# A review of recent advances in the spherical harmonics expansion method for semiconductor device simulation

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## Abstract

The Boltzmann transport equation is commonly considered to be the best semi-classical description of carrier transport in semiconductors, providing precise information about the distribution of carriers with respect to time (one dimension), location (three dimensions), and momentum (three dimensions). However, numerical solutions for the seven-dimensional carrier distribution functions are very demanding. The most common solution approach is the stochastic Monte Carlo method, because the gigabytes of memory requirements of deterministic direct solution approaches has not been available until recently. As a remedy, the higher accuracy provided by solutions of the Boltzmann transport equation is often exchanged for lower computational expense by using simpler models based on macroscopic quantities such as carrier density and mean carrier velocity. Recent developments for the deterministic spherical harmonics expansion method have reduced the computational cost for solving the Boltzmann transport equation, enabling the computation of carrier distribution functions even for spatially three-dimensional device simulations within minutes to hours. We summarize recent progress for the spherical harmonics expansion method and show that small currents, reasonable execution times, and rare events such as low-frequency noise, which are all hard or even impossible to simulate with the established Monte Carlo method, can be handled in a straight-forward manner. The applicability of the method for important practical applications is demonstrated for noise simulation, small-signal analysis, hot-carrier degradation, and avalanche breakdown.

## Keywords

Spherical harmonics Boltzmann transport equation Semiconductor device simulation Silicon Germanium Hot carrier degradation Box integration## 1 Introduction

Moment-based approaches for semiconductor device simulations are, despite their deficiencies for scaled-down devices, still the most popular methods for technology computer-aided design (TCAD). For example, the drift-diffusion and hydrodynamic models are still used to predict the device characteristics of scaled-down devices, even though their limitations and deficiencies in this regime are well known [1]. These deficiencies could not be addressed through models obtained by taking higher moments either, as closure conditions are hard to formulate and need to rely on empirical arguments [2].

Higher accuracy than that provided by moment-based methods can in principle be obtained by solving the full Boltzmann transport equation (BTE) for the carrier probability distribution function \(f(\varvec{x}, \varvec{p}, t)\), where \(\varvec{x}\) denotes the spatial coordinate, \(\varvec{p}\) momentum, and *t* time. While moment-based models only provide information about averaged quantities such as mean velocity of the particle ensemble centered at a given spatial location \(\varvec{x}\), solutions of the BTE provide full information about the distribution of carriers with respect to their momentum. This allows for a full consideration of many details such as scattering processes and high-energy effects. On the other hand, the additional momentum coordinates require a numerical resolution of the momentum space at each spatial discretization element in one way or another. Therefore, the computational effort for solving the BTE is considerably higher than for moment-based models.

The Monte Carlo method was one of the first methods used to solve the BTE for semiconductors and is still the most popular method used today. It provides several appealing advantages for practical use: First, implementations are relatively easy, hence first results can be obtained quickly. Second, many complicated physical details such as sophisticated bandstructures can be included with the Monte Carlo method. Third, the Monte Carlo method is fairly robust because it does not involve the solution of large systems of nonlinearly coupled equation, where divergence may occur, but instead relies on stochastic sampling. On the other hand, the stochastic nature of the Monte Carlo method is also responsible for major shortcomings. The first is due to the inversely proportional relationship of the accuracy with the square root of the number of particles and thus also processor cycles [3]: If the distribution function needs to be resolved over several orders of magnitude, excessive execution times are required [4]. This is the case if rare events or small currents need to be resolved, or if small-signal analysis is performed. Also, the square-root dependence mandates that scaling Monte Carlo simulations beyond the computational resources power provided by a single workstation or a single cluster yields diminishing returns when considering the additional resources invested. The second shortcoming of the Monte Carlo method is the inherent transient nature: Self-consistent device simulations require time steps on the order of femtoseconds to resolve plasma oscillations, hence simulations of time intervals in the millisecond regime or beyond become practically infeasible [5].

