Corrigendum: Full Valley and Spin Polarizations in Strained Graphene with Rashba Spin Orbit Coupling and Magnetic Barrier

Scientific Reports 6: Article number: 21590; published online: 22 February 2016; updated: 23 January 2017 This Article contains errors in Reference 15 which was incorrectly given as: Song, Y., Zhai, F. & Guo, Y. Strain-tunable spin transport in ferromagnetic graphene junctions. Appl. Phys. Lett.103, 183111 (2013).


as a gauge potential ( )
A r s 17 . Explicitly, we have δ ( ) = A r t s , where δt parametrizes the strain by its effect on the nearest neighbor hopping, δ → + t t t ( ≈ ) t 3 eV . Within a tight-binding formulation of the electronic motion 12 , the change of hopping energy can be achieved . 772 4 meV , if the bond length increase 10% under strain 9,17 . RSOC can be induced by growing graphene on a Ni surface by catalytic methods 18 , and study shows that Au intercalation at the graphene-Ni interface creates a giant spin-orbit splitting (≈ ) 100 meV in the graphene Dirac cone up to the Fermi energy. We consider the RSOC strength λ ( ) r are constant in the strained region and vanishes otherwise. The magnetic field is assumed to vary only along the x axis, which can be generated by depositing ferromagnetic metal or superconducting materials on top of the dielectric layer, as is the way in semiconductor heterostructures 19,20 . The induced magnetic field can be approximately described 21 here α = ± is the sign of the spin projection along the in-plane direction orthogonal to the propagation direction, µ = ± stands for the conduction (+ ) and valence (− ) bands. α k x is the longitudinal wave vector, , here φ 0 is the incident angle. For an electron in τ z valley with energy E and incident angle φ 0 , we denote the transmission probability as , here the indexes ( ′) = ↑ , ↓ s s specify the incoming and transmitting spin orientations respectively. The transfer matrix method 23 is adopted to calculate φ ( , ) Once the transmission probability is known, one can determine the spin-dependent conductance 24 .
, W is the width of the graphene sample in the y direction. The valley, spin resolved and total conductance are defined as We also introduce valley and spin polarizations P v and P s = −  10 , where Γ is time-reversal operator 25 . It is worth to note that in the energy ranges − . , − . [ 18 9 meV 10 2 meV] and .
, . [10 2 meV 18 9 meV], the conductance is contributed mainly by its spin-flipped component ↓↑ G from the ′ K valley, while the others conductances are greatly suppressed, even the transport channels from K valley are forbidden wholly. Such transport property leads to that there are not only full valley polarization and high spin polarization but also obvious total conductance in the energy ranges − . , − . [ 18 9 meV 10 2 meV] and .
, . [10 2 meV 18 9 meV] [seen in the green dashed lines in Fig. 2(c)]. In addition, the extremely high spin and valley polarizations also are present in the energy region − 10.2 meV < E F < 10.2 meV, but the rather small conductance makes it impossible for either detection or applications. From Fig. 2(c), one can also find that the valley and spin polarizations decrease gradually and eventually decay to zero with increasing the energy.
In this work, we propose and demonstrate that valley and spin filters can be simultaneously achieved in a strained graphene with RSOC and magnetic barrier. Such a remarkable result can be understood by the band structures in each region. For comparison, we first show the band structures near the ′ K and K points in strained graphene with only magnetic barrier [ Fig. 3(a)] and only RSOC [ Fig. 3(b)]. In Fig. 3(a), a band gap is induced, and the band gap for K valley is wider than that of ′ K valley since the total vector potential ± A A M s acting on K electrons is distinct in amplitude from its counterpart for ′ K electrons 15 . That is to say, a valley gap comes into being [see the green ellipse dashed lines in Fig. 3(a)], where only electron state from one valley is propagating wave and that from the other valley is evanescent at a fixed Fermi energy. Thus the valley polarization can be achieved as long as the Fermi energy located inside the valley gap, akin to that in the strained graphene with a periodic magnetic modulation 15 . In addition, spin polarization is absent in such strained graphene system with magnetic barrier due to spin degeneracy. In Fig. 3(b). One can clearly see that the RSOC makes the spin degeneracy lifted and produces a spin energy gap. As a result, the spin polarization for single valley can be achieved in the spin energy gap 9,10 . But such system is not able to generate a valley-polarized current because of the mirror symmetry of K and ′ K valleys 10 . So we study the band structure of strained graphene with both magnetic barrier and RSOC [ Fig. 3

