Wigner molecules in phosphorene quantum dots Tanmay Thakur and Bartłomiej Szafran AGH University of Science and Technology

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Wigner molecules in phosphorene quantum dots

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

We study Wigner crystallization of electron systems in phosphorene quantum dots with conﬁne-

ment of an electrostatic origin with both circular and elongated geometry. The large eﬀective masses

in phosphorene promote the separation of the electron charges already for quantum dots of rela-

tively small size. The anisotropy of the eﬀective mass allows for the formation of Wigner molecules

in the laboratory frame with a conﬁned charge density that has lower symmetry than the conﬁne-

ment potential. We ﬁnd that in circular quantum dots separate single-electron islands are formed

for two and four conﬁned electrons but not for three trapped carriers. The spectral signatures of

the Wigner crystallization to be resolved by transport spectroscopy are discussed. Systems with

Wigner molecule states are characterized by a nearly degenerate ground state at B= 0 and are

easily spin-polarized by the external magnetic ﬁeld. In electron systems for which the single-electron

islands are not formed, a more even distribution of excited states at B= 0 is observed, and the

conﬁned system undergoes ground state symmetry transitions at magnetic ﬁelds of the order of 1

Tesla. The system of ﬁve electrons in a circular quantum dot is indicated as a special case with two

charge conﬁgurations that appear in the ground-state as the magnetic ﬁeld is changed: one with the

single electron islands formed in the laboratory frame and the other where only the pair-correlation

function in the inner coordinates of the system has a molecular form as for three electrons. The

formation of Wigner molecules of quasi-1D form is easier for orientation of elongated quantum dots

along the zigzag direction with heavier electron mass. The smaller electron eﬀective mass along

the armchair direction allows for freezing out the transverse degree of freedom in the electron mo-

tion. Calculations are performed with a version of the conﬁguration interaction approach that uses

a single-electron basis that is pre-optimized to account for the relatively large area occupied by

strongly interacting electrons allowing for convergence speed-up.

I. INTRODUCTION

Electron gas with Coulomb interactions dominating

over the kinetic energies forms a Wigner crystal [1–4].

Its ﬁnite counterparts, e.g. Wigner molecules [5–21] are

formed in quantum dots at low electron numbers in spa-

tially large systems [5] or in a strong magnetic ﬁeld that

promotes the single-electron localization [22, 23].

The conﬁned charge density in quantum dots deﬁned

in materials with isotropic eﬀective mass reproduces the

symmetry of conﬁnement potential. For this reason in

circular quantum dots, separation of the electrons in the

Wigner phase occurs only in the inner coordinates of the

system spanned by relative electron-electron distances

[22]. For lowered symmetry, the Wigner molecules can

appear in the laboratory frame [24], with the special case

of one-dimensional systems that is studied with much of

attention [6, 9, 16, 20, 21].

Phosphorene [25–28] is a particularly interesting ma-

terial for Wigner-molecule physics due to the large elec-

tron eﬀective masses and their strong anisotropy [29–35]

Large masses reduce the kinetic energy as compared to

the electron-electron interaction energy. Lowering the

Hamiltonian symmetry by the eﬀective mass anisotropy

is promising for observation of the Wigner molecules in

the laboratory frame.

Phosphorene quantum dots [36–43] in the form of small

ﬂakes have been extensively studied, in particular from

the point of view of optical properties. In this work we

consider a clean electrostatic conﬁnement that keeps the

conﬁned electrons oﬀ the edge of the ﬂake. In ﬁnite sheets

of graphene, the edges inhibit the Wigner crystallization

[12]. Advanced phosphorene gating techniques have been

developed [28, 44–47] for e.g. fabrication of the ﬁeld-

eﬀect transistors [26, 48, 49] and experimental studies of

the quantum Hall eﬀects [50–53] are carried out. There-

fore, the formation of clean electrostatic quantum dots

[54] in phosphorene is within experimental reach.

Ordering of the electron charge in Wigner molecules of

single-electron islands in quasi 1D systems [6, 9, 16, 20,

21] reduces the electron-electron interaction energy at the

cost of increasing the kinetic energy due to the electron

localization. In GaAs systems with low electron band ef-

fective mass of 0.067m0conditions for Wigner molecule

formation occur only in very long systems of hundreds of

nanometers [16] already for four electrons. On the other

hand, the light electron mass in GaAs favors the reduc-

tion of the 2D conﬁnement to an eﬀectively 1D form with

all the electrons occupying the same state of quantiza-

tion for the transverse motion. Hence, the large eﬀective

masses in phosphorene are promising for producing the

Wigner molecules in systems of relatively small sizes, but

may inhibit formation of 1D conﬁnement.

In this paper, we consider the formation of Wigner

molecules in the laboratory frame for a few electrons con-

ﬁned in circular and elongated quantum dots for varied

conﬁnement orientation and look for spectral signatures

of Wigner crystallization to be experimentally resolved.

