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BN Quantum well
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ScienceDirect Materials Today: Proceedings 2 (2015) 4474 – 4476
Hydrogenic donor binding energy in a strained BxGa1-xN/BN quantum well 1
N.Narayana Moorthy1 and A.John Peter2* Research Scholar, Madurai Kamaraj University, Madurai-625 021. India. 2 Dept.of Physics, Govt.Arts College, Melur-625 106. Madurai. India. *Corresponding author:[email protected] Abstract
Group III nitride materials are well known for their superior thermal, mechanical and optical properties. They can be used for short-wavelength light-emitting diodes, laser diodes and optical detectors and high electron mobility transistors and they have a high melting point, high thermal conductivity, a stronger resistance to oxidation at high temperatures and transparency to a large spectrum. Group III-Nitride alloys can be easily integrated in integrated electronic systems including current amplifiers. In the present work, the binding energy of a hydrogenic donor is studied taking into account the geometrical confinement and the various Boron alloy content. GaN semiconducting material acts as the inner quantum well material whereas BN material acts as the outer barrier material. The energy eigen values and binding energy of a hydrogenic donor are obtained using variational technique within a single band effective mass approximation. Keywords: Optical Properties, Group III-Nitride, hydrogenic donor, geometrical confinement.
where r is the distance between the electron and the
I. Introduction
r
Boron nitride and Gallium nitride materials are considered to be the promising wide band gap candidates ranging from the ultraviolet to the visible regions of the spectrum [1]. Boron nitride material is given attention due to the its exotic physical properties such as extreme hardness more than diamond, high melting point, and interesting dielectric, mechanical thermal properties and high oxidation resistance. It is predicted to have piezoelectricity and it can be applied for roomtemperature hydrogen storage [2].
The taken Hamiltonian of the hydrogenic impurity in a Boron based GaN coupled quantum well, within the framework of effective mass approximation, can be written as
H
2
2
! e 2 V ( z) * Hr 2me ( E )
V z
The trial wave function for the ground state electron without the impurity is chosen as z Lw Lb / 2
A cosh(k1 z ) ° ° B ((sin(k 2 z ) tan(k 2 Lw ) cos(k 2 z )) ° ®C cosh(k1 z ) ° D((sin(k z ) tan(k L ) cos(k z )) 2 2 w 2 ° °¯0
Lw Lb / 2 d z Lb / 2 z d Lb / 2 Lb / 2 z d Lb / 2 Lw z ! Lb / 2 Lw
(2) where A, B, C and D are obtained from the continuity of the wave function at the interfaces
k1 (1)
m* (E)
due to the band offset in confinement potential the coupled BxGa1-xN/BN quantum well structure is taken from the Ref. 3.
\ (U , z)
Model and calculation
U 2 z2
e impurity, , is the energy dependent effective mass of the electron, H is the dielectric constant of the inner quantum well. V(z) is the barrier height of the quantum well. The electron
k2
2m * (VB ( z ) E0 ) / ! 2 2m * E0 / !
,
2
, E0 is the ground state subband energy. It is calculated by solving the 2214-7853 © 2015 Elsevier Ltd. All rights reserved. Selection and peer-review under responsibility of the conference committee members of the International conference on Nano Science & Engineering Applications - 2014 doi:10.1016/j.matpr.2015.10.053
4475
N. Narayana Moorthy and A. John Peter / Materials Today: Proceedings 2 (2015) 4474 – 4476
transcendental
equation
k1 tan(k 2 Lw ) tan(k 2 Lb / 2)
k2
numerically. This fixes the values of k1 and k2 for the lowest values of the subband energy values. Lb is taken as 50 Å throughout this model. The binding energy, due to the hydrogenic donor, is defined as
E0 H
Eb H
.
H
.
min
.
(3)
min is obtained by minimizing the where Hamiltonian (Eq.(1)) with respect to the variational parameters in the trail wave function. The Schrödinger equation is solved variationally by
BxGa1-xN inner semiconductor material also increases with the Boron alloy concentration. The electronic and optical properties of any heterostructure semiconductor strongly depend on their spatial confinement and its composition. In conclusion, the subband energies and the donor binding energies have been studied taking into consideration of spatial and the Boron alloy content in a BxGa1-xN/BN double quantum well. The result shows that the binding energy enhances with the reduction in dimension. The reduction in dimensionality and the high surface area to volume ratio are the important factors which are responsible for this enhancement behaviour [4]. These enhancement properties are required for the potential applications leading to fabricate some novel optical devices.
min and the binding energy of the donor finding in a quantum well is given by the difference between the energy with and without the Coulomb interaction. The subband energy, E0, is the eigenvalue of the Hamiltonian without the Coulomb interaction term.