To overcome the limited accuracy of moment-based methods on the one hand, but to avoid excessive execution times of the Monte Carlo method on the other hand, sophisticated deterministic methods for solving the BTE were developed. A full discretization of the \((\varvec{x}, \varvec{p})\)-space for solving the transient BTE using a weighted essentially non-oscillatory (WENO) scheme for stabilization was proposed by Carillo et al. for one-dimensional simulations [6] and later extended to two-dimensional device simulations (cf. [7] and references therein). Spherical coordinates were used in momentum space in order to better resolve the spherical symmetry of the analytical band structures employed [8, 9]. A related method was proposed by Galler et al. [10], where the multigroup equations are solved using a finite element-like decomposition of the full momentum space into tiny cells.

The most mature deterministic method so far and the focus of this review is the spherical harmonics expansion (SHE) method. It has been successfully employed for a much wider range of device quantities than any of the other direct solution approaches. The SHE method mathematically exploits the fact that the distribution of carrier momentum in equilibrium shows a spherical symmetry. As a consequence, the equilibrium distribution function can be represented exactly with a zeroth-order expansion. This is in contrast to moment-based methods, where higher-order moments do not vanish in equilibrium. Since expansions in spherical harmonics can be seen as an extension of Fourier series from the circle (i.e., one angular component \(\varphi \)) to the sphere (i.e., two angular components \(\theta \), \(\varphi \)), a rich mathematical foundation is available [11, 12, 13].

First numerical applications of the SHE method for solving the BTE for homogeneous semiconductors can already be found in the 1960s, for example, in the work of Baraff [14]. However, it took until the early 1990s until the SHE method was first used for device simulation [15, 16, 17]. In these early works, the SHE method was derived from a perturbation of the equilibrium state, resulting in a first-order SHE method for which promising agreement with Monte Carlo results was obtained. Subsequently, several authors refined the method: Lin et al. derived a Scharfetter-Gummel-type stabilization for the first-order SHE method [18], and later coupled it with the Poisson equation and a hole continuity equation [19]. Hennacy et al. extended the SHE method to arbitrary order [20, 21]. Schroeder et al. proposed physically sound boundary conditions [22] to address sharp boundary layers when forcing the distribution function to equilibrium at the contacts. Vecchi et al. introduced a methodology to include full-band effects [23, 24]. The same group proposed an efficient solution scheme with a multigrid-like refinement near the conduction band edge and observed a decoupling of the system of linear equations after discretization when using certain scattering processes [25]. Singh identified a regularizing behavior of inelastic scattering mechanisms over elastic ones and used this to construct a positivity-preserving discretization [26]. Rahmat et al. were the first to observe a decoupling of spatial and angular terms and presented results for a third-order discretization using an upwind scheme [27]. The Bologna group proposed a methodology for handling impact ionization [28], hot electron injections [29], and electron-electron scattering [30, 31] for first-order SHE. In order to consider certain quantum mechanical effects, Goldsman et al. applied a first-order SHE to a modified BTE taking local contributions from the Wigner equation into account [32]. At about the same time, Ben Abdallah [33, 34] and Ringhofer [35, 36, 37, 38] provided important foundations for a better mathematical understanding of the BTE in general and the SHE method in particular. This understanding was supplemented by the results of Hansen et al., who used the SHE method for the modeling of plasma physics [39].

In the following, we provide a unified presentation of recent improvements of the SHE method. We focus on contributions since the work of Jungemann et al. in 2006 [40], who introduced a sound mathematical stabilization and demonstrated the need for higher-order expansions for nanoscale devices. Section 2 introduces the SHE method at the continuous level with some details on the choice of boundary conditions. The inclusion of additional physical processes and application scenarios of the SHE method are presented in Sect. 3. Section 4 discusses the current state-of-the-art discretization and summarizes various techniques developed for further reducing overall simulation times. Rather than providing a separate results section, we present results directly at the respective point of discussion to preserve a coherent flow of discussion. Finally, we draw a conclusion and discuss possible future research directions worthwhile to pursue.

## 2 The SHE method

*Q*refers to the scattering operator. The force term may also depend on the magnetic field [41], which will be neglected in this work for the sake of conciseness. In principle, a BTE needs to be solved for each valley and each carrier type. Interactions between the different valleys and carrier types occur through intervalley scattering and generation-recombination processes. For better readability, the subsequent discussion assumes a single valley for a single carrier type unless noted otherwise and arguments are suppressed whenever appropriate.

