Non-perturbation theory of electronic dynamic conductivity for two-barrier resonance tunnel nano-structure
The non-perturbation theory of electronic dynamic conductivity for open two-barrier resonance tunnel structure is established for the first time within the model of rect angular potentials and different effective masses of electrons in the elements of nano-structure and the wave function linear over...
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Tkach, M.V. Seti, Ju.O. Voitsekhivska, O.M. 2017-06-10T19:24:06Z 2017-06-10T19:24:06Z 2011 Non-perturbation theory of electronic dynamic conductivity for two-barrier resonance tunnel nano-structure / M.V. Tkach, Ju.O. Seti, O.M. Voitsekhivska // Condensed Matter Physics. — 2011. — Т. 14, № 4. — С. 43702:1-10. — Бібліогр.: 24 назв. — англ. 1607-324X PACS: 73.21.Fg, 73.90.+f, 72.30.+q, 73.63.Hs DOI:10.5488/CMP.14.43702 arXiv:1202.4600 https://nasplib.isofts.kiev.ua/handle/123456789/120037 The non-perturbation theory of electronic dynamic conductivity for open two-barrier resonance tunnel structure is established for the first time within the model of rect angular potentials and different effective masses of electrons in the elements of nano-structure and the wave function linear over the intensity of electromagnetic field. It is proven that the results of the theory of dynamic co nductivity, developed earlier in weak signal approximation within the perturbation method, qualitatively and quantitatively correlate with the obtained results. The advantage of non-perturbation theory is that it can be extended to the case of electronic currents interacting with strong electromagnetic fields in open multi-shell reso nance tunnel nano-structures, as active elements of quantum cascade lasers and detectors. Вперше запропоновано непертурбацiйну теорiю електронної динамiчної провiдностi вiдкритої дво-бар’єрної резонансно-тунельної структури у моделi прямокутних потенцiалiв i рiзних ефективних мас електронiв у рiзних елементах наносистеми та з лiнiйною за напруженiстю електромагнiтного поля хвильовою функцiєю системи. Показано, що результати розвинутої ранiше теорiї динамiчної провiдностi у малосигнальному наближеннi (у межах теорiї збурень) якiсно i кiлькiсно корелюють з отриманими результатами. Переваги непертурбацiйної теорiї в тому, що вона може бути поширена на випадок взаємодiї потокiв електронiв з потужними електромагнiтними полями у вiдкритих багатошарових резонансно-тунельних наноструктурах, як активних елементах квантових каскадних лазерiв i детекторiв. en Інститут фізики конденсованих систем НАН України Condensed Matter Physics Non-perturbation theory of electronic dynamic conductivity for two-barrier resonance tunnel nano-structure Непертурбацiйна теорiя електронної динамiчної провiдностi двобар’єрної резонансно-тунельної наноструктури Article published earlier |
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| title |
Non-perturbation theory of electronic dynamic conductivity for two-barrier resonance tunnel nano-structure |
| spellingShingle |
Non-perturbation theory of electronic dynamic conductivity for two-barrier resonance tunnel nano-structure Tkach, M.V. Seti, Ju.O. Voitsekhivska, O.M. |
| title_short |
Non-perturbation theory of electronic dynamic conductivity for two-barrier resonance tunnel nano-structure |
| title_full |
Non-perturbation theory of electronic dynamic conductivity for two-barrier resonance tunnel nano-structure |
| title_fullStr |
Non-perturbation theory of electronic dynamic conductivity for two-barrier resonance tunnel nano-structure |
| title_full_unstemmed |
Non-perturbation theory of electronic dynamic conductivity for two-barrier resonance tunnel nano-structure |
| title_sort |
non-perturbation theory of electronic dynamic conductivity for two-barrier resonance tunnel nano-structure |
| author |
Tkach, M.V. Seti, Ju.O. Voitsekhivska, O.M. |
| author_facet |
Tkach, M.V. Seti, Ju.O. Voitsekhivska, O.M. |
| publishDate |
2011 |
| language |
English |
| container_title |
Condensed Matter Physics |
| publisher |
Інститут фізики конденсованих систем НАН України |
| format |
Article |
| title_alt |
Непертурбацiйна теорiя електронної динамiчної провiдностi двобар’єрної резонансно-тунельної наноструктури |
| description |
The non-perturbation theory of electronic dynamic conductivity for open two-barrier resonance tunnel structure is established for the first time within the model of rect angular potentials and different effective masses of electrons in the elements of nano-structure and the wave function linear over the intensity of electromagnetic field. It is proven that the results of the theory of dynamic co nductivity, developed earlier in weak signal approximation within the perturbation method, qualitatively and quantitatively correlate with the obtained results. The advantage of non-perturbation theory is that it can be extended to the case of electronic currents interacting with strong electromagnetic fields in open multi-shell reso nance tunnel nano-structures, as active elements of quantum cascade lasers and detectors.
