Zugl.: Augsburg, Univ., Diss., 2009

In der vorliegenden Arbeit werden die strukturellen und magnetischen Eigenschaften verschiedener 3d-Übergangsmetalloxidketten (TMO-Ketten) auf Ir(001) und Pt(001) untersucht. Diese weisen eine (3 × 1) Struktur mit periodisch angeordneten Ketten auf, die nur über die Sauerstoffbindung an das Substrat gekoppelt sind. Während die Struktur durch experimentelle und theoretische Untersuchungen bestätigt ist, liegen für die magnetischen Eigenschaften ausschließlich Rechnungen vor. Zur Überprüfung dieser theoretischen Vorhersagen wird die Methode der spinpolarisierten Rastertunnelmikroskopie (SP-STM) verwendet, die die Abbildung der magnetischen Ordnung mit atomarer Auflösung erlaubt. Die Untersuchungen beginnen mit der Vorstellung der Ir(001) Oberfläche, die eine (5 × 1) Rekonstruktion aufweist. Eine Aufhebung dieser Rekonstruktion erreicht man durch das Heizen des Ir-Substrats in Sauerstoffatmosphäre unter Bildung einer (2 × 1) Sauerstoffrekonstruktion. Die Qualität der Oberfläche hängt dabei von der Wachstumstemperatur T und dem verwendeten Sauerstoffdruck pOx ab. Die bei T = 550°C und pOx = 1 × 10^−8 mbar hergestellte Sauerstoffrektonstruktion dient als Ausgangspunkt für die folgenden Präparationen von CoO2, FeO2 und MnO2-Ketten. Dazu wird jeweils eine drittel Monolage (ML) des Übergangsmetalls auf die Oberfläche des Substrates gedampft und die Probe unter Sauerstoffatmosphäre ein weiteres Mal geheizt. Auf diese Weise kann die (3 × 1) Struktur der bekannten Ketten bestätigt und die Gruppe der TMO-Ketten um die CrO2-Ketten erweitert werden. In der einschlägigen Fachliteratur wurden Vorhersagen bezüglich der magnetischen Struktur der TMO-Ketten publiziert, wonach entlang und zwischen CoO2-Ketten eine ferromagnetische (FM) und für FeO2 und MnO2-Ketten eine antiferromagnetische (AFM-) Kopplung vorliegt.Während die Überprüfung dieser Vorhersagen mit SP-STM für CoO2 und CrO2-Ketten keine Hinweise auf magnetische Strukturen liefert, liegen bei FeO2 und MnO2-Ketten unterschiedliche magnetische Phasen vor. In der Tat kann mit den experimentell gefundenen Einheitszellen die AFM-Kopplung entlang beider Ketten bestätigt werden. Im Gegensatz widersprechen die Kopplungen zwischen den Ketten den Berechnungen. Bei FeO2-Ketten liegt eine stabile FM Ordnung vor, die zu einer magnetischen (3 × 2) Einheitszelle mit einer leichten Magnetisierung in Richtung der Oberflächennormalen führt (out-of-plane). Die MnO2-Ketten weichen ebenfalls von der berechneten magnetischen kollinearen Ordnung zwischen benachbarten Ketten ab und zeigen eine chirale Struktur. Durch die Rotation der Mn-Spins um 120° in der Probenebenen (in-plane) entsteht eine magnetische (9 × 2) Einheitszelle, deren Periode durch neue DFT-Rechnungen bestätigt wird. Nach diesen Berechnungen handelt es sich um eine Spinspirale, die durch die Dzyaloshinskii-Moriya (DM-) Wechselwirkung bei einem Energiegewinn von 0,3 meV pro Mn-Atom gegenüber den kollinearen FM Zustand stabilisiert wird. Diese wird ähnlich wie bei bereits publizierten Clustern und Adatomen auf Pt(111) durch die Rudermann-Kittel-Kasuya-Yosida (RKKY-) Wechselwirkung vermittelt und erklärt den experimentell gefundenen einheitlichen Drehsinn der Spiralen. Die RKKY-Wechselwirkung zeigt eine starke Abhängigkeit von der Fermi-Oberfläche des Substrats. Im folgenden Kapitel werden deshalb mit TMO-Ketten auf Pt(001) die strukturellen und magnetischen Eigenschaften auf einem weiteren Substrat analysiert, wobei zum Zeitpunkt der Arbeit nur die Existenz der CoO2-Ketten aus der Literatur bekannt war. Vergleichbar mit Ir(001) besitzt auch Pt(001) eine rekonstruierte Oberfläche, die sich aber stabil gegenüber Oxidation zeigt. Dadurch muss die drittel ML des Übergangsmetalls direkt auf die Rekonstruktion aufgedampft werden. Das Wachstum des Übergangsmetalls ist dabei von der Temperatur des Substrats abhängig und beeinflusst das Ergebnis der nachfolgenden Oxidation. Diese erfolgt analog zum Wachstum der Ketten auf Ir(001) durch das Heizen der Probe in Sauerstoffatmosphäre und resultiert nur für das Aufdampfen des Übergangsmetalls auf kalte Pt(001) Oberflächen in Ketten mit der Periode von 3aPt. Auf diese Weise kann nicht nur die (3 × 1) Struktur der CoO2-Ketten bestätigt werden, sondern auch durch atomare Auflösung die Gruppe der TMO-Ketten um MnO2-Ketten auf Pt(001) erweitert werden. Im Gegensatz dazu sind die nicht magnetischen Messungen im Fall von Fe nicht eindeutig. Zwar liegen auch hier Ketten im Abstand des dreifachen Pt Gittervektors vor, trotzdem ist die (3 × 1) Struktur nicht nachweisbar. Dies liegt an einer Korrugation mit einer Periode von 2aPt entlang der Ketten, was ein Hinweis auf eine Peierls Instabilität sein kann. Entsprechend dem Vorgehen für Ir(001) werden für die TMO-Ketten auf Pt(001) SP-STM Messungen durchgeführt und die Vorhersage einer AFM-Kopplung für CoO2-Ketten überprüft. Auch hier können, wie im Fall von CoO2-Ketten und im Widerspruch zur Vorhersage, für beide Polarisationsrichtungen der Spitze keine magnetischen Strukturen gefunden werden. Darüber hinaus verhalten sich die MnO2-Ketten auf Pt(001) mit ihrer chiralen magnetischen Struktur ähnlich zu denen auf Ir(001). Dies bestätigt die Annahme