The total absorption coefficient in spherical GaN/AlxGa1 – xN core-shell nanodots is theoretically investigated taking into account effective mass approximation. The influence of the nanosystem geometry upon the energy spectrum and transition energy δE associated to interlevel transitions is studied. We found that the energy transitions vary with the core-shell radius, and the peak position of the total absorption coefficient is greatly affected by changing of the shell radius. The possibility of tuning the resonant energies by using the geometric core shell effect of the spatial confinement can be useful in the optoelectronic devices applications. Also we observed that the magnitudes of the total absorption coefficient can be increased significatly compared traditional cases of QD, and the peaks are shifted to the lower energies.
When you are citing the document, use the following link http://essuir.sumdu.edu.ua/handle/123456789/2960

Abouelaoualim, D.

Elkadadra, A.

Oueriagli, A.

Outzourhit, A.

Electronic Sumy State University Institutional Repository

JOURNAL OF NANO- AND ELECTRONIC PHYSICS ЖУРНАЛ НАНО- ТА ЕЛЕКТРОННОЇ ФІЗИКИ Vol. 4 No 3, 03004(5pp) (2012) Том 4 № 3, 03004(4cc) (2012)   2077-6772/2012/4(3)03004(5) 03004-1  2012 Sumy State University Absorption Coefficients of GaN / AlxGa1 – xN Core-Shell Spherical Quantum Dot  D. Abouelaoualim*, A. Elkadadra, A. Oueriagli, A. Outzourhit  LPSCM,Department of Physics, Faculty of Sciences Semlalia,  Cadi Ayaad University P.O. Box 2390, 40000 Marrakech, Morocco  (Received 15 May 2012; published online 30 October 2012)  The total absorption coefficient in spherical GaN/AlxGa1 – xN core-shell nanodots is theoretically inves-tigated taking into account effective mass approximation. The influence of the nanosystem geometry upon the energy spectrum and transition energy E associated to interlevel transitions is studied. We found that the energy transitions vary with the core-shell radius, and the peak position of the total absorption coeffi-cient is greatly affected by changing of the shell radius. The possibility of tuning the resonant energies by using the geometric core shell effect of the spatial confinement can be useful in the optoelectronic devices applications. Also we observed that the magnitudes of the total absorption coefficient can be increased sig-nificatly compared traditional cases of QD, and the peaks are shifted to the lower energies.  Keywords: Quantum dot, Core-shell, Optical absorption.    PACS numbers: 73.21.Ac, 78.67.Pt                                                             * abouelaoualim_d@hotmail.com  1. INTRODUCTION  The interest in the study of the physical properties of confined quantum systems such as quantum wells [1], nanowires [2], and quantum dots (QDs) [3], has increased with the recent progress in semiconductor nanotechnology. In these structures the precise engi-neering would enable us to confine the charge carriers motion in one, two or three dimensions. Quantum dots can be used to make not just better optical devices [4], but also as optical data storage medium, where bits of information are stored as electrons or holes confined within the dots [5]. One advantage of these is that since the information is typically stored as a single (or few) electron, these devices should require a very low switching energy. The charge to be stored in the dots can be written by optical illumination, which, by wave-length selective writing or local probe techniques, can approach the situation where each quantum dot carries a bit of information. Such quantum dot optical memory devices could potentially achieve ultra-dense storage capacities of Terabit/in2. Recently, in order to aim of improving the electronic structures and optical proper-ties of QDs structure single photon detector for applica-tion as an optical data storage medium , different shapes and various confinement potentials have been widely studied, using the effective mass and parabolic one band approximations [6]. In this regard, we consid-er a GaN/AlxGa1 – xN core-shell spherical quantum dot (CSQD)and an interest in the optical phenomena based on intersubband transitions in such structure, the opti-cal. Because the confinement of quasiparticles in such structure leads to the enhancement of the oscillator strength of electron impurity excitations. Meanwhile, the dependence of the optical transition energy on the confinement strength (or dot size) allows the tunability of the resonance frequency. The outline of the paper is as follows: Hamiltonian, the relevant eigenvalues and eigenfunctions of an electron confined in a CSQD are described in Section 2. In Section 3, analytical expres-sions for the linear and nonlinear intersubband optical ACs and RI changes are obtained using the density matrix approach and the iterative method. Our numer-ical results and a brief conclusion are presented in Sec-tions 4 and 5, respectively.  2. MODEL AND THEORY   The model used in calculation and analyses is an isolated GaN/AlxGa1 – xN core-shell quantum dot with core radius R1 and shell radius R2, shown in Fig. 1.    