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Investigating Hot-Carrier Effects using the Backward Monte Carlo Method

3.2 Spherical Harmonic Expansion

Spherical harmonics are mathematical functions defined on the surface of the unit sphere. The spherical harmonics \( Y^{l,m} \) form an orthogonal basis [6, 26, 38–40]:

(3.31) \{begin}{align} \int \limits _{\theta =0}^\pi \int \limits _{\varphi =0}^{2\pi } Y^{l,m} Y^{l’,m’}\sin {(\theta )} \,\mathrm {d}\varphi \,\mathrm {d}\theta =\delta _{l,l’}\,\delta _{m,m’}, \{end}{align}

where (\( \delta _{a,b} \)) is the Kronecker delta.

A deterministic approach to solve the BTE relies on the expansion of the distribution function \( f(\vec {r},\vec {k},t) \) into spherical harmonics [22, 24, 25, 40, 50, 84, 87]:

(3.32) \{begin}{align} f(\vec {r},\vec {k},t) = \sum _{l=0}^\infty \sum _{m=-l}^l \, f_{l,m}(\vec {r},\Epsilon ,t)Y^{l,m}(\theta ,\varphi ) \{end}{align}

where the wave vector \( \vec {k} \) in the distribution function is transformed into spherical coordinates \( \Epsilon \), \( \theta \) and \( \varphi \) on equi-energy surfaces.

The elliptical valleys are transformed into spherical ones by the Herring-Vogt transformation [23, 86]. The spherical coordinates of the wave-vector \( (k,\theta ,\varphi \)) can be mapped to spherical coordinates of energy \( (\Epsilon , \theta , \varphi \)). This direct one-to-one mapping can be achieved with the non-parabolic band-structure approximation [50, 58, 86], mentioned in Section 2.1.3. Therefore, the seven-dimensional space \( (\vec {r},\vec {k},t) \) of the BTE can be reduced to a five-dimensional space \( (\vec {r},\Epsilon ,t) \). This reduces the computational expenses for the deterministic solution.

Recently, many improvements have been made in the field of the SHE method to solve the BTE. Full-band effects have been considered as well as quantum mechanical effects. Further, the treatment of three-dimensional devices, as well as carrier-carrier scattering, is possible [6, 39, 48, 106].

A drawback of this method is that it relies on the spherical symmetry of the analytical band-structure and therefore is unable to account for the fully anisotropic numerical structure. Considering only some full-band effects, this method is not very accurate in the treatment of high-energy carriers.

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