We study the impact of the size effect and the varying perpendicular magnetic field B on the transport properties of topological nodal-line semimetals (TNLSMs) by using the Landauer-Büttiker formula combined with the nonequilibrium Green’s function. We theoretically calculate the band structure, conductance G and local density of states (LDOS) of TNLSMs nanowire. The results show that both the size effect and the application of varying magnetic field play a crucial role in the electron transport of TNLSMs. We find that both the step height and width of the conductance plateau in TNLSMs are closely related to the size effect of the system. For the system with a large TNLSMs, the low energy states are dominated by flat bands, leading to a series of quantized conductance plateaus. Furthermore, it has been observed that the presence of a magnetic field leads to the formation of the Landau levels in the TNLSMs, which exhibit sensitivity to the direction of the applied magnetic field. A perpendicular magnetic field parallel to the transport direction forms a linearly dispersed Landau level in the TNLSMs system, which leads to an increase in its conductance G and LDOS. At the same time, when the magnetic field perpendicular to the transport direction is applied, a series of the Landau levels with the flat bands are formed near the Fermi energy level EF of the system, resulting in a significant decrease of its conductance G and a rapid decay of LDOS. These unique physical properties of the TNLSMs nanowire offer a wide range of the applications in magnetoelectric transport.

Andreev reflection is an important quantum tunneling phenomenon in the conductor-superconductor junction. The Andreev reflection coefficients T-AR of a hybrid system with s-wave superconductor connected by topological nodal-line semimetals (TNLSMs-SC junction system) is calculated theoretically by using the Landauer-B & uuml;ttiker formula combined with the nonequilibrium Green's function method. The results show that when the direction of the boundary state electron and the incident electron are the same, only the bulk states of the TNLSMs involve the Andreev reflection of the hybrid system, and the Andreev reflection coefficients T-AR enhance with the increase of the Fermi energy EF. We also study the effect of on-site energy epsilon(z) and mass term m on the Andreev reflection and find that the Andreev reflection in the system decreases rapidly with the increase of on-site energy epsilon(z) and mass term m. Moreover, we find that only in the presence of a mass term m, the Andreev reflection coefficients T-AR of the system changes with the rise of the Fermi energy E-F. When a perpendicular magnetic field is applied in the system, the Andreev reflection coefficients T-AR in the superconducting gap will appear a series of oscillating peaks. For a hybrid system with the large perpendicular magnetic field applied, we find that the maximum Andreev reflection coefficients TAR=7.2 at the Fermi energy EF=0.0 and the incident electron energy E=+/- 0.1. The Andreev reflection coefficients T-AR is gradually enhanced in the superconducting gap (incident energy |E|<= 0.2) when a disorder is applied to the superconductor region of the system. However, the symmetry of the Andreev reflection coefficients T-AR is broken when the perpendicular magnetic field is applied to the system. These peculiar transport properties of the TNLSMs-SC junction system are expected to provide theoretical guidance for future applications.

The spin Nernst effect (SNE) of a topological nodal-line semimetals (TNLSMs) nanowire in a four-terminal cross-bar device is studied. Based on the nonequilibrium Green’s function method combining with the tight-binding Hamiltonian, the spin Nernst coefficients Ns3z, Ns3x and Ns3y of the two thermal transport models (kz-y case and kx-y case) under a perpendicular magnetic field are obtained. The traditional SNE describes the transverse voltage and spin current caused by the longitudinal thermal gradient. The transverse spin current is generated by the spin–orbit interaction rather than the perpendicular magnetic field. Here we find that the spin Nernst coefficients are equal to zero at the zero-magnetic field, but have finite values for non-zero magnetic fields in TNLSMs. The spin Nernst coefficient is closely related to the strength of the magnetic field and temperature. With the increase of the magnetic field strength, the peak height of the spin Nernst coefficient increases. These results indicate that the SNE is induced by the perpendicular magnetic field in the present TNLSMs system, which can be considered as an anomalous spin Nernst phenomenon. In addition, we find that Ns3z displays a series of peaks when the transmission coefficient TLR jumps from one integer plateau to another, and the spin Nernst coefficient Ns3i (i=x,y,z) is an odd function of the Fermi energy EF with Ns3z/x/y(−EF)=−Ns3z/x/y(EF) in connection mode kz-y. For the kx-y connection mode, due to the presence of the mass term m opening the energy gap, a sudden jump of the transmission coefficient TLR in the vicinity of the energy gap edge causes a very larger peak for the spin Nernst coefficient Ns3z/x/y, and the symmetrical properties of the spin Nernst coefficient Ns3z/x/y are completely broken in the strong magnetic field. These results indicate that the spin Nernst coefficient in TNLSMs strongly depends on the thermal gradient direction and the transverse lead connection direction. The spin Nernst coefficient in TNLSMs has strongly spatial anisotropy, which can be used as the characterization of the experimental detection of TNLSMs.

