Abstract

Our model, treating oxide as solid annulus freely expanded from the silicon (Si) consumed due to increased molecular volume whose geometry enables closed-form expression of time as a function of thickness in constant-parameters case, was revised in non-dimensional form maintaining the appearance of the original Si radius. While this constant-parameters case describes oxide thickness decreasing with decreasing Si radius in concave cases as reported from the experiment, in convex cases thickness is instead described to increase with decreasing Si radius, contradicting published experimental observations. Performing stress analysis displacing surfaces of expanded oxide and remaining Si back to their shared interface, stress-dependent solubility, diffusivity, and reaction rate were investigated toward resolving this discrepancy between the model and reported experiments. With stress-dependent parameters, closed-form expression of time as a function of oxide thickness is no longer achieved, with numerical integration instead required to compute oxidation times. If considering solubility or diffusivity to increase with hydrostatic stress or reaction rate to decrease with increasing interface pressure radially, as hypothesized, increasing oxide thickness with decreasing original Si radius in convex cases remains predicted, in conflict with experimental reports in the literature. It is shown that the experimental observation of an oxide thickness decreasing with decreasing Si radius in convex cases is possible if considering reaction rate to instead increase with increasing interfacial pressure. The same may be possible if considering solubility or diffusivity to instead decrease with increasing hydrostatic stress, tuning activation energies describing the strength of such dependence.

Introduction

The growth of oxide layers on silicon (Si) by reaction of molecular oxidant (O2 or H2O) diffusing through the layer from environmental surroundings at its free surface, to its interface with the Si as the other reactant, is a technologically important process (e.g., micro-devices produced through such interfacial oxidation). Considering oxidant gas-phase transport from the environment to its concentration at the oxide surface, then diffusion through the growing oxide to a lower concentration at the interface with Si, and finally reaction there with the Si as three coupled serial processes, Deal and Grove [1] for flat surfaces developed a closed-form expression relating oxide thickness to growth time. This linear-quadratic kinetics expression, coincident with experimental observations, describes thickness initially increasing linearly with time when the interfacial reaction or gas-phase transport processes are rate-limiting, thereafter transitioning to increases proportional to the square-root of time when the oxide thickens and diffusion through it slows becoming the rate-limiting serial process.

While planar geometry pertains to the most prominent situation of flat Si wafers, oxidation upon cylindrically curved surfaces is also of practical significance. Not only upon circular cross-sectional posts and micro- or nanowires, this more importantly also exists at corners along straight step-height changes as trenches and walls on patterned wafers in micro-device fabrication which rather than sharp and square instead have finite radius thus existing as a sector of a cylinder. In the proximity of such features, Marcus and Sheng [2] found oxide thicknesses to differ from neighboring flat surfaces, with concave inside corners at the bottom of trenches especially experiencing severely depressed oxidation rates. To study such effects, Kao et al. [3] performed series of oxidation experiments on cylindrical Si micro-structures of various radii of curvature and found relative to flat infinite-radius surfaces that for both convex and concave cases thickness of oxides grown within a fixed time decreased with decreasing radius, with this thickness reduction being stronger on concave than convex surfaces, an example as shown in Fig. 1. Krzeminski et al. [4] also observed these effects of radius and convexity/concavity in their later experiments on cylindrical Si nanostructures. Though only studying convex cases, Liu et al. [5] likewise found at any fixed time that Si nanowires of decreasing radius produced correspondingly thinner oxide, as Liu et al. [6] also later reported for not only Si but also W nanowires. The same effect was also observed in spherical convex geometries, where Okada and Iijima [7] in experiments on small Si particles found a common 3 h time produced oxide whose thickness decreased with decreasing particle size. The experimental observations of Kao et al. [3] summarized earlier, due to its breadth not only of curvature radii investigated but especially inclusion of concave cases as well, serve as the set against whose trends previously published models and any further model developments presented here will be compared.

Fig. 1
Typical published example [3] of oxide thickness measured after fixed oxidation duration as a function of curvature (inverse of radius) of concave and convex cylindrical Si surfaces, here for 1440 min wet oxidation at 800 °C
Fig. 1
Typical published example [3] of oxide thickness measured after fixed oxidation duration as a function of curvature (inverse of radius) of concave and convex cylindrical Si surfaces, here for 1440 min wet oxidation at 800 °C
Close modal

Unlike the flat case modeled by Deal and Grove [1] where the three serial processes occur through a common cross section, in the cylindrical case depicted in Fig. 2 oxidant gas-phase transport from the environment takes place to an oxide free surface at radius b while reaction of the oxidant with the Si takes place at radius a at which the oxide interfaces with it, while oxidant diffusion through the oxide occurs at all radii between a and b, requiring the different cross-sectional areas across which these processes occur to be considered. Kao et al. [8] treated this cylindrical geometry in developing an expression for rate of oxide thickness increase dxo/dt that also includes both oxide free surface and interface radii b and a, which evolve with time. Recognizing oxide thickness xo as the difference between these two radii, b could be replaced in terms of xo and a, with the oxide growth rate expression dxo/dt being a differential equation in not only oxide thickness xo but also the radius of remaining silicon a as a second variable. To permit solution, they consider a second condition of the volume of oxide produced by the consumption of Si to an instantaneous remaining radius a, and treat this oxide as incompressible in coming to a corresponding oxide thickness. Numerically simulating oxide growth by determining increase in film thickness Δxo for each increment in time Δt, at each of three temperatures using a set of constant parameter values assumed for that temperature, while this model does predict convex radius to result in oxide thicknesses less than the flat case, the concave radius is found to instead describe an oxide thickness greater than the flat case as opposed to being thinner than even the convex case at that same radius as observed experimentally. Lemme [9] performed a similar numerical simulation for only a convex case, instead finding it to describe oxide thickness greater than the flat case, though this change from Kao et al. [8] modeling likely results from an algebraic error in treating oxide volume incompressibility.

Fig. 2
Models of cylindrical oxide with its free surface at radius b having oxidant concentration Co, as exposed to gaseous environment of imposed concentration Cg, from Ref. [10]. The dashed line at radius a indicates where the oxide would interface with the Si and have oxidant concentration Ci in (a) convex and (b) concave cases.
Fig. 2
Models of cylindrical oxide with its free surface at radius b having oxidant concentration Co, as exposed to gaseous environment of imposed concentration Cg, from Ref. [10]. The dashed line at radius a indicates where the oxide would interface with the Si and have oxidant concentration Ci in (a) convex and (b) concave cases.
Close modal

In our previous paper [10], an approach was proposed enabling integration of the oxide growth rate dxo/dt expression of Kao et al. [8] in closed-form, resulting in an explicit equation for time t to grow an oxide to thickness xo including the original radius of the convex or concave Si cylinder Ro. This was achieved by applying the dxo/dt expression on a geometry of the annulus of oxide as free-standing solid expanded from the annulus of Si consumed in its production, in which all dimension distances were expanded by a factor f accounting for the larger molecular volume of the oxide relative to that of the Si, with the radius of the oxide free surface thus at b = fRo and the interfacial radius of the oxide a differing from this now-constant b by xo, such that upon substitution of these b and a the growth rate expression dxo/dt remains sufficiently simple that it may be integrated in closed-form. While the dimensional form of the resulting equation shows the dependence on original Si cylinder radius, in transforming it to non-dimensional form to broaden applicability of output, Ro was used to normalize xo and other length quantities, as well as in the normalization of time t and reactions rate, leaving the dependence on radius of curvature in the non-dimensionalized form obscured. Thus, one objective of this work is to re-cast the non-dimensionalization of this closed-form equation relating oxidation time and thickness such that dependence on radius of curvature remains apparent.

