2.2.2.1 Choose the set-up
For a specific system, there are many aspects to choose what set-up we want to use, which would dramatically change the calculated results as well as the computational cost.
Those aspects include:
2.2.2.1.1 Unit cell formula:
“thin” film of CuO, at least as the focus of interest in thesis, is actually not “thin” enough to be considered as a “2D” material. The unit cell needs to have periodic boundary conditions in all direction to simulate infinitely repeating cells of CuO. The primitive unit cell of CuO consists of 2 Cu atoms and 2 O atoms in a triclinic crystal system. But that “primitive unit cell” is “primitive unit cell” by considering only the positions of Cu and O atoms assuming no further difference between atoms of the same element. There can be properties like spin where the Cu atoms become different, and it may require a bigger unit cell to properly express the repeating pattern of the spin. The aspect of spin will be discussed in more details in section 2.2.2.1.5.
Figure 2.11. Primitive unit cell of CuO, Cu in blue, O in red.
27 Figure 2.12. unit cell of Cu15SnO16 to simulate the Sn-doped CuO, Cu in blue, O in
red, Sn in grey.
For the Sn-doped CuO, we need to make a supercell and replace one or more Cu atoms with Sn. Figure 2.12 shows a unit cell with Cu15SnO16, which is equivalent to 10% Sn- doped CuO by weight. This would effectively assume all of the Sn atoms are arranged in a perfectly repeating pattern (because of the periodic boundary conditions).
Generally, bigger unit cells can reveal properties and mechanisms that smaller unit cell can not express because those properties and mechanisms may repeat in a much bigger pattern, or even no repeating pattern at all. But on the other hand, considering much bigger unit cells costs much more computational resources. And for example, even if from the experimental data, we can expect the configuration of spins consists of domains with the size of about 10 μm, it is infeasible to use DFT to actually calculate all of the atoms in the same domain, let alone calculating the domains together.
Choosing the unit cell formula is one of the tradeoffs between accuracy and computational cost. We may get hints from the expected properties and mechanisms to know the minimum required size. We may also try to increase or decrease the size until we have suitable results.
28 2.2.2.1.2 Functionals
As explained in the section about theory of DFT, the major problem with DFT is that, except for the free-electron gas, the exact functionals for correlation and exchange are unknown. On the other hand, we can use approximations of those functionals to calculate of certain physical quantities quite accurately.
We have been developing those functionals from both theoretical and experimental approaches, but it is still very complicated and we have no concrete evident on which functional should be used on each case.
The usually used functionals include:
LDA: Local-density approximations (LDA) are a type of approximations to the exchange–correlation (XC) energy functional. In this case, the functional depends solely on the density at the position where the functional is evaluated:
𝐸XCLDA[𝑛] = ∫ 𝜀XC(𝑛)𝑛(𝐫)d3𝐫 (2.4) In LDA, the exchange–correlation energy is usually separated into the correlation part and the exchange part: 𝜀XC= 𝜀X+ 𝜀C. The exchange part is called the Dirac exchange, or sometimes as Slater exchange; This part takes the form 𝜀X ∝ 𝑛1/3. On the other hand, the correlation part may have many mathematical forms. Based on quantum Monte Carlo simulations of jellium, accurate formulae for the correlation energy density 𝜀C(𝑛↑, 𝑛↓) have been constructed. Because of the LDA’s assumption that the density is the same everywhere, the LDA has a tendency to over-estimate the correlation energy and underestimate the exchange energy. The errors due to those two parts (correlation and exchange) tend to compensate each other to a certain degree.
GGA: To correct for the tendency to overestimate the correlation energy and underestimate the exchange energy of LDA, one of the method is to expand in terms of the gradient of the density in order to account for the non-homogeneity of the true electron density. The corrections in this case is based on the changes in density away from the coordinate. This method is generalized gradient approximations (GGA) and its exchange–correlation energyhave the equation (2.5).
𝐸XCGGA[𝑛↑, 𝑛↓] = ∫ 𝜀XC(𝑛↑,𝑛↓,∇𝑛↑,∇𝑛↓)𝑛(𝐫)d3𝐫 (2.5)
29 Hubbard model: The Hubbard model, also known as DFT+U, states that each electron experiences competing forces: one pushes it away from its neighbors, while the other pushes it to tunnel to neighboring atoms. In particular, the Hubbard correction, most commonly denoted by Ueff, is applied in first principles based simulations using DFT.
The value of Ueff is often found by trying to fit it with the experimental data. The usage of the Hubbard correction in DFT simulations can improve the prediction of electron localization and thus prevent the incorrect prediction of metallic conduction in insulating systems.
