Motivation
In this project, the energy harvesting capability of commercially available [001]oriented 68PbMg_{1/3}Nb_{2/3}O_{3}–32PbTiO_{3} (PMN–32PT) single crystal capacitors was measured experimentally. An energy density of 100 J/L/cycle, corresponding to 4.92 W/L, was obtained by successively dipping the material in oil baths at temperatures 80 and 170ºC and cycling the electric ﬁeld between 2 and 9 kV/cm. Similarly, an energy density of 55 J/L/cycle was obtained between 80 and 140ºC. An estimated 40% of this energy resulted from the strain polarization due to the rhombohedral to tetragonal phase transition. The strain from this transition disappeared when the maximum operating temperature exceeded the Curie temperature of about 150ºC. The optimal low electric ﬁeld used in the Olsen cycle maximizing the energy harvested was found to be around 2 kV/cm. In addition, the material suffered from (i) dielectric breakdown for electric ﬁelds larger than 9 kV/cm and (ii) cracking from thermal stress for operating temperature differences in excess of 90ºC. A physical model predicting the total amount of energy harvested was also derived, accounting for thermal expansion as well as temperature dependent dielectric constant and spontaneous polarization. The model predictions fell within 20% of the experimental results in the temperature range between 80 and 170ºC and electric ﬁelds ranging from 2 to 9 kV/cm. Theoretical analysis


In the absence of a phase transition, the electric displacement D of the material at temperature T is deﬁned as a function of the dielectric permittivity ε, the applied electric ﬁeld E and the spontaneous polarization P_{S} . This relation is illustrated in Figure 1.
For an unclamped crystal, it consists of two terms, namely (i) the primary pyroelectric coefficient accounting for changes in crystal dipole moment and (ii) the secondary pyroelectric coefficient due to dimensional changes in the crystal both caused by changes in temperature. The result of integrating the energy density equation is
The term d_{33}x_{33}/s_{33} accounts for the secondary pyroelectric effect which is due to thermal expansion. Here, x_{33} is the strain resulting from the thermal expansion, d_{33} is the piezoelectric coefficient and s33 is the elastic compliance. By assuming the properties do not depend on temperature, the former equation reduces to one given by Sebald et al.[1].

Figure 1. Electric displacement versus electric field for a typical pyroelectric material at temperatures T hot and T cold along with the Olsen cycle. The electrical energy generated per cycle is represented by the area enclosed by 1–2–3–4. 
Experimental Results The electric ﬁeld E_{L} was ﬁxed at 2.0 kV/cm and the cold temperature T_{cold} was maintained at 80ºC for all experiments. The largest amount of energy harvested from this sample was 100 J/L/cycle, corresponding to 4.92 mW/cm^{3}, obtained between temperatures T_{cold} = 80ºC and T_{hot} = 170ºC and electric ﬁelds E_{L}=2 kV/cm and E_{H}=9 kV/cm. Figure 2 compares the energy density obtained experimentally with of that predicted (i) by Equation (2) assuming constant properties and (ii) by Equation (3) accounting for temperature dependent properties. The constantproperty model deteriorated as the temperature difference ΔT = T_{cold} − T_{cold} increased. The effects of thermal expansion were taken into account by the temperaturedependentproperty model for T_{hot} =130 and 140ºC. However, they were neglected for T_{hot}=150ºC as the terms x_{33} and d_{33} are negligible [2]. Figure 2 shows that reasonable agreement was found between this model’s predictions and experimental data. In fact, the relative error was less than 20% for all values of temperatures and electric ﬁelds explored. In addition, the model presented here does not require knowledge of the pyroelectric coefﬁcient in order to calculate the energy density generated by a ferroelectric material subjected to the Olsen cycle. However, this model does not simulate the hysteresis behavior and is valid in the region where the displacement depends linearly on the electric ﬁeld. This is generally the case for the Olsen cycle and indeed was the case for the experiments performed on PMN–32PT with E_{L} larger than 2.0 kV/cm. Figure 2. Experimentally measured energy density harvested with PMN–32PT single crystals versus high electric field E_{H} for different values of ∆T=T_{hot}−T_{cold}. The temperature T_{cold} was maintained at 80ºC and the low electric