Test of Marcus Theory Predictions for Electroless Etching of Silicon

Article pubs.acs.org/JPCC

Test of Marcus Theory Predictions for Electroless Etching of Silicon Kurt W. Kolasinski,* Jacob W. Gogola, and William B. Barclay Department of Chemistry, West Chester University, West Chester, Pennsylvania 19383, United States

ABSTRACT: Rational design of stain etchants has led to greatly improved control of porous silicon formation and thicker, more uniform layers. It has also facilitated the quantitative evaluation of the hole injection probability per collision of various oxidants (VO2+, Fe3+, Ce4+, and IrCl62−) with a silicon surface, the absolute rate constant, and direct comparison to the predictions of Marcus theory. The absolute rate constants vary roughly from 10−33 to 10−31 m4 s−1, which indicates a maximum rate constant of 1 × 10−25 m4 s−1 for VO2+ but only 1 × 10−27 m4 s−1 for Fe3+, 6 × 10−28 m4 s−1 for Ce4+, and 7 × 10−29 m4 s−1 for IrCl62−. Therefore, the charge transfer step that limits the rate of etching induced by VO2+ is well described by a Marcus theory description of an outer shell electron transfer process with a matrix element for coupling between the oxidant and the valence band roughly equal to the upper limit previously determined by Lewis and co-workers. However, for Fe3+, Ce4+, and IrCl62− coupling is much weaker, indicating that system specific calculations of the values of the reorganization energy, the coupling matrix element, and the tunneling range parameter are required to determine the extent of kinetically significant dynamical corrections for the description of electrochemical reactions at the liquid/semiconductor interface. been demonstrated by Chidsey10 at gold electrodes and Lewis and co-workers11,12 at semiconductor electrodes. In the realm of ultrahigh vacuum surface science, the measurement of sticking coefficients has been of central importance to the development of dynamical models of molecule/surface and atom/surface interactions.13 One of the reasons for this is, as shown by Tully,14−16 that the sticking coefficient s is equivalent to κ, the transmission coefficient from transition state theory. The sticking coefficient at the gas/solid interface is dependent on molecular orientation (including the surface impact parameter) and energetic factors. It describes the deviations of the rate constant for adsorption (and by invoking microscopic reversibility also for desorption) from the simple gas kinetic collision rate. The concept of a sticking coefficient at the liquid/solid interface has received much less attention. This is perhaps in part because the kinetics of liquid/metal electrode reactions is generally diffusion limited. However, the kinetics at liquid/ semiconductor electrode interfaces is inherently slower as a result of the greatly reduced density of electronic states. The

I. INTRODUCTION The Marcus theory of charge transfer1 provides an excellent basis for the understanding of electron exchange between donor and acceptor species and has also been extended to other types of reactivity.2−6 The prediction and later confirmation of an inverted region, that is, the presence of a maximum in a plot of the rate constant for charge transfer versus exoergicity of reaction, were the crowning achievements of this theory. The description of the inverted region by Marcus theory is far from the predicted parabolic shape;7 nonetheless, this counterintuitive behavior has been confirmed. Moser and Grätzel8 suggested that the reason for slow back transfer in dye sensitized inorganic semiconductor solar cells is inverted region behavior. This behavior was subsequently confirmed by Dang and Hupp.9 The behavior of the inverted region is somewhat different at a solid electrode. Rather than decreasing beyond a certain exoergicity, the rate constant for electron transfer from an occupied band to an acceptor state should approach a maximum; however, because an occupied band always provides states that are degenerate with the acceptor once the acceptor has left the gap, a constant rather than inverted region should be observed.7 The existence of a maximum rate constant has © 2012 American Chemical Society

Received: August 2, 2012 Revised: September 12, 2012 Published: September 12, 2012 21472

dx.doi.org/10.1021/jp3076723 | J. Phys. Chem. C 2012, 116, 21472−21481

The Journal of Physical Chemistry C

Article

Figure 1. Schematic representations of inner and outer sphere processes at an electrode surface (inset left) and a Marcus theory description of charge transfer between an acceptor A and donor D. Reorganization of the equilibrium solvation shell of A (gray) to the equilibrium solvation shell of the donor (black) would come at expense of the reorganization energy λ. The splitting near the curve crossing is directly proportional to twice the coupling matrix element HDA.

solid surfaces is complicated by its inherent many-body nature as well as experimental difficulties in confirmation. Marcus22−24 and Gerischer25−27 concurrently set the basis for such a theory of interfacial electron transfer. Smith and Nozik28−32 developed first principles molecular dynamics models to treat charge transfer across a liquid/semiconductor interface. More recently, Lewis and co-workers have revisited these foundations to make significant advances in our understanding of interfacial electron transfer.11,12,33−40 Here, we use silicon etching in fluoride solutions to form porous silicon (por-Si) as a model system for investigation of the dynamics of charge transfer at the electrolyte/solid interface.41 Si atoms are etched according to the Gerischer mechanism17,42,43 and the surface maintains near perfect hydrogen termination throughout, which minimizes the effects of changing surface chemistry on the charge transfer rate. Kolasinski and co-workers have shown43−46 previously that Marcus theory provides the basis of understanding why the etching of silicon in acidic fluoride solutions is self-limiting and results in the formation of a nanocrystalline porous solid. In this work we developed the quantitative analysis of electron transfer and etch rates further. We experimentally determined the probability that a collision of an oxidant with the surface leads to the etching of a silicon atom. In so doing we were able to probe the dynamics of the collision that leads to electron transfer from the silicon electrode to a solvated oxidant. We then compared to theoretical predictions. We found that the four oxidants considered follow an outer sphere charge transfer mechanism in the weak coupling limit. Furthermore, we found evidence for dynamical corrections to the charge transfer rate. In other words, the electronic transmission coefficient varies by orders of magnitude because the coupling matrix element between a given oxidant and the silicon valence band varies greatly. More accurate values of the λ and HDA drawn from first

