Transmission of wave packets through open mesoscopic systems
Tunneling of the wave packets having rectangular and Gaussian forms through a quantum diode is investigated. By using the potential of the system the S-matrix for this structure is obtained and the analytic expressions describing the form of the transmitted pulse are calculated. These analytic solut...
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irk-123456789-1071602016-10-15T03:01:40Z Transmission of wave packets through open mesoscopic systems Ivanov, N. Skalozub, V. Section E. Phase Transitions and Diffusion Processes in Condensed Matter and Gases Tunneling of the wave packets having rectangular and Gaussian forms through a quantum diode is investigated. By using the potential of the system the S-matrix for this structure is obtained and the analytic expressions describing the form of the transmitted pulse are calculated. These analytic solutions are compared with the numeric ones. An excellent coincidence is detected. The time-delay for the Gaussian pulse tunneling through the quantum dot is investigated in details. На основе решения уравнения Липпманна-Швингера для возмущенного потенциала рассчитывается матрица рассеяния туннельного диода. Используя формализм S-матрицы и модифицированный метод седловой точки, аналитически рассчитывается форма волнового пакета, выходящего из квантовой точки при подаче на нее пакетов гауссовой и прямоугольной форм. Найдена и проиллюстрирована зависимость времени задержки сигнала гауссовой формы при туннелировании для случая квантовой точки. На основі розв’язку рівняння Ліппманна-Швінгера для збуреного потенціалу розраховано матрицю розсіювання тунельного діоду. Користуючись формалізмом S-матриці та модифікованого методу сідлової точки, аналітично розраховується форма хвильового пакету, що виходить з квантової точки при подаванні на неї пакетів гауссової та прямокутної форм. Знайдено та проілюстровано залежність часу затримки сигналу гауссової форми при тунелюванні для випадку квантової точки. 2012 Article Transmission of wave packets through open mesoscopic systems / N. Ivanov, V. Skalozub // Вопросы атомной науки и техники. — 2012. — № 1. — С. 292-295. — Бібліогр.: 5 назв. — англ. 1562-6016 PACS: 03.65.Pm, 03.65.Ge, 61.80.Mk http://dspace.nbuv.gov.ua/handle/123456789/107160 en Вопросы атомной науки и техники Dnipropetrovsk National University |
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Section E. Phase Transitions and Diffusion Processes in Condensed Matter and Gases Section E. Phase Transitions and Diffusion Processes in Condensed Matter and Gases |
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Section E. Phase Transitions and Diffusion Processes in Condensed Matter and Gases Section E. Phase Transitions and Diffusion Processes in Condensed Matter and Gases Ivanov, N. Skalozub, V. Transmission of wave packets through open mesoscopic systems Вопросы атомной науки и техники |
| description |
Tunneling of the wave packets having rectangular and Gaussian forms through a quantum diode is investigated. By using the potential of the system the S-matrix for this structure is obtained and the analytic expressions describing the form of the transmitted pulse are calculated. These analytic solutions are compared with the numeric ones. An excellent coincidence is detected. The time-delay for the Gaussian pulse tunneling through the quantum dot is investigated in details. |
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Article |
| author |
Ivanov, N. Skalozub, V. |
| author_facet |
Ivanov, N. Skalozub, V. |
| author_sort |
Ivanov, N. |
| title |
Transmission of wave packets through open mesoscopic systems |
| title_short |
Transmission of wave packets through open mesoscopic systems |
| title_full |
Transmission of wave packets through open mesoscopic systems |
| title_fullStr |
Transmission of wave packets through open mesoscopic systems |
| title_full_unstemmed |
Transmission of wave packets through open mesoscopic systems |
| title_sort |
transmission of wave packets through open mesoscopic systems |
| publisher |
Dnipropetrovsk National University |
| publishDate |
2012 |
| topic_facet |
Section E. Phase Transitions and Diffusion Processes in Condensed Matter and Gases |
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http://dspace.nbuv.gov.ua/handle/123456789/107160 |
| citation_txt |
Transmission of wave packets through open mesoscopic systems / N. Ivanov, V. Skalozub // Вопросы атомной науки и техники. — 2012. — № 1. — С. 292-295. — Бібліогр.: 5 назв. — англ. |
| series |
Вопросы атомной науки и техники |
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AT ivanovn transmissionofwavepacketsthroughopenmesoscopicsystems AT skalozubv transmissionofwavepacketsthroughopenmesoscopicsystems |
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2025-07-07T19:34:37Z |
| last_indexed |
2025-07-07T19:34:37Z |
| _version_ |
1837017992216444928 |
| fulltext |
TRANSMISSION OF WAVE PACKETS THROUGH OPEN
MESOSCOPIC SYSTEMS
N. Ivanov∗and V. Skalozub
Dnipropetrovsk National University, 49010, Dnipropetrovsk, Ukraine
(Received October 31, 2011)
Tunneling of the wave packets having rectangular and Gaussian forms through a quantum diode is investigated. By
using the potential of the system the S-matrix for this structure is obtained and the analytic expressions describing
the form of the transmitted pulse are calculated. These analytic solutions are compared with the numeric ones.
