1. A simple model for the equivalent circuit of heavily irradiated Si diodes in standard CV measurements D.Campbell, A.Chilingarov, T.Sloan Lancaster University, UK 7 th RD50 Workshop CERN, 14-16.11.05

2. A.Chilingarov, A simple model for the equivalent circuit of heavily irradiated Si diodes in standard CV measurements 2 Outline 1.Introduction 2.Generic equivalent circuit diagram 3.Choice of a minimal model 4.Results and discussion 5.Conclusions

3. A.Chilingarov, A simple model for the equivalent circuit of heavily irradiated Si diodes in standard CV measurements 3 We propose a simple equivalent circuit able to reproduce the experimental CV characteristics of a heavily irradiated Si diode in the frequency range 20 Hz – 30 kHz. The model also incorporates the following well established observations. a)The resistivity of the electrically neutral bulk is close to that of intrinsic Si (see e.g. Z.Li, NIM A342 (1994) 105). b)A kink in the IV curves occurs at much lower bias than in the CV curves (R.Wunstorf, PhD thesis,1992; L.J.Beattie, PhD thesis, 1998). c)At low bias there are two space regions in the detector adjacent to the p + and n + electrodes and they grow proportionally to the square root of bias voltage (see e.g. A.Chilingarov, 4 th RD48 Workshop,1998, CERN/LEB 98-11, p.367; G.Cramberger, PhD thesis, 2001; O.Krasel, PhD thesis, 2004). 1. Introduction

4. A.Chilingarov, A simple model for the equivalent circuit of heavily irradiated Si diodes in standard CV measurements 4 The measurements were made with the diode s62 produced by SINTEF. This is a typical PIN diode (5x5 mm 2, 335  m thick) made of a standard Si. It was irradiated by neutrons to 1 MeV equivalent fluence of 0.8 10 14 n/cm 2 and annealed to a minimum in the depletion voltage. The actual measurement temperature was -0.5 o C and the frequency varied from 20 Hz to 1 MHz. Both real and imaginary parts of the diode impedance were measured and corrected for the effects of the DC bias circuit. 1. Introduction (continued)

5. A.Chilingarov, A simple model for the equivalent circuit of heavily irradiated Si diodes in standard CV measurements 5 Experimental data for the C p normalised to its value at high bias. The U 1/2 line is to guide the eye.

6. A.Chilingarov, A simple model for the equivalent circuit of heavily irradiated Si diodes in standard CV measurements 6 2. Generic equivalent circuit diagram = … RkRk CkCk X Y For CV measurements the diode can be regarded as a capacitor filled with a material with properties that may vary along x-axis but are uniform in other directions. A natural circuit diagram for such a device is a series of modules consisting of an ideal capacitor parallel to an ideal resistor with parameters uniform also along x within each module. For each module the impedance can be described by two parameters. We will use: C k and Q k =  C k R k =  k  k  0 =  k. The serial capacitor C s and resistor R s for the whole diode are: High frequency limit  k >>1 for all k At high frequency C≈const=C 0 for any bias, therefore a natural assumption is:  =const at any bias. We assume additionally that  is the same for any frequency. These basic assumptions are used everywhere in this work. A small R s means C p =C s

7. A.Chilingarov, A simple model for the equivalent circuit of heavily irradiated Si diodes in standard CV measurements 7 = … RkRk CkCk X Y Normalise x so that x=0 at one electrode and x=1 at the other. Denote the thickness of the k-th section as  k, the total detector thickness as w and its area as A. Then 1/C k =  k /C 0, where C 0 =A  0 /w is the geometrical capacitance of the whole diode, and the above equations become: It can be proved that in this case the parallel capacitor C p of the whole diode can be only higher than C 0 and is equal to C 0 if and only if all Q k are equal. In other words for uniform  C p > C 0 means non-uniform  and vice versa. A strict proof of this is given in our paper NIM A552 (2005) 152 (Proceedings of the Florence RESMDD04). At this still quite generic level some useful predictions can already be made. 0 1 (1)

8. A.Chilingarov, A simple model for the equivalent circuit of heavily irradiated Si diodes in standard CV measurements 8 Denote minimum Q k as Q min. As follows from the eqs. (1): Since one finally gets a testable constraint: In the plot Q min is set to  i  0 with intrinsic resistivity  i at its nominal value for 0 o C of 2.6 M  cm. Conclusion: when 1/f=T<2  (  r ) min where (  r ) min =  min  0 is the minimum relaxation time, the deviation of C s,p from their minimum C 0 is suppressed.

