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Journal of Nanomaterials
Volume 2010 (2010), Article ID 160639, 27 pages
Research Article

Nanowire and Nanocable Intrinsic Quantum Capacitances and Junction Capacitances: Results for Metal and Semiconducting Oxides

Microwave Technology Branch, Naval Research Laboratory, Washington, DC 20375, USA

Received 16 August 2010; Accepted 25 October 2010

Academic Editor: Xuedong Bai

Copyright © 2010 C. M. Krowne. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


Here we calculate the intrinsic quantum capacitance of RuO2 nanowires and RuO2/SiO2 nanocables (filled interiors of nanotubes, which are empty), based upon available ab initio density of states values, and their conductances allowing determination of transmission coefficients. It is seen that intrinsic quantum capacitance values occur in the aF range. Next, expressions are derived for Schottky junction and p-n junction capacitances of nanowires and nanocables. Evaluation of these expressions for RuO2 nanowires and RuO2/SiO2 nanocables demonstrates that junction capacitance values also occur in the aF range. Comparisons are made between the intrinsic quantum and junction capacitances of RuO2 nanowires and RuO2/SiO2 nanocables, and between them and intrinsic quantum and junction capacitances of carbon nanotubes. We find that the intrinsic quantum capacitance of RuO2-based nanostructures dominates over its junction capacitances by an order of magnitude or more, having important implications for energy and charge storage.

1. Introduction

With the past work on nanowire and nanotube (or nanocable) materials, including silicon nanowires [14], carbon nanotubes (single and many walled) [519], metallic nanowires, and now nanoribbon graphene [2029], it is not surprising that so many uses have been found for nanostructured one-dimensional (1D) objects. These uses range from employment in solid state devices like Schottky and p-n diodes to transistors in electronics, to uses as catalysts, fuel cell membranes, batteries, and supercapacitors for energy applications, with solid state solar cells bridging the gap between electronic and energy uses.

Recent findings [30] for a particularly interesting metallic oxide, ruthenium dioxide RuO2, deposited as a shell on an inner silicon dioxide SiO2 core (forming a coaxial cable geometry), have shown possibilities for use as fuel cells and solar cells because of its anomalously high electronic conductivity (0.5 S/cm at 0.1% volume of RuO2 in the RuO2/SiO2 composite when deposited as nanoclusters; conductivity range is discussed in [31, 32] related to its most ordered simplest crystalline form to least ordered forms), optical transparency, negligible amount of expensive atomic Ru element used (0.3 mg of RuO2 per square of SiO2 paper—28 ¡c per square), ultrahigh surface area (90 m2 per gram RuO2), anomalously high energy storage (>700 F per gram anhydrous RuO2), and vigorous catalytic action for water splitting. Previous uses in the bulk and nanoscale forms include Schottky barrier photovoltaics [33], field emission of nanorods [34], and thick film resistors [35]. To properly make use of RuO2, its atom scale chemistry is important in addition to electronic properties [36, 37].

There are other structured metal oxides which have potential for electronic applications, including cathodoluminescent Ga2O3 nanowires [3842], electrochromic behavior materials (WO3, MoO3, TiO2, V2O3, and Sb-doped SnO2 dispersed into inert inorganic supports) [43, 44], WO3 nanowires on 𝑊 [45], resistive gas-sensor materials (like WO3, SnO2, and In2O3) [46], and indium-tin-oxide transparent conductive materials [47, 48] which can be used for solar cells and LEDs, for example. Furthermore, gadolinium-based oxides [49, 50] may hold potential for ionic nanowires. Finally, nanobeams, represent quasi-one-dimensional objects, smaller than characteristic grain sizes, that may be of interest, which have been realized in VO2, and are semiconductors below a critical temperature 𝑇𝑐=68 C with an optical bandgap 𝐸𝑔=0.6 eV [51].

The following four sections treat the intrinsic quantum capacitance of RuO2 nanowires (Section 2), quantum conductance and transmission coefficient of RuO2 nanowires (Section 3), quantum conductance and transmission coefficient of RuO2/SiO2 nanocables (Section 4), and intrinsic quantum capacitance of RuO2/SiO2 nanocables (Section 5). After these four sections, the next two sections cover semiconductor junction capacitances of nanocables (Section 6) and nanowires (Section 7). Section 6 is divided into Sections 6.1 and 6.2 examining, respectively, Schottky- and asymmetric-semiconductor junction capacitances of nanocables. Section 7 also is divided into Sections 7.1 and 7.2, investigating, respectively, symmetric- and asymmetric-semiconductor junction capacitances of nanocables.

Finally, the paper presents a discussion and comparison in Section 8 between the intrinsic quantum capacitances and the junction capacitances, in four parts, Sections 8.1, 8.2, 8.3, and 8.4, focusing on respectively, junction capacitances of nanocables and nanowires, intrinsic quantum capacitances of nanocables and nanowires, comparisons between intrinsic quantum and junction capacitances of nanocables and nanowires, and electrochemical aspects in relation to the physics of nanowires and nanocables. A short conclusion follows (Section 9). Next follow two appendices, one providing details on the Green’s function solution for Poisson’s equation in the electrostatic limit (Appendix A), the other on modifications in the nonabrupt nanocable junction potential with distance along the longitudinal axis (Appendix B).

2. Intrinsic Quantum Capacitance of RuO2 Nanowires

Intrinsic quantum capacitance of nanowires (Figure 1(a)), based upon charge storage of electron carriers, calculated from the density of states (DOSs) determined from first principles quantum simulations employing the orbital structure of the crystalline system, utilizing the unit of electron charge magnitude 𝑒, is given in the report by Amantram and Léonard [13] and the text by Leonard [14] as𝐶𝑖=𝑒2𝐷𝑙𝐸𝐹,(1) where 𝐷𝑙(𝐸𝐹) is the density of states at the Fermi level 𝐸𝐹 and the subscript 𝑙 on 𝐷𝑙 indicates that this is one dimensional density of states whose units are eV-1·nm-1. Equation (1) may be derived by finding the added energy stored by adding electrons to the system when 𝐸=𝐸(𝑘), finding for a single electron 𝛿𝐸=𝑑𝐸(𝑘)/𝑑𝑘|𝐸=𝐸𝐹𝛿𝑘=𝛿𝑘/[2𝜋𝐷(𝐸𝐹)], using the discrete level separation 𝛿𝑘=2𝜋/𝐿 for a nanowire (or nanocable) of length 𝐿, and equating this band structure energy to the capacitive energy stored 𝐸𝐶, where 𝑒=𝑄=𝐶𝑉 giving 𝑒2=𝐶(𝑒𝑉)=𝐶𝐸𝐶, making capacitance per unit length equal to 𝐶𝑙= 𝐶/𝐿= 𝑒2/𝐿𝐸𝐶. Expression (1) may also be obtained by incrementing the electrostatic potential 𝑉 of the entire nanowire (or nanocable) to 𝑉+𝛿𝑉, which adds the charge 𝛿𝑄=𝑒𝐷𝑙(𝐸)𝛿[𝑓(𝐸𝐸𝐹)]𝑑𝐸 or 𝛿𝑄=𝑒𝐷𝑙(𝐸)[𝑓(𝐸+𝑒𝛿𝑉𝐸𝐹)𝑓(𝐸𝐸𝐹)]𝑑𝐸, which yields 𝛿𝑄=𝑒2𝐷𝑙(𝐸𝐹)𝛿𝑉 [because 𝛿𝑄𝑒𝛿𝑉𝐷𝑙(𝐸)[𝑑𝑓(𝐸𝐸𝐹)/𝑑𝐸]𝑑𝐸 which makes 𝛿𝑄𝑒2𝛿𝑉𝐷𝑙(𝐸)[𝑑𝑓(𝐸𝐸𝐹)/𝑑𝐸]𝑑𝐸=𝑒2𝛿𝑉𝐷𝑙(𝐸)01𝑑𝑓] and 𝐶𝑖=𝛿𝑄/𝛿𝑉. 𝐷𝑙(𝐸𝐹) may be calculated from the bulk three-dimensional (3D) density of states 𝐷(𝐸𝐹) using the following relationship:𝐷𝑙𝐸𝐹=𝜋𝑅2𝑉unit-cell𝐷𝐸𝐹.(2) At the Fermi level, for RuO2, 𝐷(𝐸𝐹) is known to be 3.2 [52], 3.8 (rutile crystal structure [53]), 2.6 (orthorhombic crystal structure [53]), 2 [54], with [55] not providing absolute scales to extract values from but an earlier work of these authors [56] suggest a value of 1.4 (rutile structure, off of their Figure 7; [57] extracts an incorrect value of 1.7 listed in their Table 1), 2.36 [57], and 3.6 (off of Figure 4 in [58]; [57] extracts an incorrect value of 2.89 listed in their Table 1). The unit cell volume for a rutile crystal structure is 𝑉RuO2uni-cell=𝑎𝑏𝑐 and the tetragonal values of the cell sides [34] are 𝑎()=𝑏()=4.45 and 𝑐()=3.16, making 𝑉RuO2uni-cell=65.07078Å30.065071nm3. Note that the values of 𝑎=𝑏, and 𝑐 used are close to earlier reported values of 𝑎=4.4909 and 𝑐=3.1064 [59].

