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Thermalization in energy band tails
JOURNAL OF NON-CRYSTALLINESOLIDS8-10 (1972) 954-958 © North-Holland Publishing Co.
T H E R M A L I Z A T I O N I N E N E R G Y BAND T A I L S
FRANK STERN IBM Watson Research Center, Yorktown Heights, New York 10598, U.S.A.
The rate equations for carriers injected into an amorphous semiconductor are considered for a simple model with radiative recombination of injected electron and holes and with emission and absorption of optical phonons of a single energy. The approximate energy dependence of the rate coefficients is calculated for a model of the band tail states. A steady-state solution of the rate equations is compared with experimental results for luminescence and photoconductivity in amorphous As2TezSe.
1. Introduction One of the principal problems in the theory of amorphous semiconductors is the form of the density of states for specific materials. In this paper we assume the presence of band tails, and consider the dynamics of injected carriers. Our principal concern is the formulation of a model for photoconductivity and photoluminescence which can be related to other properties of the material, such as conductivity or optical absorption. The energy loss of the injected electrons to the lattice is approximated by assuming the emission of optical phonons of a single energy. This process affects the energy distribution of the carriers but does not change the total concentration of electrons and holes. The recombination process which we consider is radiative recombination of electrons and holes, with provision for a matrix element which depends on the energy of both the initial and the final states.
2. Rate equations With the assumption that only a single phonon energy is involved in the thermalization of carriers in the band tail, and with the additional assumption that the quantities of physical interest do not vary too rapidly on the scale of the phonon energy, we can replace the continous density of states by a ladder of energy levels E, separated by the phonon energy Eph. The rate of change of the population of electrons in level i can be written as a sum of terms corresponding to the various mechanisms that are operating. The three mechanisms we consider are shown in fig. 1. The first is the 954
955
THERMALIZATION IN ENERGY BAND TAILS
thermal relaxation mechanism, for which the rate of change of carriers in level i is Dif, = ti+lO,+ 1 [fi+l (1 - f~) - f~(1 - fs+l) e x p ( - E p h / K T ) ] - hOi [f/(1 - f i - 1 ) - f i - i (1 - fi) exp ( - E p h / K T ) ] .
(1)
where D i is the number of states per unit volume in level i, f i s the probability that a state is occupied by an electron, and Eph is the phonon energy. We ignore here the complications associated with spin 1) or other level degeneracies. An approximate expression for the rate coefficient t is given in the next section. p(E) 2
E
E
Fig. 1. Schematic representation of (1) thermalization by phonon emission and absorption, (2) radiative recombination, and (3) nonradiative transitions between overlapping band tail states. Radiative recombination between electrons in the conduction band tail and holes in the valence band tail, labeled 2 in fig. 1, is given by Oifi = - ~. riyO,Dj [fi (1 - f j ) - f ] ( 1 - fi) exp ( - E u / K T ) ] ,
(2)
J
where the primed quantities refer to the valence band tail, and E u = E i - E ~ . If the band tails overlap, then E~ can be less than Ej. The rate expression for that case is obtained by multiplying the right-hand side of eq. (2) by exp ( E u / K T ) . The rate for transitions between states at the same energy, labeled 3 in fig. 1, is given by D,f, = s,O,D:, (f~, - f~),
E'k = E,,
(3)
and corresponds to nonradiative transitions induced by low-energy phonons. Similar transitions can also occur with the emission or absorption of one or more optical phonons, but such processes have not been included in the present calculation.
