Journal of Modern Physics, 2013, 4, 968-973
http://dx.doi.org/10.4236/jmp.2013.47130 Published Online July 2013 (http://www.scirp.org/journal/jmp)
Energy Band Analysis of MQW Structure Based on
Kronig-Penny Model
Yu Zhang1*, Yi Wang2
1School of Physics and Engineering, Sun Yat-sen University, Guangzhou, China
2School of Applied Physics and Materials, Wuyi University, Jiangmen, China
Email: *zykelly18@163.com, yiwangll@yahoo.com.cn
Received May 1, 2013; revised June 2, 2013; accepted June 28, 2013
Copyright © 2013 Yu Zhang, Yi Wang. This is an open access article distributed under the Creative Commons Attribution License,
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
ABSTRACT
The effects of different potential well depths, well widths and barrier widths on energy band of multiple quantum well
(MQW) structures are discussed in detail based on Kronig-Penny model. The results show that if the well and barrier
width stay unchanged, the first and second band gaps increase linearly with the well depth. When the well depth is con-
stant, the first and second band gaps increase exponentially with the barrier width in a wide well. However, in narrow
well one, the second band gap saturates when the barrier width is wide enough. On condition that the well and barrier
have equal width, the first band gap decreases exponentially with well-barrier width while the second gap still shows an
exponential increase with the width. These results are insightful for the design of MQW structure optoelectronic de-
vices.
Keywords: Kronig-Penny Model; MQW Structure; Potential Well; Barrier
1. Introduction
Recent years have seen the rapid development of light-
emitting diodes (LEDs), laser diodes (LDs) and solar
cells. Particularly, LEDs have become high-performance
devices widely used in display and lighting industry [1].
The core of these devices is the MQW structure. So, it is
obvious that researches concerning features of MQW
energy band are the foundation of further development of
LED, LD devices and so on. A major obstacle for GaInN
based LED to further penetrate into the general illumina-
tion market is that their efficiency suffers a substantial
decrease as the injection current increases, which is
called “efficiency droop” [2-7]. Researches show that the
decrease of the barrier height within the MQW region
[1,8], such as applying p-type-doped barriers or a lightly
n-type-doped GaN injection layer just below the InGaN
MQWs on the n side [1,9] and the trapezoidal wells of
MQW structure [10] lead to the reduction of efficiency
droop at high injection levels. The GaInN/GaInN MQW
structure has much lower triangular barriers in the active
region which also contributes to the enhancement of
LED performances [2]. The emission wavelength of In-
GaN-Based MQW structure LD is the shortest one ever
generated by a semiconductor laser diode [11]. InGaN/
GaN MQW can also improve the behavior of solar cells.
[12,13] In this way, a profound study of different MQW
structure is essential. Kronig-Penney model (K-P model)
has been widely used in analyzing the energy band be-
havior of crystals and super lattices [14,15]. Some re-
searchers studied the spectrum, transmission and con-
ductance of electrons in bilayer graphene with K-P
model [16]. Further discussion concerning energy band
behavior by K-P model is of great importance for under-
standing the band features and electron transmission be-
havior of MQW structure. However, there are few re-
ports investigating comprehensively the quantum me-
chanics and energy band features of different MQW
structure. In this article, the energy band behavior of
MQW with different potential well depths, widths and
barrier widths is discussed using Kronig-Penney model.
With the aid of Mathematica, the efficiency of calcula-
tion and accuracy of theoretical analysis are enhanced.
2. Kronig-Penney Model
Kronig-Penney model [17] is a potential field model with
periodic array of rectangular potential wells.
The schematic Kronig-Penney model of the crystal is
shown in Figure 1. Assuming the width of potential well
is a and the barrier width is b, V0 is the depth of quantum
*Corresponding author.
C
opyright © 2013 SciRes. JMP
Y. ZHANG, Y. WANG 969
Figure 1. Kronig-penney model and real film made MQW
structure.
well or the height of quantum barrier, the periodic poten-
tial field can be described as:

0
0,
,
nc a
Vx Vn


( 1)x nc
cxnca
 

cab
(1)
where is the period of potential field.
The Hamiltonian of the system can be given as:

22
2
d
d
2
H
Vx
mx
0

In well region
x
a
ee
iKx iKx
AB

, the wave function is given
by the superposition of two wave functions propagating
toward left and right respectively, that is:
w
(2)
where K is limited by
22
2
K
m00EV

0bx 
ee
In barrier region the wave function can
be expressed as:
1
F
xFx
CD

b
(3)
with F confined by
22
0
FE
m 
2
.
According to Bloch theorem, the wave function 1b
in another barrier region can be expressed by axc

2b
as:

b

21
eikc b
x
cx



(4)
where k is the electron wave vector.
A, B, C, D can be obtained by continuity requirements
of wave functions and their first derivatives at well-bar-
rier boundaries. The determinant constituted by the coef-
ficients of A, B, C, D should be zero.


