Electronic Screening Model for the Capacitance and Dielectric of Arbitrary Shaped Nanostructured Materials

Transcription

1 Chapter 2 Electronic Screening Model for the Capacitance and Dielectric of Arbitrary Shaped Nanostructured Materials There s plenty of room at the bottom - Richard P. Feynman 73

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3 Abstract The influence of geometry/shape and size of electrode is known to cause anomalous capacitance, dielectric constant and screening length but theoretically is less understood. A theory of electronic screening in metal or semi-conducting nanostructures based on Thomas-Fermi (TF) approximation is presented. The model developed is generalized for nanostructures with arbitrary shape and topology. To demonstrate the flexibility of our approach we show how different nanostructure modeled as modulated surface, nanopits, nanohorns and idealized geometries may influence the electronic charge storing capacity. We show how the local shape and morphology influence screening capacitance, dielectric constant and screening length. Quantum electron spillover distance correction is made through theory of parallel surface. Analysis shows capacitance localization in modulated surfaces and oscillation in nanopits showing strong dependence of local surface geometry. We argue that the shape/size of nanostructures and quantum mechanical spillover of electron are fundamental factor in observing several anomalous observation in capacitance, dielectric and screening length. Finally comparison with capacitance data of nanoporous carbon show reasonable agreement indicating the electronic contribution in deciding the capacitance maximum and the rapid fall in capacitance in smaller pore size. 75

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5 1 Introduction The electronic screening of the metal is an important factor that influences the interfacial properties and various phenomena occurring at the surface [1]. Especially, at nano scale [2], the electronic screening plays a very important role in deciding the value of capacitance in high dielectric constant capacitor [3], metal oxide-field effect transistor (MOSFET), electric double layer transistors (EDLTs) [4], quantum dots, nanocapacitors [5] and small conductors [2, 6]. It is well known that nano scale properties like surface energy density, field emission, adhesion force, contact resistance depends on the size and shape of the nano objects [7, 8, 9]. Moreover, the importance of the electronic screening effects have been revealed in the capacitance of semiconductor/electrolyte hetero-junction, electric double layer [10, 11], advanced carbon nano structures like, carbide derived carbons (CDC) [12], activated carbons [13], carbon nano tubes [14] and graphene based nano scale capacitors [15]. Recently, it was found that a very large capacitance enhancement can be obtained through negative electronic compressibility in semiconductor two dimensional (2D) electronic system [16]. Also the observation of giant dielectric in an assembly of ultrafine Ag particles is also attributed to the result of quantum confinement of electronic wave-functions attaining a percolating configuration [17]. Similar giant capacitance or dielectric constant increase are seen in insulatorconductor composite [18, 19] at the percolation threshold [20, 21]. The variation of capacitance with the size of electrode is an important feature in nanoscale energy storage materials [12, 15, 22]. The electrode materials of interest include transition metal oxides nano composite, with two dimensional (2D) electron gases at interface, carbon based (activated carbon and carbide derived carbon) and graphene. The electronic screening has a long history with a wide range of theoretical model [23] (see Fig. 2.1). Semi classical approach based on the Thomas- Fermi (TF) model, is usually applied successfully to nano scale objects like carbon nano tubes, fullerene carbon pea-pods, metallic clusters [25], nanoparticles 77

6 Figure 2.1: Schematic picture of electronic screening in (a) metal and (b) semiconductor. The models are adapted from [24]. Here κ 1 T F is the electronic Thomas- Fermi screening length, r s the spillover distance of electron and r H is the compact Helmholtz layer thickness. The metal surface is at potential ϕ 0 and fall off rapidly to zero as distance increase. In case of semiconductor the space charge region is inside the material and the potential drop is ϕ = ϕ C ϕ F where ϕ C and ϕ F are the conduction band and Fermi level potential. [26] and semiconductor nanocrystals [27, 28]. It is known that variation of electron density and spillover play affect the capacitance of electric double layer in monoatomic nanowires [29], adsorption, field emission and charge transfer reaction [30] in nanoscale electrode. However, the electronic quantities like capacitance or dielectric is found to be affected by nanoscale morphology of the material. The issue of screening in metallic nanostructures or metal-like nanoobjects and the role of nanoscale morphology is still least understood. In this chapter, we focus on the influence of geometry/shape of the metal surface at nanoscale on the electronic screening. The understanding of electronic screening phenomena at nanoscale will help us to understanding how the capacitance and dielectric constant is affected by electronic properties and nanoscale 78

