Δ1Conduction-Band Minimum of Ge from High-Pressure Studies onp−nJunctions

Abstract

The effect of hydrostatic pressure on $p\ensuremath{-}n$ junctions of germanium has been investigated with a view to determining the energy of the ${\ensuremath{\Delta}}_{1}$ conduction-band minimum relative to that of the ${L}_{1}$ minimum. The pressure-induced shift in the forward bias voltage at a constant current level, which is a measure of the energy-gap change with pressure, passes through a maximum near the pressure at which the lowest conduction band shifts from the ${L}_{1}$ to the ${\ensuremath{\Delta}}_{1}$ minimum. Because of the two-band conduction in Ge at pressures above 15 kbar, the observed shift in the forward bias voltage not only represent an effective-energy-gap change determined by the density-of-state average over the two bands and their pressure coefficients, but is also modified by the change in carrier mobility with pressure. Since the junction current in the present case was dominated by electrons, the observed shifts were corrected for the electronic-mobility change with pressure, using the pressure variation of the resistivity of $n$-type Ge. From the effective-energy-gap change with pressure given by $\ensuremath{\Delta}{E}_{G\mathrm{eff}}=\ensuremath{\Delta}({\mathrm{E}}_{L}\ensuremath{-}{E}_{v})\ensuremath{-}kT\mathrm{ln}[1+(\frac{{N}_{\ensuremath{\Delta}}}{{N}_{L}})\mathrm{exp}(\ensuremath{-}\frac{\ensuremath{\Delta}E}{\mathrm{kT}})]$, and using $\frac{d{E}_{G}}{\mathrm{dV}}=3.7$ eV ($\frac{d{E}_{G}}{\mathrm{dP}}=5.0\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}3}$ eV/kbar), $\frac{{N}_{\ensuremath{\Delta}}}{{N}_{L}}=2.7$, and $\frac{d(\ensuremath{\Delta}E)}{\mathrm{dV}}=\ensuremath{-}4.6$ eV [$\frac{d(\ensuremath{\Delta}E)}{\mathrm{dP}}=\ensuremath{-}6.2\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}3}$ eV/kbar], the best fit to the experimental data yields 0.18\ifmmode\pm\else\textpm\fi{}0.01 eV for the zero-pressure energy separation $\ensuremath{\Delta}E$ between the ${L}_{1}$ and ${\ensuremath{\Delta}}_{1}$ minima. The observed shifts give initial pressure coefficients of 5.0\ifmmode\pm\else\textpm\fi{}0.1\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}3}$ and -1.3\ifmmode\pm\else\textpm\fi{}0.3\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}3}$ eV/kbar for the ${L}_{1}$ and ${\ensuremath{\Delta}}_{1}$ minima, respectively.