5 shows the normalized strain εH/ε0H versus the normalized vertical distance z/h for various values of α for r/R =0.0 and r/R=0.5, and ε0H is the strain of the cylinder under compression
Trang 1An Analytical Solution for Inhomogeneous Strain Fields
to constant strain without end friction and α= Fig 4 shows the normalized axial strain 1
inhomogeneous, and the maximum values can be more than 40% and 30% for r/R=0.0 and
r/R=0.5 respectively, comparing to that without end friction Fig 5 shows the normalized strain εH/ε0H versus the normalized vertical distance z/h for various values of α for
r/R =0.0 and r/R=0.5, and ε0H is the strain of the cylinder under compression without end friction and can be calculated according to (58) Fig.5 shows that the normalized strain
0
/ H
H
ε ε is quite inhomogeneous, and the maximum values can be 100% and 53% more
than those without end friction for r/R=0.0 and r/R=0.5 respectively Overall, the
inhomogeneous strain distributions are induced in the cylinder as long as friction exists between the end surface and the loading platens, and the larger the friction on the end surfaces, that is, the smaller the value of β, the more non-uniform inhomogeneous strain
is induced within the cylinder
9.2 The strain distributions within cylinder for different shape of cylinder
All of the numerical calculations given above are for /h R=2.0 In order to investigate the shape effect on the strain distribution within cylinder under compression with end friction, Figs 6-8 plot the normalized strains / 0
rr rr
ε ε , εθθ/εθθ0 and / 0
zz zz
ε ε versus the normalized
vertical distance z/h from the center of the cylinder for various values of h/R for r/R=0.0 and
0.0
β= Figs 6-8 show that a larger deviation may be induced for shorter cylinder For example, 35% error in / 0
H H
ε ε can be induced even at the center of the cylinder for h/R=0.5
But the strain distributions for long cylinders are more homogeneous, especially the strains are relatively uniform at the central part of the cylinder if /h R≥ So a relatively long 2cylinder should be suggested for compression test
Fig 6 The normalized strain εrr/ε0rr versus the normalized distance z/h along the axis of loading for various ratios of h/R
Trang 2Fig 7 The normalized strain εθθ/ε0θθ versus the normalized distance z/h along the axis of loading for various ratios of h/R
Fig 8 The normalized strain / 0
zz zz
ε ε versus the normalized distance z/h along the axis of loading for various ratios of h/R
10 The effect of strain on the valence-band structure of wurtzite GaN
The band structure of wurtzite GaN deserves attention since the valence bands, such as the heavy-hole, light-hole and split-off bands are close each other The strain effects on wurtzite GaN are less understand (Chuang & Chang, 1996) Based on the deformation potential theory of Luttinger-Kohn and Bir-Pikus (Bir & Pickus, 1974), the valence-band structure of the strained wurtzite GaN can be described by a 6x6Hamiltonian according to the envelope-function method, and the basis function for wurtzite GaN can be written as
Trang 3An Analytical Solution for Inhomogeneous Strain Fields
whereα=(1 / 2 )e i(3 /4 3 /2)π +φ ,β=(1 / 2 )e i( /4π +φ/2)and φ=tan (−1 k y/ )k x
The 6x6 Hamiltonian is obtained as
Trang 46 0
The exact analytical solution for the inhomogeneous strain field within a finite and
transversely isotropic cylinder under compression test with end friction is derived The
method employed Lekhnitskii's stress function in order to uncouple the equations of
Trang 5An Analytical Solution for Inhomogeneous Strain Fields
equilibrium It was found that the end friction leads to a very inhomogeneous strain distribution within cylinder, especially in the area near the end surface Numerical results show that all of the strain components, including the axial, radial, circumferential and shear strains, are inhomogeneous, both in distribution pattern and magnitude, the maximum value of the strain concentration near the end surfaces can be 100% higher than the constant strain in the case without end friction However, the strain distributions are relatively uniform at the central parts of long cylinders, say in the area of 0.5− h z< <0.5h, the magnitude of the strains can be more than 2% of that without end friction The method for analyzing the effect of the strain and end friction on the band structure of wurtzite GaN is discussed, end friction has effect on the shape of constant energy surfaces of valence bands and the band gaps between the heavy-hole, light-hole and split-off bands of wurtzite GaN