7 morphology of materials. The chapter is organized as follows: first we understand the local shape of any arbitrary surface using the concepts from differential geometry in relation to the two principal radii of the curvature of surface. Next we develop a phenomenological theory of electronic capacitance of the metal nanostructured electrodes. The theory is based on Thomas-Fermi approximation and generalized for an arbitrary surface geometry (morphologies) and topologies. Further correction of spillover distance of electron near the electrode surface is made through the theory of parallel surface and the detailed analysis of surface with modulations, idealized geometries, nanopits and nanohorns is carried out. A comparison with experimental data of nanoporous carbon electrode in ionic liquid are analyzed. Finally, the conclusions of the work are summarized. 2 Local Shape Element: Geometrical Concepts Our theoretical investigations for the surface shape/geometry is based on the notion of the surface characterization by the shape element using the concepts of differential geometry. The local surface shape at α-th point of the surface may approximated by z α = 1 ( x 2 k 1 (α) + y 2 k 2 (α) ) (2.0.1) 2 where k 1 (α) = 1/R 1 and k 2 (α) = 1/R 2 are the curvatures at a point α and R 1 and R 2 are the principal radii of curvature. The various shape configurations may be generated by varying the curvatures through k 1 and k 2. Usually a surface may be characterized by the two curvatures: mean (H) and Gaussian (K). The mean curvature is the average of the two principal curvatures (defined as H = 1/2(k 1 +k 2 )) and the Gaussian curvature is the product of the two principal radii of the curvatures (defined as K = k 1 k 2 ). Figure 2.2 illustrates the various local shapes as a function of curvatures k 1 and k 2. The shape element of the surface may be used to construct well defined geometries like sheet/plane, spherical, spheroidal, cylindrical, conical etc. The shape element can describe the geometry and morphology 79

8 of advanced nanocarbon like graphene (planar or modulated), fullerenes (spherical), hypo-/hyperfullerenes (spheroidal), nanotubes and nanorods (cylindrical), nanocones (conical), nanorings (toroidal) etc. It can also describe disorder in nanoporous carbons, activated carbons and carbide derived carbons (CDCs) by taking into account the fluctuation in morphology. Figure 2.2: Schematic shape space diagram as a function of curvatures k 1 and k 2. (1) First quadrant: cup (k 1 = k 2 ). (2) Second quadrant: saddle ( k 1 = k 2 ). (3) Third quadrant: cap ( k 1 = k 2 ). (4) Fourth quadrant: saddle (k 1 = k 2 ). The orange and brown dotted lines show the regions where k 1 = k 2 are equal. Orange with opposite sign and brown with same signs. On the axes are the parabolic surfaces. At the center where k 1 = k 2 = 0 is the plane. 3 Thomas-Fermi Screening Capacitance of Arbitrary Geometries We begin our investigation by taking a shape element Γ m as a system of collection of electron whose surface local number density ρ( r) under Thomas-Fermi (TF) 80

9 approximation [31, 32] (the derivation is shown in the appendix) as: ρ( r) = (3π 2 ) 1 {(2m/ 2 )(E f + eϕ( r))} 3/2 (3.0.1) where = h/2π and h is the Planck s constant, E f is the Fermi energy, m is the mass and e is the electronic charge of electron and ϕ( r) is the electrostatic potential at r. Now the electrostatic potential ϕ( r) arising when the electron density departs from the uniformity is given by the Poisson s equation as: where ρ 0 2 ϕ = 4πe ϵ m (ρ ρ 0 ) (3.0.2) is the uniform electron density of the surface when the electrostatic potential at surface is ϕ 0. Expanding it for small values of ϕ compared with E f, we have the linearized Thomas-Fermi (LTF) equation as: 2 ϕ = κ 2 T F ϕ (3.0.3) where κ T F = (6πn 0 e 2 /ϵ m E f ) 1/2 is the inverse TF screening length [33] where ϵ m is the background dielectric constant, E f is the Fermi energy of electrons and n 0 = (1/3π 2 )(2mE f / 2 ) 3/2 = (2/3)D F E f is the average concentration of electrons and D F is the density of states. The boundary conditions (1) interfacial potential w.r.t is ϕ = ϕ 0, and (2) the the bulk potential ϕ = 0. Equation is applicable for any arbitrary shape and size of the metal electrode. 81

10 4 Electronic Capacitance of Nanoscale Electrodes and Electron Spillover Correction The electronic capacitance density (c T F ) at α-th point of surface is (see the appendix for detailed derivation) given as: [ c E (H α, K α ) = c M 1 1 H α 1 ] (H κ T F 2κ 2 α 2 K α ) + T F (4.0.1) where the local quantities H α = (1/2)(1/R 1 (α)+1/r 2 (α)) and K α = 1/R 1 (α)r 2 (α) are the mean and Gaussian curvatures at α-th point on surface and R 1 and R 2 are the two principal radii of curvatures. Here (Hα 2 K α ) 0 and is called the local asphericity parameter which determines the local curved surface deviation from the spherical surface. The local screening capacitance of arbitrary curved surface given by Eq , has three contributing terms. The first term is dependent on the material properties (viz. ϵ m, D F ), the second term is dependent on the geometry (through H) and the third term represents the coupling between geometrical (H, K) and material properties (viz. ϵ m, D F ). Equation is our main result and reduces to the planar metal capacitance c M [36, 33] when H = 0 and K = 0 as: Here l T F = κ 1 T F c M = ϵ mϵ 0 l T F (4.0.2) is the TF screening length. For a typical system with electronic density of state D F = cm 3 ev 1 having material dielectric ϵ m = 3.28, the TF screening length is l T F = 0.3 nm and the screening capacitance c M = 9.6 µf/cm 2. Now we take the account of the relatively slow decay of the electronic density at the metal surface. This occurs at a small distance (about fraction of nanometers) due to the small mass of the electron called as electronic spillover at the metal surface. The spillover of electron is found to affect the capacitance in the nanoscale 82

