Theory and ideal operating cycle

When two materials with different work functions are electrically connected, the Fermi levels are equalized by transport of electrons from the material with lower work function to the material with higher work function (see Fig. 1a and b). This displacement of electrons produces an excess of electrons and depletion of electrons, i.e. negative and positive electric charges, on the surfaces of the materials. These electric charges, in turn, give rise to an electric field and a voltage across the gap between the materials.

In the general case the voltage across the gap between the electrodes of the work-function energy harvester can be written as V gap = u-V bi , where u is the output voltage of the WFEH, i.e. voltage between the WFEH electrodes, measured via the connecting circuit (see Fig. 1c), V bi = (ϕ 1 -ϕ 2 )/e is the built-in voltage (i.e. the voltage across the gap in the thermodynamical equilibrium shown in Fig. 1b), e is the elementary charge and ϕ 1 and ϕ 2 are the work functions of the electrode materials. The electric charge on the WFEH plates is given by Q = C gap V gap + C par u, where C gap is the capacitance between the electrodes and C par is the parasitic capacitance. The total capacitance of the WFEH can be written as C = C gap + C par . Since C par is constant, the electric current supplied or drawn by the WFEH can be written using the equations for V gap and Q as

Equation (1) is the fundamental equation describing the charging and discharging phenomena caused by the built-in voltage. The drive towards thermodynamical equilibrium (where Fermi levels are equal as shown in Fig. 1b) gives rise to the term which can be exploited in energy harvesting. The overall behaviour of the harvester device depends on the time evolution of C and the external circuit connected to the electrodes. The time evolution of the capacitance depends on the geometry and mechanics of the system and the electrostatic force (equation (8) in Methods section). Since equation (1) depends also on the time derivative of the product of C and u, highly nonlinear behaviour is expected. The simplest form of electric load is a resistor. The electric current passing through the load resistor R is given by i = –u/R and equation (1) reduces to

By considering the electric energy stored in the WFEH and neglecting all losses, the power output of an ideal WFEH can be calculated (the details are given in Section 1 of Supplementary Information). The output power depends strongly on how the electric charge is extracted from the WFEH. We consider two operation schemes: voltage and charge constrained modes. The charge-constrained mode is shown to generate much higher output power than the voltage-constrained mode. Therefore, the output power of the charge-constrained mode represents the theoretical maximum output power of the WFEH. The motion of the work-function energy harvester is assumed to be periodic with a period of Δt. During the cycle of operation the total capacitance of the WFEH is varied from the minimum value C min to the maximum value C max .

In the voltage constrained mode any excess electric charge is immediately extracted using, for example, a low impedance load circuit. The voltage constrained mode yields the average output power as

In the charge constrained mode the WFEH capacitor is charged or discharged only when the stored electric energy reaches a maximum or minimum, respectively. These extremes are reached when the WFEH capacitance is at maximum and minimum. The charging and discharging can be controlled with an electrical switch, for example. The ideal operating cycle of a WFEH operating in the charge constrained mode consist of 5 phases depicted in Fig. 1d. By combining all the phases of the cycle the average output power of a WFEH operating in the charge constrained mode can be written as

Unlike in the charge-constrained mode, the ideal voltage-constrained mode does not lose power because of the parasitic capacitor. The ideal voltage-constrained mode could also be realized more easily, but comparison of equations (3) and (4) shows that the power output of the charge-constrained mode is at least two times the power output of the voltage-constrained mode. At large values of C max /C min ratio the charge-constrained mode generates substantially more power than the voltage-constrained mode with limiting value .

