H.
YAMAMOTO
et al.: Resonant Tunneling
in
a TripleBarrier Structure
571
phys. stat.
sol.
(b)
167,
571
(1991)
Subject classification: 73.20 and 73.40
Department
of Information
Science,
Fukui
University
I)
Resonant Tunneling
in
a
Symmetrical
Rectangular TripleBarrier Structure
BY
H. YAMAMOTO, Y. KANIE,
and
K.
TANIGUCHI
Resonant tunneling
is
studied theoretically in a symmetrical rectangular triplebarrier structure.
Analytical expressions for the transmission coefficient
and
the resonance conditions are derived.
It
is
found that resonant tunneling with transmission
1
and resonant tunneling with transmission
<
I can
occur in the symmetrical triplebarrier structure.
It
is
confirmed
that
transmission <
1
at resonance
may occur when
the
outer barrier width is
less
than
about one half
of the
central barrier width.
Resonanztunneln
in
einer symmetrischen rechtwinkligen
DreifachBarrierenstruktur
wird theoretisch
untersucht. Analytische Ausdrucke fur den
Transmissionskoeffizienten
und die Rcsonanzbedingungen
werden abgeleitet.
Es
wird gefunden, da13
in
einer symmetrischen
DreifachBarriercnstruktur
Resonanz
tunneln mit
Transmissionskoeffizienten
1
und Resonanztunneln
mit
Transmission <
1
auftreten kann.
Es
wird bestatigt, daR bei Resonanz die Transmission <
1
wird,
wenn
die iiuBere Barrierenbreite
kleiner
als
die halbe Breite der Zentrdlbarriere ist.
1.
Introduction
Theoretical and experimental studies of the resonant tunneling in doublebarrier structures
have been continued for the last twenty years [l to
91.
It is well known that resonance
occurs when the incident electron energy almost coincides with the eigenenergy in the
quantum well. In the doublebarrier structure only
a
single well exists and, on the other
hand, in the triplebarrier structure double wells exist which are coupled through a central
thin barrier. It is expected that the triplebarrier structure with coupled quantum wells may
produce another tunneling characteristic which will not be obtained in the doublebarrier
structure. In the previous work the transmission coefficient and the resonance condition
were reported for a symmetrical rectangular triplebarrier structure
[
101.
In this paper, a theoretical investigation of resonant tunneling is carried out in detail for
the symmetrical, rectangular triplebarrier structure. The analytical expressions for the
transmission coefficient and the resonance condition are derived in Section
2.
In Section
3,
the transmission coefficients versus electron energy for the unity resonance are shown in
Fig.
2
and
3.
It is seen in this case that
a
resonance state is split into two sharp spectral
lines and that the transmission peaks are getting apart as the electron energy is increased
or the central barrier width becomes thinner. In Table
1
are shown for comparison the
values of the resonance energy in the symmetrical triplebarrier structure and those for the
eigenenergy in the doublewell structure. The difference between them is very subtle:
It is
confirmed that the resonance may occur when an incident electron energy coincides with
a
resonance energy which is almost equal
to
an eigenenergy in the doublewell structure.
')
391
Bunkyo,
Fukui 910, Japan.
512
H. YAMAMOTO,
Y. KANIE,
and
K. TANIGUCHI
v
0
In Section
4,
the variations of resonance energy with the outer barrier width are examined
and are depicted in Fig.
4
and
5.
It is found that the resonant tunneling with unity
transmission (the unity resonant tunneling) is changed to resonant tunneling with transmis
sion smaller than unity (the underunity resonant tunneling
)
when the outer barrier width
becomes thinner than the width determined by the critical equation (11). The curve
determined by
(1 1)
is shown by the thin solid line in Fig.
4
and
5.
If
the outer barrier width
is thicker than the critical value estimated by (l l ), the transmission coefficient at resonance
is unity and the resonance state consists
of
a
doublet. On the other hand, for the range
of
thinner outer barrier than the critical value the resonance state consists of a singlet and
the value of the transmission coefficient is always smaller than unity even at resonance and
is decreased with thinner outer barrier width. It is confirmed that the underunity resonant
tunneling may occur when the outer barrier width is less than about one half of the central
barrier width.
The results are described in Section
5.
It is believed that the derived expressions,
( 5)
and
(6), for
the transmission coefficient and the resonance condition are useful for evaluating
the energy variations of the transmission coefficient and estimating the resonance energy
in the symmetrical, rectangular triplebarrier structures and that (1
1)
may be useful for
distinguishing the unity resonance from the underunity resonance.
region
1 2 3
4
5 6 7

