Квантово-химические расчеты наноструктур на основе перовскитов тема диссертации и автореферата по ВАК РФ 02.00.01, кандидат наук Куруч Дмитрий Дмитриевич

• Куруч Дмитрий Дмитриевич
• кандидат наук
• 2017

Оглавление диссертации кандидат наук Куруч Дмитрий Дмитриевич

Введение

1 Методы и модели, использованные в расчетах

2 Результаты расчетов и их обсуждение

2.1 Расчеты объемных кристаллов

2.2 Расчеты нанослоев

2.2.1 Моделирование дефектных поверхностей

2.2.2 Моделирование прекурсорных нанослоев

2.3 Расчеты нанотрубок

2.3.1 Симметрия нанотрубок

2.3.2 Расчеты одностенных нанотрубок

2.3.3 Моделирование двустенных нанотрубок

2.3.4 Сравнение нанотрубок на основе 8гТЮ3, ВаТЮ3, Бг7г03 и Ва2г03

2.4 Адсорбция молекул воды на поверхности нанослоев

2.4.1 Выбор элементарной ячейки

2.4.2 Адсорбция воды на ВаО-терминированных поверхностях

2.4.3 Адсорбция воды на 7г02- и НГО2-терминированных поверхностях

2.4.4 Адсорбция на У-замещенной поверхности. Позиции адсорбции воды

и соответствующие энергии

2.4.5 Изучение различных положений протона и оценка энергетических барьеров миграции между ними

2.5 Адсорбция молекул воды на поверхности нанотрубок

2.5.1 Моделирование адсорбции воды

2.5.2 Адсорбция на поверхности нанотрубки при низкой плотности адсорбата

2.5.3 Адсорбция на внутренней поверхности нанотрубки при высокой плотности адсорбата

2.5.4 Электронные свойства гидратированных и негидратированных нанотрубок

Заключение

Список литературы

Введение диссертации (часть автореферата) на тему «Квантово-химические расчеты наноструктур на основе перовскитов»

Введение

Перовскиты - это сложные оксиды, преимущественно состава ABO3, где Л -это двухвалентный металл, а В - четырехвалентный (переходный) металл. В настоящее время перовскиты являются широко изучаемыми объектами, демонстрирующими уникальные электронные, магнитные и оптические свойства [1, 2], которые лежат в основе работы многих существующих и потенциальных устройств [3, 4]. Наномасштабные системы на основе БгТЮ3, ВаТЮ3, Бг7Ю3 и Ва2Ю3 реализуются в таких объектах как нанотрубки [5-10], наностержни [11-13] и нанокристаллы [11, 14, 15]. Эти наноматериалы в недалеком будущем могут значительно повысить эффективность различных электронных, электрооптических, электрохимических и электромеханических устройств и систем [16].

Перовскиты могут существовать в различных фазовых модификациях, которые проявляют, вообще говоря, различные свойства. Число фазовых модификаций зависит от конкретной комбинации катионов Л и В [1, 17]. Описание объемных кристаллических фаз будет приведено в разделе 2.1.

Существует довольно большое количество экспериментальных работ [5-10] по синтезу нанотрубок (НТ) на основе перовскитов, результаты которых трудно анализировать из-за отсутствия точных моделей, описывающих химическую структуру и морфологию синтезированных НТ. Большинство современных теоретических моделей [18] НТ основаны на сворачивании слоев, вырезанных из объемных кристаллов. Однако эта процедура обычно обеспечивает моделирование только тонкостенных НТ. В результате большинство теоретических расчетов выполняются для неорганических НТ с толщиной стенок около 2 А. Однако минимальная толщина стенок синтезированных БгТЮ3 и ВаТЮ3 НТ составляет несколько десятков ангстрем [5, 8-10].

В обзорах [18, 19] опубликованных работ по теоретическому моделированию неорганических НТ показано, что НТ на основе перовскитов фактически изучались только в нескольких работах. Так, Пискунов и Шпор [20] показали, что теория функционала плотности (ТФП) предсказывает отрицательные энергии для некоторых НТ, свернутых из тонких нанолистов состава БгТЮ3. Авторы указали, что «отрицательная энергия сворачивания минимизирует энергию образования НТ по отношению к объемному кристаллу 8гТЮ3». Следовательно, одной из возможных причин отрицательных энергий сворачивания является большая положительная энергия образования нанослоя (по отношению к объемному кристаллу), используемого для сворачивания НТ. Как правило, это связано с наличием оборванных связей на поверхности нанослоя. Расположение

оборванных связей на изогнутой поверхности может быть более благоприятным, чем на плоской поверхности нанослоя.

Многие АВ03 перовскиты при замещении катионов В примесными ионами меньшей валентности становятся кислород - дефицитными кристаллами. При повышенных температурах такие материалы обычно обладают кислородной проводимостью [21, 22]. Во влажной среде при повышенных температурах дефектные перовскиты могут поглощать молекулы воды, диссоциация которых (с последующей диффузией Н+) обуславливает возникновение протонной проводимости. Перовскито-подобные протонные проводники сохраняют достаточно высокую проводимость при средних температурах и широко используются в топливных элементах из-за низкой энергии активации миграции протона. В частности, было показано, что У-замещенный Ва2г03 [2, 23], имея высокую стабильность, во влажной атмосфере обладает высокой протонной проводимостью.

Большое количество теоретических исследований посвящено миграции протонов в замещенных кристаллах типа перовскита, в то время как только несколько авторов [24-26] дополнительно рассматривают кислородные вакансии в объемном кристалле непосредственно перед исследованием абсорбции воды. Так, в работах [24, 25] был выполнен расчет энергии замещения и энергии миграции кислорода в нескольких перовскитах. Авторы [24] рассчитали энергетический профиль при миграции кислородной вакансии между соседними кислородными позициями. В результате они получили энергию активации миграции кислородной вакансии вдоль ребра 7г06 октаэдра в У-замещенном 8йг03, которая оказалась равна 0.65 эВ. Расчеты авторов [25] показали, что У наиболее благоприятный заместитель в Ва2г03. Образование дефектов в кристаллах перовскитов находящихся в термодинамическом равновесии с газообразным кислородом, было изучено теоретически в работе [26] на основе термодинамического моделирования в рамках ТФП. Авторы [26] указывают на то, что в большинстве случаев примесные атомы предпочтительно замещают атом 7г. Результаты расчетов [26] также показали, что кислородные вакансии в кристаллах Ва2г03 термодинамически неустойчивы при низких температурах. Однако, расчеты дефектных свободных и протонированных поверхностей Ва2г03 и ВаНЮ3 отсутствуют.

Адсорбция молекул на поверхности кристаллов - это важнейший фактор, который, в конечном итоге, определяет многие физические и химические свойства «реальной» поверхности раздела между конденсированной и жидкой или газовой фазами. Так, например, адсорбция воды на поверхности оксидов металлов определяет структуру двойного электрического слоя, который отвечает за электрохимические свойства поверхности, используемые в работе электродных

сенсоров и топливных элементов. Другой важный пример, где адсорбция играет важную роль - это процесс фотокаталитического расщепления воды на поверхности оксидов металлов. Этот процесс является перспективным и экологически безопасным способом получения газообразного водорода. Очевидно, роль молекул воды должна возрастать, когда размер кристаллитов уменьшается до нанослоев, нанотрубок и наностержней из-за значительного увеличения их удельной поверхности. Адекватное описание таких систем может быть достигнуто только в том случае, если известна реальная атомная структура поверхности раздела адсорбент-адсорбат. Современная квантовая химия и молекулярно-статистическая физика предоставляют много возможностей для такого описания. В частности, квантово-химические методы ab initio обеспечивают теоретический инструмент для определения конкретных структурных факторов, способствующих преобладанию молекулярных или ионных форм воды на поверхности оксида, когда результаты экспериментальных измерений, как правило, допускают неоднозначную интерпретацию. На сегодняшний день неэмпирические расчеты адсорбции воды на поверхности кристалла были выполнены для широкого класса оксидов металлов. Среди них особый интерес представляют бинарные и тройные оксиды металлов IV группы из-за их важных применений. Имеются существенные сведения о строении свободных [18, 27-30] и гидратированных [18,31-38] поверхностей этих веществ. В частности, в недавнем обзоре [27] представлены результаты расчетов поверхностной релаксации, энергетических и химических свойств поверхностей (001), (011) и (111) перовскитов ABO3 с использованием гибридных обменно-корреляционных функционалов в рамках ТФП. Неэмпирические исследования [31, 32, 34-38] гидратации чистых и модифицированных поверхностей перовскитов (титаната, цирконата и гафната стронция) дают полезную информацию о состоянии адсорбированных молекул воды в этих системах. Тем не менее, экспериментальные и теоретические исследования адсорбции воды на поверхности BaZrO3 и BaHfO3 практически отсутствуют.

В последние годы взаимодействие молекул воды с поверхностью нанообъектов привлекает большое внимание. Адсорбция молекул воды на поверхности замещенных и незамещенных углеродных нанотрубок, графита и графена была исследована в ряде теоретических работ [39-42]. Моделирование адсорбции воды на неорганических НТ обсуждается, однако, лишь в нескольких публикациях, посвященных, в частности, НТ оксидов титана [43] и цинка [44]. Авторы первой упомянутой работы [43] рассчитали энергию адсорбции воды на однослойной НТ, свернутой из нанослоев анатаза TiO2 параллельно поверхности (101) с использованием ТФП. Молекулы воды в молекулярной или диссоциированной форме помещали на внешние или внутренние поверхности трубки. Авторы [43]

пришли к выводу, что кривизна поверхности нанотрубки играет важную роль в диссоциативной адсорбции воды и незначительную роль в молекулярной адсорбции. Авторы работы [44] провели полноэлектронные ТФП расчеты адсорбции воды на поверхности непериодических нанотрубок 7пО, расположенных между двумя золотыми пластинами. На внешнюю поверхность вышеупомянутых нанотрубок помещали от одной до четырех молекул воды. Было показано, что наличие адсорбированных молекул Н20 на поверхности приводит к уменьшению проводимости нанотрубок. Авторы [44] пришли к выводу, что нанотрубки на основе 7пО могут быть использованы для электрохимического обнаружения и мониторинга присутствия молекул воды, применяя метод смещающего напряжения.

В нескольких недавних исследованиях изучена возможность применения нанообъектов для фотокаталитического разложения воды. Например, были проанализированы ширины запрещенных зон и положения ширин запрещенных зон в стехиометрических [45] и нестехиометрических [45, 46] наностержнях, вырезанных из кубического кристалла БгТЮ3. Следует также упомянуть предварительное сообщение Весселя и Шпора [47] о моделировании нанотрубок БгТЮ3 в водной среде с использованием неэмпирической молекулярной динамики. Влияние примесных атомов на эффективность расщепления воды было недавно изучено в работе Пискунова и др. [48] для НТ состава ТЮ2 и БгТЮ3.

В этой работе сообщаются результаты неэмпирических расчетов структуры и свойств нанослоев, одностенных и двустенных нанотрубок (ОСНТ и ДСНТ), полученных на основе перовскитов, а также характеристик адсорбции молекул воды на поверхностях указанных наносистем.

Представленный выше краткий обзор теоретических работ показывает, что расчеты протонирования и адсорбции молекул воды на бездефектных и дефектных поверхностях рассматриваемых перовскитов либо малочисленны, либо отсутствуют. Лишь единичные работы посвящены изучению структуры и стабильности одностенных нанотрубок на основе перовскитов, в то время как свойства многостенных нанотрубок вообще не рассматриваются в опубликованных работах. Взаимодействие молекул воды с поверхностями нанотрубок практически не изучалось. Указанные обстоятельства и определяют актуальность данной работы.

1 Методы и модели, использованные в расчетах

Квантово-химические расчеты были проведены методом ТФП с использованием гибридного обменно-корреляционного функционала PBE0 [49, 50]. Этот функционал успешно применяется в расчетах объемных и поверхностных свойств перовскитов [18], а также хорошо воспроизводит параметры водородных связей с участием молекул воды [51].

Для выполнения расчетов использовалась компьютерная программа CRYSTAL [52, 53]. Все расчеты, представленные в этой работе, были выполнены в приближении ЛКАО (линейная комбинация атомных орбиталей).

Важным аспектом, определяющим тип квантово-химических расчетов, является выбор базиса для описания кристаллических орбиталей. Базис плоских волн (ПВ), который имеет некоторые технические преимущества по сравнению с базисом АО, используется в большинстве современных моделирований периодических систем. В то же время, метод ЛКАО также обеспечивает хорошее качество вычисляемых свойств [18]. Более того, расчеты нанообъектов в приближении ЛКАО не требуют искусственного восстановления периодичности в одном или двух направлениях [18], в отличие от расчетов в базисе ПВ, где необходимо восстанавливать трехмерную периодичность из-за неполноты 2D и 1D базисов ПВ в 3D-пространстве. Наконец, следует отметить, что широко применяемый для молекулярных системах анализ заселенности по Малликену легко обобщается для периодических систем [54, 55], если для описания кристаллических орбиталей используется ЛКАО.

Для описания взаимодействия остовных и валентных электронов атомов Ba, Y, Zr и Hf в наших ранних работах [33, 56] использовались псевдопотенциалы Штоля [57, 58], а в последующих работах [59-62] для атомов Ba, Sr, Zr и Ti -псевдопотенциалы CRENBL [63-65]. Мы использовали т.н. приближение малых остовов, в котором состояния субвалентных электронов, также как и валентных электронов вычисляются самосогласованно. Чтобы исключить квазилинейную зависимость функций базисного набора, диффузные s-, p-, d-, и f- орбитали гауссова типа с показателями экспонент меньше чем 0.1 а.е. были удалены из соответствующих базисных наборов, а экспоненты других поляризующих функций были переоптимизированы для периодических расчетов. Поляризующая f-функция атома Ba, существующая в базисном наборе, была исключена, т.к. она не оказывает заметного влияния на Ba2+ ионное состояние. Для атома О использовался полноэлектронный базис 8-411G(d) [66-68], который был скорректирован для расчетов свойств кристаллических перовскитов. Для изучения дефектной поверхности перовскитов кислородная вакансия

моделировалась как атом «призрак», содержащий «пустой» базис атома О, центрированный на положении ядра удаленного атома. Контрактованный базисный набор 6-31G(p) [68, 69] применялся для атома водорода.

В случае объемных кристаллов, интегрирование по зоне Бриллюэна было проведено с использованием сетки Монкхорста-Пака [71] размерностью 8 х 8 х 8 в ранних работах [33, 56] и 12 х 12 х 12 в последующих работах [59-62]. Число k-точек в каждом периодическом направлении уменьшалось обратно пропорционально соответствующей постоянной ячейки, чтобы обеспечить приблизительно равномерное сканирование зоны Бриллюэна для орторомбического кристалла SrZrü3, всех нанослоев и НТ. Сходимость электронного цикла считалась достигнутой, когда разность полных энергий на двух последовательных итерациях процедуры самосогласования становилась меньше 10-6 а. е. Принятая методика обеспечивает хорошее качество рассчитанных объемных свойств для всех исследованных кристаллов.

Для подготовки исходной геометрической структуры наносистем и для анализа их симметрии использовались встроенные возможности программы CRYSTAL [52, 53], а также инструменты программного пакета Materials Studio [71]. Параметры решетки и положения атомов полностью оптимизировались для всех рассмотренных систем с учетом ограничений, накладываемых симметрией.

