Pseudospin heat conductivity in the J_{\mathrm{eff}}=1/2 antiferromagnet Sr{}_{2}IrO{}_{4}

Pseudospin heat conductivity in the antiferromagnet SrIrO

Frank Steckel Leibniz-Institute for Solid State and Materials Research, IFW-Dresden, 01069 Dresden, Germany    Akiyo Matsumoto Department of Physics and Department of Advanced Materials, University of Tokyo, Hongo 113-0033, Japan    Tomohiro Takayama Max-Planck-Institute for Solid State Research, 70569 Stuttgart, Germany    Hidenori Tagaki Department of Physics and Department of Advanced Materials, University of Tokyo, Hongo 113-0033, Japan Max-Planck-Institute for Solid State Research, 70569 Stuttgart, Germany    Bernd Büchner    Christian Hess Leibniz-Institute for Solid State and Materials Research, IFW-Dresden, 01069 Dresden, Germany Center for Transport and Devices, TU Dresden, 01069 Dresden, Germany
July 27, 2019

We report the in-plane and out-of-plane heat conductivity of the antiferromagnetic spin-orbit induced Mott insulator with . Our data reveal clear-cut evidence for magnetic heat transport within the IrO planes which provides the unique possibility to analyze the thermal occupation and scattering of pseudospin excitations. The analysis of the magnetic heat conductivity yields a low-temperature ( K) magnetic mean free path  nm, consistent with boundary scattering. Upon heating towards room temperature, the mean free path strongly decreases by one order of magnitude due to thermally activated scattering of the pseudospin excitations. The latter reveals that the coupling of these excitations to the lattice is radically different from that of -excitations in cuprate analogs.

71.70.Ej, 44.10.+i, 66.70.-f

The physics of iridium oxide materials has recently moved into the focus as these materials realize a plethora of novel quantum magnetic phases based on, e.g., the square, honey-comb, and hyperkagome lattice types Kim et al. (2009, 2008); Ye et al. (2013); Singh and Gegenwart (2010); Okamoto et al. (2007) with pseudospins. This includes the sought-after possible realization of a quantum spin-liquid with peculiar elementary excitations Qi et al. (2009); Lawler et al. (2008). The magnetic heat conductivity is considered an important tool to probe quantum spin and topological excitations Qi et al. (2009), as it is sensitive to both the thermal occupation and the scattering of such quasiparticles. In the past years, this sensitivity has been exploited extensively for probing the elementary spin excitations in many different low-dimensional quantum magnets Yamashita et al. (2010, 2009); Sologubenko et al. (2001); Hess et al. (2007); Hlubek et al. (2010); Sologubenko et al. (2000); Hess et al. (2001, 2003a); Berggold et al. (2006); Hess (2007). For materials with a strong spin-orbit coupling (SOC) and pseudospins, such as the iridates, it remains however unclear whether heat transport can be used to probe the pseudospin excitations at all, because the strong SOC is likely to cause strong scattering due to phonons. Accordingly, successful experiments on iridate materials are lacking, apart from pioneering attempts Singh et al. (2013).

One of the up to the present best studied iridate materials is the compound which is a spin-orbit induced Mott insulator Kim et al. (2008) with localized electrons on the Ir ions in a state. The material possesses a very similar structure as LaCuO, i.e. a square lattice of Ir ions is formed by corner-sharing IrO plaquettes, where adjacent IrO-planes are separated by SrO layers Huang et al. (1994). A strong antiferromagnetic exchange of the order  eV couples the pseudospins giving rise to two-dimensional (2D) magnetic excitations as is revealed by resonant inelastic x-ray scattering (RIXS) Kim et al. (2012). orders long-range antiferromagnetically at  K, where a weak ferromagnetic moment occurs due to canting of the IrO octahedra Cao et al. (1998); Jackeli and Khaliullin (2009); Ye et al. (2013).

In this Letter, we report the in-plane and out-of-plane heat conductivity of . We observe a highly unusual in-plane heat conductivity with anomalous temperature dependence at which is absent for the out-of-plane direction. This is incompatible with phonon heat transport and evidences magnetic heat transport within the IrO planes. Thus, our data reveal the first example for magnetic heat transport in a compound. We analyze the low temperature magnetic heat conductivity in terms of a Boltzmann-type approach and extract the magnetic mean free path . At low temperature, is limited by boundary scattering. Upon heating to room temperature, decreases by an order of magnitude, indicating temperature-activated scattering. This scattering process is the dominating one at elevated temperature and can be assigned to strong magneto-elastic coupling.

