Photoionization cross section in a spherical quantum dot: Effects of some parabolic confining electric potentials
A theoretical investigation of the effects of spatial variation of confining electric potential on photoionization cross section (PCS) in a spherical quantum dot is presented. The potential profiles considered here are the shifted parabolic potential and the inverse lateral shifted parabolic poten...
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irk-123456789-1570002019-06-20T01:27:30Z Photoionization cross section in a spherical quantum dot: Effects of some parabolic confining electric potentials Tshipa, M. A theoretical investigation of the effects of spatial variation of confining electric potential on photoionization cross section (PCS) in a spherical quantum dot is presented. The potential profiles considered here are the shifted parabolic potential and the inverse lateral shifted parabolic potential compared with the well-studied parabolic potential. The primary findings are that parabolic potential and the inverse lateral shifted parabolic potential blue shift the peaks of the PCS while the shifted parabolic potential causes a red shift. Представлено теоретичне дослiдження впливу просторової змiни обмежуючих електричних потенцiалiв на перерiз фотоiонiзацiї в сферичнiй квантовiй точцi. Розглянутi тут профiлi потенцiалiв є змiщений параболiчний потенцiал i дзеркальне зображення змiщеного параболiчного потенцiалу, якi порiвнюються з добре вивченим параболiчним потенцiалом. Основний результат — це те, що параболiчний потенцiал i дзеркальне зображення змiщеного параболiчного потенцiалу зсувають пiки перерiзу фотоiонiзацiї у фiолетовий дiапазон, тодi як змiщений параболiчний потенцiал спричинює червоний зсув. 2017 Article Photoionization cross section in a spherical quantum dot: Effects of some parabolic confining electric potentials / M. Tshipa // Condensed Matter Physics. — 2017. — Т. 20, № 2. — С. 23703: 1–9. — Бібліогр.: 17 назв. — англ. 1607-324X PACS: 71.55.Eq, 73.21.La, 73.22.Dj DOI:10.5488/CMP.20.23703 arXiv:1706.07286 http://dspace.nbuv.gov.ua/handle/123456789/157000 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
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A theoretical investigation of the effects of spatial variation of confining electric potential on photoionization
cross section (PCS) in a spherical quantum dot is presented. The potential profiles considered here are the
shifted parabolic potential and the inverse lateral shifted parabolic potential compared with the well-studied
parabolic potential. The primary findings are that parabolic potential and the inverse lateral shifted parabolic
potential blue shift the peaks of the PCS while the shifted parabolic potential causes a red shift. |
| format |
Article |
| author |
Tshipa, M. |
| spellingShingle |
Tshipa, M. Photoionization cross section in a spherical quantum dot: Effects of some parabolic confining electric potentials Condensed Matter Physics |
| author_facet |
Tshipa, M. |
| author_sort |
Tshipa, M. |
| title |
Photoionization cross section in a spherical quantum dot: Effects of some parabolic confining electric potentials |
| title_short |
Photoionization cross section in a spherical quantum dot: Effects of some parabolic confining electric potentials |
| title_full |
Photoionization cross section in a spherical quantum dot: Effects of some parabolic confining electric potentials |
| title_fullStr |
Photoionization cross section in a spherical quantum dot: Effects of some parabolic confining electric potentials |
| title_full_unstemmed |
Photoionization cross section in a spherical quantum dot: Effects of some parabolic confining electric potentials |
| title_sort |
photoionization cross section in a spherical quantum dot: effects of some parabolic confining electric potentials |
| publisher |
Інститут фізики конденсованих систем НАН України |
| publishDate |
2017 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/157000 |
| citation_txt |
Photoionization cross section in a spherical quantum dot: Effects of some parabolic confining electric potentials / M. Tshipa // Condensed Matter Physics. — 2017. — Т. 20, № 2. — С. 23703: 1–9. — Бібліогр.: 17 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT tshipam photoionizationcrosssectioninasphericalquantumdoteffectsofsomeparabolicconfiningelectricpotentials |
| first_indexed |
2025-07-14T09:20:58Z |
| last_indexed |
2025-07-14T09:20:58Z |
| _version_ |
1837613563437383680 |
| fulltext |
Condensed Matter Physics, 2017, Vol. 20, No 2, 23703: 1–9
DOI: 10.5488/CMP.20.23703
http://www.icmp.lviv.ua/journal
Photoionization cross section in a spherical quantum
dot: Effects of some parabolic confining electric
potentials
M. Tshipa
University of Botswana, Corner of Notwane and Mobuto Road, P/Bag 00704, Gaborone, Botswana
Received January 6, 2017, in final form April 13, 2017
A theoretical investigation of the effects of spatial variation of confining electric potential on photoionization
cross section (PCS) in a spherical quantum dot is presented. The potential profiles considered here are the
shifted parabolic potential and the inverse lateral shifted parabolic potential compared with the well-studied
parabolic potential. The primary findings are that parabolic potential and the inverse lateral shifted parabolic
potential blue shift the peaks of the PCS while the shifted parabolic potential causes a red shift.
