Effects of Crystal Orientation on the Optical Properties...
Transcript of Effects of Crystal Orientation on the Optical Properties...
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Effects of Crystal Orientation on
the Optical Properties of GaN
based Devices
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Outline
Crystal orientation and polarization
k.p method for QW of arbitrary
orientation
Results for InGaN/GaN QW
LD performance
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Spontaneous Polarization
Cubic
zero net polarization
Hexagoanl
Non-zero net
polarization
N. G. Thillosen (PhD thesis: Spin-Bahn-
Wechselwirkung in niedrigdimensionalen
AlxGa1−xN/GaN-Elektronengasen)
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Piezoelectric polarization
Cubic Hexagonal
No strain Bi-axial
strainNo strain
Bi-axial
strain
N. G. Thillosen (PhD thesis: Spin-Bahn-
Wechselwirkung in niedrigdimensionalen
AlxGa1−xN/GaN-Elektronengasen)
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Alternative growth orientations
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Band structure
Conduction band is treated by
parabolic model
SzeV
kz n
rki
t
c
ntt |)(
1),(
S is spherical wavefunction, η is spin,φn(z) is
the envelope function of nth subband
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Valence band
Valence band is treated by 6x6 k.pmethod, including top three valence bands
6
1
),(1
),(i
i
t
i
m
rki
t
v
m UkzgeV
kz tt
gm is the envelope function of the mth subband
Ui are basis functions
)()(),( DkHkH
|)(|),( kEkH
For quantum well
strain
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Rotation
JQE, v43,p1175,2007
)(kHHamiltonian for arbitrary rotation
can be obtained by rotation matrix
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Strain tensor for arbitrary crystal
orientation Strain coefficients for arbitrary crystal
orientation are calculated by
minimizing the elastic energy under
the conditions for pseudo-morphic
growth.
PRB, v59, 4725, 1999
6
1
6
12
1
i j
jiij dVCW
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Anisotropic band structure
c-plane: isotropic
m-plane: anisotropic
band structure.
Smaller effective
mass in y-direction
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Optical momentum matrix
2
22
||||
||||
v
m
c
n pe
MeMnm
Optical momentum matrix elements
for transition between conduction
band n and valence band m are
defined as
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Optical matrix elements: c-plane
A,B,C are for transitions from conduction band to top
three valence bands. Integrating over (kx-ky) plane,
optical matrix elements are NOT dependent on E-field
polarization on x-y plane.
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Optical matrix elements: m-plane
A,B,C are for transitions from conduction band to top
three valence bands. Optical matrix elements are
dependent on polarization.
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Optical gain
Optical gain is calculated by
,,
2
2
00
2
)(][||)(mn
v
m
c
nnmt
r
LffMkddmcn
qg
L(ω) is broadening function. Note 2D integration is
necessary except for c-plane QWs, where the band
structure is isotropic. Similar 2D integration for
spontaneous rates as well.
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Optical gain for different
orientations
30nm-In15Ga85N/GaN
Quantum well at
carrier density of
6.67e25/m3
No polarization
charge in all cases
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Effects of crystal orientation:
from c-plane to m-plane No polarization charge
Optical gain: polarization dependent,
enhanced in y-polarization
Transport: effective mass is reduced
How do these affect the
laser performance?
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Laser performance:
finite-element simulation
Edge-emitting laser active region:
30-nm In15Ga85N/GaN QW (x3)
No polarization charges in all cases
Band diagram
L-I curve