# Effects of Two-Dimensional Material Thickness and Surrounding Dielectric
Medium on Coulomb Interactions and Excitons

F. García Flórez<EMAIL_ADDRESS>Institute for Theoretical Physics and
Center for Extreme Matter and Emergent Phenomena, Utrecht University,
Princentonplein 5, 3584 CC Utrecht, The Netherlands  Laurens D. A. Siebbeles
<EMAIL_ADDRESS>Optoelectronic Materials Section, Department of
Chemical Engineering, Delft University of Technology, Van der Maasweg 9, 2629
HZ, Delft  H. T. C. Stoof<EMAIL_ADDRESS>Institute for Theoretical Physics
and Center for Extreme Matter and Emergent Phenomena, Utrecht University,
Princentonplein 5, 3584 CC Utrecht, The Netherlands

###### Abstract

We examine the impact of quantum confinement on the interaction potential
between two charges in two-dimensional semiconductor nanosheets in solution.
The resulting effective potential depends on two length scales, namely the
thickness $d$ and an emergent length scale $d^{*}\equiv\epsilon
d/\epsilon_{\text{sol}}$, where $\epsilon$ is the permittivity of the
nanosheet and $\epsilon_{\text{sol}}$ is the permittivity of the solvent. In
particular, quantum confinement, and not electrostatics, is responsible for
the logarithmic behavior of the effective potential for separations smaller
than $d$, instead of the one-over-distance bulk Coulomb interaction. Finally,
we corroborate that the exciton binding energy also depends on the two-
dimensional exciton Bohr radius $a_{0}$ in addition to the length scales $d$
and $d^{*}$ and analyze the consequences of this dependence.

##  I Introduction

Research into two-dimensional materials has increased in recent years, driven
in particular by prospects for their use in new state-of-the-art
optoelectronic devices that convert light into electric current and vice versa
[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15]. However, understanding
these devices and making them more efficient requires a firm foundation as a
starting point. In particular, two-dimensional semiconductor nanosheets in
solution are studied in pump-probe experiments, in which electrons and holes
are created using a pump laser and their nature and properties are
subsequently characterized with a probe laser measuring the complex
conductivity [15]. Since the presence of excitons may make-or-break a
particular application for optoelectronic devices, a refined understanding of
the properties of excitons, such as their mass, average size, and binding
energy, is advantageous. The most important aspect that greatly affects the
dynamics of excitons is the attractive interaction potential between electrons
and holes that allows the bound state to form. Electron-hole interactions in
two-dimensional materials have therefore been considered for several decades,
resulting in the Rytova-Keldysh potential that has been extensively used, and
extended, in the literature thus far [16, 17, 18, 19, 20, 21, 22, 23]. Our
goal in this paper is to better understand the consequences of introducing
quantum confinement into the electron-hole interaction potential, specifically
its role regarding the short-distance logarithmic behavior expected for a
purely two-dimensional Coulomb potential, and ultimately also on the exciton
properties. Our approach in particular discusses the importance of the three
length scales involved in the exciton problem, i.e., the thickness of the
nanosheet $d$, the emergent length scale $d^{*}\equiv\epsilon
d/\epsilon_{\text{sol}}$ from electrostatics that typically is much larger
than the thickness $d$ as the permittivity $\epsilon$ of the nanosheet is much
larger than the permittivity $\epsilon_{\text{sol}}$ of the solvent, and the
two-dimensional exciton Bohr radius $a_{0}$ that is introduced by quantum
mechanics due to the relative kinetic energy of the electron-hole pair.

Figure 1:  Diagram of the system under consideration. A nanosheet of thickness
$d$ and permittivity $\epsilon$ is surrounded by an environment above and
below with permittivities $\epsilon_{1}$ and $\epsilon_{2}$ respectively. The
Rytova-Keldysh potential is the solution of the electrostatics problem for the
interaction potential between the two charges separated by an in-plane
distance $r$, and displaced along the perpendicular axis by a distance $z_{1}$
and $z_{2}$, respectively.

Initially presented in Refs. [16, 17], the electrostatic Rytova-Keldysh
potential indeed incorporates both length scales $d$ and $d^{*}$, as we will
see explicitly in a moment. More specifically, the Rytova-Keldysh potential is
the solution of the electrostatics problem that describes the electron-hole
interactions in a nanosheet of permittivity $\epsilon$ and thickness $d$,
surrounded by an environment of permittivities $\epsilon_{1}$ and
$\epsilon_{2}$. Figure 1 shows an artist impression of this configuration. An
analytic expression that approximates the Rytova-Keldysh potential and that is
widely used to describe interactions in two-dimensional materials, is obtained
in the large distance limit $r\gg d$ and $\epsilon\gg\epsilon_{1,2}$. This
analytic expression is

$\displaystyle\begin{split}V^{\mathcal{H}\text{N}}(r)&=-\frac{e^{2}}{4\pi\epsilon}\\\
&\times\frac{\pi}{d}\left[\mathcal{H}_{0}\left(\frac{\epsilon_{1}+\epsilon_{2}}{\epsilon}~{}\frac{r}{d}\right)-N_{0}\left(\frac{\epsilon_{1}+\epsilon_{2}}{\epsilon}~{}\frac{r}{d}\right)\right]~{}~{},\end{split}$
(1)

that we denote as the Struve-Neumann (“$\mathcal{H}\text{N}$”) potential. Here
$\mathcal{H}_{\nu}(x)$ is the Struve function and $N_{\nu}(x)$ is the Neumann
function, also known as the Bessel function of the second kind. In order to
expose its universal properties, we make the potential dimensionless by
dividing by the Coulomb energy $E^{C}\equiv e^{2}/4\pi\epsilon d$, which
results in the dimensionless potential

