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The chirality dependent spin filter design in the graphene-like junction
This content has been downloaded from IOPscience. Please scroll down to see the full text. 2015 J. Phys.: Condens. Matter 27 125005 (http://iopscience.iop.org/0953-8984/27/12/125005) View the table of contents for this issue, or go to the journal homepage for more Download details: IP Address: 104.239.165.217 This content was downloaded on 16/06/2017 at 21:41 Please note that terms and conditions apply.
You may also be interested in: Topological metallic states in spin–orbit coupled bilayer systems Hui Pan, Xin Li, Zhenhua Qiao et al. Manipulating interface states in monolayer–bilayer graphene planar junctions Fang Zhao, Lei Xu and Jun Zhang Topological phases in two-dimensional materials: a review Yafei Ren, Zhenhua Qiao and Qian Niu The effect of dephasing on edge state transport through p–n junctions in HgTe/CdTe quantum wells Ying-Tao Zhang, Juntao Song and Qing-Feng Sun Measurable spin-polarized current in two-dimensional topological insulators Xing-Tao An, Yan-Yang Zhang, Jian-Jun Liu et al. Multiple topological interface states in silicene S K Wang, J Wang and K S Chan Quantum anomalous Hall effect in time-reversal-symmetry breaking topological insulators Cui-Zu Chang and Mingda Li Symmetry-dependent spin-charge transport and thermopower through a ZSiNR-based FM/normal/FM junction Benliang Zhou, Benhu Zhou, Xiongwen Chen et al.
Journal of Physics: Condensed Matter J. Phys.: Condens. Matter 27 (2015) 125005 (7pp)
doi:10.1088/0953-8984/27/12/125005
The chirality dependent spin filter design in the graphene-like junction Hongyu Tian1 , Sake Wang2 , Jingguo Hu3 and Jun Wang2 1
Department of Physics, Yancheng Institute of Technology, Jiangsu 224051, People’s Republic of China Department of Physics, Southeast University, Nanjing 210096, People’s Republic of China 3 College of Physics Science and Technology, Yangzhou University, Yangzhou 225009, People’s Republic of China 2
E-mail: [email protected] and [email protected] Received 10 December 2014, revised 26 January 2015 Accepted for publication 2 February 2015 Published 19 February 2015 Abstract
We investigate the chirality-dependent spin transport in a graphene-like topological insulator (TI) TI/n junction, where a perpendicular magnetic field or an off-resonant circularlypolarized light field is applied to the normal (n) region. It is found that the coupling between the helical edge states of the TI and chiral edge states from the magnetic/light field results in a perfect spin filtering effect and only one spin species can tunnel through the junction interface. The origin is ascribed to the chirality-conservation requirement, since the two spin species have the opposite chiralities in the TI region and in the n region both of them have the same chiralities. For a TI/n superlattice structure, the spin filtering effect is enhanced and even survives in a fairly strong disorder environment. Keywords: topological insulator, spin filtering effect, chirality-conservation (Some figures may appear in colour only in the online journal)
Recently, topological insulators (TI) have attracted much attention of researchers and the peculiar surface (edge) states in the TI may have possible applications in the spintronics field [10–16]. For a 2D TI, the intrinsic strong SOC leads to opposite spin states counter propagating along a given edge of the sample [17, 18]. The helical edge states are robust against the time-reversal-symmetry conserving disorders owing to backscattering prohibition. It indicates a very long spin coherence time of electrons and this is very useful in information processing [19, 20]. Therefore, researchers devote themselves to utilizing the helical edge states to control and manipulate the spin states. Since observation of the helical edge states in the HgTe quantum wells [21], many efforts have been made to manipulate the spin states experimentally and theoretically in this compound. For example, Br¨une et al constructed an H-shaped structure to realize spin injection and detection by a purely electrical method [22]. Zeng [8] also proposed a U-shaped multiterminal device to separate and filter spins. Nevertheless, HgTe quantum well has some limitations such as toxicity, difficulty in processing and incompatibility in combination with current silicon-based electronic technology [23]. Another typical TI is the graphene with intrinsic SOC, but the very weak SOC limits
1. Introduction
Spintronics is a unique approach to information processing that utilizes the spin state of an electron, rather than its charge [1–6]. It is anticipated that the spin-based devices will hold great promise in increasing data processing speed and lowering power consumption. Considerable efforts have been made to manipulate and control spin with the ferromagnetic metals [2] or via the electric methods by using the spin–orbit coupling (SOC) in semiconductors. However, the former is still very challenging because of the spin dephasing effect and the short spin relaxation time in metal-based systems [7]. Moreover, the short screening length in metals also makes it difficult to control spin polarization via an electric field [8]. Another method to controlling spin current is to exploit the Rashba SOC in semiconductors, which can generate an effective magnetic field and split up and down spins. Datta two decades ago proposed a spin field-effect transistor by using the Rashba SOC to manipulate spin current in a two dimensional electron gas (2DEG) system [9]. However, due to the conductance mismatch between the ferromagnetic metal and the 2DEG as well as the relative small SOC strength in 2DEG, the spin current polarization is not as high as wished. 0953-8984/15/125005+07$33.00
1
© 2015 IOP Publishing Ltd Printed in the UK
J. Phys.: Condens. Matter 27 (2015) 125005
H Tian et al
its usage in reality. So tremendous efforts have been devoted to searching for the graphene-like 2D materials with large SOC such as silicene, germanene and MoS2 , et al [23–27]. Similarly, it is expected that these TI materials should have significant applications in spintronics. For instance, Rachel found that silicene nanoribbons with manipulated edges can be used as a perfect spin-filter by introducing an in-plane anti-ferromagnetic exchange field [28]. Tasi proposed a silicene-based spin filter device taking advantage of the bulk carrier rather than the helical edge currents, although it only exists at weak disorders [29]. Advancement in fabrication and compatibility with silicon-based nanotechnology makes graphene-like materials particularly interesting for spintronics. Since the helical edge states in a TI mean that the two opposite spin species in a given material boundary possess opposite chiralities, the chirality degree of freedom can in principle provide a method to control and manipulate spin when we introduce the chirality mixing effect into the TI. Thus in this work, we will investigate the spin dependent scattering properties with different chiralities in a graphenelike TI/n junction, where a perpendicular magnetic field B or an off-resonant circularly-polarized light field is applied in the n region to induce a chiral edge state. The TI/n junction is realized by tuning the Fermi energy of each region in the energy gap or beyond the gap by using a gate voltage. It is shown that for the TI/n junction, the spin which has the same chirality in the two regions can flow through the junction interface with a unit transmission, whereas the spin with opposite chiralities in each region is prohibited to tunnel through the interface. A spin filtering effect is thus produced due to this chiralitymixing effect in the TI system. An multiple TI/n junction is also studied to further restrain the spin possessing opposite chiralities in the TI and n regions and the spin filter effect is found to be enhanced significantly. This paper is organized as follows. In section 2, we present the model of the studied TI/n junction as well as formulas describing the spin filtering effect. In section 3, numerical results are given and discussions on the physics origin are presented. A brief conclusion is drawn in the last section.
