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Enhanced spin polarization in an asymmetric magnetic graphene superlattice
Solid State Communications 151 (2011) 1131–1134
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Enhanced spin polarization in an asymmetric magnetic graphene superlattice Qi-Rui Ke, Hai-Feng Lü ∗ , Xiao-Dong Chen, Xiao-Tao Zu Department of Applied Physics and State Key Laboratory of Electronic Thin Films and Integrated Devices, University of Electronic Science and Technology of China, Chengdu 610054, China
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Article history: Received 6 May 2011 Accepted 15 May 2011 by V. Fal’ko Available online 24 May 2011 Keywords: A. Magnetic graphene superlattice A. Barrier asymmetry D. Spin conductance D. Spin polarization
abstract The authors investigate the spin-resolved transport through an asymmetrical magnetic graphene superlattice (MGS) consisting of the periodic barriers with abnormal one in height. To quantitatively depict the asymmetrical MGS, an asymmetry factor has been introduced to measure the height change of the abnormal barrier. It is shown that the spin filter effect is strongly enhanced by the barrier asymmetry both in the Klein and the classical tunneling regimes. In the presence of abnormal barrier, the conductance with certain spin direction is suppressed with respect to different tunneling regimes, and thus high spin polarization with opposite sign can be achieved. Crown Copyright © 2011 Published by Elsevier Ltd. All rights reserved.
Recent progress in experimental fabrication of graphene attracts extensive interest in research on the exploration of electronic properties and novel transport phenomena in this material [1–3]. Graphene is a monolayer of carbon atoms densely packed in a honeycomb lattice, in which the underlying lattice has a special band structure with Dirac points at the corners of the Brillouin zone [4,5]. Such a band structure is expected to lead to many extraordinary electronic transport phenomena such as anomalously quantized Hall effects, absence of weak localization, existence of a minimum conductivity, and so on [2,3,6]. More importantly, it provides us with an experimental test for the Klein paradox [7,8]. All these properties are significant in the design of various graphene-based devices, and graphene is thus regarded as a perspective base for the future electronics. In particular, interesting discussions and predictions have just been reported for electron behaviors in graphene superlattices (GSLs), associated with different types of periodic potential: Kronig–Penney, [9–11] muffin-tin, [10,12] and cosine [13,14]. It is an important issue in spintronics that how to effectively control and manipulate the spin degrees of freedom in solid-state systems. Since graphene exhibits high mobility, very long spin scattering time, and small spin–orbit interaction, [15] graphene is considered as an ideal material for spin conduction [15,16]. Magnetism can be induced in graphene by doping and defects or by applying an external transverse electric field. Recently, it has been theoretically predicted and experimentally realized that the ferromagnetic correlation can be induced in graphene by the proximity effect [17]. A rough estimation of the exchange splitting in
∗
Corresponding author. Tel.: +86 13438320098. E-mail address: [email protected] (H.-F. Lü).
graphene induced by the ferromagnetic insulator EuO could be 5 meV. Experimentally, periodic potentials may be realized on graphene by different techniques such as interaction with substrate or electron beam-induced deposition of adsorbates [18,19]. The potential period may be as small as 3 nm or as large as 20 nm, while the potential amplitude is in the range of a few tenths of an electron volt [20]. Compared with the single barrier structure, it is found that the superlattice structure leads to an enhanced spin polarization [21]. In fact, due to the restriction of fabrication techniques, it is very difficult to obtain a structure with perfect symmetry. So it is necessary to investigate the spin transport in an asymmetrical MGS, which will give a more realistic scenario of the transport behavior in an MGS system. In this letter, we investigate the spin transport in NG-lead/FG/ (NG/FG)N /NG-lead superlattices in the presence of an abnormal barrier that denotes the symmetry of the MGS and its height is changeable. The transmission, the measurable quantity ballistic conductance, and the spin polarization of the ballistic conductance are indicated as functions of the gate voltages and the system’s symmetry degree. The results show that spin transport in the asymmetrical MGS with an abnormal barrier is rather different from that in the symmetrical MGS structures. It is indicated that the asymmetric structure is a favorable candidate for the spintronic devices, e.g., a highly efficient spin filter with the spin polarity tunable by structural factors. The results also may be useful for tailoring material parameters. Until now, the MGS model for the experimental system has been described in detail [22,23]. Fig. 1 shows the geometry of asymmetric magnetic superlattice which none of the former work have focused on. In order to quantitatively interpret the effect of barrier symmetry inhomogeneity of the system, the potential
0038-1098/$ – see front matter Crown Copyright © 2011 Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.ssc.2011.05.026
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Q.-R. Ke et al. / Solid State Communications 151 (2011) 1131–1134
with e−ik(j+1)x lj 0
P (j) =
×
Fig. 1. (Color online) A schematic view of the model, both the widths of the FG strip and the NG strip are L. (a) The illustration of the asymmetrical magnetization of the MGS with the second FG strip (red online) unlike the rest. (b) The profile of the height of the potential barrier corresponding to (a).
