Anisotropic spin-dependent thermopower and current in ferromagnetic graphene junctions

Physics Letters A 378 (2014) 73–76

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Physics Letters A www.elsevier.com/locate/pla

Anisotropic spin-dependent thermopower and current in ferromagnetic graphene junctions Zhi Ping Niu ∗ College of Science, Nanjing University of Aeronautics and Astronautics, Jiangsu 210016, China

a r t i c l e

i n f o

Article history: Received 15 July 2013 Received in revised form 6 October 2013 Accepted 23 October 2013 Available online 25 October 2013 Communicated by R. Wu Keywords: Anisotropic spin thermopower Pure spin current Two-dimensional graphene

a b s t r a c t We study the spin-dependent thermoelectric transport through two-dimensional normal/ferromagnetic/ normal/ferromagnetic/normal graphene (NG/FG/NG/FG/NG) junctions. It is found that both charge and spin thermopowers depend on the FG’s magnetization direction and exhibit an anisotropic behavior. Interestingly, the spin thermopower can be as large as the charge thermopower and even can exceed the latter in magnitude. Moreover, the pure spin thermopower and spin current emerge in this device. The results obtained here suggest a feasible way of enhancing thermospin effects and generating the pure spin current in two-dimensional graphene. © 2013 Elsevier B.V. All rights reserved.

1. Introduction Spin caloritronics, the combination of thermoelectrics and spintronics, focusing on heat and spin transport, has attracted much interest [1–13]. A notable recent discovery of spin caloritronics is the observation of spin Seebeck effect by Uchida et al. [1], which further inspires widely experimental and theoretical investigations. The spin Seebeck effect was observed in various materials ranging from the metallic ferromagnets Co2 MnSi [2] to the semiconducting ferromagnet (Ga,Mn)As [3], and even in the insulating magnets LaY2 Fe5 O12 [4] and (Mn,Zn)Fe2 O4 [8]. Jia and Berakdar [6] also theoretically investigated the anisotropic charge and spin thermopower across normal-metal/helical-multiferroic/ferromagnetic heterojunctions. However, the spin thermopower in these bulk samples is so weak that it may be overwhelmed by the accompanied charge thermopower of several orders larger. In addition, the temperature difference unavoidably generates also a regular voltage bias across the sample, which may preclude easy applicability in spintronic devices. On the other hand the spin-dependent thermoelectric properties of graphene have also attracted a great attention [14–19]. Thermally driven spin-polarized currents through a magnetized zigzag graphene nanoribbon(ZGNR)-based device have been reported in Refs. [14–16]. Zeng et al. [17] found a very large multivalued and controllable thermal magnetoresistance and charge Seebeck effects in a spin valve which consists of ZGNR electrodes with different magnetic configurations. Cheng [18] investigated the

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spin thermopower and thermoconductance properties in a ferromagnetic zigzag edged graphene nanoribbon. The results indicate that two mechanisms, Klein tunneling and the band selective rule, are involved to determine the thermoelectric properties. However, the results obtained for such heterostructures were shown to be very sensitive to the device geometry, e.g., the device length and the disorder effects [14–16,18,20]. So, their use for realistic applications is not realized till now due to the extreme difficulty in the experimental implementation. In this view, besides better understanding the thermoelectric transport in graphene quantum structures, the investigation of the spin-dependent thermopower based on two-dimensional (2D) graphene is desirable for the development of such materials in spin caloritronics. In this paper we study the spin-dependent thermoelectric transport through 2D normal/ferromagnetic/normal/ferromagnetic/ normal graphene (NG/FG/NG/FG/NG) junctions. Both charge and spin thermopowers depend on the magnetization direction of the right FG and become strongly anisotropic. The spin thermopower can be as large as the charge thermopower and even can exceed the latter in magnitude. Moreover, the pure spin thermopower and spin current can be obtained in this device. 2. Model and formulation The device we consider here is a 2D NG/FG/NG/FG/NG junction with a temperature bias  T = T L − T R , where T L (T R ) is the temperature in the left (right) lead (Fig. 1(a)). The graphene is parallel to the x– y plane. The NG/FG (or FG/NG) interfaces locate at x = 0, L, 2L, and 3L, respectively, where the x axis is chosen to be perpendicular to the interface. The ferromagnetism in the FG regions can be induced by doping and defect or putting

