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Persistent current in finite-width ring with surface disorder
ARTICLE IN PRESS
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, a
Department of Physics, Xiangtan University, Xiangtan 411105, Hunan, China National Laboratory of Solid State Microstructures, Nanjing University, Nanjing 210093, China
b
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, Bu¨ttiker et al. [1] predicted that even in the presence of disorder, an isolated one-dimensional (1D) metallic ring threaded by the magnetic flux F can carry an equilibrium persistent current with periodicity F0 ¼ h/e, the flux quantum. The existence of persistent currents had been confirmed by the experimental observations in single/ ensemble of isolated mesoscopic ring [2–7]. Except for the case of the nearly ballistic GaAs–AlGaAs ring [4], all the measured currents are in general one or two orders of magnitude larger than those predicted from the theory [8–13]. The diamagnetic response of the ensemble-averaged persistent current in the vicinity of the zero magnetic fields also contrasts with most predictions [10,11]. This means that the experimental results are not well understood theoretically so far. The persistent currents in mesoscopic rings are the subject of intensive research [14–17]. Metals are intrinsically disordered which tends to decrease the persistent currents in mesoscopic rings. To explore the disorder effect, many of the theoretical studies Corresponding author at: Department of Physics, Xiangtan University, Xiangtan 411105, Hunan, China. E-mail address: [email protected] (J.W. Ding).
0921-4526/$ - see front matter r 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2007.11.010
took the limit from 2D to 1D ring, for which different 1D ring Hamiltonians were used. For example, Kim et al. [18] investigated the behavior of persistent currents of 1D normal-metal rings with the impurity potential. The diamagnetic response near the zero magnetic fields was attributed to multiple backward scattering off the impurities. Also, Maiti et al. [19] built a simple 1D tight-binding Hamiltonian with diagonal disorder and long-range hopping integrals to account for the observed behavior of persistent currents in single-isolated-diffusive normal metal rings of mesoscopic size. In the experiments, however, the mean width of the sample ring was usually comparable to its mean radius. In such finite-width rings, it was found that the typical current pffiffiffiffiffiIffityp increases with the channel number M by I typ M , while the disorder-averaged current is independent of M in the ballistic regime [9], only including even Fourier components. On the other hand, confinement and surface roughness effects on the magnitude of the persistent current were analyzed in the case of the ballistic 2D metallic rings [20], which may contribute coherently to the persistent current. It was shown that the typical current increases linearly with the channel number M. These means that 1D description is oversimplified to describe quantitatively the finite-width rings in experiments.
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H.B. Chen, J.W. Ding / Physica B 403 (2008) 2015–2020
In the presence of disorder, some works had been done on the finite-width 2D ring, focusing on the influence of bulk disorder on persistent currents [21]. The bulk disorder was usually considered to be inside the material through which the wave travels. Some general characteristics of persistent current had been obtained in such bulk-disordered systems. It was shown that the typical current in the metallic regime was modified by a corrective diffusive factor and in the localized regime it decreased exponentially with the disorder strength. Due to recent advances in nanotechnology, interestingly, it is possible to fabricate mesoscopic devices in which carriers are mainly scattered by the boundaries and not by impurities or defects located inside them [22]. Based on such an actual structure, recently, a shell-doped nanowire model was proposed, from which a novel transport behavior was obtained, that is, the larger the disorder, the weaker the localization [23]. For the surface roughness or defects, similarly, there may exist a large disorder at the surface of a finite-width ring. Especially, when the system size was shrunk down to the nanometer scale, the surface-to-volume ratio becomes larger, leading to very strong quantum size effects and surface effects. In a practical implementation of surface roughness, Cuevas et al. [24] had studied the quantum chaotic dynamics by building a new model of quantum chaotic billiard. The essential feature of this model is the inclusion of diagonal disorder at the surface of the system. The obtained energy spectrum statistics shows a complex behavior, very different from that previously reported in the usual chaotic billiard model. Obviously, the similar complex energy spectrum may be expected in a surface disordered ring, indicating some new features in persistent current. How about the influence of the surface disorder on the persistent current in finite-wide rings? In this paper, taking into account the surface roughness or defects, we build a surface disordered 2D ring model. The effect of surface disorder on persistent current in such 2D ring is explored within the tight-binding frame. It is found that with the increasing disorder strength, the typical current increases in the strong disorder regime, while it decreases in the weak disorder regime. Also, the disorder induced changes in the persistent current depending strongly both on the ring width and radius, which shows up a singular quantum size effect. 2. Model and formulae We consider a 2D mesoscopic ring enclosing a magnetic flux 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 maximum number of the open channels in a structure consisting of M ring chains is equal to M. Taking a single atomic level per lattice site, the tight-binding Hamiltonian by considering
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.
