Ballistic magnon heat conduction and possible Poiseuille flow in the helimagnetic insulator Cu$_2$OSeO$_3$

TL;DR

This study reports exceptionally high magnon thermal conductivity in Cu$_2$OSeO$_3$, revealing ballistic magnon and phonon transport and suggesting Poiseuille flow, making it a key system for magnon dynamics research.

Contribution

It demonstrates ballistic magnon and phonon transport in Cu$_2$OSeO$_3$ and proposes Poiseuille flow as an explanation for enhanced magnon mean-free paths.

Findings

Magnon thermal conductivity reaches ~70 W/mK near 5 K.

Ballistic transport observed below 1 K for magnons and phonons.

Evidence of Poiseuille flow indicating superfluid-like magnon behavior.

Abstract

We report on the observation of magnon thermal conductivity $\kappa_m\sim$ 70 W/mK near 5 K in the helimagnetic insulator Cu$_2$OSeO$_3$, exceeding that measured in any other ferromagnet by almost two orders of magnitude. Ballistic, boundary-limited transport for both magnons and phonons is established below 1 K, and Poiseuille flow of magnons is proposed to explain a magnon mean-free path substantially exceeding the specimen width for the least defective specimens in the range 2 K $<T<$ 10 K. These observations establish Cu$_2$OSeO$_3$ as a model system for studying long-wavelength magnon dynamics.

Tables

Table 1. Table 1: Ranges of scattering parameters from Callaway modeling of κ L subscript 𝜅 𝐿 \kappa_{L} .

${\ell }_0$ (mm)	$v\left(\mathrm{km}/s\right)$	$A\left({10}^4{K}^{-4}\right)$	$b$	$C\left(K^{-4}\right)$
0.15	$2.06-2.15$	$1.8-2.0$	$6.6-6.9$	$10-15$
0.31	$2.15-2.35$	$1.5-1.8$	$6.0-6.6$	$80-110$
0.60	$2.06-2.3$	$1.75-2.0$	$6.3-6.6$	$34-40$

Equations

$${\kappa }_L=\left(\frac{2{\pi }^2}{15}\right){\left(\frac{k_{B}T}{\hslash }\right)}^3{k}_B⟨v^{-2}⟩{\ell }_{ph}$$

$${\kappa }_m=\frac{\zeta (3)k_B^3{\ell }_m}{4{\pi }^2\hslash D}{T}^2,$$

$$u_m=\frac{D}{16{\pi }^2}{\left(\frac{k_{B}T}{D}\right)}^{5/2}\Gamma (5/2)\zeta (5/2;1),$$

$${\kappa }_L=\frac{k_B}{2{\pi }^{2}v}{\left(\frac{k_B}{\hslash }\right)}^3{T}^3\left[{\int }_0^{{\Theta }_D/T}\frac{x^4{e}^x}{{\left(e^x-1\right)}^2}{\tau }_C(x,T)dx\right]\left(1+\frac{\overline{{\tau }_C(x,T)/{\tau }_N(x,T)}}{\overline{{\tau }_C(x,T)/{\tau }_R(x,T)}}\right),$$

$${\rm with}\quad\overline{f(T)}=\int_{0}^{\Theta_{D}/T}\frac{x^{4}e^{x}}{\left(e^{x}-1\right)^{2}}f(x,T)dx\Bigg{/}\int_{0}^{\Theta_{D}/T}\frac{x^{4}e^{x}}{\left(e^{x}-1\right)^{2}}dx,$$

$${\tau }_R^{-1}(x,T)=v/{\ell }_{ph}+A{x}^2{T}^4\exp \left(-\frac{{\Theta }_D}{bT}\right)+C{x}^4{T}^4,$$

$${\tau }_d^{-1}=\frac{n}{7}\frac{9a^3}{4\pi v^3}{\left(\frac{M_{Se}}{\overline{M}}\right)}^2{\omega }^4,$$

$${\tau }_{3N}^{-1}=2.6S\frac{k_B}{\hslash }{T}_d^2{T}_e^{-3/2}{T}^{1/2},{\tau }_{4N}^{-1}=6.1\times 10^{-4}\frac{k_B{T}^4}{S^2\hslash T_e^3},$$