 

   
c
Lb9
a+
LW
je
L b 2
++
LW
je
Lb9
+
Fig. 1. Schematic energy diagram
of
a
symmetrical rectangular triplebarrier
>
structure
2.
Transmission Coefficient and Resonance Condition
In the rectangular triplebarrier structure coupled quantum wells exist whose eigenenergy
is different from that in each well. Therefore, it is expected that the resonance characteristics
in the triplebarrier structure will be different from those in the doublebarrier structure.
In this section expressions for the transmission coefficient and the resonance condition will
be derived by using the transfer matrix method. The schematic energy diagram of the
symmetrical, onedimensional, rectangular triplebarrier structure to be studied is shown
in Fig.
1,
where
L,,
is the outer barrier width,
L,,
the central barrier width,
V,
the barrier
height, and
L,
the well width. The seven regions are specified by the coordinates (x,,
x2,
xg,
...,
x6). It is assumed that an electron with energy
E
is incident from the left and
transmits to the right along the xdirection, and the electron mass is constant (m*) for all
regions
of
the structure.
We may express the wave function in the jth region
(j
=
1,
2,
3,
...,
7)
by
yj(x)
=
Aj
exp (ikjx)
+
Bj
exp
(
ik,x),
where
kj
=
[2m*(E

l 5)]”’/h,
Resonant Tunneling
in
a Symmetrical Rectangular TripleBarrier Structure
573
12
is the reduced Planck constant,
V,
the potential energy of the jth region,
yj
the wave
function in the jth region, and the coefficients Aj and
Bj
are constants to be determined
from the boundary conditions such that the wave functions and their first derivatives are
matched at each discontinuity point xi, where the subscript
i
=
1, 2,
3
...,
6.
Using the transfer matrix, we obtain the following relation:
where
(4)
(kj
+
kj+
1)
exp
[i(

kj
+
kj +
1)
xj] (kj

kj+
,)
exp
[ i (

kj

kj +
xj]
(kj

kj+1)
exp [i(kj
+
kj+1)
Xj
(kj
+
kj +l )
exP
[i (kj

kj+1)
xjl
Rj
=
[1/(2kj)l
In the symmetrical triplebarrier structure considered here we may take
x,

x1
=
xs

xg
=
L,, ,
x4

x3
=
Lb2
>
x3

x2
=
x5

xq
=
L,
,
v,
=
v3
=
v5
=
v,
=
0,
v,
=
v,
=
v,
=
v,
,
kl
=
k3
=
k,
=
k,
=
1~
=
[2m*E]l/2/12,
and
k,
=
k4
=
k,
=
i p
=
i[2m*(V0

E)]"2/12
After a straightforward algebra we can derive the transmission coefficient
T3,
in the
symmetrical triplebarrier structure for
0
<
E
<
V,,
H,,
=
2
[cash
,8Lbl] [COS xL,]