### 2.1 Spherical harmonics expansion

*u*in energy space reads

*u*on the unit sphere with mild regularity requirements, the expansion coefficients \(u_{l,m}\) are obtained from a projection onto the respective spherical harmonic [11]:

*u*over the whole Brillouin zone \(\mathcal {B}\) for a kinetic energy \(\varepsilon \) as

*Z*is obtained from the Jacobian of the coordinate transformation as

*Z*will lead to unrelated expansion coefficients obtained from (4) and (6) in general.

*f*is a-priori unknown and only known to fulfill the BTE, it is not enough to only compute projections of the form (4) or (6). Instead, a system of equations for the unknown expansion coefficients \(f_{l,m}\) needs to be derived from the BTE. Such a system is obtained by projecting (2) onto the spherical harmonics \(Y^{l,m}\). For details of the derivation, we refer to the literature [40] and directly state the resulting set of equations:

*g*as

*f*of the BTE.

### 2.2 Boundary conditions

The system of equations (18) needs to be supplemented with suitable boundary conditions in order to fully specify the system. Homogeneous Neumann boundary conditions are imposed at spatial non-contact boundaries. Similarly, homogeneous Neumann boundary conditions are applied at the lower energy boundary at \(\varepsilon = 0\) and for the upper energy boundary at \(\varepsilon = \varepsilon _{\max }\) for some user-defined value of \(\varepsilon _{\max }\). Scattering processes with initial or final energy outside the considered energy range, including scatter events to or from the band gap, are invalid and hence ignored.

*T*denotes temperature, and

*M*is a suitable normalization factor in order to obtain the correct contact carrier density. This is in some sense similar to contact models often used for moment-based models, where a known value of the carrier density is prescribed as a Dirichlet boundary condition. At closer inspection, however, Maxwell-Boltzmann distributions as Dirichlet boundary conditions for SHE are problematic: While such a thermal equilibrium assumption is reasonable at the inflow contacts, it leads to steep gradients (so-called boundary layers) at the outflow-contact at higher bias [22]. In other words, a heated carrier distribution is forced to thermal equilibrium at the outflow-contact.

### 2.3 Stabilization and H-transform

*l*with

*H*-transformation [16] was applied in [41] and used in all subsequent publications. The essence of the

*H*-transformation is to apply a change of coordinates from kinetic energy \(\varepsilon \) to total energy \(H = \varepsilon + \varepsilon _{\mathrm {b}} + \mathrm {q}\psi (\varvec{x})\), through which the derivative with respect to energy in (18) vanishes. Overall, one obtains

*H*and the time derivative of the potential. While the term does not contribute for stationary solutions, it is important to consider the term for small-signal analysis as well as noise simulations [45].

*H*-transformation, the carrier trajectories in free flight, which are given by constant total energy

*H*, are well resolved when using a regular grid with respect to the total energy coordinate, cf. Fig. 2. The price to pay for the improved numerical stability is the dependence of the band edge on the potential; simulation regions for the conduction and valence band edges must be recomputed after each change of the potential, resulting in stability issues for transient simulations, cf. Sect. 6.3. MEDS applied to the

*H*-transformed equations results in the multiplication of equations for odd

*l*by a constant, hence this constant can also be omitted without changing the solution of the system. As discussed by Hong et al. favorable numerical properties are obtained when using the adjoint equations for the discretization instead [41, 46]. The adjoint equations can also be obtained through direct manipulation and read

## 3 Modeling

In this section, we discuss material-specific properties and physical details to extend the general description of the SHE method in Sect. 2. These details are essential for predictive device simulation and require a careful modeling of the underlying material. While most of the discussion is centered around silicon and germanium, the concepts are likely to be applicable to other materials as well, even though other processes such as polar-optical phonon scattering may play a much more important role.

### 3.1 Band structure

*k*is one-to-one, hence the term \((\hbar k)^{-1}\) in (10) can be evaluated directly. For common analytical bandstructure models, namely the parabolic bandstructure

*k*.

