Active conductivity of plane two-barrier resonance tunnel structure as operating element of quantum cascade laser or detector
Within the model of rectangular potentials and different effective masses of electrons in different elements of plane two-barrier resonance tunnel structure there is developed a theory of spectral parameters of quasi-stationary states and active conductivity for the case of mono-energetic electronic...
Saved in:
| Published in: | Condensed Matter Physics |
|---|---|
| Date: | 2011 |
| Main Authors: | , , , |
| Format: | Article |
| Language: | English |
| Published: |
Інститут фізики конденсованих систем НАН України
2011
|
| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/120004 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | Active conductivity of plane two-barrier resonance tunnel structure as operating element of quantum cascade laser or detector / M.V. Tkach, Ju.O. Seti, V.O. Matijek, O.M. Voitsekhivska // Condensed Matter Physics. — 2011. — Т. 14, № 2. — С. 23704:1-11. — Бібліогр.: 18 назв. — англ. |
Institution
Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1859812158570758144 |
|---|---|
| author | Tkach, M.V. Seti, Ju.O. Matijek, V.O. Voitsekhivska, O.M. |
| author_facet | Tkach, M.V. Seti, Ju.O. Matijek, V.O. Voitsekhivska, O.M. |
| citation_txt | Active conductivity of plane two-barrier resonance tunnel structure as operating element of quantum cascade laser or detector / M.V. Tkach, Ju.O. Seti, V.O. Matijek, O.M. Voitsekhivska // Condensed Matter Physics. — 2011. — Т. 14, № 2. — С. 23704:1-11. — Бібліогр.: 18 назв. — англ. |
| collection | DSpace DC |
| container_title | Condensed Matter Physics |
| description | Within the model of rectangular potentials and different effective masses of electrons in different elements of plane two-barrier resonance tunnel structure there is developed a theory of spectral parameters of quasi-stationary states and active conductivity for the case of mono-energetic electronic current interacting with electromagnetic field. It is shown that the two-barrier resonance tunnel structure can be utilized as a separate or active element of quantum cascade laser or detector. For the experimentally studied In₀.₅₃Ga₀.₄₇As/In₀.₅₂Al₀.₄₈As nano-system it is established that the two-barrier resonance tunnel structure, in detector and laser regimes, optimally operates (with the biggest conductivity at the smallest exciting current) at the quantum transitions between the lowest quasi-stationary states.
У моделi прямокутних потенцiалiв i рiзних ефективних мас електрона в рiзних елементах плоскої двобар’єрної резонансно-тунельної структури (ДБРТС) розвинута квантово-механiчна теорiя спектральних параметрiв квазiстацiонарних станiв i провiдностi цiєї системи для випадку моноенергетичного пучка електронiв, якi взаємодiють з електромагнiтним полем. Показано, що нано-ДБРТС може слугувати окремим елементом або активним елементом каскадного лазера чи детектора.
На прикладi експериментально дослiджуваної наносистеми In₀.₅₃Ga₀.₄₇As/In₀.₅₂Al₀.₄₈As показано, що у детекторному i лазерному режимах робота ДБРТС є оптимальною (з найбiльшою провiднiстю при найменшому струмi
збудження), коли вона працює на квантових переходах мiж найнижчими квазiстацiонарними станами.
|
| first_indexed | 2025-12-07T15:20:02Z |
| format | Article |
| fulltext |
Condensed Matter Physics, 2011, Vol. 14, No 2, 23704: 1–11
DOI: 10.5488/CMP.14.23704
http://www.icmp.lviv.ua/journal
Active conductivity of plane two-barrier resonance
tunnel structure as operating element of quantum
cascade laser or detector
M.V. Tkach∗, Ju.O. Seti, V.O. Matijek, O.M. Voitsekhivska
Fedkovych Chernivtsi National University, 2 Kotsyubinsky Str., 58012 Chernivtsi, Ukraine
Received March 4, 2011, in final form May 12, 2011
Within the model of rectangular potentials and different effective masses of electrons in different elements of
plane two-barrier resonance tunnel structure there is developed a theory of spectral parameters of
quasi-stationary states and active conductivity for the case of mono-energetic electronic current interacting
with electromagnetic field. It is shown that the two-barrier resonance tunnel structure can be utilized as a
separate or active element of quantum cascade laser or detector. For the experimentally studied
In0.53Ga0.47As/In0.52Al0.48As nano-system it is established that the two-barrier resonance tunnel structure,
in detector and laser regimes, optimally operates (with the biggest conductivity at the smallest exciting current)
at the quantum transitions between the lowest quasi-stationary states.
Key words: resonance tunnel structure, conductivity, quantum laser, quantum detector
PACS: 73.21.Fg, 73.90.+f, 72.30.+q, 73.63.Hs
1. Introduction
During the last decades, after the creation of the first nano-lasers by Faist and Capasso [1, 2]
working at the transitions between electronic levels of size-quantization, the evident success was
achieved in the improvement of quantum cascade lasers (QCLs) [3–6] and quantum cascade detec-
tors (QCDs) [7–10] with various geometric design. These devices operate effectively in the actual
terahertz range of electromagnetic waves with the frequencies getting into the known atmosphere
transparency windows. Thus, the QCLs and QCDs are constantly in the field of researchers’ vision.
The main purpose of investigations is to optimize the parameters of nano-devices which is
actually a hard task due to the absence of a consequent and complete theory of physical processes
in open nano-systems. As far as the active working element in experimentally produced QCL or
QCD were the open resonance tunnel structures (RTS), with different number of barriers and wells,
the main theoretical attention was paid to the study of static and dynamic conductivities in such
nano-systems because they determine the main QCL or QCD parameters, such as region and width
of operating frequency range, radiation intensity, exciting current and so on.
The theory of dynamic conductivity [11–17] of electrons in open RTS, as separate active el-
ement of QCL working in ballistic regime, is developed within the analytic solution of complete
Schrodinger equation using the simplified model of electron constant effective mass in all parts of
nano-system and δ-like approximation of rectangular potential barriers. In the cited and other pa-
pers, based on the simplified model of RTS, important results were obtained explaining the general
properties of conductivity in open systems, but due to the rather rough approximating models, as
it was established in reference [18], the obtained magnitudes of resonance energies and width of
electron quasi-stationary states in RTS (essentially determining the conductivity magnitude) were
manifestly overestimated compared to the more realistic model of rectangular potentials. There-
fore, in the approximated model, the problem of the optimization of QCL, QCD or the operation
of separate elements was not observed at all.
