#  Size quantization of an exciton: A toy model of the “dead layer”

M. E. Raikh  Department of Physics and Astronomy, University of Utah, Salt
Lake City, UT 84112

###### Abstract

Size-quantization levels of an exciton in large nanocrystals is studied
theoretically. For the nanocrystal size, $L$, much bigger than the Bohr
radius, $a_{B}$, the level positions do not depend on $a_{B}$. The correction
to the levels in a small parameter $a_{B}/L$ depends on the reflection phase
of the exciton from the boundary. Calculation of this phase constitutes a
three-body problem: electron, hole, and the boundary. This calculation can be
performed analytically in the limit when the hole is much heavier than the
electron. Physically, a slow motion of the hole towards the boundary takes
place in the effective potential created by the fast motion of the electron
orbiting the hole and touching the boundary. As a result, the hole is
reflected before reaching the boundary. The distance of the closest approach
of the hole to the boundary (the dead layer) exceeds $a_{B}$ parametrically.

##  I Introduction

A concept of the exciton dead layerpekar ; Hopfield  has emerged in the course
of study of the light reflection from a boundary between air and a medium with
strong exciton-photon coupling. As a result of this coupling, two (exciton-
like and photon-like) waves can propagate in the medium. Thus, in order to
find the reflection coefficient, conventional boundary conditions of the
continuity of the tangent components of electric and magnetic fields,
additional boundary condition is needed. Analysis in the pioneering paper by
Hopfield and ThomasHopfield  suggests that the form of this condition is
vanishing of the exciton polarization at certain distance, $l$, away from the
boundary. This distance was estimated in Ref. Hopfield,  as $l=2a_{B}$, where
$a_{B}$ is the Bohr radius of the exciton. In early, see e.g. Refs. 1974, ;
Bassani, ; DelSole2, ; DelSole1, ; DelSole, ; Bastard, , as well as in recent,
see e.g. Refs. Cardona, ; Ivchenko, ; Poddubny, , follow-up papers the concept
of dead layer associated with distance, $l$, was employed.

In the absence of the exciton-photon coupling, the issue of reflection of the
exciton from the surface is still important. Namely, the phase of the
reflection coefficient defines the positions of the size-quantization level of
the exciton in the quantum well.

In particular, in Ref. Bassani, , photoluminescence spectra from thick GaAs
quantum well revealed a number of peaks attributed to the size quantization of
excitons. Their positions were fit by a formula

${\cal E}_{n}=E_{x}+\frac{\pi^{2}\hbar^{2}n^{2}}{2M\bigl{(}L-2l\bigr{)}^{2}},$
(1)

where $L$ is the well thickness, $M$ is the net mass of the exciton, and
$E_{x}$ is the position of the exciton line in the bulk GaAs. For the dead-
layer thickness, the value $l=a_{B}$ was chosen.

Size quantization also manifests itself in the positions of the exciton
luminescence lines in nanocrystals. These studies, pioneered in Refs. ekimov,
; Brus, , were later conducted on wide variety of nanocrystals. The importance
of dead layer for the size-quantization in ZnO nanocrystals was studied in
Ref. Fonoberov, .

Recent revival of interest to excitons in nanocrystals is related to inorganic
perovskites, which, due to their exceptionally high photoemission rates, show
a good promise for applications. As demonstrated experimentally in Ref. nano1,
, these nanocrystals exhibit a strong size-quantization effect. Further
experiments, see e.g. Refs. nano2, ; nano3, ; nano4, ; nano5, , indicated that
nanocrystal sizes vary in the range $10$nm-$50$nm, while the exciton Bohr
radius was estimated as $a_{B}\sim 2-6$nm.nano5

On the theoretical side, microscopic description of the exciton dead layer
poses a challenge, since it is a three-body problem: electron, hole and a
boundary. In other words, while the motion of electron and hole can be
separated into the motion of the center of mass and the relative motion, this
separation is inconsistent with the boundary conditions that the two-particle
wave function turns to zero when electron and hole touch the boundary
individually. These no-escape boundary conditions constitute a microscopic
origin of the dead layer.

Below we consider a toy model which allows to derive the dead layer
analytically and confirm Eq. (1) rigorously from the Schrödinger equation. A
crucial assumption, which allows us to capture the electron-hole correlation
in the presence of the boundary, is that the hole mass, $m_{h}$, is much
bigger than the electron mass, $m_{e}$. Under this assumption, to the first
order in $m_{e}/m_{h}$, the fast motion of electron takes place in the field
of a “static” hole.DelSole  Our main conclusion is that, in the formula Eq.
(1), the thickness of the dead layer, $l_{N}$, is not constant, but depends on
the number, $N$, of the quantization level.

Figure 1: (Color online) (a) Within a toy model, one-dimensional nanocrystal
occupies a domain $|x|<\frac{L}{2}$. When the hole is much heavier than the
electron, a slow motion of the hole takes place in the effective potential
created by the fast motion of the electron. As electron approaches one of the
hard walls closer than $\frac{1}{2}a_{B}$, where $a_{B}$ is the Bohr radius,
the bound state for electron disappears. When electron is outside the yellow
regions, it “senses” the boundary only weakly, and, via the interaction,
creates a barrier for the hole; (b) Discrete energies of the exciton is
determined by the size quantization of the hole. The hole does not touch the
boundaries, it is rather reflected from the barrier of the width, $l$, (the
dead layer) created by the electron.

##  II The model

To maximally simplify the calculations, we assume that the confinement is one-
dimensional in the form of two hard walls at $x=\pm\frac{L}{2}$, see Fig. 1.
Moreover, we will replace the Coulomb attraction between the electron and hole
by a short-range attraction. Under these assumptions, the Hamiltonian of the
model reads

${\hat{H}}=\frac{{\hat{p}}_{e}^{2}}{2m_{e}}+\frac{{\hat{p}}_{h}^{2}}{2m_{h}}-v_{0}\delta(x_{e}-x_{h}),$
(2)

where $x_{e}$ and $x_{h}$ are the coordinates of the electron and hole,
respectively; $v_{0}$ is the attraction strength. The wave function
$\Psi(x_{e},x_{h})$ satisfies the boundary conditions

$\Psi\left(\pm\frac{L}{2},x_{h}\right)=0,~{}~{}~{}~{}\Psi\left(x_{e},\pm\frac{L}{2}\right)=0.$
(3)

If $m_{e}$ and $m_{h}$ were comparable, the conditions Eq. (3) would strongly
mix the motion of the center of mass and the relative motion near the
boundaries. Thus, to proceed further, we assume that the hole is much heavier
than the electron and exploit the smallness of the parameter $m_{e}/m_{h}$.

