Static bending and wave propagation analyses of a flexoelectric semiconductor nanobeam incorporating antisymmetric thickness-stretch

Static bending and wave propagation analyses of a flexoelectric semiconductor nanobeam incorporating antisymmetric thickness-stretch

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Ziwen Guo, Gongye Zhang^*, Changwen Mi^*

Acta Mechanica Sinica | 2025, 41(12) : 124203

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Acta Mechanica Sinica | 2025, 41(12): 124203

• RESEARCH PAPER •

Static bending and wave propagation analyses of a flexoelectric semiconductor nanobeam incorporating antisymmetric thickness-stretch

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Ziwen Guo, Gongye Zhang^*, Changwen Mi^*

Affiliations

Jiangsu Key Laboratory of Mechanical Analysis for Infrastructure and Advanced Equipment, School of Civil Engineering, Southeast University, Nanjing 210096, China

Published: 2025-12-01 doi: 10.1007/s10409-024-24203-x

Outline

Abstract

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We examine the electromechanical field and charge redistribution within a flexoelectric semiconductor (FS) nanobeam, accounting for bending, fundamental thickness-shear, and antisymmetric thickness-stretch deformations. The coupled governing equations include microstructure, flexoelectric, and semiconductor effects, highlighting the interplay between mechanical displacement, electric potential, and charge carriers. For applications in flexoelectronic devices, the static bending of a simply supported FS beam induced by uniform pressure and wave propagation in an unbounded FS beam are analytically addressed using the derived framework. The effects of antisymmetric thickness-stretch on mechanical displacements and electron concentration perturbation, as well as size dependence of microstructure and flexoelectric effects, are identified. An interesting finding reveals that wave frequencies of the antisymmetric thickness-stretch mode, as anticipated by the proposed model, are larger compared to those of the model neglecting flexoelectric and semiconductor effects. For the first time, the cutoff frequency of antisymmetric thickness-stretch impacted by the two features is explained mathematically. These findings are beneficial for enhancing the performance of flexoelectronic sensors and electroacoustic devices.

Key words

Flexoelectric semiconductor / Size-dependent effect / Antisymmetric thickness-stretch / Principle of virtual work

Cite this Article

Ziwen Guo, Gongye Zhang, Changwen Mi. Static bending and wave propagation analyses of a flexoelectric semiconductor nanobeam incorporating antisymmetric thickness-stretch[J]. Acta Mechanica Sinica, 2025 , 41 (12) : 124203 - . DOI: 10.1007/s10409-024-24203-x

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1.　Introduction

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Flexoelectricity, characterized by polarization due to non-uniform strain, is allowed by symmetry in all crystalline solids, rendering it a more widespread property compared to the traditional piezoelectric effect [1-10]. However, the polarization charge generated by flexoelectricity in bulk materials typically remains small, which limits its practical application. Interestingly, as strain gradients tend to increase inversely with diminishing sample size, the flexoelectric effect demonstrates a size-dependent behavior [11-17]. This feature presents a promising avenue for maximizing flexoelectric response by working at the nanoscale. As a result, flexoelectricity has garnered attention for its prospective utility in considerable applications such as nanogenerators, sensors, and actuators [3,18-21]. Notably, unlike their macroscopic counterparts, nanoscale structures exhibit a dependence on their characteristic sizes, known as the microstructure effect [22-24]. Hence, it is imperative to concurrently consider the size-dependent effects of flexoelectricity and microstructure while modeling flexoelectric devices.

Due to the absence of charge carriers, piezoelectric/flexoelectric dielectrics are limited to transmitting the function of an external electric field solely through induction. The advent of third-generation semiconductors in the last century has opened up a new possibility for integrating piezoelectric and semiconductor properties, with the goal of establishing a direct interrelation between mechanical motion and electronic devices [25,26]. Theoretical and experimental studies focusing on electronic behaviors at the interface/junction have formed a new field termed “piezotronics” [27-38]. Recently, Wang et al. [39] also demonstrated an interaction within centrosymmetric semiconductors between mechanical fields and mobile charges. This finding prompts systematic investigations into flexoelectronics. Sun et al. [40] conducted an empirical investigation into the current-voltage characteristics of a silicon-based Schottky barrier diode under various tip forces and clarified the tuning mechanism of flexoelectricity on charge motion behaviors. In pursuit of direct electromechanical functionality, Guo et al. [41] introduced the application of silicon flexoelectronic transistors. Their findings underscored that flexoelectric polarization induced in inhomogeneously strained crystals significantly affects the redistribution of charge carriers.