∗
E-mail: ktf@chnu.edu.ua
c© M.V. Tkach, Ju.O. Seti, V.O. Matijek, O.M. Voitsekhivska, 2011 23704-1
http://dx.doi.org/10.5488/CMP.14.23704
http://www.icmp.lviv.ua/journal
M.V. Tkach et al.
In the paper, using the model of rectangular potential wells and barriers and considering differ-
ent electron effective masses in different parts, there is developed a theory of active conductivity
of open plane two-barrier RTS as separate nano-detector or nano-laser. For the nano-system with
In0.53Ga0.47As wells and In0.52Al0.48As barriers the best geometric design of two-barrier RTS is
established providing its optimal work as an active element of nano-detector or nano-laser, i.e., pro-
viding the maximal active conductivity through the RTS at the minimal life times of electrons in
the operating quasi-stationary states (QSSs). Besides, it is shown that contrary to the laser where
the radiation transitions between two lowest electron QSS-s are not always optimal, the energy of
mono-energetic electron beam falling at two-barrier RTS should correspond to the energy of the
lowest QSS (from which the quantum transitions into the second resonance electron state occur
with the absorption of electromagnetic energy) for the detector to operate effectively.
2. Active conductivity of two-barrier RTS
In Cartesian coordinate system, the open two-barrier RTS is observed with geometric parame-
ters shown in figure 1. The small differences of lattice constants of RTS barriers and wells make it
possible to study the nano-system within the models of effective masses and rectangular potentials
m(z) =
{
m0 ,
m1 ,
U(z) =
{
0, reg. 0, 2, 4,
U, reg. 1, 3.
(1)
Figure 1. Geometric (a) and potential energy (b) schemes of two-barrier resonance tunnel struc-
ture.
The electronic beam moving in the direction perpendicular to the two-barrier RTS plane, im-
pinges on it from the left hand side. The electrons with energy (E) and concentration (n0) are
assumed to be uncoupled. In order to obtain the conductivity of a nano-system, determined by
the density of current flowing through it, according to the quantum mechanics, one has to find
the wave function of the electron dependent on time, interacting with the electromagnetic field
periodic in time.
In the problem under study, the movement of electrons is observed as one-dimensional (k|| = 0).
Consequently, Ψ(z, t) wave function satisfies the complete Schrodinger equation
i~
∂Ψ(z, t)
∂t
= [H +H(z, t)] Ψ(z, t), (2)
23704-2
Active conductivity of plane two-barrier resonance tunnel structure
where
H = −
~
2
2
∂
∂z
1
m (z)
∂
∂z
+ U(z) (3)
is the electron Hamiltonian in stationary case,
H(z, t) = −eǫ {z [θ (z)− θ (z − z3)] + z3θ (z − z3)}
(
eiωt + e−iωt
)
. (4)
The Hamiltonian of an electron interacts with time varying electromagnetic field with frequency
(ω) and amplitude of electric field intensity (ǫ).
The solution of equation (2) in the approximation of small signal [11–17] is written as follows:
Ψ(z, t) = Ψ0 (z) e
−iω0t +Ψ1 (z, t) , (ω0 = E/~) (5)
where Ψ0(z) function is the solution of stationary Schrodinger equation
HΨ0(z) = EΨ0(z). (6)
The first order correction in one-mode approximation is found as
Ψ1 (z, t) = Ψ+1 (z) e
−i(ω0+ω)t +Ψ−1 (z) e
−i(ω0−ω)t. (7)
Preserving the magnitudes of the first smallness order and taking into account formulas (5),
(6), (2), the equation is obtained for the both parts Ψ±1(z) of function Ψ1(z, t)
(
−
~
2
2
∂
∂z
1
m (z)
∂
∂z
+ U(z)− ~ (ω0 ± ω)
)
Ψ±1 (z) +H (z)Ψ0 (z) = 0, (8)
where
H(z) = −eǫ {z [θ (z)− θ (z − z3)] + z3θ (z − z3)} .
The solution of stationary Schrodinger problem, equation (6) is written as
Ψ0(z) = Ψ
(0)
0 (z)θ(−z) +
3
∑
p=1
Ψ
(p)
0 (z)[θ(z − zp−1)− θ(z − zp)] + Ψ
(4)
0 (z)θ(z − z3)
=
(
eik
(0)z +B(0)e−ik(0)z
)
θ(−z) +A(4)eik
(4)zθ(z − z3)
+
3
∑
p=1
(
A(p)eik
(p)z +B(p)e−ik(p)z
)
[θ(z − zp−1)− θ(z − zp)], (9)
where
k(0) = k(2) = k(4) = k = ~
−1
√
2m0E , k(1) = k(3) = ~
−1
√
2m1(E − U) ,
z0 = 0, z1 = ∆−
1 , z2 = b+∆−
1 , z3 = b+∆, ∆ = ∆−
1 +∆+
1 .
The unknown coefficients B(0), A(4), A(p), B(p) (p = 1, 2, 3) are fixed by the conditions of
wave functions and their densities of currents continuity at all nano-system interfaces
Ψ
(p)
0 (zp) = Ψ
(p+1)
0 (zp),
1
m0(1)
dΨ
(p)
0
dz
∣
∣
∣
∣
∣
z=zp
=
1
m1(0)
dΨ
(p+1)
0 (z)
dz
∣
∣
∣
∣
∣
z=zp
(p = 0, 1, 2, 3) (10)
as well as because the nano-system is an open one, from the normalizing condition for the wave
functions (at fixed k|| = 0)
∞
∫
−∞
Ψ∗
0(k
′z)Ψ0(kz)dz = δ(k − k′). (11)
23704-3
M.V. Tkach et al.
The obtained wave function Ψ0 defines the density of electronic current and, thus, the perme-
ability coefficient of the system as function of energy. It is well known [18], that the permeability
coefficient in the vicinity of their maxima is of a quasi-Lorentz shape. Consequently, the position
of maximum in the energy scale defines the resonance energy (En) and the width of Lorentz curve
at half of its height fixes the resonance width (Γn) of the corresponding QSS.
