More on the Γ-X band structures

We plot the band structures in Γ-X direction more detailed and with wider fre-quency range as Fig 2.24 (unfiltered) and Fig 2.25 (color marked). The plate modes are folded but do not open a band gap for the surface modulation has little effect to them.

And the invisible sound line that influence the plate modes also folded on the boundary of the irreducible Brillouin zone.

For the Love modes, they are hardly influenced by the in-plane plate modes but coupled with SH-type plate modes when the r/a is larger. The Rayleigh mode is found coupled with in-plane plate modes. Although it is predicted that the Love waves become leaky when they exceed the sound lines but they can still be found far above the sound lines. It indicates the loss is not much.

A complex band structure is calculated for the r/a = 0.2 case, and plotted in Fig 2.26. To know the attenuation of the evanescent waves in the first Bragg band gap, Bloch’s theorem with complex wavenumbers kx

= π/a – iα is applied to calculate the

dispersion relation with varying α. It is found that the attenuation of the evanescent wave at the band gap center frequency is 4.34 dB/a. Knowing the attenuation of Love waves in the PC gives a reference for designing of devices.

Table 2.1 Material properties of α-quartz w.r.t. crystal axes

Density (kg/m³) 2650

Elastic stiffness tensor under a constant electric field c^E in Voigt notation (Pa)

Piezoelectric coupling tensor e in Voigt notation (C/m²)

Permittivity tensor with a con-stant strain εε^S in matrix form

Table 2.2 Material properties of ST-cut quartz w.r.t. specimen axes

Euler angles

(

^{Φ Θ Ψ, ,}

) (

0 ,132 45', 0° ° °

)

Elastic stiffness tensor under a constant electric field c^E in Voigt notation (Pa)

2.749 8.603 10 1.052 10 2.749 9.663 10 4.806 10 1.344 10

8.603 10 4.806 10 1.307 10 1.838 10 1.052 10 1.344 10 1.838 1

Piezoelectric coupling tensor e in Voigt notation (C/m²)

Permittivity tensor with a con-stant strain εε^S in matrix form

0 4.0198 10 9.0869 10 0 9.0869 10 4.0055 10

−

Table 2.3 Material Properties of 90°-rotated ST-cut quartz w.r.t. specimen axes

Euler angles

(

^{Φ Θ Ψ, ,}

) (

0 ,132 45', 90° ° °

)

Elastic stiffness tensor under a constant electric field c^E in Voigt notation (Pa)

Piezoelectric coupling tensor e in Voigt notation (C/m²)

Permittivity tensor with a con-stant strain εε^S in matrix form (F/m)

11 13

11

13 11

4.0198 10 0 9.0869 10

0 3.9215 10 0

9.0869 10 0 4.0055 10

− −

Table 2.4 Material properties of the PECVD SiO

2 and aluminum.

PECVD SiO2 Aluminum

Deposition source/method Si(OC H )2 5 4(TEOS), O2/PECVD 99.999% aluminum/evaporation Optical index of refractive 1.48~1.58 (See Fig. 4.34) —

Electric conductivity (S/m) — 3.55×10⁷

Density (kg/m³) 2200 2750

Elastic stiffness (λ, µ) = (20 GPa, 26 GPa)^a (E, ν) = (70 GPa, 0.35)

Shear bulk wave speed 3438 m/s 3099 m/s

a The stiffness constants of SiO2 are obtained from experiments while others are from the literatures

x

Fig. 2.1 The specimen axes.

X

Fig. 2.2 Euler angles of the three rotations of the coordinate system, where (X, Y, Z) stand for the crystal axes while (x, y, z) stand for the specimen axes.

Z

X

Y

x y

x y

z

z

90°-rotated ST-cut

ST-cut

Fig. 2.3 ST-cut and 90°-z-rotated ST cut.

a

1

x

Fig 2.4 The bases of the direct and reciprocal lattices and the unit cells of a 2-D lattice.

a1 x

1^st Brillouin zone Wigner-Seitz cell

direct lattice reciprocal lattice

B = 2 π A

^-1

Fig 2.5 The Wigner-Seitz cell and the 1^st Brillouin zone of a 2-D square lattice.

0 2 4 6 8 0

2 4 6 8

kxh ΩhcSH,90ST

0 2 4 6 8

0 2 4 6 8

kxh ΩhcSH,90ST

Fig 2.6 Love wave dispersion (thick red lines) obtained by a thickness-fixed (10h) plate model (left) and a thickness-varying (1.6λ) model (right). The blue dashed line and the purple dot-dashed line indicate the SH bulk wave of the substrate (90ST quartz) and the S wave velocity of the guiding layer (CVD SiO2), respectively.

x y z

k u

uSH

ux

uy

uz

Fig 2.7 Determination of the SH component.

