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

Authors: C. M. Krowne
Journal: Journal of Nanomaterials
 (2010)

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## Abstract

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.


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## Body

## 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 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 [38–42], 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 Section8 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 (Figure1(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(1)Ci=e2Dl(EF),
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=e∫-∞∞Dl(E)δ[f(E-EF)]dE or δQ=e∫-∞∞Dl(E)[f(E+eδV-EF)-f(E-EF)]dE, which yields δQ=e2Dl(EF)δV [because δQ≈-eδV∫-∞∞Dl(E)[df(E-EF)/dE]dE which makes δQ≈-e2δVDl(E)∫-∞∞[df(E-EF)/dE]dE=-e2δVDl(E)∫-10df] and Ci=δQ/δV. Dl(EF) may be calculated from the bulk three-dimensional (3D) density of states D(EF) using the following relationship:(2)Dl(EF)=πR2Vunit-cellD(EF).
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 Vuni-cellRuO2=abc and the tetragonal values of the cell sides [34] are a(Å)=b(Å)=4.45 and c(Å)=3.16, making Vuni-cellRuO2=65.07078Å3≈0.065071nm3. 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].(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, thattNC≫R.
(a)(b)The final formula forCi, in units of aF/nm, is given by(3)CiRuO2=21.1R̅2(nm)DRuO2(#states/eV⋅unit cell)aF/nm.
Setting DRuO2(EF)≈3eV-1/cell, and the nanowire radius R=1nm, we find that(4)CiRuO2=63.3aF/nm.
This may be compared to a single-walled carbon nanotube intrinsic capacitance CiSWCNT=4e2/(πℏvF)=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](5)ρbulkRuO2=34μΩ⋅cm
a value considerably higher than, say a good monoatomic metal like silver, whose resistivity value is ρbulkAg=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 ρbulkRuO2=35 μΩ·cm. Using the slightly smaller value, we can write the conductivity as(6)σbulkRuO2=1ρbulkRuO2=29410Ω-1⋅cm-1=29410S/cm.
With this conductivity value, the conductance of a 1 μm long nanowire is be calculated to be(7)GNWRuO2=ANWRuO2LNWRuO2σbulkRuO2=9.2394×10-6mhos=9.2394μS.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](8)G=2e2hgspin∑m∫-∞∞Tm(E){-∂f(E)∂E}dE.
where h 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/h,gspin, and the summation over the integrals for each scattering mechanism, ImG. Then (8) may be rewritten as(9)G=G0gspin∑mImG,
where we will take the spin degeneracy to be gspin=2, and(10)ImG=∫-∞∞Tm(E){-∂f(E)∂E}dE.
We will assume that this integral can be represented approximately by the following product at the Fermi level:(11)ImG=Tm(EF)f(EF)
and inserting this into (9) yields(12)G=G0sf(EF)∑mTm(EF),
where we use G0s (G0s=154.94 μS) as the fundamental quantum conductance with spin degeneracy included, and write the summation of all modes of scattering as(13)T(EF)=∑mTm(EF)
because it is the total conductance G that is measured. Placing (13) into (12) allows that total transmission coefficient to be calculated as(14)T(EF)=2GG0s
because the Ferm-Dirac function, f(E)=1/[1+exp(E-EF)/kBT], is simply a half when evaluated at the Fermi level. For a 1 nm radius nanowire of RuO2, we find from (14) that(15)TNWRuO2(EF)=0.1193≈0.12.


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

These nanocables (see Figure1(b)), coaxial geometry with an inner SiO2 solid cylindrical core and an outer cylindrical shell of RuO2, are described in [30], as having electronic conductivity(16)σNW cableRuO2/SiO2=0.3-0.8Scm,5wt%RuO2onSiO2.
To use a formula like (7) to find the conductance, we need the inner core radius, outer shell radius, and their difference:(17)ΔR=RRuO2-RSiO2=2-3nm,RSiO2=30nmandup,RRuO2=RSiO2+ΔR.
Because the core-shell combination acts as a parallel resistor system, the conductance must be(18)GNW cableRuO2/SiO2=1LNWcableRuO2/SiO2{ASiO2σSiO2+ARuO2σRuO2}=ARuO2LNWcableRuO2/SiO2σRuO2,
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, or(19)ARuO2clust=2πRavcableΔRcable,Ravcable=(RRuO2+RSiO2)2.
For ΔR=2 nm and RSiO2=30 nm, ARuO2clust=124π nm2, and taking σNWcableRuO2/SiO2=0.5 S/cm and LNWcableRuO2/SiO2=1 μm, (20)GNWcableRuO2/SiO2=1.95×10-8mhos=0.0195μS.
This is considerably smaller than the RuO2 nanowire result (see (7)) by a factor of(21)GNWRuO2GNWcableRuO2/SiO2=474
whose value is substantially smaller than an Ag nanowire by a factor of(22)GNWAgGNWRuO2=σAgσRuO2=21.
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 that(23)ρITO={400μΩ⋅cm,150nmfilm,200μΩ⋅cm,170nmfilm
so using a value of  ρITO=300 μΩ·cm for a ITO nanowire, we see that(24)GNWRuO2GNWITO=σRuO2σITO=8.83.
The transmission coefficient is given by (14),(25)TNWcableRuO2/SiO2(EF)=2GNWcableRuO2/SiO2G0s=2.516×10-4.
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 is(26)CiRuO2/SiO2=e2DlRuO2/SiO2(EF)
with the density of states expressible for the nanocable (Figure 1 (b)) as(27)DlRuO2/SiO2(EF)=2πRavcableΔRcableVunit-cellRuO2DRuO2/SiO2(EF).
Again, inserting (27) into (26), we obtain a formula similar to (3),(28)CiRuO2/SiO2=14.262R̅avcable(nm)ΔRcableDRuO2/SiO2(#states/eV⋅unitcell)aF/nm.
Here, DRuO2/SiO can be estimated by(29)DRuO2/SiO2(EF)=GNWcableRuO2/SiO2GNWRuO2DRuO2(EF).
Referring back to Section 2 for DRuO2, and to (21) for the conductance ratio, DRuO2/SiO2 can be obtained and (28) evaluated for CiRuO2/SiO2 as(30)CiRuO2/SiO2=5.60aF/nm.
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 ofCiRuO2/SiO2(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=aorbitalextent≈2aC-C=2 (1.42Å=0.142nm), consistent with Mintmire and White [5] and Leonard [14], the area of a R=1 nm radius CNT ring or annulus is(31)ACNT=2πRtCNT≈4πaC-CR=0.568πnm2.
Equation (31) allows us to express CiRuO2/SiO2(ΔRcable=tCNT) as(32)CiRuO2/SiO2(ΔRcable=tCNT)=ACNTARuO2CiRuO2/SiO2=0.0256aF/nm
which is an order of magnitude smaller than CiSWCNT=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, CgRuO2/SiO2=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 Ci,gRuO2/SiO2=CgRuO2/SiO2,CiRuO2/SiO2 given in (30) times the overall total length L, may be equated to Ci,gRuO2/SiO2 times the total volume of the nanocables times its mass density ρcableRuO2, yielding(33)CiRuO2/SiO2=Ci,gRuO2/SiO22πRavcableΔRcableρcableRuO2.
Solving for ρcableRuO2,(34)ρcableRuO2=CiRuO2/SiO2Ci,gRuO2/SiO212πRavcableΔRcable
which when evaluated using the available capacitances, gives the remarkable density(35)ρcableRuO2=0.02052gm/cc.
Thus, the density reduction of RuO2 in the nanocable structure, allowing the high capacitive energy storage, is(36)ρcableRuO2ρcryRuO2=340
using the known value for rutile (tetragonal) crystalline structured RuO2.


