Density Functional Theory Studies for Transition Metals: Small (Fe,Co)-clusters in fcc Ag, and the Spin Density Wave in bcc Chromium.

Transcription

1 Density Functional Theory Studies for Transition Metals: Small (Fe,Co)-clusters in fcc Ag, and the Spin Density Wave in bcc Chromium. Promotor: Prof. Dr. S. Cottenier Proefschrift ingediend tot het behalen van de graad van doctor in de wetenschappen door Veerle Vanhoof 2006

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3 Contents Introduction 7 1 Introduction to ab initio calculations Many-body systems Level 1: The Born-Oppenheimer Approximation Level 2: Density Functional Theory The theorems of Hohenberg and Kohn The Kohn-Sham equations The exchange-correlation functional Spin Density Functional Theory Spin-orbit coupling Beyond the LDA/GGA-level of Density Functional Theory Level 3: Solving the equations The APW method The LAPW method The APW+lo method Wien2k Magnetic clusters in nonmagnetic host lattices Computational and experimental details Results and Discussion on magnetic clusters in Ag Fe in Ag Co in Ag FeCo in Ag Conclusions

4 4 CONTENTS 3 The Spin Density Wave in Cr: ab initio and hyperfine field considerations Computational details Choosing a XC-functional The Pure Spin Density Wave Perturbation range and node-node interaction The formation energy of node planes Conclusions The Spin Density Wave with impurities Cr/Sn multilayers Isolated Cd and Sn impurities Conclusions Summary and Conclusions 87 A Bloch s theorem 91 B Brillouin zone integration 93 C Mössbauer spectroscopy 95 D Paramagnetism and superparamagnetism 103 E Low Temperature Nuclear Orientation 107 F Cr and the Spin Density Wave 111 F.1 Structural and magnetic properties of Cr G LDA(+U) versus GGA: the extended edition 115 H LDA+U and the Spin Density Wave 121 I List of Abbreviations 125 J List of Publications 127 K Nederlandse samenvatting 129 K.1 Magnetische clusters in een niet-magnetisch gastrooster K.2 De spindichtheidsgolf in Cr

5 CONTENTS 5 K.3 Besluit Bibliography 135

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7 Introduction Since its theoretical foundation in the mid-1960 s [1, 2], Density Functional Theory (DFT) is being used in computational solid state physics to calculate structural, magnetic, hyperfine and ground state properties of real materials from first principle. In this work we will use Density Functional Theory to study two transition metal materials. A short introduction for each subject is given here, a more elaborated one at the beginning of the respective chapters. The first study is devoted to small clusters of transition metals in a nonmagnetic host lattice. They are known from Mössbauer spectroscopy to be superparamagnetic. The experimental data are, however, often rather featureless, and do not allow to determine the fluctuation frequency for the different cluster sizes. In this work, we demonstrate how the combination of ab initio calculations and Mössbauer experiments in the mk range allows one to obtain quantitative values for the fluctuation frequencies for the case of single Fe impurities and FeCo dimers in Ag. We show that the fluctuation frequency of single Fe impurities in Ag is constant between 77 K and 6 mk, while the fluctuations of a FeCo dimer in the same sample are two orders of magnitude faster at 77 K but rapidly drop to become static at 6 mk. The second research topic concerns the spin density wave (SDW) of Cr. Bcc Cr is the only light to medium-heavy element for which Density Functional Theory (DFT) predicts the wrong ground state for the pure compound: antiferromagnetic, instead of the experimentally found antiferromagnetic spin density wave (SDW) ground state. Very recently, a remarkable solution to the conflict was put forward by Uzdin and Demangeat [3]. In this work we will make the key ideas that are present in the work of Uzdin and Demangeat more explicit, and give careful arguments for these ideas, based on experiment, 7

8 8 Introduction existing and new interpretations of their model hamiltonian results, and new ab initio calculations for the pure Cr spin density wave and Cr doped with impurities. This thesis contains four major chapters. In the first chapter an introduction to ab initio calculations is given, describing Density Functional Theory and its solving methods. The application of Density Functional Theory for the hyperfine study of small Fe/Co-clusters in a nonmagnetic Ag host is discussed in the second chapter. It is followed by the study of the spin density wave in Cr in the third chapter. Finally, conclusions and an outlook are presented in the last chapter.

9 Chapter 1 Introduction to ab initio calculations The study at the quantum level of the electronic, magnetic and other properties of a periodic crystal is one of the traditional subjects of solid state physics. At present, a lot of theoretical schemes have been proposed to interpret experimental measurements, to predict new effects and to design new materials from first principle (or ab initio). A calculation is said to be ab initio if it starts from the basic equations of motion (Schrödinger or Dirac equation) without the use of any empirical parameters. In this chapter an introduction to ab initio calculations and the basic concepts behind it will be given. 1.1 Many-body systems From the microscopic point of view a solid can be seen as a collection of heavy, positively charged nuclei (n) and lighter, negatively charged electrons (e). The nuclei and electrons are treated as electromagnetically interacting point charges 9

