First Principles Studies of Functional Materials Based on Graphene and Organometallics

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(1)Digital Comprehensive Summaries of Uppsala Dissertations from the Faculty of Science and Technology 1120. First Principles Studies of Functional Materials Based on Graphene and Organometallics SUMANTA BHANDARY. ACTA UNIVERSITATIS UPSALIENSIS UPPSALA 2014. ISSN 1651-6214 ISBN 978-91-554-8869-7 urn:nbn:se:uu:diva-217175.

(2) Dissertation presented at Uppsala University to be publicly examined in Polhemsalen, Ångströmlaboratoriet, Lägerhyddsvägen 1, Uppsala, Friday, 14 March 2014 at 13:15 for the degree of Doctor of Philosophy. The examination will be conducted in English. Faculty examiner: Stefano Sanvito ( Department of Physics Trinity College Dublin). Abstract Bhandary, S. 2014. First Principles Studies of Functional Materials Based on Graphene and Organometallics. Digital Comprehensive Summaries of Uppsala Dissertations from the Faculty of Science and Technology 1120. 90 pp. Uppsala: Acta Universitatis Upsaliensis. ISBN 978-91-554-8869-7. Graphene is foreseen to be the basis of future electronics owing to its ultra thin structure, extremely high charge carrier mobility, high thermal conductivity etc., which are expected to overcome the size limitation and heat dissipation problem in silicon based transistors. But these great prospects are hindered by the metallic nature of pristine graphene even at charge neutrality point, which allows to flow current even when a transistor is switched off. A part of the thesis is dedicated to invoke electronic band gaps in graphene to overcome this problem. The concept of quantum confinement has been employed to tune the band gaps in graphene by dimensional confinement along with the functionalization of the edges of these confined nanostructures. Thermodynamic stability of the functionalized zigzag edges with hydrogen, fluorine and reconstructed edges has been presented in the thesis. Keeping an eye towards the same goal of band gap opening, a different route has been considered by admixing insulating hexagonal boron nitride (h-BN) with semimetal graphene. The idea has been implemented in two dimensional h-BN-graphene composites and three dimensional stacked heterostructures. The study reveals the possibility of tuning band gaps by controlling the admixture. Occurrence of defects in graphene has significant effect on its electronic properties. By random insertion of defects, amorphous graphene is studied, revealing a semi-metal to a metal transition. The field of molecular electronics and spintronics aims towards device realization at the molecular scale. In this thesis, different aspects of magnetic bistability in organometallic molecules have been explored in order to design practical spintronics devices. Manipulation of spin states in organometallic molecules, specifically metal porphyrin molecules, is achieved by controlling surface–molecule interaction. It has been shown that by strain engineering in defected graphene, the magnetic state of adsorbed molecules can be changed. The spin crossover between different spin states can also be achieved by chemisorption on magnetic surfaces. A significant part of the thesis demonstrates that the surface-molecule interaction not only changes the spin state of the molecule, but allows to manipulate magnetic anisotropies and spin dipole moments via modified ligand fields. Finally, in collaboration with experimentalists, a practical realization of switching surface–molecule magnetic interactions by external magnetic fields is demonstrated. Keywords: Graphene, Magnetism, Organometallics, Density functional theory, Electron correlation, Spin switching, Nanoribbons, Exchange interaction, Edge functionalization, Band gap Sumanta Bhandary, Department of Physics and Astronomy, Materials Theory, Box 516, Uppsala University, SE-751 20 Uppsala, Sweden. © Sumanta Bhandary 2014 ISSN 1651-6214 ISBN 978-91-554-8869-7 urn:nbn:se:uu:diva-217175 (http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-217175).

(3) Dedicated to My Parents and Sister.

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(5) List of papers. This thesis is based on the following papers, which are referred to in the text by their Roman numerals. I. Complex edge effects in zigzag graphene nanoribbons due to hydrogen loading. S. Bhandary, M. I. Katsnelson, O. Eriksson and B. Sanyal. Phys. Rev. B 82, 165405, (2010).. II. Controlling electronic structure and transport properties of zigzag graphene nanoribbons by mono- and difluorinated edge functionalization. S. Bhandary, G. Penazzi, J. Fransson, T. Frauenheim, O. Eriksson, and B. Sanyal. Manuscript.. III. Functionalization of edge reconstructed graphene nanoribbons by H and Fe: A density functional study. S. Haldar, S. Bhandary, S. Bhattacharjee, O. Eriksson, D. Kanhere, B. Sanyal. Solid State Comm. 152, 1719, (2012).. IV. Interpolation of atomically thin hexagonal boron nitride and graphene: electronic structure and thermodynamic stability in terms of all-carbon conjugated paths and aromatic hexagons. J. Zhu, S. Bhandary, B. Sanyal and H. Ottoson. J. Phys. Chem. C 115, 10264, (2011).. V. VI. Electronic structure of graphene/h-BN heterostructures in a quasiperiodic Fibonacci sequence. S. Bhandary, S. Haldar, O. Eriksson, and B. Sanyal. Manuscript. Disorder-induced metallicity in amorphous graphene. E. Holmström, J. Fransson, O. Eriksson, R. Lizárraga, B. Sanyal, S. Bhandary, M. I. Katsnelson. Physical Review B 84, 205414, (2011)..

(6) VII. Correlated electron behavior of metalorganic molecules: insights from density functional theory and exact diagonalization studies. S. Bhandary, T. Wehling, Patrik Thunström, Igor di Marco, Barbara Brena, O. Eriksson, and B. Sanyal. Manuscript.. VIII. Graphene as a reversible spin manipulator of molecular magnets. S. Bhandary, S. Ghosh, H. Herper, H. Wende, O. Eriksson and B. Sanyal. Phys. Rev. Lett. 107, 257202, (2011).. IX. X. XI. Manipulation of spin state of iron porphyrin by chemisorption on magnetic substrates. S. Bhandary, B. Brena, P. M. Panchmatia, I. Brumboiu, M. Bernien, C. Weis, B. Krumme, Corina Etz, W. Kuch, H. Wende, O. Eriksson, B. Sanyal. Phys. Rev. B 88, 024401, (2013). Defect controlled magnetism in FeP/graphene/Ni(111). S. Bhandary, O. Eriksson and B. Sanyal. Scientific Reports 3, 3405, (2013). Field-regulated switching of the magnetization of Co-porphyrin on graphene. D. Klar, S. Bhandary, A. Candini, L. Joly, P. Ohresser, S. Klyatskaya, M. Schleberger, M. Ruben, M. Affronte, O. Eriksson, B. Sanyal, and H. Wende. Submitted.. Reprints were made with permission from the publishers..

(7) The following papers are co-authored by me, but not included in the thesis. • Magnetocrystalline anisotropy and uniaxiality of MnAs/GaAs (100) films. J. M. Wikberg, R. Knut, S. Bhandary, I. Di Marco, M. Ottosson, J. Sadowski, B. Sanyal, P. Palmgren, C. W Tai, O. Eriksson, O. Karis, P. Svedlindh. Phys. Rev. B 83, 024417, (2011). • Route towards finding large magnetic anisotropy in nanocomposites: Application to a W1−x Rex /Fe multilayer. S. Bhandary, O. Grånäs, L. Szunyogh, B. Sanyal, L. Nordström, O. Eriksson. Phys. Rev. B 84, 092401, (2011). • First-principles study of magnetism in Pd3 Fe under pressure. B. Dutta, S. Bhandary, S. Ghosh, B. Sanyal. Phys. Rev. B 86, 024419, (2012). • Improved gas sensing activity in structurally defected bilayer graphene. Y. Hajati, T. Blom, SHM Jafri, S. Haldar, S. Bhandary, M. Z. Shoushtari, O. Eriksson, B. Sanyal, K. Leifer. Nanotechnology, 23, 505501, (2012). • Iron porphyrin molecules on Cu (001): Influence of adlayers and ligands on the magnetic properties. H. C. Herper, M. Bernien, S. Bhandary, C. F. Hermanns, A. Krüger, J. Miguel, C. Weis, C. Schmitz-Antoniak, B. Krumme, D. Bovenschen, C. Tieg, B. Sanyal, E. Weschke, C. Czekelius, W. Kuch, H. Wende, O. Eriksson. Phys. Rev. B 87, 174425, (2013). • Oxygen-tuned magnetic coupling of Fe-phthalocyanine molecules to ferromagnetic Co films. D. Klar, B. Brena, H. C. Herper, S. Bhandary, C. Weis, B. Krumme, C. Schmitz-Antoniak, B. Sanyal, O. Eriksson, H. Wende. Phys. Rev. B 88, 224424, (2013). • Fe phthalocyanine on Co(001): Influence of surface oxidation on structural and electronic properties. H. C. Herper, S. Bhandary, O. Eriksson, B. Sanyal, and B. Brena. Accepted in Phys. Rev. B.