$V^{\mathcal{H}\text{N}}(r)=-\pi\left[\mathcal{H}_{0}\left(\frac{2r}{d^{*}}\right)-N_{0}\left(\frac{2r}{d^{*}}\right)\right]~{}~{},$
(2)

where we have defined $d^{*}\equiv 2\epsilon d/(\epsilon_{1}+\epsilon_{2})$,
thus explicitly showing that the dimensionless Struve-Neumann potential only
depends on the ratio $r/d^{*}$. In other words, the dimensionless Struve-
Neumann potential does not depend on $d$ and $d^{*}$ independently but only on
the latter, consistent with the limit $r\gg d$. Note that from now on every
energy is made dimensionless in the same manner. For the purpose of simplicity
in the discussion of our results we consider only the case of nanosheets in
solution with
$\epsilon_{1}=\epsilon_{2}\equiv\epsilon_{\text{sol}}\leq\epsilon$, while
equations are given for the more general situation
$\epsilon_{1}\neq\epsilon_{2}$.

Our paper is organized as follows. Section II revisits the properties of the
Rytova-Keldysh potential, that is, both its small and large-distance behavior.
Section III presents a derivation of the potential that incorporates the
effect of quantum confinement, and it is applied to both the Coulomb
potential, which is valid for the special case
$\epsilon_{\text{sol}}=\epsilon$, and to the full Rytova-Keldysh potential for
which $\epsilon_{\text{sol}}\neq\epsilon$. Finally, Sec. IV analyzes the
exciton binding energy computed using each of the potentials presented in the
previous sections, and Sec. V concludes with a discussion about our findings.

##  II Electrostatics

The full Rytova-Keldysh potential, after dividing by the Coulomb energy
$E^{C}$, is given in momentum space by [17, 16]

$\displaystyle\begin{split}V^{\text{RK}}&(k,z_{1},z_{2})=-\frac{2\pi
d^{2}}{kd}\\\
&\times\frac{2\cosh\left[k\left(\frac{d}{2}-z_{1}\right)+\eta_{2}\right]\cosh\left[k\left(\frac{d}{2}+z_{2}\right)+\eta_{1}\right]}{\sinh\left(kd+\eta_{1}+\eta_{2}\right)}~{}~{},\end{split}$
(3)

where

$\eta_{a}\equiv\frac{1}{2}\ln\left(\frac{\epsilon+\epsilon_{a}}{\epsilon-\epsilon_{a}}\right)~{}~{},~{}~{}~{}\text{
for }a\in\\{1,2\\}~{}~{}.$ (4)

The coordinates $z_{1}$ and $z_{2}$ correspond to the position of the two
charges along the axis perpendicular to the plane as shown in Fig. 1, and are
both located in the interval $[-d/2,d/2]$. Furthermore, we define what is
usually called the Rytova-Keldysh potential as $V^{\text{RK}}(k)\equiv
V^{\text{RK}}(k,0,0)$.

Figure 2:  The Rytova-Keldysh potential compared to the three approximations
at small, intermediate, and large distances. In the regime $r\gg d$ the
Rytova-Keldysh potential is approximated by the Struve-Neumann potential,
given in Eq. (2). In the regime $r\gg d^{*}$ the Struve-Neumann potential
further reduces to the Coulomb potential of the solvent, given in Eq. (5). In
the regime $r\ll d$ the Coulomb potential of the semiconductor material is
recovered, given in Eq. (7)

To study the Rytova-Keldysh potential in real space Eq. (3) must be Fourier
transformed numerically for $z_{1}=z_{2}=0$, which Fig. 2 shows as a solid red
line. The result contains three different regions separated by the lengths $d$
and $d^{*}$, i.e., $r\ll d$, $r\gg d$, and at even larger distances $r\gg
d^{*}$, each resulting in a different approximation of the Rytova-Keldysh
potential. In the regime $r\gg d^{*}$ the Rytova-Keldysh potential reduces to
the Coulomb potential with the permittivity of the solvent, that in our units
is expressed as

$V^{\text{RK}}(r\gg d^{*})\simeq-\frac{d^{*}}{r}\equiv
V_{\text{sol}}^{\text{C}}(r)~{}~{}.$ (5)

Notice that even though $V^{\text{C}}_{\text{sol}}(r)$ seems to depend on the
thickness $d$ and material permittivity $\epsilon$, it is only but a byproduct
of scaling by $E^{C}$ — in S.I. units this potential is a function of
$\epsilon_{\text{sol}}$ alone, that is,

$V_{\text{sol}}^{\text{C}}(r)=-\frac{e^{2}}{4\pi\epsilon_{\text{sol}}}\frac{1}{r}\hskip
28.45274pt\text{ (in S.I. units)}.$ (6)

The Struve-Neumann potential approximates the Rytova-Keldysh potential for
distances $r\gg d$, as Fig. 2 shows, but because this approximation also
assumes $d^{*}/d=\epsilon/\epsilon_{\text{sol}}\gg 1$ it is not valid for
distances $r\ll d^{*}$. At small distances $r\ll d$ the Rytova-Keldysh
potential instead reduces to the Coulomb potential with the bulk permittivity
of the nanosheets, that in our units is simply

