The GW self-energy, which is a simple product in time, becomes a convolution in the frequency domain. The evaluation of
Σ(r, r0, ω) = lim η→0 i 2π Z dω0eiω0ηG(r, r0, ω + ω0)W (r, r0, ω0) (4.52) is a complicated task. Here the screened interaction is given by
W (r, r0, ω) = Z
dr00−1(r, r00; ω)ν(r00− r0) (4.53) This indicates that the interaction between quasi-particles is much weaker than the “bare” Coulomb interaction between the electrons, due to the re- pulsion of the electrons in the neighborhood of an electron, i.e. an exchange- correlation hole that is created which screens the interaction. The dielectric function in the frequency domain is
(r, r0; ω) = δ(r − r0) − Z
dr00ν(r00− r0)P (r, r0; ω), (4.54) where the irreducible polarizability (in the random phase approximation (RPA)) can be shown [23] to be equal to
P0(r, r0; ω) = X nk,n0k0 φnk(r)φ∗n0k0(r)φnk∗ (r0)φn0k0(r0) ×h fnk(1 − fn0k0) 0 nk− 0 n0k0 + ω + iη − (1 − fnk)fn0k0 0 n0k0 − 0nk+ ω − iη i (4.55)
where fnk are the Fermi factors.
We begin by solving the quasiparticle equation within a DFT approach, using a standard local approximation to the self-energy
Σ(r, r0, ω) = δ(r − r0)Vxc(r) (4.56)
the quasiparticle equation then becomes a Kohn-Sham equation
(ˆh + Vxc)ψKSi = KSi ψKSi (4.57)
where Vxc(r) is the local exchange-correlation potential. In many cases the
Kohn-Sham eigenvalues KSi already provide a reasonable estimate of the band structure and are in qualitative agreement with experiment. For sys- tems where the quasiparticle wave functions are known, one also finds that
4.6 The GW approximation in practice 63
the Kohn-Sham wave functions and quasiparticle wave functions are approx- imately equal [58] (however, it should be noted that there are notable ex- ceptions, such as Cu2O, where better calculations are necessary and a self-
consistent GW (scGW) approach is used [59]). These observations indicates that the self-energy correction is small and justifies the use of first-order perturbation theory to obtain approximate energies
i ≈ KSi + hψ KS
i |Σ(r, r 0
, i) − Vxc|ψKSi i (4.58)
This equation indicates that one does not need to calculate the complete Σ matrix, as one element is sufficient to calculate one corrected energy. This is however a nonlinear equation, as Σ depends on the quasi-particle energy. This can be simplified by using a linear expansion of Σ around the Kohn- Sham eigenenergy KS i Σ(r, r0, i) ≈ Σ(r, r0, KSi ) + (i− KSi ) ∂Σ(r, r0, KSi ) ∂i KS i (4.59) Using this in the equation above we get
i ≈ KSi + ZihψiKS|Σ(r, r 0
, i) − Vxc|ψiKSi (4.60)
where the quasi-particle renormalization factor (also called the quasi-particle weight) Zi is given by Zi = 1 − hψKSi |∂Σ(r, r 0, KS i ) ∂i KS i |ψiKSi (4.61) In practice all calculations are carried out in momentum space, so the formulations need to be transformed by applying Fourier transforms. The Fourier transform for the two variable function G0 is defined as
G0(r, r0, ω) → G0GG0(q, q0, ω) =
1 Ω
Z
drdr0e−i(G+q·r)G0(r, r0, ω)e−i(G0+q0·r0) (4.62) where G denotes a reciprocal lattice vector and q wave vectors in the irre- ducible Brillouin zone. Applying this to the real space expression for the polarizability P0 in Eq.(4.52) gives
PGG0 0(q, ω) =
2 Ω
X
nn0k
hnk|e−i(G+q)·r|n0k + qihn0k + q|e−i(G0+q)·r0|nki ×h fnk(1 − fn0k0) 0 nk − 0n0k0 + ω + iη − (1 − fnk)fn0k0 0 n0k0 − 0nk+ ω − iη i (4.63)
Similarly one obtains, using Eq.(4.40) and Eq.(4.38), GG0(q, ω) = δGG0 − 4π |G + q|2P 0 GG0(q, ω) (4.64) WGG0(q, ω) = 4π |G + q|2 −1 GG0(q, ω) (4.65)
In order to obtain the self energy at a given frequency, the convolution in Eq.(4.52) needs to be calculated. Therefore one needs to calculate −1 for different q-points and for all frequencies. This is computationally very time consuming, especially because the function P0(ω) has many poles close to
the real axis. In order to avoid this burden, a plasmon-pole model will be used in order to model the inverse dielectric function. The idea behind this approach is that this function is dominated by the peaks at which = 0, i.e. the plasmon peaks. An additional benefit of using such a plasmon-pole model is that it allows to perform the convolution analytically.
The model used in this thesis is the Godby-Needs plasmon-pole model [60], which models −1 as follows
−1GG0(q, ω) = δGG0 + Ω2 GG0(q) ω2− (˜ω GG0(q) − iη)2 (4.66) The two parameters Ω2GG0 and ˜ωGG0 are determined in ABINIT by perform-
ing the RPA calculation at two frequencies: at ω = 0 and at ω ∼ iωp, where
ωp is the classical plasmon frequency. These frequencies are convenient as
these are far away from the poles of P0, so that the fit is more stable. Al- ternative choices for determining the fitting parameters exist [61] as well as alternate plasmon-pole models (e.g. Ref. [62]).
If W is split in the bare Coulomb interaction ν and the remainder W − ν the self-energy can be split in an exchange (Σx) and a correlation (Σc) part.
