Chapter 4
Resonator Measurement and
Characterization
In this chapter, the experimental methods used to characterize the Draper resonator
prototype devices and their measurement results are presented. It begins with a summary
of the measurement of packaged resonators manufactured by TFR Technologies, Inc.
(63140 Britta St. C-106, Bend OR 97701), which was performed to validate the testing
and analysis procedures before using them on new experimental devices.
4.1
S-parameter Measurements
S-parameters are usually the measurement variables of choice when characterizing
multiport devices at RF frequencies. At these high frequencies, not only do the lumped
element approximations of circuit elements break down, true opens and shorts become
infeasible to implement, which precludes direct impedance measurement. Through
calibration and mathematical manipulations, impedance data may be acquired from sparameters. Once this data was obtained for each device, it was matched to a circuit
model and equivalent parameters were extracted.
4.1.1 Materials and Setup
The TFR devices were FBAR-type resonators packaged in alumina with two external
ground-signal-ground (GSG) probing pads. The total packaged dimensions were about
31
Network
Analyzer
Ground
Signal
Resonator
Ground
Wafer
probes
Figure 4.1: Schematic view of a single Draper resonator test structure showing
placement of testing probe tips. The background gray represents the substrate
silicon. Nickel was used for the top electrode metal and is shown in black.
The bottom electrode metal, Molybdenum, is shown in white, and as shown
is only accessible from above through one set of probe pads.
3.8mm x 2.6mm x 0.8mm. The expected resonant frequency was about 2.1 GHz.
The Draper resonators were fabricated onto a wafer and measured directly from it. Many
devices with a variety of bar dimensions and tether lengths were placed onto one wafer.
Each device was connected to a pair of GSG probe pads for measurement (Figure 4.1).
All s-parameter measurements were performed in air at room temperature. Both the TFR
and Draper devices were designed to be tested in a two-port configuration. When
measuring the TFR devices, a Hewlett-Packard 8510C Network Analyzer, Cascade
Microtech probe station, 1000 m pitch GSG probe tips from GGB Industries, Inc., and a
6 GHz GGB calibration substrate were used. For the Draper devices, measurements were
taken with a Hewlett Packard 8753E Network Analyzer and 150 m pitch GSG probe
tips on the same probe station, with a smaller calibration substrate from the same
company.
32
4.1.2 Data Collection Procedure
Before measurements can be taken, the testing apparatus needs to be calibrated on open,
load, short, and through structures with known parameters. The calibration substrates
provide these structures and the necessary parameter values, which are loaded into the
network analyzer before calibration. During calibration, the network analyzer is swept
through a given frequency range on each test structure multiple times. The data for each
structure are averaged, and with the preloaded parameter values the analyzer calculates
calibration coefficients that remove the effects of the probe tips, transmission cables
linking the probe tips to the analyzer, and the analyzer measurement channels. Each
calibration is only good for a short period of time as environmental variables such as
temperature affect the test apparatus. For these setups it was found that calibrating once
per working day was sufficient for a given frequency range. For each set of devices, data
were initially taken in a wide frequency band to locate the resonance of interest, and then
a very high resolution measurement was taken about the resonant frequency. Sparameters were measured on several consecutive sweeps then averaged and saved to
disk as complex values, to be later imported into MATLAB for analysis.
4.2
Data Analysis
The goal of the data analysis was to produce parameters characterizing the resonators and
closely associated parasitics that would be useful when employing these resonators for a
given application, like a filter. The analysis proceeded as follows: first, the s-parameters
were transformed to z-parameters. Then, a discrete element two-port network model was
applied, and the z-parameters were transformed to discrete impedance values for these
elements. Finally, this data was fitted to a lumped element circuit model for each
impedance element, and circuit element parameters were extracted. From equivalent
circuit parameters, a resonator may be implemented easily into most filter designs using
established techniques.
33
4.2.1 Data Transformation
Four S-parameters were measured for each TFR device: S11, S12, S21, S22. Sij is measured
by inputting an electrical wave at port j and measuring the output at port i while setting
the inputs at all other ports to zero and terminating them with matched loads to cancel
reflections. Sij is the ratio of the complex amplitudes of the output wave at port i and the
input at port j [20] (Figure 4.2):
Vk  0, k  j
+
V1
-
2-port
Port 2
network
V2-
V2+
Figure 4.2: Incident and reflected waves of a two-port system
S12 Data
1
Magnitude (unitless)
V1
Vi 
V j
Port 1
S ij 
0.8
0.6
0.4
0.2
0
0.5
Phase (radians)
(4.1)
0
-0.5
-1
-1.5
1.5
2
2.5
Frequency (GHz)
Figure 4.3: Typical S-parameter data
34
3
In theory, any network of passive elements should be reciprocal with S12 = S21, and
inspection of the data confirmed that these values differed by less than 1% for all devices,
so they were averaged into a composite parameter S12 (Figure 4.3).
