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The Deal–Grove model mathematically describes the growth of an oxide layer on the surface of a material. In particular, it is used to predict and interpret
thermal oxidation In microfabrication, thermal oxidation is a way to produce a thin layer of oxide (usually silicon dioxide) on the surface of a wafer. The technique forces an oxidizing agent to diffuse into the wafer at high temperature and react with it. The ra ...
of silicon in semiconductor device fabrication. The model was first published in 1965 by Bruce Deal and
Andrew Grove Andrew Stephen Grove (born András István Gróf; 2 September 193621 March 2016) was a Hungarian-American businessman and engineer who served as the third CEO of Intel Corporation. He escaped from Communist-controlled Hungary at the age of 20 ...
of Fairchild Semiconductor, building on
Mohamed M. Atalla Mohamed M. Atalla ( ar, محمد عطاالله; August 4, 1924 – December 30, 2009) was an Egyptian-American engineer, physicist, cryptographer, inventor and entrepreneur. He was a semiconductor pioneer who made important contributions to ...
's work on silicon surface passivation by thermal oxidation at
Bell Labs Nokia Bell Labs, originally named Bell Telephone Laboratories (1925–1984), then AT&T Bell Laboratories (1984–1996) and Bell Labs Innovations (1996–2007), is an American industrial research and scientific development company owned by mult ...
in the late 1950s. This served as a step in the development of
CMOS Complementary metal–oxide–semiconductor (CMOS, pronounced "sea-moss", ) is a type of metal–oxide–semiconductor field-effect transistor (MOSFET) fabrication process that uses complementary and symmetrical pairs of p-type and n-type MOSF ...
devices and the fabrication of integrated circuits.


Physical assumptions

The model assumes that the oxidation
reaction Reaction may refer to a process or to a response to an action, event, or exposure: Physics and chemistry *Chemical reaction * Nuclear reaction *Reaction (physics), as defined by Newton's third law * Chain reaction (disambiguation). Biology and ...
occurs at the interface between the oxide layer and the substrate material, rather than between the oxide and the ambient
gas Gas is one of the four fundamental states of matter (the others being solid, liquid, and plasma). A pure gas may be made up of individual atoms (e.g. a noble gas like neon), elemental molecules made from one type of atom (e.g. oxygen), or ...
. Thus, it considers three phenomena that the oxidizing species undergoes, in this order: # It diffuses from the bulk of the ambient gas to the surface. # It diffuses through the existing oxide layer to the oxide-substrate interface. # It reacts with the substrate. The model assumes that each of these stages proceeds at a rate proportional to the oxidant's concentration. In the first step, this means
Henry's law In physical chemistry, Henry's law is a gas law that states that the amount of dissolved gas in a liquid is directly proportional to its partial pressure above the liquid. The proportionality factor is called Henry's law constant. It was formulat ...
; in the second,
Fick's law of diffusion Fick's laws of diffusion describe diffusion and were derived by Adolf Fick in 1855. They can be used to solve for the diffusion coefficient, . Fick's first law can be used to derive his second law which in turn is identical to the diffusion ...
; in the third, a
first-order reaction In chemistry, the rate law or rate equation for a reaction is an equation that links the initial or forward reaction rate with the concentrations or pressures of the reactants and constant parameters (normally rate coefficients and partial reactio ...
with respect to the oxidant. It also assumes steady state conditions, i.e. that transient effects do not appear.


