Low-Temperature Electrical Resistivity of Pure Niobium

Abstract

The temperature dependence of the normal-state electrical resistivity of very pure niobium is reported here. The measurements were carried out in the temperature range from the superconducting transition (${T}_{c}=9.25\ifmmode^\circ\else\textdegree\fi{}$K) to 300\ifmmode^\circ\else\textdegree\fi{}K in zero magnetic field and from 2-22\ifmmode^\circ\else\textdegree\fi{}K in a magnetic field strong enough to quench all superconductivity. The resistance-versus-temperature data were analyzed in terms of the possible scattering mechanisms likely to occur in niobium. To fit the data a single-band model was assumed. The best fit can be expressed as ${\ensuremath{\rho}=(4.98\ifmmode\pm\else\textpm\fi{}0.7)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}5}+(0.077\ifmmode\pm\else\textpm\fi{}3.0)\ifmmode\times\else\texttimes\fi{},{10}^{\ensuremath{-}7}{T}^{2}+(3.10\ifmmode\pm\else\textpm\fi{}0.23)}{\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}7}[\frac{{t}^{3}{J}_{3}(\frac{\ensuremath{\Theta}}{T})}{7.212}]+(1.84\ifmmode\pm\else\textpm\fi{}0.26)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}10}[\frac{{T}^{5}{J}_{5}(\frac{\ensuremath{\Theta}}{T})}{124.4}]}$ where $T$ is in \ifmmode^\circ\else\textdegree\fi{}K; ${J}_{3}$ and ${J}_{5}$ are integrals occurring in the Wilson and Bloch theories, respectively; and the best value for $\ensuremath{\Theta}$, the effective Debye temperature, is (270\ifmmode\pm\else\textpm\fi{}10)\ifmmode^\circ\else\textdegree\fi{}K. The data are normalized so that $\ensuremath{\rho}(298\ifmmode^\circ\else\textdegree\fi{}\mathrm{K})=1$. The fitting scheme, which might be best described as a least-squares fractional-error fit, yielded a standard deviation of 2.6% per datum. The limits on the coefficients were obtained by varying each coefficient separately so as to make the fit worse by 1 standard deviation. In this analysis the Bloch term contributes about 55% to the total resistivity at room temperature and the Wilson term about 45%. Over most of the temperature range below 300\ifmmode^\circ\else\textdegree\fi{}K, the ${T}^{3}$ Wilson term dominates. Thus it is concluded that interband scattering is quite important in niobium. Because of the large magnitude of interband scattering, it was difficult to determine the precise amount of ${T}^{2}$ dependence in the resistivity. Several schemes for establishing an upper bound to this term were employed. The smallest, but still reliable, upper bound for the coefficient of this term $B{T}^{2}$ was found to be $B\ensuremath{\le}5.7\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}7}$ in the normalized units. This corresponds to about $B\ensuremath{\le}8\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}12}$ \ensuremath{\Omega} cm/${\mathrm{deg}}^{2}$, which is more than an order of magnitude smaller than previously thought for niobium. There is an interesting relation between the intraband term in niobium and the total resistivity of molybdenum; it is found that they are nearly equal at their respective Debye temperatures. This result is briefly discussed in terms of their respective $d$-band densities of states.