12 Acknowledgment
This work was supported by the National Natural Science Foundation of China (Grant No
11032003 and 10872033) and the State Key Laboratory of Explosion Science and Technology
13 References
Bir, G L & Pickus, G E (1974) Symmetry and Strain Induced Effects in Semiconductors, John
Wiley, New York, USA
Chau K T & Wei, X X(1999) Finite solid circular cylinders subjected to arbitrary surface
load: part I Analytic solution International Journal of Solids and Structures, Vol 37,
pp 5707-5732
Choi, S W & Shah, S P (1998) Fracture mechanism in cement-based materials subjected to
compression Journal of Engineering Mechanics ASCE, Vol 124, pp 94-102
Chuang, S L & Chang, C S (1996), k⋅p method for strained wurtzite semiconductors
Physical Review B, Vol 54, pp 2491-2504
Goroff, I & Kleinman, L (1963) Deformation potentials in silicon III effects of a general
strain on conduction and valence levels Physical Review, Vol 132, pp 1080-1084 Hasegawa, H (1963) Theory of cyclotron resonance in strained silicon crystals Physical
Review, Vol 129, pp 1029-1040
Hussein, A & Marzouk, H(2000) Finite element evaluation of the boundary conditions for
biaxial testing of high strength concrete Material Structure, Vol 33, pp 299-308
Jiang, H & Singh, J (1997) Strain distribution and electronic spectra of InAs/GaAs
self-assembled dots: An eight-band study Physical Review B, Vol 56, pp 4696-4701 Lekhnitskii, S G(1963) Theory of elasticity of an anisotropic elastic body, English translation by
P Fern , Holden~Day Inc., San Francisco, USA
Mathieu, H ; Mele, P and Ameziane, E L., et al (1979) Deformation potentials of the direct
and indirect absorption edges of GaP Physical Review B, Vol 19,pp 2209-2223 Pollak, F H & Cardona, M (1968) Piezo-Electroreflectance in Ge, GaAs, and Si Physical
Review, Vol 172, pp 816-837
Pollak, F H (1990) In Strained-Layer Superlattices, edited by T Pearsall, Semiconductors and
Semimetals, Academic, Boston, , USA
Suzuki, K & Hensel, J C (1974) Quantum resonances in the valence bands of germanium I
Theoretical considerations Physical Review B, Vool 9, pp 4184-4218
Trang 6Singh, J (1992) Physics of Semiconductors and Their Heterostructures, McGraw~Hill Higher
Education, New York, USA
Torrenti, J.M ; Benaija, E H & Boulay, C (1993) Influence of boundary conditions on strain
softening in concrete compression test Journal of Engineering Mechanics ASCE, Vol
119, pp 2369-2384
Wei, X X ; Chau, K T & Wong, R H C (1999) A new analytic solution for the axial point
load strength test for solid circular cylinders Journal of Engineering Mechanics, Vol 125, pp 1349-1357
Wei, X X(2008) Non-uniform strain field in a wurtzite GaN cylinder under compression
and the related end friction effect on quantum behavior of valence-bands Mechanics of Advanced Materials and Structures, Vol 15, pp 612-622
Wright, A F(1997) Elastic properties of zinc-blende and wurtzite AlN, GaN, and InN
Journal of Applied Physics, Vol 82, pp 2833-2839
Trang 7Application of Quaternary AlInGaN- Based Alloys
for Light Emission Devices
Sara C P Rodrigues1, Guilherme M Sipahi2, Luísa Scolfaro3
and Eronides F da Silva Jr.4
1Departamento de Física - Universidade Federal Rural de Pernambuco
2Instituto de Física de São Carlos - Universidade de São Paulo
3Department of Physics - Texas State University
4Departamento de Física - Universidade Federal de Pernambuco
the pioneer works of Nakamura et al. at Nichia Corporation in 1993 (Nakamura et al.(1995)) when the blue LEDs and pure green LEDs were invented, an enormous progress
