Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Condensed matter physics wikipedia , lookup
Density of states wikipedia , lookup
Low-energy electron diffraction wikipedia , lookup
Tight binding wikipedia , lookup
Metastable inner-shell molecular state wikipedia , lookup
Electron mobility wikipedia , lookup
Electron-beam lithography wikipedia , lookup
International Journal of High Speed Electronics and Systems cfWorld Scientic Publishing Company IMPACT IONIZATION AND HIGH FIELD EFFECTS IN WIDEBANDGAP SEMICONDUCTORS M. REIGROTZKI, J. R. MADUREIRA, A. KULIGK, N. FITZER, R. REDMER Fachbereich Physik, Universitat Rostock D-18051 Rostock, Germany and S. M. GOODNICK, M. DU Ry Department of Electrical Engineering, Arizona State University Tempe, Arizona 85287-5706, USA Received (received date) Revised (revised date, if applicable) Accepted (accepted date, if applicable) Impact ionization plays a crucial role for electron transport in wide-bandgap semiconductors at high electric elds. Therefore, a realistic band structure has to be used in calculations of the microscopic scattering rate, as well as high eld quantum corrections such as the intercollisional eld eect. Here we consider both, and evaluate the impact ionization rate for wide-bandgap materials such as ZnS. A pronounced softening of the impact ionization threshold is obtained, as found earlier for materials like Si and GaAs. This eld dependent impact ionization rate is included within a full-band ensemble Monte Carlo simulation of high eld transport in ZnS. Although the impact ionization rate itself is strongly aected, little eect is observed on measurable quantities such as the impact ionization coecient or the electron distribution function itself. 1. Introduction High eld transport in semiconductors has long been a concern in relation to the performance of semiconductor electronic and optoelectronic devices for over three decades.1 To a high degree of success, nonequilibrium transport has been described within the context of the semi-classical Boltzmann Transport Equation (BTE), with instantaneous scattering events described by Fermi's Golden rule, and uncorrelated scattering processes of carriers with the lattice and one another. Such a framework is the basis for semiconductor device simulation tools based on moments of the BTE, or through direct solution of the BTE via particle based techniques such as the Ensemble Monte Carlo (EMC) method.2 However, the question has always remained as to the limitations of this approach in terms of the underlying quantum transport equation, and the role of corrections to the BTE such as collision broadening (CB) permanent address: Instituto de Fisica, Universidade Estadual de Campinas, Unicamp, 13083-970 Campinas, S~ao Paulo, Brazil y permanent address: Siemens GmbH, An der Untergeis 8 D-36251 Bad Hersfeld, Germany 1 2 Impact ionization and high eld eects in semiconductors and the intercollisional eld eect (ICFE), which become more pronounced at high elds. In particular, the role of such eects in the performance of submicron Si and GaAs devices has been uncertain at best. In recent years, interest has developed in wide-bandgap semiconductors such as ZnS, GaN and SiC for optoelectronic and high-power, high-frequency electronic applications. In such wide-bandgap materials, elds in excess of 1 MV/cm are common, which necessitates re-evaluation of the validity of the BTE for such systems. A high-eld process of particular interest in such wide-bandgap systems is the impact ionization rate associated with electron{hole pair excitation due to energetic hot carriers in the conduction or valence bands. In ZnS and SrS thin lm electroluminescent devices for example, this mechanism is responsible for space charge formation which may result in eld clamping and suppression of luminescence.3 We have previously calculated the impact ionization rate for Si and GaAs,4 ZnS,5;6;7 GaN,8 and SrS9 using a full band structure approach, but neglecting the inuence of the electric eld on the collision term through the ICFE. In these studies, we have used the local and nonlocal empirical pseudopotential method to calculate the band structure, which has a pronounced inuence on the numerical results for the impact ionization rate compared to analytical approximations for the electronic dispersion. The ICFE should also aect the behavior of the impact ionization rate in the high-eld regime. For instance, the evaluation of the Barker{Ferry