# Electric Current Dependent Fracture in GaN Piezoelectric Semiconductor Ceramics

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## Abstract

**:**

## 1. Introduction

## 2. Experiment

#### 2.1. Material and Specimens

^{−1}(the coercive field E

_{c}= 8.19 kV cm

^{−1}). Here, it is worth noting that polarization along the length direction of a sample needs a higher voltage, which will be done in our next work. The poling temperature and time were 120 °C and 30 min. During the poling process, two poly tetra fluoro ethylene sheets (PTFE) were bond on both surfaces along the thickness direction of a sample with a thickness of PTFE t

_{s}= 0.5 mm. Then, silver paste was plated on the up and bottom surfaces of PTFE sheets. An adjustable DC high-voltage (provided by a power supply of 100 kV) was applied to silver wires that were welded on electrodes. All of the polarization jigs were placed in a plexiglass container that was filled with silicone oil (with a relative dielectric constant of 2.73) to prevent high-voltage discharging. Under an applied electric field (32 kV cm

^{−1}), carriers in a sample would be redistributed (electrons accumulate the positive pole and holes towards to the negative pole), and a new internal electric field would be generated, which can cause domain switch and finally make the sample reach the saturation polarization strength. After polarizing, the samples were cleaned by an ultrasonic cleaner and stored in a drum wind drying oven at 70 °C for 10 min.

#### 2.2. Experimental Configuration and Fracture Tests

^{−1}. The electric current was applied by a linear power that can be adjusted with a high resolution of 10

^{−4}A. The critical mechanical and electrical loads at fracture were recorded to calculate the physical field and critical intensity factors, so as to investigate the relationship between the applied current and the fracture behaviors of PSC. Fracture morphologies were investigated via an ultra deep field microscope (KEYENCE, VHX-700FC, Osaka, Japan).

## 3. Numerical Analysis

#### 3.1. Basic Equations

_{ij}are the stress components (i, j = x, y), D

_{i}and J

_{i}are the components of electric displacement vector and electrical current, respectively, q is the elementary charge, and Δn is the variation of carrier density.

_{ij}, e

_{ij}, and κ

_{ij}are the elastic, piezoelectric, and dielectric constants. n

_{0}is the initial carrier density, and μ

_{ij}and d

_{ij}are the electron mobility and diffusion, respectively.

#### 3.2. Boundary Conditions

_{a}is the voltage applied at the left end, ν

_{rec}is the thermal recombination velocity, and n

_{m}is the critical electron density that can be expressed as

_{B}is the surface barrier of GaN, T is absolute temperature, k

_{B}is the Boltzmann constant, h is the Planck constant, and m

_{e}= 1.82 × 10

^{−31}kg and N

_{c}= 2.23 × 10

^{24}m

^{−3}denotes the effective mass of a conduction band electron and the effective state density of conduction bands, respectively. V

_{bi}is the built-in voltage [34,35] that is given by

_{M}(4.26 eV) [36], and the electron affinity of GaN, qχ (4.10 eV) [37]. Therefore, the potential barrier can be calculated (0.16 V).

_{j}is the outer normal vector.

#### 3.3. Intensity Factor

_{σ}, the electric displacement intensity factor, K

_{D}, and the electric current intensity factor, and K

_{J}, for the given sample geometry and loading conditions (see Figure 3). The material parameters of polarized GaN PSCs were listed in Table 1 [39].

_{D}, that is,

_{D}= 10.5 nm by substituting the relevant constants into formula above. In finite element analysis, the minimum mesh size of 8 nm was applied near the Schottky contact surface.

_{σ}, K

_{D}, and K

_{J}), as follows:

_{σ}

^{0}, K

_{D}

^{0}, and K

_{J}

^{0}are the three intensity factors in the case of N

_{D}= 1.29 × 10

^{23}m

^{−3}, P = 4 N, and E

_{a}= 10

^{6}V m

^{−1}.

_{xx}(see Equation (2a)). Therefore, the electric load has no effect on the stress intensity factor. Similarly, due to the polarization direction along the y-axis, the piezoelectric charge moves in the y-axis direction under the applied mechanical load (piezoelectric-semiconductive effect). The electric current density in the x-direction (J

_{x}) and the electric displacement in the x-direction (D

_{x}) are not affected by the mechanical load (see Figure 5).

_{σ}, the electric displacement intensity factor, K

_{D}, and the electric current intensity factor, and K

_{J}, conform to a unique fitting function (see Figure 6a–c), that is

_{a}and J

_{a}represent the applied electric field and electric current density, respectively. f

_{i}(a/w) are the shape factors of a three-point bending specimen, and i = 1, 2, and 3. e

_{j}are the fitted coefficients, which are corresponding to the stress, electric displacement, and electric current intensity factors, respectively, as summarized in Table 2.

## 4. Results and Discussion

_{σ}

_{,C}= 0.205 ± 0.011 MPa m

^{1/2}, which is obviously lower than fracture toughness of insulating piezoelectric ceramics (around 1 MPa m

^{1/2}) [41,42,43]. As is well known, fracture toughness is a material property that can serve as a fracture criterion. Thus, we need to further investigate fracture under combined mechanical and electrical loading and verify the influence of electric current on the fracture behavior of conductive PSCs.