profile of the system can be given by Marchini and Wintterlin [18]
µ0 − µ − (λi + 1)sh i = 1, 2, . . . , n V (x) =
l(2i−1) < x < l(2i) (1)
0
otherwise
where n is the superlattice period, the suffix i is the number of the potential, s = ±1 corresponds to the majority and minority spins, µ0 and µ are the chemical potential in the ferromagnetic graphene and normal graphene, respectively, the latter tunable by gate voltage, and the local Zeeman splitting h which can be induced by the presence of a magnetic insulator in the graphene sheet remains invariant on account of only the parallel magnetization configuration being considered in this paper. The value of the asymmetry factor λi is introduced to define the varying height of the abnormal potential barrier. If λi = 0, it means that the MGS is a perfectly symmetrical structure, here, the ith potential barrier can be thus given by Vi = V0 = µ0 − µ − sh. If λi ̸= 0, it demonstrates an asymmetrical structure, such as the MGS with a second abnormal barrier as depicted in Fig. 1(a). The Dirac-like Hamiltonian which describes the charge carriers of the system has the form as the following [19]
ˆ = vF (σx px + σy py ) + V (x)ˆI , H
(2)
with the Fermi velocity vF ≈ 106 m/s in graphene, σˆ x , σˆ y are the Pauli matrices and px = −ih¯ ∂∂x , py = −ih¯ ∂∂y . Then we can obtain the general solution to the Hamiltonian, that is to say, the plane wave function of the electrons in the jth monolayer graphene strip can be given by Katsnelson et al. [7], Marchini and Wintterlin [18]
[
Ψj (x) = aj
1 αj eiθj
e
ikjx x
+ bj
1 −αj e−iθj
e
−ikjx x
]
eiky y
(3)
here θj is the angle in the jth strip, aj and bj are the transmission amplitudes, the sign αj = sign[µ0 − V (x)], and the wave vector kjF = |µ0 − V (x)| /h¯ vF , Moreover, because of the transmission invariance in the y direction, the perpendicular wave vector component [18] ky = kjF sin θj , the parallel wave vector component [18] kjx = kjF cos θj . According to the boundary conditions, the coefficients aj , bj can be obtained from the continuity of the wave functions, as a result, the transfer matrix has the form [20]
a2n+1 b2n+1
=
2n ∏ j =1
P (j) ×
a1 b1
(4)
1 αj eiθj
0 eik(j+1)x lj
1 −αj e−iθj
eikjx lj 0
1 αj+1 eiθj+1
0 e
−ikjx lj
1 −αj+1 e−iθj+1
.
