Tuning the Quasi-bound States of Double-barrier Structures: Insights from Resonant Tunneling Spectra

Abstract

In this work, we study the resonant tunneling (RT) of electrons and H atoms in double-barrier (DB) systems. Our numerical calculations directly verify the correspondence between the resonant tunneling energies and the energy levels of quasi-bound states (QBS) within the double barriers. Based on this, in-depth analyses are carried out on the modulation of QBS energy levels and numbers which show step variation with the inter-barrier spacing. The mathematical criterion for the existence of QBS is derived, and the impacts of the barrier width and barrier height on QBS levels are investigated. Taking the rectangular double-barrier as an example, we have studied the manipulation of electronic structures and optical properties of the inter-barrier region (quasi-potential well) by tuning the inter-barrier spacing (width of quasi-potential well). Atom-like optical absorption features are found in the range of infrared to visible spectrum, which can be continuously tuned by the variation of quasi-potential well width. The potential application of double-barrier nanostructures in ultrahigh-precision detection of electromagnetic radiations is demonstrated.

Full text

Tuning 1 the Quasi-bound States of Double-barrier Structures: Insights from
Resonant Tunneling Spectra
Wei Li∗
Center for Fundamental Physics, School of Mechanics and opticelectrical Physics,
Anhui University of Science and Technology, Huainan, Anhui 232001, People’s Republic of China
Yong Yang†
Key Lab of Photovoltaic and Energy Conservation Materials, Institute of Solid State Physics,
HFIPS, Chinese Academy of Sciences, Hefei 230031, China and
Science Island Branch of Graduate School, University of Science and Technology of China, Hefei 230026, China
(Dated: November 12, 2024)
In this work, we study the resonant tunneling (RT) of electrons and H atoms in double-barrier
(DB) systems. Our numerical calculations directly verify the correspondence between the resonant
tunneling energies and the energy levels of quasi-bound states (QBS) within the double barriers.
Based on this, in-depth analyses are carried out on the modulation of QBS energy levels and num-
bers which show step variation with the inter-barrier spacing. The mathematical criterion for the
existence of QBS is derived, and the impacts of the barrier width and barrier height on QBS levels
are investigated. Taking the rectangular double-barrier as an example, we have studied the manipu-
lation of electronic structures and optical properties of the inter-barrier region (quasi-potential well)
by tuning the inter-barrier spacing (width of quasi-potential well). Atom-like optical absorption fea-
tures are found in the range of infrared to visible spectrum, which can be continuously tuned by the
variation of quasi-potential well width. The potential application of double-barrier nanostructures
in ultrahigh-precision detection of electromagnetic radiations is demonstrated.
I. INTRODUCTION
Resonant tunneling (RT) is a unique phenomenon in
quantum tunneling, specically occurring in a double-
barrier (DB) system. In this scenario, an incident parti-
cle can traverse the barriers with a probability of 100%.
The exploration of RT began with the foundational the-
oretical and experimental work by Tsu, Esaki, Chang,
and others in the 1970s [1–3], gaining considerable atten-
tion in the 1980s [4–11]. Research on RT continues today
[12–19], largely due to its applications in microelectronic
devices such as resonant tunneling diodes [12, 20–22].
Recent studies have investigated dynamic RT through
quasi-bound superstates (QBSS) generated by oscillating
delta-function potentials [15].
The prevalent conceptual framework for RT suggests
that the energy of the incident particle aligns with one
of the quasi-bound states (QBS) energy levels formed
within the DBs, leading to resonance and constructive
interference of the wave functions, thus maximizing tun-
neling probability [2, 3, 7, 15, 23, 24]. However, this
picture remains largely hypothetical and has yet to be
substantiated through numerical or experimental veri-
cation. In a related work, one of the authors has per-
formed a systematic investigation of quantum tunneling
through DBs of arbitrary shape [18], establishing gen-
eral conditions for RT. This work demonstrates that RT
can be realized for any particle with incident energy less
than the barrier height by adjusting the distance between
∗Corresponding author: wliustc@aust.edu.cn
†Corresponding author: yyanglab@issp.ac.cn
the barriers. This implies that continuous tuning of the
QBS energy levels is possible through manipulation of
the barrier distance, eectively modifying the width of
the quasi-potential well.
Based on a number of DB systems, we have metic-
ulously examined in this paper the RT of electrons, as
well as the RT of H atoms whose tunneling eects have
been demonstrated experimentally [25–27]. Comprehen-
sive numerical simulations have validated the one-to-one
correspondence between the RT energies and the QBS
energy levels within the double-barrier region. Utiliz-
ing this insight, we conducted a detailed analysis of how
varying the distance between the barriers inuences the
position and abundance of QBS levels. Additionally, we
elucidated the mathematical conditions necessary for the
existence of these energy levels and explored the impacts
of barrier width and height variations.
Focusing on rectangular DB systems, we investigated
the eects of inter-barrier spacing (the width of the quasi-
potential well) on the electronic structures and optical
properties of the region. This examination highlights the
free-atom-like electronic and optical features of QBS in
DB nanostructures, which enable potential applications
in ultrahigh-precision detection of electromagnetic radi-
ation, underscoring their transformative potential in the
elds of nanoelectronics and nanophotonics.
The rest of this paper is organized as follows. Section
II demonstrates numerically the correspondence between
the RT energies and QBS levels. Section III elucidates
how the spacing of potential barriers can modulate the
energy levels of QBS and deduces the conditions that
must be met with for RT to take place. Section IV ex-
amines the optical properties of the quasi-potential well

2
region, with a particular focus on the eects of varying
the inter-barrier spacing (width of quasi-potential well)
and its potential applications in the ultrahigh-precision
detection of electromagnetic waves within the infrared
spectrum.
II. CORRESPONDENCE OF THE TUNNELING
SPECTRA AND THE QUASI-BOUND STATES
In this section, we study the one-to-one correspondence
between the QBS energy levels and the RT energy levels,
using rectangular double-barriers (DBs) as the model sys-
tems. Numerically, the QBS energy levels were obtained
by solving the Schrödinger equation in one-dimensional
(1D) systems [−ℏ2
2m
∂2
∂x2+V(x)]ψ(x) = Eψ(x)where m
denotes the particle mass, such as the electron or a hy-
drogen atom considered in this study, ℏis the reduced
Planck constant, and V(x)represents the potential func-
tion. We consider a symmetrical DB quantum-well struc-
ture illustrated in Fig. 1(a), which can be easily general-
ized to asymmetrical congurations and will not be con-
sidered here for simplicity. A potential well of width wis
located between two barriers each of which with a barrier
width a, and barrier height V0, respectively. The wave-
functions [ψ(x)] and eigenvalues (E) related to QBS can
be obtained by exact diagonalization of the Schrödinger
equation in real space, subjected to the boundary condi-
tion of ψ[±L] = 0, with x=±Lbeing the edge sites of
the DB.
The quantum tunneling across double-barriers of any
shape can be quantied using the transfer matrix method
(TMM), a powerful technique for studying the transmis-
sion properties in nonperiodic systems [1, 16–18, 28–31].
For the propagation of a quantum particle across a single
barrier V(x)with compact support (the intrinsic prop-
erty of physical interactions), the transmitted and re-
ected amplitudes (AL, BL;AR, BR) of the wave func-
tions (ψL, ψR) may be related by a transfer matrix (de-
noted by M) as follows [16–18]
AR
BR=MAL
BL=m11 m12
m21 m22  AL
BL.(1)
The incoming wave function (with incident energy E)
is expressed by ψL=ALeikx +BLe−ikx, and the out-
going wave function is ψR=AReikx +BRe−ikx , where
k=p2mE/ℏ2, and mis the particle mass. The deter-
minant |M|= 1, for systems where time-reversal sym-
metry preserves, and the transmission coecient is given
by T=1
|m11|2. In general, the matrix elements mij (the
subscripts i, j = 1, 2) are complex numbers and obey
the conjugate relations m11 =m∗
22 and m12 =m∗
21.
To numerically calculate the transmission coecient of
a quantum particle, the entire barrier is sliced to obtain
a chain of rectangular potential barriers (V1,V2, …, Vj,
…, Vn−1,Vn). Transmission through each of these rect-
angular potential barriers is similarly described using the
aforementioned Eq. (1), via a transfer matrix (Mj). The
global transfer matrix Mis obtained as follows:
M=Y
j=n...1
Mj=m11 m12
m21 m22 (2)
The transmission coecient is calculated by
Tr(E) = 



