Hydrodynamic modeling of avalanche breakdown in a gate overvoltage protection structure

Transcription

1 Solid-State Electronics 44 (2000) 1135±1143 Hydrodynaic odeling of avalanche breakdown in a gate overvoltage protection structure Martin Knaipp a, *, Werner Kanert b, Siegfried Selberherr c a Austria Mikro Systee International AG Schloû Prestatten, A-8141 Unterpre Statten, Austria b In neon Technologies AG, AIAPTDCD Balanstrasse 73, D Munich, Gerany c Institute for Microelectronics, Gusshausstrasse 25-29, A-1040 Vienna, Austria Received 30 Septeber 1999; received in revised for 19 January 2000; accepted 27 February 2000 Abstract The breakdown of an overvoltage protection structure is analyzed in the teperature range fro 298 to 523 K. The avalanche generation rates are odeled as a function of the carrier and lattice teperature. The generation rates are proportional to the carrier concentration. Careful attention is given to the pre-breakdown regie and to the breakdown process. The iportance of various generation processes to the ipact process is studied as well as the in uence on variations of the ionization threshold energy and of the energy loss during the ipact process. It is shown that the carrier generation inside the junction causes adiabatic carrier cooling, which leads to di erent carrier heating e ects at low and high lattice teperature. The behavior of carrier heating at roo teperature is strongly a ected by the asyetric eld distribution inside the junction. The reason for this is the eld dependence of the used trap assisted band to band tunneling odel and of the direct band to band tunneling odel. It is shown that at roo teperature, the onset of hole ipact ionization plays an iportant role for the electron heating. This is di erent at a teperature of 523 K, where the electrons doinate the onset of ipact ionization. Ó 2000 Elsevier Science Ltd. All rights reserved. Keywords: Ipact ionization; Hydrodynaics; Avalanche breakdown; Avalanche diodes; Zener diodes 1. Introduction In recent years, device feature sizes have been continuously scaled down to the sub-icron range. This size reduction causes an increase of the axiu eld strength inside the device, which leads to a higher probability for avalanche processes. An exaple of investigations on ipact ionization (II) is the substrate current siulation in MOSFETs, where the aount of substrate current is an iportant indicator for the aging behavior of the device [1,8]. To siulate the device breakdown, it is necessary to apply an accurate, physically otivated ipact ionization odel. The standard drift di usion (DD) * Corresponding author. Address: AMS International, Schloss Prestaetten, 8141 Unter-praestaetten, Austria. E-ail address: artin.knapp@asint.co (M. Knaipp). equations can only use a eld dependent ipact ionization odel, as only inaccurate estiations of the carrier teperatures are available [2,3]. However, the electric eld dependence is inaccurate, especially in sall devices. Non-local e ects ust be taken into account when the typical thickness of space charge regions becoes coparable with the carrier energy relaxation lengths. To account for non-locality in a ore suitable way, the ipact generation rate ust be calculated as a function of the local carrier teperatures and not of the local electric eld. The local carrier teperature can be calculated using Monte Carlo or hydrodynaic (HD) siulations. One way to use the inforation of the average carrier teperature is to copute an equivalent electric eld using the teperature versus electric eld characteristic that results fro Monte Carlo calculations for the hoogeneous case [4±6]. Finally, the equivalent electric /00/$ - see front atter Ó 2000 Elsevier Science Ltd. All rights reserved. PII: S (00)

