Transcript Conferences

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A meshless k∙p method for analyzing electronic
structures of quantum dots
Ting Mei
Email： [email protected]
The Key Laboratory of Space Applied Physics and Chemistry
Ministry of Education and Shaanxi Key Laboratory of Optical Information Technology
School of Science, Northwestern Polytechnical University
Xi'an 710072, China
Introduction
Formulation of FT-based k∙p method
Control of computation error
Case study: strain effect
Optical gain calculation
Control of Spurious solutions
Summary
Photonics Spectra Features –
QW Laser
QW LED
“Lasers Gone Dotty”
QD laser
single-photon source
Shrinking dimensions concentrates
It iselectron
crucial to
understand
electronic
structures
states
and benefits
lasing
and optical properties
• low threshold
• relatively temperature-insensitive
• quick modulation
They are all about band-edge transitions!
• good noise performance
A practical demand for electronic structure calculation
Intermixing Technique
 Core technique for fabrication of Photonic Integrated Circuits
 Local band gap modification after growth
 Control of interdiffusion process
InP / InGaAsP / InP
InP / InGaAs / InP
Laser
In
Ga
As
P
Eg0
Modulator
<
Eg1
Waveguide
<
Eg2
CB
E1
E1
Eg (as-grown) < Eg (intermixed)
Blueshift phenomina
HH1
HH1
VB
Absorption
After QWI
Before QWI
Laser modulator
Waveguide
ON OFF
op
Band gap energy
Photonic Integrated Circuits
A practical demand for electronic structure calculation
160
i-InP
n-InP
16
p-InP
RTA 240s
140
14
120
12
100
C-HH shift
C-LH shift
Blue shift (nm)
Intensity (a.u.)
180s
120s
90s
80 u-InP
LV
10
LIII
8
14
u-InP
12
60
10
40
8
20
60s
6
10
n-InP
n-InP
60
30s
p-InP
8
6
40
4
As-grown
20
2
1200 1300 1400 1200 1300 1400 1200 1300 1400
0
60
Wavelength (nm)
120
180
240
60
120
Annealing duration (s)
Deriving diffusion lengths from photoluminescence data
need calculate of electronic structures
o Appl. Phys. Lett. 91, 181111, 2007
180
240
(Å)
p-InP
Diffusion length
TE
TM
Practicability - Expected features of calculation method
 Time efficient
 Easy programming
 Allowing parametric investigation
i.e. study of Influence of structural shape
Modeling electronic structures
The first principles
atomistic, empirical
The k∙p method
continuum, empirical
applicable near the G point of
Brillouin zone (k=0)
good enough for optical
property
2D (QW)
102
103
104
105
atoms in the unit cell
Tight Binding
atoms that can be handled
atomistic, ab initial
most accurate
challenging in handling
excited states properties
3D bulk
101
1D (QWR)
106
0D (QD)
107
Nature of the Bands Near Bandedges
Fundamentals
In semiconductors we are primarily interested in the valence band and
conduction band. Moreover, for most applications we are interested in what
happens near the top of the valence band and the bottom of the conduction
band. These states originated from the atomic levels of the valence shell in
the elements making up the semiconductor
s+p (longitudinal)
s-type
p (transverse)
IV Semiconductors
E
Conduction
Band
(
)
CB
2
2
2
C 1s 2s 2p
Si 1s22s22p63s23p2
Indirect bandgap
2
2
6
2
6
2
2
Ge 1s 2s 2p 3s 3p 4s 4p
Direct bandgap
p-type
III-V Semiconductors
Ga 1s22s22p63s23p63d104s24p1
As 1s22s22p63s23p63d104s24p3
Heavy Hole Band ( HH)
Light Hole Band (LH )
Split Off Band (SO)
Outmost atomic levels are either s-type or p-type
Eg
k
∆
Fundamentals
Basic theory of k∙p method
Symmetry of bulk crystal
Crystal potential VL  R  r   y L  r 
Time-independent Schrödinger equation


2
 2m   VL  r y n  r   E  k y n  r 
 0

ik r
Bloch wave y n  r   e un ,k  r 
2 2
0
 3 T  V  2 W 
U
W U
  A 20 T2  V
k2 T  V  CB