kinetic equation10 for Si11 has indicated that the threshold energy for impact ionization is lowered due to the eld because the impacting electron is further accelerated during the collision. This results in a higher ionization rate near the threshold, whereas for higher energies of the impacting electron the eld inuence vanishes. Quade et al.12 applied a density matrix approach to carrier generation in semiconductors. Within the parabolic band approximation, they were able to give an essentially analytical result for the eld-assisted impact ionization rate which was evaluated for GaAs and Si. Again, systematic lowering of the threshold energy with the eld strength has been shown. The eld-dependence of the collision integral was also studied by means of the Green function technique, solving the Kadano{Baym equations13 in various approximations. Avoiding the conventional gradient expansion or delta-function approximation for the spectral density, an integral equation was derived for the EDF taking into account the ICFE.14;15 The Levinson16 or Barker{Ferry transport equation10 was evaluated within a saddle-point approximation for GaAs at high eld strengths, taking into account electron{phonon interactions.17 Alternatively, a gauge-invariant formulation of the Airy representation of the Kadano{Baym theory was developed.18 The Mori projection operator technique was applied to study nonlinear transport in semiconductors including the ICFE and collision broadening.19 The method of the nonequilibrium statistical operator as developed by Zubarev20 was applied to study both steady-state and transient properties in hot-electron transport.21;22 In the present work, we derive quantum kinetic equations for the EDF in Impact ionization and high eld eects in semiconductors 3 semiconductors using the Zubarev approach, and take into account the full elddependence of the collision integral. We then focus on impact ionization processes and re-derive the general, eld-dependent impact ionization rate given by Quade 12 et al. within the parabolic band approximation, and the Keldysh formula23 valid for energies near the threshold. As previously found, a softening of the threshold is obtained due to the ICFE, which is much more pronounced in the wide-bandgap materials due to the much higher onset elds required for impact ionization to occur. These eld-dependent ionization rates for ZnS are incorporated into an EMC simulation for high eld transport to evaluate the eect on observable quantities such as the electron distribution function, and the impact ionization coecient. In both cases, the role of the ICFE is found to be minimal even though the eect on the impact ionization rate itself is substantial. 2. Impact Ionization Rate 2.1. Quantum kinetic equation Quantum kinetic equations for a semiconductor in a homogeneous electric eld E~0 (t) can be derived within response theory taking into account the full elddependence of the collision term.24 The distribution function for electrons in the band 1 with momentum k1 is determined from @ f (t) + eE~ (t) @ f (t) = J ( k ; t): (1) e 1 1 0 @t 1 k1 @ k~1 1 k1 The collision term Je (1 k1 ; t) contains electron{phonon, electron{impurity and electron{electron scattering via a respective Hamilton operator. We restrict ourselves to electron{electron collisions since we are especially interested in the impact ionization rate. Dening fi ; ki g i, we get within second-order perturbation theory a Markovian form for the electron{electron collision term:25 Z t X 0 "(t ?t) cos[ 0 (1; 2; 3; 4)(t; t0)] Jeel (1; t) = ? 42 jMtot (1; 2; 3; 4)j2 "lim !0 ?1dt e h 2;3;4 ff1 (t)f2 (t)[1 ? f3 (t)][1 ? f4(t)] ? f3 (t)f4 (t)[1 ? f1 (t)][1 ? f2 (t)]g : (2) 0 Mtot is the total matrix element including direct, exchange, and umklapp processes. The collision term is given by an integral over all former times t0 which is usually interpreted as memory eect. In the limit of constant electric elds E~ 0 , canonical momenta transform the distribution functions to a gauge-invariant form: 0 (1; 2; 3; 4)(t; t0) = h1 ("3 + "4 ? "1 ? "2 )(t ? t0 ) ! ~0 ~k3 ~k4 ~k1 ~k2 e E ? 2h m + m ? m ? m (t ? t0 )2 + 31 !F3 (t ? t0 )3 : 3 4 1 2 (3) 4 Impact ionization and high eld eects in semiconductors The kinetic energies "i are given by the band structure, and mi is the eective mass of an electron in the band i at wave vector ki . The quantity !F is dened by 2 2 1 1 1 1 e E 0 3 (4) !F = 2h m + m ? m ? m : 3 4 1 2 In the process of impact ionization, a conduction band electron impact ionizes a valence band electron, i. e. 1 + 2 ! 