Quantum Dot Semiconductor Amplifier: Theoretical Model
Overview: In the following, we introduce a theoretical model of an
optoelectronic device with an active area containing semiconductor quantum dots
(QDs). Depending on the geometry and the reflectivity of the facets, the
device can work as an optical amplifier or a laser. If the bias voltage is reversed
the same model can be used to describe an optical absorber.
Contents
Theoretical Model
In this section, we briefly introduce the equations required to simulate
optoelectronic devices containing QDs as optically active material.
The equations presented here are used to model:
 the material polarisation (1.1),
 the carrier occupation probability of the QD states (1.2) and (1.3),
 the rates of QD carrier scattering (1.4)  (1.7),
 the dynamics of the 2D carrier density in the semiconductor layer adjacent to the
QDs (1.8) and (1.9),
 the 2D injection density (1.10),
 the counterpropagating optical fields in the laser cavity (1.11).
Equations (
1.1), (
1.2), (
1.3), (
1.8), (
1.9), and (
1.11) form a system
of coupled differential equations that can be integrated using the evenodd
integration scheme introduced
here.
The layout of a simulator based on the theoretical model is presented in
section 2.
1.1 Material Polarisation and QD Level Occupation Probability
The following three equations describe the dynamics of
the microscopic polarisation (
1.1) and the probability that the bound QD levels are
populated by electrons (
1.2) and holes (
1.3), respectively.
These equations are derived and discussed in more detail
here.


∂ ∂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^{±} },

  (1.1) 


∂ ∂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

, 
  (1.2) 


∂ ∂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

, 
  (1.3) 

1.2 CarrierCarrier and CarrierPhonon Scattering
The term describing intradot carrierphonon scattering in (
1.2) and
(
1.3) is given by [
2]:


∂ ∂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↓} ) },

  (1.4) 

where
c labels electrons or holes and
k is the QD level index.
More details about intradot carrier scattering can be
found
here.
Equations (
1.2) and (
1.3) also include a term related to
carrierphonon scattering with carrier exchange between the 2D semiconductor
quantum well adjacent to the QDs and bound QD states. This process is discussed
here
and can be modelled using:
[
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}  ⎤ ⎦


  (1.5) 

The terms describing QD capture rates of electrons and holes due to Auger scattering (carriercarrier scattering)
in (
1.2) and (
1.3) are discussed
here and modelled using the following equations:
 
∂ ∂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}) ,

  (1.6) 

 
∂ ∂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}) , 
  (1.7) 

1.3 2D Carrier Dynamics
The dynamics of the 2D carriers in the quantum well adjacent to the QDs is modelled using a
diffusion equation: [
2]:


∂
∂t

n^{e}_{2D} = J − D_{a}  ⎡ ⎣

∂^{2}
∂x^{2}

+ 
∂^{2}
∂z^{2}
 ⎤ ⎦

n^{e}_{2D} − Γ_{loss} − n_{QD} 
∑
i

 ⎧ ⎨
⎩

∂
∂t

n^{e}_{i}  ⎢ ⎢

e−ph
WL

+ 
∂
∂t

n^{e}_{i}  ⎢ ⎢

c−c
WL
 ⎫ ⎬
⎭

, 
  (1.8) 
 
∂
∂t

n^{h}_{2D} = J − D_{a}  ⎡ ⎣

∂^{2}
∂x^{2}

+ 
∂^{2}
∂z^{2}
 ⎤ ⎦

n^{h}_{2D} − Γ_{loss} − n_{QD} 
∑
j

 ⎧ ⎨
⎩

∂
∂t

n^{h}_{j}  ⎢ ⎢

h−ph
WL

+ 
∂
∂t

n^{h}_{j}  ⎢ ⎢

c−c
WL
 ⎫ ⎬
⎭

, 
  (1.9) 

where the first term in both equations describes the carrier injection with current
density
J.
D_{a} is the ambipolar diffusion coefficient [
5] and
n_{QD} is the QD
sheet density. The 2D carrier loss rate Γ
_{loss} includes contributions due to
nonradiative, spontaneous and Augerrecombination, respectively, and is given by:
Γ
_{loss} = γ
_{nr} n^{e}_{2D} + γ
_{sp} n^{e}_{2D} n^{h}_{2D} + γ
_{aug} n^{e}_{2D} n^{e}_{2D} n^{h}_{2D} .
The last two terms in (
1.8) and (
1.9) describe scattering between
carriers confined to QDs and 2D charge carriers.
1.4 Injection Current Density
We describe the injection current density
J
in
in terms of the applied bias voltage
V_{bias}.
Assuming an Ohmic regime, we have:
J = σ
E_{⊥} = σ
V_{bias}/
d_{⊥},
where σ is the conductivity and
d_{⊥} is the device dimension perpendicular to the
injection stripe. The conductivity is given by:
σ =
e (μ
^{e}n^{e}_{3D}
+ μ
^{h}n^{h}_{3D}),
where
e is the electron charge, μ
^{e} is the
electron mobility, μ
^{h} is the
hole mobility,
n^{e}_{3D} is the 3D electron charge density,
and
n^{h}_{3D} is the 3D hole charge density.
Additionally, we approximate the 3D charge density by:
n_{3D} =
n_{2D}/
d_{⊥}.
Since we include radiative and nonradiative electronhole
recombination explicitly, we write the term
describing carrier
injection in (
1.8) and (
1.9) as:

