A New Concept for Solving the Boltzmann Transport Equation in Ultra-last ransient Situations

A concept based on relaxation of the hydrodynamic parameters is introduced to arrive at a computational model for the extreme non-equilibrium distribution function of carriers in multi-valley bandstructure. The relaxation times are taken to describe the evolution scale of the distribution function. The developed model is able to account for transport phenomena at the momentum relaxation scale. The model, together with the Monte Carlo simulation, is applied to obtain the electron distribution function in each valley of the lower conduction band in GaAs, and to study the evolution of the distribution function in GaAs subjected to rapid changes in field.

I. THE CONCEPT OF HYDRO-KINETIC TRANSPORT THEORY It has been shown in the previous study that evolu- tion of the kinetic distribution function f(k) can be characterized by relaxation of the hydrodynamic parameters.The exact description for f(k) described by the Boltzmann transport equation (BTE) requires where n,, and k are the density, mean energy, and average momentum, respectively, and k 3 and k 4 are the higher order moments.In semiconductor, "c n > xe > Xm(relaxation times of n, e, and k, respectively), and the characteris- tic times of higher-order moments (higher than e) are assumed to be smaller than :m" Under the influence a drastic change in field, infor- mation given by the smaller-scale moments tends to vanish in a shorter time.After a sufficient time, f(k) 217 will evolve into the momentum-scale hydro-kinetic (HK) distribution function, fm(k;n,e,k) which include dynamic information at scales greater than or as small as m" Theoretically, the relaxation times at this scale are to be evaluated from fm(k;n,e,k) and therefore become e and k dependent.At a time greater than m after the change in field, k dependence in fm becomes insignificant, fm then evolves into the energy-scale HK distribution function, fe(k;n,e.)which is able to provide dynamic information at scales greater than or as small as e.The relaxation times at this scale of interest therefore become only energy dependent.In most hydrodynamic models for semiconductor devices, the relaxation times are actually taken only energy dependent [2].However, some phenomena described by the momentum relaxation in fast tran- sient or non-stationary situations might be ignored when using the energy-dependent relaxation times.At a time considerably greater than xe, e has reached its steady state value and is in equilibrium with the elec- tric field, E. fe(k;n,e) therefore evolves into the quasi-equilibrium distribution function, fE(k,n,E).
The HK transport theory offers a concept combin- ing the kinetic and hydrodynamic approaches.The HK distribution at a certain evolution stage can be chosen to approximate f(k) in devices.In non-equi- librium situations, only fe or fm are of interest.The description for fe has been introduced and applied to study electron transport in GaAs and Si in previous papers [1], [3].The approach to fm in the multi-valley bandstructure is proposed in the next section.

