The structural properties discussed in Chap. 4 show that in the case of spherical energy bands the coupling between the SHE equations is sparse. This allows for a reduction of memory requirements for the system matrix of the discretized equations from to . The proposed matrix compression scheme further reduces the memory requirements for the system matrix to , which finally results in the total memory requirements being dominated by the unknowns already at an expansion order of five. For a three-dimensional device simulation using discrete total energies and grid points (), the number of unknowns for a first-order expansion are up to , where even-order expansion coefficients enter the linear solver. On the same grid, a ninth-order expansion leads to a total of unknowns, of which enter the linear solver. Even though the matrix compression scheme ensures that the total memory requirements stay within a reasonable amount of a few gigabytes, the high computational effort due to the large number of unknowns leads to long simulation times.
With the use of unstructured grids for the SHE method as described in Chap. 5, the number of grid points in the -space can be reduced to lower numbers than for structured grids without sacrificing accuracy in critical device regions. Therefore, the total number of unknowns in the linear system is reduced from to , where and refer to the number of unknowns in -space and can be a factor two to five smaller than , cf. Sec. 5.4.
Similar to unstructured grids, which allow for a high resolution in critical regions and a lower resolution in less important regions, a higher expansion order can be chosen at locations and energies with high influence on a target quantity such as current, carrier density, or average carrier velocity. Consequently, instead of a fixed-order expansion as in (2.41), variable-order expansions of the form
The advantage of the SHE method is that the equilibrium distribution is described exactly by a zeroth-order expansion. This is in contrast to macroscopic transport models derived from moments of the BTE, where higher-order moments do not vanish. Therefore, it is reasonable to expect that for the resolution of small deviations from the equilibrium distribution, only a low expansion order is required for the SHE method in order to obtain good accuracy.
In this section the discretization of the SHE equations using variable-order expansions is presented. For a maximum expansion order in the simulation domain, a variable-order expansion can be obtained by assembling a system matrix for maximum expansion order and then imposing homogeneous Dirichlet conditions on all expansion coefficients with . However, such an approach is impractical due to the unnecessary effort of setting up a much larger system matrix than actually required. In addition, the benefit of reduced memory required for variable expansion orders is eliminated when setting up a system matrix for uniform expansion order first.
It is assumed that the unknowns are enumerated in such a way that the expansion coefficients for a particular box or a dual box are consecutive. In addition, the unknowns associated with boxes are enumerated first, then the unknowns associated with the dual boxes , which leads to the system matrix structure discussed in Sec. 4.3. The resulting structure of the system matrix for a uniform expansion order is depicted in Fig. 6.1. The block sizes for each block coupling two boxes are:
Here, and denote the number of even and odd expansion coefficients up to and including order respectively. For instance, consider the block in the rows for box and in the columns for the dual box , which is located at the interface between and . From the discrete equations for the even-order projections in steady-state (5.9) one obtains the system matrix block for a certain discrete energy directly from rewriting the discrete equations in matrix form:
where the subscript index denotes the -th component of the vector and the expansion order is odd. The matrix block results from the scattering operator and is diagonal up to couplings induced by energy displacements due to inelastic scattering mechanisms, cf. Thm. 3. A similar structure is obtained for the even-to-even, the odd-to-even, and the odd-to-odd block.
When allowing variable-order expansions, the dimensions of the coupling blocks can differ for each box and each dual box . Reconsidering the example of the matrix block for the box and the dual box from above, and assuming an even expansion order for and an odd expansion order for the dual box , the discrete system of equations is given by
This leads to the system matrix structure shown in Fig. 6.2, where the individual small blocks are now of different dimension according to the respective expansion order.
The expansion orders for the boxes and the dual boxes should not be chosen arbitrarily. Consider the case of spherical energy bands, where only a single box carries the maximum even expansion order . If all surrounding dual boxes carry an odd expansion order of at most , then the highest-order expansion coefficients for the box do not couple due to Thm. 2 with expansion coefficients of lower order. Since the right hand side of the linear equation is zero except for Dirichlet boundary values, the expansion coefficients are computed as zero. Moreover, since the size of the final linear system of equations is determined by the expansion orders of the boxes only, Thm. 2 suggests that the odd expansion order of the dual box should be larger than the even expansion orders of the two overlapped boxes and , otherwise the linear system of equations yields a less accurate solution at the same computational effort. Summing up, one obtains:
Conversely, since odd expansion orders larger than required do not lead to higher accuracy because of the same reasoning with the roles of and exchanged, the odd expansion order of a dual box is set to the maximum even expansion order of the overlapped boxes plus one, i.e.