To elucidate the complex process by which mechanical force modulates free-carrier motions within flexoelectric semiconductors (FSs), mechanics researchers have conducted insightful investigations on FS fibers and films. Zhao et al. [42] employed the drift-diffusion theory to investigate the extension coupling electromechanical properties of a GaN nanowire, exploring the synergistic impacts arising from strain gradient-induced flexoelectricity. PN hetero-junctions in an FS composite beam were numerically studied by Li et al. [43]. Qu et al. [44,45] established electromechanical coupling models of FS beams and their composite structures based on the Bernoulli-Euler theory. Moreover, Qu et al. [46] examined the interplay of acoustic waves and charge carriers in FS plates, employing diverse structural theories including Kirchhoff-Love and Mindlin-Reissner models. Taking into account the Mindlin-Medick assumption, Zhang et al. [47] realized charge redistribution through the warping of an FS composite beam.

The Mindlin-Medick assumption incorporates five deformation modes, which can be decoupled into two groups within uniform beams or plates. The first group consists of traditional extension, fundamental thickness-stretch, and (second-order) symmetric shear, while the other includes bending and fundamental thickness-shear [22]. The latter can actually be considered as a first-order truncation, wherein a higher-order deformation along the thickness direction (specifically, the antisymmetric thickness-stretch) remains unexplored. To better understand the role of antisymmetric thickness-stretch in regulating flexoelectronic performances and reveal potential new phenomena, this paper presents a main contribution to exploring the static bending and wave propagation properties of an FS nanobeam within antisymmetric thickness-stretch, where microstructure, flexoelectric, and semiconductor effects are fully investigated. The comprehension of these issues will contribute to a more accurate analysis of FS structures, thereby facilitating the optimization of flexoelectronic sensor and electroacoustic device designs.

The following outlines the structure of this paper: In Sect. 2, the governing equations and boundary conditions (BCs) for an FS nanobeam are rigorously deduced using the principle of virtual work (PVW). Section 3 presents numerical analyses focusing on two aspects―one is the static bending of a simply supported FS beam induced by a uniform pressure, and the other is the wave propagation in an unbounded FS beam. The paper concludes with summaries and key findings in Sect. 4.

2.　One-dimensional governing equations of an FS beam

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For n-type centrosymmetric semiconductors, the general constitutive relations describing strain gradient, flexoelectric, and semiconductor effects are as follows [25,45,48,49]:

(1a)

(1b)

(1c)

(1d)

where T_ij and τ_ijk are the Cauchy stress and higher-order stress tensors, respectively; D_i and

represent the electric displacement vector and electron current density vector; c_ijkl, f_ijkl, ε_ij, and g_ijklmn are the elastic constants, flexoelectric coefficients, dielectric constants, and higher-order elastic constants, respectively; the elementary charge, denoted as q, equals 1.6 × 10⁻¹⁹ C; n signifies the concentration of electrons;

and

are the carrier mobility and diffusion constant for electrons; S_kl and η_jkl are defined as the strain and strain gradient components, respectively; E_j denotes the electric field, whereas N_j refers to the gradient of concentration of electrons. These quantities can be characterized by the following kinematic relations:

(2)

with u_i and φ standing for the mechanical displacement and electrostatic potential, respectively.

We treat

, where n₀ is the doping level for electrons; in contrast, Δn and

are the concentration perturbation of electrons and concentration of ionized donors, respectively. For tiny concentration perturbations, Eq. (1d) may be linearized as

(3)

which is sufficient to describe the material behaviors.

Consider an FS beam featuring a rectangular cross-section, as illustrated in Fig. 1(a). Deformations of bending, fundamental thickness-shear, and thickness-stretch are considered and shown in Fig. 1(b)-(d). Following that, the displacement field in the FS beam can be described as

(4)

where u₃⁽⁰⁾ and u₁⁽¹⁾ are the bending and fundamental thickness-shear, respectively; u₃⁽²⁾ represents the (second-order) thickness-stretch. Note that u₃⁽²⁾ discussed here is antisymmetric (Fig. 1(d)), which is different from the symmetric thickness-stretch that coupled with extension and symmetric shear [50].

