In this work, we have investigated the effect of an external magnetic field and, for the first time, number of wells with constant total effective length to study the degeneracy in electronic energy levels. We have used constant total effective length because it is technologically important. Then we have tried to remove the n-fold degeneracy of the n-well multiple quantum well by means of the external magnetic field but the two-fold degeneracy was remain and not removed. Finally, the effect of the external magnetic field on the number of bound states and the situation of unchanging absorption coefficient in a wide magnetic field interval are also investigated.
When you are citing the document, use the following link http://essuir.sumdu.edu.ua/handle/123456789/3195

Izadifard, M.

Solaimani, M.

Electronic Sumy State University Institutional Repository

JOURNAL OF NANO- AND ELECTRONIC PHYSICS ЖУРНАЛ НАНО- ТА ЕЛЕКТРОННОЇ ФІЗИКИ Vol. 5 No 3, 03023(5pp) (2013) Том 5 № 3, 03023(5cc) (2013)   2077-6772/2013/5(3)03023(5) 03023-1  2013 Sumy State University Electronic and Optical Properties of GaN / AlN Multiple Quantum Wells under Static  External Magnetic Field  M. Solaimani, M. Izadifard  Faculty of Physics, Shahrood University of Technology, Shahrood, Iran  (Received 21 April 2013; revised manuscript received 08 May 2013; published online 12 July 2013)  In this work, we have investigated the effect of an external magnetic field and, for the first time, num-ber of wells with constant total effective length to study the degeneracy in electronic energy levels. We have used constant total effective length because it is technologically important. Then we have tried to re-move the n-fold degeneracy of the n-well multiple quantum well by means of the external magnetic field but the two-fold degeneracy was remain and not removed. Finally, the effect of the external magnetic field on the number of bound states and the situation of unchanging absorption coefficient in a wide magnetic field interval are also investigated.  Keywords: Electronic energy levels, Magnetic field, Single and multiple quantum wells, Finite difference method, Linear absorption coefficient.   PACS numbers: 68.65.Fg, 71.35. – y   1. INTRODUCTION  Semiconductor nanostructure systems like quantum wells have an essential role in the modern nanotech-nologies. Based on this importance, a great amount of investigations have devoted to this field of study. In this way, different geometrical shapes of the quantum well systems have been investigated. Some of these shapes are: hyperbolic quantum well [1], triangular quantum wells [2-3], delta-doped quantum well [4], and some others [5]. Besides, the effect of wells number in double quantum well [6-7] and triple quantum well [8] have also been studied. Magnetic field is also a flexible tool to modify or change different properties of these systems. The magnetic field changes the confinement potential, then the electronic structure of the system will be modified. Here we have used another interesting way to con-struct our quantum confinement geometry which we expect find good situation in the modern technologies. Retaining the total length of the structure constant and varying the number of wells within it in our resent studies [9-10], led to Constant Total Effective Length Multiple Quantum Wells (CTEL-MQWs). Our strategy of building multiple quantum wells may lead to smaller nano-devices with optimized physical properties and at the same time we have saved the time and material needed to fabricate the device. Since, one of the most challenging parameters in modern technologies is the device size, Here, by tuning the total effective length of the structure, smaller devices may be fabricated. In the present work, we have tried to study the effect of the number of wells on energy levels of a GaN / AlN MQW system with constant total effective length and same well and barrier thicknesses. The effect of the applied magnetic field on the ground state wave function and energy levels and the degeneracy in them, number of bound states and also the effect of the magnetic field on the linear absorption peak positions are investigated. Finally, we have used a Finite Difference method to solve our Schrödinger Hamiltonian.  2. FORMALISM  The effective Schrödinger equation in the effective mass approximation and within the envelop function method in the presence of an external magnetic field perpendicular to the growth direction is [11],  2 2 2 2 2* 2 * 2( ) ( ) ( ) ( )2 2d e B xH x V x x E xm dx m c (1)  Where m* is the effective mass, c is the speed of light in free space, e is the electron charge and B is the applied external magnetic field perpendicular to the growth direction. The eigen-energies and eigen-function have been investigated by the numerical dis-cretization methods[12]. Then the first order absorp-tion coefficient (1)( )  for intersubband can be calcu-lated by [13],  *2(1)22 21 exp ( ) /( ) ln1 exp ( ) //( ) ( / )F i BBfir eff F f Binf i inE E k Tm k TcMn L E E k TE E (2)  where  