Formulation of the Boltzmann Equation as a Multi-Mode Drift-Diffusion Equation

We present a multi-mode drift-diffusion equation as reformulation of the Boltzmann 
equation in the discrete momentum space. This is shown to be similar to the conventional drift-diffusion equation except that it is a more rigorous solution to the Boltzmann equation because the current and carrier densities are resolved into M×1 vectors, where M is the number of modes in the discrete momentum space. The mobility and diffusion coefficient become M×M matrices which connect the M momentum space modes. This approach is demonstrated by simulating electron transport in bulk silicon.


INTRODUCTION
With the continued down-scaling of semiconduc- tor devices, there is a need to develop device simulators that can treat carrier transport taking into account off-equilibrium carrier distributions by solving the Boltzmann Transport Equation as accurately as possible.Several techniques have been developed to do so-such as the Monte Carlo [1], hydrodynamic [2], spherical harmonic [3], and cellular automata methods [4], and the Scattering Matrix Approach [5,6,7].Each method has its own limitations-for example, computational bur- den, calibration of parameters, low order approx- imation for the distribution function, "artificial diffusion" of carriers and restriction to fixed spatial square grids.Ideally, a simulation method should provide all the capabilities of drift-diffusion simulators (i.e., simulations from equilibrium to high bias with smooth results at low computa- tional burden) while also resolving carrier distribu- tion and treating scattering processes rigorously.Our objective in this paper is to take a step in this direction.
Therefore, here we will describe a re-formula- tion of the scattering matrix equations which expresses the 1-D spatial Boltzmann equation as a 1-D spatial drift-diffusion equation in a dis- cretised 3-D momentum space.The current and carrier densities generalise to M vectors, where *Corresponding author.Tel.: (765) 494-3515.Fax: (765) 494-6441.E-mail: lundstro@ecn.purdue.edu.
M is the number of modes in the discrete 3-D momentum space.The mobility and diffusion co- efficient become M M matrices which connect the M momentum space modes.Solving the Boltz- mann equation, then, reduces to solving a set of M coupled drift-diffusion equations which might be done by a generalisation of the standard techni- ques for solving the conventional drift-diffusion Equation [8].

FORMULATION OF THE MULTI-MODE DD EQUATION
The one-mode method of McKelvey [9, 10] and similar observations by Shockley [11] can be generalised to M-modes in 3-D momentum space and this gives us an expression for the differential flux equations in 1-D real space for the M-modes in momentum space as follows [7] 7e(x) where the M x vectors J+ (x) and -re(x) are the fluxes discretised in positive and negative direc- tions of momentum Px respectively, at a position x.The elements of the (M M) differential matrices [{/(x)] can be interpreted as the inverse of an inter- mode mean free path in presence of all possible scattering mechanisms.In general, these terms can be difficult to calculate analytically and therefore we use an indirect procedure.The space-indepen- dent differential [{/] matrices are calculated from the matrix logarithm of the transmission matrix of a semiconductor slab divided by the slab thickness.
The transmission matrix itself is obtained from the scattering matrix calculated by Monte Carlo techniques [7].All possible information about the underlying physics of scattering (band structure, phonons, ionised impurities and electric field) that is included in the Monte Carlo simulation is automatically embedded in the scattering matrix and hence in the [so0-] matrices.
Now returning to Eq. ( 1), we find its symmetric and anti-symmetric components and relate each flux Ji(x) to its velocity vi and its population density ni(x).Thus, we obtain a multi-mode drift- diffusion equation and its associated continuity equation: dx where the diffusion, inverse Einstein and mobility matrices are defined as follows: and IV] is a diagonal matrix whose elements are the mode velocities.The coefficient matrices for the continuity equation are e Equations ( 2) and ( 3) are the key results of the paper.Note that they are very similar to the conventional drift-diffusion form.This multi- mode drift-diffusion equation has associated M xM mobility and diffusion matrices that depend only on the scattering mechanisms and momentum space discretisation.
It is important to note that the field dependence of the relevant matrices here.The discrete form of the Boltzmann Transport Equation [7] indicates that each of the differential [ij (x)] submatrices are linearly dependent on the field, for any given orientation of the field.This relation makes the [#(x)], [D(x)], [a(x)] and [/3(x)] matrices straight- forward to calculate once the invariant zero-field [0.(x)] matrices and the field-coefficient matrices are known.

