#  Influence of a squeezed prewell on tunneling properties and bound states in
heterostructures

Yaroslav Zolotaryuk and Alexander V. Zolotaryuk  Bogolyubov Institute for
Theoretical Physics, National Academy of Sciences of Ukraine, Kyiv 03143,
Ukraine

###### Abstract

A resonant tunneling effect of an extremely thin potential well on the
transmission of charged particles through a planar heterostructure with an
arbitrary potential profile is investigated in a squeezing limit as the well
width tends to zero. In this limit, the transmission probability through the
structure is shown to be blocked for all the parameter values of the well,
except a resonance set of Lebesgue measure zero. The peak-to-valley ratio is
shown to increase crucially with the squeezing of the well: the thinner is its
thickness, the resonant peaks become sharper and localized at isolated points.
Contrary, a discrete spectrum of the heterostructure (if any) does exist both
on the resonance set and beyond it, however, the squeezing scenario here turns
out to be quite interesting and sophisticated.

††journal: Annals of Physics

Keywords: one-dimensional quantum systems, multi-layer heterostructures, point
interactions, resonant tunneling

##  1 Introduction

In the fast developing field of nanoscience, the question how to control the
particle transport in nanoscale quantum devices is very important. As a
result, there are numerous theoretical studies on tunneling effects in one
dimension describing the behavior of quantum particles when going through a
potential barrier or passing across a potential well. In many cases, it is
useful to perform these studies in the zero-thickness limit when the barrier
or well width shrinks to zero. It is common knowledge that in the barrier
case, this limit is realized as Dirac’s delta function $\delta(x)$. However,
for the positive-energy solutions of a one-dimensional stationary Schrödinger
equation

$-\psi^{\prime\prime}(x)+V(x)\psi(x)=E\psi(x),$ (1)

where $\psi(x)$ is the wave function, $E$ the energy of quantum particles,
$V(x)$ the potential for particles, and the prime stands for the
differentiation over the space $x$, the similar limit of a well-shaped
potential $V(x)$ seems to have no relevance with a real structure. In the
present work, we argue that a more singular one-point limit, instead of the
$\delta(x)$ distribution, has to be applied in order to describe correctly the
transmission properties of a quantum particle over a well. In this case, the
squeezing rate of the well will exceeds that of realizing a $\delta$-barrier
and therefore one cannot expect any well-defined limit function for the
potential $V(x)$ in Eq. (1). However, this is not a necessary condition,
because the whole Schrödinger equation may admit in the squeezing limit a
finite solution for the wave function $\psi(x)$. In this case, we are dealing
with a so-called point interaction (see book [1] for details as well as some
recent works [2, 3, 4, 5, 6, 7, 8], a few to mention) that provides the
corresponding two-sided boundary conditions on the wave function $\psi(x)$ at
a point of singularity [9].

The point limit of a quantum well materialized in the present paper appears to
be similar to that used for realizing the point interactions with potentials
in the form of the derivative of Dirac’s delta function $\delta^{\prime}(x)$.
As established in the series of publications [10, 11, 12, 13, 14, 15, 16, 17],
the main feature of the point interactions based on the potential
$\gamma\delta^{\prime}(x)$, where $\gamma$ is a strength constant, is the
existence of a resonance set of isolated points in the $\gamma$-space, at
which the transmission of quantum particles through a
$\gamma\delta^{\prime}$-barrier is non-zero, whereas beyond the resonance
points, this barrier acts as a perfectly reflecting wall. A similar point
limit has recently been implemented for one-dimensional pseudospin-one systems
[18].

An interesting problem concerns the perturbation of regular background
potentials by different point interactions. One of these interactions is a so-
called $\delta^{\prime}$-interaction for which the derivative
$\psi^{\prime}(x)$ is continuous at the point of singularity, but $\psi(x)$
discontinuous (the notation adopted in book [1]). The self-adjoint Hamiltonian
of a harmonic oscillator perturbed by this interaction has rigorously been
defined in Ref. [19]. In a subsequent publication [20], this study has been
extended for the perturbation by a triple of $\delta^{\prime}$-interactions
using the Cheon-Shigehara approach [21]. Another type of the point
interactions, which were used as a perturbation of background potentials, is
materialized through the derivative of the delta-function $\delta^{\prime}(x)$
(referred in the literature to as a $\delta^{\prime}$-potential, see e.g.,
Ref. [22]). This point interaction has been used as the perturbation of a
constant electric field and a harmonic oscillator [23] as well as an infinite
square well [24]. The energy spectrum of a one-dimensional V-shaped quantum
well perturbed by three types of point interactions composed from the
$\delta$\- and $\delta^{\prime}$-potentials has been studied in the papers
[25, 26, 27]. All these systems are combined from a specific regular subsystem
plus a point interaction centered inside this subsystem. It is of interest to
examine the situation when a point interaction is found outside a background
subsystem at some finite distance.

The present work focuses on the investigation of the role of a quantum well
(referred in some publications to as a prewell, see, e.g., Refs.[28, 29, 30])
on the tunneling of quantum particles through a given background structure
being of a general form. The well is assumed to be extremely thin and located
apart from the background subsystem at some distance. For our studies, it is
convenient to use the transfer matrix approach. Some preliminaries concerning
the calculation of scattering coefficients and bound states, using the
transfer matrix for Eq. (1) with an arbitrary potential $V(x)$, are presented
in Section 2. In the next Section, a point approximation of the well-shaped
potential $V(x)$ in the limit as its support shrinks to zero, is examined. The
comparison of this approximation with the $\delta$-limit is discussed. The
transmission matrix of the bilayer composed of a rectangular well and a layer
with an arbitrary potential profile (referred to as a $B$-layer) is calculated
in Section 4. In the next Section, the influence of this squeezed well on the
tunneling through the $B$-barrier is studied. Effects of the well on the
discrete spectrum of the $B$-layer are investigated in Section 6. The next
Section is devoted to some concluding remarks.

