BIBECHANA 18 (1) (2021) 91-99 

91 
 

BIBECHANA 
ISSN 2091-0762 (Print), 2382-5340 (Online) 

Journal homepage: http://nepjol.info/index.php/BIBECHANA 

Publisher: Department of Physics, Mahendra Morang A.M. Campus, TU, Biratnagar, Nepal 

 

 

Study of tunneling  and oscillation in a 𝑮𝒂𝑨𝒔/𝑨𝒍𝟎.𝟑𝑮𝒂𝟎.𝟕𝑨𝒔 

superlattice: using transfer matrix method 
 

 

S. Shrestha1,*, N. Bhusal1, S. Byahut1,  C. K. Sarkar2 
1Central Department of Physics, Tribhuvan University, Kathmandu, Nepal 
2 Dept. of Electronics and Tele-Communication Engineering, Jadavpur University, Kolkata 700 

032, India 
*Email: sanju12np@yahoo.com 
 

 

Article Information: 

Received: January 28, 2020 

Accepted: July 7, 2020 

 

Keywords:  

GaAs/Al0.3Ga0.7As  

Superlattice  

Quantum tunneling  

High frequency oscillation  
 

 

 
 

 

ABSTRACT 

The study of superlattice (SL) system in lower dimensions, which shows 

tunneling and oscillation of particles, is very interesting from the point of view 

of fundamental physics and is important for devices. Here, we present results of 

our calculations related to tunneling/oscillations in a general SL system using a 

Symmetrical Quadruple Well (SQW) potential. This class of problems is 

handled using a transfer matrix (T-matrix), which is obtained by taking the 

solution of Schroedinger equation with appropriate boundary conditions on 

either side of SQW and then repeatedly applying it for the SL system. The 

electron wave functions in the system are found to be either in a symmetric or an 

anti-symmetric state with a very small energy difference between the two, 

leading to oscillations between these states. In this study, probability density and 

period of oscillation of the particle in SQW is calculated. The result is useful for 

high frequency operations in devices using SL. 

DOI: https://doi.org/10.3126/bibechana.v18i1.27450 

This work is licensed under the Creative Commons CC  BY-NC License. https://creativecommons.org/licenses/by-nc/4.0/ 
 

 
1. Introduction 

 

A superlattice (SL) can be modeled by a set of 

quantum wells, spaced sufficiently close, in which 

electrons can tunnel between the wells. The 

potential profile of the SL is analogous to the 

Kronig-Penney model [1,2]. The SL system has a 

variety of applications in devices such as resonant 

tunneling structure and far-infrared detector. If the 

quantum wells are periodically coupled, new 

phenomena may arise because carriers from one 

quantum well may tunnel though an intervening 

layer to the next well. By varying the number of 

wells and controlling the strength of interaction 

between wells there is a possibility of making 

semiconductor devices with tailored band gap. This 

will enable devices sensitive to and operating at 

energies in the range of our interest.  

http://nepjol.info/index.php/BIBECHANA
mailto:sanju12np@yahoo.com
https://doi.org/10.3126/bibechana.v18i1.27450
https://creativecommons.org/licenses/by-nc/4.0/


S. Shrestha et al. / BIBECHANA 18 (1) (2021) 91-99  

92 
 

The idea of semiconductor SL was first proposed 

by Esaki and Tsu in 1970 [3]. Semiconductor SL is 

formed by layered fabrication of two or more 

semiconductors of different band gaps (e.g. 𝐺𝑎𝐴𝑠 

and𝐴𝑙0.3𝐺𝑎0.7𝐴𝑠) in a nanometer scale. It has been 

studied for different applications by several groups 

for their optical and transport properties [4-6]. 

Photoluminescence measurement was performed in 

SL with parabolic bands, such as 𝐺𝑎𝐴𝑠/

𝐴𝑙𝐺𝑎𝐴𝑠[7]. 𝐴𝑙𝐺𝑎𝐴𝑠/𝐺𝑎𝐴𝑠 SL mini-band 

detectors, which operate at very low or zero bias 

voltage, have been discussed and their peak 

spectral response in the range from 4 to 10 pm has 

also been reported [8,9]. Blotch oscillations in 

AlGaAs/GaAs SL in terahertz region using Monte 

Carlo method have also been studied [10], in 

which, the effective mass approximation was used 

in two dimensional model and studied electron 

miniband transport. Renk et al. [11] have done 

experimental study on semiconductor SL oscillator 

in the frequency range of 60GHz.  
 