3
4
6 5
2 1
(b)
n
TI
Figure 1. (a) Schematic of a zigzag edge graphene-like TI/n junction. The transition region from the n region to TI region is denoted in the dashed rectangle with its length M = 4 and width N = 6. (b) Schematic of the edge states scattering phenomenon at the TI/n interface, where the magnetic field or the light field is applied in the n region. One spin species will pass through the junction while the other one will be prevented because it has opposite chiralities in the TI and n regions, the red and black line represent the edge states with different chirality.
next-nearest-neighboring hopping with respect to positive z axis, respectively. In the presence of magnetic field B, a Peierl’s phase factor φij is added in the hopping interactions j 0 with the vector potential A = and φij = i A · dl/φ (−By, 0, 0) and φ0 = h ¯ /e. When an off-resonant circularlypolarized light field is irradiated onto the silicene sheet, the corresponding electromagnetic potential could be given by A(t) = (A sin(ωt), A cos(ωt)) where ω is the frequency of light with ω > 0 for the right circulation and ω < 0 for the left circulation. The potential satisfies the time periodicity + T ) = A(t) with T = 2π/ | ω |. The modification of A(t the Hamiltonian to the Kane–Mele model [30] owing to the time-periodic perturbation can be understood as the sum of two second-order processes and summarized as [31] λ †
H = −i √ νij aiα ajβ , (2) 3 3 i,j αβ
The basic nature of graphene-like 2D materials including the SOC can be described by the Kane–Mele model [30] † † H = (iα + wiα )aiα aiα − t eiφij aiα aj α
where λ is a constant relevant with the light intensity and we set λ = λSO in our calculations. For the graphene-like TI/n junction, the onsite energy in the left or right lead, iα = En (ETI ) can be tuned by the gate voltages. In the transition region, the electrostatic potential varies linearly from En to ETI over a length M, as shown in figure 1(a). The on-site disorder energy wi is uniformly distributed in the range [−W/2, W/2]t with the disorder strength W . The size of the transition region is described by the length M and the number of the zigzag chains N , for instance, figure 1(a) depicts the system with M = 4 and N = 6. It is noted that the magnetic field or the off-resonant circularly-polarized light field exists only in the transition region as well as the n region. The application of the off-resonant light does not directly excite electrons while
i,j α
λSO iφij † z +i √ e νij aiα σαβ ajβ , 3 3 i,j αβ
2
4 3
2. Model and formalism
iα
1
(a)
(1)
† and aiα are the creation and annihilation operators where aiα at the discrete site i with spin polarization α and i, j /i, j runs over all the nearest/next-nearest-neighbor hopping sites. The first term represents the on-site energy and the random disorder, the second and third terms represent the nearest and next-nearest coupling with the effective SOC λSO . σ = (σx , σy , σz ) is the Pauli matrix of spin, νij = +1 and νij = −1 correspond to the anticlockwise and clockwise
2
J. Phys.: Condens. Matter 27 (2015) 125005
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Figure 2. The conductance G↑ and G↓ versus ETI for different length M in the n/TI junction with En = 0.16t where the magnetic field is applied in the n region. Parameters are N = 400, φ = 0 in (a) and φ = 0.04 in (b)–(d).
