Physics/Essays/Fedosin/Quantum Hall composite resonator

The Quantum Hall composite resonator (QHCR) is a topological object with quantum reactive parameters (C and L), that as expected has fixed resonant frequency, due to the odd number of electrons which are moving in the homogenous magnetic field B. QHCR in a two-dimensional system of electrons has characteristic impedance proportional to the von Klitzing constant $$R_K = h/e^2 $$ = 25,812.807449(86) Ω, the odd number of electrons ($$j = 3,5,7,...$$) and inversely to the Landau level number ($$i = 1,2,3,...$$):
 * $$ Z = \sqrt{\frac{L}{C}} = \frac{j}{i} R_K \ .$$

History
The discovery of the Quantum Hall effect (QHE) by Klaus von Klitzing in 1980 was completely unexpected, as nobody considered that a macroscopic physical value such as electrical resistance could quantized. Robert B. Laughlin proposed a simple quantum theory in 1981, which explained the quantization of resistance in terms of one-electron filling of Landau levels in a strong magnetic field which caused the electrons to move in circles. Furthermore, the semiclassical Hall current in this approach is due to the adiabatic derivative of the total system energy by magnetic flux through the closed loop. This approach, however, does not take into consideration the electric fields, which can be strong in certain conditions (especially in semiconductor devices such as MOSFETs), and does nothing with the reactive parameters of the Quantum LC circuit formed.

The situation becomes more complicated when Horst Ludwig Stormer, Daniel C. Tsui, and Arthur Gossard discovered the fractional quantum Hall effect. The simple quantum theory of Laughlin would consider the "carriers" of fractional charge, as composite fermions. Further developments was made by S.H. Simon in 1998. In this approach, Landau levels are filled by composite fermions which have fractional charges, as the QHE Laughlin model has. The Laughlin-Jain-Simon approach is the dominant view now, but some physicists strongly criticize it. For example, Keshav N. Shrivastava thought that quasiparticles form the quantum of magnetic flux ($$~\phi_0 = \frac{h}{e} $$), which could be connected to an electron forming a compound state with some magnetic and electric flux quantum. The theory of such quasiparticles could not be derived from the first principles.

Furthermore, M.I. Dyakonov described the current situation as being inconvenient: on one hand, there are many experimental facts which confirm the idea of composite fermions that takes part in the "singular" theoretical approach to the phenomenon, which hints at the physical reality of the idea, but on the other hand, nobody can theoretically justify the existence of these free quasiparticles.

However, there is another approach to the FQHE. At the beginning of the 21st century, Raphael Tsu proposed a new conception of quantum mechanics in which the characteristic impedance plays the dominant role. This dominance was illustrated in the quantum oscillator problem by Yakov Zel'dovich in 2008.

But Tsu and Zel'dovich were not the firsts. It so happens that there is a little interest in macroscopic properties such as resistance, capacitance, and inductance in the modern quantum electrodynamics. Furthermore, these properties are neglected even in classical electrodynamics, in favour of concepts such as electric and magnetic fields. For example, Maxwell equations are formulated in terms of fields and so the resulting solutions only include fields. "Field approaches" in electrodynamics that consider "point charges" leads to mathematical singularities (renormalization problems) when the interaction radius trends to zero. Furthermore, these singularities are present in quantum electrodynamics too, where power methods are developed to compensate them. In applied physics, reactive parameters such as capacitance and inductance are used instead of fields.

The present day situation (without proper development of the theory of reactive parameters such as inductance, capacitance and electromagnetic resonator) impedes developments of information technologies and quantum computing.The mechanical harmonic oscillator was studied in the early 1930s, when the quantum theory was first developed. However, the quantum consideration of the LC circuit was started only by Louisell in 1973. Since then, there were no practical examples of quantum capacitances and inductances, therefore this approach did not received much consideration. Theoretically correct introductions of the quantum capacitance for QHE, based on the density of states, were first presented by Serge Luryi in 1988. However, Luryi did not introduce the quantum inductance, and this approach was not considered in the quantum LC circuit and resonator. One year later, O.L. Yakymakha considered an example of the series and parallel QHE LC circuits (both integer and fractional) and their characteristic impedances. However he did not consider the Schrodinger equation for the quantum LC circuit.

