Physical Modeling

The SHE equations (2.34) and (2.35) incorporate material-specific properties by the velocity term , the modulus of the wave vector as a function of energy, the generalized density of states , and the scattering operator . However, the velocity term and the density of states are not independent and depend on the dispersion relation by

The second half of this chapter is devoted to the various scattering effects. Since scattering balances the energy gain of carriers due to the electric field, accurate expressions for the scattering operators are mandatory. Scattering mechanisms leading to a linear operator are discussed in Sec. 3.3, while the case of nonlinear scattering operators is investigated in Sec. 3.4.

Due to the nonuniformity of the crystal lattice in silicon with respect to a change in direction, the dispersion relation linking the particle momentum with the particle energy cannot be accurately described by the idealized setting of an infinitely deep quantum well, for which the solution of Schrödinger’s equation yields a parabolic dependence of the energy on the wave vector:

A more accurate approximation to the band structure in silicon can be obtained by a slight modification of the form

The deficiencies of the nonparabolic dispersion relation can be mitigated by a combination of four energy bands as proposed by Brunetti et. al. [11]:

The BTE has in principle to be solved in each energy band with index for a distribution function . However, since the individual distribution functions for each energy band are not of particular interest, it is preferred to have a single dispersion relation describing the total distribution function .

In order to account for the angular dependence of the energy on the wave vector in a semiconductor, the inverse dispersion relation is expanded as

A SHE of the valence band has been carried out by Kosina et al. [60] up to eV. Pham et al. [73] proposed a refined method for an expansion also including higher energies. A fitted band structure based on SHE for the conduction band has been presented by Matz et al. [69]. Due to the bijective mapping between energy and wave vector, the velocity and the density of states are not in perfect agreement with full-band data. Nevertheless, good results compared to the Monte Carlo method are obtained [45].

A comparison of the presented energy band models is given in Fig. 3.2. Slight deviations of the many-band model and the full-band density of states are due to different Monte Carlo data used in [11]. While the many-band model provides a good fit for the density of states, it fails to approximate the carrier velocity and is worse than the nonparabolic model (3.3). As expected, the fitted band model provides the best accuracy, even though approximations are less accurate at energies above eV.

Vecchi et al. [107] used full-band Monte Carlo data for the velocity and the generalized density of states for a first-order SHE. This is possible because in this case all terms with an explicit representation of the band structure, i.e. , vanish. However, higher-order SHE does not allow for a similar procedure, because the angular coupling term (2.33) does not vanish any longer and an explicit expression for the modulus of the wave vector is required.

Recently, Jin et al. [48] suggested a reformulation as follows: Consider

While the band structure links the particle energy with the particle momentum, it does not fully describe the propagation of carriers. In the presence of an electrostatic force, carriers would be accelerated and thus gain energy indefinitely unless scattering with the crystal lattice or with other carriers is included in the model. The most important scattering mechanisms are discussed in the following. The scattering operator is assumed to be given in the form

Atoms in the crystal lattice vibrate around their fixed equilibrium locations at nonzero temperature. These vibrations are quantized by phonons with energy . Acoustic vibrations refer to a coherent movement of the lattice atoms out of their equilibrium positions. Depending on the displacements with respect to the direction of propagation of the lattice wave, transversal (TA) and longitudinal (LA) acoustic modes are distinguished.

Since the change in particle energy due to acoustic phonon scattering is very small, the process is typically modelled as an elastic process [47], which does not couple different energy levels. The scattering rate can thus be written as

| (3.22) |

where the coefficient is given by

Optical phonon scattering refers to an out-of-phase movement of lattice atoms. In ionic crystals, these vibrations can be excited by infrared radition, which explains the name. Similar to acoustic phonon scattering, transversal (TO) and longitudinal (LO) modes are distinguised.

Since the involved phonon energies are rather high, cf. Tab. 3.3, optical phonon scattering is typically modeled as an inelastic process leading to a change of the particle energy. With the phonon occupation number given by the Bose-Einstein statistics

| (3.25) |

where is symmetric in and and given by

Si | Ge | ||||

Mode | |||||

TA | eV/cm | meV | eV/cm | meV | |

LA | eV/cm | meV | eV/cm | meV | |

LO | eV/cm | meV | eV/cm | meV | |

TA | eV/cm | meV | eV/cm | meV | |

LA | eV/cm | meV | eV/cm | meV | |

TO | eV/cm | meV | eV/cm | meV | |

Dopants in a semiconductor are fixed charges inside the crystal lattice. Since carriers are charged particles as well, their trajectories are influenced by these fixed charges, leading to a change of their momentum. The model by Brooks and Herring [10, 47] suggests an elastic scattering process with scattering coefficient

= ∫
_{B}s_{imp}(x,k,k^{′})(1 − cos(θ))dk^{3} |

= ∫
_{0}^{∞}∫
_{0}^{π} sinθdθZdε |

∫
_{0}^{π} sinθdθ = . |

The scattering operator in low-density approximation for single carrier processes is linear, which is very attractive from a computational point of view. Consequently, scattering processes leading to a nonlinear scattering operator are often neglected in order to avoid nonlinear iteration schemes. In the following, two such types of scattering mechanisms are considered.

