Electron Spin-Lattice Relaxation in Phosphorus-Doped Silicon

Electron spin-lattice relaxation in phosphorus-doped silicon has been investigated over a magnetic field range of 0 to 11 000 oersteds, a temperature range of 1.25\ifmmode^\circ\else\textdegree\fi{}K to 4.2\ifmmode^\circ\else\textdegree\fi{}K, and a concentration range of ${10}^{14}$ P/cc to 3\ifmmode\times\else\texttimes\fi{}${10}^{16}$ P/cc. Three distinct ${\ensuremath{\tau}}_{S}(\ensuremath{\Delta}{m}_{S}=\ifmmode\pm\else\textpm\fi{}1, \ensuremath{\Delta}{m}_{I}=0)$ relaxation mechanisms have been identified, and their functional dependences on magnetic field, temperature, and concentration have been determined. These mechanisms are characterized as follows: (a) $(\frac{1}{{\ensuremath{\tau}}_{S}})({H}^{4}, T)$ is concentration independent, and has an ${H}^{4}$ and $T$ dependence. At 3000 oersteds and 1.25\ifmmode^\circ\else\textdegree\fi{}K, $(\frac{1}{{\ensuremath{\tau}}_{S}})({H}^{4}, T)=(2.63\ifmmode\pm\else\textpm\fi{}0.10)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}5}$ ${\mathrm{sec}}^{\ensuremath{-}1}$. (b) $(\frac{1}{{\ensuremath{\tau}}_{S}})({T}^{7})$ is independent of concentration and magnetic field, and has a ${T}^{7}$ dependence. At 2.00\ifmmode^\circ\else\textdegree\fi{}K, $\frac{1}{{\ensuremath{\tau}}_{S}}({T}^{7})=(1.65\ifmmode\pm\else\textpm\fi{}0.15)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}4}$ ${\mathrm{sec}}^{\ensuremath{-}1}$. (c) $(\frac{1}{{\ensuremath{\tau}}_{S}})(\mathrm{conc}.)$ depends linearly on concentration for concentrations below ${10}^{16}$ P/cc, and has approximately an ${H}^{\ensuremath{-}\frac{1}{2}}$ and $T$ dependence. At 3000 oersteds and 1.25\ifmmode^\circ\else\textdegree\fi{}K, $\frac{1}{{\ensuremath{\tau}}_{S}}(\mathrm{conc}.)$ for a 4\ifmmode\times\else\texttimes\fi{}${10}^{15}$ P/cc sample is 3.3\ifmmode\pm\else\textpm\fi{}0.4\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}4}$ ${\mathrm{sec}}^{\ensuremath{-}1}$. In addition to these three ${\ensuremath{\tau}}_{S}$ mechanisms, the horizontal relaxation modes ($\ensuremath{\Delta}{m}_{I}=\ifmmode\pm\else\textpm\fi{}1$, $\ensuremath{\Delta}{m}_{S}=0, \ifmmode\pm\else\textpm\fi{}1, \ensuremath{\mp}1$) were investigated. $\frac{1}{{\ensuremath{\tau}}_{X}}(\ensuremath{\Delta}{m}_{I}=\ifmmode\pm\else\textpm\fi{}1, \ensuremath{\Delta}{m}_{S}=\ensuremath{\mp}1)$ at 2.16\ifmmode^\circ\else\textdegree\fi{}K is independent of concentration and magnetic field, and between 2.16\ifmmode^\circ\else\textdegree\fi{}K and 4.2\ifmmode^\circ\else\textdegree\fi{}K, has a ${T}^{6.5}$ temperature dependence; all of which strongly suggests a dominant Raman process in this temperature region. At 2.16\ifmmode^\circ\else\textdegree\fi{}K, ${\ensuremath{\tau}}_{X}=3.0\ifmmode\pm\else\textpm\fi{}0.4$ hours. At 1.25\ifmmode^\circ\else\textdegree\fi{}K, the magnetic field dependence of the horizontal modes was measured. The large errors (associated with the very long times involved) make it difficult to ascertain the dominant mechanism here. However, our results are not consistent with a quadratic magnetic field dependence of $\frac{1}{{\ensuremath{\tau}}_{X}}$. At low magnetic fields, concentration dependent ${\ensuremath{\tau}}_{N}(\ensuremath{\Delta}{m}_{I}=\ifmmode\pm\else\textpm\fi{}1, \ensuremath{\Delta}{m}_{S}=0)$ and ${\ensuremath{\tau}}_{X}$ mechanisms arise, due to an admixture of states which allows $\frac{1}{{\ensuremath{\tau}}_{S}}(\mathrm{conc}.)$ also to induce $\frac{1}{{\ensuremath{\tau}}_{N}}$ and $\frac{1}{{\ensuremath{\tau}}_{X}}$ transitions. When all the preceding mechanisms are properly superposed, their resultant agrees well with the experimental relaxation probabilities, except for a small discrepancy which shows up for dilute samples at 1.25\ifmmode^\circ\else\textdegree\fi{}K. This discrepancy (\ensuremath{\sim}2\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}5}$ ${\mathrm{sec}}^{\ensuremath{-}1}$ at 1.25\ifmmode^\circ\else\textdegree\fi{}K) can be accounted for by introducing another mechanism. There is some indication that this mechanism is associated with the amount of compensation.The theoretical origins of the mechanisms are discussed. A theory is proposed to explain the concentration dependent ${\ensuremath{\tau}}_{S}$ mechanism, according to which rapidly relaxing close pairs of phosphorus atoms, which are few in number, relax the spins of the large number of isolated phosphorus atoms via a spin diffusion process. Experiments supporting this hypothesis are presented.