*k*and the density of states

*Z*. Vecchi et al. found that for a first-order SHE the equations can be recast such that the term \(\varvec{\varGamma }_{l,m}\) as defined in (10) does not contribute and hence an explicit form is not required for

*k*[23]. Thus, even though such a one-to-one mapping is formally required for the derivation of the SHE method, full-band data without an explicit one-to-one mapping such as in Fig. 3 can be used directly. Jin et al. extended this approach to expansions of arbitrary order [53]. They observed that under the assumption of spherically symmetric dispersion relations one can write

*k*in (10). With this, Jin et al. showed that full-band data can directly be used for the group velocity

*v*and the generalized density of states

*Z*. They demonstrated good agreement of the distribution function obtained from one-dimensional device simulations using the SHE method with results from full-band Monte Carlo simulations.

Hong et al. proposed a further refinement of the approach by Jin et al. by postponing the isotropic valley approximations in earlier approaches until the last stage of the model derivation [46]. The proposed method is to use a generalized coordinate transformation to construct a model of the first conduction band for increased accuracy, while higher conduction bands are approximated using the isotropic model. This hybrid approach constitutes a good compromise between higher accuracy and lower computational cost.

### 3.2 Pauli principle

### 3.3 Carrier-carrier scattering

The inclusion of carrier-carrier scattering increases the computational effort considerably. This is due to the non-local coupling of the carrier-carrier scattering operator with respect to energy. In other words, for a fixed spatial coordinate \(\varvec{x}\), carrier-carrier scattering may occur between two carriers with arbitrary initial and final energies. In contrast, carrier-phonon scattering involves a fixed energy transfer only. As a consequence, execution times as well as memory requirements when considering carrier-carrier scattering increase by about one to two orders of magnitude depending on the resolution with respect to energy, cf. Fig. 6.

### 3.4 Generation and recombination

### 3.5 Quantum mechanical corrections

If the SHE method is used for the simulation of scaled-down devices in the deca-nanometer regime, the semi-classical nature of the BTE is not enough to account for quantum mechanical effects. In particular, quantum mechanics requires that the peak carrier concentration in the channel of a MOSFET is located a few nanometers away from the interface to the gate oxide rather than at the interface. A solution of the Boltzmann-Poisson system, however, does not reflect this fact unless special correction schemes are employed.

*f*, and outer normal vector \(\varvec{n}\) are used at semiconductor-insulator interfaces.

Simulation results show good quantitative agreement with solutions of the Schrödinger equation for one-dimensional simulations of metal-oxide-semiconductor structures, cf. Fig. 8. In practice, these corrections come at negligible cost, because the additional numerical effort for computing the quantum mechanical correction potential is tiny compared to the numerical effort required for computing the SHE coefficients.

Subband-splitting is another way of considering quantum mechanical effects. A two- or one-dimensional BTE is solved in transport direction, while the one- or two-dimensional Schrödinger equation is solved in the perpendicular confinement directions, respectively. Instead of the SHE method, Fourier expansions for solving the BTE with a two-dimensional momentum space are sufficient, hence reducing the computational effort [65, 66]. A degenerate BTE with one-dimensional momentum space can even be solved directly without any problems. As a consequence, we will not discuss these approaches in more detail, but refer to the literature for further details [46, 67].

## 4 Numerics

The presentation of the SHE method in Sect. 2 as well as the various models discussed in Sect. 3 was based on a continuous formulation of the equations. In this section, we outline the discretization of the *H*-transformed SHE equations (23), discuss solution procedures for self-consistency with Poisson’s equation, and discuss numerical tweaks to minimize execution times.

### 4.1 Discretization

The finite volume method (also known as box integration method) is an appealing choice for the discretization of the *H*-transformed SHE equations in (23), because it ensures local charge conservation properties similar to moment-based models. In a naive discretization, all expansion coefficients \(g_{l,m}\) are discretized in a conforming manner and the spatial divergence is converted to a surface integral as usual. Such a direct discretization, however, suffers from spurious numerical oscillations and instabilities.