Hybrid functionals: Hybrid functionals are a type of approximations for the exchange – correlation energy functional in DFT which incorporate a portion of exact exchange from Hartree–Fock theory with the rest of the exchange–correlation energy from other sources (ab initio or empirical). Rather than the density, the exact exchange energy functional is expressed in terms of the Kohn–Sham orbitals. Therefore, it is termed an implicit density functional. Hybrid functionals include B3LYP, PBE0, HSE, Meta- hybrid GGA, etc. Hybrid functionals often have many parameters fitted to experimental results. Exchange-correlation functional of B3LYP (Becke, 3-parameter, Lee–Yang–
Parr) is described by equation (2.6).
𝐸XCB3LYP = (1 − 𝑎)𝐸XLSDA+ 𝑎𝐸XHF+ 𝑏Δ𝐸XB+ (1 − 𝑐)𝐸CLSDA+ 𝑐𝐸CLYP (2.6) In the equation (2.6), a = 0.20, b = 0.72, and c = 0.81. 𝐸𝑥𝐵 is a generalized gradient approximation: the Becke 88 exchange functional and the correlation functional of Lee, Yang and Parr for B3LYP, and 𝐸cLSDA is the VWN local spin density approximation to the correlation functional. The three parameters (a, b, c) have been taken without modification from Becke's original fitting of the analogous B3PW91 functional to a set of atomization energies, ionization potentials, proton affinities, and total atomic energies.
2.2.2.1.3 Pseudo-potential
The pseudopotential is an method to replace the complicated effects of its nucleus and the core (i.e. non-valence) electrons of an atom with an effective potential, which is referred to as pseudopotential.
Ultra-soft and norm-conserving are the two most common types of pseudopotential used in DFT simulating electrons and atoms. They allow a basis-set with a significantly lower
30 cut-off (the frequency of the highest Fourier mode) to be used to describe the electron wavefunctions and so allow proper numerical convergence with reasonable computing resources.
Norm-conserving pseudopotential takes the form of the equation (2.7).
𝑉̂𝑝𝑠(𝑟) = ∑ ∑ |𝑌𝑙𝑚⟩𝑉𝑙𝑚(𝑟)⟨𝑌𝑙𝑚|
𝑚 𝑙
(2.7)
In the equation (2.7), the |𝑌𝑙𝑚⟩ projects a one-particle wavefunction, such as one Kohn- Sham orbital, to the angular momentum labeled by {𝑙, 𝑚}. 𝑉𝑙𝑚(𝑟) . It is the pseudopotential that acts on the projected component. Different angular momentum states would experience different potentials. Therefore, the HSC norm-conserving pseudopotential is non-local.
There are two conditions used to construct the Norm-conserving pseudopotentials:
Inside the cut-off radius, the norm of each pseudo-wavefunction be identical to its corresponding all-electron wavefunction.
All-electron and pseudo wavefunctions are identical outside cut-off radius.
Ultra-soft pseudopotentials relax the norm-conserving constraint to reduce the necessary basis-set size further at the expense of introducing a generalized eigenvalue problem.
With a non-zero difference in norms we can now define 𝑞𝐑,𝑖𝑗 by equation (2.8).
𝑞𝐑,𝑖𝑗 = ⟨𝜙𝐑,𝑖|𝜙𝐑,𝑗⟩ − ⟨𝜙̃𝐑,𝑖|𝜙̃𝐑,𝑗⟩ (2.8) And so a normalised eigenstate of the pseudo Hamiltonian now obeys the generalized equation (2.9).
𝐻̂|Ψ𝑖⟩ = 𝜖𝑖𝑆̂|Ψ𝑖⟩ (2.9) And the operator 𝑆̂ is defined in the equation (2.10).
𝑆̂ = 1 + ∑|𝑝𝐑,𝑖⟩𝑞𝐑,𝑖𝑗⟨𝑝𝐑,𝑗|
𝐑,𝑖𝑗
(2.10)
31 In the equation (2.10), 𝑝𝐑,𝑖 are projectors that form a dual basis with the pseudo reference states inside the cut-off radius, and are zero outside, as shown in equation (2.11).
⟨𝑝𝐑 ,𝑗|𝜙̃𝐑,𝑗⟩
𝑟<𝑟𝑐 = 𝛿𝑖,𝑗 (2.11)
2.2.2.1.4 Lattice parameters and position of atoms
We can input the initial lattice parameters and position of atoms. Even though those values may converge during energy optimization, a good initial set of lattice parameters and position of atoms would at least save a lot of computational resources.
