Supplementary Information Miniaturized electromechanical devices with multi-vibration modes achieved by orderly stacked structure with piezoelectric strain units

1 Electronic Materials Research Laboratory, Key Lab of Education Ministry and State Key Laboratory for Mechanical Behavior of Materials, School of Electronic Science and Engineering, Xi’an Jiaotong University, Xi’an 710049, China. 2 OPPO Guangdong Mobile Communication Co., LTD. Shenzhen 518051, China 3 School of Materials Science and Engineering, Peking University, Beijing 100871, China 4 Institute for Advanced Study, Shenzhen University, Shenzhen 518051, China

As shown in Figs. 3a-b, the normal strain and shear strain can be realized by exciting the T1 mode and B2 mode of the OSSPSU, respectively. Theoretically, the strain ελ of OSSPSU generating normal vibration modes can be estimated by the following formula: Where Lj is the outline dimension of the OSSPSU; δLj is the displacement along j-direction of the motion points PL (point on the left of the selected deformation region), C (point on the center of the selected deformation region), or PR (point on the right of the selected deformation region). According to the simulation results, the shear strain ελ of OSSPSU will be excited via synergistic effect in i-j plane.
The strain ελ of OSSPSU generating shear vibration modes can be computed as: Where li is the outline dimension of the shear strain deformation region of the OSSPSU; δli and δli ' are the displacements along i-direction of PR and PR ' , respectively; δLj ' is the displacement along j-direction of PR ' ; θij and θji are the shear angles in i-j plane shown in Fig. 3b To effectively excite the diagonal liner motion of the friction tip, the resonant frequencies (fr) of the 31-mode and the 36-mode should be as close as possible. Therefore, we used the FEM to perform modal simulation and size optimization of the piezoelectric OSSPSU stator. Considering the miniaturization design, we fixed the length (LL) of the piezoelectric stator as 5 mm, and the height (LH) as 1.3 mm. The friction tip size was fixed as φ0.8 mm l × 0.5 mm h . Through FEM simulation, the dependence of the two vibration modes on the width (LW) was obtained.
As shown in Supplementary Fig. 2a Obviously, the volume of the ceramic-OSSPSU stator is three times that of the single crystal stator As listed in Table 1, from the calculation results of experimental and simulation data (calculated by Supplementary Eq. (S5)), due to the larger bandwidth, the keff of the crystal-OSSPSU stator is about twice that of the ceramic-OSSPSU stator, indicating that the former may have more significant advantages in output efficiency as a piezoelectric motor.

Supplementary Note 3. The frequency dependence of the displacements of the OSSPSU stator
The variation of the coupled horizontal (δl1, along the 1-axis, as shown in Fig. 4a) and the coupled vertical (δL2, along the 2-axis, as shown in Fig. 4a) displacements of the friction tips for the piezoelectric OSSPSU stators were tested as a function of frequency ( Supplementary Fig. 4) under the electric field intensity of 20 V mm -1 . Obviously, the operating frequency range of the crystal-OSSPSU stator is wider than that of the ceramic-OSSPSU stator, owing to the larger bandwidth (shown in Fig.   4b-c). Moreover, the slight difference between two channels is due to the minor errors in fabrication of the piezoelectric OSSPSU stators 3 . As shown in Supplementary Fig. 4a

Supplementary Note 4. Effect of external force on the properties of OSSPSU stator
We investigated the effect of uniaxial stress on the impedance spectra of 31-36 coupled mode OSSPSU stators by a home-made uniaxial stress loading setup 4 . The stress dependence of impedance spectra of OSSPSU stators were measured by the LCR meter (Agilent E4294A). During the test, the uniaxial stress was applied perpendicular to the polarization direction, as shown in Supplementary Fig.   5a. Supplementary Figs. 5b-

