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Electronic Correlation Effects in Transition Metal Systems: from Bulk Crystals to Nanostructures

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Electronic Correlation Effects in Transition Metal Systems: from Bulk Crystals to Nanostructures Electronic Correlation Effects in Transition Metal Systems: From Bulk Crystals to Nanostructures Dissertation zur Erlangung des Doktorgrades des Fachbereichs Physik der Universität Hamburg vorgelegt von Michael Karolak aus Kutno Hamburg 2013 Gutachter der Dissertation Prof. Dr. Alexander Lichtenstein Dr. Frank Lechermann Gutachter der Disputation Prof. Dr. Alexander Lichtenstein Prof. Dr. Giorgio Sangiovanni Datum der Disputation 22.11.2013 Vorsitzender des Prüfungsausschusses Dr. habil. Michael Martins Vorsitzender des Promotionsausschusses Prof. Dr. Peter Hauschildt Dekan des Fachbereichs Physik Prof. Dr. Heinrich Graener iii I suppose it is tempting, if the only tool you have is a hammer, to treat everything as if it were a nail. Abraham H. Maslow [279] iv v Abstract Condensed matter systems containing transition metals, be it single molecules or crys- talline solids, exhibit many complex physical phenomena that warrant the constant in- terest of the scientific community. Their complex behavior emerges as a feature of the strongly correlated electrons occupying the partially filled d or f shells. From the tech- nological point of view such systems are of interest, because they often show a complex phase diagram and, furthermore, their properties can be controlled by external handles, like temperature, pressure etc. Such systems pose a challenge to theoretical physics, since the multitude of phases exhibited by them indicates many competing degrees of freedom at work. The present thesis is concerned with a successful general methodology that can be applied for the realistic description and in some cases even prediction of the properties of systems containing strongly correlated electrons. The framework that we refer to as DFT++ is a general scheme to combine realistic density functional theory calculations with model Hamiltonian approaches to produce a unified ab initio description. In the first part of this work we are concerned with the development of the general DFT++ methodology within the formalism of the projector augmented-wave method. We outline the general structure of the approach and show the specifics of an implementation in- volving only an Anderson impurity model and also self-consistent dynamical mean-field theory. Certain intricacies occurring at the interface between density functional and model Hamiltonian methods are also discussed in detail, with special focus on the so-called dou- ble counting of interaction terms. Subsequently, we apply the DFT++ method to different systems ranging from bulk crys- tals to single atoms adsorbed on a surface. We begin by examining the spin state transi- tions in LaCoO3, which have been under debate for over 50 years. We find corroborating evidence for one of the two major scenarios discussed in this context, namely the low-spin to high-spin transition, not involving the so-called intermediate-spin state. As a second application we study the impact of dynamical correlations on the electronic structure of different transition metal-benzene sandwich molecules trapped between the tips of copper nanowires as examples of molecular magnetic devices. We find that, de- pending on the chemical valence of the transition metal center and in particular the ge- ometry of the molecule the unusual orbital Kondo effect can be observed and, to some extend, controlled. Finally, we consider the series of isolated manganese, iron, cobalt and nickel atoms ad- sorbed on a silver surface and explain their excitation spectra. We report on valence electron photoemission experiments revealing the complex evolution of the electronic spectra through this series: we find a monotonous decrease in the splitting of higher en- ergy features and a non-monotonous variation of low energy spectral weight. By means of calculations using the DFT++ formalism we explain the photoemission results and show that both observations can be traced back to Hund’s exchange. vi vii Zusammenfassung Festkörpersysteme, die Übergangsmetalle entahlten, seien es einzelne Moleküle oder kristalline Festkörper, weisen viele komplexe physikalische Phänomene auf, die das kon- stante Interesse an Ihnen rechtfertigen. Das komplexe Verhalten dieser Systeme kann auf die stark korrelierten Elektronen in den teilweise gefüllten d oder f-Schalen zurückge- führt werden. Solche Systeme stellen eine Herausforderung für die Theorie dar, da die Vielzahl der möglichen Phasen auf viele konkurrierende Freiheitsgrade hinweist. Die vorliegende Arbeit beschäftigt sich mit einer allgemeinen Methode, die für die realistis- che Beschreibung und Vorhersage der Eigenschaften von Systemen mit stark korrelierten Elektronen angewendet werden kann. Die Methodik, die wir als DFT++ bezeichnen, stellt eine allgemeine Vorschrift zur kombination realistischer