###  II.1 Lowest order in $m_{e}/m_{h}$

In the limit of large $m_{h}$, we can neglect the kinetic energy of the hole.
After that $x_{h}$ enters the Hamiltonian Eq. (2) only as a parameter, which
can be viewed as fixed. Then the general form of the eigenfunction,
$\varphi_{x_{h}}(x_{e})$, which turns to zero at $x_{e}=\pm\frac{L}{2}$ is the
following

$\displaystyle\varphi_{x_{h}}(x_{e})=\alpha\sinh\Big{[}\kappa_{h}\Big{(}\frac{L}{2}+x_{e}\Big{)}\Big{]},~{}~{}~{}-\frac{L}{2}<x_{e}<x_{h}$
$\displaystyle\varphi_{x_{h}}(x_{e})=\beta\sinh\Big{[}\kappa_{h}\Big{(}x_{e}-\frac{L}{2}\Big{)}\Big{]},~{}~{}~{}x_{h}<x_{e}<\frac{L}{2}.$
(4)

The corresponding energy is
$E_{e}(\kappa_{h})=-\frac{\hbar^{2}\kappa_{h}^{2}}{2m_{e}}$. The condition of
continuity at $x_{e}=x_{h}$ reads

$\alpha\sinh\Big{[}\kappa_{h}\Big{(}\frac{L}{2}+x_{h}\Big{)}\Big{]}=\beta\sinh\Big{[}\kappa_{h}\Big{(}x_{h}-\frac{L}{2}\Big{)}\Big{]}.$
(5)

The discontinuity of the derivative at $x_{e}=x_{h}$ yields the second
condition

$\displaystyle\kappa_{h}\alpha$
$\displaystyle\cosh\Big{[}\kappa_{h}\Big{(}\frac{L}{2}+x_{h}\Big{)}\Big{]}-\kappa_{h}\beta\cosh\Big{[}\kappa_{h}\Big{(}x_{h}-\frac{L}{2}\Big{)}\Big{]}$
$\displaystyle=-\frac{2m_{e}v_{0}}{\hbar^{2}}\alpha\sinh\Big{[}\kappa_{h}\Big{(}\frac{L}{2}-x_{h}\Big{)}\Big{]}.$
(6)

Combining Eqs. (5) and (II.1), we get the equation for $\kappa_{h}$

$\sinh\kappa_{h}L=\frac{2m_{e}v_{0}}{\hbar^{2}\kappa_{h}}\sinh\Big{[}\kappa_{h}\Big{(}\frac{L}{2}-x_{h}\Big{)}\Big{]}\sinh\Big{[}\kappa_{h}\Big{(}\frac{L}{2}+x_{h}\Big{)}\Big{]}.$
(7)

This equation defines the dependence $\kappa_{h}(x_{h})$, i.e. the dependence
of the electron energy on the position of the hole.

The concept of a dead layer is meaningful only when the “nanocrystal” is big,
namely, $L$ is much bigger than the size of the exciton. In the limit
$\kappa_{h}L\gg 1$ and $\big{(}\frac{L}{2}-x_{h}\big{)}\ll L$, we can replace
$\sinh\kappa_{h}L$ by $\frac{1}{2}\exp\big{(}\kappa_{h}L\big{)}$ and
$\sinh\big{[}\kappa_{h}\big{(}\frac{L}{2}+x_{h}\big{)}\big{]}$ by
$\frac{1}{2}\exp\big{[}\kappa_{h}\big{(}\frac{L}{2}+x_{h}\big{)}\big{]}$. Then
Eq. (7) assumes the form

$\kappa_{h}=\frac{m_{e}v_{0}}{\hbar^{2}}\bigg{\\{}1-\exp\Big{[}-2\kappa_{h}\Big{(}\frac{L}{2}-x_{h}\Big{)}\Big{]}\bigg{\\}},$
(8)

where the exponent in the square brackets accounts for electron “hitting” the
right wall and does not “sense” the left wall. When the exponent is small, Eq.
(8) yields

$\kappa_{h}=\frac{m_{e}v_{0}}{\hbar^{2}}=\frac{1}{a_{B}},$ (9)

and, correspondingly, $E_{B}=\frac{\hbar^{2}}{2m_{e}a_{B}^{2}}$ for the
binding energy.

As $x_{h}$ approaches the right (or left) wall, the bound state disappears at
critical $\frac{L}{2}-x_{h}=\frac{a_{B}}{2}$. Near the threshold, the solution
of Eq. (8) behaves linearly with $x_{h}$

$\kappa_{h}=\frac{\Big{(}\frac{L}{2}-x_{h}-\frac{a_{B}}{2}\Big{)}}{\left(\frac{L}{2}-x_{h}\right)^{2}}\approx\frac{4}{a_{B}^{2}}\left[\frac{L}{2}-x_{h}-\frac{a_{B}}{2}\right].$
(10)

The corresponding normalized wave functions have the form

$\varphi_{x_{h}}(x_{e})=\frac{\left(2\kappa_{h}\right)^{1/2}}{a_{B}}\begin{cases}\bigl{(}L-2x_{e}\bigr{)},~{}x_{h}<x_{e}<\frac{L}{2},\\\
a_{B}\exp\bigl{[}\kappa_{h}(x_{e}-x_{h})\bigl{]},~{}x_{e}<x_{h}.\end{cases}$
(11)

With $E_{e}(\kappa_{h})$ depending on $x_{h}$ via Eq. (8), a hole with a
finite mass will move slowly in the effective potential created by a fast-
moving electron. We will establish the form of this potential in the next
Section.