High frequency devices have been an important 

research area for many years. The application of 

resonant tunneling diode (RTD) in 1980’s was not 

very successful in the terahertz (THz) region due to 

the extrinsic design manner, which relied on 

external circuit elements to induce oscillation. A 

new theory for the origin of intrinsic oscillation due 

to inter sub-band coupling in a multi sub-band 

system in the terahertz frequency region has been 

developed (for example, current oscillation in a 

Double Barrier Quantum Well Structure [DBQWS] 

based oscillators) [12,13]. The relationship between 

oscillation frequency and sub-band structure can be 

used as design criteria for new types of intrinsic 

oscillators. 
 

The tunneling calculations on sawtooth and 

rectangular SL of AlGaAs/GaAs using transfer 

matrix was previously done [14]. Hence, the 

concept of the transfer matrix on SL of 

AlGaAs/GaAs, as done by Forrest et al. [14] is used 

to study tunneling, oscillatory behavior and the 

frequency of oscillation. In the present work a SL 

of 𝐺𝑎𝐴𝑠 and 𝐴𝑙0.3𝐺𝑎0.7𝐴𝑠 is studied. An accurate 

form of solution is offered by transfer matrix 

method [15]. We first solve a Symmetric 

Quadruple Wells (SQW) (with three barriers in 

between) problem exactly and find the matrix 

which associates the wave function in the first well 

with the wave function of the fourth well. This 

matrix is then used to develop the Transfer Matrix 

(T-matrix) which relates the wave function in the 

first well of the SL with the wave function of the 

last well. 
 

In this paper, we derive a theoretical formalism for 

a general SL, utilizing the methods of T-matrix. 

First we find the solution of the Schrödinger 

equation as applied to an SQW with suitable 

boundary conditions and then repeating the 

obtained result for the whole SL system. In the first 

part of Section 2 we develop a general theory of a 

SL and derive T-matrix by finding first the solution 

of the Schroedinger equation as applied to a SQW 

with suitable boundary conditions and then 

repeating the obtained result for the whole SL 

system. 
 

In the second part, we use the T-matrix method to 

find solution in the SQW. A SQW is a four well 

structure bounded by vacuum levels at the either 

ends in the growth directions. Due to spatial 

symmetry the electron wave function in the system 

is either symmetric or anti-symmetric with a small 

energy difference between the levels [16]. 
 

To simplify the problem, time evolution of 

combined symmetric and anti-symmetric lowest 

lying state is taken which gives the frequency of 

oscillation in the SQW. The probability density of 

finding a particle in the SQW is also calculated and 

shown graphically. 
 

2. Theoretical Formulation   
 

2.1 A general superlattice (SL) 
 

Consider a general SL system of (𝑛 + 1) layers 

having 𝑛/2 barriers and (𝑛/2 + 1) wells, Fig. 

1(a). We explore the tunneling of electrons 

through a single barrier to find the transfer 

coefficient. 



S. Shrestha et al. / BIBECHANA 18 (1) (2021) 91-99  

93 
 

Schrodinger equations for an electron with 

energy𝐸 < 𝑉0, (V0 being the height of the barrier) 

and effective masses 𝑚𝑤and 𝑚𝑏in the well and in 

the barrier regions respectively of the SL are: 

for the well region, 

d
2𝜓𝑤(z)

dz2
+

2𝑚𝑤E

ℏ2
𝜓𝑤(z)=0           (1a)  

for the barrier region, 

d
2𝜓𝑏(z)

dz2
−

2𝑚𝑏(𝑉0−𝐸)

ℏ2
𝜓𝑏(z)=0         (1b)  

The corresponding wave vectors are: 

 𝑘𝑤 = √2𝑚𝑤𝐸 ℏ2⁄ , and𝑘𝑏 = √2𝑚𝑏(𝑉0 − 𝐸) ℏ2⁄ .  