The growth and characterization of single crystals of have been described in Ref. Kim et al., 2009. The crystal dimension in our experiment was  mm where the shortest edge is the crystallographic -direction. The thermal conductivity has been measured with a home-made device in a four-point configuration using a chip resistor as heater and a thermocouple to measure the temperature gradient parallel to the -planes Hess et al. (2003b). Due to the limited size of the thin plate-like single crystal it was impossible to perform a four-point measurement along the -direction. Nevertheless, a two-point measurement was possible. For our setup we carefully investigated the differences between the two- and four-point configuration. For the two-point configuration we found, that above  K the temperature dependence of the heat conductivity is reproduced correctly with the caveat of an uncertain absolute value. Therefore, we consider data from the two-point measurement only to search for anomalous temperature dependence in the vicinity of .

Figure 1: Heat conductivity of measured in the plane () and along the direction (). Purely phononic fits to the heat conductivity are shown as lines. The fit to the direction is given by the values  W/Km,  W/m and  K. Similarly, the phononic fit to the direction with  W/Km,  W/m. For the direction extreme fit parameters were used to find limits of the phononic temperature behavior marked as the shaded area. The upper bound is found by the values  W/Km,  W/m and , the lower bound by  W/Km,  W/m and .

Fig. 1 shows the measured heat conductivity of along the and directions, and , respectively. From resistivity measurements Korneta et al. (2010); Kini et al. (2006) and applying the Wiedemann-Franz-law it follows that has a low contribution of electrons to the heat conductivity which is three orders of magnitude lower than the measured . Therefore, we neglect the contribution of the electrons and consider the total heat conductivity in direction as the sum of a conventional phononic and a potential magnetic contribution, whereas is purely phononic.

The in-plane heat conductivity exhibits a peak at low temperature ( K), which is followed by a steep decrease that slightly levels off at around 75 K. At further increased temperature a broad step around is observed and the curve almost saturates close to room temperature. This temperature dependence is incompatible with canonical phononic heat conduction. In a simple approach the heat conductivity is proportional to the specific heat , the velocity , and the mean free path of the heat carriers:


In the case of phonons as heat carriers, the velocity and the mean free path are approximately constant at low temperature and follows the temperature dependence of the specific heat. At higher temperatures, umklapp scattering becomes important which reduces the mean free path and thus leads to the observed low-temperature peak. This process depends on the number of excited phonons and leads to . Thus, at high temperatures where approaches the Dulong-Petit constant, the phononic heat conductivity is approximated by Berman (1976).

The leveling off at  K and the broad step-like feature near are clearly inconsistent with this expected temperature dependence. Two completely different scenarios are conceivable for explaining this unexpected behavior. It is possible that enhanced scattering of phonons occurs due to critical magnetic fluctuations near . In fact, a dip structure near is often found in antiferromagnetic materials like in MnO Slack and Newman (1958) or in CoF Slack (1961). However, such critical fluctuations are unlikely to affect near 75 K, i.e. far away from . Moreover, the phonon scattering due to magnetic fluctuations typically affects the heat conductivity isotropically even in layered systems Hofmann et al. (2001). Therefore, we carefully inspected the heat conductivity parallel to the -axis in the vicinity of . At temperatures higher than 200 K, the curve is absolutely featureless and fully described by the aforementioned -law as is indicated in the figure. Thus, we can clearly rule out a phononic origin of the anomalous behavior in . On the other hand, the anomalous behavior can also arise from a 2D magnetic heat conductivity within the IrO-layers which adds to , i.e., results from the sum of phononic and magnetic contributions while is purely phononic. Indeed, such magnetic heat transport is frequently observed in low-dimensional quantum magnets Yamashita et al. (2010, 2009); Sologubenko et al. (2001); Hess et al. (2007); Hlubek et al. (2010); Sologubenko et al. (2000); Hess et al. (2001, 2003a); Berggold et al. (2006); Hess (2007). Thus, we conclude that in a magnetic contribution to the heat conductivity is present in .