##  III Effective potential for a hole

To incorporate the hole motion, we search for the solution of the Scrödinger
equation ${\hat{H}}(x_{e},x_{h})\Psi=E\Psi$ in the form

$\Psi(x_{e},x_{h})=\varphi_{x_{h}}(x_{e})\Phi(x_{h}).$ (12)

Substitution of this form into the Schrödinger equation yields

$\displaystyle-$
$\displaystyle\frac{\hbar^{2}}{2m_{h}}\left[\varphi_{x_{h}}(x_{e})\frac{\partial^{2}\Phi}{\partial
x_{h}^{2}}+2\frac{\partial\varphi_{x_{h}}(x_{e})}{\partial
x_{h}}\frac{\partial\Phi}{\partial
x_{h}}+\frac{\partial^{2}\varphi_{x_{h}}(x_{e})}{\partial
x_{h}^{2}}\Phi(x_{h})\right]$ $\displaystyle=$
$\displaystyle\Big{[}E-E_{e}(\kappa_{h})\Big{]}\varphi_{x_{h}}(x_{e})\Phi(x_{h}).$
(13)

In deriving Eq. (III) we took into account that $\varphi_{x_{h}}(x_{e})$
satisfies the equation

$\left[\frac{{\hat{p}}_{e}^{2}}{2m_{e}}-v_{0}\delta(x_{e}-x_{h})\right]\varphi_{x_{h}}(x_{e})=E_{e}(\kappa_{h})\varphi_{x_{h}}(x_{e}).$
(14)

We assume that the electron wave function $\varphi_{x_{h}}(x_{e})$ is
normalized $\int dx_{e}\Big{(}\varphi_{x_{h}}(x_{e})\Big{)}^{2}=1$. To obtain
a closed equation for the hole wave function, $\Phi(x_{h})$, we multiply Eq.
(III) by $\varphi_{x_{h}}(x_{e})$ and integrate over $x_{e}$. This yields

$-\frac{\hbar^{2}}{2m_{h}}\left[\frac{\partial^{2}\Phi}{\partial
x_{h}^{2}}+2I(x_{h})\frac{\partial\Phi}{\partial
x_{h}}+J(x_{h})\Phi\right]=\Big{[}E-E_{e}(\kappa_{h})\Big{]}\Phi,$ (15)

where the functions $I(x_{h})$ and $J(x_{h})$ are defined as

$\displaystyle I(x_{h})=$ $\displaystyle\int
dx_{e}\varphi_{x_{h}}(x_{e})\frac{\partial\varphi_{x_{h}}(x_{e})}{\partial
x_{h}}$ $\displaystyle=$ $\displaystyle\frac{\partial}{2\partial x_{h}}\int
dx_{e}\Big{(}\varphi_{x_{h}}(x_{e})\Big{)}^{2},$ (16) $\displaystyle
J(x_{h})=$ $\displaystyle\int
dx_{e}\varphi_{x_{h}}(x_{e})\frac{\partial^{2}\varphi_{x_{h}}(x_{e})}{\partial
x_{h}^{2}}$ $\displaystyle=$ $\displaystyle\frac{\partial^{2}}{2\partial
x_{h}^{2}}\int dx_{e}\Big{(}\varphi_{x_{h}}(x_{e})\Big{)}^{2}-\int
dx_{e}\left(\frac{\partial\varphi}{\partial x_{h}}\right)^{2}.$ (17)

Normalization condition ensures that $I(x_{h})=0$ and that the first term in
the right-hand side in the expression for $J(x_{h})$ is zero.

The fact that the equation for $\Phi(x_{h})$ is closed is a consequence of our
choice Eq. (12) of the wave function $\Psi(x_{e},x_{h})$ in the form of the
product. By making this choice, we neglected the excited states of the
electron wave function. This choice is justified if the typical electron
energy, $E_{e}(\kappa_{h})$, is much bigger than the size-quantization energy,
$\sim\hbar^{2}/m_{h}L^{2}$, of the hole. Upon setting $I(x_{h})=0$ we cast Eq.
(18) into the form of the Schrödinger equation

$-\frac{\hbar^{2}}{2m_{h}}\frac{\partial^{2}\Phi}{\partial
x_{h}^{2}}+V(x_{h})\Phi=E\Phi,$ (18)

with effective potential, $V(x_{h})$, representing the sum

$V(x_{h})=\frac{\hbar^{2}}{2m_{h}}\int
dx_{e}\left(\frac{\partial\varphi_{x_{h}}(x_{e})}{\partial
x_{h}}\right)^{2}-\frac{\hbar^{2}}{2m_{e}}\Big{(}k_{h}(x_{h})\Big{)}^{2}.$
(19)

##  IV Comparing contributions to the effective potential

At distances much bigger than $a_{B}$ from the boundary the second term in
$V(x_{h})$ dominates, we thus have $\kappa_{h}=\frac{1}{a_{B}}$, so that
$V(x_{h})=-E_{B}$. Upon approach to the right boundary $x_{h}=\frac{L}{2}$ the
term $\frac{\hbar^{2}\kappa_{h}^{2}}{2m_{e}}$ falls off and finally vanishes
at $x_{h}=\frac{L-a_{B}}{2}$. Near this point the first term in $V(x_{h})$
dominates the effective potential. To trace the crossover between the first
and second terms, it is sufficient to use the asymptotic expressions Eqs. (10)
and (11). Differentiating Eq. (11) with respect to $x_{e}$ we get

$\frac{\partial\varphi_{x_{h}}(x_{e})}{\partial
x_{h}}=\bigl{[}1+2\kappa_{h}(x_{e}-x_{h})\bigr{]}\exp\bigl{[}\kappa_{h}(x_{e}-x_{h})\bigr{]}\frac{d\kappa_{h}}{dx_{h}}.$
(20)

Expressing $\frac{d\kappa_{h}}{dx_{h}}$ from Eq. (10) we have

$\displaystyle\Biggl{(}\frac{\partial\varphi_{x_{h}}(x_{e})}{\partial
x_{h}}\Biggr{)}^{2}$
$\displaystyle=\frac{8}{\kappa_{h}a_{B}^{4}}\Bigl{[}1+2\kappa_{h}(x_{e}-x_{h})\Bigr{]}^{2}\exp\bigl{[}2\kappa_{h}(x_{e}-x_{h})\bigr{]}.$
(21)

Then the integration over $x_{e}$ yields

$\displaystyle\frac{\hbar^{2}}{2m_{h}}\int
dx_{e}\left(\frac{\partial\varphi_{x_{h}}(x_{e})}{\partial x_{h}}\right)^{2}$
$\displaystyle=\frac{2\hbar^{2}}{m_{h}\kappa_{h}^{2}a_{B}^{4}}=\frac{\hbar^{2}}{8m_{h}\big{[}L-a_{B}-2x_{h}\big{]}^{2}}.$
(22)

Thus, the first term in $V(x_{h})$ falls off quadratically away from the
threshold $x_{h}=\frac{1}{2}\left(L-a_{B}\right)$. It should be compared to
the second term, $\frac{\hbar^{2}\kappa_{h}^{2}}{2m_{e}}$. Using the threshold
behavior Eq. (10) of $\kappa_{h}$, we have

$\frac{\hbar^{2}\kappa_{h}^{2}}{2m_{e}}=\frac{2\hbar^{2}}{m_{e}a_{B}^{4}}\bigg{[}L-a_{B}-2x_{h}\bigg{]}^{2}.$
(23)

Comparing Eqs. (IV) and (23) we conclude that the crossover from the first to
the second term in $V(x_{h})$ takes place at

$L-a_{B}-2x_{h}=\frac{a_{B}}{2}\left(\frac{m_{e}}{m_{h}}\right)^{1/4}.$ (24)

We see that crossover takes place in the threshold region, i.e. at
$\left(L-a_{B}-2x_{h}\right)\ll a_{B}$. This is ensured by the smallness of
the ratio $m_{e}/m_{h}$. Note, that this smallness also justifies the above
use of the asymptotic expressions for $\varphi_{h}(x_{e})$ and for
$\kappa_{h}(x_{h})$.