For the first time, both quantum values, capacitance and inductance, were considered by Yakymakha in 1994, during spectroscopic investigations of MOSFETs at the very low frequencies (sound range). The flat quantum capacitances and inductances here had thicknesses about Compton wavelength of electron and its characteristic impedance – the wave impedance of free space. Three year later, Devoret presented a complete theory of the quantum LC circuit (applied to the Josephson junction). Possible application of the quantum LC circuits and resonators in quantum computations were considered by Devoret in 2004.

Quantum Hall resonator (QHR)
Magnetic length of the resonator is:
 * $$l = \sqrt{\frac{\hbar}{eB}}, \ $$

where $$\hbar \ $$ is the reduced Planck constant, $$e \ $$ is the elementary charge, and $$B  \ $$ is the external homogenous magnetic field.

Surface scaling parameter is:
 * $$S = 2\pi l^2 = \frac{h}{eB}. \ $$

Cyclotron frequency is:
 * $$\omega_c = \frac{eB}{m}, \ $$

where $$m \ $$ is the effective electron mass in solids.

QHR density of states (DOS) is:
 * $$D = \frac{1}{S \hbar \omega_c} = \frac{m}{2\pi \hbar^2}. \ $$

DOS quantum capacitance is:
 * $$C = q^2 D S, \ $$

where $$q $$ is the induced electric charge of the resonator.

DOS quantum inductance is:
 * $$L = \phi^2 D S, \ $$

where $$ \phi $$ is the induced magnetic flux.

Maximal value of the capacitance energy is:
 * $$W_C = \frac{q^2}{2C} = \frac{1}{2D S}. \ $$

Maximal value of the inductance energy is:
 * $$W_L = \frac{ \phi^2}{2L} = \frac{1}{2D S}. \ $$

QHR resonance frequency is:
 * $$\omega = \frac{1}{\sqrt{L C}} = \frac{1}{q \phi D S}. \ $$

The standard quantum oscillator has the following eigenvalues of the Hamiltonian:
 * $$W_n = \hbar \omega_0(n + 1/2), \ $$

where at $$n = 0 \ $$ we shall have zero oscillation:
 * $$W_0 = \hbar \omega_0/2. \ $$

QHR energy conservation law (at n = 0) is:
 * $$W_{QR} = \frac{1}{2}\hbar \omega = \frac{\hbar}{2q \phi D S} = \frac{1}{2D S}, \ $$

which yields the following relationship between $$q $$  and  $$\phi $$ (an analogue of uncertainty principle ):
 * $$q \phi = \hbar. \ $$

QHR characteristic impedance:
 * $$ Z = \sqrt{\frac{L}{C}} = \frac{ \phi }{q} = R_K \ $$.

The solution of these two equation yields the following values for $$q $$  and  $$\phi $$:
 * $$q = \frac{e}{\sqrt{2\pi}} \ $$
 * $$ \phi = \frac{\phi_0}{\sqrt{2\pi}}, \ $$

where $$ \phi_0= h/e \ $$ is the  magnetic monopole.


 * $$C = q^2 D S = 2 \alpha \cdot \frac{\varepsilon_0}{\lambda_0}S \ ,$$
 * $$L = \phi ^2 D S = 2 \beta \cdot \frac{\mu_0}{\lambda_0}S, \ $$

where $$\alpha \ $$  is the electric fine structure constant], $$\beta = \frac{1}{4\alpha} \ $$ is the magnetic coupling constant, $$\varepsilon_0 \ $$ is the electric constant, $$\mu_0 \ $$ is the vacuum permeability, and $$\lambda_0 = \frac{h}{ mc} \ $$ is the Compton wavelength of electron. Note that, the QHR resonance frequency takes the values of cyclotron frequency here:
 * $$\omega = \frac{1}{\sqrt{L C}} = \frac{eB}{m} = \omega_c. \ $$

Constant magnetic field
Let us consider the standard QHE for one electron filling Landau levels step by step at constant magnetic field $$B_1 = const \ $$ and increasing electric field $$E_1 = variable. \ $$ At the first Landau level surface electron concentration will be:
 * $$n_{1} = \frac{1}{S_{1}} = \frac{eB_1}{h}. \ $$

First Landau level energy is:
 * $$W_{L1} = \hbar \omega_{c1}, \ $$

where $$\omega_{c1} \ $$ is the cyclotron frequency due to the constant magnetic field $$B_1 \ $$.