The average powers of ideal work-function energy harvesters operated in the charge constrained mode are plotted in Fig. 1e–f as functions of C max /C min ratio, frequency f = 1/Δt and the built-in voltage V bi . High output power can only be reached if C max is much larger than C min . In this case the power generated in phase 5 is negligible compared to the phase 3 (see Fig. 1d). As Fig. 1e–f and equation (4) show, P idealQ is directly proportional to the operating frequency and the square of V bi . Therefore, material pairs with high built-in voltages are desired, but otherwise any materials with sufficient conductivity are suitable and in addition to metals also semiconductors can be utilised. For example, for metal pair Pt and Mg we have V bi between 1.46 V and 2.27 V and for Pt and Al between 0.86 V and 1.87 V27. Here the spread in V bi arises from the typical spread in the work function values of metals found in the literature. If the material pair is formed from the same semiconductor but of different type (p and n), then the built-in voltage is roughly equal to the energy gap of the semiconductor. For example, a built-in voltage of 1.1 V could be obtained with silicon and values above 3 V could be reached with wide band-gap semiconductors such as SiC and GaN. Availability of n and p-type diamond allows built-in-voltages of over 5 V to be reached.

Experimental verification with prototype of work-function energy harvester

In order to verify the model and to demonstrate the work function energy-harvesting concept we built a macroscopic variable capacitance prototype with 63 mm × 64 mm aluminium and copper capacitor plates (see Fig. 2a and b and Methods). In the harvesting test measurements a voltage across a load resistor was recorded against temporal change of the Al-Cu capacitor. Prior to these measurements the built-in voltage of the plates was measured using a null method, where the load and the measurement instrument is replaced by a DC voltage source U DC and the output current is measured while the capacitor plates move. In this configuration, similar to the one invented by Lord Kelvin18,19, equation (1) reduces to . The RMS of measured current i as a function of U DC is shown in Fig. 2d and the RMS current is minimized at U DC = V bi , which is given by the intersection of the lines fitted to the data. The average value of the measured built-in voltage V bi was 1.03 ± 0.08 V.

Figure 2 (a,b) Schematic picture of the experimental setup: (a) top view of the whole setup and (b) side view of the cam disk. The setup is inclined at an angle of 15 degrees in order to utilize gravitational counter force. Voltage u across the load resistor is indicated. (c) Photograph of the setup. The initial gap between the plates is adjusted with a positioner stage. (d) Example of electric current data from DC voltage sweeping measurements. The value of the built-in voltage V bi = 1.03 ± 0.08 V is determined from the intersection of the lines fitted to the data. The RMS value of the electric current is not equal to zero at u = V bi due to noise and current offset. (e, f) Time dependencies of measured and simulated voltage across a 1.00 GΩ load resistor connected to a Cu/Al parallel plate work-function energy harvester with the minimum distance between the plates d 0 of (e) 170 µm and (f) 580 µm. Time is normalized with the operating period Δt. Full size image

The time dependencies of the measured and simulated voltage across a load resistor connected to the experimental work-function energy harvester with different minimum distances between the plates d 0 (see equations (5) and (6) in Methods) and different load resistors are shown in Fig. 2e and f and Supplementary Figs. 1 and 2. In order to compare the data obtained at various operating frequencies 1/Δt in a single plot, the normalized time t/Δt is used. The total capacitance is at maximum at time t = Δt/2 and at minimum at times t = 0 and t = Δt. The data shows that the experiments and the numerical model agree well over the whole range of frequencies (10 mHz – 0.9 Hz) and values of d 0 (170 µm, 350 µm and 580 µm) and different loads (1 GΩ and 100 MΩ). The shape of the voltage signal changes from signal having two peaks with opposite polarities to smoothed triangle wave as the frequency increases and d 0 decreases. At the same time the average output power of the WFEH increases. These effects are caused by the fact that the operating period Δt approaches the electrical time constant RC max of the WFEH.

