7 OPTICAL GAIN ANALYSIS IN SEMICONDUCTOR...

123

7.1 Introduction

The gain-carrier-density and gain-current relations are considered as most important

parameters for the design and optimization of modern semiconductor diode lasers.

Therefore, many efforts have already been made to calculate the laser gain or the material

gain in bulk and low dimensional structures of semiconductor materials. The majority of

these theoretical gain models are developed considering the parabolic dispersion relations

of the conduction and valence bands [1, 2]. In reality, the band curvatures do not remain

perfectly parabolic at the energies away from the band edge. Nonetheless, the parabolic

gain model may be a good approximation for bulk GaAs- and InP based lasers used for

emitting light in the near-infrared spectral region. However, the narrow-gap InSb based

compounds have to be used for midinfrared applications, [3, 4, 5]. In these

semiconductors, the conduction band is highly nonparabolic and a reexamination of the

gain model is needed for proper investigation of laser operation. As for example, in

quantum-well structures of these materials, the valence-bands are considerablely

nonparabolic due to the band-mixing effect and the resulting gain spectra are significantly

different from those predicted by the parabolic band aassumptions [3, 6].

Moreover, as discussed in the preceeding chapters, due to the presence of

nonparabolicity in the band structure in the low dimensions the interband absorption

coefficients are seen to be significantly different when compared with their parabolic

band approximations. Higher rate of electron transitions in different subbands for photons

of same energy is expected because of availability of higher density of states arising in

non parabolic band profile. Again the separation between the energy subbands, formed

due to the quantization of energy levels in low dimensional structures, is more closely

spaced and as a result probability of transition will also be higher. Thus, the impact of the

conduction band nonparabolicity on the gain spectrum is investigated in this chapter in

semiconductor lasers of bulk state and their low dimensional structures. The interband

7 OPTICAL GAIN ANALYSIS IN

SEMICONDUCTOR LASER

124

absorption theory developed in the previous chapters for various nanostructures

considering the wave vector dependent optical transition matrix element (OME) on the

basis of three band energy model of Kane are applied to model the material gain of the

semiconductor lasers of various dimensions. The theoretical analysis is applied to diode

lasers of InAs, InSb, Hg1-xCdxTe and In1-xGaxAsyP1-y lattice matched to InP as examples

of gain medium because these compounds have been found to be the most promising

material systems for both the double heterostructure and the multiple-quantum-well lasers

operating in different ranges of infrared spectrum. A comparison with the conventional

parabolic band model is also performed. The obtained results show the necessity to

include the conduction band nonparabolicity in modeling of lasers based on narrow-gap

semiconductor materials.

7.2. Theoretical background

7.2.1. Formulation of the generalized optical gain for narrow band gap materials in

bulk state

The optical gain in a bulk semiconductor laser can, in general, be written as [2],

2 2

23

2

0

( )4

[ (

ˆ.

)

( )

( ) ( ) , ) ( ) (( ] ( ))h

bulk

o r

cvav

c c ve v

d k a p k

E k E k k E k E

eg

m cn

f f L k

(7.1)

where, is the angular frequency of the emitted or absorbed light, c is the velocity of

light in free space, r

n is the relative refractive index of the semiconducting material, 0 is

the permittivity of free space, 0m is the free electron mass, 3d k is the differential volume

element in k

space and is given by 3 24d k k dk , a is the polarization vector of the light

vector, ( )cE k

represents the energy of the conduction band (CB) with wave vector k

,

( )vE k

represents the energy of valence band (VB) with wave vector k

, respectively.

( ) ( ) ( )c vE k E k k

denotes the k

dependent energy difference between the VB and CB

and ( )cvp k

is the optical transition matrix elements (OME) , e

f and h

f are the electron

and hole Fermi-Dirac distributions while )(L k

is the Lorentzian line-shape broadening

125

function, 2 2[ / (2 )] / [( )( 2)) ( / ]k

and depends on the relaxation time ,

in .

Due to the intraband relaxation process such as the electron-electron scattering, electron-

phonon scattering, etc., broadening occurs in the gain spectrum of semiconductor lasers

[7, 8]. When the scattering relaxation is included, the delta function may he replaced by a

Lorentzian function with a line width :

2 2

/ (2 )

( ) ( /( ) ( )

( ) ( ) 2)c v

c v

E k E kE k E k

(7.2)

where 2 factor has been included such that the area under the function is properly

normalized:

( ) ( ) ( ) 1c v

E k E k d

For the three band energy model of Kane, effective optical energy gap for optical

transitions at wave vector k

, is expressed as

1/22 2

2( ) ( ) ( )c v g g

r

kk E k E k E E

m

in which gE is the band-gap energy, rm is the reduced mass and is given by

1 1 1

cr vm m m , cm the effective mass of the conduction band electron and vm the

effective mass of the heavy hole at the top of the valence band. The dispersion law of the

conduction electrons in III–V binary, ternary and quaternary materials in accordance with

the three band model of Kane is given by [9]

22

( )2 c

kE

m

(7.4)

in which ( ) ( 1)( 1) / ( 1)E E a E b E c E , 1/ , 1/ ( )g ga E b E , is the spin-

orbit splitting constant and 1/ ( 2 / 3)gc E .

