Influence of dimensional static and dynamic charges on conduction in the active zone of a quantum cascade laser
The theory of active dynamic conductivity in the three-barrier active zone of a quantum cascade laser has been developed in the model of the electron effective mass and rectangular potential in the low signal approximation. In the preceding paper, it was shown that the static charge causes an increa...
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Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України
2015
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| Zitieren: | Influence of dimensional static and dynamic charges on conduction in the active zone of a quantum cascade laser / A.M. Gryschuk, I.V. Boyko // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2015. — Т. 18, № 2. — С. 123-127. — Бібліогр.: 9 назв. — англ. |
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irk-123456789-1218022017-06-19T03:03:16Z Influence of dimensional static and dynamic charges on conduction in the active zone of a quantum cascade laser Gryschuk, A.M. Boyko, I.V. The theory of active dynamic conductivity in the three-barrier active zone of a quantum cascade laser has been developed in the model of the electron effective mass and rectangular potential in the low signal approximation. In the preceding paper, it was shown that the static charge causes an increase of the lifetime of electronic quasistationary states and the shift of the energy levels into the high-energy range without changing maximum values of the active dynamic conductivity. The dynamic charge causes redistribution of the partial components of the active dynamic conductivity without affecting the spectral parameters of electron. It has been set that the partial components of the dynamic conductivity caused by the passing through electron flow from nanostructures reduce, and the components of conductivity caused by the flow in the opposite direction increase, thus, the conductivity value remains constant. 2015 Article Influence of dimensional static and dynamic charges on conduction in the active zone of a quantum cascade laser / A.M. Gryschuk, I.V. Boyko // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2015. — Т. 18, № 2. — С. 123-127. — Бібліогр.: 9 назв. — англ. 1560-8034 DOI: 10.15407/spqeo18.02.123 PACS 73.21.Fg, 73.40.Gk, 73.63.Hs http://dspace.nbuv.gov.ua/handle/123456789/121802 en Semiconductor Physics Quantum Electronics & Optoelectronics Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України |
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The theory of active dynamic conductivity in the three-barrier active zone of a quantum cascade laser has been developed in the model of the electron effective mass and rectangular potential in the low signal approximation. In the preceding paper, it was shown that the static charge causes an increase of the lifetime of electronic quasistationary states and the shift of the energy levels into the high-energy range without changing maximum values of the active dynamic conductivity. The dynamic charge causes redistribution of the partial components of the active dynamic conductivity without affecting the spectral parameters of electron. It has been set that the partial components of the dynamic conductivity caused by the passing through electron flow from nanostructures reduce, and the components of conductivity caused by the flow in the opposite direction increase, thus, the conductivity value remains constant. |
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Gryschuk, A.M. Boyko, I.V. |
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Gryschuk, A.M. Boyko, I.V. Influence of dimensional static and dynamic charges on conduction in the active zone of a quantum cascade laser Semiconductor Physics Quantum Electronics & Optoelectronics |
| author_facet |
Gryschuk, A.M. Boyko, I.V. |
| author_sort |
Gryschuk, A.M. |
| title |
Influence of dimensional static and dynamic charges on conduction in the active zone of a quantum cascade laser |
| title_short |
Influence of dimensional static and dynamic charges on conduction in the active zone of a quantum cascade laser |
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Influence of dimensional static and dynamic charges on conduction in the active zone of a quantum cascade laser |
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Influence of dimensional static and dynamic charges on conduction in the active zone of a quantum cascade laser |
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Influence of dimensional static and dynamic charges on conduction in the active zone of a quantum cascade laser |
| title_sort |
influence of dimensional static and dynamic charges on conduction in the active zone of a quantum cascade laser |
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Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України |
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| citation_txt |
Influence of dimensional static and dynamic charges on conduction in the active zone of a quantum cascade laser / A.M. Gryschuk, I.V. Boyko // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2015. — Т. 18, № 2. — С. 123-127. — Бібліогр.: 9 назв. — англ. |
| series |
Semiconductor Physics Quantum Electronics & Optoelectronics |
| work_keys_str_mv |
AT gryschukam influenceofdimensionalstaticanddynamicchargesonconductionintheactivezoneofaquantumcascadelaser AT boykoiv influenceofdimensionalstaticanddynamicchargesonconductionintheactivezoneofaquantumcascadelaser |
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2025-07-08T20:32:44Z |
| last_indexed |
2025-07-08T20:32:44Z |
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1837112248494981120 |
| fulltext |
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2015. V. 18, N 2. P. 123-127.
