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5.2 Validity of the Boltzmann Transport Equation

As device dimensions in modern CMOS transistors are in the order of only several ten nanometers, the question of the validity of the Boltzmann transport equation,

$\displaystyle \frac{\partial f_n({\ensuremath{\mathitbf{r}}},{\ensuremath{\math...
...mathitbf{k}}},t)=\biggl(\frac{\partial f_n}{\partial t}\biggr)_\mathrm{coll}\ ,$ (5.5)

as a fundamental description of carrier transport arises. This equation was originally derived for dilute gases. In the following some of the approximations of the Boltzmann transport equation and their implications are addressed.

The solution of the Boltzmann transport equation with an external force $ {\ensuremath{\mathitbf{F}}}({\ensuremath{\mathitbf{r}}})$ provides the distribution function $ f_n({\ensuremath{\mathitbf{r}}},{\ensuremath{\mathitbf{k}}},t)$ from which macroscopic quantities can be derived. The right-hand side of (5.5) describes the changes to the distribution function induced by scattering. The particle's group velocity is determined from the semiconductor band structure $ E_n({\ensuremath{\mathitbf{k}}})$ as $ {\ensuremath{\mathitbf{v}}}_n({\ensuremath{\mathitbf{k}}}) = \hbar^{-1} \nabla_{\ensuremath{\mathitbf{k}}}
E_n({\ensuremath{\mathitbf{k}}})$. In the parabolic band approximation, $ \hbar^{-1}
\nabla_{\ensuremath{\mathitbf{k}}} E_n({\ensuremath{\mathitbf{k}}}) = \hbar {\ensuremath{\mathitbf{k}}} /\hat{m}^{\ast}$, and the particle's group velocity can be calculated from the effective mass tensor $ \hat{m}^{\ast}$.

The distribution function $ f_n({\ensuremath{\mathitbf{r}}},{\ensuremath{\mathitbf{k}}},t)\mathrm{d}^3k\mathrm{d}^3r$ defines the probability density to find a particle in $ \mathrm{d}^3k\mathrm{d}^3r$ at a given time $ t$. Obviously, such a statistical description can only be appropriate when the number of carriers is large. Extremely down-scaled devices may contain too few carriers to justify this kind of statistical treatment.

Since carriers interact through their electric fields, the distribution function $ f_n({\ensuremath{\mathitbf{r}}},{\ensuremath{\mathitbf{k}}},t)$ at a particular point in the six dimensional position-momentum (phase) space at a given time can only be determined from the knowledge of $ f_n$ in all other points. This would involve a treatment using an $ N-$particle system and an $ N-$particle distribution function. However, if the carrier-carrier correlations are weak, the $ N-$particle distribution function can be contracted to a one-particle distribution function [Harris04]. Alternatively, the influence of other carriers can be treated through the self-consistent electric field [Venturi89] and schemes where the Pauli exclusion principle is included [Bosi76,Lugli85,Yamakawa96,Ungersboeck06b].

A main assumption of the Boltzmann transport equation is that particles can be treated semiclassically, obeying Newton's law. Quantum mechanics enters the equation only through the band structure and the description of the collision term. Since both the position and the momentum of a particle are arguments of the distribution function, apparently the quantum mechanical uncertainty principle $ \Delta p \Delta r\geq \hbar/2$ is violated. Assuming a spread in particle energy of $ \ensuremath {{\mathrm{k_B}}}{} T$, one finds that the spread in position is

$\displaystyle \Delta r \geq \hbar / (2 \sqrt{2 m^\ast \ensuremath {{\mathrm{k_B}}}T}) = \lambda_\mathrm{B}/2\ .$ (5.6)

Here, $ \lambda_\mathrm{B}$ denotes the particle's thermal average wavelength. Thus, one should not attempt to localize the particle's position exactly with respect to its thermal average wavelength. If the potential varies sharply on the scale of $ \lambda_\mathrm{B}$, which is typically in the order of 10 nm to 20 nm at room temperature, condition (5.6) is not satisfied, and instead of the Boltzmann equation a wave equation must be solved to study the propagation of a carrier wave through the device.

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E. Ungersboeck: Advanced Modelling Aspects of Modern Strained CMOS Technology