sticking coefficient thus becomes a more accessible experimental quantity of measurement. As an example, Kolasinski17 has shown that the sticking coefficient of fluoride on hydrogenterminated silicon can be increased by 10 orders of magnitude by photoexcitation. The description of reactive collisions at the liquid/solid interface is inherently more complex because of the intermolecular interactions involved in solvation and the extraction of a liquid phase molecule into the adsorbed phase. A heterogeneous charge transfer process is influenced not only by the coupling of the donor and acceptor states involved; but also by the energetics of the solvation shell. Thus, the reaction probability per collision (analogous to the sticking coefficient) is not directly related to κel (the transmission coefficient for the electron in the transition state). Instead, the two system dependent parameters of paramount importance for determining the charge transfer probability per collision are the reorganization energy λ and the coupling matrix element HDA between the donor and acceptor levels |D⟩ and |A⟩, respectively. The character of the orbitals involved in charge transfer determines the magnitude of HDA as well as the related tunneling range parameter β. The reorganization energy is defined in Figure 1 as the Gibbs energy required to take the acceptor from its equilibrium solvation shell geometry to the geometry of the equilibrium solvation shell of the donor. The magnitude of HDA is also related to Figure 1. In the weak coupling limit shown, the Gibbs energy curves maintain their diabatic nature. As the coupling grows stronger the curves split into an excited state and an adiabatic ground state that smoothly connects the reactants through the transition state to the products. Great advances have been made in the ab initio description of homogeneous electron transfer reactions18,19 and λ.20,21 The development of a quantitative theory of electron transfer at 21473



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principle calculations would allow firmer conclusions to be drawn.

If the number of pores N increases linearly in time, then the surface area A of N cylinders in a film with thickness greater than their radius, h ≫ r, is given by

II. EXPERIMENTAL SECTION Formation of por-Si by stain etching has been studied with infrared spectroscopy (FTIR), photoluminescence (PL), reflectometry, and scanning electron microscopy (SEM). All substrates were Si(100) (p-type prime grade, 14−22 Ω cm resistivity). V2O5 (Fisher certified grade), FeCl3·6H2O (Fisher purified), H2IrCl6 (Sigma-Aldrich), CeF4 (Sigma-Aldrich), and HF (JT Baker 49% analytical grade) were used. All etchants were aqueous with no added alcohol or surfactant. Samples were rinsed in 1:1 water/ethanol then ethanol (three times) and dried in a stream of Ar. Scanning electron microscopy was performed with an FEI Quanta 400 ESEM. The SEM operated with integrated Oxford INCA energy dispersive X-ray spectroscopy (EDS). Fourier transform infrared spectroscopy (FTIR) was performed as described previously45 with a Nicolet Protégé 460. IR spectra were recorded with a diffuse reflectance attachment purged with dry N2 at a resolution of 4 cm−1 by averaging 512 scans. A planar hydrogen-terminated surface created by etching a polished Si(100) wafer in 49% HF for 5 min was used for a reference spectrum. Reflectometry was performed on an Ocean Optics USB2000 spectrometer with a DH-2000 deuterium tungsten halogen lamp from 360 to 1100 nm. Photoluminescence measurements were made on a Cary Eclipse fluorescence spectrophotometer with excitation at 350 nm. Roughly 1 cm2 substrates were etched in 0.121 M V2O5 + HF, 1.20 M FeCl3·6H2O + HF, 0.0201 M CeF4 + H2SO4 +H2O, or 0.0194 M H2IrCl6 + HF. A signal proportional to surface area was measured via FTIR measurements. As discussed previously,45 FTIR absorption peak area is proportional to the surface area; however, we have not determined the proportionality constant to arrive at the absolute magnitude. Film thickness via cross-sectional SEM measurements was followed as a function of etch time as described previously.44−49 In addition, white light reflectometry has been used to estimate the porosity of the thin films. Our analysis of reflectometry data was limited to a simple effective medium approximation of a homogeneous layer rather than a more complex model that allows for regions of differing porosity.50,51

A = k 2t(k 0t + k1t 2)

The constants k0 and k1 are determined uniquely from the h vs t data and the constant k2 is then determined independently from the FTIR data since the integrated area under the IR peaks associated with the Si−H bonds is proportional to the surface area. The derivative of eq 1 yields the linear etch rate R h = dh/dt = k 0 + 2k1t

(3)

The constant k0, therefore, is equal to the etch rate at time t = 0, Rh,0, which is not influenced by diffusion through the layer and represents a good quantity for quantitative comparison of the efficacy of various oxidants at inducing por-Si formation. The data for the four different oxidants are presented in Figures 2−5, in which data are only shown for films that are still

Figure 2. Plot of thickness as determined by cross sectional SEM and IR peak area in the Si−H stretch region for the etchant made with V2O5 and HF.

smooth such that the infrared scattering factor is constant.45 The second-order dependence of the film thickness on time of Eq. (1) is confirmed for all four solutions. For longer etch times in V2O5 + HF, it has been confirmed that the thickness versus time behavior converts to a linear dependence, as is expected after the number of pores saturates.45 For 0.121 M V2O5 solution we find that the initial etch rate is Rh,0 = 8.0 ± 0.3 nm s−1, for 1.20 M FeCl3·6H2O solution Rh,0 = 0.42 ± 0.07 nm s−1, for 0.0201 M CeF4 Rh,0 = 0.034 ± 0.010 nm s−1, and for 0.0194 M H2IrCl6 Rh,0 = 0.056 ± 0.011 nm s−1. In all cases, the uncertainty is listed as ± one standard deviation. The behavior of the surface area, as represented by the integrated peak area of the SiHx stretch region around 2100 cm−1, is very well reproduced by eq 2 for both V2O5 and FeCl3·6H2O solutions and fairly well for CeF4 etchants. The H2IrCl6 data does not fit this model. The poor fit to the IR data for H2IrCl6 does not in any way influence the accuracy of the etch rate data because the etch rate is determined from the cross sectional SEM data. It should also be noted that the IR absorbance is significantly smaller for the H2IrCl6 films and that