10 10 CHAPTER 1 Introduction to ab initio calculations and the exact nonrelativistic many-body hamiltonian for this system becomes: Ĥ = h2 2 i 2 Ri M i h e 2 8πɛ 0 r i r j i j i r 2 i 1 e 2 Z i m e 4πɛ 0 R i r j i,j + 1 e 2 Z i Z j 8πɛ 0 R i R j i j (1.1) M i is the mass of the nucleus at R i. The electrons have mass m e and are positioned at r i. The first and second term correspond to the kinetic energy operator of the nuclei and of the electrons respectively. The remaining terms describe the Coulomb interaction between electrons and nuclei, among electrons and among nuclei respectively. To know the state (wave functions Ψ( R, r), with R = { R i i = 1,..., N n } and r = { r i i = 1,..., N e }) of this system the corresponding Schrödinger equation has to be solved: ĤΨ( R, r) = EΨ( R, r) (1.2) Unfortunately, due to the high degree of complexity it is impossible to solve this equation without introducing some approximations. 1.2 Level 1: The Born-Oppenheimer Approximation The mass of the electrons is much smaller than the one of the nuclei while the electromagnetic forces acting on them are of comparable magnitude. Consequently, the electronic motion ( 10 6 m/s) is considerably faster than the nuclear motion ( 10 3 m/s). Because the nuclei move so slowly on the scale of velocities of relevance to the electrons it is justified to assume that at any moment the electrons will be in their ground state with respect to the instantaneous nuclear configuration. The assumption of instantaneous electronic equilibrium for every nuclear configuration implies that the electron wave function is a solution of the Schrödinger

11 1.3 Level 2: Density Functional Theory 11 equation for hamiltonian 1.1 with fixed nuclear positions: Ĥ = h2 r 2 i 2 m i e }{{} ˆT + 1 8πɛ 0 i j 1 4πɛ 0 i,j e 2 Z i R i r j } {{ } ˆV ext e 2 Z i Z j R i R j } {{ } ˆV NN + 1 8πɛ 0 i j e 2 r i r j } {{ } ˆV (1.3) This hamiltonian corresponds to the electronic motion in the external potential produced by the nuclei. Important to note here is that the nuclear repulsion ( ˆV NN ) contributes to the total energy by augmenting it with a constant amount. As a consequence, removing this term from the hamiltonian will not affect the corresponding wave function. The resulting hamiltonian can be written as: Ĥ e = h2 r 2 i 2 m i e }{{} ˆT + 1 8πɛ 0 i j 1 4πɛ 0 e 2 r i r j } {{ } ˆV i,j e 2 Z i R i r j } {{ } ˆV ext (1.4) Ĥ = Ĥe + ˆV NN (1.5) Equation 1.4 is called the electron hamiltonian, ˆT represents the kinetic energy of the electrons, ˆV the electronic repulsion and ˆV ext the electron-nucleus interaction. From now on, electrons and nuclei can be treated separately. This decoupling of the electronic and nuclear motion is known as the Born-Oppenheimer or adiabatic approximation (see also chapter 22 in [4]). 1.3 Level 2: Density Functional Theory The remaining quantum many-body problem is still too complex to solve for large systems. In order to deal with realistic materials, relevant in solid state physics, further approximations have to be made. A historically very important

12 12 CHAPTER 1 Introduction to ab initio calculations approximation method is the Hartree-Fock method (HF). It gives the best solution that satisfies the constraint that it can be written as a Slater determinant, and can be extended systematically to the exact solution. The Hartree-Fock method is, however, numerically quite complex to solve for solids. HF will not be treated here, but can be found in many condensed matter textbooks (e.g. [4]). Another breakthrough for computational physics was reached with the development of the density functional theory (DFT) by Hohenberg and Kohn [1] and Kohn and Sham [2] The theorems of Hohenberg and Kohn In 1964 Hohenberg and Kohn stated two theorems on which DFT has been built: First theorem: There is a one-to-one correspondence between the ground state density ρ( r) of a many-electron system (atom, molecule, solid) and the external potential V ext. An immediate consequence is that the ground state expectation value of any observable Ô is a unique functional of the exact ground state electron density: < Ψ Ô Ψ > = O[ρ] (1.6) Second theorem: For Ô being the hamiltonian Ĥ, the ground state total energy functional H[ρ] E Vext [ρ] is of the form E Vext [ρ] = < Ψ ˆT + ˆV Ψ > + < Ψ ˆV ext Ψ > (1.7) = F HK [ρ] + ρ( r)v ext ( r)d r (1.8) where the Hohenberg-Kohn density functional F HK [ρ] is universal for any many-electron system. E Vext [ρ] reaches its minimal value (equal to the ground state total energy) for the ground state density corresponding to V ext We will not prove the theorems here [1], but we will clarify their meaning. The one-to-one correspondence between the ground state density and the external potential has some important implications. It is obvious that, given the external potential of the system, it is possible to find a unique ground state density for the system. Solving the Schrödinger equation yields the ground state wave function out of which the ground state density can be calculated. Intuitively, the ground state density seems to contain less information than the

13 1.3 Level 2: Density Functional Theory 13 ground state wave function. If this were true, the inverse correspondence (from ground state density to external potential) would not hold. The first theorem of Hohenberg and Kohn, however, states that this correspondence holds as well. In other words: the density contains as much information as the wave function. As a consequence of the second theorem, and more precisely of the fact that the ground state density minimizes E Vext [ρ], the Rayleigh-Ritz variational method can be used to obtain the ground state density. Important to note is that E Vext [ρ] evaluated for the ground state density corresponding to V ext equals the ground state energy. Only this value of E Vext [ρ] has a physical meaning. And finally, in the second theorem the Hohenberg-Kohn-functional F HK contains no information on the nuclei and the nuclear positions. Consequently, the functional is the same for all many-electron systems (universal). Unfortunately F HK is not known and, at this level, the DFT remains a formally exact but useless theory. The second term in equation 1.8 is trivial The Kohn-Sham equations An important step towards applicability of DFT has been made by Kohn and Sham [2]. They proposed to rewrite F HK as follows: F HK [ρ] = T 0 [ρ] + V H [ρ] + (V x [ρ] + V c [ρ]) }{{} V xc[ρ] (1.9) where T 0 [ρ] is the kinetic energy functional for noninteracting electrons and V H [ρ] is the Hartree contribution, which describes the interaction with the field obtained by averaging over the positions of the remaining electrons. Although no on-site electron-electron interaction is taken into account, V H [ρ] is already a good approximation for the electron interaction. Assuming we know the exchange-correlation functional V xc [ρ], we can now write: E Vext [ρ] = T 0 [ρ] + V H [ρ] + V xc [ρ] + V ext [ρ] (1.10) Equation 1.10 can be interpreted as the energy functional of noninteracting particles submitted to two external potentials V ext [ρ] and V xc [ρ], with corre-