^{4}A m

^{−2}, the mean value of the critical stress intensity factor increases from 0.19 to 0.26 MPa m

^{1/2}, and thus fracture toughness increases by 36.8%. However, as the applied electric current further increases, fracture toughness decreases. Under an applied electric current, the corresponding fracture toughness distributions can be completely changed (see Figure 9). That is, the fracture behavior of PSCs is significantly influenced by an electric current. Therefore, under combined mechanical and electric loading, electric current should be introduced in the fracture criteria for PSCs with a single-edge crack.

_{σ,C}= 0.194 MPa m

^{1/2}, the critical electric displacement intensity factor K

_{D}

_{,C}= 3.409 × 10

^{13}C m

^{−3/2}, the critical current density intensity factor K

_{J}

_{,C}= 3.236 × 10

^{3}A m

^{2}, with the normalized fitting coefficients d = 51.65, f = 51.64, and g = −1.05. The mechanical and electrical fracture toughness of PSC GaN was obtained. Due to the theoretical simplification and scattering of the experimental data, there are differences between the theoretical and experimental results (see Figure 10). Nevertheless, it is indicated that Equation (17) is able to serve as a failure criterion for PSC specimens with a single-edge crack under a combined electric current, electric field, and mechanical load. The fracture behavior of GaN PSCs can be predicted when the geometry conditions and information of loads are available.

^{6}A m

^{−2}(see Figure 12). In case of pure mechanical load, fracture surface was flat, as shown in Figure 13a. Under combined electrical and mechanical loading, however, fracture surface was rough and its most area was melted and re-solidified. The difference in the shape of fracture surfaces is due to defects that are sensitive to the electric current. The concentrated electric current can cause discharge at the crack tip and burn the extension surface, as seen from the dark area in Figure 13b.

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 3.**(

**a**) Schematic representation of an experimental configuration and (

**b**) an actual coupling experimental loading structure.

**Figure 4.**(

**a**) The finite element mesh for a PSCs specimen and (

**b**) its locally refined meshes at the crack tip.

**Figure 5.**Normalized intensity factors versus loads under (

**a**) the mechanical loading, (

**b**) the electric field, and (

**c**) the electric current density.

**Figure 6.**The fitted empirical formulae of the intensity factors of GaN PSCs, (

**a**) stress intensity factor, (

**b**) electric displacement intensity factor and (

**c**) electric current intensity factor.

**Figure 7.**(

**a**) Critical mechanical load versus the length of a pre-crack under a pure mechanical load and (

**b**) the critical stress intensity factors of GaN PSCs.

**Figure 9.**The probability density of K

_{IC}under (

**a**) the pure mechanical loading and (

**b**) the combined electrical and mechanical loading (1.63 × 10

^{4}A m

^{−2}).

**Figure 10.**Experimental and fitting results for failure of PSCs specimens with a single-edge crack under combined mechanical and negative electrical loading.

**Figure 11.**Specimens in fracture testing (

**a**) without an applied electric current and (

**b**) under the combined electrical and mechanical loading (1.63 × 10

^{4}A m

^{−2}).

**Figure 12.**The electric current density distribution in the extension of a crack tip (see Figure 2) and the corresponding nephogram near the crack tip.

**Figure 13.**Fracture morphologies under (

**a**) the pure mechanical loading and (

**b**) the combined electrical and mechanical loading.

Elastic Stiffness (10^{9} Nm^{−2}) | Piezoelectric Constant (C m^{−2}) | Relative Dielectric Constant (k_{ij}/k_{0}) | Migration Rate (cm^{2} V^{−1} s^{−1}) | Diffusion Coefficient (cm^{2} s^{−1}) |
---|---|---|---|---|

C_{11}= 298.4 | e_{31} = −0.52 | ε_{11} = 9.5 | μ_{11} = 653 | d_{11} = 16.99 |

C_{12} = 121.0 | e_{15} = −0.31 | ε_{33} = 10.3 | μ_{33} = 982 | d_{33} = 25.53 |

C_{13} = 142.5 | e_{33} = 0.61 | |||

C_{33} = 289.2 | ||||

C_{44} = 23.1 |

**Table 2.**Fitting coefficients (e

_{j}in Equation (16)) under the different types of intensity factors.

Intensity Factors | Fitting Coefficients, e_{j} (j = 1, 2, 3, 4, 5) | ||||
---|---|---|---|---|---|

Stress intensity factor | 3.055 | −7.141 | 33.479 | −61.360 | 52.909 |

Electric displacement intensity factor | 1.921 | −0.426 | 2.157 | −3.438 | 3.206 |

Electric current intensity factor | 1.800 | −0.256 | 2.350 | −4.101 | 4.270 |

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**MDPI and ACS Style**

Qin, G.; Lu, C.; Zhang, X.; Zhao, M.
Electric Current Dependent Fracture in GaN Piezoelectric Semiconductor Ceramics. *Materials* **2018**, *11*, 2000.
https://doi.org/10.3390/ma11102000

**AMA Style**

Qin G, Lu C, Zhang X, Zhao M.
Electric Current Dependent Fracture in GaN Piezoelectric Semiconductor Ceramics. *Materials*. 2018; 11(10):2000.
https://doi.org/10.3390/ma11102000

**Chicago/Turabian Style**

Qin, Guoshuai, Chunsheng Lu, Xin Zhang, and Minghao Zhao.
2018. "Electric Current Dependent Fracture in GaN Piezoelectric Semiconductor Ceramics" *Materials* 11, no. 10: 2000.
https://doi.org/10.3390/ma11102000