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doi:10.1016/j.ph

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Physica B 403 (2008) 2015–2020

www.elsevier.com/locate/physb

Persistent current in finite-width ring with surface disorder

H.B. Chena, J.W. Dinga,b,�

aDepartment of Physics, Xiangtan University, Xiangtan 411105, Hunan, ChinabNational Laboratory of Solid State Microstructures, Nanjing University, Nanjing 210093, China

Received 26 March 2007; received in revised form 2 November 2007; accepted 17 November 2007

Abstract

We explore the surface disorder effect on the persistent current in a finite-width ring. In the strong disorder regime, the persistent

current increases with surface disorder strength, while it decreases in the weak disorder regime. The result is at variance with the

observation in bulk-disordered ring. Also, it is shown that the disorder-induced changes in the persistent current strongly depend on both

the ring width and radius, which show up a singular quantum size effect.

r 2007 Elsevier B.V. All rights reserved.

PACS: 73.23.Ra; 72.15.Rn; 73.20.�r

Keywords: Persistent current; Tight-binding; Finite-width ring; Surface disorder

1. Introduction

In a pioneering work, Buttiker et al. [1] predicted thateven in the presence of disorder, an isolated one-dimensional(1D) metallic ring threaded by the magnetic flux F can carryan equilibrium persistent current with periodicity F0 ¼ h/e,the flux quantum. The existence of persistent currents hadbeen confirmed by the experimental observations in single/ensemble of isolated mesoscopic ring [2–7]. Except for thecase of the nearly ballistic GaAs–AlGaAs ring [4], all themeasured currents are in general one or two orders ofmagnitude larger than those predicted from the theory[8–13]. The diamagnetic response of the ensemble-averagedpersistent current in the vicinity of the zero magnetic fieldsalso contrasts with most predictions [10,11]. This means thatthe experimental results are not well understood theoreti-cally so far. The persistent currents in mesoscopic rings arethe subject of intensive research [14–17].

Metals are intrinsically disordered which tends todecrease the persistent currents in mesoscopic rings. Toexplore the disorder effect, many of the theoretical studies

e front matter r 2007 Elsevier B.V. All rights reserved.

ysb.2007.11.010

ng author at: Department of Physics, Xiangtan Univer-

11105, Hunan, China.

ss: jwding@xtu.edu.cn (J.W. Ding).

took the limit from 2D to 1D ring, for which different 1Dring Hamiltonians were used. For example, Kim et al. [18]investigated the behavior of persistent currents of 1Dnormal-metal rings with the impurity potential. Thediamagnetic response near the zero magnetic fields wasattributed to multiple backward scattering off the impu-rities. Also, Maiti et al. [19] built a simple 1D tight-bindingHamiltonian with diagonal disorder and long-range hop-ping integrals to account for the observed behavior ofpersistent currents in single-isolated-diffusive normal metalrings of mesoscopic size. In the experiments, however, themean width of the sample ring was usually comparable toits mean radius. In such finite-width rings, it was foundthat the typical current Ityp increases with the channelnumber M by I typ�

ffiffiffiffiffiffiMp

, while the disorder-averagedcurrent is independent of M in the ballistic regime [9],only including even Fourier components. On the otherhand, confinement and surface roughness effects on themagnitude of the persistent current were analyzed in thecase of the ballistic 2D metallic rings [20], which maycontribute coherently to the persistent current. It wasshown that the typical current increases linearly with thechannel number M. These means that 1D description isoversimplified to describe quantitatively the finite-widthrings in experiments.

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Fig. 1. Schematic illustration of the surface disordered ring with M ¼ 4

and N ¼ 16. The open circles represent the ordered sites in the core with

ei ¼ 0, and the solid circles the disordered ones at the surface region with

randomly distributed site-energies.

H.B. Chen, J.W. Ding / Physica B 403 (2008) 2015–20202016

In the presence of disorder, some works had been doneon the finite-width 2D ring, focusing on the influenceof bulk disorder on persistent currents [21]. The bulkdisorder was usually considered to be inside the materialthrough which the wave travels. Some general charac-teristics of persistent current had been obtained in suchbulk-disordered systems. It was shown that the typicalcurrent in the metallic regime was modified by a cor-rective diffusive factor and in the localized regime itdecreased exponentially with the disorder strength. Due torecent advances in nanotechnology, interestingly, it ispossible to fabricate mesoscopic devices in which carriersare mainly scattered by the boundaries and not byimpurities or defects located inside them [22]. Basedon such an actual structure, recently, a shell-dopednanowire model was proposed, from which a noveltransport behavior was obtained, that is, the larger thedisorder, the weaker the localization [23]. For the surfaceroughness or defects, similarly, there may exist a largedisorder at the surface of a finite-width ring. Especially,when the system size was shrunk down to the nano-meter scale, the surface-to-volume ratio becomes larger,leading to very strong quantum size effects and surfaceeffects.

