Overview: In the following, we show how to describe
carrier capture and recombination in a semiconductor nanoscale quantum dot (QD)
structure. We also present equations of motions related to the material polarisation induced
by electron and holes captured into QDs.
The model is based on MaxwellBloch equations and takes into account
the following effects:
 induced recombination of electrons and holes with photon emission
 scattering of carriers between discrete QD levels (section 2.1)
 scattering of carriers between a 2D semiconductor quantum well (QW) and
the discrete QD levels (section 2.2 and 2.3)
Note: The equations describing the material polarisation can be
coupled with the waveguide model introduced
here to simulate
optoelectronic devices like lasers, optical amplifiers, absorbers, etc.
In this section, we introduce a semiclassical approach to model lightmatter interaction in a
semiconductor structure. In this approach, the light fields are described by a classical waveequation
derived from Maxwell's equations:

 ⎡ ⎣

∂ ∂t

± 
β k_{0}


c ϵ_{p}


∂ ∂z
 ⎤ ⎦

E_{T}^{±} =  ⎡ ⎣

i 2 ω_{0}

 ⎛ ⎝

c^{2} ϵ_{p}
 ⎞ ⎠


∂^{2} ∂x^{2}

+ 
i ω_{0} 2


δϵ_{a} ϵ_{p}

Γ− 
β k_{0}


c ϵ_{p}


α 2
 ⎤ ⎦

E_{T}^{±} + 
i ω_{0} 2


Γ ϵ_{0} ϵ_{p}

P_{T}^{±} + F(t)e_{x}. 
 (1.0) 
Eq.
1.0 describes the propagation of the transverse electric field
E_{T}^{±} including effects like light diffraction, waveguiding,
absorbtion, and the interaction with medium via the
material polarisation P_{T}^{±}.
For detail related to the notation and the derivation of this equation please consult sections:
Waveguiding and Optical Fields
and
QDSOA Theoretical Model.
Eq.
1.0 can be integrated numerically,
given that the material polarisation
P_{T}^{±} can be
calculated by:
P_{T}^{±} =
V^{−1}∑
_{j,i} p_{j,i}^{±} , where V is the
normalizing volume, and electron and hole states are labelled using the index
i and
j,
respectively.
The quantity
p_{j,i}^{±} represents the expectation value of the quantum
mechanical operator related to the
microscopic polarization induced by charge
carriers confined to QDs.
In the following, we derive an equation of motion describing the temporal evolution of
p_{j,i}^{±}.
The Hermitian operator related to the microscopic polarisation
in dipole approximation is defined as [
9]:
∧p_{j,i} =
b_{j} a_{i} ∆
^{∗}_{i,j} , where the following notation is used:
 
dipole matrix element, (induced by an electron in state i and a hole in state j), 
 
 
electron creation, annihilation operator, 
 
 
hole creation, annihilation operator. 
  (1.1) 

We start with Heisenberg's equation of motion for the polarisation operator, given by [
5]:


∂ ∂t


^ p

j,i

= 
i ħ

 ⎡ ⎣

^ H

, 
^ p

j,i
 ⎤ ⎦


  (1.2) 
In freecarrier approximation and assuming solely dipole interaction
with the electric field, the Hamiltonian of the physical system is given by:


^ H

= 
∑
i

ϵ^{e}_{i} a^{f}_{i} a_{i} + 
∑
j

ϵ^{h}_{j} b^{f}_{j} b_{j} − 
∑
i,j

[ a^{f}_{i} b^{f}_{j} ∆_{i,j} + b_{j} a_{i} ∆^{∗}_{i,j} ] ·E. 
  (1.3) 
The first two terms represent the energy of the bound QD electron and hole states, respectively.
The last term in (
1.3) represents the interaction between the
interlevel dipoles and the electric field. Evaluating
(
1.2) we get:



∂ ∂t


^ p

j,i

= − 
i ħ

( ϵ^{e}_{i} + ϵ^{h}_{j}) 
^ p

j,i

− 
i ħ

{∆_{ij} ⊗∆_{ij}^{∗} [ a^{f}_{i} a_{i} + b^{f}_{j} b_{j} − 1 ] ·E}, 
  (1.4) 
where `⊗' labels the tensor product and
∆
_{ij} ⊗∆
_{ij}^{∗} is the dipole matrix tensor.
In obtaining (
1.4) we have neglected terms of the form:
a^{f}_{k} a_{i} and
b^{f}_{l} b_{i}, representing
intraband crosscorrelations.
The equations describing their dynamics contain fast oscillating
contributions of the form:
i(ϵ
^{e}_{k} − ϵ
^{e}_{i})/ħ and
i(ϵ
^{h}_{l} − ϵ
^{h}_{j})/ħ,
justifying the omission of these terms.
Assuming that the polarisation is coherent with the optical fields introduced
here
and that ω
_{0} is the optical frequency, we make the following ansatz [
5]:


~

= 
^ p

± j,i

e^{±i βz − i ω0 t} 



~ E^{±}

= 
1 2

( E^{±} e^{±i βz − i ω0 t} + E^{∗±} e^{±i βz + i ω0 t} ) 
  (1.5) 

Introducing ansatz (
1.5) into (
1.4) we obtain:


∂ ∂t


^ p

± j,i

=  ⎡ ⎣

− 
i ħ

( ϵ^{e}_{i} + ϵ^{h}_{j})+ i ω_{0}  ⎤ ⎦


^ p

± j,i

− 
i 2 ħ

{∆_{ij} ⊗∆_{ij}^{∗} [ a^{f}_{i} a_{i} + b^{f}_{j} b_{j} − 1 ] ·E^{±} } 
 
 − 
i 2 ħ

{∆_{ij} ⊗∆_{ij}^{∗} [ a^{f}_{i} a_{i} + b^{f}_{j} b_{j} − 1 ] e^{ 2 ( ±i βz + i ω0 t )} ·E^{∗±}} 
  (1.6) 

The last term in (
1.6) can be neglected (rotating wave approximation).
It contains an exponential factor that oscillates at optical frequencies and
will average to zero for timescales much larger than the period of the optical field.
Equation (
1.6) describes the dynamics of the Heisenberg operator related to
the material polarisation induced by charge carriers confined to QDs. In a similar way,
starting from Heisenberg's equation of motion for the operators
a^{f}_{i} a_{i} and
b^{f}_{j} b_{j}
and using the ansatz (
1.5) one can derive [
5]:
 


i 2 ħ


∑
j


^ p

f,+ j,i

·E^{+} + 
^ p

f,− j,i

·E^{−} − 
^ p

+ j,i

·E^{∗,+} − 
^ p

− j,i

·E^{∗,−} 
 
 


i 2 ħ


∑
i


^ p

f,+ j,i

·E^{+} + 
^ p

f,− j,i

·E^{−} − 
^ p

+ j,i

·E^{∗,+} − 
^ p

− j,i

·E^{∗,−} 
  (1.7) 

Equations (
1.6) and (
1.7) refer to timedependent
operators in the Heisenberg picture.
Using the following definitions:
 

< 
^ p

± j,i

> microscopic polarisation, 
  (1.8) 
 

< a^{f}_{i} a_{i} > occupation probability of QD electron level i, 
 
 

< b^{f}_{j} b_{j} > occupation probability of QD hole level j, 
 

and taking the expectation value of (
1.6) and (
1.7), we obtain:
 

 ⎡ ⎣

− 
i ħ

( ϵ^{e}_{i} + ϵ^{h}_{j})+ i ω_{0}  ⎤ ⎦

p_{j,i}^{±} − 
i 2 ħ

{∆_{ij} ⊗∆_{ij}^{∗} [ n^{e}_{i} + n^{h}_{j} − 1 ] ·E^{±}} , 
  (1.9) 
 


i 2 ħ


∑
j

p_{j,i}^{∗,+} ·E^{+} + p_{j,i}^{∗,−} ·E^{−} − p_{j,i}^{+} ·E^{∗,+} − p_{j,i}^{−} ·E^{∗,−}, 
  (1.10) 
 


i 2 ħ


∑
i


p

∗,+ j,i

·E^{+} + 
p

∗,− j,i

·E^{−} − 
p

+ j,i

·E^{∗,+} − 
p

− j,i

·E^{∗,−}. 
  (1.11) 

