Electron Beam Induced Current (EBIC) Simulation Using ATLAS
1. Introduction
Electron Beam Induced Current (EBIC) is a semiconductor analysis technique that is used in failure analysis to identify defects or buried junctions in semiconductors [12], and examines minority carrier properties such as diffusion length, carrier lifetime, surface recombination velocity [3]. In EBIC, the electron beam is made to impinge upon the device at a distance away from the pn junction. This results in the generation of the electronhole pairs at these regions which will then diffuse to the pn junction as shown in Figure 1. When these carriers reach the junction, they are separated and thus collected by the builtin electric field at the pn junction due to the depletion region. Note that only the minority carriers i.e. electrons from the ptype material and holes from the ntype material, will be collected. These collected carriers contribute to a flow of current when the device is shortcircuited.
Figure 1. Diffusion and collection of charge carriers in the presence
of an electron beam. 
This induced current due to the scanning of the electron beam across the pn junction can be sensed with a sensitive current amplifier. This current is orders of magnitude (approximately three orders of magnitude) larger than the electron beam current that produces the effect, since the energy to produce an electronhole pair is comparatively small, therefore many electronhole pairs are produced by the highly energetic primary electron. For example, it takes only approximately 3.6 eV to produce an electronhole pair in silicon, which means that an electron beam with beam energy of 1 keV can produce approximately 300 pairs, an effective current amplification of 300.
2. Simulating Electron BeamSpecimen Interaction
The electron beamspecimen interaction of interest here is the excess charge carriers i.e. electronhole pair generation within the specimen. When kilovolt electrons are incident upon a solid material, they lose energy during penetration by scattering from valence and core electrons. The distributions of the electronhole pair generation are usually described by two types of analytical functions [45]; one represents the depthdose which is the energy loss per unit depth in the beam direction. This determines the depth resolution. The other function, the lateraldose function represents the energy deposited per unit length at rightangles to the beam direction. This is important for the lateral or “spatial” resolution of EBIC technique.
In this simulation, the electronhole pair generation was simulated in ATLAS, with the use of an analytical expression [6] given as follows,
(1)
In this equation, G_{0} is the local rate of the electronhole pair generation, F(r,y,E) is the lateraldose function and h(y,E) is the depthdose function.
The local rate of the electronhole pair generation, G_{0} is simply given by:
(2)
In this expression, E is the electron beam energy, I is the electron beam current, f is the fraction of the electron beam energy that is reflected by the sample, and q is the electronic charge. f is a small numerical correction factor (= 0.08 for silicon). The quantity e_{i} is the energy expended by the incident electrons in the formation of a single electronhole pair. Kobayashi et al. [7] has derived an empirical relation between and the bandgap energy, Eg of semiconductors, namely
(3)
For the depthdose function, h(y,E) we have used the function suggested by Everhart [4] who approximate a “universal” depth dose function by a polynomial:
(4)
where is the depth, y normalized by the electron range R(E). The electron range, R(E) is essentially the depth to which an average electron would penetrate if it suffers no large angle scattering and it is a expressed as a function the electron beam energy as follows:
(5)
where (in g cm^{3}) is the density of the material, and the expression is accurate for 5< E < 25KeV. This depthdose expression should be valid for atomic numbers 10 < Z < 15 and it has been found to give reasonably accurate results by other researchers.
For the lateraldose function, F(r,y,E) we have used the empirical expression as proposed by Donolato [5] for silicon:
(6)
where, ^{2} = 0.36^{2} + 0.11 , is the electron beam diameter and r^{2} = x^{2}in the twodimensional case.
To implement the above expressions into ATLAS, we have used the F.RADIATE of ATLAS CInterpreter function to define equations (1) to (6). Table 1 summarizes the parameters used for a beam energy, E = 10KeV with beam current, I = 10pA for Silicon material.
Figure 2 shows the definition of the generation rate for the electron beam as a function of position using equation (1) to (6). Note that the x and y position in CInterpreter is in micron, therefore, we need to convert it to centimeters before computing the generation rate which is in cm^{3} s^{1}.


Figure 2. CInterpreter function for defining the
generation rate as a function of position.

The simulated contours of the generation rate for the case of an electron beam energy, E = 10KeV and the beam current, I = 10pA is shown in Figure 3. From Figure 3, it can be seen that the high generation rate of the electronhole pairs are located near to the material surface, along the beam axis at x = 0µm. This shows that the incident of high energy electron beam has resulted in a small deviation of the high energy electron from its trajectory. As the electrons penetrate further into the material, some electrons may also scatter with a small loss of energy from the nuclei and undergo a large angular deviation from their original trajectories. The result of this scattering is the formation of a “Teardrop” shape electronhole pair generation. In addition, one can also observe from Figure 3 that the stopping range of the electron beam is approximately at 1µm.
Figure 3. Electronhole pair generation rate at beam
energy = 10keV

3. Types of Geometries For Observing EBIC
There are basically four popular types of geometries for observing EBIC and these are illustrated schematically in Figure 4. These four types of geometries can be subdivided into two groups of geometries and these are the perpendicular geometry and the planar geometry. In the perpendicular pn junction and Schottky barrier geometry as shown in Figure 4 (a) and (b) respectively, the electron beam is scanned in a direction that is perpendicular to the junction. On the other hand, in the planar pn junction and Schottky barrier geometry as shown in Figure 4 (c) and (d) respectively, the electron beam is scanned in a direction that is parallel to the junction.
Figure 4. Schematic illustration of charge collection
geometries. Diagram (a) and (b) illustrate the perpendicular pn junction
and Schottky barrier geometry respectively, in which the spacecharge
region is denoted with cross hatching. Diagram (c) and (d) illustrate
the planar pn junction and Schottky barrier geometry respectively.

