Field-dependent heat transport in the Kondo insulator SmB :
phonons scattered by magnetic impurities
The thermal conductivity of the Kondo insulator SmB was measured at low temperature, down to 70 mK, in magnetic fields up to 15 T, on single crystals grown using the floating-zone method. The residual linear term at is found to be zero in all samples, for all magnetic fields, in agreement with a previous study. There is therefore no clear evidence of fermionic heat carriers. In contrast to prior data, we observe a large enhancement of with increasing field. The effect of field is anisotropic, depending on the relative orientation of field and heat current (parallel or perpendicular). However, the anisotropy goes in opposite ways in different crystals, pointing to an extrinsic, sample-dependent mechanism. We propose that heat is carried predominantly by phonons, which are scattered by magnetic impurities.
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Samarium hexaboride (SmB) is a Kondo insulator, a material in which the interaction between the localized electrons and the conduction band gives rise to a hybridized band structure with a gap . The correlated metallic behavior at high temperature smoothly becomes insulating below 40 K with the opening of the Kondo gap, giving rise to a diverging resistance at low temperature. However, below K, a resistivity plateau is observed, the signature of a metallic state at the surface of the sample [2; 3; 4; 5; 6; 7]. This surface state may be topological in nature.
Recently, two independent studies [8; 9] reported the observation of de Haas-van Alphen oscillations (dHvA) in SmB, but with different interpretations. In the first, Li et al. attributed the quantum oscillations to a 2D Fermi surface associated with the metallic surface state . In the second, Tan et al. detected additional frequencies and attributed the quantum oscillations to a three-dimensional (3D) Fermi surface associated with the insulating bulk . Although it has been shown that dHvA oscillations can indeed occur in a band insulator like SmB , a more exotic possibility is the existence of neutral fermions.
One way to detect mobile fermions is through their ability to carry entropy, which a measurement of the thermal conductivity should in principle detect as a non-zero residual linear term in the limit, i.e. . In this article, we report low-temperature thermal conductivity measurements in high-quality single crystals of SmB down to 70 mK. In agreement with a prior study , we observe no residual linear term , at any value of the magnetic field up to 15 T. However, unlike in that prior study, we observe a large field-induced enhancement of . There are two general scenarios for this: either magnetic excitations (like magnons or spinons) carry heat, or phonons are scattered by a field-dependent mechanism, like magnetic impurities or spin fluctuations. In this Article, we argue that our data on SmB are consistent with the latter scenario, and discuss in particular the case of phonons scattered by magnetic impurities.
We studied three single crystals of SmB grown by the floating zone method at Johns Hopkins University , labeled Z1, Z2 and ZC.
Sample Z1 was prepared using in-house sources of samarium and boron, while samples Z2 and ZC were prepared using commercial sources. The two types of starting material imply different concentrations of rare earth impurities. The former predominantly contains Gd impurities , while the latter have more non-magnetic than magnetic impurities. In addition, samples contain Sm vacancies, typically at the % level. Such vacancies are known to enhance the valence of Sm in SmB from the non-magnetic Sm valence to the magnetic Sm, possibly acting like magnetic impurities . The samples were cut in the shape of rectangular platelets with the following dimensions (length width thickness, in ) : (Z1); (Z2); (ZC). The contacts were made using H20E silver epoxy. The thermal conductivity was measured in a dilution refrigerator down to mK with a standard one-heater two-thermometers technique with the heat flowing along the longest dimension. The current was injected along the (100) high-symmetry direction of the cubic crystal structure, i.e. . The magnetic field was applied either parallel () or perpendicular () to the heat current.
Note that the conductivity of the metallic surface state makes a completely negligible contribution to the sample’s thermal conductivity, so the measured is strictly a property of the insulating bulk.