##  2 Scattering coefficients and bound states defined through transmission
matrix elements

Consider a potential profile $V(x)$ of an arbitrary shape with the support on
the interval $x_{1}<x<x_{2}$, where $x_{1}$ and $x_{2}$ are arbitrary points
on the $x$-axis. The transmission matrix for Eq. (1) that connects the
boundary conditions of the wave function $\psi(x)$ and its derivative
$\psi^{\prime}(x)$ at $x=x_{1}$ and $x=x_{2}$ is defined through the matrix
equation

$\left(\begin{array}[]{ll}\psi(x_{2})\\\
\psi^{\prime}(x_{2})\end{array}\right)=\Lambda\left(\begin{array}[]{ll}\psi(x_{1})\\\
\psi^{\prime}(x_{1})\end{array}\right),\quad\Lambda=\left(\begin{array}[]{ll}\lambda_{11}~{}~{}~{}\lambda_{12}\\\
\lambda_{21}~{}~{}~{}\lambda_{22}\end{array}\right),$ (2)

where $\lambda_{ij}\in{\mathbb{R}}$ and $\det\Lambda=1$.

For positive energy solutions of Eq. (1) and the left-to-right current, the
reflection-transmission coefficients $R$ and $T$ are defined using the
following scattering setting:

$\psi(k;x)=\left\\{\begin{array}[]{ll}{\rm e}^{{\rm i}kx}+R{\rm e}^{-{\rm
i}kx}&\mbox{for}~{}-\infty<x<x_{1}\,,\\\ T{\rm e}^{{\rm
i}kx}&\mbox{for}~{}~{}x_{2}<x<\infty,\end{array}\right.\quad
k:=\sqrt{E}\,,~{}~{}E>0.$ (3)

Using then these expressions as the boundary conditions at $x=x_{1}$ and
$x=x_{2}$ in the matrix equation (2), we obtain

$R=-\left[\lambda_{11}-\lambda_{22}+{\rm
i}\left(k\lambda_{12}+k^{-1}\lambda_{21}\right)\right]D^{-1}{\rm e}^{2{\rm
i}kx_{1}},\quad T=2D^{-1}{\rm e}^{{\rm i}k(x_{1}-x_{2})},$ (4)

where $D:=\lambda_{11}+\lambda_{22}-{\rm
i}\left(k\lambda_{12}+k^{-1}\lambda_{21}\right)$. Using that $\det\Lambda=1$,
one can get the relation $|D|^{2}=4+u^{2}+v^{2}$, where

$u:=\lambda_{11}-\lambda_{22}\quad\mbox{and}\quad
v=k\lambda_{12}+k^{-1}\lambda_{21}\,.$ (5)

Thus, the reflection and transmission probabilities can be expressed in terms
of the matrix elements $\lambda_{ij}$ as follows

${\cal R}:=|R|^{2}={u^{2}+v^{2}\over 4+u^{2}+v^{2}}\quad\mbox{and}\quad{\cal
T}:=|T|^{2}={4\over 4+u^{2}+v^{2}}\,.$ (6)

Similarly, one can express the discrete energy spectrum in terms of the
elements of the $\Lambda$-matrix (2). To this end, consider the negative
energy solutions of Eq. (1) with $E=-\kappa^{2}$, where $\kappa>0$ describes
the energy levels of the system. In this case, beyond the interval
$x_{1}<x<x_{2}$ (a free-particle space), the wave function is given by

$\psi(\kappa;x)=\left\\{\begin{array}[]{ll}C_{1}{\rm
e}^{\kappa(x-x_{1})}&\mbox{for}~{}-\infty<x<x_{1}\,,\\\ C_{2}{\rm
e}^{-\kappa(x-x_{2})}&\mbox{for}~{}~{}x_{2}<x<\infty,\end{array}\right.$ (7)

where $C_{1}$ and $C_{2}$ are arbitrary constants. Using the expressions (7)
as the boundary conditions at $x=x_{1}$ and $x=x_{2}$ in the matrix equation
(2), we obtain two equations with respect to the constants $C_{1}$ and
$C_{2}$. Obeying the compatibility condition, we get a general equation for
bound states. It reads

$\lambda_{11}+\lambda_{22}+\kappa\lambda_{12}+\kappa^{-1}\lambda_{21}=0,$ (8)

where the matrix elements $\lambda_{ij}$’s in general depend on $\kappa$.

##  3 A point approximation of a potential well revisited

In general, a one-point approximation of an arbitrary potential $V(x)$ in Eq.
(1) can be realized by the replacement $V(x)\to
V_{\varepsilon}(x)=\varepsilon^{-\nu}V(x/\varepsilon),$ where $\varepsilon>0$
is a dimensionless squeezing parameter that tends to zero, and the
dimensionless parameter $\nu>0$ has to be chosen appropriately in each
particular case. In the typical case as $\nu=1$, the regular function
$V_{\varepsilon}(x)$ converges to $\alpha\delta(x)$ in the sense of
distributions, where $\delta(x)$ is Dirac’s delta function and the integral
$\alpha=\int_{-\infty}^{\infty}V(x)dx$ becomes a strength of the
$\delta$-potential. For values $\nu\neq 1$, the potential $V_{\varepsilon}(x)$
has no $\varepsilon\to 0$ limits in the sense of distributions.

Assume that the potential $V(x)$ in Eq. (1) has the profile of a rectangular
barrier or well, i.e. $V(x)\equiv V=$ const. $\neq 0$ on a finite interval
$x_{1}\leq x\leq x_{2}$ and $V(x)\equiv 0$ beyond this interval. For this
potential, the $\Lambda$-matrix (2) can be calculated explicitly, resulting in

$\Lambda=\left(\\!\\!\begin{array}[]{ll}~{}~{}~{}\cos(ql)~{}~{}~{}~{}~{}~{}~{}~{}q^{-1}\sin(ql)\\\
-q\sin(ql)~{}~{}~{}~{}~{}~{}~{}~{}\cos(ql)\end{array}\right),\quad
l:=x_{2}-x_{1}\,,\quad q=\sqrt{E-V}.$ (9)

Then, the calculation of the transmission probability ${\cal T}$ according to
the second formula (6) leads to the expression

${\cal T}=\left[1+{V^{2}\over 4E(E-V)}\sin^{2}(ql)\right]^{\\!-1}\\!.$ (10)

In the particular case of tunneling through a rectangular barrier with height
$h$ and width $l$, the delta-function approximation ($\nu=1$) is specified as
$V=\varepsilon^{-1}h$ and $l=\varepsilon a$. Then, in the limit as
$\varepsilon\to 0$, the $\delta$-potential strength is $\alpha=ha$ and the
formula (10) reduces to the standard form

${\cal T}={1\over 1+(\alpha/2k)^{2}}$ (11)

that describes a monotonic decrease of the tunneling transmission with the
growth of the strength $\alpha$.

