Abstract
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.
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Introduction
Studies on nanoscale thermoelectric devices have attracted much attention during the past a few years1,2,3,4. It is well accepted that nanoscale materials may provide an opening for the thermoelectricity in meeting the challenge of being a sustainable energy source5. Huge deviation from the Wiedemann-Franz law6,7 in the nanostructure materials5 makes new opportunities for investigating novel thermoelectric devices with high efficiency8,9. Specially, spin caloritronics (spin Seebeck effect) was observed by Uchida et al10,11. They found that the spin-polarized currents (
and 
) can be induced by a temperature gradient and flow in opposite directions. These wonderful discoveries strongly promote research on new energy of thermoelectricity12,13.
A single-molecule magnet (SMM) is a typical nanoscale material. In experiments, controlling the molecular spin14 and measuring thermopowers of molecule15,16,17,18,19,20 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 equation21,22,23,24,25. Many fantastic phenomena have been found in the experimental and theoretical studies, including negative differential conductance26,27, Berry phase blockade28, the magnetization of SMM controlled by spin-bias and thermal spin-transfer torque29. A SMM in a single spin state is necessary for generating the pure spin current21,22,29 without the magnetic field or magnetic electrodes. Meanwhile, it implies that the system temperature is limited by the blocking temperature of SMM (TB). When the symmetry of spin in the leads is broken, the angle (θ) between the easy axis of SMM and the spin orientation in the electrodes will influence the transport properties in the SMM devices. Specially, spin-bias30,31 and this angle (θ) 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-bias29,32,33 and angles (θ) 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 as29,34H = Hleads + HSMM + Ht, in which
Hleads describes the free electrons in two leads, with 
being the creation (annihilation) operator for a continuous state in the 
lead with the energy 
and spin index 
, 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 α lead is defined as 
with 
and 
and 
for 
. 
is the voltage and 
is the spin voltage. Pα denotes the polarization of α lead and is defined as 
. HSMM denotes the molecular degrees of freedom, in which 
and 
is the creation (annihilation) operators for the LUMO. 
is the single-electron energy of the LUMO level, which is tuned by a gate voltage Vg. 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 K2 is the easy-axis anisotropy of SMM. Ht describes the tunneling between the LUMO of SMM and the electrodes and θα denotes the angle between the spin orientation of lead- α and the easy-axis of the SMM (as z-axis).
In the following, we turn to numerical calculations with parameters: S = 2, J = 0.1 meV, K2 = 0.04 meV, U = 1.0 meV and 
. The tunneling parameters are set to 
. The properties of the leads are set to PR = PL = 0. Conventionally, Ic and Is are defined as charge current and spin current respectively and we set 
and the thermopower and current are scaled in the unit of 
. We can find all of the thermopowers are symmetric about θ = π because of the spatial symmetry of the sandwich structure.
Transport properties
First, we consider that the left electrode is nonmagnetic with Vs = 0.01 meV and V = 0 meV. The system temperature is lower than the anisotropy-induced energy barrier 
. Figure 1a and Figure 1b show Ic and Is as function of θ for different values of Vg with ΔT = 0.0002 meV, respectively. In this case, we can find that Is is almost ten times of Ic. The maximum or minimum value of Ic and Is depends on the gate voltage Vg and θ, but the positions of these extremums only depend on the Vg. When 
, Is is exactly equal to zero due to the coefficient 
, which leads 
. Moreover, we show Ic and Is as a function of Vg with two types of ΔT at θ = 0 in Figure 1e. One can find that Ic is extremely sensitive to the temperature difference. However, Is 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 Vs. The higher the temperature is, the less these differences are generated.
Ic and Is as a function of θ for different Vg and ΔT with parameters S = 2, 
, J = 0.1 meV, K2 = 0.04 meV, U = 1.0 meV, kB = 1, PL = PR = 0.
In Fig (a) and (b), a tiny temperature difference is considered: 
and T = 0.02 meV. In the (c) and (d), Vg is set to 0.1 meV and the average temperature is fixed: T = 0.02 meV. (e) shows the temperature difference influences on the Ic and IS with θ = 0. (f) displays the details of the constituents of the Ic and Is with θ = 0. (g) displays Ic and Is as a function of T for different Vg with TL = 0.2 meV and Vs = 0.01 meV at θ = 0. (h) shows Ic and IS as a function of TR for different Vs with TL = 0.8 meV and Vg = 0.27 meV at θ = 0. Solid lines denote charge currents and dash dot lines mark spin currents in (g) and (h).
In Figure1c and 1d, we show Ic and Is as a function ΔT for different θ at Vg = 0.1 meV and T = 0.02 meV, respectively. In this case, interesting phenomena can be observed. When θ = 0, π, 2π, Ic first increases and then decreases with decreasing of ΔT, but it decreases monotonically when 
. However, Is changes monotonically with decreasing ΔT and equals to zero at 
.