9. A.Chilingarov, A simple model for the equivalent circuit of heavily irradiated Si diodes in standard CV measurements 9 3. Choice of a minimal model RdRd CdCd C nd R nd A minimal equivalent circuit needs at least two modules: one for the non-depleted and one for the depleted part of the diode. Denote the normalised depleted thickness as d and the quality factor in the depleted (non-depleted) region as Q d (Q nd ). The C p measured for the fully depleted diode can be used as C 0. At each bias voltage there are 3 unknown parameters : d, Q nd and Q d. If one of them is fixed from elsewhere, the other two can be found from the eqs. (1). As shown in our paper NIM A 552 (2005) 152 this model cannot reproduce experimental data if the resistivity of the non-depleted part is in the vicinity of that of the intrinsic Si.

10. A.Chilingarov, A simple model for the equivalent circuit of heavily irradiated Si diodes in standard CV measurements 10 R d1 C d1 C nd C d2 R nd R d2 RdRd CdCd C nd R nd = We assumed that the resistivity of the depleted region is not uniform but can be approximated by only two regions with uniform resistivity  1 and  2 with relative thicknesses 1-  and . For uniform  the C d should be larger than its minimal theoretical value C o /d. Also it is natural to expect that  nd ≈  i and both  1 and  2 are higher than  nd. (2)

11. A.Chilingarov, A simple model for the equivalent circuit of heavily irradiated Si diodes in standard CV measurements 11 The model does not require Q to be actually uniform in two real x regions. It is just a simple approximation of the real Q(x). Note also that for non- monotonic Q(x) the x intervals can be rearranged so that to make Q monotonic. Then it can be naturally approximated by the Q uniform in two regions. Eqs. (2) contain 5 unknown parameters: d, , Q nd, Q d1, Q d2. Only two of them can be found from the experimental data at each bias point. Further assumptions are therefore needed.

12. A.Chilingarov, A simple model for the equivalent circuit of heavily irradiated Si diodes in standard CV measurements 12 R.Wunstorf, PhD thesis, 1992 L.J.Beattie, PhD thesis, 1998 It has been known for a long time that for heavily irradiated Si diodes the kink in IV occurs at much lower bias than that in the CV curves. Typically the voltage ratio is 0.4-0.5. Note also that below the kink the current grows as the square root of the bias voltage. U I /U d =0.16

13. A.Chilingarov, A simple model for the equivalent circuit of heavily irradiated Si diodes in standard CV measurements 13 x 10 _ + N eff E n+n+ p+p+ More recently it has been shown that the space charge region in heavily irradiated diode consists of two areas one adjacent to each electrode (“double junction”). Both these regions grow proportionally to the square root of bias voltage and merge at ~ 50% of the U dep found from the CV measurements at 10 kHz (see e.g. A.Chilingarov, 4 th ROSE Workshop, 1998; G.Kramberger, PhD thesis, 2001; O.Krasel, PhD thesis 2004). Based on the above we assumed that the depleted region thickness d grows as a square root of bias and reaches d=1 at U=U merge ≈ 0.4 U dep (CV@10kHz). The value of Q nd was calculated as  nd  0. Assuming a certain value for the  nd the impedance of the non-depleted region was subtracted from the measured impedance and the values of (C p ) d and Q d for the depleted region as a whole were found. This procedure eliminated two of the five unknown parameters d and Q nd.

14. A.Chilingarov, A simple model for the equivalent circuit of heavily irradiated Si diodes in standard CV measurements 14 In our case the “depletion” voltage determined from the experimental CV data at 10 kHz is 146 V. Therefore we have chosen 60 V for U merge which in our model is the voltage at which the whole diode volume becomes depleted of the free carriers. The nominal value of the intrinsic resistivity at 0 o C is  i =2.6±0.2 M  cm. However  nd =  i leads to the value of C p for the depleted region being lower than the minimum allowed value C 0 /d at low bias and high frequencies. To avoid this  nd was set at 3.2 M  cm i.e. by ~25% higher than  i. In Ref. B.Dezillie, Z.Li et al. IEEE NS-46 (1999) 221 the resistivity of heavily irradiated Si was found to be also higher than  i. In addition it was found that U merge >80 V also leads to (C p ) d <C 0 /d at high frequencies.

