- Email: [email protected]
Gate tunable spin transport in graphene with Rashba spin-orbit coupling
Superlattices and Microstructures 98 (2016) 473e491
Contents lists available at ScienceDirect
Superlattices and Microstructures journal homepage: www.elsevier.com/locate/superlattices
Gate tunable spin transport in graphene with Rashba spinorbit coupling Xiao-Dong Tan a, Xiao-Ping Liao a, Litao Sun a, b, c, * a
SEU-FEI Nano-Pico Center, Key Laboratory of MEMS of Ministry of Education, Collaborative Innovation Center for Micro/Nano Fabrication, Device and System, Southeast University, Nanjing 210096, PR China b Center for Advanced Carbon Materials, Southeast University, Jiangnan Graphene Research Institute, Changzhou 213100, PR China c Center for Advanced Materials and Manufacture, Joint Research Institute of Southeast University, Monash University, Suzhou 215123, PR China
a r t i c l e i n f o
a b s t r a c t
Article history: Received 12 July 2016 Accepted 5 September 2016 Available online 8 September 2016
Recently, it attracts much attention to study spin-resolved transport properties in graphene with Rashba spin-orbit coupling (RSOC). One remarkable finding is that Klein tunneling in single layer graphene (SLG) with RSOC (SLG þ R for short below) behaves as in bi-layer graphene (BLG). Based on the effective Dirac theory, we reconsider this tunneling problem and derive the analytical solution for the transmission coefficients. Our result shows that Klein tunneling in SLG þ R and BLG exhibits completely different behaviors. More importantly, we find two new transmission selection rules in SLG þ R, i.e., the single band to single band (S / S) and the single band to multiple bands (S / M) transmission regimes, which strongly depend on the relative height among Fermi level, RSOC, and potential barrier. Interestingly, in the S / S transmission regime, only normally incident electrons have capacity to pass through the barrier, while in the S / M transmission regime the angle-dependent tunneling becomes very prominent. Using the transmission coefficients, we also derive spin-resolved conductance analytically, and conductance oscillation with the increasing barrier height and zero conductance gap are found in SLG þ R. The present study offers new insights and opportunities for developing graphenebased spin devices. © 2016 Elsevier Ltd. All rights reserved.
Keywords: Graphene Rashba spin-orbit coupling Spin transport
1. Introduction Spin transport and control are of fundamental and practical importance in future graphene-based spintronic devices [1]. Klein tunneling is one of the most supernatural and counterintuitive transport phenomena in which an incoming electron can penetrate through a potential barrier even if its energy is lower than the barrier height [2]. Observation of this exotic phenomenon is quite difficult, because it requires extremely high energy in experiment [3]. Graphene, a single atomic layer of graphitic carbon, exhibits the liner spectrum near the Dirac points, where the carriers behave like massless Dirac fermions. This unique property makes it possible to test the relative quantum tunneling in graphene experimentally. Thus, it arouses
* Corresponding author. SEU-FEI Nano-Pico Center, Key Laboratory of MEMS of Ministry of Education, Collaborative Innovation Center for Micro/Nano Fabrication, Device and System, Southeast University, Nanjing 210096, PR China. E-mail address: [email protected] (L. Sun). http://dx.doi.org/10.1016/j.spmi.2016.09.007 0749-6036/© 2016 Elsevier Ltd. All rights reserved.
474
X.-D. Tan et al. / Superlattices and Microstructures 98 (2016) 473e491
intense interest to investigate Klein tunneling in graphene-based systems [4e13]. Most theoretical works focused mainly on the case of spin-independent tunneling, while spin-orbit coupling (SOC) was less considered. The SOC in graphene comprises intrinsic and extrinsic components, which play a key role in generating a topological insulating state and manipulating electron spins. However, the intrinsic SOC in graphene is extremely weak. Thus, various methods to induce the large extrinsic SOC have been proposed theoretically [14e16] and explored experimentally [17e19]. Rashba SOC (RSOC) originating from the space inversion symmetry breaking, has attracted tremendous research interest in various field of physics and materials science [20], owing to its controllability via an external gate voltage [21]. Marchenko et al. reported a large RSOC in the Au-intercalated graphene-Ni system [17]. Such enhancement is crucial for the development of graphene-based devices such as the Das-Datta spin field effect transistor (FET) [22]. Motivated by the experimental development, recently much attention has paid to investigate the effect of RSOC on spin transport properties of graphene [23e28]. Among these works, one remarkable finding was that Klein tunneling in single layer graphene (SLG) with RSOC (SLG þ R for short below) behaves as in bi-layer graphene (BLG) [25]. Considering the early work by Katsnelson et al. [4], such fantastic behavior seems somewhat surprising. As demonstrated in Ref. [4], chiral tunneling in SLG and BLG exhibits completely different behaviors. So, why does the appearance of RSOC repaint a new tunneling picture in SLG? Admittedly, the low-energy spectrum of SLG þ R and BLG has the same mathematical form [25], but this is not enough to declare that low-energy physics in SLG þ R and BLG behaves similarly. First, the interlayer coupling parameter g1 in BLG and the Rashba coupling l in SLG are two fundamentally different interactions. RSOC in SLG comes from the ps hybridization [29], and unlike conventional semiconduction 2D electron gases, it is independent of the momentum. The parameter g1 in BLG characterizes the nearest-neighbor hopping between the two graphene layers, and almost has no effect on SOC as demonstrated in Refs. [30,31]. Furthermore, to the best of our knowledge, carriers in BLG are massive and similar to conventional non-relativistic quasiparticles [32], which are quite different from massless Dirac fermions in SLG. In fact, it requires different equations to describe electrons in BLG and SLG þ R. As a result, for tunneling problems in BLG, there are four possible solutions for a given energy, and two of them correspond to evanescent waves [4] which do not exist in SLG þ R. Combining these facts, we think that it is necessary to reexamine whether Klein tunneling in SLG þ R behaves as in BLG. In this paper, we reconsider the tunneling problem in SLG þ R based on the effective Dirac theory. We show that the tunneling in SLG þ R and BLG exhibits different behaviors. In BLG, normally incident electrons are always completely reflected by the potential barrier [4], but this is not always true for SLG þ R case. In SLG þ R, normally incident electrons are allowed some tunneling, but are not always perfectly transmitted, consistent with the previous report in Ref. [23]. Interestingly, when the Fermi lever in SLG þ R is less than the strength of RSOC (0 < EF < l), the tunneling only occurs in normally incident electrons. We analyze this case and obtain the analytical solution for the transmission coefficient, revealing the properties of spin polarization independent of the height of potential barrier. We also discuss the case of EF > l in more detail, and intriguing behaviors of the tunneling and the spin-resolved conductance oscillations will be shown. This paper is organized as follows. In Sec. 2 we describe our model and solve it analytically. In Sec. 3 we investigate the subband-dependent transmission coefficients in SLG þ R. Then, we derive and discuss the spin-resolved conductance in Sec. 4. In Sec. 5 we discuss the dependence of spin polarization on the barrier height. Based on the spin transport properties discussed above, in Sec. 6 we propose a conceptive graphene-spin device. Our conclusions are summarized in Sec. 7.
2. Model 0
Near the Dirac points K and K , the effective Hamiltonian of SLG þ R is described by Ref. [33].
v v 1 þ l tsx Sy sy Sx ; Ht0 ¼ iZnF tsx þ sy vx vy 2
(1)
where Si and si (i ¼ x, y) are Pauli matrices denoting the real spin and the pseudospin, respectively.vF z 106m/s is the Fermi 0 velocity, and l represents RSOC strength. t ¼ ±1 describes two inequivalent Dirac cones (K and K ). Considering the RSOC, the electronic bands near K point are given by
Enm ¼
nm 2
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi l2 þ 4ðZvF kÞ2 ml ;
n; m ¼ ±1;
(2)
where the product nm specifies electron and hole subbands. The corresponding eigenstates are
3
2
nei2f 6 cnm eif 7 1 7; 6 jvm〉 ¼ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
4 5 incnm eif 2 2 1þc nm
1
(3)
X.-D. Tan et al. / Superlattices and Microstructures 98 (2016) 473e491
475
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi with f ¼ arctan(kx/ky), cnm ¼ Enm =ðZvF kÞ, and k ¼ ðk2x þ k2y Þ. Here kx and ky are the Cartesian components of electron wave vector measured from K point. The spin averages for state jnm〉 are given by
2mZvF k sinðfÞ ; 〈Sx 〉nm ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi l2 þ 4ðZvF kÞ2
2mZvF k cosðfÞ 〈Sy 〉nm ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; l2 þ 4ðZvF kÞ2
〈Sz 〉nm ¼ 0;
(4)
which are independent of n and lie in the graphene plane. Clearly, for a given momentum, the real spin of an electron occupying the state jn, m ¼ 1〉 is in the opposite polarization direction to the electron occupying the state jn, m ¼ 1〉. Moreover ! ! the in-plane spin is perpendicular to the electron momentum because of 〈 S $ k 〉nm ¼ 0, as shown in Fig. 1(a). Considering an external potential barrier,
VðxÞ ¼
U; 0 < x < D ; 0; otherwise
(5)
then the system Hamiltonian is described by
Ht ¼ Ht0 þ VðxÞ:
(6)
Here, we assume that the sample of SLG with length L and width W is big enough that boundary effect can be ignored. Thus, for the K-point Hamiltonian (6), it has the following formal solutions in regions I, II, and III [see Fig. 1(b)], respectively,
3 3 2 ið1Þnþ1 ei2fn iei2f1 7 iðk xþk yÞ 6 6 h eif1 7 iðk xþk yÞ 1 r1 hn eifn 1x y 1 7e 1x y þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 6 7 6 jKI ðx; yÞ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 4 ih eif1 5e nþ1 ifn 5 2 2 ið1Þ h e 1 2LW 1 þ hn 2LW 1 þ h1 n 1 1 3 2 i2f2 ie 6 h eif2 7 iðk xþk yÞ r2 2x y 7 6 2 þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; 4 ih eif2 5e 2 2LW 1 þ h22 1 2
2 3 3 iei2q1 iei2q1 i q i q 6 x e 1 7 iðq xþk yÞ 6 x e 1 7 iðq xþk yÞ a1 b1 1s 1x y 6 1s 6 7 1x y þ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 7 jKII ðx; yÞ ¼ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ffi 4 ix eiq1 5e
ffi 4 ix eiq1 5e 2 2 1s 1s 2LW 1 þ x1s 2LW 1 þ x1s 1 1 2 2 3 3 iei2q2 iei2q2 i q i q 2 2 6 6 7 7 a2 b2 6 x2s e 6 x2s e 7 iðq2x xþky yÞ ; 7 iðq2x xþky yÞ þ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ffi 4 ix eiq2 5e
ffi 4 ix eiq2 5e 2s 2s 2LW 1 þ x22s 2LW 1 þ x22s 1 1
(7)
2
(8)
Fig. 1. (a) The low-energy band structure of SLG þ R, and the spin configuration of electrons with arrows marking the spin directions. Blue and red lines denote n ¼ 1 and n ¼ 1 subands, respectively. The spin of an electron occupying the different n-subband has the opposite orientation, and is perpendicular to the electron momentum. (b) The tunable potential barrier with height U and width D. The Fermi level EF is denoted by the purple dashed line. The azury filled areas indicate occupied states. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
476
X.-D. Tan et al. / Superlattices and Microstructures 98 (2016) 473e491
2
2 3 3 iei2f1 iei2f2 if if 6 h e 1 7 iðk xþk yÞ 6 h e 2 7 iðk xþk yÞ t1 t2 6 1 6 2 7 1x y þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 7 2x y ; jKIII ðx; yÞ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 ih eif1 5e 4 ih eif2 5e 2 2 1 2 2LW 1 þ h1 2LW 1 þ h2 1 1
(9)
where
ky ; qn
qn ¼ arcsin hn ¼
En ; ZvF kn
knx ¼
1 ZvF
qn ¼
1 ZvF
En ¼
1 2
Uns ¼
Uns
xns ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k2n k2y ;
kn ¼
fn ¼ arcsin
ZvF qn
qnx ¼
ky ; kn
;
(10)
(11)
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi q2n k2y ;
(12)
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
E E ð1Þn l ; ðwave vector outside the barrierÞ
(13)
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ffi ðE UÞ E U ð1Þn l ; ðwave vector inside the barrierÞ
(14)
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi l2 þ 4ðZvF kn Þ2 þ ð1Þn l ;
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 s l2 þ 4ðZvF qn Þ2 þ ð1Þn l ; 2
(15)
(16)
with s ¼ sign(EU). n ¼ 1, 2 describe two different spin-split subbands as indicated in Fig. 1(a). Similarly, for the K' valley the wave functions in regions I, II, and III [see Fig. 1(b)], have the following forms:
3 3 2 ið1Þnþ1 hn eifn ih1 eif1 0 7 7 iðk xþk yÞ 6 6 r 1 1 1 1 1x y 7 iðk1x xþky yÞ þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 7 6 6 jKI ðx; yÞ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i2f1 5e 4 ið1Þn ei2fn 5e 4 ie 2 2LW 1 þ h2n 2LW 1 þ h 1 h1 eif1 hn eifn 2 3 ih2 eif2 0 6 1 7 iðk xþk yÞ r2 2x y 6 7 þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; 4 iei2f2 5e 2LW 1 þ h22 h2 eif2 2
0
3 3 2 ix1s eiq1 ix1s eiq1 0 7 iðq xþk yÞ 7 iðq xþk yÞ 6 6 a1 b1 1 1 1x y 7 1x y þ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 7 6 6 jKII ðx; yÞ ¼ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ffi 4 iei2q1 5e
ffi 4 iei2q1 5e 2 2 2LW 1 þ x1s 2LW 1 þ x1s iq1 iq1 x1s e x1s e 3 3 2 2 iq2 ix2s e ix2s eiq2 0 0 7 iðq xþk yÞ 6 6 1 7 iðq xþk yÞ a2 b2 1 2x y 7 2x y þ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 7 6 6 ; þ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ffi 4 iei2q2 5e
ffi 4 iei2q2 5e 2 2 2LW 1 þ x2s 2LW 1 þ x2s iq2 iq2 x2s e x2s e
(17)
2
0
0
3 3 2 2 ih1 eif1 ih2 eif2 0 0 7 7 iðk xþk yÞ 6 6 t t 1 1 1 2 7 iðk1x xþky yÞ þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 7 2x y ; 6 6 jKIII ðx; yÞ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i2f1 5e i2f2 5e 4 4 ie ie 2 2LW 1 þ h21 2LW 1 þ h 2 h1 eif1 h2 eif2
(18)
0
where the specific parameters are equal to the above definitions as listed in Eqs. 10e16.
