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Enhanced optical spin current injection in the hexagonal lattice with intrinsic and Rashba spin–orbit interactions
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Enhanced optical spin current injection in the hexagonal lattice with intrinsic and Rashba spin–orbit interactions
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College of Science, Hohai University, Nanjing, 210098, China
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Jianfei Zou ∗ , Chunmei Tang, Aimei Zhang
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Article history: Received 24 November 2016 Received in revised form 22 January 2017 Accepted 30 January 2017 Available online xxxx Communicated by R. Wu
We study the photo-induced spin current injection in a hexagonal lattice with both intrinsic and Rashba spin–orbit interactions which is irradiated by a polarized light beam. It is found that the spin current injection rate could be enhanced as the graphene lattice is in the topological insulator state. Furthermore, the spin current injection rate could be remarkably modulated by the degree of polarization of light and its frequency. © 2017 Elsevier B.V. All rights reserved.
Keywords: Spin current Hexagonal lattice Spin–orbit coupling Optical excitation
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1. Introduction
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Spin current, which carries information about both the flow direction of electron and spin orientation, is an important issue in spintronics. Over the past decade, the theorists and experimentalists have done a large amount of research works on generation, manipulation and detection of the spin current in semiconductor and metal structures [1–9]. So far various methods including electrical and optical mechanisms have been proposed to generate spin current. Scientists have found that the spin-orbital interaction (SOI) plays a key role in the generation of spin current. Some groups have demonstrated that the low dissipative or even dissipationless spin current could be driven by an external electric field in the spin-orbital coupling systems [2,3]. Bhat and his co-workers have found that the spin current could be produced in the noncentrosymmetric semiconductors by one-photon absorption of linearly polarized light [5]. Graphene, a two dimensional hexagonal lattice of carbon atoms, has been a revolutionary material since its first successful fabrication and characterization [10]. It is well known that the graphene, as well as other two dimensional Dirac materials such as silicene and MoS2 monolayer, possesses excellent physical properties which are quite distinct from the traditional materials [10–14].
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Corresponding author. E-mail address: [email protected] (J. Zou).
http://dx.doi.org/10.1016/j.physleta.2017.01.064 0375-9601/© 2017 Elsevier B.V. All rights reserved.
The low energy behavior of electrons near the Fermi level in graphene is described by Dirac equation. It is expectable that the spin current in graphene lattice could exhibit some characteristic phenomena. In the past few years, many works on how to generate the spin current in graphene have been reported, in which several methods including the dynamical [15–18], thermal [19,20] and optical [21] ones have been proposed. In experiments Shiraishi’s group has demonstrated that the pure spin current could be injected into a single layer graphene attached to a ferromagnetic electrode by using dynamical spin pumping method [17,18]. Rioux and Burkard have proposed a photo-excitation scheme for pure spin current generation in graphene with Rashba SOI [21]. Inglot and his co-workers have studied the efficiency of the optical spin injection in graphene with Rashba SOI in a magnetic field [22,23]. In this work we propose an optical mechanism to generate the spin current injection in a hexagonal lattice model with both intrinsic and Rashba SOIs, which is exposed to a polarized light. It is found that the excited spin current could be enhanced by the intrinsic SOI. The spin current injection is more efficient as the graphene lattice is in the topological insulator state than in the semimetal state. We also find that the spin current injection rate remarkably depends on the degree of polarization of light and its frequency. In the rest of this paper, it is organized as follows. The model and formulation for the spin current in the graphene lattice model are presented in Sec. 2. The numerical results and discussion are shown in Sec. 3. Finally we give a brief summary in Sec. 4.
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H 1 = −ev F (τ σx A x + σ y A y ).
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This periodic perturbation leads to electron transitions between the valance and conduction bands and consequently generate spin current. Although the definition of spin current in the spin–orbit coupling systems is still controversial [27–29]. Here we use the generally accepted and physically appealing definition of the symmetrized spin current operator [27]
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ˆJ i j (k) = [ S i vˆ j + vˆ j S i ]/2,
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Fig. 1. (Color online.) Schematics of band structures and interband transitions in graphene lattice model with Rashba and intrinsic SOI. (a) for the semimetal state (λ S O < λ R ). (b) for the topological insulator state (λ S O > λ R ). The spin current produced by the optical transition | E 2 → | E 4 reverses its direction as the topological phase transition from the semimetal (a) to the topological insulator (b) occurs.