3 + 4, see Fig. 1. The band indices and energies 1; 3; 4 run over the conduction bands, while 2 belongs to the valence bands. Supposing that the semiconductor is not highly excited, the conduction bands are almost empty so that the Pauli blocking factors are unity, i. e. (1 ? fi ) 1. Furthermore, the (second) in-scattering term in Eq. (2) can be neglected compared with the (rst) out-scattering term in the balance for the population of states with momentum k1 . The collision integral is then simply given by a eld-dependent impact ionization rate rii (1; E0 ) via Jeii (1; E0 ; t) = ?rii (1; E0 ; t)f1 (k1 ? eE0 t) with X rii (1; E0 ; t) = ? 42 jMtot(1; 2; 3; 4)j2 h 2;3;4 "lim ! 0 Z t ?1 dt0 e"(t ?t) cos[ 0 (1; 2; 3; 4)(t; t0)]: 0 (5) The wave-vector dependent impact ionization rate has to be evaluated considering the full eld dependence and a realistic band structure in the cosine term as well as the full momentum dependence of the matrix element Mtot including an appropriate screening function for the Coulomb interaction. conduction band 1 3 4 2 valence bands Fig. 1. Schematic impact ionization process for electrons. 2.2. Electron initiated impact ionization at zero elds The numerical evaluation of Eq. (5) is rather complex. We rst review results for the impact ionization rate within simple approximations. Neglecting the inuence of the electric eld in the collision term and considering the Markov limit t ! 1, the integral over the time t0 gives the energy conserving delta function and we have X (6) rii (1; 0) = ? 42 jMtot(1; 2; 3; 4)j2 ("3 + "4 ? "1 ? "2 ): h 2;3;4 Impact ionization and high eld eects in semiconductors 5 The integration over the momenta can easily be performed supposing a constant matrix element and spherical parabolic bands with eective masses for the valence (m2 = mv ) and conduction bands (m1 = m3 = m4 = mc ). Dening the eective mass ratio = mc =mv and the parameter = (1+2)=(1+ ), the threshold energy Eth = Eg is related to the fundamental band gap Eg , and the famous Keldysh formula23 for impact ionization is derived: rii(K) ("1 ; 0) = P0 ["1 ? Eth ]2 : (7) The prefactor P0 is often used as t parameter for the energy-dependent impact ionization rate in simulations of high eld transport in semiconductors.26 "1 = h 2 k12 =(2m1) is the kinetic energy of the impacting electron. However, previous calculations for Si and GaAs,4;27 ZnS,5;6;7 GaN,8;28;29;30 SrS,8;9 InN,31 and SiC32 have shown that the full band structure has to be considered when calculating the impact ionization rate via Eq. (6). Pronounced contributions arise from higher conduction bands, especially in wide-bandgap materials like ZnS, GaN, or SrS. The empirical pseudopotential method (EPM) is the standard tool to determine the band structure of a semiconductor material. Four to six conduction bands (and four valence bands) are usually considered for the complete numerical evaluation of the zero-eld impact ionization rate (6). The inuence of nonlocal pseudopotentials has been studied for ZnS7 and ab initio band structures have been used in calculations of the impact ionization rate of SrS.9 However, the EPM represents in most cases a reasonable compromise between desired accuracy and available computer capacity. The integrals in Eq. (6) extend over the entire Brillouin zone and are evaluated using an ecient numerical procedure developed by Sano and Yoshii.27 Making extensive use of symmetry relations imposed by the crystal structure, the integrations can be restricted to the irreducible wedge of the Brillouin zone where a large number of points can be taken into account for the numerical evaluation. For instance, using a uniform grid in wave vector space with 152 points in the irreducible wedge corresponds to 4481 points across the whole Brillouin zone;5;6;7;8;9 228 points in the irreducible wedge were also considered.28;29;30;31;32 A further increase of the number of grid points does not aect the calculated rate signicantly, except in the threshold region. The interaction between the conduction and valence electrons is described by a wave-vector dependent dielectric function derived by Levine and Louie.33 The frequency dependence of the dielectric function becomes more important as the energy of charge carriers increases. At high energies, we observe that primarily umklapp processes (which we take into account up to the sixth order) contribute to the calculated rate and, therefore, due to the large momentum transfer, the inuence of the carrier energy on the screening function is less important. Jung 34 et al. have employed a wave-vector and frequency-dependent dielectric function within the random phase approximation for the calculation of the impact ionization rate in GaAs using a Monte Carlo integration technique. Their results agree well 6 Impact ionization and high eld eects in semiconductors with our ndings which were obtained using a static dielectric function.4 The wave-vector dependent impact ionization rate rii (1; 0) is usually averaged over the entire Brillouin zone to obtain an energy-dependent rate R(E ) via R(E ) = X (E ? "1 )rii (1; 0) 1 X (E ? "1 ); (8) 1 which is shown in Fig. 2 for the wide-bandgap materials ZnS, GaN, and SrS. The general behavior is almost the same. The threshold energy relative to the conduction band minimum is given approximately by the gap energy. The numerical results according to Eq. (8) are well t by a power law relation of the form R~ (E ) = C [E ? Eth ]a ; (9) which can easily be implemented in full-band Monte Carlo simulations of electron transport in semiconductors. The prefactor C , the threshold energy Eth , and the power a are given in Table 1 for a variety of semiconductor materials. Obviously, the inuence of the band structure manifests itself in values a > 2 compared with the original Keldysh formula (7) derived for spherical parabolic bands. 10 16 10 14 10 12 10 10 −1 electron ionization rate [s ] Eth=3.7 Eth=3.8 Eth=4.0 10 8 10 6 ZnS (Ref. 5) GaN (Ref. 8) SrS (Ref. 9) GaN (Ref. 28) 4 6 8 electron energy [eV] 10 12 Fig. 2. Electron impact ionization rate for GaN, ZnS, and SrS. 2.3. Hole initiated impact ionization at zero elds A second possible electron{electron scattering process contributing to impact ionization is the relaxation of a valence band electron, passing enough energy to a second valence band electron to be ionized across the gap into the conduction band and generate an additional free carrier. This scattering process can be understood as the scattering of two holes in which a hole residing in the conduction band is ionized across the gap into the valence band, leading to an additional positive charge in the valence band and, correspondingly, to a negative free charge in the conduction Impact ionization and high eld eects in semiconductors 7 Table 1. Parameters for the interpolation formula (8) for the electron and hole initiated (see below) impact ionization rate for various semiconductor materials.35 36 ; Eth electrons: Si GaAs GaN ZnS SrS holes: GaN ZnS [eV] [1010eV?a s?1 ] C a 0.8 1.8 3.6 3.8 4.0 36.22 93.659 0.00949 5.935 59.723 3.683 4.743 7.434 5.073 3.182 3.4 3.8 0.35 0.71 5.33 6.23 band, see Fig. 3. This scattering process is interpreted as the hole initiated impact ionization, and due to the above explanations, the corresponding ionization rate can be determined using the very same numerical scheme as in the calculation of the ionization rate of electrons described above. The sole dierence is the inversion of the dispersion relation. conduction band 4 2 3 1 valence band Fig. 3. Schematic impact ionization process for holes. The respective impact ionization rates are calculated for holes in all four valence bands. Holes in the upper valence band can not initiate ionization processes. In GaN and ZnS the main contribution arises from the lowest and second lowest band. In SrS only holes from the lowest valence band are able to initiate ionization events. The corresponding energy-dependent impact ionization rate is shown in Fig. 4 for the wide-bandgap materials ZnS, GaN, and SrS.35 Previous theoretical results of Oguzman et al.29 for the threshold region of GaN using another band structure (B) are included for comparison. The shaded area covers their non-averaged, kdependent rates according to their band structure (B) via "(k). For comparison, the inset shows our non-averaged, k-dependent rates displayed in the same manner using the EPM band structures A and B (full and open circles). As can be seen, the calculated rate is relatively insensitive to modest changes in the band structure itself. The rates of Oguzman et al. are up to two orders of magnitude higher than ours. 8 Impact ionization and high eld eects in semiconductors −1 hole ionization rate [s ] This dierence is probably due to dierent integration schemes or screening models, and using dierent