The Fermi functions e

f and h

f are given by

1

1

( )) [1 exp{( ( ) ) / }]

( )) [1 exp{( ( )( )

(

/ }]

e

h

c c

vv

fc

fv

f

f

E k E k E kT

E k E k E kT

(7.5)

126

Here, the electrons and holes are assumed in equilibrium as determined by the quasi-

Fermi levels fcE and fv

E , respectively.

fcE and fv

E are related to the densities of injected electrons and holes [10]. Using the

generalized Sommerfeld’s lemma [11] and equal electron and hole injection into the

active laser medium (i.e., n p )

3/2

2 1

1 12

2(3 ) ( ) ( )c

fc fc

mn M E N E

3/2

2 1

1 12

2(3 ) ( ) ( )v

fv fv

mp M E N E

where

3/2

1( ) [ ( )]fc fcM E E

1 1

1

( ) ( ,0) ( )s

fc fc

r

N E D r M E

22 1 2

2( , ) [2( ) (1 2 ) (2 )]

r Jr r

r J

fc

D r J kT rE

where / 2h , is the reduced Planck’s constant.

Next, using the expression for OME in accordance with the three band energy

model of Kane as derived in the previous chapter (section 3.4.1 Eq. (3.72)) and carrying

out the integration the final expression for optical gain is arrived at (for the case 0 )

22[ ( )( ) ( ) ( ( )) ]

hbulk g M N eg G E ff f f (7.8)

in which ( )ef and ( )h

f are Fermi functions which correspon to injected electron and

hole concentrations given by Eqs. (7.6) and (7.7), and the coefficient

127

1/22

2

0

( ).

6 ( 2 ).

/ 3

grbulk

r g g

EmeG

nc E E

(7.9)

and

21/21/2 ' 2

' ' ' '

2 2 (( ) 1

8 ( ( (

)

) ) )()

g g g

M

g

E E Esf

E

(7.10)

21/21/2 ' 2

' ' ' '

2 2 (( ) 1

4 ( ( (

)

) ) )()

g g g

N

g

E E Etf

E

(7.11)

Fermi functions may also be approximated as

2 2 1

2 2 1

) [1 exp{ / 2 (( ) ) / ( ) / }]

) [1 exp{ / 2 ( ( ) ) / ( ) /( }]

(r c g g c fc

r v g g

e

h v fv

f

f

m m E E E E kT

m m E E E E kT

(7.12)

wherec

E and v

E are the energy at the bottom of conduction band and at the top of valence

band prior to carrier injection, respectively. The first term in the exponential represents

the effect of nonparabolicity while the second term represent shift of quasi Fermi level

due to carrier injection. It is apparent that because of the presence of squared term the

effect of nonparabolicity is dominant factor in estimating the Fermi functions which in

turn determines the gain in laser.

7.2.2. Formulation of the generalized optical gain for wide band gap semiconductor

in bulk state

For wide band gap materials (gE ) where nonparabolicity in the band structure is

less, the above formulation of optical gain can be transformed easily to the results of gain

for wide band gap materials having parabolic energy band. Under the assumption

of 0 , or ' 0 and 0s it leads to ( ) 0Mf and ( ) 1Nf . Therefore, it

follows that for wide band gap III-V compound semiconductors, where 0 , the optical

gain can be expressed as

22

[ ( )( ) ]) (hbulk bulk eg

g G E f f (7.13)

128

Under the constraints (gE or

gE ) together with the condition

fcE and 1

fvE , the injected carrier statistics given in Eqs. (7.6) and (7.7) assume the

forms

1/2 3/2

15( ) ( )

4c fc fc

kTn N F F

(7.14)

1/2 3/2

15( ) ( )

4v fv fv

kTp N F F

(7.15)

where 3/222 2 / , /c c fc fcN m kT E kT , is the nonparabolicity factor and

0( )

tF x is

the one parameter Fermi-Dirac integral of order 0t which can be written as [12]

0

0

1

0 0

1( ) (1 exp( )) ,

( 1)

t

tF x y y x dy

t

1y

where ( ) is the complete Gamma function, or for all 0t analytically continued as a

complex contour integral around the negative axis. It may be noted that for 0 , Eqs.