14 14 CHAPTER 1 Introduction to ab initio calculations sponding Kohn-Sham hamiltonian: Ĥ KS = ˆT 0 + ˆV H + ˆV xc + ˆV ext (1.11) = h2 2 i + e2 ρ( r ) 2m e 4πɛ 0 r r d r + ˆV xc + ˆV ext (1.12) with the exchange-correlation operator given by the functional derivative: ˆV xc = V xc[ρ] ρ The Kohn-Sham theorem can now be stated as follows: The exact ground state density ρ( r) of an N-electron system is (1.13) ρ( r) = N ψi ( r)ψ i ( r) (1.14) i=1 where the single-particle wave functions ψ i ( r) are the N lowest-energy solutions of the Kohn-Sham equation Ĥ KS ψ i = ɛ i ψ i (1.15) To obtain the ground state density of the many-body system the Schrödingerlike single-particle equation must be solved. The only unknown contributor to this problem is the exchange-correlation functional. Available approximations for this functional will be treated in the next section. Two additional remarks have to be made. First, one has to realize that the single-particle wave functions ψ i ( r) as well as the single-particle energies ɛ i are no electron wave functions or electron energies. They are merely mathematical functions without a physical meaning. Only the total ground state density calculated from these quasi-particles equals the true ground state density. And second, the Kohn-Sham hamiltonian depends on the electron density through the Hartree and the exchange-correlation term, while the electron density depends on the ψ i to be calculated. This means that we are actually dealing with a self-consistency problem: the solutions determine the original equation. An iterative procedure is thus needed to solve the problem. In the first iteration a pondered guess is made for the starting density. The latter allows for the construction of the initial Kohn-Sham hamiltonian. Solving the equation results in a new set of ψ i and a new electron density. The new density will differ

15 1.3 Level 2: Density Functional Theory 15 strongly from the previous one. With this density a new ĤKS can be produced etc. In the end succeeding densities will converge, as will the hamiltonians. A solution consistent with the hamiltonian has been reached. The Kohn-Sham equation proves to be a practical tool to solve many-body problems The exchange-correlation functional As mentioned in the previous section the Kohn-Sham equation can be solved if the exchange-correlation functional is known. Given the fact that an exact expression is not available, the introduction of an approximation is needed. Two such often used approximations are LDA (Local Density Approximation) and GGA (Generalized Gradient Approximation). The oldest approximation is the LDA [1, 2, 5 7] which defines the exchangecorrelation functional as: Vxc LDA [ρ] = ρ( r) ɛ xc (ρ( r)) d r (1.16) here ɛ xc (ρ( r)) stands for the exchange-correlation function (not functional) for the homogeneous electron gas with interacting electrons and is numerically known from Monte Carlo calculations. The underlying idea is very simple. At each point in space the exchange-correlation energy is approximated locally by the exchange-correlation energy of a homogeneous electron gas with the same electron density as present at that point. LDA is based on the local nature of exchange-correlation and the assumption that the density distribution does not vary too rapidly. In spite of its simplicity, LDA performs quite well even for more realistic systems. A more sophisticated approach is made with GGA [8 10]. While LDA only depends on the local density ρ( r) itself, GGA also incorporates the density gradient: Vxc GGA [ρ] = ρ( r) ɛ xc (ρ( r), ρ( r) ) d r (1.17) GGA usually performs better than LDA, but in the case of LDA a unique ɛ xc (ρ( r)) is available. For GGA however, because the density gradient can be implemented in various ways, several versions exist. Moreover, many versions of GGA contain free parameters which have to be fitted to experimental data. Strictly spoken, these GGA versions are no longer ab initio.

16 16 CHAPTER 1 Introduction to ab initio calculations Finally, a few more tendencies observed through comparison with experiments can be given. Lattice constants calculated using LDA are in general 1 3% smaller than the experimental ones. LDA also has a bad reputation concerning spin polarized calculations (next section). For many systems a wrong magnetic ground state is found. Lattice parameters calculated with GGA on the other hand match in most cases quite well with experiment. And for many magnetic systems GGA predicts the correct ground state, although magnetic moments are often overestimated. Altogether, GGA seems to be an improvement over LDA Spin Density Functional Theory All previous sections made use of the total density ρ( r). Also for magnetic systems this total density can be calculated and, as stated by the first Hohenberg- Kohn theorem, there exists a functional of ρ( r) that gives the magnetic moments of the system. Unfortunately, this functional is unknown. This problem is solved by introducing a (redundant) additional parameter m = ρ ρ into DFT. In this way one gains control over m, which is numerically more stable. In this theory, known as spin density functional theory (SDFT) [11], the total density ρ( r) can be written in function of the two so-called spin densities: ρ( r) = ρ ( r) + ρ ( r) (1.18) Assuming that the two theorems of Hohenberg and Kohn still hold, we can write: < Ψ Ô Ψ > = O[ρ, ρ ] (1.19) < Ψ ÊV ext Ψ > = E Vext [ρ, ρ ] (1.20) Or in words, every observable property is a functional of the spin densities. And the ground state spin densities can be obtained by minimizing the ground state total energy functional E Vext [ρ, ρ ]. Unfortunately, the first theorem is not proven to be true [12, 13], contrary to the original Hohenberg and Kohn theorem. But it turns out that, assuming the theorem is true, the corresponding theory describes reality almost as accurate as normal DFT. In spite of the lack of theoretical proof, magnetic systems are commonly treated with this Spin Density Functional Theory (SDFT).