Key words: photoionization cross section, electric confining potential, spherical quantum dot, hydrogenic
impurity
PACS: 71.55.Eq, 73.21.La, 73.22.Dj
1. Introduction
Recent advances in nanofabrication technology have made it possible to fabricate nanostructures
of different sizes and shapes [1–6]. Nanostructures have a wide range of applications including in
nanomedicine [7], information processing [8], energy physics [9] and gas sensing [10], to mention a
few. When fabricating these structures, it is impossible to eliminate all impurities. In some cases, it may
be advantageous to introduce impurities to improve the performance of nanodevices (doping). If the
impurity happens to be positively charged, then an electron may be bound to it. Given enough energy,
the electron can break free from the electrostatic grasp of the impurity. The excitation energy can be
in different forms, one of which is electromagnetic radiation of appropriate frequency. In this regard,
photoionization studies on nanostructures could offer an insight into the electron-impurity interaction in
a wide variety of conditions. As such, the literature is awash with investigations of PCS [11–15].
The purpose of this work is to explore the effects of some of the potentials that vary parabolically
with the radial distance on the PCS in a spherical quantum dot (SQD). This communication is organized
as follows. The theory is presented in section 2. The following section 3 entails discussions and analysis,
and concluding remarks are laid in section 4.
2. Preliminary notes
The system investigated is a hydrogenic impurity located at the centre of a spherical quantum dot of
radius R, which may be a GaAs material embedded in a GaAlAs matrix. The electric potentials inside the
spherical dot assume parabolic spatial variations with the radial distance from the centre of the quantum
dot. Photoionization is a process in which a bound charge carrier is liberated to the continuum by some
This work is licensed under a Creative Commons Attribution 4.0 International License . Further distribution
of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.
23703-1
https://doi.org/10.5488/CMP.20.23703
http://www.icmp.lviv.ua/journal
http://creativecommons.org/licenses/by/4.0/
M. Tshipa
appropriate radiation, with cross section [12]
σlm = σ0~ω
∑
f
|〈 f |®r |i〉|2δ(E f − Ei − ~ω), (2.1)
where ~ω is the photon energy and σ0 = (4π2αFSnrE2
in)/(3εE2
av). Ein is the effective incident electric
field, Eav is the average electric field in the dot of refractive index nr and dielectric constant ε . Ei and
E f are the energies associated with the initial and the final eigenstates |i〉 and | f 〉, respectively. 〈 f |®r |i〉 is
the usual matrix element coupling the initial states to the final states, αFS is the fine structure and ®r is the
position vector. The energy conserving δ function is replaced by the Lorentzian function
δ(E f − Ei − ~ω) =
~Γ
π{[~ω − (E f − Ei)]
2 + (~Γ)2}
, (2.2)
where Γ is the hydrogenic impurity linewidth.
Due to the spherical symmetry of the system, electron wave functions can be cast in the form
Ψ(θ, φ, r) = ClmYlm(θ, φ)χ(r). Ylm(θ, φ) are the usual spherical harmonics whereas χ(r) is the radial
component of the wave function satisfying the Schrödinger equation, within the effective mass approxi-
mation,
1
r2
d
dr
[
r2 d
dr
χ(r)
]
+
{
2µ
~2
[
Elm +
kee2
εr
− V(r)
]
−
l(l + 1)
r2
}
χ(r) = 0, (2.3)
where µ is the effective mass of the electron (of charge −e), ke is the Coulomb constant. Angular
momentum and magnetic quantum numbers are designated by l and m, respectively, and Clm is the
normalization constant.
The confining electric potentials considered here are the parabolic, shifted parabolic and the inverse
lateral shifted parabolic potentials, each superimposed on an infinite spherical quantumwell (ISQW). For
these calculations, the initial states are described by wave functions of bound electrons (with impurity)
while the final states are associated with wave functions for an electron in an SQD without the charged
impurity. Evaluation of the matrix elements for an SQD leads to the selection rules ∆l = ±1. Therefore,
the l values of the final and initial states will differ by unity.