In a practical implementation of surface roughness,Cuevas et al. [24] had studied the quantum chaoticdynamics by building a new model of quantum chaoticbilliard. The essential feature of this model is the inclusionof diagonal disorder at the surface of the system. Theobtained energy spectrum statistics shows a complexbehavior, very different from that previously reported inthe usual chaotic billiard model. Obviously, the similarcomplex energy spectrum may be expected in a surfacedisordered ring, indicating some new features in persistentcurrent. How about the influence of the surface disorder onthe persistent current in finite-wide rings?

In this paper, taking into account the surface rough-ness or defects, we build a surface disordered 2D ringmodel. The effect of surface disorder on persistentcurrent in such 2D ring is explored within the tight-bindingframe. It is found that with the increasing disorderstrength, the typical current increases in the strong dis-order regime, while it decreases in the weak disorderregime. Also, the disorder induced changes in the persi-stent current depending strongly both on the ringwidth and radius, which shows up a singular quantum sizeeffect.

2. Model and formulae

We consider a 2D mesoscopic ring enclosing a magneticflux line. The sample ring can be modeled by M,concentrically connected tight-binding ring chains with N

sites each ring chain, as shown in Fig. 1. The maximumnumber of the open channels in a structure consisting of M

ring chains is equal to M. Taking a single atomic level perlattice site, the tight-binding Hamiltonian by considering

non-interacting electrons is given by

H ¼X

i

�icy

i ci þX

i;j

V ði; jÞcyi cj (1)

with on-site energies ei, where i labels the coordinatesof the sites in the lattice. The hopping integrals V(i, j) arerestricted to the nearest neighbors of a site. Assumingthat the vector potential A has only an azimuthal com-ponent, we take V ði; jÞ ¼ t exp i

R j

iAdl

� �; in units of the

quantum flux F0, where l is a vector that points fromthe site i to any of its nearest neighbors. To model thesurface disorder, the on-site energies ei in the surfaceregion are taken to be randomly distributed withininterval [�W, W], W describing the disorder strength,whereas the other sites have a constant energy equal tozero.For the finite-width ring, we neglect the self-inductance

effect on the persistent current in the system. At zerotemperature, the total current can be calculated by

I ¼ �qE=qF ¼ �X

n

qEn=qF (2)

with E the total energy of the system. Here n labels thecorresponding eigenlevels. The second equality in Eq. (2) isvalid only in the absence of electron–electron interactions,which are neglected here. The current is a periodic functionof F with fundamental period F0. Usually, one is interestedin the typical current [8,20], which is defined as the squareroot of the disorder (W) and flux average of the square ofthe persistent current

I typ ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffihI2iF;W

q. (3)

To obtain good statistics, the typical currents are averagedover many realizations of the disorder configurations. Inour calculations, the number of the averaged configura-tions varies from 100 to 200, depending on the size of thesystems.

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-0.2

-0.1

-0.5 0.0 0.5-0.2

-0.1

0.0

-0.5 0.0 0.5

Ene

rgy/

t

0.0

Φ/Φ0 Φ/Φ0

a b

c d

Fig. 2. Typical energy spectra in half-filling region for the M ¼ 4 and N ¼ 100 sample ring as a function of flux at varying surface disorder strength:

(a) W ¼ 1; (b) W ¼ 4; (c) W ¼ 10; and (d) W ¼ 16.

0 10 12 14 16

0.000

0.002

0.004

0.006

I typ

[t/Φ

0]

bulk disorder, 0.5 fillingsurface disorder, 0.5 fillingsurface disorder, 0.4 fillingsurface disorder, 0.6 filling

M=4, N=100

2 4 6 8

disorder strength, W/t

Fig. 3. Typical current as a function of disorder strength W at 410, 510and 6

10

filling in a finite-width ring with surface disorder and bulk disorder.

H.B. Chen, J.W. Ding / Physica B 403 (2008) 2015–2020 2017

3. Results and discussion

Now let us calculate the energy spectra and persistentcurrents in the surface disordered sample rings. In ourcalculations, the site energies ei, disorder strength W, andenergy E are given in units of the model parameter t, andthus the persistent currents in units of t/F0.