Fig 2.8 Slowness of the bulk waves in 90°-rotated ST-cut quartz along its xy-plane

unit cellthe a. Rayleigh

mode b. 0^th Love

mode c. a plate

mode d. 1^st Love mode

Fig 2.9 Love modes selecting

1. All modes 2. Blue: selected SAW modes

3. Red: selected Love modes

Fig 2.10 (Quasi-) Love waves’ dispersions along 0° (upper), 45° (mid) and 90°

(lower) off the x-axis.

Fig 2.11 Electromechanical coupling coefficient

Fig 2.12 Penetration depth of the Love waves

Fig 2.13 DOE of the Love waves

0 2 4 6 8 0

2 4 6 8

kxh whêcSH,90ST

Fig 2.14 Love wave dispersion of the SiO2/IDT/quartz structure, with SiO2 thickness

= 0.9 µm, IDT aluminum thickness = 100nm, IDT pitch = 3 µm.

unit cell a. b.

electric field electric field

displacement displacement

Fig 2.15 Two resonant modes on the lower (a) and the upper (b) edge of the stop band.

Fig 2.16 The admittance of the IDT consists of infinite pairs around the two modes.

a. K² = 0.08%

Fig 2.17 The relation of the excited frequency vs. IDT pitch of the SiO2/IDT/quartz, with SiO2 thickness = 0.9 µm, IDT aluminum thickness = 100nm, IDT pitch = 3 µm.

Phononic SiO

₂

thin film

90°-rotated ST-cut quartz

x z y

Fig 2.18 A schematic drawing of the phononic SiO2/quartz layered structure

r a

x z y

Fig 2.19 The unit cell of the phononic SiO2/quartz layered structure

Fig 2.20 The sound lines

0

Fig 2.21 (unfiltered) calculated band structures of phononic crystals with (a) r/a = 0.1, (b) r/a = 0.2, (c) r/a = 0.4.

0 1 2 3 4

w hê c

SH,90ST

0 1 2 3 4

w hê c

SH,90ST

0 1 2 3 4

w hê c

SH,90ST

Fig 2.22 the color marked band structures of phononic crystals with (a) r/a = 0.1, (b) r/a

= 0.2, (c) r/a = 0.4.

k ω

k ω

b = 2π/a 1^st B.Z.

1^st B.Z. b = 2π/a

Fig 2.23 (a) Dispersion of a homogeneous medium. (b) Dispersion of a periodic struc-ture with vanishingly small modulation (empty lattice). (c) Dispersion of a periodic structure, mode coupling happens when two modes are close and not orthogonal.

(a)

(b)

(c)

0 500 000 1.0¥ 10⁶ 1.5¥ 10⁶ 2.0¥ 10⁶

Fig 2.24 Unfiltered dispersion data of PCs with r/a = 0.1 (left) and 0.2 (right).

0

Fig 2.25 Colorized dispersion data of PCs with r/a = 0.1 (left) and 0.2 (right).

0 P 0

1 2 3 4

ka WaocSH,90ST

0.2 0.4 0.6 0.8 Im(ka, Fig 2.26 The complex band structure of the PC with r/a = 0.2.

–

Chapter 3

Design of a One-port Resonator

In this chapter we investigate the reflection performance of the proposed phononic structure and give an optimized geometry design of the one-port Love wave resonator.

Also the design of the transmission experiment verifying the band gap existence is giv-en in the text.

3.1 Design parameters of a one-port resonator

The plane structure of the 1-port resonator is shown in Fig 3.1. The two PC arrays act as the reflecting gratings and a Love wave resonant cavity forms between them. The IDT is then placed in the cavity to excite the desired resonant mode.

The number of IDT pairs (the number of wavelengths in the cavity) is usually a few dozens. Increasing the number may help on the Q factor due to the sinω/ω fre-quency response of the IDT. If we consider the IDT as a phase array of line sources with 180° phase shift, large numbers of the IDT pairs may also reduce bulk wave loss since it has high acoustic directivity along the surface. The number also affects the electrical impedance of the device.

The aperture of the IDT can adjust the electrical impedance magnitude as well. But

it should be much larger than the wavelength to reduce the diffraction. In our resonator the aperture is set to be 100 times the wavelength. It is not but can be optimized for im-pedance matching with adjacent components in further applications, if needed.