doi: 10.15407/spqeo18.02.123
© 2015, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
123
PACS 73.21.Fg, 73.40.Gk, 73.63.Hs
Influence of dimensional static and dynamic charges
on conduction in the active zone of a quantum cascade laser
A.M. Gryschuk
1
, I.V. Boyko
2
1
Ivan Franko Zhytomyr State University,
40, Velyka Berdychivska str., Zhytomyr, phone: (0412) 372-763, e-mail: teor-caf@meta.ua
2
I. Pul’uj Ternopil National Technical University,
56, Ruska str., Ternopil, e-mail: boyko.i.v.theory@gmail.com
Abstract. The theory of active dynamic conductivity in the three-barrier active zone of a
quantum cascade laser has been developed in the model of the electron effective mass
and rectangular potential in the low signal approximation. In the preceding paper, it was
shown that the static charge causes an increase of the lifetime of electronic quasi-
stationary states and the shift of the energy levels into the high-energy range without
changing maximum values of the active dynamic conductivity. The dynamic charge
causes redistribution of the partial components of the active dynamic conductivity
without affecting the spectral parameters of electron. It has been set that the partial
components of the dynamic conductivity caused by the passing through electron flow
from nanostructures reduce, and the components of conductivity caused by the flow in
the opposite direction increase, thus, the conductivity value remains constant.
Keywords: quantum cascade laser, resonant-tunnelling structure, dynamic conductivity,
static charge, dynamic charge.
Manuscript received 04.11.14; revised version received 02.04.15; accepted for
publication 27.05.15; published online 08.06.15.
1. Introduction
Nowadays, development of semiconductor technology
and physics is closely related with researches in quantum
cascade lasers (QCL) and detectors [1-8] as well as
physical processes in them.
The active elements of the mentioned nanodevices
operating in the terahertz and infrared region of
electromagnetic field frequencies are flat resonant-
tunnelling structures (RTS), which physical and
geometrical parameters greatly determine the properties
of the mentioned nanodevices. So, to find out the
conditions of nanolaser and detector optimization, it is
important to know the essence of physical processes
occurring at the coherent electron transport through the
multilayer RTS. The theory of the electron transport
through the three-barrier RTS with an applied permanent
longitudinal electric field based on the founded self-
consistent solution of the full Schrödinger equation and
Poisson equation has been developed in this paper.
Taking as an example a three-barrier RTS as an
active zone of the experimentally realized QCL with
In1–xGaxAs wells and In1–xAlxAs barriers shows the
effect of the dimensional static and dynamic charge on
spectral parameters of quasi-stationary states (QSS) of
an electron and an active dynamic conductivity of
nanosystems.
2. The theory of dynamic conduction in the three-
barrier active zone of quantum cascade lasers
To calculate the active dynamic conduction of electrons
in the three-barrier active zone of QCL, let us consider
that the system is placed in the Cartesian coordinate
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2015. V. 18, N 2. P. 123-127.
doi: 10.15407/spqeo18.02.123
© 2015, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
124
system in such a manner that its layers are perpendicular
to the separation boundaries of nanosystems. The
constant electric field F
is applied along the normal to
the layers of RTS. The geometric parameters of the
system are known (Fig. 1).
As the difference between the lattice constant at the
interfaces layer-well and layer-barrier is negligible, the
model of an effective mass and rectangular potentials
can be applied to electrons:
,
)()()()(
2
0
122
3
1
212
p
ppb
p
ppw
zzzzm
zzzzzmzm
(1)
,)(
)(
555
2
0
122
zzzzzzzeF
zzzzUzU
p
pp
(2)
where )(z is the Heaviside function;
61 , zz ; mw and mb are the effective masses
of electron in the potential wells and in barriers of the
nanostructure.