modifies the electron bands through virtual photon absorption processes [31–34]. In the gap at Ef = 0, the edge states of the well-known Quantum spin Hall (QSH) type emerges also in the n region. When the optical field is turned on, the time-reversal symmetry of the system is broken and the QSH states shall be slightly destroyed, which depends on the strength of the applied field. The degeneracy of the QSH edge states is lifted and the two spin states of different velocities are created on each boundary. The velocities of the two spins increase or decrease further with the applied field strength. When the field strength exceeds to the SOC, the system is transformed to the Quantum Hall(QH) insulator region. Therefore, a chiral edge state forms at the boundaries of the n region [33]. When the magnetic field is turned on, the spin degeneracy of the QSH phase is lifted while the QH phase beyond the gap remains spin degenerate. As Fermi energies reside in the SOC gap, the system is in the QSH phase with opposite spin channels propagating in opposite directions at a given edge of the material, while for Fermi energies outside the gap, the system is in the QH phase and the two opposite spin channels on a given edge propagate in the same direction [35, 36]. Figure 1(b) displays the scattering phenomenon of different chiralities at the TI/n interface. In the TI region, the helical edge states whose chiralities are denoted by the black and red lines, circulate oppositely along the interface. While in the n region, the chiral edge states denoted by the two black lines flow along the interface. The spin-down species can pass through the junction because it has the same chiralities in the two sides of the junction. However, for the spin-up electron, it has different chiralities in the two
sides of the junction, so it cannot form a mixing states and the tunnel process is prohibited. The spin-dependent current through the graphene-like TI/n junction is calculated from the Landauer-B¨uttiker formula e (3) dETnTI↑(↓) (E)[fn (E) − fTI (E)], I↑(↓) = h where fn(TI) (E) = 1/{exp[(E − eVn(TI) )/kB T ] + 1} is the Fermi distribution function of the two leads. TnTI↑(↓) (E) is the spin-dependent electron transmission from the source to drain electrodes and the spin-up (down) conductance can be expressed as G↑(↓) =
e2 e2 TnTI↑(↓) (E) = Tr[ n Gr TI Ga ]↑↑(↓↓) , h h
(4)
a r where n(TI) = i( n(TI) − n(TI) ) is the line width function of a r )) the retarded (advanced) the two electrodes with n(TI) ( n(TI) self-energy. Gr (Ga ) is the retarded (advanced) Green’s r ] function and given by Gr = [Ga ]† = 1/[E −Htran − nr − TI and Htran is the Hamiltonian of the transition region.
3. Results and discussion
In the following numerical calculations, we set the hopping energy t = 1.0 eV and λSO = 0.04t, which is commensurate with silicene. Under a perpendicular magnetic √field B, the magnetic flux through a honeycomb lattice is 3 23B and the 3
J. Phys.: Condens. Matter 27 (2015) 125005
H Tian et al
Figure 3. The conductance G↑ and G↓ in the TI/n junction versus En with the light field applied in the n region. Parameters are ETI = 0 and N = 100.
Figure 4. The conductance G↑ and G↓ in the n/TI/n junction versus ETI with different disorder strength where a magnetic field is applied in the n region. Parameters are φ = 0.04, N = 400, En = 0.16t. Figures (a)–(c) correspond to the n/TI/n junction and (d) corresponds to the n/TI1/n/TI2/n junction with ETI1 = 0. The length of TI region and n region are set at M1 = 150, the length of the transition region is M = 10.
4
J. Phys.: Condens. Matter 27 (2015) 125005
H Tian et al
Figure 5. The conductance G↑ and G↓ in the n/TI/n junction versus ETI with different disorder strength where the light field is applied in the n region. Parameters are En = 0, N = 100. Figures (a)–(c) correspond to the n/TI/n junction while (d) corresponds to the n/TI1/n/TI2/n junction with ETI1 = 0. The length of the central TI region and n region are set at M1 = 80, the length of the transition region is M = 40. √
while in the right lead the two spin species circulate in the same direction with the same chiralities owing to the influence of the light. It is shown that the spin-up species pass through the junction due to the chirality-mixing effect while the spin down species are inhibited in the energy gap and filtered at a relative large M, as is shown for G↓ = 0 in a large energy scale in the gap when M = 20. In the following, we will study the chirality-dependent spin transport properties in the n/TI/n junction with the magnetic field or light field applied in the n region. The results are shown in figures 4 and 5. When W = 0, one spin species provides a unity conductance due to the chirality-mixing effect in the energy gap, while the other one is further decreased because of different chiralities in the leads and conductor (the TI/n junction). Therefore, the conductor can be operated as a spin filter. In the presence of disorders, the spin filter effect is still stable when the magnetic field is applied in the n region, for instance, it is still perfect at W = 0.04t as shown in figure 4(c). In contrast, when the light field is applied in the n region, the spin filter effect is not as perfect as figure 4. As shown in figures 5(b) and (c), G↓ grows with an increase of W and even up to the magnitude of 0.1, although G↑ always equals e2 / h in the energy gap. This indicates clearly that the spin filter effect is destroyed in the strong disorder case. However, it is shown that when a multiple-interface n/TI1/n/TI2/n junction is introduced, the spin filter effect is enhanced in all the energy gap even at W = 0.04t as shown in figures 4(d) and 5(d).
