Highly Asymmetric Nodal Semimetal in Bulk ${\mathrm{SmB}}_6$

We show that novel low temperature properties of bulk ${\mathrm{SmB}}_6$, including the sudden growth of the de Haas–van Alphen (dHvA) amplitude (and heat capacity) at millikelvin temperatures and a previously unreported linear-in-temperature bulk electrical conductivity at liquid helium temperatures, signal the presence of a highly asymmetric nodal semimetal. We show that a highly asymmetric nodal semimetal is also a predicted property of the Kondo lattice model (with dispersionless $f$-electron levels) in the presence of Sm vacancies or other defects. We show it can result from a topological transformation of the type recently considered by Shen and Fu and eliminates the necessity of a neutral Fermi surface for explaining bulk dHvA oscillations in ${\mathrm{SmB}}_6$.

Figure 1

(a) Temperature-dependent amplitude of the 330 T dHvA frequency in ${\mathrm{SmB}}_6$ (black circles) together with a fit (blue line) to $A_{\text{dHvA}}(T)=A_c{R}_{T,c}+A_f{R}_{T,f}$, in which $R_{T,c}=X_c/\sinh X_c$ and $R_{T,f}=X_f/\sinh X_f$ are the thermal damping factors, $X_c=2{\pi }^2{m}_c^{*}{k}_{B}T/\hslash eB$ and $X_f=2{\pi }^2{m}_f^{*}{k}_{B}T/\hslash eB$, showing it to stem from the superposition of conduction electronlike and $f$-electronlike channels with effective masses $m_c^{*}$ and $m_f^{*}$, respectively. (Inset) The same fit with a logarithmic $T$ axis. (b) Collapsed curves of the low $T$ region of the bulk conductance of ${\mathrm{SmB}}_6$, inferred from Fig. 2, after subtracting the surface contribution ${\sigma }_{\text{surf}}$ from each curve, with a dotted line extending to $T=0$ added as a guide to the eye. (Inset) Similar collapsed curves from Fig. 2.

Figure 2

(a) $T$ dependence of the electrical resistivity of ${\mathrm{SmB}}_6$ grown using the floating zone technique [20] together with a fit to Eq. (1) below $\approx 4\mathrm{K}$. (b) $T$-dependent resistance (with reciprocal scaling) of flux grown ${\mathrm{SmB}}_6$ crystals of different thicknesses, rescaled so that their bulk-dominated resistances are the same at 20 K [21], showing that $s_{\text{node}}T$ is invariant to thickness, indicating it to be of bulk origin. The black lines are given by Eq. (1) in which ${\sigma }_{\text{surf}}$ is adjusted to accommodate different sample thicknesses, while $s_{\text{node}}$ is held constant. (c) Similar plot in which the surface of flux grown ${\mathrm{SmB}}_6$ is radiation damaged to different depths (numbers shown for two faces) [22], revealing that the surface contribution ${\sigma }_{\text{surf}}$ is significantly impacted, while $s_{\text{node}}T$ is invariant, again indicating $s_{\text{node}}T$ to be of bulk origin.

Figure 3

(a) The real component of the reconstructed electronic dispersion ${ϵ}_{\mathbf{k}}$ for $F_0=330\mathrm{T}$ and $m_0^{*}\approx 0.18m_e$ (where $m_e$ is the free electron mass), as for the small $\rho$ ellipsoids in ${\mathrm{SmB}}_6$ [10], for which we have $v_0\approx 640000{\mathrm{ms}}^{-1}$, $k_0\approx 1.00\times {10}^9{\mathrm{m}}^{-1}$, and ${ϵ}_0\approx 21\mathrm{meV}$. We use $\mathrm{\Delta }ϵ=2\mathrm{meV}$ [26] and $\lambda =800\AA$, which yields ${\mathrm{\Gamma }}_0=\hslash |{\mathbf{v}}_0|/\lambda \approx 5.3\mathrm{meV}$. (b) An expanded view of the electronic dispersion for the same $F$, $m^{*}$, and $\mathrm{\Delta }ϵ$, but for different values of $\lambda .\lambda =800\AA$ corresponds to the situation in which ${\mathrm{\Gamma }}_0\ll 2V$, $\lambda =120\AA$ corresponds to the situation in which ${\mathrm{\Gamma }}_0\lesssim 2V$, $\lambda =80\AA$ corresponds to the situation in which ${\mathrm{\Gamma }}_0\gtrsim 2V$, and $\lambda =40\AA$ corresponds to the situation in which ${\mathrm{\Gamma }}_0\gg 2V$.

Figure 4

(a) The real component of the reconstructed electronic dispersion ${ϵ}_{\mathbf{k}}$ for the small $\rho$ ellipsoids in ${\mathrm{SmB}}_6$ [10], using $\mathrm{\Delta }ϵ=2\mathrm{meV}$ [26] and $\lambda =80\AA$, which yields ${\mathrm{\Gamma }}_0=\hslash |{\mathbf{v}}_0|/\lambda \approx 53\mathrm{meV}$. (b) The imaginary contribution for $\lambda =80\AA$. Vertical dotted lines indicate the values of $k$ (assuming circular geometry) at which Landau levels intersect ${ϵ}_{\mathbf{k}}^{\pm }$ and ${\mathrm{\Gamma }}_{\mathbf{k}}^{\pm }$ at $B=31.4\mathrm{T}$, given by all values of $k=\sqrt{(2eB/\hslash )(\nu +\frac{1}{2})}$ for which $\nu$ is an integer.