Sondheimer oscillations as a probe of non-ohmic flow in WP2 crystals

As conductors in electronic applications shrink, microscopic conduction processes lead to strong deviations from Ohm’s law. Depending on the length scales of momentum conserving (lMC) and relaxing (lMR) electron scattering, and the device size (d), current flows may shift from ohmic to ballistic to hydrodynamic regimes. So far, an in situ methodology to obtain these parameters within a micro/nanodevice is critically lacking. In this context, we exploit Sondheimer oscillations, semi-classical magnetoresistance oscillations due to helical electronic motion, as a method to obtain lMR even when lMR ≫ d. We extract lMR from the Sondheimer amplitude in WP2, at temperatures up to T ~ 40 K, a range most relevant for hydrodynamic transport phenomena. Our data on μm-sized devices are in excellent agreement with experimental reports of the bulk lMR and confirm that WP2 can be microfabricated without degradation. These results conclusively establish Sondheimer oscillations as a quantitative probe of lMR in micro-devices.


Supplementary Note 1: Other Sondheimer orbits in WP 2
As seen in Fig. 2, WP 2 has a Fermi surface (FS) with two spin-split electron and hole pockets. In the main text, we focus our attention on the larger of the dogbone-shaped electron pockets as the most likely origin of the observed Sondheimer oscillations (SO). In principle, however, each of the four pockets may contribute to SO and therefore should be investigated.
In Fig. 3, we show the results of our investigation of each of the three FS pockets not discussed in the main text. In Fig. 3a, b and e, we focus on the smaller dogbone pocket. Similarly to the larger pocket, this pocket exhibits an extended region of near-constant dA dk which may lead to SO. The average dA dk in this region is somewhat smaller, leading to a calculated thickness-dependence which makes a less accurate prediction of the experimental data. Nevertheless, we cannot exclude that this pocket also contributes to the observed SO. We do not feed back any experimental data into the calculation. Given the similarity in slope, we could not differentiate between these orbits and expect that in reality, both of them contribute. Fig. 3c, d, f and g concern the hole cylinders. Here, there are several (periodically repeated) sections of the FS with extremal values of dA dk . Using these values of dA dk to calculate the thickness dependences of the associated frequencies, we see that most of these orbits do not fit our experimental data. There is only one theoretically possible orbit on the smaller hole cylinder (Fig. 3f) which predicts frequencies in agreement with our experiment. However, it is clear that it cannot be this orbit which we observe.
Firstly, because only a narrow slice of the FS can contribute to this orbit, due to the relatively sharp peak in dA dk . The amplitude of any oscillation arising from this FS slice should therefore be approximately 40 times smaller than those arising from the dogbone pockets. Secondly, there is no sign in our data of any of the extremal orbits on the larger hole cylinder (Fig. 3g). If the extremal orbit on the smaller cylinder was observed, then the ones on the larger cylinder should be too. Based on this, it is clear that the hole cylinders play no role in the observed Sondheimer oscillations.

Supplementary Note 2: Quantum oscillations
Here we discuss the quantum oscillations (SdH) observed in our samples (see Fig. 4.) and extract a quantum lifetime τ q as well as effective masses for several orbits.
These QO can be observed as weak 1/B-periodic oscillations of the longitudinal resistivity above 14 T, shown in Fig. 4a. In order to separate the oscillatory part of the signal from the large background, we take the second derivative of the data and plot it against 1/B. From the resulting signal, we then extract the Fast Fourier Transform shown in Fig. 4b. The effective masses listed in this figure were extracted through a fitting of the peak amplitudes as a function of temperature, using the Lifshitz-Kosevich (LK) formula.
In Fig. 4c, we show a Dingle plot for a QO orbit with a frequency of 2.6 kT at a temperature of 1.5 K. This plot is made by taking the amplitudes of the peaks in the oscillatory component of the resistivity acquired after subtraction of a smooth background, ∆ρ xx , and dividing by the thermal damping term from the LK formula, R T . Plotting the logarithm of this against the inverse of the magnetic field yields a linear relation with slope 14.69 m * m e T D , where T D = 2πk B τ q . From this, we find τ q = (3 ± 1) × 10 −13 s. This is a typical value for τ q , which in our measurements ranges from 10 −13 s to 10 −12 s for different QO frequencies, temperatures and devices.
We have also studied the QO as a function of the angle between the magnetic field and the device. The results of this study are shown in Fig. 4d and are in good agreement with previously published theoretical calculations as well as data 1 .