- Email: [email protected]
Edged topological disordered quantum ring in the presence of magnetic flux
Physics Letters A 374 (2010) 1762–1768
Contents lists available at ScienceDirect
Physics Letters A www.elsevier.com/locate/pla
Edged topological disordered quantum ring in the presence of magnetic flux Edris Faizabadi ∗ , Mahboubeh Omidi Department of Physics, Iran University of Science and Technology, 16846 Tehran, Iran
a r t i c l e
i n f o
Article history: Received 9 October 2009 Received in revised form 2 February 2010 Accepted 10 February 2010 Available online 18 February 2010 Communicated by R. Wu Keywords: Quantum ring Edged topological disorder Disorder-averaged current Temperature-averaged current Magnetic susceptibility
a b s t r a c t The effects of edged topological disorder on persistent current and magnetic susceptibility are investigated in a two-dimensional quantum ring enclosed within a magnetic flux in tight binding model. In an ordered quantum ring, persistent current shows kink-like variation, while in an edged topological disordered quantum ring, disorder-averaged current gets continuous variation due to appearance of energy gaps in energy spectra. Magnetic susceptibility has delta-like paramagnetic peaks in an ordered quantum ring, while in the edged topological disordered one; the delta-like peaks become wider and begin to overlap. In addition magnetic susceptibility versus magnetic flux shows a sign reversal, which depends on the disordered configuration. In the ordered quantum ring, the temperature-averaged current decreases when the temperature increases, while in the edged topological disordered one, at first temperatureaveraged current raises somewhat and then reduces in a general thermal behavior manner. At absolute zero temperature, in strong regime; the disorder-averaged current versus magnetic flux is periodic with the period of Φ0 /2; whereas, this period is Φ0 in weak regime or in an ordered one (Φ0 is magnetic flux quantum). Within higher temperatures, the period of Φ0 is achieved only in the ordered ring and Φ0 /2 in either weak or strong regime, wide or narrow ring. © 2010 Elsevier B.V. All rights reserved.
1. Introduction During the last decade, a verity of experimental and theoretical probes was concentrated on nanoscale structures [1–8]. Physical understanding of the electronic properties of nanostructures allows us to fabricate the electronic elements which are widely applied as components of the new generation of electronic and photonic devices [1,3]. Quantum rings are fascinating structures because they provide a unique system to study quantum interference phenomena such as Aharonov–Bohm oscillations, magnetoresistence and persistent currents [5,7–11]. Primarily, Hund [12] predicted the existence of persistent currents in quantum rings, and then it was followed by Byers and Yang [13], and Bloch [14]. Also in 1983, Buttiker, Imry, and Landauer predicted the existence of persistent current in a one-dimensional (1D) quantum ring [15]. In 1990, Levy et al. [16] measured persistent current over an ensemble of approximately 107 copper rings. Other experiments were performed by Chandrasekhar et al. [17] on single diffusive gold and by Mially et al. [18] on a semiconductor single loop in the GaAs/GaAlAs system. The experimental results did not match with the theoretical researches because the disorder effects and electron–electron interactions had been neglected. However, several theoretical researches by accounting electron–electron interac-
*
Corresponding author. Tel.: +98(21)77240477; fax: +98(21)77240497. E-mail address: [email protected] (E. Faizabadi).
0375-9601/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2010.02.025
tions [19,20] or different kinds of disorders have been done. In an attempt to consider these effects, Bouzerar et al. studied persistent current in a 1D quantum ring in the presence of interacting electrons with diagonal disorder. They have found that both disorder and interactions always decrease the persistent current by localizing the electrons [21]. Maiti et al. explained the dependence of persistent current and magnetic susceptibility on diagonal disorder strength in the tight-binding model with higher-order hopping integral in the Hamiltonian [22]. Since the mean width of the sample was comparable to its mean radius in experiments, it was necessary to be investigated in a two-dimensional (2D) quantum ring. Helene Bouchiat et al. [23] inspected the effect of bulk disorder on the persistent current in 2D quantum rings. They realized that in the weak disorder limit, average current (I ave = I ) was independent of the number of channels, and in strong disorder limit, I / I 0 was affected by the order of the square of the typical current I 2 / I 02 (expressed in I 0 = ev F / L, where e is electron charge, v F is Fermi velocity, and L is the ring circumference). For the ballistic 2D metallic samples, confinement and surface roughness effects on magnitude of persistent current were investigated experimentally by Chandrasekhar et al. [17] and theoretically by Apel et al. [24]. They found that localized border states contribute coherently to the persistent current and its magnitude is enhanced with respect to their value in the absence of confinement. In another attempt Maiti have studied the effect of edge disorder on quantum transport through a finite width mesoscopic ring by using the Green’s function technique based on the tight-binding formu-
E. Faizabadi, M. Omidi / Physics Letters A 374 (2010) 1762–1768
