J. Phys. Chem. C 2007, 111, 5497-5499
5497
Analysis of CapacitancesPotential Measurements at the Silicon-Electrolyte Interface Revisited P. Allongue,* J.-N. Chazalviel, C. Henry de Villeneuve, and F. Ozanam Physique de la Matie` re Condense´ e, Ecole Polytechnique, CNRS, 91128 Palaiseau, France ReceiVed: December 15, 2006; In Final Form: February 8, 2007
The analysis of the capacitance-potential curves is revisited by using a more accurate bias dependence of the silicon space charge layer capacitance (CSC) under accumulation conditions (Tardella, A.; Chazalviel, J.-N. Phys. ReV. B 1985, 32, 2439). In the case of a well-defined stepped H-terminated Si(111) surface, we show that the improved analysis yields a more realistic CH value: ∼8 µF/cm2 (∼3.5 µF/cm2 was derived from the Poisson-Boltzmann analysis). However, in the case of samples grafted with organic chains, we show that the model used for CSC has no influence on the determination of the effective dielectric constant EFF of the organic layer.
Introduction Organic films on silicon substrates prepared by wet chemistry on H-terminated surfaces have been receiving increasing interest during the past decade because they can find application in microelectronics and biosensors.1,2 To improve the chemical stability of a given hybrid interface, one challenging issue remains in optimizing the packing density by determining the most appropriate grafting method and/or adjusting the grafting conditions for a given route. In this context, it becomes essential to quickly and quantitatively determine the resulting surface coverage. Among characterization techniques, X-ray photoelectron spectroscopy (XPS) requires time-consuming angle resolved measurements to be truly quantitative.3 Scanning tunneling microscopy provides direct information on the local molecular packing as in the case of phenyl monolayers imaged on Si(111) in the electrolytic environment4 or methyl monolayers imaged with a low-temperature UHV--STM.5 It is, however, not so easy to image long molecular chains, and the method is useless with disordered layers. More routine laboratory characterizations of the surface coverage include ellipsometry6 and attenuated total reflection (ATR) Fourier transform infrared spectroscopy (ATR-FTIR),7 even though making FTIR quantitative is more time-consuming. Routine characterizations of hybrid silicon-organic monolayerinterfacesareelectrochemicalcapacitancemeasurements.4,8-11 In this case, the surface coverage is derived from the effective dielectric constant (EFF) of the organic film, which is easily obtained under cathodic conditions at an n-type electrode so as to accumulate electrons at the surface, which prevents any interface oxidation (the same characterization would be questionable with a p-type substrate as holes would be accumulated). The method is rapid and versatile to study and monitor the interface properties after or during diverse treatments.10 The data analysis requires, however, the knowledge of the bias dependence of the silicon space charge layer capacitance (CSC) under accumulation conditions. To date, our group has used the analytical expression derived from the Poisson-Boltzmann approximation.12 However, under strong accumulation condi* Corresponding author. Tel: +33 1 69334431. Fax: +33 1 69333004. E-mail:
[email protected].
tions, a known shortcoming of this approximation is the fact that the potential well near the surface (band bending region) becomes narrow on the scale of the electron de Broglie wavelength, making incorrect an approximation relating the electronic density to the local potential value. In other words, under such conditions, the electron states are actually quantized in the z-direction perpendicular to the surface. A calculation taking into account these quantum effects was performed by Tardella and Chazalviel.13 In this calculation, the potential was determined self-consistently from Poisson’s equation, together with numerical resolution of the Schro¨dinger equation along z, the charge density being obtained by populating the electron states according to Fermi-Dirac statistics. Although that approach is not exact (exchange and correlation effects were not taken into account), it represents a significant improvement over the simple Poisson-Boltzmann approach. This paper aims, therefore, at revisiting previous analyses of capacitance measurements at H-terminated and modified silicon electrodes in light of this improved approach for CSC. CapacitancesPotential Plot Analysis The equivalent capacitance C of the interface (Figure 1) is the series combination of the capacitances corresponding to the space charge region in the silicon substrate (CSC), the monolayer (CML), and the Helmholtz layer (CH). It reads
1/C ) 1/CH + 1/CML + 1/CSC
(1)
where CSC is the only bias-dependent capacitance. Two models are considered next for CSC. Classical Poisson-Boltzmann Approximation. In this model,12 there exists an analytical expression for CSC under accumulation conditions
CSC )
[
] [ ]
q2Si0ND 2kT
1/2
VS exp -q 2kT
(2)
The corresponding accumulated charge density reads
[
VS QSC ) [2Si0kTND]1/2 exp -q 2kT
10.1021/jp068614z CCC: $37.00 © 2007 American Chemical Society Published on Web 03/22/2007
]
(3)
5498 J. Phys. Chem. C, Vol. 111, No. 14, 2007
Allongue et al.
