Reflection and transmission of quasi-plane waves at the interface of piezoelectric semiconductors with initial stresses

We examine the reflection and transmission phenomena of quasi-longitudinal plane(QP)waves in an AlN-ZnO laminated composite structure.The structure is designed under the influence of the initial stresses in which one carrier piezoelectric semiconductor(PSC)half-space is in welded contact with another PSC half-space.The secular equations in the transversely isotropic PSC material are derived from the general dynamic equation,taking the initial stresses into consideration.It is shown that the incident quasi-longitudinal wave(QP-mode)at the interface generates four types of reflected and transmitted waves,namely,QP wave,quasi-transverse(QSV)wave,electric-acoustic(EA)wave,and carrier plane(CP)wave.The algebraic equations are obtained by imposing the boundary conditions on the common interface of the laminated structure.Reflection and transmission coefficients of waves are obtained by implementing Cramer’s rule.Profound impacts of the initial stresses and exterior electric biasing field on the reflection and transmission coefficients of waves are investigated and presented graphically.

Analysis of piezoelectric semiconductor fibers under gradient temperature changes

Piezoelectric semiconductors(PSs)possess both semiconducting properties and piezoelectric coupling effects,making them optimal building blocks for semiconductor devices.PS fiber-like structures have wide applications in multi-functional semiconductor devices.In this paper,a one-dimensional(1D)theoretical model is established to describe the piezotronic responses of a PS fiber under gradient temperature changes.The theoretical model aims to explain the mechanism behind the resistance change caused by such gradient temperature changes.Numerical results demonstrate that a gradient temperature change significantly affects the physical fields within the PS fiber,and can induce changes in its surface resistance.It provides important theoretical guidance on the development of piezotronic devices that are sensitive to temperature effects.

Indentation behavior of a semi-infinite piezoelectric semiconductor under a rigid flat-ended cylindrical indenter

This paper theoretically studies the axisymmetric frictionless indentation of a transversely isotropic piezoelectric semiconductor(PSC)half-space subject to a rigid flatended cylindrical indenter.The contact area and other surface of the PSC half-space are assumed to be electrically insulating.By the Hankel integral transformation,the problem is reduced to the Fredholm integral equation of the second kind.This equation is solved numerically to obtain the indentation behaviors of the PSC half-space,mainly including the indentation force-depth relation and the electric potential-depth relation.The results show that the effect of the semiconductor property on the indentation responses is limited within a certain range of variation of the steady carrier concentration.The dependence of indentation behavior on material properties is also analyzed by two different kinds of PSCs.Finite element simulations are conducted to verify the results calculated by the integral equation technique,and good agreement is demonstrated.

Electromechanical Fields Near a Circular PN Junction Between Two Piezoelectric Semiconductors

We study electromechanical fields near the interface between a circular piezoelectric semiconductor cylinder and another piezoelectric semiconductor in which it is embedded. The cylinder is p-doped. The surrounding material is n-doped. The phenomenological theory of piezoelectric semiconductors consisting of the equations of piezoelectricity and the conservation of charge for holes and electrons is used. The theory is linearized for small carrier concentration perturbations. An analytical solution is obtained, showing the formation of a PN junction near the interface. Various electromechanical fields associated with the junction are calculated. The effects of a few physical parameters are examined.

Multi-field coupling and free vibration of a sandwiched functionally-graded piezoelectric semiconductor plate

Sandwiched functionally-graded piezoelectric semiconductor(FGPS)plates possess high strength and excellent piezoelectric and semiconductor properties,and have significant potential applications in micro-electro-mechanical systems.The multi-field coupling and free vibration of a sandwiched FGPS plate are studied,and the governing equation and natural frequency are derived with the consideration of electron movement.The material properties in the functionally-graded layers are assumed to vary smoothly,and the first-order shear deformation theory is introduced to derive the multi-field coupling in the plate.The total strain energy of the plate is obtained,and the governing equations are presented by using Hamilton’s principle.By introducing the boundary conditions,the coupling physical fields are solved.In numerical examples,the natural frequencies of sandwiched FGPS plates under different geometrical and physical parameters are discussed.It is found that the initial electron density can be used to modulate the natural frequencies and vibrational displacement of sandwiched FGPS plates in the case of nano-size.The effects of the material properties of FGPS layers on the natural frequencies are also examined in detail.

