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L6 ENGR_ECE4243-6243 Handout#1 10042016(Summary L5+HW#4) F. Jain
Absorption coefficient expressions for band-to-band direct and excitonic
transitions, indirect transitions, band filling and band tailing.
The power continuity equation gives power P at a point in semiconductor
medium:
-∇ P(x,y,z) = W*ħ (Rate of absorption)
(1)
The absorption coefficient is also defined in terms of probability of absorption P CV
when an intensity I() is on for t sec duration on a sample of volume V having bandband transitions.
α = ħω *
𝑃𝑐𝑣
𝑡
*
1
(2)
𝐼(𝑣)𝑉
PCV is expressed in terms of integral of one transition from state o in
valence band to state m in the conduction band.
∞
Pcv= ∫0 𝑃𝑚𝑜 ∗ ϱ (ℎ𝑣)𝑑𝑣
(3)
Here, ϱ is join density of states expressed as: Here, mr is the reduced mass
defined below.
3
2
1
2
ϱ (ℎ𝑣) = [V4π(2mr ) (ℎ𝑣 − 𝐸𝑔 ) ]/ℎ3
(3b)
1
1
1


mr me mh
(3C)
The probability is obtained by solving a time dependent Schrodinger
equation having photon interaction as a perturbation Hamiltonian. It is
shown [Moss et al.] that
𝑠𝑖𝑛2 (ω−ω𝑚𝑜 )∗𝑡
Pmo(t)=|𝐻𝑚𝑜 |2 *
(4)
ħ2 (ω−ω𝑚𝑜 )2
Here,  is the frequency of radiation and mo is the angular frequency
corresponding to energy difference Em-Eo.
2𝑒 2 𝐼(𝑣)(|𝒑𝑚𝑜 |)2
|𝐻𝑚𝑜 |2 =
(5)
3m2o 𝑛𝑟 𝜀𝑟 𝑐ω2
Here, pmo is the momentum matrix element which has three components.
 p x m0

*
  j  m
 0, other than k s

 0 d  
x
C , other than k  kC  kV
|𝑝𝑚𝑜 |2 = |𝑝𝑥,𝑚𝑜 |2 + |𝑝𝑦,𝑚𝑜 |2 + |𝑝𝑧,𝑚𝑜 |2
Add Eq. 7-9
Using Eqs. 5, 4 and 3,
Pcv (t ) 
2e 2 I (v)  ( mo )
3m02 nr  0 c 2 2
sin 2 1    m0  t
2
d m0
2


  m0 
0
(3.6)
𝑝 𝑖𝑠 𝑠𝑞𝑢𝑎𝑟𝑒 𝑜𝑓 𝑚𝑜𝑚𝑒𝑛𝑡𝑢𝑚
(6)

p mo
2

1
(10)
sin 2 x
in the integral, so it is taken out.
x2


sin 2 1    m 0  t
sin 2 ax
a  sin 2 ax

2
d m 0 =t
dx  ], or 
Using the standard integral, 
dx  , [ 
2
2
2
2
(
ax
)


 m0 
ax
2
0
0
0
Let us take another look at the integration:
2 1
 km   t
 2 sin
 km   t
2dx
t
2
d , let
 x , d  
2
 4 
t
2
t





