Ultrahigh spin thermopower and pure spin current in a single-molecule magnet

Using the non-equilibrium Green's function (NEGF) formalism within the sequential regime, we studied ultrahigh spin thermopower and pure spin current in single-molecule magnet(SMM), which is attached to nonmagnetic metal wires with spin bias and angle (θ) between the easy axis of SMM and the spin orientation in the electrodes. A pure spin current can be generated by tuning the gate voltage and temperature difference with finite spin bias and the arbitrary angle except of . In the linear regime, large thermopower can be obtained by modifying Vg and the angles (θ). These results are useful in fabricating and advantaging SMM devices based on spin caloritronics.

Using the non-equilibrium Green's function (NEGF) formalism within the sequential regime, we studied ultrahigh spin thermopower and pure spin current in single-molecule magnet(SMM), which is attached to nonmagnetic metal wires with spin bias and angle (h) between the easy axis of SMM and the spin orientation in the electrodes. A pure spin current can be generated by tuning the gate voltage and temperature difference with finite spin bias and the arbitrary angle except of h~1 2 p, 3 2 p. In the linear regime, large thermopower can be obtained by modifying V g and the angles (h). These results are useful in fabricating and advantaging SMM devices based on spin caloritronics.
S tudies on nanoscale thermoelectric devices have attracted much attention during the past a few years [1][2][3][4] . It is well accepted that nanoscale materials may provide an opening for the thermoelectricity in meeting the challenge of being a sustainable energy source 5 . Huge deviation from the Wiedemann-Franz law 6,7 in the nanostructure materials 5 makes new opportunities for investigating novel thermoelectric devices with high efficiency 8,9 . Specially, spin caloritronics (spin Seebeck effect) was observed by Uchida et al 10,11 . They found that the spin-polarized currents (I : and I ; ) can be induced by a temperature gradient and flow in opposite directions. These wonderful discoveries strongly promote research on new energy of thermoelectricity 12,13 .
A single-molecule magnet (SMM) is a typical nanoscale material. In experiments, controlling the molecular spin 14 and measuring thermopowers of molecule [15][16][17][18][19][20] have been realised by directly using a scanning tunneling microscope. The spin-dependent transport properties, such as tunneling magnetoresistance(TMR) and spin Seebeck effect, were investigated in the sequential, cotunneling, and Kondo regimes using Wilson's numerical renormalization group and quantum master equation [21][22][23][24][25] . Many fantastic phenomena have been found in the experimental and theoretical studies, including negative differential conductance 26,27 , Berry phase blockade 28 , the magnetization of SMM controlled by spin-bias and thermal spin-transfer torque 29 . A SMM in a single spin state is necessary for generating the pure spin current 21,22,29 without the magnetic field or magnetic electrodes. Meanwhile, it implies that the system temperature is limited by the blocking temperature of SMM (T B ). When the symmetry of spin in the leads is broken, the angle (h) between the easy axis of SMM and the spin orientation in the electrodes will influence the transport properties in the SMM devices. Specially, spin-bias 30,31 and this angle (h) are important and crucial on thermoelectric effect.
In this paper, we theoretically investigate the thermoelectric effects of a sandwich structure of NM/SMM/ NM with spin-bias 29,32,33 and angles (h) between the easy axis of SMM and the spin orientation in the electrodes. We show that, in this system, pure spin currents are observed even though the system temperature is higher than the blocking temperature due to the spin symmetry broken by spin bias. In the linear regime, both thermopower and figure of merit are dependent on the angle and spin bias. It's worth noting that the angle plays a critical role on generating spin thermopowers. The figure of merit could tend to infinity by tuning the voltage gate at special angles, which implies that this system has an ultrahigh thermoelectric efficiency.

