2.2  The Boltzmann Transport Equation

The Boltzmann Transport equation (BTE) has been originally developed by Ludwig Boltzmann to statistically describe transport of atoms and molecules (particles) of an idealized diluted gas. The BTE describes the transport by two processes, namely free streaming and scattering. In order to use the BTE to describe transport of an electron gas in a semiconductor, the BTE for diluted gases needs to be modified. First the free streaming of the charge carriers in the lattice is described by the equations of motion (Newton’s law)

(2.13)

where the relation p = ℏk and the band structure is used. Second, to describe the collisions between electrons in the electron gas, we will use quantum mechanical perturbation theory (Fermi’s Golden Rule). It has to be noted that in the classical framework of the BTE, where Heisenbergs uncertainty principle is being neglected, one could directly track position and momentum of each electron. However, this tracking of each single particle in a classical approach is currently infeasible.

Figure 2.4:

Illustration of the movement of an electron through a 2D semiconductor. The BTE treats the free streaming (blue lines) classically and the scattering events (red crosses) quantum-mechanically.

Instead we will modify the BTE as stated above, by rederiving the BTE using Equation (2.4). Although the derivation of the BTE has been covered in many works before, its derivation will nevertheless be repeated from  [13, 12] for the sake of clarity. For this purpose we need to define a distribution function f(x,k,t) of the carriers in real- and reciprocal-space. The distribution function f(x,k,t) is defined such that

(2.14)

is the number of particles (electrons or holes) in the infinitesimal small volume d³xd³k in the six-dimensional phase-space. The distribution function is a semi-classical concept, since it assumes that both position and momentum of a particle can be measured to arbitrary accuracy at the same time. In order to introduce the semi-classical concept into the Hamiltonians of Equation (2.4), we treat the electrons as classical particles governed by classical mechanics in the time intervals they do not scatter with other particles. Thus the external force exerted on a single electron is

The group velocity vνg(k) can then be calculated  [12] as

Starting from the definition of the density matrix

(2.15)

we define the distribution function as the Fourier transformed density matrix

(2.16)

This definition is compatible with the previous definition in Equation (2.14), since the density matrix describes a statistical ensemble of quantum states in real space. In order to transform the density matrix to the phase space, a Fourier transform needs to be carried out. To asses the time evolution of the density matrix we utilize the Liouville-Von-Neumann equation,

(2.20)

the operators can be linearized by assuming that each electron is sharply distributed in k-space and that the electrostatic potential changes over x are small compared to the changes of the electronic wavefunction over x

- φout(x1,t) + φout(x2,t)	≈∇xφout((x2 + x1)∕2,t)⋅(x2 + x1) = =F(x)∕q⋅x′	(2.21)
Eν(-i∇x1) - Eν(-i∇x2)	⇒ iℏvν g(-i∇x′)⋅∇x	(2.22)

(2.23)

Finally, a Fourier transformation of the equation above is carried out and one obtains after a few algebraic rearrangements the well known homogeneous BTE for the νth band

(2.24)

where {fν(x,k,t)} is the free-streaming operator. In order to incorporate scattering processes a semi-classical perturbation term {fν(x,k,t)} on the right hand side of the BTE is required

(2.25)

{fν(x,k,t)} is obtained by perturbation theory and modeled using a statistical description of each scattering process. In the BTE the operator {fν(x,k,t)} describes the classical free-streaming of the carriers in between scattering events which are described by the scattering operator {fν(x,k,t)} (cf. Figure 2.4).

The force F(x,t) due to a gradient of the electrostatic potential or the band edges in the absence of magnetic fields is given by

(2.26)

where ∥q∥ is the sign-less elementary charge q, φ(x,t) the electrostatic potential and E_C∕V the band edge energy for either electrons or holes. Since the electrostatic potential is needed in order to compute the force exerted on each charge carrier, one needs to solve Poisson’s equation and the BTE for electrons and holes self-consistently. The full system of equations thus reads,

∇⋅ = ∥q∥(n - p + C), ∂tfν n + ={fνn} = n{fν n}- Γn{fν n,fν p }, ∂tfν p + ={fνp} = p{fν n}- Γp{fν n,fν p },

(2.27)

