Spin-dependent thermoelectric effects in Fe-C6 doped monolayer MoS2

By using the non-equilibrium Green’s function with density functional theory, we have studied the thermal spin transport properties of Fe-C6 cluster doped monolayer MoS2. The results show that the device has a perfect Seebeck effect under temperature difference without gate voltage or bias voltage. Moreover, we also find the thermal colossal magnetoresistance effect, which is as high as 107%. The competition between spin up electrons and spin down holes of the parallel spin configuration leads to peculiar behavior of colossal magnetoresistance and thermo-current, which is essential for the design of thermal transistors. These results are useful in future MoS2-based multifunctional spin caloritronic devices.

exchange-correlation potential, and the valence electronic orbitals were expanded in double-polarized basis set. The thermally induced current was given by 36 here σ (=↑, ↓) denotes the spin index, and μ L/R is the electrochemical potential for source/drain. As we just considered the temperature difference without gate voltage or bias voltage, then μ L = μ R = E F (Fermi level) was set to zero.
A is spin-resolved transmission function. The system was relaxed sufficiently till the maximum force dropped below a threshold value of 0.01 eV/Å. Experimentally, the magnetization of the left and right electrodes can be aligned in parallel (P) or antiparallel (AP) spin configuration by a sufficiently strong external magnetic field. Therefore, the P and AP spin configurations were both considered. The total energy of AP spin configuration was 0.88 meV less than that of the P spin configuration per unit cell. Figure 2 displays the thermally induced currents of Fe-C 6 device versus T L with different ∆T (ΔT = T R − T L ), and these versus ∆T at different T L for P spin configuration. The trend of curve is almost identical for spin-up or spin-down state, and there are no temperature threshold. The spin-polarized currents increase rapidly in low T L region and slowly in high T L region for all ∆T (Fig. 2(a)), but the currents almost linearly increase in the whole range of ∆T for all T L ( Fig. 2(b)). The different sign of spin-up and spin-down currents implies SSE.

