Current-Induced Hall Effect

Jacob Szeftel; Jean-Claude Lévy

doi:10.20944/preprints202403.1131.v1

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Current-Induced Hall Effect

This version is not peer-reviewed.

Jacob Szeftel^*,Jean-Claude Lévy

This version is not peer-reviewed.

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Submitted:

16 March 2024

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19 March 2024

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Abstract

The properties of the current-induced Hall effect are analysed, for which the static magnetic field, supplied by an external source in the traditional experiment, is created by the current itself. The special experimental setup, needed for its observation, is described. It is shown how, combined with the skin effect, it could give access to the concentration of conduction electrons in superconductors. Besides this experiment might permit to dodge shortcomings, ensueing from the Meissner effect and limiting severely the sensitivity of the conventional Hall voltage measurement.

Keywords:

Hall effect

;

Meissner effect

;

skin effect

Subject:

Physical Sciences - Condensed Matter Physics

This preprints belongs to the Topic

Condensed Matter Physics and Catalysis

1. Introduction

When a superconducting material of type I is cooled in a static magnetic field H, starting from its normal state, H is screened[1] inside bulk material by eddy currents, while crossing the critical temperature

T_{c}

, at which superconductivity sets in. Moreover Newton’s law requires a net force accelerating the conduction electrons for the current to grow from 0 up to its final value. However, this manifestation of the Meissner effect has kindled a lasting controversy [2,3] over the very nature of this force, because H remains unaltered at

T_{c}

. In particular, quantum[4] and classical[5,6,7] origins have been called forward. Nevertheless, this debate has remained so far inconclusive, because the original experimental setup[8] does not enable one to sort out the relevant interpretation. Fortunately, a new kind of experiment has been proposed recently[4], a slightly modified version of which might offer such a possibility by taking advantage of the current induced Hall effect[7], as shown in this article.

The outline is as follows : the experimental setup, used to measure the Hall voltage, is described in section II, whereas the quantum and classical explanations are discussed in sections III and IV, respectively; an experiment, based on the skin effect, enabling one to distinguish between both views, is discussed in section V.

2. Experimental Setup

As seen in Figure 1, the superconducting sample consists in a hollow cylinder, characterized by its symmetry axis z and inner and outer radii

r_{i}, r_{o}

. It is located in a cylindrical frame with coordinates

r, θ, z

(

\Rightarrow r_{i} < r < r_{o}

). Its length

2 b

(

\Rightarrow - b < z < b

) is taken

> > r_{o}

, in order to get rid of any end effect. The material contains conduction electrons of charge e, effective mass m and concentration

c_{0}

. The current

I_{z}

, flowing along the z axis, will be kept constant. Besides, the corresponding current density

j_{z 0}

is homogeneous in the normal state (

T > T_{c}

). It then reads

j_{z 0} = \frac{I_{z}}{π (r_{o}^{2} - r_{i}^{2})} .

(1)

There is also[9]

j_{z 0} = c_{0} e v_{z 0}

with

v_{z 0}

standing for the electron mass center velocity. Thanks to the Ampère-Maxwell law, it gives rise, to an azimuthal magnetic field

H_{θ} (r)

, which in turn exerts a radial Lorentz force

F_{r} (r)

on each electron

\begin{matrix} H_{θ} (r) = \frac{r^{2} - r_{i}^{2}}{2 r} j_{z 0} \Rightarrow \\ F_{r} = - μ_{0} e v_{z 0} H_{θ} (r) = - \frac{μ_{0}}{2 c_{0}} \frac{r^{2} - r_{i}^{2}}{r} j_{z 0}^{2} \end{matrix},

(2)

with

μ_{0}

being the permeability of vacuum. Then

F_{r} (r)

displaces the conduction electrons, which gives rise to a space charge density

Q (r)

and thence to a radial electric field

E_{r 0} (r)

, such that

Q (r) = div (ϵ_{0} E_{r 0} (r))

(

ϵ_{0}

stands for the electrical permittivity of vacuum). Finally, in the steady regime,

F_{r} (r)

is offset by

e E_{r 0} (r)

, which results into the Hall voltage

U_{0}

, measured between points

A, B

in Figure 1. Thus

E_{r 0}, U_{0}

read in the normal state

\begin{matrix} E_{r 0} = - \frac{F_{r}}{e} \Rightarrow U_{0} = \int_{r_{i}}^{r_{o}} E_{r 0} (r) d r \\ = \frac{μ_{0}}{2 π^{2} e c_{0} (r_{o}^{2} - r_{i}^{2})} (\frac{1}{2} - \frac{r_{i}^{2}}{r_{o}^{2} - r_{i}^{2}} log \frac{r_{o}}{r_{i}}) I_{z}^{2} \end{matrix} .