einer indirekten DM-Wechselwirkung, wobei durch die 72° Rotation der Mn-Spins eine längere Periode der zykloidalen Spinspirale festgestellt wird. Die Erklärung dafür liegt in der Abhängigkeit der RKKY-Wechselwirkung vom Fermi-Wellenvektor des Substrats, während sich die DM-Wechselwirkung beim Übergang von Ir zu Pt nur wenig ändert. / In this work the structural and magnetic properties of various 3d transition-metal oxide (TMO) chains on Ir(001) and Pt(001) are investigated. These exhibit a (3 × 1) structure with periodically arranged chains that are only coupled to the substrate by oxygen bonds. While the structure is confirmed by experiments and theory, the magnetic coupling is merely available by theory. To verify these theoretical predictions, the method of spin-polarized scanning tunneling microscopy (SP-STM) is used, which enables the detection of magnetic nanostructures by the high lateral resolution. First the Ir(001) surface is introduced, which exhibits a (5×1) reconstruction. Removal of this reconstruction is achieved by heating the Ir substrate in oxygen atmosphere to form a (2 × 1) oxygen reconstruction. The observed surface quality depends on both, the temperature T and the oxygen pressure pOx during growth. The best result is achieved with T = 550°C and pOx = 1 × 10^−8 mbar and thus serves as the starting point for the following preparations of CoO2, FeO2 and MnO2 chains. In each case one third of the monolayer (ML) of the transition-metal is evaporated onto the substrate and the sample is heated once more under an oxygen atmosphere. After this procedure not only the (3 × 1) structure of the known chain system is confirmed, but also the group of TMO chains could be extended by the CrO2 chains. In the literature, predictions have been published regarding the magnetic structure of the TMO chains. According to these, the coupling is ferromagnetic (FM) along and between CoO2 chains and antiferromagnetic (AFM) for FeO2 and MnO2 chains. While SP—STM results do not suggest a magnetic structure for CoO2 and CrO2, the measurements reveal magnetic structures for FeO2 and MnO2 chains. Comparing the calculations with the experimentally observed magnetic order, the AFM coupling along both TMO—chains can be confirmed. Contradictory to the theoretical predictions, the interchain coupling differs. For FeO2 chains we find stable FM coupling which lead to a magnetic (3 × 2) unit cell with an easy magnetization direction parallel to the surface normal (out-of-plane). The MnO2 also deviate from the calculated collinear order between adjacent chains and show a chiral system. The in-plane rotation of the Mn spins by 120° form a magnetic (9 × 2) unit cell whose period is confirmed by new DFT calculations. According to these calculations, the spin spiral is stabilized by the Dzyaloshinskii—Moriya interaction (DMI) with an energy gain of 0,3 meV per Mn atom with respect to the collinear FM state. Similar to already published systems consisting of magnetic clusters and adatoms on a Pt(111) surface, the DMI is mediated by the Rudermann—Kittel—Kasuya—Yoshida (RKKY) interaction and explains the experimentally found uniform direction of rotation of the spirals. The RKKY interaction depends on the Fermi surface of the substrate. Based on this fact, the growth of TMO chains is investigated in the following chapter, where only the existence of CoO2 chains is known from literature at the time of this work. Similar to Ir(001), Pt(001) also has a reconstructed surface that is now stable to oxidation. As a consequence, the transition-metal is evaporated directly on the reconstructed surface. Both the growth of the transition metal on the Pt(001) and the topography after the following growth procedure of TMO chains depend on the substrate temperature during evaporation. This procedure follows the TMO growth on Ir(001) by heating the sample in oxygen atmosphere and only results in a stripe pattern with a period of 3aPt for the evaporation on cold Pt(001). In this way not only the (3 × 1) structure of the CoO2 can be confirmed, but also the group of TMO chains is extended by MnO2 chains through atomic resolution scans. In contrast, the non-magnetic measurements in the case of Fe are not obvious. Although there are chains with the expected period of 3aPt, the (3 × 1) structure is not fully resolved along the chains due to a corrugation with a period of 2aPt along the chains, which could be an indication of a Peierls instability. Following the procedure for Ir(001) SP-STM investigations are performed to verify the prediction of an AFM coupling for CoO2 chains. Indeed, no magnetic structures of CoO2 chains are observed for both directions of tip polarization identical to the same chains on Ir(001). In addition, the MnO2 chains with their chiral magnetic structure on Pt(001) behave similarly to those on Ir(001). This confirms the assumption of an indirect DM interacation, where the 72° rotation of the Mn spins indicates a longer period of the cycloidal spin spiral. The explanation for this can be found in the dependence of the RKKY interaction on the Fermi wave vector of the substrate, while the DM interaction changes only slightly due to the transition from Ir to Pt.