11 1
ee e
eee
ik ab
iKa iKaFb
ik abik
iKa iKaFb
iK iKF
e
iKiKe F


We can get the following equation by simplifying the
determinant:

22
sinh sin
2
cosh coscos
FK Fb Ka
FK
1
0
ee
ee
ik abFb
a bFb
F
F




(5)
bKa kab
(6)


The left term of Equation (6) can be plotted without
introducing infinite deep potential well approximation.
22
sinh sin
2
cosh cos
FK
KFbKa
FK
Fb Ka
f
(7)
2
0
2
2mV
F
K

cos ka b
where
The value of function on the right of
Equation (6) ranges from 1 to 1 so that K is available
only when the value of
f
K is between 1 to 1. The
graph of
f
K
0.5ab
is shown in Figure 2. AB stands for the
width of the first energy gap and CD is the width of the
second energy gap.
The influence of different well depths, well widths and
barrier widths on energy band of MQW structure is in-
vestigated as following.
3. Energy Band Features of Different MQW
Structure
3.1. The Influence of Different Well Depths on
Energy Band
3.1.1. Cases of Potential Well and Barrier with Equal
Width
On condition that the potential well and barrier have
equal width, MQW structure with equal width of well
and barrier, i.e.
ab
0
V
, and well depth taking the
value of 3, 6, 9, 12 separately are studied, where , ,
and are taken an arbitrary unit. The graphs of
KvsK
-15 -10-5051015
-2
0
2
4
6
8
are shown in Figure 3.
f
As shown in Figure 2, AB is the width of the first en-
ergy gap and CD is the width of the second energy gap.
f(K)/a.u.
K/a.u .
AB
CD
V0=6
Figure 2. Energy band structure of multiple quantum wells
(a = b = 0.6, V0 = 6).
Copyright © 2013 SciRes. JMP
Y. ZHANG, Y. WANG
Copyright © 2013 SciRes. JMP
970
5
-15 -10-50
-1
0
1
2
3
10 15
V0=3
f(K)/a.u.
K/a. u.
-15 -10-5051015
-1.5
0.0
1.5
3.0
4.5
6.0
(a)
(b)
V0=6
f(K)/a.u.
K/a.u.
5
-15 -10-50
-1.5
0.0
1.5
3.0
4.5
6.0
10 15
(c)
V0=9
f(K)/a.u.
K/a.u.
-15 -10-5051015
-3
0
3
6
9
12
15
(d)
V0=12
f(K)/a.u.
K/a.u.
0.3 2.62yx
0.13 0.19yx
0.21 1.64yx
Figure 3. MQW structure with a = b = 0.5. (a), (b), (c) and (d) correspond to well depth 3, 6, 9, 12 respectively.
Table 1. The changes of the first and second energy band
gap values with well depth (a = b = 0.5).
The values of the first and second energy band gap in
the four graphs of Figure 3 can be obtained according to
the calculation, the results of which are given in Table 1. Well depth 3 6 9 12
The first energy
band gap 3.35460 4.58822 5.431486.06226
The second energy
band gap 0.53438 0.96853 1.340051.66706
Via linear fitting, the relationship of well depth and
energy band gaps is obtained, as shown in Figure 4.
The relationship between the first energy band gap and
well depth (Figure 4(a)) can be approximated to a linear
function 1 with linearly dependent coef-
ficient of 0.9887. The relationship between the second
energy band gap and well depth (Figure 4(b)) can also
be approximated by a linear function 2
with linearly dependent coefficient of 0.9979. It can be
concluded that the first and second energy band gaps
increase linearly with the well depth when the well width
equals to the barrier width.
2 4 6 8101214
2.8
3.5
4.2
4.9
5.6
6.3
The first ene rgy g ap /a.u
W ell depth/a.u
(a)
2468101214
0.3
0.6
0.9
1.2
1.5
1.8
3.1.2. Cases of Potential Well and Barrier with
Unequal Width
MQW structure devices with different well width and
barrier width are an ubiquitous situation in practical ap-
plications. Discussing the situation of well width differ-
ing from barrier width can offer theoretical instructions
for LED, solar cell designing and study of new materials,
like graphene. Here, MQW structure with well width a =
0.8, barrier width b = 0.2 is taken into consideration. The
change of the first and second energy band gap values
with well depth is shown in Table 2.
(b)
The second energy gap /a.u
W ell depth/a.u
As shown in Figure 5, the relationship between the
first energy band gap and well depth (Figure 5(a)) can