Figure 1: (a) Basic nanowire geometry. (b) Basic nanocable geometry. In the text the kernel is developed based upon the outer shell being much thicker than the inner core, namely, that 𝑡NC𝑅.

The final formula for 𝐶𝑖, in units of aF/nm, is given by𝐶RuO2𝑖=21.1𝑅2(nm)𝐷RuO2(#states/eVunitcell)aF/nm.(3) Setting 𝐷RuO2(𝐸𝐹)3eV1/cell, and the nanowire radius 𝑅=1nm, we find that𝐶RuO2𝑖=63.3aF/nm.(4) This may be compared to a single-walled carbon nanotube intrinsic capacitance 𝐶SWCNT𝑖=4𝑒2/(𝜋𝑣𝐹)=0.4aF/nm [14]. The reason why the SWCNT formula is 𝑅 independent whereas the nanowire is not, is that for the nanowire larger 𝑅 means a larger cross-sectional area of atoms to include, whereas for the SWCNT, its thickness remains one atomic layer thick no matter what 𝑅 is. It is apparent from this calculation, that the charge storage capacity of RuO2 is over 158 times that of single-walled carbon nanotubes. This is over two orders of magnitude improvement and suggests that metal-oxide nanowires may be better for charge storage applications.

Before we can go on to calculate the intrinsic capacitance of an outer cylindrical shell of RuO2 surrounding an inner SiO2 core (coaxial cable geometry), we must study the conductivity and transmission properties of nanowires and nanocables in the next section.

3. Quantum Conductance and Transmission Coefficient of RuO2 Nanowires

At room temperature, the resistivity of RuO2 is given by [55]𝜌RuO2bulk=34𝜇Ωcm(5) a value considerably higher than, say a good monoatomic metal like silver, whose resistivity value is 𝜌Agbulk=1.62μΩ·cm [60]. The value shown in (5) is in nearly perfect agreement with that in [37], who cites the value from [61], as 𝜌RuO2bulk=35μΩ·cm. Using the slightly smaller value, we can write the conductivity as𝜎RuO2bulk=1𝜌RuO2bulk=29410Ω1cm1=29410S/cm.(6) With this conductivity value, the conductance of a 1 μm long nanowire is be calculated to be𝐺RuO2NW=𝐴RuO2NW𝐿RuO2NW𝜎RuO2bulk=9.2394×106mhos=9.2394𝜇S.(7)

From this conductance, a transmission coefficient characterizing the scattering properties of the metal-oxide material can be determined. Here is how it is derived. The conductance can be expressed fairly accurately, for small applied voltage differences to the wire ends, as [1014]𝐺=2𝑒2𝑔spin𝑚𝑇𝑚(𝐸)𝜕𝑓(𝐸)𝜕𝐸𝑑𝐸.(8) where is Planck’s constant, 𝑇𝑚 is the transmission coefficient for the 𝑚th mode, and 𝑓(𝐸) is the Fermi-Dirac distribution function providing the statistics for the particles under consideration. This equation can be broken into three factors, the quantum of conductance 𝐺0=2𝑒2/,𝑔spin, and the summation over the integrals for each scattering mechanism, 𝐼𝑚𝐺. Then (8) may be rewritten as𝐺=𝐺0𝑔spin𝑚𝐼𝑚𝐺,(9) where we will take the spin degeneracy to be 𝑔spin=2, and𝐼𝑚𝐺=𝑇𝑚(𝐸)𝜕𝑓(𝐸)𝜕𝐸𝑑𝐸.(10) We will assume that this integral can be represented approximately by the following product at the Fermi level:𝐼𝑚𝐺=𝑇𝑚𝐸𝐹𝑓𝐸𝐹(11) and inserting this into (9) yields𝐺=𝐺0𝑠𝑓𝐸𝐹𝑚𝑇𝑚𝐸𝐹,(12) where we use 𝐺0𝑠 (𝐺0𝑠=154.94μS) as the fundamental quantum conductance with spin degeneracy included, and write the summation of all modes of scattering as𝑇𝐸𝐹=𝑚𝑇𝑚𝐸𝐹(13) because it is the total conductance 𝐺 that is measured. Placing (13) into (12) allows that total transmission coefficient to be calculated as𝑇𝐸𝐹𝐺=2𝐺0𝑠(14) because the Ferm-Dirac function, 𝑓(𝐸)=1/[1+exp(𝐸𝐸𝐹)/𝑘𝐵𝑇], is simply a half when evaluated at the Fermi level. For a 1 nm radius nanowire of RuO2, we find from (14) that𝑇RuO2NW𝐸𝐹=0.11930.12.(15)

4. Quantum Conductance and Transmission Coefficient of RuO2/SiO2 Nanocables

These nanocables (see Figure 1(b)), coaxial geometry with an inner SiO2 solid cylindrical core and an outer cylindrical shell of RuO2, are described in [30], as having electronic conductivity𝜎RuO2/SiO2NWcableS=0.30.8cm,5wt%RuO2onSiO2.(16) To use a formula like (7) to find the conductance, we need the inner core radius, outer shell radius, and their difference:Δ𝑅=𝑅RuO2𝑅SiO2𝑅=23nm,SiO2=30nmandup,𝑅RuO2=𝑅SiO2+Δ𝑅.(17) Because the core-shell combination acts as a parallel resistor system, the conductance must be𝐺RuO2/SiO2NWcable=1𝐿RuO2/SiO2NWcable𝐴SiO2𝜎SiO2+𝐴RuO2𝜎RuO2=𝐴RuO2𝐿RuO2/SiO2NWcable𝜎RuO2,(18) where the second equality holds when the inner core is perfectly insulating, which we are assuming.

The area of the thin annulus of RuO2 clusters forming the outer shell, has an area equal to the difference between the inner and outer circular cross-sections, or𝐴clustRuO2=2𝜋𝑅cable𝑎𝑣Δ𝑅cable,𝑅cable𝑎𝑣=𝑅RuO2+𝑅SiO22.(19) For Δ𝑅=2 nm and 𝑅SiO2=30 nm, 𝐴clustRuO2=124𝜋 nm2, and taking 𝜎RuO2/SiO2NWcable=0.5 S/cm and 𝐿RuO2/SiO2NWcable=1μm, 𝐺RuO2/SiO2NWcable=1.95×108mhos=0.0195𝜇S.(20) This is considerably smaller than the RuO2 nanowire result (see (7)) by a factor of𝐺RuO2NW𝐺RuO2/SiO2NWcable=474(21) whose value is substantially smaller than an Ag nanowire by a factor of𝐺AgNW𝐺RuO2NW=𝜎Ag𝜎RuO2=21.(22) How does this value compare to indium tin oxide (ITO; 5 wt% SnO2 + 95% wt% In2O3) which is studied by Kim et al. [48]? They found that𝜌ITO=400𝜇Ωcm,150nmlm,200𝜇Ωcm,170nmlm(23) so using a value of 𝜌ITO=300μΩ·cm for a ITO nanowire, we see that𝐺RuO2NW𝐺ITONW=𝜎RuO2𝜎ITO=8.83.(24) The transmission coefficient is given by (14),𝑇RuO2/SiO2NWcable𝐸𝐹𝐺=2RuO2/SiO2NWcable𝐺0𝑠=2.516×104.(25) This is much smaller than that for the RuO2 nanowire, and indicates that a RuO2/SiO2 nanocable, with RuO2 nanoclusters having interfacial interconnects, will have transmission reduction due to those interfaces, beyond the already expected bulk-like and size-reduced geometrical scattering.

