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Transport properties of graphene under periodic and quasiperiodic magnetic superlattices
Physics Letters A 377 (2013) 1368–1372
Contents lists available at SciVerse ScienceDirect
Physics Letters A www.elsevier.com/locate/pla
Transport properties of graphene under periodic and quasiperiodic magnetic superlattices Wei-Tao Lu a,b,∗ , Shun-Jin Wang c , Yong-Long Wang a,b , Hua Jiang a,b , Wen Li a a b c
School of Science, Linyi University, 276005 Linyi, China Institute of Condensed Matter Physics, Linyi University, 276005 Linyi, China Department of Physics, Sichuan University, 610064 Chengdu, China
a r t i c l e
i n f o
Article history: Received 3 December 2012 Received in revised form 6 March 2013 Accepted 27 March 2013 Available online 30 March 2013 Communicated by R. Wu Keywords: Graphene Quasiperiodic magnetic superlattice Transport property
a b s t r a c t We study the transmission of Dirac electrons through the one-dimensional periodic, Fibonacci, and Thue– Morse magnetic superlattices (MS), which can be realized by two different magnetic blocks arranged in certain sequences in graphene. The numerical results show that the transmission as a function of incident energy presents regular resonance splitting effect in periodic MS due to the split energy spectrum. For the quasiperiodic MS with more layers, they exhibit rich transmission patterns. In particular, the transmission in Fibonacci MS presents scaling property and fragmented behavior with self-similarity, while the transmission in Thue–Morse MS presents more perfect resonant peaks which are related to the completely transparent states. Furthermore, these interesting properties are robust against the profile of MS, but dependent on the magnetic structure parameters and the transverse wave vector. © 2013 Elsevier B.V. All rights reserved.
1. Introduction The physical properties of graphene in the presence of inhomogeneous perpendicular magnetic fields have attracted considerable attention, since its realization. The transport and bound states of Dirac electrons in graphene were reported in various magnetic structures involving single barrier [1], several barriers [2,3], and quantum dots [4]. There also exist many theoretical works on periodic magnetic superlattices (MS) in graphene. Dell’Anna found that the Fermi velocity at Dirac points is isotropically renormalized in MS, in contrast to the case of electric superlattices, and the spectrum and the nature of the states strongly depend on the conserved longitudinal momentum and the barrier width of MS [5]. The low-energy electronic structure of graphene under a one-dimensional MS could be mapped into that of graphene under an electric superlattice or vice versa [6]. The gapped states were studied analytically in graphene under periodic magnetic and electric fields [7]. We found that the transport has a general splitting rule through MS in graphene, of which the corresponding vector potential is a periodic field, and the splitting is independent of the MS profile [8]. In Refs. [9] and [10], the electronic properties of a magnetic Kronig–Penney superlattice with δ -function barriers have been discussed in graphene, where electron trans-
*
Corresponding author at: School of Science, Linyi University, 276005 Linyi, China. E-mail address: [email protected] (W.-T. Lu). 0375-9601/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.physleta.2013.03.035
port could be understood in terms similar to light propagation in periodic stratified media. On the other hand, many works on the quasiperiodic system [11] have been performed, which is an intermediate case between periodic and disordered ones. The Fibonacci superlattices and Thue–Morse superlattices are two typical quasiperiodic systems discussed widely. Those studies have shown that the quasiperiodic systems have a highly fragmented energy spectrum, and their eigenstates can be critical states with self-similar pattern which are neither extended nor localized [12,13]. Many theoretical works on optical transmission, energy spectrum, and density of states in various quasiperiodic systems have been reported [12–17]. Recently, the transport properties through quasiperiodic electric superlattices in graphene have been investigated [18–20]. Biswas discussed the resonant tunneling through a Fibonacci superlattice in bilayer graphene [18]. Sena et al. found that the spectrum of quasibound states in Fibonacci graphene superlattice distributes as a Cantor-like set by virtue of transverse wave vector [19]. However, the results on transport of Dirac electrons in quasiperiodic MS are still lacking, and the magnetic field greatly affects the physical properties of Dirac electrons in comparison to the pure electric field, especially for the Klein tunneling. In this work, the quasiperiodic MS in graphene are considered, involving Fibonacci MS and Thue–Morse MS. The required magnetic profile can be produced by ferromagnetic stripes located on top of the graphene layer, or by virtue of other means [21]. The transmission of Dirac electron through the quasiperiodic MS is discussed theoretically based on
W.-T. Lu et al. / Physics Letters A 377 (2013) 1368–1372
1369
where the Fermi velocity v f ≈ 0.86 × 106 m/s, the pseudospin matrix σ = (σx , σ y ) is the Pauli matrix, p = ( p x , p y ) is the momentum operator, and A(x) is the vector potential. For convenience, the dimensionless units are introduced: l B = h¯ /e B 0 , E 0 = h¯ v f /l B , B (x) → B 0 B (x), A (x) → B 0 l B A (x), r → l B r , k → k/l B , and E → E 0 E. For a typical value B 0 = 0.1 T, we have l B = 81 nm and E 0 = 7.0 meV. The magnetic field and the corresponding vector potential are infinite and homogeneous along the y-direction, resulting in the conservation of the transverse wave vector k y . The vector potential A is constant in each region of the models. For given incident energy E and transverse wave vector k y , the solution in jth region of Eq. (1) could be written as Ψ j (x, y ) = ψ j (x)e ik y y with Fig. 1. The schematics of magnetic fields of building blocks P and Q indicated by the black arrows, and the corresponding vector potentials indicated by shaded areas.
