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Vibrations in solids: From first principles lattice dynamics to high temperature phase stability
Linköping University, Department of Physics, Chemistry and Biology, Nanostructured Materials. Linköping University, Faculty of Science & Engineering.
2015 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

In this thesis I introduce a new method for calculating the temperature dependent vibrational contribution to the free energy of a substitutionally disordered alloy that accounts for anharmonicity at high temperatures. This method exploits the underlying crystal symmetries in an alloy to make the calculations tractable. The validity of this approach is demonstrated by constructing the phase diagram via direct minimization of the Gibbs free energy of a notoriously awkward and technologically important system, Ti1-xAlxN. The vibrational entropy including anharmonic effects is shown to be large and comparable to the configurational entropy at high temperatures, and with its inclusion, the theoretical miscibility gap of Ti1-xAlxN is reduced from 6560 K to 2860 K, in line with atom probe experiments. A similar treatment of Zr1-xAlxN and Hf1-xAlxN alloys suggests that mass disorder has a minimal effect on phase stability compared with chemical ordering. My method is also capable of demonstrating that Hf1-xAlxN, which is dynamically unstable at room temperature, is stabilised at high temperatures. Moreover I develop a new method of computing temperature dependent elastic constants for alloys from their phonon spectra, and show that for Ti1-xAlxN, the elastic anisotropy is found to increase with temperature, helping to explain the spinodal decomposition.

The effects of lattice dynamics on phase stability, mechanical, magnetic and transport properties on other materials are also examined. Four specific systems are discussed in detail. Firstly, in the case of CrN, lattice vibrations are shown to decrease the antiferromagnetic to paramagnetic phase transition temperature from 500 K to 380 K, in line with experimental evidence. Secondly, a temperature/pressure induced phase transition in AlN becomes much more facile than in the quasiharmonic approximation, and the thermal conductivity of the rocksalt phase is shown to be much lower than that of the wurtzite phase, as a result of the increased anharmonicity in the rocksalt structure. Thirdly, the temperature dependence of elastic constants of TiN becomes more isotropic as the temperature increases. Finally, iron carbides are evaluated as potentially important phases at the Earth’s core; specifically, calculating the Gibbs free energy of a recently discovered orthorhombic phase of Fe7C3 demonstrates that it is not stable relative to the known hexagonal phase at extreme pressure and temperatures.

Abstract [sv]

I denna avhandling introducerar jag en ny metod för att beräkna de temperaturberoende vibrationernas bidrag till den fria energin för oordnade legeringar, vilket kan förklara anharmoniska effekter vid höga temperaturer. För att göra beräkningarna mer lätthanterliga utnyttjar den här metoden den inneboende strukturella symmetrin för kristaller i en legering. Hur lämpligt det här tillvägagångssättet är visas genom att konstruera ett fasdiagram genom en direkt minimering av Gibbs fria energi, vilket görs för ett notoriskt besvärligt (men teknologiskt viktigt) system, nämligen Ti1-xAlxN. Jag visar att vibrationsentropin, inkluderat de anharmoniska effekterna, är stor och jämförbar med konfigurationsentropin vid höga temperaturer. Vidare visar jag att man, genom att inkludera dem, kan reducera Ti1-xAlxNs teoretiska löslighetslucka från 6560 K till 2860 K, i linje med de experiment som gjorts med atomsondstomografi. Genom att behandla legeringarna Zr1-xAlxN och Hf1-xAlxN på samma sätt framstår det som troligt att en oordning i massan har en minimal effekt på fastemperaturen jämfört med en kemisk sammansättning. Min metod kan även visa att Hf1-xAlxN, som är instabil vid rumstemperatur, stabiliseras vid höga temperaturer. Dessutom har jag utvecklat en ny metod för att beräkna de temperaturberoende elastiska konstanterna för legeringar utifrån deras fononspektrum, och visar detta för Ti1-xAlxN. Den elastiska anisotropin visas öka avhängigt av temperaturen, vilket förklarar det spinodala sönderfallet.