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Z.P. Niu / Physics Letters A 378 (2014) 73–76

I σ = eL 0σ μσ +

e T

L 1σ  T ,

(3)

where μσ = e  V σ is the difference in the chemical potentials of the two leads in the spin-σ channel, and  V σ = ( V e + 12 σ  V s )

with  V e = [ 12 (μ R ↑ + μ R ↓ ) − 12 (μ L ↑ + μ L ↓ )]/e the charge bias. As the transmission is spin dependent, the temperature gradient may lead to a spin accumulation in the leads, which generally results in a nonzero spin-voltage bias  V s = (δ μ R − δ μ L )/e, where δ μi = δ μi ↑ − δ μi ↓ (i = L , R) with δ μi σ the electrochemical potential of the spin-σ channel in the i lead. The coefficients L nσ (n = 0, 1, 2)   ∂ f (E) are defined as L nσ = h1 dE ( E − E F )n N ( E ) dφ T σ cos φ[− ∂ E ], where f ( E ) is the Fermi–Dirac distribution function. Thus, one can introduce the spin-dependent thermopower S σ by taking I σ = 0 [6,24,25],

Sσ =

Fig. 1. (a) A schematic diagram of a 2D NG/FG/NG/FG/NG junction with a temperature bias. (b) Dispersion relation of the device in the parallel (antiparallel) magnetization configuration.

a magnetic insulator bar onto a graphene sheet [21,22]. The mag (r) = h(sin θ, 0, cos θ) with netization vector in the FG regions is h h the magnitude of exchange field. The magnetization in the left FG is assumed along the z axis, i.e., θ = 0; while that in the right FG orients along the (sin θ, 0, cos θ) direction, which can be controlled by a weak external magnetic field. The Hamiltonian of such a structure simply reads [23]



 ˆ = v F σ0 ⊗ (τ · p ) + σ · h (r) ⊗ τ0 , H

(1)

 = where v F ≈ 10 m/s is the Fermi velocity, τ = (τx , τ y ) and σ (σx , σ y ) are Pauli matrices in pseudospin and spin space, and σ0 6

and τ0 are unit matrices. Dispersion relation of the device in the NG and FG regions is illustrated in Fig. 1(b). Consider a spin-σ (σ =↑, ↓, or ±) electron incident from the left lead on the NG/FG interface at an angle φ to the interface normal. The wave functions are given as follows. In the left and right leads Ψ L = ψσ+ + r↑σ ψ↑− + r↓σ ψ↓− , and Ψ R =

t ↑σ ψ↑+ + t ↓σ ψ↓+ ; In the middle NG region Ψ M = p ψ↑+ + qψ↓+ +

mψ↑− + nψ↓− . ψσ+ and ψσ− are ψ↑δ = (1, δ e δ i φ , 0, 0) T e δ ikx and ψ↑δ = (0, 0, 1, δ e δ i φ )T e δ ikx , where the superscript T denotes transposition and δ = ±1. In the left (right) FG region we have Ψ F = a L ( R ) ψ F+↑ + b L ( R ) ψ F+↓ + c L ( R ) ψ F−↑ + d L ( R ) ψ F−↓ , where the wave functions are

ψ Fδ ↑ = (cos(θ1 ), δ cos(θ1 )e i δφ↑ , sin(θ1 ), δ sin(θ1 )e i δφ↑ )T e i δk↑ x and ψ Fσ↓ = (− sin(θ1 ), −δ sin(θ1 )e i δφ↓ , cos(θ1 ), δ cos(θ1 )e i δφ↓ )T e i δk↓ x with θ1 = 0 (θ/2) in the left (right) FG region. The wave vec tors are k = E cos φ/¯h v F , kσ =

( E + σ h)2 − (¯h v F k y )2 /¯h v F , and k y = E sin φ/¯h v F = ( E + σ h) sin φσ /¯h v F . By matching the wave functions at the NG/FG interfaces, we can obtain the reflection and transmission coefficients r↑σ and t ↑σ (t ↓σ and r↓σ ) in the spin-up (spin-down) channel. The current can be given as