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We consider the electronic states near the Dirac point in the graphene model with both intrinsic and Rashba SOIs, which is described by the effective Dirac Hamiltonian as follows [24,25]
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I ivj→c (ω) = (1)
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E μν (k) = μh¯ λR + ν h¯
v 2F k2 + (λR − μλSO )2
(2)
with μ = ±1 and ν = ±1. We rewrite the energy bands as E n (k) where the index n = 1 ∼ 4 label the bands in the order of increasing energy. It has been known that the graphene model is a semimetal as λ S O < λ R , while it is a topological insulator as λ S O > λ R [24]. The energy band structures for the two states are shown in Fig. 1. In this work, we apply a beam of polarized light perpendicularly to irradiate onto the graphene lattice. The vector potential describing the monochromatic wave is of the form
A (t ) = A 0 (cos φ cos ωt , sin φ sin ωt )
(3)
with A 0 = E 0 /ω where ω is the frequency of the light and E 0 is the amplitude of the electric field component. Here the parameter φ is used to specify the polarization of the light. For φ = 0 (π /2), the optical field represented by the vector potential A (t ) is linearly polarized in the x ( y ) direction. For φ = π /4 (3π /4), it is left (right) circularly polarized. For the general values of φ , the field A (t ) describes elliptically polarized light. The effect of the electromagnetic field on the graphene lattice can be incorporated into the Hamiltonian of Eq. (1) by the minimal coupling substitution h¯ k → h¯ k − e A (t ). The consequential Hamiltonian simply reads H T = H + H 1 , where the interaction term of the graphene model with the light field takes the form
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The injection rate of spin current originating from the intersubband transitions between the valance band v and the conduction band c can be calculated by using the Fermi’s golden rule. It is written as
The first term in Eq. (1) represents the free Hamiltonian in graphene lattice around K valley (τ = 1) or K’ valley (τ = −1), where v F is the Fermi velocity, k = (kx , k y ) is the in-plane wavevector relative to the Dirac point, and σ = (σx , σ y , σz ) are the Pauli matrices acting on the sublattice space. The second term describes the Rashba SOI where λR is the Rashba coupling parameter controlled by an external electric field or substrate material and S = ( S x , S y , S z ) are the Pauli matrices for electrons’ real spin. The last term in Eq. (1) describes the intrinsic SOI in the graphene model where λSO is its strength, which could be modified in graphene via adatom deposition [26]. The energy spectrum of the graphene model can by obtained by diagonalizing the above Hamiltonian. It consists of four energy bands, which are written as
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where the velocity operator is given by vˆ = h¯ −1 ∂ H T /∂ k, the subscript i = x, y , z represent the spin components and j = x, y stand for the transfer directions of spin carriers. The spin current for a given state |nk is defined by
2. Model and formulation
+ h¯ λSO τ σz S z .
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J i j ,n = nk| ˆJ i j |nk.
H = h¯ v F (τ σx kx + σ y k y ) + h¯ λR (τ σx S y − σ y S x )
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(5)
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(4)
2π
(6)
d2 k
h¯
|ck| H 1 | vk|2 J i j ,c − J i j , v
(2π )2 × δ( E ck − E vk − h¯ ω) f ( E vk ) [1 − f ( E ck )],
I i j (ω) =
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(7)
(8)
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cv
Considering the symmetry of the graphene model, we can obtain that only one independent nonzero component of the spin current exists: I xy = − I yx , while the others are zero. Hereafter we calculate the spin current injection rate I xy along y axis with the spin polarized along x direction. Furthermore, we emphasize that the electron with state |k, ↑ and its counterpart with state | − k, ↓ are pumped by the light with equal probability due to the symmetry of band structures. The pair has opposite velocity such that it contributes null charge current. Consequently, the photo-induced spin current in our system is pure. We study the undoped graphene model at zero temperate. In such circumstances, there exist four kinds of valance-to-conduction transition contributed to the total spin-current injection, which are (1) | E 2 → | E 3 , (2) | E 2 → | E 4 , (3) | E 1 → | E 3 and (4) | E 1 → | E 4 (shown in Fig. 1). The four contributions are written in order as
(ω + 4λ R )(a + bη) , I 1 (ω) = I 0 (ω) (ω + 2λ R )[(ω + 2λ R )4 − 16λ2S O λ2R ]
I 3 (ω) = − I 0 (ω)
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I ivj→c (ω).
I 2 (ω) = − I 0 (ω)
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where E v (c )k is the eigenvalue of the valance (conduction) band v (c ). f ( E v (c )k ) is the Fermi distribution function. The total injection rate of the spin current is the sum of all allowed transitions
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8(λ R − λ S O ) (2 + η)
(9a)
,
(9b)
8(λ R + λ S O ) (2 + η)
ω
I 4 (ω) = − I 1 (ω, λ R → −λ R )
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ω3
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,
(9c) (9d)
with a = 2ω[(ω + 2λ R )3 + 8λ R λ2S O ], b = (ω + 4λ R )[(ω + 2λ R )3 − 8λ R λ2S O ], and I 0 (ω) = E 02 e 2 v F /(16h¯ ω) including the valley degeneracy. Here we define the degree of the polarization of light η = (Ix − I y )/(Ix + I y ) = cos 2φ , where Ix (I y ) is the intensities of linear polarized component in the x ( y ) direction. η = ±1 denote the linear polarized radiation in the x and y directions, respectively. It should be emphasized that I i (ω) (i = 1 ∼ 4) have the same formations for the semimetal and topological insulator
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states. The detailed derivation of Eqs. (9) is presented in the Appendix. In the case of the semimetal state, i.e., λ S O < λ R , one obtains the total spin-current injection rate
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I xy (ω) = I 1 (ω) + I 2 (ω) [ω − 2(λ R − λ S O )]
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(10)
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where (x) is the Heaviside step function. Nevertheless, the total spin-current injection rate for the case of the topological insulator state (λ S O > λ R ) is given by
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+ I 3 (ω) [ω − 2(λ R + λ S O )] + I 4 (ω) (ω − 4λ R ),
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I xy (ω) = [ I 1 (ω) + I 2 (ω)] [ω − 2(λ S O − λ R )]
+ [ I 3 (ω) + I 4 (ω)] [ω − 2(λ S O + λ R )].
