Curvature-Induced Anomalous Enhancement in the Work Function of

Letter pubs.acs.org/JPCL

Curvature-Induced Anomalous Enhancement in the Work Function of Nanostructures Jasmin Kaur and Rama Kant* Complex Systems Group, Department of Chemistry, University of Delhi, University Road, New Delhi 110007, India S Supporting Information *

ABSTRACT: An analytical theory to estimate the electronic work function in curved geometries is formulated under Thomas−Fermi approximation. The work function is framed as the work against the electrostatic self-capacitive energy. The contribution of surface curvature is characterized by mean and Gaussian curvature (through multiple scattering expansion). The variation in work function of metal and semimetal nanostructures is shown as the consequence of surface radius of curvature comparable to electronic screening length. For ellipsoidal particles, the maximum value of work function is observed at the equator and poles for oblate and prolate particles, respectively, whereas triaxial ellipsoid shows nonuniform distribution of the work function over the surface. Similarly, theory predicts manifold increase in the work function for a particle with atomic scale roughness. Finally, the theory is validated with experimental data, and it is concluded that the work function of a nanoparticle can be tailored through its shape.

U

cluster, star, fern, cube, or many more morphologies. The surface acquired by the nanomaterials due to small size and shape is attributed to the varying curvature at various sites; however, curvature effects are most effectively evident when the particle size or local surface features are comparable to the electronic screening length of a material. The screening length is altered by a change in surface composition; recently, it has been shown that by altering the surface composition one can tune the work function of electrode.29,30 Here we show that the work function of materials can be tuned by altering the shape or roughness of the nanoparticle. We develop a theoretical route to trace the curvature dependence of electronic work function of nanoparticles within the framework of Thomas−Fermi approximation. In general, surface curvature at point α of the surface is characterized by mean curvature (Hα) and Gaussian curvature (Kα). The theory is designed by inculcating the multiple scattering expansion in the curvature9 of a material into work function through selfcapacitance of that material. The self-capacitance of curved geometries is known for spherical conducting particles,31 cylindrical pore,12 and arbitrary geometries.9 Following the mathematical formulation of theory, variation of work function of ellipsoid nanoparticles with their aspect ratio (σ) is explained to have its origin in the prominence of their curvature at various sites, with decreasing size. This effect is attributed to the comparability of any of the semiprinciple axis of the structure and the electronic screening length. The spherical structure of the particle emerges as a limiting case in transition from oblate (σ < 1) to prolate (σ > 1) structures. The local sites over the

nderstanding of size-dependent electronic and atomic properties of a material facilitate better insight into the phenomena at fundamental levels. Nanomaterials, due to their relatively higher abundance of surface atoms, enhanced surface curvature and heterogeneity, have several unique properties such as enhanced charge transfer, 1,2 superior catalytic activity,3−8 altered electronic properties,9 reduced photoelectric threshold,10,11 and so on. The anomaly in capacitance of subnanopores,12,13 size- and shape-dependent catalytic and dielectric properties of nanoparticles, altered adsorption kinetics at nanoscale,14 and so on are some of the experimental observations that direct toward the significant participation of material’s electronic properties at phenomenological level. One of such electronic properties of extreme importance is the work function. It is the electronic threshold energy that is required to remove an electron from the surface of a metal at 0 K.15 For metals the densities of state are high, and thus the electronic screening distance is of the atomic dimensions only. The influence of small changes in density due to impurities or surface structure fluctuation or curvature therefore is evident on the nanoscale. Thus, work function of nanostructured materials is a crucial property and hence needs a thorough insight. Several attempts have been made to establish a theoretical route of estimating metal work function, viz. image force approach for metal16 and insulators,17,18 variational approach for small metal clusters,19 exchange correlation formulation, and so on. Relating it to the lattice energy in the crystal,20−22 heat of sublimation of univalent metal,23 atomic volume,24 hydrogen overpotential,25,26 and ionization potential27,28 are some of the other perspectives. Nanostructures can be viewed as the colonies of several atoms or nanoparticles fused together to generate different morphologies. These could be linear bead morphology (e.g., modulated nanorod) or agglomeration to form spherical © XXXX American Chemical Society

Received: June 6, 2015 Accepted: July 4, 2015

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DOI: 10.1021/acs.jpclett.5b01197 J. Phys. Chem. Lett. 2015, 6, 2870−2874