The electric potential and electron concentration perturbation can be estimated using

(5)

where φ⁽¹⁾ and n⁽¹⁾ represent the first-order electric potential and first-order electron concentration, respectively. It is seen that Eq. (5) is limited to describing the variation of electric potential and electron concentration anti-symmetrically along the thickness direction of the beam. Note that a higher-order description is required for more complex distributions of φ and Δn [51]. Subsequently, the substitution of Eqs. (4) and (5) into Eq. (2) leads to the component forms of strain, strain gradient, electric field intensity, and electron concentration gradient in the FS beam.

Based on the PVW approach, the field equations and corresponding BCs of an arbitrary FS structure can be obtained from [52]

(6)

where δW₍_d₎, δW_{(_c)}, and δW_{(_i)} are defined as the virtual work of actions at a distance, contact forces, and internal forces [50]. In this circumstance, no surface phenomena are applied to the beam, i.e., δW₍_c₎ = 0. δW₍_d₎ and δW_{(_i)} are

(7a)

(7b)

with V being the region occupied by the structure; ρ stands for mass density; λ is a constant satisfying all terms related to concentrations of charge carriers; f_i is the mechanical body force. The first- and second-time derivatives are represented by the overhead “·” and “··” in this instance and the sequel

Using Eqs. (4) and (5) into Eq. (7a) gives

(8)

where

(9)

are the body force, mass density, and elementary charge resultants across the cross-section area A of the beam, respectively.

According to Eqs. (2), (4), (5), and (7b), we have

(10)

where

(11)

are the stress, higher-order stress, electric displacement, and electron electric current resultants on the beam cross-section, respectively.

Substituting Eqs. (8) and (10) into Eq. (6) and applying the fundamental lemma of the calculus of variations results in

(12)

Note that Eq. (12) represents the coupled field equations for the established FS beam model. These equations hold true at any moment inside the temporal interval [t₀, t₁]. The associated BCs are given as

(13)

with the overhead bars representing the prescribed value. For specific constraints, natural BCs or essential BCs can be chosen from these general options.

It is vital to emphasize that the field Eq. (12) and the BCs (13) are applicable to arbitrarily anisotropic FS beams. In the subsequent development, the beam is taken to be made from a centrosymmetric semiconductor with m3m point group. Then, on the basis of material matrices provided in Appendix A, the constitutive relations in terms of field variables can be obtained as

(14)

(15)

(16)

(17)

Note that in Eq. (15), the relation g_ijklmn = ℓ²c_ijlmδ_kn is employed through simplifying the general form of the strain gradient theory [53], as demonstrated by Altan and Aifantis [54]. Here, ℓ denotes the sole length scale parameter that adequately captures the microstructure effect.

Substituting Eqs. (14)-(17) into Eq. (11), the corresponding resultants may be written in terms of u₃^(0), u₁⁽¹⁾, u₃⁽²⁾, φ⁽¹⁾, and n⁽¹⁾, i.e.,

(18)

(19)

(20)

(21)

where I₂=bh³/12 and I₄=bh⁵/80 represent the second and fourth moment of area, respectively.

Using Eqs. (18)-(21) in Eq. (12) then leads to the governing equations for the proposed FS beam as

(22a)

(22b)

(22c)

(22d)

(22e)

The governing equations in Eqs. (22a)-(22e), together with the related BCs described in Eq. (13), formally define the boundary-initial value problem for determining u₃⁽⁰⁾, u₁⁽¹⁾, u₃⁽²⁾, φ⁽¹⁾, and n⁽¹⁾. Furthermore, this issue requires initial conditions to be provided at instance t = 0. By disregarding the antisymmetric thickness-stretch deformation u₃⁽²⁾, Eqs. (22a)-(22e) reduce to

(23)

which is a Timoshenko beam model that incorporates microstructure, flexoelectric, and semiconductor effects.

Furthermore, we introduce several non-dimensional variables as follows:

(24)

where T is an arbitrary time interval. Hence, the non-dimensional governing equations become

(25a)

(25b)

(25c)

(25d)

(25e)

where the nondimensional coefficients are

(26a)

(26b)

(26c)

(26d)

(26e)

(26f)

3.　Examples: static bending and wave propagation analyses

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To comprehensively investigate the role of antisymmetric thickness-stretch in the FS nanobeam model developed in the previous section, two sample problems addressing different aspects are examined here. One involves the static bending of a simply supported FS beam induced by uniform pressure, and the other examines the wave motions in an unbounded FS beam. Together, these problems provide a detailed description of electromechanical interactions in the FS nanobeam when antisymmetric thickness-stretch is taken into account.