##  V Energy levels

Semiclassical quantization condition for the particle with mass, $m_{h}$,
moving in the potential, $V(x_{h})$, see Fig. 1, reads

$\displaystyle
2\left(\frac{2m_{h}}{\hbar^{2}}\right)^{1/2}\int\limits_{0}^{x_{t}}dx_{h}\Bigg{[}E_{N}+\frac{\hbar^{2}\bigl{(}\kappa_{h}(x_{h})\bigr{)}^{2}}{2m_{e}}\Bigg{]}^{1/2}$
$\displaystyle=\pi\left(N+\frac{1}{2}\right).$ (25)

where $x_{t}$ is the turning point defined as
$\hbar\kappa_{h}(x_{t})=\left(2m_{e}|E_{N}|\right)^{1/2}$. Using Eq. (8) we
get the following expression for $x_{t}$

$x_{t}=\frac{L}{2}+\frac{a_{B}}{2}\Biggl{(}\frac{E_{B}}{|E_{N}|}\Biggr{)}^{1/2}\ln\Biggl{[}1-\biggl{(}\frac{|E_{N}|}{E_{B}}\biggr{)}^{1/2}\Biggr{]}.$
(26)

Since $\kappa_{h}$ changes only in the vicinity of the upper limit and is
equal to $\frac{1}{a_{B}}$ otherwise, it is convenient to isolate the constant
part

$\displaystyle\Bigg{[}E_{N}+\frac{\hbar^{2}\kappa_{h}^{2}}{2m_{e}}\Bigg{]}^{1/2}=\Bigg{[}E_{N}+\frac{\hbar^{2}}{2m_{e}a_{B}^{2}}\Bigg{]}^{1/2}$
$\displaystyle+\frac{\frac{\hbar^{2}}{2m_{e}}\bigl{(}\kappa_{h}^{2}-\frac{1}{a_{B}^{2}}\bigr{)}}{\biggl{(}E_{N}+\frac{\hbar^{2}\kappa_{h}^{2}}{2m_{e}}\biggr{)}^{1/2}+\biggl{(}E_{N}+\frac{\hbar^{2}}{2m_{e}a_{B}^{2}}\biggr{)}^{1/2}}.$
(27)

Integration of the first term is elementary. To perform integration of the
second term it is convenient to switch from the variable $x_{h}$ to the
dimensionless variable $u=\kappa_{h}a$. To do so, we differentiate both sides
of Eq. (8). This yields

$\frac{du}{2u^{2}}\bigg{[}\ln\left(1-u\right)+\frac{u}{1-u}\bigg{]}=-\frac{dx_{h}}{a_{B}}.$
(28)

Then the condition Eq. (V) takes the form

$\displaystyle\frac{\pi}{2}\left(N+\frac{1}{2}\right)\Biggl{(}\frac{m_{e}}{m_{h}}\Biggr{)}^{1/2}$
$\displaystyle=\left(1-q_{N}\right)^{1/2}\Bigg{[}\frac{L}{2a_{B}}+\frac{\ln\left(1-q_{N}^{1/2}\right)}{2q_{N}^{1/2}}\Bigg{]}-I(q_{N}).$
(29)

where the function $I(q_{N})$ is defined as

$\displaystyle I(q_{N})=$
$\displaystyle\int\limits_{q_{N}^{1/2}}^{1}\frac{du}{2u^{2}}\bigg{[}\ln\left(1-u\right)+\frac{u}{1-u}\bigg{]}\frac{1-u^{2}}{\left(u^{2}-q_{N}\right)^{1/2}+\left(1-q_{N}\right)^{1/2}}.$
(30)

The parameter

$q_{N}=\frac{|E_{N}|}{E_{B}}$ (31)

is the dimensionless exciton energy, so that $1-q_{N}$ has a meaning of the
dimensionless size-quantization energy. In the limit of large $L$ the size-
quantization energy is much smaller than $E_{B}$, so that
$\left(1-q_{N}\right)\ll 1$. This, in turn, means that the lower limit in the
integral Eq. (V) is close to $1$. Then the integral $I(q_{N})$ can be
evaluated asymptotically in small parameter $1-q_{N}$, which yields

$I(q_{N})=\frac{\left(1-q_{N}\right)^{1/2}}{2}\ln 2.$ (32)

Substituting this expression in the right-hand side of Eq. (V), we arrive at a
closed equation for $q_{N}$, which can be also viewed as the equation for the
electron energies, $E_{N}$. Solving this equation in the limit of large $L$,
we get

$E_{N}=-E_{B}+\frac{\hbar^{2}\pi^{2}n^{2}}{2m_{h}\bigl{(}L-l_{N}\bigr{)}^{2}},$
(33)

where $l_{N}$ is given by

$\displaystyle
l_{N}=a_{B}\ln\Biggl{[}\frac{E_{B}}{\frac{\pi^{2}\hbar^{2}}{2m_{h}L^{2}}\Bigl{(}N+\frac{1}{2}\Bigr{)}^{2}}\Biggr{]}$
$\displaystyle=2a_{B}\Biggl{[}\ln\frac{L}{\pi
a_{B}\Bigl{(}N+\frac{1}{2}\Bigr{)}}+\frac{1}{2}\ln\frac{m_{h}}{m_{e}}\Biggr{]}.$
(34)

We see that the result Eq. (33) essentially reproduces Eq. (1) but with the
width of the dead layer specified. We see that the width, $l_{N}$, grows with
$L$, which is somewhat non-trivial.

