Thermal capture of electrons in silicon

Abstract

A theory for thermal transition probalities of an electron trapped in a semiconductor is developed and applied to capture of electrons by donors in Si. For the interaction of the electron with the lattice vibrations we use the Bardeen-Shockley deformation potential. As the donor ionization energy is less than the Debye energy, acoustic phonons are important and contributions by optical phonons are not taken into account. The calculation is done in the Born-Oppenheimer approximation and only linear terms in the lattice variables for the perturbing operator and the electronic energy levels are kept. Multi-phonon processes are possible if the equilibrium position of the lattice atoms is different for the electronic states between which the transition takes place (Frank-Condon effect). An expansion of the transition probability in terms of the number of participating phonons is given. In Si at low temperatures only one-phonon processes are important and then the Born-Oppenheimer approximation gives the same result that would be obtained using the Born-Hartree approximation. For capture directly into the ground state of donors in Si the energy band structure is of great importance. There are six energy minima, located along the (100) directions. The electronic wave functions are superpositions of functions belonging to each minimum, these functions being products of Bloch functions associated with the k vectors of the energy minima and solutions of a Wannier equation. Transitions between two wave functions of this type contain intervalley contributions as well as intravalley contributions. The former are of importance if the vector connecting the two energy minima is about equal to the propagation vector of a phonon having an energy equal to the electronic energy difference between the two states. If the energy minima are located at about three-quarters of the way to the Brillouin zone face, then in Si this condition can best be satisfied for an umpklapp process leading from the (100) to the (100) valley. For phosphorus donors this process yields a capture cross section of σ = |u f| 2( 20 T ) × 10 −12 cm 2 , where u f is the Fourier coefficient of exp ( i K·r) in the expansion of the periodic parts of the Bloch functions and K is the reciprocal lattice vector associated with the umklapp process. We estimate | u f | 2 to be larger than ( 1 65 ). For deep traps, transition probabilities and capture cross sections are small compared to those for shallow traps. Furthermore, they depend sensitively on the value of the electron-lattice interaction constants and the trapped state wave function. Relatively small changes in the wave function can change the calculated capture cross section by as much as 10 8. The calculated cross sections are of order 10 −28 to 10 −20 cm 2.