Predictive model for scanned probe oxidation kinetics

Previous descriptions of scanned probe oxidation kinetics involved implicit assumptions that one-dimensional, steady-state models apply for arbitrary values of applied voltage and pulse duration. These assumptions have led to inconsistent interpretations regarding the fundamental processes that contribute to control of oxide growth rate. We propose a model that includes a temporal crossover of the system from transient to steady-state growth and a spatial crossover from predominantly vertical to coupled lateral growth. The model provides an excellent fit of available experimental data.

͑Received 24 January 2000; accepted for publication 13 March 2000͒   Previous descriptions of scanned probe oxidation kinetics involved implicit assumptions that one-dimensional, steady-state models apply for arbitrary values of applied voltage and pulse duration.These assumptions have led to inconsistent interpretations regarding the fundamental processes that contribute to control of oxide growth rate.We propose a model that includes a temporal crossover of the system from transient to steady-state growth and a spatial crossover from predominantly vertical to coupled lateral growth.The model provides an excellent fit of available experimental data.© 2000 American Institute of Physics.͓S0003-6951͑00͒03119-3͔ For scanned probe microscope ͑SPM͒ oxidation 1 to become established as a routine and reliable technique for nanodevice 2 and nanostructure 3 fabrication, a predictive, quantitative model embodying a realistic physical description of the growth mechanism is essential.There has been considerable recent interest in interpreting experimental kinetic data for SPM oxidation [4][5][6][7][8][9][10][11][12] and general agreement has emerged on the dependence of oxide height, h(t), width, W(t), and aspect ratio, h/W, on exposure parameters, i.e., voltage applied between SPM tip and substrate, V ͓Ϸ5 -20 V͔, and the pulse duration, t͓Ϸ10 Ϫ3 -10 3 s͔.In 1995 Gordon 7 suggested that Cabrera-Mott theory 13 was appropriate, with significant progress in this direction reported in 1997 by Stievenard 8 and Avouris. 9Stievenard arrived at the inverse-log form, 1/h(t)ϭk(V)Ϫlog t, beginning with the Cabrera-Mott assumptions and by introducing a thickness-dependent cutoff field E L ϭV/h L .Avouris proposed a direct-log form, h(t)ϭkЈ(V)•log t, instead.It is noteworthy that Stievenard and Avouris achieved apparently satisfactory empirical fits of their data based on these alternate forms of the growth, since each equation is derived from very different assumptions according to discussions of Fehlner and Mott. 14ecognizing that the direct-log form is the appropriate one for SPM oxidation, we examine the assumptions used to derive it.Uhlig 15 obtained the direct-log form, h(t) ϭkЉ(V)•log(k U tϩ1), by assuming that the rate-controlling step involves interaction of oxyanions with electronic species-holes or electrons-and defects at a metal-oxide interface ͑here, the Si/SiO x interface͒.In particular, Uhlig's model unifies key concepts recognized by Stievenard and Avouris with our conjectures about the influence of space charge on SPM oxide growth.According to this model, production of charged defects 16 leads to a buildup of space charge within the oxide, inhibiting further growth over long pulse times. 10,12If we follow Uhlig in attributing the thickness-related change in activation energy to space charge rather than stress as we have suggested in previous papers, we reach full agreement with Avouris on the form of the rate law for SPM oxide growth.We can also invoke Uhlig's theory to provide a physical basis for Stievenard's cutoff field, since E L is simply the potential at which space charge in the oxide nulls the externally applied field.As Uhlig has indicated, this is the point at which fast oxide growth ͑in the vertical dimension͒ ceases.
