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

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.

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 ¡ c 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].
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).

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 e, is given in the report by Amantram and Léonard [13] and the text by Leonard [14] as where D l (E F ) is the density of states at the Fermi level E F and the subscript l on D l 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=EF δk = δk/[2πD(E F )], 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 In the text the kernel is developed based upon the outer shell being much thicker than the inner core, namely, that t NC R.
of the entire nanowire (or nanocable) to V + δV, which adds the charge 2 δVD l (E) 0 −1 df ] and C i = δQ/δV.D l (E F ) may be calculated from the bulk three-dimensional (3D) density of states D(E F ) using the following relationship: At the Fermi level, for RuO 2 , D(E F ) 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 The final formula for C i , in units of aF/nm, is given by Setting D RuO2 (E F ) ≈ 3 eV −1 /cell, and the nanowire radius R = 1 nm, we find that This may be compared to a single-walled carbon nanotube intrinsic capacitance C SWCNT i = 4e 2 /(πDv F ) = 0.4 aF/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 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.

Quantum Conductance and Transmission Coefficient of RuO 2 Nanowires
At room temperature, the resistivity of RuO 2 is given by [55] ρ RuO2 bulk = 34 μΩ • cm (5) a value considerably higher than, say a good monoatomic metal like silver, whose resistivity value is ρ Ag bulk = 1.62 μΩ•cm [60].The value shown in ( 5) is in nearly perfect agreement with that in [37], who cites the value from [61], as ρ RuO2 bulk = 35 μΩ•cm.Using the slightly smaller value, we can write the conductivity as With this conductivity value, the conductance of a 1 μm long nanowire is be calculated to be .2394 × 10 −6 mhos = 9.2394 μS.(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 [10][11][12][13][14] where h is Planck's constant, T m 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 G 0 = 2e 2 /h, g spin , and the summation over the integrals for each scattering mechanism, I mG .Then (8) may be rewritten as where we will take the spin degeneracy to be g spin = 2, 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 G 0s (G 0s = 154.94μS) 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 G 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

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 (18) 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 ΔR = 2 nm and R SiO2 = 30 nm, A clust RuO2 = 124π nm 2 , and taking σ RuO2/SiO2 NW cable = 0.5 S/cm and L RuO2/SiO2 NW cable = 1 μm, 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 200 μΩ • cm, 170 nm film (23) so using a value of ρ ITO = 300 μΩ•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.

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, D RuO2/SiO can be estimated by Referring back to Section 2 for D RuO2 , and to (21) for the conductance ratio, D RuO2/SiO2 can be obtained and ( 28) evaluated for C RuO2/SiO2 i 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 C RuO2/SiO2 i (ΔR cable = t CNT ) 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 t CNT = a orbital extent ≈ 2a C-C = 2 (1.42 Å = 0.142 nm), consistent with Mintmire and White [5] and Leonard [14], the area of a R = 1 nm radius CNT ring or annulus is Equation (31) allows us to express C RuO2/SiO2 i (ΔR cable = t CNT ) as which is an order of magnitude smaller than C SWCNT i = 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 a orbital extent , 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, C RuO2/SiO2 g = 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 RuO 2 in the RuO 2 /SiO 2 nanocables.Therefore, setting (30) times the overall total length L, may be equated to times the total volume of the nanocables times its mass density ρ RuO2 cable , yielding Solving for ρ RuO2 cable , 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 .

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][39][40][41][42], display cathodoluminescence.These oxides have a large bandgap, experimentally determined to be E g = 4.8 eV [38], whereas the theoretically determined value is about 5.8 eV [39].When doped with Sn, a deep donor level E d = 0.96 eV below the conduction band arises [38], sufficient for allowing the measured cathodoluminescent properties.Because k B T = 0.026 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.