−1
(5)
Here, a1 = 1, b2n+1 = 0, b1 is the reflection coefficient, a2n+1 is the transmission coefficient, assume both the widths of the FM and the spacing between these strips are L, then lj = (j − 1)L. For λi = 0 (i = 1, 2, . . . , n), the transfer matrix is reduced to the symmetric case in which the formulation of the transfer matrix above can be expressed as in Ref. [23]. Suppose the incident angle θ1 = θ , then the transmission probability T↑↓ (θ ) = |a2n+1 |2 can be obtained; in addition, the dimensionless spin-resolved conductivity can be calculated by Niu π /2 et al. and Castro Neto et al. [21,24] G↑↓ = G0 −π /2 T↑↓ (θ) cos θ dθ ,
where G0 = e2 mvF w/ h¯ 2 with w the width of the graphene strip along the y direction, G↑ and G↓ are the spin-up and spin-down conductance components, respectively. Finally, we can evaluate the electron spin-polarization effect in the spin transport process, the spin polarization of G is defined as χ = (G↑ − G↓ )/(G↑ + G↓ ). Fig. 2 demonstrates the spin-resolved transmission probability Tσ versus incidence angle θ for several different asymmetrical factor λ, which corresponds to the changeable height of the abnormal barrier. When λ = 0, the structure of the MGS is symmetric, the separation between the spin-up and spin-down transmission probability is slight as displayed in Fig. 2(a). Perfect transmission of Tσ = 1 is observed at θ = 0, which is independent of the width and height of the barriers. This is a unique feature of massless Dirac fermions and related to the Klein paradox. The transmission remains nearly perfect in a wide range around θ = 0. For relative large angles, the transmission coefficients Ts deviate largely from unity, and exhibit oscillatory behavior. The positions of the oscillation peaks are spin-resolved due to different local Zeeman splittings. When the barrier asymmetry is considered (λ ̸= 0), the values and peaks of the spin-down transmission probability T↓ dramatically decrease and even disappear at non-zero incidence angle while the profile of the spin-up transmission probability T↑ is almost not affected. Correspondingly, the separation between the spin-up and spin-down transmission probability is evidently broadened, which may lead to the potential applications for modulating the features of resonant peaks in graphene-based electronic devices. As the increase of λ, the effective barrier height for the spin-down electron becomes higher, and the transmission area of T↓ is thus narrowed. When λ increases on to the case of µ − Vs < 0, tunneling of the spin-down electrons is changed from the case of classical motion to the Klein tunneling, and T↓ is increased with λ in the latter case (as shown in Fig. 2(d)). To understand the spin transport through the MGS with the changeable abnormal barrier more straightforwardly, we present the dependence of spin-resolved conductance Gσ on the gate voltage for different asymmetric parameter λ in Fig. 3. Here two different tunneling regimes, Klein tunneling region and classical motion region, are separately discussed. Electron tunneling in these two regimes can be manipulated by adjusting the gate voltage, where ξ = µ/µ0 > 0 corresponds to the classical motion region and ξ < 0 corresponds to another one approximately. In the Klein tunneling region (ξ < 0 i.e. µ + sh < 0), the effective barrier for the spin-up electrons becomes lower in the presence of asymmetry, which is not in favor of the tunneling processes. In this region, the spin-down conductance G↓ is slightly affected by the structure asymmetry, while the spin-up conductance G↑ is suppressed considerably. However, the results are opposite in the
Q.-R. Ke et al. / Solid State Communications 151 (2011) 1131–1134
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Fig. 2. (Color online) Spin-up (solid line) transmission probability and spin-down (dashed line) transmission probability versus incidence angle θ for different asymmetrical factor λ2 in the parallel magnetization configuration. Here, we choose the superlattice periods n = 10, the other parameters are L = 50 nm, h = 20 meV, µ0 = µ = 100 meV.
Fig. 3. (Color online) Spin-up (solid line) and spin-down (dashed line) conductance as a function of gate voltage ξ = µ/µ0 for different asymmetrical factor λ2 . The parameters chosen are the same as Fig. 2.