(8) Comments on my participation. In papers I-III, V and VI-X, my participation was in all three parts, planning the research, calculations and writing the paper. In the calculation part, there were contributions from MIK in paper I, GP and JF in paper II, SH and SB in paper III, SH in paper V, SG in paper VIII and BB, PMP, IB in paper IX. I contributed with major part of calculations with contribution from JZ and manuscript writing in paper IV. The IPR and the Green’s function calculations were performed by me in paper VI. The experimental work in paper IX was performed by the group of HW and WK. Paper X is also a combined work with experimental group led by HW. I have performed theoretical calculations and contributed in the paper writing..

(9) Contents. 1. Introduction. ................................................................................................. 11. 2. Theoretical background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Many body problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Density functional theory (DFT) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.2 The Kohn-Sham formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Exchange and correlation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Exchange-correlation approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Local density approximation (LDA) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Generalized-gradient approximation (GGA) . . . . . . . . . . . . . . . . . 2.4 Beyond LDA: LDA+U . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 DFT++ Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Electronic structure of periodic solids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7 Plane waves and pseudopotentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7.1 The projector augmented wave (PAW) method . . . . . . . . . . . . .. 15 15 17 17 18 19 19 20 21 22 23 24 26. 3. Magnetic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Exchange interactions and orbital dependence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Spin dipole moment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Magnetic anisotropy energy (MAE) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Spin-orbit interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 30 31 32 35 36. 4. Modification of graphene in bulk and nanostructures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Edge functionalization of graphene nanoribbons . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1 Complex edge effect in hydrogen passivated ZGNR . . . . 4.1.2 Controlling electronic structure and transport properties by edge fluorination of ZGNRs . . . . . . . . . . . . . . . . . . . . . 4.1.3 Reconstructed edge GNRs and hydrogen passivation . . . 4.2 Chemical alloying in graphene . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 h-BN-graphene composites: stability issues . . . . . . . . . . . . . . . . . . 4.2.2 Variation of band gaps in h-BN-Graphene composites . 4.3 h-BN graphene heterostructures in Fibonacci sequence . . . . . . . . . . . . . . 4.4 Deviation from crystallinity: disorder induced metallicity in amorphous graphene . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 39 39 39. 5. 42 46 48 49 50 52 54. Magnetism in organometallics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 5.1 Bistability in iron porphyrin (FeP) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 5.1.1 The spin crossover scenario . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58.

(10) 5.2. 5.3. 5.4. 5.5. 5.6. Spin state manipulation with strain engineering . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 Graphene as a substrate and the role of defects . . . . . . . . . . . . . 5.2.2 Strain induced reversible spin switch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Spin state manipulation by chemisorption on magnetic surfaces 5.3.1 Surface orientations and adsorption sites . . . . . . . . . . . . . . . . . . . . . . . 5.3.2 Chemisorption and induced high spin state . . . . . . . . . . . . . . . . . . . 5.3.3 Molecule-surface magnetic exchange interaction . . . . . . . . . 5.3.4 Spin state detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Controlling magnetism in FeP with defects in graphene deposited on Ni (111) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.1 Induced magnetism in graphene . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.2 Magnetic properties of adsorbed molecules . . . . . . . . . . . . . . . . . . Magnetization switching of CoP on graphene/Ni(111) with applied magnetic field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.1 Intermolecular-layer and surface molecule magnetic interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.2 Field regulated switching of magnetization . . . . . . . . . . . . . . . . . . . Spin dipole moments in porphyrin molecules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 6. Summary and Outlook. 7. Sammanfattning på svenska. 8. Acknowledgements. 9. References. ............................................................................... 62 62 63 64 64 66 66 67 67 68 69 71 71 72 74 77. ....................................................................... 80. .................................................................................... 83. ................................................................................................... 85.

(11) 1. Introduction. The conventional way of miniaturization and yet increased productivity of modern day devices is facing a challenge, be it transistors, sensors or storage devices. A further downscaling is limited by the amount of heat dissipation, leakage between devices, doping problem etc.. The requirement at this moment is to improve on the limiting points of top-down approach by searching for new materials or technology. The rise of graphene [1, 2]seems to bring an immediate solution for the problems in silicon based electronics. Graphene, the thinnest material with an extremely stable 2D structure, fits perfectly in the demand for the nano device fabrications. Moreover, the charge carrier mobility in graphene is extremely high (200×103 cm2 V−1 s−1 ) even at ambient conditions and remains unaffected with temperature, chemical or electrical doping (even at charge concentration, n > 1011 cm−2 ). Moreover, an ambipolar field effect, i. e. possibilities of tuning charge carriers continuously from electron to hole makes graphene particularly interesting. The charge carriers in graphene mimic relativistic particles and hence are described by Dirac equation with zero rest mass and effective Fermi velocity, vF ≈106 m s−1 . The consequence of the relativistic nature is exhibited in fascinating phenomena like anomalous quantum Hall effects (QHE) [3, 4], minimum quantum conductivity [5, 6] at vanishing charge carrier concentration and Klein tunneling [7]. From the application perspective, high carrier mobility and extremely long mean free path without scattering promote graphene to have ballistic transport feasible in sub-micrometer length. On the other hand, the occurrence of minimum quantum conductivity restricts graphene as a potential candidate for electronic application. The modification of pristine graphene, hence, has to take place in order to induce a semiconductor gap, which can be engineered in several ways. One of the most promising route is to translate quantum confinement effects in semiconductor physics to graphene by restricting its dimensionality. Induced band gaps in quasi one dimensional graphene nanoribbons show a unique dependence on edge orientations and band gaps are found inversely proportional to the ribbon widths [8]. This is in particular advantageous as the implementation of integrated circuits can be realized on a single sheet of graphene with properly patterned areas acting as metallic electrodes and semiconducting active regions [9]. Experimental realizations of nanoribbons [10–13] have enhanced this possibility. In practice, the device realization is obstructed because of less control over the edge orientation and stability issues at the edges. 11.

(12) Manipulating properties of pristine graphene with the aid of chemical modification is another avenue to achieve the same goal. A successful step forward has been taken with hydrogen [14] and fluorine [15, 16] adsorption on graphene surfaces inducing a metal to insulator transition. Also graphene can be alloyed with large band gap materials, that can induce band gaps. Such a composite with hexagonal boron nitride, has been synthesized [17] opening up a great possibility of chemically controlled band gaps. Looking forward to the possibilities, one aim of the thesis is to study the edge stability issues of graphene nanoribbons. Thermodynamical stability of chemically modified edges have been investigated with the aid of first principles density functional theory. Edge functionalization brings in an immediate effect on the electronic properties of ribbons, including band gap modification, edge magnetism etc., which have been addressed in details. Alternative possibilities of band gap opening have also been explored with chemically alloyed 2D and stacked 3D heterostructures. Drifting beyond crystallinity by disorder induced effects in graphene properties have been studied as well. Going beyond the conventional route, the information processing at the molecular scale can be achieved by employing quantum features of organic molecules. The realm of molecular electronics relies on proper sensing of change in electrical current in organic molecules. In addition, another intrinsic degree of freedom, the electron spin, can as well be exploited for information processing. The two fluid theory of Mott [18], which states that the velocities of electrons with different spins in a magnet are different, forms the basis of spin dependent electronics [19, 20]. Organic molecules have certain advantages in this respect apart from its purity by having smaller spin dephasing compared to its inorganic counterpart [22] and the coexistence of different functional properties like variable conductivity [23], magnetism [24] etc.. Employing the general idea of external stimuli controlled spin response, spintronic devices like molecular spin valve, spin transistors have been realized both from the theory and experiments. For instance, a spin valve (SV) is formed with a molecule or molecular complexes in the core and two electrodes, where at least two components consist of magnetic element to control spin current via magnetic interaction between them [21]. A detailed study of molecular spin valve with non magnetic single dots [25–27], magnetic multi dots [28, 29] and spin transistors with quantum multidots [30, 31] and single molecule magnet [34, 35] have been performed from both theoretical and experimental aspects, establishing their practical realizations. A distinctly different approach towards information processing can be realized by utilizing molecular spin states. The instrumental idea behind this approach is the transformation of spin state information to electrical current. Reading out of magnetic states in terms of current has been realized with a theoretical investigation with Mn12 single molecule magnet [36]. To be potentially employed in this aspect of molecular spintronics, the molecules have to exhibit different feasible spin states, which can be controlled by external 12.

