Thermal resistivity and hydrodynamics of the degenerate electron fluid in antimony

Detecting hydrodynamic fingerprints in the flow of electrons in solids constitutes a dynamic field of investigation in contemporary condensed matter physics. Most attention has been focused on the regime near the degeneracy temperature when the thermal velocity can present a spatially modulated profile. Here, we report on the observation of a hydrodynamic feature in the flow of quasi-ballistic degenerate electrons in bulk antimony. By scrutinizing the temperature dependence of thermal and electric resistivities, we detect a size-dependent departure from the Wiedemann-Franz law, unexpected in the momentum-relaxing picture of transport. This observation finds a natural explanation in the hydrodynamic picture, where upon warming, momentum-conserving collisions reduce quadratically in temperature both viscosity and thermal diffusivity. This effect has been established theoretically and experimentally in normal-state liquid 3He. The comparison of electrons in antimony and fermions in 3He paves the way to a quantification of momentum-conserving fermion-fermion collision rate in different Fermi liquids.


A. Supplementary Note 1 : Magnetoresistance and mobility.
The high mobility of charge carriers in Sb leads to a very large magnetoresistance, as reported in Ref. [8]. The samples presented in this study confirm this. As an example, the magnetoresistance of sample S6 at T = 2K and B = 9T is shown in Supplementary Figure 1.a. This large magnetoresistance translates into a suppression of the electronic thermal conductivity through the Wiedemann-Franz law. As a consequence, the separation of lattice and electronic contributions of κ becomes straightforward. The mobility and the magnetoresistance of the samples used in this study are shown in Supplementary Figure 1.b and compared to other semi-metals. One can see that carriers in Sb are extremely mobile compared to most other semi-metals. Supplementary Table 1 compares the electronic properties of Sb with a few other semi-metals. Supplementary Figure  2 shows the magnitude of the electrical T 2 -resistivity prefactor A 2 in four different semi-metals. One can see that A 2 decreases with increasing Fermi temperature, as previously noted in the case of numerous dilute metals [9]. The correlation between A 2 and E 2 F is an extension of the Kadowaki-Woods correlation [10] to low-density systems [11].

B. Supplementary Note 2 : Estimation of the electronic mean-free-path
In the Drude picture, the measured residual resistivity, ρ 0 is related to the scattering time of electrons and holes and their masses by Supplementary Eq. (1): In Sb, the compensation between electron and hole densities holds with an accuracy of 10 −4 and one has: n = p = 5.5 × 10 19 cm −3 [8]. However, electrons and hole pockets have different shapes, significant mass anisotropy and are not aligned parallel to each other. Their associated scattering time is unlikely to be identical. The mean-free-path of the samples given in table 1 of the main text was extracted from their residual resistivity using a conservative and crude approximation. If the Fermi surface is composed of z h spheres for hole-like and z e spheres for electron-like carriers, then the average Fermi wave-vector for both k e F = (3π 2 (n/z e )) 1/3 and k h F = (3π 2 (p/z h )) 1/3 . Now neglecting the possibility that for holes the valleys may be connected to each other (Fig.2 of the main text), we took z e = z h = 3 and found k h F = k e F = 0.82nm −1 . Depending on the orientation, the actual and anisotropic k F resides between 0.45 and 2.4nm −1 [2]. In this approximation, the mean-free-path can be evaluated using the Drude formula and becomes Supplementary Eq.(2): The 0 values given given in Table 1 of the main text, has been extracted using this equation with z e = z h = 3. In this approximation, the residual resistivity times the average diameter s has a lower boundary set by the carrier concentration ( Supplementary Eq.(3)).

C. Supplementary Note 3 : Dingle mobility
Quantum oscillations have been used to map the Fermi surface of Sb [13]. As seen in Supplementary Figure 3.a, they are easily observable in our crystals. The Dingle analysis yields a mobility, which is much lower than the mobility extracted from residual resistivity. Moreover, as one can see in in Supplementary Figure 3.b, they barely change in four different samples, in spite of their ten-fold variation in residual resistivity. While 0 in sample S1 is 10 times shorter than in sample S3, the mobility is only 1.2 time larger.
Such a large discrepancy have been found in other dilute metals [6,14]. In all three cases, the quasi-particle lifetime extracted from transport is orders of magnitude longer than the Dingle scattering time. Our cleanest samples show a 2000-fold discrepancy, which is to be compared to what was reported for the cleanest sample in Cd 2 As 3 (r ≈ 10000) and in WP 2 (r ≈ 5000).
The most plausible explanation is to assume that disorder comes with a variety of length scales. There is a broad distribution of the effective size of the scattering centers. The mean-free-path according to residual resistivity is long, because point-like defects (such as extrinsic atoms) do not efficiently scatter a carrier whose wavelength extends over 10 interatomic distances. The mean-free-path according to quantum oscillations is short, because such defects can affect the phase of the travelling electron. They are therefore capable of broadening Landau levels.
This interpretation would also explain the equality of Dingle mobilities in contrast to the difference in residual resistivities. The impurity content of all samples is expected to be identical, because they were grown from an identical melt, but this is not the case of dislocation density and other extended scattering centers, which can be removed by heat treatment.
The amplitude of the magnetoresistance is set by µ 0 extracted from residual resistivity and not by µ D . The cleaner the sample, the larger its magnetoresistance (see Supplementary Figure 1).