(7.14) and (7.15) assume the forms

1/2 ( )c fc

n N F , 1/2 ( )v fv

p N F

and Fermi functions in (7.12) are reduced to

1

1

) [1 exp{ / ( ) ( ) / }]

) [1 exp{ / ( ) ( ) / }

(

](

r c g c fc

r vh v

e

g v f

f

f

m m E E E kT

m m E E E kT

(7.16)

Hence the final expression for optical gain can be simplified to

22 1

1

( ) [1 exp{ / ( ) ( ) / }]

[1 exp{ / ( ) (

[

) / }] ]

g r c g c fc

r v vg fv

g G E m m E E E kT

m m E E E kT

(7.17)

where 1/22 2

0/ 6( ) /r r gcG e n m E , which is in agreement with the well known

results for parabolic band [1, 2].

7.2.3. Formulation of the generalized optical gain for QW structure of narrow band

gap semiconductors

The gain in a quantum well laser may be written as

129

2

2

0

22

,

( )2

[

ˆ. ( )

( )( ( ) ,) ( )] ( ) (( ) )

cvwell av

c

well t

no r z

e h vnn cn vn

d k a p k

E k E

eg

m cn d

f f Lk k E k E k

(7.18)

where 2

td k is the differential elemental area in k

space and is given by 2 2t t td k k dk , a

is the polarization vector of the light vector, ( )cnE k

represents the nth

subband energy of

the conduction band (CB) with wave vector k

, ( )vnE k

represents the nth

subband energy

of valence band (VB) with wave vector k

, respectively. ( ) ( ) ( )cn vn nE k E k k

denotes

the k

dependent energy difference between the VB and CB, e

f and h

f are the electron

and hole Fermi-Dirac distributions while )(L k


function. The k

dependent effective band gap

1/2

2 2 ( )( ) ( ) ( ) c

n cn vn g g

r

m Ek E k E k E E

m

where ( )E and the corresponding subband energy levels n

E in the conduction band are

given by Eqs. (4.4) and (4.5) in chapter 4. The Fermi functions e

f and h

f are given by

1

1

( )) [1 exp{( ( ) ) / }]

( )) [1 exp{( ( ) ) / }

(

( ]

e

h vn vn

cn cn fcn

fvn

f

f

E k E k E kT

E k E k E kT

Here, as before, the electrons and holes are assumed in equilibrium determined by the

quasi-Fermi levels fcnE and fvn

E , respectively.

fcnE and fvn

E are related to the densities of injected electrons and holes[10]. Using the

generalized Sommerfeld’s lemma [11]

max

2 2 221

( , ) ( , )z

z

n

c

D fcn z fcn z

n

mn M E n N E n

Since for equal electron and hole injection 2 2D Dn p

130

max

2 2 221

( , ) ( , )z

z

n

vD fvn z fvn z

n

mp M E n N E n

where

2

2

2

( ( )2

) z

c z

fcn fcn

n

m dM E E

2 2

1

( , ) ( ,0) ( , )s

fcn z fcn z

r

N E n D r M E n

where fcn

E is the quasi Fermi energy in the presence of size quantization as measured

from the edge of the conduction band in the vertically upward direction. Inserting next

the expression for OME in accordance with the three band energy model of Kane as

derived in the previous chapter (section 4.4.1 Eq. (3.72) and carrying out the integration

the final expression for optical gain for a well grown in the z-direction is arrived at (for

the case 0 )

2 2( ) sin ( ) 4 [ ( ) ( )]cos ( )hwell M N

n

well eGg ff ff (7.22)

in which ( )e

f and ( )h

f are Fermi functions to be corresponding to injected electron and

hole concentrations and ( )M

f , ( )N

f are as defined earlier.

Then,

2

0

( )( )

(8 )( 2 / (3 ))

g

well

z r gd

e EG

Cn E

, (7.23)

Fermi functions may be also be approximated as

2 2 1

2 2 1

) [1 exp{ / 2 (( ) ) / (0) ( ) / }]

) [1 exp{ / 2 ( ( ) ) / (0) (

(

( ) / }]

r c n n cn fcn

r v n n fvn

e

h vn

f

f

m m E E kT

m m E E kT

(7.24)

wherecn

E and vn

E are the subband energy at the bottom of conduction band and at the top

of the valence band prior to carrier injection. The first term in the exponential represents


131

due to carrier injection. It is apparent for the expressions that the effect of nonparabolicity

plays dominant role in estimating the Fermi functions which in turn determines the gain

in laser.