V
(
r
)

k

p
u
(
r
)

E
(
k
)

)
  02mA 2 W  U  L  3 T m
0 n ,k T  V  n 2 T  V 2 m W uUn ,k (rCB
V 
0
0
0 




LH
*
*
*
*
*
*
 cc
 cc

 cc

 cc
cc
cc
cc
P  Q
cc
cc
cc
cc
cc
cc
cc
cc
cc
cc
cc
cc
Hamiltonian equations
H
0
3 2S
R
 2R
S*
1 2S *
n, j
cc
j
plane wave

 HH
1 2S 
 HH
2R* 
expansion
3 2S *  LH

0
 SO
j ,k 0 0
Z
 SO
CB
E
 2Q
Luttinger-Kohn
P  Q
2Q
Z a ( k )u
un ,cck (r )  
j
cc
unit cell function
*
*
 cc cc
 P  Qat k =0
S
R
Hamiltonian
Perturbation

cc
P  Q
0
H
( k )   cc cc
Center
of Brillouin
zone
term

y  R  r   y r 
Wave function
( k ) a j  En ( k ) a n
Eg
k
(r )
HH AlGaAs
GaAs
∆ AlGaAs
LH
j
Knowing uj,0 and En(k=0)  Solving En(k) and yn
o
o
o
o
SO
N. W. Ashcroft and N. D. Mermin, Solid state physics. New York: Holt, Rinehart and Winston, 1976
J. Callaway, Quantum theory of the solid state. Boston: Academic Press, 1991
J. M. Luttinger, and W. Kohn, Phys. Rev. 97, 869 (1955)
E. O. Kane, J. Phys. Chem. Solids. 1, 249 (1957).
Bulk crystal  low-dimensional structure
Bulk crystal
y n  r   eikr un ,k  r 
Low-dimensional structure
Envelope function theory
unit cell function
plane wave
un ,k (r )   a j ( k )u j ,k0 0 (r )
V
y (r )   F j  r  u j ,0  r 
j
j
Number of bands
Envelope function
Bloch basis
Band index
Position-dependent Hamiltonian
H (r; k ) :
k  i ;  k   ( )
2
3D mesh
for QDs
Coupling equations on the 3D mesh
AlGaAsGaAsAlGaAs
Finite element method (FEM)
Finite difference method (FDM)
o C. Pryor, Phys. Rev. B. 57, 7190 (1998)
o S. F. Tsay, et al. Phys. Rev. B. 56, 13242 (1997)
Numerical approaches for the k∙p method
•
Finite element •
method (FEM) •
•
•
Finite difference •
method (FDM) •
•
Can We?
Cumbersome mesh process
Finite element to deal with coupling equations
Numerical integration
More serious
Computationally intensive
issue in common –
poor control of
spurious solutions
Simple structures only
8 couple equations on every point
Numerical differentiation
Programming challenge for QDs
CB
•
•
•
•
discard the 3D mesh
skip numerical integration/differentiation
ease the programming job
have better control on spurious solutions
VB
Fourier-transform k∙p method (FTM)
i z
H( z; k || , kˆz )F( z )  EF( z )
Real space
FT
Momentum space
(Fourier domain)
H( k z )
Differential equation
FT
k z F(k z )
Linear equation
Analytical expression or
FFT of the shape function
Mesh for dealing with coupling equations
Numerical integration
Numerical differentiation
truncation excellent control of spurious solutions!
The use of Fourier series 
o
o
o
o
T. Mei, J. App. Phys. 102, 053708 (2007)
Q. J. Zhao, and T. Mei, J. Appl. Phys., 109, 063101 (2011)
Q. J. Zhao, T. Mei, D. H. Zhang, and O. Kurniawan, Opt. Quant. Electron., 42, 705(2011)
Q. J. Zhao, T. Mei, and D. H. Zhang, J. Appl. Phys., 111,053702 (2012)
Detailed FTM formulation for QW superlattice
Envelope function in Fourier series expansion

