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Tekspenuh

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92

CHAPTER 5

Results on Energy Gap for ZnS

x

Se

1-x

: The Effect of Spin-Orbit Interaction

5.1 INTRODUCTION

Solutions to the energy-eigenvalue problem of a semiconductor crystal yields the band structure in the form of "E−k" diagram, or the ‗dispersion‘-curve of the semiconductor. Electronic, optical, and magnetic phenomena in semiconductors can be understood by looking at a small portion of the band structure. These portions of the band structure are the lowest level in the conduction band and the highest level in the valence band. The highest point of the valence bands are known as the -point, and constitute the (kx = 0, ky = 0, kz = 0) point in the k-space. In most compound semiconductors, the maximum of the valence band and the minimum of the conduction band occur at the same point in the k-space i.e. at the -point. Such semiconductors are called direct band gap semiconductors and form the core of most optical devices.

Spin-orbit splitting occurs in semiconductors in the valence band, because the valence electrons are very close to the nucleus. In quantum-mechanical description, the wave equation depends on the spin of the particles. The usual Schrödinger equation applies to the spin-0 particles in the non-relativistic domain, while the Klein–Gordon equation is the relativistic equation appropriate for spin-0 particles. The spin-1/2 particles are governed by the relativistic Dirac equation which, in the non-relativistic limit, leads to the Schrödinger–Pauli equation [Bjorken and Drell, 1964; Davydov,

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93

1965; Messiah, 1968]. In the case of particles with spin 1 (i.e., bosons), only relativistic equations are considered [Berestetskii, 1989]. A charged particle with non-zero spins couples to an external magnetic field as if, in addition to its electric charge, it had a magnetic dipole moment. In the case of a spin-1/2 charged particle, the relation between the magnitudes of the magnetic dipole moment and of the intrinsic angular momentum given by the Dirac or the Schrödinger–Pauli equation does not coincide with that of a uniformly charged rotating body given by classical physics, but somewhat surprisingly it does coincide with that of a rotating charged black hole in the Einstein–Maxwell theory [Debney et al., 1969; Newman, 2002]. The k.pperturbation method [Nag, 1980;

Kane, 1966] is based on the fact that the cell periodic functions for the electrons for any wave number k in different bands form a complete set and the expression of the wave functions for electrons are in terms of the functions for the minima and maxima (i.e.

HOMO-LUMO bands).

The calculation of Eg by solving the Schrödinger-Pauli equation with the effect of spin-orbit interaction will be obtained.

5.2 COUPLING OF SPIN AND ORBITAL ANGULAR MOMENTUM

The energy of an electron due to coupling between its spin (s) and orbital angular momentum (l) can be derived as follows. The magnetic field B generated by an electron travelling with momentum p in the electrostatic field Ẽ created by nucleus plus core electrons is given as [Cohen-Tannodji et al., 1977]

2

0

B 1 p

m c  (5.1) where m0 is the free electron mass and c is the velocity of light. Since e  Vso

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94

2

0

B 1 V p

 em c   (5.2)

where e is the charge of an electron, and V is the effective potential. The intrinsic magnetic moment of the electron

0 2 0

s

es e

Mmm  (5.3)

interacts with B. This interaction energy is given by

2 2

 

0

. .

SO s 2

H M B V p

m c

     (5.4)

where σ = 2s/ћ is the Pauli spin matrices.

Another relativistic correction due to the precession of the spin angular momentum vector relative to the laboratory frame gives an additional factor of 1/2 [McGlynn et al., 1969]. Including this correction factor, the total relativistic contribution to the Hamiltonian due to the spin-orbit coupling is given by [Herman et al., 1963]

2 2

0

[ ].

SO 4

H V p

m c

   (5.5)

In a central field potential, VV r( )and

1 dV .

V r

 r dr (5.6)

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95

Consequently, Eqn. (5.5) can be written as

 

2 2 0

2 2 0

4 . 4 . .