Having established this main experimental finding, namely the first observation of magnetic heat conductivity in a system, we move on to its quantitative analysis by extracting the magnetic mean free path. A phenomenological approach is used to model the phononic heat conductivity at high temperatures (cf. the solid line in Fig. 1). To estimate the uncertainty of the phononic background in , we performed extreme phononic fits for determining lower and upper bounds as is indicated by the shaded area 111Note that the lower bound found from a fit resembles the high temperature behavior found by a fit to the Callaway model Callaway (1959)..

In the temperature range between and  K the measured exceeds the expected phononic heat conductivity remarkably, corroborating our conclusion of a significant . We subtracted the phononic fit from the measured and obtain as shown in Fig. 2. increases with increasing temperature up to  K with a maximum value of about  W/Km, and decreases for higher temperatures. If one considers the coarse generic behavior of heat conductivity given by Eq. (1), the low-temperature increase arises from the thermal occupation of pseudospin excitations (termed magnons hereafter). The peak at 150 K and the high-temperature decrease of cannot be related to the maximum of the specific heat of 2D spin excitations, as the latter is expected around  K Sengupta et al. (2003) (with the exchange constant in a 2D Heisenberg model), i.e. at much higher temperatures than considered here. Instead, the decrease must be primarily related to a temperature dependence of the mean free path since the average magnon velocity is unlikely to change strongly at the rather low temperatures (, with  eV Cetin et al. (2012); Hirata et al. (2013); Kim et al. (2012)) considered here.

Figure 2: Magnetic heat conductivity of resulting from the difference between and the phononic fit (dots). The line is the low-temperature fit with constant magnetic mean free path and a constant shift from Eq. (2). The uncertainty from the phononic fit is spanned by the shaded area.

For investigating our result for further, we follow an approach that has previously been used to analyze of the analog LaCuO Hess et al. (2003a), i.e., we extend the simple kinetic description of Eq. (1) by accounting for possible momentum dependencies in two dimensions Hess et al. (2003a); Hess (2007), i.e. with the specific heat ( and are the energy and the Bose occupation function of the mode ), the velocity and the mean free path of a magnon with wave vector .

In , the dispersion is a steeply increasing function with band maxima at Kim et al. (2012). Thus, at the low temperatures considered here, primarily modes with small momenta are relevant for the heat transport. For simplicity, we therefore assume a temperature independent mean free path . We approximate the dispersion Hess et al. (2003a), where  meV is the experimental anisotropy gap revealed by ESR Bahr et al. (2014) and  m/s the small- magnon velocity extracted from the RIXS single magnon dispersion Kim et al. (2012). Thus, considering two magnon branches, we get Hess et al. (2003a):


with and  Å,  Å the relevant lattice parameters Crawford et al. (1994); Fujiyama et al. (2012), and the Debye temperature for magnons.

We use Eq. 2 to model the low temperature by assuming a temperature independent for a certain temperature range. This corresponds to the physical picture of dominating magnon boundary scattering. A corresponding fit is shown in Fig. 2 where we account for a possible offset due to an uncertain phonon background at low temperature Hess et al. (2003a). The fit describes the data well up to 75 K and yields a low temperature mean free path of  nm. This value corresponds to times of the Ir-Ir distance .

We now address the apparent deviation from a temperature-independent mean free path that becomes evident for higher temperatures  K. In this regime, the afore used simple low-temperature Debye approach cannot be employed anymore because it fails to properly describe the magnetic specific heat at elevated temperatures. We therefore use the theoretical result of of the Heisenberg antiferromagnet on a square lattice Sengupta et al. (2003); Hofmann et al. (2003) with the exchange coupling  eV Cetin et al. (2012); Hirata et al. (2013); Kim et al. (2012) and the above for calculating the temperature dependent based on the kinetic expression (1) for a 2D system. At low temperature (c.f. Fig. 3), is roughly constant up to  K reflecting the low-temperature boundary scattering as revealed by the afore analysis. For increasing temperature, decreases strongly where the change amounts up to an order of magnitude near room temperature. At the highest temperature accessible in this experiment, seems to saturate close to 1.2 nm, i.e., one order of magnitude above the Ir-Ir distance which constitutes a natural minimum value of .