2 Результаты расчетов и их обсуждение

2.1 Расчеты объемных кристаллов

Перовскиты АВ03 могут существовать в различных фазовых модификациях. Кубическая структура, присущая большинству перовскитов, характеризуется пространственной группой Рт-3т. Многие перовскиты, в частности Бг7г03 и ВаТЮ3 демонстрируют наличие других фаз. Последовательность фазовых превращений в 8йг03 при повышенных температурах насчитывает три перехода: из орторомбической фазы РЬпт в орторомбическую Стст при 970 К, затем в тетрагональную 14/тст при 1100 К, и, наконец, в кубическую фазу Рт-3т при 1400 К. Высоко симметричная кубическая фаза Рт-3т кристалла ВаТЮ3 также устойчива при высоких температурах и демонстрирует серию из трех фазовых переходов при понижении температуры: в тетрагональную 14/тст при 393 К, орторомбическую Атт2 при 278 К и ромбоэдрическую К3т при 183 К. В этой работе мы изучили наноструктуры, главным образом, основанные на кубических фазах рассмотренных кристаллов. Для кристалла 8йг03, была рассмотрена и орторомбическая модификация с пространственной группой РЬпт. Объемные кристаллы использовались прежде всего как тестовые системы для проверки надежности выбранного расчетного метода. Рассчитанные параметры решетки, модули упругости, энергии атомизации и ширины запрещенных зон приведены в таблице 1. Для оценки модулей упругости рассмотренных кристаллов использовалось уравнение состояния Мурнагана [72], параметры которого определялись методом наименьших квадратов исходя из найденной зависимости полной энергии от объема элементарной ячейки кристалла. Полученные значения опубликованы нами в работах [33, 56, 59-62] и хорошо согласуются с имеющимися экспериментальными данными (таблица 1).

Для оценки полных энергий Е(7г02), £(НЮ2) и Е(У203) в связи с исследованием кислородных вакансий, были полностью оптимизированы элементарные ячейки моноклинных (Р21/с) фаз 7г02 и НГО2, а также кубический (1а-3) фазыУ203. Рассчитанные свойства объемных кристаллов 7г02, НГО2 и У203 сравниваются с экспериментальными данными в таблице 1.

Таблица 1. Рассчитанные значения параметров решетки, модуля упругости и ширины запрещенных зон объемных кристаллов.

Группа симметрии Параметры решетки, А Модуль Энергия Ширина

упругости, ГПа атомизации, эВ запрещенной зоны, эВ

ВаИЮ3 Рт-3т 4.189 [33] (4.180 [73])* 4.194 [33] 175 [33] 33.3 [33] 5.8 [33] 5.4 [33]

Ва2г0э Рт-3т 4.203 [61] (4.190 [74)] 172 [33] 32.8 [33] (5.3 [75])

ВаТЮ3 Рт-3т 3.986 [59] (3.996 [76]) 201 [59] (162 [77]) 29.7 [59] (31.6 [77]) 3.7 [59] (3.2 [77])

Рт-3т 4.123 [60] (4.154 [78]) 206 31.6 5.49

БйЮэ

РЬпт 5.750, 5.819, 8.173 [60] (5.786, 5.815, 8.196 [78]) 197 - 6.01

БгТЮэ Рт-3т 3.885 [62] (3.900 [79]) 226 (183 [80]) 29.6 3.8 [62] (3.3 [81])

2г02 Р21/с 5.152, 5.208, 5.307; 99.5 [56] 22.0 [56] 6.4 [56]

(5.150, 5.208, 5.317; 99.2 [82]) (22.7 [83]) (5.8 [84])

НГО2 Р2/с 5.154, 5.195, 5.308, 99.3 22.5 6.94

(5.117, 5.175, 5.292, 99.2 [85]) (22.0 [86]) (5.8 [87])

У203 1а-3 10.576 [56] (10.596 [88]) 34.8 [56] (36.1 [83]) 7.2 [56] (6.0 [84])

*Экспериментальные данные приведены в скобках.

2.2 Расчеты нанослоев

2.2.1 Моделирование дефектных поверхностей

Чередующиеся (001) атомные плоскости в объемном кристалле перовскита АВ03 имеют разный состав. Следовательно, могут существовать как А0, так и В02-терминированные поверхности для (001) кубических и (001) и (110) орторомбических 2Э структур перовскитов. Во всех исследуемых перовскитах А0 и В02 слои имеют нейтральный заряд, поэтому оба типа (001) поверхностей формально неполярные. В работе [56], посвященной моделированию дефектных поверхностей кубических Ва2г03 и ВаНГО3, мы рассматривали нестехиометрические нанослои, состоящие из 9-ти чередующихся (001) атомных плоскостей. В последующих работах [59-62] тонкие стехиометрические нанослои, состоящие из 2, 4 или 6 чередующихся (001) атомных плоскостей Ва2г03, ВаТЮ3, Бг7г03 или БгТЮ3 изучались как исходные структуры при моделировании нанотрубок.

При расчетах симметрично-терминированных (нестехиометрических) 9-ти слойных пластин кристаллов Ва2г03 и ВаНГО3 использовалась инверсионная симметрия. Такая симметрия обеспечивает эквивалентность верхних и нижних поверхностей нанослоя. В процессе оптимизации структуры поверхности, позиции всех атомов подвергались релаксации за исключением атомов средней плоскости, которые были зафиксированы симметрией. Энергия образования каждой из двух разных терминаций не может быть определена отдельно без дополнительных допущений. Однако, средняя энергия образования поверхности Епов имеет ясный физический смысл, так как обе терминации появляются одновременно при образовании поверхности. Для моделей с нечетным числом атомных плоскостей Епов может быть определена следующим образом:

Епов = (ЕАО + ЕВ02 -пЕкрист) / (^НсХ (1)

где ЕАО и ЕВ02 полная энергия А0- и В02-терминированных нанослоев на их ячейку, Екрист - полная энергия объемного кристалла на формульную единицу, а п общее количество формульных единиц в обоих нанослоях. Множитель 4 перед БНС - площадь поверхности 2Э элементарной ячейки, соответствует образованию 4-х граней при раскалывании кристалла, с образованием двух разно-терминированных пластин. Наши расчеты дают близкие средние энергии

_Л _Л

образования (001) поверхностей Ва2г03 и ВаНЮ3 - 1.21 Дж-м и 1.27 Дж-м , соответственно.

Для изучения атомной и электронной структуры дефектов поверхностей Ва2г03 и ВаНГО3 были исследованы У-замещенные нанослои. Известно [2, 25], что атомы У чаще всего замещают атомы в положении В, поэтому В02-

терминированные (001) поверхности наиболее подходят для моделирования поверхностных вакансий. Предположительно, атом Y должен быть расположен в окрестности вакансии из-за эффекта ассоциации примесный атом-вакансия [24, 25].Девяти плоскостные пластины состоят из 4 BaO и 5 ZrO2 или HfO2 чередующихся плоскостей. Для создания вакансии один из поверхностных атомов кислорода удаляется и два соседних атома Zr или Hf на поверхности элементарной ячейки заменяются атомами Y. При использовании расширенных 2D ячеек 2 х 2 и 3 х 3, были построены две модели: (I) без инверсии и с вакансией только на одной поверхности пластины; (II) с вакансиями на противоположных сторонах пластины (в данном случае инверсионная симметрия обеспечивает эквивалентность обеих поверхностей). При этом рассматривались различные положения атомов заместителей вблизи вакансии для обеих моделей.

Образование вакансии на поверхности описывается реакцией:

Sur + Y2O3 ^ Sur''Y2^ + 2МО2, (M = Zr, Hf), (2)

где Sur - это бездефектный ZrO2- или НЮ2-терминированный нанослой BaZrO3 или BaHfO3, Sur''Y2^ - та же самая система с кислородной вакансией и двумя атомами Zr или Hf, замещенными на два атома Y в элементарной ячейке. Все включенные в уравнение 2 фазы (поверхности и объемные кристаллы) могут быть рассчитаны в рамках одного и того же уровня теории, что обеспечивает равную точность вычисляемых величин и/или сокращение возможных ошибок. Это подтверждается близкими значениями энергий образования Еобр вакансии в рассматриваемых моделях, которые рассчитываются относительно энергий устойчивых фаз ZrO2, HfO2 и Y2O3 по уравнению:

Еобр = (1/«)[E(Sur''Y2n^n) + 2«E(MO2) - E(Sur) - nE(Y2O3)], (M = Zr, Hf), (3)

где n = 1 или 2 для моделей I и II, соответственно.

Для установления наиболее выгодных поверхностных конфигураций примесь-вакансия, мы рассмотрели различные положения атомов Y на внешней и следующей за ней атомных плоскостях. На рисунке 1 показаны четыре возможных симметрийно неэквивалентных положения атома Y относительно кислородной вакансии на внешнем слое поверхности расширенной ячейки 2х 2. В этом случае соотношение Y/Zr или Y/Hf на поверхности равно 1/1, а концентрация кислородной вакансии 1/8.

В таблице 2 приведены энергии образования вакансии с использованием обеих моделей поверхности, полученные как с использованием базиса в центре вакансии, так и без него. Добавление базиса в центре вакансии снижает значение энергии образования на ~10 кДж-моль-1 и не влияет на релаксацию атомов. В соответствии с таблицей 2, наиболее выгодной является структура 0v1 для

Ба2г03, а для ВаНЮз - структура 0v0 (объяснение использованных обозначений дано во втором столбце таблицы 2).

0v0

0v1

-1v1

v01

Рисунок 1. Симметрийно неэквивалентные положения атомов У вблизи вакансии на верхнем слое поверхности расширенной ячейки 2x2 Ба2г03 или ВаНГОз (вид сверху). Большая светлая сфера - вакансия; желтые сферы - атомы О; синие сферы - атомы 2г или Н; фиолетовые сферы - атомы У.

Из таблицы 2 видно, что модели I и II дают близкие результаты. Это говорит о том, что выбор пластины, состоящий из 9-ти слоев, достаточно хорошо передает свойства дефектной поверхности кристаллов Ба2г03 и ВаНЮ3. Это также означает, что любая из двух моделей может быть использована для воспроизведения свойств дефектной поверхности. Все последующие результаты получены на основе модели II.

Таблица 2. Энергии образования (Еобр, кДж-моль-1) кислородных вакансий в верхней плоскости У-замещенной (001) поверхности нанослоя Ба2г03 и ВаНГО3.

Кристалл, модель, ячейка

BaZrO3

ВаНГОз

Положения атомов Y I, 2x2 II, 2x2 II, 3x3 II, 2x2 II, 3x3

0v0: Y(1)(-1/2,0,0) Y(2)(1/2,0,0) 152 (161)* 154 (163) 140 146 131

0v1: Y(1)(-1/2,0,0) Y(2)(1/2,1,0) 144 (155) 145 (155) 130 149 136

-1v1: Y(1)(-1/2,-1,0) Y(2)(1/2,1,0) 179 131 197 148

v01: Y(1)(1/2,0,0) Y(2)(1/2,1,0) 205 212

*Энергии образования вакансии, полученные без использования базиса в центре вакансии, приведены в скобках.

Для расширенной 2D ячейки Эх 3 соотношение Y/Zr и Y/Hf равно 2/7, а концентрация кислородной вакансии - 1/18. В этом случае были рассчитаны только три структуры, указанные в таблице 2. Увеличение ячейки с 2 х 2 до 3 х 3 приводит к снижению Еобр на 15 кДж-моль"1 или больше. По-видимому, это объясняется ослаблением взаимодействия вакансия-вакансия и Y-Y за счет

снижения их поверхностной концентрации. Тем не менее, для расширенной ячейки 3x3 наиболее выгодными остаются те же структуры, что и для расширенной ячейки 2 х 2, а именно, 0у1 для Ба2г03 и 0v0 для ВаНГО3. В случае структуры 0у1один атом У располагается в 2.3 А, а второй в 4.6 А от центра вакансии. Как видно из рисунка 2, замещение атомов 7г и Н атомами Y на 7Ю2-терминированной или НГО2-терминированной поверхности, сопровождается заметной реконструкцией поверхности в окрестности кислородной вакансии. Было найдено, что релаксация атомов верхнего слоя поверхности в окрестности вакансии не зависит существенно от размера расширенной ячейки, 2x2 или 3x3. При этом ближайшие атомы кислорода сдвинуты к центру вакансии, в то время как атомы 7г, Н и Y - в противоположном направлении (рисунок 2). Заметим, что на бездефектной поверхности атомы в процессе релаксации испытывают в основном только вертикальные смещения.

(а) (б)

Рисунок 2. Атомная релаксация в окрестности вакансии 0у1 на верхнем слое (001) поверхности Ба2г03 и ВаНГОз с использованием расширенной ячейки 3 х 3. (а) Вид вдоль направления (001); (б) вид вдоль направления(010). См. легенду к рисунку 1.

Чтобы изучить возможность миграции вакансии, был рассчитан энергетический барьер между эквивалентными положениями вакансии на дефектных поверхностях Y-замещенных Ва2Ю3 и ВаНГО3. Переходное состояние было получено в результате оптимизации с ограничением по симметрии (С2/т) (при этом дополнительный базис кислорода на вакансии не использовался). Кислород, в найденном переходном состоянии, занимает промежуточное положение между двумя эквивалентными позициями вакансий (рисунок 3) и приподнят над поверхностью примерно на 0.4А. Если убрать ограничение по симметрии, то при небольшом сдвиге кислород при оптимизации мигрирует в начальное или конечное состояние. Рассчитанные энергетические барьеры 32 кДж-моль-1 для Ба2г03 и 22 кДж-моль-1 для ВаНЮ3 относительно невелики, что указывает на

Список литературы диссертационного исследования кандидат наук Куруч Дмитрий Дмитриевич, 2017 год

Список литературы

[1] Lines, M. E., Glass, A. M. Principles and applications of ferroelectric and related materials / M. E. Lines, A. M. Glass. — Oxford: Clarendon, 2001. — 694 p.

[2] Kreuer, K. D. Proton-conducting oxides / K. D. Kreuer // Annual review of materials research. — 2003. — Vol. 33.— p. 333.

[3] Iwahara, H. High temperature proton conductors based on perovskite-type oxides / P. Colomban (Ed.) Proton conductors: Solids, membranes and gels—materials and devices. — Cambridge: Cambridge university press, 1992. — Ch. 8.

[4] Iwahara, H., Asakura, Y., Katahira, K., Tanaka, M. Prospect of hydrogen technology using proton-conducting ceramics / H. Iwahara, Y. Asakura, K. Katahira, M. Tanaka // Solid state ionics. — 2004. — Vol. 168. — p. 299.

[5] Mao, Y., Banerjee, S., Wong, S. S. Hydrothermal synthesis of perovskite nanotubes / Y. Mao, S. Banerjee, S. S. Wong // Chemical communications. — 2003. — Vol. 3. — p. 408.

[6] Chen, Y. Y., Yu, B. Y., Wang, J. H., Cochran, R. E., Shyue, J. J. Template-based fabrication of SrTiO3 and BaTiO3 nanotubes / Y. Y. Chen, B. Y. Yu, J. H. Wang, R. E. Cochran, J. J. Shyue // Inorganic chemistry. — 2009. — Vol. 48. — p. 681.

[7] Yang, Y., Wang, X., Sun, Ch., Li, L. Phase transition in BaTiO3 nanotube arrays / Y. Yang, X. Wang, Ch. Sun, L. Li // Journal of applied physics. — 2011. — Vol. 109. — 014109.