4. Using The Perpendicular PN Junction Geometry for Extraction of Minority Carrier Diffusion Length
In this section, we will use the perpendicular pn junction geometry to study the extraction of the minority carrier diffusion length by 2D ATLAS simulation. A simple simulation configuration is as shown in Figure 5. In this configuration, the simulation structure consists of a pwell in the nsubstrate. The pwell region takes a Gaussian doping concentration of 10^{17}cm^{3} while the nsubstrate region has a uniform doping concentration of 10^{15}cm^{3}. We have defined a fixed minority carrier lifetime, of 3ns for this simulation and a constant mobility, m of 150cm^{2}/(V . s). Therefore, the minority carrier diffusion length, L can be calculated to be approximately 1.08µm. In addition, the surface recombination velocity at the semiconductor/insulator interface is set to zero in this simulation.
From Figure 5, we can see that the pn junction of the structure is located at about 9µm with a junction depth of 2µm. The electron beam with a beam energy of 10keV is then scanned in the direction which is perpendicular to the vertical pn junction starting from x = 2µm to 9µm. A beam step size of x = 0.5mm is used for the electron beam.
Figure 5. EBIC Simulation using the perpendicular
pn junction geometry.

ATLAS makes the EBIC simulation more convenient with the use of the DBINTERNAL feature of DeckBuild. We have first created a template input deck file to set the location of the electron beam as variable using the SET statement and then solve for the EBIC current. The beam position can be shifted by using the CInterpreter functions. We have replaced the parameter x in equation (6) with (x + 2), (x + 3)... etc for different beam positions and then saved as different .lib such as “ebic2.lib” for beam position, x = 2µm, “ebic3.lib” for x = 3µm, and so on. Finally, a simple Design of Experiment (DOE) input deck is created to simulate a series of EBIC current at different beam locations using the DBINTERNAL.
Figure 6 shows the hole concentration of the structure when the electron beam is at x = 5µm. From this plot, one can observe that minority carrier i.e. hole in the Ntype region increases tremendously with a peak concentration of about 3 10^{15} cm^{3} after the impingement of the electron beam at x = 5µm. This is about three times higher than the majority carrier concentration in the Ntype region. This high concentration of the minority carriers in the Ntype region will then diffuse to the junction and be collected at the junction. The collected EBIC current at different beam positions is then plotted as shown in Figure 7.
Figure 6. Hole concentration of the structure with
the electron beam at x = 5mm.

Figure 7. Graph of collected EBIC current, I_{EBIC} versus
beamtojunction distance, x_{b}.

In Figure 7, note that we have plotted the EBIC current, I_{EBIC} against the beamtojunction distance, x_{b} rather than the beam position, x so as to be consistent with other EBIC literatures. Therefore, x_{b} = 0mm corresponds to the electron beam position exactly at the pn junction.
From this graph, it can be observed that the peak collected EBIC current occurs at x_{b} = 0mm (i.e. when the electron beam incidents directly on top of the pn junction) is about 52nA. Since the beam current used is 10pA, this means that the collected EBIC current is about 5200 times that of the beam current. As the electron beam scans away from the junction, the collected EBIC current begins to decay exponentially. This is in good agreement will the theoretical equation [3] that shows the EBIC current, I_{EBIC} takes the following exponential relationship:
(7)
where K is a constant, = 0 for surface recombination velocity, v_{s} = 0 and a = 1/2 for v_{s} = , L is the minority carrier diffusion length.
By taking the natural logarithm on both sides of equation (7) and rearranging, we can obtain the following equation:
(8)
From the above equation (8), it shows that by plotting ln(I_{EBIC} / x_{b}) against x_{b}, this will yield a straight line with a gradient, 1/L. Thus, by measuring the gradient of the straight line, we can determine the minority carrier diffusion length, L. To verify this, we have take the natural logarithm of I_{EBIC} of Figure 7 and reproduce as shown in figure 8. Note that we have defined the surface recombination velocity, vs in the simulation as zero. Therefore, the term x_{b} in equation (8) becomes unity.
From Figure 8, it is verified that the natural logarithm of I_{EBIC} yields a straight line. The gradient of the straight line is measured using the Ruler feature of TonyPlot which is found to be 0.945. Hence, the extracted minority carrier diffusion length, L is found to be 1.06µm. This is close to that specified in our simulation which is 1.08µm.
Figure 8. Natural logarithm ofI_{EBIC} versus beamtojunction distance, x_{b}.. 
5. Conclusion
It has been shown that ATLAS can be use to simulate the effect of Electron Beam Induced Current (EBIC). The generation rate of the electron beam has been successfully implemented into the simulator through the use of ATLAS CInterpreter function. We have demonstrated that by stepping the electronhole pair generation of the electron beam in the direction perpendicular to the pn junction, the simulated EBIC current can be use to extract the minority carrier diffusion length of a device.
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