Ab-initio calculations

Based on Density Functional (DFT) Theory, ab-initio calculations were performed using the Vienna ab-initio simulation package (VASP)^17,18, a sophisticated package for

calculating ab-initio quantum-mechanical molecular dynamics simulations. It uses either pseudopotentials or projector-augmented wave (PAW) method and a plane wave basis set to calculate forces and the full stress tensors so that atoms can be relaxed into their instantaneous ground-state^103,104.

3.2.1 Density functional theory

Originally, in the calculation of the many-body structure, the nucleus is regarded as stationary (Born-Oppenheimer approximation)¹⁰⁵, so that electrons can be regarded as moving in the electrostatic potential V generated by the nucleus. States of electrons can be described by Schrödinger equation: Where N is the electron number, U is the interaction potential of the electron. The

operators T and U are pervasive operators, which are the same in all systems, while the operator V is dependent on the system and is not universality. However, solution of Schrödinger equation needs too much work, which is not suitable for complex systems.

In contrast, the density functional theory transforms the many-body problem with U into a single-body problem without U. The electron density 𝑛(𝑟 ) is given by:

𝑛(𝑟 ) = 𝑁 ∫ 𝑑³𝑟₂∫ 𝑑³𝑟₃… ∫ 𝑑³𝑟_𝑁Ψ^∗(𝑟⃗⃗⃗ , 𝑟₁ ⃗⃗⃗ , 𝑟₂ ⃗⃗⃗ , … , 𝑟₃ ⃗⃗⃗⃗ )𝛹(𝑟_𝑁 ⃗⃗⃗ , 𝑟₁ ⃗⃗⃗ , 𝑟₂ ⃗⃗⃗ , … , 𝑟₃ ⃗⃗⃗⃗ )_𝑁 (14) Currently, the most successful DFT was developed by Kohn and Sham, which brought the Nobel prize to Kohn in 1998.^106-108

3.2.2 Exchange functional

Kohn-Sham density functional theory is widely used to calculate the properties of atoms, molecules¹⁴, and solids’ ground-state using a self-consistent-field¹⁰⁹. The major challenge of this theory is that the exchange-correlation energy:

𝐸_𝑋𝐶 = 𝐸_𝑋 + 𝐸_𝐶

(

15

)

27

must be approximated. The simplest approximation is a local-density approximation (LDA)¹⁴, which can be gained from Thomas-Fermi model^110,111. It depends solely upon the value of the electronic density at each point in space, which is represented as:

𝐸_𝑥𝑐^𝐿𝐷𝐴

[

𝜌

]

=

∫

𝜌

(

𝒓

)

𝜀

(

𝜌

)

𝑑𝒓

(

16

)

where ρ is the electronic density and

ε

xc is the exchange-correlation energy per particle of a homogeneous electron gas of charge density ρ.

Generalized Gradient Approximation (GGA)¹⁵ brings an improvement in the accuracy provided by the LDA. LDA focuses the density value at a certain point. However, GGA takes its gradient into account, where the Perdew-Burke-Ernzerhof (PBE) functional is a form of GGA functionals¹⁶, which contains experimental parameters whose values have been fitted to reproduce experiment or more accurate calculations.

Both LDA and GGA-PBE have been employed in the structural and elastic properties section to compare with experimental values. In bandgap calculations, it was found that, hybrid functional obtained more accurate results.^112-115 Heyd–Scuseria–Ernzerhof-06 (HSE06)¹¹⁶ hybrid functional is modified from the conventional GGA-PBE. The difference between them is that HSE06 replaces the previous exchange energy in PBE with a new one mixed Hartree-Fock and PBE’s exchange energy for a certain range:

𝐸_𝑋^𝐻𝑆𝐸 = α𝐸_𝑋^{𝐻𝐹,𝑆𝑅}

(

𝜇

)

+

(

1 − 𝛼

)

𝐸_𝑋^{𝑃𝐵𝐸,𝑆𝑅}

(

𝜇

)

+ 𝐸_𝑋^{𝑃𝐵𝐸,𝐿𝑅}

(

𝜇

) (

17

)

Where α is the mixing parameter between short-range HF exchange energy 𝐸_𝑋^{𝐻𝐹,𝑆𝑅}(𝜇) and the short-range PBE energy 𝐸_𝑋^{𝑃𝐵𝐸,𝑆𝑅}(𝜇), usually α = 0.25; 𝜇 controls the screening, which can define the range of HF correction, normally 𝜇 = 0.2 Å^-1 in HSE06 functional.

3.2.3 Computational procedure

Before precise calculations were performed, internal coordinates of the perturbed

structures were relaxed. A conjugate-gradient algorithm is used to relax the ions into their instantaneous ground state. The cut-off energy of the plane wave basis was set at 450eV to keep calculations both accurate and efficient. Using a Gaussian smearing method¹¹⁷, the smearing width was set to 0.2eV. To make the relaxed structure resemble the experimental one, only ions and cell volume were changed to maintain crystal shape.

28

After the relaxation, the structure turns to a steady state that is of low energy. This is followed the self-consist calculation. This calculation aims to improve the structural model to obtain the lowest system energy by forcing strict wave function and charge density values. Accurate Fermi level location can be computed in the self-consist calculation and based on this result, the crystal's electronic structure, such as band gap and density of state, can be further calculated with realistic expectations.

29 this study, the PBE and LDA method were employed to relax the lattice constants of modeled structures, and the results are listed in Table 2 and Figure 4-1.

It was found that, compared with previous experimental values¹⁰, the

constant a calculated by PBE is slightly larger for Si-like compositions, while the band gap increases with increased Ge content. However, it should be noted that LDA figures are always lower than the experimental one¹⁰. The gap gradually decreases to quite small

values with structure tending to be more Ge-like.

4.1.2 Elastic properties

To understand the elastic properties of the GeSi alloy system, calculations were

performed for the values of three independent second-order elastic constants (Cij) using first-principle calculations based on "stress-strain" method, data is shown at Figure 4-2 and Table 3 below. For the cubic phase, the stability condition is C11 > 0, C44 > 0, C11 >

C12, C11 + 2C11 > 0, C12 < B< C11118,119. In this study, the elastic constants of all structures meet the condition, which demonstrates that the structures are mechanical stable in the cubic phase. These elastic constants were calculated by PBE and LDA method, most of

Si Ge1Si7 Ge1Si3 Ge3Si5 Ge1Si1 Ge5Si3 Ge3Si1 Ge7Si1 Ge PBE 5.468 5.504 5.539 5.576 5.616 5.655 5.696 5.739 5.783 LDA 5.403 5.431 5.458 5.487 5.517 5.548 5.580 5.613 5.647 EXP 5.431 5.457 5.483 5.509 5.537 5.567 5.596 5.627 5.658 Figure 4-1 GexSi(1-x)'s lattice parameter after structual

Table 2 GexSi1-x's experimental and structural relaxed lattice parameters

In document Junjie Li Bachelor of Materials Engineering (Page 37-40)