J = e ( 
μ

e

n^{e}_{2D} + 
μ

h

n^{h}_{2D}) 
V_{bias} − (C^{e}_{pot} − C^{h}_{pot})
d^{2}_{⊥}

, 
  (1.10) 
where C
^{e}_{pot} = C
^{e}_{pot}(
n^{e}_{2D})
and C
^{h}_{pot} = C
^{h}_{pot}(
n^{h}_{2D})
is the chemical potential of 2D electrons in the
conduction band and 2D holes in the valence band,
respectively. Note: The chemical potential is usually denoted by the greek letter μ. Here we chose C
_{pot} to avoid confusion
with the electron and hole
mobility.
1.5 Propagation of the Optical Fields Inside the Laser Cavity
The optical fields propagating inside the laser cavity are modelled
using (
1.34) an equation derived in the section called 'Waveguiding and Optical Fields'.
Spontaneous emission has been included in (
1.2)
and (
1.3) as a loss channel. However, spontaneous emission also couples to the
laser radiation field, inducing amplitude and phase fluctuations (noise).
In order to model the impact of spontaneous emission noise on the optical fields, we add
Langevin noise sources
F(
t) to the 2D waveequation (
1.34):

 ⎡ ⎣

∂ ∂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}. 
 (1.11) 
We have assumed that the transverse
electric field is polarised in xdirection. As a result the noise term
F(
t) is
multiplied with the unity vector in xdirection
e_{x}.
The Langevin noise term is given by
[
3]
[
4]
[
5]:

F(t) = 
1 2

 ⎡ ⎣

1 dt


∑
i,j

Γ^{sp}_{j,i} n^{h}_{j}(t)
n^{e}_{i}(t)  ⎤ ⎦

[1/2]

 ⎡ ⎣

ħ ω_{0} ϵ_{0} ϵ_{p}
V
 ⎤ ⎦

[1/2]

exp(i2πϕ), 
 (1.12) 
where
V is the volume of the optical resonator, and ϕ is a
random Gaussian variable with zero mean and unity variance.
The quantity √[(ħ ω
_{0})/(ϵ
_{0} ϵ
_{p} V)]
represents the 'electric field per photon' [
2].
To calculate the coefficient of spontaneous emission Γ
^{sp}_{j,i}, we use
the approximation [
6]:

Γ^{sp}_{j,i} = 
ω_{j,i}^{3} n_{eff} 3πϵ_{0}ħ c^{3}


3 ∑
k=1;l=1

e_{k} [ ∆_{j,i} ⊗∆_{j,i}^{∗} ] e_{l}, 
 (1.13) 
where we have multiplied the dipole matrix tensor ∆
_{j,i} ⊗∆
_{j,i}^{∗}
from the left and right with the standard unity vectors
and summed over all spatial coordinates.
The quantity ω
_{j,i} in
1.13 is the optical frequency
corresponding to the QD transition form level
i to level
j, while
n_{eff} is the effective refractive index defined as:
n_{eff} = √ε
_{p} and ε
_{p} is the
effective dielectric constant
.
Simulator Layout
In the following, we present the layout of a device simulator based on
the theoretical model introduced in the section above.
 Open output streams/files.
 Initialise physical parameters:
 length, width, height, and shape of the optical cavity,
 reflectivity of front and rear facet, wavelength, reflective index of active area and cladding layer
 QD energy levels, QD dipole elements, coefficients related to inhomogeneous and homogeneous broadening,
 material and geometrical parameters describing the semiconductor heterostructure (matrix,QWell,QD),
 temperature,
 calculate QWell wavefunctions,
and subband energy levels,
 calculate the
effective mass of carriers in the (doped) QWell structure,
 set up numerical grid,
 initialise inhomogeneously broadened QD ensemble,
 calculate QDQWell carrier scattering rates,
 calculate
(relative) effective dielectric constant and
optical confinement factor,
 initialise differential operators.
 Set initial boundary conditions.
 Start numerical integration loop and repeat until final timepoint is reached:
 Close output streams/files.
Bibliography
[1]
Y. Suematsu and A.R. Adams.
Handbook of Semiconductor Lasers
and Photonic Integrated
Circuits.
Chapman and Hall,London, 1994.
[2]
E. Gehrig and O. Hess. Mesoscopic spatiotemporal theory for quantum dot lasers.
Physical Review A (Atomic, Molecular, and Optical Physics),65(3):033804, 2002
[3]
W. Chow, S. W. Koch, and M. Sargent.
SemiconductorLaser Physics.
SpringerVerlag, 1994.
[4]
H. Haug and S. W. Koch.
Quantum Theory Of The Optical And Electronic Properties Of
Semiconductors.
Singapore, World Scientific, 1998.
[5]
Ahmed, M.; Yamada, M.; Saito, M.; , Numerical modeling of intensity and phase noise in semiconductor
lasers,
Quantum Electronics, IEEE Journal of , vol.37, no.12, pp.16001610, Dec 2001
[6]
J. P. Loehr.
Physics of Strained Quantum Well Lasers.
Kluwer Academic Publishers, 1998.