High-field transport and electroluminescence in ZnS phosphor layers

Transcription

1 JOURNAL OF APPLIED PHYSICS VOLUME 83, NUMBER 6 15 MARCH 1998 High-field transport and electroluminescence in ZnS phosphor layers Manfred Dür a) and Stephen M. Goodnick Department of Electrical Engineering, Arizona State University, Tempe, Arizona Shankar S. Pennathur Altera Corporation, 101 Innovation Drive, San Jose, California John F. Wager Department of Electrical and Computer Engineering, Center for Advanced Materials Research, Oregon State University, Corvallis, Oregon Martin Reigrotzki and Ronald Redmer Universität Rostock, Fachbereich Physik, Universitätsplatz 3, D Rostock, Germany Received 20 October 1997; accepted for publication 25 November 1997 A full-band Monte Carlo simulation of the high-field electron transport in the ZnS phosphor layer of an alternating-current thin-film electroluminescent device is performed. The simulation includes a nonlocal empirical pseudopotential band structure for ZnS and the relevant scattering mechanisms for electrons in the first four conduction bands, including band-to-band impact ionization and impact excitation of Mn 2 luminescent centers. The steady-state electron energy distribution in the ZnS layer is computed for phosphor fields from 1 to 2 MV/cm. The simulation reveals a substantial fraction of electrons with energies in excess of the Mn 2 impact excitation threshold. The computed impact excitation yield for carriers transiting the phosphor layer exhibits an approximately linear increase with increasing phosphor field above threshold. The onset of Mn 2 impact excitation coincides with the onset of band-to-band impact ionization of electron-hole pairs which prevents electron runaway at high electric fields American Institute of Physics. S I. INTRODUCTION Alternating-current thin-film electroluminescent ACT- FEL devices are used in the production of high-resolution, flat-panel displays. An ACTFEL device essentially consists of a wide band gap semiconductor such as ZnS referred to as the phosphor layer sandwiched between two insulating layers subject to a large ac basis, as illustrated in Fig. 1. As shown in this figure, carriers are field emitted under bias into the phosphor layer from trap states at the semiconductorinsulator interface. The injected electrons are accelerated under the influence of high electric fields in the phosphor on the order of 1 to 2 MV/cm. At sufficiently high fields, the energetic electrons traversing the phosphor layer undergo subsequent scattering events, as shown in Fig. 1, including impact excitation of intentionally doped luminescent centers. Luminescence occurs as the excited states in these luminescent centers radiatively decay to their ground state. An understanding of high-field carrier transport in the phosphor layer and the physics of the different threshold processes such as band-to-band impact ionization and impact excitation of luminescent impurities is essential for ACTFEL device design, especially in the development of new phosphors for fullcolor EL displays. 1 Prior research in the area of high-field transport in ZnS phosphors has considered various approaches, ranging from lucky-drift models, 2 to Monte Carlo calculations using parabolic conduction bands, 3 nonparabolic conduction bands, 4 and full-band simulation based on a local pseudopotential calculation for the first two conduction bands. 5 The use of a single parabolic conduction band model 3 is clearly inadequate, since electrons subject to the high fields typical of ACTFEL operation readily gain sufficient energy to transfer to higher bands. A nonparabolic conduction band model provides a somewhat better estimate of the energy distribution of the carriers in the phosphor layer. 4 However, the actual conduction band density of states in typical zincblende structure materials decreases past a certain energy in contrast to the monotonically increasing density of states of parabolic and nonparabolic band models. In order to account for a more accurate representation of the band picture at higher energies, a full-band calculation needs to be incorporated into the transport model. Brennan 5 utilized the local pseudopotential band structure of Cohen and Bergstresser 6 for ZnS in a Monte Carlo simulation for electric fields only up to 1 MV/cm. His results for the electron distribution function up to this field seem to indicate a relatively small probability of electrons existing at energies above the threshold for impact excitation of Mn 2 impurities approximately 2.1 ev. However, due to the large density of states in this region, the actual number of electrons above the threshold may be much larger. Also, as pointed out by Brennan, the local pseudopotential band structure may be inadequate, and a full nonlocal calculation should be performed. In a previous article, 7 in the following quoted as article I, we presented the results of a nonlocal empirical pseudopoa Electronic mail: duer@asu.edu /98/83(6)/3176/10/$ American Institute of Physics