 km
2 


t 2 2 sin 2 x
t
sin 2 x
=

dx =(t/2)*2*t

dx
2  x 2
4 t  x 2
ρ is assumed to be varying much slowly than
2e 2 I (v)  ( mo ) t
Pcv (t ) 
3m02 n r
 0 c 
2
2
p mo
2
(11)
E. O. Kane has shown that
2 2
2 𝑚𝑜 𝑃
|𝑝𝑚𝑜 | = 2
ħ
(12)
Where,
2
𝑃 =(
P2 =
1
𝑚𝑒
2
ħ Eg
2𝑚𝑒
−
1
𝑚𝑛
)
3ħ2 Eg (Eg +Δs0 )
(13)
2(3Eg +2Δs0 )
, if 𝑚𝑒 ≪ 𝑚𝑛
𝐸𝑔 >
2
3
𝛥𝑠𝑜
(14)
Equation 14 and 12 gives
𝑚𝑜2 Eg
2
|𝑝𝑚𝑜 |
𝐾=𝐾𝑜
=
(15)
2𝑚𝑒
Substitute equation 15 into equation 5 to obtain 𝐻𝑚𝑜 |2 :
𝑚𝑜2 Eg
2𝑒 2 𝐼(𝑣)
2
|𝐻𝑚𝑜 | = 2
∗[
]
3mo 𝑛𝑟 𝜀𝑟 𝑐ω2
2𝑚𝑒
Substitute equation 15b into equation 4 to obtain Pmo:
𝑚𝑜2 Eg
2𝑒 2 𝐼(𝑣)
𝑠𝑖𝑛2 (ω−ω𝑚𝑜 )∗𝑡
Pmo(t) = 2
∗
[
]
*
3mo 𝑛𝑟 𝜀𝑟 𝑐ω2
2𝑚𝑒
ħ2 (ω−ω𝑚𝑜 )2
Substitute Pmo from Eq. 15C in Eq. 3 to obtain PCV
2𝑒 2 𝐼(𝑣)ϱ(ω𝑚𝑜 )πt 𝑚02 Eg
𝑃𝑐𝑣 (𝑡) =
∗
3𝑚02 𝑛𝑟 𝜀0 𝑐ħ2 ω2
2𝑚𝑒
Subsisting for density of states 
2
(15b)
(15c)
(16A)
4𝜋
𝑃𝑐𝑣 (𝑡) =
3
2𝑒 2 𝐼(𝑣)𝑉[ 3 ](2𝑚𝑟 )2 (ℎ𝑣−Eg )1/2 πt
ℎ
3𝑚02 𝑛𝑟 𝜀0 𝑐ħ2 ω2
4π
𝛂=
ħω2e2 [ 3 ][2mr ]3/2
ħ
[
] [hv
6m20 nr 𝜀0 𝑐ħω2
−
∗
𝑚02 Eg
(16B)
2𝑚𝑒
1/2
2
1/2 mo Eg
Eg ] [
]
2me
(17)
𝛂 = A (h – Eg)1/2
3 

2  4 
2
1
  2e   3 2mv    m 2 E  2
 
 o g  ,
Where A= 
2
2
 6m0 nr  0 c
  2me 




A= 3.38 x 10-7 (nr)-1 (me/mo)1/2 in units of m-1 eV-1/2
(17B)
Excitonic transitions:
For exciton transitions in direct band gap semiconductor, the absorption
confident is derived by Elliott (1957).
e 
,
 ex     AR 
sinh 
1
2
γ=(
𝑅
ħω−Eg
1/2
)
,
(Eq. 2 page 164)
Here, R is the exciton binding energy Eex.
It is simplified to as photon energy approaches Eg or below.
𝛂𝐞𝐱 = AR1/2 π
eπγ
(eπγ −e−πγ )/2
≈ 2πAR1/2
1
(1−e−2πγ )
≈ 2πAR1/2 , at h=Eg.
(X, page 165 Eq. 4B)
The plot is shown below (Moss et al.)
Excitonic peak at (h-Eg)/R =-1. That is, when h is Eex or R below Eg.
3
Case II: When photon energy is equal or less than band gap Eg.
h  E g ,
 