It is considered that the monochromatic electron
flow with the energy E and initial concentration n0 falls
along the normal to the layers of open RTS on the left.
The electron flow is considered to be one-dimensional
and can be described by the wave function ),( tz ,
satisfying the full Schrödinger equation.
),,(),()(
)(
1
2
),(
2
tztzHzU
zzmz
t
tz
i
(3)
where
),()(
)()()(),( 55
tzeee
zzzzzzeЄtzH
titi
(4)
is the Hamiltonian, which first summand describes
interaction of electron with alternating electromagnetic
field with the frequency ω and the tension amplitude of
its electric component Є. The second summand
describes interaction of electron with the dimensional
space charge, which potential ),( tz can be found from
the Poisson equation:
),(4
),(
)( tzen
z
tz
z
z
, (5)
where
2
0
122
3
1
212)()(
p
ppb
p
ppw
zzzz
zzzzzz
(6)
is the dielectric permittivity of three-barrier RTS, εw and
εb are dielectric permittivity of layer materials in
potential wells and barriers, and
2
0 ),(),( tzntzn (7)
is the variable in the electron density space.
It is evident from the structure of equations (3) and
(5), considering the Hamiltonian form (4) and the
equation (7), that they form a self-consistent system.
The solution of the full Schrödinger equation (3)
with the Hamiltonian (4) in the weak signal
approximation looks like:
./,)(
)()(),(
0
)(
1
)(
10
0
00
Eez
ezeztz
ti
titi
(8)
Substituting the equivalence (7) in the Poisson
equation (5), considering (8), with keeping the
summands of the first degree, we have got the following
equation:
,)()()(4
),(
)(
00
titi ezezzen
z
tz
z
z
(9)
where
.)()(
,)()()()()(
,)()(
*
1010
2
00
zz
zzzzz
zz
(10)
For any p-layers from RTS, the solution of the
equation (9) looks like that:
.
)()()(),(
1
5
1
)()()(
pp
p
tiptipp
st
zzzz
ezezztz
(11)
The following equations (12), (13) are got from (9),
considering (11) after equating the values of the same
order of smallness.
)(
4)( )(
0
0
2
)(2
z
en
z
z p
p
p
st
, (12)
)(
4)( )(
0
2
)(2
z
en
z
z p
p
p
, (13)
and its solutions are as follows:
,)()(
4
)(
)(
21
)(
1
0 0
212
)(
0
0)(
1
p
p
p
z z
p
p
p
st
CzzCdzdzz
en
z
(14)
.)(
)(
4
)(
)(
21
)(
1
0 0
212
)(0)(
1
p
p
p
z z
p
p
p
CzzC
dzdzz
en
z
(15)
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2015. V. 18, N 2. P. 123-127.
doi: 10.15407/spqeo18.02.123
© 2015, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
125
Fig. 1. Geometrical and energetic schemes of the three-barrier RTS (a) and the renormalized potential profile of the three-barrier
RTS, caused by the static charge from the dependence on z with the electron energy stEE 3 and charge carrier concentrations
,cm102 317 n 5∙1017, 1018 (b).
They determine the potential )()( zp
st caused by the
static space charge and potentials determined by the
dimensional dynamic charge in the case of electronic
transition with the absorption of )()( zp
and emission
of photons )()( zp
.
All the unknown coefficients
)(
2
)(
1
)(
2
)(
1 ;;;
pppp
CCCC
are uniquely determined from
the continuity conditions of the potential ),( tzp and
electric displacement field within all RTS.
Substituting (11) and (8) from the full Schrödinger
equation (5), after equating the coefficients of tie to
the variables of the zero degree, we have got the
equation:
,0)()()(
)(
1
2
0
2
zEzezU
zzmz
st
(16)
).()()()()(
)()()()(
)(
1
2
055
10
2
zzzzzzzzªe
zzezU
zzmz
st
(17)
The resulting Schrödinger equations (16), (17)
together with the Poisson equations (12), (13) form a set
of mutually agreed equations.