In Fig. 1, the thermal conductivity of sample Z1 is plotted as vs , for various values of the field applied perpendicular to the heat current (). At K, the effect of the field is non-monotonic: decreases with at first, and then increases. At K (Fig. 1a), the behavior is simpler: the magnetic field enhances . This is true for our three zone-grown samples, as shown in Fig. 2. Preliminary data on a flux-grown crystal reveal a similar field-induced enhancement of . It is not clear to us why no enhancement was observed in the previous study by Xu and co-workers , performed on a flux-grown crystal (see Fig. 2d).
To examine whether part of the heat transport in SmB is carried by fermionic particles, we first look for a residual linear term. Simple extrapolation of the data in Fig. 1a yields at all fields. The same is true for all samples (Fig. 2), as also found by Xu and co-workers . Indeed, a linear fit to vs at describes the data well below 0.3 K, but it yields a negative value for . This means that must go over to a higher power of at very low , as expected for phonon conduction, which must go as in the limit . Moreover, the large enhancement with field does not generate a residual linear term. Indeed, the data at T also extrapolate to zero as , and are consistent with below 0.15 K. Because in all samples at all fields, we are left with no direct evidence of fermionic carriers of heat. Note, however, that we cannot entirely exclude them, as they could be present but thermally decoupled from the phonons that bring the heat into the sample .
In Fig. 3, the low-temperature thermal conductivity at T is shown for a field parallel and perpendicular to the heat current. At the lowest temperatures, the effect of a field is isotropic. However, above a certain temperature (0.4 K in Z1, 0.2 K in Z2),an anisotropy develops. Remarkably, this anisotropy goes in opposite ways for sample Z1, where is larger when , compared to sample Z2, where is larger when . This sample dependence points to an extrinsic mechanism for the field dependence of in SmB.
Here we discuss a possible scenario for the complex behavior of heat transport in SmB, where we take the conservative view that the heat is carried entirely by phonons. The question is what scatters those phonons. In the absence of electrons, since SmB is a bulk insulator, there are two kinds of scattering processes. The first kind is independent of magnetic field. It includes sample boundaries, dislocations, grain boundaries, vacancies, and non-magnetic impurities. Now because in SmB is strongly field dependent, there must be a second kind of scattering process, which depends on field, involving either low-energy magnetic excitations, such as magnons, or magnetic impurities. Recent experiments suggest the existence of intrinsic sub-gap excitations, either in the form of bosonic excitations as observed via inelastic neutron scattering , or persistent spin dynamics that extend to very low temperatures as measured in muon spin relaxation experiments .
Because SmB samples are known to contain significant levels of rare-earth impurities and Sm vacancies, phonons are certainly scattered by those impurities. Even at the 1 % level, magnetic impurities can cause a major suppression of the phonon thermal conductivity in insulators at low temperature . Phonons will scatter most strongly when their energy matches the difference between the atomic energy levels of the impurity. Applying a magnetic field will split some of those energy levels. Increasing the field can therefore make phonons at low less and less scattered. This is our proposal for why in SmB increases with field at low .
With decreasing temperature, when the phonon mean free path grows to reach the sample boundaries, it becomes constant and equal to , where is the cross-sectional area of the sample. (This is strictly true only for scattering off rough surfaces. Smooth surfaces can cause specular reflection of phonons, yielding a temperature-dependent (wavelength-dependent) mean free path that exceeds the sample dimensions .)
In that regime, the phonon thermal conductivity is given by :
where is the inverse square of the sound velocity averaged over three acoustic branches in all directions.
An estimate of the appropriate mean sound velocity may be obtained in terms of the longitudinal () and transverse sound velocities ( and )  :
Given the dimensions of sample Z1, mm, our estimate of the boundary-limited phonon conductivity is with , where the error bar reflects the uncertainty on and on sample dimensions. In Fig. 1a and 4a, we see that the data for sample Z1 at T are consistent with below 0.15 K, with , a value close to our estimate from Eq. (1). At , however, is well below that. Our hypothesis is that a magnetic scattering process present in zero field lowers in SmB, and this process is quenched or gapped by a field, until it is essentially inactive at T.