Now we can see from equation (4b):
 ex  h   2 AR 2  2 AE ex2
(6, page 166)
We see that Equation (2) expresses both direct band-to-band at energy above band gap and slightly
below band gap.
Case I:
When photon energy is much larger than the band gap ℎ𝑣 ≫ Eg,
1
𝛂𝐞𝐱 =
2A(ħω−Eg )1/2 γπ
2πγ
1
, this simplifies to 𝛂𝐞𝐱 = A(ħω − Eg )1/2 ,
which is 𝛂𝐞𝐱 = α(ħω)
(band-to-band)
h
Eg
Absorption coefficient as a funciton h Absorption with excitonic contribution shown in
the form of peak below Eg.
(without excitonic contributions).
In the case of quantum wells, the excitonic binding energy Eex or R increases. In addition the direct
band-to-band absorption coefficient is higher due to increased joint density of states magnitude.
4
Calculation of exciton binding energy in bulk layers: (LED Notes, part II)
The binding energy of an exciton ranges from 0.004 eV to 0.04 eV. In general the exciton binding
energy Eex is:
4
m
rq
(1)
E ex = 2 2 2
8o r h
Where:, mr = reduced mass of the exciton given by
1
1
1
= +
(2)
m r mh me
q = electron charge, εr = dielectric constant of semiconductor or carbon nanotube etc.
h = Plank's constant
Binding energy can also be viewed as the dissociation energy. The latter being the energy needed
to make the electron and hole free, overcoming the Columbic attraction.
Quantitatively, Eex = (13.6/r2)*(mr/mo); the unit is in electron Volt. This comes from the fact that
4
q
the ionization potential of an hydrogen atom is 13.6 eV = mo2 2 . Semiconductors that have small
8o h
dielectric constant has larger exciton binding energy.
Bound and free excitons:
Excitons are free to move in semiconductor layer or they are bound to certain impurity sites. For
example in p-type GaP (dopted with zinc), when electrons are injected they form excitons if GaP
is doped with oxygen or nitrogen. In the case of oxygen, Zn-O locations provides sites for excitons
to remain localized at those sites as energy levels facilitates exciton formation.
Bound exciton at Zn- O site in p-GaP
When the electron and hole forming an exciton recombine, this recombination results in photon
emission. The probability of photon emission is as high (if not higher) as in the case of direct
transitions. The energy of photon hv emitted upon the decay of an exciton at the Zn-O site is:
h = E g - 0.3 - Eex .
(3)
if Eex = 0.04 eV
h = 2.24 - 0.3 - .04 = 1.9eV
(4)
or wavelength of emission is =1.24/h = 0.65 μm or 6500Å.
Bound exciton at nitrogen sites in p-GaP
An injected electron gets trapped at the nitrogen level, and subsequently binds a hole to form an
exciton. The decay of this exciton results in a photon emission. The energy of the photon is
h = E g - 0.08 - Eex
(6)
If Eex = 0.004eV
h = 2.24 - .08 - .004 = 2.156eV
(7)
1.24
=
= 0.575 m = 5750 
(8)
2.156
Although GaP is an indirect energy gap material, the electron-hole recombination via excitonic decay
is almost like a vertical (direct) transition. This is due to momentum rule relaxation as Heisenberg
uncertainty principle (x*p ~h) provides larger latitude in momentum change (p) as x is very
small due to bound excitons.
5
Calculation of exciton binding energy in quantum wells:
In quantum wells, the heavy and light hole bands split. In unstrained wells and compressive
strained wells, the heavy holes are dominant where as in tensile strained quantum wells, light holes
are dominant. Exciton binding is higher for smaller width quantum wells than the larger width
wells for a given material/semiconductor layer.
The electron and hole levels are
computed (like HW).
Ee1
Ehh1
Fig. 1(a) E = 0
Fig.2(a) Reverse biased p-n diode with MQWs.
Change in electro-absorption and exciton peak shift due to E-field:
The exciton binding energy as well as absorption plot is a funciton of electric field which is applied
by an external radio frequency source. One way to apply is to make a p-n junction diode and
reverse biasing it. This way the p and n- layers sandwich the multiple quantum well layers.
Electron wave
function
E’e
1g
E’hh
1
Hole wave function
Fig. 1(b) Quantum well in the presence of
E
Electric field moves the electron and holes in Fig. 2(b) Absorption/responsivity as a
different positions. This reduces absorption as funciton of applied voltage or electric field.
well as excitonic binding energy.
The electric field also changes the electron E’e1and hole E’hh1energy levels. The change in electron
energy levels and change in exciton binding energy E’ex results in the photon energy value at which
the exciton absorption peaks. See Fig. 2(b).
Note that the band-to-band absorption due to free carriers is not shown in Fig. 2(b). It is to the
right of plots shown here. It varies as 𝛂
= A (h – Eg)1/2.
6
Change in electro-refraction in multiple quantum wells due to E-field:
Index of refraction nc is complex when there are losses as photons travel in the medium.
nc= nr – i k
Where extinction coefficient is which is k = α/4, here alpha is the absorption coefficient. The
square root of permittivity  is nc.
nc= nr – i k = (ri)1/2
The imaginary part of the permittivity is given by:
The imaginary part i = 2nr , alpha is absorption coefficient, lambda is the wavelength and
nr the real part of index of refraction.