is the angular frequency of the excitation electromagnetic field, EF represents the Fermi energy, Ei and Ef denote the quantized energy levels for the initial and final states, respectively, kB is the Boltz-mann constant, c is the free space speed of light, Leff is the effective spatial extent of electrons, nr is the refrac-tive index, τin is the intersubband relaxation time (which we have used the numerical value of 0.14 ps [14]). Moreover in these equations Mfi is the dipole ma-trix element which is defined as [13],   *( )| | ( )fi f iM z e z z dz  (3)  In order to evaluate the Fermi energy we have used the 2D-electron density,   M. SOLAIMANI, M. IZADIFARD J. NANO- ELECTRON. PHYS. 5, 03023 (2013)   03023-2 *2 221 1ln(1 exp( ( ) / ) )n nD D Bj F j Bj jm k Tn n E E k T  (4)  which was taken from [15]. In this equation j repre-sents the subband state number. In order to find the Fermi Energy we have used the n2D  3.0  1016 [14] and then by varying the Fermi energy we have tried to find the true EF.  3. RESULTS AND DISCUSSION   In this work we have applied an external magnetic field perpendicular to the growth direction of a GaN / AlN MQWs structure including of 1 to 7 numbers of wells. For this purpose, we have used m*  0.15 m0 (where m0 is the free electron mass), the barrier height Vconf  1.28 eV [15], Leff  600 Å. In our calculations we have used equally spaced wells in a MQWs with the same well and barrier widths. For example in a triple quantum well we have three wells with well width of (600 / (3 + 4)) Å. Thus the well width for a MQW with n wells can be found by (600 / (2n + 1)) Å. In the Fig. 1 we have tried shown the subband  energies versus an external magnetic field B perpen-dicular to the growth direction for MQWs (1 6n ). In parts 2 (a-f) two, three, four, five and six lowest states for these MQWs have been shown and within the inset the sixteen lowest subband is presented. The insets is presented to find out that the intersec-tion of subband will be occurred or not and of course for more illustrations. The n-fold degeneracy at B  0 is clearly from these figures. By increasing of the well number n > 2 and the magnetic field some degeneracy    Fig. 1 – Effect of an external magnetic field B perpendicular to the growth direction on subband energies of some MQWs with different number of wells  ELECTRONIC AND OPTICAL PROPERTIES OF GAN / ALN… J. NANO- ELECTRON. PHYS. 5, 03023 (2013)   03023-3   Fig. 2 – Effect of the magnetic field on the ground state wave function of a 3-well MQW  are removed. For double quantum well by increasing of the applied magnetic field the degeneracy is almost remains. For MQWs with an odd number of wells the ground state is not degenerate but for MQWs with even well numbers we have two fold degenerate states. For example in the 3-well MQWs the ground state has not degeneracy in the presence of the magnetic field but the first excited state has two-fold degeneracy. But for 4-well MQWs both ground and excited states have two fold degenerate states. This behavior is also seen by calculating of other excited states (inset figures). In general the magnetic field can split the n-fold degener-ate subband of an n-well MQWs (n  even and B  0 T) into n / 2 two fold degenerate subbands. The intersec-tion of the subband energies in the presence of the external magnetic field (about 25 T) will take place for MQWs with n > 2. The underplaying reason for the de-   Fig. 3 – Effect of an external magnetic field B and number of wells on the number of bound states of the quantum system    M. SOLAIMANI, M. IZADIFARD J. NANO- ELECTRON. PHYS. 5, 03023 (2013)   03023-4 generacy removal property of the external magnetic field is that, the magnetic field adds a parabolic term to the confinement potential and thus magnetic field changes it. Now the existing wells in a multiple quan-tum well have not exactly the same geometrical shape and thus the energy levels are not degenerate like prior to the application of the magnetic field (as we know, a multiple quantum well with n identical wells the wells have n-fold degeneracy[10]). But the parabolic potential which magnetic field adds is a symmetric potential with respect to the centre of the central well. Now n / 2 of the wells have same geometrical shape as the other n / 2 wells thus the magnetic field splits the n-fold de-generate subband of an n-well MQWs (n  even and B  0 T) into n / 2 two fold degenerate subbands. In the Fig. 2 we have presented the effect of the magnetic field on the normalized ground state wave functions of MQWs including of 3 wells.  This figure show that at zero magnetic field there 3 peak which are localized within three wells. As it can be seen the central well is slightly more probable to include the electron. Increasing of the magnetic field up to 100 G leads to the localization of the wave function within the central well. It could be due to the fact that the external magnetic field adds a parabolic term around the origin to the potential of the system.  