However, in the case of the transport of particles across a potential well,
instead of the monotonic law (11), we are dealing with an oscillating behavior
of the transmission (10) owing to the factor $\sin^{2}(ql)$ where $q$ is real.
Therefore the $\delta$-well approximation by the point potential
$V(x)=-\alpha\delta(x)$, where $\alpha=ad$ ($d$ is a well depth), fails to
describe adequately the transmission given by the same formula (11). Instead,
one can use in the formula (10) a more singular approximation (using $\nu=2$),
namely $V=-\varepsilon^{-2}d$, replacing $q$ and $l$ defined in (9) with

$q=q_{\varepsilon}=\sqrt{E+\varepsilon^{-2}d}\quad\mbox{and}\quad
l=\varepsilon a.$ (12)

Within this approximation, the transmission probability

${\cal T}={\cal T}_{w,\varepsilon}=\left[1+{d^{2}\over
4\varepsilon^{2}E(\varepsilon^{2}E+d)}\sin^{2}\\!\left(\sqrt{\varepsilon^{2}E+d}\,a\right)\right]^{\\!-1}$
(13)

exhibits an oscillating behavior for $\varepsilon>0$, where the resonant peaks
become more pointed as $\varepsilon\to 0$. The $\varepsilon\to 0$ limit of
(13) reads

${\cal T}={\cal T}_{w}=\lim_{\varepsilon\to 0}{\cal
T}_{w,\varepsilon}=\left\\{\begin{array}[]{ll}1&\mbox{if}~{}~{}\sqrt{d}\,a=n\pi,\\\
0&\mbox{if}~{}~{}\sqrt{d}\,a\neq n\pi,\end{array}\right.\quad
n=1,\,2,\,\ldots.$ (14)

Thus, the transmission probability (14) appears to be non-zero only on some
lines in the $\\{d,a\\}$-space forming a set of Lebesgue measure zero given by

$\Sigma:=\\{d,a~{}|~{}\sqrt{d}\,a=n\pi,~{}n\in{\mathbb{N}}\\}$ (15)

and referred from now on a resonance set. Beyond this set, the transmission is
identically zero. Therefore the zero-range approximation with the scaled
function

$V_{\varepsilon}(x)=-\left\\{\begin{array}[]{ll}\varepsilon^{-2}d&\mbox{for}~{}~{}0\leq
x\leq l=\varepsilon a,\\\ 0&\mbox{otherwise}\end{array}\right.$ (16)

turns out to be more adequate than the $\delta$-approximation because it
exhibits peaks and valleys existing in the realistic case ($\varepsilon=1$),
as shown in Fig. 1. Contrary to the formula (11), this figure illustrates the
spire-like convergence of the transmission probability ${\cal
T}_{w,\varepsilon}$ as $\varepsilon\to 0$.

Figure 1:  The transmission probability ${\cal T}_{w,\varepsilon}$ as a
function of the thickness parameter $a$ plotted according to formula (13) with
$E=0.01$ eV and $d=0.2$ eV. The transmission probability for the unsqueezed
(realistic) potential ($\varepsilon=1$) is shown by black line. The squeezing
($\varepsilon\to 0$) scenario in the case $\nu=2$ is illustrated by the lines
plotted for $\varepsilon=0.1$ (blue) and $\varepsilon=0.01$ (red). The green
line together with $\blacklozenge$ corresponds to the $\delta$-approximation
($\nu=1$) with $\varepsilon=0.01$, calculated according to formula (10), where
$E=0.01$ eV, $V=\varepsilon^{-1}h$, $h=0.2$ eV, and $l=\varepsilon a$. The
calculations have been carried out in the units for which
$\hbar^{2}/2m^{*}=1$. Here, $m^{*}$ is an effective electron mass chosen to
equal $0.1\,m$, where $m$ is the free electron mass, so that 1 eV = 2.62464
nm-2.

Summary 1: Let the Schrödinger equation (1), in which the shape of the
potential $V(x)$ is a rectangular well with depth $d$ and width $a$, is used
for the description of the transmission of quantum particles across the well.
For realizing a well-defined and adequate point interaction, the rectangular
approximation (16) has to be applied instead of the $\delta$-well
approximation. For the approximation (16), in the squeezing limit as
$\varepsilon\to 0$, the spire-like convergence of the transmission ${\cal
T}_{w,\varepsilon}$ leads to the existence of a non-zero (being perfect)
transmission only on isolated lines forming the resonance set (15). Beyond
this set, the well acts as a perfectly reflecting wall.

##  4 The special case of a $W$–$B$ bilayer structure

In the following we will consider a bilayer heterostructure composed of a
layer with the potential in the form of a rectangular well (also called in
some papers a prewell, see, e.g., Refs. [28, 29, 30]) and a background layer
with the profile of an arbitrary shape. Schematically, this structure is shown
in Fig. 2

Figure 2:  Sketch of the two cases of a $W$–$B$ bilayer system. (a) A
rectangular well (emitter prewell) is located in front of the $B$-layer with
an arbitrary potential profile. (b) A rectangular well (collector prewell) is
found behind the $B$-layer.

regarding the two situations: the prewell (also referred to as a $W$-layer or
a $W$-subsystem) is incorporated either in front of the background layer
(called a $B$-layer or a $B$-subsystem) or behind it.

Assume that the transmission matrix $\Lambda=\Lambda_{w}$ given by Eq. (9)
corresponds to the $W$-layer with the potential profile $V_{w}(x)\equiv V<0$
defined on the interval $x_{1}\leq x\leq x_{2}$ and $V_{w}(x)\equiv 0$ beyond
this interval. Suppose further that the potential $V_{b}(x)$ for the $B$-layer
is defined on another interval $y_{1}\leq x\leq y_{2}$ and the elements of the
corresponding transmission matrix $\Lambda_{b}$ are denoted by
$\lambda_{ij}$’s. Let the distance $\rho$ between the $W$\- and $B$-layers be
non-zero. Then the transmission matrix that corresponds to the free space
between the subsystems is

$\Lambda_{0}=\left(\begin{array}[]{cc}~{}~{}~{}\cos(k\rho)~{}~{}~{}~{}~{}k^{-1}\sin(k\rho)\\\
-k\sin(k\rho)~{}~{}~{}~{}~{}~{}~{}\cos(k\rho)\end{array}~{}\right),\quad
k:=\sqrt{E}\,.$ (17)

