Derivation of Absorption Coefficient in Germanium Like Materials
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Transcript of Derivation of Absorption Coefficient in Germanium Like Materials
8/12/2019 Derivation of Absorption Coefficient in Germanium Like Materials
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The absorption coefficient determines how far into a material light of a particular wavelength can penetrate before it is absorbed.The absorption coefficient is defined as the rate of decrease in the
intensity of the light as it is passed through a certain material. Mathematically it is defined as the
depth in the material up to which light intensity decreases to e-1. Its unit is cm-1..
Absorption coefficient can be expressed as
)()()( 0 E E E high E ……………….(1)
where
α=absorption coefficient in cm-1.
E=Photon Energy in eV.
The first term (α E 0 (E )) describes the absorption for the lowest direct band gap (E0) and the
second term accounts for the absorption above the band gap. Indirect transition below the
indirect band gap are not considered. E0 transition involves heavy hole and light hole valence
bands and s like conduction band at the centre of brillouin zone (Г =0).
With assumption of parabolic dispersion (E-K relation) the E0 absorption can be expressed as
)()()()()()()( 000 E f E f E E f E f E E clh
clh
chh
chh E ……………….(2)
In equation 2
where
E= Energy band gap in eV.
α0cv is the absorption coefficient for an empty conduction band and a full valence band.(v=lh/hh)
Superscript chh(clh) is for transition between heavy hole (light hole) and conduction band.
f c(E) and f v(E) is the occupation probability for the conduction band states and valence bandstates separated by energy E.
absorption coefficient can be expressed in terms of 1 (real part) and cv
2 (imaginary part)
of empty band dielectric function.
8/12/2019 Derivation of Absorption Coefficient in Germanium Like Materials
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)(
)()(
1
20
E c
E E E
cvcv
……………….. (3)
is Planck’s constant.
The real part of the dielectric constant changes a little over the band gap. It is fitted in a equationon the basis of experimental data of real part of dielectric function in germanium. As real part
changes negligible so strain and compositional effect neglected for this value.
62.2
88.1103.11)(1
E E ………………(4)
For each of the heavy hole and light hole transitions the imaginary part of the dielectric constant
can be written as
)()()()(2 E S E E E
cvcv
f
cv
x
cv
……………(5) [ref. V. R. D’Costa, Y. Fang, J. Mathews, R.Roucka, J. Tolle, J. Menéndez, and J.Kouvetakis, Semicond. Sci. Technol. 24, 115006 (2009).]
cv
x ( below band gap excitonic contributions) andcv
f is the dielectric function for free
uncorrelated hole-electron pairs multiplied by )( E S cv sommerfield enhancement factor.
[J. A. Burton, Physica 20, 845 (1954)]
Below band gap excitonic contributions given by:
n
nn
cvcvcv
x E E
nm E
Re P
13
0222
242
)(116
……………….(5)
[J. A. Burton, Physica 20, 845 (1954)]
P is the momentum matrix element, e is the electronic charge, cv is reduced electron hole
mass. 0 is the static dielectric constant, m is the electron rest mass and is Planck’s
constant. cv R is Rydberg constant which is mathematically defined as)2(
2
0
2
4
e R cv
cv ……(6).
With this definition excitonic energy (En) can be written as
20 n
R E E cv
vn
, wherev
E 0
is the
direct band gap for heavy holes, v=hh (light holes, v=lh).
The expression for cv
f is given by [J. A. Burton, Physica 20, 845 (1954)]
)()(3
240
5.
022
5.122
vv
cvcv
f E E E E E m
P e
…………..(7)
8/12/2019 Derivation of Absorption Coefficient in Germanium Like Materials
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)( x is unit step function. Sommerfield enhancement factor given by
cv
cvcvcve
E S
sinh)( ……………..(8)
5.
0
v
cv
cv E E
R ……………..(9)
cv is transition time for heavy hole (light hole electrons).
The heavy hole and light hole
direct band gap can be calculated by considering strain using
standard deformation potential
theory.0 E is the strain shift in the band edge. 001 E is strain shift in 001 direction. Expression for strain
shifts are given by
11
120 12
C
C a E h
1
22
11
12
001C
C b E
ha and b are hydrostatic potential and shear deformation potential respectively. is the in plane
strain coefficient which is given by
0
0
a
aa
Where a is strained lattice constant and a0 is relaxed lattice constant.
The direct band gap E0 plays a critical role due to dependence on composition of Sn. The
expression given by
)1()1()( 000 xbx x E x E x E SnGe
WhereGe
E 0 =0.8eV,Sn
E 0 = -0.4eV, b=Bowing parameter which has value -2.5eV.
The occupation probability for conduction band and heavy hole/light hole band given by
001000
0010
2
0
2
00100100
00
2
1
)(4
9
2
1
4
1
2
E E E E
E E E E E E
hh
lh
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T K
E E E f
B
fce
c
exp1
1)(
T K
E E E f
B
hh f
hh
exp1
1)(
e E is the electron energy in conduction band and hh E is heavy hole energy. The expressions for
these energy is given by [Semiconductors and Semimetals, Volume 39, Richard K et al.,
Academic Press ,1993]
)( g
e
r
e E E m
m E
)( g
e
r hh E E
m
m E
r m , em and hhm are the reduced mass for electron and the hole.
hhe
hhe
r mm
mmm
The material parameters reported for Germanium given by [V. R. D’Costa, Y. Y. Fang, J. Tolle,
J. Kouvetakis, and J. Menéndez, Thin Solid Films 518, 2531 (2010).] E0(eV) P /2m
m
clh
m
chh
0 0
(eV)ha (eV) b(eV) C12/C11
0.803 12.61 0.01803 0.0300 16.2 0.297 -9.64 -1.88 0.3755
Calculation of Matrix Element( P )[Kane,1957]: matrix element can be expressed by
2
2
02 p
m P
. p is the interband momentum matrix parameter. Kane estimated its expression
which is used for direct band gap structures.
0
0
*
0
*
022
232
3
g
g
e
e
E
E
mm
mm p
0 , *
0m , g E and 0m are the spin-orbit splitting energy, effective mass of electron
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in the conduction band, band gap and electron rest mass.
So the matrix element is given by
g
g
g
e
e
E
E
E mm
m
m P
0
0*
0*
02
232
3
Different material parameters for Ge, Si, and α-Sn which used in the calculations of Absorption
coefficients as reported in literature are[Ref. :N.Yahyaoui et al., Journal of Applied Physics
115,033109 (2014).
a C11 C12 Ev,av 0 b ac av γ1 γ2
Si 5.4311 1.675 0.690 -0.47 0.04 -2.1 1.98 2.46 4.22 0.39Ge 5.6579 1.315 0.494 0 0.30 -2.9 -8.24 1.24 13.38 4.24α-Sn 6.4892 6.9 2.9 0.69 0.80 -2.7 -5.33 1.55 -15 -11.45
a=lattice constant in Å.
C11C12=elastic stiffness coefficient in 1012dyne/cm2.
Ev,av= valence band average energy in eV.
0 =spin- orbit splitting in eV.
b,ac,av=deformation potential in eV.
γ1, γ2=kohn luttinger parameters.