eV)
Results and discussion
defined as
R*y
m e4 / 2! 2H 2
Bohr radius is given by a
y (m nerg
The subband energy of the electron in the conduction band and thereby the binding energy of the donor in Boron gallium nitride semiconducting material is calculated taking into consideration of spatial confinement. The hydrogenic donor binding energy is obtained by choosing a suitable trail wave function which contains variational parameters. Numerical calculations are done using atomic units in which the electronic charge and the Planck’s constant are assumed as unity. The effective Rydberg energy is
(B) -- x=0.2 (C) -- x=0.3 (D) -- x=0.4
200
ing e
160
Bind
120 80 40
B C
and the effective
D
! 2H / m e 2 .
The variation of binding energy as a function of well width in a BGaN/BN quantum well for various Boron alloy concentration is presented in Fig.1. It is observed that the confined barrier height increases with the Boron alloy content in BxGa1-xN ternary material. It is because the band gap of B xGa1-xN increases with the Boron alloy content and the quantum confinement of electrons. The enhancement of binding energy is observed when the well is reduced in all the cases. However, for the smaller well size the donor binding energy seems to decrease due to the squeezing of the wave function into the barrier. On contrary, the binding energy approaches the bulk value of the barrier for the larger well width. The enhancement of effective Bohr radius and the reduction of effective Rydberg are noticed when the Boron alloy content is increased. The band gap of
5
10 20 40 60 1002005001000 --
Well width ()
Fig.1 Variation of binding energy as a function of well width in a BxGa1-xN/BN quantum well for various Boron alloy concentration.
.
4476
N. Narayana Moorthy and A. John Peter / Materials Today: Proceedings 2 (2015) 4474 – 4476
References [1] [2]
X.Luo et al., J. Phys. Chem. C. 112 (2008) 9516. S. H. Jhi and Y. K. Kwon , Phys. Rev. B. 69 (2004) 245407 .
[3] [4] [5]
N.Narayana Moorthy, A.John Peter and Chang Woo Lee, Superlatt. Microstruct. 70 (2014) 13. N.S.Minimala, A.John Peter and Chang Kyoo Yoo, Phase Transition, 86 (2013) 824. S.Baskoutas, W.Schommers, A.F.Terzis, M.Rieth, V.Kapaklis and C.Politis, Phys.Lett.A 308 (2003) 219.
ScienceDirect Materials Today: Proceedings 2 (2015) 4474 – 4476
Hydrogenic donor binding energy in a strained BxGa1-xN/BN quantum well 1
N.Narayana Moorthy1 and A.John Peter2* Research Scholar, Madurai Kamaraj University, Madurai-625 021. India. 2 Dept.of Physics, Govt.Arts College, Melur-625 106. Madurai. India. *Corresponding author:[email protected] Abstract
Group III nitride materials are well known for their superior thermal, mechanical and optical properties. They can be used for short-wavelength light-emitting diodes, laser diodes and optical detectors and high electron mobility transistors and they have a high melting point, high thermal conductivity, a stronger resistance to oxidation at high temperatures and transparency to a large spectrum. Group III-Nitride alloys can be easily integrated in integrated electronic systems including current amplifiers. In the present work, the binding energy of a hydrogenic donor is studied taking into account the geometrical confinement and the various Boron alloy content. GaN semiconducting material acts as the inner quantum well material whereas BN material acts as the outer barrier material. The energy eigen values and binding energy of a hydrogenic donor are obtained using variational technique within a single band effective mass approximation. Keywords: Optical Properties, Group III-Nitride, hydrogenic donor, geometrical confinement.
where r is the distance between the electron and the
I. Introduction
r
Boron nitride and Gallium nitride materials are considered to be the promising wide band gap candidates ranging from the ultraviolet to the visible regions of the spectrum [1]. Boron nitride material is given attention due to the its exotic physical properties such as extreme hardness more than diamond, high melting point, and interesting dielectric, mechanical thermal properties and high oxidation resistance. It is predicted to have piezoelectricity and it can be applied for roomtemperature hydrogen storage [2].
The taken Hamiltonian of the hydrogenic impurity in a Boron based GaN coupled quantum well, within the framework of effective mass approximation, can be written as
H
2
2
! e 2 V ( z) * Hr 2me ( E )
V z
The trial wave function for the ground state electron without the impurity is chosen as z Lw Lb / 2
A cosh(k1 z ) ° ° B ((sin(k 2 z ) tan(k 2 Lw ) cos(k 2 z )) ° ®C cosh(k1 z ) ° D((sin(k z ) tan(k L ) cos(k z )) 2 2 w 2 ° °¯0
Lw Lb / 2 d z Lb / 2 z d Lb / 2 Lb / 2 z d Lb / 2 Lw z ! Lb / 2 Lw
(2) where A, B, C and D are obtained from the continuity of the wave function at the interfaces
k1 (1)
m* (E)
due to the band offset in confinement potential the coupled BxGa1-xN/BN quantum well structure is taken from the Ref. 3.
