Contents
1 Introduction 3
1.1 Preamble . . . 4
2 Density Functional Theory 6 2.1 Introduction . . . 7
2.2 The Hohenberg-Kohn theorem and the Kohn-Sham represen-tation . . . 8
2.3 The exchange-correlation potential . . . 12
2.4 Density Functional Theory in practice . . . 13
2.5 The Augmented Plane Waves method . . . 15
3 Optical properties 17 3.1 Theory . . . 18
3.2 Examples . . . 23
3.3 The Kubelka-Munk model . . . 27
4 Many-body perturbation theory 30 4.1 Introduction . . . 31
4.2 Hedin’s equations and the GW approximation . . . 33
4.3 GW in practice . . . 36
5 The metallic red photocatalyst SrNbO3 38 5.1 Introduction . . . 39
5.2 State of the art: origin of the color . . . 40
5.3 Computational details . . . 42
5.4 Calculation for the ideal crystal . . . 44
5.5 Effects of Sr-vacancies on SrNbO3 . . . 51
5.6 Effects of O-vacancies on SrNbO3 . . . 56
5.7 A simple model for O-vacancies . . . 60
6 Conclusions and Perspectives 65
Chapter 1
Introduction
1.1
Preamble
The understanding of the properties of solids has considerably increased since Bloch theory set the basis for a quantum description of crystals. This is due in part to a technological developement since the time needed for a calcu-lation has been reduced by means of faster and faster computers. On the other side the theoretical investigation and the discovery of new methods has played a fundamental role; it would be hard to imagine to reach the present precision without Density Functional Theory, a milestone in con-densed matter physics. The higher computational resources together with the development of more sophisticated techniques have made possible fully ab-initio calculations. No more knowledge than the structure of the crystal is necessary and in certain cases even the geometry of the crystal structure can be theoretically determined. From this ”microscopic” knowledge we are able to derive macroscopic properties like mechanical, transport and optical properties.
Despite this remarkable improvement there are still open problems in the field. In particular, some classes of materials are not well described by the standard DFT methods. Among them we have the transition metal oxides. 3d metal oxides can be considered as prototypes of so-called strong correlated systems. The intense study of these materials ended up with the invention of new theoretical tools which aim at fixing the deficiencies of the standard methods. The interest in these materials is not purely theoretical since their properties, and the possibility to tune them, can find real technological ap-plications.
While the 3d series has been intensely studied, works on the 4d series are considerably fewer. This is partially due to the fact that they do not present such a theoretical challenge like the more correlated 3d materials. Never-theless, there are many reasons to investigate these materials. Due to the electronic and structural similarities between the 3d and 4d compounds we are in the position to identify and evaluate the genuine effects of correlation which are often entangled with spurious effects of impurities and defects. A typical kind of defects in oxides are, for example, the Oxygen vacancies. It has already been proven, and it will be proven in this work as well, that a realistic description of oxides cannot neglect the possible presence of Oxygen vacancies since they may have important consequences on the macroscopic properties of these compounds. An analysis based on the common features and carried onto the simpler 4d systems can then be useful even for the understanding of the more complicated 3d systems.
In this work we decided to focus our attention on a particular 4d metal-lic compound, SrNbO3. In addition to the reasons discussed above we have
chosen this material because of its potential applications. Its efficiency as photocatalyst for water splitting can have important applications for Hydro-gen production. Despite of the fact that photocatalysts are often semicon-ductors, its capability to absorb visible light (which accounts for about 43% of the incoming solar light) with enough energy to break the water molecule would give a clean and ”low-cost” way to obtain Hydrogen. Our work will focus on the nature of the fundamental feature for the photocatalysis, namely the absorption of visible light and on the identification of the characteristics which give rise to it.
We have structured this Thesis work as follows. The first three chapters are about the methods. In particular, Chapter 2 is about Density Functional Theory which nowadays is the standard method for electronic structure cal-culations. The essential theory is described and the practical aspects are dis-cussed. Since the optical properties cover an important part of this Thesis we have decided to dedicate Chapter 3 to them. Here, the theory together with some explicative examples are discussed. Moreover the basic theory of diffuse reflectance and color is discussed within the simple Kubelka-Munk model. In Chapter 4 we describe the Green’s function approach and the perturbation theory. Hedin’s equations and the GW approximation are introduced, again with a discussion about the practical calculations. In Chapter 5 we show and discuss our results about the material under investigation, SrNbO3. Our
Chapter 2
2.1
Introduction
Density Functional Theory (DFT) is nowadays a standard method for elec-tronic structure calculations. Even if in some cases DFT results are known to be unsatisfactory it can be the starting point for refinements. In this chapter we present the basic theory and point out the various approximations that are usually done. A brief description of how DFT is used in practice is also given.
When we deal with an electronic system, once we have fixed the position of the ions in the solid (Born-Oppeineimer approximation), the most general Hamiltonian we would like to solve is
H = N X i=1 pi2 2m + N X i=1 vext(ri) + X i6=j e2 |ri − rj| (2.1) The first two terms in this Hamiltonian, the kinetic term and the external potential, are one-particle operators meaning that they act separately on the coordinates of each electron. The external potential specifies the system we are dealing with. In the case of interest for us the external potential is due to the interaction between the electrons and the ions which form the lattice. What makes the Hamiltonian in (2.1) a formidable problem to solve is the electron-electron interaction term. From a physical point of view it is this term that is responsible for many of the most interesting phenomena in condensed matter physics but, from a computational point of view, this term complicates the task of diagonalizing the Hamiltonian as it makes the Hamiltonian not separable. An exact solution to the problem is possible only for few simple examples and direct numerical solutions can be computed only for systems with a small number of electrons.
An other possible approach is to try to find an approximate solution. A famous example of this kind of approach is the Hartree-Fock theory. The fundamental approximation of this theory is done on the wavefunction; the true wavefunction is replaced with the separable one
Ψ(r1, ...rN) ≈ A[φ1(r1)...φN(rN)] (2.2)
where A is the antisymmetrisation operator imposing the correct symme-try of the wavefunction. The minimization of the energy with respect to the single-particle wavefunctions φi(r) leads to the Hartree-Fock equations
[−h¯
2
2m∇
2+ V
where in addition to the kinetic term we have a mean-field term (Hartree potential) VH(r) = Z n(r 1) |r − r1| dr1 (2.4)
and the exchange potential, which is a consequence of the anti-symmetry of the wavefunction [Vexcφi](r) = − X j Z φj∗(r1) e2 |r − r1| φi(r1)φi(r)dr1 (2.5)
Here the wavefuntions φi(r) can be thought of as single-particle
wave-functions for the electron immersed in the mean-field due to the external potential and the one created by all the others electrons.
We stress the fact that the approximation is done at the level of the wave-function. In assuming the form of eq. (2.2) we are not taking into account correlations beyond the exchange between the electrons. Moreover this treat-ment has the disadvantage of generating a non-local potential, namely the exchange potential. More importantly this approach usually gives unsatis-factory results because it neglects electronic screening and thus overestimate the exchange. The next step in improving the method could be to use as a trial function a linear combination of such wavefunctions. This is in princi-ple the exact way to tackle the problem but it is in practice feasible only for small systems.
DFT takes a completely different approach to the problem. We do not try to approximate directly the wavefunction but we exploit an exact con-nection between the energy of the ground state of the system and the density which makes possible to compute in principle in an exact way ground-state properties while working on an auxilary non-interacting system.
2.2
The Hohenberg-Kohn theorem and the
Kohn-Sham representation
The first and most fundamental step in DFT is to identify the electronic density as the only necessary variable determining ground state properties. This point is not trivial at all if we realize that the informations relevant for the ground state encoded in the wavefunction, a complex function of 3N variables, can be expressed by means of real function of just 3 variables.
