The radjative and non-radiative transitions of a trapped elearon in a crystal are discussed with the aid of the generating fimaion method. 1 l2l Assuming a linear form for the elearon-latrice interaCtion Hamiltonian, one can calculate the generating functions of these two processes for a. general electronlattice system, that is : not only. the equilibrium lattice positions but also the vibrational frequencies and the axes of the normal modes are different for the two electronic states between which the transition takes place.The generating funaion for radiative transition, from which one can calculate the shape of the absorption band by the inverse formula for Laplace transformation, faCtorizes into three functions, the fir·st of which represents the effea of the difference in equilibrium positions, the ·second due to the difi"enmce in frequency tensors ( N-diinensional, in general, where N is the number of modes) and the third due to the variation of the transition dipole moment with lattice configuration. All these effeas COntribute additively to the moments of the absorption curve such as peak shift, broadening and asymmetry, and one can estimate the order of magnitude of each contribution for aaual examples. The third faaor is important when one discusses a transition which is forbidden but for the lattice vibration.The generating function for non-radiative transition also turns out to be a produa of three parts, the two of which are identical with the first tWo faaors for the radiative transition stated above. We can derive, in a general form, the low and high tempera· ture features 1 l of the transition probability, that is : the temperature dependences of the probability are given by exp( -Eo/kT) and exp( -E*/kT) for the two limiting cases, respectively, where Eo is the energy difference of the minimum points of the two adiabatic potentials, while E* is the minimum point of the inter·seaion of the two adiabatic potentials. Th1s means that at low temperatures the transition occurs primarily as a tunneling elfea while at high temperatures the dominant process is the jumps over the aCtivated states. The result of Huang and Rhys"l is obtained as a special case. The method used here, however, permits one to calculate the thermal ionization probability of a trapped eleCtron or hole in non-polar crystals such as silicon and germanium. The result at high temperatures is written in the form -m*c2 (kT)2where Eo is the depth of trapping, m* is the effeCtive mass of the electron or the hole, c is the longitudinal sound velocity in the crystal and r is a dimensionless constant which contains the well-known eleCtron lattice interaction constant C. The aCtivation energy E* is given by E*= (1 + r/2)2/2r·.co;;:;; Eo· For the crystals cited above r ranges from 0.1 to 0.4.The cross-seaion u t for retrapping process is also calculable as we can relate it with W, on the basis of detailed balance theorem. The numerical values of the ionization rate and the cross·seaion are very sensitive to r which depends on the elfeaive mas...
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