## Abstract

The shape and size are the two important geometrical factors that affect the electronic screening in nano-materials. Here, we develop an analytical theory for electronic capacitance based on Thomas–Fermi screening in conjunction with ‘multiple scattering method’ for arbitrary-shaped nanostructures including electronic spillover correction. We relate the electronic capacitance of the material to the curvature correction expressed in terms of ratio of electronic screening length to principal radii of curvature. Electronic capacitance of various nanostructures is obtained showing geometrical shape- and size-dependent electronic screening in nanostructures that manifest important consequences in charge storage enhancement or reduction.

## 1. Introduction

Owing to miniaturization and the advent of nanoscale materials, electronic screening effects are found to have an increasing role in technology [1,2]. Understanding the role of shape and size on electronic properties of nanostructured materials is very important since it influences the interfacial properties and various phenomena occurring at surface [3–5]. In the last few years, electronic properties especially capacitance have received increasing attention from the energy storage [1,2,6–9] point of view. In particular, how the non-regular shape and size of materials is going to affect the electronic capacitance is an interesting question.

In this article, we develop an electronic screening model for an arbitrary shape nanostructure. We show how the shape and size of materials have important impact on the electronic screening and its charge storage property. We obtained analytical results solving the Poisson equation for electrostatics in the Thomas–Fermi (TF) approximation [10–12]. Further correction of electron spillover [13,14] near the electrode surface is made. The result for the capacitance is obtained which is valid for arbitrary shape/size geometries of a surface. The influence of local shape on electronic capacitance is analysed for four surface models: arbitrary-shaped surface, modulated surface, Gaussian-shaped surface and idealized surface geometries. We found strong influence of local geometry resulting in localization and oscillation of charge/capacitance in modulated surfaces and Gaussian-shaped nanostructures.

## 2. Theory

### (a) Local shape analysis

Our theoretical investigation on the influence of surface shape/geometry is based on the notion of surface characterized by shape element using the concepts of differential geometry. The local surface shape at the *α*th point of the surface *Γ*_{m} may be approximated as
2.1where *k*_{1}(*α*)=1/*R*_{1} and *k*_{2}(*α*)=1/*R*_{2} are the curvatures at a point *α*; *R*_{1} and *R*_{2} are the principal radii of curvature (table 1). Figure 1 illustrates various shape elements that may be generated by varying *k*_{1} and *k*_{2}.

Let us consider the shape element as a system of collection of electrons whose surface local number density *ρ*(** r**) under TF approximation [10,11] is
2.2where ;

*h*is the Planck's constant,

*E*

_{f}the Fermi energy,

*m*the mass of an electron and

*e*the electronic charge. Now the electrostatic potential

*ϕ*(

**) arising when the electron density departs from uniformity is given by Poisson's equation (PE), ∇**

*r*^{2}

*ϕ*=−(4

*πe*/

*ϵ*

_{m})(

*ρ*−

*ρ*

_{0}), where

*ρ*

_{0}is the uniform electron density of the surface when the electrostatic potential at the surface is

*ϕ*

_{0}. Now expanding equation (2.2) for small values of

*ϕ*compared with

*E*

_{f}, from PE we have the linearized TF equation 2.3The inverse TF screening length [15]

*κ*

_{TF}=(6

*πn*

_{0}

*e*

^{2}/

*ϵ*

_{m}

*E*

_{f})

^{1/2}, where

*ϵ*

_{m}is the background dielectric constant,

*E*

_{f}the Fermi energy of electrons and the average concentration of electrons and

*D*

_{F}the density of states. Physically,

*κ*

^{−1}

_{TF}=

*l*

_{TF}characterizes the extent of electronic screening towards an external electric field, and it represents a distance (few angstroms for metals and fraction of nanometres for semi-metals), where the potential difference approaches zero, i.e.

*ϕ*=0 outside the space charge layer in material. The potential difference at the surface is

*ϕ*=

*ϕ*

_{0}. When equation (2.3) reduces to the classical electrostatic Laplace equation ∇

^{2}

*ϕ*=0.