While our closed-form expression [10] affords simple algebraic implementation, it deviates from experimental observations of Kao et al. [3] though in a manner complementary to their own constant-parameter numerical model [8], describing at any time an oxide thickness that decreases from the flat case with decreasing radius of curvature in concave cases as observed, but in convex cases instead increasing in thickness from the flat case with such decreasing radius of curvature contrary to observation. It has been hypothesized that such deviations may result from parameter values such as solubility, diffusivity, and reaction rate as constant, when they may instead be stress-dependent, with these stresses arising from the swelling expansion of oxide compared to the Si from which it is formed while still maintaining an interface, due to their differences in molecular volume, as well as differing extents of deformation that must occur to accommodate such oxide expansion depending on the extent of convex or concave curvature. Hsueh and Evans [11] performed stress analyses of such interfacial as well as surface oxidation situations, considering cases where expansion of the oxide may occur by either uniaxial or dilational strain, initially demonstrating an elastic analysis to describe instantaneous stress state within an oxidized cylindrical body, though not further coupling such stress analysis with the oxidation kinetics via stress-dependent parameters to model the evolving oxide growth. Yoshikawa et al. [12] also took an elastic stress analysis approach, furthermore implementing a strain energy-dependent reaction rate which they couple to the kinetics of oxidation at convex and concave cylindrically-rounded corners of Si trenches at their top and bottom edges to describe the evolution of oxide geometries there. While Yoshikawa et al. [12] appear to be considering the oxide expansion during interfacial oxidation to occur in a manner more nearly classified by Hsueh and Evans [11] as uniaxial (radial) strain, the geometry of our prior oxidation kinetics modeling [10] considering the oxide as a free-standing annulus expanded from the original annulus of Si consumed is more nearly classified by Hsueh and Evans [11] as a dilational strain mechanism. Their stress analysis in such a situation is performed by bringing the oxide and remaining Si back into continuity across their interface, much like a problem of thermal expansion mismatch or press-fit interference. As another objective, and the primary contribution and novelty of this paper, its later portion will investigate coupling elastic stress analysis of such a cylindrical oxidation to kinetics of the oxidation, considering a stress dependence of not only reaction rate as Yoshikawa et al. [12] but furthermore exploring such for oxidant solubility and diffusivity parameters as well.

Model Development

As developed by Kao et al. [8], Eq. (1) describes the rate of increase in oxide thickness xo with increasing time t in a cylindrically curved case of interfacial oxidation, where oxidant from a surrounding environment is diffusing from the oxide's free surface at radius b through its thickness to its interface with the Si instantaneously at radius a where the oxidant and metal react as shown in Fig. 2, producing additional oxide while consuming Si with N indicating the number of oxidant molecules consumed per unit oxide volume produced
(1)

The oxidizing severity of the surrounding environment is characterized by the concentration Cg of oxidant it is capable of imposing at the oxide's exposed free surface under equilibrium conditions of full solubility. From here, three processes in series share the same rate of oxidant molecules per unit time: gas-phase transport from surrounding environment to instantaneous oxidant concentration Co on oxide free surface as described by transport coefficient h; diffusion through the oxide from its free surface to its interface with the Si as described by diffusivity D; and reaction at this interface oxidizing the Si as described by reaction rate k. Each of these serial processes is represented by a term in the denominator of Eq. (1) containing its corresponding parameter (h, D, k), and if a low value of any of these parameters when coupled with instantaneous values of oxide radii b and a causes its corresponding term in the denominator to be especially the largest, it becomes the rate-limiting of the three serial processes. The upper and lower signs appearing before the diffusion term represent the convex (Fig. 2(a)) and concave (Fig. 2(b)) cases, respectively.

The single Eq. (1) appears to contain not only oxide thickness xo as a variable, but also the oxide's radius a at which it interfaces with the Si that must evolve as the metal is correspondingly consumed. However, as the oxide thickness is the magnitude of the difference between the radii of the oxide's free and Si-interfacing surfaces b and a it may additionally be stated that
(2)
where again upper and lower signs refer to convex and concave cases, respectively, which upon substitution allows differential Eq. (1) in a single variable xo.
Upon oxidation, the molecular volume of oxide per mole of Si contained Ωo is greater than the original molecular volume of Si alone Ωsi, so in its free-standing state the oxide would be of expanded dimensions compared to the annulus of original Si consumed in its production. Considering the oxide as solid and its dimensions in the above analysis as those in its expanded free-standing state, as in our prior analysis [10], the radius b of the free surface of the expanded free-standing oxide may be related to the radius of the original Si from which it was produced Ro by
(3)
where f is an expansion factor by which linear dimensions are increased, for example, the cube-root of the molecular volume of the oxide Ωo in ratio to that of the original silicon ΩSi. As such, the differential equation describing oxide growth becomes
(4)
To broaden applicability of any calculation performed with it, Eq. (4) is rearranged to non-dimensional form that lessens the number of free parameters appearing within it. Rearrangement can be performed such that length dimensions appear in ratio to the quantity D/k so non-dimensional oxide thickness and original Si radius become xo*=xo/(D/k) and Ro*=Ro/(D/k), respectively while gas-phase transport coefficient is normalized to reaction rate h*=h/k resulting in time being normalized as t*=t/(DNk2Cg). The seven free parameters in Eq. (4) are thus reduced to only Ro*, h*, and f in the resulting non-dimensional form
(5)

It should be noted this form has been modified from that in our prior work [10] where length dimensions were instead normalized to Ro, which as a shortcoming caused original Si radius as a free parameter of primary importance to no longer be evident in the non-dimensional equation form. The h* in this new form is preferable to its inverse that was termed k* in our prior work as it appears in the denominator term associated with gas-phase transport and thus is better having a symbol associated correspondingly with gas-phase transport rather than reaction rate. Of course, in turn the non-dimensionalization of time here would differ from that of our prior work.

With oxide growth rate expressed as a function of oxide thickness itself xo* in Eq. (5), the amount of time required to grow an oxide can be found by integrating the inverse of that oxide growth rate
(6)
from the initial oxide thickness (usually zero starting from bare Si) to a final thickness of interest as the upper limit.
In studying the effect of original Si curvature Ro*, an important baseline case for comparison would be a flat surface with radius approaching infinity. Causing the xo*/(fRo*) term in the argument of the “ln” in Eq. (5) to become correspondingly small relative to 1 to which it is being added, by series approximation this ln may be expressed as xo*/(fRo*) itself, with the flat baseline case in turn
(7)
with an oxidation rate starting at (1+1h*)1 on bare Si when xo*=0.