##  VI Discussion

It is seen from Eq. (V) that the arguments of both logarithms in the right-
hand side are large. This suggests that the size of the dead layer exceeds the
exciton radius parametrically. On the other hand, it follows from Eq. (IV)
that the effective repulsive potential for a hole diverges at
$x_{h}=\frac{1}{2}\left(L-a_{B}\right)$, i.e. at a distance $\frac{a_{B}}{2}$
from the “hard wall” at $x_{h}=\frac{L}{2}$.

The fact that the turning point, $x_{t}$, for the hole motion, given by Eq.
(26), is parametrically further away from the boundary than $a_{B}$ suggests
the following picture of the exciton reflection. As a hole slowly moves
towards the boundary, it is orbited by a fast-moving electron. The energy of
the system is $-E_{B}$. Upon the approach to the boundary, the electron
“senses” the boundary by virtue of the no-escape boundary condition. As a
result, the energy of the system electron+hole increases. This increase acts
as a barrier for the hole, from which the hole is reflected being accompanied
by the electron.

Our conclusion that $l\gg a_{B}$ suggests that the above assumption about the
short-range character of the electron-hole attraction can be relaxed.
Moreover, we can extend the above consideration to the 3D case. In the
Appendix we derive the form of potential barrier, ${\tilde{V}}(x_{h})$, which
enters into the quantization condition

$\displaystyle
2\left(\frac{2m_{h}}{\hbar^{2}}\right)^{1/2}\int\limits_{0}^{x_{t}}dx_{h}\Bigg{[}E_{N}+E_{B}-{\tilde{V}}(x_{h})\Bigg{]}^{1/2}$
$\displaystyle=\pi\left(N+\frac{1}{2}\right).$ (35)

With the help of Eq. (A) we find the positions of the turning points

$x_{t}=\frac{L}{2}+\frac{a_{B}}{2}\ln\Biggl{(}1-\frac{|E_{N}|}{E_{B}}\Biggr{)},$
(36)

which appears to be similar to the 1D expression Eq. (26). Repeating the steps
in the previous section, it can be readily shown that our main result Eq. (V)
for the width of the dead layer remains valid in three dimensions.

##  Appendix A Effective potential in 3D

In this Appendix we calculate the correction to the ground state energy of
electron in the field of a hole due to the presence of a boundary at
$x=\frac{L}{2}$. We start by writing down the Schrödinger equations with and
without the boundary

$-\frac{\hbar^{2}}{2m_{e}}\Delta\psi_{0}-\frac{e^{2}}{|{\bf r}_{e}-{\bf
r}_{h}|}\psi_{0}=-E_{B}\psi_{0},$ (37)
$-\frac{\hbar^{2}}{2m_{e}}\Delta\psi-\frac{e^{2}}{|{\bf r}_{e}-{\bf
r}_{h}|}\psi+W\big{(}x_{e}-\frac{L}{2}\big{)}\psi=E\psi,$ (38)

where the function $W\big{(}x_{e}-\frac{L}{2}\big{)}$ describes a barrier
which ensures that electron does not penetrate into the region
$x_{e}>\frac{L}{2}$.

Multiplying Eq. (37) by $\psi({\bf r})$ and Eq. (38) by $\psi_{0}({\bf r})$,
subtracting the two, and integrating over the domain $x_{e}<\frac{L}{2}$, we
find

$\displaystyle E(x_{h})+E_{B}=$
$\displaystyle-\frac{\hbar^{2}}{2m_{e}}\int\limits_{x_{e}=\frac{L}{2}}dy_{e}dz_{e}\Bigg{(}\psi\frac{\partial\psi_{0}}{\partial
x_{e}}-\psi_{0}\frac{\partial\psi}{\partial x_{e}}\Bigg{)}$
$\displaystyle\approx\frac{\hbar^{2}}{4m_{e}}\int\limits_{x_{e}=\frac{L}{2}}dy_{e}dz_{e}\frac{\partial\psi_{0}^{2}}{\partial
x_{e}}.$ (39)

In Eq. (A) we took into account that the first term in the brackets is zero,
since $\psi({\bf r})=0$ at the boundary. We have also set $\psi({\bf
r})=\psi_{0}({\bf r})$ in the second term. Integration over the plane
$x_{e}=\frac{L}{2}$ in the right-hand side yields the correction to the
binding energy of the exciton due to the presence of the boundary.

The ground state wave function of an electron in the field of a hole which is
located at $(x_{h},0,0)$ has the form

$\psi_{0}({\bf r}_{e})=\frac{1}{\left(\pi
a_{B}^{3}\right)^{1/2}}\exp\Bigg{\\{}-\frac{\Bigl{[}y_{e}^{2}+z_{e}^{2}+\left(x_{e}-x_{h}\right)^{2}\Bigr{]}^{1/2}}{a_{B}}\Bigg{\\}}.$
(40)

With $\psi_{0}({\bf r}_{e})$ given by Eq. (40), the integration in Eq. (A) can
be performed explicitly in polar coordinates

$\displaystyle E(x_{h})+E_{B}$
$\displaystyle=\frac{\hbar^{2}\big{(}\frac{L}{2}-x_{h}\big{)}}{2m_{e}a_{B}^{4}}\int\limits_{0}^{\infty}d\rho^{2}\frac{\exp\Big{[}-\frac{2}{a_{B}}\Big{(}\rho^{2}+\big{(}\frac{L}{2}-x_{h}\big{)}^{2}\Big{)}^{1/2}\Big{]}}{\Big{[}\rho^{2}+\big{(}\frac{L}{2}-x_{h}\big{)}^{2}\Big{]}^{1/2}}$
$\displaystyle=\frac{\hbar^{2}\big{(}\frac{L}{2}-x_{h}\big{)}}{2m_{e}a_{B}^{3}}\exp\Big{[}-\frac{2}{a_{B}}\Big{(}\frac{L}{2}-x_{h}\Big{)}\Big{]}={\tilde{V}}(x_{h}).$
(41)

The result Eq. (A) defines the form of the barrier from which the exciton is
reflected.

##  Appendix B Acknowledgements

The work was supported by the Department of Energy, Office of Basic Energy
Sciences, Grant No. DE- FG02- 06ER46313.