in this field was observed which has been reviewed by several authors (Ambacher (1998);Nakamura et al (2000)) The rapid advances in the hetero-epitaxy of the group-III nitrides(Fernández-Garrido et al (2008); Kemper et al (2011); Suihkonen et al (2008)) have facilitatedthe production of new devices, including blue and UV LEDs and lasers, high temperature andhigh power electronics, visible-blind photodetectors and field-emitter structures (Hirayama(2005); Hirayama et al (2010); Tschumak et al (2010); Xie et al (2007); Zhu et al (2007)).There has been recent interest in the AlxIn1−x−yGayN quaternary alloys due to potentialapplication in UV LEDs and UV-blue laser diodes (LDs) once they present high brightness,high quantum efficiency, high flexibility, long-lifetime, and low power consumption (Fu et al.(2011); Hirayama (2005); Kim et al (2003); Knauer et al (2008); Liu et al (2011); Park et al.(2008); Zhmakin (2011); Zhu et al (2007)) The availability of the quaternary alloy offers anextra degree of freedom which allows the independent control of the band gap and latticeconstant Another interesting feature of the AlGaInN alloy is that it gives rise to higheremission intensities than the ternary AlGaN alloy with the absence of In (Hirayama (2005);Wang et al (2007)) An important issue is related to white light emission, which can beobtained by mixing emissions in different wavelengths with appropriate intensities (Roberts(1997); Rodrigues et al (2007); Xiao et al (2004))
Highly conductive p-type III-nitride layers are of crucial importance, in particular, for the production of LEDs Although the control of p-doping in these materials is still subject of
discussion, remarkable progress has been achieved (Hirayama (2005); Zhang et al (2011)) and
14
Trang 8recently reported experimental results point towards acceptor doping concentration high as
≈1019cm −3(Liu et al (2011); Zado et al (2011); Zhang et al (2011))
The group-III nitrides crystallize in both, the stable wurtzite (w) phase and the metastablecubic (c) phase Unlike for the hexagonal w-structure, the growth of cubic GaN is morecomplicated due to the thermodynamically unstable nature of the structure In hexagonalGaN inherent spontaneous and piezoelectric polarization fields are present along the c-axisbecause of the crystal symmetry Due to these fields, non-polar and semi-polar systemshave attracted interest One method to produce real non-polar materials is the growth ofthe c-phase Considerable advances in the growth of c-nitrides, with the aim of getting acomplete understanding of the c-nitride-derived heterostructures have been observed (As(2009); Schörmann et al (2007)) Successful growth of quaternary c-AlxIn1−x−yGayN layerslattice matched to GaN has been reported (Kemper et al (2011); Schörmann et al (2006)).The absence of polarization fields in the c-III nitrides may be advantageous for some deviceapplications Besides, it has been shown that these quaternary alloys can be doped easily asp-type, and due to the wavelength localization the optical transition energies are higher in thealloys than in GaN (Wang et al (2007))
However, the exact nature of the optical processes involved in the alloys with In is a subject ofcontroversy Different mechanisms have been proposed for the origin of the carriers’ localizedstates in the quantum well devices One is related to the low solubility of InN in GaN, leading
to the presence of nanoclusters inside the alloy, which can be suppressed by biaxial strain
as predicted and measured in c-InGaN samples (Marques et al (2003); Scolfaro et al (2004);Tabata et al (2002)) The second mechanism proposes that the recombination occurs throughthe quantum confined states (electron-hole pairs or excitons) inside the well
In this chapter we show the results of detailed studies of the theoretical photoluminescence(PL) and absorption spectra for several systems based on nitride quaternary alloys, using the
k · p theory within the framework of effective mass approximation, in conjunction with the
Poisson equation for the charge distribution Exchange-correlation effects are also includedwithin the local density approximation (Rodrigues et al (2002); Sipahi et al (1998)) Allsystems are assumed to be strained, so that the optical transitions are due to confinementeffects The theoretical method will be described in section 2 Through these calculations
the possibility of obtaining light emission from undoped (see section 3) and p-doped (see
section 4)quaternary AlXIn1−X−YGaYN/AlxIn1−x−yGayN superlattices (SLs) is addressed
By properly choosing the x and y contents in the wells and the acceptor doping concentration
N A as well X and Y in the barriers, it is shown to be possible to achieve light emission which
covers the visible spectrum from violet to red The investigation is also extended to doublequantum wells (DQWs), as described in section 5, confronting the results with experimentaldata reported on these systems (Kyono et al (2006)) The results are compared with regard