The solutions of them are as follows:
.)()()( 00
0
1
zzz
(18)
Here,
,)()(
][
)()()(
1
4
1 1
)(),(
0
)(),(
0
5
)()6(
0
)0(
00
1
),(
1
),(
5
)6()0(
ll
p
N
l
zziKlpzziKlp
zzikzik
zzzz
eBeA
zzeAzeBz
l
lp
l
lp
(19)
,)()()()(
)(
)()(
5
)6(
0
5
1
4
1 1
)(
0
2
)6(
00
zzz
eЄz
zzzz
dz
zd
m
eЄ
z
eЄz
z
ll
p
N
l
l
l
(20)
where
barrier the
.))()((2
barrier the
,))()((2
)(
;))((2
01
00
),(),(
00
1)6()0(
out
zeUm
in
zem
zKK
mkk
l
l
l
lplp
(21)
The formula to calculate the electric currents
density through RTS is as follows:
.
),(
),(
),(
),(
2
),(
1
1
1
1
0
dz
zEd
zE
dz
zEd
zE
m
nie
zEj
w
(22)
And it is proportional to the real parts of the
corresponding active dynamic conductivities ),( E .
The full active dynamic conductivity of RTS
),( E can be determined as the sum of two partial
components:
,),(),(),( EEE (23)
where
,
2
),(
2
)6(
1
)6(
1
2
)6(
1
)6(
12
50
0
AkAk
ªzm
n
E
2
)0(
1
)0(
1
2
)0(
1
)0(
12
50
0
2
),( BkBk
ªzm
n
E
. (24)
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2015. V. 18, N 2. P. 123-127.
doi: 10.15407/spqeo18.02.123
© 2015, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
126
Fig. 2. Dependences of the transparency coefficient on the energy without (dotted line) and with the dimensional charge (solid
lines) of the first three quasi-stationary states at concentrations n0 = 1017, 317 cm102 (a) and dependences of their resonance
energies st
3
st
2
st
1 ,, EEE and widths st
3
st
2
st
1 ,, on the concentration n0 (b).
3. Discussion of the results
The calculation of the active dynamic conductivity and
spectral parameters of quasi-stationary electronic states
was done using the active zone of the experimental QCL
as an example, which was described in the work [9]
(Fig. 1). RTS contains In0.53Ga0.47As wells and
In0.52Ga0.48As barriers and can be described by the
following geometrical Δ1 = 4.5 nm, Δ2 = 1.0 nm, Δ3 =
2.4 nm, b1 = 8.0 nm, b2 = 5.7 nm and physical mw =
0.041me, mb = 0.082me, U = 516 meV, εw = 13.899, εb =
12.726 parameters. Fig. 1b describes the potential profile
of the studied RTS renormalized by the static charge that
was calculated for different values of the concentration
of electrons (n = 2∙10
17
cm
–3
, 5∙10
17
, 10
18
). It is obvious
from Fig. 1b that the increase of electron concentration
causes deformation of the potential profile of
nanosystem, which influences spectral characteristics of
electronic QSS. The mentioned changes reflect the
results of the calculation of the transparency coefficient
D(E) (Fig. 2a) within the limits of energies of the first
three QSS, the resonance energies st
nE , and the lifetimes
of electrons st
n (Fig. 2a) depending on the concentration
of the charge n0, considering the space charge.
As seen from Fig. 2a, the dimensional charge
deforms the Lorentz shape to the wedge shape, causing
the increase in values of all resonant energies ( st
nE ) and
widths ( st
n ) (Fig. 2b). The values of the maxima of
D(E) with augmentation of the concentration increase.
As the electron lifetimes )( st
n in the corresponding QSS
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2015. V. 18, N 2. P. 123-127.
doi: 10.15407/spqeo18.02.123
© 2015, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
127
are related with resonant widths ( st
n ), as stst
nn , so
the augmentation of electron concentration causes its
incensement. And it is the significant factor for the great
values of n0.
Fig. 3 shows the results of calculating the
logarithms of the dynamic conductivity appearing in the
quantum transition 3→2 and its partial components
without (
323232 ,, ) and considering
(
323232
~,~,~ ) the influence of the dimensional charge.
Fig. 3 demonstrates that in the case of conductivity
that was calculated excluding the influence of the
dimensional charge, the value of the component of the
conduction determined by the electron flow to the output
from nanosystems (
) is bigger than the component of
the conductivity (
) defined by the flow in the
opposite direction, i.e.
