Aharonov-Bohm oscillations in a quantum ring: Eccentricity and electric-field effects

Aharonov-Bohm oscillations in a quantum ring: Eccentricity and electric-field effects

A. Bruno-Alfonso*

Departamento de Matemática, Faculdade de Ciências, UNESP-Universidade Estadual Paulista, Avenida Luiz Edmundo Carrijo Coube sn, 17033-360, Bauru-São Paulo, Brazil

A. Latgé

Instituto de Física, UFF-Universidade Federal Fluminense, Avenida Litorânea sn, 24210-340, Niterói-RJ, Brazil 共Received 6 October 2004; published 15 March 2005兲

The effects of an in-plane electric field and eccentricity on the electronic spectrum of a GaAs quantum ring in a perpendicular magnetic field are studied. The effective-mass equation is solved by two different methods: an adiabatic approximation and a diagonalization procedure after a conformal mapping. It is shown that the electric field and the eccentricity may suppress the Aharonov-Bohm oscillations of the lower energy levels. Simple expressions for the threshold energy and the number of flat energy bands are found. In the case of a thin and eccentric ring, the intensity of a critical field which compensates the main effects of eccentricity is determined. The energy spectra are found in qualitative agreement with previous experimental and theoretical works on anisotropic rings.

DOI: 10.1103/PhysRevB.71.125312 PACS number共s兲: 73.21.⫺b, 73.23.⫺b, 73.63.⫺b

I. INTRODUCTION

Quantum rings 共QRs兲 are doubly connected mesoscopic systems where the ballistic motion of charge carriers may take place. Fortunately, many properties of QRs can be ex-plained with single-electron theory.1–3_{In particular, the} elec-trons in a perfectly circular ring threaded by a perpendicular magnetic field may have a well-defined projection of the angular momentum Lz=បl in the direction of the field. As a function of the threading magnetic flux␾, the energy of the state with l = 0 gives an upward parabola with the vertex at

␾= 0. Moreover, a horizontal shift of that curve in −l␾0, where␾0= 2␲ប/e, is obtained for other values of l. Hence, when the levels are indexed as energy increases, each of them performs periodic oscillations with period␾0. Such os-cillations are a result of a quantum-interference phenomenon which is known as the Aharonov-Bohm共AB兲 effect, and may be detected by transport measurements.4 _Optical experiments5 _{has also detected the AB effect on a charged} particle in a nanoscale QR.

Actually, the circular shape is an idealization of grown or fabricated solid-state rings, where imperfections of the struc-tures often occur.4_{In order to study the manifestation of the} AB effect in a less symmetric ring, a two-dimensional共2D兲 annular ellipse with smoothly varying width has been considered.6 The asymmetry of the system was shown to produce the opening of gaps and the localization of some states in the wider regions, thus leading to the flattening of the corresponding energy levels. Also, the effect of the vari-able curvature of an elliptical QR of uniform width on the electron-energy spectrum in a magnetic field has been studied.7_{The expected AB oscillations were found, but} en-ergy gaps appeared due to the confinement of the electron in the regions with larger curvature. Moreover, the influence of the anisotropy of a semiconductor QR, as due to an applied electric field, on the AB oscillations and the optical spectrum has been analyzed.8 _{It was found that in the presence of} threading magnetic field, the electric field destroys the

rota-tional invariance of the QR and suppresses the AB oscilla-tions of the lower energy levels.

In the absence of the magnetic field, the electronic states in QRs subjected to an in-plane electric field9_{and including} eccentricity effects10 _{have been studied. We have calculated} the electronic spectrum of eccentric rings by a combination of a conformal mapping with a matrix-diagonalization procedure.10 _{In particular, we have shown that the} ground-state polarization due to eccentricity may be compensated by an in-plane electric field. Also, 2D QRs of constant width and arbitrary centerline, where the effects of a varying cur-vature play a central role, have been considered.11