{
(2E

V,)/[E(V,

E)]'l2) [sinh pL,J [sin xL,]
,

{(2E

V,)/[E(V~

E)]"~} [sinh /?&] [sin XL,]
.
(7)
(8)
It is understood that
T3,
becomes unity when W,,
=
0:
W,,
=
0
is the resonance condition
in the symmetrical rectangular triplebarrier structure. The detailed investigation
of
the
transmission coefficient and the resonance condition will be given in the following sections,
3
and 4.
H2,
=
2
[cash
PLb2] [COS xL,]
574
H.
YAMAMOTO,
Y. KANIE,
and K.
TANIGUCHI
Fig.
2.
Transmission coefficient vs. electron energy
in
a triplebarrier structure
(Lb1
=
6 nm, L,,
=
4
nm,
and L,v
=
7
nm)
3.
Unity
Resonant
Tunneling
In the above section the expression for the transmission coefficient
T3s
has been derived.
It is seen from
( 5)
that
T3s
becomes unity when
W,,
=
0:
W,,
=
0
is the resonance condition
and gives the values of the resonance energy. Here let
us
investigate the energy variation
of the transmission coefficient. As a typical example, the electron energy dependence of the
transmission coefficient in the GaAs/AlGaAs symmetrical triplebarrier structure
is
shown
in Fig.
2
and
3,
where
V,
=
0.3228
eV and
m*
=
(0.067
x 0.092)''2m,
=
0.0785m0 (ma
is
0.1
0.2
0.3
ELECTRON ENERGY
(eV)

Fig. 3. Transmission coefficient
vs.
electronenergy in
a
triplebarrier structure
(Lbl
=
6
nm,
L,,
=
5
nm,
and
L,
=
7nm)
Resonant Tunneling in
a
Symmetrical Rectangular TripleBarrier Structure
575
the free electron mass) are employed
[lo,
111:
L,,
=
6
nm,
L,
=
7 nm, and
L,,
=
4 nm
are used in Fig. 2 and
L,,
=
6
nm,
L,
=
7 nm, and
L,,
=
5 nm in Fig.
3.
Here it must be
noted that
L,,/L,,
>
1
(>
0.5).
It
is seen from these figures that the transmission coefficient
at resonance is unity and the resonance state is split into two sharp spectral lines, i.e., the
resonance energies consist of a doublet, and that the transmission peaks are getting apart
as the electron energy is increased or the central barrier width is thinner. In the offresonance
energy range the values of the transmission coefficient become smaller with increasing
central barrier width or total barrier width.
=
L,,
=
L,,
then
(6)
becomes
The expression for the resonance condition has been given by W,,
=
0.
If
w3S
=
[sinh
PLbl [H1SH2S

3
(9)
H,,
=
H,,
=
H,
which leads to the resonance condition in the identical rectangular
triplebarrier structure, i.e.,
H
=
1,
while
H
=
0
is the resonance condition in the
doublebarrier structure
[7,
121.
When
L,,
f
03,
the condition W,,
=
0
is reduced to the equation determining the value
of the eigenenergy in the symmetrical doublewell structure:
2
[cos
xL,]