III. RESULTS Kolasinski and Yadlovskiy have previously analyzed the increase in surface area and thickness of por-Si films formed in V2O5 + HF solutions.49 Here we expanded their data set to include stain etchants made with Fe(H2O)63+, Ce4+, and IrCl62− in place of the V2O5. They developed a model similar to that of Brumhead et al.,52 who proposed that film formation begins when pores nucleate and rapidly propagate from the surface into the bulk. The propagation rate slows during the nonlinear growth rate phase, while the number of pores increases until the density reaches a saturation value that achieves carrier depletion in the nanoscale-width pore walls. We model the porous layer as being composed of an array of cylinders oriented perpendicular to the exterior surface of the film. The thickness of the film is equal to the height of the cylinders h. During the initial portion of film formation, the thickness is assumed to increase in a second-order fashion according to h = k 0t + k1t 2

(2)

(1) 21474



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Figure 3. Plot of thickness as determined by cross sectional SEM and IR peak area in the Si−H stretch region for the etchant made with FeCl3·6H2O and HF.

Figure 5. Plot of thickness and IR peak area in the Si−H stretch region for the etchant made with H2IrCl6 and HF. The thickness was determined by reflectometry for the first six samples and by cross sectional SEM for the final point.

Figure 4. Plot of thickness as determined by cross sectional SEM and IR peak area in the Si−H stretch region for the etchant made with CeF4 dissolved in H2SO4 and H2O.

Figure 6. Reflectogram of a por-Si thin film produced by etching for 1800 s in 0.0194 M H2IrCl6 in 50% HF(aq). The fit to the data assumes a linear porosity gradient from 0.80 at the top to 0.25 at the bottom of the film.

these were the only films that were not photoluminescent, which may indicate a different microstructure for these films. White light reflectometry was used as a probe of porosity, thickness and film homogeneity. Porosity is defined as the void fraction of the film, i.e., a porosity of 0.8 corresponds to a film with only 20% Si and 80% “empty” pores. In all cases it was found that the fit of spectra to interference by a single homogeneous layer was close but not that good. Nonetheless, the films all have uniform colorations. Therefore, we conclude that the layers have a uniform optical path length defined by a uniform thickness and porosity gradient. As shown in Figure 6, some films exhibited reflectance spectra that were well fitted by a linear gradient in the porosity. This was true for all but one of the samples etched with an H2IrCl6 solution. The films made in the other etchants were all substantially thicker than the H2IrCl6 etched samples and all of these exhibited a more complex porosity gradient. A more thorough optical analysis

will be required to determine the nature of this gradient more precisely. Porosity inhomogeneity is not unexpected. Our previous work has shown that the upper part of stain etched films typically is less porous than the bottom. This can be observed directly in cross sectional SEM for thick films (>several μm) for which exfoliation of the upper portion of the layer is often observed.46−48 For the H2IrCl6 etched samples, the thickness could only be independently confirmed with SEM for the thickest sample. For all other samples, the thickness shown in Figure 5 was determined by reflectometry. Even with the presence of a porosity gradient, a fit to the reflectance spectra using the thickness determined by SEM can be used to estimate the mean porosity of the films. The H2IrCl6 etched samples have a mean porosity of 0.5−0.6 for the 21475



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shortest etch times. The two longest etched films, which have thicknesses near 100 nm, have a porosity of ∼0.8−0.9. The CeF4 etched films have a porosity of ∼0.7, whereas the V2O5 and FeCl3·6H2O etched samples exhibit a porosity of ∼0.8. These values are consistent with our previous estimates based on the observation of cracks in thick films.47 It is unclear whether the low porosity of the thinnest H2IrCl6 etched films is unique to this etchant or to sub-100 nm films, as we did not probe films of similar thickness in the other etchants. If the number of pores is increasing toward an asymptotic limit as suggested by our model, then the porosity should also increase as etching progresses.

Zw = cox(RT /2πM )1/2

of an oxidant with concentration cox and molar mass M held at temperature T. R is the gas constant. Here we assume nonspecific adsorption and that the charge transfer event is an outer-sphere electron transfer from the silicon valence band to a solvated oxidant in solution. From this standpoint the mobile species is taken to be hexacoordinated metal in the appropriate oxidation state, e.g. [V(V)O2·4H2O]+, [Fe(III)·6H2O]3+, [Ce(IV)·6H2O]4+, and [Ir(IV)Cl6]2−. In shorthand notation we will refer to these as VO2+, Fe3+, Ce4+, and IrCl62− below. From the values in Table 1, we see that charge transfer is a very improbable event for all of the oxidants. This conclusion