The transmission matrix of the whole $W$–$A$ system is the product

$C=\left(\begin{array}[]{ll}c_{11}~{}~{}~{}c_{12}\\\
c_{21}~{}~{}~{}c_{22}\end{array}\right)=\left\\{\begin{array}[]{ll}C_{wb}:=\Lambda_{b}\Lambda_{0}\Lambda_{w}&\mbox{for}~{}x_{1}<x_{2}<y_{1}<y_{2}\,,\\\
C_{bw}:=\Lambda_{w}\Lambda_{0}\Lambda_{b}&\mbox{for}~{}y_{1}<y_{2}<x_{1}<x_{2}\,,\end{array}\right.$
(18)

where the matrix $C_{wb}$ corresponds to the ($w-b$)-configuration (the
prewell is located in front of the $B$-potential and $y_{1}-x_{2}=\rho$) and
the matrix $C_{bw}$ to the ($b-w$)-configuration (the prewell is found behind
the $B$-potential and $x_{1}-y_{2}=\rho$). Explicitly, for the
($w-b$)-configuration, the $c_{ij}$-elements are given by

$\begin{array}[]{llll}\vskip 6.0pt plus 2.0pt minus
2.0ptc_{11}=c_{11;\,wb}&=&\lambda_{11}\left[\cos(ql)\cos(k\rho)-(q/k)\sin(ql)\sin(k\rho)\right]\\\
\vskip 6.0pt plus 2.0pt minus
2.0pt&-&\lambda_{12}\left[q\sin(ql)\cos(k\rho)+k\cos(ql)\sin(k\rho)\right],\\\
\vskip 6.0pt plus 2.0pt minus
2.0ptc_{12}=c_{12;\,wb}&=&\lambda_{11}\left[q^{-1}\sin(ql)\cos(k\rho)+k^{-1}\cos(ql)\sin(k\rho)\right]\\\
\vskip 6.0pt plus 2.0pt minus
2.0pt&+&\lambda_{12}\left[\cos(ql)\cos(k\rho)-(k/q)\sin(ql)\sin(k\rho)\right],\\\
\vskip 6.0pt plus 2.0pt minus
2.0ptc_{21}=c_{21;\,wb}&=&\lambda_{21}\left[\cos(ql)\cos(k\rho)-(q/k)\sin(ql)\sin(k\rho)\right]\\\
\vskip 6.0pt plus 2.0pt minus
2.0pt&-&\lambda_{22}\left[q\sin(ql)\cos(k\rho)+k\cos(ql)\sin(k\rho)\right],\\\
\vskip 6.0pt plus 2.0pt minus
2.0ptc_{22}=c_{22;\,wb}&=&\lambda_{21}\left[q^{-1}\sin(ql)\cos(k\rho)+k^{-1}\cos(ql)\sin(k\rho)\right]\\\
\vskip 6.0pt plus 2.0pt minus
2.0pt&+&\lambda_{22}\left[\cos(ql)\cos(k\rho)-(k/q)\sin(ql)\sin(k\rho)\right].\end{array}$
(19)

For the ($b-w$)-configuration, the matrix $C_{bw}$ is obtained from the matrix
elements (19) by the replacements
$\lambda_{11}\longleftrightarrow\lambda_{22}$ as follows

$C_{bw}=\left(\begin{array}[]{lr}\vskip 6.0pt plus 2.0pt minus
2.0ptc_{22;\,wb}(\lambda_{22}\to\lambda_{11})~{}~{}~{}c_{12;\,wb}(\lambda_{11}\to\lambda_{22})\\\
c_{21;\,wb}(\lambda_{22}\to\lambda_{11})~{}~{}~{}c_{11;\,wb}(\lambda_{11}\to\lambda_{22})\end{array}\right).$
(20)

Below we will study the influence of a squeezed prewell on a $B$-subsystem in
two situations. First, it is of interest to know how the presence of a prewell
will act on the transmission of quantum particles through a potential barrier.
Second, the influence of a prewell on the structure of bound states in a
$B$-well is to be investigated. To implement the squeezing limit, in Eqs. (19)
and (20), we use the $\varepsilon$-dependence (12). In this limit, the
distance $\rho$ and the width $l_{b}:=y_{2}-y_{1}$ are accepted to be fixed.

##  5 Influence of a squeezed prewell on the tunneling through an arbitrary
barrier

Assume in this section that the potential $V_{b}(x)$ has a non-zero barrier
part and a rectangle-like prewell is incorporated at distance $\rho$ either in
front of the $B$-barrier or behind it. Intuitively, due to the resonant
transmission (14), one can expect that the squeezed prewell will allow a net
tunneling current through the barrier only on the resonance set $\Sigma$. More
rigorously, based on Eqs. (19) and (20), this can be proven as follows.
Setting in these equations $q$ and $l$ given by (12), we find that the
elements $c_{11}$ and $c_{21}$ in general diverge as $\varepsilon\to 0$.
However, due to the factor $\sin(ql)$, this divergence is suppressed on the
set $\Sigma$. More precisely, in the limit as $\varepsilon\to 0$, we have

$\\!q|\sin(ql)|=\varepsilon^{-1}\sqrt{\varepsilon^{2}E+d}\,\Big{|}\sin\\!\left(\\!\sqrt{\varepsilon^{2}E+d}\,a\right)\\!\Big{|}\to\left\\{\begin{array}[]{ll}\,0&\mbox{for}~{}(d,a)\in\Sigma,\\\
\infty&\mbox{for}~{}(d,a)\notin\Sigma.\end{array}\right.\\!\\!$ (21)

As a result, for the expressions (5), where $u=u_{\varepsilon}$ and
$v=v_{\varepsilon}$ are used for the whole system, i.e., $\lambda_{ij}$’s are
replaced respectively by $c_{ij}$’s given by Eqs. (19) and (20), we obtain the
$\varepsilon\to 0$ limits

