Topological thermal Hall effect of magnons in magnetic skyrmion lattice

Topological transport of fermions is governed by the Chern numbers of the energy bands lying below the Fermi energy. For bosons, e.g., phonons and magnons in a crystal, topological transport is dominated by the Chern number of the lowest energy band when the bandgap is comparable with the thermal energy. Here, we demonstrate the presence of topological transport by bosonic magnons in a lattice of magnetic skyrmions—topological defects formed by a vortexlike texture of spins. We find a distinct thermal Hall signal in the magnetic skyrmion phase of an insulating polar magnet ${\mathrm{GaV}}_4{\mathrm{Se}}_8$, identified as the topological thermal Hall effect of magnons governed by the Chern number of the lowest energy band of the magnons in a triangular lattice of magnetic skyrmions. Our findings lay a foundation for studying topological phenomena of other bosonic excitations via thermal Hall probe.

Figure 1

Magnetic skyrmion phase in ${\mathrm{GaV}}_4{\mathrm{Se}}_8$. Crystal structure of ${\mathrm{GaV}}_4{\mathrm{Se}}_8$ viewed (a) parallel and (b) perpendicular to the plane of ${\left({\mathrm{V}}_4{\mathrm{Se}}_4\right)}^{5+}$ clusters drawn by vesta software [55]. The double arrow in (a) shows the interlayer distance $d$. (c) Illustration of experimental setup. One heater and three thermometers ($T_{\mathrm{H}}$, $T_{\mathrm{L}1}$, and $T_{\mathrm{L}2}$) are attached to detect both the longitudinal and transverse temperature gradients (shown by the exaggerated color gradation for clarity) in the sample fixed on the LiF heat bath (see Methods). Temperature dependence of (d) magnetization ($M$) at 0.1 T and (e) longitudinal thermal conductivity (${\kappa }_{xx}$) at zero field. The structural transition temperature ($T_{\mathrm{S}}$) and magnetic ordering temperature ($T_{\mathrm{C}}$) are shown. The inset shows an enlarged view near $T_{\mathrm{C}}$. Color plots of thermal Hall conductivity (${\kappa }_{xy}/T$) measured in (f) magnetizing ($\left|B\right|$ up) and (g) demagnetizing ($\left|B\right|$ down) procedures. Paramagnetic (Para), cycloidal (Cyc), magnetic skyrmion (SkX), intermediate (Int) [31], and forced-ferromagnetic (FFM) phases are indicated. Red solid lines and black circles show the magnetic phase boundary determined by the previous measurements [30, 31] and magnetization measurements (top panels of Fig. 2), respectively.

Figure 2

Thermal Hall effect by magnetic skyrmions. Top, middle, and bottom panels show the field dependence of the magnetization ($M$), field-induced deviation of the longitudinal thermal conductivity $[{\kappa }_{xx}\left(B\right)-{\kappa }_{xx}\left(0\right)]$ normalized by the zero-field value, and thermal Hall conductivity divided by the temperature (${\kappa }_{xy}/T$), respectively. The field differential of magnetization $\partial M/\partial B$ is shown in arbitrary units in the top panels (solid lines). Data obtained in magnetizing ($\left|B\right|$ up) and demagnetizing ($\left|B\right|$ down) processes are shown in red and blue, respectively. (d)–(f) To show the magnetic hysteresis in the saturation field, $B_{\mathrm{sat}}$ in the $\left|B\right|$ up and $\left|B\right|$ down processes is marked separately at lower temperatures. For clarity, data in the middle panels are multiplied by a constant indicated in the panel. Data of ${\kappa }_{xy}/T$ at 0.2 K are multiplied by 0.5 by the domain volume effect in the different cooling process (see Appendix pp4). Gray solid lines in ${\kappa }_{xy}/T$ show an estimation of ${\kappa }_{xy}^{\mathrm{mag}}$.

Figure 3

Field dependence of ${\kappa }_{xy}/T$ up to 15 T at (a) 20 K, (b) 12 K, (c) 8 K, and (d) 2 K. (e) Temperature dependence of ${\kappa }_{xy}/T$ (left) and ${\kappa }_{xx}/T$ [right, the same data shown in Fig. 1]. Black circles show ${\kappa }_{xy}/T$ at 15 T [from (a) to (d)] multiplied by $-4.5$, which scales well to ${\kappa }_{xx}/T$.

Figure 4

Theoretical calculations of topological Hall effects of magnons. (a) Schematic illustration of the topological thermal Hall effects (THEs) in a lattice of Néel-type skyrmions. Magnon thermal currents are bent by the Berry phase effect given by the skyrmion lattice, giving rise to the topological THE of magnons (${\kappa }_{xy}^{\mathrm{topo}}>0$). The backaction of the topological THE produces the skyrmion Hall effect on the skyrmion lattice (${\kappa }_{xy}^{\mathrm{sky}}<0$) [26]. Color plots of the Berry curvature of the (b) lowest and (c) second lowest energy bands of the magnons in a triangular lattice of the Néel-type magnetic skyrmions. (d) Energy bands along high-symmetry points. Color denotes the Berry curvature that is normalized in each band. (e) Temperature dependence of ${\kappa }_{xy}^{2\mathrm{D}}/T$ (left) and $M$ (right) obtained by calculation (solid lines) and experiments (symbols). Experimental data obtained at 0.36 T in the magnetizing ($\left|B\right|$ up) and 0.2 T in the demagnetizing ($\left|B\right|$ down) processes are shown in red and blue circles, respectively. To fit with the theoretical result, the horizontal axis of the experimental ${\kappa }_{xy}^{2\mathrm{D}}/T$ data is normalized by $T_{\mathrm{C}}=20\mathrm{K}$. The vertical axis of ${\kappa }_{xy}^{2\mathrm{D}}/T$ in the variable temperature insert (VTI) [dilution refrigerator (DR)] measurements is multiplied by 10 (5) to further include the domain volume effect (see Appendix pp4).