[8] Singh, S., Krupanidhi, S. B. Sol-Gel template synthesis, structural characterization and growth mechanism of barium zirconate nanotubes / S. Singh, S. B. Krupanidhi // Current nanoscience. — 2009. — Vol. 5. — p. 339.

[9] Calleja, A., Granados, X., Ricart, S., Oró, J., Arbiol, J., Mestres, N., Carrillo, A. E., Palmer, X., Cano, F., Tornero, J. A., Puiga, T., Obradors, X. High temperature transformation of electrospun BaZrO3 nanotubes into nanoparticle chains / A. Calleja, X. Granados, S. Ricart, J. Oró, J. Arbiol, N. Mestres, A. E. Carrillo, X. Palmer, F. Cano, J. A. Tornero, T. Puiga, X. Obradors // CrystEngComm. — 2011. — Vol. 13. — p. 7224.

[10] Zhu, Y. F., Zhang, L., Natsuki, T., Fu, Y. Q., Ni, Q. Q. Facile synthesis of BaTiO3 nanotubes and their microwave absorption properties / Y. F. Zhu, L. Zhang, T. Natsuki, Y. Q. Fu, Q. Q. Ni // ACS Applied materials and interfaces. — 2012. — Vol. 4. — p. 2101.

[11] Mao, Y., Banerjee, S., Wong, S. S. Large-scale synthesis of single-crystalline perovskite nanostructures / Y. Mao, S. Banerjee, S. S. Wong // Journal of the American chemical society. — 2003. — Vol. 125. — p. 15718.

[12] Im, B., Joshi, U. A., Lee, K. H., Lee, J. S. Growth of single crystalline barium titanate nanowires from TiO2 seeds deposited on conducting glass / B. Im, U. A. Joshi, K. H. Lee, J. S. Lee // Nanotechnology. — 2010. — Vol. 21. — 425601.

[13] Karthauser, S., Vasco, E., Dittmann, R., Waser, R. Fabrication of arrays of SrZrO3 nanowires by pulsed laser deposition / S. Karthauser, E. Vasco, R. Dittmann, R. Waser // Nanotechnology. — 2004. — Vol. 15. — p. S122.

[14] Nakashima, K., Fujii, I., Wada, S. Low-temperature synthesis of SrZrO3 nanocubes by the composite-hydroxide-mediated approach / K. Nakashima, I. Fujii, S. Wada // Journal of crystal growth. — 2013. — Vol. 376. — p. 35.

[15] Leonard, K. J., Sathyamurthy, S., Paranthaman, M. P. Characterization of BaZrO3 nanoparticles prepared by reverse micelle synthesis / K. J. Leonard, S. Sathyamurthy, M. P. Paranthaman // Chemistry of materials. — 2005. — Vol. 17. — p. 4010.

[16] Ghosez, Ph., Junquera, J. in Handbook of theoretical and computational nanotechnology / M. Rieth, W. Schommers (Ed.). — CA: American scientific publisher, Stevenson Ranch, 2006. — Ch. 134.

[17] Woodward, P. M. Octahedral tilting in perovskites. II. Structure stabilizing forces / P. M. Woodward // Acta crystallographica. B. — 1997. — Vol. 53. — p. 44.

[18] Evarestov, R. A. Quantum chemistry of solids. LCAO first principles treatment of crystals and nanostructures, 2nd ed.; Springer series in Solid-state sciences Vol. 153 / R. A. Evarestov. — Berlin, Heidelberg: Springer, 2012. — 734 p.

[19] Evarestov, R. A. Theoretical modeling of inorganic nanostructures. symmetry and ab initio calculations of nanolayers, nanotubes and nanowires; Springer series in NanoScience and Technology / R. A. Evarestov. — Berlin, Heidelberg: Springer, 2015. — 672 p.

[20] Piskunov, S., Spohr, E. SrTiO3 Nanotubes with negative Strain energy predicted from first principles / S. Piskunov, E. Spohr // The journal of physical chemistry letters. — 2011. — Vol. 2. — p. 2566.

[21] Wang, W., Virkar, A. V. Ionic and electron-hole conduction in BaZr0 93Y007O3-5 by 4-probe dc measurements / W. Wang, A. V. Virkar // Journal of power sources. — 2005. — Vol. 142. — p. 1.

[22] Chen, X., Rieth, L., Miller, M. S., Solzbacher, F. High temperature humidity sensors based on sputtered Y-doped BaZrO3 thin films / X. Chen, L. Rieth, M. S. Miller, F. Solzbacher // Sensors and actuators B. — 2009. — Vol. 137. — p. 578.

[23] Steele, B. C. H., Heinzel, A. Materials for fuel-cell technologies / B. C. H. Steele, A. Heinzel // Nature. — 2001. — Vol. 414. — p. 345.

[24] Davies, R. A., Islam, M. S., Gale, J. D. Dopant and proton incorporation in perovskite-type zirconates / R. A. Davies, M. S. Islam, J. D. Gale // Solid state ionics. — 1999. — Vol. 126. — p. 323.

[25] Islam, M. S., Slater, P. R., Tolchard, J. R., Dinges, T. Doping and defect association in AZrO3 (A = Ca, Ba) and LaMO3 (M = Sc, Ga) perovskite-type ionic conductors / M. S. Islam, P. R. Slater, J. R. Tolchard, T. Dinges // Journal of the chemical society, Dalton transactions. — 2004. — Vol. 19. — p. 3061.

[26] Sundell, P. G., Björketun, M. E., Wahnström, G. Thermodynamics of doping and vacancy formation in BaZrO3 perovskite oxide from density functional calculations / P. G. Sundell, M. E. Björketun, G. Wahnström // Physical review B.

— 2006. — Vol. 73. — 104112.

[27] Padilla, J., Vanderbilt, D. Ab initio study of SrTiO3 surfaces / J. Padilla, D. Vanderbilt // Surface science. — 1998. — Vol. 418. — p. 64.

[28] Heifets, E., Eglitis, R. I., Kotomin, E. A., Maier, J., Borstel, G. Ab initio modeling of surface structure for SrTiO3 perovskite crystals / E. Heifets, R. I. Eglitis, E. A. Kotomin, J. Maier, G. Borstel // Physical review B. — 2001. — Vol. 64. — 235417.

[29] Evarestov, R. A., Bandura, A. V., Alexandrov, V. E. Hybrid HF-DFT comparative study of SrZrO3 and SrTiO3 (001) surface properties / R. A. Evarestov, A. V. Bandura, V. E. Alexandrov // Physica status solidi (b). — 2006.

— Vol. 243. — p. 2756.

[30] Eglitis, R. I. Ab initio calculations of SrTiO3, BaTiO3, PbTiO3, CaTiO3, SrZrO3, PbZrO3 and BaZrO3 (001), (011) and (111) surfaces as well as F centers, polarons, KTN solid solutions and Nb impurities therein / R. I. Eglitis // International journal of modern physics B. — 2014. — Vol. 28. — 1430009.

[31] Evarestov, R. A., Bandura, A. V., Alexandrov, V. E. Hybrid HF-DFT modelling of water adsorption on (001) surface of orthorhombic and cubic SrHfO3 / R. A. Evarestov, A. V. Bandura, V. E. Alexandrov // Surface science. — 2007. — Vol. 601. — p. 1844.

[32] Shi, W. J., Xiong, S. J. Ab initio study of water adsorption on TiO2-terminated (100) surface of SrTiO3 with and without Cr doping / W. J. Shi, S. J. Xiong // Surface science. — 2010. — Vol. 604. — p. 1987.

[33] Bandura, A. V., Evarestov, R. A., Kuruch, D. D. Hybrid HF-DFT modeling of monolayer water adsorption on (001) surface of cubic BaHfO3 and BaZrO3 crystals / A. V. Bandura, R. A. Evarestov, D. D. Kuruch // Surface science. — 2010. — Vol. 604. — p. 1591.

[34] Hinojosa, B. B., van Cleve, T., Asthagiri, A., A first-principles study of H2O adsorption and dissociation on the SrTiO3 (100) surface / B. B. Hinojosa, T. van Cleve, A. Asthagiri // Molecular simulaion. — 2010. — Vol. 36. — p. 604.

[35] Guhl, H., Miller, W., Reuter, K. Water adsorption and dissociation on SrTiO3 (001) revisited: A density functional theory study / H. Guhl, W. Miller, K. Reuter // Physical review B. — 2010. — Vol. 81. — 155455.

[36] Raghavan, S., Carvalho, A., Le Formal, F., Setter, N., Oberg, S., Briddon, P. R. Adsorbate-localized states at water-covered (100) SrTiO3 surfaces / S. Raghavan, A. Carvalho, F. Le Formal, N. Setter, S. Oberg, P. R. Briddon // Applied physics letters. — 2011. — Vol. 98. — 012106.

[37] Evarestov, R. A., Bandura, A. V., Blokhin, E. N., The water adsorption on the surfaces of SrMO3 (M= Ti, Zr, and Hf) crystalline oxides: quantum and classical modeling / R. A. Evarestov, A. V. Bandura, E. N. Blokhin // Journal of physics: conference series. — 2007. — Vol. 93. — 012001.

[38] Evarestov, R. A., Bandura, A. V., Blokhin, E. N. Hybrid HF-DFT modelling of water adsorption on (001) surface of orthorhombic and cubic SrHfO3 / R. A. Evarestov, A. V. Bandura, E. N. Blokhin // Integrated ferroelectrics. — 2009. — Vol. 108. — p. 37.

[39] Pati, R., Zhang, Y., Nayak, S. K., Ajayan, P. M. Effect of H2O adsorption on electron transport in a carbon nanotube / R. Pati, Y. Zhang, S. K. Nayak, P. M. Ajayan // Applied physics letters. — 2002. — Vol. 81. — p. 2638.

[40] Oubal, M., Picaud, S., Rayez, M. T., Rayez, J. C. Interaction of water molecules with defective carbonaceous clusters: an ab initio study / M. Oubal, S. Picaud, M. T. Rayez, J. C. Rayez // Surface Science. — 2010. — Vol. 604. — p. 1666.

[41] Dai, J. Y., Yuan, J. M. Modulating the electronic and magnetic structures of P-doped graphene by molecule doping / J. Y. Dai, J. M. Yuan // Journal of physics: condensed matter. — 2010. — Vol. 22. — p. 225501.

[42] Ambrosetti, A., Silvestrelli, P. L. Adsorption of rare-gas atoms and water on graphite and graphene by van der Waals-corrected density functional theory / A. Ambrosetti, P. L. Silvestrelli // Journal of physical chemistry C. — 2011. — Vol. 115. — p. 3695.

[43] Liu, H., Tan, K. Adsorption of water on single-walled TiO2 nanotube: A DFT investigation / H. Liu, K. Tan // Computational and theoretical chemistry. — 2012. — Vol. 991. — p. 98.

[44] Duan, M. Y., Shi, G. S., Wang, C. L., Zhou, L. P., Chen, X. R., Fang, H. P. A Theoretical study of a single-walled ZnO nanotube as a sensor for H2O molecules / M. Y. Duan , G. S. Shi, C. L. Wang, L. P. Zhou, X. R. Chen, H. P. Fang // Communications in theoretical physics. — 2012. — Vol. 58. — p. 275.

[45] Bandura, A. V., Evarestov, R. A., Zhukovskii, Y. F. Energetic stability and photocatalytic activity of SrTiO3 nanowires: ab initio simulations / A. V. Bandura, R. A. Evarestov, Y. F. Zhukovskii // RSC Advances. — 2015. — Vol. 5. — p. 24115.

[46] Fu, Q., He, T., Li, J. L., Yang, G. W. Band-engineered SrTiO3 nanowires for visible light photocatalysis / Q. Fu, T. He, J. L. Li, G. W. Yang // Journal of applied physics. — 2012. — Vol. 112. — 104322.

[47] Wessel, M. Spohr, E. in Book of Abstracts of Joint 12th Russia/CIS/Baltic/Japan Symposium on Ferroelectricity and 9th International Conference Functional Materials and Nanotechnologies / M. Wessel, E. Spohr. — Riga, 2014. — p. 285.

[48] Piskunov, S., Lisovski, O., Begens, J., Bocharov, D., Zhukovskii, Y. F., Wessel, M., Spohr, E. C-, N-, S-, and Fe-doped TiO2 and SrTiO3 nanotubes for visible-light-driven photocatalytic water splitting: prediction from first principles / S. Piskunov, O. Lisovski, J. Begens, D. Bocharov, Y. F. Zhukovskii, M. Wessel, E. Spohr // Journal of [hysical chemistry C. — 2015. — Vol. 119. — p. 18686.

[49] Perdew, J. P., Ernzerhof, M., Burke, K. Rationale for mixing exact exchange with density functional approximations / J. P. Perdew, M. Ernzerhof, K. Burke // The journal chemical physics. — 1996. — Vol. 105. — p. 9982.

[50] Adamo, C., Barone, V. Toward reliable density functional methods without adjustable parameters: The PBE0 model / C. Adamo, V. Barone // The journal chemical physics. — 1999. — Vol. 110. — p. 6158.

[51] Tuma, C., Boese, A. D., Handy, N. C. Predicting the binding energies of H-bonded complexes: A comparative DFT study / C. Tuma, A. D. Boese, N. C. Handy // Physical chemistry chemical physics. — 1999. — Vol. 1. — p. 3939.

[52] Dovesi, R., Saunders, V. R., Roetti, C., Orlando, R., Zicovich-Wilson, C. M., Pascale, F., Civalleri, B., Doll, K., Harrison, N. M., Bush, I. J., D'Arco, P., Llunell, M. CRYSTAL09 user's manual / R. Dovesi, V. R. Saunders, C. Roetti, R. Orlando, C. M. Zicovich-Wilson, F. Pascale, B. Civalleri, K. Doll, N. M. Harrison, I. J. Bush, P. D'Arco, M. Llunell. — Turin: University of Turin, 2013. — 307 p.

[53] Dovesi, R., Saunders, V. R., Roetti, C., Orlando, R., Zicovich-Wilson, C. M., Pascale, F., Civalleri, B., Doll, K., Harrison, N. M., Bush, I. J., D'Arco, P., Llunell, M., Causa, M., Noel, Y. CRYSTAL14 user's manual / Dovesi, V. R. Saunders, C. Roetti, R. Orlando, C. M. Zicovich-Wilson, F. Pascale, B. Civalleri, K. Doll, N. M. Harrison, I. J. Bush, P. D'Arco, M. Llunell, M. Causa, Y. Noel. — Turin: University of Turin, 2014. — 382 p.

[54] Pisani, C. Quantum-mechanical Ab initio calculations of the properties of crystalline materials, lecture notes in chemistry, Vol. 67 / C. Pisani (Ed.). — Berlin: Springer, 1996. — 332 p.

[55] Veryazov, V. A., Leko, A. V., Evarestov, R. A. Local characteristics of crystal electronic structure in the Hartree-Fock method / V. A. Veryazov, A. V. Leko, R. A. Evarestov // Physics of the solid state. — 1999. — Vol. 41. — p. 1286.

[56] Bandura, A. V., Evarestov, R. A., Kuruch, D. D. LCAO Calculations of (001) Surface Oxygen Vacancy Structure in Y-Doped BaZrO3 / A. V. Bandura, R. A. Evarestov, D. D. Kuruch // Integrated Ferroelectrics. — 2010. — Vol. 123. — p. 1.

