Investigation of the Size-Scaling Behavior of Spatially Nonuniform

small, low barrier height regions on the surface of an otherwise high barrier ...... an unusually quick rise in current at small forward biases (ΦB,e...
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J. Phys. Chem. B 2001, 105, 12303-12318

12303

Investigation of the Size-Scaling Behavior of Spatially Nonuniform Barrier Height Contacts to Semiconductor Surfaces Using Ordered Nanometer-Scale Nickel Arrays on Silicon Electrodes Robert C. Rossi and Nathan S. Lewis* DiVision of Chemistry and Chemical Engineering, 127-72, California Institute of Technology, Pasadena, California 91125 ReceiVed: May 15, 2001

Nanosphere lithography has been used to prepare a series of ordered, periodic arrays of low barrier height nanometer-scale n-Si/Ni contacts interspersed among high barrier height n-Si/liquid contacts. To form the arrays, ordered bilayers of close-packed polystyrene spheres were deposited onto (100)-oriented n-type single crystal Si surfaces. The spheres formed a physical mask through which Ni was evaporated to produce regularly spaced and regularly sized Si/Ni contacts. By varying the diameter of the latex spheres from 174 to 1530 nm, geometrically self-similar Si/Ni structures were produced having triangular Si/Ni regions with edge dimensions of 100-800 nm. The resulting Si surfaces were used as electrodes in contact with a methanolic solution of LiClO4 and 1,1′-dimethylferrocene+/0. The current-voltage and photoresponse properties of these mixed barrier height contacts were strongly dependent on the size of the Ni regions, even though the fraction of the Si surface covered by Ni remained constant. Electrodes formed from large-dimension Si/Ni and Si/electrolyte contacts behaved as expected for two area-weighted Schottky diodes operating independently and in parallel, whereas electrodes having nanoscale Si/Ni regions surrounded by Si/liquid contacts behaved in accord with effective barrier height theories that predict a “pinch-off” effect for mixed barrier height systems of sufficiently small physical dimensions.

I. Introduction The electrical behavior of semiconductor/metal (SC/M) contacts having spatially inhomogeneous barrier heights has attracted much theoretical,1-10 computational,11-14 and experimental attention.15-36 Of specific interest is the regime in which the scale length of the barrier height nonuniformities is comparable to or smaller than the depletion width of the semiconductor. In this regime, analytic theories1,2 and numerical simulations11-13 have indicated that the current density through small, low barrier height regions on the surface of an otherwise high barrier height SC/M contact should be a strong function of the band bending in the semiconductor and of the spatial dimensions of the low barrier height regions. Specifically, the “effective” barrier height of such regions is predicted to be much higher than the barrier height observed at an otherwise identical SC/M contact of larger spatial dimension. Freeouf et al. discussed this effect in 198211 and coined the term “pinch-off” to describe the modulation of electric potential profiles behind nanometer-scale barrier height spatial nonuniformities in a SC/M contact. Tung and co-workers subsequently performed more detailed numerical simulations12 and Tung derived a set of analytic equations1,2 to describe the current density-voltage (J-V) behavior of pinched-off systems. On the basis of these results, Tung and co-workers argued that microscopic variations in local barrier heights could explain several widely observed anomalies of ostensibly homogeneous SC/M contacts, including the To effect,37 the frequent observation of diode quality (ideality) factors appreciably greater than unity,1 and the discrepancies typically observed between Schottky barrier * Author to whom correspondence should be addressed.

heights measured by differential capacitance-voltage and J-V techniques.27 Subsequent ballistic electron emission microscopy (BEEM) studies have clearly indicated that the barrier height across SC/M contacts is often spatially inhomogeneous,15-19 and with few exceptions14,38 the pinch-off model developed by Tung1,2 has been found to provide an adequate explanation of the behavior of such contacts.7,20-23,39 Efforts have also been made to verify directly the existence of the pinch-off phenomenon using BEEM40 and dark J-V measurements at different temperatures24 at intentionally inhomogeneous Schottky contacts prepared using two distinct contacting metals (SC/M1|M2 interfaces).19 These studies provided strong qualitative support for the pinch-off hypothesis, but were quantitatively limited by a lack of information about the physical extent of the low barrier height contacts and their unperturbed barrier heights. Electrode surfaces possessing lateral variations in local barrier height, usually arising from the deposition of metal particles onto semiconductor photoelectrodes, have concurrently attracted significant attention in the electrochemical literature.41-54 Heller and co-workers investigated the electrocatalytic55 and optical56 properties of noble metal deposits on InP electrodes in contact with aqueous acidic electrolytes, and concluded that the J-V behavior of such contacts was strongly affected by a modification of the work function of the metal particles upon exposure to H2(g).57,58 Nakato and co-workers have exploited Pt deposits to obtain improved stability and extraordinarily large photovoltages from n-Si/liquid contacts.59-61 Light-induced charging of the metal particles and elevated zero-point energies in quantized potential wells,62-64 as well as electrostatic arguments analogous to the pinch-off effect,41,61 were invoked to account for the J-V behavior of such systems. Meissner and co-workers