AR
AL



2
×KR
KL
=|M|2
|m22|2×KR
KL
(3)
where AL, AR;KL, KRare the incident amplitude, the
transmitted amplitude, the incident wave vector and the
transmitted wave vector, respectively; |M|is the deter-
minant of M. It follows that the condition when the
transmission coecient Tr(E) = 1 corresponds to RT.
The studies on the one-to-one correspondence of RT
and QBS levels in DB systems have been carried out for
electrons and hydrogen (H) atoms, to demonstrate the
universality of the picture and to highlight the eects of
particle mass. The electron QBS levels and wave func-
tions, as determined by the exact diagonalization method
[32], and the RT levels calculated by the TMM are pre-
sented in Fig. 1. It is clearly seen that the QBS are
predominantly conned within the quasi-potential well,
with only a small portion extending into the barrier re-
gion [Figs. 1(c-h)]. If the potential barrier is suciently
large, the penetration depth may be characterized by
d∼ℏ
2√2m(V0−E), under the WKB approximation. It
is evident that as the energy increases, the penetration
depth also becomes larger. The ground-state wave func-
tion exhibits even parity, while the rst excited state
shows odd parity, and the second excited state exhibits
even parity, and so forth: An even-odd-even-odd alter-
nating parity pattern of QBS wave functions present.
Based on the tunneling spectra (Fig. 1), the incident
energies corresponding to RT, or the RT levels are read-
ily obtained.
Furthermore, the full-width-at-half-maximum
(FWHM) of each RT peak (i.e., the energy broad-
ening, denoted by σ) can be deduced and used to
estimate the lifetime of RT levels and consequently the
lifetime of QBS levels. The calculated energy levels
using the two methods are listed in Table I, along with
the energy broadening and parities of wave functions.
The DB system contains 15 QBS of electrons with the
following parameters: barrier height V0= 3 eV, barrier
width a= 10 Å, and well width w= 50 Å. The results
regarding the quantum nature of H atoms in a DB
system, i.e., the QBS levels and wave functions, and
quantum tunneling as a quantum particle, are depicted
in Fig. 2 and compared in Table II. By examining the
characteristics of wave functions displayed in Figs. 1
and 2, the following facts are evidenced: There is no
node for the ground-state wave function, there is one
node for the rst excited state, and more generally, the
nth QBS wave function ψnhas (n−1) nodes. Such an
observation is in line with Sturm’s theorem.

3
log
FIG. 1. Correspondence of the quasi-bound states (QBS) energy levels of electrons (15 levels) and the resonant tunneling
(RT) energies for electrons. The parameters for the DBs system are: barrier height V0= 3 eV, barrier width a= 10 Å, well
width w= 50 Å. Top panels (a-h): The results derived from the exact diagonalization method. (a) A schematic diagram of
the double-barrier system, with the quasi-potential well shaded by shallow pink; (b) Distribution of energy levels, with a blue
dashed lines indicating the positions of the barrier heights V0= 3.0 eV. For graphical clarity, the data lines representing distinct
QBS levels along the the vertical axis, have been scaled by their respective energy level coecients. (c, e, g) Distribution of
wave functions for n= 1 −12 and the corresponding probability distributions in (d, f, h), respectively. Bottom panels: On the
left, the transmission spectra calculated by the transfer matrix method (TMM), with the transmission probability plotted on
a logarithmic scale. The energy levels are list in Table I. The bottom right panel, shows the relationship between the number
of QBS and the barrier spacing (the width of the well) w. The red straight dashed line represents the linear t n=Aw +B
with A= 0.2806, B = 0.4576.
As shown in right bottom part of both Figs. 1 and 2,
the number of QBS levels shows a stepwise increase with
the width of the potential well w, which can be approxi-
mated by the formula n=Aw +B, a relationship that is
inherently determined by the de Broglie wavelength λd
of the particle. In the case of an ideal innite square po-
tential well, stationary wave solutions are possible only
when the width of the well is an integer or half-integer
multiple of the de Broglie wavelength for a given particle
energy. Likewise, in the scenario involving a double po-
tential barrier, the condition wn∝nλdis still satised
[18].
From the precise RT levels and the QBS levels enumer-
ated in Table I for electrons and Table II for H atoms,
it is clearly seen that the QBS levels obtained by ex-
act diagonalization correspond one-to-one to the RT lev-
els in the transmission spectrum, numerically conrm-
ing the aforementioned physical picture directly. On the
other hand, due to the large dierence of particle mass
(me/mH∼1/1837), the energy scale to show remark-
able quantum eects at similar spatial scale is dierent.
This is evidenced from the barrier height of DB systems
under investigation: V0= 3.0eV vs V0= 0.05 eV. The
dierence is understandable from the de Broglie wave-
length λd=h
√2mE , which requires that the energy ratio
EH/Ee=me/mH∼1/1837 for the same λd. Such a
magnitude of barrier height can be encountered in the
diusion of H atoms on some realistic systems such as
Pt(111) surface [17] or the graphene surface [33] where
the van der Waals interactions are dominant.

4
TABLE I. Correspondence of the RT energies and the QBS levels of electrons. σ: energy broadening of RT peaks (equals to
FWHM). The parities of the wave functions of the QBS P(ψn)are listed. The parameters for the DB system are the same as
in Fig. 1.
nRT (eV)σ(eV) QB (eV) P(ψn)
1 0.01386653 7.29270116×10−12 0.01376915 even
2 0.05545493 6.32678284×10−11 0.05506559 odd
3 0.12473090 2.86959844×10−10 0.12385562 even
4 0.22163489 7.84563775×10−10 0.22008077 odd
5 0.34607818 1.79778014×10−90.34365395 even
6 0.49793645 4.77584527×10−90.49445318 odd
7 0.67703949 1.54043764×10−80.67231165 even
8 0.88315491 4.62366565×10−80.87700186 odd
9 1.11596083 1.32685825×10−71.10820927 even
10 1.37499794 3.47897480×10−71.36548613 odd
11 1.65957780 1.27538303×10−61.64816373 even
12 1.96858512 5.41751317×10−61.95516505 odd
13 2.29997023 2.62593281×10−52.28453108 even
14 2.64903860 2.33287659×10−42.63186215 odd
15 2.99971227 2.71227891×10−32.98244627 even
TABLE II. Similar to Table I but for H atoms, with V0= 0.05 eV,a= 10 Å, w= 10 Å.
nRT (eV)σ(eV) QB (eV) P(ψn)
1 0.00019028 1.29597178×10−48 0.00018891 even
2 0.00076102 1.98128875×10−47 0.00075554 odd
3 0.00171186 1.67422238×10−46 0.00169953 even
4 0.00304220 1.26036784×10−45 0.00302030 odd
5 0.00475115 1.70737103×10−44 0.00471696 even
6 0.00683749 2.26009733×10−43 0.00678833 odd
7 0.00929958 5.64252190×10−42 0.00923278 even
8 0.01213524 1.30591663×10−40 0.01204818 odd
9 0.01534157 7.63734659×10−39 0.01523170 even
10 0.01891467 7.65211908×10−37 0.01877952 odd
11 0.02284908 1.21086224×10−34 0.02268632 even
12 0.02713692 5.70105862×10−32 0.02694444 odd
13 0.03176596 6.25819418×10−29 0.03154206 even
14 0.03671522 3.20658167×10−25 0.03645906 odd
15 0.04194171 1.61833083×10−20 0.04165478 even
16 0.04731380 1.03883988×10−13 0.04700944 odd

