Raman Spin-Lattice Relaxation of Shallow Donors in Silicon (1963)

This work reports both experimental measurements and calculations of two-phonon Raman spin-lattice relaxation times for shallow donors in silicon. Saturation, line broadening, and adiabatic fast passage techniques were used to measure ${T}_{s} (\ensuremath{\Delta}{M}_{s}=\ifmmode\pm\else\textpm\fi{}1,\ensuremath{\Delta}{M}_{I}=0)$ between 2 and 30\ifmmode^\circ\else\textdegree\fi{}K for P, As, Sb, and Bi at 3300 G. ${T}_{x} (\ensuremath{\Delta}{M}_{s}=\ifmmode\pm\else\textpm\fi{}1,\ensuremath{\Delta}{M}_{I}=\ensuremath{\mp}1)$ was measured in the liquid He range for As, Sb, and Bi at 3300 G. The calculations of the spin-lattice relaxation rates $\frac{1}{{T}_{s}}$ and $\frac{1}{{T}_{x}}$ for Raman processes were made using only the $1S$ doublet and triplet valley-orbit excited states. The calculations were made for both the power-law temperature-dependent region and the exponential temperature-dependent region.$\frac{1}{{T}_{s}}$ for P and As showed a power-law temperature dependence ${T}^{n}$ with $n$ closer to 9 than to 7 between 2.5 and 6\ifmmode^\circ\else\textdegree\fi{}K for P and between 3 and 11\ifmmode^\circ\else\textdegree\fi{}K for As. In this range there is no magnetic field-crystal axis orientation dependence and also no magnetic field dependence between 3300 and 8000 G as shown by Wilson and Feher. These facts suggest a spin-orbit spin-flip interaction which would have a "Van Vleck cancellation." No ${T}^{13}$ Raman temperature-dependent $\frac{1}{{T}_{s}}$ was observed experimentally. P, As, Sb, and Bi all show an exponential temperature-dependent $\frac{1}{{T}_{s}}$ above 6, 11, 4, and 26\ifmmode^\circ\else\textdegree\fi{}K, respectively. The rate constant $E (\frac{1}{{T}_{s}}\ensuremath{-}E{e}^{\ensuremath{-}\frac{\ensuremath{\Delta}}{\mathrm{kT}}})$ gets larger as one goes to the higher $Z$ donor impurities.The ratio $\frac{{T}_{s}}{{T}_{x}}$ was measured between 2 and 5\ifmmode^\circ\else\textdegree\fi{}K for ${\mathrm{As}}^{75}$, the result showing $\frac{{T}_{s}}{{T}_{x}}\ensuremath{\propto}{T}^{\ensuremath{-}2}$ above 2.2\ifmmode^\circ\else\textdegree\fi{}K consistent with ${T}_{s}\ensuremath{\propto}\ensuremath{-}8.5$ and ${T}_{s}\ensuremath{\propto}{T}^{\ensuremath{-}6.5}$ near 5\ifmmode^\circ\else\textdegree\fi{}K. Experimental values of ${T}_{x}$ were obtained for ${\mathrm{Sb}}^{121}$ and ${\mathrm{Bi}}^{209}$ at 2.2 and 4.2\ifmmode^\circ\else\textdegree\fi{}K, respectively.Use of the Hasegawa-Roth mechanism for the calculated Raman $\frac{1}{{T}_{s}}$ for both the power-law and the exponential temperature-dependent regions gives a result nearly four orders of magnitude smaller than experiment. In the short-wavelength phonon region the $1S$ triplet can be equally as important as the $1S$ doublet via intervalley Umklapp processes. For As it is shown that intervalley Umklapp processes will predominate over intravalley processes. In the exponential case the calculated $\frac{1}{{T}_{s}}$ rate can have a maximum value when the excited states have level widths equal $2g{\ensuremath{\mu}}_{B}H$.The calculated Raman $\frac{1}{{T}_{x}}$ rates for the different donors show good agreement with the experimental values. The $\frac{1}{{T}_{x}}$ rate can have an exponential temperature-dependent term but it will be much smaller than the ${T}^{7}$ term for the temperature range considered here.Calculations have been made of the level widths of the P and As $1S$ excited states due to spontaneous phonon emission. It is shown for As that the dominant intervalley Umklapp spontaneous emission will be sharply peaked along the [100] axis.