2 1136 M. Knaipp et al. / Solid-State Electronics 44 (2000) 1135±1143 Noenclature C electron and hole eld enhanceent factors de c ; de v band edge displaceents due to band gap narrowing j n electron heat conductivity l n electron obility l T L ; N tot electron and hole obilities depending on the lattice teperature and the total dopant concentration (independent of theral carrier energies) l HD T L ; T electron and hole hydrodynaic obilities (depending on the theral carrier energies) ˆ n; p electron and hole concentrations w local potential s w; electron and hole energy relaxation ties s T L electron and hole lifeties (for the used recobination odel) s 0 electron and hole lifeties at 300 K (for the used recobination odel) s TL ;~E electron and hole lifeties depending on the lattice teperature and the electric eld D n electron di usion coe cient E c electron band edge energy E g local band gap E t energy level for the recobination centers threshold energy for ipact ionization E thr G SRH;TBB G BB G II H n generation rate for the cobined Shockley±Read±Hall and trap assisted band to band tunneling processes generation rate for the band to band tunneling process electron and hole generation rates for the ipact ionization process electron heat source ter ~J n electron current density N c ; N v densities of states in the conducting and valence band N t concentration of recobination centres N tot total dopand concentration N ref reference concentration for the band gap narrowing odel ~S n electron energy ux electron teperature T n T L c n c 0 lattice teperature electron scattering coe cient electron and hole capture rates (for the used recobination odel) k B Boltzann constant n i intrinsic concentration q eleentary charge ~v sat; electron and hole saturation velocities w n electron average theral energy eld is used instead of the local electric eld to calculate the carrier generation rates. This is often perfored in cobination with a conventional DD odel to achieve better convergence copared to a fully HD siulation [7]. The siulations in this paper have been carried out by applying a direct HD-II odel, where the ionization rate per carrier depends only on the carrier teperature. For the siulations, the general purpose device siulator MINIMOS-NT is used where a consistent HD equation set is ipleented. The usage of one type of siulator without any device speci c Monte Carlo inforation is a coon practise in industry. The advantage of the HD-II odel copared to a DD- II odel is its non-locality. This allows us to obtain the axiu generation rates per carrier at di erent device locations. The calculated generation rate per carrier and tie depends on the local theral energy, which results fro an energy integration up to the local point. As entioned a typical application for ipact ionization siulation is the reliability analysis of MOS- FETs. The ain di erence between the reliability siulation and the breakdown siulation of a diode concerns the high ultiplication factor of the diode after breakdown. The large nuber of generated carriers has in uence on the potential distribution, which changes the device behavior. As a consequence, a robust nuerical iteration schee is necessary to achieve sooth generation rates over the device area. The high local elds require a high doping, which results in a steep junction. Therefore, additional odels such as band gap narrowing, Shockley±Read±Hall generation (SRH), trap assisted band to band tunneling (TBB) and direct band to band tunneling (BB) have to be taken into account. After breakdown Auger recobination becoes iportant. In the pre-breakdown regie, the SRH, TBB and BB odels generate free carriers in the depletion zone. This results in carrier cooling, which has an in- uence on the breakdown voltage. After breakdown, these odels provide recobination as np > n 2 i : 1 This paper is organized as follows:

3 In Section 2, the overvoltage protection structure is presented. Section 3 describes the used HD equation set together with the used odels. Section 4 gives a discussion of the siulation results. In Section 5, conclusions are drawn. M. Knaipp et al. / Solid-State Electronics 44 (2000) 1135± The investigated device The structure is widely used to protect electronic coponents, e.g. gates of MOSFETs, against voltage spikes. During operation, the anode contact is connected with the gate etal while the source is connected with the cathode. In the case of a gate voltage spike, the breakdown of the eitter p protects the MOSFETs against overvoltage. The top view of the eitter p junction has an octagonal layout with an area of about 120 l 2. The syetrical line is shown on the left side of Fig. 1. The generated current is transferred through the p-well