low-dimensional quantum magnets 71
3.5
Experimental observation of anisotropic heat trans-
port in low-dimensional quantum magnets
The theoretical work on the energy dynamics of low-dimensional quantum magnets to be presented in Chap. 3.7 is strongly motivated by the ex- perimental observation of anisotropic heat transport in materials that are known to show strong signatures of low-dimensional quantum magnets. Even though the focus of the theoretical analysis will be on chains, we first discuss the so-called telephone number compounds (Sr,Ca,La)14Cu24O41, which are spin ladder materials. They played an important role in realizing that the magnon contribution to thermal transport in magnetic insulators can out- weigh the phonon contribution at elevated temperatures (see Ref. [3] for a review). Furthermore, all-optical measurement techniques, that resemble a non-equilibrium setup, have been applied to Sr9La5Cu24O41 [54, 55]. Second, we discuss recent results for the spin chain material SrCuO2where an increase in sample purity has led to a drastic increase of the magnetic contribution to the thermal conductivity [154].
The anisotropic heat transport of Sr14Cu24O42 has first been reported in- dependently in Refs. [168] and [182], after a thorough investigation of these compounds as part of the search for high-TC superconductors [183]. In copper based quantum magnets such as (Sr,Ca,La)14Cu24O41the electronic transport properties are dominated by the Cu2+ super exchange along the Cu-O-Cu bonds in the CuO layers. In this configuration the Heisenberg type (S~·S~) su- perexchange interaction is mediated via hybridization with the intermediate oxygen atoms [18]. Specifically, in (Sr,Ca,La)14Cu24O41, every second plane of the crystal lattice consists of Cu2O3 ladders. The inter-ladder-coupling is frustrated while the inter-plane coupling is small, effectively isolating the spin ladders [11, 12]. The chains, which constitute the intermediate planes of the crystal lattice, were found to be nearly dispersionless and as such do not contribute to thermal transport [184, 185, 186, 187, 188].
In the following we concentrate on one specific example of the spin ladder compounds, namely Sr9La5Cu24O41. Replacing Sr with La to a certain degree reduces the intrinsic hole doping of the ladder planes, leading to very clean spin ladders [168]. The left panel of Fig 9 shows the heat conductivity κ of Sr9La5Cu24O41 as a function of temperature measured along the three different axis of a three dimensional bulk crystal [168]. The two curves with overall smaller values, are obtained from measurements along the a and b axis of the crystal. They can be understood as the standard phonon contribution [168]. However, if the thermal conductivity is measured along the c-axis of
Figure 9: Anisotropic heat conductivity in Sr9La5Cu24O41. Left: Anisotropic thermal conductivity, taken from Ref. [168] with permission from the author. Right: Snapshots of the time resolved spreading of heat in Sr9La5Cu24O41us- ing the microthermal imaging technique [54], taken from [55] with permission from the author.
the crystal the thermal conductivity is broadly peaked around 175K. The fit based on the phononic conductivity can only describe the behavior at very low temperatures in this case. However, it can be used to isolate the anisotropic contribution by substracting the phononic part and studying κmag = κ− κphonon. The high temperature peak is nowadays understood as the magnon contribution to heat transport [155, 168].
Exhibiting the largest magnetic contribution to the thermal conductivity for a long time, Sr9La5Cu24O41 was prone to be the best candidate for new experiments. Otter et al. used two techniques to measure the time-resolved heat propagation through a bulk sample [54], as opposed to the established steady-state techniques. The so called micro-thermal imaging is especially interesting as far as it motivates the approach to non-equilibrium transport presented in Chap. 3.6 and 3.7. The experimental protocol can be briefly summarized as follows: A bulk sample of Sr9La5Cu24O41 is coated with a flu- orescent substance and subsequently a spot of 40µm2 on the sample surface is heated up using an 488nm argon laser. While the heat spreads out in the sample, the photoluminescence is collected and imaged by a CCD camera with an integration time of 20−30µs. The right panel of Fig. 9 shows the latest result of this technique [55], the time resolved anisotropic spreading of heat. While these non-equilibrium type-of-experiments are conceptually interesting, it is currently under discussion how to interpret the results to re-
3.5 Experimental observation of anisotropic heat transport in
low-dimensional quantum magnets 73
Figure 10: Ballistic heat transport in very pure samples of SrCuO2. Left: Anisotropic heat conductivity along the three axis of the crystal as a function of temperature. Right: Average mean free path of the magnetic excitations as a function of temperature, calculated based on the data forκusing Eq (42). Taken from Ref. [154] with permission from the author
produce the thermal conductivity as known from steady state measurements [189].
Among the spin chain materials, significant progress in has been made for SrCuO2. During the early stages of the field of heat transport in low- dimensional quantum magnets SrCuO2 showed promisingly large magnon contributions to the thermal conductivity but the magnon and phonon con- tribution are hard to separate [3, 155]. Yet, by preparing samples of previ- ously unachieved purity Hlubek et al. recently measured the highest κmag to that date and a macroscopic mean free path for the magnonlmag & 1µm [154]. The left panel of Fig. 10 shows their result for the thermal conductiv- ity measured along the three axis of the crystal and a sketch of the relevant crystal structure. The black dots where reproduced from [155] and illustrate how close the phononic and magnonic contribution are in magnitude in this case, the high-temperature peak being only a shoulder on the low temper- ature phonon peak (open symbols for a and b axis). For the high purity sample (red dots) the maximal value of the thermal conductivity doubles, and now the phonon fit shows clear magnon contributions, exceeding even those of Sr9La5Cu24O41. The right panel of Fig. 10 shows the mean free path derived from κmag within a phenomenological model for the magnon heat
conductivity, introduced in Ref. [190]: lmag =
3 πNskb2T
κmag, (42)
where Ns is the number of spin chains per unit area. As real bulk samples always have a finite defect density the authors interpret the macroscopical mean free path as an important phenomenological measure of ballistic trans- port. It is an open question whether or not the data relates to the ballistic heat transport in the Heisenberg chain.