Вперше запропоновано непертурбацiйну теорiю електронної динамiчної провiдностi вiдкритої дво-бар’єрної резонансно-тунельної структури у моделi прямокутних потенцiалiв i рiзних ефективних мас електронiв у рiзних елементах наносистеми та з лiнiйною за напруженiстю електромагнiтного поля хвильовою функцiєю системи. Показано, що результати розвинутої ранiше теорiї динамiчної провiдностi у малосигнальному наближеннi (у межах теорiї збурень) якiсно i кiлькiсно корелюють з отриманими результатами. Переваги непертурбацiйної теорiї в тому, що вона може бути поширена на випадок взаємодiї потокiв електронiв з потужними електромагнiтними полями у вiдкритих багатошарових резонансно-тунельних наноструктурах, як активних елементах квантових каскадних лазерiв i детекторiв.
|
| issn |
1607-324X |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/120037 |
| citation_txt |
Non-perturbation theory of electronic dynamic conductivity for two-barrier resonance tunnel nano-structure / M.V. Tkach, Ju.O. Seti, O.M. Voitsekhivska // Condensed Matter Physics. — 2011. — Т. 14, № 4. — С. 43702:1-10. — Бібліогр.: 24 назв. — англ. |
| work_keys_str_mv |
AT tkachmv nonperturbationtheoryofelectronicdynamicconductivityfortwobarrierresonancetunnelnanostructure AT setijuo nonperturbationtheoryofelectronicdynamicconductivityfortwobarrierresonancetunnelnanostructure AT voitsekhivskaom nonperturbationtheoryofelectronicdynamicconductivityfortwobarrierresonancetunnelnanostructure AT tkachmv neperturbaciinateoriâelektronnoídinamičnoíprovidnostidvobarêrnoírezonansnotunelʹnoínanostrukturi AT setijuo neperturbaciinateoriâelektronnoídinamičnoíprovidnostidvobarêrnoírezonansnotunelʹnoínanostrukturi AT voitsekhivskaom neperturbaciinateoriâelektronnoídinamičnoíprovidnostidvobarêrnoírezonansnotunelʹnoínanostrukturi |
| first_indexed |
2025-11-25T22:33:24Z |
| last_indexed |
2025-11-25T22:33:24Z |
| _version_ |
1850566836810153984 |
| fulltext |
Condensed Matter Physics, 2011, Vol. 14, No 4, 43702: 1–10
DOI: 10.5488/CMP.14.43702
http://www.icmp.lviv.ua/journal
Non-perturbation theory of electronic dynamic
conductivity for two-barrier resonance
tunnel nano-structure
M.V. Tkach∗, Ju.O. Seti, O.M. Voitsekhivska
Chernivtsi National University, 2 Kotsyubinsky Str., 58012 Chernivtsi, Ukraine
Received August 29, 2011, in final form November 9, 2011
The non-perturbation theory of electronic dynamic conductivity for open two-barrier resonance tunnel struc-
ture is established for the first time within the model of rectangular potentials and different effective masses of
electrons in the elements of nano-structure and the wave function linear over the intensity of electromagnetic
field. It is proven that the results of the theory of dynamic conductivity, developed earlier in weak signal approx-
imation within the perturbation method, qualitatively and quantitatively correlate with the obtained results. The
advantage of non-perturbation theory is that it can be extended to the case of electronic currents interacting
with strong electromagnetic fields in open multi-shell resonance tunnel nano-structures, as active elements of
quantum cascade lasers and detectors.
Key words: resonance tunnel nano-structure, conductivity, non-perturbation theory
PACS: 73.21.Fg, 73.90.+f, 72.30.+q, 73.63.Hs
1. Introduction
The experimental produce of nano-lasers and nano-detectors and, further, quantum cascade
lasers and detectors [1–3] stimulates the intensive development of the theory of dynamic conduc-
tivity for nano-heterosystems as active elements of these unique devices. In spite of the twenty
years period of investigating the interaction between electromagnetic field and electronic currents
in open nano-structures, the respective theory is far from being completed. One of the reasons
is the mathematical problems arising at the quantum mechanical research of physical processes
caused by the interaction of quasi-particles with classic and quantized (phonons) fields in open
nano-structures.
The theory of electronic conductivity for the two- and three-barrier resonance tunnel structures
(RTS) [4–10] is rather complicated and mathematically sophisticated even without taking into
account the dissipative processes (scattering at the phonons, impurities, imperfections). Therefore,
the maximally simplified model is used in the above mentioned and other papers [11–13]: δ-like
approximation of potential barriers for the electrons and weak signal approximation equivalent to
the first order of perturbation theory (PT) over the intensity of electromagnetic field interacting
with electronic current in RTS.
We must note that δ-like approximation of potential barriers essentially simplifies the model of
nano-structure. Herein, the electron is automatically characterized by the unitary effective mass
within the whole system, which permits to calculate the RTS dynamic conductivity [4–13] using
the PT iteration method and in such a way, quit the frames of linear approximation over the field
intensity.
Further, in references [14, 15] it was shown that δ-like approximation of potential barriers
correctly described the qualitative properties of spectral parameters of quasi-stationary states
of electrons and the dynamic conductivity of nano-structures but the magnitudes of resonance
energies were overestimated by tens per cent and resonance widths by ten times with respect to
∗
E-mail: ktf@chnu.edu.ua
c© M.V. Tkach, Ju.O. Seti, O.M. Voitsekhivska, 2011 43702-1
http://dx.doi.org/10.5488/CMP.14.43702
http://www.icmp.lviv.ua/journal
M.V. Tkach, Ju.O. Seti, O.M. Voitsekhivska
their magnitudes in a more realistic model of rectangular potentials and different effective masses
of electron in the elements of nano-structure. It was displayed that in any RTS, the δ-barrier
model strongly underestimated the electrons life times in all quasi-stationary states and, thus, the
magnitude of the dynamic conductivity became the orders smaller even for the structures with
weak electromagnetic field.
