Evidence for non-s-wave symmetry of theπgap inMgB2from intermodulation distortion measurements

Abstract

We present low-temperature low-power intermodulation-distortion (IMD) measurements of high-quality ${\text{MgB}}_{2}$ 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 ${T}_{c}$. 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 $T⪡{T}_{c}/2$ varies as ${T}^{\ensuremath{-}2}$, 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 ${\text{MgB}}_{2}$. Most likely, this admixture is prominent for the $\ensuremath{\pi}$ gap. Within the constraints of the hexagonal crystal symmetry of ${\text{MgB}}_{2}$, the best fit with our IMD measurements is obtained with a gap $\ensuremath{\Delta}(\ensuremath{\phi},T)={\ensuremath{\Delta}}_{0}(T)\text{sin}(6\ensuremath{\phi})$, where $\ensuremath{\phi}$ is the azimuthal angle in the $\stackrel{ˆ}{ab}$ plane, and ${\ensuremath{\Delta}}_{0}(T)$ 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.