3.3.3 Non-MAXWELLian Distributions

The FERMI-DIRAC or MAXWELL-BOLTZMANN distribution functions are frequently used to describe the distribution of carriers in equilibrium since they are the solution of BOLTZMANN's transport equation for the case of vanishing applied electric field. In the channel region of a MOSFET, however, the energy distribution deviates from the ideal shape implied by expressions (3.15) or (3.20). Carriers gain energy by the electric field in the channel, and they experience scattering events. Models to describe the distribution function of such hot carriers have been studied by numerous authors [101,102,103]. One possibility to describe the distribution of hot carriers is to use a heated MAXWELLian distribution function

$$f({\mathcal{E}})=A\exp\left(-\frac{{\mathcal{E}}}{{\mathrm{k_B}}T_{n}}\right) \ ,$$ (3.24)

where $T_{n}$ denotes the electron temperature and $A$ is a normalization constant. The validity of this approach, however, is limited. Fig. 3.4 shows in the left part the contour lines of the heated MAXWELLian distribution function at the Si-SiO$_2$ interface in comparison to results^3.4 for a MOSFET with a gate length of $\ensuremath {L_\mathrm{g}}=180\,$nm and a thickness of the gate dielectric of 1.8nm at a bias of $\ensuremath{V_\mathrm{DS}}= \ensuremath{V_\mathrm{GS}}= 1\,V$. It is evident that the heated MAXWELLian distribution (full lines) yields only poor agreement with the results (dashed lines). The distribution function at two points near the middle of the channel (point A) and near the drain contact (point B) are shown in the right part of this figure. Particularly the high-energy tail in the middle of the channel is heavily overestimated by the heated MAXWELLian model. This is unsatisfactory since a correct description of the high energy tail is crucial for the evaluation of hot-carrier injection at the drain side used for programming and erasing of EEPROM devices.

Figure 3.4: Comparison of the heated MAXWELLian distribution (full lines) with the results from a Monte Carlo simulation (dotted lines) in a turned-on 180 nm MOSFET. Neighboring lines differ by a factor of 10. The distributions at point A and B are compared with a cold MAXWELLian distribution in the right figure.

To obtain a better prediction of hot-carrier effects, CASSI and RICCÓ presented an expression to account for the non-MAXWELL ian shape of the electron energy distribution function [101]

$$f({\mathcal{E}}) = A \exp\left(-\frac{\chi{\mathcal{E}}^3}{E^{1.5}}\right) \ ,$$ (3.25)

with $\chi$ as fitting parameter and $E$ being the local electric field in the channel. This local-field dependence was soon questioned by other authors such as FIEGNA et al. [104] who replaced the electric field with an effective field calculated from the average electron energy to model the EEPROM writing process. HASNAT et al. used a similar form for the distribution function [105]

$$f({\mathcal{E}}) = A \exp\left(-\frac{{\mathcal{E}}^\xi}{\eta({\mathrm{k_B}}T_{n})^\nu}\right) \ .$$ (3.26)

They obtained values of $\xi = 1.3$, $\eta = 0.265$, and $\nu = 0.75$ by fitting simulation results to measured gate currents. However, these values fail to describe the shape of the distribution function along the channel when compared to results [106]. A quite generalized approach for the EED has been proposed by GRASSER et al.

$$f({\mathcal{E}}) = A\exp\left( - \left( \frac{{\mathcal{E}}}{{\mathcal{E}_\mathrm{ref}}} \right) ^ b\right) \ .$$ (3.27)

In this expression the values of ${\mathcal{E}_\mathrm{ref}}$ and $b$ are mapped to the solution variables $T_{n}$ and $\beta_n$ of a six moments transport model [107]. Expression (3.27) has been shown to appropriately reproduce results in the source and the middle region of the channel of a turned-on MOSFET. However, this model is still not able to reproduce the high energy tail of the distribution function near the drain side of the channel because it does not account for the population of cold carriers coming from the drain. This was already visible in the right part Fig. 3.4 near the drain side of the channel: The distribution consists of a cold MAXWELLian, a high-energy tail, and a second cold MAXWELLian at higher energies. Expression (3.27) cannot reproduce the low-energy MAXWELL ian. A distribution function accounting for the cold carrier population near the drain contact was proposed by SONODA et al. [103], and an improved model has been suggested by GRASSER et al. [106]:

$$f({\mathcal{E}}) = A\left(\exp\left( - \left( \frac{{\mathcal{E}}... ... + c \exp\left(-\frac{{\mathcal{E}}}{{\mathrm{k_B}}T_\mathrm{L}}\right)\right).$$ (3.28)

Here the pool of cold carriers in the drain region is correctly modeled by an additional cold MAXWELLian subpopulation. The values of ${\mathcal{E}_\mathrm{ref}}$, $b$, and $c$ are again derived from the solution variables of a six moments transport model [106]. Fig. 3.5 shows again the results from simulations in comparison to the analytical model. A good match between this non-MAXWELLian distribution and the results can be seen.

Figure 3.5: Comparison of the non-MAXWELLian distribution (full lines) with the results from a Monte Carlo simulation (dotted lines) in a turned-on 180 nm MOSFET. Neighboring lines differ by a factor of 10. The distributions at point A and B are compared with a cold MAXWELLian distribution in the right figure.

