Theoretical Nanoarchitectonics of GaN Nanowires for Ultraviolet Irradiation-Dependent Electromechanical Properties

In this paper, we propose a one-dimensional model that combines photoelectricity, piezoelectricity, and photothermal effects. The influence of ultraviolet light on the electromechanical coupling properties of GaN nanowires is investigated. It is shown that, since the ultraviolet photon energy is larger than the forbidden gap of GaN, the physical fields in a GaN nanowire are sensitive to ultraviolet. The light-induced polarization can change the magnitude and direction of a piezoelectric polarization field caused by a mechanical load. Moreover, a large number of photogenerated carriers under photoexcitation enhance the current density, whilst they shield the Schottky barrier and reduce rectifying characteristics. This provides a new theoretical nanoarchitectonics approach for the contactless performance regulation of nano-GaN devices such as photoelectric sensors and ultraviolet detectors, which can further release their great application potential.


Introduction
As a direct wide band gap semiconductor material, GaN exhibits high-power density, superior electrical and thermal conductivities, strong radiation resistance, and high breakdown voltage [1]. Owing to the size effect, GaN nanostructures exhibit a smaller Young's modulus and a higher quality factor, which makes them have obvious advantages in the application of nano-mechanical systems [2,3]. In addition, because of the piezoelectric and semiconductive properties, piezoelectric potential generated in a crystal can effectively regulate the carrier transport capacity of an interface/unction region under mechanical loading [4,5]. Such a unique synergistic effect makes the piezoelectric potential produce a similar "gate circuit", and hence, many novel modern electromechanical coupling devices have been developed, such as piezoelectric charge-coupled devices [6][7][8] and energy conversion supplies [9][10][11][12][13]. Inevitably, there is an urgent need for the active regulation of device performance.
GaN is a natural optoelectronic material. Due to the fact that the ultraviolet photon energy is larger than the forbidden gap of GaN (the energy band difference between conduction and valence bands [14]), the physical fields in a GaN nanowire are sensitive to ultraviolet. Irradiation can stimulate the generation, separation, transport, and recombination of carriers in GaN [15]. The available studies have mainly focused on regulating the electrical transport characteristics of piezoelectric semiconductors (PSCs) under mechanical loads [16] and doping [17]. There are few reports on regulating their electromechanical properties by means of light excitation [18][19][20][21]. For example, based on the photoelectric experiment of ZnO nanowires, Wang and Zhang found that ultraviolet light can weaken Schottky's rectification characteristics [22][23][24]. To the best of our knowledge, however, it is still a lack of quantitative analysis from the theoretical and numerical aspects. In practical applications, however, a device is inevitably affected by light irradiation, which can deteriorate the electrical properties of GaN materials through the generation-recombination of carriers. That is, radiation causes a certain disturbance of electrical properties. Therefore, it is necessary to investigate the influence of light irradiation on the electromechanical properties of GaN structures. It is expected that a new approach can be developed for the performance regulation of GaN devices without doping.
In this paper, GaN nanowires under a combined photoexcitation and electrical load are comprehensively investigated by using both theoretical and numerical methods. The paper is organized as follows. In Section 2, a one-dimensional (1-D) thermo-piezoelectric theory is first proposed, including the photoconductive and photothermal effects. Then, in Section 3, the influence of ultraviolet irradiation is analyzed on the physical fields of GaN nanowires, and photoexcitation regulation of the electrical transport properties is discussed in an Ag-GaN Schottky junction. Finally, several main conclusions are summarized in Section 4.