We study the thermoelectric transport of the graphene p - n junction under the perpendicular magnetic field. The Seebeck coefficient S c , the thermoelectrical figure of merit ZT and the power-generation efficiency n are obtained by the Landauer-B & uuml;ttiker formula combined with the nonequilibrium Green's function method. Compared to the perfect graphene system, the graphene p-n - n junction has a zero-transport coefficient plateau (or the transport gap). The sudden jump of the transmission coefficient near the transport gap edge lead to very larger peaks of the Sc c and ZT . Especially in the presence of a magnetic field, the perpendicular magnetic field applied to the p-n - n junction strongly suppresses the conductance, and enhances the Seebeck coefficient Sc c and increases the ZT . Moreover, it is found that the Seebeck coefficient Sc c and ZT are strongly dependent on the applied perpendicular magnetic field cent , the potential drop in the center region of the p-n - n junction and the center region length M of the p-n - n junction. This means that the thermoelectric performance of the graphene p - n junction can be easily regulated by changing the magnetic field and the center region lengths of the p - n junction. Finally, the power-generation efficiency n of the graphene p - n junction as a power generator is calculated. It is found that when the Carnot power-generation efficiency is greater than 30%, ZTM M can still be greater than 10. The large ZTM M value also maintains a high power-generation efficiency, which indicates that the graphene p - n junction has potential applications as thermoelectric devices.

The hybrid topological superconductor (TSC) with multiple Majorana edge states can be formed in a system where a magnetic topological insulator (MTI) thin film is coupled with two s-wave superconductors in proximity. The band structure and thermal transport properties of the hybrid TSC hosting chiral and helical Majorana edge states are investigated. By using the nonequilibrium Green’s function method combined with the Landauer–Büttiker formula, the scattering coefficients (i.e., the normal tunneling T, the local Andreev reflection TLAR, the crossed Andreev reflection TCAR) and two-terminal electric thermal conductance κe are calculated. With the change of the exchange field Mz, the system goes through a series of topological phase transitions. The chiral Majorana edge states, the helical Majorana edge states and the multiple Majorana edge states possessing both chiral and helical edge modes are induced at the boundary of TSC. At the boundary of the central TSC, a single Majorana fermion edge state is equivalent to half an ordinary fermion to carry heat, leading to the generation of a half-integer quantized thermal conductance plateau. Thus, half-integer (i.e., 1/2, 3/2) and integer (i.e., 1, 2) quantized thermal conductance plateaus appear in different topological phases. These quantized plateaus that survive well over a certain range of temperatures can be used experimentally to detect chiral and helical Majorana fermions.

The Nernst effect is the transverse mode of thermoelectric transport, in which a longitudinal thermal gradient induces a transverse current in the conductor while under a perpendicular magnetic field. Here the Nernst effect in a mesoscopic topological nodal-line semimetals (TNLSMs) system of four-terminal cross-bar with the spin-orbit coupling under a perpendicular magnetic field is studied. The Nernst coefficient N-c in two non-equivalen connection modes (k(z)-y mode and k(x)-y mode) is calculated based on the tight-binding Hamiltonian combined with the nonequilibrium Green's function method. When the magnetic field is absent with phi= 0.0, the Nernst coefficient N-c = 0 is exactly regardless of the temperature. When the magnetic field is not zero, the Nernst coefficient exhibits a series of densely oscillating peaks. The height of peak strongly depends on the magnetic field, and the Nernst coefficient is an even function of the Fermi energy EF satisfying the symmetrical property N-c(-EF) = N-c(EF). The Nernst coefficient is also closely related to the temperature T. When the temperature is very low (or T. 0), the Nernst coefficient depends linearly on temperature. In the presence of a strong magnetic field, the Nernst coefficient shows peaks when the Fermi energy crosses the Landau levels. Under the weak magnetic field, the influence of spin-orbit coupling in TNLSMs materials on Nernst effect is very obvious. In the presence of the mass term, the PT-symmetry of the system is destroyed, the nodal ring of TNLSMs is broken and an energy gap will be opened. The Nernst coefficient N-c has a large value in the energy gap, which is very promising for the application of the transverse thermoelectric transport.