# Aharonov-Bohm oscillations in phosphorene quantum rings: mass anisotropy
compensation by confinement potential

Tanmay Thakur and Bartłomiej Szafran  AGH University of Science and
Technology, Faculty of Physics and Applied Computer Science,
al. Mickiewicza 30, 30-059 Kraków, Poland

###### Abstract

We consider the Aharonov-Bohm (AB) effect on a confined electron ground state
in a quantum ring defined electrostatically within the phosphorene monolayer.
The strong anisotropy of effective masses in phosphorene quenches ground-state
oscillations for a circular ring because of interrupted persistent current
circulation around the ring. An elliptic deformation of the confinement
potential can compensate for the anisotropy of the effective masses and
produce ground-state parity transformations with the AB periodicity. Moreover,
a specific ratio of the semiaxes is determined for which the spectrum becomes
identical to that of a circular quantum ring and an isotropic effective mass.
We identify a generalized angular momentum operator which commutes with the
continuum Hamiltonian for the chosen ratio of the semi-axes that closes the
avoided crossings of energy levels for states of the same parity and spin.
Ground-state oscillations for the two-electron ground state are also
discussed.

##  I Introduction

Phosphorene [1] or a monolayer form of black phosphorus [2, 3, 4] is
extensively studied for optics [5], field effect transistors [2, 6, 7] and
quantum Hall effects [9, 8, 10]. Unlike the half-metallic graphene,
phosphorene is a direct gap semiconductor that can host the electrostatic
lateral confinement of electrons. The electrostatic fields produce a clean
confinement in gated two-dimensional systems for investigation of the single-
electron and interaction effects [11] in carriers traps. A particular form of
the lateral confinement that attracts a lot of attention is the quantum ring
[12]. The annular confinement allows for persistent current circulation in the
presence of an external magnetic field, with the spectrum and magnetic
response which is periodic with the Aharonov-Bohm periodicity [13]. The
periodicity of the spectrum of a phosphorene ring defined as a rectangular
flake of the crystal with a central opening has been studied in Ref. [14]
including the effect of the zigzag and armchair edges of the crystal. The
purpose of this paper is to investigate a clean quantum ring defined within
phosphorene by an external potential that keeps the electrons off the edges of
the crystal and is not affected by its details.

The anisotropy of the phosphorene crystal structure [4] results in a strongly
anisotropic electron effective mass [2, 15, 16, 17, 18] that is much larger
along the zigzag chains of ions [16] than in the perpendicular direction. The
anisotropy prevents the persistent current flow in the electron ground state
confined in a circular ring. However, current circulation can be restored by
deforming the confinement potential to an elliptic form. Then, the spectrum
acquires a braided pattern of even and odd parity energy levels, which cross
with the Aharonov-Bohm period. Moreover, we propose a geometry for the
elliptic confinement in which a modified angular momentum operator, with one
of the Cartesian coordinates rescaled, commutes with the Hamiltonian and the
energy spectrum becomes similar to the one of an electron in a circular
quantum ring with isotropic effective mass. For a confined electron pair
interacting with the Coulomb potential, the operator no longer commutes with
the Hamiltonian, but the Aharonov-Bohm oscillations of the ground-state energy
appear for a tuned confinement potential.

##  II Theory

###  II.1 Tight-binding model

We work with the phosphorene monolayer (see Fig. 1) using the Hamiltonian,

$\displaystyle
H_{TB}=\sum_{kl}t_{kl}p_{kl}c_{k}^{\dagger}c_{l}+\sum_{k}V_{k}c^{\dagger}_{k}c_{k}+{g\mu_{B}B}\sigma_{z}/2,$
(1)

where the first sum describes the hopping between the neighboring atoms. The
values for $t_{kl}$ (see Table 1) are taken from the five-parameter effective
tight-binding Hamiltonian of Ref. [17] . The positions of the ions in the
phosphorene crystal [3] are plotted in Fig.1 with the zigzag chains oriented
along the $y$ direction. In Eq. (1), $p_{kl}$ are the Peierls phase shifts
that the electron acquires from the vector potential along the line between
$k$ and $l$ ions,
$p_{kl}=e^{i\frac{e}{\hbar}\int_{\vec{r_{k}}}^{\vec{r_{l}}}\vec{A}\cdot\vec{dl}}$.
We consider the magnetic field perpendicular to the monolayer $(0,0,B)$ with
the vector potential taken in the symmetric gauge ${\bf
A}=(-\frac{By}{2},\frac{Bx}{2},0)$. In Eq. (1), $V_{k}$ stands for the
external potential on the ion $k$. The spin Zeeman effect is introduced by the
last term of the Hamiltonian.

The $g$-factor for phosphorene of $g\approx 2.03$ was determined using
$\textbf{k}\cdot\textbf{p}$ theory by Junior et al. [19]. Zhou et al. [20]
indicated the value of $g=2.14$ for monolayer black phosphorus. On the other
hand, experiments have determined the value of $g=2\pm 0.1$ [21] and $g\approx
1.8-2.7$ [22] and some have even reported $g=5.7\pm 0.7$ at low filling
factors. Nevertheless, the spin Zeeman term produces only a linear shift in
the energy defining the spin splitting. The magnetic field promotes the spin-
down energy levels to the ground state sooner or later on the magnetic field
scale. We focus on avoided crossings or crossings of the states of the
polarized spin. The absence or presence of Aharonov-Bohm oscillations will not
be affected by the value of the $g$-factor. Therefore, the spin Zeeman term is
calculated with the value $g=2$.