(13) stimuli. Also the properties have to be retained in contact with the electrodes. Manipulation of spin state thus provides the key to access this emerging field and spin crossover (SCO) molecules are foreseen to play an instrumental role in this regard. Information processing can be accessed by exploiting magnetic (or optical) bistability of a molecule or molecular assembly. This makes molecules extremely appealing not only in the field of spintronics, but also in high density data storage and quantum computation. Where the SCO is concerned, organic molecules consisting of transition metal (TM) centers show exotic magnetic behavior. The feasibility of SCO can be realized in complexes having ions with dn (4 ≥ n ≤ 7) configuration [37]. Among all these complexes, the SCO is particularly abundant in Fe ions in 2+ and 3+ charge states. A proper balance of spin pairing energy [38] and ligand field strength favors this phenomenon to occur [39]. Moreover, the spin states can be perturbed by external stimuli, like temperature [40], light [41], ligand addition (removal) [42], pressure [43], electric field [44], etc., which induce SCO. Within the scope of this thesis, molecules belonging to TM centered porphyrin and phthalocyanine are considered to explore this property under different chemical environments. Magnetic ordering between organometallic molecules is rather feeble and seldom exhibits a paramagnetic behavior. From spintronics perspective, this is a hindrance as one of the basic demand of information processing is the intrinsic magnetic ordering, i. e., existence of remnant magnetization on single molecule. Moreover, for practical device realization, this property has to be retained on metallic surfaces. This has very recently been realized with Fe4 SMM on gold [45] and TbPc2 on graphite [46]. A single ion anisotropy is quite crucial in this respect and a significant part of the thesis is devoted to analyze magnetic anisotropies of the molecules. External magnetic field or magnetic exchange with ferromagnetic surface, on the other hand, can induce a magnetic ordering. This in particular is interesting from technological aspects of spintronic devices, such as magnetic switches. The requirements for the practical realization of this kind of device are : (1) an exchange mechanism for magnetic ordering of the molecules. (2) easy switching possibilities with external magnetic fields. Developments in element specific magnetization study, X-ray magnetic circular dicroism (XMCD) and spin polarized scanning tunneling microscopy (STM) and spectroscopy (STS) important development in this regard. Successful magnetic ordering of paramagnetic molecules on ferromagnetic surfaces has been achieved recently, [47–49] with reasonable control over exchange coupling strengths. But plenty of space yet to be filled when it comes to switching possibilities. The objective of the other part of the thesis is hence two fold : (1) exploring spin cross over possibilities in organometallics, where a dedicated study is performed to understand the mechanism and its limitations, effects of external stimuli on it and to achieve a control over the manipulation of it. It has also been observed that spin dipole moment has a significant contribution in 13.

(14) determining the magnetic states of molecules. (2) understanding the surfacemolecule interaction with fundamental details of coupling and its dependence on surface orientations, effect of surface modifications, defects etc. and explore the possibilities of switching with external perturbations.. 14.

(15) 2. Theoretical background. 2.1 Many body problem Determination of the electronic structure of atoms, molecules, and solids has always been in focus of condensed matter physics and chemistry. The solution, however, is nontrivial because of two fundamental reasons. Electrons in matter behave as quantum particles and the interaction between them is quantum mechanically correlated. Secondly, the complexity of describing electrons in a quantum mechanical system increases dramatically with the number of electrons involved in it, resulting in so called ‘many-body problem’. The properties of a quantum mechanical system can be obtained by solving wave equations. The state, where the many particle system resides is described by wavefunction, ψ({ri },t), and the dynamics is governed by the Schrödinger equation for non relativistic particles. Hψ = i¯h. ∂ψ ∂t. (2.1). where the energy operator H, known as the Hamiltonian, takes the following form for a system containing atomic nuclei and electrons, H =−. ZI e2 h¯ 2 2 h¯ 2 2 − − ∑ 2mI I ∑ |ri − RI | i 2me ∑ i I i,I +. e2 ZI ZJ e2 1 1 + ∑ ∑ 2 i= j |ri − r j | 2 I=J |RI − RJ |. (2.2). where ri and me represent the position and mass of the ith electron and RI , ZI and mI correspond to position, atomic number, and mass of the I th nucleus respectively. The terms in Eqn. 2.2, (from left) involve kinetic energies of electrons and nuclei, Coulomb interaction between electrons and nuclei, electronelectron Coulomb interactions and Coulomb interactions between nuclei. The Hamiltonian does not involve any explicit time dependent term, which allows to write the wavefunction in a separable form with spatial and time dependent parts, ψ({ri },t) = φE ({ri }) exp(−iEt). The Eqn. 2.1, hence, takes the simple form, Hψ = Eψ. (2.3). E is the energy of the system. By minimizing the energy in a variational way the ground state of the system can be obtained.The nuclei are ∼ 103 times 15.

(16) massive compared to the electrons and have much lower velocities than electrons. The electrons, hence, respond almost instantaneously to the dynamics of nuclei, which has led to the frozen nuclei approximation, known as the BornOpenheimar Approximation [50]. This makes the Hamiltonian convenient by separating out nuclei part from the electronic part, which can be expressed as,. H =−. ZI e2 e2 1 h¯ 2 2i − ∑ + ∑ ∑ 2me i 2 i= j |ri − r j | i,I |ri − RI |. (2.4). The total energy of the system is obtained by adding the nuclei-nuclei in2 Je teraction, 12 ∑I=J |RZIIZ−R , which is treated classically by the Ewald method. J| The solution of the Schrödinger equation still remains nontrivial because the motion of an electron in solid is affected by other electrons via electron e2 . A proper treatment electron correlaelectron correlation term, 12 ∑i= j |ri −r j| tion requires the many-body wavefunction to contain 3N variables, which in solid (with N∼ 1023 electrons) is practically not feasible. Mean field schemes hence have been devised to approximate the exact many-body problem. Hartree approached the problem by writing many electron wavefunction as a simple product of single electron wavefunctions. Proceeding with the variational method this allows to find single particle Hamiltonian equations, which are similar to Schrödinger equation but with an effective potential. The potential, namely the Hartree potential can be thought of an interaction of the electron with time averaged distribution of all other electrons. The approach, however, lacks antisymmetric description of the fermionic wavefunction, which consequences to its failure. This has been incorporated in Hartree-Fock formalism, where many electron wave function is expressed in a Slater determinant form. The immediate consequence is the Pauli exclusion principle, states the occurrence of two electron at same space with same quantum numbers is forbidden. The variational approach with this wavefunction leads to the similar Hamiltonian equations but introduces an exchange potential alongside the Hatree potential. The Hatree-Fock approach recovers the shortcomings of the Hartree formalism but fails in crystal as it does not incorporate any electron correlation. In small finite systems, e. g. molecules, where many-body effect is not prominent Hartree-Fock method is rather successful. Nevertheless, both the approaches are wavefunction based, which makes it computationally nontrivial for large system size (expense O(N5 -N7 ) for system size N). Electron density on the other hand is more tractable quantity with drastically reduced degrees of freedom and significantly reduces computational expenses (expense O(N3 )). 16.