The solutions of inhomogeneous equations (8) are the super-positions of functions
Ψ±1 (z) = Ψ± (z) + Φ± (z) , (12)
where Ψ± (z) are the solutions of homogeneous and Φ± (z) are solutions of inhomogeneous equa-
tions (8).
The solutions of homogeneous equations (8) are found as
Ψ±(z) = Ψ
(0)
± (z)θ(−z) +
3
∑
p=1
Ψ
(p)
± (z)[θ(z − zp−1)− θ(z − zp)] + Ψ
(4)
± (z)θ(z − z3)
= B
(0)
± e−ik
(0)
±
zθ(−z) +A
(4)
± eik
(4)
±
(z−z3)θ(z − z3)
+
3
∑
p=1
[
B
(p)
± e−ik
(p)
±
(z−zp−1) +A
(p)
± eik
(p)
±
(z−zp−1)
]
[θ(z − zp−1)− θ(z − zp)], (13)
where
k
(0)
± = k
(2)
± = k
(4)
± = ~
−1
√
2m0(E ± ~ω) , k
(1)
± = k
(3)
± = ~
−1
√
2m1 [(E − U)± ~ω] . (14)
Exact partial solutions of equations (8) are known
Φ±(z) =
3
∑
p=1
[
∓
eǫ
~ω
zΨ
(p)
0 (z) +
eǫ
mpω2
dΨ
(p)
0 (z)
dz
]
[θ(z − zp−1)− θ(z − zp)]
∓
eǫ
~ω
z3Ψ
(4)
0 (z3) θ(z − z3). (15)
Thus, the general solutions of these equations can be written as
Ψ±1 (z) = Ψ
(0)
±1 (z) θ(−z) +
3
∑
p=1
Ψ
(p)
±1 (z) [θ(z − zp−1)− θ(z − zp)] + Ψ
(4)
±1 (z) θ(z − z3). (16)
The conditions of the equations of wave functions (16) and the respective continuity of currents
at all interfaces of nano-systems
Ψ
(p)
±1 (zp) = Ψ
(p+1)
±1 (zp) ,
dΨ
(p)
±1 (z)
m0(1)dz
∣
∣
∣
∣
∣
z=zp
=
dΨ
(p+1)
±1 (z)
m1(0)dz
∣
∣
∣
∣
∣
z=zp
(p = 0, 1, 2, 3) (17)
lead to the system of eight inhomogeneous equations determining all eight unknown coefficients
B
(0)
± , A
(4)
± , B
(p)
± , A
(p)
± (p = 1, 2, 3). Thus, now Ψ±(z) functions, the first order correction – Ψ1(z, t)
and, consequently, the whole Ψ(z, t) wave function are completely defined.
According to the quantum mechanics, the density of current of uncoupling electrons with con-
centration n0 is given by the formula
j(z, t) =
ie~n0
2m(z)
[
Ψ(z, t)
∂
∂z
Ψ∗(z, t)−Ψ∗(z, t)
∂
∂z
Ψ(z, t)
]
. (18)
Taking into account the small sizes of two-barrier RTS compared to the electromagnetic wavelength,
in quasi-classic approximation [11–17] the calculation of the guided current density is performed,
determining the real part of nano-system conductivity
σ(ω) = σ+(ω) + σ−(ω) =
~
2ωn0
2z3m0ǫ2
[
k+
(
∣
∣
∣
B
(0)
+
∣
∣
∣
2
+
∣
∣
∣
A
(4)
+
∣
∣
∣
2
)
− k−
(
∣
∣
∣
B
(0)
−
∣
∣
∣
2
+
∣
∣
∣
A
(4)
−
∣
∣
∣
2
)]
. (19)
23704-4
Active conductivity of plane two-barrier resonance tunnel structure
Here σ+(ω), σ−(ω) are the components of conductivity throughout the nano-system and in the
opposite direction, respectively, caused by the electronic currents from the nano-system after the
interaction with electromagnetic field therein.
3. Discussion of the results
The operating characteristics of a separate nano-laser, nano-detector or quantum cascade nano-
devices at their base are determined by the properties of active conductivity, depending on spectral
parameters (resonance energies and widths) of electron QSSs. The latter characteristics, in turn,
are defined by material and geometric parameters of a nano-system. Thus, one has to study, first
of all, the spectral parameters of electron QSSs for further analyzing and understanding the main
properties of the active conductivity. For example, the paper studies the widely experimentally
investigated [1–3, 7–10] plane two-barrier RTS (figure 1), consisting of In0.53Ga0.47As wells (m0 =
0.046 me) and In0.52Al0.48As barriers (m1 = 0.089 me). The difference of electron potential energy
between the barrier and the well is U = 516 meV. This structure well satisfies the developed theory
conditions due to the close magnitudes of lattice constants (a0 = 0.5867 nm, a1 = 0.5868 nm),
wells (ε0 = 14.2) and barriers (ε1 = 12.7) dielectric constants.
1 2 3 4 5 6 7 8
0
100
200
300
400
-6
-4
-2
0
2
4
b=10.8 nm
=9 nm
E
n
,
m
e
V
+
, nm
a
ln
n
E
3
E
2
E
1
ln
3
ln
2
ln
1
0 5 10 15 20
0
100
200
300
400
-4
-2
0
2
b
II
b
I
ln
3
ln
2
ln
1
--
=6 nm
+
=3 nm
ln
n
E
n
,
m
e
V
b, nm
b
E
II
2
E
II
3
E
1
E
2
21
E
3
E
I
2
E
I
1
32
Figure 2. Dependencies of resonance energies
En and logarithms of resonance widths ln Γn :
(a) on output barrier width ∆+ at ∆ = 9 nm
and (b) on well width b.