956
E.STERN
Finally, we note that the term in the rate equation which takes into account generation by light of photon energy E is D,f~ = Gr,kDiD/, ( f k -- f ) / ~
rtmD,D" ( f " - f ) ,
(4)
I
where G is the total generation rate per unit volume, E ~ , = E ~ - E , and E " = E l - - E. Note that monochromatic light can generate carriers in a band of energies when there is no selection rule. The combined rate equations are obtained by adding together all the processes which can change the population of the levels. In the steady state, the equations reduce to a set of simultaneous non-linear equations in the occupation probabilitiesfi a n d f j . If the bands are symmetric, the number of equations is halved. We have assumed symmetric bands for our numerical work, and have solved the equations iteratively. 3. Rate coefficients
The coefficients r and t which enter in (1) and (2) have been evaluated using a model for the density of states and for the wave functions which has been described elsewhere 2). The radiative rate coefficients are riJ = 2 N e 2 ]Eij [ p 2 1 m e , v ( E i ' E j,) l , 2v / a h 4 c 3 ,
(5)
where IMla2vis the integral of the product of the envelope wave functions of the state with energy Ei in the conduction band and the state with energy Ej in the valence band, averaged over all relative positions of the states involved, N is the index of refraction, and P is h / m times the matrix element of the momentum operator taken between the periodic parts of the Bloch functions for the band edges in the crystal. We do not have a quantitative model for the band tail states of amorphous chalcogenides from which to calculate the matrix element for these materials. But the same matrix element enters in the radiative recombination and in the optical absorption, so that one can check whether a given model treats these two effects consistently. The thermalization rate t has been calculated with the same model2), assuming that the wave functions are localized and have spatial decay constants fli. The relevant matrix element is obtained from eq. (8c) of ref. 2 by taking fli ~-fli-1 for fie and flv, taking k c and kv to be zero, and replacing k¢o by the phonon wavevector. We assume that the material is polar, and that 1/ff is the difference between the reciprocal high-frequency and low-frequency dielectric constants. Then ti ~- 25rc2eED~- 1/3~,hflifli - l"
(6)
Our result differs from that of Hindleya), calculated for states near the
THERMALIZATION
IN ENERGY BAND TAILS
957
mobility edge, by the numerical factor 251~2/21f12a2, where a 3 =I/N is the atomic volume. We have averaged over all relative positions of the initial and final states in calculating the matrix element. A calculation which takes spatial correlations into account is likely to lead to a thermalization rate which falls off faster than (6) for states far in the band tail.
4. Steady-state results We have applied the model described above to the photoconductivity and luminescence results for As2Te2Se4,5). We used a density of states of the Kane 8) form with r/=90 meV, a nominal energy gap and a mobility gap of 1.05-5 x 10 -4 T eV, and a phonon energy of 25 meV. The rate coefficient s in (3) was varied over several orders of magnitude near the value of t at the same energy, but did not influence the results. The recombination rate coefficient r was chosen to be consistent with the measured average value 1.8 × 10-lO cm 3 sec-1 found from the photoconductivity near room temperature 4). This leads to an unphysically large optical absorption constant and suggests that nonradiative recombination is important near room temperature, consistent with the low luminescence efficiency above 150 °KS). Some important aspects of the results are these: (1) we were able to fit the position of the luminescence peak 5), and to describe the temperature and excitation-intensity dependence of the photoconductivity 4) using the same rate coefficients; (2) the calculated luminescence width is 60 meV, less than half of the measured width, but the calculated peak shift is 40 meV, more than twice as large as the measured shift ( < 15 meV) for a light intensity change of a factor of 10 at the highest experimental intensities used. A possible explanation for this is the effect of Coulomb interactions between the recombining carriers, as discussed by Weiser 7). An interesting qualitative result of our calculations is that at sufficiently low temperatures there is substantial departure from quasi-equilibrium for the carriers above the recombination edge 4, 5, 7). Thus experiments on photoconductivity at low temperatures may give direct information on the magnitude of the thermalization rates. When we take the recombination and thermalization rates to have the form indicated in (5) and (6), with physically reasonable coefficients, we find that the thermalization rate is much larger than the recombination rate over most of the energy range of interest. The recombination edge is then determined by the rapid falloff in the density of states. As discussed above, we believe that the thermalization rate falls off faster in the tail than indicated by (6), so that the recombination edge may be determined by the crossover of the thermalization and recombination rates, as postulated previously S).
958
F.STERN
Lattice polarization can lead to an activation energy for thermalization of carriers in highly localized states, which would also contribute to the rapid decrease of the thermalization rate in the tail, especially at low temperatures. A better theory of correlation and polarization effects is needed. There is nevertheless reason to hope that we shall eventually be able to treat nonequilibrium properties of band tail states on the same basis as conductivity and optical absorption.
Acknowledgements I am indebted to M. H. Brodsky, R. Fischer, L. Friedman, N. F. Mott, and K. Weiser for discussions relating to this work.
References 1) 2) 3) 4)
M. H. Cohen and D. L. Johnson, J. Non-Crystalline Solids 3 (1970) 271. F. Stern, Phys. Rev. B 3 (1971) 2636. N. K. Hindley, J. Non-Crystalline Solids 5 (1970) 31. K. Weiser, R. Fischer and M. H. Brodsky, in: Proc. Tenth Intern. Conf. on the Physics of Semiconductors, Cambridge, Mass., 1970 (U.S. Atomic Energy Commission, Oak Ridge, Tennessee, 1970) p. 667. 5) R. Fischer, U. Heim, F. Stern and K. Weiser, Phys. Rev. Letters 26 (1971) 1182. 6) E. O. Kane, Phys. Rev. 131 (1963) 79, eq. (48). 7) K. Weiser, J. Non-Crystalline Solids 8--10 (1972) 922.