3.1　Static bending

According to Eq. (13), the specific BCs at x₁ = 0 and x₁ = L for a simply supported FS beam associated with Ohmic contact conditions (see Fig. 2) are

(27a)

(27b)

(27c)

(27d)

(27e)

(27f)

(27g)

(27h)

Using Eqs. (18) and (19) in Eqs. (27b), (27e), (27f) yields

(28)

The non-dimensional BCs are

(29)

In the case of static bending, a uniformly transverse load ϑ (i.e., f₃⁽⁰⁾=ϑ) is exerted on the upper surface of the beam, while other body forces and all the time derivatives vanish. Then the boundary value problem can be analytically addressed using the subsequent Fourier series solutions:

(30)

where ξ_k=kπh/L;

and

are the unknown Fourier coefficients. It can be easily seen that U₃⁽⁰⁾, U₁⁽¹⁾, U₃⁽²⁾, Ψ, and N satisfy the BCs in Eq. (29) for any

and

The non-dimensional mechanical load

in Eq. (25a) can similarly be expanded as

(31a)

(31b)

Inserting Eqs. (30) and (31a) into Eqs. (25a)-(25e) yieds

(32)

where [K_ij] is a 5 × 5 matrix detailed in Eq. (B1) in Appendix B. By solving the system of Eq. (32) will yield

These can then be substituted into Eq. (30) to derive the accurate solutions of U₃⁽⁰⁾, U₁⁽¹⁾, U₃⁽²⁾, Ψ, and N in the static bending issue of the simply supported FS beam displayed in Fig. 2.

For illustrative purposes, the geometrical parameters of the nanobeam are considered to be moderately thick with h = 20 nm, b = 2h, and L = 5h. The beam is chosen to be silicon. The mass density, elastic constants, dielectric coefficient, electron mobility, and diffusion constant are provided in Sze and Ng’s work [55]. The flexoelectric coefficients are f₁₁ = 1.3 nC/m and f₁₄ = f₁₁₁ = 0.4 nC/m [39].

The total terms present in each expansion within Eq. (30) can be regulated by adjusting the parameter k, necessitating a convergence analysis. To ensure that the analytical solution is accurate enough, the charge accumulation amount Q on the upper half of the beam

varying with k is calculated. It is found that the numerical results obtained with k = 25 are identical to those calculated with higher k values (up to k = 40), matching the third decimal place for Q (= –8.654 × 10^–23 C). Considering the linear coupling relationship between mechanical displacements, electric potential, and electron concentration perturbation within the beam, as well as the correspondence between Q and Δn, it can be concluded that utilizing the first 25 terms in Eq. (30) is adequate for the convergence of all the field variables.

We place greater emphasis on the effect of the antisymmetric thickness-stretch of the beam on electromechanical interactions. Figure 3(a) and (b) illustrates the bending and fundamental thickness-shear at the top surface of the beam where x₃ = h/2. It can be observed that when u₃⁽²⁾ is excluded (Timoshenko beam model), both bending and fundamental thickness-shear deformations slightly diminish compared to those in the current beam model.

Figure 4(a) and (b) depicts the electron concentration perturbation Δn of the FS beam in a 3D view, resulting from the uniformly distributed stress ϑ. It is evident that electrons fluctuate towards the bottom of the beam, resulting in a negative charge accumulation in that area. Along the same cross-section, charge redistribution displays antisymmetry across the neutral layer of the beam. The charge carriers exhibit near uniformity along the surfaces of the FS beam, making it easy to drain or inject via top and bottom electrodes when serving as a circuit element. The resulting numerical findings provide valuable insights for elucidating the mechanism of charge redistribution within the FS beam.

Furthermore, it is showcased that the concentration perturbation of electrons predicted by the two beam models differ significantly (with a discrepancy of more than tenfold), indicating that the flexoelectric polarization produced by u₃⁽²⁾ is much greater than that of u₃⁽⁰⁾ but in the opposite direction. This underscores the importance of incorporating antisymmetric thickness-stretch in the examination of flexoelectronic properties within FS structures.