10.1021/jp011861c CCC: $20.00 © 2001 American Chemical Society Published on Web 11/16/2001

12304 J. Phys. Chem. B, Vol. 105, No. 49, 2001 SCHEME 1: Plot of the Conduction Band Energy Minimum as a Function of Position in an n-Type Semiconductor behind a Mixed Barrier Height Contacta

a The effective barrier height at a pinched-off low barrier height region, ΦB,eff, is slightly smaller than that behind a spatially large homogeneous high barrier height region, ΦoB, but is much larger than the barrier height behind a spatially large low barrier height region, ΦB,patch.

studied the effects of colloidal gold deposits on aqueous p-GaAs electrodes65,66 and observed results that could not be interpreted in terms of two types of semiconductor contacts behaving independently and in parallel. More recently, Hiesgen and Meissner have used photocurrent scanning tunneling microscopy to investigate the local barrier heights around copper particles electrodeposited onto WSe2.67 These measurements demonstrated a size dependence in the barrier heights of nanoscopic WSe2/Cu Schottky contacts, though the nature of the WSe2/Cu junctions was not fully understood and particle charging was again suggested as a possibility. The pinch-off effect could potentially be quite advantageous in the application of semiconductor electrodes to photoelectrochemical energy conversion.61,66 Photoelectrode stability is typically enhanced when continuous layers of metal are coated onto semiconductor surfaces; however, recombination is concomitantly increased due to facile thermionic emission of majority carriers over the SC/M barrier.68 Reducing the coverage of metal reduces recombination but also reduces the fraction of the surface that is kinetically stabilized toward corrosion or passivation processes. Dense arrays of nanometer-sized metallized regions operating in the pinch-off regime should retain most of their catalytic functionality without significantly disrupting the rectification properties of a high barrier height semiconductor/electrolyte (SC/E) contact covering the majority of a semiconductor surface.41 At a pinched-off SC/M|E contact, majority carriers would not experience the low barrier height of a macroscopic SC/M contact, but would instead experience the relatively high barrier height imposed by the surrounding SC/E contact (Scheme 1). Additionally, minority carriers will be effectively attracted to the high barrier height locations in the regions near the pinched-off metal contacts. Rapid interfacial charge flow should then occur from the semiconductor to the metal and through the metal to the redox species of interest. In this fashion, the pinched-off contact should provide a facile path for directing minority carriers to participate in desirable, possibly multielectron transfer, interfacial redox processes. If the pinchoff effect can be exploited, semiconductor electrodes coated with nanoscopic metal islands could thereby be made to effectively direct minority carriers toward catalytic metal sites without incurring the majority carrier recombination effects that would otherwise deleteriously affect the properties of a semiconductor interface having an appreciable fraction of its area covered with (low barrier height) metal contacts.61,66

Rossi and Lewis The approach described herein involves the formation of ordered, periodic arrays of low barrier height Ni contacts on an etched (100)-oriented crystalline Si surface. “Nanosphere” (or “natural”) lithography provides a facile method for producing self-similar patterns of metal that scale with the size of the microspheres employed in the lithographic process.69,70 The lithographic step is followed by the formation of a Si/liquid contact, which produces high barrier height regions on the unmetallized portions of the surface and which also forms a massively parallel electrical contact to the Ni features. n-Si/Ni diodes are poor rectifiers, having room-temperature exchange current densities, Jo, on the order of 10-4 A cm-2; these exchange currents are dominated by thermionic emission over a 0.6 V barrier height.71,72 The ability of Ni to form silicide contacts at low temperatures73 provides strong mechanical adherence, stability in electronic behavior over time, and reproducibility in barrier height over multiple experiments.74 A solution of 1,1′-dimethylferrocene/dimethylferrocenium (Me2Fc+/0) in methanol forms a high (≈1.0 V) barrier height contact at n-Si75-77 and provides a convenient second component to form the structured nonuniform barrier height electrode. Me2Fc+/0 competes effectively with oxidation of n-Si under illumination,78 providing an electrochemical contact that is stable over long time periods when care is taken to exclude water and oxygen from the nonaqueous electrochemical cell. In the absence of interaction between the two different barrier height regions, one would expect a Ni contact covering as little as 1% of an n-Si surface to have a significant and deleterious effect on the photoresponse of an n-Si electrode in contact with CH3OH1.0 M LiClO4-Me2Fc+/0. In contrast, if the pinch-off effect were operative, then theoretically >10% of the surface could be covered with metal without significantly affecting the majority carrier recombination current density at the semiconductor surface. II. Theoretical Expectations Analytic expressions have been derived for the electric potential profile of a Schottky (SC/M) contact to an n-type semiconductor in which a circular patch (or “dot”) of radius Ro having a low barrier height, ΦB,patch, is located in a uniformly high barrier height field, ΦoB.1,2 In cylindrical coordinates, (F,θ,z), with the z axis perpendicular to the semiconductor surface and F ) 0 located at the center of the low barrier height patch, the boundary condition for the electric potential, V(F,θ,z), at the surface is therefore