2 J. Appl. Phys., Vol. 83, No. 6, 15 March 1998 Dür et al FIG. 2. Comparison of the density of states for ZnS calculated using local and nonlocal band structures. FIG. 1. Illustration of carrier injection 1 and carrier transport 2 in the phosphor layer of an ACTFEL device. Energetic electrons impact excite luminescent impurities 3 as well as generate electron-hole pairs by impact ionization 4, before they are trapped at the phosphor-insulator interface 5. tential calculation derived for cubic ZnS. In this article, a full-band calculation of the band-to-band impact ionization rate was reported based on a special integration scheme over the whole Brillouin zone. Comparison was made to the experimental impact ionization coefficient data for ZnS reported by Thompson and Allen 8 incorporating this band structure into a full-band Monte Carlo code. In the present article, we extend this calculation to investigate high-field transport and impact excitation of luminescent impurities in ZnS with the goal of understanding the limiting efficiency mechanisms of ACTFEL devices. The high-field carrier distribution is calculated in the normal range of ACTFEL operation, 1 to 2 MV/cm, and impact excitation of Mn 2 impurities is included in the simulation. The results show that the onset of impact excitation of luminescent impurities occurs at approximately 1 MV/cm, in good agreement with experimental results for ACTFEL devices. In Sec. II, we review the nonlocal band structure and impact ionization rate calculation reported in article I. In Sec. III, we detail the Monte Carlo model used, together with the various scattering mechanisms considered. Section IV presents the results for high-field transport in ZnS based on this Monte Carlo simulation including the high-field electron distribution for various fields. Section V then presents the model used for impact excitation of Mn impurities, and the calculated impact excitation yield. II. NONLOCAL BAND STRUCTURE AND IMPACT IONIZATION In the phosphor layer of ZnS-based ACTFEL devices, the electric fields are typically much higher than those experienced by charge carriers in conventional electronic materials due to the large band gap (E g 3.7 ev) and hence higher breakdown fields. For electric fields on the order of 1 to 2 MV/cm, electrons populate several conduction bands in the phosphor material. Thus, a complete description of high-field transport including impact ionization utilizing the full band structure of ZnS is critical. One of the standard techniques of moderate computational effort for the calculation of realistic band structures is the empirical pseudopotential method 9 EPM. In this method, the Bloch functions representing the crystal wave function are expanded in terms of a plane wave basis. The matrix elements of the crystal potential are treated as adjustable parameters which are fit to experimental data such as optical gaps. In a local EPM calculation, the dependence of the pseudopotential on the angular momentum components, l, present in the atomic core states is ignored. Nonlocal correction terms may be added to the local pseudopotential to account for this dependence. Local EPM parameters for ZnS have been reported by Walter and Cohen. 10 We have extended their local EPM calculation by also including nonlocal contributions for l2 to the pseudopotential. Gaussian potentials have been chosen to model the nonlocal part of the pseudopotential. The new nonlocal parameter set was determined by comparing the resulting band structure to available optical data for this material as discussed in detail in article I. Figure 2 compares the density of states for the local and nonlocal EPM calculations, including the lowest four conduction bands. As can be seen, the main difference is that the intervalley separation energies between the valley and the higher lying X and L valleys are considerably smaller for the nonlocal versus the local case. Since we are interested primarily in the high-field transport properties of ZnS, we neglect the spin-orbit interaction in the calculation of the impact ionization scattering rate discussed below. Its inclusion changes the results only slightly, as is evident from the corresponding gap energies listed in Table II of article I. The plane wave basis with the spin-orbit term omitted greatly reduces the computational burden to calculate the scattering matrix elements. The multiplication of carriers by impact ionization is of central importance for understanding the high-field transport in semiconductor devices both as a limiting mechanism and as a basis for device functionality. Of particular interest are the threshold behavior and the orientation dependence of the

3 3178 J. Appl. Phys., Vol. 83, No. 6, 15 March 1998 Dür et al. ionization rate, which have been investigated for Si, GaAs, 13,15 ZnS, 16,17 and GaN. 18 In article I, we have studied the influence of impact ionization on the high-field transport in ZnS. The scattering rate for impact ionization is determined from Fermi s golden rule as rk 1,n 1 2 V n 2,n 3,n 4 d M tot 1,2;3,4 2 E n1 k 1 3 k 2 d 3 k 3 d 3 k 4 E n2 k 2 E n3 k 3 E n4 k 4, where V is the crystal volume, and k i and n i are the wave vector and the band index, respectively, labeling Bloch electron state i1,2,3,4 with energy E ni (k i ). M tot denotes the matrix element for impact ionization including direct, exchange, and umklapp processes. Here, the initial electron states i1,2 correspond to an electron in the conduction band and a second electron in the valence band. The final states i3,4 are for both electrons in the conduction band. The integrals extend throughout the entire Brillouin zone and are evaluated using an efficient numerical procedure developed by Sano and Yoshii. 