In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
and
physics
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
, surface growth refers to models used in the
dynamical
In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a ...
study of the growth of a surface, usually by means of a
stochastic differential equation
A stochastic differential equation (SDE) is a differential equation in which one or more of the terms is a stochastic process, resulting in a solution which is also a stochastic process. SDEs are used to model various phenomena such as stock pr ...
of a
field
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grass ...
.
Examples
Popular growth models include:
*
KPZ equation
*
Dimer model
In geometry, a domino tiling of a region in the Euclidean plane is a tessellation of the region by dominoes, shapes formed by the union of two unit squares meeting edge-to-edge. Equivalently, it is a perfect matching in the grid graph formed by p ...
*
Eden growth model
The Eden growth model describes the growth of specific types of clusters such as bacterial colonies and deposition of materials. These clusters grow by random accumulation of material on their boundary. These are also an example of a surface fra ...
* SOS model
*
Self-avoiding walk
In mathematics, a self-avoiding walk (SAW) is a sequence of moves on a lattice (a lattice path) that does not visit the same point more than once. This is a special case of the graph theoretical notion of a path. A self-avoiding polygon (SAP) ...
*
Abelian sandpile model
The Abelian sandpile model (ASM) is the more popular name of the original Bak–Tang–Wiesenfeld model (BTW). BTW model was the first discovered example of a dynamical system displaying self-organized criticality. It was introduced by Per Bak, ...
*
Kuramoto–Sivashinsky equation
In mathematics, the Kuramoto–Sivashinsky equation (also called the KS equation or flame equation) is a fourth-order nonlinear partial differential equation. It is named after Yoshiki Kuramoto and Gregory Sivashinsky, who derived the equation ...
(or the flame equation, for studying the surface of a flame front)
They are studied for their
fractal
In mathematics, a fractal is a geometric shape containing detailed structure at arbitrarily small scales, usually having a fractal dimension strictly exceeding the topological dimension. Many fractals appear similar at various scales, as illu ...
properties,
scaling
Scaling may refer to:
Science and technology
Mathematics and physics
* Scaling (geometry), a linear transformation that enlarges or diminishes objects
* Scale invariance, a feature of objects or laws that do not change if scales of length, energ ...
behavior,
critical exponent
Critical or Critically may refer to:
*Critical, or critical but stable, medical states
**Critical, or intensive care medicine
*Critical juncture, a discontinuous change studied in the social sciences.
*Critical Software, a company specializing in ...
s,
universality class
In statistical mechanics, a universality class is a collection of mathematical models which share a single scale invariant limit under the process of renormalization group flow. While the models within a class may differ dramatically at finite s ...
es, and relations to
chaos theory
Chaos theory is an interdisciplinary area of scientific study and branch of mathematics focused on underlying patterns and deterministic laws of dynamical systems that are highly sensitive to initial conditions, and were once thought to have co ...
,
dynamical system
In mathematics, a dynamical system is a system in which a Function (mathematics), function describes the time dependence of a Point (geometry), point in an ambient space. Examples include the mathematical models that describe the swinging of a ...
, non-equilibrium / disordered / complex systems.
Popular tools include
statistical mechanics
In physics, statistical mechanics is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities. It does not assume or postulate any natural laws, but explains the macroscopic be ...
,
renormalization group
In theoretical physics, the term renormalization group (RG) refers to a formal apparatus that allows systematic investigation of the changes of a physical system as viewed at different scales. In particle physics, it reflects the changes in the ...
,
rough path theory, etc.
Kinetic Monte Carlo surface growth model
Kinetic Monte Carlo
The kinetic Monte Carlo (KMC) method is a Monte Carlo method computer simulation intended to simulate the time evolution of some processes occurring in nature. Typically these are processes that occur with known transition rates among states. It ...