15. A.Chilingarov, A simple model for the equivalent circuit of heavily irradiated Si diodes in standard CV measurements 15 Below U merge both parameters describing the depleted region: normalised (C p ) d and Q d, were found to change relatively weakly with bias (see examples at this slide).

16. A.Chilingarov, A simple model for the equivalent circuit of heavily irradiated Si diodes in standard CV measurements 16 Of the remaining 3 unknown parameters  was chosen to describe the region with higher Q and tuned so that the ratio Q high /Q low was minimal to reach the maximum uniformity of Q (and thus  ) in the depleted region. Note the following features of the results. a) Both high,  H, and low,  L, resitivities remain ~constant below U merge and then grow with bias; b) The ratio  H /  L is always ~10 except for the highest bias, where  L becomes almost as high as  H ; c) Both  H and  L decrease with frequency; d) The  L is everywhere above  nd and approaches it at U~U merge for the highest frequency 30kHz. 4. Results and discussion

17. A.Chilingarov, A simple model for the equivalent circuit of heavily irradiated Si diodes in standard CV measurements 17 The same parameters, plus the fraction of the depleted region with lower resistivity  L =1- , as a function of frequency at fixed bias voltages. Note that while the  L decreases with frequency approaching  nd its fraction  L also decreases. For high bias it drops below 10%.

18. A.Chilingarov, A simple model for the equivalent circuit of heavily irradiated Si diodes in standard CV measurements 18 Several comments 1) Increase of both  L and  H with bias can be related to the decrease of the concentrations of electrons and holes in the depleted bulk with bias. 2) Decrease of the resitivities with frequency indicates the presence of so called AC losses when dielectric, which is a good insulator under static conditions, consumes considerable energy in the AC fields. This phenomenon is also described as the existence of the imaginary part in the dielectric constant:  =    -j   (see e.g. J.Kraus “Electromagnetics: with applications” McGraw-Hill Book Co - Singapore, 1999). 3) The ratio  H /  L is of order of 10 and to a first approximation does not depend on frequency. However the depleted region partition between low and high resistivity does change with frequency: at low frequencies it is occupied mostly by  L well above the  nd, and at high frequency mostly by  H. In total the resistivity of the depleted region is always significantly higher than that of the non-depleted part. 4) The model does not show anything special for 10 kHz frequency routinely used for the finding of the U d from the CV curves.

19. A.Chilingarov, A simple model for the equivalent circuit of heavily irradiated Si diodes in standard CV measurements 19 5. Conclusions 1.A simple equivalent circuit for an irradiated Si diode is proposed in the form of three modules with ideal C parallel to ideal R, one for the non-depleted region and two for the depleted one. 2.The model reproduces the experimental CV curves in the frequency range 20 Hz – 30 kHz. The resistivity of the non-depleted bulk is close to that of the intrinsic Si. The full depletion occurs at the bias where the IV curve has a kink and two “double junction” space charge regions merge. At higher bias the excess capacitance is attributed to the non-uniformity of the depleted region resistivity, which is always higher than  nd. 3.The clear frequency dependence of the depletion region resistivities and partitioning indicates the presence of the frequency dependent AC losses in the space charge region.

20. A.Chilingarov, A simple model for the equivalent circuit of heavily irradiated Si diodes in standard CV measurements 20 4.The model is as simple as possible and thus can only grasp the main features of the electrical characteristics of heavily irradiated Si diode. Theoretical input is required for more realistic model. 5.The model is the first attempt to combine coherently several well established facts about the IV and CV curves and the electric field profile. According to this model the diode becomes essentially depleted of the free carriers at much lower voltage than U d found from the CV measurements at ~10 kHz. The latter however is in a reasonable agreement with the methods based on the CCE measurements (see e.g. G.Lindström et al. NIM A466(2001)308). Thus the problem of a proper understanding and definition of the U d remains unresolved and further efforts are needed. Conclusions (continued)