12 The interaction between conduction and valence electrons is described by a wave vector dependent dielectric function derived by Levine and Louie. 19 The wave vector dependent impact ionization rate r(k 1,n 1 ) is averaged over the entire Brillouin zone to obtain an energy dependent rate, which is well fit by a power law R EPEEth a. 2 An improvement of the nonlocal band structure for ZnS used in article I has led to new parameters P s 1 ev a, E th 3.8 ev, and a This fit formula is included in the Monte Carlo model presented in the next section. We have neglected phonon-assisted impact ionization processes as well as deep level ionization in our present calculations. These effects may be important at high fields. Furthermore, other high-field effects such as collision broadening or the intracollisional field effect have also to be considered for a more comprehensive description of the transport process of hot electrons in semiconductors. 20,21 Further studies are necessary to ascertain the influence of the calculated anisotropy of the impact ionization rate on the ionization coefficient, where an additional effect due to the band structure is expected. 22 In addition, the influence of dynamic screening, which was found to be substantial for Si, 23 should also be investigated for wide band gap materials such as ZnS and GaN III. MONTE CARLO MODEL A. Ensemble Monte Carlo simulation Monte Carlo methods are used extensively in modeling high-field transport in semiconducting materials. 25,26 Basically, the technique consists of numerically following in time the trajectories of charge carriers subject to instantaneous scattering events. The time between scattering events is generated stochastically using the calculated scattering cross sections of all possible mechanisms and the computer random number generator, as are the momentum and energy after scattering. Between scattering events, carriers accelerate due to the presence of external fields according to the Bloch theorem. An ensemble of particles may be simultaneously simulated, over which averages may be taken to obtain the time dependent single particle distribution function, and the associated dynamical information about the system. The ensemble Monte Carlo code presented in this article includes the full band structure information of the phosphor material and all the pertinent scattering mechanisms to solve the semiclassical Boltzmann transport equation under highfield conditions. Our approach essentially follows the model developed by the Illinois group, which is described in detail elsewhere. 27 The nonlocal energy dispersion relation for ZnS obtained in Sec. II is used in the particle dynamics during acceleration by the applied electric field and after collisions occur. However, the scattering rates due to phonons and impurities are generated using a density of states approach in which the rates based on a nonparabolic band model are renormalized by the actual density of states at the final energy after scattering to account for deviations from the freeelectron like density of states. The justification for this approach is that at high energies the optical deformation potential interaction governs electron scattering, as will be discussed below. B. Scattering mechanisms The scattering rates for various mechanisms are calculated based on first-order time dependent perturbation theory. For energies below the L and X valley minima, the scattering rates are computed according to a nonparabolic band model. 4 The effective mass and nonparabolicity parameter of the valley entering into these scattering rates were extracted directly from the EPM calculation of the band structure. The scattering mechanisms taken into account include scattering due to polar optical phonons, acoustic, and optical phonons via the deformation potential interaction, and ionized impurities. At energies above the L and X thresholds, electron scattering is dominated by the optical deformation potential mechanism. To the lowest order, this mechanism is isotropic in nature and depends only on the density of final states. Therefore, the density of states shown in Fig. 2 is used directly to determine the scattering rate at high energy. The main parameters entering the deformation potential scattering rate are the density of states and the deformation potential. Experimental measurement of the optical deformation potentials for inter- and intravalley scattering in ZnS does not exist to our knowledge. However, information regarding the high-field distribution may be indirectly inferred from measurement of carrier multiplication due to band-toband impact ionization. Such measurements were performed for ZnS on n-type Schottky diodes by Thompson and Allen 8 in which the electron and hole impact ionization coefficients were extracted by assuming these quantities are the same for

