Condensed Matter II
Problem set #8
Spring 2023
Donor impurities in GaAs
Background
We consider the direct band gap semiconductor GaAs, with Nd = 1016.cm−3 hydrogenic
donor impurities. The following properties are given:
• direct band gap: Eg = 1.40 eV.
• m∗
e = 0.07m0 for the conduction band effective mass.
• mh = 0.7m0 for the valence band holes effective mass.
• ϵGaAs = 15
Questions
Temperature dependence of the Fermi level
(i) Compute the concentration of intrinsic carriers in the material at room temperature,
and compare it with the concentration of impurities.
(ii) Compute the ionization energy Ed of the impurities.
(iii) Establish the expression of the density of carriers, as a function of T, Ed, Nd, EF .
(iv) From the previous expression, deduce the expression for EF (T). Calculate the value
of EF (T) for T = 300K and T = 30K.
Transport properties
(i) Compute the electron and hole carrier concentrations at room temperature and at
30K.
(ii) What is the critical doping, above which an impurity band would appear as a result
of the presence of substitutional impurities in GaAs? (you may use the given value
of the Bohr radius a0 = 5.3 × 10−11 m). What is the main difference between the
impurity band scenario and the the isolated impurity scenario?
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