Figure 1: Ions in phosphorene monolayer placed on two planes separated by a distance of 0.213 nm. The lines link the neighbor ions with the largest hopping energies (in-plane neighbors, green lines, hopping energy $-1.22$ eV) and (neighbors of separate planes, red lines, hopping energy $3.665$ eV), see Table I.  $r_{kl}($nm$)$ |  $t_{kl}($eV$)$
---|---
$0.222$ | -1.22
$0.224$ | 3.665
$0.334$ | -0.205
$0.347$ | -0.105
$0.423$ | -0.055
Table 1: Hopping energies according to Ref. [17] for a single black phosphorus
layer. The left column shows the distance between the ions and the right
column the hopping energy $t_{kl}$ applied in the tight-binding Hamiltonian
(1).

###  II.2 Effective-mass Hamiltonian

Part of the results of this work is obtained in the continuum approximation to
the tight-binding Hamiltonian. We use a single-band effective-mass operator,

$\displaystyle H_{em}$ $\displaystyle=$
$\displaystyle{\left(-i\hbar\frac{\partial}{\partial
x}+e{A_{x}}\right)^{2}}/{2m_{x}}+{\left(-i\hbar\frac{\partial}{\partial
y}+e{A_{y}}\right)^{2}}/{2m_{y}}$ (2) $\displaystyle+$ $\displaystyle
V(x,y)+{g\mu_{B}B}\sigma_{z}/2.$

with the effective mass parameters derived by fitting the tight-binding
spectrum to the harmonic oscillator spectrum in Ref. [24] with mass about five
times heavier for the carrier motion along the zigzag chains of the crystal
(see Fig. 1), $m_{x}=0.17037m_{0}$ and $m_{y}=0.85327m_{0}$. The effective
mass Hamiltonian is diagonalized using the finite-difference technique.

###  II.3 Confinement potential

We attempt to compensate for the anisotropy of the effective masses by the
anisotropy of the confinement potential. For that purpose, we use the
following external potential

$V(x,y)=\frac{1}{2}m_{x}\omega^{2}(\rho(x,y)-R)^{2},$ (3)

with $\rho(x,y)=\sqrt{x^{2}+y^{2}/\alpha}$, where $\alpha$ is a parameter that
controls the anisotropy of the potential. The potential vanishes for points
$(x,y)$ forming an ellipse

$\frac{x^{2}}{R^{2}}+\frac{y^{2}}{\alpha R^{2}}=1.$ (4)

We take the confinement energy $\hbar\omega=6$ meV. Changing $\alpha$ we keep
the area within the ellipse fixed taking $R=\alpha^{-1/4}R_{c}$, with
$R_{c}=30$ nm, so that the number of magnetic flux quanta threading the
ellipse is the same for a given magnetic field $B$ independent of $\alpha$.
For $R_{c}=30$ nm a flux quantum threads the ring at 1.43 T, which is the
period of the energy spectrum on the $B$ scale for a strictly 1D circular
quantum ring. We use $\alpha\leq 1$ so that the half length of the major axis
$a$ of the ellipse is oriented along the $x$ axis $a=R=\alpha^{-1/4}R_{c}$ and
the half length of the minor axis is $b=\sqrt{\alpha}R=\alpha^{1/4}R_{c}$.

In this work, we focus on the confined electron states of the conduction band.
In the continuum Hamiltonian, the bottom of the conduction band is set as the
reference energy level. The tight-binding spectrum produces the conduction and
valence bands extrema spaced by the energy gap. For a finite flake, the
spectrum contains also in-gap states that are localized at the edge of the
flake and the spectrum is not symmetric with respect to the center of the
energy gap [14]. Ref. [14] which used the same tight-binding parameterization
[17] provides the lowest conduction-band state energy level of $\simeq 0.4$ eV
for a square flake with a side length of 8 nm in the absence of external
potential. In this paper, we are interested in states confined in the external
potential that are independent of the details of the edge and thus correspond
to an infinite crystal. However, the calculations are carried out in an
elliptical flake for which the position of the ions satisfies the condition
$x^{2}+y^{2}/\alpha<R_{s}^{2}$. We take the flake large enough to contain all
the discussed states within the confinement potential so that the results are
independent of $R_{s}$. However, for $V=0$, the conduction band states occupy
the entire flake, and the results depend on $R_{s}$ [25]. The dependence on
the lowest energy level in $R_{s}$ is well approximated by the dependence
$E_{0}(R_{s})=C/R_{s}^{2}+E_{\infty}$, where $C=673.31$(meV nm2) and
$E_{\infty}=340$ meV. The $R_{s}^{-2}$ dependence is due to the finite-size
effect, that is, the kinetic energy due to localization in a finite flake, and
$E_{\infty}=340$ meV is the estimated position of the bottom of the conduction
band for an infinite crystal. In the results presented in the following we
shift down the tight-binding energies by $E_{\infty}$.