numbers of k points in the irreducible wedge. The corresponding parameters of the interpolation formula according to Eq. (9) are given in Table 1.36 10 20 10 18 10 16 10 14 10 12 10 10 14 10 12 10 10 10 10 10 8 10 6 8 3.5 4.5 5.5 6.5 ZnS GaN (A) GaN (B) SrS GaN (Ref. 29) 2 4 6 8 10 12 14 16 hole energy [eV] 18 20 22 Fig. 4. Hole impact ionization rate for GaN, ZnS, and SrS.35 Compared with the electron initiated impact ionization rates displayed in Fig. 2, two special features have to be noticed for the hole rates. First, the threshold energy for hole ionization is in general slightly lower than for electron ionization.9 This result is due to the at shape of the top of the upper valence bands, which allows holes to initiate ionization events as soon as they have reached a kinetic energy equal to the gap energy. In the case of SrS, the width of the upper valence band is too narrow to allow holes to gain enough kinetic energy to initiate ionization events due to energy conservation. For this reason, only holes in the lowest valence band contribute to the ionization rate and, therefore, SrS does not show the usual threshold behavior near the gap energy. Instead, the ionization rate sets in only above 14.7 eV and immediately jumps to very high values. Second, the hole initiated impact ionization rates vanish in a certain energy range completely, i. e. between about 5 eV and 12 eV for ZnS and 7 eV and 19.5 eV for GaN. For higher energies, they set in again immediately as discussed already for SrS. This special behavior is due to the occurrence of an energy gap between the upper three valence bands and the lowest one and, therefore, no states are available for energetic holes. Thus, the hole initiated impact ionization rate shows a very dierent behavior compared with the respective electron rate except for the region near the threshold energy, if hole ionization occurs there at all. Since the hole scattering rates are as low as the electron rates near the threshold energy, and vanish completely in the medium energy range, we conclude that hole initiated impact ionization processes can be neglected compared with the electron processes for both ZnS and SrS. Impact ionization and high eld eects in semiconductors 9 2.4. Field-dependent impact ionization rate The evaluation of the eld-dependent impact ionization rate (5) is important for high eld strenghts when the electrons are further accelerated during the collision. This is usually denoted as intra-collisional eld eect (ICFE). Quade et al.12 were able to evaluate the integrations over the momenta in Eq. (5) essentially analytically in the Markov limit t ! 1 considering the full, statically screened Coulomb matrix element, but performing the eective mass approximation. We obtain with the denition of the Airy function Z 1 ? Ai x d cos a 3 x = (3a)1=3 (3a)1=3 0 the general result: rii ("1 ; E0 ) = PQ (Q) Z 1 dE S ("1; "1 ? E ) 1ii Ai Eth ? "ii1 + E ; EF EF 2 1=3 0) : EFii = h!Fii ; !Fii = (1 +8m)(eE h c 0 (10) (11) !Fii is the electro-optical frequency. We follow the denition of the other quantities in Eq. (11) given in Quade et al.12 Performing the constant matrix element approximation but taking into account the full eld-dependence of the collision integral (5), a modied impact ionization rate can be derived: 2 Z 1 1 Ai Eth ? "1 + E : (12) r(K) (" ; E ) = P dE E ii 1 0 K 0 Eth EFii EFii The zero-eld limit yields the original Keldysh formula for impact ionization (7). Comparing Eqs. (7), (9), and (12), we have proposed recently24 a new t formula for impact ionization that considers the inuence of an applied electric eld and the full band structure according to a Z 1 E 1 E th ? "1 + E (F) r (" ; E ) = C dE : (13) Ai ii 1 0 0 Eth EFii EFii The parameters C , Eth , and a are already given in Table 1 for the zero-eld electron initiated impact ionization rates. We show as an example the eld dependent impact ionization rate for the widebandgap material ZnS in Fig. 5; the behavior of GaN and SrS is very similar. The ICFE inuences only the direct threshold region and leads to a systematic lowering of the threshold energy with increasing eld strength. The sensitivity of this lowering is related to the eective masses which are introduced as material parameters in the quantity EFii in Eq. (11) for the parabolic band result. The analytical form of Eq. (11) is also adapted for the t formula (13) that reects the full band structure via the parameter a and, again, the quantity EFii . About 0.5 10 Impact ionization and high eld eects in semiconductors 13 10 11 -1 rate [s ] 10 9 10 2.0 MV/cm 1.5 MV/cm 1.0 MV/cm 0.5 MV/cm 0.1 MV/cm Zero Field 7 10 5 10 3.0 3.5 4.0 4.5 5.0 energy [eV] 5.5 6.0 Fig. 5. Field-dependent electron impact ionization rate for ZnS.24 eV (1 eV) above the threshold energy, the impact ionization rate becomes already independent of the eld for the narrow (wide) band gap materials. 