Constant-Parameters Baseline Case

If model parameters (k, D, f, etc.) are treated as constant, and in turn, h* and Ro* in oxide growth rate in Eq. (5) are not varying with increasing oxide thickness, as shown in our earlier study [10], it inverse may be integrated in closed-form per Eq. (6) from a lower limit of zero, yielding an analytical expression for the time to grow on a bare cylindrically curved Si surface to a thickness of interest xo*
(8)
In the case of flat Si substrate, the inverse of oxide growth rate expressed in Eq. (7) is easily integrated to yield the well-known linear-parabolic oxidation kinetics in such a flat case
(9)
where required oxidation time initially increases linearly with oxide thickness at small oxide thicknesses xo*1 before eventually adopting an increase proportional instead to the square of oxide thickness upon thicker oxides xo*1. Even in cylindrically curved cases having some finite radius Ro*, such linear then parabolic kinetics as described in Eq. (9) will still initially exist, since small values of oxide thickness xo*Ro* in Eq. (5) allow the same series approximation of the ln term as in the flat case, leading to oxide growth rate again approximated by Eq. (7) during early stages until the curvature effects described by Eq. (5) begin to cause more substantial effect at larger oxide thicknesses.

Figure 3 illustrates example behavior predicted for this constant-parameter case for various original Si radii Ro* both convex and concave, as well as the flat baseline case, with f ≈ 1.31 approximated from molecular volumes of Ωo = 0.045 nm3 and ΩSi = 0.020 nm3 for the oxide and the Si from which it is formed, respectively. Considering values of gas-phase transport coefficient h ≈ 108μm/h approximated by Deal and Grove [1] and in turn reaction rate k ≈ 1800 μm/h they calculated from wet O2 oxidation data, a large h* ≈ 55,600 is adopted here. Thus, in this example, the gas-phase transport term in the denominator of Eq. (5) remains small and not rate-limiting, to such an extent that Kao et al. [8] consider it negligible and discard it in their earlier modeling efforts.

Fig. 3
Non-dimensional presentation of constant-parameters case at various convex and concave original Si radii Ro* including infinite (flat) as a function of oxide thickness: (a) oxide growth rate, (b) inverse oxide growth rate, and (c) oxidation time. For oxide, expansion factor f = 1.31 and gas-phase transport h* = 55,600. X indicates point at which Si is completely consumed for convex solid cylinders.
Fig. 3
Non-dimensional presentation of constant-parameters case at various convex and concave original Si radii Ro* including infinite (flat) as a function of oxide thickness: (a) oxide growth rate, (b) inverse oxide growth rate, and (c) oxidation time. For oxide, expansion factor f = 1.31 and gas-phase transport h* = 55,600. X indicates point at which Si is completely consumed for convex solid cylinders.
Close modal

In Fig. 3(a) depicting oxidation rate as a function of oxide thickness xo* per Eq. (5), Si of all original radii Ro* display an initial rate (1+1h*)1 just slightly less than one. In the flat case (infinite radius) as described by Eq. (7), the oxidation rate decreases monotonically as oxide thickness increases as its diffusion term becomes increasingly rate-limiting, while in concave cases, for smaller radii Ro* slightly more extreme monotonic decreases of oxidation rate with increasing oxide thickness are noted than for the flat infinite-radius baseline case. For convex cases, a solid Si cylinder will be completely consumed, and oxidation will cease when the oxide thickness reaches xo*=fRo*. At this oxide film thickness, Eq. (5) shows the oxidation rate to approach one just as the convex Si is reaching full consumption. Thus, for convex cases, the initial reduction of oxidation rate with increasing oxide thickness eventually transitions to increasing oxidation rate back up to this final one value. Smaller convex radii experience lesser decreases in oxidation rate from the initial value, and are fully consumed at smaller values of oxide thickness. Each concave case is shown as continuing on through indefinitely larger oxide thicknesses as if a cylindrical hole within bulk Si, however if the original Si was instead an annulus of finite wall thickness oxidizing only on its inner concave surface then it would instead be completely consumed when the oxide film thickness reaches f times its original wall thickness.

Figure 3(b) depicts inverse of the oxidation rate (i.e., the time per unit increase in oxide thickness) as a function of oxide thickness. Thus, the time t* to reach any oxide thickness per Eq. (6) is the area under such a curve up to that xo* value, for which numerical approximation is straightforward, though in this constant-parameter case such integration can be done in closed-form with resulting Eq. (8) behavior plotted in Fig. 3(c). At any fixed time, for concave cases the model describes oxide thickness that decreases with decreasing radius as experimentally observed by Kao et al. [3] and depicted in Fig. 1. For convex cases, however, this constant-parameter model describes oxide thickness at any fixed time that instead increases with decreasing radius, rather than decreasing though to a lesser extent than the concave cases as observed experimentally. It is hypothesized due to the molecular volume expansion of the oxide film relative to the Si substrate from which it is produced that a resultant stress state will lead to stress-dependent variation of oxidation parameters, and this may be the source of the discrepancy between published experimental observations of behavior in convex cases [3] and that described by a constant-parameter model.

Stress-Dependent Parameters Case

At the interface between oxide film and Si substrate, as oxidant diffusing from the surrounding environment through the oxide is reacting with a differential thickness of Si there and expanding its thickness, this expansion and causal oxidation reaction is anticipated to be impeded by any pressure p (negative of radial stress in a curved case) that might exist across this interface, which we represent by reaction rate k=koexp{(Vk/KT)p} with Arrhenius dependence characterized by reaction activation volume Vk. It is also anticipated that oxidant solubility at the oxide free surface would be reduced by increasingly compressive (negative) hydrostatic stress in the oxide σhox at that location Cg=Cgoexp{(VCg/KT)σhox} per solubility activation volume VCg, and diffusivity would similarly be reduced at all locations through the oxide thickness D=Doexp{(VD/KT)σhox} by hydrostatic stress there per diffusivity activation volume VD. Non-dimensionalizations xo*, Ro*, h*, and t* described above in the constant-parameter case are now instead performed using the stress-free values of reaction rate ko, solubility Cgo, and diffusivity Do. If furthermore pressure and hydrostatic stress are normalized to Young's modulus of the oxide Eox and activation volumes are normalized to KT/Eox, substituting stress-dependent k, Cg, and D into Eq. (5) yields oxidation rate expression
(10)

Interfacial pressure p* and oxide hydrostatic stress σhox* are determined from considering the oxide as an annulus with all dimensions in its free-standing state expanded by factor f from that of the Si from which it was produced. Axisymmetric stress analysis is performed adopting a standard interference “press” fit treatment radially, for example, as in Ref. [13], where the surface of radius a of the oxide most recently formed and the surface of instantaneous radius Ri of the remaining Si from which that most recent oxide has formed are deflected from their free-standing states and brought together to form the oxide/metal interface while also deflecting the oxide and Si longitudinally to a shared length. The Si and oxide here are furthermore treated as “long,” such that their longitudinal deformation states are approximated as uniform and not a function of radial position. The stress analysis is detailed in the  Appendix with resultant equations summarized below in non-dimensionalized form, where the longitudinal stresses are also normalized to the oxide Young's modulus, as too is the Young's modulus of the silicon ESi*=ESi/Eox. Treating the oxide as a hollow cylinder that is either internally pressurized in convex case or externally pressurized in concave case, the dependences of oxide radial and circumferential stresses on radial position are such that in their sum these radial position dependencies cancel, resulting in a hydrostatic stress upon further summing in of the uniform longitudinal oxide stress that is independent of radial position. The notable result is a diffusivity D that, while hydrostatic stress-dependent, is nonetheless uniform through the oxide's thickness at any instant of time.

Convex Stress Analysis.