classical motion region. For ξ > 0, the gap of the spin-resolved conductance become broaden with the increases of λ, and the spinup conductance has less shifts than that of the spin-down. It is obvious that the combined effects of broken symmetry and the enhancement of effective potential will result in the much lower spin-down conductance. In addition, for the spin-down electrons, the increased height of the effective barrier enlarged the orbital energy of the spin-up electrons. As shown in Fig. 3, the spin filter effect is strongly enhanced for a relative large λ both in the Klein and the classical tunneling regimes. By modulating the height of the abnormal barrier, the spin-down (up) electrons can be obtained in the Klein (classical) tunneling regime. It is shown in Fig. 3 that barrier asymmetry suppresses the conductance of certain spin with respect to different tunneling regimes. Moreover, the spin polarization of electron transport through the MSL with a changeable abnormal barrier, is plotted in Fig. 4. It is displayed in Fig. 4 that the spin polarization χ in both regimes is considerably strengthened in the presence of structure
asymmetry. In the classical motion region, the polarization plateau with high χ can be formed and shifts to a large gate voltage with the increase of λ. While in the Klein tunneling regime, the spin polarization oscillates periodically as a function of gate voltage for λ = 0. This behavior arises from the phase coherence of the electron wave functions in the transport process. When the variable barrier is considered, the phase coherence between electrons is destroyed and periodic oscillations of the polarization peaks are broaden as the polarization plateau with high χ . Similar to the classical motion case, the polarization plateau shifts to a low gate voltage with the increase of λ. In the former studies, effect of barrier asymmetry on the spin transport in a magnetic superlattice based on semiconductor systems has been discussed [25]. The asymmetry could increase the barrier height of spin-down electrons (for h > 0), which leads the suppression of the spin-up conductance. Therefore, the spin polarization is considerably strengthened, and the asymmetric structure is considered as a favorable candidate for the spintronic
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Q.-R. Ke et al. / Solid State Communications 151 (2011) 1131–1134
Therefore, effective spin filter effect and the spin current with opposite polarization sign can be realized. Acknowledgments This project was supported by the Foundation for Innovative Research Groups of the NSFC under Grant (No. 60721001), the NSFC (Nos 11004022 and 61006081), and the Fundamental Research Funds for the Central Universities (No. ZYGX2009J042). References
Fig. 4. (Color online) Spin polarization of conductance as a function of gate voltage ξ = µ/µ0 for different asymmetrical factor λ2 . The parameters chosen are the same as Fig. 2.
devices. However, spin transport in the MGS demonstrates more abundant and novel tunneling properties in the presence of barrier asymmetry. By adjusting the gate voltage, one can make the tunneling electron in a classical or Klein area. In these two regimes, the asymmetric factor suppresses the electron transport with respect to different spin channels, as a result of the unique tunneling feature of relativistic particles. Therefore, effective spin filter effect and the spin current with opposite polarization sign can be realized. In summary, we have investigated the spin transport in NGlead/FG/(NG/FG)N/NG-lead superlattices in the presence of an abnormal barrier that denotes the symmetry of the MGS and its height is changeable. The transmission, the ballistic conductance, and the spin polarization of the conductance are indicated as functions of the gate voltages and the system’s symmetry degree. The results indicate that the asymmetric structure is a favorable candidate for the spintronic devices, e.g., a highly efficient spin filter with the spin polarity tunable by structural factors. In the classical or Klein tunneling regimes, which can be controlled by an external gate voltage, the conductance with certain spin direction is suppressed with respect to different tunneling regimes.