Fig. 1 – Core-shell GaN-AlxGa1 – xN quantum dot system show-ing (a) core and shell layers; and (b) layer potential profile  Suppose the core-shell quantum dot forms a shell well and a centric barrier for the two kinds of materials have different potentials. The potential of the core is chosen to be referent zero point of energy, and the band-gap of GaN is wider than that of AlxGa1 – xN, thus VC. On the premise of the effective mass approxima-tion, the steady Schrödinger equation for electron can be written as   22222 21sinsin21( ) ( ),sinii nlmrr rm rV r E r                               (1)  where, 2h  , h is Planck’s constant; E the energy  D. ABOUELAOUALIM, A. ELKADADRA, A. OUERIAGLI… J. NANO- ELECTRON. PHYS. 4, 03004 (2012)   03004-2 eigenvalue; and nml(r) the responding eigenfunction. n is the principal quantum number, and l and m are the angular momentum quantum numbers. im  is the ef-fective mass of electron in the ith region, (r)  constant of core-shell QDs materiels and Vi(r) the potential. They are relative to the position in the model and ex-pressed as follows   1 12 1 2,< ,im r Rmm R r R   (2)    1, 1 22,0, 0<r ,<r ,, ri cRV r V R RR    (3)  Due to the fact that the mass and the potential are symmetric spherically, the separation of radial and angular coordinates leads to nml(r)  Rnml(r)Ynml(, ), where Rnlm(r) is the radial wave function, and Ynlm(,) is the spherical harmonics n is the principal quantum number, and l and m are the angular momentum quan-tum numbers. The radial eigenfunction, Rnl(r) consists of three parts according to the electron position. Two cases need to be considered for the solution of Rnl(r). In regions where electron eigenenergy E > Vc, the solution is a linear combination of spherical Bessel function jl and Neumann function nl, written as           1 ,1 1 ,1 1,2 ,2 2 ,2 1 2,2,,,0,l nl l nlnl l nl l nlA J k r B n k r r RR r A J k r B n k r R r Rr R      (4)  with   2,1 12 / ,nlk m E  (5)     2,1 22 / ,nl ck m E V   (6)  A1, A2, B1 and B2 are normalized constants. In re-gions where cE V , the radial wave function is           1 12' ',1 ,1 1,',2 1 2,2,,,0,ll nl l nlnl nlA J k r B J k r r RR r A h k r R r Rr R     (7)  with     2,2 22 / ,nl ck m V E   (8)  1' '2,A A  and1'B  are normalized constants. Let us consider that the system is excited by a mon-ochromatic electromagnetic field,   . .,i tE t Ee c c   which is polarized along the Z-direction. Therefore, the time evolution of the density operator of the system is given by [7]    01, ,H ME tt i             (9)  where H is the Hamiltonian of the system in the ab-sence of the electromagnetic field, M ez  is the dipole operator along the Z-axis, 0  is the unperturbed densi-ty operator and   is the damping operator due to the electron-phonon interaction and the other collision pro-cesses. The above equation can be solved using the per-turbation method by expanding    as [8]     ,nn    (10)  with          1111, ,,nn nijijijnij ijH M E tt ii               (11)  On the other hand, the electric polarization of the system can be written as         0 01,i t i tP t Ee Eetrace MV         (12)  where V is the volume of the system, 0 is the permit-tivity of free space, and trace means the summation over the diagonal elements of the matrix. Using the same density matrix formalism, the analytic expres-sions of the linear and the third order nonlinear sus-ceptibilities for a two-level quantum system are given and '  is the real part of the permittivity [9]:      210 ,v ijij ijME i     (13)             2 2230 2 224,v ij ijij ijij ijff jjfj if fj ifM E ME i EM ME i E i             (14)  where  is the carrier density, fj f iE E E   is the en-ergy interval of the two level system. Mfj is the dipole matrix element, which is defined by fi f iM er   ,   , 1,2f i   and Г shows the damping due to electron- phonon interaction. The absorption coefficient is given by       0' Im       (15)  The total absorption coefficient  ,I  can be writ- ABSORPTION COEFFICIENTS OF AGN/ALXGA1 – XN CORE-SHELL… J. NANO- ELECTRON. PHYS. 4, 03004 (2012)   03004-3 ten as            1 3, , ,I I I        (16)  where I is the incident optical intensity and is de-fined as     '22I E  (17)  Here  is the permeability of the system, and ' is the real part of the permittivity.  3. NUMERICAL RESULTS AND ANALYSIS  In the numerical calculations carried out for a core-shell quantum dot, the following parameter values have been used [13]: characterize the electrical and optical properties of the physical parameters used in our numerical work for GaN  and AlxGa1 – xN are taken from ref [10].The effective mass is 0.19GaN em m  for GaN   1 00.19 1 0.33x xAl Ga Nm x m    for AlxGa1 – xN. The real part of permittivity is chosen to be . The band-gap is    6.13 1 3.42 1 1.0cE x eV x eV x x eV      . The conduction band offset is chosen as     0.75 0b g gV E x E  . The dephasing constant cor-responding to elastic scattering is 13 0.14ps   and the incident optical intensity is 31 /cmI MW . By virtue of the importance of intersubband transi-tion, we have studied the variation of energy difference between the ground state to the exited state as a func-tion of CSQDs size, with a GaN core and a Al1 – x GaxN shell, both of which were assumed to have a wurtzite structure. Such dots have a potential barrier of VC = 2.77 eV  for both electrons and