Vidare undersöks gitterdynamikens effekt på andra materials fasstabilitet, samt deras mekaniska, magnetiska och transportegenskaper. Fyra specifika system diskuteras i detalj. Den här studien visar, för det första, att vibrationerna i gittret för CrN sänker övergångstemperaturen mellan den antiferromagnetiska och paramagnetiska fasen från 500 K till 380 K, vilket överensstämmer med existerande experimentella bevis. För det andra visar den att en fasövergång inducerad av temperatur eller tryck i AlN blir betydligt smidigare än i den kvasiharmoniska approximeringen, och att värmeledningsförmågan i bergsaltsfasen blir betydligt lägre som ett resultat av den ökade anharmoniciteten i bergsaltsstrukturen. För den tredje blir temperaturberoendet av TiNs elastiska konstanter mer isotropisk när temperaturen ökar. Slutligen utvärderas järnkarbider som potentiellt viktiga faser i jordkärnan; mer specifikt visas den, genom att beräkna Gibbs fria energi för en nyligen upptäckt ortorombisk fas av Fe7C3, inte vara stabil relativ till den redan kända hexagonala fasen vid tryck och temperatur på extrema nivåer.

Abstract [de]

In der vorliegenden Arbeit stelle ich eine neu entwickelte Methode zur Berechnung der temperaturabhängigen Vibrationsbeiträge zur freien Energie von ungeordneten Legierungen unter Berücksichtigung nicht- harmonischen Verhaltens bei hohen Temperaturen vor. Diese Methode nutzt bei der Berechnung die in der jeweiligen Legierung vorhandenen Kristallsymmetrien aus. Die Gültigkeit dieses Ansatzes wird durch die Konstruktion des Phasendiagrams des technologisch wichtigen komplexen Systems Ti1-xAlxN unter direkter Minimierung der freien Gibbs Energie gezeigt. Es zeigt sich, dass die Vibrationsentropie durch Berücksichtigung nicht-harmonischer Effekte in ihrer Größe vergleichbar der Konfigurationsentropie bei hohen Temperaturen ist, und dass durch ihre Miteinbeziehung die theoretische Mischungslücke von 6560 K auf 2860 K reduziert wird, in Einklang mit experimentellen Atomsonden Messungen. Vergleichbares Anwenden der Methode auf die Legierungen Zr1-xAlxN und Hf1-xAlxN zeigt, dass eine Unordnung der atomaren Masse im Vergleich zu chemischer Unordnung nur einen minimalen Effekt auf die Stabilität der Phase hat.

Die von mir entwickelte Methode ist des Weiteren im Stande zu zeigen, dass Hf1-xAlxN (dynamisch instabil bei Raumtemperatur) sich bei hohen Temperaturen stabilisiert. Weiterhin wurde eine neue Methode zur Berechnung von temperaturabhängigen elastischen Konstanten aus Phonon Spektren von Legierungen entwickelt und gezeigt, dass für Ti1-xAlxN die elastische Anisotropie mit der Temperatur ansteigt und so die spinodale Entmischung erklärt wird. Die Effekte von Gitterdynamik auf Phasenstabilität, mechanische-, magnetische- und Transport Eigenschaften werden zusätzlich für vier spezifische Systeme im Detail untersucht, und es wird gezeigt, dass

  • in Einklang mit experimentellen Ergebnissen die Gitterschwingungen in CrN die Temperatur des Phasenüberganges von antiferro- magnetischer- zu paramagnetischer Phase von 500 K auf 380 K reduziert;
  • mit der hier vorgestellten Methode ein Druck/Temperatur induzierter Phasenübergang in AlN müheloser beschrieben werden kann als in der quasiperiodischen Näherung. Weiterhin wird gezeigt dass die thermische Leitfähigkeit in AlN als Resultat aus erhöhter nicht-Harmonizität in der NaCl-artigen Gitterstruktur herabgesetzt wird;
  • die Temperaturabhängigkeit der elastischen Konstanten von TiN mit steigender Temperatur mehr und mehr isotrop wird;
  • die mit der vorgestellten Methode berechnete freie Gibbs Energie der kürzlich entdeckten orthorhombischen Phase Fe7C3 nahelegt, dass sie gegenüber der bekannten hexagonalen Phase nicht stabil ist.
Place, publisher, year, edition, pages
Linköping: Linköping University Electronic Press, 2015. , 94 p.
Series
Linköping Studies in Science and Technology. Dissertations, ISSN 0345-7524 ; 1718
National Category
Physical Sciences
Identifiers
URN: urn:nbn:se:liu:diva-122949DOI: 10.3384/diss.diva-122949ISBN: 978-91-7685-911-7 (print)OAI: oai:DiVA.org:liu-122949DiVA: diva2:875208
Public defence
2015-12-04, Schrödinger, Fysikhuset, Campus Valla, Linköping, 10:15 (English)
Opponent
Supervisors
Available from: 2015-11-30 Created: 2015-11-30 Last updated: 2015-12-01Bibliographically approved
List of papers
1. Temperature dependence of TiN elastic constants from ab initio molecular dynamics simulations
Open this publication in new window or tab >>Temperature dependence of TiN elastic constants from ab initio molecular dynamics simulations
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2013 (English)In: Physical Review B. Condensed Matter and Materials Physics, ISSN 1098-0121, E-ISSN 1550-235X, Vol. 87, no 9Article in journal (Refereed) Published
Abstract [en]