[1] K.S. Novoselov, A.K. Geim, S.V. Morozov, D. Jiang, Y. Zhang, S.V. Dubonos, I.V. Grigorieva, A.A. Firsov, Science 306 (2004) 666. [2] Y. Zhang, Y.W. Tan, H.L. Stormer, P. Kim, Nature (London) 438 (2005) 201. [3] K.S. Novoselov, A.K. Geim, S.V. Morozov, D. Jiang, M.I. Katsnelson, I.V. Grigorieva, S.V. Dubonos, A.A. Firsov, Nature (London) 438 (2005) 197. [4] P.R. Wallace, Phys. Rev. 71 (1947) 622. [5] For a review, see T. Ando, J. Phys. Soc. Japan 74 (2005) 777. [6] V.P. Gusynin, S.G. Sharapov, Phys. Rev. Lett. 95 (2005) 146801. [7] M.I. Katsnelson, K.S. Novoselov, A.K. Geim, Nat. Phys. 2 (2006) 620. [8] O. Klein, Z. Phys. 53 (1929) 157. [9] C. Bai, X. Zhang, Phys. Rev. B 76 (2007) 075430. [10] C.H. Park, L. Yang, Y.W. Son, M.L. Cohen, S.G. Louie, Nat. Phys. 4 (2008) 213. [11] M. Barbier, F.M. Peeters, P. Vasilopoulos, J.M. Pereira, Phys. Rev. B 77 (2008) 115446. [12] C.H. Park, L. Yang, Y.W. Son, M.L. Cohen, S.G. Louie, Phys. Rev. Lett. 101 (2008) 126804. [13] J.H. Ho, Y.H. Chiu, S.J. Tsai, M.F. Lin, Phys. Rev. B 79 (2009) 115427. [14] L. Brey, H.A. Fertig, Phys. Rev. Lett. 103 (2009) 046809. [15] E.W. Hill, A.K. Geim, K. Novoselov, F. Schedin, P. Blake, IEEE Trans. Magn. 42 (2006) 2694. [16] N. Tombros, C. Jozsa, M. Popinciuc, H.T. Jonkman, B.J. Van Wees, Nature (London) 448 (2007) 571; V.V. Cheianov, V.I. Fal’ko, Phys. Rev. B 74 (2006) 041403(R). [17] H. Haugen, D.H. Hernando, A. Brataas, Phys. Rev. B 77 (2008) 115406. [18] S.G.S. Marchini, J. Wintterlin, Phys. Rev. B 76 (2007) 075429. [19] A.L.V. de Parga, F. Calleja, B. Borca, J.M.C. Passeggi, J. Hinarejos, F. Guinea, R. Miranda, Phys. Rev. Lett. 100 (2008) 056807. [20] J.C. Meyer, C.O. Girit, M.F. Crommie, A. Zettl, Appl. Phys. Lett. 92 (2008) 123110. [21] Z.P. Niu, F.X. Li, B.G. Wang, L. Sheng, D.Y. Xing, Eur. Phys. J. B 66 (2008) 245. [22] M. Razavy, Quantum Theory of Tunneling, World Scientific, 2003, Chap. 6. [23] M.R. Masir, P. Vasilopoulos, F.M. Peeters, Phys. Rev. B 79 (2009) 035409. [24] A.H. Castro Neto, F. Guinea, N.M.R. Peres, K.S. Novoselov, A.K. Geim, Rev. Mod. Phys. 81 (2009) 109. [25] R.Y. Yuan, R.Z. Wang, K. Xue, J.S. Wei, X.M. Song, B. Wang, H. Yan, Phys. Rev. B 74 (2006) 024417.
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Solid State Communications journal homepage: www.elsevier.com/locate/ssc
Enhanced spin polarization in an asymmetric magnetic graphene superlattice Qi-Rui Ke, Hai-Feng Lü ∗ , Xiao-Dong Chen, Xiao-Tao Zu Department of Applied Physics and State Key Laboratory of Electronic Thin Films and Integrated Devices, University of Electronic Science and Technology of China, Chengdu 610054, China
article
info
Article history: Received 6 May 2011 Accepted 15 May 2011 by V. Fal’ko Available online 24 May 2011 Keywords: A. Magnetic graphene superlattice A. Barrier asymmetry D. Spin conductance D. Spin polarization
abstract The authors investigate the spin-resolved transport through an asymmetrical magnetic graphene superlattice (MGS) consisting of the periodic barriers with abnormal one in height. To quantitatively depict the asymmetrical MGS, an asymmetry factor has been introduced to measure the height change of the abnormal barrier. It is shown that the spin filter effect is strongly enhanced by the barrier asymmetry both in the Klein and the classical tunneling regimes. In the presence of abnormal barrier, the conductance with certain spin direction is suppressed with respect to different tunneling regimes, and thus high spin polarization with opposite sign can be achieved. Crown Copyright © 2011 Published by Elsevier Ltd. All rights reserved.
Recent progress in experimental fabrication of graphene attracts extensive interest in research on the exploration of electronic properties and novel transport phenomena in this material [1–3]. Graphene is a monolayer of carbon atoms densely packed in a honeycomb lattice, in which the underlying lattice has a special band structure with Dirac points at the corners of the Brillouin zone [4,5]. Such a band structure is expected to lead to many extraordinary electronic transport phenomena such as anomalously quantized Hall effects, absence of weak localization, existence of a minimum conductivity, and so on [2,3,6]. More importantly, it provides us with an experimental test for the Klein paradox [7,8]. All these properties are significant in the design of various graphene-based devices, and graphene is thus regarded as a perspective base for the future electronics. In particular, interesting discussions and predictions have just been reported for electron behaviors in graphene superlattices (GSLs), associated with different types of periodic potential: Kronig–Penney, [9–11] muffin-tin, [10,12] and cosine [13,14]. It is an important issue in spintronics that how to effectively control and manipulate the spin degrees of freedom in solid-state systems. Since graphene exhibits high mobility, very long spin scattering time, and small spin–orbit interaction, [15] graphene is considered as an ideal material for spin conduction [15,16]. Magnetism can be induced in graphene by doping and defects or by applying an external transverse electric field. Recently, it has been theoretically predicted and experimentally realized that the ferromagnetic correlation can be induced in graphene by the proximity effect [17]. A rough estimation of the exchange splitting in
∗
Corresponding author. Tel.: +86 13438320098. E-mail address: [email protected] (H.-F. Lü).