In the Fig. 3 we have presented the effect of an ex-ternal magnetic field B and number of wells on the number of bound states of our quantum systems. In all of these systems, increasing of the external magnetic field enhanced the number of bound states. But as it is clear from the figure it increased in a stepwise manner. This can be expected from the fact that the energy lev-els of a quantum system are discretized.  When we in-crease the amount of the external magnetic field (and thus the confinement of the free carriers), then in some special values of the external magnetic field a new  energy level fall within the bound states region. Thus in this interval of the magnetic field the number of bound states remain constant. On the other hand, at zero magnetic field, the larger the number of wells can the more number of bound states. In the meantime in some special number of wells there is a rather wide magnetic field interval within which the numbers of bound states do not change. This has direct effect of the optical properties on the system like absorption coeffi-cient we have presented in the next figure. In the Fig. 4 we have presented the effect of an  external magnetic field B and incident photon energy on the linear absorption coefficient of our quantum systems. Each point the plot is the absorption peak position which occurred in a special incident photon energy and external magnetic field. As we can see in this figure, increasing of the exter-nal magnetic field leads to the peak positions which occur in the greater incident photon energy.  If we put aside the single quantum well, then the slop of the dia-gram decreases by increasing of the number of wells. As we saw in the previous figure, in some special num-ber of wells there was a rather wide magnetic field in-terval within which the numbers of bound states did not changed. The systems which had wider unchanging magnetic field interval in the number of bound state diagram, now have wider magnetic field interval with-in which the apportion peak position do not change.     Fig. 4 – Effect of an external magnetic field B and incident photon energy on the linear absorption coefficient of the quan-tum system  If we need a system which we like to have unchanging absorption coefficient in a wide magnetic field interval we have to use multiple quantum wells with greater number of wells.  4. CONCLUSION  In this study, the effect of external magnetic field and number of wells on the electronic structure and optical properties MQWs is investigated. In the n-well MQWs with number of wells greater than 6, the n-fold degeneracy is not seen and we had a supperlattice. Thus the splitting of the subbands was not so large which is due to the depth of the well which is approxi-mately high. It tried to remove the degeneracy of the n-well MQWs (2 6n ) by an external magnetic field B perpendicular to the growth direction. The degenerate states split into some 2-fold degenerate states. Increas-ing of the external magnetic field enhanced the number of bound states. In some special number of wells there was a rather wide magnetic field interval within which the numbers of bound states did not change. Increasing of the external magnetic field led to the peak positions which occurred in the greater incident photon energy. If we need a system which we like to have unchanging absorption coefficient in a wide magnetic field interval we had to use multiple quantum wells with greater number of wells.  AKNOWLEDGEMENTS  We are grateful for the financial support of the Ira-nian nanotechnology initiative council (INIC) and shahrood University of Technology supports.       ELECTRONIC AND OPTICAL PROPERTIES OF GAN / ALN… J. NANO- ELECTRON. PHYS. 5, 03023 (2013)   03023-5 REFERENCES  1. P. Reimann, Phys. Rep. 361, 57 (2002). 2.  I. Vurgaftman, J.M. Hinckley, J. Singh, IEEE J. Quan-tum Elect. 30, 75 (1994). 3. Z. Yang, R. Li, Q. Wei, T. Yu, Y. Zhang, W. Chen, X. Hu, Appl. Phys. Lett. 94, 061120 (2009). 4.  R.B. Santiago, L.G. Guimaraes, Solid-State Electron. 46, 89 (2002). 5. E. Ozturk, I. Sokmen, Solid-State Commun. 126, 605 (2003). 6. N. Eseanu, Phys. Lett. A 375, 1036 (2011). 7. A.K. Ghatak, I.C. Goyal, R.L. Gallawa, IEEE J. Quantum Elect. 26, 305(1990).  8. K.-M. Wong, D.W.E. Allsopp, Semicond. Sci. Technol. 24, 045018 (2009). 9. A. Latge, M. Pacheco, Z. Barticevic, Semicond. Sci. Tech-nol. 17, 952 (2002). 10. M. Solaimani, M. Izadifard, H. Arabshahi, M.R. Sarkardei, J. Lumin. 134, 699 (2013). 11. M. Solaimani, M. Izadifard, H. Arabshahi, M.R. Sarkardei, J. Lumin. 134, 88 (2013).  12. E. Ozturk, I. Sokmen, Superlattice. Microst. 48, 312 (2010).  13. S. Sheppard, M.V.K. Chari, Numerical Methods in Elec-tromagnetism (Academic Press: 1999). 14. D. Ahn, S.L. Chuang, Phys. Rev. B 35, 4149 (1987). 15.  D. Ahn, S.L. Chuang, IEEE J. Quantum Elect. 23, 2196 (1987). 16. O. Manasreh, Semiconductor Heterojunctions and Nanostructures (McGraw Hill: 2005). 17. www.netlib.org. 18. ftp://ftp.caam.rice.edu/pub/software/ARPACK. 19. I. Vurgaftman, J.R. Meyer, L.R. Ram-Mohan, J. Appl. Phys. 89, 5815 (2001).  