(3)

Hence measuring the Hall voltage

U_{0}

gives access to the electron concentration

c_{0}

. At last, several remarks are in order

due to cylindrical symmetry, $U_{0}$ is independent from $θ$ and in addition proportional to $I_{z}^{2}$ , unlike the usual Hall voltage proportional to $I_{z}$ , because the field $H_{θ}$ itself is proportional to the current $I_{z}$ ;
since the sign of $U_{0}$ cannot be predicted[9] in general, the experiment sketched above should be done first in the normal state, i.e. at $T > T_{c}$ , for calibration purposes, because $c_{0}$ is well known;
due to different sample shape, i.e. parallelepiped versus cylinder, the superficial charge, giving rise to the electric field, lies at the outer edges of the sample in the traditional measurement, whereas it is a space charge, distributed over the whole bulk matter, in the experiment discussed here.

Figure 1. Cross-section of the sample; the current

I_{z}

and the electric field

E_{r i}

(

i = 0, 1, 2

), aroused by the Hall effect, are respectively parallel to the

z, r

axes, whereas the magnetic field

H_{θ}

, induced by

I_{z}

, is along the azimuthal axis, that is normal to the

z, r

axes;

i = 0

refers to the case of the normal metal, whereas

i = 1, 2

correspond respectively to the quantum and classical analyses for a superconductor; the matter (dotted area), making up the cylinder of radius

r_{i}

, delineated by the dashed lines, is to be carved out to measure the Hall voltage

U_{i = 0, 1, 2} (z)

between

A, B

Figure 1. Cross-section of the sample; the current

I_{z}

and the electric field

E_{r i}

(

i = 0, 1, 2

), aroused by the Hall effect, are respectively parallel to the

z, r

axes, whereas the magnetic field

H_{θ}

, induced by

I_{z}

, is along the azimuthal axis, that is normal to the

z, r

axes;

i = 0

refers to the case of the normal metal, whereas

i = 1, 2

correspond respectively to the quantum and classical analyses for a superconductor; the matter (dotted area), making up the cylinder of radius

r_{i}

, delineated by the dashed lines, is to be carved out to measure the Hall voltage

U_{i = 0, 1, 2} (z)

between

A, B

3. Quantum Approach

Let the sample, depicted in Figure 1, be cooled down with

I_{z}

kept constant for time

t > 0

. Actually, it does not become superconducting until

t = t_{f}

, at which

T (t_{f}) < T_{c}

is low[10] enough, so as to fulfil

j_{z 0} < j_{c} (T (t_{f}))

with

j_{c} (T)

referring to the critical current density. Note that, due to irreversible Joule dissipation resulting from the finite ac resistivity in the superconducting state[7,11], there is no one to one correspondence between

H_{θ}

and

j_{c}

, so that, contrary to an ubiquitous misconception[1,15], the phrase “critical magnetic field” is meaningless. Conversely, the critical current is well-defined, because the current density together with temperatureT do characterise[10] the thermodynamical state of every superconductor, as recalled below in section V.

It is then argued[4] that, once the sample becomes superconducting, the Lorentz force, combined with some undefined quantum force[4], push the electrons outward along the radial axis, which causes the current density

j_{z 1}

to become r dependent but to remain independent from z for

|z| < < b

in the steady regime, defined by

T (t > t_{f}) = T (t_{f})

. Then

j_{z 1}

reads[4]

j_{z 1} (r) = j_{z 1} (r_{o}) e^{\frac{r - r_{o}}{λ_{L}}}, λ_{L} = \sqrt{\frac{m}{μ_{0} c_{1} e^{2}}},

(4)

for which

λ_{L} < < r_{i}, c_{1}

refer to London’s length[1,2] and the concentration of superconducting electrons, respectively. In addition,

j_{z 1} (r_{o})

can be assigned to its value by requiring

I_{z} = 2 π \int_{r_{i}}^{r_{o}} j_{z 1} (r) r d r \Rightarrow j_{z 1} (r_{o}) = \frac{I_{z}}{2 π λ_{L} r_{o}} .

(5)

By proceeding furthermore as in the previous section, the magnetic field

H_{θ}

, the Lorentz force

F_{r}

, the electric field

E_{r 1}

and the resulting Hall voltage

U_{1}

are worked out to read

\begin{matrix} H_{θ} (r) = \int_{r_{i}}^{r} j_{z 1} (u) \frac{u}{r} d u = \frac{e^{\frac{r - r_{o}}{λ_{L}}}}{2 π r_{o}} (1 - \frac{r_{i}}{r} e^{\frac{r_{i} - r}{λ_{L}}}) I_{z} \\ \Rightarrow F_{r} (r) = - \frac{μ_{0}}{c_{1}} H_{θ} (r) j_{z 1} (r) \Rightarrow E_{r 1} (r) = - \frac{F_{r} (r)}{e} \\ \Rightarrow U_{1} = \int_{r_{i}}^{r_{o}} E_{r 1} (r) d r = \frac{μ_{0}}{8 e c_{1} {(π r_{o})}^{2}} I_{z}^{2} \end{matrix} .

(6)

Hence

c_{1}

can be known by measuring

U_{1}

owing to Eq.(6).

However several issues appear questionable in this analysis[4]:

it relies upon London’s equation[2]

$B_{z} (r) + μ_{0} λ_{L}^{2} curl (j_{θ} (r)) = 0,$

(7)

with $B_{z}, j_{θ}$ being the magnetic induction, parallel to the z axis, and the persistent current density, parallel to the azimuthal axis. Actually, the validity of Eq.(7) has been ascertained[7], but for infinite electrical conductivity, whereas the ac conductivity has been proved[11] to be finite on the basis of low-frequency susceptibility data[12,13], which thereby invalidates Eq.(7). As another consequence of finite ac conductivity, the thermodynamical state, characterising the superconducting sample in the steady regime $t > t_{f}$ , depends[14] upon $T (t \in [0, t_{f}])$ . At last, though Eq.(7) has been worked out[2,7] for a magnetic fieldparallel to the z axis, and azimuthal current, it is applied[4], with no justification, to a physical case, characterised by an azimuthal field and current flowing along the z axis. Accordingly, since Eq.(7) ensues[7] from solving Maxwell’s equations for a transient regime ( $\Rightarrow \frac{\partial B_{z}}{\partial t} (t \in [0, t_{f}]) \neq 0, \frac{\partial j_{θ}}{\partial t} (t \in [0, t_{f}]) \neq 0$ ), the claim, that it applies as well to a steady regime[4] ( $\Rightarrow \frac{\partial B_{θ}}{\partial t} (t > t_{f}) = \frac{\partial j_{z}}{\partial t} (t > t_{f}) = 0$ ) with undefined $B_{θ} (t < t_{f}), j_{z} (t < t_{f})$ , is groundless;
the origin of the quantum force is self-admittedly[4] unknown, as is the expression of the potential, which it is supposed to derive from, so that this very phrase is actually misleading. In addition, the matching conditions[4] at $z = \pm b$ require a persistent radial current $j_{r}$ to flow in the steady regime $t > t_{f}$ . Then the azimuthal magnetic field $H_{θ}$ , acting on $j_{r}$ , will give rise to a Lorentz force, parallel to the z axis, which will cause eventually $I_{z}$ to grow till the critical value is reached and the sample goes thereby normal, which contradicts the assumption[4] of a stable, steady regime;
a key assumption[4] says that the current should be carried by holes in the normal metal. This entails that the Fermi energy $E_{F}$ ought to be close to the Van Hove singularity[9], located at the upper edge of the one-electron band $ϵ_{u}$ . Consequently, there is $ρ (E_{F}) \propto \sqrt{ϵ_{u} - E_{F}}$ for a three-dimensional sample[9], with $ρ (ϵ), ϵ$ standing for the one-electron density of states and energy, respectively. Then it implies finally $ρ^{'} (E_{F}) = \frac{d ρ}{d ϵ} (E_{F}) < 0$ . However such a condition will be proved below to be at loggerheads with a thermodynamical rationale, showing the opposite, namely a stable superconducting phase requires $ρ^{'} (E_{F}) > 0$ .