Dichtefunktional-Rechnungen mit Modellansätzen dar, um eine ab initio Beschreibung zu realisieren. Im ersten Teil dieser Arbeit sind wir mit der Entwicklung der allgemeinen DFT++ Methodik, innerhalb des Formalismus der Projektor Augmentierten Ebenen Wellen- Methode, befasst. Wir skizzieren die Struktur des Ansatzes und zeigen die Besonder- heiten einer Implementierung, die ein Störstellenmodell oder auch eine selbstkonsistente dynamische Molekularfeldtheorie beinhaltet. Bestimmte Feinheiten an der Schnittstelle zwischen der Dichtefunktionaltheorie und dem Modellansatz werden ebenfalls aus- führlich diskutiert. Anschließend wenden wir obige Methode auf verschiedene Systeme an, die sowohl kristalline Festkörper, als auch einzelne Atome auf einer Oberfläche beinhalten. Wir beginnen mit der Untersuchung der Spin-Übergänge in LaCoO3. Wir finden Hinweise für eines der beiden Szenarien, die in diesem Zusammenhang diskutiert werden, nämlich den “low-spin” zu “high-spin”- Übergang, der nicht den sogenannten “intermediate-spin”- Zustand beinhaltet. Als zweite Anwendung untersuchen wir die Auswirkungen der dynamischen Korrela- tionen auf die elektronische Struktur von verschiedenen Übergangsmetall-Benzol Sand- wich Molekülen zwischen den Spitzen eines Kupfernanodrahtes als Beispiele für eine molekulare magnetische Nanostruktur. Wir finden, dass, beeinflusst durch die chemische Valenz des Übergangsmetallzentrums und insbesondere die Geometrie des Moleküls, der ungewöhnliche orbitale Kondoeffekt beobachtet und kontrolliert werden kann. Schließlich betrachten wir isolierte Atome von Mangan, Eisen, Kobalt und Nickel auf einer Silberoberfläche und erklären ihre Anregungsspektren. Wir berichten über Photoe- missionsexperimente, die die komplexe Entwicklung der elektronischen Spektren zeigen: wir finden eine monotone Abnahme der Aufspaltung von Charakteristika bei höheren Energien und eine nicht-monotone Variation des spektralen Gewichts bei niedrigen An- regungsenergien. Durch Berechnungen mit Hilfe des DFT++ Formalismus erklären wir die Messungen und zeigen, dass das beobachtete Verhalten zum Hundschen Austausch zurückverfolgt werden kann. viii Contents Introduction 1 1 Theory and Methodology 6 1.1 Ab initio theory of condensed matter systems . 6 1.2 Density Functional Theory . 8 1.2.1 Kohn-Sham Formalism . 12 1.2.2 Exchange and Correlation. 16 1.3 Implementations of DFT . 21 1.3.1 Gaussian Basis Sets . 22 1.3.2 Plane-Wave basis set . 24 1.3.3 The Projector Augmented-Wave Method . 25 1.4 Model Hamiltonians and Dynamical mean-field theory. 30 1.4.1 Green functions and Coherent State Path Integrals . 30 1.4.2 Hubbard Model . 38 1.4.3 Anderson Impurity Model . 41 1.4.4 Dynamical Mean-Field Theory . 45 1.5 DFT++ Framework. 50 1.5.1 General idea of DFT++ . 51 1.5.2 The Coulomb interaction tensor . 53 1.5.3 DFT+U . 59 1.5.4 DFT+DMFT . 62 1.5.5 DFT+Σ for equilibrium and coherent transport. 67 1.6 Impurity Solvers . 70 1.6.1 Hirsch-Fye Quantum Monte Carlo Method . 71 1.6.2 Continuous-Time Quantum Monte Carlo . 75 1.6.3 Non-Crossing / One-Crossing Approximation . 79 1.6.4 Exact Diagonalization and Lanczos method . 83 1.7 Analytic Continuation . 87 x Contents 2 DFT++ in the PAW framework 92 2.1 Wannier Functions . 93 2.2 Implementation within PAW . 95 2.2.1 Wannier Hamiltonian and tight-binding quantities . 101 2.2.2 Comparison between projection formalism and formalism relying explicitly on the Wannier Hamiltonian . 101 2.2.3 Charge Self-consistency . 104 2.2.4 Transformations of the basis . 105 2.2.5 DFT++ for inequivalent atoms and layered systems . 107 2.2.6 DFT++ for vacuum states . 109 2.3 Benchmark: Cubic Perovskite SrVO3 ..................................110 2.3.1 Tight-binding discussion of the three band case . 111 2.3.2 Comparison of different PLO methods . 112 2.3.3 DFT+DMFT calculations . 114 2.4 Charge Self-consistency: The Volume of paramagnetic bcc Iron. .117 2.5 DFT+Σ for vacuum states: Co on Cu(111) . 120 2.6 Summary and Conclusions . 122 3 Double Counting in DFT+DMFT 124 3.1 NiO — a charge transfer insulator . 125 3.2 Methodology . 127 3.3 Double counting approaches . 129 3.4 Results . 133 3.5 Summary and Conclusions . 140 4 Spin State Transition in LaCoO3 142 4.1 Spin state Transitions in the ionic Picture . 146 4.2 DFT and Projections . 149 4.3 DFT+DMFT. .153 4.4 Summary and Conclusions . 159 5 Correlation Effects in Transition Metal Benzene Molecules 160 5.1 The Kondo effect in metals and nanoscopic devices . 161 5.2 Transition Metal Benzene Complexes . 167 5.3 Theoretical Setup . 170 5.4 General Observations . 173 5.5 Electronic structure on the DFT level. 176 5.6 Discussion of LDA Hybridization Functions . 180 5.7 DFT+Σ results. .182 5.7.1 Orbital Kondo Effect in CoBz2 and VBz2 . 182 5.7.2 ScBz2, TiBz2 and NiBz2 .......................................188 Contents xi 5.8 Spin and magnetism in DFT and DFT++ . 190 5.9 Summary and Conclusions . 193 6 Excitation spectra of transition metal atoms on the Ag (100) surface 195 6.1 Experiment . 196 6.2 Theoretical setup . ..
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