V

y (r )   F j  r  u j ,0  r  Fz  r   1
j
L eikz z  c n ein z
Waver vector   2  k z   2
Fourier Frequency
c n  c1 , c2 ,..., c8 n
  2 L
T
n
Periodical length
Boundary condition for N QW superlattice periods
Hamiltonian in Fourier
series expansion
Complex exponential
functions’ orthogonality
H  H  zz   z 
Eigen-equation
Transformation
of Hamiltonian

2

2 j
,0 j N
NL
 H z   z  k z  H0  z 

 2 

   H  zz   q  eiq z  k z    H  z   q  eiq z  k z   H  0  q  eiq z
q
 q

 q

Operator:
H Q    ei k z m  z Q ei k z n  z
q
2
kz
kz  
H( )
H( )
  m   n  k  H z q  m  n


z
uv 
2


 m  k z  n  k z  H uv zz   q  m  n 
 M st ct   E ct 
Order of Fourier truncation
Dimension of M: V
s  j  V (m  N tr )
t  j ' V (n  N tr )
 2 N tr  1 V  2 N tr  1
T. Mei, J. App. Phys. 102, 053708 (2007)
H( )
H uv 0  q  m  n 
j, j '  1,2, V
m, n   N tr , 0,
q  2 N tr ,
0,
N tr
2 N tr
Hamiltonian matrix for QD (SYM & BF operator ordering)
1
i  n x x x  n y  y y  n z  z z 
ik x x  ik y y  ik z z
Wavefunction
F r  
e
cnx n y nz e

d xd yd z
n x n y nz
Burt-Foreman (BF) Operator Ordering
H  H  zz  k z  H L z  k z  k z H R z   H  0
2
Mc  E c
Eigen equation
Symmetrical (SYM) Operator Ordering
H  H  zz  k z  H  z  k z  H 0
2
Hamiltonian matrix (SYM)
Hamiltonian matrix (BF)
M st 
M um ,vn 
 H jj ',qi mi ni
 H uv(0),qi mi ni
  k   n     k  n    
 
1
nd order
 
H
2

uv ,qi  mi  ni  k  m     k   n   


H
 jj ',qi mi ni   k  n  k  n    ,
2  ,
   
  
  