lation [5]. A novel transport phenomenon is observed which give the enhancement of the current amplitude with the increase of the edge disorder strength in the strong disorder regime, while the amplitude decreases in the weak disorder regime. Also it was shown that the transport properties are significantly influenced by the radius and the width of the ring. The influence of surface disorder on the persistent current in finite width rings was investigated by Chen and Ding [25]. They found that in the weak disorder strength, the persistent current decreased with surface disorder strength, while it increased in the strong disorder strength. Also they showed that the disorderinduced changes in the persistent current strongly depended on both the ring width and its radius. The fabricated quantum rings do not have perfect shapes; the contact between the ring edges and its surrounding causes a kind of disorder in the shapes of the edges which affects on the ring properties. In inverse view, it is possible to manipulate quantum rings with artificial desired edged topological disorder to design new semiconductor materials and devices. Band gap engineering is an effective technique for the design of new semiconductor devices. Heterojunctions and contemporary growth techniques allow band diagrams with nearly arbitrary and continuous band gap variations to be made. The transport properties of electrons can be continuously tuned for a special application. Here by applying artificial desired edged topological disorder on two-dimensional quantum ring a new generation of devices with unique capabilities is emerging. In this work, by using exact diagonalization technique, the edged topological disorder effects in a two-dimensional quantum ring are investigated. The study is performed in strong and weak regimes. In the weak regime, the distance between the edged chain rings (channels) and inner channels is supposed to be large and only hopping strengths of the sites in the edged channels are random. In the other case, the distance between the edged channels and the inner channels is sufficiently small and the hopping strengths of the sites between the edged channels and nearestneighbor channels are also random. The Letter is organized as follows. The Hamiltonian and formalism are discussed in the next section. In Section 3 the results of numerical calculations for Fermi energy, energy spectra, disorder-averaged current and magnetic susceptibility as a function of magnetic flux are presented. Temperature-averaged current versus temperature and magnetic flux is also discussed. Finally, we end the Letter with a brief conclusion. 2. Method and formalism The effects of edged topological disorder on a typical 2D quantum ring threaded by a magnetic flux which is depicted in Fig. 1 are investigated. This system consists of M tight-binding ring chains, simply pointed out as M channels which are concentrically connected. Each channel has N sites of similar atoms. The Hamiltonian of the system in the tight binding model can be described as follows [26]
H=
εi c i† c i +
i
i , j ,m,m
i t mm ij e
j i
.dl † A ci c j
+ c.c.,
(1)
† ci
where and c i are the creation and destruction operators for an electron at the site i; the index j labels the nearest-neighbors of i-th site, and εi and t mm are on-site energy and the hopping ij
is the vector potential in azimuthal diintegral, respectively. A rection and l is a vector that points from the site i to any of its nearest-neighbors. The phase is measured in units of magnetic flux quantum Φ0 = hc /e. Channels are indicated by m and m with respective values of 1, 2, . . . and M with the innermost channel being 1. The hopping integral equals to a constant value of t in
1763
Fig. 1. The schematic plot of a 2D quantum ring with edged topological disorder is threaded by a magnetic flux which includes M channels with N sites.
channel 1 in the ordered quantum ring. The nearest-neighbors of each site in both its channel and its nearest-neighbor channels are considered. The constant t mm i j s between sites in channel m is attained with the relation −1/(1 + (2π (m − 1))/ N )2 . It is supposed that the distance between two successive sites in two sequential channels is the same as the distance between two successive sites in channel 1 of the ordered quantum ring. The edged topological disorder is investigated in two different regimes; weak and strong. In the weak regime, called ETDO quantum ring, the hopping strengths of the sites within channel 1 besides channel M are randomly chosen within the interval of [−1.5, −0.5] in units of t; whereas, in the strong regime, called ETD quantum ring, the hopping strengths of the sites within the channel 1, within channel M, between channels 1 and 2, and also between channels M − 1 and M are accidentally picked out in the same domain. In order to model the edged topological disorder in the weak regime, t i11 and t iMj M are randomly picked out within the j
aforementioned interval, while in ETD quantum ring, all t mm i j s except t i11 , t i12 , t iMj M −1 , and t iMj M are constant; in other words, not j j
and t iMj M but also all t i12 and t iMj M −1 are randomly seonly t i11 j j lected in the strong regime. Since the magnetic flux is confined to a small domain, changing in phase is negligible. By using exact diagonalization technique [27] the eigenenergies of the system are evaluated. Persistent current at zero temperature in the absence of electron–electron interaction can be also written as
I =−
∂ En ∂E =− , ∂Φ ∂Φ n
(2)
where E is the total energy of the system, Φ is the magnetic flux, and E n is eigenenergy. The average current can be evaluated as
I ave = I c ,
(3)
where . . .c is the ensemble average on different realizations of disorder. The temperature-averaged current at none-zero temperature is calculated by
IT =
− ∂ E n e −β E n ∂Φ
n
Z
,
(4)
where Z is the partition function that can be written as
Z=
e −β E n ,
n
where β = 1/( K B T ) and K B is Boltzmann constant.
(5)
1764
E. Faizabadi, M. Omidi / Physics Letters A 374 (2010) 1762–1768
Fig. 2. Typical energy spectra as a function of magnetic flux for 2D quantum ring with N = 100, M = 4, εi = 0.5, constant hopping strengths = −1, and random hopping strengths are distributed within [−1.5, −0.5]. (a) Ordered quantum ring, (b) ETDO quantum ring, (c) ETD quantum ring.
E. Faizabadi, M. Omidi / Physics Letters A 374 (2010) 1762–1768
1765
Magnetic susceptibility is given by derivative of the persistent current with respect to the magnetic flux.
χ (Φ) =
n 3 16π 2
∂ I (Φ) , ∂Φ
(6)
where n is the number of sites. The disorder-averaged susceptibility can be achieved by averaging over the disorder configurations as
χave = χ c .