A two-port network can also be fully described by four z-parameters, defined as in Eqns.
4.2 and 4.3 and Figure 4.4:
(4.2)
V1  Z11I1  Z12 I 2
(4.3)
V2  Z21I1  Z22 I 2
Once again, for a passive network the cross parameters should be equal, so Z12 is set
equal to Z21. Z-parameters may be transformed from s-parameters using the following
equations, assuming a reciprocal network where S12 = S21 and Z12 = Z21 [20]:
(4.4)
Z11  Z 0
2
(1  S11 )(1 - S 22 )  S12
2
(1  S11 )(1 - S 22 ) - S12
(4.5)
Z12  Z 0
2S12
2
(1  S11 )(1 - S22 ) - S12
(4.6)
Z 22
2
(1  S11 )(1  S 22 )  S12
 Z0
2
(1  S11 )(1 - S 22 ) - S12
I1
V1
-
2-port
network
Port 2
+
Port 1
For all measurements, the line impedance Z0 was 50.
+
V2
-
Figure 4.4: Port voltage and current conventions for a two-port system
35
I2
4.2.2 Network Impedance Model
Zb
Za
Zc
Figure 4.5: Two-port impedance -model
A two-port “” network was used to model each device as a collection of discrete
impedances (Figure 4.5). Applying Kirchoff’s Laws to this model and combining with
Eqns. 4.2 and 4.3 results in the following expressions for each impedance:
(4.7)
Z11 
Z a Z b  Z c 
Za  Z b  Zc
(4.8)
Z12 
Za Zc
Za  Zb  Zc
(4.9)
Z 22 
Z c Z a  Z b 
Za  Z b  Zc
This model is over-simplified; its advantage is that the fitting procedure is greatly
facilitated since this model only has as many independent impedance values as the
number of independent two-port S-parameter values. However, its use can only be
justified if the resulting error is not too significant. The fitting with this model proved
acceptable for the TFR devices, and should be sufficiently accurate for the Draper
devices as well because test structures will be included on-wafer to allow direct
measurement of the major unmodeled parasitic elements. Now, the equations giving the
port characteristics in terms of the internal impedances can be inverted to yield explicit
expressions for each impedance, where |Z| ≡ Z11Z22-Z122:
(4.10) Z a 
(4.11) Z b 
Z
Z 22 - Z12
Z
Z12
36
(4.12) Z c 
Z
Z11 - Z12
4.2.3 Parameter Extraction
The transformations described in the previous section were performed on the measured
data for each device to give Za, Zb, and Zc as a function of frequency. Zb, which models
the resonator block, was compared to the equivalent impedance generated by the
Butterworth Van-Dyke model (Section 3.3) for a set of circuit element values, which
were varied by the MATLAB routine lsqcurvefit to get the best fit. The initial values for
the fit routine were calculated as follows: the maximum and minimum impedance
magnitudes (“|Zp|” and “|Zs|”) and the frequencies at which they occur (“fp” and “fs”) were
extracted from the data for Zb. From analysis of the BVD transfer function, the following
relations are obtained [19]:
(4.13) ws  2πf s 
1
LC
(4.14) w p  2πf p  ws 1 
C
C0
(4.15) Z s 
R
R
1  jws RC 0
(4.16) Z p 
1 - jw p RC 0
2
p
w RC
2
0

1
w RC 02
2
p
Using these equations, a very good approximation for the desired parameters R, L, C, and
C0 is obtained from |Zp|, |Zs|, fp, and fs. When fitting to the sharp resonance peak, an
accurate initial guess was necessary to obtain a good final fit, and these approximations
proved sufficient.
37
4.3
Measurement Results
4.3.1 TFR Devices
Measurement and analysis of ten TFR devices was carried out while awaiting fabrication
of the Draper resonator prototypes. Figure 4.6 plots Zb for one measured device alongside
the equivalent BVD impedance calculated from fitted parameters. The matching is
excellent, with an average per-point error less than 2% for frequencies near or below the
resonance. The error is about 15% for higher frequencies as the lines begin to deviate,
partly due to unmodeled parasitics whose effects grow more pronounced at higher
frequencies, and partly because the BVD model only contains one resonance while the
actual resonators exhibit higher-order modes.
The dominating behavior expected of the shunt impedances Za and Zc is of a capacitor,
from consideration of the physical makeup of these devices. They were fit to a series
RLC circuit instead of only one capacitor to allow for a slightly different slope in
magnitude, and to account for the real component of impedance present in the data.