be approximated by a linear function 1
with linearly dependent coefficient of 0.98991; and the
Figure 4. Linear fitting of well depth and energy band gaps
with equal well and barrier width. (a) and (b) correspond to
the first and second energy band gap respectively.
Y. ZHANG, Y. WANG 971
Table 2. The first and second e nergy band gaps as the func-
tion of well depths (a = 0.8, b = 0.2).
Well depth 3 6 9 12
The first energy
band gap 2.12240 2.90948 3.45674 3.87540
The second energy
band gap 0.32891 0.61104 0.85702 1.07386
2 468
1.6
2.0
2.4
2.8
3.2
3.6
4.0
101214
(a)
The first energy g ap / a.u
W ell depth/a.u
10 12
2 4 6 8
0.2
0.4
0.6
0.8
1.0
1.2 (b)
The second energy gap /a.u
Well depth/a.u
0.10 0.10yx
Figure 5. Linear fitting of well depth and energy band gaps
when well width is unequal with barrier width. (a), (b) cor-
respond to the first and second energy band gap respec-
tively.
second energy band gap as the function of well depth
(Figure 5(b)) can also be given as a linear function
2 with linearly dependent coefficient of
0.99827. The analyses and calculation reveal that the first
and second energy band gaps increase linearly with the
well depth in the case of unequal width of well and bar-
rier.
According to the theoretical analyses above, we can
see that the first and second band gaps increase linearly
with the depth of quantum well if the well and barrier
widths are kept constant. The result suggests that the
deep potential well is equivalent to the case of narrow
quantum well structure, in which the quantum mechanics
effect is enhanced. For an infinite depth potential well,
the quantum effect of the first level energy is getting
enhanced, thus leads to the widening of the first energy
band gap. Meanwhile, due to the increase of quantum
tunneling effect, the quantum effect of the second level
energy is also getting strengthened, and consequently
leads to the widening of the second energy band gap.
3.2. The Influence of Different Well and Barrier
Widths
3.2.1. The Influence of Barrier Width on Wide Well
MQW
Wide potential well structure is the key feature for many
microelectronic devices, such as solar cells [12,18]. In
order to understand the mechanism of different MQW
structure for a desired electronic device designing, it is
important to investigate the wide well band features. For
a MQW structure with well depth 0
V and well
width a = 0.8, barrier width b varying from 0.2 to 1, the
energy gaps of the first band and second band are shown
in Table 3.
6
The first and second energy band gaps as a function of
barrier width are shown in Figure 6. The relationship
between the first energy band gap and barrier width
(Figure 6(a)) can be fitted to an exponential function
0.32
12.71e 4.35
x
y
 ; while the second energy band
Table 3. The change of the first and second energy band
gaps with barrier width (well depth V0 = 6, well width a =
0.8).
Well width a 0.8 0.8 0.8 0.8 0.8
Barrier width b 0.2 0.4 0.6 0.8 1.0
The first energy
band gap 2.909483.59544 3.93764 4.129444.24182
The second energy
band gap 0.611040.96156 1.14882 1.245431.29350
0.20.40.60.81.0
2.8
3.2
3.6
4.0
4.4 (a)
The first energ y gap /a.u
Barrier width/a.u
0.2 0.4 0.6 0.8 1.0
0.4
0.6
0.8
1.0
1.2
1.4
(b)
The second energy gap /a.u
Barrier wi d th/a .u
energy band gap respectively.
Figure 6. Energy gaps VS barrier width in a wide well
MQW structure. (a), (b) correspond to the first and second
Copyright © 2013 SciRes. JMP
Y. ZHANG, Y. WANG
972
gap as a function of barrier width (Figure 6(b)) can be 0.19
13.46e 6.04y ; The relationship of the second
energy band gap and b
given as 0.31
41e 1.35. It can be concluded that
nd second energy ban
3.2.2. The Influence of Barrier Widths on a Narrow
Narroll MQW is also widely used in many
from
and
se
able 4. The first and second energy band gaps vs barrier
