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Investigating Hot-Carrier Effects using the Backward Monte Carlo Method

2.3 Scattering

Carriers, characterized by Bloch waves, propagate through the ideal lattice according to the dispersion relation without any perturbation. Many effects such as lattice vibrations, impurities, and high energy particles, create a scattering potential \( U_S(\vec {r},t) \) and therefore cause a perturbation of the carrier state [65]. These perturbations induce an instant change of the particle’s wave-vector \( \vec {k} \) into \( \vec {k’} \), as illustrated in Fig 2.9.

(-tikz- diagram)

Figure 2.9: Scattering of the wave-vector \( \vec {k} \) into \( \vec {k’} \) by the perturbation potential \( U_S(\vec {r},t) \)

The transition probability from \( \vec {k} \) to \( \vec {k’} \) can be calculated using Fermi’s Golden Rule, which is derived from the time-dependent perturbation theory of the first order [102]. The heart of the Golden Rule is the so-called matrix element of the perturbation potential [102]:

(2.24) \{begin}{align} \braket {\vec {k’}|U_S|\vec {k}}=\int _{\Omega } \Psi ^*_{\vec {k’}}(\vec {r}) \, U_S(\vec {r}) \, \Psi _{\vec {k}}(\vec {r}) \, \mathrm {d}^3\vec {r} , \label {eq:matrix} \{end}{align}

where \( \Psi \) represents the wave function of the carrier and \( \Omega \) is the volume of the crystal in which the wave functions are normalized. The transition rate from state \( \vec {k} \) to state \( \vec {k’} \) is expressed by Fermi’s Golden Rule [102]:

(2.25) \{begin}{align} S(\vec {k},\vec {k’}) = \frac {2\pi }{\hbar } \left | \braket {\vec {k’}|U_S|\vec {k}}\right |^2 \, \delta \left (\Epsilon (\vec {k’}) - \Epsilon (\vec {k}) \mp \hbar \omega \right ) , \{end}{align}

where the delta-function expresses conservation of energy.

The following section should illustrate which scattering mechanisms are at play in carrier transport. A more detailed mathematical description can be found in [51, 65, 102].

2.3.1 Impurity Scattering

Carriers in a semiconductor device are usually supplied or removed through doping. In doped regions, the carrier motion is significantly disturbed by scattering due to ionized impurities, which are distributed randomly.

The electrostatic potential due to a point charge in vacuum is coulombic. However, the potential due to an impurity charge in a crystal is more or less screened depending on how many free carriers are present. Scattering due to the screened coulomb potential has been evaluated mainly with the Brooks-Herring approach [51, 102].

2.3.2 Phonon Scattering

Bloch states are the eigenstates of a perfect crystal. Therefore, electrons are not scattered by the purely periodic potential associated with the array of ions constituting the crystal. However, electrons are scattered by lattice vibrations propagating in the crystal because the periodicity of the crystal is disturbed. A small displacement of an ion in the crystal causes a small change in the crystal potential. Hence, the deviation of the crystal potential from pure periodicity may be expressed, theoretically, by the amplitude of the lattice vibrations. However, because of the difficulty of knowing the crystal potential itself, this deviation is expressed in a rather phenomenological way, such as the deformation potential method. Since the lattice vibrations can be quantized as phonons, the influence of lattice vibrations on electron motions is referred to the electron-phonon interaction [102].

This type of interaction is one of the dominant scattering processes in semiconductor devices at room temperature. Carriers in small-sized devices can acquire high energies from the high electric field applied. Therefore, scatterings based on the spontaneous emission of phonons take place even though there are only a few phonons present at low temperature [102].

There exists two types of phonon modes: acoustic and optical. For acoustic mode phonons, neighboring atoms displace in the same direction, and hence the changes in lattice spacing are produced by the strain or differential displacement. For optical phonons, neighboring atoms displace in opposite direction. Hence, the displacement produces the change in lattice spacing directly. Since the acoustic and optical phonon scatterings can be expressed by a deformation potential, which relates lattice vibrations to changes in the band energies, they are referred to as deformation potential scattering [102].

2.3.3 Carrier-Carrier Scattering

There are two types of carrier-carrier scattering processes. One is a binary scattering in which two carriers collide, and the other is a scattering due to the excitation of the collective motion of carriers, also known as plasma scattering. In this work, the first process will be described in Chapter 5.

The main difficulty in the calculation of binary scattering arises from the lack of knowledge of the distribution function, which comes into the calculation in three ways: one is through the screening factor of the interaction potential; the second is the Boltzmann scattering operator which contains a product of the distribution function and is thus non-linear. The third is by the fact that the scattering is restricted by the distribution function via Pauli’s exclusion principle. Because of screening, the collisions between carriers will be less frequent with increasing carrier density [P5, 102].

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