Objetivos específicos

5.1 COMPUTATION METHOD

Since Ni vacancies have two elementary paths for diffusion, we performed the Kinetic Monte Carlo (KMC) simulations based on the harmonic transition state theory [49] to evaluate the diffusivity of Ni vacancies in NiO crystal between 1200 K and 1700 K.

The main idea of KMC is simulating the time evolution of some processes occurring in nature by using transition rate to repeat random sampling. In this thesis, the rate constant for diffusion can be calculated using a simplified transition state theory by the following equation:

𝑘_𝑇 = 𝑛_𝑝 𝑣 exp (−∆𝐸𝑚

𝑘𝑇 ) (7)

where 𝑛_𝑝 is the number of possible jump directions; ν is the harmonic frequency and ∆𝐸_𝑚 is the migration energy of this jump. Because there are two diffusion path with different migration energy and different frequency, the rate of two processes can be combined to obtain the total rate and the time step is calculated as inverse of the total rate. At each time step one of the two possible processes is randomly selected with probability that is the product of the time step and the rate of the individual process [50].

In long term, the diffusion distance and diffusion time have the correlation that: 𝑡 = 𝑥2

where 𝑡 is the diffusion time; 𝑥 is the diffusion distance and 𝐷 is the diffusion coefficient. By tracing the vacancies in the KMC simulation, we can obtain the diffusion time 𝑡 and the diffusion distance 𝑥. Then we can fit the diffusion coefficient 𝐷.

According to Arrhenius Law:

𝐷 = 𝑓 exp(− ∆𝐸 𝑘𝑇⁄ ) (1) using the diffusion coefficient 𝐷 simulated by KMC from 1200K to 1700K, we can fit the effective migration energy ∆𝐸.

5.2 RESULTS AND DISCUSSION

From the Arrhenius plot (as shown in Fig. 13, an example of 𝑉_Ni0 ) of our calculated diffusion coefficients, we predicted that the effective migration energy for the Ni diffusion through exchanges with 𝑉_Ni0, 𝑉_Ni1− and 𝑉_Ni2− were 2.16 eV, 1.89 eV and 1.84 eV, respectively.

We can see that the migration energy of 𝑉_Ni0 is much higher than it of 𝑉_Ni1− and 𝑉_Ni2−. These results agree with the conclusion in Peterson’s work [9] that 𝑉_Ni2− _{moves with the smaller}

activation enthalpy than 𝑉_Ni1−. In summary, we found that 𝑉_Ni2− led to the fastest diffusion of Ni in NiO crystal.

6.0 CONCLUSION

In this thesis, we presented our result of the formation energy and migration energy of Ni vacancies and O vacancies with different charge states in intrinsic NiO crystal calculated by DFT+U. Detailed diffusion paths for charged vacancies in NiO crystal are investigated. We predicted migration energy of vacancy diffusion varied with charge states of the vacancies, especially for O vacancies in NiO. Further analysis reveals that defect level shift and different number of electrons on defect level contributes to migration energy difference for different charged vacancies. This work provides a view on controlling diffusion kinetics by tuning charge states of the vacancies or changing defect level shift through some processes such as doping.

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