The solutions of the equations above are: 

For the well region:    

    𝜓𝑤(z = 𝐴𝑤sin(𝑘𝑤𝑧) +  𝐵𝑤cos(𝑘𝑤𝑧)   (2a) 

For the barrier region:  

  𝜓𝑏(z) = 𝐴𝑏sinh(𝑘𝑏𝑧) +  𝐵𝑏cosh(𝑘𝑏𝑧) (2b) 

The applicable boundary conditions are continuity 

of the wave function and equality of the effective 

mass-weighted slope of the wave functions (Ben 

Daniel Duke condition) at the interface [17]. 

Applying this for the (p-1)th well, pth barrier and 

(p+1)th well we get 

(
𝐴𝑝+1

𝐵𝑝+1
)

= 𝑀𝑤
−1(𝑧)|𝑧=𝑧𝑝

𝑀𝑏(𝑧)|𝑧=𝑧𝑝
𝑀𝑏

−1(𝑧)|
𝑧=𝑧𝑝−1

𝑀𝑤(𝑧)|𝑧=𝑧𝑝−1
(

𝐴𝑝−1

𝐵𝑝−1
) 

   = 𝑇𝑝+1←𝑝−1 (
𝐴𝑝−1

𝐵𝑝−1
)   (3) 

where, 

𝑀𝑤(𝑧)|𝑧=𝑧𝑝
= [

sin(𝑘𝑤𝑧𝑝) cos(𝑘𝑤𝑧𝑝)

(
𝑘𝑤

𝑚𝑤
) cos(𝑘𝑤𝑧𝑝) (−

𝑘𝑤

𝑚𝑤
) sin(𝑘𝑤𝑧𝑝)

]    (4a)

    

𝑀𝑏(𝑧)|𝑧=𝑧𝑝
= [

sinh(𝑘𝑏𝑧𝑝) cosh(𝑘𝑏𝑧𝑝)

(
𝑘𝑏

𝑚𝑏
) cosh(𝑘𝑏𝑧𝑝) (

𝑘𝑏

𝑚𝑏
) sinh(𝑘𝑏𝑧𝑝)

]      (4b) 

and  
 𝑇𝑝+1←𝑝−1 =

𝑀𝑤
−1(𝑧)|𝑧=𝑧𝑝

𝑀𝑏(𝑧)|𝑧=𝑧𝑝
𝑀𝑏

−1(𝑧)|𝑧=𝑧𝑝−1
𝑀𝑤(𝑧)|𝑧=𝑧𝑝−1

           

                (5) 
 

𝑇𝑝+1←𝑝−1 is the transfer matrix, which is a product 

of four matrices and connects the coefficients of the 

wave functions of the (𝑝 + 1)𝑡ℎand (𝑝 − 1)𝑡ℎwell.  

 

The idea of transfer matrix can be extended to 

envelope the whole SL system. Therefore, 

(
𝐴2𝑛+1

𝐵2𝑛+1
) = 𝑇(2𝑛+1)←1 (

𝐴1

𝐵1
)   (6) 

 

2.2 Symmetric quadruple well (SQW) 
 

Consider a SQW, a system formed by four alternate 

wells and three barriers between them using, for 

example, alternating layers of GaAs and 

Al0.3Ga0.7As in such a way that the central barrier 

divides the system into symmetric left and right 

halves as shown in Fig. 1(b). R and L in the Fig. 

1(b) are the extreme right and extreme left high 

potential barriers going into vacuum level. The 

wave functions in R and L regions have a single 

exponentially decreasing term. The width of the 

barrier and the wells are taken to be 𝑙𝑏and 𝑙𝑤 

respectively. The odd Greek numerals as subscript 

on 𝜓 represent, from right, the four wells and even 

numbered subscripts give the three barriers. 
 

Since the potential is symmetric, the solutions of 

the Schroedinger equations 2(a) and 2(b) separate 

into symmetric and anti-symmetric wave functions. 
 

Symmetric solutions are given by  

𝜓𝑅(𝑧) = 𝐷𝑅𝑒−𝑘𝐷𝑧 

𝜓𝑉𝐼𝐼
𝑠 (𝑧) = 𝐴7

𝑠 sin(𝑘𝑤𝑧) + 𝐵7
𝑠 cos(𝑘𝑤𝑧) 

𝜓𝑉𝐼
𝑠 (𝑧) = 𝐴6

𝑠 sinh(𝑘𝑏𝑧) + 𝐵6
𝑠 cosh(𝑘𝑏𝑧)                   (7) 

𝜓𝑉
𝑠 (𝑧) = 𝐴5

𝑠 sin(𝑘𝑤𝑧) + 𝐵5
𝑠 cos(𝑘𝑤𝑧) 

𝜓𝐼𝑉
𝑠 (𝑧) = 𝐶𝑠 cosh(𝑘𝑏𝑧) 

 

Here, 𝑘𝑤and 𝑘𝑏were defined earlier and 𝑘𝐷 =

√2𝑚𝑤(𝑉𝐷 − 𝐸) ℏ2⁄ , where 𝑉𝐷is the vacuum level. 