$${\tau }_{3U}^{-1}=1.4\times 10^3\frac{S{k}_B{T}_d^2}{\hslash (T_{e}T{)}^{1/2}}\exp (-12T_e/T),{\tau }_{4U}^{-1}=\frac{2}{S^2}\frac{k_B{T}^{3/2}}{\hslash T_e^{1/2}}\exp (-12T_e/T),{\tau }_i^{-1}=0.4c\frac{k_B}{\hslash }\frac{k_B{T}^{5/2}}{T_e^{3/2}},$$

$$T_d=\frac{(\text{textsl}g{\mu }_B{)}^2}{k_B{a}^3},T_e=\frac{2SJ}{k_B}.$$

$${\kappa }_m=\frac{1}{3}{C}_m{v}_m\left[{\ell }_R^B(1-\Sigma )+{\ell }_{R}F(L_{eff})\Sigma \right],$$

$$F(L_{eff})=\frac{1}{2{\pi }^2}{\left(\frac{L_{eff}}{\ell }\right)}^2\left(\sqrt{1+4{\pi }^2{\left(\frac{\ell }{L_{eff}}\right)}^2}-1\right),$$

Full text

I Introduction

Spin-mediated heat conduction in ferromagnetic materials has been of interest for decades, but a dearth of suitable ferro- or ferrimagnetic insulators exhibiting magnonic heat conduction has limited investigation FriedbergDouthett ; McCollum ; Luthi ; Douglass ; BhandariVerma ; WaltonRives ; Pan ; BoonaHeremans . The most widely studied example is yttrium-iron garnet (YIG), for which a small magnonic thermal conductivity is well-established at low temperatures. Magnon heat conduction and energy exchange between magnons and phonons have attracted renewed attention recently because of their importance for the burgeoning fields of spin caloritronics SpinCaloritronics and magnon spintronics MagnonSpintronics wherein thermally-driven spin currents induce electrical signals. Essential to the development of related technologies is a deeper understanding of magnon heat conduction and magnon-phonon interactions generally, and identifying suitable materials for realizing practical devices.

Here we report magnon thermal conductivities $\kappa_{m}\sim 70$ W/mK near 5 K in single crystals of the helimagnetic insulator Cu2OSeO3, far exceeding those observed previously in any other ferro- or ferrimagnets (including YIG). Distinguished in applied magnetic field, both the magnon and phonon ($\kappa_{L}$) thermal conductivities exhibit ballistic behavior below 1 K, with mean-free paths (mfps) limited by the specimen boundaries and $\kappa_{m}\propto T^{2}$, $\kappa_{L}\propto T^{3}$. At $T>1$ K, $\kappa_{m}$ for clean specimens increases substantially faster than $\propto T^{2}$ and reaches values twice as large as expected from spin-wave theory. We consider both magnon-phonon drag and Poiseuille flow of magnons as potential mechanisms for this enhancement, and present analysis supporting the latter.

Cu2OSeO3 is a cubic material CrystalSymmetry ; CrystalSymmetry2 (space group P213), consisting of a three-dimensional distorted pyrochlore (approximately fcc) lattice of corner-sharing Cu tetrahedra. Inequivalence of the copper sites and strong magnetic interactions within tetrahedra lead to a 3-up-1-down, spin $S=1$ magnetic state Palstra ; Belesi that persists above the long-range magnetic ordering temperature tetrahedronGS ; ESR . Weaker interactions between tetrahedra lead to their ferromagnetic ordering below $T_{C}\simeq 58$ K. Dzyaloshinsky-Moriya interactions induce a long-wavelength, incommensurate helical spin structure and promote a Skyrmion lattice phase Skyrmion ; SpecHeat near $T_{C}$ that has attracted considerable attention. At low temperatures the low-field state is helimagnetic wherein the atomic spins rotate within a plane perpendicular to the helical axis with a wavelength $\lambda_{h}\simeq 62$ nm; mutliple domains with helices aligned along $\langle 100\rangle$ directions characterize this phase. At $H\gtrsim 300$ Oe the helices of individual domains rotate along the field to form a single-domain, conical phase in which spins rotate on the surface of a cone. Further increasing the field narrows the conical angle until $H\gtrsim 1$ kOe where the ferrimagnetic, collinear-spin state emerges.