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ω + 4λ R O 1 (ω) = O 0 (ω) , ω + 2λ R O 2 (ω) = O 0 (ω) O 3 (ω) = O 0 (ω) O 4 (ω) = O 0 (ω)
4(λ R − λ S O )2
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ω2 4(λ R + λ S O )2
ω
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ω − 4λ R , ω − 2λ R
(12a)
, ,
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(11)
These are the main results of this paper. By now let us analyze the above expressions for the spin current injection. It is obvious that the injection rate of spin current is dependent on the polarization of light. Moreover, the spin current injection does not vanish as the light is perfectly circular polarized, which differs from the result of Rioux and Burkard’s work [21]. Actually the spin current injection rate for the circular polarized light is the sum of those for x and y-linear polarized light, i.e., I xy ,c = ( I xy ,x + I xy , y )/2, where the third subscript letters c, x and y stand for the circular, x and y-linear polarization respectively. This is because a circular polarized light can be decomposed as a combination of x and y-linear polarized beams of light. Now let us focus on the sign of the four contributions of the spin current injection. We note that the sign of the first contribution I 1 is opposite to that of the other three as the graphene model is in the semimetal state. In other words, the spin current injection I 1 from the subband optical transition | E 2 → | E 3 will partly canceled out by the contributions of the other subband transitions. As the topological phase transition of the graphene model from the semimetal (λ S O < λ R ) to the topological insulator (λ S O > λ R ) takes place, the second contribution I 2 reverses its sign due to the band inversion, while the others remain their signs unchanged. It means that the spin current injection I 1 from the transition | E 2 → | E 3 and I 2 from the transition | E 2 → | E 4 are constructive. So it is expectable that the spin current injection rate will be enhanced when the graphene model is in the topological insulator state. We will show the conclusions in the next section. We also calculate the associated light absorption using the Fermi’s golden rule. The expressions for the total light absorption O t (ω) is similar to that of the spin current injection rate Eq. (10) and Eq. (11). The four contributions of light absorption are given by
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3
(12b) (12c) (12d)
where O 0 (ω) = E 02 e 2 /(4h¯ 2 ω) including the spin and valley degeneracy. It is clear that the light absorption is independent of the light polarization. These expressions for the light absorption reduce to the form given by the previous work as the intrinsic SOI is absent [21]. In addition, we know that the robust edge states emerge at the boundary of sample as the graphene lattice is in the state of the topological insulator [24]. The edge states do contribute to the optical absorption and spin current injection. However, in this work,
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Fig. 2. (Color online.) (a) Spin-current injection rate J xy as functions of light frequency for circular (c) and linear (in the x and y directions) polarizations when the graphene model is in a semimetal state. (b) Spectra of the total light absorption and its intersubband contributions. Here we set λ S O = 0.
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the proportion of the optical excitation resulting from the edge states is ignorable due to the large area of the sample. It could be inferred that the effect of the edge states on the optical excitation is significant as the graphene in the topological state is cut into the nanoribbons.
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3. Results and discussion
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Now we come to show some numerical results about the spincurrent injection rate and the associated light absorption in the graphene model. In our calculations we set the Rashba frequency as a unit. It should be noted that our hexagonal lattice model could be realized on a graphene or silicene sheet. In experiments, the Rashba and intrinsic SOI strengths in graphene and silicene can be enhanced to the order of magnitude of meV [30–32]. The corresponding light irradiating on the graphene sheet should be terahertz.
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3.1. In the semimetal state
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First, let us consider the graphene model in the state of semimetal, i.e., the case of λ S O < λ R . In Fig. 2 (a), we plot the spin-current injection rate as functions of the light frequency for the linear and circular polarizations. It clearly demonstrates that the spin-current injection rate remarkably depends on the light frequency and polarizations. To explain this phenomena, we show the corresponding total optical absorption and its four individual contributions in Fig. 2 (b). It is found that the major contributions of the optical absorption originate from the interband transitions between the top of the valence bands and the bottom of the conduction bands. So does the spin-current injection rate, when the light frequency ω < 2λ R , it only exits the first valance-toconduction transition: | E 2 → | E 3 , while the other transitions are forbidden due to the energy gap. It should be noted that, in this condition, the rate of the spin-current injection is positive for the linear polarization in the x-direction, while it is negative for that in the y-direction as shown in Fig. 2 (a). As the frequency increase over 2λ R , the second and third transitions emerge which result in the spin-current injection rate changing dramatically. Especially the spin-current change its direction for the circular and x-directional linear polarization since the direction of the spincurrent from the first transition is opposite to the others (see
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Fig. 3. (Color online.) Linear dependence of spin-current injection rate I xy on the degree of polarization of light for different frequency when the graphene model is in a semimetal state. We set λ S O = 0.
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Eqs. (9)). As the frequency increases, all of the four transitions will take part in such that the total spin current approaches to vanish. Fig. 3 displays the spin-current injection rate as a function of the degree of polarization of light for different radiation frequency. It explicitly reveal its linear dependence on the degree of polarization which also can be concluded from Eqs. (9). Furthermore, when there is only the first interband transition | E 2 → | E 3 (ω = 0.1λ R in Fig. 3), the variation of the spin-current injection as changing the polarization is the most remarkable. The spin-current injection rate reaches its maximum value 4.0I 0 (ω) as the light is perfectly linearly polarized in x direction. In the most situations with a fixed frequency, the direction of the photo-induced spin-current can be switched by modulating the degree of the polarization.
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Fig. 4. (Color online.) (a) Spin-current injection rate I xy as functions of light frequency for circular (c) and linear (in the x and y directions) polarizations as the graphene model becomes a topological insulator. (b) Spectra of the total light absorption and its intersubband contributions. Here we set λ S O = 2λ R .