The magnitude of φ⁽¹⁾ and n⁽¹⁾ under varying ϑ and n₀ are explored as part of the parameter studies, as illustrated in Figs. 5 and 6. These figures reveal that increasing load magnitude significantly enhances both the electric potential and electron concentration. Particularly, the relationship between load magnitude and the related fields is linear at any specific point within the beam, as indicated by the linear differential equation (32). Figure 6(a) demonstrates that doping levels have a minor inhibitory influence on the electric potential. This behavior is attributed to the redistribution of electrons, which shield the flexoelectric polarization and result in a decrease in the electric potential’s amplitude. The electric potential is expected to reach its peak when the semiconductor effect disappears. In contrast, when the value of n₀ is increased, the electromechanical interactions in the beam change more quickly. As a result, the electron concentration becomes highly responsive to changes in n₀, as shown in Fig. 6(b).

We extended our research to explore the tuning mechanism of microstructure and flexoelectric effects on mechanical displacements in the suggested beam model. Figure 7(a)-(c) visualizes the variations of bending, fundamental thickness-shear, and antisymmetric thickness-stretch at the top surface of the beam along the axial direction x₁ with different length scale parameter ℓ, where the displacement curves neglecting both microstructure and flexoelectric effects are also plotted for comparison (depicted by the black solid line). Note that when the flexoelectric effect diminishes, the absence of the electric field hinders the interaction between mobile charges and displacements, thereby resulting in the disappearance of the semiconductor effect. Our observations bring to light that considering length scale parameters or flexoelectric coefficients will reduce all the displacements. This stiffening phenomenon can be attributed to the size-dependent effect at the micro and nanometer scales [13,56,57]. Furthermore, Fig. 7(a)-(c) illustrates that the microstructure effect significantly influences u₃⁽⁰⁾, u₁⁽¹⁾, and u₃⁽²⁾ of the beam, while the impact of the flexoelectricity is negligible for the specific material properties and geometry parameters examined in this study.

3.2　Wave propagation

In the second example study, we examine the propagation characteristics of straight-crested waves with infinite boundaries (L = ∞) along the x₁ direction. Without body forces, the non-dimensional field variables in the beam can be assumed in the following form:

(33)

where

and

are non-dimensional wave amplitudes, i is the imaginary unit satisfying i² = −1, while ζ and Ω are denoted as the non-dimensional wave number and angular wave frequency, respectively.

Substituting Eq. (30) back into Eqs. (25a)-(25e) yields

(34)

where [M_ij] is a 5 × 5 matrix detailed in Eq. (B2) in Appendix B.

For non-zero solutions, the determinant of the coefficient matrix in Eq. (34) must vanish. This determinant is stated as a power function that depends on the wave frequency and the wave number. In actual application, a numerical approach has been constructed to determine the dispersion relations. By supplying a sequence of wave numbers, the associated wave frequencies are determined to match the zero-determinant requirement. Consequently, this technique permits the discovery of a total of nine dispersion relations. On the other hand, for any given combination of wave number and wave frequency satisfying a certain dispersion relation, the ratios among the five unknown wave amplitudes may be ascertained using Eq. (34).

Figure 8 displays the dispersion relations resulting from Eq. (34). The material parameters employed are consistent with those detailed in Sect. 3.1. Given the specified material and structure properties, only three groups of the nine dispersion relations demonstrate positive real wave frequencies, as electric waves are unable to propagate independently. In ascending sequence, they are designated as bending, fundamental thickness-shear, and antisymmetric thickness-stretch waves.

In Fig. 8, it becomes apparent that the wave frequency values corresponding to the antisymmetric thickness-stretch mode anticipated by the present model surpass those of the model lacking flexoelectric and semiconductor effects (“without FS”). Conversely, the frequencies associated with the other two waves remain consistent between the two models. To distinguish between the mechanisms of flexoelectric and semiconductor effects on the antisymmetric thickness-stretch waves, an analytical expression for the cutoff frequency of antisymmetric thickness-stretch in the proposed FS beam model is given by

(35)

where W^ATS is the cutoff frequency of antisymmetric thickness-stretch, and γ stands for the flexoelectric and semiconductor effects. Recall that the cutoff frequency here relates to the boundary frequency below which the waves cannot pass. From Eq. (35), it becomes evident that the incorporation of the flexoelectric effect (with f₁₁ ≠ 0) would always result in a higher W^ATS. Nevertheless, it is notable that W^ATS predicted by the current model diminishes as the semiconductor effect increases (i.e., as n₀ ↑). The comprehension of the wave motion mechanism is crucial for the optimization design of electroacoustic devices.