##  III Results and discussion

(a) |  (b)
---|---
(c) |  (d)
Figure 2: The tight binding (a) and continuum Hamiltonian (b) eigenvalues for $\alpha=1$. The blue and red lines show the spin-down and spin-up energy levels respectively. The tight-binding energy levels in (a) were shifted down by $E_{\infty}=340$ meV (see text in II.B). Panel (c) show the confinement potential and panel (d) the ground-state charge density for $B=0$ calculated using the continuum model. (a) |  (b)
---|---
(c) |  (d)
Figure 3: Same as Fig. 2 only for $\alpha=1.2{\frac{m_{x}}{m_{y}}}$.  (a) |  (b)
---|---
Figure 4: Magnified view of the low-energy part of the tight-binding spectrum for $\alpha=1.2{\frac{m_{x}}{m_{y}}}$ (a) [Fig. 4(a)] and $\alpha={\frac{m_{x}}{m_{y}}}$ (b) [Fig. 6(a)]. Letters ’e’ and ’o’ near the energy levels in (a) mark the even and odd parity energy levels that are eigenstates of the parity operator with the eigenvalues $+1$ and $-1$, respectively. $\Delta_{0}$ is the even-odd parity splitting at $B=0$ and $\Delta_{e}$ is the width of the avoided crossing between the lowest even parity energy levels for $B\simeq 7.75$ T. The red arrow in (b) shows the avoided crossing $\Delta_{e}$, here of the width of 13$\mu$eV. (see Table II). (a) |  (b)
---|---
(c) |  (d)
Figure 5: Same as Fig. 2 only for $\alpha={\frac{m_{x}}{m_{y}}}$. In panel (b)
the colors indicate the eigenvalue of the angular momentum $l_{z}^{\prime}$
operator, and in panel (a) the colors describe the spin as in Fig. 2.

###  III.1 Single-electron solutions

The low-energy spectrum of a circular ring ($\alpha=1$) is plotted in Fig.
2(a) (Fig. 2(b)) for the tight-binding model (continuum model). The TB results
[Fig. 2(a)] here and below are shifted down by 340 meV. The results of both
approaches are nearly identical. The ground state of a circular ring is
localized in two islands near the $y$ axis on the opposite sides of the ring
center [Fig. 2(c)]. The variation of the wave function implies a contribution
to the kinetic energy which is large on the $x=0$ axis due to the low value of
mass $m_{x}$. In consequence the ground-state wave function is far from the
$x$ axis, and no persistent current circulation is possible in the ground-
state.

The lowest spin-up and spin-down energy levels of Fig. 2(a,b) are two-fold
degenerate with respect to the parity. Degeneracy results from the lack of
tunnelling of the wave function across the $x$ axis (Fig. 2(d)). In the
excited state part of the spectrum, one can see pairs of energy levels that
cross in a braid-like pattern. The corresponding states have opposite
parities, hence the crossings of the levels. These oscillations are
reminiscent of the angular momentum transitions for a circular ring with
isotropic effective mass [12, 26].

With $\alpha<1$ the confinement area on the $y$ axis becomes thinner and the
one along the $x$ axis wider [see Fig.3(c) for
$\alpha=1.2{\frac{m_{x}}{m_{y}}}$]. The confinement energy along the $y$-axis
increases and that along the $x$-axis decreases. The $x$ axis is now
accessible for the ground-state electron [Fig. 3(d)]. The ground-state
degeneracy is lifted, and the ground-state crossings of energy levels of
opposite parity are observed [Fig. 3(a,b)].

In Fig. 4(a) we plotted a magnified view of the low-energy part of the
spectrum. By $\Delta_{0}$ we denote the energy splitting of the lowest even
and odd parity energy levels taken at $B=0$, which defines the range of the
ground-state energy oscillations as functions in the external magnetic field.
The braided two energy levels cross with the Aharonov-Bohm period. The second
quantity marked in Fig. 4(a) by $\Delta_{e}$ is the width of the avoided
crossing of even-parity energy levels taken at $B\simeq 7.75$ T.

Decreasing the anisotropy parameter to $\alpha={\frac{m_{x}}{m_{y}}}$ we find
that the ground-state charge density [Fig. 5(d)] is constant along the
confinement potential minimum Fig. 5(c)]. The continuum spectrum [Fig. 5(b)]
contains crossings of energy levels in the entire spectrum. For the tight-
binding model – see also Fig. 4(b) – we find the parity-related crossings of
the energy levels in the ground state as in Fig. 3, and only narrow avoided
crossings are found between the first and second excited energy levels of the
same parity. The one marked by the red arrow in Fig. 4(b) corresponds to
$\Delta_{e}=13$ $\mu$eV. $\Delta_{e}$ attains its minimal value for
$\alpha=\frac{m_{x}}{my}$. Values of $\Delta_{0}$ and $\Delta_{e}$ calculated
with the tight binding approach for varied $\alpha$ are summarized in Table
II.

$\alpha$ | $\alpha/({m_{x}/m_{y}})$ |  $\Delta_{0}$ (meV) |  $\Delta_{e}$ (meV)
---|---|---|---
1 | 2.238 | 0.001 | 0.673
0.894 | 2 | 0.012 | 0.414
0.536 | 1.2 | 0.061 | 0.137
0.469 | 1.05 | 0.096 | 0.046
0.447 | 1 | 0.113 | 0.013
0.424 | 0.95 | 0.104 | 0.028
Table 2: The spacing $\Delta_{0}$ between the lowest energy levels of even and
odd parity for $B=0$ (see Fig. 4(a)) and the $\Delta_{e}$ width of the avoided
crossing between the two lowest energy levels of even parity (Fig. 4(a)) as a
function of the eccentricity parameter $\alpha$ for $B\simeq 7.75$ T. The
results are calculated with the tight binding approach.

