Mixed-state surface impedance of Nb-Ti: Flux flow, pinning, and creep

We present a thorough study of the surface impedance <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:msub> <a:mi>Z</a:mi> <a:mi>s</a:mi> </a:msub> </a:math> in the mixed state of sputtered Nb-Ti coatings on insulating substrates, with the aim to assess and quantify the mechanisms governing vortex motion. Measurements are performed by means of a dual mode dielectric-loaded resonator allowing the simultaneous determination of <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"> <c:msub> <c:mi>Z</c:mi> <c:mi>s</c:mi> </c:msub> </c:math> at 16.5 and 26.7 GHz, which enable to derive the various relevant vortex dynamics parameters. A static magnetic field <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"> <e:msub> <e:mi mathvariant="normal">μ</e:mi> <e:mn>0</e:mn> </e:msub> <e:mi>H</e:mi> <e:mo>≤</e:mo> <e:mn>1.25</e:mn> <e:mtext> </e:mtext> <e:mi mathvariant="normal">T</e:mi> </e:math> is applied normally to the sample surface, in the <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"> <i:mrow> <i:mi>T</i:mi> <i:mo>=</i:mo> <i:mn>5</i:mn> <i:mtext> </i:mtext> <i:mi mathvariant="normal">K</i:mi> <i:mo>−</i:mo> <i:msub> <i:mrow> <i:mi>T</i:mi> </i:mrow> <i:mrow> <i:mi>c</i:mi> </i:mrow> </i:msub> </i:mrow> </i:math> temperature range. The measured pinning frequency is <l:math xmlns:l="http://www.w3.org/1998/Math/MathML" display="inline"> <l:mo>≈</l:mo> <l:mn>4.5</l:mn> <l:mrow> <l:mtext> </l:mtext> <l:mi>GHz</l:mi> </l:mrow> </l:math> at 5 K and 1 T, with a pronounced decrease with field. A scaling analysis of the pinning constant <n:math xmlns:n="http://www.w3.org/1998/Math/MathML" display="inline"> <n:msub> <n:mi>k</n:mi> <n:mi>p</n:mi> </n:msub> <n:mo stretchy="false">(</n:mo> <n:mi>H</n:mi> <n:mo>,</n:mo> <n:mi>T</n:mi> <n:mo stretchy="false">)</n:mo> </n:math> dependence shows that pinning is dominated by point pins. Thermal creep is negligibly small at the lowest <r:math xmlns:r="http://www.w3.org/1998/Math/MathML" display="inline"> <r:mi>T</r:mi> </r:math> , possibly becoming finite only near <t:math xmlns:t="http://www.w3.org/1998/Math/MathML" display="inline"> <t:msub> <t:mi>T</t:mi> <t:mi>c</t:mi> </t:msub> </t:math> . Flux flow resistivity is found to be consistent with the time-dependent Ginzburg-Landau predictions for a moderately dirty superconductor.