TIME-DEPENDENT CARRIER VELOCITIES IN III-V COMPOUNDS CALCULATED BY THE LEGENDRE-POLYNOMIAL ITERATIVE METHOD S.C. Van Someren Greve, Th. G. Van de Roer To cite this version: S.C. Van Someren Greve, Th. G. Van de Roer. TIME-DEPENDENT CARRIER VELOCITIES IN III- V COMPOUNDS CALCULATED BY THE LEGENDRE-POLYNOMIAL ITERATIVE METHOD. Journal de Physique Colloques, 1981, 42 (C7), pp.c7-343-c7-348. <10.1051/jphyscol:1981742>. <jpa-00221679> HAL Id: jpa-00221679 https://hal.archives-ouvertes.fr/jpa-00221679 Submitted on 1 Jan 1981 HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.
JOURNAL DE PHYSIQUE Colloque C7, supplément au n 10, Tome 42, octobre 1981 page C7-343 TIME-DEPENDENT CARRIER VELOCITIES IN 111-V COMPOUNDS CALCULATED BY THE LEGENDRE-POLYNOMIAL ITERATIVE METHOD S.C. van Someren Greve and Th. G. van de Roer Eindhoven University of Technology, Department of Electrical Engineering, P.O. Box 513, Eindhoven, The Netherlands Résumé. Une méthode semi-analytique rapide est présentée pour la résolution de 1 équation de Boltzmann par la décomposition en polynômes orthogonaux de Legendre. Cette méthode est basée sur l'introduction d'un terme "self scatterlng". Cette méthode, si on la compare à l'intégration numérique est directe et ne souffre pas des difficultés dues aux instabilités numériques. Elle permet aussi de calculer la réponse de la répartition des vitesses sur un champ électrique variable dans le temps. Abstract. A fast semi-analytical method is described for solving the time-dependent Boltzmann equation expanded in Legendre polynomials. The method is based upon the introduction of a self-scattering term. Compared with direct numerical integration this method is straightforward and has no difficulties due to numerical instabilities. It also allows calculation of the timeresponse of the distribution function to a varying electric field. 1. Introduction. Different methods to calculate the carrier distribution function in semiconductors have been developed. The Monte Carlo method [1], [2] which is based on the simulation of motion of one or a certain number of electrons in space allows complicated band structures and scattering mechanisms to be taken into account. The method however is time-consuming especially when a great number of carriers is considered in order to model time-space dependent phenomena. Iterative methods are much faster. The iterative method of Rees [3] is based on transforming the Boltzmann equation into an integral form. This method can be modified so that each iteration step becomes equivalent to a time step of the physical system. To reduce the number of points which represent the distribution function Hammar [4] has modified the iterative method by expanding the distribution function in Legendre polynomials. According to Hammar et least 100 terms in this expansion are required to accurately represent the distribution function but a two-term expansion already gives quite an accurate result for the quantities of physical interest. Four or six terms give these quantities with high precision. The set of equations obtained cannot be integrated in a straightforward manner due to numerical instabilities connected with the homogeneous solutions. This problem can however be circumvented by subtracting unwanted solutions in each step. In this paper it is shown that the set of equations upon the introduction of self scattering can be Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1981742
C7-344 JOURNAL DE PHYSIQUE transformed into an integral form. Each iteration step now will be equivalent to a time step of the physical system. The simple form of the integrals makes it possible to perform each iteration very fast. 2. Basic equations. The time and space independent Boltzmann equation is: where v(&) =!s(k,kt)+' g(k) = Is(&','f)f (5' )E. The iterative technique proceeds in the following way: first the current approximation of f(5) is used to calculate the function g(k). With this function the next approximation of f(5) is obtained by solving equation (2.1) which now is a linear differential equation. It can be shown C51 that f(5) converges to thestationarydistribution function. In the method of Rees C31 eq. (2.1) is converted into an integral equation. To simplify his method Rees introduced a fictitious "self scattering" process which leaves the state of a particle unchanged. Defining the self-scattering rate by: where I' is a constant, the function g*(&) is defined by g*(k) =!CS(~',E) + S' (F',%)If(5' )dk. Equation (2.1) is then transformed into - ee. V f(5) + Tf(k) = g*(k). a k It was shown furthermore by Rees that by choosing T much greater than the values of v(&) at energies at which a substantial amount of electrons is present, the iterations model the time-dependent distribution function that develops from the 1 initial function f (k). Each iteration step then is equivalent to a time step 0 - Following Hammar C41 it is assumed that the direction of the applied electric field forms a symmetry axis for the distribution function which then can be expanded in orthogonal Legendre polynomials: m where k = [kl and X = cos8, 8 being the angle between and E. The advantageous property of S(k,kl,x) being diagonal remains true for S(k,k',x) + S1(k,k',x). g*(%) can be expanded in the same way with the coefficients g*(k) given by:
Inserting the expansion (2.7) in eq. (2.6) an infinite set of coupled first-order differential equations can be obtained: m 1 - afm= 1 {%A eemn cg*-rf n n )--B kmnn f 1. (2.9) ak n=o The values of A and B* have been given by Hammar C41. mn For practical evaluation the set has to be truncated at a certain number N which has to be even, otherwise the last equation will be decoupled from the others. 3. The particular solution. The differential equations given by (2.9) have besides the particular solution also homogeneous solutions which either have a singularity at k=o or grow exponentially as k+m. It is therefore important to have a method to find only the particular solution. Consider the set of differential equations: The fundamental matrix?(k) is formed by columns which are independent solutions of the homogeneous set: The particular solution then can be written as C61: where An analytical method has been found to calculate the matrices 2 and rank N. Full detailsofthe method will be published elsewhere. Their general form is: for arbitrary (n,m,i=o,l,..., N-l). The X. are roots of the equation: fir P N c-) eexi = 0; (PN is N~~ Legendre polynomial). since N is even, the h. occur in pairs with opposite Signs. L The ordering of the homogeneous solutions is such that:
C7-346 JOURNAL DE PHYSIQUE A useful property is then: The particular solution (3.3) has to obey the boundary conditions: lim fo(k) = f (0), k-to lim fr(k) = 0 (00), k+o lim fa(k) = 0. k- The condition (3.9a) is fulfilled if the components of the vector _v(ko) obey the relation at k = 0: (3.9a) (3.9~) It turns out that condition (3.9b) is then fulfilled automatically. The condition (3.9~) can be fulfilled if the components v (0) are chosen in the following,way: 2n N-l v 2n (0) =-I 1 Y2n,i(~)g:(~)ds. (3.11) o i=o Defining : the components V ~~+~(O) take the form: The particular solution can then be expressed as: By calculating first the integrals: Ak N-l, 12,(k) = ex~(-x~~ak)i~~(k+ak) + exp(-x2,t) 1 'an, i (k+t)gz(k+t)dt, (3.15a) o i = o and Ak N-l, I~n+l(~) = exp(-x2nak)12n+l(k-ak) + o i=o Y ~ ~ i(k-t + ~ )g;(k-t)dt,, (3.15b) l exp(-iznt) 1
where I (0) = IZn+l(o), the numerical computation of f.(k) can be performed very 2n J fast. 4. Results and conclusions. Hammar C41 has found that at room temperature a two-term expansion already gives quite good results for the macroscopic quantities. From our work we conclude that when the electrical field is not too high the same holds true for the time-dependent velocity: For instance, fig. (4.1) shows the response of the average electron velocity in GaAs to a field step from zero to 6 kv/cm and back to 5 kv/cm. Thelatter part compares very well to the results of Rees 131. Also, results for field stepsup to 10 kv/cm show very good agreement with Monte Carlo results of Kaszynski C71. In fig. (4.2) a similar curve of Ga(0.47)In(0.53)As is given where polar optical phonon, intervalley and alloy scattering are taken into account. The alloy scattering potential was put at 0.4 ev. The other parameters have been taken from Fauquembergue [S]. The striking feature of this curve is the large velocity overshoot which indicates.that this is an excellent material for short-channel field effect transistors. At higher fields the two-term expansion gives a much less satisfactory description of the time-dependent behaviour. Fig. (4.3) shows the response of the electron velocity in GaAs to a field step from zero to 20 kv/cm using two- and four-term expansions. Only the four-term expansion gives good agreement with Monte Carlo results C?]. However, it can be seen that the two-term expansion gives quite good estimate for the amplitude and time duration of the velocity overshoot, as well as for the static velocity. It is especially the central valley which is described insufficiently by two terms. Here the main scattering process below the energy at which intervalley scattering starts is polar optical phonon scattering which is strongly dependent on the angle between the wave vectors before and afterscattering. This scattering mechanism leads to a strongly elongated distribution in the field direction. The higher valleys are dominated by intervalley scattering which randomizes the directions of the velocities much stronger. In these valleys a twoterm expansion gives a good description even at high fields. After a certain time most of the carriers will occupy these higher valleys, so errors in the central valley will be less influential which explains why the stationary drift velocity is calculated well even using a two-term expansion. * The time steps used in the calculations were around 10-l4 seconds
C7-348 JOUNAL DD PHYSIQUE References. [l] Kurosawa, T., Proc. Int. Conf. Phys. Semicond. Kyoto (1966). l21 Kurosawa, T., J. Phys. Soc. Japan Suppl. 21, 424 (1966). C31 Rees, H.D., IBM J. Res. Dev. 13, 537-542 (1969). C41 Hammar, C.. J. Phys. C. g, 70-78 (1973). l51 Vassell, M.O., J. of Math. Phys. 11, 408-412 (1970). C61 Coddington, A. and Levinson, N., Theory of ordinary diff.eq.,mcgraw-hill(1955). C71 Kaszynski, A., These Universitb de Lille, 1979. C81 Fauquembergue, R., private communication. GaAs 300 K... N.2 N.4 M. Carlo ref.171 2. 1 (ooo)valley all valleys D i ; '(PS) : i. \ I" 2 i '.c.