$V^{\text{RK}}(r\ll d)\simeq-\frac{d}{r}\equiv V^{\text{C}}(r)$ (7)

and which in S.I. units reads

$V^{\text{C}}(r)=-\frac{e^{2}}{4\pi\epsilon}\frac{1}{r}\hskip 28.45274pt\text{
(in S.I. units)}.$ (8)

In order to present a more physical picture of the Rytova-Keldysh potential,
it may be interpreted as having a space-dependent dielectric function that
connects the behavior at small and large distances, as

$V^{\text{RK}}(r)=-\frac{\epsilon}{\epsilon(r)}\frac{d}{r}~{}~{}.$ (9)

where $\epsilon(r)$ is the dielectric function. Equations (5), (7), and (9)
suggest that $\epsilon(r\gg d)/\epsilon\rightarrow
d/d^{*}=\epsilon_{\text{sol}}/\epsilon$ and $\epsilon(r\ll
d)/\epsilon\rightarrow 1$, as is explicitly shown in Fig. 3. Physically, at
small distances charges only feel the permittivity of the material as opposed
to the surrounding solvent at large distances. Note that the crossover in the
dielectric function roughly takes place at $r\simeq d$, as expected
physically. Notice also that Eqs. (5) and (7) do not explicitly depend on the
permittivities of the semiconductor material nor of the solvent, thus by using
the variables $r/d$ and $r/d^{*}$ the behavior of the potential is universal
for small and large distances respectively. Figure 2 shows the Rytova-Keldysh
potential and its approximations for each regime and Fig. 4 shows such
universal behavior for several values of
$d^{*}/d=\epsilon/\epsilon_{\text{sol}}$.

Figure 3:  Real-space dependence of the dielectric function divided by the
permittivity of the nanosheet $\epsilon(r)/\epsilon$, obtained from the
Rytova-Keldysh potential. At small distances $r/d\ll 1$ it saturates to one,
which means that $\epsilon(r)$ reduces to the permittivity of the nanosheet.
At large distances it saturates to $d/d^{*}=\epsilon_{\text{sol}}/\epsilon$,
recovering the Coulomb potential of the solvent. The horizontal lines mark the
corresponding saturation value.  Figure 4:  Real-space dependence of the
Rytova-Keldysh potential obtained by numerically Fourier transforming Eq. (3)
for $z_{1}=z_{2}=0$. The red lines (lowest one on the left, highest on the
right) correspond to $d/d^{*}=1$, in which case the Rytova-Keldysh potential
reduces to the Coulomb potential, and are the same on both plots. Vertical
dashed lines mark the values $d/d^{*}=\epsilon_{\text{sol}}/\epsilon$ (left)
and $d^{*}/d=\epsilon/\epsilon_{\text{sol}}$ (right). (Left) The Rytova-
Keldysh potential is shown in terms of $r/d^{*}$, thus the universal behavior
is present for $r/d^{*}\gg 1$. (Right) The Rytova-Keldysh potential is shown
in terms of $r/d$, thus now the universal behavior is present for $r/d\ll 1$.

Confining the electric field to two dimensions, as obtained by solving the
purely two-dimensional Poisson equation, modifies the usual $1/r$ behavior of
the Coulomb potential to a logarithmic behavior [24]. Since nanosheets are
(quasi) two-dimensional one could expect that the interaction potential
exhibits this logarithmic behavior at very small distances to a good
approximation. However, Figs. 2 and 4 clearly show that the Rytova-Keldysh
potential does not present such behavior, since electrostatics alone does not
incorporate any two-dimensional confinement at small distances, and
consequently it does not correctly describe the interaction between charges in
this regime.

##  III Quantum confinement

As a consequence of the Rytova-Keldysh potential lacking a logarithmic
behavior at small distances, it appears that the electrostatic approach alone
does not incorporate the complete physics of the problem. An important
omission from this picture are the quantum-mechanical corrections to the
potential, that is, the effect of the confined wave function of the
interacting charges in the $z$ direction, which become significant when the
inter-particle distance is of the order of the nanosheet thickness $d$ or
less.

Consider a charge confined to the nanosheet in the direction of the axis
perpendicular to the plane by an infinite well of size equal to the thickness
$d$. Solving for the $z$-component of the wave function in the Schrödinger
equation with such a confining potential yields

$u_{n}(z)=\sqrt{\frac{2}{d}}\cos\left(\frac{n\pi z}{d}\right)~{}~{}\text{, for
{\hbox{n = 1, 3, 5, ...}}}~{}~{},$ (10)

and

$u_{n}(z)=\sqrt{\frac{2}{d}}\sin\left(\frac{n\pi z}{d}\right)~{}~{}\text{, for
{\hbox{n = 2, 4, 6, ...}}}~{}~{},$ (11)

with the corresponding energy

$E_{n}=\frac{\hbar^{2}}{2m}\left(\frac{n\pi}{d}\right)^{2}~{}~{},$ (12)

where $n$ is a positive integer. If the energy difference between $E_{1}$ and
$E_{2}$ is large enough compared to the interaction energy, i.e.,
$E_{2}-E_{1}\gg E^{C}(d^{*}/d)$, then excited states are not populated and
only the ground state with wave function $u(z)\equiv u_{1}(z)$ alone
determines the effect of quantum confinement. In that case the thickness of
the nanosheet has to satisfy

$d\ll\frac{4\pi\epsilon_{\text{sol}}}{e^{2}}\frac{3\pi^{2}\hbar^{2}}{m}\equiv
d_{\text{max}}~{}~{}.$ (13)

for the system to be (quasi) two dimensional.

