ARTICLE IN PRESS

Physica E 41 (2009) 1393–1402

Contents lists available at ScienceDirect

Physica E

1386-94

doi:10.1

� Corr

E-m

journal homepage: www.elsevier.com/locate/physe

Quantum rings with tunnel barriers in a threading magnetic field:Spectra, persistent current and ballistic conductance

S. Bellucci �, P. Onorato

INFN, Laboratori Nazionali di Frascati, P.O. Box 13, 00044 Frascati, Italy

a r t i c l e i n f o

Article history:

Received 9 February 2009

Received in revised form

30 March 2009

Accepted 7 April 2009Available online 17 April 2009

PACS:

73.21.�b

73.23.�b

73.63.�b

Keywords:

Quantum rings

Aharonov–Bohm oscillations

Ballistic conductance

77/$ - see front matter & 2009 Elsevier B.V. A

016/j.physe.2009.04.003

esponding author. Tel.: +39 6 94032888; fax:

ail address: bellucci@lnf.infn.it (S. Bellucci).

a b s t r a c t

The electronic states of isolated quantum rings with some tunnel barriers in a threading magnetic field

are calculated. We show that the presence of obstacles analogously to smooth and tiny variations in the

width may strongly affect the Aharonov–Bohm (AB) oscillations by producing a quenching of magnetic

dependent oscillating behavior of the lower energy levels. This quenching reflects the localization of the

quantum states and can be justified in terms of metallic-insulating transition induced by the barriers.

We discuss how the behavior of the energy levels has effects both on the persistent charge currents in

isolated ring and on the ballistic conductance of a conducting ring symmetrically coupled to two leads.

& 2009 Elsevier B.V. All rights reserved.

1. Introduction

In recent years both experimental and theoretical physicscommunities have devoted a great deal of attention to thenanostructures (e.g. in the field of quantum electronics [1]). Inparticular a big effort has been devoted to the study and therealization of devices with a ring geometry at nanometric scale.The latter devices, known as quantum rings (QRs), have played anessential role in observing quantum-interference effects and havesome analogies with superconducting ring by showing thepresence of a persistent current, a magnetically induced electriccurrent that flows undiminished in a superconducting materialor circuit.

The interference phenomena, first of all the Aharonov–Bohmeffect can be typically observed in the transport measurements ofopen QRs connected to leads. The phenomena related to thepersistent current can be revealed, on the contrary, by studyingclosed (or isolated) QRs.

Quantum interference and Aharonov–Bohm effect: The ringconductors have played an essential role in observing howcoherent superpositions of quantum states on the mesoscopicscale leave imprints on measurable transport properties [2]. In

ll rights reserved.

+39 6 94032716.

fact open QRs, as the one depicted in Fig. 1 (left top), represent asolid state realization of a two-slit experiment, where an electronentering the ring can propagate in two possible directions. Inthese devices superpositions of quantum states are sensitive tothe acquired topological phases in external fields: a magnetic[Aharonov–Bohm [3] (AB) effect] or an electric [Aharonov–Casher[4] (AC) effect for particles with spin]. The field’s variationsgenerate an oscillatory pattern of the ring conductance [5]. Theseeffect can be seen as a paradigm of quantum mechanical phasecoherence [6,7].

The AB effect was first observed in normal metal rings [5] (AuQR with a radius, R of �0:5mm and a width, W , of �50 nm) whilein more recent years semiconductor QRs have been the subjectof intense experimental [8–14] and theoretical [15–22] works(e.g. rings patterned in a GaAs heterostructure).

Persistent current: In a seminal paper Buttiker and coworkers[23] showed as small and strictly one-dimensional (1D) rings ofnormal metal, driven by an external magnetic flux, act likesuperconducting rings with a Josephson junction, except that 2e isreplaced by e. Thus in Ref. [24] the authors established aconnection between these AB-like effects by demonstrating thatthe transmission probability of the ring, which determines theelectric resistance, exhibits resonances near the energies of theelectronic states of the closed ring. It is the flux dependence ofthe resonances which gives rise to the strong oscillatory behaviorof the electric resistance. Thus since from the 1980s there is a vast

ARTICLE IN PRESS

Fig. 1. (Left top) Scheme of a circular ring with two leads. The radius R is assumed

in the range 1002400 nm, while the width W�0:1R. (Left bottom) The one-

dimensional ring symmetrically coupled to conducting leads and interrupted by a

tunnel barrier (quantum point contact) in the lower arm. (Right) Theoretical

spectrum of a single mode ring. (Top) The parabolas with constant l have a

minimum at l ¼ m. The red–blue zig-zag line corresponds to the lth energy level

from the ground state i.e. to a Coulomb peak after the charging energy has been

subtracted in the constant interaction model. (Bottom) Schematic spectrum when

an imperfection (impurity or tunnel barrier is present). (For interpretation of the

references to color in this figure legend, the reader is referred to the web version of

this article.)

S. Bellucci, P. Onorato / Physica E 41 (2009) 1393–14021394

literature of theoretical [23–28] and experimental [29–31] studieson persistent currents. Measurements of the persistent chargecurrent have been performed both in an ensemble of metallicrings [29] and on single rings realized in nanostructured (two-dimensional) 2D electron systems [30,31]. The energy spectra ofclosed rings [32] are the basis for the prediction of persistentcurrents [23] and related experiments [29–31] and these spectrahave recently been analyzed by using optical techniques [33,34].However, the energy spectra of closed devices can be also deducedfrom the measurement of Coulomb blockade (CB) resonances thatmark configurations in which a bound state in the ring aligns withthe Fermi level in source and drain at fixed magnetic field.Recently magnetotransport experiments on a ring-shaped semi-conductor quantum dot in the Coulomb blockade regime [35]were done [10]. The measurements allowed to extract the discreteenergy levels of a realistic ring, which were found to agree wellwith theoretical expectations. Such an agreement, so far onlyfound for few-electron quantum dots, was extended to a many-electron system [36].