There are also cases with materials having several phases. Those phases would be corresponding to different preparation process. This means there are many very stable local-minima-energy configurations; and the experimental results may be corresponding to any of those very stable local-minima-energy configurations. In DFT calculation, assuming everything else work perfectly, the configuration after energy optimization would depend heavily on the initial set. And even if we can hypothetically find the global-minimum-energy configuration, it may not be corresponding to the experimental results we are trying to compare to.
Therefore, we may use some experimental results to give us some guidance on which values of initial lattice parameters and position of atoms we should input; and in some cases, we may need to try starting the calculation from multiple sets of initial lattice parameters and position of atoms.
2.2.2.1.5 Spin
Other than having to use bigger unit cells as mentioned above, the aspect of spin also has a problem with having many local minima as there can be many high-energy configurations of spin in between. This can theoretically be solved by try start ing from many different starting configurations. But that would cost a lot of computational resources.
Both PHASE and Material Project let us set the initial configuration of spin. For each Cu atom, it is not only about up or down, it is specified by zeta, which is defined in the equation (2.12).
32 𝜁 =(𝑛↑−𝑛↓)
(𝑛↑+𝑛↓) (2.11)
Theoretically, zeta may converge during energy optimization. But the results may still depend on the input values of zeta as the energy optimization may not have run long enough.
We can rely on some experimental results to have an idea of what configuration of spin the system would have.
According to B.X. Yang et al. (1988), CuO has the configuration of spin as shown in Figure 2.13 [58].
Figure 2.13. The magnetic structure of CuO. The circles denote the Cu ions in the (x,1
4,z) plane, and the squares denote those in the (x,3
4,z) plane. The filled symbols denote the ions with spins pointing to [111] direction, while the open symbols denote
the spins pointing to the opposite direction. Within each plane, the Cu2+ spins are arranged completely antiferromagnetically along the Cu-O-Cu chains running in [101]
direction. The dashed line illustrates a magnetic unit cell [58].
It is also possible to reproduce the antiferromagnetic property in a smaller cell, as long as half of them point in each way. It is worth trying because smaller unit cell is much faster to be calculated. Figure 2.14 shows an antiferromagnetic structure with a unit cell of Cu4O4.
33 Figure 2.14. Cu4O4 unit cell of antiferromagnetic CuO. The AFM configuration was
set as indicated by arrows.
2.2.2.1.6 Vacancy
We can simulate vacancies by removing some atoms from the unit cell. This would change the calculated band structure and may reveal some properties of the material.
But we also have to take note that if we only do this, it does not include the movement of the vacancies, it is only about calculating properties like band structure of the system with certain vacancies. It is possible to use DFT to calculate the energy difference (and thus formation energy) to support the dynamic and/or Monte Carlo methods, but it would be too far from the scope of this thesis.
2.2.2.2 Optimizing structure for minimal energy
The next step is gradually changing the values of lattice parameters and position of atoms to reach a state with minimal energy.
The experimental measurement always have uncertainty, and even a small difference of values of lattice parameters and position of atoms can lead to a big change of band structure.
There are also problem with position of atoms being difficult to measure experimentally.
In the unit cell of CuO, while the location of Cu atoms would be described by simple rational number, the position of O atoms seems to be an irrational number.
34 From the initial set of initial lattice parameters and position of atoms, we have to try performing energy optimization to get at least a more consistent configuration, in which the force between the atoms should be near zero. The program would do this by finding the small changes of lattice parameters and position of atoms which lead to lower energy.
With this algorithm, we may or may not reach the global minimum as the optimization may get stuck at a local minimum. The optimization usually ends when the gradient becomes flat enough.
2.2.2.3 Calculate DoS and band structure
For each certain structure and other aspects of computational details (functional, pseudo - potential, etc.), we can calculate the self-consistent field.
Self-consistent field (SCF) methods include both Hartree-Fock (HF) theory and Kohn- Sham (KS) density functional theory (DFT). Self-consistent field theories only depend on the electronic density matrices, and are the simplest level of quantum chemical models. In this step, the program tries to find the electron density distribution that has the minimal energy.
We can set the tolerance for the difference which we may accept as small enough to finish the calculation of Self-consistent field.
35 Figure 2.15. The process of self-consistent field calculation.
After we have the self-consistent field, the program can apply the theory of DFT to calculate the density of states.
We can specify the band k-point mesh, or just leave it for auto suggestion, we can calculate the band structure of the system from the results of the self-consistent field calculation.
36