3. Ensemble Monte Carlo Simulation and Impact Ionization Coecient In order to understand the role that the ICFE plays in high eld transport, we include the eld dependent rates of the previous section into a full band EMC simulation for ZnS described in detail elsewhere.7 Basically, an EMC simulation is a particle based simulation technique where the particle trajectories under the inuence of external elds and random scattering events are tracked.2 Instantaneous random scattering events in the crystal are generated stochastically using the random number generator.2;37 As such, the EMC provides a direct solution to the Boltzmann equation for the one-particle distribution functions for electrons and holes and macroscopic averages derived from them. The full band dispersion relation E (k) for ZnS is taken into account using the EPM band structure. In the present work we only simulate electron transport and neglect the role of any holes generated by impact ionization processes. The Monte Carlo simulation includes polar optical phonon scattering, scattering due to acoustic phonons, optical deformation potential scattering, ionized impurity scattering, and band-to-band impact ionization using the modied rates of the previous section. As discussed elsewhere,7 we employ deformation potential scattering only above the intervalley threshold, where the density of states is used to renormalize the scattering rate to account for full band eects in the scattering rate. Two deformation potential constants are used to model the scattering rate due to optical phonons: D(1; 2) = (1 109; 9 108) eV/cm.36 Simulations are performed for an electron concentration of 1016 cm?3 in the conduction band and a temperature of 300 K. Impact ionization and high eld eects in semiconductors 11 Fig. 6 compares the electron distribution function versus energy calculated from the EMC simulation for three dierent eld strengths above 1 MV/cm. As can be seen, there appears to be no signicant eect due to the inclusion of the ICFE at any eld strength. electron distribution function 3 10 without ICFE with ICFE 2.0 2 10 1.5 1 10 1.0 0 10 0 1 2 3 4 5 electron energy [eV] 6 7 8 Fig. 6. Electron distribution function for three dierent electric eld strengths (in MV/cm) with (solid lines) and without (broken lines) the inclusion of the ICFE. A measurable quantity directly associated with the impact ionization rate is the impact ionization coecient which represents the number of electron{hole pairs created by an energetic particle per unit path length in the crystal. The impact ionization coecient is directly related to carrier multiplication in a reverse biased junction, and hence can be obtained from current multiplication versus bias data. In the EMC simulation, this quantity is calculated by tabulating the number of impact ionization events per carrier traversing a 0.5 m thick layer of ZnS. Fig. 7 shows the calculated result with and without the inclusion of the ICFE. The result without the ICFE corresponds to the results reported earlier by us for this quantity.35 As can be seen from this gure, there is only little dierence in the impact ionization coecient despite the rather dramatic eect on the impact ionization rate near the threshold energy shown in Fig. 5. 4. Discussion and Conclusions The main results from the present study are the calculated eect of the ICFE on the impact ionization rate in ZnS and other wide-bandgap materials, and its subsequent eect on the electron distribution function and impact ionization coecient. While the eect on the bare impact ionization rate is substantial, the resulting eect on the distribution function itself and the associated impact ionization coecient is negligible. The reason for the lack of any substantial inuence on transport may be understood from Fig. 5. When the eld is relatively low (below 0.1 MV/cm) the ICFE is negligible. At high elds close to the threshold for impact ionization to occur, 12 Impact ionization and high eld eects in semiconductors 4 impact ionization coefficient [10 /cm] 10 8 without ICFE with ICFE 6 4 2 1 0.50 0.52 0.54 0.56 0.58 0.60 0.62 0.64 inverse electric field [cm/MV] Fig. 7. Calculated impact ionization coecient versus inverse electric eld strength for EMC simulations with (solid line) and without (broken line) the inclusion of the ICFE. the threshold energy is broadened, but only for the region of energy over which the scattering rate is relatively low, below 1010 s?1 . The strongest contributions to the total scattering must come from the higher energy tail of the distribution where the impact ionization rate is comparable to the electron{phonon scattering rate, for any eect to be observed in the distribution function itself. However, the eect of the ICFE in this high energy region is small. Likewise, in terms of the impact ionization coecient itself, the signicant contributions to band-to-band impact ionization come from electrons well above threshold where the scattering rate is large. Conversely, in this region, there is little eect again due to the ICFE, hence the inuence on the impact ionization coecient is negligible. The role of collision broadening on the impact ionization rate has been studied recently using nonequilibrium Green functions for the derivation of an appropriate kinetic equation.38 It has been shown that collisions between the particles lead to a broadened one-particle spectral function which is of non-Lorentzian shape in contrast to former assumptions.11 A further lowering of the threshold energy for impact ionization is obtained, and the rate itself is increased in the threshold region. Collision broadening is already eective for eld strengths below 0.5 MV/cm, while the ICFE determines the behavior of the impact ionization rate for higher elds. For impact energies of about 1 eV above threshold, the collision broadening is almost negligible similar to the behavior found for the ICFE, see Fig. 5. Therefore, collision broadening has also no pronounced inuence on the EDF and the ionization coecient. Acknowledgements We thank K. Brennan (Atlanta), D. Ferry (Tempe), M. Fischetti (Yorktown Heights), T. Kuhn (Munster), V. Morozov (Moscow), W. Schattke (Kiel), E. Scholl (Berlin) Impact ionization and high eld eects in semiconductors 13 and P. Vogl (Munchen) for stimulating discussions. This work was supported by the Deutsche Forschungsgemeinschaft under contract No. RE 882/9-2 and by DARPA under the Phosphor Technology Center of Excellence, Grant No. MDA 972-93-10030. 1. Quantum Transport in Semiconductors, edited by C. Jacoboni, L. Reggiani, and D. K. Ferry, Plenum Press, New York, 1992. 2. C. Jacoboni and P. Lugli, The Monte Carlo Method for Semiconductor Device Simulation, Springer-Verlag, Berlin, 1989. 3. W. M. Ang, S. S. Pennathur, L. Pham, J. F. Wager, S. M. Goodnick, and A. A. Douglas, \Evidence for band-to-band impact ionization in evaporated ZnS:Mn alternatingcurrent thin-lm electroluminescent devices", J. Appl. Phys. 77, 2719 (1995). 4. M. Stobbe, R. Redmer, and W. Schattke, "Impact ionization rate in GaAs", Phys. Rev. B 49, 4494 (1994). 5. M. Reigrotzki, M. Stobbe, R. Redmer, and W. Schattke, \Impact ionization rate in ZnS", Phys. Rev. B 52, 1456 (1995). 6. M. Reigrotzki, R. Redmer, I. Lee, S. S. Pennathur, M. Dur, J. F. Wager, S. M. Goodnick, P. Vogl, H. Eckstein, and W. Schattke, \Impact ionization rate and high-eld transport in ZnS with nonlocal band structure", J. Appl. Phys. 80, 5054 (1996). 7. M. Dur, S. M. Goodnick, S. S. Pennathur, J. F. Wager, M. Reigrotzki, and R. Redmer, \High-eld transport and electroluminescence in ZnS phosphor layer", J. Appl. Phys. 83, 3176 (1998). 8. M. Reigrotzki, M. Dur, W. Schattke, N. Fitzer, R. Redmer, and S. M. Goodnick, \HighField Transport and Impact Ionization in Wide Bandgap Semiconductors", phys. stat. sol. (b) 204, 528 (1997). 9. M. Dur, S. M. Goodnick, R. Redmer, M. Reigrotzki, M. Stadele, and P. Vogl, Journal of the Society for Information Displays (accepted for publication). 10. J. R. Barker and D. K. Ferry, \Self-Scattering Path-Variable Formulation of highField, time-Dependent, Quantum Kinetic Equations for Semiconductors in the FiniteCollision-Duration Regime", Phys. Rev. Lett. 42, 1779 (1979); see also D. K. Ferry, Semiconductors, Macmillan, New York, 1991, Chapter 15. 11. J. Bude, K. Hess, and G. J. Iafrate, \Impact ionization in semiconductors: Eects of high electric elds and high scattering rates", Phys. Rev. B 45, 10958 (1992). 