Optimization of quantum cascade laser operation by geometric design of cascade active band in open and closed models
Using the effective mass and rectangular potential approximations, the theory of electron dynamic conductivity is developed for the plane multilayer resonance tunnel structure placed into a constant electric field within the model of open nanosystem, and oscillator forces of quantum transitions with...
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Інститут фізики конденсованих систем НАН України
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| Цитувати: | Optimization of quantum cascade laser operation by geometric design of cascade active band in open and closed models / M.V. Tkach, Ju.O. Seti, I.V. Boyko, O.M. Voitsekhivska // Condensed Matter Physics. — 2013. — Т. 16, № 3. — С. 33701:1-10. — Бібліогр.: 12 назв. — англ. |
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nasplib_isofts_kiev_ua-123456789-1208362025-02-09T10:03:45Z Optimization of quantum cascade laser operation by geometric design of cascade active band in open and closed models Оптимiзацiя роботи квантового каскадного лазера геометричним дизайном каскаду у вiдкритих i закритих моделях Tkach, M.V. Seti, Ju.O. Boyko, I.V. Voitsekhivska, O.M. Using the effective mass and rectangular potential approximations, the theory of electron dynamic conductivity is developed for the plane multilayer resonance tunnel structure placed into a constant electric field within the model of open nanosystem, and oscillator forces of quantum transitions within the model of closed nanosystem. For the experimentally produced quantum cascade laser with four-barrier active band of separate cascade, it is proven that just the theory of dynamic conductivity in the model of open cascade most adequately describes the radiation of high frequency electromagnetic field while the electrons transport through the resonance tunnel structure driven by a constant electric field. У наближеннi ефективних мас i прямокутних потенцiалiв розвинута теорiя електронної динамiчної провiдностi плоскої багатошарової резонансно-тунельної структури у постiйному електричному полi в моделi вiдкритої наносистеми та сил осциляторiв квантових переходiв у моделi закритої системи. На прикладi експериментально реалiзованого квантового каскадного лазера з чотирибар’єрною активною зоною окремого каскаду показано, що саме теорiя динамiчної провiдностi у моделi вiдкритого каскаду найбiльш адекватно описує процес випромiнювання високочастотного електромагнiтного поля при проходженнi електронiв крiзь резонансно-тунельну структуру у постiйному електричному полi. 2013 Article Optimization of quantum cascade laser operation by geometric design of cascade active band in open and closed models / M.V. Tkach, Ju.O. Seti, I.V. Boyko, O.M. Voitsekhivska // Condensed Matter Physics. — 2013. — Т. 16, № 3. — С. 33701:1-10. — Бібліогр.: 12 назв. — англ. 1607-324X PACS: 73.40.Gk, 85.30.Mn, 81.07.St DOI:10.5488/CMP.16.33701 arXiv:1310.1231 https://nasplib.isofts.kiev.ua/handle/123456789/120836 en Condensed Matter Physics application/pdf Інститут фізики конденсованих систем НАН України |
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English |
| description |
Using the effective mass and rectangular potential approximations, the theory of electron dynamic conductivity is developed for the plane multilayer resonance tunnel structure placed into a constant electric field within the model of open nanosystem, and oscillator forces of quantum transitions within the model of closed nanosystem. For the experimentally produced quantum cascade laser with four-barrier active band of separate cascade, it is proven that just the theory of dynamic conductivity in the model of open cascade most adequately describes the radiation of high frequency electromagnetic field while the electrons transport through the resonance tunnel structure driven by a constant electric field. |
| format |
Article |
| author |
Tkach, M.V. Seti, Ju.O. Boyko, I.V. Voitsekhivska, O.M. |
| spellingShingle |
Tkach, M.V. Seti, Ju.O. Boyko, I.V. Voitsekhivska, O.M. Optimization of quantum cascade laser operation by geometric design of cascade active band in open and closed models Condensed Matter Physics |
| author_facet |
Tkach, M.V. Seti, Ju.O. Boyko, I.V. Voitsekhivska, O.M. |
| author_sort |
Tkach, M.V. |
| title |
Optimization of quantum cascade laser operation by geometric design of cascade active band in open and closed models |
| title_short |
Optimization of quantum cascade laser operation by geometric design of cascade active band in open and closed models |
| title_full |
Optimization of quantum cascade laser operation by geometric design of cascade active band in open and closed models |
| title_fullStr |