2.1. Parabolic potential
When the parabolic potential, which has the form
V(r) =
1
2
µω2
0r2, (r < R), (2.4)
is inserted into the Schrödinger equation (2.3) in the presence of the donor impurity, then the second
order differential equation is solvable in terms of the Heun Biconfluent function [16, 17]
χ(r) = C1lmeg1(r)r lHeunB
(
2l+1, α, β, γ, g2(r)
)
+C2lmeg1(r)r−(l+1)HeunB
(
−(2l+1), α, β, γ, g2(r)
)
(2.5)
with
α = 0, β = −
2Elm
~ω0
, γ =
4kee2
~ε
√
−
µ
~ω0
, (2.6)
and the arguments
g1(r) =
µω0
2~
r2, g2(r) =
√
2g1(r). (2.7)
For a solid quantum dot, the coefficient corresponding to the second linearly independent solution,
C2lm, should be taken as zero due to the divergent nature of this solution at the centre of the SQD.
Applying the boundary condition, concerning the continuity of the wave function at the walls of the SQD
(r = R), leads to the energy spectrum of an electron in an SQD with a parabolic potential as
Elm = −
1
2
βR~ω0 , (2.8)
23703-2
Photoionization cross section in a spherical quantum dot
with βR satisfying the condition
HeunB
(
2l + 1, α, βR, γ, g1(R)
)
= 0. (2.9)
In the absence of the impurity, the radial part of the wave function is in terms of the hypergeometric
function
χ0(r) = C0
lme−g1(r)r
l
2−
3
4 M
(
[a0], [b], 2g1(r)
)
, (2.10)
where C0
lm
= C0
1lm is the normalization constant and
a0 =
l
2
+
3
4
−
E0
lm
2~ω0
, b = l +
3
2
. (2.11)
Here, g1(r) is given by equation (2.7). The demand that the wave function vanishes at the walls of the
nanostructure yields the energy dispersion relation
E0
lm =
(
l +
3
2
− 2a0
R
)
~ω0 , (2.12)
where a0
R gives the condition M
(
[a0
R], [b], 2g1(R)
)
= 0.
2.2. Shifted parabolic potential
This potential is convex: maximum at the centre, and decreases parabolically to assume a minimum
value (here taken as zero) at the radius;
V(r) =
1
2
µω2
0(r − R)2, (r < R). (2.13)
The solution to the radial component of the Schrödinger equation (2.3) corresponding to this potential is
also in terms of the Heun Biconfleunt function [equation (2.5)] but with
α = 2
√
−
µω0R2
~
, g1(r) =
µω0
2~
(r − 2R)r, g2(r) = −i
√
µω0
~
r . (2.14)
β and γ are the same as those for the parabolic potential with the donor impurity. The energy spectrum
is given by the usual boundary conditions at the walls of the SQD as
Eml = −
βR
2
~ω0 , (2.15)
where βR is the value of β that satisfies the condition given in equation (2.9).
Without the donor impurity, the solution to the radial part of the Schrödinger equation is of the same
functional form as that in equation (2.5) with all the parameters having the same expressions as for the
case with the impurity except for β = β0 and γ which are
β0 = −
2E0
lm
~ω0
and
γ = 0.
The energy eigenvalues for a shifted parabolic confining potential without the impurity can be written
as
E0
lm = −
β0
R
2
~ω0 , (2.16)
where β0
R is the value of β = β0 that satisfies equation (2.9).
23703-3
M. Tshipa
2.3. Inverse lateral shifted parabolic potential
This potential has a concave increase in the radial distance from the centre of the SQD, with a
maximum value at the walls of the SQD:
V(r) =
1
2
µω2
0[(2R − r)r], (r < R). (2.17)
The radial wave functions corresponding to this potential configuration are given by equation (2.5),
with
β =
(2Elm − µω
2
0R2)
i~ω0
and
g2(r) = i
(
−
µ2ω2
0
~2
)1/4
r,
with α, γ and g1(r) being identical to those for the shifted parabolic potential in the presence of the
impurity. Again, imposing the boundary conditions at the walls of the SQD yields the equation for the
determination of the energy eigenvalues;
Elm =
i
2
βR~ω0 +
1
2
µω2
0R2, (2.18)
where βR is the value of β that satisfies the condition set in equation (2.9).
In the absence of the impurity, the solution to the radial part of the Schrödinger equation is of the
same form as in equation (2.5), but with γ = 0 and
β = β0 =
(2E0
lm
− µω2
0R2)
i~ω0
.
