PHYSICAL REVIEW E VOLUME 53, NUMBER 6 JUNE 1996 Double-layer capacitance on a rough metal surface L. I. Daikhin School of Chemistry, Tel Aviv University, 69978 Tel Aviv, Israel A. A. Kornyshev* and M. Urbakh† Institut fu¨r Energieverfahrenstechnik, Forschungszentrum Ju¨lich GmbH (KFA), 52425 Ju¨lich, Germany ~Received 30 November 1995! An expression for the double layer capacitance of rough metal-electrolyte, metal-semiconductor, or semiconductor-electrolyte interfaces is derived which shows the interplay between the Debye length and the lengths characterizing roughness. Different dependencies of the capacitance, as compared to the flat interface, on the concentration of charge carriers in electrolyte or semiconductor are predicted. Examples of the typical roughness spectra are considered. The cases of Euclidean roughness show weak dependence on the particular form of the roughness spectrum, being sensitive only to its main parameters: the random mean square height of roughness and correlation length. A method is proposed for the in situ characterization of surface rough- ness: the measurement of surface roughness with a ‘‘Debye ruler,’’ based on the conventional measurements of the double layer capacitance. @S1063-651X~96!05305-3# PACS number~s!: 68.45.2v, 41.20.Cv I. INTRODUCTION problem, giving rise to different functional dependencies on electrolyte concentration and potential. The Gouy-Chapman theory of electrolyte plasma near a In this article we show how the competition between the flat charged wall @1,2#, which appeared a decade earlier than Debye length and the correlation length of roughness modi- the Debye theory of bulk electrolytes @3#, is the basis of fies the Gouy-Chapman result. It is obvious, a priori, that the many successful constructions in electrochemistry @4#, col- limiting value of capacitance at short Debye lengths should loid science @5#, biophysics @6#, and semiconductor science follow Eq. ~1! but with S replaced by Sreal5RS. In the limit and technology @7#. In the low voltage limit, the Gouy- of long Debye lengths the roughness would not be mani- Chapman theory gives a transparent result for the space fested in the capacitance which would obey the native Eq. charge capacitance ~1!. How does the crossover between these two limits occur? One may expect to recover the whole curve, modifying Eq. C5CGC[«kS/4p, ~1! ~1!, 21 ˜ where k is the Gouy ~5Debye! length, « the dielectric C5R~k!CGC , ~18! constant of the solvent, and S the area of the flat interface. As it should be, the capacitance is inversely proportional to where the roughness function˜ R~k! varies between ˜R~0!51 the separation between the charge and counter charge, in and ˜R(`)5R.1. The problem for the theory is, then, to find plasma provided by the Debye length. this function. For the case of a weak roughness, we derive A long period in electrochemistry was associated with the the general expression for the roughness function, establish studies performed on the liquid mercury drop electrode, and its limiting behavior, and study the cases of different surface later on Ga, InGa, and GaTl alloys @8#. Providing an ideally morphologies ~sinusoidal corrugation, random Gaussian smooth interface between the metal and electrolyte, liquid roughness, and self-affine fractal structures!. electrodes allowed a set of classical results in electrocappil- In electrolyte solutions the Debye length can easily be lary phenomena, adsorption and electrochemical kinetics. varied by changing the electrolyte concentration with no Studies on solid electrodes ~Cd, Pb, Bi, Sn, Cu! faced prob- effect on surface roughness. Experimental data on ˜ lems associated with a nonsmooth character of the interface R(k)5C/CGC may then be used for probing the roughness @9#. The concept of the geometrical roughness factor of the metal surface in contact with electrolytes. The same R5Sreal/S, i.e., the ratio of the true surface to the apparent idea can be put into the basis of an evaluation of the rough- surface ~flat cross-section area! became common @9#. How- ness of metal-semiconductor or semiconductor-electrolyte ever, in many cases, experimental data cannot be rationalized interfaces. Debye length in semiconductors can be varied by in terms of this parameter only. The latter is not surprising. radiation or temperature-induced excitation of charge carriers Roughness may be responsible for additional characteristic into the conduction band @10#~doping will require a creation lengths, which may compete with other typical lengths in the of a new junction with an uncontrollable effect on the struc- ture of the contact!. The idea of a competition between the characteristic *Corresponding author. Electronic address: rbe016@djukfa11 scales of roughness and the optical and acoustic wavelengths †Permanent address: School of Chemistry, Tel Aviv University, was explored in the classical works of Rayleigh @11# and 69978 Tel Aviv, Israel. Electronic address: [email protected] Fano @12#, and in the subsequent studies in interfacial optics 1063-651X/96/53~6!