previous up next contents Previous: Effective Charge of Up: 3.3.3 Electronic Stopping Process Next: 3.3.4 Thermal Lattice Vibration Empirical Electronic Stopping Model

On the basis of the Lindhard electronic stopping model Hobler et al. proposed an empirical stopping model [36] for the implantation into crystalline silicon where the local electron densities in the target are not explicitly considered for the calculation of the stopping power. Instead an impact parameter of the ion is defined.

The model is composed of a non-local part which dominates in case of large impact parameters (channeling ions), and a local part which describes the close collisions.

$\displaystyle dE = dE_{nl} + dE_{loc}$ (3.126)

The non-local part $ dE_{nl}$ is derived from the simplified velocity proportional stopping power calculated by Lindhard (3.119).

\begin{displaymath}\begin{split}dE_{nl} &= N\cdot k_{corr}\cdot k\cdot \sqrt{E}\...\cdot \exp\left(-\frac{p_{max}}{a}\right)\right] \end{split}\end{displaymath} (3.127)

The coefficient $ k$ considers as well the properties of the ion species as of the target material. A value proposed for $ k$ in [51] is

$\displaystyle k =8\cdot \pi\cdot \hbar\cdot a_0\cdot \sqrt{2}\cdot\frac{Z_1^{\f...
...\left(Z_1^{\frac{2}{3}}+Z_2^{\frac{2}{3}}\right)^{\frac{3}{2}}\cdot \sqrt{M_1}}$ (3.128)

The local part $ dE_{loc}$ is exponentially proportional to the impact parameter as proposed by Oen and Robinson [61]. It is related to a single collision event and therefore a discontinuous contribution to the total energy loss.

$\displaystyle dE_{loc} = x_{loc}\cdot \frac{k_{corr}\cdot k\cdot \sqrt{E}}{2\cdot a^2\cdot \pi}\cdot \exp\left(-\frac{p}{a}\right)$ (3.129)

$ N$ is the target atom density, $ Z_2$ is the charge of the target atoms, $ Z_1$ and $ M_1$ are the charge and the mass of the implanted particle, $ a_0$ is the Bohr radius, and $ E$ the energy of the particle. $ k_{corr}$, $ x_{nl}$, $ x_{loc}$, $ p_{max}$ and $ a$ are empirical parameters.

As indicated by Lindhard the electronic stopping power is only proportional to the ion velocity $ v$ for low ion velocities. A velocity of

$\displaystyle v < Z_1^{2/3}\cdot \frac{e^2}{\hbar} = 2,44\cdot 10^6\cdot Z_1^{2/3} (cm/s)$ (3.133)

can be considered as a limit. For higher energies the stopping power reaches a maximum before it decreases again. Worth mentioning is that experimentally determined stopping power functions do not yield a stopping power which is proportional to the ion velocity for very low ion energies $ E$. Instead an electronic stopping power being proportional to $ E^{0.3}$ up to $ E^{0.45}$ [2], [18] is reported. This fact has to be considered as a source for small errors when applying the electronic stopping power law derived by Lindhard. Fig. 3.8 schematically shows the functional behavior of the electronic stopping power over the whole energy range.

Figure 3.8: Functional behavior of the electronic stopping power as a function of the ion velocity.
\psfrag{Electronic stopping power}[c][c]{\LARGE\sf ...

Table 3.3: The electronic stopping power is only proportional to the velocity up to a maximal energy. These maximal energies are summarized for several ion species.
Ion species Energy limit (MeV)
Boron 2.2
Nitrogen 4.6
Fluorine 8.7
Phosphorus 28.0
Arsenic 194.1
Indium 495.5

As a model for energies around the maximum of the stopping power an interpolation between the velocity proportional stopping power $ S_{ep}$ and a stopping formula derived by Bethe and Bloch $ S_{eb}$ [1], [4], [6], [39], which is valid for high ion energies, is suggested in [75].

$\displaystyle S_e = (S_{ep}^{-c} + S_{eb}^{-c})^c$ (3.134)

According to the Bethe-Bloch theory the electronic stopping power for high ion velocities is determined by ([73])

$\displaystyle S_{eb} = \frac{8\cdot \pi\cdot Z_1^2\cdot e^4}{I_0\cdot \epsilon_b}\cdot \ln\left(\epsilon_b+1+\frac{5}{\epsilon_b} \right)$ (3.135)

$\displaystyle \epsilon_b = \frac{4\cdot E\cdot\frac{m_e}{M_1\cdot m_0}}{Z_2\cdot I_0}.$ (3.136)

$ Z_1$ and $ Z_2$ are the core charges of the ion and of the target, $ e$ is the elementary charge, $ m_e$ is the electron mass, $ m_0$ the atomic mass unit, $ M_0$ the relative atom mass of the ion, and $ I_0$ the so called Bloch constant determined by (3.137) in units of eV.

$\displaystyle I_0 = \left\{\begin{array}{cl} 12 + 7\cdot Z_2^{-1} & \text{\norm...
... 58,5\cdot Z_2^{-1.19} & \text{\normalsize for $Z_2 \ge 13$} \end{array}\right.$ (3.137)

The exponent $ c$ used for the interpolation around the stopping power maximum allows to determine the height of the maximum of the total stopping power $ S_e$. Besides a modification of the law for the non-local stopping power also a modification of the energy dependence of the ratio between local and the non-local part (3.132) is suggested for higher ion energies.

$\displaystyle x_{nl} = y_{nl}\cdot\frac{S_e}{k}^{2\cdot q}$ (3.138)

The major drawback of the empirical model is that many empirical parameters have to be determined especially for crystalline materials and therefore a lot of SIMS measurements are necessary to extend this model to new material types. But according to [31] the accuracy of the empirical model is better than the model based on just the local electron density as has been demonstrated for the case of boron implantations in crystalline silicon. Additionally it requires less computational effort, because just the impact parameter related to a certain collision event has to be determined to be able to calculate the electronic stopping power related to a single collision event.

The determination of the impact parameter requires no additional computational effort, because it is also used to model the nuclear stopping power process. In contrast the calculation of the local electron density along the flight path of the ion is a very computational intensive task. Additionally, a numerical integration of the electronic stopping power has to be performed in the local electron density model, because the electron density varies significantly when the ion passes an atom of the target.

previous up next contents Previous: Effective Charge of Up: 3.3.3 Electronic Stopping Process Next: 3.3.4 Thermal Lattice Vibration

A. Hoessiger: Simulation of Ion Implantation for ULSI Technology