non-interacting electrons is given by H¼
X
i cyi ci þ
i
X
V ði; jÞcyi cj
(1)
i;j
with on-site energies ei, where i labels the coordinates of the sites in the lattice. The hopping integrals V(i, j) are restricted to the nearest neighbors of a site. Assuming that the vector potential A has only R j an azimuthal component, we take V ði; jÞ ¼ t exp i i A dl ; in units of the quantum flux F0, where l is a vector that points from the site i to any of its nearest neighbors. To model the surface disorder, the on-site energies ei in the surface region are taken to be randomly distributed within interval [W, W], W describing the disorder strength, whereas the other sites have a constant energy equal to zero. For the finite-width ring, we neglect the self-inductance effect on the persistent current in the system. At zero temperature, the total current can be calculated by X I ¼ qE=qF ¼ qE n =qF (2) n
with E the total energy of the system. Here n labels the corresponding eigenlevels. The second equality in Eq. (2) is valid only in the absence of electron–electron interactions, which are neglected here. The current is a periodic function of F with fundamental period F0. Usually, one is interested in the typical current [8,20], which is defined as the square root of the disorder (W) and flux average of the square of the persistent current I typ ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi hI 2 iF;W .
(3)
To obtain good statistics, the typical currents are averaged over many realizations of the disorder configurations. In our calculations, the number of the averaged configurations varies from 100 to 200, depending on the size of the systems.
ARTICLE IN PRESS H.B. Chen, J.W. Ding / Physica B 403 (2008) 2015–2020
0.0
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-0.1
-0.2 -0.5
0.0
0.5 -0.5
0.0
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Φ/Φ0
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.
3. Results and discussion Now let us calculate the energy spectra and persistent currents in the surface disordered sample rings. In our calculations, the site energies ei, disorder strength W, and energy E are given in units of the model parameter t, and thus the persistent currents in units of t/F0. In Fig. 2, we show the typical energy spectra in halffilling region for the M ¼ 4 and N ¼ 100 sample ring as a function of flux at varying surface disorder strength. The four values of W are chosen by a steady increase to be W ¼ 1, 4, 10, and 16. For a given W, one disorder configuration can be obtained randomly. But, the results of energy spectra are qualitatively unchanged. In the presence of surface disorder, the disorder-induced energy gaps are observed at the crossing points in energy spectrum, as shown in Fig. 2. The result is similar to that of the bulk disordered system [9]. Interestingly, an extreme value Wc of disorder strength is obtained from the changes in energy spectra with increasing disorder. For weaker disorder (WoWcE4), the energy levels become more isolated with W increasing and thus the energy curves change more smoothly with flux [shown in Fig. 2 (a) W ¼ 1 and (b) W ¼ 4]. At W ¼ Wc, especially, there exist even large gaps in the energy spectra and the slopes of the energy curves tend to vanish. Obviously, a small slope rate will lead to less contribution to the current in carried by En. This means that the total current should dramatically decrease with W increasing, just as previous predictions. In the regime of stronger disorder (W4Wc), however, some substantial changes appear in the energy spectra of the surface disordered ring. From Fig. 2 (c) W ¼ 10 and (d) W ¼ 16,
Ityp [t/Φ0]
0.006
M=4, N=100 bulk disorder, 0.5 filling surface disorder, 0.5 filling surface disorder, 0.4 filling surface disorder, 0.6 filling
0.004
0.002
0.000 0
2
4
6
8
10
12
14
16
disorder strength, W/t 4 5 6 Fig. 3. Typical current as a function of disorder strength W at 10 , 10 and 10 filling in a finite-width ring with surface disorder and bulk disorder.
it is seen that the slopes of the energy curves recover and increase with W increasing. This indicates a rise of persistent current even in the case of strong disorder, different from the results predicted from the bulkdisordered systems. To understand the surface disorder effect on persistent current, Fig. 3 shows the W dependence of the typical current for the same sample ring in Fig. 2. For a comparison, the results of the bulk-disordered ring with the same size are also presented. In the case of bulk disorder, as expected, Ityp decreases continuously with the disorder strength W and diminishes to zero at WE3 (WoWc). This can be easily understood from the theory of
ARTICLE IN PRESS H.B. Chen, J.W. Ding / Physica B 403 (2008) 2015–2020
2018
Ityp [t/Φ0]
0.008 N = 100 M=3 M=4 M=5
0.004
0.000 0.008
Ityp [t/Φ0]
M=4 N = 100 N = 128 N = 150
0.004
0.000 0
2
4
6
8
10
12
14
16
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.