17 1.3 Level 2: Density Functional Theory Spin-orbit coupling Thus far a nonrelativistic approach was used. When the electron velocity is of the same order as the speed of light, however, relativistic corrections are needed. Although dipolar interactions may contribute, the most important contribution comes from the interaction between spin and orbital moments, S and L. Hund s third rule gives a description of the coupling between the two moments. The first and second rule imply a maximization of the two moments separately. While, according to the third rule, L and S couple to form a total angular momentum J. Or more precisely: for less than half filled shells J = L S, while J = L + S for a more than half filled shell. In a nonrelativistic description, using the Schrödinger equation, spin-orbit coupling is not taken into account. But the interactions appear in the fully relativistic description of the system, given by the Dirac equation which may be found in several textbooks [14]. In general this description requires a relativistic four-compound wave function, but for sufficiently light atoms (which corresponds to the low-energy limit) the wave function can almost completely be described with two components. This allows for an approximation of the Dirac equation: Ĥ sr Eψ = Ĥsr ψ ξ( σ l)ψ (1.21) is the scalar relativistic hamiltonian which contains the nonrelativistic Schrödinger equation, the relativistic mass correction and the Darwin shift. The last term in equation 1.21 describes the spin-orbit coupling for a spherically symmetric potential with l the orbital angular momentum and ξ the spin-orbit parameter. The spin-orbit coupling is especially important for core electrons of heavy atoms. A relativistic calculation is therefore needed for these core electrons. For valence electrons the spin-orbit coupling is less crucial and it is often sufficient to start with a scalar relativistic calculation. The spin-orbit coupling is added afterwards in a second variational step [15] Beyond the LDA/GGA-level of Density Functional Theory Although exact DFT should be capable of obtaining ground state properties correctly, the LDA (and GGA) are not successful for all systems. LDA (and in many cases also GGA) fails in describing the electronic structure and the

18 18 CHAPTER 1 Introduction to ab initio calculations conduction properties of strongly correlated materials which usually contain transition metal or rare-earth metal ions with partially filled d (or f) shells. Such transition metal oxides and rare-earth metal compounds normally have well-localized d (f) electrons and a sizable energy gap between occupied and unoccupied subbands. When such a system is treated with LDA, which has an orbital-independent potential, a partially filled d (f) band with a metallic-type electronic structure and itinerant d (f) electrons is found. This behavior arises because in LDA the spin and orbital polarization are driven by the exchange interactions of the homogeneous electron gas instead of the screened on-site Coulomb interactions [16,17]. As a consequence, LDA fails in describing orbital polarization correctly. In figure 1.1 the density of states for d- and f-electrons obtained by LDA is compared to the one for LDA+U. Where LDA+U is an improvement on LDA that incorporates the missing on-site Coulomb interaction U, i.e. the Coulomb-energy needed to place two electrons at the same site. LDA+U induces a minor shift to the left of the peak at the Fermi energy (0 ev ) in the LDA spectrum, while it spreads out the f-peak at the Fermi energy over a broad energy range. A more elaborated description of LDA+U is given in the next section. Several attempts have been made to improve on LDA in order to take into account strong correlations. For example: the self-interaction correction [18, 19], the Hartree-Fock method [20] with configuration interaction, the GW approximation [21], etc. In the remainder of this section we will discuss the two approaches which are relevant for this work. LDA+U The main idea behind the first approach, the LDA+U method, is to treat the correlation of the localized d (f) electrons explicitly. This is done by adding the relevant electron-electron interaction terms to the total energy functional, meanwhilest avoiding double counting of the interaction that was already present in LDA. An expression which includes on-site Coulomb and exchange interaction was proposed by Anisimov et al. [16, 22]: E LDA+AMF = E LDA + 1 U(n mσ n 0 )(n m 2 σ n 0 ) m,m,σ m,m,m m,σ (U J)(n mσ n 0 )(n m σ n 0 ) (1.22)

19 1.3 Level 2: Density Functional Theory number of electrons LDA d-electrons LDA f-electrons LDA+U d-electrons LDA+U f-electrons Energy (ev) Energy (ev) Figure 1.1: The effect of LDA+U (AMF version) on the density of states of d- and f-electrons. For pure Cr and for an Er impurity in Fe the DOS of the d-electrons and f-electrons respectively are shown for an LDA and an LDA+U calculation. For the d-electrons a minor shift to the left of the peak at the Fermi energy (0 ev ) is visible between the LDA and LDA+U spectrum, while the f-peak at the Fermi energy is spread out over a broad energy range.

20 20 CHAPTER 1 Introduction to ab initio calculations where n mσ, with m = 2,..., +2 (d) or 3,..., +3 (f) and σ the spin, are occupation numbers of the localized orbitals, n 0 1 = 2(2l+1) mσ n mσ and AMF stands for around mean field. U and J are screened Coulomb and exchange parameters. The functional in equation 1.22 allows for a development of the spin and/or orbital polarization around and beyond the mean field LDA solution. Improvements on this formula have been made by Czyżyk et al. [23]. They replaced n 0 by n 0 1 σ = 2(2l+1) m n mσ. In this way the average is taken for each spin separately. To be consistent E LSDA is used instead of E LDA. As a consequence of this change additional degrees of freedom for spin-split solutions are introduced. As a second improvement they replaced U and J by matrices U mm and J mm. This makes the model more realistic and partially accounts for the multiplet splitting. The energy functional corresponding to these changes reads: E LSDA+AMF = E LSDA U mm (n mσ n 0 σ)(n m σ n 0 σ) m,m,σ m,m,m m,σ (U mm J mm )(n mσ n 0 σ)(n m σ n 0 σ) (1.23) To make this formula generally valid for different l values and symmetries a general expression for U mm and J mm is used in terms of Slater integrals and Gaunt s numbers [24] (which are related to the much more common Clebsch- Gordan coefficients). By taking the functional derivative of the total energy over the charge density of a particular orbital one obtains the effective potential for that orbital: V LSDA+AMF mσ ( r) = V LSDA mσ ( r) + m U mm (n m σ n 0 σ) + m,m m (U mm J mm )(n m σ n 0 σ) (1.24)