Die vorliegende Arbeit untersucht mit Rastertunnelmikroskopie (RTM) und -spektroskopie (RTS) die Korrelation von strukturellen, elektronischen und magnetischen Eigenschaften auf metallischen Oberflächen. Zuerst wird der spin-aufgespaltene Oberflächenzustand des Ni(111) analysiert. Anschließend geht der Fokus über auf dünne Eisenfilme, die auf Rh(001) gewachsen wurden. Zuletzt wird die CePt$_5$/Pt(111)-Oberflächenlegierung untersucht. Nickel ist ein bekannter Ferromagnet und die (111)-Oberfläche war in der Vergangenheit schon mehrfach das Objekt theoretischer und experimenteller Studien. Trotz intensiver Bemühungen wurden inkonsistente Ergebnisse veröffentlicht und ein klares, konsistentes Bild ist noch nicht vorhanden. Aus diesem Grund wird die Ni(111)-Oberfläche mittels RTM und RTS erforscht, die den Zugang sowohl zu besetzten als auch unbesetzten Zuständen ermöglicht. Mit der Methode der Quasiteilcheninterferenz wird eine detailierte Beschreibung der Banddispersion erhalten. Die Austauschaufspaltung zwischen Minoritäts- und Majoritätsoberflächenzustands wird zu ∆E$_{ex}$ = (100 ± 8) meV ermittelt. Der Ansatzpunkt des Majoritätsbandes liegt bei E − E$_F$ = −(160 ± 8)meV und die effektive Masse beträgt m^* = +(0,14 ± 0,04)me. Des Weiteren liegt der Ansatzpunkt der Oberflächenresonanz der Majoritätladungsträger energetisch bei E−E$_F$ = −(235±5)meV mit einer effektiven Masse von m^* = +(0,36±0,05)m$_e$. Um unmissverständlich den dominierenden Spin-Kanal in der RTS zu identifizieren, wurden hexagonale Quantentröge durch reaktives Ionenätzen hergestellt und mit der Hilfe eines eindimensionalen Quantentrogmodells interpretiert. Die sechs Kanten eines Hexagons erscheinen unterschiedlich. Atomar aufgelöste Messungen zeigen, dass gegenüberliegende Kanten nicht nur eine unterschiedliche Struktur haben sondern auch unterschiedliche spektroskopische Eigenschaften, die durch einen alternierend auftauchenden oder abwesenden spektroskopischen Peak charakterisiert sind. Magnetische Messungen ergeben allerdings keine endgültigen Ergebnisse bezüglich des Ursprungs des Beobachtungen. Das zweite experimentelle Kapitel dreht sich um dünne Eisenfilme, die auf eine saubere Rh(001)-Oberfläche aufgebracht und diese dann mit RTM, RTS und spin-polarisierter (SP- )RTM untersucht werden. Eine nahezu defektfreie Rh(001)-Oberfläche ist notwendig, um ein Wachstum der Eisenfilme mit wenigen Defekten zu erhalten. Dies ist relevant, um das magnetische Signal korrekt interpretieren zu können und den möglichen Einfluss von Adsorbaten auszuschließen. Die erste atomare Lage Fe ordnet sich antiferromagnetisch in einer c(2 × 2)-Struktur an mit der leichten Magnetisierungsachse senkrecht zur Probenoberfläche. Die zweite und dritte Lage verhält sich ferromagnetisch mit immer kleiner werdenden Domänen für steigende Bedeckung. Ab 3,5 atomaren Lagen kommt es vermutlich zu einer Änderung der leichten Magnetisierungsrichtung von vertikal zu horizontal zur Probenebene. Dies wird durch kleiner werdende Domänengrößen und den gleichzeitig breiter werdenden Domänenwänden signalisiert. Temperaturabhängige spin-polarisierter RTM erlaubt es die Curietemperatur der zweiten Lage auf 80 K zu schätzen. Zusätzlich wurde bei dieser Bedeckung eine periodische Modulation der lokalen Zustandsdichte gemessen, die mit steigender Periodizität auch auf der dritten und vierten Lage erscheint. Temperatur- und spannungsabhängige Messungen unterstützen eine Interpretation der Daten auf der Grundlage einer Ladungsdichtewelle. Ich zeige, dass die beiden für gewöhnlich konkurrierende Ordnungen (Ladungs- und magnetische Ordnung) koexistieren und sich gegenseitig beeinflussen, was theoretische Rechnungen, die in Zusammenarbeit mit F. P. Toldin und F. Assaad durchgeführt wurden, bestätigen können. Im letzten Kapitel wurde die Oberflächenlegierung CePt$_5$/Pt(111) analysiert. Diese System bildet laut einer kürzlich erschienenen Veröffentlichung ein schweres Fermionengitter. Von der sauberen Pt(111)-Oberfläche ausgehend wurde die Oberflächenlegierung CePt$_5$/Pt(111) hergestellt. Die Dicke der Legierung (t in u.c.) lässt sich durch die aufgedampfte Menge an Cer variieren und die erzeugte Oberfläche wurde mit RTM und RTS für verschiedene Dicken unter- sucht. RTM-Bilder und LEED (engl.: low energy electron diffraction)-Daten zeigen konsistente Ergebnisse, die in Zusammenarbeit mit C. Praetorius analysiert wurden. Für Bedeckungen unter einer atomaren Lage Cer konnte keine geordnete Struktur mit dem RTM beobachtet werden. Für 2 u.c. wurde eine (2 × 2)-Rekonstruktion an der Oberfläche gemessen und für 3 u.c. CePt$_5$ wurde eine (3√3×3√3)R30◦-Rekonstruktion beobachtet. Der Übergang von 3 u.c. CePt5 zu 5 u.c. CePt$_5$ wurde untersucht. Mit Hilfe eines Strukturmodells schließe ich, dass es weder zu einer Rotation des atomaren Gitters noch zu einer Rotation des Übergitters kommt. Ab einer Bedeckung von 6 u.c. CePt5 erscheint eine weitere Komponente der CePt$_5$-Oberflächenlegierung, die keine Rekonstruktion mehr besitzt. Das atomare Gitter verläuft wieder entlang der kris- tallographischen Richtungen des Pt(111)-Kristalls und ist somit nicht mehr um 30^° gedreht. Für alle Bedeckungen wurden Spektroskopiekurven