We use the conﬁguration interaction approach [55–59]

that requires an optimized single-electron basis [60–62]

arXiv:2210.02705v2 [cond-mat.mes-hall] 24 Oct 2022

for convergence due to the strong electron-electron inter-

action eﬀects [63] in phosphorene.

This paper is organized as follows. In the Theory sec-

tion we describe the applied computational approach. In

the Results section we ﬁrst describe the results for circu-

lar quantum dots and next the Wigner molecules in quasi

one-dimensional conﬁnement oriented along the zigzag

and armchair crystal directions. Section IV contains the

discussion of the experimentally accessible signatures of

the Wigner molecule formation in the laboratory frame.

Summary and conclusions are given in Section V. In the

Appendix we include details on the single-electron wave

functions used for optimization of the basis, the choice

of the computational box, and the spectra without the

Zeeman interaction.

II. THEORY

In this section we describe the ﬁnite diﬀerence method

applied to the continuum Hamiltonian of a single elec-

tron in phosporene (subsection II.A), the model potential

(II.B) and the conﬁguration interaction approach (II.C)

with optimization of the single-electron basis allowing for

faster convergence of the conﬁguration interaction ap-

proach. Subsection II.D describes the formula for extrac-

tion of the charge density and pair correlation functions

for discussion of the Wigner crystallization of the con-

ﬁned system.

A. Single-electron Hamiltonian

We use the single-band approximation for Hamiltonian

describing the electrons of the conduction band of mono-

layer phosphorene [35, 64]

H0=−i~∂

∂x +eAx2

/2mx+−i~∂

∂y +eAy2

/2my

+W(x, y) + gµBBσz/2,(1)

where W(x, y)is the conﬁnement potential. In Eq. (1)

we use the eﬀective masses mx= 0.17037m0for the arm-

chair crystal direction (x) and my= 0.85327m0for the

zigzag direction (y). The values for the masses were

determined in Ref. [35] by ﬁtting the conﬁned energy

spectra of the continuum single-band Hamiltonian to the

results of the tight-binding method. A detailed compar-

ison of the spectra as obtained by the continuum model

to the tight-binding ones is given in Ref. [35] for the

harmonic oscillator potential and in Ref. [64] for the

annular conﬁnement. In Eq. (1) we take the Landé fac-

tor g=g0= 2 after the k·ptheory of Ref. [65]. The

values of experimentally extracted g-factors vary; in par-

ticular an increase with respect to g0was reported [63] at

low ﬁlling factors, which is attributed [63, 65] to strong

electron-electron interaction eﬀects in black phosphorus.

The electron-electron interaction in this work is treated

in an exact manner. The spin Zeeman term leads to the

spin polarization of the conﬁned system. The exact value

of the magnetic ﬁeld producing the spin polarization is

aﬀected by the adopted g-factor value, but no qualitative

eﬀect for the Wigner crystallization of the charge density

is expected as long as the spin-orbit coupling is absent.

The spectral features of Wigner crystallization for g= 0

are discussed in the Appendix.

We work with a square mesh with a spacing ∆xin both

the xand ydirections. The Hamiltonian acting on the

wave function Ψµ,η = Ψ(xµ, xη) = Ψ(µ∆x, η∆x)in the

ﬁnite-diﬀerence approach reads

H0Ψµ,η ≡~2

2mx∆x22Ψµ,η −CyΨµ,η−1−C∗

yψµ,η+1

+~2

2my∆x2(2Ψµ,η −CxΨµ−1,η −C∗

xψµ+1,η)

+Wµ,ηΨµ,η +gµBB

2σz,(2)

where Cx= exp(−ie

~∆xAx)and Cy= exp(−ie

~∆xAy)

introduce the Peierls phases [66] for the description of

the orbital eﬀects of the perpendicular magnetic ﬁeld

(0,0, B). For calculation of the phase shifts, we use the

symmetric gauge A= (Ax, Ay, Az)=(−By/2, Bx/2,0).

Hamiltonian (2) is diagonalized in a ﬁnite computational

box with the inﬁnite quantum well set at the end of the

box (see the Appendix).

B. Model potential

For evaluation of a realistic conﬁnement potential W

we use a simple model with a phosphorene plane em-

bedded in a Al2O3dielectric that ﬁlls the area between

two parallel electrodes [Fig. 1(a,b)]. A higher (lower)

potential energy for electrons is introduced at the top

(bottom) electrode. The bottom electrode is grounded

and contains a protrusion that approaches the phospho-

rene layer. As a result, the electrostatic potential within

phosphorene forms a cavity that traps the electrons of the

conduction band. The model is a variation [67, 68] of a

gated GaAs quantum dot of Ref. [69]. Below we use two

models: one with a circular protrusion (Fig. 1(a)) and

the other with a rectangular one (Fig. 1(b)). The latter

is used in the following to study the case close to the

1D conﬁnement. The conﬁnement potential to be used

in the Hamiltonian is given by W(x, y) = −eV (x, y, zp),

with the electrostatic potential Vthat we evaluate by

solving the Laplace equation −∇2V= 0 and zpis the

coordinate of the phosphorene layer. For evaluation of

the potential we use the ﬁnite element method similar to

the one applied in Ref. [68] for a charge-neutral phospho-

rene plane. The conﬁnement potential at the monolayer

is plotted in Fig. 1(c) for the circular protrusion of Fig.