i=2nre2/(m2o2)]*(Density of states)*(transition matrix element)*(polarization
factor)*(Line shape funciton).
Ref. W. Huang, UCONN Ph.D. thesis 1995]
Gaussian Line shape function: L(E)=[1/()1/2]*exp[-(ho-h)2/2], =h/(ln2)-1/2,=1.54*10-13 s.
The density of states changes with quantum well, quantum wire and dots. Transition matrix
element involves wave functions in one, two or three dimensional confinement.
The real part r is related to imaginary part i as
The imaginary and real and index of refraction are:
Use i and  relation above to recognize various terms in the following gain expression using
excitons: Note gain coefficient g = - (fc + fv -1); here fc and fv are the Fermi distributions for
electrons and holes.
2 e2
[| M b|2  2  1/2 | ex (0) |2
g ex (  ) =

c

L
L
 o n r mo
y z l,h
 (| e ( y )h ( y )dy |2 | e ( z )h ( z )dz |2 )   ex  L( E ex )  ( f c + f v - 1)]
Exciton Binding Energy in Quantum Wires (OPTIONAL)
(Ref. W. Huang, UCONN Ph.D. thesis, 1995)
7
8
Indirect band-to-band transitions: summary (optional)
(A) Steps:  (h ) 
PCV (t )
I (h )Vt
Definition
Initial state ‘0’, intermediate state ‘i’ or ‘i’’, and final state ‘m’. Indirect transition can take place
in two ways:
0  i 
(a)
  Phonon assisted Direct
i   m
Process (a) is dominant if E 0  E m
The probability that an electron makes a transition from the initial ‘0’ to the final ‘m’ state is Pm0
, which is:
(B) Pm 0 
4 H i0
2
  i 0   
4
2
H
 2
mi

sin 2  m 0     p  t

  p 
2
2
m0
(2)
Eq.2 is obtained using second order perturbation theory.
Note that Pi 0 
H i0

2

2
sin 2  m 0   i 0  t
 m0   i 0 
2
2(b)
2
2
(Direct transition)
Hi 0
2
2 e2 I ( ) Pi 0

3m02 nr 0c 2
2
Hmi is the matrix element which represents electron-phonon interaction. ‘+’ sign represents
phonon emission and the ‘-‘ is for phonon absorption.
Qualitatively,




[here, '1' accounts for phonon emission]

 2

and Hmi is proportional to N p V

Hmi
2
is proportional to N p  1 V
(3)
N p is the number of phonons in a particular mode in the crystal of volume V.
By definition, 4)
5) N p  1 
Np 
1
e
h q kT
1
1
e
1 
hq kT
e
1
h q kT
e
(or the occupancy of the qth mode)
11
h q kT
1

1
1 e
 E p kT
The energy balance equations are:
h  E m  E 0  E p

(6)

h  E m  E 0  E f
(phonon absorption) 
(C) PCV is obtained by summing Pm0 over all pairs of initial states (in the valence band) and all
final states (in the conduction band).
(phonon emission)
9
PCV  PmV  Pm 0
7)

and
PmV   Pm 0  v Ev d  m 0 
0
  E p  E g
P
8) PCV (t ) 
mv
 c Ec dEc
0
Where
 E p for phonon absorption
and (- E p ) for phonon emission
Eq.2, 7 & 8 give:

2 H i 0 H mi
2
PCV (t ) 
2
 2 E0  h 
2
  E p  Eg
  E  E
t
c
c
v
c
   E p dEc
0
Density of states (in volume V)
4 2mc 
c  Ec   V
h3



12 
32
V 4 2mv  E g  Ev 
v  Ev  

h3
32
Ec
m

m is the CB (10) minima

Conservation of energy – just like joint density of states
 c ( E c ) and  v ( E c    E p ) in the integrand give
12 12
 Ec Ec dEc 
E c2

2
1
2
  E
 Eg 
2
p
Thus, Eq.9 integral is
 mc  me 


 m  m 
n
 v
8V 2 m 3 mc mv 
h6
32
  E
 Eg 
2
p
2

Pcv (t ) 2 H i 0 H mi
8V 2 m 3
mc mv 3 2    E p  E g 2
and
 2

2
6
t
h
 E 0   
2
Or for phonon emission
2 H i 0
Pcv (t )
B 
1
 2
 C 
2
E
t
 E0    V  1  e p
2
mc mv 3 2   E p  E g 2
kT
 8V 2 m 3

h6

Combining Eq.1 with Eq.11, we get
12)
10
(9)
 h  
C   E p  E g 
2
C   E p  E g 
2