aufgenommen, die keinen Hinweis auf ein kohärentes schweres Fermionensystem geben. Eine Erklärung hierfür kommt aus einer LEED-IV Studie, die besagt, dass jede gemessene Oberfläche mit einer Pt(111)-Schicht terminiert ist. Das RTM ist sensitiv für die oberste Schicht und somit wäre der Effekt eines kohärenten schweren Fermionensystems nicht unbedingt messbar. / The present work investigates the correlation of structural, electronic, and magnetic properties at metal surfaces by scanning tunneling microscopy (STM) and spectroscopy (STS). First I analyze the spin-split surface state of Ni(111). Subsequently the focus goes on iron thin films grown on Rh(001). Finally the heavy-fermion candidate CePt5/Pt(111) is investigated. Nickel is a well-known ferromagnet and its (111) surface has been the subject of several theoretical and experimental studies in the past. Despite intensive efforts, inconsistent results have been reported and a clear consistent picture is still missing. For this reason, the Ni(111) surface has been probed by STM and STS, which give access to both occupied and unoccupied states. By quasi-particle interference mapping a detailed description of the band dispersion is obtained. The exchange splitting between minority and majority spin states amounts to ∆E$_{ex}$ = (100 ± 8) meV. The onset of the majority band is located at E − E$_F$ = −(160 ± 8)meV and its effective mass is m^* = +(0,14 ± 0,04)me. Furthermore, the onset of the majority spin surface resonance is energetically located at E−E$_F$ = −(235±5)meV and with an effective mass equal to m^* = +(0,36±0,05)m$_e$. To unequivocally identify which spin channels dominate the STS signal, hexagonal quantum wells have been created by sputtering, and interpreted using a one-dimensional quantum well model. The six edges of the hexagon result to be unequal. Atomically resolved measurements show that adjacent edges have not only a different structure, but also different spectroscopic signatures characterized by an alternating sequence of presence and absence of an additional spectroscopic peak. Spin-dependent (SP-STM) measurements did not give any definite conclusion on the origin of this observation. The second experimental section deals with thin iron films deposited on a clean Rh(001) surface and examined by STM, STS and SP-STM. A nearly defect-free Rh(001) is necessary to obtain a growth of iron films with few defects. This is required to correctly interpret the magnetic signal excluding the possible influence of contaminants. The first atomic layer of Fe orders antiferromagnetically in a c(2 × 2)-structure with the easy magnetization axis perpendicular to the surface plane. The second and third layer behaves ferromagnetically with domains sizes which get progressively smaller by increasing the coverage. Above 3.5 atomic layers, a reorientation of the easy magnetization direction from out-of-plane to in-plane takes place. This is signaled by the size of magnetic domains which become smaller while at the same time domain walls become larger. Temperature-dependent SP-STM measurements allow to estimate a Curie temperature of approximatelly 80K for the second layer. At this coverage an additional periodic modulation of the local density of states is detected and persists, although with a shorter wavelength, in the third and fourth layer. Temperature and voltage-dependent measurements support an interpretation of these data based on the existence of a charge density wave. I show that these two usually competing orders (charge and magnetic order) coexist and influence each other, as also confirmed by theoretical calculations performed in collaboration with F. P. Toldin and F. Assaad. In the final chapter the CePt5/Pt(111) intermetallic surface compound has been analyzed. This system has been recently reported to give rise to a heavy Fermion lattice. Starting from the clean Pt(111) surface, the intermetallic surface compound CePt5/Pt(111) is prepared. The thickness of the alloy (t in u.c.) can be varied by the evaporated amount of cerium and the surface produced is examined with STM and STS for various thicknesses. STM images and LEED patterns analyzed in collaboration with C. Praetorius provide consistent results. For coverages below one atomic layer cerium no ordered structure with the STM was observed. For 2 u.c. a (2 × 2) surface structure and for 3 u.c. CePt5 a (3√3×3√3)R30◦-structure was observed. The transition from 3 u.c. CePt5 to 5 u.c. CePt5 was investigated. Supported by structural modelling I conclude that neither a rotation of the atomic lattice nor a rotation of the superstructure was observed. Starting at a coverage of 6 u.c. CePt5 the CePt5 intermetallic surface compound evolves into a different structure. The high symmetry direction is aligned with the underlying Pt(111) crystal and no longer rotated by 30. For all coverages spectroscopic data are acquired, which give no indication of a coherent heavy Fermion system. One explanation is based on a LEED-IV study, which says that any measured surface is terminated with a Pt(111)-layer. The STM is sensitive to the uppermost layer, and thus the effect of a coherent heavy Fermion system would not necessarily measurable.