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The Journal of Physical Chemistry Letters

scattering expansion in curvature has been presented to elaborate the curvature effects in several properties of the material.12,32 Kant et al. have presented an analytical theory for curvature effects on electronic capacitance of various nanostructures through Thomas−Fermi screening approach in association with multiple scattering method.9,12 The areaspecific electronic capacitance is expressed through a convergent expansion in powers of inverse Thomas−Fermi screening length (κTF = l−1 TF) and characteristic curvatures of the surface9

surface correspond to sites with varying work functions pertaining to their curvatures. Final segment models the work function for spherical nanoparticles with random roughness. This is important in nanodimensional materials because they have high density of structural defects with high indices facets exposed at the surface along with low coordination sites, which is indicative of atomic scale roughness with high curvature sites. The work function, ϕE, is the amount of work against the self-capacitive energy of the material and is defined as ϕE =

e2 2C E

⎡ H (Hα2 − Kα) ⎤ ⎥ C E = C E0⎢1 − α − 2 κTF 2κTF ⎦ ⎣

(1)

where e is the electronic charge and CE = cEAF is the area specific electronic self-capacitance.9 AF is the area of a circular disc of radius, lTF, that is, Thomas−Fermi electronic screening length defined as l TF

⎛ 2ϵE ⎞1/2 F ⎟ =⎜ 2 ⎝ 3n0e ⎠

(3)

C0E

is the electronic self-capacitance of the plane surface defined as C0E = c0EAF, wherein c0E = ϵ/lTF. The finite probability of electron density delocalization outside the metal is attributed to the electronic spillover over the surface. The electronic spillover up to distance δ from the surface and its correction in the mean and Gaussian curvatures is attained through Sterner’s formula defined as9

(2)

Here ϵ is the permittivity of the material, EF is the Fermi energy, and n0 is the average number density of electrons. Selfcapacitance indicates the capacity of a surface to hold charge on its surface per unit potential of zero charge. Therefore, selfcapacitive energy denotes the energy by which electrons are held at the surface. The interfacial region with gradient of electron density is characterized by electronic screening length (lTF). The work function is obtained as the amount of work against the self-capacitive energy of the disc with radius, lTF, over the surface. (See Figure 1.) Thus, this disc signifies the

Hα′ =

Hα − Kαδ 1 − 2Hαδ + Kαδ

Kα′ =

2

Kα 1 − 2Hαδ + Kαδ 2 (4)

Combining eqs 1 and 3, the curvature-dependent work function ϕE emerges as ϕE = ϕE0[1 −

Hα′ −1 l TF

−

(Hα′ 2 − Kα′ ) −2 2l TF

]−1 (5)

Here ϕ0E denotes the electronic work function for the plane surface defined as

ϕE0 =

e2 2π ϵl TF

(6)

Thus the first term in curvature expansion denotes the material’s contribution to the electronic self-capacitive energy (work function), the second term is the purely geometrydependent contribution, and the third term is the coupling of material’s geometric and electronic properties. For a given shape, curvature varies with the size of a material, and the electronic work function of idealized geometries can be expressed in the ratio of the characteristic Thomas−Fermi screening length and size of the material with spillover correction (ra = r ± δ)9 as

Figure 1. Schematic illustration of the concept of work function through self-capacitance due to electronic screening. Work function is electrostatic capacitive energy of a disc of radius lTF.

−1 ⎛ l 2 ⎞⎤ ⎛ l TF ⎞ TF ⎥ + a ⎜ ⎟ + b⎜ 2 ⎟ ⎝ ra ⎠ ⎝ ra ⎠⎥⎦ ⎣⎢

⎡

ϕE =

ϕE0⎢1

(7)

where a and b are the shape-dependent constants: plane surface (0, 0), tube (1/2, −1/8), spherical cavity (1, 0), sphere (−1, 0), and rod (−1/2, −1/8). lTF emerges as the key characteristic parameter through which the material contributions are regulated in its electronic properties like work function. It is through this length that the variations in electronic density of states and Fermi energy are exhibited and further incorporation of the curvature terms highlight the variation in these properties depending upon the site over the surface. Values of the work function obtained using eq 6 for planar surfaces correspond closely to the experimentally available value. These calculated work function values for 27 elements are listed in the Supporting Information.

