3.3.4 Thermal Lattice Vibration

Although the wafer temperature during ion implantation is normally below 400 K, the lattice vibrations can influence the trajectories of the implanted ions. Especially the probability for scattering an ion out of a channel (de-channeling) is increased by increasing the wafer temperature. Due to the fact that the knowledge of the atomic lattice behavior is required for the simulation it is necessary to apply a temperature dependent lattice vibration model.

The thermal motion of the lattice atoms can be appropriately modeled by a spherically symmetric Gaussian function $f(\vec{\Delta x})$ which determines the probability for finding an atomic core at the position $\vec{\Delta x}$ when the corresponding lattice site is located at the origin.

$$f(\vec{\Delta x}) = \frac{1}{\sqrt[3]{2\cdot \pi\cdot \sigma^2}}\cdot\exp\left( -\frac{\vert\vec{\Delta x}\vert^2}{2\cdot \sigma^2} \right)$$ (3.139)

The straggling of this distribution function can be modeled by the Debye model of a solid [5].

$$\sigma^2 = 12,1\;\left[{\mathrm{\AA\sqrt{K}}}\right]\cdot\sqrt{\frac{\frac{x_D}{\Phi(x_D)} + \frac{1}{4}}{M_2\cdot \Theta_D}}$$ (3.140)

$$x_D = \frac{\Theta_D}{T}$$ (3.141)

$$\Phi(x_D) = \frac{1}{x_D}\cdot \int\limits_0^{x_D} \frac{t\cdot \;dt}{\exp(t)-1}$$ (3.142)

$M_2$ is the relative mass of the atom of the solid. The Debye temperature $\Theta_D$ can be used as an empirical parameter, which can be determined for instance by measurements of the specific heat or by channeling experiments [17], [41] as summarized in Tab. 3.4. Fig. 3.9 shows the average vibration amplitude of silicon atoms as a function of the temperature for three different Debye temperatures.

Table 3.4: Experimentally determined Debye temperatures.

Experimental method	Debye temperature
Specific heat	645 K
Channeling experiments [17]	490 K
Implantation simulation [41]	450 K

Figure 3.9: Amplitude of lattice vibration as a function of the temperature for three Debye temperatures.