In this work, the conduction-electron states in GaAs quan-tum rings in the presence of a perpendicular magnetic field are calculated within the effective-mass approximation. The in-plane motion of the electron is restricted to the region between two nonconcentric circles. Attention is focused on the effects of the eccentricity and in-plane electric fields, which break the rotational symmetry of the ring.8–10_The in-terplay between the electric field and the eccentricity is ex-plained in terms of simple equations for each transversal mode, within an adiabatic approximation. In particular, we determine the number of flat energy levels and the spacing between them. For a comparison, we have calculated the same energy levels by adapting the method of Lavenère et

al.,10 to include the effects of the threading magnetic flux. The energy spectra are found in qualitative agreement with previous experimental and theoretical works on anisotropic rings.4,8

II. THEORETICAL FRAMEWORK

To study the electronic spectrum of a GaAs QR in crossed magnetic and electric fields, a parabolic-band scheme within the effective-mass approximation is used. Hence, the station-ary envelope functions satisfy the equation

Hˆ3D␺3D共rជ兲 = E3D␺3D共rជ兲 共1兲 with

Hˆ3D=

关− iបⵜជ + eAជ共rជ兲兴2

2m* + eFជ· rជ+ U3D共rជ兲, 共2兲 where U3D共rជ兲 is the QR-confinement potential, Aជ共rជ兲 is the vector potential of the magnetic field, and Fជ is the electric-field intensity. Also, −e is the electron charge and m*_{is the} effective mass 共m*_{= 0.067m}

0 for conduction electrons in GaAs, where m0is the electron mass兲.

The QR confinement is taken as U3D共rជ兲=V共z兲+U2D共x,y兲, where V共z兲 gives the vertical confinement and

U2D共x,y兲 =

再

0, if共x − ⌬兲

2_{+ y}2_{艌 a}2_{and x}2_{+ y}2_{艋 b}2

⬁, otherwise

冎

共3兲 defines a doubly-connected region R in the xy plane. Actually,R is a ring with circular boundaries, where a 共b兲 is the internal 共external兲 radius and ⌬ is the eccentricity 共with 兩⌬兩⬍b−a兲. The vector potential is chosen as

Aជ共rជ兲=共Bជ⫻rជ兲/2, where Bជ=共0,0,B兲 is the magnetic field, and the electric field is taken as Fជ=共F,0,0兲. The 2D ring R and the electromagnetic-field configuration are depicted in Fig. 1. The envelope wave function may be written as

␺3D共r→兲=␹共z兲␺2D共x,y兲, where the in-plane wave function sat-isfies

Hˆ2D␺2D共x,y兲 = E2D␺2D共x,y兲, 共4兲 with Hˆ2D= − ប2 2m*ⵜ共x,y兲 2 −ieបB 2m*

冉

− y ⳵ ⳵x+ x ⳵ ⳵y

冊

+ e2B2 8m*共x 2_{+ y}2_兲 + eFx + U2D共x,y兲, 共5兲

being the in-plane Hamiltonian, which contains the effects of

the eccentricity and the electric and magnetic fields. More-over, the vertical wave function␹共z兲 satisfies

冋

− ប 2

2m*

d2

dz2+ V共z兲

册

␹共z兲 = 共E3D− E2D兲␹共z兲. 共6兲

The potential V共z兲 is not specified here, but it is supposed to have a discrete set of low-energy levels.

Two different methods are used, in what follows, to solve Eq. 共4兲: 共i兲 an adiabatic approximation leading to a one-dimensional共1D兲 problem and 共ii兲 a 2D approach involving a diagonalization procedure after a conformal mapping.10_As those approaches involve coordinate transformations be-tween the rectangular coordinates 共x1, x2兲=共x,y兲 and a couple共q1, q2兲 of suitable curvilinear coordinates, it is con-venient to establish a curvilinear version of Eq.共4兲.

The Laplace operator is transformed as12,13

ⵜ_共x,y兲2 =1 J

兺

h=1 2 ⳵ ⳵qh

冉

J

兺

j=1 2 G−1_jh ⳵ ⳵qj

冊

, 共7兲

where J = det共M兲 is the Jacobian of the transformation, the Jacobian matrix satisfies

Mhj=

⳵xh

⳵qj

, 共8兲

and G = MTM is the covariant metric tensor.13_{The curvilinear} coordinates are chosen to guarantee that J is always positive. Also, the partial derivatives are transformed as

⳵ ⳵xh =

兺

j=1 2 M−1_jh ⳵ ⳵qj . 共9兲

The probability of finding the electron in 关x1, x1+ dx1兴 ⫻关x2, x2+ dx2兴 is given by 兩␺2D共x1, x2兲兩2dx1dx2 =兩␩共q1, q2兲兩2dq1dq2, with

␩共q1,q2兲 =␺2D共x1,x2兲

冑

J, 共10兲

and the orthonormality relations for ␩共q1, q2兲 apply to

␺2D共x1, x2兲 as well.

The differential equation for␩共q1, q2兲 is

Tˆ␩共q1,q2兲 = E2D␩共q1,q2兲, 共11兲 with Tˆ = − ប 2 2m* 1

冑

J

兺

h=1 2 ⳵ ⳵qh

冉

J

兺

j=1 2 G−1_jh ⳵ ⳵qj 1

冑

J

冊

−ieបB 2m*

冑

J_h=1

兺

2 共− 1兲h_x 3−h

冉

兺

j=1 2 M_jh−1 ⳵ ⳵q_j 1

冑

J

冊

+e 2_B2 8m*共x1 2_{+ x} 2 2_{兲 + eFx1+ U2D共x1,x2兲, 共12兲} where x1 and x2 are functions of q1 and q2. Equation 共4兲 is then solved for ␺2D共x,y兲 in the region R, where U2D= 0, with the Dirichlet boundary condition␺2D共x,y兲=0.

FIG. 1. Schematic view of a 2D ring under crossed electric Fជ and magnetic Bជ fields, with a and b being the internal and external radii, respectively. The shift⌬ is the eccentricity.

III. THE ADIABATIC APPROXIMATION

If the ring R is thin enough, i.e., b−aⰆb, then it can be treated as a quasi-one-dimensional strip in the

xy plane. Since the inner circle can be parameterized as

共x,y兲=关⌬+a cos共␪兲,a sin共␪兲兴, with −␲艋␪艋␲, and the outer circle is 共x,y兲=关b cos共␪兲,b sin共␪兲兴, a mean circle may be defined as 共x,y兲=关⌬/2+r cos共␪兲,r sin共␪兲兴, with

r =共a+b兲/2 being the mean radius. The curvilinear

coordi-nates共q1, q2兲=共u,␪兲 are implicitly defined by

冉

x y

冊

=

冋

共1/2 − u兲⌬ + 共r + uw兲cos共␪兲

共r + uw兲sin共␪兲

册

, 共13兲 with w = b − a being the mean width, and −1 / 2艋u艋1/2. Hence,␪ and u are the longitudinal and transversal coordi-nates, respectively. The Jacobian J =共r+uw兲关w−⌬ cos共␪兲兴 of this transformation is a positive function for

u苸关−1/2,1/2兴 and arbitrary␪.

After a lengthy but straightforward calculation, the opera-tor Tˆ in Eq.共12兲 is obtained in terms of the small parameters ⍀=w/r and␰=⌬/w. A first-order power expansion of Tˆ in terms of ⍀ and␰ is performed to obtain the Hermitian op-erator Tˆ0共not shown, for the sake of brevity兲. The adiabatic approximation leads to a variational solution of

Tˆ0␩共u,␪兲 = E2D␩共u,␪兲, 共14兲 where ␩共u,␪兲=

冑

2 sin共n␲u + n␲/ 2兲␺1D共␪兲, with n=1,2,3,… corresponding to different transversal modes. The longitudi-nal mode␺1D共␪兲 satisfies

Hˆ_1D共n兲␺1D共␪兲 =

冉

E_2D−ប 2_n2_␲2

2m*w2

冊

␺1D共␪兲, 共15兲

with the cyclic boundary condition ␺1D共␪+ 2␲兲=␺1D共␪兲 for all values of␪. The Hamiltonian is

