Efficient Evaluation of Ionized-Impurity Scattering in Monte Carlo Transport Calculations

Transcription

1 H. Kosina: Ionized-Impurity Sattering in Monte Carlo Transport Calulations 475 phys. stat. sol. (a) 163, 475 (1997) Subjet lassifiation: 72.2.Dp; 72.2.Fr; S5.11 Effiient Evaluation of Ionized-Impurity Sattering in Monte Carlo Transport Calulations H. Kosina Institute for Miroeletronis, TU Vienna, Gusshausstrasse 27±29, A-14 Vienna, Austria (phone: ++43/1/ , fax: ++43/1/559224, (Reeived June 3, 1997; in revised form July 9, 1997) Carrier sattering by ionized impurities is strongly anisotropi, and events with small sattering angles are highly preferred. On the Monte Carlo tehnique applied to semiondutor devie modeling this behavior imposes several problems whih are disussed. We present a method whih redues the amount of low-angle sattering very effetively. Instead of the anisotropi sattering mehanism an equivalent isotropi mehanism is defined whih gives the same momentum relaxation time. Depending on doping onentration and arrier energy the equivalent sattering rate is up to four orders of magnitude lower. By analyzing the Boltzmann transport equation a ondition is derived on whih the use of the equivalent sattering model is justified. The equivalene of the anisotropi and the isotropi sattering models is also demonstrated empirially by means of Monte Carlo alulations. The plain Brooks-Herring model is employed as well as an improved model inluding momentum-dependent sreening and oherent multi-potential sattering. An empirial orretion to attenuate the peak of the sattering rate at low energies is presented, and the statistial sreening model of Ridley is ritially disussed. 1. Introdution To simulate arrier transport in submirometer-sale semiondutor devies the Monte Carlo tehnique has found widespread appliation [1 to 4]. Beause of high doping onentrations in suh devies arrier mobilities are onsiderably redued by sattering off ionized impurities. The large range of Coulomb fores makes the sattering ross setion of a single ion very large, or even infinite if no sreening is assumed. Therefore, Coulomb sattering is a strongly anisotropi proess with a high probability for small-angle sattering events. While sattering events hanging the momentum very little are predited to our frequently their effet on momentum relaxation is small. This property of Coulomb sattering, though being physially sound, poses some problems on the Monte Carlo tehnique. A great many of low-angle sattering events have to be proessed thus onsuming omputation time. Very short free-flight times are obtained whih further degrade the effiieny of the Monte Carlo proedure. Fig. 1 illustrates this problem. As an example, the low-field mobility as a funtion of the ionized-impurity onentration, N I, has been alulated by the Monte Carlo method. The standard Brooks-Herring model was employed (see, e.g. [5][6]). One an observe that at low impurity onentrations about 93% of all sattering events are of Coulomb type. Phonon sattering onstitutes the rest. The urve in Fig. 1 then dereases slightly with inreasing N I and at about 1 2 m 3 starts to inrease. The latter effet is due to degeneray when the power law of the sreening parameter b s hanges from N 1=2 I to 32*

2 476 H. Kosina Fig. 1. Relative frequeny of Coulomb sattering as a funtion of doping onentration. Beause of the large range of Coulomb fores the frequeny is very high even at low onentrations N 1=6 I. At low impurity onentrations we observe the paradoxial situation that on the one hand ionized-impurity sattering is the most frequent proess, and that on the other hand it has nearly no effet on the mobility. This disrepany ours beause the ollisional and the momentum ross setions for the onsidered proess an differ by several orders of magnitude. We present a method whih allows to redue the amount of low-angle sattering very effiiently without altering the underlying transport problem. Instead of using the highly anisotropi sattering ross setion of sreened Coulomb interation we use an equivalent ross setion, whih is isotropi, and whih yields the same the momentum relaxation rate as the anisotropi ross setion. A ondition is derived on whih the use of the equivalent ross setion is justified. Another advantage of the new sattering model is that it is well suited for full-band Monte Carlo alulations. In these alulations it is problemati to randomly selet a momentum transfer q aording to a non-onstant probability density funtion. This speifi problem does not arise if an equivalent isotropi model is used instead of the original anisotropi one. 2. Mobility Definition The transport properties of arriers in semiondutors an be well desribed by the Boltzmann transport equation [7]. In the following we onsider the sattering integral at the right-hand side of the Boltzmann equation whih takes for the non-degenerate ase the ˆ V 2p 3 fp k ;k f k P k;k f k g dk : 1 k Here, V denotes the volume of the rystal, f is the eletron distribution funtion and k the eletron wave vetor. Sattering of harge arriers is a quantum mehanial proess