21 1.3 Level 2: Density Functional Theory 21 Czyżyk et al. [23] went one step further by rewriting the total energy functional 1.23 in a different form: E LSDA+U = E LSDA m,m,σ m,m,m m,σ U mm n mσ n m σ (U mm J mm )n mσ n m σ 1 2 UN(N 1) JN (N 1) JN (N 1) }{{} E at lim (1.25) They argued that, in order to correct the LDA in the description of localized d (f) electrons which are embedded in the reservoir of delocalized electrons, one should subtract the term E at lim obtained in the atomic limit from the total energy functional instead of a term corresponding to the mean field while adding an explicit electron-electron interaction term. The effective potential now becomes: V LSDA+U mσ ( r) = V LSDA+AMF mσ ( r) (U J)(n 0 σ 1 2 ) (1.26) This potential reveals clearly the effect of the LDA+U method. The first term, Vmσ LSDA+AMF ( r), allows for orbital and/or spin polarization around the LSDA solution. On top of this the last term will shift the center of the orbital level depending on its average occupation. For a completely empty state the level is moved upwards by an amount 1 2 (U J), while a completely filled state is shifted downward by an amount 1 2 (U J). One last comment is in order here: for surveyability site indices in U, J, n mσ and n 0 σ have been dropped as well as the summation over all sites. The index l, which distinguishes between different localized levels (d or f) was dropped as well. Orbital polarization Not only LDA+U but also the Orbital Polarization (OP) method takes Hund s rules into account. In the OP method this is done by adding an extra term to

22 22 CHAPTER 1 Introduction to ab initio calculations the effective potential [25, 26]: V OP = c OP < L z > l z (1.27) where c OP is the orbital polarization parameter, < L z > is the projection of the orbital momentum on the magnetization direction and l z is the z-component of the single electron orbital momentum parallel to M. As in the case of LDA+U this leads to corrections in the total energy. It is in principle possible to calculate the LDA+U parameters U and J and the OP parameter c OP from first principle. In many cases, however, no unique U and J exist and only an interval can be found. U and J are more often used as fit parameters to fit the calculated results to experimental results. In such a case one tries to approach experimental results as good as possible by adjusting the parameters manually until a good match is found. One has to realize that, by tuning these parameters, these methods are no longer truly ab initio methods. 1.4 Level 3: Solving the equations At this point we are left with one final task: solving the Kohn-Sham equation that resulted from DFT: ( h2 2 2m m + e2 ρ( r ) ) e 4πɛ 0 r r d r + V xc + V ext ψ m ( r) = ɛ m ψ m ( r) }{{} Ĥ KS (1.28) An important step towards the final solution will be to expand the singleparticle wave functions in a suitable basis set, say {φ b p} p=1,...,p : ψ m = P c m p φ b p (1.29) p=1 In principle this basis set is infinite (P = ). But in practice, of course, the basis set must be finite. Note however that the use of a limited basis set makes it impossible to describe ψ m exactly. It is therefore important to search for suitable limited basis sets, from which good approximations of ψ m can be constructed. More details on such basis sets will be given further on in this chapter.

23 1.4 Level 3: Solving the equations 23 By substituting 1.29 into equation 1.28 an eigenvalue problem appears: c.. < φ b i Ĥsp φb j > ɛ m < φ b i φb j >. m =.. (1.30) The P eigenfunctions and eigenvalues can be found by diagonalizing this matrix. Increasing P will increase the number of eigenfunctions (and eigenvalues) and the accuracy of the approximation. Such a matrix diagonalization is rather easy to implement on a computer. The many-body problem of equation 1.1 has thus been reduced to a solvable problem. As mentioned before, the choice of a good basis set will be very important. The accuracy of the approximation as well as the needed computation time will strongly depend on the basis set. The calculated ψ m will approach the real solutions better if a larger basis set is used. But this will increase the computation time as well because of the increased matrix size. Moreover, not only the number of basis functions, but also the shape of these functions has a major influence. One can imagine using basis functions that look a lot like the eventual solution. Hence, less basis functions are needed to reach the same level of accuracy. A smaller basis set also means a smaller matrix and consequently less computation time. However, this kind of basis set has as a major drawback that it is not generally applicable: every system needs its own optimized basis set. Such a basis is very efficient for a specific system, but the resulting wave functions adopt many properties from the basis, therefore it is called biased. In practice, good basis sets are simultaneously efficient and unbiased. We will treat some important examples in 1.4.1, and c m p The APW method A first, historically important, example of an efficient and unbiased basis is the Augmented Plane Wave basis (APW), proposed by Slater in 1937 [27]. This basis set is not of any practical use anymore, but is worth mentioning because it is the predecessor of the other basis sets to be discussed. At first sight, one might think of using a plane wave basis set, since, according to the Bloch theorem, eigenfunctions of a periodic hamiltonian can be expanded in a plane wave basis (see Appendix A). Moreover, we can