323232 . From the
calculated dependences of the dynamic conductivity
with considering the dimensional charge, it is obvious
that with increasing the concentration of electrons n0, the
total value of conductivity 32
~ increases and the partial
conduction component, determined by the direct electron
flow ( 32
~ ) decreases, but the component in the opposite
direction ( 32
~ ) increases. Thus, the dimensional charge
causes redistribution of partial components in the total
value of conductivity with increasing the concentration.
4. Conclusions
The quantum-mechanical theory of the spectral
parameters of quasi-stationary states and dynamic
conductivity of three-barrier RTS with a constant
electric field as an active zone of QCL, considering the
variable dimensional charge appearing in the process of
electron transport through the nanostructure in a variable
electromagnetic field has been developed in this paper.
The self-consistent solution of the Schrödinger and
Poisson equations for different random electrons has
been obtained in the model of effective masses and
rectangular potentials.
The calculation of resonant energies, resonance
widths, active dynamic conductivity and its partial
components in the nanosystem model that corresponds
to the experimental values of QCL was made in this
paper. The value of the resonant energies that correlates
with the experimental data was calculated with the
accuracy not less than 5%. It has been shown that with
incensement of the concentration of electrons, the energy
of laser radiation in a quantum transition 3→2 decreases,
and the total value of active dynamic conductivity
increases, thus the contribution of the partial component
of conductivity determined by flow and directed
opposite to the exit of nanosystems increases in it.
References
1. J.M. Wolf, A. Bismuto, M. Beck, and J. Faist,
Distributed-feedback quantum cascade laser emitting
at 3.2 μm // Opt. Exp. 22(2), p. 2111-2118 (2014).
2. D. Bachmann, M. Rösch, C. Deutsch, M. Krall,
G. Scalari, M. Beck, J. Faist, K. Unterrainer and
J. Darmo, Spectral gain profile of a multi-stack
terahertz quantum cascade laser // Appl. Phys. Lett.
105(18), 181118-1-181118-4 (2014).
3. A. Buffaz, M. Carras, L. Doyennette, A. Nedelcu,
X. Marcadet and V. Berger, Quantum cascade
detectors for very long wave infrared detection //
Appl. Phys. Lett. 96(17), 172101-1-172101-3 (2010).
4. D. Hofstetter, F.R. Giorgetta, E. Baumann,
Q. Yang, C. Manz and K. Kohler, Midinfrared
quantum cascade detector with a spectrally broad
response // Appl. Phys. Lett. 93(22), 221106-1-
221106-3 (2008).
5. M.V. Tkach, Ju.O. Seti, I.V. Boyko, Effect of nonlinear
electron-electron interaction on electron tunneling
through an asymmetric two-barrier resonance tunnel
structure // Ukr. J. Phys. 57(8), p. 849-859 (2012).
6. Ju.O. Seti, M.V. Tkach, I.V. Boyko, Influence of
non-linear electrons interaction at their transport
through the symmetric two-barrier resonance nano-
system // J. Optoelectron. Adv. Mater. 14(3-4),
p. 393-400 (2012).
7. X. Gao, D. Botez, and I. Knezevic, Phonon
confinement and electron transport in GaAs-based
quantum cascade structures // J. Appl. Phys. 103(7),
073101-1-073101-9 (2008).
8. M.V. Tkach, Ju.O. Seti, Ju.B. Grynyshyn,
Influence of confined polarization phonons on the
electron spectrum in the three-barrier active zone of
a quantum cascade detector // Ukr. J. Phys. 59(12),
p. 1191-1200 (2014).
9. C. Sirtori, J. Faist, F. Capasso, D. L. Sivco, Mid-
infrared (8.5 μm) semiconductor lasers operating at
room temperature // IEEE Photonics Technology
Lett. 9(3), p. 294-296 (1997).
Fig. 3. Dependences of logarithms of dynamic conductivity,
resulting in laser transitions 3→2 and their partial
components, calculated considering the dimensional charge
(
323232
~,~,~ ) and without consideration
(
323232 ,, ), on the concentration of charge carriers n0.
|