( ( 2E

Vo)/[E(Vo

E)]”2}
[sin
xL,]
=
k
V,[exp
(&2)]
[sin xL,]/[E(Vo

E)]l”
.
We will compare the values of the resonance energy evaluated from W3,
=
0
with the values
of the eigenenergy evaluated from
(10).
The values of resonance energy in the triplebarrier
and those of the eigenenergy in the symmetrical doublewell structure are shown for the
GaAs/AlGaAs system in Table
1
:
Lb1
=
6
nm,
L,,
=
5 nm, and
L,
=
7 nm are used for
the triplebarrier and
L,,
=
5
nm and
L,
=
7 nm in
(10)
for the doublewell structure.
Only subtle differences are seen between them, though it is recognized that the difference
between them
is
slightly increased with increasing energy. Therefore, it is understood that
when an electron of energy
E
is incident on a triplebarrier structure, the transmission
coefficient becomes unity if
E
is equal to the resonance energy which is almost equal to
the eigenenergy in the doublewell structure.
It is obtained for the resonance with unity transmission (unity resonant tunnel
ing) that a resonance state consists of a doublet and that the transmission peaks are getting
apart as the electron energy is increased or the central barrier width is thinner. The value
of the transmission coefficient becomes smaller with increasing central barrier width or
total barrier width in the offresonance energy range. The unity resonant tunneling may
Ta bl e 1
Values
of
resonance energy and eigenenergy
energy level resonance energy eigenenergy
number
El
1
0.052209 0.052212
El2
0.053384 0.053386
E,,
0.194 578 0.194553
0.203440 0.203404
E 2 2
V,
=
0.3228 eV,
m*
=
0.0785m0, L,
=
7
nm,
Lb,
=
6
nm, and
Lbz
=
5
nm.
evaluated by
W,,
=
0
(eV)
evaluated by (10) (eV)
576
H.
YAMAMOTO,
Y.
KANIE,
and
K.
TANIGUCHI
occur for L,,/L,,
>
0.5, which will be discussed in addition to the case of L,,/Lb,
5
0.5
in the following section.
4.
UnderUnity Resonant Tunneling
In Section
3,
the energy variations of the transmission coefficient have been examined. For
L,,
=
6
nm
(Lbl/Lb,
>
0.5)
the unity resonant tunneling is seen and the resonance state
is composed of a doublet, which is shown in Fig.
2
and
3.
In this section the underunity
resonant tunneling (the resonant tunneling with underunity transmission) will be in
vestigated.
First, the resonance energy values with unity transmission versus outer barrier width are
shown by the solid line for
L,
=
7 nm and 17 nm when L,,
=
5
nm in Fig.
4
and
5,
respectively. It is seen in Fig.
4
that two resonance states with unity transmission coefficient
exists for
a
moderately wide outer barrier (i.e., for L,,
>
2.5
nm) as in Fig.
2
and
3.
However,
for a thinner outer barrier width than about one half
of
the central barrier width (here
<
2.3
nm) no resonance level exists with unity transmission coefficient. In Fig.
5,
five
resonance states are seen for L,
=
17nm and are composed of a doublet with unity
transmission for about L,,
>
2.5 nm, but no resonance level with unity transmission is
seen for
a
considerable thin outer barrier
(Lbl
<
1.9
nm). It is considered from
Fig. 4
and
5
that the transmission coefficient becomes smaller than unity when the outer barrier width
is thinner than the critical one L,, at which the doublet spectrum
is
reduced to
a
singlet
keeping unity transmission. It seems that the critical
L,,
value at each resonance level will
be the boundary between the unity resonance and the underunity resonance. We will
investigate this boundary later.
Secondly, let
us
investigate the energy variation
of
the transmission coefficient for the
underunity resonance. In Fig.
6,
the energy variation
of
the transmission coefficient is
shown for
=
2.3
nm, L,,
=
5
nm, and
L,
=
17 nm. The five resonance states are seen.
The lowest (first) resonance state
El
and the highest (fifth) state
E,
are composed of a

a,
w



&
0.2

U

w
z
W
z
0
U





I
I
I
I
3
4
OUTER
BARRIER
WIDTH
Lbl
(nm)

I
Fig.
4.
Resonance energy level vs. outer barrier width i n
a
triplebarrier structure (Lb2
=
5
nm
and
L,
=
7 nm)
Resonant Tunneling in a Symmetrical Rectangular TripleBarrier Structure
n,
>

t
0.3:

0.2
U
W
z
w
z
0
U
c
1
W
Y
0.1
577


I




Q
"


I
I
I
I
I
I
I

I




I
I
I
I
I
I
I
OUTER BARRIER
WIDTH
Lb!
(nm)

Fig.
5.
Resonance energy level
vs.
outer barrier width in
a
triplebarrier structure
(Lb2
=
5 nm and
L,
=
17nm)
doublet whose peak
is
unity. However, the second state E,, the third state
E,,
and the
fourth one E, are singlets and their peaks are smaller than unity, although the peak of
E,
is
almost
unity.
As
seen in Fig.
5,
it
is
considered that the states of E, and
E,
belong to
the unity resonance condition and the states of
E,,
E,, and E, belong to the underunity
resonance condition. Therefore, it is understood that the unity resonance or the underunity
resonance may occur corresponding to a resonance energy level for
L,,
=
2.3
nm
(Lb1
JLb,
=
0.46
<
0.5).
In Fig.
7
the energy variation of the transmission coefficient is shown for
0.1 0.2
0.3
ELECTRON ENERGY
(eV)

