2 p-side of the Detector

The p-side is simpler so we will look at models for that side first. The simplest model ignores $C_{is}$, which effectively ignores adjacent strips. This simple model is shown in Fig. 2. The $R_{big}$ resistor represents the leakage to the other side of the detector. However, it is really there because SPICE gives an error if it's not there for some unknown reason.

Figure 2: Simple model ignoring adjacent strips

The frequency response of the simple circuit given by SPICE is plotted with the measured data of strip 120 on the p-side of the 1B3 part with full bias voltage of 40V as shown Fig. 3. The simulation does not match well with the data. However, we can see the important features of the model. The initial downward slope is due to $C_{AC}$. When the magnitude of the impedance of $C_{AC}$ given by $\vert 1/\omega C_{AC}\vert$ equals $R_b$, the bias resistor begins to dominate and we see the flat region. When $R_b$ equals $\vert 1/\omega C_b\vert$, then the impedance starts to turn downwards again. At very high frequencies beyond 1 MHz and off the plot, we see another flat region where $R_b/n$ dominates. A few things to note about the plot are that there is some sort of interference, perhaps a radio station, at 800kHz and 250kHz and possibly the regions around those frequencies.

Figure 3: Simulation of the simple model vs. measurements on a normal strip

At this point, there are two things that we can do. The first is to assume that the parasitic elements due to the probe station is in parallel with the model and try to extract the data that is just the detector. The assumption that the parasitics are in parallel with the detector is a bad one at higher frequencies when the interstrip capacitances come into effect and we no longer have a parallel model. The second thing to try is to model the raw data by adding parasitic elements to the simple SPICE model that we have. Of course none of this has taken into account the adjacent strips yet.

First, we will take a look at modeling the raw data by adding extra elements into the original simple model. We put a capacitor in series with an element consisting of a resistor in parallel with a series RC (see Fig. 4). The values of $C_1$, $C_2$, $R_1$, and $R_2$ are arbitrary and chosen to fit the data (see Fig. 5). It is not known at this point whether this model is meaningful in any way, and it fails at high frequencies. The values for the

$$\begin{array}{ccc} C_1 &=& 90 \textrm{pF}\\ C_2 &=& 50 \t... ...\textrm{k}\Omega\\ R_2 &=& 60 \textrm{k}\Omega\\ \end{array}$$ (4)

Other values can fit the phase data better at frequencies around 1MHz, but fit the impedance worse. For example the values

$$\begin{array}{ccc} C_1 &=& 90 \textrm{pF}\\ C_2 &=& 20 \t... ... \textrm{k}\Omega\\ R_2 &=& 5 \textrm{k}\Omega\\ \end{array}$$ (5)

are plotted as well in Fig. 6. Also of note is that $R_b/n$ and $C_b$ can be removed with little change in the plot. The parasitic element dominates over the coupling to the backplane.

Figure 4: Model of one strip with additional parasitics

Figure 5: Simulation of a model with parasitic elements vs. measurements of a normal strip

Figure 6: Simulation of a model with parasitic elements vs. measurements of a normal strip

Now we take a look at a circuit that only includes the parasitic part (see Fig. 7). This SPICE model is compared with the data of a plucked strip 108 on the p-side of the 1B3 part at 40V bias (see Fig. 8 and 9). From this, we can see if the model of the parasitic elements matches the data on a plucked strip. The first plot is for the first set of values for $C_1$, $C_2$, $R_1$, and $R_2$ and the second plot is for the second set of values listed above.

Figure 7: A plucked strip should only see this part of the circuit

Figure 8: Simulation of the parasitic model vs. measurements of a plucked strip

Figure 9: Simulation of the parasitic model vs. measurements of a plucked strip

The second thing to try is to assume that the parasitic element is in parallel with what is being measured. Let $Z_T$ be the total impedance of a normal strip. If we measure the impedance $Z_p$ of a strip that is plucked between the upilex and the wafers, then we can subtract out the effect of $Z_p$, the parasitics due to the probe and possibly the upilex. What we are then left with is $Z_s$, the impedance of the silicon detector. The extraction process works as follows. The total impedance is given by

$$\frac{1}{Z_T} = \frac{1}{Z_p} + \frac{1}{Z_s}$$ (6)

Let us define the following

$$Z_T = \vert Z_T\vert e^{i\phi_T}$$ (7)

$$Z_p = \vert Z_p\vert e^{i\phi_p}$$ (8)

$$Z_s = \vert Z_s\vert e^{i\phi_s}$$ (9)

so that

$$\frac{1}{\vert Z_s\vert} = \frac{1}{\vert Z_T\vert} e^{i(\phi_s-\phi_T)} - \frac{1}{\vert Z_p\vert} e^{i(\phi_s-\phi_p)}$$ (10)