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3.2. In the topological insulator state
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Now we turn to the topological insulator state, i.e., the case of λ S O > λ R . Fig. 4 (a) exhibits the variation of the spin-current injection rate with respect to the light frequency for the linear and circular polarizations as λ S O = 2λ R . The corresponding optical absorption is shown in Fig. 4 (b). It is also found that the direction and the injection rate of the spin-current can be effectively controlled by changing the light frequency and its polarization. However, this is quiet different from the results for the case of the semimetal state. The spin current injection rate in the case of the topological insulator state (in the frequency range [2λ R , 6λ R ]) is much larger than that in the semimetal state when the linear polarized light along x-direction is applied. Moreover, the photo-induced electron transition does not arise until the light frequency exceeds the energy gap (ω > 2λ R in Fig. 4). When the frequency varies in the range [2λ R , 6λ R ], there are two kinds of valance-to-conduction transition contributions, i.e., the first transition | E 2 → | E 3 and the second | E 2 → | E 4 . Both of this two contributions to the spin-current are positive such that the total spin-current is constructive and always positive regardless of the light polarization. As the light frequency is larger than 6λ R , the other two transitions arise, which is destructive to the spincurrent produced by the previous contributions. Consequently, the total spin-current rapidly becomes faint and changes its sign. Fig. 5 plots the dependence of the spin current injection rate on the degree of polarization of light for different radiation frequency when λ S O = 2λ R . It is noted that the linear dependence of the spin current injection rate is distinct from that for the semimetal graphene. When it exits only the first two interband transitions: | E 2 → | E 3(4) (ω = 2.1λ R in Fig. 5), the spin current injection is always positive and achieves its maximum value 7.0I 0 (ω) as the light is linear polarized in the x-direction. In fact, we can obtain
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Fig. 5. (Color online.) Linear dependence of spin-current injection rate I xy on the degree of polarization of light for different frequency as the graphene model becomes a topological insulator. We set λ S O = 2λ R .
the maximal injection rate of the spin current (6 + 3λ R /λ S O ) I 0 (ω) according to Eq. (11) as ω = 2(λ S O − λ R ) and η = 1. The maximal injection rate for the topological insulator state is relatively greater than that for the semimetal state. As the frequency increase over 6λ R , all of the four transitions emergent. Consequently, the dependence of the spin-current injection rate on the degree of polarization becomes not noticeable. By the way, let us briefly discuss another case: the topologically trivial insulator state. It is found that the graphene lattice model without the intrinsic SOI is a topologically trivial insulator when a staggered sublattice potential is introduced in the model [24,25]. The velocity of electrons is decreased by the energy gap which is opened by the staggered sublattice potential. Consequently, the spin current injection will suppressed in the conventional insulator state.
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4. Summary
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We have studied the photo-induced spin current in graphene lattice model with intrinsic and Rashba SOIs. It is found that the spin-current injection rate remarkably depends on the degree of
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polarization of light, frequency and the states of the graphene lattice. The spin current injection could be enhanced by the intrinsic SOI, which is more prominent as the graphene model is in the topological insulator state than in the semimetal state. Besides, the spin current injection rate shows linear relationship with the degree of polarization and reaches a maximum when the light is perfectly linearly polarized. Furthermore, the direction of the spincurrent can be switched by changing the polarization of light as the irradiation with appropriate frequency. The injection of spin current could be detected via electrical currents [18], spin accumulation [33,34], or pump-probe techniques [35]. We ignore the effects of the edge states and the spin current relaxation in this work, which are interesting and needed further investigation in the future.
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This work was supported by the Fundamental Research Funds for the Central Universities (Grant No. 2013B00514), the National Natural Science Foundation of Jiangsu (Grant No. BK20140839) and the National Natural Science Foundation of China (Grant Nos. 11347127, 61404044, 11404091). Appendix A. The derivation of the spin current injection rate The procedures of the derivation of Eqs. (9) for the spin current injection are similar. For brevity, we present a detailed derivation of Eq. (9b) as an example. First, for the spin current injection rate I xy originating from the transition | E 2 → | E 4 , Eq. (7) is rewritten as
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2→4 I xy (ω) =
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∞
e 2 E 02 2π h¯ ω
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2π
2 dθ k v x,42 cos φ − i v y ,42 sin φ
dk
2 0
0
× J xy ,4 − J xy ,2 δ( E 4 − E 2 − h¯ ω) f ( E 2 )[1 − f ( E 4 )].
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(A.1)
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Here the matrix elements of the velocity operators vˆ = v F (σx , σ y ) for the K valley, written in the eigenstate basis {| E 1 , | E 2 , | E 3 , | E 4 }, take the forms
v F (λ R − λ S O ) cos θ v x,42 = E 4 | vˆ x | E 2 = , v 2F k2 + (λ R − λ S O )2
(A.2)
v F (λ R − λ S O ) sin θ v y ,42 = E 4 | vˆ y | E 2 = . v 2F k2 + (λ R − λ S O )2
(A.3)
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The diagonal matrix elements of the spin current operator ˆJ xy
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written in the same eigenstate basis are given by
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J xy ,2(4)
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2→4 I xy (ω) = I 2 (ω) (ω − E 2 )
=
h¯ v F −2(λ R − λ S O )( E 2(4) /h¯ − λ S O ) +
v 2F k2 (cos 2θ
4 v 2F k2 + (λ R − λ S O )( E 2(4) /h¯ − λ S O )
− 1)
, (A.4)
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(A.5)
where I 2 (ω) is just the Eq. (9b) and E 2 is the energy gap between the subbands | E 2 and | E 4 which equals 2(λ R − λ S O ) for the case λ R > λ S O or 2(λ S O − λ R ) for the case λ S O > λ R . In the same way, the others in Eqs. (9) can be derived. By adding up the four parts, one gets the Eqs. (10) and (11) for the total spin current injection rate. Equations (12) for the light absorption can be obtained by using the analogous approach. References
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where the eigenenergy E 2(4) should be expressed in the polar coordinate (k, θ). By now we substitute the Eqs. (A.2)∼(A.4) into Eq. (A.1), let f ( E 2 )[1 − f ( E 4 )] = 1 for the undoped graphene model at zero temperate and then integrate the Eq. (A.1). After simplification one obtains
Acknowledgements
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Enhanced optical spin current injection in the hexagonal lattice with intrinsic and Rashba spin–orbit interactions
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College of Science, Hohai University, Nanjing, 210098, China
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Jianfei Zou ∗ , Chunmei Tang, Aimei Zhang
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Article history: Received 24 November 2016 Received in revised form 22 January 2017 Accepted 30 January 2017 Available online xxxx Communicated by R. Wu
We study the photo-induced spin current injection in a hexagonal lattice with both intrinsic and Rashba spin–orbit interactions which is irradiated by a polarized light beam. It is found that the spin current injection rate could be enhanced as the graphene lattice is in the topological insulator state. Furthermore, the spin current injection rate could be remarkably modulated by the degree of polarization of light and its frequency. © 2017 Elsevier B.V. All rights reserved.