numerical results and compared with the ones in periodic MS. It is found that the transmission as a function of incident energy is fragmented and has common structure in quasiperiodic MS, and the features are quite different from the results for quasiperiodic electric superlattices in graphene [18–20]. The nature of these features originates from the quasiperiodicity of the systems, analogous to the optical transmission spectrum in quasiperiodic photonic structures. The Letter is organized as follows. In Section 2, we introduce the periodic MS, Fibonacci MS, and Thue–Morse MS, and the transfer-matrix method is used. We show the numerical results and discussions in Section 3. Finally, we draw conclusions in Section 4. 2. Model and method The Fibonacci structure can be realized by juxtaposing the two basic building blocks P and Q in Fibonacci sequence, and the nth generation of the process S n is given by the recursive rule S n = S n−1 S n−2 , for n 1, starting with S 0 = P and S −1 = Q . The Fibonacci generations are S 1 = P Q , S 2 = P Q P , S 3 = P Q P P Q , etc. The total number of building blocks P and Q in each sequence is equal to the Fibonacci number F n = F n−1 + F n−2 with F 0 = F −1 = 1. The Thue–Morse structure based on Thue–Morse sequence can be defined by the recursive relation U n = U n−1 U n−1 , for n 1, with U 0 = P Q and U 0 = Q P , where U n is the complement of U n . The Thue–Morse generations are U 1 = P Q Q P , U 2 = P Q Q P Q P P Q , etc., and the total number of building blocks in each sequence is equal to 2n+1 . We shall consider the one-dimensional periodic and quasiperiodic MS perpendicular to the plane of graphene. The magnetic field is assumed to be uniform along the y-direction and to vary along the x-direction. The quasiperiodic Fibonacci MS and Thue–Morse MS can be realized by two magnetic blocks P and Q arranged in Fibonacci and Thue–Morse sequences, respectively. In addition, three periodic MS are considered, i.e., ( P )m , ( P Q )m , and ( P Q Q )m with the period number m, where P , P Q , and P Q Q are the unit cells of the three periodic MS, respectively. Fig. 1 depicts the profiles of magnetic blocks P and Q , each of which is made up of two opposite magnetic δ -function barriers. In the Landau gauge, both corresponding vector potentials A P and A Q have rectangular shapes with barrier widths d P / Q and well widths l P / Q . Thus, all the vector potential fields are superlattices, and their structures are the same as their corresponding MS, which play a key role to the following transport properties. At low energy, the electron in graphene could be described by an effective massless Dirac equation with a linear energy dispersion. In the presence of a magnetic field perpendicular to the plane, the equation reads as
v f σ · p + eA(x) Ψ = E Ψ,
(1)
ψ j (x) = a j
1
q j +ik j E
e iq j x + b j
1
−q j +ik j
e −iq j x .
(2)
E
Here, k j = k y + A j , q j is the longitudinal wave vector satisfying
q2j + (k y + A j )2 = E 2 .
(3)
In order to calculate the transmission probability and the energy levels of quasibound states for periodic and quasiperiodic MS, the transfer-matrix method Eq. (2) can be rewrit is employed.
ten as ψ j (x) = G j H j Hj =
e
iq j x
0
0
e
−iq j x
aj
bj
, where G j =
1
1
(q j +ik j )/ E (−q j +ik j )/ E
and
. Based on the continuity condition of the wave
functions at the interface x = x j between jth and ( j + 1)th regions,
a
a
1 , and M j = H − (x )G −j+11 G j H j (x j ). j +1 j Thus, the total transfer-matrix for MS with n regions can be written as M = M n−1 · · · M j · · · M 1 . Then the transmission probability can be obtained from T = 1 − | M 21 |2 /| M 22 |2 , and M i j is the matrix element of M. Assuming that the eigenstates decay exponentially in the vector potential barriers at both extremities of the MS, one may get the condition for quasibound states [22]:
one can get:
j +1
b j +1
= Mj
M 22 = 0,
j
bj
(4)
in the energy region k y < E < ( A + k y ) where the eigenstates are evanescent inside the barriers and propagating inside the wells. 3. Results and discussions In this section, the transport properties of Dirac electron through periodic MS, Fibonacci MS, and Thue–Morse MS in graphene are studied numerically. The widths of barriers and wells for both magnetic blocks are the same and fixed as d P / Q = l P / Q = 0.5 in units of l B in the following results. First, the transmission probability as a function of incident energy for periodic MS ( P )5 , ( P Q )5 , and ( P Q Q )5 is shown in Figs. 2(a)–(c) at A P = 3.0 and A Q = 1.0. Obviously, the transmission exhibits a new kind of resonance, which is not the Klein tunneling due to the suppression of the Klein tunneling in this energy region. From Fig. 2(a) we can see that the transmission exhibits 4-fold resonance splitting in ( P )5 , and the splitting is (m − 1)-fold in ( P )m [8]. Quite differently, for periodic MS ( P Q )5 of which the unit cell is arranged with two magnetic blocks, the transmission presents two resonant domains in the considered energy region, and there are 4-fold resonant peaks in each domain, as shown in Fig. 2(b). Compared with ( P )5 , the resonant domains in ( P Q )5 become narrow, due to the narrowed energy band of quasibound states which will be shown later. Fig. 2(c) shows that the transmission presents three resonant domains in ( P Q Q )5 of which the unit cell is arranged with three magnetic blocks. Thus, it can be concluded that the transmission would present n resonant domains and (m − 1)-fold resonant peaks in each domain, for the periodic MS of which the period number is m and the unit cell is arranged with n magnetic blocks. Furthermore, the position of
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Fig. 2. Transmission probability versus incident energy in periodic MS. Here A P = 3.0, A Q = 1.0, the solid curves for k y = 0.0, and dashed curves for k y = 0.7. The periodic MS (a) ( P )5 , (b) ( P Q )5 , and (c) ( P Q Q )5 .