In figure 2 there are shown the resonance en-
ergies (En) and logarithms of resonance widths
(ln Γn , in units Γ0 = 1 meV) of the three low-
est electron QSSs depending on geometric pa-
rameters of two-barrier RTS: i.e., on the width
of the output barrier ∆+ (figure 2 (a)) at con-
stant well width b = 10.8 nm with sum barriers
width ∆ = ∆− + ∆+ = 9 nm; potential well
width b (figure 2 (b)) at the fixed barriers widths
∆− = 6 nm, ∆+ = 3 nm. The resonance energy
(En) of electron n-th QSS is fixed by the po-
sition of maximum of n-th peak of permeability
coefficient and the resonance width (Γn) – as the
width of the same peak at the half of its height
in energy scale.
From figure 2 it is clear that the change of
both widths of barriers (∆−, ∆+) almost does
not change the spectrum of resonance energies
(En), as square function of quantum number n.
That is: En = E1n
2, like the spectrum of quasi-
particle in infinitely deep potential well. Here E1
is the resonance energy of the first electron QSS
in two-barrier RTS (fixed, as it was mentioned
above, by the position of the first maximum of
permeability coefficient in energy scale).
Contrary to the resonance energies (En), the
magnitudes of resonance widths (Γn) essentially
depend on the ratio between the barrier widths.
It is clear (figure 2 (a)) that the increasing
width of the output barrier (∆+) and the re-
spective decreasing width of the input barrier
(∆− = ∆ − ∆+) causes a linear decrease of all
ln Γn , approaching the minima magnitudes at
∆+ = ∆− = 4.5 nm. At further ∆+ increase, the magnitudes of ln Γn linearly increase. Such a
behavior of the width of electron QSSs (Γn) in 0 < ∆± < ∆/2 range can be described by the typ-
ical, for the quantum-barrier systems, dependence: Γn = Γ0
n exp(−γn∆
±), where Γ0
n is the width
23704-5
M.V. Tkach et al.
of n-th virtual QSS, existing at ∆± = 0, ∆∓ = ∆ 6= 0, b 6= 0. The magnitudes γn characterize the
speed of resonance widths decrease when the barriers width (∆±) increases.
The dependence of electron QSSs spectral parameters (En , Γn) on the well width (b) is shown
in figure 2 (b). In the figure one can see that for the increasing well width, the resonance energies
(En) and logarithms of resonance widths (ln Γn) shift into the region of smaller energies. Herein,
En ∼ b−3/2, contrary to the energy spectrum in infinitely deep potential well, where En ∼ b−2.
The decreasing character of resonance widths is caused by a decrease of resonance energies which
is equivalent to the increase of effective potential barriers above them.
The numerical calculations prove that the established properties of spectral parameters (En ,
Γn) of electron QSSs are equitable at any magnitudes of two-barrier RTS geometric parameters
(b, ∆±, ∆). Further, the active conductivity of nano-system is studied as a function of energy
(E) of mono-energetic electronic beam impinging upon two-barrier RTS and electromagnetic field
frequency (ω). The concentration of electronic current is assumed to be small (n0 ≈ 1016 cm−3),
which allows us to neglect the coupling between electrons.
In order to establish the optimal geometric design of two-barrier RTS, when the system operates
in the required range of electromagnetic wave frequencies as separate nano-detector or nano-laser,
one has to analyze the behavior of active conductivity (σ) together with its components (σ−, σ+)
formed by the input and output currents, depending on the nano-system geometric parameters. It
is clear that the effectiveness of any nano-device operation would be better at the maximal absolute
value of conductivity (σ) at the demanded condition |σ+| ≫ |σ−| and minimal life time (τn = ~/Γn)
of electrons in the operating QSSs (minimizing the negative effect of dissipative processes).
Now let us study the two-barrier RTS conductivity, considering that the mono-energetic beam
of uncoupling electrons impinges on it from the left, in the direction perpendicular to the planes
of the layers. The energy of electrons is the same as the resonance energy (En) of n-th QSS.
The electronic current interacts with the electromagnetic field in such a way that the quantum
transitions take place in the nano-system. As a result, the active conductivity is formed, being,
as it is well known [17], positive (detector) for the transitions accompanied by the absorption of
electromagnetic field energy, and negative (laser) with the energy radiation.
The numeric calculations prove that the active conductivity is mainly formed by quantum
transitions between two nearest electron QSSs, because the transitions between the states with
equal parity are forbidden and the transitions into the far resonance states are weak. Therefore,
studying the conductivity and its components, we consider only the transitions between the nearest
states.
In order to detect (or radiate) the electromagnetic field with frequency ω due to the quantum
transitions between two QSSs with the resonance energies En and En±1 , it is necessary that
the energy (E) of electrons impinging upon the system should correspond to the energy (En)
of those n-th state from which the transition occurs. Then, the electromagnetic field energy is
defined as: ~ωn,n±1 = |En −En±1| = (2n± 1)E1 . Consequently, the estimation of electron ground
QSS energy is E1 = ~ωn,n±1/(2n ± 1) and thus, the energy of the impinging electronic current
is: En = ~ωn,n±1n
2/(2n ± 1). The latter, due to the established properties (figure 2 (a), (b)),
weakly depends on the barrier widths (∆−, ∆+) and is mainly determined by the well width (b)
of two-barrier RTS.
Thus, when the energy (~ω) of the detected (or radiated) electromagnetic field is known, first
of all the estimation of the required well width b0 = π~n(2m0E1)
−1/2 = (~π2(2n±1)/2m0ωnn′)1/2
can be obtained within the model of infinitely deep potential well. Then, a more exact magnitude
of the two-barrier RTS well width (b) is found as its variation in b0 vicinity.
The developed theory for two-barrier RTS active conductivity is valid for any type of quantum
transitions (laser or detector). Further, to optimize the design of the system under research as
an active element of nano-laser or nano-detector, there is performed the calculation of electron
QSSs life times and maxima of active conductivity together with its components (σ−, σ+). The
latter are formed by detector transitions with absorption (figure 3) and by laser transitions with
radiation (figure 4) of electromagnetic field energy (~ω) with the respective wave-lengths (λ) shown
in figures 3 (a), (c) and 4 (a), (c). The widths of the wells: b = 10.8 nm, 21.6 nm and the widths
23704-6
Active conductivity of plane two-barrier resonance tunnel structure
of the barriers: ∆ = ∆− + ∆+ = 6 nm, 9 nm, 12 nm were used as typical ones [3, 7–10] for the
experimentally investigated QCDs or QCLs with the range of operating frequencies being in the
sub-infrared windows of atmosphere transparency.