Figure 3: Magnetic mean free path of (dots) resulting from the phononic fit to along . The shaded area marks the range spanned by the lower and upper bound for the phononic heat conductivity. At low temperatures the constant magnetic mean free path is found to be  nm. The dashed line is the phenomenological fit to by Eq. (3) with  K and . The inset shows the magnetic mean free path together with the spin-spin correlation length (open squares) determined by RIXS Fujiyama et al. (2012) on a semilogarithmic scale.

The strong decrease at elevated temperatures clearly signals the onset of a temperature-activated scattering process. We assume that both the low-temperature boundary scattering and the temperature-activated process are independent of each other. Following Matthiessen’s rule, the mean free path can then be written as


where the second term is an empirical formula for temperature-activated scattering in magnetic heat transport that has been successfully used in one-dimensional systems Sologubenko et al. (2001); Hlubek et al. (2010, 2011, 2012) (with the characteristic energy scale of the temperature-activated scattering process and a proportionality factor). As can be seen in the figure, this formula fits the experimental quite well. The fit yields  K 222We estimate the uncertainty of the fit result by separately fitting the upper and lower bounds of the grey shaded area, which yield  K with and  K with . which should be considered as a very coarse estimate of the energy of the most important scattering mode. The value roughly lies in the energy range of Ir-O-Ir bond bending modes, which have been suggested to strongly couple to the electronic structure Moon et al. (2009); Cetin et al. (2012). Thus, the primary cause of the temperature-activated scattering may be ascribed to the scattering of the magnons off these phonons.

It is instructive to compare the mean free path with the spin-spin correlation length measured by resonant x-ray diffusive scattering Fujiyama et al. (2012), see inset of Fig. 3. This quantity is a conceivable natural upper limit for . In the long-range ordered phase below , the spin-spin correlation length is infinitely large and thus unimportant for the magnetic heat transport. For higher temperatures, decreases rapidly with increasing temperature but remains still more than an order of magnitude larger than . Therefore, it plays only a minor role in limiting the magnetic heat conductivity above , if any. This suggests that the seeming anomaly in around (cf. Fig. 1) is mostly unrelated to the onset of magnetic ordering but rather connected with the growing importance of temperature-activated magnon scattering. Note, however, that a faint change of slope is discernible in the semilogarithmic representation of shown in the inset of Fig. 3.

Figure 4: Magnetic heat conductivity of in comparison with that of LaCuO (reproduced from Ref. Hess et al., 2003a).

The strong magnon-phonon scattering evident in our data reveals a qualitative difference of the pseudospin heat transport of and the spin heat transport of the almost isostructural and thus closely related -system LaCuO Hess et al. (2003a), as is inferred from a direct comparison, cf. Fig. 4. Both the spin wave velocity of LaCuO Hayden et al. (1991), and the low-temperature magnon mean free path ( nm) of the sample considered in the figure Hess et al. (2003a) are approximately twice larger than those of our sample. Considering these parameters and Eq. (2), the almost identical low-temperature increase of both curves at  K can be understood as the consequence of dominating magnon-boundary scattering in both cases. However, upon further increasing , a strong suppression of of occurs while that of LaCuO continues to increase up to room temperature. Apparently, the magnon-phonon scattering in is dramatically stronger than that in LaCuO, despite similar phonon spectra in both compounds Moon et al. (2009); Cetin et al. (2012); Pintschovius et al. (1989). This unambiguously evidences a peculiar and particularly strong nature of the magneto-elastic coupling in , arising from the large SOC and the resulting entanglement of spin and orbital degrees of freedom Jackeli and Khaliullin (2009).

In conclusion, our data provide the first experimental result for low-dimensional pseudospin heat transport in an iridate compound. Our data show that the magnetic heat conductivity remains a valuable tool to probe the generation and the scattering of magnetic excitations also in these systems. At low temperatures  K, the magnetic heat transport is dominated by magnon scattering off static boundaries and thus comparable with that of 2D systems. However, at higher temperatures unusual strong magnon-phonon scattering becomes increasingly important, highlighting the peculiar nature of the pseudospin moments and excitations.

This work was supported by the Deutsche Forschungsgemeinschaft through SFB 1143. Discussions with V. Kataev and W. Brenig are gratefully acknowledged.


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