3.3.3.3 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.

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

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

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

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

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

• $k_{corr}$ is the Lindhard correction factor to empirically adopt the strength of the electronic stopping.
• $a$ is a screening factor for the local contribution. A value of $\frac{a_{U}}{0.3}$ is proposed in [61] where $a_{U}$ is the screening length used for the calculation of the interatomic screening potential (3.92). In [36] this length is multiplied by the empirical screening pre-factor $f$.

  $$a = f\cdot\frac{a_{U}}{0.3}$$ (3.130)

  
  

• $p_{max}$ is the maximum considered impact parameter for the local stopping part, while contributions resulting from impact parameters larger than $p_{max}$ are considered by (3.127) for the non-local stopping part.
• $x_{nl}$ and $x_{loc}$ are the non-local and the local portion of the total electronic stopping power which requires that

  $$x_{nl}+x_{loc} = 1.$$ (3.131)

  
  
  The ratio between $x_{nl}$ and $x_{loc}$ depends on the energy of the ion especially for light ions like boron.

  $$x_{nl} = y_{nl} \cdot E^{q}$$ (3.132)

  
  
  The empirical parameters $y_{nl}$ and $q$ are called the non-local pre-factor and non-local power.

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

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

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].

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

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

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

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

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

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