T H E R M A L I Z A T I O N I N E N E R G Y BAND T A I L S
FRANK STERN IBM Watson Research Center, Yorktown Heights, New York 10598, U.S.A.
The rate equations for carriers injected into an amorphous semiconductor are considered for a simple model with radiative recombination of injected electron and holes and with emission and absorption of optical phonons of a single energy. The approximate energy dependence of the rate coefficients is calculated for a model of the band tail states. A steady-state solution of the rate equations is compared with experimental results for luminescence and photoconductivity in amorphous As2TezSe.
1. Introduction One of the principal problems in the theory of amorphous semiconductors is the form of the density of states for specific materials. In this paper we assume the presence of band tails, and consider the dynamics of injected carriers. Our principal concern is the formulation of a model for photoconductivity and photoluminescence which can be related to other properties of the material, such as conductivity or optical absorption. The energy loss of the injected electrons to the lattice is approximated by assuming the emission of optical phonons of a single energy. This process affects the energy distribution of the carriers but does not change the total concentration of electrons and holes. The recombination process which we consider is radiative recombination of electrons and holes, with provision for a matrix element which depends on the energy of both the initial and the final states.
2. Rate equations With the assumption that only a single phonon energy is involved in the thermalization of carriers in the band tail, and with the additional assumption that the quantities of physical interest do not vary too rapidly on the scale of the phonon energy, we can replace the continous density of states by a ladder of energy levels E, separated by the phonon energy Eph. The rate of change of the population of electrons in level i can be written as a sum of terms corresponding to the various mechanisms that are operating. The three mechanisms we consider are shown in fig. 1. The first is the 954
955
THERMALIZATION IN ENERGY BAND TAILS
thermal relaxation mechanism, for which the rate of change of carriers in level i is Dif, = ti+lO,+ 1 [fi+l (1 - f~) - f~(1 - fs+l) e x p ( - E p h / K T ) ] - hOi [f/(1 - f i - 1 ) - f i - i (1 - fi) exp ( - E p h / K T ) ] .
(1)
where D i is the number of states per unit volume in level i, f i s the probability that a state is occupied by an electron, and Eph is the phonon energy. We ignore here the complications associated with spin 1) or other level degeneracies. An approximate expression for the rate coefficient t is given in the next section. p(E) 2
E
E
Fig. 1. Schematic representation of (1) thermalization by phonon emission and absorption, (2) radiative recombination, and (3) nonradiative transitions between overlapping band tail states. Radiative recombination between electrons in the conduction band tail and holes in the valence band tail, labeled 2 in fig. 1, is given by Oifi = - ~. riyO,Dj [fi (1 - f j ) - f ] ( 1 - fi) exp ( - E u / K T ) ] ,
(2)
J
where the primed quantities refer to the valence band tail, and E u = E i - E ~ . If the band tails overlap, then E~ can be less than Ej. The rate expression for that case is obtained by multiplying the right-hand side of eq. (2) by exp ( E u / K T ) . The rate for transitions between states at the same energy, labeled 3 in fig. 1, is given by D,f, = s,O,D:, (f~, - f~),
E'k = E,,
(3)
and corresponds to nonradiative transitions induced by low-energy phonons. Similar transitions can also occur with the emission or absorption of one or more optical phonons, but such processes have not been included in the present calculation.