2.2.1  Scattering

In this section a short introduction into modelling the scattering term {fν(x,k,t)} on the right hand side of Equation (2.25) is given. While traveling through the lattice the electrons interact with the lattice or other electrons. Such interactions, where the electrons change their momentum, are termed scattering events. Considering an infinitesimal small volume d³kd³x in phase space (cf. Figure 2.5), the electrons can either scatter into or out of this volume. Thus the scattering operator is normally split into an in-scattering and out-scattering operator as follows:

The transitions in and out of an infinitesimal small volume d³kd³x in phase space do not occur instantly. Nevertheless they are often modeled as an instant transition via a rate of expected scattering charge carriers per second. Additionally it is assumed that the scattering occurs at a certain point in real space and is a local event. For each flavor of scattering there is a rate of transitions S^ν(k,k′,t) from k to k′ in the phase space. An incomplete list of interactions considered in this work can be found in Table 2.1. Note that in a homogeneous semiconductor the scattering rate does not depend on the spatial location of the electron. This simplifies the evaluation of the scattering terms. In order for an electron to scatter, the initial state must be occupied by an electron and the final state must be empty by virtue of Pauli’s exclusion principle. With all of the above the scattering operator is often modelled as

{fν(x,k,t)} = ∑ ν ∫	in{fν(x,k,t)}
	-out{fν(x,k,t)}d3k′,	(2.28)

(2.29)

where EF is the Fermi-level and equals the chemical potential μ known from thermodynamics  [12]. The computational burden to evaluate the integral in Equation (2.28) can strongly depend on how the bandstructure is resolved and becomes easier to evaluate with the parabolic band approximation. Nevertheless, upon solving the BTE the term f^ν(x,k′,t)(1 - f^ν(x,k,t)) in Equation (2.28) might prove to be too challenging for a certain numerical BTE solver. Thus the Pauli exclusion principle is often dropped and the scattering operator reduces to

{fν(x,k,t)} = ∑ ν ∫	Sν(k′,k,t)fν(x,k′,t)
	- Sν(k,k′,t)fν(x,k,t)d3k′	(2.30)

(2.31)

The integral in Equation (2.30) is easier to evaluate than Equation (2.28), especially when using the parabolic band approximation and forms the basis of many important approximations, such as the relaxation time approximation (RTA)  [22]. Dropping Pauli’s exclusion principle is clearly justified if (1 - f^ν(x,k,t)) ≈ 1 and thus f^ν(x,k,t)) ≪ 1 , which is only true for low electron (hole) concentrations and non degenerate semiconductors. Thus dropping Pauli’s exclusion principle in Equation (2.28) is often termed low density approximation.

2.2.2  Recombination and Generation

The scattering operator as described in Section 2.2.1 does not consider electron-hole recombination (cf. Equation (2.25)). In order to be able to consider recombination/generation processes the BTE needs to be split into a BTE for electrons and holes respectively. Since recombination/generation models can be quite complex  [29], we first restrict this section to the most simple two-state defect (cf. Figure 2.6), where only a single function over time, namely the trap occupancy fνT(t) and trap level E_T are needed to describe the state of the defect. Additionally it will be assumed that there are enough electrons and holes that are either spin up or spin down, such that the trap occupancy is independent of the electron spin. In case the spin cannot be neglected, the trap occupancy needs to be split into its spin components, fνT(t) ⇒ f_T,up^ν + f_T,down^ν. With these assumptions the macroscopic rate equation for the trap occupancy reads  [12, 30]

∂tfν T(t)	= ∑ ν ∫ (1 - fν T(t))r21	(2.32)
	+ fν T(t)r12d3k,

Putting all of the above together, the recombination operators Γ^p∕n{fνn,fνp} for electrons and holes, which have been introduced in Equation (2.27), read

Γn{fν n,fν p }	= ∑ ν ∫ Gνn(k)fν T(t)(1 - fν n) - Rνn(k)(1 - fν T(t))fν nd3k,	(2.33)
Γp{fν n,fν p }	= ∑ ν ∫ Gνp(k)(1 - fν T(t))(1 - fν p ) - Rνp(k)fν T(t)fν p d3k,	(2.34)

(2.35)

where σⁿ and σ^p are experimentally determined capture cross sections and vⁿ and v^p reaction velocities. From detailed balance  [31, 12] the generation rates can be determined

(2.36)