Electronic and Optical Properties of GaN / AlN Multiple Quantum Wells under Static External Magnetic Field

SocioEconomic Challenges Journal (Sumy State University)

JOURNAL OF NANO- AND ELECTRONIC PHYSICS ЖУРНАЛ НАНО- ТА ЕЛЕКТРОННОЇ ФІЗИКИ 
Vol. 5 No 3, 03023(5pp) (2013) Том 5 № 3, 03023(5cc) (2013) 
 
 
2077-6772/2013/5(3)03023(5) 03023-1  2013 Sumy State University 
Electronic and Optical Properties of GaN / AlN Multiple Quantum Wells under Static  
External Magnetic Field 
 
M. Solaimani, M. Izadifard 
 
Faculty of Physics, Shahrood University of Technology, Shahrood, Iran 
 
(Received 21 April 2013; revised manuscript received 08 May 2013; published online 12 July 2013) 
 
In this work, we have investigated the effect of an external magnetic field and, for the first time, num-
ber of wells with constant total effective length to study the degeneracy in electronic energy levels. We 
have used constant total effective length because it is technologically important. Then we have tried to re-
move the n-fold degeneracy of the n-well multiple quantum well by means of the external magnetic field 
but the two-fold degeneracy was remain and not removed. Finally, the effect of the external magnetic field 
on the number of bound states and the situation of unchanging absorption coefficient in a wide magnetic 
field interval are also investigated. 
 
Keywords: Electronic energy levels, Magnetic field, Single and multiple quantum wells, Finite difference 
method, Linear absorption coefficient. 
 
 PACS numbers: 68.65.Fg, 71.35. – y 
 
 
1. INTRODUCTION 
 
Semiconductor nanostructure systems like quantum 
wells have an essential role in the modern nanotech-
nologies. Based on this importance, a great amount of 
investigations have devoted to this field of study. In 
this way, different geometrical shapes of the quantum 
well systems have been investigated. Some of these 
shapes are: hyperbolic quantum well [1], triangular 
quantum wells [2-3], delta-doped quantum well [4], and 
some others [5]. Besides, the effect of wells number in 
double quantum well [6-7] and triple quantum well [8] 
have also been studied. Magnetic field is also a flexible 
tool to modify or change different properties of these 
systems. The magnetic field changes the confinement 
potential, then the electronic structure of the system 
will be modified. 
Here we have used another interesting way to con-
struct our quantum confinement geometry which we 
expect find good situation in the modern technologies. 
Retaining the total length of the structure constant and 
varying the number of wells within it in our resent 
studies [9-10], led to Constant Total Effective Length 
Multiple Quantum Wells (CTEL-MQWs). Our strategy 
of building multiple quantum wells may lead to smaller 
nano-devices with optimized physical properties and at 
the same time we have saved the time and material 
needed to fabricate the device. Since, one of the most 
challenging parameters in modern technologies is the 
device size, Here, by tuning the total effective length of 
the structure, smaller devices may be fabricated. In the 
present work, we have tried to study the effect of the 
number of wells on energy levels of a GaN / AlN MQW 
system with constant total effective length and same 
well and barrier thicknesses. The effect of the applied 
magnetic field on the ground state wave function and 
energy levels and the degeneracy in them, number of 
bound states and also the effect of the magnetic field on 
the linear absorption peak positions are investigated. 
Finally, we have used a Finite Difference method to 
solve our Schrödinger Hamiltonian. 
 