Results

Given these assumptions, the flux of oxidant through each of the three phases can be expressed in terms of concentrations, material properties, and temperature. :J_ = h_g (C_g- C_s) :J_ = D_ \frac :J_ = k_i C_i By setting the three fluxes equal to each other J_ = J_ = J_, the following relations can be derived: :\frac = \frac :\frac = \frac Assuming a diffusion controlled growth i.e. where J_ determines the growth rate, and substituting C_i and C_s in terms of C_g from the above two relations into J_ and J_ equation respectively, one obtains: :J_ = J_ = \frac If ''N'' is the concentration of the oxidant inside a unit volume of the oxide, then the oxide growth rate can be written in the form of a differential equation. The solution to this equation gives the oxide thickness at any time ''t''. :\frac = \frac = \frac :x^2 + Ax = Bt + ^2 + Ax_i :x^2 + Ax = B(t+\tau) where the constants A and B encapsulate the properties of the reaction and the oxide layer respectively, and x_i is the initial layer of oxide that was present at the surface. These constants are given as: :A=2 D_ (\frac + \frac) :B= \frac :\tau = \frac where C_s = H P_g , with H being the gas solubility parameter of the
Henry's law In physical chemistry, Henry's law is a gas law that states that the amount of dissolved gas in a liquid is directly proportional to its partial pressure above the liquid. The proportionality factor is called Henry's law constant. It was formulat ...
and P_g is the partial pressure of the diffusing gas. Solving the quadratic equation for ''x'' yields: :x(t) = \frac Taking the short and long time limits of the above equation reveals two main modes of operation. The first mode, where the growth is linear, occurs initially when t+\tau is small. The second mode gives a ''quadratic'' growth and occurs when the oxide thickens as the oxidation time increases. :t+\tau \ll \frac \Rightarrow x(t) = \frac(t+\tau) :t+\tau \gg \frac \Rightarrow x(t) = \sqrt The quantities ''B'' and ''B/A'' are often called the ''quadratic'' and ''linear reaction rate constants''. They depend exponentially on temperature, like this: :B = B_0 e^; \quad B/A = (B/A)_0 e^ where E_A is the
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 ...
and k is the
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 constan ...
in eV. E_A differs from one equation to the other. The following table lists the values of the four parameters for single- crystal silicon under conditions typically used in industry (low doping,
atmospheric An atmosphere () is a layer of gas or layers of gases that envelop a planet, and is held in place by the gravity of the planetary body. A planet retains an atmosphere when the gravity is great and the temperature of the atmosphere is low. A s ...
pressure). The linear rate constant depends on the orientation of the crystal (usually indicated by the
Miller indices Miller indices form a notation system in crystallography for lattice planes in crystal (Bravais) lattices. In particular, a family of lattice planes of a given (direct) Bravais lattice is determined by three integers ''h'', ''k'', and ''� ...
of the crystal plane facing the surface). The table gives values for <100> and <111> silicon.


Validity for silicon

The Deal–Grove model works very well for single-crystal silicon under most conditions. However, experimental data shows that very thin oxides (less than about 25 nanometres) grow much more quickly in O_2 than the model predicts. In silicon nanostructures (e.g.,
silicon nanowire Silicon nanowires, also referred to as SiNWs, are a type of semiconductor nanowire most often formed from a silicon precursor by etching of a solid or through catalyzed growth from a vapor or liquid phase. Such nanowires have promising applications ...
s) this rapid growth is generally followed by diminishing oxidation kinetics in a process known as self-limiting oxidation, necessitating a modification of the Deal–Grove model. If the oxide grown in a particular oxidation step greatly exceeds 25 nm, a simple adjustment accounts for the aberrant growth rate. The model yields accurate results for thick oxides if, instead of assuming zero initial thickness (or any initial thickness less than 25 nm), we assume that 25 nm of oxide exists before oxidation begins. However, for oxides near to or thinner than this threshold, more sophisticated models must be used. In the 1980s, it became obvious that an update to the Deal-Grove model is necessary to model the aforementioned thin oxides (self-limiting cases). One such approach that more accurately models thin oxides is the Massoud model from 1985 The Massoud model is analytical and based on parallel oxidation mechanisms. It changes the parameters of the Deal-Grove model to better model the initial oxide growth with the addition of rate-enhancement terms. The Deal-Grove model also fails for polycrystalline silicon ("poly-silicon"). First, the random orientation of the crystal grains makes it difficult to choose a value for the linear rate constant. Second, oxidant molecules diffuse rapidly along grain boundaries, so that poly-silicon oxidizes more rapidly than single-crystal silicon.
Dopant A dopant, also called a doping agent, is a trace of impurity element that is introduced into a chemical material to alter its original electrical or optical properties. The amount of dopant necessary to cause changes is typically very low. Whe ...
atoms strain the silicon lattice, and make it easier for silicon atoms to bond with incoming oxygen. This effect may be neglected in many cases, but heavily doped silicon oxidizes significantly faster. The pressure of the ambient gas also affects oxidation rate.


References


Bibliography

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External links


Online Calculator including pressure, doping, and thin oxide effects
{{DEFAULTSORT:Deal-Grove model Semiconductor device fabrication Chemical engineering Nanomaterials Nanoelectronics