Nucleation theory in growth modeling of nanostructures V.G. Dubrovskii St. Petersburg Academic University & Ioffe Physical Technical Institute RAS, St.-Petersburg, Russia Repino, 13- July 2013, Lecture # 1 Plan: Introduction Epitaxy techniques Semiconductor quantum dots and nanowires Elements of nucleation theory Zeldovich nucleation rate Gibbs-Thomson effect and Laplacian pressure Nucleation on laterally confined facets

Modeling of nanostructure formation Growth theory Nucleation Theory of nanostructure formation Quantum dots Nanowires Epitaxial techniques (MBE, MOCVD…) InAs/GaAs(100) QDs GaAs/GaAs(111)B-Au NWs Main goals of modeling: Understanding Prediction Optimization New morphology New structure New materials

Size-dependent quantum effects in nanostructures DOS of nanostructures: Effect on optical properties: SE: DOS: Bulk:

Transformation of QD distribution function into DOS High uniformity High density (?) Controlled composition Controlled morphology Controlled crystal structure Required properties of NS ensembles: Morphology of nanostructure ensembles depends on growth process !!!

Alfred Cho – the father of MBE

Technologies of nanostructure formation: MBE and CVD 1. Molecular beam epitaxy = MBE Developed in early 70s Now widely used to produce high-quality layers of different compound semiconductors with very abrupt interfaces and good control of thickness, doping and composition Materials are deposited in a form of molecular beams on a heated substrate Molecular beams are originated from thermally evaporated elemental sources (effusion cells) Growth rates are typically of order of several angstroms per second MBE system consists of 3 main vacuum chambers: -Growth chamber -Buffer chamber (preparation and storage of samples) -Load lock (to bring samples in and out of the vacuum environment) Rotating samples (manipulator) Pressure gauge (ion gauge) Nitrogen cooler Cryo-pumps, ion pump, turbo pumps to remove gases, residual pressure is typically less than Torr Substrates holders made from Ta, Mo or pyrolytic boron nitride

Scheme of typical MBE system Monitor residual gases, source beams In situ growth control Sample rotation Deposition Example for GaAs: As (As 4 or As 2 through a cracker Ga Al In Be (p-doping) Si (n-doping)

In situ monitoring by RHEED

In situ monitoring by RHEED (continued …) Physical nature of RHHED oscillations

Modern MBE reactors Riber 49 GaAs growth 6 x 3 inch substrates Growth rate 1-3 A/s 10 sources As cracking Two parallel loading systems RHEED QMA Cryo-panel 4 standard HEMT processes daily

MOCVD Metal organic chemical vapor deposition (MOCVD) = MOVPE is being used for crystal growth from 1960 and in 1980s was applied for the fabrication of compound semiconductor – based materials and devices For example, LED structures are grown almost exceptionally by MOCVD MOCVD systems contain: - the gas handling system to meter and mix reactants - the reactor (vertical or horizontal in design) - the pressure control system -the exhaust facilities Basic principle is the deposition of the required growth species with precursors at ~ atmospheric pressure of a carrier gas and chemical reaction in the temperature field of a heated substrate Group III sources are trimethylgallium (TMGa), TMAl, TMIn Group V species are typically hydride gases such as arsine (AsH 3 ) and phoshpine (PH 3 ), or NH 3 for GaN Very high V/III ratios (50-100) because the incorporation of group V elements Is self-limited (very high partial pressure of group V species) Growth rate and composition is controlled by partial pressures of the species and by the substrate temperature

Chemistry of MOCVD growth process for GaAs Source of a metal-organic compound (liquid or solid state) H2H2 Hydrides (gaseous) Chemical reaction Radiofrequency generator (~450 kHz) Heating up to С Growth of compound semiconductor on a crystal substrate Example of chemical reaction for the GaAs epitaxy: (CH 3 ) 3 Ga + AsH C H2H2 GaAs + 3CH 4 Vapors in H 2 Exhaust of gases

Modern MOCVD reactors (1-x)Ga(CH3) 3 + xIn(CH3) 3 + NH 3 -> In x Ga 1-x N + 3CH 4 Reactor Aixtron 2000/HT (2003): GaN growth 6 x 2-inch substrates Productivity > 500 blue LED structures monthly Each wafer contains ~ LED chips 0.35*0.35 mm

Heterostructres for blue-green and white LEDs Main technological stages: Wafers Al 2 O 3 Materials (TMGa, TMAl,TMIn, gases) Epitaxial growth of LED heterostructure Processing and production of chips Packaging Fabrication of final device Increasing In concentration in InGaN => larger wavelength

Direct formation of Stranski-Krastanow QDs Relaxation of elastic stress in the island – main driving force for 2D-3D transition SK growth mode 20 nm

Direct formation of QDs (continued …) Critical thickness h 1c for 2D-3D transition Coherent stained islands At h=h 1c, RHEED pattern changes from strikes to spots ε 0 >2% Dislocations 2 ML InAs/GaAs

VLS growth of whiskers by Wagner & Ellis and Givargizov High temperature (T ~ C) CVD experiments of s with micrometer diameters Пар-жидкость-кристалл или ПЖК (в английской литературе vapor-liquid- solid VLS)) механизм роста одномерных структур, таких как нановискеры в процессе химического осаждения из газовой фазы.нановискеры химического осаждения из газовой фазы Wagner & Ellis, APL 1964

Formation of vertical nanowires on activated surfaces by MBE 1-st stage (MBE chamber): oxide desorption from GaAs substrate and buffer layer growth GaAs wafer 3-st stage (MBE chamber): formation of Au-Ga alloy droplets; deposition of GaAs – growth of NW GaAs wafer GaAs NW 2-st stage (Vacuum or MBEchamber): Au deposition on a GaAs substrate surface GaAs wafer Au film GaAs/GaAs(111)B-Au