An excellent coincidence is detected. The time-delay for the Gaussian pulse tunneling through the quantum dot is
investigated in details.
PACS: 03.65.Pm, 03.65.Ge, 61.80.Mk
1. INTRODUCTION
The S-matrix formalism is commonly used in scat-
tering theory for describing an association between a
state after interaction with the one before interaction.
This object contains a complete information on a sys-
tem. For calculation of the S-matrix elements, either
a perturbation theory is used or the methods based
on studying of the S-matrix general characteristics
are applied. The method for calculation of S-matrix
elements was worked out in Ref. [1]. This approach
is based on the establishing of the relationship be-
tween the final state and the state before interaction,
presented in terms of the solutions for the Lippmann-
Shwinger equation with a perturbed potential. By
splitting a system potential in the perturbed and un-
perturbed parts and finding Green’s functions for the
former part, one can construct a solution. It gives a
possibility to find the R-matrix of the scattering sys-
tem. After that, using the relation between R- and
S-matrices, one is able to determine the values of the
scattering matrix elements.
The common feature of quantum dots, double-
well diodes, quantum tunneling transistors is the ex-
istence of potential wells with discreet energy levels.
At the same time, this results in the resonance con-
ductivities of these systems. The problem of the time-
delay determination (for pulse tunneling) is very im-
portant (see Refs. [2, 4]). Its solution, in particular,
gives a possibility to find out for which parameters
of the pulse or a quantum system the speed of the
tunneling becomes maximal. Also it gives an oppor-
tunity for miniaturization and increasing the produc-
tivity of the microcircuit with quantum elements.
It is reasonable to begin with determining the
characteristics of a wave packet best related with
the quantum system. The approach applied here is
based on the S-matrix formalism and modified saddle
point method developed in Refs. [1,3]. It completely
formalizes the solution of the wave packet tunneling
problem and gives a possibility to define the form
and the time-delay for arbitrary-form wave packets
tunneling through a quantum system with resonance
levels. The parameters of tunneling have to be ex-
pressed in terms of the incident pulse.
Using the approach of Ref. [5] let us obtain the
S-matrix for the diode, which energy potential is rep-
resented in Fig. 1.
Em-dE
Em
a+eU
a
I II III IV V VI
0 Z1 Z2Z4 Z3 Z
+a
eUI
2
E
eФв
Fig. 1. Potential energy of diode
At first, we represent the potential eliminating the
perturbation parts (from z1 to z3). Then we obtain
the form of the diode potential (Fig. 2),
In this way we split the potential in the perturbed
ΔV (z) and unperturbed V0(z) parts, respectively.
The wave functions for the scattering state coming
from the left are:
∗Corresponding author E-mail address: na ivanov@dsu.dp.ua
292 PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY, 2012, N1.
Series: Nuclear Physics Investigations (57), p.292-295.
Em-dE
Em
a+eU
a
I II III IV V VI
0 Z1 Z2Z4 Z3 Z
+a
eUI
2
E
eФв
Fig. 2. Potential energy without perturbed parts
ΨL
1 = eik1z + Ae−ik1z, z < 0,
ΨL
2 = Beik2z + Ce−ik2z, 0 < z < z1,
ΨL
3 = Deik3z + Ee−ik3z , z1 < z < z4,
ΨL
4 = Feik4z + Ge−ik4z , z4 < z < z2,
ΨL
5 = Heik5z + Ne−ik5z, z2 < z < z3,
ΨL
6 = Meik6z , z3 < z,
and for the one coming from the right are:
ΨR
1 = e−ik6z + A′eik6z, z3 < z,
ΨR
2 = B′e−ik5z + C′eik5z, z2 < z < z3,
ΨR
3 = D′e−ik4z + E′eik4z , z4 < z < z2,
ΨR
4 = F ′e−ik3z + G′eik3z , z1 < z < z4,
ΨR
5 = H ′e−ik2z + N ′eik2z, 0 < z < z1,
ΨR
6 = M ′e−ik1z, z < 0.