Zb Magnitude and Phase
Impedance Phase (radians)
Impedance Magnitude (dB)
60
50
40
30
20
data
model
10
0
2
1
0
-1
-2
1.5
2
2.5
Frequency (GHz)
Figure 4.6: Measured and modeled Zb. Simulation parameters:
R = 2.76 , L = 91.6 nH, C = 0.061 pF, C0 = 1.54 pF
38
3
Impedance Magnitude (dB)
Za Magnitude and Phase
50
45
40
35
data
model
30
Impedance Phase (radians)
25
-1
-1.5
-2
-2.5
1.5
2
2.5
3
Frequency (GHz)
Zc Magnitude and Phase
Impedance Phase (radians)
Impedance Magnitude (dB)
50
45
40
35
data
model
30
25
1
0
-1
-2
-3
1.5
2
2.5
3
Frequency (GHz)
Figure 4.7: Measured and modeled impedance data for Za and Zc
Pictured in Figure 4.7 are the measured impedances Za and Zb for the same device as in
Figure 4.6, plotted with their fitted RLC impedances. There is some coupling to the
resonance evident, which is due to unmodeled factors. Nine of the ten identical devices
measured had similarly-shaped data: the tenth matched remarkably well with a single
39
uncoupled capacitance. The fitting routine proceeded very smoothly for all ten sets of
data, due to strong matching with the model.
4.3.2 Draper Resonator Measurements
Fabrication of the first Draper resonators was delayed due to difficulty in obtaining high
quality Aluminum Nitride (AlN) films. Since the goal of the first fabrication run was to
assess basic device functionality and frequency scaling behavior, two fabrication steps
were identified that could be postponed till later runs. Their removal had a significant
impact on the frequency response of these initial resonators, so the nature of the
imperfections will be briefly explained before presenting the results.
Impact of Incomplete Fabrication
The major fabrication steps are described by Figure 4.8. The steps skipped were d and e
in the figure. Their purpose is to reduce the extra capacitance that would otherwise be
placed in parallel with the resonator, which is a very significant amount as the probe pad
is on the order of 1000 times the area of the resonator bar. As it turns out, an additional
parallel capacitance of this magnitude mostly overwhelms any resonant peaks, even with
Q as high as 104, according to simulation (Figure 4.9). Exacerbating this difficulty, the Q
of these initial resonators is not expected to be over 1000 as the processing has yet to be
optimized. Thus, the resistance at resonance is competing with a very small impedance
shorting out the resonator due to this large capacitance. The shunting is more severe at
higher frequencies because the capacitor impedance is inversely proportional to
frequency, and for the majority of the devices on the wafer no resonance could be
detected at the expected frequency. Fortunately, a few bars made with extra-large
dimensions had resonant frequencies low enough to be detectable over the capacitor
shunting. Also, the increased bar cross-sectional area meant reduced impedance level
overall, which also helped these larger resonators to have detectable resonances.
40
(a)
(b)
(c)
(d)
(e)
Figure 4.8: Overview of Draper resonator test structure fabrication. The material
depicted in white is Molybdenum, gray is AlN, hatched is Nickel, and black
is a conductive weld. (a) Overhead view of resonator with ground probe
strips. (b) Sideview of initial stack as deposited on silicon substrate. This
stack pattern represents all three GSG strips. (c) Ni and AlN layers are
removed from the left probe pads of all three strips to allow contact with the
bottom electrode. (d) The tethers on the resonator strip only have their
electrodes selectively etched to reduce capacitance. (e) All three strips have
the top and bottom electrodes shorted together to eliminate stray capacitance.
41
0
S21 (dB)
-20
Cthru=2pF
Cthru=0
-40
-60
-80
-100
-120
780
790
800
810
820
830
(a)
-15
S21 (dB)
-16
-17
-18
-19
-20
188
190
192
194
196
Frequency (MHz)
198
200
(b)
Figure 4.9: Shunting effect of a large parallel capacitance “Cthru” on resonator response.
(a) 6 m bar length, f0 = 800 MHz, Q = 104
(b) 25 m bar length, f0 = 193 MHz, Q = 1000
Measurement and Analysis of Longitudinal Mode Resonances
Longitudinal-mode resonances were detected for devices of two bar lengths: 25 m and
30 m. Typical S21 data for both lengths are plotted in Figure 4.10. Note that on this
scale, the resonant peaks are hardly discernible. However it is clear that the wideband
characteristics are very similar, although quite a bit different than that of a single
capacitor. High resolution data was taken for four devices around their resonances, two of
each length. Figures 4.11 and 4.12 show Zb for all these devices after data transformation.