###  III.2 Angular momentum in the rescaled space

The crossings of the energy levels in the spectrum of Fig. 5(b) suggest that
an additional symmetry is present in addition to the parity. With the
substitution $y^{\prime}={y}/\sqrt{\alpha}$ and $\alpha={\frac{m_{x}}{m_{y}}}$
the Hamiltonian (2) becomes

$\displaystyle H_{em}$ $\displaystyle=$
$\displaystyle-\frac{\hbar^{2}}{2m_{x}}\left(\frac{\partial^{2}}{\partial
x^{2}}+\frac{\partial^{2}}{\partial y^{\prime
2}}\right)+\frac{e^{2}B^{2}}{8m_{y}}(x^{2}+y^{\prime 2})$ (5) $\displaystyle+$
$\displaystyle\frac{eB}{2\sqrt{m_{x}m_{y}}}l_{z}^{\prime}+V(\rho^{\prime}-R)+{g\mu_{B}B}\sigma_{z}/2,$

with $\rho^{\prime}=\sqrt{x^{2}+y^{\prime 2}}$ and the $z$ component angular
momentum operator in the deformed space
$l_{z}^{\prime}=i\hbar\left(y^{\prime}\frac{\partial}{\partial
x}-x\frac{\partial}{\partial y^{\prime}}\right)$. The confinement potential
acquires circular symmetry upon rescaling of the $y$ coordinate and the
Hamiltonian commutes with $l_{z}^{\prime}$ operator. The crossings of the
eigenstates of the effective mass Hamiltonian are due to the symmetry which
upon rescaling of the $y$ coordinate is no longer hidden. The $l_{z}^{\prime}$
eigenvalues are given by color in Fig. 5(b). The ground state undergoes
$l_{z}^{\prime}$ angular momentum transitions similar to the ones found for
circular quantum rings with isotropic electron effective mass [26]. Note that
the applicability of $l^{\prime}_{z}$ operator is not limited to quantum
rings, but it can also be used to any potential profile which is radially
symmetric for the rescaled $y$ coordinate.

###  III.3 1D limit

For a narrow radial confinement (large $\omega$) the low-energy part of the
spectrum occupy the same state of radial quantization and there is essentially
one degree of freedom of motion along the ring. The Hamiltonian (5) put in
circular coordinates reads

$\displaystyle H_{em}$ $\displaystyle=$
$\displaystyle-\frac{\hbar^{2}}{2m_{x}}\left(\frac{1}{\rho^{\prime}}\frac{\partial}{\partial\rho^{\prime}}+\frac{\partial^{2}}{\partial\rho^{\prime
2}}-\frac{l_{z}^{\prime 2}}{\rho^{\prime
2}}\right)+\frac{B}{2\sqrt{m_{x}m_{y}}}l_{z}^{\prime}+$ (6)
$\displaystyle+\frac{e^{2}B^{2}}{8m_{y}}\rho^{\prime 2}+V(\rho^{\prime}-R),$

where we neglected the spin Zeeman term. For strong confinement (large
$\omega$) the radial profile of the wave function no longer depends on
$l_{z}^{\prime}$ or $B$. Then, the terms with the derivatives with respect to
$\rho^{\prime}$ and the external potential produce the same energy
contribution for all the states involved. With this contribution set as the
reference energy level, we obtain the energy spectrum of the form,

$E(l_{z}^{\prime},B)=\frac{\hbar^{2}l_{z}^{2}}{2m_{x}R^{2}}+\frac{e^{2}B^{2}}{8m_{y}}R^{2}+\frac{eB}{2\mu}l_{z}^{\prime},$
(7)

where $\mu=\sqrt{m_{x}m_{y}}$. With
$R=\left(\frac{m_{y}}{m_{x}}\right)^{1/4}R_{c}$ one obtains an expression that
is symmetric in the effective masses,

$\displaystyle E(l_{z}^{\prime},B)$ $\displaystyle=$
$\displaystyle\frac{\hbar^{2}l_{z}^{\prime 2}}{2\mu
R_{c}^{2}}+\frac{e^{2}B^{2}}{8\mu}R_{c}^{2}+\frac{eB}{2\mu}l_{z}^{\prime}$ (8)
$\displaystyle=$ $\displaystyle\frac{\hbar^{2}}{2\mu
R_{c}^{2}}\left(\frac{\Phi}{\Phi_{0}}+l_{z}^{\prime}\right)^{2},$

where $\Phi_{0}=\frac{h}{e}=\frac{2\pi\hbar}{e}$ is the flux quantum and
$\Phi=B\pi R_{c}^{2}$. The final result with the geometric average of the
effective masses is identical to that of the circular ring with an isotropic
effective mass [26].

Figure 6 shows the 2D continuum Hamiltonian (6) spectra with the spin Zeeman
effect excluded for $\hbar\omega=6$ meV in Fig. 6(a), $\hbar\omega=120$ meV in
Fig. 6(b) and the results of the 1D formula in Fig. 6(c). In Fig. 6(a) we see
a diamagnetic shift of the spectrum to higher energy. The period of the
Aharonov-Bohm oscillations is slightly larger than in the 1D results due to
compression of the wave function by the external magnetic field that decreases
average $\rho^{\prime}$ below $R_{c}$. For the radial wave function confined
stronger around $R_{c}$ in Fig. 6(b), we see the results approach the results
for the analytical formula (8).

(a)
---
(b
(c)
Figure 6:  The energy spectrum of Hamiltonian (5) with $g=0$ for
$\hbar\omega=6$ meV (a), $\hbar\omega=120$ meV (b), and the 1D formula (7).
The color of the lines gives the $l_{z}^{\prime}$ operator eigenvalue

###  III.4 Two-electron spectrum

The electron-electron interaction potential does not commute with the
$l_{z}^{\prime}$ operator, so the presence of the ground-state oscillations is
not given a priori. We calculated the two-electron energy spectrum in the
continuum approach using the Hamiltonian

$H_{2e}=H_{em}({\bf r}_{1})+H_{em}({\bf
r}_{2})+\frac{e^{2}}{4\pi\epsilon_{0}\epsilon}\frac{1}{r_{12}},$ (9)

where $H_{em}$ is the single-electron Hamiltonian (2), and $\epsilon=12$ is
taken. The two-electron Hamiltonian is diagonalized in the basis of up of the
two-electron Slater determinants constructed from the 30 lowest-energy single-
electron eigenfunctions of Hamiltonian (2).