region of electronic influence over the surface, dictating the site-specific or local-site-dependent work function. Through lTF, the work function is thus a characteristic property based on material’s characteristic electronic parameters. In real systems, the structure of the particle has some level of deviations with respect to the reference ideal surface. Here we attempt to trace the effect of surface curvature of nanostructured material on the work function, by characterizing the structure as the collection of electrons distributed under Thomas−Fermi approximation. The relation is designed through the curvature-dependent electronic capacitance, CE, for nanostructured materials obtained through multiple scattering expansion12,32 in curvature terms. The multiple 2871

DOI: 10.1021/acs.jpclett.5b01197 J. Phys. Chem. Lett. 2015, 6, 2870−2874

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Figure 2. (A) Effect of aspect ratio (σ) on maximun and minimum value of work function of metal (dotted lines) and semimetal (solid lines) nanoparticle. Here x = y = 7α, z = σx, and α = 1 for semimetal and α = 4.5 for metal. (B) Distribution of work function of an ellipsoidal semimetal particle: (a) prolate, x = y = 3.5, z = 7.5; (b) oblate, x = y = 11, z = 3.5; and (c) triaxial ellipsoid, x = 10, y = 5, z = 3.5 and δ = 0.5. x, y, z, and δ are in units of respective lTF (ϕ*E = ϕE/ϕ0E).

particle can be customized. The maximum limit of curvature that can be achieved is that of a single atom evident at the kinks and pits over the surface. The size-dependent work function for ultrafine Ag particles observed by Schmidt-Ott, Schurtenberger, and Siegmann34 has been interpreted on the basis of image potential, which is the principle interaction beyond a few angstroms above the surface, by Wood.35 The size-dependent work function of spherical particles has been generally expressed as a linear combination of planar work function and inverse dependence on size of the particle.18,35−39 From eq 5, the curvature-dependent work function for a spherical particle emerges as

From the theoretical point of view, the most widely studied is a nanoparticle with spherical geometry; however, elongation of a nanoparticle along any axis or compression of growth on the other can result in ellipsoidal structures. This structure is of interest because it can help to visualize varying curvatures over a single particle surface. Formulas for mean and Gaussian curvatures of ellipsoid are provided in the Supporting Information. An ellipsoid has convex surface throughout. Work function is thus regulated by the extent of convexity at a particular site over the particle surface, which, in turn, is controlled by its aspect ratio. With varying aspect ratios, starting from an oblate particle, several local sites of maximum and minimum work function are generated. (See Figure 2A.) At σ = 1, spherical limit is reached and the intrinsic uniform curvature contribution from a spherical particle attributes to the uniform work function; however, it remains higher from that of the planar surface due to positive contributions from the curvature. For a prolate structure, the maximum curvature is at pole region, whereas the equator region exhibits lower curvatures. At the former position, the work function is observed to reach a high value comparative to the planar surface work function, and at latter positions it is close to the planar value, Figure 2B(a), whereas in the case of oblate structure the maximum curvature is generated at equator region of the particle and thus work function acquires higher value at these sites, Figure 2B(b). For a triaxial ellipsoid, surfaces exhibit deformations along various directions and to varying extents. Consequently, the curvature is nonuniform and manifest asymmetric distribution of work function at the surface, Figure 2B(c). Experimentally observed work function of metals and semimetals lies within the range of 2.1−5.1 eV.33 As exhibited in Figure 2, the work function value of a nanostructure went up to twice the value it has in planar structure for semimetal such as carbon material with well-customized size and shape. Thus, a single particle behaves as if there are different materials at different sites over the structure. This effect is induced merely by curvature of the material. Therefore, by optimizing the curvature of surface structure sites, the work function of a

ϕEs = ϕE0 +

1 ⎛ e2 ⎞ ⎜ ⎟ 2πϵ ⎝ ra ⎠

(8)

With reducing size of the nanoparticle, the curvature contribution to the work function is increased. Because of the convex surface of spherical particle, the effective work function increases due to positive contribution from curvature with reducing particle size, as depicted in Figure 2A. Because sphere is a symmetric structure, the effect of curvature over the surface is isotropic. Owing to a wide range of defects that can originate in the structure of nanoparticles, different kinds of surface distortions can be evident. Even for very fine nanoparticles several local sites with even finer curvatures can persist over the surface attributing to the intrinsic roughness features. So, most of the spherical particles are not ideally spherical in reality. If the size of these features is comparable to the electronic screening length of material, the curvature influence will prevail in properties of the particle even when size of the particle itself is comparatively large. The random surface fluctuations are characterized as ensemble average of local deviations in mean and Gaussian curvatures at each surface site to account for intrinsic roughness features of the surface.32 The roughness is characterized through deviations from the reference curvature such that the ensemble average mean values ⟨H − ⟨H⟩⟩ = ⟨K − ⟨K⟩⟩ = 0 and variance values, ⟨(H − ⟨H⟩)2⟩ and ⟨(K − ⟨K⟩)2⟩, 2872