This model for the distribution function, however, requires to calculate the third even moment of the distribution function, the kurtosis $\beta_n$. As an approximation $\beta_n$ can be calculated by an expression obtained for a bulk semiconductor where a fixed relationship between $\beta_n$, $T_{n}$, and the lattice temperature $T_\mathrm{L}$ exists:

$$\beta_\mathrm{Bulk}(T_{n}) =\frac{T_\mathrm{L}^2}{T_{n}^2} + 2 \f... ...athcal{E}}} \frac{\mu_S}{\mu_n} \left(1 - \frac{T_\mathrm{L}}{T_{n}}\right) \ .$$ (3.29)

In this expression $\tau_{\mathcal{E}}$, $\tau_\beta$, $\mu_n$, and $\mu_S$ are the energy relaxation time, the kurtosis relaxation time, the electron mobility, and the energy flux mobility, respectively. The value of $\tau_\beta \mu_S/ \tau_{\mathcal{E}}\mu_n$ can be approximated by a fit to data [106]. Estimating the kurtosis from (3.29), the distribution (3.27) can be used within the energy-transport or hydrodynamic model. For a parabolic band structure, the expressions

$$T_{n}=\displaystyle \frac{2}{3}\frac{\Gamma\left( \displaystyle \... ...style \frac{3}{2b}\right)}\frac{{\mathcal{E}_\mathrm{ref}}}{{\mathrm{k_B}}} \ ,$$ (3.30)

$$\beta_n =\frac{3}{5}\frac{\Gamma\left(\displaystyle\frac{3}{2b}\r... ...playstyle\frac{7}{2b}\right)}{\Gamma\left( \displaystyle \frac{5}{2b}\right)^2}$$ (3.31)

are found [107], where $\Gamma(x)$ denotes the Gamma function

$$\Gamma(x) = \int_0^\infty \exp(-\alpha) \alpha^{x-1}\,\ensuremath {\mathrm{d}}\alpha \ .$$ (3.32)

While (3.30) can easily be inverted to obtain ${\mathcal{E}_\mathrm{ref}}(T_{n})$, the inversion of (3.31) to find $b(T_{n})$ at $\beta_n(b) = \beta_\mathrm{Bulk}(T_n)$ cannot be given in a closed form. Instead, a fit expression

$$b(T_n) = 1 + b_0 \left( 1 - \frac{T_\mathrm{L}}{T_{n}}\right)^{b_1} + b_2 \left( 1 - \frac{T_\mathrm{L}}{T_{n}}\right)^{b_3}$$ (3.33)

with the parameters $b_0$=38.82, $b_1$=101.11, $b_2$=3.40, and $b_3$=12.93 can be used. Using ${\mathcal{E}_\mathrm{ref}}(T_n)$ and $b(T_n)$ the distribution can be approximated without knowledge of $\beta_n$. Fig. 3.6 shows simulation results for a 500nm MOSFET using the heated MAXWELLian distribution (3.24), the non-MAXWELLian distribution (3.28), and the non-MAXWELLian distribution (3.27) using (3.30) and (3.33) to calculate the values of ${\mathcal{E}_\mathrm{ref}}$ and $b$. It can be seen that the fit to the results from simulations is good. However, the emerging population of cold carriers near the drain end of the channel leads to a significant error in the shape of the distribution at low energy. This is important for certain processes, while in the case of tunneling the high-energy tail is more crucial.

With expression (3.27) for the distribution function and the assumption of a FERMI-DIRAC distribution in the polysilicon gate, the supply function (3.14) becomes

$$N({\mathcal{E}}) = A_1\frac{{\mathcal{E}_\mathrm{ref}}}{b} \,\Gam... ...emath {{\mathcal{E}}_\mathrm{c}}}{{\mathrm{k_B}}T_\mathrm{L}}\right)\right) \ ,$$ (3.34)

where $\Gamma_\mathrm{i}(\alpha, \beta)$ denotes the incomplete gamma function

$$\Gamma_\mathrm{i}(x,y) = \int_y^\infty \exp(-\alpha) \alpha^{x-1}\,\ensuremath {\mathrm{d}}\alpha \ .$$

In (3.34) the explicit value of the FERMI energy was replaced by the shift of the two conduction band edges $\Delta \ensuremath {{\mathcal{E}}_\mathrm{c}}$. Assuming a MAXWELL ian distribution in the polysilicon gate, the supply function can be further simplified to

$$N({\mathcal{E}}) = A_1\frac{{\mathcal{E}_\mathrm{ref}}}{b} \,\Gam... ... \ensuremath {{\mathcal{E}}_\mathrm{c}}}{{\mathrm{k_B}}T_\mathrm{L}}\right) \ .$$ (3.35)

Using the accurate shape of the distribution (3.28), the expressions for the supply function become

$$N({\mathcal{E}}) = A_1\frac{{\mathcal{E}_\mathrm{ref}}}{b} \,\Gam... ...nsuremath {{\mathcal{E}}_\mathrm{c}}}{{\mathrm{k_B}}T_\mathrm{L}}\right)\right)$$

for a FERMI-DIRAC distribution, and

$$N({\mathcal{E}}) = A_1\frac{{\mathcal{E}_\mathrm{ref}}}{b} \,\Gam... ...elta \ensuremath {{\mathcal{E}}_\mathrm{c}}}{{\mathrm{k_B}}T_\mathrm{L}}\right)$$ (3.36)

assuming a MAXWELLian distribution in the polysilicon gate.

Figure 3.6: The heated MAXWELLian distribution (3.24), the non-MAXWELLian distribution (3.28), and the non-MAXWELLian distribution (3.27) with (3.30) and (3.33) compared to Monte Carlo results (from top to bottom).

A. Gehring: Simulation of Tunneling in Semiconductor Devices