Thus the electronic properties of nanorings have attractedconsiderable theoretical studies [37]. In the most of these worksthe model of QR has a circular shape and uniform width. However,distorted rings must be considered because imperfections occur ingrowth and fabrication processes [8,11]. The effects of oneimperfection along ring on the single particle spectrum aresummarized in Fig. 1(bottom right) as a function of magneticfield: for rings with defects, such as impurities or barriers inside,in general, the degeneracies are removed, the AB effect is reduced,and the sawtooth-type behavior is rounded. Moreover, distortedrings have interesting electronic spectra in a threading magneticfield, with anticrossings and quenched oscillation of levels as afunction of the field strength. This occurs in elliptic QRs [38,39], incircular rings with varying widths in the presence of an in-planeelectric field [40], and in QRs with structural distortions [41].Rings of arbitrary shape in the presence of a threading magneticfield have also been considered, e.g. in Ref. [42] persistent andradiation induced currents were calculated in QRs of arbitraryshape, uniform width, and finite-barrier transverse confinement.

In summary the strength of interference phenomenon can bemodulated by tailoring the shape and the size of the QR [43].

In this paper we focus on the properties of a QR in the presenceof some obstacles (tunnel barriers or quantum wells) along thepath of the electrons in the arms. On the system acts a transverseuniform magnetic field, B. The obstacles can be seen as attractiveor repulsive impurities, widening or restrictions in the channel’swidth.

We calculate the properties of both open and closed QR whichcan be experimentally investigated by measuring the conductancein two different electron transport regimes, the ballistic and theCB; the first regime emphasize the interferometric properties(as the AB oscillations), the second one is strictly related to thespectra of the isolated QR and allows a discussion about thepersistent current.

In Section 2, we introduce the Hamiltonian for the Q1D ring, inorder to calculate the single particle spectrum of an isolatedclosed QR. We discuss the effects of the presence of some (5)tunnel barriers, focusing on the role of the barrier strength. Wealso consider the presence of attractive barriers. Thus we discussthe modifications of the persistent currents in various cases.

In Section 3 we analyze the transport properties for an open QRattached to two leads and the quantum-interference effects byanalyzing the oscillations in the transmission. The density plot ofthe ballistic conductance as a function of magnetic field and Fermienergy reveals some effects quite similar to the ones analyzed forthe single energy spectrum of a closed ring.

In the Appendix we present the theoretical grounds of theelectron transport in the CB and ballistic regimes.

2. Closed quantum rings and energy spectrum

Here we consider low-dimensional electron systems formed byquasi-one-dimensional (Q1D) QR patterned in a two-dimensionalelectron gas (2DEG). Due to the band offsets at the interface oftwo different materials (e.g. AlGaAs-GaAs heterostructures) theelectrons are confined by forming a 2DEG (in the xy-plane) and aQR can be realized by using gate patterned with varioustechniques [10]. We assume that the radius R of the QR rangesfrom 100 to 400 nm while the width, W , can be assumed as �0:1R

in agreement with Refs. [44,45].

2.1. The model of Q1D ring

Here we outline briefly the derivation of the Hamiltoniandescribing the motion of an electron in a realistic Q1D ring [46].We consider 2D electrons in the x2y plane that are furtherconfined to move in a ring by a radial potential VcðrÞ.

The full single-electron Hamiltonian reads

H ¼ð~p� e~AÞ2x þ ð~p� e~AÞ2y

2m�þ VcðrÞ, (1)

where ~A ¼ ð�By=2;Bx=2;0Þ is the vector potential of an externalmagnetic field applied in the z direction and m� is the effectiveelectron mass in the 2DEG. Due to the circular symmetry of theproblem, it is natural to rewrite the Hamiltonian in polarcoordinates [46]:

H ¼ �_2

2m�@2

@r2þ

1

r

@

@r�

1

r2i@

@jþFF0

� �2" #

þ VcðrÞ,

where F is the magnetic flux threading the ring, F0 the fluxquantum. In the case of a thin ring, i.e. when the radius R of thering is much larger than the radial width of the wave function, it isconvenient to project the Hamiltonian on the eigenstates of

ARTICLE IN PRESS

S. Bellucci, P. Onorato / Physica E 41 (2009) 1393–1402 1395

H0 ¼ ð�_2=2m�Þ½@2=@r2 þ ð1=rÞ@=@r� þ VcðrÞ. To be specific, we use

a parabolic radial confining potential,

VcðrÞ ¼ 12m�o2ðr � RÞ2 (2)

for which the radial width of the wave function is given bylo ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi_=m�o

p. In the following, we assume lo=R51 and neglect

contributions of order lo=R. In this limit, H0 reduces to

H0ðr; prÞ ¼ �_2

2m�@2

@r2

" #þ

1

2m�o2ðr � RÞ2. (3)

Thus

H�H0ðr; prÞ þ _oR i@

@jþFF0

� �2

, (4)

where _oR ¼ _2=2m�R20.

The corresponding bandstructure is given by

�n;m ¼ _o nþ1

2

� �þ mþ F

F0

� �2

_oR. (5)

Usually R�10!o [45], i.e. _ob_oR, thus we can limit to thelowest subband n ¼ 0 i.e. to a 1D system.