(c)]. It is shown that valley gap is induced in the energy range
[see the green ellipse dashed lines in Fig. 3(c)]. Moreover, we also find that only single spin band at ′ K valley contributions to the current in that energy range as shown in Fig. 3(c), which leads to only one spin-flipped conductance for one valley is dominate [ Fig. 2(a,b)]. Thus, high valley-and spin-polarization currents can be expected at those energies [ Fig. 2(c)]. In addition, because the above band structure only for = k 0 y , while the overall conductance is contributed from electrons with not only = k 0 y , but also ≠ k 0 y , which lead the current can flow, even when the Fermi energy is located in the band gap near the ′ K points [as seen in Fig. 2(a,c)]. The band structure also shows that the valley and single spin band gaps disappear for high Fermi energy, which lead to the valley and spin polarizations become lower and eventually converge to zero with increasing the Fermi energy. As a result, to observe valley/spin polarization current in the concerned structure, it is highly desirable to keep the system with valley gap and single spin band gap.
The valley and single spin band gaps can be deceived by the extreme points of energy band (A, B, C and D points) [see in Fig. 3(c)]. In Fig. 4(a-c), we plot the extreme points of energy band as functions of strain, magnetic barrier and RSOC strengths. With increasing the strain and magnetic strengths, we find that the valley gap firstly widens and then has no obvious change, and the location of the valley gap is shifted to high-energy region [ Fig. 4(a,b)]. While the increase of RSOC strength makes the valley gap narrow and the location of the valley gap move toward low-energy region gradually [ Fig. 4(c)]. In addition, it is shown that the single spin band gap coincides with valley gap for all Fermi energy with the increasing of the strain strength [ Fig. 4(a)]. With increasing the magnetic strength, the single spin band gap firstly coincides with valley gap and then is narrower than the valley gap [ Fig. 4(b)]. And the increase of RSOC strength causes that the single spin band gap firstly is smaller than valley gap and then coincides with valley gap [Fig. 4(c)]. The gap characters reflect on the valley/spin polarization [ Fig. 4(d-i)]. In addition to the high valley/spin polarization in transmission gap, we can also find that the high valley polarization is mainly appears in the valley gap [ Fig. 4(d-f)], and the high spin polarization mostly occur in the single spin band gap [ Fig. 4(g-i)]. However, effective valley/spin filtering also requires a high transmitted valley/spin current. From Fig. 4(j-l), one can find in the transmission gap region, the total conductance is small. While within the valley gap or single spin band gap region, the total conductance can be larger than G 2 0 , allowing for a remarkable valley/spin current, which confirming that such a strained graphene with valley gap and single spin band gap is an effective valley/spin filtering device. Moreover, more abundant spin polarization features of the system can be find in Fig. 4(g,h), where both high positive and negative spin polarizations are present with increasing the strain and magnetic strengths.

Summary
In conclusion, we have demonstrated that strained graphene with RSOC and magnetic barrier can generate valley-and spin-polarization currents. To observe such polarized current in the concerned structure, it is highly desirable to keep the system with valley gap and single spin band gap. The full valley-polarization current mainly appears in the valley gap and full spin-polarization one mostly occurs in the single spin band gap. The valley and  single spin band gaps can be modulated by strain, RSOC and magnetic strength. In addition, under the strain and magnetic manipulation, the spin polarization can be switched by adjusting the strain and magnetization strengths. The full valley-and spin-polarization currents provide the desirable routines to construct spin/valley filter for graphene-based logic applications. Our results would be also applicable to other spin and valley coupled systems, such as monolayers of silicene and MoS 2 , where low energy physics are governed by massive Dirac fermion.