[57] Dolg, M. Effective core potentials / J. Grotendorst (Ed.) Modern methods and algorithms of quantum chemistry, proceedings, 2nd edition, NIC Series Vol. 3. — Jülich: John von Neumann Institute for Computing, 2000. — p. 507.

[58] Dolg, M., Peterson, K. A., Schwerdtfeger, P., Stoll, H. Stuttgart/Cologne energy-consistent (ab initio) pseudopotentials suitable for wavefunction-based and density functional calculations [Electronic resource] / M. Dolg, K. A. Peterson, P. Schwerdtfeger, H. Stoll // Institute for theoretical chemistry, University of Stuttgart, Stuttgart. — 2014. — URL: http://www.tc.uni-koeln.de/PP/index.en.html.

[59] Evarestov, R. A., Bandura, A. V., Kuruch, D. D. BaTiO3-based nanolayers and nanotubes: First-principles calculations / R. A. Evarestov, A. V. Bandura, D. D. Kuruch // Journal of computational chemistry. — 2013. — Vol. 34. — p. 175.

[60] Bandura, A. V., Evarestov, R. A., Kuruch, D. D. Nanotubes folded from cubic and orthorhombic SrZrO3: First-principles study / A. V. Bandura, R. A. Evarestov, D. D. Kuruch // IOP Conference Series: Materials Science and Engineering. — 2013. — Vol. 49. — 012009.

[61] Bandura, A. V., Kuruch, D. D., Evarestov, R. A. Ab initio modeling of wall structure and shape in perovskite-based nanotubes / A. V. Bandura, D. D. Kuruch, R. A. Evarestov // Computational material science. — 2015. — Vol. 96. — p. 124

[62] Bandura, A. V., Kuruch, D. D., Evarestov, R. A. Quantum Chemical Study of Water Adsorption on the Surfaces of SrTiO3 Nanotubes / A. V. Bandura, D. D. Kuruch, R. A. Evarestov // ChemPhysChem. — 2015. — Vol. 16. — p. 2192.

[63] Hurley, M. M., Pacios, L. F., Christiansen, P. A., Ross, R. B., Ermler, W. C. Ab initio relativistic effective potentials with spin-orbit operators. II. K through Kr / M. M. Hurley, L. F. Pacios, P. A. Christiansen, R. B. Ross, W. C. Ermler // The journal of chemical physics. — 1986. — Vol. 84. — p. 6840.

[64] LaJohn, L. A., Christiansen, P. A., Ross, R. B., Atashroo, T., Ermler, W. C. Ab initio relativistic effective potentials with spin-orbit operators. III. Rb through Xe / L. A. LaJohn, P. A. Christiansen, R. B. Ross, T. Atashroo, W. C. Ermler // The journal of chemical physics. — 1987. — Vol. 87. — p. 2812.

[65] Ross, R. B., Powers, J. M., Atashroo, T., Ermler, W. C., LaJohn, L. A., Christiansen, P. A., Ab initio relativistic effective potentials with spin-orbit operators. IV. Cs through Rn / R. B. Ross, J. M. Powers, T. Atashroo, W. C. Ermler, L. A. LaJohn, P. A. Christiansen // The journal of chemical physics. — 1990. — Vol. 93. — p. 6654.

[66] Bredow, T., Jug, K., Evarestov, R. A. Electronic and magnetic structure of ScMnO3 / T. Bredow, K. Jug, R. A. Evarestov // Physica status solidi (b) . — 2006. — Vol. 243. — p. R10.

[67] Schäfer, A., Huber, C., Ahlrichs, R. Fully optimized contracted Gaussian basis sets of triple zeta valence quality for atoms Li to Kr / A. Schäfer, C. Huber, R. Ahlrichs // The journal of chemical physics. — 1994. — Vol. 100. — p. 5829.

[68] CRYSTAL Basis Sets Library, Theoretical Chemistry Group [Electronic resource]. — Torino: University of Torino. — URL: http: //www.crystal. unito .it/basis-sets. php.

[69] Hariharan, P. C., Pople, J. A. The influence of polarization functions on molecular orbital hydrogenation energies / P. C. Hariharan, J. A. Pople // Theoretica chimica acta. — 1973. — Vol. 28. — p. 213.

[70] Monkhorst, H. J., Pack, J. D. Special points for Brillouin-zone integrations / H. J. Monkhorst, J. D. Pack // Physcal review B. — 1976. — Vol. 13. — p. 5188.

[71] Materials Studio software package [Electronic resource], San Diego: Accelrys Software Inc. — 2012. — URL: http://accelrys.com/products/materials-studio/index.html.

[72] Murnaghan, F. D. The compressibility of media under extreme pressures / F. D. Murnaghan // Proceedings of the National Academy of Sciences. — 1944. — Vol. 30. — p. 244.

[73] Feteira, A., Sinclair, D. C., Rajab, Kh. Z., Lanagan, M. T. Crystal structure and microwave dielectric properties of alkaline-earth hafnates, AHfO3 (A=Ba, Sr, Ca) / A. Feteira, D. C. Sinclair, Kh. Z. Rajab, M. T. Lanagan // Journal of American ceramic society. — 2008. — Vol. 91. — p. 893.

[74] Yamanaka, S., Fujikane, H., Hamaguchi, T., Muta, H., Oyama, T., Matsuda, T., Kobayashi, S., Kurosaki, K. Thermophysical properties of BaZrO3 and BaCeO3 / S. Yamanaka, H. Fujikane, T. Hamaguchi, H. Muta, T. Oyama, T. Matsuda, S. Kobayashi, K. Kurosaki // Journal of alloys compounds. — 2003. — Vol. 359. — p. 109.

[75] Robertson, J. Band offsets of wide-band-gap oxides and implications for future electronic devices / J. Robertson // Journal of vacuum science and technology B. — 2000. — Vol. 18. — p. 1785.

[76] Edwards, J. W., Speiser, R., Johnston, H. L. Structure of barium titanate at elevated temperatures / J. W. Edwards, R. Speiser, H. L. Johnston // Journal American chemical society. — 1951. — Vol. 73. — p. 2934.

[77] Hellwege, K. H., Hellwege, A. M. Ferroelectrics and Related Substances; Landolt-Börnstein, New Series, Group III / K. H. Hellwege, A. M. Hellwege, (Ed.). — Berlin: Springer Verlag, 1981. — Vol. 16.

[78] Kennedy, B. J., Howard, Ch. J., Chakoumakos, B. C. High-temperature phase transitions in SrZrO3 / B. J. Kennedy, Ch. J. Howard, B. C. Chakoumakos // Physical review B. — 1999. — Vol. 59. — p. 4023.

[79] Abramov, Yu. A., Tsirelson, V. G., Zavodnik, V. E., Ivanov, S. A., Brown, I. D. The chemical bond and atomic displacements in SrTiO3 from X-ray diffraction analysis / Yu. A. Abramov, V.G. Tsirelson, V.E. Zavodnik, S.A. Ivanov, I.D. Brown // Acta crystallographica B. — 1995. — Vol. 51. — p. 942.

[80] Mitsui, T. Nomura, S. Numerical Data and Functional Relations in Science and Technology - Crystal and Solid State Physics, Landolt-Bornstein, New Series, Group III / T. Mitsui, S. Nomura (Ed.). — Berlin: Springer. — 1982. — Vol. 16.

— Part A.

[81] van Benthem, K., Elsasser, C., French, R. H. Bulk electronic structure of SrTiO3: experiment and theory / K. van Benthem, C. Elsasser, R. H. French // Journal of applied physics. — 2001. — Vol. 90. — p. 6156.

[82] Yashima, M., Hirose, T., Katano, S., Suzuki, Y., Kakihana, M., Yoshimura, M. Structural changes of ZrO2-CeO2 solid solutions around the monoclinic-tetragonal phase boundary / M. Yashima, T. Hirose, S. Katano, Y. Suzuki, M. Kakihana, M. Yoshimura // Physical review B. — 1995. — Vol. 51. — p. 8434.

[83] Glushko, V. P., Medvedev, V. A., Gurvich, L. V. Thermal Constants of Substances / V. P. Glushko, V. A. Medvedev, L. V. Gurvich (Ed.). — New York: Wiley. — 1999. — 6592 p.

[84] Robertson, J. High dielectric constant oxides / J. Robertson // The European physical journal applied physics. — 2004. — Vol. 28. — p. 265.

[85] Hann, R.E., Suitch, P.R., Pentecost, J.L. Monoclinic Crystal Structures of ZrO2 and HfO2 Refined from X-ray Powder Diffraction Data / R.E. Hann, P.R. Suitch, J.L. Pentecost // Journal of the American ceramic society. — 1985. — Vol. 68. — p. 285.

[86] Gurvich, L. V., Veits, I. V. Thermodynamic Properties of Individual Substances / V. P Glushko (Ed). — Moscow: Nauka, 1978. — 568 p.

[87] Lim, S. G., Kriventsov, S., Jackson, T. N., Haeni, J. H., Schlom, D. G., Balbashov, A. M., Uecker, R., Reiche, P., Freeouf, J. L., Lucovsky, G. J. Dielectric functions and optical bandgaps of high-K dielectrics for metal-oxide-semiconductor field-effect transistors by far ultraviolet spectroscopic ellipsometry / S. G. Lim, S. Kriventsov, T. N. Jackson, J. H. Haeni, D. G. Schlom, A. M. Balbashov, R. Uecker, P. Reiche, J. L. Freeouf, G. J. Lucovsky // Journal of applied physics. — 2002. — Vol. 91. — p. 4500.

[88] Baldinozzi, G., Berar, J-F., Calvarin, G. Rietveld refinement of two-phase Zr-doped Y2O3 / G. Baldinozzi, J-F. Berar, G. Calvarin // Materials science forum.

— 1998. — Vol. 278. — p. 680.

[89] Damnjanovic, M., Nicolic, B., Milosevic, I. Symmetry of nanotubes rolled up from arbitrary two-dimensional lattices along an arbitrary chiral vector / M.

Damnjanovic, B. Nicolic, I. Milosevic // Physical review B. — 2007. — Vol. 75.

— 033403.

[90] Damnjanovic, M., Milosevic, I., Vukovic, T., Sredanovic, R., Full symmetry, optical activity, and potentials of single-wall and multiwall nanotubes / M. Damnjanovic, I. Milosevic, T. Vukovic, R. // Physical review B . — 1999. — Vol. 60. — p. 2728.

[91] Bandura, A. V., Evarestov, R. A. Ab initio structure modeling of ZrO2 nanosheets and single-wall nanotubes / A. V. Bandura, R. A. Evarestov // Computational materials science. — 2012. — Vol. 65. — p. 395.

[92] Evarestov, R. A., Bandura, A. V. Symmetry and non-empirical calculations of structure and properties of single- and double-wall SrTiO3 nanotubes / Nanodevices and nanomaterials for ecological security, NATO Science for peace and security; Series B: Physics and biophysics. — Dordrecht: Springer, 2012. — Ch. 7. — p. 75.

[93] Shannon, R. D. Revised effective ionic radii and systematic studies of interatomic distances in halides and chalcogenides / R. D. Shannon // Acta crystallogrphica A. — 1976. — Vol. 32. — p. 751.

[94] Wang, L. Q., Ferris, K. F., Azad, S., Engelhard, M. H. Adsorption of water on (001) surface of SrTiO3 and SrZrO3 cubic perovskites: Hybrid HF-DFT LCAO calculations / L.-Q. Wang, K. F. Ferris, S. Azad, M. H. Engelhard //The journal of physical chemistry B. — 2005. — Vol. 109. — p. 450

[95] Evarestov, R. A., Bandura, A. V. LCAO calculation of water adsorption on (001) surface of Y-doped BaZrO3 / R. A. Evarestov, A. V. Bandura // Solid state ionics.

— 2011. — Vol. 188. — p. 25. 7.

[96] Björketun, M. E., Sundell, P. G., Wahnström, G. Effect of acceptor dopants on the proton mobility in BaZrO3: A density functional investigation / M. E. Björketun, P. G. Sundell, G. Wahnström // Physical review B. — 2007. — Vol. 76. — 054307.

[97] van Duin, A. C. T., Merinov, B. V., Han, S. S., Dorso, C. O., Goddard III, W. A. ReaxFF Reactive Force Field for the Y-Doped BaZrO3 Proton Conductor with Applications to Diffusion Rates for Multigranular Systems / A. C. T. van Duin, B. V. Merinov, S. S. Han, C. O. Dorso, W. A. Goddard III // The journal physical chemistry A. — 2008. — Vol. 112. — 11414.

[98] Merinov, B., Goddard III, W. Proton diffusion pathways and rates in Y-doped BaZrO3 solid oxide electrolyte from quantum mechanics / B. Merinov, W. Goddard III // The journal of chemical physics. — 2009. — Vol. 130. — 194707.

[99] Münch, W., Kreuer, K. D., Seifert, G., Maier, J.Proton diffusion in perovskites: comparison between BaCeO3, BaZrO3, SrTiO3, and CaTiO3 using quantum

molecular dynamics / W. Münch, K. D. Kreuer, G. Seifert, J. Maier // Solid state ionics. — 2000. — Vol. 183. — p. 136.

[100] Memming, R. Semiconductor Electrochemistry / R. Memming. — Weinheim: Wiley-VCH, 2001. — 399 p.

[101] Catlow, C. R. A., Guo, Z. X., Miskufova, M., Shevlin, S. A., Smith, A. G. H., Sokol, A. A., Walsh, A., Wilson, D. J., Woodley, S. M. Advances in computational studies of energy materials / C. R. A. Catlow, Z. X. Guo, M. Miskufova, S. A. Shevlin, A. G. H. Smith, A. A. Sokol, A. Walsh, D. J. Wilson, S. M. Woodley // Philosophical transactions of the royal society A. — 2010. — Vol. 368. — p. 3379.

SAINT-PETERSBURG STATE UNIVERSITY

Manuscript Copy

KURUCH

Dmitry Dmitrievich

Quantum-chemical calculations of perovskite-based nanostructures

Qualification: 02.00.01 - Inorganic Chemistry

A dissertation submitted in conformity with the requirements for the degree of PhD in

Chemistry

Academic Supervisor:

Professor,

Doctor of physical and mathematical sciences, Honored Worker of Science of the Russian Federation

R. A. Evarestov

Saint-Petersburg 2017

Contents

Introduction.....................................................................................................................71

1 Computational methods............................................................................................75

2 Results and Discussion.............................................................................................77

2.1 Bulk Perovskite Structure...................................................................................77

2.2 Calculations of nanolayers..................................................................................79

2.2.1 Modeling of doped surfaces.........................................................................79

2.2.2 Modeling of precursor nanolayers................................................................82

2.3 Nanotube calculations.........................................................................................85

2.3.1 Symmetry of nanotubes................................................................................85

2.3.2 Calculations of single-walled nanotubes......................................................86

2.3.3 Modeling of double-walled nanotubes.........................................................92

2.3.4 Comparison of SrTiO3-, BaTiO3-, SrZrO3- and BaZrO3-based nanotubes .. 95

2.4 Adsorption of water molecules on nanolayer surfaces.......................................97

2.4.1 Unit cell choice.............................................................................................97

2.4.2 Water adsorption at BaO-terminated surfaces.............................................97

2.4.3 Water adsorption at ZrO2- and HfO2-terminated surfaces.........................100

2.4.4 Adsorption on Y-doped surface. Water adsorption sites and corresponding energies...........................................................................................102

2.4.5 Study of different proton positions and estimation of energy barriers......106

2.5 Adsorption of water molecules on nanotube surface.......................................109

2.5.1 Water adsorption models............................................................................109

2.5.2 Adsorption on nanotube surface at low adsorbate density.........................111

2.5.3 Inner surface adsorption at high adsorbate density....................................114

2.5.4 Electronic properties of hydrated and non-hydrated nanotubes.................116

Conclusions...................................................................................................................119

References.....................................................................................................................123

Introduction

The ABO3 perovskite-type crystals are the crystalline solids, where A is a divalent cation and B is tetravalent transition-metal atom. Presently, ABO3 type perovskites are widely studied objects that demonstrate unique electronic, magnetic and optical properties [1, 2], which are the basis for many existing and potential applications [3, 4]. Nanoscale systems based on SrTiO3 and BaTiO3, as well as on SrZrO3 and BaZrO3 are observed in a wide variety of realizations such as nanotubes [5-10], nanorods [11-13], and nanocrystals [11, 14, 15]. These nanomaterials can significantly improve the performance of a variety of electronic, electro-optical, electrochemical, and electromechanical devices and systems [16].