The number of the PC columns is adequate for about 5~10. It shows good reflec-tion even with only 3 columns of PCs, which is shown in the next secreflec-tion. Too much of the PCs increase the total size of the device only.

And the delay distance between the PCs and the IDT is a critical dimension in the resonator. There will be strong performance degeneration if there is a little bias on the delay distance. It can be optimized by the numerical method described in the upcoming sections.

3.2 Reflection on the PC border

We conducted a FEM simulation to investigate the reflection, transmission or scat-tering phenomenon on the PC border and find an effective reflective plane. The geome-try is shown in Fig 3.2. We had 3 models with 3, 5, or 10 columns of PCs. We took only one row with periodic boundary conditions along the y-direction to reduce the calcula-tion amount. The electric potential was applied on the interface of SiO2 and quartz just as two pairs of IDT, as the green and the purple areas in Fig 3.2. They are away from the PC for a certain delay distance. Here the mechanical loading effect of the IDT is not considered so the pitch is determined by the Love wave velocity and the center fre-quency of the band gap. Then an AC power at the gap center frefre-quency was driven, with sweeping the delay distance through a range. Perfectly matched layer (PML) was placed around the structure to reduce the unwanted reflected waves on the boundaries.

We first plot the displacement amplitude of the most left section on the interface of the model as Fig 3.3. Since the IDT generates both left and right propagating waves,

and part of the right propagating wave is reflected by the PC and interfered with the left propagating wave. From the interfered displacement amplitude the reflection coefficient could be obtain. Assume that there is an “effective reflective plane” around the border of the PC, with distance d away from the source. Part of the incident wave is reflected at this plane with reflection coefficient α, as shown in Fig 3.4. Then the interfered wave on the left is therefore

e^+ikx

(

1+α ⋅e^ik2d

)

^{. (3.1)}

The absolute value of the numbers in the parentheses indicates the interference amplitude, and is plotted with different α in Fig 3.5. It is dependent on d, where the d just differs from our “delay distance” for a constant. The amplitude has a maximum 1+α (the most constructive interference) and a minimum 1–α, the most destructive in-terference. So we can obtain the reflection coefficients of 3, 5, 10 rows of PCs from the maximums and minimums in Fig 3.3 by the equation

α = Max.− min.

Max.+ min.. (3.2)

And they are 96.5%, 97.5% and 96.6%, respectively. It is surprising that the PC with only 3 rows has good reflection as well. We notice that in Fig 3.3 the second peak is slightly higher than the first one. This may due to that the bulk waves are also generated by the source and detected on the interface. The bulk waves have amplitude inversely proportional to the distance from the source. Since the estimated reflection coefficient is affected by the presence of bulk waves, the true reflection coefficient of PC for Love waves should be larger than the values mentioned above. We also plotted the interfered displacement amplitude at positions 0.5 and 1 times the wavelength beneath the inter-face to detect the skimming bulk wave, as shown in Fig 3.6. We can see the amplitude is much smaller and with lower reflection coefficients. The wavenumber of the

skim-ming bulk wave is not matched with the IDT pitch, so it shows much smaller amplitude.

And the skimming bulk wave is not reflected by the surface PC thus lead to a lower re-flection coefficient. A series of plots of the simulation are shown in Fig 3.7 to Fig 3.9, where the color indicates the displacement amplitude in both linear and log scales. First the constructive and destructive interferences are obvious. And focusing on the log scale plots, for example Fig 3.7 (c), we see a yellow beam toward the left with shallow skim-ming and this is obviously the SH-type bulk wave. And in Fig 3.7 (d) the yellow part looks like a sector, it is the color of both the Love wave and the bulk wave. Although it looks like leakiness but we know that the Love wave is NOT leaky as its phase velocity is slower than that of the SH bulk wave. And if we look at Fig 3.9 (d), we see the skim-ming bulk wave can propagate beneath the PC. The bulk wave can be reduced by in-creasing the number of IDT pairs, since their wavelength are not matched and the Love wave is non-leaky.

Back to the reflection/transmission problem. The incident wave amplitude can be obtained as (Max.+min.)/2, thus the transmission coefficient can be found by the ratio of the amplitude on the right of PC to the incident amplitude. And the scattered amount into the bulk is obtained by 1 – reflection coefficient – transmission coefficient.