The symmetric solutions 𝜓𝐼𝐼𝐼
𝑠 (𝑧), 𝜓𝐼𝐼

𝑠 (𝑧),𝜓𝐼
𝑠(𝑧) 

and 𝜓𝐿
𝑠(𝑧)are obtained by changing 𝑧 to – 𝑧 in 

𝜓𝑉
𝑠 (𝑧), 𝜓𝑉𝐼

𝑠 (𝑧), 𝜓𝑉𝐼𝐼
𝑠 (𝑧)and 𝜓𝑅

𝑠 (𝑧) respectively 

and without changing the sign of  𝜓(𝑧).  
 

The anti-symmetric solutions are obtained in 

an analogous manner and are denoted by 

superscript ‘a’. The solution in the central 

barrier is given by a sinh  rather than a cosh. 

Change of sign of 𝑧 is accompanied by 

simultaneous change in the sign of the 

corresponding 𝜓. 



S. Shrestha et al. / BIBECHANA 18 (1) (2021) 91-99  

94 
 

 
 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Fig. 1: (a) Rectangular superlattice of equal barrier 

height Vo. (b) Spatially Symmetric Quadruple Well 

(SQW) with a barrier in the center.  

The boundary conditions are applied at each point 

separating a barrier and a well and also at R and at 

L to obtain expressions for the energy levels 

corresponding to the symmetric and anti-

symmetric states. All the coefficients in different 

regions can be related to the coefficient 𝐶𝑠or𝐶𝑎 in 

the central region (barrier region IV). After some 

tedious algebra we get  
 

tan(𝑘𝑤
𝑠 𝑧4) =

(−
𝐵1

𝑠

𝐴1
𝑠 )[1+(

𝑘𝑤
𝑠

𝑘𝐷
)(

𝑚𝑏
𝑚𝑤

)
𝐴1

𝑠

𝐵1
𝑠 ]

[1−(
𝑘𝑤

𝑠

𝑘𝐷
)(

𝑚𝑏
𝑚𝑤

)
𝐵1

𝑠

𝐴1
𝑠 ]

= (−
𝐵1

𝑠

𝐴1
𝑠) 𝐹 (𝐴1

𝑠 , 𝐵1
𝑠 ,

𝑘𝑤
𝑠

𝑘𝐷
), 

which gives, 𝑘𝑤
𝑠 𝑧4 = 𝑛𝜋 +

tan−1 [(−
𝐵1

𝑠

𝐴1
𝑠 ) 𝐹 (𝐴1

𝑠 , 𝐵1
𝑠 ,

𝑘𝑤
𝑠

𝑘𝐷
)]   (8) 

Since VD is very large in comparison to V0, the 

wave vector ratio 
𝑘𝑤

𝑠

𝑘𝐷
 is small. This leads the 

function 𝐹 (𝐴1
𝑠 , 𝐵1

𝑠,
𝑘𝑤

𝑠

𝑘𝐷
) to converge to 1[14]. The 

final relation is, 

𝑘𝑤
𝑠 (

3

2
𝑙𝑏 +  2 𝑙𝑤) = 𝑛𝜋 + tan−1 (−

𝐵1
𝑠

𝐴1
𝑠) = 𝑛𝜋 + tan−1 (−

𝛽1
𝑠

𝛼1
𝑠)    

(9) 

where   
𝐵𝑛

𝑠

𝐴𝑛
𝑠 =

𝛽𝑛
𝑠

𝛼𝑛
𝑠  

Left hand side relation in (9) is a function of energy 

assuming the barrier parameters are known and the 

right hand side is accessible in terms of energy. 