11 quantum systems like cylindrical carbon nanotube capacitor [38]. The excess charge due to the spillover is located in front of the jellium surface [37]. The effective electron boundary from the electrode surface or the penetration length may be approximated by x = ϵ m /κ 0 T F. Here (κ0 T F ) 1 is the TF screening length of free electron gas and ϵ m is the background dielectric constant. The background dielectric constant may be related to the optical refractive index n as ϵ m n 2. The electric field dependent spillover length can also be estimated [39, 40, 41]. The typical value of spillover length in parallel-plate capacitor of a pair of graphite sheet is 0.1 nm [38]. In order to calculate the electronic capacitance at the spillover plane, we have to take modified curvature of the spillover surface parallel to the original surface. The relation of the area element ds, the mean curvature H and the Gaussian curvature K on the parallel surface compared with the original surface with area ds and the mean and Gaussian curvatures H and K is given (as derived in Chapter 1 appendix) as: ds = ds(1 2Hr s + Krs) 2 (4.0.3) H = H Kr s 1 2Hr s + Krs 2 (4.0.4) and K = K 1 2Hr s + Krs 2 (4.0.5) When r s 0, we have H = H and K = K. Substituting H and K in Eq , we calculate the electronic capacitance with spillover correction (c s E ) as: [ c s E = c M 1 1 H α 1 ] (H κ T F 2κ 2 α 2 K α) + T F (4.0.6) Equation is of general in nature and applicable to any arbitrary geometry. 83

12 5 Models for Surface 5.1 Arbitrary Shaped Surface Figure 2.3 (a) shows the dependence of the capacitance c = c s E /c M with respect to the surface shape variation when the two principal radii of the curvatures are changed. One can see from the plots (a & b) that there are four different capacitance variations in the four quadrants (marked as I, II, III and IV respectively). I represents capacitance density variation for the surface shape element in which the surface is like cavity but the electronic charge is on the concave side and both R 1 and R 2 is positive. This situation corresponds to a charged nanorod in which the electronic charge is inside of the surface (note the calculation of capacitance are in shape space where the electronic charge variation is present and the curvature are opted accordingly). II represents for a shape element which is saddle like in shape where the electronic charge is inside the material. The signs of R 1 and R 2 in shape space is negative and positive, respectively. III represents for a shape element in which the surface is cap like in shape and where the electronic charge is on the concave side. The sign of R 1 and R 2 is negative in shape space. IV represents for a shape element in which the surface is saddle like in shape and corresponds to an opposite situation as in case of II. In this case, the signs of R 1 and R 2 in shape space is positive and negative, respectively. All the four quadrants are separated by the region (blue strip in Fig 2.3(a) and white stripe in Fig. 2.3(b)) representing breakdown due to the singular curvatures or unphysical prediction of the theory. This singular curvatures can be corrected with induction of the higher order terms in the curvature. 5.2 Surface with Modulations So far we have considered surface with no surface modulation, disorder or roughness. But realistic surfaces like graphite shows fluctuation (may be translation 84

13 Figure 2.3: (a) 3D plots of normalized capacitance with spillover correction rs = lt F /5. The plots were generated in units of lt F. (b) contour plots. Here R1 = R1 /lt F and R2 = R2 /lt F are scaled principal radii of curvature. The labels in contour plot represent the relative value of capacitance w.r.t plane whose value is one. 85

14 Figure 2.4: (a) A model of the electrode with surface modulation. (b) Contour plot of local capacitance density of a modulated surface. The contour label shows the magnitude of capacitance enhanced w. r. t planar capacitance cm. The plots were generated for h = 1 and L = 7 in units of lt F. Here x = x/lt F and y = y/lt F are the dimensionless coordinates. 86

15 or rotational), turbo-static disorder and roughness (heterogeneity or polycrystallinity). Surfaces like nanoscrolls, graphene sheets, nanoribbons have fluctuations and distribution of surface curvatures. In order to understand the effect of surface morphological fluctuation on the electronic capacitance, we take a surface with modulation as: ζ(x, y) = hcos(2πx/l)cos(2πy/l) (5.2.1) where h is the amplitude of the surface modulation, L is the separation between the two peaks. We derived the curvatures: mean and Gaussian by using the Monge formula (given in the appendix) and calculated the capacitance. Figure 2.4 (b) represents a contour plot of the local capacitance density of a fluctuating surface profile. The plot shows the capacitance density is localized in certain regions of the shape space. There are regions of high (enhanced) and low (reduced) electronic charges. This is an interesting result showing that fluctuation in surface can induced localization of the electronic charge. The charge is preferentially enhanced where the surface curvature is maximum. The symmetry of charge density localization in the contour plots of enhanced and reduced regions are quite different. 5.3 Gaussian Morphology In order to understand the effect of surface curvature distribution, we consider a model of Gaussian distribution function given by ζ(x, y) = i ( ) h ((x iµ) + (y jµ)) Exp 2πσ 2 2σ 2 j (5.3.1) where h is the amplitude (corresponding to height/depth of nanohorn/nanopits), σ the variance (corresponding to the width of nanohorn/nanopits) and µ is the mean (corresponding to separation between the two horn/pits). Figure 2.5 shows 87