V(F,θ,z) )

{

ΦB,patch for 0 e F e Ro for F > Ro ΦoB

(1)

The potential in the semiconductor behind such a patch is independent of θ and is given by1,79

(

V(F,z) ) Vbb 1 ∆ 2π

∫02π ∫0R

o

z 2 W

)

F1z

[z2 + F2 + F12 - 2FF1 cos(θ1)]3/2

dF1 dθ1 (2)

where ∆ ≡ ΦoB - ΦB,patch and W and Vbb are the depletion width and the band bending, respectively, behind an unperturbed region of the high barrier height contact (i.e., for a point with F > Ro). The scalar quantity Vbb is defined as80

Vbb ≡ ΦoB - Vn + Vapp

(3)

Spatially Nonuniform Barrier Height Contacts to SC Surfaces with Vapp the external potential applied across the junction, and with the value of Vn deduced from the energy difference between the Fermi level and the conduction band energy minimum in the bulk of the n-type semiconductor:

Vn )

()

kT NC ln e ND

J. Phys. Chem. B, Vol. 105, No. 49, 2001 12305 SCHEME 2: Calculated Equipotential Contours within a Semiconductor as a Function of Scale for Parallel Contacts of Differing Barrier Height on a Semiconductor Surface, Based on Eq 2 and Assuming Simple Superposition of the Electric Potential Perturbations Introduced by Each Contacta

(4)

In eq 4, ND and NC equal the n-type semiconductor’s donor density and effective density of states in the conduction band, respectively. Pinch-off occurs when the condition1

W∆ 2Vbb

Ro
2Ro total defect area fraction (upper bound)f,i

dislocations 0.0031

point vacancies 0.0037

point vacancies 0.0040

point vacancies 0.0091

a Manufacturers’ reported physical diameters as obtained by transmission electron or optical array microscopy. b Radius of a circular dot having the same area as the observed triangular Ni patch. c These values were obtained by tapping-mode atomic force microscopy. The uncertainty reflects one standard deviation in the measured value. d Distance between centers of nearest-neighbor particles on patterned surface. e Average effective radius of the largest Ni particle produced by a point vacancy defect. f Area fraction of electrode covered by defect-related particles of effective radius > 2Ro. g The density of this one-dimensional defect is reported as the length of defect expected per unit area of nanopatterned substrate. h The edge size of the grains, employing eq 20 and assuming that the grains are squares. i Total does not include grain boundary defects, which produce negligible defect area contributions for all values of Ds.

are essentially identical to those of the independent contact model (eq 15). Additionally, for both dot sizes, the forward bias current density observed for the metallized n-Si electrodes was much larger than the current density observed for a bare n-Si surface in contact with the CH3OH-Me2Fc+/0 solution. For the smaller dot sizes, current densities decreased with the size of the metal dots, even at essentially constant fractional

area coverage of metal on the n-Si. For the smallest (Ds ) 174 nm, Ro ) 27.5 nm) metal dot structures (Figure 8d), the currents were within a factor of 2 of those observed for the bare n-Si electrodes, even though ≈ 8% of the surface was covered with Ni. Figure 8 also presents the predictions of the analytic SC/ M|E interacting contact (pinch-off) model (eqs 8 and 11) for

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Figure 6. Effective radii, Ro, of the Ni particles associated with dislocation defects on nanopatterned Si surfaces prepared using nanospheres of Ds ) (a) 1530 nm, (b) 760 nm, (c) 365 nm, and (d) 174 nm. These results are based on analysis of total image areas of approximately (a) 30000 µm2, (b) 2000 µm2, (c) 800 µm2, and (d) 100 µm2, respectively.