*l*and \(l^\prime \) are of the same parity [37, 40]. Moreover, due to the derivatives with respect to the angles in (17), there holds

*l*even) are thus associated with the discrete control volumes, while odd-order unknowns are associated with the interfaces between control volumes. For a discretization based on kinetic energy, the odd-order unknowns are associated with the corners of the control volume interfaces in order to account for the additional energy derivative [40]. A discretization of the

*H*-transformed equations, however, directly associates the odd-order unknowns with each interface at the same total energy

*H*. The reason is that derivatives with respect to energy are absent and thus a staggered grid with respect to energy is not appropriate [68].

The most commonly used finite volume scheme for semiconductor device simulation is vertex-based. Control volumes (“boxes”) are taken from the dual Voronoi grid of a Delaunay mesh so that each box can be associated with a vertex and vice versa. Densities are then associated with each vertex and fluxes between boxes are associated with the edge connecting the two vertices, cf. Fig. 9. A drawback of this vertex-based scheme is the requirement of Delaunay meshes, which are very challenging to generate [69]. Rupp et al. proposed a cell-centered discretization scheme, where the cells (triangles, tetrahedra, etc.) are taken as boxes and hence the method is suitable for arbitrary meshes [70]. However, to account for the wide-spread use of vertex-based discretizations, we will consider a vertex-based discretization of the SHE equations in the following.

*i*, and \(B_{i,j}\) the box associated with the dual box obtained from combining the contributions of the boxes \(B_i\) and \(B_j\) associated with the edge joining the vertices

*i*and

*j*, cf. Fig. 10. This results in a conforming decomposition of the simulation domain for structured and unstructured grids, i.e., both \(\cup _i B_i\) and \(\cup _{i,j; i < j} B_{i,j}\) exactly cover the simulation domain if the underlying mesh is sufficiently regular. For a discretization of both the even-order and odd-order equations, the velocity \(\varvec{v}\) and the density of states

*Z*(and thus also \(\hat{\varvec{j}}_{l,m}^{l^\prime , m^\prime }\) and \(\varvec{\varGamma }_{l,m}^{l^\prime , m^\prime }\)) are assumed to be piecewise constant with respect to the spatial coordinate in each box \(B_{i,j}\). Similarly, the unknown expansion coefficients \(g_{l^\prime ,m^\prime }\) are assumed to be constant over their associated boxes \(B_i\) and \(B_{i,j}\) and energy interval \([H^-, H^+]\), respectively. Furthermore, we only need to focus our attention on the discretization of the free streaming operator, because the scattering operator does not contain any spatial derivatives and is thus not affected by the

*H*-transformation [40, 41]. Integration of the free streaming operator in the even-order equations (23) over the box \(B_i\) and the energy range \([H^-, H^+]\) leads to

The discrete forms (38) and (40) can be used for structured as well as unstructured meshes on which a Voronoi-based finite volume scheme is possible. On structured grids, they exactly result in the discrete equations derived by Hong et al. [41] using a dimensional splitting.

### 4.2 The role of spherical symmetry

*l*and

*m*, both coupling terms can take nonzero values only if \(l^\prime = l \pm 1\) and \(m^\prime = \pm \vert m \vert \pm 1\). This greatly simplifies the coupling structure for isotropic compared to anisotropic band structures.

*s*in (29) after a transformation to \((\varepsilon , \theta , \varphi )\) coordinates in general is

*l*,

*m*) and \((l^\prime , m^\prime )\) is solely determined by the free streaming operator.

### 4.3 Self-consistency

*n*, hole density

*p*, and net doping \(\mathcal {C}\) together with the BTE provides accurate values for quantities such as the current density.

*Gummel method*[74]. It relies on an iterated solution of the Poisson equation and the drift-diffusion equation or the BTE for each carrier type. For unipolar devices, the second carrier type may also be ignored. The distribution function obtained from a solution of the BTE is translated into a carrier density via the relation

The second method for achieving self-consistency is Newton’s method, through which quadratic convergence close to the solution is obtained. Newton’s method requires the solution of a system described by the full Jacobi matrix of the coupled equations in each step. While partial derivatives of the densities with respect to the SHE coefficients are easily obtained from (45), additional care needs to be taken when computing the partial derivatives of the terms in the BTE with respect to the potential \(\psi \). In a formulation based on kinetic energy, only the terms involving the force \(\varvec{F}\) lead to additional contributions to the Jacobi matrix. On the other hand, a formulation based on total energy *H* needs to account for the dependence of the total energy *H* on the potential. In particular, \(\varvec{j}_{l,m}^{l^\prime , m^\prime }\) and \(\varvec{\varGamma }_{l,m}^{l^\prime , m^\prime }\) depend on \(\psi \) through the total energy *H*.