3.3 Optimization for the delay distance

In the previous section, a delay distance can be found such that a most constructive interference is achieved. However with the mechanical loading effect of the IDT, the interference amplitude becomes complex as Fig 3.11. This may be due to the internal reflection of the metal gratings. Instead of finding the delay distance for the strongest interference, we directly made a resonator model as Fig 3.12, and optimize the delay distance such that a greatest resonance is in the cavity. In this model we have 10.5 pairs

of IDT in the resonator and 10 columns of PCs as the reflector, where the IDT here is with full consideration of the mechanical loading effect, as we did in Sec.2.4. Similarly we take only one row with periodic conditions. We also applied the symmetric condi-tion to reduce the degree of freedom in FEM calculacondi-tion. We plot the resonant ampli-tude vs. the delay distance in Fig 3.13. The optimized delay distance is where the reso-nant amplitude achieves a maximum. We found maximums show repeatedly with spac-ing about half the wavelength, but not exactly. Also the response rapidly deteriorates as the delay distance slightly deviate from the optimized value.

3.4 Resonator evaluation

The frequency response of the optimized resonator is calculated, and the electrical admittance Y = G + iB is plotted in Fig 3.14. The Q factor of the resonator is about 1400 estimated by the fraction of the resonant frequency over the half-conductance band-width. And the resonant displacement amplitude is plotted in Fig 3.15. To find out where the energy leaks, we plotted the mechanical power flux P_i =T_iju!_jas Fig 3.16 and the out flux component on the model boundary as Fig 3.17. In Fig 3.17 we see green color on the right boundary of the guiding layer. It means no energy loss from here for the PC reflects almost all Love wave. But a little lower we see a red region. It is the skimming bulk wave traveling under the PC and the energy leaves the resonator, it can be reduced with large number of IDT pairs. On the bottom boundary we see red and blue colors. The energy loss here may due to the finite-depth model with PML, which is different from the real situation.

In the simulation the material damping and the ohmic loss have not been included which may make the Q factor decrease more or less.

3.5 Experiment setup

There are two sub-experiments in the thesis: The transmission-of-PC experiment that verifies the existence of the partial band gap, and the GHz range resonator perfor-mance experiment. The setups are described below.

Fabrication ability should be considered when designing the dimensions. We chose the resonant frequency of the resonator (i.e., center frequency of the band gap) to be 1.25 GHz, thus the geometry dimensions are determined from the dimensionless band structure (Fig 2.22) as follows: lattice constant a = 1.5 µm; PC hole radius r = 300 nm;

CVD SiO2 film thickness h = 900 nm. It seems comfortable with the NEMS process containing electron beam lithography.

In the transmission-of-PC experiment we use slanted finger IDTs (SFITs) to gen-erate and receive broad band Love waves by electrical signals, and the PC array is placed between the 2 SFITs thus the transmission of Love waves through the PC can be measured from the insertion loss of the two ports. Since the calculated partial band gap is located from 1.09 ~ 1.40 GHz, the excited frequency of the SFIT is set from 0.95 ~ 1.56 GHz, which the bandwidth is twice the gap width. From the frequency-pitch rela-tion plot (Fig 2.17) we found the SFIT pitch should be from 1.15 ~ 2.12 µm. The corre-sponding minimum line width is 570 nm. It is reported that asymmetric arrangement in the number of SFIT pairs helps reject the side lobe responses, so the numbers of SFIT pairs are set to be 32/40 pairs or 24/30 pairs. The aperture is set to be 100 times the maximum wavelength (423 µm) to reduce the diffraction effect. The maximum tilt an-gle (40 pairs SFIT) is 5°. And the propagation distance is designed long enough such that the triple transit echo is not mixed with the direct signal in time domain in order to apply the time gating technique when analyzing the measured signals. The SFIT

geom-etry parameters are listed in Table 3.1. Transmission of PCs with 10, 15, 10 columns were measured. Also SFIT signal without PCs were measured for reference.

As for the resonator we have 10.5, 30.5, 60.5 pairs of IDT as Table 3.2, combined with 10, 15, 20 columns of PC reflector, totally 9 versions.

Table 3.1 SFIT parameters.

#1 #2

# of Pairs 32.5/40.5 24.5/30.5

Pitch 1.15 ~ 2.12 µm 1.15 ~ 2.12 µm

Aperture 423 µm 423 µm

Maximum tilt angle 4°/5° 3°/4°

Propagation length^a 332 µm 251 µm Aluminum thickness 100 nm 100 nm

a The propagation length is measured from center to center of the SFITs.

Table 3.2 IDTs for resonators.