(KMC) is a form of computer simulation in which atoms and molecules are allowed to interact at given rate that could be controlled based on known
physics
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
. This simulation method is typically used in the micro-electrical industry to study crystal surface growth, and it can provide accurate models surface morphology in different growth conditions on a time scales typically ranging from micro-seconds to hours. Experimental methods such as
scanning electron microscopy (SEM),
X-ray diffraction
X-ray crystallography is the experimental science determining the atomic and molecular structure of a crystal, in which the crystalline structure causes a beam of incident X-rays to diffract into many specific directions. By measuring the angles ...
, and
transmission electron microscopy (TEM), and other computer simulation methods such as
molecular dynamics (MD), and
Monte Carlo simulation (MC) are widely used.
How KMC surface growth works
1. Absorption process
First, the model tries to predict where an atom would land on a surface and its rate at particular environmental conditions, such as temperature and vapor pressure. In order to land on a surface, atoms have to overcome the so-called
activation energy
In chemistry and physics, activation energy is the minimum amount of energy that must be provided for compounds to result in a chemical reaction. The activation energy (''E''a) of a reaction is measured in joules per mole (J/mol), kilojoules pe ...
barrier. The frequency of passing through the activation barrier can by calculated by the
Arrhenius equation
In physical chemistry, the Arrhenius equation is a formula for the temperature dependence of reaction rates. The equation was proposed by Svante Arrhenius in 1889, based on the work of Dutch chemist Jacobus Henricus van 't Hoff who had noted in 18 ...
:
where A is
thermal frequency of
molecular vibration
A molecular vibration is a periodic motion of the atoms of a molecule relative to each other, such that the center of mass of the molecule remains unchanged. The typical vibrational frequencies range from less than 1013 Hz to approximately 1014 Hz ...
, k is
Boltzmann constant
The Boltzmann constant ( or ) is the proportionality factor that relates the average relative kinetic energy of particles in a gas with the thermodynamic temperature of the gas. It occurs in the definitions of the kelvin and the gas constant, ...
.
2. Desorption process
When atoms land on a surface, there are two possibilities. First, they would
diffuse
Diffusion is the net movement of anything (for example, atoms, ions, molecules, energy) generally from a region of higher concentration to a region of lower concentration. Diffusion is driven by a gradient in Gibbs free energy or chemical p ...
on the surface and find other atoms to make a cluster, which will be discussed below. Second, they could come off of the surface or so-called
desorption
Desorption is the physical process where a previously adsorbed substance is released from a surface. This happens when a molecule gains enough energy to overcome the activation barrier of the bounding energy that keeps it in the surface.
There ...
process. The desorption is described exactly as in the
absorption
Absorption may refer to:
Chemistry and biology
* Absorption (biology), digestion
**Absorption (small intestine)
*Absorption (chemistry), diffusion of particles of gas or liquid into liquid or solid materials
*Absorption (skin), a route by which ...
process, with the exception of a different activation energy barrier.
For example, if all positions on the surface of the crystal are energy equivalent, the rate of growth can be calculated from
Turnbull formula Turnbull may refer to:
People
*See Turnbull (surname)
*Malcolm Turnbull, former Prime Minister of Australia
Places
*Turnbull High School in Bishopbriggs, Scotland
* Turnbull National Wildlife Refuge, located near Spokane, Washington, USA
* Turnbu ...
:
where, ∆G = E
in – E
out, A
out, A
o out are frequencies to go in or out of crystal for any given molecule on the surface, h - height of the molecule in the growth direction, C
o concentration of the molecules in direct distance from the surface.
3. Diffusion process on surface
Diffusion process can also be calculated with Arrhenius equation:
where, D is
diffusion coefficient
Diffusivity, mass diffusivity or diffusion coefficient is a proportionality constant between the molar flux due to molecular diffusion and the gradient in the concentration of the species (or the driving force for diffusion). Diffusivity is enco ...
, E
d is
diffusion activation energy.