16 Figure 2.5: Model of the nanoelectrodes (a) array of nanohorns (b) array of nanopits and (c) array of fused nanohorns. 88

17 a surface with (a) array of nanohorns, (b) array of nanopits and (c) fused array of nanohorns Nanopits Electrodes Figure 2.6 shows the contour plots of nanopits for four different cases of surface: (a) single isolated nanopit, (b) two fused nanopits, (c) four fused nanopits and (d) array on nine fused nanopits. Figure 2.6(a) shows contour plot of local capacitance density for single isolated nanopits. The plot shows clearly that, capacitance is enhanced maximum corresponding to the surface where the curvature is maximum as like in the case of modulated surface. The enhancement is 1.6 times that of planar surface (whose value is one in contour plot). The symmetry of the contour is similar to that of the local surface. As we see there are regions where contour label are less than 1 indicating reduced capacitance due to the surface geometry. These regions are locally concave and of reduced capacitance. Figure 2.6(b) shows the case where two nanopits are fused. The plot clearly shows the saddle like surface in between the two nanopits where they are fused. The capacitance density clearly shows the capacitance localization and the symmetry of contour is changed as the local topography of the surface is varied. Figure 2.6(c) shows the case when four nanopits are fused. Here also the capacitance is localized. The enhancement is seen where the surface curvature is maximum in nanopits. There is one to one correspondence of the surface feature to the contour of capacitance density plot. Figure 2.6 (d) shows an array of fused nanopits Effect of Concavity and Convexity of Local Geometry Figure 2.7 is the contour plots and shows the effect of local concavity and convexity of morphology and its influence on the electronic capacitance. The nanopits (a) shows higher capacitance compared to the nanohorns (b). The labels in the contour plots indicate the relative enhancement or reduction in the electronic capacitance w. r. t. planar geometry which has a value of 1 in the plot. Figure

18 Figure 2.6: Contour plots of local capacitance density of: (a) single (b) two fused (c) four fused and (c) array of nanopits. The contour label show the magnitude of enhancement from the planar TF capacitance. The values for σ, µ, h for (a) single is 2, 5σ and 5σ and for (b) two fused (c) four fused (d) array is 2, 2.5σ and 5σ. The plots were generated in units of l T F. 90

19 (a) & (b) clearly show that, the local concavity and convexity have opposite effect on the electronic capacitance Capacitance Localization and Oscillation in Nanopits Figure 2.8 shows the variation of capacitance with the depth of nanopits. As we increase the depth of the nanopits, we observe a gradual change in the electronic capacitance in the shape space. A capacitance peak emerge when h = 9σ. When the depth of nanopit is large (h = 13σ ) we see localization and bifurcation of capacitance. At the region of highest curvature there is a capacitance peak followed by a region of low capacitance density forming a ring. After that there is a region of enhanced capacitance forming concentric rings. 5.4 Fused Nanoparticles and Modulated Pores Here we analyze the case of fused nanoparticles connected through necks and nanoporous electrodes with pore size modulation. The local radius r may be represented as a cosine function of the axial coordinate z as: r(z) = b + acos(kz). The parameters a is the amplitude, b is the mean size of particle/pore and k is the period (of surface modulation) and may be selected in such a way to obtain the shape and size of the surface required. The maximum size of the particle is given by (b + a) and the minimum size (at neck) of the fused particles is (b a). Obtaining the curvatures: mean and Gaussian (formula given in the appendix) we obtained normalized electronic capacitance c through Equation Figure 2.9 (c) shows the local capacitance density of nanostructures with modulations. Blue solid lines represent the case of fused nanoparticles and blue dashed lines correspond to nanostructures with corrugated cylindrical shaped pores. The plots show that, the local capacitance is localized and enhanced in the neck region in case of the fused nanoparticles. The localized and enhanced capacitance density and reduced capacitance density are characterized by the value of capacitance greater than or less than one (the value of capacitance for perfectly smooth 91

20 Figure 2.7: Contour plot of capacitance of (a) nanopit and (b) nanohorn. The plots were generated for σ = 2, µ = 2σ, h = 5σ and l T F = 1 nm. 92

21 Figure 2.8: 3D plot of capacitance enhancements in nanopits and localization as the depth of nanopits is varied. The plots were generated using σ = 2, a = 2σ and the depth of nanopits h is varied as (a) h = 5σ, (b) h = 9σ and (c) h = 13σ. The plots are in scales of lt F. 93