holes to tunnel through in order to reach the shell from the core. A summary of these calculations is shown in Fig. 2(a), (b). In these graphs, when holding the shell thickness fixed, the energy levels behave similarly to those of QDs, Fig. 2(a). With the core radius fixed, increasing the shell size produces an entirely different trend-both the states Fig. 2(b), move down in energy. This causes the transition energy between those states to be roughly constant. These results imply a simple rule of thumb for this system- the transition energies are primarily determined by the shell radius. In Fig. 3, we display the total AC as function of pump photon energy   for different core  radius with a fixed shell radius when the transition energy levels selected in the region where E < Vc, namely inside the core well. It is seen from this figure that for the GaN / Al1 – x GaXN core-shell QDs the resonant magnitudes are 6.5 × 107 /m for R1 = 42 Å, Fig. 3(a) . It can be seen that each curve has only one peak which is due to single photon resonance. Fur-thermore we can find that the total AC increase and the peak shift to lower energy. And the larger the R1 is, the smaller the peak shift is. These conclusions regarding the AC spectrum can be physically explained as follows: as a consequence of the quantum size effect, energy distances between   a  b  Fig. 2 – Energy levels of GaN / AlxGa1 – xN core-shell quantum dot  with (a) fixed shell radius and (b) fixed core radius  electronic states in the conduction band become smaller when R1 increases, Fig. 2(a), the larger the size is, the smaller the energy distance is; the increase of the dipole matrix element becomes stronger with the increase of the radius. When R2 increases from 42 Å to 52 Å with a fixed R1, the peak shift amount to 7 × 107 /m, Fig. 3(b). The reason for this phenomenon is related to electron confinement. When the electron eigenenergy E < VC, electrons are completely confined in the core well and there is nearly no electron distributing in the shell bar-rier, the probability distribution for the electronic wave function mainly concentrates in the core well. Even though shell layer widens much, contributes a little to electronic wave function and the peak shift as long as the size of core layer keeps a constant. Effect of barrier mole fraction (as a parameter) on the total AC of the system is illustrated in Fig. 4. We plot the total optical AC as functions of the photon en-ergy and x, with I = 1MW/cm2, R1 = 50°Å and R2 = 62 Å. It is shown that with increase of mole fraction, the ab-sorption peak decreases. This is related to the fact that  D. ABOUELAOUALIM, A. ELKADADRA, A. OUERIAGLI… J. NANO- ELECTRON. PHYS. 4, 03004 (2012)   03004-4  a  b  Fig. 3 – Total AC is plotted as functions of the photon energy and (a) core radius R1 with R2 = 62 Å (b) shell radius R2 with, R1 = 48 Å;  with I = 1MW/cm2 and x = 0.15, when E < Vc    Fig. 4 – Total optical AC as functions of the photon energy and Aluminium concentration x, with I = 1MW/cm2 and R1 = 48 Å and R2 = 62 Å  the transition energy E . When Aluminum concentra-tion increase, the barrier increases, this enhances the confinement of electrons. As a result, the overlap be-tween electron wave function at two subbands increases, and thus the optical intersubband transition also en-hances its intensity.  4. CONCLUSIONS  The total absorption coefficient (AC) of GaN / AlxGa1 – xN core-shell spherical quantum dot have been investigated, using the compact-density ma-trix formalism and an iterative method. We have found that the size double-layered spherical quantum dot parameter has a significantly influence on the total optical AC, and the peaks shift, crease and toward the lower energies as R1 (R2) decreases. Also, the Alumi-num concentration (the potential height) has signifi-cant effects on the optical AC. We expect that this work will be of great help for describing the correct behavior of optical properties in DSQD with different size, which may be useful in technological applications as quantum cryptography or quantum information processing using single photon detector.  REFERENCES  1. H. Carrere, V.G. Truong, X. Marie, R. Brenot, G.de Valicourt, F. Lelarge, T. Amand, Appl. Phys. Lett. 97, 121101 (2010). 2. Hassan H. Hassan, Harold N. Spector, Phys. Rev. B 33, 5456 (1986). 3. D. Abouelaoualim, Acta Phys. Pol. A 112, 49 (2007). 4. Mohammadreza Khorasaninejad, Simarjeet Singh Saini, Opt. Express 18, 23442 (2010). 5. M.R.K. Vahdani, G. Rezaei, Phys. Lett. A 373, 3079 (2009). 6. D. O'Brien, S.P. Hegarty, G. Huyet, A.V. Uskov, Opt. Lett. 29, 1072 (2004). 7. J.C. Blakesley, P. See, A.J. Shields, B.E. Kardynał, P. Atkinson, I. Farrer, D.A. Ritchie, Phys. Rev. Lett. 94, 067401 (2005). 8. H.W. Li, B.E. Kardynał, P. See, A.J. Shields, P. Simmonds, H.E. Beere, D. Ritchie, Appl. Phys. Lett. 91, 073516 (2007). 9. V. Nikolaev, E.A. Avrutin, Phys. Rev. B 70, 125319 (2004). 10. G. Alvarez, Luis G. G. V. Dias da Silva, E. Ponce, E. Dagotto, Phys. Rev. E 84, 056706 (2011). 11. Anders M.N. Niklasson, Matt Challacombe, Phys. Rev. Lett. 92, 193001 (2003). 12. T. Takagahara, Phys. Rev. B 36, 9293 (1987). 13. Yen-Kuang Kuo, Wen-Wei Lin, Jpn. J. Appl. Phys. 41, 73 (2002).    