-150 -100 -50 0 50 100 150

-150

-100

-50

0

50

100

150

y (n

m)

0 20 40 60 80 100� (unit �R)

0

5

10

15

20

N (�

)

U = 0; �(B)/�0 = n, n±1+4, n+1/2

x (nm)

Fig. 2. (Color online) (Top) Scheme of a circular ring and energy levels as a function of

which is marked by the dashed line in, is 150 nm. (Bottom left) In the plot of Nð�Þwe repo

large scale approximation is shown while (Bottom Right) the DOS exhibits peaks typic

2.2. Isolated quantum ring

It follows that for fixed values of the Fermi energy, �F , and ofthe band n there are eigenstates which have the general form

Cn;mðr;jÞ ¼ unðr � R0Þeimj, (6)

where unðxÞ are the eigenstates of the 1D harmonic oscillator.In the case of an isolated QR after imposing the periodic

boundary conditions we have m ¼ m 2 Z and the related band-structure is shown in Fig. 2.

Here, we study QRs with a uniform width W ¼ 15 nm and limitourselves to the transversal mode with n ¼ 0 (with energy_o0�25 meV if we assume a GaAs based device). The longi-tudinal energy � will be given in units of _oR which correspondsfor a ring with a radius of 150 nm in GaAs, to an energy0.63 meV.

The results reported in Fig. 2 are displayed for energy levels asa function of the magnetic flux in the range 0pFðBÞ=F0p2. If weassume a circular the 150 nm radius ring, since the area enclosedby the centerline is S�0:071mm2, a threading flux quantumcorresponds to a magnetic field of 58.5 mT. The typical ABoscillations of the energy levels, with a period F0, are clearlyobserved in Fig. 2(right).

0.5 1 1.5 2�(B) / �0

2

4

6

8

10

� (u

nit

�R

)

U = 0; �(B)/�0 = n, n±1+4, n+1/2

0 20 40 60 80 100� (unit �R)

0

0.2

0.4

0.6

0.8

1

1.2

DO

S

the magnetic flux. The ring width is W ¼ 15 nm and the radius of the centerline,

rt the 1D–0D properties of the QR: a behavior similar to the one of a 1D system in a

al of 0D systems.

ARTICLE IN PRESS

S. Bellucci, P. Onorato / Physica E 41 (2009) 1393–14021396

Density of states: The density of states (DOS) of a systemdescribes the number of states at each energy level that areavailable to be occupied. The calculation for DOS starts bycounting the Nð�Þ allowed states at a certain energy � that arecontained within ½�; �þ d��. If we assume V the volume of thesystem the DOS, g, is given by

gDð�Þ ¼1

V

dNð�Þd�

,

where D labels the dimensionality of the electron system. In a 1Dstructure, the values of � are quantized in two of the threedirections and g1ð�Þ / ��1=2 while Nð�Þ / �1=2.

In a 0D structure, the values of � are quantized in all directions.All the available states exist only at discrete energies describedand g0 can be represented by a sum of delta functions. Thus Nð�Þ isgiven by a step function.

In Fig. 2 we report the plot of Nð�Þ which clarifies the 1D–0Dproperties of the QR, which shows a behavior similar to the one ofa 1D system in a large scale approximation, while as is clear fromthe plot of the DOS that the peaks typical of 0D are present.

The effect of magnetic field on the DOS reveals that theelectrons are free to move along the ring and the DOS reflect theAB oscillations of the energy levels.

2.3. Spectrum calculation

2.3.1. Theoretical approach to the calculation of the electronic states

with an additional potential

Here we consider an additional potential, VIðjÞ, which cancorrespond to the barriers (wells) or to the variations of the QR’swidth. We now calculate matrix elements of the total Hamilto-nian,

HT¼ H þ VIðjÞ,

where H comes from Eq. (1) in the basis of eigenfunctions of Eq.(6) that correspond to quasi-1D radial subbands, labeled here bythe quantum number n. Next we assume n ¼ 0 and m ¼ m. Thediagonal matrix elements are given by

HTm;m ¼ hC0;mjHjC0;mi ¼ �0;m.

The off-diagonal matrix elements are given by

HTm;m0 ¼ hC0;mjVIðjÞjC0;m0 i. (7)

Thus we can diagonalize the matrix, HTij corresponding to the

whole Hamiltonian.

2.3.2. Rings with obstacles

The energy spectrum and AB effect in a two-dimensionalnanoring interrupted by Nb identical barriers are studied.

It is found that the magnetic flux can modify both the phaseand amplitude of wave functions also in the presence of thebarriers. The latter obstacles also affect the AB oscillations, asdiscussed in many previous papers [19,47] .

The model Hamiltonian for an electron in a 2D ring with Nb

identically barriers placed at the positions ji is obtained byintroducing an additional potential

VIðjÞ ¼ UXNb

i¼1

dðj�jiÞ.

Here we focus on a QR where Nb ¼ 5 barriers are placedaccording the plot in Fig. 3(top left) in jn ¼ np=3 with n 2 ½�2;2�.The results reported in Fig. 3 show clearly that the presence ofobstacles produces gaps, also very large, and a flattening of someenergy levels [47].

In order to emphasize this behavior we can introduce the DOS.The presence of strong barriers (see Fig. 4(top)) produce a stronglocalization and a corresponding large gap opening. The 1Dbehavior also of the Nð�Þ function is not more evident while thesingle level peaks in the DOS are grouped in pseudo-bands. Theresponse to magnetic field of the peaks reveals that some of thesepseudo-bands are insulating (i.e. because of the stronglocalization the AB oscillations disappear) while other bands aremetallic (the electrons behave as free carriers and the energy hassignificant AB oscillations).

In detail: in the case of small strength of the barriers(U ¼ 0:5_oR) we have a small gap between the lowest (flat) sixlevels and the other ones. In the case of small attractive barriers(U ¼ �0:5_oR) a large gap is opened between the lowest (flat) fivestates and the more oscillating levels while in the middle of thegap a quite flat one (the 6th) is present. When the barrier is large(U ¼ 2:5 and 4_oR) we have a large semiconducting pseudo-bandwith isolated localized states and a large gap separates theinsulating states from the metallic ones.