(19)
X.-D. Tan et al. / Superlattices and Microstructures 98 (2016) 473e491
477
3. Klein tunneling in SLG þ R Now, let us discuss in detail the role of RSOC in the tunneling problem. We define Tnm (n, m ¼ 1, 2) as the tunneling probability of an electron initial in subband n and outgoing in subband m. Then, the total transmission ability of each incident electron is given by
tot Tn ¼ Tn1 þ Tn2 ;
n ¼ 1; 2
(20)
Note that the Hamiltonian (6) obeys the time-reversal symmetry. Thus, the transmission coefficients are identical for K and 0 0 K valleys. Indeed, straightforward calculations also show that the unsettled parameters in wave functions for K and K valleys, satisfy the same equation set (A1), as shown in the Appendix A. So, in the following we only need to discuss the K valley. 3.1. The case of 0 < EF < l First, we consider the case of 0 < EF < l. According to the boundary conditions that require the continuity of wave function, the transmission coefficients T1tot and T2tot in this case are calculated as
8 > T~ > < 11 tot T1 ðE; l; U; D; f1 Þ ¼ ~0 > T > : 11 0 T2tot ðE; l; U; D; f2 Þ ¼ 0;
ðf1 ¼ 0; 0 < U < EF Þ ðf1 ¼ 0; EF < U < EF þ lÞ ; ðf1 ¼ 0; U > EF þ lÞ ð p=2 f1 < 0; 0 < f1 p=2; U > 0Þ
ð p=2 f2 p=2; U > 0Þ;
(21)
(22)
where T~ 11 is given by
1 1 ; T~ 11 ¼ 1 þ ðr1 þ 1=r1 2Þsin2 ðQ1 Þ 4
(23)
with
r1 ¼
EðE U þ lÞ ; ðE UÞðE þ lÞ
Q1 ¼
(24)
D qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðE UÞðE U þ lÞ:
(25)
ZvF
It deserves to be mentioned that in this case only normal incident electrons have capacity to pass through the barrier. Moreover, the electrons always occupy the same subband before and after penetrating through the barrier, thus suggesting a single band to single band (S / S) transmission regime. This special tunneling behavior is very different from the case without considering RSOC [4]. In Ref. [4], it was shown that the transmission probability of an electron depends on incident angles at a given potential barrier. Moreover, the tunneling at normal incidence is always 100%. However, in the presence of RSOC, only if the resonance conditions (Q1 ¼ Np, N ¼ 0, 1,…) are satisfied, can normally incident electrons penetrate through the barrier perfectly according to Eq. (23). Therefore, RSOC plays a significant role in the S / S transmission regime. 3.2. The case of EF > l Then, let us discuss the case of EF > l. In this case, the tunneling depends not only on the S / S transmission regime but also on the single band to multiple bands (S / M) transmission regime. Thus, the transmission should be discussed in five different regions according to the barrier height. In the S / M transmission regime, the subband-dependent transmission coefficients are limited by the critical incident angle fcn . If jfn j fcn , we have Tnm ¼ 0 (n, m ¼ 1, 2), while jfn j < fcn the transmission coefficients are calculated as
det H CA1 M 2 11 ; T11 ¼ det D CA1 B
T12 ¼
h2 cosðf2 Þ det H12 CA1 F h1 cosðf1 Þ det D CA1 B
(26) 2 ;
(27)
478
X.-D. Tan et al. / Superlattices and Microstructures 98 (2016) 473e491
h cosðf1 Þ det H21 CA1 M T21 ¼ 1 h2 cosðf2 Þ det D CA1 B T22
2 ;
(28)
det H CA1 F 2 22 ; ¼ det D CA1 B
(29)
where the specific parameters are shown in the Appendix A. i) For 0 < U < EFl (S / M transmission regime),
T1tot ðE; l; U; D; f1 Þ ¼
T2tot ðE; l; U; D; f2 Þ ¼
8 > > < > > :
0; T11 þ T12 ;
8 > > < > > :
0;
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! ðE UÞðE U lÞ ¼ arcsin EðE þ lÞ ;
(30)
otherwise sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! ðE UÞðE U lÞ c p=2 jf2 j f2 ¼ arcsin EðE lÞ :
(31)
p=2 jf1 j
fc1
T21 þ T22 ;
otherwise
ii) For EFl < U < EF (S / S transmission regime),
T1tot ðE; l; U; D; f1 Þ ¼
f1 ¼ 0 ; T~ 11 ; 0; otherwise
T2tot ðE; l; U; D; f2 Þ ¼ 0;
ð p=2 f2 p=2Þ;
(32) (33)
where T~ 11 is given by Eq. (23). iii) For EF < U < EFþl (S/S transmission regime),
T1tot ðE; l; U; D; f1 Þ ¼ 0; ð p=2 f1 p=2Þ; ~ f2 ¼ 0 ; T2tot ðE; l; U; D; f2 Þ ¼ T 22 ; 0; otherwise
(34) (35)
where
1 1 ; T~ 22 ¼ 1 þ ðr2 þ 1=r2 2Þsin2 ðQ2 Þ 4
(36)
with
r2 ¼ Q2 ¼
EðE U lÞ ; ðE UÞðE lÞ
D qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðE UÞðE U lÞ:
ZvF
iv) For EFþl < U < 2EF (S/M transmission regime),
(37) (38)
X.-D. Tan et al. / Superlattices and Microstructures 98 (2016) 473e491
T1tot ðE; l; U; D; f1 Þ ¼
T2tot ðE; l; U; D; f2 Þ ¼
8 > > < > > : 8 > > < > > :
0; T11 þ T12 ; 0;
479
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! ðE UÞðE U þ lÞ ¼ arcsin EðE þ lÞ ;
(39)
otherwise sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! ðE UÞðE U þ lÞ c p=2 jf2 j f2 ¼ arcsin EðE lÞ ;
(40)
p=2 jf1 j
fc1
T21 þ T22 ;
otherwise
v) For U > 2EF (S / M transmission regime),
T1tot ðE; l; U; D; f1 Þ ¼
8 > > < > > :
0;
p=2 jf1 j
T11 þ T12 ;
T2tot ðE; l; U; D; f2 Þ ¼ T21 þ T22 ;
fc1
sffiffiffiffiffiffiffiffiffiffiffi ! El ¼ arcsin Eþl ;
(41)
otherwise ðjf2 j p=2Þ:
(42)
According to Eqs. (30), (31), (39)e(42), we have Tntot ¼ 0 when jfn j fcn , (n ¼ 1, 2). So, in the following we only need to discuss the nontrivial region of jfn j < fcn , (n ¼ 1, 2). Figs. 2e4 show examples of angle- and subband-dependent transmission probabilities in the S / M transmission regime. Here, the Fermi energy of incident electrons is chosen as EF ¼ 80 meV that is most typical in experiments with graphene [4]. When U ¼ 15 meV, as can be seen in Fig. 2(a), the transmission probability T12 is very weak, and T11 plays a preponderant role in the tunneling. Moreover, the total transmission probability almost reaches to T1tot ¼ 1 for all incident angles within jf1 j < fc1 z17:83+ . By contrast, T22 strongly depends on the incident angle of electron, as shown in Fig. 2(b). When the incident angle is toward to the critical value fc2 z39:6+ , T22 decreases fast, and only in the range of about jf2j <30 , T22 is comparable with T11. We can also see that T12 and T21 display the similar behavior, as shown in the insets of Fig. 2(a) and (b). When we increase the barrier height to U ¼ 145 meV, the S / M transmission becomes remarkable. We can see in Fig. 3(a) that the small-angle-incident electrons prefer tunneling with a probability T11. When the incident angle is located in the interval of about 9.3 < jf1j < 12.8+, we can see T12 > T11, suggesting that there is no simple dominant relation between T11 and T12. When an incoming electron occupying the subband n ¼ 2, it is prone to penetrate through the barrier with a probability T22, but T21 holds an advantage over T22 near the fc2 z39:6+ [see Fig. 3(b)]. In addition, unlike the discontinuous behavior of T11 and T12 (T21 and T22) at the critical angle fc1 z17:83+ (fc2 z39:6+ ) [see Figs. 2 and 3],
Fig. 2. Angle- and subband-dependent transmissions through a barrier of wide D ¼ 110 nm and height U ¼ 15 meV in SLG þ R with EF ¼ 80 meV and l ¼ 50 meV. (a) Transmissions within jf1 j < fc1 z17:83+ for the incident electron in subband n ¼ 1. (b) Transmissions within jf2 j < fc2 z39:6+ for the incident electron in subband n ¼ 2. The insets show T12 (top) and T21(bottom).
480
X.-D. Tan et al. / Superlattices and Microstructures 98 (2016) 473e491
Fig. 3. Angle- and subband-dependent transmissions through a barrier of wide D ¼ 110 nm and height U ¼ 145 meV in SLG þ R with EF ¼ 80 meV and l ¼ 50 meV. (a) Transmissions within jf1 j < fc1 z17:83+ for the incident electron in subband n ¼ 1. (b) Transmissions within jf2 j < fc2 z39:6+ for the incident electron in subband n ¼ 2.
Fig. 4. Angle- and subband-dependent transmissions through a barrier of wide D ¼ 110 nm and height U ¼ 200 meV in SLG þ R with EF ¼ 80 meV and l ¼ 50 meV. (a) Transmissions within jf1 j < fc1 z28:71+ for the incident electron in subband n ¼ 1. (b) Transmissions within jf2j 90+ for the incident electron in subband n ¼ 2.
the corresponding transmission coefficients for U ¼ 200 meV are continuous at the critical angle fc1 z28:71+ (fc2 ¼ 90+ ), because the transmission probabilities just vanish at this critical incident angle (see Fig. 4). In order to further reveal the influences of angle (fn), subbands (n ¼ 1, 2), and potential barrier (U) on transmission probabilities, we depict Tntot , Tn1, and Tn2, (n ¼ 1, 2) as a function of U and fn in Fig. 5, which intuitively exhibits the global distributions of the transmission probabilities at a given l ¼ 50 meV. For lower barrier (0 < U < 30 meV) electrons in subband n ¼ 1 almost perfectly pass through the barrier for all incident angles within jf1 j < fc1 [see Fig. 5(a)]. When the barrier is located in the range of 30 meV < U < 80 meV, only normal incident electrons are allowed tunneling with a probability T11 [see Fig. 5(b)]. When 80 meV < U < 130 meV, the transmissions are forbidden for electrons in subband n ¼ 1, but only the normally incident electrons occupying the subband n ¼ 2 are allowed tunneling with a probability T22 [see Fig. 5(e)]. For a higher barrier (130 meV < U < 200 meV) tunneling is very sensitive to the small-angle incidence, where T11 plays a significant role [see Fig. 5(b)], compared with T12 [see Fig. 5(c)]. We also see in Fig. 5(c) that T12 has no contribution to T1tot in the case of 0 < U < 130 meV, where the similar behaves are also found in T21 [see Fig. 5(f)]. When it comes to T22 [see Fig. 5(e)], similar to T11, it is the main contributor to total transmission probability T2tot , compared with T21. So, on the whole, the electrons prefer tunneling between the same spin-split subbands. In order to get further insight into the effect of RSOC on electron
X.-D. Tan et al. / Superlattices and Microstructures 98 (2016) 473e491 Fig. 5. Transmission probabilities (a) T1tot , (b) T11, (c) T12, (d) T2tot , (e) T22, and (f) T21 through a 110-nm-wide barrier as a function of incident angle f and barrier height U in SLG þ R with EF ¼ 80 meV and l ¼ 50 meV.