Nonlinear free vibration of piezoelectric semiconductor doubly-curved shells based on nonlinear drift-diffusion model

In this paper, the nonlinear free vibration behaviors of the piezoelectric semiconductor(PS) doubly-curved shell resting on the Pasternak foundation are studied within the framework of the nonlinear drift-diffusion(NLDD) model and the first-order shear deformation theory. The nonlinear constitutive relations are presented, and the strain energy, kinetic energy, and virtual work of the PS doubly-curved shell are derived.Based on Hamilton's principle as well as the condition of charge continuity, the nonlinear governing equations are achieved, and then these equations are solved by means of an efficient iteration method. Several numerical examples are given to show the effect of the nonlinear drift current, elastic foundation parameters as well as geometric parameters on the nonlinear vibration frequency, and the damping characteristic of the PS doublycurved shell. The main innovations of the manuscript are that the difference between the linearized drift-diffusion(LDD) model and the NLDD model is revealed, and an effective method is proposed to select a proper initial electron concentration for the LDD model.

The mechanism to reform dynamic performance of an elastic wave-front in a piezoelectric semiconductor by the wave-carrier interaction induced from static biasing fields

The propagation of an elastic wave(EW)in a piezoelectric semiconductor(PSC)subjected to static biasing fields is investigated.It is found that there exist two coupling waves between electric field and charge carriers.One is stimulated by the action of the polarized electric field in the EW-front on charge carriers(EFC),and the other is stimulated by the action of initial electric field in biasing fields on dynamic carriers(IEC).Obviously,the latter is a man-made and tunable wave-carrier interaction.A careful study shows that IEC can play a leading role in remaking dynamic performance of the wave-front and an inter-medium role in transferring energy from biasing fields to EW-fronts.Hence,a method is proposed to reform the EW performance by biasing-fields:reforming the dispersivity of EW-fronts by promoting competition between IEC and EFC and inverting the dissipation by the IEC to transfer energy from biasing fields to EWfronts.The corresponding tuning laws on the phase-frequency characteristics of an EW show that the wave velocity can be regulated smaller than the pure EW velocity at a lowfrequency and larger than the pure piezoelectric wave velocity at a high-frequency.As for regulating the amplitude-frequency characteristics of the EW by the IEC,analyses show that EWs can obtain amplification only for those with relatively high vibration frequencies(small wave lengths).The studies will provide guidance for theoretical analysis of waves propagating in PSCs and practical application and design of piezotronic devices.

Nonlinear analysis on electrical properties in a bended composite piezoelectric semiconductor beam

In this paper,the interactions between the transverse loads and the electrical field quantities are investigated based on the nonlinear constitutive relation.By considering a composite beam consisting of a piezoelectric semiconductor and elastic layers,the nonlinear model is established based on the phenomenological theory and Euler’s beam theory.Furthermore,an iteration procedure based on the differential quadrature method(DQM)is developed to solve the nonlinear governing equations.Before analysis,the convergence and correctness are surveyed.It is found that the convergence of the proposed iteration is fast.Then,the transverse pressure induced electrical field quantities are investigated in detail.From the calculated results,it can be found that the consideration of nonlinear constitutive relation is necessary for a beam undergoing a large load.Compared with the linear results,the consideration of the nonlinear constitutive relation breaks the symmetry for the electric potential,the electric field,and the perturbation carrier density,and has little influence on the electric displacement.Furthermore,the non-uniform pressures are considered.The results show that the distributions of the electric field quantities are sensitively altered.It indicates that the electrical properties can be manipulated with the design of different transverse loads.The conclusions in this paper could be the guidance on designing and manufacturing electronic devices accurately.