4 J. Appl. Phys., Vol. 83, No. 6, 15 March 1998 Dür et al FIG. 3. Calculated impact ionization coefficient vs inverse electric field for ZnS. The solid curve passes through the experimental data of Thompson and Allen see Ref. 8, while the three other curves represent theoretical fits to these data for different sets of optical deformation potentials. both types of carriers. We have used a set of two optical deformation potentials, one characterizing the lowest conduction band, and the other characterizing the upper bands, to bring our simulated results for the impact ionization coefficient as a function of electric field into agreement with Thompson and Allen s data. These deformation potentials are indicated as a pair of numbers in Fig. 3 for D I, D II, and D III which represent three different fits that bracket the experimental data from either side. The details of this choice of proper deformation potentials were discussed in article I. Figure 4 compares the optical deformation potential scattering rate in the first and the upper conduction bands associated with parameter set D II for the local and the nonlocal EPM calculations. As one can see, the scattering rates strongly reflect the variations in the density of states shown in Fig. 2. Renormalization of the scattering rates due to collision broadening and the intracollisional field effect are ignored in the present work except for a constant energy broadening used in choosing the final state after scattering. Although these effects are in principle important for such high scattering rates, the systematic inclusion of broadening effects is still an open issue. Further, without experimental measurement or an ab initio calculation of the electronphonon coupling constants, the semiempirical approach used herein is deemed adequate. FIG. 4. Optical deformation potential scattering rate at 300 K for the local and the nonlocal band structure using parameter set D II shown in Fig. 3 for a the lowest conduction band and b the upper conduction bands. order to achieve sufficient accuracy in simulating rare events such as impact ionization and impact excitation. Figure 5 shows the transient average drift velocity of electrons in the ZnS layer for two different applied electric fields at lattice temperatures T300 and 77 K. The simulations have been performed for one choice of optical deformation potentials, set D II. The drift velocity increases initially as the electrons rapidly gain kinetic energy from the field. When electrons reach energies where intervalley scattering dominates, they are transferred to other valleys and a steady-state population is established. Since the electron system reaches steady state within several tenths of a picosec- IV. HIGH-FIELD TRANSPORT In the ensemble Monte Carlo simulations presented in this work, we have included the first four conduction bands for ZnS. The full band structure is represented using 916 points in the irreducible wedge of the first Brillouin zone. Typically, an ensemble of particles is followed in FIG. 5. Transient average drift velocity of electrons in ZnS for two different phosphor fields at 300 and 77 K. Shown are the results for deformation potential parameter set D II.

5 3180 J. Appl. Phys., Vol. 83, No. 6, 15 March 1998 Dür et al. FIG. 6. Time evolution of the average electron energy in ZnS for different phosphor fields at 300 K using deformation potential parameter set D II. FIG. 7. Steady-state drift velocity of electrons in ZnS vs electric field for lattice temperatures 300 and 77 K using parameter set D II. The Monte Carlo data of Brennan see Ref. 5 at 300 K are also shown. ond, the carriers traverse almost the entire phosphor layer on the order of 0.5 m in steady state. It is also noted from this figure that, for a given temperature, steady state is established more quickly at higher electric fields. Figure 6 shows the time evolution of the average electron energy for three different electric fields with impact ionization scattering included or omitted in the Monte Carlo simulation. The calculations are made at 300 K using deformation potential parameter set D II. With the inclusion of band-to-band impact ionization scattering, a steady-state average energy is reached in less than 0.7 ps in all cases and increases with increasing electric field. When impact ionization is omitted, a significant number of electrons gain energy from the field at a faster rate than they lose energy to the crystal lattice for phosphor fields in excess of 1.5 MV/cm, which leads to electron runaway to extremely high energies. The average energy increases monotonically with time and a steady state is never reached. This effect was observed in the simplified simulations based on a parabolic band model. 3 Contrarily, electron runaway was not observed using a nonparabolic multivalley model 4 because of the increase of the density of states and therefore the scattering rates in such a model. Correcting the optical deformation potential scattering rate using the full band density of states results in a decreasing rate past a certain energy, allowing runaway. However, inclusion of impact ionization scattering in the full band model acts as a stabilizing mechanism, helping the electron system reach a steady state at high electric fields. Figure 7 shows the variation of the steady-state drift velocity as a function of the applied electric field for a density of /cm 3 ionized impurities at lattice temperatures 300 and 77 K. Deformation potential set D II has been used in the simulations. Brennan s 5 steady-state results at 300 K are also included for comparison. The calculated velocity-field curve at 300 K is in agreement with Brennan s result at high fields but gives much smaller values at low fields since ionized impurity scattering dominates the low-field mobility of electrons for impurity densities in excess of /cm 3, as was shown in Ref. 4. At 77 K, the reduced electron-phonon interaction