(7)
3. Results By using the method which is developed in the previous section, the effects of weak and strong edged topological disorders on all the Fermi energy, energy spectra, persistent current, magnetic susceptibility, and temperature-averaged current in the 2D quantum ring are investigated. In our calculations, the energies are given in units of the hopping integral t, Φ is measured in fundamental flux Φ0 , current is in unit of t /Φ0 , and the magnetic susceptibility is in unit of t /Φ02 . We consider M = 4, N = 100, the number of electrons in the ring ( N e ) = 200 (half filling region), εi = 0.5, random hopping strengths are distributed within the interval [−1.5, −0.5] and the constant hopping energy is equal to −1. The most electronic properties of the system can be explained with the energy spectra. In Fig. 2 the energy spectra as a function of magnetic flux are depicted. In an ordered quantum ring, the energy levels cross each other and there are many degeneracies without any energy gap in the spectrum. In a strictly one-dimensional (1D) quantum ring the energy levels have a close relation with the energy levels of one-dimensional Bloch problem. It is possible to think that our system is composed of M 1D chain rings whose discrete energy levels are mixed to construct the schemes depicted in Fig. 2(a). In other words, the energy spectra which are illustrated in Fig. 2(a) are a blend of these discrete energy levels. In an edged topological disordered quantum ring, the energy levels repel each other and energy gaps appear in the energy spectra. These energy gaps in the spectrum can be interpreted as a back-scattering happening due to the boundaries of the sample with edged topological disorder. The Fermi energy versus magnetic flux is shown in Fig. 3. It varies periodically with the period of Φ0 which arises from Aharonov–Bohm effect. If an electron moves in a field-free region that is not simply connected, but surrounds a hole containing magnetic flux Φ , then upon completing a circuit, the electron acquires an additional phase factor e ieΦ/c , under the condition that the electron wave function is single-valued; therefore, the phase factor is unity, implying that the enclosed flux is quantized [28]. In the edged topological disordered quantum ring, the Fermi energy tends towards continuous variation, which is due to back-scattering occurrence at disordered boundaries of the system. Also the variation of Fermi energy as a function of magnetic flux (Fig. 3) shows that Fermi energy level decreases in the presence of edge topological disorder. In ETD quantum ring (Fig. 3(c)) Fermi energy level is lower than this level in ETDO quantum ring. The system with disordered edge as a result of lowering Fermi energy is more stable and more compatible with nature. The persistent current can be derived by differentiating energy with respect to magnetic flux, and the average current can be obtained by averaging over all disorder configurations. In Fig. 4(a) the variation of persistent current versus magnetic flux is illustrated for an ordered quantum ring. Also the results for the dependence of disorder-averaged current on the magnetic flux over 500 configurations in ETDO and ETD quantum ring are depicted in Figs. 4(b) and 4(c) in the half filling region, respectively. In the
Fig. 3. Fermi energy versus magnetic flux for 2D quantum ring with N = 100, M = 4, εi = 0.5, constant hopping strengths = −1 and random hopping strengths are distributed within [−1.5, −0.5] in half filling region. (a) Ordered quantum ring, (b) ETDO quantum ring, (c) ETD quantum ring.
ordered quantum ring (Fig. 4(a)), persistent current has kink-like variation due to the different contributions of the energy levels in the mixed energy band region. In the edged topological disordered quantum rings (Figs. 4(b) and 4(c)), the current varies continuously with magnetic flux as a result of edged topological disorder which breaks the symmetries and help to lift most of the degeneracies. In addition, the extremum points are shifted with respect to the ordered quantum ring. Lifting degeneracies and the displacement of extremum points can be described as a back-scattering occurrence at disordered edges of the system. As disorder strengthens, the localized states increase, and the magnitude of disorder-averaged current will be less. These are illustrated in Figs. 4(b) and 4(c) in comparison to Fig. 4(a) that has not any disorder. In a narrow ETD quantum ring; the disorder-averaged current versus magnetic flux is periodic with the period of Φ0 /2; whereas, this period is Φ0 in an ETDO or in an ordered one. In ETDO quantum ring, just the hopping strengths in innermost and outermost channels are random, so the disorder hoppings do not affect the
1766
E. Faizabadi, M. Omidi / Physics Letters A 374 (2010) 1762–1768
Fig. 4. (a) Persistent current versus magnetic flux in the ordered quantum ring in half filling region; (b) Disorder-averaged current versus magnetic flux in the ETDO quantum ring in half filling region; (c) Disorder-averaged current versus magnetic flux in the ETD quantum ring in half filling region.
inner quantum channels. But in the ETD quantum ring, since the hopping strengths of the sites between channels 1 and 2, and also M and M − 1 are random, disordered effects influence on the whole system and affect on changing on all different disordered configurations diagrams. This process leads to the period of Φ0 /2 that agrees with Levy’s experimental result [15]. It is necessary to ∂E mention that I = − ∂Φ and M = − ∂∂ BE to relate our calculation to Levy’s experiment. In fact, the period of Φ0 /2 is a disordered statistical phenomenon, which emerges as a result of configurational averaging over strong disorderliness (at T = 0). In the absence of edged topological disorder, the period becomes Φ0 because our ensemble contains identical rings. Besides, in the weak regime; we also obtain the same result as our ensemble includes almost similar rings. The period approaches to Φ0 /2 only in the strong regime within limit of narrow rings, in which our ensemble contains dissimilar rings. Fig. 5(a) represents variation of magnetic susceptibility as a function of magnetic flux in the ordered quantum ring. In addition, the variation of disorder-averaged susceptibility by averaging over 500 disorder configurations is depicted in Figs. 5(b)
Fig. 5. (a) Magnetic susceptibility as a function of magnetic flux in the ordered quantum ring; (b) Disorder-averaged susceptibility as a function of magnetic flux in the ETDO quantum ring; (c) Disorder-averaged susceptibility as a function of magnetic flux in the ETD quantum ring.
and 5(c) for the edged topological disordered quantum ring. The magnetic susceptibility has delta-like paramagnetic peaks in ordered quantum ring and these peaks are related to the crossing of the single-electron states. In the presence of edged topological disorder, the delta-like peaks become wider and the overlap between these broadened peaks is observed in Figs. 5(b) and 5(c). They can also be described by the back-scattering incident at the boundaries with edged topological disorder. Moreover, increasing of localized states decreases magnitude of disorder-averaged susceptibility in edged topological disordered quantum ring. In addition, it is illustrated that magnetic susceptibility as a function of magnetic flux shows a sign reversal which depends on the disorder configurations. Figs. 6(a), 6(b) and 6(c) show the dependence of temperatureaveraged current (I T ) on temperature and magnetic flux for the ordered quantum ring, ETDO quantum ring and ETD quantum ring, respectively. The system consists of 400 spinless electrons and Figs. 6(b) and 6(c) are averaged over 10 disorder configurations. In
E. Faizabadi, M. Omidi / Physics Letters A 374 (2010) 1762–1768
Fig. 6. Temperature-averaged current versus magnetic flux and temperature. (a) Ordered quantum ring, (b) ETDO quantum ring, (c) ETD quantum ring.