to the PL emissions for the different systems , also when an external electric field is present.Finally it is shown that by adopting appropriated combinations of SLs is possible to obtain thebest conditions in order to get white-light emission This fact is fundamental in the design ofnew optoelectronic devices
2 Theoretical band structure and luminescence spectra calculations
During the last few years, the super-cell k · p method has been adapted to quantum wells and
superlattices (SLs) ( Rodrigues et al (2002); Sipahi et al (1996)) Using this approach, one canself-consistently solve the 8×8 Kane multiband effective mass equation (EME) for the charge
Trang 9Application of Quaternary AlInGaN- Based Alloys for Light Emission Devices 3
distribution ( Sipahi et al (1998)) The results below are calculated assuming an infinite SL ofsquared wells along <001> direction
The multiband EME is represented with respect to plane waves with vectors K=(2π/d)l (l
being an integer and d the SL period) equal to the reciprocal SL vector The rows and columns
of the 8×8 Kane Hamiltonian refer to the Bloch-type eigenfunctions| jm j k)ofΓ8heavy- and
light-hole bands,Γ7spin-orbit-split-hole band andΓ6conduction band; k denotes a vector in
the first SL Brillouin zone (BZ)
By expanding the EME with respect to plane waves(z | K)one is able to represent this equationwith respect to Bloch functions( r | jm j k+K e z) For a Bloch-type eigenfunction(z | E k)of the
SL of energy E and wavevector k, the EME takes the form:
where H0 is the effective kinetic energy operator, generalized for a heterostrucures H STis
the strain operator originated from the lattice mismatch, V HET is the potential that arises
from the band offset at the interfaces, which is diagonal with respect to jm j , j m j , V XC is theexchange-correlation potential for carriers taken within Local Density Approximation (LDA),
V A is the ionized acceptor charge distribution potential, and V H is the Hartree potential orone-particle potential felt by the carrier from the carriers charge density So the Coulomb
potential, V C given by contribution of V A and V Hpotentials, can be obtained by means of theself-consistent procedure, where the Poisson equation stands, in the reciprocal space as,
with ε being the dielectric constant, e the electron charge, N A(z) the ionized acceptors
concentration, and p(z)being the holes charge distribution, which is given by
where s is the spin coordinate.
The next term in the Hamiltonian is the strain potential, V ST The kind of strain in thesesystems is biaxial, so it can be decomposed into two terms, a hydrostatic term and anuniaxial term ( Rodrigues et al (2001)) Since the hydrostatic term changes the gap energy,thus not affecting the valence band potential depth, only the uniaxial strain componentwill be considered ( Rodrigues et al (2001)) This latter may be calculated by the followingexpression:
= − 2/3D u xx(1+2C12/C11), (4)where− 2/3D u is the shear deformation potential, C11and C12are the elastic constants, and
xxis the lattice mismatch which is given by:
357
Application of Quaternary AlInGaN- Based Alloys for Light Emission Devices
Trang 10with a barrier and a well being the lattice parameters of the barrier and well materials,respectively.
Through these definitions one can calculate the Fourier coefficients of the strain operator
(K | (z ) | K )and express the strain term of the Hamiltonian V STas follows:
(1997)) The band shift potential V HET is diagonal with respect to jm j , j m j, and is defined by
where(K | V HET | K )are the Fourier coefficients of V HETalong the growth direction
From the calculated eigenstates, one can determine the luminescence and absorption spectra
of the SL by using the following general expression ( Sipahi et al (1998))
πE n e ( k ) − E n q ( k ) − ¯h ω2+γ2
where m0 is the electron mass, ω is the incident radiation frequency, γ is the emission
broadening (assumed as constant and equal to 10 meV), E n e and E n qare the energies associated
to n e and n q, respectively, the electron and hole states involved in the transition The
occupation functions N n
e kand[1− N n
q k]are the Fermi-like occupation functions for states
in the conduction- and valence-band, respectively
For the calculation of luminescence (absorption) spectra, the sum in Eq ( 9) is performed overthe occupied states in the conduction (valence) band, and unoccupied states in the valence(conduction) band ( Sipahi et al (1998))