2. FORMALISM 
 
The effective Schrödinger equation in the effective 
mass approximation and within the envelop function 
method in the presence of an external magnetic field 
perpendicular to the growth direction is [11], 
 
2 2 2 2 2
* 2 * 2
( ) ( ) ( ) ( )
2 2
d e B x
H x V x x E x
m dx m c
 (1) 
 
Where m* is the effective mass, c is the speed of 
light in free space, e is the electron charge and B is the 
applied external magnetic field perpendicular to the 
growth direction. The eigen-energies and eigen-
function have been investigated by the numerical dis-
cretization methods[12]. Then the first order absorp-
tion coefficient (1)( )  for intersubband can be calcu-
lated by [13], 
 
*
2(1)
2
2 2
1 exp ( ) /
( ) ln
1 exp ( ) /
/
( ) ( / )
F i BB
fi
r eff F f B
in
f i in
E E k Tm k Tc
M
n L E E k T
E E
 (2) 
 
where  is the angular frequency of the excitation 
electromagnetic field, EF represents the Fermi energy, 
Ei and Ef denote the quantized energy levels for the 
initial and final states, respectively, kB is the Boltz-
mann constant, c is the free space speed of light, Leff is 
the effective spatial extent of electrons, nr is the refrac-
tive index, τin is the intersubband relaxation time 
(which we have used the numerical value of 0.14 ps 
[14]). Moreover in these equations Mfi is the dipole ma-
trix element which is defined as [13], 
 
 *( )| | ( )fi f iM z e z z dz  (3) 
 
In order to evaluate the Fermi energy we have used 
the 2D-electron density, 
 
 M. SOLAIMANI, M. IZADIFARD J. NANO- ELECTRON. PHYS. 5, 03023 (2013) 
 
 
03023-2 
*
2 2
2
1 1
ln(1 exp( ( ) / ) )
n n
D D B
j F j B
j j
m k T
n n E E k T  (4) 
 
which was taken from [15]. In this equation j repre-
sents the subband state number. In order to find the 
Fermi Energy we have used the n2D  3.0  1016 [14] 
and then by varying the Fermi energy we have tried to 
find the true EF. 
 
3. RESULTS AND DISCUSSION  
 
In this work we have applied an external magnetic 
field perpendicular to the growth direction of a 
GaN / AlN MQWs structure including of 1 to 7 numbers 
of wells. For this purpose, we have used m*  0.15 m0 
(where m0 is the free electron mass), the barrier height 
Vconf  1.28 eV [15], Leff  600 Å. In our calculations we 
have used equally spaced wells in a MQWs with the 
same well and barrier widths. For example in a triple 
quantum well we have three wells with well width of 
(600 / (3 + 4)) Å. Thus the well width for a MQW with n 
wells can be found by (600 / (2n + 1)) Å. 
In the Fig. 1 we have tried shown the subband  
energies versus an external magnetic field B perpen-
dicular to the growth direction for MQWs (1 6n ). In 
parts 2 (a-f) two, three, four, five and six lowest states 
for these MQWs have been shown and within the inset 
the sixteen lowest subband is presented. 
The insets is presented to find out that the intersec-
tion of subband will be occurred or not and of course for 
more illustrations. The n-fold degeneracy at B  0 is 
clearly from these figures. By increasing of the well 
number n > 2 and the magnetic field some degeneracy  
 
 
Fig. 1 – Effect of an external magnetic field B perpendicular to the growth direction on subband energies of some MQWs with 
different number of wells 
 ELECTRONIC AND OPTICAL PROPERTIES OF GAN / ALN… J. NANO- ELECTRON. PHYS. 5, 03023 (2013) 
 