The action mechanism of the work done by the electric field force on moving charges to stimulate the emergence of carrier generation/recombination in a PN junction

It is discovered that the product of the current and the electric field in a PN junction should be regarded as the rate of work(power)done by the electric field force on moving charges(hole current and electron current),which was previously misinterpreted as solely a Joule heating effect.We clarify that it is exactly the work done by the electric field force on the moving charges to stimulate the emergence of non-equilibrium carriers,which triggers the novel physical phenomena.As regards to Joule heat,we point out that it should be calculated from Ohm’s law,rather than simply from the product of the current and the electric field.Based on this understanding,we conduct thorough discussion on the role of the electric field force in the process of carrier recombination and carrier generation.The thermal effects of carrier recombination and carrier generation followed are incorporated into the thermal equation of energy.The present study shows that the exothermic effect of carrier recombination leads to a temperature rise at the PN interface,while the endothermic effect of carrier generation causes a temperature reduction at the interface.These two opposite effects cause opposite heat flow directions in the PN junction under forward and backward bias voltages,highlighting the significance of managing device heating phenomena in design considerations.Therefore,this study possesses referential significance for the design and tuning on the performance of piezotronic devices.

Electronic band energy of a bent ZnO piezoelectric semiconductor nanowire

The electric band energy variation in a bent piezoelectric semiconductor(PSC) nanowire of circular cross-section induced by the mechanical force is analyzed based on a six-band k · p method. The electric-mechanical fields are first obtained analytically in a cantilever bent PSC nanowire by solving the fully-coupled electro-mechanical equations. Then, the band energy is acquired numerically via the six-band Hamiltonian.By considering further the nonlinear coupling between the piezoelectric and semiconducting quantities, the contribution of the redistribution carriers to the electric field is analyzed from the Gauss’ s law. Numerical examples are carried out for an n-type Zn O nanowire in different locations induced by an applied concentrated end force. They include the electric potential, heavy hole(HH), light hole(LH), spin-orbit split-off(SO),and conduction band(CB) edges along the axial and thickness directions. Our results show that the applied force has a significant effect on the band energies. For instance, on the bottom surface along the axial direction, the bandgaps near the fixed end are greater than those near the loading end, and this trend is reversed on the top surface. Moreover,at a fixed axial location, the energy level of the lower side can be enhanced by applying a bending force at the end. The present results could be of significant guidance to the electronic devices and piezotronics.

Application of the homotopy analysis method to nonlinear characteristics of a piezoelectric semiconductor fiber

Based on the nonlinear constitutive equation,a piezoelectric semiconductor(PSC)fiber under axial loads and Ohmic contact boundary conditions is investigated.The analytical solutions of electromechanical fields are derived by the homotopy analysis method(HAM),indicating that the HAM is efficient for the nonlinear analysis of PSC fibers,along with a rapid rate of convergence.Furthermore,the nonlinear characteristics of electromechanical fields are discussed through numerical results.It is shown that the asymmetrical distribution of electromechanical fields is obvious under a symmetrical load,and the piezoelectric effect is weakened by an applied electric field.With the increase in the initial carrier concentration,the electric potential decreases,and owing to the screen-ing effect of electrons,the distribution of electromechanical fields tends to be symmetrical.

One-dimensional dynamic equations of a piezoelectric semiconductor beam with a rectangular cross section and their application in static and dynamic characteristic analysis

Within the framework of continuum mechanics, the double power series ex- pansion technique is proposed, and a series of reduced one-dimensional （1D） equations for a piezoelectric semiconductor beam are obtained. These derived equations are universal, in which extension, flexure, and shear deformations are all included, and can be degen- erated to a number of special cases, e.g., extensional motion, coupled extensional and flexural motion with shear deformations, and elementary flexural motion without shear deformations. As a typical application, the extensional motion of a ZnO beam is analyzed sequentially. It is revealed that semi-conduction has a great effect on the performance of the piezoelectric semiconductor beam, including static deformations and dynamic be- haviors. A larger initial carrier density will evidently lead to a lower resonant frequency and a smaller displacement response, which is a little similar to the dissipative effect. Both the derived approximate equations and the corresponding qualitative analysis are general and widely applicable, which can clearly interpret the inner physical mechanism of the semiconductor in the piezoelectrics and provide theoretical guidance for further experimental design.