Basic Equations for a GaN Nanowire under Light Irradiation
Uniform light irradiation can cause temperature rising in a semiconductor structure. According to the principle of heat balance [25], the change in temperature versus time yields where C T is the heat capacity, and G T = H g A g represents the heat exchange coefficient, with the air thermal convection coefficient H g = 4 W K −1 m −2 and A g the contact area between the semiconductor and air [26]. T p and T 0 are the initial and room temperatures, respectively. S denotes the illumination area, and I is the light intensity. At the beginning of illumination, the amount of variation in temperature ∆T(0) = 0. Solving Equation (1), the temperature change can be obtained as where ∆T opt = S I/G T is the maximum change of temperature in a steady state, and τ θ is the thermal time constant. Here, it is worth noting that, when the illumination time t τ θ , the temperature change ∆T (t) = ∆T opt = S I/G T . The electron-hole pairs in a semiconductor can be excited when the photon energy is higher than the band gap width [22]. That is, GaN can be excited by ultraviolet light to produce non-equilibrium carriers. Under uniform illumination, the non-equilibrium carrier concentrations in a steady state are described as [13] ∆n opt = βαP opt λ/hce −αd τ n , where β is the internal quantum efficiency, representing the number of photocarrier pairs excited by each photon. α is the absorption coefficient, P opt is the illumination intensity, λ is the wavelength, and h and c are the Planck constant and light velocity, respectively. d represents the incident depth, and τ n and τ p are the lifetimes of non-equilibrium electrons and holes, respectively. As illustrated in Figure 1, let us assume that photoexcited carriers gradually decay with the transmission depth of incident light. In the case of a 1-D nanowire with a radius of r, the average concentration of photo-generated carriers on its cross-section can be expressed as ∆n opt = ( x 2 +y 2 ≤r 2 ∆n opt dxdy)/πr 2 , terials 2023, 16, x FOR PEER REVIEW 3 of 11 of r, the average concentration of photo-generated carriers on its cross-section can be expressed as Taking a 1-D GaN nanorod with a length of 2L as an example (see Figure 2), the caxis is along the z-direction, with a uniform beam of ultraviolet light vertically irradiated on the upper surface. Its physical and mechanical behaviors are governed by the motion equation, electrostatics Gauss's law, and the current continuity equation [27][28][29][30], that is 0, where σzz, Dz, J n z , and J p z denote the stress tensor, electric displacement, electron concentration density, and hole current density. N + D and N − A represent the ionization degrees of donor and acceptor impurities, respectively. q is the unit charge (1.602 × 10 −19 C), and n and p are the electron and hole doping concentrations. Un and Up are the net recombination rates of free electrons and holes. Here, the generation and recombination of free electrons and holes are in a dynamic equilibrium state, that is, Un and Up are equal to 0. Taking a 1-D GaN nanorod with a length of 2L as an example (see Figure 2), the c-axis is along the z-direction, with a uniform beam of ultraviolet light vertically irradiated on the upper surface. Its physical and mechanical behaviors are governed by the motion equation, electrostatics Gauss's law, and the current continuity equation [27][28][29][30], that is where σ zz , D z , J n z , and J p z denote the stress tensor, electric displacement, electron concentration density, and hole current density. N + D and N − D represent the ionization degrees of donor and acceptor impurities, respectively. q is the unit charge (1.602 × 10 −19 C), and n and p are the electron and hole doping concentrations. U n and U p are the net recombination rates of free electrons and holes. Here, the generation and recombination of free electrons and holes are in a dynamic equilibrium state, that is, U n and U p are equal to 0.  For a 1-D PSC with the polarization direction along the z-axis, the constitutive equation in Cartesian coordinates can be written as [31][32][33][34] where ε zz is the strain tensor, E z is the electric field strength, c 33 and e 33 are the elastic and piezoelectric coefficients, κ 33 is the dielectric constant, λ 33 is the thermal expansion coefficient, and p 33 is the pyroelectric coefficient.
where k B is Boltzmann's constant and T 0 is the reference temperature. The strain ε zz and the electric field E z are related to the mechanical displacement u and the electric potential ϕ, respectively, that is where u z and ϕ are the mechanical displacement and electric potential, respectively. Substituting Equation (6) into Equation (4), the governing equations are obtained by c 33 For an n-type GaN nanowire, the concentrations of acceptor and donor impurities are N + D = 1 × 10 23 m −3 and N - where N i is the concentration of intrinsic carriers. Other relevant material constants are listed in Table 1 [36][37][38][39][40]. Generally speaking, an analytic solution for such a nonlinear model is difficult to be obtained. Hence, to solve the photoexcitation physical problem, a numerical iterative method is adopted by using the PDE module in COMSOL Multiphysics software. Here it is worth noting that Guo and Yang obtained the approximate analytical solution of 1-D piezoelectric semiconductors by a perturbation method, and in comparison with the results from COMSOL, it can be applied to verify the reliability and accuracy of our calculation [41,42].