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] ϕ annulus (0, 0, z 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 contactn 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 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 ρ p → ∞, the infinitely asymmetric case, and the exactly symmetric case, when ρ p = ρ n .
Here ρ = eN d .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 z 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 z (z W, δ), 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 V NC j by where sign of V bias 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 α NC .The junction voltage, enlisting (44), is For the case where W δ, Because we would expect W/R 1 for nanowires and nanocables, taking the limiting form of (50) for W/R → ∞ is reasonable and yields allowing W to be expressed as For an unbiased device, the junction voltage may be replaced by the built-in voltage, giving where the 4π part is simply indicative that we are using cgs units.
The single permittivity ε NC 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][67][68][69][70][71].It should be noted that a more accurate form of C NC can be found by using the equality in (50) and taking the derivative of both sides of that equation with respect to V bias , solving for dW/dV bias , and inserting that into the capacitive expression of (46).When this is done, one finds that To obtain C Schottky NC in terms of V bi , set V NC j = V bi on the lefthand side of( 50) and solve for W(V bi ), and insert this into (55).
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 and leads to the capacitance per unit area of (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 L NC D , given by Let us evaluate C NC for a carbon nanotube, using (60), noting that for small bias voltages and a built-in voltage V bi = 0.42 V typical of a SWCNT, α NC ≈ 1, at room temperature, the square root factor reduces to 1/ V bi .Set R = 1 nm and t NC = t NT = 0.284 nm, A NC = A CNT = 2πRt NT , giving from ( 60) (last equality in (62) follows from (53), and it yields the form C ), 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 capacitance which corresponds to a fraction f = 10 −2 of C atoms contributing electrons (N d = f N CNT = 0.5/nm 3 ), if the number of atoms in a volumetric sense is approximated as N CNT = 5 × 10 22 /cc, a value consistent with Avogadro's number and other atomic densities [75].Using the volume in a l CNT = 1 nm length, V CNT = l CNT A CNT = 1.784 nm 3 , the number of doped atoms is N doped atoms CNT = V CNT N d = 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), the number of carbon atoms in this cell N hex = 2, and find the volume per atom as 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 N d yields The N d 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 N d = 1.344 × 10 21 /cc for f = 10 −2 fractional doping.(Even f = 10 −4 could yield N d = 1.344 × 10 19 /cc.)What we learn from examining the capacitance results of ( 58) and ( 59) which rely upon a linear junction voltagedepletion 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 C
Schottky NC using (57), the simpler Schottky junction capacitance formula, versus using (55), is shown in Figure 3, where the normalized capacitance C Schottky NC /(Rε NC ) 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/R → 10, 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).are plotted.Also, as a comparison is a result for a bulk-like rendition given by (62). of the nanocable.Equation ( 42) must be replaced by

Asymmetric-Semiconductor p-n Junction
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 z 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, ρ p → ∞, (70) and (75) will reduce V NC asym, 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 ndoping.
For the situation of a symmetric junction, when ρ p = ρ n = ρ, and (70) becomes and ( 74) reduces to (One notes that the symmetry properties W) are satisfied by ( 67) and ( 71), and cause V NC sym, j = 2V NC sym (W).)Consider the limit of expression (77) when W/R → ∞.The nanocable junction voltage reduces to 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, dV bias (79) we see that solving for W in( 78) and taking the derivative, yields The exponent γ in (80), for low bias voltages having V j = V bi , and for the values used for the RuO 2 /SiO 2 nanocables before, namely, V bi = 0.42 V and R → R cable Capacitance can be determined exactly by taking the derivative of (77), finding To obtain C sym NC in terms of V bi , set V NC j = V bi on the lefthand side of (77): and solve for W(V bi ), and insert this into (83).One might wonder what form is obtained for C sym NC by taking the limit W/R → ∞ 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 C sym NC 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.
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 Rε i (this product's units is Farads) where i = NC, NW.That is, the capacitance is provided as C/(Rε i ).

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.

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 trans- 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, where the change of variables (Note, care is required in selecting correct branch cuts, and we use − √ X 2 − R 2 in the third line of (90), and (90).)The potential from the n-side of the junction will be In ( 90) and ( 91), the ∓ symbol has its "−" and "+" signs refer to, respectively, z ≥ W and z ≤ −W .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 V NW sym, p (z) and V NW sym, n (z) 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 ρ(z ) = ρ = const 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 C NW can be found by directly using formula in (97) and taking the derivative of both sides of that equation with respect to V bias , utilizing (48), solving for dW/dV bias , 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 Equation ( 95) does not allow W to be expressed as, say ), which for an unbiased device, replacing the junction voltage by the builtin voltage, gives form is reminiscent of the two-sided abrupt junction depletion width seen in [66], namely, W = sqrt(4ε s V bi /[qN B ]) (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 A NW = πR 2 , will be using (98), 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 = e 1/2 ≈ 1.65 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 C p-n NW , employ (97) and take its derivative on both sides of that equation with respect to V bias , utilizing (48), solving for dW/dV bias , and inserting that into a capacitive expression like (79), namely, the first line of( 99) To obtain C p-n NW in terms of V bi , set V NW j = V bi on the lefthand side of (97): and solve for W(V bi ), 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 result Evaluating (102) for a small radius nanowire like the previously examined carbon nanotube, with R = 1 nm, gives Figure 6 shows the dependence of nanowire junction capacitance C p-n NW 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.