With increasing filling number $$i_1 = 1, 2, 3,... \ $$ we shall have the following parameters of QHR:

Surface electron concentration is:
 * $$n_{i} = i_1 n_{1}. \ $$

Landau level energy is:
 * $$W_{Li} = i_1 W_{L1}. \ $$

Density of states is:
 * $$D_{i} = \frac{n_{i}}{W_{Li}} = D_{1}. \ $$

Induced effective electric charge is:
 * $$q_{i} = i_1 q_{1}. \ $$

DOS quantum capacitance is:
 * $$C_{i} = \frac{ q_{i}^2}{ W_{Li} } = i_1 C_{1}. \ $$

So, we have the parallel connection of “elementary resonator” quantum capacitances in that case.

Induced magnetic flux is:
 * $$ \phi_i = \phi_1. \ $$

DOS quantum inductance is:
 * $$L_{i} = \frac { \phi_1^2}{ W_{Li} } =\frac{1}{i_1}L_{1}. \ $$

So, we have the parallel connection of “elementary resonator” quantum inductances in that case.

Characteristic impedance is:
 * $$ Z_{i} = \sqrt{\frac{L_{i}}{C_{i}}} = \frac{1}{i_1}\sqrt{\frac{L_{1}}{C_{1}}} = \frac{1}{i_1} Z_{1}. \ $$

Resonance frequency is:
 * $$\omega_{i} = \frac{1}{\sqrt{L_{i}C_{i}}} = \frac{1}{\sqrt{L_{1}C_{1}}} = \frac{eB_1}{m} = \omega_{c1}. \ $$

Thus, in the constant homogenous magnetic field with one electron filling Landau level we shall have the following conservating parameters:
 * $$D_{i}= const, \quad \omega_{i} = const. \ $$

Constant electric field
Let us consider the other case when electric field is constant ($$E_2 = (E_1)_{max} = const \ $$), and increasing magnetic field:
 * $$B_1 \le B_2 \le j_2\cdot B_1 = 2B_1, \ $$

where $$j_2 \ $$ is the maximal number of electron on Landau level.

The maximum number of Landau level is taken as:
 * $$i_2 = (i_1)_{max} = k = 3. \ $$

For the sake of the unitary consideration it is need to redefine parameters of the first stage future composite resonator. Surface electron concentration is:
 * $$n_{i2} = n_{i1} = k\cdot n_{1}. \ $$

Landau level energy is:
 * $$W_{Li2} = k\cdot W_{L1}. \ $$

Density of states is:
 * $$D_{i2} = D_{i1} = D_{1}. \ $$

Induced charge is:
 * $$q_{i2} = k\cdot q_{1}. \ $$

DOS quantum capacitance is:
 * $$C_{i2} = C_{i1} = k\cdot C_{1}. \ $$

Induced magnetic flux is:
 * $$ \phi_{i2} = \phi_{i1} = \phi_1. \ $$

DOS quantum inductance is:
 * $$L_{i2} = k\cdot L_{1}. \ $$

Characteristic impedance is:
 * $$ Z_{i2} = \frac{1}{k} Z_{1}. \ $$

Resonance frequency is:
 * $$\omega_{i2} = \omega_{i1} = \omega_{c1}. \ $$

Thus, we are ready to consider the second case at $$j_2 =2 \ $$, $$i_2 = 2 \ $$ and $$B_2 = j_2B_1 = 2B_1 \ $$. Note that with increasing magnetic field to $$B_2 = 2B_1 \ $$ the second electron from the second Landau level goes to the first Landau level filled by the first electron, and the third electron goes to the second Landau level. So, we have the composite resonator with two filled Landau levels.

