
. t
-
7/24/2019 Calculation of Transmission Tunneling Current Across Arbitrary
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an electron with energy E moving normally on the barrier, is
given by
r (x)
=
Aj
expUkjx)
+
OJ exp( - ikjx)
, 2)
where
k
j
=
J [2mtCE
-10)] If i,
(3)
and
is the
reduced Planck s constant.
From the
continuity
of lh (x) and
1/
mJ')
d1h
I dx)
at
each boundary,Il the
detenniningA
j
and OJ in Eqs. 2)
can
be reduced
to
the multiplication of the following
N
+ 1
2X2) matrices:
(4)
4
=
[
+ SI
)exp[ - i(k
l
+ I -
kl )xI]
2 1
-
S[
)exp[i(k/+
1 +
kJ )x/]
l -S / )exp[
i k
l
+ I +k[)Xd]
1
+S/)exp[i k
l
+
l
- k / ) x
1
]
(5)
and
mt+l
k[
S/= .
(6)
m? k
l
+
1
By setting Ao
=
1 and B N + I
=
0 in Eq. 4) for j
=
N + 1,
we can calculate the transmission amplitude AN+- 1 and the
transmission probability D(E) as foHows:
7)
and
m* k
D(E) _ ~ I 2
- ' k N + II
,
m
N
+
1
0
8)
where
(9)
B
Transmission
current
calculation
The band diagram, used for calculating the transmission
current-voltage
I-V)
characteristics for
the
potential bar
rier, is shown in Fig. 2. The solid line denotes the potential
function for
the
barrier, whereas
the
broken line denotes
the
approximated potential function. As shown in Fig. 2, the
U x,V)
qVa
o
Ef+qVa-qV
t
>
x
qVb
t
FIG. 2. Energy band diagram for the potential barrier un der biasing condi
tions.
1498
J.
Appl. Phys., Vol. 61,
No.4,
15 February 1987
i
total applied voltage
V
is expressed as
V =
Va
+ Vb + V
d
,
where Va Vb and Vd are the voltage drop values in the
accumulation layer, the barrier, and the depletion layer, re
spectively. These values
and the
space charge ns
per
unit
area in the depletion layer, which is equal
to the
net charge in
the accumulation layer, are determined using fonowing
equations
2;
exp qValkn
- q V a l k T - l =q2
n
;/2okTN
D
, 10)
Vb =
lLb[qnJ X)]dX,
Vd =
qn;/2N t - N
D
,
11 )
12)
where N D
is the
donor concentration in
the
semiconductor
at both contacts, Lb is the barrier thickness, q is the elec
tronic charge,
k
is
the
Boltzmann s constant,
and T
is
the
temperature. The Boltzmann distribution and the depletion
layer approximation are assumed for
Egs. 10) and 12),
respectively.
The potential function U(x) in the barrier is determined
by the
superimposition of
the
zero-bias potential
and
poten
tial drop
(x)
=
[qn, l
(x) ] dx. The transmission proba
bility D(E
x
, V) through the barrier region is calculated as
described above. The accumulation and depletion regions
are assumed not to affect the transmission probability. As
suming the dependence of transmission probability only on
longitudinal electron energy Ex for oblique incidence, the
current
density is given by 13
q m ~
J(E
x
)dEx
=
--::::23 D(E ,
Vb)
21Tfi
xL: [ /o(E) -IN+1CE)JdEdE
x
(13a)
where
10
and f N + I
are
the distribution functions in
the
left
contact
and
in
the
right contact, respectively. Assuming
the
Fermi-Dirac distribution, Eq. (13a) can be rewritten as
fol
lows:
q m ~ k
JCE ) =
21T
2
fz3 D(Ex , Vb)
Xln .
1 + exp Ef +
qV; -
Ex ) lkT )
1 + exp E
r
+ qVa - qV - Ex
) lkT
130)
Here, the
Fermi
level in the accumulation region is treated as
to be raised by qVa .
