Energy Band Analysis of MQW Structure Based on Kronig-Penny Model

doi:10.4236/jmp.2013.47130

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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: *[email protected], [email protected]

Received May 1, 2013; revised June 2, 2013; accepted June 28, 2013

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.

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:



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:



mx



In well region

a

iKx iKx





, the wave function is given

by the superposition of two wave functions propagating

toward left and right respectively, that is:



(2)

where K is limited by

m00EV





0bx 



In barrier region ，the wave function can

be expressed as:

xFx







(3)

with F confined by

m 

.

According to Bloch theorem, the wave function 1b



in another barrier region can be expressed by axc





as:



eikc b









(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

iKiKe F









We can get the following equation by simplifying the

determinant:



sinh sin

cosh coscos

FK Fb Ka

ik abFb

a bFb











(5)

bKa kab



(6)







The left term of Equation (6) can be plotted without

introducing infinite deep potential well approximation.

sinh sin

cosh cos

KFbKa

Fb Ka







(7)

2mV

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





K is between −1 to 1. The

graph of





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

, 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

are shown in Figure 3.

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 .

V0=6

Figure 2. Energy band structure of multiple quantum wells

(a = b = 0.6, V0 = 6).

Y. ZHANG, Y. WANG

970

-15 -10-50

-1

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.

-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

(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

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

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

able 4. The first and second energy band gaps vs barrier

2 0.2 0.2 0.2

21.y



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

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 .



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



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

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.

Y. ZHANG, Y. WANG

973

[3] M

. H. Kim, M. F. Schubert, Q. Dai, J. K. Kim, E. F.

Schubert, J. Piprek and Y. Park, Applied Physics Letters,

Vol. 91, 2007, Article ID: 183507.

doi:10.1063/1.2800290

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

[4] H. Morkoç, “Handbook of Nitride Semiconductors and

Devices,” Vol. 3, John Wiley & Sons Inc., New York,

2008.

8(b)) can be given as function 0.47

65e 1.55. It

21.y

 [5] M. F. Schubert, et al., Applied Physics Letters, Vol. 91,

2007, Article ID: 231114. doi:10.1063/1.2822442

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

[6] M. F. Schubert, J. Xu, J. K. Kim, E. F. Schubert, M. H.

Kim, S. Yoon, S. M. Lee, C. Sone, T. Sakong and Y. Park,

Applied Physics Letters, Vol. 93, 2008, Article ID: 041102.

doi:10.1063/1.2963029

[7] M. R. Krames, O. B. Shchekin, R. Mueller-Mach, G. O.

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rease of well-barrier width is equivalent to the

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,

the electrons leap across the barrier more easily, thus

leading to the enhancement of correlating effect of elec-

trons between the well and the barrier which further re-

sults in the broadening of the second band gap.

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4. Conclusion

features of MQW with different well

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