For both $\alpha=1$ (Fig. 7(a)) and $\alpha={\frac{m_{x}}{m_{y}}}$ (Fig. 7(c))
at $B=0$ the ground state is four-fold degenerate, with singlet and triplet
energy levels of the same energy. The exchange interaction is zero since the
electrons form single-electron islands that are completely separated. For
$\alpha=1$ the localization of the charge density formed a single-electron
island already without interaction (Fig. 2(d)). For
$\alpha={\frac{m_{x}}{m_{y}}}$ the electron-electron interaction separates the
electron density to the opposite ends of the longer semiaxis of the ellipse
(Fig. 7(d)). A more or less uniform electron distribution is obtained for
$\alpha=1.7{\frac{m_{x}}{m_{y}}}$ (Fig. 7(f)). In this case, the ground state
for $B=0$ is a singlet which is not degenerate with the triplet. The field of
about 0.5 T promotes the triplet to the ground state. Periodic avoided
crossings are observed in the ground state, which are similar to the ones
found for a system with isotropic effective mass but in an anisotropic quantum
ring [28]. For the circular ring, the triplet-energy levels correspond to odd
values of the total angular momentum ($L$) [29] that correspond to the
negative parity $(-1)^{L}$. In our two-electron system, only the parity is a
good quantum number. In Fig. 7(e) we see a series of avoided crossings between
the ground state and the first excited state. The avoided crossings obtained
for the spin-polarized two-electron levels are observed due to the same – odd
– parity of these levels, which is in contrast to the single-electron ground
state where the crossings of energy levels corresponding to opposite parity
were observed in the ground state.

(a) |  (b)
---|---
(c) |  (d)
(e) |  (f)
Figure 7: The two-electron energy spectrum (a,c,e) and the ground-state charge
density (b,d,f) calculated using continuum model for $B=0$ and $\alpha=1$
(a,b), $\alpha={\frac{m_{x}}{m_{y}}}$ (c,d) and for
$\alpha=1.7{\frac{m_{x}}{m_{y}}}$(e,f). In (a,c,e) the color of the lines
corresponds to the $z$ component of the total spin.

###  III.5 Discussion

The spectra of confined quantum rings are experimentally studied for annular-
shaped traps defined in gated two-dimensional electron gas [30]. The current
transport for electron traps weakly coupled to the electron reservoirs is
governed by the Coulomb blockade [31, 11], with the single-electron current
passing through the system only when the chemical potential of the confined
$N$-electron system falls within the transport window defined by the Fermi
levels of the source and drain. The technique can also be used for detection
of the excited part of the spectra when the corresponding energy level enters
the transport window [31, 30]. The transport spectroscopy allows for
reconstruction of the energy spectra with a precision of the order of a few
$\mu$eV [30] with the Aharonov-Bohm periodicity as the signature of the
angular confinement. The gated quantum rings can exhibit an elliptically
deformed confinement potential as in Ref. [30] in particular.

The gating techniques for phosphorene have been developped [32, 33, 34] and
applied for fabrication of the field-effect transistors. The ring-like
potential can be defined electrostatically in a plane plate capacitor system
with a tubular electrode protruding from one of the plate electrodes with the
phosphorene layer embedded in a dielectric [35, 36]. For a similar gating
system defined for electrons on liquid helium surface, see e.g. Ref. [27].

The results of this paper indicate the way to observe the Aharonov-Bohm effect
for the system confined by the external potential in phosphorene. For circular
quantum rings the Aharonov-Bohm oscillations can only be observed in the
excited part of the spectrum since in the degenerate ground state the electron
density forms separated islands and the persistent current circulation is
interrupted. For an elliptical deformation of the confinement potential, the
oscillations of the ground-state parity appear with the Aharonov-Bohm
periodicity. The amplitude of these oscillations has been determined (see
$\Delta_{0}$ in Table II). Based on results of Ref. [30] one can expect that
for $\alpha\leq 1.2\frac{m_{x}}{m_{y}}$ the ground-state oscillation should
enter the experimental resolution.

For a specifically chosen eccentricity parameter $\alpha$ one can reduce the
spectra to those that are characteristic to a circular quantum ring with an
isotropic effective mass. We explained this effect analytically in the
effective mass approximation that indicates an additional symmetry found for a
value of $\alpha$ and the spectra that agree with the tight-binding ones up to
an avoided crossing in the excited energy spectra found in the latter, which
is minimal for the optimal value of $\alpha=\frac{m_{x}}{m_{y}}$ (see
$\Delta_{e}$ in Table II). The width of the avoided crossing should also be
accessible for an experimental study.

##  IV Summary and Conclusions

We have studied the ground-state energy oscillations in a quantum ring
potential defined within monolayer black phosphorus with the tight-binding and
effective mass models. In a circular quantum ring, the strong anisotropy of
the effective mass produces a ground state localized along the axis related to
the heavier mass. The current circulation around is possible for an elliptic
ring. A braided pattern of even and odd-parity energy levels is then observed
in the ground state with crossings appearing with the Aharonov-Bohm
periodicity. In particular, for the ellipse with the ratio of the semiaxes
equal to the effective masses ratio, the electron density becomes uniform
along the ring. Then, the single-electron energy spectrum becomes similar to
that of a circular quantum ring with an isotropic effective mass equal to the
geometric average of the effective masses along the two crystal directions. We
demonstrated that the angular momentum in the rescaled space $l_{z}^{\prime}$
is definite in the single-electron Hamiltonian eigenstates. We provided an
analytical formula for the spectrum in the 1D limit. The appicability of the
$l^{\prime}_{z}$ operator exceeds the quantum rings and can be used for
modeling other confined systems in phoshorene. For two electrons, an
elliptical deformation of the ring produces avoided crossings in the ground-
state due to the same parity of low-energy spin-polarized states appearing
periodically on the magnetic field scale.