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Figure 3. (A) Size-dependent scaled (average) work function of rough nanosphere. Parameters used for metal (red), lTF = 0.13 nm, and for semimetal (green), lTF = 0.27 nm. ⟨δH2⟩ = 35/l2TF and ⟨ϕsE⟩* = ⟨ϕsE⟩/ϕsE. (B) Experimental work function for Ag nanoparticles34 (scaled with calculated value of work function for planar surface). Parameters: lTF = 0.13 nm, ⟨δH2⟩ = 1.25/l2TF. In both graphs, solid and dashed lines denote scaled work function of rough and smooth nanospheres from theory, respectively.

are finite. So, the average work function with random surface fluctuation contribution is ⟨ϕEs⟩ = ϕEs + ⟨δϕEs⟩

feature of the particle or the size of the particle itself. For ellipsoidal particles, the maximum value of work function is observed at the equator and poles for oblate and prolate particles, respectively, whereas triaxial ellipsoid shows nonuniform distribution of the work function over the surface. Hence, we conclude that the work function of the nanoparticle can be tailored through its shape. Similarly, theory predicts manifold increase in the work function for a particle with atomic scale roughness. We have shown that inclusion of roughness of nanoparticles is essential in prediction of correct work function, and this fact is validated through available data. It is well-known that the work function influences many surface phenomena, viz. charge transfer, catalytic activity, adsorption, photoelectric activity, and so on. In particular, the heterogeneous kinetics is exponentially related to the work function of the material. Hence, the alteration in geometry and morphology on the subnano- and nanoscale will enormously influence its kinetics. Further consideration of the influence of surface curvature on charge transfer kinetics has been taken elsewhere.

(9)

where ⟨δϕsE⟩ is the overall contribution due to fluctuation in surface curvature. The average work function of a rough sphere with average size ⟨r⟩ is obtained as 2 ⎛ ⎞ ⟨δH2⟩l TF ⎟ ⟨ϕEs⟩ = ϕEs⎜1 + (1 − l TF/⟨r ⟩)3 ⎠ ⎝

(10)

where ⟨δH2⟩ is the ensemble average of variance of mean curvature fluctuations. For semimetals, lTF is thicker compared with metals. In metals, high density of electronic charges attribute to small lTF. As is evident in Figure 3A, the influence of structural features are strongly manifested as incremented in normalized work function for particles with r < 20 nm; however, for nanoparticles with larger sizes, the persistence of curvature effects exhibits scaled work function greater than 1 for both materials, and it asymptotically merges into the planar limit. Roughness features of the particle surface manifest manifold increase in the scaled (average) work function. It is a consequence of several roughness features at the surface comparable to the characteristic screening length. For the same extent of surface fluctuations and same particle size, semimetal nanospheres exhibit higher variation than metal nanospheres. This result can be attributed to the extent of comparability of lTF for either material to the surface roughness features at a particular particle size. In Figure 3B, experimental values of work function for Ag nanoparticles34 exhibit an excellent coherence with the theory when considered as rough particles, while smooth particle model underestimates ϕsE. It is concluded that the electronic work function of metal and semimetal can be tuned by altering the surface curvature of nanostructures and characteristic material property, the Thomas−Fermi electronic screening length. Our theory shows anomalous rise in work function by enhancing convex features in the nanostructure. The effect of altering geometry and morphology is pronounced when the characteristic electronic screening length is comparable to surface local

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ASSOCIATED CONTENT

S Supporting Information *

Calculated electronic work functions for 27 elements. Parametric coordinates of the ellipsoidal geometry along with mean and Gaussian curvature, which has been used to study the distribution of work function as a function of curvature of ellipsoidal nanoparticle. The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpclett.5b01197.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS R.K. thanks DST (SB/S1/PC-021/2013) and University of Delhi for financial support under “Scheme to strengthen R&D 2873

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doctoral research program”. J.K. thanks UGC, New Delhi for Senior Research Fellowship (SRF).

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