II Experimental Methods

Phase pure, single crystals of Cu2OSeO3 were grown by chemical vapor transport Growth . Cu2OSeO3 powder was first synthesized by three stoichiometric (2:1 CuO:SeO2) heat treatments at 600 *∘*C, each followed by quenching and grinding. The resulting powder was placed in an evacuated fused-silica tube with a temperature gradient of 640 *∘*C - 530 *∘*C, with NH4Cl as the transport additive. After six weeks, single crystals with typical sizes of 75-125 mm3 were seen, and seed crystals were also added to increase yield. Purity of single crystals were verified by magnetization and X-ray diffraction experiments, showing reproducibility of physical property behavior and good crystallinity.

Specimens were cut from single-crystal ingots, oriented by x-ray diffraction, and polished into thin parallelopipeds. We focus in this work on specimens with heat flow along the $[111]$ direction and perpendicular magnetic field applied along $[1\bar{1}0]$ for which our data is most extensive. Data for other orientations of heat flow and applied field will be presented elsewhere PrasaiUnpub . A two-thermometer, one-heater method was employed to measure the thermal conductivity in applied magnetic fields up to 50 kOe. Specimens were suspended from a Cu heat sink with silver epoxy and affixed with a 1 k$\Omega$ chip heater on the free end. A matched pair of RuO bare-chip sensors, calibrated in separate experiments and mounted on thin Cu plates, were attached to the specimen through 0.125 mm diameter Au-wire thermal links bonded to the Cu plates and specimen with silver epoxy. Measurements were performed in a 3He “dipper” probe with integrated superconducting solenoid.

A total of 5 different crystals were studied with transverse dimensions, $\ell_{0}\equiv 2\sqrt{{\rm\emph{\large a}}/\pi}$ (a is the cross-sectional area) ranging from 0.15-0.60 mm. Three of these ($\ell_{0}=0.15,0.31,0.60$ mm) are the primary focus of this work. A fourth crystal for which data is less complete, was cut from the same ingot as $\ell_{0}=0.15$ mm and appears in Fig. 2. Data for the fifth crystal appears in Appendix D, Fig. 7.

III Results and Discussion

III.1 Zero-field thermal conductivity

Figure 1 (a) shows $\kappa(T)$ for $H=0$ on three crystals labeled by their transverse dimension ($\ell_{0}$). Notable is the magnitude which reaches $\sim 400$ W/mK (for $\ell_{0}=0.60$ mm) at the maximum near $T=8$ K, exceptional for a complex oxide. $\kappa$ is also strongly sample dependent for $T<10$ K, scaling with $\ell_{0}$ at the lowest $T$, but not in the region of the maxima. As we discuss further below, the latter feature is attributable to differing point-defect concentrations to which $\kappa_{L}$ is sensitive near its maximum. Here we note the likely defects are Se vacancies (common in Se compounds SeDefects ) and numerical modeling of $\kappa_{L}$ (Appendices D, E, Fig. 6) implies vacancy concentrations per f.u. of 5.6$\times 10^{-4}$, $1.6\times 10^{-3}$, and $4.1\times 10^{-3}$ for the specimens with $\ell_{0}=0.15$ mm, 0.60 mm, and 0.31 mm, respectively.

We assume the measured thermal conductivity to be a sum of lattice (phonon) and magnon contributions, $\kappa=\kappa_{L}+\kappa_{m}$, valid in the boundary scattering regime ($T\lesssim 3$ K as discussed below) when the phonon-magnon relaxation time ($\tau_{ph-m}$) exceeds, but is comparable to, the phonon-boundary scattering time ($\tau_{b}$) SandersWalton . Assuming the $q=0$ relaxation to be representative of the magnon system, an estimate, $\tau_{ph-m}\sim 3\times 10^{-8}$ s at 30 K, can be inferred from intrinsic ferromagnetic resonance linewidths Seki . Since the magnon density declines as $T^{3/2}$, $\tau_{ph-m}$ should increase to $\sim 10^{-7}-10^{-6}$ s at $T\lesssim 3$ K where $\tau_{b}=\ell_{0}/v_{ph}\sim 10^{-7}$ s (using $v_{ph}\approx 2$ km/s); thus the assumption is justified.