Elastic properties of cubic TiN are studied theoretically in a wide temperature interval. First-principles simulations are based on ab initio molecular dynamics (AIMD). Computational efficiency of the method is greatly enhanced by a careful preparation of the initial state of the simulation cell that minimizes or completely removes a need for equilibration and therefore allows for parallel AIMD calculations. Elastic constants C11, C12, and C44 are calculated. A strong dependence on the temperature is predicted, with C11 decreasing by more than 29% at 1800 K as compared to its value obtained at T=0 K. Furthermore, we analyze the effect of temperature on the elastic properties of polycrystalline TiN in terms of the bulk and shear moduli, the Young's modulus and Poisson ratio. We construct sound velocity anisotropy maps, investigate the temperature dependence of elastic anisotropy of TiN, and observe that the material becomes substantially more isotropic at high temperatures. Our results unambiguously demonstrate the importance of taking into account finite temperature effects in theoretical calculations of elastic properties of materials intended for high-temperature applications.

Place, publisher, year, edition, pages
American Physical Society, 2013
National Category
Natural Sciences Engineering and Technology
Identifiers
urn:nbn:se:liu:diva-78754 (URN)10.1103/PhysRevB.87.094114 (DOI)000316791600001 ()
Note

Funding Agencies|Swedish Research Council|621-2008-5535621-2011-4426|Swedish Foundation for Strategic Research (SSF) programs|10-0026|project Designed Multicomponent Coatings (MultiFilms)||Erasmus Mundus doctoral program DocMase||Ministry of Education and Science of the Russian Federation within the framework of Program Research and Pedagogical Personnel for Innovative Russia|14.B37.21.089010.09.2012|

Available from: 2012-06-20 Created: 2012-06-20 Last updated: 2017-12-07Bibliographically approved
2. Vibrational free energy and phase stability of paramagnetic and antiferromagnetic CrN from ab initio molecular dynamics
Open this publication in new window or tab >>Vibrational free energy and phase stability of paramagnetic and antiferromagnetic CrN from ab initio molecular dynamics
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2014 (English)In: Physical Review B. Condensed Matter and Materials Physics, ISSN 1098-0121, E-ISSN 1550-235X, Vol. 89, no 17, 174108- p.Article in journal (Refereed) Published
Abstract [en]

We present a theoretical first-principles method to calculate the free energy of a magnetic system in its high-temperature paramagnetic phase, including vibrational, electronic, and magnetic contributions. The method for calculating free energies is based on ab initio molecular dynamics and combines a treatment of disordered magnetism using disordered local moments molecular dynamics with the temperature-dependent effective potential method to obtain the vibrational contribution to the free energy. We illustrate the applicability of the method by obtaining the anharmonic free energy for the paramagnetic cubic and the antiferromagnetic orthorhombic phases of chromium nitride. The influence of lattice dynamics on the transition between the two phases is demonstrated by constructing the temperature-pressure phase diagram.