## 6. Semiconductor Junction Capacitances of Nanocables

As mentioned in Section1, there may be metallic oxides, which when properly doped, that may act as semiconductors. We already know that Ga2O3 nanowires [38–42], 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 semiconductorn-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 distributionρ(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](37)φannulus(0,0,z)=1εav∫-∞∞dz′∫0Rsds∫02πdϕρ(s,ϕ,z′)s2+(z-z′)2=2πεav∫-∞∞dz′∫0Rsρ(s,z′)dss2+(z-z′)2≈2πεav∫-∞∞dz′Rρ(z′)tNCR2+(z-z′)2
leading to the kernel (or Green’s function)(38)K(z,z′)=2πRtNCεNC1R2+(z-z′)2.
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).(a) Schottky junction between a planar metal contact and a semiconductorn-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.
(a)(b)Potential due to then-side of a metal planar contact-n nanocable junction is given, using (38), by [64, page 81, 2.261, c>0](39)Vnring,cable(z)=∫0WK(z,z′)ρ(z′)dz′=ρ2πRtNCεNC∫0Wdz′R2+(z-z′)2=ρ2πRtNCεNC∫-zW-zdz̅′R2+z̅′2=ρ2πRtNCεNCln⁡[2(R2+z̅′2+z̅′)]|-zW-z=ρ2πRtNCεNCln⁡[R2+(W-z)2+W-zR2+z2-z]
caused by the charge density depletion separation (assumed abrupt for simplicity nonabruptness is addressed in Appendix B) in the annulus volume(40)ρrs(z)={ρ,0<z<W-Wδρ,-δ<z<0,0,z<-δorz>W.
Here ρ=eNd. Equation (40) has incorporated the condition of charge neutrality,(41)ANC∫0Wρ(z′)dz′=-ANC∫-δ0ρ(z′)dz′.
For the image charge in the metal (which is negative), its potential contribution is(42)Vmetalring,cable(z)=∫-δ0K(z,z′)ρ(z′)dz′=-ρWδ2πRtNCεNC∫-δ0dz′R2+(z-z′)2=-ρWδ2πRtNCεNC∫-δ-z-zdz̅′R2+z̅′2=-ρWδ2πRtNCεNCln⁡[2(R2+z̅′2+z̅′)]|-δ-z-z=-ρWδ2πRtNCεNCln⁡[R2+z2-zR2+(δ+z)2-δ-z].
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)(43)VNC=Vmetalring,cable+Vnring,cable
or(44)VNC=ρ2πRtNCεNC×{-Wδln⁡[R2+z2-zR2+(δ+z)2-δ-z]+ln⁡[R2+(W-z)2+W-zR2+z2-z]}.
We note that for large z (z≫W,δ),(45)VNC=ρ2πRtNCεNC{1+δW}(Wz)2.The capacitance must be given by(46)CNC=±dQNCdVbias,
where the differential element is taken of(47)QNC=ρ2πRavNCΔRNCW
and the bias voltage across the junction is related to the junction voltage VjNC by(48)VjNC=Vbi±Vbias-αNCkBTe,
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), is(49)VjNC=VNC(W)-VNC(-δ)=ρ2πtNCεNC×{-Wδln⁡[R2+W2-WR2+(δ+W)2-δ-W]+ln⁡[RR2+W2-W]+Wδln⁡[R2+δ2+δR]-ln⁡[R2+(W+δ)2+W+δR2+δ2+δ]}.
For the case where W≫δ,(50)VjNC|W≫δ=ρ2πtNCεNC{2WR-11+(R/W)2}.
Because we would expect W/R≫1 for nanowires and nanocables, taking the limiting form of (50) for W/R→∞ is reasonable and yields(51)VjNC|W≫δ→0W/R≫1≈ρ4πtNCεNC(WR)
allowing W to be expressed as(52)WNC|W/R≫1≫δ/R→0=εNCVjNC|W≫δ→0,W/R≫1eNdR4πtNC.
For an unbiased device, the junction voltage may be replaced by the built-in voltage, giving(53)WbiNC|W/R≫1≫δ/R→0=εNCVbieNdR4πtNC,
where the 4π part is simply indicative that we are using cgs units.The single permittivityεNC 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(54)εNC≈wdielinnerεdielinner+wshellsemiεshellsemi+wdielouterεdielouterwdielinner+wshellsemi+wdielouter.
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 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 that(55)CNCSchottky=RεNC2×[1-1211+(W/R)2+12(WR)21[1+(W/R)2]3/2]-1.
To obtain CNCSchottky in terms of Vbi, set VjNC=Vbi on the left-hand side of(50)(56)Vbi=ρ2πtNCεNC{2WR-11+(R/W)2}
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 is(57)CNCSchottky=12RεNC
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, then(58)CNCSchottky=CCNTSchottky=0.0563aF,εCNT=1.0.
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 CNCSchottky=CCNTSchottky=4.5 aF.For our RuO2/SiO2 nanocable dimensions, replacing R→Ravcable=31 nm makes the capacitance(59)CNC,SchottkyRuO2/SiO2=6.44aF,εNT=3.69
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 WbiNC|W/R≫1≫δ/R→0=(RεNCVbi)/[4πtNCeNd], and leads to the capacitance per unit area of(60)CNCua=±1ANCdQNCdVbias=±eNddW(Vbias)dVbias=eεNCNd2(Vbi±Vbias-αNCkBTe)-1/2R8πtNC=εNC2LDNC(βVbi±βVbias-αNC)-1/2
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 LDNC, given by(61)LDNC=LDbulk8πtNCR,LDbulk=εNCeNdβ,β=ekBT.Let us evaluateCNC 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, αNC≈1, 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)(62)CNCSchottky=RRtNCeεNCNdπ4Vbi=R24εNCWbiNC
(last equality in (62) follows from (53), and it yields the form CNC,uaSchottky=(εNC/WbiNC)(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 capacitance(63)CNCSchottky=CCNT=0.033aF,Nd=5×1020/cc,εNT=1.0
which corresponds to a fraction  f=10-2 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 NCNTdopedatoms=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 (64)Vatom=AhexNhextCNT=Ahex22aC-C=323aC-C3=7.439×10-3nm3
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 NCNTatoms=VCNT/Vatom=239.82, making NCNTdopedatoms=fNCNTatoms=2.40.For our RuO2/SiO2 nanocables dimensions, replacing R→Ravcable=31 nm and tNC→δRcable=2 nm in (62), we find(65)CNC,SchottkyRuO2/SiO2=60.46aF,Nd=5×1020/cc,εNT=3.69
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 yields(66)CNC,SchottkyRuO2/SiO2=6.05aF,Nd=5×1018/cc,εNT=3.69.
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=10-2 fractional doping. (Even f=10-4 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 evaluatingCNCSchottky using (57), the simpler Schottky junction capacitance formula, versus using (55), is shown in Figure 3, where the normalized capacitance CNCSchottky/(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/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).Figure 3
Normalized nanocable Schottky junction capacitance againstW/R. Normalization is to RεNC. Formulas (55) and (57) providing CNCSchottky are plotted. Also, as a comparison is a result for a bulk-like rendition given by (62).