The average series resonant frequencies of 148.3 MHz and 125.3 MHz for each bar
length are about as expected when electrode loading is taken into account (which was
42
-5
S21 (dB)
-10
-15
-20
-25
-5
S21 (dB)
-10
-15
-20
-25
100
150
200
250
300
350
400
Frequency (MHz)
450
500
550
600
Figure 4.10: S21 magnitude data for 25 m (top) and 30 m (bottom) long devices.
Longitudinal length mode resonances are circled.
Magnitude ()
595
580
590
585
570
580
575
560
570
565
Phase (degrees)
550
-81
-83
-82
-84
-83
-85
-84
-86
-85
146
-87
147
148
149
Frequency (MHz)
148
148.5
149
Frequency (MHz)
Figure 4.11: Transformed impedance Zb for two 25 m long resonators.
43
149.5
Magnitude ()
900
960
880
940
860
920
900
840
880
Phase (degrees)
820
-71
-72
-73
-74
-75
121.5 122 122.5 123 123.5 124
Frequency (MHz)
-76.5
-77
-77.5
-78
-78.5
-79
-79.5
-80
121.5
122
122.5 123 123.5
Frequency (MHz)
124
Figure 4.12: Transformed impedance Zb for two 30 m long resonators.
ignored in the prior derivation for the resonant frequency). Inspection of the plots shows
a discrepancy between the data and the ideal BVD impedance, most evidently in the
phase behavior. A BVD resonator has an almost entirely imaginary impedance offresonance, so its impedance phase should be about 90 degrees plus some multiple of 180.
The deviation from one of these values indicates an additional unmodeled component in
the through impedance of these devices which adds a real component to the impedance,
possibly additional parasitic resistance in series with Zb. In the case of the 30 m bars,
there is clearly a more complex unmodeled behavior affecting the data as the phase
shows a significant slope as well. The net result of these deviations is that algorithmic
fitting of the BVD model to this data is nearly impossible, and could only be attempted
with a model modified to better fit this data. Since these prototypes were fabricated with
a large masking parasitic that hindered measurement, which should be removed from
future runs, it was decided premature to develop a more complex model at this point.
Instead, parameters were fitted by hand to the BVD model for the prototype devices. The
results and equivalent BVD parameters are shown in Figure 4.13. The additional
44
590
simulated
data
Magnitude ()
580
570
560
550
-80
simulated
data
Phase (degrees)
-82
-84
-86
-88
-90
145.5
146
146.5
147
147.5
148
148.5
149
149.5
150
Frequency (MHz)
Figure 4.13: Fitted and measured Zb impedance of a 25 m resonator. R = 11500 W, L =
6.7 mH, C = 0.1718 fF, C0 = 1.89 pF
capacitance basically replaces the C0 of the resonator, since parallel capacitances add and
the parasitic capacitance is about 1000 times the natural value of C0. The fitted value for
this capacitance is about what was expected from the area of the probe pads. The value of
Q estimated from these values (which is not affected by C0 except through the difficulty
in measurement and fitting presented by the shunting effect) is 542, once again
approximately what was expected from this first round of fabricated devices. The Q
values for future devices should be much improved as various aspects of the fabrication
process are optimized.
As mentioned above, the primary goal of the prototype fabrication round was to confirm
resonator functionality. By finding longitudinal resonances that varied as expected with
length, proper function of these devices was verified.
45
Probe Pad Parasitics
The transformed data for the probe pad parasitic impedances Za and Zc exhibited some
Impedance Magnitude ()
15000
10000
5000
0
100
150
200
250
300
350
400
Frequency (MHz)
450
500
550
600
450
500
550
600
(a)
1600
Impedance Magnitude ()
1400
1200
1000
800
600
400
100
150
200
250
300
350
400
Frequency (MHz)
(a)
Figure 4.14: Transformed impedances Za (a) and Zc (b).
46
interesting behavior. One representative set of data is shown in Figure 4.14. The first
important detail is that the port impedances are not symmetric, which makes sense since
the pads themselves are not symmetric: on one side the probe touched down on a single
layer of metal on the substrate, while on the other side the probe touched down on top of
the three-layer stack. Unfortunately, the pad orientation of the measurement ports was not
recorded, so it is not certain which probe pad corresponds to which port impedance.
Some general observations about the impedance data can still be made. First, the origin of
the impedance should primarily be due to fringe capacitance between the probe signal
pads and the two neighboring ground pads in each G-S-G probe pad set. One impedance
looks very much like a capacitor while the other looks like a much smaller capacitive
response with a good deal of complex behavior added in. A possible explanation for this
is that the fringe capacitance for one side is much higher than the other, so the stronger
capacitance dominates its modeled impedance response while the weaker capacitance on
the other side allows more subtle, complex behaviors to become noticeable. The presence
of the piezoelectric stack on only one set of probe pads could change the effective
dielectric constant of the fringe capacitance and explain the different capacitance values.
47
48