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Enhanced Valley-Resolved Thermoelectric Transport in a Magnetic Silicene Superlattice

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Enhanced Valley-Resolved Thermoelectric Transport in a Magnetic Silicene Superlattice Home Search Collections Journals About Contact us My IOPscience Enhanced valley-resolved thermoelectric transport in a magnetic silicene superlattice This content has been downloaded from IOPscience. Please scroll down to see the full text. 2015 New J. Phys. 17 073026 (http://iopscience.iop.org/1367-2630/17/7/073026) View the table of contents for this issue, or go to the journal homepage for more Download details: IP Address: 62.204.246.50 This content was downloaded on 13/08/2015 at 09:21 Please note that terms and conditions apply. New J. Phys. 17 (2015) 073026 doi:10.1088/1367-2630/17/7/073026 PAPER Enhanced valley-resolved thermoelectric transport in a magnetic OPEN ACCESS silicene superlattice RECEIVED 19 April 2015 Zhi Ping Niu, Yong Mei Zhang and Shihao Dong REVISED College of Science, Nanjing University of Aeronautics and Astronautics, Jiangsu 210016, Peopleʼs Republic of China 8 June 2015 ACCEPTED FOR PUBLICATION E-mail: [email protected] 18 June 2015 Keywords: valleytronics, magnetic silicene superlattice, thermoelectric transport PUBLISHED 22 July 2015 Content from this work Abstract may be used under the Electrons in two-dimensional crystals with a honeycomb lattice structure possess a valley degree of terms of the Creative Commons Attribution 3.0 freedom in addition to charge and spin, which has revived the field of valleytronics. In this work we licence. investigate the valley-resolved thermoelectric transport through a magnetic silicene superlattice. Since Any further distribution of this work must maintain spin is coupled to the valley, this device allows a coexistence of the insulating transmission gap of one attribution to the author(s) and the title of valley and the metallic resonant band of the other, resulting in a strong valley polarization Pv. Pv the work, journal citation oscillates with the barrier strength V with its magnitude greatly enhanced by the superlattice structure. and DOI. In addition, a controllable fully valley polarized transport and an on/off switching effect in the conductance spectra are obtained. Furthermore, the spin- and valley-dependent thermopowers can be controlled by V, the on-site potential difference between A and B sublattices and Fermi energy, and enhanced by the superlattice structure. Enhanced valley-resolved thermoelectric transport and its control by means of gate voltages make the magnetic silicene superlattice attractive in valleytronics applications. 1. Introduction In addition to manipulating the charge or spin of electrons, another way to control electric current in two- dimensional (2D) materials with a honeycomb lattice structure is by using the valley degree of freedom (DOF) [1–3]. This leads to valleytronics, in which information is encoded by the valley quantum number of the electron [1–12]. A fundamental goal in valleytronics is to generate and detect a controllable valley-polarized current. The seminal proposal of valley filters [4] relies on perfect graphene zigzag nanoribbons, which would harden its experimental realization. Several schemes of valley filters have been proposed based on bulk graphene, which utilize the valley dependence of trigonal band warping [13, 14], strain-induced pseudomagnetic fields [15, 16], and line defects [17]. However, the presence of inversion symmetry in the crystal structure of pristine graphene makes control of the valley DOF difficult. In contrast, silicene that possesses a staggered honeycomb lattice structure is inversion-asymmetric. Although the valley- and spin-polarized currents have been investigated in several works in simple normal/ferromagnetic junctions with one or two barriers [18–20], none of the previous works focuses on the valley-polarized transport of 2D magnetic silicene superlattices. It is well known that superlattices possess many interesting electronic transport properties and band structures, which are mainly determined by the periodicity of the barrier potential rather than by the properties of the individual potential barrier. Thus we expect that a 2D silicene monolayer under a periodic magnetic modulation may be a promising candidate to invent valleytronic devices. Recently, spin caloritronics [21], the combination of thermoelectrics and spintronics, has received much attention and caused the renaissance of thermoelectricity in spintronic devices. Spin caloritronics covers physical phenomena [22] that are classified as independent electron (such as spin-dependent Seebeck), collective (such as spin Seebeck), and relativistic (such as spin-Hall) effects. Spin caloritronics has been studied extensively in graphene-based devices [23], and a giant charge thermoelectric coefficient was reported in a graphene superlattice [24]. However, the effect of the valley DOF on spin caloritronics in a 2D silicene superlattice has not yet been considered. © 2015 IOP Publishing Ltd and Deutsche