The theory of dynamic conductivity established in references [16–19] for the open two- and
three-barrier RTS is based, as a rule, on a more realistic model but it is so complicated compared
to the δ-barrier model that it is practically impossible to leave the framework of weak signal
approximation. Nevertheless, the development of experimental capabilities makes the problem of
strong interaction of electronic currents and electromagnetic field in RTS more and more urgent.
Therefore, it is necessary to develop the non-perturbation theory (NPT) of conductivity for
open RTS where the intensity of electromagnetic field would not play such a critical role as in PT.
The motivation for the positive expectations regarding the NPT existed because the solution of
complete Schrodinger equation with Hamiltonian describing the interaction between electrons and
varying in time electromagnetic field was known [20]. However, in spite of the known analytical
expression for the exact wave function, the theory of conductivity for the open RTSs was not
successfully developed.
The possible approach to the solution of this problem for the dynamic conductivity of the two-
barrier RTS is proposed in our paper for the first time. We develop the NPT for the electronic
dynamic conductivity using the wave function which is the exact solution of complete Schrodinger
equation in the linear approximation over the field. Being convinced that the results obtained in
the first order of PT for electronic conductivity correlate with the results of the herein developed
NPT, the established approach can be used in developing a general theory of electronic current
interacting with strong electromagnetic field in open multi-shell nano-structures, being the active
elements of quantum cascade lasers and detectors.
2. Hamiltonian of the system. Finding the wave function from the complete
Schrodinger equation
The open two-barrier RTS is studied in the Cartesian coordinate system with OZ axis perpen-
dicular to the planes of nano-structure, figure 1. The small difference between the lattice constants
of the nano-structure wells and barriers makes it possible to use the effective masses
m(z) = m0
2
∑
p=0
[θ(z − z2p−1)− θ(z − z2p)] +m1
1
∑
p=0
[θ(z − z2p)− θ(z − z2p+1)] (1)
and rectangular potentials
U(z) = U
1
∑
p=0
[θ(z − z2p)− θ(z − z2p+1)]. (2)
Here θ(z) is Heaviside step function; z−1 → −∞, z4 → +∞.
It is assumed that mono-energetic electron current with the energy (E), density of current
(J+
0 ∼
√
E) and concentration (n0) moving perpendicularly to the planes of two-barrier RTS falls
at it from the left side. The electronic movement can be considered as one-dimensional (
−→
k || = 0).
According to the numeric evaluations, the velocity within the nano-structure is by 3–4 orders
smaller than the velocity outside. Thus, the interaction between electrons and electromagnetic
field with frequency (ω) and intensity of electric field (ǫ) is essential only within the two-barrier
RTS and can be neglected outside it.
The electron wave function has to satisfy the complete Schrodinger equation
i~
∂Ψ(E,ω, z, t)
∂t
= [H0(z) +H1(z, t)]Ψ(E,ω, z, t), (3)
43702-2
Non-perturbation theory of electronic dynamic conductivity
Figure 1. Energy scheme for the electrons and geometry of two-barrier RTS.
where
H0(z) = −~
2
2
∂
∂z
1
m(z)
∂
∂z
+ U(z) (4)
is the Hamiltonian of the electron without interaction with the field.
The electron interaction with the electromagnetic field varying in time, is described by the
Hamiltonian
H1(z, t) = −2eǫz[θ(z)− θ(z − z3)] cosωt. (5)
The both linearly independent exact solutions of complete Schrodinger equation with Hamil-
tonian H0(z) in the potential well are known: exp(±ikz − iω0t) [17, 19], ω0 = E~
−1. The both
linearly independent exact solutions of equation (3) with Hamiltonian H(z, t) = H0(z) +H1(z, t),
taking into account the electron-electromagnetic field interaction in linear approximation over the
electric intensity are also known:
exp
[
i
(
±kz − ω0t+
2eǫz
~ω
sinωt± 2eǫk
mω2
cosωt
)]
,
where k is electron quasi-momentum [20]. Thus, using the exact solutions of equation (3) for the
case of two-barrier RTS, the electron wave function is written as
Ψ(E,ω, z, t) = Ψ0(E, z, t)θ(−z)+
3
∑
p=1
Ψp(E,ω, z, t)[θ(z−zp−1)−θ(z−zp)]+Ψ4(E, z, t)θ(z−z3). (6)
In the outer media of two-barrier RTS, where the interaction with electromagnetic field is
neglected, the wave functions are
Ψ0(E, z, t) = (a0e
ikz + b0e
−ikz)e−iω0t, (7)
Ψ4(E, z, t) = a4e
ikz−iω0t, (8)
where
k = ~
−1
√
2m0E .
Here it is taken into account that the mono-energetic electronic current impinges at RTS from
the left hand side, since in the left hand media there is both a falling and a reflected wave while
in the right one there is the wave moving towards the infinity only.