Fig. 4. Transmission probability versus incident energy at A P = 3.0, A Q = 1.0, and k y = 0.0. (a) The solid curve for periodic MS ( P )4 with 4 layers, dashed curve for the 3rd generation Fibonacci MS with 5 layers, and dotted curve for the 1st generation Thue–Morse MS with 4 layers. (b) The solid curve for periodic MS ( P )8 with 8 layers, dashed curve for the 4th generation Fibonacci MS with 8 layers, and dotted curve for the 2nd generation Thue–Morse MS with 8 layers.
Fig. 3. The distribution of the energy levels for the periodic MS as a function of the period number m, at A P = 3.0, A Q = 1.0, and k y = 0.0. The periodic MS (a) ( P )m , (b) ( P Q )m , and (c) ( P Q Q )m .
resonant domain moves to the right in the energy region with the increase of transverse wave vector k y . This regular resonant transmission is the reflection of energy spectrum for the periodic MS, which is determined by its structure. Due to the coupling of eigenstates via tunneling in MS, the degenerate energy levels of the independent wells become nondegenerate and split. Then these split levels redistribute themselves into groups around their unperturbed position and form bands, and each band contains the same amount of energy levels. The energy levels for MS can be obtained by solving Eq. (4). As shown in Figs. 3(a)–(c), one energy band is formed in ( P )m , two bands in ( P Q )m , and three bands in ( P Q Q )m . With the increase of period number m, the energy levels are distributed symmetrically. The transmission resonance occurs when the incident energy coincides with the energy level of quasibound state. Thus, as an application, the resonant peaks can be used to probe the energy spectrum of graphene-based interacting multi-well structures. In contrast to periodic MS, the transmission in quasiperiodic MS exhibits more complicated resonant features. Figs. 4(a) and (b) display the transmission probability as a function of incident energy for periodic MS, Fibonacci MS, and Thue–Morse MS with fewer layers. From Fig. 4(b) we can see that, for all the three
Fig. 5. Transmission probability versus incident energy for Fibonacci MS at A P = 2.0, A Q = 1.0, and k y = 0.0. (a) The 8th generation with 110 layers, (b) the 10th generation with 288 layers, and (c) the 12th generation with 754 layers.
MS with 8 layers, in comparison to that in periodic MS, the total number of resonant peaks in Fibonacci MS and Thue–Morse MS decreases and their distribution becomes irregular. Most resonant peaks in Thue–Morse MS are unity, which does not hold in Fibonacci MS. In addition, the transmission for the 1st generation Thue–Morse MS U 1 = P Q Q P presents regular resonant peaks, due to its symmetrical structure, as shown in Fig. 4(a). The complicated transmission in quasiperiodic MS implies its less regular energy spectrum. However, for Fibonacci MS with more layers, the transmission presents rich and relatively regular structure. Figs. 5(a)–(c) show the transmission probability as a function of incident energy for the 8th, 10th, and 12th generations Fibonacci MS, respectively, at A P = 2.0, A Q = 1.0 and k y = 0.0. One can see that all the transmissions display similar structure and fractal property in energy space, i.e., the transmission zones and transmission gaps for different Fibonacci MS are arranged in a similar way. With
W.-T. Lu et al. / Physics Letters A 377 (2013) 1368–1372
Fig. 6. Transmission probability versus incident energy for the 13th generation Fibonacci MS with 1220 layers. The parameters are the same as those in Fig. 5.
the increase of the number of layers, more transmission zones disappear gradually, meanwhile, more resonant peaks appear, and the fractal property becomes more evident. This fractal property arises from the fragmented energy spectrum of Fibonacci MS, regardless of the value of transverse wave vector k y . In order to show the self-similarity of transmission for Dirac electron in Fibonacci MS, the transmission probability as a function of incident energy for the 13th generation Fibonacci MS is shown in Figs. 6(a)–(c). Fig. 6(a) shows the transmission in the energy region 0.5 < E < 3.0. Figs. 6(b) and (c) show the results of Fig. 6(a) in the reduced energy regions 1.48 < E < 1.98 and 1.67 < E < 1.77, respectively. Comparing these figures, one can readily observe the self-similar behavior between the whole and the local transmission spectra. This feature is analogous to the optical transmission spectrum in quasiperiodic photonic structures arranged in a Fibonacci sequence [15,17], to some extent, which is the refection of the self-similarity in the corresponding energy spectrum. The self-similarity is considered as a fundamental property of the transmission spectrum for quasiperiodic structure, not found in other parts of the full energy region. Furthermore, the energy center of self-similarity is E = 1.723, approximately equal to the energy level of the 1th Fibonacci MS, nearby which the transmission spectrum is distributed symmetrically, as shown in Fig. 6(c), implying the effect of periodicity in Fibonacci MS. In addition, the transmission of Dirac electron in Fibonacci MS also has the scaling property with respect to the generation number. To illustrate the scaling property, the transmission probability as a function of incident energy for the 7th, 8th, 13th, and 14th generations Fibonacci MS is shown in Figs. 7(a)–(d). It can be seen that, the transmission profiles for the 7th and 8th generations in the energy region 1.5 < E < 2.0 are the same as the ones for the 13th and 14th generations in the energy region 1.714 < E < 1.733, respectively, and the transmission profile is recovered after six generations, implying the six-cycle feature. The transmission spectrum is almost symmetrical about E = 1.723. The scale factor which indicates a reduction of energy is approximately equal to 26, the same as the result on optical transmission spectrum [17]. It is noteworthy that the self-similarity and the scaling property of Dirac electron discussed above are sensitive to the structure of MS, but weakly dependent on its profile. Finally, we study the transmission of Dirac electron in Thue– Morse MS with more layers. Figs. 8(a)–(c) show the transmission
1371
Fig. 7. Transmission probability versus incident energy for Fibonacci MS, and the parameters are the same as those in Fig. 5. (a) The 7th generation with 68 layers, (b) the 8th generation with 110 layers, (c) the 13th generation with 1220 layers, and (d) the 14th generation with 1974 layers.