24 24 CHAPTER 1 Introduction to ab initio calculations label ψ m (equation 1.29) with the quantum numbers of the Bloch theorem: ψ m = ψ n k. However, too many plane wave basis functions are needed to describe the rapidly oscillating behavior of the eigenfunctions near the nucleus, which makes it a very time consuming and in practice unusable method. For this reason another approach is needed for the region around atomic nuclei. The APW method is based on the knowledge that the strongly varying, nearly spherical potential and wave functions near an atomic nucleus are very similar to those of an isolated atom. In the region between the atoms the potential is almost constant and hence the wave functions are better described by plane waves which are the solution of the Schrödinger equation for a constant potential. Based on this observation, space is divided in two regions where different basis expansions are used. Centered around the atomic nuclei (α) nonoverlapping muffin-tin (MT) spheres (S MT,α ) of radius R MT,α are constructed. The region in between the spheres is called the interstitial region (I). We can now define an APW basis function as follows: φ k K ( r, E) = 1 V e i( k+ K). r r I l,m Aα, k+ K lm u α l (r i, E) Y l m(ˆr i ) r S MT,α (1.31) with k a vector in the first Brillouin zone, K a reciprocal lattice vector and V the unit cell volume. The Ym(r l i ) are spherical harmonics with {l, m} an angular momentum index and r i = r r α where r α is the atomic position within the unit cell of atom α. A α, k+ K lm are expansion coefficients and u α l (r i, E) is a solution of the radial Schrödinger equation with spherical averaged crystal potential V (r) centered on the atom, at given energy E: [ ] d 2 l(l + 1) + dr2 r 2 + V (r) E ru α l (r, E) = 0 (1.32) Imposing continuity of u α l (r i, E) and the corresponding plane wave on the muffin-tin sphere determines the coefficients A α, k+ K lm. Unfortunately, the APW method has an important drawback: the energy dependence of u α l (r i, E). In order to describe an eigenfunction ψ m of the Kohn- Sham equation 1.28 properly, the corresponding eigenvalue ɛ m must be used for E. Since ɛ m is not known yet a guess must be made for the value of E. For this value the APW basis can be constructed and the Kohn-Sham equation

25 1.4 Level 3: Solving the equations 25 can be solved. The guessed E should be a root of this equation. If not, a new value for E must be tried until the chosen value turns out to be an eigenvalue of the equation. This procedure has to be repeated for every eigenvalue and is therefore very time consuming. A general solution to this problem consists in some kind of enhancement of the basis in the muffin-tin spheres in order to remove the energy dependence The LAPW method Regular LAPW In the APW method one had to construct the functions u α l (r i, E) by using the unknown eigenvalue ɛ n k of the searched eigenstate ψ n k for E. These basis functions lack variational freedom to deal with even small deviations of the band energy ɛ n k from the trial value for E. A solution to this problem consists of making the APW s energy independent within a certain energy region, as is done in the Linearized Augmented Plane Wave method (LAPW) [28 30]. Where Linearized points at the fact that the resulting secular equation 1.30 will be linear in E. An LAPW basis function has the same form as an APW basis function, but the part of the basis function in the muffin-tin region, the augmentation, has been adapted: φ k K ( r) = 1 V e i( k+ K). r [ l,m A α, k+ K lm ] u α l (r i, E 0 ) + B α, k+ K lm u α l (r i, E 0 ) Ym(ˆr l i ) r I r S MT,α (1.33) The APW augmentation has been replaced by a linear combination of the original function u α l and its energy derivative u α l uα l (ri,e) E, evaluated at a fixed linearization energy E 0. One can interpret the new term between square brackets as a first order Taylor expansion around a fixed energy E 0 : u α l (r i, E) = u α l (r i, E 0 ) + (E E 0 ) u α l (r i, E 0 ) + O(E E 0 ) 2 (1.34) If the energy E 0 differs slightly from the true band energy E, such a linear combination will reproduce the APW radial function at the band energy. This yields a basis set that is flexible enough to represent all eigenstates in a region around E 0.

26 26 CHAPTER 1 Introduction to ab initio calculations The remaining coefficients A α, k+ K lm and B α, k+ K lm can be determined by imposing continuity of the LAPW on the muffin-tin sphere both in value and slope. One cannot fix the coefficients through a Taylor expansion because this would re-introduce the energy dependence. At this stage E 0 is still the same for all values of l. We can go one step further by choosing a different E 0, say El α, for every l-value of atom α. In this case El α is normally chosen at the center of the corresponding band with l-like character. This gives the final definition for an LAPW: φ k K ( r) = 1 V e i( k+ K). r ( l,m A α, k+ K lm r I u α l (r i, E α l ) + Bα, k+ K lm u α l (r i, E α l ) ) Y l m(ˆr i ) r S MT,α (1.35) If the energy parameters El α are carefully chosen for each value of the angular momentum, a single diagonalization will yield an entire set of accurate energy bands for the corresponding k-point k. This is a major improvement in comparison with the APW method, where a diagonalization is needed for every energy band. The LAPW basis set as defined in formula 1.35, however, is not of any practical use yet. Because the basis set as well as the basis functions defined in formula 1.35 are infinitely large two additional parameters have to be introduced to limit these sizes. The first parameter, l max, controls the size of the LAPW augmentation which consists of an infinite sum over angular momenta l. This summation will be truncated at l max. While the second parameter, the plane wave cutoff K max, determines the size of the basis set. Only those basis functions with a K that satisfies the condition K K max are included in the basis set. As a consequence, l max and K max control the accuracy of the calculation (completeness and quality). Good choices for these parameters are therefore very important. Moreover, l max and K max are not completely independent: because of the imposed boundary conditions l max and K max must be tuned in such a way that the truncations still match at the boundary. This can be done by noting that, for a given l max, Y l max m has at most 2l max nodes along a large circle around the muffin-tin sphere, i.e. on a circle with a contour equal to 2πR MT. This corresponds to l max /(πr MT ) nodes per unit length. On the other hand K max