## References

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# Connecting Everyday Objects with the Metaverse:
A Unified Recognition Framework

Liming Xu1
Andrew P. French
1This work was completed while the first author was a Ph.D. student at
University of Nottingham Ningbo China. Department of Engineering
University of Cambridge
Cambridge, United Kingdom
<EMAIL_ADDRESS>School of Computer Science
University of Nottingham
Nottingham, United Kingdom
<EMAIL_ADDRESS>Dave Towey2
Steve Benford 2Corresponding author. School of Computer Science
University of Nottingham Ningbo China
Ningbo, China
<EMAIL_ADDRESS>School of Computer Science
University of Nottingham
Nottingham, United Kingdom
<EMAIL_ADDRESS>
###### Abstract

The recent Facebook rebranding to Meta has drawn renewed attention to the
metaverse. Technology giants, amongst others, are increasingly embracing the
vision and opportunities of a hybrid social experience that mixes physical and
virtual interactions. As the metaverse gains in traction, it is expected that
everyday objects may soon connect more closely with virtual elements. However,
discovering this “hidden” virtual world will be a crucial first step to
interacting with it in this new augmented world. In this paper, we address the
problem of connecting physical objects with their virtual counterparts,
especially through connections built upon visual markers. We propose a unified
recognition framework that guides approaches to the metaverse access points.
We illustrate the use of the framework through experimental studies under
different conditions, in which an interactive and visually attractive
decoration pattern, an Artcode, is used as the approach to enable the
connection. This paper will be of interest to, amongst others, researchers
working in Interaction Design or Augmented Reality who are seeking techniques
or guidelines for augmenting physical objects in an unobtrusive, complementary
manner.

###### Index Terms:

Artcode, augmented reality, interaction, metaverse, visual marker

(a) Barcode

(b) QR code

(c) Data matrix

(d) Rohs’ code

(e) ARTag

(f) ARToolkit

(g) ReacTIVision

(h) D-touch

(i) Artcode

Figure 1: Visual marker examples.

##  I Introduction

Attending events virtually has become a normalized part of our everyday life,
due partly to the COVID-19 pandemic [1]. Increasingly, events are held online,
or support attendance through avatars, on platforms such as Zoom, and Gather
Town. This form of virtual engagement may well continue beyond COVID-19.
Moreover, Facebook’s recent rebranding to Meta and Microsoft’s announcement of
launching into the metaverse strengthen the likelihood of this being part of
our new normal [2]. It is therefore reasonable to expect that our future will
include a physical world even more augmented by a wide variety of virtual
worlds. These virtual worlds may require unobtrusive and easy-to-use access
points to a massive integrated network of virtual worlds or metaverse.
Attainment of a fully-realized, immersive metaverse will require efforts and
advances in multiple areas, including computer graphics, display hardware, and
communication networks [3]. In this paper, we address the issue of connections
between the physical and the virtual worlds, proposing a conceptual framework
for recognizing access points that may be hidden or camouflaged visual
markers.

The term “metaverse” was coined in 1992 by Neal Stephenson in his science-
fiction novel _Snow Crash_ [4], depicting a 3D virtual world where people can
interact with each other, and with intelligent agents, through their avatars
[5]. 30 years later, and the development of metaverse is arguably still in its
infancy, still with no generally accepted definition [6, 7, 5]. The
development framework of the metaverse, and its characteristics, have been
studied in the literature. Benford [5], for example, listed five metaverse
properties: a virtual world; a virtual reality; persistence; connection to the
real world; and other people. In contrast to the industrial seven-layer
metaverse value chain described by Radoff [8], Duan et al. [7] proposed a
three-layer metaverse development architecture, representing the physical
world, interaction, and the virtual world. In spite of the lack of consensus
on definition, there does appear to be general agreement that three basic
metaverse properties are: (i) a physical world; (ii) a virtual world; and
(iii) the connection between these two worlds.

Although various devices have been designed for accessing virtual elements or
virtual worlds, a map showing the presence of access points to these virtual
worlds would guide the connection (and potentially enhance the experience). If
this could be provided in an explicit and straightforward manner, for example,
through an annotation indicating the presence of such entrances to virtual
worlds, then even better! In contexts requiring aesthetic-awareness, such at
art galleries, implicit markers integrated into a part of the environment
(such as in the surface pattern of an object) may be more appealing. In other
environments, like in a corridor or hallway, both implicit and explicit visual
markers may be acceptable. In this paper, we report on the use of such surface
visual markers for connecting everyday objects with digital materials — such
as digital footprints, a virtual world, or a metaverse. We propose a unified
recognition framework (URF) for bridging the physical and virtual worlds
through visual decorations.

The main contributions of this paper are threefold, summarised as follows:

  * •

We report on the use of visual markers as clues to prompt interaction with
virtual worlds.

  * •

We generalize a URF for identifying the presence of access points in public
spaces.

  * •

We report on experimental studies conducted using one type of visual marker
(Artcodes [9, 10]), illustrating how the proposed URF works.

The rest of this paper is organized as follows. Section II briefly reviews the
related work on visual markers in augmented (AR) and virtual reality (VR).
Section III introduces the URF and the preliminaries pertaining to this work.
Section IV describes experimental studies evaluating the use of Artcodes as
access points to virtual elements. Section V includes discussion of the
implications of this study. Finally, Section VI concludes this paper and
describes future work.

Figure 2: A unified recognition framework (URF) for visual markers.

##  II Related work on visual markers

A variety of visual markers (see examples in Figure 1), both human-readable
and not, have been proposed [11, 9], with two of the most well-known being
barcodes [12] (Figure 1a) and QR codes (Quick Response codes, Figure 1b) [13].
The barcode was among the earliest methods of representing data in a visual,
machine-readable form, initially patented in 1952 [9]. While barcodes mainly
appear in the retail sector, QR codes have become a ubiquitous feature [9].
Barcodes and QR codes were designed to be reliably read by machines, with no
error occurring when they are scanned. However, this reliability comes at a
cost of limited aesthetics: Neither are visually meaningful to humans, and it
can be difficult to distinguish different codes though visual inspection
alone.

Many other visual marker systems have similar characteristics to barcodes and
QR codes, often with their information being encoded within a matrix of black
and white dots, and usually with some form of error detection and correction
mechanisms. Examples of such marker systems include the Data Matrix [14]
(Figure 1c) and the Rohs visual code [15] (Figure 1d). While these visual
markers are effective for encoding data, they were not intended for camera
pose estimation and calibration, and are thus not appropriate for use as
fiducials in AR systems — a fiducial is a type of marker mounted within an
environment to enable estimation of the relative pose between the camera and
object. Some example fiducial systems are: ARTag (Figure 1e) [16]; ARToolkit
(Figure 1f) [17]; and reacTIVison (Figure 1g) [18]. ARTag markers employ a
square border for marker localization, connectivity and perimeter analysis.
They have a large library of patterns inside the border and use edge-detection
approaches to achieve reliability [16]. ARToolkit markers consist of a thick
square black border with a variety of patterns in the interior — the black
outline allows for marker localisation and _homography_ 111An isomorphism in
projective spaces that is used to calibrate camera pose. calculation. The
reacTIVision markers are automatically generated by fiducial recognition
engines such as Amoeba and D-touch [19]: They have compact geometry and offer
a limited space for users to adjust their aesthetic aspects [18].