2.4. Persistent charge currents

In order to emphasize the role of the barriers and the producedflattening of the energy levels we can also analyze the presence inthe QR of a persistent charge current. Having calculated thesingle-particle electronic properties of the ring, we proceed toevaluate the persistent charge current. At zero temperature, it isgiven by [23]

I ¼ �@Egs

@F¼ �

Xi2occupied

@�i

@F, (8)

where Egs is the ground state energy, and Ei are the single particleenergy eigenvalues. Here i stands for a set of quantum numbersused to label corresponding eigenstates, including here the spinprojection in the local spin frame. The second equality in Eq. (8) isvalid only in the absence of electron–electron interactions, whichwe neglect here. The zero-temperature formula applies when thethermal energy kBT is smaller than the energy difference betweenthe last occupied state and the first unoccupied one. In thefollowing, we will always consider the number N of electrons inthe ring to be fixed, i.e. work in the canonical ensemble. This is therelevant situation for an isolated ring. When N is large enough,the persistent charge current in units of I0 ¼ _oaN=F0 has auniversal behavior independent of N.

The flux dependence of the persistent charge current isdistinctly different for the following cases, corresponding toorbital degeneracy of the QRs:

(i)

N ¼ 4N, (ii) N ¼ 4N þ 2,

(iii)

N ¼ 2N þ 1,

where N denotes a positive integer. Here (i) corresponds to half-filled shells, (ii) to filled shells and (iii) to one unpaired electron inthe open shell.

We start from the case of vanishing U corresponding to the QRwithout barriers. In case (i) where N ¼ 4N the jumps of thepersistent current at F=F0 ¼ n, with n being integer, are reported.In case (ii) where N ¼ 4N þ 2 the jumps of the persistent currentat F=F0 ¼ nþ 1=2 are shown. In case (iii) where N ¼ 2nþ 1 thejumps of the persistent current at F=F0 ¼ n=2 are reported.

This universal behavior of the persistent current, based on theone of the vanishing barrier case, is the asymptotic behavior of themetallic states. When the barriers are introduced in the I versus B

curve, the profile is strongly smoothed.

ARTICLE IN PRESS

-150 -100 -50 0 50 100 150x (nm)

-150

-100

-50

0

50

100

150y

(nm

)

0 0.5 1 1.5 2� (B) /�0

0

5

10

15

20

�R

)

Nb = 5;U = -0.5 �R

Nb = 5; U = -0.5 �R Nb = 5; U = -0.5 �R

0 0.5 1 1.5 2� (B)/�0

0

5

10

15

20

25

�R

)

0 0.5 1 1.5 2� (B) /�0

0

5

10

15

20

25

30

�R

)� (

� (

� (

Fig. 3. (Color online) (Top left) Scheme of a circular ring with a radius of 150 nm and width W0 ¼ 15 nm. Five barriers are placed according in jn ¼ np=3 with n 2 ½�2;2�.

Energy levels as a function of the magnetic flux for three different values (U ¼ �0:5;0:5; 2:5_oR) of the strength of the barriers.

S. Bellucci, P. Onorato / Physica E 41 (2009) 1393–1402 1397

In Fig. 5 we report the flux dependence of persistentcharge currents circling in ballistic quasi-one-dimensional ringswith five barriers and N electrons (N between 3 and 13). Theresults are reported for a set of values of the potential strength,repulsive with U between U ¼ 0 and 2:5 and attractive (bottomleft panel) with U ¼ �0:5 (in unit _oR). For a uniform ring wefound that the system shows odd–even parity in the persistentcurrent density.

In this subsection the non-interacting electrons are considered.As it is known the presence of Coulomb interaction (in detail theshort range component of the electron electron repulsion) canintroduce some interesting effects (as the fractional AB effect [48]or other correlation effects). Here we just want to focuson the smoothing of the IðBÞ curves due to the presence of thebarriers and we do not investigate the effects of correlation.However, the short range interaction can be relevant in smallradius QRs.

3. Ballistic transport through the ring

3.1. Theoretical treatment of the scattering

We start from the bandstructure in Eq. (5). It follows that forfixed values of the Fermi energy, �F , and of the band n, that weassume n ¼ 0 in the quasi-1D limit, there are two differenteigenstates C0;m, i.e. particles with fixed Fermi energy can gothrough the ring with two different wavenumbers mþ and m�,depending on the direction of motion (�):

m� ¼ �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�F � _o=2

_oR

s�

FF0¼ � ~m� F

F0, (9)

where we define

~m ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�F � _o=2

_oR

s¼

ffiffiffiffiffi~�F

p.

ARTICLE IN PRESS

0 10 20 30 40 50� (unit �R)

0 10 20 30 40 50� (unit �R)

0 10 20 30 40 50� (unit �R)

0 10 20 30 40 50� (unit �R)

� (unit �R) � (unit �R)

02468

101214

N (�

)

0

0.2

0.4

0.6

0.8

1

DO

S

02468

101214

N (�

)

0

0.2

0.4

0.6

0.8

1

DO

S

0 20 40 60 80 1000

2.55

7.510

12.515

17.5

N (�

)

0 20 40 60 80 1000

0.20.40.60.8

11.2

DO

SU = -0.5; �(B)/�0 = n, n±1/4, n+1/2 U = -0.5; �(B)/�0 = n, n±1/4, n+1/2

U = 0.5; �(B)/�0 = n, n±1/4, n+1/2 U = 0.5; �(B)/�0 = n, n±1/4, n+1/2

U = 2.5; �(B)/�0 = n, n±1/4, n+1/2 U = 2.5; �(B)/�0 = n, n±1/4, n+1/2

0.5

20 40 60 80 100

1

1.5

2

DO

S

U = 4 ; �(B)/�0 = n, n±1/4, n+1/2�R

s (unit �R)