27 1.4 Level 3: Solving the equations 27 describes a plane wave with K max /π nodes per unit length. Equating these suggests a R MT K max = l max. A good value for l max can thus be chosen if K max is known. In practice, l max is fixed to a value of 10 and a well converged basis is obtained for R MT K max 7 9 for most systems. LAPW with Local Orbitals (LAPW+LO) An even more efficient method is the Local Orbital (LO) extension to the LAPW method [29, 31]. It has not been mentioned so far for which electron states the LAPW method is used. The electron states located closest to the nucleus (called core states) are strongly bound to the nucleus and behave almost as if they were free atom states. They do not participate in chemical bonding and are entirely contained inside the muffin-tin sphere. Therefore, these states can be treated as free atom states subjected to the potential of all the other states. States that cross the muffin-tin sphere and thus participate in chemical bonding are called valence states. Unlike the core state, these valence states are treated with LAPW. The LAPW method works very well for describing valence states as long as all states have a different l-value. But one can imagine a situation in which 2 valence states exist with the same l quantum number, say l, but different principle quantum number n. In such a case the state with the lowest n-value lies far below the Fermi level and is a core-like state that is not completely contained inside the muffin-tin sphere. It is called a semi-core state. Constructing a suitable LAPW basis set for this case is difficult, because it is not clear how to choose El α. This problem is solved in the LAPW+LO method where a new type of basis function, a Local Orbital, is added to the LAPW basis: φ lm α,lo( r) = 0 r S MT,α ( A α,lo lm uα l (r i, E1,l α ) + Bα,LO lm uα l (r i, E1,l α ) +C α,lo lm uα l (r i, E α 2,l ) ) Y l m(ˆr i ) r S MT,α (1.36) While each LAPW is connected to a vector k and contains a sum over atoms and l-characters, a Local Orbital is independent of k and K. It belongs to only one atom (α) and has a specific l-character. For a specific l-value 2l + 1 Local Orbitals are added (m = l, l + 1,..., l). The Local Orbitals are local in the

28 28 CHAPTER 1 Introduction to ab initio calculations sense that they are identically zero outside the muffin-tin spheres. Imposing the constraint that a Local Orbital is normalized and becomes 0 in value and slope on the muffin-tin sphere determines the three coefficients A α,lo lm, Bα,LO lm and C α,lo lm. The uα l (r i, E1,l α ) and uα l (r i, E1,l α ) are the same as in the regular LAPW method with as linearization energy E1,l α an energy value for the highest of the two valence states. Since the lowest valence state, or semi-core state, resembles a free atom state it will be sharply peaked at an energy E2,l α. A single radial function u α l (r i, E2,l α )) is therefore included in the Local Orbital, which is sufficient to describe this state. Adding Local Orbitals will increase the basis set size and consequently also the computation time. The number of Local Orbitals added is, however, always low compared to the number of basis functions in an LAPW basis set. Moreover, the slight increase in computation time is largely compensated by the gain in accuracy The APW+lo method Regular APW+lo method In general, the electron wave functions behave rather smooth in the interstitial region. Consequently, this part of the wave function can be described with a relatively small set of plane waves. The number of augmented plane waves needed to describe the wave function will, therefore, most often be determined by the representation of the wave function in the muffin-tin region. A more efficient augmentation will result in a smaller basis set size. The LAPW method undermines the efficiency of the APW augmentation by introducing the derivative u α l and imposing additional boundary conditions (continuity in slope). This leads to a larger basis set size for the LAPW basis compared to the more efficient APW basis. An alternative way for linearizing the APW method that does maintain the APW efficiency is available: the augmented plane wave + local orbital method (APW+lo) [32, 33]. The APW+lo basis set is a combination of two complementary basis sets. The principal basis set consists of the original APW basis functions, with the functions u α l evaluated at fixed linearization energies

29 1.4 Level 3: Solving the equations 29 E α l (for each l-value a suitable energy is chosen): φ k K ( r, E) = 1 V e i( k+ K). r r I l,m Aα, k+ K lm u α l (r i, E l ) Y l m(ˆr i ) r S MT,α (1.37) As mentioned for the APW method, such a basis set can only describe eigenfunctions that have eigenenergies in the immediate vicinity of the E α l. One therefore adds a new set of local orbitals (abbreviated by lo, to avoid confusion with the LO s used in LAPW+LO) to provide enough variational flexibility in the radial basis functions: φ lm α,lo( r) = 0 r S MT,α ( ) A α,lo lm uα l (r i, El α) + Bα,lo lm uα l (r i, El α) Ym(ˆr l i ) r S MT,α (1.38) The coefficients A α,lo lm and Bα,lo lm are determined by the requirement that the lo is zero on the muffin-tin sphere and normalized. By choosing the same linearization energies for the lo s as for the principal APW basis functions, the number of energy parameters will be limited to one E l per l-quantum number. This restriction is not strictly needed, but one can argue that the results of a linearized method should not depend on the exact choice of the linearization energies, as long as they are reasonable. For the APW+lo basis set a size comparable to the APW basis set size is required to obtain the same level of accuracy. An APW+lo basis set is thus smaller than a corresponding LAPW basis set. Moreover, as in the LAPW method, a single diagonalization is needed per k-point. Semi-core states in the APW+lo method As in the case of the LAPW method, semi-core states will be hard to describe with the regular APW+lo method. These states can be treated in the same