The visual appearance of marker systems that rely on geometrical features for
localization and encoding is strongly constrained. In the majority of cases,
the shape (the geometry) of the markers is automatically generated, allowing
little freedom of design. In contrast, another type of visual markers, such as
D-touch and its variant Artcodes [9, 20], offer much more flexibility in
geometrical form, both for the outline shape and the interior elements.
D-touch encodes information through the topological structure of the markers —
the adjacency information of connected components, represented in a region
adjacency tree [21]. This supports users’ creation of their own readable
markers that are both aesthetic and meaningful [11]. Artcode implements and
extends the D-touch approach, refining their drawing rules, and introducing
human-meaningful (but machine-irrelevant) embellishments and aesthetic style
guidelines. The Artcode approach provides the creative freedom to produce
visually appealing _and_ machine-readable markers (patterns) that are
meaningful to humans, and that resemble free-form images.

In addition to these visual marker technologies based on geometry or topology,
conventional image recognition technologies have also been employed to relate
information to a much wider variety of images. Blippar [22] and Google Lens
[23], for example, make use of image recognition techniques to embed data into
images. However, because these techniques often use neural networks and vector
matching for encoding and decoding information, it is challenging (or
impossible) to explain and interpret how the system works to non-technical
designers or users. More recently, new systems that use deep-generative
networks to automatically generate markers have been proposed, including
learnable visual markers [24], E2ETag [25] and DeepFormableTag [26].

##  III Unified Recognition Framework (URF)

As AR and metaverse applications become more pervasive, we will live in a
world with dispersed access points to connect with virtual elements. There
will be an increasing number of entrances to these elements within our
surrounding environment, through a variety of virtual markers, both visible
and “hidden”. Identifying the probable existence of these entrances will be
the first step to triggering the follow-up interaction. Considering the many
types of entrance that may co-exist, a unified recognition framework (URF)
will be needed. In this section, we present such a conceptual URF for general
visual marker presence recognition and identification.

Given the number of extant visual markers, both in academia and in industry,
and the high likelihood of many more systems emerging in the future,
attempting to explicitly include all in this URF would be unrealistic. We
therefore only include a selection of some typical markers to show the basic
URF components. The left part of Figure 2 shows a common scene, an indoor area
of a building with various visual markers (highlighted in the picture). Not
all of the annotated objects are readable — some are explicitly-placed
readable Artcodes (in red boxes), while others (in blue boxes) are commonplace
objects that could be enhanced as visual markers.

As shown in Figure 2, the URF involves three stages: marker presence
detection; marker identification; and marker decoding. The _detection_ stage
involves detecting visual markers in the surrounding environment. Given the
scenario in the left part of Figure 2), for example, this stage would detect
the possible presence of visual markers using image processing and computer
vision techniques, and would output a set of localized candidate visual
markers. This output set is then passed to the _identification_ stage (the
middle of Figure 2) to determine if they _are_ markers, and, if so, what class
of markers they belong to (Artcodes, QR codes, Blippar images, etc.). A key
component of the identification stage is a _multi-label classifier_ that
accepts the candidate markers, and outputs their corresponding classes or
labels. The final stage is the _decoding_ , which includes a _decoder pool_
from within which the corresponding decoder identifies and decodes the
embedded message in the visual marker.

Once the data (codes) carried by the visual marker are identified, the
connected visual information (labelled by the visual marker) can be triggered.
In this URF, visual marker detection and identification are two independent
stages, but in reality, these two things are often done together. Although the
URF is a conceptual framework, describing the essential components and a
feasible pipeline to bridge the physical and virtual worlds, the concrete
implementation may differ from one scenario to another. A possible URF
implementation may be an _all-in-one_ brokering system that recognizes the
presence of all (or most) of the visible or hidden visual markers, then calls
the corresponding decoders or identifiers, and then steps into the embedded
virtual worlds.

The next section presents experimental studies examining discovery of the
presence of visual markers using a concrete marker system, Artcode [9, 27].

##  IV Experimental studies

The URF proposed in the last section includes the two primary elements: visual
marker discovery and identification, with discovery of the markers being a
_prerequisite_ to the follow-up identification. Moreover, providing hints and
clues to the location of (camouflaged) access points to virtual worlds may
encourage people to explore those connections, thus creating new interaction
opportunities. Given the importance of visual marker discovery in the URF
pipeline, we conducted two case studies into how digital clues can be provided
to guide users with devices (such as AR headsets) to approach the object and
enter the metaverse. Artcodes, which are both meaningful to humans, and
readable by scanners, were selected as the marker system.

###  IV-A The Artcode approach

Artcodes222https://www.artcodes.co.uk/ are human-designable topological visual
markers, developed based on the D-touch system [11]. By incorporating
additional drawing constraints and aesthetic embellishments, Artcodes enable
more visually pleasing and interactive patterns than d-touch [9]. Figures 1h
and 1i show examples of d-touch and Artcode markers. A valid Artcode consists
of two parts: a recognizable foreground (the food image in Figure 1i); and
some image-based background (the text in Figure 1i). The foreground is
intended for reading by machines, but the background can be designed for human
consumption. Artcodes can be beautiful, interactive motifs that can decorate
the surface of everyday objects without impacting the aesthetics of the object
in the way that QR codes would.

Because of their unobtrusive and non-obvious properties, the presence of an
Artcode is not usually obvious: Close inspection may be needed to discover an
Artcode when there are no visual clues. Detection of Artcodes through their
general visual features, identifying their probable locations by means of a
_heat map_ , is therefore a meaningful approach. Given the space limitations
of this article, interested readers are referred to the literature for more
information about Artcodes, including their design, detection, and
identification [9, 19, 27, 20, 28].

###  IV-B Experimental setting

We conducted experiments to explore Artcode detection in an environment, and
deliver clues to guide the subsequent interaction. We assumed a realistic
interaction scenario, in which users may wear or carry devices in a physical
space, standing far away from the Artcodes: When they discover the presence of
an Artcode, they can follow clues to approach the target for further
interaction. Rather than fully simulating this scenario, we simplified it
while maintaining its core characteristics: Users gain increasing amounts of
details as they approach the target.