TABLE 3: Best Fit Parameters Obtained by Application of Diode Equation (eq 17) to Dark J-V Curves sphere diameter used in mask (Ds), nm

174

365

760

1530

number of SC/E electrodes included in average (n) SC/E temperature (T), C° SC/E exchange current density (Jo,l), A cm-2 SC/E effective barrier height (Φobs,l), V SC/E diode quality factor (Al)

13 25.7 ( 1.2 (5.2 ( 1.5) × 10-7 0.790 ( 0.008 2.02 ( 0.15

9 26.2 ( 1.1 (8.8 ( 2.8) × 10-8 0.837 ( 0.009 1.90 ( 0.16

4 25.6 ( 0.5 (1.4 ( 0.4) × 10-7 0.822 ( 0.010 1.75 ( 0.13

8 24.5 ( 2.4 (1.1 ( 0.4) × 10-7 0.825 ( 0.011 1.92 ( 0.17

number of SC/M|E electrodes included in average (n) SC/M|E temperature (T), C° SC/M|E exchange current density (Jo), A cm-2 SC/M|E effective barrier height (Φobs), V SC/M|E diode quality factor (A)

6 26.4 ( 1.1 (9.9 ( 3.0) × 10-7 0.775 ( 0.008 1.83 ( 0.13

3 25.8 ( 0.1 (2.5 ( 0.8) × 10-6 0.749 ( 0.008 2.13 ( 0.20

5 25.5 ( 0.9 (5.5 ( 1.7) × 10-6 0.728 ( 0.008 1.73 ( 0.15

7 26.5 ( 1.0 (4.8 ( 3.2) × 10-5 0.675 ( 0.017 1.8 ( 0.6

number of SC/M electrodes included in average (n) SC/M temperature (T), C° SC/M exchange current densitya(Jo,M), A cm-2 SC/M effective barrier heighta(ΦB,M), V SC/M diode quality factora(AM)

14 25.6 ( 1.1 (4 ( 21) × 10-4 0.62 ( 0.15 1.6 ( 7.8

12 26.1 ( 0.3 (8.1 ( 11.4) × 10-4 0.60 ( 0.04 2.0 ( 2.0

4 25.2 ( 0.5 (2.6 ( 1.6) × 10-5 0.688 ( 0.017 1.2 ( 0.3

7 26.7 ( 1.1 (1.5 ( 7.8) × 10-3 0.59 ( 0.14 2.0 ( 9.1

a The large uncertainty in these values reflects the limited number of data points that could be collected before the mass transport limit was reached.

each set of nanopatterned electrodes, except for those with Ds ) 1530 nm for which the model does not apply because the pinch-off criterion (eq 5) is not satisfied. The model had no adjustable parameters and all input quantities (Table 4) were determined experimentally. The predictions are in excellent agreement with experimental observations, and the data for the small Ro dot patterns differs significantly from what would be expected for a system that did not display the pinch-off phenomenon. Table 5 presents the values of Ro or Φ/B that provided the optimal least-squares fit of the interacting contact model to the experimental data when the specified parameter was allowed to float. For comparison, Figure 8 also includes the predictions of the independent contact model.

A substantial number of nanopatterned electrodes exhibited an unusually quick rise in current at small forward biases (ΦB,eff on the order of those observed at n-Si/Ni contacts and unphysically large A values), only to return to “normal” behavior at large forward biases. This local inflection behavior was particularly common and pronounced on the electrodes patterned with the smallest spheres, i.e., those having metal features expected to be almost completely pinched-off. This behavior was eventually associated with the disproportionate effect of a small number of large-area Ni defects on those electrodes, usually located at the edges of the electrodes where the epoxy did not completely cover the unpatterned regions of the substrates. Microscopic investigation of some of these electrodes

Spatially Nonuniform Barrier Height Contacts to SC Surfaces

J. Phys. Chem. B, Vol. 105, No. 49, 2001 12313

Figure 7. Effective radii, Ro, of Ni particles on nanopatterned Si surfaces prepared using nanospheres of Ds ) (a) 1530 nm, (b) 760 nm, (c) 365 nm, and (d) 174 nm. The empty bars include only nondefect particles, while the filled bars represent the statistical results based on all observed particles, including those resulting from defects in the masks. The total areas of the images employed in these analyses were (a) 2100 µm2, (b) 416 µm2, (c) 148 µm2, and (d) 20 µm2.

