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EE 232: Lightwave Devices
Lecture #14 – Excitons
Instructor: Seth A. Fortuna
Dept. of Electrical Engineering and Computer Sciences
University of California, Berkeley
4/4/2019
2Fortuna – E3S Seminar
Two-particle effective mass equations
e−
h+
R
err
hr
( , ) ( ) ( )G =R r R r
2 22
*( ) 0
2 4r r
r
qE
m
− =
−
r
r
22
* *( ) 0
)2(RR
e hm mE G−
−
+=
R
(motion of a “free” particle)
(relative motion of the “hydrogen-like” molecule)
X g r RE E E E= ++
(two-particle envelope wave-function)
(total energy)
So far, we have ignored any attraction between the negative electron and positive hole. In reality, the Coulombic attraction results in new bound electron-hole states, or excitons.
Like many two-body problem in physics, we can break up the problem into one describingthe center of mass motion of the two-particle system and the internal motion of theindividual particles.
3Fortuna – E3S Seminar
Exciton energy
Exciton radius
: Rydberg energy (13.6 eV)
R : Exciton Rydberg energy
H
X
R
2 20
* r H X
r
nr nm
am
n a= =
11: Bohr radius (5.29 )10Ha −
22
2R
e h
EM
=
+=
K
K k k
Exciton center of mass moves through the crystal as plane wave
4 *
2 2 2
0
*
2 2 2
0
1 1
(4 ) 2
1
rr
r
r H X
r
q mE
n
R R
n n
m
m
= −
−
− =
=
Exciton binding (internal) energy
Exciton “kinetic” energy
4Fortuna – E3S Seminar
Exciton energy
22
2
2
2
XX g
Xg
ER
n M
n
E
RE
+−
−
=K
Since, as usual, optk is small1n =
2n =
3n =
n =
K
XE
0
gE
Black dot represents theground state (electron invalence band).
Photon withcreates an electron-holebound pair with energy andK-vector on one of the curves(see Blood Appendix C)
2
g XR nE −=
Exciton binding energy forcommon semiconductors
5Fortuna – E3S Seminar
Absorption with inclusion of excitonic effect
Refer to Chuang Ch. 14 for details of the derivation
22
0 )2 ( 0)ˆ( ) (cv r
n
n gC Ere E = + −= p
e.g. Bulk (without excitonic effect)
( )1 1( ) (0)x y zi
n neV V
+ +
=→=k k k r
r
2 2
*2n
r
kE
m=
2 2 2 y y z z
y z
x x
x
k n k n k nL L L
= = =
This is a more general expression for absorption that can account for both free carrier and excitonic absorption.First, let us check that we recover our result for free carrier absorption
2
0
2
0
2
0
ˆ( ) ( )
ˆ ( ) ( )
ˆ ( )
1v r
n
r
c g
cv r n g
cv r g
e E
e E E E dE
C EV
C
C e E
=
=
=
+
+
−
−
−
p
p
p
(Assume here that ) 1vf =
Sum over all states
6Fortuna – E3S Seminar
Absorption with inclusion of excitonic effect
(See Chuang Ch. 3)
2
2
3 3
2
3
(0) Bound states
1
+ continuum
1
4 sinh(
states
(0)
)(0)
r X
B
X
E R
C
r
X
X x
n
e
R a E R
a
=
=
=
Bound states
Continuum states
(Bound states)
(Continuum states)
7Fortuna – E3S Seminar
2 2
0 3 3ˆ( ) ( 1 )
2B cv
n x X
e nCR a n
= +p
Absorption with inclusion of excitonic effect
2 2
0 3 3ˆ( ) ( )
2B cv x g
n X
C En
e R na
= +− −p
Bound states
Continuum states
2
0 3
2
0 3
ˆ( ) ( )
Let ( )
ˆ( ) ( )
n
2 sinh( )
2 si h( )
r X
r X
cv g
X x X
g X
c
E R
C r
r
E R
C r
r
v x
X x X
e E
E R
e E
dE eC E
R a E R
dE eC R
R a E R
=
= −
=
+ −
+
p
p
( ) g XE R= −
8Fortuna – E3S Seminar