Two studies were conducted, both involving five images sequences (Figures 3a
and 4a) captured with a smartphone moving from far away to close proximity to
an Artcode. The size of the Artcode gradually increases as the smartphone
moves towards to the target, from top to bottom in the left-most column of the
figures (Figures 3a and 4a). Recognition is more challenging from further
away. Apart from this, the two studies other settings differed as follows: The
first study, Figure 3, used a simple Artcode design, in good lighting, with an
uncluttered scene, and an unoccluded Artcode. The second study, Figure 4,
involved a more difficult scenario, using a complex Artcode design, shaded
lighting, a cluttered scene, and a partially occluded Artcode.

Considering space limitations, and the focus of this paper, the technical
details for building the Artcodes-detection machine-learning model are
omitted. Similarly, the details underlying the various elements in Figures 3
and 4 (including generation of the proposals and presence maps) are also
omitted. Interested readers are again referred to the literature for more
information [9, 27, 20].

()

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()

()

()

()

(a) Input

(b) Proposal

(c) Gray

(d) Fused

Figure 3: Simple Artcode detection study in clean background, good lighting.

()

()

()

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()

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(a) Input

(b) Proposal

(c) Gray

(d) Fused

Figure 4: Complex Artcode detection study in cluttered background, poor
lighting.

###  IV-C Results

Figures 3 and 4 contain the content and results of the two studies. The four
columns in each figure, from left to right, are: (a) the input images (b) the
Artcode proposals, annotated with yellow rectangles; (c) the gray Artcode
presence heat map; and (d) the fused image (created by combining the input
image (a) with the heat map (c)). The red boxes indicate the ground-truth
Artcodes.

TABLE I: Decoding results for the images in Figures 3a and 4a. Decoded Image | 1st (top) | 2nd | 3rd | 4th | 5th (bottom)
---|---|---|---|---|---
1st study (Figure 3) | $\times$ | $\times$ | $\surd$ | $\surd$ | $\surd$
2nd study (Figure 4) | $\times$ | $\times$ | $\times$ | $\times$ | $\times$

In addition to the presence detection results in Figures 3 and 4, Table I
presents the decoding results (generated according to Artcode decoding
procedures [9]). Ticks and crosses in the table indicate whether the given
image was successfully decoded or not, with ticks (“$\surd$”) indicating
success; and crosses (“$\times$”) indicating failure.

It is clear that the detection proposals in both studies cover the actual
marker areas — the penguins in Figure 3, and the fish in Figure 4 — in all
image sequences, with dense accumulation of the proposal rectangles centering
around the target markers. This is further evidenced in the presence maps
(gray and fused), where the marker areas are distinctly visible as heat spots
(the bright areas in the 3rd and 4th columns of Figures 3 and 4). The Artcode
proposals in all five of the first study images center around the true Artcode
areas, identified by the red boxes: In the second study, in contrast, although
the Artcode proposals cover the true Artcode areas, there are multiple
proposals that are not around the actual target, especially for the images
that were captured from a greater distance (in the top three rows of Figure
4a).

The cluttered scene in the second study affects the detection, increasing the
number of false positives: Many non-Artcode objects in this scene may look
like Artcodes, with their generic visual features potentially causing the
classifier to label them as Artcodes. However, although redundant heat spots
were generated, the actual target Artcodes are also identified: Figures 4c and
4d show multiple detections (indicated by heat spots), but one of them does
contain the actual target Artcode. Heat spots in the presence maps can alert
the user to the possible existence of access points to the metaverse,
encouraging the user to come close for follow-up examination and
identification.

According to the decoding results (Table I), the top two images in the first
study (those captured from the furthest distance) could not be decoded, due to
the low resolution and loss of details. The closer three input images in the
first study, however, ware successfully identified and decoded, opening up the
“hidden” virtual worlds. This represents a simplified realistic interaction,
where the users often come closer to a target after first getting the general
impression (the hint or clue).

The more complicated environment in the second study, including a more
sophisticated Artcode, poorer lighting, clutter, and occlusion (with a
chopstick in the way), resulted in none of the five images being successfully
decoded. This also represents a common, real-world situation, where the target
image may be obscured from certain angles. In this case, the presence maps
should motivate the user to get nearer, and to remove the obstruction, or to
explore new viewing angles for better identification. The explorative
interaction process allowed by the proposed URF would enable various designs
(e.g., design for serendipity [29, 30]), and open up new interaction
opportunities for connecting to the metaverse.

##  V Discussion and implications

The two studies present a simplified and concrete implementation of the
proposed framework, illustrating the key steps of detecting and identifying
visual markers before decoding them, and accessing the metaverse. Currently,
implementing the proposed URF for all known visual markers may not be feasible
— partly due to the ever-expanding set of such markers, and the regular
emergence of new interaction devices. However, this investigation using
Artcodes as a representative marker provides evidence for the URF’s
applicability. This paper, and the URF generally, can also serve as guidance
for metaverse access point design, using visual markers (especially in an
unobtrusive but explorative manner). The proposed framework also includes a
mixed interaction manner, combining physical movement and digital engagement
in an augmented physical world with ubiquitous connection access points.

##  VI Conclusion and future work

In this paper, we have explored the problem of connecting with virtual worlds
(or the metaverse) in an augmented physical world. We have presented a unified
recognition framework (URF) consisting of three components for designing and
implementing an explorative access point. A concrete implementation of this
URF using Artcodes as access points was used to illustrate the process. An
example of visual markers, Artcodes are both machine-readable and human-
meaningful decorative patterns that represent the kind of access tool that
will become increasingly commonplace in the future. The initial discovery of
the presence of markers (indicated by a heat map) and the follow-up, closer
inspection and detection were demonstrated by the two studies in the paper.
The URF would enable the design of a kind of brokering system that can invoke
appropriate recognition algorithms to deal with different types of access
points, and may inspire interaction design in the metaverse age. While this
study used smartphones and Artcodes, our future work will include the
investigation of other AR devices and other visual markers.

## Acknowledgments

This work is supported by the Natural Science Foundation of China (project no.
61872167). The authors acknowledge the financial support from the Artificial
Intelligence and Optimisation Research Group (AIOP), the Faculty of Science
and Engineering (FoSE), the International Doctoral Innovation Centre, Ningbo
Education Bureau, Ningbo Science and Technology Bureau, and the University of
Nottingham.

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