SO

H dV r p

m c r dr dV l m c r dr

l s

 

(5.7)

where 2 2

0

1 2

dV m c r dr

  ,and

s2.

HSO is the one-electron spin-orbit coupling operator for one atom. In a solid this operator is summed over all atoms.

5.3 THE k.pPERTURBATION THEORY WITH THE EFFECT OF SPIN-ORBIT INTERACTION

Let as assume that the conduction band minimum and the valence band maximum are at the zone center and that the valence band is triply degenerate. Define

2 2

' 2 0

E  E k m (5.8)

where E is the energy eigenvalue, is Plank‘s constant divided by 2π, and m0 is the free electron mass. On the basis on this consideration and including the spin vector we may choose iS↑, (X+iY)↓, Z↑, (X-iY)↓, iS↓, Z↓, (X+iY)↑ as the base vectors for ψ. Here X, Y and Z represent the x, y and z axis of the Brillouin zone. The periodic wave function can be written as

nk

r,

eik r.unk

r,

(5.9)
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96

where u is periodic. α = ± 1 is a (pseudo)spin index; σ represents a vector of the Pauli

spin matrices 0 1 , 0 , 1 0

1 0 0 0 1

x y z

i

    i     , andi 1. Using nk from Eqn. (5.9) in the Schrödinger-Pauli equation gives

2 0

2 2 2

 

0

2 [ ].

m 4 V p V r E

m c   

 

      

 

  (5.10)

Substituting Eqn. (5.9) into Eqn. (5.10) gives

2 0

2 2 2

 

.

 

.

 

0

2 [ ]. , ,

4

ik r ik r

nk nk

m V p V r e u r Ee u r

m c

 

      

 

  (5.11)

that finally leads to

       

   

 

2 . 2 . 2 .

0

. 2 2

0 .

2 , 2 , ,

[ ]. ,

4

,

ik r ik r ik r

nk nk nk

ik r nk

ik r nk

m e u r ike u r k e u r

V p V r e u r m c

Ee u r

  

 

 

      

 

    

 

(5.12) After factoring the term eik r. and replacing pkwe get

     

 

2 2 2

2 2

0 2 2 2 2

0 0 0 0

2 . [ ]. [ ]. ,

2 4 4

,

nk

nk

m V r k p k V p V k u r

m m m c m c

Eu r

  

   

          

   

  

(5.13)

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97

Note that the total Hamiltonian is given by

0 2

2 0

2 ( )

SO

SO

H H H

V r H m

 

     (5.14)

Taking the k vector in the z direction and consider the Hamiltonian corresponding to the terms of Eqn. (5.13), the mutual interaction of the conduction and valence bands leaves the band doubly degenerate. We take as a basis

   

   

, 2 , , 2 ,

2 , , 2

iS X iY Z X iY iS

X iY Z X iY

      

      (5.15)

The first four functions are respectively degenerate with the last four. The 8  8 interaction matrix may be written as 0

0 H

H

 

 

  where

0 0

0 3 2 3 0

2 3 0

0 0 0 3

c v

v

v

E kP

H E

kP E

E

 

 

  

 

   

   

 

(5.16)

The positive constant ∆ which is the spin-orbit splitting of the valence band, and the real quantity P is defined by [Kane, 1957] as

P i

m0

S p Zz (5.17)
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98

and

2 2

0

3

4 y x

i V V

X p p Y

m c x y

 

  

  (5.18)

Ec and Ev refer to the eigenvalues of the Hamiltonian H0. Ec corresponds to the conduction band and Ev to the valence band. Symmetry properties have been used. We should note that H E and H given by Eq. (5.16).