Perovskites can exist in different phase modifications depending on combination of cations A and B [1, 17] and their properties are often symmetry dependent. The description of bulk crystal phases will be given in the section 3.1.

Though a relatively large number of experimental data were obtained [5-10] for synthesized perovskite-based nanotubes (NTs), it is difficult to analyze these data due to lack of models accurately describing the chemical structure and morphology of the perovskite-based NTs. Most of the modern theoretical models [18] of the rolled-up NTs are created through folding of layers cut from bulk crystals. However this procedure usually provides NTs with thin (monolayer) walls only. As a result, most of the theoretical calculations are performed for the inorganic NTs with the wall thickness of about 2 A. Nevertheless, the minimum wall thickness of synthesized SrTiO3 and BaTiO3 NTs was found to be about several tens of Angstroms [5, 8-10].

The reviews [18, 19] of published papers on theoretical modelling of inorganic NTs shows that perovskite NTs were studied only in a few works. Thus, Piskunov and Spohr [20] demonstrated that the density functional theory (DFT) predicted negative strain energies for some NTs folded from thin 2D SrTiO3 nanosheets. The authors pointed out that "negative strain energy minimizes the NT formation energy with respect to the SrTiO3 bulk". Hence, one of the possible reasons for negative calculated strain energies is the large positive (relative to the bulk crystal) formation energy of the nanolayer used for NT folding. As a rule, this is due to the presence of dangling bonds on the nanolayer surface. The arrangement of the dangling bonds on the bended surface may be more favorable than on the flat nanolayer surface.

Many ABO3-type perovskite oxides when doped, typically on the B-site by a lower-valent cation become oxygen-deficient. Such materials often exhibit significant oxygen-ion conductivity at elevated temperatures [21, 22]. When the defective perovskite surface is exposed to a water vapor containing atmosphere at elevated temperatures, it can absorb water into the structure and consequently may become a proton conductor. Perovskite-type proton conductors have been studied [2] as electrolyte materials for

intermediate temperature operating solid oxide fuel cell, because of the lower activation energy for proton migration. In particular, there has been recent interest in BaZrO3, which, when doped with lower-valent cations on the Zr4+ sites and exposed to a humid atmosphere, is known to become a protonic conductor [2, 23]. As with other perovskites, doping is essential to the uptake of protons.

A great deal of theoretical investigations was devoted to the proton migration in defective bulk perovskites, while only a few works [24-26] also consider the oxygen vacancies themselves prior to the water absorption. Thus, in the works of Davies et al. [24], and Islam et al. [25] the atomistic simulation techniques were used to investigate the energetics of defect and dopants and of oxygen migration in several perovskites. Authors of Ref [24] had calculated the energy profile for vacancy migration between adjacent oxygen sites. They found [24] the activation energy of 0.65 eV for the oxygen migration along the edges of the ZrO6 octahedra in Y-doped SrZrO3. Calculations of Islam et al [25] predict Y to be one of the most favorable dopants in BaZrO3. Theoretical study [26] of defect formation in the perovskite oxides in equilibrium with an oxygen-containing atmosphere was based on the density functional calculations and thermodynamic modeling. Authors [26] found that most of the investigated dopants preferably incorporate on Zr sites in the lattice. The interaction between pairs of dopants was found to be repulsive and correlated with the ionic radius of the dopant, while the dopant-oxygen vacancy interaction was found attractive in agreement with the conclusions [25] about the stability of the dopant-vacancy clusters. Results of Sundell et al [26] also show that oxygen vacancies in bulk perovskites are not thermodynamically stable at low temperatures. To the best of our knowledge, there were no calculations of the defective bare and protonated surfaces of BaZrO3 and BaHfO3 crystals.

The adsorption of water on the surface of a metal oxide controls the structure of the electric double layer which is also responsible for the functionality of the electrode sensors and fuel cells. Another important example where absorption plays an important role is the process of photocatalytic decomposition of water at the oxide surface. This process is a promising and environmentally friendly method for production of gaseous hydrogen. The role of the water molecules should obviously increase when the dimension of the crystallites decreases up to nanolayers, nanotubes, and nanorods due to a significant rise of their specific surface area. An adequate description of such systems can be achieved only if the actual atomic structure of the adsorbent-adsorbate interface is known. Modern quantum chemistry and molecular statistical physics offer a lot of possibilities for such description. In particular, ab initio quantum chemical methods provide a tool to identify specific structural factors contributing to the prevalence of molecular or ionic forms of water on the oxide surface, where the results of experimental measurements are generally allow ambiguous interpretation. To date, ab

initio studies of water adsorption on the crystal surface were made for a wide class of metal oxides. Among them the binary and ternary oxides of group IV metals are of particular interest because of their important applications. The significant information on the structure of the bare [18, 27-30] and hydrated [18,31-38] surfaces of these substances is available. In particular, the recent review [27] presents the results of calculations of surface relaxations, energetics, and bonding properties for ABO3 perovskite (001), (011) and (111) surfaces using predominantly a hybrid description of exchange within a scope of the density functional theory (DFT). The ab initio studies [31, 32, 34-38] of hydration of pure and modified surfaces of perovskites (strontium titanate, zirconate and hafnate) provide the useful information about the state of adsorbed water molecules in these systems. However, both experimental and theoretical investigations of water adsorption on BaZrO3 and BaHfO3 surface are practically absent.

The interaction of water molecules with the surface of nano-objects attracts significant attention in recent years. The adsorption of water molecules on the surface of substituted and unsubstituted carbon nanotubes, graphite and graphene has been explored in a number of theoretical studies [39-42]. Simulation of water adsorption on inorganic nanotubes is discussed however only in a few publications which deal particularly with nanotubes of titanium [43] and zinc [44] oxides. The authors of the first mentioned work [43] calculated the energy of water adsorption on a single-walled TiO2 nanotube rolled up from nanolayers parallel to the (101) surface of anatase using DFT. Water in either molecular or dissociated form was placed on the outer or inner surfaces of the tube. The authors concluded [43] that the curvature of the nanotube surface plays an important role in the dissociative adsorption of water and a minor role in the adsorption of molecular water. The authors of Ref. [44] performed all-electron DFT calculations of water adsorption on the surface of non-periodic ZnO nanotubes which are placed between two gold plates. One to four water molecules was placed on the external surface of the aforementioned nanotubes. It has been shown that the presence of adsorbed H2O molecules on the surface leads to a decrease in nanotubes conductivity. Authors concluded [44] that ZnO-based nanotubes can be used to detect and monitor the presence of water molecules by applying bias voltages.

The possibility of application of nano-objects for the photocatalytic decomposition of water is investigated in several recent studies. For example, the band gaps and band edge positions in the stoichiometric [45] and non-stoichiometric [45, 46] nanorods cut from a cubic SrTiO3 crystal were analyzed. The preliminary reports by Wessel and Spohr [47] on modeling of SrTiO3 nanotubes in an aqueous environment using the ab initio molecular dynamics should be mentioned among the latest studies. The effect of doping of TiO2 and SrTiO3 nanotubes on water splitting was recently studied in the work by Piskunov et al. [48].

In this work we report the results of the first-principles computations of the structure and properties of bulk perovskite crystals, nanolayers, single- and double-walled nanotubes, as well as the characteristics of water adsorption on different surfaces of nanolayers and nanotubes.

The brief review presented above has demonstrated that theoretical modeling of the protonation and adsorption of water molecules on the perfect and defect perovskite surfaces were very rare or absent. There were only a few research articles reporting the study of the structure and stability of single-walled perovskite-based nanotubes, while the multiwalled nanotubes were not considered at all in the available literature. Practically no studies have been conducted on the interaction of water molecules with the surface of nanotubes. These important concerns determine the relevance of present work.

1 Computational methods

Quantum chemical calculations have been carried out within the periodic DFT using a hybrid exchange-correlation functional PBE0 [49, 50]. This functional had been successfully applied for calculation of both perovskite bulk and surface properties [18], as well as the parameters of hydrogen bonds involving water molecules [51]. To perform the first-principles calculations, the CRYSTAL computer code [52, 53] was used. All calculations presented in this work were performed within the LCAO approximation. The one important aspect which determines the type of quantum-chemical calculations is the choice of the basis set for the crystalline orbitals expansion. PW basis which has some technical advantages over LCAO approximation is used in the most of present-day simulations of periodic systems. At the same time, the periodic LCAO method also provides the good quality of calculated properties [18]. Moreover, the LCAO calculations of the nanoobjects do not require artificial repeating of the systems in one or two directions [18], in contrast to PW calculations where it is necessary to restore 3D periodicity due to incompleteness of 2D and 1D PW basis in 3D space. Finally, it should be noted, that widely applied for molecular systems Mulliken population analysis is easily generalized for the periodic systems [54, 55] if LCAO expansion of crystal orbitals is employed.

The Stuttgart [56, 57] small-core pseudopotentials have been used for Ba, Y, Zr and Hf core-valence electron interactions in early [33, 58] works and the CRENBL [59-61] small-core pseudopotentials for the Ba, Sr, Zr and Ti atoms in succeeding [62-65] works. We used the small-core pseudopotential, which replaces only inner core electrons, whereas the orbitals for subvalence electrons as well as for valence electrons are calculated self-consistently. To prevent the quasilinear dependences, the diffuse s-, p-, d-, f- Gaussian type orbitals with exponents less than 0.1 a.u. were removed from the original basises, and the exponents of other polarization functions have been reoptimized for the periodic calculations. The polarizing f-function for barium atom existing in the initial basis set has been excluded due to negligible contribution in the

9-1-

case of Ba ionic state. For O atom we used all-electron 8-411G(d) basis set [66-68] which was adjusted for calculations of crystalline perovskites. To study the defective perovskite surface the oxygen vacancies can be modeled as ghost atoms, by deleting the nuclear charge and shell electron charges, but leaving the basis set centered at the atomic position. Contracted 6-31G(p) basis set [68, 69] has been used for hydrogen atom. The reciprocal space integration was performed by sampling the Brillouin zone of the bulk crystals with an 8 x 8 x 8 Monkhorst-Pack mesh [70] in early [33, 58] works and 12 x 12 x 12 Monkhorst-Pack mesh [70] in succeeding [62-65] works. A number of k-points in each periodic dimension was reduced inversely to the corresponding cell

constant to provide approximately uniform Brillouin zone sampling for the orthorhombic bulk SrZrO3, all nanolayers and NTs. The convergence criterion has been chosen to be the total energy difference less than 10-6 a.u. in the two successive cycles of SCF procedure. The method chosen provides a good quality of the calculated bulk properties for all studied crystals.

The built-in capabilities of CRYSTAL computer code [52, 53] and the Surface Builder module of the Materials Studio software package [71] were used for preparing the initial structure of nanosystems and for symmetry analysis. The lattice parameters and fractional positions of all the atoms in the considered systems were fully optimized.

2 Results and Discussion

2.1 Bulk Perovskite Structure

Perovskites ABO3 can exist in different phase modifications. The cubic structure inherent to all perovskite crystals is characterized by the space group Pm-3m. SrZrO3 and BaTiO3 crystals also exhibit other phases. The sequence of phase transitions in SrZrO3 passes from orthorhombic Pbnm to orthorhombic Cmcm at 970 K, then to tetragonal I4/mcm at 1100 K, and lastly, to the cubic Pm-3m phase at 1400 K. The high symmetry cubic Pm-3m bulk phase of BaTiO3 crystal is stable at high temperatures and displays a series of three transitions to the tetragonal I4/mcm at 393 K, orthorhombic Amm2 at 278 K and rhombohedral R3m at 183 K. In this work we have studied all the cubic crystals and orthorhombic SrZrO3 phase which is characterized by the space groups Pbnm. The lattice constants of all crystals were optimized and obtained values have been used in all surface simulations. Calculated lattice parameters, bulk modules, cohesion energies, and band gaps are given in Table 1.

The Murnaghan [72] equation of state is used to estimate the bulk modules of the considered crystals. Calculated values are published in our papers [33, 58, 62-65], and agree well with the available experimental data (Table 1).

In connection with oxygen vacancy study the monoclinic (P21/c) ZrO2 and HfO2 and cubic (Ia-3) Y2O3 crystal structures were fully optimized to estimate the energies E(ZrO2), E(HfO2) and ET2O3). Calculated bulk properties of ZrO2, HfO2 and Y2O3 crystals are compared with the experimental data in Table 1.

Table 1. Calculated lattice parameters, bulk modules, cohesive energies and band gaps for bulk crystals.

Crystal Space group Lattice parameters, A Bulk modulus, GPa Cohesion energy, eV Band gap, eV

BaHfOs Pm-3m 4.189 [33] (4.180 [73]) 175 [33] 33.3 [33] 5.8 [33]

4.194 [33] 5.4 [33] (5.3 [75])

BaZrOs Pm-3m 4.203 [64] (4.190 [74)] 172 [33] 32.8 [33]

BaTiOs Pm-3m 3.986 [62] (3.996 [76]) 201[62] (162 [77]) 29.7 [62] (31.6 [77]) 3.7 [62] (3.2 [77])

SrZrOs Pm-3m 4.123 [63] (4.154 [78]) 206 31.6 5.49

Pbnm 5.750, 5.819, 8.173 [63] (5.786, 5.815, 8.196 [78]) 197 - 6.01

SrTiOs Pm-3m 3.885 [65] (3.900 [79]) 226 (183 [80]) 29.6 3.8 [65] (3.3 [81])

ZrO2 P2i/c 5.152, 5.208, 5.307; 99.5 [58] 22,0 [58] 6,4 [58]

(5.150, 5.208, 5.317; 99.2 [82]) (22.7 [83]) (5.8 [84])

HfO2 P2i/c 5.154, 5.195, 5.308, 99.3 22,5 6,94

(5.117, 5.175, 5.292, 99.2 [85]) (22.0 [86]) (5.8 [87])

Y2O3 Ia-3 10.576 [58] (10.596 [88]) 34.8 [58] (36.1 [83]) 7,2 [58] (6.0 [84])

*Experimental values are given in parentheses.

2.2 Calculations of nanolayers

2.2.1 Modeling of doped surfaces.

Alternating (001) atomic planes in bulk perovskites have a different composition. Consequently, the AO and the BO2-terminated surfaces can exist for the (001) cubic and (001) and (110) orthorhombic perovskite 2D structures. In all considered perovskites, the AO and BO2 layers are charge neutral, so that both (001) surface types are formally nonpolar. In early [33, 58] works we have studied thick non-stoichiometric nanolayers composed of 9 alternating (001) atomic planes from cubic BaHfO3 and BaZrO3. In succeeding [62-65] works the thin stoichiometric nanolayers composed of 2, 4 or 6 alternating (001) atomic planes from BaZrO3, BaTiO3, SrZrO3 and SrTiO3 were investigated as precursors for nanotube generation.