Absorption Coefficients of GaN / AlxGa1 – xN Core-Shell Spherical Quantum Dot

SocioEconomic Challenges Journal (Sumy State University)

JOURNAL OF NANO- AND ELECTRONIC PHYSICS ЖУРНАЛ НАНО- ТА ЕЛЕКТРОННОЇ ФІЗИКИ 
Vol. 4 No 3, 03004(5pp) (2012) Том 4 № 3, 03004(4cc) (2012) 
 
 
2077-6772/2012/4(3)03004(5) 03004-1  2012 Sumy State University 
Absorption Coefficients of GaN / AlxGa1 – xN Core-Shell Spherical Quantum Dot 
 
D. Abouelaoualim*, A. Elkadadra, A. Oueriagli, A. Outzourhit 
 
LPSCM,Department of Physics, Faculty of Sciences Semlalia,  
Cadi Ayaad University P.O. Box 2390, 40000 Marrakech, Morocco 
 
(Received 15 May 2012; published online 30 October 2012) 
 
The total absorption coefficient in spherical GaN/AlxGa1 – xN core-shell nanodots is theoretically inves-
tigated taking into account effective mass approximation. The influence of the nanosystem geometry upon 
the energy spectrum and transition energy E associated to interlevel transitions is studied. We found that 
the energy transitions vary with the core-shell radius, and the peak position of the total absorption coeffi-
cient is greatly affected by changing of the shell radius. The possibility of tuning the resonant energies by 
using the geometric core shell effect of the spatial confinement can be useful in the optoelectronic devices 
applications. Also we observed that the magnitudes of the total absorption coefficient can be increased sig-
nificatly compared traditional cases of QD, and the peaks are shifted to the lower energies. 
 
Keywords: Quantum dot, Core-shell, Optical absorption. 
 
