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Revisiting Meissner effect
arXiv:1407.1231v2 [cond-mat.supr-con] 13 Jan 2015
Jacob Szeftel1 and Nicolas Sandeau2
1
2
ENS Cachan, LPQM, 61 avenue du Président Wilson, 94230 Cachan, France
Aix Marseille Université, CNRS, Centrale Marseille, Institut Fresnel UMR 7249, 13397, Marseille, France
PACS
74.25.Ha – First pacs description
Abstract –Using a classical treatment based on Newton and Maxwell’s equations, Meissner effect
is shown to stem from persistent eddy currents. This work makes no use of London’s equation.
The penetration depth of the magnetic field is found to differ markedly from London’s length and
to depend on experimental conditions. The properties of a superconductor, cooled in a magnetic
field, are accounted for within the same framework. Hall effect is predicted and calculated. Two
experiments, allowing to check the validity of this analysis and to measure London’s length, are
presented.
introduction. – Superconductivity is characterized
by two prominent properties [1, 2]: persistent currents in
vanishing electric field and Meissner effect [3], conveying
the rapid decay of an applied magnetic field within bulk
matter, provided the field is lower than the critical field
Hc or Hc1 in a superconductor of type I or II, respectively.
Whereas a compelling explanation for the existence of persistent currents is still lacking [1, 2] more than a century
after their discovery, some insight was achieved into Meissner effect thanks to London’s conjecture [4]
B + µ0 λ2L curlj = 0 ,
at T = 0. Actually all experimental data indicate [6–12]
indeed λM > λL . Besides as all attempts [2, 6, 7, 13] at
accounting for this discrepancy have been of phenomenological nature so far, the validity of Eq.(1) could never be
ascertained, which has resulted into a muddled state of
affairs :
• whereas some authors [5, 14, 15] attempted to justify
Eq.(1) by a classical treatment, another school
of mind claimed conversely that Meissner effect
stemmed from some unknown quantum effect [16],
possibly related with the BCS theory [17] and Cooper
pairs [18];
(1)
where µ0 , j, λL stand for the magnetic permeability of vacuum, the persistent current, induced by the magnetic induction B and London’s length, respectively. Eq.(1), combined with Newton and Maxwell’s equations, entails [1,2,4]
that the penetration depth of the magnetic field λM is
equal to λL , inferred [1, 2, 4, 5] to read as
r
m
,
λL =
µ0 ρe2
where e, m, ρ stand for the charge, effective mass
and concentration of superconducting electrons. Because
ρ is supposed to decrease from its maximum value at
T = 0, down to ρ = 0 at the critical temperature Tc , its
upper bound is equal to the nominal concentration of
conduction electrons in the normal state. A numerical
application yields λL ≈ 200Å in Al, assuming that all
conduction electrons are superconducting and m is equal
to the free electron mass. However this estimate turns
out to be much smaller than λM = 1.6µm, measured [2,5]
• whenever a superconducting material is cooled in a
magnetic field H, starting from its normal state, the
latter is expelled [3] from inside the material, while
crossing the critical temperature Tc (H), at which superconductivity sets in. This further manifestation
of Meissner effect has kindled a murky and inconclusive debate over the alleged distinction between a real
world superconductor and a fictitious perfect conductor [1, 2, 4, 19].
This work [20] is aimed at dispelling the current confusion, by working out a consistent theory of Meissner effect,
resorting solely to classical tools but refraining from using Eq.1. It will be shown that λM 6= λL . Besides Hall
effect is predicted to take place. The outline is as follows: Section I and II deal with Meissner effect; the case
of the field cooled superconductor is addressed in Section
p-1
Jacob Szeftel1 Nicolas Sandeau2
law in superconductors undergoing an oscillating electric
field ∝ eiωt . However because the physical meaning of τ
is quite different from its counterpart in a normal metal
rc
[1], i.e. the average time between two successive scattering
events, undergone by a conduction electron, it is in order
to work out Ohm’s law for a superconductor.