1767
1768
E. Faizabadi, M. Omidi / Physics Letters A 374 (2010) 1762–1768
the ordered quantum ring I T decreases by increasing temperature as a result of randomization of occupation of the energy levels, so the current components averagely annul each other. In the disordered quantum ring, as a result of symmetry breaking, the degeneracies in energy spectra are lifted, and by increasing the allowed levels close to Fermi energy as a result of splitting of degenerate level, at first I T increases to some extent and then the general loss initiates. In an ordered quantum ring temperature-averaged current versus magnetic flux is periodical with the period of Φ0 while in ETDO and ETD quantum rings, the period of oscillations is Φ0 /2. This can be understood by the scenario that within higher temperature respect to absolute zero, whole the energy levels of any ring will be effective in the temperature-averaged current because all the system energy levels have an occupation probability. So, edged topological disorder affects the temperature-averaged current any way. The period of Φ0 is achieved only in the absence of edged topological disorder and Φ0 /2 in either weak or strong regime, wide or narrow ring. 4. Summery and conclusion To summarize, the effects of edged topological disorder is investigated in two different regimes. In the weak regime, the distance between channels 1 and M and their nearest neighbors is supposed to be sufficiently large, i.e. the edged topological disorder does not affect inner channels, and this is true for ETDO quantum rings. The strong regime refers to edged topological disorder in which the distance between channels 1 and M and their nearest neighbors is small enough to penetrate to interior channels (the ETD quantum rings). It is shown that energy spectra show crossing of the energy levels in the ordered quantum ring; and with appearing of edged topological disorder the energy levels repel each other and the energy gaps appear. In contrast to the ordered quantum ring, in the presence of disorder, the Fermi energy tends to vary continuously as a function of magnetic flux. It is also shown that the disorderaveraged current varies continuously as a function of magnetic flux in the edged topological disordered quantum ring. At absolute zero temperature, the period of oscillation of disorder-averaged current is remained Φ0 in the ETDO quantum ring while it approaches to Φ0 /2 in the strong regime within limit of narrow rings, in which our ensemble contains dissimilar rings. This variation is due to influence of disordered effects on the whole system in the ETD quantum ring. In addition, magnetic susceptibility versus magnetic flux has delta-like paramagnetic peaks
in ordered quantum ring. In the presence of edged topological disorder, these delta-like peaks become wider and begin to overlap. In addition, it is illustrated that magnetic susceptibility as a function of magnetic flux shows a sign reversal, which depends on the disorder configurations. In the ordered quantum ring, temperatureaveraged current decreases by raising the temperature, and in edged topological disordered quantum ring, this current at first begins to increase to some extent and then decreases in the same way as general thermal behavior. In addition, in edged topological disordered quantum ring, the period of oscillations of I T becomes Φ0 /2. Overall, the unique condition for the period of Φ0 /2 in absolute zero temperature is the strong regime situation within the limit of narrow rings; whereas, the presence of edged topological disorder (weak or strong) is enough within higher temperatures whether ring is narrow or not. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28]
Jianxin Zhong, G. Malcolm Stocks, Nano Lett. 6 (2006) 128. Jianxin Zhong, G. Malcolm Stocks, Phys. Rev. B 75 (2007) 033410 (4 pages). C.Y. Yang, J.W. Ding, N. Xu, Physica B 394 (2007) 69. Santanu K. Maiti, J. Nanosci. Nanotech. 9 (9) (2009) 5664. Santanu K. Maiti, Slolid State Sci. 10 (2008) 1456. Santanu K. Maiti, Chem. Phys. Lett. 446 (2007) 365. J.H. Dai, Y.L. Lin, S.C. Lee, IEEE Photon. Technol. Lett. 19 (2007) 1511. J.H. Dai, J.H. Lee, Y.L. Lin, S.C. Lee, Jpn. J. Appl. Phys. 47 (2008) 2924. Y. Aharonov, D. Bohm, Phys. Rev. 115 (1959) 485. R.A. Webb, S. Washburn, C.P. Umbach, R.B. Laibowitz, Phys. Rev. Lett. 54 (1985) 2696. J.M. García, G. Medeiros-Ribeiro, K. Schmidt, T. Ngo, J.L. Feng, A. Lorke, J. Kotthaus, P.M. Petroff, Appl. Phys. Lett. 71 (1997) 2014. F. Hund, Ann. Phys. (Leipzig) 32 (1938) 102. N. Byers, C.N. Yang, Phys. Rev. Lett. 7 (1961) 46. F. Bloch, Phys. Rev. 137 (1965) A787. M. Buttiker, Y. Imry, R. Landauer, Phys. Lett. A 96 (1983) 365. L.P. Lévy, G. Dolan, J. Dunsmuir, H. Bouchiat, Phys. Rev. Lett. 64 (1990) 2074. V. Chandrasekhar, R.A. Webb, M.J. Brady, M.B. Ketchen, W.J. Gallagher, A. Kleinsasser, Phys. Rev. Lett. 67 (1991) 3578. D. Mailly, C. Chapelier, A. Benoit, Phys. Rev. Lett. 70 (1993) 2020. H. Kato, D. Yoshioka, Phys. Rev. B 50 (1994) 4943. A. Muller-Groeling, H.A. Weidenmuller, Phys. Rev. B 49 (1994) 4752. G. Bouzerar, D. Poilblanc, G. Montambaux, Phys. Rev. B 49 (1994) 8258. S.K. Maiti, J. Chowdhury, S.N. Karmakar, J. Phys.: Condens. Matter 18 (2006) 5349. H. Bouchiat, G. Montambaux, J. Phys. (Paris) 50 (1989) 2695. V.M. Apel, G. Chiappe, M.J. Sanchez, Phys. Rev. Lett. 85 (2000) 4152. H.B. Chen, J.W. Ding, Physica B 403 (2008) 2015. E. Faizabadi, M. Omidi, Phys. Lett. A 373 (2009) 1469. H.Q. Lin, J.E. Gubernatis, J. Tobochnik, Comput. Phys. 7 (1993) 401. S. Gasiorowicz, Quantum Physics, second edition, John Wiley & Sons, 1996.