Place, publisher, year, edition, pages
American Physical Society, 2014
National Category
Physical Sciences
Identifiers
urn:nbn:se:liu:diva-110985 (URN)10.1103/PhysRevB.89.174108 (DOI)000341308600001 ()
Note

Funding Agencies|Erasmus Mundus Joint European Doctoral Programme DocMASE; SECO Tools AB; Swedish Research Council [621-2011-4426, 621-2011-4417]; Swedish Foundation for Strategic Research (SSF) programs SRL [10-0026]; project Designed Multicomponent Coatings (MultiFilms); Knut and Alice Wallenberg Foundation (KAW)

Available from: 2014-10-01 Created: 2014-10-01 Last updated: 2017-12-05Bibliographically approved
3. First-principles calculations of properties of orthorhombic iron carbide Fe7C3 at the Earths core conditions
Open this publication in new window or tab >>First-principles calculations of properties of orthorhombic iron carbide Fe7C3 at the Earths core conditions
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2015 (English)In: Physical Review B. Condensed Matter and Materials Physics, ISSN 1098-0121, E-ISSN 1550-235X, Vol. 91, no 21, 214112Article in journal (Refereed) Published
Abstract [en]

A recently discovered phase of orthorhombic iron carbide o-Fe7C3 [Prescher et al., Nat. Geosci. 8, 220 (2015)] is assessed as a potentially important phase for interpretation of the properties of the Earths core. In this paper, we carry out first-principles calculations on o-Fe7C3, finding properties to be in broad agreement with recent experiments, including a high Poissons ratio (0.38). Our enthalpy calculations suggest that o-Fe7C3 is more stable than Eckstrom-Adcock hexagonal iron carbide (h-Fe7C3) below approximately 100 GPa. However, at 150 GPa, the two phases are essentially degenerate in terms of Gibbs free energy, and further increasing the pressure towards Earths core conditions stabilizes h-Fe7C3 with respect to the orthorhombic phase. Increasing the temperature tends to stabilize the hexagonal phase at 360 GPa, but this trend may change beyond the limit of the quasiharmonic approximation.

Place, publisher, year, edition, pages
American Physical Society, 2015
National Category
Condensed Matter Physics
Identifiers
urn:nbn:se:liu:diva-120164 (URN)10.1103/PhysRevB.91.214112 (DOI)000356791500001 ()
Note

Funding Agencies|Swedish Research Council (VR) [621-2011-4426]; SSF programs "Multifilms"; SRL [10-0026]; Erasmus Mundus Joint European Doctoral Programme DocMASE; Ministry of Education and Science of the Russian Federation [14.Y26.31.0005]; Tomsk State University Academic D. I. Mendeleev Fund Program [8.1.18.2015]; Swedish Foundation for Strategic Research (SSF) program FUNCASE

Available from: 2015-07-13 Created: 2015-07-13 Last updated: 2017-12-04
4. Impact of anharmonic effects on the phase stability, thermal transport, and electronic properties of AlN
Open this publication in new window or tab >>Impact of anharmonic effects on the phase stability, thermal transport, and electronic properties of AlN
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2016 (English)In: Physical Review B, ISSN 2469-9950, E-ISSN 2469-9969, Vol. 94, no 10, 104305Article in journal (Refereed) Published
Abstract [en]

Wurtzite aluminium nitride is a technologically important wide band gap semiconductor with an unusually high thermal conductivity, used in optical applications and as a heatsink substrate. Many of its properties depend on an accurate description of its lattice dynamics, which have thus far only been captured in the quasiharmonic approximation. In this work, we demonstrate that anharmonicity has a considerable impact on its phase stability and transport properties, since anharmonicity is much stronger in the rocksalt phase. We compute a pressure-temperature phase diagram of AlN, demonstrating that the rocksalt phase is stabilised by increasing temperature, with respect to the wurtzite phase. We demonstrate that including anharmonicity, we can recover the thermal conductivity of the wurtzite phase (320 Wm−1K−1 under ambient conditions), and compute the hitherto unknown thermal conductivity of the rocksalt phase (96 Wm−1K−1). We also show that the electronic band gap decreases with temperature. These findings provide further evidence that anharmonic effects cannot be ignored in high temperature applications.

National Category
Condensed Matter Physics
Identifiers
urn:nbn:se:liu:diva-122955 (URN)10.1103/PhysRevB.94.104305 (DOI)000384061100003 ()
Note

Funding agencies; Swedish Research Council (VR programs) [2015-04391, 621-2012-4401]; Swedish Foundation for Strategic Research (SSF program) [SRL10-0026]; VINNOVA [M-Era.net Project] [2013-02355 (MC2)]; Swedish Research Council VR program [637-2013-7296]; Swedish Foundati

Available from: 2015-11-30 Created: 2015-11-30 Last updated: 2017-12-01Bibliographically approved
5. Anharmonicity changes the solid solubility of an alloy at high temperatures
Open this publication in new window or tab >>Anharmonicity changes the solid solubility of an alloy at high temperatures
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2015 (English)Manuscript (preprint) (Other academic)
Abstract [en]