5
3
0 5 10 15 20 25 30
x
(
Å
)
0.00
0.01
0.02
0.03
0.04
0.05
V
(eV
)
(a)
0.00 0.02 0.04 0.06
E
n
(eV)
0.0
0.2
0.4
0.6
0.8
1.0 (b)
0 5 10 15 20 25 30
x
(
Å
)
-0.4
-0.2
0.0
0.2
0.4
Ψ
(
x
)(a
u
.
)
(c)
0 5 10 15 20 25 30
x
(
Å
)
0.000
0.025
0.050
0.075
0.100
0.125
0.150
0.175
0.200
ρ
(
x
)
(d)
0 5 10 15 20 25 30
x
(
Å
)
-0.4
-0.2
0.0
0.2
0.4
Ψ
(
x
)
(
au
.
)
(e)
0 5 10 15 20 25 30
x
(
Å
)
0.000
0.025
0.050
0.075
0.100
0.125
0.150
0.175
0.200
ρ
(
x
)
(f)
0 5 10 15 20 25 30
x
(
Å
)
-0.4
-0.2
0.0
0.2
0.4
Ψ
(
x
)(a
u
.
)
(g)
0 5 10 15 20 25 30
x
(
Å
)
0.000
0.025
0.050
0.075
0.100
0.125
0.150
0.175
0.200
ρ
(
x
)
(h)
0.00 0.02 0.04 0.06
En
(eV)
-80
-60
-40
-20
0
log
10(
Tr
(
E
))
0 2 4 6 8 10
well width w
(
Å
)
0
5
10
15
n
n
n
= 1
.
5542
w
+ 0
.
3904
FIG. 2: Similar to Fig. 1 but for H atoms. The parameters for the double-barriers system are:
V0= 0.05 eV, a = 10 Å, w = 10 Å. The red dashed line of bottom panel represents the linear t
n= 1.5542w+ 0.3904.
x
x x x
xx
FIG. 2. Similar to Fig. 1 but for H atoms. The parameters for the double-barriers system are: V0= 0.05 eV,a= 10 Å, w= 10
Å. The red dashed line of bottom panel represents the linear t n= 1.5542w+ 0.3904.
Aside from the good agreement between the RT and
QBS levels, there are still minor dierences as seen from
Table I and II . The exact diagonalization method can
in principle provide exact solutions for the system, in-
cluding energy levels and wave functions which can be
used to calculate the transfer probability between dif-
ferent states. Practically, high-precision numerical re-
sults from exact diagonalization require substantial com-
putational resources. In the results presented in Fig. 1,
the variation step size along the x-axis is 0.01 Å, and
the dimension of the matrix to be diagonalized is about
10000 ×10000. For simple potential barrier structures
as that considered in this study, the TMM oers signif-
icantly higher computational eciency for lower energy
spacing. Nevertheless, to accurately determine the posi-
tions of resonant energy levels, it is necessary to employ
specialized computational techniques to approximate the
energy points where the transmission probability is 1.
Taking the rst data row in Table I as an example, the
FWHM of the lowest RT is σ= 7.29270116 ×10−12 eV
which is very small compared to the energy levels of the
system. This implies that the transmission probability
drops very rapidly even a tiny deviation from the RT
levels, hence a high energy resolution is required to ac-
curately determine the exact numerical values of the RT
levels.
In our calculations, we have achieved a very high
level of precision, by setting a convergence criterion of
|1−T r(E)|<10−60, which strongly ensures that the
system has experienced RT. The results of listed in Ta-
ble II indicate that, compared to electrons, much more
stringent condition has to be met with for the RT of H
atoms, with the energy window (i.e., energy broadening
σ) of the transmission spectrum being much narrower
than in the case of electrons. For the lowest tunneling
level (E∼0.00019 eV), H atoms can still completely tra-
verse the double potential barriers, albeit requiring very
precise incident energy with a broadening of σ∼10−48
eV, comparing to the case of electron RT via the lowest
QBS level (σ∼10−12 eV). From both Table I and II, it

6
is also seen that the energy broadening for RT increases
with higher levels, relaxing gradually the constraint on
the monochromaticity of incident energy. Such a critical
condition of RT provides an opportunity for measuring
the energy of incident particles with ultrahigh accuracy.
III. DEPENDENCE OF QUASI-BOUND STATES
ON THE DOUBLE-BARRIER GEOMETRIES
In this section a detailed analysis is carried out to
study the dependence of QBS on the key parameters de-
scribing the double-barrier (DB) geometries: The inter-
barrier spacing, the width of single barrier, and the bar-
rier height.
A. The Variation of QBS Levels with Inter-barrier
Spacing
It is evident that the QBS levels and/or RT levels are
closely related to the inter-barrier spacing, that is, the
quasi-well width. Before presenting the numerical re-
sults, we rst conduct some theoretical analysis. The
transmission across a single rectangular barrier (the di-
agonal matrix element m11 in Eq. (1)) can be expressed
as follows [16–18]
m11 = 2γe−ika [i(k2−β2) sinh(βa)+2kβ cosh(βa)],(4)
where k=p2mE/ℏ2,β=p2m(V0−E)/ℏ2, and γ=
1
4βk ,Cm=q2m
ℏ2. The Eq. (4) can be rewritten as
m2
11 = 4γ2σ2ei2(α−Cma√E),(5)
where σ=√A2+B2,A= (k2−β2) sinh(βα),
B= 2βk cosh(βα), and the angle α= arctan( A
B) =
arctan(δtanh(Cmap(V0−E))),Cm≡q2m
ℏ2,δ≡(β
k−
k
β). Drawing upon the theorem presented in Ref. [18] the
corresponding width of well is w=wn=nπ
k−π+θ+2ka
2k,
where θ= arg(m2
11). The number of QBS levels nscales
stepwise with the width of the potential well wwith dif-
ferent step for a given incident energy E.
The dependence of the QBS (energies and level counts
n) on the inter-barriers spacing at dierent barrier
heights for electrons and H atoms are shown in Fig. 3
and 4, respectively. Despite the dierent order of mag-
nitudes, the QBS levels of electrons and H atoms exhibit
a remarkably similar variation trend with inter-barriers
spacing at various barrier heights.
When the term π+θ+2ka
2kis negligible with comparison
to the inter-barrier spacing (quasi-well width w), i.e.,
π+θ+2ka
2k≪wn=w, one approximately has w≈nπ
k,
which yields E≈n2π2ℏ2
2mw2. This indicates that the QBS
levels decrease monotonically with w, reducing the en-
ergy gap between each of the QBS, and eventually con-
verge to the exact bound levels in an innite-depth square
potential well where En=n2π2ℏ2
2mw2. This is independent
of the geometries of the potential barriers. The variation
trend is clearly demonstrated in Fig. 5, where the rst
ve QBS levels (Ei) of electron and H atom are shown
along with the linearly tted data lines as a function of
the integer n2. Furthermore, compared to the RT levels
across a nite-depth square potential well, such a math-
ematical expression diers only by a constant (i.e., the
well depth V0) [34]. The asymptotic coincidence of the
QBS levels in arbitrarily shaped DBs with the exact so-
lutions for an innite-depth square potential well, and its
similarity to the RT levels for a nite-depth square po-
tential well reveal the intrinsic connection between these
quantum systems.
The range of wave vectors for incident particles dif-
fers with varying barrier heights, leading to a variation
in the number of energy levels and the linear coe-
cient associated with the quasi-well width, which is di-
rectly proportional to the maximum value of QBS en-
ergy (√Emax ∼√V0). Consequently, the width of the
energy level steps is inversely proportional to height of
barrier, as evidenced from Fig. 3. As pointed out in Ref.
[18], the resonant states (equivalently, the QBS) appears
periodically with inter-barrier spacing (quasi-well width)
wvia the variation step ∆w=π
k∼h
2√2mV0, which is
just the step width associated with the variation of QBS
numbers. In the case of H atoms, the much larger parti-
cle mass and consequently the much smaller ∆wleads to
an almost linear variation of QBS numbers with w(see
lower panels of Fig. 4).
B. Parameter Space for QBS
Upon examining Figs. 3 and 4, it becomes evident
that RT of quantum particles is not ubiquitous across
varying barrier heights and quasi-well widths. Speci-
cally, it can be expected that QBS are absent when the
quasi-well is narrow enough or the barrier height is not
large enough. Such a constraint is intrinsically contained
in the mathematical expression of wn, which gives that
w1=(π−θ)ℏ
2√2mV0−a. It requires that w≥w1to have RT
to take place in a DB system, or equivalently, to guar-
antee at least one QBS presents in the quasi-well region.
The necessary and sucient condition to have only one
QBS level is therefore w1≤w≤∆w=π
k. The results
of more generalized analysis are shown in Fig. 6, where
a boundary line can be drawn in the parameter space
of the barrier height (V0) versus well width (w), for the
existence/absence of QBS of electrons and H atoms. It
is evident that RT is possible only when the system’s
parameters are positioned above the boundary line. On
the boundary, there exists exactly one QBS, and its en-
ergy level is close to the barrier height. Utilizing this
diagram, we are able to ascertain the specic parame-
ters related to DB systems that allow the occurrence of
RT. The V0−wparameter constraint on the existence
of QBS levels is signicantly dierent from the case of