### 6.2. Asymmetric-Semiconductorp-n Junction Capacitance of Nanocables

For the asymmetric abruptp-n junction (Figure 2(b)), unequal doping occurs in the p- and n-sides of the nanocable. Equation (42) must be replaced by(67)Vasym,pring,cable(z)=∫-zp0K(z,z′)ρ(z′)dz′=-ρp2πRtNCεNC∫-zp0dz′R2+(z-z′)2=-ρp2πRtNCεNC∫-zp-z-zdz̅′R2+z̅′2=-ρp2πRtNCεNCln⁡[2(R2+z̅′2+z̅′)]|-zp-z-z=-ρp2πRtNCεNCln⁡[R2+z2-zR2+(zp+z)2-zp-z]
caused by the charge density depletion separation in the annulus volume(68)ρasymNC(z)={ρn,0<z<zn-ρp,-zp<z<0,0,z<-zporz>zn.
Charge neutrality demands that the condition of (41) be generalized for arbitrary depletion widths,(69)ANC∫0znρ(z′)dz′=-ANC∫-zp0ρ(z′)dz′
which because of the unequal but constant doping densities assumed in (68), allows one depletion width to be determined in terms of the other:(70)zp=ρnρpzn.
Similarly, (39) must be replaced with(71)Vasym,nring,cable(z)=∫0znK(z,z′)ρ(z′)dz′=ρ2πRtNCεNC∫0zndz′R2+(z-z′)2=ρ2πRtNCεNC∫-zzn-zdz̅′R2+z̅′2=ρ2πRtNCεNCln⁡[2(R2+z̅′2+z̅′)]|-zzn-z=ρ2πRtNCεNCln⁡[R2+(zn-z)2+zn-zR2+z2-z].The total potential along the nanocable lengthz will then be a superposition of both the acceptor and donor and charge potentials in (67) and(71)(72)VasymNC=Vasym,pring,cable+Vasym,nring,cable
or(73)VasymNC(z)=2πRtNCεNC×{-ρpln⁡[R2+z2-zR2+(zp+z)2-zp-z]+ρnln⁡[R2+(zn-z)2+zn-zR2+z2-z]}
which makes the junction potential difference(74)Vasym,jNC=VNC(zn)-VNC(-zp)=2πRtNCεNC×{-ρpln⁡[R2+zn2-znR2+(zp+zn)2-zp-zn]+ρnln⁡[RR2+zn2-zn]+ρpln⁡[R2+zp2+zpR]-ρnln⁡[R2+(zn+zp)2+zn+zpR2+zp2+zp]}=2πRtNCεNC×{-ρpln⁡[R2+zn2-znR2+zp2+zpRR2+(zp+zn)2-zp-zn]+ρnln⁡[RR2+(zn+zp)2+zn+zpR2+zp2+zpR2+zn2-zn]}.If we define the total depletion width of the nanocable as(75)WasymNC=zp+zn=(ρnρp+1)zn
then when we examine the case when the p-region doping density gets large, ρp→∞, (70) and (75) will reduce Vasym,jNC 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, whenρp=ρn=ρ, and (70) becomes(76)zp=ρnρpzn=zn=W
and (74) reduces to(77)Vsym,jNC=VNC(W)-VNC(-W)=ρ2πRtNCεNC{2ln⁡[R2+W2+WR2+W2-W]+ln⁡[R2+4W2-2WR2+4W2+2W]}.
(One notes that the symmetry properties Vsym,nNC(W)=-Vsym,pNC(-W) and Vsym,nNC(-W)=-Vsym,pNC(W) are satisfied by (67) and (71), and cause Vsym,jNC=2VsymNC(W).) Consider the limit of expression (77) when W/R→∞. The nanocable junction voltage reduces to(78)Vsym,jNC|W/R≫1≈ρ4πRtNCεNWln⁡(WR)
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,(79)CNCsym=±dQNCsymdVbias=±eNdANCdW(Vbias)dVbias
we see that solving for W in(78)(80)W(Vj)=ReεNCVj/(4πρRtNC)
and taking the derivative, yields(81)CNCsym=12εNCW.
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 R→Ravcable=31 nm and tNC→ΔRcable=2 nm, with NNC=Nd=1019/cc (ρ=eNd), γ=0.13905 making W=1.149 R. Clearly, this does not satisfy W≫R well. But if Nd was an order of magnitude smaller, then W=4.0169 R, which is respectable. For the larger doping value, we calculate(82)CNC,symRuO2/SiO2=6.95aF,Nd=1.0×1019/cc,εNC=3.69.Capacitance can be determined exactly by taking the derivative of (77), finding(83)CNCsym=εNC2R2+W22-(R2+W2)/(R2+4W2).
To obtain CNCsym in terms of Vbi, set VjNC=Vbi on the left-hand side of (77):(84)Vbi=ρ2πRtNCεNC{2ln⁡[R2+W2+WR2+W2-W]+ln⁡[R2+4W2-2WR2+4W2+2W]}
and solve for W(Vbi), and insert this into (83). One might wonder what form is obtained for CNCsym by taking the limit W/R→∞ in (83) after its formula has been derived:(85)CNCsym=13εNCW
which differs slightly from (81), making the capacitance somewhat smaller,(86)CNC,symRuO2/SiO2=4.64aF,Nd=1.0×1019/cc,εNC=3.69.Comparison of the most general formula (83) for capacitance of a symmetric nanocable junction CNCsym 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 symmetricp-n semiconductor junction capacitance against W/R. Normalization is to RεNC. Formulas (81), (83), and (85) providing CNCsym are plotted.Table1 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.
FormulaFormula(C/[Rε])TypeLimit(55)12{1-0.5/1+(W/R)2+(1/2)(W/R)2/[1+(W/R)2]3/2}-1Nanocable-SchottkyR≫tNC(57)12Nanocable-SchottkyW≫R(62)14RWPlanar Equiv. NCW≫R(81)12WRSymmetricp-n NCW≫R(83)121+(W/R)21+(2W/R)221+(2W/R)2-1+(W/R)2Symmetricp-n NCR≫tNC(85)13WRSymmetricp-n NCW≫R(100)14[1+(WR)2-1+(2WR)2+(W/R)+(R/W)1+(R/W)2-(2W/R)+(R/2W)1+(R/2W)2+2WR]-1Symm.p-n NWW≫R(102)WRSymm.p-n NWNone(115)12RWAsym.p-n NWρp→∞


## 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 distributionρ(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](37)φannulus(0,0,z)=1εav∫-∞∞dz′∫0Rsds∫02πdϕρ(s,ϕ,z′)s2+(z-z′)2=2πεav∫-∞∞dz′∫0Rsρ(s,z′)dss2+(z-z′)2≈2πεav∫-∞∞dz′Rρ(z′)tNCR2+(z-z′)2
leading to the kernel (or Green’s function)(38)K(z,z′)=2πRtNCεNC1R2+(z-z′)2.
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).(a) Schottky junction between a planar metal contact and a semiconductorn-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.
(a)(b)Potential due to then-side of a metal planar contact-n nanocable junction is given, using (38), by [64, page 81, 2.261, c>0](39)Vnring,cable(z)=∫0WK(z,z′)ρ(z′)dz′=ρ2πRtNCεNC∫0Wdz′R2+(z-z′)2=ρ2πRtNCεNC∫-zW-zdz̅′R2+z̅′2=ρ2πRtNCεNCln⁡[2(R2+z̅′2+z̅′)]|-zW-z=ρ2πRtNCεNCln⁡[R2+(W-z)2+W-zR2+z2-z]
caused by the charge density depletion separation (assumed abrupt for simplicity nonabruptness is addressed in Appendix B) in the annulus volume(40)ρrs(z)={ρ,0<z<W-Wδρ,-δ<z<0,0,z<-δorz>W.
Here ρ=eNd. Equation (40) has incorporated the condition of charge neutrality,(41)ANC∫0Wρ(z′)dz′=-ANC∫-δ0ρ(z′)dz′.
For the image charge in the metal (which is negative), its potential contribution is(42)Vmetalring,cable(z)=∫-δ0K(z,z′)ρ(z′)dz′=-ρWδ2πRtNCεNC∫-δ0dz′R2+(z-z′)2=-ρWδ2πRtNCεNC∫-δ-z-zdz̅′R2+z̅′2=-ρWδ2πRtNCεNCln⁡[2(R2+z̅′2+z̅′)]|-δ-z-z=-ρWδ2πRtNCεNCln⁡[R2+z2-zR2+(δ+z)2-δ-z].
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)(43)VNC=Vmetalring,cable+Vnring,cable
or(44)VNC=ρ2πRtNCεNC×{-Wδln⁡[R2+z2-zR2+(δ+z)2-δ-z]+ln⁡[R2+(W-z)2+W-zR2+z2-z]}.
We note that for large z (z≫W,δ),(45)VNC=ρ2πRtNCεNC{1+δW}(Wz)2.The capacitance must be given by(46)CNC=±dQNCdVbias,
where the differential element is taken of(47)QNC=ρ2πRavNCΔRNCW
and the bias voltage across the junction is related to the junction voltage VjNC by(48)VjNC=Vbi±Vbias-αNCkBTe,
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), is(49)VjNC=VNC(W)-VNC(-δ)=ρ2πtNCεNC×{-Wδln⁡[R2+W2-WR2+(δ+W)2-δ-W]+ln⁡[RR2+W2-W]+Wδln⁡[R2+δ2+δR]-ln⁡[R2+(W+δ)2+W+δR2+δ2+δ]}.
For the case where W≫δ,(50)VjNC|W≫δ=ρ2πtNCεNC{2WR-11+(R/W)2}.
Because we would expect W/R≫1 for nanowires and nanocables, taking the limiting form of (50) for W/R→∞ is reasonable and yields(51)VjNC|W≫δ→0W/R≫1≈ρ4πtNCεNC(WR)
allowing W to be expressed as(52)WNC|W/R≫1≫δ/R→0=εNCVjNC|W≫δ→0,W/R≫1eNdR4πtNC.
For an unbiased device, the junction voltage may be replaced by the built-in voltage, giving(53)WbiNC|W/R≫1≫δ/R→0=εNCVbieNdR4πtNC,
where the 4π part is simply indicative that we are using cgs units.The single permittivityεNC 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(54)εNC≈wdielinnerεdielinner+wshellsemiεshellsemi+wdielouterεdielouterwdielinner+wshellsemi+wdielouter.
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 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 that(55)CNCSchottky=RεNC2×[1-1211+(W/R)2+12(WR)21[1+(W/R)2]3/2]-1.
To obtain CNCSchottky in terms of Vbi, set VjNC=Vbi on the left-hand side of(50)(56)Vbi=ρ2πtNCεNC{2WR-11+(R/W)2}
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 is(57)CNCSchottky=12RεNC
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, then(58)CNCSchottky=CCNTSchottky=0.0563aF,εCNT=1.0.
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 CNCSchottky=CCNTSchottky=4.5 aF.For our RuO2/SiO2 nanocable dimensions, replacing R→Ravcable=31 nm makes the capacitance(59)CNC,SchottkyRuO2/SiO2=6.44aF,εNT=3.69
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 WbiNC|W/R≫1≫δ/R→0=(RεNCVbi)/[4πtNCeNd], and leads to the capacitance per unit area of(60)CNCua=±1ANCdQNCdVbias=±eNddW(Vbias)dVbias=eεNCNd2(Vbi±Vbias-αNCkBTe)-1/2R8πtNC=εNC2LDNC(βVbi±βVbias-αNC)-1/2
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 LDNC, given by(61)LDNC=LDbulk8πtNCR,LDbulk=εNCeNdβ,β=ekBT.Let us evaluateCNC 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, αNC≈1, 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)(62)CNCSchottky=RRtNCeεNCNdπ4Vbi=R24εNCWbiNC
(last equality in (62) follows from (53), and it yields the form CNC,uaSchottky=(εNC/WbiNC)(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 capacitance(63)CNCSchottky=CCNT=0.033aF,Nd=5×1020/cc,εNT=1.0
which corresponds to a fraction  f=10-2 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 NCNTdopedatoms=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 (64)Vatom=AhexNhextCNT=Ahex22aC-C=323aC-C3=7.439×10-3nm3
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 NCNTatoms=VCNT/Vatom=239.82, making NCNTdopedatoms=fNCNTatoms=2.40.For our RuO2/SiO2 nanocables dimensions, replacing R→Ravcable=31 nm and tNC→δRcable=2 nm in (62), we find(65)CNC,SchottkyRuO2/SiO2=60.46aF,Nd=5×1020/cc,εNT=3.69
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 yields(66)CNC,SchottkyRuO2/SiO2=6.05aF,Nd=5×1018/cc,εNT=3.69.
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=10-2 fractional doping. (Even f=10-4 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 evaluatingCNCSchottky using (57), the simpler Schottky junction capacitance formula, versus using (55), is shown in Figure 3, where the normalized capacitance CNCSchottky/(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/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).Figure 3
Normalized nanocable Schottky junction capacitance againstW/R. Normalization is to RεNC. Formulas (55) and (57) providing CNCSchottky are plotted. Also, as a comparison is a result for a bulk-like rendition given by (62).