Thus, by plotting left hand and right hand side 

functions with energy the intersecting points would 

give the symmetric energy eigen values. The 

evaluation of the right hand side function can be 

done using the value of  
𝐵1

𝑠

𝐴1
𝑠  in (8).  

 

Let T be the transfer matrix that connects the wave 

function coefficients of region I to region III. 

(
A3

s

B3
s) = [

T11 T12

T21 T22
] (

A1
s

B1
s) 

On solving and rearranging the terms we arrive at 

the following relation, 

(
𝐵1

𝑠

𝐴1
𝑠 ) =

[𝑇11−𝑇21
𝐴3

𝑠

𝐵3
𝑠 ]

[𝑇22
𝐴3

𝑠

𝐵3
𝑠 −𝑇12]

                                                       (10) 

Furthermore, the evaluation of 𝐶𝑆 can be done by 

normalization condition for the system as, 

|𝐶𝑠|2 {∫ |−𝛼1
𝑠sin(𝑘𝑤𝑧) + 𝛽1

𝑠cos(𝑘𝑤𝑧)|2𝑑𝑧
−𝑧3

−𝑧4
+

∫ |−𝛼2
𝑠sinh(𝑘𝑏𝑧) + 𝛽2

𝑠cosh(𝑘𝑏𝑧)|2𝑑𝑧
−𝑧2

−𝑧3
+

∫ |−𝛼3
𝑠sin(𝑘𝑤𝑧) + 𝛽3

𝑠cos(𝑘𝑤𝑧)|2𝑑𝑧
−𝑧1

−𝑧2
+

∫ |cosh(𝑘𝑏𝑧)|2𝑑𝑧
𝑧1

−𝑧1
+ ∫ |𝛼3

𝑠sin(𝑘𝑤𝑧) +
𝑧2

𝑧1

𝛽3
𝑠cos(𝑘𝑤𝑧)|2𝑑𝑧 + ∫ |𝛼2

𝑠sinh(𝑘𝑏𝑧) +
𝑧3

𝑧2

𝛽2
𝑠cosh(𝑘𝑏𝑧)|2𝑑𝑧 + ∫ |𝛼1

𝑠sin(𝑘𝑤𝑧) +
𝑧4

𝑧3

𝛽1
𝑠cos(𝑘𝑤𝑧)|2𝑑𝑧} = 1                                        (11) 

 

The replication of similar technique for anti-

symmetric states leads to the solution of energy 

value and wave function coefficient 𝐶𝑎for the 

particles in this state. Calculations on concrete 

systems, detailed later, show that consecutive 

symmetric and anti-symmetric energy levels are 

very close, resulting in oscillations between the 

levels.   
 

The total travelling wave solution in the SQW is   

𝜓(𝑧, 𝑡) =
1

√2
[𝜓s exp (−𝑖

2𝜋

ℎ
𝐸𝑠𝑡) + 𝜓a exp (−i

2π

h
Eat)] 

          (12) 

 



S. Shrestha et al. / BIBECHANA 18 (1) (2021) 91-99  

95 
 

where 𝐸𝑠 and 𝐸𝑎 are nearby symmetric and anti-

symmetric energy values. 
 

The probability to find particle at any time 𝑡 is 

given by: 

𝑃(𝑧, 𝑡) = |𝜓(𝑧, 𝑡)|2 

𝑃(𝑧, 𝑡) =
1

2
[𝜓𝑠2

+ 𝜓𝑎2
+ 𝜓s𝜓a (exp (−𝑖

2𝜋

ℎ
(𝐸𝑎 − 𝐸𝑠)𝑡) + exp (𝑖

2𝜋

ℎ
(𝐸𝑎 − 𝐸𝑠)𝑡))] 

𝑃(𝑧, 𝑡) =
1

2
[𝜓𝑠2 + 𝜓𝑎2 + {𝜓s𝜓a (exp (−𝑖

2𝜋

ℎ
∆𝐸𝑡) + exp (𝑖

2𝜋

ℎ
∆𝐸𝑡))}] 

 

Therefore,  

𝑃(𝑧, 𝑡) =
1

2
[𝜓𝑠2

+ 𝜓𝑎2
+ 2𝜓s𝜓acos (

2𝜋

ℎ
∆𝐸𝑡)]

 

This equation gives the probability to find the 

particle at time ‘t’ in the neighborhood of spatial 

region z and can be used to study the time 

evolution of system including its tunneling 

behavior in SQW. 