12. W. Quade, E. Scholl, F. Rossi, and C. Jacoboni, \Quantum theory of impact ionization in coherent high-eld semiconductor transport", Phys. Rev. B 50, 7398 (1994). 13. L. P. Kadano and G. Baym, Quantum Statistical Mechanics, Benjamin, New York, 1962. 14. A. P. Jauho and J. W. Wilkins, \Theory of high-electric-eld quantum transport for electron-resonant impurity systems", Phys. Rev. B 29, 1919 (1984). 15. S. K. Sarker, \Quantum transport theory for high electric elds", Phys. Rev. B 32, 743 (1985). 16. I. B. Levinson, Zh. Eksp. Teor. Fiz. 47, 660 (1969) [Sov. Phys.-JETP 30, 362 (1970)]. 17. P. Lipavsky, F. S. Khan, F. Abdosalami, and J. W. Wilkins, \High-eld transport in semiconductors. I. Absence of the intra-collisional-eld eect", Phys. Rev. B 43, 4885 (1991). 18. R. Bertoncini and A. P. Jauho, \Quantum transport theory for electron-phonon systems in strong electric elds", Phys. Rev. Lett. 68, 2826 (1992). 19. J. Y. Ryu and S. D. Choi, \Quantum-statistical theory of high-eld transport phenomena", Phys. Rev. B 44, 11 328 (1991). 20. D. N. Zubarev, Nonequilibrium Statistical Thermodynamics, Consultants Bureau, New 14 Impact ionization and high eld eects in semiconductors York, 1974. 21. D. Y. Xing and C. S. Ting, \Green's-function approach to transient hot-electron transport in semiconductors under a uniform electric eld", Phys. Rev. B 35, 3971 (1987). 22. D. Y. Xing, P. Hu, and C. S. Ting, \Balance equations for steady-state hot-electron transport in the approach of the nonequilibrium statistical operator", Phys. Rev. B 35, 6379 (1987). 23. L. V. Keldysh, \Kinetic theory of impact ionization in semiconductors", Zh. Exp. Theor. Phys. 37, 713 (1959) [Sov. Phys.{JETP 37, 509 (1960)]. 24. R. Redmer, J. R. Madureira, N. Fitzer, S. M. Goodnick, W. Schattke, and E. Scholl, \Field eect on the impact ionization rate in semiconductors", J. Appl. Phys. 87, 781 (2000). 25. D. N. Zubarev, V. Morozov, and G. Ropke, Statistical Mechanics of Nonequilibrium Processes, Vol. 1: Basic Concepts, Kinetic Theory, Akademie-Verlag, Berlin, 1996. 26. M. V. Fischetti and S. E. Laux, \Monte Carlo analysis of electron transport in small semiconductor devices including band-structure and space-charge eects", Phys. Rev. B 38, 9721 (1988). 27. N. Sano and A. Yoshii, \Impact-ionization theory consistent with a realistic band structure of silicon", Phys. Rev. B 45, 4171 (1992). 28. J. Kolnik, I. H. Oguzman, K. F. Brennan, R. Wang, and P. P. Ruden, \Calculation of the wave-vector-dependent interband impact-ionization transition rate in wurtzite and zinc-blende phases of bulk GaN", J. Appl. Phys. 79, 8838 (1996). 29. I. H. Oguzman, E. Bellotti, K. F. Brennan, J. Kolnik, R. Wang, and P. P. Ruden, \Theory of hole initiated impact ionization in bulk zincblende and wurtzite GaN", J. Appl. Phys. 81, 7827 (1997). 30. E. Bellotti, K. F. Brennan, R. Wang, and P. P. Ruden, \Calculation of the electron initiated impact ionization transition rate in cubic and hexagonal phase ZnS", J. Appl. Phys. 82, 2961 (1997). 31. E. Bellotti, B. K. Doshi, K. F. Brennan, J. D. Albrecht, and P. P. Ruden, \Ensemble Monte Carlo study of electron transport in wurtzite InN", J. Appl. Phys. 85, 916 (1999). 32. E. Bellotti, H.-E. Nilsson, K. F. Brennan, P. P. Ruden, and R. Trew, \Monte Carlo calculation of hole initiated impact ionization in 4H phase SiC", J. Appl. Phys. 87, 3864 (2000). 33. Z. H. Levine and S. G. Louie, \New model dielectric function and exchange-correlation potential for semiconductors and insulators", Phys. Rev. B 25, 6310 (1982); see also M. S. Hybertson and S. G. Louie, \First-principles theory of quasiparticles: Calculation of band gaps in semiconductors and insulators", Phys. Rev. Lett. 55, 1418 (1985); \Electron correlation in semiconductors and insulators: Band gaps and quasiparticle energies", Phys. Rev. B 34, 5390 (1986). 34. H. K. Jung, K. Taniguchi, and C. Hamaguchi, \Impact ionization model for full band Monte Carlo simulation in GaAs", J. Appl. Phys. 79, 2473 (1996). 35. M. Reigrotzki, R. Redmer, N. Fitzer, S. M. Goodnick, M. Dur, and W. Schattke, \Hole initiated impact ionization in wide band gap semiconductors", J. Appl. Phys. 86, 4458 (1999). 36. M. Reigrotzki, PhD thesis, University of Rostock, 1998. 37. C. Moglestue, Monte Carlo Simulations of Semiconductor Devices, Chapman and Hall, New York, 1993. 38. J. R. Madureira, D. Semkat, M. Bonitz, and R. Redmer, \Impact ionization rates of semiconductors in an electric eld: The eect of collisional broadening", in preparation.