TABLE 4: Parameter Values Employed in Application of Equations 11 and 16 Ds, nm

174

365

760

1530

Al AM Jo,l, A cm-2 ND, cm-3 Ro, nm T, K ΦB,M, V factual

2.02 ( 0.15 1.6 ( 7.8 (5.2 ( 1.5) × 10-7 (7.9 ( 0.4) × 1014 27.5 ( 6.7 299.5 ( 1.1 0.62 ( 0.15 0.071 ( 0.007

Measured Quantities 1.90 ( 0.16 2.0 ( 2.0 (8.8 ( 2.8) × 10-8 (7.6 ( 0.2) × 1014 56 ( 12 299.0 ( 0.1 0.60 ( 0.04 0.090 ( 0.014

1.75 ( 0.13 1.2 ( 0.3 (1.4 ( 0.4) × 10-7 (7.7 ( 0.4) × 1014 106 ( 25 298.7 ( 0.9 0.688 ( 0.017 0.071 ( 0.010

1.92 ( 0.17 2.0 ( 9.1 (1.1 ( 0.4) × 10-7 (8.4 ( 0.3) × 1014 260 ( 60 299.6 ( 1.0 0.59 ( 0.14 0.090 ( 0.011

Fdots, cm-2 Vbb (Vapp ) 0), V Vinv, V Vn, V W (Vapp ) 0), nm ∆, V Φ/B, V

3.0 × 109 0.563 0.563 0.278 968 0.221 0.842

Computed Quantities 9.2 × 108 0.560 0.560 0.280 984 0.240 0.840

2.0 × 108 0.560 0.560 0.280 978 0.152 0.840

4.4 × 107 0.567 0.567 0.277 940 0.256 0.843

allowed the large metal defects to be identified and covered with epoxy, and when this was done the local inflections in the J-V curves of the treated electrodes became far less pronounced. Because not all the nanopatterned electrodes exhibited this behavior, those that did exhibit it were eliminated from our analysis. It is possible that the local inflection is a result of the mass-transport flux limit being reached at the electrolyte-metal interface of unpinched-off, low-effective-barrier height defects, saturating the current they can contribute to the electrode under large forward biases. As the defects become mass-transport limited and unable to provide any more current, the behavior of such electrodes comes to more closely resemble that of an electrode lacking any large area defects and covered only with Ni particles of the intended size.

C. Photoresponse of Nanopatterned Electrodes. Figure 9 shows the photoresponses, obtained by plotting the short-circuit current densities, Jsc ) Isc/Stot, versus Voc, for each set of electrodes. The data are again grouped by the diameter of the spheres employed in the preparation of the nanopatterned electrodes of each set. This figure also contains the predicted photoresponses, based on the measured dark J-V properties and eq 16, overlaid onto the experimental photoresponse data. A clear effect, correlated with the size of the metal dots, is readily apparent in the data. Even though the total coverage of metal was nearly identical for the various nanopatterned electrodes, those covered with metal dots having the smallest spatial dimensions exhibited the highest open-circuit voltages at a given short-circuit photocurrent density. The effect was

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Figure 8. Forward bias J-V responses of n-Si/E, n-Si/Ni|E, and n-Si/Ni electrodes in contact with CH3OH-1.0 M LiClO4-0.10 M Me2Fc-0.015 M Me2Fc+ in the dark. The data are grouped by the diameter of the nanospheres, Ds, used in preparing the electrodes and represent the geometric mean current density obtained at several (Table 3) identically prepared electrodes. Applied potentials were corrected for series resistance and concentration overpotential losses. The error bars indicate the standard deviation in the current values measured across the electrodes. Thin lines indicate fits to the diode equation (eq 17, Table 3), whereas the thicker lines indicate the predictions of the interacting contact model (solid, calculated using eqs 8 and 11), where applicable, and the independent contact model (dashed, calculated using eqs 11 and 15), using the experimentally measured input parameters of Table 4. Results obtained for: (a) Ds ) 1530 nm, (b) Ds ) 760 nm, (c) Ds ) 365 nm, (d) Ds ) 174 nm.