the Anderson localization [25]. The larger the disorder is, the stronger the localization. Therefore, the strong disorder may result in carrier localization and thus lead to a disappearing current in the bulk disordered thin ring. For surface disorder ring, it is seen from Fig. 3 that the overall currents exceed that in bulk-disordered ring. Interestingly, it is found that Ityp does not vary monotonically with the disorder strength W, different from the bulk-disordered system. At the half-filling, the extreme value Wc of disorder strength is obtained to be WcE4, consistent with that in the 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-doped nanowire [23], where a localization/quasi-delocalization transition was observed at a critical disorder strength. Below and above the half-filling, we also calculate the 4 6 typical currents at 10 and 10 filling, as shown in Fig. 3. The similar anomalous behavior has also been obtained. For other cases (far from the half-filling), the enhancement of the current amplitude may be small, while the typical current still increases at very high disorder. This may be due to the fact that far from the half-filling, the quasi-ideal states existing at about the spectrum center [23,24] have less contribution to the current. As for the extreme value Wc, it shows the onset of an atypical behavior. To the best of our knowledge this anomalous behavior has not been pointed out in previous discussions of persistent current. This behavior can be understood by the following consideration. The surface
disordered system can be regarded as a coupled system comprising of two subsystems, the ordered core and the disordered surface. The Hamiltonian equation (1) can be divided into two modified sub-Hamiltonian. Without the coupling between the two subsystems, the states in the inner core are extended, while those in the surface region are localized. In the weaker disorder regime, the electron states in the central regime are scattered by the surface disorder, and thus tend to be localized so that the current decreases with increasing W. In the stronger disorder regime, however, the influence of the surface disorder on the inner perfect core becomes weak. Actually, the surface disorder scattering effect on the ordered core is represented by the modified term in the sub-Hamiltonian of the ordered core, just as that derived in a shell-doped nanowire [23]. The modified term is inversely proportional to W and tends to zero for infinite disorder. Therefore, the typical current increases with W increasing beyond Wc. In the limit of infinite disorder, one may expect a rather simple scenario in which bulk and surface are decoupled. Consequently, the ordered states would lie on inner sites whereas localized states would be located at surface sites. Then the persistent current comes only from the perfect cluster extended states. This is a trivial limit and the most interesting situations are of 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 for three different ring widths and radii. At a given N (radius), the ratio of the surface to the central region will decrease with M (width) increasing, and the channels in the latter
ARTICLE IN PRESS H.B. Chen, J.W. Ding / Physica B 403 (2008) 2015–2020
contribute more to the persistent current than in the former. As a result, the typical current increases with increasing M, as shown in Fig. 4(a). However, it is noticed that the changes in typical current with M depend strongly on disorder strength. From Fig. 4(a), a little variation is obtained in weaker disorder regime, while large changes in the stronger disorder regime. For M ¼ 3, 4 and 5, for instance, the interval of the amplitude of typical currents approximates to 0.001 at W ¼ 2, while it tends to 0.002 at W 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 the changes also depends on the disorder strength, increasing with W. Even in the weaker disorder regime, a distinct decrease of Ityp is also observed with increasing N. Therefore, the quantum size effects play an important role in determining the persistent current in a surface disordered 2D ring, which should be considered quantitatively to describe the experiments performed in finite-width rings. In the present model, we neglect the effect of electron–electron interactions on the persistent current. Actually, the role of the interactions in bulk disorder systems is still unclear, which is an open subject. Some results indicated that both long-range [26,27] and short-range [28] electron–electron interactions suppress the persistent current. On the other hand, it was indicated that the persistent current was enhanced by the electron–electron interactions [15,29,30]. Present studies for noninteracting electrons can be extended to interacting electron system, which will help to understand the different effects of surface disorder and electron–electron