(17) 2.1.1 Density functional theory (DFT) The foundation of DFT relies on the fact that any property of an interacting many-body system is a unique functional of the ground state density n0 (r). The very first illustration of DFT of quantum systems was proposed by Thomas and Fermi [51,52] based on non-interacting homogenous electron gas density. The simple approximation certainly lacked the accurate description of electrons in a many body system, which consequences its failure. The DFT found a strong foundation with Hohenberg-Kohn [53] formalism, which introduced an exact theory of interacting many-body systems. The approach relies on the two following theorems: Theorem I For any system of interacting particles in an external potential Vext , the potential can uniquely be determined except for a constant, by the ground state particle density, n0 (r). Theorem II A universal functional for the energy E[n] in terms of density n(r) can be defined for any external potential Vext . For any particular Vext , the ground state energy of the system is the global minimum of the energy functional and the density n(r) which minimizes the functional is the exact ground state density n0 (r). The total energy of the system can be written as, E[n] = F[n] +. . d 3 rVext (r)n(r). (2.5). where the functional F, describes the kinetic energy and all the electronelectron interactions. This function is universal as it does not depend on the external potential, so it has to be ideally the same for any system. But the theorem does not provide any means to determine the exact form of the functional and hence has to be approximated in order to be applied in practical calculations.. 2.1.2 The Kohn-Sham formalism The scheme for practical realization of DFT was proposed by Kohn and Sham [54]. The underpinned idea of Kohn-Sham approach was to replace the interacting many-body system with an auxiliary system of non interacting particles having the same ground state. The total energy functional can be written as: E[n] =. . d 3 rVext (r)n(r) + TS [n] +. 1 2. . d 3 rd 3 r. . . n(r)n(r ) + Exc |r − r |. (2.6). where, Vext is the external potential, TS is the kinetic energy term of the hypothetical non-interacting electrons. The third term is the classical electrostatic 17.

(18) energy (Hartree) of the electrons and all the many-body effects are grouped in to the exchange correlation energy, Exc . The minimization of the KohnSham energy functional in Eqn. 2.6, with respect to the density n(r) results in a Schrödinger-like Kohn-Sham (KS) equation, 1 2 (2.7) HKS (r)ψi (r) = − +VKS (r) ψi (r) = εi ψi (r). 2 The potential VKS , in Eqn. 2.7 is an effective potential, consisting of external potential Vext and Hartree potential VH = d 3 r n(r) and exchange correlation |r−r |. δ Exc [n] δ n(r) ,. where electrons move independently. ψi are the eigenpotential, Vxc = functions corresponding to eigenvalues εi . The ground state electron density hence is obtained as: occ. n(r) = ∑ |ψi (r)|2. (2.8). i=1. The Kohn-Sham equation in 2.7 is an independent particle equation and does depend on the choice of Exc . The potential is achieved self consistently with the resulting density. The kinetic energy of the non-interacting system can be written as: N. TS [n] = ∑ εi −. . d 3 rVe f f (r)n(r). (2.9). i=1. which yields following expression of the total energy, N. E = ∑ εi − i=1. 1 2. . . d 3 rd 3 r. . n(r)n(r ) − |r − r |. . d 3 rVxc (r)n(r) + Exc [n]. (2.10). The formalism is exact and will lead to the exact ground state of interacting many-body system, if the form of Exc is exactly known.. 2.2 Exchange and correlation The key achievement of Kohn-Sham approach is to explicitly separate out the independent single particle kinetic energy and the classical Hartree term from the remaining interacting part exchange-correlation functional, which can be approximated in different ways. A formal description of exchange correlation functional is given by adiabatic connection method. A coupling is established between interacting and non-interacting system by a scaling factor λ , which defines the strength of the electron-electron interaction. This is varied from 0 (non-interacting) to 1(interacting system), keeping the density unchanged (hence adiabatic). This results to, 1 Exc [n(r)] = 2 18. . . n(r)dr. nxc (r, r ) dr |r − r |. (2.11).

(19) where nxc (r, r ) is the coupling factor averaged exchange correlation hole. . nxc (r, r ) =. 1 0. nλxc (r, r )dλ. (2.12). And thus from Eqns. 2.11 and 2.12, one can define the exchange correlation density : 1 εxc [n(r)] = 2. . nxc (r, r ) dr |r − r |. (2.13). The exchange-correlation hole can be divided in to two parts, exchange and correlation holes, known as Fermi and Coulomb hole, respectively. nxc (r, r ) = nx (r, r ) + nc (r, r ). (2.14). nx (r, r ) = nxc,λ =0 (r, r ) nc (r, r ) = nxc (r, r ) − nx (r, r ). (2.15) (2.16). where,. The exchange hole nx can be described by the Hatree-Fock expression of the energy. Ex [n(r)] =. 1 2. . . n(r)dr. nx (r, r ) dr |r − r |. (2.17). The exchange-correlation functional can thus be written as, Exc =. . n(r)εxc [n(r)]dr. (2.18). Having known the exact form of the exchange-correlation density, the exact ground state can be reached.. 2.3 Exchange-correlation approximations In reality, unknown form of exchange correlation density directs us to model it. A proper sampling of the density surrounding each electron is required to construct εxc and different levels of approximations appear there, as described below.. 2.3.1 Local density approximation (LDA) The first ever form of exchange-correlation energy was proposed by Hohenberg and Kohn in their original DFT paper [53], where exchange-correlation 19.

(20) energy of a system is approximated with the same associated with the homogeneous electron gas. This is the simplest form of exchange-correlations functional, which works surprisingly well for many systems. LDA Exc [n(r)] =. . hom n(r)εxc [n(r)]dr. (2.19). εxc represents the exchange-correlation energy density of a homogeneous electron gas with density n(r). As mentioned earlier, the energy corresponds to the exchange hole, nx , can be derived from exact analytic formula given by Dirac [55]. 3 9 11 0.4582 Ex [n(r)] = − ( 2 ) 3 ≈ − 4 4π rs rs. (2.20). where rs is the Seitz radius. The other contribution from the correlation hole is obtained by an interpolation formula, that basically connects the known forms of εchom at low and high density limits. An interpolation formula, which is used commonly, was prescribed by Perdew and Zunger [56], where interpolation coefficients are derived from the quantum Monte Carlo data of homogeneous electron gas. In spite of its simplicity, LDA seems to reproduce structural properties, e.g, lattice constants are accurate within 3%, error in bulk moduli are on a bit higher side (>10%) but that is not unfamiliar in transition metals. But binding energies, for example, comes out to be relatively higher (over binding). It is also found not so impressive in describing crystal phase stability, for example, the high pressure phase in SiO2 is more favorable than the zero pressure phase. LDA also underestimates the phase transition pressure in diamond, Si, Ge, etc... 2.3.2 Generalized-gradient approximation (GGA) An improvement over LDA was suggested by Hohenberg and Kohn [53] by considering higher order density gradient expansion terms which is known as the gradient expansion approximation (GEA). However, the GEA fails as exchange hole does not integrate to -1 i. e. violation of the sum rule. In spite of it’s failure, it paved a way to construct GGA exchange correlation hole by real space cutoff of GEA exchange hole. With the introduction of an analytic function, known as the enhancement function, Fxc [n(r), Δn(r)], GGA exchange correlation energy can be obtained by modifying the LDA energy density : GGA Exc [n(r)] =. . hom n(r)εxc [n(r)]Fxc [n(r), Δn(r)]dr. (2.21). A parameter free form of exchange enhancement factor was provided by Perdew and Wang [57], which was later on modified to give a simplified form, 20.

(21) known as PBE after the names of Perdew, Burke and Ernzerhof, who devised the formula. The GGA correlation is also constructed with a paramatrized form of the homogeneous electron gas correlation energy and a gradient dependent term [58, 59]. The over binding problem of LDA is corrected by GGA along with improved results in atomic and molecular energies, bulk phase stability, magnetic properties etc.. However, both LDA, GGA schemes are found inadequate to describe, for example, band gap of Mott insulators like transition metal, rare earth compounds.. 2.4 Beyond LDA: LDA+U The problem with LDA lies in the dependence of energy functional E(N) on number of electrons, N. E(N) and its derivative ∂ E/∂ N, both are continuous for an integral value of N. For an exact functional, the derivative has a discontinuity, which has a significant contribution in the band gap. LDA also fails to reproduce orbital energies(εi = ∂ E/∂ ni , ni is the orbital occupation numbers) and is often found in bad agreement with experiment. A correction to the LDA energy functional has been brought in by incorporating explicit Coulomb interaction, U, of localized electrons in a Hatree-Fock (HF) like approach, namely the LDA+U correction. The key intention of this approach is to divide electrons in two subsystems, e.g., localized electrons for which explicit Coulomb interaction is taken in to account while LDA describes the rest of the wide band electrons. Instead of density, the corrected energy functional can be defined in terms of density matrix elements {ρ}, E LDA+U [nσ (r), {ρ σ }] = E LDA [nσ (r)] + E U [{ρ σ }] − Edc [{ρ σ }]. (2.22). nσ (r). where, is the charge density for electrons with spin σ . The first term is the Kohn-Sham energy functional, while the second term describes the HF correction to the functional, expressed as, 1 σ −σ E U [{ρ}] = ρm m ∑ { m, m |Vee |m , m ρmm 2 {m},σ σ σ +( m, m |Vee |m , m − m, m |Vee |m , m )ρmm ρm m }. (2.23). where the matrix elements of the screened coulomb interaction, Vee , among the correlated electrons can be expressed in terms of Slater integrals F k and the spherical harmonics. m, m |Vee |m , m = ∑ ak (m, m , m , m )F k. (2.24). k. where, ak (m, m , m , m ) =. k 4π ∗ lm|Ykq |lm lm |Ykq |lm. ∑ 2k + 1 q=−k. (2.25) 21.