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 z = z n , −z p , with a total nanowire depletion width W T = W NW asym = z p + z n we find Now consider the case when the p-side becomes highly doped, ρ p → ∞.Formula (111) becomes, if one also allows when simplifying notation W T → z n = W.Because this is similar to the altered form of (51) when we studied the effect then of a quadratic W dependence, we may write W as and giving for an unbiased device, The capacitance will be, enlisting (113), In mks units this would yield 2πR 2 ε NW /W, and the per-unit area capacitance would be C NW,ua asym | ρp → ∞, W/R 1 = 2ε NW /W, displaying the familiar form [62] which is inversely proportional to W. Evaluating (115) for a large radius nanowire like the previously examined RuO 2 /SiO 2 nanocable, with R = 31 nm, gives Figure 7 shows the dependence of the asymmetric nanowire junction capacitance C NW asym | ρp → ∞, W/R 1 on the W/R ratio, as given by (115).
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 Rε NW (this product's units is Farads).The last three rows pertain to the nanowire results.

Discussion and Comparison of Quantum
Capacitances and Junction Capacitances results plotted shows that although there is a declining dependence with W/R as noted before, it is slower than the simple planar bulk-like behavior seen for 3D junctions one is familiar with.This arises from the different junction voltage dependence seen in ( 50), which may be approximated less drastically than (51) to better understand how it controls the roll-off seen in Figure 3. Instead of dropping the second term of ( 50), but expanding it, we find that giving an effective junction voltage dependence only occurred when the most crude approximation is made, namely, letting the voltage-W/R ratio be perfectly linear, as in (51).
We also found that an asymmetric p-n nanocable junction, in the limit that the p type doping becomes huge, limits to the just discussed metal-semiconductor junction for nanocables.Now consider the other Schottky case treated, that for the nanowire.It was studied by considering an asymmetric p-n junction, and then taking the limit as its p side doping approaches a value much in excess of the finite doped nside.The result is (112), a quadratic dependence on W/R, which we know gives that classic rapid roll-off in capacitance C Schottky NW , displayed in Figure 7. Thus, for both the nanocable and the nanowire, when considering a junction voltage dependence of the form V Schottky j ∝ (W/R) 1+Δ , we see that 0 < Δ < 1.One naturally asks how can it be that for these special cases we have some mimicking of 3D behavior.The answer may reside in the fact that the huge charge storage capacity of a nearly infinitely doped side is somewhat equivalent to a planar surface unlimited in its 2D sheet extent.So a parallel plate capacitor, which is a feature of the abrupt planar 3D junction analysis, may be roughly satisfied for Schottky nanocable and nanowire junctions.
Clearly this is not so for finite doped asymmetric or symmetrically doped nanocable or nanowire junctions.Examination of the exact expressions for voltage-depletion width ratio, in ( 77) and (97), does not allow one to readily pick out an easily recognizable behavior.However, the approximations allow one to see that logarithmic or modified logarithmic dependences occur in (78) and (98).These dependences cause the capacitances to vary roughly linearly ((81) or (85) for the nanocable, and (102) for the nanowire).Therefore, in the form V p-n sym, j ∝ (W/R) 1+Δ , Δ < 0 must hold, or the dependence is sublinear.Not surprisingly, the capacitance behavior against W/R does not decline, but rather increases, as seen in Figures 4 and 6.
The increasing capacitances with W/R may arise from the fact that the charge, when changing from a 3D parallel plate configuration, has in the nanocable or nanowire realizations, had the majority of the infinite sheet folded down and onto a

Formula
Formula (C/[Rε]) Type Limit finite extent cylinder.Thus, as junction voltage is increased, the charge is now stored along the cylinder, whereas before it could be placed largely on the infinite sheets.The net effect of this change in charge storage placement, is that no longer does the capacitance provided by a p-n junction appear as a simple parallel plate capacitor with a separation between plates equal to W. So instead of the capacitance declining with W as in the simple parallel plate picture, the capacitance may actually go up because of the extra charge swept through when the distance is shifted along the nanocable or nanowire cylinder.

