# Quantum capacitance

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{{Short description|Type of capacitance}}
'''Quantum capacitance''',<ref name="Luryi">{{cite journal|author=Serge Luryi|date=1988|title=Quantum capacitance devices|url=http://www.ece.sunysb.edu/~serge/63.pdf|journal=Applied Physics Letters|volume=52|issue=6|pages=501–503|bibcode=1988ApPhL..52..501L|doi=10.1063/1.99649}}</ref> also known as '''chemical capacitance'''<ref name=Bisquert/> is defined as the variation of [electrical charge](/source/Electric_charge) <math>q</math> with respect to the variation of internal [chemical potential](/source/chemical_potential) <math>\mu</math>, i.e., <math>C_{q} = \frac{dq}{d\mu}</math>.<ref name="Luryi"/><ref name=Bisquert/><ref name=":0" /> It was first introduced theoretically by Serge Luryi (1988).<ref name="Luryi" />

In the simplest example, if a [parallel-plate capacitor](/source/parallel-plate_capacitor) is made so that one or both of the plates has a low [density of states](/source/density_of_states), then the capacitance is ''not'' given by the normal formula for parallel-plate capacitors, <math>C_e</math>. Instead, the capacitance is lower, as if there was another capacitor in series, <math>C_q</math>. This second capacitance, related to the [density of states](/source/density_of_states) of the plates, is the quantum capacitance and is represented by <math>C_q</math>.  The combination of regular geometric capacitance with chemical/quantum capacitance is called '''electrochemical capacitance'''<ref name=":0">{{Cite journal|last1=Miranda|first1=David A.|last2=Bueno|first2=Paulo R.|date=2016-09-21|title=Density functional theory and an experimentally-designed energy functional of electron density|journal=Phys. Chem. Chem. Phys.|language=en|volume=18|issue=37|pages=25984–25992|doi=10.1039/c6cp01659f|pmid=27722307|issn=1463-9084|bibcode=2016PCCP...1825984M}}</ref> <math>\frac{1}{C_{\bar{\mu}}} = \frac{1}{C_e} + \frac{1}{C_q}</math>.

Quantum capacitance is especially important for low-density-of-states systems, such as a [2-dimensional electronic system](/source/2DEG) in a semiconductor surface or interface or [graphene](/source/graphene), and can be used to construct an experimental energy functional of [electron density](/source/electron_density).<ref name=":0" />

== Overview ==

When a voltmeter is used to measure an electronic device, it does not quite measure the pure [electric potential](/source/electric_potential) (also called [Galvani potential](/source/Galvani_potential)). Instead, it measures the [electrochemical potential](/source/electrochemical_potential), also called "[fermi level](/source/fermi_level) difference", which is the ''total'' free energy difference per electron, including not only its [electric potential energy](/source/electric_potential_energy) but also all other forces and influences on the electron (such as the [kinetic energy](/source/kinetic_energy) in its [wavefunction](/source/wavefunction)). For example, a [p-n junction](/source/p-n_junction) in equilibrium, there is a galvani potential (built-in potential) across the junction, but the "voltage" across it is zero (in the sense that a voltmeter would measure zero voltage).

In a capacitor, there is a relation between charge and voltage, <math>Q=CV</math>. As explained above, we can divide the voltage into two pieces: The [galvani potential](/source/galvani_potential), and everything else.

In a traditional metal-insulator-metal capacitor, the galvani potential is the ''only'' relevant contribution. Therefore, the capacitance can be calculated in a straightforward way using [Gauss's law](/source/Gauss's_law).

However, if one or both of the capacitor plates is a [semiconductor](/source/semiconductor), then galvani potential is ''not'' necessarily the only important contribution to capacitance. As the capacitor charge increases, the negative plate fills up with electrons, which occupy higher-energy states in the band structure, while the positive plate loses electrons, leaving behind electrons with lower-energy states in the band structure. Therefore, as the capacitor charges or discharges, the voltage changes at a ''different'' rate than the galvani potential difference.

In these situations, one ''cannot'' calculate capacitance merely by looking at the overall geometry and using Gauss's law. One must also take into account the band-filling / band-emptying effect, related to the density-of-states of the plates. The band-filling / band-emptying effect alters the capacitance, imitating a second capacitance in series. This capacitance is called '''quantum capacitance''', because it is related to the energy of an electron's quantum wavefunction.

Some scientists refer to this same concept as '''chemical capacitance''', because it is related to the electrons' [chemical potential](/source/chemical_potential).<ref name=Bisquert>{{Cite journal
| doi = 10.1021/jp035395y
| volume = 108
| issue = 7
| pages = 2313–2322
| last = Bisquert
| first = Juan
|author2=Vyacheslav S. Vikhrenko
 | title = Interpretation of the Time Constants Measured by Kinetic Techniques in Nanostructured Semiconductor Electrodes and Dye-Sensitized Solar Cells
| journal = The Journal of Physical Chemistry B
| date = 2004
| citeseerx = <!--10.1.1.626.2494-->
}}</ref>

The ideas behind quantum capacitance are closely linked to [Thomas–Fermi screening](/source/Thomas%E2%80%93Fermi_screening) and [band bending](/source/band_bending).

== Theory ==
Take a capacitor where one side is a metal with essentially-infinite density of states. The other side is the low density-of-states material, e.g. a [2DEG](/source/2DEG), with density of states <math>\rho</math>. The geometrical capacitance (i.e., the capacitance if the 2DEG were replaced by a metal, due to galvani potential alone) is <math>C_\text{geom}</math>.