 

3. Results and Discussion 
 

We consider 𝐺𝑎𝐴𝑠 and 𝐴𝑙𝑥𝐺𝑎(1−𝑥)𝐴𝑠  SL system 

with SQW potential in which the width of the well 

and barrierare 4 nm and 5 nm respectively. The 

height of barrier depends on the concentration of 𝐴𝑙 

in 𝐴𝑙𝑥𝐺𝑎(1−𝑥)𝐴𝑠 and is given by 0.067 + 0.083𝑥. 

With 𝑥 = 0.3,the height of barrier is taken to be 

228 𝑚𝑒𝑉[17]. Effective masses of electrons in the 

well (𝑚𝑤) and in the barrier (𝑚𝑏) regions are 

taken as 0.067𝑚𝑒 and 0.091𝑚𝑒 [18] respectively.  

Fig. 2(a) shows the graphical solution for energy 

levels in symmetric state corresponding to the value 

of the right and of the left sides of the Eq. (9). The 

plot of𝑌1 = 𝑘𝑤
𝑠 (

3

2
𝑙𝑏 +  2 𝑙𝑤) is in solid (black) line 

and𝑌2 = 𝑛𝜋 + tan−1 [(−
𝐵1

𝑠

𝐴1
𝑠 )] is plotted in dash 

(red), dot (blue),dash dot(magenta) and short dash 

dot (navy) styles color (color online) for n=1, 2, 3 

and 4 respectively. The eigen values of energy for 

the symmetric states for n=1, 2 and 3, in eV, are 

0.0147000839, 0.136077023 and 0.156888216 

respectively. Fig. 2(b) shows similar plots of 𝑌1 and 

𝑌2 in the same colors and styles for n=1 to 4 for 

anti-symmetric states. The anti-symmetric energy 

eigen values are 0.0147488590, 0.137090183 and 

0.15718252 in eV the states n=1,2 and 3 

respectively. 
 

For simplicity, only the lowest symmetric and anti-

symmetric state for n=1 are considered in the 

calculation. The energy of the lowest anti-

symmetric state is slightly higher as compared to 

the symmetric state, which is clearly seen in inset 

of the Fig. 2(c). The difference ∆𝐸 is 

0.0000487751 𝑒𝑉 and is also referred as ‘energy 

ting’[16]. 
 

Fig. 3 shows the variation of the lowest (𝑛 = 1) 

lying anti-symmetric dash (red) and symmetric dot 

(blue) wave functions as a function of the width of 

the SQW. It shows the amplitude of the two waves, 

which are in phase in the right half and amplitude 

peaks in this region. They are out of phase in the 

left hand side and cancel each other. 
 

The time evolution of the combined wave function 

is displayed in Fig. 4.(a), (b) and (c), which show 

the probability of finding a particle with position 

within the SQW at different instants of time.  At 

t=0, Fig. 4(a) shows higher probability in the right 

half of the system. It gradually decreases and 

becomes nearly equal in both half at t = T/4, as 

shown in Fig. 4(b). Then, the peak starts to increase 

in the left half of the SQW. t= T/2, the probability 

of finding the particle in the left half of the system  

 



S. Shrestha et al. / BIBECHANA 18 (1) (2021) 91-99  

96 
 

 

0.0 0.1 0.2

0

4

8

12

l
b
=5 nm m

w
=0.067 m

e

m
b
=0.091 m

e

(a) V
o
=228 meV

l
w
=4 nm

F
u

n
c

ti
o

n
 y

1
 a

n
d

 y
2

Energy in eV

 

. 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Fig. 2: (a) Plot ofY1 = kw
s (

3

2
lb +  2 lw)in solid (black) line andY2 = nπ + tan−1 [(−

B1
s

A1
s)] in dash (red), 

dot (blue), dash dot (magenta) and short dash dot(navy) styles color (color online) for n=1, 2, 3 and 4 

respectively for symmetric states. The intersecting values of abscissa give the energy of symmetric states. 

(b) Similar plot of Y1 in solid (black) line and Y2 in different style and color line showing the energy of 

anti-symmetric states. (c) Expanded view of the lowest symmetric in dash (red) and anti-symmetric in 

solid  (blue) states (online) of the curvesare shown in the inset view. 