2 0.2 0.2 0.2
21.y
x
the first ad gaps increase exponent-
tially with the barrier width in wide well MQW structure
when the well depth is kept constant.
Well MQW
w potential we
electronic devices, such as LEDs [19]. For a narrow well
MQW with well depth 06V and well width a = 0.2,
the barrier widthbvarying 0.1 to 1, the energy gaps
of first band and second band are shown in Table 4.
Figure 7 demonstrates the relation between first
cond energy band gaps and barrier width in a narrow
well MQW structure. The relationship between the first
energy band gap and barrier width (Figure 7(a)) can be
approximated by an exponential function
T
width (V0 = 6, a = 0.2).
Well width a 0.2 0.
Barrier width b
3.97338 4.849765.30120 5.57940 5.90510
d energy 0.30910 0.464530.53600 0.56535 0.56746
0.1 0.2 0.3 0.4 0.6
The first energy
band gap
The secon
band gap
0.0 0.1 0.2 0.3 0.4
3.5
4.0
4.5
5.0
5.5
6.0
6.5
0.50.60.7
(a)
The first energy gap /a.u
Barrier width/a.u
.4 0.5 0.60.1 0.2 0.3 0
0.24
0.30
0.36
0.42
0.48
0.54
0.60
(b)
The second energy gap /a.u
Barrier width/a.u
Figure 7. Energy band gaps of narrow well MQW vs bar-
rier width. (a), (b) correspond to the first and second energy
band gap respectively .
x
arrier width (Figure 7(b)) can be
described by an exponential function
0.11
20.67e 0.58y. The graphs show that the first
and second energy ban
3.2.3. The Influence of Equal Well-Barrier Width on
For a understanding of band features for
x
d gaps increase exponentially with
the increase of barrier width in narrow well MQW struc-
ture when the well depth is kept constant. However, the
second band gap value becomes saturated when the bar-
rier width b is comparatively large, and it no longer in-
creases with the barrier width.
Energy Band
comprehensive
different MQW structures, the MQW structure with
equal width well-barrier is also taken into consideration.
For a certain well depth 06V, we assume the values of
equal well-barrier width
d vary from 0.2 to 1,
the calculated results are gble 5.
ab
iven in Ta
Figure 8 presents the relationship between the first
able 5. The first and second energy band gaps with equal
0.2 0.4 0.6 0.8 1.0
T
well-barrier wi dth (V0 = 6).
Equal well-barrier
width (d)
The first energy
band gap
The secon
4.84976 4.70040 4.45296 4.12944 3.77426
d energy
band gap 0.46453 0.82983 1.08173 1.24543 1.34076
0.2 0.4 0.6 0.8 1.0
3.6
3.9
4.2
4.5
4.8
5.1 (a)
The first energy gap /a.u
Barrier width/a.u
0.2 0.4 0.6 0.8 1.0
0.3
0.6
0.9
1.2
1.5
(b)
The second energy gap /a.u
Barrier width/a.u
Figure 8. Energy band gaps as the function of equal well-
barrier width. (a), (b) correspond to the first and second
energy band gap respectively.
Copyright © 2013 SciRes. JMP
Y. ZHANG, Y. WANG
Copyright © 2013 SciRes. JMP
973
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and second energy band gaps and the equal well-barrier
width d. The first energy band gap as the functions of
equal well-barrier width (Figure 8(a)) can be described
as 0.18
10.5e 5.51y, and the second relation (Figure
x
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65e 1.55. It
21.y
x
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can be concluded from Figur rgy e 8 that the first ene
band gap decreases exponentially with the well-barrier
width d while the second energy band gap still shows an
exponential increase with the well-barrier width for
aequilate MQW structure when the well depth is kept
constant.
The inc
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diminution of well depth. It leads to weakened quantum
effect and the decrease of the first energy band gap width.
On the other hand, due to the diminution of well depth,
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4. Conclusion
features of MQW with different well
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