To investigate the (001) surfaces of cubic BaZrO3 and BaHfO3 we have studied symmetrically terminated (non-stoichiometric) 9-layer slabs. Inversion symmetry provides the equivalence of upper and lower nanolayer surfaces. During the surface structure optimization, the positions of all atoms were fully relaxed except atoms in the slab middle plane which was fixed at the bulk geometry. The formation energy of two different terminations cannot be determined separately without the additional assumptions. However, average surface energy Esurf has a clear physical meaning because both terminations appear simultaneously upon the surface creation. For the models with odd number of atomic planes surface energy can be defined by the relation:

Esurf = (EAO + EBO2 -nEbulk) / ^SNlX (1)

where EAO and EBO2 are nanolayer total energies per cell for AO and BO2 terminations, respectively, Ebulk - bulk crystal total energy per formula unit, and n is the overall number of formula units per 2D unit cell in both nanolayers. The multiplier 4 before SNL - 2D unit cell surface area, corresponds to formation of 4 faces upon crystal cleaving.

Our calculations give the close average surface energies for (001) surfaces of BaZrO3

_'j _'j

and BaHfO3 - 1.21 J-m and 1.27 J-m , respectively.

To study the atomic and electronic structure of the surface defects in BaZrO3 and BaHfO3 the Y-doped nanolayers have been considered. It is known [2, 25] that Y dopant is largely incorporated onto the B rather than A site, so BO2-terminated (001) surface is most suitable for the vacancy creation. Presumably, Y atom should be located in the vacancy vicinity due to effect of dopant-vacancy association [24, 25].

To model the defective (001) ZrO2- or HfO2-terminated surface of BaZrO3 and BaHfO3, correspondingly, we have studied the symmetrically ended (non-stoichiometric) 9-plane nanolayers consisting of alternating 4 BaO and 5 ZrO2 or HfO2 planes. To create vacancy one of the surface oxygens is removed and two neighboring Zr or Hf atoms are replaced by Y atoms in the surface unit cell. Using 2 x 2 and 3 x 3

surface supercells, two models have been constructed: (I) with a vacancy on one side of nanolayer only; (II) with the vacancies on the both sides of slab (the inversion symmetry provides the equivalence of the both surfaces in this case). The different locations of dopant atoms near the vacancy are examined in our work. Coordinates of all atoms were optimized, except the positions of middle-plane atoms in the model II.

The formation of the surface vacancy was described by the following reaction:

Sur + Y2O3 ^ Sur''Y2^ + 2MO2, (M = Zr, Hf), (2)

where Sur is BaZrO3 or BaHfO3 model system (nanolayer) with pristine ZrO2 or HfO2 surface, Sur''Y2^-the same system with one oxygen removed and two Zr or Hf atoms substituted by Y dopants in the surface unit cell. All involved in Eq. 2 (surface and bulk) phases can be treated within the same level of theory that provides equal precision of calculated quantities and/or the cancellation of possible errors. This is confirmed by the close formation energy values for the different models in consideration. Accordingly, the vacancy formation energy Eform relative the stable ZrO2, HfO2 and Y2O3 bulk phases is calculated by the equation:

Eform = (1/«)[E(Sur''Y2n^n) + 2«E(MO2) - E(Sur) - nE(Y2O3)], (M = Zr, Hf), (3)

where n = 1 or 2 for models I and II, respectively.

We considered the different positions of Y atom in the top and next-to-top atomic planes to establish the most favorable surface vacancy-dopant configurations. In Figure 1 we show four possible symmetry-nonequivalent vacancy arrangements with both Y atoms in the top layer of 2x2 surface supercell (neighboring atoms are shown only for view from top). In this case Y/Zr or Y/Hf ratio in the surface layer is equal to 1/1, whereas oxygen vacancy concentration is 1/8.

0v0 0v1 -1v1 v01

Figure 1. Symmetry-nonequivalent positions of Y atoms near the vacancy in the top layer of 2x2 supercell of BaZrO3 or BaHfO3. Legend: large light sphere - vacancy; middle yellow spheres - O atoms; small blue spheres - Zr or Hf atoms; small violet spheres - Y atoms.

In Table 2 we present the formation energies obtained with and without the ghost vacancy basis set using both slab models. The addition of ghost basis at the vacancy site reduces the formation energy by about 10 kJ-mol-1 but does not affect the surface relaxation. According to data in Table 2, the most favorable structure is 0v1 for BaZrO3,

and for BaHfO3 - 0v0 structure (explanation of this label is given in the second column of Table 2). It is seen in Table 2 that Model I and Model II produce close results. This justifies the sufficiency of the 9-plane nanolayer to represent the defective surface in BaZrO3 or BaHfO3 crystals. It also means that any of two models can be used for reproducing the properties of defective surface. The most of subsequent results reported below are obtained basing on Model II.

Table 2. Formation energies (Eform, kJ-mol-1) of surface oxygen vacancy surrounded by Y atoms and located in the top (001) surface layer of BaZrO3 and BaHfO3.

Crystal, model, cell

BaZrO3 BaHfO3

Vacancy label and Y atom positions I, 2x2 II, 2x2 II, 3x3 II, 2x2 II, 3x3

0 Y(1)(-1/2,°,0) : Y(2)(1/2,0,0) 152 (161) 154 (163) 140 146 131

Y(1)(-1/2,0,0) : Y(2)(1/2,1,0) 144 (155) 145 (155) 130 149 136

Y(1)(-1/2,-1,0) : Y(2)(1/2,1,0) 179 131 197 148

Y(1)(1/2,0,0) v01: Y(2)(1/2,1,0) 205 212

*Formation energies obtained without ghost basis set are given in parentheses.

In the case of 3 x 3 supercell the Y/Zr or Y/Hf ratio in the surface layer is equal to 2/7, and vacancy concentration is 1/18. We recalculated three selected structures using this extended cell. Obtained formation energies are also given in Table 2. The cell increasing from 2 x 2 to 3 x 3 leads to the decrease of Eform by 15 kJ-mol-1 or more. Supposedly, this can be explained by weakening of vacancy-vacancy and Y-Y interactions due to reducing of their surface concentration. However, the most favorable structure for 3 x 3 cell is the same as was obtained for 2 x 2 cell, namely, 0v1 for BaZrO3 and 0v0 for BaHfO3. In the structure 0v1 the first Y atom is located at 2.3 A, whereas the second one - at 4.6 A from the vacancy center. As can be seen from Figure 2, the incorporation of Y atoms in ZrO2- or HfO2-terminated surface is accomplished by a noticeable reconstruction of the surface structure around the oxygen vacancy. We found that the relaxation of top surface layer near the vacancy does not depend considerably on using 2 x 2 or 3 x 3 supercell. The nearest oxygen atoms shift towards the vacancy center, whereas Zr, Hf and Y atoms shift in opposite direction.

(a) (b)

Figure 2. Atomic relaxations in the vicinity of oxygen vacancy 0v1 in the top layers of (001) surface of BaZrO3 and BaHfO3 with using of 3 x 3 supercell. (a) in (001) plane; (b) in (010) plane. See Figure 1 for legend.

To study the possibility of the vacancy migration we have estimated the energy barrier between equivalent vacancy sites at the defective Y-doped perovskite surface. A saddle point on the potential energy surface was obtained by the structure optimization with C2/m symmetry constraints (no ghost basis has been added to vacancy site). In the transition state the oxygen atom occupies an intermediate position (Figure 3) between two equivalent vacancy sites and it is lifted above the surface by approximately 0.4A.

Figure 3. Atomic structure of transition state for oxygen vacancy migration in top layer of (001) surface of BaZrO3 and BaHfO3. Top view on the 2 x 2 supercell. See Figure 1 for legend.

2.2.2 Modeling of precursor nanolayers.

When allowed to relax, this structure moves to the final (or initial) vacancy state. Calculated energy barriers of 32 kJ-mol-1 for BaZrO3, and of 22 kJ-mol-1 for BaHfO3 are not large, and migration of surface oxygen vacancy is very probable at high temperatures. Moreover, we can suppose that energy barrier for surface vacancy

migration is much smaller than that in the bulk phase due to reducing of steric restrictions. Thus, we have made the estimation of energy barrier for the oxygen vacancy migration from the top (ZrO2 or HfO2) to the next (BaO) layer. This corresponds to the oxygen migration down along the edge of the top YO6 semioctahedra. We obtained the value about 77 kJ-mol-1 for this barrier which is more than twice larger than barrier for the surface vacancy migration.

Stoichiometric nanosheets composed of alternating (001) AO and BO2 (A = Sr or Ba, B = Zr or Ti) atomic planes from the cubic SrZrO3, BaZrO3 and BaTiO3 phase, (001) or (110) atomic layers from orthorhombic SrZrO3 phase have been constructed and studied as nanotube precursors. The nanolayer formation and surface energies were calculated using the equations:

where Enl is nanolayer total energy per 2D unit cell, Ebulk - bulk crystal total energy per formula unit, n is the number of formula units in the nanolayer 2D unit cell, and the multiplier 2 before SNL (which is 2D unit cell surface area) corresponds to formation of two faces upon crystal cleaving.

The lattice parameters and atomic structure of initial nanolayers have been optimized before folding. It has been proved that the obtained lattice constants of two-dimensional square or rectangular lattices are only slightly decreased compared to the bulk values (Table 3). However, nanolayers consisting of 4-6 planes exhibit noticeable relaxation. Atomic shifts, generally, depend on the type of the layer, nevertheless, the oxygen atoms basically relax outward, while the metal atoms relax inward.

Eform = (ENL/n - ^ulkX Esurf = (ENL - nEbulk)/(^»SNlX

(4)

(5)

Table 3. Calculated lattice parameters, formation and surface energies of nanosheets cut from SrZrOs, BaZrOs and BaTiOs.

Parent phase and surface index

Value System SrZrOs, BaZrOs, BaTiOs, SrZrOs,

Pm-3m Pm-3m Pm-3m Pbnm

001 001 110

Lattice 2 layer slab 4.115 4.141 3.880 a=5.782, b=5.867 a/V2=5.802, b/V2=5.854

constants, 4 layer slab 4.104 4.157 3.920 a=5.766, b=5.778 a/V2=5.704, b/V2=5.813

a, b, Â 6 layer slab 3.940

Formation 2 layer slab 282.2 245.7 211.2 276.7 275.6

energy, 4 layer slab 145.7 142.8 124.2 157.2 156.8

kJ-mol"1 6 layer slab 86.2

Surface 2 layer slab 1.22 1.19 1.16 1.s6 1.s5

energy, J^m"2 4 layer slab 6 layer slab 1.44 1.s7 1.34 1.39 1.57 1.57

2.3 Nanotube calculations

2.3.1 Symmetry of nanotubes

The initial structure of the nanotubes can be obtained as a result of the so-called layer folding [19, 89], which means the construction of the cylindrical surfaces of nanotubes by rolling up the two-periodic (2D) crystalline layers. In this approach, the two vectors should be defined for the particular nanotube chirality (n1, n2): rolling vector R and translation vector L. The vector R determines chirality and diameter of the nanotube, turning into circumference of the cylindrical surface upon nanotube rolling up. The vector L is the smallest 2D translation vector normal to R, and it determines the 1-D periodicity along the nanotube axis. Both vectors in the basis (a1, a2) of layer unit cell (UC) are represented by integer coefficients: R = n1a1+n2a2, L = l1a1+l2a2. Possible chirality types (n1, n2) for the given 2D layer symmetry are determined by the orthogonality condition between R and L [89] The nanotube symmetry is described by line groups. Line group families compatible with the given layer group are established by Damnjanovic et al. [89] and discussed in details in monography by Evarestov [19].

The cubic phase of the crystals provides (001) stoichiometric nanolayer with a 2D periodic group of P4mm. After the rolling procedure [89] this layer can generate (n, 0) and (n, n) single-walled nanotubes (SWNTs) with the line symmetry group belonging to line group family 11 (point symmetry Dnh) and to family 13 (point symmetry D2nh), respectively. The presence of the helical axis (2n)n distinguishes the latter chirality from the former. The (001) nanolayers obtained from the orthorhombic SrZrO3 crystal have a rectangular cell with a 2D symmetry of Pb11 and can be folded with two different types of chirality depending on the direction of the rolling vector. The NTs with (n1, 0) chirality belong to the line group family 2 (S2n), and NTs with (0, n2) chirality belong to family 7 (Cm) [89].

Line symmetry group of double-walled nanotubes (DWNTs) can be found as an intersection of symmetry groups of its singlewall (SW) constituents [90]. Let (n1, n2) and (n1', n2) define the chiral vectors of SW constituents of DWNT. The DWNT axial point group CN is the principal axis subgroup of DWNT line group with N =G(n, n') = G(n1, n2, n1', n2), where G is the greatest common divisor of n1, n2, nv and n2. Only NTs composed exclusively of either (n, n) or (n, 0) SW constituents may have additional mirror and glide planes, as well as two-fold axes perpendicular to the tube axis. The DWNTs (n1, 0)@(n2, 0), (0, n1)@(0, n2) and (n1, n1)@(n2, n2) are commensurate, if the minimal translation period for such DWNTs is the same as for the constituents. The symmetry group of (n1, 0)@(n2, 0) or (0, n1)@(0, n2) DWNTs belongs to the same family as its SW constituents. The two possibilities [90] arise for DWNTs (n1, n1)@(n2, n2) generated from the cubic phase: (a) both n1/N and n2/N are odd; (b) one or both n1/N

and n2/N are even. In the case (a), the line symmetry groups are the same as for their SW constituents and belong to family 13 (the order of screw axis is 2N). In the case (b), the rotations around the screw axis of order 2N are replaced by rotations around the pure rotation axis of order N and the DWNT line symmetry group belongs to family 11.

2.3.2 Calculations of single-walled nanotubes

We consider stoichiometric single-walled ABO3 NTs of different diameters and chiralities. The initial structures of the nanotubes were constructed by rolling up the relaxed stoichiometric ABO3 nanolayers consisting of alternating AO and BO2 atomic planes. We have considered the chiralities (n, 0) and (n, n) for cubic phase nanolayers, and the chiralities (n1, 0) and (0, n2) for layers of the SrZrO3 orthorhombic phase. Then, two different types of NTs can be constructed with any chirality: (I) with AO outer shell and (II) with BO2 outer shell. The inner shell has alternate composition because of the choice of an even number of planes in the stoichiometric nanolayer.

To estimate the SWNT stability relative to the precursor nanolayer structure we use the NT strain energy Estr:

Estr(NT) = E/(NT)// - Em(NL)/m, (6)

where E/(NT) is the total energy of NT unit cell consisting of / bulk formula units, and Em(NL) is the total energy of the optimized nanolyer unit cell containing m bulk formula units.