Optimization of quantum cascade laser operation by geometric design of cascade active band in open and closed models |
| title_full_unstemmed |
Optimization of quantum cascade laser operation by geometric design of cascade active band in open and closed models |
| title_sort |
optimization of quantum cascade laser operation by geometric design of cascade active band in open and closed models |
| publisher |
Інститут фізики конденсованих систем НАН України |
| publishDate |
2013 |
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https://nasplib.isofts.kiev.ua/handle/123456789/120836 |
| citation_txt |
Optimization of quantum cascade laser operation by geometric design of cascade active band in open and closed models / M.V. Tkach, Ju.O. Seti, I.V. Boyko, O.M. Voitsekhivska // Condensed Matter Physics. — 2013. — Т. 16, № 3. — С. 33701:1-10. — Бібліогр.: 12 назв. — англ. |
| series |
Condensed Matter Physics |
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| first_indexed |
2025-11-25T15:46:20Z |
| last_indexed |
2025-11-25T15:46:20Z |
| _version_ |
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| fulltext |
Condensed Matter Physics, 2013, Vol. 16, No 3, 33701: 1–10
DOI: 10.5488/CMP.16.33701
http://www.icmp.lviv.ua/journal
Optimization of quantum cascade laser operation by
geometric design of cascade active band in open and
closed models
M.V. Tkach∗, Ju.O. Seti, I.V. Boyko, O.M. Voitsekhivska
Chernivtsi National University, 2 Kotsyubinsky St., 58012 Chernivtsi, Ukraine
Received November 20, 2012
Using the effective mass and rectangular potential approximations, the theory of electron dynamic conductivity
is developed for the plane multilayer resonance tunnel structure placed into a constant electric field within the
model of open nanosystem, and oscillator forces of quantum transitions within the model of closed nanosystem.
For the experimentally produced quantum cascade laser with four-barrier active band of separate cascade, it is
proven that just the theory of dynamic conductivity in the model of open cascade most adequately describes the
radiation of high frequency electromagnetic field while the electrons transport through the resonance tunnel
structure driven by a constant electric field.
Key words: resonance tunnel nanostructure, conductivity, quantum cascade laser
PACS: 73.40.Gk, 85.30.Mn, 81.07.St
1. Introduction
It is well known that the operation of quantum cascade laser (QCL) [1–4], quantum cascade detector
(QCD) [5–7] and other appropriately operating nanodevices is based on the transport properties of open
multilayer nanostructures. In spite of the long period of investigations into electron transport through
the resonance tunnel structures (RTS), taking into account the interaction of electronic current with con-
stant electric and high frequency electromagnetic fields, the current theory is still far from being well
correlated with experimental data.
The main problems in elaborating a consistent theory of electron transport through the RTS are the
mathematical difficulties arising when solving the non-stationary Schrodinger equation with Hamiltoni-
ans for even comparatively simple models with open boundaries which allow the infinite movement of
quasi-particles. In order to avoid these difficulties, the evaluations obtained for the closed analogues of
open RTS with rectangular potential wells and barriers were used in early papers [2–4] for the theory of
electron transport through the QCL active bands. The closed models did not make it possible to study the
currents due to the stationary electron states but they satisfactorily described the electron spectrum and,
thus, the energies of electromagnetic radiation and wave functions which were used for the calculation
of dipole moments of quantum transitions.
Active bands of QCL, such as open two- and three-barrier RTS, were theoretically studied in [8–10].
In these papers, the non-stationary one-dimensional Schrodinger equation describing the electron trans-
port through the RTS with δ-like potential barriers was solved taking into account the interaction with
constant electric and high frequency electromagnetic fields. The simplified model of a constant effective
mass of an electron along the whole nanosystem and δ-barrier approximation of the potential made it
possible to calculate and investigate the electronic currents and, consequently, to calculate the dynamic
conductivity in ballistic regime when the biggest lifetimes in operating quasi-stationary states weremuch
∗E-mail: ktf@chnu.edu.ua
©M.V. Tkach, Ju.O. Seti, I.V. Boyko, O.M. Voitsekhivska, 2013 33701-1
http://dx.doi.org/10.5488/CMP.16.33701
http://www.icmp.lviv.ua/journal
M.V. Tkach et al.
smaller than the relaxation times for the electron energy due to the dissipative processes (phonons, im-
purities and so on).