481
482
X.-D. Tan et al. / Superlattices and Microstructures 98 (2016) 473e491
Fig. 6. Angle- and subband-dependent total transmission through a barrier of wide D ¼ 110 nm and height U ¼ 200 meV, for several different l, in SLG þ R with EF ¼ 80 meV. (a) shows T1tot versus f1, and (b) shows T2tot versus f2.
transmissions, we depict the total transmission Tntot , for several different l, as a function of the incident angle fn (n ¼ 1, 2). As can be seen in Fig. 6, when l is very small it shows T1tot zT2tot , which is in agreement with the previous results [4,34]. As we increase l, T1tot and T2tot exhibit a big difference. This can be explained by the fact that when l / 0 the spin-up and the spindown states of electrons are approximately degenerate, but when l is increased, the degeneracy is gradually removed. Thus, Klein tunneling becomes very complicated and subtle when the RSOC is noticeable. 4. Spin-resolved conductance 4.1. In the case of S / M transmission regime For each transmitted electron in the state,
3 ið1Þnþ1 ei2fn 7 iðk xþk yÞ 6 1 hn eifn 7e nx y 6 j ¼ jn; kn 〉 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 nþ1 ifn 5 2 ið1Þ h e 2LW 1 þ hn n 1 2
ðn ¼ 1; 2Þ;
(43)
the contribution to the current density along the x-direction is given by y j± nx ¼ evF j sx j ¼
ehv2F cosðfn Þkn Tn±
; pLW 2E ð1Þn l
(44)
where n denotes the original subband in which an incident electron occupies, and
Tn± ¼
Tn1 þ Tn2 h1 cosðf1 ÞTn1 h2 cosðf2 ÞTn2 H ± ; 2 1 þ h21 1 þ h22
(45)
is the probability of an electron in subband n throughout the barrier with the spin up (þ) or the spin down () state of the real spin operator Sy (for details see Appendix B). The total current density is determined by carriers with all possible values of kx ! ! ! and ky. The number of electrons in range of k / k þ d k is determined by the product of its state density and occupied possibility,
dnelec ¼ 2 f ðE; mε Þ
LW LW kn dkn dfn ¼ 2E ð1Þn l f ðE; mε ÞdEdfn ; 4p2 h2 v2F 0
where the factor 2 arises because K and K valleys have the same contribution to the current density. Here
(46)
X.-D. Tan et al. / Superlattices and Microstructures 98 (2016) 473e491
f ðE; mε Þ ¼
1 ; 1 þ eðEmε Þ=kB T
483
(47)
is the Fermi-Dirac distribution. Note that there exist two reservoirs. The left reservoir injects carriers up to a quasi-Fermienergy m1 and the right reservoir emits carriers up to a quasi-Fermi-energy m2. The two baths compete with each other and finally the electrons obtain equilibrium distribution with a certain chemical potential mL and mR. Then the total net current density following from the left reservoir to the right reservoir is given by c
± Jnx
Zfn
e ¼ h
Zm1 cosðfn Þdfn
p
m2
fcn
kn
½f ðE; mL Þ f ðE; mR ÞTn± dE;
(48)
where fcn is the critical incidence angle, and Tn± exists only if jfn j < fcn . The carrier densities can be characterized by the chemical potential mL and mR, and their respective levels are between m1 and m2. If the bias is chosen to be small enough, then we can obtain the following approximation,
vf vf f ðE; mL Þ f ðE; mR Þz ðmL mR Þ ¼ eV0 : vm vE
(49)
At very low temperature, we have
vf zdðE EF Þ: vE
(50)
As a result, the spin-dependent current density along the x-direction is given by c
± Jnx
e2 V ¼ 0 h p
Zfn
kn ðEF ; lÞcosðfn ÞTn± ðEF ; l; U; D; fn Þdfn :
(51)
fcn
According to the definition of conductance (G ¼ I/V), we obtain that ± Inx ¼
ZW
0
± ± Jnx dy ¼ Jnx W ¼ Gn± V0 :
(52)
0
Then, the effective conductance (conductance per unit length) is given by 0
G± n ¼
± Gn± Jnx ¼ : W V0
(53)
Thus, we explicitly obtain the spin-dependent conductance in the S / M transmission regime, c
e2 1 G± n ¼ h p
Zfn
kn ðEF ; lÞcosðfn ÞTn± ðE; l; U; D; fn Þdfn :
(54)
fcn
Since we are interested in the states of outgoing electrons, we define the conductance for outgoing spin up electrons as þ GS/M ¼ Gþ [ 1 þ G2 ;
(55)
and similarly, GS/M ¼ G Y 1 þ G2 :
Then the total conductance in the S / M transmission regime is given by
(56)
484
X.-D. Tan et al. / Superlattices and Microstructures 98 (2016) 473e491
Fig. 7. Total and spin-resolved conductances as a function of barrier height U in SLG þ R with EF ¼ 30 meV, l ¼ 50 meV, and D ¼ 110 nm. The first six ticks on the U axis correspond to the above six solid band diagrams in the barrier region.
GS/M ¼ GS/M þ GS/M : tot [ Y
(57)
4.2. In the case of S / S transmission regime In this case, the single electron current density along the x-direction is given by
~j± ¼ nx
± ehv2F kn T~ n
: pLW 2E ð1Þn l
(58)
with
h T~ nn T~ nn ± ±ð1Þn n 2 : T~ n ¼ 2 1 þ hn
(59)
The number of normally incident electrons in range of kx / kxþdkx is given by
~ elec ¼ 2 dn
2pL 2E ð1Þn l L f ðE; mε Þdkn ¼ f ðE; mε ÞdE; 2p h2 v2F kn
(60)
The total net current density is calculated as
±
J e nx ¼
e2 2V0 ~ ± T ðE ; l; U; DÞ: h W n F
Then, at very low temperature the current intensity is given by
(61)
X.-D. Tan et al. / Superlattices and Microstructures 98 (2016) 473e491
485
Fig. 8. Total and spin-resolved conductances as a function of barrier height U in SLG þ R with EF ¼ 80 meV, l ¼ 50 meV, and D ¼ 110 nm. The ten ticks on the U axis correspond to the above ten solid band diagrams in the barrier region. The inset shows the total conductance Gtot, for several different l, as a function of U in SLG þ R with EF ¼ 80 meV.
~I ± ¼ nx
ZW
~ ±V : ~J ± dy ¼ ~J ± W ¼ G nx nx n 0
(62)
0
Thus, the spin-dependent conductance in the S / S transmission regime is given by 2
~ ± ¼ e 2T~ ± ðE ; l; U; DÞ: G n F n h
(63)
Conformance to definitions in Eqs. 55e57, it is clear to give the following expressions
GS/S [
GS/S [
8 > ~þ; > ð0 < EF < l; U > 0Þ l; E l < U < E Þ ; ¼ G F F F 1 > > :G ~ þ ; ðE > l; E < U < E þ lÞ F F F 2 8 > ~; > ð0 < EF < l; U > 0Þ l; E l < U < E Þ ; ¼ G F F F 1 > > ~ ; ðE > l; E < U < E þ lÞ :G F F F 2
S/S GS/S þ GS/S : tot ¼ G[ Y
(64)
(65)
(66)
486
X.-D. Tan et al. / Superlattices and Microstructures 98 (2016) 473e491
Fig. 9. Spin polarization as a function of barrier height U in SLG þ R with EF ¼ 80 meV and D ¼ 110 nm, for several different l.
Fig. 10. Cross-sectional configuration of a graphene-spin device.
4.3. Numerical results and discussions We are now in a position to discuss the spin-resolved conductance. First, we consider the nontrivial case of 0 < EF < l. As can be seen in Fig. 7, the total conductance Gtot is almost dominated by GY. At lower barrier, Gtot approximately keeps a maximal constant 2e2/h. When the barrier lies in the interval of EF < U < EFþl, Gtot is equal to zero. While for the remaining interval of U, the spin-resolved conductance displays oscillating behaviors. These remarkable features show that fabrication of a graphene-based FET with a noticeable on-off ratio is feasible in the presence of a sizeable RSOC (l > EF). Next, we discuss the conductance for the case of EF > l. We can see in Fig. 8 that for a given l ¼ 50 meV the spin-resolved conductance G[ (GY) is monotone decreasing in the range of 0 < U < EFl. When we increase U to the interval of EFl < U < EF, spin-up electrons are completely reflected and only spin-down electrons have contribution to conductance. While for the range of EF < U < EFþl, it is just the opposite. Further increasing U, the conductance G[ (GY) displays oscillating behaviors. For l ¼ 50 meV, the total conductance Gtot does not exist a zero-gap along the U-axis. We also compared Gtot for several different l, as shown in the inset of Fig. 8. We can see that when U falls into the region EFl < U < EFþl, Gtot is low and fluctuates with U. Out of this region, Gtot tends to decrease with increasing l. More detailed calculation shows that in the range of EFl < U < EFþl the maximal Gtot is no more than 2e2/h, while the minimum is not zero except some special values of U. Thus, it is difficult to achieve considerable on-off ratios for graphene-based FETs in the case of EF > l.
X.-D. Tan et al. / Superlattices and Microstructures 98 (2016) 473e491
487
5. Spin polarization in the transmitted region According to the above analysis, the transmitted electrons can be spontaneously spin polarized along the y-direction. So, it is necessary to quantify the spin polarization of electrons in the transmitted region. Following Ref. [35], the spin polarization can be defined as
P¼
G[ GY : G[ þ GY
(67)
In the case of S / S transmission regime, it is easy to obtain
P¼
GS/S GS/S [ Y GS/S þ GS/S [ Y
8 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 EF ðEF þ lÞ > > > ; ð0 < EF < l; U > 0Þ > > 2EF þ l > > > > p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi < 2 EF ðEF þ lÞ ¼ ; ðEF > l; EF l < U < EF Þ ; > 2EF þ l > > > pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > > > 2 EF ðEF lÞ > > : ; ðEF > l; EF < U < EF þ lÞ 2EF l
(68)
which is a constant independent of U when EF and l are given. For EF ¼ 30 meV and. l ¼ 50 meV, we have P z 0.8907. When EF > l, the high spin polarization also occurs in the range of EFl < U < EFþl. Moreover, the direction of spin polarization is opposite near the critical point U ¼ EF. For example, when we fix EF ¼ 80 meV and l ¼ 50 meV, we have P z 0.9712 for 30 meV < U < 80 meV, and P z 0.8907 for 80 meV < U < 130 meV (see Fig. 9). This remarkable result indicates that one can realize spin switches in the region EFl < U < EFþl. When U is located in the range of 0 < U < EFl, the spin polarization increases with increasing U on the whole. In the range of U > EFþl, spin polarization is weak and displays oscillating behaviors with increasing U. By comparing the spin polarization for several different l, it clearly shows that RSOC plays a significant role in spin polarized transport in SLG þ R. 6. Potential applications Graphene PN and PNP junctions provide an effective platform for investigating Klein tunneling [4] and other relevant quantum electrodynamics phenomena. They are also the most important basic building blocks in spintronics devices. Based on the above discussions, we propose a conceptive graphene-spin device. Most works adopted the top-gated graphene-based device. However, impurities and organic residue caused by fabrication processes degrade the quality of graphene layer [36,37]. In Ref. [38], it presents a new method to fabricate a graphene device with neither deposition of dielectric material on the graphene nor electron-beam irradiation. Unlike top-gated devices, in this scheme the pre-defined local gate (poly-silicon) is embedded into a SiO2 substrate, as shown in Fig. 10. This local gate is used to create a tunable potential barrier, and the back gate controls the Fermi lever of SLG but does not affect the carrier density in the local-gate region [38]. On the other hand, the considerable SOC is a crucial element in the development of graphene-based spin FETs and realization of topological insulators. To enhance the SOC strength, one can deposit the graphene sheet on some special substrates such as tungsten disulfide (WS2) and Au/Ni(111). As reported in Refs. [18,19], strong SOC can be induced in these systems. By tuning local and back gate voltages, both control of barrier height and Fermi lever in SLG can be achieved, thereby manipulating spin transport properties of carriers. According to the detailed theoretical analyses in the present study, we think that this device may promisingly qualify as a spin FET. At least, it might realize the functions such as spin filtration, spin switch, and electron beam collimation. 7. Conclusions In summary, we investigated the spin transport properties in SLG þ R. The transmission coefficients through a potential barrier were obtained analytically. Results show that Klein tunneling in SLG þ R and BLG exhibits completely different behaviors. The tunneling at normal incidence in BLG shows a perfect reflection, while in SLG þ R it displays more dramatic behaviors. In the S / S transmission regime that mainly occurs in the case of 0 < EF < l, only normally incident electrons have capacity to pass through the barrier in SLG þ R. Moreover, the tunneling between different spin-subbands is forbidden in this regime, and consequently outgoing electrons are highly spin polarized along y-direction. As reported in Ref. [4], in the absence of RSOC, the barrier always remains perfectly transparent for electrons close to the normal incidence. However, in the presence of RSOC, this perfect tunneling is not always allowed unless the resonance conditions are satisfied. In the S / M transmission regime that only exists in the case of EF > l, the electron transmission depends not only on subbands but also strongly on incident angles. In this regime, the tunneling could happen at not only normal but also oblique incidence. Based on the transmission coefficients, we also derived spin-resolved conductance analytically. The most significant finding is that when 0 < EF < l the conductance appears a gap in the interval of EF < U < EFþl. This remarkable features indicates that