The Hamiltonian
ˆ
has un ”universal” part which is the kinetic and electron-eletron inter-action term, and a term which is characteristic of the system, ˆVext. Given
a certain ˆVext we could in principle diagonalize the Hamiltonian, obtain the
ground-state wavefunction (non-degenerate systems), and then compute the ground-state density ng(r). We have established the relation
ˆ
Vext→ ng(r) (2.7)
The Hohenberg-Kohn (HK) theorem[14, 20] states that even the converse is true. More precisely it states that a given ground state density can stem from only one external potential. Then, given the ground state density we can in principle obtain the potential which created it
ng(r) → ˆVext (2.8)
In order to make this property useful we still need two other steps. First we need to turn the traditional Rayleigh-Ritz variational principle on the wavefunction into a variational principle on the density and second we need a convenient way to do this variation.
The first is achieved by means of the so-called constrained search algo-rithm due to Levy[30]. The HK theorem states that for every V-representable density1, it is well defined the functional
EV0[n] = hΨg|H0|Ψgi = hΨg|T + Vee+ V0
ext|Ψgi = F [n] +
Z
Vext0 (r)n(r)dr (2.9) where Vext0 is the unique external potential that gives n(r) as ground state density. In particular F [n], the expectation value of kinetic and e-e terms, is a universal functional of the density. Following Levy, we can define an other functional
Q[n] = min
Ψ→n hΨ|T + Vee|Ψi (2.10)
where the minimum must be searched among all the wavefunctions that give n(r) as density. (Now the density must only be N-representable which means Ψ - representable). Then it can be easily proven the equality
min
n Q[n] +
Z
Vext(r)n(r)dr = Eg (2.11)
1a density n is said to be V-representable if exists an external potential which gives n
This is a quite fundamental step in DFT formulation. This tells us that we can really use the density as a variational parameter. Of course an acceptable density must be normalized to N, the number of electrons in the system but this can be taken into account by a Lagrange multiplier.
The next step is to find a convenient way to carry on the minimization procedure. In a work of Kohn and Sham[21] the problem was converted to the solution of a set of mean-field like equations. In order to explain this, be ng(r) the ground state density of the true interacting Hamiltonian. We
consider know an auxiliary non-interacting system (KS system) defined by the Hamiltonian
HKS = ˆT + ˆVKS(r) (2.12)
where ˆT is the kinetic term and ˆVKS is an external, local potential chosen
such that the ground state density of the KS system coincides with the true system one. For the ground state energy of the KS system we have the expression
Eg,KS = Tf[n] +
Z
VKS(r)n(r)dr (2.13)
where Tf[n] is the kinetic term computed in the non-interacting ground state
kinetic energy. The fact that is a functional of the density is still a conse-quence of the HK theorem. The ground-state density for the KS ground-state wavefunction is easily written
n(r) =
N
X
i=1
|φi(r)|2 (2.14)
where φi(r) are the one-particle wavefunctions. Let us now write the energy
functional for the interacting system Eg = Tf[n] + e2 2 Z Z n(r0)n(r) |r0− r| drdr 0 + Exc[n] + Z Vext(r)n(r)dr (2.15)
where we have simply written the functional F [n] in term of more familiar terms, namely the kinetic term for non interaction systems and the Hartree mean-field term, and the so called exchange-correlation functional (which is actually defined by this equation) which contains all the contributions beyond the mean field. Now we realize that we can carry on the minimization
procedure equivalently on the two equations (2.13) and (2.15) if we identify VKS(r) = δ δn(r)[ e2 2 Z Z n(r0)n(r00) |r0− r00| drdr 0 + Exc[n] + Z Vext(r)n(r)dr] = VH(r) + Vext(r) + Vxc(r) (2.16) where we have defined the exchange-correlation potential
Vxc(r) =
δ
δn(r)Exc[n] (2.17)
which is by definition a local potential.
The minimization procedure leads us to the so-called Kohn-Sham equa-tions [−¯h 2 2m∇ 2 + VH(r) + Vext(r) + Vxc(r)]φi(r) = εiφi(r) (2.18)
Where the εi are just Lagrange multipliers which keep the one-particle
wavefunctions normalized. We have then succeded in mapping the originary interacting problem into a non-interacting one. We obtained N mean-field like equations with a local potential which is defined only by the density. We want to stress that even if it seems to be the case, these equations are not to be interpreted as mean-field equations. The KS orbitals have been used just to mimick the true density and in principle have no other physical meaning. Similarly the KS eigenvalues εi are just Lagrange multipliers (except for the
highest one which correctly gives the Fermi level ref.).
Equations (2.18) must be solved iteratively. We start with a trial density (or equivalently, a trial set of single-particle wavefunctions) and we build the KS potential. Then we solve the KS equations and use the obtained wavefunctions to build the new potentials. We reiterate the process until the cenvergence has been reached. Once we get a consistent set of wavefunctions we can compute the density and the ground-state energy
ng(r) = X i |φi(r)|2 (2.19) Eg = X i εi− e2 2 Z Z n(r0)n(r00) |r0− r00| drdr 0 + Exc[n(r)] − Z Vxc(r)n(r)dr (2.20)
Even if everything that has been done so far is rigorously correct, once we start to work in the KS framework we have limited ourselves to the cal-culations of these two observables. From a logical point of view we should look at the DFT formalism as a two steps procedure: The first is the HK theorem (and the constrained search algorithm) and the second the KS solu-tion in term of single particle wavefuncsolu-tions. The HK theorem tells us that every property or quantity of the ground-state is a functional of the density. This means that even the true N-particles ground-state wavefunction and all observables’ expectation value on it can be thought as a functional of the density. Unfortunately (and obviously...) this functional is unknown and even unconceivable. Conversely it is a quite easier task to write down the expectation value of the Hamiltonian as a functional of the density which can in principle exactly evaluated by means of the KS representation.
From this discussion it should be clear that the wavefunction obtained from the KS equations is by no means an approximation to the true wave-function. Every use of it beyond the calculation of the ground-state density and energy should be considered to be out of the logical scheme of DFT.
2.3
The exchange-correlation potential
Even if we have succeeded in turning the problem of solving the true N-particle problem into the one of solving a set of self-consistency differential equations, we still can not proceed to solve them. In fact the exchange-correlation functional, where we have hidden all the interactions between electrons beyond the mean-field term is unknown. Even though some exact properties of this functional are known we have to approximate it in order to proceed. Concerning the exact properties we refer to the literature[11], here we will briefly discuss the most used approximate form for the functional.
The first and simplest functional is the so-called Local Density Approxi-mation (LDA). The assumption behind this approxiApproxi-mation is that the total exchange-correlation energy is the sum of local contributions. In formula:
Exc=
Z
n(r)xc[n(r)]dr (2.21)
where xc[n(r)] is the exchange-correlation energy per particle and it
de-pends on the local density n(r) only. Once we have assumed this form for the potential we are somehow forced, because of the universality of the functional, to replace xc(n(r)) with the exchange-correlation energy of the uniform
true system with an homogeneous electron gas with the same local density n(r) for which we are able to compute the correlation energy and then we add all the local terms.
A natural way to try to improve the LDA approach is to include in the xc-potential density fluctuations. But a naif expansion of the functional in terms of small density variation leads in general to even worse results[11]. The reason for this is that the potential (actually the xc- hole) violates an important sum rule from the energetic point of view. The correct way to include fluctuations is to use a cut-off in order to satisfy the sum rule. We do not go through the details and technicalities and simply refer to Refs.[23, 39] for a complete explanation.