Fig. 8. Transmission probability versus incident energy for Thue–Morse MS, and the parameters are the same as those in Fig. 5. (a) The 4th generation with 64 layers, (b) the 6th generation with 256 layers, and (c) the 8th generation with 1024 layers.
probability as a function of incident energy for the 4th, 6th, and 8th generations Thue–Morse MS, respectively. One may find that all the transmissions are fragmented and have common structure, similar to Fibonacci MS. With the increase of the generation, more and more perfect resonant peaks with T = 1 are presented, which are related to the completely transparent states originating from a special positional correlation between two magnetic blocks in Thue–Morse MS [13,14]. However, different from Fibonacci MS, the transmission in Thue–Morse MS does not possess self-similarity and scaling property. The above results indicate that Fibonacci systems behave more regularly than Thue–Morse systems as far as the transmission is concerned, in agreement with previous claim [16]. 4. Conclusion In this Letter, we have discussed the transport properties of Dirac electron through periodic and quasiperiodic MS in graphene, with the help of transfer-matrix method. For periodic MS, the transmission presents regular resonance splitting effect in energy
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space. Both transmissions in Fibonacci MS and Thue–Morse MS with fewer layers present irregular oscillation. However, with the increase of number of layers, the quasiperiodic MS exhibit rich transmission patterns, such as the self-similarity and scaling property in Fibonacci MS, and the perfect resonant peaks in Thue– Morse MS, which originate from the quasiperiodicity of the systems and the suppression of Klein tunneling. Furthermore, the transmission is sensitive to the incident energy, transverse wave vector, and magnetic structure parameters. The above transport properties should also exist in the more realistic MS with smooth vector potential in experiment. In fact, the results obtained in this Letter could be extended to the case where the magnetic blocks are of different widths and the same strengths. We hope that the results may be useful for the understanding on physical property of Dirac electron in quasiperiodic systems, even for the potential technological applications in future [23]. Acknowledgements This work was supported by the National Natural Science Foundation of China (Grant Nos. 10974137 and 11047020), and the Natural Science Foundation of Shandong Province, China (Grant Nos. ZR2012AM022 and ZR2010AQ025). References [1] A. De Martino, L. Dell’Anna, R. Egger, Phys. Rev. Lett. 98 (2007) 066802; A. De Martino, L. Dell’Anna, R. Egger, Solid State Commun. 144 (2007) 547. [2] F. Zhai, K. Chang, Phys. Rev. B 77 (2008) 113409.
[3] M.R. Masir, P. Vasilopoulos, A. Matulis, F.M. Peeters, Phys. Rev. B 77 (2008) 235443; M.R. Masir, P. Vasilopoulos, F.M. Peeters, Appl. Phys. Lett. 93 (2008) 242103. [4] A.V. Rozhkov, G. Giavaras, Y.P. Bliokh, V. Freilikher, F. Nori, Phys. Rep. 503 (2011) 77. [5] L. Dell’Anna, A. De Martino, Phys. Rev. B 79 (2009) 045420; L. Dell’Anna, A. De Martino, Phys. Rev. B 83 (2011) 155449. [6] L.Z. Tan, C.H. Park, S.G. Louie, Phys. Rev. B 81 (2010) 195426. [7] I. Snyman, Phys. Rev. B 80 (2009) 054303. [8] W.T. Lu, W. Li, Y.L. Wang, C.Z. Ye, H. Jiang, J. Appl. Phys. 112 (2012) 083712. [9] M.R. Masir, P. Vasilopoulos, F.M. Peeters, New J. Phys. 11 (2009) 095009; M.R. Masir, P. Vasilopoulos, F.M. Peeters, J. Phys.: Condens. Matter 22 (2010) 465302. [10] S. Ghosh, M. Sharma, J. Phys.: Condens. Matter 21 (2009) 292204; M. Sharma, S. Ghosh, J. Phys.: Condens. Matter 23 (2011) 055501. [11] D. Schechtman, I. Blech, D. Gratias, J.W. Cahn, Phys. Rev. Lett. 53 (1984) 1951. [12] M. Kohmoto, L.P. Kadanoff, C. Tang, Phys. Rev. Lett. 50 (1983) 1870; E. Macia, F. Dominguez-Adame, Phys. Rev. Lett. 76 (1996) 2957. [13] C.S. Ryu, G.Y. Oh, M.H. Lee, Phys. Rev. B 46 (1992) 5162; C.S. Ryu, G.Y. Oh, M.H. Lee, Phys. Rev. B 48 (1993) 132. [14] N.H. Liu, Phys. Rev. B 55 (1997) 3543. [15] M. Kohmoto, B. Sutherland, K. Iguchi, Phys. Rev. Lett. 58 (1987) 2436. [16] P. Carpena, V. Gasparian, M. Ortuno, Phys. Rev. B 51 (1995) 12813. [17] E.L. Albuquerque, M.G. Cottam, Phys. Rep. 376 (2003) 225. [18] S. Mukhopadhyay, R. Biswas, C. Sinha, Phys. Status Solidi B 247 (2010) 342. [19] S.H.R. Sena, J.M. Pereira Jr, G.A. Farias, M.S. Vasconcelos, E.L. Albuquerque, J. Phys.: Condens. Matter 22 (2010) 465305. [20] P.L. Zhao, X. Chen, Appl. Phys. Lett. 99 (2011) 182108; T.X. Ma, C. Liang, L.G. Wang, H.Q. Lin, Appl. Phys. Lett. 100 (2012) 252402. [21] S.J. Lee, S. Souma, G. Ihm, K.J. Chang, Phys. Rep. 394 (2004) 1. [22] H. Chau Nguyen, M. Tien Hoang, V. Lien Nguyen, Phys. Rev. B 79 (2009) 035411. [23] E. Macia, Appl. Phys. Lett. 73 (1998) 3330; R.W. Peng, X.Q. Huang, F. Qiu, M. Wang, A. Hu, S.S. Jiang, M. Mazzer, Appl. Phys. Lett. 80 (2002) 3063.
