Quantum Well States
Overview: The first part in this section introduces a general description of 2D WL states using a
finite confinement quantum well approach and applying the effective mass approximation.
An implicit equation for determining the chemical potential of 2D carriers in quasiequilibrium is
presented in section
1.1. In section
1.2, we present equations
used to model the dynamics of 2D carrier carriers in a quantum well.
Contents
Description of Quantum Well States
In a quantum well, charge carriers are confined in one spatial direction.
The confinement can be achieved by enclosing a thin sheet of semiconductor
material within a different semiconductor material with a larger bandgap
e.g. a strained sheet of In
_{0.2}Ga
_{0.8}As enclosed in a GaAs matrix.
Assumptions: Due to strain the
energy degeneracy at the Γpoint of the heavy hole and light hole
conduction band is lifted [
4].
To model scattering involving holes, we assume that the main
contribution is due to scattering of heavy holes and we neglect contributions from
the light hole and splitoff conduction bands.
We also assume that the 2D carriers in the
WL are in quasiequilibrium and can be described by specifying the chemical potential of
the conduction and valence band, respectively.
The energy values of the quantum well subbands are calculated using the
effective mass approximation. Hereby, we follow the approach of Loehr [
4].
The total wavefunction is assumed to be of the form:

Ψ_{n,kt}(r) = 
1
√Ω

e^{ikt ·R} u_{c}(r)f_{n}(y), 
  (1.1) 
where
k_{t} is the transverse crystal momentum vector referred to the
confinement direction
y,
R is a lattice vector, the function
u_{c}(
r) has the periodicity of the lattice and Ω is a
normalisation constant. In order to determine
f_{n}(
y) , the part of
the wavefunction characterising the confined 2D states with subband index n,
the following equation has to be solved:

 ⎡ ⎣

− 
ħ^{2}
2


d
dy


1
m^{∗}(y)


d
dy

+V_{0}(y)  ⎤ ⎦

f_{n}(y) = E_{n} f_{n}(y), 
  (1.2) 
where
V_{0} , the confinement potential, is given by the band offset at the
semiconductor interfaces (see Fig. ),
m^{∗}(
y)
is the effective mass in the different semiconductor regions and
E_{n} is
the subband energy. The energy dispersion relation for electrons in the
conduction band is parabolic in first order in
k^{2}_{t} and reads [
4]:

E^{e}_{n}(k_{t}^{e}) = E^{e}_{n} + 
ħ^{2}  k_{t}^{e} ^{2}
2 m^{∗,e}_{n}

where 
1
m^{∗,e}_{n}

=  ⌠ ⌡

∞
−∞

f_{e,n}(y)^{2} 
1
m^{∗,e}(y)

dy. 
  (1.3) 
A similar dispersion equation holds for heavy holes in the valence band:

E^{h}_{n}(k_{t}^{h}) = E^{h}_{n} − 
ħ^{2}  k_{t}^{h} ^{2}
2 m^{∗,h}_{n}

where 
1
m^{∗,h}_{n}

=  ⌠ ⌡

∞
−∞

f_{h,n}(y)^{2} 
1
m^{∗,h}(y)

dy. 
  (1.4) 
Taking into consideration that the confinement potential
V_{0}(
y) in (
1.2)
depends on the band offsets, the applied bias voltage,
and the doping concentration of the
WL and the surrounding semiconductor layers, the 2D wavefunctions
f_{n}(
y)
can be calculated numerically. More details on this subject are found
here.
1.1 Chemical Potential of Doped 2D Semiconductor Structures
In the following, we assume that electrons and holes
in the 2D layer surrounding the QDs are in quasiequilibrium
and follow the FermiDirac distributions for electrons and holes, respectively [
3]:
 f_{e}(E,μ_{e}) = 
1
exp( 
E − μ_{e}
k_{B} T

) + 1 

and f_{h}(E,μ_{h}) = 
1
exp( 
μ_{h} − E
k_{B} T

) + 1 


 '  (1.5) 

where μ
_{e} / μ
_{h} is the chemical potential of electrons/holes,
k_{B} is the Boltzmann constant and T is the temperature. The approximation can be used
since carriercarrier and carrierphonon scattering will lead to the relaxation of
any nonequilibrium distribution to a quasiFermi distribution on
a femtosecond timescale [
1].
Using this assumption the charge density for electrons
n^{e}_{2D}
and holes
n^{h}_{2D} can be expressed as [
4]:
 n^{e}_{2D} = 
ν_{e} ∑
n=1


k_{B} T m^{∗}_{n,e}
πħ^{2}

ln  ⎡ ⎣

1 + exp  ⎛ ⎝

μ_{e} − E_{n,e}
k_{B} T
 ⎞ ⎠
 ⎤ ⎦


  (1.6) 

 n^{h}_{2D} = 
ν_{h} ∑
n=1


k_{B} T m^{∗}_{n,h}
πħ^{2}

ln  ⎡ ⎣

1 + exp  ⎛ ⎝

E_{n,h} − μ_{h}
k_{B} T
 ⎞ ⎠
 ⎤ ⎦


  (1.7) 

where the sum runs over the total number of 2D electron and hole subbands, respectively.
The effective
mass for each subband is calculated using (
1.4).
In the intrinsic case, the charge density of electrons equals the charge density of holes.
If the semiconductor structure is doped, the condition of charge neutrality can be formulated as:
n_{2D}^{h} +
N_{D} =
n_{2D}^{e} +
N_{A}, where
N_{D} and
N_{A} are the concentration of
ionized donors and acceptors, respectively.
Inserting (
1.6) and (
1.7) into the charge neutrality equation
and substituting the chemical potential
of the valence band μ
_{h} using: μ
_{e} − μ
_{h} =
eV_{bias}, we obtain:


ν_{h} ∑
n=1

 ⎧ ⎨
⎩

k_{B} T m^{∗}_{n,h}
πħ^{2}

+ 
E_{n,h} + eV_{bias} − μ_{e}
2 k_{B} T

+ln  ⎡ ⎣

2 cosh  ⎛ ⎝

E_{n,h} + eV_{bias} − μ_{e}
2k_{B} T
 ⎞ ⎠
 ⎤ ⎦

+ N_{D}  ⎫ ⎬
⎭

= 
 
 
ν_{e} ∑
n = 1

 ⎧ ⎨
⎩

k_{B} T m^{∗}_{n,e}
πħ^{2}

+ 
μ_{e} − E_{n,e}
2k_{B} T

ln  ⎡ ⎣

2 cosh  ⎛ ⎝

μ_{e} − E_{n,e}
2k_{B} T
 ⎞ ⎠
 ⎤ ⎦
 ⎫ ⎬
⎭

+ N_{A}, 
  (1.8) 

where we have used the relation:
ln[ 1 + exp(
x)] =
x/2 + ln[ 2cosh(
x/2)] in order to increase the numerical accuracy.
The chemical potential of the conduction band μ
_{e} can be determined
numerically using (
1.8). The numerical procedure is detailed
here.
Finally, we can determine the charge density of the conduction and valence band, respectively
using (
1.6) and (
1.7).
1.2 2D Carrier Density Dynamics
In section
1.1, we have outlined
how to determine the quasiequilibrium
charge density of electrons and holes for a
given 2D semiconductor structure subject to a bias voltage.
If we apply a forward bias voltage, carriers are injected into the
device and depleted by electronhole recombination processes or
carrier scattering with the surrounding semiconductor structure
(in our case the QDs and the 3D bulk semiconductor medium).
The figure on the right shows the bandstructure of a QD inclosed by a QWell
that in turn is surrounded by bulk semiconductor material. The applied forward
voltage (indicated by a slanting of the band profile) leads to an accummulation
of electrons in the conduction band (CB) and holes in the valence band (VB) of the
3D bulk medium, respectively.
From here carriers cascade via a series of relaxation
processes (involving scattering with the lattice and scattering with other carriers)
towards lower lying energy states. This is indicated by white arrows.
The yellow arrow represents spontaneous electronhole recombination with emission
of a photon (wiggly red curve).
Depending on the 2D carrier density in the WL and the population of the QDs
with carriers QD⇔WL scattering
may lead to carrier capture into the QDs or ejection of
carriers from the QDs. At a sufficiently high 2D charge carrier density (of the order
of 10
^{11} cm
^{−2}) the inscattering of charge
carriers dominates.
A quasiequilibrium steady state is reached when carrier
injection and carrier loss balance.
Taking into account scattering processes between QD and WL described
so far (see page: Bloch Equations sections
2.2 and
2.3), the dynamics
of the WL carrier density is modelled by the following set of equations:
[
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
 ⎫ ⎬
⎭

, 
  (1.9) 
 
∂
∂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
 ⎫ ⎬
⎭

, 
  (1.10) 

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 (
1.9) and (
1.10) describe scattering between
carriers confined to QDs and 2D charge carriers.
We describe the injection current density 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 (
1.9) and (
1.10) as:

J = e ( 

μ

e

n^{e}_{2D} + 

μ

h

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

, 
  (1.11) 
where μ
_{e} = μ
_{e}(
n^{e}_{2D}) and μ
_{h} = μ
_{h}(
n^{h}_{2D}) is the chemical potential of 2D electrons in the
conduction band and 2D holes in the valence band,
respectively.
Adjusting the bias voltage (as a model parameter) allows us to set the 2D carrier density (at steady state)
and implicitly the occupation probability
of the QD states with charge carriers.
Bibliography
 [1]

W. Chow, S. W. Koch, and M. Sargent.
SemiconductorLaser Physics.
SpringerVerlag, 1994.
 [2]

Edeltraud Gehrig and Ortwin Hess.
Mesoscopic spatiotemporal theory for quantumdot lasers.
Physical Review A (Atomic, Molecular, and Optical Physics),
65(3):033804, 2002.
 [3]

Richard Liboff.
Introductory Quantum Mechanics.
AddisonWesley, Upper Lake River, New Jersey, fourth edition, 2003.
 [4]

J. P. Loehr.
Physics of Strained Quantum Well Lasers.
Kluwer Academic Publishers, 1998.
 [5]

Y. Suematsu and A.R. Adams.
Handbook of Semiconductor Lasers and Photonic Integrated
Circuits.
Chapman and Hall,London, 1994.
File translated from T_{E}X by
T_{T}Hgold,version 4.00.