488
X.-D. Tan et al. / Superlattices and Microstructures 98 (2016) 473e491
fabrication of a graphene-based FET with a high on-off ratio is feasible if 0 < EF < l. In the case of EF > l, the conductance decays with increasing U if 0 < U < EFl. When EFl < U < EFþl, the electrons are highly spin polarized, and spin switch effect emerges near the critical point U ¼ EF. We also found that the conductance displays oscillation behaviors in this interval as well as the interval of U > EFþl. For EF > l, except some special values of U, it is difficult to pinch off the conducting channel, which limits achievable on-off ratios for graphene-based FETs in this case, but it can serve as a spin switch. Acknowledgments We acknowledge X.G. Wan for useful discussions. This work was supported by the National Natural Science Foundation of China (Nos. 51420105003, 11327901, 61274114, and 11525415). Appendix A. angle- and subband-dependent tunneling probabilities Based on the boundary conditions, jI ðx ¼ 0Þ ¼ jII ðx ¼ 0Þ and jII ðx ¼ DÞ ¼ jIII ðx ¼ DÞ, we obtain the following equation set
A C
B D
x1 x2
¼
0 ; ε2
0 ε1
(A1)
where the specific parameters are listed below
2
eiðq1x D2q1 Þ 6 x1s eiðq1x Dq1 Þ A¼6 4 x eiðq1x Dq1 Þ 1s
eiq1x D
eiðq1x D2q1 Þ x1s eiðq1x Dq1 Þ x1s eiðq1x Dq1 Þ eiq1x D
2
ei2q1 6 x eiq1 1s C¼6 4 x eiq1 1s 1 2
ei2f1 6 h eif1 1 D¼6 4 h eif1 1
2
1 3
a1
6 a2 7 7 x1 ¼ 6 4 b1 5;
b2
ei2q1 x1s eiq1 x1s eiq1 1
ei2q2 x2s eiq2 x2s eiq2 1
ei2f2 0 h2 eif2 0 h2 eif2 0 1 0 2 3 r1 6 r2 7 6 x2 ¼ 4 7 ; t1 5 t2
3 eiðq2x D2q2 Þ iðq2x Dq2 Þ 7 x2s e 7; x2s eiðq2x Dq2 Þ 5 eiq2x D
3 eiðk2x D2f2 Þ iðk2x Df2 Þ 7 h2 e 7; h2 eiðk2x Df2 Þ 5 eik2x D
0 0 eiðk1x D2f1 Þ 6 0 0 h1 eiðk1x Df1 Þ B¼6 4 0 0 h eiðk1x Df1 Þ 1 0 0 eik1x D 2
eiðq2x D2q2 Þ x2s eiðq2x DDq2 Þ x2s eiðq2x Dq2 Þ eiq2x D
(A3)
3 ei2q2 iq2 7 x2s e 7 ; x2s eiq2 5 1
(A4)
3 0 07 7; 05 0
3 ei2f1 if 6h e 1 7 1 7 6 1 ε1 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 h eif1 5; 2 1 2 1 þ h1 1 2
(A2)
(A5)
(A6) 3 ei2f2 if 6 h e 2 7 1 7 6 2 ε2 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 h eif2 5: 2 2 2 1 þ h2 1 2
(A7)
Using the Cramer's rule and block matrix skill, straightforward calculation then gives the expressions of the transmission coefficients
det H CA1 M2 11 ; T11 ¼ det D CA1 B
T12 ¼
h2 cosðf2 Þ det H12 CA1 F h1 cosðf1 Þ det D CA1 B
(A8) 2 ;
(A9)
X.-D. Tan et al. / Superlattices and Microstructures 98 (2016) 473e491
h cosðf1 Þ det H21 CA1 M T21 ¼ 1 h2 cosðf2 Þ det D CA1 B T22
2 ;
489
(A10)
det H CA1 F 2 22 ; ¼ det D CA1 B
(A11)
with
2
H11
ei2f1 6 h eif1 1 ¼6 4 h eif1 1 1 2
H21
ei2f1 6 h eif1 1 ¼6 4 h eif1 1 1 2
0 60 M¼6 40 0
0 0 0 0
ei2f2 h2 eif2 h2 eif2 1
ei2f1 h1 eif1 h1 eif1 1
ei2f2 h2 eif2 h2 eif2 1
ei2f2 h2 eif2 h2 eif2 1
3 0 eiðk2x D2f2 Þ iðk2x Df2 Þ 7 0 h2 e 7; 0 h2 eiðk2x Df2 Þ 5 0 eik2x D
2
3 0 07 7; 05 0
H12
3 0 07 7; 05 0
ei2f1 6 h eif1 1 ¼6 4 h eif1 1 1 2
H22
2
0 60 F¼6 40 0
ei2f1 6 h eif1 1 ¼6 4 h eif1 1 1
ei2f2 h2 eif2 h2 eif2 1 ei2f2 h2 eif2 h2 eif2 1
0 eiðk1x D2f1 Þ 0 h1 eiðk1x Df1 Þ 0 h1 eiðk1x Df1 Þ 0 eik1x D
3 0 07 7: 05 0
3 0 ei2f1 if1 7 0 h1 e 7; 0 h1 eif1 5 0 1 3 0 ei2f2 if2 7 0 h2 e 7; 0 h2 eif2 5 0 1
(A12)
(A13)
(A14)
Here, the inverse of the matrix A is calculated as
2
A1
a 6 a* ¼6 4 p* p
b b* m* m
d d* g* g
3 a * 7 a 7 ; p* 5 p
(A15)
where the parameters are given by
b ¼ au* ;
d ¼ au;
(A16)
with a ¼ secðq1 Þeiðq1x Dq1 Þ =4 and u ¼ cosðq1 Þ=x1s þ i sinðq2 Þ=x2s , and
g ¼ pc* ;
m ¼ p c;
(A17)
with p ¼ secðq2 Þeiðq2x Dq2 Þ =4 and c ¼ i sinðq1 Þ=x1s þ cosðq2 Þ=x2s . Appendix B. spin-resolved transmission probabilities In the basis of fuA[ 〉; uAY 〉; uB[ 〉; uBY 〉g, the real spin matrix Sy reads
2
0 i 6i 0 15Sy ¼ 6 40 0 0 0
3 0 0 0 0 7 7: 0 i 5 i 0
(B1)
Its eigenstates for the eigenvalue sy ¼ 1 are
3 i 6 1 1 7 7; j[〉1 ¼ pffiffiffi 6 24 0 5 0 2
3 0 6 1 0 7 7; j[〉2 ¼ pffiffiffi 6 4 i 5 2 1
and for the eigenvalue sy ¼ 1 are
2
(B2)
490
X.-D. Tan et al. / Superlattices and Microstructures 98 (2016) 473e491
2 3 i 7 1 6 1 7; jY〉1 ¼ pffiffiffi 6 5 4 0 2 0
2 3 0 1 6 07 7: jY〉2 ¼ pffiffiffi 6 4 2 i5 1
(B3)
When a transmitted electron in the state
3 ið1Þnþ1 ei2fn 7 iðk xþk yÞ 6 1 hn eifn 7e nx y ; 6 j ¼ jn; kn 〉 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 5 nþ1 if n hn e 2LW 1 þ h2n ið1Þ 1 2
ðn ¼ 1; 2Þ;
(B4)
one can detect the spin-up electron occupying subband n with a probability,
Pnþ ¼ j1 〈[jn; kn 〉j2 þ j2 〈[jn; kn 〉j2 ¼
1 h cosðfn Þ þ ð1Þn n ; 2 1 þ h2n
(B5)
and the spin-down electron with a probability,
Pn ¼ j1 〈Yjn; kn 〉j2 þ j2 〈Yjn; kn 〉j2 ¼
1 h cosðfn Þ ð1Þn n : 2 1 þ h2n
(B6)
Thus, in the S / M transmission regime, the probability that one observes an electron originally occupying subband n and going out the barrier with spin up (þ) or spin down () state is given by
Tn± ¼ Tn1 P1± þ Tn2 P2± ¼
Tn1 þ Tn2 h1 cosðf1 ÞTn1 h2 cosðf2 ÞTn2 H ± : 2 1 þ h21 1 þ h22
(B7)
Similarly, one can obtain the spin-dependent probability in the S / S transmission regime, i. e,
h cosðfn ÞT~ nn T~ nn ± ±ð1Þn n : T~ n ¼ T~ nn Pn± ¼ 2 1 þ h2n
(B8)
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22]
H. Wei, R.K. Kawakami, G. Martin, F. Jaroslav, Graphene spintronics, Nat. Nanotechnol. 9 (2014) 794e807. O. Klein, Die Reflexion von Elektronen an einem Potentialsprung nach der relativistischen Dynamik von Dirac, Z. Phys. 53 (1929) 157e165. W. Greiner, B. Müller, J. Rafelski, Quantum Electrodynamics of Strong Fields, Springer, Berlin, 1985. M.I. Katsnelson, K.S. Novoselov, A.K. Geim, Chiral tunnelling and the Klein paradox in graphene, Nat. Phys. 2 (2006) 620e625. J.A. Sulpizio, N. Stander, K. Todd, B. Yang, D. Goldhaber-Gordon, B. Huard, Transport measurements across a tunable potential barrier in graphene, Phys. Rev. Lett. 98 (2007) 236803. R.V. Gorbachev, A.S. Mayorov, A.K. Savchenko, D.W. Horsell, F. Guinea, Conductance of p-n-p structures with “Air-Bridge” top gates, Nano Lett. 8 (2008) 1995. A.F. Young, P. Kim, Quantum interference and Klein tunnelling in graphene heterojunctions, Nat. Phys. 5 (2009) 222e226. S. Sutar, E.S. Comfort, J. Liu, T. Taniguchi, K. Watanabe, J.U. Lee, Angle-dependent carrier transmission in graphene p-n junctions, Nano Lett. 12 (2012) 4460. R.N. Sajjad, S. Sutar, J.U. Lee, A.W. Ghosh, Manifestation of chiral tunneling at a tilted graphene p-n junction, Phys. Rev. B 86 (2012) 155412. B. Van Duppen, F.M. Peeters, Klein paradox for a pn junction in multilayer graphene, Europhys. Lett. 102 (2013) 27001. T. Yokoyama, Spin and valley transports in junctions of Dirac fermions, New J. Phys. 16 (2014) 085005. Z. Rashidian, F.M. Mojarabian, P. Bayati, G. Rashedi, A. Ueda, T. Yokoyama, Conductance and Fano factor in normal/ferromagnetic/normal bilayer graphene junction, J. Phys. Condens. Matter 26 (2014) 255302. Q. Wilmart, S. Berada, D. Torrin, V.H. Nguyen, G. Fve, J.M. Berroir, P. Dollfus, B. Plaais, A Klein-tunneling transistor with ballistic graphene, 2d Mater. 1 (2014) 11006. A.H. Castro Neto, F. Guinea, Impurity-induced spin-orbit coupling in graphene, Phys. Rev. Lett. 103 (2009) 26804. S. Konschuh, M. Gmitra, J. Fabian, Tight-binding theory of the spin-orbit coupling in graphene, Phys. Rev. B 82 (2010) 245412. J.S. Jeong, J. Shin, H.W. Lee, Curvature-induced spin-orbit coupling and spin relaxation in a chemically clean single-layer graphene, Phys. Rev. B 84 (2011) 195457. D. Marchenko, A. Varykhalov, M.R. Scholz, G. Bihlmayer, E.I. Rashba, A. Rybkin, A.M. Shikin, O. Rader, Giant Rashba splitting in graphene due to hybridization with gold, Nat. Commun. 3 (2012) 1232. A. Avsar, J.Y. Tan, T. Taychatanapat, J. Balakrishnan, G.K. Koon, Y. Yeo, J. Lahiri, A. Carvalho, A.S. Rodin, E.C. O'Farrell, Spin-orbit proximity effect in graphene, Nat. Commun. 5 (2014) 4875. Z. Wang, D.K. Ki, H. Chen, H. Berger, A.H. Macdonald, A.F. Morpurgo, Strong interface-induced spineorbit interaction in graphene on WS2, Nat. Commun. 6 (2015) 8339. A. Manchon, H.C. Koo, J. Nitta, S.M. Frolov, R.A. Duine, New perspectives for Rashba spin-orbit coupling, Nat. Mater. 14 (2015) 871. M. Gmitra, S. Konschuh, C. Ertler, C. Ambrosch-Draxl, J. Fabian, Band-structure topologies of graphene: spin-orbit coupling effects from first principles, Phys. Rev. B 80 (2009) 235431. S. Datta, B. Das, Electronic analog of the electro-optic modulator, Appl. Phys. Lett. 56 (1990) 665e667.
X.-D. Tan et al. / Superlattices and Microstructures 98 (2016) 473e491 [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38]
491
D. Bercioux, A. De Martino, Spin-resolved scattering through spin-orbit nanostructures in graphene, Phys. Rev. B 81 (2010) 165410. M. Rataj, J. Barnas, Graphene p-n junctions with nonuniform Rashba spin-orbit coupling, Appl. Phys. Lett. 99 (2012) 162107. M.H. Liu, Spin-dependent Klein tunneling in graphene: role of Rashba spin-orbit coupling, Phys. Rev. B 85 (2012) 85406. J.F. Liu, K.S. Chan, Y-shaped spin filter in graphene with Rashba spin-orbit coupling, J. Phys. Soc. Jpn. 82 (2013) 74711. Q. Zhang, Z. Lin, K.S. Chan, Spin polarization switching in monolayer graphene through a Rashba multi-barrier structure, Appl. Phys. Lett. 102 (2013) 142407. K. Shakouri, M.R. Masir, A. Jellal, E.B. Choubabi, F.M. Peeters, Effect of spin-orbit couplings in graphene with and without potential modulation, Phys. Rev. B 88 (2013) 115408. H. Min, J.E. Hill, N.A. Sinitsyn, B.R. Sahu, L. Kleinman, A.H. Macdonald, Intrinsic, Rashba Spin-orbit, Interactions in graphene sheets, Phys. Rev. B 74 (2006) 165310. M. Gmitra, D. Kochan, J. Fabian, S. Konschuh, Theory of spin-orbit coupling in bilayer graphene, Phys. Rev. B 85 (2012) 115423. nyos, Intrinsic and substrate induced spin-orbit interaction in chirally stacked trilayer graphene, Phys. Rev. B 87 (2013) 45419. G. Burkard, A. Korma E. Mccann, V.I. Fal'Ko, Landau-level degeneracy and quantum Hall effect in a graphite bilayer, Phys. Rev. Lett. 96 (2005) 86805. C.L. Kane, E.J. Mele, Quantum spin hall effect in graphene, Phys. Rev. Lett. 95 (2005) 226801. A.H.C. Neto, F. Guinea, N.M.R. Peres, K.S. Novoselov, A.A.K. Geim, The electronic properties of graphene, Rev. Mod. Phys. 81 (2007) 109e162. G. Schmidt, Concepts for spin injection into semiconductorsda review, J. Phys. D. App. Phys. 38 (2005) R107eR122. B. Fallahazad, S. Kim, L. Colombo, E. Tutuc, Dielectric thickness dependence of carrier mobility in graphene with HfO2 top dielectric, Appl. Phys. Lett. 97 (2010) 123105. M. Ishigami, J.H. Chen, W.G. Cullen, M.S. Fuhrer, E.D. Williams, Atomic structure of graphene on SiO2, Nano Lett. 7 (2007) 1643e1648. N. Seung-Geol, K. Dong-Keun, P. Jong Wan, K. Youngwook, K. Jun Sung, L. Hu-Jong, Ballistic transport of graphene pnp junctions with embedded local gates, Nanotechnology 22 (2011) 415203.
