Extraction of Doping Density Distributions from C-V Curves
Hartmut F.-W. Sadrozinski
SCIPP, Univ. California Santa Cruz, Santa Cruz, CA 95064 USA
1. Connection between C, Neff, V
Start with Poisson equation
d 2V
e
(1)
= − N eff
2
ε
dx
and the equation for capacitance of a parallel capacitor
A
C =ε
(2)
x
. where x is the thickness of the depleted region. We measure C-V, i.e. the capacitance as
a function of voltage, which using eq. 2 can be turned into V(x). According to eq. 1,
differentiating this function twice results in the depletion voltage.
Successive approximation works quite well: divide the capacitor of thickness d into slabs
of equal thickness t, which each have a capacitance
d
(3)
Ct = Co .
t
The reciprocal of these constant capacitances can be added up to make up the capacitance
measured and the corresponding voltages ΔtV(x) across the capacitors are adjusted to
match the observed C-V curve. Assuming that the doping density does not change across
the distance t, the voltage ΔtV(x) can be expressed in term of the local doping density
x
dV ( x)
e
= − ∫ N eff ( x)dx
(4)
ε0
dx
x +t x
e
e
N eff ( x)(( x + t ) 2 − x 2 )) = − N eff ( x) * x * t
ε x 0
ε
2ε
Need good fits for C vs. V curves for this, and a student is working on this.
Δ tV ( x) = −
e
∫ ∫N
eff
( x)dx'dx = −
(5)
2. Simulations using Neff distributions
One approach is to assume a trial Neff distribution and adjust the parameters such that
the simulated C-V curves fit the experimental ones.
x
dV ( x)
e
e
= − ∫ N eff ( x)dx = − N eff (1) * ( x − xx0) + E0
ε0
ε
dx
E ( 0) = E 0
e
Ec = − N eff (1) * ( xx1 − xx0) + E0
ε
e
E0 = + N eff (1) * ( xx1 − xx0) + Ec
ε
V ( x) = −
e
e
N eff (1) * [( x) 2 − ( xx0) 2 ] − N eff (1) * xx0 * [ x − xx0] + E0 * [ x − xx0] − V0
2ε
ε
For un-irradiated detectors or detectors where the doping densities are uniform or at least
monotonic functions of the depth, this is a straight forward procedure: one slowly
increases the depletion depth with a step size of about one micron, and calculates the
corresponding fields and voltages depending on the trial Neff distributions.
For trial distributions, we follow the model from Fig. 3.b of our Carmel paper (M.
Bruzzi et al.) shown in Table I and Fig. 1, where three regions with constant Neff’s are
defined; one in front with Neff(front), a constant field region in the center (Neff=0) and one
with the opposite sign in the back, Neff(back). These distributions were derived from TCT
data of b-type MCz detectors irradiated to higher fluences. The simulation varies the
boundaries of these three regions and the value of the Neff’s.
Table I: Trial Neff Distributions
Protons
Neutrons
x [um]
0
150
190
299
E [V/cm]
5000
1500
1500
33000
Neff [cm-3]
1.51E+12
1.51E+12
-1.88E+13
-1.88E+13
x [um]
0
200
240
299
E [V/cm]
24000
2000
2000
10000
Neff [cm-3]
7.11E+12
7.11E+12
-8.76E+12
-8.76E+12
+
Electric Field [V/cm]
4x10
4
+
p
n
15
3x10
4
2x10
4
1x10
4
-2
24 GeV protons 2.1x10 cm
14
-2
Reactor neutrons 5x10 cm (b)
0
0
50
100
150 200
x [μm]
250
300
Fig. 1 Electric field profile as determined by TCT in n-type MCz silicon after irradiation with
24 GeV protons up to 2x1015cm-2 and with 5x1014cm-2 1MeV neutrons.
For irradiated detectors, which might exhibit a double junction (DJ), one can try
to start depletion from both sides. The question is then which step size to choose on the
two sides. The only choice of step size which let to a reasonable agreement between
simulation and data was that the step size at the back contact Δxb was reduced in the ratio
of the respective Neff’s with respect to the front side Δxf:
Δxb = N eff ( front ) / N eff (back ) * Δx f
Since in all cases considered the back side Neff was much larger, the results were
consistent with starting the depletion only from the front. In many cases, starting the
depletion from the front leads to better fits of the data, as shown in Fig. 2. The extracted
distributions of the doping densities are qualitatively quite similar, and in the following
we will consider the simulations starting from the front only.
