Journal of Nanomaterials

Volume 2010 (2010), Article ID 160639, 27 pages

http://dx.doi.org/10.1155/2010/160639

## 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.

#### Abstract

Here we calculate the intrinsic quantum capacitance of RuO_{2} nanowires and RuO_{2}/SiO_{2} 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 RuO_{2} nanowires and RuO_{2}/SiO_{2} nanocables demonstrates that junction capacitance values also occur in the aF range. Comparisons are made between the intrinsic quantum and junction capacitances of RuO_{2} nanowires and RuO_{2}/SiO_{2} nanocables, and between them and intrinsic quantum and junction capacitances of carbon nanotubes. We find that the intrinsic quantum capacitance of RuO_{2}-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 [1–4], carbon nanotubes (single and many walled) [5–19], metallic nanowires, and now nanoribbon graphene [20–29], 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 RuO_{2}, deposited as a shell on an inner silicon dioxide SiO_{2} 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 RuO_{2} in the RuO_{2}/SiO_{2} 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 RuO_{2} per square of SiO_{2} paper—28 per square), ultrahigh surface area (90 m^{2} per gram RuO_{2}), anomalously high energy storage (>700 F per gram anhydrous RuO_{2}), 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 RuO_{2}, 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 Ga_{2}O_{3} nanowires [38–42], electrochromic behavior materials (WO_{3}, MoO_{3}, TiO_{2}, V_{2}O_{3}, and Sb-doped SnO_{2} dispersed into inert inorganic supports) [43, 44], WO_{3} nanowires on [45], resistive gas-sensor materials (like WO_{3}, SnO_{2}, and In_{2}O_{3}) [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 VO_{2}, and are semiconductors below a critical temperature C with an optical bandgap eV [51].

The following four sections treat the intrinsic quantum capacitance of RuO_{2} nanowires (Section 2), quantum conductance and transmission coefficient of RuO_{2} nanowires (Section 3), quantum conductance and transmission coefficient of RuO_{2}/SiO_{2} nanocables (Section 4), and intrinsic quantum capacitance of RuO_{2}/SiO_{2} 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 RuO_{2} 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 , is given in the report by Amantram and Léonard [13] and the text by Leonard [14] as
where is the density of states at the Fermi level and the subscript on 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 , finding for a single electron , using the discrete level separation for a nanowire (or nanocable) of length , and equating this band structure energy to the capacitive energy stored , where giving , making capacitance per unit length equal to *= **= *. Expression (1) may also be obtained by incrementing the electrostatic potential of the entire nanowire (or nanocable) to , which adds the charge or , which yields [because which makes ] and . may be calculated from the bulk three-dimensional (3D) density of states using the following relationship:
At the Fermi level, for RuO_{2}, 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 and the tetragonal values of the cell sides [34] are and , making . Note that the values of , and used are close to earlier reported values of and [59].

The final formula for , in units of aF/nm, is given by
Setting , and the nanowire radius , we find that
This may be compared to a single-walled carbon nanotube intrinsic capacitance [14]. The reason why the SWCNT formula is independent whereas the nanowire is not, is that for the nanowire larger means a larger cross-sectional area of atoms to include, whereas for the SWCNT, its thickness remains one atomic layer thick no matter what is. It is apparent from this calculation, that the charge storage capacity of RuO_{2} 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 RuO_{2} surrounding an inner SiO_{2} 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 RuO_{2} Nanowires

At room temperature, the resistivity of RuO_{2} is given by [55]
a value considerably higher than, say a good monoatomic metal like silver, whose resistivity value is *μ*Ω·cm [60]. The value shown in (5) is in nearly perfect agreement with that in [37], who cites the value from [61], as *μ*Ω·cm. Using the slightly smaller value, we can write the conductivity as
With this conductivity value, the conductance of a 1 *μ*m long nanowire is be calculated to be