7
1 0
2215 21
FIG. 3. The dependence of the QBS on the inter-barriers spacing (w) at dierent barrier height V0= 3.0,1.0,0.5 eV(from left
to right) for electrons. Top panels: The rst ve energy levels (E1to E5). Bottom) panel: The number of QBS levels as a
function of w. The red dashed line represents the linear t n=Aw +B.
40
45
50
0.0000
0.0001
0.0002
0.0003
40
45
50
0.0000
0.0001
0.0002
0.0003
40
45
50
0.0000
0.0001
0.0002
0.0003
FIG. 4. Similar to Fig. 3 but for H atoms at dierent barrier height V0= 0.2,0.1,0.05 eV(from left to right).
a nite-depth square potential well, where at least one
even-parity bound state exists in spite of the width and
depth of the potential well [34].
C. Eects of Boundary Conditions on QBS Levels
In one-dimensional systems, the Schrödinger equation
is an ordinary dierential equation for which the bound-

8
0 5 10 15 20 25
0.0
0.2
0.4
0.6
0.8
2
(a) electron
= 60
= 50
= 40
= 30
0 5 10 15 20 25
0.0000
0.0002
0.0004
0.0006
= 60
= 50
= 40
= 30
2
(b) H
FIG. 5. Dependence of QBS levels of electrons (left) and H atoms (right) on the square of the principle quantum number n,
for a number of quasi-well width w(in units of Å).
FIG. 6. V0−wparameter space for the RT of electrons (left) and H atoms (right) at a given barrier width (a= 10 Å).
ary conditions play a nontrivial role. In this subsec-
tion, we study how the boundary conditions and double-
barrier geometries would aect the eigenvalues (QBS lev-
els). Specially, we study the dependence of QBS levels on
barrier width, for a symmetrical DB with a xed inter-
barrier spacing (quasi-well width) w. As illustrated in
Fig. 7, we pay attention to the investigation of three
types of boundary conditions and delve into the eects
on the QBS levels of the DB systems. Table III summa-
rizes the QBS levels obtained by the exact diagonaliza-
tion at dierent DB boundary conditions. Despite the
small magnitude of variations, the impact of dierent
barrier widths on the position of QBS energy levels is
generally negligible.
IV. TUNABLE OPTICAL PROPERTIES OF THE
DOUBLE-BARRIER SYSTEMS
In this section, we study the light absorption proper-
ties of the system. The response functions, i.e., the di-
electric function ϵ=ϵ1+iϵ2and the optical conductivity
σ=σ1+iσ2, are pivotal in characterizing the interac-
tions between applied electromagnetic elds and materi-
als. These functions encompass both real and imaginary
components, which are crucial for understanding the ma-
terial’s response to electromagnetic waves. In general,
ϵ(ω)and σ(ω), represent complex-valued functions of an-
gular frequency ω. The real component of the optical
conductivity, denoted as σ1, is instrumental in determin-
ing the absorption within the medium, as it inuences
the imaginary part of the dielectric function, ϵ2. Con-

9
TABLE III. The QBS levels of electrons in rectangular DB, calculated using dierent types of boundary conditions (BC) as
schematically shown in Fig. 7: ψ[±(a+w
2)] = 0 for BC1, ψ[±(2a+w
2)] = 0 for BC2 and ψ[±∞]=0for BC3. The parameters
describing the DB system: V0= 3.0 eV,w= 50 Å.
n EBC 1(eV)EBC 2(eV)EBC 3(eV)
1 0.013768 0.013766 0.013771
2 0.055064 0.055052 0.055071
3 0.123852 0.123826 0.123868
4 0.220074 0.220027 0.220103
5 0.343644 0.343570 0.343697
6 0.494438 0.494332 0.494548
7 0.672291 0.672147 0.672531
8 0.876975 0.876787 0.877498
9 1.108175 1.107937 1.109278
10 1.365443 1.365151 1.367682
11 1.648110 1.647758 1.652489
12 1.955096 1.954681 1.963466
13 2.284425 2.283923 2.300354
14 2.631483 2.632941 2.662871
15 2.973093 2.983424 3.050714
8
FIG. 7: Schematic diagrams for dierent boundary conditions of DB systems, where varying barrier
widths present.
FIG. 7. Schematic diagrams for dierent boundary conditions
of DB systems, where varying barrier widths present.
versely, the imaginary component of the optical conduc-
tivity, σ2, contributes to the real part of the dielectric
function, ϵ1, which in turn aects the polarization of the
material [35]. The components ϵ1and ϵ2are critical for
understanding how the material inuences the propaga-
tion of light. Specically, ϵ1provides insights into the
retardation of light’s velocity, while ϵ2accounts for the
absorption and loss of light energy due to polarization
as it traverses across the material medium. A compre-
hensive understanding of the dielectric function is vital
for the analysis and application of materials in various
optical and electronic devices.
For simplicity without losing generality of the results,
we consider the situation in which the incident light wave
is a monochromatic plane wave, to investigate the optical
properties of DB systems based on electric dipole tran-
sition. From the picture of medium absorption [36], the
imaginary part of the corresponding dielectric function
ϵ2(ω)can be given as follows (refer to the details in Ap-
pendix A)
ϵ2(ω) = πe2
6ℏϵ0Ω0|⟨k|r|n⟩|2Jnk(ω) = πe2
6ℏϵ0w|⟨k|r|n⟩|2
S0
δ(En−Ek−ℏω)(6)

10
where ϵ0is the vacuum permittivity, eis elementary elec-
tric charge, Ω0is the volume of the well region, wis the
width of the barrier, ris the electron coordinate in real
space and S0is the cross section area of the quasi-well
region within the DB system. Jnk (ω) = δ(En−Ek−ℏω)
is the joint density of states (JDOS), representing the
energy level distribution corresponding to the quantum
transition between the nth and kth level with an energy
dierence of Enk =ℏω.In semiconducting systems, it cor-
responds to the JDOS as determined by the valence and
conduction bands [37]. The term |⟨k|r|n⟩|2determined
by the energy level distribution and the transition matrix,
reects the constraints imposed by the selection rules on
the light absorption spectra of the electron bound states.
Utilizing the wave functions obtained from extract di-
agonalization, we have directly computed the matrix el-
ements |⟨k|r|n⟩|2and the complex dielectric function ϵ2
numerically. In the calculations, we take k= 1,2,3(oc-
cupied states), n= 2,3, . . . , N (unoccupied states); N
represents the total number of QBS levels. For the quasi-
well width w= 10,20,30,40,50,60 Å, with a cross-
sectional area S0=a2= 100 Å2, the quantities Jnk(ω)
and ϵ2,nk(ω)have been calculated separately. This ap-
proach will provide a direct insight into how the spac-
ing of the potential barriers inuences the distribution of
electronic energy levels and, consequently, the modula-
tion of optical absorption characteristics. The numerical
results are presented in Tables IV-V and Tables VI-IX (
see Appendix B) and Figs. 8-9.
First, we study the characteristics of JDOS and show
the eects of optical selection rules. In the context of
electric dipole transitions, the electric dipole operator