graphene induced by the ferromagnetic insulator EuO could be 5 meV. Experimentally, periodic potentials may be realized on graphene by different techniques such as interaction with substrate or electron beam-induced deposition of adsorbates [18,19]. The potential period may be as small as 3 nm or as large as 20 nm, while the potential amplitude is in the range of a few tenths of an electron volt [20]. Compared with the single barrier structure, it is found that the superlattice structure leads to an enhanced spin polarization [21]. In fact, due to the restriction of fabrication techniques, it is very difficult to obtain a structure with perfect symmetry. So it is necessary to investigate the spin transport in an asymmetrical MGS, which will give a more realistic scenario of the transport behavior in an MGS system. In this letter, we investigate the spin transport in NG-lead/FG/ (NG/FG)N /NG-lead superlattices in the presence of an abnormal barrier that denotes the symmetry of the MGS and its height is changeable. The transmission, the measurable quantity ballistic conductance, and the spin polarization of the ballistic conductance are indicated as functions of the gate voltages and the system’s symmetry degree. The results show that spin transport in the asymmetrical MGS with an abnormal barrier is rather different from that in the symmetrical MGS structures. It is indicated that the asymmetric structure is a favorable candidate for the spintronic devices, e.g., a highly efficient spin filter with the spin polarity tunable by structural factors. The results also may be useful for tailoring material parameters. Until now, the MGS model for the experimental system has been described in detail [22,23]. Fig. 1 shows the geometry of asymmetric magnetic superlattice which none of the former work have focused on. In order to quantitatively interpret the effect of barrier symmetry inhomogeneity of the system, the potential
0038-1098/$ – see front matter Crown Copyright © 2011 Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.ssc.2011.05.026
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Q.-R. Ke et al. / Solid State Communications 151 (2011) 1131–1134
with e−ik(j+1)x lj 0
P (j) =
×
Fig. 1. (Color online) A schematic view of the model, both the widths of the FG strip and the NG strip are L. (a) The illustration of the asymmetrical magnetization of the MGS with the second FG strip (red online) unlike the rest. (b) The profile of the height of the potential barrier corresponding to (a).
profile of the system can be given by Marchini and Wintterlin [18]
µ0 − µ − (λi + 1)sh i = 1, 2, . . . , n V (x) =
l(2i−1) < x < l(2i) (1)
0
otherwise
where n is the superlattice period, the suffix i is the number of the potential, s = ±1 corresponds to the majority and minority spins, µ0 and µ are the chemical potential in the ferromagnetic graphene and normal graphene, respectively, the latter tunable by gate voltage, and the local Zeeman splitting h which can be induced by the presence of a magnetic insulator in the graphene sheet remains invariant on account of only the parallel magnetization configuration being considered in this paper. The value of the asymmetry factor λi is introduced to define the varying height of the abnormal potential barrier. If λi = 0, it means that the MGS is a perfectly symmetrical structure, here, the ith potential barrier can be thus given by Vi = V0 = µ0 − µ − sh. If λi ̸= 0, it demonstrates an asymmetrical structure, such as the MGS with a second abnormal barrier as depicted in Fig. 1(a). The Dirac-like Hamiltonian which describes the charge carriers of the system has the form as the following [19]
ˆ = vF (σx px + σy py ) + V (x)ˆI , H
(2)
with the Fermi velocity vF ≈ 106 m/s in graphene, σˆ x , σˆ y are the Pauli matrices and px = −ih¯ ∂∂x , py = −ih¯ ∂∂y . Then we can obtain the general solution to the Hamiltonian, that is to say, the plane wave function of the electrons in the jth monolayer graphene strip can be given by Katsnelson et al. [7], Marchini and Wintterlin [18]
[
Ψj (x) = aj
1 αj eiθj
e
ikjx x
+ bj
1 −αj e−iθj
e
−ikjx x
]
eiky y
(3)
here θj is the angle in the jth strip, aj and bj are the transmission amplitudes, the sign αj = sign[µ0 − V (x)], and the wave vector kjF = |µ0 − V (x)| /h¯ vF , Moreover, because of the transmission invariance in the y direction, the perpendicular wave vector component [18] ky = kjF sin θj , the parallel wave vector component [18] kjx = kjF cos θj . According to the boundary conditions, the coefficients aj , bj can be obtained from the continuity of the wave functions, as a result, the transfer matrix has the form [20]
a2n+1 b2n+1
=
2n ∏ j =1
P (j) ×
a1 b1
(4)
1 αj eiθj
0 eik(j+1)x lj
1 −αj e−iθj
eikjx lj 0
1 αj+1 eiθj+1
0 e
−ikjx lj
1 −αj+1 e−iθj+1
.