In Fig. 2, we show the typical energy spectra in half-filling region for the M ¼ 4 and N ¼ 100 sample ring as afunction of flux at varying surface disorder strength. Thefour values of W are chosen by a steady increase to beW ¼ 1, 4, 10, and 16. For a given W, one disorderconfiguration can be obtained randomly. But, the results ofenergy spectra are qualitatively unchanged. In the presenceof surface disorder, the disorder-induced energy gaps areobserved at the crossing points in energy spectrum, asshown in Fig. 2. The result is similar to that of the bulkdisordered system [9]. Interestingly, an extreme value Wc ofdisorder strength is obtained from the changes in energyspectra with increasing disorder. For weaker disorder(WoWcE4), the energy levels become more isolated withW increasing and thus the energy curves change moresmoothly with flux [shown in Fig. 2 (a) W ¼ 1 and (b)W ¼ 4]. At W ¼Wc, especially, there exist even large gapsin the energy spectra and the slopes of the energy curvestend to vanish. Obviously, a small slope rate will lead toless contribution to the current in carried by En. This meansthat the total current should dramatically decrease with W

increasing, just as previous predictions. In the regime ofstronger disorder (W4Wc), however, some substantialchanges appear in the energy spectra of the surfacedisordered ring. From Fig. 2 (c) W ¼ 10 and (d) W ¼ 16,

it is seen that the slopes of the energy curves recover andincrease with W increasing. This indicates a rise ofpersistent current even in the case of strong disorder,different from the results predicted from the bulk-disordered systems.To understand the surface disorder effect on persistent

current, Fig. 3 shows the W dependence of the typicalcurrent for the same sample ring in Fig. 2. For acomparison, the results of the bulk-disordered ring withthe same size are also presented. In the case of bulkdisorder, as expected, Ityp decreases continuously with thedisorder strength W and diminishes to zero at WE3(WoWc). This can be easily understood from the theory of

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0.000

0.004

0.008

0 8 10 12 14 16

0.000

I typ

[t/Φ

0]

N = 100M = 3

M = 4

M = 5I ty

p [t

/Φ0]

M = 4

N = 100

N = 128

N = 1500.004

0.008

2 4 6

disorder strength, W/t

Fig. 4. Typical current as a function of disorder strength W for various size rings at half-filling: (a) N ¼ 100, and M ¼ 3, 4, and 5; (b) M ¼ 4, and N ¼ 100,

128, and 150.

H.B. Chen, J.W. Ding / Physica B 403 (2008) 2015–20202018

the Anderson localization [25]. The larger the disorder is,the stronger the localization. Therefore, the strong disordermay result in carrier localization and thus lead to adisappearing current in the bulk disordered thin ring. Forsurface disorder ring, it is seen from Fig. 3 that the overallcurrents exceed that in bulk-disordered ring. Interestingly,it is found that Ityp does not vary monotonically with thedisorder strength W, different from the bulk-disorderedsystem. At the half-filling, the extreme value Wc of disorderstrength is obtained to be WcE4, consistent with that inthe energy spectra. In the stronger disorder regime(W4Wc), the typical current increases with increasing W,while it decreases in the weaker disorder regime (WoWc).The behavior is very similar to that in a shell-dopednanowire [23], where a localization/quasi-delocalizationtransition was observed at a critical disorder strength.Below and above the half-filling, we also calculate thetypical currents at 4

10and 6

10filling, as shown in Fig. 3. The

similar anomalous behavior has also been obtained. Forother cases (far from the half-filling), the enhancement ofthe current amplitude may be small, while the typicalcurrent still increases at very high disorder. This may bedue to the fact that far from the half-filling, the quasi-idealstates existing at about the spectrum center [23,24] have lesscontribution to the current.