In addition to these two ”classical” potentials we have to mention also the so-called hybrid potentials. They consist in a mixture of a ”classic” potential (LDA for example) and the HF exchange potential. Even if there is a rationale behind this artificial procedure[40] the hybrid potentials are not rigorously justified and more importantly are not parameter free.
2.4
Density Functional Theory in practice
In the previous section we have given the theory establishing exact rela-tionships between ground-state properties and the density. The only ap-proximation introduced so far is done on the exchange-correlation potential. However if we wanted to strictly stick to the logics of the HK theorem and the KS representation, we could compute, as already said, only the energy and the density of the ground-state. For solids we could for example study the ground-state energy as a function of lattice parameter and geometrical structure in order to predict the most stable atomic configuration. More, we could get charge-density maps and compare them with the experimental ones obtained by X-ray spectroscopy (see Ref.[19] for a review about DFT results).
What is left out of the DFT formalism is the study of excited states. The relevance of excited states of a system is evident; important physical proper-ties of matherials are in fact response properproper-ties, examples include: optical properties (which are 2-particles excitations) and photoelectron spectroscopy (which are 1-particle excitations). The correct way to treat excited states would be to ”extend” the HK theorem beyond the ground-state. This is pos-sible within Time-Dependent Density Functional Theory (TDDFT) [41, 36] and to some extent finite temperature DFT [33]. We refer the interested reader to the references. Here we will discuss how DFT results are used in practice.
In the Bloch theory of crystals the electrons are treated as non-interacting particles immersed in a external static potential, the so-called crystal poten-tial, which has the same periodicity of the lattice and accounts for electron-electron interactions at the mean-field level. In this case we can obtain a set of 1-particle states with relative eigenvalues, which can be labelled with a well defined crystal momentum k and a band index n. Plotting these eigen-values as a function of k varying in the first Brillouin zone for all n we obtain the so-called band structure plot. We could say that the problem of every ab-initio calculation method is twofold: first, the problem of building the crystal potential and second, recognizing and including the effects that are not caught by the mean-field approach. The very meaning of a bandstructure plot should be questioned in the so-called strong correlated system where the main properties are due to correlations beyond the mean-field.
Assuming that the ”mean-field” approach is an acceptable approximation and that the system can be described in terms of single-particle states we are naturally led to ask ourselves what is the best wavefunction, product of single-particle states, to represent the system. If ”the best” means the one which minimizes the ground-state energy, the Hartree-Fock wavefunction (see Introduction) is by definition what we are looking for. We can say that the HF formalism naturally leds to a band structure characterization of the crystal. Once we have the band structure we can compute the optical response on the crystal (see Chpater 3) and all the other quantities we are interested in. Indeed we have done an approximation but we know what we have neglected, and we can say that the approximation is under control.
It is not clear at all how to get such a band structure characterization of the crystal from the DFT formalism. Actually, we have to realize that this is by no means the goal of DFT. In spite of this, what all DFT based codes do is simply to take the KS eigenvalues and plot them in order to get a band strucure plot. Even the matrix element necessary for the calculation of the optical response are computed using the KS eigenfunctions. We see that what is done in practice is just use the KS wavefunction as an approximation to the real wavefunction despite it has not been designed to do this job. It could then seem unexpected that not only for the calculation of ground state properties of the system but also for of all the properties involving excited states, as well as for the band structure characterization of the system, KS eigenvalues and eigenfunctions are in general much better then the HF ones. At this level we could say that the HF approximation is too rough and it misses too many important effects which are on the contrary included in DFT. We will come on this problem again in Chapter 4, where will make use of the Green’s function and GW formalism to further try to explain why KS wavefunction ends up to be such a good approximation.
2.5
The Augmented Plane Waves method
Once we have built the crystal potential with the methods we have described in the previous sections, we have to solve the KS equations. Several methods have been designed for this purpose and we refer to Ref.[32] for a general discussion. Here we will describe only the method on which the code we have used is based, namely the (linearized) Augmented Plane Waves ((l)APW) method.
The basic idea behind this method is the fact that we can think the crystal volume as divided in two regions: the region close to the atoms and the region between them. We can expect to find a quickly varying potential close to the atoms while a smoother one in the interstitial region. Correspondingly, we can expect a quickly varying or a smoother wavefunction respectively. It is then convenient to use two different basis sets to expand the wavefunction in the two regions. In particular, we can use a set of atomic orbital-like functions for the region close to the atoms and a set of plane waves for the interstitial region. This is precisely what is done by the WIEN2k code[6]. First we choose how divide the space specifying the so-called ”muffin-tin” radius RM T for each atomic species. In the region made up of all the spheres
of radius RM T centered in the atomic positions we expand the wavefunction
as
φkn(r) =
X
l,m
[Al,mul(r, εl) + Bl,m˙ul(r, εl)]Yl,m(ˆr) (2.22)
where Yl,m(ˆr) are spherical harmonics, ul(r, εl) is the solution of the radial
equation for energy εl and ul(r, ε˙ l) is its energy derivative. This last term
is due to the linearization of the dependence of ul(r, εl) on the energy. The
coefficients Al,m and Bl,m are found matching the function and its derivative
on the surface of the muffin-tin sphere.
In the interstitial region plane waves are used
φkn(r) =
1 √
V e
iknr (2.23)
where kn = k + Kn and Kn is a reciprocal lattice vector. Finally the
Bloch wavefunction is written in the form
ψk(r) =
X
n
The parameter which controls the convergence of the basis is RM TKmax
Chapter 3
3.1
Theory
In this section we will give some details about the calculation of the optical properties that have been done. We will briefly discuss the theory and we will give more informations about the actual calculations done by the code used. The theory can be more correctly formulated in terms of the Green’s functions (see Chapter 4) but here we will present it in a more ”physical” way. We will work out the theory for the longitudinal response as it is more easy to deal with scalar quantity and because it is equal in the long-wavelength limit to the transverse one, which is the interesting case for practical applications. The situation under study consists of un unperturbed system governed by the unperturbed Hamiltonian H0, which is supposed to be exactly solvable
for which we have a complete set of eigenvectors and eigenvalues, φn and εn.
We perturb the system with an external time dependent potential resulting in the total Hamiltonian
H(t) = H0+ Vext(t) (3.1)
A convenient way to describe the linear response of a system to an applied external potential can be done in terms of the dielectric function ε. This is by definition the proportinality constant relating the external potential (externally controlled) and the actual potential in the system.
Vef f(r; ω) =
Z
ε−1(r, r1; ω)Vext(r1; ω)dr1 (3.2)
Where Vef f is the sum of the external potential and the one due to the
polarization of the system in response to the perturbation
Vef f(r; ω) = Vind(r; ω) + Vext(r; ω) (3.3)
In the previous two equations we have already Fourier transformed the time variable taking advantage of the time traslational invariance. This means that we are studing the response of the system to a external peri-odic perturbation.
