Thermal conductivity of local moment models with strong spin-orbit coupling

We study the magnetic and lattice contributions to the thermal conductivity of electrically insulating strongly spin-orbit coupled magnetically ordered phases on a two-dimensional honeycomb lattice using the Kitaev-Heisenberg model. Depending on model parameters, such as the relative strength of the spin-orbit induced anisotropic coupling, a number of magnetically ordered phases are possible. In this work, we study two distinct regimes of thermal transport depending on whether the characteristic energy of the phonons or the magnons dominates, and focus on two different relaxation mechanisms, boundary scattering and magnon-phonon scattering. For spatially anisotropic magnetic phases, the thermal conductivity tensor can be highly anisotropic when the magnetic energy scale dominates, since the magnetic degrees of freedom dominate the thermal transport for temperatures well below the magnetic transition temperature. In the opposite limit in which the phonon energy scale dominates, the thermal conductivity will be nearly isotropic, reflecting the isotropic (at low temperatures) phonon dispersion assumed for the honeycomb lattice. We further discuss the extent to which thermal transport properties are influenced by strong spin-orbit induced anisotropic coupling in the local moment regime of insulating magnetic phases. The developed methodology can be applied to any 2D magnon-phonon system, and more importantly to systems where an analytical Bogoliubov transformation cannot be found and magnon bands are not necessarily isotropic.

Figure 1

Honeycomb lattice with bond labels $\gamma =\{x,y,z\}$ used for the Kitaev terms in Eq. (1).

Figure 2

Phase diagram of the Kitaev-Heisenberg model with the parametrization of Eq. (2). A variety of magnetic and nonmagnetic “liquid” phases are present as a function of the angle $\phi$ [23]. A schematic of the various ordered states is shown. The magnetic unit cell for the zigzag and the stripy phase is shown as a dashed rectangle.

Figure 3

Lowest order magnon-phonon scattering diagrams used for the calculation of the transport relaxation times in the regime in which thermal transport is phonon-dominated. Wavy lines represent phonon propagators whereas straight lines are magnon propagators. Panel (a) represents C processes which involve two magnon creations or annihilations, whereas panel (b) represents R processes that involve phonon emission or absorption.

Figure 4

Lowest order magnon-phonon scattering diagrams used for the calculation of the transport relaxation times in the regime in which thermal transport is magnon-dominated. Straight lines represent magnon propagators whereas wavy lines are phonon propagators. Panels (a) and (b) represent R processes which involve phonon emissions or absorptions. Panel (c) represents C processes which involve phonon emission or absorption.

Figure 5

${\kappa }_{xx}$ component of the total fully ballistic thermal conductivity per unit area, for each ordered phase (see the legend of each panel), for different relative strengths of the Debye energy $E_D$ to the magnon characteristic energy $SA$ (given on the top of each panel), versus temperature. The temperature region is well below the lowest of the two characteristic energy scales (phononic or magnonic). Notice that the conductivity components are measured in the units given by the prefactor on the right-hand side of Eq. (47). The spatial direction $x$ is defined as in Fig. 11, Appendix pp1.

Figure 6

${\kappa }_{yy}$ component of the total ballistic thermal conductivity per unit area, for each ordered phase (see the legend of each panel), for different relative strengths of the Debye energy $E_D$ to the magnon characteristic energy $SA$ (given on top of each panel), versus temperature. The temperature region is well below the lowest of the two characteristic energy scales (phononic or magnonic). Notice that the conductivity components are measured in the units given by the prefactor on the right-hand side of Eq. (47). The spatial direction $y$ is defined as in Fig. 11, Appendix pp1.

Figure 7

Phonon-dominated transport: ${\kappa }_{xx}$ and ${\kappa }_{yy}$ component of the phononic thermal conductivity per unit area, for each ordered phase, for three different subregimes: ballistic, intermediate, and diffusive (see the legend of each panel) as well as pure boundary scattering, versus temperature. The spatial directions $x$ and $y$ are defined as in Fig. 11, Appendix pp1.

Figure 8

Phonon-dominated transport: ${\kappa }_{xx}/T^2$ and ${\kappa }_{yy}/T^2$ component of the phononic thermal conductivity per unit area, for each ordered phase, for three different subregimes: ballistic, intermediate, and diffusive (see the legend of each panel) as well as pure boundary scattering, versus temperature. The spatial directions $x$ and $y$ are defined as in Fig. 11, Appendix pp1.

Figure 9

Magnon-dominated transport: ${\kappa }_{xx}$ and ${\kappa }_{yy}$ component of the magnonic thermal conductivity per unit area, for each ordered phase, for three different subregimes: ballistic, intermediate, and diffusive (see the legend of each panel) as well as pure boundary scattering, versus temperature. The spatial directions $x$ and $y$ are defined as in Fig. 11, Appendix pp1.