Intrinsic Quantum Capacitances of Nanocables and
Nanowires.There are two widely different perspectives on the configurational placement and consequent charge storage access the nanocables and nanowires will have when considered for energy/charge storage uses such as in fuel cells, batteries, and supercapacitors, versus uses as single or finite numbered electrodes, channels, and other electronic device applications.
In the energy storage/charge storage model, the nanocable or nanowire has direct access to the enveloping medium, which may be a liquid electrolyte, and this provides parallel access charge pathways to the structures.This is not the case, for example, in many electronic devices, for example, when a single nanocylinder is being used as an FET channel, and an electrode below, above, or surrounding it, is acting as the gate.In this case, the nanostructure may be modeled as a cylinder over a ground plane (or back gate), and its classically determined capacitance is C grd-pl = πε/sinh −1 (2h/R) which for h R, is C grd-pl ≈ πε/ ln(2h/R) [77, page 1182, (10.1.11),page 1211, (10.1.44)],where R is again merely the cylinder radius, and h is the cylinder center-to-ground separation.One calculates C grd-pl ≈ 0.005 aF/nm for R = 0.75 nm and h = 100 nm when ε = 1.For a much smaller h = 10 nm when ε = 4, C grd-pl ≈ 0.034 aF/nm.Comparing this value to the intrinsic quantum capacitance mentioned earlier in Section 2 for single-walled carbon nanotubes, C SWCNT i = 0.4 aF/nm, shows that C grd-pl < C SWCNT i .Because the gate potential voltage influencing the nanostructure is in a series circuit (the charge on the gate must flow, even if it is displacement current given by dD/dt in Maxwell's equation, from the gate through the intervening dielectric medium, and then into the nanostructure), the actual capacitive influence of the gate on the nanostructure will be It is clear that for the selected values for the classical capacitance, C grd-pl C SWCNT i and C gate ≈ C grd-pl .That is, the intrinsic quantum capacitance has little effect on the gate-channel biasing relationship.This may not be true for much larger permittivities, as seen for materials like HfO 2 , whose dielectric constant is ε = 16 − 22 [78][79][80][81][82], which could push C grd-pl up to C grd-pl ≈ 0.187 aF/nm, and then only about a factor of two smaller than C SWCNT i .Another instance where the order of dominance switches, is that of the liquid gating, which has been used to achieve strong field effect action on carbon nanotubes when the nanotube is immersed in solution.The circuit is still essentially one in series, but because some liquids like water (which has ε = 80) have very large dielectric constants compared to conventional materials like SiO 2 or Si 2 N 4 , this can enormously enhance capacitance.Leonard [14] estimates the capacitance per unit length to be C lg ≈ 10 aF/nm using the approximation C lg ≈ 2πε/ ln(2l V /Rγ E ) Journal of Nanomaterials (assumes R l V ), with the potential screening distance in the electrolyte l V = εk B T/[Z 2 e 2 N I ], ion valence Z, N I ion density per unit volume, and γ E Euler's constant.Reason why C lg ≈ 10 aF/nm is so large compared to C grd-pl is due not only to the large ε, but also because both estimation formulas have logarithmic arguments, with vastly different normalized distances.For C grd-pl , that distance is 2h, which we had chosen to be between 200 nm down to 20 nm, whereas, for C lg ≈ 10 aF/nm, it is 2l V /γ E , both normalized with respect to R. For water, Leonard indicates 2l V /γ E ≈ 1 nm [14].With the normalizing distance R = 0.75 nm, S lg-bg ≈ ln(2h/R)/ ln(2l V /Rγ E ), making S lg-bg ≈ ln(266.7)/ln(1.333), and acknowledging that despite R l V not being well satisfied, going ahead anyway to get the rough estimate S lg-bg ≈ 5.586/0.2877= 19.42,for the h = 100 nm case.If one takes the old C grd-pl ≈ 0.005 aF/nm value and multiplies by a factor of 80 for permittivity, and 20 for the logarithmic length rescaling, one would obtain 1600 times the old value, giving C lg ≈ 8 aF/nm.For h = 10 nm, S lg-bg ≈ 3.283/0.2877= 11.41, and taking a factor of ten for the logarithmic length rescaling, gives C lg ≈ 4 aF/nm.Both results show that C lg is not only much larger than C grd-pl , but considerably in excess of C SWCNT i , making the intrinsic quantum capacitance the controlling storage element in these series types of circuit arrangements found in FET like structures.