Within the two-barrier RTS, where the electron-electromagnetic field interaction is essential,
the wave function is found as linear combinations of eigen wave functions of the Hamiltonian H(z, t)
Ψ1(E,ω, z, t) =
[
a1f
→
1 (E,ω, z, t)eχz + b1f
←
1 (E,ω, z, t)e−χz
]
e−iω0t, (9)
Ψ2(E,ω, z, t) =
[
a2f
→
2 (E,ω, z, t)eikz + b2f
←
2 (E,ω, z, t)e−ikz
]
e−iω0t, (10)
Ψ3(E,ω, z, t) =
[
a3f
→
1 (E,ω, z, t)eχz + b3f
←
1 (E,ω, z, t)e−χz
]
e−iω0t, (11)
43702-3
M.V. Tkach, Ju.O. Seti, O.M. Voitsekhivska
where
f⇄
p (E,ω, z, t) = exp(iα sinωt± iβp cosωt) , (p = 1, 2) (12)
α =
2eǫz
~ω
; β1 =
2ieǫχ
m1ω2
; β2 =
2eǫk
m0ω2
; χ = ~
−1
√
2m1(U − E) . (13)
Further, using the known [21] expansion of exponential functions into Fourier range over all the
harmonics, the functions f⇄
p (E,ω, z, t) are written as
f⇄
p (E,ω, z, t) =
∞
∑
n1=−∞
∞
∑
n2=−∞
i±n2jn1
(α)jn2
(βp)e
i(n1∓n2)ωt , (14)
where jn are the cylindrical Bessel functions of the whole order.
The quantum transitions, accompanied by energy radiation or absorption, occur between elec-
tron quasi-stationary states with odd number under the effect of electromagnetic field. The most
intensive transitions arise between the neighbouring resonance states. Thus, the expression (14) for
the functions f⇄
p (E,ω, z, t) can be essentially simplified by leaving only zero and first harmonics
from the whole infinite range. Then, in a one-mode approximation for the functions f⇄
p (E,ω, z, t),
the following expression is obtained
f⇄
p (E,ω, z, t) = C⇄
p (E,ω, z) +
{
Dp(E,ω, z) + i
[
F⇄
p (E,ω, z) +G⇄
p (E,ω, z)
]}
e−iωt
+
{
−Dp(E,ω, z) + i
[
F⇄
p (E,ω, z)−G⇄
p (E,ω, z)
]}
eiωt, (15)
with
C⇄
p (E,ω, z) = j0(α)j0(βp) + 2
∞
∑
n1=1
[j4n1
(α)j4n1
(βp)− j4n1−2(α)j4n1−2(βp)], (16)
Dp(E,ω, z) =
∞
∑
n1=1
{j4n1−1(α)[j4n1
(βp) + j4n1−2(βp)]− j4n1−3(α)[j4n1−2(βp) + j4n1−4(βp)]}, (17)
G⇄
p (E,ω, z) = ±
∞
∑
n1=1
{j4n1
(α)[j4n1−1(βp)−j4n1+1(βp)]+j4n1−2(α)[j4n1−1(βp)−j4n1−3(βp)]}. (18)
Within the framework of the linear Hamiltonian over the field intensity (ǫ), the rather compli-
cated formulas (16)–(18) correctly define the electron wave function in a one-mode approximation
independently of the intensity magnitude. In the case of small intensity when the condition
min[α(E), β1(E), β2(E)] ≪ 1 (19)
is fulfilled, expanding the Bessel functions into a series and preserving the linear term over the
field, the expressions for coefficients are simplified
C⇄
p = 1; Dp = −α/2; F⇄
p = ±βp/2; G⇄
p = 0. (20)
Thus, the wave function is also obtained in a convenient analytical form
Ψ(E,ω, z, t) = (a0e
ikz + b0e
−ikz)e−iω0tθ(−z) + a4e
ikze−iω0tθ(z − z3)
+
3
∑
p=1
e−iω0t
{
ape
Kpz
[
1 +
1
2
(iβp + α)eiωt +
1
2
(iβp − α)e−iωt
]
+ bpe
−Kpz [1− 1
2
(iβp − α)eiωt − 1
2
(iβp + α)e−iωt]
}
[θ(z − zp−1)− θ(z − zp)] ,(21)
where
K1 = K3 = χ; K2 = ik.
Using the obtained wave function Ψ(E,ω, z, t) one can perform the calculation of the perme-
ability coefficient for the two-barrier RTS, obtaining the spectral parameters of electron quasi-
stationary states and active dynamic conductivity of nano-structure.