## 6.2. Asymmetric-Semiconductorp-n Junction Capacitance of Nanocables

For the asymmetric abruptp-n junction (Figure 2(b)), unequal doping occurs in the p- and n-sides of the nanocable. Equation (42) must be replaced by(67)Vasym,pring,cable(z)=∫-zp0K(z,z′)ρ(z′)dz′=-ρp2πRtNCεNC∫-zp0dz′R2+(z-z′)2=-ρp2πRtNCεNC∫-zp-z-zdz̅′R2+z̅′2=-ρp2πRtNCεNCln⁡[2(R2+z̅′2+z̅′)]|-zp-z-z=-ρp2πRtNCεNCln⁡[R2+z2-zR2+(zp+z)2-zp-z]
caused by the charge density depletion separation in the annulus volume(68)ρasymNC(z)={ρn,0<z<zn-ρp,-zp<z<0,0,z<-zporz>zn.
Charge neutrality demands that the condition of (41) be generalized for arbitrary depletion widths,(69)ANC∫0znρ(z′)dz′=-ANC∫-zp0ρ(z′)dz′
which because of the unequal but constant doping densities assumed in (68), allows one depletion width to be determined in terms of the other:(70)zp=ρnρpzn.
Similarly, (39) must be replaced with(71)Vasym,nring,cable(z)=∫0znK(z,z′)ρ(z′)dz′=ρ2πRtNCεNC∫0zndz′R2+(z-z′)2=ρ2πRtNCεNC∫-zzn-zdz̅′R2+z̅′2=ρ2πRtNCεNCln⁡[2(R2+z̅′2+z̅′)]|-zzn-z=ρ2πRtNCεNCln⁡[R2+(zn-z)2+zn-zR2+z2-z].The total potential along the nanocable lengthz will then be a superposition of both the acceptor and donor and charge potentials in (67) and(71)(72)VasymNC=Vasym,pring,cable+Vasym,nring,cable
or(73)VasymNC(z)=2πRtNCεNC×{-ρpln⁡[R2+z2-zR2+(zp+z)2-zp-z]+ρnln⁡[R2+(zn-z)2+zn-zR2+z2-z]}
which makes the junction potential difference(74)Vasym,jNC=VNC(zn)-VNC(-zp)=2πRtNCεNC×{-ρpln⁡[R2+zn2-znR2+(zp+zn)2-zp-zn]+ρnln⁡[RR2+zn2-zn]+ρpln⁡[R2+zp2+zpR]-ρnln⁡[R2+(zn+zp)2+zn+zpR2+zp2+zp]}=2πRtNCεNC×{-ρpln⁡[R2+zn2-znR2+zp2+zpRR2+(zp+zn)2-zp-zn]+ρnln⁡[RR2+(zn+zp)2+zn+zpR2+zp2+zpR2+zn2-zn]}.If we define the total depletion width of the nanocable as(75)WasymNC=zp+zn=(ρnρp+1)zn
then when we examine the case when the p-region doping density gets large, ρp→∞, (70) and (75) will reduce Vasym,jNC 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, whenρp=ρn=ρ, and (70) becomes(76)zp=ρnρpzn=zn=W
and (74) reduces to(77)Vsym,jNC=VNC(W)-VNC(-W)=ρ2πRtNCεNC{2ln⁡[R2+W2+WR2+W2-W]+ln⁡[R2+4W2-2WR2+4W2+2W]}.
(One notes that the symmetry properties Vsym,nNC(W)=-Vsym,pNC(-W) and Vsym,nNC(-W)=-Vsym,pNC(W) are satisfied by (67) and (71), and cause Vsym,jNC=2VsymNC(W).) Consider the limit of expression (77) when W/R→∞. The nanocable junction voltage reduces to(78)Vsym,jNC|W/R≫1≈ρ4πRtNCεNWln⁡(WR)
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,(79)CNCsym=±dQNCsymdVbias=±eNdANCdW(Vbias)dVbias
we see that solving for W in(78)(80)W(Vj)=ReεNCVj/(4πρRtNC)
and taking the derivative, yields(81)CNCsym=12εNCW.
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 R→Ravcable=31 nm and tNC→ΔRcable=2 nm, with NNC=Nd=1019/cc (ρ=eNd), γ=0.13905 making W=1.149 R. Clearly, this does not satisfy W≫R well. But if Nd was an order of magnitude smaller, then W=4.0169 R, which is respectable. For the larger doping value, we calculate(82)CNC,symRuO2/SiO2=6.95aF,Nd=1.0×1019/cc,εNC=3.69.Capacitance can be determined exactly by taking the derivative of (77), finding(83)CNCsym=εNC2R2+W22-(R2+W2)/(R2+4W2).
To obtain CNCsym in terms of Vbi, set VjNC=Vbi on the left-hand side of (77):(84)Vbi=ρ2πRtNCεNC{2ln⁡[R2+W2+WR2+W2-W]+ln⁡[R2+4W2-2WR2+4W2+2W]}
and solve for W(Vbi), and insert this into (83). One might wonder what form is obtained for CNCsym by taking the limit W/R→∞ in (83) after its formula has been derived:(85)CNCsym=13εNCW
which differs slightly from (81), making the capacitance somewhat smaller,(86)CNC,symRuO2/SiO2=4.64aF,Nd=1.0×1019/cc,εNC=3.69.Comparison of the most general formula (83) for capacitance of a symmetric nanocable junction CNCsym 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 symmetricp-n semiconductor junction capacitance against W/R. Normalization is to RεNC. Formulas (81), (83), and (85) providing CNCsym are plotted.Table1 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.
FormulaFormula(C/[Rε])TypeLimit(55)12{1-0.5/1+(W/R)2+(1/2)(W/R)2/[1+(W/R)2]3/2}-1Nanocable-SchottkyR≫tNC(57)12Nanocable-SchottkyW≫R(62)14RWPlanar Equiv. NCW≫R(81)12WRSymmetricp-n NCW≫R(83)121+(W/R)21+(2W/R)221+(2W/R)2-1+(W/R)2Symmetricp-n NCR≫tNC(85)13WRSymmetricp-n NCW≫R(100)14[1+(WR)2-1+(2WR)2+(W/R)+(R/W)1+(R/W)2-(2W/R)+(R/2W)1+(R/2W)2+2WR]-1Symm.p-n NWW≫R(102)WRSymm.p-n NWNone(115)12RWAsym.p-n NWρp→∞


## 7. Semiconductorp-n Junction Capacitances of Nanowires

The symmetricp-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 asymmetricp-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 Semiconductorp-n Junction Capacitance of Nanowires