(22) and 0 ≤ k ≤ 2l. Slater integrals are often paramatrized and expressed in terms of onsite Coulomb and exchange parameters, U and J, respectively. The energy functional given by LDA already consists of a contribution from the electron-electron interaction. The third term in Eq. 2.27, known as double counting, is hence needed to be subtracted from the total energy functional. 1 1 Edc [{ρ σ }] = Untot (ntot − 1) − J[n↑ (n↑ − 1) + n↓ (n↓ − 1)] 2 2. (2.26). where, ntot = Tr[ρ] and nσ = Tr[σ ρ]. A simplified expression of the LDA+U energy functional was proposed by Drudarev et al. [61], E LDA+U [nσ (r), {ρ σ }] = E LDA [nσ (r)] (U − J) σ σ σ + ) − ( ∑ ρmm ρm m )] ∑[(∑ ρmm 2 σ m m,m. (2.27). In this approach U and J do not enter individually while a difference between them has the key significance.. 2.5 DFT++ Method A different route to treat strongly interacting nature of the electrons was pioneered by Andersen [62], where lattice problem is mapped to an impurity model including essential physics of the many-body problem. The single impurity Andersen model recasts the problem by considering a single impurity embedded in a metallic host and can be described as, H = ∑ εk c†k ck + ∑ εidj di† d j k. +. i, j. 1 Ui jkl di† d †j dk dl + ∑(Vik c†k di + H.c..) 2 i,∑ j,k,l ik. (2.28). where the first term provides the energy of conduction electrons. εk is the band energy and ck , (c†k ) is the annihilation (creation) operators for Bloch states. εidj describes all the onsite energies of impurity including crystal field and spin-orbit coupling and i, j, k, l = (m, σ ) represents combined orbital and spin indices. di are the annihilation operators, while Ui jkl represents the local Coulomb interaction. The last term describes interaction with surrounding atoms. The hopping matrix element Vik appears in hybridization function, represented as : VikVk j (2.29) Δi j (ε) = ∑ k ε + iδ − εk 22.

(23) The hybridization function can be obtained from the local impurity Green’s function, (2.30) G−1 imp (ε) = ε + iδ − εc − Δ(ε) where εc is the static crystal field. An impurity Green’s function is expressed in terms of local orbitals by projection on to the Kohn-Sham Green’s function. In general Lehman representation the KS Green’s function can be expressed as, |ψnk ψnk | (2.31) GKS (ε) = ∑ ε nk + iδ − εnk where ψnk and εnk are the Bloch eigenfunctions and eigenvalues. A projection of full KS Green’s function on to local orbital, χm , yields, ∗. Gimp (ε) = mm. m Pm Pnk ∑ ε + iδ nk− εnk nk. (2.32). ∗. m = χ | ψ , Pm = ψ | χ are projections. where Pnk m nk nk m nk Provided the hybridization from DFT, the impurity problem is solved with an exact diagonalization solver.. 2.6 Electronic structure of periodic solids In a single particle scenario, provided by the KS equation, electrons feel an effective potential, VKS , which follows the lattice periodicity in solids. HKS (r)ψi (r) = [−. h¯ 2 2 ∇ +VKS (r)]ψi (r) = εi ψi (r). 2me. (2.33). where, VKS (r + R) = VKS (r),. (2.34). where R is the lattice periodicity. In a periodic crystal Bloch theorem implies the crystal momentum k to be a good quantum number and sets a boundary condition for the KS wavefunction, ψk , ψk (r + R) = eik.R ψk (r). (2.35). where ψk (r) is the Bloch wavefunction, ψk (r) = eik.r uk (r). (2.36). uk (r) follows the lattice periodicity. To solve the KS equation, one can expand single particle wavefunction in a complete basis set φi,k (r), which satisfies Bloch’s criteria for periodic boundary condition. ψnk (r) = ∑ ci,nk φi,k (r). (2.37). i. 23.

(24) From Eqns. 2.33 and 2.37, one can write,. ∑[hφi,k | HKS |φ j,k i − εnk hφi,k |φ j,k i]c j,nk = 0.. (2.38). j. where the first term in Eq. 2.38 describes the effective Hamiltonian matrix element and the second term represents the overlap matrix element. By solving a secular equation, we can obtain the eigenvalues εnk and the expansion coefficients ci,nk . det[hφi,k | H |φ j,k i − εnk hφi,k |φ j,k i] = 0.. (2.39). The transformation of many electron Schrödinger equation to a single particle one, namely KS equation has a big impact on electronic structure calculation which makes many problem tractable. In practice, the implementation still faces considerable amount of numerical difficulties due to distinctly different signatures of wavefunction in different regions of space. Closer to the nucleus, in the core region, the wavefunction is rapidly oscillatory due to strong attractive coulomb potential. Expansion in atomic like orbitals will provide a better description of wavefunction in this region. An extremely dense grid is required for the numerical accuracy. On the other hand, in the bonding region between two atoms, kinetic energy of the system is small and impact of chemical environment is significant. This makes the wavefunction flexible and respond strongly to the external changes. A complete basis set is thus needed to describe this flexibility of the wavefunction. Numerous techniques have been developed based on these requirements, which have their own limitations and advantages.. 2.7 Plane waves and pseudopotentials There are several possible choices for the basis set, e.g, gaussian functions, localized atomic like orbitals (muffin-tin orbitals, atomic orbitals), plane waves etc.. Results, presented in this thesis are obtained from plane wave based methods, where the eigenfunction can be expressed as: 1 ψnk (r) = ∑ cnk (G) × √ ei(k+G).r Ω G. (2.40). where cnk are the expansion coefficients of the wavefunction in a plane wave basis ei(k+G).r and G are √ the reciprocal lattice vectors. k is Bloch wave vector and the pre-factor 1/ Ω preserves the normalization of the wavefunction. The basis set is complete with infinite number of plane waves. In practice, plane wave expansion is truncated with an energy cut off, defined as h¯ 2 Ecut = 2m |k + Gmax |2 . 24.

(25) The key idea behind the introduction of the pseudopotential was to avoid an explicit core description and to get rid of the nodal structure of wavefunction. The core electrons screen the external potential and provide a softer potential to valence electrons. This can be described with a smoother wavefunction. This key concept of orthogonalized plane wave (OPW) method plays the instrumental role in the construction of pseudopotentials. A schematic representation of pseudopotential formalism is presented in Fig. 2.1.. Φ. Φ. ps. rc. ae. . ps. v. . Figure 2.1. Schematic representation of pseudopotential and corresponding pseudo wavefunction. A nodeless pseudo wavefunction φ ps (red line) matches with all electron wavefunction φ ae (blue line) at cut-off radius rc . This introduces a much softer pseudopotential v ps compared to all electron potential vae ∼ −Z r .. Ab-inito pseudopotential was introduced by Hamman, Schlüter and Chiang in 1979 [63] with norm conserving pseudo potentials (NCPP). The formalism involves generation of nodeless pseudo wavefunction, φlps which conserves the norm of a true all electron wavefunction, φlae , and preserves the all electron eigenvalues, εlae for each quantum numbers l, for a given reference configuration. An orbital dependent potential is generated by inverting the Shrödinger equation at εlae h¯ 2 2 ∇ + vl (r) − εlps ]φlps (r) = 0 2me 1 h¯ 2 2 ps ∇ φl (r) ⇒ vl (r) = εlps + ps φl (r) 2me [−. (2.41) (2.42). Unscreening of the orbital dependent potential vl , i.e, removing the Hartree and the exchange correlation contribution from the pseudo charge density 25.