22 Figure 2.9: (a) Model of the electrode with fused nanoparticles. (b) Model of pore with modulation. (c) local capacitance density of modulated corrugated cylindrical surface whose radius of varies periodically. The plots were generated for b = 5, k = 0.9 and a = 2k. The solid and dashed lines represent capacitance of the fused particle and modulated pore respectively. The length scales are operated in terms of electronic screening length l T F. 94

23 cylindrical surface with no modulation). In the neck region, the capacitance is maximum as the curvature is highest and negative. Where as the capacitance is minimum at the diameter of the particle. But in the case of porous system with modulation we see capacitance is reduced at the neck region and enhanced at the diameter of the pore. The situation is opposite to that of the fused particles. 5.5 Fused Particles and Modulated Pores and With Spillover Correction c * zhnml Figure 2.10: Comparison of local capacitance of nano fused (blue lines) particles and nanopores (red lines) with modulation in presence of spillover of electron. The plots were generated for b = 5, k = 0.9 and a = 1.5k and spillover correction distance r s = l T F /5. Figure 2.10 shows comparative plots of local capacitance of fused nanoparticles and nanopore with modulation. The blue line represents local capacitance density of fused nanoparticles (with corrugated like modulation in cylinder). The blue dashed lines are with the spillover correction. The red lines represent the local capacitance of nanopore with the modulation. The red dashed lines correspond to the capacitance with spillover correction. The spillover correction enhance the de- 95

24 pendence of the electronic capacitance on the local curvature of the surface and the shows reduction in electronic capacitance either at highest (b+a) and lowest (b-a) rod or pore size. The capacitance localization at the position of highest curvature (convex local geometry) shows largest enhancement. The spillover correction leads reduced capacitance in the fused particles and nanopores with modulation. 5.6 Idealized Geometries In this section, we focus on idealized nanosize structure of the surface. We assume that the thickness of electron gas due to the electronic spillover [10, 11, 37] from the edge of the electrode surface r s is approximately a fraction of l T F. The effective position of the material surface due to spillover correction is estimated through attuned radius r a = (r ± r S ), where + and sign is assigned when the electron spillover is in convex and concave side of the materials respectively and r a = r r S for spherical cavity and cylindrical rod and r a = r + r S for sphere and cylindrical rod. Taking the curvature of metal surface at the spillover plane (after spillover distance correction, see Table 2.1), we may write the TF capacitance from Eq as: c E = c M ( 1 + a l T F r a + b l2 T F r 2 a ) (5.6.1) where (a, b) are the pair of dimensionless numbers depending on the type of geometry concerned. The values of (a,b) of plane is (0, 0); sphere ( 1, 0); tube (1/2, 1/8); spherical cavity (1, 0); and for rod is ( 1/2, 1/8). The first term in Eq is the electronic space charge capacitance (material contribution). The second term is the geometric capacitance corresponding to shape/size of the material and the third term is coupling capacitance which depends on both shape/size and the relative ratio of the two lengths i.e. the space charge screening length and the size of electrode. Figure 2.11 shows the plot of electronic capacitance of nanoscale idealized 96

25 Table 2.1: List of mean (H) and Gaussian (K) curvatures of idealized geometries along with corrected mean (H ) and Gaussian (K ) curvatures* geometry H K H K cylindrical nanorod 1/2r 0 1/2(r r s ) 0 cylindrical nanopore 1/2r 0 1/2(r + r s ) 0 nanocavity 1/r 1/r 2 1/(r + r s ) 1/(r + r s ) 2 nanosphere 1/r 1/r 2 1/(r r s ) 1/(r r s ) 2 *Note the curvatures H and K are in electronic charge layer inside the material and H and K are at the spillover surface outside the material. Figure 2.11: TF capacitance as a function of shape and size of electrode. The plots were generated by using the following parameters: ϵ 0 = F/cm, ϵ m = 5 and D F = cm 3 /ev 1. A(sphere), B(rod), C(planar sheet), D( tube), E(spherical cavity). The plots were generated with a spillover correction x = l T F /4. 97

26 geometries. The lines marked as A, B, C, D and E corresponds to sphere, rod, sheet, tube and cavity nanoscale geometries, respectively. The TF capacitance decreases as the electrode size decreases for the convex geometries (sphere and rod) whereas the TF capacitance increases with decrease in the electrode size for concave geometries (tube and cavity). The results in Fig suggest that electrode shape/size is a very important factor affecting the electrostatic TF capacitance of metal nanostructures. An important feature is the emergence of capacitance maximum when r = r s for concave geometries (tube and cavity). The capacitance show steep fall from maximum at smaller pore size i.e. r < r s. Note that the effective distance is negative when r < r s in the case of concave (tube and cavity) geometry but this is not in case of the convex (rod and sphere) geometry. Such negative distance can occur in nanoscale quantum system [42, 37]. 6 Effect of Spillover and Screening Length Variation To understand clearly the importance of spillover of electronic density and the density of states through l T F, we confine our analysis to a case of a cylindrical pore model. Figure 2.12(a) shows the effect of variation of spillover distance. The spillover correction have three different effects on capacitance of the pore. It shows enhanced capacitance when the pore size is decreased upto 2r = l T F + 2r s where a maximum in capacitance is obtained. Smaller pore size results in rapid fall in capacitance. When pore size is larger (2r > l T F + 2r s ), it shows enhanced capacitance effect where as smaller (2r < l T F + 2r s ) shows reduced capacitance. The magnitude of enhancement of the maximum is not affected by the variation of spillover distance. The dependence of capacitance on the spillover of electron density is reported in double-walled carbon nanotube capacitor [38]. Figure 2.12(b) shows the plot of local capacitance density for different screening lengths. The plots show that increasing the screening length results in higher 98