III.2 Ballistic lattice and magnon thermal conductivities distinguished in applied field

The magnetic field dependence of $\kappa$ through the various spin phases [Fig. 1 (b)], allows for distinguishing $\kappa_{L}$ and $\kappa_{m}$. The key features of $\kappa(H)$: (1) abrupt changes of $\kappa$ at the phase boundaries, (2) a suppression of $\kappa$ with increasing field in the collinear phase and saturation at the highest fields (50 kOe) and lowest $T$. Behavior (2) is typical of $\kappa_{m}$ in ferro- and ferrimagnets McCollum ; Luthi ; Douglass ; BhandariVerma ; WaltonRives ; Pan ; BoonaHeremans – spin-wave excitations are depopulated (gapped) for fields such that ${\textsl{g}}\mu_{B}H\gg k_{B}T$ (Fig. 4 in Appendix A shows that the field at which $\kappa(H)$ saturates corresponds to ${\textsl{g}}\mu_{B}H/k_{B}T\simeq 6$). With g-factor ${\textsl{g}}\simeq 2.1$ the magnon gap is $\sim 0.14$ K/kOe, such that $\kappa(50\ {\rm kOe})\simeq\kappa_{L}$ for $T\lesssim 1.2$ K.

We find $\kappa_{L}\propto T^{n}$ [triangles, Fig. 1 (c)] with $n=2.7-3$, consistent with phonon mfps limited by the specimen boundaries [Fig. 1 (d)] and nearly diffuse scattering.

The Casimir expression for diffuse scattering, boundary-limited thermal conductivity can be used to determine the phonon mean-free path ($\ell_{ph}$) Casimir ,

[TABLE]

where $\langle v^{-2}\rangle=[(1/3)(1/v_{LA}^{3}+2/v_{TA}^{3})]^{2/3}$ is the Debye averaged sound velocity. A fit of the low-$T$ $\kappa_{L}(T)$ data [Fig. 1 (c)] to the form $\kappa_{L}=AT^{n}$ yields $A=1.52,2.32,5.62$ and $n=2.96,2.80,2.70$, respectively, for the specimens with $\ell_{0}=0.15,0.31,0.60$ mm. The power of $T$ slightly less than 3 is common in insulators Ziman , indicating some specularity to the boundary scattering. Consistent with observations, the $\ell_{0}=0.60$ mm specimen ($n=2.70$) was polished on one of its large faces with finer abrasive (1 $\mu{\rm m}$) than the other specimens (5 $\mu{\rm m}$). Longitudinal and traverse sound velocities for the $[111]$ direction from ultrasonic measurements Inosov are $v_{LA}\simeq 3.3$ km/s and $v_{TA}\simeq 1.85$ km/s, respectively. Combining these parameters in the above equation yields $\ell_{ph}\simeq 0.16,0.24,0.59$ mm, in good agreement with the effective transverse dimension of the specimens.

The corresponding $\kappa_{m}$ in the helical and conical phases computed by subtraction [vertical arrows and dashed lines, Fig. 1 (b)], are $\propto T^{2}$ for $T\leq 1$ K, consistent with constant magnon mfps (Fig. 2; $\kappa_{m}^{col}$ is omitted for clarity). For boundary-limited spin-wave heat conduction we have FriedbergDouthett ,

[TABLE]

where $\zeta(3)\simeq 1.202$. A fit of the $\kappa_{m}^{con}(T)$ data [Fig. 2 (b)] at $T<1$ K to the form $BT^{2}$ gives $B=1.25,2.3,2.6$ W/mK3, respectively, for the specimens with $\ell_{0}=0.15,0.31,0.60$ mm; the equation above implies $\ell_{m}\simeq 0.14,0.25,0.28$ mm. The value of $\ell_{m}$ for the $\ell_{0}=0.60$ mm specimen is significantly smaller than the specimen dimension, suggesting a maximum magnetic domain size. Similarly, a value of $\ell_{m}\sim 0.34$ mm for this specimen is inferred from a plot of $\kappa_{m}$ vs $\ell_{0}$ [Fig. 1 (d)]. Within the multi-domain helical phase, values for $\ell_{m}$ are roughly half as large.

The ballistic character of the magnon transport in the $T^{2}$ regime is further corroborated by using kinetic theory to convert $\kappa^{con}_{m}$ (or $\kappa^{col}_{m}$) to magnetic specific heat ($C_{m}$) and then comparing the latter to expectations of spin-wave theory. We have $C_{m}=3\kappa_{m}/(v_{m}\ell_{m})$, where $v_{m}=(2/\hbar)Dq$, $D=52.6$ meV Å2 is the spin-wave stiffness Portnichenko (the dispersion at low energy is well-described DipoleNote by $E=Dq^{2}$). The dominant magnons for boundary-limited $\kappa_{m}$ have dominantmagnons $q_{dom}=(2.58k_{B}T/D)^{1/2}$ such that $v_{m}\simeq 1040T^{1/2}$ m/s. Assuming diffuse scattering of magnons at the crystal (or domain) boundaries, the computed $C_{m}$ for all crystals agrees well with linear spin-wave theory (Appendix B, Fig. 5).