leads to higher drift velocities and a lower threshold field for intervalley transfer given by the position of the peak of the velocity-field curve. Figure 8 shows the simulated steady-state particle distribution i.e., the number of particles in a given energy range at 300 K using deformation potential set D II. The distribution is plotted for four different phosphor electric fields, 0.5, 1.0, 1.5, and 2.0 MV/cm, both on a linear and a semilogarithmic scale. It is clear from this plot that there are almost no electrons with energy above the threshold for impact excitation of Mn 2 approximately 2.1 ev for 0.5 MV/cm, and that this situation rapidly changes as one goes from 1 to 2 MV/cm, in qualitative agreement with the brightness-voltage characteristics of this phosphor material when used in ACTFEL devices. At 2.0 MV/cm, we see that there are FIG. 8. Calculated steady-state particle distribution in ZnS for four different electric fields at 300 K using parameter set D II. The distribution is plotted on a a linear and b a semilogarithmic scale.

6 J. Appl. Phys., Vol. 83, No. 6, 15 March 1998 Dür et al FIG. 9. Comparison of the steady-state particle distributions in ZnS calculated using local and nonlocal band structures. The calculations have been performed for an electric field of 1 MV/cm at 300 K employing deformation potential set D II. some electrons with energies as high as 9 ev, although these carriers represent less than one part in 10 4 compared to the peak of the distribution. The secondary peak in the particle distribution at about 4 ev is directly related to the minimum in the density of states shown in Fig. 2, which translates to a reduced scattering rate there, and therefore a pile-up of carriers. Figure 9 compares the particle distributions in steady state calculated using the local and nonlocal band structure in the Monte Carlo simulation. The results are shown for an electric field of 1 MV/cm at 300 K employing deformation potential set D II. The distribution based on the local EPM calculation is shifted towards higher energies with respect to the nonlocal distribution. This shift is mainly a consequence of the larger intervalley separation energies between the valley and the X and L valleys in the local versus the nonlocal case, since the density of states, and therefore the deformation potential scattering rate, starts to increase rapidly beyond these valley thresholds. This can be seen from Figs. 2 and 4, respectively. In general, in comparing to full-band simulations for other wide band gap materials, we find that the dominant peak in the particle distribution is correlated to the first maximum in the density of states, with a pile-up of carriers at energies below this maximum due to the high scattering rate there. In Fig. 10, we have plotted the steady-state particle distribution for the same phosphor electric fields and lattice temperature as in Fig. 8b, but for another choice of deformation potentials, set D III. One can see by comparison with Fig. 8b that, for a given electric field, the main peak in the distribution slightly broadens and shifts to higher energies due to the lower deformation potential scattering rate in the first band associated with set D III, thus leading to a hotter electron population. FIG. 10. Calculated steady-state particle distribution in ZnS for four different electric fields at 300 K using deformation potential parameter set D III. electron radiatively decays back to its ground state. The emission colors of ACTFEL devices can be controlled by adding different luminescent centers to the phosphor layer. In ZnS-based devices, Mn 2 gives yellow emission 2.1 ev with the highest luminance. The impact excitation rate of Mn 2 luminescent impurities in ZnS used in the Monte Carlo simulations has been calculated from the transition rate derived by Bringuier 28 based on the exchange scattering process. For Mn 2 impurities embedded in ZnS, exchange scattering is known to be the dominant excitation path. The calculation of the excitation rate is outlined in the Appendix. To simplify matters, the ground and the excited state of the impurity atom have been assumed to be in the form of Gaussian orbitals. The spatial extent of the orbitals was adjusted to fit the measured average cross section 29,30 which is on the order of cm 2. Figure 11 shows the simulated band-to-band impact ionization yield and luminescent center impact excitation yield for electrons transiting a ZnS layer of 0.5 m thickness for various electric fields. The yields are calculated from the number of impact ionization and impact excitation events that are recorded during the simulation in steady state over a time interval corresponding to the average transit time of electrons through the phosphor layer. We have performed the calculations at 300 K for two of the deformation potential V. IMPACT EXCITATION OF LUMINESCENT IMPURITIES Impact excitation is the process in which a hot electron interacts with a luminescent center in the host phosphor to excite a ground state electron to an excited state, losing energy in this process. Luminescence results as the excited FIG. 11. Simulated impact ionization and impact excitation yields i.e., number of events per electron traversing a 0.5-m-thick ZnS layer as a function of electric field. Shown are the results for two of the deformation potential parameter sets listed in Fig. 3.

