Неустойчивость тонких пленок, самоорганизация на поверхности презентация

Нестабильность механически напряженной пленки
Instability of mechanically stressed films
Механизм развития нестабильности - поверхностная диффузия
Mechanism of instability development is surface diffusion
Причина нестабильности – избыточная упругая энергия
Origin of instability is elastic energy

Малые периодические изменения толщины
Small periodic variation of thickness
ω=2π/λ

Малые периодические изменения толщины
Small periodic variation of thickness

Плотность энергии и химический потенциал
Energy and chemical potential
Увеличение площади поверхности квадратично по a / λ (change in surface area)
Химический потенциал (Chemical potential)

Критическая длина стабильности
Critical length of stability
Пусть a зависит от времени
Изменение свободной энергии, усредненное по периоду λ:
Free energy over a period
Нестабильность Азаро-Тиллера—Гринфельда
Asaro-Tiller-Grinfeld instability

Нестабильность напряженной пленки
Asaro-Tiller--Grinfeld instability
ν = 0.3
G = 0.67 1011 Pa
γ = 1 J/m2
ε = 0.007
Asaro-Tiller--Grinfeld instability
M = E/(1- ν)
E = 2G(1+ν)

Нестабильность пленки GeSi на Si
Asaro-Tiller--Grinfeld instability of GeSi on Si
Transmission electron microscopy cross-sectional image of a Si0:81Ge0:19 alloy film grown epitaxially on a Si substrate (a). The ridges are aligned with a <100> crystallographic direction. While the TEM image appears to represent a fully two-dimensional configuration, the planview images of the film surface (b) shows that the regular ordering has a relatively short range. The normal distance between parallel lines in the lower images is the peak-to-peak distance in the upper image, or about 300nm. Reproduced from Cullis et al. (1992). εm=0.66%
a

Возмущения второго порядка
Second order disturbation

Применимость приближения малых флуктуаций
Applicability of small perturbation approach
The dependence of change in surface energy and elastic energy of a solid due to sinusoidal perturbation of surface shape versus the amplitude-to-wavelength ratio of that perturbation. The results show that the small slope approximation is reliable for values of a/λ up to 0.1, and is a fair approximation for significantly larger values.

Домашнее задание (Homework) 4
Пленка Ge толщиной 4 нм выращена эпитаксиально на подложке Si с ориентацией (111) и толщиной 400 мкм.
4-nm-thick Ge epitaxial film was grown over 400-μm-thick Si substrate with (111) orientation.
Определить (determine)
1. Анизотропию кривизны структуры по Стони Anisotropy of Stoney curvature (if any?)
2. Критическую длину Asaro-Tiller-Grinfeld нестабильности пленки (γ = 1 J/m2). Asaro-Tiller-Grinfeld critical length (γ = 1 J/m2).

Начальная стадия самоорганизации КТ InAs на поверхности GaAs
Initial stage of self-organization of InAs QDs on GaAs surface

КТ InAs на поверхности GaAs
InAs quantum dots on GaAs surface
AFM image

Причина формирования КТ – выигрыш в упругой энергии
The reason for self-organization is a release of elastic energy
Effect of island shape on the volume elastic relaxation of a coherently strained island. The dark area is that with a large elastic strain energy density: (a) island with height-to-width
ratio h/L<<1 is weakly relaxed; (b) island with height-to-width ratio h/L>>1 is nearly completely relaxed.

Механизм Странского-Крастанова
Stranski-Krastanow mechanism
Размерный параметр задачи
Dimensional parameter
γf~1J/m2, Mf~1011Pa, εm~0.03

Механизм Странского-Крастанова
Stranski-Krastanow mechanism
dimensionless parameter

Соотношение вертикального и латерального размеров
Optimal aspect ratio
Free energy change per unit volume of island of material due to reorganization
of uniformly strained film material to a nonuniformly strained conical island.
The parameter Z = ζ/V1/3 represents the size of the island relative to the natural system length ζ.