We will look at the symmetric semiconductor nanowirep-n junction here (Figure 5(a)). First specify the depletion region charge density, which in the abrupt approximation, changes from (40) to(87)ρsymNW(z)={ρ,0<z<W-ρ,-W<z<0,0,|z|>W.
By inspecting the integral formula for a vacuum potential solution [62], the on axis value for the nanowire is (use transformation u=s2+(z-z′)2,du=2sds)(88)φNW(0,0,z)=1εNW∫-∞∞dz′∫0Rsds∫02πdϕρ(s,ϕ,z′)s2+(z-z′)2=2πεNW∫-∞∞dz′∫0Rsdsρ(s,z′)s2+(z-z′)2=2πεNW∫-∞∞ρ(z′)dz′∫0Rsdss2+(z-z′)2=2πεNW∫-∞∞ρ(z′)[R2+(z-z′)2-(z-z′)]dz′
leading to the kernel (∣ ∣ = abs( ) operator is chosen leading to the correct branch cuts)(89)K(z,z′)=2πεNW[R2+(z-z′)2-|z-z′|]
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](90)Vsym,pNW(z)=∫-W0K(z,z′)ρ(z′)dz′=2πεNW∫-W0ρ(z′)[R2+(z-z′)2+(z-z′)]dz′=-ρ2πεNW{∫X-WX0X2dXX2-R2+zW(1+12Wz)}=-ρ2πεNW×{(z+W)R2+(z+W)2-zR2+z22-R22ln⁡[R2+(z+W)2-(z+W)R2+z2-z]∓zW(1+12Wz)(z+W)R2+(z+W)2-zR2+z22}.
where the change of variables X2=R2+(z-z′)2, -(z-z′)dz′=XdX was used. (Note, care is required in selecting correct branch cuts, and we use -X2-R2 in the third line of (90), and X=-R2+(z-z′)2 and S=+X2-R2 in the fourth line of (90).) The potential from the n-side of the junction will be(91)Vsym,nNW(z)=∫0WK(z,z′)ρ(z′)dz′=ρ2πεNW∫0W[R2+(z-z′)2-(z-z′)]dz′=ρ2πεNW{[XS2+R22ln⁡|X+S|]|X0XW-zW(1-12Wz)}=ρ2πεNW×{(W-z)R2+(z-W)2+zR2+z22-R22ln⁡[R2+(z-W)2+z-WR2+z2+z]∓zW(1-12Wz)(W-z)R2+(z-W)2+zR2+z22}.
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  Vsym,pNW(z)  and Vsym,nNW(z) satisfy the physical symmetry and limiting conditions(92)Vsym,nNW(W)=-Vsym,pNW(-W),Vsym,nNW(-W)=-Vsym,pNW(W),(93)lim⁡z→∞Vsym,nNW(z)=-lim⁡z→-∞Vsym,pNW(z)=ρπR2W/z.(a) Symmetric semiconductorp-n nanowire junction. (b) Asymmetric semiconductor p-n nanowire junction. This is examined for the limiting case when ρp→∞.
(a)(b)The total potential due to both thep- 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),(94)VsymNW(z)=Vsym,pNW(z)+Vsym,nNW(z)
or(95)VsymNW(z)=-ρ2πεNW{12[(z+W)R2+(z+W)2-(W-z)R2+(z-W)2]-zR2+z2-12R2×ln⁡[R2+(z+W)2-(z+W)R2+(z-W)2+z-WR2+z2+zR2+z2-z]∓(z+W)R2+(z+W)2W2}.
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(96)Vsym,jNW=VNW(W)-VNW(-W)=2[VpNW(W)+VnNW(W)]=2VsymNW(W).(97)Vsym,jNW=ρ4πεNW{WR2+W2-WR2+(2W)2+R22×ln⁡⁡[(2W)2+R2-2WRW2+R2+WW2+R2-W]+WR2+W2-WR2+(2W)2+R22W2WR2+W2}.
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 (98)Vsym,jNW|W/R≫1≈ρπR2εNW[1+2(RW)2ln⁡WR].
Equation (95) does not allow W to be expressed as, say WNW|W/R≫1=εNWVsym,jNW|W/R≫1/(4πeNd), which for an unbiased device, replacing the junction voltage by the built-in voltage, gives WbiNW|W/R≫1=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 thep-n nanowire junction with area ANW=πR2, will be using (98),(99)CNWp-n=±dQNWdVbias=±πR2ρdW(Vbias)dVbias=12εNWW(W/R)21-2ln⁡(W/R).
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=e1/2≈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 ofCNWp-n, 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)(100)CNWp-n=RεNW4[1+(WR)2-1+4(WR)2+W/R1+(R/W)2-2W/R1+(R/2W)2+11+(W/R)2-11+(2W/R)2+2WR]-1.
To obtain CNWp-n in terms of Vbi, set VjNW=Vbi on the left-hand side of (97):(101)Vbi=ρ4πεNW×{WR2+W2-WR2+4W2+R22ln⁡⁡[4W2+R2-2WRW2+R2+WW2+R2-W]+WR2+W2-WR2+4W2W2}
and solve for W(Vbi), and insert this into (100).An approximation to the nanowirep-n junction capacitance capacitance may be examined for large W/R by taking the limit of (100) for W/R→∞, with the result(102)CNWp-n=εNWW.Evaluating (102) for a small radius nanowire like the previously examined carbon nanotube, with R=1 nm, gives(103)CNWp-n=0.113aF,εNW=1.
Alternatively, evaluating (102) for a large radius nanowire like the previously examined RuO2/SiO2 nanocable, with R=31 nm, gives(104)CNWp-n=12.7aF,εNW=3.69.Figure6 shows the dependence of nanowire junction capacitance CNWp-n 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 symmetricp-n semiconductor junction capacitance against W/R. Normalization is to RεNW. Formulas (100) and (102) providing CNWp-n are plotted.


### 7.2. Asymmetric Semiconductorp-n Junction Capacitance of Nanowires

The asymmetric nanowire semiconductorp-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(105)ρasymNW(z)={ρn,0<z<zn-ρp,-zp<z<0,0,z<-zporz>zn.
Potential contribution from the p-side of the nanowire junction, using (105), will be(106)Vasym,pNW(z)=∫-zp0K(z,z′)ρ(z′)dz′=2πεNW∫-zp0ρ(z′)[R2+(z-z′)2+(z-z′)]dz′=-ρp2πεNW{∫X-zpX0X2dXX2-R2+zzp(1+12zpz)}=-ρp2πεNW{(z+zp)R2+(z+zp)2-zR2+z22-R22ln⁡[R2+(z+zp)2-(z+zp)R2+z2-z]∓zzp(1+12zpz)(z+zp)R2+(z+zp)2-zR2+z22}.
Likewise, the potential contribution from the n-side of the nanowire junction will be(107)Vasym,nNW(z)=∫0znK(z,z′)ρ(z′)dz′=ρn2πεNW∫0zn[R2+(z-z′)2-(z-z′)]dz′=ρn2πεNW{[XS2+R22ln⁡|X+S|]|X0Xzn-zzn(1-12znz)(zn-z)R2+(z-zn)2+zR2+z22}=ρn2πεNW×{(zn-z)R2+(z-zn)2+zR2+z22-R22ln⁡[R2+(z-zn)2+z-znR2+z2+z]∓zzn(1-12znz)(zn-z)R2+(z-zn)2+zR2+z22}.
Total potential due to both the p- and n-sides of the asymmetric nanowire junction depletion region will be(108)VasymNW(z)=Vasym,pNW(z)+Vasym,nNW(z)
and inserting into this (106) and (107) yields(109)VasymNW(z)=2πεNW{ρpzR2+z2-(z+zp)R2+(z+zp)22+ρpR22ln⁡[R2+(z+zp)2-(z+zp)R2+z2-z]±ρpzzp(1+12zpz)+ρn(zn-z)R2+(z-zn)2+zR2+z22-ρnR22ln⁡[R2+(z-zn)2+z-znR2+z2+z]∓ρnzzn(1-12znz)zR2+z2-(z+zp)R2+(z+zp)22}.
Junction voltage is calculated from(110)Vasym,jNW=VasymNW(zn)-VasymNW(-zp)
and when evaluating (109) at z=zn,-zp, with a total nanowire depletion width WT=WasymNW=zp+zn we find
(111)Vasym,jNW=2πεNW{-WTR2+WT22(ρp+ρn)+znρpR2+zn2+zpρnR2+zp22+R22{ρpln⁡[R2+WT2-WTR2+zn2-zn]+ρnln⁡[R2+WT2-WTR2+zp2-zp]}+znzp(ρp+ρn)-zpρpR2+zp2-znρnR2+zn22-R22{ρpln⁡[R2+WT2-WTR2+zn2-zn]ρnln⁡[RR2+zp2+zpRR2+zn2+zn]+ρpln⁡[RR2+zp2+zp]}}.Now consider the case when thep-side becomes highly doped, ρp→∞. Formula (111) becomes, if one also allows zn≫R (implies WT>zn≫R),(112)VasymNW|ρp→∞W/R≫1=ρnπW2εNW
when simplifying notation WT→zn=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
(113)WasymNW|ρp→∞W/R≫1=εNWVasym,jNW|ρp→∞,W/R≫1πeNd
and giving for an unbiased device,
(114)WbiNW|ρp→∞W/R≫1=εNWVbiπeNd.
The capacitance will be, enlisting (113),
(115)CasymNW|ρp→∞W/R≫1=±ANWeNddW(Vbias)dVbias=12RεNWRW.
In mks units this would yield CasymNW|ρp→∞,W/R≫1=2πR2εNW/W, and the per-unit area capacitance would be CasymNW,ua|ρp→∞,W/R≫1=2εNW/W, displaying the familiar form [62] which is inversely proportional to W. Evaluating (115) for a large radius nanowire like the previously examined RuO2/SiO2 nanocable, with R=31 nm, gives(116)CasymNW|ρp→∞W/R≫1=23.189aF,Nd=5×1018/cc,εNW=3.69.
However, if evaluated for a R=1 nm radius,(117)CasymNW|ρp→∞W/R≫1=0.0241aF,Nd=5×1018/cc,εNW=3.69.
Figure 7 shows the dependence of the asymmetric nanowire junction capacitance CasymNW|ρp→∞,W/R≫1 on the W/R ratio, as given by (115).Figure 7
Normalized nanowire Schottky junction capacitance againstW/R. Normalization is to RεNW. Formula (115) for CasymNW|ρp→∞,W/R≫1 is plotted.Table1 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 RεNW (this product’s units is Farads). The last three rows pertain to the nanowire results.