Physikalische Gesellschaft New J. Phys. 17 (2015) 073026 Z P Niu et al Figure 1. (a) Schematic view of the magnetic silicene superlattice and (b) illustration of the electron energy profile in a given spin and valley channel. In this work we study the valley-resolved thermoelectric transport through a 2D magnetic silicene superlattice by using the transfer matrix method. Since spin is coupled to the valley, we observe the valley and spin-resolved transmission, leading to a strong valley polarization Pv. Pv oscillates with the barrier strength, with its magnitude greatly enhanced by the superlattice structure. In addition, a controllable fully valley-polarized transport and an on/off switching effect in the conductance spectra are obtained. Furthermore, the spin- and valley-dependent thermopowers can be controlled by the barrier strength, the on-site potential difference between A and B sublattices and Fermi energy, and enhanced by the superlattice structure. 2. Model and Formulation The geometry of a magnetic superlattice is a 2D silicene under a periodic magnetic modulation (figure 1(a)). Effectively, the system can be regarded as consisting of normal silicene (NS) with thickness L and ferromagnetic silicene (FS) with thickness D arranged alternately, finally connected to two semi-infinite NS electrodes. The coordinate of the jth interface between NS and FS is marked by lj. Only a parallel magnetization configuration will be considered. The potential profile of the system is a multiple step-like quantum well structure with Vσ =−Vhσ for l21jj− <<xl 2(j =…1, 2, , N), with the barrier strength V controlled by the gate voltage; Vσ = 0 otherwise. h stands for the ferromagnetic exchange field, which can be induced by a proximity effect between a ferromagnetic insulator and silicene, as proposed and realized for graphene [25, 26]. If the superlattice has N periods, it contains N − 1 NS and N FS, finally coupled to two NS electrodes. The low-energy effective Hamiltonian simply reads [18, 20, 27, 28] =−−+ˆˆτητΔτ Hvppησ,,F ()x x y yz ησ V σ (1) ˆ with pxy()=−i ∂xy()and Δησ, =−ησλso Δ z. vF is the Fermi velocity, and τ⃗ = (,τττxyz , )are Pauli matrices in pseudospin space. η =±1corresponds to the K and K′ valleys, and σ =±1denotes the spin indices. λso stands for the spin–orbit coupling. Δz is the on-site potential difference between A and B sublattices and can be efficiently tuned by a perpendicular electric field applied perpendicular to the plane. The eigenvalues of the Hamiltonian are given by 2 2 EvkVησ, =±()() F +Δ ησ, + σ,(2) 22 where kkk=+xy. The bandgap is located at the K and K′ points and given by 2 ∣∣Δησ, . Thus, different gaps can arise in different spins and valleys. The energy profile of electrons in the superlattice is illustrated in figure 1(b). In the NS we set Vσ=0, so from equation (2) one finds that E++, and E−−, (or E+−, and E−+, ) are degenerate, while in the FS the degeneracy is broken. Due to the coupling between the valley and spin DOF, it is possible to simultaneously manipulate the valley and spin DOF in silicene. With the general solutions of equation (1), the wave functions in the NS and FS are respectively given by ⎡ ⎛ ⎞ ⎛ ⎞ ⎤ vkF + −vk− ⎢ ⎜ ⎟ ikxxxy⎜ F ⎟ −iikx⎥ ky ΨφNS =+aησ, eee,(3)b ησ, − ⎣ ⎝ EN ⎠ ⎝ EN ⎠ ⎦ 2 New J. Phys. 17 (2015) 073026 Z P Niu et al and ⎡ ⎛ ⎞ ⎛ ⎞ ⎤ vkF ′ −vk′ ⎢ ⎜ +⎟ ikxxx′ ⎜ F −⎟ −′iikx⎥ kyy ΨFS = cησ, ed+ ησ, ⎝ ⎠ ee (4) ⎣ ⎝ EF ⎠ EF ⎦ ()′′ () where kk± =±x iη ky, EN =+E Δησ, , and EF =+EVΔησ, − σ. The wave vectors are 2 2 2 2 2 2 2 kExF=−Δαησ, cos v, kEVx′ =−−−()σ Δησ, () vkvFy F, and kEyF=−Δαησ, sin vwith α the incident angle. For the electron’s transmission through the superlattice, consisting of N potential barriers via the wave function continuity at the NS/FS interfaces, we get the relationship of ⎛ ⎞ −N N ⎜⎟tησ, ⎡ ⎤ ⎡ ⎤ 1 = ⎣RRTW (d)⎦ ⎣ W (d) (0)⎦ (5) ⎝ 0 ⎠ ()rησ, −−11 − 1 with T(0)= RWW ( DM ) MRBB ( DM ) B MW [29] and d =+DL, where ⎛ ⎞ ⎛ ⎞ ⎛ ikx ⎞ ⎛ ikx′ ⎞ vkFF+−− vk vkFF+−′ − vk′ e0x x M = ⎜ ⎟,,()M = ⎜ ⎟ Rx= ⎜ ⎟ and R ()x = ⎜e0⎟. W B W −ikx B −ikx′ ⎝ EENN⎠ ⎝ EEFF⎠ ⎝ 0ex ⎠ ⎝ 0ex ⎠ 2 After the transmission probability for spin σ and valley η electrons Ttησ,,=∣ ησ ∣ is derived from the above equation, the current can be written as +∞ π e 2 ⎡ ⎤ Iησ,,=−d()dcosEN Eαα T ησ⎣ f ()(), E f E ⎦ (6) ∫∫π LR h −∞ − 2 EW2 − Δ 2 where N ()E = ησ, with W the width of the silicene sheet is the carrier density of states (DOS) of the left vF −1 NS electrode. fβ ()EEEkT=+ {1 exp[( −FB )β ]} (β = LR, ) stands for the Fermi distribution function in the β electrode. The conductance at zero temperature can be obtained as 2 π e 2 GEησ, ()= NET() ησ, cosd.αα (7) ∫ π h − 2 By using equation (7), we can define the valley and spin polarization Pv and Ps: PvKKKK=−()()GGGG′′ + σ and Ps =−()()GGGG↑↓↑↓ +, where Gσσ=+GGKK,,′ σis the conductance in the spin- channel, while Gηη=+GG,,↑↓ ηrepresents the conductance in the η valley. In the linear response regime, i.e., TTTLR≈=, we calculate the valley- and spin-resolved thermopower Sησ, [27, 30]: 1 L1,ησ , Sησ, =− (8) eT L0,ησ , with L =−1 d(EE E )n NE () dαα cos T [ −∂ f ()] E .
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