Figure 1. Equivalent capacitance of the silicon-organic layerelectrolyte interface. The different capacitances correspond to the silicon space charge (CSC), the organic monolayer (CMOL), and the Helmholtz layer (CH).
where VS ( 0.15 V (we recall that VS < 0). In the quantum model, the variations of CSC are smooth with a downward curvature (the calculated curve can be fitted with CSC ) ∼|VS|R with R < 1, for |VS| > 0.25 V), while the variations of CSC are exponential in the classical model. This difference in behavior is due to the increasing error made with the classical approximation as the potential well becomes narrower. For practical use, Table S1 (Supporting Information) gives CSC and QS as a function of VS. Simulation of Experimental Data. CSC and QS were first tabulated as a function of VS (using eqs 2 and 3 within the classical approach or using Table S1 within the quantum approach). The potential distribution across the interface was then calculated at each potential using
∆V ) V - Vfb ) VS + ∆VMOL,H
(4)
where ∆VMOL,H ) QS/CMOL,H is the total potential drop across the organic layer and the Helmholtz layer [CMOL,H ) CMOLCH/ (CMOL + CH)]. One then determines C(∆V) as the series combination of CSC and CMOL,H (see eq 1). To scale the
calculated plot in the absolute scale of potential V, one just needs to define Vfb since V ) Vfb + ∆V (eq 4). A fit of the C-V curve of a modified electrode is therefore realized with two independent parameters Vfb and CMOL,H. A similar procedure is applied for the H-terminated surface (in this case ∆VMOL,H becomes ∆VH ) QS/CH), which gives the corresponding Vfb and CH. Combining both results and assuming that the CH value is the same at the H-terminated surface and at the modified surface (a reasonable hypothesis for two hydrophobic surfaces), the capacitance CMOL was calculated from eq 1, and an effective dielectric constant EFF was obtained by inverting the expression CMOL ) EFF0/dCHAIN. Results and Discussion In a recent paper,10 we showed that alkyl monolayers prepared by thermal reaction with 1-alkene on a well-defined Hterminated surface constitute a model system. For a reaction time longer than 20 h, we found that water is excluded at the nanometer scale from the organic film because the fraction of hydrogen atoms substituted by alkyl chains was above 0.42 (i.e., close to the maximum theoretical value of 0.5) from quantitative angle resolved XPS measurements.3 In other words, the molecular surface density is ∼3.3 × 1014 cm-2 (there are 7.8 × 1014 sites/cm2 on an ideal Si(111) surface). A similar surface density was also derived from quantitative FTIR.14 No detectable traces of interfacial oxide were found from either XPS or FTIR after exposure to different aqueous solutions. In addition, the density of the interface gap states was estimated to be ∼3.3 × 1010 defects/cm2 from electrochemical measurements,10 which was supported by first principle calculation of the electronic structure at the interface between Si(111) and alkyl monolayer.15 Hence, the H-terminated sample and the optimized grafted sample fulfill requirements for modeling with a surface charge arising only from electron accumulation in the conduction band (see CapacitancesPotential Plot Analysis). Figure 3 compares experimental data (symbols) taken from ref 10 with the two kinds of simulations (solid and dotted lines). For the Hterminated surface (Figure 3a), the classical model (dotted line) is in very good agreement with experimental data for -1 V < V < -0.6 V only. At more negative potentials, calculation and experiment strongly deviate from each other. The quantum model (solid line) appears much better because the calculated curve overlaps very well with the experimental one over the entire range of potentials. This is because of the smoother variations of CSC with band bending (Figure 2). Calculations yield CH ) 3.5 µF/cm2 with the classical model and 8 µF/cm2 with the quantum model. This large difference arises from the fact that CSC is greatly overestimated in the classical approximation (see Figure 2a). In the case of a silicon surface grafted with an alkyl monolayer (12 carbons), Figure 3b shows that
Figure 2. (a) Calculated CSC-VS plots and (b) corresponding variations of the charge QS within the classical model (dashed line) and quantum model (solid line) for 1 Ω cm n-type Si(111) [ND ) 2 × 1015 cm-3].
Silicon-Electrolyte Interface Measurements
Figure 3. C-V plots of n-type Si(111) electrodes recorded in 0.1 M H2SO4. Symbols are experimental data. The solid and dashed lines are simulations according to the quantum model and classical model, respectively. (a) H-terminated surface. Curves are calculated with Vfb ) -0.63 V and CH ) 3.5 µF/cm2 (classical model) or 8 µF/cm2 (quantum model). (b) Optimized alkylated surface (12 carbons per chain). See text. The two curves are calculated with EFF ) 2.15, Vfb ) -0.98 V, and the relevant CH value determined for the bare surface.
the two models give closely similar curves. In addition, the agreement between calculation and experiment is now quite good over an extended range of potentials. The relevant CH value was used in each case. In both cases, we used Vfb ) -0.98 V and EFF ) 2.15 (with dCHAIN ) 13.9 Å for a 12 carbon chain3). Note that the value EFF ) 2.15 is very close to the value of polyethylene (2.3), which is in perfect accordance with the high surface coverage derived from XPS and FTIR. The classical approximation becomes quite acceptable here because the equivalent interfacial capacitance is dominated by the small capacitance of the monolayer (CMOL ) ∼1.36 µF/cm2). In addition, this series capacitance prevents the system from reaching as deep accumulation conditions as those obtained at the hydrogenated surface. We also reconsidered the analysis of other types of modified silicon samples, including those grafted with alkyl chains bearing an acid terminal group.14 We found that the results are not affected by the model used for CSC. The same parameters Vfb and EFF are obtained with the two approaches. We also verified the case of less ideal samples for which limited interface oxidation was suspected from the presence of a small peak before the rise of the capacitance (see, i.e., Figure 3 in ref 10). Under strong accumulation conditions, the charge stored in the interface states remains, however, nearly constant as the energy levels are totally filled up. The corresponding surface charge is typically