Typical transient effects in a piezoelectric semiconductor nanofiber under a suddenly applied axial time-dependent force

Based on the mechanical motion equation,Gauss’s law,and the current continuity condition,we study a few typical transient effects in a piezoelectric semiconductor(PS)fiber to realize the startup and turning-off functions of common piezotronic devices.In this study,the transient extensional vibration induced by a suddenly applied axial time-dependent force is examined in a cantilevered n-type ZnO nanofiber.Neither the magnitude of the loadings nor the doping concentration significantly affects the propagation caused by disturbance of the axial displacement.However,both of the factors play an important role in the propagation caused by disturbance of the electron concentrations.This indicates that the electromechanical coupling effect can be expected to directly determine the electronic performance of the devices.In addition,the assumption of previous simplified models which neglect the charge carriers in Gauss’s law is discussed,showing that this assumption has a little influence on the startup state when the doping concentration is smaller than 1021 m-3.This suggests that the screening effect of the carriers on the polarized electric field is much reduced in this situation,and that the state is gradually transforming into a pure piezoelectric state.Nevertheless,the carriers can provide a damping effect,which means that the previous simplified models do not sufficiently describe the turning-off state.The numerical results show that the present study has referential value with respect to the design of newly multifunctional PS devices.

Transient analysis on surface heated piezoelectric semiconductor plate lying on rigid substrate

Based on the thermo-electro-elastic coupling theory,the mathematical model for a surface heated piezoelectric semiconductor(PS)plate is developed in the time domain.Applying the direct and inverse Laplace transformations to the established model,the mechanical and electrical responses are investigated.The comparison between the analytical solution and the finite element method(FEM)is conducted,which illustrates the validity of the derivation.The calculated results show that the maximum values of the mechanical and electrical fields appear at the heating surface.Importantly,the perturbation carriers tend to concentrate in the zone near the heating surface under the given boundary conditions.It can also be observed that the heating induced elastic wave leads to jumps for the electric potential and perturbation carrier density at the wavefront.When the thermal relaxation time is introduced,all the field quantities become smaller because of the thermal lagging effect.Meanwhile,it can be found that the thermal relaxation time can describe the smooth variation at the jump position.Besides,for a plate with P-N junction,the effect of the interface position on the electrical response is studied.The effects of the initial carrier density on the electrical properties are discussed in detail.The conclusions in this article can be the guidance for the design of PS devices serving in thermal environment.

Effects of an attached functionally graded layer on the electromechanical behaviors of piezoelectric semiconductor fibers

In this paper,we propose a specific two-layer model consisting of a functionally graded(FG)layer and a piezoelectric semiconductor(PS)layer.Based on the macroscopic theory of PS materials,the effects brought about by the attached FG layer on the piezotronic behaviors of homogeneous n-type PS fibers and PN junctions are investigated.The semi-analytical solutions of the electromechanical fields are obtained by expanding the displacement and carrier concentration variation into power series.Results show that the antisymmetry of the potential and electron concentration distributions in homogeneous n-type PS fibers is destroyed due to the material inhomogeneity of the attached FG layer.In addition,by creating jump discontinuities in the material properties of the FG layer,potential barriers/wells can be produced in the middle of the fiber.Similarly,the potential barrier configuration near the interface of a homogeneous PS PN junction can also be manipulated in this way,which offers a new choice for the design of PN junction based devices.

Nonlinear magneto-mechanical-thermo coupling characteristic analysis for transport behaviors of carriers in composite multiferroic piezoelectric semiconductor nanoplates with surface effect

In this paper,to better reveal the surface effect and the screening effect as well as the nonlinear multi-field coupling characteristic of the multifunctional piezoelectric semiconductor(PS)nanodevice,and to further improve its working performance,a magneto-mechanical-thermo coupling theoretical model is theoretically established for the extensional analysis of a three-layered magneto-electro-semiconductor coupling laminated nanoplate with the surface effect.Next,by using the current theoretical model,some numerical analyses and discussion about the surface effect,the corresponding critical thickness of the nanoplate,and the distributions of the physical fields(including the electron concentration perturbation,the electric potential,the electric field,the average electric displacement,the effective polarization charge density,and the total charge density)under different initial state electron concentrations,as well as their active manipulation via some external magnetic field,pre-stress,and temperature stimuli,are performed.Utilizing the nonlinear multi-field coupling effect induced by inevitable external stimuli in the device operating environment,this paper not only provides theoretical support for understanding the size-dependent tuning/controlling of carrier transport as well as its screening effect,but also assists the design of a series of multiferroic PS nanodevices.