A high population of the conduction band can lead to the case that the distribution function takes large values near the band edge, thus the term cannot be approximated with any longer:

| (3.31) |

Repeating the steps from Sec. 2.2, one finally obtains for the projected in-scattering operator assuming velocity randomization and a scattering rate independent of the angles

Numerical results in [43] confirm that the low-density approximation does not have a high impact on macroscopic quantities such as the electron density or carrier velocities, but notable differences in the distribution function are obtained near the band edge. The carrier population is then shifted towards higher energies, because all states at lower energies are already populated.

The linear scattering operators in Sec. 3.3 stem from the scattering of carriers with noncarriers. Very important for particularly the high energy tail of the distribution function is carrier-carrier scattering [79]. A carrier-carrier scattering mechanism requires that the two source states are occupied, and the two final states after scattering are empty. This leads to a scattering operator of the form

| (3.34) |

where the scattering coefficient now depends on the spatial location and on two pairs of initial and final states. With a low-density approximation, the nonlinearity of degree four of the carrier-carrier scattering operator reduces to second order:

| (3.35) |

The scattering coefficient can be derived to be of the form [96, 99]

Carrier-carrier scattering, particularly electron-electron scattering, has so far been discussed for first-order SHE only [108, 110]. In a joint work with Peter Willibald Lagger [62], the author has recently extended the method to arbitrary-order SHE, and the derivation is given in the following.

The scattering operator (3.35) is again split into an in-scattering term and an out-scattering term as in Sec. 2.2. Inserting (3.36) into (3.35), one integration can be carried out due to the momentum conservation. An integration with respect to yields

| (3.39) |

where and similarly for and . A projection onto the spherical harmonic yields

| (3.40) |

Up to now, no approximations have been applied. However, a direct evaluation of these nested integrals at each node in the simulation domain is certainly prohibitive for the use within a simulator due to excessive execution times. Consequently, the further derivation is based on the following two assumptions:

- can be taken as an external parameter. In principle, the energy of the second particle before scattering depends due to energy conservation on all variables, i.e. . Because the asymptotically exponential distribution of carriers with respect to energy, it is plausible, yet heuristic, that the second particle has an energy close to the band gap. Since no additional information about the second particle involved is known, the average energy is taken for , which is accessible due to the nonlinear iteration scheme required for the solution of the discrete set of equations.
- The scattering coefficient can be expanded into spherical harmonics with
respect to and . This can for example be achieved by using the isotropic
approximation (3.30) in order to obtain an isotropic scattering rate with equal
macroscopic relaxation time. As an alternative, one may directly compute a spherical
projection in order to obtain the expansion coefficients of the
expansion

With these assumptions, one can split the integrands to

| (3.42) |

An expansion of into spherical harmonics, the use of the delta distribution for an elimination of the integral over , and the assumption of spherical energy bands leads to

| (3.43) |

Summing up, the projected in-scattering operator is given by

| (3.44) |

For the more general case of an expansion of the scattering coefficient (3.41), one obtains again with the assumption of spherical energy bands

| (3.45) |

The projection of the out-scattering operator starts with the same steps as for the in-scattering operator in order to arrive at

| (3.46) |

The angles and depend in a complicated way on the other angles, therefore rather crude simplifications are applied. Scattering with another carrier is to a first approximation determined by the density of carriers at the particular location inside the device. Consequently, the distribution function of the second particle, which is described by and , is approximated by the isotropic part of the distribution function only, since it fully describes the carrier density. With and considering again as the average energy, (3.46) simplifies to

| (3.47) |

The more general case of an expansion of the scattering coefficient (3.41) results in

| (3.48) |

For the full projected scattering operator, one thus obtains

| (3.49) |

With the use of an expansion of the scattering coefficient (3.41), one obtains

| (3.50) |

One can immediately see that the full scatter operator vanishes for the equilibrium case, where is given by a Maxwell distribution. Therefore, even though simplifications have been used for the separate derivation of the projected equations for the in- and the out-scattering operators, the resulting expressions are consistent.