Wir stellen eine Untersuchung einer Kombination zwischen HQET und relativistischer QCD vor, die das Ziel hat, die b-Quark Masse und die Zerfallskonstante des Bs-Mesons aus Gitter-Simulationen, unter Nichtbeachtung virtueller Fermionenschleifen, zu gewinnen. Wir beginnen mit einem kleinen Volumen, in dem man das b-Quark direkt simulieren kann, und stellen die numerische Verbindung mit einem großen Volumen, wo ``finite-size'''' Effekte vernachlässigbar sind, mit Hilfe einer ``finite-size'''' Methode her. Diese besteht aus zum Kontinuum extrapolierten Schritten, wobei der Massenpunkt, der der physikalischen b-Quark Masse entspricht, durch eine Interpolation erreicht wird. In diese Interpolation fliessen die in der HQET erzielten Resultate ein. Mit dem durch die Sommersche Skale r0 bestimmten Gitterabstand und den experimentalen Werten für die Bs- und K-Massen erhalten wir die Endergebnisse für die renormierungsgruppeninvariante Masse Mb = 6.88(10) GeV, äquivalent zu mb(mb) = 4.42(6) GeV in dem MSbar-Schema und fBs = 191(6) MeV für die Zerfallskonstante. Eine Renormierungsbedingung für den Chromo-magnetischen Operator, der in führender Ordnung der Entwicklung in der schweren Quarkmasse in HQET für die Massenaufspaltung zwischen dem pseudoskalaren und dem vektoriellen Kanal mesonischer schwer-leicht gebundener Zustände verantwortlich ist, wird auf der Basis von Gitter-Korrelationsfunktionen bereitgestellt. Dies eignet sich gut für eine nicht-störungstheoretische Rechnung, welche einen großen Bereich der Renormierungsskala umfasst und keine Valenz-Quarks beinhaltet. Die Zwei-Schleifen Ordnung der entsprechenden anomalen Dimension im Schrödinger-Funktional-Schema wird mit Hilfe von veröffentlichten Ergebnissen berechnet; dies erforderte eine neue Ein-Schleifen Rechnung im SF-Schema mit einem nicht verschwindenden Hintergrundfeld. Die Gitterartefakte bezüglich der Skalenentwicklung des Renormierungsfaktors werden zur Ein-Schleifen Ordnung untersucht, und es wird von nicht-störungstheoretischen Simulationen, unter Nichtbeachtung virtueller Fermionenschleifen, bestätigt, dass sie für die gegenwärtige verfügbare numerische Präzision vernachlässigbar sind. / We present a study of a combination of HQET and relativistic QCD to extract the b-quark mass and the Bs-meson decay constant from lattice quenched simulations. We start from a small volume, where one can directly simulate the b-quark, and compute the connection to a large volume, where finite size effects are negligible, through a finite size technique. The latter consists of steps extrapolated to the continuum limit, where the b-region is reached through interpolations guided by the effective theory. With the lattice spacing given in terms of the Sommer''s scale r0 and the experimental Bs and K masses, we get the final results for the renormalization group invariant mass Mb = 6.88(10) GeV, translating into mb(mb) = 4.42(6) GeV in the MSbar scheme, and fBs = 191(6) MeV for the decay constant. A renormalization condition for the chromo-magnetic operator, responsible, at leading order in the heavy quark mass expansion of HQET, for the mass splitting between the pseudoscalar and the vector channel in mesonic heavy-light bound states, is provided in terms of lattice correlations functions which well suits a non-perturbative computation involving a large range of renormalization scales and no valence quarks. The two-loop expression of the corresponding anomalous dimension in the Schrödinger functional (SF) scheme is computed starting from results in the literature; it requires a one-loop calculation in the SF scheme with a non-vanishing background field. The cutoff effects affecting the scale evolution of the renormalization factors are studied at one-loop order, and confirmed by non-perturbative quenched computations to be negligible for the numerical precision achievable at present.