 

 

 

 

0.0 0.1 0.2

0

4

8

12

l
b
=5 nm m

w
=0.067 m

e

m
b
=0.091 m

e

V
o
=228 meV

l
w
=4 nm

(b) 

F
u

n
c

ti
o

n
 Y

1
 a

n
d

 Y
2

Energy in eV

 

0.0147000 0.0147001
2.49950

2.49975

2.50000

0.00 0.05 0.10 0.15

0

2

4

6

F
u

n
c
ti
o

n
 Y

1
 a

n
d

 Y
2

Energy in eV

(c) 

Symmetric Antisymmetric

Energy in eV

E



S. Shrestha et al. / BIBECHANA 18 (1) (2021) 91-99  

97 
 

 

 

 

 

 

 

 

 

 

 

 

Fig. 3: The lowest symmetric and anti-symmetric wave functions, shown by solid (blue) and dash (red) 

line (color online)respectively versus the net width of the SQW. Here, V0=228 meV, width of each barrier 

region (lb)=5 nm and width of each well region (lw)=4 nm. 

 

 

 

 

 

 

 

 

 

 

 

Fig. 4: Probability distribution of finding a charged particle in different spatial region within the SQW 

with height of barrier region V0= 228 meV, width of barrier region lb=5nm at time(t)=0, t= T/4 and t=T/2 

shown by solid (black), dash (red) and dash-dot (blue) (color online) line respectively. 

 

-10 0 10

-40

-20

0

20

40

Lowest Anti-symmetric 
a

 

Lowest 

Symmetric 
s

W
a

v
e

 a
m

p
li

tu
d

e
 


Position (z) in nm

 

 

-10 0 10
0.0

0.5

1.0

0.0

0.5

1.0

time (t)=51

time (t)=0time (t)=102

P
ro

b
a

b
il

it
y

 a
m

p
li

tu
d

e
 P

(z
,t

)

Position (z) in nm

 



S. Shrestha et al. / BIBECHANA 18 (1) (2021) 91-99  

98 
 

becomes the largest, as shown in Fig. 4(c). Thus, 

the particle which initially was in the left region of 

system tunneled through all the barriers to reach to 

the right part of the system, giving rise to 

tunneling-oscillation phenomenon. The time taken 

by particle to tunnel through the barrier is found to 

be 102 units of time, which is half of the time 

period of oscillation measured in the units of 

(
ℎ

𝑒∗10−2). The total time period of oscillation is 𝑇 =

2 ∗ 102 ∗ (
ℎ

𝑒∗10−2) i.e. 8.479 ∗ 10−11 𝑠. It 

corresponds to the tunneling frequency equal 

to11.79 𝐺𝐻𝑧. 
 

The tunneling behavior of electron in symmetric 

double well has been studied by E.P. Lopez 2006 

[16] with the barrier height(𝑉0) as 108 𝑚𝑒𝑉 and 

barrier width (𝑙𝑏) as 0.2 A0 and the width of the 

well(𝑙𝑤) as0.6 𝐴0also. The authors have calculated 

the frequency of oscillations to be 25.356 𝐺𝐻𝑧. 
 

It is found that as the barrier height is increased, the 

difference in energy values decreases. This 

increases the time period and the time taken by the 

particle to tunnel through it. For sufficiently high 

barriers the particle may not tunnel through, 

because the frequency of tunneling decays to a very 

small value. In case of very high barrier quantum 

well structures in which the frequency of oscillation 

is almost zero. 

This model of tunneling oscillation is expected to 

be useful for the study of the strength of the barrier 

and oscillation behavior of electron in symmetric 

quantum well/barrier systems. In addition, the fast 

switching semiconductor devices and high 

frequency oscillators can be studied by the 

proposed method and using the related results. 
 

4. Conclusion 
 

A method of exact solution for energy level and 

wave function of the particle in SL is studied using 

T-matrix. The tunneling behavior of the electron in 

SQW is described with the help of time evolution.  

The value of energy ting and tunneling frequency 

for the electron described in SQW are estimated to 

be∇𝐸1 = 0.0000487751𝑒𝑉 and  𝜈𝑇 = 11.79 𝐺𝐻𝑧 

respectively. The method can be extended for 

larger number of wells as well for fast switching 

semiconductor devices in 𝐺𝐻𝑧 region. 
 

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