The coupled differential equations (
1.9)  (
1.11)
describe the dynamics of the carrier occupation probability of
bound QD states and the corresponding dipole moment.
Since our starting point was
the Hamiltonian (
1.3) in freecarrier approximation
these equations do not include
terms describing relaxation and dephasing processes. Moreover, equations
(
1.9)  (
1.11) refer to one single QD.
A manybody description of the QDWL system, including Coulomb and carrierphonon [
12] interaction,
would provide a selfconsistent way of describing dephasing and energy shifts due to scattering processes.
Additionally, it would lead to a dependence of ω
_{j,i}^{e,h} on the QD level occupation [
3],
[
6].
While such a description is
beyond the scope of this model, numerical calculations have
shown that, due to the strong confinement regime, Coulomb effects on the optical
properties of the QD structure presented here are relatively weak [
13].
In order to include carrier capture and relaxation processes, equations (
1.9)  (
1.11)
are extended by terms describing carriercarrier and carrierphonon scattering, respectively.
Carrier scattering leads to a fast decay of the interlevel polarisation
and will be included via an effective dephasing rate
[
5]. In our model, the effective polarisation dephasing rate, γ
_{j,i}, is
linked to the homogeneous broadening of the QD transitions.
We also take into account the spectral broadening ∆γ
_{j,i}
and spectral shift ∆ω
_{j,i} of the QD transition due
to elastic scattering with 2D charge carriers in the wetting layer [
16].
The following definitions are introduced:
 

( ϵ^{e}_{i} + ϵ^{h}_{j} )ħ^{−1} , 
 
 

n_{2D}^{e} A_{j,i}^{e} + n_{2D}^{h} A_{j,i}^{h}, 
 
 

n_{2D}^{e} B_{j,i}^{e} + n_{2D}^{h} B_{j,i}^{h}, 
  (1.12) 

where ω
_{j,i}^{e,h} is the angular frequency of a photon emitted
by recombination of an electron
in QD state
i and a hole in state
j,
n_{2D}^{e} is the electron charge density,
and
n_{2D}^{h} is the
hole charge density in the 2D WL. The coefficients
A_{j,i} and
B_{j,i}
describing spectral broadening and shift, respectively,
are calculated according to [
16].
Starting from the differential equations (
1.9)  (
1.11)
and including terms related to
spontaneous emission and scattering processes, we obtain:


∂ ∂t

p_{j,i}^{±} = − (γ_{p} + ∆γ_{j,i})p_{j,i}^{±} + i(ω_{j,i}^{e,h} − ω_{0} − ∆ω_{j,i})p_{j,i}^{±} 
 

− 
i 2 ħ

{∆_{ij} ⊗∆_{ij}^{∗} [ n^{e}_{i} + n^{h}_{j} − 1 ] ·E^{±} }, 
  (1.13) 


∂ ∂t

n^{e}_{i} = 
i 2 ħ


∑
j

{ p_{j,i}^{∗,+} ·E^{+} + p_{j,i}^{∗,−} ·E^{−} − p_{j,i}^{+} ·E^{∗,+} − p_{j,i}^{−} ·E^{∗,−} }− 
∑
j

Γ^{sp}_{i,j} n^{h}_{j} n^{e}_{i} 
 

+ 
∂ ∂t

n^{e}_{i}  ⎤ ⎦

c−ph
QD

+ 
∂ ∂t

n^{e}_{i}  ⎢ ⎢

c−c
QD ↔ WL

+ 
∂ ∂t

n^{e}_{i}  ⎢ ⎢

c−ph
QD ↔ WL

, 
  (1.14) 


∂ ∂t

n^{h}_{j} = 
i 2 ħ


∑
i

 ⎧ ⎨
⎩

p

∗,+ j,i

·E^{+} + 
p

∗,− j,i

·E^{−} − 
p

+ j,i

·E^{∗,+} − 
p

− j,i

·E^{∗,−}  ⎫ ⎬
⎭

− 
∑
i

Γ^{sp}_{i,j} n^{h}_{j} n^{e}_{i} 
 
 + 
∂ ∂t

n^{h}_{j}  ⎢ ⎢

c−ph
QD

+ 
∂ ∂t

n^{h}_{j}  ⎢ ⎢

c−c
QD ↔ WL

+ 
∂ ∂t

n^{h}_{j}  ⎢ ⎢

c−ph
QD ↔ WL

, 
  (1.15) 