5. Intrinsic Quantum Capacitance of RuO2/SiO2 Nanocables

Following formula (1), the intrinsic quantum capacitance of RuO2/SiO2 nanocables is𝐶RuO2/SiO2𝑖=𝑒2𝐷RuO2/SiO2𝑙𝐸𝐹(26) with the density of states expressible for the nanocable (Figure 1 (b)) as𝐷RuO2/SiO2𝑙𝐸𝐹=2𝜋𝑅cable𝑎𝑣Δ𝑅cable𝑉RuO2unit-cell𝐷RuO2/SiO2𝐸𝐹.(27) Again, inserting (27) into (26), we obtain a formula similar to (3),𝐶RuO2/SiO2𝑖=14.262𝑅cable𝑎𝑣(nm)Δ𝑅cable𝐷RuO2/SiO2(#states/eVunitcell)aF/nm.(28) Here, 𝐷RuO2/SiO can be estimated by𝐷RuO2/SiO2𝐸𝐹=𝐺RuO2/SiO2NWcable𝐺RuO2NW𝐷RuO2𝐸𝐹.(29) Referring back to Section 2 for 𝐷RuO2, and to (21) for the conductance ratio, 𝐷RuO2/SiO2 can be obtained and (28) evaluated for 𝐶RuO2/SiO2𝑖 as𝐶RuO2/SiO2𝑖=5.60aF/nm.(30) This result is over an order of magnitude lower compared to the RuO2 nanowire capacitance found in (4), not a totally unexpected result.

One might wonder what the value of 𝐶RuO2/SiO2𝑖(Δ𝑅cable=𝑡CNT) might be if the RuO2/SiO2 nanocable had the same cross-sectional area as a single-walled carbon nanotube. Using the SWCNT thickness 𝑡CNT=𝑎orbitalextent2𝑎C-C=2 (1.42=0.142nm), consistent with Mintmire and White [5] and Leonard [14], the area of a 𝑅=1 nm radius CNT ring or annulus is𝐴CNT=2𝜋𝑅𝑡CNT4𝜋𝑎C-C𝑅=0.568𝜋nm2.(31) Equation (31) allows us to express 𝐶RuO2/SiO2𝑖(Δ𝑅cable=𝑡CNT) as𝐶RuO2/SiO2𝑖Δ𝑅cable=𝑡CNT=𝐴CNT𝐴RuO2𝐶RuO2/SiO2𝑖=0.0256aF/nm(32) which is an order of magnitude smaller than 𝐶SWCNT𝑖=0.4 aF/nm for the SWCNT. This result is not entirely unexpected, since the electron conduction mediated by the orbitals perpendicular to the plane of the carbon nanotube cylindrical wall, whose physical extent is given by 𝑎orbitalextent, is known to be extremely large, quasiballistic in fact.

Because we know the measured value of the capacitance per gram of the RuO2/SiO2 nanocables, 𝐶RuO2/SiO2𝑔=700 F/gm, it is possible, if we assume all this capacitance comes from the intrinsic quantum capacitance, and no junction capacitances to be discussed in the next sections, to find the effective density of RuO2 in the RuO2/SiO2 nanocables. Therefore, setting 𝐶RuO2/SiO2𝑖,𝑔=𝐶RuO2/SiO2𝑔,𝐶RuO2/SiO2𝑖 given in (30) times the overall total length 𝐿, may be equated to 𝐶RuO2/SiO2𝑖,𝑔 times the total volume of the nanocables times its mass density 𝜌RuO2cable, yielding𝐶RuO2/SiO2𝑖=𝐶RuO2/SiO2𝑖,𝑔2𝜋𝑅cable𝑎𝑣Δ𝑅cable𝜌RuO2cable.(33) Solving for 𝜌RuO2cable,𝜌RuO2cable=𝐶RuO2/SiO2𝑖𝐶RuO2/SiO2𝑖,𝑔12𝜋𝑅cable𝑎𝑣Δ𝑅cable(34) which when evaluated using the available capacitances, gives the remarkable density𝜌RuO2cable=0.02052gm/cc.(35) Thus, the density reduction of RuO2 in the nanocable structure, allowing the high capacitive energy storage, is𝜌RuO2cable𝜌RuO2cry=340(36) using the known value for rutile (tetragonal) crystalline structured RuO2.

6. Semiconductor Junction Capacitances of Nanocables

As mentioned in Section 1, there may be metallic oxides, which when properly doped, that may act as semiconductors. We already know that Ga2O3 nanowires [3842], display cathodoluminescence. These oxides have a large bandgap, experimentally determined to be 𝐸𝑔=4.8 eV [38], whereas the theoretically determined value is about 5.8 eV [39]. When doped with Sn, a deep donor level 𝐸𝑑=0.96 eV below the conduction band arises [38], sufficient for allowing the measured cathodoluminescent properties. Because 𝑘𝐵𝑇=0.026 eV at room temperature, this will not be a material useful for ordinary semiconducting applications. However, the metallic oxide Ga2O3, and others such as WO3, MoO3, TiO2, V2O3, SnO2, In2O3, and VO2, and with other stoichiometric atomic combinations more favorable for obtaining suitable bandgaps, with available donor or acceptor species, may be found. Finally, like the metallic RuO2/SiO2 nanocables studied in Section 4, there may be semiconducting analogs.

In the next subsection, we will first look at one of the simplest cases, the junction between a planar metal contact and a semiconductor n-type nanocable. For that Schottky junction, its junction potential difference as a function of its n-type depletion width will be found, and from it the capacitance. After that, the much more complicated, but general case of an abrupt asymmetric p-n semiconductor nanocable junction will be addressed in the second subsection. Here the junction potential difference as a function of its p- and n-type depletion widths will be found, and for the two limiting cases of an infinitely high p-type doping density and symmetric doping densities, capacitances will be determined.

The following section, then, addresses nanowire junctions.

6.1. Schottky-Semiconductor Junction Capacitance of Nanocables

The nanocable potential functions can be found by an integral expression over a volume which accounts for nanocable annulus, radius, and length. For examination of the potential along the longitudinal axis of the cable, given a charge distribution 𝜌(𝑟,𝜙,𝑧) in the junction region between a metal contact and an n-doped thin annulus region (Figure 2(a)), this Schottky junction potential can be found [62]𝜑annulus=1(0,0,𝑧)𝜀𝑎𝑣𝑑𝑧𝑅0𝑠𝑑𝑠02𝜋𝜌𝑑𝜙𝑠,𝜙,𝑧𝑠2+(𝑧𝑧)2=2𝜋𝜀𝑎𝑣𝑑𝑧𝑅0𝑠𝜌𝑠,𝑧𝑑𝑠𝑠2+(𝑧𝑧)22𝜋𝜀𝑎𝑣𝑧𝑑𝑧𝑅𝜌𝑡NC𝑅2+(𝑧𝑧)2(37) leading to the kernel (or Green’s function)𝐾𝑧,𝑧=2𝜋𝑅𝑡NC𝜀NC1𝑅2+(𝑧𝑧)2.(38) Electrostatic Green’s function basis for (37) and (38) can be found in [63], for example, and is discussed in Appendix A (Green’s function solution of Poisson’s equation for electrostatic approach to field solution).

Figure 2: (a) Schottky junction between a planar metal contact and a semiconductor n-type nanocable. (b) Asymmetric semiconductor p-n nanocable junction. Two limits are examined in detail, the case when 𝜌𝑝, the infinitely asymmetric case, and the exactly symmetric case, when 𝜌𝑝=𝜌𝑛.