As shown elsewhere[16], there is

$\begin{matrix} \frac{d E_{F}}{d T} (T) = - \frac{{(π k_{B})}^{2}}{3} \frac{ρ^{'} (E_{F})}{ρ (E_{F})} T \\ \Rightarrow ρ^{'} (E_{F}) \frac{d E_{F}}{d T} (T) < 0 \end{matrix},$

wherein $k_{B}$ is Boltzmann’s constant and $\frac{d E_{F}}{d T}$ has been reckoned at fixed normal electron concentration $c_{n}$ . Hence $E_{F} (T, c_{n})$ at fixed T looks as plotted in Figure 2a, Figure 2b, for $ρ^{'} (E_{F}) > 0$ and $ρ^{'} (E_{F}) < 0$ , respectively. The infinite slope $\frac{\partial E_{F}}{\partial c_{n}} (c_{n} \to 0) \to \infty$ is typical of a 3 dimensional Van Hove singularity[9], associated with the bottom of the conduction band. Note that Figure 2 displays $\frac{\partial μ}{\partial c_{1}} < 0$ ( $μ (c_{1})$ refers to the chemical potential of superconducting electrons[10,17]), which has been shown to be a prerequisite for persistent currents[17], thermal equilibrium[10], occurrence of superconductivity[16,18] and the Josephson effect[19]. The two-fluid system, comprising normal and superconducting electrons in respective concentration $c_{n}, c_{1}$ (charge conservation requires $c_{n} + c_{1} = c_{0}$ ), is at thermal equilibrium at $T \leq T_{c}$ , provided the following equation is fulfilled[16,17,18]

$E_{F} (T, c_{n}) = μ (c_{1}) .$

(8)

Eq.(8) means that the free energy of the whole electron system is stationary at fixed T. However, as shown elsewhere[17], Figure 2a and Figure 2b depict, respectively, the case of stable ( $ρ^{'} (E_{F}) > 0 \Rightarrow \frac{\partial E_{F}}{\partial c_{n}} + \frac{\partial μ}{\partial c_{1}} > 0$ ) and unstable ( $ρ^{'} (E_{F}) < 0 \Rightarrow \frac{\partial E_{F}}{\partial c_{n}} + \frac{\partial μ}{\partial c_{1}} < 0$ ) equilibrium. This statement of fact completes the proof that the hole-driven superconductivity[4] cannot be observed because it implies $ρ^{'} (E_{F}) < 0$ . Likewise, for the reader’s convenience, we recall[16,18] the necessary conditions for a second order transition to occur at $T_{c}$

$\begin{matrix} E_{F} (T_{c}, c_{0}) = μ (0), ρ^{'} (E_{F} (T_{c}, c_{0})) > 0 \\ \frac{\partial E_{F}}{\partial c_{n}} (T_{c}, c_{0}) = - \frac{\partial μ}{\partial c_{1}} (0) \end{matrix} .$

4. Classical Approach

The magnetic susceptibility not being continuous at

T_{c}

has been shown[7] to be responsible for the Meissner effect taking place in a superconductor, cooled in a static magnetic field. Since no paramagnetic contribution has ever been observed in the superconducting state[1,9], the latter is deemed to be in a macroscopic singlet spin state, so that its susceptibility

χ_{s}

is determined entirely by Lenz’s law[7]

\Rightarrow χ_{s} < 0

. Meanwhile the magnetic susceptibility of normal electrons

χ_{n}

comprises[9] two components, respectively paramagnetic (Pauli) and diamagnetic (Landau), and is in general

> 0

. Thus the local, magnetic inductions

B_{s} (r), B_{n} (r)

, both being parallel to the azimuthal axis, read, respectively, in the superconducting and normal states

B_{s} (r) = μ_{0} (1 + χ_{s}) H_{θ} (r), B_{n} (r) = μ_{0} (1 + χ_{n}) H_{θ} (r) .