interactions. In addition, a new type of carbon structure, multi-walled carbon nanotorus, had been recently fabricated by different techniques [31,32], which indicates the existence and possibility of a real system of surface disordered ring device in 3D case. Such a device can be devised by the surface adsorption and/or doping into its outer wall of the nanotorus, while the inner wall is kept clean. Despite the simplicity of our model, it allows the study of several situations of physical interest including the case of 2D graphite ring and 3D carbon nanotorus with surface disorder. Furthermore, it would be interesting and possibly relevant for real systems to know what happens in the case of a smooth decrease of the impurity density with the distance from the surface, which is under our further consideration. 4. Summary In summary, tight-binding Hamiltonian model of surface disordered ring is proposed, in which the diagonal disorder is considered to exist only on the surface region of a finitewidth ring. The surface disorder effect on the energy spectra and thus the persistent current are explored in such finite-width rings. Our results indicate that the typical current shows a complex behavior with the strength of the surface disorder. The typical current decreases in the weak disorder regime, a minimum existing at intermediate
2019
disorder, while it increases in the strong disorder regime. This manner is contrast to the case of bulk disorder, in which the typical current decreases monotonously and tends to become zero with the increasing disorder strength. The anomalous scenario is a consequence of the interplay between the ordered and disordered subsystem, which strongly depends on the surface disorder strength. Also, it is shown that the variations in the typical current with the disorder strength strongly depend on the ring width and radius, which show up a singular quantum size effect. Acknowledgments This work was supported by the National Natural Science Foundation of China (no. 10674113), Program for New Century Excellent Talents in University (NCET-060707), Foundation for the Author of National Excellent Doctoral Dissertation of China (Grant no. 200726), Hunan Provincial Natural Science Foundation of China (no. 06JJ50006), and partially by the Scientific Research Fund of Hunan Provincial Education Department (no. 06A071). References [1] M. Bu¨ttiker, Y. Imry, R. Landauer, Phys. Lett. A 96 (1983) 365. [2] (a) L.P. Le´vy, G. Dolan, J. Dunsmuir, H. Bouchiat, Phys. Rev. Lett. 64 (1990) 2074; (b) L.P. Le´vy, Physica B 169 (1991) 245. [3] V. Chandrasekhar, R.A. Webb, M.J. Brady, M.B. Ketchen, W.J. Gallagher, A. Kleinsasser, Phys. Rev. Lett. 67 (1991) 3578. [4] D. Mailly, C. Chapelier, A. Benoit, Phys. Rev. Lett. 70 (1993) 2020. [5] B. Reulet, M. Ramin, H. Bouchiat, D. Mailly, Phys. Rev. Lett. 75 (1995) 124. [6] E.M.Q. Jariwala, P. Mohanty, M.B. Ketchen, R.A. Webb, Phys. Rev. Lett. 86 (2001) 1594. [7] R. Deblock, R. Bel, B. Reulet, H. Bouchiat, D. Mailly, Phys. Rev. Lett. 89 (2002) 206803. [8] (a) H.F. Cheung, Y. Gefen, E.K. Riedel, W.H. Shih, Phys. Rev. B 37 (1988) 6050; (b) H.F. Cheung, E.K. Riedel, Y. Gefen, Phys. Rev. Lett. 62 (1989) 587. [9] (a) H. Bouchiat, G. Montambaux, J. Phys. (Paris) 50 (1989) 2695; (b) G. Montambaux, H. Bouchiat, D. Sigeti, R. Friesner, Phys. Rev. B 42 (1990) 7647. [10] V. Ambegaokar, U. Eckern, Phys. Rev. Lett. 65 (1990) 381. [11] A. Schmid, Phys. Rev. Lett. 66 (1991) 80. [12] F. Von. Oppen, E. Riedel, Phys. Rev. Lett. 66 (1991) 84. [13] B.L. Altshuler, Y. Gefen, Y. Imry, Phys. Rev. Lett. 66 (1991) 88. [14] N.C. Yu, M. Fowler, Phys. Rev. B 45 (1992) 11795. [15] T. Giamarchi, B.S. Shastry, Phys. Rev. B 51 (1995) 10915. [16] S.K. Maiti, Physica E 31 (2006) 117. [17] U. Eckern, P. Schwab, J. Low Temp. Phys. 126 (2002) 1291 and references therein. [18] M.D. Kim, C.K. Kim, K. Nahm, Phys. Rev. B 72 (2005) 085333. [19] S.K. Maiti, J. Chowdhury, S.N. Karmakar, J. Phys.: Condens. Matter 18 (2006) 5349. [20] V.M. Apel, G. Chiappe, M.J. Sa´nchez, Phys. Rev. Lett. 85 (2000) 4152. [21] T. Guhr, A. Mu¨ller-Groeling, H.A. Weidenmu¨ller, Phys. Rep. 299 (1998) 189. [22] L.P. Kouwenhoven, C.M. Marcus, P.L. McEuen, S. Tarucha, R.M. Westervelt, N.S. Wingreen, in: L.L. Sohn, et al. (Eds.), Mesoscopic Electron Transport: Proceedings of the NATO Advanced
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[23] [24] [25] [26] [27]
H.B. Chen, J.W. Ding / Physica B 403 (2008) 2015–2020 Study Institutes (NATO Advanced Study Institute, Series E: Applied Sciences), vol. 345, Kluwer, Dordrecht, 1997. J.X. Zhong, G.M. Stocks, Nano. Lett. 6 (2006) 128. E. Cuevas, E. Louis, J.A. Verges, Phys. Rev. Lett. 77 (1996) 1970. E. Abrahams, P.W. Anderson, D.C. Licciardello, T.V. Ramakrishnan, Phys. Rev. Lett. 42 (1979) 673. M. Abraham, R. Berkovits, Phys. Rev. Lett. 70 (1993) 1509. H. Kato, D. Yoshioka, Phys. Rev. B 50 (1994) 4943.