Potential due to the n-side of a metal planar contact-n nanocable junction is given, using (38), by [64, page 81, 2.261, 𝑐>0]𝑉ring,cable𝑛(𝑧)=𝑊0𝐾𝑧,𝑧𝜌𝑧𝑑𝑧=𝜌2𝜋𝑅𝑡NC𝜀NC𝑊0𝑑𝑧𝑅2+(𝑧𝑧)2=𝜌2𝜋𝑅𝑡NC𝜀NC𝑊𝑧𝑧𝑑𝑧𝑅2+𝑧2=𝜌2𝜋𝑅𝑡NC𝜀NC2ln𝑅2+𝑧2+𝑧||||𝑊𝑧𝑧=𝜌2𝜋𝑅𝑡NC𝜀NCln𝑅2+(𝑊𝑧)2+𝑊𝑧𝑅2+𝑧2𝑧(39) caused by the charge density depletion separation (assumed abrupt for simplicity nonabruptness is addressed in Appendix B) in the annulus volume𝜌𝑟𝑠𝑊(𝑧)=𝜌,0<𝑧<𝑊𝛿𝜌,𝛿<𝑧<0,0,𝑧<𝛿or𝑧>𝑊.(40) Here 𝜌=𝑒𝑁𝑑. Equation (40) has incorporated the condition of charge neutrality,𝐴NC𝑊0𝜌𝑧𝑑𝑧=𝐴NC0𝛿𝜌𝑧𝑑𝑧.(41) For the image charge in the metal (which is negative), its potential contribution is𝑉ring,cablemetal=(𝑧)0𝛿𝐾𝑧,𝑧𝜌𝑧𝑑𝑧𝑊=𝜌𝛿2𝜋𝑅𝑡NC𝜀NC0𝛿𝑑𝑧𝑅2+(𝑧𝑧)2𝑊=𝜌𝛿2𝜋𝑅𝑡NC𝜀NC𝑧𝛿𝑧𝑑𝑧𝑅2+𝑧2𝑊=𝜌𝛿2𝜋𝑅𝑡NC𝜀NC2ln𝑅2+𝑧2+𝑧||||𝑧𝛿𝑧𝑊=𝜌𝛿2𝜋𝑅𝑡NC𝜀NCln𝑅2+𝑧2𝑧𝑅2+(𝛿+𝑧)2.𝛿𝑧(42) The total potential along the nanocable length 𝑧 will then be a superposition of both the donor depletion and metal image ring charge potentials in (39) and(42)𝑉NC=𝑉ring,cablemetal+𝑉ring,cable𝑛(43) or𝑉NC=𝜌2𝜋𝑅𝑡NC𝜀NC×𝑊𝛿ln𝑅2+𝑧2𝑧𝑅2+(𝛿+𝑧)2𝛿𝑧+ln𝑅2+(𝑊𝑧)2+𝑊𝑧𝑅2+𝑧2.𝑧(44) We note that for large 𝑧 (𝑧𝑊,𝛿),𝑉NC=𝜌2𝜋𝑅𝑡NC𝜀NC𝛿1+𝑊𝑊𝑧2.(45)

The capacitance must be given by𝐶NC=±𝑑𝑄NC𝑑𝑉bias,(46) where the differential element is taken of𝑄NC=𝜌2𝜋𝑅NC𝑎𝑣Δ𝑅NC𝑊(47) and the bias voltage across the junction is related to the junction voltage 𝑉NC𝑗 by𝑉NC𝑗=𝑉𝑏𝑖±𝑉bias𝛼NC𝑘𝐵𝑇𝑒,(48) where sign of 𝑉bias is associated with, respectively, reverse or forward bias and the last term is an approximate correction due to the mobile majority carrier spatial distribution tail, discussed in Sze with related references [65]. The tail correction is based upon bulk arguments, and it is expected to be somewhat different by a proportional factor 𝛼NC. The junction voltage, enlisting (44), is𝑉NC𝑗=𝑉NC(𝑊)𝑉NC(𝛿)=𝜌2𝜋𝑡NC𝜀NC×𝑊𝛿ln𝑅2+𝑊2𝑊𝑅2+(𝛿+𝑊)2𝑅𝛿𝑊+ln𝑅2+𝑊2+𝑊𝑊𝛿ln𝑅2+𝛿2+𝛿𝑅ln𝑅2+(𝑊+𝛿)2+𝑊+𝛿𝑅2+𝛿2.+𝛿(49) For the case where 𝑊𝛿,𝑉NC𝑗||𝑊𝛿=𝜌2𝜋𝑡NC𝜀NC2𝑊𝑅11+(𝑅/𝑊)2.(50) Because we would expect 𝑊/𝑅1 for nanowires and nanocables, taking the limiting form of (50) for 𝑊/𝑅 is reasonable and yields𝑉NC𝑗||𝑊𝛿0𝑊/𝑅1𝜌4𝜋𝑡NC𝜀NC𝑊𝑅(51) allowing 𝑊 to be expressed as𝑊NC||𝑊/𝑅1𝛿/𝑅0=𝜀NC𝑉NC𝑗||𝑊𝛿0,𝑊/𝑅1𝑒𝑁𝑑𝑅4𝜋𝑡NC.(52) For an unbiased device, the junction voltage may be replaced by the built-in voltage, giving𝑊NC𝑏𝑖||𝑊/𝑅1𝛿/𝑅0=𝜀NC𝑉𝑏𝑖𝑒𝑁𝑑𝑅4𝜋𝑡NC,(53) where the 4𝜋 part is simply indicative that we are using cgs units.

The single permittivity 𝜀NC characterizing the nanocable takes into account the field penetration from the semiconducting shell into both the dielectric core and the outside medium, often air but it could be another surrounding dielectric. It might be estimated by𝜀NC𝑤innerdiel𝜀innerdiel+𝑤semishell𝜀semishell+𝑤outerdiel𝜀outerdiel𝑤innerdiel+𝑤semishell+𝑤outerdiel.(54) Equation (53) enables the use of a single permittivity, which was the basis of developing a tractable kernel or electrostatic Green’s function approach. Without this assumption, a much more complicated field matching approach must be utilized, involving continuity conditions at cylindrical interfaces implying Bessel function type solutions [6671]. It should be noted that a more accurate form of 𝐶NC can be found by using the equality in (50) and taking the derivative of both sides of that equation with respect to 𝑉bias, solving for 𝑑𝑊/𝑑𝑉bias, and inserting that into the capacitive expression of (46). When this is done, one finds that𝐶SchottkyNC=𝑅𝜀NC2×11211+(𝑊/𝑅)2+12𝑊𝑅211+(𝑊/𝑅)23/21.(55) To obtain 𝐶SchottkyNC in terms of 𝑉bi, set 𝑉NC𝑗=𝑉𝑏𝑖 on the left-hand side of(50)𝑉bi=𝜌2𝜋𝑡NC𝜀NC2𝑊𝑅11+(𝑅/𝑊)2(56) and solve for 𝑊(𝑉bi), and insert this into (55).

Taking the derivative of (52) with respect to 𝑉bias, and inserting into (46), the capacitance is𝐶SchottkyNC=12𝑅𝜀NC(57) which is amazingly independent of explicit doping dependence. However, (53) shows that the depletion width does have this dependence. If we evaluate the capacitance for radaii typical of a carbon nanotube, say at 𝑅=1 nm, then𝐶SchottkyNC=𝐶SchottkyCNT=0.0563aF,𝜀CNT=1.0.(58) Equation (58) assumes most of the volume is air. If instead, the CNT was surrounded by a high dielectric constant like water with 𝜀H2O=80.1 at 20 C, then 𝐶SchottkyNC=𝐶SchottkyCNT=4.5 aF.

For our RuO2/SiO2 nanocable dimensions, replacing 𝑅𝑅cable𝑎𝑣=31 nm makes the capacitance𝐶RuO2/SiO2NC,Schottky=6.44aF,𝜀NT=3.69(59) using a relative permittivity 𝜀NC=3.69 which is a compromise between the core 𝜀SiO2=2.15 at 546.1 nm [72], and that of RuO2 with 𝜀RuO2(𝐸>2eV)3,4<|𝜀RuO2(𝐸=1eV)|<6, and 2<𝜀RuO2,<3, where 𝜀RuO2, is the high frequency dielectric constant [57, 73, 74]. This choice will be utilized throughout the remainder of the paper for the nanocable.