30 30 CHAPTER 1 Introduction to ab initio calculations way as before by adding an extra set of Local Orbitals (LO) to the basis set: 0 r S MT,α φ lm α,lo( r) = ( ) A α,lo lm uα l (r i, E1,l α ) + Cα,LO lm uα l (r i, E2,l α ) Ym(ˆr l i ) r S MT,α (1.39) and Cα,LO lm follow from the same requirements as before: zero value at the muffin-tin sphere and normalized. A α,lo lm One has to realize that the two kinds of local orbitals, lo and LO, are constructed for completely different reasons. While the former is constructed to improve the overall description of valence states over a wide energy range, the latter is added primarily in systems where high-laying semi-core states are present. Mixed LAPW/APW+lo method As mentioned before an APW+lo basis set is smaller than an LAPW basis set, but the gain from using an APW+lo basis set will decrease for each l added in the complementary basis set of local orbitals. Since adding one extra l to the basis set corresponds to adding 2l + 1 local orbitals, high l-values will have a significant influence on the basis set size. Furthermore, the LAPW method needs a larger basis set mainly because of a few states that are difficult to treat with LAPW. In general, LAPW has some difficulties with valence d and f states and with states in atoms with a small sphere size. For these reasons it can be interesting to use a mixed LAPW and APW+lo basis [34] within one calculation. In such an approach the states that are difficult for LAPW are treated with APW+lo and all the others are calculated using ordinary LAPW s. For efficiency reasons the use of a mixed basis set is common practice. To conclude this section, an overview of the three basis sets is given in Table 1.1. In the APW method and the APW+lo method the same minimal K max is required to obtain a well converged basis set, while a larger K max is needed for the LAPW method. Therefore, the APW and APW+lo method have equal basis set sizes. However, because the APW method is energy dependent, the equations have to be solved several times (N) to find a single eigenvalue. As a consequence, the APW method is very time consuming and not in use anymore. The APW+lo method, on the other hand, is about 10 times faster than the

31 1.5 Wien2k 31 Table 1.1: Comparison of the different basis sets. The parameter N represents the number of times an equation has to be solved in order to find a single eigenvalue. APW LAPW APW+lo energy dependent yes no no minimal K max a a a 0 basis set size (P) ± 130 ± 200 ± 130 # diagonalizations per k-point P 1 1 additional Local Orbitals (LO) no yes yes computation time w.r.t. APW+lo NP 10 1 LAPW method, due to its smaller basis set size. But as explained before, the APW+lo basis set size grows rapidly when adding additional Local Orbitals. In practice, one uses an APW+lo basis set or a mixed LAPW/APW+lo basis set for these reasons. 1.5 Wien2k In the previous sections, the theory behind ab initio calculations has been clarified. Of course, to perform an actual calculation a software package is needed in which this theory is implemented. All calculations performed for this thesis work were executed with the Wien2k-code, an Augmented Plane Wave + Local Orbitals Program for Calculating Crystal Properties. Figure 1.2 shows the iterative flowchart of Wien2k. As can be seen on the flowchart (figure 1.2) the code is divided in two parts: the first part (top row of figure 1.2) processes the input files while the second part (remaining part of figure 1.2) performs a self-consistent calculation. The input routine starts from a structure file containing information on the atomic configuration of the system: lattice parameters, atomic species, atomic positions, muffin-tin radii, etc. Based on this file the subroutines NN, SGROUP and SYMMETRY check for overlap between the different muffin-tin spheres and determine the symmetry of the unit cell. In the next step LSTART calculates the atomic densities for all atoms in the

32 32 CHAPTER 1 Introduction to ab initio calculations NN check for overlapping spheres SYMMETRY LSTART atomic calculations Hψ nl = E nl ψ nl atomic densities input files DSTART superposition of atomic densities SGROUP struct files input files ρ struct files KGEN k-mesh generation LAPW0 2 V C = 8πρ Poisson V XC (ρ) LDA ORB LDA+U, OP potentials V = V C + V XC V LAPW1 [ 2 + V ]ψ k = E k ψ k V MT LCORE atomic calculation Hψ nl = E nl ψ nl E k ψk ρcore Ecore LAPWSO add spin-orbit interaction LAPW2 ρ val = Σψ k ψ k E k < E f ρ val ρ old MIXER ρnew = ρ old (ρ val + ρcore) LAPWDM calculated density matrix ρnew STOP yes converged? no Figure 1.2: Program flow in WIEN2k

33 1.5 Wien2k 33 unit cell which KGEN uses in combination with the other input files to determine a suitable k-mesh (see Appendix B). And in the final initialization step, DSTART, a starting electron density ρ is constructed based on a superposition of the atomic densities. During this initialization all the necessary parameters are fixed as well: the exchange-correlation approximation (LDA, GGA, LSDA), R MT K max, l max and the energy parameter that separates the core states from the valence states. For an optimal use of computation time a good choice of R MT K max and the k-mesh is needed: the calculation should be accurate, but not more accurate than needed and fast, but not faster than the required accuracy permits. Once the starting density is generated the self-consistent calculation can start. This process is divided into several subroutines which are repeated over and over until convergence is reached and the calculation is self-consistent. LAPW0 starts with calculating the Coulomb and the exchange-correlation potential. Next LAPW1 solves the secular equation for all the k-values in the k- mesh and finds by diagonalization of the Kohn-Sham equation the eigenvalues and eigenfunctions of the valence states. The following subroutine, LAPW2, determines the Fermi-energy which separates filled from unfilled states. Once this energy is known the eigenfunctions resulting from LAPW1 can be used to construct a valence density: ρ val ( r) = φ k,i ( r)φ k,i ( r) (1.40) ɛ k,i <E F The states and energies of the core electrons are calculated separately in a regular atomic calculation in the LCORE subroutine, which results in a total core density ρ core. Both densities, ρ core and ρ val, combined give the total density, ρ tot = ρ core + ρ val. Since this density often differs a lot from the old density, ρ old, they are mixed by MIXER to avoid large fluctuations between iterations that would lead to divergence: ρ new = ρ old (ρ core + ρ val ). Once the end of the cycle is reached Wien2k checks for convergence between the old and the new density. If they differ from each other a new iteration is started with the new density as input density. This procedure is repeated until the old and new density are consistent. When this is the case, the self-consistency cycle ends and the self-consistent solution of the equation equals ρ new. This program flow corresponds to the trajectory in full lines shown in figure 1.2. The subroutines reached by the dashed lines are optional subroutines.