27 HaL HbL c * c * 1.2 r S 1.2 l TF rHnmL rHnmL Figure 2.12: Local electronic capacitance density as a function of shape and size of electrode. (a) Variation of spillover distance. The plots were generated for R S = l T F /4, l T F /3, l T F /2 where l T F = 0.7 nm. (b) Variation of TF screening length. The plots were generated for l T F (nm) = 0.1, 0.3, 0.7 with a spillover correction. capacitance with simultaneous change in the location of maximum. The enhancement is more rapid than in the case of spillover variation and no change in the magnitude of the maximum is observed. 7 Shape and Size Dependent Dielectric Constant and Effective Screening Length As said before giant capacitance or dielectric constant are also observed in metaloxides, nancomposite [18] materials near percolation threshold [20]. There are arguments that, geometrical effects lead to increase the dielectric constants [43, 44]. We may rewrite Eq as: c T F = (ϵ 0 ϵ eff /l T F ) where ϵ eff is the effective dielectric constant which depends on the morphology (through curvatures) as: ( ϵ eff = ϵ M 1 l ) T F 2 H l2 T F 2 (H2 K) (7.0.1) 99

28 The terms in Eq may be looked upon as shape anisotropy dependent quantities. The first term is the independent of the local geometry, the second term depends on the local mean curvature of the surface and the third term depends of the local asphericity (H 2 K) of the surface. Also we may also define Eq as c T F = (ϵ 0 ϵ M /l eff ) where ( l eff = l T F 1 l ) 1 T F 2 H l2 T F 2 (H2 K) (7.0.2) is effective screening length depending on morphology of the surface. Hence we may understand more clearly, the origin of dependence of shape and the size of nanostructure on the capacitance and dielectric through the effective screening length. This effective length also defines the relationship between the actual distance r and the edge of jellium electrode after spillover at attuned size r a = (r±r s ) The + and signs are taken for convex and concave geometries, respectively. It is important to note that r a < 0 when r s > r and r a > 0 when r s < r in concave geometry, while r a > 0 when r s < r and r s > r in convex geometry. 8 Capacitance of Electrostatic Nanocapacitors Recently, nanocapacitors with metal and thin films resulting in metal-films-metal configuration are usually designed for energy storage devices like supercapacitors [46]. Such interface formed at a metal involved oxides or dielectric. The capacitance c mi of such capacitor configuration may be written in series of dielectric capacitance (c die ) and the interface capacitance (c T F ) as: c 1 mi = c 1 T F + c 1 die = (ϵ 0 ϵ d /t) (1 + (ϵ d /ϵ m )(l eff /t)) 1 (8.0.1) where c die = (ϵ 0 ϵ d /t) and ϵ d is the dielectric constant with thickness of t. It reduces to the planar case when both H and K are zero. Similar results exits in 100

29 context of high dielectric constant capacitors [3], metal-films-metal capacitor [45] and nanocapacitors [5]. The thickness dependence of dielectric constant of thin films in metal/insulator (ceramic) may also arise because of the term (ϵ d /ϵ m )l eff in the denominator in Eq However, it will be significant for t (ϵ d /ϵ m )l eff. The correction due to field penetration into the metal and the electronic spillover correction is important for ultra-thin films and the dielectric layer dominates for thicker films. Moreover the dependence of the thickness of dielectric on the capacitance is due to the morphological features (H and K) of surface through l eff. 9 Comparison with Anomalous Capacitance Experimental Data Recently, Gogotsi and coworker have observed the anomalous increase in capacitance in nanoporous CDC electrodes having pore size less than 1 nm. Huang et al. [49] have suggested that the ions lines up along the pore axis forming an electric wire-in-cylinder capacitor (EWCC). Adopting the EWCC as a model for Helmholtz layer the specific capacitance may be written as: c H = ϵ H ϵ 0 /rln(r/r H ) where r is the cylindrical pore radius. Now the total capacitance c may be obtained as: c 1 = c 1 E + c 1 H (9.0.1) Figure 2.13 shows the comparison of theoretical prediction with experimental data. The capacitance data for Ti-CDC in (a) ionic liquid (EMI-TFSI) and (b) 1.5 M TEABF 4 in CH 3 CN were fitted using density of state value D f = cm 3 ev 1 and D F = cm 3 ev 1 and material dielectric constant ϵ m = 3.28 (typically of graphitic system [51]). The Helmholtz layer dielectric ϵ m = 1 and thickness r H = 0.35 nm (approx. equal to interlayer separation of two graphene layer in graphite). The fit values of density of states are in typical orders 101