IV. DISCUSSION Previously we have demonstrated that V2O5, FeCl3·6H2O, and CeF4 can be used to make nanocryrstalline por-Si thin films that exhibit visible photoluminescence.45,46,48,49 Here we have shown that H2IrCl6 can also be used as an oxidant in a stain etchant. Based on the photoluminescence behavior, there must be differences in film structure depending on the etchant used to produce it. Whereas V2O5 and FeCl3·6H2O etched samples are immediately photoluminescent in the visible even while still in the etchant, CeF4 etched samples only exhibit visible PL after exposure to air for several days.45,46,48,49 The H2IrCl6 etched samples do not exhibit PL even after exposure to air for months. Nonetheless they are still hydrogen-terminated and highly uniform, as are all of the thin films regardless of the nature of the oxidant. For all of the films, energy dispersive spectroscopy (EDS) confirms that only Si and adventitiously adsorbed O and C are detected above the sensitivity limit. No metals or halogens are detected within the limited sensitivity of EDS on thin films. A quantitative discussion of the data presented in Figures 2−5 begins by evaluating the etch rate per unit area RA and from this, determining the probability sR that a collision of an oxidant with the surface leads to the etching of a Si atom. In the following section, we develop an expression to describe the rate of charge transfer from the Si electrode to the oxidant Rinj and show how this quantity is related to the maximum rate constant kmax, the reorganization energy λ and the coupling matrix element HDA. The calculation of λ requires knowledge of the kinetics of the self-exchange reaction and is detailed in section IV.3. Finally in section IV.4, we derive the rate constant for etching k from RA and compare it to kmax. Within the accuracy of our calculation of λ, we make quantitative conclusions about HDA and whether the charge transfer reactions studied are described by an outer sphere electron transfer mechanism. IV.1. Etch Rate RA and Etch Probability sR. The measured linear etch rates Rh reported above can be converted into etch rates per unit surface area RA according to RA = R hρA ε (4)

Table 1. Values of the Linear Etch Rate Rh, the Etch Rate Per Unit Area RA, and the Reactive Charge Transfer Coefficient sR species VO2+ Fe3+ Ce4+ IrCl62−

Rh/nm s−1 8.0 0.42 0.034 0.056

± ± ± ±

RA/m−2 s−1

0.3 0.07 0.010 0.011

(3.2 (1.7 (1.2 (1.7

± ± ± ±

0.5) 0.4) 0.4) 0.3)

× × × ×

1020 1019 1018 1018

sR (4.3 (4.7 (2.5 (4.6

± ± ± ±

0.6) 1.0) 0.9) 0.9)

× × × ×

10−8 10−10 10−9 10−9

validates our kinetics approach in that such a low probability and concomitantly slow etch rate ensures that the collision rate is controlled by the bulk solution concentration rather than a diffusion gradient. The question now is whether a hole injection probability of only 10−10 to 10−8 per collision is a reasonable number. IV.2. Hole Injection Rate Rinj. To answer this we turn to Marcus theory as extended to electron exchange with a semiconductor surface by Lewis and co-workers.11,12,33,36−40 The rate of hole injection into the valence band can be written R inj = coxNVk maxW (E)

(7) −3

where NV = 1.04 × 10 m is the density of states in the valence band, kmax is the maximum rate constant, and W(E) is the Marcus hop probability factor that depends on the relative position of the acceptor level compared to the energy of the valence band edge EV. If the oxidant with Nernst potential Eox has an acceptor level that lies above the valence band edge 25

⎡ ⎛ (E − E + λ)2 ⎞⎤ ox ⎟⎥ W (E) = exp⎢ −⎜ V ⎢⎣ ⎝ 4λkBT ⎠⎥⎦

(8)

whereas when the acceptor level is resonant with states below the valence band maximum W (E) = exp( −λ /4kBT )

(9)

λ is the solvent reorganization energy and kB the Boltzmann constant. kmax occurs when the activation Gibbs energy ΔG‡ vanishes as defined by

where ρA is the atomic density of silicon and ε is the porosity of the film. The rate of etching is controlled by the rate at which holes are injected into the silicon valence band.17,43,45,46 Virtually all of the holes are injected when the oxidant collides with the surface. Thus we can define a reactive charge transfer coefficient sR, analogous to a reactive sticking coefficient, as the probability that a collision leads to hole injection according to

sR = RA /zZw

(6)

ΔG‡ =

(E V − Eox + λ)2 4λkBT = 0

(10)

This condition for the optimum reorganization energy defines the onset of the inverted region for homogeneous electron transfer. However, as mentioned in the introduction, the inverted region has a different form for heterogeneous electron transfer due to the resonant energetic overlap of the

(5)

where z is the number of holes transferred per collision (=1 for all oxidants considered here) and Zw is the collision rate 21476



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on a value for λse, the self-exchange reorganization energy. It can be used to calculate the value of the reorganization energy at a semiconductor electrode interface, λsc. The outer sphere contribution in self-exchange is given by

valence band with acceptor levels of an appropriate energy. Lewis and co-workers,11,12,36,38,39 have measured a value of kmax ≈ 10−25 to 10−24 m4 s−1 with Si, ZnO, InP, and GaAs electrodes. They have also shown that this value is consistent with the maximum magnitude of kmax calculated quantum mechanically from k max =

l 2π |HDA|2 β −1(4πλkBT )1/2 2/3 Si 1/3 ℏ ρA (6/π )

λse,out =

(12)

Since β ≈ 1 Å, the diameter of a water molecule is 2.76 Å and the radius of a typical hexaaquo transition metal ion is ∼3−4 Å, >90% of electron transfer occurs by tunneling from the valence band into the acceptor level as it collides with the surface and moves from a distance of about 6 Å away from the surface (the distance of the solvated ion core with an intervening water molecule) and 3 Å (the minimum distance of the ion core from the surface). For Fe3+ (E° = 0.771 V) and IrCl62− (E° = 0.8665 V), the acceptor level lies above the valence band edge at standard state. However, it is the Nernst potential E E = E° −

RT ⎛ ared ⎞ ln⎜ ⎟ zF ⎝ aox ⎠

(14)

In eq 14, a is the ionic radius across which the electron transfer occurs, the value of which includes the ion radius (1.55 Å for VO2+ across the O atom but only 0.68 Å for the V(V) ions from the ionic radii of Shannon57) and the diameter of a water molecule (2.76 Å ref 58). An exception is IrCl62− since the chlorides form the inner shell and the water radius does not need to be added. In eq 14, ε0 is the vacuum permittivity, e is the elementary charge, n is the refractive index of the solvent, and εs is the static dielectric constant (relative permittivity) of the solvent. Both the redox couple in solution and the image charge in the semiconductor electrode contribute to the total reorganization energy. The outer-sphere reorganization energy of a redox couple at a semiconductor electrode, λsc,out, is expected to be less than λse,out of the couple in homogeneous solution because of screening effects.12,23,56 For the one-electron processes discussed here, a theoretical value for λsc,out can be calculated from