### (b) Surface-dependent potential–analytical solution through Green's function method

For a domain *Ω*_{M} bounded by an arbitrary-shaped surface *Γ*_{M}, which is held at constant potential (Dirichlet boundary), *ϕ*^{+}|_{S}=*ϕ*_{0} Green's function (GF) *G* for *l*_{TF} equation satisfies
2.4with the homogeneous boundary conditions at surface *S* and far away from surface (, *viz*. *G*|_{S}=0. The solution of equation (2.4) for the potential, *ϕ*(*r*) can be written as [16–18] (R. Kant, M. B. Singh, unpublished data. 2012 Generalization of linearized Gouy–Chapman–Stern model of electric double layer for nanostructured and porous electrodes: deterministic and stochastic morphology (http://arXiv.org/abs/1202.5827)
2.5We seek to express *G* in terms of Yukawa-like potential. The free space Green's function satisfying equation (2.4) for the entire infinite domain is *G*_{0}(*r*,*r*′)=(1/4*π*|*r*−*r*′|) e^{−κTF|r−r′|}. Now the surface-dependent Green's function *G* is expressed as [17]
2.6

### (c) Thomas–Fermi screening capacitance of arbitrary surface geometry

The local surface charge density *σ*_{M} may be obtained from Gauss's law as
2.7where d*S* is the area element, *E* is the electric field and equal to inward normal derivative of potential to the surface . The differential capacitance *c* is obtained by differentiation of surface charge with respect to surface potential *ϕ*_{0} as
2.8where *ϕ*_{0} is the potential difference applied at the surface and is the inward normal derivative of the potential to the surface.

Using equations (2.5) and (2.6) and the fundamental singularity at the boundary [16–18] (R. Kant, M. B. Singh, unpublished data. 2012 Generalization of linearized Gouy–Chapman–Stern model of electric double layer for nanostructured and porous electrodes: deterministic and stochastic morphology, http://arXiv.org/abs/1202.5827), the differential capacitance density is
2.9The terms in equation (2.9) can be looked upon as one-, two- and three-scattering terms as
2.10
2.11
and
2.12Equation (2.10) may be expanded through local surface coordinates *α*, *β* and *γ*. For a weakly curved surface where TF screening length is much smaller than the smallest scale of curvature the scattering kernel ∂*G*_{0}(*β*^{+},*α*)/∂*n*_{α} is expressed through a local coordinate system [16,17] with the *z*-axis being parallel to inward normal vector *n*_{α} and a tangent plane on which projection is made. The local equation of surface, *S*, in terms of curvature radii, *R*_{1}(*α*) and *R*_{2}(*α*): is introduced to equation (2.12) through surface area element where *g*=1+(∇*z*_{α}(*x*,*y*))^{2}. Using ∂/∂*n*_{α}≡−∂/∂*z*, the kernel ∂*G*_{0}(*α*,*β*)/∂*n*_{α} under planar approximation [16] reads to first order as (∂*G*_{0}(*α*,*β*)/∂*n*_{α})=−(*z*/*ρ*)(∂*G*_{0}(*ρ*)/∂*ρ*), where *ρ*=|*α*′−*β*′|=(*x*^{2}+*y*^{2})^{1/2} is the distance in tangent plane, the Green's function in the tangent plane. Now, we can rewrite the one-scattering integral equation (2.10) as
2.13which is further simplified using the angular averages
2.14and
2.15Substituting equations (2.15) to (2.13) and integrating over *ρ*, we finally get the one-scattering term as
2.16where . Similarly on iteration in equations (2.11) and (2.12), two- and third-scattering terms are obtained as
2.17Now substituting equations (2.16) and (2.17) to (2.9), after simplification the capacitance density at position *α* is
2.18where *c*_{M}=*ϵ*_{0}*ϵ*_{m}*κ*_{TF} is the planar surface electronic capacitance and . The local quantities *H*_{α}=(*k*_{1}+*k*_{2})/2 and *K*_{α}=*k*_{1}*k*_{2} are, respectively, the mean and Gaussian curvature at the *α*th point on *Γ*_{m} and related to the local surface shape element (see equation (2.1)). Here, and is called the local asphericity parameter. It determines the local curved surface deviation from a spherical shape element. Equation (2.18) reduces to the planar capacitance *c*_{M} [12,15] when *H*=0 and *K*=0. Equation (2.18) has three contributing terms. The first term is dependent on the electronic properties of the material (*ϵ*_{m},*D*_{F},*E*_{f}), the second term is purely geometry-dependent (through *H*) and the third term represents the coupling between geometrical (*H*,*K*) and electronic properties of the material.