2 4 6 8 10
0
10
20
30
E
2
=162.2 meV
23
=202.4 meV
23
=6.1 m
6 nm
=9 nm
=12 nm
ln
+
23
ln
--
23
ln
± 2
3
b=10.8 nm
, nm
2 4 6 8 10
0
10
20
30
E
2
=52.1 meV
23
=65.6 meV
23
=18.9 m
6 nm
=9 nm
=12 nm
ln
+
23
ln
--
23
ln
± 2
3
b=21.6 nm
, nm
2 4 6 8 10
0
5
10
15
-5
0
5
10
15
20
25
ln
23
ln
(
+ 2
3
-
-- 2
3
)
ln
2
3
b=10.8 nm
, nm
=9 nm
6 nm
=12 nm
ln( - )
=12 nm
=9 nm
6 nm
2 4 6 8 10
0
5
10
15
-5
0
5
10
15
20
25
6 nm
=9 nm
=9 nm
ln
23
ln
2
3
b=21.6 nm
, nm
=12 nm
ln
(
+ 2
3
-
-- 2
3
) ln( - )
=12 nm
6 nm
2 4 6 8 10
0
10
20
30
E
1
=40.1 meV
12
=122.1 meV
12
=10.1 m
6 nm
=9 nm
=12 nm
ln
+
12
ln
--
12
ln
± 1
2
b=10.8 nm
, nm
2 4 6 8 10
0
10
20
30
E
1
=13 meV
12
=39 meV
12
=31.8 m
6 nm
=9 nm
12 nm
ln
+
12
ln
--
12
ln
± 1
2
b=21.6 nm
, nm
2 4 6 8 10
0
5
10
15
-5
0
5
10
15
20
25
ln
12
ln
(
+ 1
2
-
-- 1
2
ln
1
2
b=10.8 nm
, nm
6 nm
=9 nm
=12 nm
6 nm
ln( - )
=12 nm
=9 nm
2 4 6 8 10
0
5
10
15
-5
0
5
10
15
20
25
=9 nm
ln
12
=9 nm
ln
1
2
b=21.6 nm
, nm
=12 nm
6 nm
ln
(
+ 1
2
-
-- 1
2
) ln( - )
=12 nm
6 nm
b
c
d
Figure 3. Dependencies of conductivity logarithms lnσ+
23 , ln σ−
23 , ln σ+
12 , lnσ−
12 , ln(σ+
23 − σ
−
23),
ln(σ+
12 − σ
−
12) and life times ln τ23 , ln τ12 on output barrier width ∆+ at different well widths b
and sum ∆, for the two-barrier RTS operating as detector.
23704-7
M.V. Tkach et al.
2 4 6 8 10
0
10
20
30
E
3
=364.6 meV
32
=202.4 meV
32
=6.1 m
6 nm
=9 nm
=12 nm
ln|
+
32
|
ln
--
32
|
ln
± 3
2
|
b=10.8 nm
, nm
2 4 6 8 10
0
10
20
30
E
3
=117.7 meV
32
=65.6 meV
32
=18.9 m
6 nm
=9 nm
=12 nm
ln
+
32
|
|ln
--
32
|
ln
± 3
2
|
b=21.6 nm
, nm
2 4 6 8 10
0
5
10
15
-5
0
5
10
15
20
25
ln
32
ln
|
+ 3
2
-
-- 3
2
|
ln
3
2
b=10.8 nm
, nm
=9 nm
6 nm
=12 nm
ln| - |
=12 nm
=9 nm
6 nm
2 4 6 8 10
0
5
10
15
-5
0
5
10
15
20
25
6 nm
=9 nm
=9 nm
ln
32
ln
3
2
b=21.6 nm
, nm
=12 nm
ln
|
+ 3
2
-
-- 3
2
| ln| - |
=12 nm
6 nm
b
2 4 6 8 10
0
10
20
30
E
2
=162.2 meV
=122.1 meV
=10.1 m
6 nm
=9 nm
=12 nm
ln
+
21
|
ln
--
21
|
ln
± 2
1
|
b=10.8 nm
, nm
2 4 6 8 10
0
10
20
30
E
2
=52 meV
=39 meV
=31.8 m
6 nm
=9 nm
12 nm
ln
+
21
|
ln
--
21
|
ln
± 2
1
|
b=21.6 nm
, nm
c
2 4 6 8 10
0
5
10
15
-5
0
5
10
15
20
25
ln
21
ln
|
+ 2
1
-
-- 2
1
|
ln
2
1
b=10.8 nm
, nm
6 nm
=9 nm
=12 nm
6 nm
ln| - |
=12 nm
=9 nm
2 4 6 8 10
0
5
10
15
-5
0
5
10
15
20
25
=9 nm
ln
21
=9 nm
ln
2
1
b=21.6 nm
, nm
=12 nm
6 nm
ln
|
+ 2
1
-
-- 2
1
| ln| - |
=12 nm
6 nm
d
Figure 4. Dependencies of conductivity logarithms ln |σ+
32| , ln |σ
−
32| , ln |σ
+
21| , ln |σ
−
21| , ln |σ
+
32 −
σ
−
32| , ln |σ
+
21−σ
−
21| and life times ln τ32 , ln τ21 on output barrier width ∆+ at different well width
b and sum ∆, for the two-barrier RTS operating as laser.
The calculated conductivity logarithms: forward ln |σ+
12(21)|, ln |σ
+
23(32)| and backward ln |σ−
12(21)|,
ln |σ−
23(32)| and their differences ln |σ+
12(21) − σ−
12(21)|, ln |σ
+
23(32) − σ−
23(32)| (in units σ0 = 1 S/cm)
formed by the detector (1 → 2, 2 → 3) and laser (2 → 1, 3 → 2) quantum transitions; and loga-
23704-8
Active conductivity of plane two-barrier resonance tunnel structure
rithms of electron life times sum in operating QSSs ln τn,n±1 , where τn,n±1 = τn + τn±1 (in units
τ0 = 1 s) are presented in figures 3, 4 respectively, as functions of output barrier width ∆+ at the
condition of the constant sum of barriers widths: ∆ = ∆− +∆+ = const (6 nm, 9 nm, 12 nm).
