^{1}

^{2}

Monte Carlo simulation of paths of a large number of impinging electrons in a multi-layered solid allows defining area of spreading electrons (A) to capture overall behavior of the solid. This parameter “A” follows power law with electron energy. Furthermore, change in critical energies, which are minimum energies loses corresponding to various electrons, as a function of variation in lateral distance also follows power law nature. This power law behavior could be an indicator of how strong self-organization a solid has which may be used in monitoring efficiency of device fabrication.

The interaction of beam of energetic electrons with the target solid material technique, which is electron beam lithography, is of great interest in probing material properties (chemical, electrical, physical etc.) [

One of the most important properties of real networks, ranging from social to biological protein-protein networks [

The path traversed by impinging energetic electrons in solid is based on the electron transport within the stochastic formalism of scattering process of electrons with the solid along their trajectory [

We consider impinging electrons suffering elastic collisions from scattering centres distributed in the single or multi-layered thin solid film, where, differential cross section can be described by classical screened Rutherford’s formula [

where, e is electronic charge, Z is the atomic number of the material and

(2)

where,

where, _{1} in [0, 1], and azimuthal angle

Thus,

The energy of the electron, suffering interaction from the scattering centres distributed randomly along its path, continuously loses its kinetic energy and can be calculated using Bethe’s continuous slowing down approximation model [

where, M is the atomic weight of the target material; _{A} is the Avogadro’s number and C is a constant (C®1; C < 1). The mean ionization energy J can be obtained from the empirical formula [

where .

The mean free path for single layered system, calculated using Equation (1), can be extended for multi-layered system by defining a probability P_{m}(u) that the electron once scattered from first layer is not scattered until mth layer [

where, _{m} is the distance between mth and (m + 1) th layers along z-axis. This

free path for single layer system is calculated as

by,

in the same way, mean free path for m layered system can be calculated using,

From Equation (8) one can able to calculate u_{m} of impinging electron in mth layered material system. Now, starting from an initial vector

Thus the path of the penetrating electron in m-layered material system can be traced using the recursive procedure in (9) with average energy loss

We consider single layered GaP and double layered system of GaP with resist PMMA (C_{5}H_{8}O_{2}), and simulated using the Monte Carlo simulation procedure described in the previous section to trace the trajectory of impinging energetic electrons in the systems and energy loss. The initial positions of the penetrating 500 electrons are taken as the same as ^{−}^{8} metre in the double layered calculation.

The paths of the impinging electrons (500 electrons’ paths) in single layer GaP system with different energies ([2 - 100] keV) are calculated (

The straight line is the fitting curve to the calculated data. The value of

We then calculated the As in double layered material system (1 mm thickness for PMMA and rest for GaP) for various electron energies [2 - 100] keV (

The energy loss of impinging 15000 electrons in single layer systems is calculated as a function of lateral distance R of the electron trajectories for different energies (

E_{c} and R_{c} for different Es in the range [10 - 100] keV. Then we calculated possible changes in these critical energies (_{c}, R_{c}), and error bars are standard deviations of the thicknesses of the drop curves (lines drawn parallel to E axis in

The power law fit to the data gives the power law exponent to be

We now calculated

The path and spread of the trajectory of any energetic electron is proportional to the energy it possesses and material in which the electron is penetrating. Even though the impinging electrons trajectories are stochastic zig-zag nature, the overall behavior of the large number of electrons exhibits the nature of the material’s characteristics. The area of the impinging electrons in single layered system follows power law as a function of electrons’ energy.

The power law behavior is found to various systems, starting from social systems to brain protein-protein networks which indicate important functional and organizational characteristics. This behavior indicates that the properties of the system are independent of scale of the system. Since this law also reflects the fractal nature of the system, it characterizes as an indicator of self-organized behavior of the material system. The impinging energetic electrons experience the self-organized behavior of the system which is reflected in the parameters calculated using Monte Carlo simulation procedure. This idea of fractal nature could be used as an order parameter in the fabrication of multi-layered device with proper efficiency.

Singh, M.S. and Singh, R.K.B. (2017) Power Law Nature in Electron Solid Interaction. Advances in Materials Physics and Chemistry, 7, 11-18. http://dx.doi.org/10.4236/ampc.2017.71002