Fig. 4. (Color online) DOS versus energy for different values of the strength of the barriers. In each plot the AB oscillations of the energy levels are reported for three

significant values of the magnetic field corresponding to FðBÞ=F0 ¼ n;nþ 1=4;nþ 1=2 with integer n. (Top) Density of states for quite strong tunnel barriers. A pseudo-band

of localized states is evident at �t10_oR . Thus a large gap (D��20_oR) separates a metallic pseudo-band. In the middle of the gap we can found a localized state. Above the

energy of the first metallic pseudo-band a second gap is opened (D��30_oR) interrupted by an insulating (localized) state. The same behavior is shown by the small

strength attractive impurities which also produce a nearly insulating pseudo-band with a significant gap interrupted by a localized state. For U ¼ 0:5_oR a gap is opened

between two metallic pseudo-bands. For U ¼ 2:5_oR a strong barrier regime analogous to the one reported in the top panel is realized. In the left panels we show the Nð�Þplot that reproduce a transition from the ideal 1D case to an evident staircase function obtained for U ¼ 2:5_oR.

S. Bellucci, P. Onorato / Physica E 41 (2009) 1393–14021398

ARTICLE IN PRESS

Fig. 5. Flux dependence of persistent charge currents circling in ballistic quasi-

one-dimensional rings with five barriers located in jn ¼ np=3 with n 2 ½�2;2�. The

results are reported for a set of values of the potential strength U. The total number

of electrons is set to N between 3 and 13 according the fame labels. In panel top left

U ¼ 0 corresponds to the ring without barriers. In panel top right U ¼ 0:5. In panel

bottom right U ¼ 2:5 while in panel bottom left the case of attractive potential is

presented. The persistent current is measured in units of I0 ¼ _oaN=F0.

S. Bellucci, P. Onorato / Physica E 41 (2009) 1393–1402 1399

Thus we introduce an additional potential in the Hamiltonianwhich describes Nb identically barriers placed at the positions ji.

We approach this scattering problem using the quantumwaveguide theory [49]. For the strictly one-dimensional ring thewave functions in different regions for each value of the spin aregiven below

cL ¼ eikx1 þ re�ikx1 ,

ci ¼ Aieimþj þ Bie

�im�j,

cR ¼ teikx2 ,

where we can assume the wavevector of the incident propagatingelectrons in the leads

k ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2m�

�F � _o=2

_2

s¼emR

.

As we shown in Fig. 6 i labels a sector between two nodes,where a node is given by a barrier or a lead–ring intersection.

Thus we use the Griffith boundary condition [50], which statesthat the wave function is continuous and that the current densityis conserved at each intersection. Thus we obtain the transmissioncoefficients.

3.2. Ballistic conductance

We can show the ballistic conductance as a function of themagnetic flux at a fixed Fermi energy. The periodicity in B is givenby the interferometric effect that reproduces the typical ABoscillations. In absence of the barriers the peaks positions of theballistic conductance (white line in Fig. 6, top right) reproducewell the single particle spectrum reported in Fig. 2.

The presence of barriers gives a kind of suppression of the ABoscillations in the ballistic conductance (compare the flattening inthe energy levels reported in Figs. 2 and 3 and the one observed inthe conductance in the B2� plane, reported in Fig. 6).

Moreover, by a comparison of Fig. 6(top right) and the otherpanels in the same figure it is clear that also in the ballistic regimethe usual ‘‘metallic’’ QR (for each value of the Fermi energy thereare values of the magnetic field for which the conductance has amaximum value, i.e. the electrons are totally transmitted)becomes ‘‘insulating’’ (there are values of the Fermi energy wherethe conductance is fully suppressed for all the values of themagnetic field, i.e. the electrons are totally reflected).

In Fig. 6 we show the ballistic conductance as a function ofmagnetic flux and the ‘‘kinetic energy’’ e�F for a three barriers ring.The white lines corresponding to the conductance peaks(G ¼ 2e2=h) have a dependence on the magnetic field quite similarto the one reported in Fig. 3 where the isolated ring spectra wereshown.

The gaps and a flattening of the lower energy levels can berevealed also by measuring the ballistic conductance.

4. Discussion

We have calculated the electronic states, the energy spectra,the persistent current density and the ballistic conductance atzero temperature, of quantum rings in a threading magnetic field.We focused our attention on the effects of the presence of someobstacles, as tunnel barriers or quantum wells (repulsive orattractive barriers).

A simple approach, based on the numerical diagonalization ofthe exact Hamiltonian, was used in order to obtain the energylevels in the quasi-1D limit. The quantum waveguide theory wasused for obtaining the ballistic conductance in the same limit.

ARTICLE IN PRESS

0 0.5 1 1.5 2

0

10

20

30

40

�F

0 0.5 1 1.5 20

5

10

15

20

�F

U = 2.5

0 0.5 1 1.5 2

0

10

20

30

40

�FU = 5

� (B)/�0

� (B)/�0 � (B)/�0

U = 0

Fig. 6. Density plot of the conductance as a function of magnetic flux and the ‘‘kinetic energy’’ e�F in unit _oR for a set of values of the potential strength U ¼ 0;2:5;5_oR.

Here we assume that in the ring Nb ¼ 3 identical barriers are present.