The doubly degenerate wave functions which result from the diagonalization of the Hamiltonian of Eqn. (5.16) may be written as

1

 

, ' ' ' ' ' '

k k 2 k

X iY

u k ra is  b   c Z

  (5.19) and

2

 

, ' ' ' ' ' '

k k 2 k

X iY

u k ra is  b   c Z  (5.20)

where the coefficients ak,, bk,, ck are obtained by applying the normalization condition

(i.e. 2 2 2 1

k k k

abc  ), and are given by [Haga and Kimura, 1964]

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99

      

 

 

 

 

  

 

2 1 2

0 0 0 0

2 2

0 0 0

1 2

0 2 2 *

1 2 2

2 2

0 0

2 2 2 1

0 0 0

2 0

2

0 0

' 6 2 3

, ,

' 6 9 4

, 2 ,

2 '

4 ,

3 6 9 4

' 6 9 4 ,

6 2 3

, 6 9 4

k k

k

k k

k

k k

k k

Eg Eg Eg Eg

a Eg Eg Eg

Eg k m

b Eg Eg

Eg Eg Eg

c t t Eg

Eg Eg

 

 

  

 

 

     

 

 

      

 

    

  

 

      

     

   

   

 

1 2

2

 

 

 

 

s is the s-type atomic orbital (i.e. in conduction band) in both unprimed and primed coordinates (i.e. X, X', Y, Y', and Z, Z') and ', ' indicates the spin-up and spin-down function in the primed coordinates, X',Y'and Z' are the p-type atomic orbital‘s in the primed coordinates.

If the k vector is not in the z direction, the Hamiltonian is more complicated but it can be transformed to the form of Eq. (5.14) by a rotation of the basis function.

   

   

2 2

'

2 2

'

cos 2 sin 2

sin 2 cos 2

i i

i i

e e

e e

 

 

 

   

  

   

   

    ,

' ' '

cos cos cos sin sin

sin cos 0

sin cos sin sin cos

X X

Y Y

Z Z

    

 

    

      

      

     

       

 

,

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100

besides, the spin vector can be written as

s 2 [Ghatak et al., 2008].

The angles θ and  are the usual polar angles of the k vector referred to the crystal symmetry axes x, y, and z: with θ measured from z and  measured from x. This transformation would be obvious if the functions X, Y, Z transformed like the spherical harmonics x, y, z under the full spherical group rather than just under the tetrahedral group.

From the above, it can be written

P i

m0

s p Zz  i

m0

p k

 

and

     

     

     

   

     

     

 

1 2

' ' ' '

' ' ' '

' ' ' '

' ' ' '

' '

, ,

' '

2

' ' '

2

' ' ' ' ' ' '

2 2

' ' ' '

2

' '

k k

k k

k k

k k

k k k k

k k

k k

k k

p k u k r p u k r

a a is p is a b is p X iY

a c is p Z b a X iY p is

b b b c

X iY p X iY X iY p Z

c a Z p is c b Z p X iY

c c Z p Z

      

      

        

      

  

Hence, we introduce

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101

     

 

   

 

   

 

     

1 2

' ' ' '

' ' ' '

' ' ' '

, ,

' ' '

2

' ' '

2

' ' ' ' ' '

2 2

k k

k k

k k

k k

k k k k

p k u k r p u k r

b a X iY p is c a Z p is

a b is p X iY a c is p Z

b c c b

X iY p Z Z p X iY

      

      

       

(5.21)

Since we are interested for the effect of spin orbit, the last two terms on Eqn. (5.21) may be neglected since there are no spin term, is. Hence from Eqn. (5.21), we can write

     

* *

' '

' ' ' '

' '

X Y X

X iY p is X p is iY p is i u ps iu piu i X p s Y p s

   

  

 

 

and for X’, Y’ and Z’, we get

' cos cos cos sin sin

' sin cos

' sin cos sin sin cos

X X Y Z

Y X Y

Z X Y Z

    

 

    

  

  

  

Then

X p s' cos cos  X p s cos sin  Y p s sin Z p sprˆ where

rˆ1iˆcos cos  ˆjcos sin kˆsin1, and

Y p s'  sin X p s cos Y p sprˆ2, where

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102

rˆ2  iˆsin ˆjcos, so that

X'iY'

p sp ir

ˆ1rˆ2

Thus

' '

' '

ˆ1 ˆ2

' '

2 2

k k

a b a b

X iY p s ir r

          

