Quantum mechanical modeling of magnon-phonon scattering heat transport across three-dimensional ferromagnetic/nonmagnetic interfaces

Magnon-phonon scattering (MPS) has attracted widespread attention in quantum heat/spin transport across the ferromagnetic/nonmagnetic (F/N) interfaces, with the rapid progress of experiments on spin caloritronics in recent years. However, the lack of theoretical methods, accounting for the MPS rigorously, has seriously hindered investigations on the quantum heat transport in magnetic nanostructures with broken translational symmetry, such as F/N interfaces. In this paper, we propose a theoretical formalism of the nonequilibrium Green function to incorporate the MPS into the quantum heat transport for three-dimensional ferromagnetic nanostructures, rigorously, through a diagrammatic perturbation analysis. A computational scheme is developed for the first-principles simulation of quantum heat transport in practical magnetic nanostructures, and a generalized formalism of heat flow is presented for the analysis of the elastic and inelastic process of heat transport. A thermal rectification driven by MPS is observed in the numerical simulation of heat transport across the F/N interface based on the ${\text{CrI}}_3$ monolayer, which is consistent with recent studies. In this paper, we open the gate to first-principles investigations of quantum heat transport in magnetic nanostructures and pave the way for the theoretical design of magnetic thermal nanodevices.

Figure 1

Schematics of the physical model in the present three-dimensional (3D) nonequilibrium Green function (NEGF) formalism. (a) Ferromagnetic devices with a transverse periodicity. (b) Ferromagnetic/nonmagnetic (F/N) interfaces with a transverse periodicity; the magnetization ($M$) of the ferromagnetic region is assumed to point in the $+z$ direction. An external magnetic field along the $+z$ direction ($h_z$) is applied to all regions. The magnon-phonon scattering (MPS) is only considered in the central region (C). In the case of F/N interfaces, the energy of magnons can only be transferred from the left contact (Contact L) to the right contact (Contact R) via MPS.

Figure 2

Feynman diagrams for the magnon-phonon scattering self-energies in energy-momentum space of (a) magnons and (b) phonons under the self-consistent Born approximation (SCBA). The straight and wavy lines represent the full Green's functions of magnons and phonons, respectively.

Figure 3

Schematic of the energy exchange in the present magnon-phonon scattering (MPS) devices. The solid and dashed lines represent elastic and inelastic heat flow, respectively. The wavy line represents the energy exchange between magnons and phonons through MPS. The MPS acts as a virtual contact (Contact M) to exchange the heat energy between magnons and phonons.

Figure 4

Heat transport through the present ferromagnetic/nonmagnetic (F/N) interface based on ${\mathrm{CrI}}_3$ monolayer (ML). (a) Schematic of the numerical model; the transverse direction ($y$ direction) of this device is periodic, and an external magnetic field pointing in the $+z$ direction is applied to the whole device; the nonmagnetic region (Contact R) is assumed to have the same site configuration as ${\mathrm{CrI}}_3$ but to be nonmagnetic. (b) Schematic of the heat flow across the present F/N interface device; the dashed (or solid) straight line with arrow and the dashed (or solid) wavy line with arrow denote the inelastic (or elastic) heat flow of magnons and phonons, respectively. (c) Schematic of the temperature deviation of the magnons as induced by the nonequilibrium phonons; the dashed and dotted lines denote the effective local temperature (ELT) of magnons without and with magnon-phonon scattering, respectively, and the solid line denotes the ELT of phonons.

Figure 5

The temperature, temperature difference, and external magnetic field dependence of (a)–(c) inelastic heat flow and (d)–(f) phonon drag temperature deviation of magnons. A temperature difference of 1 K is used in (a), (c), (d), and (f). A magnetic field of 0.0 meV is adopted in (a) and (d). A central temperature of 50 K is used in (b), (c), (e), and (f).

Figure 6

The local density of states (LDOS) of magnons (a) at different central temperatures with $h_z=0$ meV; (b) under different magnetic field $h_z$ at 50 K. (c) shows the $h_z$-dependent heat flow density of magnons at 50 K. (d)–(f) show the LDOS of phonons donated by (e) Cr and (f) I atoms, respectively, at different temperature with $h_z=0$ meV.

Figure 7

Schematic for the micro-process of energy transfer from magnons to phonons in ferromagnetic/nonmagnetic (F/N) interfaces. A central temperature ($T$) of 50 K, a temperature difference ($\mathrm{\Delta }T$) of 1 K, and an external magnetic field ($h_z$) of 0.0 meV are adopted here. The thick lines in the diagrammatic representation of magnon-phonon scattering mean the accumulation of particles in the central region of the device, where magnons with higher energy ($\varepsilon >6$ meV) transfer energy to phonons, and turn to be magnons with lower energy ($\varepsilon <6$ meV).

Figure 8

Schematic of the simplified physical model for a three-dimensional (3D) ferromagnetic nanostructure with a transverse ($yz$ plane) periodicity. An external magnetic ($h_z$) along the $+z$ direction is applied to the entire system. The straight lines connecting two atoms denote the force constants $K$, the curves with double-arrow between two spins denote exchange coupling constants $J$, and the wavy lines between $J$ and $K$ denote the magnon-phonon coupling constants $M$.

Figure 9

The equivalent connections for the magnon-phonon scattering self-energy of (a) magnons and (b) phonons. The exchange of ${\tau }_1$ and ${\tau }_2$ produces two equivalent connected diagrams and cancels the second-order expansion coefficient ($\frac{1}{2!}$) of time evolution operator $S_C$.

Figure 10

(a) The dispersion relations of magnon (thin solid lines) and phonon (bold solid lines) along the transverse direction ($y$-axis); phonons with higher energy ($\hslash \omega >15$ meV) are forbidden to participate in the MPS process; (b) the temperature dependence of heat flow without (hollow circle) and with MPS (others); a temperature difference of 1 K is adopted and an external magnetic field of 0.0 meV is used; MPS impedes phonon transport obviously at high temperature.