DOS quantum capacitance in that case will be:
 * $$C_{22} = \frac{2}{3}C_{11} = q_{22}^2 D_{22}\frac{1}{n_{22}} = \frac{q_{22}^2}{W_{L22}}, \ $$

where
 * $$C_{11} = q_{11}^2 D_{11}\frac{1}{n_{11}} = \frac{q_{11}^2}{W_{L11}}. \ $$

The total energy of this composite resonator will be:
 * $$W_{L22} = \frac{3}{2}(\frac{q_{22}}{q_{11}})^2 W_{L11} = \frac{3^3}{2}W_{L11}, \ $$

at $$q_{22} = 3q_{11}. \ $$

Density of states will be:
 * $$D_{22} = \frac{n_{22}}{W_{L22}} = \frac{2n_{22}}{27W_{L11}} = \frac{2}{27}\frac{n_{22}}{n_{11}}D_{11}. \ $$

The natural supposition for surface electron density is:
 * $$n_{22} = 3^2\cdot n_{11} \ $$

yields the following value for DOS in that case:
 * $$D_{22} = \frac{2\cdot 9}{27}D_{11} = \frac{2}{3}D_{11}. \ $$

DOS quantum inductance is:
 * $$L_{22} = \frac{3}{2}L_{11}, \ $$

where induced magnetic flux is considered:
 * $$ \phi_{22} = \frac{9}{2} \phi_{11}. \ $$

Characteristic impedance in the case will be:
 * $$ Z_{22} = \sqrt{\frac{L_{22}}{C_{22}}}= \frac{3}{2} Z_{11}. \ $$

Resonance frequency is:
 * $$\omega_{22} = \frac{1}{\sqrt{L_{22}C_{22}}} = 2\omega_{11} = \frac{e}{m}2B_1 = \omega_{c2}. \ $$

Now we are ready to consider the third case at $$j_2 = 3 \ $$, $$i_2 = 1 \ $$ and $$B_2 = j_2 B_1 = 3B_1 \ $$. Note that, with increasing magnetic field to $$B_2 = 3B_1 \ $$ the third electron from the second Landau level goes to the first Landau level filled by the first and second electrons. So, we have the composite resonator with one filled Landau level by three electrons. DOS quantum capacitance in that case will be:
 * $$C_{33} = \frac{1}{3}C_{11} = q_{13}^2 D_{13}\frac{1}{n_{13}} = \frac{q_{13}^2}{W_{L13}}. \ $$

The total energy of this composite resonator will be:
 * $$W_{L13} = 3(\frac{q_{13}}{q_{11}})^2 W_{L11} = 3^3W_{L11}, \ $$

at $$q_{13} = 3q_{11}. \ $$

Density of states will be:
 * $$D_{13} = \frac{n_{13}}{W_{L13}} = \frac{n_{13}}{27W_{L11}} = \frac{1}{27}\frac{n_{13}}{n_{11}}D_{11}. \ $$

The natural supposition for surface electron density is:
 * $$n_{13} = 3\cdot n_{22} = 3^3\cdot n_{11} \ $$

yields the following value for DOS in that case:
 * $$D_{13} = \frac{n_{13}}{W_{L13}} = \frac{n_{11}}{W_{L11}} = D_{11}. \ $$

So, the new composite system returns to the natural 2D- case.

DOS quantum inductance is:
 * $$L_{13} = \frac{3}{1}L_{11}, \ $$

where induced magnetic flux is considered:
 * $$ \phi_{13} = \frac{9}{1} \phi_{11}. \ $$

Characteristic impedance in that case will be:
 * $$ Z_{13} = \sqrt{\frac{L_{13}}{C_{13}}}= \frac{3}{1}\rho_{11}. \ $$

Resonance frequency is:
 * $$\omega_{13} = \frac{1}{\sqrt{L_{13}C_{13}}} = 3\omega_{11} = \frac{e}{m}3B_1 = \omega_{c3}. \ $$

Thus, the composite quantum Hall resonator is formed.

The energy per Landau level is:
 * $$W_{L13} = \hbar \omega_{c3} = 3\hbar \omega_{c1}. \ $$

The energy gap of QHCR:
 * $$\Delta W_{gap} = 3^2\cdot \hbar \omega_{c3} = 3^3\cdot \hbar \omega_{c1}. \ $$

Now the QHCR can be considered as elementary quantum object for the external magnetic and electric fields, independently of its intrinsic structure. Note that, this object is connected to three electrons and could change its state simultaneously.

QHCR and fractional QHE
Now we can consider the s.c. fractional QHE by the composite resonator model formed by three electrons. As in the case of one electron resonator model we should to fix magnetic field:
 * $$B_3 = 3B_1 = const \ ,$$

and increase electric field:
 * $$(E_1)_{max} \le E = variable. \ $$

So, the filling Landau level number $$i_3 = 1, 2, 3,... \ $$ will be connected with three electrons for every Landau level.

