Peripheral Capacitance Correction Factor for Square

For an accurate determination of Neff from C-V data it is also necessary to correct for the peripheral capacitance. Under the depletion approximation it is assumed that when a p+-n junction is reverse biased, the depletion region extends into the n-type bulk with a cross section equal to that of the junction area i.e. on the basis of plane parallel geometry. Lateral expansion of the depletion region around the peripheral of the junction is ignored.

In devices where the junction area is large in comparison to the depletion depth, which is the case for devices based on low resistivity silicon, this effect can largely be ignored. For high resistivity substrates, such as in the detector structures studied here, the effect is significant and requires correction.

Goodman [182] approximated the capacitance due to the lateral expansion effect in circular junctions as: 2 p C o si p π ε ε = (3.6)

where: p = perimeter of the contact.

An improved approximation was provided by Copeland [183]. The peripheral capacitance was represented by:

rb

C

=ε

ε

π

where: r = the radius of the circular contact, b = a constant.

b was found to be equal to 1.5. By comparison, the equivalent value of b for the Goodman

model is π.

The Copeland correction was applied to the measured C-V data. Figure 3.7 shows the uncorrected data along with the corrected data in a C-2 versus V plot. Significant improvement to the linearity can be observed. Comparison of the corrected data to a straight line revealed that linearity was still not observed. It appears that an additional component of capacitance must be subtracted.

The validity of using the Copeland peripheral capacitance correction was questioned. It was initially derived for circular junction areas and does not give account for square junction areas. It was hypothesised that the additional capacitance is associated with the lateral expansion of the depletion region from the corners of the square junction. The Copeland correction was not formulated to account for this contribution.

Figure 3.7: C-2 _{versus V}

R for data uncorrected for peripheral capacitance and data corrected for peripheral capacitance using the Copeland expression.

A direct experimental observation of the lateral expansion of the depletion region about the corner of the junction was done using the technique of Ion Beam Induced Charge Collection (IBICC). In this technique a narrow beam of heavy ions is made incident on the surface of the detector structure. The beam is then scanned across the device with sub micron accuracy. The e- h pairs generated within the silicon as a result of the ion’s energy loss are collected under the action of the applied electric field. A charge sensitive preamplifier and pulse height processing system allows the detector charge collection characteristics to be studied. The additional data regarding the spatial incidence of the ion beam allows a map of the relative charge collection efficiency across the scanned region to be constructed.

20 40 60 80 100 Reverse voltage (V) 0 100 200 300 400 (Capacitance) -2 (pF ⋅cm -2 ) -2 × 10 6

Uncorrected for peripheral capacitance

For this experiment the IBICC technique was used to profile the electric field around the edge and corner of a detector in order to directly observe the lateral expansion effect. The IBICC measurements were performed using the Nuclear Microprobe system of the Microanalytical Research Centre within the School of Physics at the University of Melbourne. This facility has been able to achieve a beam spot size of 0.05 µm for 2.4 MeV He+ ions [184].

The U4 and U5 devices were not used. It was considered that the heavy ion beam could cause radiation damage to the detector bulk and jeopardise the integrity of the detectors for the subsequent radiation hardness study. An alternative detector with a square junction was obtained. A schematic of the detector used is shown in Figure 3.8. The detector had standard p+nn+ structure with an implant area of 5 × 5 mm2. The junction was surrounded by a guard ring structure. The guard ring and detector junction edge were separated by a distance of 100 µm. A layer of SiO2 had been grown over the interlaying region.

The IBICC measurements were performed using 2.8 MeV He+ _{ions at approximately 1500}

ions per second. Typical spot size was 0.1 µm. The LET for this ion in silicon is approximately 1 MeV⋅cm2⋅mg-1. Penetration depth is about 10 µm. Two types of scans were performed. In the first a line scan was made from a point outside the detector substrate to a point approximately 0.3 mm inside the detector window region. The step between each measurement was 0.5 µm. Scans were done with the detector reverse biased at voltages of 0, -2, -5, -10, -50, -100 and -120 V. The full depletion voltage for this detector was -170 V. Soft breakdown at a reverse voltage of approximately -140 V prevented measurements at full depletion. In the second scan a square region of area 1000 µm × 1000 µm located about a detector corner region was made. In this case the detector was biased with a reverse voltage of -120 V. For both measurements the guard ring was left floating. The results of the line scan are shown in Figure 3.9.

Figure 3.8: Layout of the detector used in the IBICC measurements. Note the narrow

gap (~ 10 µm) between the detector Al metallisation and SiO2 layer. The regions over

which the IBICC area scan and line scan were performed are shown. Detector junction (metallised with Al) SiO2 (100 µm)

Guard rail, 200 µm wide.

(metallised with Al) Outer detector

substrate (SiO2)

~ 10 µm gap between SiO2

and Al metallisation.

Region over which the IBICC line scan was performed.

Region over which the IBICC area scan was performed.

Figure 3.9: Horizontal line scans performed using 2.8 MeV He+ ions.