Keywords: Spin current Hexagonal lattice Spin–orbit coupling Optical excitation
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1. Introduction
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Spin current, which carries information about both the flow direction of electron and spin orientation, is an important issue in spintronics. Over the past decade, the theorists and experimentalists have done a large amount of research works on generation, manipulation and detection of the spin current in semiconductor and metal structures [1–9]. So far various methods including electrical and optical mechanisms have been proposed to generate spin current. Scientists have found that the spin-orbital interaction (SOI) plays a key role in the generation of spin current. Some groups have demonstrated that the low dissipative or even dissipationless spin current could be driven by an external electric field in the spin-orbital coupling systems [2,3]. Bhat and his co-workers have found that the spin current could be produced in the noncentrosymmetric semiconductors by one-photon absorption of linearly polarized light [5]. Graphene, a two dimensional hexagonal lattice of carbon atoms, has been a revolutionary material since its first successful fabrication and characterization [10]. It is well known that the graphene, as well as other two dimensional Dirac materials such as silicene and MoS2 monolayer, possesses excellent physical properties which are quite distinct from the traditional materials [10–14].
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Corresponding author. E-mail address: [email protected] (J. Zou).
http://dx.doi.org/10.1016/j.physleta.2017.01.064 0375-9601/© 2017 Elsevier B.V. All rights reserved.
The low energy behavior of electrons near the Fermi level in graphene is described by Dirac equation. It is expectable that the spin current in graphene lattice could exhibit some characteristic phenomena. In the past few years, many works on how to generate the spin current in graphene have been reported, in which several methods including the dynamical [15–18], thermal [19,20] and optical [21] ones have been proposed. In experiments Shiraishi’s group has demonstrated that the pure spin current could be injected into a single layer graphene attached to a ferromagnetic electrode by using dynamical spin pumping method [17,18]. Rioux and Burkard have proposed a photo-excitation scheme for pure spin current generation in graphene with Rashba SOI [21]. Inglot and his co-workers have studied the efficiency of the optical spin injection in graphene with Rashba SOI in a magnetic field [22,23]. In this work we propose an optical mechanism to generate the spin current injection in a hexagonal lattice model with both intrinsic and Rashba SOIs, which is exposed to a polarized light. It is found that the excited spin current could be enhanced by the intrinsic SOI. The spin current injection is more efficient as the graphene lattice is in the topological insulator state than in the semimetal state. We also find that the spin current injection rate remarkably depends on the degree of polarization of light and its frequency. In the rest of this paper, it is organized as follows. The model and formulation for the spin current in the graphene lattice model are presented in Sec. 2. The numerical results and discussion are shown in Sec. 3. Finally we give a brief summary in Sec. 4.
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H 1 = −ev F (τ σx A x + σ y A y ).
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This periodic perturbation leads to electron transitions between the valance and conduction bands and consequently generate spin current. Although the definition of spin current in the spin–orbit coupling systems is still controversial [27–29]. Here we use the generally accepted and physically appealing definition of the symmetrized spin current operator [27]
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ˆJ i j (k) = [ S i vˆ j + vˆ j S i ]/2,
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Fig. 1. (Color online.) Schematics of band structures and interband transitions in graphene lattice model with Rashba and intrinsic SOI. (a) for the semimetal state (λ S O < λ R ). (b) for the topological insulator state (λ S O > λ R ). The spin current produced by the optical transition | E 2 → | E 4 reverses its direction as the topological phase transition from the semimetal (a) to the topological insulator (b) occurs.