One may wonder what happens to capacitance, if in formula (51) the power of the (𝑊/𝑅) factor was 2, not 1. (This actually happens for Schottky nanowires—see Section 7.2, and here occurs by dropping the second term in (50).) This makes 𝑊NC𝑏𝑖|𝑊/𝑅1𝛿/𝑅0=(𝑅𝜀NC𝑉𝑏𝑖)/[4𝜋𝑡NC𝑒𝑁𝑑], and leads to the capacitance per unit area of𝐶𝑢𝑎NC1=±𝐴NC𝑑𝑄NC𝑑𝑉bias=±𝑒𝑁𝑑𝑉𝑑𝑊bias𝑑𝑉bias=𝑒𝜀NC𝑁𝑑2𝑉𝑏𝑖±𝑉bias𝛼NC𝑘𝐵𝑇𝑒1/2𝑅8𝜋𝑡NC=𝜀NC2𝐿NC𝐷𝛽𝑉𝑏𝑖±𝛽𝑉bias𝛼NC1/2(60) and looks like the classical bulk form with planar junction modified by the last factor in the third line or has a newly defined Debye length 𝐿NC𝐷, given by𝐿NC𝐷=𝐿bulk𝐷8𝜋𝑡NC𝑅,𝐿bulk𝐷=𝜀NC𝑒𝑁𝑑𝛽𝑒,𝛽=𝑘𝐵𝑇.(61)

Let us evaluate 𝐶NC for a carbon nanotube, using (60), noting that for small bias voltages and a built-in voltage 𝑉bi=0.42 V typical of a SWCNT, 𝛼NC1, at room temperature, the square root factor reduces to 1/𝑉bi. Set 𝑅=1 nm and 𝑡NC=𝑡NT=0.284 nm, 𝐴NC=𝐴CNT=2𝜋𝑅𝑡NT, giving from  (60)𝐶SchottkyNC=𝑅𝑅𝑡NC𝑒𝜀NC𝑁𝑑𝜋4𝑉bi=𝑅24𝜀NC𝑊NCbi(62) (last equality in (62) follows from (53), and it yields the form 𝐶SchottkyNC,𝑢𝑎=(𝜀NC/𝑊NC𝑏𝑖)(𝑅/𝑡NC)(1/2)(1/[4𝜋]), with a typical bulk like factor [65], modified by the nanocable parameters). Equation (62) arises if the first term in the (𝑊/𝑅)2 contribution of (50) is dropped. Anyway, using (62) gives for the CNT capacitance𝐶SchottkyNC=𝐶CNT𝑁=0.033aF,𝑑=5×1020/cc,𝜀NT=1.0(63) which corresponds to a fraction 𝑓=102 of C atoms contributing electrons (𝑁𝑑=𝑓𝑁CNT=0.5/nm3), if the number of atoms in a volumetric sense is approximated as 𝑁CNT=5×1022/cc, a value consistent with Avogadro’s number and other atomic densities [75]. Using the volume in a 𝑙CNT=1 nm length, 𝑉CNT=𝑙CNT𝐴CNT=1.784 nm3, the number of doped atoms is 𝑁dopedatomsCNT=𝑉CNT𝑁𝑑=0.892 atoms, which is quite believable. An even more accurate way to estimate this number is to use the unwrapped flat graphene hexagonal unit cell size determined in terms of the C-C distance found in (31), 𝐴hex=(3/2)3𝑎C-C2, the number of carbon atoms in this cell 𝑁hex=2, and find the volume per atom as 𝑉atom=𝐴hex𝑁hex𝑡CNT=𝐴hex22𝑎C-C=323𝑎C-C3=7.439×103nm3(64) corresponding to 𝑁CNT=1/𝑉atom=1.344×1023/cc and 𝑁𝑑=𝑓𝑁CNT=1.34/nm3. The number of atoms in the solid annulus of the CNT then would be 𝑁atomsCNT=𝑉CNT/𝑉atom=239.82, making 𝑁dopedatomsCNT=𝑓𝑁atomsCNT=2.40.

For our RuO2/SiO2 nanocables dimensions, replacing 𝑅𝑅cable𝑎𝑣=31 nm and 𝑡NC𝛿𝑅cable=2 nm in (62), we find𝐶RuO2/SiO2NC,Schottky=60.46aF,𝑁𝑑=5×1020/cc,𝜀NT=3.69(65) using the same doping density we had for the carbon nanotube. That may not be entirely reasonable, and using a value two orders of magnitude lower for 𝑁𝑑 yields𝐶RuO2/SiO2NC,Schottky=6.05aF,𝑁𝑑=5×1018/cc,𝜀NT=3.69.(66) The 𝑁𝑑 employed in the last calculation is commonly seen for ordinary semiconductors, and avoids the high value enlisted in the CNT calculation, which as we had seen may even be higher, approaching 𝑁𝑑=1.344×1021/cc for 𝑓=102 fractional doping. (Even 𝑓=104 could yield 𝑁𝑑=1.344×1019/cc.)

What we learn from examining the capacitance results of (58) and (59) which rely upon a linear junction voltage-depletion width relationship, and (63), (65), and (66) which uses a planar bulk-like quadratic behavior, is that the values are quite sensitive to the details of the nanostructure geometry and associated derivation details.

Effect of evaluating 𝐶SchottkyNC using (57), the simpler Schottky junction capacitance formula, versus using (55), is shown in Figure 3, where the normalized capacitance 𝐶SchottkyNC/(𝑅𝜀NC) is plotted against the ratio 𝑊/𝑅. Also, formula (62) resulting from dropping a term, is also plotted. It is seen that agreement between (55) and (57) becomes very close as 𝑊/𝑅10, whereas for 𝑊/𝑅=1, the error is noticeable at 17.7%. Formula (62) has a declining trend, but is way off in magnitude from the accurate expression (55).

Figure 3: Normalized nanocable Schottky junction capacitance against 𝑊/𝑅. Normalization is to 𝑅𝜀NC. Formulas (55) and (57) providing 𝐶SchottkyNC are plotted. Also, as a comparison is a result for a bulk-like rendition given by (62).
6.2. Asymmetric-Semiconductor p-n Junction Capacitance of Nanocables

For the asymmetric abrupt p-n junction (Figure 2(b)), unequal doping occurs in the p- and n-sides of the nanocable. Equation (42) must be replaced by𝑉ring,cableasym,𝑝(𝑧)=0𝑧𝑝𝐾𝑧,𝑧𝜌𝑧𝑑𝑧=𝜌𝑝2𝜋𝑅𝑡NC𝜀NC0𝑧𝑝𝑑𝑧𝑅2+(𝑧𝑧)2=𝜌𝑝2𝜋𝑅𝑡NC𝜀NC𝑧𝑧𝑝𝑧𝑑𝑧𝑅2+𝑧2=𝜌𝑝2𝜋𝑅𝑡NC𝜀NC2ln𝑅2+𝑧2+𝑧||||𝑧𝑧𝑝𝑧=𝜌𝑝2𝜋𝑅𝑡NC𝜀NCln𝑅2+𝑧2𝑧𝑅2+𝑧𝑝+𝑧2𝑧𝑝𝑧(67) caused by the charge density depletion separation in the annulus volume𝜌NCasym𝜌(𝑧)=𝑛,0<𝑧<𝑧𝑛𝜌𝑝,𝑧𝑝<𝑧<0,0,𝑧<𝑧𝑝or𝑧>𝑧𝑛.(68) Charge neutrality demands that the condition of (41) be generalized for arbitrary depletion widths,𝐴NC𝑧𝑛0𝜌𝑧𝑑𝑧=𝐴NC0𝑧𝑝𝜌𝑧𝑑𝑧(69) which because of the unequal but constant doping densities assumed in (68), allows one depletion width to be determined in terms of the other:𝑧𝑝=𝜌𝑛𝜌𝑝𝑧𝑛.(70) Similarly, (39) must be replaced with𝑉ring,cableasym,𝑛=(𝑧)𝑧𝑛0𝐾𝑧,𝑧𝜌𝑧𝑑𝑧=𝜌2𝜋𝑅𝑡NC𝜀NC𝑧𝑛0𝑑𝑧𝑅2+(𝑧𝑧)2=𝜌2𝜋𝑅𝑡NC𝜀NC𝑧𝑛𝑧𝑧𝑑𝑧𝑅2+𝑧2=𝜌2𝜋𝑅𝑡NC𝜀NC2ln𝑅2+𝑧2+𝑧||||𝑧𝑛𝑧𝑧=𝜌2𝜋𝑅𝑡NC𝜀NCln𝑅2+𝑧𝑛𝑧2+𝑧𝑛𝑧𝑅2+𝑧2.𝑧(71)