−1
(5)
Here, a1 = 1, b2n+1 = 0, b1 is the reflection coefficient, a2n+1 is the transmission coefficient, assume both the widths of the FM and the spacing between these strips are L, then lj = (j − 1)L. For λi = 0 (i = 1, 2, . . . , n), the transfer matrix is reduced to the symmetric case in which the formulation of the transfer matrix above can be expressed as in Ref. [23]. Suppose the incident angle θ1 = θ , then the transmission probability T↑↓ (θ ) = |a2n+1 |2 can be obtained; in addition, the dimensionless spin-resolved conductivity can be calculated by Niu π /2 et al. and Castro Neto et al. [21,24] G↑↓ = G0 −π /2 T↑↓ (θ) cos θ dθ ,
where G0 = e2 mvF w/ h¯ 2 with w the width of the graphene strip along the y direction, G↑ and G↓ are the spin-up and spin-down conductance components, respectively. Finally, we can evaluate the electron spin-polarization effect in the spin transport process, the spin polarization of G is defined as χ = (G↑ − G↓ )/(G↑ + G↓ ). Fig. 2 demonstrates the spin-resolved transmission probability Tσ versus incidence angle θ for several different asymmetrical factor λ, which corresponds to the changeable height of the abnormal barrier. When λ = 0, the structure of the MGS is symmetric, the separation between the spin-up and spin-down transmission probability is slight as displayed in Fig. 2(a). Perfect transmission of Tσ = 1 is observed at θ = 0, which is independent of the width and height of the barriers. This is a unique feature of massless Dirac fermions and related to the Klein paradox. The transmission remains nearly perfect in a wide range around θ = 0. For relative large angles, the transmission coefficients Ts deviate largely from unity, and exhibit oscillatory behavior. The positions of the oscillation peaks are spin-resolved due to different local Zeeman splittings. When the barrier asymmetry is considered (λ ̸= 0), the values and peaks of the spin-down transmission probability T↓ dramatically decrease and even disappear at non-zero incidence angle while the profile of the spin-up transmission probability T↑ is almost not affected. Correspondingly, the separation between the spin-up and spin-down transmission probability is evidently broadened, which may lead to the potential applications for modulating the features of resonant peaks in graphene-based electronic devices. As the increase of λ, the effective barrier height for the spin-down electron becomes higher, and the transmission area of T↓ is thus narrowed. When λ increases on to the case of µ − Vs < 0, tunneling of the spin-down electrons is changed from the case of classical motion to the Klein tunneling, and T↓ is increased with λ in the latter case (as shown in Fig. 2(d)). To understand the spin transport through the MGS with the changeable abnormal barrier more straightforwardly, we present the dependence of spin-resolved conductance Gσ on the gate voltage for different asymmetric parameter λ in Fig. 3. Here two different tunneling regimes, Klein tunneling region and classical motion region, are separately discussed. Electron tunneling in these two regimes can be manipulated by adjusting the gate voltage, where ξ = µ/µ0 > 0 corresponds to the classical motion region and ξ < 0 corresponds to another one approximately. In the Klein tunneling region (ξ < 0 i.e. µ + sh < 0), the effective barrier for the spin-up electrons becomes lower in the presence of asymmetry, which is not in favor of the tunneling processes. In this region, the spin-down conductance G↓ is slightly affected by the structure asymmetry, while the spin-up conductance G↑ is suppressed considerably. However, the results are opposite in the
Q.-R. Ke et al. / Solid State Communications 151 (2011) 1131–1134
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Fig. 2. (Color online) Spin-up (solid line) transmission probability and spin-down (dashed line) transmission probability versus incidence angle θ for different asymmetrical factor λ2 in the parallel magnetization configuration. Here, we choose the superlattice periods n = 10, the other parameters are L = 50 nm, h = 20 meV, µ0 = µ = 100 meV.