## 7.1. Symmetric Semiconductorp-n Junction Capacitance of Nanowires

We will look at the symmetric semiconductor nanowirep-n junction here (Figure 5(a)). First specify the depletion region charge density, which in the abrupt approximation, changes from (40) to(87)ρsymNW(z)={ρ,0<z<W-ρ,-W<z<0,0,|z|>W.
By inspecting the integral formula for a vacuum potential solution [62], the on axis value for the nanowire is (use transformation u=s2+(z-z′)2,du=2sds)(88)φNW(0,0,z)=1εNW∫-∞∞dz′∫0Rsds∫02πdϕρ(s,ϕ,z′)s2+(z-z′)2=2πεNW∫-∞∞dz′∫0Rsdsρ(s,z′)s2+(z-z′)2=2πεNW∫-∞∞ρ(z′)dz′∫0Rsdss2+(z-z′)2=2πεNW∫-∞∞ρ(z′)[R2+(z-z′)2-(z-z′)]dz′
leading to the kernel (∣ ∣ = abs( ) operator is chosen leading to the correct branch cuts)(89)K(z,z′)=2πεNW[R2+(z-z′)2-|z-z′|]
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](90)Vsym,pNW(z)=∫-W0K(z,z′)ρ(z′)dz′=2πεNW∫-W0ρ(z′)[R2+(z-z′)2+(z-z′)]dz′=-ρ2πεNW{∫X-WX0X2dXX2-R2+zW(1+12Wz)}=-ρ2πεNW×{(z+W)R2+(z+W)2-zR2+z22-R22ln⁡[R2+(z+W)2-(z+W)R2+z2-z]∓zW(1+12Wz)(z+W)R2+(z+W)2-zR2+z22}.
where the change of variables X2=R2+(z-z′)2, -(z-z′)dz′=XdX was used. (Note, care is required in selecting correct branch cuts, and we use -X2-R2 in the third line of (90), and X=-R2+(z-z′)2 and S=+X2-R2 in the fourth line of (90).) The potential from the n-side of the junction will be(91)Vsym,nNW(z)=∫0WK(z,z′)ρ(z′)dz′=ρ2πεNW∫0W[R2+(z-z′)2-(z-z′)]dz′=ρ2πεNW{[XS2+R22ln⁡|X+S|]|X0XW-zW(1-12Wz)}=ρ2πεNW×{(W-z)R2+(z-W)2+zR2+z22-R22ln⁡[R2+(z-W)2+z-WR2+z2+z]∓zW(1-12Wz)(W-z)R2+(z-W)2+zR2+z22}.
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  Vsym,pNW(z)  and Vsym,nNW(z) satisfy the physical symmetry and limiting conditions(92)Vsym,nNW(W)=-Vsym,pNW(-W),Vsym,nNW(-W)=-Vsym,pNW(W),(93)lim⁡z→∞Vsym,nNW(z)=-lim⁡z→-∞Vsym,pNW(z)=ρπR2W/z.(a) Symmetric semiconductorp-n nanowire junction. (b) Asymmetric semiconductor p-n nanowire junction. This is examined for the limiting case when ρp→∞.
(a)(b)The total potential due to both thep- 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),(94)VsymNW(z)=Vsym,pNW(z)+Vsym,nNW(z)
or(95)VsymNW(z)=-ρ2πεNW{12[(z+W)R2+(z+W)2-(W-z)R2+(z-W)2]-zR2+z2-12R2×ln⁡[R2+(z+W)2-(z+W)R2+(z-W)2+z-WR2+z2+zR2+z2-z]∓(z+W)R2+(z+W)2W2}.
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(96)Vsym,jNW=VNW(W)-VNW(-W)=2[VpNW(W)+VnNW(W)]=2VsymNW(W).(97)Vsym,jNW=ρ4πεNW{WR2+W2-WR2+(2W)2+R22×ln⁡⁡[(2W)2+R2-2WRW2+R2+WW2+R2-W]+WR2+W2-WR2+(2W)2+R22W2WR2+W2}.
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 (98)Vsym,jNW|W/R≫1≈ρπR2εNW[1+2(RW)2ln⁡WR].
Equation (95) does not allow W to be expressed as, say WNW|W/R≫1=εNWVsym,jNW|W/R≫1/(4πeNd), which for an unbiased device, replacing the junction voltage by the built-in voltage, gives WbiNW|W/R≫1=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 thep-n nanowire junction with area ANW=πR2, will be using (98),(99)CNWp-n=±dQNWdVbias=±πR2ρdW(Vbias)dVbias=12εNWW(W/R)21-2ln⁡(W/R).
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=e1/2≈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 ofCNWp-n, 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)(100)CNWp-n=RεNW4[1+(WR)2-1+4(WR)2+W/R1+(R/W)2-2W/R1+(R/2W)2+11+(W/R)2-11+(2W/R)2+2WR]-1.
To obtain CNWp-n in terms of Vbi, set VjNW=Vbi on the left-hand side of (97):(101)Vbi=ρ4πεNW×{WR2+W2-WR2+4W2+R22ln⁡⁡[4W2+R2-2WRW2+R2+WW2+R2-W]+WR2+W2-WR2+4W2W2}
and solve for W(Vbi), and insert this into (100).An approximation to the nanowirep-n junction capacitance capacitance may be examined for large W/R by taking the limit of (100) for W/R→∞, with the result(102)CNWp-n=εNWW.Evaluating (102) for a small radius nanowire like the previously examined carbon nanotube, with R=1 nm, gives(103)CNWp-n=0.113aF,εNW=1.
Alternatively, evaluating (102) for a large radius nanowire like the previously examined RuO2/SiO2 nanocable, with R=31 nm, gives(104)CNWp-n=12.7aF,εNW=3.69.Figure6 shows the dependence of nanowire junction capacitance CNWp-n 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 symmetricp-n semiconductor junction capacitance against W/R. Normalization is to RεNW. Formulas (100) and (102) providing CNWp-n are plotted.