where Γ
^{sp}_{i,j} is the rate of spontaneous emission and the last two three
in both (
1.14) and (
1.15) refer to carrierphonon and
carriercarrier scattering, respectively.
In our theoretical model, QDs are refilled with charge carriers via carrier capture
involving emission of phonons and via carriercarrier (Auger) capture processes.
Depending on the 2D carrier density in the wetting layer, the population of
QDs with carriers, and the temperature, scattering processes may lead to caputer of carriers into the QDs or
excitation of carriers out of bound QD states.
At sufficiently high 2D charge carrier densities (in the range of
10
^{11} cm
^{−2} − 10
^{12} cm
^{−2}), capture processes transferring charge carriers
from the sourounding semiconductor medium into discrete QD states dominate.
Fast QW↔QD carrier capture and intradot relaxation
are essential for the ground state operation of QDLs and QDSOAs
(e.g the amplification of a train of short pulses in a QDSOA [
14]).
In our theoretical model, we describe carrier scattering between 2D states and
discrete QDs mediated by carrierphonon and Auger processes, respectively.
Depending on the operating conditions, scattering processes can also
lead to the depletion of QD states. E.g. holes in excited QD states have a
high probability of being ejected into 2D WL states due to the low confinement
energy. It has been shown that Auger processes represent a major nonradiative
loss channel in long wavelength devices [
11].
In general, carrierphonon and Auger scattering rates depend strongly on the
transition energy. An inhomogeneous broadened QD ensemble is characterized by
dottodot variations of the confinement energy w.r.t. the 2D continuum
and the level structure of bound states. In our approach, we calculate scattering
rates for each QD in the statistical ensemble (for details see section ).
In section
1.2, we present the calculation of LOphonon mediated
intradot scattering rates.
The description of LOphonon and Auger mediated QW↔QD
scattering rates is presented in section
1.3 and
1.4, respectively.
This type of process includes scattering of charge carriers between discrete QD states.
We model intradot scattering of carriers coupled to LOphonon modes.
The scattering rates are calculated according to [
10]
under the assumption that LOphonons decay into acoustic phonons
(due to an anharmonic coupling term).
The electronphonon coupling strength in [
10] is calculated
for boxlike QDs, though the authors point out
that the final result (relaxation rate) does not depend sensitively
on the coupling constant. For simplicity
we use the same scattering rates to model holeLOphonon scattering.
As seen in Fig.
1.1 the LOphonon intradot scattering rates
depend on the dephasing between the
transition energy and the energy of the LOphonon. (The frequency of
LOphonons in the InGaAs alloy is taken from
[
1], [
2], [
7].)
Figure 1.1:
Rate of intradot electron scattering (at 300 K)
with emission or absorption
of one 'GaAslike' LOphonon as a function of the transition energy.
The term describing intradot carrierphonon scattering in (
1.14) and
(
1.15) is given by [
8]:


∂ ∂t

n^{c}_{k}  ⎢ ⎢

c−ph
QD

= 
∑
k_{↑} > k

{ γ^{em}_{k,k↑} n^{c}_{k↑} ( 1 − n^{c}_{k} ) − γ^{abs}_{k,k↑} n^{c}_{k} ( 1 − n^{c}_{k↑} ) } 
 
 + 
∑
k_{↓} < k

{ γ^{abs}_{k,k↓} n^{c}_{k↓} ( 1 − n^{c}_{k} ) − γ^{em}_{k,k↓} n^{c}_{k} ( 1 − n^{c}_{k↓} ) }, 
  (1.16) 

where
c labels electrons or holes and
k is the level index.
γ
_{em}^{ph}(
k,
k_{↑})
and γ
_{abs}^{ph}(
k,
k_{↑}) are the scattering rates for the absorption
and emission of one LOphonon, respectively.
n^{c}_{k} is the
occupation probability of QD state k.
The
first summation on the right side of (
1.16) describes the relaxation of
carriers from a higher level
k_{↑} to a lower level
k, the second summation describes
the excitation of carriers from a lower level
k_{↓} to a higher level
k.
Another scattering process involving the emission or absorption of LOphonons is the
capture of carriers from the
WL into the QDs or the escape of carriers from the QDs to the 2D wetting
layer. The scattering rates for this type of process
are modelled using the following equation [
4]:
 