Упругая энергия при изменении отношения высоты островка к его основанию
Elastic energy vs. aspect ratio
Total, normalized energy as a function of island aspect ratio (one with a log scale on the
abscissa is shown to convey more detail in the transition regime). Data points represent a mix of spherical caps and pucks; all fall along the same curve. For aspect ratios larger than approximately 0.25, the energy (when scaled as such) becomes independent of aspect ratio. Jonsdottir et al, 2006

Начальные стадии формирования пирамид Ge на Si
Initial stages of formation of Ge islands on Si
STM images of the surface evolution during growth of a Ge layer on Si (001). In panels (a)–(d), the Ge coverage grows from 2.8 to 4.0 ML’s. During growth (a)–(c) first mounds
(prepyramids) develop, then these convert (d) to {105} facetted pyramids. From Vailionis et al., 2000.

Формы островков Ge на Si
Shapes of Ge islands on Si
The size distribution of several island types occurring during deposition of Ge on Si(001). The inset shows the time evolution of the island density for different types. Adapted
from Vailionis et al., 2000.

Распределение по размерам
Size distribution
0
1
2
Size distributions of domes and pyramids calculated by means of equilibrium statistics (lines) for two different Ge coverages Q, and obtained from AFM (black dots). From Rudd et al., 2003

Эволюция формы
Shape evolution
Shape transition of Ge or SiGe islands grown on Si(001) during growth solid arrows, postgrowth annealing (dotted arrow), and Si capping (dashed arrow). The solid curves represent the critical volumes for pyramids and domes. From Rastelli, Kummer, and von Kaenel, 2002.

Формы островков
Shape evolution
Typical Ge island shapes obtained by STM during Si capping of Ge domes grown on Si(001): (a),(b), domes; (c) pyramids; (d)–(f) prepyramids. The Si coverages are 0, 1, 2, 4, 8, and 16 ML’s for panels (a)–(f). From Rastelli et al., 2001.

Геометрия островков InAs на GaAs и Ge на Si
Geometry of InAs islands on GaAs and Ge on Si
Geometry of an InAs pyramid over an InAs wetting layer deposited on a GaAs(001) surface.
3D STM image (a) and schematic top and side views (b) of a 3D island formed during the deposition of pure Ge on Si(001). All faces of the island are {105} facets. The height of the island is 2.8 nm while the base dimensions are of the order of 20 and 40 nm. Due to frequently observed elongation of the islands—that is either in [100] or [010] direction—the island’s shape resembles that of a “hut”. The vertical scale of the STM image is exaggerated: the islands actually are fairly Kat with an 11.3° tilt angle of the facets as it is shown in (b) (after Mo et al. and Teichert et al.).

Нуклеация пирамидальных островков
Nucleation of pyramidal islands
The volume dependence of the energy necessary to create a pyramidal island, calculated for three different orientations of the facets. Energy and volume are normalized to the critical values of {105} facetted pyramids. Adapted from Tersoff and LeGoues, 1994; Brunner et al., 2000.
Энергия создания пирамидального островка с гранями под углом θ
A = σm2 (1- ν)/(2πG)

Фасетирование островков InAs на GaAs
Facetting of InAs islands on GaAs
Facetting of InAs islands (a) sketch of facets of an InAs islands on GaAs(001) obtained by STM; (b) and (c) zoom of the reconstructed {-1-37} facet; (d) structure model of facet reconstruction. From Marquez et al., 2003.

Фасетирование островков
Facetting of islands
Зубчатая фасетированная поверхность. Эффективные силы противоположного знака приложены к соседним углам. (Forces on facetted surface)
Полная энергия (Total energy)
The balance of forces acting on a crystal edge. Forces caused by the intrinsic surface stress τ are balanced by the force G acting from the bulk of the crystal. According to Newton’s third law, the reaction F=-G is acting from the surface layer on the bulk of the crystal, resulting in the strain field. The inset in the middle depicts the force balance, which is similar for a liquid and for a crystal.