Consider two neighboring boxes and with expansion orders and , . Denote the dual box overlapping and with . From Thm. 2 and Guideline 2 it follows that the expansion coefficients defined for couple with expansion coefficients in up to order . These expansion coefficients in couple according to Thm. 2 with expansion coefficients in up to order , thus there is no gain in accuracy for larger than . This leads to the second guideline:
In the previous section the efficient assembly of SHE equations using variable expansion orders has been discussed. However, the variable expansion orders were assumed to be given for each box and each dual box . For every-day TCAD purposes, such a selection should be based on an adaptive strategy until a prescribed accuracy for a certain target quantity is reached. This section thus discusses strategies to automatically pick suitable expansion orders within the simulation domain. Due to Guideline 1, it suffices to select expansion orders for the the boxes in .
What is considered to be a suitable or good expansion order for a box clearly depends on the target quantity. Macroscopic quantities are typically obtained by moments of the distribution function such as
In the following, three different strategies for the adaptive choice of expansion orders are presented. The first approach is based on an analytical result for the SHE of a function in dependence of the smoothness of . The second approach increases the expansion order particularly in regions with high weight on one or more target quantities. The third approach is a rather classical residual-based technique, which is common in finite element and finite volume methods [1, 8, 56]. Even though the three strategies are presented separately, in practice they should be combined in order to obtain best results.
The SHE can be seen as the three-dimensional extension of Fourier series. Since the latter is more widely known than the former, the motivation for the adaptive strategy for SHE is first given for Fourier series. The transition from Fourier series to SHE does not impose additional complications then.
A -periodic function can be expanded into trigonometric functions as
For SHE, a similar result holds for a function defined on the unit sphere [31, 18]:
Since the spherically symmetric distribution function in equilibrium is described exactly by a zeroth-order expansion, higher-order expansion coefficients provide a measure for the distortion of from equilibrium. This is in contrast to macroscopic models derived from moments of the BTE such as those described in Sec. 1.1, where even moments of the distribution function in equilibrium do not vanish.
Given a numerical solution of the SHE equations for certain expansion orders , an adaptive adjustment of the expansion order can be based on (6.13). Division by and taking the logarithm leads for to
Since due to Guideline 1 the highest available expansion order is associated with dual boxes in , the rate of decay for a box can be based either on the highest expansion order for the box , or on the dual boxes overlapping . In the latter case, one obtains for the error estimator for the box
The exact evaluation of the the term is a maximization problem over all angles and leads to high overall numerical effort, because such a maximization problem needs to be solved at every grid point . A numerically very cheap approximation to the maximum is
The case in (6.16) requires additional treatment since evaluates to zero. Replacing by resolves these problems and partly accounts for the overestimation of by (6.17). For larger values of , the replacement of by does not cause a significant difference anyway. This leads to the final form of the estimator:
The estimator (6.18) is based on the rate of decay of the expansion coefficients only and does not consider the exponential decay of the distribution function at higher energies. Therefore, the estimator treats regions with large values of the distribution function as being equally important as regions with very low values of the distribution function. This is a disadvantage if the target quantity is a macroscopic quantity such as the average carrier velocity, for which only large values of the distribution function lead to significant contributions. In such situations, an additional term may be added to (6.18) in order to penalize low-probability regions of the distribution function. In the special case , this modification is equivalent to dropping the term from (6.18). It should be noted that this modification towards higher weight at regions with high probability is similar in spirit to a target quantity driven control of the expansion order discussed next.
Another approach to the adaptive control of the expansion order is based on one or more dedicated target quantities, for which the expansion orders are chosen such that a prescribed accuracy is obtained. For instance, consider as target quantity the average particle energy
An increase of the expansion order is most appropriate in regions where the weights are high. On the contrary, regions with a weight of several orders of magnitude below the boxes of highest weight can be kept at lowest expansion order, because the contribution to the target quantity is very small anyway.
A comparison of the weight functions for the calculation of the average carrier energy of a Maxwellian distribution and a shifted and heated Maxwellian distribution is depicted in Fig. 6.4. For the Maxwellian distribution it is sufficient to consider only energies up to about eV for the computation of the average energy at an accuracy of about one percent, because the weights at higher energies are below already. However, for the shifted and heated Maxwellian distribution, contributions up to an energy of about eV need to be considered for comparable accuracy.
An expansion order control based solely on the weight function does not account appropriately for long-range influences of regions of the distribution function with low probability on regions with higher probability. This is a particular concern in short devices at high electric fields, where strong variations of the distribution function with respect to energy are observed. Similar to the estimation of the rate of decay in Sec. 6.2.1, a residual-based approach provides a better monitor for the need for higher expansion orders than the target quantity driven approach in the previous section.
Let denote the numerical solution of the SHE equations for a possibly variable expansion order obtained from the solution of the system
It is important to keep in mind that the residual is not invariant with respect to transformations of the linear system. In particular, if the -th equation is multiplied with a certain factor, then also the residual is multiplied with the same factor, even though the solution of the linear system is unchanged. Consequently, it is advisable to first apply normalizations, for instance by multiplication of each equation in the system with the reciprocal of the respective diagonal entry.