The ground-state wave function determines the quantum-confined potential
$V_{\text{qc}}^{\mathcal{V}}(k)$ as

$V_{\text{qc}}^{\mathcal{V}}(k)=\int_{-\frac{d}{2}}^{\frac{d}{2}}\text{d}z_{1}\int_{-\frac{d}{2}}^{\frac{d}{2}}\text{d}z_{2}~{}u^{2}(z_{1})u^{2}(z_{2})V^{\mathcal{V}}(k,z_{1},z_{2})~{}~{}.$
(14)

Note that $V^{\mathcal{V}}(k,z_{1},z_{2})$ corresponds to the in-plane Fourier
transform of a three-dimensional potential, that is

$V^{\mathcal{V}}(k,z_{1},z_{2})=\int\text{d}^{2}\bm{r}~{}V^{\mathcal{V}}(r,z_{1},z_{2})e^{i\bm{k}\cdot\bm{r}}~{}~{}.$
(15)

Here $\mathcal{V}=\text{C},~{}\text{RK}$ denotes either the Coulomb potential
or the full Rytova-Keldysh potential, respectively.

###  III.1 Coulomb potential

Let us first apply this procedure to the Coulomb potential for the purpose of
better understanding quantum confinement by itself, with the added advantage
that our findings carry over to the Rytova-Keldysh potential case as Sec.
III.2 presents. Using the Coulomb potential means that the nanosheet and the
solvent have the same permittivity, i.e.,
$\epsilon_{1}=\epsilon_{2}=\epsilon$, which is not experimentally realizable,
as the dielectric constant of a semiconductor typically obeys
$\epsilon\gg\epsilon_{\text{sol}}$, that is, $d^{*}\gg d$, but sheds light on
the mechanism that introduces the short-distance logarithmic behavior. Note
that in this case there is only one length scale, the thickness $d$, which is
equivalently expressed as $d^{*}=d$. Introducing the in-plane Fourier
transform of the Coulomb potential

$V^{\text{C}}(k,z_{1},z_{2})=-\frac{2\pi d^{2}}{kd}~{}e^{-k|z_{1}-z_{2}|}$
(16)

into Eq. (14) yields

$\displaystyle\begin{split}V_{\text{qc}}^{\text{C}}(k)&=-\frac{2\pi
d^{2}}{kd}~{}\frac{3(kd)^{3}+20\pi^{2}}{\left[(kd)^{2}+4\pi^{2}\right]^{2}}\\\
&\phantom{=}~{}-\frac{2\pi
d^{2}}{kd}~{}\frac{32\pi^{3}}{(kd)^{2}}\frac{kd-1+e^{-kd}}{\left[(kd)^{2}+4\pi^{2}\right]^{2}}\\\
&\equiv V_{<}(k)+V_{>}^{\text{C}}(k)~{}~{}.\end{split}$ (17)

The potential is separated into $V_{<}(k)$ and $V_{>}^{\text{C}}(k)$, each
contribution dominating in the regimes $r\ll d$ and $r\gg d$ respectively.
Notice that the Fourier transform of any of the terms contained in
$V_{>}^{\text{C}}(k)$ diverges if integrated separately from the others.

In the large-distance regime $r\gg d$ the quantum-confined Coulomb potential
reduces to the Coulomb potential, the reason being that quantum confinement is
significant only at small distances. The long-distance behavior is obtained by
expanding in powers of $kd$ and keeping the lowest-order term, yielding

$V_{\text{qc}}^{\text{C}}(kd\ll 1)\simeq-\frac{2\pi
d^{2}}{kd}=V^{\text{C}}(k)~{}~{}.$ (18)

In the small-distance regime $r\ll d$ the contribution $V_{<}(r)$ dominates,
up to a constant given by $V_{>}^{\text{C}}(r=0)$. An exact analytic
expression for $V_{<}(r)$ is derived from Fourier transforming Eq. (17), which
results in

$V_{<}(r)=-3K_{0}\left(\frac{2\pi r}{d}\right)+\frac{2\pi
r}{d}K_{1}\left(\frac{2\pi r}{d}\right)~{}~{},$ (19)

where $K_{\nu}(z)$ is the modified Bessel function of the second kind. As a
consequence of the asymptotic behavior of $V_{<}(r)$ that tends to zero as
$(r/d)^{\frac{1}{2}}e^{-r/d}$ for $r\rightarrow\infty$, its contribution is
significant at small distances and exponentially suppressed otherwise. The
limit $K_{\nu}(r\rightarrow 0)$ reveals the short-distance logarithmic
behavior as

$V_{<}(r\ll d)\simeq-3\left[\ln\left(\frac{\pi
r}{d}\right)+\gamma_{E}\right]-1~{}~{},$ (20)

where $\gamma_{E}$ is the Euler-Mascheroni constant. More physically, this
logarithmic behavior is a direct consequence of the fact that the effective
potential is obtained by averaging the bulk Coulomb potential over the
$z$-position of the electron and the hole.

Equation (20) shows some similarities between the quantum-confined Coulomb
potential and the Rytova-Keldysh potential at small distances, since both
depend on the ratio $r/d$ alone, however $V_{\text{qc}}^{\text{C}}(r)$
incorporates the expected logarithmic behavior whereas the Rytova-Keldysh
potential does not. In order to explicitly display the universality of the
quantum-confined Coulomb potential, Fig. 5 presents the result of numerically
Fourier transforming Eq. (17) to real space, as a function of $r/d$.