As depicted in Fig. 4(a), in the convex case, we are considering the Si to be a solid cylinder with an annular oxide around it. Longitudinally, as the oxide is expanded by factor f compared to the Si, upon deforming them to a common shared length the oxide will experience a negative longitudinal strain ϵlox while that of the silicon ϵlSi will be positive such that
(11)
while a correspondingly compressive longitudinal stress in the oxide σlox* over its cross section provides an axial force balance with the tensile longitudinal stress in the silicon σlSi* over its cross section
(12)
Fig. 4
Models of Si cylinder and oxide formed from it when expanded and free-standing for stress analysis. Original Si surface at Ro forms oxide free surface at b while instantaneous Si surface at Ri and oxide surface formed from it at a must be brought together to form Si/oxide interface: (a) convex case of a solid Si cylinder and (b) concave case of a cylindrical hole through a bulk Si body.
Fig. 4
Models of Si cylinder and oxide formed from it when expanded and free-standing for stress analysis. Original Si surface at Ro forms oxide free surface at b while instantaneous Si surface at Ri and oxide surface formed from it at a must be brought together to form Si/oxide interface: (a) convex case of a solid Si cylinder and (b) concave case of a cylindrical hole through a bulk Si body.
Close modal
As in their free-standing states, the inner radius of the oxide a is expanded from and thus greater than the outer radius of the remaining silicon Ri by factor f, bringing these surfaces together to form an interface is the case of a negative “interference” thus with a negative value of pressure p* across the interface. Considering the oxide as a hollow cylinder of Poisson's ratio νox internally pressurized by p* to get expressions for its radial and circumferential stresses then substitution into a Hooke's law expression for longitudinal strain provides
(13)
while the Si as a solid cylinder of Poisson's ratio νSi externally pressurized by the same p* thus having radial and circumferential stresses that are both the negative of that external pressure upon substitution into a Hooke's law expression for longitudinal strain provides
(14)
Treating radial deflections at the inner surface of the hollow cylindrical oxide and the outer surface of the solid Si cylinder to transform their interference to an interface
(15)
provides a system of five Eqs. (11)(15) from which the values of five unknowns ϵlSi, ϵlox, σlSi*, σlox*, p* may be determined at any thickness of oxide xo*. With σlox* and p* now known, the resultant hydrostatic stress being an average of longitudinal, radial, and circumferential stresses becomes uniform through the oxide due to the radial position dependencies of radial and circumferential stresses canceling as shown in the  Appendix yielding the expression
(16)
so that at any oxide thickness xo*, the value of its instantaneous growth rate dxo*/dt* may be calculated using Eq. (10), and the time t* to attain this thickness xo* numerically approximated as the area under a plot of the inverse of growth rate as a function of thickness up to that specific xo* of interest. Solution of simultaneous stress analysis equations as well as trapezoidal integration of the inverse growth rate to calculate the oxidation time to reach any oxide thickness were performed by python code employing SciPy module.

Concave Stress Analysis.

As depicted in Fig. 4(b), in the concave case we consider the Si to be a bulk infinite body with a cylindrical hole, within which an oxide annulus is produced from the surface of the Si hole. Both the Si cylindrical hole and oxide annulus are again considered to be “long” and each thus with longitudinal deformation state approximated as uniform and not a function of radial position. With the Si as bulk of large axial cross section bringing its longitudinal stress toward zero as well rendering it rigid axially, and thus with the longitudinal expansion of the free-state oxide relative to it by factor f therefore being accommodated solely by the negative longitudinal strain of the oxide when bringing it back to a shared length interfacing with the Si, it may be approximated that ϵlox(f1). As in their free-standing states, the outer radius of the oxide a is expanded from and thus greater than the inner radius Ri of the remaining Si by factor f, bringing these surfaces together to form an interface is now the usual case of a positive interference thus with a positive value of pressure p* across the interface. Considering the oxide as a hollow cylinder now externally pressurized by p* to get expressions for its radial and circumferential stresses, then substitution into a Hooke's law expression for longitudinal strain provides
(17)
Treating radial deflections at the outer surface of the hollow cylindrical oxide and the inner surface of the cylindrical hole through a bulk Si body (i.e., hollow cylindrical Si with infinite outer radius) to transform their interference to an interface provides
(18)
Using the two Eqs. (17) and (18), at any oxide thickness the values of interface pressure p* and oxide longitudinal stress σlox* may be solved, and in turn, the hydrostatic stress again found to be uniform through the oxide and expressed as
(19)

Once again, in such a curved case, at any oxide thickness xo*, the value of its instantaneous growth rate dxo*/dt* may be calculated using Eq. (10), and the time t* to attain this thickness xo* numerically approximated as the area under a plot of the inverse of growth rate as a function of thickness up to that specific xo* value of interest, again with all such calculations performed by python code.

Flat Case Stress Analysis.

The flat baseline case is approximated by allowing the Si original radius Ro* in either of the curved cases above to approach infinity. Selecting the simpler and compact concave curved case to work from, and rearranging Eq. (17) to an expression for interfacial pressure
(20)
It can be seen for an oxide film growing on a flat semi-infinite bulk Si substrate that no pressure develops across the metal/oxide interface, as may have been intuited. Furthermore, the seeming singularities within Eqs. (18) and (19) due to (1+xo*fRo*)21 denominator as Ro* goes to infinity is instead resolved upon substituting the above expression for p*. Upon thereafter also allowing Ro* toward infinity, Eq. (18) yields an expression for oxide longitudinal stress as this flat case is approached, which upon substitution into Eq. (19) provides an expression for oxide hydrostatic stress in the flat baseline asymptote case
(21)
Substituting this oxide hydrostatic stress and interfacial pressure pflat*=0 into Eq. (10) yields
(22)
Since the oxide hydrostatic stress in this flat asymptote is constant and not a function of xo*, the integration of Eq. (6) may be performed in closed-form, again providing an analytical linear-parabolic kinetics expression for time t* to grow an oxide of thickness xo* atop bare flat Si
(23)

Stress Analysis Summary.

In oxidation of silicon with νox = 0.17 and f ≈ 1.31, for the flat baseline case the summary Fig. 5 shows the oxide hydrostatic stress σhox*0.220 and interfacial pressure p* = 0 both constant and independent of increasing oxide thickness. When curved, considering νSi = 0.27 and ESi*=2.286, for convex cases the oxide hydrostatic stress varies from an initial σhox*0.220 toward zero while interfacial pressure in a complementary fashion varies from an initial zero value toward p* ≈ − 0.220 as oxide thickness increases, with these variations of stress occurring more sharply with oxide thickness in cases of smaller original Si radius Ro*. This trend may be understood by recalling that the convex solid Si is completely consumed when oxide thickness xo* reaches fRo*, at which the p* ≈ − 0.220 value has been reached but must thereafter step back to zero as an interface no longer exists. Thus, for smaller values of concave Ro*, this full variation of stress values must be completed within a correspondingly smaller range of xo*. For concave cases, the interfacial pressure p* instead increases from the initial zero to increasingly positive values while the oxide hydrostatic stress becomes increasingly negative from its initial σhox*0.220 as oxide thickness increases, with these variations again occurring more sharply with oxide thickness in cases of smaller Ro*.