Hˆ_1D共n兲= 2

冕

−1/2

1/2

sin共n␲u + n␲/2兲Tˆ0sin共n␲u + n␲/2兲du

= ប 2 2m*r2

冋

− d2 d␪2− 2i␾ ␾0 d d␪ + ␾2 ␾0 2+ ␣n 2 cos共␪兲

册

− ប 2 8m*r2+ eF⌬ 2 , 共16兲

where ␾=␲r2_{B is the magnetic flux over the mean circle,}

␾0= 2␲ប/e, and ␣n= 4m*r2 ប2

冉

eFr + ប2_n2_␲2_⌬ m*w3

冊

共17兲

is a parameter which measures the anisotropy of the system 共due to the eccentricity and the applied electric field兲 for the

nth transversal mode.

It is worth noting that the four terms in the parentheses of Eq.共16兲 contain most of the physics of the problem. In fact, the first three are responsible for the Aharonov-Bohm oscil-lations of the energy levels associated to the nth transversal mode, while the fourth term can suppress the oscillations of the energy levels below a certain threshold. Moreover, the

last two terms in Eq.共16兲 lead to a shift of the energy levels that does not depend on the magnetic field.

The anisotropy parameter in Eq.共17兲 depends linearly on the electric-field intensity and the eccentricity. One may no-tice that the eccentricity effect is stronger for:共i兲 bigger and thinner rings共larger r and smaller w兲 and 共ii兲 higher trans-versal modes共larger n兲, but it does not depend on the effec-tive mass or the electric charge of the particle. On the other hand, the effect of the electric field is stronger for:共i兲 bigger rings and共ii兲 larger effective mass m*_{, but it does not depend} on the mean width w or the transversal-mode index n. Also, the electric-field term has opposite sign for a positive charge, i.e., for a hole. Moreover, ␣n vanishes for the electric-field intensity

Fn= −

ប2_n2_␲2_⌬

em*_w3_r . 共18兲

It means that it is possible to repair the eccentric ring by applying an in-plane electric field, avoiding the suppression of the Aharonov-Bohm oscillations. However, the critical field-intensity Fn depends on the transversal-mode index n and the effective mass m*_{, and has opposite signs for positive} and negative charges. In this sense, it is not possible to mend the whole electronic spectrum of the eccentric ring.

The eigenfunctions of Hˆ_1D共n兲in Eq.共16兲 are

␺1D共j,n兲共␪兲 = 1

冑

2 exp

冉

− i ␾ ␾0 ␪

冊

⫻

冋

Aj,nC

冉

4␭j,n,␣n, ␪ 2

冊

+ Bn,jS

冉

4␭j,n,␣n, ␪ 2

冊

册

共19兲

with j = 1 , 2 , 3 , . . ., and C共a,q,z兲 and S共a,q,z兲 being the even and odd共in the variable z兲 Mathieu functions, respectively, which are linearly independent solutions of the differential equation14,15

y

⬙

共z兲 + 关a − 2q cos共2z兲兴y共z兲 = 0. 共20兲

Here, C共a,q,z兲 关S共a,q,z兲兴 is the solution with the initial conditions y共0兲=1 and y

⬘

共0兲=0 关y共0兲=0 and y

⬘

共0兲=1兴. Ac-cording to the condition ␺1D共␪+ 2␲兲=␺1D共␪兲, the coeffi-cients Ajand Bjsatisfy