###  III.2 Rytova-Keldysh potential

With a better understanding of quantum confinement, let us explore the more
realistic electrostatics situation described by the Rytova-Keldysh potential.
Due to the disparate permittivities of nanosheet and solvent, the behavior of
the quantum-confined Rytova-Keldysh potential depends on the two distinct
length scales $d$ and $d^{*}$. Introducing the full Rytova-Keldysh potential
into Eq. (14) yields

$\displaystyle\begin{split}V_{\text{qc}}^{\text{RK}}(k)&=V_{<}(k)-\frac{2\pi
d^{2}}{kd}~{}\frac{32\pi^{4}}{(kd)^{2}}\\\
&\phantom{=V_{<}(k)}~{}\times\frac{1}{\left[(kd)^{2}+4\pi^{2}\right]^{2}}\left(kd-\frac{\sinh(kd+\eta_{1})\sinh(\eta_{2})+\sinh(kd+\eta_{2})\sinh(\eta_{1})-2\sinh(\eta_{1})\sinh(\eta_{2})}{\sinh(kd+\eta_{1}+\eta_{2})}\right)\\\
&\equiv V_{<}(k)+V_{>}^{\text{RK}}(k)~{}~{},\end{split}$ (21)

Figure 5:  Real-space dependence of the quantum-confined Coulomb potential as
a function of $r/d$, computed by Fourier transforming Eq. (17). The two
contributions $V_{<}(r)$ and $V_{>}^{\text{C}}(r)$ are represented by the
dotted and dashed lines, respectively. The thin solid line corresponds to
$V^{\text{C}}(r)$. The vertical dashed line correspond to the point $r/d=1$.

which is separated again into $V_{<}(k)$ and $V_{>}^{\text{RK}}(k)$,
dominating in the small and large-distance regimes respectively. Since
$V_{<}(k)$ is identical as for the quantum-confined Coulomb potential, given
in Eq. (17), it follows that the logarithmic behavior is in fact introduced by
quantum confinement regardless of the potential.

The tail of the Rytova-Keldysh potential reemerges in the large-distance
regime $r\gg d$ by expanding $V_{>}^{RK}(k)$ in powers of $kd$, while keeping
only the lowest-order term, which results in

$\displaystyle\begin{split}V_{\text{qc}}^{\text{RK}}(kd\ll
1)\simeq&-\frac{2\pi
d^{2}}{kd}~{}\frac{\cosh(\eta_{1})\cosh(\eta_{2})}{\cosh(\eta_{1}+\eta_{2})}\\\
&\times\frac{2}{kd+\tanh(\eta_{1}+\eta_{2})}~{}~{},\end{split}$ (22)

further simplifying to

$V_{\text{qc}}^{\text{RK}}(kd\ll
1)\simeq-\frac{2\pi(d^{*})^{2}}{kd^{*}}\frac{1}{\frac{kd^{*}}{2}+1}=V^{\mathcal{H}\text{N}}(k)$
(23)

in the case of $d^{*}/d=\epsilon/\epsilon_{\text{sol}}\gg 1$. Equation (23)
shows that the quantum-confined Rytova-Keldysh potential at large distances is
approximated by the Struve-Neumann potential that only depends on the length
scale $d^{*}$. In the small distance regime $r\ll d$ the logarithmic behavior
of $V_{<}(r)$ takes over, while $V_{>}^{\text{RK}}(r)$ saturates to a constant
at $r=0$, similarly to the quantum-confined Coulomb potential. Note, however,
that in the truly two-dimensional limit $d\rightarrow 0$ the quantum-confined
Rytova-Keldysh potential reduces to the Coulomb potential of the solvent as
given in Eq. (5). The same also occurs for the Rytova-Keldysh potential and
the Struve-Neumann potential. Physically, this demonstrates that in the limit
$d\rightarrow 0$ the effective permittivity is no longer determined by the
semiconductor nanosheets, but only by the solvent.

Figure 6 shows the real-space dependence of the quantum-confined Rytova-
Keldysh potential as a function of $r/d^{*}$ (on the left) and $r/d$ (on the
right), analogously to Fig. 4, and presents the universal behavior in the
regime $r/d^{*}\gg 1$ and $r/d\ll 1$ respectively. Because $V_{<}(r)$ only
depends on $r/d$, on the right side of Fig. 6 it shows as only one line.

Figure 6:  Real-space dependence of the quantum-confined Rytova-Keldysh
potential as a function of $r/d^{*}$ (left) and $r/d$ (right), computed by
Fourier transforming Eq. (21). The two contributions $V_{<}(r)$ and
$V_{>}^{\text{RK}}(r)$ are represented by dotted and dashed lines,
respectively. The thin solid lines correspond to $V^{\text{RK}}(r)$, for each
value of $d/d^{*}=\epsilon_{\text{sol}}/\epsilon$ (left) and
$d^{*}/d=\epsilon/\epsilon_{\text{sol}}$ (right). The red line (bottom one on
the left, top one on the right, $d/d^{*}=1$) is computed using the quantum-
confined Coulomb potential, and thus is the same as that shown in Fig. 5.
(Left) The vertical dashed lines correspond to the points $r/d^{*}=d/d^{*}$.
(Right) The vertical dashed magenta line corresponds to $r/d=1$.