The δ-barrier model for an open RTS [11, 12] essentially overestimates the resonance widths of oper-
ating quasi-stationary states compared to themore adequatemodel of rectangular potentials. It explained
some properties of electron transport but could not be used as a reliable base to be compared with the
experimental data and, thus, to optimize the geometric design of QCL active band.
According to the abovementioned and using the effective mass and rectangular potential approxima-
tions, in this paper we develop the theory of quasi-stationary spectrum and dynamic conductivity of the
electrons interacting with high frequency electromagnetic field within the model of open multilayer RTS
and stationary spectrum together with oscillator forces of quantum transitions within themodel of closed
RTS in a constant electric field. We use the obtained theoretical results in order to calculate the energy
of electromagnetic radiation for the experimentally produced QCL [3] with a four-barrier active band of
a separate cascade. The comparison of numerical and experimental data illustrates the capabilities of
different models in optimizing the active band geometric design.
2. Theory of dynamic conductivity of a resonance tunnel cascade with
four-barrier active band and oscillator forces of quantum transitions
in closed model
The separate cascade of QCL such as RTS, containing a four-barrier active band and injector consisting
of a certain number of plane nanolayers (wells and barriers) having fixed sizes, figure 1, is studied within
two models: open (o) and closed (c). The constant electric field with intensity F is applied perpendicularly
to the RTS planes. For the open model, we assume that the monoenergetic current of non-interacting
electrons having energy E and concentration n0 impinges at RTS from the left hand side, perpendicularly
to its planes. Under these conditions and taking into account the small difference between the lattice
constants of wells and barriers, the problem settles to the study of one-dimensional electron transport
using the models of effective mass and rectangular potentials.
Taking the coordinate system as it is shown in figure 1, the effective mass and potential energy of an
Figure 1. The energy scheme of separate cascadewith four-barrier active region and injector. The widths
of the barriers (∆p ): 5.0, 1.5, 2.2, 3.0, 2.3, 2.2, 2.0, 2.3, 2.8 and widths of the wells (bp ): 0.9, 4.7, 4.0, 2.3, 2.2,
2.0, 2.0, 1.9, 1.9 are presented from the left to the right in nm units.
33701-2
Optimization of quantum cascade laser operation
electron in open (o) or closed (c) RTS (without the field) is conveniently written as
m{o
c
}(z) =
{
m0
m1
}
[θ(−z)+θ(z −b)]+m0
NW
∑
p=1
[
θ(z − z2p−1)−θ(z − z2p )
]
+ m1
NB−1
∑
p=0
[
θ(z − z2p )−θ(z − z2p+1)
]
, (1)
U {o
c
}(z)=
{
0
U
}
[θ(−z)+θ(z −b)]+U
NB−1
∑
p=0
[
θ(z − z2p )−θ(z − z2p+1)
]
, (2)
where NW, NB are the numbers of wells and barriers in the RTS which correspond to the active band or
to the whole cascade, depending on the model.
In order to observe the electron transport through the RTS, the latter should be obligatory an open
one. Thus, a dynamic conductivity arises when quasi-stationary states are present in a nanosystem. De-
veloping the theory of RTS active conductivity within the open model, we study the properties of electron
stationary spectrum and oscillator forces of quantum transitions within the closed model in order to be
compared. The widths of outer barriers of active band (or cascade) in the closed model limit to the phys-
ical infinity and the constant electric field is applied inside the nanosystem only. The reason to study the
closed model is that the similar model is, evidently, the theoretical base of the choice of experimental geo-
metric design of QCL cascade with active band and injector [3].We are going to compare the experimental
data with our results obtained within the model of open RTS.
