Solid State Physics group
Tutorial - Theory of IBIC

This example provides a simple graphical visualization of the Shockley-Ramo theorem for the charge and current induced in a fully depleted device by the drift of a charged particle along the electric field lines. In the sketched 1-dimensional configuration the linear electrostatic potential and the constant electric field are presented. In the plot at the left the red circle represents the charged particle moving towards the left electrode. On the right side of the screen, time-dependent values of induced charge and current are shown. As the total induced charge is given by the difference in weighting potential between the initial and final position of the moving carrier, the induced charge plot presents the rise of the signal as a function of the elapsed time from generation point. Since the device is fully depleted, the weighting potential is obtained simply by rescaling to unity the maximum electric potential. The linearity of the curve implies the linear rise-time of the induced signal, if a constant electric field is assumed. Similarly, according to the Shockley-Ramo theorem, the constant electric field (and therefore the constant weighting field) leads to a constant induced current along the whole path of the particle towards the collecting electrode.

In the second example, the effect of carrier recombination on the induced pulse is presented. The example simulates the time-dependent signal associated with the drift of 300 charged particles in the same planar condenser geometry presented in Example 1. The carriers are generated at the time step t=10, and their time-evolution is subsequently simulated. At each time step, a constant recombination probability is given to each charge carrier. The decrease in the number of charged particles is monitored. As an effect of the carriers recombination, the cumulative induced charge saturates at a lower value than that expected from the previous example. If the distance between the electrode and the generation position is known, the exponential decay affects the charge collection efficiency of the capacitor according to Hecht's formula:



where v is the drift velocity of the charged particle q, d is the distance between the electrodes and τ is the carrier lifetime. Similarly, the induced current by each charge carrier is constant at any point in the device; therefore, the total current induced at the sensitive electrodes has the same exponentially decreasing behavior as that of the number of particles.

The the third example is devoted to the qualification of the contribution of carriers' drift and diffusion to the induced charge signal. In this example, a partially depleted diode is considered, and the electrostatics of the device are described (top left plots). The point-like generation of 300 minority carriers at a position external to the depletion region is defined at the initial time t=0. The drift-diffusion motion of charge carriers is then followed in time. Carriers outside the depleted region can diffuse at each time step with same probability in both horizontal and vertical directions. The diffusion of minority carriers in the neutral region implies an increase in time of the spread in the particle distribution; consequently, part of generated carriers can diffuse into the depletion region where they drift towards the collecting electrode inducing a signal. Contrarily to "Example 2" the carrier lifetime in the depletion region has been considered as much greater than the drift time. Hence, the total induced charge at the final instant of the simulation is then proportional to the number of minority carriers injected in the depletion region.

The fourth simulation describes a lateral IBIC experiment performed on the partially depleted diode presented in the previous example. In this case, both majority and minority carriers are considered, each having a given, constant lifetime. Simulation involves the point-like generation of 300 electron-hole pairs at a position representing the ion strike location. The experiment consists of a series of simulations, similar to those in "Example 3", at increasing distance between the generation point from the collecting electrode. If the generation point is within the depletion region, both carrier species contribute to the induced signal, according to Shockley-Ramo theorem; assuming a recombination time larger than the carriers drift time in the depletion region, a 100% CCE is then achieved. On the other hand, when the generation point is external to the depletion region, the electric field prevents the majority carriers to enter into the depleted volume, and therefore the signal is given only by the injection of diffused minority carriers, according to the mechanism presented in "Example 3". The exponential tail of the CCE corresponds to the decreasing probability of minority carriers to diffuse into the depletion region before recombining in the neutral region.

The simulation in "Example 5" schematically presents the effect of the presence of a highly damaged layer on the carriers transport in a fully depleted device. The geometry and the electrostatics of the device are assumed to be the same as those in "Example 2"; in this case, an highly damaged region (highlighted in red in the plot at the left) is considered by defining an high recombination probability region for the charge carriers. The time-evolution of current and cumulative induced charge follows the Hecht's formula until the carriers reach the damaged layer, where an abrupt decrease in the number of free charge carriers is observed. The sudden recombination of most of charge carriers leads to a decrease in the induced current and to a saturation of the cumulative induced charge at the value reached before the charged particles encountered the damaged layer.

The last example provides a schematic representation of the exploitation of frontal IBIC microscopy to investigate the non-homogenous transport properties of a polycrystalline device. At the left, an optical image of a polycrystalline sample is shown; grain boundaries are clearly visible. The scan of an ion microbeam onto the sample area allows for the mapping of the CCE. At the right, the associated inhomogeneous CCE map is presented, indicating different transport properties associated with different grains. At the bottom, the CCE profile corresponding to the current horizontal beam scan line is shown.