Direct and indirect band gaps: Difference between revisions

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'''Quantum capacitance (density)''' is a physical value first introduced by Serge Luryi (1988)<ref name =Luryi>Serge Luryi (1988). "Quantum capacitance devices". ''Appl.Phys.Lett.'' 52(6). [http://www.ece.sunysb.edu/~serge/63.pdf Pdf]</ref> to describe the [[2DEG|2D-electronic systems]] in silicon surfaces and [[Heterojunction|GaAs 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]].
 
== Theory ==
In the general case 2D-density of states in a solid could be defined by the following:
 
:<math>D_{2D} = \frac{m^*}{\pi \hbar^2}</math>,
 
where <math>m^* = \xi m_0 </math> is a current carrier's effective mass in a solid, <math>m_0 </math> is the electron mass, and <math>\xi </math> is a dimensionless parameter which considers the zone structure of a solid. So, the quantum capacitance can be defined as follows:
 
:<math>C_{QL} = e^2\cdot D_{2D} = \xi \cdot C_{Q0}</math>,
 
where <math>C_{Q0} = 8\pi \alpha \cdot C_{QY} \ </math> - the ‘‘ideal value’’ of quantum capacitance at <math>\xi = 1 \ </math> and another ideal quantum capacitance:
 
:<math>C_{QY} = \frac{\epsilon_0}{\lambda_0} = 3.649\;2417\;\mathrm{F/m^2}</math>,
 
where <math>\epsilon_0 - \ </math> [[Vacuum permittivity|dielectric constant]], <math>\alpha - \ </math> [[fine structure constant]] and <math>\lambda_0 - \ </math> [[Compton wavelength|Compton wave length]] of electron, first defined by Yakymakha (1994) )<ref name =Yakym1>Yakymakha O.L., Kalnibolotskij Y.M. (1994). "Very-low-frequency resonance of [[MOSFET]] [[amplifier]] parameters". ''Solid- State Electronics'' 37(10),1739-1751 [http://dx.doi.org/10.1016/0038-1101(94)00152-6 pdf]</ref> in the spectroscopic investigations of the silicon MOSFETs.
 
== Experiments ==
=== 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).<ref name =Liu>Qingmin Liu and Alan Seabaugh (2001). "New Physical Understanding of the Resonant Tunneling Diode Small-Signal Equivalent Circuit". Notre Dame University Publications. [http://www.nd.edu/~nano/publications/3a0623LiuDRC.pdf Notre Dame University Publications][http://arxiv.org/abs/0812.3927 Pdf]</ref> They investigated GaAs heterostructure tunnel junctions. It is evident that tunnel junction capacity is defined by the metallurgic tunnel junction surface:
 
:<math>C_{TD} = \frac{\epsilon_0 S_{TD}}{a_{TD}} \approx 6 \times 10^{-14}\;\mathrm{F} = 60\;\mathrm{fF}</math>
 
where <math>S_{TD} = (1.6 \times 10^{-6}\;\mathrm{m})^2 - \ </math>
AlAs/InGaAs/AlAs- metallurgic tunnel junction surface, <math>a_{TD} = 3.84\times 10^{-10}\;\mathrm{m} - \ </math> 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:
 
:<math>C_{TDQ} = \frac{\epsilon_0 S_{TD}}{\lambda_0} \approx 9.34 \times 10^{-12}\;\mathrm{F} = 9.32\;\mathrm{pF} \ </math>.
 
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<ref name =Chen>Zhihong Chen, Joerg Appenzeller (2008). "Mobility Extraction and Quantum Capacitance Impact in High Performance Graphene Field-effect Transistor Devices". Electron Devices Meeting, 2008. IEDM 2008. IEEE International. San Francisco, CA, USA. ISBN 978-1-4244-2377-4 [http://arxiv.org/abs/0812.3927]</ref> on the high quality [[Graphene]] field-effect transistor devices. In this paper the Luryi definition of quantum capacitance was used:
 
:<math>D_{Gr} = \frac{aE}{\pi \hbar^2v_F^2}, a = 1 \ </math>
 
:<math>C_{QL} = e^2D_{Gr} = \frac{e^2m^*}{\pi\hbar^2}</math>,
where <math>m^* = \frac{E}{v_F^2} = \frac{\hbar}{v_F}\cdot \sqrt{\pi n_{2D}} - \ </math> 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):
 
:<math>C_{exp} \approx 4\times 10^{-3}\;\mathrm{F/m^2} \ </math>.
 
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),<ref name =Yakym2>Yakymakha O.L.(1989). ''High Temperature Quantum Galvanomagnetic Effects in the Two- Dimensional Inversion Layers of MOSFET's'' (In Russian). Kiev: Vyscha Shkola. p.91. ISBN 5-11-002309-3. [http://eqworld.ipmnet.ru/ru/library/books/Yakimaha1989ru.djvu  djvu]</ref> the 2D-system with the particles of two sorts have zone structure with the minimal particles concentration about the intrinsic value:
 
:<math>n_{2D}(T_c) = 3.03\times 10^{14}\;\mathrm{m^{-2}} \ </math>.
 
Using this value for the cyclotron mass, we obtain its minimum value:
 
:<math>m^*(n_{2D}(T_c)) =\frac{2\hbar}{\alpha c}\sqrt{\pi n_{2D}(T_c)} \approx 2.9767\times 10^{-33}\;\mathrm{kg} \ </math>.
 
Then, dimensionless parameter will be approximated as:
 
:<math>\xi = \frac{m^*(n_{2D}(T_c))}{m_0} \approx 3.2677 \times 10^{-3} \ </math>.
 
From above, we can find out the minimal value of the quantum capacitance in graphene:
 
:<math>C_{QL} = \xi C_{Q0} \approx 2.187\times 10^{-3}\;\mathrm{F/m^2} \ </math>.
 
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 “[[Electronic band structure|band structure]]” in graphene near the Dirac point with nonzero value of the “[[Band gap|forbidden band]]”.
 
==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]
*[http://www.nature.com/nphys/journal/v2/n10/abs/nphys412.html "Measurement of the quantum capacitance of interacting electrons in carbon nanotubes" ''Nature'']
 
{{DEFAULTSORT:Quantum Capacitance}}
[[Category:Quantum models]]
[[Category:Quantum mechanics]]

Revision as of 09:47, 7 November 2013

Quantum capacitance (density) is a physical value first introduced by Serge Luryi (1988)[1] to describe the 2D-electronic systems in silicon surfaces and GaAs 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.

Theory

In the general case 2D-density of states in a solid could be defined by the following:

D2D=m*π2,

where m*=ξm0 is a current carrier's effective mass in a solid, m0 is the electron mass, and ξ is a dimensionless parameter which considers the zone structure of a solid. So, the quantum capacitance can be defined as follows:

CQL=e2D2D=ξCQ0,

where CQ0=8παCQY - the ‘‘ideal value’’ of quantum capacitance at ξ=1 and another ideal quantum capacitance:

CQY=ϵ0λ0=3.6492417F/m2,

where ϵ0 dielectric constant, α fine structure constant and λ0 Compton wave length of electron, first defined by Yakymakha (1994) )[2] in the spectroscopic investigations of the silicon MOSFETs.

Experiments

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).[3] They investigated GaAs heterostructure tunnel junctions. It is evident that tunnel junction capacity is defined by the metallurgic tunnel junction surface:

CTD=ϵ0STDaTD6×1014F=60fF

where STD=(1.6×106m)2 AlAs/InGaAs/AlAs- metallurgic tunnel junction surface, aTD=3.84×1010m 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:

CTDQ=ϵ0STDλ09.34×1012F=9.32pF.

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[4] on the high quality Graphene field-effect transistor devices. In this paper the Luryi definition of quantum capacitance was used:

DGr=aEπ2vF2,a=1
CQL=e2DGr=e2m*π2,

where m*=EvF2=vFπn2D 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):

Cexp4×103F/m2.

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),[5] the 2D-system with the particles of two sorts have zone structure with the minimal particles concentration about the intrinsic value:

n2D(Tc)=3.03×1014m2.

Using this value for the cyclotron mass, we obtain its minimum value:

m*(n2D(Tc))=2αcπn2D(Tc)2.9767×1033kg.

Then, dimensionless parameter will be approximated as:

ξ=m*(n2D(Tc))m03.2677×103.

From above, we can find out the minimal value of the quantum capacitance in graphene:

CQL=ξCQ02.187×103F/m2.

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”.

References

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External links

  1. Serge Luryi (1988). "Quantum capacitance devices". Appl.Phys.Lett. 52(6). Pdf
  2. Yakymakha O.L., Kalnibolotskij Y.M. (1994). "Very-low-frequency resonance of MOSFET amplifier parameters". Solid- State Electronics 37(10),1739-1751 pdf
  3. Qingmin Liu and Alan Seabaugh (2001). "New Physical Understanding of the Resonant Tunneling Diode Small-Signal Equivalent Circuit". Notre Dame University Publications. Notre Dame University PublicationsPdf
  4. Zhihong Chen, Joerg Appenzeller (2008). "Mobility Extraction and Quantum Capacitance Impact in High Performance Graphene Field-effect Transistor Devices". Electron Devices Meeting, 2008. IEDM 2008. IEEE International. San Francisco, CA, USA. ISBN 978-1-4244-2377-4 [1]
  5. Yakymakha O.L.(1989). High Temperature Quantum Galvanomagnetic Effects in the Two- Dimensional Inversion Layers of MOSFET's (In Russian). Kiev: Vyscha Shkola. p.91. ISBN 5-11-002309-3. djvu