Figure 5

Magnetic-field poling (MP) effect. Field dependence of (a) $M$, (b) $\left[{\kappa }_{xx}\left(B\right)-{\kappa }_{xx}\left(0\right)\right]/{\kappa }_{xx}\left(0\right)$, and (c) ${\kappa }_{xy}/T$ at 5 K of sample 2. Field differential of magnetization $\partial M/\partial B$ is shown in arbitrary units as solid lines. These measurements were done in the $\left|B\right|$ down process after cooling through $T_{\mathrm{S}}$ under zero field [zero MP (ZMP), gray and black symbols] and a finite field (MP, red symbols). See Appendix pp5 for other MP data at different temperatures.

Figure 6

Field dependence of the field-induced deviation of the longitudinal thermal conductivity $[{\kappa }_{xx}\left(B\right)-{\kappa }_{xx}\left(0\right)]$ normalized by the zero-field value (top panels), and the thermal Hall conductivity divided by the temperature (${\kappa }_{xy}/T$, bottom panels) observed in the (a) magnetization ($\left|B\right|$ up) and (b) demagnetization ($\left|B\right|$ down) measurements of sample 2. Data obtained in the first run zero magnetic-field poling (ZMP), second run ZMP, and second run magnetic-field poling (MP) are shown by gray, black, and red circles, respectively.

Figure 7

Field dependence of the magnetization up to high field at 5 K. The magnetic field was applied along [111] of the sample. The inset shows an enlarged view of the low field data.

Figure 8

(a) Temperature dependence of ${\kappa }_{xx}$ of sample 2 in the zero magnetic-field poling (ZMP) and magnetic-field poling (MP) measurements. (b) Temperature dependence of ${\kappa }_{xy}/T$ in the $\left|B\right|$ up (0.36 T, triangles) and $\left|B\right|$ down (0.2 T, reversed triangles) processes. These measurements were done in sample 2 after cooling through $T_{\mathrm{S}}$ under zero field in the first and second runs (ZMP, gray and black symbols) and a finite field the second run (MP, red symbols).

Figure 9

Color plots of the thermal Hall conductivity divided by the temperature (${\kappa }_{xy}/T$) of sample 2 measured in the (a) magnetizing ($\left|B\right|$ up) and (b) demagnetizing ($\left|B\right|$ down) procedures. Red solid lines show the magnetic phase boundary determined by the previous measurement [30]. Black circles show the phase boundary of sample 2 determined by the magnetization measurement (top panels of Fig. 10).

Figure 10

(a)–(e) Field dependence of the magnetization ($M$, top panels), field-induced deviation of the longitudinal thermal conductivity $[{\kappa }_{xx}\left(B\right)-{\kappa }_{xx}\left(0\right)]$ normalized by the zero-field value $\left[{\kappa }_{xx}\left(0\right)\right]$ (middle panels), and thermal Hall conductivity divided by the temperature (${\kappa }_{xy}/T$, bottom panels) at different temperatures of sample 2. The field differential of magnetization $\partial M/\partial B$ is shown in arbitrary units in the top panels (solid lines). Data obtained in the magnetizing ($\left|B\right|$ up) and demagnetizing ($\left|B\right|$ down) processes are shown in red and blue, respectively. For clarity, data in the middle panels are multiplied by a constant indicated in the panel. Gray solid lines in ${\kappa }_{xy}/T$ show an estimation of ${\kappa }_{xy}^{\mathrm{mag}}$.

Figure 11

Field dependence of the transverse temperature difference $\mathrm{\Delta }{T}_y=T_{\mathrm{L}1}-T_{\mathrm{L}2}$. (a) Field procedure of thermal Hall measurement. (b) Field dependence of $\mathrm{\Delta }{T}_y/Q$ of sample 1 observed at 10 K. (c) Field dependence of $\mathrm{\Delta }{T}_y^{\mathrm{asym}}/Q$ obtained for magnetizing (red) and demagnetizing (blue) processes.

Figure 12

Temperature dependence of (a) magnetization ($M$) and (b) longitudinal thermal conductivity (${\kappa }_{xx}$) of samples 1 and 2.

Figure 13

Temperature dependence of ${\kappa }_{xy}/T$ at 0.36 T ($\left|B\right|$ up) and 0.2 T ($\left|B\right|$ down, blue circles) of samples 1 and 2. Dilution refrigerator (DR) data of sample 1 are multiplied by 0.5, as described in Appendix pp4. Data of sample 2 are multiplied by 5 for a comparison.

Figure 14

Temperature dependence of ${\kappa }_{xx}$ of sample 1 at different magnetic fields. Data were measured in the $\left|B\right|$ up process in the dilution refrigerator (DR).

Figure 15

(a)–(d) Field dependence of ${\kappa }_{xy}/T$ of sample 1 in the dilution refrigerator (DR) measurements (triangles). The magnitude of ${\kappa }_{xy}/T$ in the DR measurements is almost twice as large as that observed in the previous variable temperature insert (VTI) measurements (circles).

Figure 16

Thermal current ($Q$) dependence of the longitudinal ($\mathrm{\Delta }{T}_x$, right axis) and the transverse ($\mathrm{\Delta }{T}_y^{\mathrm{asym}}$, left axis) temperature difference of sample 1 at 10 K. Note that $\mathrm{\Delta }{T}_x$ data of $\left|B\right|$ up and $\left|B\right|$ down overlap each other.