A transfer of energy from the spin system to the lattice as the magnon gap opens is implied, given the near-adiabatic conditions of the specimens during measurement. The corresponding increase in the average temperature of the sample ($T_{avg}$) in the high-field regime [solid curves, right ordinates in Fig. 1 (b)] should reflect only a fraction of the total spin energy, since much of it must be distributed within thermometers, thermal links, and heater. As a further self-consistency check on our analysis, this fraction is determined (Appendix C) to be $\sim 4\%\ (30\%)$ at $T=0.67$ K ($5.2$ K).

III.3 Determining the magnon thermal conductivites at higher $T$

Given that the phonon mfps are boundary-limited at $T\leq 1$ K, the abrupt increase in $\kappa$ at the helical-conical transition [$H\approx 250$ Oe in Fig. 1 (b)] must be attributed to an increase in $\kappa_{m}$ associated with the approximate doubling of $\ell_{m}$ noted above. It is significant that this jump, $\Delta\kappa=\kappa_{m}^{con}-\kappa_{m}^{hel}$ [Fig. 2 (a)], exhibits the same $\propto T^{2}$ behavior for magnon boundary scattering at low $T$ as found for both $\kappa_{m}^{hel}$ and $\kappa_{m}^{con}$ computed by subtracting $\kappa_{L}$ (Fig. 2). Since $\Delta\kappa$ is independent of any assumptions regarding $\kappa_{L}$, it validates the implicit assumption that $\kappa_{L}$ is independent of field.

At $T>1.2$ K where the applied field is insufficient to fully suppress $\kappa_{m}$, $\Delta\kappa$ represents a lower bound on $\kappa_{m}^{con}$ [Fig. 2 (b)] since we expect $\kappa_{L}<\kappa_{m}^{hel}$ as is clear in the data of Fig. 1 (b) at $T=5.2$ K. Very similar results for $\Delta\kappa(T)$ were found for a specimen with $[110]$ heat flow and perpendicular field along $[1\bar{1}0]$, thus a large $\kappa_{m}$ is not restricted to the $[111]$ direction PrasaiUnpub . The sharp decline of $\Delta\kappa$ at $T\gtrsim 7$ K, and its disappearance for $T\gtrsim 12$ K, indicate that $\kappa_{m}$ has a maximum at $T\sim 5-6$ K and becomes negligible for $T\gtrsim 12$ K. The latter is supported by recent spin-Seebeck measurements Aqeel indicating a sharp decline in spin-polarized heat current in the same temperature regime.

To estimate $\kappa_{m}^{con}$ at higher $T$, this behavior of $\kappa_{m}$ and the low-$T$ $\kappa_{L}$ are exploited as strong constraints on calculations of $\kappa_{L}(T)$ at $T\geq 1.2$ K using the Callaway model (Appendix D, Fig. 6). This procedure, dictates the error bars on $\kappa_{m}^{con}$ in Fig. 2 (b) and, as noted above, provides estimates of specimen defect (Se vacancy) concentrations (Appendix E).

III.4 Anomalous $T$ dependence for $\kappa_{m}$ and possible Poiseuille flow

A most striking feature of both $\Delta\kappa(T)$ and $\kappa_{m}^{con}(T)$, aside from unprecedented magnitudes, is their increase, for the two least defective specimens, with a substantially higher power of $T$ than $\propto T^{2}$ at $T\geq 1$ K (Fig. 2). An additional contribution to $C_{m}$ from spin-wave "optic" modes cannot be expected in this temperature regime since those sufficiently dispersive to contribute to $\kappa_{m}$ have energies exceeding $25$ meV Portnichenko . We are aware of only two possible mechanisms that can potentially explain this observation: (1) magnon-phonon drag, (2) Poiseuille flow of magnons. Theory suggests that for momentum-independent magnon relaxation time $\tau_{m}$, an additive phonon-magnon drag contribution should take the general form Chernyshev , $\kappa_{drag}\sim(1/3)C_{L}v_{m}^{2}\tau_{m}\propto T^{4}\tau_{m}$, thus offering a stronger $T$ dependence. The relevant magnon-phonon interactions are normal, momentum-conserving processes.