Fig. 5
Non-dimensionalized stress analysis summary showing hydrostatic stress within the oxideσhox* and pressure p* across the oxide/Si interface as a function of oxide thickness at various convex and concave original Si radii Ro* including infinite (flat). For case of f = 1.31, ESi* = 2.286, Si = 0.27, ox = 0.17, where X indicates point at which Si is completely consumed for convex solid cylinders (and any interfacial pressure p* will correspondingly drop back to zero).
Fig. 5
Non-dimensionalized stress analysis summary showing hydrostatic stress within the oxideσhox* and pressure p* across the oxide/Si interface as a function of oxide thickness at various convex and concave original Si radii Ro* including infinite (flat). For case of f = 1.31, ESi* = 2.286, Si = 0.27, ox = 0.17, where X indicates point at which Si is completely consumed for convex solid cylinders (and any interfacial pressure p* will correspondingly drop back to zero).
Close modal

Stress-Dependent Model Output Discussion

Runs of the model for several values of original Si radius (Ro*=7.5, 10, 15) both convex and concave were performed with only one of the parameters solubility Cg, diffusivity D, or reaction rate k at a time having a stress dependence considered. The same magnitude value of 8 was arbitrarily selected for non-dimensional activation volume V* for each of these parameters Cg, D, k with these stress dependency expressions written in anticipation of a positive activation volume, though model runs were also run with a negative activation volume V* = − 8 to further investigate the event that the stress effect on a parameter was opposite that anticipated. While closed-form expressions are provided for time t* to grow an oxide thickness xo* atop a bare Si for all baseline cases of flat Si substrate and/or constant parameters, for any case with stress-dependent parameters at a specific curvature radius value Ro*, at each oxide thickness value a system of equations must be solved simultaneously to determine the corresponding interfacial pressure and oxide hydrostatic stress values, and subsequently oxide growth rate dxo*/dt*. Creating from this a plot of the inverse of oxide growth rate as a function of oxide thickness, the growth time t* corresponding with any oxide thickness is approximated by numerically integrating the area under this inverse oxide growth rate plot up to the thickness of interest xo*. Records of these calculated pairs of thickness and growth time may then be plotted, most conventionally with thickness xo* as a function of time t*, even though for the baseline cases the closed-form expressions are instead explicit in t*.

Studying the effect of stress-dependent solubility alone in Fig. 6, it is first noted that the collection of constant solubility (VCg*=0) curves in the middle of the plot are the same as the constant-parameters plot of Fig. 3 but with axes swapped. While describing at any fixed time a thickness that decreases with decreasing radius in concave cases, it also describes in convex cases an oxide thickness that instead increases with decreasing radius to values even greater than the flat case, in conflict with what has been reported experimentally. In the case of stress-dependent solubility with VCg*=+8, as hydrostatic stresses in the oxide are negative per Fig. 5 and thus solubility reduced, the thicknesses at any time are reduced from the constant solubility VCg*=0 case. Furthermore, as concave cases have oxide hydrostatic stress that is increasingly negative and in turn solubility that is further reduced with decreasing radius, while convex cases have oxide hydrostatic stress that is decreasingly negative and in turn solubility that is less reduced with decreasing radius, the rank ordering at any fixed time of oxide thickness with curvature remains the same as that in the constant-solubility case. Interestingly, if the effect of oxide hydrostatic stress on solubility was opposite that anticipated, as indicated for example by VCg*=8, while of course the negative hydrostatic stresses would instead cause higher solubility and greater oxide thicknesses at any fixed time, furthermore the increasingly negative hydrostatic stresses in the concave cases of lesser radius in this case cause solubility so much more greatly increased that thicknesses are now greater than the flat case. Meanwhile, convex cases of lesser radius and thus less negative hydrostatic stress have their solubility increased the least which results in thicknesses further reduced from the flat case. The effect is sufficiently strong that trends of curvature have thus been reversed from those predicted in the constant-parameters case, now with the concave thicknesses in excess of the flat case, conflicting with experimental observation.

Fig. 6
Non-dimensional presentation of oxide thickness as a function of time for cases focusing on the effect of stress-dependent oxidant solubility at various convex and concave original Si radii Ro* including infinite (flat). Shown are examples where solubility increases (VCg* = +8) and decreases (VCg* = −8) with increasingly tensile oxide hydrostatic stress, as well as the constant-parameters case (VCg* = 0) for comparison.
Fig. 6
Non-dimensional presentation of oxide thickness as a function of time for cases focusing on the effect of stress-dependent oxidant solubility at various convex and concave original Si radii Ro* including infinite (flat). Shown are examples where solubility increases (VCg* = +8) and decreases (VCg* = −8) with increasingly tensile oxide hydrostatic stress, as well as the constant-parameters case (VCg* = 0) for comparison.
Close modal

The effect of stress-dependent diffusivity alone in Fig. 7 is much like that discussed for stress-dependent solubility alone in Fig. 6 above. This similarity is largely because for both diffusivity and solubility, it is the same hydrostatic component within the oxide that provides their stress dependence, and we are considering the same values of activation volume describing the strength of that dependence. Furthermore, examining the oxide growth rate expression of Eq. (10), in the example case we are studying of extremely large h* = 55,600 causing the gas-phase transport (second) term of the denominator to be negligible consistent with the analysis of Kao et al. [8], the interfacial reaction (first) term of the denominator is rate-limiting and controls behavior only over a brief initial period, after which oxide of sufficient thickness through which oxidant must travel from free surface to Si interface has grown such that the diffusion (third) term of the denominator becomes predominant. At this point, multiplying the whole growth rate expression by a hydrostatic stress-dependent Arrhenius factor in a single solubility-based term in the numerator is similar to dividing a third but predominating diffusion term in the denominator by such a hydrostatic stress-dependent Arrhenius factor with any slight difference due to the interfacial reaction first term of the denominator briefly rate-controlling during early initial periods. Though unanticipated, a negative value of diffusivity activation volume of this magnitude VD*=8 again causes the rank ordering of oxide thicknesses at fixed time t* to reverse from that observed at constant parameters (VD*=0) or with positive activation volume, with concave cases of lesser radius having thickness increasingly greater than the flat case, contrary to experimental observation.

Fig. 7
Non-dimensional presentation of oxide thickness as a function of time for cases focusing on the effect of stress-dependent oxidant diffusivity at various convex and concave original Si radii Ro* including infinite (flat). Shown are examples where diffusivity increases (VD* = +8) and decreases (VD* = −8) with increasingly tensile oxide hydrostatic stress, as well as the constant-parameters case (VD* = 0) for comparison.
Fig. 7
Non-dimensional presentation of oxide thickness as a function of time for cases focusing on the effect of stress-dependent oxidant diffusivity at various convex and concave original Si radii Ro* including infinite (flat). Shown are examples where diffusivity increases (VD* = +8) and decreases (VD* = −8) with increasingly tensile oxide hydrostatic stress, as well as the constant-parameters case (VD* = 0) for comparison.
Close modal