$\left\\{\begin{array}[]{ll}\vskip 3.0pt plus 1.0pt minus
1.0ptu_{\varepsilon}\\\
v_{\varepsilon}\end{array}\right\\}\to(-1)^{n}\\!\left\\{\\!\begin{array}[]{ll}\vskip
3.0pt plus 1.0pt minus
1.0pt\left[(\lambda_{11}-\lambda_{22})\cos(k\rho)\mp\left(k\lambda_{12}+k^{-1}\lambda_{21}\right)\sin(k\rho)\right]\\\
\left[(k\lambda_{12}+k^{-1}\lambda_{21})\cos(k\rho)\pm(\lambda_{11}-\lambda_{22})\sin(k\rho)\right]\end{array}\\!\right\\}$
(22)

on the set $\Sigma$, whereas $|u_{\varepsilon}|$ and
$|v_{\varepsilon}|\to\infty$ outside the set $\Sigma$. Here, the upper signs
in the square brackets belong to the $(w-b)$-configuration and the lower ones
to the $(b-w)$-configuration. From both these equations we immediately obtain
the same expression for $u^{2}+v^{2}$ if the prewell would be absent. Thus,
the total tunneling transmission through the whole system coincides with the
transmission across the barrier described by the $\Lambda_{b}$-matrix,
however, this occurs only on the resonance set $\Sigma$ for the squeezed
prewell.

Summary 2: Let ${\cal T}_{wb}$ or ${\cal T}_{bw}$ be the transmission
probability of quantum particles tunneling through the system composed of a
squeezed (as $\varepsilon\to 0$) prewell located at any distance $\rho$ from a
barrier (in front of the barrier or behind it). This transmission is ${\cal
T}_{wb}={\cal T}_{bw}={\cal T}_{w}\cdot{\cal T}_{b}$ where ${\cal T}_{w}$ is
defined by (14). Moreover, it does not depend on the distance $\rho$.

This conclusion can be illustrated on the simple example of a barrier
subsystem. The transmission probability for this subsystem is a monotonically
decreasing function given by the formula (11). The presence of an additional
well crucially changes this transmission making it resonant and dependent on
the well parameters as demonstrated by the contour plot for the
($b-w$)-configuration in Fig. 3.

Figure 3:  The transmission probability ${\cal T}_{bw,\varepsilon}$ as a
function of widths $a$ and $l_{b}$ for the system composed of prewell with
depth $d=0.2$ eV and barrier with height $V_{b}=0.1$ eV, which has been
calculated for two values: (a) $\varepsilon=1$ and (b) $\varepsilon=0.1$.
Here, $E=0.02$ eV and $\rho=10$ nm. Panel (a) corresponds to $\varepsilon=1$,
while panel (b) to $\varepsilon=0.1$. The calculations have been performed
using the same units as in Fig. 1.

Here the widths $a$ and $l_{b}$ are variables, and distance $\rho$ is
arbitrary and fixed. The plots are present for the whole system at two values
of $\varepsilon$ shown in Fig. 3(a) for $\varepsilon=1$ (unsqueezed prewell)
and in Fig. 3(b) for $\varepsilon=0.1$ (squeezed prewell). The figure clearly
demonstrates that the presence of a prewell essentially improves the resonance
properties of the transmission through a barrier.

##  6 Effects of a squeezed prewell on the discrete spectrum of bound states
in the $B$-layer of an arbitrary profile

In this section, we consider the situation when Eq. (1) with a potential
$V_{b}(x)$ (the prewell is absent) admits the existence of negative-energy
solutions with $E=-\kappa^{2}$, where $\kappa>0$ describes the discrete
spectrum of bound states. For this case, we assume that the equation for the
bound state levels $\kappa$ is given by Eq. (8), where the elements
$\lambda_{ij}$’s correspond to the potential $V_{b}(x)$ and consequently to
the $\Lambda_{b}$-matrix. Similarly, for the bound states of the whole $W$–$A$
system defined on the interval $(0,\,l+\rho+l_{b})$, we have the same Eq. (8),
in which $\lambda_{ij}$’s are replaced with $c_{ij}$’s given by Eqs. (19) and
(20) where $k={\rm i}\kappa$, $q$ and $l$ are determined by Eqs. (12) with
$E=-\kappa^{2}$, i.e., $q=q_{\varepsilon}=\sqrt{\varepsilon^{-2}d-\kappa^{2}}$
and $l=\varepsilon a$. Explicitly, for the ($w-b$)-configuration, Eq. (8) with
respect to the unknown $\kappa$ becomes

$\displaystyle\\!\\!\\!\left(\lambda_{11}+\lambda_{22}+\kappa\lambda_{12}+{\lambda_{21}\over\kappa}\right)(1+\tau)+\left({\kappa\over
q_{\varepsilon}}\lambda_{11}-{q_{\varepsilon}\over\kappa}\lambda_{22}-q_{\varepsilon}\lambda_{12}+{\lambda_{21}\over
q_{\varepsilon}}\right)t_{\varepsilon}$ (23) $\displaystyle+$
$\displaystyle\\!\\!\\!\left({\kappa^{2}\over
q_{\varepsilon}}\lambda_{12}-{q_{\varepsilon}\over\kappa^{2}}\lambda_{21}-{q_{\varepsilon}\over\kappa}\lambda_{11}+{\kappa\over
q_{\varepsilon}}\lambda_{22}\right)t_{\varepsilon}\tau=0,$

where
$t_{\varepsilon}:=\tan(q_{\varepsilon}l)=\tan\\!\left(\sqrt{d-(\varepsilon\kappa)^{2}}\,a\right)$
and $\tau:=\tanh(\kappa\rho)$. The equation for the ($b-w$)-configuration is
obtained from Eq. (23) by the replacements
$\lambda_{11}\longleftrightarrow\lambda_{22}$ . On the resonance set $\Sigma$,
we have $t_{\varepsilon}\to 0$ and therefore Eq. (23) reduces to Eq. (8) that
describes the discrete spectrum of the unperturbed $B$-subsystem. Beyond the
set $\Sigma$, we have $\lim_{\varepsilon\to 0}t_{\varepsilon}\neq 0$, however,
in this case, $q_{\varepsilon}\to\infty$ and therefore Eq. (23) asymptotically
reduces to

$\lambda_{11}\tau+\lambda_{22}+\kappa\lambda_{12}+\kappa^{-1}\lambda_{21}\tau=0$
(24)

for the ($w-b$)-configuration. Replacing here
$\lambda_{11}\longleftrightarrow\lambda_{22}$ , one obtains the equation for
the ($b-w$)-configuration. Therefore, in the $\varepsilon\to 0$ limit, outside
the resonance set $\Sigma$, the bound state energies depend only on the
distance $\rho$ through the factor $\tau$. For sufficiently large distances
$\rho$, we have $\tau\to 1$ and therefore Eq. (24) coincides with Eq. (8).
This means that the squeezed prewell does not affect the discrete spectrum of
the $B$-subsystem if the $W$\- and $B$-subsystems are separated by sufficient
large $\rho$. Note that the squeezed $W$-well itself has no bound states.