Since

 

1 2

' ' ' '

ˆ ˆ

is p X iY i s p X s p Y p ir r

  

 

we can write

a bk2k

is p X

'iY'

 ' '

 a bk2k p ir

ˆ1rˆ2

 ' '

Similarly,

 

3

' '

ˆ ˆ ˆ

sin cos sin sin cos

ˆ Z p is i Z p s

ip i j k

ipr

    

  

 where

rˆ3iˆsin cos  ˆjsin sin kˆcos Thus

c ak k Z p is' c a iprk k ˆ3  ' ' and

c ak k is p Z' c a iprk k ˆ3  ' ' Therefore

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103

 

     

   

' ' ' '

' ' ' '

1 2

' ' ' '

2 2

ˆ ˆ

2

k k k k

k k k k

a b a b

X iY p s is p X iY

p a b a b ir r

      

       

(5.22)

Similarly

 

' ' ' '

' '

3

' '

ˆ

k k k k

k k k k

c a Z p is c a is p Z ip c a c a r

    

    (5.23)

Combining, Eqn. (5.22) and (5.23), we can write

 

2

ˆ1 ˆ2

   

' '

 

' '

ˆ3

 

' '

cv k k k k k k k k

p kp irr a b    b a   ipr c a c a  

(5.24) From the above relations, we can write

   

   

' 2 2

' 2 2

cos 2 sin 2

sin 2 cos 2

i i

i i

e e

e e

 

 

    

      (5.25) Therefore,

     

     

' ' 2

2

sin 2 cos 2 cos 2

sin 2 sin 2 cos 2

i

x x x

i

x x

e e

  

  

        

      (5.26)

Since 0

x x

      , and 1

2

x x

      , so from Eqn. (5.26) we get

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104

   

       

         

 

 

' ' 2 2

2 2

2 2

1 cos 2 sin 2

2

1 cos sin cos 2 cos sin sin 2

2

1 cos sin cos 2 cos sin 1 cos 2

2

1 2 cos cos 1 cos sin 2

1 cos cos sin 2

i i

x e e

i i

i i

i i

     

     

   

  

     

 

     

 

      

     

 

(5.27) Similarly

' ' 1

cos sin cos

2

y i   

   

and ' ' 1sin

2

z

    .

Therefore,

     

   

   

' ' ˆ ' ' ˆ ' ' ˆ ' '

1 cos cos sin ˆ cos sin cos ˆ sin ˆ

2

1 cos cos ˆ sin cos ˆ sin ˆ ˆsin ˆcos

2

x y z

i j k

i i i j k

i j k i i j

      

      

          

    

 

       

(5.28)

Similarly, we write

' ' 1 ˆsin cos ˆsin sin ˆcos 1ˆ3

2i   j   k 2r

      

and ' ' 1 ˆ3 2r

    .

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105

Using the above results and following Eqn. (5.24) we can write

           

       

' ' ' ' ' '

1 2 3

3 1 2

ˆ ˆ ˆ

2

ˆ ˆ ˆ

2 2

cv k k k k k k k k

k k k k

k k k k

p k p ir r a b b a ipr c a c a

a b b a

pr ir r c a c a

          

   

 

     

  

 

(5.29) Hence,

 

2 ˆ ˆ3

1 ˆ2

2 2

k k

cv k k k k

b b

p kpr irr a  c a  c  (5.30)

We can write rˆ1rˆ2rˆ3 1, also prˆ3pxsin cos iˆpysin sin  ˆjpzcoskˆ, where

   

 

 

* 3

,

0, 0,

0 , 0

c vX

cvX cvY

p s p X s p Y s p Z

s p X u r pu r d r p

s p Y p

and s p ZpcvZ

 

0 .

Then from Eqn. (5.30)

k pˆ. cv

 

k

kˆ.2pr irˆ ˆ3

1rˆ2

akbk2 ckakbk2 ck (5.31)
(15)

106

where ‗.‘ is the dot product.