7.2.4. Formulation of the generalized optical gain for QW structure of wide band

gap semiconductors

Under the assumption of 0 or ' 0 and 0s it leads to

( ) 0M

f and ( ) 1N

f . Therefore, it follows that for wide band gap III-V compound

semiconductors, where 0 , the optical gain can be expressed as

2( ) 4 [s ( ) ( )]coh

z

wewel l

n

el lg G f f (7.25)

Under the constraints (gE or

gE ), the injected carrier statistics given in (7.20)

and (7.21) assume the forms

max

2 2 0 1(1 2 ) ( ) 2 ( )n

D c cn fcn fcn

n

n N E F kTF (7.26)

max

2 2 0 1(1 2 ) ( ) 2 ( )n

D v vn fvn fvn

n

p N E F kTF (7.27)

where 2

2 / ( ) , ( ) /c c fcn fc cnN m kT E E kT , is the nonparabolicity factor and

( )j fnF is the one parameter Fermi-Dirac integral of order j .

It may be noted that for 0 , Eqs. (7.26) and (7.27) assume the forms

max

2 2 0 ( )n

D c fcn

n

n N F ,max

2 2 0 ( )n

D v fvn

n

p N F

while Fermi function in (7.24) reduces to

1

1

) [1 exp{ / ( ) ( ) / }]

) [1 exp{ / ( ) ( ) ]

(

( / }

r c n cn fcn

r v nh n

e

vn fv

f

f

m m E E kT

m m E E kT

(7.28)


2 1

1

1

( ) 4 cos [1 exp{ / ( ) ( ) / }]

[

[

1 exp{ / ( ) ( ) / }] ]vn

well r c n cn fcn

n

r v n fvn

g G m m E E kT

m m E E kT

(7.29)

132

where 2

1 0( ) / (8 )z rG e Cd n , which is in agreement with the well known results

reported for parabolic band [13- 16].

7.2.5. Formulation of the generalized optical gain for QWR structure of narrow

band gap semiconductors

The gain in a quantum wire laser may be written as

2

2

0

2

,

4( )

[ ( ) ( )

ˆ. ( )

( ) ( ) , ) (] ( ) ( )

wire l

m no r x y

e h vmn

cvwire av

cmn cmn vmn

eg

m cn d ddk a p k

E k E k kf f E k E kL

(7.30)

where l

dk is the differential elemental width in k

space, a is the light polarization

vector, ( )cmnE k

represents the subband energy level of the conduction band (CB) with

wave vector k

, ( )vmnE k

represents the subband energy level of valence band (VB) with

wave vector k

having subband indices m and n. ( ) ( ) ( )cmn vmn mnE k E k k

denotes the

k

dependent energy difference between the VB and CB, e

f and h

f are the electron and

hole Fermi-Dirac distributions while )(L k


function. Again, the effective band gap can be written as

1/2

2 2 ( )( ) ( ) ( ) c

mn cmn vmn g g

r

m Ek E k E k E E

m

(7.31)

where ( )E and the corresponding subband energy levels mn

E in the conduction band are

given by Eqs. (5.4) and (5.5) in chapter 5. The Fermi functions e

f and h

f are given by

1

1

( )) [1 exp{( ( ) ) / }]

( )) [1 exp{(

(

( ) / }( ) ]

cmn cmn fcmn

f

e

h vmn vmn vmn

f

f

E k E k E kT

E k E k E kT

Here, as before, the electrons and holes are in equilibrium as determined by the quasi-

Fermi levels fcmnE and fvmnE , respectively. fcmnE and fvmnE are related to the densities of

133

injected electrons and holes [10] and may be expressed using the generalized

Sommerfeld’s lemma [11]

max max

1 3 3

1 1

2 2( , , ) ( , , )

y y

x y

n n

c

D fcmn x x fcmn z x

n n

mn M E n n N E n n

and

max max

1 3 3

1 1

2 2( , , ) ( , , )

y y

x y

n n

v

D fvmn x x fvmn z x

n n

mp M E n n N E n n

where

222

3

2

2)

2( ) (

yx

c

f

x c

cmn fcmn

y

nn

m dE

m dM E

3 3

1

( , , ) ( ,0) ( , , )s

fcmn x y fcmn x y

r

N E n n L r M E n n

where fcmnE is the quasi Fermi energy in presence of size quantization as measured from

the edge of the conduction band in the vertically upward direction. Inserting next the

expression for OME in accordance with the three band energy model of Kane as derived

in the previous chapter (section 4.4.1 Eq. (3.72) and carrying out the integration the final

expression for optical gain for QWR grown in the z-direction is arrived at ( for the case

0 )

2

2

2

2

( )sin 2cos ( )( )

([ ( ) ( )]

) (0)h

U V

w

w

ire

n

ire

e

mnm

f fg

Gf f

(7.35)

in which ( )e

f and ( )h

f are Fermi functions corresponding to the injected electron and

hole concentrations and ( )U

f , ( )V

f are defined as earlier.