The feasibility of using the simple load resistor as the harvesting circuit can be estimated by comparing the average output power of the system P ave to the output power of an ideal WFEH operating in the charge-constrained mode, P idealQ (equation (4)), as this mode generates the highest obtainable power. The measured and simulated normalized powers P ave /P idealQ are plotted in Fig. 3a as functions of the operating frequency. At low frequencies and large values of d 0 the measured output powers are slightly larger than the simulated powers due to noise. The highest measured power was 10% of the ideal power. In Section 3.2 of Supplementary information it is shown that a WFEH with load resistor is characterized by the normalized time constant RC max /Δt. Normalizing the operating frequencies of the measured data (see Fig. 3b) shows, that the features of the power curves indeed fall on almost the same locations on the RC max /Δt axis. The differences between the curves are caused by different values of C max /C min , which range from 1.9 to 4.0.

Figure 3 Measured and simulated average output power P ave of a Cu/Al parallel plate work-function energy harvester with a load resistor R as a function of (a) the operating frequency 1/Δt and (b) the normalized time constant RC max /Δt. The average output powers are normalized to the ideal charge-constrained output power P idealQ . The value of the load resistor and the minimum distance between the plates d 0 are indicated in the legend. Simulated data of a WFEH with a load varying between on-state (10 Ω) and off-state (100 GΩ) is also shown in (a). Full size image

Optimisation of output power using variable load

The temporal variation of capacitance has a large effect on the performance of a WFEH (see Section 3.3 of Supplementary information). For a specific temporal dependence of the capacitance there are optimal values for RC max /Δt and C max /C min . A WFEH with a resistor as the load generates less output power than both the charge-constrained and voltage-constrained operating cycles. This is caused by non-optimal charging and discharging of the WFEH capacitor. For maximum power the WFEH capacitor should be fully charged when the capacitance is at the maximum and fully discharged when the capacitance is at the minimum. At low frequencies the WFEH capacitor is discharged too early because of the slow variation of the WFEH capacitance. At high frequencies, in turn, the WFEH operates too quickly to be adequately charged when the WFEH capacitance is at its maximum. Fig. 3 illustrates well this behaviour. The determining factor is the ratio of the electrical time constant, RC max and the period of the mechanical motion, Δt. Between these two extremes, when ratio RC max /Δt is near 0.2, the optimal operation is reached.

The ideal charge-constrained cycle defines the maximum output power of the work-function energy harvesters. This ideal cycle requires use of variable load in the harvesting circuit. Variable load can be realized using, for example, electrical switches. Such switching circuits28,29,30 have already been utilized in electrostatic energy harvesters, the operation of which is similar to the WFEHs. The switching ensures that the WFEH capacitor is fully charged and discharged during the operation cycle similarly as in the ideal cycle. In the simplest case the WFEH can be charged by shorting the capacitor electrodes. This can be done using a mechanical switch in parallel with the load resistor. The timing of the switching is critical as the switch should only be closed when the capacitance is at its maximum, otherwise the output power can even be much lower than the power obtained without switching. Despite this difficulty, we managed to demonstrate over 2-fold increase in power output of our experimental setup using the switching operation (see the details in Section 2.2 of Supplementary information).

The power increase due to the use of a variable load in work-function energy harvesters estimated with our numerical model is shown in Fig. 3a. The normalized power decreases with decreasing frequency because the off-state resistance (100 GΩ) is not high enough to avoid leakage current. The simulations with variable loads yield much higher power than the simulations with fixed load: With the minimum plate distance of 170 µm the simulated variable load WFEH produces 58% of the ideal power at maximum, whereas the simulated fixed load WFEH produces only 13% of the ideal power at maximum.