Contents lists available at ScienceDirect
Superlattices and Microstructures journal homepage: www.elsevier.com/locate/superlattices
Gate tunable spin transport in graphene with Rashba spinorbit coupling Xiao-Dong Tan a, Xiao-Ping Liao a, Litao Sun a, b, c, * a
SEU-FEI Nano-Pico Center, Key Laboratory of MEMS of Ministry of Education, Collaborative Innovation Center for Micro/Nano Fabrication, Device and System, Southeast University, Nanjing 210096, PR China b Center for Advanced Carbon Materials, Southeast University, Jiangnan Graphene Research Institute, Changzhou 213100, PR China c Center for Advanced Materials and Manufacture, Joint Research Institute of Southeast University, Monash University, Suzhou 215123, PR China
a r t i c l e i n f o
a b s t r a c t
Article history: Received 12 July 2016 Accepted 5 September 2016 Available online 8 September 2016
Recently, it attracts much attention to study spin-resolved transport properties in graphene with Rashba spin-orbit coupling (RSOC). One remarkable finding is that Klein tunneling in single layer graphene (SLG) with RSOC (SLG þ R for short below) behaves as in bi-layer graphene (BLG). Based on the effective Dirac theory, we reconsider this tunneling problem and derive the analytical solution for the transmission coefficients. Our result shows that Klein tunneling in SLG þ R and BLG exhibits completely different behaviors. More importantly, we find two new transmission selection rules in SLG þ R, i.e., the single band to single band (S / S) and the single band to multiple bands (S / M) transmission regimes, which strongly depend on the relative height among Fermi level, RSOC, and potential barrier. Interestingly, in the S / S transmission regime, only normally incident electrons have capacity to pass through the barrier, while in the S / M transmission regime the angle-dependent tunneling becomes very prominent. Using the transmission coefficients, we also derive spin-resolved conductance analytically, and conductance oscillation with the increasing barrier height and zero conductance gap are found in SLG þ R. The present study offers new insights and opportunities for developing graphenebased spin devices. © 2016 Elsevier Ltd. All rights reserved.
Keywords: Graphene Rashba spin-orbit coupling Spin transport
1. Introduction Spin transport and control are of fundamental and practical importance in future graphene-based spintronic devices [1]. Klein tunneling is one of the most supernatural and counterintuitive transport phenomena in which an incoming electron can penetrate through a potential barrier even if its energy is lower than the barrier height [2]. Observation of this exotic phenomenon is quite difficult, because it requires extremely high energy in experiment [3]. Graphene, a single atomic layer of graphitic carbon, exhibits the liner spectrum near the Dirac points, where the carriers behave like massless Dirac fermions. This unique property makes it possible to test the relative quantum tunneling in graphene experimentally. Thus, it arouses
* Corresponding author. SEU-FEI Nano-Pico Center, Key Laboratory of MEMS of Ministry of Education, Collaborative Innovation Center for Micro/Nano Fabrication, Device and System, Southeast University, Nanjing 210096, PR China. E-mail address: [email protected] (L. Sun). http://dx.doi.org/10.1016/j.spmi.2016.09.007 0749-6036/© 2016 Elsevier Ltd. All rights reserved.
474
X.-D. Tan et al. / Superlattices and Microstructures 98 (2016) 473e491
intense interest to investigate Klein tunneling in graphene-based systems [4e13]. Most theoretical works focused mainly on the case of spin-independent tunneling, while spin-orbit coupling (SOC) was less considered. The SOC in graphene comprises intrinsic and extrinsic components, which play a key role in generating a topological insulating state and manipulating electron spins. However, the intrinsic SOC in graphene is extremely weak. Thus, various methods to induce the large extrinsic SOC have been proposed theoretically [14e16] and explored experimentally [17e19]. Rashba SOC (RSOC) originating from the space inversion symmetry breaking, has attracted tremendous research interest in various field of physics and materials science [20], owing to its controllability via an external gate voltage [21]. Marchenko et al. reported a large RSOC in the Au-intercalated graphene-Ni system [17]. Such enhancement is crucial for the development of graphene-based devices such as the Das-Datta spin field effect transistor (FET) [22]. Motivated by the experimental development, recently much attention has paid to investigate the effect of RSOC on spin transport properties of graphene [23e28]. Among these works, one remarkable finding was that Klein tunneling in single layer graphene (SLG) with RSOC (SLG þ R for short below) behaves as in bi-layer graphene (BLG) [25]. Considering the early work by Katsnelson et al. [4], such fantastic behavior seems somewhat surprising. As demonstrated in Ref. [4], chiral tunneling in SLG and BLG exhibits completely different behaviors. So, why does the appearance of RSOC repaint a new tunneling picture in SLG? Admittedly, the low-energy spectrum of SLG þ R and BLG has the same mathematical form [25], but this is not enough to declare that low-energy physics in SLG þ R and BLG behaves similarly. First, the interlayer coupling parameter g1 in BLG and the Rashba coupling l in SLG are two fundamentally different interactions. RSOC in SLG comes from the ps hybridization [29], and unlike conventional semiconduction 2D electron gases, it is independent of the momentum. The parameter g1 in BLG characterizes the nearest-neighbor hopping between the two graphene layers, and almost has no effect on SOC as demonstrated in Refs. [30,31]. Furthermore, to the best of our knowledge, carriers in BLG are massive and similar to conventional non-relativistic quasiparticles [32], which are quite different from massless Dirac fermions in SLG. In fact, it requires different equations to describe electrons in BLG and SLG þ R. As a result, for tunneling problems in BLG, there are four possible solutions for a given energy, and two of them correspond to evanescent waves [4] which do not exist in SLG þ R. Combining these facts, we think that it is necessary to reexamine whether Klein tunneling in SLG þ R behaves as in BLG. In this paper, we reconsider the tunneling problem in SLG þ R based on the effective Dirac theory. We show that the tunneling in SLG þ R and BLG exhibits different behaviors. In BLG, normally incident electrons are always completely reflected by the potential barrier [4], but this is not always true for SLG þ R case. In SLG þ R, normally incident electrons are allowed some tunneling, but are not always perfectly transmitted, consistent with the previous report in Ref. [23]. Interestingly, when the Fermi lever in SLG þ R is less than the strength of RSOC (0 < EF < l), the tunneling only occurs in normally incident electrons. We analyze this case and obtain the analytical solution for the transmission coefficient, revealing the properties of spin polarization independent of the height of potential barrier. We also discuss the case of EF > l in more detail, and intriguing behaviors of the tunneling and the spin-resolved conductance oscillations will be shown. This paper is organized as follows. In Sec. 2 we describe our model and solve it analytically. In Sec. 3 we investigate the subband-dependent transmission coefficients in SLG þ R. Then, we derive and discuss the spin-resolved conductance in Sec. 4. In Sec. 5 we discuss the dependence of spin polarization on the barrier height. Based on the spin transport properties discussed above, in Sec. 6 we propose a conceptive graphene-spin device. Our conclusions are summarized in Sec. 7.
2. Model 0
Near the Dirac points K and K , the effective Hamiltonian of SLG þ R is described by Ref. [33].
v v 1 þ l tsx Sy sy Sx ; Ht0 ¼ iZnF tsx þ sy vx vy 2
(1)
where Si and si (i ¼ x, y) are Pauli matrices denoting the real spin and the pseudospin, respectively.vF z 106m/s is the Fermi 0 velocity, and l represents RSOC strength. t ¼ ±1 describes two inequivalent Dirac cones (K and K ). Considering the RSOC, the electronic bands near K point are given by
Enm ¼
nm 2
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi l2 þ 4ðZvF kÞ2 ml ;
n; m ¼ ±1;
(2)
where the product nm specifies electron and hole subbands. The corresponding eigenstates are
3
2
nei2f 6 cnm eif 7 1 7; 6 jvm〉 ¼ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
4 5 incnm eif 2 2 1þc nm
1
(3)
X.-D. Tan et al. / Superlattices and Microstructures 98 (2016) 473e491
475
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi with f ¼ arctan(kx/ky), cnm ¼ Enm =ðZvF kÞ, and k ¼ ðk2x þ k2y Þ. Here kx and ky are the Cartesian components of electron wave vector measured from K point. The spin averages for state jnm〉 are given by
2mZvF k sinðfÞ ; 〈Sx 〉nm ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi l2 þ 4ðZvF kÞ2
2mZvF k cosðfÞ 〈Sy 〉nm ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; l2 þ 4ðZvF kÞ2
〈Sz 〉nm ¼ 0;
(4)
which are independent of n and lie in the graphene plane. Clearly, for a given momentum, the real spin of an electron occupying the state jn, m ¼ 1〉 is in the opposite polarization direction to the electron occupying the state jn, m ¼ 1〉. Moreover ! ! the in-plane spin is perpendicular to the electron momentum because of 〈 S $ k 〉nm ¼ 0, as shown in Fig. 1(a). Considering an external potential barrier,
VðxÞ ¼
U; 0 < x < D ; 0; otherwise
(5)
then the system Hamiltonian is described by
Ht ¼ Ht0 þ VðxÞ:
(6)
Here, we assume that the sample of SLG with length L and width W is big enough that boundary effect can be ignored. Thus, for the K-point Hamiltonian (6), it has the following formal solutions in regions I, II, and III [see Fig. 1(b)], respectively,
3 3 2 ið1Þnþ1 ei2fn iei2f1 7 iðk xþk yÞ 6 6 h eif1 7 iðk xþk yÞ 1 r1 hn eifn 1x y 1 7e 1x y þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 6 7 6 jKI ðx; yÞ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 4 ih eif1 5e nþ1 ifn 5 2 2 ið1Þ h e 1 2LW 1 þ hn 2LW 1 þ h1 n 1 1 3 2 i2f2 ie 6 h eif2 7 iðk xþk yÞ r2 2x y 7 6 2 þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; 4 ih eif2 5e 2 2LW 1 þ h22 1 2
2 3 3 iei2q1 iei2q1 i q i q 6 x e 1 7 iðq xþk yÞ 6 x e 1 7 iðq xþk yÞ a1 b1 1s 1x y 6 1s 6 7 1x y þ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 7 jKII ðx; yÞ ¼ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ffi 4 ix eiq1 5e
ffi 4 ix eiq1 5e 2 2 1s 1s 2LW 1 þ x1s 2LW 1 þ x1s 1 1 2 2 3 3 iei2q2 iei2q2 i q i q 2 2 6 6 7 7 a2 b2 6 x2s e 6 x2s e 7 iðq2x xþky yÞ ; 7 iðq2x xþky yÞ þ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ffi 4 ix eiq2 5e
ffi 4 ix eiq2 5e 2s 2s 2LW 1 þ x22s 2LW 1 þ x22s 1 1
(7)
2
(8)
Fig. 1. (a) The low-energy band structure of SLG þ R, and the spin configuration of electrons with arrows marking the spin directions. Blue and red lines denote n ¼ 1 and n ¼ 1 subands, respectively. The spin of an electron occupying the different n-subband has the opposite orientation, and is perpendicular to the electron momentum. (b) The tunable potential barrier with height U and width D. The Fermi level EF is denoted by the purple dashed line. The azury filled areas indicate occupied states. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
476
X.-D. Tan et al. / Superlattices and Microstructures 98 (2016) 473e491
2
2 3 3 iei2f1 iei2f2 if if 6 h e 1 7 iðk xþk yÞ 6 h e 2 7 iðk xþk yÞ t1 t2 6 1 6 2 7 1x y þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 7 2x y ; jKIII ðx; yÞ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 ih eif1 5e 4 ih eif2 5e 2 2 1 2 2LW 1 þ h1 2LW 1 þ h2 1 1
(9)
where
ky ; qn
qn ¼ arcsin hn ¼
En ; ZvF kn
knx ¼
1 ZvF
qn ¼
1 ZvF
En ¼
1 2
Uns ¼
Uns
xns ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k2n k2y ;
kn ¼
fn ¼ arcsin
ZvF qn
qnx ¼
ky ; kn
;
(10)
(11)
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi q2n k2y ;
(12)
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
E E ð1Þn l ; ðwave vector outside the barrierÞ
(13)
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ffi ðE UÞ E U ð1Þn l ; ðwave vector inside the barrierÞ
(14)
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi l2 þ 4ðZvF kn Þ2 þ ð1Þn l ;
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 s l2 þ 4ðZvF qn Þ2 þ ð1Þn l ; 2
(15)
(16)
with s ¼ sign(EU). n ¼ 1, 2 describe two different spin-split subbands as indicated in Fig. 1(a). Similarly, for the K' valley the wave functions in regions I, II, and III [see Fig. 1(b)], have the following forms:
3 3 2 ið1Þnþ1 hn eifn ih1 eif1 0 7 7 iðk xþk yÞ 6 6 r 1 1 1 1 1x y 7 iðk1x xþky yÞ þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 7 6 6 jKI ðx; yÞ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i2f1 5e 4 ið1Þn ei2fn 5e 4 ie 2 2LW 1 þ h2n 2LW 1 þ h 1 h1 eif1 hn eifn 2 3 ih2 eif2 0 6 1 7 iðk xþk yÞ r2 2x y 6 7 þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; 4 iei2f2 5e 2LW 1 þ h22 h2 eif2 2
0
3 3 2 ix1s eiq1 ix1s eiq1 0 7 iðq xþk yÞ 7 iðq xþk yÞ 6 6 a1 b1 1 1 1x y 7 1x y þ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 7 6 6 jKII ðx; yÞ ¼ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ffi 4 iei2q1 5e
ffi 4 iei2q1 5e 2 2 2LW 1 þ x1s 2LW 1 þ x1s iq1 iq1 x1s e x1s e 3 3 2 2 iq2 ix2s e ix2s eiq2 0 0 7 iðq xþk yÞ 6 6 1 7 iðq xþk yÞ a2 b2 1 2x y 7 2x y þ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 7 6 6 ; þ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ffi 4 iei2q2 5e
ffi 4 iei2q2 5e 2 2 2LW 1 þ x2s 2LW 1 þ x2s iq2 iq2 x2s e x2s e
(17)
2
0
0
3 3 2 2 ih1 eif1 ih2 eif2 0 0 7 7 iðk xþk yÞ 6 6 t t 1 1 1 2 7 iðk1x xþky yÞ þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 7 2x y ; 6 6 jKIII ðx; yÞ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i2f1 5e i2f2 5e 4 4 ie ie 2 2LW 1 þ h21 2LW 1 þ h 2 h1 eif1 h2 eif2
(18)
0
where the specific parameters are equal to the above definitions as listed in Eqs. 10e16.