In the vast realm of inorganic materials, the Cu2+-containing cuprates form one of the richest classes. Due to the combined effect of crystal-field, covalency and strong correlations, all undoped cuprates are magnetic insulators with well-localized spins S=1/2, whereas the charge and orbital degrees of freedom are frozen out. The combination of the spin-only nature of their magnetism with the unique structural diversity renders cuprates as excellent model systems. The experimental studies, boosted by the discovery of high-temperature superconductivity in doped La2CuO4, revealed a fascinating variety of magnetic behaviors observed in cuprates. A digest of prominent examples should include the spin-Peierls transition in CuGeO3, the Bose-Einstein condensation of magnons in BaCuSi2O6, and the quantum critical behavior of Li2ZrCuO4. The magnetism of cuprates originates from short-range (typically, well below 1 nm) exchange interactions between pairs of spins Si and Sj, localized on Cu atoms i and j. Especially in low-dimensional compounds, these interactions are strongly anisotropic: even for similar interatomic distances |Rij|, the respective magnetic couplings Jij can vary by several orders of magnitude. On the other hand, there is an empirical evidence for the isotropic nature of this interaction in the spin space: different components of Si are coupled equally strong. Thus, the magnetism of cuprates is mostly described by a Heisenberg model, comprised of Jij(Si*Sj) terms. Although the applicability of this approach to cuprates is settled, the model parameters Jij are specific to a certain material, or more precisely, to a particular arrangement of the constituent atoms, i.e. the crystal structure. Typically, among the infinite number of Jij terms, only several are physically relevant. These leading exchange couplings constitute the (minimal) microscopic magnetic model. Already at the early stages of real material studies, it became gradually evident that the assignment of model parameters is a highly nontrivial task. In general, the problem can be solved experimentally, using elaborate measurements, such as inelastic neutron scattering on large single crystals, yielding the magnetic excitation spectrum. The measured dispersion is fitted using theoretical models, and in this way, the model parameters are refined. Despite excellent accuracy of this method, the measurements require high-quality samples and can be carried out only at special large-scale facilities. Therefore, less demanding (especially, regarding the sample requirements), yet reliable and accurate procedures are desirable. An alternative way to conjecture a magnetic model is the empirical approach, which typically relies on the Goodenough-Kanamori rules. This approach links the magnetic exchange couplings to the relevant structural parameters, such as bond angles. Despite the unbeatable performance of this approach, it is not universally applicable. Moreover, in certain cases the resulting tentative models are erroneous. The recent developments of computational facilities and techniques, especially for strongly correlated systems, turned density-functional theory (DFT) band structure calculations into an appealing alternative, complementary to the experiment. At present, the state-of-the-art computational methods yield accurate numerical estimates for the leading microscopic exchange couplings Jij (error bars typically do not exceed 10-15%). Although this computational approach is often regarded as ab initio, the actual procedure is not parameter-free. Moreover, the numerical results are dependent on the parameterization of the exchange and correlation potential, the type of the double-counting correction, the Hubbard repulsion U etc., thus an accurate choice of these crucial parameters is a prerequisite. In this work, the optimal parameters for cuprates are carefully evaluated based on extensive band structure calculations and subsequent model simulations. Considering the diversity of crystal structures, and consequently, magnetic behaviors, the evaluation of a microscopic model should be carried out in a systematic way. To this end, a multi-step computational approach is developed. The starting point of this procedure is a consideration of the experimental structural data, used as an input for DFT calculations. Next, a minimal DFT-based microscopic magnetic model is evaluated. This part of the study comprises band structure calculations, the analysis of the relevant bands, supercell calculations, and finally, the evaluation of a microscopic magnetic model. The ground state and the magnetic excitation spectrum of the evaluated model are analyzed using various simulation techniques, such as quantum Monte Carlo, exact diagonalization and density-matrix renormalization groups, while the choice of a particular technique is governed by the dimensionality of the model, and the presence or absence of magnetic frustration. To illustrate the performance of the approach and tune the free parameters, the computational scheme is applied to cuprates featuring rather simple, yet diverse magnetic behaviors: spin chains in CuSe2O5, [NO]Cu(NO3)3, and CaCu2(SeO3)2Cl2; quasi-two-dimensional lattices with dimer-like couplings in alpha-Cu2P2O7 and CdCu2(BO3)2, as well as the 3D magnetic model with pronounced 1D correlations in Cu6Si6O18*6H2O. Finally, the approach is applied to spin liquid candidates --- intricate materials featuring kagome-lattice arrangement of the constituent spins. Based on the DFT calculations, microscopic magnetic models are evaluated for herbertsmithite Cu3(Zn0.85Cu0.15)(OH)6Cl2, kapellasite Cu3Zn(OH)6Cl2 and haydeeite Cu3Mg(OH)6Cl2, as well as for volborthite Cu3[V2O7](OH)2*2H2O. The results of the DFT calculations and model simulations are compared to and challenged with the available experimental data. The advantages of the developed approach should be briefly discussed. First, it allows to distinguish between different microscopic models that yield similar macroscopic behavior. One of the most remarkable example is volborthite Cu3[V2O7](OH)2*2H2O, initially described as an anisotropic kagome lattice. The DFT calculations reveal that this compound features strongly coupled frustrated