Capacitance of a highly ordered array of nanocapacitors: Model and microscopy

Capacitance of a highly ordered array of nanocapacitors: Model and microscopy A. Cortés, C. Celedón, P. Ulloa, D. Kepaptsoglou, and P. Häberle Citation: J. Appl. Phys. 110, 104312 (2011); doi: 10.1063/1.3660683 View online: http://dx.doi.org/10.1063/1.3660683 View Table of Contents: http://jap.aip.org/resource/1/japiau/v110/i10 Published by the American Institute of Physics. Related Articles Ultrafast electrical measurements of polarization dynamics in ferroelectric thin-film capacitors Rev. Sci. Instrum. 82, 124704 (2011) Decoupling electrocaloric effect from Joule heating in a solid state cooling device Appl. Phys. Lett. 99, 232908 (2011) Demonstration of interfacial charge transfer in an organic charge injection device APL: Org. Electron. Photonics 4, 261 (2011) Demonstration of interfacial charge transfer in an organic charge injection device Appl. Phys. Lett. 99, 223304 (2011) Reduction of leakage currents with nanocrystals embedded in an amorphous matrix in metal-insulator-metal capacitor stacks Appl. Phys. Lett. 99, 222905 (2011) Additional information on J. Appl. Phys. Journal Homepage: http://jap.aip.org/ Journal Information: http://jap.aip.org/about/about_the_journal Top downloads: http://jap.aip.org/features/most_downloaded Information for Authors: http://jap.aip.org/authors

JOURNAL OF APPLIED PHYSICS 110, 104312 (2011) Capacitance of a highly ordered array of nanocapacitors: Model and microscopy A. Cortés, 1 C. Celedón, 1 P. Ulloa, 1 D. Kepaptsoglou, 2 and P. Häberle 1,a) 1 Universidad Técnica Federico Santa María, Valparaíso, Chile 2 Departament of Physics, University of Oslo, Oslo NO-0316, Norway (Received 12 August 2011; accepted 5 October 2011; published online 29 November 2011) This manuscript describes briefly the process used to build an ordered porous array in an anodic aluminum oxide (AAO) membrane, filled with multiwall carbon nanotubes (MWCNTs). The MWCNTs were grown directly inside the membrane through chemical vapor deposition (CVD). The role of the CNTs is to provide narrow metal electrodes contact with a dielectric surface barrier, hence, forming a capacitor. This procedure allows the construction of an array of 10 10 parallel nano-spherical capacitors/cm 2. A central part of this contribution is the use of physical parameters obtained from processing transmission electron microscopy (TEM) images, to predict the specific capacitance of the AAOs arrays. Electrical parameters were obtained by solving Laplace s equation through finite element methods (FEMs). VC 2011 American Institute of Physics. [doi:10.1063/1.3660683] I. INTRODUCTION Highly ordered capacitors have been proven useful in the fabrication of sensor, energy storage devices, and detectors. 1 Anodic aluminum oxide (AAO) membranes are dielectric materials with self-ordered and uniform pore distributions. 2 This property can be used to manufacture nanoscale semispherical capacitors. The patterned self-ordered hexagonal array structure have a density close to 10 10 pores/cm 2. 3 Through a procedure similar to the one described by Gösele, 3 we have produced an array of parallel nano-spherical capacitors. 1 The membrane physical dimensions can be controlled by the anodization procedure. One of the electrodes in the parallel capacitor arrangement is a close tip multiwall carbon nanotube (MWCNT) contained inside the alumina pores. Most MWNTs synthesized from nanoparticle seeds are closed at both extremes with a combination of hexagon and pentagon atomic arrangements. In the case of MWCNTs prepared inside open-ended AAO membranes with no catalytic particles, the resulting tubes are free of a closing cap. In our case, one of the alumina membrane ends is closed, hence, carbon is deposited at this end forming a closed tube. The other alumina extreme is open, therefore the MWCNTs are also open ended. The MWCNT shape is like a standard macroscopic glass test tube, but with nanoscale dimensions. The AAO barrier layer is the dielectric, which is covered with a thin Au layer to form the second electrode. The MWCNTs are also Au covered to achieve electrical contact (Fig. 1(a)). Our experimental measurements of the specific capacitance (nf/cm 2 ) were compared with the corresponding values calculated from the electric fields obtained by solving Laplace s equation by finite element methods (FEMs), using physical parameters obtained from TEM. We also compared our experimental results with the simpler spherical capacitor model. a) Electronic mail: patricio.haberle@usm.cl. II. EXPERIMENT The AAO template was prepared following a standard prescription, 2 which produces the porous structure and a remnant aluminum layer without electrolyzing. This metallic Al is chemically removed to obtain a metal-free alumina mold. The mold has open pores in one side, because a barrier-type oxide layer is formed during the electrolysis process at the other end. The MWCNTs were synthesized inside the AAO mold by decomposing acetylene at 650 C in a CVD-type arrangement. 4 The CNTs grow uniformly along the pores mold. The thickness of the tubes is controlled by the synthesis time. Thin Au layers (50 nm) are deposited on both sides of the device to form the electrical contacts; one Au layer over the alumina surface barrier and the other one over the free end of the metallic MWCNTs. Experimental measurements of capacitance were performed over different size samples using a Promax MZ505 m operated at a single frequency of 120 Hz. The size of the samples were obtained from digitized optical images of each capacitor. A TEM cross section of the metal insulator metal (MIM) nano-capacitor is shown in Fig. 2. The actual pore pore distance can be obtained from the TEM cross section of the sample by a simple geometrical correction, assuming hexagonal symmetry. This procedure is required because the samples are not necessarily cut along the hexagonal membrane main directions. The distance is determined using the geometrical relations shown in Fig. 3 and Eq. (1), x 0 is the inter-pore distance, as seen directly from the TEM plane of view and x is the separation between x 0 middle point and the second-layer pore center. The inter-pore distance (D p ) is then determined by the simple relation: rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi D p ¼ x 02 þ 4 3 x2 : (1) 0021-8979/2011/110(10)/104312/4/$30.00 110, 104312-1 VC 2011 American Institute of Physics