To explore this further, we plot the data as vs in Fig. 4a. We see that all curves lie below an upper bound given by the straight line with . As seen from the data plotted as vs in Fig. 4b, at K the field increases until it hits that line, at T, above which it saturates. At K, saturation occurs above T (Fig. 4b). This behavior is consistent with a scattering process that is gapped by the field.
As seen in Fig. 5, the same situation is observed in our three samples, whereby at low is confined below , where is consistent with the value estimated from Eq. 1 given the particular sample dimensions.
We have focused so far on the low-temperature regime below 0.3 K or so. At higher temperatures, there is a strong but complex field dependence of , as seen in Figs. 1b and 5. In order to make sense of it, it is instructive to look at the incremental effect of increasing the field by 1 T, as a function of temperature up to 5 K. This is shown in Fig. 6a. The difference between and reveals a peak at some temperature. As we increase the field, the peak moves up to higher and higher temperature. In Fig. 6b, the peak position is seen to scale with in a roughly linear manner. This is again consistent with a scenario of magnetic scattering being gapped by the field, with the gap growing with . Note, however, that we are now dealing with phonons of much higher energy than before.
It is interesting to look at the energy scales. Given that the phonons which dominate the thermal conductivity have an energy , Fig. 6b tells us that those phonons are scattered by a mechanism whose characteristic energy . If the scattering is associated with the Zeeman splitting of atomic levels, so that , we get from the condition , close to the value of for Sm vacancies.
One of the most puzzling features of the field dependence is its anisotropy relative to current direction. In Fig. 3 we saw that the effect of a field in sample Z1 is larger for than for , while the opposite is true for Z2. This suggests that it is not an intrinsic property of pristine SmB, but would instead come from different impurities in different samples. Note that below a certain temperature – 0.4 K in sample Z1 and 0.2 K in sample Z2 – there is no anisotropy at T (Fig. 3). This is consistent with our interpretation that once the field has gapped the magnetic scattering process, the phonon mean free path becomes independent of field as it is limited by the sample boundaries, irrespective of whether or .
In summary, we propose that rare-earth impurities in SmB, including Sm vacancies, known to be present at significant levels in even the best samples , scatter phonons, and this scattering process is suppressed by a magnetic field, at low temperature. Theoretical calculations are needed to understand the temperature and field dependence, as well as the anisotropy of the process.
We have measured the thermal conductivity of SmB down to 70 mK, in three zone-grown crystals. No residual linear term was observed in any sample, neither in zero field nor in any field up to 15 T. This means that there is no concrete evidence of fermionic heat carriers in SmB. However, the field produces a significant enhancement of in all samples. Preliminary data on a flux-grown sample are consistent with those findings.
We interpret our data in a scenario where phonons are the only carriers of heat, and they are scattered by a magnetic mechanism that is gapped by the field, such that by 15 T the phonon mean free path grows to reach the sample boundaries at the lowest temperatures. The fact that the effect of field depends on its orientation relative to the heat current in ways that are opposite for different samples points to an extrinsic mechanism. We propose that phonons are scattered by magnetic rare-earth impurities or vacancies. The fact that the field-induced enhancement of shifts linearly to higher with increasing is consistent with the Zeeman splitting of atomic levels responsible for the impurity scattering.
We thank S. Fortier for his assistance with the experiments. L.T. acknowledges support from the Canadian Institute for Advanced Research (CIFAR) and funding from the Institut quantique, the Natural Sciences and Engineering Research Council of Canada (NSERC; PIN:123817), the Fonds de recherche du Québec - Nature et Technologies (FRQNT), the Canada Foundation for Innovation (CFI), and a Canada Research Chair. Research at the University of Maryland was supported by AFOSR through Grant No. FA9550-14-1-0332 and the Gordon and Betty Moore Foundationâs EPiQS Initiative through Grant No. GBMF4419. Work at the Institute for Quantum Matter (IQM) was supported by the U.S. Department of Energy, Office of Basic Energy Sciences, Division of Materials Sciences and Engineering through Grant No. DE-FG02-08ER46544. Partial funding for this work was provided by the Johns Hopkins University Catalyst Fund.
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