3 Ionized-Impurity Sattering in Monte Carlo Transport Calulations 477 whih is frequently treated by quantum mehanial perturbation theory. P k; k in (1) represents the transition probability from an initial state k to a final state k. A marosopi momentum balane equation is obtained by multiplying both sides of the Boltzmann equation by the eletron momentum, p ˆ hk, and integrating over k. By applying this proedure to (1) one obtains the average rate of loss of eletron momentum due to ˆ k dk 1 B V ˆ h f k 2p 3 k k P k ;k dk Adk k k ˆ h f k kt 1 m k dk: k In order to proeed from (3) to (4) one has to assume that P depends only on q, the angle between k and k, and not on the azimuthal angle j, P k; k P k; k ; os q : This assumption is fulfilled for any isotropi sattering potential. The ondition (5) ensures that the vetor-valued integral over k in (3) is parallel to k, and the proportionality fator between these two vetors is given by the mirosopi momentum relaxation time, t m. Now we introdue as defining relation for the mobility (see, e.g. [8][9]) J ˆ : In this general form, m is a tensor quantity. Due to an anisotropi band struture and an arbitrarily distorted off-equilibrium distribution funtion the urrent density and the momentum loss rate are not neessarily parallel vetors. The eletron urrent density is defined as J ˆ e v g k f k dk: 7 k A salar mobility an finally be extrated from (6) by using the magnitudes of the involved vetors, e vg k f k dk m ˆ h k t 1 m k f k dk : 8 In the following we adopt this mobility definition beause its derivation involves only very few approximations. Other mobility definitions found in the literature rely on muh more stringent assumptions, suh as the relaxation time approximation, onstant effetive mass, and/or a perturbative solution of the Boltzmann equation at very low eletri fields

4 478 H. Kosina 3. The Equivalent Sattering Cross Setion Anisotropi and elasti sattering an be disussed in terms of the differential sattering ross setion, s k; os q. For a given wave number k, the differential sattering ross setion an be interpreted as a probability density funtion of the sattering angle q. In the following the notation notation z ˆ os q will be adopted. For ionized-impurity sattering the differential sattering ross setion s and the quantum mehanial transition probability P are related by (see, e.g. [5]) 1 V s k; z ˆ N I v g k 2p 3 1 P k; k ;z k 2 dk : 9 N I denotes the onentration of the impurity enters, and v g is assumed to be derived from an isotropi band struture, v g The total sattering rate l and the momentum relaxation time, whih was introdued in (4), an be expressed in terms of the differential sattering as follows: l k ˆ2pN I v g k t m k ˆ 2p N I v g k 1 s k; z dz; 1 1 z s k; z dz: 11 To takle the problem of small-angle sattering we onstrut an equivalent sattering ross setion ~s that fulfills two requirements: 1. The equivalent ross setion is k; z ; 2. The equivalent and the original ross setion yield idential momentum relaxation times: ~t m k t m k : 13 With (13) it is ensured that the main transport parameter, namely the non-equilibrium mobility (6), is not altered as long as the distribution funtion remains unhanged. The sattering ross setion ~s and the total sattering rate ~ l of the equivalent model an readily be obtained by ombining (1), (11) and (12), (13) ~s k; z ~s k ˆ z s k; z dz ; ~l k ˆ 1 15 t m k In a Monte Carlo proedure ionized-impurity sattering an now be treated as an elasti and isotropi proess. The sattering rate for this proess is given by (15). The omputation of after-sattering state, k', is onsiderably simplified due to the isotropiity of the 14