104312-2 Cortés et al. J. Appl. Phys. 110, 104312 (2011) revolution instead of the hexagonal prism to simplify both the equations and boundary conditions. The individual cells considered in the calculation of the fields are approximated by a circular edge instead of the actual hexagon. The radius (q e ) of the effective cylinder is defined by conserving the hexagonal prism s volume. pq 2 e ¼ 2 pffiffi 3 g 2 M ; (2) FIG. 1. (Color online) (a) Diagram of a nano-capacitor array based on an AAO membrane filled with MWCNTs. (b) Hexagonal unit cell for the AAOs. The microscopic image in Fig. 2 is consistent with an insulator barrier of 33-nm, pore diameter of 53 nm, and inter-pore distance D p of 94 nm. The TEM porosity 3 calculated for this array turns out to be 29%. III. ELECTRIC CAPACITANCE Because of the regularity of the pore positions, the array has been considered as formed by identical hexagonal prisms (Fig. 1(b)). Each one of them consists of a pore filled with a conductor (MWCNTs) embedded and supported in a dielectric oxide (alumina) substrate. Because of the arrangement symmetry, the electric field along the boundary surface of the unit cell does not have a perpendicular component. The electric flux lines emanating from the cathode never cross the side walls of the prism before reaching the anode. In this model, the total capacitance is then the parallel sum of the individual prisms capacitance. The FEM (Ref. 5) calculations were performed using an effective volume of with g M ¼ D p /2 ¼ 47 nm; the half-distance between nearest pores. This approach makes use of the TEM measurements. The surface of the spherical sections shown in Fig. 1 are assumed to be centered around the hexagonal symmetry point C in real space. The calculation involves solving Laplace s equation for the electrical potential inside the alumina volume. Because of the rotational symmetry of the configuration, the problem reduces to a potential differential equation (PDE) in two variables, in cylindrical coordinates (q and z have been chosen as variables): @ 2 U @q 2 þ 1@U q@q þ @2 U ¼ 0: (3) @z2 Neumann s boundary condition for this geometry becomes: @U @q ¼ 0; fq ¼ 0; q ¼ q eg: (4) The Dirichlet boundary conditions are normalized in a way that the potentials are equal to 0 V outside the surface barrier and 1 V inside the alumina-pore edge (MWNT). We used the FEM to solve the differential equation, using a triangular six-nodes by element mesh (with 2425 nodes, partitioned in 1152 elements). A higher density of nodes was used near the tip and close to the Au substrate, to obtain a more precise description of the fields. A simpler and alternative picture, is to consider an equivalent capacitance formed by sections of two spherical electrodes 1 (Eq. (5)). TEM results were used to determine the barrier layer height and the inner and outer radius of the FIG. 2. TEM cross section of barrier layer, MWCNTs, and surface Au electrode. The contrast even allows the determination of the MWNT s radius. FIG. 3. (Color online) Geometrical diagram used to obtain the true pore distance and pore diameter from a TEM cross section. Based on these measurements, even the correct porosity of the membrane can be estimated assuming a hexagonal array for the membrane. The expression for the interpore distance, as shown in Eq. (1), can be directly obtained from this image.