Contents lists available at ScienceDirect
Physics Letters A www.elsevier.com/locate/pla
Edged topological disordered quantum ring in the presence of magnetic flux Edris Faizabadi ∗ , Mahboubeh Omidi Department of Physics, Iran University of Science and Technology, 16846 Tehran, Iran
a r t i c l e
i n f o
Article history: Received 9 October 2009 Received in revised form 2 February 2010 Accepted 10 February 2010 Available online 18 February 2010 Communicated by R. Wu Keywords: Quantum ring Edged topological disorder Disorder-averaged current Temperature-averaged current Magnetic susceptibility
a b s t r a c t The effects of edged topological disorder on persistent current and magnetic susceptibility are investigated in a two-dimensional quantum ring enclosed within a magnetic flux in tight binding model. In an ordered quantum ring, persistent current shows kink-like variation, while in an edged topological disordered quantum ring, disorder-averaged current gets continuous variation due to appearance of energy gaps in energy spectra. Magnetic susceptibility has delta-like paramagnetic peaks in an ordered quantum ring, while in the edged topological disordered one; the delta-like peaks become wider and begin to overlap. In addition magnetic susceptibility versus magnetic flux shows a sign reversal, which depends on the disordered configuration. In the ordered quantum ring, the temperature-averaged current decreases when the temperature increases, while in the edged topological disordered one, at first temperatureaveraged current raises somewhat and then reduces in a general thermal behavior manner. At absolute zero temperature, in strong regime; the disorder-averaged current versus magnetic flux is periodic with the period of Φ0 /2; whereas, this period is Φ0 in weak regime or in an ordered one (Φ0 is magnetic flux quantum). Within higher temperatures, the period of Φ0 is achieved only in the ordered ring and Φ0 /2 in either weak or strong regime, wide or narrow ring. © 2010 Elsevier B.V. All rights reserved.
1. Introduction During the last decade, a verity of experimental and theoretical probes was concentrated on nanoscale structures [1–8]. Physical understanding of the electronic properties of nanostructures allows us to fabricate the electronic elements which are widely applied as components of the new generation of electronic and photonic devices [1,3]. Quantum rings are fascinating structures because they provide a unique system to study quantum interference phenomena such as Aharonov–Bohm oscillations, magnetoresistence and persistent currents [5,7–11]. Primarily, Hund [12] predicted the existence of persistent currents in quantum rings, and then it was followed by Byers and Yang [13], and Bloch [14]. Also in 1983, Buttiker, Imry, and Landauer predicted the existence of persistent current in a one-dimensional (1D) quantum ring [15]. In 1990, Levy et al. [16] measured persistent current over an ensemble of approximately 107 copper rings. Other experiments were performed by Chandrasekhar et al. [17] on single diffusive gold and by Mially et al. [18] on a semiconductor single loop in the GaAs/GaAlAs system. The experimental results did not match with the theoretical researches because the disorder effects and electron–electron interactions had been neglected. However, several theoretical researches by accounting electron–electron interac-
*
Corresponding author. Tel.: +98(21)77240477; fax: +98(21)77240497. E-mail address: [email protected] (E. Faizabadi).
0375-9601/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2010.02.025
tions [19,20] or different kinds of disorders have been done. In an attempt to consider these effects, Bouzerar et al. studied persistent current in a 1D quantum ring in the presence of interacting electrons with diagonal disorder. They have found that both disorder and interactions always decrease the persistent current by localizing the electrons [21]. Maiti et al. explained the dependence of persistent current and magnetic susceptibility on diagonal disorder strength in the tight-binding model with higher-order hopping integral in the Hamiltonian [22]. Since the mean width of the sample was comparable to its mean radius in experiments, it was necessary to be investigated in a two-dimensional (2D) quantum ring. Helene Bouchiat et al. [23] inspected the effect of bulk disorder on the persistent current in 2D quantum rings. They realized that in the weak disorder limit, average current (I ave = I ) was independent of the number of channels, and in strong disorder limit, I / I 0 was affected by the order of the square of the typical current I 2 / I 02 (expressed in I 0 = ev F / L, where e is electron charge, v F is Fermi velocity, and L is the ring circumference). For the ballistic 2D metallic samples, confinement and surface roughness effects on magnitude of persistent current were investigated experimentally by Chandrasekhar et al. [17] and theoretically by Apel et al. [24]. They found that localized border states contribute coherently to the persistent current and its magnitude is enhanced with respect to their value in the absence of confinement. In another attempt Maiti have studied the effect of edge disorder on quantum transport through a finite width mesoscopic ring by using the Green’s function technique based on the tight-binding formu-
E. Faizabadi, M. Omidi / Physics Letters A 374 (2010) 1762–1768