Results and Discussion
Under the ultraviolet light with a wavelength of 350 nm and the mechanical conditions as illustrated in Figure 2, when there is no applied current across (in and out) the nanorod, the boundary conditions at the two ends can be written as Here, n and p satisfy the following electrical neutral conditions At the position z = 0, the conditions of displacement and potential are Due to the photoconductive effect, a large number of photogenerated carriers are produced with the increase in light intensity, which obviously enhances the concentration of free electrons and holes (see Figure 3a,b). In addition, ultraviolet light changes the distribution of carriers, especially at both ends. This is attributed to the synergy of piezoelectric and pyroelectric effects. That is, due to the photothermal effect, temperature increases under irradiation of light, which leads to the separation of positive and negative ions in GaN nanowires, resulting in pyroelectric charges. The polarity of pyroelectric polarization is opposite to that of piezoelectric polarization. With the increase of irradiation intensity, pyroelectric polarization becomes dominant, which changes the distribution of piezoelectric polarization charges (see Figure 4a). Similarly, because the direction of pyroelectric potential is opposite to the piezoelectric field generated by pressure, the piezoelectric potential is weakened and the comprehensive potential is even reversed (see Figure 4b,c). Consistent with the potential, the piezoelectric field decreases with the light intensity. When the light intensity reaches a certain value, the pyroelectric field plays a dominant role, and the comprehensive polarization field in GaN is opposite to that without ultraviolet light (see Figure 4d). distribution of piezoelectric polarization charges (see Figure 4a). Similarly, because the direction of pyroelectric potential is opposite to the piezoelectric field generated by pressure, the piezoelectric potential is weakened and the comprehensive potential is even reversed (see Figure 4b,c). Consistent with the potential, the piezoelectric field decreases with the light intensity. When the light intensity reaches a certain value, the pyroelectric field plays a dominant role, and the comprehensive polarization field in GaN is opposite to that without ultraviolet light (see Figure 4d).   Let us take the Ag electrodes at both ends of a GaN nanowire as an example, w forms a double Schottky contact. As illustrated in Figure 2, the boundary condition  Let us take the Ag electrodes at both ends of a GaN nanowire as an example, which forms a double Schottky contact. As illustrated in Figure 2, the boundary conditions are where V a is an applied bias voltage, V bi is the built-in voltage, and V b is the piezoelectric potential (i.e., the potential difference between the two ends caused by photoexcitation). V n rec and V p rec denote the thermal recombination velocities of electrons and holes at the Schottky interface, respectively. n m and p m are the critical electron and hole concentrations, which can be represented as [43][44][45][46][47] where Φ B is the GaN surface barrier when the electron energy is equal to the Fermi level, and T is the absolute temperature. m e = 1.82 × 10 −31 kg is the effective mass of conduction band electrons, and N c = 2.23 × 10 24 m −3 denotes the effective density of states of conduction bands. The built-in voltage, V bi is defined by [43][44][45] V bi = Φ B − k B T/q ln(N c /n 0 ), (15) and the Schottky contacts barrier height qΦ B can be represented as [43,44] which is the difference between the working function of silver (qΦ M = 4.26 eV) [23] and the electron affinity (qχ = 4.1 eV) of GaN [38], leading to the potential difference, Φ B = 0.16 eV. It is seen from Figure 5a that ultraviolet light can significantly change the I-V characteristics of a GaN Schottky junction. In the absence of light, there is an obvious rectification characteristic due to the Schottky barrier between GaN nanowires and Ag electrode. However, with the increase of ultraviolet power density, the Schottky contact rectification weakens and gradually shifts to the Ohmic contact. This is due to the pyroelectric polarization caused by the photothermal effect, which reduces the barrier height of a junction region. In addition, a large number of carriers generated by photoexcitation have a shielding effect on characteristics of the Schottky rectifier. When the photo-generated carriers increase to a particular concentration, the Schottky barrier is completely shielded, and the rectifying characteristics may be lost. That is, ultraviolet excitation can be applied to regulate the electrical transport behavior of GaN nanodevices.  (15) and the Schottky contacts barrier height qΦB can be represented as [43,44] , BM q q q  =  − (16) which is the difference between the working function of silver (qΦM = 4.26 eV) [23] and the electron affinity (qχ = 4.1 eV) of GaN [38], leading to the potential difference, ΦB = 0.16 eV.
It is seen from Figure 5a that ultraviolet light can significantly change the I-V characteristics of a GaN Schottky junction. In the absence of light, there is an obvious rectification characteristic due to the Schottky barrier between GaN nanowires and Ag electrode. However, with the increase of ultraviolet power density, the Schottky contact rectification weakens and gradually shifts to the Ohmic contact. This is due to the pyroelectric polarization caused by the photothermal effect, which reduces the barrier height of a junction region. In addition, a large number of carriers generated by photoexcitation have a shielding effect on characteristics of the Schottky rectifier. When the photo-generated carriers increase to a particular concentration, the Schottky barrier is completely shielded, and the rectifying characteristics may be lost. That is, ultraviolet excitation can be applied to regulate the electrical transport behavior of GaN nanodevices. As shown in Figure 5b, the current increases gradually with the light intensity, regardless of a positive or a negative bias voltage. However, the increasing amplitude becomes smaller and tends to be saturated. That is mainly due to the photoconductivity effect, producing a large number of photo-generated carriers that increases the current density. In addition, the photothermal effect increases with the optical power density and produces a thermal electric field, which enhances the Schottky barrier and the shielding effect of the Schottky junction, and thus, the current is gradually saturated [47,48].

Conclusions
By using a 1-D PSC model, we have investigated the photoexcitation-dependent As shown in Figure 5b, the current increases gradually with the light intensity, regardless of a positive or a negative bias voltage. However, the increasing amplitude becomes smaller and tends to be saturated. That is mainly due to the photoconductivity effect, producing a large number of photo-generated carriers that increases the current density. In addition, the photothermal effect increases with the optical power density and produces a thermal electric field, which enhances the Schottky barrier and the shielding effect of the Schottky junction, and thus, the current is gradually saturated [47,48].