Thermal transport through a spin-phonon interacting junction: A nonequilibrium Green's function method study

Using the nonequilibrium Green's function method, we consider heat transport in an insulating ferromagnetic spin chain model with spin-phonon interaction under an external magnetic field. Employing the Holstein-Primakoff transformation to the spin system, we treat the resulted magnon-phonon interaction within the self-consistent Born approximation. We find the magnon-phonon coupling can change qualitatively the magnon thermal conductance in the high-temperature regime. At a spectral mismatched ferromagnetic-normal insulator interface, we also find thermal rectification and negative differential thermal conductance due to the magnon-phonon interaction. We show that these effects can be effectively tuned by the external applied magnetic field, a convenient advantage absent in anharmonic phonon and electron-phonon systems studied before.

Figure 1

Model diagram of a 1D insulating ferromagnetic spin junction under an external magnetic field. Spin-phonon interaction is included only in the central region. The two leads are modeled by semi-infinite chains with different temperatures.

Figure 2

Diagram representations for the self-energies of (a) magnons and (b) phonons under the SCBA. The double wavy line and single wavy line are for the full and noninteracting GFs of phonons, respectively. The double straight line is for the full GF of magnons.

Figure 3

Thermal conductances of the 1D ferromagnetic Heisenberg junction for magnons and phonons under an external magnetic field. Parameters of the central region are $J_{12}^C=1.5\mathrm{meV}$, $K_{12}^C=6\times {10}^4{\mathrm{meV}}^2$. For the leads, $J_{ij}^L$ and $J_{ij}^R$ are $2\mathrm{meV}$; $K_{ij}^L$ and $K_{ij}^R$ are $1\times {10}^5{\mathrm{meV}}^2$. Couplings of the central region with the leads are $J_{01}^{LC}=J_{23}^{CR}=J_{12}^C$, $K_{01}^{LC}=K_{23}^{CR}=K_{12}^C$. The magnon-phonon coupling constant is $M_0=14\mathrm{meV}$. The external magnetic field is $h_z=4\mathrm{meV}$ for all regions.

Figure 4

Thermal conductance for the magnons with different magnon-phonon coupling constants. The other parameters are the same as those in Fig. 3.

Figure 5

Energy current $I_L$ of an FN insulator interface device as a function of the temperature bias $\mathrm{\Delta }T$ with $T_{L,R}=T_0\pm \mathrm{\Delta }T/2$. $T_0$ is the average temperature of the two electrodes. The strength of Heisenberg ferromagnetic exchange interactions is 2 meV for all the regions on the left side of atom 2 in Fig. 1, and they are effectively zero in the right region. The magnitude of the dynamic matrix elements is $K_0=1\times {10}^5{\mathrm{meV}}^2$ for all the regions on the right side of atom 1, and they are very large for its left region. The magnon-phonon coupling strength is $M_0=20\mathrm{meV}$. The external magnetic field is $h_z=2\mathrm{meV}$ for the magnetic part.

Figure 6

Local density of states for (a) magnons and (b) phonons of atom 2. The parameters are the same with those of Fig. 5.

Figure 7

(a) The energy current $I_L$ as a function of the temperature bias. (b) The corresponding local density of states for the magnons of atom 2. The average temperature of the two electrodes is $T_0=60\mathrm{K}$. The other parameters are the same as those of Fig. 5.

Figure 8

(a) Total energy current out of the left lead. (b) Energy current via the magnon-phonon channel. We set $J_{23}^{CR}=J_{ij}^R$ and $h_z=2\mathrm{meV}$ for the whole region. The average temperature is $T_0=60\mathrm{K}$. The other parameters are the same as those of Fig. 5.

Figure 9

Local temperatures for (a) magnons and (b) phonons of atom 2 as a function of the temperature bias. (i) Left lead temperature and (ii) Right lead temperature are given for comparison. (iii), (iv), and (v) correspond to the local temperatures of (a) magnons or (b) phonons with $J_{23}^{CR}=0.05$, 0.1, and $0.2\mathrm{meV}$, respectively. The other parameters are the same as in Fig. 8 except for the added probes. One probe is for measuring $T_2^J$ and consists of a semi-infinite chain, with the exchange interaction $J_{ij}^P=20\mathrm{meV}$ in the lead. The other probe is for measuring $T_2^{\text{ph}}$, with the dynamic matrix element $K_{ij}^P=2\times {10}^5{\mathrm{meV}}^2$. The two probes couple very weakly to the measured atom.

Figure 10

Ballistic thermal conductances for magnons and phonons of a ferromagnetic spin chain under an external magnetic field in the low-temperature range. The magnetic field is $h_z=4\mathrm{meV}$.