Electrons confined to a long

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 degenerate electron gas, finished

Problem

Electrons confined to a long, thin conducting wire behave as a one-dimensional free-electron gas with spin-summed density of states per unit length

D(E)=\dfrac{1}{\pi\hbar}\sqrt{\dfrac{2m_e}{E}}

for

E > 0

. The wire carries a linear number density

n_1=5.0\times10^{8}\,\mathrm{m^{-1}}

of conduction electrons. ((a)) Starting from the grand partition function of a single non-degenerate level that can hold zero or one electron, derive

f(E)=\left[e^{(E-\mu)/k_BT}+1\right]^{-1}

. ((b)) Show that the Fermi energy is

E_F=(n_1\pi\hbar)^2/(8m_e)

and evaluate

E_F

and the Fermi temperature

T_F

numerically. ((c)) Apply the Sommerfeld expansion to the number equation to find

\mu(T)-E_F

to order

T^2

for this density of states, evaluate the shift at

T=4.0\,\mathrm{K}

, and state whether

\mu

rises or falls, explaining the sign from the shape of

D(E)

. ((d)) Give

\gamma

per unit length using

\gamma=\tfrac{\pi^2}{3}k_B^2D(E_F)

and evaluate it numerically.

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