corresponding Peierl’s phase is φ = 32φ3B . In the presence of 0 disorders, the conductance is averaged over up to 1000 random configurations. Firstly, we investigate the interplay of the spin-dependent chiralities of electrons between the Hall edge states and the helical edge states in the presence of the magnetic field (figure 2) or the light field (figure 3) in the n region. Figure 2 shows the up-spin conductance (G↑ ) and down-spin one (G↓ ) versus the Fermi energy of the right lead ETI at En = 0.16t with different transition region lengths M and φ. When the magnetic field B = 0 as shown in figure 2(a), G↑(↓) = e2 / h in the energy gap because there is one spin-up transverse mode and one spin-down transverse mode in the right lead, although there are more transverse subbands in the left lead. In the presence of magnetic field B, the subbands evolve into the Landau levels and the chiral quasiparticles circulate in the same direction along the interface, as shown in figure 1(b). Due to the chirality-mixing effect, the spin-down species penetrate the junction, while the spin-up species are inhibited. This can be seen clearly from figure 2(b), where G↓ = e2 / h in the energy gap and G↑ = 0 at −λ < ETI < 0 while it increases with ETI at ETI > 0, this results from the fact that the quasiparticles are all holes in the two leads when ETI > 0 [37]. Moreover, the inhibition is dependent on the length of the transition region, for instance, it is more prominent at M = 4 (figure 2(b)) than M = 16 (figure 2(d)). In figure 3 we present G↑ and G↓ versus the Fermi energy of the right lead En at ETI = 0. In the left lead the helical edge states circulate oppositely along the interface 5
J. Phys.: Condens. Matter 27 (2015) 125005
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which makes the spin filter effect even survive in a strong disorder case. We also show that the filtered spin species can be controlled by the bipolar property of graphene as well as the chirality of the magnetic or light field. Due to the simple geometry and successful fabrication of the graphenelike materials in experiment, it is expected that the studied two-terminal device can be experimentally fabricated and our results may shed light on the control and manipulation of spin in the helical edge state of a TI. Acknowledgments
We thank support from NSFC (Grant No. 11174242, 11274059, 11447216, 11447218) and NSF of Jiangsu Province (Grant No. BK20131284). Figure 6. The conductance G↑ and G↓ in the n/TI/n junction versus En where the magnetic field is applied in the n region. Parameters are ETI = 0, N = 400, φ = 0.16. The length of the center TI region and transition region is M1 = 200 and zero.
References [1] Fert A 2008 Rev. Mod. Phys. 80 1517 [2] Wolf S A, Awschalom D D, Buhrman R A, Daughton J M, ´ S, Roukes M L, Chtchelkanova A K and von MolnAr Treger D M 2001 Science 294 1488 ˇ c I, Fabian J and Sarma S D 2004 Rev. Mod. Phys. 76 323 [3] Zuti´ [4] Donarini A, Begemann G and Grifoni M 2009 Nano Lett. 9 2897 [5] McCallum A T and Johnson M 2009 Nano Lett. 9 2350 [6] Mourachkine A, Yazyev O V, Ducati C and Ansermet J P 2008 Nano Lett. 8 3683 [7] Tian H Y, Yang Y H and Wang J 2012 Eur. Phys. J. B 85 252 [8] Zeng M G and Liang G C 2012 J. Appl. Phys. 112 073707 [9] Datta S and Das B 1990 Appl. Phys. Lett. 56 665 [10] Bernevig B A, Hughes T L and Zhang S C 2006 Science 314 1757 [11] B¨uettner B et al 2011 Nat. Phys. 7 418 [12] Cha J J, Williams J R, Kong D, Meister S, Peng H, Bestwick A J, Gallagher P, Goldhaber-Gordon D and Cui Y 2010 Nano Lett. 10 1076 [13] Hong S S, Kundhikanjana W, Cha J J, Lai K J, Kong D S, Meister S, Kelly M A, Shen Z X and Cui Y 2010 Nano Lett. 10 3118 [14] Kong D S et al 2011 Nat. Nanotechnol. 6 705 [15] Zhao Y H, Hu Y B, Liu L, Zhu Y and Guo H 2011 Nano Lett. 11 2088 [16] Steinberg H, Gardner D R, Lee Y S and Jarillo-Herrero P 2010 Nano Lett. 10 5032 [17] Hasan M Z and Kane C L 2010 Rev. Mod. Phys. 82 3045 [18] Qi X L and Zhang S C 2010 Phys. Today 63 33 [19] Jiang H, Cheng S G, Sun Q F and Xie X C 2009 Phys. Rev. Lett. 103 036803 [20] Zhang L B, Chang K, Xie X C, Buhmann H and Molenkamp L W 2010 New J. Phys. 12 083058 [21] Roth A, Br¨une C, Buhmann H, Molenkamp L W, Maciejko J, Qi X L and Zhang S C 2009 Science 325 294 [22] Br¨une C, Roth A, Buhmann H, Hankiewicz E M, Molenkamp L W, Maciejko J, Qi X L and Zhang S C 2012 Nat. Phys. 8 485 [23] Liu C C, Feng W X and Yao Y G 2011 Phys. Rev. Lett. 107 076802 [24] Seixas L, Padilha J E and Fazzio A 2014 Phys. Rev. B 89 195403 [25] Vargiamidis V, Vasilopoulos P and Hai G Q 2014 J. Phys.: Condens. Matter 26 345303 [26] Wang J and Fischer S 2014 Phys. Rev. B 89 245421 [27] Feng B J, Ding Z J, Meng S, Yao Y G, He X Y, Cheng P, Chen L and Wu K H 2012 Nano. Lett. 12 3507 [28] Rachel S and Ezawa M 2014 Phys. Rev. B 89 195303