The total potential along the nanocable length 𝑧 will then be a superposition of both the acceptor and donor and charge potentials in (67) and(71)𝑉NCasym=𝑉ring,cableasym,𝑝+𝑉ring,cableasym,𝑛(72) or𝑉NCasym(𝑧)=2𝜋𝑅𝑡NC𝜀NC×𝜌𝑝ln𝑅2+𝑧2𝑧𝑅2+𝑧𝑝+𝑧2𝑧𝑝𝑧+𝜌𝑛ln𝑅2+𝑧𝑛𝑧2+𝑧𝑛𝑧𝑅2+𝑧2𝑧(73) which makes the junction potential difference𝑉NCasym,𝑗=𝑉NC𝑧𝑛𝑉NC𝑧𝑝=2𝜋𝑅𝑡NC𝜀NC×𝜌𝑝ln𝑅2+𝑧2𝑛𝑧𝑛𝑅2+𝑧𝑝+𝑧𝑛2𝑧𝑝𝑧𝑛+𝜌𝑛𝑅ln𝑅2+𝑧2𝑛𝑧𝑛+𝜌𝑝ln𝑅2+𝑧2𝑝+𝑧𝑝𝑅𝜌𝑛ln𝑅2+𝑧𝑛+𝑧𝑝2+𝑧𝑛+𝑧𝑝𝑅2+𝑧2𝑝+𝑧𝑝=2𝜋𝑅𝑡NC𝜀NC×𝜌𝑝ln𝑅2+𝑧2𝑛𝑧𝑛𝑅2+𝑧2𝑝+𝑧𝑝𝑅𝑅2+𝑧𝑝+𝑧𝑛2𝑧𝑝𝑧𝑛+𝜌𝑛𝑅ln𝑅2+𝑧𝑛+𝑧𝑝2+𝑧𝑛+𝑧𝑝𝑅2+𝑧2𝑝+𝑧𝑝𝑅2+𝑧2𝑛𝑧𝑛.(74)

If we define the total depletion width of the nanocable as𝑊NCasym=𝑧𝑝+𝑧𝑛=𝜌𝑛𝜌𝑝𝑧+1𝑛(75) then when we examine the case when the p-region doping density gets large, 𝜌𝑝, (70) and (75) will reduce 𝑉NCasym,𝑗 to the form of (49). That is, a Schottky junction consisting of a perfect infinite metal plane contacting an n-doped nanocable is equivalent to an asymmetric p-n junction when the p-doping becomes extremely large compared to the n-doping.

For the situation of a symmetric junction, when 𝜌𝑝=𝜌𝑛=𝜌, and (70) becomes𝑧𝑝=𝜌𝑛𝜌𝑝𝑧𝑛=𝑧𝑛=𝑊(76) and (74) reduces to𝑉NCsym,𝑗=𝑉NC(𝑊)𝑉NC(𝑊)=𝜌2𝜋𝑅𝑡NC𝜀NC2ln𝑅2+𝑊2+𝑊𝑅2+𝑊2𝑊+ln𝑅2+4𝑊22𝑊𝑅2+4𝑊2.+2𝑊(77) (One notes that the symmetry properties 𝑉NCsym,𝑛(𝑊)=𝑉NCsym,𝑝(𝑊) and 𝑉NCsym,𝑛(𝑊)=𝑉NCsym,𝑝(𝑊) are satisfied by (67) and (71), and cause 𝑉NCsym,𝑗=2𝑉NCsym(𝑊).) Consider the limit of expression (77) when 𝑊/𝑅. The nanocable junction voltage reduces to𝑉NCsym,𝑗||𝑊/𝑅1𝜌4𝜋𝑅𝑡NC𝜀NW𝑊ln𝑅(78) which is a very different form of junction voltage dependence on 𝑊 than that for the Schottky nanocable junction seen in (51). It has gone from a linear to a logarithmic dependence. Using a formula like in (46) for the capacitance, namely,𝐶symNC=±𝑑𝑄symNC𝑑𝑉bias=±𝑒𝑁𝑑𝐴NC𝑉𝑑𝑊bias𝑑𝑉bias(79) we see that solving for 𝑊 in(78)𝑊𝑉𝑗=Re𝜀NC𝑉𝑗/(4𝜋𝜌𝑅𝑡NC)(80) and taking the derivative, yields𝐶symNC=12𝜀NC𝑊.(81) The exponent 𝛾 in (80), for low bias voltages having 𝑉𝑗=𝑉bi, and for the values used for the RuO2/SiO2 nanocables before, namely, 𝑉bi=0.42 V and 𝑅𝑅cable𝑎𝑣=31 nm and 𝑡NCΔ𝑅cable=2 nm, with 𝑁NC=𝑁𝑑=1019/cc (𝜌=𝑒𝑁𝑑), 𝛾=0.13905 making 𝑊=1.149 R. Clearly, this does not satisfy 𝑊𝑅 well. But if 𝑁𝑑 was an order of magnitude smaller, then 𝑊=4.0169 R, which is respectable. For the larger doping value, we calculate𝐶RuO2/SiO2NC,sym=6.95aF,𝑁𝑑=1.0×1019/cc,𝜀NC=3.69.(82)

Capacitance can be determined exactly by taking the derivative of (77), finding𝐶symNC=𝜀NC2𝑅2+𝑊22𝑅2+𝑊2/𝑅2+4𝑊2.(83) To obtain 𝐶symNC in terms of 𝑉bi, set 𝑉NC𝑗=𝑉bi on the left-hand side of (77):𝑉bi=𝜌2𝜋𝑅𝑡NC𝜀NC2ln𝑅2+𝑊2+𝑊𝑅2+𝑊2𝑊+ln𝑅2+4𝑊22𝑊𝑅2+4𝑊2+2𝑊(84) and solve for 𝑊(𝑉bi), and insert this into (83). One might wonder what form is obtained for 𝐶symNC by taking the limit 𝑊/𝑅 in (83) after its formula has been derived:𝐶symNC=13𝜀NC𝑊(85) which differs slightly from (81), making the capacitance somewhat smaller,𝐶RuO2/SiO2NC,sym=4.64aF,𝑁𝑑=1.0×1019/cc,𝜀NC=3.69.(86)

Comparison of the most general formula (83) for capacitance of a symmetric nanocable junction 𝐶symNC with either (81) or (85) in Figure 4, shows that the less accurate formulas seem to bracket it, with (81) almost always being greater than it, whereas (85) is always slightly less. Equation (81) diverges from (83) noticeably as the 𝑊/𝑅 ratio increases.

Figure 4: Normalized nanocable symmetric p-n semiconductor junction capacitance against 𝑊/𝑅. Normalization is to 𝑅𝜀NC. Formulas (81), (83), and (85) providing CsymNC are plotted.

Table 1 summarizes the nanocable capacitance formulas found in the last subsection and in this subsection. The formulas are given in unitless form because each capacitance is normalized to 𝑅𝜀𝑖 (this product’s units is Farads) where 𝑖=NC,NW. That is, the capacitance is provided as 𝐶/(𝑅𝜀𝑖).

Table 1: Capacitance formulas for nanocables and nanowires, under different limiting conditions.

7. Semiconductor p-n Junction Capacitances of Nanowires

The symmetric p-n junction for semiconductor nanowires is a basic building block of nanowire devices, and would be of great interest to determine its capacitance. The nanocable potential functions cannot be used because they only include a thin annulus of semiconducting cross-section, while the nanowire has a disk cross-section.