 
03023-3 
 
 
Fig. 2 – Effect of the magnetic field on the ground state wave 
function of a 3-well MQW 
 
are removed. For double quantum well by increasing of 
the applied magnetic field the degeneracy is almost 
remains. For MQWs with an odd number of wells the 
ground state is not degenerate but for MQWs with even 
well numbers we have two fold degenerate states. For 
example in the 3-well MQWs the ground state has not 
degeneracy in the presence of the magnetic field but 
the first excited state has two-fold degeneracy. But for 
4-well MQWs both ground and excited states have two 
fold degenerate states. This behavior is also seen by 
calculating of other excited states (inset figures). In 
general the magnetic field can split the n-fold degener-
ate subband of an n-well MQWs (n  even and B  0 T) 
into n / 2 two fold degenerate subbands. The intersec-
tion of the subband energies in the presence of the 
external magnetic field (about 25 T) will take place for 
MQWs with n > 2. The underplaying reason for the de- 
 
 
Fig. 3 – Effect of an external magnetic field B and number of wells on the number of bound states of the quantum system 
 
 
 M. SOLAIMANI, M. IZADIFARD J. NANO- ELECTRON. PHYS. 5, 03023 (2013) 
 
 
03023-4 
generacy removal property of the external magnetic 
field is that, the magnetic field adds a parabolic term to 
the confinement potential and thus magnetic field 
changes it. Now the existing wells in a multiple quan-
tum well have not exactly the same geometrical shape 
and thus the energy levels are not degenerate like prior 
to the application of the magnetic field (as we know, a 
multiple quantum well with n identical wells the wells 
have n-fold degeneracy[10]). But the parabolic potential 
which magnetic field adds is a symmetric potential 
with respect to the centre of the central well. Now n / 2 
of the wells have same geometrical shape as the other 
n / 2 wells thus the magnetic field splits the n-fold de-
generate subband of an n-well MQWs (n  even and 
B  0 T) into n / 2 two fold degenerate subbands. 
In the Fig. 2 we have presented the effect of the 
magnetic field on the normalized ground state wave 
functions of MQWs including of 3 wells.  
This figure show that at zero magnetic field there 3 
peak which are localized within three wells. As it can 
be seen the central well is slightly more probable to 
include the electron. Increasing of the magnetic field up 
to 100 G leads to the localization of the wave function 
within the central well. It could be due to the fact that 
the external magnetic field adds a parabolic term 
around the origin to the potential of the system.  
In the Fig. 3 we have presented the effect of an ex-
ternal magnetic field B and number of wells on the 
number of bound states of our quantum systems. In all 
of these systems, increasing of the external magnetic 
field enhanced the number of bound states. But as it is 
clear from the figure it increased in a stepwise manner. 
This can be expected from the fact that the energy lev-
els of a quantum system are discretized.  When we in-
crease the amount of the external magnetic field (and 
thus the confinement of the free carriers), then in some 
special values of the external magnetic field a new  
energy level fall within the bound states region. Thus 
in this interval of the magnetic field the number of 
bound states remain constant. On the other hand, at 
zero magnetic field, the larger the number of wells can 
the more number of bound states. In the meantime in 
some special number of wells there is a rather wide 
magnetic field interval within which the numbers of 
bound states do not change. This has direct effect of the 
optical properties on the system like absorption coeffi-
cient we have presented in the next figure. 
In the Fig. 4 we have presented the effect of an  
external magnetic field B and incident photon energy 
on the linear absorption coefficient of our quantum 
systems. Each point the plot is the absorption peak 
position which occurred in a special incident photon 
energy and external magnetic field. 
As we can see in this figure, increasing of the exter-
nal magnetic field leads to the peak positions which 
occur in the greater incident photon energy.  If we put 
aside the single quantum well, then the slop of the dia-
gram decreases by increasing of the number of wells. 
As we saw in the previous figure, in some special num-
ber of wells there was a rather wide magnetic field in-
terval within which the numbers of bound states did 
not changed. The systems which had wider unchanging 
magnetic field interval in the number of bound state 
diagram, now have wider magnetic field interval with-
in which the apportion peak position do not change.  
 