to an adjacent p area, where the cathode contact is located. During device operation, the n-doped substrate is reverse biased. The applied cathode-substrate voltage is 10 V. Due to the high applied substrate voltage and the relatively low doped regions, the substrate p-well and substrate p space charge region are uch larger copared to the steep eitter p junction. The easured currents which are scaled according to the siulated device area are shown in Fig. 2. In a rst step, we have to extract the aount of leakage current fro the epitaxy/p-well and the epitaxy/ p space charge regions. This is necessary to estiate the in uence of the SRH contribution in the breakdown junction. The pure SRH contribution is given in an operation regie, where the easured leakage current is x Section A Anode A' n + (Eitter) + p-well + p p n Epitaxy Layer n + Substrate Substrate Contact FOX Cathode Fig. 1. The siulated overvoltage protection structure. The arrows indicate the break-down region. During operation the substrate bias is 10 V. The doping pro le of Section A±A 0 is used for the one-diensional siulations. The arrow arked with x gives the positive x-direction. Fig. 2. Measured currents in a teperature range fro 298 to 523 K. The current is scaled to l 2 according to the size of the siulated device. alost independent of the applied voltage. This situation occurs in the high lattice teperature regie as shown in Fig. 2. The used SRH odel is described in Ref. [9] with a lattice teperature dependence as given in Section 2. The SRH current in the epitaxy/p-well and epitaxy/p space charge regions is only one part of the coplete leakage current. The second part coes fro the inority carrier concentration, which controls the di usion current over the reverse pn-junction. This concentration is deterined by a relatively sall SRH contribution, which is generated outside the space charge regions and by a larger part, which is caused by the intrinsic concentration. It should be entioned that the extraction of the SRH contribution was done by a coplete two-diensional siulation with appropriate boundary odeling using a device structure according to Fig. 1. The siulation results give best agreeent with the easured 523 K data at a trap concentration of c 3. This concentration is then used for the one-diensional device siulation of the eitter p breakdown junction. The result of the subtraction of the SRH coponent fro the easured data is shown in Fig. 3. The leakage currents in the pre-breakdown regie are now eld dependent and nearly independent of the lattice teperature. It should be entioned that the calibrated SRH contribution in the high teperature regie slightly overestiates the low teperature SRH contribution as described in Ref. [9]. This overestiation copensates the intrinsic current contribution, which nally leads to pure eld dependent leakage current coponents. The eld dependent leakage current in the pre-breakdown regie is therefore caused by the TBB, BB, and HD-II processes.

4 1138 M. Knaipp et al. / Solid-State Electronics 44 (2000) 1135±1143 Fig. 3. Measured currents in a teperature range fro 298 to 523 K with the sub-tracted SRH contribution which is calculated fro a two-diensional device siulation. The current is scaled to l 2 according to the size of the siulated device. 3. Models and hydrodynaic equations The Poisson and continuity equations for the HD equation set are straight forward and can be found, e.g. in Ref. [10]. Iportant for the considered proble is the forulation of the carrier energy ux equation, which reads for electrons (analog for holes) div ~S n ˆ grad E c q w ~J n 3 2 k Bn T n T L s w;n H n : 2 The energy ux ~S n is the su of a conductive and convective ter ~S n ˆ j n grad T n ~ J n q w n k B T n : 3 For the heat conductivity j we use the Wiedeann± Franz law j n ˆ 5=2 c n k2 B q l nnt n ˆ 5=2 c n k B D n n: 4 For the siulation the scattering coe cients c n and c p are assued to be zero. The variable H n in Eq. (2) is a heat source ter of the carrier energy syste. The ter describes the net energy loss caused by the generation/recobination processes. In our siulations, it is assued that in the pre-breakdown regie the SRH, TBB and BB odels generate carriers with an energy close to the band edge. The generated carriers therefore have a teperature near the lattice teperature, which iplies that the heat source ter can be neglected for these processes. Only the continuity equations are involved without any energy loss in the carrier heat ux equation (2). After breakdown, the generation processes are replaced by the appropriate recobination processes and the question is how to split the recobination heat between the carrier and the lattice syste. At least the in uence of the recobination heat is sall in the vicinity of the pnjunction because of the sall density of recobination centers, hence the in uence of this carrier heating process can be neglected. The iportant contribution to H is given by the II process itself, where the threshold energy E thr has to be supplied by the carriers perforing the ionization process. The carrier energy loss depends on the II generation rates for electrons and holes G II : H ˆ G II E g 5 In Eq. (5), the threshold energy E thr is replaced by the local band gap energy E g. The odel for the band gap depends on the lattice teperature according to Ref. [11] and accounts for band gap narrowing [12] p de v ˆ de c ˆ E ref h h 2 0:5 ; 6 h ˆ ln N tot =N ref with N ref ˆ c 3 and E ref ˆ 0:009 ev. For the carrier type, a teperature dependent obility odel is used [13] l HD T L ; T ˆ l T L ; N tot 1 a T T L ; a ˆ 3 2 k B l T L ; N tot s w; ~v 2 sat; q : 7 The ost iportant paraeters, which have in uence on the carrier teperatures are the energy relaxation ties (ERT) s w;. To estiate the in uence of these ties, we used two di erent odels during the siulations. In the rst odel, we used the constant values of s w;n ˆ 0:35 ps for electrons and s w;p ˆ 0:4 ps for holes [14]. The second odel evaluates the electron ERT, according to Ref. [15] with a lattice teperature dependence as described in Ref. [16]. In this odel, the electron ERT increases fro s w;n ˆ 0:4 ps for T n ˆ 300 K up to ore than s w;n ˆ 0:8 ps for T n > 6000 K. The used lattice teperature dependence is described in Ref. [16] and results in a slightly ERT decrease at higher lattice teperatures. The equation for the electron ERT in picoseconds reads " 2 T n s w;n T n ; T L ˆ1:0 0:538 exp 0: K 0:09 T n 300 K 0:17 T L 300 K # : 8

5 M. Knaipp et al. / Solid-State Electronics 44 (2000) 1135± The ERT for holes was extracted fro Ref. [15] with the indirect ethod described in Ref. [16]. The results show that the hole ERT is nearly independent fro the hole energy and the lattice teperature. During our siulations, we used a hole ERT of s w;p ˆ 0:3 ps. The siulation results in Section 4 refer to the initial ERT odels, where constant ERTs are used. The results of the siulations with the electron ERT of odel 2 show a siilar behavior. Section 4 shows that the average electron teperature is about 2500 K, which corresponds to an electron ERT of 0.6 ps. The section explains that the axiu carrier heating occurs at higher lattice teperatures, which results in a bigger carrier heating di erence when di erent ERT odels are used. The used coe cients for the HD-II odel with respect to the two ERT odels are suarized in Table 1. The lattice teperature dependence of the SRH odel is given in the exponents of the auxiliary variables n 1 and p 1, where the trap level E t is assued to equal the intrinsic level (E t E i ˆ 0 ev). In the pre-breakdown regie the nuber of available traps is uch larger than the nuber of carriers involved in the generation process. Therefore the tie of readjustent of an electron in a trap, once it is trapped, has been assued to be negligible [3]. The odel for the generation rate reads G SRH;TBB np n 2 i ˆ s p n n 1 s n p p 1 ; n 1 ˆ N c T L e Ec Et =kt L ; p 1 ˆ N v T L e Et Ev =kt L : 9 The lattice teperature dependence on the SRH lifeties is expressed as a power law [17]. The lifeties s are calculated using the lifeties s 0 at 300 K s 0 ˆ 1 1:5 c 0 N ; s T L ˆs : 10 t T L The trap assisted band to band tunneling odel odi es the SRH lifeties in Eq. (9) with the so-called eld enhanceent factors C n and C p. In case of strong local electric elds, the lifeties in Eq. (9) are odi ed to Table 1 The coe cients of the HD-II odel a First ERT odel Second ERT odel C s s 1 C C C a The coe cients refer to the ERT odels as explained in Section 3. s T L ; ~E ˆ s T L 1 C T L ; ~E : 11 The factors C n and C p are evaluated according to Refs. [18,19]. In the pre-breakdown regie at a low lattice teperature, the BB process has a ajor in uence. The odel for the generation rate reads [19] G BB ˆ BD n; p; n i j ~Ej 2:5 E 1=2 g exp E 1Eg 3=2 j~ej! : 12 The dependence on the lattice teperature is introduced via the band gap energy. The paraeters B and E 1 were calibrated to the easured data and have the values B ˆ 1: s 1 c 3 and E 1 ˆ 2: Vc 1. The relative population function D n; p; n i in Eq. (12) decides whether generation or recobination occurs D n; p; n i ˆ np n 2 i n n i p n i : 13 The used HD-II odel was rst proposed in Refs. [20,21]. The generation rate G T ; depends on the carrier teperature T and is proportional to the carrier concentration G II T ; ˆA u 1 erfc p u ˆ kbt E thr : 1 p 1 u exp 2 u u ; 14 The lattice teperature dependence is introduced by rewriting Eq. (14) as G II T ; T L ; ˆA T L ; u u 1 erfc p u 1 p 1 u exp ; 2 u k B T u ˆ E g T L ; N tot : 15 For the pre-factor A, we take the relation, T L A T L ; u ˆC 1 exp C 2 T 0 E thr T L T 0 exp C 3 C 4 u T : k B T 0 T 0 16 The coe cients C 1 to C 4 are suarized in Table 1. In Eq. (16), the reference teperature is T 0 ˆ 300 K, and the ionization energy is E thr ˆ 1:12 ev.

6 1140 M. Knaipp et al. / Solid-State Electronics 44 (2000) 1135± Siulation results With the odels described in the previous section, we siulated the one-diensional device according to the vertical doping pro le section A±A 0 shown in Fig. 1. When we copare the siulation results of Fig. 4 with those of Fig. 3, a di erence in the pre-breakdown regie is observed, which results fro neglecting the SRH and the intrinsic contribution of the breakdown junction leakage current in Fig. 3. One iportant question arises, why does the breakdown voltage increase with the lattice teperature? It can be shown that the breakdown condition is delayed when the lattice teperature is increased. The breakdown condition occurs when both carrier types reach a certain teperature so that avalanche generation can start. The ajor part of this section explains the process of carrier heating within the breakdown junction. To understand the processes in the junction, it is convenient to look rst at a typical eld distribution. The electric eld before breakdown with an applied bias of 6.3 V is shown in Fig. 5. The higher n-doped side (on the left) causes an asyetric eld inside the junction. In Fig. 6, the carrier teperature distributions of the junction are shown at lattice teperatures of 298 and 523 K. To analyze the di erent generation odels, we Fig. 5. The carrier concentrations at 298 K. The curves arked with triangles denote the concentrations before breakdown (bias 6.3 V). The curves with circles denote the concentrations after breakdown (bias 6.4 V). The position of axiu electron generation is at 0.41 l and cause the increase in the hole concentration. The increase in the electron concentration at 0.46 l is caused by the axiu hole generation rate at this position. The eld strength at the bias of 6.3 V is also shown. Fig. 6. Carrier heating of the investigated device using an anode (left side) bias of 4 V. Only SRH generation is considered. The electrons ove fro the right to the left. The donor doping is arked with the squared arks while the acceptor doping with the circled arks. Fig. 4. Siulated currents with the described HD odel. The currents in the pre-breakdown regie are saller copared to Fig. 2 because the SRH contribution fro epitaxy/p-well and the epitaxy/p space charge regions is not included. On the other hand, the currents in the pre-breakdown regie are larger copared to Fig. 3 because the SRH contribution and the intrinsic contribution of the breakdown junction leakage currents are neglected in Fig. 3. The current is scaled to l 2 according to the size of the siulated device. rst show the siulation results with the SRH odel, while the TBB, BB and HD-II odels are neglected. It can be seen that the carrier teperatures at 523 K are higher than at the roo teperature. This e ect can be explained by coparing the di usion current coponent and the SRH current coponent, which is generated in the junction space charge region. At a lattice teperature of 298 K, the SRH current is about 92 ties larger than the di usion current. The carrier teperature difference between 298 and 523 K can therefore be ex-

7 M. Knaipp et al. / Solid-State Electronics 44 (2000) 1135± plained by a carrier cooling at 298 K inside the space charge region of the junction. This process is purely adiabatic as no energy loss occurs in the considered carrier syste. The resulting carrier heating is stronger when the entering current coponent gets higher. In contrast, the resulting carrier heating decrease when the generation of carrier inside the junction increase. At the beginning of the space charge region, the carrier concentration caused by SRH generation is higher copared to the carrier density fro the incoing di usion current. This leads to a teperature decrease at the beginning of the carrier heating inside the junction (denoted by the arrows in Fig. 6). At a higher lattice teperature, the situation is copletely di erent. As the teperature dependence of the di usion current is proportional to n 2 i, the di usion current increases uch faster at higher teperatures than the SRH current, as the SRH current is only proportional to n i (9). At 523 K, alost the total current is due to di usion, so that nearly all carriers have to pass the whole space charge region, which nally leads to a higher carrier teperature. In the next siulation, the process of TBB generation is switched on together with the SRH process. The results for 298 K are shown in Fig. 7. The axiu of the TBB generation rate is about 10 ties higher copared to the axiu SRH rate. Due to the asyetric junction, the eld dependent process is located closer to the n-doped side. This asyetry has an iportant in- uence on the nal teperature dependence. The electrons are accelerated fro the right to the left. When they arrive at the region where TBB occurs, they are cooled down by the generated electrons. The axiu electron teperature is therefore saller copared to the pure SRH process. The situation is copletely different for holes. When they enter the junction region, they are cooled by the generated TBB holes. The cooling process of the holes has its axiu at the beginning of their way through the junction. Therefore, the relative part caused by the generated SRH holes in relation to the total nuber of generated holes is saller copared with the result where only SRH generation is considered. For that reason, the holes are heated ore copared to the pure SRH process as shown in Fig. 6. The results of the siulations explain that the process of adiabatic cooling can nally lead to higher carrier teperatures at other locations. In the following siulations, the additional process of BB generation is switched on. The results for 298 K are shown in Fig. 8. The axiu BB rate is about four orders of agnitude higher copared with the axiu TBB rate and uch ore localized on the left side of the junction. The generation process leads to a strong electron cooling and an additional hole heating. Until now, the siulation results overestiated the nal teperature di erence between electrons and holes because of the strong eld dependence of the TBB and BB odels. When the carrier teperature dependent HD-II odel is switched on, the situation gets reversed because the holes start generating electron hole pairs at the right part of the pn-junction. The generated holes leave the junction at the right side. The generated electrons have to pass nearly the whole junction, which results in a rise of the electron teperature as shown in Fig. 8. The breakdown nally starts when the hole ipact ionization has heated up the electron syste above the breakdown teperature. Fig. 7. Carrier heating of the investigated device using an anode (left side) bias of 4 V. Only SRH and TBB generation is considered. The electrons ove fro the right to the left. The linear shape of the TBB generation rate is also shown. Fig. 8. Carrier heating of the investigated device using an anode (left side) bias of 4 V. The results correspond to odel sets, which accounts for the SRH/TBB/BB and SRH/TBB/BB/II processes. The electrons ove fro the right to the left. The linear shape of the BB generation rate is also shown.