# of Pairs 10.5, 30.5, 60.5

Pitch 1.50 µm

Aperture 300 µm

Aluminum thickness 100 nm

RF

Delay Distance

Ape rtur e

Fig 3.1 A schematic drawing of the layout of the one-port resonator

Fig 3.2 Geometry of the FEM simulation investigating PC reflection

Fig 3.3 interfered displacement amplitude

effective reflective plane source

d x

Fig 3.4 A schematic drawing shows the reflection at the effective plane

Fig 3.5 Interfered amplitude vs. d/wavelength with different α.

Fig 3.6 interfered displacement amplitude at different depth

(a) (b)

(c) (d)

Fig 3.7 Reflection simulation on 3 rows of PC. (a), (c): destructive interference. (b), (d):

constructive interference. (a), (b): colors indicate the y-displacement in linear scale. (c), (d): colors indicate the y-displacement in log scale.

(a) (b)

(c) (d)

Fig 3.8 Reflection simulation on 5 rows of PC. (a), (c): destructive interference. (b), (d):

constructive interference. (a), (b): colors indicate the y-displacement in linear scale. (c), (d): colors indicate the y-displacement in log scale.

(a) (b)

(c) (d)

Fig 3.9 Reflection simulation on 5 rows of PC. (a), (c): destructive interference. (b), (d):

constructive interference. (a), (b): colors indicate the y-displacement in linear scale. (c), (d): colors indicate the y-displacement in log scale.

Fig 3.10 Geometry of the FEM simulation

Fig 3.11 interfered displacement amplitude (with mechanical loading effect of IDT)

Fig 3.12 Geometry of the FEM simulation (resonator)

Fig 3.13 Resonant displacement amplitude (with mechanical loading effect of IDT)

Fig 3.14 Simulated admittance of the resonator

Fig 3.15 Transverse displacement amplitude at resonance

Fig 3.16 Mechanical power flux at resonance

Fig 3.17 Mechanical power flux on the model boundary at resonance

Port 1 Port 2

Fig 3.18 Experiment setup of PC transmission experiment.

Chapter 4 Fabrications

In this chapter, the fabrication process is demonstrated in detail. Fig. 4.1 shows a schematic picture and Fig. 4.2 shows a section view of the device – a phononic SiO2/IDT/quartz structure. Electron beam (EB) lithography was utilized to generate sub-micrometer patterns for IDTs and PCs. The patterns of aluminum wire and the con-tact pad window on SiO2 film were defined by photolithography. All metal depositions were done through the evaporation method for good smoothness and ease for lift-off.

SiO2 film was deposited by a plasma-enhanced chemical vapor deposition (PECVD) method for the high deposition rate and the low residual stress. And the two SiO2 etch-ing step were accomplished through the reactive ion etchetch-ing (RIE) process instead of wet etching method to reach vertical side walls. Full process charts are shown from Fig.

4.4 to Fig. 4.10.

4.1 Brief of EB lithography

In the fabrication process, three times of EB lithography were carried out. Some basic concepts of EB lithography and the methodology the author implemented will be briefed in this section.

4.1.1 Why EB lithography

The resolution of lithography is limited by the wavelength of the exposure source and the resist thickness. Since the EB have a much higher momentum than the ultra-violet (UV) light used in photolithography, the wavelength of EB is much smaller de-duced from the famous de Broglie relation

λ

=

h p

/ , where λ, h, and p represent the wavelength, Plank’s constant and the momentum, respectively. Therefore EB lithogra-phy can achieve much better resolution than photolithogralithogra-phy does. In practical, pat-terns with size 10 nm can be generated by EB lithography. [55]

4.1.2 Anti-charging

However, some difficulties were to be dealt with. The first was the electron charging effect. Since the substrate of our device was an insulating material (quartz), the emitted electrons from the electron gun accumulated near the top surface of the sub-strate, formed a negative charge, and made the exposure EB deflected, finally resulted in pattern distortion and field stitching error. A schematic picture of the charging effect is shown in Fig. 4.11. A commercial anti-charging chemical ESpacer 300Z was coated on the top of EB resist to solve this problem. With a spin rate of 2000 rpm, one can ob-tain a thin conductive layer of thickness about 200 Å which does not affect the exposure mechanism. Since the chemical is very expansive ($3000 for 100 ml), it was taken by a micropipette. After exposure, ESpacer was removed by water rinsing before develop-ment. Fig. 4.15 shows the microscope picture of the developed patterns without and with coating of ESpacer.

4.1.3 Dose determination and the proximity effect

The second was determination of the appropriate dose value and correction of the

proximity effect. The EB lithography system used was JEOL JBX-5000LS – a Gaussian

proximity effect. The EB lithography system used was JEOL JBX-5000LS – a Gaussian

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