43702-4
Non-perturbation theory of electronic dynamic conductivity
3. Permeability coefficient and dynamic conductivity of two -barrier RTS
Now, we can find the dynamic conductivity caused by the quantum transitions of electrons from
the quasi-stationary state with the energy E into the states with the energies E+~ω or E−~ω due
to the effect of the periodical electromagnetic field with intensity ǫ and frequency ω. Therefore,
we have to define the densities of electron currents: J(E + ~ω) and J(E − ~ω), flowing out of
the RTS with the respective energies. In such approach, the complete wave function is written as
linear combination of wave functions describing the electron states with the energies E, E + ~ω
and E − ~ω. The functions Ψ(E ± ~ω, ω, z, t) can be obtained from the expression (21) for the
already known function Ψ(E,ω, z, t) using the substitution E → E ± ~ω. Then,
Ψ(E ± ~ω, ω, z, t) = b±0 e
−ik±ze−i(ω0±ω)tθ(−z) + a±4 e
ik±ze−i(ω0±ω)tθ(z − z3)
+
3
∑
p=1
e−iω0t
{
a±p e
K±
p
z
[
e∓iωt +
1
2
(iβ±p ± α) +
1
2
(iβ±p ∓ α)e∓2iωt
]
+ b±p e
−K±
p
z
[
e∓iωt − 1
2
(iβ±p ∓ α)− 1
2
(iβ±p ± α)e∓2iωt
]}
× [θ(z − zp−1) + θ(z − zp)] , (22)
where
K±1 = K±3 = χ± = ~
−1
√
2m1[U − (E ± ~ω)] ; K±2 = ik± = i~−1
√
2m0(E ± ~ω) ; (23)
β±1 =
2ieǫχ±
m1ω2
; β±2 =
2eǫk±
m0ω2
. (24)
The mono-energetic electron current falls at RTS with the energy E = ~ω0. Under the effect
of electromagnetic field there occur quantum transitions into higher (with the energy E + ~ω) or
lower (with the energy E − ~ω) electron quasi-stationary states, the currents from which produce
the dynamic conductivity of a nano-structure. In order to describe this physical process correctly,
we have to leave the terms containing only the first harmonic (±ω) in formula (22) for the wave
function. Thus,
Ψ(E ± ~ω, ω, z, t) = b±0 e
−ik±ze−i(ω0±ω)tθ(−z) + a±4 e
ik±ze−i(ω0±ω)tθ(z − z3)
+
3
∑
p=1
e−i(ω0±ω)t(a±p e
K±
p
z + b±p e
−K±
p
z)[θ(z − zp−1) + θ(z − zp)]. (25)
The complete wave function Φ(E,E − ~ω,E + ~ω, ω, z, t) can be written at the base of super-
position (linear combination) of wave functions (21) and (25). It depends on electron energies E,
E − ~ω, E + ~ω and electromagnetic field frequency ω. For a convenient presentation, it is further
written as Φ(E,ω, z, t).
Φ(E,ω, z, t) = Φ0(E,ω, z, t)θ(−z)
+
3
∑
p=1
Φp(E,ω, z, t)[θ(z − zp−1)− θ(z − zp)] + Φ4(E,ω, z, t)θ(z − z3), (26)
43702-5
M.V. Tkach, Ju.O. Seti, O.M. Voitsekhivska
where
Φ0(E,ω, z, t) = e−iω0t
(
A0e
ikz +B0e
−ikz +B+
0 e−ik
+ze−iωt +B−0 e−ik
−zeiωt
)
, (27)
Φp(E,ω, z, t) = e−iω0t
{
Ape
Kpz
[
1 +
1
2
(iβp + α)eiωt +
1
2
(iβp − α)e−iωt
]
+Bpe
−Kpz
[
1− 1
2
(iβp − α)eiωt − 1
2
(iβp + α)e−iωt
]
+
(
A+
p e
K+
p
z +B+
p e
−K+
p
z
)
e−iωt +
(
A−p e
K−
p
z +B−p e−K
−
p
z
)
eiωt
}
, (28)
Φ4(E,ω, z, t) = e−iω0t
(
A4e
ikz +A+
4 e
ik+ze−iωt +A−4 e
ik−zeiωt
)
. (29)
The two-barrier RTS under study is an open one, consequently the wave function Φ(E,ω, z, t)
at any moment of time has to satisfy the normality condition
∞
∫
−∞
Φ∗(k′, ω, z, t)Φ(k, ω, z, t)dz = δ(k − k′). (30)
The wave function and its density of current should be continuous at all nano-structure interfaces
Φp(E,ω, zp, t) = Φp+1(E,ω, zp, t);
∂Φp(E,ω, z, t)
m0(1) ∂z
∣
∣
∣
∣
z=zp
=
∂Φp+1(E,ω, z, t)
m1(0) ∂z
∣
∣
∣
∣
z=zp
. (31)
The coefficients at zero harmonics: B0, Ap, Bp, A4 are definitely obtained from the system of ho-
mogeneous equations (31) through the coefficient A0. This is, in turn, related to the density of start
electron current impinging at RTS: J+
0 = en0
√
2Em−10 |A0|2, where n0 is the concentration of elec-
trons in this current, e is electron charge. Coefficients at the first harmonics: B±0 , A±p , B
±
p , A±4 are
defined through the now known coefficients at zero harmonics of function Φ(E,ω, z, t). According
to the quantum mechanics [22], the permeability coefficient for the two-barrier RTS is
D(E) = |A4/A0|2. (32)
It is well known [14, 23] that the permeability coefficient D(E) determines the spectral param-
eters: resonance energies (En) and resonance widths (Γn) of quasi-stationary states of electrons.
The positions of D(E) maxima in the energy scale fix the resonance energies, while their widths
at the halves of maximal heights D(En) fix the resonance widths of these quasi-stationary states.