D=−er is an odd function of coordinates, therefore
non-zero matrix elements |⟨k|r|n⟩|2= 0 is possible only
for transitions between the states with opposite parity
of wave functions. The corresponding selection rule is
therefore
Odd parity state ⇆Even parity state (7)
For the one-dimensional systems discussed herein, the
wave functions are plane waves along the xaxis, it is only
necessary to consider the parity selection rule. Addition-
ally, the matrix element representing the transition prob-
ability is largely determined by the overlap of the wave
functions of the initial and nal states. A larger dier-
ence in principal quantum numbers can lead to a more
signicant overlap, especially in cases where the electron
hops to a more diuse orbital (see the wave functions
in Fig. 1). Our numerical data corroborate this asser-
tion. Taking results from Table IV as an example, we
calculated the optical absorption properties for transi-
tions from the k= 1 state to the nth states when the
well width w= 10,20 and 30 Å. The results demonstrate
that eective absorption only occurs between the k= 1
state and states whose quantum number nis an even
number, where a non-zero ϵ2presents. It is evident that
this is determined by the parity of the wave functions of
QBS, adhering to the parity selection rule. As presented
in Fig. 1 and Table I, the principle quantum number
ndesignating the QBS levels exhibits consistent parity
characteristics with the wave functions.
The time scale related to the light absorption process
can be investigated through the JDOS Jnk(ω). By ap-
proximating the delta function δ(En−Ek−ℏω)with a
normalized Gaussian function 1
√2πΓne−(En−Ek−ℏω
Γn)2and
leveraging the energy-time uncertainty relationship, the
inverse of broadening Γnreects the lifetime of opti-
cal transition. The broadening of the lowest QBS en-
ergy level σ1is minimal and signicantly smaller than
that of higher energy levels (See Table I). Therefore,
it is reasonable to approximate the broadening of the
JDOS Γnwith the broadening of the nal state σn. The
JDOS at ℏω=En−Ek=Enk can be rewritten as
Jnk(ω) = 1
√2πΓn. Clearly, the level broadening σnin-
tensies with the increment of n, the peaks of the JDOS
decrease rapidly, and the associated absorption lifetime
reduces correspondingly. From Table IV and V, the
JDOS varies from 102to 1010 eV−1. In natural units,
1eV−1≈6.582 fs, hence the lifetime of the light absorp-
tion process spans the range of ∼0.1ps to 10 µs. This
timescale is considerably longer than the typical value
(∼ps) observed for optical transitions in semiconductors
[37–39].
The imaginary part of the dielectric function ϵ2for a
number of transition processes are listed in Tables IV -
IX (Tables VI-IX, see Appendix B). Obviously, a larger
dierence in the principal quantum number n, corre-
sponds to the absorption/emission of higher-energy (and
typically shorter wavelength) photons, which is a gen-
eral feature observed in the gaseous phase of atoms or
molecules. This principle is employed in the design of
solar cells and light-emitting diodes (LEDs), where the
absorption and emission of light are critical. By adjusting
the Fermi level, the DB systems based on real materials
can be designed to absorb a specic range of the solar
spectrum more eciently or emit light at desired wave-
lengths. Considering the extraordinary light absorption
capabilities and the unprecedented energy resolution of
DB systems, it is possible to engineer systems that can
accurately measure the energy of electromagnetic waves
across the domain of infrared/THz and even into the
visible spectrum. With an energy resolution as low as
10−10 eV, these systems oer an unprecedented level of
sensitivity and precision. Such renement would be in-
valuable for a spectrum of scientic and technological
applications, including sophisticated spectroscopic anal-
ysis, quantum optical experiments, and the innovation of
cutting-edge optoelectronic devices [40–42].
To elucidate the role of selection rules in the light ab-
sorption process and to showcase the inuence of vary-
ing well widths more eectively, Figs. 8 and 9 depict
schematic diagrams of the k= 1 →ntransitions, along
with the JDOS and absorption spectra. The left pan-
els employ red arrows to indicate permissible transitions,
while blue arrows with red crosses symbolize the prohib-

11
FIG. 8. The optical absorption properties of DB system of electrons. Left panels: Selection rules for transitions between
QBS energy levels (k→n). Red arrows denote permitted transitions, whereas blue arrows marked with a red cross indicate
forbidden transitions. Right panels: The JDOS Jnk (ω)and the imaginary part of the dielectric function ϵ2,nk (ω)corresponding
to transitions between dierent energy levels (k→n). The absorption of photon energy ℏω=Enk =En−Ek, which
corresponds to the energy level dierences, indicated by wavelength λ≈1240/∆E(nm), is represented by dierent colors from
red to purple. The color scale corresponds to an energy range of [0.0 - 3.0]eV or a wavelength range [∞- 413.3] nm. The
parameters same as in Table IV.

12
FIG. 9. The optical absorption properties of DB system of electrons at dierent well widths w= 40,50,60 Å, similar to Fig.
8.
ited transitions, providing a clear visual representation
of the selection rules as illustrated by Eq.(7). The ab-
sorption spectra on the right, which closely resembles
the absorption spectrum of elements, with discrete spec-
tral lines representing transitions from k= 1 to various
nal states n. It is evident that the JDOS Jnk on the left
vertical axis always present for any n, while the imagi-
nary part of the dielectric function ϵ2on the right vertical
axis exhibits spectral lines only for transitions that are
allowed by the selection rules. To enhance the spectral
line features, we have utilized a coloring scheme akin to
that of elemental absorption spectra, with the hue of each
line determined by its energy. As seen from Figs. 8 and
9, and the data listed in Tables IV-IX (Tables VI-IX, see
Appendix B), the imaginary part of the dielectric func-
tion ϵ2,nk(ω)for the transition between the low-lying en-

13
TABLE IV. The optical absorption properties of DB system of electrons at dierent well widths w= 10,20,30 Å. For each
well width, from left to right, they are as follows: The absorption of photon energy ℏω=Enk =En−Ek, the FWHM of
resonant peak Γn, the JDOS Jnk (ω)and the imaginary part of the dielectric function ϵ2,nk(ω)corresponding to transitions
between dierent energy levels (k= 1 →n). The symbol ”—” indicates forbidden transitions. The units of ℏωand Γnare eV
and the unit of Jnk(ω)is eV−1. Parameters : the barrier height V0= 3.0eV, the width of the barrier a= 10 Å, the inter-barrier
potential well width (from top to bottom) w= 10,20,30 Å, the initial QBS number k= 1.
w10 20 30
nℏωΓnJnk ϵ2ℏωΓnJnk ϵ2ℏωΓnJnk ϵ2
2 0.7334 3.1721×10−71.2577×1066.0601×1050.2282 3.5565×10−91.1217×1088.5992×1070.1090 6.2676×10−10 6.3652×1086.8027×108
3 1.8676 4.4571×10−58.9507×103— 0.6056 3.4509×10−81.1560×107— 0.2903 3.3568×10−91.1884×108—
4 1.1261 4.3497×10−79.1718×1054.5738×1030.5430 1.2890×10−83.0949×1072.1280×105
5 1.7771 7.8591×10−65.0762×104— 0.8660 6.9619×10−85.7304×106—
6 2.5225 2.9148×10−41.3687×1035.1509×10−11.2571 4.4837×10−78.8977×1054.6845×102
7 1.7128 3.3706×10−61.1836×105—
8 2.2251 4.6616×10−58.5580×1037.6659×10−1
9 2.7703 9.3803×10−44.2530×102—
TABLE V. The optical absorption properties of DB system of electrons at dierent well widths w= 40,50,60 Å and k= 1,
similar to Table IV.
w40 50 60
nℏωΓnJnk ϵ2ℏωΓnJnk ϵ2ℏωΓnJnk ϵ2
2 0.0636 1.8253×10−10 2.1856×1093.0029×1090.0416 6.3268×10−11 6.3056×1091.0595×1010 0.0293 3.1975×10−11 1.2477×1010 2.4790×1010
3 0.1694 8.2558×10−10 4.8323×108— 0.1109 2.8696×10−10 1.3902×109— 0.0781 1.0581×10−10 3.7703×109—
4 0.3174 2.3578×10−91.6920×1081.4913×1060.2078 7.8456×10−10 5.0849×1085.4746×1060.1465 2.7503×10−10 1.4505×1091.8459×107
5 0.5071 8.4105×10−94.7434×107— 0.3322 1.7978×10−92.2191×108— 0.2342 7.2055×10−10 5.5366×108—
6 0.7383 3.3829×10−81.1793×1077.9204×1030.4841 4.7758×10−98.3533×1076.8405×1040.3414 2.0686×10−91.9286×1081.8651×105
7 1.0103 1.2283×10−73.2479×106— 0.6632 1.5404×10−82.5898×107— 0.4680 3.8962×10−91.0239×108—
8 1.3222 3.7955×10−71.0511×1061.2019×1020.8693 4.6237×10−88.6283×1061.2000×1030.6138 9.4192×10−94.2354×1076.9485×103
9 1.6726 2.0394×10−61.9562×105— 1.1021 1.3269×10−73.0067×106— 0.7788 2.0561×10−81.9403×107—
10 2.0588 1.0996×10−53.6279×1041.0670×1001.3611 3.4790×10−71.1467×1064.1012×1010.9628 5.0196×10−87.9476×1063.3483×102
11 2.4751 1.0379×10−43.8439×103— 1.6457 1.2754×10−63.1280×105— 1.1656 1.1066×10−73.6050×106—
12 2.9027 1.4433×10−32.7642×1022.5154×10−31.9547 5.4175×10−67.3639×1048.7362×10−11.3870 2.9400×10−71.3570×1061.8959×101
13 2.2861 2.6259×10−51.5192×104— 1.6266 9.1721×10−74.3495×105—
14 2.6352 2.3329×10−41.7101×1037.8839×10−31.8838 2.9447×10−61.3548×1057.4663×10−1
15 2.9858 2.7123×10−31.4709×102— 2.1578 1.3265×10−53.0076×104—
16 2.4467 5.1916×10−57.6844×1031.8845×10−2
17 2.7466 3.7245×10−41.0711×103—
ergy levels (e.g., 1→2,1→4,1→6,2→3,2→5)
varies from the order of 102to 1010 for photon energies
ℏω∼0.02 to 0.7 eV, which is one to nine orders of magni-
tude higher than that of typical semiconductors [37, 43].
For photons with an energy ℏω=Enk =En−Ekthat
excites eective transition from the kth state to the nth
state with a sum energy broadening of ∆Enk ∼(Γn+Γk),
the variation range of frequency (∆ν) and wavelength
(∆λ) is therefore |∆νnk |
νnk =|∆λnk|
λnk =|∆Enk|
Enk . In the case of
1−→ 2transitions, the relative resolution for frequency
and wavelength can be as high as 10−8to 10−10 (see
e.g., Table IV). Such intense optical absorption features
point to the possibility of ultrahigh-precision detection
of infrared or THz radiations based on nano-sized DB
systems [44].
Now we study the impact of varying quasi-well widths.
As shown in Fig. 10, for a given (k, n), as the well width
increases, an increase in the well width correlates with a
decrease in the energy of the absorbed light (resulting in
longer wavelengths), and the corresponding level broad-
ening becomes smaller. Generally, for the optical transi-
tion between two states with principal quantum numbers
kand n(k < n), the resulted change in band gap due to a
small variation of quasi-well width ∆wmay be estimated
as follows: ∆Eg∼ −(n2−k2)ℏ2π2
2mw2×2( ∆w
w) = −2Eg∆w
w.
Therefore, a reduced quasi-well width leads to an en-