134

Then

1/22

016 2 / 3

g g

wire

r r g

E EeG

n m EWc

(7.36)

Fermi functions may also be approximated as

2 2 1

2 2 1

) [1 exp{ / 2 (( ) ) / (0) ( ) / }]

) [1 exp{ / 2 ( ( ) ) / (0) (( ) }]

(

/

e

h vmn

r c mn mn cmn fcmn

r v mn mn fvmn

f

f

m m E E kT

m m E E kT

(7.37)

wherecmn

E and vmn

E are the subband energy at the bottom of conduction band and at the

top of valence band prior to carrier injection. The first term in the exponential represents


due to carrier injection.

7.2.6. Formulation of the generalized optical gain for QWR structure of wide band

gap semiconductors

Under the assumption of 0 , ' 0 and 0s it leads to

( ) 0U

f and ( ) 1V



2

22

2[

co( ) ( )]

s( )

( ) (0)hwire

m nmn

wireefg

Gf

(7.38)

For the case of materials having (gE or

gE ), the injected carrier statistics given

in Eqs. (7.33) and (7.34) assume the forms

max max

1 1 0 1/2 1/2

0

1(1 (3 / 2) ) ( ) (3 / 4) ( )

n m

D c fcmn fcmn

n m

n N b F kTFa

(7.39)

and

max max

1 1 0 1/2 1/2

0

1(1 (3 / 2) ) ( ) (3 / 4) ( )

n m

D v fvmn fvmn

n m

p N b F kTFa

(7.40)

where

135

1

0 0

222

1 0

2

( , )

[1 ( , )]; ( , )[1 ( , )] ;

2 2 / ( ) , ( /

2 2

)c c fcmn fcmn

yx

c x c y

m n

a m n b m n m n

N m kT E

nn

m d m d

b kT

,

is the nonparabolicity factor and ( )j fnF is the one parameter Fermi-Dirac integral of

order j .

It may be noted that for 0 , Eqs. (7.39) and (7.40) assume the forms

max max

1 1 1/2 ( )n m

D c fcmn

n m

n N F ,max max

1 1 1/2 ( )n m

D v fvmn

n m

p N F

1

1

) [1 exp{ / ( ) ( ) / }]

) [1 exp{ / ( ) ( ) / }

(

( ]

r c mn cmn fcmne

h vmr v mn fn vmn

f

f

m m E E kT

m m E E kT

(7.41)


1

2 2

2

2

1

2 cos( ) [1 exp{ / ( ) ( ) / }]

( ) (0)

[1 exp{ /

[

]( ) ( ) / }]

wire r c mn

vmn

cmn fcmn

n mmn

r v mn fvmn

Gg m m E E kT

m m E E kT

(7.42)

where

1/22

2

016

g

r r

E

Wc

eG

n m

, which is in agreement with the well known results for

parabolic band [17, 18] reported earlier.

7.2.7. Formulation of the generalized optical gain for QD structure of narrow band

gap semiconductors

The gain in a quantum dot laser may be written as

22

2,

0

8ˆ. ( , . )

4

( , , ) ( , , ) ( )

( ) (

1( )

[ ) , ) ( ) ( )( ) ( )] (

dot

m n

cvdot av

lr x y z

cmnl vmnl mnl

cmn ce h vm mn vmn n

ea p m n l

m n d d d

E m n l E m n l k k

E k E k k E k E k

gc

f f L

(7.43)

where x

d , yd ,z

d are the dimensions of the dot respectively in the x , y and z direction

and m , n , l are the respective indices for the quantized k

vector of the electron and

136

mnlk simply denotes the value of the wave vector corresponding to sublevel energy when

transition occurs.cmnl

E ,vmnl

E are the energy sublevels in the conduction and valence

band, respectively, a is the light polarization vector ande

f and h

f are the electron and

hole Fermi-Dirac distributions while )(L k


function. The effective band gap

1/2

2 2 ( )( , , ) ( , , ) ( , , ) c mnl

mnl cmnl vmnl g g

r

m Em n l E m n l E m n l E E

m

(7.44)

where ( )mnl

E and the corresponding subband energy levels mnl

E in the conduction band

are given by Eq. (6.4) in chapter 6. The Fermi functions e

f and h

f are given by

1

1

( , , )) [1 exp{( ( , , ) ) / }]