iAj,nsin

冉

␲␾ ␾0

冊

C

冉

4␭j,n,␣n, ␲ 2

冊

= Bj,ncos

冉

␲␾ ␾0

冊

S

冉

4␭j,n,␣n, ␲ 2

冊

共21兲 and Aj,ncos

冉

␲␾ ␾0

冊

C

⬘

冉

4␭j,n,␣n, ␲ 2

冊

= iBj,nsin

冉

␲␾ ␾0

冊

S

⬘

冉

4␭j,n,␣n, ␲ 2

冊

, 共22兲 where C

⬘

共a,q,z兲=Cz共a,q,z兲 and S

⬘

共a,q,z兲=Sz共a,q,z兲. Hence,␭j,n=␭j共␣n,␾兲 with ␭j共␣,␾兲 being the jth root of

cos2

冉

␲␾ ␾0

冊

C

⬘

冉

4␭,␣,␲ 2

冊

S

冉

4␭,␣, ␲ 2

冊

= − sin2

冉

␲␾ ␾0

冊

C

冉

4␭,␣,␲ 2

冊

S

⬘

冉

4␭,␣, ␲ 2

冊

, 共23兲 and the corresponding value of E_2D is

E_2D共j,n兲=ប 2_n2_␲2 2m*w2 + ប2 2m*r2

冋

␭j共␣n,␾兲 − 1 4

册

+ eF⌬ 2 . 共24兲 From Eq.共23兲, it is straightforward to show that the spec-trum is even关␭j共␣, −␾兲=␭j共␣,␾兲兴 and periodic with period

␾0关␭j共␣,␾+␾0兲=␭j共␣,␾兲兴 as a function of the magnetic flux ␾. Moreover, ␭_j共−␣,␾兲=␭_j共␣,␾兲, due to symmetry. Then, one can limit the numerical calculations to the ranges

␣艌0 and 0艋␾艋␾0/ 2, where␭j共␣,␾兲 is a monotonic func-tion

of ␾. In a wider range of ␾ values, ␭j共␣,␾兲 exhibits AB oscillations as a function of ␾ with amplitude

␦j共␣兲=兩␭j共␣,␾0/ 2兲−␭j共␣, 0兲兩. In particular, for ␣= 0 the bandwidth is given by␦_j共0兲=共2j−1兲/4.

Figure 2 displays the eigenvalues ␭_j共␣,␾兲 as a function of the anisotropy parameter ␣ for j = 1 , . . . , 20 and 0艋␾艋␾0/ 2. The solid 共dashed兲 lines correspond to ␾= 0 共␾=␾0/ 2兲 and the gray bands are for 0⬍␾⬍␾0/ 2. One can clearly distinguish between the bands above and below the line␭=␣/ 2, which corresponds to the maximum value of the potential␣cos共␪兲/2 in Eq. 共16兲. The higher bands are wide and very small gaps separate them. This means that such bands show strong Aharonov-Bohm oscillations. Instead, the lower bands are very narrow with large gaps between them. Those flatbands occur because the asymmetry of the system leads to a confinement of the particle around

␪=␲+ 2k␲ 共␪= 2k␲兲 for ␣⬎0 共␣⬍0兲, with k being an integer. In fact, the second-order Taylor expansion of the potential ␣cos共␪兲/2 in Eq. 共16兲 around its minima is 兩␣兩 关共␪−␪k兲2/ 4 − 1 / 2兴, where ␪k=␲+ 2k␲ 共␪k= 2k␲兲 for

␣⬎0 共␣⬍0兲. Hence, for sufficiently large 兩␣兩, the states

with small index j correspond to a coupled set of linear oscillators.16The associated bands are almost flat and can be given in terms of an isolated oscillator as

␭j共␣,␾兲 ⬇ − 兩␣兩

2 +

冉

j − 1

2

冊

冑

兩␣兩. 共25兲 Since the line␭=兩␣兩/2 separates narrow and wide bands, Eq. 共25兲 may be used to estimate the number of flatbands as

N_␣⬇

冑

兩␣兩 +1₂. 共26兲

Of course, one should take the nearest positive integer to N_␣. However, the Eq.共26兲 underestimates the value of N_␣, since the spacing between consecutive bands decreases as j in-creases共see Fig. 2兲.

