Electrons confined in a GaAs quantum well form a two-dimensional electron gas (2DEG)…

PHYSICS · Multi-part · Fermi University Physics · Thermal, Statistical & Complex Systems · Condensed Matter, Materials & Soft Matter · Quantum statistical mechanics and quantum gases

Solid State Physics, Ch The degenerate electron gas, finished

Problem

Electrons confined in a GaAs quantum well form a two-dimensional electron gas (2DEG) with effective mass

m^*=0.067\,m_e

and areal density

n_s=5.0\times10^{15}\,\mathrm{m^{-2}}

(both spin states counted). Treated as a two-dimensional free-electron gas, the density of states per unit area is

D(E)=m^*/(\pi\hbar^2)

, a constant independent of

E

, for

E > 0

and zero below. ((a)) Starting from the grand partition function of a single non-degenerate level that can hold zero or one electron, derive the Fermi-Dirac occupation function

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

. ((b)) Apply the Sommerfeld expansion to the number equation

n_s=\int_0^\infty D(E)f(E)\,dE

for this constant

D(E)

to show that

\mu(T)=E_F

to order

T^2

, and give the one-line reason why no shift appears at this order. ((c)) Apply the Sommerfeld expansion to

U=\int_0^\infty E\,D(E)f(E)\,dE

to show that

C_V/A=\gamma T

with

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

, and evaluate

E_F

, the Fermi temperature

T_F

, and

\gamma

(per unit area) numerically for this 2DEG.

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