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 [1-2], 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 p-n junction. This results in the generation of the electron-hole pairs at these regions which will then diffuse to the p-n junction as shown in Figure 1. When these carriers reach the junction, they are separated and thus collected by the built-in electric field at the p-n junction due to the depletion region. Note that only the minority carriers i.e. electrons from the p-type material and holes from the n-type material, will be collected. These collected carriers contribute to a flow of current when the device is short-circuited.

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 p-n 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 electron-hole pair is comparatively small, therefore many electron-hole pairs are produced by the highly energetic primary electron. For example, it takes only approximately 3.6 eV to produce an electron-hole 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 Beam-Specimen Interaction

The electron beam-specimen interaction of interest here is the excess charge carriers i.e. electron-hole 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 electron-hole pair generation are usually described by two types of analytical functions [4-5]; one represents the depth-dose which is the energy loss per unit depth in the beam direction. This determines the depth resolution. The other function, the lateral-dose function represents the energy deposited per unit length at right-angles to the beam direction. This is important for the lateral or “spatial” resolution of EBIC technique.

In this simulation, the electron-hole pair generation was simulated in ATLAS, with the use of an analytical expression [6] given as follows,

(1)

In this equation, G₀ is the local rate of the electron-hole pair generation, F(r,y,E) is the lateral-dose function and h(y,E) is the depth-dose function.

The local rate of the electron-hole pair generation, G₀ 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 electron-hole pair. Kobayashi et al. [7] has derived an empirical relation between and the bandgap energy, Eg of semiconductors, namely

(3)

For the depth-dose 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 depth-dose 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 lateral-dose function, F(r,y,E) we have used the empirical expression as proposed by Donolato [5] for silicon:

(6)

where, ² = 0.36² + 0.11 , is the electron beam diameter and r² = x²in the two-dimensional case.

To implement the above expressions into ATLAS, we have used the F.RADIATE of ATLAS C-Interpreter 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 C-Interpreter is in micron, therefore, we need to convert it to centimeters before computing the generation rate which is in cm^-3 s^-1.

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 electron-hole 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 electron-hole 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. Electron-hole 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 sub-divided into two groups of geometries and these are the perpendicular geometry and the planar geometry. In the perpendicular p-n 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 p-n 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 p-n junction and Schottky barrier geometry respectively, in which the space-charge region is denoted with cross hatching. Diagram (c) and (d) illustrate the planar p-n junction and Schottky barrier geometry respectively.

4. Using The Perpendicular P-N Junction Geometry for Extraction of Minority Carrier Diffusion Length

In this section, we will use the perpendicular p-n 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 p-well in the n-substrate. The p-well region takes a Gaussian doping concentration of 10¹⁷cm^-3 while the n-substrate region has a uniform doping concentration of 10¹⁵cm^-3. We have defined a fixed minority carrier lifetime, of 3ns for this simulation and a constant mobility, m of 150cm²/(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 p-n 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 p-n 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 p-n 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 C-Interpreter 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 N-type region increases tremendously with a peak concentration of about 3 10¹⁵ 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 N-type region. This high concentration of the minority carriers in the N-type 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, IEBIC versus beam-to-junction distance, xb.

In Figure 7, note that we have plotted the EBIC current, I_EBIC against the beam-to-junction 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 p-n 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 p-n 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 ofIEBIC versus beam-to-junction distance, xb..

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 C-Interpreter function. We have demonstrated that by stepping the electron-hole pair generation of the electron beam in the direction perpendicular to the p-n junction, the simulated EBIC current can be use to extract the minority carrier diffusion length of a device.

References

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