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We consider the electronic states near the Dirac point in the graphene model with both intrinsic and Rashba SOIs, which is described by the effective Dirac Hamiltonian as follows [24,25]
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I ivj→c (ω) = (1)
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E μν (k) = μh¯ λR + ν h¯
v 2F k2 + (λR − μλSO )2
(2)
with μ = ±1 and ν = ±1. We rewrite the energy bands as E n (k) where the index n = 1 ∼ 4 label the bands in the order of increasing energy. It has been known that the graphene model is a semimetal as λ S O < λ R , while it is a topological insulator as λ S O > λ R [24]. The energy band structures for the two states are shown in Fig. 1. In this work, we apply a beam of polarized light perpendicularly to irradiate onto the graphene lattice. The vector potential describing the monochromatic wave is of the form
A (t ) = A 0 (cos φ cos ωt , sin φ sin ωt )
(3)
with A 0 = E 0 /ω where ω is the frequency of the light and E 0 is the amplitude of the electric field component. Here the parameter φ is used to specify the polarization of the light. For φ = 0 (π /2), the optical field represented by the vector potential A (t ) is linearly polarized in the x ( y ) direction. For φ = π /4 (3π /4), it is left (right) circularly polarized. For the general values of φ , the field A (t ) describes elliptically polarized light. The effect of the electromagnetic field on the graphene lattice can be incorporated into the Hamiltonian of Eq. (1) by the minimal coupling substitution h¯ k → h¯ k − e A (t ). The consequential Hamiltonian simply reads H T = H + H 1 , where the interaction term of the graphene model with the light field takes the form
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The injection rate of spin current originating from the intersubband transitions between the valance band v and the conduction band c can be calculated by using the Fermi’s golden rule. It is written as
The first term in Eq. (1) represents the free Hamiltonian in graphene lattice around K valley (τ = 1) or K’ valley (τ = −1), where v F is the Fermi velocity, k = (kx , k y ) is the in-plane wavevector relative to the Dirac point, and σ = (σx , σ y , σz ) are the Pauli matrices acting on the sublattice space. The second term describes the Rashba SOI where λR is the Rashba coupling parameter controlled by an external electric field or substrate material and S = ( S x , S y , S z ) are the Pauli matrices for electrons’ real spin. The last term in Eq. (1) describes the intrinsic SOI in the graphene model where λSO is its strength, which could be modified in graphene via adatom deposition [26]. The energy spectrum of the graphene model can by obtained by diagonalizing the above Hamiltonian. It consists of four energy bands, which are written as
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where the velocity operator is given by vˆ = h¯ −1 ∂ H T /∂ k, the subscript i = x, y , z represent the spin components and j = x, y stand for the transfer directions of spin carriers. The spin current for a given state |nk is defined by
2. Model and formulation
+ h¯ λSO τ σz S z .
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J i j ,n = nk| ˆJ i j |nk.
H = h¯ v F (τ σx kx + σ y k y ) + h¯ λR (τ σx S y − σ y S x )
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(2π )2 × δ( E ck − E vk − h¯ ω) f ( E vk ) [1 − f ( E ck )],
I i j (ω) =
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(7)
(8)
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Considering the symmetry of the graphene model, we can obtain that only one independent nonzero component of the spin current exists: I xy = − I yx , while the others are zero. Hereafter we calculate the spin current injection rate I xy along y axis with the spin polarized along x direction. Furthermore, we emphasize that the electron with state |k, ↑ and its counterpart with state | − k, ↓ are pumped by the light with equal probability due to the symmetry of band structures. The pair has opposite velocity such that it contributes null charge current. Consequently, the photo-induced spin current in our system is pure. We study the undoped graphene model at zero temperate. In such circumstances, there exist four kinds of valance-to-conduction transition contributed to the total spin-current injection, which are (1) | E 2 → | E 3 , (2) | E 2 → | E 4 , (3) | E 1 → | E 3 and (4) | E 1 → | E 4 (shown in Fig. 1). The four contributions are written in order as
(ω + 4λ R )(a + bη) , I 1 (ω) = I 0 (ω) (ω + 2λ R )[(ω + 2λ R )4 − 16λ2S O λ2R ]
I 3 (ω) = − I 0 (ω)
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I ivj→c (ω).
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where E v (c )k is the eigenvalue of the valance (conduction) band v (c ). f ( E v (c )k ) is the Fermi distribution function. The total injection rate of the spin current is the sum of all allowed transitions
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8(λ R − λ S O ) (2 + η)
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ω
I 4 (ω) = − I 1 (ω, λ R → −λ R )
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with a = 2ω[(ω + 2λ R )3 + 8λ R λ2S O ], b = (ω + 4λ R )[(ω + 2λ R )3 − 8λ R λ2S O ], and I 0 (ω) = E 02 e 2 v F /(16h¯ ω) including the valley degeneracy. Here we define the degree of the polarization of light η = (Ix − I y )/(Ix + I y ) = cos 2φ , where Ix (I y ) is the intensities of linear polarized component in the x ( y ) direction. η = ±1 denote the linear polarized radiation in the x and y directions, respectively. It should be emphasized that I i (ω) (i = 1 ∼ 4) have the same formations for the semimetal and topological insulator
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states. The detailed derivation of Eqs. (9) is presented in the Appendix. In the case of the semimetal state, i.e., λ S O < λ R , one obtains the total spin-current injection rate
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I xy (ω) = I 1 (ω) + I 2 (ω) [ω − 2(λ R − λ S O )]
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where (x) is the Heaviside step function. Nevertheless, the total spin-current injection rate for the case of the topological insulator state (λ S O > λ R ) is given by
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+ I 3 (ω) [ω − 2(λ R + λ S O )] + I 4 (ω) (ω − 4λ R ),
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I xy (ω) = [ I 1 (ω) + I 2 (ω)] [ω − 2(λ S O − λ R )]
+ [ I 3 (ω) + I 4 (ω)] [ω − 2(λ S O + λ R )].