7 3182 J. Appl. Phys., Vol. 83, No. 6, 15 March 1998 Dür et al. FIG. 12. Simulated impact ionization and impact excitation yields as a function of electric field for two different lattice temperatures using parameter set D II. material, a comparison of the calculated ionization coefficient with different choices of deformation potentials to experimental carrier multiplication data is shown, and the resulting deformation potentials are used to calculate the highfield distribution and impact excitation yield. Very few electrons are found above the threshold for impact excitation of Mn for fields less than about 1 MV/cm, in agreement with the experimental behavior of ACTFEL devices. Our results confirm the expected trends in impact excitation yield with temperature. The impact excitation yield is found to be sensitive to the optical deformation potential, which suggests further investigation of the electron-phonon coupling in phosphor materials is needed in order to better understand their high-field behavior. parameter sets shown in Fig. 3, D II and D III. For both sets of parameters, the impact ionization yield is larger than the corresponding excitation yield for a density of /cm 3 Mn atoms. The comparison reveals that for an approximately 40% change in the deformation potential, there is a corresponding change of a factor of 2 in the impact excitation yield, which is consistent with the fact that the square of the deformation potential enters into the scattering rate. 4 The onset of impact excitation of luminescent centers occurs at about 1 MV/cm. This threshold is in excellent agreement with the experimental threshold of 1.04 MV/cm deduced by photoinduced luminescence PIL of evaporated ZnS:Mn ACTFEL devices. 31 The impact excitation yield exhibits an approximately linear increase with increasing phosphor field after threshold is reached. Figure 11 also suggests that the threshold for electroluminescence in ZnS:Mn ACTFEL devices should occur in the same field range as band-to-band impact ionization of electron-hole pairs. Recall from Fig. 6 that band-to-band impact ionization helps to stabilize the hot electron distribution at high electric fields so that runaway does not occur. Additionally, band-to-band impact ionization provides electron multiplication, and hence gain, and is also the creation mechanism for holes, which can be subsequently trapped in the ZnS phosphor layer, giving rise to positive space charge. 32 Figure 12 compares the impact ionization yield and the impact excitation yield for deformation potential set D II at two different lattice temperatures: T300 and 77 K. Impact excitation as well as impact ionization increases with decreasing temperature since the scattering of electrons due to phonons is reduced. At the lower temperature, the electron system reaches a higher average energy in steady state so that more electrons are capable of impact exciting luminescent impurities or generating electron-hole pairs by band-toband impact ionization. VI. CONCLUSIONS We have simulated high-field transport in ZnS using an ensemble Monte Carlo method including a nonlocal band structure, full-band impact ionization rates, and a simplified model for impact excitation of Mn 2 impurities. Due to uncertainties in the electron-phonon coupling constants in this ACKNOWLEDGMENTS We thank Peter Vogl of Walter Schottky Institute, Munich, for his assistance in the nonlocal band structure calculation. This work was supported by the U.S. Army Research Office under Contract No. DAAH04-94-G-0324, by the Advanced Research Projects Agency under the Phosphor Technology Center of Excellence, Grant No. MDA , and by the Deutsche Forschungsgemeinschaft under Grant No. Re 882/6-2. APPENDIX: EXCHANGE IMPACT EXCITATION RATE In the appendix, a derivation is given of the impact excitation rate of luminescent impurities used in the Monte Carlo simulations based on the transition rate for exchange scattering derived by Bringuier. 