30 15 15 chmfêcm 2 L 10 5 æ æ æ æ æ æ HaL æ chmfêcm 2 L 10 5 æ æ æ æ æ HbL æ æ Ti-CDCHEMI-TFSIL æ Ti-CDCH1.5 M TEABF 4 L rHnmL rHnmL Figure 2.13: Comparison of experimental capacitance data (a) 1-ethylmethylimidazoliumbis(trifluroromethanesulfonyl)imide ionic liquid (EMI-TFSI) (b) 1.5 M TEABF 4 in CH 3 CN. The data were taken from ref. [22] and [49]. as in activated carbon system (D f = ev 1 cm 3 ) [51, 13]. The possible contribution of diffuse electron density and spillover in capacitance of nanoporous carbon is suggested by Shim and Kim [47]. The theoretical plots show reasonable agreement with the nature of capacitive data. This shows that the peak value of capacitance results from the contribution of capacitance from both the Helmholtz layer and the electronic capacitance. This is very different from earlier proposed models [49, 47, 52] which have neglected the contribution from the electronic screening layer and the spillover of electron. The fall of capacitance on the left side of capacitance maximum is due to the electronic capacitance and shows the importance and the possible contribution of capacitance due to the spillover of electron. 10 Conclusions In conclusion, we have developed a theory for electronic screening capacitance based on the Thomas-Fermi approximation. The model is generalized for arbitrary 102

31 shape nanoscale surface morphologies and topologies. Electronic capacitance of various surfaces with modulation, nanopits and idealized geometries are calculated showing the usefulness of the developed theory in handling complex shape surfaces. Capacitance localization is found due to the geometry of the surface. Locally concave and convex surface are found to have different capacitance. The concave surface are found to have larger capacitance due to increase in the curvature. The model is able to fit the experimental capacitance data of micropore of nanoporous CDC. Especially the model predicts the capacitance maximum and rapid fall in capacitance as pore size is decreased. Thus, providing a possible explanation of Gogotsi-Simon effect of the anomalous capacitance in nanoporous CDC. We argue that the electronic screening capacitance with the electronic spillover contribution are important factors in deciding the overall value of capacitance at nanosacle electrodes. This work suggest that in nanoscale energy storage material electronic effects are importanct and hence must be accounted in understanding the charge storage capacity. The electronic effects also contribute to the surface stress of material. Also the surface stress induced dimensional changes are observed in nanoporous materials [53]. Morphological changes in electrode shape and size are observed due to nanoscale electric double layer forces as the potential is changed. We hope that our results will inspire to investigate the contribution of electronic screening free energy in nanoscale energy storage devices like supercapacitors and batteries. For example, it will be interesting to study the effect of electronic stress with electric double layer surface tension to understand the change in geometry in nanoporous supercapacitive system. 103

32 Appendices A Derivation of Electronic Number Density of Electrons in Metals The number of electron density of metal from periodic boundary is [54] ρ(k) = (4π/3)k3 (2π/L) 3 2 = V k3 3π 2 (A.1) where V = L 3 is the volume and k is the wave number of electron wave. Now using the Thomas-Fermi approximation, we have the ground state energy, E f of electron as: E f = 2 k 2 f 2m (A.2) where k f is the Fermi wave vector. Substituting the value of k f in the Eq. A.1 we have the electronic density of Thomas-Fermi model as: ρ(k) = V ( ) 3/2 2mEf (A.3) 3π 2 2 This is the lowest energy electron density. This electron density is modified by the presence of charge so the new maximum kinetic energy is E f + eφ( r). Now the electronic density for a unit volume is given by ρ(k) = (3π 2 ) 1 {(2m/ 2 )(E f + eφ( r))} 3/2 (A.4) 104

33 B Analytical Solution of Linearized Thomas-Fermi (LTF) Equation for Potential 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 the Green s function G for LTF equation satisfies ( 2 κ 2 T F )G(r, r ) = δ 3 (r r ) (B.1) with the homogeneous boundary conditions at surface S and far away from surface (r ), viz. G S = 0. The solution of Eq. B.1 for the potential, ϕ(r) can be written [34, 35] as: ϕ(r) = ϕ 0 S = ϕ 0 ds β G(r, β + ) n β d 3 r 2 G(r, r ) V ( ) = ϕ 0 1 κ 2 d 3 r G(r, r ) V (B.2) We seek to express G in terms of Yukawa like potential. The free space Green s function satisfying Eq. B.1 for the entire infinite domain as: G 0 (r, r ) = (1/4π r r )e κ T F r r (B.3) Now the surface dependent Green s function G is expressed [35] as: G(r, r ) = G 0 (r, r ) G 0 (r, α) G 0(α, r ) ds α n α G 0 (β, r ) ds α ds β n α n β G 0 (r, α) G 0(α, β) + (B.4) 105