 H uv,qi mi ni ,L  k  n    
st order
1

1

 
  H jj ',qi mi ni  2k   m  n    
 
   H uv ,q  m  n , R  k  m    
2 


i
i
i
0
Dimensions of Hamiltonian V
o
o
 2 N  1 V   2 N  1  ,   x, y, z 



Q. J. Zhao, and T. Mei, J. Appl. Phys., 109, 063101 (2011)
Q. J. Zhao, T. Mei, and D. H. Zhang, J. Appl. Phys., 111,053702 (2012)
0th order
Finite Difference Method (FDM) v.s. Fourier Transform Method (FTM)
QW: Position-dependent Hamiltonian
H  H ( zz ) k z2  H ( z ) k z  H (0)
ˆ  i 
Differential operator:
k
z
z
FDM: finite difference
FTM: Fourier components
in momentum space
H  H ( zz ) kˆ 2  H ( z ) kˆ  H (0)
in real space
H   H ( zz ) 2z  iH ( z ) z  H (0)
  z  H
( zz )
z f 
z
 H
zi
z
zz 
 q  eiq z kˆz2
q
 H 
i
  H ( z )  z f   z H ( z ) f 
zi
2
 H (0)
z
 q  eiq z kˆz
0
q
q
 H 
zi
q
Mesh points:
0
o
o
1
2
…
i-1 i i+1
…
N-1 N
X. Cartoxia, “Theoretical Methods for Spintronics in Semiconductor with Applications”, Doctor
of Philosophy, Carliforlian Institute of Technology, Pasadena(2003)
T. Mei, J. Appl. Phys. 102, 053708(2007).
Finite Difference Method (FDM) v.s. Fourier Transform Method (FTM)
FDM
FTM
•
Finite difference
•
Real space
•
Mesh points
•
•
•
Fourier transform
Solving domain
•
Fourier domain
Matrix dimension
•
Order of Fourier truncation
Banded distribution
Matrix sparse (nonzero)
•
Highly sparse
Mesh points
Computation accuracy
•
Mesh points & Order of
Fourier truncation
Numerical technique
Simple (either FFT or analytic expression)
Simplifies optical property computation
Much smaller matrix dimension
High demand for computer memory
Mainly determined by the latter
o
o
o
o
X. Cartoxia, et al J. Appl. Phys. 93, 3974(2003
T. Mei, J. Appl. Phys. 102, 053708(2007).
B. Lassen, et al, Commun. Comput. Phys. 6, 699 (2009)
W. Liu, et al, J. Appl. Phy. 104, 053119 (2008)
Control of Computation errors in FTM (isolated QDs)
Ground state energies E1 and HH1(eV)
Ground state energies E1E1
andand
HH1(eV)
HH1 (eV)
Crosstalk
error
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
-0.1
-0.2
-0.3
-0.4
-0.5
-0.6
-0.7
-0.8
Truncation
error
Pyramidal QD
L = 3Lx is chosen by Andreev & Gunawan
Lx=Ly=136 Å h=60 Å
0.7 1.0
0.6
0.5
0.4
0.3
0.2
0.1
0.0
-0.1
-0.2
-0.3
-0.4
-0.5