5 Ionized-Impurity Sattering in Monte Carlo Transport Calulations 479 proess k ˆ k; j ˆ2pr 1 ; os q ˆ 2r 2 1 ; where r 1 and r 2 are two random numbers evenly distributed between and 1, and spherial polar oordinates are assumed. For an anisotropi sattering proess with a high preferene for forward sattering the momentum relaxation rate t 1 m is always smaller than the total sattering rate l. Therefore, using the equivalent sattering rate (15) in a Monte Carlo proedure has the advantage that ionized-impurity sattering beomes a less frequent sattering proess. The effet, that the equivalent sattering model requires a onsiderably lower number of Coulomb sattering events to be proessed, an be interpreted as a gathering of many sattering events eah with small momentum transfer to one sattering event with large momentum transfer. Now the question arises to what extent the transport problem is altered when the equivalent sattering model is used. We assume that the transport problem is desribed by a Boltzmann equation. Beause any hange of the sattering ross setion will not hange the left-hand side of this equation, we only have to investigate how the righthand side is affeted. In the Appendix we show that the sattering integrals are 16 if two prerequisites are fulfilled. First, the transition probability P has to aount for exat energy onservation as it is stated by the d±funtion in the golden rule of Fermi. Seond, the distribution funtion has to be of the following form whih is usually referred to as diffusion approximation f k; os q ˆf S k f 1 k os q : 17 The impat of this restrition is supposed to be pratially negligible for the here onsidered problems. One should be aware of the fat that l and P in the sattering integral (1) omprise ontributions of many sattering proesses, and that only the ontribution of Coulomb sattering is eventually modified but not those of the dominant phonon proesses. Furthermore, Coulomb sattering plays an important role espeially at low eletri fields, where the approximation (17) is ertainly very aurate. At high fields, however, where it might beome neessary to inlude higher powers of os q in (17) the influene of Coulomb sattering diminishes rapidly. For instane, the saturation veloity does not depend on the doping onentration and hene not on impurity sattering. For the reasons disussed so far it an be expeted that replaing a partiular anisotropi Coulomb sattering model by the orresponding isotropi one will introdue an error whih is virtually negligible in most pratial ases. 4. Ridley's Statistial Sreening Model One subjet to be disussed in the ontext of the problems related to small-angle sattering is the statistial sreening model introdued by Ridley [1]. In a form suited for Monte Carlo alulations [11] this model has beome very popular, and many authors have employed it in their Monte Carlo odes.

6 48 H. Kosina Formally, the statistial sreening model is obtained by modifying the sattering ross setion of the Brooks-Herring model as follows: s mod k; z ˆs BH k; z exp RN I pb 2 z ; 18 R denotes the average distane between the ions, and b is the impat parameter as funtion of the sattering angle, z ˆ os q. This model an be interpreted to ut off the longrange part of the sreened Coulomb interation. As a onsequene the amount of lowangle sattering is effetively redued, whih is probably the main reason for the popularity of this model. However, it is to note that aording to (11) and (6) suh modifiation of the ross setion inevitably results in a modified mobility. The original intention was to get an impurity sattering model that is appliable over a wide range of onditions. However, in a semiondutor at finite temperature we have a more speifi ondition in that free arriers responsible for sreening are always available. Therefore, in the ase of semiondutors it is not neessary to bridge the gap between the Conwell-Weisskopf (see, e.g. [5][6]) and the Brooks-Herring models, and the latter model an always be used without faing any divergene problems. We now shall briefly disuss the basi assumptions of the Brooks-Herring model and the additional assumption introdued by the onept of statistial sreening. For the Brooks-Herring model the transition probability is derived for one sreened impurity enter in the Born approximation. Then the situation in a doped semiondutor is desribed by simply multiplying the single-enter transition probability by the total number of impurities, N I V. Espeially this proedure of weighting all enters equally is not adopted in Ridley's model. Ridley argues that the random loation of the impurities leads to some extent to statistial anellation of the many impurity potentials. With the sattering ross setion a weight has to be assigned whih dereases exponentially with the distane of the eletron from the impurity enter ((18)). A measure for this distane is the impat parameter. However, the introdution of suh an exponential weight funtion appears somewhat artifiial, and we think of two indiations that suggest that all the impurity enters should be weighted equally. First, it seems that, instead of leading to weaker sattering due to statistial sreening, the random loation of the enters is a prerequisite to have effetive Coulomb interation. In [12] experimental evidene is reported that sattering beomes weaker if part of the ionized enters are orrelated. If all impurities formed some regular pattern there would be no impurity sattering at all. Therefore, assuming that non-orrelated impurities lead to weaker sattering seems not logial. Seond, one an onsider the potential aused by all impurities. Within the linear sreening theory the sreened Coulomb potentials V s of the single enters an be superimposed in a straightforward manner, V r ˆV s r r 1 V s r r 2... Using the Born approximation the Fourier transform of the potential has to be alulated and the absolute value of the squared potential. The linear translations of the potentials only hange the phase. V q ˆV s q exp iq r 1 V s q exp iq r 2... ; 2 jv q j 2 ˆjVs 2 2 q j jvs q j... 2jV2 s q j os q r 2 r 1... : 21 19

7 Ionized-Impurity Sattering in Monte Carlo Transport Calulations 481 Obviously, the single-ion ontributions in (21) do not depend on any spatial oordinate. Only the interferene terms ontaining the osine funtion depend on the spatial loations of the ions. These terms desribe oherent two-ion sattering. Both the Brooks- Herring model and Ridley's model expliitly treat sattering as two-body interations. Therefore, when disussing these models we have to onsider only the single-ion terms in (21) and an exlude the two-ion terms from the disussion. We onlude from (21) that for any single-ion sattering model the single-ion potentials have to be superimposed with unity weight. 5. Ionized Impurity Sattering We start with the Brooks-Herring model and then disuss some physial refinements of this model. For the Brooks-Herring model the Fourier transform of the sattering potential is of the form jv BH q j 2 ˆ Ze 2 1 e e r q 2 b 2 ; s 2 22 where e r denotes the relative permittivity of the semiondutor, and b s is the inverse Fermi-Thomas sreening length, b 2 s ˆ e2 n F 1=2 h e e r k B T n F 1=2 h : 23 Here, n represents the eletron onentration, T n the eletron temperature, F j the Fermi integral of order j, and h is the redued Fermi energy. Employing the potential (22) in Fermi's golden rule and performing the integration over the final states one ends up with a sattering rate l BH and a momentum relaxation rate, whih defines the equivalent sattering rate ~ l BH, l BH k ˆC k 1 b 2b 2 1 b ; 24 s ~l BH k ˆC k 1 4k 2 ln 1 b b : 25 1 b In these equations we have set b ˆ 4k 2 =b 2 s, and the energy-dependent prefator C is of the form C k ˆ N I Z 2 e 4 2ph 2 e e r 2 v g k : 26 Fig. 2 shows the ratio ~ l BH =l BH as a funtion of the doping onentration for a lowenergeti eletron E ˆ :1k B T, a thermal eletron E ˆ k B T and a hot eletron E ˆ 1k B T. Depending on the doping onentration and the eletron energy the sattering rate of the isotropi model an be several orders of magnitude lower than that of the anisotropi model. This results in a onsiderable saving of time expended at the proessing of Coulomb sattering events in the ourse of a Monte Carlo alulation.

8 482 H. Kosina Fig. 2. Ratio of the isotropi sattering rate, whih is equal to the momentum relaxation rate, and the anisotropi sattering rate as it results from the Brooks-Herring model (Eqs. (24) and (25)). The parameter denotes the eletron energy normalized by k B T, with T ˆ 3 K The Brooks-Herring model is well suited to demonstrate the peuliarities of ionizedimpurity sattering. To get quantitative agreement with measured mobility urves, however, the model needs to be extended to inlude additional physial effets. We found that momentum-dependent sreening and a orretion term for oherent multi-ion sattering have a strong impat on the mobility. How these effets an be aounted for in an unompliated fashion will be published elsewhere. The resulting sattering potential is given by jv q j 2 ˆ Ze 2 1 e e r q 2 b 2 1 s G q 2 sin qr qr : 27 Here, R denotes the average distane between ions whih is defined as R ˆ 2pN I 1=3. The sreening funtion G is approximated by a rational funtion [13]. The model of ionized-impurity sattering defined by (27) an be employed in a Monte Carlo proedure very effiiently by resorting to internal self-sattering. Within this framework sattering is treated as if it were a Brooks-Herring problem. The Brooks-Herring sattering rate is multiplied by some amplifiation fator B, whih is hosen suh that the following inequality is fulfilled in the interval q 2 ;2kŠ: BjV BH q j 2 jv q j 2 : 28 In this way we get a Brooks-Herring type sattering rate whih is larger than the sattering rate ~ l resulting from the potential (27), B ~ l BH k ~ l k : Note that both ~ l BH and ~ l are rates of isotropi sattering proesses. Now there is a welldefined probability P of aepting the seleted sattering event: P ˆ ~l=b ~ l BH. This prob- 29