Y. Ando and T Itoh 1498[This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to
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III, EXAMPLES
Various potential barriers fabricated with
GaAsl
Alx
Gal x As structures were analyzed.
The
conduction-band
offset t:.Ec
was taken to
be 60% of
GaAs and Alx
Gal x As
r
band-gap
difference. 14 In
the
fol
lowing, the electron concentration in
GaAs
was taken to be
n
=
1 X 10
18
em
-
3,
equal to N
D
and
the
Fermi energy was
presumed
to
be
0.05
eVat
77 K. The
parameters used are
listed be1ow
I5
:
1::.E
= {O.75X eV)
c O.7Sx + O.69 x - 0.45)2 (eV)
m*lmo =
0.067
+ O.083x
lEo =
13.1 1 -
x) +
10.
Ix
,
for
xO.45 ,
14a)
(14b)
Cl4c)
where
ma and Eo
are
the
free electron mass
and
the vacuum
static dielectric constant, respectively.
Ao Transmission
probability across
quantum barrier
structures
The
transmission probability is calculated in semicon
ductor-insulator-semiconductor SIS) structures with
a
rec
tangular barrier
and
with a parabolic barrier as shown in
Figs.
3(a) and 3(b),
respectively.
Figure 4(a) shows
the
transmission probability
D
for
x =
0.05 eV across
the
rectangular barrier [in Fig. 3
(a)]
with respect
to VI> In
these calculations,
the barrier
is divid
ed into
N
segments with
N
values ranging between 10 and 80.
There
is only a slight difference in solutions for
N;;.40 and
they converge
to the
solid curve
[in
Fig. 4(a)] for
N;;.80.
For
this case,
the Airy
functions can give
the
exact solution,
6
which coincides with the solid curve.
The
oscillatory behav
ior ofD
Vb
) is presumed
to
be due
to
resonance through the
virtual states above
the
barrier.
16
Electron wavelengths
at
the
resonant states are about 80 A for
Vb =
1 V
and
50 A for
a)
Rectangular Barrier
(b)
Parabolic Barrier
AlxGal-xAs
x ~ O - O _ 5 - 0 )
f ~ 7 e v
~ E
GaAs
~ - - -
350A
--0. GaAs
FIG. 3.
Analyzed single-barrier structure
band
diagrams. (a) Rectangular
barrier. (b) Parabolic barrier.
1499
J. Appl. Phys., VoL
61
,
No.4,
15
February 1987
.,...
'"
10-2--- - 0 _ - - -
_____
_ I
a)
Rectangular
Barrier L\j
r
Ex:=:O 050V
/1
j
1
I
1
p J
J ~
~ _ J
N=20
N=40,80
0.5 Hi
1.5
Voltage (v)
-- -
~
b)
Parabolic
Barrier
Vb=O
1 0
I' i
i\
i \ /
. \ \ I.
N=9
N=15
.. \
. \ I . '
, v i.j I.
\ 1
0.0 L
- l N _ = _ 1 _ 9 _ . 3 _ 9 ~ - - _ . _____
L__J
ao
Q4
Q8
Energy eV)
FIG. 4. Transmission probability plot la)
vs
voltage for the rectangul
barrier shown in Fig.
3(aJ.
where
Nvalues
range from
10
to 80,
and
(h)
electron energy calculated for the parabolic
barrier
sh )wn in Fig.
3(b
where N values range from 9 to 39.
Vb
=
2 V whereas the width
of
each segment is about 9 A
for
N
= 40. Thus, in the present method, the exact solutio
can be obtained by choosing a segment
width
sufficient
smaller
than the
electron wavelengths
at
the resonant state
Fignre
4
b) shows transmission probability
D
acros
the
parabolic barrier
[in
Fig. 3
(b)] at thermal
equilibrium
with respect to
E
for
N
values ranging between 9 and 3
The
flat
structure
for the transmission probability profiles
a notable difference from the
structure
for a rectangular ba
rier.
