Theoretical study of enhancing the piezoelectric nanogenerator’s output power by optimizing the external force’s shape

Utilizing piezoelectric nanogenerators (PENGs)¹ to harvest the abundant environmental mechanical energies and convert them into electricity is a promising technology to provide self-powered supplies for distributed sensor network, implantable medical devices, and so on. Now, most of the self-powered signal detections are realized by taking the PENG as both a power unit and a sensing unit.^2–6 Utilizing PENG to power commercial sensors is rarely reported. As for the implantable medical devices, most of the studies remain on proving the PENGs’ potential to be used as power supplies for implantable medical devices.^7–11 The critical factor that limits the PENG’s further development is the PENG’s low electrical output. In order to improve the PENG’s output, numerous methods have been reported such as using materials with high piezoelectric coefficient^12–15 and structural optimization.^16–21 To date, the highest output power density was achieved by a multilayered curved PENG featuring an instantaneous maximum power density of 17 mW/cm².²¹ However, the PENGs’ average power relating to practical applications is much smaller than the maximum instantaneous power. So, further improving the PENG’s output is still an urgent task to broaden the PENGs’ application areas.

Besides the piezoelectric coefficient and device’s structure, the PENG’s output is also affected by the external force. PENG driven by larger amplitude and higher frequency will generate higher output.²² In environment, besides the amplitude and frequency, forces are also different with each other in their shapes. Forces generated by finger tapping, water dropping, and heartbeat can be categorized into square shaped forces which increase and decrease rapidly. Forces generated by respiration, muscle stretching, and wrist bending can be categorized into triangular shaped forces which increase and decrease slowly. Forces generated by sound and machine’s vibrating can be categorized into sinusoidal shaped forces which vary according to a sinusoidal form. The output signal generated by the PENGs is closely related to how the external force is applied. Therefore, the shape is also a key factor that determines the PENG’s output. Optimizing the shape of the external force could provide another solution to improve the PENG’s output further.

Here, we demonstrated that the PENG driven by square shaped force has the dominating superiority to power devices with small resistance and low driving frequencies over that driven by triangular shaped and sinusoidal shaped forces. When the load resistance is 100 MΩ and the driven forces have a magnitude of 19.6 nN and a frequency of 10 Hz, the average power of PENG driven by square shaped force is six orders of magnitude higher than that driven by triangular shaped and sinusoidal shaped forces.

The working mechanism of a PENG is shown in Fig. 1(a). When forces are exerted on the top surface of the c axis-oriented lead zirconate titanate (PZT) nanowire poling along the z axis (left part of Fig. 1(a)), the piezopotential generated across the PZT nanowire along the z axis will drive the electrons in the external circuit flow to generate electricity. A simple resistor-capacitor (RC) circuit was used to describe the PENG’s electrical characteristics.²³ The equivalent circuit is shown in Fig. 1(b). The nanowire can be considered as a voltage source V, which relies on the piezopotential to drive the electrons flow through the circuit. C_PENG is the PENG’s capacitance, and R is the resistance of the external load. In order to determine the PENG’s output current and voltage, the voltage source’s voltage dependence on the external force is determined first. At a given static force, the PENG’s piezoelectric voltage can be accurately obtained by solving the fully coupled governing equation through finite element method or approximately obtained by solving a first order approximation of the coupled governing equation.²⁴ Fig. 1(c) shows the potential distribution in the PZT nanowire obtained by the above two mentioned methods.