(11)

where lSi is the effective coupling length in the semiconductor, β is the tunneling range parameter, and HDA, as before, is the coupling matrix element between the donor and acceptor levels |D⟩ and |A⟩. The coupling matrix element HDA depends on orbital overlap and symmetry. Thus, it varies exponentially with distance r from the distance of closest approach rm according to ⎡ (r − rm) ⎤ |HDA|2 = ⟨D|H |A⟩2 = V0 exp⎢β ⎥ ⎣ ⎦ 2

e2 ⎛ 1 1⎞ ⎜ − ⎟ 2 ⎝ 8πε0a n ε⎠

λsc,out =

⎡ 2 2 e2 ⎢ 1 ⎛ 1 1⎞ 1 ⎛⎛ nsc − n ⎞ 1 ⎜ ⎟ − ⎜ ⎟ ⎜ − ⎜ ε ⎠ 2re ⎝⎝ nsc 2 + n2 ⎠ n2 8πε0 ⎢⎣ a ⎝ n2 ⎛ ε 2 − ε 2 ⎞ 1 ⎞⎤ ⎟ ⎟⎟⎥ − ⎜ sc 2 2 ⎝ εsc + ε ⎠ ε ⎠⎥⎦

(13)

(15)

where n is the refractive index (3.49 for Si, 1.25 for 49% HF, 1.34 for 0.5 M H2SO4 and 1.41 for 64% H2SO4). re is the distance from the acceptor to the electrode (re = a). The static dielectric constant of Si is 11.9 at 25 °C and 80.8, 78.7, and 83.8 for 49% HF, 0.5 M H2SO4, and 64% H2SO4, respectively. These values are either taken directly from tables59 or are calculated using mole fractions from pure substance values. The inner-sphere reorganization energy at a semiconductor electrode is half of the value of λse,in, since half as many molecules participate in each electron-transfer event. The total reorganization energy for a redox couple at a semiconductor electrode is therefore given by

where R is the gas constant, T the temperature, z = 1 is the valence, F the Faraday constant, and ared and aox are the activities of the reduced and oxidized species, respectively, that determines the charge transfer rate. Initially ared ≈ 0 and we have chosen sufficiently high concentrations and solution volumes such that aox changes by less than 1% during the etching. Therefore, throughout our experiments E lies below the valence band maximum for all oxidants. Note that the combination of eqs 7, 8, and 13 imply that, particularly for species with an E° value above the valence band edge, an extent of reaction (i.e., concentration) dependent rate of charge injection should be observed for reactions that are run toward completion. IV.3. Calculation of λ. The solvent reorganization energy λ is calculated as the sum of inner sphere and outer sphere contributions. The outer sphere contribution is easily calculated within a continuum model framework, whereas the inner sphere contribution requires knowledge of the changes in structure that occur upon reduction.53−55 The effects of screening in a metal or semiconductor electrode reduce the values of λ as compared to those found for homogeneous electron transfer.12,23,56 The only experimental values we have found to compare to are those of Memming and Möllers, who measured the relaxation energy at a semiconducting SnO2 electrode for Fe3+ and Ce4+ to be 1.2 ± 0.2 and 1.7 ± 0.2 eV, respectively. Their electrolyte contained 0.5 M H2SO4. We calculated values of λ for each of our oxidants. As an alternative to calculating the inner sphere contribution directly, Hamann et al.12 have proposed an approach that relies

λsc = λsc,in + λsc,out =

(λse − λse,out) 2

+ λsc,out

(16)

In Table 2, we see that appropriate values were available for all oxidants used here except VO2+. The Fe3+ self-exchange system has been particularly well characterized and was the subject of detailed calculations by Rosso et al.21 Thus, we used it as the basis to make estimations for the other ions. For Fe3+, we know that λse = 2.11 eV. Assuming an Arrhenius form for kse and the fact that the activation Gibbs energy Δ‡G = λse/4kBT for self-exchange kse = A exp( −Δ‡G /kBT ) = A exp(−λse /4kBT )

(17)

allowed us to calculate the pre-exponential factor

A = kseexp(λse /4kBT )

(18)

The prefactor depends on the oxidant according to55 21477



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Table 2. Self-Exchange Rate Parametersa a/Å

rμ/Å

Fe Ce4+ IrCl62− ion

3.44 3.45 3.77 4.194 νn/s−1

5.30 5.32 5.81 6.47

VO2+ Fe3+ Ce4+ IrCl62−

2.28 1.19 6.62 1.17

ion VO2+ 3+

× × × ×

1013 1013 1012 1013

kse/M−1 s−1

νox/cm−1

4.2 4.4 2 × 105 w/J mol−1

400 475 265 410 I/M

0 8.70 × 103 1.18 × 104 4.56 × 103

1.5 0.55 1.2 1.5

This, in turn is related to the electronic coupling matrix element HDA by

νred/cm−1 249 296 165 370 KA/M−1 1.77 5.34 1.81 4.19

× × × ×

HDA 2 =

where h is the Planck constant. From known values or Fe we calculated A, KA, νn, κel, νel, and HDA. Then assuming the value of HDA = 7.69 × 10−3 eV, which was calculated for Fe3+, is the same for all of the oxidants and an estimated value of λse, we calculated νel and κel for each oxidant. An optimization loop is set up to obtain values for κel, νel and λse that are self-consistent with the experimental value of kse and the calculated values of KA and νn. Equations 14−16 were then used to calculate the values of the inner and outer sphere reorganization energies and the total reorganization energy in front of the Si electrode. These values are given in Table 3. For VO2+, we calculated the outer sphere contribution directly and we estimated the inner sphere contribution to be the mean of Fe3+ and Ce4+ values.