S. Bellucci, P. Onorato / Physica E 41 (2009) 1393–14021400

Regarding these quantum rings, we have obtained interestingresults:

(i)

we proved that the presence of attractive or repulsive barriersin the ring may produce quenching of the AB oscillations ofthe lower energy levels analogously to the smooth and tinyvariations in the width;

(ii)

we show that the discussed quenching of the AB oscillationsreflects the localization of the quantum states and we discusshow this behavior can be seen as a metallic-insulatingtransition induced by the barriers;

(iii)

we analyze how these properties of the energy levels can bealso seen in the persistent current.

These results are in agreement with the ones obtained in theprevious papers about similar systems. However, in previous

papers the role of the barriers strength and their positions was notaccurately investigated while the effects of the obstacle on theposition of the conductance peaks were just marginally discussed.In fact usually the barriers studied in the QR were assumed asimpurities or disorder [16].

In this paper we propose a device where the strength and theposition of the intentionally placed obstacles play a central role. Infact the symmetry of the system reproduces the one of theperiodic lattice of a crystal. The latter lattice produces somepseudo-bands with corresponding bandgaps, while by acting onthe strength and on the positions of the barriers it is possible tomodulate the energy gap, the flatness and the number of theinsulating states. We study the N ¼ 5 barriers because we want toproduce significant lattice effects, such as the presence of a visibleinsulating band, and we want also to avoid the even symmetry.Moreover, in the case of the open QR, we want at least one barrier

ARTICLE IN PRESS

S. Bellucci, P. Onorato / Physica E 41 (2009) 1393–1402 1401

in each arm, and no even symmetry. It follows that we need atleast three barriers. Thus, in order to produce a correspondencebetween the case of open and closed QR we introduce five barriersin the closed QR and three barriers and two intersections in theopen QR.

Moreover, one of the main results proposed in this paper is thecorrespondence between the conductance peaks in the ballisticregime of an open (two leads) QR and the energy spectrum of anisolated one. Thus the artificial lattice produces effects which canbe observed in the plot of the conductance as a function of theFermi energy and magnetic field. The insulating (low energy)states (where the localization plays a central role) can beidentified because the strong AB oscillations in the B2� planeare strongly suppressed while some pseudo-metallic states arepresent at higher energies.

The discussed phenomenology can have relevant application inthe engineering of novel devices. In fact by acting on the strengthand the positions of the barriers it is possible to modulate theenergy shift between the conductance peaks and to open anarbitrary gap between the peaks (i.e. it is possible to operate athigher temperature without any superposition of the peaks).

From the model to experiment: In this paper we discussed thecase of an ideal QR. This model corresponds to the experimentaldevice of a ring-shaped nanostructure which we supposepatterned in a two-dimensional electron gas. Using the typicaltechniques for the production of nanodevices in semiconductorheterostructures quantum rings with the considered parameters(radius, width, magnetic fields) are within the experimentallyfeasible range.

The tunnel barriers, corresponding to restrictions, localized inthe arms of the ring can be experimentally realized by quantumpoint contacts [51,52] (QPCs), constrictions defined in the plane ofa 2DEG, with a width of the order of the electron Fermiwavelength and a length much smaller than the elastic meanfree path. The quantum wells, corresponding to widenings(attractive obstacles), could be experimentally realized by smalllateral quantum dots (QDs). These artificial obstacles can berealized in split-gate devices, for example, which offer thepossibility to tune the effective width of the constriction andwidening, and thus the number of occupied 1D levels, via theapplied bias voltage. Moreover, by using the split-gate techniques,the experimentalists can be able to localize these obstacles.

Therefore, the experimental observation of the discussedbehaviors is a feasible task. In fact the experiments like the onereported in Ref. [10] show how the proposed geometries can berealized (e.g. the QPCs) so as the transport measurements in theCB regime, useful to a direct investigation of the energy spectra,can be realized.

Appendix A. The quantum ring and the transport regimes

The transport properties of a QR at very low temperature arestrongly dependent on the coupling between the external leadsand the ring.

A.1. Strong coupling limit: ballistic regime

In a Q1D device, if the width is smaller than the mean free pathand comparable to the Fermi wavelength, the electrons move in aperfectly coherent way throughout the device i.e. the current cantravel adiabatically in any state without scattering into any otherone, so we can apply the ballistic theory.

Here we consider the case where the 1D ring is symmetricallycoupled to two contact leads (Fig. 1, left) in order to study the

transport properties of the system subject to a constant, low biasvoltage (linear regime).

Moreover, in this simple approach we assume perfect coupling(strong coupling limit) between leads and ring (i.e. fullytransparent contacts), neglecting backscattering effects leadingto resonances.

A general model for near-equilibrium transport is due to theLandauer and Buttiker [53] contributions that are condensed inthe so-called Landauer formula. This formula expresses theconductance of a system at very low temperatures (the so-calledzero temperature conductance) and very small bias voltagescalculated directly from the energy spectrum by relating it to thenumber of forward propagating electron modes at a given Fermienergy

G � Gð�F Þ ¼e2

h

Xs;n

Xs0 ;n0jts;s0 ;n;n0 ð�F Þj

2, (A.1)

where Ts0sn0n ¼ jts;s0 ;n;n0 ð�F Þj

2 denotes the quantum probability oftransmission between incoming (n;s) and outgoing (n0;s0)asymptotic states defined on semi-infinite ballistic leads. Thelabels n;n0 and s;s0 refer to the corresponding mode and spinquantum numbers, respectively.

Next we limit our analysis to the case of just one modeinvolved: n ¼ n0 ¼ 0 (1D limit) and neglect effects of spin splitting,thus we can replace the formula in Eq. (A.1) with the simplifiedone,

G ¼2e2

hTð�F ;FðBÞÞ,

where the transmission amplitude depends on the Fermi energy,�F , and on the magnetic field.