Contents lists available at SciVerse ScienceDirect
Physics Letters A www.elsevier.com/locate/pla
Transport properties of graphene under periodic and quasiperiodic magnetic superlattices Wei-Tao Lu a,b,∗ , Shun-Jin Wang c , Yong-Long Wang a,b , Hua Jiang a,b , Wen Li a a b c
School of Science, Linyi University, 276005 Linyi, China Institute of Condensed Matter Physics, Linyi University, 276005 Linyi, China Department of Physics, Sichuan University, 610064 Chengdu, China
a r t i c l e
i n f o
Article history: Received 3 December 2012 Received in revised form 6 March 2013 Accepted 27 March 2013 Available online 30 March 2013 Communicated by R. Wu Keywords: Graphene Quasiperiodic magnetic superlattice Transport property
a b s t r a c t We study the transmission of Dirac electrons through the one-dimensional periodic, Fibonacci, and Thue– Morse magnetic superlattices (MS), which can be realized by two different magnetic blocks arranged in certain sequences in graphene. The numerical results show that the transmission as a function of incident energy presents regular resonance splitting effect in periodic MS due to the split energy spectrum. For the quasiperiodic MS with more layers, they exhibit rich transmission patterns. In particular, the transmission in Fibonacci MS presents scaling property and fragmented behavior with self-similarity, while the transmission in Thue–Morse MS presents more perfect resonant peaks which are related to the completely transparent states. Furthermore, these interesting properties are robust against the profile of MS, but dependent on the magnetic structure parameters and the transverse wave vector. © 2013 Elsevier B.V. All rights reserved.
1. Introduction The physical properties of graphene in the presence of inhomogeneous perpendicular magnetic fields have attracted considerable attention, since its realization. The transport and bound states of Dirac electrons in graphene were reported in various magnetic structures involving single barrier [1], several barriers [2,3], and quantum dots [4]. There also exist many theoretical works on periodic magnetic superlattices (MS) in graphene. Dell’Anna found that the Fermi velocity at Dirac points is isotropically renormalized in MS, in contrast to the case of electric superlattices, and the spectrum and the nature of the states strongly depend on the conserved longitudinal momentum and the barrier width of MS [5]. The low-energy electronic structure of graphene under a one-dimensional MS could be mapped into that of graphene under an electric superlattice or vice versa [6]. The gapped states were studied analytically in graphene under periodic magnetic and electric fields [7]. We found that the transport has a general splitting rule through MS in graphene, of which the corresponding vector potential is a periodic field, and the splitting is independent of the MS profile [8]. In Refs. [9] and [10], the electronic properties of a magnetic Kronig–Penney superlattice with δ -function barriers have been discussed in graphene, where electron trans-
*
Corresponding author at: School of Science, Linyi University, 276005 Linyi, China. E-mail address: [email protected] (W.-T. Lu). 0375-9601/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.physleta.2013.03.035
port could be understood in terms similar to light propagation in periodic stratified media. On the other hand, many works on the quasiperiodic system [11] have been performed, which is an intermediate case between periodic and disordered ones. The Fibonacci superlattices and Thue–Morse superlattices are two typical quasiperiodic systems discussed widely. Those studies have shown that the quasiperiodic systems have a highly fragmented energy spectrum, and their eigenstates can be critical states with self-similar pattern which are neither extended nor localized [12,13]. Many theoretical works on optical transmission, energy spectrum, and density of states in various quasiperiodic systems have been reported [12–17]. Recently, the transport properties through quasiperiodic electric superlattices in graphene have been investigated [18–20]. Biswas discussed the resonant tunneling through a Fibonacci superlattice in bilayer graphene [18]. Sena et al. found that the spectrum of quasibound states in Fibonacci graphene superlattice distributes as a Cantor-like set by virtue of transverse wave vector [19]. However, the results on transport of Dirac electrons in quasiperiodic MS are still lacking, and the magnetic field greatly affects the physical properties of Dirac electrons in comparison to the pure electric field, especially for the Klein tunneling. In this work, the quasiperiodic MS in graphene are considered, involving Fibonacci MS and Thue–Morse MS. The required magnetic profile can be produced by ferromagnetic stripes located on top of the graphene layer, or by virtue of other means [21]. The transmission of Dirac electron through the quasiperiodic MS is discussed theoretically based on