In the next subsection, we will first look at the high symmetry case of equal doping on either side of the semiconductor nanowire junction. After that, the much more complicated case of an abrupt asymmetric p-n semiconductor nanowire junction will be addressed in the second subsection. Here the junction potential difference as a function of its p- and n-type depletion widths will be found, and the limiting case of an infinitely high p-type doping density will be studied. That Schottky-like capacitance will be determined.

7.1. Symmetric Semiconductor p-n Junction Capacitance of Nanowires

We will look at the symmetric semiconductor nanowire p-n junction here (Figure 5(a)). First specify the depletion region charge density, which in the abrupt approximation, changes from (40) to𝜌NWsym(𝑧)=𝜌,0<𝑧<𝑊𝜌,𝑊<𝑧<0,0,|𝑧|>𝑊.(87) By inspecting the integral formula for a vacuum potential solution [62], the on axis value for the nanowire is (use transformation 𝑢=𝑠2+(𝑧𝑧)2,𝑑𝑢=2𝑠𝑑𝑠)𝜑NW=1(0,0,𝑧)𝜀NW𝑑𝑧𝑅0𝑠𝑑𝑠02𝜋𝜌𝑑𝜙𝑠,𝜙,𝑧𝑠2+(𝑧𝑧)2=2𝜋𝜀NW𝑑𝑧𝑅0𝜌𝑠𝑑𝑠𝑠,𝑧𝑠2+(𝑧𝑧)2=2𝜋𝜀NW𝜌𝑧𝑑𝑧𝑅0𝑠𝑑𝑠𝑠2+(𝑧𝑧)2=2𝜋𝜀NW𝜌𝑧𝑅2+(𝑧𝑧)2𝑧𝑧𝑑𝑧(88) leading to the kernel (∣ ∣ = abs( ) operator is chosen leading to the correct branch cuts)𝐾𝑧,𝑧=2𝜋𝜀NW𝑅2+(𝑧𝑧)2||𝑧𝑧||(89) which is used for calculating the potential from the p-side of the junction [76], see [76, page 59, 260.01 and page 60, 262.01], and also [64, page 86, 2.271,  3., with 𝐼1 for 𝑐>0]𝑉NWsym,𝑝(𝑧)=0𝑊𝐾𝑧,𝑧𝜌𝑧𝑑𝑧=2𝜋𝜀NW0𝑊𝜌𝑧𝑅2+(𝑧𝑧)2+𝑧𝑧𝑑𝑧=𝜌2𝜋𝜀NW𝑋0𝑋𝑊𝑋2𝑑𝑋𝑋2𝑅21+𝑧𝑊1+2𝑊𝑧=𝜌2𝜋𝜀NW×(𝑧+𝑊)𝑅2+(𝑧+𝑊)2𝑧𝑅2+𝑧22𝑅22ln𝑅2+(𝑧+𝑊)2(𝑧+𝑊)𝑅2+𝑧21𝑧𝑧𝑊1+2𝑊𝑧.(90) where the change of variables 𝑋2=𝑅2+(𝑧𝑧)2, (𝑧𝑧)𝑑𝑧=𝑋𝑑𝑋 was used. (Note, care is required in selecting correct branch cuts, and we use 𝑋2𝑅2 in the third line of (90), and 𝑋=𝑅2+(𝑧𝑧)2 and 𝑆=+𝑋2𝑅2 in the fourth line of (90).) The potential from the 𝑛-side of the junction will be𝑉NWsym,𝑛=(𝑧)𝑊0𝐾𝑧,𝑧𝜌𝑧𝑑𝑧=𝜌2𝜋𝜀NW𝑊0𝑅2+(𝑧𝑧)2𝑧𝑧𝑑𝑧=𝜌2𝜋𝜀NW𝑋𝑆2+𝑅22||||||||ln𝑋+𝑆𝑋𝑊𝑋01𝑧𝑊12𝑊𝑧=𝜌2𝜋𝜀NW×(𝑊𝑧)𝑅2+(𝑧𝑊)2+𝑧𝑅2+𝑧22𝑅22ln𝑅2+(𝑧𝑊)2+𝑧𝑊𝑅2+𝑧21+𝑧𝑧𝑊12𝑊𝑧.(91) In (90) and (91), the symbol has its “−” and “+” signs refer to, respectively, 𝑧𝑊 and 𝑧𝑊. Thus the electrostatic Green’s function for the bounded or partitioned problem in 𝑧-space is specified in two out of its three spatial regions. Since we will not be making evaluations in the interior depletion charged region 𝑊<𝑧<𝑊, it is not supplied here, although it can also be determined. Branch cuts selected for 𝑉NWsym,𝑝(𝑧) and 𝑉NWsym,𝑛(𝑧) satisfy the physical symmetry and limiting conditions𝑉NWsym,𝑛(𝑊)=𝑉NWsym,𝑝𝑉(𝑊),NWsym,𝑛(𝑊)=𝑉NWsym,𝑝(𝑊),(92)lim𝑧𝑉NWsym,𝑛(𝑧)=lim𝑧𝑉NWsym,𝑝(𝑧)=𝜌𝜋𝑅2𝑊/𝑧.(93)

Figure 5: (a) Symmetric semiconductor p-n nanowire junction. (b) Asymmetric semiconductor p-n nanowire junction. This is examined for the limiting case when 𝜌𝑝.

The total potential due to both the p- and n-sides of the junction depletion region will be (mobile carriers are neglected, which would migrate to the outer part of the nanowire cylinder, and not allow 𝜌(𝑧)=𝜌=const to be extracted from the integration), enlisting (90) and (91),𝑉NWsym(𝑧)=𝑉NWsym,𝑝(𝑧)+𝑉NWsym,𝑛(𝑧)(94) or𝑉NWsym(𝑧)=𝜌2𝜋𝜀NW12(𝑧+𝑊)𝑅2+(𝑧+𝑊)2(𝑊𝑧)𝑅2+(𝑧𝑊)2𝑧𝑅2+𝑧212𝑅2×ln𝑅2+(𝑧+𝑊)2(𝑧+𝑊)𝑅2+(𝑧𝑊)2+𝑧𝑊𝑅2+𝑧2+𝑧𝑅2+𝑧2𝑧𝑊2.(95) Junction voltage is found from a relationship like in (72), using either the general formula (95), or more simply, from the symmetry conditions in (92), which make𝑉NWsym,𝑗=𝑉NW(𝑊)𝑉NW𝑉(𝑊)=2NW𝑝(𝑊)+𝑉NW𝑛(𝑊)=2𝑉NWsym(𝑉𝑊).(96)NWsym,𝑗=𝜌4𝜋𝜀NW𝑊𝑅2+𝑊2𝑊𝑅2+(2𝑊)2+𝑅22×ln(2𝑊)2+𝑅22𝑊𝑅𝑊2+𝑅2+𝑊𝑊2+𝑅2𝑊+𝑊2.(97) Again it should be noted that a more accurate form of 𝐶NW can be found by directly using formula in (97) and taking the derivative of both sides of that equation with respect to 𝑉bias, utilizing (48), solving for 𝑑𝑊/𝑑𝑉bias, and inserting that into a capacitive expression like that of (55). We will not do that here first, but obtain a simpler form instead by considering the limiting form for 𝑊/𝑅, which allows 𝑉NWsym,𝑗||𝑊/𝑅1𝜌𝜋𝑅2𝜀NW𝑅1+2𝑊2𝑊ln𝑅.(98) Equation (95) does not allow 𝑊 to be expressed as, say 𝑊NW|𝑊/𝑅1=𝜀NW𝑉NWsym,𝑗|𝑊/𝑅1/(4𝜋𝑒𝑁𝑑), which for an unbiased device, replacing the junction voltage by the built-in voltage, gives 𝑊NWbi|𝑊/𝑅1=𝜀NW𝑉bi/(4𝜋𝑒𝑁𝑑), whose form is reminiscent of the two-sided abrupt junction depletion width seen in [66], namely, 𝑊=sqrt(4𝜀𝑠𝑉bi/[𝑞𝑁𝐵]) (Factor of 1/[16𝜋] under the square root operator, has 1/[4𝜋] of it due to the use of cgs units.)