## 7.2. Asymmetric Semiconductorp-n Junction Capacitance of Nanowires

The asymmetric nanowire semiconductorp-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(105)ρasymNW(z)={ρn,0<z<zn-ρp,-zp<z<0,0,z<-zporz>zn.
Potential contribution from the p-side of the nanowire junction, using (105), will be(106)Vasym,pNW(z)=∫-zp0K(z,z′)ρ(z′)dz′=2πεNW∫-zp0ρ(z′)[R2+(z-z′)2+(z-z′)]dz′=-ρp2πεNW{∫X-zpX0X2dXX2-R2+zzp(1+12zpz)}=-ρp2πεNW{(z+zp)R2+(z+zp)2-zR2+z22-R22ln⁡[R2+(z+zp)2-(z+zp)R2+z2-z]∓zzp(1+12zpz)(z+zp)R2+(z+zp)2-zR2+z22}.
Likewise, the potential contribution from the n-side of the nanowire junction will be(107)Vasym,nNW(z)=∫0znK(z,z′)ρ(z′)dz′=ρn2πεNW∫0zn[R2+(z-z′)2-(z-z′)]dz′=ρn2πεNW{[XS2+R22ln⁡|X+S|]|X0Xzn-zzn(1-12znz)(zn-z)R2+(z-zn)2+zR2+z22}=ρn2πεNW×{(zn-z)R2+(z-zn)2+zR2+z22-R22ln⁡[R2+(z-zn)2+z-znR2+z2+z]∓zzn(1-12znz)(zn-z)R2+(z-zn)2+zR2+z22}.
Total potential due to both the p- and n-sides of the asymmetric nanowire junction depletion region will be(108)VasymNW(z)=Vasym,pNW(z)+Vasym,nNW(z)
and inserting into this (106) and (107) yields(109)VasymNW(z)=2πεNW{ρpzR2+z2-(z+zp)R2+(z+zp)22+ρpR22ln⁡[R2+(z+zp)2-(z+zp)R2+z2-z]±ρpzzp(1+12zpz)+ρn(zn-z)R2+(z-zn)2+zR2+z22-ρnR22ln⁡[R2+(z-zn)2+z-znR2+z2+z]∓ρnzzn(1-12znz)zR2+z2-(z+zp)R2+(z+zp)22}.
Junction voltage is calculated from(110)Vasym,jNW=VasymNW(zn)-VasymNW(-zp)
and when evaluating (109) at z=zn,-zp, with a total nanowire depletion width WT=WasymNW=zp+zn we find
(111)Vasym,jNW=2πεNW{-WTR2+WT22(ρp+ρn)+znρpR2+zn2+zpρnR2+zp22+R22{ρpln⁡[R2+WT2-WTR2+zn2-zn]+ρnln⁡[R2+WT2-WTR2+zp2-zp]}+znzp(ρp+ρn)-zpρpR2+zp2-znρnR2+zn22-R22{ρpln⁡[R2+WT2-WTR2+zn2-zn]ρnln⁡[RR2+zp2+zpRR2+zn2+zn]+ρpln⁡[RR2+zp2+zp]}}.Now consider the case when thep-side becomes highly doped, ρp→∞. Formula (111) becomes, if one also allows zn≫R (implies WT>zn≫R),(112)VasymNW|ρp→∞W/R≫1=ρnπW2εNW
when simplifying notation WT→zn=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
(113)WasymNW|ρp→∞W/R≫1=εNWVasym,jNW|ρp→∞,W/R≫1πeNd
and giving for an unbiased device,
(114)WbiNW|ρp→∞W/R≫1=εNWVbiπeNd.
The capacitance will be, enlisting (113),
(115)CasymNW|ρp→∞W/R≫1=±ANWeNddW(Vbias)dVbias=12RεNWRW.
In mks units this would yield CasymNW|ρp→∞,W/R≫1=2πR2εNW/W, and the per-unit area capacitance would be CasymNW,ua|ρp→∞,W/R≫1=2εNW/W, displaying the familiar form [62] which is inversely proportional to W. Evaluating (115) for a large radius nanowire like the previously examined RuO2/SiO2 nanocable, with R=31 nm, gives(116)CasymNW|ρp→∞W/R≫1=23.189aF,Nd=5×1018/cc,εNW=3.69.
However, if evaluated for a R=1 nm radius,(117)CasymNW|ρp→∞W/R≫1=0.0241aF,Nd=5×1018/cc,εNW=3.69.
Figure 7 shows the dependence of the asymmetric nanowire junction capacitance CasymNW|ρp→∞,W/R≫1 on the W/R ratio, as given by (115).Figure 7
Normalized nanowire Schottky junction capacitance againstW/R. Normalization is to RεNW. Formula (115) for CasymNW|ρp→∞,W/R≫1 is plotted.Table1 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 RεNW (this product’s units is Farads). The last three rows pertain to the nanowire results.


## 8. Discussion and Comparison of Quantum Capacitances and Junction Capacitances

### 8.1. Junction Capacitances of Nanocables and Nanowires

Examination of the Schottky nanocable formulas and graphic results plotted shows that although there is a declining dependence withW/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(118)VjNC|W≫δ=ρ2πtNCεNC{2WR+12(RW)2-1}
giving an effective junction voltage dependence VjNC|W≫δ=(4πρtNC/εNC)(W/R)1+Δ on W/R  which is slightly faster than linear because Δ>0. It is this super linear behavior which causes CNCSchottky to decline with W/R. Flat dependence for CNCSchottky 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 asymmetricp-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 asymmetricp-n junction, and then taking the limit as its p side doping approaches a value much in excess of the finite doped n-side. The result is (112), a quadratic dependence on W/R, which we know gives that classic rapid roll-off in capacitance  CNWSchottky, displayed in Figure 7.Thus, for both the nanocable and the nanowire, when considering a junction voltage dependence of the formVjSchottky∝(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 Vsym,jp-n∝(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 withW/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 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.


### 8.2. 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 isCgrd-pl=πε/sinh-1(2h/R) which for h≫R, is  Cgrd-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 Cgrd-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, Cgrd-pl≈0.034 aF/nm. Comparing this value to the intrinsic quantum capacitance mentioned earlier in Section 2 for single-walled carbon nanotubes, CiSWCNT=0.4 aF/nm, shows that Cgrd-pl<CiSWCNT. 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 Cgate=Cgrd-plCiSWCNT/(Cgrd-pl+CiSWCNT). It is clear that for the selected values for the classical capacitance, Cgrd-pl≪CiSWCNT and Cgate≈Cgrd-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 HfO2, whose dielectric constant is ε=16-22 [78–82], which could push Cgrd-pl up to Cgrd-pl≈0.187 aF/nm, and then only about a factor of two smaller than CiSWCNT. 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 SiO2 or Si2N4, this can enormously enhance capacitance. Leonard [14] estimates the capacitance per unit length to be Clg≈10 aF/nm using the approximation Clg≈2πε/ln(2lV/RγE) (assumes R≪lV), with the potential screening distance in the electrolyte lV=εkBT/[Z2e2NI], ion valence Z, NI ion density per unit volume, and γE Euler’s constant. Reason why Clg≈10 aF/nm is so large compared to Cgrd-pl is due not only to the large ε, but also because both estimation formulas have logarithmic arguments, with vastly different normalized distances. For Cgrd-pl, that distance is 2h, which we had chosen to be between 200 nm down to 20 nm, whereas, for Clg≈10 aF/nm, it is 2lV/γE, both normalized with respect to R. For water, Leonard indicates 2lV/γE≈1 nm [14]. With the normalizing distance R=0.75 nm,  Slg-bg≈ln(2h/R)/ln(2lV/RγE), making Slg-bg≈ln(266.7)/ln(1.333), and acknowledging that despite  R≪lV not being well satisfied, going ahead anyway to get the rough estimate  Slg-bg≈5.586/0.2877=19.42, for the h=100 nm case. If one takes the old Cgrd-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 Clg≈8 aF/nm. For h=10 nm, Slg-bg≈3.283/0.2877=11.41, and taking a factor of ten for the logarithmic length rescaling, gives Clg≈4 aF/nm. Both results show that Clg is not only much larger than Cgrd-pl, but considerably in excess of CiSWCNT, making the intrinsic quantum capacitance the controlling storage element in these series types of circuit arrangements found in FET like structures.


### 8.3. 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 Figure8 according to (4), (30), and (32) for, respectively, the RuO2 nanowire, the RuO2/SiO2 nanocable, and the thin walled shell RuO2/SiO2 nanocable. To plot against L/R, each formula must be rescaled to read CiT=RCi(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 single-walled 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 log10-linear display. The nearly 4 orders of magnitude variation becomes apparent in the Figure 9 plot.Figure 8
Intrinsic quantum capacitanceCiT(aF) against L/R according to (4), (30) and (32) and the SWCNT for, respectively, the RuO2 nanowire, the RuO2/SiO2 nanocable, the thin walled shell RuO2/SiO2 nanocable, and the carbon nanotube.Figure 9
Intrinsic quantum capacitanceCiT(aF) against L/R curves in Figure 8 replotted in a log10-linear display.Similarly to the previous comparison plots between the various intrinsic quantum capacitances, comparison plots of the various junction capacitances are provided in Figure10. The most accurate formulas (8), (13), (18), and (22), are utilized giving, respectively, the junction capacitances for the Schottky RuO2/SiO2 nanocable, the symmetric p-n RuO2/SiO2 nanocable, the symmetric p-n RuO2 nanowire, and the Schottky RuO2 nanowire. To plot against W/R, each formula must be rescaled to read CjT=Cjn(Rε) in aF (attoFarads), where the normalized junction capacitances Cjn previously plotted are unnormalized here. These curves are then replotted in a log10-linear display in Figure 11, which pulls the Schottky RuO2 nanowire off of the abscissa when it asymptotes.Figure 10
Junction capacitanceCj against W/R for most accurate formulas (55), (83), (100), and (115) giving, respectively, the results for the Schottky RuO2/SiO2 nanocable, the symmetric p-n RuO2/SiO2 nanocable, the symmetric p-n RuO2 nanowire, and the Schottky RuO2 nanowire.Figure 11
Junction capacitanceCj (aF) against W/R curves in Figure 8 replotted in a log10-linear display.The other reason for replotting the junction capacitance curves on a log10-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 RuO2 and RuO2/SiO2 nanostructures. Only for the ultra thin shelled nanocables, namely the SWCNT and the very thin walled RuO2/SiO2 nanocable, is this not true for the quantum capacitances. For the very thin walled RuO2/SiO2 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.Figure 12
Intrinsic quantum capacitanceCiT (aF) and junction capacitance Cj (aF) replotted in a log10-linear display, against L/R or W/R as a single abscissa.