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 [10–14]
where is Planck’s constant, is the transmission coefficient for the th mode, and 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 , and the summation over the integrals for each scattering mechanism, . Then (8) may be rewritten as
where we will take the spin degeneracy to be , and
We will assume that this integral can be represented approximately by the following product at the Fermi level:
and inserting this into (9) yields
where we use ( *μ*S) as the fundamental quantum conductance with spin degeneracy included, and write the summation of all modes of scattering as
because it is the total conductance that is measured. Placing (13) into (12) allows that total transmission coefficient to be calculated as
because the Ferm-Dirac function, , is simply a half when evaluated at the Fermi level. For a 1 nm radius nanowire of RuO_{2}, we find from (14) that

#### 4. Quantum Conductance and Transmission Coefficient of RuO_{2}/SiO_{2} Nanocables

These nanocables (see Figure 1(b)), coaxial geometry with an inner SiO_{2} solid cylindrical core and an outer cylindrical shell of RuO_{2}, are described in [30], as having electronic conductivity
To use a formula like (7) to find the conductance, we need the inner core radius, outer shell radius, and their difference:
Because the core-shell combination acts as a parallel resistor system, the conductance must be
where the second equality holds when the inner core is perfectly insulating, which we are assuming.

The area of the thin annulus of RuO_{2} clusters forming the outer shell, has an area equal to the difference between the inner and outer circular cross-sections, or
For nm and nm, nm^{2}, and taking S/cm and *μ*m,
This is considerably smaller than the RuO_{2} nanowire result (see (7)) by a factor of
whose value is substantially smaller than an Ag nanowire by a factor of
How does this value compare to indium tin oxide (ITO; 5 wt% SnO_{2} + 95% wt% In_{2}O_{3}) which is studied by Kim et al. [48]? They found that
so using a value of *μ*Ω·cm for a ITO nanowire, we see that
The transmission coefficient is given by (14),
This is much smaller than that for the RuO_{2} nanowire, and indicates that a RuO_{2}/SiO_{2} nanocable, with RuO_{2} 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 RuO_{2}/SiO_{2} Nanocables

Following formula (1), the intrinsic quantum capacitance of RuO_{2}/SiO_{2} nanocables is
with the density of states expressible for the nanocable (Figure 1 (b)) as
Again, inserting (27) into (26), we obtain a formula similar to (3),
Here, can be estimated by
Referring back to Section 2 for , and to (21) for the conductance ratio, can be obtained and (28) evaluated for as
This result is over an order of magnitude lower compared to the RuO_{2} nanowire capacitance found in (4), not a totally unexpected result.

One might wonder what the value of might be if the RuO_{2}/SiO_{2} nanocable had the same cross-sectional area as a single-walled carbon nanotube. Using the SWCNT thickness (), consistent with Mintmire and White [5] and Leonard [14], the area of a nm radius CNT ring or annulus is
Equation (31) allows us to express as
which is an order of magnitude smaller than 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 , is known to be extremely large, quasiballistic in fact.

Because we know the measured value of the capacitance per gram of the RuO_{2}/SiO_{2} nanocables, 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 RuO_{2} in the RuO_{2}/SiO_{2} nanocables. Therefore, setting given in (30) times the overall total length , may be equated to times the total volume of the nanocables times its mass density , yielding
Solving for ,
which when evaluated using the available capacitances, gives the remarkable density
Thus, the density reduction of RuO_{2} in the nanocable structure, allowing the high capacitive energy storage, is
using the known value for rutile (tetragonal) crystalline structured RuO_{2}.

#### 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 Ga_{2}O_{3} nanowires [38–42], display cathodoluminescence. These oxides have a large bandgap, experimentally determined to be eV [38], whereas the theoretically determined value is about 5.8 eV [39]. When doped with Sn, a deep donor level eV below the conduction band arises [38], sufficient for allowing the measured cathodoluminescent properties. Because eV at room temperature, this will not be a material useful for ordinary semiconducting applications. However, the metallic oxide Ga_{2}O_{3}, and others such as WO_{3}, MoO_{3}, TiO_{2}, V_{2}O_{3}, SnO_{2}, In_{2}O_{3}, and VO_{2}, 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 RuO_{2}/SiO_{2} 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 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] leading to the kernel (or Green’s function) 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).