field E_{L} was set at 2.0 kV/cm. The dashed and solid lines correspond to predictions by Equations (2) and (3), respectively. Comparing DE loops with the Olsen cycle The strain x3 resulting from thermal expansion between 80 and 120ºC was found to be 0.11% but was negligible when no phase transition occurred [2]. Between T_{cold}=80ºC and T_{hot}=130ºC, the PMN–32PT samples underwent rhombohedral to tetragonal phase transition. However, only those with T_{hot}>T_{Curie} transitioned from tetragonal to cubic structures. Figure 3 shows the displacement versus electric ﬁeld for bipolar hysteresis curves at temperatures T_{cold}=80ºC and T_{hot} equals to 130, 140, 150, and 170ºC for electric ﬁelds E_{L}= 3 kV/cm and E_{H} = 6 kV/cm. It also plots the Olsen cycle operating between these two temperatures. Furthermore, it establishes that for T_{hot} = 130 and 140ºC, the electric displacement observed during the Olsen cycle was larger than the change in displacement between the two D–E curves at temperatures T_{hot} and T_{cold}. This can be attributed to the secondary pyroelectric effect which contributed about 40% of the total change in displacement. However, when T_{hot} was set larger than T_{Curie} = 150ºC, the strain polarization due to thermal expansion vanished. According to Li et al.[3], a [001]poled PMN–32PT has two different strains depending on the operating conditions. When the sample is heated from 80ºC to temperature below TCurie, the strain due to tetragonal to rhombohedral phase transition is 0.11% [3]. In contrast, when the PMN–32PT is cooled from above T_{Curie} down to 80ºC, the strain due to tetragonal to rhombohedral transition is 0.02% [3]. Increasing E_{H} led to an increase in the amount of energy harvested as illustrated in Figure 3. This parameter was limited to 9 kV/cm corresponding to approximately the dielectric strength of the material. Figure 3. D–E curves and Olsen cycle performed at temperatures T_{cold} and T_{hot}. Blue lines correspond to D–E curves at T_{cold} = 80ºC. Red lines correspond to D–E loops for T_{hot} equal to 130, 140, 150 and 170ºC. Solid lines correspond to the Olsen cycles performed between 80ºC and T_{hot}. The Olsen cycle is displaced to coincide with D–E curve at T_{hot.} Effect of electric fields E_{L} and E_{H} Figure 4 shows four different Olsen cycles in the D–E diagram performed with E_{L} = 0.0, 1.5, 2.0, and 3.0 kV/cm. The cycles are shown superimposed to facilitate their comparison. For the three cases when E_{L} was equal to 1.5, 2.0, and 3.0 kV/cm, the material reached approximately the same electric displacement when the electric field was raised to E_{L} at T_{cold}. The optimal value of E_{L} was found to be 2.0 kV/cm with an energy density of 24.8 J/L/cycle. The energy density harvested when E_{L} = 0 kV/cm was 0.5 J/L. This was due to crossover of the hot and cold D–E curves.
Figure 4. Experimental Olsen cycle diagrams performed on [001] poled PMN–32PT between T_{cold} = 80ºC and T_{hot} = 140ºC for E_{H}= 6 kV/cm and E_{L} varying from 0 to 3.0 kV/cm. Conclusion This study presented experimental measurements of energy density generated by commercial PMN–32PT capacitors subjected to the Olsen cycle. A physical model was also derived to predict the energy density generated during the cycle. The maximum energy density harvested with PMN–32PT was found to be 100 J/L between temperatures T_{cold} = 80ºC and T_{hot} = 170ºC and electric field spanning from 2.0 to 9.0 kV/cm. Future study should be performed to further validate the model and its applicability to different materials. References [1] Sebald G., Seveyrat L., Guyomar D., Lebrun L., Guiffard B., and Pruvost S., 2006. Electrocaloric and pyroelectric properties of 0.75Pb(Mg1/3Nb2/3)O3–0.25PbTiO3 single crystals. J. Appl. Phys., Vol. 100, pp. 124112.
[2] Yasuda N., Banno T., Fujita K., Ohwa H., Matushita M, Yamashita Y., Iwata M., and Ishibashi Y., 2007. Piezoelectric properties of relaxor ferroelectric solid solution single crystals PMN–PT and PZN–PT near MPB under pressures, Ferroelectrics Vol. 347, pp. 44–49. [3] Li Z., Xu Z., Xi Z., Xiang F., and Yao X., 2007. Thermal expansion characteristics in [001]oriented PMN–0.32PT single crystals, Ferroelectrics, Vol. 355, pp. 245–251. Publications R. Kandilian, A. Navid, and L. Pilon, 2011. Pyroelectric Energy Harvesting Capabilities of PMNPT Near the Morphotropic Phase Boundary, Smart Materials and Structures, Vol. 20, No. 5, 055020. doi: 10.1088/09641726/20/5/055020. 