-0.6
-0.7
-0.8
1.0
1.5
2.0
2.5
Cuboidal QD
3.0
3.5
4.0
4.5
Volume of
computation
Periodical length
L
Selection of Periodical length L
Ntr=1
Ntr=2
Ntr=3
Ntr=4
Ntr=5
Ntr=6
Ntr=7
LLx
Fourier truncation
Ntr
Truncation
error
Crosstalk
error
Pyramidal QD
•
5.0
L/Lx
Ntr=7 by trading off the demand of higher
order of Fourier truncation and the limit of
computer capacity
Cuboidal QD
Lx
•
Ntr=1
Ntr=2
Ntr=3
Ntr=4
Ntr=5
Ntr=6
L
A smaller Ntr (= 6) is enough for smooth
structures
 Smooth structures possess narrow-span
Fourier spectrum (Mei)
Selection of Fourier truncation Ntr
ax=ay=az=136 Å
Computation accuracy & Computer capacity
1.5
2.0
2.5
3.0
L/LL/a
x x
3.5
o
o
o
4.0
4.5
5.0
A. D. Andreev,et al J. Appl. Phys., 86, 297(1999)
O. Gunawan,et al, Phys. Rev. B, 71, 205319(2005)
T. Mei, J. App. Phys. 102, 053708 (2007)
Fundamentals
Strain definition & influence
∆x
x
P ' Q ' PQ u

PQ
x
u du
P : e  lim

x  0  x
dx
1  u u
QDs : eij   i  j
2  x j xi
 i , j  1,2,3 
PQ :
(a)
P
O
∆x+∆u
x+∆u
(b)
Q
Q’
P’
O
Deformation of an extendible string:
(a) unstreched and (b) stretched
(a) unstrained
(b) strained
CB
Compression
tension
a
a
as
as
as
as
Epitaxial growth of a material layer on a
substrate
o
o
•
•



•
•
Not a property of crystals
A response of external force
(Lattice mismatch)
A balance of internal force
(stress)
Displacement
 e11 e12
e
e22
 13
e
 23 e13
E
e13 
e23 

e33 
k
HH
LH
eh  e11  e22  e33
eb  e33   e11  e22  2
SO
Strain influence on band structure
J. F. Nye, Physical properties of crystals: their representation by
tensors and matrices, Oxford University Press, Oxford, 1985
S. L. Chuang, Physics of optoelectronic device, Wiley, New York,
1995, p. 144-154.
Fundamentals
Strain computation
Continuum Mechanical (CM) theory
•
•
A microscopic physical theory
Green’s function method
–
Real space
Green’s tensor is used to represent the
response of the external force
Real space – integration
Fourier domain – Fourier-transform
–
–
Eshelby’s inclusions theory
Stress
•
Green’s tensor
•
QD shape function
•
Young’s modulus
•
Poisson’s ratio
Fourier domain
Displacement
•
Green’s tensor
•
QD shape function
•
Elastic constants
Strain tensor of a single QD in the Fourier domain
1