According to Eq. 共24兲, the energy of the flatbands with small j and corresponding to the nth transversal mode can be given as E_2D共j,n兲⬇ប 2_n2_␲2 2m*w2 − ប2 2m*r2

冉

兩␣n兩 2 + 1 4

冊

+ eF⌬ 2 +ប 2

冑

_兩␣ n兩 2m*r2

冉

j − 1 2

冊

, 共27兲

which is a sequence of uniformly spaced levels separated by

sn= ប2

冑

_兩␣

n兩

2m*_r2 . 共28兲

This is an important result which allows one to obtain the magnitude 兩␣_n兩 of the anisotropy parameter in terms of the mean radius r, after the experimental determination of the level spacing. A further relation between the ring parameters is given by the threshold energy which separates narrow and wide bands. This energy is

E_thr共n兲=ប 2_n2_␲2 2m*w2 + ប2 2m*r2

冉

兩␣n兩 2 − 1 4

冊

+ eF⌬ 2 , 共29兲 but the experimental error for E_thr共n兲is of the order of sn. In-stead, the energy of the lower longitudinal mode E_2D共1,n兲may be found more accurately. The latter establishes an important relation between the mean width w, the mean radius r, the eccentricity ⌬ and the electric-field intensity F 关see Eq. 共17兲兴.

Figure 2 also shows that, below the threshold line ␭=␣/ 2, the bandwidth␦j共␣兲 decreases as the anisotropy pa-rameter␣ increases. The actual width of the thinner bands, however, is not shown in the figure, because the thickness of the edge curves in the artwork is fixed. To better illustrate the evolution of the bandwidths, Fig. 3 displays the depen-dence of the rate ␦j共␣兲/␦1共0兲 with the anisotropy ␣, for

j = 1 , . . . , 20. One clearly sees that lower bands are thinner

and that the bandwidths decrease exponentially for suffi-ciently large values of ␣. The number of flatbands, N_␣, is also inferred from the figure. This is done by following a criterium in which a band is flat if it is thinner than a half of

␦1共0兲. With this idea in mind, N␣ is the number of curves FIG. 2. Energy spectrum given by the roots␭_j共␣,␾兲 as a

func-tion of the anisotropy parameter ␣ for the lower 20 bands

共j=1, ... ,20兲 and 0艋␾艋␾0/ 2. The solid 共dashed兲 lines corre-spond to the magnetic flux␾=0 共␾=␾0/ 2兲 and the gray bands are for 0⬍␾⬍␾0/ 2.

crossing the dashed line共depicted in the figure兲 at anisotropy parameters shorter than␣. That number may be estimated by a simple fitting as

N_␣⬇ − 0.322 + 1.153

冑

兩␣兩, 共30兲

where the nearest positive integer to N_␣ should be taken. This number is relevant for the study of persistent currents in QRs, since the flatbands do not contribute to the current. Hence, the number of electrons in the ring should be greater than N_␣for the AB oscillations to be detected in an equilib-rium transport measurement.

It is worth noting that for a concentric GaAs QR with

r = 400 Å, w⬃20 Å, and F=10 kV/cm, the anisotropy

pa-rameter of Eq.共17兲 is ␣⬇225.1 for all values of n. Hence, according to the Eq. 共30兲, there are 17 flatbands for all the transversal modes. Moreover, the Eq.共28兲 gives the spacing between the lower bands sn⬇5.33 meV. These results are in good agreement with the work of Barticevic et al.8

IV. CONFORMAL MAPPING AND DIAGONALIZATION PROCEDURE

To simplify the solution of the Eq.共4兲, a conformal map-ping which transforms the eccentric ringR in a concentric ring R

⬘

is used.10 _{Considering the complex variables ␻= x} + iy and␻

⬘

= x

⬘

+ iy

⬘

, a suitable conformal mapping is

␻

⬘

= f共␻兲 = ␻− x0 1 − x0␻/b2

, 共31兲

where x0= b2共1+L1L2− L3兲/共2⌬兲, with L3=

冑

共1−L12兲共1−L22兲,

L1=共⌬−a兲/b, and L2=共⌬+a兲/b. In fact, the function f maps 兩␻−⌬兩=a and 兩␻兩=b onto 兩␻

⬘

−⌬兩=a

⬘

and 兩␻

⬘

兩=b, respec-tively. The external radius of the concentric ringR

⬘

is b and its internal radius is a

⬘

= b2_共1−L1L

2− L3兲/共2a兲.