TABLE 5: Numeric Results of Interacting Contact (Pinch-Off) Model Applied to n-Si/Ni|E Electrodesa sphere diameter (Ds), nm nmb

geometrically expected effective particle radius, measured effective particle radius (Ro), nmb critical effective particle radius ((∆W)/(2Vbb)), nmc best fit effective particle radius (Ro,fit), nmb,d calculated solution barrier height equivalent (Φ/B), Vf / ), Vd best fit solution barrier height equivalent (ΦB,fit analytic model estimate of effective barrier height (ΦB,eff), V point dipole estimate of effective barrier height (ΦB,eff), V dipole layer estimate of effective barrier height (ΦB,eff), V

174

365

760

1530

17.3 27.5 ( 6.7 190 30.7 0.842 0.832 0.750 0.752 0.758

36.3 56 ( 12 211 63 0.840 0.819 0.692 0.696 0.711

75.6 106 ( 25 133 95 0.840 0.869 0.644 0.652 0.697

152 260 ( 60 212 e 0.843 e e e 0.590g

a Where applicable, values are for equilibrium conditions (V app ) 0, J ) 0) and employ the measured value of the effective particle radius, Ro. Radius of a circular dot having the same area as the triangular Ni dot formed. c The largest particle radius satisfying eq 5; particles with larger effective radii are not expected to exhibit the pinch-off effect. d The optimal value obtained in fitting the experimental data with eq 11, when this variable is allowed to float and all others are held constant at the values indicated in Table 4. e The interacting contact model is not applicable to these cases because the particles are not pinched off, viz., the measured Ro exceeds the critical effective particle radius. f As per Table 4. g The dipole layer model (correctly) predicts that this contact will not be pinched-off. The potential maximum along the central axis of the dipole layer occurs at the semiconductor surface, as shown in Figure 10a.

b

significant in that, at Jph ≈ Jsc ) 10 mA cm-2, Voc increased from 250 mV to 400 mV as the effective radius of the Ni dots decreased from 255 to 27.5 nm. At the smallest Ni particle radii, the Voc values were nearly equal to those of the unmetallized electrodes, indicating that recombination had been significantly suppressed even though ≈ 8% of the surface was still covered with metal. Figure 9 also shows the predicted photoresponse properties obtained from measurements of the dark responses in conjunction with eq 16. For the Ni dots having Ro ) 27.5 nm, Voc was within 100 mV of the value obtained at a bare electrode, for which bulk recombination/diffusion has been shown to deter-

mine the current-voltage properties at sufficiently large forward bias voltages and/or illumination intensities.49 V. Discussion A. Growth of Nanosphere Masks. The process employed in this work for preparing extended (many mm2) crystalline structures was roughly analogous to Czochralski crystal growth, and entailed the slow extraction of a hydrophilic substrate from an aqueous sol of monodispersed-diameter nanospheres under a controlled-humidity ambient.89 Because the substrate and ionically functionalized spheres are both very hydrophilic, sol is constantly drawn onto the nonsubmerged portion of the

Spatially Nonuniform Barrier Height Contacts to SC Surfaces

J. Phys. Chem. B, Vol. 105, No. 49, 2001 12315

Figure 9. Plots of the photoresponses of the same electrodes studied in Figure 8. The thin lines indicate the predicted photoresponses (solid, n-Si/E and dashed, n-Si/Ni) obtained from the diode equation (eq 17 with -Voc replacing Vapp and Jsc replacing J), based on the parameters extracted from the dark responses (Table 3). The thick lines indicate the predictions of the interacting contact model (solid, calculated using eqs 8 and 16), where applicable, and the independent contact model (dashed, calculated using eqs 15 and 16) using the experimentally measured input parameters (Table 4). Results obtained for: (a) Ds ) 1530 nm, (b) Ds ) 760 nm, (c) Ds ) 365 nm, (d) Ds ) 174 nm.

substrate, where the water evaporates and the nanospheres are left behind.90 Once a seed crystal forms on the substrate surface, the rate of evaporation from the water-covered crystal approaches steady state. Thus, the rate of sol flow into the nonsubmerged substrate region becomes constant, such that the crystal grows at a fixed rate and to a fixed thickness if the sample is extracted from the sol at a properly selected constant velocity. Growth conditions (Table 1) were optimized to obtain bilayer crystals. The two-dimensional nanosphere crystals consisted of two hexagonally close-packed layers, stacked on top of each other in an AB fashion. A point vacancy defect in a singlelayer nanosphere mask leads to the formation of a metal patch equal in size to the spheres used in the mask itself. In contrast, a similar defect in one layer of a bilayer mask leads to a much smaller metal defect, provided that the defects in the two layers are not correlated. Bilayer masks also have the advantage of providing a nanopattern with relatively large spaces between the Ni dots, making it possible to treat the dots independently and thus allowing application of Tung’s analytic model1 to the resulting nanostructured SC/M|E interfaces. B. Properties of Nanosphere Masks. TMAFM observations of n-Si/Ni|E electrode surfaces after their use in electrochemical experiments verified that the Ni nanopatterns were robust and survived the electrode preparation, etching, and electrochemical characterization processes. The patterns were regular and produced consistently sized Ni patches having a controlled spacing on the Si surface (Figures 3 and 4). Although dislocations and grain boundaries were observed to occur in both the upper and lower layers of the nanosphere bilayer crystals, point vacancies were almost exclusive to the upper of the two layers. This is consistent with the suggestion