In Figure 4 we depict the dependence of the 2-layer NT strain energy on its average diameter DNT calculated as the sum of Rmin and Rmax, which are the distances from NT axis to the nearest and the most remote atoms, correspondingly. Figures. 4a and 4b show that the strain energy of AO-terminated NTs and ZrO2-terminated BaZrO3 NTs decreases with an increase of the size of NTs, whereas the strain energy of ZrO2-terminated SrZrO3 and BaO-terminated BaTiO3 NTs goes up with such an increase. In the case of BaO-terminated BaTiO3 and ZrO2-terminated SrZrO3 NTs the negative strain energy is a consequence of the large positive formation energy of the initial nanolayers. On the other hand, the AO-terminated 2-layer NTs have positive or zero strain energies because of unfavorable interaction between O and Sr atoms on the external SrO surface due to its distension. The formation energy of the 2-layer cubic BaZrO3 nanolayer is less than that of SrZrO3 by about 40 kJ-mol-1. As a result, the strain energy of the ZrO2-terminated BaZrO3 NTs is higher than that of the analogous SrZrO3 NTs. The difference between the strain energies of ZrO2 and BaO terminations is not as large as in the case of SrZrO3 NTs. Figures. 4a and 4b demonstrate that the strain energy of the BaO-terminated BaTiO3 NTs is negative and exhibits weak dependence on the tube diameter. In Figure 4c we can see that the values of strain energies of (n1, 0) and (0, n2) NTs generated from the orthorhombic phase are very close, and thus indicate a

small structural difference between these tubes. In all the cases considered, the strain energy tends to zero with an increase of the NT diameter, since the total energy (per formula unit) of the NT tends to the energy of the corresponding nanolayer.

Figure 4. Dependence of strain energy Estr on average nanotube diameter Dav for 2-layer SrZrO3, BaZrO3 and BaTiO3 NTs folded from (001) layers of the cubic phase with chirality (n, 0) (a) and (n, n) (b), for SrZrO3 NTs folded from (001) layers of the orthorhombic phase with chirality (n1, 0) and (0, n2) (c).

The atomic positions on the boundary NT surfaces exhibit a noticeable relaxation. Nevertheless, NTs folded from 2-layer cubic nanolayers, generally, keep the initial structure (Figure 5) provided that the original rotohelical symmetry is maintained during the optimization procedure. However, if the symmetry decreases due to removing of the screw rotations, the SrO-terminated NTs generated from the cubic phase with (n, n) chirality show an additional relaxation accompanied by a significant decrease of energy. The ZrO2 shells in these tubes separate into two atomic (O and Zr) subshells, as it takes place in the NTs generated from the orthorhombic SrZrO3 phase. The SrO-terminated NTs folded from (001) layers of the orthorhombic phase (Figure 5b) with chirality (ni, 0) or (0, n2) provide equal structures with almost zero strain energies (Figure 4c).

(b)

S*o

O Q,

id °o °0

0*5°oO; •o 1

%

, „ 0«0 w

0« °o

Ct-OI^O O.O.

(c) (d)

Figure 5. Cross-sectional view of optimized structure of cylindrical SrZrO3 NTs: (a) ZrO2-terminated 2-layer NT (20, 0) folded from layers of cubic phase; (b) SrO-terminated 2-layer NT (14, 0) folded from layers of orthorhombic phase; (c) ZrO2-terminated 4-layer NT (16, 16) folded from layers of cubic phase; (d) SrO-terminated 4-layer NT (32, 0) folded from layers of cubic phase. Legend: small green spheres - Sr or Ba atoms; small blue spheres - Ti or Zr atoms; middle yellow spheres - O atoms.

Note that the rolling vectors (n1, 0) and (0, n2) for (001) layers of the orthorhombic phase are analogous to the rolling vectors (n1, n1) and (-n2, n2) for (001) layers of the cubic phase. Indeed the orthorhombic primitive cell can be regarded as a slightly

deformed cubic 21/2acubx21/2acubx2acub supercell. Because of this, the orthorhombic [100] and [010] directions approximately correspond to [110] and [-110] directions in the cubic phase. Therefore, the orthorhombic phase manifests itself in SrO-terminated NTs folded from 2-layer slabs of both the cubic and orthorhombic SrZrO3 phases.The nanotubes folded from 2-layer slabs have very thin walls and cannot represent the structure of experimentally observed perovskite NTs which have the wall thickness of about 30 A or greater [5, 8-10]. More realistic models can be obtained by folding the thicker layers. However, due to a large difference between the circumference of the inner and outer surfaces, great overstrains may appear in the folded thick nanolayer (see discussion below) which leads to a break of interatomic bonds and overall reorganization or even disintegration of the NT walls upon structure optimization [62]. In the case of perovskites it means that the 6-layer or thicker slabs are unfavorable for the construction of the rolled-up NTs. One of the alternatives is to consider merging of walls of double- or multiwalled NTs.

Let us first consider the results obtained by rolling up of 4-layer slabs cut from SrZrO3, BaZrO3 and BaTiO3 bulks. If the maximal possible order of the rotation axis is preserved, the AO and BO2 external surfaces of (n, 0) NTs and BO2 external surfaces of (n, n) NTs undergo a prominent relaxation while maintaining the initial cylindrical shape (Figure 5c and 5d). The atomic relaxation in ZrO2 and TiO2 shells is accompanied by the breaking of some transverse B-O bonds followed by formation of B=O bonds that are shorter than bonds in the bulk phases (1.9 A vs. 2.1 A). Thus, the relaxed structure of (n, n) NT external BO2 shell is composed of the isolated BO2 fragments connected with the underneath oxygen atoms (Figure 5c). The formation energy of such NTs is relatively high (> 200 kJ-mol-1) as well as their strain energy.

In the case of NTs with the (n, n) chirality, the spontaneous splitting of the AO-terminated (A = Sr, Ba) 4-layer NTs into two separated shells was observed at n < 16 for SrZrO3 and BaZrO3 NTs. The internal shell consists of one ZrO2 layer, and the external shell consists of three AO-ZrO2-AO layers (Figure 6a). The internal zigzag ZrO2 SWNT has a hexagonal morphology. Such a splitting is possible due to close translational periods tNT of zirconate-based (n, n) NTs (tNT -5.8 A) and zirconia-based (n, 0) NTs (tNT -5.6 A [91]). To estimate the stability of split DWNTs the atomic positions in each of the two SW constituents were optimized separately and the binding energies Ebind between two NTs were calculated:

Ebind(NT) = [E(DWNT) -E1(SWNT)-E2(SWNT)]/n, (7)

where E(DWNT) is the total energy of DWNT, E1(SWNT) and E2(SWNT) are the total energies of NTs that comprise DWNT, and n is the number of bulk formula units in DWNT.

(a) (b)

Figure 6. Cross-sectional view of AO-terminated 4-layer (16, 16) NTs: (a) split structure of SrZrO3; (b) non-split structure of BaZrO3. See Figure 5 for legend.

The negative binding energies of SW constituents suggest that the split DWNTs are energetically more favorable than two isolated SWNTs (Table 4). At n >16 AO-terminated (n, n) NTs are not fragmented and exhibit higher formation energies. However, the inner ZrO2 subshell of those NTs has a structure comparable to that of the split case (Figure 6b). The splitting behavior was not observed in BaTiO3 NTs, presumably because of incommensurability of the periods of (n, 0) titania-based and (n, n) titanate-based NTs.

Table 4. Formation energies (Eform) of DWNTs generated by splitting of 4-layer SWNTs, binding energies (Ebind) of their two shells (kJ-mol-1), and corresponding band gaps (Eg, eV).

Initial chirality SrZrO3 BaZrO3

Eform Ebind Eg Eform Ebind Eg

(12, 12) 189 -58.7 5.3 179 -26.8 3.8

(14, 14) 195 -49.3 5.1 181 -16.3 2.6

(16, 16) 210 -33.2 4.5 209* - 4.8

*

Splitting does not occur in (16, 16) BaZrO3 4-layer SWNT.

The NT wall splitting is one of the ways to reduce the bending strain for NTs folded from 4-layer slabs. There are other ways to do so by symmetry reducing. In this work we have studied two possibilities: (1) decrease of the rotation axis order; and (2) rolling up of layers of the orthorhombic phase (SrZrO3 only).

When the rotation or screw axis order is decreased, a prominent reconstruction can be expected. Cross-section of the NTs is no longer a circumference and resembles a polygon, consisting of 4-layer blocks which are partially split (Figure 7a) in AO-

terminated NTs, or slightly bent (Figure 7b) in BO2-terminated NTs. The number of blocks is equal to the order of the adopted rotation axis. Formation energy of low-symmetry NTs is noticeably less (by 50 kJ-mol-1 in some cases, see Table 5) than formation energy of the tubes with a full set of symmetry operations. The lower the rotational axis order and the wider the 4-layer block, the smaller is the formation energy of the polyhedral NTs. As follows from Table 5, BO2-terminated NTs have in general higher formation energies and bigger diameters than AO-terminated NTs.

(c) (d)

Figure 7. Cross-sectional view of SrZrO3 4-layer NTs: (a, b) reconstructed structure of NTs folded from the layers of cubic phase with chirality (16, 16); (c, d) structure of NTs folded from the layers of orthorhombic phase with chirality (16, 0). (a, c) SrO-terminated; (b, d) ZrO2-terminated. See Figure 5 for legend.

The rolling up layers of the orthorhombic phase (Figure 7c and 7d) can also provide relatively stable NTs maintaining the cylindrical shape. SrO-terminated NTs shown in Figure 7c are more favorable and, in general, retain the wall structure of the initial 4-layer slab. The calculated band gaps in 4-layer NTs (Table 5) vary from 4.5 to 6.1 eV. Reconstruction of NT walls does not lead to unequivocal change of the band gaps; relatively high values of Eg were obtained for SrZrO3 NTs, and particularly low values were found for BaZrO3 NTs (Tables 4 and 5).

Table 5. Comparison of translational periods (tNT, A), average diameters (DNT, A), formation energies (Eform, kJ-mol-1) and band gaps (Eg, eV) of cylindrical and reconstructed 4-layer SWNTs consisting of 64 SrZrO3 formula units.

Chirality, termination Cylindrical Reconstructed

(parent phase) ¿NT dnt Eform Eg Figurea ¿NT DNT Eform Eg Figurea

(32, 0) SrO-terminated (cubic) 4.0 44.0 208 6.0 5d 4.1 42.4 187 5.9 -

(16, 16) SrO-terminated (cubic) 5.7 24.5 210 4.5 6ab 5.7 26.4 160 5.4 7a

(16, 0) SrO-terminated (orthorhombic) 5.8 26.4 186 4.5 7c - - - - -

(32, 0) ZrO2-terminated (cubic) 3.9 46.3 192 6.1 - 4.0 44.1 188 5.5 -

(16, 16) ZrO2-terminated (cubic) 5.9 30.1 256 5.7 5c 5.8 30.2 199 4.9 7b

(16, 0) ZrO2-terminated (orthorhombic) 5.9 30.4 214 6.0 7d - - - - -

aThe reference to corresponding image is given, if it exists.

bSplit structure.

2.3.3 Modeling of double-walled nanotubes

We constructed DWNT initial structures by combining two 2-layer SWNTs of the same chirality with different diameters so that the initial intertube distance Aw at original rolled-up geometry changes from 3 to 6 A. Each of the adopted DWNTs was composed of about 300 atoms altogether. We attempted to use the chiralities (n1, 0)@(n2, 0) and (n1, n1)@(n2, n2) with n1 and n2 having the greatest common divisor N of the sufficiently large value to preserve the rotational symmetry as much as possible. To study the stability of DWNTs we used the binding energy Ebind which was the energy gain (per formula unit) upon combining of two constituent SWNTs. The calculated binding energies of selected SrZrO3 and BaZrO3 DWNTs of both terminations are given in Table 6.

After structure optimization, the SW components exhibit almost the same structure as their constituents if the initial intertube distance Aw is greater than 6 A. The walls preserve a cylindrical shape with retaining symmetry properties. However, if Aw < 6.0 A, the final structure may differ from the initial DWNT structure drastically because of two walls merging into consolidated single wall. The actual threshold of Aw depends on the SWNT chirality. Thus, the SrO-terminated NT (24 0)@(36 0) simulated from the cubic SrZrO3 crystal merges into the SWNT despite a rather large initial interwall distance (5.9 A), whereas the analogous BaZrO3 and SrTiO3 [92] DWNTs of both terminations and ZrO2-terminated SrZrO3 DWNT exhibit regular cylindrical structure for approximately the same initial interwall distance. The cylindrical two-wall structure was also found for ZrO2-termianted DWNTs at chirality (12 12)@(20 20) with the initial intertube distance of 5.2 to 5.4 A. As it can be seen from Table 6, the binding energy for cylindrical DWNTs is close to zero. This indicates a weak interaction between the constituents. In all the other DWNTs considered the two original SWNTs merge into one SW tubular object upon the geometry optimization (Figure 8). Note that stability of AO outer shell polyhedral tubes is greater than stability of BO2 outer shell tubes.

Table 6. Binding energies £,bind of SWNT components and initial intertube distances (Aw) for cylindrical and merged DWNTs simulated from layers of cubic SrZrO3 and BaZrO3 crystals.

Initial chirality Type of NT Type of NT Aw, A -Ebind, kJ-mol-1

SrZrO3 BaZrO3 SrZrO3 BaZrO3

SrO or BaO outer shell

(24, 0)@(32, 0) merged 3.2 3.2 -136 -101

(24, 0)@(36, 0) cylindrical/merged 5.9 5.9 -114 0.0

(12, 12)@(18, 18) merged 3.6 3.6 -142 -105

(12, 12)@(20, 20) merged 5.5 5.4 -132 -102

ZrO2 outer shell

(24, 0)@(32, 0) merged 3.1 3.1 -81 -71

(24, 0)@(36, 0) cylindrical 5.7 5.8 0.0 0.0

(12, 12)@(18, 18) merged 3.4 3.4 -59 -45

(12, 12)@(20, 20) cylindrical 5.2 5.3 0.0 0.0

Analogous behavior was observed for (n1, 0)@(n2, 0) NTs generated from the orthorhombic SrZrO3 crystal. Those NTs are similar to (n1, n1)@(n2, n2) NTs simulated from the cubic SrZrO3 crystal. Moreover, the final merged structures are practically identical for the both initial phases. This can be explained by the fact mentioned above that the rolling vector (n, 0) for (001) layers of the orthorhombic phase is almost equivalent to the rolling vector (n, n) for (001) layers of the cubic phase.

Cross-section of the obtained tubular objects resembles a polygon. The merged walls consist of 4-layer blocks that have a deformed cubic perovskite structure. The number of blocks is equal to the order of the rotation axis which is the common divisor N of the initial SWNT chiralities. The blocks are connected to each other by strongly reconstructed zones with partially broken Zr-O bonds. All the polyhedron-shaped NTs are remarkably stable relative to the 2-layer SWNT components; their binding energy varies from -50 to -100 kJ-mol-1, and their formation energy is generaly lower than the formation energy of SWNTs folded from 4 layer slabs considered in the preceding section.

(c) (d)

Figure 8. Cross-sectional view of DWNTs with merged walls: (a, b) BaZrO3 (12, 12)@(18, 18); (c, d) SrZrO3 (24, 0)@(32, 0). (a, c) AO-terminated; (b, d) ZrO2-terminated. See Figure 5 for legend.