9 Ionized-Impurity Sattering in Monte Carlo Transport Calulations 483 ability an also be expressed as P ˆ 2k B 2k jv q j 2 q 3 dq jv BH q j 2 q 3 dq : 3 Instead of solving these integrals diretly one an think of solving them by means of Monte Carlo integration. Keeping this in mind the internal self-sattering algorithm an be defined as follows. A random number q r 2 ;2kŠis hosen aording to the probabil- Fig. 3 Comparison of the a) relative frequenies and b) low-field mobilities obtained from the anisotropi Brooks-Herring model and the equivalent isotropi model; T ˆ 3 K and E ˆ 7 V/m

10 484 H. Kosina ity density funtion jv BH q j 2 q 3. Then a random number p r is hosen evenly distributed between and BjV BH q r j 2.Ifp r <jv q r j 2 then the sattering event is aepted, otherwise it is rejeted and self-sattering is performed instead. By means of the desribed method expliit integration of the potential (27) an be avoided, a task whih an be very time-onsuming depending on the omplexity of the used sreening funtion. 6. Results and Disussion The low-field mobility in unompensated silion will be disussed. In addition to ionizedimpurity sattering the transport model omprises aousti intra-valley sattering, six different types of phonon inter-valley sattering [6] and plasmon sattering. We adopt a non-paraboli and isotropi band struture using an effetive mass of m* ˆ :32 and a non-paraboliity oeffiient of a ˆ :5 ev 1. Room temperature is assumed. A single-partile Monte Carlo program is employed, and steady-state averages are omputed by sampling the quantities of interest at the before-sattering states [6]. Due to the high arrier onentrations onsidered the Pauli exlusion priniple has to be aounted for. A rejetion tehnique [6] is used assuming equilibrium Fermi-Dira statistis for the oupation probabilities of the final states. The eletron mobility presented in the following is given by m ˆ v d =E, where the drift veloity, v d ˆhv g i, is obtained by averaging the eletron group veloity, and the assumed magnitude of the eletri field was 7 V/m. It should be noted that (6), whih was used to theoretially justify the new approah, will onsistently simplify to m ˆ v d =E if a spatially uniform ondition is assumed. In Fig. 3 the anisotropi Brooks-Herring model (24) and the equivalent isotropi model (25) are ompared. Fig. 3a shows that the isotropi sattering model onsiderably Fig. 4. Results of the Brooks-Herring model and the extended Coulomb sattering model ompared with experimental data [14]. The extended model agrees better with the experimental data than the plain Brooks-Herring model. Both, the anisotropi version and the isotropi version of the extended model yield the same results