The conformal transformation give the rectangular coor-dinates 共x,y兲 in terms of the curvilinear coordinates 共q1, q2兲=共x

⬘

, y

⬘

兲. The Jacobian of this mapping is J=g−2, with g =共1 + x0x

⬘

/b 2_兲2_{+ x} 0 2_共y

_⬘

_兲2_/b4 共1 − x02 /b2兲 , 共32兲

and the Jacobian matrix is

M = 1 共1 − x02_/b2_兲g2

冉

␮ ␯ −␯ ␮

冊

, 共33兲 with␮=共1+x0x

⬘

/ b2_兲2_{− x} 0 2_共y

_⬘

_兲2_{/ b}4 _{and ␯= 2x} 0y

⬘

共1+x0x

⬘

/ b2兲/ b2_.

Equation共11兲 is solved in the region R

⬘

with the Dirichlet boundary condition␩共x

⬘

, y

⬘

兲=0, and

Tˆ = ប 2 2m*

冉

Nˆ + 2␾ r2␾0 Lˆ + ␾ 2 r4␾0 2Qˆ

冊

+ eFXˆ , 共34兲 with Nˆ =−gⵜ_共x2_⬘,y⬘兲g, Qˆ =x2+ y2, Xˆ =x, and Lˆ = i 共1 − x02 /b2兲

再

2x0y

⬘

b2 + y

⬘

冉

1 + 2x0x

⬘

b2 + x₀2 b2

冊

⳵ ⳵x

⬘

+ −

冋

共x

⬘

+ x0兲

冉

1 +x0x

⬘

b2

冊

− x0共y

⬘

兲2 b2

册

⳵ ⳵y

⬘

冎

. 共35兲

For this purpose, the eigenfunctions are expressed as

␩共x

⬘

,y

⬘

兲 =

兺

l,m

c_m,l␩_m,l共0兲共x

⬘

,y

⬘

兲, 共36兲 where l takes integer values, m = 1 , 2 , 3 , . . ., and␩_m,l共0兲共x

⬘

, y

⬘

兲 is defined in our previous work.10,17_{For each of the electronic} states, the energy E2Din Eq.共11兲 and the corresponding co-efficients cm,lare found by diagonalization of Tˆ in the ortho-normal base␩_m,l共0兲共x

⬘

, y

⬘

兲. In the calculations, the base is trun-cated, and the accuracy of the results may be increased by taking a base with a larger number of elements.

V. FURTHER RESULTS AND DISCUSSION

Figure 4 displays the lower 2D energy levels as functions of the mean magnetic flux ␾, for conduction electrons in GaAs QRs with a = 900 Å, b = 1100 Å, and different combi-nations of eccentricity ⌬ and electric-field intensity F. The solid lines are the eigenvalues obtained from Eq.共34兲 using

m = 1 , 2 , 3 and l = −40, . . . , 40, whereas the dashed lines are

the results from Eq.共24兲 with n=1, following the adiabatic approximation. The field intensity F = −84.2 V / cm has been taken from the Eq.共18兲, for ⌬=6 Å and n=1. A good agree-ment between the 2D approach and the adiabatic approxima-tion共solid and dashed lines兲 is apparent in the figure.

The energy spectrum corresponding to a concentric QR in the absence of electric fields is shown in Fig. 4共a兲, where the Aharonov-Bohm oscillations are clearly seen. Figure 4共b兲 shows the effect of applying the in-plane field

F = −84.2 V / cm to the concentric QR. The oscillations of the

lower levels are nearly suppressed,8 _{since this field pushes} the electron to the right-hand side of the ring. Following Eq. 共17兲, the value of the anisotropy parameter is␣1⬇−29.61. According to Eq.共30兲 the number of flat levels is 6, as can be seen in Fig. 4共b兲. Also, the results show that flat levels are

FIG. 3. The bandwidth␦j共␣兲, in units of the width␦1共0兲 of the

first band in a concentric ring with electric field F = 0, as a function of the anisotropy parameter ␣ for j=1, ... ,20. The band index j increases from the left-bottom to the right-top corner. The dashed line is for␦_j共␣兲=␦1共0兲/2.

separated by s1⬇0.31 meV. A QR with eccentricity ⌬=6 Å is considered in the bottom panels of Fig. 4. Panel 共c兲, where F=0 V/cm, shows that the eccentricity leads to the same effects as the in-plane electric field. In this case the localization of the electron occurs in the wider region,6_and the picture resembles Fig. 4共b兲 since ␣1⬇29.61. Instead, panel共d兲 for F=−84.2 V/cm is similar to Fig. 4共a兲, since the main effects of eccentricity and electric field are compen-sated, i.e.,␣1⬇0.