(19)
X.-D. Tan et al. / Superlattices and Microstructures 98 (2016) 473e491
477
3. Klein tunneling in SLG þ R Now, let us discuss in detail the role of RSOC in the tunneling problem. We define Tnm (n, m ¼ 1, 2) as the tunneling probability of an electron initial in subband n and outgoing in subband m. Then, the total transmission ability of each incident electron is given by
tot Tn ¼ Tn1 þ Tn2 ;
n ¼ 1; 2
(20)
Note that the Hamiltonian (6) obeys the time-reversal symmetry. Thus, the transmission coefficients are identical for K and 0 0 K valleys. Indeed, straightforward calculations also show that the unsettled parameters in wave functions for K and K valleys, satisfy the same equation set (A1), as shown in the Appendix A. So, in the following we only need to discuss the K valley. 3.1. The case of 0 < EF < l First, we consider the case of 0 < EF < l. According to the boundary conditions that require the continuity of wave function, the transmission coefficients T1tot and T2tot in this case are calculated as
8 > T~ > < 11 tot T1 ðE; l; U; D; f1 Þ ¼ ~0 > T > : 11 0 T2tot ðE; l; U; D; f2 Þ ¼ 0;
ðf1 ¼ 0; 0 < U < EF Þ ðf1 ¼ 0; EF < U < EF þ lÞ ; ðf1 ¼ 0; U > EF þ lÞ ð p=2 f1 < 0; 0 < f1 p=2; U > 0Þ
ð p=2 f2 p=2; U > 0Þ;
(21)
(22)
where T~ 11 is given by
1 1 ; T~ 11 ¼ 1 þ ðr1 þ 1=r1 2Þsin2 ðQ1 Þ 4
(23)
with
r1 ¼
EðE U þ lÞ ; ðE UÞðE þ lÞ
Q1 ¼
(24)
D qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðE UÞðE U þ lÞ:
(25)
ZvF
It deserves to be mentioned that in this case only normal incident electrons have capacity to pass through the barrier. Moreover, the electrons always occupy the same subband before and after penetrating through the barrier, thus suggesting a single band to single band (S / S) transmission regime. This special tunneling behavior is very different from the case without considering RSOC [4]. In Ref. [4], it was shown that the transmission probability of an electron depends on incident angles at a given potential barrier. Moreover, the tunneling at normal incidence is always 100%. However, in the presence of RSOC, only if the resonance conditions (Q1 ¼ Np, N ¼ 0, 1,…) are satisfied, can normally incident electrons penetrate through the barrier perfectly according to Eq. (23). Therefore, RSOC plays a significant role in the S / S transmission regime. 3.2. The case of EF > l Then, let us discuss the case of EF > l. In this case, the tunneling depends not only on the S / S transmission regime but also on the single band to multiple bands (S / M) transmission regime. Thus, the transmission should be discussed in five different regions according to the barrier height. In the S / M transmission regime, the subband-dependent transmission coefficients are limited by the critical incident angle fcn . If jfn j fcn , we have Tnm ¼ 0 (n, m ¼ 1, 2), while jfn j < fcn the transmission coefficients are calculated as
det H CA1 M 2 11 ; T11 ¼ det D CA1 B
T12 ¼
h2 cosðf2 Þ det H12 CA1 F h1 cosðf1 Þ det D CA1 B
(26) 2 ;
(27)
478
X.-D. Tan et al. / Superlattices and Microstructures 98 (2016) 473e491
h cosðf1 Þ det H21 CA1 M T21 ¼ 1 h2 cosðf2 Þ det D CA1 B T22
2 ;
(28)
det H CA1 F 2 22 ; ¼ det D CA1 B
(29)
where the specific parameters are shown in the Appendix A. i) For 0 < U < EFl (S / M transmission regime),
T1tot ðE; l; U; D; f1 Þ ¼
T2tot ðE; l; U; D; f2 Þ ¼
8 > > < > > :
0; T11 þ T12 ;
8 > > < > > :
0;
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! ðE UÞðE U lÞ ¼ arcsin EðE þ lÞ ;
(30)
otherwise sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! ðE UÞðE U lÞ c p=2 jf2 j f2 ¼ arcsin EðE lÞ :
(31)
p=2 jf1 j
fc1
T21 þ T22 ;
otherwise
ii) For EFl < U < EF (S / S transmission regime),
T1tot ðE; l; U; D; f1 Þ ¼
f1 ¼ 0 ; T~ 11 ; 0; otherwise
T2tot ðE; l; U; D; f2 Þ ¼ 0;
ð p=2 f2 p=2Þ;
(32) (33)
where T~ 11 is given by Eq. (23). iii) For EF < U < EFþl (S/S transmission regime),
T1tot ðE; l; U; D; f1 Þ ¼ 0; ð p=2 f1 p=2Þ; ~ f2 ¼ 0 ; T2tot ðE; l; U; D; f2 Þ ¼ T 22 ; 0; otherwise
(34) (35)
where
1 1 ; T~ 22 ¼ 1 þ ðr2 þ 1=r2 2Þsin2 ðQ2 Þ 4
(36)
with
r2 ¼ Q2 ¼
EðE U lÞ ; ðE UÞðE lÞ
D qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðE UÞðE U lÞ:
ZvF
iv) For EFþl < U < 2EF (S/M transmission regime),
(37) (38)
X.-D. Tan et al. / Superlattices and Microstructures 98 (2016) 473e491
T1tot ðE; l; U; D; f1 Þ ¼
T2tot ðE; l; U; D; f2 Þ ¼
8 > > < > > : 8 > > < > > :
0; T11 þ T12 ; 0;
479
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! ðE UÞðE U þ lÞ ¼ arcsin EðE þ lÞ ;
(39)
otherwise sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! ðE UÞðE U þ lÞ c p=2 jf2 j f2 ¼ arcsin EðE lÞ ;
(40)
p=2 jf1 j
fc1
T21 þ T22 ;
otherwise
v) For U > 2EF (S / M transmission regime),
T1tot ðE; l; U; D; f1 Þ ¼
8 > > < > > :
0;
p=2 jf1 j
T11 þ T12 ;
T2tot ðE; l; U; D; f2 Þ ¼ T21 þ T22 ;
fc1
sffiffiffiffiffiffiffiffiffiffiffi ! El ¼ arcsin Eþl ;
(41)
otherwise ðjf2 j p=2Þ:
(42)
According to Eqs. (30), (31), (39)e(42), we have Tntot ¼ 0 when jfn j fcn , (n ¼ 1, 2). So, in the following we only need to discuss the nontrivial region of jfn j < fcn , (n ¼ 1, 2). Figs. 2e4 show examples of angle- and subband-dependent transmission probabilities in the S / M transmission regime. Here, the Fermi energy of incident electrons is chosen as EF ¼ 80 meV that is most typical in experiments with graphene [4]. When U ¼ 15 meV, as can be seen in Fig. 2(a), the transmission probability T12 is very weak, and T11 plays a preponderant role in the tunneling. Moreover, the total transmission probability almost reaches to T1tot ¼ 1 for all incident angles within jf1 j < fc1 z17:83+ . By contrast, T22 strongly depends on the incident angle of electron, as shown in Fig. 2(b). When the incident angle is toward to the critical value fc2 z39:6+ , T22 decreases fast, and only in the range of about jf2j <30 , T22 is comparable with T11. We can also see that T12 and T21 display the similar behavior, as shown in the insets of Fig. 2(a) and (b). When we increase the barrier height to U ¼ 145 meV, the S / M transmission becomes remarkable. We can see in Fig. 3(a) that the small-angle-incident electrons prefer tunneling with a probability T11. When the incident angle is located in the interval of about 9.3 < jf1j < 12.8+, we can see T12 > T11, suggesting that there is no simple dominant relation between T11 and T12. When an incoming electron occupying the subband n ¼ 2, it is prone to penetrate through the barrier with a probability T22, but T21 holds an advantage over T22 near the fc2 z39:6+ [see Fig. 3(b)]. In addition, unlike the discontinuous behavior of T11 and T12 (T21 and T22) at the critical angle fc1 z17:83+ (fc2 z39:6+ ) [see Figs. 2 and 3],
Fig. 2. Angle- and subband-dependent transmissions through a barrier of wide D ¼ 110 nm and height U ¼ 15 meV in SLG þ R with EF ¼ 80 meV and l ¼ 50 meV. (a) Transmissions within jf1 j < fc1 z17:83+ for the incident electron in subband n ¼ 1. (b) Transmissions within jf2 j < fc2 z39:6+ for the incident electron in subband n ¼ 2. The insets show T12 (top) and T21(bottom).
480
X.-D. Tan et al. / Superlattices and Microstructures 98 (2016) 473e491
Fig. 3. Angle- and subband-dependent transmissions through a barrier of wide D ¼ 110 nm and height U ¼ 145 meV in SLG þ R with EF ¼ 80 meV and l ¼ 50 meV. (a) Transmissions within jf1 j < fc1 z17:83+ for the incident electron in subband n ¼ 1. (b) Transmissions within jf2 j < fc2 z39:6+ for the incident electron in subband n ¼ 2.
Fig. 4. Angle- and subband-dependent transmissions through a barrier of wide D ¼ 110 nm and height U ¼ 200 meV in SLG þ R with EF ¼ 80 meV and l ¼ 50 meV. (a) Transmissions within jf1 j < fc1 z28:71+ for the incident electron in subband n ¼ 1. (b) Transmissions within jf2j 90+ for the incident electron in subband n ¼ 2.
the corresponding transmission coefficients for U ¼ 200 meV are continuous at the critical angle fc1 z28:71+ (fc2 ¼ 90+ ), because the transmission probabilities just vanish at this critical incident angle (see Fig. 4). In order to further reveal the influences of angle (fn), subbands (n ¼ 1, 2), and potential barrier (U) on transmission probabilities, we depict Tntot , Tn1, and Tn2, (n ¼ 1, 2) as a function of U and fn in Fig. 5, which intuitively exhibits the global distributions of the transmission probabilities at a given l ¼ 50 meV. For lower barrier (0 < U < 30 meV) electrons in subband n ¼ 1 almost perfectly pass through the barrier for all incident angles within jf1 j < fc1 [see Fig. 5(a)]. When the barrier is located in the range of 30 meV < U < 80 meV, only normal incident electrons are allowed tunneling with a probability T11 [see Fig. 5(b)]. When 80 meV < U < 130 meV, the transmissions are forbidden for electrons in subband n ¼ 1, but only the normally incident electrons occupying the subband n ¼ 2 are allowed tunneling with a probability T22 [see Fig. 5(e)]. For a higher barrier (130 meV < U < 200 meV) tunneling is very sensitive to the small-angle incidence, where T11 plays a significant role [see Fig. 5(b)], compared with T12 [see Fig. 5(c)]. We also see in Fig. 5(c) that T12 has no contribution to T1tot in the case of 0 < U < 130 meV, where the similar behaves are also found in T21 [see Fig. 5(f)]. When it comes to T22 [see Fig. 5(e)], similar to T11, it is the main contributor to total transmission probability T2tot , compared with T21. So, on the whole, the electrons prefer tunneling between the same spin-split subbands. In order to get further insight into the effect of RSOC on electron
X.-D. Tan et al. / Superlattices and Microstructures 98 (2016) 473e491 Fig. 5. Transmission probabilities (a) T1tot , (b) T11, (c) T12, (d) T2tot , (e) T22, and (f) T21 through a 110-nm-wide barrier as a function of incident angle f and barrier height U in SLG þ R with EF ¼ 80 meV and l ¼ 50 meV.
