Quantum resonances in the valence bands of germanium. II. Cyclotron resonances in uniaxially stressed crystals

Abstract

As a consequence of the degeneracy of the Ge valence bands at $\stackrel{\ensuremath{\rightarrow}}{k}=0$, the lowest Landau levels are anomalously spaced, so the cyclotron resonances of holes under "quantum" conditions, $\frac{\ensuremath{\hbar}\ensuremath{\omega}}{k\ensuremath{\Theta}}>1$ ($\ensuremath{\Theta}$ being the temperature), give rise to complex line spectra. We report here measurements of these "quantum" spectra taken at 53 GHz and 1.2\ifmmode^\circ\else\textdegree\fi{}K in samples of Ge subjected to uniaxial, compressive stresses which lower the cubic symmetry and remove the valence-band degeneracy. The effect of the stress $T$ on the positions of the quantum lines permits their identification and also provides a direct measurement of the uniaxial deformation potentials: ${D}_{u}=3.32\ifmmode\pm\else\textpm\fi{}0.20$ eV and ${D}_{u}^{\ensuremath{'}}=3.81\ifmmode\pm\else\textpm\fi{}0.25$ eV. At large stresses, as the energy surfaces assume ellipsoidal shape, the quantum lines arrange themselves into four line series identifiable by ${M}_{J}=\ifmmode\pm\else\textpm\fi{}(\frac{1}{2}),\ifmmode\pm\else\textpm\fi{}(\frac{3}{2})$ for $T\ensuremath{\parallel}[001]$ and $T\ensuremath{\parallel}[111]$. With increasing stress the lines converge to two series limits corresponding to the "classical" effective masses of the two split bands, $|{M}_{J}|=(\frac{1}{2}), (\frac{3}{2})$. For stress along each of the principal crystallographic directions---[001], [111], and [110]---the positions of two quantum lines, the "fundamental" transitions, are approximately invariant under stress and lie one at each series limit. From measurements of these extremely sharp lines in the geometry ${\stackrel{\ensuremath{\rightarrow}}{H}}_{0}\ensuremath{\parallel}T$ we have determined the valence-band inverse-mass parameters to a "spectroscopic" precision: ${\ensuremath{\gamma}}_{1}=13.38\ifmmode\pm\else\textpm\fi{}0.02$, ${\ensuremath{\gamma}}_{2}=4.24\ifmmode\pm\else\textpm\fi{}0.03$, and ${\ensuremath{\gamma}}_{3}=5.69\ifmmode\pm\else\textpm\fi{}0.02$. Less detailed but corroborative experiments were also done in the geometry ${\stackrel{\ensuremath{\rightarrow}}{H}}_{0}\ensuremath{\perp}T$. From the measurements for $T\ensuremath{\parallel}[110]$ we read out directly the ratio of the deformation potentials, $\frac{{D}_{u}^{\ensuremath{'}}}{{D}_{u}}=1.15\ifmmode\pm\else\textpm\fi{}0.02$. The strain interaction between the valence band edge and the spin-orbit-splitt-off valence-band results in a small linear shift of the fundamental transitions. Surprisingly, the deformation potentials, ${D}_{w}=2.31\ifmmode\pm\else\textpm\fi{}0.17$ eV and ${D}_{w}^{\ensuremath{'}}=2.81\ifmmode\pm\else\textpm\fi{}0.20$ eV, measured from this interaction are significantly smaller than those given above which were measured directly from the gross strain decoupling of the valence bands. The difference is ascribed to the existence of spin-dependent deformation potentials which contribute differently to the two processes. The quantum-resonance line shapes are governed largely by strain and ${k}_{H}$ broadening; however, for the narrowest lines, viz., the fundamental transitions, relaxation-time effects are in evidence and have been briefly investigated.