## Acknowledgments

This work was supported by the National Science Centre (NCN) according to
decision DEC-2019/35/O/ST3/00097. Calculations were performed on the PLGrid
infrastructure.

## References

  * [1] H. Liu, A.T. Neal, Z.Zhu, Z. Luo, X. Xu, D. Tomanek, and P.D. Ye, ACS Nano 8, 4033 (2014).
  * [2] L. Li, Y. Yu, G. J. Ye, Q. Ge, X. Ou, H. Wu, D. Feng, X. H. Chen, and Y. Zhang, Nat. Nanotechnol. 9, 372 (2014).
  * [3] S. Fukuoka, T. Taen, and T. Osada, J. Phys. Soc. Jpn. 84, 121004 (2015).
  * [4] M. Akhar, G. Anderson, R.Zhao, A. Alruqi, J.E. Mroczkowska, G. Sumanasekera, J.B. Jasinski, npj 2D Mater Appl 1, 5 (2017).
  * [5] G. Zhang, S. Huang, F. Wang, and H. Yan, Laser Photonics Rev. 15, 2000399 (2021).
  * [6] D. He, Y. Wang, Y. Huang, Y. Shi, X. Wang, and X. Duan, Nano Lett. 19, 331 (2019).
  * [7] X.Li, Z. Yu, X. Xiong, T. Li, T. Gao, R. Wang, R. Huang, and Yanqing Wu, Sci. Adv. 5 eaau3194 (2019).
  * [8] G. Long, D. Maryenko, J. Shen, S. Xu, J. Hou, Z. Wu, W.K. Wong, T. Han, J. Lin, Y. Cai, R. Lortz, and N. Wang, Nano Lett. 16, 7768 (2016).
  * [9] G. Long, D. Maryenko, S. Pezzini, S. Xu, Z. Wu, T. Han, J. Lin, C. Cheng, Y. Cai, U. Zeitler, and N. Wang, Phys. Rev. B 96, 155448 (2017).
  * [10] J. Yang, S. Tran, J. Wu, S. Che, P. Stepanov, T. Taniguchi, K. Watanabe, H. Baek, D. Smirnov, R. Chen, and C. N. Lau, Nano Lett. 18, 229 (2018).
  * [11] R. Hanson, L. P. Kouwenhoven, J. R. Petta, S. Tarucha, and L. M. K. Vandersypen, Rev. Mod. Phys. 79, 1217 (2007).
  * [12] V.M. Fomin, Physics of Quantum Rings, Springer Nature, Cham (2018).
  * [13] Y. Aharonov and D. Bohm, Phys. Rev. 115, 485 (1959).
  * [14] L.L. Li, D. Moldovan, P. Vasilopoulos,and F. M. Peeters, Phys. Rev. B 95, 205426 (2017).
  * [15] R. Schuster, J. Trinckauf, C. Habenicht, M. Knupfer, and B. Büchner, Phys. Rev. Lett. 115, 026404 (2015).
  * [16] J. Qiao, X. Kong, Z.X. Hu, F. Yang, W. Ji, Nat. Commun. 5, 4475 (2014).
  * [17] A. N. Rudenko and M. I. Katsnelson, Phys. Rev. B 89, 201408(R) (2014).
  * [18] A. N. Rudenko, S. Yuan and M. I. Katsnelson, Phys. Rev. B 92, 085419(R) (2015).
  * [19] P. Faria Junior, M. Kurpas, M. Gmitra, and J. Fabian, Phys. Rev. B 100, 115203 (2019).
  * [20] X. Zhou, , W.K. Lou, D. Zhang, F. Cheng, G. Zhou, and Chang, K. Phys. Rev. B 95, 045408 (2017).
  * [21] L. Li, F.Yang, G.J. Ye, Z. Zhang, Z. Zhu, W. Lou, X. Zhou, L. Li, K. Watanabe, T. Taniguchi, and K. Chang, Nat. Nano. 11, 593–597 (2016).
  * [22] N. Gillgren, D. Wickramaratne, Y. Shi, T. Espiritu, J. Yang, J. Hu, J. Wei,X. Liu, Z. Mao, K. Watanabe, and T. Taniguchi, 2D Materials 2, 011001 (2014).
  * [23] F. Yang, Z. Zhang, N.Z. Wang, G.J. Ye, W. Lou, X. Zhou, K. Watanabe,T. Taniguchi, K. Chang, X.H. Chen, and Y. Zhang, Nano Letters, 18, 6611 (2018).
  * [24] B. Szafran, Phys. Rev. B 101, 235313 (2020).
  * [25] With $\alpha=\frac{m_{x}}{m_{y}}$ for $R_{s}$=25, 30, 35, …, 70 (in nm) we obtain the lowest conduction band energy level at 341.066, 340.764, 340.559, 340.415, 340.328, 340.266, 340.184, 340.161, and 340.138 (in meV), respectively.
  * [26] S. Viefers, P. Koskinen, P.S. Deo, and M. Manninen, Physica E, 21 1 (2004).
  * [27] B. Szafran, Phys. Rev. B 104, 235402 (2021).
  * [28] V. M. Fomin, V. N. Gladilin, J. T. Devreese, N. A. J. M. Kleemans, and P. M. Koenraad, Phys. Rev. B 77, 205326 (2008).
  * [29] J.-L. Zhu, Z. Dai, and X. Hu, Phys. Rev. B 68, 045324 (2003).
  * [30] A. Fuhrer, S. Luescher, T. Ihn, T. Heinzel, K. Ensslin, W. Wegscheider, and M. Bichler, Nature 413, 822 (2001).
  * [31] Kouwenhoven L.P., Marcus C.M., McEuen P.L., Tarucha S., Westervelt R.M., Wingreen N.S. (1997) Electron Transport in Quantum Dots. In: Sohn L.L., Kouwenhoven L.P., Schön G. (eds) Mesoscopic Electron Transport. NATO ASI Series (Series E: Applied Sciences), vol 345. Springer, Dordrecht.
  * [32] L. Li, Y. Yu, Q. Ge, X. Ou, H. Wu, D. Feng, X. H. Chen, and Y. Zhang, Nat. Nanotechnol. 9, 372 (2014).
  * [33] D. He, Y. Wang, Y. Huang, Y. Shi, X. Wang, and X. Duan, Nano Lett. 19, 331 (2019).
  * [34] X. Li, Z. Yu, X. Xiong, T. Li, T. Gao, R. Wang, R. Huang, and Y. Wu, Sci. Adv. 5, eaau3194 (2019)
  * [35] W. Dickerson, V. Tayari, I. Fakih, A. Korinek, M. Caporali, M. Serrano-Ruiz, M. Peruzzini, S. Heun, G. A. Botton, and T. Szkopek, App. Phys. Lett. 112, 173101 (2018).
  * [36] B. Deng, V. Tran, Y. Xie, H. Jiang, L. Cheng, Q. Guo, X. Wang, H Tian, S.J. Koester, H. Wang, J.J. Cha, Q. Xia, L. Yang, and F. Xia, Nat. Commun. 8, 14474 (2017).