In order to examine the bound states of the whole system as functions of the
prewell parameters and to study the $\varepsilon\to 0$ behavior if $\tau<1$,
we have solved numerically Eq. (23) in the special case when the $B$-potential
has the form of a rectangular well with depth $V_{b}<0$. In this case, we are
dealing with a double-well potential defined on the interval $0\leq x\leq
l+\rho+l_{b}$ . A qualitative analysis about the existence of bound states is
similar to that given in Ref. [31]. The results of the numerical analysis of
the solutions to Eq. (23), where the elements $\lambda_{ij}$’s correspond to a
rectangular $B$-well (given by Eq. (9) with $q=\sqrt{|V_{b}|-\kappa^{2}}$ and
$l=l_{b}=y_{2}-y_{1}$), are illustrated by Figs. 4 and 5. Numerically, the
unperturbed spectrum of the $B$-well, consisting of the solutions to Eq. (8),
has been chosen to have three levels of $\kappa$ ($N=3$): $\kappa_{1}^{0}$,
$\kappa_{2}^{0}$ and $\kappa_{3}^{0}$
($\kappa_{1}^{0}>\kappa_{2}^{0}>\kappa_{3}^{0}$). In general, for sufficiently
small $\tau<1$, the number of solutions to Eq. (24) that represent the
$\varepsilon\to 0$ limit spectrum of the whole $W$–$B$ structure beyond the
resonance set $\Sigma$, is reduced by one becoming $N-1$. Therefore, in our
numerical case with $N=3$, the parameter $\tau$ has been chosen in such a way
that Eq. (24) admits two solutions $\kappa_{1}$ and $\kappa_{2}$
($\kappa_{1}>\kappa_{2}$).

Figure 4:  Dependence of the discrete spectrum in the case $N=3$ on the
thickness parameter $a$ of the $W$-well for $\varepsilon=0.01$, which is shown
by solid (black) curves. The parameter values are $d=0.2$ eV, $\rho=0.5$ nm,
$V_{b}=-\,0.5$ eV and $l_{b}=7$ nm. Solutions to Eq. (8):
$\kappa_{1}^{0}=1.08819,\,\kappa_{2}^{0}=0.90138,\,\kappa_{3}^{0}=0.50528$ are
indicated by blue horizontal straight lines. Between the origin $a=0$ and the
first resonance point $a=a_{1}$, the spectrum consists of three levels
$\kappa_{1,\varepsilon}$ , $\kappa_{2,\varepsilon}$ and
$\kappa_{3,\varepsilon}$
($\kappa_{1,\varepsilon}>\kappa_{2,\varepsilon}>\kappa_{3,\varepsilon}$). At
point $a=a_{1}$, a new level $\kappa_{4,\varepsilon}$ detaches from $\kappa=0$
as shown in the inset. Similarly, a new level $\kappa_{5,\varepsilon}$
detaches from $\kappa=0$ at $a=a_{2}$. Visually, horizontal line
$\kappa_{1}^{0}$ coincides with $\kappa_{2,\varepsilon}$ on interval
$0<a<a_{1}$ , with $\kappa_{3,\varepsilon}$ on interval $a_{1}<a<a_{2}$ and
with $\kappa_{4,\varepsilon}$ for $a>a_{2}$. The inset shows how these lines
are distinguished.

In more detail, Figs. 4 and 5 represent the $\kappa$-levels as numerical
solutions to Eq. (23) at fixed depth $d$ and distance $\rho$. At a fixed
sufficiently small $\varepsilon$, Fig. 4 illustrates the dependence of the
$\kappa$-solutions on the thickness parameter $a$. Here, at the resonance
points $a=a_{n}:=n\pi/\sqrt{d}$ , the new levels are successively detaching
from the origin $\kappa=0$, so that there are three levels on the interval
$a_{0}<a<a_{1}$, the additional fourth level appears on the interval
$a_{1}<a<a_{2}$, and the fifth level appears for $a>a_{2}$ . In the vicinity
of the resonant values $a_{n}$’s, the avoided crossings are clearly observed
as can be clearly seen in the inset of Fig. 4. Instead of the new levels, the
lowest levels, which depend on $\varepsilon$, tend to infinity as
$\varepsilon\to 0$. This limit behavior is shown in Fig. 5, where the
calculations have been carried out for three values of $a$. Here, one value is
resonant ($a=a_{1}$) and the other two are chosen at the middles of the non-
resonance intervals $0<a<a_{1}$ and $a_{1}<a<a_{2}$ .