The oscillator strength, f n e n q ( k), is given by
Trang 11Application of Quaternary AlInGaN- Based Alloys for Light Emission Devices 5
GaN InN AlN
γ1 2.96 3.77 1.54
γ2 0.90 1.33 0.42
γ3 1.20 1.60 0.64
Δso(meV) 17 3 19a(Å) 4.552 5.030 4.380
Table 1 Values of the parameters used in the self-consistent calculations of the p-doped cubic
(Al0.20In0.05Ga0.75)N/(AlxIn1−x−yGay)N SLs Data extracted from Refs (Ramos et al (2001);Rodrigues et al (2000; 2002); Schörmann et al (2006))
where p x is the dipole momentum in the x-direction, σ e andσ q denote the spin values forelectron and holes, respectively
All the parameters used in this analysis are shown in Table I For the quaternary (AlxIn1−x−y
Gay )N band gap energy dependence on the alloy contents, x and y, was used the expression
provided in Ref.( Marques et al (2003)) For all the other quantities, linear interpolationswere taken using the values for the binaries, AlN, GaN, InN The temperature dependence
of bandgap energies was evaluated through the Varshni analytical expression as applied forGaN ( Kohler et al (2002))
3 Undoped cubic AlxInyGa1−x−yN systems
In order to analyze the effects of the use of quaternary alloys in the electronic transitions, Fig
1 presents the theoretical PL spectra at T= 2 K calculated for strained undoped In0.2Ga0.8N/AlxGayIn1−x−yN SLs with x=0.03, 0.10, and 0.20 and y=0.40, 0.47, and 0.51, respectively The
barriers, constituted by the ternary alloy, have width d1 =60 nm, while the wells have width
d2=3 nm It is important to remark that all systems are strained, so the luminescence cannotarise from nanoclusters created during the growth In all cases in this section the first peakseen in the PL spectra corresponds to the first electronic transition E1-HH1 (first electron levelE1 and first heavy-hole level HH1) ( Rodrigues et al (2005))
From Fig 1 one can observe that with the appropriate choice of parameters it is possible toreach wavelengths from the red to the blue region One can also see that, by changing the wellwidth as depicted in Fig 2, the peaks in the PL spectra exhibit larger variations As the wellwidth decreases, the transition energy gets closer to the red region This occurs because of thechanges in the energies caused by the confinement and strain effects, which become stronger
as the In content increases
As the results described above are from systems where InGaN represents the barriers andthe quaternary alloy is in the wells, one can change the picture and start analyzing systemswhere the barriers correspond to the quaternary alloys while the InGaN alloy forms the wells
359
Application of Quaternary AlInGaN- Based Alloys for Light Emission Devices
Trang 12Fig 1 Theoretical normalized PL spectra for strained undoped
In0.2Ga0.8N/AlxGayIn1−x−yN SLs, with x=0.03 (solid line), 0.10 (dashed line), and 0.20
(dotted line) and y= 0.40, 0.47, and 0.51, respectively, barrier width d1= 60 nm, well width d2
= 3 nm
Fig 2 PL peaks as a function of the well width d2for the same systems of Fig 1
Fig 3 presents calculated SL systems with the same configurations as Fig 1, but usingGaN as barriers instead of AlxGayIn1−x−yN It presents calculated theoretical PL spectra,
at T = 2 K, for Al0.10Ga0.47In0.43N/ In0.55Ga0.45N, Al0.17Ga0.47In0.36N/ In0.42Ga0.68N, and
Al0.25Ga0.47In0.28N/ In0.25Ga0.75N SLs (solid lines) The figure presents also, for comparison,the systems of Fig 1 (dashed lines) A similar behavior, as obtained in Fig.1, is seen alsofor InGaN barriers, with the possibility of light emission covering the entire visible spectra
Trang 13Application of Quaternary AlInGaN- Based Alloys for Light Emission Devices 7
Fig 3 Theoretical normalized PL spectra for the strained undoped SLs
Al0.10Ga0.47In0.43N/In0.55Ga0.45N (solid line), Al0.17Ga0.47In0.36N/In0.42Ga0.68N (dashed line)and Al0.25Ga0.47In0.28N/In0.25Ga0.75N (dotted line) The barrier width is d1= 60 nm and the
well width is d2= 3 nm For comparison, we show the results for the SLs GaN/In0.55Ga0.45N(dash-dotted line), GaN/GaN/In0.42Ga0.68N (dash-dot-dotted line), and GaN/In0.25Ga0.75N(short-dashed line) systems