Typical RHEED patterns during the wire growth 200 nm GaAs/GaAs(111)B 200 nm GaAs/Si(100 )

ZB and WZ phase of III-Vs All III-V NWs, except nitrides, have STABLE ZB cubic phase in BULK FORM In GaAs: Difference in cohesive energies = – 24 meV per pair at zero ambient pressure. T.Akiyama et al, Jpn.J.Appl.Phys, 2006; M.I.McMahon and R.J.Nelmes, PRL, 2005 Bulk ZB GaAs becomes unstable at pressure ~ 80 GPa !!! Most of ZB III-V nanowires contain WZ phase: A.I.Person et al., Nature Materials 2004, Au-assisted MOVPE of III-V/III-V J.C.Harmand et al., APL 2005, Au-assisted MBE of GaAs/GaAs I.P.Soshnikov et al., Phys. Sol. State 2006, Au-assisted MBE of GaAs/GaAs P.Mohan et al., Nanotechnology 2005, selective area catalyst free growth of III-Vs C.Chang-Hasnain group, Au-assisted MOCVD of III-V/Si AND MANY OTHERS! ABC=ccc=3C= ABA=hhh=2H=(11)

Hexagonal WZ phase in III-V NWs !!! LPN CNRS: GaAs NWs on GaAs InAs NWs on InAs C. Chang-Hasnain, group: TEM image [ ] zone axis FFT of TEM image InP NWs on Si APL 2005 APL 2007

ZB-WZ transition in GaAs NWs (Ioffe & LPN) Au-assisted MBE of GaAs on the GaAs(111)B substrate WZ ZB Switching from WZ to ZB at the end of growth I.P.Soshnikov et al, Phys. Sol. State 2005 Switching from ZB to WZ at the beginning of growth F.Glas et al., Phys. Rev. Lett 2007 ZB phase systematically appears at low supersaturation !

Nucleation Gibbs free energy of 2D island formation (fixed T, P, N): (1a) Difference in chemical potentials (energetically favorable) Surface term (energetically unfavorable) i γ – solid-vapor surface energy per unit area (J/m 2 ) Δμ – difference of chemical potentials (J) Normally, a is a large parameter ~ several tens h A= i Consider 2D island of ML height h, area A=c 1 r 2 and perimeter P=c 2 r, r = radius in k B T units Surface energy constant

Gibbs free energy n=10 -3, a=15 =0.75 (1), 1 (2), 1.5 (3) and 2 (4). Activation barrier for nucleation: Critical number of atoms: Half-width near maximum: F icic F and i c decrease as supersaturation increases !!!

A story about Zeldovich and nucleation theory Я.Б. Зельдович ФИЗИЧЕСКИЕ ОСНОВЫ ТЕОРИИ ФАЗОВЫХ ПРЕВРАЩЕНИЙ ВЕЩЕСТВА (КУНИ Ф.М., 1996), ФИЗИКА Сформулированы цели современной теории фазовых превращений, введены понятия о стабильных и нестабильных фазах вещества, образовании зародышей стабильной фазы в недрах метастабильной, вероятностно-статистическое представление о потоке зародышей как о ведущей кинетической характеристике фазового превращения. Описана временная зависимость фазового превращения (уравнение Зельдовича ???).

Nucleation rate F i ic-Δicic-Δic ic+Δicic+Δic icic exp(F)>>1 I II III Region 1: Equilibrium size distribution Region 2: Fluctuations [ flux I] I – nucleation rate [1/cm 2 s] Region 3: Growth di c /dt=0 f(i,t) – island size distribution [1/cm 2 ] Kinetic equation for size distribution in region II: Boundary conditions:

Nucleation rate (continued…) Stationary solution at J=const with the 2 nd boundary condition: i-1 i i+1 J=0 equilibrium J=const steady state To meet the 1 st boundary condition, I should equal: Laplace method General Zeldovich formula for 2D islands

Gibbs-Thomson effect and Laplacian pressure R PLPL PVPV Consider liquid (L) spherical drop of radius R in equilibrium with vapor (V) Find P L -P V, P L and P V Solution: 1) System at fixed T, V and μ => maximum of at constant volume For a sphere with For a cylindrical isotropic solid with yields γ Laplacian surface pressure

GT effect and Laplaciam pressure (continued …) 2) At finite R, equilibrium state is defined by At R, equilibrium state is defined by Subtract (1) from (2); take into account that liquid is incompressible and that vapor is ideal Liquid: Vapor: (1) (2)

Mononuclear and polynuclear growth I – nucleation rate, v=dr/dt – 2D island growth rate, R – face radius I and v are time-independent during growth (constant supersaturation) Kashchiev interpolation formula: V L = vertical growth rate of facet of radius R due to 2D nucleation Generally, V L =f(I,v,R) RR Polynuclear growth is generally faster ! Dependence on the nucleation barrier: VLVL

A story about Kolmogorov-Mehl- Johnson-Avrami model A. Kolmogorov Уравнение Джонсона Мела Аврами Колмогорова (англ. Johnson Mehl Avrami Kolmogorov equation, JMAK) описывает процесс фазового перехода при постоянной температуре. Изначально оно было получено для случая кристаллизации расплавов в 1937 году А. Н. Колмогоровым, и независимым образом в 1939 году Р. Ф. Мелом и У. Джонсоном, а также было популяризировано в серии статей М. Аврами в годах.англ.фазового переходакристаллизации расплавовА. Н. КолмогоровымР. Ф. Мелом Википедия: