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We have reviewed the use of QMC methods for the calculation of energy gaps in atoms, molecules, and crystalline solids. Although the quasiparticle gap does not formally satisfy a variational principle, we have argued that in practice the fixed-node error in the quasiparticle energy gap is overwhelmingly likely to be positive. Reoptimisation of trial wave functions for systems in which electrons have been added or removed can be expected to improve the calculated quasiparticle energy gaps. For neutral excitations this is not necessarily the case, as was shown in Sec. 4.1.1, and reoptimisation may result in the formation of a pathological excited-state trial nodal surface. Unless the neutral excitation results in a trial wave function that transforms as a 1D irrep of the full symmetry group of the

system and the target state is the lowest-energy eigenstate transforming as that irrep, reoptimisation of the free parameters in the excited-state wave function should not be attempted. Since Jastrow factors do not affect the nodal surface and hence DMC energy, there is little to be gained by reoptimising Jastrow factors in excited states; on the other hand, reoptimising backflow functions in states in which electrons have been added to or removed from the neutral ground state can significantly improve DMC quasiparticle energy gaps.

The use of larger-than-typical DMC time steps for excitation calculations has been shown to be a major source of possible computational savings in DMC energy gap calculations. Time-step bias appears to cancel extraordinarily well in energy gaps. In Si we have made computational savings of a factor of four by using larger time steps in backflow calculations.

Our calculations employing multideterminant trial wave functions for Si at the Γ point show that, even where bands are exactly degenerate, it is not necessarily the case that a few-determinant excited-state wave function comprised of contribu- tions from all possible combinations of degenerate single-particle orbitals performs any better than the single-determinant alternative. On the other hand, such a multideterminant wave function significantly lowers the energy of the singlet first- excited state of O2. The need for multideterminant wave functions appears to be more of an issue in studies of excitations in molecules than those in crystals.

We have evaluated energy gaps in atomic, molecular, and crystalline systems using the VMC and DMC methods with single-determinant SJ and SJB trial wave functions. In atomic Ne, where vibrational and finite-size effects are not present, we have achieved highly accurate ionization potentials in comparison with experi- mental data from which relativistic effects have been removed. The MAE across all of our SJB-DMC calculated ionization potentials for Ne is 0.34%, demonstrating the intrinsic high accuracy achieved by the SJB-DMC method.

In various molecules, where vibrational effects may be present, but finite-size effects never are, we have repeatedly achieved energies which are in reasonable

agreement with their experimental counterparts, with differences attributable to vibrational corrections. We have investigated using DFT to relax excited-state geometries. It too is important, having the largest impact in the H2 (∼ 0.8 eV) and O2 (∼0.5 eV) dimers, of all the molecules we have studied. For the parahy- drogen molecule we performed DMC calculations of the ionization potential with the protons treated as distinguishable quantum particles, demonstrating excellent agreement with highly-accurate experimental results. This makes clear the funda- mental importance of geometrical and vibrational effects when comparingab initio

gaps with experiment.

We have probed the effects of fixed-node errors in SJ-DMC energy gap calcu- lations for atoms, molecules, and crystalline solids, finding that the inclusion of backflow functions generally improves DMC energy gaps in these systems (espe- cially in solids, where backflow lowers gaps by 0.1–0.2 eV). We have shown that, in the case of Si, the use of backflow functions reoptimised in anionic and cationic states is crucial in order to achieve reasonable agreement with experiment. Resid- ual overestimates (_O(0.5 eV) for first-row atoms) are expected in solids due to the presence of vibrational effects, which are the dominant remaining source of un- certainty when it comes to comparison with experiment. We have also performed gap calculations for free-standing monolayer phosphorene, showing that systematic finite-size effects are qualitatively different in 2D materials, and that an explicit treatment of two-dimensional screening is important in forming corrections (see also Sec. 5.1.3).

Chapter 5

Electron gases in doped

two-dimensional semiconductors

There are a number of ways in which the two-dimensional electron gases in electron- doped semiconducting TMDs and in metallic two-dimensional materials differ from those which have been previously realised, e.g. in Si MOSFET inversion layers, at III-V semiconductor heterostructure interfaces, at ZnO/ZnMgO inter- faces [259,260], and possibly that at the LaAlO3/SrTiO3 interface [261].1 The key difference is that in the atomically flat limit, charge carriers in two-dimensional materials experience much less electrostatic screening, as “the bulk” no longer ex- ists in the same sense. In two-dimensional semiconductors, this lack of out-of-plane screening coupled with the fact that the layers themselves are polarisable leads to the realisation of the already discussed Keldysh interaction (See 3.1.1).

The focus of this chapter will be on the study of 2DEGs wherein charge-carriers interact via a screened interaction which (in the low-density limit) is of Keldysh form. The phase-diagram of the two-dimensional homogeneous electron gas has previously been studied by a variety of authors [262,263], but none of those cases are directly comparable to the physical situation realised in (electron-doped) two- dimensional semiconductors or two-dimensional metals. Modelling such a scenario requires the development of a version of the Keldysh interaction which is compat-

ible with periodic boundary conditions, which is where this chapter begins.

5.1

Periodic Keldysh interaction

In order to study electron gases in which the polarisation field of the remainder of a two-dimensional material acts to screen the interaction of the conduction electrons, under the Keldysh model, one must find means of dealing with lattice sums, for particles atri and rj, of the form

v(ri,rj) = X

R

vK(|ri−rj−R|), (5.1)

where vK is the Keldysh interaction, and R denotes the set of two-dimensional lattice vectors of the periodic system in question. Such sums are conditionally convergent by the same arguments as those for the Coulomb interaction. Ulti- mately, it is the behaviour of the sum at long-range which is problematic, but at distances much greater than the Keldysh interaction screening parameter, r?,

the Keldysh interaction reduces to the Coulomb interaction (inheriting the same problematic long-range behaviour). The Ewald method [264] is one way of over- coming the difficulty posed by such conditionally convergent lattice sums, and is routinely used in electronic structure codes to evaluate lattice sums over the Coulomb potential.

Here, we will present an original derivation of an Ewald-like version of the Keldysh interaction, whose lattice sums are absolutely convergent.