/6192~8!/$10.0053 6192 © 1996 The American Physical Society 53 DOUBLE-LAYER CAPACITANCE ON A ROUGH METAL SURFACE 6193 @13–15#. Similar methods were used in the theory of friction between consecutive peaks and valleys on the rough surface. @16# and quartz microbalance @17#. The interplay between the We will also assume that h,k21. roughness spectra and diffusion lengths was intensively in- Solving Eq. ~2!, it is convenient to Fourier transform the vestigated in the context of the anomalous frequency depen- potential and the surface profile function from tangential co- dence of the electrochemical impedance @18–21# and diffu- ordinates R5(x,y) to the corresponding wave vectors sion to the surface @22,23#. The competition between the K5(Kx ,Ky)asf~K!5*dRf~R!exp~2iK•R!. Equation ~2! roughness scale and the Debye length has been explored then transforms to @24–28# in the context of the surface stability and surface 2 forces. However, the features of the space charge ~double d 2K22k2 f~K,z!50. ~4! layer! capacitance, the key quantity in electrochemistry, were H dz2 J not considered. We address mainly solid metal-liquid electrolyte systems, According to the Rayleigh approximation @11# the solution but the basic results can be extended on other electrified of Eq. ~4! in the half space z,j(x,y), has the form interfaces, such as metal-semiconductor, semiconductor- K,z 5A K exp 2q z , 5 electrolyte, chargeable biological interfaces, and metal-solid f~ ! ~ ! ~ K ! ~ ! electrolyte contacts @29#. 2 2 1/2 where qK5(k 1K ) . This approximation neglects the terms proportional exp(qKz) which would be important in- II. BASIC EXPRESSION FOR CAPACITANCE side deep protrusions and grooves in the metal surface, ig- nored in our consideration. The boundary condition 3 leads A. Boundary problem for potential ~ ! to the integral equation on the prefactor A~K!, Consider a rough metal surface in contact with an electro- lyte. We take the z axis pointing towards the electrolyte and dK A~K! dR exp@2q j~x,y!# describe the interface by the equation z5j(x,y). The plane E ~2p!2 E K z50 is chosen such that the average value of the function 2 j(x,y) over the surface is equal to zero. 3exp@2i~K82K!•R#5~2p! f0d~K8!, ~6! In the Gouy-Chapman theory, the distribution of the elec- trostatic potential f~r! in the electrolyte is described by the where d~K! is the two-dimensional Dirac d function. nonlinear Poisson-Boltzmann equation. As a first step we Since we are bound to the case of weak roughness, restrict our consideration by its linearized version, valid for u¹j~x,y!u!1, hk!1, ~7! low electrode potentials f,kBT/e: the standard perturbation technique @11–17,20,23,25# may be ~¹22k2!f~r!50. ~2! applied to find A~K!. The first exponential in Eq. ~6! is ex- For a 1-1 binary electrolyte solution, the Debye length, panded into the series, 21 2 1/2 k 5(«kBT/8pne ) , where n is the electrolyte concentra- tion, « the dielectric constant of the solvent, e charge of E dR exp@2qKj~x,y!#exp@2i~K82K!•R# electron, T the temperature, and kB the Boltzmann constant @for multivalent ions n must be replaced by the ‘‘ionic 2 2 .~2p! d~K2K8!2qKj~K82K! strength’’ I5n(1/2)(z i ni where zi and ni are the valence and the stoichiometric coefficients of the ion of sort i#. For 1 dK 2 9 nondegenerate semiconductors with one sort of charge carri- 1 qK 2 j~K82K2K9!j~K9!, ~8! 21 2 1/2 2 E ~2p! ers, k 5(«kBT/4pne ) where n is the density of charge carriers and « is the high frequency dielectric constant of the and Eq. ~6! may be solved by iterations. In order to deter- semiconductor. An expression for k exists for solid electro- mine the first nonvanishing correction to the capacitance lytes with the same type of dependence on mobile ions con- caused by roughness, we must find A~K! up to the second centration and temperature @30#. order in h. Hence The solution of Eq. ~2! must satisfy the boundary condi- tion which fixes the potential at the metal-electrolyte inter- A~K!5A0~K!1A1~K!1A2~K!, ~9a! face where f x,y,z5j~x,y! 5f ~3! „ … 0 2 A0~K!5~2p! f0d~K!, ~9b! relative to the zero level in the bulk of the electrolyte: f~z `!50. A1~K!5f0q0h~K!, ~9c! ) 1 dK8 B. Perturbation theory A ~K!5 f q h~K2K8!h~K8!~2q 2q !. 2 2 0 0 E ~2p!2 K8 0 Consider weakly rough surfaces for which h, the charac- ~9d! teristic size of roughness in the z direction, is less than the tangential one, l.