Various Comparisons of Quantum and Junction Capacitances of Nanocables and Nanowires.
In order to compare the various intrinsic quantum capacitances with each other, they are plotted against the length over radius ratio in Figure 8 according to (4), (30), and (32) for, respectively, the RuO 2 nanowire, the RuO 2 /SiO 2 nanocable, and the thin walled shell RuO 2 /SiO 2 nanocable.To plot against L/R, each formula must be rescaled to read C iT = RC i (L/R) in aF (attoFarads), and it is this rescaling that causes the nanocable curve to exceed in magnitude (and slope) the nanowire curve.The thin walled nanocable and the singlewalled carbon nanotube (a hollow nanocable) provide stark comparisons, because they are so small and nearly overlap the abscissa.For proper viewing of the SWCNT curve and the thin walled nanocable curve, the results of Figure 8 are replotted in Figure 9 in a log 10 -linear display.The nearly 4 orders of magnitude variation becomes apparent in the Figure 9 plot.
Similarly to the previous comparison plots between the various intrinsic quantum capacitances, comparison plots of the various junction capacitances are provided in Figure 10.The most accurate formulas (8), ( 13), (18), and (22), are utilized giving, respectively, the junction capacitances for the Schottky RuO 2 /SiO 2 nanocable, the symmetric p-n RuO 2 /SiO 2 nanocable, the symmetric p-n RuO 2 nanowire, and the Schottky RuO 2 nanowire.To plot against W/R, each formula must be rescaled to read C jT = C n j (Rε) in aF (attoFarads), where the normalized junction capacitances C n j previously plotted are unnormalized here.These curves are then replotted in a log 10 -linear display in Figure 11, which pulls the Schottky RuO 2 nanowire off of the abscissa when it asymptotes.4), ( 30) and ( 32) and the SWCNT for, respectively, the RuO 2 nanowire, the RuO 2 /SiO 2 nanocable, the thin walled shell RuO 2 /SiO 2 nanocable, and the carbon nanotube.The other reason for replotting the junction capacitance curves on a log 10 -linear display, is that a direct comparison of the junction capacitances can be made to all of the intrinsic quantum capacitance results.This is accomplished in Figure 12, where all of the intrinsic quantum capacitance results and all of the junction capacitance results are shown together.There is an implicit equivalencing of the length L for the nanostructures used in the intrinsic quantum capacitance calculations and the depletion width W in the junction capacitance calculations, namely, that L ↔ W, allowing one abscissa coordinate to be used.The combined plot shows immediately that the intrinsic quantum capacitance by about one order of magnitude exceeds all of the other capacitances,  for the considered RuO 2 and RuO 2 /SiO 2 nanostructures.Only for the ultra thin shelled nanocables, namely the SWCNT and the very thin walled RuO 2 /SiO 2 nanocable, is this not true for the quantum capacitances.For the very thin walled RuO 2 /SiO 2 nanocable, its quantum capacitance is always less than all of the quantum capacitances and all of the junction capacitances.For the SWCNT quantum capacitance, this is true up to L/R = W/R = 4, when it crosses the Schottky nanowire junction capacitance, and exceeds it beyond that value, staying below the Schottky nanocable junction capacitance.is anhydrous [30], versus being hydrous [83], and this is sure to affect the potential distributions beyond the coaxial cylinder for the nanocable, and the distributions for a simple continuous core in a RuO 2 nanowire, in the past sections the focus has been on the calculation of properties based upon the physics.Here we would like to insert some assessment of possible electrochemical factors, from the perspective of understanding supercapacitors, variously also known as ultracapacitors, and their relation to batteries.For the moment, consider a capacitance C under discussion, to be a constant with voltage.Then, considering the simple arrangement of a parallel plate, as charge is piled up on the plates, with increasing charge accumulated, it becomes increasingly harder to add additional charge.This is why the energy stored on the capacitor will be E cap = CV 2 /2 = QV/2, with the factor of 1/2 included, for V being the final potential difference on the plates.However, this is not the case for the electrochemical charging of a cell in a battery obeying a Nerstian relation.For the battery, E batt = CV 2 = QV , and this relation was obtained noting the fact that for an ideal battery, the voltage on the cell remains constant as more charge is accumulated [84].