### 8.4. Electrochemistry Meets Physics: Implications for Nanowires and Nanocables

Although we mentioned the RuO2 shell 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 RuO2 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 Ecap=CV2/2=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, Ebatt=CV2=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].IfC is not a constant, then it is suggested that C=dQ/dV instead of C=Q/V, which may occur in the case of double-layer or pseudocapacitance at electrodes [84]. The double-layer capacitance Cdl ubiquitously arises at all electrode interfaces, having higher values for aqueous electrolytes than for nonaqueous solvents or organic surfactants. The double-layer 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 double-layers, 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/cm2, can result in very large capacitive densities of 250 F/g, using an effective area per gram of 1000 m2/g as for carbon, for example. This is consistent with the large values seen for anhydrous or hydrous RuO2 (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 Cϕ, 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 underpotential-deposited species.In view of Section8.2, 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 CQC, linked on the porous nanostructure by a resistance Rp. The quantum capacitances come off of this trunk, pointing perpendicular to it, branching in a series fashion, into a parallel arrangement of the double-layer Cdl and pseudo Cϕ-capacitances. Resistances RF and RD are, respectively, the Faradaic charging and discharging resistances. RD and Cϕ are in parallel, in series with RF, and the entire parallel arrangement in series with the solution resistance RS. 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 Clg in Section 8.2, as inserted in place of the double-layer capacitance Cdl, or the pseudocapacitance Cϕ, depending upon the specific electrochemical interactions taking place.Figure 13
Circuit diagram of the intrinsic quantum capacitance embedded in the double-layer/pseudocapacitive subcircuit.


## 8.1. Junction Capacitances of Nanocables and Nanowires

Examination of the Schottky nanocable formulas and graphic results plotted shows that although there is a declining dependence withW/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(118)VjNC|W≫δ=ρ2πtNCεNC{2WR+12(RW)2-1}
giving an effective junction voltage dependence VjNC|W≫δ=(4πρtNC/εNC)(W/R)1+Δ on W/R  which is slightly faster than linear because Δ>0. It is this super linear behavior which causes CNCSchottky to decline with W/R. Flat dependence for CNCSchottky 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 asymmetricp-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 asymmetricp-n junction, and then taking the limit as its p side doping approaches a value much in excess of the finite doped n-side. The result is (112), a quadratic dependence on W/R, which we know gives that classic rapid roll-off in capacitance  CNWSchottky, displayed in Figure 7.Thus, for both the nanocable and the nanowire, when considering a junction voltage dependence of the formVjSchottky∝(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 Vsym,jp-n∝(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 withW/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 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.


## 8.2. 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 isCgrd-pl=πε/sinh-1(2h/R) which for h≫R, is  Cgrd-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 Cgrd-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, Cgrd-pl≈0.034 aF/nm. Comparing this value to the intrinsic quantum capacitance mentioned earlier in Section 2 for single-walled carbon nanotubes, CiSWCNT=0.4 aF/nm, shows that Cgrd-pl<CiSWCNT. 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 Cgate=Cgrd-plCiSWCNT/(Cgrd-pl+CiSWCNT). It is clear that for the selected values for the classical capacitance, Cgrd-pl≪CiSWCNT and Cgate≈Cgrd-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 HfO2, whose dielectric constant is ε=16-22 [78–82], which could push Cgrd-pl up to Cgrd-pl≈0.187 aF/nm, and then only about a factor of two smaller than CiSWCNT. 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 SiO2 or Si2N4, this can enormously enhance capacitance. Leonard [14] estimates the capacitance per unit length to be Clg≈10 aF/nm using the approximation Clg≈2πε/ln(2lV/RγE) (assumes R≪lV), with the potential screening distance in the electrolyte lV=εkBT/[Z2e2NI], ion valence Z, NI ion density per unit volume, and γE Euler’s constant. Reason why Clg≈10 aF/nm is so large compared to Cgrd-pl is due not only to the large ε, but also because both estimation formulas have logarithmic arguments, with vastly different normalized distances. For Cgrd-pl, that distance is 2h, which we had chosen to be between 200 nm down to 20 nm, whereas, for Clg≈10 aF/nm, it is 2lV/γE, both normalized with respect to R. For water, Leonard indicates 2lV/γE≈1 nm [14]. With the normalizing distance R=0.75 nm,  Slg-bg≈ln(2h/R)/ln(2lV/RγE), making Slg-bg≈ln(266.7)/ln(1.333), and acknowledging that despite  R≪lV not being well satisfied, going ahead anyway to get the rough estimate  Slg-bg≈5.586/0.2877=19.42, for the h=100 nm case. If one takes the old Cgrd-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 Clg≈8 aF/nm. For h=10 nm, Slg-bg≈3.283/0.2877=11.41, and taking a factor of ten for the logarithmic length rescaling, gives Clg≈4 aF/nm. Both results show that Clg is not only much larger than Cgrd-pl, but considerably in excess of CiSWCNT, making the intrinsic quantum capacitance the controlling storage element in these series types of circuit arrangements found in FET like structures.


## 8.3. 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 Figure8 according to (4), (30), and (32) for, respectively, the RuO2 nanowire, the RuO2/SiO2 nanocable, and the thin walled shell RuO2/SiO2 nanocable. To plot against L/R, each formula must be rescaled to read CiT=RCi(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 single-walled 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 log10-linear display. The nearly 4 orders of magnitude variation becomes apparent in the Figure 9 plot.Figure 8
Intrinsic quantum capacitanceCiT(aF) against L/R according to (4), (30) and (32) and the SWCNT for, respectively, the RuO2 nanowire, the RuO2/SiO2 nanocable, the thin walled shell RuO2/SiO2 nanocable, and the carbon nanotube.Figure 9
Intrinsic quantum capacitanceCiT(aF) against L/R curves in Figure 8 replotted in a log10-linear display.Similarly to the previous comparison plots between the various intrinsic quantum capacitances, comparison plots of the various junction capacitances are provided in Figure10. The most accurate formulas (8), (13), (18), and (22), are utilized giving, respectively, the junction capacitances for the Schottky RuO2/SiO2 nanocable, the symmetric p-n RuO2/SiO2 nanocable, the symmetric p-n RuO2 nanowire, and the Schottky RuO2 nanowire. To plot against W/R, each formula must be rescaled to read CjT=Cjn(Rε) in aF (attoFarads), where the normalized junction capacitances Cjn previously plotted are unnormalized here. These curves are then replotted in a log10-linear display in Figure 11, which pulls the Schottky RuO2 nanowire off of the abscissa when it asymptotes.Figure 10
Junction capacitanceCj against W/R for most accurate formulas (55), (83), (100), and (115) giving, respectively, the results for the Schottky RuO2/SiO2 nanocable, the symmetric p-n RuO2/SiO2 nanocable, the symmetric p-n RuO2 nanowire, and the Schottky RuO2 nanowire.Figure 11
Junction capacitanceCj (aF) against W/R curves in Figure 8 replotted in a log10-linear display.The other reason for replotting the junction capacitance curves on a log10-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 RuO2 and RuO2/SiO2 nanostructures. Only for the ultra thin shelled nanocables, namely the SWCNT and the very thin walled RuO2/SiO2 nanocable, is this not true for the quantum capacitances. For the very thin walled RuO2/SiO2 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.Figure 12
Intrinsic quantum capacitanceCiT (aF) and junction capacitance Cj (aF) replotted in a log10-linear display, against L/R or W/R as a single abscissa.


## 8.4. Electrochemistry Meets Physics: Implications for Nanowires and Nanocables

Although we mentioned the RuO2 shell 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 RuO2 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 Ecap=CV2/2=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, Ebatt=CV2=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].IfC is not a constant, then it is suggested that C=dQ/dV instead of C=Q/V, which may occur in the case of double-layer or pseudocapacitance at electrodes [84]. The double-layer capacitance Cdl ubiquitously arises at all electrode interfaces, having higher values for aqueous electrolytes than for nonaqueous solvents or organic surfactants. The double-layer 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 double-layers, 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/cm2, can result in very large capacitive densities of 250 F/g, using an effective area per gram of 1000 m2/g as for carbon, for example. This is consistent with the large values seen for anhydrous or hydrous RuO2 (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 Cϕ, 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 underpotential-deposited species.In view of Section8.2, 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 CQC, linked on the porous nanostructure by a resistance Rp. The quantum capacitances come off of this trunk, pointing perpendicular to it, branching in a series fashion, into a parallel arrangement of the double-layer Cdl and pseudo Cϕ-capacitances. Resistances RF and RD are, respectively, the Faradaic charging and discharging resistances. RD and Cϕ are in parallel, in series with RF, and the entire parallel arrangement in series with the solution resistance RS. 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 Clg in Section 8.2, as inserted in place of the double-layer capacitance Cdl, or the pseudocapacitance Cϕ, depending upon the specific electrochemical interactions taking place.Figure 13
Circuit diagram of the intrinsic quantum capacitance embedded in the double-layer/pseudocapacitive subcircuit.


## 9. 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 RuO2 nanowires (63 aF) and RuO2/SiO2 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 1020/cc of a 1 nm tube radius) and RuO2/SiO2 nanocables (6 aF for a 30 nm inner dielectric core radius). RuO2 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 1018/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 [87–91], nanowires [92, 93] and including V3O7·H2O 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.



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*Source: 160639-2011-02-13.xml*