An other important quantity is the density-density response function χnn
which connects the external potential and the change in the density induced by it
δn(r; ω) = Z
Linear response theory (see, for example, Ref.[11]) gives to us an exact representation for χnn. The dielectric function is connected to the
density-density response function by the formula
ε−1(r, r1; ω) = δ(r − r1) +
Z e2
|r − r2|
χnn(r2, r1; ω)dr2 (3.5)
In the case of a crystal, we can make explicit the spatial translational symmetry by Fourier transforming also the spatial coordinates. Equations (3.5) and (3.2) now read
ε−1G,G0(k; ω) = δG,G0+ vG(k)χG,G0(k; ω) (3.6) and Vef f,G(k; ω) = X G0 ε−1G,G0(k; ω)Vext,G0(k; ω) (3.7)
where fG(k) is the Fourier transform of f (r) at q = G + k, G being a
reciprocal lattice vector and k being in the first Brillouin Zone. In particular we see in equation (3.7) that a perturbation of given momentum q will in general generate charge fluctuations of momentum k + G, for all reciprocal lattice vectors G. This in contrast to the case of the homogeneus electron gas where te excitation has the same momentum as the perturbation. These higher momentum excitations are the so-called local field effects and are a consequence of the periodicity of the crystal. For the energy range of interest the external potential momentum is small compared to the dimensions of the Brillouin zone; this means that we can actually forget about the G0 dependence of the external potential and set G0 = 0. Moreover we are often interested only in the macroscopic dielectric function defined to be the proportionality constant between the external potential and the effective one averaged in the unit cell[1]. This means for a potential in the form of eq (3.7), now including the spatial dependance
< V (r; t) >u.c.=<
X
G
VG(k; ω)ei(k+G)r >u.c. = V0(k; ω)eikr (3.8)
and eq (3.7) now reads
and we define the macroscopic dielectric function
εM(k; ω) =
1
ε−10,0(k; ω) (3.10)
A typical approximation done (and that we will do) is to neglect the local field effects which are still present in the definition of eq (3.10). In formulae
εM(k; ω) ≈ ε0,0(k; ω) (3.11)
We stress the fact that this approximation is different from the long-wavelenght limit (k → 0) which we will make later on.
Now we are ready to give a formula for the calculation of the dielec-tric function. We will follow here the derivation of Ref.[12] which we think gives a quite immediate physical meaning to the formula. This derivation is equivalent to the RPA approximation[7] and a brief treatment using Green’s function will be give in a subsequent section (see GW chapter).
The starting point is a set of crystal eigenvalues and eigenfunctions (of course, in our calculations we will use the DFT ones, but see section ”DFT in practice”). These are solutions of the effective one-particle Hamiltonian
H0φn,k(r) = εn,kφn,k(r) (3.12)
Now we perturb the system with a time-dependent potential
H(t) = H0+ (V (q; ω)ei(qr−ωt)+ h.c.) (3.13)
we make use of the Fermi Golden Rule to compute the transition between two states labelled with α = (n, k) and β = (n0, k0). For the α to β transition we have Pα→β = 2π ¯ h
hφβ|Vext(q; ω)eiqr|φαi
2
δ(εβ− εα− ¯hω) (3.14)
And similarly for the β to α transition
Pβ→α =
2π ¯ h
hφα|Vext(q; ω)e−iqr|φβi
2
We now multiply Eq. (3.14) by f (εα)(1 − f (εβ)) and Eq. (3.15) by
f (εβ)(1−f (εα)), where f (ε) is the temperature-dependent Fermi distribution
function. We then subtract the obtained expressions and sum over all states to get the net number of transitions per unit time
W (q; ω) = 2π ¯ h X α,β
hφα|Vext(q; ω)e−iqr|φβi
2
δ(εα− εβ + ¯hω)(f (εα) − f (εβ))
(3.16) The conservation of the crystal momentum restrics the summation to states with k0 = k + q. We can now estabilish the connection with the dielectric function by noting that that W (q; ω) multiplied by ¯hω is the power delivered to the system and that this same quantity can be espressed in terms of ε by means of Maxwell’s equations
P (q; ω) = ω 2π q2 e2|Vext(q; ω)| 2 Im[ε(q; ω)] = ¯hωW (q; ω)) (3.17) Obtaining Im[ε(q; ω)] = 4π 2e2 q2 X n,k,n0 hφn,k|e−iqr|φn0,k+qi 2 δ(εn0,k+q− εn,k− ¯hω)(f (εn,k) − f (εn0,k+q)) (3.18) The real part of ε can be computed by making use of the Kramers-Kronig relations, obtaining ε(k; ω) = 1 + 4πe 2 q2 X n,k,n0 | hφn,k|e−iqr|φn0,k+qi|2 (εn0,k+q− εn,k− ¯hω) − iη (f (εn,k) − f (εn0,+q0)) (3.19) We can now proceed to tak the long-wavelength limit which we will dis-cuss in connection to what the OPTIC program[2] included in the WIEN2K package[6] actually computes. We start by noting that in the expression (3.19) both intraband (n = n0) and interband (n 6= n0) transitions are in-cluded. We will treat them separately as the OPTIC program treat them in different ways. We start with the interband transition where we can easily take the T → 0 limit; in this limit the Fermi distribution simply restricts the summation to transtions from fully occupied states to completely unfilled
states. Labelling these bands to which they belong with v and c respectively, in the q → 0 limit for the matrix elements we have:
hφc,k|e−iqr|φv,k+qi ≈ − ¯ h m pc,v,k εc,k− εv,k q (3.20)
where pc,v,k is the momentum matrix element between the two states of
momentum k but belonging to different bands (the transitions are vertical in the band structure plot). The limit depends on which direction the limit is taken. This defines the dielectric function to be a (diagonal in this case) matrix, one entry for each polarization. Ultimately the quantity which is computed is the imaginery part of ε which reads
Im[εi(ω)] = ¯ h2e2 πm2 X c,v Z [pi;c,v,k]2δ(εc,k− εv,k− ω)dk (3.21)
where of course the summations are restricted to available transitions. An important quantity related to expression (3.21) is the so-called joint density of states. It is defined simply by setting the matrix elements to 1.
J (ω) = ¯h 2e2 πm2 X c,v Z δ(εc,k− εv,k− ω)dk (3.22)
This quantity tells us how many transitions are allowed by energy and momentum conservation disregarding the effects of the matrix elements. This can be useful in recognizing the forbidden transitions, namely the transitions that do not occure because of a zero matrix element.
The treatment of the intraband transitions is more delicate. Even if an equivalent result can be directly obtained from eq (3.21) [11], we prefer to go through the conductivity. Again, the goal is to explain how these quantities are computed by the OPTIC program. Starting from the Maxwell’s equations we can express the power in terms of the conductivity
P (q; ω) = 2Re[σ(q; ω)]q
2
e2|Vext(q; ω)| 2
= ¯hωW (qω) (3.23)
Similarily to what we have done before, we then get the imaginary part by means of the Kramers-Kronig relations. In taking the q → 0 limit the only difference compared to the previous calculation is in the fermi function which has to be written
f (εn,k+q) − f (εn,k) ≈
∂f
We obtain σi(ω) = e2 8π3m2 τ 1 − iτ ω X n Z p2i;n,nδ(εn,k− εF)dk (3.25)
Where we have renamed η = ¯hτ−1. The delta function comes from the derivative of the fermi distribution and restricts the sums over states at states at the Fermi level. Recalling the expression for the static conductivity from the Drude model we can identify the plasma frequency
ω2p,i = h¯ 2 e2 πm2 X n Z p2i;n,nδ(εn,k− εF)dk (3.26)
This is the quantity computed by the program. The output is then pre-sented under the form of a dielectric function for a Drude model with the computed plasma frequency
i(ω) =
ωp,i2
ω(ω + i/τ ) (3.27)
The parameter τ is an external parameter and can be thought as a broad-ening parameter. The OPTIC program allows us to study separately the contributions of the intra- and inter-band transitions and then to sum them in order to get the total dielectric function.
3.2
Examples
In order to make explicit the meaning of the various constants introduced so far we show some example plots for the dielectric function in the case of inter- and intra-band transition. This will help us to immediately recognize their values by just looking at the plot. We plot the conductivity and the absorption coefficient as well (see Appendix) in order to make explicit the connection between the various optical constants.