34 C Thomas-Fermi Screening Capacitance of Arbitrary Surface Geometry The local surface charge density σ M may be obtained from Gauss s law as: σ M = ϵ 0 ϵ m S E.dS (C.1) where ds is the area element, E is the electric field and equal to inward normal derivative of potential to the surface E = ˆn. ϕ. The differential capacitance c is obtained by differentiation of surface charge with respect to the surface potential ϕ 0 as: c = dσ ( ) M d ϕ = ϵ m ϵ 0 dϕ 0 dϕ 0 S n.ds (C.2) where ϕ 0 is the potential difference applied at the surface and ϕ/ n = ˆn. ϕ is the inward normal derivative of the potential to the surface. Using Eqs. B.2 and B.4 and the fundamental singularity at the boundary [34, 35], the differential capacitance density is c(α) = ϵ mκ 2 T F 4π d 3 r G(α+, r V n α [ d 3 r 2 G 0(α, r ) V n α 2 2 G0 (α, β) G 0 (β, r ) ds β n α n β +2 3 G0 (α, β) G 0 (β, r) G 0 (γ, r ) ds β ds γ n α n β n γ ] = ϵ mκ 2 T F 4π (C.3) The terms in Eq. C.3 can be looked upon as one-, two- and three-scattering terms as: Σ 1 = 2 d 3 r G 0(α, r ) n α 106 (C.4)

35 Σ 2 = 2 2 G0 (α, β) n α G 0 (β, r ) n β ds β d 3 r (C.5) Σ 3 = 2 3 G0 (α, β) n α G 0 (β, r) n β G 0 (γ, r ) n γ ds β ds γ d 3 r (C.6) Equation C.4 may be expanded through local surface coordinates α, β, γ. For a weakly curved surface, where Thomas-Fermi screening length κ 1 T F is much smaller than smallest scale of curvature, the scattering kernel G 0 (β +, α)/ n α is expressed through a local coordinate system [34, 35] with the z-axis 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) : z α = (1/2) (x 2 /R 1 (α) + y 2 /R 2 (α)) +... is introduced to Eq. C.6 through surface area element ds β = gdxdy where g = 1 + ( z α (x, y)) 2. Using / n α / z the kernel G 0 (α, β)/ n α under planar approximation [34] 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 distance in tangent plane, G 0 = exp( ρ)/4πρ the Green s function in tangent plane. Now we can rewrite the one-scattering integral Eq. C.4 as: Σ 1 = 2 [ dxdy G 0 (ρ) + 1 z 2 2 ρ ] G 0 ρ (C.7) which is further simplified using the angular averages as: 1 ρ z = 1 2π ( ) cos 2 θ dθ R 1 (α) + sin2 θ = π ( 1 R 2 (α) 2 R 1 (α) + 1 ) R 2 (α) (C.8) 107

36 1 z 2 = 1 2π ρ 4 4 = 3π 8 0 ( cos 2 θ dθ R 1 (α) + sin2 θ R 2 (α) ) + ( 1 R 1 (α) R 2 (α) 2 ) 2 π 4R 1 (α)r 2 (α) (C.9) Substituting Eq. (C.8) to (C.4) and integrating over ρ we finally get the onescattering term as: Σ 1 = 1 κ T F 1 κ 3 T F ( 3 2R 2 α ) ( ) O 2R 1 (α)r 2 (α) Rα 3 (C.10) where 1/R α = (1/2)(1/R 1 (α) + 1/R 2 (α)). Similarly on iteration in Eq.?? and Eq. C.6 second and third-scattering terms are obtained as: Σ 2 = 1 κ 2 T F ( ) O, Σ R α Rα 3 3 = 1 κ 3 T F 1 R 2 α ( ) 1 + O R 3 α (C.11) Now substituting Eq. C.10 and C.11 to C.3, after simplification the capacitance density at position α is [ c T F (H α, K α ) = c M 1 1 H α 1 ] (H κ T F 2κ 2 α 2 K α ) + T F (C.12) which is Eq in the text. D Curvatures of Surface The local curvature κ(x) of a curve ζ(x) is given by κ(x) = ζ xx (x) (1 + ζ 2 x(x)) 3/2 (D.1) where ζ x = ζ/ x and ζ xx = 2 ζ/ x 2 are the first and second derivatives of the local surface profile. But for a general arbitrary surface the degree of freedom of height modulation could be in x and y directions. The surface related curvatures: 108

37 mean H and Gaussian K may be obtain from Monge representation (note here the convention used is opposite to that of as in ref. [50]) H = 2ζ xζ y ζ xy (1 + ζ 2 y)ζ xx (1 + ζ 2 x)ζ yy 2(1 + ζ 2 x + ζ 2 y) 3/2 (D.2) and K = ζ xxζ yy ζ 2 xx (1 + ζ 2 x + ζ 2 y) 2 (D.3) where the partial derivative ζ xx = 2 ζ/ x 2 etc. 109

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