Absorption with inclusion of excitonic effect
Continuum states (cont’d)
2
0 3 2
2
0
20 3 2
2
33 2
ˆ( )
ˆ
ˆ) ( )
( )
2 sinh( )
2
2 1
(2
cv
X x
cv
X x
C cv D
X x
C
eC
R a
CR a e
CR
e
e
ea
S
−=
−
=
=
p
p
p
is the Sommerfieldenhancementfactor
Let’s expand this expression
( )3
3/
2
0 32
2
0 33 22
ˆ( ) ( )
ˆ ( )
2
2
g x
cv D
X x
g
cv
X x
C
D
E RC
R a
Ca
e S
Ee S
R
−=
−=
p
p
3DS
( )g XE R= −
9Fortuna – E3S Seminar
Absorption with inclusion of excitonic effect
Continuum states (cont’d)2
0 3324 *
20
2 2
0
3 2
2
3 2
*
*2
0 32 2
3
ˆ( ) ( )
ˆ ( )
) ( )
42
(4 ) 2
21
2
(
g
c cv D
rr
r r
rcv g D
free D
EC e S
q m
q m
mC e E S
S
−=
= −
=
p
p
We see that the continuum absorption is free carrier absorption multiplied by the Sommerfield enhancement factor. Evidently, theelectrostatic attraction between electron and hole increases thetransition strength.
10Fortuna – E3S Seminar
Summary - Absorption with excitonic effects for bulk semiconductor
2 2
0 3 3ˆ( ) ( 1 )
2B cv
n x X
e nCR a n
= +p
3( ) ) ( )(free DC S =
2 2
0 33 3)
2
( ) ( ) ( )
ˆ( ) ( 1 ) ) ((
B C
cv f
Xn
ree D
x
e n SR a
Cn
= +
= + +p
231
)2
(De
S
−
=−
Boundstates
Continuumstates
( ) g XE R= −
Totalabsorption
2
XX gE E
R
n= −Exciton Energy
Absorption
11Fortuna – E3S Seminar
Summary - Absorption with excitonic effects for bulk semiconductor
1n =
2n =
3n =
continuum
free
Linewidth broadening not included
GaAs bulk
Absorption is step-likeat bandedge!
12Fortuna – E3S Seminar
Experimental data - Bulk semiconductor
M.D. Sturge. Phys. Rev. 127, 768 (1963).
13Fortuna – E3S Seminar
Experimental data - Quantum well
Source: S. Schmitt-Rink et al. Advances in Physics, 38:2, 89-188
T=300K
We observe exciton peaks due to bound states and enhancement of the absorption in the continuum. Exciton peak observed near the beginning of each subbandtransition. Quantization gives higher binding energy allowing for clear observation of exciton peaks even at room temperature. (See Chuang Ch 14 for more details)
14Fortuna – E3S Seminar
Excitons at high carrier injection
Low-density
High-density
separation > Xa
Separation ~ Xa
At high carrier injection, the Coulombic potential is “screened-out” and excitons do not form.
The exciton density at which excitons beginto dissociate is called the Mott density.
3
1 1
4Exciton volume
3
Mott
X
N
a
=
Ex. For GaAs, this estimation gives. Photonic devices
are often operated with carrier density exceeding the Mott density. In this case,the exciton effects (Coulombic enhancements) may be reduced.
17 -310 cmMottN
15Fortuna – E3S Seminar
Excitons at high carrier injection
Absorption fromn=1 exciton statedecreases withincreased carrierinjection
Fox. Optical Properties of Solids.
16Fortuna – E3S Seminar
Excitons at high carrier injection
Haug and Koch. Phys. Rev. A 39, 1887 (1989).
Coulomb enhancement of opticaltransition probability for GaAs quantum well (T=300K)
12 -2105 cmn =
12 -2102 cmn =
12 -2101 cmn =
211 -5 0 m1 cn = To a reasonable first approximation,Coulombic enhancement canbe ignored under typical operatingconditions for a semiconductor laserat room temperature.