Surface electron concentration in the case will be:
 * $$n_{i3} = i_3 n_{13} \ .$$

Density of states will be:
 * $$D_{i3} = \frac{n_{i3}}{W_{i3} } = \frac{i_3n_{13}}{i_3W_{13}} = D_{13}. \ $$

Induced electric charge is:
 * $$q_{i3} = i_3 q_{13} = variable \ . $$

DOS quantum capacitance is:
 * $$C_{i3} = i_3^2q_{13}^2D_{i3}\frac{1}{n_{i3}} = i_3 C_{13} \ .$$

Induced magnetic flux is:
 * $$ \phi_{i3} = \phi_{13} = const. \ $$

DOS quantum inductance is:
 * $$L_{i3} = > \phi_{i3}^2D_{i3}\frac{1}{n_{i3}} = \frac{1}{i_3}L_{13} \ .$$

Characteristic impedance in that case will be:
 * $$ Z_{i3} = \sqrt{\frac{L_{i3}}{C_{i3}}} = \frac{1}{i_3}\sqrt{\frac{L_{13}}{C_{13}}} = \frac{1}{i_3} Z_{13} = \frac{3}{i_3} Z_{11} \ $$

where $$ Z_{11} = R_K \ .$$

QHCR resonance frequency:
 * $$\omega_{i3} = \frac{1}{\sqrt{L_{i3} C_{i3} }} = \frac{1}{\sqrt{L_{13}C_{13}}} = \omega_{13} = 3\omega_{11} \ . $$

Note that, in the general case filling number $$i_3 = 1, 2, 3,... \ $$ could be connected with any odd quantity of filling electrons per Landau level ($$k = 3, 5, 7, 9,... \ $$). However, in those cases we shall have other values for characteristic impedance and resonance frequency:
 * $$ Z_k = \frac{k}{i_3} Z_{11}$$
 * $$ \omega_k = k \omega_{11}. \ $$

Quantum antidot
Resonance electron tunnelling in the Goldman quantum antidot can be investigated experimentally by measuring of the periodic quantization (oscillation) of magnetic and electric fields:
 * $$\Delta B_{ik}\cdot S = \frac{\Delta B_{ik}h}{eB} = \Delta \phi_{ik} \ $$
 * $$\Delta D_{ik}\cdot S = \frac{\Delta D_{ik}h}{eB} = \Delta q_{ik} \ $$

where $$i = 1, 2, 3,... \ $$ is the Landau level number, and $$k = 3, 5, 7,... \ $$ is the electron number of composite resonator.

The electric field in the quantum antidot can be represented through applied voltage:
 * $$\Delta D_{ik} = \varepsilon_0 \varepsilon_r \Delta E_{ik} = \frac{\varepsilon_0 \varepsilon_r}{d_x} \Delta V_{ik} = C_x\Delta V_{ik}, \ $$

where $$\varepsilon_0 \ $$ is the electric constant, $$\varepsilon_r  \ $$ is the relative permittivity, $$d_x \ $$ is the heterojunction thickness, and $$C_x  \ $$ is the heterojunction mesoscopic capacitance per unit area.

The induced electric charge and induced magnetic flux quantization is due to the properties of composite resonator formed in the quantum antidot:
 * $$\Delta q_{ik} = iq_0 = ie \ $$
 * $$\Delta \phi_{ik} = k \phi_0 = k \frac{h}{e}. \ $$

Characteristic impedance of composite resonator is:
 * $$Z_{ik} = \frac{\Delta \phi_{ik}}{\Delta q_{ik}} = \frac{k}{i}R_K = \frac{\Delta B_{ik}}{C_x\Delta V_{ik}}. \ $$

Now we can find out the relationship between electric voltage and magnetic field quantization:
 * $$\Delta V_{ik} = \frac{i}{k}\frac{\Delta B_{ik}}{C_xR_K}. \ $$

Thus, there are no any fractional electric charges in the quantum antidot. The number of tunnelling electron is due to the Landau level number ($$i \ $$). But the number of magnetic charge quantum is due to the electron in the composite resonator ($$k \ $$).