28 Gaussian orbitals are assumed for the ground and excited state of the impurity center which allow analytic solution for a single parabolic energy band. The initial and final states of Bloch electrons scattered off an impurity center in the phosphor can be written in the form n,k r 0 1 V expik r 0u n,k r 0, A1a n,k r 0 1 V expik r 0u n,k r 0, A1b where n,k (n,k) are the band index and wave vector of the Bloch state before after scattering. The initial and final states of the atomic impurity are given by g (r 1,r 2,...,r ) and e (r 1,r 2,...,r ), corresponding to the ground and excited state of the impurity. r 1,r 2,...,r represent the coordinates of the valence electrons involved in the transition. The matrix element for the impact excitation transition due to exchange scattering may be written according to Eq. 22 of Bringuier s article as

8 J. Appl. Phys., Vol. 83, No. 6, 15 March 1998 Dür et al Mn,k;n,k m1 d r dr 0 e *r 1,...,r m r 0,...,r n * r,k 0 r m dq e 2 0 q,q expiq r 0 r m g r 1,...,r n,k r 0, A2 where d r denotes dr 1...dr and (q,) is the dielectric function of the host material. Due to the exchange process, the coordinates r 0 of the Bloch electron have been exchanged with the atomic coordinates r m after the transition. Consider one term in the sum over m, say m1. Substituting A1 into A2, we define the form factors F n,gkq,r 2,...,r 1 dr 1 g r 1,r 2,...,r u n * r V,k 1 expikq r 1, F n,e kq,r 2,...,r A3a 1 V dr 0 e *r 0,r 2,...,r u n,k r 0 expikq r 0, and rewrite the term m1 of the sum in A2 as M 1 n,k;n,k dq dr 2...dr F n,e kq,r 2,...,r F n,gkq,r 2,...,r e 2 0 q,q 2. A3b A4 Now, at this point, one has to do something about the integration over the position coordinates of the other 1 valence electrons. To a first approximation, we assume that the wave function of the impurity system is just a product function of independent atomic wave functions. Then the integrations over dr 2...dr would be over orthonormal functions giving unity if the initial and final state are the same, or zero if they are different. Therefore, M 1 n,k;n,k dqf n,e kq F n,gkq e 2 0 q,q 2. A5 If we assume further that the matrix element does not depend on which m we choose in the summation, then the sum in A2 gives just M 1. Since the Bloch functions appearing in the form factors A3 are periodic in the direct lattice, they can be expanded in a Fourier series involving reciprocal lattice vectors u n,k r 0 G U G n,kexpig r 0, A6a u n * r,k 1 G U G * n,kexpig r 1. A6b Inserting these series expansions into A3, A5 is then given by M 1 n,k;n,k G,G F g kq,g U G n,ku G * n,k dqf e kq,g e 2 0 q,q 2, A7 where the form factors F g and F e for the ground and excited state of the impurity center are the Fourier transforms of the atomic wave functions F g kq,g 1 F e kq,g 1 V dr 1 g r 1 expikqg r 1, V dr 0 e *r 0 expikqg r 0. A8a A8b We now consider a model form for the atomic wave functions. Since in the exchange process, the atomic wave functions of the initial and final state are not integrated over the same position variable, the exact symmetry of the atomic orbitals is not too important, rather their localization in space. Let us assume that the ground state is described by a Gaussian orbital which is often used in ab initio pseudopotential calculations g ra g expr/a g 2, A9 where a g is the effective radius of the orbital, and A g is the normalization constant given by A g 2 3/4 a 3/2 g. A10 When we substitute A9 in A8, the ground state form factor becomes F g kq,gv 1/2 3/2 A g a g 3 expq 2 a g 2 /4, A11 where qkqg. Likewise, we find for the excited state form factor F e kq,gv 1/2 3/2 A e a e 3 expq 2 a e 2 /4, A12 where qkqg. Thus, the transition matrix element A7 can be written as