Thus

 

2 32 1 22 2

2

2 2

ˆ. ˆ. ˆ . ˆ ˆ

2 2 2

1 cos .

4 2 2

k k

cv k k k k

k k

Z k k k k

b b

k p k k pr ir r a c a c

b b

pa c a c

    

        

    

       

(5.32)

So, the average value of k pˆ. cv

 

k 2 over the entire angle θ is

 

2 2 2 0

0 2

2

1 1 1

ˆ. cos 2

4 2 2 2 2

8 2 2 .

k k

cv Z k k k k

av

k k

Z k k k k

b b

k p k p a c a c d d

b b

p a c a c

  

 

    

          

   

   

    

       

 

(5.33)

Now the equations that gives the energy eigenvalues may be obtained from Eqn. (5.14), keeping only the terms corresponding to conduction band and the degenerate valence bands and neglecting all other terms [Nag, 1980],

( ' ) 0,

( ' 2 3) 2 3 0,

2 3 ( ' 3) 0,

( ' ) 0,

k c k

k v k

k k k v

k v

a E E c pk

b E E c

a pk b c E E

d E E

  

     

      

 

(5.34)

The energy eigenvalues are given by equating the determinant of the coefficient of ak ,bk , ck to zero i.e.

(16)

107

 

 

( ' ) 0

0 ' 2 3 2 3 0

2 3 ' 3

c

v

v

E E pk

E E

pk E E

 

     

    

(5.35)

and the equation giving the values is

( 'EEc)( 'EEv)( 'EEv  ) p k E2 2( 'Ev 2 3)0 (5.36)

Solving Eqn. (5.36) we get

 

  

 

  

2 2

2 2

' 2 3

' ' '

' 2 3

' ' '

c c

c c

c c

c c

p k E E Eg E E

E E Eg E E Eg E E Eg

E E

E E Eg E E Eg p k

   

  

     

   

 

     

 

 

 

(5.37)

WhenE'Ec, we may neglect E'Ec in comparison to Eg where Eg=Ec-Ev. Egn.

(5.37) can be simplified to

 

 

2 2

2 3

c ' E E Eg

Eg Eg p k

   

 

 

 

 

(5.38)

In a similar way, to get Ev

 

  

2 2

' 2 3

' ' '

v v

v v

p k E E E E

E E Eg E E

  

 

    

and hence

 

  

2 2

' 2 3

' ' '

v v

v v

p k E E

E E

E E Eg E E

  

       (5.39)

(17)

108

When E' = Ev, the dispersion relation for two valence band having wave functions (X+iY) and (X-iY) is

2 2

0

2 .

v

E E k

  m (5.40)

When Ec = 0 this leads as to Eg = -Ev, hence we can write Eqn. (5.36) as

 

 

2 2

' 2 3

' .

' '

v

p k E Eg E E

E E Eg

  

 

   (5.41)

Consequently

 

   

 

2 2 ' 2 3 2 3

' ' .

g c v

E Eg Eg

E E E p k

E E Eg Eg Eg

      

           (5.42)

Now if the band edge effective mass is m*, so from Eqn. (5.38) we get

   

 

2 2 '

2 3 E Ec Eg Eg

p k Eg

  

  

which can be written as

 

 

 

 

 

 

2 2

2 0

2

2 2 2 2

*

0 2

2

* 0

2 2 3

2 2

2 3

1 1

2 3 2 .

c

Eg E E k Eg

p m

Eg k

k k

Eg Eg

m m

Eg k

Eg Eg

Eg m m

 

   

 

 

  

 

  

 

 

  

   

     

(5.43)

(18)

109

From Eqn. (5.42) we have

   

 

2 2 ' ' '

' 2 3

E Eg E E Eg

p k E Eg

   

    (5.44)

Hence from Eqn. (5.42) and Eqn. (5.43) we have

  

   

 

2 2

* 0

' ' 2 3

1 1

' .