Now we equate the real and imaginary parts to get

$$\frac{\vert Z_T\vert}{\vert Z_u\vert} = \frac{\sin{\phi_s-\phi_T}}{\sin{\phi_s-\phi_p}}$$ (11)

and

$$\vert Z_s\vert = \frac{1}{\frac{1}{\vert Z_T\vert} e^{i(\phi_s-\phi_T)} - \frac{1}{\vert Z_p\vert} e^{i(\phi_s-\phi_p)}}$$ (12)

The first equation, we can solve numerically for $\phi_s$ with Mathematica or Excel and plug into the second equation.

Now we can attempt to plot the data given by SPICE for the simple model (see Fig. 2), against this newly extracted data. The plots are shown in Fig. 10. We can see that at low frequencies, the extraction process works well. However, when the neighboring strips become important and $C_{is}$ comes into play, we get a deviation from the model.

Figure 10: Simulation of the simple model vs. extracted data

To go even further, we can add the neighboring strips into the model, as well as break up the strip itself into a series of capacitors and resistors. We put in the interstrip capacitances and in addition, we put in an implant resistance $R_{impl}$, which has a value of about 54 k$\Omega$/cm which gives about $R_{impl}=220k\Omega$, and a upilex interstrip capacitance $C_u$ which is about 0.5 pF/cm giving a value of about 3pF.

At this point, we need to get into the details of how the probe works. The probe has 256 pins. 252 of them are all connected together and float. Of the four remaining, one is the test strip and the other three are floating. The test strip is always next to one floating strip on one side and two floating strips on the other side.

The only way to get anything close to the measured data is to include the parasitic elements shown previously in Fig. 7. We find that the impedance and phase no longer change after adding about 48 neighboring strips. We also find no change after dividing the strip up into four segments, each with their own interstrip and AC coupling capacitances. In this model, we have to alter the values that we used previously for $C_1$, $C_2$, $R_1$, and $R_2$ to get a good fit of the data. Also, a different value of $C_u$ was used other than 3pF. This might have to do with the fact that there is also an interstrip capacitance between upilex strips that are 2 apart in addition to the nearest neighbor interstrip capacitance. Additionally, the value of $R_{big}$ is important at very low frequencies. $R_{big}$ actually represents the leakage to the backplane and to get a better fit around 50 Hz, we set that value as well. The values used to produce the plot in Fig. 11 are

$$\begin{array}{ccc} C_1 &=& 80 \textrm{pF}\\ C_2 &=& 30 \t... ...\textrm{pF}\\ R_{big} &=& 150 \textrm{M}\Omega\\ \end{array}$$ (13)

Figure 11: Simulation of a complicated network model vs. data

Another parameter that we can vary is the bias voltage $V_b$. As the bias voltage drops, the capacitance to the backplane should increase as the charge carriers get closer to each other and the depletion region shrinks. The capacitance $C_b$ should be proportional to $\frac{1}{\sqrt{V_b}}$. The question is if this effect can be seen, and if we can quantitatively extract the value of $C_b$ by taking measurements. One thing to be careful of is the amplitude of the input frequency. If it is comparable to the bias voltage, then we could see unwanted effects. So for this data, the bias voltage was at 100 mV. Because of noise, we begin taking data at 200 Hz rather than 20 Hz. We expect that $C_b$ affects only higher frequencies and so this shouldn't be a problem. Fig. 12 is a plot of the impedance and phase at various bias voltages.

Figure 12: Impedance and phase for various bias voltages

We can see that there is some dependence on $V_b$ for lower bias voltages. Now we pick fixed frequencies of 1kHz, 10kHz, and 100kHz and vary $V_b$ (see Fig. 13). If $C_b$ dominates at a certain frequency, then $\vert Z\vert\sim \frac{1}{\omega C_b}$. So if we plot $\vert Z\vert^2$ vs $V_b$, we should get a straight line since $C_b \propto \frac{1}{\sqrt{V_b}}$, or equivalently, $V_b \propto \frac{1}{C_b^2}$. The plots are shown in Fig. 14. There is a region for each of the plots where the behavior is somewhat linear. It is not know yet if this effect is completely due to $C_b$ or if other capacitances such as $C_{is}$ are also somehow varying and changing these values.

Figure 13: Bias voltage sw for various frequencies

Figure 14: Plots of $\vert Z\vert^2$ for various frequencies