Estimation of performance of MEMS device

The performance of a MEMS-based WFEH device can be estimated by using MEMS-scale parameter values of electrostatic energy harvesters. Here we adopt the parameters of Ref. 31. In the MEMS simulations (see Methods) the energy harvester was connected to either constant or variable load and excited with sinusoidal vibration with an amplitude of a = 32.5 m/s2 around the mechanical resonance frequency f r = 1868 Hz. With this kind of excitation and a charging voltage of 9 V the electrostatic harvester gives an output power of 1.2 µW31. The dependence of the output power of the simulated WFEH device on the frequency of the vibration is shown in Fig. 4a. Note that by setting the built-in voltage of the WFEH and the external power supply of electrostatic harvester equal we find that the WFEH gives equal or higher power output (see Section 5 of Supplementary information). The output power of the WFEH increases with increasing value of the built-in voltage V bi . The device produces the highest power when the vibration frequency matches with the mechanical resonance frequency of the device. Comparison of the data from the devices with constant and variable loads shows that the output power increases when switching is used. For example, at the built-in voltage of 1 V the increase in output power is over one order of magnitude. The power maxima vs. frequency in the variable load case are wider than in the fixed load case. In other words, the harvesting efficiency in the variable load case is less sensitive to matching the mechanical resonance frequency to the vibration frequency, which underlines the importance of electrostatic force (equation 8 in Methods section) in the dynamics.

Figure 4 Simulated output power of MEMS work-function energy harvesters with a load varying between 100 Ω and 50 GΩ and a constant 200 kΩ load as (a) functions of the vibration frequency at three values of V bi and (b) function of the built-in voltage V bi at vibration frequency of 1868 Hz. The parameters of the simulated device were taken from an existing electrostatic energy harvester31. The amplitude of the vibration is 32.5 m/s2. Full size image

The dependence of the output power on the built-in voltage is shown in Fig. 4b. In the case of constant 200 kΩ load, the power increases quadratically with V bi , as suggested by equations (3) and (4), but in the case of variable load the dependence is weaker (the second derivative of the curve is negative). This is likely a consequence of electrostatic spring softening, which shifts the resonance off the exciting vibration frequency. In the constant load case this effect is not observed because the duration of the peak electrostatic force per cycle is shorter than in the variable load case.

Work-function charge pump

In the most effective part of the charge-constrained ideal cycle (Fig. 1d) the WFEH acts essentially as a charge pump. It pushes the excess charge to an external circuit. This can be exploited in an elegant manner to charge a storage capacitor C sto , which, in turn, can supply power to an electric load. Such a work-function charge pump can be constructed from a shuttle moving between the fixed electrodes as sketched in Fig. 5a. During the operation of this device (see Fig. 5b) the shuttle first makes electric contact with the lower electrode, thus charging the shuttle. Next, the shuttle moves up and the electric energy in the system increases as the capacitance between the shuttle and the lower electrode decreases. Finally, the shuttle touches the upper electrode and the excess electric charge in the shuttle flows to C sto . Note that the work-function charge shuttle has a captivating analogy with a nanomechanical single-electron shuttle32. The shuttle operation can be adapted for WFEH in general or, on the other hand, a work-function charge pump can be realized without a shuttle using external switches (see Fig. 5c) that mimic the phases of Fig. 1d. In principle, the switch configuration should exhibit characteristics identical to that of the shuttle device, but the shuttle device can provide significantly larger maximum capacitance.

Figure 5 (a) Schematic picture of a work-function charge pump, where a shuttle moving between two fixed electrodes acts simultaneously as a built-in switch. The shuttle can be fixed to a frame of reference, e.g., by an elastic beam (not shown). The shuttle and the upper electrode are made of the same material (1) and the lower electrode of material with dissimilar work function (2). This charge pump is connected to storage capacitor C sto , which supplies power to an electric load. (b) The operation phases of the work-function charge pump of (a): Charging of the shuttle (1) and the flow of charge to C sto (2). (c) A work-function charge pump based on external switches S1 and S2: A work-function energy harvester with total capacitance C connected to storage capacitor C sto , which supplies power to a load circuit. u sto is the voltage across C sto . (d) Dependence of the normalized electric energy stored in C sto as a function of the number of charging cycles N normalized with the number of cycles N mpp needed to reach the maximum power point. The stored electric energy, E sto is normalized with the maximum electric energy which can be stored in the storage capacitor, . (e) Dependence of the normalized charging power of a work-function charge pump on the normalized voltage u sto /u sat calculated with various values of C sto /C min . The charging power is normalized with , which is twice the maximum average power available from the WFEH. Full size image