lation [5]. A novel transport phenomenon is observed which give the enhancement of the current amplitude with the increase of the edge disorder strength in the strong disorder regime, while the amplitude decreases in the weak disorder regime. Also it was shown that the transport properties are significantly influenced by the radius and the width of the ring. The influence of surface disorder on the persistent current in finite width rings was investigated by Chen and Ding [25]. They found that in the weak disorder strength, the persistent current decreased with surface disorder strength, while it increased in the strong disorder strength. Also they showed that the disorderinduced changes in the persistent current strongly depended on both the ring width and its radius. The fabricated quantum rings do not have perfect shapes; the contact between the ring edges and its surrounding causes a kind of disorder in the shapes of the edges which affects on the ring properties. In inverse view, it is possible to manipulate quantum rings with artificial desired edged topological disorder to design new semiconductor materials and devices. Band gap engineering is an effective technique for the design of new semiconductor devices. Heterojunctions and contemporary growth techniques allow band diagrams with nearly arbitrary and continuous band gap variations to be made. The transport properties of electrons can be continuously tuned for a special application. Here by applying artificial desired edged topological disorder on two-dimensional quantum ring a new generation of devices with unique capabilities is emerging. In this work, by using exact diagonalization technique, the edged topological disorder effects in a two-dimensional quantum ring are investigated. The study is performed in strong and weak regimes. In the weak regime, the distance between the edged chain rings (channels) and inner channels is supposed to be large and only hopping strengths of the sites in the edged channels are random. In the other case, the distance between the edged channels and the inner channels is sufficiently small and the hopping strengths of the sites between the edged channels and nearestneighbor channels are also random. The Letter is organized as follows. The Hamiltonian and formalism are discussed in the next section. In Section 3 the results of numerical calculations for Fermi energy, energy spectra, disorder-averaged current and magnetic susceptibility as a function of magnetic flux are presented. Temperature-averaged current versus temperature and magnetic flux is also discussed. Finally, we end the Letter with a brief conclusion. 2. Method and formalism The effects of edged topological disorder on a typical 2D quantum ring threaded by a magnetic flux which is depicted in Fig. 1 are investigated. This system consists of M tight-binding ring chains, simply pointed out as M channels which are concentrically connected. Each channel has N sites of similar atoms. The Hamiltonian of the system in the tight binding model can be described as follows [26]
H=
εi c i† c i +
i
i , j ,m,m
i t mm ij e
j i
.dl † A ci c j
+ c.c.,
(1)
† ci
where and c i are the creation and destruction operators for an electron at the site i; the index j labels the nearest-neighbors of i-th site, and εi and t mm are on-site energy and the hopping ij
is the vector potential in azimuthal diintegral, respectively. A rection and l is a vector that points from the site i to any of its nearest-neighbors. The phase is measured in units of magnetic flux quantum Φ0 = hc /e. Channels are indicated by m and m with respective values of 1, 2, . . . and M with the innermost channel being 1. The hopping integral equals to a constant value of t in
1763
Fig. 1. The schematic plot of a 2D quantum ring with edged topological disorder is threaded by a magnetic flux which includes M channels with N sites.
channel 1 in the ordered quantum ring. The nearest-neighbors of each site in both its channel and its nearest-neighbor channels are considered. The constant t mm i j s between sites in channel m is attained with the relation −1/(1 + (2π (m − 1))/ N )2 . It is supposed that the distance between two successive sites in two sequential channels is the same as the distance between two successive sites in channel 1 of the ordered quantum ring. The edged topological disorder is investigated in two different regimes; weak and strong. In the weak regime, called ETDO quantum ring, the hopping strengths of the sites within channel 1 besides channel M are randomly chosen within the interval of [−1.5, −0.5] in units of t; whereas, in the strong regime, called ETD quantum ring, the hopping strengths of the sites within the channel 1, within channel M, between channels 1 and 2, and also between channels M − 1 and M are accidentally picked out in the same domain. In order to model the edged topological disorder in the weak regime, t i11 and t iMj M are randomly picked out within the j
aforementioned interval, while in ETD quantum ring, all t mm i j s except t i11 , t i12 , t iMj M −1 , and t iMj M are constant; in other words, not j j
and t iMj M but also all t i12 and t iMj M −1 are randomly seonly t i11 j j lected in the strong regime. Since the magnetic flux is confined to a small domain, changing in phase is negligible. By using exact diagonalization technique [27] the eigenenergies of the system are evaluated. Persistent current at zero temperature in the absence of electron–electron interaction can be also written as
I =−
∂ En ∂E =− , ∂Φ ∂Φ n
(2)
where E is the total energy of the system, Φ is the magnetic flux, and E n is eigenenergy. The average current can be evaluated as
I ave = I c ,
(3)
where . . .c is the ensemble average on different realizations of disorder. The temperature-averaged current at none-zero temperature is calculated by
IT =
− ∂ E n e −β E n ∂Φ
n
Z
,
(4)
where Z is the partition function that can be written as
Z=
e −β E n ,
n
where β = 1/( K B T ) and K B is Boltzmann constant.
(5)
1764
E. Faizabadi, M. Omidi / Physics Letters A 374 (2010) 1762–1768
Fig. 2. Typical energy spectra as a function of magnetic flux for 2D quantum ring with N = 100, M = 4, εi = 0.5, constant hopping strengths = −1, and random hopping strengths are distributed within [−1.5, −0.5]. (a) Ordered quantum ring, (b) ETDO quantum ring, (c) ETD quantum ring.
E. Faizabadi, M. Omidi / Physics Letters A 374 (2010) 1762–1768
1765
Magnetic susceptibility is given by derivative of the persistent current with respect to the magnetic flux.
χ (Φ) =
n 3 16π 2
∂ I (Φ) , ∂Φ
(6)
where n is the number of sites. The disorder-averaged susceptibility can be achieved by averaging over the disorder configurations as
χave = χ c .
