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Supplementary material
The relation between k.p theory, self-consistent k.p theory and device simulation
models
1. Device simulation models:
The basic equations to describe the electrical behavior of device are the following:
a) Poisson’s equation of electric potential V (field)
-（
 0 dc
q
V）=  n  p  N D (1  f D )  N A f A   Ntj ( j  ftj )  Nit ,
j
E  V
q : basic electric charge;
 0 ,  dc : vacuum permittivity and relative permittivity of the material;
n, p : electron and hole concentration, usually by injection;
N D , N A : donor and acceptor concentration;
f D , f A : Fermi-Dirac distribution function for donor and acceptor;
N tj : defect density for deep trap of j level;
 j : Dirac Delta function for deep trap of j level;
f tj : Fermi-Dirac distribution function for deep trap of j level;
N it : interface charge density, for example, polarization charge;
E : electric field;
b) Current continuity equations for electrons and holes
 J n   Rntj  Rsp  Rst  Rauger  Gopt (t ) 
j
f
n
 ND D
t
t
 J p   R tjp  Rsp  Rst  Rauger  Gopt (t )  
j
f
p
 NA A
t
t
J n , J p : electrons and holes current flux density;
Rntj , Rtjp : nonradiative recombination term of deep trap of j level for electrons and
holes;
Rsp : spontaneous recombination term;
Rst : stimulated recombination term;
Rauger : Auger recombination term;
Gopt (t ) : generation term by optical pumping, only important in photoluminescence;
t : time variable;
The basic equations to describe optical behavior are:
c) Wave equation for light:
 2W ( x, y )  k02 ( ( x, y )   2 )W ( x, y )  0
x, y : space variable;
W : optical wave function;
k0 : wave vector in vacuum;
 : complex eigenvalue of wave equation or propagation constant;
d) Photon rate equation
c
1
[ gm 
1
1
ln( )]S  cm  Rsp dv  0
2 L rm
c : light velocity in vacuum;
1 : real part of propagation constant;
gm : modal gain;
L : cavity length;
rm : mirror reflectivity;
S : total number of photons;
cm : fraction of spontaneous emission that couples into lasing mode;
v : volume;
Equation d) is considered in devices such as lasers where cavity effect is obvious.
The basic equation to describe thermal behavior is:
3) Heat transport equation
cp 
T
  T  H
t
c p : specific heat capacity;
 : mass density of the material;
T : heat function;
t : time variable;
 : thermal conductivity;
H : heat sources including Joule heat, generation/recombination heat, and Peltier and
Thomson heating terms (thermoelectric terms);
For light emitting devices, light output power ( LOP ) is very important. Basically,
LOP  Rsp , where Rsp is the spontaneous recombination term. In a simple ABC model
based on classical theory, Rsp  Bn2 , but it usually requires quantum mechanical
calculation of band structure in quantum wells (this is exactly what have been done in
our simulation by k.p theory).
k.p theory is a way to solve the Schrödinger equation in semiconductors to obtain
band structure, wave function and recombination terms. k.p theory for nitride device
was recently developed by S. L. Chuang [1]. The k.p theory for nitride devices is
slightly different from that for GaAs-based ones. One key issue is how to deal with
polarization field, which was represented by the method by Bernardini, F. and V.
Fiorentini [2].
2. Self-consistent k.p theory
The self-consistent k.p theory just means a simultaneous solution of Schrödinger
equation and Poisson equation. So our model is in nature a self-consistent k.p theory
incorporated with more advanced models for nitride devices.
3. The methodology of using no polarization field as a reference in polar QWs
The methodology of using no polarization field as a reference in polar QWs has
been accepted in many groups in the GaN community [3]-[5].
4. Polarization charge, polarization field and screening effect
Polarization field includes piezoelectric polarization and also spontaneous
polarization. Polarization field was equivalent to interface charges through Poisson
equation:
 0 dc
q
 E =Nit
N it stands for polarization charge at the interface. As there are defects, deep level
traps, donor and acceptor (as shallow level traps) in the material. Even at zero bias
where there is no current injection, the field is still screened. Therefore, the equation
changes to:
 0 dc
q
 E =Nit  N defects
N defects stands for charge from defects. According to experiment, the screen ratio is
20%-80%, and we set it at 50% in our work. Note that N it and N defects generally do
not move in the material and also do not change with current injection (so the screen
ratio is relatively unchanged), that is
Nit  N0 (x  x it ), j
N 0 is a constant and  (x  x it ) stands for the Dirac Delta function of space variable.
When current density increases, free carriers such as electrons and holes increase,
which can be calculated by current continuity equation:
 J n   Rntj  Rsp  Rst  Rauger  Gopt (t ) 
j
f
n
 ND D
t
t
In the steady state, electron or hole concentration (n(x, j); p(x, j)) is a function of
space variable and current density. Because there exists the electric field, we have
n(x, j)  p(x, j),  x
Instead
 n(x, j)   p(x, j), j
x
x
Above is the charge neutrality condition in the device. So now the Poisson equation is
 0 dc
q
 E =Nit  N defects -n(x, j)  p(x, j),
We see how the screen ratio changes with current injection.
 0 dc
q
 0 dc
q
 0 dc
q
screened by defects,deep/shallow level traps
 E =N it 

further screened by carrier/current injection
 E =N it  N defects 

 E =N it  N defects -n(x, j)  p(x, j)
From both the current continuity equation and Poisson equation, generally we have
j , n , E 
This is the exactly way how we deal with this effect. Just as the Poisson equation
introduced at the beginning:
-（
 0 dc
q
V）=  n  p  N D (1  f D )  N A f A   Ntj ( j  ftj )  Nit ,
j
E  V
It should be emphasized that we make no hypothesis that the polarization charge Nit
decreases with current injection.
From our previous work [6], we get a formula between E and j as a parabolic fit.
The fit is empirical, and may be valid for only certain devices. A standard and
self-consistent way is to solve the Poisson equation and current continuity equation.
Reference:
[1] S. L. Chuang, and C. S. Chang, “k⋅p method for strained wurtzite
semiconductors”, Phys. Rev. B, 54, 2491 (1996).
[2] F. Bernardini and V. Fiorentini, “Nonlinear macroscopic polarization in III-V
nitride alloys”, Phys. Rev. B, 64, 085207 (2001).
[3] J. Piprek, R. Farrell, S. DenBarrs, and S. Nakamura, “Effects of built-in
polarization on InGaN-GaN vertical-cavity surface-emitting lasers”, IEEE Photon.
Technol. Lett., 18, 7 (2006).
[4] J.-R. Chen, C.-H. Lee, T.-S. Ko, Y.-A. Chang, T.-C. Lu, H.-C. Kuo, Y.-K. Kuo,
and S.-C. Wang, “Effects of built-in polarization and carrier overflow on InGaN
quantum-well lasers with electronic blocking layers”, J. Lightwave Technol., 26,
329 (2008).
[5] Z. Yang, R. Li, Q. Wei, T. Yu, Y. Zhang, W. Chen, and X. Hu, “Analysis of optical
gain property in the InGaN/GaN triangular shaped quantum well under the
piezoelectric field,” Appl. Phys. Lett., 94, 061120 (2009).
[6] Z. Gong, N. Y. Liu, Y. B. Tao, D. Massoubre, E. Y. Xie, X. D. Hu, Z. Z. Chen,
G. Y. Zhang, Y. B. Pan, M. S. Hao, I. M.Watson, E. Gu, and M. D. Dawson,
“Electrical, spectral and optical performance of yellow–green and amber
micro-pixelated InGaN light-emitting diodes”, Semicond. Sci. Technol., 27,
015003 (2012).