∂ ∂t

n^{c}_{k}  ⎢ ⎢

c−ph
WL

= 
m^{∗}_{c} 4 πħ^{2}


e^{2} ω_{LO} ϵ_{0}

 ⎛ ⎝

1 ϵ_{∞}

− 
1 ϵ_{stat}
 ⎞ ⎠

F(E_{Q})  ⎡ ⎣
 ⎛ ⎝

n_{LO} + 1  ⎞ ⎠

f_{Q}  ⎛ ⎝

1−n^{c}_{k}  ⎞ ⎠

− n_{LO}  ⎛ ⎝

1−f_{Q}  ⎞ ⎠

n^{c}_{k}  ⎤ ⎦


  (1.17) 

where
m^{∗}_{c} is the effective mass of the 2D carriers in the WL and
ω
_{LO} is the LO phonon frequency. ϵ
_{0} is the
permittivity of vacuum, ϵ
_{∞} and ϵ
_{stat} are the high
frequency and static dielectric constants.
n_{LO}=[exp(ħω
_{LO}/
k_{B}T)−1]
^{−1}
is the Bose occupation probability of LO phonons.
f_{Q}(
n^{c}_{2D},
E_{Q}) is the Fermi
occupation probability of the 2D carriers at the transition energy:
E_{Q}=
E_{QD}+ħω
_{LO}.
F(
E_{Q}) is a formfactor that depends on
the transition energy and on the wavefunctions of the 0D and 2D states.
Figure 1.2:
Rate of carrier capture (due to emission of one 'GaAslike' LOphonon)
from a 2D WL state to a QD state as a function of the transition energy.
Temperature: 300 K.
Fig.
1.2 shows the dependence of the inscattering rates from the 2D wetting
layer to a QD state on the transition energy.
The scattering rates are calculated for cylindrical QDs [
4] and
depend on the confinement energies of the QD states,
the geometry of QDs and WL, the temperature, the 2D charge density,
and the occupation probability of the QD states.
The derivation of (
1.17) is based on Fermi's golden rule using the single state energies and considering
scattering events involving one LOphonon only. This is reflected in the cutoff of the relaxation rates
for transition energies larger than the LOphonon energy.
A quantum kinetic treatment of
carrierphonon interaction of the QDWL system in the polaron picture [
12] shows that fast
intradot scattering of carriers is relatively insensitive to the detuning between transition energy and
LOphonon energy. For scattering from the 2D wetting layer to the localized QD states a stronger dependence
on the detuning is found. Direct relaxation from the WL to the QD GS can still occur even for a detuning
exceeding double the LOphonon energy, but with a reduced scattering rate. Keeping the limitations
in mind, simulation of QDWL scattering of carriers can still be described
by (
1.17) as long as at least one excited QD electron/hole state has a confinement energy with respect
to the wetting layer states of less than ħω
_{LO}.
Carriercarrier scattering processes with capture of one carrier into discrete QD states have been proposed
as a possible relaxation path that is especially effective at high carrier densities.
QD capture rates of electrons and holes due to Auger scattering (including carrier
saturation effects) are modelled by the following equations [
15]:
 
∂ ∂t

n^{e}_{i}  ⎢ ⎢

c−c
WL

= C_{ee} n^{e}_{2D} n^{e}_{2D} (1 − n^{e}_{i}) + C_{eh} n^{e}_{2D} n^{h}_{2D}(1 − n^{e}_{i}) + 
∑
j

B_{eh} n^{h}_{2D} n^{e}_{i} (1 − n^{h}_{j}) , 
  (1.18) 

 
∂ ∂t

n^{h}_{j}  ⎢ ⎢

c−c
WL

= C_{hh} n^{h}_{2D} n^{h}_{2D} (1 − n^{h}_{j}) + C_{he} n^{h}_{2D} n^{c}_{2D} (1 − n^{h}_{j}) − 
∑
i

B_{eh} n^{h}_{2D} n^{e}_{i} (1 − n^{h}_{j}) , 
  (1.19) 

where
B_{he} is the Auger coefficient for a scattering event in which a 2D electron
interacts with a 0D hole and is captured into a discrete QD state, whereas the hole is scattered to
the WL.
C_{ee} is the Auger coefficient related to the interaction of two electrons
from the WL and subsequent capture of one of the electrons into the QD, whereas the other is
scattered into a 2D energy state of higher energy. The Auger coefficients are labelled according to
the convention that the first index denotes the captured carrier and the second index
denotes the carrier scattered to a 2D state. The Auger coefficients
C_{ee},
C_{eh},
B_{eh},
C_{he},
and
C_{hh}
refer to cylindrical QDs [
15] and depend on the transition energy.
In our model, they are calculated for every QD in the (inhomogeneously broadened) ensemble.