Results and Discussion
We now discuss the charge currents I C (I up + I dn ) and spin currents I S (I up − I dn ) as a function of T L to describe the SSE. As can be seen from Fig. 3, I S is about 10 times larger than I C for each of ∆T. When ∆T ranges from 20 K to 60 K, I S increases. It is noteworthy that I C shows unusual behavior. We take ∆T = 60 K as an example. As T L increases, I C declines to negative value, and reaches to its minimum −0.7 nA at T L = 100 K, after that, I C begins to increase to almost zero (about −0.09 nA) at 140 K. Whereafter, I C continues to increase with increasing T L , and finally reaches the maximum value 7.21 nA at T L = 370 K, then decreases again. The novel property of I C is due to the competition between I up and I dn , videlicet, the result of the competition between the spin up hole carriers and the spin down electron carriers in the heat transport. The nonlinear response of I C is very necessary for the design of thermal transistors. At ΔT = 60 K, T L = 140 K, the value of I C is −0.09 nA, but I S is −8.87 nA. Although at T L = 150 K, I C is very small (0.24 nA), but the absolute value of I S is also large (I S = −44.62 nA). The above results indicate that the net spin current is produced at ΔT = 60 K within the T L region (140 K, 150 K), and the total charge current is well suppressed. Thus the perfect SSE occurs.
Here, the electrode leads are the same material, and the left electrode is colder than the right electrode. When the source and drain have different temperature, the behavior of carrier is determined not only by transmission function, but also by the R which is only related to the temperature T L/R of source/drain. If the transmission functions of spin-up and spin-down channels are strong symmetric around Fermi level, the electron current (I e ) and hole current (I h ) will be cancelled each other and the device will not have total charge current, only net spin current is produced. In Fe-C 6 device, the spin down carriers (electrons) flow from drain to source, producing the positive current, while the current generated by the spin-up carriers (holes) flowing from source to drain is negative, then there is a nonzero spin current in the P configuration.
To elucidate the physical mechanism for SSE, we analyze the band structure of Fe-C 6 doped monolayer MoS 2 system and the spin dependent transmission function of Fe-C 6 device, as shown in Fig. 4(a). The mechanism of the current generated from temperature difference credits to the electron band structure. The left panel of Fig. 4(a) manifests that the spin-up and spin-down bands are split and have a good symmetry near the Fermi level, thus the Fe-C 6 doped system is a spin semiconductor. The transmission function (the middle panel of Fig. 4(a)) shows that the spin-up and spin-down transport channels are both opened near the Fermi level for P spin configuration, due to the match of the bands of the left and right electrode. Accordingly, transmission peaks near the Fermi level present in the energy range (−0.124 eV, −0.045 eV) and (0.055 eV, 0.114 eV) for spin-up and spin-down channels, respectively. Because they are very close to the Fermi level, the temperature threshold is small. These two peaks break the electron-hole symmetry in the transmission function, resulting in the nonzero net thermal spin currents. Additionally, the transmission peaks of both channels are almost symmetrical about the Fermi level, thereby, Fe-C 6 device produces a perfect SSE. The spin splitting of the density of states (DOS) in Fig. 4(b) verifies the spin-polarized transport. Furthermore, the spin up and down DOS are nearly symmetric about the Fermi level, therefore, there should be SSE. The lower panel of Fig. 4(b) reveals that it is the d-states of Fe, p-states of the nearest neighbor C, and d-states of the second-nearest neighbor Mo contributing to the thermal transport properties. Now, we analyze the current spectra ( (Fig. 4(c)), which reflects the value of current. The spin-up and spin-down current spectra both augment with the increasing of T L or ∆T, and they are almost equal in area, resulting in numerically approximate equal of spin up and spin down currents with opposite signs. These phenomena further confirm the emergence of SSE.
When the device is transformed from P spin configuration to AP spin configuration, the CMR effect of the device appears, . Figure 5(a) display that the total charge current of AP spin configuration (I C AP ) is smaller than that of P spin configuration (I C P ), owing to the transport channel of P spin configuration is closer to the Fermi level than that of AP spin configuration. I C P changes from negative value to positive value with increasing T L , but its absolute value at most T L points is much larger than I C AP . Figure 5(b) denotes that the CMR reaches 10 7 % (or −10 7 %) in wide range of T L and is accompanied by the change of symbol, which has special significance and will be widely used in the logic devices. The reason for above phenomenon is that I C P varies from negative to positive sign with increasing T L (see Fig. 5(a)) and is much larger than I C AP which is negative and almost zero, thus positive and negative magnetoresistances emerge as indicated by the insert of Fig. 5(b).
The thermal magneto-resistance behavior of Fe-C 6 doped MoS 2 is very similar to that of Fe-doped MoS 2 37 , but the mechanism is disparate. The negative magnetoresistance of Fe-C 6 doped MoS 2 system is the result of   competition between spin up holes and spin down electrons of P spin configuration. The negative magnetoresistance of Fe-doped MoS 2 system is caused by the sign change of total charge current of AP spin configuration 37 . The CMR shows a strong regularity within several ∆T and its value gradually moves close to zero until finally reaches to −10 4 % in Fe-C 6 device. For convenience, we also call the negative magnetoresistance phenomenon 'Zigzag' . The behavior of 'Zigzag' shifts to low T L with increasing ∆T. Negative magnetoresistance appears in the lower T L with a higher ∆T, which is in agreement with that of Fe-doped MoS 2 system. However, the maximum negative magneto-resistance value begins to increase when ∆T decreases. Namely, the magnetoresistance of ∆T = 60 K is less than that of ∆T = 40 K, which is also less than that of ∆T = 20 K. Based on temperature difference and magnetic field, one can control carriers transport to change the magnetoresistance of Fe-C 6 doped monolayer MoS 2 two-probe system, the thermal CMR is higher than 10 7 % and accompanied with the conversion between negative value and positive value. Our findings may provide a good reference to experiment.
We also consider the size effect on the thermally transport. For P spin configuration, the phenomenon of the thermal transport is almost not affected by the length of the scattering region, and the perfect SSE arises yet. The tendency of thermally induced CMR is not changed when T L increases, but in many of the T L points, its values augment nearly two orders of magnitude when the length of the scattering region is enlarged from two doped units to four doped units, reaching 10 10 % because of the decrease of I AP . Meanwhile, the 'zigzag' phenomenon of CMR still exists. It can be concluded that the thermal transport properties are enhanced by aggrandizing the length of scattering region.

Conclusions
In conclusion, a new configuration of spin caloritronic device based on Fe-C 6 cluster doped monolayer MoS 2 was presented, and its thermal transport properties had been investigated by NEGF-DFT approach. Spin-polarized currents can be produced only by applying temperature gradient between the left and the right electrodes. The SSE is observed in the thermal electron current. An intriguing thermally induced CMR without gate regulating occurs in the device, which has different sign in distinct T L region while ∆T is fixed. The peculiar behavior of CMR and the nonlinear response regime of I c arise from the competition between spin-up electrons and spin-down holes of the P spin configuration. Additionally, we find that the thermal transport properties of enlarged scattering region are enhanced. The results manifest that the MoS 2 -based materials have potential application in spin caloritronics and spintronics.