956
E.STERN
Finally, we note that the term in the rate equation which takes into account generation by light of photon energy E is D,f~ = Gr,kDiD/, ( f k -- f ) / ~
rtmD,D" ( f " - f ) ,
(4)
I
where G is the total generation rate per unit volume, E ~ , = E ~ - E , and E " = E l - - E. Note that monochromatic light can generate carriers in a band of energies when there is no selection rule. The combined rate equations are obtained by adding together all the processes which can change the population of the levels. In the steady state, the equations reduce to a set of simultaneous non-linear equations in the occupation probabilitiesfi a n d f j . If the bands are symmetric, the number of equations is halved. We have assumed symmetric bands for our numerical work, and have solved the equations iteratively. 3. Rate coefficients
The coefficients r and t which enter in (1) and (2) have been evaluated using a model for the density of states and for the wave functions which has been described elsewhere 2). The radiative rate coefficients are riJ = 2 N e 2 ]Eij [ p 2 1 m e , v ( E i ' E j,) l , 2v / a h 4 c 3 ,
(5)
where IMla2vis the integral of the product of the envelope wave functions of the state with energy Ei in the conduction band and the state with energy Ej in the valence band, averaged over all relative positions of the states involved, N is the index of refraction, and P is h / m times the matrix element of the momentum operator taken between the periodic parts of the Bloch functions for the band edges in the crystal. We do not have a quantitative model for the band tail states of amorphous chalcogenides from which to calculate the matrix element for these materials. But the same matrix element enters in the radiative recombination and in the optical absorption, so that one can check whether a given model treats these two effects consistently. The thermalization rate t has been calculated with the same model2), assuming that the wave functions are localized and have spatial decay constants fli. The relevant matrix element is obtained from eq. (8c) of ref. 2 by taking fli ~-fli-1 for fie and flv, taking k c and kv to be zero, and replacing k¢o by the phonon wavevector. We assume that the material is polar, and that 1/ff is the difference between the reciprocal high-frequency and low-frequency dielectric constants. Then ti ~- 25rc2eED~- 1/3~,hflifli - l"
(6)
Our result differs from that of Hindleya), calculated for states near the
THERMALIZATION
IN ENERGY BAND TAILS
957
mobility edge, by the numerical factor 251~2/21f12a2, where a 3 =I/N is the atomic volume. We have averaged over all relative positions of the initial and final states in calculating the matrix element. A calculation which takes spatial correlations into account is likely to lead to a thermalization rate which falls off faster than (6) for states far in the band tail.
4. Steady-state results We have applied the model described above to the photoconductivity and luminescence results for As2Te2Se4,5). We used a density of states of the Kane 8) form with r/=90 meV, a nominal energy gap and a mobility gap of 1.05-5 x 10 -4 T eV, and a phonon energy of 25 meV. The rate coefficient s in (3) was varied over several orders of magnitude near the value of t at the same energy, but did not influence the results. The recombination rate coefficient r was chosen to be consistent with the measured average value 1.8 × 10-lO cm 3 sec-1 found from the photoconductivity near room temperature 4). This leads to an unphysically large optical absorption constant and suggests that nonradiative recombination is important near room temperature, consistent with the low luminescence efficiency above 150 °KS). Some important aspects of the results are these: (1) we were able to fit the position of the luminescence peak 5), and to describe the temperature and excitation-intensity dependence of the photoconductivity 4) using the same rate coefficients; (2) the calculated luminescence width is 60 meV, less than half of the measured width, but the calculated peak shift is 40 meV, more than twice as large as the measured shift ( < 15 meV) for a light intensity change of a factor of 10 at the highest experimental intensities used. A possible explanation for this is the effect of Coulomb interactions between the recombining carriers, as discussed by Weiser 7). An interesting qualitative result of our calculations is that at sufficiently low temperatures there is substantial departure from quasi-equilibrium for the carriers above the recombination edge 4, 5, 7). Thus experiments on photoconductivity at low temperatures may give direct information on the magnitude of the thermalization rates. When we take the recombination and thermalization rates to have the form indicated in (5) and (6), with physically reasonable coefficients, we find that the thermalization rate is much larger than the recombination rate over most of the energy range of interest. The recombination edge is then determined by the rapid falloff in the density of states. As discussed above, we believe that the thermalization rate falls off faster in the tail than indicated by (6), so that the recombination edge may be determined by the crossover of the thermalization and recombination rates, as postulated previously S).
958
F.STERN
Lattice polarization can lead to an activation energy for thermalization of carriers in highly localized states, which would also contribute to the rapid decrease of the thermalization rate in the tail, especially at low temperatures. A better theory of correlation and polarization effects is needed. There is nevertheless reason to hope that we shall eventually be able to treat nonequilibrium properties of band tail states on the same basis as conductivity and optical absorption.
Acknowledgements I am indebted to M. H. Brodsky, R. Fischer, L. Friedman, N. F. Mott, and K. Weiser for discussions relating to this work.
References 1) 2) 3) 4)
M. H. Cohen and D. L. Johnson, J. Non-Crystalline Solids 3 (1970) 271. F. Stern, Phys. Rev. B 3 (1971) 2636. N. K. Hindley, J. Non-Crystalline Solids 5 (1970) 31. K. Weiser, R. Fischer and M. H. Brodsky, in: Proc. Tenth Intern. Conf. on the Physics of Semiconductors, Cambridge, Mass., 1970 (U.S. Atomic Energy Commission, Oak Ridge, Tennessee, 1970) p. 667. 5) R. Fischer, U. Heim, F. Stern and K. Weiser, Phys. Rev. Letters 26 (1971) 1182. 6) E. O. Kane, Phys. Rev. 131 (1963) 79, eq. (48). 7) K. Weiser, J. Non-Crystalline Solids 8--10 (1972) 922.