Hence, by using the matching conditions we have
found all the amplitudes for these states.
Then we obtain the Green-functions which are so-
lutions of inhomogeneous Shrödinger equation[
h̄2
2m∗
d2
dz2
− V0(z) + υ
]
Γ(z, z′, υ) = δ(z − z′) (1)
for all of the potential sections (υ is the particle en-
ergy). The solution for the Shrödinger equation with
initial potential can be represented as the superposi-
tion of the unperturbed solution (Ψ0 = Ψ0(z, υ)) and
the solution for the perturbation:
Ψ = Ψ0 +
∫
dz′Γi(z, z′, υ)ΔV (z′)Ψ(z′, υ), (2)
where i is a number of matching points and
Γi(z, z′, υ) corresponds to a potential block Green-
function. This equation represents the solution of
the Lippmann-Shwinger equation. For each of this
blocks we can obtained the transmission and reflec-
tion coefficients taking into account the consecutive
order of matching points and the corresponding to
them Green’s functions. After that we use the well-
known relationship between these coefficients and the
S-matrix elements. In this way we find the S-matrix
for the resonant system.
2. FORM OF THE TRANSMITTED
WAVE-PACKET
The incident and outgoing pulses are bound by the
next expression:
Ψout
a =
1
2π
∑
b
∫ ∞
−∞
Ψin
b Sb,adk, (3)
which is integrated over all the k-space. The para-
meters of the outgoing pulse wave function are: x is
the coordinate variable, t is the time variable, k and
k0 are momentum and center of incident wave packet
in the momentum space. kj are the positions of S-
matrix poles. We consider the incident pulse width a
in the real space and all calculations will be realized
in terms of the unperturbed pulse.
Let us make use the next dimensionless variables:
q′ = x
a ; τ = t
ta
; z = a(k − k0); lj = akj ;
ρj = a
Γj
2 ; l0 = ak0.
Moreover, to simplify further expressions we intro-
duce the parametrization:
β =
1 + iτ
2
, q0j = l0 − lj + iρj , q = q′ − l0τ. (4)
Here, the coordinate variable becomes q, time – τ ,
ta = ma2/h̄, lj and l0 are the localization poles and
center of wave packet respectively, and ρj is the width
of j-th resonance level. These variables give us a
sufficient number of parameters to describe the wave
packet tunneling through a quantum system with res-
onance levels. Now we make use of a modified saddle-
point method to obtain a transmitted solution [1].
Time interval τ after which we observe the outgoing
wave packet should be larger than ta. This is the nec-
essary condition for the existence of the resonance .
A saddle point position is defined by the stationarity
conditions:
dg(z)
dz
= 0, �g(z) = const, (5)
�g(z) < �g(zk),
where zk is the k-th saddle point. The resulting part
of the resonance wave function amplitude near the
stationarity point is
Ψ(q > 0, τ) ∼ 1
a
eil0(q− 1
2 l0τ) ×
×
∑
jk
egres
j (xs
k)
√
− 1
egres′′
j (xs
k)
f res
j (xs
k). (6)
This formula gives the asymptotic representation for
the wave packet passed though a resonance quantum
system.