The fully coupled equation was solved by using the COMSOL software package. The parameters used in the COMSOL software are as follows: radius r = 25 nm, height h = 500 nm, force f = 19.6 nN, material PZT-5H with piezoelectric coefficient d₃₃ = 593 pC/N, and dielectric constant $ε=3400$⁠. As for the boundary conditions, the PENG is fixed and grounded at its bottom. The normal displacement on the sidewalls of the PENG was set to be zero. Free tetrahedrons are used to mesh the PENG and the number of degrees of freedom is 34 452. As shown from the simulation result in Fig. 1(c), the potential increases linearly with the z coordinate of the nanowire. The voltage drop between the ends of this nanowire is 117.8 mV. Next, the piezoelectric potential distribution and voltage drop were also calculated by a first order approximation of the coupled governing equation. In this approximation, the voltage drop V and external force f can be related by a simple equation V = d₃₃f/C_PENG. C_PENG is 9.85 × 10⁻¹⁷ F, which is obtained by the COMSOL simulation. Substituting the relevant values into the expression of V, the obtained voltage drop is 117.6 mV. This value is just 0.19% smaller than the result obtained by solving the fully coupled governing equation. Therefore, using the first approximation of the governing equation has a rather high accuracy in obtaining the open circuit voltage of the PNEG.

After figuring out the relationship between the external force and voltage source, the PENG’s electric output is studied using the first approximation of the governing equation as shown in Fig. 2. In this study, the frequency of the external force is 10 Hz, and the resistance of the external load is 100 MΩ. The maximum voltage and current of the PENG driven by the square shaped, triangular shaped, and sinusoidal shaped external forces are 115 mV, 1.15 nA (Fig. 2(a)), 46.6 nV, 0.46 fA (Fig. 2(b)), and 73.1 nV, 0.73 fA (Fig. 2(c)), respectively. The average power of the PENGs driven by the square shaped, triangular shaped, and sinusoidal shaped external forces are 3.5 × 10⁻¹⁷ W, 2.68 × 10⁻²³ W, and 2.16 × 10⁻²³ W, respectively. The average power of the PENG driven by the square shaped force is six orders higher than that driven by sinusoidal shaped and triangular shaped forces, which show that the square wave shaped force has the dominating superiority in driving the PENG.

In order to figure out the reasons that cause the large variances of the average power of the PENG driven by different shaped forces. The PENG’s output was examined from the equivalent circuit. In the equivalent circuit of the PENG, the external resistance and nanowire’s capacitance are fixed. The remaining factor that can influence the PENG’s output is the voltage source whose characteristics is directly related with the external force by a simple relation V = d₃₃f/C_PENG. From this equation, it can be found that the behavior of PENG under external forces is similar to charging or discharging a capacitor through external load. As for the sinusoidal shaped force, the voltage of the external load can be expressed by $VR=d33f*sin(wt-δ)/(CPENGζ)$⁠, where $ζ=(1+1/(R2CPENG2ω2))1/2$ and $δ=arctan(1/(ωCPENGR))$⁠. Substituting the capacitance, resistance, and sinusoidal shaped force’s frequency into ζ, a relative large value of 1.6 × 10⁶ was obtained for ζ. This large denominator leads to a much small voltage drop across the external resistance. Compared with the sinusoidal shaped force (Fig. 3(a)), the square shaped force contains many high frequency sinusoidal components as shown in Fig. 3(b). These high frequency components will largely reduce the value of denominator ζ. Therefore, a larger voltage drop across the resistance will be obtained. As for the triangular shaped wave, the high frequency sinusoidal components are much smaller than that of the square shaped force. More importantly, the adjacent sinusoidal components have opposite signs as shown in Fig. 3(c). Therefore, the voltage generated by different sinusoidal components will cancel out each other, and the voltage across the external resistance is even smaller than the PENG driven by the sinusoidal shaped force.