1(a) and in Fig. 1(d) for the rectangular protrusion of

Fig. 1(b).

(a) (b)

100

150

0 50 100 150

y (nm)

x (nm)

130

135

140

145

150

155

-eV (meV)

(c)

100

150

0 50 100 150

y (nm)

x (nm)

100

105

110

115

120

125

-eV (meV)

(d)

FIG. 1. Schematics of model systems for evaluation of the conﬁnement potential. The phosphorene monolayer is embedded

in a dielectric that ﬁlls the space between electrodes of the plane capacitor conﬁguration. The lower metal electrode contains

a circular (a) or rectangular (b) protrusion. The upper electrode is kept at a higher potential energy −eV2= 0.25 eV for the

electrons than the lower one −eV1= 0. The protrusion introduces an inhomogeneity of the electric ﬁeld within the capacitor

that forms the conﬁnement potential for the electrons of the conduction band. We use h1+h2+h3= 150 nm and h1= 50 nm.

For the circular protrusion (a) we take h2= 50 nm and for the rectangular one (b) h2= 30 nm. The radius of the protrusion

in (a) is R= 50 nm, and the sides of the rectangle in (b) have lengths l1= 80 nm and l2= 30 nm. The conﬁnement potential

on the phosphorene plane is plotted in (c) and (d) for the circular and rectangular protrusions, respectively. The origin in (c,d)

is the symmetry center of the protrusion.

C. Diagonalization of the N-electron Hamiltonian

The system of N-conﬁned electrons is described with

the Hamiltonian

HN=

i=1

H0(i) +

j>i

4π0 rij

.(3)

We take the dielectric constant = 9 assuming that the

phosphorene is embedded in Al2O3.

In the standard conﬁguration-interaction method [55–

59] the N-electron Hamiltonian is diagonalized on the

basis of Slater determinants constructed with the single-

electron Hamiltonian H0eigenstates. Each of the Slater

determinants deﬁnes a conﬁguration, e.g. a distribution

of electrons over the single-electron states. The number

of Slater determinants to be used in the calculation is es-

tablished by a study of the convergence of the energy es-

timates. Reaching convergence in the present calculation

is challenging because of the strong electron-electron in-

teraction in phosphorene [63]. The energy of the ground-

state estimated for N= 4 at B= 0 in the circular poten-

tial of Fig. 1(c) is plotted with the black line in Fig. 2(a)

as a function of the number of the lowest-energy single-

electron states νthat span the Slater determinant basis.

The Hamiltonian HNcommutes with the operator of the

zcomponent of the total spin and also with the parity

operator due to the point symmetry of the potential. The

symmetries allow for a few-fold reduction of the number

of basis elements. In Fig. 2 the four-electron ground

state at B= 0 is the spin singlet Sz= 0 of an even

spatial parity. Only Slater determinants of these symme-

tries contribute to the ground-state wave function. For

ν= 60 the Slater determinants set counts 60

4=487 635

elements, of which only about 94 500 determinants cor-

respond to Sz= 0 and either even or odd parity. The

right vertical axis in Fig. 2(a) shows the number of Slater

determinants of the spin-parity symmetry that are com-

patible with and contribute to the ground state.

The convergence of the CI method using the H0single-

electron eigenstates (black line in Fig. 2(a)) is slow. The

electron-electron interaction has a pronounced eﬀect on

the electron localization, since the electrons in phospho-

rene are quite heavy as compared to those in e.g. GaAs,

and the deformation of the charge density in terms of the

single-electron energy is cheap. This in turn results in a

high numerical cost of the convergent calculations that

require a large number of single-electron states to be in-

cluded in the convergent basis. Therefore, due to the

strong electron-electron interaction, the set of H0eigen-

states is not the best starting point for a convergent CI

calculation. The literature indicates a number of meth-

ods to speed-up the convergence, including the HF+CI

method [60–62] where the basis for the CI method is

based on the Hartree-Fock single-electron spin-orbitals.

In the HF+CI approach, the mean-ﬁeld eﬀects of the

electron-electron interaction are accounted for already in

the single-electron basis and the CI is responsible only

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WignermoleculesinphosphorenequantumdotsTanmayThakurandBartªomiejSzafranAGHUniversityofScienceandTechnology,FacultyofPhysicsandAppliedComputerScience,al.Mickiewicza30,30-059Kraków,PolandWestudyWignercrystallizationofelectronsystemsinphosphorenequantumdotswithconne-mentofanelectrostaticoriginwithboth...

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