W.-T. Lu et al. / Physics Letters A 377 (2013) 1368–1372
1369
where the Fermi velocity v f ≈ 0.86 × 106 m/s, the pseudospin matrix σ = (σx , σ y ) is the Pauli matrix, p = ( p x , p y ) is the momentum operator, and A(x) is the vector potential. For convenience, the dimensionless units are introduced: l B = h¯ /e B 0 , E 0 = h¯ v f /l B , B (x) → B 0 B (x), A (x) → B 0 l B A (x), r → l B r , k → k/l B , and E → E 0 E. For a typical value B 0 = 0.1 T, we have l B = 81 nm and E 0 = 7.0 meV. The magnetic field and the corresponding vector potential are infinite and homogeneous along the y-direction, resulting in the conservation of the transverse wave vector k y . The vector potential A is constant in each region of the models. For given incident energy E and transverse wave vector k y , the solution in jth region of Eq. (1) could be written as Ψ j (x, y ) = ψ j (x)e ik y y with Fig. 1. The schematics of magnetic fields of building blocks P and Q indicated by the black arrows, and the corresponding vector potentials indicated by shaded areas.
numerical results and compared with the ones in periodic MS. It is found that the transmission as a function of incident energy is fragmented and has common structure in quasiperiodic MS, and the features are quite different from the results for quasiperiodic electric superlattices in graphene [18–20]. The nature of these features originates from the quasiperiodicity of the systems, analogous to the optical transmission spectrum in quasiperiodic photonic structures. The Letter is organized as follows. In Section 2, we introduce the periodic MS, Fibonacci MS, and Thue–Morse MS, and the transfer-matrix method is used. We show the numerical results and discussions in Section 3. Finally, we draw conclusions in Section 4. 2. Model and method The Fibonacci structure can be realized by juxtaposing the two basic building blocks P and Q in Fibonacci sequence, and the nth generation of the process S n is given by the recursive rule S n = S n−1 S n−2 , for n 1, starting with S 0 = P and S −1 = Q . The Fibonacci generations are S 1 = P Q , S 2 = P Q P , S 3 = P Q P P Q , etc. The total number of building blocks P and Q in each sequence is equal to the Fibonacci number F n = F n−1 + F n−2 with F 0 = F −1 = 1. The Thue–Morse structure based on Thue–Morse sequence can be defined by the recursive relation U n = U n−1 U n−1 , for n 1, with U 0 = P Q and U 0 = Q P , where U n is the complement of U n . The Thue–Morse generations are U 1 = P Q Q P , U 2 = P Q Q P Q P P Q , etc., and the total number of building blocks in each sequence is equal to 2n+1 . We shall consider the one-dimensional periodic and quasiperiodic MS perpendicular to the plane of graphene. The magnetic field is assumed to be uniform along the y-direction and to vary along the x-direction. The quasiperiodic Fibonacci MS and Thue–Morse MS can be realized by two magnetic blocks P and Q arranged in Fibonacci and Thue–Morse sequences, respectively. In addition, three periodic MS are considered, i.e., ( P )m , ( P Q )m , and ( P Q Q )m with the period number m, where P , P Q , and P Q Q are the unit cells of the three periodic MS, respectively. Fig. 1 depicts the profiles of magnetic blocks P and Q , each of which is made up of two opposite magnetic δ -function barriers. In the Landau gauge, both corresponding vector potentials A P and A Q have rectangular shapes with barrier widths d P / Q and well widths l P / Q . Thus, all the vector potential fields are superlattices, and their structures are the same as their corresponding MS, which play a key role to the following transport properties. At low energy, the electron in graphene could be described by an effective massless Dirac equation with a linear energy dispersion. In the presence of a magnetic field perpendicular to the plane, the equation reads as
v f σ · p + eA(x) Ψ = E Ψ,
(1)
ψ j (x) = a j
1
q j +ik j E
e iq j x + b j
1
−q j +ik j
e −iq j x .
(2)
E
Here, k j = k y + A j , q j is the longitudinal wave vector satisfying
q2j + (k y + A j )2 = E 2 .