In figures 3 (a), (c) and 4 (a), (c) one can see that independently of the well width (b) and sum
of barriers widths (∆) all ln |σ±
12|, ln |σ
±
23| and ln |σ±
21|, ln |σ
±
32| magnitudes qualitatively similarly
depend on the output barrier width (∆+). Herein, it is evident that ln |σ±
21| > ln |σ±
12| > ln |σ±
32| >
ln |σ±
23|. Their main properties are as follows.
The increase of ∆+ (and the corresponding decrease of ∆−) in the range 0 < ∆+ < ∆/2
causes the linear increase of ln |σ+
12(21)|, ln |σ
+
23(32)| and ln |σ−
12(21)|, ln |σ
−
23(32)| magnitudes. Herein,
there are always satisfied the inequalities: ln |σ+
12(21)| > ln |σ−
12(21)|, ln |σ
+
23(32)| > ln |σ−
23(32)|. In the
range of output barrier widths ∆/2 < ∆+ < ∆, on the contrary, these magnitudes are linearly
decreasing with ln |σ−
12(21)| > ln |σ+
12(21)|, ln |σ−
23(32)| > ln |σ+
23(32)|. In the symmetric two-barrier
RTS, when ∆+ = ∆− = ∆/2, we have ln |σ+
12(21)| = ln |σ−
12(21)|, ln |σ
+
23(32)| = ln |σ−
23(32)| which is
completely understandable from physical considerations because the electronic current, after the
interaction with the electromagnetic field, flows from the two-barrier RTS by two currents equal
over magnitudes and opposite over directions.
The established properties of active conductivity (σ) and its components (σ+
12(21), σ
−
12(21) and
σ+
23(32), σ
−
23(32)) lead to the evident conclusion. Now it is clear that the condition |σ+
12(21)| ≫ |σ−
12(21)|
or |σ+
23(32)| ≫ |σ−
23(32)|, at which the two-barrier RTS optimally operates as detector or laser, is
fulfilled only in the interval of widths ∆+ < ∆/2 (∆/2 < ∆− = ∆ −∆+), i.e., at the ascending
parcels of lnσ+
12(21) , lnσ+
23(32) and lnσ−
12(21) , lnσ−
23(32) as functions of ∆+. There is performed a
calculation of ln |σ+
12(21)−σ−
12(21)|, ln |σ
+
23(32)−σ−
23(32)| and ln τ12 , ln τ23 as functions of ∆+ in order
to establish the best ratio between ∆+ and ∆− magnitudes (at fixed ∆ = 6 nm, 9 nm, 12 nm).
The results are shown in figures 3 (b), (d) and 4 (b), (d).
From figures 3 and 4 one can see that at any ∆, maximum magnitudes of ln |σ+
12(21) − σ−
12(21)|,
ln |σ+
23(32)−σ−
23(32)| are approached at such widths ∆+ = ∆+
m , which are in the small vicinity from
the left hand side of ∆+ = ∆/2. It is clear that as far as the output barrier width ∆+
m is only
a little bit smaller than the input barrier width (∆−
m = ∆ − ∆+
m), hence, the forward electronic
current through the thinner output barrier is much bigger than the backward current through the
thicker input barrier, observed in figure 3 (b), (d) and 4 (b), (d).
For the optimal design of two-barrier RTF and coherent character of electromagnetic wave (in
case of laser), the evident physical requirements must be fulfilled: the life time (τn,n±1) of electron
in both operating QSSs should not be bigger than the time (τd) of dissipative processes (interaction
with phonons, impurities and so on). According to the known estimations [3] τd ≈ 20 s or ln τd ≈ 3.
Now, there are obtained the consequent estimations of optimal widths of both barriers ∆+ and
∆− at which, at the conditions |σ+
12(21)| ≫ |σ−
12(21)|, |σ
+
23(32)| ≫ |σ−
23(32)|, the magnitudes σ12(21) ,
σ23(32) are maximum due to the linear dependence of ln |σ+
12(21) − σ−
12(21)|, ln |σ
+
23(32) − σ−
23(32)| and
ln |σ±
12(21)|, ln |σ
±
23(32)| maxima on ∆ (figure 3, 4) at fixed well width (b). The magnitudes of all
evaluated characteristics are shown in figures 3, 4. From the figure one can see that when the
well width (b) increases, the optimal sum of both barriers widths (∆ = ∆+ + ∆−), confined by
dissipative processes scattering time (τd) , decreases.
The detection or radiation of electromagnetic field with certain frequency can be obtained
due to quantum transitions between QSSs in different two-barrier RTSs with respective widths of
potential wells. There arises a question: which one of the two RTSs is more optimal – the one with
smaller well width and operating at the transitions between lower levels or the one with bigger
well width and operating at the respective transitions between higher levels.
There is an important property of two-barrier RTS, which is really clear from physical consid-
erations. When the electron energy is equal to the resonance energy of any level except the ground
one, the quantum transitions into the lower QSSs (figure 4) are more probable than into the higher
ones (3). Consequently, the nano-system has a negative conductivity and operates in the regime
of electromagnetic field radiation. Thus, in a detector regime, the two-barrier RTS operates only
23704-9
M.V. Tkach et al.
when the energy of electronic current is equal to the resonance energy of ground QSS and quan-
tum transition into the second state occurs with the absorption of electromagnetic field energy.
We should note that all the known experimentally utilized nano-detectors operate exactly in this
regime [7–10].
As far as the laser regime is concerned, this is realized within the quantum transitions between
the neighbouring levels from arbitrary higher into arbitrary lower states. For example, we can
compare the advantages and disadvantages of two different RTS, radiating the electromagnetic field
in one of the ranges of atmosphere transparency windows (λ = 8− 14 µm or ~ω = 89− 155 meV).