In order to calculate the dynamic conductivity within the open model and oscillator forces within the
closed one, we first solve the stationary Schrodinger equations
H{o
c
}(z)Ψ{o
c
}(z)= EΨ{o
c
}(z) (3)
with the Hamiltonian of an electron in RTS driven by a constant electric field
H{
o
c
} =−
ħ
2
2
∂
∂z
m−1
{
o
c
}(z)
∂
∂z
−eF
(
z[θ(z)−θ(z −b)]−
{
b
0
}
θ(z −b)
)
. (4)
The solutions of equations (3) are written as follows:
Ψ{o
c
}(z) = Ψ
(0)
{o
c
}(z)θ(−z)+
NW+NB
∑
p=1
Ψ
(p)
{o
c
}(z)
[
θ(z − zp−1)−θ(z − zp )
]
+ Ψ
(NW+NB+1)
{o
c
} (z)θ(z −b), (5)
where the wave functions
Ψ
(0)
{o
c
}(z) = A(0)
{o
c
}e
{
ik
χ
}
z
+B (0)
{o
c
}e
−
{
ik
χ
}
z
, (6)
Ψ
(p)
{o
c
}(z) = A
(p)
{o
c
}Ai(ξp (z))+B
(p)
{o
c
}Bi(ξp (z)), [p = 1÷ (NW +NB)], (7)
Ψ
(NW+NB+1)
{o
c
} (z) = A
(NW+NB+1)
{o
c
} e
{
iK
χ
}
z
+B
(NW+NB+1)
{o
c
} e
−
{
ik
χ
}
z
(8)
are the superpositions of the exact linearly independent solutions of equations (3) in the respective ranges
of z variable. Here, we introduced the notations
k =ħ
−1
√
2m0E , χ=ħ
−1
√
2m1(U −E ), K =ħ
−1
√
2m0(E +V ), V = eF b,
ξp (z)=
+
(
2m1 V b2
ħ2
)
1
3
(
U −E
V
−
z
b
)
, p = 1, 3, 5, . . . ,
−
(
2m0 V b2
ħ2
)
1
3
(
E
V
−
z
b
)
, p = 2, 4, 6, . . . .
(9)
33701-3
M.V. Tkach et al.
Ai(ξ), Bi(ξ) are the Airy functions.
The conditions of a wave function and its density of current continuity should be fulfilled at all
nanosystem interfaces in the both models
Ψ
(p)
{o
c
}(zp ) =Ψ
(p+1)
{o
c
} (zp ),
dΨ
(p)
{o
c
}(z)
m{o
c
}(z)dz
∣
∣
∣
∣
∣
∣
∣
z=zp−ε
=
dΨ
(p+1)
{o
c
} (z)
m{o
c
}(z)dz
∣
∣
∣
∣
∣
∣
∣
z=zp+ε
, (10)
p = 0÷ (NW +NB), ε→+0.
The wave functions tend to zero at z →±∞ in the closed model, since B (0)
(c)
= A
(NW+NB+1)
(c)
= 0. Thus, the
system of equations (10) brings us to the dispersion equation, consistently determining the energy spec-
trum (En) and all coefficients A
(p)
(c)
, B
(p)
(c)
through one of them. The latter is obtained from the normality
condition
∞
∫
−∞
Ψ
∗
(c)n(z)Ψ(c)n′(z)dz = δnn′ . (11)
Now, the electron wave functions Ψ(c)n(z) and energies (En) of all stationary states are defined in the
closed model. Using them, the oscillator forces of quantum transitions between the states n and n′ can be
calculated within the formula
fnn′ =
2(En −En′)m(c)
ħ2
∣
∣
∣
∣
∣
∣
∞
∫
−∞
Ψ
∗
(c)n(z) zΨ(c)n′(z)dz
∣
∣
∣
∣
∣
∣
2
. (12)
For the open model, there should be no backward wave from the right of a nanosystem, since
B
(NW+NB+1)
(o)
= 0. All coefficients A
(p)
(o)
, B
(p)
(o)
of the wave function Ψ(o)(z) are found from the condition (10)
through one of them, in its turn, defined by the incident density of the current impinging at RTS from the
left hand side. In this case, the electron spectrum is the quasi-stationary one with the resonance energies
(En) and resonance widths (Γn = ħτ−1
n ) where τn is the lifetime in the n-th quasi-stationary state. The
resonance energies are fixed by the maxima of a probability distribution function of an electron inside
RTS (in energy scale E )
W (E )=
1
b
b
∫
0
∣
∣Ψ(o)(E , z)
∣
∣
2
dz. (13)
The resonance widths (Γn) are fixed by the widths of this function at the halves of its maxima placed at
the respective resonance energies En .
The quantum transitions between the quasi-stationary states occur when the electrons transport
through the open RTS placed into the electric field. Consequently, the electromagnetic field with the re-
spective frequency arises. Its intensity is proportional to the magnitude of the dynamic conductivity. In
the quantum transitions accompanied by the absorption of electromagnetic energy, the positive dynamic
conductivity is formed and, during the radiation of electromagnetic energy, the negative dynamic con-
ductivity of RTS is formed.