A more intriguing alternative is that magnons undergo Poiseuille flow, predicted 50 years ago for both phonons and magnons Gurzhi ; GuyerKrumhansl ; ForneyJackle , but observed only for phonons and only in exceptionally clean materials (e.g. crystalline 4He He4 ). When the mfp for normal scattering ($\ell_{N}$) is much shorter than both the transverse dimension ($\ell_{0}$) and the mfp for bulk resistive scattering processes ($\ell_{R}$), quasiparticles undergo many momentum-conserving scattering events before losing their momentum at the specimen boundaries. Under the stringent conditions $\ell_{N}<\ell_{0}/2<(\ell_{N}\ell_{R})^{1/2}$, the effective mfp approaches that for a particle undergoing random walk with step size $\ell_{N}$, mfp$\sim\ell_{0}^{2}/4\ell_{N}\gg\ell_{0}$. We pursue this scenario further since all of the relevant scattering rates for magnons have been computed Akheizer ; ForneyJackle for a Heisenberg ferromagnet in the low-$T$ regime, and interactions with phonons which underlie phonon-drag are predicted to be significantly weaker.

Forney and Jäckle ForneyJackle calculated rates for normal and umklapp magnon scattering and elastic magnon-impurity scattering (non-magnetic defects). The expressions contain three parameters (Appendix F), two of which are set by the lattice constant and exchange coupling. The only remaining free parameter is the defect concentration. Figure 3 (a) shows the relevant mfp’s employed for the least defective crystal ($\ell_{0}=0.15$ mm). The conditions for Poiseuille flow are met in the shaded region. $\kappa_{m}^{con}$ is computed [solid curves, Fig. 3 (b)] from the kinetic theory expression with a mfp described by an interpolation formula [eq. (F1)] that yields the conventional resistive scattering length well outside the Poiseuille window, $\ell_{R}^{B}=(1/\ell_{0}+1/\ell_{3U}+1/\ell_{4U}+1/\ell_{i})^{-1}$, and tends toward $\ell_{0}^{2}/\ell_{N}$ within the Poiseuille regime. Interpolation is controlled by “switching factors” GuyerKrumhansl ; deTomas related to the ratio $\ell_{N}/\ell_{R}$ (Appendix F and Fig. 8). The data are well-described by the model (with defect concentrations 12, 22, 62 ppm for $\ell_{0}=0.15$ mm, 0.60 mm, and 0.31 mm), though the computed maxima for the more defective specimens deviate from experiment, a consequence of the Poiseuille window being shifted to lower $T$ as the impurity scattering mfp decreases. This may signal inadequacy of the magnon-impurity scattering model, perhaps because spin defects in the present system may be associated with Se vacancies as suggested by a correlation between the defect concentrations inferred for magnons and phonons (Appendix F, Fig.9).

IV Summary

Our observations reveal Cu2OSeO3 to be a model system for further study of long-wavelength magnon dynamics, e.g. our proposal that magnons undergo Poiseuille flow implies that magnon “second sound” might also be observed. Since both the conical and collinear-phase magnon heat conductivities are similar in magnitude, helical magnetism is evidently not the origin of its unusually large $\kappa_{m}$. Since long-wavelength magnons play a prominent role in the spin-Seebeck effect BoonaHeremans ; SpinCaloritronics the results presented here also make it possible to investigate interfacial spin-current transfer using calibrated magnon heat currents and to explore the possible role of the spin phases on transfer efficiency.

V Acknowledgments

The authors acknowledge helpful comments from A. L. Chernyshev. This material is based upon work supported by U.S. Department of Energy (DOE), Office of Basic Energy Sciences (BES) Grant No.’s DEFG02-12ER46888 (University of Miami) and DEFG02-08ER46544 (Johns Hopkins University).

Appendix A Additional low-$T$ $\kappa(H)$ data

Figure 4 shows additional low-$T$ $\kappa(H)$ data showing suppression of the magnon contribution at high fields where we infer $\kappa=\kappa_{L}$. We also plot the field $H_{sat}$ at which $\kappa$ becomes field-independent against temperature.