 2 C  C  2  2 
2
 
C

2
C



an i



11
12
i j
  44
eijs  ξ    0   ξ   ij 

2
3



1
p



1   C12  C44  
2
2 
2 


p 1 C44  Can p 
 2  C44  Can  j    

ξ
: Cartesian coordinates in Fourier domain
Cij : Elastic constants Can  C11  C12  2C44
  ξ  : QD shape function in the Fourier domain
0
: Initial lattice constant
Analytical strain expression with linear elasticity
eij 
 2 
3
o
o
 e  ξ  exp iξ  r 
d1d 2d 3 n1 ,n2 ,n3
s
ij
n
n
o
o
J. R. Downes, D. A. Faux, and E. P. O’Reily,
J. Appl. Phys. 81, 6700 (1997).
G. S. Pearson and D. A. Faux, J. Appl. Phys.
88, 730 (2000).
D. A. Faux and U. M. Christmas, J. Appl.
Phys. 98, 033534 (2005).
D. Andreev, et al, J. Appl. Phys.86, 297
(1999).
Effects of dimension and strain – Bandedge profiles
(a)
(b)
Band-edge profile along [100] direction (eV)
0.4
0.2
CB
0
Solid lines: with strain
-0.4 Dashed lines: without strain
HH
-0.6
-1.2 (a)
-150
0.8
-100
LH
-50
0
50
Distance along z direction(Å)
100
Solid lines: with strain
-0.4 Dashed lines: without strain
HH
LH
-0.8
-1
SO
-100
-50
0
50
Distance along x direction(Å)
0
Solid lines: with strain
Dashed lines: without strain
-0.6
No sharp
features
-0.8flatten top
due to
HH
LH
-1
SO
-100
-50
0
50
Distance along z direction(Å)
z (Å)
o
o
o
100
150
100
150
0.6
0.4
Enlargement0.2of bandgap
Reduction
of CB&LH potential wells
CB
0
Increase of HH potential
well
0.2
-1.2
-150
-0.2
0.8
0.4
-0.4
CB
-1.2 (b)
-150
150
0.6
-0.2
0.4
Sharp features due to
concentrated-0.6strain
SO
-1
0.6
with strain 0.2
without strain 0
-0.2
Crossover of HH
-0.8
and LH
wells
(ev)
along
Bandedge
Band-edge profile along
[001] [001]
direction (eV)
Bandedge along [100] (ev)
InAs/GaAs QD
Band-edge profile along [100] direction (eV)
0.6
0.8
Bandedge along [100] (ev)
[001]
along
Bandedge
Band-edge profile
along [001]
direction(ev)
(eV)
0.8
-0.2
-0.4
CB
Solid lines: with strain
Dashed lines: without strain
HH
-0.6
LH
-0.8
-1
-1.2
-150
SO
-100
-50
0
50
Distance along x direction(Å)
Q. J. Zhao et al, J. Appl. Phys., 109, 063101-063113 (2011)
C. Pryor, Phys. Rev. B 57, 7190(1998)
A. Schliwa, et al. Phys. Rev. B 76, 205324(2007)
x (Å)
100
150
Effects of dimension and strain – Ground state energies
Variations of E1 , HH1, and LH1 v.s. truncation factor
Strain effect (E1 and LH1) is more significant at
small f (concentrated strain)
•
Strain effect does not change HH1 much
due to little change of band-edge profile
•
The main contribution of CB-VB is the
variation of E1 (consistent with the tightbinding work)
Ground energy states (eV)
•
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
-0.1
-0.2
-0.3
-0.4
-0.5
-0.6
-0.7
-0.8
CB
Solid circles: with strain
Hollow circles: without strain
HH
LH
0.0
0.1
0.2 0.3
0.4
0.5
0.6 0.7
Truncation factor f
o
o
Q. J. Zhao et al, J. Appl. Phys., 109, 063101-063113 (2011)
R. Santoprete, et al., Phys. Rev. B 68,235311(2003)
0.8
0.9
Effects of dimension and strain – 3D PDFs
3D Probability density functions (PDFs) of a pyramidal InAs/GaAs QD structure
(a)
Band-edge profile along [001] direction (eV)
0.8
0.6
0.4
0.2
CB
0
-0.2
Solid lines: with strain
-0.4 Dashed lines: without strain
HH
-0.6
LH
-0.8
SO
-1
-1.2
-150
-100
-50
0
50
Distance along z direction(Å)
100
150
(b)
Band-edge profile along [100] direction (eV)
0.8
0.6
0.4
0.2
-0.2
Solid lines: with strain
-0.4 Dashed lines: without strain
HH
-0.6
LH
-0.8
-1
-1.2
-150
•
•
•
CB
0
SO
-100
-50
0
50
Distance along x direction(Å)
100
Variation of CB’s major PDF component is not as obvious as that of VB’s
Reduction of band mixing( CB and VB, HH and LH)
Variations of PDFs’ size, shape, and position
Q. J. Zhao et al, J. Appl. Phys., 109, 063101-063113 (2011)
150
Effects of dimension and strain – 2D PDFs
2D Probability density functions (PDFs) of a pyramidal InAs/GaAs QD structure
(a) With strain
•
•
•
(b) Without strain
Variations of PDFs’ size, shape, position, and value
Weaker confinement of carriers
Consistent with band-edge profiles
Q. J. Zhao et al, J. Appl. Phys., 109, 063101-063113 (2011)
Fundamentals
Optical gain calculation for QWs
1
Lorentzian/Sech line-shape function gain spectrum
Band mixing effect
2
Spontaneous emission coefficient gain spectrum
g spe     C0  
dk x dk y
4 2
nc ,nv