Supplementary Note 5. Theoretical derivation of motion velocity
To experimentally verify the actual performance of the OSSPSU stator (with the size of 5 mm Length × 1.3mm Height × 1.06 mm Width ), we proposed a linear ultrasonic OSSPSU motor as shown in Fig. 5a.
Which consists of a base, a piezoelectric OSSPSU stator with a friction tip (zirconia, with the size of φ0.8 mm l × 0.5 mm h ), a stator holder, a slider with a friction plate made of zirconia, and a preloading structure composed of springs and screw. When a sinusoidal alternating current (AC) voltage excitation signal ( ) 2 ( sin ft  ) was applied to CH1 or CH2, the friction tip forms the continuous microscopic oblique liner motion, which in turn pushes the slider to perform macroscopic linear motion through friction force. To describe the output properties of the piezoelectric ultrasonic motor, the OSSPSU stator is considered to be a Timoshenko beam 5,6 . When the friction tip is in contact with the friction plate, the motion velocity of the slider (vs) along 1-axis can be expressed as where t is the motion time, M is the mass of slider, μs, Ff, μp, FT are the friction coefficient and the friction force of the liner guide of the slider, and between the friction tip and the friction plate, respectively. Furthermore, is the total displacement of the friction tip along 1-axis, and δl1 is the maximum displacement along 1-axis. Combined with Supplementary Eqs.
(S2), and (S6)-(S8), the vs can be calculated as follows: Where θ12 and θ21 are the shear angles on the plane perpendicular to the 3-axis at 31-36 coupled vibration mode. Therefore, the slider motion speed is closely related to the stator piezoelectric performance, the copuled displacement along 1-axis (δl1) and 2-axis (δL2) (i.e., the strains of the artificial 31-36 vibration mode), preload force, friction coefficient, and slider mass.
To determine to the optimal preload force of the OSSPSU stator, we designed a home-made test setup, as shown in Supplementary Fig. 6. By this test setup, the optimal preload force of PIMNT crystal-OSSPSU stator is found to be about 300 mN. Supplementary Fig. 6 Home-made preload force test setup of the OSSPSU stator.

Supplementary Note 6. The step resolution measurement of the micro liner ultrasonic OSSPSU motors.
In addition, the detailed bidirectional motion steps of the OSSPSU motors under different number of period of pulse signals are shown in Supplementary Fig. 7. To sum up, for crystal-OSSPSU stator, when the pulse signal voltage is 150 VP-P and the period is less than 5 cycles, and for ceramic-OSSPSU stator, when the pulse signal voltage is 250 VP-P and the period is less than 26 cycles the average horizontal force of the friction tip cannot overcome the static friction of the slider, resulting in the slider remaining in still (which called the deadzone area of the ultrasonic motor) 4,5 . Especially, the measurement of step resolution is in a home-made noise reduction cavity ( Supplementary Fig. 9), where the ambient noise in it is around 2nm. Hence, the step displacement cannot be further reduced.  Supplementary Fig. 10). The total mass of the slider and lens is about 1.5 g. Supplementary Fig. 11a shows the relationship (measured at 167 kHz) between the up-and-down motion velocities of AF OSSPSU motor and the applied voltage. It is clearly seen that the maximum speeds of AF OSSPSU motor moving upward and downward are 19.4 mm s -1 and 25.4 mm s -1 , respectively. This is because the slider and the lens mass act as resistance when moving up and provide power when moving down, resulting in asymmetrical speeds of the two channels. Similarly, the step precision of the two channels of the OSSPSU motor is asymmetrical ( Supplementary Fig. 11b).

Supplementary
Obviously, the step displacements reduce linearly with the decrease of the cycle of pulse signals, while nd the minimum step is 17 nm, higher than the step shown in Figure 6b. This is because of the inability to use the noise reduction cavity in this test, the ambient noise is relatively large (about 15nm). The results indicate that the miniature nanostep OSSPSU motor based on the [001]-PIMNT single crystal-OSSPSU stator has potential application in lens module, endoscope and other fields.