All other parameters remain the same as for the case of this potential with the donor impurity. The
application of the boundary conditions gives the energy eigenvalues in the absence of the impurity for an
inverse lateral shifted parabolic potential as
Elm =
i
2
β0
R~ω0 +
1
2
µω2
0R2, (2.19)
where β0
R is the value of β0 that satisfies the condition set in equation (2.9).
3. Main results
The parameters used in these calculations are relevant to GaAs quantum dots: effective electronic
mass µ = 0.067me, me being the free electron mass and ε = 12.5. The impurity linewidth has been taken
such that ~Γ = 0.2 meV. Figure 1 displays the effects of these potential geometries on the normalised
ground state radial probability density [Rlm(r) = r2 χ(r)2/R2] in the presence of the hydrogenic impurity
for an SQD of radius R = 200 Å. The plots with circles represent an infinite spherical square well, the
dashed curve (solid plot with dots) corresponds to the parabolic potential (inverse lateral shifted parabolic
potential) while the solid plot represents the shifted parabolic potential. The parabolic potential and the
inverse lateral shifted parabolic potential have the propensity to shift peaks of the radial probability
density towards the centre of the SQD, which decreases the electron-impurity distance of separation. The
influence of the inverse lateral shifted parabolic potential is slightly more pronounced than that of the
parabolic potential because, even though the two are equal at the centre and at the walls of the SQDs, the
former is always greater in the region between the centre and the walls. The shifted parabolic potential,
on the other hand, shifts the peaks of the radial probability density towards the outer regions of the SQD,
increasing the electron-impurity distance of separation.
23703-4
Photoionization cross section in a spherical quantum dot
Figure 1. (Color online) The effect of different potentials on the normalised ground state radial probability
density for an SQD of radius R = 200 Å. The curve with circles represents an infinite spherical square
well while the dashed corresponds to the parabolic potential, the solid plot is associated with the shifted
parabolic potential and the inverse lateral shifted parabolic potential is represented by the solid line with
dots, each of strength ~ω0 = 20 meV.
The l > 0 electrons are localized towards thewalls of the SQD.As such, the radial position expectation
values of such electrons are in the regions where both parabolic and inverse lateral shifted parabolic
potentials are higher than for the ground state. This gives these potentials the proclivity to affect the
higher l valued electrons more than the lower l valued electrons. As a result, transition energies increase
with increasing strengths of the potentials. On the other hand, the shifted parabolic potential decreases
the transition energies. This is because this potential affects the lower l valued electrons more than it does
the higher l valued electrons, since the lower l valued electrons spend most of their time in the regions
where this potential is greater. This is depicted in figure 2, which shows transition energies as functions
of strengths of the three potentials, in an SQD of radius 200 Å. In the figure, the parabolic potential is
represented by dashed plots, the solid curves are associated with the shifted parabolic potential while
the inverse lateral shifted parabolic potential is represented by the solid plots with dots. Inevitably, the
energy needed for photoionization is modified as the strengths of these potentials are varied. Thus, peaks
of the photoionization cross section undergo shifts as strengths of these potentials increase, according to
the dependence of the transition energies on the strengths of different potentials.
In the absence of the impurity, the first order transition energies (s → p), ∆Esp, are less than those
of the second order (p → d), ∆Epd, for an ISQW for all values of nanodot radius. In the presence
of the impurity, there exists a radius at which the first order and the second order transition energies
coincide. In these calculations, this radius is in the neighbourhood of R0 = 171 Å. For SQDs with radii
less (greater) than R0, the second order transition energies are more (less) than the first order transition
energies. The parabolic and the inverse lateral shifted parabolic potentials reduce the value of this radius
as they intensify. On the contrary, increasing the strength of the shifted parabolic potential increases R0,
sending it to infinity as it intensifies further. In this case, ∆Esp and ∆Epd would never coincide and
∆Epd > ∆Esp.
In figure 3, which depicts the s → p and p → d transitions PCSs, the SQD radius is greater than
R0, thus the s→ p peak occurs at larger beam energies than the second order peak. Increasing strengths
of the parabolic and the inverse lateral shifted parabolic potentials blue shifts the peaks of the PCS,
simultaneously moving them apart (figure 3). This can be beneficial in cases where transitions between
different states (for example, the s → p and the p → d transitions) need to be isolated and distinct, for
23703-5
M. Tshipa
Figure 2. (Color online) The dependence of the first and second order transition energies on the strengths
of different potentials. The solid (dashed) curves represent the inverse lateral shifted parabolic potential
(parabolic potential), while the plots with dots are associated with the shifted parabolic potential, for
R = 200 Å.