Since Newton’s method may fail with a poor initial guess, the initial nonlinear iterations are often carried out using Gummel’s method. When the current iterate is closer to the actual solution, Newton’s method is then used, ultimately resulting in quadratic convergence. In certain scenarios, the SHE method may also exhibit higher numerical stability than moment-based methods: Jungemann et al. reported superior numerical stability of the SHE method during their study of impact ionization effects [75].

### 4.4 Adaptive variable-order scheme

From a computational standpoint, it is desirable to use a small maximum expansion order \(l_{\max }\) to minimize the numerical complexity. On the other hand, first-order expansions provide, despite their appealing properties discussed in Sect. 2, insufficient accuracy for scaled-down devices under quasi-ballistic transport conditions. Certain regions of a device, for example deep in the bulk, do not provide any significant contributions to carrier transport, hence the additional computational effort for high-order expansions may not be necessary. Similarly, a high-order expansion may not be necessary at high energies where the distribution function takes very small values.

Rupp et al. developed a variable-order scheme to select appropriate expansion orders across the device [76]. Their scheme allows for the specification of the maximum expansion order depending on the location in \((\varvec{x}, H)\)-space, i.e., \(l_{\max } = l_{\max }(\varvec{x}, H)\). Alternatively, the scheme may also be interpreted as selecting \(l_{\max }\) fixed throughout the whole simulation domain, but certain expansion coefficients are a-priori set to zero because they are expected (or known) to be insignificant.

Since the manual specification of expansion orders is impractical for engineering purposes, Rupp et al. also proposed adaptive schemes for automatically selecting the expansion order in a bootstrap procedure [76]: Starting from a first-order SHE, the expansion order is increased to third order in regions where the SHE truncation error is large (Fig. 12). Three schemes have been proposed for the detection of these regions: One is based on the relative weights for the computation of a target quantity such as the current density, the second monitors the decay of the expansion orders with respect to *l* for fixed \((\varvec{x}, H)\), and the third is a residual-based scheme similar to those typically used with finite element methods. After the expansion order is locally increased, another solution of the Boltzmann-Poisson system is computed and the adaption procedure repeated until convergence.

An adaptive variable-order scheme is particularly beneficial when used with structured grids in two or three spatial dimensions. The reason is that the tensor construction of structured grids enforces that high resolutions in one part of the device also result in high resolution in other, possibly less important, parts of the device. The adaptive variable-order scheme will then select a low expansion order in these less important parts of the device. Conversely, if the savings in computational cost for unstructured grids instead of structured grids are already high, the additional savings from an adaptive variable-order scheme are smaller [77].

### 4.5 Parallelization

Iterative methods are preferred over direct methods for the solution of large systems of linear equations such as those obtained in each nonlinear iteration step when using the SHE method. At the same time, the use of iterative methods typically requires good preconditioners to accelerate the convergence process. Jungemann et al. reported successful convergence using preconditioners based on incomplete LU factorizations for a formulation based on kinetic energy [40]. Vecchi et al. observed a decoupling of the *H*-transformed SHE equations into several subsystems depending on the inelastic scattering mechanisms employed and on the grid spacing in energy direction [25]. Rupp et al. extended these ideas to a general block preconditioning scheme, where the preconditioner can be built and applied in parallel for each discrete total energy [78]. The approach is based on the observation that scaled-down devices are increasingly dominated by quasi-ballistic transport. Therefore, the action of the full system matrix is captured in good approximation by a system matrix without inelastic scattering events. In the absence of inelastic scattering, the system matrix decouples into independent subsystems for each discrete total energy. Therefore, Rupp et al. proposed to build a parallel block-preconditioner from a system without inelastic scattering events in order to solve the full system including inelastic scattering. Since the number of discrete energies is in the hundreds, enough parallelism is available even for massively parallel architectures such as GPUs. Performance gains of up to an order of magnitude over a single-threaded implementation were reported on a shared memory system, cf. Fig. 13 [78]. These gains partly stem from the smaller computational effort in computing the preconditioner due to the absence of inelastic scattering, and partly from a better utilization of the underlying hardware.