All three processes strongly depend on
surface morphology at a certain time. For example, atoms tend to lend at the edges of a group of connected atoms, the so-called island, rather than on a flat surface, this reduces the total energy. When atoms diffuse and connect to an island, each atom tends to diffuse no further, because activation energy to detach itself out of the island is much higher. Moreover, if an atom landed on top of an island, it would not diffuse fast enough, and the atom would tend to move down the steps and enlarge it.
Simulation methods
Because of limited computing power, specialized simulation models have been developed for various purposes depending on the time scale:
''a) Electronic scale simulations'' (density function theory, ab-initio molecular dynamics): sub-atomic length scale in femto-second time scale
''b) Atomic scale simulations (MD)'': nano to micro-meter length scale in nano-second time scale
''c) Film scale simulation (KMC)'': micro-meter length scale in micro to hour time scale.
''d) Reactor scale simulation (phase field model)'': meter length scale in year time scale.
Multiscale modeling
Multiscale modeling or multiscale mathematics is the field of solving problems which have important features at multiple scales of time and/or space. Important problems include multiscale modeling of fluids, solids, polymers, proteins, nucleic ac ...
techniques have also been developed to deal with overlapping time scales.
How to use growth conditions in KMC
The interest of growing a smooth and defect-free surface requires a combination set of physical conditions throughout the process. Such conditions are
bond strength
In chemistry, bond energy (''BE''), also called the mean bond enthalpy or average bond enthalpy is the measure of bond strength in a chemical bond. IUPAC defines bond energy as the average value of the gas-phase bond-dissociation energy (usually a ...
, temperature,
surface-diffusion limited and
supersaturation
In physical chemistry, supersaturation occurs with a solution when the concentration of a solute exceeds the concentration specified by the value of solubility at equilibrium. Most commonly the term is applied to a solution of a solid in a liqu ...
(or impingement) rate. Using KMC surface growth method, following pictures describe final surface structure at different conditions.
1. Bond strength and temperature
Bond strength and temperature certainly play important roles in the crystal grow process. For high bond strength, when atoms land on a surface, they tend to be closed to atomic surface clusters, which reduce total energy. This behavior results in many isolated cluster formations with a variety of size yielding a
rough surface
Surface roughness, often shortened to roughness, is a component of surface finish (surface texture). It is quantified by the deviations in the direction of the normal vector of a real surface from its ideal form. If these deviations are large, ...
. Temperature, on the other hand, controls the high of the energy barrier.
Conclusion: high bond strength and low temperature is preferred to grow a smoothed surface.
2. Surface and bulk diffusion effect
Thermodynamically, a smooth surface is the lowest ever configuration, which has the smallest
surface area
The surface area of a solid object is a measure of the total area that the surface of the object occupies. The mathematical definition of surface area in the presence of curved surfaces is considerably more involved than the definition of arc ...
. However, it requires a kinetic process such as surface and bulk diffusion to create a perfectly flat surface.
Conclusion: enhancing surface and bulk diffusion will help create a smoother surface.
3. Supersaturation level
Conclusion: low impingement rate helps creating smoother surface.
4. Morphology at different combination of conditions
With the control of all growth conditions such as temperature, bond strength, diffusion, and saturation level, desired morphology could be formed by choosing the right parameters. Following is the demonstration how to obtain some interesting surface features:
See also
*
Domino tiling
In geometry, a domino tiling of a region in the Euclidean plane is a tessellation of the region by dominoes, shapes formed by the union of two unit squares meeting edge-to-edge. Equivalently, it is a perfect matching in the grid graph formed by p ...
*
Diffusion-limited growth
*
Stochastic partial differential equation
Stochastic partial differential equations (SPDEs) generalize partial differential equations via random force terms and coefficients, in the same way ordinary stochastic differential equations generalize ordinary differential equations.
They have ...
References
Kinetic Monte Carlo
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*
*
*
*
*
* {{cite web, url=http://www.roentzsch.org/SurfDiff/ , title=Vicinal surface diffusion, author= Lars Röntzsch, access-date=23 May 2019
Surface science
Semiconductor device fabrication
Thin film deposition
Coatings