Figure 3. (Color online) The sum of the first and second PCSs as functions of beam energy for the ISQW
(~ω0 = 0 meV) (solid curve) and for the parabolic potential of strength ~ω0 = 10 meV (dashed curve)
superimposed upon the ISQW, for an SQD of radius R = 200 Å.
research or practical purposes. The inverse lateral shifted parabolic potential is generally greater than
the parabolic potential. Therefore, it has more propensity to blue shift and separate the PCS peaks than
the latter. The shifted parabolic potential red shifts the peaks of the PCSs, with the first order peak
experiencing more shifting until it equals and surpasses the p → d peak in being red shifted (figure 4).
This implies that the shifted parabolic potential can be utilized to have the s → p and the p → d
transitions having the same photon energies of excitation, or even to have control over which transition
23703-6
Photoionization cross section in a spherical quantum dot
Figure 4. (Color online) The influence of the shifted parabolic potential on the variation of normalized
PCS with the perturbing field energy for an SQD of radius R = 200 Å. The curves are for different
potential strengths: ~ω0 = 0 meV (solid plot), ~ω0 = 7 meV (dashed curve) and ~ω0 = 12 meV (solid
plot with squares), superimposed upon the ISQW.
exactly is to have a lower photon energy of excitation. The PCS in figures 3 and 4 for an infinite cylindrical
quantum well (~ω0 = 0 meV) are concurrent with those in the literature [14].
Figure 5 shows the normalized PCSs for the s → p transition as functions of the strengths of the
three confining electric potentials. The dashed plots correspond to the parabolic potential, the shifted
Figure 5. (Color online) The dependence of the s→ p transition PCS on strengths of different potentials:
the parabolic (inverse lateral shifted parabolic potential) is represented by the dashed curves (plots with
dots) while the shifted parabolic potential is associated with the solid lines. The two bundles of graphs
are generated for the beam energies η1 = ~ω = 30 meV and η2 = ~ω = 30.5 meV, all for an SQD of
radius 200 Å.
23703-7
M. Tshipa
parabolic potential is represented by the solid curves while the plots with dots are associated with the
inverse lateral shifted parabolic potential. The PCSs are plotted for two beam energies, one is less than the
ISQW transition energy ∆E0 (transition energies corresponding to ~ω0 = 0), while the other is slightly
greater than ∆E0. Since the parabolic potential and the inverse lateral shifted parabolic potential enhance
transition energies as they intensify (figure 2), an increase in the strengths of the two potentials capacitates
photoionization only when the beam energy is greater than ∆E0. Contrarily, an increase in the strength of
the shifted parabolic potential will capacitate photoionization for beam energies that are less than ∆E0,
owing to its proclivity to decrease transition energies (also in figure 2).
4. Conclusions
Electron states in a spherical quantum dot have been obtained within the effective mass regime, which
were utilized to calculate the photoionization cross section associated with a centred donor impurity. In
particular, the effect of intrinsic confining electric potentials that have parabolic dependence on the radial
distance was interrogated. The primary findings are that the parabolic potential and the inverse lateral
shifted parabolic potential blue shift the peaks of the PCS, while the shifted parabolic potential red shifts
the peaks. Thus, the three electric potentials can be used to tune nanodevices, without necessarily having
to alter the dimensions of the SQDs.
Acknowledgements
The author wishes to humbly thank M. Masale and T.G. Motsumi for the invaluable discussions we
have had.
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Photoionization cross section in a spherical quantum dot
Перерiз фотоiонiзацiї в сферичнiй квановiй точцi: Ефекти
деяких параболiчних обмежуючих електричних потенцiалiв
M. Тшiпа
Унiверситет Ботсвани, м. Габороне, Ботсвана
Представлено теоретичне дослiдження впливу просторової змiни обмежуючих електричних потенцiалiв
на перерiз фотоiонiзацiї в сферичнiй квантовiй точцi. Розглянутi тут профiлi потенцiалiв є змiщений па-
раболiчний потенцiал i дзеркальне зображення змiщеного параболiчного потенцiалу, якi порiвнюються
з добре вивченим параболiчним потенцiалом. Основний результат — це те, що параболiчний потенцi-
ал i дзеркальне зображення змiщеного параболiчного потенцiалу зсувають пiки перерiзу фотоiонiзацiї у
фiолетовий дiапазон, тодi як змiщений параболiчний потенцiал спричинює червоний зсув.
Ключовi слова: перерiз фотоiонiзацiї, електричний обмежуючий потенцiал, сферична квантова точка,
воднеподiбна домiшка
23703-9
Introduction
Preliminary notes
Parabolic potential
Shifted parabolic potential
Inverse lateral shifted parabolic potential
Main results
Conclusions
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