It is clear from figure 2 (b) that the radiation with the field energy ~ω = 122 meV can be realized
by two two-barrier RTSs with equal barrier widths (∆− = 6 nm, ∆+ = 3 nm) but different well
widths: I) at the transition 2 → 1 (bI = 10.8 nm) τI1 = 14.6 ps, τI2 = 1.5 ps, τI21 = τI2 +τI1 = 16.1 ps,
EI
2 = 162 meV, |σ+
21| = 1200 S/cm, |σ−
21| = 1 S/cm; II) at the transition 3 → 2 (bII = 15 nm)
τII2 = 4.7 ps, τII3 = 1.1 ps, τII32 = τII3 + τII2 = 5.8 ps, EII
2 = 219 meV, |σ+
32| = 811 S/cm,
|σ−
21| = 2 S/cm. Comparing the both nano-systems, one can see, even if the life-time (τI12) of
electron in the operating QSS for the system I is almost three times bigger than the life time (τII32 )
for the system II, but, herein, the starting electron current (jI0 ∼
√
EI
2 ) at system I is 1.2 times
smaller than the one (jII0 ∼
√
EII
3 ) at the system II. The intensity of radiation, proportional to the
conductivity, for the system I is about 1.5 times bigger than for the II system. Thus, the increase
of QSSs life times for the system I is not bigger than the scattering times of electrons due to the
dissipative processes (phonons, impurities and so on) destroying the coherence. Thus, the optimal
two-barrier RTS is the one operating at quantum transitions between two lowest QSSs.
Finally, we should note that the theory for the active conductivity of two-barrier RTS developed
within the framework of simple rectangular potentials model and different electron effective masses
approximation can be used in future as a basis of investigation and optimization of complicated
open RTS with bigger number of wells and barriers, intensively utilized as operating elements of
quantum cascade nano-lasers and nano-detectors.
References
1. Faist J., Capasso F., Sivco D.L., Sirtori C., Hutchinson A.L., Cho A.Y., Science, 1994, 264, 553;
doi:10.1126/science.264.5158.553.
2. Faist J., Capasso F., Sirtori C., Appl. Phys. Lett., 1995, 66, 538; doi:10.1063/1.114005.
3. Gmachl C., Capasso F., Sivco D.L., Cho A.Y., Rep. Prog. Phys., 2001, 64, 1533;
doi:10.1088/0034-4885/64/11/204.
4. Scalari G., Ajili L., Faist J., Beere H., Linfield E., Ritchie D., Davies G., Appl. Phys. Lett., 2003, 82,
3165; doi:10.1063/1.1571653.
5. Diehl L., Bour D., Corzine S., Zhu J., Hofler G., Loncar M., Troccoli M., Capasso F., Appl. Phys.
Lett., 2006, 88, 201115; doi:10.1063/1.2203964.
6. Wang Q.J., Pflug C., Diehl L., Capasso F., Edamura T., Furuta S., Yamanishi M., Kan H., Appl. Phys.
Lett., 2009, 94, 011103; doi:10.1063/1.3062981.
7. Hofstetter D., Beck M., Faist J., Appl. Phys. Lett., 2002, 81, 2683; doi:10.1063/1.1512954.
8. Gendron L., Carras M., Huynh A., Ortiz V., Koeniguer C., Berger V., Appl. Phys. Lett., 2004, 85,
2824; doi:10.1063/1.1781731.
9. Giorgetta F.R., Baumann E., Hofstetter D., Manz C., Yang Q., Kohler K., Graf M., Appl. Phys. Lett.,
2007, 91, 111115; doi:10.1063/1.2784289.
10. Hofstetter D., Giorgetta F.R., Baumann E., Yang Q., Manz C., Kohler K., Appl. Phys. Lett., 2008,
93, 221106; doi:10.1063/1.3036897.
11. Elesin V.F., J. Exp. Theor. Phys., 1997, 85, 264; doi:10.1134/1.558273.
12. Elesin V.F., Kateev I.Yu., Podlivaev A.I., Semiconductors, 2000, 34, 1321; doi:10.1134/1.1325431.
13. Elesin V.F., J. Exp. Theor. Phys., 2002, 94, 794; doi:10.1134/1.1477905.
14. Belyaeva I.V., Golant E.I., Pashkovskii A.B., Semiconductors, 1997, 31, 103; doi:10.1134/1.1187090.
15. Gel’vich E.A., Golant E.I., Pashkovski A.B., Sazonov V.P., Tech. Phys. Lett., 1999, 25, 382;
doi:10.1134/1.1262490.
16. Golant E.I., Pashkovskii A.B., Semiconductors, 2000, 34, 327; doi:10.1134/1.1187981.
23704-10
http://dx.doi.org/10.1126/science.264.5158.553
http://dx.doi.org/10.1063/1.114005
http://dx.doi.org/10.1088/0034-4885/64/11/204
http://dx.doi.org/10.1063/1.1571653
http://dx.doi.org/10.1063/1.2203964
http://dx.doi.org/10.1063/1.3062981
http://dx.doi.org/10.1063/1.1512954
http://dx.doi.org/10.1063/1.1781731
http://dx.doi.org/10.1063/1.2784289
http://dx.doi.org/10.1063/1.3036897
http://dx.doi.org/10.1134/1.558273
http://dx.doi.org/10.1134/1.1325431
http://dx.doi.org/10.1134/1.1477905
http://dx.doi.org/10.1134/1.1187090
http://dx.doi.org/10.1134/1.1262490
http://dx.doi.org/10.1134/1.1187981
Active conductivity of plane two-barrier resonance tunnel structure
17. Tkach M.V., Makhanets O.M., Seti Ju.O., Dovganiuk M.M., Voitsekhivska O.M., Acta Phys. Pol. A,
2010, 117, 965.
18. Tkach N.V., Seti Yu.A., Low Temp. Phys., 2009, 35, 556; doi:10.1063/1.3170931.
Теорiя активної електронної провiдностi плоскої
двобар’єрної наносистеми як робочого елемента квантового
каскадного лазера чи детектора
М.В. Ткач, Ю.О. Сетi, В.О. Матiєк, О.М. Войцехiвська
Чернiвецький нацiональний унiверситет iменi Юрiя Федьковича,
58012 Чернiвцi, вул. Коцюбинського 2
У моделi прямокутних потенцiалiв i рiзних ефективних мас електрона в рiзних елементах плоскої
двобар’єрної резонансно-тунельної структури (ДБРТС) розвинута квантово-механiчна теорiя спект-
ральних параметрiв квазiстацiонарних станiв i провiдностi цiєї системи для випадку моноенерге-
тичного пучка електронiв, якi взаємодiють з електромагнiтним полем. Показано, що нано-ДБРТС
може слугувати окремим елементом або активним елементом каскадного лазера чи детектора.