According to electrodynamics [24], in a quasi-static approximation, the energy (E), got by
the electrons from the field during the period T = 2π/ω, is related to the real part of dynamic
conductivity (σ)
E =
4πz3ǫ
2
ω
σ(E,ω). (33)
The same energy is defined by the electron currents flowing out of the nano-structure through
the densities of currents of uncoupling electrons
E =
~ωT
e
{[J(E + ~ω, z3)− J(E + ~ω, 0)]− [J(E − ~ω, z3)− J(E − ~ω, 0)]} . (34)
According to quantum mechanics [22], the density of current is defined by the wave function
Φ(E,ω, z, t)
J(E, z) =
ie~n0
2m(z)
[
Φ(E,ω, z, t)
∂Φ∗(E,ω, z, t)
∂z
− Φ∗(E,ω, z, t)
∂Φ(E,ω, z, t)
∂z
]
. (35)
Thus, as a result of analytical calculations, the final expression for the dynamic conductivity of
two-barrier RTS is obtained:
σ(E,ω) =
~
2ωn0
2z3m0ǫ2
[k+(|B+
0 |2 + |A+
4 |2)− k−(|B−0 |2 + |A−4 |2)] . (36)
43702-6
Non-perturbation theory of electronic dynamic conductivity
It is evident that in linear approximation over the electromagnetic field intensity, the coefficients
B±0 , A±4 are also linear. Consequently, in this approximation, (the same in PT [4–13, 15–19]) the
dynamic conductivity is independent of ǫ.
The calculations of spectral parameters of electron quasi-stationary states and dynamic
conductivity of nano-structure were performed at the base of the developed NPT for
In0.52Al0.48As/In0.53Ga0.47As two-barrier RTS with physical parameters: m0 = 0.046 me, m1 =
0.089 me, U = 516 meV, n0 = 1016 cm−3 and typical geometrical ones: b = 10.8 nm, ∆+ +∆− =
6 nm, ∆+ = 2 ÷ 4.5 nm. The same calculations were performed in the frames of the previously
developed PT [19] in the first order over the field intensity for comparison.
The results obtained for positive or negative conductivities σ(E,Ω = ~ω) produced by the quan-
tum transitions of electrons interacting with electromagnetic field in the processes of absorption
(1 → 2) or radiation (2 → 1) are shown in figures 2 (a), (b) and figures 2 (d), (e), respectively.
The spectral parameters (resonance energies and resonance widths) of electron quasi-stationary
states, defined from the permeability coefficient [14, 23], do not depend on the method of calcu-
lation. Their magnitudes, obtained for the two-barrier RTS with b=10.8 nm, ∆+ = ∆−=3 nm
are presented in figures 2 (a), (b), (d), (e). The figures prove that the functions σ(E,Ω = const)
and σ(E = const,Ω) are of the shape of Lorentz curve in both methods (NPT and PT). How-
ever, herein, it is clear that the magnitudes σ(E,Ω) in PT are overestimated in the detector and
underestimated in laser quantum transitions at any E and ω, comparing to the exacter NPT.
In the both methods, the positive conductivity maximum: σ12 = max σ12(E,Ω), caused by
the detector (accompanied by electromagnetic wave absorption) quantum transitions between the
first and second quasi-stationary states is calculated at the plane (E, Ω) in the point: E = E1,
Ω = Ω12 = E2 − E1. For the two-barrier RTS under study we obtained: σPT
12 =20557 S/cm, and
σNPT
12 =16740 S/cm. It means that the PT gives the magnitude at 22.8 % bigger than the NPT.
It is also shown that in the both methods, the widths (ΓΩ) of σ(E1,Ω) functions are almost
coinciding and in Ω scale coincide to the resonance width of the second quasi-stationary state
(ΓNPT
Ω ≈ ΓPT
Ω ≈ Γ2). The widths (ΓE) of σ(E,Ω12) functions coincide in E scale and with resonance
width of the first quasi-stationary state (ΓNPT
E ≈ ΓPT
E ≈ Γ1).
In the both methods, the negative conductivity minimum: σ21 = minσ21(E,Ω), caused by the
laser (accompanied by electromagnetic wave radiation) quantum transitions between the second
and first quasi-stationary states is placed at the plane (E, Ω) in the point: E = E2, Ω = Ω21 = E2−
E1. For the two-barrier RTS under study, we obtained: σPT
21 = −41300 S/cm, σNPT
21 = −49990 S/cm.
Contrary to the positive conductivity, the widths of the negative one in both scales (E, Ω) are
close to each other and to the resonance width Γ1, i.e. ΓNPT
E ≈ ΓPT
E ≈ ΓNPT
Ω ≈ ΓPT
Ω ≈ Γ1.
In figures 2 (c), (f) the dependences of maximal (minimal) magnitudes of positive (negative)
conductivities on the width of the outer barrier (∆+) at a fixed sum width of both barriers (∆+ +
∆− = 6 nm) are shown for NPT and PT. It is clear that the functions in the transitions 1 ⇆ 2 and
2 ⇆ 3 are located close to each other in both methods not only qualitatively but also quantitatively.