10−1 10−3 10−3 10−2

Ionic radii plus water diameter a from Shannon57 except for IrCl62− from Yusenko et al.60 Self exchange rate constants kse from Campion et al.61 for Ce4+ and IrCl62−, and from Sutin55 for Fe3+. Vibrational wavenumbers of the oxidized νox and reduced νred forms are used to calculate the frequency factor of the preexponential term νn as described in the text. w is the work to bring ox and red together to an electron exchange distance rm = 1.542 a in a solution with ionic strength I. KA is the precursor formation equilibrium constant. (19)

Table 3. Values of the Electron Transfer Frequency νel, the Electronic Transmission Coefficient κel, and Preexponential Factor A Derived from the Self Exchange Rate Constanta

Γ is the nuclear tunneling factor and is unity for all species considered here. The frequency factor νn is determined by the stretch mode wavenumbers of the inner sphere (ν̃ox and ν̃red), which is a M−OH2 stretch for all of the oxidants with the exception of the Ir−Cl stretch mode for IrCl62−/3−. The values of the stretch wavenumbers are taken from Benes,62 Jarzec̨ ki,63 Jayasree,64 and Groth65 for VO2+/VO2+, Fe3+/2+, Ce4+/3+, and IrCl62−/3−, respectively. The value of νn is given by 21 ⎡ ((cν ̃ )2 + (cν ̃ )2 ) ⎤1/2 ox red ⎥ νn = ⎢ 2 ⎦ ⎣

VO2 Fe3+ Ce4+ IrCl62− ion VO2+ Fe3+ Ce4+ IrCl62−

(20)

2[1 − exp( −νel /2νn)] 2 − exp( −νel /2νn)

(21)

1.14 1.14 1.04 0.93

× 10−2 × 10−2 × 10−2 λsc,in(Si)

0.78 0.78 0.58 0.64

0.49 0.49 0.48 0.12

3.50 × 109 2.11 1.18 × 109 1.99 35.4 × 109 1.24 λsc λsc(SnO2)/eV 1.27 1.27 1.06 0.80

1.2 ± 0.2 1.7 ± 0.2

Our calculated value of λsc for Fe3+ agrees well with the value obtained experimentally for a SnO2 electrode.66 However, our calculated value for Ce4+ is significantly lower than the previously obtained experimental result. IV.4. Relating RA to Rh, λ, and HDA. The rate constant k for the etching reaction is defined by RA = coxNVk

(26)

Calculated values of k can be found in Table 4. Lewis and co-workers have determined that kmax is in the range 10−25−10−24 m4 s−1 for an outer sphere process at a semiconductor surface involving ZnO with cobalt trisbipyridine, ruthenium pentaamine pyridine, cobalt bis-1,4,7trithiacyclononane, and osmium bis-dimethyl bipyridine bis-

(22)

Table 4. Rate Constant for Etching k and Maximum Value of the Electron Transfer Rate Constant kmax Derived from Our Experimental Etch Rates

(23)

Thus the frequency of electron transfer within the activated complex νel is ⎞ ⎛ κ −1 ⎟ νel = −2νn ln⎜⎜ 1 el ⎟ ⎝ 2 κel − 1 ⎠

6.85 × 1011 5.53 7.04 × 1011 9.85 8.93 × 1011 7.21 λse,out/eV λsc,out/eV

Values of reorganization energy λ (se for the case of self exchange, sc for the case of a semiconductor electrode). λsc(SnO2) are the experimental values reported in ref 66.

where B = [2NAe2/ε0εskBT]1/2, and I is the ionic strength with units of mol m−3 for B and rm expressed in SI units. Calculated thusly, w has units of J mol−1. This allows us to calculate the values of w and KA for all four species. The electronic transmission coefficient κel is given by55 κel =

λse/eV

a

z1z 2e 2NA 4πε0εsrm(1 + BrmI1/2)

A/M−1

κελ

+

where NA is the Avogadro constant, rm is the distance between the reactants at closest approach in meters, δr = 0.833 Å is the thickness of the region over which charge transfer occurs. Rosso et al. found that rm is greater than a by a factor of 1.542 for the Fe3+/2+ system and we use that correction factor here for all species. The work required to bring the reactants together is calculated from w=

νel/s−1

ion

The equilibrium constant for the formation of the precursor complex KA is calculated from KA = 4πNArm 2δr exp( −w/RT )

(25) 3+/2+

a

A = KAκelvn Γ

1/2 hνel ⎛ λkBT ⎞ ⎜ 3 ⎟ 2 ⎝ π ⎠

ion VO2+ 3+

Fe Ce4+ IrCl62−

(24) 21478

k/m4 s−1 (2.1 ± 0.3) (2.2 ± 0.5) (9 ± 3) (1.1 ± 0.3)

× × × ×

kmax/m4 s−1 −31

10 10−33 10−33 10−32

9.9 1.1 5.7 6.6

× × × ×

10−26 10−27 10−28 10−29



Article

uncertainty in the calculated value of λsc may be as large as 0.2 eV, the values of kmax in Table 4 should only be taken as order of magnitude estimates. From the values in Table 4 we can conclude the following: (1) VO2+: The etch rate in this system is described accurately by a Marcus theory treatment of an outer sphere process with a reorganization energy of 1.27 eV. This agreement implies that the electron is transferred from the Si valence band into V(V) ion across a coordinating water rather than across one of the V−O bonds because this distance corresponds to the distance used in our calculation for the ionic radius. Electron transfer across the V−O bond would lead to a value of kmax that is 4 orders of magnitude larger, which is unrealistic. It also implies that the electron transfer is the rate determining step whereas the subsequent reactive steps embodied in the half reaction