Fig. 3. (Color online) Spin-up (solid line) and spin-down (dashed line) conductance as a function of gate voltage ξ = µ/µ0 for different asymmetrical factor λ2 . The parameters chosen are the same as Fig. 2.
classical motion region. For ξ > 0, the gap of the spin-resolved conductance become broaden with the increases of λ, and the spinup conductance has less shifts than that of the spin-down. It is obvious that the combined effects of broken symmetry and the enhancement of effective potential will result in the much lower spin-down conductance. In addition, for the spin-down electrons, the increased height of the effective barrier enlarged the orbital energy of the spin-up electrons. As shown in Fig. 3, the spin filter effect is strongly enhanced for a relative large λ both in the Klein and the classical tunneling regimes. By modulating the height of the abnormal barrier, the spin-down (up) electrons can be obtained in the Klein (classical) tunneling regime. It is shown in Fig. 3 that barrier asymmetry suppresses the conductance of certain spin with respect to different tunneling regimes. Moreover, the spin polarization of electron transport through the MSL with a changeable abnormal barrier, is plotted in Fig. 4. It is displayed in Fig. 4 that the spin polarization χ in both regimes is considerably strengthened in the presence of structure
asymmetry. In the classical motion region, the polarization plateau with high χ can be formed and shifts to a large gate voltage with the increase of λ. While in the Klein tunneling regime, the spin polarization oscillates periodically as a function of gate voltage for λ = 0. This behavior arises from the phase coherence of the electron wave functions in the transport process. When the variable barrier is considered, the phase coherence between electrons is destroyed and periodic oscillations of the polarization peaks are broaden as the polarization plateau with high χ . Similar to the classical motion case, the polarization plateau shifts to a low gate voltage with the increase of λ. In the former studies, effect of barrier asymmetry on the spin transport in a magnetic superlattice based on semiconductor systems has been discussed [25]. The asymmetry could increase the barrier height of spin-down electrons (for h > 0), which leads the suppression of the spin-up conductance. Therefore, the spin polarization is considerably strengthened, and the asymmetric structure is considered as a favorable candidate for the spintronic
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Therefore, effective spin filter effect and the spin current with opposite polarization sign can be realized. Acknowledgments This project was supported by the Foundation for Innovative Research Groups of the NSFC under Grant (No. 60721001), the NSFC (Nos 11004022 and 61006081), and the Fundamental Research Funds for the Central Universities (No. ZYGX2009J042). References
Fig. 4. (Color online) Spin polarization of conductance as a function of gate voltage ξ = µ/µ0 for different asymmetrical factor λ2 . The parameters chosen are the same as Fig. 2.
devices. However, spin transport in the MGS demonstrates more abundant and novel tunneling properties in the presence of barrier asymmetry. By adjusting the gate voltage, one can make the tunneling electron in a classical or Klein area. In these two regimes, the asymmetric factor suppresses the electron transport with respect to different spin channels, as a result of the unique tunneling feature of relativistic particles. Therefore, effective spin filter effect and the spin current with opposite polarization sign can be realized. In summary, we have investigated the spin transport in NGlead/FG/(NG/FG)N/NG-lead superlattices in the presence of an abnormal barrier that denotes the symmetry of the MGS and its height is changeable. The transmission, the ballistic conductance, and the spin polarization of the conductance are indicated as functions of the gate voltages and the system’s symmetry degree. The results indicate that the asymmetric structure is a favorable candidate for the spintronic devices, e.g., a highly efficient spin filter with the spin polarity tunable by structural factors. In the classical or Klein tunneling regimes, which can be controlled by an external gate voltage, the conductance with certain spin direction is suppressed with respect to different tunneling regimes.
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