Question

Semiconductors can become conductive if their temperature is raised sufficiently to populate
the (empty) conduction band from the highest filled levels in the valence band. The ratio of
the populations in the highest level of the conduction band to that of the lowest level in the
conduction band is nconduction/nvalance = gconduction/gvalance e–ΔE/kT where ΔE is the band gap, which is
0.661 eV for Ge and 5.5 eV for diamond. Assume for simplicity that the ratio of the
degeneracies is one and that the semiconductor becomes sufficiently conductive when
nconduction/nvalance = 1.0×10–6. At what temperatures will germanium and diamond become
sufficiently conductive? Given that the most stable form of carbon at normal pressures is
graphite and that graphite sublimates near 3700 K, could you heat diamond enough to make
it conductive and not sublimate it?

Answer 1

For Ge

n_conduction/n_valence = 1 X e^-E/kT

1 X 10^-6 = 1 X e^-E/kT

ln (1 X 10^-6 ) = - E / kT

T = - E / k X ln (1 X 10^-6 )

T = - 0.661 x 1.602 x 10^-19 /1 eV/ 1.380 X 10^-23 X ln (1 X 10^-6 )

T = - 0.661 x 1.602 x 10^-19 J /1 eV/ 1.380 X 10^-23 J/K X -13.815

T = 938 K

For Diamond

n_conduction/n_valence = 1 X e^-E/kT

1 X 10^-6 = 1 X e^-E/kT

ln (1 X 10^-6 ) = - E / kT

T = - E / k X ln (1 X 10^-6 )

T = - 5.5 x 1.602 x 10^-19 /1 eV/ 1.380 X 10^-23 X ln (1 X 10^-6 )

T = - 5.5 x 1.602 x 10^-19 J /1 eV/ 1.380 X 10^-23 J/K X -13.815

T = 4610 K

Ge become sufficiently conductive at 938 K and diamond become conductive at 461 K.

Diamond is very good semiconductor. When we heat it it firstly converts into graphite then it vapourizes around 4000K. Just above 10 GPa, molten carbon becomes diamond when we cool down it by maintaining the pressure. When we make the diamond we can again depressurize it slowly without changing its phase. So its possible to heat the diamond enough to make it conductive and not sublimate it by maintaning the pressure and temperature.