Besides the spin filter effect in the n/TI/n junction, the n/TI/n junction could also be acted as a spin selectivity by changing the Fermi energy of the n region under the perpendicular magnetic field B, the result is shown in figure 6, which corresponds to the sharp interface. When En < −λ, G↑ = e2 / h while G↓ approaches about zero due to the chirality-mixing effect, the opposite case occurs at En > λ. Up to now, we only consider the zigzag-edge material. In fact, it has very similar behaviors for the armchair edge: the spin species with the same chiralities can pass through the TI/n junction while the other spin species will be prohibited because it has different chiralities in different leads. Moreover, this kind of spin species could also be filtered through n/TI/n junction but the other spin species could be transmitted perfectly even in the strong disorder case. However, for the armchair-edge case, the two spin species were fully prohibited near the Dirac point because the edge states are also gapped due to the much longer damping length [38] of the edge states. Recently, silicene sheets and ribbons have been demonstrated through synthesis on metal surfaces [39–41], while the MoS2 crystals in high magnetic fields have been measured in experiment [42], the proposed spin filter design is likely to be realized in these graphene-like materials. 4. Conclusion
In summary, we have investigated the spin filter effect in the graphene-like TI/n junction with an external magnetic field or light field applied in the n region. By using the nonequilibrium Green’s function method and a tight-binding model, we numerically demonstrated that the chirality-conservation can lead to a conspicuous spin filter effect in the TI/n junction, the spin with the same chiralities in the TI and n regions is allowed to transport whereas the opposite-chirality spin is thoroughly prohibited. For a multiple TI/n junction, the transmitted one is not affected at all due to the conservation of chirality while the transmission of the opposite spin is further suppressed, 6
J. Phys.: Condens. Matter 27 (2015) 125005
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[37] Taskin A A, Ren Z, Sasaki S, Segawa K and Ando Y 2011 Phys. Rev. Lett. 107 016801 [38] Ezawa M and Nagaosa N 2013 Phys. Rev. B 88 121401 [39] Firoz Islam S K and Ghosh T K 2014 J. Phys.: Condens. Matter 26 335303 [40] Vogt P, Padova P D, Quaresima C, Avila J, Frantzeskakis E, Asensio M C, Resta A, Ealet B and Lay G L 2012 Phys. Rev. Lett 108 155501 [41] Aufray B, Kara A, Vizzini S, Oughaddou H, L´eandri C, Ealet B and Lay G L 2010 Appl. Phys. Lett. 96 183102 [42] Goto T, Kato T, Uchida K and Miura N 2000 J. Phys.: Condens. Matter 12 6719
[29] Tsai W F, Huang C Y, Chang T R, Lin H, Jeng H T and Bansil A 2013 Nat. Commun. 4 1500 [30] Kane C L and Mele E J 2005 Phys. Rev. Lett. 95 226801 [31] Ezawa M 2013 Phys. Rev. Lett. 110 026603 [32] Kitagawa T, Oka T, Brataas A, Fu L and Demler E 2011 Phys. Rev. B 84 235108 [33] Quelle A and Smith C M 2014 Phys. Rev. B 90 195137 [34] Zhai X C and Jin G J 2014 Phys. Rev. B 89 235416 [35] Shevtsov O, Carmier P, Petitjean C, Groth C, Carpentier D and Waintal X 2012 Phys. Rev. X 2 031004 [36] De Martino A, H¨utten A and Egger R 2011 Phys. Rev. B 84 155420
7
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The chirality dependent spin filter design in the graphene-like junction
This content has been downloaded from IOPscience. Please scroll down to see the full text. 2015 J. Phys.: Condens. Matter 27 125005 (http://iopscience.iop.org/0953-8984/27/12/125005) View the table of contents for this issue, or go to the journal homepage for more Download details: IP Address: 104.239.165.217 This content was downloaded on 16/06/2017 at 21:41 Please note that terms and conditions apply.
You may also be interested in: Topological metallic states in spin–orbit coupled bilayer systems Hui Pan, Xin Li, Zhenhua Qiao et al. Manipulating interface states in monolayer–bilayer graphene planar junctions Fang Zhao, Lei Xu and Jun Zhang Topological phases in two-dimensional materials: a review Yafei Ren, Zhenhua Qiao and Qian Niu The effect of dephasing on edge state transport through p–n junctions in HgTe/CdTe quantum wells Ying-Tao Zhang, Juntao Song and Qing-Feng Sun Measurable spin-polarized current in two-dimensional topological insulators Xing-Tao An, Yan-Yang Zhang, Jian-Jun Liu et al. Multiple topological interface states in silicene S K Wang, J Wang and K S Chan Quantum anomalous Hall effect in time-reversal-symmetry breaking topological insulators Cui-Zu Chang and Mingda Li Symmetry-dependent spin-charge transport and thermopower through a ZSiNR-based FM/normal/FM junction Benliang Zhou, Benhu Zhou, Xiongwen Chen et al.