We start with the intraband contribution to the dielectric function. As we can see in Eq. (3.27) we have only two parameters, namely the plasma frequency ωp and the relaxation time τ . The real and imaginary part of the
dielectric function are shown in Fig. 3.1 for three different models ( i.e. three different sets of parameters).
We see that the value at which the real part of ε crosses the zero axis is determined by both parameters but it is equal to the plasma frequency ωp
-10 -5 0 5 10 0 1 2 3 4 5 6 7 8 [eV] Im Re p=4 =0.5 p=6 =0.5 p=4 =1
Figure 3.1: Drude-like real (full line) and imaginary (dashed line) part of the dielectric function for three different set of parameters: ωp=4 τ =0.5 (red),
ωp=6 τ =0,5 (green), ωp=4 τ =1 (blue).
be in this limit we can take as plasma frequency the value at which the real part of is zero.
We can compute the corresponding absorption coefficient starting from the real and maginary part of the dielectric function. The result is shown in Fig. 3.2 with the same sets of parameters.
We observe that the value of the relaxation time determines the height of the peak while the cut-off frequency mostly depends on the plasma frequency ωp. In the Drude model the electron gas can absorb only light with frequency
smaller then the plasma frequency, while for larger frequncies it is almost transparent.
We proceed now to give similar examples for an interband transition. We will perform the calculations in a simple case: we assume a direct gap between two parabolic bands. The valence band is supposed fully occupied and the conduction band completely empty. We start from the derived equation
0 2 4 6 8 10 0 1 2 3 4 5 6 7 8 absorption (a.u.) [eV] p=4 =0.5 p=6 =0.5 p=4 =1
Figure 3.2: Drude-like absorption coefficient computed starting from the dielectric functions of Fig. 3.1. The colors have been chosen in the same way. (3.21) for the imaginary part of the dielectric function. In this simple case we obtain Im[ε(ω)] = ¯h 2 e2 πm2 Z [pi;c,v,k]2δ(εc,k− εv,k− ω)dk (3.28)
Now we have to make an assumption for the matrix elements in order to carry on the calculation. We assume it to be non-zero for k=0 and approx-imable to a constant in the energy range of interest.
[pi;c,v,k] ≈ M (3.29)
Obtaining the expression
Im[ε(ω)] = ¯h 2 e2|M |2 πm2 Z δ(εc,k− εv,k− ω)dk (3.30)
We can evaluate the integral if we write the energy dispersion relation in the simple way
εc,k= εc+ ¯ h2k2 2mc (3.31) εv,k = εv− ¯ h2k2 2mv (3.32) where mc and mv are the effective masses of the conduction and valence
band respectively. Introducing the reduced mass
1 µ = 1 mc + 1 mv (3.33) the argument of the delta function becames
εG+
¯ h2k2
2µ − ¯hω (3.34)
where we have intoduced the energy gap εG = εc− εv. At this point the
integration is straightforward and we obtain
Im[ε(ω)] = 2e 2 m2|M | 2 (2µ ¯ h2) 3 2 1 ω2(¯hω − εG) 1 2 (3.35)
for ¯hω > εG and zero otherwise. As expected we have no absorption until
the photon energy is equal to the energy gap. Unfortunately the approxima-tion is too rough to make use of the Kramers-Kronig relaapproxima-tions (which need the knowledge over all the frequency axes) to get the real part.
For the general case, even if the precise behaviour of the imaginary part of ε depends on the shape and the filling of the bands involved in the transition, this calculation clearly shows that traces of an allowed interband transition can be found in the imaginary part of ε. The more realistic case can be expected to be a combination of interband and intraband transitions which have to be analyzed case by case.
3.3
The Kubelka-Munk model
In this section we address the problem of deriving the color of a material starting from the knowledge of its microscopic optical properties. First, we have to realize that the most part of the light that reaches our eyes has not been directly reflected by the objects but it is actually diffused light. If we want to compute the diffuse reflectance we should take in account the actual composition of the material which is diffusing the light.
The most simple way to compute the diffuse reflectance is by means of the Kubelka-Munk model[25]. The system under study consists of a layer of thickness L made up of closely packed particles possibly immersed in a matrix with concentration c. The particles are characterized by two quantities: the absoprtion coefficient α(ω) and the backscattering coefficient β(ω). The first is an intrinsic parameter and can be computed using the methods described in the previous section. The latter gives the amount of light which is scattered to the direction opposed to the incident one which depends on the actual shape of the particles. It is in principle a frequency dependent parameter but it is found for several materials to be approximately constant in the visible light range and to lie in the range 10 − 500mm−1[29]. The angular dependence of the diffuse reflectance is completely neglected and we end up with a simple one dimensional model. At the bottom of the layer we have a substrate characterized by the reflectance RS.
Calling J+(x) and J−(x) the two light fluxes propagating downward and
upward respectively along the x direction we have the set of differential equa-tions
dJ+(x)
dx = −2αJ+(x) − β(J+(x) + J−(x)) (3.36)
dJ−(x)
dx = 2αJ−(x) + β(J+(x) + J−(x)) (3.37) From these equations we obtain for the reflectance at x R(x) = −J+(x)/J−(x)
the equation dR(x) R = dxβ(R + 1 R − 2(1 + 2α β )) (3.38)
Integrating x from 0 to L with R(0) = RS we have for the reflectance at
R(L) = R −1 ∞(RS− R∞) − R∞(RS− R−1∞) exp(βL(R−1∞− R∞)) (RS− R∞) − (RS− R−1∞) exp(βL(R−1∞− R∞)) (3.39) where R∞ = 1 + 2α β − r (1 + 2α β ) 2− 1 (3.40)
is the diffuse reflectance for the case of an infinitly thick layer.
The model can be made more sophisticated to take in account more effects and we refer to Ref.[28] for a more complete explanation.
We will make use of the KM model in the simple case of an infinite layer Eq. (3.40). Taken in this form the KM model simply tells that all the light which is not absorbed is diffused. This can be seen taking the limit α = 0 which gives R = 1 and α → ∞ which gives R = 0.
Figure 3.3: Color matching functions of the CIE 1931 XYZ color space. They reproduce the sensitivity of the human eye to the various colors. Picture taken from Ref.[51]
We will make use of the computed diffuse reflectance to compute the color in a similar way to what is done in Ref.[47]. The diffuse reflectance is sampled with the so-called color matching functions of the CIE 1931 XYZ color space[44]. The three color matching functions Fig. 3.3 reproduce the sensibility of the human eyes to red, green and blue. Starting from the
knowledge of the diffuse reflectance R(ω) and of the spectral distribution of the light source S(ω) we can compute the XYZ coordinates with the formula
(X, Y, Z) = C Z
dωR(ω)S(ω)(x(ω), y(ω), z(ω)) (3.41) where C is a normalization constant chosen such that CR dωS(ω)y(ω) = 1. In the XYZ values we find information about the color and the bright-ness. While the brightness depends on the value of β, the color is almost independent of it as long we assume β to be constant with ω.
Chapter 4
Many-body perturbation
theory
4.1
Introduction
The Density Functional Theory since its formulation has succeeded in giving a correct description of the electronic structure of several crystals. However DFT and the approximation for the exchange-correlation potential, in partic-ular LDA, are known to fail in some specific cases. The most famous weakness of DFT is its capability to correctly predict the gap for insulators[38]; when comparing DFT results with experimental data we find a systematic under-extimation for the gap. In some cases DFT gives even qualitatively wrong results predicting some insulators to be metallic. This is the case, for exam-ple, of the Mott-Hubbard insulators[15] and f-electron systems[8]. Strictly speaking these are necessarily not problems of DFT. As already discussed in section (DFT in practice) we have two kinds of approximation in practical uses of DFT: the first concerning the exchange-correlation potential, and its approximations 1, the second, less controlled, connected to the fact that we
often compute quantity which involve excited states of the system, which are out of the DFT formalism. In particular what is hard to accept logically is the interpretation of the KS eigenvalues as excitation energy of the system.