Now suppose that ''N'' electrons (a charge of <math>Q=N e</math>) are moved from the metal to the low-density-of-states material. The Galvani potential changes by <math>\Delta V_\text{galvani} = Q/C_\text{geom}</math>. Additionally, the [internal chemical potential](/source/internal_chemical_potential) of electrons in the 2DEG changes by <math>\Delta \mu_\text{internal} = N/\rho = Q/(\rho e)</math>, which is equivalent to a voltage change of <math>\Delta V_\text{quantum} = (\Delta \mu_\text{internal}) / e = Q/(\rho e^2)</math>.

The total voltage change is the sum of these two contributions. Therefore, the total effect is ''as if'' there are two capacitances in series: The conventional geometry-related capacitance (as calculated by Gauss's law), and the "quantum capacitance" related to the density of states. The latter is:

{{Equation box 1
|indent=:
|equation=<math>C_\text{quantum} = \rho e^2</math>
|cellpadding
|border
|border colour = #0073CF
|background colour=#F5FFFA}}

In the case of an ordinary 2DEG with parabolic dispersion,<ref name=Luryi />

:<math>C_\text{quantum} = \frac{g_v m^* e^2}{\pi \hbar^2}</math>

where <math>g_v</math> is the valley degeneracy factor, and ''m''* is [effective mass](/source/effective_mass_(solid-state_physics)).

==Applications==

The quantum capacitance of [graphene](/source/graphene) is relevant to understanding and modeling gated graphene.<ref>{{Cite journal
| doi = 10.1007/s11671-009-9515-3
| pmid = 20672092
| volume = 5
| issue = 3
| pages = 505–511
| last = Mišković
| first = Z. L.
|author2=Nitin Upadhyaya
 | title = Modeling Electrolytically Top-Gated Graphene
| journal = Nanoscale Research Letters
| date = 2010
|arxiv = 0910.3666 |bibcode = 2010NRL.....5..505M
 | pmc=2894001}}</ref> It is also relevant for [carbon nanotubes](/source/carbon_nanotubes).<ref>
{{Cite journal |doi = 10.1038/nphys412 |volume = 2 |issue = 10 |pages = 687–691 |last = Ilani |first = S. |author2=L. a. K. Donev |author3=M. Kindermann |author4=P. L. McEuen  |title = Measurement of the quantum capacitance of interacting electrons in carbon nanotubes |journal = Nature Physics |date = 2006 |url=https://www.lassp.cornell.edu/lassp_data/mceuen/homepage/Publications/QuantumCapacitance_NaturePhysics.pdf|bibcode = 2006NatPh...2..687I |doi-access = free }}</ref>

In modeling and analyzing [dye-sensitized solar cells](/source/dye-sensitized_solar_cells), the quantum capacitance of the [sintered](/source/sintered) [TiO<sub>2</sub>](/source/TiO2) [nanoparticle](/source/nanoparticle) electrode is an important effect, as described in the work of [Juan Bisquert](/source/Juan_Bisquert).<ref name=Bisquert/><ref>{{Cite journal| doi=10.1039/B310907K | journal = Phys. Chem. Chem. Phys. | title = Chemical capacitance of nanostructured semiconductors: its origin and significance for nanocomposite solar cells | author = Juan Bisquert | date=2003 | volume=5 | issue = 24 | page = 5360|bibcode = 2003PCCP....5.5360B }}</ref><ref>{{cite book |title=Nanostructured Energy Devices: Equilibrium Concepts and Kinetics |author=Juan Bisquert |date=2014 |publisher=CRC Press |url=https://www.crcpress.com/Nanostructured-Energy-Devices-Equilibrium-Concepts-and-Kinetics/Bisquert/p/book/9781439836026 |isbn=9781439836026 |access-date=2017-01-09 |archive-date=2016-11-23 |archive-url=https://web.archive.org/web/20161123054056/https://www.crcpress.com/Nanostructured-Energy-Devices-Equilibrium-Concepts-and-Kinetics/Bisquert/p/book/9781439836026 |url-status=dead }}</ref>

Luryi proposed a variety of devices using 2DEGs, which only work because of the low 2DEG density-of-states, and its associated quantum capacitance effect.<ref name=Luryi/> For example, in the three-plate configuration metal-insulator-2DEG-insulator-metal, the quantum capacitance effect means that the two capacitors interact with each other.

Quantum capacitance can be relevant in [capacitance–voltage profiling](/source/capacitance%E2%80%93voltage_profiling).

When [supercapacitor](/source/supercapacitor)s are analyzed in detail, quantum capacitance plays an important role.<ref>{{Cite journal|last=Bueno|first=Paulo R.|date=2019-02-28|title=Nanoscale origins of super-capacitance phenomena|journal=Journal of Power Sources|volume=414|pages=420–434|doi=10.1016/j.jpowsour.2019.01.010|issn=0378-7753|bibcode=2019JPS...414..420B|s2cid=104416995 |hdl=11449/190051|hdl-access=free}}</ref>

==References==
{{reflist}}

==External links==
*[http://nano.ece.ubc.ca/pub/john04a.pdf D.L. John, L.C. Castro, and D.L. Pulfrey "Quantum Capacitance in Nanoscale Device Modeling" ''Nano Electronics Group Publications''.]
*[https://nanohub.org/resources/695/ ECE 453 Lecture 30: Quantum Capacitance]

{{DEFAULTSORT:Quantum Capacitance}}
Category:Quantum models
Category:Capacitance

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Adapted from the Wikipedia article [Quantum capacitance](https://en.wikipedia.org/wiki/Quantum_capacitance) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Quantum_capacitance?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.