that the nanospheres roll along the smooth substrate surface in forming the lower layer, but must roll over the corrugated lower layer in forming the upper layer.90 The increase in point vacancy density (on a per-lattice-point basis) with increasing sphere size (Table 2) can be readily understood within the same framework. The Ni particles constituting the double-layer nanopatterns prepared in our work were appreciably larger and more triangular than idealized geometric considerations would lead one to expect.69 This behavior might be ascribed to scattering of thermally evaporated Ni atoms off the sides of the nanospheres that form the upper layer of the mask. The choice of the minimum threshold used in the sizing algorithm was to a certain extent necessarily subjective because the particle crosssections were essentially triangular and their edges not perfectly distinct. The relative height cutoffs were kept consistent between images and across the various nanosphere masks, producing reasonable precision in the measurements at the cost of a moderate (20-40%) systematic underestimation of the effective dot radii. The consistency in particle shape and cross-section observed across the range of particle sizes suggests that tip convolution effects were not substantial in these images. Nonetheless, a slight overestimation is incorporated into the tabulated particle sizes as a result of not employing a tip convolution correction,91 which offsets to some extent the underestimation associated with the height cutoff method used in sizing the particles. Table 2 includes data for the fractional area of each nanopatterned surface statistically expected to be covered by particles which result from point vacancy and dislocation defects and which have effective radii larger than 2Ro. These values are based on the measured defect densities (Table 2) and the observed particle size distributions (Figures 5 and 6). In no case

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Figure 10. Predictions of the point dipole model compared with those of the dipole layer model for V(0,z) when the Ni patches are formed using spheres of diameter: (a) Ds ) 1530 nm, (b) 760 nm, (c) 365 nm, or (d) 174 nm. The point dipole model systematically underestimates the saddle point potential, with the magnitude of the error clearly increasing as Ds increases. In case (a), the point dipole model incorrectly predicts a saddle potential whereas the dipole layer model correctly predicts no local maximum in V(0,z). This comparison reinforces the need to ensure that the constraint of eq 5 is satisfied before using the pinch-off model to predict the potential field in the semiconductor.

did the defect area fraction exceed 1% of the total surface area; moreover, the defect fraction remained nearly constant for the three smallest values of Ds. While the defect density for the Ds ) 1530 nm spheres did approach 1% of the total surface area, large-area defects are not expected to behave any differently than the “normal” nanopattern particles at these sphere sizes, for which the electrode behavior is expected to depend solely on the fraction of the surface covered with Ni. The relatively small defect densities therefore justify the use of eqs 11 or 16, incorporating a single effective radius value, in predicting the I-V behavior of these systems. C. Dark Electrical Properties versus Size of Metal Dots. The electrical measurements appear to demonstrate clearly the effects of pinch-off in the dark J-V behavior of nanopatterned n-Si/Ni|E parallel junctions. The observation of this electrical behavior at small forward biases in the dark suggests that it is electrostatic in origin, as opposed to resulting from hydrogen evolution,57,58 the formation of an oxide layer,44,92 or of metal particle charging associated with slow electron transfer out of the metal45,62,63 under illumination conditions. Quantitative evaluation of the J-V data indicates that Tung’s analytic model,1,2 formulated to describe transport across pinched-off inhomogeneous SC/M contacts, can also be adapted to the description of SC/M|E contacts. The results described above indicate that the modified model of eqs 8 and 11 provides excellent agreement with experimental results for the n-Si/Ni|E system.75 The calculations made in this work implicitly assume that the n-Si/Ni contacts studied are homogeneous, which is questionable in light of BEEM observations reported in the solidstate literature.17 Nonetheless, it is clear from the observed