In order to calculate the negative conductivity of open RTS operating in a laser regime, one has to
obtain the wave functions of electrons interacting with the electromagnetic field. It is found from the
time-dependent Schrödinger equation:
iħ
∂Ψ(z, t)
∂t
=
[
H(o)(z)+H(z, t)
]
Ψ(z, t), (14)
where H(o)(z) is the Hamiltonian (4) for the electrons in RTS without the electromagnetic field and
H(z, t) =−eE {z [θ (z)−θ (z −b)]+bθ (z −b)}
(
eiωt
+e−iωt
)
(15)
is the Hamiltonian of electrons interacting with time-dependent electromagnetic field characterized by
the frequency ω and its electric field intensity E .
33701-4
Optimization of quantum cascade laser operation
Assuming the amplitude of a high frequency electromagnetic field to be small, we find the solution of
equation (14) in a one-mode approximation using the perturbation theory
Ψ (z, t) =
+1
∑
s=−1
Ψs (z) e−i(ω0+sω)t , (ω0 = E/ħ), (16)
where Ψs=0(z)≡Ψ(o)(z).
Preserving the first order magnitudes in equation (14), we obtain inhomogeneous equations for the
corrections Ψ±1(z) to the wave functions
[
H(o) (z)−ħ (ω0 ±ω)
]
Ψ±1 (z)−eE {z [θ (z)−θ (z −b)]+bθ (z −b)}Ψ0 (z) = 0. (17)
Their solutions are the superpositions of functions
Ψ±1 (z) =Ψ± (z)+Φ± (z) . (18)
The functions Ψ± (z), being the solutions of homogeneous equations, are written as
Ψ±(z) = Ψ
(0)
±
(z)θ(−z)+
NW+NB
∑
p=1
Ψ
(p)
±
(z)
[
θ(z − zp−1)−θ(z − zp )
]
+Ψ
(NW+NB+1)
±
(z)θ(z −b)
= B (0)
±
e−ik±zθ(−z)+ A
(NW+NB+1)
±
eiK±zθ(z −b)
+
NW+NB
∑
p=1
[
A
(p)
±
Ai(ξ
(p)
±
)+B
(p)
±
Bi(ξ
(p)
±
)
]
[
θ(z − zp−1)−θ(z − zp )
]
, (19)
where
k± =ħ
−1
√
2m0(E ±Ω), K± =ħ
−1
√
2m0[(E ±Ω)+V ], Ω=ħω,
ξ
(p)
±
(z)=
+
(
2m1 V b2
ħ2
)
1
3
[
U − (E ±Ω)
V
−
z
b
]
, p = 1, 3, 5, . . . ,
−
(
2m0 V b2
ħ2
)
1
3
(
E ±Ω
V
−
z
b
)
, p = 2, 4, 6, . . . .
(20)
The partial solutions of inhomogeneous equations (17) have the exact analytical form
Φ±(z) = π
E
F
NW+NB
∑
p=1
Bi(ξ
(p)
±
)
ξ(p)
∫
1
[
η−κ
2
3 (z)
U (z)−E
V
]
Ai
(
η∓κ
2
3 (z)
Ω
V
)
Ψ
(p)
(o)
(
η
)
dη
− Ai(ξ
(p)
±
)
ξ(p)
∫
1
[
η−κ
2
3 (z)
U (z)−E
V
]
Bi
(
η∓κ
2
3 (z)
Ω
V
)
Ψ
(p)
(o)
(
η
)
dη
(21)
×
[
θ(z − zp−1)−θ(z − zp )
]
∓
eE b
Ω
Ψ
(NW+NB+1)
(o) (b)θ(z −b), (22)
where
κ(z)=ħ
−1
√
2m(o)(z)b2V . (23)
The conditions of the wave function Ψ(z, t) and its density of current continuity at all RTS interfaces
bring us to the fitting conditions similar to the equations (10) for the functions Ψ±1 (z). Also, these equa-
tions define the unknown coefficients B
(p)
±
, A
(p)
±
[p = 0÷ (NW +NB +1)], and, consequently, the complete
wave function Ψ(z, t).