spin chains, thus a completely different type of magnetic frustration is realized. Second, the developed approach is capable of providing accurate estimates for the leading magnetic couplings, and consequently, reliably parameterize the microscopic Hamiltonian. Dioptase Cu6Si6O18*6H2O is an instructive example showing that the microscopic theoretical approach eliminates possible ambiguity and reliably yields the correct parameterization. Third, DFT calculations yield even better accuracy for the ratios of magnetic exchange couplings. This holds also for small interchain or interplane couplings that can be substantially smaller than the leading exchange. Hence, band structure calculations provide a unique possibility to address the interchain or interplane coupling regime, essential for the magnetic ground state, but hardly perceptible in the experiment due to the different energy scales. Finally, an important advantage specific to magnetically frustrated systems should be mentioned. Numerous theoretical and numerical studies evidence that low-dimensionality and frustration effects are typically entwined, and their disentanglement in the experiment is at best challenging. In contrast, the computational procedure allows to distinguish between these two effects, as demonstrated by studying the long-range magnetic ordering transition in quasi-1D spin chain systems. The computational approach presented in the thesis is a powerful tool that can be directly applied to numerous S=1/2 Heisenberg materials. Moreover, with minor modifications, it can be largely extended to other metallates with higher value of spin. Besides the excellent performance of the computational approach, its relevance should be underscored: for all the systems investigated in this work, the DFT-based studies not only reproduced the experimental data, but instead delivered new valuable information on the magnetic properties for each particular compound. Beyond any doubt, further computational studies will yield new surprising results for known as well as for new, yet unexplored compounds. Such "surprising" outcomes can involve the ferromagnetic nature of the couplings that were previously considered antiferromagnetic, unexpected long-range couplings, or the subtle balance of antiferromagnetic and ferromagnetic contributions that "switches off" the respective magnetic exchange. In this way, dozens of potentially interesting systems can acquire quantitative microscopic magnetic models. The results of this work evidence that elaborate experimental methods and the DFT-based modeling are of comparable reliability and complement each other. In this way, the advantageous combination of theory and experiment can largely advance the research in the field of low-dimensional quantum magnetism. For practical applications, the excellent predictive power of the computational approach can largely alleviate designing materials with specific properties.

In the vast realm of inorganic materials, the Cu2+-containing cuprates form one of the richest classes. Due to the combined effect of crystal-field, covalency and strong correlations, all undoped cuprates are magnetic insulators with well-localized spins S=1/2, whereas the charge and orbital degrees of freedom are frozen out. The combination of the spin-only nature of their magnetism with the unique structural diversity renders cuprates as excellent model systems. The experimental studies, boosted by the discovery of high-temperature superconductivity in doped La2CuO4, revealed a fascinating variety of magnetic behaviors observed in cuprates. A digest of prominent examples should include the spin-Peierls transition in CuGeO3, the Bose-Einstein condensation of magnons in BaCuSi2O6, and the quantum critical behavior of Li2ZrCuO4. The magnetism of cuprates originates from short-range (typically, well below 1 nm) exchange interactions between pairs of spins Si and Sj, localized on Cu atoms i and j. Especially in low-dimensional compounds, these interactions are strongly anisotropic: even for similar interatomic distances |Rij|, the respective magnetic couplings Jij can vary by several orders of magnitude. On the other hand, there is an empirical evidence for the isotropic nature of this interaction in the spin space: different components of Si are coupled equally strong. Thus, the magnetism of cuprates is mostly described by a Heisenberg model, comprised of Jij(Si*Sj) terms. Although the applicability of this approach to cuprates is settled, the model parameters Jij are specific to a certain material, or more precisely, to a particular arrangement of the constituent atoms, i.e. the crystal structure. Typically, among the infinite number of Jij terms, only several are physically relevant. These leading exchange couplings constitute the (minimal) microscopic magnetic model. Already at the early stages of real material studies, it became gradually evident that the assignment of model parameters is a highly nontrivial task. In general, the problem can be solved experimentally, using elaborate measurements, such as inelastic neutron scattering on large single crystals, yielding the magnetic excitation spectrum. The measured dispersion is fitted using theoretical models, and in this way, the model parameters are refined. Despite excellent accuracy of this method, the measurements require high-quality samples and can be carried out only at special large-scale facilities. Therefore, less demanding (especially, regarding the sample requirements), yet reliable and accurate procedures are desirable. An alternative way to conjecture a magnetic model is the empirical approach, which typically relies on the Goodenough-Kanamori rules. This approach links the magnetic exchange couplings to the relevant structural parameters, such as bond angles. Despite the unbeatable performance of this approach, it is not universally applicable. Moreover, in certain cases the resulting tentative models are erroneous. The recent developments of computational facilities and techniques, especially for strongly correlated systems, turned density-functional theory (DFT) band structure calculations into an appealing alternative, complementary to the experiment. At present, the state-of-the-art