Figure 10

Magnon-dominated transport: ${\kappa }_{xx}/T^n$ and ${\kappa }_{yy}/T^n$ component of the magnonic thermal conductivity per unit area, for each ordered phase, for three different subregimes: ballistic, intermediate, and diffusive (see the legend of each panel) as well as pure boundary scattering, versus temperature. The appropriate temperature exponent $n$ that should divide ${\kappa }_{xx}$ and ${\kappa }_{yy}$ such that the pure boundary scattering results are represented by horizontal straight lines (at least at low temperatures) is given in the nearby yellow inset. The exponents can slightly vary for the spatial directions $x$ and $y$, as defined in Fig. 11, Appendix pp1.

Figure 11

Zigzag magnetic phase: A magnetic unit cell consists of four magnetic moments labeled as A, B, C, and D, and is represented by the gray-shaded rectangle shown in the figure. The translation vectors of the periodic magnetic structure are the vectors $\mathit{a}$ and $\mathit{b}$. The translation vectors of the chemical periodic structure are the vectors ${\mathit{t}}_{\mathit{1}}$ and ${\mathit{t}}_{\mathit{2}}$, and a chemical unit cell is represented by any dashed parallelogram. For the Néel and the ferromagnetic states the magnetic unit cell coincides with the chemical unit cell (that is common to all phases).

Figure 12

Lower spin wave dispersion relations of the zigzag phase, as given by Eqs. (A2). The yellow surface corresponds to ${\omega }_1\left(\mathit{k}\right)$ and the blue surface to ${\omega }_2\left(\mathit{k}\right)$. Notice that the magnon wave vector components $k_x$ and $k_y$ are measured in units of $\frac{2\pi }{a}$ and $\frac{2\pi }{b}$, respectively, and the spin wave energy is measured in units of $\frac{SJ}{2\hslash }$. The shaded hexagon within the Oxy plane is the first Brillouin zone (1BZ) of the honeycomb lattice. The plot is for $K/J=-2.65$ and $\alpha =2\pi /3$.

Figure 13

Upper spin wave dispersion relations of the zigzag phase, as given by Eqs. (A3). The yellow surface corresponds to ${\omega }_3\left(\mathit{k}\right)$ and the blue surface to ${\omega }_4\left(\mathit{k}\right)$. Notice that the magnon wave vector components $k_x$ and $k_y$ are measured in units of $\frac{2\pi }{a}$ and $\frac{2\pi }{b}$, respectively, and the spin wave energy is measured in units of $\frac{SJ}{2\hslash }$. The shaded hexagon within the Oxy plane is the 1BZ of the honeycomb lattice. The plot is for $K/J=-2.65$ and $\alpha =2\pi /3$.

Figure 14

Lower spin wave dispersion relations of the stripy phase, as given by Eqs. (A6). The yellow surface corresponds to ${\omega }_1\left(\mathit{k}\right)$ and the blue surface to ${\omega }_2\left(\mathit{k}\right)$. Notice that the magnon wave vector components $k_x$ and $k_y$ are measured in units of $\frac{2\pi }{a}$ and $\frac{2\pi }{b}$, respectively, and the spin wave energy is measured in units of $\frac{SJ}{2\hslash }$. The shaded hexagon within the Oxy plane is the 1BZ of the honeycomb lattice. The plot is for $K/J=-1$ and $\alpha =2\pi /3$.

Figure 15

Upper spin wave dispersion relations of the stripy phase, as given by Eqs. (A7). The yellow surface corresponds to ${\omega }_3\left(\mathit{k}\right)$ and the blue surface to ${\omega }_4\left(\mathit{k}\right)$. Notice that the magnon wave vector components $k_x$ and $k_y$ are measured in units of $\frac{2\pi }{a}$ and $\frac{2\pi }{b}$, respectively, and the spin wave energy is measured in units of $\frac{SJ}{2\hslash }$. The shaded hexagon within the Oxy plane is the 1BZ of the honeycomb lattice. The plot is for $K/J=-1$ and $\alpha =2\pi /3$.

Figure 16

Spin wave dispersion relations of the Néel phase, as given by Eqs. (A10). The yellow surface corresponds to ${\omega }_1\left(\mathit{k}\right)$ and the blue surface to ${\omega }_2\left(\mathit{k}\right)$. Notice that the magnon wave vector components $k_x$ and $k_y$ are measured in units of $\frac{2\pi }{a}$ and $\frac{2\pi }{b}$, respectively, and the spin wave energy is measured in units of $\frac{SJ}{2\hslash }$. The shaded hexagon within the Oxy plane is the 1BZ of the honeycomb lattice. The plot is for $K/J=1$ and $\alpha =2\pi /3$.

Figure 17

Spin wave dispersion relations of the ferromagnetic phase, as given by Eqs. (A14). The yellow surface corresponds to ${\omega }_1\left(\mathit{k}\right)$ and the blue surface to ${\omega }_2\left(\mathit{k}\right)$. Notice that the magnon wave vector components $k_x$ and $k_y$ are measured in units of $\frac{2\pi }{a}$ and $\frac{2\pi }{b}$, respectively, and the spin wave energy is measured in units of $\frac{SJ}{2\hslash }$. The shaded hexagon within the Oxy plane is the 1BZ of the honeycomb lattice. The plot is for $K/J=1$ and $\alpha =2\pi /3$.