B. V characteristics for
double-barrier structures
The
transmission probability and
the -V
characteristi
are calculated for double-barrier structures with a rectang
lar
well
and
with a parabolic well, as
shown
in Figs. 5
(a) an
5(b),
respectively.
Y.
Ando and
T.
Itoh
149[This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to
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7/24/2019 Calculation of Transmission Tunneling Current Across Arbitrary
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a) Rectangular
WeI
AlAs
AlAs
GaAs
I lo ev
~ - - - - - - - - Ec
GaAs ' '30A
iOOA
S OA GaAs
b) Parabolic Well
AlxGa1-xAs
x=1-0-1)
r
O.956eV
---- --
Ec
GaAs
I 100A
- GaAs
FIG.
5
Analyzed double-barrier structure band diagrams.
a)
Rectangular
well. b) Parabolic well.
Figures
6 a)
and
6 b)
show D Ex) for a rectangular
well [in Fig. 5 a)] with
N
= 32
and
for
a
parabolic well [ in
Fig.
5 b)]
with N
=
41, respectively. The peaks of D E
x
),
En n
=
0,1,2, ... ), separa te at regular intervals for a parabol
ic weIll? as shown in Fig.
6 b),
in contrast to that for a
rectangular well shown in
Fig. 6 a),
These results agree wen
with the concept that boundary state energies are expressed
as En =
n
+
112)00
for the quantum
wen
expressed as
U x) =
1/2)m*w
2
x
2
,4
Calculated J- V characteristics for both structures
at
77
K are shown in Figs.
7 a) and 7 b).
Figures
7 a) and 7 b)
correspond
to
the rectangular well
and the
parabolic wen,
respectively. In these figures, the solid lines denote the
suIts, including the effects of accumulating and depleting,
whereas the broken lines denote the results without these
effects. With the accumulation and depletion t aken into ac
count,
the current
density increases and
the
voltage shifts.
These results show
that
voltage drops at contact layers seri
ously affect J-V characteristics,
and
hence, should be taken
into
account in analyzing and designing quantum size de
vices.
IV. APPLICATION-ANALYSIS OF HETs HOT
ELECTRON TRANSISTORS)
The HET, one
of
the quantum size effect devices utiliz
ing electron tunneling through potentia] barriers, was ana
lyzed
to
show
the
feasibility of this method.
A. AnalysiS procedure for
HETs
Figure 8 is
a
band diagram
of the HET
proposed by
Heiblum. 18 The
J-
V
characteristics
ofHETs
can be analyzed
applying the present calculation to the potential function
U X,VBE,V
CB
shown in Fig. 8), where
VEE
is the voltage
applied between emitter and base, and VeE is the voltage
between the base and collector. The present calculation is the
1500
J.
Appl. Phys., Vol. 61,
No.4,
15
February 1987
10
0
0.0
0.0
(a) Rectangular Well
0.5
1.0
Energy (eV)
(b)
Parabolic Well
0.5
Energy (eV)
1.0
FIG. 6 Transmission probability vs electron energy plot
a)
calculated for
the rectangular well structure shown in Fig. 5
a)
and b) calculated for the
parabolic well struc ture shown ill Fig. 5 b )
extension
of that
used for
MOMOM
Cmetal-oxide-metal
oxide-metal) devices. 18
The transmi.ssion probability DE Ex, VEE across the
emitter barrier and Dc Ex,
V
CB ) across the collector barrier
can be calculated, as described in Sec.
II
A. Current density
JE between the emitter and
base,
was calculated as de
scribed
in
Sec. II B That is,
qm*
1
E E
x
) = ~ D E E x ) [ iE E ) iB E)]dE ,
21117 Ex
15)
Y. Ando and
T.
Itoh
1500[This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to
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6
E
D
--
4
-;J
o-E
(/)
c
OJ
0
-
2
::J
,)
0
0.0
6
E
()