The insertions in figures 2 (c), (f) prove that the errors (η = 1 − σPT/σNPT,%) of conductivities
calculated within PT with respect to NPT weakly depend on the relationship between the widths
of both barriers.
Finally, we should note that in the both methods the dependences of σ on ∆+ at ∆− +∆+ =
∆ = const are not only of the same shape (figures 2 (c), (f)) but their magnitudes at any ∆+ are
close to each other. The behaviour of the function is clearly explained by physical considerations.
In reference [19] it was proven that the magnitude of dynamic conductivity is proportional to the
electron life times in those quasi-stationary states between which the quantum transitions occur due
to the interaction with electromagnetic field. If the difference between the widths of both barriers
|∆+−∆−| is big, the life times in all quasi-stationary states are small because the electrons rapidly
quit the two-barrier RTS through the thinner barrier. If the barriers widths correlate, the life times
in all quasi-stationary states increase, approaching the maximal magnitude at ∆− = ∆+ = ∆/2.
From figures 2 (c), (f) one can see that σ dependence on ∆+ is qualitatively similar to the above
described evolution of life times with the only difference that maxσ(∆+) is approached not at
∆− = ∆+ = ∆/2 but at ∆+
0 > ∆/2. This is also clear, because, contrary to the electron life times
in quasi-stationary states independent on the number of electrons in two-barrier RTS (in the frames
43702-7
M.V. Tkach, Ju.O. Seti, O.M. Voitsekhivska
of the model neglecting electron-electron interaction), the conductivity depends on this factor due
to the interaction with electromagnetic field. Thus, at the increase of input barrier width (∆−), the
number of electrons reflected from RTS increases. Therefore, at ∆− = ∆+∆+
0 < ∆/2 the number
of electrons in the RTS is bigger than at ∆− = ∆+ = ∆/2 and, consequently, σ(∆+
0 ) > σ(∆/2).
120 121 122 123 124
0
5
10
15
20
NPT
0.9 meV
PT
0.91 meV
NPT
PT
12
, meV
(E
1
,
)·
1
0
3
,
S
/c
m
a)
E
2
=162.2 meV
2
=0.9 meV
-50
-40
-30
-20
-10
0
121.8 122.0 122.2 122.4
PT
0.09 meV
NPT
0.09 meV
1
=0.09 meV(E
2
,
)·
1
0
3
,
S
/c
m
NPT
PT
12
, meV
d)
39.8 40.0 40.2 40.4
0
5
10
15
20
NPT
PT
NPT
E
0.09 meV
PT
E
0.09 meVE
1
=40.1 meV
1
=0.09 meV
E, meV
(E
,
1
2
)·
1
0
3
,
S
/c
m
E
1
b)
-50
-40
-30
-20
-10
0
161.8 162.0 162.2 162.4 162.6
NPT
PT
PT
E
0.1 meV
NPT
E
0.1 meV
2
=0.9 meV(E
,
1
2
)·
1
0
3
,
S
/c
m
E, meVE
2
e)
2.0 2.5 3.0 3.5 4.0 4.5
0
1
2
3
0
0.1
0.2
0.3
2 3 4
0
0.2
0.4
+
, nm
,
%
2 3
1 2
2 3
2
3
·1
0
4
,
S
/c
m
1 2
1
2
·1
0
4
,
S
/c
m
+
, nm
c)
-6
-4
-2
0
2.0 2.5 3.0 3.5 4.0 4.5
-0.6
-0.4
-0.2
0
2 3 4
0
0.2
0.4
+
, nm
,
%
3 2
2 1
3
2
·1
0
4
,
S
/c
m
3 2
2
1
·1
0
4
,
S
/c
m
+
, nm
2 1
f)
Figure 2. Dependences of maximal magnitudes of positive (a, b, c) and negative (d, e, f) con-
ductivities σ on electromagnetic field energy Ω = ~ω (a, d) and electron energy E (b, e) in
two-barrier RTS at ∆
+
= ∆
−=3 nm. Dependences of maximal magnitudes of positive and neg-
ative conductivities on the relationship between the both barrier widths (c, f) at ∆+
+∆
−= 6 nm
obtained within the non-perturbation theory (bold solid and dashed curves) and perturbation
theory (thin solid and dashed curves).
43702-8
Non-perturbation theory of electronic dynamic conductivity
4. Conclusions
1. The non-perturbation theory of active dynamic conductivity for the open two-barrier RTS,
preserving the terms linear over the electromagnetic field in an electron wave function, is proposed
for the first time.
2. It is shown that the properties of positive and negative conductivities of two-barrier RTS,
shown earlier within the linear approximation over the field perturbation theory, in the so-called
weak signal approximation are not only qualitatively similar but quantitatively correlate to the
results of a more exact non-perturbation theory proposed.
3. The developed non-perturbation theory of dynamic conductivity for the two-barrier nano-
structure can be used for the multi-shell RTS and generalized for the physically and technically
important case of electron currents interacting with strong electromagnetic fields in quantum cas-
cade lasers and 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. Golant E.I., Pashkovskii A.B., Phys. Tehn. Poluprovodnikov, 1994, 28, 954 (in Russian).
