Theory of relaxation oscillations in semiconductor quantum dot

Theory of relaxation oscillations in semiconductor quantum dot

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Rate equations for photon and
electron/ hole occupation in
wetting layer (WL) and quantum
dot (QD) states
Microscopic approach to
determine Coulomb scattering
rates in InAs / GaAs QD lasers
Strong relaxation oscillations
(RO) on ps / ns timescale as
observed in experiment
Rate equations for electron density
photon density in QD states:
, hole density
cavity loss, optical confinement factor, Quantum Dot density, spontaneous emission coefficient
'
z-component and Bloch function '
.
72
Scattering rates '*;'= - /
4 5
Spontaneous emission: Normalization Area:
Einstein coefficient: Induced emission: , and
+;?7@ , 0 5 , 4 0
Timescales restricted by applicability of Markov approximation:
occupation probability in e / h QD state
Coulomb scattering rates with sum over WL state occupations with Evaluate matrix elements function and energy levels byby
eigenfunctions
harm.
approximated
approximated
eigenfunctionsofof2D
2D
harm.
oscillator
oscillator and
and orthogonal
orthogonal plane
plane waves
waves
0
.
20
7
20
7
:,>+9 9-2 /
(
' (
'
strong
strongconfinement
confinementby
byinfinite
infinitebarrier
barriereffective
effective
well
wellwidth
widthapproximation
approximation
by separation of QD–WL system's wave
:
into in-plane component ,
2
72
(
'*;'= - /
0
calculated up to 2nd order Coulomb interaction
Lindhard
equation
screening:
screening:2D
2Dstatic
staticlimit
limitofofdynamic
dyn. Lindhard
equation
2
background dielectric constant
2
3
:,>+9 9-2 /
'
2
Calculation of steady states of equations (1)-(5)
'
'
3
7
/
2
2
scattering rates artificially
10 - times reduced
with and (a) Photon density
2
70
72
8 .
WL occupation
2
9
2
70
3
<
.
2
#
6
2
2
3
5
.
6
#
2
3
4
<
2
-/
< 4
2
5
6
#
3
2
2
-/
-/
turn-on delay of 0.8 ns
12.5 GHz relaxation oscillations
0
(b) QD carrier density
(c) measured RO in QD structure
3
(Kuntz, Bimberg 2005)
strongly damped RO
4
4
@
7
$ + -2 /
@
0
good agreement with simulations
0
1
.
2
0
1
<
3
<
5
.
30
3
2
2
2.1
2.@
2<
2<2
2
2<
7
6 +-2 /
Parameters :
#
6
72
2
7
8-2 /
4
5
0
9
7
equations (1)-(5) with injection current
(
2
8-2 /
7
72
7
rate equations for WL electron density and hole density :
in QD fed by
70
= > , 5 hole and
electron levels
?86(8-/-2 /
/
7
6 -2
0
$ +-2 /
(
2
characteristics
2
2
7
$ +
diode – laser
typical
:,76 +-
'
2
6 -2 /
7
0
-2 /
Laser: weak δ - like perturbation
'*;- 72
/
system (1) – (3) with fixed scattering rates
0
.
1
2
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