Fig.
6.
Transmission coefficient vs. electron energy in
a
triplebarrier structure
(.&I
=
2.3
nm,
Lhz
=
5nm, and
L,
=
17nm)
H.
YAMAMOTO,
Y. KANIE,
and
K. TANICUCH~
0.1 0.2
0.3
ELECTRON
ENERGY
(eV)

Fig.
7. Transmission coefficient vs. electron energy in a triplebarrier structure
(Lbl
=
2
nm,
LbZ
=
5 nm, and
L,
=
17 nm)
Lbl
=
2
nm,
L,,
=
5
nm, and
L,
=
17
nm. It is seen that
El
is still composed of
a
doublet
and the peak is unity, but that each transmission spectrum for the states
of
E,, E,,
E,,
and
E,
is a singlet and its peak is smaller than unity: It is obtained that
El
becomes a singlet
spectrum for
<
1.9 nm. Moreover, the peak values of
E,, E,,
and
E,
are decreased
compared to those in Fig.
6.
It is confirmed that the peak value
of
the transmission coefficient
decreases with decreasing outer barrier width for the underunity resonance. It is considered
that this phenomenon (the underunity resonance) may occur when the outer barrier width
is thinner and the coupling effect between the two wells becomes weaker than that between
regions
1
and
3
(the lefthandside well) or than that between regions
5
(the righthand
sidewell) and
7:
In other words, the confining effect by the two barriers of regions
2
and
6
is decreased.
Thirdly, as mentioned above, the critical
L,,
value at each resonance level may separate
the unity resonance from the underunity resonance. Theoretically,
it
is impossible to derive
the critical(boundary) equation which passes through all critical
L,,
points for any well
width. However, we can derive an approximate equation for this boundary in the following
manner. It can be seen that the curve composed
of
all critical
L,,
points is almost coincident
with a tangent to the curve of the unity transmission resonance level versus outer barrier
width. The equation for this tangent curve can be derived in the following procedure.
Mathematically, at
a
resonance for
L,,
>
L,,/2 two values of the well width are given due
to two wells. At a certain energy value these two values for the well are reduced to
a
single
value when the outer barrier width is decreased and the unity resonance is changed to
underunity resonance. We regard
W,,
(6)
as
a
function of
L,
and assume that the equation
W,,
=
0
has a unique root for variable
L,.
Thus, the approximate equation for the boundary
is
derived,
Resonant Tunneling
in
a Symmetrical Rectangular TripleBarrier
Structure
579
The curve represented by
(1
1) is shown by the thin solid line (the boundary curve) in Fig.
4
and 5. The curve always passes through the fixed point
( E
=
V0/2, L,,
=
L,,/2) which
corresponds to the maximum L,, value for the underunity resonance and simultaneously
the minimum L,, value for the unity transmission when
E
=
V,/2: This point is shown as
the dot
Q
in Fig. 4 and 5.
It is considered important that (11) passes through the fixed point
Q
which represents
the maximum Lb1 value for the underunity resonance, because the resonant tunneling with
underunity transmission occurs only for L,,
<
L,, 12. Though the boundary equation
cannot be obtained strictly, we have derived the approximate equation
(11)
whose curve
is tangent to the curves
of
unity transmission resonance levels versus outer barrier width,
which will be confirmed to be an excellent approximation to the boundary as follows. The
curve represented by (11) does not always pass through all the critical L,, points for unity
transmission in each resonance state except for point
Q
as stated and seen in Fig. 4 or 5.
The difference between the critical value and that
of
(11) for a given energy value
becomes larger as the value of the resonance energy deviates from the value of V,/2. When
the resonance energy is increased from V0/2 and approaches V,, the maximum value of this
difference is estimated to be about
1.3%
in the GaAs/AlGaAs symmetrical triplebarrier
structure. On the other hand, when the value of the resonance energy is decreased from
V0/2 to zero, the difference is increased at first and then decreased gradually. The maximum
difference is about
0.0003%
in the GaAs/AlGaAs system. Thus, these differences may be