Intrinsic quantum capacitance
If C is not a constant, then it is suggested that C = dQ/dV instead of C = Q/V , which may occur in the case of doublelayer or pseudocapacitance at electrodes [84].The doublelayer capacitance C dl ubiquitously arises at all electrode interfaces, having higher values for aqueous electrolytes than for nonaqueous solvents or organic surfactants.The doublelayer capacitance values can be quite large, and are caused by the small double-layer charge separations on the order of 3 Å for a compact double-layer.Extremely diffuse doublelayers, on the other hand, can have charge distributions over an extent of 0.1 μm.Utilization of the double-layer at electrodes, with typical values 15-50 μF/cm 2 , can result in very large capacitive densities of 250 F/g, using an effective area per gram of 1000 m 2 /g as for carbon, for example.This is consistent with the large values seen for anhydrous or hydrous RuO 2 (whose value can be 3 times larger).The double-layer capacitance is affiliated with electrostatic charge separation.
Pseudocapacitance is distinguished from double-layer capacitance by its origin-it arises in cases where Faradaic charge transfer processes lead to passage of charge that depend on thermodynamic factors and the potential.Conway et al. [84] designate this capacitance as C φ , and associate it with redox reactions for which the potential is a logarithmic function of the ratio of activities of the oxidized and reduced species, or with a process of progressive occupation of the surfaces sites on an electrode by underpotentialdeposited species.
In view of Section 8.2, here is proposed a circuit model for an extended length nanowire or nanocable, or an electrode made thereof, based upon the equivalent circuits in Sections 3.2-3.4 of [84].In the circuit diagram, shown in Figure 13, we see the intrinsic quantum capacitance, denoted by C QC , linked on the porous nanostructure by a resistance R p .The quantum capacitances come off of this trunk, pointing perpendicular to it, branching in a series fashion, into a parallel arrangement of the doublelayer C dl and pseudo C φ -capacitances.Resistances R F and R D are, respectively, the Faradaic charging and discharging resistances.R D and C φ are in parallel, in series with R F , and the entire parallel arrangement in series with the solution resistance R S .One terminal B, is connected off of these solution resistances, whereas the other terminal A is attached to the nanostructures.One may view the previously discussed liquid gating capacitance C lg in Section 8.2, as inserted in place of the double-layer capacitance C dl , or the pseudocapacitance C φ , depending upon the specific electrochemical interactions taking place.

