Quantum capacitance
Encyclopedia
Quantum capacitance is a physical value first introduced by Serge Luryi (1988) to describe the 2D-electronic systems
in silicon surfaces and AsGa junctions
. This capacitance was defined through standard density of states in the solids. Quantum capacitance could be used in the quantum Hall effect (integer and fractional) investigations as a new approach which uses quantum LC circuit
.
, (1)
where
is a current carrier's effective mass in a solid, 
is the electron mass, and 
is a dimensionless parameter which considers the zone structure of a solid. So, the quantum capacitance can be defined as follows:
, (2)
where
- the ‘‘ideal value’’ of quantum capacitance at 
and another ideal quantum capacitance:
, (3)
where
dielectric constant, 
fine structure constant and 
Compton wave length
of electron, first defined by Yakymakha (1994) ) in the spectroscopic investigations of the silicon MOSFETs.
(4)
where
AlAs/InGaAs/AlAs- metallurgic tunnel junction surface,
junction thickness (its value could be estimated by heterostructure lattice constant). For the aim of comparison, the quantum capacitance (Yakymakha) in this case should be:
.
Therefore this value is significantly greater than the experimental value obtained by Seabaugh. Thus, in the general case of tunnel junction, neither Yakymakha, nor Luryi (there are no 2D- density of states in the 1D- dimension) approaches could be used.
field-effect transistor devices. In this paper the Luryi definition of quantum capacitance was used:
(5)
, (6)
where
current carriers cyclotron mass. The authors obtained the linear mirrors dependence of the quantum capacitance on the gate voltage, with the minimal value at the ‘‘Dirac point’’ about (Fig.7):
.
In the case of the multi-layer graphene there is the constant value (independent of the gate voltage) of the quantum capacitance, equal to the minimal value of the mono-layer graphene. Further the authors supposed, that in the "ideal case" the quantum capacitance of graphene should have zero value at the Dirac point. This isn’t true. According to Yakymakha (1989), the 2D-system with the particles of two sorts have zone structure with the minimal particles concentration about the intrinsic value:
. (7)
Using this value for the cyclotron mass, we obtain its minimum value:
.(8)
Then, dimensionless parameter will be approximated as:
. (9)
From above, we can find out the minimal value of the quantum capacitance in graphene:
. (10)
This value is two times lesser than the experimental value. Nothing strange is there. Actually, we used above only one type of current carriers during estimation procedure. However, at the Dirac point in graphene, we have the two type conductivity, due to the quasi-electron and quasi-holes. Therefore, on practice, we have two quantum capacitances due to the quasielectron and quasiholes, connected in the parallel circuit. These two quantum capacitances confirms indirectly the existence of the “band structure
” in graphene near the Dirac point with nonzero value of the “forbidden band
”.
2DEG
A two-dimensional electron gas is a gas of electrons free to move in two dimensions, but tightly confined in the third. This tight confinement leads to quantized energy levels for motion in that direction, which can then be ignored for most problems. Thus the electrons appear to be a 2D sheet...
in silicon surfaces and AsGa junctions
Heterojunction
A heterojunction is the interface that occurs between two layers or regions of dissimilar crystalline semiconductors. These semiconducting materials have unequal band gaps as opposed to a homojunction...
. This capacitance was defined through standard density of states in the solids. Quantum capacitance could be used in the quantum Hall effect (integer and fractional) investigations as a new approach which uses quantum LC circuit
Quantum LC circuit
An LC circuit can be quantized using the same methods as Quantum harmonic oscillator. An LC circuit is a variety of resonant circuit or tuned circuit and consists of an inductor, represented by the letter L, and a capacitor, represented by the letter C...
.
Theory
In the general case 2D-density of states in a solid could be defined by the following:, (1)where
is a current carrier's effective mass in a solid, is the electron mass, and is a dimensionless parameter which considers the zone structure of a solid. So, the quantum capacitance can be defined as follows:, (2)where
- the ‘‘ideal value’’ of quantum capacitance at and another ideal quantum capacitance:, (3)where
dielectric constant, fine structure constant and Compton wave lengthCompton wavelength
The Compton wavelength is a quantum mechanical property of a particle. It was introduced by Arthur Compton in his explanation of the scattering of photons by electrons...
of electron, first defined by Yakymakha (1994) ) in the spectroscopic investigations of the silicon MOSFETs.
Heterostructure tunnel junctions
The first attempt of quantum capacitance experimental confirmation in the 21st century were made by Qingmin Liu and Alan Seabaugh from the Notre Dame University (2001). They investigated GaAs heterostructure tunnel junctions. It is evident that tunnel junction capacity is defined by the metallurgic tunnel junction surface: (4)where
AlAs/InGaAs/AlAs- metallurgic tunnel junction surface,
junction thickness (its value could be estimated by heterostructure lattice constant). For the aim of comparison, the quantum capacitance (Yakymakha) in this case should be:.Therefore this value is significantly greater than the experimental value obtained by Seabaugh. Thus, in the general case of tunnel junction, neither Yakymakha, nor Luryi (there are no 2D- density of states in the 1D- dimension) approaches could be used.
Graphene MOSFETs
One publication on the theme was made by Zhihong Chen and Joerg Appenzeller on the high quality GrapheneGraphene
Graphene is an allotrope of carbon, whose structure is one-atom-thick planar sheets of sp2-bonded carbon atoms that are densely packed in a honeycomb crystal lattice. The term graphene was coined as a combination of graphite and the suffix -ene by Hanns-Peter Boehm, who described single-layer...
field-effect transistor devices. In this paper the Luryi definition of quantum capacitance was used:
(5), (6)where
current carriers cyclotron mass. The authors obtained the linear mirrors dependence of the quantum capacitance on the gate voltage, with the minimal value at the ‘‘Dirac point’’ about (Fig.7):.In the case of the multi-layer graphene there is the constant value (independent of the gate voltage) of the quantum capacitance, equal to the minimal value of the mono-layer graphene. Further the authors supposed, that in the "ideal case" the quantum capacitance of graphene should have zero value at the Dirac point. This isn’t true. According to Yakymakha (1989), the 2D-system with the particles of two sorts have zone structure with the minimal particles concentration about the intrinsic value:
. (7)Using this value for the cyclotron mass, we obtain its minimum value:
.(8)Then, dimensionless parameter will be approximated as:
. (9)From above, we can find out the minimal value of the quantum capacitance in graphene:
. (10)This value is two times lesser than the experimental value. Nothing strange is there. Actually, we used above only one type of current carriers during estimation procedure. However, at the Dirac point in graphene, we have the two type conductivity, due to the quasi-electron and quasi-holes. Therefore, on practice, we have two quantum capacitances due to the quasielectron and quasiholes, connected in the parallel circuit. These two quantum capacitances confirms indirectly the existence of the “band structure
Electronic band structure
In solid-state physics, the electronic band structure of a solid describes those ranges of energy an electron is "forbidden" or "allowed" to have. Band structure derives from the diffraction of the quantum mechanical electron waves in a periodic crystal lattice with a specific crystal system and...
” in graphene near the Dirac point with nonzero value of the “forbidden band
Band gap
In solid state physics, a band gap, also called an energy gap or bandgap, is an energy range in a solid where no electron states can exist. In graphs of the electronic band structure of solids, the band gap generally refers to the energy difference between the top of the valence band and the...
”.