 
 
Fig. 4 – Effect of an external magnetic field B and incident 
photon energy on the linear absorption coefficient of the quan-
tum system 
 
If we need a system which we like to have unchanging 
absorption coefficient in a wide magnetic field interval 
we have to use multiple quantum wells with greater 
number of wells. 
 
4. CONCLUSION 
 
In this study, the effect of external magnetic field 
and number of wells on the electronic structure and 
optical properties MQWs is investigated. In the n-well 
MQWs with number of wells greater than 6, the n-fold 
degeneracy is not seen and we had a supperlattice. 
Thus the splitting of the subbands was not so large 
which is due to the depth of the well which is approxi-
mately high. It tried to remove the degeneracy of the n-
well MQWs (2 6n ) by an external magnetic field B 
perpendicular to the growth direction. The degenerate 
states split into some 2-fold degenerate states. Increas-
ing of the external magnetic field enhanced the number 
of bound states. In some special number of wells there 
was a rather wide magnetic field interval within which 
the numbers of bound states did not change. Increasing 
of the external magnetic field led to the peak positions 
which occurred in the greater incident photon energy. 
If we need a system which we like to have unchanging 
absorption coefficient in a wide magnetic field interval 
we had to use multiple quantum wells with greater 
number of wells. 
 
AKNOWLEDGEMENTS 
 
We are grateful for the financial support of the Ira-
nian nanotechnology initiative council (INIC) and 
shahrood University of Technology supports. 
 
 
 
 
 
 ELECTRONIC AND OPTICAL PROPERTIES OF GAN / ALN… J. NANO- ELECTRON. PHYS. 5, 03023 (2013) 
 
 
03023-5 
REFERENCES 
 
1. P. Reimann, Phys. Rep. 361, 57 (2002). 
2.  I. Vurgaftman, J.M. Hinckley, J. Singh, IEEE J. Quan-
tum Elect. 30, 75 (1994). 
3. Z. Yang, R. Li, Q. Wei, T. Yu, Y. Zhang, W. Chen, X. Hu, 
Appl. Phys. Lett. 94, 061120 (2009). 
4.  R.B. Santiago, L.G. Guimaraes, Solid-State Electron. 46, 
89 (2002). 
5. E. Ozturk, I. Sokmen, Solid-State Commun. 126, 605 
(2003). 
6. N. Eseanu, Phys. Lett. A 375, 1036 (2011). 
7. A.K. Ghatak, I.C. Goyal, R.L. Gallawa, IEEE J. Quantum 
Elect. 26, 305(1990).  
8. K.-M. Wong, D.W.E. Allsopp, Semicond. Sci. Technol. 24, 
045018 (2009). 
9. A. Latge, M. Pacheco, Z. Barticevic, Semicond. Sci. Tech-
nol. 17, 952 (2002). 
10. M. Solaimani, M. Izadifard, H. Arabshahi, M.R. Sarkardei, 
J. Lumin. 134, 699 (2013). 
11. M. Solaimani, M. Izadifard, H. Arabshahi, M.R. Sarkardei, 
J. Lumin. 134, 88 (2013).  
12. E. Ozturk, I. Sokmen, Superlattice. Microst. 48, 312 
(2010).  
13. S. Sheppard, M.V.K. Chari, Numerical Methods in Elec-
tromagnetism (Academic Press: 1999). 
14. D. Ahn, S.L. Chuang, Phys. Rev. B 35, 4149 (1987). 
15.  D. Ahn, S.L. Chuang, IEEE J. Quantum Elect. 23, 2196 
(1987). 
16. O. Manasreh, Semiconductor Heterojunctions and 
Nanostructures (McGraw Hill: 2005). 
17. www.netlib.org. 
18. ftp://ftp.caam.rice.edu/pub/software/ARPACK. 
19. I. Vurgaftman, J.R. Meyer, L.R. Ram-Mohan, J. Appl. 
Phys. 89, 5815 (2001). 
 


English

https://essuir.sumdu.edu.ua/bitstream/123456789/31955/1/Solaimani_Linear%20absorption%20coefficient.pdf

Electronic and Optical Properties of GaN / AlN Multiple Quantum Wells under Static External Magnetic Field

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