In principle, the block preconditioning scheme can also be used on smaller-sized clusters. For large-scale simulations, preconditioners based on incomplete LU factorizations are known to scale rather poorly. Hence, better parallel preconditioners, particularly multigrid preconditioners, are desirable, but have not been investigated for the SHE method yet.

## 5 Selected applications

### 5.1 Noise

The ongoing interest in further improving the noise performance of semiconductor devices is hampered by the inability of the Monte Carlo method to simulate the noise behavior of devices at technically relevant frequencies in the lower GHz range [75]. Jungemann has demonstrated that the SHE method is well suited for the simulation of noise by solving the Langevin-Boltzmann equation in the frequency domain [79, 80]. The deterministic nature of the SHE method also allows for the accurate simulation of rare events and slow processes, which for example occur in the case of deep traps. For these reasons, Dinh et al. used the SHE method as a reference to benchmark a commercial noise solver based on the drift-diffusion and hydrodynamic models [81].

### 5.2 Small-signal analysis

Small-signal analysis requires that the BTE is linearized together with the Poisson equation at the bias point under inspection, hence fluctuations of the electrostatic potential play a role [46]. Such fluctuations can be considered directly through the force term when using a formulation based on kinetic energy \(\varepsilon \) as in (18), but additional attention is required when using the *H*-transformation. Since the location of the band edge in \((\varvec{x}, H\))-space depends on the electrostatic potential, a naive application of the SHE method for small-signal analysis yields time-varying coefficients. Lin et al. proposed to fix the stationary part of the electrostatic potential and to keep an additional derivative with respect to total energy for the linearization [82]. This resolves the problems with time-varying coefficients, but results in an additional coupling of adjacent discrete energies.

### 5.3 Hot carrier degradation

High electric fields, as they are common in the pinch-off region of a MOSFET, lead to a strong acceleration of carriers. A few carriers may reach energies up to several electron volts, which is sufficient for creating electron–hole pairs or for surpassing the oxide energy barrier. These so-called *hot carriers* are of utmost interest for the study of device degradation phenomena [83, 84].

The need for a high resolution of high-energy tails has long- hampered scientific progress because of excessive execution times obtained with the Monte Carlo method. First results for a long-channel MOSFET were reported only recently [87]. As a remedy, simplified versions of the model are used in practice [83, 88]. With the availability of the SHE method, these simplifications are no longer necessary.

### 5.4 Avalanche breakdown

The abrupt onset and strong nonlinear behavior makes the simulation of avalanche breakdown during the switching of power devices numerically very challenging. Also, the breakdown is not immediate: At typical breakdown voltages of several tens of Volts, the breakdown may need hundreds of picoseconds to fully develop. With a time step restriction of a femtosecond or less, the Monte Carlo method is therefore not suitable for the simulation of avalanche breakdown.

Jabs et al. developed a continuation method to deal with these challenges and presented simulation results using the SHE method for the avalanche breakdown of a 2D vertical power MOSFET and a *pn*-diode at a reverse biases of up to 39 Volt [58, 89]. They introduced a penalty parameter through which they controlled the current and avoided divergence in the numerical solver. Also, to address the ill-conditioning of the full system matrix, they introduced a splitting of the system matrix into a contribution from the BTE without impact ionization (matrix \(\varvec{B}\)) and a contribution from impact ionization (matrix \(\varvec{Q}\)). By using \(\varvec{B}\) as a preconditioner for a Richardson iteration to solve the full system described by the matrix \(\varvec{B} - \varvec{Q}\), they obtained a robust numerical scheme.

