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The Pseudopotential-Density-Functional Method Applied to Semiconducting Crystals

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The Pseudopotential-Density-Functional Method Applied to Semiconducting Crystals The pseudopotential-density-functional method applied to semiconducting crystals Citation for published version (APA): Denteneer, P. J. H. (1987). The pseudopotential-density-functional method applied to semiconducting crystals. Technische Universiteit Eindhoven. https://doi.org/10.6100/IR263952 DOI: 10.6100/IR263952 Document status and date: Published: 01/01/1987 Document Version: Publisher’s PDF, also known as Version of Record (includes final page, issue and volume numbers) Please check the document version of this publication: • A submitted manuscript is the version of the article upon submission and before peer-review. There can be important differences between the submitted version and the official published version of record. People interested in the research are advised to contact the author for the final version of the publication, or visit the DOI to the publisher's website. • The final author version and the galley proof are versions of the publication after peer review. • The final published version features the final layout of the paper including the volume, issue and page numbers. Link to publication General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. • Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain • You may freely distribute the URL identifying the publication in the public portal. If the publication is distributed under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license above, please follow below link for the End User Agreement: www.tue.nl/taverne Take down policy If you believe that this document breaches copyright please contact us at: [email protected] providing details and we will investigate your claim. Download date: 05. Oct. 2021 THE PSEUDOPOTENTIAL-DENSITY-FUNCTIONAL METI-tOD APPLIED TO SEMICONDUCTING CRYSTALS P.J.H. DENTENEER THE PSEUDOPOTENTIAL-DENSITY -FU NCTIONAL METHOD APPLIED TO SEMICONDUCTING CRYSTALS PROEFSCHRIFT ter verkrijging van de graad van doctor aan de Technische Universiteit Eindhoven, op gezag van de rector magnificus, Prof. dr. F.N. Hooge, voor een commissie aangewezen door het college van de!kanen in het openbaar te verdedigen op vrijdag 5 juni 1987 te 16.00 uur door PETER JAN HENDRIK DENTENEER geboren te Brunssum Druk: Dissertatiedrukkerij WiblO, Helmond. Dit proefschrift is goedgekeurd door de promotoren: Prof.dr. W. van Haeringen en Prof.dr. J.T.L. Devreese Aan mijn ouders White timorous knowtedge stands considering, auàacious ignorance hath done the deed. Samuel Daniel ABSTRACf A detailed description is given of the pseudopotential-density­ functional metbod to accurately calculate from first principles the electronic and atomie structure of the ground state of crystals. Density-functional theory necessitates the self-consistent solution of the one-electron Schrödinger equation, wbereas pseudopotentials allow for tbe inclusion in the calculation of valenee electrans only and for the expansion of tbe functions of interest in plane waves. All necessary formulae are given to obtain tbe self-consistent density of valenee electrons, screening potential, and energy of tbe ground state. P.articular emphasis is placed on tbe application of tbe technique of "special points in tbe first Brillouin zone" to perform necessary integrations over reelprocal space. The exploi tation of space-group symmetry in tbe solution of tbe Scbrödinger equation is discussed and illustrated for tbe case of expansion of tbe wave function in plane waves. Furtbermore, characteristic features of tbe calculational scbeme connected witb self-consistency and finite cutoffs are pointed out and utilized to reduce the computational work. Results of calculations for silicon. diamond, and two structurally extreme polytypes of silicon carbide illustrate tbe metbod and techniques described. Finally, tbe applicability of tbe metbod to surfaces, interfaces, superlattices, and polytypes is briefly discussed. page Chapter 1 Introduetion 1 Chapter 2 The pseudopotential-density-functional metbod in momentum spa.ce 7 2.1 Density-functional theory 8 2.2 Pseudopotential theory 13 2.3 Momenturn-space formalism for self-consistent pseudopotentlal calculations 20 2.3.1 Totalenergyin direct space 21 2.3.2 Total energy in momenturn space 24 2.3.3 Self-consistent solution of Kohn-Sham equations in momenturn space 31 2.4 Matrix elements of norm-conserving pseudopotentials 36 2.5 Cutoff parameters 40 Chapter 3 Special points in the first Brillouin zone 45 3.1 General theory