Equation (2.18) is the expression of electronic capacitance applicable to nanostructure of any shape and size. It may be written as *c*_{TF}(*H*_{α},*K*_{α})=*ϵ*_{0}*ϵ*_{m}/*l**_{TF} or , where effective screening length and dielectric constant *ϵ**_{m}=*ϵ*_{m}*Σ*_{c}. Hence, electronic screening length and dielectric constant may be interpreted as geometric shape- and size-dependent quantities. Equation (2.18) relates the electronic capacitance to the surface shape through the mean and Gaussian curvature.

Now, we take account of the relatively slow decay of the electronic density at the metal surface that occurs at a small distance (about fraction of nanometres) because of the small mass of the electron called as ‘electronic spillover’ [13,14,19,20] at the metal surface. The spillover of electron is found to affect the capacitance of electric double layer in monoatomic nanowires [21], cylindrical carbon nanotube capacitor [22], adsorption, field emission and charge transfer reaction [23] in nanoscale electrode. The excess charge owing to spillover is located in front of the electrode (jellium) surface [13,14]. The effective electron boundary from the electrode surface is shifted in a fraction of electronic screening length. Typical value of spill length in parallel-plate capacitor of a pair of graphite sheets is approximately 0.1 nm [22]. In order to calculate the electronic capacitance at the spillover plane [13,14,20,24], we have to take the modified mean *H*′ and Gaussian *K*′ curvatures which are, respectively, [25,26] (see the appendix for derivation of curvatures)
2.19where *δ* is the electron spillover distance. For small *δ*, we have *H*′≈*H* and *K*′≈*K*. Now the electronic capacitance of arbitrary nanostructured material with spillover correction is
2.20

## 3. Results and discussion

### (a) Arbitrary shape

In figure 2, we show the capacitance of four regions: I, II, III and IV corresponding to different geometrical shape space regions as in figure 1. Region I represents capacitance density variation for elliptic concave region (cavity surface shape element but the electronic charge is on the convex side and both *R*_{1} and *R*_{2} are ‘+’). This situation corresponds to a charged nanorod in which the electronic charge is inside the material. Region II represents a shape element which is saddle-like where the electronic charge is inside the material and signs of *R*_{1} and *R*_{2} in shape space are ‘−’ and ‘+’, respectively. Region III represents a shape element in which the surface is cap-like in shape and where the electronic charge is on the concave side where both signs of *R*_{1} and *R*_{2} are ‘−’ in shape space. Region IV represents a shape element in which the surface is saddle-like and corresponds to an opposite situation as in region II. Here, in this case, the signs of *R*_{1} and *R*_{2} in shape space are ‘+’ and ‘−’, respectively. All the four quadrants are separated by a region (blue strip) representing the breakdown owing to singular curvatures or unphysical prediction of the theory, which can be corrected with induction of the higher-order terms in curvature. Maximum enhancement in capacitance is shown in region III corresponding to elliptic convex, and an enhancement of it is usually larger than the planar electronic capacitance value. The other regions *viz.*, I, II and IV show an overall lowering in the magnitude of capacitance. This finding clearly suggests that electronic capacitance will relatively be larger in nanoconfined system where the electronic screening is on the convex side of the materials.

### (b) Modulated nanostructures

In figure 3, we show a contour plot of capacitance for a two-dimensional surface with modulation where *h* is the amplitude and *L* the distance between two peaks of modulations. Using curvatures obtained through the Monge formula (given in the appendix), the model predicts the capacitance density (calculated using equation (2.20)) localization in certain regions of the shape space. There are regions of high (enhanced) and low (reduced) electronic capacitance. The contour label shows the amount of capacitance enhanced/reduced w.r.t. planar surface whose value is 1. The capacitance is preferentially enhanced where the surface curvature is maximum. The symmetry of capacitance density localization in the contour plots of enhanced and reduced regions are quite different.