History
First, the unconventional integer QHE was discovered in graphene by Novoselov et. al. and Zhang et al.(2005). Theoretical interpretation was proposed by Sharapov et.al. The point is that conductance quantization has anomaly at the first Landau level, due to a twice smaller degeneracy. Furthermore, the first Landau level is filled by electron and hole quasiparticles simultaneously. Similar phenomena were presented in the classical case in silicon MOSFETs, which are working in the weak inversion mode first considered by Swanson (1972) and further modernized by Yakymakha (1981). The only difference between graphene and silicon devices is that MOSFETs are the unipolar devices (due to the p-n- junctions of source and drain).

The room-temperature quantum galvanomagnetic effects in the 2D-systems consisting of the two sort particles was predicted by Yakymakha (1989)< /ref name =Yakym1>

First experimental confirmation of the room-temperature QHE were obtained in graphene by Novoselov et.al. (2007) . This is due to the highly unusual nature of charge carriers in graphene, which behave as massless relativistic particles (Dirac fermions) and move with little scattering under ambient conditions.

Graphene QHE composite resonator
The standard characteristic impedance of the electron quantum Hall resonator in graphen is:
 * $$ Z_{Rn} = \sqrt{\frac{L_{Rn}}{C_{Rn}}} = \frac{R_K}{g_n}, \ $$

where $$g_n $$ is the electron degeneration number due to the band structure of graphene. The standard characteristic impedance of the hole quantum Hall resonator in graphen is:
 * $$ Z_{Rp} = \sqrt{\frac{L_{Rp}}{C_{Rp}}} = \frac{R_K}{g_p}, \ $$

where $$g_p = g_n = g = 4 $$ is the hole degeneration number due to the band structure of graphene.

At the first Landau level both, electrons and holes are filling the same 2D surface (something like chessboard).

So, the elementary composite resonator here will be formed by one electron and one hole. The total inductance of this resonator will be consisted of the electron and hole inductances connected in series:
 * $$L_{RT} = L_{Rn} + L_{Rp} = 2L_R. \ $$

By analogy, the total capacitance of this resonator will be consisted of the electron and hole capacitances connected in series:
 * $$C_{RT}^{-1} = C_{Rn}^{-1} + C_{Rp}^{-1} = 2C_R^{-1}. \ $$

Thus, the total characteristic impedance of the composite resonator will be as follows:
 * $$ Z_{RT} = \sqrt{\frac{L_{RT}}{C_{RT}}} = \sqrt{\frac{4L_{R}}{C_{R}}} = 2\frac{R_K}{g}, \ $$

where $$g = 4 $$.

The second Landau level is filled by monopolar particles (electrons or holes). Therefore, the total characteristic impedance of the second Landau level will be consisted of the first composite resonator and second standard resonator connected in parallel:
 * $$ Z_{RT2}^{-1} = Z_{RT}^{-1} + Z_{R}^{-1} = \sigma_{Hg}(1 + 1/2), \ $$

where $$\sigma_{Hg} = g/R_K $$ is the quantum conductance for graphene.

For the third Landau level and higher we would have the following total characteristic impedance:
 * $$ Z_{RTi}^{-1} = \sigma_{Hg}(i - 1/2), \quad \quad i = 1,2,3,.. \ $$

where $$i $$ is the Landau level number.

Reference Books

 * Stratton J.A.(1941). Electromagnetic Theory. New York, London: McGraw-Hill.p.615. djvu
 * Детлаф А.А., Яворский Б.М., Милковская Л.Б.(1977). Курс физики. Том 2. Электричество и магнетизм (4-е издание). М.: Высшая школа, "Reference Book on Electricity"  djvu
 * Гольдштейн Л.Д., Зернов Н.В. (1971). Электромагнитные поля. 2- издание. Москва: Советское Радио. 664с. "Electromagnetic Fields" djvu
 * Zyun F. Ezawa: Quantum Hall Effects - Field Theoretical Approach and Related Topics. World Scientific, Singapore 2008, ISBN 978-981-270-032-2
 * Sankar D. Sarma, Aron Pinczuk: Perspectives in Quantum Hall Effects. Wiley-VCH, Weinheim 2004, ISBN 978-0-471-11216-7
 * Composite Fermions, Edited by O. Heinonen, World Scientific, Singapore (1998).