На прикладi експериментально дослiджуваної наносистеми In0.53Ga0.47As/In0.52Al0.48As показано,
що у детекторному i лазерному режимах робота ДБРТС є оптимальною (з найбiльшою провiднiстю
при найменшому струмi збудження), коли вона працює на квантових переходах мiж найнижчими
квазiстацiонарними станами.
Ключовi слова: резонансно-тунельна структура, провiднiсть, квантовий лазер, квантовий
детектор
23704-11
http://dx.doi.org/10.1063/1.3170931
Introduction
Active conductivity of two-barrier RTS
Discussion of the results
|
| id | nasplib_isofts_kiev_ua-123456789-120004 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1607-324X |
| language | English |
| last_indexed | 2025-12-07T15:20:02Z |
| publishDate | 2011 |
| publisher | Інститут фізики конденсованих систем НАН України |
| record_format | dspace |
| spelling | Tkach, M.V. Seti, Ju.O. Matijek, V.O. Voitsekhivska, O.M. 2017-06-10T17:26:39Z 2017-06-10T17:26:39Z 2011 Active conductivity of plane two-barrier resonance tunnel structure as operating element of quantum cascade laser or detector / M.V. Tkach, Ju.O. Seti, V.O. Matijek, O.M. Voitsekhivska // Condensed Matter Physics. — 2011. — Т. 14, № 2. — С. 23704:1-11. — Бібліогр.: 18 назв. — англ. 1607-324X PACS: 73.21.Fg, 73.90.+f, 72.30.+q, 73.63.Hs DOI:10.5488/CMP.14.23704 arXiv:1106.5129 https://nasplib.isofts.kiev.ua/handle/123456789/120004 Within the model of rectangular potentials and different effective masses of electrons in different elements of plane two-barrier resonance tunnel structure there is developed a theory of spectral parameters of quasi-stationary states and active conductivity for the case of mono-energetic electronic current interacting with electromagnetic field. It is shown that the two-barrier resonance tunnel structure can be utilized as a separate or active element of quantum cascade laser or detector. For the experimentally studied In₀.₅₃Ga₀.₄₇As/In₀.₅₂Al₀.₄₈As nano-system it is established that the two-barrier resonance tunnel structure, in detector and laser regimes, optimally operates (with the biggest conductivity at the smallest exciting current) at the quantum transitions between the lowest quasi-stationary states. У моделi прямокутних потенцiалiв i рiзних ефективних мас електрона в рiзних елементах плоскої двобар’єрної резонансно-тунельної структури (ДБРТС) розвинута квантово-механiчна теорiя спектральних параметрiв квазiстацiонарних станiв i провiдностi цiєї системи для випадку моноенергетичного пучка електронiв, якi взаємодiють з електромагнiтним полем. Показано, що нано-ДБРТС може слугувати окремим елементом або активним елементом каскадного лазера чи детектора. На прикладi експериментально дослiджуваної наносистеми In₀.₅₃Ga₀.₄₇As/In₀.₅₂Al₀.₄₈As показано, що у детекторному i лазерному режимах робота ДБРТС є оптимальною (з найбiльшою провiднiстю при найменшому струмi збудження), коли вона працює на квантових переходах мiж найнижчими квазiстацiонарними станами. en Інститут фізики конденсованих систем НАН України Condensed Matter Physics Active conductivity of plane two-barrier resonance tunnel structure as operating element of quantum cascade laser or detector Теорiя активної електронної провiдностi плоскої двобар’єрної наносистеми як робочого елемента квантового каскадного лазера чи детектора Article published earlier |
| spellingShingle | Active conductivity of plane two-barrier resonance tunnel structure as operating element of quantum cascade laser or detector Tkach, M.V. Seti, Ju.O. Matijek, V.O. Voitsekhivska, O.M. |
| title | Active conductivity of plane two-barrier resonance tunnel structure as operating element of quantum cascade laser or detector |
| title_alt | Теорiя активної електронної провiдностi плоскої двобар’єрної наносистеми як робочого елемента квантового каскадного лазера чи детектора |
| title_full | Active conductivity of plane two-barrier resonance tunnel structure as operating element of quantum cascade laser or detector |
| title_fullStr | Active conductivity of plane two-barrier resonance tunnel structure as operating element of quantum cascade laser or detector |
| title_full_unstemmed | Active conductivity of plane two-barrier resonance tunnel structure as operating element of quantum cascade laser or detector |
| title_short | Active conductivity of plane two-barrier resonance tunnel structure as operating element of quantum cascade laser or detector |
| title_sort | active conductivity of plane two-barrier resonance tunnel structure as operating element of quantum cascade laser or detector |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/120004 |
| work_keys_str_mv | AT tkachmv activeconductivityofplanetwobarrierresonancetunnelstructureasoperatingelementofquantumcascadelaserordetector AT setijuo activeconductivityofplanetwobarrierresonancetunnelstructureasoperatingelementofquantumcascadelaserordetector AT matijekvo activeconductivityofplanetwobarrierresonancetunnelstructureasoperatingelementofquantumcascadelaserordetector AT voitsekhivskaom activeconductivityofplanetwobarrierresonancetunnelstructureasoperatingelementofquantumcascadelaserordetector AT tkachmv teoriâaktivnoíelektronnoíprovidnostiploskoídvobarêrnoínanosistemiâkrobočogoelementakvantovogokaskadnogolazeračidetektora AT setijuo teoriâaktivnoíelektronnoíprovidnostiploskoídvobarêrnoínanosistemiâkrobočogoelementakvantovogokaskadnogolazeračidetektora AT matijekvo teoriâaktivnoíelektronnoíprovidnostiploskoídvobarêrnoínanosistemiâkrobočogoelementakvantovogokaskadnogolazeračidetektora AT voitsekhivskaom teoriâaktivnoíelektronnoíprovidnostiploskoídvobarêrnoínanosistemiâkrobočogoelementakvantovogokaskadnogolazeračidetektora |