To further validate the accuracy of the results, the dispersion curves calculated based on linear elasticity theory (without flexoelectric and semiconductor effects) are compared with the exact solutions obtained using the multi-dimensional moduli ratio convergence method [58-60], as depicted in Fig. 9. It is evident that within the displayed range of wave numbers and wave frequencies, the approximate solution aligns closely with the exact one. Notably, the dispersion curves for bending waves in both cases are nearly identical, demonstrating the correctness of the theoretical model established in this study.

4.　Concluding remarks

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A new FS beam model integrating bending, fundamental thickness-shear, and antisymmetric thickness-stretch is established. The governing equations and complete BCs are developed through the PVW approach for FS, where microstructure, flexoelectric, and semiconductor effects are simultaneously treated in a cohesive framework. Numerical results in the static bending case underscore the critical significance of considering antisymmetric thickness-stretch in analyzing the flexoelectronic properties of nanobeams. The analytical investigation into wave propagation reveals that the wave frequencies of the antisymmetric thickness-stretch mode anticipated by the current model are higher than those of the elastic model ignoring flexoelectric and semiconductor effects. Moreover, our analysis explores the intricate interplay between these effects on the cutoff frequency, elucidating their distinct contributions to wave motions.

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Year 2025 volume 41 Issue 12

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doi: 10.1007/s10409-024-24203-x

Receive Date：2024-06-25
Online Date：2026-03-24
Published：2025-12-01

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Received：2024-06-25
Accepted：2024-10-09

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Jiangsu Key Laboratory of Mechanical Analysis for Infrastructure and Advanced Equipment, School of Civil Engineering, Southeast University, Nanjing 210096, China

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^* E-mail addresses: gyzhang@seu.edu.cn (Gongye Zhang);

mi@seu.edu.cn (Changwen Mi)

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表12种不同金属材料的力学参数

科 Family	属数 Number of genus	种数 Number of species	占总种数比例 Percentage of total species (%)	属 Genus	种数 Number of species	占总种数比例 Percentage of total species (%)
鹅膏菌科Amanitaceae	2	11	5.26	鹅膏菌属 Amanita	10	4.78
小菇科 Mycenaceae	2	12	5.74	丝盖伞属 Inocybe	5	2.39
多孔菌科 Polyporaceae	8	14	6.70	蜡蘑属 Laccaria	5	2.39
红菇科 Russulaceae	3	23	11.00	小皮伞属 Marasmius	6	2.87
				小菇属 Mycena	11	5.26
				光柄菇属 Pluteus	5	2.39
				红菇属 Russula	17	8.13
				栓菌属 Trametes	5	2.39

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Figure 1 FS beam and corresponding deformation modes.

Figure 2 Sketch of a simply supported FS beam and the coordinate system.

Figure 3 Effect of antisymmetric thickness-stretch on (a) bending and (b) fundamental thickness-shear (ℓ = 1 × 10^–9 m, n₀ = 1 × 10²¹ m^–3, ϑ = 1 N/m).

Figure 4 Effect of antisymmetric thickness-stretch on electron concentration perturbation: (a) current beam model and (b) Timoshenko beam model (ℓ = 1 × 10^–9 m, n₀ = 1 × 10²¹ m^–3, ϑ = 1 N/m).

Figure 5 Effect of load magnitudes on (a) φ⁽¹⁾ and (b) n⁽¹⁾ (ℓ = 1 × 10^–9 m, n₀ = 1 × 10²¹ m^–3).

Figure 6 Effect of doping levels on (a) φ⁽¹⁾ and (b) n⁽¹⁾ (ℓ = 1 × 10^–9 m, ϑ = 1 N/m).

Figure 7 Variations of the mechanical displacements: (a) bending, (b) fundamental thickness-shear, (c) antisymmetric thickness-stretch, and (d) corresponding deformed shapes (n₀ = 1 × 10²¹ m^–3, ϑ = 1 N/m).

Figure 8 Dispersion relation of bending, fundamental thickness-shear, and antisymmetric thickness-stretch (ℓ = 1 × 10^–9 m, n₀ = 1 × 10²¹ m^–3, b = 2h = 40 nm).

Figure 9 Dispersion relation of bending, fundamental thickness-shear, and antisymmetric thickness-stretch of the present approximate solution compared to the exact solution (without flexoelectric and semiconductor effects).

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