Lastly, let us briefly consider the momentum-space dependence of each
potential with the aim of understanding their differing small and large-
distance behaviors from a new angle. As an additional benefit introducing
screening effects due to free charges is more straightforward in momentum
space, as Ref. [25] extensively shows. Figure 7 shows the momentum-space
dependence of the quantum-confined Rytova-Keldysh potential, the Struve-
Neumann potential, and the Rytova-Keldysh potential, all divided by
$V^{\text{C}}(k)$ as given in Eq. (18). In the regime $kd\ll 1$ every
potential saturates to $d^{*}/d=\epsilon/\epsilon_{\text{sol}}$, which
physically corresponds to recovering the Coulomb potential of the solvent at
large distances. On the opposite regime $kd\gg 1$ the tails differ
significantly: for the Rytova-Keldysh potential it tends to one, while for the
quantum-confined Rytova-Keldysh potential and the Struve-Neumann potential it
tends to zero. In the case of the Rytova-Keldysh potential the tail tends to
one at large momenta, which means that the Coulomb potential of the material
reappears at small distances. Since the logarithmic behavior in real space
appears due to Fourier transforming a term $\propto 1/k^{2}$, expressing the
Rytova-Keldysh potential in momentum space further shows that it does not
incorporate a logarithmic tail at small distances. For the quantum-confined
Rytova-Keldysh potential the tail does indeed tend towards zero as $1/k^{2}$
due to $V_{<}(k)$, thus resulting in a logarithmic behavior at small
distances. Because the Struve-Neumann potential is only but an approximation
of the Rytova-Keldysh potential for $kd\ll 1$, the behavior in the regime
$kd\gg 1$ is in principle not valid as it is used outside of the region of
applicability of this approximation, even though it resembles the quantum-
confined Rytova-Keldysh potential.

Figure 7:  Momentum-space dependence of the quantum-confined Rytova-Keldysh
potential (solid), the Struve-Neumann potential (dotted), and the Rytova-
Keldysh potential (dashed), divided by the Coulomb potential of the material,
as a function of $kd$. The quantum-confined Rytova-Keldysh potential is
computed from Eq. (21), the Struve-Neumann potential from Eq. (23), and the
Rytova-Keldysh potential from Eq. (3).

##  IV Excitons

Studying the small and large-distance behavior of the potentials does not by
itself present the whole picture of the exciton wave functions as quantum
mehcanics introduces another length scale into the problem due to the relative
kinetic energy of an electron-hole pair. Hence, we next compute the energy
level of an exciton bound state that forms due to the quantum-confined Rytova-
Keldysh potential, the Struve-Neumann potential, and the Rytova-Keldysh
potential with the intent of better understanding their differences and
appropriately comparing our results with the literature. Furthermore, we
determine not only the ground-state exciton energy level but also that of the
first several $s$-wave states, thus presenting a more complete analysis of our
findings.

For the purpose of relating our results to an experimentally realizable case,
we shift our focus towards CdSe nanoplatelets in hexane solvent at room
temperature [15, 25]. As a consequence, we use for the electron and hole
masses the values $m_{e}=0.27~{}m_{0}$ and $m_{h}=0.45~{}m_{0}$, where $m_{0}$
is the fundamental electron mass, corresponding to a thickness of 4.5 CdSe
monolayers having $d=1.37$ nm, given as $n=4$ in Ref. [26]. The permittivity
of the hexane solvent is $\epsilon_{\text{sol}}=2\epsilon_{0}$, where
$\epsilon_{0}$ is the vacuum permittivity. In an effort to present a more
general discussion, the ratio of the material and the hexane permittivities
$d^{*}/d=\epsilon/\epsilon_{\text{sol}}$ is treated as a free parameter, and
the exciton energy levels are computed for several thicknesses. Note that the
effect that the thickness has on the effective masses is neglected here for
simplicity without any impact on the qualitative behavior of our results.
Furthermore, any effects due to the finite lateral sizes of the nanoplatelets
are also neglected.

Finding the exciton wavefunction $u_{m}(r)$ and accompanying energy
$\mathcal{E}$ involves solving the radial part of the Schrödinger equation,
given by

$\left(-\frac{\text{d}^{2}}{\text{d}r^{2}}+\frac{m^{2}-1/4}{r^{2}}+\frac{d^{*}}{a_{0}}\frac{d^{*}}{d}\left(V(rd^{*})-\mathcal{E}\right)\right)u_{m}(r)=0~{}~{},$
(24)

with $m=0$ corresponding to the $s$-wave solution of the exciton ground state
on which we exclusively focus from now on. Here we have scaled the equation by
$E^{C}$, used $r\rightarrow rd^{*}$, and introduced the exciton Bohr radius
$a_{0}$. For a given material of permittivity $\epsilon$ the two-dimensional
Bohr radius $a_{0}$ is [27]

$a_{0}\equiv\frac{2\pi\epsilon}{e^{2}}\frac{\hbar^{2}}{m_{r}}~{}~{},$ (25)

where $m_{r}\equiv m_{e}m_{h}/(m_{e}+m_{h})$ is the reduced mass of the
exciton problem. Hence in general, the two dimensionless parameters that the
exciton energy turns out to depend on are $d^{*}/a_{0}$ and
$d^{*}/d=\epsilon/\epsilon_{\text{sol}}$. Notice that if the potential $V(r)$
is a function of the ratio $r/d^{*}$ alone, such as the Coulomb potential of
the solvent or the Struve-Neumann potential, then the dimensionless exciton
energy $\mathcal{E}$ depends only on the product
$(d^{*}/d)\cdot(d^{*}/a_{0})$. Lastly, note that the ratio $d^{*}/a_{0}$ does
not depend on the dielectric constant of the material $\epsilon$, and thus it
is equal to $d/a_{0}(\epsilon_{\text{sol}})$, i.e., the ratio of $d$ and the
exciton Bohr radius in the Coulomb potential of the solvent.