14
larged band gap and vice versa. On the other hand, the
the JDOS increases with well width, which in turn en-
hances the strength of light absorption. The width of the
potential barrier spacing (the quasi-well width), plays a
pivotal role in modulating the optical properties of the
well region in a quantum system. By engineering a DB
structure with tailored well widths and barrier heights,
one can precisely control the energy of the absorbed light
to fall within a specic range. For example, with a bar-
rier height V0= 3.0eV and well width of w= 24 Å, the
absorption spectrum is tuned to fall into the spectrum of
the visible light.
V. CONCLUSIONS
In this study, we have conducted an extensive investi-
gation into the RT of electrons and hydrogen atoms in DB
systems. Our numerical computations provide direct evi-
dence for the one-to-one correspondence between the RT
energies and QBS levels. Detailed analyses reveal how
inter-barrier spacing modulates the emergence and pro-
liferation of QBS, as well as key quantum characteristics
such as wave function parity and energy level distribu-
tion. Notably, the asymptotic behavior of the QBS levels
aligns surprisingly well with ideal quantum systems, i.e.,
the one-dimensional nite and innite square potential
wells. We observed a stepwise increase in the number of
QBS with increasing inter-barrier spacing, accompanied
by reduced energy gaps between adjacent levels. Ad-
ditionally, we identied critical thresholds essential for
QBS existence and examined how variations in barrier
width and height aect the number and position of these
levels. Using rectangular DBs as prototype systems, we
explored the inuence of boundary conditions and how
the electronic conguration and optical characteristics of
the well region can be nely tuned by adjusting inter-
barrier spacing (the quasi-well width). The investigated
DB systems exhibited atom-like optical absorption spec-
tra, which are continuously adjustable through manipu-
lation of inter-barrier spacing. Our ndings highlight the
intricate relationship between the geometric parameters
of the barriers and their resultant electronic and opti-
cal properties, oering valuable insights into the design
principles for nanostructures with tunable functionalities.
Moreover, by elucidating the free-atom-like characteris-
tics of their electronic structures and optical responses,
we demonstrated the potential of DB nanostructures as
ultrahigh-sensitivity detectors for electromagnetic radia-
tion. The ability to nely tune energy levels and optical
responses within these nanostructures opens avenues for
applications in ultrahigh-precision measurements, where
accurate detection and analysis of light absorption at the
nanoscale are critical.
ACKNOWLEDGMENTS
We acknowledge the nancial support from Na-
tional Natural Science Foundation of China (Grants No.
12074382, 11474285), and the Scientic Research Foun-
dation for High-level Talents of Anhui University of Sci-
ence and Technology under Grant YJ20240002. We are
grateful to the Hefei Advanced Computing Center for
support of supercomputing facilities. We also thank Pro-
fessors Yugui Yao and Wenguang Zhu for their reading
and helpful comments on the manuscript.
Appendix A: Derivation of dielectric function
In this appendix, we derive the mathematical expres-
sion of the imaginary part of dielectric function, for the
optical absorption of QBS. For the sake of simplicity,
we consider the incident light wave as a monochromatic
plane wave. The transition rate from the kth energy level
to the nth energy level, which is the probability of tran-
sition per unit time, can be represented as [34]
|Wnk|=π
6ℏ2|Dnk|2E2
0δ(ωn−ωk−ω),(A1)
where E0is the vibrational amplitude of the correspond-
ing electric eld E=E0cos(ωt),Ek=ℏωk,En=ℏωn,
are the kth energy level and the nth energy level, respec-
tively. The transition matrix |Dnk|2≡ |⟨k| − er|n⟩|2=
e2|⟨k|r|n⟩|2can be derived from the perturbation Hamil-
tonian H′=−eϕ =−erE0cos(ωt) = DE0cos(ωt) =
Wcos(ωt), with D=−er, involving the electron charge
eand the operator of real space coordinates r. It is ev-
ident that the parity (evenness or oddness) of the wave
functions corresponding to the kth and nth energy levels
determines whether the integral evaluates to zero or not.
The transition rate can also be expressed as
Wnk =πE2
0
6ℏ2|Dnk|2δ(ωn−ωk−ω) = πe2E2
0
6ℏ2|⟨k|r|n⟩|2Jnk(ω) = πe2E2
0
6ℏ2Ank(ω).(A2)
Here, the transition rate Jnk (ω) = δ(ωn−ωk−ω)rep-
resents the distribution of energy levels corresponding to
the transition energy dierence ℏω, which is associated
with the joint density of states (JDOS) of the valence and
conduction bands in semiconductor system. The function
Ank(ω) = |⟨k|r|n⟩|2δ(ωn−ωk−ω) = |⟨k|r|n⟩|2Jnk(ω)is
determined by the energy level distribution and the tran-
sition matrix, reecting the constraints of the selection
rules on the light absorption spectrum of the electron
bound states.

15
FIG. 10. Schematic diagram of the regulation of the optical properties of the well region by the inter-barrier spacing (quasi-well
width). (left) The black arrow signies that an increase in the well width leads to a reduction in the QBS levels; the blue
arrow denotes that a decrease in the well width results in a higher energy of the absorbed light for transitions involving states
with identical principal quantum numbers. (right) the energy dierence between QBS energy levels with k= 1 and n= 2,4,6
(from top to bottom). The color scale corresponds to a energy range of [0.0 - 3.0]eV, or a wavelength range [∞- 413.3]nm .
The height of the barrier V0= 3.0eV.
Next, we calculate the imaginary part of the dielectric
function ϵ2(ω)from the perspective of medium absorp-
tion [36]. For degenerate energy levels in the ground
state, the absorption power per unit time is given by
Pnk = 2ℏω×Wnk (A3)
From the perspective of medium absorption, the absorp-
tion power per unit volume is
ρnk =σ¯
E2= 2ωϵ2(ω)ϵ0E2
0.(A4)
In the potential well region, assuming the cross-sectional
area of the double potential barrier is S0, the correspond-
ing volume Ω0=wS0. Then the absorption power is
Pnk = Ω0ρnk = 2Ω0ωϵ2(ω)ϵ0E2
0.(A5)
From Eqs.(A2), (A3) and (A5), we have
2ℏω×πE2
0
6ℏ2|Dnk|2δ(ωn−ωk−ω) = 2Ω0ωϵ2(ω)ϵ0E2
0.(A6)
Simplifying to get
ϵ2(ω) = πe2
6ℏϵ0Ω0|Dnk|2δ(ωn−ωk−ω).(A7)
That is
ϵ2(ω) = πe2
6ℏϵ0Ω0|⟨k|r|n⟩|2δ(ωn−ωk−ω)
=πe2
6ℏϵ0Ω0|⟨k|r|n⟩|2Jnk(ω).(A8)
This is a dimensionless quantity, directly corresponding
to the experimentally observable light absorption prop-
erties of the DB nanostructure.
Appendix B: Tables of Data
In this appendix, the data for initial state label k= 2
and k= 3 in Figs. 8 and 9 are listed in Tables VI-IX.