For the region greater than 500 µm a plateau in the pulse height can be seen at all voltages. As the voltage is increased the plateau height increases. This corresponds to the expansion of the depletion region into the substrate bulk and the corresponding increase of carrier drift through the device. A net movement of charge through the device induces charge to flow in the external circuit. The pulse height of the signal produced will be proportional to the distance through which the charge carriers move in the device. i.e. charge collection efficiency increases towards 100% as the applied voltage approaches that required for full depletion. To the left of the plateau region a small peak can be seen at all voltages. The width of the peak is approximately 10 µm. In terms of the detector structure this region corresponds to the gap

250 300 350 400 450 500 550 600 650 700 750 800

Distance from detector edge (µm)

0 100 200 300 400 500 600 700 800 900 1000

Pulse height (arbitary units)

0 V -2 V -5 V -10 V -50 V -100 V -120 V

between the detector window Al metallisation and the region of SiO2. The lack of an interfacial

layer in this region permits all of the He+ ion energy to be deposited within the silicon bulk. Therefore more charge is available for collection and a higher pulse height is observed. This result is also the first indication of the effect of lateral expansion of the depletion region beyond the edge of the detector window (the edge of which is bounded by the Al metallisation). To the right of this peak, and at low voltages, the pulse height degrades as the beam approaches the edge of the device substrate. This result is almost certainly the effect of charge diffusion from the region of zero electric field into the depletion region where the electrical field exists and is able to act on the charge and produce the observed signal. At higher voltages, for example at - 120 V, in the region to the right of the narrow peak exists a second plateau of approximate width 40 µm. This region corresponds to the area over which the SiO2 layer has been deposited.

The excellent charge collection in this region, as demonstrated by the large signal pulse height, indicates that the transport process in this region must be one of drift under the influence of an electric field. Extension of the electric field into this region can only be explained as a result of lateral expansion. The slightly higher pulse height in this region as opposed to the pulse height for beams incident on the detector junction region indicates that less energy is lost in the SiO2

layer than within the Al metallisation and inactive p+ region. The effect of lateral expansion can be seen to increase with increasing bias.

The results of the area scan are shown in Figure 3.10. The relative charge collection efficiency as a function of position can be compared to the line scan result. The narrow band of highest pulse height, shown in grey, is located around the perimeter of the detector window. This corresponds to the 10 µm wide peak seen in the line scan measurements. The lateral expansion of the depletion region about the corner of the detector can also be seen. The additional capacitance due to this region must therefore be acknowledged in the calculation of the correct peripheral capacitance correction for square junction areas. The literature does not advise on the exact form of such a correction.

Figure 3.10: IBICC area scan (1000 × 1000 µm) centred over a corner region of the detector using 2.8 MeV He+ ions.

To account for this effect the following procedure was undertaken. It was assumed in a first approximation that the magnitude of the peripheral capacitance for the square contact was proportional to the perimeter of the contact:

pk

C

=ε

ε

Charge collection efficiency 100 %

The magnitude of k was found by numerical solution. C-V data was taken from a series of detectors with square geometry and areas of 1 cm2, 0.25 cm2 and 0.09 cm2. Three detectors of each size were used. A correction for the MOS capacitor effect was only required for the detectors with area 0.09 cm2. The other detectors did not have a bonding pad. The size of the peripheral capacitance was found by applying a small correction, Cp, to the measured capacitance, Cm, of the form:

p m

C

C

C

=

−

where: Cp = is the small capacitance correction associated with the peripheral capacitance .

Since a linear relationship should exist between C-2 and VR, the first derivative of C-2 with respect to VR should be equal to a constant. To determine the size of Cp the first derivative of C-2 with respect to VR was plotted as a function of VR. The value of Cp was then optimised to give a slope to this curve of 0. This corresponds to a constant value for the first derivative as a function of VR. An alternate means of determining Cp was to determine the 2nd derivative of C-2 with respect to VR and optimising Cp so that the 2nd derivative was equal to zero.

Both methods were employed yielding the same results for each detector. The results are summarised in Table 3.1. The optimal peripheral capacitance was then used to determine k from Equation 3.8. The equivalent value of b from the Copeland equation was also determined.

The average value of k was found to be 0.876. The equivalent value of b was 1.98. The result was consistent for the detectors with perimeters from 12 mm to 40 mm. By comparison, the value of b for the circular geometry junction was 1.50 [183]. For all future C-V measurements this new peripheral capacitance correction formula was applied.

Table 3.1: Peripheral capacitance correction results. Detector Area (cm2₎ Peripheral capacitance (pF) k b 1.00 3.72 0.883 1.99 1.00 3.72 0.883 1.99 1.00 3.71 0.880 1.99 0.25 1.85 0.878 1.98 0.25 1.87 0.887 2.00 0.25 1.83 0.868 1.96 0.09 1.10 0.870 1.96 0.09 1.11 0.878 1.98 0.09 1.09 0.862 1.95