Figure 4*a* shows corrugated cylindrical nanorod and (*b*) a nanopore with undulation, whose local radius *r* may be represented as a cosine function of the axial coordinate *z* as . The parameter *h* is the amplitude, *r*_{0} the mean size of the rod/pore and *k* the period of surface modulation. The corresponding parametric equations are: , and *z*=*z* where −*π*≤*θ*≤*π*. The maximum and minimum (neck) size of a particle or a pore is given by (*r*_{0}+*h*) and (*r*_{0}−*h*), respectively. Figure 4*c* shows normalized electronic capacitance *c** of nanostructures (a) (represented as solid lines) and (b) (dashed lines). The plot shows that local capacitance is localized and enhanced in the neck region in (a), where the curvature is highest and negative. This contributes positively to electronic capacitance, whereas the capacitance is minimum at the diameter of the particle as the curvature is least and positive. This contributes negatively to electronic capacitance. The situation is quite opposite in the case of (b), where capacitance is reduced at the neck region (where curvature is positive) and enhanced at the diameter of a pore (where curvature is negative).

### (c) Gaussian nanopits and nanohorns

Figure 5 shows the Gaussian-shaped (*a*) nanopits and (*c*) nanohorn whose surface is characterized by where *h* is the amplitude (height/depth), *σ* the variance (width), *μ* the mean size of horn/pits, and ‘+’ is opted for nanopits and ‘−’ for nanohorn. The respective contour plots of capacitance shown in (*b*) and (*d*) show that the local concavity (in pit) enhances, whereas local convexity (in horn) resulted in a decrease in capacitance. Hence, we conclude that surfaces with the same area can have different capacitances.

Figure 6 shows the variation of capacitance as the aspect ratio of the nanopits is changed. As we increased the depth of the nanopits, we observed a gradual change from the three-dimensional ‘Mexican hat’-shaped electronic capacitance to a ring region of low capacitance density followed by a region of capacitance peak enclosed by two concentric regions of low capacity. Thus, geometric features in nanopits may lead to localization and oscillation of capacitance.

### (d) Idealized geometries

Now the electronic capacitance of simple idealized geometries (obtained from equation (2.20)) at the spillover plane is
3.1where (*a*,*b*) is a pair of dimensionless numbers depending on the type of geometry concerned. The effective position of the material surface owing to spillover correction [13,14,20,24] is estimated through attuned radius *r*_{a}=(*r*±*δ*), where + and − signs are assigned when the electron spillover is on convex and concave sides of the material, respectively. The values of (*a*,*b*) of various geometries are as follows: plane (0,0); sphere (−1,0); tube ; spherical cavity (1,0) and rod (see the appendix for a list of corrected curvature).

Equation (3.1) shows that the electronic capacitance has three contributing terms. The first term represents contribution to electronic capacitance from the material. The second term is purely the geometry-dependent self-capacitance and the third term is the coupling between the geometry and electronic properties of material. Equation (3.1) for a spherical electrode is written as
3.2under the condition *r*_{a}>*l*_{TF}. The first term in equation (3.2) is the TF capacitance. The second term in equation (3.2) corresponds to self-capacitance of an isolated sphere in classical electrostatics [27]. Similarly for a spherical cavity electrode the electronic capacitance is given as
3.3For a large TF screening length, equation (3.3) reduces to the self-capacitance of a spherical conducting surface surrounded by a medium of dielectric *ϵ*_{m}.