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 θ for different values of Vg respectively. At the point Vg = 0.1 meV, Sc and Ss 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 θ = 0, 2π, the contributions of the spin-up and spin-down electrons are the same weight leading to 
and Ss = 0, no matter spin bias exists or not. Sc and Ss reach the maximum values for each Vg when θ = π. In Figure 2c, we plot all transport coefficients as a function of θ with Vg = −0.05 at T = 0.02. One can observe that electron thermal conductivity (ke) is zero at special θ, which may cause figure of merit ZcT or ZST tend to be infinite. Furthermore, we plot the ke as function of Vg and θ at T = 0.02 meV and find that zero ke exists only under special conditions of Vg and θ in Figure 2d. It is interesting that one can have ultrahigh spin thermopower in a single molecular magnetism through manipulating the angle θ between the easy axis of SMM and the spin orientation of the electrodes and also by tuning Vg.
Here, we consider spin-bias only exists at the left lead.
(a) and (b)show the Sc and Ss as a function of θ for different Vg respectively. (c) displays the thermopowers as a function of θ with Vg = −0.05 meV at T = 0.02 meV. (d) shows conventional thermal conductance (ke) as a function of Vg and θ at T = 0.02 meV. The other parameters are chosen as same as that in Figure 1.
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 θ 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 θ influences the redistributions of different spin currents. It is amazing that pure spin currents can be obtained by tuning Vg and ΔT with a finite spin bias and arbitrary θ except of 
. In the linear regime, infinite figure of merit can be generated by tuning Vg at special angles θ with spin bias. Specially, when the angle θ is equal to zero or 2π, 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 θ = 0 are shown in Figure 1f. When Vg is equal to 0.1 meV, the lowest-energy states of the isolated SMM are four-fold degenerate: 
and 
and the second-lowest-level state is a double degeneracy: 
. The energy level difference between the lowest and next-lowest levels is 0.0839 meV. The currents are mainly contributed by the transitions of 
and 
. According to Eq. (9), we can approximate 
and 
. At the electron-hole symmetry point, 
are only controlled by VsL and temperature TL.
However, 
depend on the temperatures of the two leads, VsL and the energy-difference between the lowest and the second-lowest state. From Figure 1f, one can find that Is is mainly contributed by 
, but Ic is decided by all transitions. It is amazing that pure spin currents can be obtained at high temperatures. In Figure 1g, it is clear that Ic first increases and then decreases with the increasing of the average temperature T at Vg = 0.27 meV. But IS 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 VsL.
Interestingly, the numerical results show ke can be equal to zero with changing Vg and θ in Figure 2d. It means that ZsT(ZcT) may be infinite when Sc(Ss) and 
are finite. The exact choices of Vg and θ are related to the details of the system's parameters. But it is necessary that Ss must be larger than Sc. 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(ΔT) and spin bias (VsL) 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 ke is zero.
Methods
Non-equilibrium Hubbard Green function has been used to solve the thermoelectric transport in the sequential and linear response regime35. The system Hamilton can be rewritten by the transition operator36,37,38,39, i.e. 
with 
for 
, 
, 
and 
. For large spin, the operator can be expressed as40:
where S is the spin quantum number, 
and 
. Finally, the retarded Green Function is written as
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, 
are the electron self-energy in the second-order approximation and the formulas for calculation are
where
Here 
is a chemical potential with 
and 
and 
for 
. Pα denotes the polarization of α lead and is defined as 
. 
is the Fermi-Dirac distributions of α lead. 
for 
and 
. The voltage is defined as 
and the spin voltage is written as 
.In the Hubbard operator representation, the eigenenergies of the unperturbed SMM can be obtained exactly, so the equation (5) is rigorous41. Following the Landauer-Buttikier formula, the expressions for the currents42 are
where
denotes the transmission coefficient of spin-
electrons
with
We can directly obtain the same formula for 
with
, depicting electric current 
and heat current
. 
and 
denote the charge current and spin current respectively. It is noticeable that our formulas (Eq. 9 and 10) are different from the conventional expressions41,42, in which θ and Fermi-Dirac function are coupled to each other.
In linear response regime, the thermoelectric coefficients are expressed as
Here, Sc and Ss denote the charge Seebeck and spin Seebeck respectively. 
is 
- electron conductance. ke is the conventional thermal conductance. 
is the
-electric current induced by a temperature difference at zero voltage bias and zero spin bias. 
is defined as temperature difference and the average temperature is expressed as
. Finally, The spin figure of merit 
and charge figure of merit 
can be calculated. 
is the thermal conductance with contributions from both electrons
and phonons
43,44,45. In our model, the phonon transport is not considered due to large mismatch of vibrational spectra between the SMM and leads.
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Acknowledgements
This work was supported by the National Natural Science Foundation of China under Grants Nos.11074081, 11274130, 11304107 and 1371015. Numerical calculations presented in this paper were carried out using the High Performance Computing Center experimental tested in SCTS/CGCL (see http://grid.hust.edu.cn/hpcc).
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K.Y. put forwards the idea and supervised the whole work. B.L. performed the numerical calculations and wrote the manuscript. B.L. and J.L. entered the discussions and analyzed the results. J.T.L. and J.H.G. made discussion on the referees' comments and revised the manuscript. All authors reviewed the manuscript.
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Luo, B., Liu, J., Lü, JT. et al. Ultrahigh spin thermopower and pure spin current in a single-molecule magnet. Sci Rep 4, 4128 (2014). https://doi.org/10.1038/srep04128
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DOI: https://doi.org/10.1038/srep04128
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