Extremely high conductivity observed in the triple point topological metal MoP

Weyl and Dirac fermions have created much attention in condensed matter physics and materials science. Recently, several additional distinct types of fermions have been predicted. Here, we report ultra-high electrical conductivity in MoP at low temperature, which has recently been established as a triple point fermion material. We show that the electrical resistivity is 6 nΩ cm at 2 K with a large mean free path of 11 microns. de Haas-van Alphen oscillations reveal spin splitting of the Fermi surfaces. In contrast to noble metals with similar conductivity and number of carriers, the magnetoresistance in MoP does not saturate up to 9 T at 2 K. Interestingly, the momentum relaxing time of the electrons is found to be more than 15 times larger than the quantum coherence time. This difference between the scattering scales shows that momentum conserving scattering dominates in MoP at low temperatures.

show the calculated diffraction pattern of the 6 ̅ 2 hexagonal-space group. After identifying the direction, the crystals were cut along desired crystallographic axes for further measurements.   Temperature dependent measured total thermal conductivity,  (total), calculated phononic (phonon) part from heat capacity and the subtracted electronic part from the measured data. The error bar in the total thermal conductivity data is the standard deviation of systematic error estimated from the uncertainty in thermal gradient and heater voltages.

Supplementary Note 5
Electrical resistivity. After characterization of two batches of MoP crystals, seven crystals were cut in similar bar shapes but in desired crystallographic directions (see Supplementary Table 1) and four contacts for resistivity and five contacts for Hall resistivity were made using 25 m Pt wire with Ag paint. Supplementary Table 1 summarizes the crystals' dimension, direction of current, , magnetic field, , residual resistivity at T = 2 K, residual resistivity ratio (300K/3K), mobility, , and charge carrier density, n, for each crystal. We can see that  is independent from the crystallographic direction showing isotropic transport in MoP.
Consequently, the phononic contribution of  at 300 K is: ph (300 K) = total (300 K)-el (300 K) = 1132-89 = 1043 WK -1 m -1 From the kinetic relation, ph (T) is written as: where  is the sound velocity and lph the phonon mean free path. For simplicity, we assume that lph is independent of temperature. However, lph is generally increasing with decreasing temperature. By this assumption, we can now calculate the temperature-independent 1/3  2 tph = ph (300 K) / Cph(300 K).  Fig. 13b and 13c, respectively.
Since the Dingle term varies exponentially with field (Eq. (1) Supplementary Fig. 13d. The linear slope divided by 14.69m* yields the value of , which is (1.60  0.15) K for the largest Fermi surface. The oscillatory deviation from the straight line fit is due to the presence of beating caused by the two close spin-split  frequencies.
Absence of extrinsic effects in the determined Dingle temperature. Independent of the method used to determine the value of , the oscillations must be free from extrinsic factors such as field inhomogeneity, sweep rate of the magnetic field and magnetic impurities within the sample.
From our sample characterizations, we can exclude magnetic impurities in our sample. The magnets used in this measurements are highly precise and homogeneous. The homogeneity of the used 18 T superconducting magnet over the sample is better than 0.2 mT at 18 T, while it is 10 -4 for the 35 T magnet. It should be noted that the spatial field dependence follows approximately a parabolic function as shown in Supplementary Fig. 14b. In general, the field inhomogeneity, H for a given frequency F should be 6 In order to observe the dHvA frequency of 14600 T in a field of 12 T (average field of our analysis), H should be much smaller than 1.6 mT. The field profile of our 18 T magnet was measured by use of a Hall sensor ( Supplementary Fig. 14b). At full field the field inhomogeneity at the sample position, with the sample having dimensions of 1  0.7  0.2 mm 3 , is less than 0.2 mT (that means it is below the resolution of our Hall sensor). Correspondingly, at 12 T the inhomogeneity is less than 0.13 mT. This indeed is much smaller (by more than a factor of 10) than the estimated 1.6 mT.
Another factor which affects the value of is the sampling rate, i.e., the number of data points taken per oscillation period. By adjusting the sweep rate stepwise according to the magnetic field value (0.005 T/min at 4.5 T to 0.1 T/min at 18 T), we ensured the acquisition of at least 12 data points per oscillation period. To estimate the influence of the sweep rate on the dHvA amplitude, the artificial damping due to the limited sampling rate was calculated by generating a sine function of constant amplitude and simulating the loss of amplitude due to decreasing data-point density and averaging effects ( Supplementary Fig. 14a). The data measured in the 35 T magnet were taken at a rather fast sweep rate. At the highest field, only 4-5 data 21 points per period were recorded, which caused an extra damping. Therefore, we refrained from extracting TD from the data measured in the 35 T magnet.
We conclude that using the data obtained in out 18 T magnet the value of is neither affected by magnetic impurities nor field inhomogeneity nor by a too fast magnetic-field sweep rate.