9 3184 J. Appl. Phys., Vol. 83, No. 6, 15 March 1998 Dür et al. M 1 n,k;n,k e2 A g A e a g 3 a e V U G n,ku G * n,k G,G dq expq2 a g 2 q 2 a e 2 /4 q,q 2. A13 To reduce this further, we first assume that is sufficiently large so that we can replace the dielectric function (q,) by its high frequency limit. Next we expand the exponential as expq 2 a g 2 q 2 a e 2 /4 expkg 2 a g 2 kg 2 a e 2 /4 expq 2 a g 2 a e 2 /4expq X, A14 where X(kG)a 2 g (kg)a 2 e /2. Then, the integral over q in A13 is readily evaluated, yielding dq expq2 a 2 g a 2 e /4expq X q 2 23/2 a l0 2l1!! 2l1! X a 2l. A15 Here we have put 2a 2 a g 2 a e 2. For simplicity, it is further assumed that a g a e a. Combining A13 with A14 and A15, the exchange matrix element becomes M 1 n,k;n,k e2 a 2 0 V G,G U G n,ku G * n,k expkg 2 kg 2 a 2 /4 2l1!! Ka 2l 2l1! 2 A16 l0 with KkGkG. If we take the limit that the atomic wave functions are highly localized in space compared to the spread of the Bloch electrons (kga 0,k Ga 0), then the exponential is approximately unity and the lowest order term in the series expansion dominates. Consequently, A16 reduces to M 1 n,k;n,k e2 a 2 U 0 V G n,ku* G n,k. G,G A17 Since, from A6, the sum over G and G is simply the cell periodic part of the Bloch wave function, we can write the matrix element as M 1 n,k;n,k e2 a 2 0 V u n,k0u n * 0. A18,k The scattering rate is given by Fermi s golden rule Sn,k n,k 2 2 M 1 n,k;n,k 2 E n k E n ke e E g ], A19 where E e and E g are the energy of the impurity center in the excited and ground state, respectively. Using A18 and introducing the impact excitation threshold energy E th E e E g, we have Sn,k n,k 2 2e4 a V 2 u n,k0 2 u n * 0,k 2 E n ke n ke th. A20 Thus, the total rate including all the scattering centers with the usual assumption that they are uniformly distributed is given by S n k 2 2e4 a 4 N li V E n ke n ke th, u n,k0 2 u n * 0,k 2 n,k A21 where N li is the density of luminescent impurities. As an approximation, we can take the limit of free electrons in a single parabolic band with effective mass m. This means that the cell periodic factors become unity. Then 2 E V n ke n ke th DEE th, A22 k where the density of states D(E) for a parabolic band is DE &m3/2 2 3 E. A23 Hence, the total scattering rate becomes SE 2 &e4 a 4 m 3/2 N li EE 0 th 1/2. A24 This expression is used in the calculation of the impact excitation yield in the present simulations rather than the fullband result A21 due to the uncertainty in the atomic radius a which is essentially fit to experiment. 1 Y. A. Ono, Electroluminescent Displays World Scientific, Singapore, B. K. Ridley, J. Phys. C: Solid State Phys. 16, H.-J. Fitting and A. von Czarnowski, Phys. Status Solidi A 93, K. Bhattacharyya, S. M. Goodnick, and J. F. Wager, J. Appl. Phys. 73, K. Brennan, J. Appl. Phys. 64, ; 75, M. L. Cohen and T. K. Bergstresser, Phys. Rev. 141, M. Reigrotzki, R. Redmer, I. Lee, S. S. Pennathur, M. Dür, J. F. Wager, S. M. Goodnick, P. Vogl, H. Eckstein, and W. Schattke, J. Appl. Phys. 80, T. D. Thompson and J. W. Allen, J. Phys. C: Solid State Phys. 20, L M. L. Cohen and J. R. Chelikowsky, Electronic Structure and Optical Properties of Semiconductors Springer, Berlin, J. P. Walter and M. L. Cohen, Phys. Rev. 183, J. Bude and K. Hess, J. Appl. Phys. 72, N. Sano and A. Yoshii, Phys. Rev. B 45, M. Stobbe, R. Redmer, and W. Schattke, Phys. Rev. B 49, N. Sano and A. Yoshii, J. Appl. Phys. 75, Y. Wang and K. F. Brennan, J. Appl. Phys. 71, M. Reigrotzki, M. Stobbe, R. Redmer, and W. Schattke, Phys. Rev. B 52,

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