8 1142 M. Knaipp et al. / Solid-State Electronics 44 (2000) 1135±1143 To copare the ionization process at roo teperature with the process at 523 K, it is iportant to analyze the current parts fro the applied generation odels without the ionization odel. At roo teperature, the quotient of I D;SRH current and the pure di usion current I D is about I D;SRH =I D ˆ 92 at a bias of 4 V. The TBB current is about a factor I D;SRH;TBB = I D;SRH ˆ 12:4 higher. The largest current part coes fro the BB process, which raises the current further by a factor of about I D;SRH;TBB;BB =I D;SRH;TBB ˆ 1098: At 523 K, the situation is quite di erent. Again, we analyze the situation without an ionization odel. The quotient of the total current and the di usion current is I D;SRH;TBB;BB =I D ˆ 1:06, when we apply the sae bias of 4 V. This eans that the carrier teperature distribution inside the junction is deterined by the di usion current, and therefore the shape of the carrier teperature distribution is siilar to that in Fig. 6. For this reason, the in uence of carrier cooling inside the junction is sall. An additional di erence is given by the in uence of the applied odels. As the BB generation stays in the sae order of agnitude, the SRH generation increases with the intrinsic concentration n i in the order of four agnitudes. The TBB current part increases in the sae way like the SRH current, because the electric eld stays nearly the sae as for 298 K. Therefore, the TBB current gives the largest contribution of all generation odels. When we switch on the HD-II odel, a current increase by a factor of 1.37 can be observed even at the low bias of 4 V. This strong in uence can be explained by the high carrier concentration inside the junction, which is caused by the raised intrinsic concentration. Even at low biases, the high ipact generation rates cool down the carrier systes below the breakdown teperature. Note that Fig. 6 is the result without any ionization odel. The shape of the carrier teperature distributions is siilar to Fig. 6, which deonstrates that the electrons have the higher teperature and start to ionize. The necessary breakdown teperature, which has to be obtained by both carrier systes, and the uch higher carrier concentrations are responsible for the higher eld dependent leakage current before breakdown starts. It should be noted that with the paraeter set in Table 1, the avalanche teperature coe cient a T ˆ DV =DT L ˆ 2:66 VK 1 is reproduced quite well. During the siulation, the breakdown teperature at 298 K is about 8500 K using a band-gap of 1.08 ev. At 523 K, the breakdown teperature is about 7200 K with a corresponding band gap of 1.02 ev. 5. Conclusion In this paper, we presented hydrodynaic breakdown siulations of an overvoltage protection structure with an asyetric pn-junction. The lattice teperature covers a range fro 298 to 523 K. It could be shown that within the HD odel, the pre-breakdown generation processes are copletely di erent at roo teperature copared to the processes at 523 K. This is ainly caused by the eld dependent generation processes in cobination with an adiabatic cooling of the carriers. These cooling processes are an inherent liit of the HD device siulation if only one energy syste is used for each carrier type. References [1] Habas P. Modeling approaches in studying the hot-carrier aging of MOSFETs. Microel Rel 1994;34(11). [2] Chynoweth AG. Ionization rates for electrons and holes in silicon. Phys Rev 1958;109:1537±40. [3] Selberherr S. Analysis and siulation of seiconductor devices. New York: Springer, [4] Fischetti MV, Laux SE. Monte Carlo analysis of electron transport in sall seiconductor devices including bandstructure and space-charge e ects. Phys Rev B 1988; 38(14):9721±45. [5] Tang JY, Hess K. Ipact ionization of electrons in silicon (steady state). J Appl Phys 1983;54(9):5139±44. [6] Wang XL, Chandraouli V, Maziar CM, Tasch AF. An e cient ulti-band Monte Carlo odel with anisotropic ipact ionization. In: Int Electron Dev Meet p. 725±8. [7] Slotboo JW, Streutker G, Dort MJv, Woerle PH, Pruijboo A, Gravesteijn DJ. Non-local ipact ionization in silicon devices. In: Int Electron Dev Meet p. 127±30. [8] Jungeann C, Meinerzhagen B, Decker S, Keith S, Yaaguchi S, Goto H. Is physically sound and predictive odeling of nos substrate currents possible? In: Solid- State Electron p. 647±55. [9] Knaipp M, Silinger T, Kanert W, Selberherr S. Analysis of leakage currents in sart power devices. 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