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ω + 4λ R O 1 (ω) = O 0 (ω) , ω + 2λ R O 2 (ω) = O 0 (ω) O 3 (ω) = O 0 (ω) O 4 (ω) = O 0 (ω)
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(11)
These are the main results of this paper. By now let us analyze the above expressions for the spin current injection. It is obvious that the injection rate of spin current is dependent on the polarization of light. Moreover, the spin current injection does not vanish as the light is perfectly circular polarized, which differs from the result of Rioux and Burkard’s work [21]. Actually the spin current injection rate for the circular polarized light is the sum of those for x and y-linear polarized light, i.e., I xy ,c = ( I xy ,x + I xy , y )/2, where the third subscript letters c, x and y stand for the circular, x and y-linear polarization respectively. This is because a circular polarized light can be decomposed as a combination of x and y-linear polarized beams of light. Now let us focus on the sign of the four contributions of the spin current injection. We note that the sign of the first contribution I 1 is opposite to that of the other three as the graphene model is in the semimetal state. In other words, the spin current injection I 1 from the subband optical transition | E 2 → | E 3 will partly canceled out by the contributions of the other subband transitions. As the topological phase transition of the graphene model from the semimetal (λ S O < λ R ) to the topological insulator (λ S O > λ R ) takes place, the second contribution I 2 reverses its sign due to the band inversion, while the others remain their signs unchanged. It means that the spin current injection I 1 from the transition | E 2 → | E 3 and I 2 from the transition | E 2 → | E 4 are constructive. So it is expectable that the spin current injection rate will be enhanced when the graphene model is in the topological insulator state. We will show the conclusions in the next section. We also calculate the associated light absorption using the Fermi’s golden rule. The expressions for the total light absorption O t (ω) is similar to that of the spin current injection rate Eq. (10) and Eq. (11). The four contributions of light absorption are given by
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(12b) (12c) (12d)
where O 0 (ω) = E 02 e 2 /(4h¯ 2 ω) including the spin and valley degeneracy. It is clear that the light absorption is independent of the light polarization. These expressions for the light absorption reduce to the form given by the previous work as the intrinsic SOI is absent [21]. In addition, we know that the robust edge states emerge at the boundary of sample as the graphene lattice is in the state of the topological insulator [24]. The edge states do contribute to the optical absorption and spin current injection. However, in this work,
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Fig. 2. (Color online.) (a) Spin-current injection rate J xy as functions of light frequency for circular (c) and linear (in the x and y directions) polarizations when the graphene model is in a semimetal state. (b) Spectra of the total light absorption and its intersubband contributions. Here we set λ S O = 0.
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the proportion of the optical excitation resulting from the edge states is ignorable due to the large area of the sample. It could be inferred that the effect of the edge states on the optical excitation is significant as the graphene in the topological state is cut into the nanoribbons.
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3. Results and discussion
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Now we come to show some numerical results about the spincurrent injection rate and the associated light absorption in the graphene model. In our calculations we set the Rashba frequency as a unit. It should be noted that our hexagonal lattice model could be realized on a graphene or silicene sheet. In experiments, the Rashba and intrinsic SOI strengths in graphene and silicene can be enhanced to the order of magnitude of meV [30–32]. The corresponding light irradiating on the graphene sheet should be terahertz.
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3.1. In the semimetal state
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First, let us consider the graphene model in the state of semimetal, i.e., the case of λ S O < λ R . In Fig. 2 (a), we plot the spin-current injection rate as functions of the light frequency for the linear and circular polarizations. It clearly demonstrates that the spin-current injection rate remarkably depends on the light frequency and polarizations. To explain this phenomena, we show the corresponding total optical absorption and its four individual contributions in Fig. 2 (b). It is found that the major contributions of the optical absorption originate from the interband transitions between the top of the valence bands and the bottom of the conduction bands. So does the spin-current injection rate, when the light frequency ω < 2λ R , it only exits the first valance-toconduction transition: | E 2 → | E 3 , while the other transitions are forbidden due to the energy gap. It should be noted that, in this condition, the rate of the spin-current injection is positive for the linear polarization in the x-direction, while it is negative for that in the y-direction as shown in Fig. 2 (a). As the frequency increase over 2λ R , the second and third transitions emerge which result in the spin-current injection rate changing dramatically. Especially the spin-current change its direction for the circular and x-directional linear polarization since the direction of the spincurrent from the first transition is opposite to the others (see
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Fig. 3. (Color online.) Linear dependence of spin-current injection rate I xy on the degree of polarization of light for different frequency when the graphene model is in a semimetal state. We set λ S O = 0.
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Eqs. (9)). As the frequency increases, all of the four transitions will take part in such that the total spin current approaches to vanish. Fig. 3 displays the spin-current injection rate as a function of the degree of polarization of light for different radiation frequency. It explicitly reveal its linear dependence on the degree of polarization which also can be concluded from Eqs. (9). Furthermore, when there is only the first interband transition | E 2 → | E 3 (ω = 0.1λ R in Fig. 3), the variation of the spin-current injection as changing the polarization is the most remarkable. The spin-current injection rate reaches its maximum value 4.0I 0 (ω) as the light is perfectly linearly polarized in x direction. In the most situations with a fixed frequency, the direction of the photo-induced spin-current can be switched by modulating the degree of the polarization.
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Fig. 4. (Color online.) (a) Spin-current injection rate I xy as functions of light frequency for circular (c) and linear (in the x and y directions) polarizations as the graphene model becomes a topological insulator. (b) Spectra of the total light absorption and its intersubband contributions. Here we set λ S O = 2λ R .