From an experimental point of view there are results showingthat the approximations of our model for the ballistic transport(transport is ballistic and one dimensional, i.e. the finite width ofthe ring-wire was not taken into account) can give validdescriptions of actual physical systems under specific experi-mental situations. Currently, high mobility samples have becomeavailable such that at cryogenic temperatures transport is foundto be ballistic over tens of microns while phase coherence andspin coherence lengths have been found in the same range ofdistance.

A.2. Weak coupling limit: Coulomb blockade regime

In nanometric devices, when

(i)

the thermal energy kBT is below the energy for adding anadditional electron to the device (mN ¼ EðNÞ � EðN � 1Þ),where EðNÞ is the ground state energy for N electrons),

(ii)

the thermal energy kBT is above the ‘‘coupling energy’’ _Gl

corresponding to the tunneling to the leads (l ¼ S Source, l ¼ D

Drain),

low bias (small Vsd) transport is characterized by a current carriedby successive discrete charging and discharging of the device(next also Quantum Dot, QD) with a just one electron.

This phenomenon, known as single electron tunneling (SET orquantized charge transport), was observed in many experimentsin vertical QDs at very small temperature [54]. In this regime theground state energy determines strongly the conductance and theperiod in Coulomb Oscillations (COs). COs correspond to the peaksobserved in conductance as a function of gate potential (Vg) andare crudely described by the CB mechanism [55]: the N-thconductance peak occurs when [56] aeVgðNÞ ¼ mN where a ¼Cg=CS is the ratio of the gate capacitance to the total capacitance

ARTICLE IN PRESS

S. Bellucci, P. Onorato / Physica E 41 (2009) 1393–14021402

of the device. The peaks and their shape strongly depend on thetemperature as explained by the Beenakker formula for theresonant tunneling conductance [55]

GðVgÞ ¼e2

4kBT

GSGD

GSþ GD

!X1q¼1

Vg � mq

kBT sinhVg � mq

kBT

� � (A.2)

here m1; . . . ;mN represent the positions of the peaks.Coulomb blockade in QR: Here we want to resume some of the

properties of the conductance of a QR in CB regime and theirtheoretical explanation. We follow the discussion reported in Ref. [10].

We start the discussion from the energy spectrum of aninfinitely thin perfect ring of radius R enclosing F=F0 flux quanta:

Em;l ¼_2

2m%R2lþ

FF0

� �2

.

Here m% is the mass of the particle, l is the angular momentumquantum number. For a given angular momentum state theenergies as a function of magnetic field (or flux F=F0) lie on aparabola with its apex at F=F0 ¼ �l, as depicted in Fig. 1(top right).According to this simple picture a single CB peak should oscillate asa function of B along a zig-zag line (blue curve) with the AB periodDB. The h=e-periodic modulation of the CB peak amplitude can beexplained in terms of the single particle spectrum as we shown inFig. 1(top right). A conductance peak in CB regime follows a line ofconstant electron number (red–blue line in figure). The current issuccessively carried by states ðlÞ; ð�l� 1Þ; ðl� 1Þ; ð�l� 2Þ; ðl� 2Þ . . .when the magnetic field B is increased from zero, that is, a change instate occurs every half flux quantum.

Comparison with the measurement in Ref. [10] shows thatindeed some peaks move along a zig-zag line, but others showbarely any B-dependence, a behavior which can be explained interms of the presence of one (or more) obstacles along theelectron path or in terms of a not uniform width (compare the topand bottom panel of Fig. 1, right).

The effects of electron electron interaction in this system canbe neglected, as we can argue by a comparison between thetheoretical results and the experimental ones.

References

[1] T. Ando, Y. Arakawa, K. Furuya, S. Komiyama, H. Nakashima (Eds.), MesoscopicPhysics and Electronics, Springer, Berlin, 1998.

[2] S. Washburn, R.A. Webb, Rep. Prog. Phys. 55 (1992) 1311.[3] Y. Aharonov, D. Bohm, Phys. Rev. 115 (1959) 485.[4] Y. Aharonov, A. Casher, Phys. Rev. Lett. 53 (1984) 319.[5] A.F. Morpurgo, J.P. Heida, T.M. Klapwijk, B.J. van Wees, G. Borghs, Phys. Rev.

Lett. 80 (1998) 1050.[6] R.A. Webb, S. Washburn, C.P. Umbach, R.B. Laibowitz, Phys. Rev. Lett. 54

(1985) 2696.[7] G. Timp, et al., Phys. Rev. Lett. 58 (1987) 2814.[8] G.E. Philipp, J.A.M. Galeana, C. Cassou, P.D. Wang, C. Guasch, B. Vogele, M.C.

Holland, C.M. Sotomayor Torres, in: S.W. Pang, O.J. Glembocki, F.H. Pollak, F.Celli, C.M. Sotomayor Torres (Eds.), Diagnostic Techniques for SemiconductorMaterials Processing II, MRS Symposia Proceedings No. 406, MaterialsResearch Society, Pittsburgh, 1996, p. 307.

[9] S. Pedersen, A.E. Hansen, A. Kristensen, C.B. Sorensen, P.E. Lindelof, Phys. Rev.B 61 (2000) 5457.

[10] A. Fuhrer, S. Luscher, T. Ihn, T. Heinzel, K. Ensslin, W. Wegscheider, M. Bichler,Nature 413 (2001) 822.

