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Investigating Hot-Carrier Effects using the Backward Monte Carlo Method

3 Transport Modeling Approaches

The BTE is commonly used to describe the carrier transport in the semi-classical regime. Over the last century, different methods have been developed and improved to describe carrier transport based on the BTE [86]. This chapter covers a short introduction on methods based on the moments of the BTE, a deterministic approach with spherical harmonics expansion, and a stochastic approach with the Monte Carlo method. Furthermore, the advantages and the drawbacks of all these methods are discussed.

3.1 Method of Moments

This method is based on the principle, that every term of the BTE is multiplied with a weight function and subsequently integrated over the Brillouin zone (BZ). This integration leads to a saturation of the coordinates in $\vec {k}$ space and leaves a set of differential equations in $(\vec {r},t)$ space. Thus, some information about the original distribution function is lost. But in many cases the equations in $(\vec {r},t)$ space are sufficient. The weight functions are often chosen as powers of $\vec {k}$ with some scaling factors to achieve physically meaningful quantities. The moments of the distribution function are defined as [2, 8, 30, 98]

$(3.1) \{begin}{align} M_j = \braket {\phi _j} = \int \phi _j\,f\,\mathrm {d}^3 k \„ \{end}{align}$

where $\phi _j$ represents the weight functions, which are scalars for even orders and vectors for odd orders of $\vec {k}$ . This work will only cover the basic principle of the moments method, and therefore only moments up to the third order will be considered. A more detailed description of this method can be found in [30]. The weight functions up to order three read:

$(3.2–3.5) \{begin}{align} \phi _0 &= 1\\ \boldsymbol {\phi }_1 &= \vec {p} = \hbar \vec {k}\\ \phi _2 & = \Epsilon = \frac {\hbar ^2k^2}{2m^*}\\ \boldsymbol {\phi }_3 &= \vec {v}\Epsilon = \frac {\hbar ^3k^2\vec {k}}{2(m^*)^2} \{end}{align}$

where $\vec {p}$ is the momentum, $\vec {k}$ is the wave-vector, $\Epsilon$ is the kinetic energy and $m^*$ is the effective mass.

The weight function $\phi _2$ considers only a band-structure with one isotropic and parabolic valley, as described in (2.17) [30]. Applying the method of moments to the BTE for electrons the moment equation can be written as [30]:

$(3.6–3.7) \{begin}{align} \partial _t \braket {\phi _j} + \nabla _r\cdot \braket {\vec {v}\phi _j} + q\vec {E}\cdot \braket {\nabla _p\,\phi _j} &= \int \phi _j\, \mathcal {Q}\,\mathrm {d}^3 k \hspace {1cm}\text {for even }j,\label {eq:moments_even}\\ \partial _t \braket {\boldsymbol {\phi }_j} + \nabla _r\cdot \braket {\vec {v}\otimes \boldsymbol {\phi }_j} + q\vec {E}\cdot \braket {\nabla _p\otimes \boldsymbol {\phi }_j} &= \int \phi _j\, \mathcal {Q}\,\mathrm {d}^3 k \hspace {1cm}\text {for odd }j,\label {eq:moments_odd} \{end}{align}$

where $\mathcal {Q}$ represents the scattering integral of the BTE, which is the right hand side of (2.26).

Equations (3.6) and (3.7) contain gradients of the weight functions, which can be calculated as [30]:

$(3.8–3.11) \{begin}{align} \nabla _p\,\phi _0 &= 0\\ \nabla _p\otimes \vec {\phi }_1 &= \mathbb {1}\\ \nabla _p\,\phi _2 &=\vec {v}\\ \nabla _p\otimes \vec {\phi }_3 &= \frac {\Epsilon }{m^*} \mathbb {1} + \vec {v}\otimes \vec {v} \{end}{align}$

Here, $\mathbb {1}$ is the unit matrix in three dimensions. The integrate over the scattering integral, on the other hand, can be modeled with the relaxation time approximation [30, 72]:

$(3.12) \{begin}{align} \int {\phi _j}\, \mathcal {Q}\,\mathrm {d}^3 k \approx -\frac {\braket {{\phi _j}} - \braket {{\phi _j}}_0}{\tau _{{\phi _j}}}, \{end}{align}$

where the index $0$ represents an average over the equilibrium distribution function. This approach assumes, that the moment $\braket {{\phi _j}}$ decays exponentially towards its equilibrium value with the time constant $\tau _{{\phi _j}}$ after the field is switched off [65]. Applying these approximations to the equations of moments (3.6) and (3.7), a set of equations can be obtained [30]:

$(3.13–3.16) \{begin}{align} \phi _0: \hspace {1cm}&\partial _t\braket {1} +\nabla \cdot \braket {\vec {v}} &&= 0\label {eq:moment_0}\\ \vec {\phi }_1: \hspace {1cm}& \nabla \cdot \braket {\vec {v}\otimes \vec {p}} + q\vec {E}\braket {\mathbb {1}} &&= -\frac {\braket {\vec {p}}}{\tau _m}\label {eq:moment_1}\\ \phi _2: \hspace {1cm}&\partial _t\braket {\Epsilon } +\nabla \cdot \braket {\vec {v}\Epsilon } + q\vec {E}\braket {\vec {v}} &&= -\frac {\braket {\Epsilon } - \braket {\Epsilon }_0}{\tau _{\Epsilon }}\label {eq:moment_2}\\ \vec {\phi }_3: \hspace {1cm}& \nabla \cdot \braket {\vec {v}\otimes \vec {v}\Epsilon } + q\vec {E}\braket {\frac {\Epsilon }{m^*} \mathbb {1} + \vec {v}\otimes \vec {v}} &&= -\frac {\braket {\vec {v}\Epsilon }}{\tau _S}\label {eq:moment_3}, \{end}{align}$

where $\tau _m$ , $\tau _\Epsilon$ and $\tau _s$ are the relaxation times for momentum, energy and energy flux, respectively. The equations above contain statistical averages of a symmetric tensor of the form $\braket {\vec {v}\otimes \vec {v}}$ . These averages can be evaluated with the diffusion approximation [90] which leads to a diagonal tensors with all dialog elements being equal [30]:

$(3.17–3.18) \{begin}{align} \braket {\vec {v}\otimes \vec {v}} &= \frac {\braket {v^2}}{3} \mathbb {1}\„\\ \braket {\vec {v}\otimes \vec {v}\Epsilon } &= \frac {\braket {\Epsilon \,v^2}}{3} \mathbb {1}\,. \{end}{align}$

The statistical averages in the equations (3.13), (3.14), and (3.15), are commonly expressed by the electron concentration $n$ , the electron temperature $T_n$ and the electrical current density $\vec {J}$ , respectively:

$(3.19–3.22) \{begin}{align} \phi _0: \braket {1} &= n\label {eq:avg_0}\\ \vec {\phi }_1: \braket {\vec {v}} &= \frac {\vec {J}}{n\,q}\label {eq:avg_1}\\ \phi _2: \braket {\Epsilon } &= \frac {3}{2}\,\text {k}_B\,n\,T_n\label {eq:avg_2}\\ \vec {\phi }_3: \braket {\vec {v}\,\Epsilon } &= \vec {S}_n\label {eq:avg_3}, \{end}{align}$

The averages of $\phi _0$ and $\phi _2$ represent densities, whereas $\phi _1$ and $\phi _3$ represent fluxes. With these expressions the equations of moments up to the third order can be written in the final from [30]:

$(3.23–3.26) \{begin}{align} &\phi _0: &\partial _t\,n -\frac {1}{q}\nabla \cdot &\vec {J_n} = 0\label {eq:final_0}\\ &\boldsymbol {\phi }_1:& &\vec {J_n} =\frac {q\,\tau _m}{m^*} \left ( \nabla \left (k_B\,n\,T_n\right ) +q\,\vec {E}\,n\right )\label {eq:final_1} \\ &\phi _2: &\frac {3}{2}\,k_B\partial _t\,(n\,T_n) + \nabla \cdot &\vec {S}_n - \vec {E}\cdot \vec {J_n} = \frac {3}{2} k_B\,n \frac {T_n - T_L}{\tau _\Epsilon }\label {eq:final_2}\\ &\boldsymbol {\phi }_3:& &\vec {S_n} = -\tau _S \left ( \frac {1}{3} \nabla \braket {\phi _4} +\frac {5}{2} \frac {q\,k_B}{m^*}\,\vec {E}\,n\,T_n\right )\label {eq:final_3} \{end}{align}$

One characteristic of the method of moments is that the transport equation of the order $i$ contains the moment of order $i+1$ . This highest moment has to be approximated, which is commonly referred to as the closure of the hierarchy of moment equations [30].

3.1.1 The Drift Diffusion Model

The drift-diffusion equation is obtained from the equations of moments by considering only the two moments (3.23) and (3.24). The term

$(3.27) \{begin}{align} \frac {q\,\tau _m}{m^*} = \mu _n \{end}{align}$

represents the electron mobility. The assumption that the carriers have the same temperature as the lattice $\left (T_n=T_L\right )$ , also known as thermal equilibrium approximation [3, 70], gives the closure relation

$(3.28) \{begin}{align} \braket {\phi _2}=\frac {3}{2}\,\text {k}_B\,n\,T_L, \{end}{align}$

With this relation the drift-diffusion transport model is obtained [30, 33, 82, 91, 93]:

$(3.29–3.30) \{begin}{align} \nabla \cdot \vec {J_n}&=q\,\partial _t\,n\\ \vec {J_n}&=\mu _n\,\text {k}_B\left (\nabla \left (n\,T_L\right )+\frac {q}{\text {k}_B}\,\vec {E}\,n\right ) \{end}{align}$

This model considers local quantities only. Therefore, it neglects non-local transport effects which occur, for example, in a sudden variation of the electric field. In order to deal with non-equilibrium effects, field-dependent mobility models were introduced [30].

More accurate macroscopic transport models include the average carrier energy. The energy transport model and the hydrodynamic model are derived from the first four moments of the BTE [30, 87].

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