E kT
 E kT
e p 1
1 e p
[Phonon Absorption ] [Phonon Emission]
Jα
C has contribution from the 0  i  m route.
13)
2 H i 0
BC 8V 2 m 3

mc mv 3 2
C
 2


I ( )V  E 0   2 V
h6
2
Eg Eg+EP
C
2
Substiutin g for H i 0 using Eq.2(b)
 2e 2 I ( ) Pi 0 2 
2  2
2 
 3m0 nr  0 c  BC 8V 2 M 3

mc mv 3 2
C



2
3
I ( )V
V
h
E0   
32
Pi 0 BC
M mc mv  e 2
or C 

2 7
2
6  m0 nr  0 c E 0   2
This is:
2
14)
32
Pmi BV
M mc mv  e 2
C

6 2  7 m02 nr  0 c E m   2
2
Exciton transitions in indirect gap semiconductors
Jα
Eg Eg+EP
C
Band-to band transitions in heavily doped semiconductors: band filling and
band tailing:
Band filling takes place in low energy gap semiconductors such as InSb where effective mass for
electrons and holes is very small as compared with Si and GaAs.
11
Fabry-Perot Cavity (Modulators and Lasers):
Cr-Au Contact
W = 3 -> 5 um
p+ GaAs cap
Excited Region
Top Contact Stripe
P-Al0.3Ga0.7As
SiO2
50µm
1 um - 2 um
p-Ga0.93As0.07As
n - Al0.2Ga0.8As
(Cladding)=ND>9.7x1018cm-3
d=0.2um
p-Ga0.83As0.17As~0.1µm
p-GaAs
1 um - 2 um
p - Al0.12Ga0.88As
(Active)=1016cm-3
Output
n - Al0.2Ga0.8As
(Cladding)=ND>9.7x1018cm-3
n-GaAs substrate
Au-Ge-Ni contact
Laser output
p-GaAs
Active Layer~0.1 to 0.5µm
n-Ga0.7Al0.3As-0.2µm
+
n - GaAs Substrate (100 150 um) ND>1019cm-3
y
Ohmic Contact
Fig .48 Distributed feedback (DFB) edgeemitting laser (first order grating).
Fig .28. An edge-emitting cavity laser with ridge.
La
P01
Lg
Lp
P02
P02
r1
Lg1
r2
Leff
rg
La+Lp LDBR
La
LDBR
r1
Fig. 55 Cavity formed by one conventional mirror and one
distributed Bragg reflector (DBR).
Fig. 56 (right panel): In a vertical cavity surface emitting laser
(VCSEL) the cavity thickness is multiple of half wavelengths and
DBRs are quarter wavelengths with high and low index.
Lg1

Fig. 56 VCSEL with DBR reflectors.
P01
Condition of oscillations:
-jL
t1 t 2 e
Eo =
- 2jL
)
E i (1 - r1 r 2 E

2
1 - r1 r 2 e-2j  n r  + j
Gain condition
1 - r1 r 2 e(g- )L = 0
1  1 
g =  + ln 

L  r1 r 2 
Phase condition
- j2m
e
=e
-4 jnrL

; L=
(g- ) 
L
2 
m
2 nr
Fig. 1a. Electric field strength after many passes in a cavity of length L.
12
=0
ECE 4243-6243 10042016 L6: F. Jain HW6B
Waveguide Equations for light waves: One dimensional confinement along x-axis.
Three slab waveguide structure. Waveguide thickness is d.
Waveguide equation is like Schrodinger equation for electrons.
 2Ey
5.
+
 2Ey
=  0
 2Ey
 x2
 z2
 t2
Wave propagating in the z-direction


E y (x,z,t) = A ¢ coskx + A ¢ sin kx e j  wt  βz 
18.
e
o
EVEN TE MODES
Solution to the Eigenvalue equation: find k and  by solving eq. 38 and 39.
β  n k 
tank d / 2 = =
k n k  β 
2 2 12
1 0
2 2
2 12
2 0

38.
2
Substitution(15),(23)
k 2 = n 22 k 02  β 2
as
and
(equation 15)
 =β 
and
(equation 23)
Add equations 15 and 23:
2
k 2 +  2 = n22  n12 k 02 , multiplying by d / 2
2
k
2
+
2
2
d 2 = 
2
n 22

n12

 k0 d 


 2 

2

2
 kd 
 d 
2
2 k d 
  +   = n 2  n1  0  Rewriting Equation 27 as:
 2 
 2
 2 
2
39.
n12 k 02
2
13
14