11 Ionized-Impurity Sattering in Monte Carlo Transport Calulations 485 redues the amount of Coulomb sattering events, espeially at low doping onentrations. Despite the large differene in the frequeny of Coulomb sattering Fig. 3b demonstrates that the resulting onentration-dependent mobilities are virtually idential. This result onfirms the theoretial investigations in Setion 3. The mobilities resulting from the sattering potential (27) are plotted in Fig. 4. This potential gives signifiantly better agreement with experimental data [14] than the plain Brooks-Herring model, whih onsiderably overestimates the mobility at intermediate and high doping onentrations. Sattering aused by the potential (27) has been implemented both as an anisotropi model and as an equivalent isotropi model. The algorithm for the isotropi model was further simplified by using internal self-sattering as reported in the previous setion. Fig. 4 shows that both ways of implementing the sattering model yields idential results. Finally, we shall disuss the peuliar energy dependene of Coulomb sattering and how to overome the related problems. In Fig. 5 the Brooks-Herring sattering rate is plotted as a funtion of the energy. Charateristi of Coulomb sattering is the peak at low energies whih is loated around E b =4 and inreases with dereasing doping onentration. E b defined as E b ˆ h 2 b 2 s =2m * is on the order of.2 mev at N I ˆ 1 16 m 3. This energy is by more than four orders of magnitude smaller than the thermal energy (k B T ˆ 25:8 mev at room temperature). Aording to the thermal distribution there will be only few eletrons that are affeted by this peak of the sattering rate. Unfortunately, not only the sattering rate but also the momentum relaxation rate shows suh a peak at low energies (Fig. 5). We introdue an empirial orretion to attenuate this peak and hene to remove the neessity of dealing with extraordinarily high sattering frequenies in Monte Carlo simulations. We start with the statistial sreening model of Ridley. The modified ross Fig. 5 Sattering rate aording to the Brooks-Herring model and the orresponding momentum relaxation rate at N I ˆ 1 15 m 3 and T ˆ 3 K. An empirial orretion is used to attenuate the peak at low energies

12 486 H. Kosina setion (18) results in the following modified sattering rate [11]: R l mod k ˆvg k 1 exp l BH k : 31 R v g k This rate does not show the problemati behavior at low energies. One an think that the expression v g k =R defines some maximum rate l max and that (31) is a rule that ensures that l mod is always smaller than l max. Eq. (31) satisfies the following inequality: l mod k Minfl max k ; l BH k g : 32 We now hose some maximum rate l max to artifiially limit the physially orret sattering rate l. A simple rule whih fulfills ondition (32), and whih is symmetri with respet l and l max, is the harmoni mean 1 ˆ 1 1 l mod l max l : 33 Obviously, the larger l max is taken the more l mod resembles the physially orret rate l. l max is an artifiial quantity whih has to be hosen suh that the effet on measured quantities suh as the mobility is negligible. Purely empirially we assume l max k ˆWv g k b s : 34 Monte Carlo simulations revealed that relative to the plain Brooks-Herring model the inrease in low-field mobility due to the modifiation (33) is less than 1:5% for W ˆ 1 and less than 2:3% for W ˆ 5. For omparison, Ridley's statistial sreening model yields low-field mobility enhanements up to 9%, depending on the doping onentration. Therefore, to keep any empirial modifiation of the mobility as low as possible one should use the modified rate (33) with an appropriate hoie for W instead of the rate (31). 7. Conlusion A method has been presented whih effetively redues the amount of low-angle sattering events that have to be proessed during a Monte Carlo transport alulation. The inverse momentum relaxation time serves as sattering rate for an equivalent sattering model. The equivalent model, whih is isotropi, exhibits an up to four orders of magnitude lower sattering rate, depending on doping onentration and arrier energy. An analysis of the sattering integral of the Boltzmann transport equation indiates that using the equivalent isotropi model instead of the anisotropi one has negligible influene on the transport problem of harge arriers in semiondutors. Monte Carlo alulations demonstrated the equivalene of both types of sattering models empirially. The assumptions of the statistial sreening model of Ridley are ritially examined, and we found that the preditions regarding mobility made by this model go into the wrong diretion. To deal with the peak of the sattering rate at low energies we presented an empirial orretion whose effet on the mobility an be ontrolled by a free parameter. Aknowledgements The author would like to thank G. Kaiblinger-Grujin for many fruitful disussions, and S. Selberherr for ontinued enouragement.