Further, considering the energy of electron-electromagnetic field interaction to be the sum of energies
of electron waves, coming out of the both sides of RTS, we calculate, in quasi-classic approximation, the
33701-5
M.V. Tkach et al.
real part of dynamic conductivity through the densities of currents of electron waves coming out of the
both sides of a nanosystem
σ(Ω,E ) =
Ω
2beE 2
{
[
j (E +Ω, z = b)− j (E −Ω, z = b)
]
−
[
j (E +Ω, z = 0)− j (E −Ω, z = 0)
]
}
. (24)
According to the quantum mechanics, the densities of currents are determined by the wave function
j (E , z)=
ieħn0
2m(o)(z)
[
Ψ(E , z)
∂
∂z
Ψ
∗(E , z)−Ψ
∗(E , z)
∂
∂z
Ψ(E , z)
]
. (25)
The real part of dynamic conductivity can be expressed as a sum of two terms
σ−(Ω,E ) =
ħΩn0
2b m0E
2
(
k+
∣
∣
∣B
(0)
+
∣
∣
∣
2
−k−
∣
∣B (0)
−
∣
∣
2
)
,
σ+(Ω,E ) =
ħΩn0
2b m0E
2
(
K+
∣
∣
∣A
(NW+NB+1)
+
∣
∣
∣
2
−K−
∣
∣A(NW+NB+1)
−
∣
∣
2
)
. (26)
The physical sense of these partial terms [σ±(Ω,E )] is evident. They are caused by the electronic currents
interacting with high frequency electromagnetic field in RTS and flowing out of it in forward (σ+) and
backward (σ−) direction with respect to the incident one.
3. Discussion of the results
Using the developed theory we display such amodel of plane nano-RTS, which describes the quantum
transitions and transport properties of electrons best of all. It makes possible to optimize the operation of
QCL by the geometric design of separate cascade active band. The numeric calculations were performed
for four nanosystems: two closed models(four-barrier active band and complete cascade)and two open
models(four-barrier active band and complete cascade).
In order to compare with the experimental data [3] we used the following physical parameters: U =
516 meV, F = 68 kV/cm, n0 = 2 ·1017 cm−3 and geometrical ones, shown in figure 1. We should note that
as far as almost equal sizes of all layers in the experimentally investigated cascade in the cited paper
contain a small number of unitary cells (2–4) of its composition elements, the approximation of effective
masses in different layers of a nanosystemwould be a rough one. At the same time, thewhole active band,
the whole injector or the whole cascade contains dozens of unitary cells in composition elements. Thus,
one can expect that the effective mass of an electron (m = 0.08me )averaged over all three composition
elements (GaAs, AlAs, InAs) is more adequate in the present physical situation.
In order to study the effect of a geometric design of a separate cascade on the operation of QCL we
calculated the energy spectrum (En) and oscillator forces of quantum transitions ( fnn′ ) within the closed
model, while resonance energies (En), lifetimes (τn), active conductivity (σnn′ ) and its partial terms
(σ±
nn′ ) — within the open model. The results are presented in figure 2, depending on the position (b1)
of inner two-barrier element between two outer barriers of an active band at the fixed sizes of all other
elements of a cascade, the same as in paper [3].
The calculations prove that the dependences of electron spectra on b1 in closed and open systems
differ not more than by 0.1%. Besides, from figures 2 (a), (e) it is clear that the first three resonance
energies, as functions of b1, calculated within the model of an open four-barrier active band [figure 2 (a)]
coincide with the respective resonance energies (E1, E2, E3) of those three states, calculated within the
model of an open complete cascade [figure 2 (e)] where the electron with maximal probability is located
in the space of an active band. In figure 2 (e) one can also see the resonance energies Ei1 ÷Ei4 (thin
curves) of quasi-stationary states, where the electron with a bigger probability is located in the injector
part of a cascade.
Using the closed model, we calculated the oscillator forces of quantum transitions ( f32 and f31) as
functions of b1. The results are presented for the four-barrier active band [figure 2 (d)] and for the com-
plete cascade [figure 2 (h)]. The condition of optimal QCL operation is fulfilled when the oscillator force
of quantum transition f32 approaches its maximal magnitude. This transition occurs between the states
33701-6
Optimization of quantum cascade laser operation
0 1 2 3 4
0
100
200
300
32= 245.98
E
n
[m
eV
]
b
1
[nm]b
1 exp
E1
E2
E3
(a)
0 1 2 3 4
0
100
200
300
Ei2
32= 246.24
E
n
[m
eV
]
b
1
[nm]b
1 exp
Ei1
Ei3
Ei4
E1
E2
E3
(e)
0 1 2 3 4
-4
-2
0
2
b
1 exp
ln
2
ln
3
ln
1
b
1
[nm]
ln
n
(b)
0 1 2 3 4
-2
0
2
4
6
b
1 exp
b
1
[nm]
ln
n
ln
3
ln
2
ln
1
(f)
0 1 2 3 4
-8
-6
-4
-2
0
2
4
6
8
b
1 exp
b
1
[nm]
lnln ln
ln lnln
32
ln
n
n
',
ln
± n
n
'
(c)
0 1 2 3 4
-8
-6
-4
-2
0
2
4
6
8
ln
n
n
',
ln
± n
n
'
b
1 exp
b
1
[nm]
(g)
lnln
ln
32
lnlnln
31
0 1 2 3 4
0.0
(d)
b
1 exp
b
1
[nm]
f31
f32
1.0
0.8
0.6
0.4
f n
n
'
0.2
0 1 2 3 4
f n
n
b
1
[nm]b
1 exp
f31f32
0.0
0.2
0.4
0.6
0.8
1.0
(h)
Figure 2. Electron energy spectrum (a, e), lifetimes (b, f), oscillator forces (d, h) and active conductivities
(c, g) as functions of an input well width (b1) of QCL active region in different models.