293
3. TUNNELING OF PACKETS THROUGH
A QUANTUM DOT
Let us apply the developed method to investigate the
tunneling of the squared-form wave packet through
the quantum dot with one resonance level. The
packet is described as the difference of two step-
functions:
Ψ(x) = Θ(x) − Θ(x − a), (7)
where a is the packet width. We use the known
Fourier-transform for Θ-function, obtained in Appen-
dix (Eq. (14)). Applying the matching conditions (5)
we find two saddle points. For outgoing wave packet,
considering the two saddle points, we get superposi-
tion of each result (Fig. 4). For the Gaussian-form in-
cident wave packet tunneling through this quantum
system, like in the case with square-form packet, we
obtain (Fig. 3):
Ψ(z) = Ψ0 exp(−z2/2), (8)
where Ψ0 is the amplitude, Ψ(z) is the form of the
incident wave packet. In this way the exponential
argument becomes:
g(z) = izq′ − βz2 − ln(z − q0). (9)
Hence we find the saddle point:
z1 =
1
2
(
q0 +
iq′
2β
[
(q0 − iq′
2β
)2 − 1
2
]1/2
)
. (10)
For τ → ∞ with the accuracy up-to-the-second order
we obtain:
Ψres(q, τ) =
iρjaj√
2π
e
− (q−l0τ)2
2+2τ2 +i arg(Ψ) × (11)
× (1 + τ2)1/4
[(q + ρi − liτ)2 + (ρiτ + li − l0)2]1/2
,
where the argument is
arg(Ψ) =
1
2
arctg(τ) +
(q − l0τ)2τ
2 + 2τ2
−
− arctg
[
ρiτ + li − l0
q + ρi − liτ
]
. (12)
0.005
0.010
0.015
0.020
0.025
0 50 100 150 200 250
q
Fig. 3. Wave-function amplitude for the Gaus-
sian pulse tunneling trough the quantum dot,
for the saddle point method (dashed line)
and an exact integration (continues line) for
τ = 100, a1 = 1, l0 = l1 = 1, ρ = 0.09
0
0.02
0.04
0.06
0.08
20 40 60 80 100 120 140
q
0 20 40 60 80 100 120 140
q
0.5
0.4
0.3
0.2
0.1
Fig. 4. Amplitude for the step-form pulse tunneling
trough the QD, for the saddle point method (dashed
line) and an exact integration (continues line) for
τ = 100, a1 = 1, l0 = l1 = 1, ρ = 0.04 (top) and
ρ = 0.007 (bottom)
7.948
7.95
7.952
7.954
7.956
40 800 120 160 200
r0
0
Fig. 5. Time-delay for the Gaussian pulse trans-
mission (dependency from energy level width) with
a = 1, ta = 1, t = 100, x = 300, k0 = 1, ki = 1
The wave function of the outgoing pulse differs
from the incident wave packet by some phase factor.
This factor depends on the kinetic energy of the in-
cident pulse, the potential energy of a barrier and
time-delay. The delay can be calculated (Figs. 5-7)
following Ref. [2]:
�t =
d arg(Ψ)
dE
. (13)
Hence the advantage of analytic (asymptotic) meth-
ods is obvious. The time-delay depends on the pulse
and system parameters (the dependencies are shown
in Figs. 5-7).
294
10
20
30
40
50
60
2 4 6 10 12 14 16 18 20
k
8
I
Fig. 6. Time-delay for the Gaussian pulse trans-
mission (dependency from s-matrix poles position)
with ta = 1, t = 100, a = 1, x = 300, k0 = 1, Γ = 1
0
4
8
12
16
20
24
28
2 4 8 10
0k 6
Fig. 7. Time-delay for transmission gaussian pulse
(dependency from incident pulse momentum) with
ta = 1, t = 100, a = 1, x = 300, ki = 1, Γ = 1
4. CONCLUSIONS
Tunneling of the wave-packets having the Gaussian
and the rectangular form is investigated. The results
obtained show that the usage of the S-matrix formal-
ism and the saddle point method gives enough pa-
rameters to describe the dynamics of the scattering
processes. This approach gives a possibility to cal-
culate a time-delay of the wave packet transmitting
through quantum systems like quantum dots, double-
well diodes, transistors. It was shown, in particular,
that for some specific values of the system and the
pulse parameters a full internal scattering is realized.
This means that an outgoing pulse is absent, tunnel-
ing does not happen.
5. APPENDIX
For completeness, let us present the outgoing wave-
function for the step-form pulse tunneling through
the quantum dot:
Ψ(q, τ) =
1
a
√
2π
eil0(q− 1
2 l0τ)
∫ ∞
−∞
dz
{ a
iz
+ πδ(
z
a
)−
− e−iz
[ a
iz
+ πδ(
s
a
)
]}
eizq′−(β− 1
2 )z2
. (14)
References
1. U. Wulf, V.V. Skalozub, and A. Zaharov.
Multi-saddle-point approximation for pulse prop-
agation in resonant tunneling // Phys. Rev. 2008,
v. B77, p. 045318-045325.
2. M. Razavy. Quantum Theory of Tunneling.
“World Scientific”, 2003, p. 351-375.
3. N.A. Ivanov, V.V. Skalozub. Propagation of
wave packets through resonant quantum systems
// Theoretical and Mathematical Physics. 2010,
v. 168 (2), p. 1096-1104.
4. L. Brillouin. Wave Propagation and Group Ve-
locity. New York: “Academic Press”, 1960.
5. P.N. Racec. Ph. D. Thesis. Cottbus: University
of Technology, 2002.
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