13 Ionized-Impurity Sattering in Monte Carlo Transport Calulations 487 Appendix The sattering integral (1) an be rewritten for a j-independent transition ˆ l k f k V 1 1 2p 2p 3 k 2 dk dz P k; k ;z dj f k ;z ;j : 1 35 Furthermore, for the transition probability P Fermi's golden rule shall hold P k ; k ˆ2p h jm q j2 d E E : 36 The isotropi ross setion: As a first step we evaluate (35) for the equivalent sattering model whih is isotropi. From (9), (14) and the golden rule (36) one an derive some artifiial matrix element ~M whih an be assigned to the equivalent sattering model. ~M 2 k ˆ z M 2 q k; z dz: 37 For an elasti sattering mehanism as onsidered here the momentum transfer is given by q 2 k; z ˆ2k 2 1 z. Keeping in mind that the transition probability ~P is independent of z the sattering integral (35) an be written as g@f ˆ ~ l k f k V 1 1 ~P k 2p 3 ;k k 2 dk 1 dz 2p dj f k ;z ;j : 38 The integral of f over the unit sphere yields the spherially symmetri part of f, f S k ˆ 1 4p 1 1 dz 2p djf k; z; j : The d-funtion within ~P allows one to put f S k in front of the integral, g@f ˆ ~ l k f k 4pf S k 1 V ~P k 2p 3 ;k k 2 dk : 39 4 The remaining integral evaluates to ~ l so that we end up with g@f ˆ ~ l k f k f S k : 41 Without loss of generality the distribution funtion an be deomposed into a spherially symmetrial and an anisotropi part, f k ˆf S k f A k 42

14 488 H. Kosina With (15) and (42) we get the following sattering integral for the equivalent model: ˆ f A k t m k : 43 The anisotropi ross setion: To be able to evaluate the sattering integral (35) for the anisotropi ross setion we have to make the assumption (17) on the shape of the distribution funtion. We shall now introdue a spherial polar oordinate system with the polar axis parallel to k. The solid angle of the eletri field is denoted by W ˆ q;j, and that of k by W ˆ q ;j. The angle between the eletri field and k an then be expressed as os g ˆ os q os q os j j sin q sin q : 44 Setting z ˆ os q and z ˆ os q we ˆ l k f k; z V 1 2p 1 2p 3 k 2 dk dj dz f S k f 1 k os g P k; k ;z 1 After integration over j and after putting f S and f 1 in front of the integral by employing the d-funtion in P one arrives ˆ l k f k; z f S k 1 V 1 2p 2 k 2 dk dz P k; k ;z 1 f 1 k os q V 1 1 2p 2 k 2 dk 1 dz z P k; k ;z : The first integral on the right-hand side evaluates to l k, and the seond one to l k t 1 m k. In the final result only the anisotropi part of f ˆ f 1 k os q : 47 t m k This results oinides with (43) if the anisotropi part of the distribution funtion is of the form f A k ˆf 1 k os q Referenes [1] M. Fishetti and S. Laux, Phys. Rev. B 38, 9721 (1988). [2] J. Higman, K. Hess, C. Hwang, and R. Dutton, IEEE Trans. Eletron Devies 36, 93 (1989). [3] F. Venturi, R. K. Smith, E. C. Sangiorgi, M. R. Pinto, and B. Rio, IEEE Trans. Computer-Aided Design 8, 36 (1989).

15 Ionized-Impurity Sattering in Monte Carlo Transport Calulations 489 [4] H. Kosina and S. Selberherr, IEEE Trans. Computer-Aided Design 12, 21 (1994). [5] D. Chattopadhyay and H. Queisser, Rev. Mod. Phys. 53, 745 (1981). [6] C. Jaoboni and P. Lugli, The Monte Carlo Method for Semiondutor Devie Simulation, Springer-Verlag, Wien/New York [7] S. Selberherr, Analysis and Simulation of Semiondutor Devies, Springer-Verlag, Wien [8] S. Bandyopadhyay, M. E. Klausmeier-Brown, C. M. Maziar, S. Datta, and M. Lundstrom, IEEE Trans. Eletron Devies 34, 392 (1987). [9] S. Lee and T. Tang, Solid State Eletronis 35, 561 (1992). [1] B. Ridley, J. Phys. C 1, 1589 (1977). [11] T. Van de Roer and F. Widdershoven, J. Appl. Phys. 59, 813 (1986). [12] E. Buks, M. Heiblum, and H. Shtrikman, Phys. Rev. B 49, 1479 (1994). [13] H. Kosina and G. Kaiblinger-Grujin, Solid State Eletronis (1997), in print. [14] G. Masetti, M. Severi, and S. Solmi, IEEE Trans. Eletron Devies 3, 764 (1983). 33 physia (a) 163/2

16 33*