Journal of Physics: Condensed Matter J. Phys.: Condens. Matter 27 (2015) 125005 (7pp)
doi:10.1088/0953-8984/27/12/125005
The chirality dependent spin filter design in the graphene-like junction Hongyu Tian1 , Sake Wang2 , Jingguo Hu3 and Jun Wang2 1
Department of Physics, Yancheng Institute of Technology, Jiangsu 224051, People’s Republic of China Department of Physics, Southeast University, Nanjing 210096, People’s Republic of China 3 College of Physics Science and Technology, Yangzhou University, Yangzhou 225009, People’s Republic of China 2
E-mail: [email protected] and [email protected] Received 10 December 2014, revised 26 January 2015 Accepted for publication 2 February 2015 Published 19 February 2015 Abstract
We investigate the chirality-dependent spin transport in a graphene-like topological insulator (TI) TI/n junction, where a perpendicular magnetic field or an off-resonant circularlypolarized light field is applied to the normal (n) region. It is found that the coupling between the helical edge states of the TI and chiral edge states from the magnetic/light field results in a perfect spin filtering effect and only one spin species can tunnel through the junction interface. The origin is ascribed to the chirality-conservation requirement, since the two spin species have the opposite chiralities in the TI region and in the n region both of them have the same chiralities. For a TI/n superlattice structure, the spin filtering effect is enhanced and even survives in a fairly strong disorder environment. Keywords: topological insulator, spin filtering effect, chirality-conservation (Some figures may appear in colour only in the online journal)
Recently, topological insulators (TI) have attracted much attention of researchers and the peculiar surface (edge) states in the TI may have possible applications in the spintronics field [10–16]. For a 2D TI, the intrinsic strong SOC leads to opposite spin states counter propagating along a given edge of the sample [17, 18]. The helical edge states are robust against the time-reversal-symmetry conserving disorders owing to backscattering prohibition. It indicates a very long spin coherence time of electrons and this is very useful in information processing [19, 20]. Therefore, researchers devote themselves to utilizing the helical edge states to control and manipulate the spin states. Since observation of the helical edge states in the HgTe quantum wells [21], many efforts have been made to manipulate the spin states experimentally and theoretically in this compound. For example, Br¨une et al constructed an H-shaped structure to realize spin injection and detection by a purely electrical method [22]. Zeng [8] also proposed a U-shaped multiterminal device to separate and filter spins. Nevertheless, HgTe quantum well has some limitations such as toxicity, difficulty in processing and incompatibility in combination with current silicon-based electronic technology [23]. Another typical TI is the graphene with intrinsic SOC, but the very weak SOC limits
1. Introduction
Spintronics is a unique approach to information processing that utilizes the spin state of an electron, rather than its charge [1–6]. It is anticipated that the spin-based devices will hold great promise in increasing data processing speed and lowering power consumption. Considerable efforts have been made to manipulate and control spin with the ferromagnetic metals [2] or via the electric methods by using the spin–orbit coupling (SOC) in semiconductors. However, the former is still very challenging because of the spin dephasing effect and the short spin relaxation time in metal-based systems [7]. Moreover, the short screening length in metals also makes it difficult to control spin polarization via an electric field [8]. Another method to controlling spin current is to exploit the Rashba SOC in semiconductors, which can generate an effective magnetic field and split up and down spins. Datta two decades ago proposed a spin field-effect transistor by using the Rashba SOC to manipulate spin current in a two dimensional electron gas (2DEG) system [9]. However, due to the conductance mismatch between the ferromagnetic metal and the 2DEG as well as the relative small SOC strength in 2DEG, the spin current polarization is not as high as wished. 0953-8984/15/125005+07$33.00
1
© 2015 IOP Publishing Ltd Printed in the UK
J. Phys.: Condens. Matter 27 (2015) 125005
H Tian et al
its usage in reality. So tremendous efforts have been devoted to searching for the graphene-like 2D materials with large SOC such as silicene, germanene and MoS2 , et al [23–27]. Similarly, it is expected that these TI materials should have significant applications in spintronics. For instance, Rachel found that silicene nanoribbons with manipulated edges can be used as a perfect spin-filter by introducing an in-plane anti-ferromagnetic exchange field [28]. Tasi proposed a silicene-based spin filter device taking advantage of the bulk carrier rather than the helical edge currents, although it only exists at weak disorders [29]. Advancement in fabrication and compatibility with silicon-based nanotechnology makes graphene-like materials particularly interesting for spintronics. Since the helical edge states in a TI mean that the two opposite spin species in a given material boundary possess opposite chiralities, the chirality degree of freedom can in principle provide a method to control and manipulate spin when we introduce the chirality mixing effect into the TI. Thus in this work, we will investigate the spin dependent scattering properties with different chiralities in a graphenelike TI/n junction, where a perpendicular magnetic field B or an off-resonant circularly-polarized light field is applied in the n region to induce a chiral edge state. The TI/n junction is realized by tuning the Fermi energy of each region in the energy gap or beyond the gap by using a gate voltage. It is shown that for the TI/n junction, the spin which has the same chirality in the two regions can flow through the junction interface with a unit transmission, whereas the spin with opposite chiralities in each region is prohibited to tunnel through the interface. A spin filtering effect is thus produced due to this chiralitymixing effect in the TI system. An multiple TI/n junction is also studied to further restrain the spin possessing opposite chiralities in the TI and n regions and the spin filter effect is found to be enhanced significantly. This paper is organized as follows. In section 2, we present the model of the studied TI/n junction as well as formulas describing the spin filtering effect. In section 3, numerical results are given and discussions on the physics origin are presented. A brief conclusion is drawn in the last section.
3
4
6 5
2 1
(b)
n
TI
Figure 1. (a) Schematic of a zigzag edge graphene-like TI/n junction. The transition region from the n region to TI region is denoted in the dashed rectangle with its length M = 4 and width N = 6. (b) Schematic of the edge states scattering phenomenon at the TI/n interface, where the magnetic field or the light field is applied in the n region. One spin species will pass through the junction while the other one will be prevented because it has opposite chiralities in the TI and n regions, the red and black line represent the edge states with different chirality.