[28] G. Bouzerar, D. Poilblanc, G. Montambaux, Phys. Rev. B 49 (1994) 8258. [29] A. Mu¨ller-Groeling, H.A. Weidenmu¨ller, Phys. Rev. B 49 (1994) 4752. [30] E. Gambetti-Ce´sare, D. Weinmann, R.A. Jalabert, P. Brune, Europhys. Lett. 60 (2002) 120. [31] J.F. Colomer, L. Henrard, E. Flahaut, G. Van Tendeloo, A.A. Lucas, P. Lambin, Nano. Lett. 3 (2003) 685. [32] H. Yu, Q.F. Zhang, G.H. Luo, F. Wei, Appl. Phys. Lett. 89 (2006) 223106.
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, a
Department of Physics, Xiangtan University, Xiangtan 411105, Hunan, China National Laboratory of Solid State Microstructures, Nanjing University, Nanjing 210093, China
b
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, Bu¨ttiker et al. [1] predicted that even in the presence of disorder, an isolated one-dimensional (1D) metallic ring threaded by the magnetic flux F can carry an equilibrium persistent current with periodicity F0 ¼ h/e, the flux quantum. The existence of persistent currents had been confirmed by the experimental observations in single/ ensemble of isolated mesoscopic ring [2–7]. Except for the case of the nearly ballistic GaAs–AlGaAs ring [4], all the measured currents are in general one or two orders of magnitude larger than those predicted from the theory [8–13]. The diamagnetic response of the ensemble-averaged persistent current in the vicinity of the zero magnetic fields also contrasts with most predictions [10,11]. This means that the experimental results are not well understood theoretically so far. The persistent currents in mesoscopic rings are the subject of intensive research [14–17]. Metals are intrinsically disordered which tends to decrease the persistent currents in mesoscopic rings. To explore the disorder effect, many of the theoretical studies Corresponding author at: Department of Physics, Xiangtan University, Xiangtan 411105, Hunan, China. E-mail address: [email protected] (J.W. Ding).
0921-4526/$ - see front matter r 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2007.11.010
took the limit from 2D to 1D ring, for which different 1D ring Hamiltonians were used. For example, Kim et al. [18] investigated the behavior of persistent currents of 1D normal-metal rings with the impurity potential. The diamagnetic response near the zero magnetic fields was attributed to multiple backward scattering off the impurities. Also, Maiti et al. [19] built a simple 1D tight-binding Hamiltonian with diagonal disorder and long-range hopping integrals to account for the observed behavior of persistent currents in single-isolated-diffusive normal metal rings of mesoscopic size. In the experiments, however, the mean width of the sample ring was usually comparable to its mean radius. In such finite-width rings, it was found that the typical current pffiffiffiffiffiIffityp increases with the channel number M by I typ M , while the disorder-averaged current is independent of M in the ballistic regime [9], only including even Fourier components. On the other hand, confinement and surface roughness effects on the magnitude of the persistent current were analyzed in the case of the ballistic 2D metallic rings [20], which may contribute coherently to the persistent current. It was shown that the typical current increases linearly with the channel number M. These means that 1D description is oversimplified to describe quantitatively the finite-width rings in experiments.
ARTICLE IN PRESS 2016
H.B. Chen, J.W. Ding / Physica B 403 (2008) 2015–2020
In the presence of disorder, some works had been done on the finite-width 2D ring, focusing on the influence of bulk disorder on persistent currents [21]. The bulk disorder was usually considered to be inside the material through which the wave travels. Some general characteristics of persistent current had been obtained in such bulk-disordered systems. It was shown that the typical current in the metallic regime was modified by a corrective diffusive factor and in the localized regime it decreased exponentially with the disorder strength. Due to recent advances in nanotechnology, interestingly, it is possible to fabricate mesoscopic devices in which carriers are mainly scattered by the boundaries and not by impurities or defects located inside them [22]. Based on such an actual structure, recently, a shell-doped nanowire model was proposed, from which a novel transport behavior was obtained, that is, the larger the disorder, the weaker the localization [23]. For the surface roughness or defects, similarly, there may exist a large disorder at the surface of a finite-width ring. Especially, when the system size was shrunk down to the nanometer scale, the surface-to-volume ratio becomes larger, leading to very strong quantum size effects and surface effects. In a practical implementation of surface roughness, Cuevas et al. [24] had studied the quantum chaotic dynamics by building a new model of quantum chaotic billiard. The essential feature of this model is the inclusion of diagonal disorder at the surface of the system. The obtained energy spectrum statistics shows a complex behavior, very different from that previously reported in the usual chaotic billiard model. Obviously, the similar complex energy spectrum may be expected in a surface disordered ring, indicating some new features in persistent current. How about the influence of the surface disorder on the persistent current in finite-wide rings? In this paper, taking into account the surface roughness or defects, we build a surface disordered 2D ring model. The effect of surface disorder on persistent current in such 2D ring is explored within the tight-binding frame. It is found that with the increasing disorder strength, the typical current increases in the strong disorder regime, while it decreases in the weak disorder regime. Also, the disorder induced changes in the persistent current depending strongly both on the ring width and radius, which shows up a singular quantum size effect. 2. Model and formulae We consider a 2D mesoscopic ring enclosing a magnetic flux 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 maximum number of the open channels in a structure consisting of M ring chains is equal to M. Taking a single atomic level per lattice site, the tight-binding Hamiltonian by considering
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.