Энергия фасетированной поверхности
Energy of facetted surface
The energy of a periodically faceted surface per unit surface area vs the period D. There always exists an optimum period of faceting Dopt due to the logarithmic dependence of the elastic relaxation energy on the period D.
Выигрыш в упругой энергии
Release of elastic energy
Полная энергия поверхности
Total energy of facetted surface
Оптимальный период Optimal period

Равновесность системы островков
The energy of a dilute array of 3D coherently strained islands per one atom vs size of the island. The parameter α is the ratio of the change in the surface energy due to the formation of islands, Esurf, and of the contribution of the edges to the elastic relaxation energy, |Eelastic|.
Удельная энергия на 1 атом в пирамидальном островке размером L x L
E0 – характерная энергия
L0 – характерная длина

Самоорганизация QD на поверхности:
Кинетика или термодинамика?
Self-organization: kinetics or thermodynamics?
Формирование квантовых точек на поверхности обычно проходит в неравновесных условиях
Formation happens under non-equilibrium conditions
Термодинамические модели в целом описывают наблюдаемые явления
Thermodynamics describes the process reasonably well
Роль кинетики зависит от конкретной системы и условий формирования квантовых точек
Role of kinetics is specific for systems and conditions
Для большого числа практически важных полупроводниковых гетеросистем разработаны рецепты воспроизводимого получения массивов квантовых точек на поверхности
Technology has been developed for reliable production of quantum dots in different systems

Роль условий роста
Role of growth conditions
Effect of As pressure on InAs/GaAs quantum dot arrays. Plan-view TEM and photoluminescence spectra for five values of arsenic pressure are given, PAs0 = 2 106 Torr. The dense array of 3D coherent dots existing for P=PAs0 undergoes a reversible transition to a planar morphology with lowering of pressure to P=1/6 PAs0. An increase of pressure results in Ostwald ripening, formation of dislocated clusters at P=3PAs0, and complete disappearance of coherent dots at P=5PAs0.

Самоорганизация квантовых точек на поверхности
Self-organization of quantum dots on a growth surface
Технология молекулярно-лучевой эпитаксии (MBE) или газофазной эпитаксии с использованием металлоорганических соединений (MOVPE, MOCVD).
Механизмы роста:
Франка—ван дер Мерве (Frank—van der Merve, FM)
Фолмера—Вебера (Volmer—Weber, VW)
Странского—Крастанова (Stranski—Krastanow, SK)

Режимы роста гетероэпитаксиальных систем
Growth modes in heteroepitaxial systems
Основные параметры:
Энтропия пересыщения
Энергии поверхности
подложки - γs
пленки - γf
границы раздела - γsf
Параметры решетки
подложки - as
пленки - af
рассогласование εm = (af - as)/af
В гетеросистеме с малым рассогласованием εm ≈ 0 реализуется
Режим Франка—ван дер Мерве (FM), если γf + γsf < γs,
Режим Фолмера—Вебера (VW), если γf + γsf > γs
Для Е=100GPa, γ=1J/m2, εm=0.02, a=0.25 nm

Фазовая диаграмма режимов роста
Phase diagram of growth modes
Равновесная фазовая диаграмма гетероэпитаксиальной системы с рассогласованием решеток. Daruka and Barabasi (1997).
Q — количество осажденного материала; εо — рассогласование решеток.

Домашнее задание (Homework) 5
Пленка Ge толщиной 4 нм выращена эпитаксиально на подложке Si с ориентацией (111) и толщиной 400 мкм.
4-nm-thick Ge epitaxial film was grown over 400-μm-thick Si substrate with (111) orientation.
Определить (determine)
4.2. Критическую длину Asaro-Tiller-Grinfeld нестабильности пленки (γ = 1 J/m2). Asaro-Tiller-Grinfeld critical length (γ = 1 J/m2).
5. Характерный размер квантовых точек Ge
Characteristic size of Ge quantum dots

Латеральное упорядочение островков
Энергия латерального взаимодействия островков
R – расстояние между островками;
V – объем островка;
(mx ; my) – единичный вектор на поверхности;
B, C – константы;
для изотропной среды В2 = 0, изотропное дипольное отталкивание;
для анизотропной среды возможно притяжение по “мягким” направлениям, обычно <100>.

Энергия латерального взаимодействия островков на поверхности
Normalized interaction energy as a function of island separation, for islands with aspect ratio AR = V/A3/2 = 1/3. Jonsdottir et al, 2006

Латеральное упорядочение островков InAs на GaAs
Ordering of quantum dots:
(a) plan-view transmission electron microscopy (TEM) micrograph of a single sheet of InAs dots grown in molecular beam epitaxy by four-monolayer deposition of InAs. Dots are preferentially aligned in rows parallel to <100>.
(b) Histogram of the direction between the nearest neighboring dots.