Symbolically this is
Σ = iGKS0 W = iGKS0 ν + iGKS0 (W − ν) = Σx+ Σc (4.67) The exchange part of the self-energy reads
Σx(r, r0, ω) = i 2π
Z
dω0GKS0 (r, r0, ω + ω0)ν(r, r0)eiω0η (4.68) which can be evaluated analytically. This leads to the Hartree-Fock exchange term hψnk|Σx|ψnki = − occ. X mq Z drdr0ψ ∗ nk(r)ψmq(r)ψmq∗ (r 0)ψ nk(r0) |r − r0| (4.69)
4.6 The GW approximation in practice 65
Similarly the correlation part of the self-energy is given by Σc(r, r0, ω) = lim η→0 i 2π Z dω0GKS0 (r, r0, ω+ω0)[W (r, r0, ω)−ν(r, r0)]eiω0η (4.70) In order to give an expression for the matrix elements of these self-energy parts, it is convenient to define the matrix elements MGνc(k, q) between the occupied valence (n = v) and the unoccupied conduction (n = c) states
MGνc(k, q) = Z d3rψνk−q∗ (r)e−i(q+G)·rψck =X G0 c∗νk−q(G0− G)cck(G0) (4.71)
Using this notation the matrix elements of the exchange part can be written in momentum space as hψnk|Σx|ψmki = − 4π Ω occ. X l X q X G MGln(k, q)[MGln(k, q)]∗ |G + q|2 (4.72)
The correlation part, using the Godby-Needs approximation is given by hψnk|Σc|ψmki = − 2π Ω X lq X GG0 Mln G(k, q)[MGln(k, q)] ∗ |G + q||G0+ q| × Ω 2 GG0(q) ωGG0(q)(ω − lk−q+ sgn(µ − lk−q)ωGG0(q)) (4.73)
The sgn function in combination with the electron chemical potential µ is 1 when considering a valence band and -1 for a conduction band. With these self-energy matrix elements calculated, one can calculate the GW corrections to the eigenvalues by using Eq.(4.60).
Chapter 5
ZnO: quasiparticle corrections
to the band gap
5.1
Introduction
Having introduced the GW technique to calculate the band gap, we will now start to apply it to the TCO materials. ZnO offers a natural choice to start with1. It is an important material in optoelectronic device technology
and is used in a number of applications such as visual displays, solar cells and so on. It has also been emerged as an economical alternative to In2O3
which is an expensive material due to limited availability of In. Thus ab initio first-principles calculations are highly desired to design new devices cost effectively. To reach this goal, it is necessary to have reliable ab initio techniques for the prediction of the band gap. Therefore here a thorough study of the quasiparticle energy corrections to the DFT band gap of ZnO is presented. Although the ground state of ZnO has the wurtzite (WZ) struc- ture, the metastable zincblende (ZB) and high pressure rocksalt (RS) phases have growing experimental interest [64, 65]. The zincblende phase of ZnO can be grown using molecular beam epitaxy. The zincblende phase may solve the challenge of controlling p-type conductivity in the optoelectronic devices [64]. Thus, in this chapter we present the quasiparticle corrections to the
1This chapter is based on the work published as - 1] H. Dixit, R. Saniz, D. Lamoen
and B. Partoens: “The quasiparticle band structure of zincblende and rocksalt ZnO ”, J. Phys.: Condens. Matter 22, 125505 (2010) 2] H. Dixit, R. Saniz, D. Lamoen and B. Partoens: “Accurate pseudopotential description of the GW bands tructure of zincblende and rocksalt ZnO ”, Computer Physics Communications 182, 2029 (2011) 3] M. Stankovski, G. Antonius, D. Waroquiers, A. Miglio, H. Dixit, K. Sankaran, M. Giantomassi, X. Gonze, M. Cˆot´e, and G.-M. Rignanese: “The G0W0 band gap of ZnO: effects of plasmon-pole models”, Phys. Rev. B 84, 241201(R) (2011).
band gap for these three crystal structures of ZnO. We use the pseudopoten- tial plane-wave method to calculate the GW band gap at the Γ point. The DFT-LDA wavefunctions are used to construct the Green’s function.
In the case of IIB-V I semiconductor materials, it is important to include the localized ‘d’ orbitals in the cation pseudopotential (PP). The localized ‘d’ orbitals play an important role in bonding, hence their inclusion as va- lence orbital is essential for a correct structure [42]. However the DFT band structure, calculated using the pseudopotential with semicore ‘d’ states and the plane wave basis, shows serious underestimation of the band gap as dis- cussed earlier in chapter 3. We further observe that the calculated quasi- particle band gap using the GW approximation, is also underestimated for ZnO. This observation is in agreement with the literature and is discussed below in detail. Thus although use of the PP with semicore ‘d’ states yields the correct structure, it results in a poor quasiparticle correction questioning the validity of such PP. We analyze this problem with a systematic study of the PP used and corresponding GW corrections to the band gap for the ZnO. Further, the different phases of ZnO considered allow us to compare between the crystal symmetry and the effect of p-d hybridization on the quasiparticle corrections. We also consider ZnS in the zincblende structure that provides an interesting comparison due to change of chemical species from oxygen to sulfur. Thus this chapter is organized as follows. First, we discuss the poor quasiparticle correction with the standard 12-electron Zn PP followed by a systematic study of self-energy dependence on the num- ber of valence electrons used in the PP for two polymorphs - zincblende and rocksalt ZnO. We find that a modified 20-electron PP of Zn, which treats the entire semi-core shell as the valence, is essential for adequate treatment of the self-energy operator. Thereafter we discuss the accuracy of the newly gener- ated 20-electron PP. Finally in section 5.5, we also analyze the effect of the plasmon-pole model used within the GW approximation on the calculated GW band gap of wz-ZnO.