5. Belyaeva I.V., Golant E.I., Pashkovskii A.B., Semiconductors, 1997, 31, 103; doi:10.1134/1.1187090.
6. Pashkovskii A.B., JETP Lett., 2005, 82, 210; doi:10.1134/1.2121816.
7. Elesin V.F., J. Exp. Theor. Phys., 1997, 85, 264; doi:10.1134/1.558273.
8. Elesin V.F., J. Exp. Theor. Phys., 2002, 94, 794; doi:10.1134/1.1477905.
9. Elesin V.F., J. Exp. Theor. Phys., 2005, 100, 116; doi:10.1134/1.1866204.
10. Tkach M.V., Makhanets O.M., Seti Ju.O., Dovganiuk M.M., Voitsekhivska O.M., J. Phys. Stud., 2010,
14, 3703 (in Ukrainian).
11. Diez E., Sanchez A., Dominguez-Adame F., Phys. Lett. A, 1996, 215, 103;
doi:10.1016/0375-9601(96)00202-2.
12. Rapedius K., Korsch H.J., J. Phys. A: Math. Theor., 2009, 42, 425301;
doi:10.1088/1751-8113/42/42/425301.
13. Kraynov V.P., Ishkhanyan H.A., Phys. Scr., 2010, 140, 014052;
doi:10.1088/0031-8949/2010/T140/014052.
14. Tkach N.V., Seti Yu.A., Low Temp. Phys., 2009, 35, 556; doi:10.1063/1.3170931.
15. Tkach M.V., Seti Yu.O., Ukr. J. Phys., 2010, 55, 798.
16. Tkach M.V., Makhanets O.M., Seti Ju.O., Dovganiuk M.M., Voitsekhivska O.M., Acta Phys. Pol. A,
2010, 117, 965.
17. Tkach N.V., Seti Ju.A., Semiconductors, 2011, 45, 376; doi:10.1134/S1063782611030195.
18. Seti Ju., Tkach M., Voitsekhivska O., Condens. Matter Phys., 2011, 14, 13701;
doi:10.5488/CMP.14.13701.
19. Tkach M.V., Seti Ju.O., Matijek V.O., Voitsekhivska O.M., Condens. Matter Phys., 2011, 14, 23704;
doi:10.5488/CMP.14.23704.
20. Golant E.I., Pashkovskii A.B., Semiconductors, 2000, 34, 327; doi:10.1134/1.1187981.
21. Jeffreys H., Swirles B., Methods of Mathematical Physics (3rd ed.). Cambridge University Press, 2000.
22. Landau L.D., Lifshitz E.M., Quantum Mechanics: Non-Relativistic Theory. Pergamon Press, 1977.
23. Tkach N.V., Seti Ju.A., Phys. Solid State, 2011, 53, 590; doi:10.1134/S1063783411030322.
24. Landau L.D., Lifshitz E.M., Pitaevskii L.P., Electrodynamics of Continuous Media (2nd ed.).
Butterworth-Heinemann, 1984.
43702-9
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/10.1134/1.1187090
http://dx.doi.org/10.1134/1.2121816
http://dx.doi.org/10.1134/1.558273
http://dx.doi.org/10.1134/1.1477905
http://dx.doi.org/10.1134/1.1866204
http://dx.doi.org/10.1016/0375-9601(96)00202-2
http://dx.doi.org/10.1088/1751-8113/42/42/425301
http://dx.doi.org/10.1088/0031-8949/2010/T140/014052
http://dx.doi.org/10.1063/1.3170931
http://dx.doi.org/10.1134/S1063782611030195
http://dx.doi.org/10.5488/CMP.14.13701
http://dx.doi.org/10.5488/CMP.14.23704
http://dx.doi.org/10.1134/1.1187981
http://dx.doi.org/10.1134/S1063783411030322
M.V. Tkach, Ju.O. Seti, O.M. Voitsekhivska
Непертурбацiйна теорiя електронної динамiчної провiдностi
двобар’єрної резонансно-тунельної наноструктури
М.В.Ткач, Ю.О.Сетi, О.М.Войцехiвська
Чернiвецький нацiональний унiверситет iм. Ю.Федьковича, вул. Коцюбинського, 2, 58012 Чернiвцi,
Україна
Вперше запропоновано непертурбацiйну теорiю електронної динамiчної провiдностi вiдкритої дво-
бар’єрної резонансно-тунельної структури у моделi прямокутних потенцiалiв i рiзних ефективних
мас електронiв у рiзних елементах наносистеми та з лiнiйною за напруженiстю електромагнiтного
поля хвильовою функцiєю системи. Показано, що результати розвинутої ранiше теорiї динамiчної
провiдностi у малосигнальному наближеннi (у межах теорiї збурень) якiсно i кiлькiсно корелюють
з отриманими результатами. Переваги непертурбацiйної теорiї в тому, що вона може бути поши-
рена на випадок взаємодiї потокiв електронiв з потужними електромагнiтними полями у вiдкритих
багатошарових резонансно-тунельних наноструктурах, як активних елементах квантових каскадних
лазерiв i детекторiв.
Ключовi слова: резонансно-тунельна наноструктура, провiднiсть, непертурбацiйна теорiя
43702-10
Introduction
Hamiltonian of the system. Finding the wave function from the complete Schrodinger equation
Permeability coefficient and dynamic conductivity of two-barrier RTS
Conclusions
|