computational methods yield accurate numerical estimates for the leading microscopic exchange couplings Jij (error bars typically do not exceed 10-15%). Although this computational approach is often regarded as ab initio, the actual procedure is not parameter-free. Moreover, the numerical results are dependent on the parameterization of the exchange and correlation potential, the type of the double-counting correction, the Hubbard repulsion U etc., thus an accurate choice of these crucial parameters is a prerequisite. In this work, the optimal parameters for cuprates are carefully evaluated based on extensive band structure calculations and subsequent model simulations. Considering the diversity of crystal structures, and consequently, magnetic behaviors, the evaluation of a microscopic model should be carried out in a systematic way. To this end, a multi-step computational approach is developed. The starting point of this procedure is a consideration of the experimental structural data, used as an input for DFT calculations. Next, a minimal DFT-based microscopic magnetic model is evaluated. This part of the study comprises band structure calculations, the analysis of the relevant bands, supercell calculations, and finally, the evaluation of a microscopic magnetic model. The ground state and the magnetic excitation spectrum of the evaluated model are analyzed using various simulation techniques, such as quantum Monte Carlo, exact diagonalization and density-matrix renormalization groups, while the choice of a particular technique is governed by the dimensionality of the model, and the presence or absence of magnetic frustration. To illustrate the performance of the approach and tune the free parameters, the computational scheme is applied to cuprates featuring rather simple, yet diverse magnetic behaviors: spin chains in CuSe2O5, [NO]Cu(NO3)3, and CaCu2(SeO3)2Cl2; quasi-two-dimensional lattices with dimer-like couplings in alpha-Cu2P2O7 and CdCu2(BO3)2, as well as the 3D magnetic model with pronounced 1D correlations in Cu6Si6O18*6H2O. Finally, the approach is applied to spin liquid candidates --- intricate materials featuring kagome-lattice arrangement of the constituent spins. Based on the DFT calculations, microscopic magnetic models are evaluated for herbertsmithite Cu3(Zn0.85Cu0.15)(OH)6Cl2, kapellasite Cu3Zn(OH)6Cl2 and haydeeite Cu3Mg(OH)6Cl2, as well as for volborthite Cu3[V2O7](OH)2*2H2O. The results of the DFT calculations and model simulations are compared to and challenged with the available experimental data. The advantages of the developed approach should be briefly discussed. First, it allows to distinguish between different microscopic models that yield similar macroscopic behavior. One of the most remarkable example is volborthite Cu3[V2O7](OH)2*2H2O, initially described as an anisotropic kagome lattice. The DFT calculations reveal that this compound features strongly coupled frustrated spin chains, thus a completely different type of magnetic frustration is realized. Second, the developed approach is capable of providing accurate estimates for the leading magnetic couplings, and consequently, reliably parameterize the microscopic Hamiltonian. Dioptase Cu6Si6O18*6H2O is an instructive example showing that the microscopic theoretical approach eliminates possible ambiguity and reliably yields the correct parameterization. Third, DFT calculations yield even better accuracy for the ratios of magnetic exchange couplings. This holds also for small interchain or interplane couplings that can be substantially smaller than the leading exchange. Hence, band structure calculations provide a unique possibility to address the interchain or interplane coupling regime, essential for the magnetic ground state, but hardly perceptible in the experiment due to the different energy scales. Finally, an important advantage specific to magnetically frustrated systems should be mentioned. Numerous theoretical and numerical studies evidence that low-dimensionality and frustration effects are typically entwined, and their disentanglement in the experiment is at best challenging. In contrast, the computational procedure allows to distinguish between these two effects, as demonstrated by studying the long-range magnetic ordering transition in quasi-1D spin chain systems. The computational approach presented in the thesis is a powerful tool that can be directly applied to numerous S=1/2 Heisenberg materials. Moreover, with minor modifications, it can be largely extended to other metallates with higher value of spin. Besides the excellent performance of the computational approach, its relevance should be underscored: for all the systems investigated in this work, the DFT-based studies not only reproduced the experimental data, but instead delivered new valuable information on the magnetic properties for each particular compound. Beyond any doubt, further computational studies will yield new surprising results for known as well as for new, yet unexplored compounds. Such "surprising" outcomes can involve the ferromagnetic nature of the couplings that were previously considered antiferromagnetic, unexpected long-range couplings, or the subtle balance of antiferromagnetic and ferromagnetic contributions that "switches off" the respective magnetic exchange. In this way, dozens of potentially interesting systems can acquire quantitative microscopic magnetic models. The results of this work evidence that elaborate experimental methods and the DFT-based modeling are of comparable reliability and complement each other. In this way, the advantageous combination of theory and experiment can largely advance the research in the field of low-dimensional quantum magnetism. For practical applications, the excellent predictive power of the computational approach can largely alleviate designing materials with specific properties.:List of Figures List of Tables List of Abbreviations 1. Introduction 2. Magnetism of cuprates 3. Experimental methods 4. DFT-based microscopic modeling 5. Simulations of a magnetic model 6. Model spin systems: challenging the computational approach 7. Kagome lattice compounds 8. Summary and outlook Appendix Bibliography List of publications Acknowledgments