next-nearest-neighboring hopping with respect to positive z axis, respectively. In the presence of magnetic field B, a Peierl’s phase factor φij is added in the hopping interactions j 0 with the vector potential A = and φij = i A · dl/φ (−By, 0, 0) and φ0 = h ¯ /e. When an off-resonant circularlypolarized light field is irradiated onto the silicene sheet, the corresponding electromagnetic potential could be given by A(t) = (A sin(ωt), A cos(ωt)) where ω is the frequency of light with ω > 0 for the right circulation and ω < 0 for the left circulation. The potential satisfies the time periodicity + T ) = A(t) with T = 2π/ | ω |. The modification of A(t the Hamiltonian to the Kane–Mele model [30] owing to the time-periodic perturbation can be understood as the sum of two second-order processes and summarized as [31] λ †
H = −i √ νij aiα ajβ , (2) 3 3 i,j αβ
The basic nature of graphene-like 2D materials including the SOC can be described by the Kane–Mele model [30] † † H = (iα + wiα )aiα aiα − t eiφij aiα aj α
where λ is a constant relevant with the light intensity and we set λ = λSO in our calculations. For the graphene-like TI/n junction, the onsite energy in the left or right lead, iα = En (ETI ) can be tuned by the gate voltages. In the transition region, the electrostatic potential varies linearly from En to ETI over a length M, as shown in figure 1(a). The on-site disorder energy wi is uniformly distributed in the range [−W/2, W/2]t with the disorder strength W . The size of the transition region is described by the length M and the number of the zigzag chains N , for instance, figure 1(a) depicts the system with M = 4 and N = 6. It is noted that the magnetic field or the off-resonant circularly-polarized light field exists only in the transition region as well as the n region. The application of the off-resonant light does not directly excite electrons while
i,j α
λSO iφij † z +i √ e νij aiα σαβ ajβ , 3 3 i,j αβ
2
4 3
2. Model and formalism
iα
1
(a)
(1)
† and aiα are the creation and annihilation operators where aiα at the discrete site i with spin polarization α and i, j /i, j runs over all the nearest/next-nearest-neighbor hopping sites. The first term represents the on-site energy and the random disorder, the second and third terms represent the nearest and next-nearest coupling with the effective SOC λSO . σ = (σx , σy , σz ) is the Pauli matrix of spin, νij = +1 and νij = −1 correspond to the anticlockwise and clockwise
2
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Figure 2. The conductance G↑ and G↓ versus ETI for different length M in the n/TI junction with En = 0.16t where the magnetic field is applied in the n region. Parameters are N = 400, φ = 0 in (a) and φ = 0.04 in (b)–(d).
modifies the electron bands through virtual photon absorption processes [31–34]. In the gap at Ef = 0, the edge states of the well-known Quantum spin Hall (QSH) type emerges also in the n region. When the optical field is turned on, the time-reversal symmetry of the system is broken and the QSH states shall be slightly destroyed, which depends on the strength of the applied field. The degeneracy of the QSH edge states is lifted and the two spin states of different velocities are created on each boundary. The velocities of the two spins increase or decrease further with the applied field strength. When the field strength exceeds to the SOC, the system is transformed to the Quantum Hall(QH) insulator region. Therefore, a chiral edge state forms at the boundaries of the n region [33]. When the magnetic field is turned on, the spin degeneracy of the QSH phase is lifted while the QH phase beyond the gap remains spin degenerate. As Fermi energies reside in the SOC gap, the system is in the QSH phase with opposite spin channels propagating in opposite directions at a given edge of the material, while for Fermi energies outside the gap, the system is in the QH phase and the two opposite spin channels on a given edge propagate in the same direction [35, 36]. Figure 1(b) displays the scattering phenomenon of different chiralities at the TI/n interface. In the TI region, the helical edge states whose chiralities are denoted by the black and red lines, circulate oppositely along the interface. While in the n region, the chiral edge states denoted by the two black lines flow along the interface. The spin-down species can pass through the junction because it has the same chiralities in the two sides of the junction. However, for the spin-up electron, it has different chiralities in the two
sides of the junction, so it cannot form a mixing states and the tunnel process is prohibited. The spin-dependent current through the graphene-like TI/n junction is calculated from the Landauer-B¨uttiker formula e (3) dETnTI↑(↓) (E)[fn (E) − fTI (E)], I↑(↓) = h where fn(TI) (E) = 1/{exp[(E − eVn(TI) )/kB T ] + 1} is the Fermi distribution function of the two leads. TnTI↑(↓) (E) is the spin-dependent electron transmission from the source to drain electrodes and the spin-up (down) conductance can be expressed as G↑(↓) =
e2 e2 TnTI↑(↓) (E) = Tr[ n Gr TI Ga ]↑↑(↓↓) , h h
(4)
a r where n(TI) = i( n(TI) − n(TI) ) is the line width function of a r )) the retarded (advanced) the two electrodes with n(TI) ( n(TI) self-energy. Gr (Ga ) is the retarded (advanced) Green’s r ] function and given by Gr = [Ga ]† = 1/[E −Htran − nr − TI and Htran is the Hamiltonian of the transition region.
3. Results and discussion
In the following numerical calculations, we set the hopping energy t = 1.0 eV and λSO = 0.04t, which is commensurate with silicene. Under a perpendicular magnetic √field B, the magnetic flux through a honeycomb lattice is 3 23B and the 3
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Figure 3. The conductance G↑ and G↓ in the TI/n junction versus En with the light field applied in the n region. Parameters are ETI = 0 and N = 100.
Figure 4. The conductance G↑ and G↓ in the n/TI/n junction versus ETI with different disorder strength where a magnetic field is applied in the n region. Parameters are φ = 0.04, N = 400, En = 0.16t. Figures (a)–(c) correspond to the n/TI/n junction and (d) corresponds to the n/TI1/n/TI2/n junction with ETI1 = 0. The length of TI region and n region are set at M1 = 150, the length of the transition region is M = 10.
