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Thin films of $ \beta- $tungsten host superconductivity in the presence ofstrong spin-orbit coupling. This non-equilibrium crystalline phase of tungstenhas attracted considerable attention in recent years due to its giant spin Halleffect and the potential promise of exotic superconductivity. However, morethan 60 years after its discovery, superconductivity in this material is stillnot well understood. Using time-domain THz spectroscopy, we measure thefrequency response of the complex optical conductivity of $ \beta- $tungstenthin film with a T$_c$ of 3.7 K in its superconducting state. At temperaturesdown to 1.6 K, we find that both the superconducting gap and the superfluidspectral weight are much smaller than that expected for a weakly coupledsuperconductor given the T$_c$. The conclusion of a small gap holds up evenwhen accounting for possible inhomogeneities in the system, which could comefrom other crystalline forms of tungsten (that are not superconducting at thesetemperatures) or surface states on $ \beta- $tungsten grains. Using detailedX-ray diffraction measurements, we preclude the possibility of significantamount of other tungsten allotropes, strongly suggesting the topologicalsurface states of $ \beta- $tungsten play the role of inhomogeneity in thesefilms. Our observations pose a challenge and opportunity for a theory ofstrongly anisotropic normal metals with strong spin-orbit coupling to describe.

Anomalously small superconducting gap in the strong spin-orbit coupled superconductor: $ \beta- $Tungsten