Further, the PENG’s average power dependence on external load and frequency was studied. During the study of the average power’s resistance dependence, the frequency of the external force is fixed at 10 Hz. The calculated results are shown in the left part of Figs. 4(a)–4(c). As for the PENG driven by triangular shaped and sinusoidal shaped forces, there exists an optimal resistance and the average power begins to decrease after getting over the optimal resistance as shown in the left part of Figs. 4(b) and 4(c). The optimal resistance and average power corresponding to the PENGs driven by triangular shaped wave and sinusoidal shaped waves are 140.7 TΩ, 0.015 fW and 140.7 TΩ, 0.021 fW, respectively. As for the PENG driven by the square shaped force, the high frequency sinusoidal components in the square shaped forces have reduced the capacitive reactance to a negligible value. Therefore, the average power is insensitive to the external resistance and keep fixed at 0.055 fW until the external load reaches a value of 80 TΩ. After that, due to the large resistance of the external circuit, only a part of the electrons can arrive at the electrode, so the average power begins to decrease. From the comparison of left part of Figs. 4(a)–4(c), when the external load’s resistance decreases from 80 TΩ, the average power of the PENG driven by triangular shaped and sinusoidal forces decreases accordingly but the average power of PENG driven by the square shaped force is unchanged. So, the PENG driven by the square shaped force has the dominating superiority to drive devices with small resistance.

Next, fixing the load resistance at 100 MΩ, the relationship between the average power and the driving frequency is studied. The calculated results are shown in the right part of Figs. 4(a)–4(c). The saturated average power of the PENG driven by square shaped, triangular shaped, and sinusoidal shaped forces are 138.8 pW, 46.6 pW, and 69.8 pW, respectively. This saturated behavior is in contrast with the average power’s dependence on external resistance. As for sinusoidal shaped force, the relationship between the average power and frequency can still be described by the mentioned expression $Paverage=f2d332/(2CRPENG2+2/(ω2R))$⁠. From this expression, we can see that the average power is a monotone increasing function with a frequency, and the limiting average power can be expressed by P_limit = f²d₃₃²/(2C_PENG²R). The limiting average power is proportional to the square of the sinusoidal force’s magnitude. From the spectral analysis of these different kinds of forces, the sinusoidal component at 10 Hz of the square shaped, triangular shaped, and sinusoidal shaped forces are 25 nN, 15.9 nN, and 19.6 nN, respectively. Analysis based on this single sinusoidal component, the PENG driven by square shaped and triangular shaped forces are 113.9 pW and 45.9 pW. These values are close to the calculated value of 138.8 pW and 46.6 pW. Therefore, at the high frequency, the average power of the PENG is dominated by the base sinusoidal component. Before the saturated frequency, there exists a good linearity between the frequency and the average power of PENGs driven by square shaped, triangular shaped, and sinusoidal shaped forces. Although the saturated average power between the PENG driven by square shaped, triangular shaped, and sinusoidal shaped forces is not larger, compared with the right part of Figs. 4(b)–4(c), the larger slope of right part of Fig. 4(a) leads to the giant differences between the average power of the PENGs driven by square shaped forces and that driven by the other two shaped forces at lower frequencies. When the external force’s frequency is 2 Hz, the average power of the PENG driven by the square shaped force (1.0 × 10⁻¹⁷ W) is seven orders of magnitude larger than that driven by triangular shaped (8.6 × 10⁻²⁵ W) and sinusoidal shaped (1.0 × 10⁻²⁴ W) forces. So, the PENG driven by the square shaped force has also the dominating superiority to drive devices in lower driving frequencies.

In summary, the PENG’s output under different forms of driving forces was theoretically studied. After demonstrating the accuracy of the first approximation of the governing equation, the voltage source’s character was related with the external force in a simple way. Using the equivalent circuit of the PENG, the output of the PENG driven by square shaped, triangular shaped, and sinusoidal shaped forces was calculated and compared. When the load resistance is 100 MΩ and the driven forces have a magnitude of 19.6 nN and a frequency of 10 Hz, the average power of PENG driven by the square shaped force is six orders of magnitude higher than that driven by triangular shaped and sinusoidal shaped forces. The PENG driven by the square shaped force has the dominating superiority to power devices with small resistance low driving frequencies over that driven by triangular shaped and sinusoidal shaped forces.

We sincerely appreciate the support from NSFC (Nos. 51322203, 51472111, and 51302120), the National Program for Support of Top-notch Young Professionals, and the Fundamental Research Funds for the Central Universities (No. lzujbky-2016-k02).