imidazole;67 for n-type Si(100) terminated with CH3OH in contact with seven different bipyridinium compounds;38 and nSi/CH3OH with dimethylferrocenium-dimethylferrocene, nGaAs/CH3CN-ferrocenium-ferrocene, and p-InP/CH3CN-cobaltocenium-cobaltocene.39 Furthermore, they have argued that a value larger than this range is physically unrealistic for an outer sphere process. Consequently, there is an expectation that the coupling matrix element HDA is bounded from above for the metal ions used here. That is, the maximum rate constant for the metal ions used here should be less than or equal to this range. To calculate a value of kmax, we need to know how to relate the hole injection rate into the silicon valence band Rinj to the etch rate RA. Anodic etching of silicon to form porous silicon follows the Gerischer mechanism.17,42,43 Etching is initiated by hole injection into a valence band state localized in a Si−Si backbond. Si + h+ → Si+

VO+2 + 2H+ + e− → VO2 + + H 2O

(27)

occur so rapidly that they are kinetically irrelevant. (2) Fe3+: The measured etch rate corresponds to a value of kmax that is 2 orders of magnitude lower than the upper bound. The only comparable value of λsc we have found is that of 1.2 ± 0.2 eV reported by Memming and Mö llers66 at a SnO2 electrode in 0.5 M H2SO4. This value is consistent with our value of 1.27 eV. This supports the conclusion that electron transfer to Fe3+ from the Si valence band follows an outer sphere mechanism. However, the square of the coupling matrix element |HDA|2, or more accurately the product |HDA|2 β−1, is lower by 2 orders of magnitude compared to the systems studied by Lewis and co-workers and VO2+. (3) Ce4+: The measured etch rate corresponds to a value of kmax that is 2 orders of magnitude lower than the upper limit. We conclude, just as for Fe3+, that electron transfer to Ce4+ from the Si valence band follows an outer sphere mechanism, and that |HDA|2 β−1 is comparable to that of Fe3+. It should be noted, however, that nonideality may play a role in the deviations noted for Ce4+. The solvent used in this case is highly concentrated H2SO4; therefore, the comparison to Memming and Möllers66 experimental value of λ = 1.7 ± 0.2 eV at a ZnO electrode in 0.5 M H2SO4 is not entirely appropriate and some of the assumptions we have made in our calculations may need refining. (4) IrCl62−: The measured etch rate leads to a value of kmax that is almost 4 orders of magnitude lower than the upper limit. Therefore, electron transfer to IrCl62− from the Si valence band is consistent with an outer sphere mechanism with a value of |HDA|2 β−1 lower by 4 orders of magnitude compared to the systems studied by Lewis and co-workers and VO2+. Again a cautionary note is in order. The observation of precipitation of IrF4 in this system leads to some uncertainty in the concentration of IrCl62−, both because some of the Ir is precipitated and because F− may be substituting some of the Cl− ligands, which would alter the energetics and kinetics of an outer sphere process.

−

During subsequent steps of the addition of F , HF and/or HF2−, which have a net stoichiometry of Si+ + 3HF → HSiF3 + 2H+ + e−

(28)

an electron is injected into the Si conduction band. The etch product HSiF3 then reacts further in solution to form the final etch products. HSiF3 + 3HF → H 2SiF6 + H 2

(29)

The conduction band electron is consumed by hydrogen reduction.

H + + e− →

1 H2 2

(30)

If this mechanism is transferred directly to etching induced by an oxidant, then the hole transfer in the first step is mediated by the oxidant, Si + Ox + → Si+ + Ox

(31)

and it is expected that one Si atom is etched per hole injected, that is, one mole of Si is etched for every one mole of oxidant that is reduced. We measured the amount of Si etched gravimetrically and used UV/vis absorption spectroscopy to determine the conversion of VO2+ to VO2+ as well as Fe3+ to Fe2+. These data will be reported on in detail elsewhere. We found that in both systems, approximately 2 moles of oxidant are consumed per mole of Si etched. We believe that the first mole is used to inject holes into Si to initiate the reaction, while the second mole is consumed by reduction with the injected conduction band electron. Our data also indicate a reduced amount of H2 evolved per Si atom etched consistent with the oxidant competing favorably with hydrogen reduction for the conduction band electrons. With this knowledge that two electrons must be transferred to the oxidant for each Si atom etched, we can relate the etch rate to the rate of hole injection by RA =

1 R inj. 2

(33)

V. CONCLUSIONS The etching of silicon in a fluoride solution is initiated when an oxidant collides with the surface and extracts an electron from the valence band. The probability that a collision of VO2+ with Si leads to etching is 4 × 10−8, which corresponds to a maximum rate constant of 1 × 10−25 m4 s−1 for electron transfer. The etching reaction involving VO2+ is initiated by nonspecific adsorption and follows an outer sphere charge

(32)

The rate of hole injection is given by eq7, which we used to calculate kmax from our experiments. In Table 4 we see that the value of kmax lies at or below the upper bound of Lewis and coworkers for all four species considered here. Because the 21479