(7)
3. Results By using the method which is developed in the previous section, the effects of weak and strong edged topological disorders on all the Fermi energy, energy spectra, persistent current, magnetic susceptibility, and temperature-averaged current in the 2D quantum ring are investigated. In our calculations, the energies are given in units of the hopping integral t, Φ is measured in fundamental flux Φ0 , current is in unit of t /Φ0 , and the magnetic susceptibility is in unit of t /Φ02 . We consider M = 4, N = 100, the number of electrons in the ring ( N e ) = 200 (half filling region), εi = 0.5, random hopping strengths are distributed within the interval [−1.5, −0.5] and the constant hopping energy is equal to −1. The most electronic properties of the system can be explained with the energy spectra. In Fig. 2 the energy spectra as a function of magnetic flux are depicted. In an ordered quantum ring, the energy levels cross each other and there are many degeneracies without any energy gap in the spectrum. In a strictly one-dimensional (1D) quantum ring the energy levels have a close relation with the energy levels of one-dimensional Bloch problem. It is possible to think that our system is composed of M 1D chain rings whose discrete energy levels are mixed to construct the schemes depicted in Fig. 2(a). In other words, the energy spectra which are illustrated in Fig. 2(a) are a blend of these discrete energy levels. In an edged topological disordered quantum ring, the energy levels repel each other and energy gaps appear in the energy spectra. These energy gaps in the spectrum can be interpreted as a back-scattering happening due to the boundaries of the sample with edged topological disorder. The Fermi energy versus magnetic flux is shown in Fig. 3. It varies periodically with the period of Φ0 which arises from Aharonov–Bohm effect. If an electron moves in a field-free region that is not simply connected, but surrounds a hole containing magnetic flux Φ , then upon completing a circuit, the electron acquires an additional phase factor e ieΦ/c , under the condition that the electron wave function is single-valued; therefore, the phase factor is unity, implying that the enclosed flux is quantized [28]. In the edged topological disordered quantum ring, the Fermi energy tends towards continuous variation, which is due to back-scattering occurrence at disordered boundaries of the system. Also the variation of Fermi energy as a function of magnetic flux (Fig. 3) shows that Fermi energy level decreases in the presence of edge topological disorder. In ETD quantum ring (Fig. 3(c)) Fermi energy level is lower than this level in ETDO quantum ring. The system with disordered edge as a result of lowering Fermi energy is more stable and more compatible with nature. The persistent current can be derived by differentiating energy with respect to magnetic flux, and the average current can be obtained by averaging over all disorder configurations. In Fig. 4(a) the variation of persistent current versus magnetic flux is illustrated for an ordered quantum ring. Also the results for the dependence of disorder-averaged current on the magnetic flux over 500 configurations in ETDO and ETD quantum ring are depicted in Figs. 4(b) and 4(c) in the half filling region, respectively. In the
Fig. 3. Fermi energy versus magnetic flux for 2D quantum ring with N = 100, M = 4, εi = 0.5, constant hopping strengths = −1 and random hopping strengths are distributed within [−1.5, −0.5] in half filling region. (a) Ordered quantum ring, (b) ETDO quantum ring, (c) ETD quantum ring.
ordered quantum ring (Fig. 4(a)), persistent current has kink-like variation due to the different contributions of the energy levels in the mixed energy band region. In the edged topological disordered quantum rings (Figs. 4(b) and 4(c)), the current varies continuously with magnetic flux as a result of edged topological disorder which breaks the symmetries and help to lift most of the degeneracies. In addition, the extremum points are shifted with respect to the ordered quantum ring. Lifting degeneracies and the displacement of extremum points can be described as a back-scattering occurrence at disordered edges of the system. As disorder strengthens, the localized states increase, and the magnitude of disorder-averaged current will be less. These are illustrated in Figs. 4(b) and 4(c) in comparison to Fig. 4(a) that has not any disorder. In a narrow ETD quantum ring; the disorder-averaged current versus magnetic flux is periodic with the period of Φ0 /2; whereas, this period is Φ0 in an ETDO or in an ordered one. In ETDO quantum ring, just the hopping strengths in innermost and outermost channels are random, so the disorder hoppings do not affect the
1766
E. Faizabadi, M. Omidi / Physics Letters A 374 (2010) 1762–1768
Fig. 4. (a) Persistent current versus magnetic flux in the ordered quantum ring in half filling region; (b) Disorder-averaged current versus magnetic flux in the ETDO quantum ring in half filling region; (c) Disorder-averaged current versus magnetic flux in the ETD quantum ring in half filling region.
inner quantum channels. But in the ETD quantum ring, since the hopping strengths of the sites between channels 1 and 2, and also M and M − 1 are random, disordered effects influence on the whole system and affect on changing on all different disordered configurations diagrams. This process leads to the period of Φ0 /2 that agrees with Levy’s experimental result [15]. It is necessary to ∂E mention that I = − ∂Φ and M = − ∂∂ BE to relate our calculation to Levy’s experiment. In fact, the period of Φ0 /2 is a disordered statistical phenomenon, which emerges as a result of configurational averaging over strong disorderliness (at T = 0). In the absence of edged topological disorder, the period becomes Φ0 because our ensemble contains identical rings. Besides, in the weak regime; we also obtain the same result as our ensemble includes almost similar rings. The period approaches to Φ0 /2 only in the strong regime within limit of narrow rings, in which our ensemble contains dissimilar rings. Fig. 5(a) represents variation of magnetic susceptibility as a function of magnetic flux in the ordered quantum ring. In addition, the variation of disorder-averaged susceptibility by averaging over 500 disorder configurations is depicted in Figs. 5(b)
Fig. 5. (a) Magnetic susceptibility as a function of magnetic flux in the ordered quantum ring; (b) Disorder-averaged susceptibility as a function of magnetic flux in the ETDO quantum ring; (c) Disorder-averaged susceptibility as a function of magnetic flux in the ETD quantum ring.
and 5(c) for the edged topological disordered quantum ring. The magnetic susceptibility has delta-like paramagnetic peaks in ordered quantum ring and these peaks are related to the crossing of the single-electron states. In the presence of edged topological disorder, the delta-like peaks become wider and the overlap between these broadened peaks is observed in Figs. 5(b) and 5(c). They can also be described by the back-scattering incident at the boundaries with edged topological disorder. Moreover, increasing of localized states decreases magnitude of disorder-averaged susceptibility in edged topological disordered quantum ring. In addition, it is illustrated that magnetic susceptibility as a function of magnetic flux shows a sign reversal which depends on the disorder configurations. Figs. 6(a), 6(b) and 6(c) show the dependence of temperatureaveraged current (I T ) on temperature and magnetic flux for the ordered quantum ring, ETDO quantum ring and ETD quantum ring, respectively. The system consists of 400 spinless electrons and Figs. 6(b) and 6(c) are averaged over 10 disorder configurations. In
E. Faizabadi, M. Omidi / Physics Letters A 374 (2010) 1762–1768
Fig. 6. Temperature-averaged current versus magnetic flux and temperature. (a) Ordered quantum ring, (b) ETDO quantum ring, (c) ETD quantum ring.