In 1995 Teuschler 6 demonstrated that an empirical power law, h(t)ϭa 0 (VϪV th )•t ␥ , provides a remarkably good fit to the experimental data, with subsequent work indicating that ␥ ranges from 0.12 to 0.4.͓a 0 and V th are constants.͔8][19] An exponential relaxation process, exp(Ϫt/), is ''stretched out'' over a longer time scale by a loss mechanism, so that exp(Ϫt/)→exp(Ϫt/) ␥ , with 0Ͻ␥р1.Since oxidation involves the transport of ionic species across the growing oxide film and subsequent reaction at one or the other of ambient/oxide and oxide/substrate interfaces, the process constitutes a reaction-diffusion system which may exhibit relaxation-dissipation behavior.In particular, note that Uhlig, in deriving the direct-log form did so by neglecting a second-order integration term of the Poisson equation justified on the assumption that h(t)Ӷh L .͓C.f., Equations ͑7͒-͑11͒ in Ref. 15.͔ Direct experimental evidence indicates that h L is on the order of 10 nanometers for SPM oxidation, with oxide thickness approaching or exceeding this value. 8,10,12A numerical integration of the linear, ͐exp(a•h)dhϭ͐dt, and the quadratic expressions, ͐exp(a•h 2 ϩb•hϩc)dhϭ͐dt, indicates that addition of the quadratic term increases the time required to achieve a given thickness by a factor of 2.5, thus ␥ϭ(2.5)Ϫ1 , or 0.4, for a retarding potential.Substituting t 0. A single overall rate constant, k U ϭ1/, in the Uhlig expression does not provide us with sufficient insight into how oxyanions, trapping sites, and electronic species interact under varying SPM oxidation conditions.However, we can augment Uhlig's basic concept by introducing space charge buildup concepts.Let's assume that silicon oxidation consists of a ''direct'' irreversible process, A→C, and an ''indirect'' one, A→B→C, according to reactions and rate constants defined in Table I. ͑See the discussions of Jonscher 17 and Mashkov 19 for examples.͒Integrated rate expressions for A(t), B(t), and C(t) have been published by Alberty and Miller 20 and are also given in Table I.The time evolution of the reactant and product concentrations defined by the Alberty-Miller scheme is displayed in Fig. 1, upper panel.Reaction of an initial concentration of oxyanions is represented by ͓A(t)͔.The density of fixed charge traps at the growing Si/SiO x interface and their dissipative effect on the oxidation rate is described by ͓B(t)͔.Evolution of the SiO x concentration, the symbol ͓C(t)͔, exhibits a growth curve that is evidently shaped by competition between the direct and indirect oxidation processes.If the Alberty-Miller equations provide us with a useful model for SPM oxidation under nondissipative conditions, then for the actual conditions encountered during SPM oxidation, h(t)ϰ͓C(t 0.4 ;k i )͔.The lower panel of Fig. 1 presents a comparison of Alberty-Miller, Uhlig, and Teuschler fits.Note that for tϽ1 s, all three expressions are equivalent.
Growth rates calculated using the Alberty-Miller and Uhlig ͑dot-dash͒ expressions are compared with experimental SPM oxidation data in Fig. 2, left-hand side.͓Experimental SPM data in this figure are replotted from Avouris, Fig. 2͑a͒.͔In fitting experimental data, values for the rate constants are not arbitrarily chosen.An initial value for the steady-state rate constant k 3 was determined directly by fitting the long-time kinetic data, using Equation ͑278͒ of Fromhold. 21A satisfactory fit for the steady-state portion of the growth curve leads to an underestimate of oxide growth at short time, which can be subtracted from the experimental data to obtain an initial estimate of k 4 .We can then figure out what k 1 ought to be by fitting the Alberty-Miller expression to the entire span of the experimental data.This leads to expected ordering of the magnitude of the rate constants, k 1 Ͼk 4 Ͼk 3 , which says that initially the direct (A→C) oxidation process occurs, but that over time the trapped state (A→B→C) builds up and interferes with growth.From the linear-log oresentation in Fig. 2, upper panel, we see that the simple Uhlig direct-log function does not adequately describe the modulated growth observed in the experimental results at long times.The origin of these modulated features in terms of competing rates, becomes evident from a glance at how ͓C(t)͔ evolves in Fig. 1.A survey of SPM oxidation literature yields numerous examples of previously overlooked modulation features in the published data-as long as a representative sampling of the steady-state regime is included.For gaining a fundamental understanding of SPM  oxidation, the transition point ͑or region͒ is an essential concept.In Fig. 2 a transition from transient growth, characterized by a t 0.4 power-law dependence, to steady-state growth, characterized by a t 0.167 dependence, is clearly observed.Figure 2, right-hand side, presents calculated and recently published experimental oxide growth kinetics for scanning electron-beam lithography ͑SEBL͒ oxidation. 