A single GaAs/AlGaAs heterojunction carries a sheet density of electrons in GaAs ( , )

PHYSICS · Multi-part · Fermi University Physics · Quantum Mechanics · Condensed Matter, Materials & Soft Matter · Wave mechanics in one dimension

Solid State Physics, Ch Heterostructures, quantum wells and the two-dimensional electron gas

Problem

A single GaAs/AlGaAs heterojunction carries a sheet density

n_s=4.0\times10^{11}\,\mathrm{cm^{-2}}

of electrons in GaAs (

m^*=0.067\,m_e

\varepsilon_s=12.9

). Someone estimates the ground subband energy this way: "The electrons are confined near the interface, so I'll model the confinement as an infinite square well of width

L=8\,\mathrm{nm}

, a typical confinement thickness. Then

E_1=\dfrac{\pi^2\hbar^2}{2m^*L^2}

." Working this through gives

E_1\approx 88\,\mathrm{meV}

. ((a)) Explain what is wrong with this reasoning. ((b)) Give a corrected estimate of

E_1

, using the depletion-charge field

F=en_s/2\varepsilon_0\varepsilon_s

and

E_n\approx\left(\dfrac{\hbar^2}{2m^*}\right)^{1/3}\left[\dfrac{3}{2}\pi eF\left(n-\dfrac14\right)\right]^{2/3}

Related theory

Quantum Mechanics

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