non-interacting electrons is given by H¼
X
i cyi ci þ
i
X
V ði; jÞcyi cj
(1)
i;j
with on-site energies ei, where i labels the coordinates of the sites in the lattice. The hopping integrals V(i, j) are restricted to the nearest neighbors of a site. Assuming that the vector potential A has only R j an azimuthal component, we take V ði; jÞ ¼ t exp i i A dl ; in units of the quantum flux F0, where l is a vector that points from the site i to any of its nearest neighbors. To model the surface disorder, the on-site energies ei in the surface region are taken to be randomly distributed within interval [W, W], W describing the disorder strength, whereas the other sites have a constant energy equal to zero. For the finite-width ring, we neglect the self-inductance effect on the persistent current in the system. At zero temperature, the total current can be calculated by X I ¼ qE=qF ¼ qE n =qF (2) n
with E the total energy of the system. Here n labels the corresponding eigenlevels. The second equality in Eq. (2) is valid only in the absence of electron–electron interactions, which are neglected here. The current is a periodic function of F with fundamental period F0. Usually, one is interested in the typical current [8,20], which is defined as the square root of the disorder (W) and flux average of the square of the persistent current I typ ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi hI 2 iF;W .
(3)
To obtain good statistics, the typical currents are averaged over many realizations of the disorder configurations. In our calculations, the number of the averaged configurations varies from 100 to 200, depending on the size of the systems.
ARTICLE IN PRESS H.B. Chen, J.W. Ding / Physica B 403 (2008) 2015–2020
0.0
a
b
c
d
2017
Energy/t
-0.1
-0.2 0.0
-0.1
-0.2 -0.5
0.0
0.5 -0.5
0.0
0.5
Φ/Φ0
Φ/Φ0
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.
3. Results and discussion Now let us calculate the energy spectra and persistent currents in the surface disordered sample rings. In our calculations, the site energies ei, disorder strength W, and energy E are given in units of the model parameter t, and thus the persistent currents in units of t/F0. In Fig. 2, we show the typical energy spectra in halffilling region for the M ¼ 4 and N ¼ 100 sample ring as a function of flux at varying surface disorder strength. The four values of W are chosen by a steady increase to be W ¼ 1, 4, 10, and 16. For a given W, one disorder configuration can be obtained randomly. But, the results of energy spectra are qualitatively unchanged. In the presence of surface disorder, the disorder-induced energy gaps are observed at the crossing points in energy spectrum, as shown in Fig. 2. The result is similar to that of the bulk disordered system [9]. Interestingly, an extreme value Wc of disorder strength is obtained from the changes in energy spectra with increasing disorder. For weaker disorder (WoWcE4), the energy levels become more isolated with W increasing and thus the energy curves change more smoothly with flux [shown in Fig. 2 (a) W ¼ 1 and (b) W ¼ 4]. At W ¼ Wc, especially, there exist even large gaps in the energy spectra and the slopes of the energy curves tend to vanish. Obviously, a small slope rate will lead to less contribution to the current in carried by En. This means that the total current should dramatically decrease with W increasing, just as previous predictions. In the regime of stronger disorder (W4Wc), however, some substantial changes appear in the energy spectra of the surface disordered ring. From Fig. 2 (c) W ¼ 10 and (d) W ¼ 16,
Ityp [t/Φ0]
0.006
M=4, N=100 bulk disorder, 0.5 filling surface disorder, 0.5 filling surface disorder, 0.4 filling surface disorder, 0.6 filling
0.004
0.002
0.000 0
2
4
6
8
10
12
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16
disorder strength, W/t 4 5 6 Fig. 3. Typical current as a function of disorder strength W at 10 , 10 and 10 filling in a finite-width ring with surface disorder and bulk disorder.