481
482
X.-D. Tan et al. / Superlattices and Microstructures 98 (2016) 473e491
Fig. 6. Angle- and subband-dependent total transmission through a barrier of wide D ¼ 110 nm and height U ¼ 200 meV, for several different l, in SLG þ R with EF ¼ 80 meV. (a) shows T1tot versus f1, and (b) shows T2tot versus f2.
transmissions, we depict the total transmission Tntot , for several different l, as a function of the incident angle fn (n ¼ 1, 2). As can be seen in Fig. 6, when l is very small it shows T1tot zT2tot , which is in agreement with the previous results [4,34]. As we increase l, T1tot and T2tot exhibit a big difference. This can be explained by the fact that when l / 0 the spin-up and the spindown states of electrons are approximately degenerate, but when l is increased, the degeneracy is gradually removed. Thus, Klein tunneling becomes very complicated and subtle when the RSOC is noticeable. 4. Spin-resolved conductance 4.1. In the case of S / M transmission regime For each transmitted electron in the state,
3 ið1Þnþ1 ei2fn 7 iðk xþk yÞ 6 1 hn eifn 7e nx y 6 j ¼ jn; kn 〉 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 nþ1 ifn 5 2 ið1Þ h e 2LW 1 þ hn n 1 2
ðn ¼ 1; 2Þ;
(43)
the contribution to the current density along the x-direction is given by y j± nx ¼ evF j sx j ¼
ehv2F cosðfn Þkn Tn±
; pLW 2E ð1Þn l
(44)
where n denotes the original subband in which an incident electron occupies, and
Tn± ¼
Tn1 þ Tn2 h1 cosðf1 ÞTn1 h2 cosðf2 ÞTn2 H ± ; 2 1 þ h21 1 þ h22
(45)
is the probability of an electron in subband n throughout the barrier with the spin up (þ) or the spin down () state of the real spin operator Sy (for details see Appendix B). The total current density is determined by carriers with all possible values of kx ! ! ! and ky. The number of electrons in range of k / k þ d k is determined by the product of its state density and occupied possibility,
dnelec ¼ 2 f ðE; mε Þ
LW LW kn dkn dfn ¼ 2E ð1Þn l f ðE; mε ÞdEdfn ; 4p2 h2 v2F 0
where the factor 2 arises because K and K valleys have the same contribution to the current density. Here
(46)
X.-D. Tan et al. / Superlattices and Microstructures 98 (2016) 473e491
f ðE; mε Þ ¼
1 ; 1 þ eðEmε Þ=kB T
483
(47)
is the Fermi-Dirac distribution. Note that there exist two reservoirs. The left reservoir injects carriers up to a quasi-Fermienergy m1 and the right reservoir emits carriers up to a quasi-Fermi-energy m2. The two baths compete with each other and finally the electrons obtain equilibrium distribution with a certain chemical potential mL and mR. Then the total net current density following from the left reservoir to the right reservoir is given by c
± Jnx
Zfn
e ¼ h
Zm1 cosðfn Þdfn
p
m2
fcn
kn
½f ðE; mL Þ f ðE; mR ÞTn± dE;
(48)
where fcn is the critical incidence angle, and Tn± exists only if jfn j < fcn . The carrier densities can be characterized by the chemical potential mL and mR, and their respective levels are between m1 and m2. If the bias is chosen to be small enough, then we can obtain the following approximation,
vf vf f ðE; mL Þ f ðE; mR Þz ðmL mR Þ ¼ eV0 : vm vE
(49)
At very low temperature, we have
vf zdðE EF Þ: vE
(50)
As a result, the spin-dependent current density along the x-direction is given by c
± Jnx
e2 V ¼ 0 h p
Zfn
kn ðEF ; lÞcosðfn ÞTn± ðEF ; l; U; D; fn Þdfn :
(51)
fcn
According to the definition of conductance (G ¼ I/V), we obtain that ± Inx ¼
ZW
0
± ± Jnx dy ¼ Jnx W ¼ Gn± V0 :
(52)
0
Then, the effective conductance (conductance per unit length) is given by 0
G± n ¼
± Gn± Jnx ¼ : W V0
(53)
Thus, we explicitly obtain the spin-dependent conductance in the S / M transmission regime, c
e2 1 G± n ¼ h p
Zfn
kn ðEF ; lÞcosðfn ÞTn± ðE; l; U; D; fn Þdfn :
(54)
fcn
Since we are interested in the states of outgoing electrons, we define the conductance for outgoing spin up electrons as þ GS/M ¼ Gþ [ 1 þ G2 ;
(55)
and similarly, GS/M ¼ G Y 1 þ G2 :
Then the total conductance in the S / M transmission regime is given by
(56)
484
X.-D. Tan et al. / Superlattices and Microstructures 98 (2016) 473e491
Fig. 7. Total and spin-resolved conductances as a function of barrier height U in SLG þ R with EF ¼ 30 meV, l ¼ 50 meV, and D ¼ 110 nm. The first six ticks on the U axis correspond to the above six solid band diagrams in the barrier region.
GS/M ¼ GS/M þ GS/M : tot [ Y
(57)
4.2. In the case of S / S transmission regime In this case, the single electron current density along the x-direction is given by
~j± ¼ nx
± ehv2F kn T~ n
: pLW 2E ð1Þn l
(58)
with
h T~ nn T~ nn ± ±ð1Þn n 2 : T~ n ¼ 2 1 þ hn
(59)
The number of normally incident electrons in range of kx / kxþdkx is given by
~ elec ¼ 2 dn
2pL 2E ð1Þn l L f ðE; mε Þdkn ¼ f ðE; mε ÞdE; 2p h2 v2F kn
(60)
The total net current density is calculated as
±
J e nx ¼
e2 2V0 ~ ± T ðE ; l; U; DÞ: h W n F
Then, at very low temperature the current intensity is given by
(61)
X.-D. Tan et al. / Superlattices and Microstructures 98 (2016) 473e491
485
Fig. 8. Total and spin-resolved conductances as a function of barrier height U in SLG þ R with EF ¼ 80 meV, l ¼ 50 meV, and D ¼ 110 nm. The ten ticks on the U axis correspond to the above ten solid band diagrams in the barrier region. The inset shows the total conductance Gtot, for several different l, as a function of U in SLG þ R with EF ¼ 80 meV.
~I ± ¼ nx
ZW
~ ±V : ~J ± dy ¼ ~J ± W ¼ G nx nx n 0
(62)
0
Thus, the spin-dependent conductance in the S / S transmission regime is given by 2
~ ± ¼ e 2T~ ± ðE ; l; U; DÞ: G n F n h
(63)
Conformance to definitions in Eqs. 55e57, it is clear to give the following expressions
GS/S [
GS/S [
8 > ~þ; > ð0 < EF < l; U > 0Þ
S/S GS/S þ GS/S : tot ¼ G[ Y
(64)
(65)
(66)
486
X.-D. Tan et al. / Superlattices and Microstructures 98 (2016) 473e491
Fig. 9. Spin polarization as a function of barrier height U in SLG þ R with EF ¼ 80 meV and D ¼ 110 nm, for several different l.
Fig. 10. Cross-sectional configuration of a graphene-spin device.
4.3. Numerical results and discussions We are now in a position to discuss the spin-resolved conductance. First, we consider the nontrivial case of 0 < EF < l. As can be seen in Fig. 7, the total conductance Gtot is almost dominated by GY. At lower barrier, Gtot approximately keeps a maximal constant 2e2/h. When the barrier lies in the interval of EF < U < EFþl, Gtot is equal to zero. While for the remaining interval of U, the spin-resolved conductance displays oscillating behaviors. These remarkable features show that fabrication of a graphene-based FET with a noticeable on-off ratio is feasible in the presence of a sizeable RSOC (l > EF). Next, we discuss the conductance for the case of EF > l. We can see in Fig. 8 that for a given l ¼ 50 meV the spin-resolved conductance G[ (GY) is monotone decreasing in the range of 0 < U < EFl. When we increase U to the interval of EFl < U < EF, spin-up electrons are completely reflected and only spin-down electrons have contribution to conductance. While for the range of EF < U < EFþl, it is just the opposite. Further increasing U, the conductance G[ (GY) displays oscillating behaviors. For l ¼ 50 meV, the total conductance Gtot does not exist a zero-gap along the U-axis. We also compared Gtot for several different l, as shown in the inset of Fig. 8. We can see that when U falls into the region EFl < U < EFþl, Gtot is low and fluctuates with U. Out of this region, Gtot tends to decrease with increasing l. More detailed calculation shows that in the range of EFl < U < EFþl the maximal Gtot is no more than 2e2/h, while the minimum is not zero except some special values of U. Thus, it is difficult to achieve considerable on-off ratios for graphene-based FETs in the case of EF > l.
X.-D. Tan et al. / Superlattices and Microstructures 98 (2016) 473e491
487
5. Spin polarization in the transmitted region According to the above analysis, the transmitted electrons can be spontaneously spin polarized along the y-direction. So, it is necessary to quantify the spin polarization of electrons in the transmitted region. Following Ref. [35], the spin polarization can be defined as
P¼
G[ GY : G[ þ GY
(67)
In the case of S / S transmission regime, it is easy to obtain
P¼
GS/S GS/S [ Y GS/S þ GS/S [ Y
8 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 EF ðEF þ lÞ > > > ; ð0 < EF < l; U > 0Þ > > 2EF þ l > > > > p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi < 2 EF ðEF þ lÞ ¼ ; ðEF > l; EF l < U < EF Þ ; > 2EF þ l > > > pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > > > 2 EF ðEF lÞ > > : ; ðEF > l; EF < U < EF þ lÞ 2EF l
(68)
which is a constant independent of U when EF and l are given. For EF ¼ 30 meV and. l ¼ 50 meV, we have P z 0.8907. When EF > l, the high spin polarization also occurs in the range of EFl < U < EFþl. Moreover, the direction of spin polarization is opposite near the critical point U ¼ EF. For example, when we fix EF ¼ 80 meV and l ¼ 50 meV, we have P z 0.9712 for 30 meV < U < 80 meV, and P z 0.8907 for 80 meV < U < 130 meV (see Fig. 9). This remarkable result indicates that one can realize spin switches in the region EFl < U < EFþl. When U is located in the range of 0 < U < EFl, the spin polarization increases with increasing U on the whole. In the range of U > EFþl, spin polarization is weak and displays oscillating behaviors with increasing U. By comparing the spin polarization for several different l, it clearly shows that RSOC plays a significant role in spin polarized transport in SLG þ R. 6. Potential applications Graphene PN and PNP junctions provide an effective platform for investigating Klein tunneling [4] and other relevant quantum electrodynamics phenomena. They are also the most important basic building blocks in spintronics devices. Based on the above discussions, we propose a conceptive graphene-spin device. Most works adopted the top-gated graphene-based device. However, impurities and organic residue caused by fabrication processes degrade the quality of graphene layer [36,37]. In Ref. [38], it presents a new method to fabricate a graphene device with neither deposition of dielectric material on the graphene nor electron-beam irradiation. Unlike top-gated devices, in this scheme the pre-defined local gate (poly-silicon) is embedded into a SiO2 substrate, as shown in Fig. 10. This local gate is used to create a tunable potential barrier, and the back gate controls the Fermi lever of SLG but does not affect the carrier density in the local-gate region [38]. On the other hand, the considerable SOC is a crucial element in the development of graphene-based spin FETs and realization of topological insulators. To enhance the SOC strength, one can deposit the graphene sheet on some special substrates such as tungsten disulfide (WS2) and Au/Ni(111). As reported in Refs. [18,19], strong SOC can be induced in these systems. By tuning local and back gate voltages, both control of barrier height and Fermi lever in SLG can be achieved, thereby manipulating spin transport properties of carriers. According to the detailed theoretical analyses in the present study, we think that this device may promisingly qualify as a spin FET. At least, it might realize the functions such as spin filtration, spin switch, and electron beam collimation. 7. Conclusions In summary, we investigated the spin transport properties in SLG þ R. The transmission coefficients through a potential barrier were obtained analytically. Results show that Klein tunneling in SLG þ R and BLG exhibits completely different behaviors. The tunneling at normal incidence in BLG shows a perfect reflection, while in SLG þ R it displays more dramatic behaviors. In the S / S transmission regime that mainly occurs in the case of 0 < EF < l, only normally incident electrons have capacity to pass through the barrier in SLG þ R. Moreover, the tunneling between different spin-subbands is forbidden in this regime, and consequently outgoing electrons are highly spin polarized along y-direction. As reported in Ref. [4], in the absence of RSOC, the barrier always remains perfectly transparent for electrons close to the normal incidence. However, in the presence of RSOC, this perfect tunneling is not always allowed unless the resonance conditions are satisfied. In the S / M transmission regime that only exists in the case of EF > l, the electron transmission depends not only on subbands but also strongly on incident angles. In this regime, the tunneling could happen at not only normal but also oblique incidence. Based on the transmission coefficients, we also derived spin-resolved conductance analytically. The most significant finding is that when 0 < EF < l the conductance appears a gap in the interval of EF < U < EFþl. This remarkable features indicates that