[11] T. Raz, D. Ritter, G. Bahir, Appl. Phys. Lett. 82 (2003) 1706.[12] T. Mano, et al., Nano Lett. 5 (2005) 425.[13] T. Kuroda, T. Mano, T. Ochiai, S. Sanguinetti, K. Sakoda, G. Kido, N. Koguchi,

Phys. Rev. B 72 (2005) 205301.[14] N.A.J.M. Kleemans, et al., Phys. Rev. Lett. 99 (2007) 146808.[15] Z. Barticevic, M. Pacheco, A. Latge, Phys. Rev. B 62 (2000) 6963.[16] J. Splettstoesser, M. Governale, U. Zulicke, Phys. Rev. B 68 (2003) 165341.[17] J. Simonin, C.R. Proetto, Z. Barticevic, G. Fuster, Phys. Rev. B 70 (2004) 205305.[18] A. Bruno-Alfonso, T.A. de Oliveira, Braz. J. Phys. 36 (2006) 434.[19] J. Planelles, J.I. Climente, F. Rajadell, Physica E 33 (2006) 370.[20] M. Aichinger, S.A. Chin, E. Krotscheck, E. Rasanen, Phys. Rev. B 73 (2006)

195310.[21] V.M. Fomin, V.N. Gladilin, S.N. Klimin, J.T. Devreese, N.A.J.M. Kleemans,

P.M. Koenraad, Phys. Rev. B 76 (2007) 235320.[22] M. Amado, R.P.A. Lima, C. Gonzalez-Santander, F. Domınguez-Adame, Phys.

Rev. B 76 (2007) 073312.[23] M. Buttiker, Y. Imry, R. Landauer, Phys. Lett. 96A (1983) 365.[24] M. Buttiker, Y. Imry, M.Ya. Azbel, Phys. Rev. A 30 (1984) 1982.[25] H.F. Cheung, Y. Gefen, E.K. Riedel, W.H. Shih, Phys. Rev. B 37 (1988) 6050.[26] D. Loss, P. Goldbart, Phys. Rev. B 43 (1991) 13762.[27] H. Bouchiat, G. Montambaux, J. Phys. France 50 (1989) 2695.[28] G. Montambaux, H. Bouchiat, D. Sigeti, R. Friesner, Phys. Rev. B 42 (1990)

7647.[29] L.P. Levy, G. Dolan, J. Dunsmuir, H. Bouchiat, Phys. Rev. Lett. 64 (1990) 2074.[30] V. Chandrasekhar, R.A. Webb, M.J. Brady, M.B. Ketchen, W.J. Gallagher,

A. Kleinsasser, Phys. Rev. Lett. 67 (1991) 3578.[31] D. Mailly, C. Chapelier, A. Benoit, Phys. Rev. Lett. 70 (1993) 2020.[32] N. Byers, C.N. Yang, Phys. Rev. Lett. 7 (1961) 46.[33] A. Lorke, et al., Phys. Rev. Lett. 84 (2000) 2223.[34] R.J. Warburton, et al., Nature 405 (2000) 926.[35] L.P. Kouwenhoven, et al., in: L.P. Kouwenhoven, G. Schon, L.L. Sohn (Eds.),

Electron Transport in Quantum Dots. Nato ASI Conference Proceedings,Kluwer, Dordrecht, 1997, pp. 105–214.

[36] S. Tarucha, D.G. Austing, T. Honda, R.J. van der Haage, L.P. Kouwenhoven, Phys.Rev. Lett. 77 (1996) 3613.

[37] H. Hu, J.-L. Zhu, D.J. Li, J.J. Xiong, Phys. Rev. B 63 (2001) 195307;T.V. Shahbazyan, I.E. Perakis, M.E. Raikh, Phys. Rev. B 64 (2001) 115317.

[38] L.I. Magarill, D.A. Romanov, A.V. Chaplik, JETP 83 (1996) 361.[39] D. Berman, O. Entin-Wohlman, M.Y. Azbel, Phys. Rev. B 42 (1990) 9299.[40] L.A. Lavenere-Wanderley, A. Bruno-Alfonso, A. Latge, J. Phys. Condens. Matter

14 (2002) 259.[41] J. Planelles, F. Rajadell, J.I. Climente, Nanotechnology 18 (2007) 375402.[42] Y.V. Pershin, C. Piermarocchi, Phys. Rev. B 72 (2005) 195340.[43] W.-C. Tan, J.C. Inkson, Phys. Rev. B 60 (1999) 5626.[44] F. Carillo, G. Biasiol, D. Frustaglia, F. Giazotto, L. Sorba, F. Beltram, Physica E 32

(2006) 53.[45] A. Bruno-Alfonso, A. Latge, Phys. Rev. B 77 (2008) 205303.[46] F.E. Meijer, A.F. Morpurgo, T.M. Klapwijk, Phys. Rev. B 66 (2002) 033107.[47] Jia-Lin Zhu, Xiquan Yu, Zhensheng Dai, Xiao Hu, Phys. Rev. B 67 (2003)

075404.[48] K. Niemela, P. Pietilainen, P. Hyvonen, T. Chakraborty, Europhys. Lett. 36

(1996) 533.[49] J.B. Xia, Phys. Rev. B 45 (1992) 3593;

P.S. Deo, A.M. Jayannavar, Phys. Rev. B 50 (1994) 11629.[50] S. Griffith, Trans. Faraday Soc. 49 (1953) 345;

S. Griffith, Trans. Faraday Soc. 49 (1953) 650.[51] T.J. Thornton, Rep. Prog. Phys. 58 (1995) 311.[52] N.T. Bagraev, et al., Physica B 378–380 (2006) 894.[53] R. Landauer, Philos. Mag. 21 (1970) 863;

M. Buttiker, Y. Imry, R. Landauer, S. Pinhas, Phys. Rev. B 31 (1985) 6207.[54] S. Tarucha, D.G. Austing, T. Honda, R.J. van der Hage, L.P. Kouwenhoven, Phys.

Rev. Lett. 77 (1996) 3613.[55] C. Beenakker, Phys. Rev. B 44 (1991) 1646.[56] O. Klein, et al., in: Kramer (Ed.), Quantum Transport in Semiconductor

Submicron Structures, 1996.