(26) n ps (r) = ∑l fl |φlps |2 , for fl being the occupation number, yields orbital dependent pseudopotential vlps . Obtained pseudopotential in this formalism is semi local, which is good for transferability but computationally expensive. Kleinman and Bylander [64] re-casted the expression in a separable non local form which yields, v ps → vkb = vloc + vnl ps ps |δ vlps φlm. φlm δ vlps | vnl = ∑ ps ps φlm | δ vlps |φlm. lm. (2.43) (2.44). δ vlps = vlps (r) − vkb (r), vloc = vloc + vkb and vkb is an arbitrary local function. The achievement of NCPP is in its accuracy but with a cost of smoothness. The potentials are still considerably "hard" which require large energy cutoff. The norm conservation constraint is relaxed in ultra soft pseudopotentials [65], which introduces additional softness to it. To compensate the valence charge deficit, an augmentation charge, Qlm (r), is needed. For valence wavefunctions ψi , n(r) = ∑ |ψi (r)|2 + ∑ ∑ ψi | βl Qlm (r) βm | ψi. i. (2.45). i lm. where, ∗. Qlm (r) = φl∗ (r)φm (r) − φlus (r)φmus (r). (2.46). And a fully non-local form is used for Vnl , vus = vloc + ∑ Dlm |βl βm |. (2.47). lm. |βl are projector functions, usually expressed in spherical harmonics and Dlm are the matrix elements. |φl , |φlus are the atomic and the pseudo wavefunctions, respectively. Ultra soft potential has reduced dramatically the requirements of the size of basis set without sacrificing the accuracy, which makes it extremely useful for modern day calculations.. 2.7.1 The projector augmented wave (PAW) method A generalized method that admixes the instrumental idea behind the augmented wave method and the pseudopotential method elegantly, was introduced by Blöchl in 1994 [66]. The approach conserves the versatility of augmented wave method but with a simpler energy and potential independent basis, as adapted in pseudopotential method. The key idea behind the PAW method lies in a transformation ℑ that links a computationally convenient auxiliary wavefunction |ψñ to an oscillatory true all electron single particle KS 26.

(27) wavefunction |ψn . |ψn = ℑ|ψñ. (2.48). where, the index n is a cumulative index for band, k-point and spin. The transformed KS equation, from the variation principle with respect to auxiliary wavefunction, ℑ† Hℑ|ψñ = ℑ† ℑ|ψñ εn. (2.49). where H˜ = ℑ† Hℑ is the pseudo Hamiltonian and O˜ = ℑ† ℑ is the overlap operator. The aim is to avoid the nodal structure of a true wavefunction close to the nucleus, which otherwise is smooth beyond a certain distance from the core, rc . ℑ is needed to modify the wavefunction in the core region and thus can be defined as : ℑ = 1 + ∑ SR. (2.50). R. R is the atom site index and SR is the difference between auxiliary and true single particle KS wavefunction, which acts within an augmented space defined by a cutoff radius, rc ∈ R. The core wavefunction does not expand beyond augmented region and thus is treated separately. The energy and the electron density of core electrons in a solid or molecule can be approximated with an isolated atom calculations, which is addressed as frozen-core approximation. The operator ℑ, hence, acts on valence wavesfunction which can be expressed within augmented region as : ψ(r) = ∑ φi (r)ci. (2.51). i∈R. where, φi (r) are partial wave solutions for Schrödinger equation for an isolated atom and ci are the expansion coefficients. Each partial wave |φi is locally mapped to one auxiliary partial wave |φ˜i by transformation ℑ. |φi = (1 + SR ) |φ˜i. SR |φ˜i = |φi − |φ˜i. f or i ∈ R (2.52). As the transformation is local, this implicitly demands the partial waves |φi. and |φ˜i to be pairwise identical beyond rc ∈ R : φi (r) = φ˜i (r). f or i ∈ R and |r − RR | > rc,R. (2.53). Any arbitrary auxiliary wavefunction can be formed with the auxiliary partial wave basis within the augmented region, ˜ ˜ ψ(r) = ∑ φ˜i (r)ci = ∑ φ˜i (r) p˜i | ψ. i∈R. (2.54). i∈R. 27.

(28) where the projector operator | p˜i satisfies the following completeness and orthogonality relations.. ∑ |φ˜i p˜i | = 1. (2.55). i∈R. φ˜i | p˜ j = δi, j. f or i, j ∈ R. (2.56). The form of transformation operator can be expressed in terms of auxiliary and the true partial waves, (2.57) ℑ = 1 + ∑ |φ − |φ˜i p˜i | i. where the sum goes over the partial waves corresponding to all atoms. Thus the true wavefunction can be regained as, ˜ = |ψ. ˜ + ∑ |ψR1 − |ψ˜ R1. ˜ + ∑ |φ − |φ˜i p˜i |ψ. |ψ = |ψ. (2.58) i. R. where, ˜ |ψR1 = ∑ |φi p˜i |ψ. (2.59). ˜ |ψ˜ R1 = ∑ |φ˜i p˜i |ψ. (2.60). i∈R. i∈R. The achievement of this transformation is that wavefunction is separated out in to different parts of the space. Inside core region, wavefunction is expressed with partial waves, that have nodal structure, i.e, |ψ˜ R = |ψ˜ R1 , which yields the true wavefunction |ψR merging to |ψ˜ R1 . Beyond augmentation, the auxiliary wavefunction and true wavefunctions are identical, i.e, |ψR = |ψ˜ R . Within the frozen-core approximation, the expectation value of an operator can be obtained as: nc. A = ∑ fn ψn | A |ψn + ∑ φnc | A |φnc. (2.61). = ∑ fn ψ˜ n | ℑ† Aℑ |ψ˜ n + ∑ φnc | A |φnc. (2.62). n. n=1 nc. n. n=1. where fn is the band occupation number. Electron density is another important quantity in DFT as any observable is a functional of density either explicitly or implicitly. This can be achieved by taking an expectation value of the real space projector operator |r r|. n(r) = n(r) ˜ + ∑(n1R (r) − n˜ 1 (r)) n(r) ˜ =∑ n. 28. R ∗ fn ψ˜ n (r)ψ˜ n (r) + n˜ c (r). (2.63).

(29) n1R (r) =. ∑. Di, j φ j∗ (r)φi (r) + nc,R (r). ∑. Di, j φ˜ j∗ (r)φ˜i (r) + n˜ c,R (r). i, j∈R 1. n˜ (r) =. (2.64). i, j∈R. nc,R and n˜ c,R are core density and smooth auxiliary core density, respectively, for the particular atom. They match exactly after the core region. Di, j is the single site density matrix, defined as : Di j = ∑ fn hψ˜ n | p˜ j i h p˜i | ψ˜ n i. (2.65). n. The PAW method is accurate within DFT framework, without few approximations like frozen-core approximation, expansion of auxiliary wavefunction with finite plane waves, convergence of partial wave expansion etc.. In spite of these facts, the PAW method gives an access to the full wavefunction, charge and spin densities with a much simpler basis set. A significant part of the results, presented in this thesis, is calculated using Vienna Ab-initio Simulation Package (VASP) [67, 68], employing PAW method.. 29.

(30) 3. Magnetic properties. per atom (P % ) Magnetic moment B. So far a general approach towards the calculation of the electronic properties is discussed with the aid of density functional theory and beyond. The treatment of magnetism in a system is brought in by breaking the spin symmetry of the system. The collective behavior of electron’s intrinsic spin moment and orbital moment, due to orbital motion describes the magnetism in a system. In a quantum mechanical system, spin moments are described by the expectation value of spin operator Sz and the orbital moment is calculated from expectation value of the angular momentum operator Lz in presence of spin-orbit coupling. An exchange interaction between spins defines the magnetic behavior of a material and provide parallel (ferromagnetic) or antiparallel (antiferromagnetic) alignment of spins. The ab-initio density functional theory has been successful. Expt. bcc fcc hcp. 2. 1. Fe. alloy. Co. Figure 3.1. Calculated and experimental spin magnetic moments of Fe-Co alloys in body centered cubic (bcc) , face centered cubic( fcc) and hexagonal closed packed (hcp) structures. The violet, green and red lines correspond to theoretical calculations for bcc, fcc and hcp phases, respectively. The experimental observations, presented by violet circle, green triangle, and red diamond for bcc, fcc and hcp phases, respectively agree perfectly with theoretical observations. The figure is redrawn from reference [69].. to describe the magnetic properties of materials in the form of bulk, surfaces 30.