and application to charge-density calculations 46 3.2 Description and computerization of the Monkhorst-Pack scheme 53 3.2.1 MP-sets for face-centred cubic lattices 54 3.2.2 MP-sets for hexagonal lattices 57 3.3 Converganee of energy-band integrations using special points 60 3.4 Equivalent special-point sets for structurally different crystals 64 Chapter 4 Exploitation ol crystal symmetry lor electronic energy bands and states 71 4.1 Construction of symmetrised plane waves: theory 72 4.2 Construction of symmetrised plane waves: illustrative examples 78 4.3 Unfolding of symmetrised electron states 85 Chapter 5 Beerets de cuisine: capita selecta 89 5.1 Non-self-consistency correction 89 5.2 Numerical noise on total-energy-versus-volume curve 93 5.3 Accuracy of energy-band integrations using special points 98 Chapter 6 ApPlication to silicon, diamond, and silicon carbide 101 6.1 Self-consistent valenee-charge density of silicon and diamond 104 6.2 Ground-state properties of silicon and cubic SiC 110 6.3 Valenee-charge density and band structure of cubic SiC 120 6.4 Accurate energy differences and equivalent special-point sets 124 6.5 Wurtzite SiC: mapping and relaxation 128 Chapter 7 Outlook: Towards a fundamental deseription of erystals with limited periodicity 135 Relerences 141 Samenvatting 146 Curriculum Vitae 14S CRAPTER 1 INTRODUCfiON The study of the condensed state of matter -solids and liquids­ constitutes one of the largest subfields of modern physics. In view of its link to society (materials science), the importance of this field is obvious. From a more scientific point of view the purpose of solicl-state physics is. of course, to understand the properties of solids starting from basic notions; Why is one solid different from another? An increased understanding of the properties of solids immediately leads to a more systematic search for materials that have desirabie properties. There is interest, for instance, in (i) solids that are as ductile and malleable as common metals, but are corrosion­ resistant, (ii) solids with the hardness and chemica! inertness of diamond, but not as costly, (iii} semiconductors with a band gap that is direct and corresponds toa desirabie frequency (color). for use in light-emitting diodes, lasers, and photo-detectors, (iv) semicon­ ductors with high electron mobilities, which have a higher potential eperating speed in electronic devices, to mention a few. To this end, experiments are needed to determine the properties of solids. We also need theories that tell us why solids have the properties they have. These theories should preferably start from elementary ingredients. Regarding solids, these elementary ingredients are the properties of the nuclei, the electrans, and their inter­ actions. The latter category of theories are called first-principLes theories or ab-initia theories. Quanturn mechanics and statistica! mechanics are such theories, which should in principle suffice to determine the properties of solids from first-principles. In practice, however. these general theories alone almost invariably generate a calculational scheme that is too complex to actually carry out. By making approximations that are not too drastic, it is possible to obtain theories that may still be called first-principles theories, but lead to practical schemes of calculation. The approximations of course must be carefully investigated for their appropriateness and should not vialate the basic laws of quanturn mechanics and statistica! mechanics. Theories that need experimental data as input. e.g., in 1 order to determine the values of parameters in the theory, are called empirica! theories. Such theories are in fact less fundamental wi th regard to predictive purposes. In what is called the scientific method, theories, irrespective of whether they are empirica! or start from first principles, are first tested to reproduce the results of experiments and are subsequently tested to pred.iet the resul ts of experiments. Only in the last decade i t bas become possible to employ first­ principles theories in the computation of solid-state properties and to reliably predict experiments. This is partly due to the steady advance made in the development of' theories, the most important reason, however, lies in the increase in computing power of the generations of digital computers that rapidly sneeeed each other. The latter development bas led some people to discern a third way to study physics, in-between expertmental and theoretica! physics, namely that of computational ph.ysics [2]. Al though computational physics bas descended from theoretica! physics historically, its approach is more akin
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