Higher transversal electronic modes may also be apparent in experimental studies.4_{Figure 5 displays some high-energy} levels for the system of Fig. 4共d兲. The solid lines are the eigenvalues of Eq.共34兲, with m=1,2,3 and l=−40, ... ,40, whereas the dashed lines are given by the Eq. 共24兲, with

n = 1 or n = 2. The diamondlike structure with⬃3 meV high

pieces are associated to the first transversal mode, and clearly shows the AB oscillations. Moreover, the flat curves and the diamondlike structure with ⬃1 meV high pieces corre-spond to the n = 2 transversal mode. For the latter states the anisotropy parameter is ␣2⬇88.82, which leads to 11 flat-bands. Also, the spacing between the lower flatbands is

s2⬇0.54 meV. This shows that the in-plane electric field

F = −84.2 V / cm, which recovers the oscillations of the n = 1

level in Fig. 4共d兲, does not repair the n=2 states.

VI. CONCLUSIONS

The conduction-electron states in GaAs quantum rings in the presence of a perpendicular magnetic field have been

calculated within the effective-mass approximation, with special attention on the effects of in-plane electric fields and eccentricity. The calculation of the 2D states was performed by two methods: an adiabatic approximation and a conformal transformation combined with a diagonalization procedure.

The adiabatic approximation, which is suitable for suffi-ciently thin rings with small eccentricity leads to a 1D prob-lem for each transversal mode. The main effects of the elec-tric field and eccenelec-tricity are contained in the anisotropy parameter. We have shown that the eccentricity effect is stronger for bigger and thinner rings and higher transversal modes, but it does not depend on the effective mass or the electric charge of the particle. Moreover, the effect of the electric field is stronger for bigger rings and heavier particles and has an opposite sign for a positive charge and does not depend on the mean width of the ring or the transversal mode. Simple expressions have been derived for the thresh-old energy separating wide from narrow bands, the number of flat levels, the distance between the lower ones, and the energy of the ground level. Furthermore, the width of the flat energy bands, as functions of the threading magnetic flux, has been shown to decrease exponentially as the anisotropy parameter increases.

The 2D approach based on the combination of a suitable conformal mapping and a diagonalization procedure is more complete, since it includes the coupling between different transversal modes and remains valid for a wider range of ring dimensions and eccentricity values. A good agreement be-tween the two approaches was obtained for thin rings with small eccentricity. As the electric field and the eccentricity may affect the electronic spectrum of GaAs quantum rings in the presence on a perpendicular magnetic field, we expect this work to be useful in magnetization, transport, and optical studies of such systems.

ACKNOWLEDGMENTS

The authors are grateful to the Brazilian Agencies FAPESP, FAPERJ, and CNPq for financial support.

FIG. 4. The lower 2D energy levels as functions of the mean magnetic flux ␾, for conduction electrons in GaAs QRs with

a = 900 Å, b = 1100 Å, and 共a兲 ⌬=0 Å and F=0 V/cm, 共b兲 ⌬=0 Å and F=−84.2 V/cm, 共c兲 ⌬=6 Å and F=0 V/cm, and 共d兲 ⌬=6 Å and F=−84.2 V/cm. Solid 共dashed兲 lines are the results of

the 2D approach with the conformal mapping 共the adiabatic approximation兲.

FIG. 5. Some 2D energy levels as functions of the mean mag-netic flux ␾, for conduction electrons in a GaAs QRs with

a = 900 Å, b = 1100 Å,⌬=6 Å, and F=−84.2 V/cm. Solid 共dashed兲

lines are the results of the 2D approach with the conformal mapping

*Electronic address: alexys@fc.unesp.br

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