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.

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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) 

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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.