values of the Lagrange function. Thus, statement (1) holds. Assume that
$p(x)>p^{i}(x)$ for all $x\in[0,\overline{X}]$, then the solution is
$q_{i}=0$, violating the market-clearing condition. Therefore, statement (2)
holds. Finally, it is never optimal to choose $q_{i}(x)>0$ for $x$ such that
$p(x)>p^{i}(x)$. Thus, $q_{i}\big{(}[0,\overline{X}]\backslash
B_{i}\big{)}=0$. The market-clearing condition implies $q_{i}(B_{i})=1$.
$\square$

Consider

$g_{jk}(x)~{}~{}\equiv~{}~{}p^{j}(x)-p^{k}(x)~{}~{}=~{}~{}\frac{1}{\xi_{j}}u_{j}(x)-\frac{1}{\xi_{k}}u_{k}(x)-\left(\frac{\eta_{j}}{\xi_{j}}-\frac{\eta_{k}}{\xi_{k}}\right)$

From the proof of Lemma 1, it follows that $g_{jk}(x)$ is strictly quasi-
concave for $j<k$. Denote by
$\underline{x}_{j}=\textrm{inf}\big{(}\textrm{supp}(q_{j})\big{)}$ and
$\overline{x}_{j}=\textrm{sup}\big{(}\textrm{supp}(q_{j})\big{)}$.

Claim D2. If $k>j$, then
$q_{k}\big{(}[\underline{x}_{j},\overline{x}_{j}]\big{)}=0$
Proof. Let $x\in(\underline{x}_{j},\overline{x}_{j})$. There exist
$x^{\prime},x^{\prime\prime}$ such that
$\underline{x}_{j}<x^{\prime}<x<x^{\prime\prime}<\overline{x}_{j}$ and
$p^{j}(x^{\prime})=p(x^{\prime})$,
$p^{j}(x^{\prime\prime})=p(x^{\prime\prime})$. Otherwise, Claim D1 would imply
that $q_{j}\big{(}(\underline{x}_{j},x)\big{)}=0$ or
$q_{j}\big{(}(x,\overline{x}_{j})\big{)}=0$, contradicting the definition of
$\underline{x}_{j}$ and $\overline{x}_{j}$. By Claim D1, we get
$p^{k}(x^{\prime})\leq p(x^{\prime})=p^{j}(x^{\prime})$ and
$p^{k}(x^{\prime\prime})\leq p(x^{\prime\prime})=p^{j}(x^{\prime\prime})$.
Hence, $g_{jk}(x^{\prime})\geq 0$ and $g_{jk}(x^{\prime\prime})\geq 0$. Since
$g_{jk}(x)$ is strictly quasi-concave, $g_{jk}(x)>0$. It follows that
$p^{k}(x)<p^{j}(x)\leq p(x)$. Therefore, $q_{k}(x)=0$ by Claim D1. $\square$

Claim D3. For any $k=1,...,N$, if $(q,p)$ is an equilibrium, then
$\underline{x}_{k}<\underline{x}_{k-1}<...<\underline{x}_{1}<\overline{x}_{1}<...<\overline{x}_{k-1}<\overline{x}_{k}$,
$q^{1}=f~{}\big{|}~{}[\underline{x}_{1},\overline{x}_{1}]$ and
$q_{j}=f~{}\big{|}~{}[\underline{x}_{j},\underline{x}_{j-1}]\cup[\overline{x}_{j-1},\overline{x}_{j}]$
for $j\in\\{2,...,k\\}$.
Proof. We prove the Claim by induction on $k$.