Similar to other expansion order control strategies outlined above, an increase of the expansion order can finally be carried out for boxes with highest residuals. An additional expansion order smoothing step after the increase ensures that Guidelines 1 and 2 are obeyed.
Average carrier velocities along a nm -diode with a bias of Volt and intrinsic region between nm and nm are compared for different uniform expansion orders as well as for the three adaptive expansion order schemes after one (maximum SHE order 3), two (maximum SHE order 5) and three (maximum SHE order 7) adaption steps. The total energy range spans eV using an energy spacing of meV. A first-order expansion is kept directly at the band edge, because it has been observed that it improves numerical stability at high expansion orders. At the contacts, a Maxwell distribution is imposed as a Dirichlet boundary condition, thus the expansion is kept at first order there. The resulting velocity curves are depicted in Fig. 6.5 and show rather small differences between the different expansion orders. Nevertheless, the logarithmic plots of the relative errors in Fig. 6.9 provide full insight.
The adaptive strategy based on the decay of expansion coefficients as described in Sec. 6.2.1 is illustrated in Fig. 6.6. In order to emphasize refinement in high-probability regions, the term is added to (6.18) as discussed above. The resulting indiciator values are shifted such that the highest value is zero. The threshold for an expansion order increase is set to . The indicator is not computed at the contact, but evaluated to in Fig. 6.6. During the adaption procedure, the adaptive control stays at a first-order expansion in the left region, where the distribution function is still close to equilibrium. The expansion order is then increased in the intrinsic region and away from the band edge at higher energies after the intrinsic regions. In summary, the adaptive strategy based on the decay of the expansion coefficients increases the expansion order mostly along the high energy tail and close to the band edge inside the intrinsic region. The increase of the expansion order near the right contact stems from the use of Dirichlet boundary conditions and can be considered to be an artifact, because the distribution function is forced from a heated distribution to a Maxwell distribution at the contact, leading to an unphysical boundary layer as discussed by Schroeder et al. .
A different behavior is observed for the target quantity based control shown in Fig. 6.7. The indicator is obtained by taking the logarithm of the contribution of the respective box in the simulation domain and shifting the values such that the highest contribution leads to an indicator value of zero. A threshold value of has been used for increasing the expansion order. Since the distribution function changes its shape only mildly at higher energies, the indicator is essentially unchanged during the adaption, leading to higher expansion orders near the band edge only. Note that the high-energy tail of the distribution function right after the intrinsic region is resolved by the increased expansion order. The error plot in Fig. 6.9(b) shows that virtually the same accuracy as for uniform expansions is obtained. However, the expansion order at later adaption steps abruptly changes from first-order to highest order as quickly as possible without violating Guidelines 1 and 2, thus it is reasonable to expect that a less abrupt change of the expansion order can preserve the accuracy using a lower number of unknowns.
Fig. 6.8 depicts the effect of the residual based expansion order control. The residual has been normalized first by scaling the uniform system such that a unit diagonal is obtained. The indicator is then obtained by taking the logarithm of the residual values and applying a suitable shift such that the highest residual value leads to an indicator value of zero. The expansion order is increased if the indicator is larger than . In principle, one may assign less weight to higher energies such as for the adaptive strategy based on the rate of decay of expansion coefficients. For illustration purposes, the unmodified indicator is shown. It should be noted that the strategy based on the decay leads to qualitatively similar indicator values if no modification at higher energies is applied. The pure residual based strategy increases the expansion order to third-order up to high energies in the right half of the device. However, fifth-order expansions are used in a very small region in the device only. A seventh-order expansion is only assigned right above the band edge inside the intrinsic region. The error plot in Fig. 6.9(c) is very similar to the error plot for the strategy based on the decay of expansion coefficients, hence the same conclusions can be drawn.
Fig. 6.9 further shows that the SHE method converges to a solution as the expansion order increases. As reference, a uniform seventh-order expansion is used. A first-order expansion shows an average error of about over the device with respect to the seventh-order expansion taken as reference. The average error of a third-order expansion is around , and approximately for a fifth-order expansion. A similar exponential decay of the error with the SHE order has also been observed by Jungemann et al.  for the collector current of a bipolar junction transistor, even though at a lower rate.
The number of unknowns obtained with uniform and three adaptive expansions are depicted in Fig. 6.10. Savings of a factor of almost three are obtained with the target quantity based scheme, which yields virtually the same accuracy as uniform expansions. Memory requirements and execution times are consequently reduced by the same factor. Typically, higher savings for the execution times are obtained in practice, because linear solvers with optimal complexity are not available in general. A combination of the three different schemes proposed may give slightly higher savings, which become significant at very high expansion orders only.