As opposed to studying the hydrostatic stress dependence of solubility or diffusivity alone and the effect of the strength of this dependence via various activation volumes, which causes the hydrostatic stress-dependent baseline flat case about which the convex and concave cases cluster to shift in the (t*, xo*) space per Eq. (23), when studying the pressure dependence of reaction rate alone through various values of Vk*, the baseline flat case does not shift as it does not develop any interfacial pressure p* with increasing oxide thickness. As the stress-independent (Vk*=0) case at various curvature radii are already depicted as the middle cluster in Figs. 6 and 7, to avoid overlap the Vk*=+8 and Vk*=8 cases are presented for comparison in separate Figs. 8(a) and 8(b), respectively. For the Vk*=+8 case, the plots at various curvatures appear to spread more broadly from the baseline flat case than when stress-independent, while at Vk*=8 the plots for various curvatures appear instead to tighten more closely to the baseline flat case. In the stress-independent case, decreasing convex radii already appear to create thicker oxide, but additionally, they develop increasingly negative interfacial pressure p* (Fig. 5) such that in the case of a positive Vk*=+8 activation energy the reaction (first) term of the denominator in oxide growth rate Eq. (10) decreases and becomes less rate-limiting during the initial growth period allowing even greater spread to thicker oxides. Conversely, the stress-independent cases of decreasing concave radii already appear to create thinner oxide and additionally they develop instead increasingly positive interfacial pressure p*, such that in the case of a positive activation energy the reaction term of the denominator in oxide growth rate Eq. (10) increases and becomes more rate-limiting during the initial growth period allowing even greater spread to thinner oxides. In contrast, though not what is anticipated, in the case of a negative reaction rate activation volume as Vk*=8, decreasing convex radii with their increasingly negative interfacial pressure p* cause the reaction term of the denominator in the growth rate to increase and become more rate-limiting during initial growth, countering the tendency in stress-independent cases of decreasing convex radii to have thicker oxides and causing the spread of these convex cases to tighten back toward the flat baseline case, with the converse effect on the concave cases also tightening them back toward the flat baseline case.

Fig. 8
Non-dimensional presentation of oxide thickness as a function of time for cases focusing on the effect of stress-dependent reaction rate at various convex and concave original Si radii Ro* including infinite (flat). Shown are examples where reaction rate (a) decreases (Vk* = +8) and (b) increases (Vk* = −8) with increasingly positive pressure across the oxide/Si interface, with (c) focusing in on the Vk* = −8 case. The constant-parameters case (Vk* = 0) may be taken from Fig. 6 or Fig. 7 for comparison as it is too heavily overlapped by the stress-dependent reaction rate cases to be plotted again here.Non-dimensional presentation of oxide thickness as a function of time for cases focusing on the effect of stress-dependent reaction rate at various convex and concave original Si radii Ro* including infinite (flat). Shown are examples where reaction rate (a) decreases (Vk* = +8) and (b) increases (Vk* = −8) with increasingly positive pressure across the oxide/Si interface, with (c) focusing in on the Vk* = −8 case. The constant-parameters case (Vk* = 0) may be taken from Fig. 6 or Fig. 7 for comparison as it is too heavily overlapped by the stress-dependent reaction rate cases to be plotted again here.
Fig. 8
Non-dimensional presentation of oxide thickness as a function of time for cases focusing on the effect of stress-dependent reaction rate at various convex and concave original Si radii Ro* including infinite (flat). Shown are examples where reaction rate (a) decreases (Vk* = +8) and (b) increases (Vk* = −8) with increasingly positive pressure across the oxide/Si interface, with (c) focusing in on the Vk* = −8 case. The constant-parameters case (Vk* = 0) may be taken from Fig. 6 or Fig. 7 for comparison as it is too heavily overlapped by the stress-dependent reaction rate cases to be plotted again here.Non-dimensional presentation of oxide thickness as a function of time for cases focusing on the effect of stress-dependent reaction rate at various convex and concave original Si radii Ro* including infinite (flat). Shown are examples where reaction rate (a) decreases (Vk* = +8) and (b) increases (Vk* = −8) with increasingly positive pressure across the oxide/Si interface, with (c) focusing in on the Vk* = −8 case. The constant-parameters case (Vk* = 0) may be taken from Fig. 6 or Fig. 7 for comparison as it is too heavily overlapped by the stress-dependent reaction rate cases to be plotted again here.
Close modal

In fact, in this tightening of various radius cases back toward the flat baseline by a negative activation volume, the magnitude of Vk*=8 is sufficient to cause convex cases to be limited to oxide thicknesses even less than that when flat. Focusing more closely upon times t* > 41.5 in Fig. 8(c), this negative value of reaction rate activation volume causes the model to describe trends of radius-dependent behavior that are similar to those observed experimentally by Kao et al. [3] with concave as well as convex cases, both having oxide thicknesses at a fixed time t* that decrease with decreasing radius and, furthermore, with concave cases having thickness more greatly reduced from the flat baseline case than convex cases. This similarity of model predictions to published experimental data such as revisited in Fig. 1 is limited to qualitative trends, with any quantitative correlation seemingly lacking strength as the model describes oxide thicknesses at fixed time that decrease with decreasing radius of curvature from the flat case by only a few percent whereas in experiment they are observed to decrease instead by several tens of percent. Any more detailed quantitative comparison is hampered by Kao et al. [3] not reporting the original Si radii, but instead radii remaining after oxidation and measured as interfacing with the oxide at the end of the experiment.

The demonstration of such predicted behavior by Vk*=8 has occurred by chance, and if the effect was strengthened by an increasingly negative reaction rate activation volume, it is anticipated that concave cases would instead begin to show thicknesses greater than the baseline flat case as V* = − 8 has already been shown capable of producing as an activation volume on solubility and diffusivity in Figs. 6 and 7, respectively. It is hypothesized that fine-tuning of activation volume to values slightly less negative than V* = − 8 when studying stress-dependent solubility or diffusivity alone can also generate periods of time where the predicted effect of Si radius of curvature also trends qualitatively toward that observed experimentally [3], with smaller Si radii generating thinner oxide and this effect being more pronounced in concave than convex cases. In turn, it is anticipated that combinations of reaction rate, solubility, and diffusivity that are each stress-dependent could qualitatively also provide such experimentally observed trends, including the possibility of a positive activation energy, provided that others have negative activation energies such that they are predominant. Despite the model's ability to make such predictions, it is concluded that this model is ultimately not able to realistically describe the experimentally observed behaviors of Kao et al. [3] unless it is allowed to contradict at least one of these expected behaviors: interfacial oxidation reaction rate that is hindered by increasingly positive interfacial pressure; oxidant solubility into the oxide's free surface that is hindered by increasingly negative hydrostatic stress; oxidant diffusivity through the oxide that is hindered by increasingly negative hydrostatic stress.

In response to the initial coupling of an elastic stress analysis by Yoshikawa et al. [12] to an oxidation kinetics model, Kao et al. [8] posited that the deformation of the oxide which must occur is far too large to be accommodated by strain that remains elastic, such as here too where it is often considered that the magnitudes of the oxide and Si strains must sum to account for the large ∼31% difference in dimensions of a free-standing SiO2 cylindrical annulus compared to that of the Si consumed in its formation. On a related note, with the correspondingly large reaction-related Vk = 25*10−30 m3 molecular volume change to SiO2 from Si [8], the non-dimensional Vk*=Vk/(KT/Eox) would instead be approximately two orders-of-magnitude higher than the |Vk*|=8 employed here in demonstrating feasibility of model output with similarity at least qualitatively to experimentally observed trends. Thus any seeming correlation to experimental observation of model output is further weakened by realization that the Vk* = − 8 demonstrating qualitative similarity here is not only opposite in sign but also much smaller in magnitude than that in actuality. Coffin et al. [14] summarized SiO2 to behave as an elastic solid only below ∼800 °C. Beyond an initial elastic stress analysis, Hsueh and Evans [11] already also further developed viscoelastic stress analyses for such cylindrical oxidation situations with creep deformations capable of accommodating larger strains, but again, they did not couple this into kinetics modeling of growing oxide. EerNisse [15] found that deformation of SiO2 may fully transform to viscous flow above ∼960 °C, and as such when revising their preliminary constant-parameter model Kao et al. [8] treated the oxide as a viscous fluid when coupling into the oxidation kinetics model an analysis of the stress state upon which its parameters may depend.