2
1/C n MCz 187-4 (protons)
1.E-03
1/C^2
1.E-04
1.E-05
n MCz 187 -4 n(protons) -19C 400Hz
sim Front
1.E-06
sim Scaled
1.E-07
0
100
200
300
400
500
Bias Voltage
Fig. 2 Logarithmic plot of the 1/C2 – V distribution of a n-type MCz silicon detector after irradiation with
24 GeV protons up to 1.4x1015cm-2 with simulations starting either at the front or both on the ftont and the
back (“scaled”).
3. Devices
We used the C-V curves of the six SMART SSD shown in Table II. The un-irradiated ptype FZ SSD 68-1 was used to verify that the procedure resulted in the expected
distributions based on a uniform Neff.
Table II: SMART SSD used
SSD
Wafer
187-2
187-2
253-4
n MCz
300 μm
p MCz
300 μm
p MCz
300 μm
n FZ
300 μm
p FZ
200 μm
66-8
1256-3
14-5
Fluence
[neq/cm-2]
1.4*1014
protons
~1.4*1014
protons
~4*1014
neutrons
~4*1014
neutrons
~4*1014
neutrons
Temperature
[oC]
-10
Frequency
[Hz]
400
CCE
-10
400
Yes
Yes
Yes
-20
250
Yes
-20
250
RT
10,000
68-1
p FZ
200 μm
Pre-rad
RT
10,000
4. Results
4.1 C-V Normalisation:
Fig. 3 shows that the normalization of the capacitance is well understood: we get
agreement between the different SSD at depletion. The two p MCz detectors with roughly
the same fluence in neutrons and protons agree very well in the limited voltage range
where they can be compared: this opens up the possibility to make predictions for
depletion profile even for detectors which show early breakdown, once we gain more
experience with the C-V curves. The p-type FZ detectors have 200 μm thickness, and it
will be interesting to compare the LTLF data of 14-5 with the pre-rad data of 68.1.
2
1/C p&n -type
1.8E-04
1.6E-04
1.2E-04
2
-2
1/C [pF ]
1.4E-04
1.0E-04
253-4- p MCz (10deg) 400Hz protons
66-8 p Mcz (-20C)
250Hz neutrons
187-4 n MCz (-10C)
400 Hz protons
1256 n Fz (-20C) 250
Hz neutrons
14-5 p FZ (RT) 10 kHz
8.0E-05
6.0E-05
4.0E-05
2.0E-05
68-1 p FZ (RT) 10 kHz
0.0E+00
0
100
200
300
400
Voltage
500
600
700
Fig. 3 Measured 1/C2 – V curves at the optimal frequencies for the temperatures selected.
4.2 Simulation Results
Fig. 4 shows the results on the simulations of the two MCz detectors which could be
fully depleted. The agreement between data and simulations are quite good, and the fits
could be improved even more by using a double-junction model with two non-uniform
Neffs. As control, the simulation of the un-irradiated p-type FZ detector 68-1results in a
straight-line 1/C2 behavior and a constant doping density of 3.02*1012 cm-3.
2
1/C p MCz W66-8 (neutrons) & W253-4 (protons)
2.0E-04
1/C^2
1.5E-04
253-4- p MCz (10deg) 400Hz protons
1.0E-04
66-8 p Mcz (-20C)
250Hz neutrons
5.0E-05
sim
0.0E+00
0
100
200
300
400
500
600
700
800
900
1000
Voltage
2
1/C n MCz 187-4 (protons)
2.0E-04
1/C^2
1.5E-04
n MCz 187 -4
n(protons) -19C
400Hz
sim
1.0E-04
5.0E-05
0.0E+00
0
100
200
300
400
500
Voltage
600
700
800
900
Fig. 4 Measured and simulated 1/C2 curves. The fits are quite good, but could be improved using a doublejunction model with two non-uniform Neffs,
The fits yield distributions of the effective donor concentration in the assumed double
junction model. They are shown in Table III and Fig. 5. A comparison of the electric field
profiles for the two MCz detectors is shown in Fig. 6. As in Fig. 1, a double junction is
clearly seen with a relative low-filed region in the detector center, indicating a very high
resistivity bulk. The relative strength of the front to back junctions is clearly different in
the two samples, and we have to await the results on n MCz irradiated with neutrons to
decide if this is a property of the material or due to the irradiating particle species as
suggested by the extraction of the filed by TCT (Fig. 1). The n-type MCz detector shows
a much larger extend and much lower strength of the electrical field in the intrinsic bulk.