considered to be trivial. Therefore, we may conclude that (1
1)
is the critical equation between
unity resonance and underunity resonance.
Moreover, roughly we may regard Lb1
=
&/2 as another approximate boundary
between unity resonance and underunity resonance: The vertical line through the point
Q
for Lb1
=
Lb2/2
is shown by the broken line in Fig.
4
and 5. This may be easily understood
from the above discussions.
In this section it is concluded that the unity resonant tunneling (the resonance with unity
transmission coefficient) may occur for Lb,/&
>
0.5
(here Lb1
>
2.5
nm for L,,
=
5 nm),
while for Lbl/Lb,
<
0.5 (here Lb1
<
2.5nm) the underunity resonant tunneling (the
resonance with underunity transmission coefficient) may occur.
5.
Conclusions
A
theoretical investigation
of
the resonant tunneling in the symmetrical rectangular
triplebarrier structure has been carried out in detail. The analytical expressions for the
transmission coefficient and the resonance condition have been derived: The transmission
coefficient is given by
( 5)
and the resonance condition by W,,
=
0,
while W,,
is
given by
(6).
It is obtained that the resonant tunneling with unity transmission occurs for
L,,
>
L,,/2.
The energy variations of the transmission coefficient in this case are shown in Fig.
2
and
3.
It is seen that a resonance state is split into two spectral lines and that the transmission
peaks are getting apart with increasing electron energy or with thinner central barrier.
It is confirmed that the transmission coefficient becomes unity when the incident elec
tron energy
E
is equal to the resonance energy which
is
almost equal to the value
of the eigenenergy in the doublewell structure with central barrier and two wells
of the same size as in the symmetrical triplebarrier structure. The values of the
resonance energy and those for the eigenenergy in the doublewell structure are given
in Table 1.
580
H.
YAMAMOTO et al.: Resonant Tunneling
in
a TripleBarrier Structure
It
is obtained that the resonant tunneling with underunity transmission coefficient may
occur when L,, <
L,,/2:
This phenomenon may appear when the outer barrier width
becomes thinner and the coupling effect between the two wells becomes weaker than that
between regions
3
(lefthandside well) and
1
or than that between regions
5
(righthandside
well) and
7.
In Fig.
4
and
5,
energy dependences of the unity resonant transmission on the
outer barrier width are shown. It is found that the underunity resonance may be
distinguished from the unity resonance by the critical curve whose equation is derived
analytically and is given by (11). In Fig. 6 and
7,
the energy variations of the transmission
coefficient for L,,
=
2.3 nm and
2
nm
(<
L,,/2)
are shown. It is recognized for the
underunity resonance that the resonance state is singlet, the transmission coefficient is
always smaller than unity, and the peak value
of
the transmission coefficient becomes
smaller as the outer barrier width is decreased.
As a result, it is concluded that the expressions for the transmission coefficient
( 5)
and
the resonance condition
W,,
=
0
may be useful for evaluating the energy variation of the
transmission coefficient and estimating the resonance energies in the symmetrical, rectangu
lar triplebarrier structure and that
(11)
may be useful for distinguishing unity and
underunity resonances.
Acknowledgement
This work is supported in part by the Murata Science Foundation.
References
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L.
ESAKI, Appl. Phys. Letters 22, 562 (1973).
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L.
G.
SOLLNER, W. D. GOODHUE, P.
E.
TANNENWALD,
C. D.
PARKER, and
D.
D. PECK, Appl.
[3]
B.
Rrcco and
M.
YA. AZBEL, Phys. Rev.
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29,
1970 (1984).
141 L.
L.
CHANC and
K.
PLOOC (Ed.), Molecular Beam Epitaxy and Heterostructures, Martinus
[5]
H.
YAMAMOTO, phys. stat.
sol.
(b) 138, K71 (1986).
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E. MENDEZ and
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