Iσ =

e h





dE N ( E )





dφ cos φ T σ f L ( E ) − f R ( E )

(2)

with the transmission probability in the spin-σ channel T σ = |t σ σ |2 + |t σ σ¯ |2 , N ( E ) = | E +h¯ vVF|W and W the width of the graphene sheet. f L ( R ) ( E ) = {1 + exp[( E − E F )/k B T L ( R ) ]}−1 stands for the Fermi distribution function in the L (R) lead and E F is the Fermi energy. Utilizing the linear response assumption, i.e., T L ≈ T R = T , the spin-dependent current can be rewritten as

V σ 1 L 1σ =− . T eT L 0σ

(4)

The charge thermopower S c and the spin thermopower S s are calculated as

Sc =

1 2

( S ↑ + S ↓ ),

(5)

( S ↑ − S ↓ ).

(6)

and

Ss =

1 2

3. Results and discussions We perform numerical calculations for h = 5 meV, L = 500 nm and T = 10 K. In Fig. 2(a), the charge thermopower S c and spin thermopower S s are plotted as a function of the Fermi energy E F . As shown in Fig. 2(a), S c in both parallel θ = 0 and antiparallel θ = π magnetization configurations vary rather sharply in the symmetry point E F = 0, where S c changes sign and reaches the maxima on one side of E F = 0 and the minima on the other side. When E F is far away from zero, | S c | decreases with E F . However, in contrast to S c , S s strongly depends on the magnetization configuration. For the antiparallel magnetization configuration S s is always zero (dash dot line in Fig. 2(a)). While for the parallel configuration S s is an even function of E F and can reach a minimum at E F = 0, where S c is zero, so we can obtain a pure spin thermopower. It should be pointed that S s can be as large as S c and even can exceed the latter in magnitude. This is very different from the results in Refs. [1–6], where S s is so weak that it may be overwhelmed by the accompanied S c of several orders larger. In order to explain the underlying physics, we plot the spindependent thermopower S σ in the parallel magnetization configuration as a function of E F in Fig. 2(b). It is found that S σ has the following symmetry S σ ( E F ) = − S σ¯ (− E F ). At E F = 0, we obtains S ↑ (0) = − S ↓ (0), so a pure spin thermopower but no charge counterpart can be generated. The behaviors of S σ can be understood from the dispersion relation of the device we consider (Fig. 1(b)). For the parallel magnetization configuration, the dispersion relations for the spin-up and spin-down channels are symmetrical about E F = 0. S σ can be rewritten as S σ = −| S eσ | + S hσ , where −| S eσ | (S hσ ) is the contribution of electrons (holes) with energy near the Fermi energy to S σ . Due to the symmetry of the dispersion relations we observe | S eσ ( E F )| = S hσ¯ (− E F ) and | S eσ¯ (− E F )| = S hσ ( E F ), which leads to the relation S σ ( E F ) = − S σ¯ (− E F ). However, for the antiparallel configuration we can find the thermopower S σ is spin-independent and S s is zero. This can be understand as follows. For the antiparallel configuration, the transport of σ spin from left to right is equal to that for σ¯ spin from right to left by the structure symmetry, so the transmission probabilities have the relation

Z.P. Niu / Physics Letters A 378 (2014) 73–76

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Fig. 2. S c , S s in both parallel and antiparallel magnetization configurations (a) and the spin-dependent thermopower S σ (b) as a function of E F . Fig. 3. S c (a) and S s (b) as a function of θ with different E F .