In trying to improve the results given by DFT we can proceed in two ways: we can try to obtain a better exchange-correlation potential or we can totally abandon the DFT formalism searching for completely different methods. The many body perturbation method we are going to describe falls in the second category. In this chapter we will briefly introduce the theoretical apparatus of the general many body perturbation techinique and the we will focus on the Hedin’s scheme. The very physical meaning of these quantities is provided by the Landau’s Fermi liquid theory (random references) in terms of quasiparticle excitation energies.
The most foundamental tool in dealing with electronic excitations in solids is the Green’s function. Given the Hamiltonian of the system H, the zero temperature Green’s function is defined
G(r1, t1; r2, t2) = − i ¯ hT (ψ(r1, t1)ψ †(r 2, t2) (4.1) where ψ(t), ψ†(t) are the time evolved annihilation and creation operators, T is the time ordering operator and the average is taken on the exact ground state of the system. For t1 > t2 it represents the probability amplitude for
finding a particle in 1 = (r1, t1) after the addition of a particle at 2 = (r2, t2)
while for t2 > t1 is the probability amplitude to find a hole in 2 after the
1In particular we can mention the derivative discontinuity problem[11] that can be
removal of a particle at 1. Given the Hamiltonian, now written with the second quantization formalism
H = Z ψ†(r)[−¯h 2 2m∇ 2+ V ext(r)]ψ(r)dr + 1 2 Z Z ψ†(r1)ψ†(r2) e2 |r1− r2| ψ(r2)ψ(r1)dr1dr2 (4.2) Once we have Fourier transformed the time variable taking advantage of the time translational invariance of the system, it can be shown (see next section) that the Green’s function satisfies the equation
[ε + ¯h 2 2m∇ 2 1− Vext(r1) − VH(r1)]G(r1, r2; ε) − Z Σ(r1, r3; ε)G(r3, r2; ε)dr3 = δ(r1− r2) (4.3) where VH, the Hartree mean-field term, has been already ”extracted”
from the electron-electron interaction term. Σ is the so-called self-energy operator and contains by definition all the correlations between electrons. In general this operator is complex, non local and energy dependent. To clarify the meaning of the self-energy operator, let’s define the ”non-interacting” Green’s function the solution of the equation2
[ε + ¯h
2
2m∇
2
1− Vext(r1) − VH(r1)]G0(r1, r2; ε) = δ(r1− r2) (4.4)
Then it easy to show the connection (Dyson equation) with the interacting one
G(r1, r2; ε) = G0(r1, r2; ε) +
Z
G0(r1, r3; ε)Σ(r3, r4; ε)G(r4, r2; ε)dr3dr4
(4.5) This formula is the basis for perturbation theory and will be of practical use when we will discuss the computational techniques.
An important representation for the Green’s function can be obtained by inserting the identity operator written as a sum of a complete set of projectors directly in the definition eq (4.1). We obtain the Lehmann representation
G(r1, r2; ε) = X εn≥µ hN, 0|ψ(r1)|N + 1, ni hN + 1, n|ψ†(r2)|N, 0i ε − (EN +1,n− EN,0) + iη (4.6) +X εn<µ hN, 0|ψ†(r 2)|N − 1, ni hN − 1, n|ψ(r1)|N, 0i ε − (EN,0− EN −1,n) − iη (4.7) where µ = EN +1,0− EN,0 is the chemical potential and η is an arbitrary
small positive quantity.
Plugging this expression in the equation of motion for the Green’s func-tion (4.3) we see that the eigenfuncfunc-tions fn(r) = hN, 0|ψ(r)|N + 1, ni and
eigenvalues εn = EN +1,n − EN,0 for εn ≥ µ (and similarly for εn < µ) are
solutions of the equation
[−¯h 2 2m∇ 2 1+ Vext(r1) + VH(r1)]fn(r) + Z Σ(r, r1; εn)fn(r1)dr1 = εnfn(r) (4.8) The functions fn have to be interpreted as quasiparticles wavefunctions
and the eigenvalues εn are the one-particle excitation energies.
This is in principle the exact way to tackle the problem. In this approach we can study both ground state properties and excited states of the system. Of course the difficult part is the avaluation of the self-energy Σ for which we will have to find some approximations. A perturbative expansion, formally derived from the Dyson’s equation, in term of the bare Coulomb interaction is not straighforward. On the other hand, the method due to Hedin expands the self-energy in terms of the weaker screened Coulomb interaction.
4.2
Hedin’s equations and the GW
approxi-mation
In this section we present the Hedin’s equations which are the basis for a perturbative expansion in term of the screened Coulomb interaction. From these equations it is then easy to derive the GW approximation which is the form of the equations which is actually used in real computations. We will use the same notation as in Ref.[13]. We start by simply writing down the equations. For the derivation we refer the interested reader to Refs.[13, 3]
Σ(1, 2) = i¯h Z
P (1, 2) = −i¯h Z G(2, 3)G(4, 2+)Γ(3, 4; 1)d(3, 4) (4.10) Γ(1, 2; 3) = δ(1 − 2)δ(2 − 3) + Z δΣ(1, 2) δG(4, 5)G(4, 6)G(7, 5)d(4, 5, 6, 7) (4.11) W (1, 2) = v(1, 2) + Z v(1, 3)P (3, 4)W (4, 2)d(3, 4) (4.12) G(1, 2) = G0(1, 2) + Z G0(1, 3)Σ(3, 4)G(4, 2)d(3, 4) (4.13)
We have included the Dyson’s equation (4.13) because it is useful to set up the perturbative scheme. In addition to the self energy operator already introduced in the previous section, we have introduced three more quantities that must be discussed: the polarization P , the vertex function Γ and the screened interaction W . The polarization is similar to the density-density reponse function introduced in Chapter 3 with the difference that it connects the density variation to the total effective potential (external plus induced potential). In the same notation of Chapter 3 we have
δn(r; ω) = Z
P (r, r1; ω)Vef f(r1; ω)dr1 (4.14)
The external field considered here is just a useful tool for the derivation of the equations and is set to zero at the end; we refer to the cited references for a more detailed derivation. The screened interaction W is defined as a corrected Coulomb interaction. If we simply replace W with v we have
W (1, 2) = v(1, 2) + Z
v(1, 3)P (3, 4)v(4, 2)d(3, 4) (4.15) where the second term is a correction which takes in account the polariza-tion of the system and reduces the strnght of the interacpolariza-tion. At this point we can describe the interaction process as a three steps process: (1) particle 1 interacts with the system (2) the system responds with a polarization (3) the polarized system interacts with particle 2. We can realize that at this stage the particles are treated as external to the system (in other words, W is a test charge-test charge interaction). To take in account the fact that the two particles actually belong to the system we have to include the vertex
Figure 4.1: Schematic represantation of the Hedin’s equation scheme. The GW approximation is obtained by putting Γ = 1. The integral symbols are implied.
function Γ which enters in the equations for Σ and P . We can summarise the scheme given by Hedin’s equations as in Fig. 4.1 (where the integral symbols are implied)
We can now describe the GW approximation which is simply the first order approximation for the self energy derived from the Hedin’s equations. We have to ”build” the screened interaction and to do this, at the lowest order we can take
Γ(0)(1, 2; 3) = δ(1 − 2)δ(2 − 3) (4.16)
P(0)(1, 2) = −i¯h Z
G(2, 3)G(4, 2+)Γ(0)(3, 4; 1)d(3, 4) = −i¯hG(1, 2+)G(2, 1) (4.17) Inserting this last expression in the equation for W we have the lowest order approximation for W
W (1, 2) = v(1, 2) + Z
v(1, 3)P(0)(3, 4)v(4, 2)d(3, 4) (4.18)
In this way we obtain the so called GW approximation for the self energy operator
Σ(1)(1, 2) = i¯hG(1, 2)W (1+, 2) (4.19) At this point we can use the G0 Green’s function defined by Eq. (4.4) to
compute the polarization and the screened Coulomb interaction and finally the self-energy operator. We are then able to ”update” the Green’s function via the Dyson’s equation. In practical calculation the choice of G0 is not that
trivial and this subject will be briefly discussed in the next section.