( , , )) [1 exp{( ( ,

(

( , ) ) / }]

e

h vmnl vmn

cmnl cmnl fcmnl

fvmnll

f

f

E m n l E m n l E kT

E m n l E m n l E kT

Here, as before, the electrons and holes are in equilibrium as determined by the quasi-

Fermi levels fcmnlE and fvmnlE , respectively. fcmnlE and fvmnlE are related to the densities of

injected electrons and holes [10] and expressed using the generalized Sommerfeld’s

lemma [11] as

max max max

0 1

1 1 1

2( )

y y z

x y z

n n n

D fcmnl

n n nx y zd d

Fd

n

and

max max max

0 1

1 1 1

2( )

y y z

x y z

n n n

D fvmnl

n n nx y zd dF

dp

where ( ) /fcmn fcmnl cmnlE E kT

22 22 2 2

2)

2 2(

yx z

c x c y

fcmn

c z

nn n

m d m dE

m d

137

and fcmnl

E is the quasi Fermi energy in presence of size quantization as measured from the

edge of the conduction band in the vertically upward direction. Inserting next the

expression for OME in accordance with the three band energy model of Kane as derived

in the previous chapter (section 4.4.1 Eq. (3.72) and carrying out the integration the final

expression for optical gain for a quantum dot is arrived at (for the case 0 )

2 2( ) ( )sin 2cos [( ) ( ) ( )]

hdot U mnl V mnl

n l

t

m

do eg f ffG f (7.48)

in which ( )e

f and ( )h

f are the Fermi functions corresponding to injected electron and

hole concentrations and ( )U

f , ( )V

f are as defined earlier. Then

1/22

08 2 / 3

g g

x y z r r g

dot

E EeG

d d n mcd E

, (7.49)

Fermi functions may be also be approximated as

2 2 1

2 2 1

) [1 exp{ / 2 (( ) ) / (0) ( ) / }]

) [1 exp{ / 2 ( ( ) ) / (0) ( ) ]

(

}( /

r c mnl cme mnl nl fcmnl

r v mnl mnh l v n fvmm l nl

f

f

m m E E kT

m m E E kT

(7.50)

wherecmn

E and vmn

E are the subband energy at the bottom of conduction band and at the

top of valence band prior to carrier injection. The first term in the exponential represents


due to carrier injection.

7.2.8. Formulation of the generalized optical gain for QD structure of wide band gap

semiconductors

Under the assumption of 0 , ' 0 and 0s it leads to

( ) 0U

f and ( ) 1V



22( ) [ ( ) (cos )]hdot

m n l

dot eg G f f (7.51)

Also for the materials satisfying the condition (gE or

gE ), the injected carrier

statistics given in (7.46) and (7.47) assume the forms

138

max max max

0 1

1 1 1

2( )

y y z

x y z

n n n

D fcmnl

n n nx y zd dF

dn

(7.52)

and

max max max

0 1

1 1 1

2( )

y y z

x y z

n n n

D fvmnl

n n nx y zd dF

dp

(7.53)

where

22 22 2 2

2 2 2(1 )

( ) /

yx z

c x c

fcmnl fcmnl

fcmnl fcmnl cmnl

y c z

nn n

m d m d mE E

E E kT

d

(7.54)

is the nonparabolicity factor and ( )j fnF is the one parameter Fermi-Dirac integral of

order j .

It may be noted that for 0 , Eqn. (7.50) assumes the form

1

1

) [1 exp{ / ( ) ( ) / }]

) [1 exp{ / ( ) ( )( ]

(

/ }

r c mnl cmnl fcmnl

r v mnl fvm

e

h v nl nlm

f

f

m m E E kT

m m E E kT

(7.55)


2 1

1

3( ) cos [1 exp{ /2 ( ) ( ) / }]

[1 exp{ / ( ) (

[

/ ]) }]

dot r c mnl cmnl fcmnl

m n l

r v mn vml fnl vmnl

g m m E E kT

m m E

G

E kT

(7.56)

where

1/22

3

08

g

x y z r r

EeG

d d d nc m

, which is in agreement with the well known

results for parabolic band [19, 20] found in the literature.

These results thus obtained are applied to a number of III-V compound

semiconductors and their alloys having nonparabolic energy band and the effect of

nonrabolicity on gain has been estimated numerically.