4
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Figure 5. The conductance G↑ and G↓ in the n/TI/n junction versus ETI with different disorder strength where the light field is applied in the n region. Parameters are En = 0, N = 100. Figures (a)–(c) correspond to the n/TI/n junction while (d) corresponds to the n/TI1/n/TI2/n junction with ETI1 = 0. The length of the central TI region and n region are set at M1 = 80, the length of the transition region is M = 40. √
while in the right lead the two spin species circulate in the same direction with the same chiralities owing to the influence of the light. It is shown that the spin-up species pass through the junction due to the chirality-mixing effect while the spin down species are inhibited in the energy gap and filtered at a relative large M, as is shown for G↓ = 0 in a large energy scale in the gap when M = 20. In the following, we will study the chirality-dependent spin transport properties in the n/TI/n junction with the magnetic field or light field applied in the n region. The results are shown in figures 4 and 5. When W = 0, one spin species provides a unity conductance due to the chirality-mixing effect in the energy gap, while the other one is further decreased because of different chiralities in the leads and conductor (the TI/n junction). Therefore, the conductor can be operated as a spin filter. In the presence of disorders, the spin filter effect is still stable when the magnetic field is applied in the n region, for instance, it is still perfect at W = 0.04t as shown in figure 4(c). In contrast, when the light field is applied in the n region, the spin filter effect is not as perfect as figure 4. As shown in figures 5(b) and (c), G↓ grows with an increase of W and even up to the magnitude of 0.1, although G↑ always equals e2 / h in the energy gap. This indicates clearly that the spin filter effect is destroyed in the strong disorder case. However, it is shown that when a multiple-interface n/TI1/n/TI2/n junction is introduced, the spin filter effect is enhanced in all the energy gap even at W = 0.04t as shown in figures 4(d) and 5(d).
corresponding Peierl’s phase is φ = 32φ3B . In the presence of 0 disorders, the conductance is averaged over up to 1000 random configurations. Firstly, we investigate the interplay of the spin-dependent chiralities of electrons between the Hall edge states and the helical edge states in the presence of the magnetic field (figure 2) or the light field (figure 3) in the n region. Figure 2 shows the up-spin conductance (G↑ ) and down-spin one (G↓ ) versus the Fermi energy of the right lead ETI at En = 0.16t with different transition region lengths M and φ. When the magnetic field B = 0 as shown in figure 2(a), G↑(↓) = e2 / h in the energy gap because there is one spin-up transverse mode and one spin-down transverse mode in the right lead, although there are more transverse subbands in the left lead. In the presence of magnetic field B, the subbands evolve into the Landau levels and the chiral quasiparticles circulate in the same direction along the interface, as shown in figure 1(b). Due to the chirality-mixing effect, the spin-down species penetrate the junction, while the spin-up species are inhibited. This can be seen clearly from figure 2(b), where G↓ = e2 / h in the energy gap and G↑ = 0 at −λ < ETI < 0 while it increases with ETI at ETI > 0, this results from the fact that the quasiparticles are all holes in the two leads when ETI > 0 [37]. Moreover, the inhibition is dependent on the length of the transition region, for instance, it is more prominent at M = 4 (figure 2(b)) than M = 16 (figure 2(d)). In figure 3 we present G↑ and G↓ versus the Fermi energy of the right lead En at ETI = 0. In the left lead the helical edge states circulate oppositely along the interface 5
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which makes the spin filter effect even survive in a strong disorder case. We also show that the filtered spin species can be controlled by the bipolar property of graphene as well as the chirality of the magnetic or light field. Due to the simple geometry and successful fabrication of the graphenelike materials in experiment, it is expected that the studied two-terminal device can be experimentally fabricated and our results may shed light on the control and manipulation of spin in the helical edge state of a TI. Acknowledgments
We thank support from NSFC (Grant No. 11174242, 11274059, 11447216, 11447218) and NSF of Jiangsu Province (Grant No. BK20131284). Figure 6. The conductance G↑ and G↓ in the n/TI/n junction versus En where the magnetic field is applied in the n region. Parameters are ETI = 0, N = 400, φ = 0.16. The length of the center TI region and transition region is M1 = 200 and zero.
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Besides the spin filter effect in the n/TI/n junction, the n/TI/n junction could also be acted as a spin selectivity by changing the Fermi energy of the n region under the perpendicular magnetic field B, the result is shown in figure 6, which corresponds to the sharp interface. When En < −λ, G↑ = e2 / h while G↓ approaches about zero due to the chirality-mixing effect, the opposite case occurs at En > λ. Up to now, we only consider the zigzag-edge material. In fact, it has very similar behaviors for the armchair edge: the spin species with the same chiralities can pass through the TI/n junction while the other spin species will be prohibited because it has different chiralities in different leads. Moreover, this kind of spin species could also be filtered through n/TI/n junction but the other spin species could be transmitted perfectly even in the strong disorder case. However, for the armchair-edge case, the two spin species were fully prohibited near the Dirac point because the edge states are also gapped due to the much longer damping length [38] of the edge states. Recently, silicene sheets and ribbons have been demonstrated through synthesis on metal surfaces [39–41], while the MoS2 crystals in high magnetic fields have been measured in experiment [42], the proposed spin filter design is likely to be realized in these graphene-like materials. 4. Conclusion
In summary, we have investigated the spin filter effect in the graphene-like TI/n junction with an external magnetic field or light field applied in the n region. By using the nonequilibrium Green’s function method and a tight-binding model, we numerically demonstrated that the chirality-conservation can lead to a conspicuous spin filter effect in the TI/n junction, the spin with the same chiralities in the TI and n regions is allowed to transport whereas the opposite-chirality spin is thoroughly prohibited. For a multiple TI/n junction, the transmitted one is not affected at all due to the conservation of chirality while the transmission of the opposite spin is further suppressed, 6
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