Appendix B Magnetic specific heat computed from $\kappa_{m}$

As noted in Ref. Portnichenko, , the Cu4 tetrahedra of Cu2OSeO3 approximate an fcc lattice, the primitive cell of which is 4 times smaller than that of the simple cubic cell. Thus the standard low-temperature form of the magnetic specific heat per volume becomes, $C_{m}=(0.113/4)k_{B}\left(k_{B}T/D\right)^{3/2}$ (this factor of $1/4$ also appears in expressions for the spin-wave thermal conductivity). Values of $C_{m}$ (as described in the text) were computed from the measured $\kappa_{m}^{col}$ (or $\kappa_{m}^{con}$) using kinetic theory and $\ell_{m}=\ell_{0}$ for the four crystals from Fig. 2 (a), with the exception of the $\ell_{0}=0.60$ mm crystal for which we used $\ell_{m}=0.34$ mm based on the effective length inferred from Fig. 1 (d). Theory and experiment agree well (Fig. 5).

Appendix C Energy transfer from spins to lattice at high field

We estimate the fraction of total spin energy transferred to the lattice of the $\ell_{0}=0.15$ mm specimen at $T=0.67$ K, upon gapping out the spin waves in maximum field [Fig. 1 (b)], as $Q/u_{m}$ where $Q=C_{L}\Delta T$ is the heat transferred per volume, computed from the lattice specific heat ($C_{L}$) and change in $T_{avg}$ induced by applied field ($\Delta T$), and $u_{m}$ is the total energy per volume in the spin system,

[TABLE]

$\Gamma(5/2)=3\pi^{1/2}/4$ and $\zeta(5/2;1)\simeq 1.341$. With $\Delta T=0.043$ K (Fig. 1b) and using the $T^{3}$ fit to the measured specific heat (dashed line, Fig. 5) to compute $C_{L}$, we find $Q=0.14$ J/${\rm m}^{3}$ and $u_{m}\simeq 3.8$ J/${\rm m}^{3}$, such that $Q/u_{m}\approx 0.036$. At $T=5.2$ K a similar analysis yields $Q/u_{m}\approx 0.30$.

Appendix D Calculations of $\kappa_{L}(T)$

The Callaway model Berman , incorporating its recent update Allen , was employed to compute $\kappa_{L}(T)$ for each of the crystals, with parameter ranges restricted by the following constraints: (1) $\kappa_{L}$ fits the low-$T$, high-field data (where $\kappa_{L}$ is inferred directly) and the $T\geq 12$ K, zero-field data (where $\kappa_{m}$ is inferred to be negligible by the vanishing of $\Delta\kappa$), (2) the maximum in $\kappa_{con}$, computed by subtracting $\kappa_{L}$ from $\kappa$ measured at the conical-collinear transition, should occur at $T\approx 5-6$ K where $\Delta\kappa$ has its maximum, (3) $\kappa_{L}\lesssim\kappa(H=50{\rm kOe})$.

The integral expression for $\kappa_{L}$ is,

[TABLE]

[TABLE]

where $v$ is the Debye averaged sound velocity (see above), $\Theta_{D}=(\hbar v/k_{B})(6\pi^{2}N/V)^{1/3}$ the Debye temperature, $x=\hbar\omega/k_{B}T$ the reduced phonon energy, $\tau^{-1}_{C}(x,T)=\tau^{-1}_{N}(x,T)+\tau^{-1}_{R}(x,T)$, and $\tau^{-1}_{N}(x,T)$ and $\tau^{-1}_{R}(x,T)$ are phonon scattering rates for normal (momentum conserving) and resistive (momentum non-conserving) processes, respectively. $\tau^{-1}_{R}(x,T)$ included terms for scattering from boundaries, other phonons (Umklapp scattering), and point-like defects (Rayleigh),

[TABLE]

where $\ell_{ph}=\ell_{0}$ is the boundary-limited phonon mean-free path and $A$, $b$, $C$ are constants. The normal scattering rate was taken to have the same frequency dependence as for Umklapp scattering Allen , but without the exponential $T$ dependency, $\tau_{N}^{-1}(x,T)=\gamma Ax^{2}T^{4}$, with $\gamma$ a constant. A broad range for $\gamma$ was explored in the fitting and it was found that only for $\gamma\leq 1/50$ were the constraints satisfied. $\gamma=1/50$ implies a normal scattering rate that begins to exceed that for Umklapp scattering at $T\lesssim 10$ K. Phonon-magnon scattering was assumed to be substantially weaker than other scattering.