  PACS numbers: 73.21.Ac, 78.67.Pt 
 
 
                                                          
*
 abouelaoualim_d@hotmail.com  
1. INTRODUCTION 
 
The interest in the study of the physical properties 
of confined quantum systems such as quantum wells 
[1], nanowires [2], and quantum dots (QDs) [3], has 
increased with the recent progress in semiconductor 
nanotechnology. In these structures the precise engi-
neering would enable us to confine the charge carriers 
motion in one, two or three dimensions. Quantum dots 
can be used to make not just better optical devices [4], 
but also as optical data storage medium, where bits of 
information are stored as electrons or holes confined 
within the dots [5]. One advantage of these is that 
since the information is typically stored as a single (or 
few) electron, these devices should require a very low 
switching energy. The charge to be stored in the dots 
can be written by optical illumination, which, by wave-
length selective writing or local probe techniques, can 
approach the situation where each quantum dot carries 
a bit of information. Such quantum dot optical memory 
devices could potentially achieve ultra-dense storage 
capacities of Terabit/in2. Recently, in order to aim of 
improving the electronic structures and optical proper-
ties of QDs structure single photon detector for applica-
tion as an optical data storage medium , different 
shapes and various confinement potentials have been 
widely studied, using the effective mass and parabolic 
one band approximations [6]. In this regard, we consid-
er a GaN/AlxGa1 – xN core-shell spherical quantum dot 
(CSQD)and an interest in the optical phenomena based 
on intersubband transitions in such structure, the opti-
cal. Because the confinement of quasiparticles in such 
structure leads to the enhancement of the oscillator 
strength of electron impurity excitations. Meanwhile, 
the dependence of the optical transition energy on the 
confinement strength (or dot size) allows the tunability 
of the resonance frequency. The outline of the paper is 
as follows: Hamiltonian, the relevant eigenvalues and 
eigenfunctions of an electron confined in a CSQD are 
described in Section 2. In Section 3, analytical expres-
sions for the linear and nonlinear intersubband optical 
ACs and RI changes are obtained using the density 
matrix approach and the iterative method. Our numer-
ical results and a brief conclusion are presented in Sec-
tions 4 and 5, respectively. 
 
2. MODEL AND THEORY  
 
The model used in calculation and analyses is an 
isolated GaN/AlxGa1 – xN core-shell quantum dot with 
core radius R1 and shell radius R2, shown in Fig. 1. 
 
  
Fig. 1 – Core-shell GaN-AlxGa1 – xN quantum dot system show-
ing (a) core and shell layers; and (b) layer potential profile 
 
Suppose the core-shell quantum dot forms a shell 
well and a centric barrier for the two kinds of materials 
have different potentials. The potential of the core is 
chosen to be referent zero point of energy, and the 
band-gap of GaN is wider than that of AlxGa1 – xN, thus 
VC. On the premise of the effective mass approxima-
tion, the steady Schrödinger equation for electron can 
be written as 
 
 
2
2
2
2
2 2
1
sin
sin2
1
( ) ( ),
sin
i
i nlm
r
r rm r
V r E r

  
 

         
            
 
   
  
 (1) 
 
where, 2h  , h is Planck’s constant; E the energy 
 D. ABOUELAOUALIM, A. ELKADADRA, A. OUERIAGLI… J. NANO- ELECTRON. PHYS. 4, 03004 (2012) 
 
 
03004-2 
eigenvalue; and nml(r) the responding eigenfunction. n 
is the principal quantum number, and l and m are the 
angular momentum quantum numbers. 
i
m  is the ef-
fective mass of electron in the ith region, (r)
 
 constant 
of core-shell QDs materiels and Vi(r) the potential. 
They are relative to the position in the model and ex-
pressed as follows 
 
 
1 1
2 1 2
,
< ,
i
m r R
m
m R r R



 
 

 (2) 
 
  
1
, 1 2
2,
0, 0<r ,
<r ,
, r
i c
R
V r V R R
R
 

 

 
 (3) 
 
Due to the fact that the mass and the potential are 
symmetric spherically, the separation of radial and 
angular coordinates leads to nml(r)  Rnml(r)Ynml(, ), 
where Rnlm(r) is the radial wave function, and Ynlm(,)
 
is the spherical harmonics n is the principal quantum 
number, and l and m are the angular momentum quan-
tum numbers. The radial eigenfunction, Rnl(r) consists 
of three parts according to the electron position. Two 
cases need to be considered for the solution of Rnl(r). In 
regions where electron eigenenergy E > Vc, the solution 
is a linear combination of spherical Bessel function jl 
and Neumann function nl, written as  
 
  
   