Figure 5:  The $\varepsilon\to 0$ scenario of the discrete spectrum for the
case when the $B$-subsystem has the form of the rectangular well that admits
three bound state levels $\kappa_{1}^{0}$ , $\kappa_{2}^{0}$ ,
$\kappa_{3}^{0}$ ($\kappa_{1}^{0}>\kappa_{2}^{0}>\kappa_{3}^{0}$). These
levels are shown by red horizontal straight lines. The parameter values are
$E=0.01$ eV, $d=0.2$ eV, $\rho=10$ nm. In panel (a), at $a=\pi/2\sqrt{d}$ (the
middle point of the non-resonance interval $0<a<a_{1}=\pi/\sqrt{d}$), three
levels $\kappa_{1,\varepsilon}$, $\kappa_{2,\varepsilon}$,
$\kappa_{3,\varepsilon}$
($\kappa_{1,\varepsilon}>\kappa_{2,\varepsilon}>\kappa_{3,\varepsilon}$) are
plotted. Their $\varepsilon\to 0$ limits are
$\kappa_{1,\varepsilon}\to\infty$, $\kappa_{2,\varepsilon}\to\kappa_{1}$,
$\kappa_{3,\varepsilon}\to\kappa_{2}$ , where $\kappa_{1}$ and $\kappa_{2}$
are solutions to Eq. (24), shown also in the inset. In panel (b), at the
resonance point $a=a_{1}=\pi/\sqrt{d}$ , because of detaching a new level from
$\kappa=0$, the $\varepsilon$-dependent spectrum consists of four levels
($\kappa_{1,\varepsilon}>\kappa_{2,\varepsilon}>\kappa_{3,\varepsilon}>\kappa_{4,\varepsilon}$).
Their $\varepsilon\to 0$ limits are $\kappa_{1,\varepsilon}\to\infty$,
$\kappa_{2,\varepsilon}\to\kappa_{1}^{0}$,
$\kappa_{3,\varepsilon}\to\kappa_{2}^{0}$ ,
$\kappa_{4,\varepsilon}\to\kappa_{3}^{0}$ , where $\kappa_{1}^{0}$,
$\kappa_{2}^{0}$ , $\kappa_{3}^{0}$ , are solutions to Eq. (8), which are also
shown in the inset. In panel (c), the $\varepsilon\to 0$ behavior of this
four-level spectrum is plotted for $a=3\pi/2\sqrt{d}$ (the middle point of the
non-resonance interval $a_{1}<a<a_{2}=2\pi/\sqrt{d}$). The $\varepsilon\to 0$
limits of these four levels are $\kappa_{1,\varepsilon}\to\infty$,
$\kappa_{2,\varepsilon}\to\infty$, $\kappa_{3,\varepsilon}\to\kappa_{1}$ ,
$\kappa_{4,\varepsilon}\to\kappa_{2}$ , where again $\kappa_{1}$ and
$\kappa_{2}$ are the solutions to Eq. (24).

The results depicted in both Figs. 4 and 5 can be extended for an arbitrary
number of bound state levels $N$ as follows. Assume that the spectrum of bound
states for the $B$-subsystem alone (i.e., at $a=0$) consists of $N$ levels:
$\\{\kappa_{1}^{0}\,,\ldots,\kappa_{N}^{0}\\}$
($\kappa_{1}^{0}>\ldots>\kappa_{N}^{0}$), which are roots of Eq. (8). When a
prewell is “switched on”, this spectrum on the interval $0<a<a_{1}$ is
modified to depend on $\varepsilon$, resulting in the transition
$\kappa_{1}^{0}\to\kappa_{1,\varepsilon}\,,\ldots,\,\kappa_{N}^{0}\to\kappa_{N,\varepsilon}$
. As illustrated by Fig. 5(a), on the interval $0<a<a_{1}$ , we have the
following limits as $\varepsilon\to 0$:
$\kappa_{1,\varepsilon}\to\infty,\,\kappa_{2,\varepsilon}\to\kappa_{1}\,,\ldots,\,\kappa_{N,\varepsilon}\to\kappa_{N-1}$
, so that on this interval the limit spectrum becomes
$\\{\kappa_{1}\,,\ldots,\,\kappa_{N-1}\\}$ consisting of the solutions to Eq.
(24).

Next, while approaching the next resonance point $a=a_{2}$ , a new smallest
level $\kappa_{N+1,\varepsilon}$ detaches from the origin $\kappa=0$, as shown
in the inset of Fig. 4. As a result, at $a=a_{2}$, we have $N+1$ levels and
the $\varepsilon\to 0$ limit is realized as
$\kappa_{1,\varepsilon}\to\infty,\,\kappa_{2,\varepsilon}\to\kappa_{1}^{0}\,,\,\ldots,\,\kappa_{N,\varepsilon}\to\kappa_{N-1}^{0}\,,\,\kappa_{N+1,\varepsilon}\to\kappa_{N}^{0}$
, resulting in the spectrum $\\{\kappa_{1}^{0},\,\ldots,\,\kappa_{N}^{0}\\}$.
This limiting behavior is illustrated by Fig. 5(b) for the case $N=3$.

Similarly, the transition from the point $a=a_{1}$ to the interval
$a_{1}<a<a_{2}$ leads to the following limit behavior as $\varepsilon\to 0$:
$\kappa_{1,\varepsilon}\to\infty,\,\kappa_{2,\varepsilon}\to\infty,\,\kappa_{3,\varepsilon}\to\kappa_{1}\,,\,\ldots,\,\kappa_{N+1,\varepsilon}\to\kappa_{N-1}$
. For the particular case $N=3$, this scenario is demonstrated by Fig. 5(c),
where the limiting discrete spectrum consists of two levels $\kappa_{1}$ and
$\kappa_{2}$ . Thus, the results described above can be formulated as follows.

Summary 3: Assume that an unperturbed $B$-subsystem has a discrete spectrum
$\\{\kappa_{1}^{0},\,\kappa_{2}^{0},\,\ldots,\,\kappa_{N}^{0}\\}$ composed of
$N$ bound state levels, which are arranged as
$\kappa_{1}^{0}>\kappa_{2}^{0}>\ldots>\kappa_{N}^{0}$, being the solutions to
Eq. (8). Suppose further that $\tau<1$ and the distance $\rho$ provides $N-1$
solutions of Eq. (24), so that the spectrum of these solutions becomes
$\\{\kappa_{1},\,\kappa_{2},\,\ldots,\,\kappa_{N-1}\\}$
($\kappa_{1}>\kappa_{2}>\ldots>\kappa_{N-1}$). Then in the presence of a
squeezed rectangular $W$-well, the spectrum of the whole $W$–$B$ system
preserves on the resonance set $\Sigma$, consisting of the points
$a=a_{n}=n\pi/\sqrt{d}$, $n\in{\mathbb{N}}$, exactly the same form as in the
case without the well, while beyond this set, the $W$–$B$ spectrum transforms
to $\\{\kappa_{1},\,\kappa_{2},\,\ldots,\,\kappa_{N-1}\\}$
($\kappa_{1}>\kappa_{2}>\ldots>\kappa_{N-1}$).