D. Supplementary Note 4 : Low field & low temperature recovery of the Wiedemann-Franz law
Supplementary Figure 4 shows the thermal conductivity plotted as κ/T as a function of T 2 in sample S4 in the low temperature region (where we showed the WF law to be satisfied in the main text) for three different magnetic fields. The arrows point to the value of L 0 /ρ B . We observe that the arrow and y-axis intercept of the linear fit match for the three magnetic fields : the WFL is recovered under the effect of these three fields. Furthermore, the slope of the linear fit to κ/T (T 2 ) remains similar for the different fields. This implies that the magnetic field does not affect the lattice thermal conductivity.

E. Supplementary Note 5 : Thermal conductivity and the third Onsager coefficient
What we have measured is the thermal conductivity measured in absence of charge current. It is to be distinguished from the thermal conductivity measured in absence of electric field, which is a pure diagonal Onsager coefficient [15]. However, in our case, the distinction is totally negligible. The heat current density, J Q and the particle flow density, J N are Onsager fluxes responding to Onsager forces : ∇ 1 T and 1 T ∇µ in Supplementary Eq.(4,5).
The thermal conductivity, κ, in absence of charge current (J e = 0) and the one, κ in absence of potential gradient (∇µ = 0 ) are to be distinguished. The latter is inversely proportional to the Onsager coefficient L 22 as shown in Supplementary Eq.(6) The former is a combination of all three Onsager coefficients and its magnitude is given by Supplementary Eq. (7): In our case, since S < 5 × 10 −6 V/K and L ∼ L 0 = 2.45 × 10 −8 V 2 /K 2 , one has S 2 L < 0.001, implying a negligible difference.

F. Supplementary Note 6 : Ambipolar Thermal Conductivity
The electronic thermal conductivity of a semi-metal includes monopolar contributions from both electrons (κ e ) and holes (κ h ) as well as an ambipolar one associated with electron-hole pairs (κ eh ). This last contribution is negligible in Sb at T T F . Heremans et al. showed that the ambipolar contribution to thermal conductivity κ eh can be written as Supplementary Eq.(8) [16]. σ e and σ h are respectively the partial electrical conductivities associated with electrons and holes while E F,e and E F,h are the Fermi energies respectively associated with electrons and holes.
In the temperature range of interest of the present study, T < 10K, the Fermi energy of holes and electrons in Sb (featured in Supplementary Table 1) leads to (k B T /E F,i ) 2 ≈ 10 −4 . This implies, at best, an ambipolar correction to the Lorenz number L eh = 5.10 −4 L 0 at T = 10K. Such a correction falls within the experimental error bars of this study and is consequently neglected in our discussion. The small magnitude of the Seebeck coefficient confirms this conclusion.

G. Supplementary Note 7 : Viscosity, thermal conductivity and quasi-particle lifetime in Fermi liquids
Abrikosov and Khatalnikov [17] in their 1959 seminal paper calculated the viscosity of a Fermi liquid given in Supplementary Eq. (9) : Here < W η > is a temperature-independent parameter representing the angular average of scattering amplitude for viscosity, η, expected to decrease with warming as T −2 . The same collisions lead to a thermal conductivity expressed as in Supplementary Eq. (10).
< W κ >, like < W η >, is neither dimensionless nor universal. The amplitude of both depends on the strength and the anisotropy of interaction and, in the case of 3 He, strongly depends on the spin components of the overlapping wave-functions. Numerous experiments confirmed that η ∝ T −2 [18][19][20] and κ ∝ T −1 [21,22]. In the case of thermal conductivity, the most elaborate set of measurements performed by Greywall [22] found that at zero pressure, the asymptotic value for κT is κT | 0 = 2.9 × 10 −4 W.m −1 . This is about 0.6 of the theoretical value of calculated by Brooker and Sykes (5 × 10 −4 W.m −1 ) [23].
Calkoen and van Weert [24] showed that in the zero temperature limit, one can write Supplementary Eq. (11).
In this equation, the notation takes = 1. In our equation 2 of the main text, in order to enhance clarity, we have introduced the dimensionless parameter B 0 , which is simply proportional to A 2 as shown in Supplementary Eq.(12).
Calkoen and van Weert [24] found that in 3 He, a nearly ferromagnetic liquid, the magnitude of A and its variation with pressure is compatible with the Landau parameters extracted from specific heat data [25]. The fundamental reason behind the temperature dependence of η and κ is the quadratic temperature dependence of the relaxation time, which can be written as Supplementary Eq.(13) [26]: Here < A > θ,φ represents the angular averages of quasi-particle scattering amplitudes for transition between spin singlet and spin triplet states [26]. In the case of 3 He, measurements of viscosity [19] and thermal conductivity [22] have found values for τ κ T 2 and τ η T 2 close to each other. τ κ T 2 can be extracted from the heat capacity per volume C v , using Supplementary Eq. (14) : As in the case of 3 He, we have used the electronic specific heat of Sb (γ = 0.105mJ.mol −1 .K −2 ) [27] and the average Fermi velocity to calculate τ κ T 2 in Sb.