Editing Capacitance (section)

===Few-electron devices===
The derivation of a "quantum capacitance" of a few-electron device involves the thermodynamic chemical potential of an ''N''-particle system given by
<math display="block">\mu(N) = U(N) - U(N-1),</math>

whose energy terms may be obtained as solutions of the Schrödinger equation. The definition of capacitance,
<math display="block">{1\over C} \equiv {\Delta V\over\Delta Q},</math>
with the potential difference
<math display="block">\Delta V = {\Delta \mu \,\over e} = {\mu(N + \Delta N) -\mu(N) \over e}</math>

may be applied to the device with the addition or removal of individual electrons,
<math display="block">\Delta N = 1</math> and <math display="block">\Delta Q = e.</math>

The "quantum capacitance" of the device is then<ref>{{cite journal
 |author1=G. J. Iafrate |author2=K. Hess |author3=J. B. Krieger |author4=M. Macucci |year=1995
 |title=Capacitive nature of atomic-sized structures
 |journal=Phys. Rev. B
 |volume=52
 |issue=15
 |pages=10737–10739 |doi=10.1103/physrevb.52.10737
|pmid=9980157 |bibcode = 1995PhRvB..5210737I }}</ref>
<math display="block">C_Q(N) = \frac{e^2}{\mu(N+1)-\mu(N)} = \frac{e^2}{E(N)}.</math>

This expression of "quantum capacitance" may be written as
<math display="block">C_Q(N) = {e^2\over U(N)},</math>
which differs from the conventional expression described in the introduction where <math>W_\text{stored} = U</math>, the stored electrostatic potential energy,
<math display="block">C = {Q^2\over 2U},</math>
by a factor of {{sfrac|2}} with <math>Q = Ne</math>.

However, within the framework of purely classical electrostatic interactions, the appearance of the factor of {{sfrac|2}} is the result of integration in the conventional formulation involving the work done when charging a capacitor,
<math display="block"> W_\text{charging} = U = \int_0^Q \frac{q}{C} \, \mathrm{d}q,</math>

which is appropriate since <math>\mathrm{d}q = 0</math> for systems involving either many electrons or metallic electrodes, but in few-electron systems, <math>\mathrm{d}q \to \Delta \,Q= e</math>. The integral generally becomes a summation. One may trivially combine the expressions of capacitance 
<math display="block">Q=CV</math> 
and electrostatic interaction energy,
<math display="block">U = Q V ,</math>
to obtain
<math display="block">C = Q{1\over V} = Q {Q \over U} = {Q^2 \over U},</math>

which is similar to the quantum capacitance. A more rigorous derivation is reported in the literature.<ref>{{cite journal
 |author1     = T. LaFave Jr
 |author2     = R. Tsu
 |date        = March–April 2008
 |title       = Capacitance: A property of nanoscale materials based on spatial symmetry of discrete electrons
 |url         = http://www.pagesofmind.com/FullTextPubs/La08-LaFave-2008-capacitance-a-property-of-nanoscale-materials.pdf
 |access-date = 12 February 2014
 |journal     = Microelectronics Journal
 |volume      = 39
 |issue       = 3–4
 |pages       = 617–623
 |doi         = 10.1016/j.mejo.2007.07.105
 |url-status  = dead
 |archive-url = https://web.archive.org/web/20140222131652/http://www.pagesofmind.com/FullTextPubs/La08-LaFave-2008-capacitance-a-property-of-nanoscale-materials.pdf | archive-date = 22 February 2014}}</ref> In particular, to circumvent the mathematical challenges of spatially complex equipotential surfaces within the device, an ''average'' electrostatic potential experienced by each electron is utilized in the derivation.

Apparent mathematical differences may be understood more fundamentally. The potential energy, <math>U(N)</math>, of an isolated device (self-capacitance) is twice that stored in a "connected" device in the lower limit <math>N = 1</math>. As <math>N</math> grows large, <math>U(N)\to U</math>.<ref name=LaFave-DCD/> Thus, the general expression of capacitance is
<math display="block">C(N) = {(Ne)^2 \over U(N)}.</math>

In nanoscale devices such as quantum dots, the "capacitor" is often an isolated or partially isolated component within the device. The primary differences between nanoscale capacitors and macroscopic (conventional) capacitors are the number of excess electrons (charge carriers, or electrons, that contribute to the device's electronic behavior) and the shape and size of metallic electrodes. In nanoscale devices, [[nanowires]] consisting of metal atoms typically do not exhibit the same conductive properties as their macroscopic, or bulk material, counterparts.