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3.2. In the topological insulator state
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Now we turn to the topological insulator state, i.e., the case of λ S O > λ R . Fig. 4 (a) exhibits the variation of the spin-current injection rate with respect to the light frequency for the linear and circular polarizations as λ S O = 2λ R . The corresponding optical absorption is shown in Fig. 4 (b). It is also found that the direction and the injection rate of the spin-current can be effectively controlled by changing the light frequency and its polarization. However, this is quiet different from the results for the case of the semimetal state. The spin current injection rate in the case of the topological insulator state (in the frequency range [2λ R , 6λ R ]) is much larger than that in the semimetal state when the linear polarized light along x-direction is applied. Moreover, the photo-induced electron transition does not arise until the light frequency exceeds the energy gap (ω > 2λ R in Fig. 4). When the frequency varies in the range [2λ R , 6λ R ], there are two kinds of valance-to-conduction transition contributions, i.e., the first transition | E 2 → | E 3 and the second | E 2 → | E 4 . Both of this two contributions to the spin-current are positive such that the total spin-current is constructive and always positive regardless of the light polarization. As the light frequency is larger than 6λ R , the other two transitions arise, which is destructive to the spincurrent produced by the previous contributions. Consequently, the total spin-current rapidly becomes faint and changes its sign. Fig. 5 plots the dependence of the spin current injection rate on the degree of polarization of light for different radiation frequency when λ S O = 2λ R . It is noted that the linear dependence of the spin current injection rate is distinct from that for the semimetal graphene. When it exits only the first two interband transitions: | E 2 → | E 3(4) (ω = 2.1λ R in Fig. 5), the spin current injection is always positive and achieves its maximum value 7.0I 0 (ω) as the light is linear polarized in the x-direction. In fact, we can obtain
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Fig. 5. (Color online.) Linear dependence of spin-current injection rate I xy on the degree of polarization of light for different frequency as the graphene model becomes a topological insulator. We set λ S O = 2λ R .
the maximal injection rate of the spin current (6 + 3λ R /λ S O ) I 0 (ω) according to Eq. (11) as ω = 2(λ S O − λ R ) and η = 1. The maximal injection rate for the topological insulator state is relatively greater than that for the semimetal state. As the frequency increase over 6λ R , all of the four transitions emergent. Consequently, the dependence of the spin-current injection rate on the degree of polarization becomes not noticeable. By the way, let us briefly discuss another case: the topologically trivial insulator state. It is found that the graphene lattice model without the intrinsic SOI is a topologically trivial insulator when a staggered sublattice potential is introduced in the model [24,25]. The velocity of electrons is decreased by the energy gap which is opened by the staggered sublattice potential. Consequently, the spin current injection will suppressed in the conventional insulator state.
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4. Summary
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We have studied the photo-induced spin current in graphene lattice model with intrinsic and Rashba SOIs. It is found that the spin-current injection rate remarkably depends on the degree of
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polarization of light, frequency and the states of the graphene lattice. The spin current injection could be enhanced by the intrinsic SOI, which is more prominent as the graphene model is in the topological insulator state than in the semimetal state. Besides, the spin current injection rate shows linear relationship with the degree of polarization and reaches a maximum when the light is perfectly linearly polarized. Furthermore, the direction of the spincurrent can be switched by changing the polarization of light as the irradiation with appropriate frequency. The injection of spin current could be detected via electrical currents [18], spin accumulation [33,34], or pump-probe techniques [35]. We ignore the effects of the edge states and the spin current relaxation in this work, which are interesting and needed further investigation in the future.
15 16 17 18 19 20 21 22 23
28 29 30 31
This work was supported by the Fundamental Research Funds for the Central Universities (Grant No. 2013B00514), the National Natural Science Foundation of Jiangsu (Grant No. BK20140839) and the National Natural Science Foundation of China (Grant Nos. 11347127, 61404044, 11404091). Appendix A. The derivation of the spin current injection rate The procedures of the derivation of Eqs. (9) for the spin current injection are similar. For brevity, we present a detailed derivation of Eq. (9b) as an example. First, for the spin current injection rate I xy originating from the transition | E 2 → | E 4 , Eq. (7) is rewritten as
32 33 34
2→4 I xy (ω) =
35
∞
e 2 E 02 2π h¯ ω
36
2π
2 dθ k v x,42 cos φ − i v y ,42 sin φ
dk
2 0
0
× J xy ,4 − J xy ,2 δ( E 4 − E 2 − h¯ ω) f ( E 2 )[1 − f ( E 4 )].
37
(A.1)
38 39 40 41 42 43 44 45 46 47 48 49
Here the matrix elements of the velocity operators vˆ = v F (σx , σ y ) for the K valley, written in the eigenstate basis {| E 1 , | E 2 , | E 3 , | E 4 }, take the forms
v F (λ R − λ S O ) cos θ v x,42 = E 4 | vˆ x | E 2 = , v 2F k2 + (λ R − λ S O )2
(A.2)
v F (λ R − λ S O ) sin θ v y ,42 = E 4 | vˆ y | E 2 = . v 2F k2 + (λ R − λ S O )2
(A.3)
50
The diagonal matrix elements of the spin current operator ˆJ xy
51
written in the same eigenstate basis are given by
52 53
J xy ,2(4)
54 55 56 57
2→4 I xy (ω) = I 2 (ω) (ω − E 2 )
=
h¯ v F −2(λ R − λ S O )( E 2(4) /h¯ − λ S O ) +
v 2F k2 (cos 2θ
4 v 2F k2 + (λ R − λ S O )( E 2(4) /h¯ − λ S O )
− 1)
, (A.4)
67 68 69 70 71 72 73
(A.5)
where I 2 (ω) is just the Eq. (9b) and E 2 is the energy gap between the subbands | E 2 and | E 4 which equals 2(λ R − λ S O ) for the case λ R > λ S O or 2(λ S O − λ R ) for the case λ S O > λ R . In the same way, the others in Eqs. (9) can be derived. By adding up the four parts, one gets the Eqs. (10) and (11) for the total spin current injection rate. Equations (12) for the light absorption can be obtained by using the analogous approach. References
26 27
where the eigenenergy E 2(4) should be expressed in the polar coordinate (k, θ). By now we substitute the Eqs. (A.2)∼(A.4) into Eq. (A.1), let f ( E 2 )[1 − f ( E 4 )] = 1 for the undoped graphene model at zero temperate and then integrate the Eq. (A.1). After simplification one obtains
Acknowledgements
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5
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