The two-band model couples the conduction ( -like) and valence ( -like) states at the…

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

Solid State Physics, Ch Semiconductors: masses, statistics, defect levels and mobility

Problem

The two-band

\mathbf{k}\cdot\mathbf{p}

model couples the conduction (

s

-like) and valence (

p

-like) states at the Brillouin zone centre

\Gamma

through the matrix element

P=\langle c|p_x|v\rangle

. With the valence-band edge taken as the zero of energy (taking

E_V=0

for this problem) and the conduction edge at

E_g

, the

2\times2

Hamiltonian for a state of wavevector

k

along

x

H(k)=\begin{pmatrix} E_g+\dfrac{\hbar^2k^2}{2m_e} & \dfrac{\hbar k P}{m_e} \\[4pt] \dfrac{\hbar k P}{m_e} & \dfrac{\hbar^2k^2}{2m_e}\end{pmatrix}.

((a)) Diagonalise

H(k)

, expand the upper (conduction) eigenvalue for small

k

, and show that the conduction-band effective mass

m^*

obeys

m_e/m^*\approx 1+E_P/E_g

, where the Kane energy is defined as

E_P\equiv 2P^2/m_e

. ((b)) InSb has

E_g=0.17\,\mathrm{eV}

and a measured electron mass

m^*=0.014\,m_e

. Use the relation of part (a) to extract

E_P

for InSb. ((c)) Using that same

E_P

, predict the electron mass of GaSb (

E_g=0.73\,\mathrm{eV}

) and compare it with the measured value

0.041\,m_e

, quoting the percentage by which the two-band prediction misses the measurement.

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