Properties of the electron-hole liquid in Si: Zero stress to the high-stress limit

Abstract

We have measured the properties of the electron-hole liquid (EHL) in Si as a function of stress from $\ensuremath{\sigma}=0 \mathrm{to} 163$ kg/${\mathrm{mm}}^{2}$ which is well into the high-stress limit where the valence bands are decoupled. These measurements provide a useful test of many-body theories for EHL in crystals with a simple band structure. To produce these high stresses and to achieve some unique experimental advantages not realized in uniform-stress geometries, we have employed a well-characterized Hertzian stress geometry to strain ultrapure crystals of Si. This geometry creates an electronic potential minimum which confines both electrons and holes inside the crystal volume. At low temperature the $e\ensuremath{-}h$ pairs spatially condense into a small volume (${10}^{\ensuremath{-}6}$ ml) of EHL at the bottom of the well. Because the liquid volume is small the EHL exists in a region of effectively uniform stress. This idea is confirmed by separate uniform-stress experiments which are included as a part of our study. We study the EHL properties by analyzing its photoluminescence with spectral, spatial, and time resolution under conditions in which stress, temperature, and photoexcitation level are varied over wide limits. In our spectral analysis, which incorporates stress-dependent band structure, we determine the Fermi energies and $e\ensuremath{-}h$ pair density as it decreases from 3.5 \ifmmode\times\else\texttimes\fi{} ${10}^{18}$ ${\mathrm{cm}}^{\ensuremath{-}3}$ at zero stress to 3.5 \ifmmode\times\else\texttimes\fi{} ${10}^{17}$ ${\mathrm{cm}}^{\ensuremath{-}3}$ in the high-stress limit. Corresponding measurements of the intrinsic EHL time decay show that the lifetime increases from 0.14 to 3.0 \ensuremath{\mu}s over the same stress range. Combining the density and lifetime measurements, we are able to investigate the density dependence of the enhancement factor ${g}_{\mathrm{eh}}(0)$ and the Auger process which dominates $e\ensuremath{-}h$ recombination. We have measured a phase diagram for excitonic gas-EHL at high stress and find a critical point of ${T}_{c}\ensuremath{\approx}20 \mathrm{K}$, ${n}_{c}\ensuremath{\approx}2.5\ifmmode\times\else\texttimes\fi{}{10}^{17} {\mathrm{cm}}^{\ensuremath{-}3}$. In the high-stress limit the measured EHL binding energy is in the range $\ensuremath{\varphi}=1\ensuremath{-}1.4$ meV. Additional spatial data reveal the presence of a repulsive force between droplets, possibly a "phonon wind," which inhibits large droplet formation. In general, the stress dependence of the ground-state properties shows trends which are in agreement with existing theories. However, some interesting discrepancies between our data and these theories are found.