16
TABLE VI. The optical absorption properties of DB system of electrons at dierent well widths w= 10,20,30 Å, similar to
Table IV. The initial state label k= 2.
w10 20 30
nℏωΓnJnk ϵ2ℏωΓnJnk ϵ2ℏωΓnJnk ϵ2
3 1.1342 4.4571×10−58.9507×1035.5547×1030.3773 3.4509×10−81.1560×1071.0461×1070.1812 3.3568×10−91.1884×1081.4869×108
4 0.8979 4.3497×10−79.1718×105— 0.4340 1.2890×10−83.0949×107—
5 1.5489 7.8591×10−65.0762×1044.1951×1020.7570 6.9619×10−85.7304×1066.4477×104
6 2.2942 2.9148×10−41.3687×103— 1.1481 4.4837×10−78.8977×105—
7 1.6038 3.3706×10−61.1836×1051.2505×102
8 2.1161 4.6616×10−58.5580×103—
9 2.6613 9.3803×10−44.2530×1028.3380×10−2
TABLE VII. The optical absorption properties of DB system of electrons at dierent well widths w= 40,50,60 Å, similar to
Table IV. The initial state label k= 2.
w40 50 60
nℏωΓnJnk ϵ2ℏωΓnJnk ϵ2ℏωΓnJnk ϵ2
3 0.1059 8.2558×10−10 4.8323×1087.7562×1080.0693 2.8696×10−10 1.3902×1092.7268×1090.0488 1.0581×10−10 3.7703×1098.7420×109
4 0.2538 2.3578×10−91.6920×108— 0.1662 7.8456×10−10 5.0849×108— 0.1172 2.7503×10−10 1.4505×109—
5 0.4436 8.4105×10−94.7434×1076.8183×1050.2906 1.7978×10−92.2191×1083.8918×1060.2049 7.2055×10−10 5.5366×1081.1471×107
6 0.6748 3.3829×10−81.1793×107— 0.4425 4.7758×10−98.3533×107— 0.3121 2.0686×10−91.9286×108—
7 0.9468 1.2283×10−73.2479×1064.3608×1030.6216 1.5404×10−82.5898×1074.2328×1040.4387 3.8962×10−91.0239×1081.9749×105
8 1.2587 3.7955×10−71.0511×106— 0.8277 4.6237×10−88.6283×106— 0.5845 9.4192×10−94.2354×107—
9 1.6090 2.0394×10−61.9562×1055.0937×1011.0605 1.3269×10−73.0067×1069.5077×1020.7495 2.0561×10−81.9403×1077.2308×103
10 1.9952 1.0996×10−53.6279×104— 1.3195 3.4790×10−71.1467×106— 0.9335 5.0196×10−87.9476×106—
11 2.4116 1.0379×10−43.8439×1032.7921×10−11.6041 1.2754×10−63.1280×1052.7821×1011.1363 1.1066×10−73.6050×1063.7740×102
12 2.8392 1.4433×10−32.7642×102— 1.9131 5.4175×10−67.3639×104— 1.3577 2.9400×10−71.3570×106—
13 2.2445 2.6259×10−51.5192×1044.7707×10−11.5973 9.1721×10−74.3495×1051.6119×101
14 2.5936 2.3329×10−41.7101×103— 1.8545 2.9447×10−61.3548×105—
15 2.9443 2.7123×10−31.4709×1021.7495×10−32.1285 1.3265×10−53.0076×1044.6143×10−1
16 2.4174 5.1916×10−57.6844×103—
17 2.7173 3.7245×10−41.0711×1037.4866×10−3
TABLE VIII. The optical absorption properties of DB system of electrons at dierent well widths w= 10,20,30 Å, similar to
Table IV. The initial state label k= 3.
w10 20 30
nℏωΓnJnk ϵ2ℏωΓnJnk ϵ2ℏωΓnJnk ϵ2
4 0.5205 4.3497×10−79.1718×1058.8337×1050.2527 1.2890×10−83.0949×1074.0557×107
5 1.1716 7.8591×10−65.0762×104— 0.5757 6.9619×10−85.7304×106—
6 1.9169 2.9148×10−41.3687×1031.3609×1010.9669 4.4837×10−78.8977×1051.1974×104
7 1.4225 3.3706×10−61.1836×105—
8 1.9349 4.6616×10−58.5580×1031.1992×101
9 2.4801 9.3803×10−44.2530×102—

17
TABLE IX. The optical absorption properties of DB system of electrons at dierent well widths w= 40,50,60 Å, similar to
Table IV. The initial state label k= 3.
w 40 50 60
nℏωΓnJnk ϵ2ℏωΓnJnk ϵ2ℏωΓnJnk ϵ2
4 0.1479 2.3578×10−91.6920×1082.8348×1080.0969 7.8456×10−10 5.0849×1081.0398×1090.0683 2.7503×10−10 1.4505×1093.5046×109
5 0.3377 8.4105×10−94.7434×107— 0.2213 1.7978×10−92.2191×108— 0.1561 7.2055×10−10 5.5366×108—
6 0.5689 3.3829×10−81.1793×1072.0170×1050.3732 4.7758×10−98.3533×1071.7401×1060.2633 2.0686×10−91.9286×1084.7424×106
7 0.8409 1.2283×10−73.2479×106— 0.5523 1.5404×10−82.5898×107— 0.3899 3.8962×10−91.0239×108—
8 1.1528 3.7955×10−71.0511×1061.8671×1030.7584 4.6237×10−88.6283×1061.8614×1040.5357 9.4192×10−94.2354×1071.0772×105
9 1.5032 2.0394×10−61.9562×105— 0.9912 1.3269×10−73.0067×106— 0.7007 2.0561×10−81.9403×107—
10 1.8894 1.0996×10−53.6279×1041.3503×1011.2503 3.4790×10−71.1467×1065.1777×1020.8846 5.0196×10−87.9476×1064.2242×103
11 2.3057 1.0379×10−43.8439×103— 1.5348 1.2754×10−63.1280×105— 1.0875 1.1066×10−73.6050×106—
12 2.7333 1.4433×10−32.7642×1022.8712×10−21.8439 5.4175×10−67.3639×1049.9171×1001.3088 2.9400×10−71.3570×1062.1499×102
13 2.1752 2.6259×10−51.5192×104— 1.5484 9.1721×10−74.3495×105—
14 2.5243 2.3329×10−41.7101×1038.4143×10−21.8057 2.9447×10−61.3548×1057.9539×100
15 2.8750 2.7123×10−31.4709×102— 2.0796 1.3265×10−53.0076×104—
16 2.3686 5.1916×10−57.6844×1031.9295×10−1
17 2.6685 3.7245×10−41.0711×103—
[1] R. Tsu and L. Esaki, Tunneling in a nite superlattice,
Applied Physics Letters 22, 562 (1973).
[2] L. L. Chang, L. Esaki, and R. Tsu, Resonant tunneling in
semiconductor double barriers, Applied Physics Letters
24, 593 (1974).
[3] L. Esaki and L. L. Chang, New Transport Phenomenon in
a Semiconductor ”Superlattice”, Physical Review Letters
33, 495 (1974).
[4] S. Luryi, Frequency limit of double-barrier resonant-
tunneling oscillators, Applied Physics Letters 47, 490
(1985).
[5] M. Tsuchiya, H. Sakaki, and J. Yoshino, Room Tem-
perature Observation of Dierential Negative Resis-
tance in an AlAs/GaAs/AlAs Resonant Tunneling Diode,
Japanese Journal of Applied Physics 24, L466 (1985).
[6] A. D. Stone and P. A. Lee, Eect of Inelastic Processes on
Resonant Tunneling in One Dimension, Physical Review
Letters 54, 1196 (1985).
[7] F. Capasso, K. Mohammed, and A. Cho, Resonant tun-
neling through double barriers, perpendicular quantum
transport phenomena in superlattices, and their device
applications, IEEE Journal of Quantum Electronics 22,
1853 (1986).
[8] E. H. Hauge, J. P. Falck, and T. A. Fjeldly, Transmission
and reection times for scattering of wave packets o
tunneling barriers, Phys. Rev. B 36, 4203 (1987).
[9] T. Weil and B. Vinter, Equivalence between resonant
tunneling and sequential tunneling in double-barrier
diodes, Applied Physics Letters 50, 1281 (1987).
[10] M. Jonson and A. Grincwajg, Eect of inelastic scatter-
ing on resonant and sequential tunneling in double bar-
rier heterostructures, Applied Physics Letters 51, 1729
(1987).
[11] A. Zaslavsky, V. J. Goldman, D. C. Tsui, and J. E.
Cunningham, Resonant tunneling and intrinsic bistabil-
ity in asymmetric double-barrier heterostructures, Ap-
plied Physics Letters 53, 1408 (1988).
[12] J. Encomendero, F. A. Faria, S. M. Islam, V. Protasenko,
S. Rouvimov, B. Sensale-Rodriguez, P. Fay, D. Jena, and
H. G. Xing, New Tunneling Features in Polar III-Nitride
Resonant Tunneling Diodes, Physical Review X 7, 041017
(2017).
[13] J. Encomendero, V. Protasenko, B. Sensale-Rodriguez,
P. Fay, F. Rana, D. Jena, and H. G. Xing, Broken Sym-
metry Eects due to Polarization on Resonant Tunneling
Transport in Double-Barrier Nitride Heterostructures,
Physical Review Applied 11, 034032 (2019).
[14] B. Tao, C. Wan, P. Tang, J. Feng, H. Wei, X. Wang,
S. Andrieu, H. Yang, M. Chshiev, X. Devaux, T. Hauet,
F. Montaigne, S. Mangin, M. Hehn, D. Lacour, X. Han,
and Y. Lu, Coherent Resonant Tunneling through Double
Metallic Quantum Well States, Nano Letters 19, 3019
(2019).
[15] G. Zangwill and E. Granot, Dynamic resonant tunneling
via a quasibound superstate, Physical Review A 106,
032201 (2022).
[16] C. Bi, Q. Chen, W. Li, and Y. Yang, Quantum nature
of proton transferring across one-dimensional potential
elds*, Chinese Physics B 30, 046601 (2021).
[17] C. Bi and Y. Yang, Atomic Resonant Tunneling in the
Surface Diusion of H Atoms on Pt(111), The Journal of
Physical Chemistry C 125, 464 (2021).
[18] Y. Yang, Penetration of arbitrary double potential bar-
riers with probability unity: Implications for testing the
existence of a minimum length, Physical Review Research
6, 013087 (2024).
[19] C. Lin, K. Futamata, T. Akiho, K. Muraki, and T. Fuji-
sawa, Resonant Plasmon-Assisted Tunneling in a Double
Quantum Dot Coupled to a Quantum Hall Plasmon Res-
onator, Physical Review Letters 133, 036301 (2024).