Let us analyze the exciton energies of CdSe in hexane in S.I. units that Fig.
8 presents, obtained from computing the exciton ground-state energy as a
function of the parameter $d^{*}/d=\epsilon/\epsilon_{\text{sol}}$, for
several values of $d^{*}/a_{0}$. Consider first the exciton energy obtained
from the Rytova-Keldysh potential $\mathcal{E}^{\text{RK}}$, represented by
the dashed lines. Because the Rytova-Keldysh potential reduces to the Coulomb
potential if $d^{*}/d=1$, the exciton energy reduces to the hydrogen-like
result

$\mathcal{E}^{\text{C}}(\epsilon)=-\frac{2m_{r}e^{4}}{(4\pi\epsilon\hbar)^{2}}~{}~{},$
(26)

that we define for the solvent as
$\mathcal{E}^{\text{C}}_{\text{sol}}\equiv\mathcal{E}^{\text{C}}(\epsilon_{\text{sol}})$,
and is represented by an empty magenta dot. Furthermore, still focusing on the
$d^{*}/d=1$ case, notice that the quantum-confined Rytova-Keldysh potential
analogously reduces to the quantum-confined Coulomb potential, thus resulting
in a smaller exciton energy
$|\mathcal{E}^{\text{RK}}_{\text{qc}}|<\left|\mathcal{E}^{\text{C}}_{\text{sol}}\right|$
solely due to quantum confinement. In order to satisfy the upper limit on the
thickness set by Eq. (13) the parameter $d^{*}/a_{0}$ has to be small enough,
which turns out to be $d\ll d_{\text{max}}\simeq 3.8$ nm for CdSe
nanoplatelets. This implies that we can consider only nanoplatelets with a
thickess of at most 10 CdSe layers. This condition is indeed satisfied for
every curve shown in Fig. 8. Regarding the Struve-Neumann potential, it should
be noticed that the resulting energy is not physically meaningful in the
$d^{*}/d=1$ case because it strictly speaking does not satisfy the assumption
$d^{*}/d\gg 1$ used in the derivation of Eq. (23).

Moving on to the regime $d^{*}/d>1$, $\mathcal{E}^{\text{RK}}$ separates from
$\mathcal{E}^{\text{C}}_{\text{sol}}$ as it becomes closer to zero. Note that
as the ratio $d^{*}/a_{0}$ grows, $\mathcal{E}^{\text{RK}}$ tends to
$\mathcal{E}^{\text{C}}(\epsilon)$ due to the Rytova-Keldysh potential
reducing to the Coulomb potential of the material in the limit
$d\rightarrow\infty$. Furthermore, when $d^{*}/d\gg d^{*}/a_{0}$, i.e.,
$a_{0}\gg d$, every one of the potentials considered in Fig. 8 results in the
same exciton energy, meaning that the small-distance behavior $r\ll d^{*}$ is
no longer significant.

In the context of the dependence on the ratio $d^{*}/a_{0}$ we similarly
identify two regimes. For $d^{*}/a_{0}\ll 1$ the electric field is mostly
outside of the nanoplatelet, and consequently the exciton energy is close to
that obtained using the Coulomb potential of the solvent
$\mathcal{E}^{\text{C}}_{\text{sol}}$. Only in the limit $d^{*}/d\gg 1$ the
effect of the permittivity of the nanoplatelet significantly impacts the
resulting energy. In the case of $d^{*}/a_{0}\gg 1$ the opposite is true: the
permittivity of the nanoplatelet has a very significant effect on the exciton
energy for any value of $d^{*}/d=\epsilon/\epsilon_{\text{sol}}$.

Figure 8:  Exciton ground-state energy computed using the quantum-confined
Rytova-Keldysh potential $\mathcal{E}^{\text{RK}}_{\text{qc}}$ (solid), the
Struve-Neumann potential $\mathcal{E}^{\mathcal{H}\text{N}}$ (dotted), and the
Rytova-Keldysh potential $\mathcal{E}^{\text{RK}}$ (dashed). The filled dots
mark the points for which $d^{*}/d=1$, while the empty magenta dot marks
$\mathcal{E}^{\text{RK}}=\mathcal{E}^{\text{C}}_{\text{sol}}$. The line
labelled with “CdSe” corresponds to the thickness $d=1.37$ nm
($d^{*}/a_{0}\simeq 4.37$), and the empty light-blue dot corresponds to the
ratio $d^{*}/d=\epsilon/\epsilon_{\text{sol}}\simeq 6.99$ that results in the
exciton ground-state energy of $-193$ meV and exciton Bohr radius of
$a_{0}\simeq 2.19$ nm, representing the CdSe nanoplatelets considered in Refs.
[15, 25]. Lower values of $d^{*}/a_{0}$ result in more negative energies.