(31) and even lower dimensional structures. In Fig. 3.1, an almost perfect agreement between theoretically calculated and experimentally observed magnetic moments can be seen for all Fe-Co alloy compositions. This establishes robustness of DFT to describe magnetic properties of a system. Orbital moments can also be extracted from element specific spectroscopy measurements, like X-ray magnetic circular dichroism (XMCD) with a sum rule analysis and DFT has also succeeded to reproduce the orbital moment to a satisfactory level. In this chapter, a particular focus will be on the mechanisms that determine the magnetic behavior of the systems of interest within the scope of this thesis. The orbital resolved exchange interaction is discussed, which is used to characterize surface molecular magnetic interactions. The spin dipole moment and the magnetic anisotropy on the other hand are crucial for magnetic molecules, in terms of proper analysis of magnetic states and spontaneous quantization direction, which are essential parts of this thesis.. 3.1 Exchange interactions and orbital dependence The exchange integral, that appears in the Heisenberg model can be extracted from ab − initio simulations by considering the total energies properly. H =−. 1 J i j sî · sˆ j ∑ 2 i= j. (3.1). where, J i j is the exchange integral and sî is the direction of the ith spin. The usual approach to extract this parameter involves explicit total energy calculations for explicit magnetic states. The method, proposed by Lichstenstein et al. [70], avoids this route and allows to find exchange parameter from single magnetic state. In this way J i j can be expressed in terms of two site Green’s function and single site potential functions, Ji j =. 1 Tr 4π. EF. j,↑ ji,↓ Im[δi (E)Gimm (E)δ j (E)Gm m (E)]dE,. (3.2). m, m being the magnetic orbital quantum numbers (ml ) while Gi j,σ denotes the Green’s function between i and j atoms for the spin channel σ (σ =↑, ↓). δi = Pi,↑ − Pi,↓ is the spin dependent potential function at site i which yields a local exchange splitting. The potential function corresponding to each l can be expressed in a parametrized form and can be written [71] as, Pli (E) = (E −Cli )μl Sl. (3.3). where, Cli , is the center of l band, Sl is the Wigner-Seitz radius and μl is the band mass. Hence, δi can be written as, δi = δCli μl Sl. (3.4) 31.

(32) δCi is the local exchange splitting, which can be expressed in terms of the local moment mi and Stoner parameter I. δCi = mi I. (3.5). To have more insight of the exchange mechanism, the orbital dependent contribution of exchange parameter can also be extracted by avoiding the trace over ml , ij Jmm . 1 = 4π. EF. j,↑ ji,↓ Im[δi (E)Gimm (E)δ j (E)Gm m (E)]dE,. (3.6). δi (E) is then proportional to a constant shift provided by the local exchange splitting. More over, the orbital resolved character of the exchange is obtained by integrating upto Fermi energy EF , which is obtained from a self-consistent calculation. This can as well be defined as a function of EF by limiting ourselves in a rigid band model, which yields, ij Jmm (EF ). ∝. EF. j,↑ ji,↓ Im[Gimm (E)Gm m (E)]dE,. (3.7). where, the integrand is essentially the susceptibility [72]. The orbital projected two-site Green’s function can be derived by taking projection of Kohn-Sham Green’s function onto spherical/tesseral harmonics around each atom in a similar approach mentioned in section 2.5. The exchange integral, obtained in this way depicts not only orbital character of it but also anticipate the behaviour of J i j for charge doped systems, within the limitation of the rigid band model.. 3.2 Spin dipole moment The prediction of dichroism occurrence in X-ray adsorption spectra (XAS) by Erskine and Stern [73] opened up a new route to have a better insight of the element specific properties. The X-ray magnetic circular dichroism (XMCD) measures the difference in XAS spectra for left and right circularly polarized light for L2,3 edges of 3d transition metal compounds and M4,5 edges of rare earth compounds. The element specific orbital ( Lz ) and spin magnetic moment (me f f ) can be extracted from XMCD spectra with the analysis of sum rule, introduced by Thole and Carra [74, 75]. . + − L2,3 (μ − μ )dE (μ + + μ 0 + μ − )dE. Lz = −2nh . (3.8). L2,3. . me f f = 2 Sz + 7 Tz = 3nh. (. L3 (μ. + − μ − )dE − 2. . L2,3. 32. . + − L2 (μ − μ )dE) (μ + + μ 0 + μ − )dE. (3.9).

(33) where μ + , μ − , μ 0 are the absorption cross-sections for right, left and linear polarized light along z-axis and nh is the number of holes. The problem with the sum rule to find the expectation values of the spin effective moment is the unknown value of spin dipole moment( Tz ). The contributions from Tz. were mostly neglected, mainly due to its extremely small contribution to the effective moment for high symmetric structures. Recent studies [76] suggest that even for octahedral systems, one can expect large magnetic dipole contribution. Along with that, this contribution can even lead to a contribution comparable to isotropic spin moment in reduced symmetry structures, like surfaces, molecules, clusters etc.. The origin of the spin dipole moment lies in the asphericity in the spindensity, which is predominantly governed by the crystal field effect and the spin-orbit interaction. For heavy rare earth elements the spin orbit interaction is much stronger compared to the crystal field and band formation is much weaker, hence, behave like free ions. In transition metal (TM) compounds, on the other hand, spin orbit interaction effect is much weaker compared to the crystal field effect. A symmetric charge density of free TM ion is thus broken in presence of neighboring point charges. Eigenfunction of the crystal field Hamiltonian thus can no more be described with pure spherical harmonics. A linear combination of opposite magnetic orbital quantum numbers (ml ) becomes the eigenfunction, quenching the orbital moment. The charge density around the TM ion then has a multipolar expansion. The charge quadrupole appears in spin dipole expression as shown in Eqn. 3.10 below. Considering only the crystal field for the time being, a system with cubic symmetry can never have quadrupole in the charge expansion due to the symmetry, thus can not possess a spin dipole moment. But in the reduced symmetry structures this can contribute significantly in effective moment. The spin dipole operator can be expressed as [77] T = ∑ Q(i) s(i). (3.10). i. where, si and Q(i) are the spin moment and the charge quadrupolar tensor for ith electron. Q(i) can be described as : (i). (i) (i). Qαβ = δαβ − 3ˆrα rˆβ. (3.11). Every component of T can be written in second quantized form as : T± = Tx ± iTy = ∑ T ± a†γ aγ Tz = ∑ T z a†γ aγ γγ. . γγ. γγ. . γγ. (3.12). where a†γ (aγ ) are creation (annihilation) operators. The matrix elements can be written as, √ √ (3.13) T ± = γ| c20 s± − 6c2±2 s∓ ± 6c2±1 sz |γ. γγ. 33.

(34) Tz γγ. = γ| −. 3 2 c s+ + 2 −1. . 3 2 c1 s− − 2c20 sz |γ. 2. (3.14). where |γ = |lm, σ , lm being the orbital quantum numbers and σ is the spin.. 4π Ylm . clm are renormalized spherical harmonics, clm = 2l+1 The expectation value of Tz can be obtained as Tz =Tr[ρTz ], where ρ is the density matrix. Without spin-orbit coupling, the matrix elements of T z can γγ. contribute via 2c20 sz . The triangular rule and the expectation value of sz operator fix the criteria that only diagonal elements of the density matrix (ρ) will contribute to the value of Tz . In this circumstance, there are two conditions to obtain a large magnetic dipole moment : (1) Non-zero magnetic moment and (2) non-equivalent charge distribution on all orbitals. The first condition is required to have a non equivalent charge distribution in different spin channels. As Tr[Tz ] is zero, the second condition requires different occupations for different orbitals. The angular dependence of the XMCD signal (made by photon beam with surface normal), gives quite useful information regarding the anisotropic behavior of system properties, such as magnetic anisotropy. Resolving the angular dependence of the spin dipole operator hence becomes important. With the recipe provided by Thole and van der Laan [78], one can find the relation between the directions of sample magnetization M, the polarization P of photon beam and the surface normal ε with XMCD intensity. A schematic diagram of particular XMCD measurement is presented in Fig. 3.2, where particular relations between M, P, ε are specified. The angle dependence of the spin dipole operator can be expressed as, 7T (ε, M, P) =. 1 7Tz [cos(τ) + 3cos(τ − 2ν)] 4. (3.15). where τ = M, P, ν = ε, M. In a special case when M || P and both parallel to the photon propagation vector (k), the angular dependence of the operator becomes: 7T (τ) =. 1 7Tz (3cos2 (τ) − 1) 2. (3.16). The expectation value of the operator becomes zero at an angle of 54.5◦ , known as the magic angle, where effective moment consists of only its spin origin. Deviation of the parallel relation between M and P introduces much more complex angular dependence, which can explicitly be calculated by taking proper description of the spin quantization axis and the angular form of spin-dipole operator. This gives us a playground to manipulate the effective mangetization by controlling the spin dipole contribution. For a low symmetry system the spin dipole moment can be quite large and a factor of 7 in Eqn. 3.24, hence, makes this contribution considerable. Moreover, the sign of Tz can also be opposite 34.