7.3. Results and discussions

Using Eqs. (7.8) and (7.13) together with parameters as given in Table 1, the optical

gain spectra at T=300 K have been plotted in Figure 7.1, for InAs as function of carrier

139

density (in cm-3

) and photon energy (in eV) where the solid curve represents the

three-band model of Kane and the dashed curve exhibits the same dependence for

isotropic parabolic band. The curves are labeled as a, b and c for carrier

concentrations 18 31.5 10 cm , 18 32 10 cm

and 18 32.5 10 cm , respectively. Figures 7.2,

7.3 and 7.4 are the plots using parameters of Table 1 for all the cases of Figure 7.1 for

InSb, Hg1-xCdxTe and In1-xGaxAsyP1-y lattice matched to InP, respectively. A noticeable

difference between the results of the two models is seen from Figures 7.1-7.4. The gain

spectrum calculated with the parabolic band model is rather wide and its maximum is

shifted towards the higher photon energies, for all the materials. Such a behavior is due to

the larger energy difference between the quasi-Fermi levels resulting from the parabolic

band approximation. The nonparabolicity of the conduction band leads to a smaller quasi-

Fermi level for electrons, hence, the crossover between gain and absorption takes place at

lower photon energies.

It appears from Figure 7.1 that the peak gain values increases with the increase in

carrier densities for both three band model of Kane as well as the parabolic band. With

increase of density, the peak gain values for three band model of Kane become larger

than those for parabolic bands because of the larger reduced density of states available

due to nonparabolicity of band structure. From Figures 7.1 and 7.2, it appears that the

separation between the peak gain value for a given carrier density (say, 18 32 10 cm ), in

accordance with the two models is wider for InSb as compared with that of InAs. From

Figures 7.3 and 7.4 it is observed that the separation is greatest for Hg1-xCdxTe and least

for In1-xGaxAsyP1-y lattice matched to InP. The effect is more prominent in case of

materials having lower band gap and higher spin-splitting factor. Hg1-xCdxTe is one

classic example of this whereas In1-xGaxAsyP1-y lattice matched to InP having just the

opposite band structure characteristics shows negligible effect. The results of Figures 7.1

to 7.4 also show that the carrier density dependence of the emission wave length is

significantly different for three band model in comparison to parabolic band model and

the difference is more for the materials having narrow band gap. Thus the effective band

structure model of the relevant materials plays critical role in determining gain of diode

lasers.

140

In Figures 7.5-7.8 the gain spectrum of QW of the same materials for both three band

model of Kane and the parabolic band approximations are plotted at T=300 K and for

carrier densitiy 18 33.0 10 cm . Similar dependence is observed for QWR and QD of

various materials and are plotted in Figures 7.9-7.12 and in Figures 7.13-7.16,

respectively, at same temperature and the carrier density (i.e., at T=300 K and for carrier

densities 18 33.0 10 cm ). It is found that the separation between the peak gain value for a

given carrier density ( 18 33.0 10 cm ) is greatest for Hg1-xCdxTe and least for In1-

xGaxAsyP1-y lattice matched to InP for all low dimensional cases. Moreover, the peak gain

values increases with the three band model of Kane compared to the parabolic band

approximation as the band gap decreases due to the increasing effect of nonparabolicity

and valence band coupling for all cases of low dimensional structures. It is noted that for

all cases of low dimensions the gain spectrum calculated with the parabolic band model is

rather wide and its maximum is shifted towards the higher photon energies, for all the

materials. Such a behavior is due to the larger energy difference between the quasi-Fermi

levels resulting from the parabolic band approximation. The nonparabolicity of the

conduction band leads to a smaller quasi-Fermi level for electrons, hence, the crossover

between gain and absorption takes place at lower photon energies for low dimensions as

well as the bulk state.

For a relative assessment the gain spectrum for bulk as well as the low

dimensional cases of the individual materials taking into effect of nonparabolicity is

plotted as a function of the photon energies in Figures 7.17-7.20 at same temperature and

carrier density as earlier. From these figures it can be concluded that the peak gain value

increases with the decrease in band gap due to increasing effect of nonparabolicity and

valence band mixing in these low band gap materials. From Figures 7.17-7.20 it is

observed that the peak gain is greatest for Hg1-xCdxTe and least for In1-xGaxAsyP1-y lattice

matched to InP. The effect is more prominent in case of materials having lower band gap

and higher spin-splitting factor. Hg1-xCdxTe satisfies both of these conditions well

whereas In1-xGaxAsyP1-y lattice matched to InP having just the opposite band structure

characteristics shows negligible effect.

The generalized formulation developed here for all low dimensions can be utilized

effectively for the determination of optical gain of all types of diode lasers of III –V

141

compound semiconductors and their ternary and quaternary alloys. Besides this, the

influence of energy band models and the various band parameters on the optical gain of

these materials are also studied. The numerical results presented in this paper would be

different for other lasing materials but the nature of variation should be same. The

theoretical results developed here can be extended in analysing various related

parameters such as threshold current density, line width, lasing frequency and others.