Bending Analysis of Circular Piezoelectric Semiconductor Plates Incorporating Flexoelectricity

Based on the three-dimensional(3D)basic equations of piezoelectric semiconductors(PSs),we establish a two-dimensional(2D)deformation-polarization-carrier coupling bending model for PS structures,taking flexoelectricity into consideration.The analytical solutions to classical flexure of a clamped circular PS thin plate are derived.With the derived analytical model,we numerically investigate the distributions of electromechanical fields and the concentration of electrons in the circular PS thin plate under an upward concentrated force.The effect of flexoelectricity on the multi-field coupling responses of the circular PS plate is studied.The obtained results provide theoretical guidance for the design of novel PS devices.

Surface Elastic Effects on Electromechanical Responses of a Piezoelectric Semiconducting Nanobeam

Piezoelectric semiconductors(PSCs)find extensive applications in modern smart electronic devices because of their dual properties of being piezoelectric and semiconductive.With the increasing demand for miniaturization of these devices,the performance of their components needs to be carefully designed and optimized,especially when reduced to nanosize.It has been shown that surface elastic properties play a substantial role in the mechanical performance of nanoscale materials and structures.Building on this understanding,the surface elastic effects,encompassing surface residual stress,surface membrane stiffness,and surface bending stiffness,are comprehensively taken into account to explore the electromechanical responses of a PSC nanobeam.Additionally,the flexoelectric effect on their responses is also systematically studied.The results of this work reveal that surface elastic properties predominantly influence mechanical performance,while the flexoelectric effect plays a more dominant role in electric-related quantities at the nanoscale.Notably,the significance of surface bending rigidity,which was often underestimated in the earlier literature,is demonstrated.Furthermore,owing to the flexoelectric effect,the linear distribution of electric potential and charge carriers along the length transforms into a nonlinear pattern.The distributions of electric potential and charge carriers across the cross section are also evidently impacted.Moreover,the size-dependent responses are evaluated.Our findings may provide valuable insights for optimizing electronic devices based on nanoscale PSCs.

Analysis of Piezoelectric Semiconductor Structures Considering Both Physical and Geometric Nonlinearities

Piezoelectric semiconductors(PSs),such as ZnO and GaN,known as the third-generation semiconductors,have promising applications in electronic and optoelectronic devices due to the coexistence and interaction of piezoelectricity and semiconductor properties.Theoretical modeling of PS structures under external loads,such as thermal and mechanical loads,plays a crucial role in the design of PS devices.In this work,we propose a nonlinear fully coupling theoretical model and investigate the multi-field coupling behaviors of PS structures and PN junctions under thermal and mechanical loads,considering physical and geometric nonlinearities.The electromechanical and semiconducting behaviors of a PS rod-like structure with flexural deformations under different combinations of temperature changes and mechanical loads are evaluated.The tuning effect of temperature changes and mechanical loads on multi-field coupling behaviors of PSs is revealed.The current–voltage characteristics of PS PN junctions are studied under different combinations of temperature changes and mechanical loads.The obtained results are helpful for the development of novel PS devices.

Electric potential and energy band in ZnO nanofiber tuned by local mechanical loading

Recent success in strain engineering has triggered tremendous interest in its study and potential applications in nanodevice design. In this paper, we establish a coupled piezoelectric/semiconducting model for a wurtzite structure ZnO nanofiber under the local mechanical loading. The energy band structure tuned by the local mechanical loading and local length is calculated via an eight-band k·p method, which includes the coupling of valance and conduction bands. Poisson's effect on the distribution of electric potential inversely depends on the local mechanical loading. Numerical results reveal that both the applied local mechanical loading and the local length exhibit obvious tuning effects on the electric potential and energy band. The band gap at band edges varies linearly with the applied loading. Changing the local length shifts the energy band which is far away from the band edges. This study will be useful in the electronic and optical enhancement of semiconductor devices.