behavior of the n-Si/Ni|E electrodes patterned with the largest sphere diameters that at least a portion of the microscopic n-Si/ Ni contacts formed in this work have extremely low barrier heights that would dominate the behavior of a dot-covered electrode if not for the pinch-off effect. Numerical simulations12 have indicated that interface nonplanarity can have an appreciable effect on pinch-off even when the nonplanarity occurs on a scale much smaller than the depletion width. The theory used herein implicitly assumes that the silicon substrates of our n-Si/Ni|E electrodes were planar. The required etch of the nanopatterned samples prior to their electrochemical characterization necessarily removes a surface layer of oxidized silicon (associated with the nanosphere crystal growth process) from the bare regions while leaving the oxide in place under the Ni. The pinch-off effect would be enhanced if the silicon surface were somewhat raised underneath the metal particles and lowered on the bare regions of the nanopatterned electrodes.12 In contrast, movement of the effective n-Si/Ni interface below the plane of the n-Si/E interface, as a result of silicide formation,93 would reduce the pinch-off effect. We do not believe either phenomenon occurred to an extent sufficient to have a quantitatively significant effect on our results. The theory described herein is also built around the assumption of a round low barrier height patch. Describing the triangular metal regions in terms of an equivalent radius is appropriate because the triangles are then reduced to the same point dipole as would be produced by a circular patch.2 The analytic interacting contact model of eqs 8 and 11 fits the data extremely well in the regime where the model is applicable. The saddle point potentials agree to within 1% for Ds ) 174 nm spheres and to within 5% for Ds ) 365 nm spheres.

Spatially Nonuniform Barrier Height Contacts to SC Surfaces However, the point dipole model becomes increasingly imprecise at predicting the saddle point potential, ΦB,eff, as the extent of pinch-off diminishes (Figure 10). For situations where eq 5 is not satisfied, the independent contact model of eqs 11 and 15 applies, in accord with the experimental observations. In the transition region, where eq 5 is barely satisfied, neither of the models is truly theoretically sound. D. Photoresponse Properties versus Size of Metal Dots. The good agreement between the dark J-V data and the photoresponse results suggests that the photoresponse of SC/ M|E electrodes can be predicted using the analytic interacting contact (pinch-off) model. These results are consistent with a qualitative model proposed by Nakato and Tsubomura41,61 to describe the behavior of SC/M|E electrodes. The use of nanopatterned contacts seems promising for the investigation of mixed barrier height phenomena in both solidstate and SC/E systems. Exploitation of the pinch-off effect is especially attractive for SC/M|E contacts because the pinchoff effect provides a mechanism to introduce catalytically active metal sites on a semiconductor surface, thus promoting desirable minority carrier-based multielectron-transfer reactions, without suffering deleterious majority-carrier-based recombination.61,66 With proper tuning of the redox potential of the electrolyte solution, appreciable concentrations of nanoscopic particles of a catalyst material can thus be placed on semiconductor surfaces without promoting thermionic emission to a position of dominance over the total recombination current. These conditions simultaneously satisfy the conditions expected to make such catalyst particles optically transparent,56 suggesting a possible strategy for the preparation of highly efficient SC/M|E electrodes capable of converting light energy into chemical fuels.42,46,55,61 VI. Conclusions Bilayer masks formed from sols of monodisperse polystyrene spheres allowed the fabrication of regular, well-defined nanometer-scale patterns of thermally evaporated Ni on Si electrode surfaces. The dark J-V and photoresponse properties of such intentionally prepared inhomogeneous contacts immersed in electrochemical solutions capable of inducing strong inversion in the semiconductor were found to exhibit a strong dependence on the size of the low barrier height metal particles, with Ni particles of effective radius between 25 and 100 nm behaving in quantitative accord with the predictions of an analytic theory describing the electrostatic pinch-off phenomenon developed by Tung and modified herein to describe strongly inverted SC/ M|E contacts. The behaviors of n-Si/Ni|E electrodes incorporating the largest Ni particles were found to fall outside the pinchoff regime and were thus well-described as the net effect of many independent n-Si/Ni and n-Si/E devices operating in parallel. Nanostructured metal deposits therefore offer a potentially attractive approach to obtaining novel photoelectrochemical device properties while minimizing the deleterious effects of metal-induced charge carrier recombination at such surfaces. Acknowledgment. The National Science Foundation is acknowledged for supporting this work through grant CHE9974562 and for providing R. Rossi with a Graduate Research Fellowship. We also thank Prof. R. P. Van Duyne and J. C. Hulteen of Northwestern University for discussions regarding nanosphere lithography, G. Kumaraswamy of Caltech for guidance with TMAFM, and Prof. K. Kavanagh and Dr. B. A. Morgan of University of California, San Diego, for helpful discussions regarding BEEM and SC/M contact stability.

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