Due to

χ_{s} \neq χ_{n}

, the magnetic induction undergoes a finite step at

T_{c}

\frac{δ B (r)}{δ t} = \frac{B_{s} (r) - B_{n} (r)}{δ t} = μ_{0} \frac{χ_{s} - χ_{n}}{δ t} H_{θ} (r),

where

δ t

refers to the time needed in the experimental procedure for T to cross

T_{c}

. Owing to the Faraday-Maxwell equation,

δ B / δ t

may induce two transient, electric fields

E_{r} (z), E_{z} (r)

, respectively parallel to the

r, z

axes, such that

\frac{\partial E_{r}}{\partial z} - \frac{\partial E_{z}}{\partial r} = - \frac{δ B}{δ t} .

Let us begin with showing

E_{z} (r) = 0

. The proof is by contradiction. As a matter of fact,

E_{z} (r) \neq 0

implies that the voltage drop

δ U = U (z = b) - U (z = - b) = - \int_{- b}^{b} E_{z} (r) d z

is r-dependent too. But this is inconsistent with

δ U

being r-independent at

z = \pm b

, i.e. at the interface with the leads, straddling the superconducting sample, because they are made out of normal metals, which entails

U (z)

being r-independent for any z. Q.E.D.

Meanwhile,

E_{r} (z) = - \frac{δ B}{δ t} z

will give rise to a transient z-dependent voltage

δ U (z) = E_{r} (z) (r_{o} - r_{i}) .

Observing

δ U (z)

could buttress the validity of this analysis.

Hence, due to the current density remaining r-independent in the steady regime

t > t_{f}

, the calculation of

E_{r 2}, U_{2}

proceeds as that of

E_{r 0}, U_{0}

in Eq.(3), except for the concentration of superconducting electrons

c_{2}

showing up instead of

c_{0}

\begin{matrix} U_{2} = \int_{r_{i}}^{r_{o}} E_{r 2} (r) d r = \\ \frac{μ_{0}}{2 π^{2} e c_{2} (r_{o}^{2} - r_{i}^{2})} (\frac{1}{2} - \frac{r_{i}^{2}}{r_{o}^{2} - r_{i}^{2}} log \frac{r_{o}}{r_{i}}) I_{z}^{2} \end{matrix} .

(9)

Comparing Eq.(6) with Eq.(9) leads to

c_{1} \neq c_{2}

. Therefore an independent determination of the concentration of superconducting electrons is needed to assess the respective validity of the quantum and classical analyses. This is the purview of the next section.

5. Skin Effect

An electromagnetic field of frequency

ω

, shone on a metal, remains confined[20] within a thin layer of thickness

δ (ω)

, called the skin depth, and located at the outer edge of the conductor, for

ω < ω_{p}

, where

ω_{p} \approx 10^{16}

Hz stands for the plasma frequency. This stems from the real part of the dielectric constant being negative for

ω < ω_{p}

. Drude’s conductivity

σ

and

δ

have been shown to read[7,21]

\begin{matrix} σ = \frac{c_{s} e^{2} τ}{m} \\ δ (ω < < ω_{c}) = \frac{λ_{L}}{\sqrt{2 ω τ}}, δ (ω > > ω_{c}) = \frac{λ_{L}}{\sqrt{2}} \end{matrix} .

c_{s}, τ

are the concentration of superconducting electrons and their associated scattering time[9,20], respectively, and

ω_{c} = \frac{1}{τ}

. Note that the normal electrons play no role, because their conductivity is

< < σ

. Then, thanks to

λ_{L} = \sqrt{\frac{m}{μ_{0} c_{s} e^{2}}}

, measuring

δ (ω < < ω_{c}), δ (ω > > ω_{c})

would give access to

c_{s}, τ

, provided m is known. Nevertheless several remarks are in order, concerning the measurement :

there being a one to one correspondence for conductivity $σ \leftrightarrow c_{s}$ and current density[10] $j \leftrightarrow c_{s}$ , illustrated by