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(18) Voorhis, T. V.; Kowalczyk, T.; Kaduk, B.; Wang, L. P.; Cheng, C. L.; Wu, Q. Annu. Rev. Phys. Chem. 2010, 61, 149−170. (19) Sit, P. H.-L.; Cococcioni, M.; Marzari, N. Phys. Rev. Lett. 2006, 97, 028303. (20) Klimkans, A.; Larsson, S. Chem. Phys. 1994, 189, 25−31. (21) Rosso, K. M.; Rustad, J. R. J. Phys. Chem. A 2000, 104, 6718− 6725. (22) Marcus, R. A. J. Phys. Chem. 1991, 95, 2010−2013. (23) Marcus, R. A. J. Phys. Chem. 1990, 94, 1050−1055. (24) Marcus, R. A. J. Chem. Phys. 1965, 43, 679−699. (25) Gerischer, H. Z. Phys. Chem. N. F 1960, 26, 233−247. (26) Gerischer, H. Z. Phys. Chem. N. F 1961, 27, 40−79. (27) Gerischer, H. Principles of Electrochemistry. In The CRC Handbook of Solid State Electrochemistry; Gellings, P., Bouwmeester, H., Eds.; CRC Press: Boca Raton, FL, 1997; pp 9−73. (28) Smith, B. B.; Halley, J. W.; Nozik, A. J. Chem. Phys. 1996, 205, 245−267. (29) Smith, B. B.; Nozik, A. J. Chem. Phys. 1996, 205, 47−72. (30) Smith, B. B.; Nozik, A. J. J. Phys. Chem. B 1997, 101, 2459− 2475. (31) Meier, A.; Selmarten, D. C.; Siemoneit, K.; Smith, B. B.; Nozik, A. J. J. Phys. Chem. B 1999, 103, 2122−2141. (32) Smith, B. B.; Nozik, A. J. J. Phys. Chem. B 1999, 103, 9915− 9932. (33) Lewis, N. S. Inorg. Chem. 2005, 44, 6900−6911. (34) Gstrein, F.; Michalak, D. J.; Royea, W. J.; Lewis, N. S. J. Phys. Chem. B 2002, 106, 2950−2961. (35) Gao, Y. Q.; Georgievskii, Y.; Marcus, R. A. J. Chem. Phys. 2000, 112, 3358−3369. (36) Royea, W. J.; Fajardo, A. M.; Lewis, N. S. J. Phys. Chem. B 1998, 102, 3653−3653. (37) Lewis, N. S. J. Phys. Chem. B 1998, 102, 4843−4855. (38) Fajardo, A. M.; Lewis, N. S. J. Phys. Chem. B 1997, 101, 11136− 11151. (39) Pomykal, K. E.; Fajardo, A. M.; Lewis, N. S. J. Phys. Chem. 1996, 100, 3652−3664. (40) Lewis, N. S. Annu. Rev. Phys. Chem. 1991, 42, 543−580. (41) Kolasinski, K. W. New Approaches to the Production of Porous Silicon by Stain Etching. In Nanostructured Semiconductors: from basic research to applications; Granitzer, P., Rumpf, K., Eds.; Pan Stanford Publishing: Singapore, 2012; in press. (42) Gerischer, H.; Allongue, P.; Costa Kieling, V. Ber. Bunsen-Ges. Phys. Chem. 1993, 97, 753−756. (43) Kolasinski, K. W. Surf. Sci. 2009, 603, 1904−1911. (44) Kolasinski, K. W.; Gogola, J. W. ECS Trans. 2011, 33, 23−28. (45) Kolasinski, K. W.; Hartline, J. D.; Kelly, B. T.; Yadlovskiy, J. Mol. Phys. 2010, 108, 1033−1043. (46) Kolasinski, K. W. J. Phys. Chem. C 2010, 114, 22098−22105. (47) Dudley, M. E.; Kolasinski, K. W. Phys. Status Solidi A 2009, 206, 1240−1244. (48) Dudley, M. E.; Kolasinski, K. W. Electrochem. Solid State Lett. 2009, 12, D22−D26. (49) Kolasinski, K. W.; Yadlovskiy, J. Phys. Status Solidi C 2011, 8, 1749−1753. (50) Pettersson, L. A. A.; Hultman, L.; Arwin, H. Appl. Opt. 1998, 37, 4130−4136. (51) Fried, M.; Polgar, O.; Lohner, T.; Strehlke, S.; Levy-Clement, C. J. Lumin. 1998, 80, 147−152. (52) Brumhead, D.; Canham, L. T.; Seekings, D. M.; Tufton, P. J. Electrochim. Acta 1993, 38, 191−197. (53) Marcus, R. A. J. Chem. Phys. 1957, 26, 867−871. (54) Hale, J. M. The rates of reactions involving only electron transfer at metal electrodes. In Reactions of Molecules at Electrodes; Hush, N. S., Ed.; Wiley-Interscience: London, 1971. (55) Inorganic Reactions and Methods. Electron-transfer and electrochemical reactions, photochemical and other energized reactions; Zuckerman, J. J., Ed.; VCH: Weinheim, Germany, 1986; Vol. 15. (56) Smith, B. B.; Koval, C. A. J. Electroanal. Chem. 1990, 277, 43− 72.

transfer process. This value of the maximum rate constant lies close to the highest physically reasonable value previously determined for a number of oxidants by Lewis and coworkers.11,12,36,38,39 On the other hand, it is clear that the kinetics of charge transfer to Fe3+, Ce4+ and IrCl62− are described by much lower values of kmax and, hence, by a much lower value of the product |HDA|2 β−1 between the coupling matrix element and the tunneling range parameter. This is good evidence for the need to consider dynamical corrections to the rate constant, that is, individual systems may exhibit significantly slower kinetics than the upper bound of kmax. It is abundantly clear that accurate, system specific calculations of the values of λ, |HDA|2, and β are required to determine the extent of kinetically significant dynamical corrections for the description of electrochemical reactions at the liquid/semiconductor interface.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Author Contributions

The manuscript was written primarily by the corresponding author author (K.W.K.). All authors were involved in the design of experiments and the acquisition of data and have given approval to the final version of the manuscript. Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS Supported by Vesta Sciences, West Chester University, Pennsylvania State System of Higher Education, and the Center for Microanalysis and Imaging, Research and Training (CMIRT) at WCU. We gratefully acknowledge the expert technical assistance of Frederick Monson, Caroline Schauer for making available a thin film reflectometry system, and Eckart Hasselbrink for valuable discussions.

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REFERENCES

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Test of Marcus Theory Predictions for Electroless Etching of Silicon

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