Figure 7*a* shows the capacitance of idealized geometries: sphere (A), rod (B), sheet (C), tube (D) and cavity (E). The plot shows a decrease in capacitance as the electrode size decreases for convex geometries (sphere and rod). By contrast, the capacitance increases with a decrease in the electrode size for concave geometries (tube and cavity) while for a planar sheet-like electrode it is constant. The geometric (second term in equation (3.1)) factor contributes differently for different geometries. It contributes positively in a pore/cavity because *r*_{a}<*r* resulting in enhancement of capacitance, whereas its contribution is negative in the case of a rod/sphere as *r*_{a}>*r* and results in a reduction of capacitance. An interesting feature is the emergence of a capacitance maximum for concave geometries (tube). The maximum of capacitance is located at the pore size 2*r*=*l*_{TF}+2*δ* and show approximately 50% enhancement for a typical model system (see figure 7*b*). The cavity geometry shows a large singularity in capacitance at pore size *r*=*δ* and rapid fall in capacitance pore size *r*<*δ*. The effective distance is negative [13,14,29] when *r*<*δ* in the case of concave (tube and cavity) geometry but this is not so in convex (rod and sphere) geometry. In such cases, the geometric capacitance (second term in equation (3.1)) of the system (tube and cavity) has negative contribution to the total capacitance resulting in the fall of capacitance at pore size *r*<*δ*. This is purely a geometric effect of confinement of electronic spillover charge in nanoconfined system. The capacitance maximum is a function of electronic screening length and the density of states of the surface as shown in figure 7*b*. The capacitance maximum position changes towards smaller pore size as the density of states is increased. Similar capacitance maximum is seen in experimental capacitance data of a nanoporous supercapacitor [7,8] but its origin is still unknown.

## 4. Conclusions

A generalized TF theory for the electronic capacitance for complex shape surfaces is developed. A major conclusion is that in general an elliptic convex region in shape space will show enhanced electronic capacitance with a maximum at an optimal sub-nanometre scale morphology and all other shapes lead to depletion in the capacitance. Our model predicts anomalous capacitance maximum in tubular nanopore, capacitance localization in a modulated surface and capacitance oscillation in high aspect ratio nanopits. Since the size-dependent electronic effects also contribute to the surface stress, which induce dimensional changes in nanostructured materials [30], we hope that our results will inspire future studies in nanoscale actuation.

## Appendix A. Supporting materials

**(a) Curvature of a curve**

The local curvature *κ*(*x*) of a curve *ζ*(*x*) is given by
A1where *ζ*_{x}=∂*ζ*/∂*x* and *ζ*_{xx}=∂^{2}*ζ*/∂*x*^{2} are, respectively, the first and second derivatives of a local curve profile. But for a general arbitrary surface, the curvatures mean *H* and Gaussian *K* may be obtained from the Monge formula as follows:
A2and
A3where the partial derivative *ζ*_{xx}=∂^{2}*ζ*/∂*x*^{2}, etc.

**(b) Curvature and area correction of parallel surfaces at spillover plane [25,26] (Sterner's formula)**

In order to obtain the curvature and area correction at the spillover surface, let us consider *Γ*_{m} a convex body in the three-dimensional Euclidean space *R*^{3}, with regular boundary ∂*Γ*_{m}∈*C*^{2} and principal radii of curvature *R*_{1} and *R*_{2}. The surface area *A* of *Γ*_{m} is given by
A4since the area element d*S* on ∂*Γ*_{m} is related with the area element d*σ* of the spherical image of *Γ*_{m} under the Gaussian map by d*S*=*R*_{1}*R*_{2}d*σ*. The convex body *Γ*_{δ} of *Γ*_{m} is defined as the set of points with distances to *Γ*_{m} smaller than *δ* (electron spillover distance), i.e. The radii of curvature of ∂*Γ*_{δ} are *R*_{1}−*δ* and *R*_{2}−*δ*. Then the local specific area d*S*′(*Γ*_{m},*δ*) of the parallel body is
A5Now let us transform the parallel surface to (*δ*−*t*) the area of transform parallel surface d*S*′(*Γ*_{m},*δ*−*t*) is
A6The relative area increase is
A7where
A8are the modified mean and Gaussian curvatures at the spillover surface. Now, we can calculate the corrected curvatures of idealized nanoscale geometry using equation (A8). Table 1 shows list of curvatures used in calculating the electronic capacitance of idealized shape shown in figure 7*a*.

- Received March 8, 2013.
- Accepted July 24, 2013.

- © 2013 The Author(s) Published by the Royal Society. All rights reserved.