(35) ε. -. P. ε . + +. k. M. P M. Surface. Figure 3.2. Sketch of an experimental setup for XMCD measurement. M, P are the magnetization direction and polarization of photon beam, respectively. ε is the surface normal.. to the spin moment, affecting the magnetization significantly. Transition metal clusters (specifically Fen and Con clusters, with n number of atoms), for example, have been studied [79] in both gas phase and on surface (Ni(001) and Au(111)). Apart from the comparable values of the spin moment and 7 Tz , a non monotonous variation of 7 Tz values are observed for each atom in the cluster depending on the site and its local coordinations. A similar behavior is observed in one of our studies of free Fen clusters and Fen clusters on vacancy defected graphene. 7 Tz can reach upto 36% of spin moment for an individual moment depending on the coordinations and the structure of the cluster. Sign of Tz also varies, which proves that this contribution can certainly not be ignored. The reduction of symmetry in bulk systems results in considerable contribution in Tz . Verwey transition in magnetite has recently been studied for magnetite thin films and nano particles by us where Tz controls the magnetic transition as the system reduces its symmetry from cubic to monoclinic. Spin dipole moments in the transition metal centered single molecular magnets can change effective moments dramatically which is going to be discussed in details in later chapters.. 3.3 Magnetic anisotropy energy (MAE) Magnetic anisotropy is a phenomenon that dictates the anisotropic preference of magnetization axis in a system. Internal free energy of a system is lowered for a spontaneous spin quantization axis (easy axis), compared to another (hard 35.

(36) axis). Magnetic anisotropy energy is the measure of this anisotropic magnetic behaviour of a system. This breaking of the rotationally isotropic spin quantization axis originates from magnetic dipole-dipole interaction and spin-lattice coupling, namely spin orbit coupling. The former one introduces energy the dependence of magnetization orientation on the shape of the system and hence referred as shape anisotropy. The later one is an intrinsic contribution, which determines the energy dependence of magnetization with respect to crystalline axes, thus called mangetocrystalline anisotropy. Because of the dependence of magnetic anisotropy energy on magnetization axis, it can be expanded in the power series of direction cosine of magnetization vector, M (Mx , My , Mz ). For example, for uniaxial anisotropy MAE can be expanded as: E = K0 + K1 (Mx2 + My2 ) + K2 (Mx2 + My2 )2 + K3 (Mx2 + My2 )3 + .... (3.17). where K’s are the anisotropy constants. In spherical co-ordinate system, (Mx2 + My2 ) = sin2 (θ ), with θ being the angle between M and z axis. This will take the following form, E = K0 + K1 sin2 (θ ) + K2 sin4 (θ ) + K3 sin6 (θ ) + .... (3.18). Only even powers of direction cosines appear, as MAE is invariant under time-reversal and magnetization itself changes its sign with time reversal. In general, magnetic anisotropy energy follows the system symmetry. For hexagonal crystals, the major contributor is K1 , while for cubic crystal the first anisotropy constant, that appears is K2 .. 3.3.1 Spin-orbit interaction The magnetocrystalline anisotropy originates from the relativistic nature of the electrons when the electron spins start to interact with magnetism due to the orbital motion of electrons around the nucleus, namely spin orbit (SO) interaction. Relativistic nature of electrons is described by the Dirac equation, which in the low velocity region (correction of order v2 /c2 ) reduces to : H =. p2 e¯h e¯h2 p4 − eΦ − 3 2 + 2 2 divE + 2 2 σ · (E × p) 8m

(37) c 8m c. 4m c

(38). 2m. (3.19). where p, Φ, and E are the momentum operator, potential and the electric field, respectively and e is the electron charge. The first two terms represent the non-relativistic Hamiltonian, consisting of the kinetic energy and the external potential, the third term is the correction over kinetic energy and the fourth one is the Darwin term. The later two are spin independent and added to the nonrelativistic part of the Hamiltonian yielding ‘scalar relativistic’ Hamiltonian. 36.

(39) The last term is spin orbit interaction, Hso . Close to the nucleus, the potential is spherical and thus the electric field can be written as E=−. r dΦ r dr. (3.20). and the spin-orbit Hamiltonian takes the following form, Hso = −. e¯h dΦ σ · (r × p) 4m2 c2 r dr e¯h dΦ L·S =− 2 2 4m c r dr = ξL·S. (3.21). where ξ is the spin-orbit coupling constant, obtained by taking the radial average of ξ (r) = − 4me¯2hc2 r dΦ dr . In transition metal compounds, the ligand field strength is of the order of 1 eV while spin-orbit coupling constant is around 50-100 meV [80]. For metal ions, hybridization provides the similar effect and bandwidth is much larger compared to spin orbit coupling. A natural choice, hence, is to consider SO interaction as perturbation. The perturbative spin-orbit coupling Hamitonian can be described as : Hso = ξ L · S = ξ (Lx Sx + Ly Sy + Lz Sz ). (3.22). For a magnetization direction n, ˆ described by the polar angles (θ , φ ), the general form of SO Hamiltonian, Hso (n) ˆ = ξ (Lx {Sx cosφ cosθ − Sy sinφ + Sz cosφ sinθ } +Ly {Sx sinφ cosθ + Sy cosφ + Sz sinφ sinθ } +Lz {−Sx sinθ + Sz cosθ }). (3.23). So, the spin-orbit contribution to the energy, from 2nd order perturbation theory, for magnetization vector along nˆ : occ Eso (n) ˆ = − ∑ ∑ nocc ikms n jkm s. ×. ki j {m} ms| Hso (n) ˆ |m s m s | Hso (n) ˆ |ms. εk js − εkis. (3.24). where i, j correspond to occupied and unoccupied states in the basis of |lms , where l, m, s are orbital, magnetic and spin quantum numbers respectively and {m} = {m, m , s, s }. εkis and nocc iks are energy eigenvalues and occupations at the sampling point k. From Eqn. 3.24, it is obvious that the SO contribution increases as the energy difference between occupied and unoccupied energy eigenvalues decreases, which gives us an opportunity to manipulate the orbital 37.

(40) Hext. z’. z. θ. yy’ y x’ x. x. ϕ. Figure 3.3. Rotated spin quantization frame (x ,y ,z ) and its relation with coordinates system (x,y,z) of sample. Spin quantization axis ( n) ˆ is varied with an external field and specified by θ and φ ) in the (x,y,z) frame [81].. character and energy positions of valence and conduction bands to control MAE. Thus energy difference between two different magnetization directions nˆ 1 and nˆ 2 induced by SO coupling, is ΔE = Eso (nˆ 1 ) − Eso (nˆ 2 ). (3.25). From Eqn. 3.23, if we assume a significant contribution from Lz Sz cosθ and substitute that in Eqn. 3.24 and 3.25, the 2nd order term in the phenomenological expansion of MAE in Eqn. 3.18 is obtained. The higher order terms in Eqn. 3.18 can be realized with higher order perturbation theory. The advantage of perturbative approach is that one can achieve anisotropy constant without going through the total energy calculation along each magnetization direction. On the other hand, one also compromises with the accuracy as the higher order contributions are neglected. Also for the degenerate levels, this approach fails. But this analysis provides a good insight along with an acceptable accuracy in anisotropy energy.. 38.

(41) Applications.

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