A newly grown metal has a Fermi-surface sheet with cyclotron mass

PHYSICS · Multi-part · Fermi University Physics · Condensed Matter, Materials & Soft Matter · Free electrons, energy bands and electronic structure · Free electrons, energy bands and electronic structure

Solid State Physics, Ch The Fermi surface: semiclassical dynamics and its measurement

Problem

A newly grown metal has a Fermi-surface sheet with cyclotron mass mc=1.0 mem_c = 1.0\,m_e, where mem_e is the free-electron mass. Its resistivity at 300 K300\,\mathrm{K} corresponds to a collision time τ300K=2.0×10−14 s\tau_{300\mathrm{K}} = 2.0\times10^{-14}\,\mathrm{s}, and the magnet available for the measurement reaches B=5 TB=5\,\mathrm{T}. Copper's belly orbit has ℏωc/kB=9.6 K\hbar\omega_c/k_B = 9.6\,\mathrm{K} at 10 T10\,\mathrm{T}, and its thermal damping factor X/sinh⁡XX/\sinh X (with X=2π2kBT/ℏωcX=2\pi^2 k_BT/\hbar\omega_c) reaches 0.530.53 at T=1 KT=1\,\mathrm{K}, a value normally taken as amply large for a clean oscillation signal. ((a)) Assuming the same fractional temperature T/(ℏωc/kB)T/(\hbar\omega_c/k_B) is needed to reach a comparable damping factor, estimate the temperature below which this metal's oscillations would be similarly clear at B=5 TB=5\,\mathrm{T}. ((b)) Taking the low-temperature collision time to scale with the residual resistivity ratio (RRR) as τlow=RRR×τ300K\tau_{\rm low} = \mathrm{RRR}\times\tau_{300\mathrm{K}}, estimate the minimum RRR needed to satisfy ωcτlow>1\omega_c\tau_{\rm low}>1 at B=5 TB=5\,\mathrm{T}.

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