While not able to arrive at an activation volume value to consistently describe any stress dependence of diffusivity, the coupled model of Kao et al. [8] does consider not only stress-dependent solubility and reaction rate within the kinetics but now additionally a pressure dependence of the viscosity required of the fluid mechanics analysis. Within such a fluid stress-dependent parameters model, they claim to successfully predict oxide thicknesses at any fixed time that decrease with decreasing curvature radius from the infinite-radius flat case in both concave and convex cases, with the effect being stronger in concave cases as observed experimentally. However, again, the Kao et al. [8] model's descriptions of oxidation time required to attain corresponding oxide thickness are not in closed-form but instead require numerical integration. They have since revised their model, for example, adding in a stress-dependent diffusivity while removing the stress dependence of solubility, as well as tuning volumes used in describing dependence of the reaction rate away from an initial value argued to be the difference between the molecular volumes of the oxide and original metal [16,17]. Such viscous fluid models have since seen broader adoption to model the stresses and in turn stress-dependent parameters describing the deformation of growing oxide films on curved surfaces both cylindrical and spherical, for example, Refs. [9,14,1820]. As such, we next propose to investigate modifying our stress analysis presented here from an elastic to a viscous model, aspiring to descriptions of oxide growth with dependence on radius of curvature both convex and concave that more fully align with experimental observations of Kao et al. [3] as well as anticipated directions of effects on parameter values by tensile or compressive stresses.

Conclusions

  1. The non-dimensionalization of our model for interfacial oxidation of cylindrically curved surfaces, treating oxide as a solid annulus freely expanded from that of the Si consumed in its production with diffusion analysis performed using this expanded geometry, was revisited so as to maintain the visibility of original Si radius of curvature in resulting equations rather than its use to normalize other terms that previously obscured it. In the case of constant parameters (solubility, diffusivity, reaction rate), the closed-form explicit analytical expression as a function of oxide thickness of the time to reach it describes at any fixed time that thickness decreases with decreasing original Si radius in concave cases as has been previously reported experimentally, but that in convex cases thickness would increase with decreasing original Si radius contradictory to reported experiments.

  2. As a result of molecular volume of oxide increased from the original Si and associated stress state, and the hypothesis that solubility and diffusivity of the oxidant should actually increase with increasing hydrostatic stress of the oxide while reaction rate should decrease with increasing pressure (negative of radial stress) across the oxide/Si interface, stress analysis was performed displacing the surfaces of the freely expanded oxide annulus and the remaining Si back together to form their interface. For this geometry, hydrostatic stress is negative and found to be uniform through the oxide. As oxide grows, the hydrostatic stress becomes increasingly negative in concave cases while instead approaching zero in convex cases, while for interfacial pressure that is initially zero, it becomes increasingly positive in concave cases and increasingly negative in convex cases as oxide grows. These variations with increasing oxide thickness become more rapid at lesser values of radius of curvature of the original Si.

  3. In such a stress-dependent parameters case, this model no longer results in closed-form analytical expressions relating oxide growth time to oxide thickness. Nonetheless, as a function of increasing oxide thickness with instantaneous stress state and corresponding parameter values, the resultant oxide growth rate may be calculated and the area under a plot of its inverse when integrated numerically up to any thickness yields the time to achieve it. When individually considering either solubility or diffusivity to increase with increasing hydrostatic stress, the oxide thickening is slowed, yet the trend of thickness variation with original Si radius remains the same as that in the constant-parameters case. However, when considering reaction rate to decrease with increasing interfacial pressure, the primary effect appears to be just a further spreading of thickness variation with original Si radius while maintaining the same trend as the constant-parameters case. As such, these stress-dependent parameter analyses each still result in describing at any fixed time for convex cases an oxide thickness that increases with decreasing original Si radius, remaining in conflict with reported experimental observation of Kao et al.

  4. If rather allowing reaction rate to increase with increasing interfacial pressure, it has been shown possible to describe periods of time for convex surfaces where oxide thickness instead decreases with decreasing original Si radius, though with that decrease being less strong than for concave surfaces, as reported for experiments on Si oxidation. It appears that the same can be achieved by allowing solubility or diffusivity to instead decrease with increasing hydrostatic stress upon tuning of activation volume describing the strength of this dependence. If none of the above reversals of direction of parameter variation with stress is allowable, stress analysis instead considering some viscous fluid aspect of the oxide may be required to achieve concurrence of model description of the effect of Si curvature with reported experimental observation.

Acknowledgment

T.A. Blanchet acknowledges support during the period of this work by the National Science Foundation Grant DMR-1713670.

Conflict of Interest

There are no conflicts of interest.

Data Availability Statement

The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.

Appendix: Interference Fit Stress Analysis

As shown in Fig. 4, with the dimensions in its free-standing state of the oxide annulus expanded beyond the dimensions of the annulus of Si from which it was produced, a radial interference aRi exists between the radii a and Ri of the oxide and instantaneous remaining Si that must be brought together to form their interface. Such a radial interference is accommodated by a difference of the radial displacements of the silicon at Ri and the oxide at a, respectively, where these radial displacements each are the product of the radius and the circumferential strain evaluated at that radius (the circumferential displacements divided by 2π) thus
(A1)
The circumferential strains are described by application of Hooke's law
(A2)
The circumferential and radial stresses are provided from consideration of pressurized annular cylinders covered in standard solid mechanics design texts as [13] and summarized below, while longitudinal stresses are also described by application of Hooke's law
(A3)
where for long cylinders, it may be assumed that the longitudinal strains of Si and oxide are each uniform not varying with radial position and furthermore must have a difference ϵlSiϵlox equal to f–1 as per Eq. (11) where f is the dimensional expansion factor of the oxide. Once the longitudinal, circumferential, and radial stresses are determined through the additional analysis below, the hydrostatic stress σh is simply their average (σl+σθ+σr)/3.

Convex Case

As shown in Fig. 4(a), in this case, the hollow oxide annulus of inner and outer radii a and b respectively is considered to be internally pressurized by p and thus have radial and circumferential stresses as a function of position r described by
(A4)
(A5)
The Si in this treatment is considered to be a solid cylinder of instantaneous radius Ri externally pressurized by p, thus with radial and circumferential stresses both simply as
(A6)
where in this case, the oxide outer cylinder as expanded in turn produces an interfacial pressure p which is negative, as will be seen. Applying Eq. (A3) to the oxide yields
(A7)
where the dependencies of radial and circumferential stresses on position r have canceled leaving a longitudinal stress σlox that like longitudinal strain in this long cylinder case is uniform and not varying with radial position, while Eq. (A3) application to the Si yields
(A8)

Equations (13) and (14) in the text are simply Eqs. (A7) and (A8) non-dimensionalized. The radial and circumferential stresses of Eqs. (A4)(A6) upon insertion into Eq. (A2) and in turn Eq. (A1) results in text Eq. (15) in rearranged non-dimensional form.