$\begin{matrix} σ (c_{s}) = \frac{c_{s} e^{2} τ}{m} \\ j_{s} (c_{s}) = c_{s} e \sqrt{\frac{2}{m} (E_{F} (T, c_{0} - c_{s}) - μ (c_{s}))} \end{matrix},$

entails that $σ$ depends on j, so that measurements of the skin depth and the Hall voltage should be performed with the same current $I_{z}$ . Besides, as for the quantum case[4], the r-dependent $j_{z 1} (r)$ leads to a r-dependent concentration $c_{1} (r)$ , which complicates further the interpretation of the Hall effect experiment;
in pure materials, $ω_{c}$ is likely to lie in the microwave range ( $ω_{c} < 10^{12}$ Hz), while it might rather belong in the IR one ( $ω_{c} > 10^{12}$ Hz) for poor conductors such as high- $T_{c}$ compounds.

6. Conclusion

The properties of the current-induced Hall effect have been presented. It has been argued that it might shed light on the very nature of the force at work in the Meissner effect, i.e. either quantum or classical. The quantum explanation fails seemingly to account for the force giving rise to the eddy currents at

T_{c}

, whereas the electrons, making up the eddy currents, are driven merely by Faraday’s electric field, induced by the magnetic susceptibility varying at

T_{c}

within the classical interpretation.

Experiments, combining the current-induced Hall effect and the skin effect, have been discussed, which might enable one to assess the respective validity of each view. At last the skin depth measurement might help to chart the so far unknown

c_{s} (j)

, which is of great significance for the superconducting to normal transition[10].

Since the Hall voltage grows like the inverse of the electron concentration (see Eqs.(3,6,9)), it is in general much weaker in metals than in semiconductors. This hurdle is even more daunting in type I superconductors [22], because the Meissner effect prevents the magnetic field from entering the sample. Thus the corresponding Hall voltage, equal to the circulation of the Hall electric field along the tiny London length, is likely to lie under the detection threshold. Conversely the experiment, discussed in section IV, does not suffer from this drawback, because the current induced magnetic field penetrates indeed into the whole bulk matter. Consequently, it could be performed even in high-T_c compounds at relatively high currents densities, yet giving rise to a magnetic field <H_c1 in the whole bulk matter (to avoid vortices building up) and thereby ensuring good sensitivity.

References

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Figure 2. schematic plots of

E_{F} (T_{c}, c_{n})

E_{F} (0, c_{n})

and

μ (c_{1})

as solid, dotted and dashed lines, respectively; cases

a, b

correspond to

ρ^{'} (E_{F}) > 0

and

ρ^{'} (E_{F}) < 0

, respectively;

\frac{\partial μ}{\partial c_{s}}

has been taken to be constant for simplicity; the origin

E_{F} = μ = 0

is set at the bottom of the conduction band; the crossing points

A, B

E_{F} (T_{c}, c_{n}), E_{F} (0, c_{n})

with

μ (c_{1})

in Figure 2a,b exemplify stable and unstable solutions of Eq.8, respectively; the tiny differences

E_{F} (T_{c}, c_{n}) - μ (c_{0} - c_{n})

and

E_{F} (0, c_{n}) - μ (c_{0} - c_{n})

have been hugely magnified for the reader’s convenience

Figure 2. schematic plots of

E_{F} (T_{c}, c_{n})

E_{F} (0, c_{n})

and

μ (c_{1})

as solid, dotted and dashed lines, respectively; cases

a, b

correspond to

ρ^{'} (E_{F}) > 0

and

ρ^{'} (E_{F}) < 0

, respectively;

\frac{\partial μ}{\partial c_{s}}

has been taken to be constant for simplicity; the origin

E_{F} = μ = 0

is set at the bottom of the conduction band; the crossing points

A, B

E_{F} (T_{c}, c_{n}), E_{F} (0, c_{n})

with

μ (c_{1})

in Figure 2a,b exemplify stable and unstable solutions of Eq.8, respectively; the tiny differences

E_{F} (T_{c}, c_{n}) - μ (c_{0} - c_{n})

and

E_{F} (0, c_{n}) - μ (c_{0} - c_{n})

have been hugely magnified for the reader’s convenience

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Current-Induced Hall Effect

Abstract

Keywords:

Subject:

1. Introduction

2. Experimental Setup

3. Quantum Approach

4. Classical Approach

5. Skin Effect

6. Conclusion

References

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