4.3
GW in practice
In the previous section estabilshed the theoretical background for the GW approximation. The Hedin’s equation represent a logically closed scheme. In practice the task is complicated by the enormous computational time de-manded for a full solution of the Hedin’s equations. In practical calculations we have further approximate in order to make the calculations feasible. In this section we will describe these approximations in relation to the actual calculations performed by the program we have used.
The first step is to choose a starting Green’s function. In the previous section we have defined the G0 operator as
[ε + h¯
2
2m∇
2
1− Vext(r1) − VH(r1)]G0(r1, r2; ε) = δ(r1− r2) (4.20)
which means that G0 is the Green’s function which exactly solves the
”Hartree problem”. This could be obtained as a result of a previous calcu-lation and then used to compute the self-energy. Even if this could seem the most natural way to proceed, this is not what is done in practice. Our starting point is the DFT Green’s function: we start from a well converged DFT calculation and we identify the Kohn-Sham eigenvalues and eigenfunc-tions as the quasiparticle energies and wavefunceigenfunc-tions. This allow us to build the DFT Green’s function which is used as zeroth order Green’s function for the GW calculation. Then we can compute the DFT polarization and we
obtain a similar result of what we obtain through the procedure of chapter 3
3. After the computation of W we can finally build the self-energy operator
Σ. The final results are presented as a shift of the DFT eigenvalues, the new eigenvalues being given by
εGWn,k = εDF Tn,k + φGWn,k Re[Σ(εDF Tn,k )] − Vxc
φGWn,k (4.21) where we have subtracted the contribution of the DFT exchange-correlation potential to avoid the double-counting and we have empolyed the quasi-particle approximation.
The reader can realize that the procedure is not selfconsistent. The oper-ators P , W and Σ are defined starting from the true Green’s funtion, G, not G0. In order to reach the consistency we should, after the computation of
Σ, update the Green’s function via Dyson’s equation. Then we could go all around the circle of Fig. (with Γ taken to be the identity!) and reiterate the entire procedure up to convergence. On the contrary, due to the demanding computational time, we stop at the very first calculation of the self-energy (”one shot GW”). The rationale behind this procedure is connected to the problem of choosing G0. We can think that if we ran a full consistent
calcula-tion we should not worry about the starting Green’s funccalcula-tion since for every choice, the final result result should be independent of the starting point and all the G0 should converge to the true (in the sense of GW approximation)
Green’s function G. The fact that we can not practically preform such a cal-culation forces us to choice the ”best” G0. It is generally true that the DFT
results are better then the Hartree (or Hartree-Fock) that is to say a better G. Consequently we have a better description of the polarization (which is connected to the screening), and of W . We can then expect to find a better self-energy if we start from the DFT result. This would provide the rationale behind the procedure exposed, and should clarify the reason why we can believe the one shot procedure to be meaningful. As always the quality of the method must be discussed in comparison with experimental results and it can not be taken as a safe method.
Chapter 5
The metallic red photocatalyst
5.1
Introduction
In this chapter we describe our results about the material under study ob-tained with the methods we have described previously. The transition metal oxide, SrNbO3 can be considered interesting under various aspects. From the
point of view of applications and technologies, it has recently proven to be a good photocatalyst[52] ; it is able to absorb visible light and use this energy to break the water molecule in oxygen and hydrogen. From a theoretical point of view it is interesting to study an oxide with the 4d transition metal Nb: while the 3d row has been intensively studied (especially materials with V or Ti[48]) the 4d row has not been studied that sistematically. This is due to the fact that the 3d metal based materials are known to be an example of correlated materials and they provide a benchmark for these kind of systems. Due to the larger extent of the 4d orbitals we expect the strong correlation effects to be small or negligible in SrNbO3, making possible to study this
material with the standard techniques, namely DFT and GW.
The employment of the GW approximation has been mostly used in the study of gapped systems mainly for the correct prediction of the energy gap. So far there are only few works about metals treated with the GW technique and then this work can be considered even as a test of the GW method in a metallic system.
The most interesting property of SrNbO3 is the capability of absorbing
the visible light. As we will discuss in the next section, it strongly absorbs the light with energy above 2 eV; what is left of the visible light in the range 1.8/2 eV gives to this material the red color (at least when it is in powder form). Our work will focus on this property. Several models have been proposed in order to explain the optical properties of SrNbO3 and we
will investigate these models as well.
In the next section we discuss the works on the subject that have been done so far. Then we briefly discuss some computational details about our caluclations. We start then to describe our results. First we study the perfect crystal case, with no defects in the structure, then we consider the effects of two kind of defects in the structure, Oxygen and Strontium vacancies. While the Strontium vacancies have been reported in some cases, it seems that the stoichiometry of the Oxygen is well under control. Nevertheless, Oxygen’s vacancies are a known problem in oxide and have already proven to have important consequences and we think it is worthy to investigate this kind of defects as well. For each case we study the electronic structure and the optical properties derived from it. The Kubelka-Munk model (see Chapter 3) is used to check if the obtained optical properties are compatible with the observed red color.
The results will be summarized in the next chapter.
5.2
State of the art: origin of the color
The previous works about the 4d transition metal SrNbO3 are both
theo-retical and experimental. The first samples synthesized have been obtained in powder form via solid state reaction [31, 17, 52, 37]. All these sample are paramagnetic and exhibit a metallic behaviour with a resistence which is comparable to the other 3d and 4d isostructural compounds. The structure is found to be a cubic perovskite with a lattice constant of 4.023 ˚A[37] but small distortion of the ideal structure have been reported. A common feature of all these sample is the Sr-deficiency; all the samples have stoichiometric formula SrxNbO3 with x ranging from 0.75 to 0.97 and the Sr concentration
in the final product seems to be sensitive to the preparation details. When reported (Refs.[31, 52] and Supplementary Information of Ref.[50] the color is said to be red or brownish. Thin film samples have been synthesized as well via pulsed laser deposition [50, 35] and even if they seem to be closer to the stoichiometry the process is quite delicate and sensitive to the syn-thesis paramters (in Ref.[35] the authors clearly point out the difficulties encountered in the process). It seems there is a difference in the color of the material between the powder and the thin film sample; in particular the film of Ref.[50] is transparent[42] but when it is ground in powder to probe the photocatalytic activity, it acquires the typical red color. This suggests that the contact with air, facilitated by the higher surface/volume ratio of the powder, would cause some reaction and a change in the constitution of the sample. To our knowledge no single crystal has been produced so far.