2 ' 2 3

E Eg E Eg Eg

k E

m m E Eg Eg Eg

     

 

 

      

  (5.45)

If E is small in comparison to Eg, the relation can be simplified to the following

2 2

* (1 )

2

k E E

m   (5.46)

where

  

*

0

1 1 1

3 2 3

m Eg

Eg m Eg Eg

             (5.47)

as given by Cohen-Tannodji et al., [1977].

Now near a characteristic point k = 0 with n=l, the energy eigenvalues may be expressed as

     

   

2 2 2 2

2

0 0

. 0

0 2 0 0

cv

n n

c v

k k P

E k E

m m E E

  

(5.48)

(19)

110

and near a characteristic point k0 can be expressed as

         

   

2 2

2 2

0 0

0

0 2

0 0 0 0

. 2

cv

n n

c v

k k P k k k

E k E k

m m E k E k

 

  

 (5.49) Using Eqn. (5.49), the deviations of E(k) near a critical point k0 can be written as

 

0 2

0

2 2 2

0

2

 

0 2

0 0 2

0 0

1 4

2 2 2

k k Pcv k k k

E k k Eg Eg

m m

 

      (5.50)

in terms of the band gap Eg0. The effective mass m* can be expressed as

   

 

2 0

* 2

0 0

2 2 3

1 1 pcv k Eg

m m m Eg Eg

   

  (5.51)

which is concluded from Eqn. (5.43) and Eqn. (5.45). Note that in case of ∆=0 we can reach the expression for the effective mass in the absence of spin-orbit as in chapter 4.

Now following Eqn. (5.50) we can estimate

2

 

2 02 0

0

* 0

ˆ. 0

4 2

3

Z cv

Eg Eg p k p m

Eg

   

  

 

 

(5.52)

Using Eqn. (5.51) in Eq. (5.49) we obtain

 

0 2

0

2 2 2

0

2 0

0

0 0 *

0

0

1 .

2 2 2 2

3

k k k k Eg Eg

E k k Eg Eg

mEg

   

     

  

 

 

(5.53)

(20)

111

Assuming that the conduction band minimum and the valence band maximum are at the zone center, then we can write

 

0 2 2 0 2 2

0

*

0 0

0

1 2

2 2 2

3

c

Eg k Eg k Eg

E k m Eg

Eg

      

  

 

 

(5.54)

and

 

0 2 2 0 2 2

0

*

0 0

0

1 .

2 2 2 2

3

v

Eg k Eg k Eg

E k m Eg

Eg

      

  

 

 

(5.55)

From Eqn. (5.54) and (5.55),

     

 

 

2 2

0

0 *

0 0

1 2

2 2

0

0 2 *

0 0

2 2

0

0 2 *

0

1 2

3

1 4

2 3

2 .

2 3

c v

k Eg E k E k Eg

Eg Eg

Eg Eg

D Eg Eg Eg Eg

D Eg

    

  

 

 

 

   

 

 

 

   

  

 

   

  

 

 

(5.56) This can be written as

     

 

   

2 2

0

0 *

2

0

2 2

0

0 2 1.76

0 0

2

2 3 2

0.124 2

3

c v

c c

c

Eg E k E k Eg Eg

D m Eg m Eg Eg

D Eg m Eg

     

    

   

 

   

  

 

 

(5.57)

using the assumption

*

0.124 0

1.76

c

m Eg

  (5.58)

(21)

112

Eqn. (5.57) represents the energy gap for alloys with the effect of spin orbit interaction.

5.4 RESULTS

We exploit Newton interpolation relation to estimate the values of spin orbit constant ∆ for the crystal structures in the ZnSxSe1-x system, which can be expressed by

∆=0.16 x+0.27 for sphalerite crystal structure (Figure 5.1), and ∆=0.328 x+0.092 for wurtzite crystal structure (Figure 5.2). When x=0 we find out that ∆=0.27 eV which gives the spin orbit constant for ZnS in case of sphalerite structure and ∆=0.092 eV in case of wurtzite structure which is given in [Nag, 1980]. We find that the values for spin orbit constant increases with x. From these relationships when x=1, the spin orbit constant ∆=0.43 eV for ZnSe in case sphalerite structure and ∆=0.42 eV in case of wurtzite structure for ZnSe.