II. MULTI-VALLEY MOMENTUM SCALE HYDRO-KINETIC MODEL
The probability density function in k space, gi(k), is taken to derive a description for the Xm-scale HK dis- tribution function in the ith valley of multi-valley bandstructure.The normalized distribution function is defined to be gi(k)= fi(k)/ni, and gi(k)dk 1.The xe-scale and Xm-scale probability density functions are therefore expressed as gei (k;n,e) and gmi (k;n,e,k) respectively.The evolution from gmi into gei is taken to be a relaxation process at each time step influenced by relaxation of ei, and k i, and the change in field.The HK evolution equation for gmi can be written as where 1/Thi is the relaxation rate for gmi evolving toward gei, and )kmi is the shifted amount for gmi in k space.For a relaxation process, the difference between gmi and gei tends to reduce at each time step due to scattering which is accounted for through exp(At/hi).In addition, gmi also shifts in k space by )kmi_to take into consideration the variations in field and k i, within At.On the other hand, gei during the evolution remains in equilibrium with mean energy.gmi can be determined from Eq.(1) if )kmi and Xhi are solved at each time step.
Evolution equations for mean energy and momen- tum can be obtained by taking the moments of HK evolution equation given in Eq. (1) emi -ei %-.emi %-)mihkmi ei exp(-At/,l/2), (2a)   mi ki + mi + mi i exp e+l/ where the parameters with the subscript and m are determined from gi and gmi, respectively.Eqs.(2a) and (2b) can be solved together with the hydrody- namic equations to evaluate mi and hi" Iterations are required to determine the and k dependent relax- ation times using gmi, and will lead to more accurate solutions for hydrodynamic parameters and gmi" III.APPLICATION TO GaAs USING A 3-VALLEY MODEL The computational model proposed in Eq.( 1) for gmi(k;n,E,k) is applied to study electron dynamics in n-type GaAs at different scales of variations in field.gi(k) is obtained from Monte Carlo simulation to ver- ify the validity of the HK concept.For simplicity, space independence is assumed.In addition, the relaxation times are taken to be only energy dependent when solving the hydrodynamic equations.A 3-valley parabolic band model is used in the Monte Carlo and hydro-kinetic methods.The average elec- The solid symbols are velocities at the instants (indicated inside the parentheses) when the distribution functions are calculated and given in Figs. 2 and 3. Electric field changes between 4 and 50 kV/cm tron velocity in each valley is given in Fig. at a pulse field varying between 4 and 50 kV/cm with a rise/fall time of 0.5 ps.The distribution function illus- trated in Fig. 2 clearly indicate that at this applied field geF evolves considerably more slowly than gmF and gF, particularly during the velocity overshoot/under- shoot interval.However, gmF that includes effects of velocity relaxation is in very good agreement with gF except at time near the overshoot peak (t 0.5 ps) or undershoot minimum (t 0.21 ps).The pronounced overshoot/undershoot behavior enhances effects of velocity relaxation.The incomplete information pro-vided by the e-dependent relaxation times in the hydrodynamic equations therefore leads to evident difference between a9 r, obtained from hydrodynamic and Monte Carlo methods near the overshoot peak and undershoot minimum (see Fig. 1).This therefore results in the discrepancy between gmF and gF near the peak and minimum (see Fig. 2at  0.488, 1.97, and 2.39 ps).However, the dependences of the higher moments that are excluded in gm might be in part responsible for the discrepancy.As discussed in Sec.II, iterations can be used to improve the accuracy for gmF and -vr'.g is obtained from Monte Carlo simulation.Electric field changes between 4 and 50 kV/cm Effects of velocity relaxation are small in the upper valleys, as shown in Fig. 3 where ge, gm and g are in good agreement, since velocity overshoot is not evi- dent in the upper valleys.However, it still shows that the evolution of geL is slightly slower than that of geL and g/: It is interesting that, although the L-valley velocity undershoot (near 2ps) is small, discrep- ancy between gmL and gL at 1.97 ps is still observed.
A pulse change in field between 4 and 20 kV/cm with a rise/fall time of 2ps is also applied to illustrate the influence of velocity relaxation on gin.The tran- sient velocity and distribution function at this slower variation in field are shown in Figs.4a-4e.The a9 v overshoot/undershoot is weaker than that shown in Fig. 1, and the difference between geF and gmF also reduces at this slower change in field.Again, geF evolves considerably more slowly than gmF" On the contrary, gmF in this case can respond to electric field almost as fast as g] even near the peak/minimum of the overshoot/undershoot (see Figs. 4b and c).In addition, gL, gmL, and g/ are in excellent agreement as shown in Figs.4d and e. Fluctuations of Monte Carlo results in the X valley is too large for field  Electric field changes between 4 and 50 kV/cm below 20kV/cm, and the X-valley distribution func- tions are therefore not included.

IV. CONCLUSIONS
The HK transport theory offers a concept that evolu- tion of the distribution function can be characterized by relaxation of the hydrodynamic parameters.A sim- ple, accurate and efficient computational model is proposed in this paper for the Xm-scale HK distribu- tion in multi-valley bandstructure in extreme non-equilibrium situations.Studies show that, in situ- ations where velocity overshoot/undershoot is not very pronounced, the "Q-scale HK distribution func- tion can provide a good description for the kinetic dis- tribution.However, in cases of strong velocity overshoot or undershoot, effects of velocity relaxation are substantially enhanced, and the "Cm-scale HK dis- tribution function is needed to accounts for this extreme non-equilibrium behavior.Same representation of symbols and lines as Fig. 2. Electric field changes between 4 and 20 kV/cm )= f(k;n,z,fc,k3,k 4

FIGURE
FIGURE Transientvelocity in each valley.Lines and symbols denote results from the hydrodynamic and Monte Carlo methods.The solid symbols are velocities at the instants (indicated inside the parentheses) when the distribution functions are calculated and given in Figs.2 and 3. Electric field changes between 4 and 50 kV/cm

FIGURE 2
FIGURE 2  Normalized distribution function in the F valley.Dashed lines, solid lines, and symbols denote ge, gm, and g, respectively.

FIGURE 3
FIGURE 3 Normalized distribution function in the L and X valley.Same representation of symbols and lines as Fig.2.Electric field changes between 4 and 50 kV/cm

FIGURE 4
FIGURE 4 (a) Transient velocity.Same presentation of symbols and lines as Fig. 1. (b)-(e) Normalized distribution function in each valley.Same representation of symbols and lines as Fig.2.Electric field changes between 4 and 20 kV/cm