104312-3 Cortés et al. J. Appl. Phys. 110, 104312 (2011) porous cross section. A similar calculation has been performed in the past using phenomenological parameters instead of actual measurements as described in Ref. 1: ab C ¼ k 4pe AAO e 0 n: (5) ðb aþ IV. EXPERIMENTAL RESULTS AND DISCUSSION Figures 4(b) and 4(c), show the FEM results for the electrostatic potential and the energy density of the field configuration. In Fig. 4(a), we describe the parameters used in the different calculations. Based on these measurements, a precise estimation of the specific capacitance can be obtained. In Fig. 5, we show different specific capacitances for this particular system; one of them is directly measured, as indicated in the experimental section, the others are calculated through the methods described above, namely FEM and the spherical capacitor approximation (Eq. (5)). It is worth noting that relevant device design information is provided by the FEM calculation, for example, the position and intensity of the electric field singularities present at points (I) and (II) as indicated in Fig. 4(c). There is indeed a considerable contribution to the specific capacitance associated to this particular field configuration. In practical applications, the field intensities at these points are relevant information, because of dielectric breakdown, they eventually limit the energy-storage capacity. The simpler spherical capacitor calculation considers both electrodes extending up to D p /2. No singular fields are present in this case and the effective electrode surfaces are slightly different to the FEM calculation; the combination of these approximations can explain why this model anticipates a smaller specific capacitance than experimentally measured. All specific capacitances were calculated using a dielectric constant of e AAO ¼ 7.0. 6 FIG. 4. (Color online) (a) The approximate spherical shape of the Al 2 O 3 barrier layer Au interface electrode. Each valley minimum coincides with the axis of a pore (C symmetry point). The radial coordinate is q and the vertical scale z. (b) FEM electric iso-potentials measured between 0 and 1 V. (c) The corresponding normalized electrostatic energy density. (I) and (II) are high-energy density points. FIG. 5. (Color online) Specific capacitance vs capacitance obtained under different conditions: experimental measurement, - - - capacitance calculated with data from Ref. 1, equivalent spherical capacitance, FEM calculation. Even though the specific capacitance was measured at a single fixed frequency, the same value can be expected for different frequencies. Sohn et al. 1 have presented a detailed discussion of capacitance measurements at different frequencies and found no frequency dependence in the capacitance of these devices, even considering the small dimensions they display. The expressions used for the ideal energy stored by unit of surface area and the corresponding FEM value are: U ¼ C s V 2 =2 ¼ 0:92 10 07 J=cm 2 : V. CONCLUSION FEM calculations provide numerical values of the specific capacitance, which are closer to the experimental measurements than other approximation methods. FEM calculations combined with TEM are indeed the appropriate means to estimate accurate results for the electric properties in these systems. Calculations based on treated TEM images are more precise than the empirical approach 7 to estimate electrical characteristics. If different acids, temperatures, and potential differences are used during synthesis, microscopy is by far the best method to determine geometrical parameters of the resulting nanoscale device. The largest amount of electrostatic energy is indeed accumulated in regions closest to the tip of the pore, as confirmed by FEMs. This effect is not detected in the simpler spherical capacitor calculations. We have only considered a single set of physical dimensions in this particular study, nevertheless different physical paramenters (i.e., porous radius, porosity) are possible to obtain in the fabrication of nanoscale capacitors based on AAOs. 2 The same methodology portrayed in this paper can be used to describe these devices. ACKNOWLEDGMENTS This research has been partially funded by the following Grant Nos.: MECESUP, No. FSM0605; FONDECYT, Nos. 1100672 and 1110935 and Basal CEDENNA No. FB0807, Chile. We recognize helpful discussions with R. Segura

104312-4 Cortés et al. J. Appl. Phys. 110, 104312 (2011) regarding CVD processing of the AAOs. Figures were prepared with the helpful contribution of Ms. Leticia González. 1 J. I. Sohn, J.-S. Kim, C. Nam, B. K. Cho, and T.-Y. Seong, Appl. Phys. Lett. 87, 123115 (2005). 2 L. da F. Costa, G. Riveros, H. Gomez, A. Cortes, M. Gilles, E. A. Dalchiele, and R. E. Marotti, Characterizing the hexagonality of anodic aluminum oxide nanoporous arrays, Appl. Phys. Lett. (to be published). 3 U. Gösele, Nano Lett. 2(7), 677 (2002). 4 R. A. Segura, W. Ibañez, R. Soto, and P. Häberle, J. Nanosci. Nanotechnol. 6, 1945 (2006). 5 P. Monk, Finite Element Methods for Maxwell s Equations (Oxford University Press, New York, 2003). 6 V. Gianneta, A. G. Nassiopoulou, C. A. Krontiras, and S. N. Georga, Phys. Status Solidi C 5, 3686 (2008). 7 T. T. Xu, F. T. Fisher, L. C. Brinson, and R. S. Ruoff, Nano Lett. 3(8), 1135 (2003).