7.4 Conclusion

In this chapter, simplified theory of optical gain in nonparabolic semiconductor laser

is developed considering the wave vector dependence of OME in accordance with the

three energy band model of Kane. Effect of nonparabolicity in the band structure, which

is the characteristics of such lower band gap materials, on gain using this model is

demonstrated. The effect of nonparabolicity being the dominant factor in the estimation

of Fermi function, it is rigorously taken into account in calculating laser gain of narrow

band gap semiconductors. Under certain limiting conditions, the well known results for

wide gap materials having parabolic energy bands are obtained as a special case of the

present analysis indicating the validity of the model. The results show a significant

deviation to the gain spectrum for lasers of narrow-gap materials having three band

model in comparison with the parabolic band model approximations. Moreover, carrier

density dependence on both the peak gain and the emission wavelength is also observed.

However, for relatively wide band gap materials the deviation of optical gain differs little

even if three band model is used instead of parabolic band model. Thus, inclusion of the

band nonparabolicity is necessary in modeling of laser diodes made of narrow-gap

materials. It may finally be noted that the purpose of the present chapter is not solely to

demonstrate the effect of nonparabolicity in optical gain but to formulate it in its most

generalized form.

142

Figure 7.1. Gain spectra in InAs at T=300 K for carrier densities: (a)18 31.5 10 cm

,

(b)18 32 10 cm

, and (c)18 32.5 10 cm

in accordance with three band model of Kane.

The dashed curves result from parabolic band.

Figure 7.2. Gain spectra in InSb at T=300 K for carrier densities: (a)18 31.5 10 cm

,

(b)18 32 10 cm

, and (c)18 32.5 10 cm



143

Figure 7.3. Gain spectra in Hg1-xCdxTe at T=300 K for carrier densities: (a)18 31.5 10 cm

,

(b)18 32 10 cm

, and (c)18 32.5 10 cm



Figure 7.4. Gain spectra in In1-xGaxAsyP1-y lattice matched to InP at T=300 K for carrier

densities: (a)18 31.5 10 cm

, (b)18 32 10 cm

, and (c)18 32.5 10 cm

in accordance with

three band model of Kane. The dashed curves result from parabolic band.

144

Figure 7.5. Gain spectra of QW in Hg1-xCdxTe at T=300 K for carrier densitiy18 33.0 10 cm

in

accordance with three band model of Kane. The dashed curve results from parabolic band.

Figure 7.6. Gain spectra of QW in InSb at T=300 K for carrier densitiy 18 33.0 10 cm

in


145

Figure 7.7. Gain spectra QW in InAs at T=300 K for carrier densitiy 18 33.0 10 cm

in


Figure 7.8. Gain spectra of QW in In1-xGaxAsyP1-y lattice matched to InP at T=300 K for carrier

densitiy 18 33.0 10 cm

in accordance with three band model of Kane. The dashed curve

results from parabolic band.

146

Figure 7.9. Gain spectra of QWR in Hg1-xCdxTe at T=300 K for carrier densitiy 18 33.0 10 cm

in


Figure 7.10. Gain spectra of QWR in InSb at T=300 K for carrier densitiy 18 33.0 10 cm

in


147

Figure 7.11. Gain spectra of QWR in InAs at T=300 K for carrier densitiy 18 33.0 10 cm

in


Figure 7.12. Gain spectra of QWR in In1-xGaxAsyP1-y lattice matched to InP at T=300 K for

carrier densitiy 18 33.0 10 cm

in accordance with three band model of Kane. The dashed

curve results from parabolic band.

148

Figure 7.13. Gain spectra of QD in Hg1-xCdxTe at T=300 K for carrier densitiy 18 33.0 10 cm

in


Figure 7.14. Gain spectra of QD in InSb at T=300 K for carrier densitiy 18 33.0 10 cm

in


149

Figure 7.15. Gain spectra of QD in InAs at T=300 K for carrier densitiy 18 33.0 10 cm

in


Figure 7.16. Gain spectra of QD in In1-xGaxAsyP1-y lattice matched to InP at T=300 K for carrier

densitiy 18 33.0 10 cm

in accordance with three band model of Kane. The dashed curve

results from parabolic band.

150

Figure 7.17. Comparative assessment of theoretical optical gain spectra of low-dimensional

quantum structures of Hg1-xCdxTe.


quantum structures of InSb.

151


quantum structures of InAs.


quantum structures of In1-xGaxAsyP1-y lattice matched to InP.

152

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