Evidence for non-s-wave symmetry of the pi gap in MgB[subscript 2] from intermodulation distortion measurements

Abstract

We present low-temperature low-power intermodulation-distortion (IMD) measurements of high-quality MgB[subscript 2] thin films that are inconsistent with presumed s-wave symmetry of the order parameter. The measurements were carried out in a stripline resonator at approximately 2 GHz between 1.8 K and T[subscript c]. The IMD arises from the nonlinear Meissner effect in which the penetration depth is dependent on the RF magnetic field. Specifically, the observed IMD vs temperature T for T≪T[subscript c]/2 varies as T[superscript −2], while for an s-wave gap symmetry in the clean limit, the low-temperature IMD decreases exponentially with decreasing temperature. We calculate the IMD from first principles for different order-parameter symmetries using a Green’s function approach and compare the results with the measured data. We propose that the observed upturn in the low-temperature IMD implies an admixture of an order parameter with nodal lines into the energy gaps of MgB[subscript 2]. Most likely, this admixture is prominent for the π gap. Within the constraints of the hexagonal crystal symmetry of MgB[subscript 2], the best fit with our IMD measurements is obtained with a gap Δ(ϕ,T)=Δ[subscript 0](T)sin(6ϕ), where ϕ is the azimuthal angle in the abˆ plane, and Δ[subscript 0](T) is the amplitude, weakly temperature dependent at low temperatures. This gap symmetry entails six nodal lines. We also present low-temperature penetration-depth measurements that are consistent with the proposed nodal gap symmetry. To relate our proposition with existing literature, we review other low-temperature probes of the order-parameter symmetry. The literature presents conflicting results, some of which are in direct support of the symmetry proposed here.