1767
1768
E. Faizabadi, M. Omidi / Physics Letters A 374 (2010) 1762–1768
the ordered quantum ring I T decreases by increasing temperature as a result of randomization of occupation of the energy levels, so the current components averagely annul each other. In the disordered quantum ring, as a result of symmetry breaking, the degeneracies in energy spectra are lifted, and by increasing the allowed levels close to Fermi energy as a result of splitting of degenerate level, at first I T increases to some extent and then the general loss initiates. In an ordered quantum ring temperature-averaged current versus magnetic flux is periodical with the period of Φ0 while in ETDO and ETD quantum rings, the period of oscillations is Φ0 /2. This can be understood by the scenario that within higher temperature respect to absolute zero, whole the energy levels of any ring will be effective in the temperature-averaged current because all the system energy levels have an occupation probability. So, edged topological disorder affects the temperature-averaged current any way. The period of Φ0 is achieved only in the absence of edged topological disorder and Φ0 /2 in either weak or strong regime, wide or narrow ring. 4. Summery and conclusion To summarize, the effects of edged topological disorder is investigated in two different regimes. In the weak regime, the distance between channels 1 and M and their nearest neighbors is supposed to be sufficiently large, i.e. the edged topological disorder does not affect inner channels, and this is true for ETDO quantum rings. The strong regime refers to edged topological disorder in which the distance between channels 1 and M and their nearest neighbors is small enough to penetrate to interior channels (the ETD quantum rings). It is shown that energy spectra show crossing of the energy levels in the ordered quantum ring; and with appearing of edged topological disorder the energy levels repel each other and the energy gaps appear. In contrast to the ordered quantum ring, in the presence of disorder, the Fermi energy tends to vary continuously as a function of magnetic flux. It is also shown that the disorderaveraged current varies continuously as a function of magnetic flux in the edged topological disordered quantum ring. At absolute zero temperature, the period of oscillation of disorder-averaged current is remained Φ0 in the ETDO quantum ring while it approaches to Φ0 /2 in the strong regime within limit of narrow rings, in which our ensemble contains dissimilar rings. This variation is due to influence of disordered effects on the whole system in the ETD quantum ring. In addition, magnetic susceptibility versus magnetic flux has delta-like paramagnetic peaks
in ordered quantum ring. In the presence of edged topological disorder, these delta-like peaks become wider and begin to overlap. In addition, it is illustrated that magnetic susceptibility as a function of magnetic flux shows a sign reversal, which depends on the disorder configurations. In the ordered quantum ring, temperatureaveraged current decreases by raising the temperature, and in edged topological disordered quantum ring, this current at first begins to increase to some extent and then decreases in the same way as general thermal behavior. In addition, in edged topological disordered quantum ring, the period of oscillations of I T becomes Φ0 /2. Overall, the unique condition for the period of Φ0 /2 in absolute zero temperature is the strong regime situation within the limit of narrow rings; whereas, the presence of edged topological disorder (weak or strong) is enough within higher temperatures whether ring is narrow or not. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28]
Jianxin Zhong, G. Malcolm Stocks, Nano Lett. 6 (2006) 128. Jianxin Zhong, G. Malcolm Stocks, Phys. Rev. B 75 (2007) 033410 (4 pages). C.Y. Yang, J.W. Ding, N. Xu, Physica B 394 (2007) 69. Santanu K. Maiti, J. Nanosci. Nanotech. 9 (9) (2009) 5664. Santanu K. Maiti, Slolid State Sci. 10 (2008) 1456. Santanu K. Maiti, Chem. Phys. Lett. 446 (2007) 365. J.H. Dai, Y.L. Lin, S.C. Lee, IEEE Photon. Technol. Lett. 19 (2007) 1511. J.H. Dai, J.H. Lee, Y.L. Lin, S.C. Lee, Jpn. J. Appl. Phys. 47 (2008) 2924. Y. Aharonov, D. Bohm, Phys. Rev. 115 (1959) 485. R.A. Webb, S. Washburn, C.P. Umbach, R.B. Laibowitz, Phys. Rev. Lett. 54 (1985) 2696. J.M. García, G. Medeiros-Ribeiro, K. Schmidt, T. Ngo, J.L. Feng, A. Lorke, J. Kotthaus, P.M. Petroff, Appl. Phys. Lett. 71 (1997) 2014. F. Hund, Ann. Phys. (Leipzig) 32 (1938) 102. N. Byers, C.N. Yang, Phys. Rev. Lett. 7 (1961) 46. F. Bloch, Phys. Rev. 137 (1965) A787. M. Buttiker, Y. Imry, R. Landauer, Phys. Lett. A 96 (1983) 365. L.P. Lévy, G. Dolan, J. Dunsmuir, H. Bouchiat, Phys. Rev. Lett. 64 (1990) 2074. V. Chandrasekhar, R.A. Webb, M.J. Brady, M.B. Ketchen, W.J. Gallagher, A. Kleinsasser, Phys. Rev. Lett. 67 (1991) 3578. D. Mailly, C. Chapelier, A. Benoit, Phys. Rev. Lett. 70 (1993) 2020. H. Kato, D. Yoshioka, Phys. Rev. B 50 (1994) 4943. A. Muller-Groeling, H.A. Weidenmuller, Phys. Rev. B 49 (1994) 4752. G. Bouzerar, D. Poilblanc, G. Montambaux, Phys. Rev. B 49 (1994) 8258. S.K. Maiti, J. Chowdhury, S.N. Karmakar, J. Phys.: Condens. Matter 18 (2006) 5349. H. Bouchiat, G. Montambaux, J. Phys. (Paris) 50 (1989) 2695. V.M. Apel, G. Chiappe, M.J. Sanchez, Phys. Rev. Lett. 85 (2000) 4152. H.B. Chen, J.W. Ding, Physica B 403 (2008) 2015. E. Faizabadi, M. Omidi, Phys. Lett. A 373 (2009) 1469. H.Q. Lin, J.E. Gubernatis, J. Tobochnik, Comput. Phys. 7 (1993) 401. S. Gasiorowicz, Quantum Physics, second edition, John Wiley & Sons, 1996.