Results
Effective hamiltonian. The general Hamiltonian is expressed as 29 H leads describes the free electrons in two leads, with c { a,s 0 k c a,s 0 k ð Þbeing the creation (annihilation) operator for a continuous state in the a[L, R lead with the energy j a,s 0 k and spin index s 0~z { ð Þ, which denotes spin-majority (spin-minority) electrons. In this paper, wideband approximation is adopted and the density of states of the leads does not depend on the energy of the two leads. The chemical potential of a lead is defined as m a s 0~+ eV a =2z ð the voltage and V s~VsL {V sR is the spin voltage. P a denotes the polarization of a lead and is defined as is the creation (annihilation) operators for the LUMO. 0 is the single-electron energy of the LUMO level, which is tuned by a gate voltage V g . U is the on-site Coulomb repulsion. J describes the Hund's rule coupling between the giant spin S of SMM and the electron spin in the LUMO, and parameter K 2 is the easy-axis anisotropy of SMM. H t describes the tunneling between the LUMO of SMM and the electrodes, and h a denotes the angle between the spin orientation of lead-a and the easy-axis of the SMM (as z-axis).
In the following, we turn to numerical calculations with parameters: S 5 2, J 5 0.1 meV, K 2 5 0.04 meV, U 5 1.0 meV and 0~0 :2 meV. The tunneling parameters are set to C~C L~CR0 :001 meV. The properties of the leads are set to P R 5 P L 5 0. Conventionally, I c and I s are defined as charge current and spin current respectively, and we set e~k B~ h~1 and the thermopower and current are scaled in the unit of I 0~1 2 C. We can find all of the thermopowers are symmetric about h 5 p because of the spatial symmetry of the sandwich structure.
Transport properties. First, we consider that the left electrode is nonmagnetic with V s 5 0.01 meV and V 5 0 meV. The system temperature is lower than the anisotropy-induced energy barrier K 2 S 2~0 :16meV À Á . Figure 1a and Figure 1b show I c and I s as function of h for different values of V g with DT 5 0.0002 meV, respectively. In this case, we can find that I s is almost ten times of I c . The maximum or minimum value of I c and I s depends on the gate voltage V g and h, but the positions of these extremums only depend on the V g . When h~p=2,3p=2, I s is exactly equal to zero due to the coefficient A L s,s 0~1=2, which leads I : :I ; . Moreover, we show I c and I s as a function of V g with two types of DT at h 5 0 in Figure 1e. One can find that I c is extremely sensitive to the temperature difference. However, I s only has a little change. Conventionally, the Fermi-Dirac distributions of spin-up and spin-down electron in the electrode are different due to finite V s . The higher the temperature is, the less these differences are generated.
In Figure1c and 1d, we show I c and I s as a function DT for different h at V g 5 0.1 meV and T 5 0.02 meV, respectively. In this case, interesting phenomena can be observed. When h 5 0, p, 2p, I c first increases and then decreases with decreasing of DT, but it decreases monotonically when h~p=2. However, I s changes monotonically with decreasing DT and equals to zero at h~p=2,3p=2.
Thermoelectric coefficients. Next, we focus on the thermopower phenomena in the linear response regime and assume spin-bias only exists at the left electrode. Figure 2a and Figure 2b display charge-Seebeck and spin-Seebeck coefficient as a function of h for different values of V g respectively. At the point V g 5 0.1 meV, S c and S s are zero due to the same weight and the opposite transmission direction of the currents and energy carried by electrons and holes. At the special points h 5 0, 2p, the contributions of the spin-up and spin-down electrons are the same weight leading to G :~G; and S s 5 0, no matter spin bias exists or not. S c and S s reach the maximum values for each V g when h 5 p. In Figure 2c, we plot all transport coefficients as a function of h with V g 5 20.05 at T 5 0.02. One can observe that electron thermal conductivity (k e ) is zero at special h, which may cause figure of merit Z c T or Z S T tend to be infinite. Furthermore, we plot the k e as function of V g and h at T 5 0.02 meV and find that zero k e exists only under special conditions of V g and h in Figure 2d. It is interesting that one can have ultrahigh spin thermopower in a single molecular magnetism through manipulating the angle h between the easy axis of SMM and the spin orientation of the electrodes, and also by tuning V g .
Thermoelectric coefficients have been investigated through solving the non-equilibrium Green's function (NEGF) in detail. The general formulas are derived to calculate the currents which depend on the angle h between the easy axis of SMM and the spin orientation in the electrodes and spin bias. The spin bias destroys the SU(2) symmetry of electron-spin in nonmagnetic electrodes, which leads the fact that the angle h influences the redistributions of different spin currents. It is amazing that pure spin currents can be obtained by tuning V g and DT with a finite spin bias and arbitrary h except of h~1 2 p, 3 2 p. In the linear regime, infinite figure of merit can be generated by tuning V g at special angles h with spin bias. Specially, when the angle h is equal to zero or 2p, spin thermopowers vanish identically even though spin bias exists. These phenomenons may provide a new approach for the design of SMM devices based spin caloritronics.

Discussion
The details of the constituents of the currents at h 5 0 are shown in Figure 1f. When V g is equal to 0.1 meV, the lowest-energy states of the isolated SMM are four-fold degenerate: 5=2 : +5=2 j i and 0,2 : +2 j i , and the second-lowest-level state is a double degeneracy: 1,3=2 : +3=2 j i { . The energy level difference between the lowest and next-lowest levels is 0.0839 meV. The currents are mainly contributed by the transitions of 3=2 : +3=2 j i { u 0,2 : +2 j iand 1,5=2 : j +5=2iu 0,2 : +2 j i . According to Eq. (9), we can approximate  0.27 meV. But I S only has little change and is not equal to zero. Due to the spin splitting induced by spin bias, pure spin currents can be generated at arbitrary temperatures. In Figure 1h, the system temperature is five times as the anisotropy-induced energy barrier and pure spin currents can be obtained by increasing V sL .
Interestingly, the numerical results show k e can be equal to zero with changing V g and h in Figure 2d. It means that Z s T(Z c T) may be infinite when S c (S s ) and G s s~:; ð Þare finite. The exact choices of V g and h are related to the details of the system's parameters. But it is necessary that S s must be larger than S c . It is well-known that electrons move from high temperature to low temperature, and it is not related to electronic spin. However, the spin-up and spin-down electrons under the spin bias can move in the opposite direction, and carry different energy according the Eq.10. Due to the competition between the temperature difference(DT) and spin bias (V sL ), and the scattering between spin-up and spin-down electrons induced by the nonlinear spin exchange, it is possible that there are non-zero amount for thermopower and electrical conductance when thermal conductivity k e is zero.

Methods
Non-equilibrium Hubbard Green function has been used to solve the thermoelectric transport in the sequential and linear response regime 35 By using the Dyson equation and the Keldysh forum, the retarded(advanced) and the lesser(greater) Green's function can be compactly expressed as respectively Here, S r=a=v leads,s are the electron self-energy in the second-order approximation and the formulas for calculation are whereT s v ð Þ denotes the transmission coefficient of spin-s electrons Here, S c and S s denote the charge Seebeck and spin Seebeck respectively. G s is selectron conductance. k e is the conventional thermal conductance. DI s is thes-electric current induced by a temperature difference at zero voltage bias and zero spin bias. DT~T L {T R is defined as temperature difference and the average temperature is expressed asT~T L zT R ð Þ =2. Finally, The spin figure of merit Z s T~G : {G ; À Á S 2 s T k and charge figure of merit Z c T~G : zG ; À Á S 2 c T k can be calculated. k~k e zk ph is the thermal conductance with contributions from both electrons k e ð Þand phonons k ph À Á [43][44][45] . In our model, the phonon transport is not considered due to large mismatch of vibrational spectra between the SMM and leads.