Previously we discussed the universality of the Rytova-Keldysh potential and
quantum-confined Rytova-Keldysh potential at small and large distances, made
explicit when expressed in terms of the variables $r/d$ and $r/d^{*}$
respectively. Even though this universality is also reproduced by the exciton
energy, it is not immediately obvious in Fig. 8. For the purpose of studying
the universal behavior, the exciton energy is made dimensionless by
$E^{C}\equiv e^{2}/4\pi\epsilon d$ which results in a dimensionless exciton
energy that we compute as a function of $(d^{*}/d)\cdot(d^{*}/a_{0})$. Figure
9 shows the exciton energies from Fig. 8 altered by these transformations.
Despite the fact that in general $\mathcal{E}$ depends on
$d^{*}/d=\epsilon/\epsilon_{\text{sol}}$ and $d^{*}/a_{0}$ separately, in the
limit $d^{*}/d\gg 1$ the exciton energy turns out to be the same regardless of
the potential used — hence it depends on a single variable: the product of the
two parameters as they appear in Eq. (24). Consequently
$\mathcal{E}_{\text{qc}}^{\text{RK}}$ shows a data collapse in the regime
$d^{*}/d\gg d^{*}/a_{0}$, that is, the exciton Bohr radius is much larger than
the thickness $a_{0}\gg d$.

Figure 9:  Exciton energy, scaled by $E^{C}\equiv e^{2}/4\pi\epsilon d$,
computed using the quantum-confined Rytova-Keldysh potential (solid) and the
Rytova-Keldysh potential (dashed). Because the Coulomb potential of the
solvent and the Struve-Neumann potential only depend on $d^{*}$, the resulting
energies $\mathcal{E}^{\text{C}}_{\text{sol}}$ and
$\mathcal{E}^{\mathcal{H}\text{N}}$ (thin and thick magenta solid lines,
respectively) do not depend on the ratio $d^{*}/a_{0}$. The filled dots mark
the points for which $d^{*}/d=1$, while the empty dots mark
$\mathcal{E}^{\text{RK}}=\mathcal{E}^{\text{C}}_{\text{sol}}$. Higher values
of $d^{*}/a_{0}$ result in more negative energies.

For a better comparison with other experimental results for CdSe in hexane
solvent, Fig. 10 shows not only the ground-state energy but also the energy of
the first several excited $s$-wave states at the fixed ratio
$d^{*}/a_{0}\simeq 4.37$ corresponding to the experiment studied in Ref. [15,
25]. Furthermore, each energy is obtained from the quantum-confined Rytova-
Keldysh potential, the Struve-Neumann potential, and the Rytova-Keldysh
potential. Notice that the excited-states energies as a function of
$d^{*}/d=\epsilon/\epsilon_{\text{sol}}$ generally follow the same trend as
that of the ground state. However, the right side of Fig. 10 shows that using
the Rytova-Keldysh potential results in an energy closer to that obtained from
the quantum-confined Rytova-Keldysh potential than from the Struve-Neumann
potential.

Figure 10:  Energy of the first several $n$s states computed using the
quantum-confined Rytova-Keldysh potential (solid), the Struve-Neumann
potential (dotted), and the Rytova-Keldysh potential (dashed). Computed for
the thickness $d=1.37$ nm ($d^{*}/a_{0}\simeq 4.37$). On the left, the
energies of the exciton ground state and the first four excited $n$s states.
The empty dots mark the value of $d^{*}/d=\epsilon/\epsilon_{\text{sol}}\simeq
6.99$ that results in the exciton ground-state energy of $-193$ meV,
representing the CdSe nanoplatelets considered in Refs. [15, 25]. The energies
of the excited states are $-96.1$ meV, $-58.6$ meV, $-39.2$ meV, and $-27.9$
meV. On the right, a zoom-in on the energy of the $3$s state, as an example,
better showing the differences between potentials.

##  V Summary and conclusion

In an effort to elucidate the behavior of the Rytova-Keldysh potential, we
have presented a comprehensive discussion on the mechanism responsible for
quantum confinement, alongside the two-dimensional short-distance logarithmic
behavior of the interaction potential. Section II began by presenting the
actual behavior of the Rytova-Keldysh potential at small distances, which led
us to develop a method to incorporate quantum confinement into the potential.
Our approach is based on the charges being confined by an infinite well, but
can easily be generalized to other situations that may be described by using a
different confinement potential, in a similar direction as in Ref. [22]. Of
course, this would quantitatively affect our results for the quantum-confined
potentials at small distances $r<d$, but not qualitatively. With the quantum-
confined Rytova-Keldysh potential in hand, we analyzed its behavior in terms
of the length scales $d$ and $d^{*}=\epsilon d/\epsilon_{\text{sol}}$ by
comparing either the inter-particle distance $r$ or the two-dimensional
exciton Bohr radius $a_{0}$ to these length scales. Note that we have
implicitly assumed that the exciton Bohr radius $a_{0}$, as well as every
other length scale, is much larger than the lattice spacing of the material,
that is, we only treat Wannier excitons, not Frenkel excitons. In order to
contrast our results with the literature, we provided an in-depth analysis of
the exciton energy obtained using the quantum-confined Rytova-Keldysh
potential, in terms of the variables $d^{*}/d=\epsilon/\epsilon_{\text{sol}}$
and $d^{*}/a_{0}$. In the future we aim to present a similar analysis for
charges in the bulk and on the surface of a topological insulator.

## Acknowledgments

We would like to thank an anonymous referee of Ref. [25] for the helpful
discussion that sparked the development of this paper. This work is part of
the research programme TOP-ECHO with project number 715.016.002, and is also
supported by the D-ITP consortium. Both are programmes of the Netherlands
Organisation for Scientific Research (NWO) that is funded by the Dutch
Ministry of Education, Culture and Science (OCW).

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