As for the extreme value Wc, it shows the onset of anatypical behavior. To the best of our knowledge thisanomalous behavior has not been pointed out in previousdiscussions of persistent current. This behavior can beunderstood by the following consideration. The surface

disordered system can be regarded as a coupled systemcomprising of two subsystems, the ordered core and thedisordered surface. The Hamiltonian equation (1) can bedivided into two modified sub-Hamiltonian. Without thecoupling between the two subsystems, the states in theinner core are extended, while those in the surface regionare localized. In the weaker disorder regime, the electronstates in the central regime are scattered by the surfacedisorder, and thus tend to be localized so that the currentdecreases with increasing W. In the stronger disorderregime, however, the influence of the surface disorder onthe inner perfect core becomes weak. Actually, the surfacedisorder scattering effect on the ordered core is representedby the modified term in the sub-Hamiltonian of the orderedcore, just as that derived in a shell-doped nanowire [23].The modified term is inversely proportional to W and tendsto zero for infinite disorder. Therefore, the typical currentincreases with W increasing beyond Wc. In the limit ofinfinite disorder, one may expect a rather simple scenario inwhich bulk and surface are decoupled. Consequently, theordered states would lie on inner sites whereas localizedstates would be located at surface sites. Then the persistentcurrent comes only from the perfect cluster extended states.This is a trivial limit and the most interesting situations areof course expected for finite W values.To explore the quantum size effect on persistent current,

in Fig. 4 we show typical currents as a function of W forthree different ring widths and radii. At a given N (radius),the ratio of the surface to the central region will decreasewith M (width) increasing, and the channels in the latter

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contribute more to the persistent current than in theformer. As a result, the typical current increases withincreasing M, as shown in Fig. 4(a). However, it is noticedthat the changes in typical current with M depend stronglyon disorder strength. From Fig. 4(a), a little variation isobtained in weaker disorder regime, while large changes inthe stronger disorder regime. For M ¼ 3, 4 and 5, forinstance, the interval of the amplitude of typical currentsapproximates to 0.001 at W ¼ 2, while it tends to 0.002 atW ¼ 10. At a given M (width), also it is seen from Fig. 4(b)that the typical current decreases with increasing N (radius)within the overall disorder range. The amplitude of thechanges also depends on the disorder strength, increasingwith W. Even in the weaker disorder regime, a distinct de-crease of Ityp is also observed with increasing N. Therefore,the quantum size effects play an important role in deter-mining the persistent current in a surface disordered 2Dring, which should be considered quantitatively to describethe experiments performed in finite-width rings.

In the present model, we neglect the effect of electro-n–electron interactions on the persistent current. Actually,the role of the interactions in bulk disorder systems is stillunclear, which is an open subject. Some results indicatedthat both long-range [26,27] and short-range [28] electro-n–electron interactions suppress the persistent current.On the other hand, it was indicated that the persistentcurrent was enhanced by the electron–electron interactions[15,29,30]. Present studies for noninteracting electrons canbe extended to interacting electron system, which will helpto understand the different effects of surface disorder andelectron–electron interactions. In addition, a new type ofcarbon structure, multi-walled carbon nanotorus, had beenrecently fabricated by different techniques [31,32], whichindicates the existence and possibility of a real system ofsurface disordered ring device in 3D case. Such a device canbe devised by the surface adsorption and/or doping into itsouter wall of the nanotorus, while the inner wall is keptclean. Despite the simplicity of our model, it allows thestudy of several situations of physical interest including thecase of 2D graphite ring and 3D carbon nanotorus withsurface disorder. Furthermore, it would be interesting andpossibly relevant for real systems to know what happens inthe case of a smooth decrease of the impurity density withthe distance from the surface, which is under our furtherconsideration.

4. Summary

In summary, tight-binding Hamiltonian model of surfacedisordered ring is proposed, in which the diagonal disorderis considered to exist only on the surface region of a finite-width ring. The surface disorder effect on the energyspectra and thus the persistent current are explored in suchfinite-width rings. Our results indicate that the typicalcurrent shows a complex behavior with the strength of thesurface disorder. The typical current decreases in the weakdisorder regime, a minimum existing at intermediate

disorder, while it increases in the strong disorder regime.This manner is contrast to the case of bulk disorder, inwhich the typical current decreases monotonously andtends to become zero with the increasing disorder strength.The anomalous scenario is a consequence of the interplaybetween the ordered and disordered subsystem, whichstrongly depends on the surface disorder strength. Also, itis shown that the variations in the typical current with thedisorder strength strongly depend on the ring width andradius, which show up a singular quantum size effect.

Acknowledgments

This work was supported by the National NaturalScience Foundation of China (no. 10674113), Program forNew Century Excellent Talents in University (NCET-06-0707), Foundation for the Author of National ExcellentDoctoral Dissertation of China (Grant no. 200726), HunanProvincial Natural Science Foundation of China (no.06JJ50006), and partially by the Scientific Research Fundof Hunan Provincial Education Department (no. 06A071).

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