Fig. 6 shows $\kappa(H=0,T)$ data for the three specimens from Fig. 1 along with two $\kappa_{L}$ curves for each (solid and dash-dotted curves). These curves border the ranges (shading) defined by the constraints noted above. Data points for $\kappa_{m}$ in Fig. 2 (b) correspond to the middle of these ranges with error bars equal to the width of the shaded region. A summary of the scattering parameters is provided in Table I.

In Fig. 7 we compare $\kappa(T)$ at $H=0$ for the most defective $\ell_{0}=0.31$ mm specimen from Fig.’s 1-3 with a less defective crystal having the same $\ell_{0}$. Consistent with expectations, Callaway-model parameter sets for $\kappa_{L}$ (solid curves, right panel) differ principally in the defect concentration ($C$).

Appendix E Estimate of Se vacancy concentration from point-defect fitting parameters for $\kappa_{L}$

Interpreting the point-defect phonon scattering rate (Table I above) as entirely due to Se vacancies, the vacancy concentration can be estimated using RatsKlem ,

[TABLE]

where $n$ is the concentration of vacancies on the Se sublattice, $a=1.22$ Å is the Se atomic radius, $v=2060$ m/s is the sound velocity, and $M_{Se}/\overline{M}\simeq 2.05$ is the ratio of the Se mass to the average mass. Using values $C=13,37,95\ {\rm K}^{-4}$ from Table I for the $\ell_{0}=0.15,0.60,0.31$ mm crystals yields concentrations per f.u., $5.6\times 10^{-4}$, $1.6\times 10^{-3}$, $4.1\times 10^{-3}$, respectively.

Appendix F Magnon scattering rates and modeling of Poiseuille flow

Forney and Jäckle ForneyJackle computed the thermally averaged 3-magnon and 4-magnon normal ($3N$, $4N$) and umklapp ($3U$, $4U$) scattering rates and magnon-impurity scattering rate ($i$) for a quadratic magnon dispersion within the Born approximation, valid for small impurity concentration, $T\ll T_{C}$ and $T\gg k_{B}\Delta$, where $\Delta$ is the energy gap ($\sim 12\ \mu$eV for Cu2OSeO3):

[TABLE]

[TABLE]

where

[TABLE]

We initially re-scaled the values $T_{d}=0.012$ K and $T_{e}=1.0$ K employed in Ref. ForneyJackle, for EuS ($T_{C}=16.5$ K) using the ratio of lattice constants and $T_{C}$ (as a surrogate for $J$). These gave $T_{d}=0.004$ K and $T_{e}=3.5$ K. Subsequently we settled on $T_{e}=4.2$ K which provided better agreement with the data for the least defective specimen. The scattering rates were adopted without modification with the exception of the exponent of the Umklapp scattering rates (we used 10 rather than 12 as above) and the prefactor of $\tau_{4U}^{-1}$ (we decreased it by a factor 380). As noted in Ref. ForneyJackle, , these changes put our four-magnon Umklapp scattering rate in better agreement with that computed by Schwabel and Michel SchwablMichel , and produced better agreement with the data. With these modifications, the only remaining adjustable parameter was the impurity concentration ($c$).

The scattering rates were incorporated into an interpolation formula for the magnon thermal conductivity using the function described in Ref. deTomas, and derived by Alvarez and Jou AlvarezJou :

[TABLE]

[TABLE]

where $\Sigma=1/(1+\ell_{N}/\ell_{R})$, $L_{eff}=\pi\ell_{0}/(2\sqrt{2})$, $\ell\equiv\sqrt{\ell_{N}\ell_{R}}$, $\ell_{R}=(1/\ell_{3U}+1/\ell_{4U}+1/\ell_{i})^{-1}$, and $\ell_{R}^{B}=(1/\ell_{0}+1/\ell_{R})^{-1}$. We used $\Sigma^{2}$ in place of $\Sigma$ in the above expression as it provided a better interpolation $\to 0$ at low-$T$ (Fig. 8).

The impurity scattering concentrations ($c$) employed to produce the curves in Fig. 3 correlate with those found for phonon-defect scattering (Fig. 9) in the Callaway analysis of $\kappa_{L}$ (Table I).