12Although exposure conditions for SPM and SEBL are vastly different, similarity of the characteristic self-limiting kinetics suggests generation of charged defects appears to be a common control factor for both processes. 10,12͓Experimental SEBL data in this figure are replotted from Wei, Fig. 2͑b͒ of Ref. 12.͔ The Alberty-Miller expression, with ␥ϭ1.0 ͑solid line͒, is shown in comparison to a representative curve calculated with ␥ϭ0.4 ͑dashed line͒ which is the dilation factor used for SPM oxidation.These results give us a degree of confidence that the Alberty-Miller equations describe the most significant aspects of silicon oxidation, at least for low temperatures.͑See Wolters 18 for a somewhat related approach to thermal oxidation of silicon.͒For the fits of experimental SPM and SEBL kinetic data reported here, we have emphasized only voltage and time dependence of the oxide growth rate.Geometrical factors, the SPM tip relative humidity, and tipsubstrate distance, for instance, are not described.These factors may weakly affect the relative magnitude of the rate constants k i .We have evaluated all relevant SPM kinetic data in an effort to quantitatively model the subtle, yet significant, ͓three-dimensional ͑3D͔͒ differences in oxidation kinetics as a function of scan speed, voltage modulation, and contact versus noncontact modes.These results will be reported elsewhere.Figure 2 should make it clear, however, that future efforts to interpret SPM oxidation kinetics data must include a sufficient range of time and voltage, along with additional factors like tip radius and humidity, in order that the true functional dependence of the growth rate may be described correctly.
FIG.1.Upper panel: SPM oxidation model based on the Alberty-Miller integrated rate equations.A direct pathway for reaction of oxyanions with silicon at the Si/SiO 2 interface is given by (A→C) and an indirect reaction pathway, mediated by trapped charge defects at the interface, is given by (A→B) followed by (B→C).Lower panel: Log-log plot of SPM oxide feature height vs pulse time comparing the Teuschler power law, Uhlig direct-log, and Alberty-Miller expressions.The Uhlig rate constant is k U ϭ0.5 and the Alberty-Miller rate constants, as defined in TableI, are k 1 ϭ0.85, k 3 ϭ0.0035,and k 4 ϭ0.125.Notice that all three forms exhibit identical power law behavior, t 0.4 , in the transient regime, i.e., for tр500 ms.

FIG. 2 .
FIG. 2. Left-hand side: Calculated and experimental SPM oxidation kinetics.Data are presented in linear-log form in the upper panel, and in log-log form in the lower panel.Exposure conditions are given in terms of applied bias voltage, V dc ϭ5, 10, and 20 V, for pulse duration of 10 Ϫ2 to 10 3 s in contact mode.͓Experimental SPM data, open circles, etc., are taken from Avouris, Fig. 2͑a͒ of Ref. 9.͔ SPM growth rates, calculated using the Alberty-Miller expression with ␥ϭ0.4,͑heavy solid line͒ and the Uhlig direct-log form ͑dot-dash line͒ are compared to power-law curves ͑light solid lines͒.Although both follow the transition from transient to steadystate growth, only the Alberty-Miller form reproduces the modulated growth observed in the steady-state regime, i.e., for tϾ1 s.Right-hand side: Calculated and experimental SEBL oxidation kinetics.Exposure conditions are given in terms of electron accelerating voltage, V acc ϭ10, 20, 30, and 40 keV, for dwell times of 10 Ϫ1 to 10 s. ͓Experimental data, open circles, etc., are taken from Wei, Fig. 2͑b͒ of Ref. 12.͔ The Alberty-Miller expression with ␥ϭ1.0 ͑solid line͒, is shown in comparison to a representative curve calculated with ␥ϭ0.4 ͑dashed line͒.

TABLE I .
SPM oxidation kinetics represented as a set of integrated rate equations in the form derived by Alberty and Miller ͑Ref.20͒.Notation for the individual reaction steps and rate constants are as they appear in Alberty-Miller, Equations ͑5͒, ͑16͒-͑8͒.Rate equations for species A, B, and C are derived assuming that the initial concentrations are AϭA o , Bϭ0, Cϭ0.The reaction sequence symbolized by (A→B) is based upon the discussion of Poindexter.