(3)
In order to calculate the transmission probability and the energy levels of quasibound states for periodic and quasiperiodic MS, the transfer-matrix method Eq. (2) can be rewrit is employed.
ten as ψ j (x) = G j H j Hj =
e
iq j x
0
0
e
−iq j x
aj
bj
, where G j =
1
1
(q j +ik j )/ E (−q j +ik j )/ E
and
. Based on the continuity condition of the wave
functions at the interface x = x j between jth and ( j + 1)th regions,
a
a
1 , and M j = H − (x )G −j+11 G j H j (x j ). j +1 j Thus, the total transfer-matrix for MS with n regions can be written as M = M n−1 · · · M j · · · M 1 . Then the transmission probability can be obtained from T = 1 − | M 21 |2 /| M 22 |2 , and M i j is the matrix element of M. Assuming that the eigenstates decay exponentially in the vector potential barriers at both extremities of the MS, one may get the condition for quasibound states [22]:
one can get:
j +1
b j +1
= Mj
M 22 = 0,
j
bj
(4)
in the energy region k y < E < ( A + k y ) where the eigenstates are evanescent inside the barriers and propagating inside the wells. 3. Results and discussions In this section, the transport properties of Dirac electron through periodic MS, Fibonacci MS, and Thue–Morse MS in graphene are studied numerically. The widths of barriers and wells for both magnetic blocks are the same and fixed as d P / Q = l P / Q = 0.5 in units of l B in the following results. First, the transmission probability as a function of incident energy for periodic MS ( P )5 , ( P Q )5 , and ( P Q Q )5 is shown in Figs. 2(a)–(c) at A P = 3.0 and A Q = 1.0. Obviously, the transmission exhibits a new kind of resonance, which is not the Klein tunneling due to the suppression of the Klein tunneling in this energy region. From Fig. 2(a) we can see that the transmission exhibits 4-fold resonance splitting in ( P )5 , and the splitting is (m − 1)-fold in ( P )m [8]. Quite differently, for periodic MS ( P Q )5 of which the unit cell is arranged with two magnetic blocks, the transmission presents two resonant domains in the considered energy region, and there are 4-fold resonant peaks in each domain, as shown in Fig. 2(b). Compared with ( P )5 , the resonant domains in ( P Q )5 become narrow, due to the narrowed energy band of quasibound states which will be shown later. Fig. 2(c) shows that the transmission presents three resonant domains in ( P Q Q )5 of which the unit cell is arranged with three magnetic blocks. Thus, it can be concluded that the transmission would present n resonant domains and (m − 1)-fold resonant peaks in each domain, for the periodic MS of which the period number is m and the unit cell is arranged with n magnetic blocks. Furthermore, the position of
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Fig. 2. Transmission probability versus incident energy in periodic MS. Here A P = 3.0, A Q = 1.0, the solid curves for k y = 0.0, and dashed curves for k y = 0.7. The periodic MS (a) ( P )5 , (b) ( P Q )5 , and (c) ( P Q Q )5 .
Fig. 4. Transmission probability versus incident energy at A P = 3.0, A Q = 1.0, and k y = 0.0. (a) The solid curve for periodic MS ( P )4 with 4 layers, dashed curve for the 3rd generation Fibonacci MS with 5 layers, and dotted curve for the 1st generation Thue–Morse MS with 4 layers. (b) The solid curve for periodic MS ( P )8 with 8 layers, dashed curve for the 4th generation Fibonacci MS with 8 layers, and dotted curve for the 2nd generation Thue–Morse MS with 8 layers.
Fig. 3. The distribution of the energy levels for the periodic MS as a function of the period number m, at A P = 3.0, A Q = 1.0, and k y = 0.0. The periodic MS (a) ( P )m , (b) ( P Q )m , and (c) ( P Q Q )m .
resonant domain moves to the right in the energy region with the increase of transverse wave vector k y . This regular resonant transmission is the reflection of energy spectrum for the periodic MS, which is determined by its structure. Due to the coupling of eigenstates via tunneling in MS, the degenerate energy levels of the independent wells become nondegenerate and split. Then these split levels redistribute themselves into groups around their unperturbed position and form bands, and each band contains the same amount of energy levels. The energy levels for MS can be obtained by solving Eq. (4). As shown in Figs. 3(a)–(c), one energy band is formed in ( P )m , two bands in ( P Q )m , and three bands in ( P Q Q )m . With the increase of period number m, the energy levels are distributed symmetrically. The transmission resonance occurs when the incident energy coincides with the energy level of quasibound state. Thus, as an application, the resonant peaks can be used to probe the energy spectrum of graphene-based interacting multi-well structures. In contrast to periodic MS, the transmission in quasiperiodic MS exhibits more complicated resonant features. Figs. 4(a) and (b) display the transmission probability as a function of incident energy for periodic MS, Fibonacci MS, and Thue–Morse MS with fewer layers. From Fig. 4(b) we can see that, for all the three
Fig. 5. Transmission probability versus incident energy for Fibonacci MS at A P = 2.0, A Q = 1.0, and k y = 0.0. (a) The 8th generation with 110 layers, (b) the 10th generation with 288 layers, and (c) the 12th generation with 754 layers.
MS with 8 layers, in comparison to that in periodic MS, the total number of resonant peaks in Fibonacci MS and Thue–Morse MS decreases and their distribution becomes irregular. Most resonant peaks in Thue–Morse MS are unity, which does not hold in Fibonacci MS. In addition, the transmission for the 1st generation Thue–Morse MS U 1 = P Q Q P presents regular resonant peaks, due to its symmetrical structure, as shown in Fig. 4(a). The complicated transmission in quasiperiodic MS implies its less regular energy spectrum. However, for Fibonacci MS with more layers, the transmission presents rich and relatively regular structure. Figs. 5(a)–(c) show the transmission probability as a function of incident energy for the 8th, 10th, and 12th generations Fibonacci MS, respectively, at A P = 2.0, A Q = 1.0 and k y = 0.0. One can see that all the transmissions display similar structure and fractal property in energy space, i.e., the transmission zones and transmission gaps for different Fibonacci MS are arranged in a similar way. With
W.-T. Lu et al. / Physics Letters A 377 (2013) 1368–1372
Fig. 6. Transmission probability versus incident energy for the 13th generation Fibonacci MS with 1220 layers. The parameters are the same as those in Fig. 5.