   
1 ,1 1 ,1 1,
2 ,2 2 ,2 1 2,
2,
,
,
0,
l nl l nl
nl l nl l nl
A J k r B n k r r R
R r A J k r B n k r R r R
r R
  


   


 (4) 
 
with 
 
 
2
,1 12 / ,nlk m E
  (5) 
 
   2,1 22 / ,nl ck m E V
   (6) 
 
A1, A2, B1 and B2
 
are normalized constants. In re-
gions where 
c
E V , the radial wave function is  
 
  
   
   
1 1
2
' '
,1 ,1 1,
'
,2 1 2,
2,
,
,
0,
l
l nl l nl
nl nl
A J k r B J k r r R
R r A h k r R r R
r R

  


  


 (7) 
 
with 
 
   2,2 22 / ,nl ck m V E
   (8) 
 
1
' '
2
,A A  and
1
'B  are normalized constants. 
Let us consider that the system is excited by a mon-
ochromatic electromagnetic field,   . .,i tE t Ee c c   
which is polarized along the Z-direction. Therefore, the 
time evolution of the density operator of the system is 
given by [7] 
 
  
01
, ,H ME t
t i

  
          
 (9) 
 
where H is the Hamiltonian of the system in the ab-
sence of the electromagnetic field, M ez  is the dipole 
operator along the Z-axis, 
0
  is the unperturbed densi-
ty operator and   is the damping operator due to the 
electron-phonon interaction and the other collision pro-
cesses. The above equation can be solved using the per-
turbation method by expanding    as [8]  
 
 
 
,
n
n
    (10) 
 
with 
 
 
 
     
 
1
1
1
1
, ,
,
n
n nij
ijij
n
ij ij
H M E t
t i
i

 




     
      
  
 (11) 
 
On the other hand, the electric polarization of the 
system can be written as 
 
 
     
 
0 0
1
,
i t i tP t Ee Ee
trace M
V
      

  

 (12) 
 
where V is the volume of the system, 0 is the permit-
tivity of free space, and trace means the summation 
over the diagonal elements of the matrix. Using the 
same density matrix formalism, the analytic expres-
sions of the linear and the third order nonlinear sus-
ceptibilities for a two-level quantum system are given 
and '  is the real part of the permittivity [9]: 
 
 
   
2
1
0 ,
v ij
ij ij
M
E i

  


  
 (13) 
 
 
   
   
 
   
2 22
3
0 2 2
2
4
,
v ij ij
ij ij
ij ij
ff jj
fj if fj if
M E M
E i E
M M
E i E i

  
 



     


    
 (14) 
 
where  is the carrier density, fj f iE E E   
is the en-
ergy interval of the two level system. Mfj is the dipole 
matrix element, which is defined by fi f iM er   ,  
 , 1,2f i   and Г shows the damping due to electron- 
phonon interaction. The absorption coefficient is given 
by 
 
     0' Im

     

  (15) 
 
The total absorption coefficient  ,I  can be writ-
 ABSORPTION COEFFICIENTS OF AGN/ALXGA1 – XN CORE-SHELL… J. NANO- ELECTRON. PHYS. 4, 03004 (2012) 
 
 
03004-3 
ten as 
 
          1 3, , ,I I I        (16) 
 
where I is the incident optical intensity and is de-
fined as  
 
  
'
2
2I E



  (17) 
 
Here  is the permeability of the system, and 
' is 
the real part of the permittivity. 
 
3. NUMERICAL RESULTS AND ANALYSIS 
 
In the numerical calculations carried out for a core-
shell quantum dot, the following parameter values 
have been used [13]: characterize the electrical and 
optical properties of the physical parameters used in 
our numerical work for GaN  and AlxGa1 – xN are taken 
from ref [10].The effective mass is 0.19GaN em m
  for 
GaN   
1 0
0.19 1 0.33
x xAl Ga N
m x m

    for AlxGa1 – xN. 
The real part of permittivity is chosen to be . The 
band-gap is    6.13 1 3.42 1 1.0cE x eV x eV x x eV      . 
The conduction band offset is chosen as 
    0.75 0b g gV E x E  . The dephasing constant cor-
responding to elastic scattering is 13 0.14ps   and the 
incident optical intensity is 31 /cmI MW . 
By virtue of the importance of intersubband transi-
tion, we have studied the variation of energy difference 
between the ground state to the exited state as a func-
tion of CSQDs size, with a GaN core and a Al1 – x GaxN 
shell, both of which were assumed to have a wurtzite 
structure. Such dots have a potential barrier of 
VC = 2.77 eV  for both electrons and holes to tunnel 
through in order to reach the shell from the core. A 
summary of these calculations is shown in Fig. 2(a), (b). 
In these graphs, when holding the shell thickness fixed, 
the energy levels behave similarly to those of QDs, 
Fig. 2(a). With the core radius fixed, increasing the 
shell size produces an entirely different trend-both the 
states Fig. 2(b), move down in energy. This causes the 
transition energy between those states to be roughly 
constant. These results imply a simple rule of thumb 
for this system- the transition energies are primarily 
determined by the shell radius. In Fig. 3, we display 
the total AC as function of pump photon energy   for 
different core  radius with a fixed shell radius when the 
transition energy levels selected in the region where 
E < Vc, namely inside the core well. It is seen from this 
figure that for the GaN / Al1 – x GaXN core-shell QDs 
the resonant magnitudes are 6.5 × 107 /m for R1 = 42 Å, 
Fig. 3(a) . It can be seen that each curve has only one 
peak which is due to single photon resonance. Fur-
thermore we can find that the total AC increase and 
the peak shift to lower energy. And the larger the R1 is, 
the smaller the peak shift is. 
These conclusions regarding the AC spectrum can 
be physically explained as follows: as a consequence of 
the quantum size effect, energy distances between  
 
a 
 
b 
 
Fig. 2 – Energy levels of GaN / AlxGa1 – xN core-shell quantum 
dot  with (a) fixed shell radius and (b) fixed core radius 
 
electronic states in the conduction band become smaller 
when R1 increases, Fig. 2(a), the larger the size is, the 
smaller the energy distance is; the increase of the dipole 
matrix element becomes stronger with the increase of 
the radius. When R2 increases from 42 Å to 52 Å with a 
fixed R1, the peak shift amount to 7 × 107 /m, Fig. 3(b). 
The reason for this phenomenon is related to electron 
confinement. When the electron eigenenergy E < VC, 
electrons are completely confined in the core well and 
there is nearly no electron distributing in the shell bar-
rier, the probability distribution for the electronic wave 
function mainly concentrates in the core well. Even 
though shell layer widens much, contributes a little to 
electronic wave function and the peak shift as long as 
the size of core layer keeps a constant. 
Effect of barrier mole fraction (as a parameter) on 
the total AC of the system is illustrated in Fig. 4. We 
plot the total optical AC as functions of the photon en-
ergy and x, with I = 1MW/cm2, R1 = 50°Å and R2 = 62 Å. 
It is shown that with increase of mole fraction, the ab-
sorption peak decreases. This is related to the fact that 
 D. ABOUELAOUALIM, A. ELKADADRA, A. OUERIAGLI… J. NANO- ELECTRON. PHYS. 4, 03004 (2012) 
 
 
03004-4 
 
a 
 
b 
 
Fig. 3 – Total AC is plotted as functions of the photon energy 
and (a) core radius R1 with R2 = 62 Å (b) shell radius R2 with, 
R1 = 48 Å;  with I = 1MW/cm2 and x = 0.15, when E < Vc 
 
 
 
Fig. 4 – Total optical AC as functions of the photon energy 
and Aluminium concentration x, with I = 1MW/cm2 and 
R1 = 48 Å and R2 = 62 Å 
 
the transition energy E . When Aluminum concentra-
tion increase, the barrier increases, this enhances the 
confinement of electrons. As a result, the overlap be-
tween electron wave function at two subbands increases, 
and thus the optical intersubband transition also en-
hances its intensity. 
 
4. CONCLUSIONS 
 
The total absorption coefficient (AC) of 
GaN / AlxGa1 – xN core-shell spherical quantum dot 
have been investigated, using the compact-density ma-
trix formalism and an iterative method. We have found 
that the size double-layered spherical quantum dot 
parameter has a significantly influence on the total 
optical AC, and the peaks shift, crease and toward the 
lower energies as R1 (R2) decreases. Also, the Alumi-
num concentration (the potential height) has signifi-
cant effects on the optical AC. We expect that this work 
will be of great help for describing the correct behavior 
of optical properties in DSQD with different size, which 
may be useful in technological applications as quantum 
cryptography or quantum information processing using 
single photon detector. 
 
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Absorption Coefficients of GaN / AlxGa1 – xN Core-Shell Spherical Quantum Dot

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