We have developed a method to accurately and efficiently determine the vibrational free energy as a function of temperature and volume for substitutional alloys from first principles. Taking Ti1−xAlxN alloy as a model system, we calculate the isostructural phase diagram by finding the global minimum of the free energy, corresponding to the true equilibrium state of the system. We demonstrate that the anharmonic contribution and temperature dependence of the mixing enthalpy have a decisive impact on the calculated phase diagram of a Ti1−xAlxN alloy, lowering the maximum temperature for the miscibility gap from 6560 K to 2860 K. Our local chemical composition measurements on thermally aged Ti0.5Al0.5N alloys agree with the calculated phase diagram.

National Category
Condensed Matter Physics
Identifiers
urn:nbn:se:liu:diva-122956 (URN)
Available from: 2015-11-30 Created: 2015-11-30 Last updated: 2015-11-30Bibliographically approved
6. Temperature-dependent elastic properties of Ti1−xAlxN alloys
Open this publication in new window or tab >>Temperature-dependent elastic properties of Ti1−xAlxN alloys
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2015 (English)In: Applied Physics Letters, ISSN 0003-6951, E-ISSN 1077-3118, Vol. 107, no 23Article in journal (Refereed) Published
Abstract [en]

Ti1−xAlxN is a technologically important alloy that undergoes a process of high temperature age-hardening that is strongly influenced by its elastic properties. We have performed first principles calculations of the elastic constants and anisotropy using the newly developed symmetry imposed force constant temperature dependent effective potential method, that include lattice vibrations and therefore the effects of temperature, including thermal expansion and intrinsic anharmonicity. These are compared with in situ high temperature x-ray diffraction measurements of the lattice parameter. We show that anharmonic effects are crucial to the recovery of finite temperature elasticity. The effects of thermal expansion and intrinsic anharmonicity on the elastic constants are of the same order, and cannot be considered separately. Furthermore, the effect of thermal expansion on elastic constants is such that the volume change induced by zero point motion has a significant effect. For TiAlN, the elastic constants soften non-uniformly with temperature: C11 decreases substantially when the temperature increases for all compositions, resulting in an increased anisotropy. These findings suggest that an increased Al content and annealing at higher temperatures will result in a harder alloy.

National Category
Condensed Matter Physics Physical Sciences
Identifiers
urn:nbn:se:liu:diva-122957 (URN)10.1063/1.4936896 (DOI)000367010800017 ()
Note

Funding agencies: Swedish Research Council (VR) [621-2011-4426, 621-2012-4401, 637-2013-7296]; Swedish Foundation for Strategic Research (SSF) [RMA08-0069, SRL10-0026]; VINNOVA [2013-02355(MC2)]; Erasmus Mundus Joint European Doctoral Program DocMASE; Ministry of Education

Vid tidpunkten för disputation förelåg publikation som manuskript

Available from: 2015-11-30 Created: 2015-11-30 Last updated: 2017-12-01Bibliographically approved
7. Influence of vibrational free energy on the phase stability of alloys from first principles
Open this publication in new window or tab >>Influence of vibrational free energy on the phase stability of alloys from first principles
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2015 (English)Manuscript (preprint) (Other academic)
Abstract [en]

We have developed a method to accurately and efficiently determine the vibrational free energy as a function of temperature and pressure for substitutional alloys from first principles. Taking the example of the technologically important hard coating alloy Ti1-xAlxN as an example, we investigate the effect on the vibrational free energy of substituting Ti for other group IV elements. By constructing the phase diagrams for these three alloys, we show why Zr1-xAlxN and Hf1-xAlxN are so difficult to experimentally synthesise in a metastable solid solution: both have solubility regions that span only a small low-AlN concentration range at temperatures above 1500 K. Moreover, Hf1-xAlxN is dynamically unstable at low temperatures and across most of the concentration range. We also show the chemical and thermal expansion effects dominate mass disorder in the Gibbs free energy of mixing.

National Category
Condensed Matter Physics Physical Sciences
Identifiers
urn:nbn:se:liu:diva-122958 (URN)
Available from: 2015-11-30 Created: 2015-11-30 Last updated: 2015-11-30Bibliographically approved

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