the increase of the number of layers, more transmission zones disappear gradually, meanwhile, more resonant peaks appear, and the fractal property becomes more evident. This fractal property arises from the fragmented energy spectrum of Fibonacci MS, regardless of the value of transverse wave vector k y . In order to show the self-similarity of transmission for Dirac electron in Fibonacci MS, the transmission probability as a function of incident energy for the 13th generation Fibonacci MS is shown in Figs. 6(a)–(c). Fig. 6(a) shows the transmission in the energy region 0.5 < E < 3.0. Figs. 6(b) and (c) show the results of Fig. 6(a) in the reduced energy regions 1.48 < E < 1.98 and 1.67 < E < 1.77, respectively. Comparing these figures, one can readily observe the self-similar behavior between the whole and the local transmission spectra. This feature is analogous to the optical transmission spectrum in quasiperiodic photonic structures arranged in a Fibonacci sequence [15,17], to some extent, which is the refection of the self-similarity in the corresponding energy spectrum. The self-similarity is considered as a fundamental property of the transmission spectrum for quasiperiodic structure, not found in other parts of the full energy region. Furthermore, the energy center of self-similarity is E = 1.723, approximately equal to the energy level of the 1th Fibonacci MS, nearby which the transmission spectrum is distributed symmetrically, as shown in Fig. 6(c), implying the effect of periodicity in Fibonacci MS. In addition, the transmission of Dirac electron in Fibonacci MS also has the scaling property with respect to the generation number. To illustrate the scaling property, the transmission probability as a function of incident energy for the 7th, 8th, 13th, and 14th generations Fibonacci MS is shown in Figs. 7(a)–(d). It can be seen that, the transmission profiles for the 7th and 8th generations in the energy region 1.5 < E < 2.0 are the same as the ones for the 13th and 14th generations in the energy region 1.714 < E < 1.733, respectively, and the transmission profile is recovered after six generations, implying the six-cycle feature. The transmission spectrum is almost symmetrical about E = 1.723. The scale factor which indicates a reduction of energy is approximately equal to 26, the same as the result on optical transmission spectrum [17]. It is noteworthy that the self-similarity and the scaling property of Dirac electron discussed above are sensitive to the structure of MS, but weakly dependent on its profile. Finally, we study the transmission of Dirac electron in Thue– Morse MS with more layers. Figs. 8(a)–(c) show the transmission
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Fig. 7. Transmission probability versus incident energy for Fibonacci MS, and the parameters are the same as those in Fig. 5. (a) The 7th generation with 68 layers, (b) the 8th generation with 110 layers, (c) the 13th generation with 1220 layers, and (d) the 14th generation with 1974 layers.
Fig. 8. Transmission probability versus incident energy for Thue–Morse MS, and the parameters are the same as those in Fig. 5. (a) The 4th generation with 64 layers, (b) the 6th generation with 256 layers, and (c) the 8th generation with 1024 layers.
probability as a function of incident energy for the 4th, 6th, and 8th generations Thue–Morse MS, respectively. One may find that all the transmissions are fragmented and have common structure, similar to Fibonacci MS. With the increase of the generation, more and more perfect resonant peaks with T = 1 are presented, which are related to the completely transparent states originating from a special positional correlation between two magnetic blocks in Thue–Morse MS [13,14]. However, different from Fibonacci MS, the transmission in Thue–Morse MS does not possess self-similarity and scaling property. The above results indicate that Fibonacci systems behave more regularly than Thue–Morse systems as far as the transmission is concerned, in agreement with previous claim [16]. 4. Conclusion In this Letter, we have discussed the transport properties of Dirac electron through periodic and quasiperiodic MS in graphene, with the help of transfer-matrix method. For periodic MS, the transmission presents regular resonance splitting effect in energy
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space. Both transmissions in Fibonacci MS and Thue–Morse MS with fewer layers present irregular oscillation. However, with the increase of number of layers, the quasiperiodic MS exhibit rich transmission patterns, such as the self-similarity and scaling property in Fibonacci MS, and the perfect resonant peaks in Thue– Morse MS, which originate from the quasiperiodicity of the systems and the suppression of Klein tunneling. Furthermore, the transmission is sensitive to the incident energy, transverse wave vector, and magnetic structure parameters. The above transport properties should also exist in the more realistic MS with smooth vector potential in experiment. In fact, the results obtained in this Letter could be extended to the case where the magnetic blocks are of different widths and the same strengths. We hope that the results may be useful for the understanding on physical property of Dirac electron in quasiperiodic systems, even for the potential technological applications in future [23]. Acknowledgements This work was supported by the National Natural Science Foundation of China (Grant Nos. 10974137 and 11047020), and the Natural Science Foundation of Shandong Province, China (Grant Nos. ZR2012AM022 and ZR2010AQ025). References [1] A. De Martino, L. Dell’Anna, R. Egger, Phys. Rev. Lett. 98 (2007) 066802; A. De Martino, L. Dell’Anna, R. Egger, Solid State Commun. 144 (2007) 547. [2] F. Zhai, K. Chang, Phys. Rev. B 77 (2008) 113409.
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