## 6 Outlook

In the following, we discuss possible future enhancements and applications of the SHE method. Based on our own experience, we consider an extension of the SHE method to more materials, the possibility to run large-scale simulations, and the solution of the transient BTE using the SHE method to be the most promising topics for future exploration.

### 6.1 More materials

The use of the SHE method for semiconductor device simulations has been focused on silicon and silicon-germanium devices. An exception to this observation is reported by Ramonas and Jungemann, who investigated the electron–phonon interaction in gallium-nitride high-electron-mobility transistors using the SHE method [90, 91]. Kargar et al. reported the use of the SHE method coupled with the Poisson and Schrödinger equations for gallium arsenide [92]. Extensions to other popular materials or material combinations such as silicon-carbide will increase the overall attractiveness and versatility of the SHE method.

### 6.2 Large-scale simulations

The additional energy coordinate implies that the SHE method requires about two to three orders of magnitude more memory than macroscopic models such as the drift-diffusion model. Today’s machines with tens of Gigabytes of main memory provide enough resources to run spatially one- and two-dimensional device simulation using the SHE method. Even fully three-dimensional device simulations are possible, yet only at moderate resolution and without carrier-carrier scattering [77]. Consequently, there is clear benefit of employing the SHE method on distributed memory machines, including supercomputers. The added benefit of such large-scale simulations is that devices can not only be simulated at higher accuracy, but for a given resolution one can also obtain shorter execution times through the use of more cores and memory channels. This is particularly interesting in an engineering environment, where short turnaround times are of importance.

### 6.3 SHE for the transient BTE

The SHE method has so far been employed for the stationary BTE only, yet a solution of the transient case would allow for a study of the long-time behavior in devices at an unprecedented level of detail. While solvers for the transient BTE are readily available in other application areas such as the simulation of rarefied gas flows, the BTE for semiconductors does not allow for a direct application of the techniques in these other areas. The primary reason is that the external force term in the BTE vanishes in other application areas. Therefore, numerical instabilities are less a concern there, as there is no *H*-transformation required and thus no dependence of the simulation domain on an external potential is encountered.

The application of a time discretization to the SHE equations using the *H*-transformation requires the transfer of the current solution at time step *k* to the next time step \(k+1\). Since in general the electrostatic potential changes from time step *k* to time step \(k+1\), an interpolation of the current solution is necessary due to the shift of the band edge. This interpolation, however, results in interpolation errors, which ultimately prevent charge conservation. It is not yet clear whether and how these issues can be addressed. A possible path forward is to relax or even drop the *H*-transformation and work with discretizations based on kinetic energy.

## 7 Summary

The SHE method has reached a level of maturity where it is not only an attractive alternative to the established Monte Carlo method, but at the same time allows for conducting research on phenomena which cannot be simulated with a stochastic method. The absence of stochastic fluctuations enables simulations of noise and exact small- signal analysis at an unprecedented accuracy and for a much larger range than ever before. While the high dimensionality of the BTE implies that the SHE method is still very demanding in terms of memory consumption, the method is considerably less costly when compared to other direct approaches.

A drawback of the SHE method for wide-spread adoption is the fact that the method is fairly complex in terms of the mathematics involved. The development of a SHE solver from scratch easily takes weeks or months of concentrated effort. However, the availability of a free open source simulator (ViennaSHE^{1}) lowers the entry barrier considerably. Also, commercial implementations are available which enable the use of the SHE method without any coding effort at all.

While the SHE method provides more insight than moment-based methods, it is unlikely that the SHE method will ever fully replace moment-based methods. Instead, the SHE method provides another method in a full hierarchy of different solution approaches. For a given application it is thus advisable to select the fastest method which fulfills the requirements on accuracy. If the particular application requires the fast computation of carrier distribution functions in one way or another, the SHE method is likely to be the best choice.

## Footnotes

## Notes

### Acknowledgments

Open access funding provided by TU Wien (TUW). K. R., M. B., T. G., and A. J. acknowledge partial support from the Austrian Science Fund (FWF), Grants P23598, P24304, and W1245. S.-M. H. was supported by the EDISON Program (Grant No. 2012M3C1A6035304) through the National Research Foundation of Korea funded by the Ministry of Science, ICT, & Future Planning.

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