Crystal growth is a non-equilibrium process whose simulation is a non-trivial problem that we are not going to tackle or even discuss in depth here. The inter- ested reader could start, e.g., from Landau and Binder (2000) and the references therein. To generate disordered samples of Si:P we run a simple Monte Carlo (MC) simulation using the Metropolis method: we generate an initial configura- tion with a fixed number of impurities and compute the total energy from the effective potentialU. We then swap the position of an impurity with a neighbour- ing Si site and calculate the change in energy∆E. Finally, we generate a random numberα ∈ (0,1)and, ifα < e−β∆E, we accept the move. Hereβ−1=k
BT, where
2.4. DEFECT PAIR FORMATION AND DOPING TECHNIQUES 35 P S Pa ir fo rm a on e ne rg y [e V ] -2.0 -1.9 -1.8 -1.7 -1.6 0.0 0.2 0.4
Distance between impuri es [a0]
4 6 8 10 12 14
Figure 2.8: Energy needed to form a defect pair at a specific distance, for both P (red circles) and S (blue squares), obtained from ONETEP runs with 4096 atoms. For S we only show the results from the relaxed-geometry runs used in the later chapters (so up to the fourth nearest neighbours). The ordinates axis is broken in order to better show the difference in scale. Both data series are obtained while keeping the lattice fixed, which means that the points for S are an upper bound to the minimum total energy.
is happening. By repeating this procedure several times over all impurities in the system, we should reach a ground-state configuration.
As mentioned before, we assume that the hyper-doping technique used to grow S-doped samples is able to place those impurities randomly in the host. For this reason, we generate disordered realisations of this material by just changing the atom assigned to randomly picked sites from Si to S.
Chapter 3
From
ab initio
to effective models
In this Chapter we talk about the second part of the work flow, namely how we use the output from theab initiosimulations to construct effective models for many disorder realisations.
We start by recalling notions about the tight-binding model and connecting it to the Anderson model and the DFT description of the material. With these concepts, we can introduce our effective models, the underlying approximation and their technical implementation. We also highlight the aspects that our model retains from an atomistic description and set it apart from the paradigm of the Anderson model.
Finally, we discuss how to diagonalize the matrices to obtain eigenvalues and eigenvectors. We also point out that the several layers of complexity in our model make it a much harder problem than the Anderson model.
3.1
The tight-binding model
The tight-binding model (TBM) is used, as the name suggests, to describe elec- trons bound to a nucleus via a strong potential, strong enough that the interaction with the rest of the lattice can be considered a negligible perturbation (Diu et al., 1989). For this reason, the TBM has been widely used to study the electronic structure of deep centres, as mentioned in Sec. 2.3.2. The reader can find a dis- cussion of the TBM in most textbooks on condensed-matter physics, including Kittel (2005), which we follow in the rest of the section.
An electron in said potential is described by an atomic orbitalφ(r−rj)centred on the nucleus atrj, namely its wave function decays exponentially on length
scales much smaller than interatomic distances. The assumption behind the TBM, then, is that we can write the wave function of an electronin a crystalas a linear combination of said atomic orbitals:¹
ψk(r)= √1 N Õ j eik·rjφ(r −r j). (3.1)
This is a Bloch function with wave vectork for a crystal ofN atoms, since we can show thatψk(r +R)=exp(ik·R)ψk(r), whereRis a translation from one lattice point to another.
Switching to bra-ket notation, we now defineψk(r)= hr|ψkiandφ(r−rj)=
hr|φji. The expectation value of the HamiltonianH is therefore
hHi=N−1hψk|H|ψki=N−1
Õ
j,m
eik·(rm−rj)
hφj|H|φmi . (3.2)
Since we are working with a lattice, we can defineρj =rj −rm and simplify the double sum as hHi=Õ j e−ik·ρj ∫ φ∗(r −ρj)Hφ(r)dr. (3.3)
At this point, it is customary to distinguish the diagonal elementsε(whereρj =0) and the off-diagonalt(ρj)as
ε =
∫
φ∗(r)Hφ(r)dr and t(ρj)=
∫
φ∗(r −ρj)Hφ(r)dr. (3.4)
These terms of the TBM are also called, respectively, “self-energies” and “hopping” terms.
The Anderson model of Ch. 1 is the TBM of a simple-cubic crystal of hydro- gen nuclei, i.e. of a material with only 1s atomic orbitals. In this case, however, we allow the diagonal elementsεj to be different for each site, and we consider constant off-diagonalHi j = t if sitesi,j are first nearest neighbours hi,ji and 0 otherwise. The corresponding Hamiltonian hence reads
H =Õ j εj |φji hφj|+t Õ hi,ji |φji hφi| . (3.5)
In the Anderson model, we usually sett = 1, whileεj is a random variable
1The TBM is also known, for this reason, as the Linear Combination of Atomic Orbitals
3.1. THE TIGHT-BINDING MODEL 39
Figure 3.1: Wave functions for an exemplary systems of 4067 Si atoms and 29 S impurities. On the left we show a localised state deep in the impurity band and on the right an extended state aboveεF. We have represented the top 90%
wave function valuesψiwith spheres of volume proportional to|ψi|2. Opacity and colour are proportional to−logL|ψi|2, withL = 16 here, so that lower (higher) values are in red transparent (violet solid). The box sizes are in units ofa.
drawn from a uniform distribution over the interval [−W/2;W/2], where the
parameterW is the disorder strength. At a critical valueWc ∼ 16.5 (Slevin and
Ohtsuki, 1999) the wave function at the band centreE = 0 undergoes a transi- tion from extended (W <Wc) to localised (W > Wc). While localisation can be defined more rigorously (del Rio et al., 1995), for this work we are happy with the commonly accepted form
|ψ(r)| ∝e−|r−ξr0|, (3.6)
wherer0is a localisation centre andξ is the localisation length introduced in Ch. 1.
States that do not satisfy (3.6) are called interchangeably extended or delocalised. Finally, we observe here that the Kohn-Sham Hamiltonian (2.7), written in the basis of NGWFs defined in (2.9), has the same structure as the TBM. The most important difference here is that each site is described by nine orbitals, instead of one. This implies that the Hamiltonian is divided in 9×9-sized submatrices of
self-energies (of and between the single NGWFs) and hopping terms (between NGWFs of the same or different sites). The overlap matrix has an analogous structure.