However, this is not possible using GaN in the barriers, since we have a limitation imposed bythe fixed gap energy value for GaN Another finding refers to the transition energies appearinghigher when the quaternary alloys constitute the barriers, when compared with the case inwhich InGaN is in the barriers This can be explained by the effective mass values which arehigher in the AlxGayIn1−x−yN alloys than in InGaN
It is also very important to investigate the influence of an external electrical field on thetransition energies and how the results compare with those for the wurtzite phase structures
In Fig 4, the theoretical PL and electroluminescence (EL) spectra were depicted at T= 2 Kcalculated for strained undoped In0.1Ga0.9N/ AlxGayIn1−x−yN SLs with x=0.03, 0.10 and 0.20,
and y=0.50 For these calculations the barrier width is d1=8 nm and the well width is d2 =
3 nm The magnitude of the electric field was 1.6 MV/cm for the EL spectra calculations Theresults indicate that the electric field enhances the shift seen in the spectra towards the redregion, as compared with the PL spectra This fact can be better visualized in Fig 5, whichshows the reduction in the transition energy as the electric field increases Such behavior isattributed to the fact that the potentials become deeper as the electrical field increases Themain consequence is the presence of more levels occupied near the bottom of the potentialwells Comparing with wurtzite structures, which have intrinsic built-in electric fields, thesituation described here is very similar, however in cubic systems higher efficiencies arepredicted ( Rodrigues et al (2005))
361
Application of Quaternary AlInGaN- Based Alloys for Light Emission Devices
Trang 14Fig 4 Theoretical normalized PL (solid line) and electroluminescence (dashed line) spectrafor strained undoped In0.1Ga0.9N/AlxGayIn1−x−yN SLs, with x = 0.03, 0.10, and 0.20, and y =
0.50, respectively, barrier width d1= 8 nm and well width d2= 3 nm The electric field usedfor EL was 1.6 MV/cm
Fig 5 PL peaks as a function of the magnitude of the electric field for systems with the samequaternary alloy contents as the ones in Fig 4
4 Doped cubic AlxInyGa1−x−yN systems
An important aspect to be analyzed is the effect of the acceptor doping on the electronictransitions Fig 6 presents the PL spectra at T = 2 K for strained p-type doped
Al0.20Ga0.05In0.75N/ AlxInyGa1−x−yN SLs, for which x and y are varied as described in Table
Trang 15Application of Quaternary AlInGaN- Based Alloys for Light Emission Devices 9
2 The ionized acceptor doping concentration considered to be uniformly distributed in
the barriers and fully ionized, is also varied assuming values of N A = 5×1018cm −3 and
N A =10×1018cm −3 These values of N Aallow us to envisage what happens in the rangefrom very low hole concentrations up to concentrations as high as≈1019cm −3 The undopedsystem is also presented for comparison The barriers widths are 8 nm and the wells widthsare 3 nm ( Rodrigues et al (2007)) The choice of values for x and y, the Al and In alloy contentswas such to reach all the visible-UV wavelength region
c-(Al0.20In0.05Ga0.75)N/(AlxIn1−x−yGay )N x y 1 − x − y
red 0.00 0.35 0.65green 0.02 0.40 0.58blue 0.08 0.45 0.47blue-violet 0.10 0.50 0.40violet 0.15 0.55 0.30
Table 2 Values used for the alloy contents x and y in the p-doped
c-(Al0.20In0.05Ga0.75)N/(AlxIn1−x−yGay)N SLs, properly chosen to attain light emission inthe electromagnetic spectral regions indicated in the left column
Fig 6 Calculated normalized photoluminescence (PL) spectra, at T = 2 K, for
Al0.20In0.05Ga0.75N/AlxIn1−x−yGay N SLs, for x and y values as shown in Table 2, for ionized acceptor concentrations of N A=0, N A=5×1018cm −3 , and N A=10×1018cm −3 Theenergy range covers the electromagnetic spectrum from red to violet
In Fig 7 the PL peaks are depicted as a function of the acceptor doping concentration for thefirst electronic transition E1-HH1 As NAincreases a red-shift in energy is observed for allregions investigated, except for the red region which presents a second electronic transitionE1-HH2 (first electron level E1 and second occupied heavy-hole level HH2) for NA= 0 and
5×1018cm −3 This behavior is directly related to the transition probabilities in such systemsand the potential profile due to the charges distribution The later is determined by the balancebetween the Coulomb and exchange-correlation potentials contribution which defines thepotential bending
363
Application of Quaternary AlInGaN- Based Alloys for Light Emission Devices