Potential due to the *n*-side of a metal planar contact-*n* nanocable junction is given, using (38), by [64, page 81, 2.261, ]
caused by the charge density depletion separation (assumed abrupt for simplicity nonabruptness is addressed in Appendix B) in the annulus volume
Here . Equation (40) has incorporated the condition of charge neutrality,
For the image charge in the metal (which is negative), its potential contribution is
The total potential along the nanocable length will then be a superposition of both the donor depletion and metal image ring charge potentials in (39) and(42)
or
We note that for large (),

The capacitance must be given by where the differential element is taken of and the bias voltage across the junction is related to the junction voltage by where sign of 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 . The junction voltage, enlisting (44), is For the case where , Because we would expect for nanowires and nanocables, taking the limiting form of (50) for is reasonable and yields allowing to be expressed as For an unbiased device, the junction voltage may be replaced by the built-in voltage, giving where the part is simply indicative that we are using cgs units.

The single permittivity 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 by 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 [66–71]. It should be noted that a more accurate form of can be found by using the equality in (50) and taking the derivative of both sides of that equation with respect to , solving for , and inserting that into the capacitive expression of (46). When this is done, one finds that To obtain in terms of , set on the left-hand side of(50) and solve for , and insert this into (55).

Taking the derivative of (52) with respect to , and inserting into (46), the capacitance is 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 nm, then Equation (58) assumes most of the volume is air. If instead, the CNT was surrounded by a high dielectric constant like water with at 20 C, then aF.

For our RuO_{2}/SiO_{2} nanocable dimensions, replacing nm makes the capacitance
using a relative permittivity which is a compromise between the core at 546.1 nm [72], and that of RuO_{2} with , and , where 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 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 , and leads to the capacitance per unit area of 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 , given by

Let us evaluate for a carbon nanotube, using (60), noting that for small bias voltages and a built-in voltage V typical of a SWCNT, , at room temperature, the square root factor reduces to . Set nm and nm, , giving from (60)
(last equality in (62) follows from (53), and it yields the form , with a typical bulk like factor [65], modified by the nanocable parameters). Equation (62) arises if the first term in the contribution of (50) is dropped. Anyway, using (62) gives for the CNT capacitance
which corresponds to a fraction of C atoms contributing electrons (nm^{3}), if the number of atoms in a volumetric sense is approximated as cc, a value consistent with Avogadro’s number and other atomic densities [75]. Using the volume in a nm length, nm^{3}, the number of doped atoms is 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), , the number of carbon atoms in this cell , and find the volume per atom as
corresponding to cc and nm^{3}. The number of atoms in the solid annulus of the CNT then would be , making .

For our RuO_{2}/SiO_{2} nanocables dimensions, replacing nm and nm in (62), we find
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 yields
The 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 cc for fractional doping. (Even could yield 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 using (57), the simpler Schottky junction capacitance formula, versus using (55), is shown in Figure 3, where the normalized capacitance is plotted against the ratio . Also, formula (62) resulting from dropping a term, is also plotted. It is seen that agreement between (55) and (57) becomes very close as , whereas for , the error is noticeable at 17.7%. Formula (62) has a declining trend, but is way off in magnitude from the accurate expression (55).