The experimental data about the optics of SrNbO3 that have been
re-ported amount to a photoemission spectrum in Ref.[16] and few optical spec-tra: in Ref.[52] the authors reported an absorbance spectrum obtained from a powder sample, which clearly shows an absorption onset at about 2 eV. This value has been obtained for samples with different x values (x=0.8, 0.85, 0.8) and the onset seems to be insensitive to Sr concentration. In this reference the authors provided the first evidence of the phocatalytic activity of SrNbO3, the central feature of this material.
An other, more complete study of this material is the one of Ref.[50] where in addition to the absorbance and trasmittance spectrum we have the reflectivity and the refractive and extinction coefficients, measured via spec-troscopic ellipsometry on a thin film sample. It is interesting to compare the two absorbance spectra of Refs.[52] and [50] ; the first (powder) is consistent with the red color since all the visible light but red is absorbed (see
Kubelka-Munk model) while in the second (film) we have just a peak centered in the red region while it is almost transparent for the other colors. Moreover, the extinction coefficient shown in Ref.[50] has no features in the energy range 2/4 eV. This is compatible with the transparency communicated by the au-thor [42].
We have already pointed out in the introduction that the most interesting property of SrNbO3 is the absorption of energy coming from the visible light
which is fundamental for the photocatalytic process. From the theoretical point of view the challenge is to find an explanation for this absorption: with our computational tools we should be able to derive the optical properties from the very basic information about the structure. The understanding of the mechanism behind the observed absorption could clear the way for a controlled tuning of the important properties and moreover, due to the common structure of the perovskite oxide compounds, we could extend the conclusions to other materials. Several theoretical studies share this goal. The first DFT based calculation is the one of Ref.[52]. Their calculation predicts an energy difference between the t2g bands (see next sections) where
it is found the Fermi level, and the first unoccupied band which fits with the position of the absorption peak. This hypothesis is supported by the calculations performed by the authors of Ref.[55] who have investigated the nature of the matrix elements as well and found the Sr unoccupied states to be important for the transition. A different scenario is supposed in Ref.[45]; here the authors have investigated the effects of the Sr-vacancies in the structure which have been reported in the literature. In their calculation the presence of Sr-vacancies would greatly narrow the ”gap” between the Oxygen p states and the t2g band to the value of 1.7 eV with the first allowed transition at
about 2eV . Even if this calculation is the only attempt we have found in the literature to investigate the effects of possible defects, we believe that this result could depend on the utilisation of an hybrid functional and on the assumption of a magnetic ordering which has not been observed. Finally, a third scenario is proposed in Ref.[50] where a measured surface plasmon mode of energy 1.8 eV would be responsible for the absorption. We can summarize the three scenarios with the help of Fig. 5.1: (i) an interband transition from the t2g manifold and some unoccupied bands, (ii) an intraband transition
between O-p states and the t2g bands in presence of Sr-vacancies and (iii) a
surface plasmon at 1.8 eV.
With our methods we can and we will fully investigate and check out the first two proposed scenarios. On the contrary it is more complicate to inves-tigate theoretically the surface plasmon. In our work we will limit ourselves in the calculation of the plasma frequency with the methods presented in Chapter 3 . We also have to say that this proposal could even coexist with
Figure 5.1: Schematic picture of the electronic structure of SrNbO3 and the
three proposed scenarios for the absorption: (i) interband transition from t2g
to eg/Sr states, (ii) interband transition from O to t2g states in presence of
Sr-vacancies and (iii) surface plasmon absorption. the others.
5.3
Computational details
In this section we will give the essential computational details of the calcu-lations we have performed. We have considered three different structures: a perfect cubic perovskite structure, a structure with Sr-vacancies and one with O-vacancies. For the ideal crystal calculation we used the experimental lattice parameter of a = 4.023 ˚A [37]. For the vacancies cases we have used a supercell approach; we repeated the unit cell and we took out one Sr atom for the Sr-vacancy case and one or two O atoms for the O-vacancy case. A supercell of 2 × 2 × 2 unit cells was used for the one Sr-vacancy and one O-vacancy cases while a 2 × 2 × 3 supercell was used for the two O vacancies case (see Fig. 5.2). For the two O-vacancies case this particular configura-tion, with two aligned vacancies, was found to be the only one which shows substantially different features in comparison to the one vacancy case (see section 5.6). All the structures have been relaxed using the pseudopoten-tial DFT based code VASP[24, 22] while keeping the volume of the supercell
Figure 5.2: The different types of crystal structures of SrNbO3investigated in
this study: (a) The stoichiometric crystal structure SrNbO3, with the
exper-imental lattice constant a = 4.023 ˚A[37], (b) one oxygen vacancy (indicated by the blue atom) in a 2 × 2 × 2 supercell, corresponding to SrNbO2.875, and
(c) two oxygen vacancies in a 2×2×3 supercell, corresponding to SrNbO2.833.
For the supercell structures all internal atomic positions were relaxed within Density Functional Theory (DFT) while keeping the unit cell volume con-stant.
constant and with a 8 × 8 × 8 discretization of the Brillouin zone. We have studied the Sr-vacancies case in the so-called Virtual Crystal Approximation (VCA) as well, where we used a unit cell but reduced the number of the electrons of the Sr atom to mimick the hole doping caused by the vacancy.
All the other DFT based calculations were performed using the all-electron full-potential WIEN2k code[6] where a PBE-GGA[23] approximation for the exchange-correlation potential was used. We have used a k-mesh of 5000 and 2000 points for the unit cell calculation and the supercell respectively. The RT M parameter equal to 7 was used and was found to be satisfactory.
Based on the DFT results we have derived the optical properties of the vari-ous cases studied employing the methods we have described in Chapter with OPTIC package [2]. Generally, the number of k-points was increased up to convergence for the calculation of the optical spectra.
We also performed GW calculations on top of the DFT outcome using the FHI-gap code[18] but due to the large computational time, we have further studied within the GW approximation only the structures made up of only one unit cell, namely the ideal crystal and the Sr-vacancy (within the VCA) cases. A 4 × 4 × 4 discretization of the Brillouin zone was used.
The results about the electronic structure mainly consist of Density of States (DOS) and band structure plots. Further informations are given by the orbital projected Density of States (pDOS) defined by
n0(ε) =
X
n,k
|hψ0|φn,ki|2δ(εn,k− ε) (5.1)
where the state ψ0 is an atomic orbital and ψn,k is a Bloch state. The
pDOS gives us informations about the nature of the Bloch states at a given energy ε and is often useful to have a more physical picture, closer to the atomistic one, of the crystal.
Generally the final results are obtained increasing the nuber of k-points and running a non self-consistent calculation. This means that once we have a fully converged self-consistent calculation, we build the potentials (or the self-energy in the case of a GW calculation) and we compute the eigenvalues on a more refined grid without further update the potentials. This standard procedure is meant just to reduce the effects of the discretization on the final results.
5.4
Calculation for the ideal crystal
We start the study of the material assuming a perfect infinite crystal. We show the atom projected Density of States in Fig. 5.3. The DOS at the Fermi level,chosen as the origin of energies is non zero which means that DFT correctly predicts this material to be metallic. We gain more information by looking at the character of the states. We see that the character of the occupied states in the range between -8 and -4 eV is mainly of O nature while for the unoccupied states we have mixed Sr and Nb character. The states at the Fermi level come from Nb atom. From a molecular point of view this result confirms the valence states of -2 and +2 for O and Sr respectively and we can think of the O atoms as having a completely filled 2p shell and the Sr with a 4p filled shell (we remark that the first unoccupied states coming from Sr are of d character). We are left with a single electron which will occupy the 4d state of Nb at the Fermi level, leaving the Nb formally in a 4d configuration In this way we have an octahedron of O ions surrounding