RESULTS
In this section, we present some preliminary results for (111) electric fields in bulk Si.A simple spherical, non-parabolic energy band structure was assumed for calculating scattering rates and the 3-D momentum space was discretised into 400 modes 20 modes in transverse ptla._nd 20modes in longitudinal px(this implies 20 J+ fluxes for +Px and 20 J-fluxes for -Px).Scattering matrices were computed by Monte Carlo simula- tion [5][6][7] and the [ij(X)] matrices were extracted using the procedure described in Section 2. Higher levels of accuracy with respect to the underlying band-structure and scattering can be attained just by using more sophisticated Monte Carlo techni- ques.The distribution of the carriers was assumed to be uniform across the modes, which is adequate to demonstrate proof-of-concept here.
Having obtained the [0.(x)] matrices, we then calculated the mobility and diffusion matrices and examined their structure.These are shown in Figure 1.Calculating these matrices involves taking matrix logarithms and inverses which produces a large number of elements that are numerically near zero.If we ignore these small elements, the matrices are very sparse ( 1-2% ignoring elements < 0.1% of the largest element) and have a simple structure.The most significant elements of the mobility matrix are the diagonal elements and off-diagonal elements only where the th mode is connected to the j th mode by field shift.The diffusion matrix is strongly diagonal and has significant but small off-diagonal elements only where the th mode is connected to the j th mode by scattering.In order to test the formulation, we used the above matrices to simulate electron transport in bulk Si with (111) electric fields.The solution in bulk is simple because there are no spatial gradients in Eqs. ( 2) and (3).Substituting the expressions for [c], [/3] and [#] from Eqs. ( 7), ( 8) and ( 6) respectively, we get [--[11]  The solution for the above Eq.( 9) is a straight- forward solution to a null-space problem (using svd factorisation in MATLAB) and it gives us the complete bulk carrier distribution function for any field 8x.Some calculated bulk distributions for (111) electric fields are shown in Figure 2. Taking the average of the mobility and the diffusion matrices over the carrier distributions so obtained we see that macroscopic mobility, diffusion coeffi- cients and electron temperature so obtained have the expected behaviour with electric field.The low- field mobility is 30% too low because of the assumed uniform intra-mode distribution.Highfield results which do not suffer from this constraint are therefore more accurate-e.g.9.915 106 cm/s at 105 V/cm.Note that the field dependence of (/z) and (D) is a consequence of the field dependence of the distribution function in this approach. 4

DISCUSSION
To illustrate how the equations would be solved under spatially varying conditions, we present a simple linear scheme using finite differences.By discretising df(x)/dx on a uniform grid of size h, we obtain system whose solution is the position dependent carrier distribution function.However, the Schar- fetter-Gummel method is normally the preferred scheme for discretising the conventional drift- diffusion equation because it is stable when the potential drop between adjacent nodes on a grid spacing h is greater than 2kBTe/e.A corresponding result must be developed for the multi-mode semiconductor equations.
With regard to future applications, it is clear to see that finding the transitions rates for the fluxes in two (and three) spatial dimensions will give similar differential flux equations for two (and three) spatial dimensions.Therefore the multi- mode method will still hold in the most general case of transport.As a final note, we should point out that the multi-mode drift-diffusion/continuity equation formulation is formally equivalent to the differential flux equations, Eq. ( 1).These equa- tions could, alternatively be integrated across the device in order to solve for carrier transport.The numerical advantages of one form over the other are not clear yet.

SUMMARY
In the simplest case, we could discretise the current equations using Eq. ( 2) and finite differences as follows 37(0-e[#(i)] ( The result is a tridiagonal matrix form for the carrier distribution function across the device (i from to N) where the elements of the tridiagonal matrix are now not scalars but MM matrices and the variables are M vectors.This is a large linear To summarise, we have presented a 1-D spatial multi-mode drift-diffusion equation as reformula- tion of the 1-D spatial Boltzmann equation in a discrete 3-D momentum space.Although, the numerical aspects in this paper were not optimised for the best accuracy, the multi-mode drift- diffusion equation was solved in the bulk for the carrier distribution function and all the macro- scopic properties that are incorporated as phenomenological models in conventional drift- diffusion were shown to fall out as a natural consequence of solving the multi-mode drift- diffusion.
The potential of this method lies in its close connection to conventional drift-diffusion-notably the equivalence of a one-mode case to conventional drift-diffusion, easy reduction to macroscopic quantities and similarity in solution techniques.The key issue now is to formulate the problem for solution on a general 2-D spatial grid.Both deterministic and stochastic solution techni- ques for solving the resulting equations should be examined.From the results so far, the multi-mode drift-diffusion formulation of the Boltzmann equation promises to be a powerful approach and may overcome some of the limitations of the scattering matrix approach.
"Alternative approach to the solution of added carrier transport problems in semiconductors", Phys.Rev. B, 123, 51.
FIGURE 3a Macroscopic bulk mobility versus (111) electric field in Si.