The $\varepsilon\to 0$ scenario of reconstructing the $W$–$B$ spectrum under
the addition of the $W$-well develops as follows. At $a=a_{0}=0$, the spectrum
is $\\{\kappa_{1}^{0},\ldots,\kappa_{N}^{0}\\}$. In the presence of the
squeezed $W$-well, this spectrum is modified by adding at each $a=a_{n}$
successively a new smallest level that detaches from the origin $\kappa=0$.
Then beyond the set $\Sigma$, on each interval $a_{n-1}<a<a_{n}$
($n\in{\mathbb{N}}$), the spectrum of bound states becomes

$\\{\kappa_{1,\varepsilon}\,,\,\ldots\,,\,\kappa_{N-1,\varepsilon}\,,\,\kappa_{N,\varepsilon}\,,\,\ldots\,,\,\kappa_{n+N-1,\varepsilon}\\}.$

In the limit as $\varepsilon\to 0$, we have the following convergence process:

$\begin{array}[]{ccccccccccccccc}\kappa_{1,\varepsilon}&>&\kappa_{2,\varepsilon}&>&\ldots&>&\kappa_{n,\varepsilon}&>&\kappa_{n+1,\varepsilon}&>&\ldots&>&\kappa_{n+N-1,\varepsilon}\\\
\downarrow&&\downarrow&&&&\downarrow&&\downarrow&&&&\downarrow\\\
\infty&&\infty&&&&\infty&&\kappa_{1}&&&&\kappa_{N-1}\end{array}$

resulting in the limit spectrum
$\\{\kappa_{1}\,,\,\ldots\,,\,\kappa_{N-1}\\}$. On the other hand, at each
resonant value $a=a_{n}$ , the spectrum is

$\\{\kappa_{1,\varepsilon}\,,\,\ldots\,,\,\kappa_{n,\varepsilon}\,,\,\kappa_{n+1,\varepsilon}\,,\,\ldots\,,\,\kappa_{N+n,\varepsilon}\\}$

and its $\varepsilon\to 0$ limit reads

$\begin{array}[]{ccccccccccccccc}\kappa_{1,\varepsilon}&>&\kappa_{2,\varepsilon}&>&\ldots&>&\kappa_{n,\varepsilon}&>&\kappa_{n+1,\varepsilon}&>&\ldots&>&\kappa_{n+N,\varepsilon}\\\
\downarrow&&\downarrow&&&&\downarrow&&\downarrow&&&&\downarrow\\\
\infty&&\infty&&&&\infty&&\kappa_{1}^{0}&&&&\kappa_{N}^{0}\end{array}$

leading finally to the limit spectrum
$\\{\kappa_{1}^{0}\,,\,\ldots\,,\,\kappa_{N}^{0}\\}$.

##  7 Concluding remarks

Thus, we have investigated the influence of an extremely thin potential well
on the spectrum of the background layer with the potential profile of an
arbitrary shape. For these studies to be implemented, it was necessary to
determine correctly the zero-range limit of a thin well in the sense of point
interactions. More precisely, while the typical $\varepsilon\to 0$ limit for
the barrier potential $V(x)$ in Eq. (1) is
$\varepsilon^{-\nu}V(x/\varepsilon)$ with $\nu=1$, in the case of a well, it
turns out to be at $\nu=2$. This is a key point of our approach for the study
of the role of a squeezed potential well in the transmission of quantum
particles through a barrier. The other interesting question is how the
presence of such a well disturbs the spectrum of bound states in any
background system located at some finite (sufficiently close) distance.

In order to implement the $\varepsilon\to 0$ limit explicitly, we have chosen
the potential well $V(x)$ in a rectangle-like form. In the squeezing limit as
$\varepsilon\to 0$, both the cases $\nu=1$ and $\nu=2$ demonstrate in Fig. 1
the crucial difference of the transmission across this well, from which it
immediately follows that the approximation with $\nu=2$ turns out to be much
more appropriate than the typical $\delta$-like representation with $\nu=1$.
Note that the potential $\varepsilon^{-2}V(x/\varepsilon)$ has no limit as
$\varepsilon\to 0$. However, the resulting point interaction is well-defined
because it satisfies the two-sided (at $x=\pm\,0$) boundary conditions on the
wave function $\psi(x)$ and its derivative $\psi^{\prime}(x)$ [9]. Indeed, on
the resonance set $\Sigma$, the corresponding connection matrix reads
$(-1)^{n}I$, where $I$ is the identity matrix and $n=1,\,2,\,\ldots\,$. Beyond
the set $\Sigma$, the boundary conditions are of the Dirichlet type:
$\psi(\pm\,0)=0$.

The whole system studied in the present paper consists of two separated (by
some distance) parallel homogeneous layers, where an extremely thin potential
well (referred to as a $W$-subsystem or a prewell) is auxiliary to a
background regular structure (called a $B$-subsystem or a $B$-layer), the
potential profile of which has an arbitrary shape. It is remarkable, as
illustrated by Fig. 3, that the decrease in prewell thickness makes the
resonant transmission through the $B$-layer more pointed and significantly
reduced between the resonance points. Therefore, adjusting the size parameters
of the prewell, it is possible to control the net current in the whole system.
This effect can be realized as a filter in nanodevices.

An interesting scenario as $\varepsilon\to 0$ has been observed when a
squeezing prewell is incorporated nearby the regular $B$-layer having a non-
empty spectrum of bound states. At each point $a_{n}=n\pi/\sqrt{d}$,
$n=1,\,2,\,\ldots\,,$ from the resonance set $\Sigma$ that corresponds to the
squeezed prewell, a new bound state level in the whole $W$–$B$ system is
detaching from the origin $\kappa=0$ at finite values of $\varepsilon>0$.
However, in the limit as $\varepsilon\to 0$, equally the same number of the
lowest levels escape to infinity. As a result, at resonant points of the set
$\Sigma$, the prewell does not affect the spectrum of the $B$-subsystem.
Outside $\Sigma$, the spectrum is modified obeying Eq. (23) and being the same
along all the intervals $a_{n-1}<a<a_{n}$ , $n=1,\,2,\,\ldots\,.$

## CRediT authorship contribution statement

Yaroslav Zolotaryuk: Conceptualization, Formal analysis, Investigation,
Software, Validation, Visualization, Writing – review & editing. Alexander V.
Zolotaryuk: Conceptualization, Formal analysis, Investigation, Methodology,
Writing – original draft, Writing – review & editing.

## Declaration of competing interest

The authors declare that they have no known competing financial interests or
personal relationships that could have appeared to influence the work reported
in this paper.

## Data availability

Data will be made available on request.

## Acknowledgements

We would like to thank the Armed Forces of Ukraine for providing security to
perform this work. The authors acknowledge ﬁnancial support from the National
Academy of Sciences of Ukraine by projects No. 0122U000887 (Y.Z.) and No.
0122U000888 (A.V.Z.). This work was also supported by the Simons Foundation.

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