Concave Case

As shown in Fig. 4(b), in this case, the hollow oxide annulus of inner and outer radii now b and a, respectively, is considered to instead be externally pressurized by p and thus have radial and circumferential stresses as a function of position r described by
(A9)
(A10)
The remaining Si in this treatment is considered to be a bulk body with a cylindrical hole through it (i.e., a cylinder of inner radius Ri that is internally pressurized by p with an infinite outer radius) and thus radial and circumferential stresses described as
(A11)
(A12)
Due to the bulk of the Si body in this case having large longitudinal cross section to distribute any longitudinal force over thus considering σlSi0 furthermore to have negligibly small strain longitudinally such that the oxide longitudinal strain alone must accommodate its dimensional expansion in the free state ϵlox=(f1). Application of Eq. (A3) to the oxide yields
(A13)
with again the dependencies of radial and circumferential stresses on r canceling and leaving an oxide longitudinal stress that is uniform with radial position. This Eq. (A13) upon substitution of oxide longitudinal strain and non-dimensionalization is Eq. (17) in the text while radial and circumferential stresses of Eqs. (A9)(A12) into Eq. (A2) and in turn Eq. (A1) upon non-dimensionalization and rearrangement becomes text Eq. (18).

References

1.
Deal
,
B. E.
, and
Grove
,
A. S.
,
1965
, “
General Relationship for the Thermal Oxidation of Silicon
,”
J. Appl. Phys.
,
36
(
12
), pp.
3770
3778
.
2.
Marcus
,
R. B.
, and
Sheng
,
T. T.
,
1982
, “
The Oxidation of Shaped Silicon Surfaces
,”
J. Electrochem. Soc.
,
129
(
6
), pp.
1278
1282
.
3.
Kao
,
D.-B.
,
McVittie
,
J. P.
,
Nix
,
W. D.
, and
Saraswat
,
K. C.
,
1987
, “
Two-Dimensional Thermal Oxidation of Silicon—I. Experiments
,”
IEEE Trans. Electron Devices
,
34
(
5
), pp.
1008
1017
.
4.
Krzeminski
,
C. D.
,
Han
,
X.-L.
, and
Larrieu
,
G.
,
2012
, “
Understanding of the Retarded Oxidation Effects in Silicon Nanostructures
,”
Appl. Phys. Lett.
,
100
(
26
), p.
263111
.
5.
Liu
,
H. I.
,
Biegelsen
,
D. K.
,
Ponce
,
F. A.
,
Johnson
,
N. M.
, and
Pease
,
R. F. W.
,
1994
, “
Self-Limiting Oxidation for Fabricating Sub-5 nm Silicon Nanowires
,”
Appl. Phys. Lett.
,
64
(
11
), pp.
1383
1385
.
6.
Liu
,
M.
,
Jin
,
P.
,
Xu
,
Z.
,
Hanaor
,
D. A. H.
,
Gan
,
Y.
, and
Chen
,
C.
,
2016
, “
Two-Dimensional Modeling of the Self-Limiting Oxidation in Silicon and Tungsten Nanowires
,”
Theoret. Appl. Mech. Lett.
,
6
(
5
), pp.
195
199
.
7.
Okada
,
R.
, and
Iijima
,
S.
,
1991
, “
Oxidation Property of Silicon Small Particles
,”
Appl. Phys. Lett.
,
58
(
15
), pp.
1662
1663
.
8.
Kao
,
D.-B.
,
McVittie
,
J. P.
,
Nix
,
W. D.
, and
Saraswat
,
K. C.
,
1988
, “
Two-Dimensional Thermal Oxidation of Silicon—II. Modeling Stress Effects in Wet Oxides
,”
IEEE Trans. Electron Devices
,
35
(
1
), pp.
25
37
.
9.
Lemme
,
B. D.
,
2009
, “
Non-Planar Silicon Oxidation: An Extension of the Deal-Grove Model
,”
M.S. thesis
,
Kansas State University
,
Manhattan, KS
.
10.
Blanchet
,
T. A.
,
2022
, “
Closed-Form General Relationship Model for the Interfacial Oxidation of Cylindrically Curved Surfaces
,”
ASME J. Eng. Mater. Technol.
,
144
(
3
), p.
031009
.
11.
Hsueh
,
C. H.
, and
Evans
,
A. G.
,
1983
, “
Oxidation Induced Stresses and Some Effects on the Behavior of Oxide
,”
J. Appl. Phys.
,
54
(
11
), pp.
6672
6686
.
12.
Yoshikawa
,
K.
,
Nagakubo
,
Y.
, and
Kanzaki
,
K.
,
1984
, “
Two Dimensional Effect on Suppression of Thermal Oxidation Rate
,”
16th Conference on Solid State Devices and Materials
,
Kobe, Japan
,
Aug. 30–Sept. 1
, pp.
475
478
.
13.
Collins
,
J. A.
,
2003
,
Mechanical Design of Machine Elements and Machines: A Failure Prevention Perspective
, 1st ed.,
Wiley
,
New York
, pp.
372
374
.
14.
Coffin
,
H.
,
Bonafos
,
C.
,
Schamm
,
S.
,
Cherkasin
,
N.
,
Assayag
,
G. B.
,
Claverie
,
A.
,
Respaud
,
M.
,
Dimitrakis
,
P.
, and
Normand
,
P.
,
2006
, “
Oxidation of Si Nanocrystals Fabricated by Ultralow-Energy Ion Implantation in Thin SiO2 Layers
,”
J. Appl. Phys.
,
99
(
4
), p.
044302
.
15.
EerNisse
,
E. P.
,
1977
, “
Viscous Flow of Thermal SiO2
,”
Appl. Phys. Lett.
,
30
(
6
), pp.
290
293
.
16.
Sutardja
,
P.
,
Oldham
,
W. G.
, and
Kao
,
D.-B.
,
1987
, “
Modeling of Stress-Effects in Silicon Oxidation Including the Non-Linear Viscosity of Oxide
,”
Proceedings of International Electron Devices Meeting
,
Washington, DC
,
Dec. 6–9
, Vol.
33
, pp.
264
267
.
17.
Sutardja
,
P.
, and
Oldham
,
W. G.
,
1989
, “
Modeling of Stress-Effects in Silicon Oxidation
,”
IEEE Trans. Electron Devices
,
36
(
11
), pp.
2415
2421
.
18.
Chen
,
Y.
,
2000
, “
Modeling of the Self-Limiting Oxidation for Nanofabrication of Si
,”
Technical Proceedings of the 2000 International Conference on Modeling and Simulation of Microsystems
,
San Diego, CA
,
Mar. 27–29
, pp.
56
58
.
19.
Chen
,
Y.
, and
Chen
,
Y.
,
2001
, “
Modeling Silicon Dots Fabrication Using Self-Limiting Oxidation
,”
Microelectron. Eng.
,
57–58
(
1
), pp.
897
901
.
20.
Omachi
,
J.
,
Nakamura
,
R.
,
Nishiguchi
,
K.
, and
Oda
,
S.
,
2001
, “
Retardation in the Oxidation Rate of Nanocrystalline Silicon Quantum Dots
,”
MRS Symposium Proceedings
,
Boston, MA
,
Nov. 26–30
, Vol.
638
.