\ (U , z)
Model and calculation
U 2 z2
e impurity, , is the energy dependent effective mass of the electron, H is the dielectric constant of the inner quantum well. V(z) is the barrier height of the quantum well. The electron
k2
2m * (VB ( z ) E0 ) / ! 2 2m * E0 / !
,
2
, E0 is the ground state subband energy. It is calculated by solving the 2214-7853 © 2015 Elsevier Ltd. All rights reserved. Selection and peer-review under responsibility of the conference committee members of the International conference on Nano Science & Engineering Applications - 2014 doi:10.1016/j.matpr.2015.10.053
4475
N. Narayana Moorthy and A. John Peter / Materials Today: Proceedings 2 (2015) 4474 – 4476
transcendental
equation
k1 tan(k 2 Lw ) tan(k 2 Lb / 2)
k2
numerically. This fixes the values of k1 and k2 for the lowest values of the subband energy values. Lb is taken as 50 Å throughout this model. The binding energy, due to the hydrogenic donor, is defined as
E0 H
Eb H
.
H
.
min
.
(3)
min is obtained by minimizing the where Hamiltonian (Eq.(1)) with respect to the variational parameters in the trail wave function. The Schrödinger equation is solved variationally by
BxGa1-xN inner semiconductor material also increases with the Boron alloy concentration. The electronic and optical properties of any heterostructure semiconductor strongly depend on their spatial confinement and its composition. In conclusion, the subband energies and the donor binding energies have been studied taking into consideration of spatial and the Boron alloy content in a BxGa1-xN/BN double quantum well. The result shows that the binding energy enhances with the reduction in dimension. The reduction in dimensionality and the high surface area to volume ratio are the important factors which are responsible for this enhancement behaviour [4]. These enhancement properties are required for the potential applications leading to fabricate some novel optical devices.
min and the binding energy of the donor finding in a quantum well is given by the difference between the energy with and without the Coulomb interaction. The subband energy, E0, is the eigenvalue of the Hamiltonian without the Coulomb interaction term.
eV)
Results and discussion
defined as
R*y
m e4 / 2! 2H 2
Bohr radius is given by a
y (m nerg
The subband energy of the electron in the conduction band and thereby the binding energy of the donor in Boron gallium nitride semiconducting material is calculated taking into consideration of spatial confinement. The hydrogenic donor binding energy is obtained by choosing a suitable trail wave function which contains variational parameters. Numerical calculations are done using atomic units in which the electronic charge and the Planck’s constant are assumed as unity. The effective Rydberg energy is
(B) -- x=0.2 (C) -- x=0.3 (D) -- x=0.4
200
ing e
160
Bind
120 80 40
B C
and the effective
D
! 2H / m e 2 .
The variation of binding energy as a function of well width in a BGaN/BN quantum well for various Boron alloy concentration is presented in Fig.1. It is observed that the confined barrier height increases with the Boron alloy content in BxGa1-xN ternary material. It is because the band gap of B xGa1-xN increases with the Boron alloy content and the quantum confinement of electrons. The enhancement of binding energy is observed when the well is reduced in all the cases. However, for the smaller well size the donor binding energy seems to decrease due to the squeezing of the wave function into the barrier. On contrary, the binding energy approaches the bulk value of the barrier for the larger well width. The enhancement of effective Bohr radius and the reduction of effective Rydberg are noticed when the Boron alloy content is increased. The band gap of
5
10 20 40 60 1002005001000 --
Well width ()
Fig.1 Variation of binding energy as a function of well width in a BxGa1-xN/BN quantum well for various Boron alloy concentration.
.
4476
N. Narayana Moorthy and A. John Peter / Materials Today: Proceedings 2 (2015) 4474 – 4476
References [1] [2]
X.Luo et al., J. Phys. Chem. C. 112 (2008) 9516. S. H. Jhi and Y. K. Kwon , Phys. Rev. B. 69 (2004) 245407 .
[3] [4] [5]
N.Narayana Moorthy, A.John Peter and Chang Woo Lee, Superlatt. Microstruct. 70 (2014) 13. N.S.Minimala, A.John Peter and Chang Kyoo Yoo, Phase Transition, 86 (2013) 824. S.Baskoutas, W.Schommers, A.F.Terzis, M.Rieth, V.Kapaklis and C.Politis, Phys.Lett.A 308 (2003) 219.