M nec ,nv  k x , k y  f ncc  k x , k y  1  f nvv  k x , k y 
  
Lorentzian line-shape function
 Enc  k x , k y   Env  k x , k y        2
v
 c

o
2
o
o
Exact envelope function theory
y (r )   F j  r  u j ,0  r 
ik x x ik y y
c  z
n
j n
n
Basis function
n  z   1
Envelope function
M
W. W. Chow et al., Semiconductor-Laser Fundamentals. Berlin:
Springer, 1999
S. L. Chuang et al, IEEE J. Quantum Elect., 32, 1791(1996)
D. Gershoni et al., IEEE J. Quantum Elect., 29, 2433(1993)
Envelope function of QWs
Fj (k )  e
j
e
nc ,nv
Spontaneous emission
2
 y nv eˆ  p y nc
o
o
Plane wave expansion
Fourier series
L exp i  k z  n 2 L  z 
nv ,n *

   c j   c nj c' ,n  u j eˆ  p u j '


j
j' n 
W. J. Fan, et al., J. Appl. Phys., 80, 3471(1996)
W. Liu, et al, J. Appl. Phys. 104, 053119 (2008)
Optical gain calculation result by six-band FTM
Relationship between gain and spontaneous coefficients (Chuang)


   F  
  1  exp 
；  Separation of quasi  Fermi level  F  Fc  Fv

 k BT  

Band energies En
conveniently
Optical gain
FTM results
Fourier series Cn
3.60
3.65
3.70
GaN-Al0.3Ga0.7N QW:
3.75
50Å/50Å
3.80
8000
7000
6000
6000
5000
5000
4000
4000
3000
TE
TE
3000
2000
2000
TM
1000
1000
TM
0
0
-1000
-2000
3.50
-1000
3.55
3.60
3.65
3.70
Photon energy(eV)
•
3.75
-2000
3.80
3.50
8000
Optical gain spectrum (1/cm)
Optical gain spectrum (1/cm)
7000
3.55
Spontaneous emission rate (a.u.)
3.50
8000
accurately
3.55
3.60
3.65
3.70
7000
3.75
26Å/62Å
6000
3.80
8000
7000
6000
5000
5000
TE
4000
4000
TE
3000
3000
2000
2000
TM
1000
1000
0
0
TM
-1000
-2000
3.50
3.55
3.60
3.65
3.70
Photon energy(eV)
S. L. Chuang et al, IEEE J. Quantum Elect., 32, 1791(1996)
-1000
3.75
-2000
3.80
Spontaneous emission rate (a.u.)
g    g
e
sp
Fundamentals
Spurious solutions issue
Reasons
•
•
•
•
Perturbative nature and incomplete set of
basis
Adoption of SYM operator ordering
Fitting of bulk material parameters to
experimental data
Satisfaction of boundary condition
•
•
•
•
Nonmonotonic behavior of CB as k
increases
Not a necessary condition for Hermitian
Ellipticity of differential operator is difficult
to be satisfied
Components are outside the 1st Brillouin Zone
Solutions
•
•
Modifying Hamiltonian matrix
Adoption of BF operator ordering
•
•
•
Cut-off Method(Plane wave expansion)
•
Cumbersome implement & Subsequent fixes
Effective for spurious solutions in VB but not
CB in QWs & modification of some parameters
Simple and effective
(Plane wave expansion)
Cut-off
FTM
(Fourier Transform)
Potential of FTM to eliminate
spurious solutions
Lassen et al
o
o
o
o
o
o
R. Eppenga, et al, Phys. Rev. B 36, 1554(1987)
K. I. Kolokolov, et al, Phys. Rev. B 68, 1613081
(2003)
W. Yang, et al, Phys. Rev. B 72, 233309(2005)
M. G. Burt, J, Phys. Condens. Mat. 4, 6651(1992)
B. A. Foreman, et al, Phys. Rev. B
56,
R12748(1997)
o
o
o
o
R. G. Veprek, et al, Phys. Rev. B
76,
165320(2007)
B. Lassen, et al, Commun. Comput. Phys. 6, 699
(2009)
M. Holm, et al, J. Appl. Phys. 92, 932 (2002)
S. R. White et al, Phys. Rev. Lett. 47, 879 (1981)
A .T. Meney et al, Phys. Rev. B 50, 10893(2003)
Eight-band calculation (QW)
Spurious solutions turn up in eight-band k∙p computation (InAs/GaAs QW)
BF operator ordering
SYM & BF
2.5
Ntr=20
Energy(eV)
 H cc iPk x
 ik P H
x
xx
H4  
 ik y P H yx
 ik P H
zx
 z
Ntr=106
2
Ntr=107
Ntr=127
1.5
1
iPk y
H xy
H yy
H zy
0.5
H xy  k x N ' k y  k y N ' k x
0
H yx  k y N ' k x  k x N ' k y
-0.5
0
0.05
0.1
0.15
0.2
iPk z 
H xz 