Conclusions
We have derived and calculated the intrinsic quantum capacitances and transmission coefficients based upon invoking ab initio first principle density of states values.These capacitances are on the order of attofarads (aF's), and have been found for RuO 2 nanowires (63 aF) and RuO 2 /SiO 2 nanocables (6 aF).Comparison to single-walled carbon nanotubes (SWCNTs) has been made (0.4 aF).We have also calculated the Schottky, unsymmetric and symmetric junction capacitances of nanocables, and evaluated the formulas for both SWCNTs (0.06 aF for doping exceeding 10 20 /cc of a 1 nm tube radius) and RuO 2 /SiO 2 nanocables (6 aF for a 30 nm inner dielectric core radius).RuO 2 and CNT nanowire capacitances have been calculated for symmetric and unsymmetric p-n junctions, and found to be as low as 0.024 aF for doping exceeding 10 18 /cc (and a 1 nm radius) and as high as 13 aF for a 30 nm nanowire radius.(Note that the results here were more carefully and thoroughly developed than in two earlier works [85] and [86, (53) and ( 57) are incorrect, and are correctly stated here as ( 97) and (101)], with entirely new aspects presented for the first time.) With the continuing current pursuit of employing nanotubes [87][88][89][90][91], nanowires [92,93] and including V 3 O 7 •H 2 O nanowires and Si/a-Si core/shells or nanocables [94], and nanocables [95] in various components from electronics (diodes, transistors, photodetectors, and photovoltaic cells) to chemistry (sensors, membranes, catalysts, batteries, fuel cells, and supercapacitors), there is little doubt that knowledge of the intrinsic quantum capacitances obtained here will be useful.That is also true for the fundamental junction capacitances found here for these nanoscopic structures.In fact, the idea of obtaining high energy and charge storage is very much uppermost in the minds of researchers and technologists these days, as is readily evidenced, for example, by the recent result on supercapacitors fabricated using SWCNTs [96].This trend of enhancing capacitance in terms of its uses and quantitative charge and energy storage is sure to endure in the future.
The Fourier transform of the kernel is determined as follows.Enlisting (38) for K(z, z ), and a similar transform pair to (B.13), where the variable z = z/R and K 0 is the modified Bessel function of integer order n = 0 [98, page 376, 9.6.21for qR > 0].For the small K 0 argument limit, an approximation derivable from small argument limits of the Bessel functions of the first and second kinds, J 0 and N 0 , may be found utilizing [99], K 0 (u) ≈ − ln(γu/2) ≈ − ln(u), reducing K(q) in (B.16) to This is a reasonable argument when thinking about large z values.
Nanocable Fourier transform potential V NC (q) evaluation by (B.15) necessitates obtaining the Fourier transform of V α 0 (q).If one takes V 0 (q) as a first approximation to V α 0 (q), then in principle V NC (q) can be found from the simplified form of (B.15), where V 0 (q) is Placing these last expressions into (B.20)provides the formula using the normalization z = qz.If one takes W ≈ 1/q in (B23), then it becomes where the I cn integral is Clearly this integral is not well posed, as it has a troublesome singularity at z = 0.This arose from making many simplifications in arriving at its final simplified form.The value of the highly simplified form of (B.25) is that it demonstrates that the form of V 0 (q) is roughly an odd function of q, which allows V NC (z) to be obtained from the inverse Fourier transform of (B.18) in a direct manner, where the third equality comes from V NC (q) being an odd function of q, [ln(qR) = log(|q|R) for q > 0 and ln(qR) = log(|q|R) + iπ for q < 0, so that for small q, ln(qR) → −∞ ≈ ln(|q|R)], the 7th from assuming an upper limit to capture much of the integral, q c ≈ 1/z, and the 8th from the small angle approximation for the sin function.
Last integral in (B.26) may be treated by a normalization q = q/q c and a sign absorption from the prefactor: − qc 0 1 ln qR dq = −q c 1 0 dq ln q + ln q c R = q c 1 0 dq − ln q − ln q c R = q c 1 0 dq a − ln q = q c e a Ei(−a) = q c e − ln(qcR) Ei ln q c R .For the constant a in the argument of the Ei function being a large number (z is large, q c therefore is tiny), the function may be examined in this limit, which does display the V NC (z) ∝ 1/ ln(R/z) logarithmic dependence seen by Leonard [14].

Figure 1 :
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 t NC R.
and the tetragonal values of the cell sides [34] are a( Å) = b( Å) = 4.45 and c( Å) = 3.16, making V RuO2 uni-cell = 65.07078Å3 ≈ 0.065071 nm 3 .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].

2 /
) assumes most of the volume is air.If instead, the CNT was surrounded by a high dielectric constant like water with ε H2O = 80.1 at 20 C, then C SiO 2 nanocable dimensions, replacing R → R cable av = 31 nm makes the capacitance

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

Figure 9 :
Figure 9: Intrinsic quantum capacitance C iT (aF) against L/R curves in Figure 8 replotted in a log 10 -linear display.

Figure 11 :
Figure 11: Junction capacitance C j (aF) against W/R curves in Figure 8 replotted in a log 10 -linear display.

8. 4 .Figure 12 :
Figure 12: Intrinsic quantum capacitance C iT (aF) and junction capacitance C j (aF) replotted in a log 10 -linear display, against L/R or W/R as a single abscissa.

29 )
Placing the (B.29) result into the V NC (z) formula from (B.28), noting that zq c ≈ 1, 344 × 10 23 /cc and N d = f N CNT = 1.34/nm 3 .The number of atoms in the solid annulus of the CNT then would be N atoms CNT = V CNT /V atom = 239.82,making N (65).46 aF, N d = 5 × 10 20 /cc, ε NT = 3.69(65) which is slightly faster than linear because Δ > 0. It is this super linear behavior which causes C

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