Table III: Simulated Neff Distributions
187-2 n MCz (protons)
66-8 p MCz (neutrons)
x [um]
0
40
183
300
E [V/cm]
18,000
3,000
3,000
46,000
Neff [cm-3]
2.4E+13
2.4E+13
-2.7E+13
-2.7E+13
x [um]
0
119
257
300
E [V/cm]
44,000
11,500
11,500
23,000
Neff [cm-3]
1.8E+13
1.8E+13
-1.3E+14
-1.3E+14
Neff vs. depth
4.0E+13
2.0E+13
Neff [cm^-3]
0.0E+00
-2.0E+13
0
50
100
150
200
250
300
-4.0E+13
-6.0E+13
-8.0E+13
-1.0E+14
n MCz 187-4 (protons)
p MCz W068-8 (neutrons)
-1.2E+14
-1.4E+14
x [um]
Fig. 5 Extracted doping density profile Neff vs. detector depth for both n- and p-type MCz detectors.
E Field
5.E+04
n MCz W187-4 (protons)
p MCz w68-8 (neutrons)
E [V/cm]
4.E+04
3.E+04
2.E+04
1.E+04
0.E+00
0
50
100
150
200
250
300
Depleted Depth x [um]
Fig. 6 Extracted electric field distribution vs. detector depth for both n- and p-type MCz detectors. The
relative size of the fields at front (x ≈ 0) and back. (x ≈ 300) is different fro n-type and p-type.
4.3 Charge Collection
From the fits the depletion characteristics of the detectors can de extracted. The
depleted depth is proportional to the reciprocal capacitance. Fig. 7 shows the depth of the
depleted region as a function of voltage for the investigated p-type n-type, and describes
the same data as Fig. 3. The most striking feature is that the n-type MCz has very
different depletion characteristics from all other wafer types. We will soon have data
from all detector types, irradiated with both neutrons and protons, which will allow us to
determine the dependence of the depletion characteristics on wafer and particle type.
Depleted Region
Depleted Region x [um]
300
200
n MCz W187-4 (protons)
P MCz W66-8 (n+p)
100
p FZ W68-1 pre-rad
n FZ W1256-2 (neutrons)
p FZ 14-2 (neutron)
0
0
100
200
300
400
Bias Voltage [V]
500
600
700
Fig. 7 Depletion characteristics extracted from the fits. The depleted region is proportional to the reciprocal
capacitance.
A critical test of the simulation procedure is the comparison of the C-V data with
charge collection (CC-V). Good agreement of CC-V with the expectation (Fig. 7) would
allow substituting relatively easy and fast electrical measurements like C-V for the much
more involved charge collection procedure to characterize the performance of irradiated
detectors. Differences between C-V and CC-V might be interpreted as an indication for
trapping.
n MCz W187-2 Median & norm 1/C
Karlsruhe ( 26 MeV protons) 1.4*10
4
14
2
neq/cm
Collected charge [fC]
3
2
Sim 1/C -20C, 250 Hz
1
CCE 5 mon @ 60C
Sim Front
0
0
200
400
600
800
Bias Voltage [V]
P MCz low 66-8 Median & Norm. 1/C
Louvain neutrons 4*10
14
2
neq/cm
4
Median [fC]
3
2
1
CCE 10 min @60C
-20C, 250 Hz
0
0
200
400
600
800
Bias Voltage [V]
Fig. 8 Measured median charge collected vs. bias voltage (CC-V) and the prediction from the fits to the CV data. The normalization between the different curves is accurate to about 10%. The difference between
the CC-V and reciprocal C-V curves might be explained by trapping in the neutral bulk.