it is seen that the slopes of the energy curves recover and increase with W increasing. This indicates a rise of persistent current even in the case of strong disorder, different from the results predicted from the bulkdisordered systems. To understand the surface disorder effect on persistent current, Fig. 3 shows the W dependence of the typical current for the same sample ring in Fig. 2. For a comparison, the results of the bulk-disordered ring with the same size are also presented. In the case of bulk disorder, as expected, Ityp decreases continuously with the disorder strength W and diminishes to zero at WE3 (WoWc). This can be easily understood from the theory of
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2018
Ityp [t/Φ0]
0.008 N = 100 M=3 M=4 M=5
0.004
0.000 0.008
Ityp [t/Φ0]
M=4 N = 100 N = 128 N = 150
0.004
0.000 0
2
4
6
8
10
12
14
16
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.
the Anderson localization [25]. The larger the disorder is, the stronger the localization. Therefore, the strong disorder may result in carrier localization and thus lead to a disappearing current in the bulk disordered thin ring. For surface disorder ring, it is seen from Fig. 3 that the overall currents exceed that in bulk-disordered ring. Interestingly, it is found that Ityp does not vary monotonically with the disorder strength W, different from the bulk-disordered system. At the half-filling, the extreme value Wc of disorder strength is obtained to be WcE4, consistent with that in the 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-doped nanowire [23], where a localization/quasi-delocalization transition was observed at a critical disorder strength. Below and above the half-filling, we also calculate the 4 6 typical currents at 10 and 10 filling, as shown in Fig. 3. The similar anomalous behavior has also been obtained. For other cases (far from the half-filling), the enhancement of the current amplitude may be small, while the typical current still increases at very high disorder. This may be due to the fact that far from the half-filling, the quasi-ideal states existing at about the spectrum center [23,24] have less contribution to the current. As for the extreme value Wc, it shows the onset of an atypical behavior. To the best of our knowledge this anomalous behavior has not been pointed out in previous discussions of persistent current. This behavior can be understood by the following consideration. The surface
disordered system can be regarded as a coupled system comprising of two subsystems, the ordered core and the disordered surface. The Hamiltonian equation (1) can be divided into two modified sub-Hamiltonian. Without the coupling between the two subsystems, the states in the inner core are extended, while those in the surface region are localized. In the weaker disorder regime, the electron states in the central regime are scattered by the surface disorder, and thus tend to be localized so that the current decreases with increasing W. In the stronger disorder regime, however, the influence of the surface disorder on the inner perfect core becomes weak. Actually, the surface disorder scattering effect on the ordered core is represented by the modified term in the sub-Hamiltonian of the ordered core, just as that derived in a shell-doped nanowire [23]. The modified term is inversely proportional to W and tends to zero for infinite disorder. Therefore, the typical current increases with W increasing beyond Wc. In the limit of infinite disorder, one may expect a rather simple scenario in which bulk and surface are decoupled. Consequently, the ordered states would lie on inner sites whereas localized states would be located at surface sites. Then the persistent current comes only from the perfect cluster extended states. This is a trivial limit and the most interesting situations are of 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 for three different ring widths and radii. At a given N (radius), the ratio of the surface to the central region will decrease with M (width) increasing, and the channels in the latter
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contribute more to the persistent current than in the former. As a result, the typical current increases with increasing M, as shown in Fig. 4(a). However, it is noticed that the changes in typical current with M depend strongly on disorder strength. From Fig. 4(a), a little variation is obtained in weaker disorder regime, while large changes in the stronger disorder regime. For M ¼ 3, 4 and 5, for instance, the interval of the amplitude of typical currents approximates to 0.001 at W ¼ 2, while it tends to 0.002 at W 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 the changes also depends on the disorder strength, increasing with W. Even in the weaker disorder regime, a distinct decrease of Ityp is also observed with increasing N. Therefore, the quantum size effects play an important role in determining the persistent current in a surface disordered 2D ring, which should be considered quantitatively to describe the experiments performed in