##### 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 by
caused by the charge density depletion separation in the annulus volume
Charge neutrality demands that the condition of (41) be generalized for arbitrary depletion widths,
which because of the unequal but constant doping densities assumed in (68), allows one depletion width to be determined in terms of the other:
Similarly, (39) must be replaced with

The total potential along the nanocable length will then be a superposition of both the acceptor and donor and charge potentials in (67) and(71) or which makes the junction potential difference

If we define the total depletion width of the nanocable as
then when we examine the case when the *p*-region doping density gets large, , (70) and (75) will reduce 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 , and (70) becomes
and (74) reduces to
(One notes that the symmetry properties and are satisfied by (67) and (71), and cause .) Consider the limit of expression (77) when . The nanocable junction voltage reduces to
which is a very different form of junction voltage dependence on 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,
we see that solving for in(78)
and taking the derivative, yields
The exponent in (80), for low bias voltages having , and for the values used for the RuO_{2}/SiO_{2} nanocables before, namely, V and nm and nm, with cc (), making R. Clearly, this does not satisfy well. But if was an order of magnitude smaller, then R, which is respectable. For the larger doping value, we calculate

Capacitance can be determined exactly by taking the derivative of (77), finding To obtain in terms of , set on the left-hand side of (77): and solve for , and insert this into (83). One might wonder what form is obtained for by taking the limit in (83) after its formula has been derived: which differs slightly from (81), making the capacitance somewhat smaller,

Comparison of the most general formula (83) for capacitance of a symmetric nanocable junction 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 ratio increases.

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 (this product’s units is Farads) where . That is, the capacitance is provided as .

#### 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) to
By inspecting the integral formula for a vacuum potential solution [62], the on axis value for the nanowire is (use transformation )
leading to the kernel (∣ ∣ = abs( ) operator is chosen leading to the correct branch cuts)
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 for ]
where the change of variables , was used. (Note, care is required in selecting correct branch cuts, and we use in the third line of (90), and and in the fourth line of (90).) The potential from the -side of the junction will be
In (90) and (91), the symbol has its “−” and “+” signs refer to, respectively, and . Thus the electrostatic Green’s function for the bounded or partitioned problem in -space is specified in two out of its three spatial regions. Since we will not be making evaluations in the interior depletion charged region , it is not supplied here, although it can also be determined. Branch cuts selected for and satisfy the physical symmetry and limiting conditions

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 to be extracted from the integration), enlisting (90) and (91),
or
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 make
Again it should be noted that a more accurate form of can be found by directly using formula in (97) and taking the derivative of both sides of that equation with respect to , utilizing (48), solving for , 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 , which allows
Equation (95) does not allow to be expressed as, say , which for an unbiased device, replacing the junction voltage by the built-in voltage, gives , whose form is reminiscent of the two-sided abrupt junction depletion width seen in [66], namely, (Factor of under the square root operator, has of it due to the use of cgs units.)

The capacitance of the *p-n* nanowire junction with area , will be using (98),
Formula (99) is really meant to be used for large ratios, but often the trend can be found for modest values. However, we see here that a singularity occurs at and that even if we kept the factor in the denominator, the expression would be useless at low values near one.

To obtain a more accurate form of , employ (97) and take its derivative on both sides of that equation with respect to , utilizing (48), solving for , and inserting that into a capacitive expression like (79), namely, the first line of(99) To obtain in terms of , set on the left-hand side of (97): and solve for , and insert this into (100).

An approximation to the nanowire *p-n* junction capacitance capacitance may be examined for large by taking the limit of (100) for , with the result

Evaluating (102) for a small radius nanowire like the previously examined carbon nanotube, with nm, gives
Alternatively, evaluating (102) for a large radius nanowire like the previously examined RuO_{2}/SiO_{2} nanocable, with nm, gives

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

##### 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), to
Potential contribution from the *p*-side of the nanowire junction, using (105), will be
Likewise, the potential contribution from the n-side of the nanowire junction will be
Total potential due to both the *p*- and *n*-sides of the asymmetric nanowire junction depletion region will be
and inserting into this (106) and (107) yields
Junction voltage is calculated from
and when evaluating (109) at , with a total nanowire depletion width we find