488
X.-D. Tan et al. / Superlattices and Microstructures 98 (2016) 473e491
fabrication of a graphene-based FET with a high on-off ratio is feasible if 0 < EF < l. In the case of EF > l, the conductance decays with increasing U if 0 < U < EFl. When EFl < U < EFþl, the electrons are highly spin polarized, and spin switch effect emerges near the critical point U ¼ EF. We also found that the conductance displays oscillation behaviors in this interval as well as the interval of U > EFþl. For EF > l, except some special values of U, it is difficult to pinch off the conducting channel, which limits achievable on-off ratios for graphene-based FETs in this case, but it can serve as a spin switch. Acknowledgments We acknowledge X.G. Wan for useful discussions. This work was supported by the National Natural Science Foundation of China (Nos. 51420105003, 11327901, 61274114, and 11525415). Appendix A. angle- and subband-dependent tunneling probabilities Based on the boundary conditions, jI ðx ¼ 0Þ ¼ jII ðx ¼ 0Þ and jII ðx ¼ DÞ ¼ jIII ðx ¼ DÞ, we obtain the following equation set
A C
B D
x1 x2
¼
0 ; ε2
0 ε1
(A1)
where the specific parameters are listed below
2
eiðq1x D2q1 Þ 6 x1s eiðq1x Dq1 Þ A¼6 4 x eiðq1x Dq1 Þ 1s
eiq1x D
eiðq1x D2q1 Þ x1s eiðq1x Dq1 Þ x1s eiðq1x Dq1 Þ eiq1x D
2
ei2q1 6 x eiq1 1s C¼6 4 x eiq1 1s 1 2
ei2f1 6 h eif1 1 D¼6 4 h eif1 1
2
1 3
a1
6 a2 7 7 x1 ¼ 6 4 b1 5;
b2
ei2q1 x1s eiq1 x1s eiq1 1
ei2q2 x2s eiq2 x2s eiq2 1
ei2f2 0 h2 eif2 0 h2 eif2 0 1 0 2 3 r1 6 r2 7 6 x2 ¼ 4 7 ; t1 5 t2
3 eiðq2x D2q2 Þ iðq2x Dq2 Þ 7 x2s e 7; x2s eiðq2x Dq2 Þ 5 eiq2x D
3 eiðk2x D2f2 Þ iðk2x Df2 Þ 7 h2 e 7; h2 eiðk2x Df2 Þ 5 eik2x D
0 0 eiðk1x D2f1 Þ 6 0 0 h1 eiðk1x Df1 Þ B¼6 4 0 0 h eiðk1x Df1 Þ 1 0 0 eik1x D 2
eiðq2x D2q2 Þ x2s eiðq2x DDq2 Þ x2s eiðq2x Dq2 Þ eiq2x D
(A3)
3 ei2q2 iq2 7 x2s e 7 ; x2s eiq2 5 1
(A4)
3 0 07 7; 05 0
3 ei2f1 if 6h e 1 7 1 7 6 1 ε1 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 h eif1 5; 2 1 2 1 þ h1 1 2
(A2)
(A5)
(A6) 3 ei2f2 if 6 h e 2 7 1 7 6 2 ε2 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 h eif2 5: 2 2 2 1 þ h2 1 2
(A7)
Using the Cramer's rule and block matrix skill, straightforward calculation then gives the expressions of the transmission coefficients
det H CA1 M2 11 ; T11 ¼ det D CA1 B
T12 ¼
h2 cosðf2 Þ det H12 CA1 F h1 cosðf1 Þ det D CA1 B
(A8) 2 ;
(A9)
X.-D. Tan et al. / Superlattices and Microstructures 98 (2016) 473e491
h cosðf1 Þ det H21 CA1 M T21 ¼ 1 h2 cosðf2 Þ det D CA1 B T22
2 ;
489
(A10)
det H CA1 F 2 22 ; ¼ det D CA1 B
(A11)
with
2
H11
ei2f1 6 h eif1 1 ¼6 4 h eif1 1 1 2
H21
ei2f1 6 h eif1 1 ¼6 4 h eif1 1 1 2
0 60 M¼6 40 0
0 0 0 0
ei2f2 h2 eif2 h2 eif2 1
ei2f1 h1 eif1 h1 eif1 1
ei2f2 h2 eif2 h2 eif2 1
ei2f2 h2 eif2 h2 eif2 1
3 0 eiðk2x D2f2 Þ iðk2x Df2 Þ 7 0 h2 e 7; 0 h2 eiðk2x Df2 Þ 5 0 eik2x D
2
3 0 07 7; 05 0
H12
3 0 07 7; 05 0
ei2f1 6 h eif1 1 ¼6 4 h eif1 1 1 2
H22
2
0 60 F¼6 40 0
ei2f1 6 h eif1 1 ¼6 4 h eif1 1 1
ei2f2 h2 eif2 h2 eif2 1 ei2f2 h2 eif2 h2 eif2 1
0 eiðk1x D2f1 Þ 0 h1 eiðk1x Df1 Þ 0 h1 eiðk1x Df1 Þ 0 eik1x D
3 0 07 7: 05 0
3 0 ei2f1 if1 7 0 h1 e 7; 0 h1 eif1 5 0 1 3 0 ei2f2 if2 7 0 h2 e 7; 0 h2 eif2 5 0 1
(A12)
(A13)
(A14)
Here, the inverse of the matrix A is calculated as
2
A1
a 6 a* ¼6 4 p* p
b b* m* m
d d* g* g
3 a * 7 a 7 ; p* 5 p
(A15)
where the parameters are given by
b ¼ au* ;
d ¼ au;
(A16)
with a ¼ secðq1 Þeiðq1x Dq1 Þ =4 and u ¼ cosðq1 Þ=x1s þ i sinðq2 Þ=x2s , and
g ¼ pc* ;
m ¼ p c;
(A17)
with p ¼ secðq2 Þeiðq2x Dq2 Þ =4 and c ¼ i sinðq1 Þ=x1s þ cosðq2 Þ=x2s . Appendix B. spin-resolved transmission probabilities In the basis of fuA[ 〉; uAY 〉; uB[ 〉; uBY 〉g, the real spin matrix Sy reads
2
0 i 6i 0 15Sy ¼ 6 40 0 0 0
3 0 0 0 0 7 7: 0 i 5 i 0
(B1)
Its eigenstates for the eigenvalue sy ¼ 1 are
3 i 6 1 1 7 7; j[〉1 ¼ pffiffiffi 6 24 0 5 0 2
3 0 6 1 0 7 7; j[〉2 ¼ pffiffiffi 6 4 i 5 2 1
and for the eigenvalue sy ¼ 1 are
2
(B2)
490
X.-D. Tan et al. / Superlattices and Microstructures 98 (2016) 473e491
2 3 i 7 1 6 1 7; jY〉1 ¼ pffiffiffi 6 5 4 0 2 0
2 3 0 1 6 07 7: jY〉2 ¼ pffiffiffi 6 4 2 i5 1
(B3)
When a transmitted electron in the state
3 ið1Þnþ1 ei2fn 7 iðk xþk yÞ 6 1 hn eifn 7e nx y ; 6 j ¼ jn; kn 〉 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 5 nþ1 if n hn e 2LW 1 þ h2n ið1Þ 1 2
ðn ¼ 1; 2Þ;
(B4)
one can detect the spin-up electron occupying subband n with a probability,
Pnþ ¼ j1 〈[jn; kn 〉j2 þ j2 〈[jn; kn 〉j2 ¼
1 h cosðfn Þ þ ð1Þn n ; 2 1 þ h2n
(B5)
and the spin-down electron with a probability,
Pn ¼ j1 〈Yjn; kn 〉j2 þ j2 〈Yjn; kn 〉j2 ¼
1 h cosðfn Þ ð1Þn n : 2 1 þ h2n
(B6)
Thus, in the S / M transmission regime, the probability that one observes an electron originally occupying subband n and going out the barrier with spin up (þ) or spin down () state is given by
Tn± ¼ Tn1 P1± þ Tn2 P2± ¼
Tn1 þ Tn2 h1 cosðf1 ÞTn1 h2 cosðf2 ÞTn2 H ± : 2 1 þ h21 1 þ h22
(B7)
Similarly, one can obtain the spin-dependent probability in the S / S transmission regime, i. e,
h cosðfn ÞT~ nn T~ nn ± ±ð1Þn n : T~ n ¼ T~ nn Pn± ¼ 2 1 þ h2n
(B8)
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22]
H. Wei, R.K. Kawakami, G. Martin, F. Jaroslav, Graphene spintronics, Nat. Nanotechnol. 9 (2014) 794e807. O. Klein, Die Reflexion von Elektronen an einem Potentialsprung nach der relativistischen Dynamik von Dirac, Z. Phys. 53 (1929) 157e165. W. Greiner, B. Müller, J. Rafelski, Quantum Electrodynamics of Strong Fields, Springer, Berlin, 1985. M.I. Katsnelson, K.S. Novoselov, A.K. Geim, Chiral tunnelling and the Klein paradox in graphene, Nat. Phys. 2 (2006) 620e625. J.A. Sulpizio, N. Stander, K. Todd, B. Yang, D. Goldhaber-Gordon, B. Huard, Transport measurements across a tunable potential barrier in graphene, Phys. Rev. Lett. 98 (2007) 236803. R.V. Gorbachev, A.S. Mayorov, A.K. Savchenko, D.W. Horsell, F. Guinea, Conductance of p-n-p structures with “Air-Bridge” top gates, Nano Lett. 8 (2008) 1995. A.F. Young, P. Kim, Quantum interference and Klein tunnelling in graphene heterojunctions, Nat. Phys. 5 (2009) 222e226. S. Sutar, E.S. Comfort, J. Liu, T. Taniguchi, K. Watanabe, J.U. Lee, Angle-dependent carrier transmission in graphene p-n junctions, Nano Lett. 12 (2012) 4460. R.N. Sajjad, S. Sutar, J.U. Lee, A.W. Ghosh, Manifestation of chiral tunneling at a tilted graphene p-n junction, Phys. Rev. B 86 (2012) 155412. B. Van Duppen, F.M. Peeters, Klein paradox for a pn junction in multilayer graphene, Europhys. Lett. 102 (2013) 27001. T. Yokoyama, Spin and valley transports in junctions of Dirac fermions, New J. Phys. 16 (2014) 085005. Z. Rashidian, F.M. Mojarabian, P. Bayati, G. Rashedi, A. Ueda, T. Yokoyama, Conductance and Fano factor in normal/ferromagnetic/normal bilayer graphene junction, J. Phys. Condens. Matter 26 (2014) 255302. Q. Wilmart, S. Berada, D. Torrin, V.H. Nguyen, G. Fve, J.M. Berroir, P. Dollfus, B. Plaais, A Klein-tunneling transistor with ballistic graphene, 2d Mater. 1 (2014) 11006. A.H. Castro Neto, F. Guinea, Impurity-induced spin-orbit coupling in graphene, Phys. Rev. Lett. 103 (2009) 26804. S. Konschuh, M. Gmitra, J. Fabian, Tight-binding theory of the spin-orbit coupling in graphene, Phys. Rev. B 82 (2010) 245412. J.S. Jeong, J. Shin, H.W. Lee, Curvature-induced spin-orbit coupling and spin relaxation in a chemically clean single-layer graphene, Phys. Rev. B 84 (2011) 195457. D. Marchenko, A. Varykhalov, M.R. Scholz, G. Bihlmayer, E.I. Rashba, A. Rybkin, A.M. Shikin, O. Rader, Giant Rashba splitting in graphene due to hybridization with gold, Nat. Commun. 3 (2012) 1232. A. Avsar, J.Y. Tan, T. Taychatanapat, J. Balakrishnan, G.K. Koon, Y. Yeo, J. Lahiri, A. Carvalho, A.S. Rodin, E.C. O'Farrell, Spin-orbit proximity effect in graphene, Nat. Commun. 5 (2014) 4875. Z. Wang, D.K. Ki, H. Chen, H. Berger, A.H. Macdonald, A.F. Morpurgo, Strong interface-induced spineorbit interaction in graphene on WS2, Nat. Commun. 6 (2015) 8339. A. Manchon, H.C. Koo, J. Nitta, S.M. Frolov, R.A. Duine, New perspectives for Rashba spin-orbit coupling, Nat. Mater. 14 (2015) 871. M. Gmitra, S. Konschuh, C. Ertler, C. Ambrosch-Draxl, J. Fabian, Band-structure topologies of graphene: spin-orbit coupling effects from first principles, Phys. Rev. B 80 (2009) 235431. S. Datta, B. Das, Electronic analog of the electro-optic modulator, Appl. Phys. Lett. 56 (1990) 665e667.
X.-D. Tan et al. / Superlattices and Microstructures 98 (2016) 473e491 [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38]
491
D. Bercioux, A. De Martino, Spin-resolved scattering through spin-orbit nanostructures in graphene, Phys. Rev. B 81 (2010) 165410. M. Rataj, J. Barnas, Graphene p-n junctions with nonuniform Rashba spin-orbit coupling, Appl. Phys. Lett. 99 (2012) 162107. M.H. Liu, Spin-dependent Klein tunneling in graphene: role of Rashba spin-orbit coupling, Phys. Rev. B 85 (2012) 85406. J.F. Liu, K.S. Chan, Y-shaped spin filter in graphene with Rashba spin-orbit coupling, J. Phys. Soc. Jpn. 82 (2013) 74711. Q. Zhang, Z. Lin, K.S. Chan, Spin polarization switching in monolayer graphene through a Rashba multi-barrier structure, Appl. Phys. Lett. 102 (2013) 142407. K. Shakouri, M.R. Masir, A. Jellal, E.B. Choubabi, F.M. Peeters, Effect of spin-orbit couplings in graphene with and without potential modulation, Phys. Rev. B 88 (2013) 115408. H. Min, J.E. Hill, N.A. Sinitsyn, B.R. Sahu, L. Kleinman, A.H. Macdonald, Intrinsic, Rashba Spin-orbit, Interactions in graphene sheets, Phys. Rev. B 74 (2006) 165310. M. Gmitra, D. Kochan, J. Fabian, S. Konschuh, Theory of spin-orbit coupling in bilayer graphene, Phys. Rev. B 85 (2012) 115423. nyos, Intrinsic and substrate induced spin-orbit interaction in chirally stacked trilayer graphene, Phys. Rev. B 87 (2013) 45419. G. Burkard, A. Korma E. Mccann, V.I. Fal'Ko, Landau-level degeneracy and quantum Hall effect in a graphite bilayer, Phys. Rev. Lett. 96 (2005) 86805. C.L. Kane, E.J. Mele, Quantum spin hall effect in graphene, Phys. Rev. Lett. 95 (2005) 226801. A.H.C. Neto, F. Guinea, N.M.R. Peres, K.S. Novoselov, A.A.K. Geim, The electronic properties of graphene, Rev. Mod. Phys. 81 (2007) 109e162. G. Schmidt, Concepts for spin injection into semiconductorsda review, J. Phys. D. App. Phys. 38 (2005) R107eR122. B. Fallahazad, S. Kim, L. Colombo, E. Tutuc, Dielectric thickness dependence of carrier mobility in graphene with HfO2 top dielectric, Appl. Phys. Lett. 97 (2010) 123105. M. Ishigami, J.H. Chen, W.G. Cullen, M.S. Fuhrer, E.D. Williams, Atomic structure of graphene on SiO2, Nano Lett. 7 (2007) 1643e1648. N. Seung-Geol, K. Dong-Keun, P. Jong Wan, K. Youngwook, K. Jun Sung, L. Hu-Jong, Ballistic transport of graphene pnp junctions with embedded local gates, Nanotechnology 22 (2011) 415203.