ensuring the needed energy of electromagnetic field radiation (in our case Ω32 = E3 −E2). Herein, the
oscillator force of quantum transition f31 should be an order smaller. Figure 2 (d) proves that for the
four-barrier active band model, one can see two regions (I and II) of b1 varying [toned at the figure 2 (d)]
where the abovementioned condition fulfills. Herein, the I region is better than the II region because
f I
32 > f I I
32 . It is clear that the experimental point b1exp nearly corresponds to the maximal magnitude of
f32 as a function of b1. In a closed model of a complete cascade [figure 2 (h)] there is one region of b1
varying (I) where f32 > f31 and one region (II) where f31 > f32. Herein, the experimental point b1exp also
nearly coincides with b1 corresponding to the maximal magnitude of f32. The width and location of an
optimal region for b1 in this model are almost the same as in the model of a four-barrier active band.
33701-7
M.V. Tkach et al.
Also, we should note that in the model of a complete cascade it seems that the QCL can operate due to the
quantum transition 3 → 1 in the II region where f31 > f32. However, as it is proven further on within the
more adequate open model, this is impossible.
The resonance energies (En , Ein ), lifetimes (τn), active conductivities (σ32,σ31) and their partial terms
(σ±
32, σ±
31) are calculated as functions of b1 for the two openmodels: active band [figures 2 (a), (b), (c)] and
complete cascade [figures 2 (e), (f), (g)]. Before analysing the figures we should note that in themodel of an
active band the dynamic conductivities and their terms were calculated through the averaging over those
energy regions where the levels of an injector band (Ei4 −Ei1) are uniformly located. The calculations of
these conductivities for the model of a separate complete cascade were performed taking into account
that the electrons leave the previous cascade with the energy E1, shifted at the magnitude E3−E1−eF b ,
with respect to the resonance energy E3 of the cascade studied.
Contrary to the closed model, the open one allows for a detailed and adequate analysis of the con-
ditions optimizing the QCL operation due to the geometric design of an active band. Within the open
models, one can evaluate the magnitude of dynamic conductivity in the needed quantum transition (for
example, 3 → 2). Besides, this conductivity would be much bigger than the conductivity of the close over
the energy transition (for example, 3 → 1) under the condition that the partial term of conductivity (σ+
32)
in forward direction would be much bigger than the partial term (σ−
31) in backward current. The calcu-
lated lifetimes (τn) of the operating electron quasi-stationary states make it possible to guide the natural
physical condition: these lifetimes should not exceed the relaxation times of dissipative processes due
to the scattering of electrons at the impurities, phonons, imperfections of media interfaces and other
factors, which, according to the evaluations [4], are not bigger than twenty picoseconds.
The results of numeric calculations of conductivities and lifetimes of electrons in the quasi-stationary
states (n = 1, 2, 3) obtained within the model of active band and complete cascade as functions of b1 are
presented in figures 2 (c), (b) and figures 2 (g), (f), respectively. Analysis of τn , σnn′ , σ±
nn′ dependences on
b1 [figures 2 (b), (c)] proves that there are two regions of b1 for the model of a four-barrier active band,
where: I — the optimal is the quantum transition 3 → 2 (experimentally observed), II — the optimal is
the quantum transition 3 → 1. The latter transition is possible [figure 2 (c)] the same as in the model of
a closed cascade. In this narrow region (II) for b1, the lifetime in the first quasi-stationary state is rather
small (τ1 É 1÷3ps< 10ps).