18
[20] S. Mandrà, J. Schrier, and M. Ceotto, Helium Isotope
Enrichment by Resonant Tunneling through Nanoporous
Graphene Bilayers, The Journal of Physical Chemistry A
118, 6457 (2014).
[21] T. A. Growden, W. Zhang, E. R. Brown, D. F. Storm,
K. Hansen, P. Fakhimi, D. J. Meyer, and P. R. Berger,
431 kA/cm2 peak tunneling current density in GaN/AlN
resonant tunneling diodes, Applied Physics Letters 112,
033508 (2018).
[22] H. Saito, S. K. Narayananellore, N. Matsuo, N. Doko,
S. Kon, Y. Yasukawa, H. Imamura, and S. Yuasa, Tunnel-
ing Magnetoresistance and Spin-Dependent Diode Per-
formance in Fully Epitaxial Magnetic Tunnel Junctions
With a Rocksalt Zn O / Mg O Bilayer Tunnel Barrier,
Physical Review Applied 11, 064032 (2019).
[23] B. Ricco and M. Ya. Azbel, Physics of resonant tunnel-
ing. The one-dimensional double-barrier case, Physical
Review B 29, 1970 (1984).
[24] G. Zangwill and E. Granot, Spatial vibrations suppress-
ing resonant tunneling, Physical Review A 101, 012109
(2020).
[25] L. J. Lauhon and W. Ho, Direct Observation of the Quan-
tum Tunneling of Single Hydrogen Atoms with a Scan-
ning Tunneling Microscope, Physical Review Letters 85,
4566 (2000).
[26] X. Meng, J. Guo, J. Peng, J. Chen, Z. Wang, J.-R. Shi,
X.-Z. Li, E.-G. Wang, and Y. Jiang, Direct visualization
of concerted proton tunnelling in a water nanocluster,
Nature Physics 11, 235 (2015).
[27] J. Guo, J.-T. Lü, Y. Feng, J. Chen, J. Peng, Z. Lin,
X. Meng, Z. Wang, X.-Z. Li, E.-G. Wang, and Y. Jiang,
Nuclear quantum eects of hydrogen bonds probed by
tip-enhanced inelastic electron tunneling, Science 352,
321 (2016).
[28] P. Mello, P. Pereyra, and N. Kumar, Macroscopic ap-
proach to multichannel disordered conductors, Annals of
Physics 181, 290 (1988).
[29] P. Pereyra, Resonant Tunneling and Band Mixing in Mul-
tichannel Superlattices, Physical Review Letters 80, 2677
(1998).
[30] P. Pereyra and E. Castillo, Theory of nite periodic sys-
tems: General expressions and various simple and illus-
trative examples, Physical Review B 65, 205120 (2002).
[31] P. Pereyra, The Transfer Matrix Method and the Theory
of Finite Periodic Systems. From Heterostructures to Su-
perlattices, physica status solidi (b) 259, 2100405 (2022),
arXiv:2109.11640 [cond-mat.mtrl-sci].
[32] R. H. Landau, M. J. Páez, and C. C. Bordeianu, Compu-
tational Physics: Problem Solving with Computers, 2nd
ed. (Wiley-VCH, Weinheim : Chichester, 2007).
[33] H. González-Herrero, E. C.-d. Río, P. Mallet, J.-Y.
Veuillen, J. J. Palacios, J. M. Gómez-Rodríguez, I. Bri-
huega, and F. Ynduráin, Hydrogen physisorption channel
on graphene: A highway for atomic H diusion, 2D Ma-
terials 6, 021004 (2019).
[34] J. Y. Zeng, Quantum Mechanics, 4th ed. (Science Press,
2007).
[35] M. S. Dresselhaus, SOLID STATE PHYSICS PART II
(MIT Online Course, 2001, 2001).
[36] K. Huang and R. Q. Han, Solid States Physics (Higher
Education Press, 1998).
[37] P. Y. Yu and M. Cardona, Fundamentals of Semiconduc-
tors: Physics and Materials Properties, Graduate Texts
in Physics (Springer, Berlin, Heidelberg, 2010).
[38] C. Trovatello, F. Katsch, N. J. Borys, M. Selig, K. Yao,
R. Borrego-Varillas, F. Scotognella, I. Kriegel, A. Yan,
A. Zettl, P. J. Schuck, A. Knorr, G. Cerullo, and
S. D. Conte, The ultrafast onset of exciton formation
in 2D semiconductors, Nature Communications 11, 5277
(2020).
[39] R. Chen, Z. Gong, J. Chen, X. Zhang, X. Zhu, H. Chen,
and X. Lin, Recent advances of transition radiation: Fun-
damentals and applications, Materials Today Electronics
3, 100025 (2023).
[40] R. R. Jones, D. C. Hooper, L. Zhang, D. Wolverson,
and V. K. Valev, Raman Techniques: Fundamentals and
Frontiers, Nanoscale Research Letters 14, 231 (2019).
[41] C. Huang, S. Duan, and W. Zhang, High-resolution time-
and angle-resolved photoemission studies on quantum
materials, Quantum Frontiers 1, 15 (2022).
[42] M. S. Munsi and H. Chaoui, Energy Management Sys-
tems for Electric Vehicles: A Comprehensive Review of
Technologies and Trends, IEEE Access 12, 60385 (2024).
[43] E. Brown, W.-D. Zhang, T. Growden, P. Fakhimi, and
P. Berger, Electroluminescence in Unipolar-Doped In
0.53 Ga 0.47 As / Al As Resonant-Tunneling Diodes: A
Competition between Interband Tunneling and Impact
Ionization, Physical Review Applied 16, 054008 (2021).
[44] S. W. Eaton, A. Fu, A. B. Wong, C.-Z. Ning, and
P. Yang, Semiconductor nanowire lasers, Nature Reviews
Materials 1, 1 (2016).