The experimentally observed SrTiO3, BaTiO3 [5, 10], and BaZrO3 [8, 9] NTs have rather thick walls, w >20 A, with the ratio w/D (where D is the maximal diameter) of about 0.3. Obviously, tubes of this type cannot be rolled up from a single slab. The difference between the outer and inner circumferences of such a tube would be so big that interatomic bonds in the outer shell should be two times longer than those in the inner shell, which is impossible. By this reason SrZrO3 or BaZrO3 NTs with a large w/D ratio should either be multiwall or composed of nanoblocks. The last case was actually found in our simulations, as discussed in previous section. It was also found experimentally that perovskite NT walls could consist of polycrystalline phases [6, 8, 9] which might appear in the form of cube-like nanocrystals of various sizes. Hence, we can suppose that the polyhedron-shaped NTs obtained in this study can be used as the simplest model of real SrZrO3 and BaZrO3 tubes with sufficiently thick walls.

2.3.4 Comparison of SrTiO3-, BaTiO3-, SrZrO3- and BaZrO3-based nanotubes

An analysis of the obtained results allows us to make some conclusions about the structure and stability of NTs generated from the four perovskites, namely from SrTiO3, BaTiO3, SrZrO3, and BaZrO3.

In the case of perovskite-based SWNTs with thin 2-layer walls three main factors govern strain energy: (1) relative stability of precursor nanolayers; (2) tension of metal-oxygen bonds in external shells; and (3) ratio Rn/RIV of the effective radii of II- and IV-group metal ions. Thus SrZrO3 NTs generated from the cubic phase have large formation energies due to the unfavorable formation energy of the 2-layer slab. On the other hand, BO2-terminated NTs are more favorable than AO-terminated NTs in the cases of SrTiO3, SrZrO3 and BaZrO3 because of the unfavorable interaction between O and A atoms on the external AO surface. Oppositely, in the case of BaTiO3 the TiO2-terminated external surface is less stable due to large tension arising in the external TiO2 shell which is hardly able to adapt to large Ba-O distances in the internal shell.

Our calculations show that ionic radii ratio RII/RIV may be used as empirical criteria of NT stability. This ratio can be estimated from the ionic radii [93] as 2.0, 2.3, 1.6, and 1.9 for SrTiO3, BaTiO3, SrZrO3 and BaZrO3, correspondingly. Accordingly, SrTiO3 and BaZrO3 exhibit similar behavior with respect to formation and strain energies. The formation energy of the analogous NTs is minimal for BaTiO3 and maximal for SrZrO3.

We have found that the chirality (n, 0) is much more favorable than the chirality (n, n) for cylindrical SWNTs folded from the 4-layer perovskite slabs with maintaining the maximal possible order of the rotation axis. The TiO2 or ZrO2 composition of the external surface is preferable for (n, 0) tubes. The existence of non-cubic phases (BaTiO3, SrZrO3) leads to the reduction of the formation energy of 4-shell cylindrical SWNTs.

Reconstruction of the NT structure (splitting, fragmentation, or wall merging) affects its stability considerably. For the majority of the considered systems the reconstructed NTs with AO external surfaces are preferable. As mentioned above, the polyhedron-shaped NTs with merged walls prove to be the most stable for SrTiO3, BaTiO3, SrZrO3 and BaZrO3 perovskites. Formation energies obtained for NTs with chiralities (24, 0)@(32, 0) and (12, 12)@(18, 18) are compared in Table 7. The values listed in Table 7 show that the relative stability of these tubular objects increases in the following order: SrZrO3 < BaZrO3 ~ SrTiO3 < BaTiO3 which correlates with the rise of the above-mentioned ratio Rn/Riv of ionic radii of group II and IV metals. The NTs obtained by merging of the components of DWNTs with chirality (n1, n1)@(n2, n2) are slightly preferable over those with chirality (n1, 0)@(n2, 0).

Table 7. Formation energy (kJ-mol-1) of polyhedral SWNT generated by merging of walls in cylindrical DWNTs for different perovskite phases (A = Sr, Ba; B = Ti, Zr)._

Crystal

Initial chirality, termination -

SrZrO3 BaZrO3 SrTiO3 BaTiO3

(24, 0)@(32, 0), AO 174 156 165 142

(24, 0)@(32, 0), BO2 184 175 176 149

(12, 12)@(18, 18), AO 165 149 155 127

(12, 12)@(18, 18), BO2 186 207 211 159

2.4 Adsorption of water molecules on nanolayer surfaces

2.4.1 Unit cell choice

The adsorption of water molecules on the surface of the crystals and nano-sized crystallites is a physical phenomenon that ultimately determines many physical and chemical properties of the "real" surface of these objects, as well as their stability. Calculations of water adsorption on the surfaces of BaHfO3 and BaZrO3 have been performed with the use of both primitive and extended 42x42 unit cells. The latter corresponds to 2D square lattice frame with translation vectors rotated by 45° about z-axis. Unlike the primitive cell, the doubled cell includes two metal ions in the surface plane in both terminations. This means that there are two electropositive adsorption centers per unit cell and, hence, two surface coverages can be regarded depending on whether one (half-monolayer) or both (full-monolayer) adsorption centers are occupied by water molecules. Moreover, the number of non-equivalent oxygens are also doubled during the cell rotation. As result, the number of water molecule degrees of freedom is increasing as well as the number of possible local minima on potential energy surface. Consequently, that gives the chance to a more favorable orientation of water molecules and one might expect the stabilization of the adsorption structure at any coverage. Our computations confirm this conclusion.

2.4.2 Water adsorption at BaO-terminated surfaces

Monolayer adsorption is only possible on the surfaces which maintain the 2D translational symmetry corresponding to cubic phases. Generally, water molecule can be adsorbed in molecular or dissociative form [31, 37, 38]. Taking into account previous experience [37], we can assume that dissociative adsorption is preferable in comparison with a molecular one for cubic perovskites on the BaO-terminated surface. It is obviously that covalently unsaturated surface terminal oxygen atom attracts one of the water hydrogen via the strong H-bond thus forcing the breaking of water molecule. During the water dissociation the mentioned H atom goes to the surface terminal oxygen forming a short (less than 1.5 A) hydrogen bond with oxygen to which it was previously attached.

We have studied the various initial positions of water molecule relative the surface of cubic phases (Figure 9), and in the most cases the geometry optimization leads to the same dissociative structure (Figure 9b). However, we found a single structure in which water save its molecular form. In this structure (Figure 9a) water hydrogens form two equal H-bonds with the surface terminal oxygens in the perpendicular to surface plane.

The adsorption energy of molecular adsorption is twice less than that of dissociative adsorption for both regarded perovskites (Table 8) if the primitive 2D unit cell is used.

(a)

(b)

(c)

(d)

(e)

Figure 9. Most stable structures for water adsorption at BaO-terminated (001) surface of BaZrO3 or BaHfO3 using primitive (top) and extended (bottom) 2D unit cell: (a, c) molecular case; (b, d) dissociative case; (e) mixed adsorption case. Legend: small green spheres - Ba atoms; small dark blue spheres - Zr or Hf atoms; middle yellow spheres and sticks - O atoms; blue sticks - H atoms.

Using of an doubled unit cell, as expected, leads to considerable decreasing of water adsorption energies (Table 8). As in case of the primitive surface unit cell, it was difficult to obtain the structures with a molecular adsorption on BaO-terminated surface because of a spontaneous dissociation of water molecules. However, for half-monolayer coverage the symmetric structure with two hydrogen bonds has been obtained similar to that which was found for primitive cell. In this structure the water molecule is located almost equidistantly from both surface terminal oxygens forming two short hydrogen bonds. Nevertheless, the energy of this structure is lower than that for monolayer coverage with primitive cell due to relaxing of terminal oxygens H-bonded to the water molecule. At monolayer coverage (with doubled unit cell) the most favorable case of molecular adsorption is presented by the structure with two molecules of water bonded

by weak hydrogen bonds (2.2 A) in infinite chains over a slab surface laying between the terminal Ba-O bonds (Figure 9c). This structure is also stabilized by hydrogen bonds between the water protons and the terminal oxygen atoms. The molecular adsorption energies are close for both perovskites and they decrease in the following order: monolayer (primitive) < half-monolayer (extended) < monolayer (extended).

Table 8. Water adsorption energy (kJ-mol"1) in the most stable configurations on (001) surfaces of BaHfO3 (upper values) and BaZrO3 (lower values).

Half-monolayer Monolayer

Cell Molecular Dissociative Molecular Dissociative Mixed

adsorption adsorption adsorption adsorption adsorption

BaO"terminated surface

Primitive -66.6 -121.1

- -

-66.9 -119.9

-97.2 -141.1 -109.4 -125.6 -122.1

Doubled -96.9 -142.2 -108.0 -124.6 -121.5

ZrO2- or HfO2-terminated surface

Primitive Doubled

-97.1 -90.6

-124.0 -120.0

-98.1 -93.3 -98.1 -93.3

-90.0 -82.8 137.1 ■129.5

136.2 -129.3

Water adsorption energy was calculated by equation.:

AEads = -(ENL+nH2O-ENL-nEH2O)/n (8)

where ENL+n^2O is the energy of the nanolayer with n adsorbed water molecules, ENL is nanolayer total energy, EH2O is the energy of isolated water molecule. The energy of the isolated water molecule was calculated with same level of approximation as it was applied for slab systems but without the periodic boundary conditions.

The dissociative adsorption structure using the extended cell (Figure 9d) is qualitatively similar to that using primitive cell. However, at smaller coverage the surface hydroxyls are arranged in such a way that the repulsion between them is minimal. This case exhibits the lowest adsorption energy per one water molecule obtained in this study (about -140 kJ-mol-1) for both perovskites. Also, two (for half-monolayer coverage) or four (for monolayer coverage) hydroxyls surround Ba2+ cation with Ba-O distance equal or less than 2.75 A. So, the formation of surface hydroxide Ba(OH)2 is supposed to be the result of water hydrolysis.

The simulation of the mixed adsorption is only possible using the extended unit cell. In this case one of two water molecules (per unit cell) keeps its molecular form, while the other molecule dissociates. For the BaO-terminated surface we have found two stable structures of such kind. One of them is energetically and structurally similar to the molecular adsorption with infinite chains of H-bonded molecules and hydroxide

ions over the nanolayer surface. Another structure (Figure 9e) is stabilized by hydrogen bonds both between bare terminal oxygen and proton of water molecule on the one side, and between proton of terminal hydroxyl and oxygen of "free" hydroxyl on the other side, but it does not contain hydrogen bonds between adsorbed species itself. Thus, it is similar to a dissociative adsorption at the BaO-terminated surface. The second structure is preferable by 10 kJ-mol-1.

2.4.3 Water adsorption at ZrO2- and HfO2-terminated surfaces

The adsorption at ZrO2- and HfO2-terminated surfaces qualitatively differs from that on BaO-terminated surfaces. Molecular adsorption is basically defined by the partially covalent character of interaction between the surface zirconium or hafnium atoms with the water oxygen. As result, the adsorption structure (Figures 10a and 10c) and energy (Table 8) essentially do not depend on a degree of a coverage and on the size of the unit cells. The distances between the water oxygen and ^-element atom is close to 2.35 A in all optimized structures. One of water protons interacts with the surface oxygen via the bent H-bond causing water molecule to tilt towards the surface plane (Figure 10a). Similar structures have been found for water molecule interacting with BO2-terminated surfaces of other perovskites [31, 37, 38, 94]. The molecular adsorption energy on HfO2-terminated surface is slightly lower (by 5 kJ-mol-1) than corresponding energy on ZrO2-terminated surface.

Although, the water molecule does not spontaneously dissociates when placing on the ZrO2- or HfO2-terminated surfaces, the relaxation of "manually" prepared dissociative structures leads to the low energy state. Two different hydroxyls are created in these structures. The first one is attached to former fivefold coordinated B atom (B = Zr or Hf) with the B-O bond length close to 2.0 A. The second (bridging) hydroxyl is formed via the proton appending to twofold coordinated oxygen lying in the surface plane. Generally, this leads to enlarging of B-O bond and distortion of B-O-B angle. We found that the correct description of dissociative water adsorption on ZrO2- and HfO2-terminated surfaces is possible using the extended unit cell only. The extension of unit cell provides a more effective relaxation of dissociated structures and the decrease of adsorption energy by almost 50 kJ-mol-1. Thus, as can be seen in Figure 10d, the protonated surface oxygens move up and unprotonated ones move down, causing the surface octahedra BO6 to tilt. So, one can speak about the orthorhombic surface reconstruction of cubic perovskites upon the water dissociative adsorption at ZrO2- or HfO2-terminated surfaces. While a cubic surface also undergoes a meaningful deformation during dissociation adsorption (Figure 10b), a translational-symmetry-constrained distortion is unprofitable and thus makes the dissociative adsorption structures energetically less favorable than the molecular one by about 10 kJ-mol-1.

Accordingly, molecular adsorption becomes more preferable in the case of primitive cell, and the dissociative one - in the extended cell case.

The dissociative adsorption energies at HfO2-terminated surfaces (as do molecular adsorption energies) are lower than those at ZrO2-terminated surfaces by about 7-8 kJ-mol-1. Consequently, we can conclude that HfO2-terminated surface is more hydrophilic than ZrO2-terminated surface. We can suppose a slightly greater ionicity of barium hafnate as compared to barium zirconate as the possible explanation of indicated result.

(c) (d) (e)

Figure 10. Most stable structures for water adsorption at BO2-terminated (001) surface of BaZrO3 and BaHfO3 using primitive (top) and extended (bottom) 2D unit cell: (a, c) molecular case; (b, d) dissociative case; (e) mixed adsorption case. See Figure 9 for legend.

The most favorable mixed adsorption structure at ZrO2- and HfO2-terminated surface (Figure 10e) is partially resemble that for the dissociative case and has approximately the same energy (Table 8). However, there is a short H-bond between the water proton and terminal hydroxyl oxygen, causing them to be tilted one to another. The length of B-O hydroxyl bond is increased, but on the other hand B-O(H2) distance is decreased compared to lengths in purely dissociative and molecular cases, correspondingly. The orthorhombic-like surface reconstruction is well recognized in this case (Figure 10e).

2.4.4 Adsorption on Y-doped surface. Water adsorption sites and corresponding energies

Trivalent Y3+ dopant can be incorporated into the BaZrO3 and BaHfO3 by aliovalent substitution at

Zr4+

or Hf4+ sites with forming of charge compensating oxygen vacancies. The protonic defect created by water uptake onto the vacancy sites is described as a interstitial proton strongly associated with the neighboring oxide ion thus forming a hydroxyl group. Y-doped BaZrO3-based oxides are demonstrated [23, 24] to combine high stability with high bulk proton conductivity. However, the limiting issue is its poor grain boundary conductivity [25, 26].

To study the water adsorption on Y-doped BaZrO3 and BaHfO3 surface, two models have been constructed: II.1Y (Figure 11a) with one Zr or Hf atom substituted by one Y and one H atom on each side of nanolayer unit cell, and II.2Y (Figure 11b) with two Zr or Hf atoms substituted by two Y atoms and two H atoms on the both sides of nanolayer unit cell. The second model can also be obtained by adding two water molecules to the system with one oxygen vacancy and two Y substituents on each side of the nanolayer unit cell. Coordinates of all atoms were optimized, except the positions of middle BO2-plane atoms. The inversion symmetry provides the equivalence of the both surfaces in this case.

(a) (b)

Figure 11. Relaxed structure of 2*2 surface unit cells. (a) model II.1Y with proton bridging position; (b) model II.2Y with oxygen vacancy. Top BYO2 layer (B = Zr or Hf) is only shown. Legend: small green spheres - Ba atoms; small dark blue spheres - Zr or Hf atoms; small violet spheres - Y atoms; middle yellow spheres and sticks - O atoms; blue sticks - H atoms.