T σ ( L → R ) = T σ¯ ( R → L ). If T σ ( L → R ) is not equal to T σ ( R → L ), we can obtain an equilibrium current without the thermal (or electric) bias between two leads, which is not physical. So we obtain T σ ( L → R ) = T σ ( R → L ). Thus we have T σ ( L → R ) = T σ¯ ( R → L ) = T σ ( R → L ) = T σ¯ ( L → R ). Due to the symmetry, we can find L nσ is spin-independent, which lead to spin-independent thermopower S σ and zero S s . It should pointed that the results obtained here are different from those in Ref. [18], where a finite S s is found in a zigzag edged ferromagnetic graphene nanoribbon with the magnetization directions of the two leads antiparallel. This is because in Ref. [18] both Klein tunneling and the band selective rule are involved to determine the thermoelectric properties, but in our paper in 2D junction the band selective rule is not involved. As discussed above, the thermopowers, especially S s , are different for the parallel and antiparallel configurations. Fig. 3(a) shows the dependence of S c on θ . At E F = 0 due to the symmetry of the dispersion relations, S ↑ and S ↓ have the same magnitude but opposite signs, so they cancel each other out, leading to zero S c . When E F is not zero, S c shows a strongly anisotropic behavior. This stems from the fact that due to the combined effect of the exchange field and the Dirac dispersion of graphene the transmission probability T σ are strongly dependent on θ , which has been discussed in Ref. [23]. Therefore L nσ becomes anisotropic, which result in an anisotropic S c . However, for large E F (for example, E F = 20 meV), S c slightly depends on θ , which is similar to the result in Ref. [6]. This is because for E F h the spin polarization in the FGs becomes so small that the influence of h on S c is very weak. Next we investigate S s as a function of θ in Fig. 3(b). Sim-

ilar to S c S s is also strongly anisotropic. Interestingly, it is shown that S s can be as large as S c . For E F = 0, we have S ↑ = − S ↓ , so we obtain a pure S s . For small E F S ↑ is negative and play a determine role, so S s can be as large as S c (see Fig. 2(b) for the parallel configuration case). Unlike S c , the magnitude of S s is maximal at zero E F and decreases with the increase of E F . Finally, we study the thermally induced spin-dependent current through a NG/FG/NG/FG/NG junction by applying temperature difference T = 10 K between two NG leads. In order to consider the effect of the thermal bias on the current in detail we take zero electric bias between two leads. The charge current I c = I ↑ + I ↓ is an even function of E F , while the spin current I s = I ↑ − I ↓ is an odd function of E F . As shown in Fig. 4(a), for the parallel configuration at the point of E F = 0, a pure spin current without accompanying charge current is obtained. However, for the antiparallel configuration only I c is found and I s is always zero. As we can see from the inset of Fig. 4(a) for the parallel configuration at E F = 0 I ↑ = − I ↓ , so they can cancel each other out, resulting into a pure spin current. We also find at the point A (B) only the current in spin-down (spin-up) channel is finite, so a 100% spinpolarized current is obtained. The behavior of the spin-dependent current can be understood from the symmetry of the dispersion relations, as discussed above for the spin-dependent thermopower. Fig. 4(b) shows I c as a function of θ with different E F . Similar to S c , for zero E F due to the symmetry of the dispersion relations I ↑ and I ↓ have the same magnitude but move along the opposite directions, so I c is always zero. When E F is not zero, I c increases with E F . This is because with increasing E F the density of states

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antiparallel configuration I s is always zero, which is independent of E F . For θ = 0, a pure spin current without accompanying charge current appears. For E F = 20 meV, because of the weak spin polarization in the FGs I s is almost independent of θ and close to zero. 4. Summary In summary we have investigated the spin-dependent thermopower and current in a two-dimensional NG/FG/NG/FG/NG junction. The charge and spin thermopowers depend on the magnetization direction of the right FG, and exhibit a strongly anisotropic behavior. Interestingly, the spin thermopower can be as large as the charge thermopower and even can exceed the latter in magnitude. When the Fermi energy is zero, a pure spin thermopower but no charge counterpart can be obtained. Moreover, a pure spin current emerges in this device. This device can be realized with current technologies and may have practical use in caloritronics applications and quantum information. Acknowledgements This work is supported by the Fundamental Research Funds for the Central Universities, NO. NS2012123. References

Fig. 4. (a) The charge and spin current in both parallel and antiparallel magnetization configurations as a function of E F . The charge current (b) and spin current (c) as a function of θ . The inset in (a) shows the spin-dependent current in the parallel magnetization configuration.

of the NG leads increase and more electrons carry the current. For fixed E F with increasing θ I c decreases and reaches a minimum at θ = π , then begins to increase. For very large E F , due to the weak spin polarization in the FGs I c is almost independent of θ . As shown in Fig. 4(c), I s is also strongly anisotropic. It is found for the

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