finite-width rings. In the present model, we neglect the effect of electron–electron interactions on the persistent current. Actually, the role of the interactions in bulk disorder systems is still unclear, which is an open subject. Some results indicated that both long-range [26,27] and short-range [28] electron–electron interactions suppress the persistent current. On the other hand, it was indicated that the persistent current was enhanced by the electron–electron interactions [15,29,30]. Present studies for noninteracting electrons can be extended to interacting electron system, which will help to understand the different effects of surface disorder and electron–electron interactions. In addition, a new type of carbon structure, multi-walled carbon nanotorus, had been recently fabricated by different techniques [31,32], which indicates the existence and possibility of a real system of surface disordered ring device in 3D case. Such a device can be devised by the surface adsorption and/or doping into its outer wall of the nanotorus, while the inner wall is kept clean. Despite the simplicity of our model, it allows the study of several situations of physical interest including the case of 2D graphite ring and 3D carbon nanotorus with surface disorder. Furthermore, it would be interesting and possibly relevant for real systems to know what happens in the case of a smooth decrease of the impurity density with the distance from the surface, which is under our further consideration. 4. Summary In summary, tight-binding Hamiltonian model of surface disordered ring is proposed, in which the diagonal disorder is considered to exist only on the surface region of a finitewidth ring. The surface disorder effect on the energy spectra and thus the persistent current are explored in such finite-width rings. Our results indicate that the typical current shows a complex behavior with the strength of the surface disorder. The typical current decreases in the weak disorder regime, a minimum existing at intermediate
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disorder, while it increases in the strong disorder regime. This manner is contrast to the case of bulk disorder, in which the typical current decreases monotonously and tends to become zero with the increasing disorder strength. The anomalous scenario is a consequence of the interplay between the ordered and disordered subsystem, which strongly depends on the surface disorder strength. Also, it is shown that the variations in the typical current with the disorder strength strongly depend on the ring width and radius, which show up a singular quantum size effect. Acknowledgments This work was supported by the National Natural Science Foundation of China (no. 10674113), Program for New Century Excellent Talents in University (NCET-060707), Foundation for the Author of National Excellent Doctoral Dissertation of China (Grant no. 200726), Hunan Provincial Natural Science Foundation of China (no. 06JJ50006), and partially by the Scientific Research Fund of Hunan Provincial Education Department (no. 06A071). References [1] M. Bu¨ttiker, Y. Imry, R. Landauer, Phys. Lett. A 96 (1983) 365. [2] (a) L.P. Le´vy, G. Dolan, J. Dunsmuir, H. Bouchiat, Phys. Rev. Lett. 64 (1990) 2074; (b) L.P. Le´vy, Physica B 169 (1991) 245. [3] V. Chandrasekhar, R.A. Webb, M.J. Brady, M.B. Ketchen, W.J. Gallagher, A. Kleinsasser, Phys. Rev. Lett. 67 (1991) 3578. [4] D. Mailly, C. Chapelier, A. Benoit, Phys. Rev. Lett. 70 (1993) 2020. [5] B. Reulet, M. Ramin, H. Bouchiat, D. Mailly, Phys. Rev. Lett. 75 (1995) 124. [6] E.M.Q. Jariwala, P. Mohanty, M.B. Ketchen, R.A. Webb, Phys. Rev. Lett. 86 (2001) 1594. [7] R. Deblock, R. Bel, B. Reulet, H. Bouchiat, D. Mailly, Phys. Rev. Lett. 89 (2002) 206803. [8] (a) H.F. Cheung, Y. Gefen, E.K. Riedel, W.H. Shih, Phys. Rev. B 37 (1988) 6050; (b) H.F. Cheung, E.K. Riedel, Y. Gefen, Phys. Rev. Lett. 62 (1989) 587. [9] (a) H. Bouchiat, G. Montambaux, J. Phys. (Paris) 50 (1989) 2695; (b) G. Montambaux, H. Bouchiat, D. Sigeti, R. Friesner, Phys. Rev. B 42 (1990) 7647. [10] V. Ambegaokar, U. Eckern, Phys. Rev. Lett. 65 (1990) 381. [11] A. Schmid, Phys. Rev. Lett. 66 (1991) 80. [12] F. Von. Oppen, E. Riedel, Phys. Rev. Lett. 66 (1991) 84. [13] B.L. Altshuler, Y. Gefen, Y. Imry, Phys. Rev. Lett. 66 (1991) 88. [14] N.C. Yu, M. Fowler, Phys. Rev. B 45 (1992) 11795. [15] T. Giamarchi, B.S. Shastry, Phys. Rev. B 51 (1995) 10915. [16] S.K. Maiti, Physica E 31 (2006) 117. [17] U. Eckern, P. Schwab, J. Low Temp. Phys. 126 (2002) 1291 and references therein. [18] M.D. Kim, C.K. Kim, K. Nahm, Phys. Rev. B 72 (2005) 085333. [19] S.K. Maiti, J. Chowdhury, S.N. Karmakar, J. Phys.: Condens. Matter 18 (2006) 5349. [20] V.M. Apel, G. Chiappe, M.J. Sa´nchez, Phys. Rev. Lett. 85 (2000) 4152. [21] T. Guhr, A. Mu¨ller-Groeling, H.A. Weidenmu¨ller, Phys. Rep. 299 (1998) 189. [22] L.P. Kouwenhoven, C.M. Marcus, P.L. McEuen, S. Tarucha, R.M. Westervelt, N.S. Wingreen, in: L.L. Sohn, et al. (Eds.), Mesoscopic Electron Transport: Proceedings of the NATO Advanced
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