Mesoscopic models for the mechanically modulated electrical conductivity of piezoelectric semiconductors

• Applicants: Privatdozent Dr. Erion Gjonaj; Professorin Dr.-Ing. Bai-Xiang Xu
• Subject Area: Mechanical Properties of Metallic Materials and their Microstructural Origins ; Electronic Semiconductors, Components and Circuits, Integrated Systems, Sensor Technology, Theoretical Electrical Engineering
• Term: from 2016 to 2021
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 317661385

• Final Report Year: 2021

Final Report Abstract

1. In summary, we signified the influence of the internal electric field on barrier height of the varistor like boundaries and therefore presented an extended phenomenological model of the barrier height. It is shown that the influence of electric field can be characterized by a dimensionless coefficient f. The extended model is well reduced to the previous model 6,4 if f=0. Results show that the piezoelectric charge induced by the internal field tends to alter the grain boundary charge and lowers the barrier height. The sensitivity of barrier height on both the mechanical loading and the applied voltage decreases, if the extended model is used. We believe that the inverse piezoelectric effect has a similar impact, independent of the model used for the trap charge at the grain boundary. 2. The previous models have been widely used to fit the experimental data of barrier heights with estimated parameters on grain boundary charge densities. If our extended model is applied to fit the same experimental data, the determined grain boundary trap charge densities are expected to be higher than those determined by the previous model. Moreover, the high gradient of the electric field near the grain boundary may lead to an additional impact on the strain via the reverse flexoelectric effect, which can be an interesting topic in the future. Besides, The inverse piezoelectric effect can have a general impact on the III-V or II-VI semiconductor junction devices e.g. the thin GaN films, AlGaN/AlN/GaN diodes, since they are also piezoelectric materials37,38. Therefore, the results of this letter are not only limited to potential barriers at grain boundaries of ZnO, but also have practical consequences on the study of other junction devices. 3. The reason that the results from the FEM and the MPM are different stems from three main points. Firstly, at 𝑇 = 300 K, the ionized donor states from FEM results show a trapezoidal shape rather than a box shape as assumed in the MPM at 𝑇 = 0 K. Secondly, in the FEM, there are no additional assumptions for the potential at the edge of the depletion region. Thirdly, the width of the GB for the FEM is 1 nm rather than 0 nm assumed by the MPM. For these reasons, the FEM is physically more meaningful. 4. Unlike the MPM, in FEM the barrier height can reach below 0 eV and the flux breakdown is not relevant to the sudden drop of the barrier height. The other way around, breakdown of the flux is controlled by the gradient of the band structure and it happens always accompanied by a very small barrier height. 5. A model for calculating the stress dependent electrical characteristics of ZnO varistors is developed. The model is based on an equivalent electrical network description of the varistor microstructure. The piezoelectric effect at the grain boundaries is taken into account by incorporating a self-consistent model for the grain boundary potential and the piezoelectrically induced grain boundary charge. This modelling approach is applied for polycrystalline varistor structures in 2D as well as in 3D geometry. The simulations reveal the strong stress sensitivity of the IV-characteristics and, furthermore, provide insight on the current flow behaviour within the material in the presence of mechanical stress. On the macroscopic scale, the effect of mechanical stress essentially shifts the IV-characteristic to a lower switching voltage. Furthermore, the effective nonlinearity of the material is reduced. These results, as well as the computed gauge factors for a typical varistor sample, are consistent with recent experimental findings for industrial ZnO varistors. The investigation of microstructurally inhomogeneous varistors and the simulation of a bent thick film varistor under shear stress demonstrate the capability of the model to cope with realistic varistor geometries. The model is also applied in the case of residual stresses arising within microstructures due to the thermal expansion anisotropy of the crystal. Polycrystalline structures with different residual thermal stresses resulting from sintering at different temperatures were analysed.

Publications

• “Mesoscopic Modelling of Mechanically-Modulated Electrical Conductivity in Zinc Oxide Varistors”, Ph.D Thesis
  Kyle A. Taylor
  
• “An Extended Grain Boundary Barrier Height Model Including the Impact of Internal Electric Field,” AIP Addvances, 8, 115126, 2018
  Bai-Xiang Xu, Zi-Qi Zhou, Peter Keil and Till Frömling
  
• “Coupled Simulation of Current Flow and Residual Thermal Stress in ZnO Varistors,” IEEE Trans. Mag., 56, 1-4, 2019
  Kyle A. Taylor, Erion Gjonaj and Herbert De Gersem
  
• “Finite Element Simulations on Piezoelectric Modulation of ZnO Grain Boundary Height,” J. Appl. Phys., 126, 205101, 2019
  Zi-Qi Zhou, Kyle A. Taylor, Erion Gjonaj, Till Frömling, and Bai-Xiang Xu
  
• “Mesoscopic Modelling of ZnO Varistors,” In Proceedings of IX ECCOMAS Thematic Conference on Smart Structures and Materials, 8-11 Juli, Paris, France (2019), pp. 1726-1737
  Kyle A. Taylor, Erion Gjonaj and Herbert De Gersem
  
• “Mesoscopic Modeling of the Mechanically Tunable Electrical Conductivity of ZnO Varistors,” J. Appl. Phys., 127, 155104 (2020)
  Kyle A. Taylor, Erion Gjonaj, Zi-Qi Zhou and Bai-Xiang Xu
  
• “Microscopic Simulation of the Mechanically Modulated Electrical Conductivity of Piezoelectric Semiconductors”, Ph.D Thesis
  Ziqi Zhou