The model of an open complete cascade [figures 2 (f), (g)] is the best one for the description of the
properties of an electron current through the RTS of QCL with the electromagnetic radiation accompany-
ing quantum transitions. Figure 2 (g) proves that in this model, the same as in the model of an open active
band, there are also two regions for b1 varying (I and II) where the conditions of optimal QCL operation
fulfill well (the transitions 3 → 2 and 3 → 1, respectively). The location and sizes of these regions are
nearly the same as in the model of an open active band and in the model of a closed complete cascade.
However, the analysis of lifetimes τ1, τ2, τ3 [figure 2 (f)] shows that in fact, the region II is not optimal
because at such a geometric design the lifetime τ1 Ê 10 ps approaches the time of dissipative processes,
breaking the coherent regime of QCL. Thus, the model of an open cascade proves that in the experimental
QCL [3] it is only one narrow region (I) (0.55 nm É b1 É 1.2 nm) of the position of the inner two-barrier
structure between the outer barriers of an active band, where the laser operates in an optimal regime.
Only this configuration ensures that the active conductivity σ32 in direct current is much bigger than
the other conductivities. Herein, not only the lifetimes of both operating quasi-stationary states are small
(τ3,τ2 É 2 ps) but the lifetime of the first quasi-stationary state (τ1 É 10 ps) through which the electrons
flow into the next cascade due to the interaction with phonons [3, 4], is minimal.
The geometric and physical parameters for the numeric calculations within four theoretical models
were taken the same as in paper [3] in order to compare the theoretical and experimental data. These
parameters are presented in figure 1 and figure 2. The numeric calculations show that in all four models
the energies (E1, E2, E3) of operating quasi-stationary states differ between each other not more than
by 0.1%. Thus, in all models the theoretical magnitude of the energy of laser radiation Ω32 = E3 −E2 =
246 meV differs from the experimental one Ω
exp
32 = 238.8 meV by 3% and the difference of the energies
E3 −E2 = 34 meV nearly coincides with the phonon energy in [3].
Finally, we should note that the experimental geometric design of QCL cascade [3] with the input well
width b1 = 0.9 nm of a four-barrier active band correlates well with the magnitude b1 in all theoreti-
cal models because it corresponds to the close to maximal oscillator forces in closed models or dynamic
33701-8
Optimization of quantum cascade laser operation
conductivities in the open models. However, only the model of an open cascade is the most appropri-
ate because it does not contain those geometric configurations of an active band, inherent to the other
models, which do not ensure the optimal regime of QCL operation.
4. Conclusions
1. We developed the theory of dynamic conductivity of electrons in open multibarrier RTS driven by
the constant electric field taking into account the interaction between electrons and high frequency
electromagnetic field.
2. Comparing with the experimentally produced QCL having a four barrier active band of separate
cascade [3] we reveal that only the model of a complete open cascade confidently ensures the opti-
mal geometric design of the active band.
3. The developed theory of dynamic conductivity of electrons through the multibarrier RTS, after
modification, can be further used to optimize the operation of QCL, QCD and other nanodevices by
means of the choice of their geometric design.
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33701-9
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M.V. Tkach et al.
Оптимiзацiя роботи квантового каскадного лазера
геометричним дизайном каскаду у вiдкритих
i закритих моделях
М.В. Ткач, Ю.О. Сетi, I.В. Бойко, О.М. Войцехiвська
Чернiвецький нацiональний унiверситет iм. Ю. Федьковича,
вул. Коцюбинського, 2, 58012 Чернiвцi, Україна
У наближеннi ефективних мас i прямокутних потенцiалiв розвинута теорiя електронної динамiчної про-
вiдностi плоскої багатошарової резонансно-тунельної структури у постiйному електричному полi в моделi
вiдкритої наносистеми та сил осциляторiв квантових переходiв у моделi закритої системи. На прикла-
дi експериментально реалiзованого квантового каскадного лазера з чотирибар’єрною активною зоною
окремого каскаду показано, що саме теорiя динамiчної провiдностi у моделi вiдкритого каскаду найбiльш
адекватно описує процес випромiнювання високочастотного електромагнiтного поля при проходженнi
електронiв крiзь резонансно-тунельну структуру у постiйному електричному полi.
Ключовi слова: резонансно-тунельна наноструктура, провiднiсть, квантовий каскадний лазер
33701-10
Introduction
Theory of dynamic conductivity of a resonance tunnel cascade with four-barrier active band and oscillator forces of quantum transitions in closed model
Discussion of the results
Conclusions
|