For $k=1$, Claim D2 and the market-clearing condition imply that
$q^{1}=f~{}\big{|}~{}[\underline{x}_{1},\overline{x}_{1}]$.

Suppose the statement holds for $k$. By Claim D2,
$q_{k+1}\big{(}[\underline{x}_{k},\overline{x}_{k}]\big{)}=0$. Therefore,
there are three possible cases.

First, suppose
$\underline{x}_{k+1}<\overline{x}_{k+1}\leq\underline{x}_{k}<\overline{x}_{k}$.
Any agent prefers lottery $q_{k+1}$ to lottery $q_{k}$—by the market-clearing
condition, $q_{m}\big{(}[\underline{x}_{m},\overline{x}_{m}]\big{)}=1$ for any
$m$. Since all agents have the same endowment, $q_{k+1}$ is feasible for a
type-$k$ agent, contradicting the optimality of $q_{k}$.

Second, suppose
$\underline{x}_{k}<\overline{x}_{k}\leq\underline{x}_{k+1}<\overline{x}_{k+1}$.
As in the first case, agent $k+1$ makes a suboptimal choice.

Therefore, it must be that
$\underline{x}_{k+1}<\underline{x}_{k}<\overline{x}_{k}<\overline{x}_{k+1}$.
Claim D2 and the market-clearing condition implies that
$q_{k+1}=f~{}\big{|}~{}[\underline{x}_{k+1},\underline{x}_{k}]\cup[\overline{x}_{k},\overline{x}_{k+1}]$.
$\square$

Claim D3 for $k=N$ and the market-clearing condition imply the proposition.
$\blacksquare$

## References

  * Akbarpour et al. (2020) Akbarpour, M., S. Li, and S. O. Gharan (2020): “Thickness and information in dynamic matching markets,” _Journal of Political Economy_ , 128, 783–815.
  * Alatas et al. (2016) Alatas, V., R. Purnamasari, M. Wai-Poi, A. Banerjee, B. A. Olken, and R. Hanna (2016): “Self-targeting: Evidence from a field experiment in Indonesia,” _Journal of Political Economy_ , 124, 371–427.
  * Austen-Smith and Banks (2000) Austen-Smith, D. and J. S. Banks (2000): “Cheap talk and burned money,” _Journal of Economic Theory_ , 91, 1–16.
  * Baccara et al. (2020) Baccara, M., S. Lee, and L. Yariv (2020): “Optimal dynamic matching,” _Theoretical Economics_ , 15, 1221–1278.
  * Baccara and Yariv (2021) Baccara, M. and L. Yariv (2021): “Dynamic matching,” in _Online and Matching-Based Market Design_ , vol. forthcoming.
  * Bloch and Cantala (2017) Bloch, F. and D. Cantala (2017): “Dynamic assignment of objects to queuing agents,” _American Economic Journal: Microeconomics_ , 9, 88–122.
  * Dejarnette et al. (2020) Dejarnette, P., D. Dillenberger, D. Gottlieb, and P. Ortoleva (2020): “Time lotteries and stochastic impatience,” _Econometrica_ , 88, 619–656.
  * DellaVigna and Malmendier (2004) DellaVigna, S. and U. Malmendier (2004): “Contract design and self-control: Theory and evidence,” _Quarterly Journal of Economics_ , 119, 353–402.
  * Dimakopoulos and Heller (2019) Dimakopoulos, P. D. and C.-P. Heller (2019): “Matching with waiting times: The german entry-level labor market for lawyers,” _Games and Economic Behavior_ , 115, 289–313.
  * Eliaz and Spiegler (2006) Eliaz, K. and R. Spiegler (2006): “Contracting with diversely naive agents,” _The Review of Economic Studies_ , 73, 689–714.
  * Ely and Szydlowski (2020) Ely, J. C. and M. Szydlowski (2020): “Moving the Goalposts,” _Journal of Political Economy (forthcoming)_ , mimeo Northwestern University.
  * Fernandez et al. (2021) Fernandez, M., K. Rudov, and L. Yariv (2021): “Centralized matching with incomplete information,” _American Economic Review: Insights_ , forthcoming.
  * Gauthier and Laroque (2014) Gauthier, S. and G. Laroque (2014): “On the value of randomization,” _Journal of Economic Theory_ , 151, 493–507.
  * H.Lieb and Loss (2001) H.Lieb, E. and M. Loss (2001): _Analysis, second edition_ , vol. 14.
  * Hylland and Zeckhauser (1979) Hylland, A. and R. Zeckhauser (1979): “The efficient allocation of individuals to positions,” _Journal of Political economy_ , 87, 293–314.
  * Kattwinkel (2020) Kattwinkel, D. (2020): “Allocation with Correlated Information: Too good to be true,” in _Proceedings of the 21st ACM Conference on Economics and Computation_ , 109–110.
  * Leshno (2019) Leshno, J. (2019): “Dynamic matching in overloaded waiting lists,” Mimeo.
  * Maskin and Riley (1984) Maskin, E. and J. Riley (1984): “Optimal auctions with risk averse buyers,” _Econometrica: Journal of the Econometric Society_ , 1473–1518.
  * Roth (1989) Roth, A. E. (1989): “Two-sided matching with incomplete information about others’ preferences,” _Games and Economic Behavior_ , 1, 191–209.
  * Rothschild and Stiglitz (1976) Rothschild, M. and J. Stiglitz (1976): “Equilibrium in competitive insurance markets: An essay on the economics of imperfect information,” _Quarterly Journal of Economics_ , 90, 629–649.
  * Schummer (2021) Schummer, J. (2021): “Influencing waiting lists,” _Journal of Economic Theory_ , 195, 105263.