H yz 
H zz 
B. A. Foreman et al, Phys. Rev. B 56,R12748(1997)
k (1/Å)
t
•
•
•
•
No spurious solutions appear in VB
Spurious solutions arise in CB since Ntr=107
BF operator ordering fails to resist spurious solutions (CB and VB coupling)
Intrinsic cut-off in FTM makes up this deficiency of BF operator ordering
No requirement for manipulating material parameters
No setting for boundary conditions
Q. J. Zhao et al, J. Appl. Phys., 111, 053702-053708 (2012)
Control of spurious solutions in FTM
Influence of order of Fourier truncation Ntr and mesh point N (InAs/GaAs QW)
1
(a)
N=1024
N=512
0.9
Comparison between k||=0 and k||=0.2
•
Spurious solutions turn up ealier in CB at large wave
vector (k||=0)
Spurious solutions in VB only present at large wave
vector (k||=0.2)
It is the Ntr control the occurrence of spurious solutions
in FTM but not step size of discretization in FDM
(Cartoxia)
0.8
k||=0
Energy(eV)
0.7
0.6
•
0.5
0.4
•
0.3
0.2
Eg
0.1
0
-0.1
-0.2
10 30
50 70
Energy(eV)
Rule of thumb
90 110 130 150 170 190 210 230 250
Order of Fourier truncation(N tr )
2.1
(b)
1.9
k||=0.2
1.7
1.5
1.3
1.1
0.9
0.7
0.5
0.3
Eg
0.1
-0.1
-0.3
-0.5
10 30 50 70 90 110 130 150170 190210 230 250
Order of Fourier truncation(Ntr)
•
Trade off (computation accuracy and elimination of
spurious solutions)
In practice, sufficient accuracy has been achieved
before spurious solutions take place
A smaller Ntr is demanded to resist spurious solutions
for heterostructures with sharp interfaces, i.e., sharp
geometry or drastic difference of material parameters
•
•
o
o
Q. J. Zhao et al, J. Appl. Phys., 111, 053702-053708 (2012)
X. Cartoxia, et al J. Appl. Phys. 93, 3974(2003)
Signature of Spurious solutions in FTM (QW)
0.6
(a) Ntr=40
Real
Imag
0.4
0.2
0
-0.2
-0.4
-1
0.2
-0.5
0
kt (1/Å)
0.5
1
(b) Nt=115
0.3
Real
Imag
(c) Nt=127
0.2
0.1
0
-0.1
-0.2
-0.3
-0.4
-2
0
kt (1/Å)
2
Real
Imag
0.1
• Ntr=115 is the turning point
• The wild-spreading spectrum of the Fourier
expansion coefficient Cn can be taken as
the signature of spurious solutions
0
-0.1
-0.2
Fourier series Cn of envelop function
Fourier series Cn of envelope function Fourier series Cn of envelope function
Fourier expansion coefficients in FTM of CB component : InAs/GaAs QW
-2
-1
0
1
2
k (1/Å)
t
o
o
Q. J. Zhao et al, J. Appl. Phys., 111, 053702-053708 (2012)
B. Lassen, et al, Commun. Comput. Phys. 6, 699 (2009)
Six-band calculation (QWR)
Spurious solutions turn up in six-band k∙p computation (GaN/AlN QWR)
(a)
Fig. (a) SYM
0
•
•
•
Energy(eV)
-0.05
-0.1
-0.15
Solutions are not stable as varying Ntr
Spurious solutions turn up quickly even at k||=0
Ntr=5 is a true solution with very poor
computation accuracy
Ntr=5
Ntr=10
-0.2
Ntr=11
Fig. (b) BF
Ntr=14
-0.25
0
0
0.05
0.1
kIIt (1/Å)
0.15
(b)
0.2
Ntr=5
-0.05
•
Ntr=10
Energy(eV)
Ntr=20
-0.1
•
Ntr=40
Convergent solutions as varying Ntr, even at
quite high order of Fourier truncation.
BF operator ordering can resist the spurious
solutions without any control of Ntr
-0.15
BF operator ordering is powerful to eliminate
spurious solutions in the six-band calculation for
any kind of heterostructure
-0.2
-0.25
0
0.05
0.1
0.15
0.2
kII (1/Å)
t
Q. J. Zhao et al, J. Appl. Phys., 111, 053702-053708 (2012)
Signature of Spurious solutions in FTM (GaN/AlN QWR)
Ntr=10 (Major PDF component)
Eh=-0.0705eV
Ntr=10 (Cn)
Ntr=11 (Major PDF component)
Eh=-0.0535eV
Ntr=11 (Cn)
• Ntr=11 is the turning point
• The wild-spread distribution of Cn can be taken as
the signature of spurious solutions
Q. J. Zhao et al, J. Appl. Phys., 111, 053702-053708 (2012)
Why is it easy for FTM to deal with spurious solution?
Spurious solution is the innate issue of the k∙p method
The k∙p method is developed in the BULK scenario, while
has to work on heterostructures with discontinuity at interface
AlGaAs GaAs AlGaAs
?
but the k∙p method is continuum model and not atomistic model
Operating order reformation is effective for some case only(e.g. BF for 6-band)
Truncation of high-order Fourier frequencies eases the discontinuity issue
•
•
•
•
Merits of FTM with BF operator ordering
Robust capability to resist spurious solutions
No change to any material parameters
No specific boundary condition
Simple control using Ntr, which is much more
convenient than other approaches
Summary
Pros of FTM
• No need for meshing
•
• No numerical differential or integration
process
• Easy programming
• Natural control of spurious solutions •
• Convenience for optical gain
calculation
• Flexibility
― Arbitrary QD shapes
•
― Superlattices / isolated structures
― Parametric variation
Cons of FTM
Careful selection of Ntr
― Memory limit Ntr
― Spurious solutions  limit Ntr
― Accuracy large Ntr
Careful selection of periodical
length, considering
- Truncation error
- Crosstalk error
Hunger for computer memory in
QD calculation
but not a problem nowadays
32G128G960G
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