Figure 5.1: Spin orbit splitting constant with various x for ZnSxSe1-x for sphalerite crystal structure.

0.26 0.28 0.3 0.32 0.34 0.36 0.38 0.4 0.42 0.44

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

x

(22)

113 Figure 5.2: Spin orbit splitting constant with various x for ZnSxSe1-x for wurtzite crystal structure.

Figure 5.3, 5.4 shows the effect of spin-orbit splitting constant upon applying the 1.7 and 1.66 correction factors in the case of sphalerite and wurtzite crystal structures comparing with the case of energy gap without spin respectively. The effect of spin- orbit increases the value of energy gap and the values of energy gap upon applying the 1.7 correction factor are greater than the values of energy gap upon applying the 1.66 correction factor.

0.05 0.09 0.13 0.17 0.21 0.25 0.29 0.33 0.37 0.41 0.45

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

x

(23)

114 Figure 5.3: Eg as a function of a concentration x for ZnSxSe1-x upon applying the 1.7 correction factor in case of wurtzite crystal structure, sphalerite crystal structure, and Eg without spin.

Figure 5.4: Eg as a function of a concentration x for ZnSxSe1-x upon applying the 1.66 correction factor in case of wurtzite crystal structure, sphalerite crystal structure, and Eg without spin.

In Figure 5.5 shows the energy band gap Eg with the effect of spin-orbit which is plotted as a function of the concentration x upon applying the 1.7 and 1.66 correction factors in the case of sphalerite and wurtzite crystal structures comparing with the experimental results reported by Larach et al. [1957] and Abo Hassan et al. [2005a].

2.7 2.8 2.9 3 3.1 3.2 3.3 3.4 3.5

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Eg (wurtzite using correction factor 1.7)

Eg (sphalerite using correction factor 1.7)

Eg (eV)

x

2.6 2.7 2.8 2.9 3 3.1 3.2 3.3

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Eg (wurtzite using correction factor 1.66)

Eg (sphalerite using 1.66 correction factor)

Eg (eV)

x

(24)

115

Figure 5.5: Eg as a function of a concentration x for ZnSxSe1-x upon applying the 1.7 and 1.66 correction factors in case of wurtzite crystal structure, sphalerite crystal structure, Eg without spin, experimental results reported by Larach et al. [1957] and Abo Hassan et al. [2005a].

5.5 SUMMARY

The theoretical results for the energy gap for ZnSxSe1-x alloys with the effect of spin orbit interaction were represented in Eqn. (5.56)

     

 

 

2 2

0

0 *

0 0

1 2 2 2

0

0 2 *

0 0

2 2

0

0 2 *

0 0

1 2

3

1 2

2 3

2 .

2 3

c v

k Eg E k E k Eg

Eg Eg

Eg Eg

D Eg Eg Eg Eg

D Eg Eg

    

  

 

 

 

   

 

 

 

   

  

 

   

  

 

 

2.5 2.65 2.8 2.95 3.1 3.25 3.4 3.55 3.7

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Eg (wurtzite using correction factor 1.7) Eg (sphalerite using correction factor 1.7) Eg (wurtzite using correction factor 1.66) Eg (sphalerite using 1.66 correction factor) Eg without spin using 1.7 correction factor Eg without spin using 1.66 correction factor Larach et al. [1957]

Abo Hassan et al [2005a]

Eg (eV)

x

(25)

116

and Eqn. (5.57)

     

 

   

2 2

0

0 *

2

0

2 2

0

0 2 1.76

0 0

2

2 3

2 .

0.124 2

3

c v

c c

c

Eg E k E k Eg Eg

D m Eg m Eg Eg

D Eg m Eg

     

    

   

 

   

  

 

 

by using the empirical relationship given in Eqn. (5.58)

 

*

1.76

0.124 0 c

m Eg

  .

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