The capacitance of the p-n nanowire junction with area 𝐴NW=𝜋𝑅2, will be using (98),𝐶𝑝-𝑛NW=±𝑑𝑄NW𝑑𝑉bias=±𝜋𝑅2𝜌𝑉𝑑𝑊bias𝑑𝑉bias=12𝜀NW𝑊(𝑊/𝑅)2.12ln(𝑊/𝑅)(99) Formula (99) is really meant to be used for large 𝑊/𝑅 ratios, but often the trend can be found for modest 𝑊/𝑅 values. However, we see here that a singularity occurs at 𝑊/𝑅=𝑒1/21.65 and that even if we kept the ln(𝑊/𝑅) factor in the denominator, the expression would be useless at low 𝑊/𝑅 values near one.

To obtain a more accurate form of 𝐶𝑝-𝑛NW, employ (97) and take its derivative on both sides of that equation with respect to 𝑉bias, utilizing (48), solving for 𝑑𝑊/𝑑𝑉bias, and inserting that into a capacitive expression like (79), namely, the first line of(99)𝐶𝑝-𝑛NW=𝑅𝜀NW4𝑊1+𝑅2𝑊1+4𝑅2+𝑊/𝑅1+(𝑅/𝑊)22𝑊/𝑅1+(𝑅/2𝑊)2+11+(𝑊/𝑅)211+(2𝑊/𝑅)2+2𝑊𝑅1.(100) To obtain 𝐶𝑝-𝑛NW in terms of 𝑉bi, set 𝑉NW𝑗=𝑉bi on the left-hand side of (97):𝑉bi=𝜌4𝜋𝜀NW×𝑊𝑅2+𝑊2𝑊𝑅2+4𝑊2+𝑅22ln4𝑊2+𝑅22𝑊𝑅𝑊2+𝑅2+𝑊𝑊2+𝑅2𝑊+𝑊2(101) and solve for 𝑊(𝑉bi), and insert this into (100).

An approximation to the nanowire p-n junction capacitance capacitance may be examined for large 𝑊/𝑅 by taking the limit of (100) for 𝑊/𝑅, with the result𝐶𝑝-𝑛NW=𝜀NW𝑊.(102)

Evaluating (102) for a small radius nanowire like the previously examined carbon nanotube, with 𝑅=1 nm, gives𝐶𝑝-𝑛NW=0.113aF,𝜀NW=1.(103) Alternatively, evaluating (102) for a large radius nanowire like the previously examined RuO2/SiO2 nanocable, with 𝑅=31 nm, gives𝐶𝑝-𝑛NW=12.7aF,𝜀NW=3.69.(104)

Figure 6 shows the dependence of nanowire junction capacitance 𝐶𝑝-𝑛NW on 𝑊/𝑅. The approximate formula (102) and the more accurate formula (100) increasingly diverge from each other as 𝑊/𝑅 increases, with the approximate relation overestimating capacitance in excess of a factor of two at 𝑊/𝑅=10.

Figure 6: Normalized nanowire symmetric p-n semiconductor junction capacitance against 𝑊/𝑅. Normalization is to 𝑅𝜀NW. Formulas (100) and (102) providing 𝐶𝑝-𝑛NW are plotted.
7.2. Asymmetric Semiconductor p-n Junction Capacitance of Nanowires

The asymmetric nanowire semiconductor p-n junction (Figure 3(b)) is considerably more involved than the previous symmetric nanowire case. Depletion region charge density of the nanowire is generalized from (87), as was the nanocable in (66), to𝜌NWasym𝜌(𝑧)=𝑛,0<𝑧<𝑧𝑛𝜌𝑝,𝑧𝑝<𝑧<0,0,𝑧<𝑧𝑝or𝑧>𝑧𝑛.(105) Potential contribution from the p-side of the nanowire junction, using (105), will be𝑉NWasym,𝑝=(𝑧)0𝑧𝑝𝐾𝑧,𝑧𝜌𝑧𝑑𝑧=2𝜋𝜀NW0𝑧𝑝𝜌𝑧𝑅2+(𝑧𝑧)2+𝑧𝑧𝑑𝑧=𝜌𝑝2𝜋𝜀NW𝑋0𝑋𝑝𝑧𝑋2𝑑𝑋𝑋2𝑅2+𝑧𝑧𝑝11+2𝑧𝑝𝑧=𝜌𝑝2𝜋𝜀NW𝑧+𝑧𝑝𝑅2+𝑧+𝑧𝑝2𝑧𝑅2+𝑧22𝑅22ln𝑅2+𝑧+𝑧𝑝2𝑧+𝑧𝑝𝑅2+𝑧2𝑧𝑧𝑧𝑝11+2𝑧𝑝𝑧.(106) Likewise, the potential contribution from the n-side of the nanowire junction will be𝑉NWasym,𝑛(𝑧)=𝑧𝑛0𝐾𝑧,𝑧𝜌𝑧𝑑𝑧=𝜌𝑛2𝜋𝜀NW𝑧𝑛0𝑅2+(𝑧𝑧)2𝑧𝑧𝑑𝑧=𝜌𝑛2𝜋𝜀NW𝑋𝑆2+𝑅22||||||||ln𝑋+𝑆𝑋𝑧𝑛𝑋0𝑧𝑧𝑛112𝑧𝑛𝑧=𝜌𝑛2𝜋𝜀NW×𝑧𝑛𝑧𝑅2+𝑧𝑧𝑛2+𝑧𝑅2+𝑧22𝑅22ln𝑅2+𝑧𝑧𝑛2+𝑧𝑧𝑛𝑅2+𝑧2+𝑧𝑧𝑧𝑛112𝑧𝑛𝑧.(107) Total potential due to both the p- and n-sides of the asymmetric nanowire junction depletion region will be𝑉NWasym(𝑧)=𝑉NWasym,𝑝(𝑧)+𝑉NWasym,𝑛(𝑧)(108) and inserting into this (106) and (107) yields𝑉NWasym=(𝑧)2𝜋𝜀NW𝜌𝑝𝑧𝑅2+𝑧2𝑧+𝑧𝑝𝑅2+𝑧+𝑧𝑝22+𝜌𝑝𝑅22ln𝑅2+𝑧+𝑧𝑝2𝑧+𝑧𝑝𝑅2+𝑧2𝑧±𝜌𝑝𝑧𝑧𝑝11+2𝑧𝑝𝑧+𝜌𝑛𝑧𝑛𝑧𝑅2+𝑧𝑧𝑛2+𝑧𝑅2+𝑧22𝜌𝑛𝑅22ln𝑅2+𝑧𝑧𝑛2+𝑧𝑧𝑛𝑅2+𝑧2+𝑧𝜌𝑛𝑧𝑧𝑛112𝑧𝑛𝑧.(109) Junction voltage is calculated from𝑉NWasym,𝑗=VNWasym𝑧𝑛𝑉NWasym𝑧𝑝(110) and when evaluating (109) at 𝑧=𝑧𝑛,𝑧𝑝, with a total nanowire depletion width 𝑊𝑇=𝑊NWasym=𝑧𝑝+𝑧𝑛 we find 𝑉NWasym,𝑗=2𝜋𝜀NW𝑊𝑇𝑅2+𝑊𝑇22𝜌𝑝+𝜌𝑛+𝑧𝑛𝜌𝑝𝑅2+𝑧𝑛2+𝑧𝑝𝜌𝑛𝑅2+𝑧𝑝22+𝑅22𝜌𝑝ln𝑅2+𝑊𝑇2𝑊𝑇𝑅2+𝑧𝑛2𝑧𝑛+𝜌𝑛ln𝑅2+𝑊𝑇2𝑊𝑇𝑅2+𝑧𝑝2𝑧𝑝+𝑧𝑛𝑧𝑝𝜌𝑝+𝜌𝑛𝑧𝑝𝜌𝑝𝑅2+𝑧𝑝2𝑧𝑛𝜌𝑛𝑅2+𝑧𝑛22𝑅22𝜌𝑛𝑅ln𝑅2+𝑧𝑛2+𝑧𝑛+𝜌𝑝