The superconducting state, carrying no current, is asr0
sumed to comprise two subsets of equal concentration ρ/2,
r1
moving in opposite directions with respective mass cenB
A
ter velocity v, −v, which ensures jθ = p = 0, where p
refers to the average electron momentum. The driving
field Eθ causes δρ/2 of electrons to be transferred from
r
one subset to the other, so as to give rise to a finite current jθ = δρev = eδp/m, where δp stands for the electron
Fig. 1: Cross-section of the superconducting sample (dotted) momentum variation. The friction force is responsible for
and the coil (hatched); Eθ , jθ are both normal to the r, z plane; the reverse mechanism, whereby electrons are transferred
vertical arrows illustrate the r dependence of Bz (r); rc has from the majority subset of concentration ρ+δρ back to the
2
been magnified for the reader’s convenience; Eq.(18) has been
). It ensues from flux quantization and
minority one ( ρ−δρ
2
integrated from A (Bz (r0 + 2rc ) = 0) to B; the matter between
Josephson’s effect [1, 2, 23] that the elementary transfer
the dashed-dotted lines should be carved out to carry out the
process involves a pair rather than a single electron. Hence
Hall effect experiment
if τ −1 is defined as the transfer probability per time unit of
one electron pair, the net electron transfer rate is equal to
III; an experiment, enabling one to assess the validity of ρ+δρ−(ρ−δρ)
= δρ
2τ
τ . By virtue of Newton’s law, the resultthis analysis, is discussed in Section IV while Hall effect is ing friction force is equal to mvδρ/τ = δp/τ ∝ jθ /τ , which
θ
analysed in Section V and another experiment yielding λL validates Eq.(2). Besides Eq.(2) applied with ∂j
∂t = 0 alis described in Section VI. Finally the new achievements lows to retrieve Ohm’s law as
of this work are summarized in the conclusion.
ρe2 τ
A sample of cylindrical shape, characterized by its axis
.
jθ = σEθ , σ =
m
z and radius r0 , is taken to contain superconducting electrons of charge e, effective mass m and concentration ρ, Although the conductivity σ has the same expression as in
and to undergo a time t dependent electric field Eθ (t, r), the normal state [1], noteworthy is that σ has been meaexpressed in cylindrical coordinates r, θ, z. Eθ is alined sured [8–11] to be ≈ 300 times larger than in the normal
along the ortho-radial axis so that divEθ (t, r) = 0 (see state.
Fig.1). Furthermore it is assumed that Eθ 6= 0, only if
Eθ induces a magnetic induction Bz (r, t), taken to be
t ∈]0, t0 [, which defines a transient regime (0 < t < t0 ) alined along the z axis. Bz is given by the first Maxwell
and a permanent one (t > t0 ).
equation as
I-transient regime. – Eθ induces a current jθ (t, r ∈
∂Eθ
Eθ
∂Bz
.
(3)
= −curlEθ = −
+
[0, r0 ]) along the ortho-radial axis, as given by Newton’s
∂t
r
∂r
law
djθ
ρe2
jθ
The displacement vector D, alined along the ortho-radial
=
Eθ −
,
(2)
axis, is defined in a metal as
dt
m
τ
z
2
jθ
where ρe
m Eθ , − τ are respectively proportional to the
driving force accelerating the conduction electrons and the
friction force due to the lattice, as inferred from Ohm’s
law. The equation of motion, used by London [4] to
surmise Eq.(1), differs from Eq.(2), insofar as it forgoes
the friction term ∝ jτθ . The first to question the validity of Eq.(1) was apparently Pippard [6, 7] who investigated the effect of impurities on electromagnetic energy
absorption at microwave frequencies in superconducting
Sn. Although his phenomenological interpretation relied
heavily on the anomalous skin effect [21, 22], observed in
normal metals, and did not consider at all the possibility
of non vanishing resistivity in superconductors carrying
time-varying currents, there is by now a large trove of
experimental data [8–11] attesting the validity of Ohm’s
D = ρeuθ + ǫ0 Eθ
,
where ǫ0 , uθ refer to the electric permittivity of vacuum
and displacement coordinate of the conduction electron
center of mass, parallel to the ortho-radial axis. Because
divEθ = 0 entails that divDθ = 0, Poisson’s law warrants
the lack of charge fluctuation around ρe. Thence since
θ
there is by definition jθ = ρe du
dt , the displacement current
reads
∂Eθ
∂Dθ
= jθ + ǫ0
.
∂t
∂t
Finally the magnetic field Hz (t, r), parallel to the z axis,
is given by the second Maxwell equation as
p-2
curlHz = −
∂Eθ
∂Dθ
∂Hz
= jθ +
= 2jθ + ǫ0
∂r
∂t
∂t
.
(4)
Revisiting Meissner effect
1
Eθ (t, r) , jθ (t, r) , Bz (t, r) , Hz (t, r) can be recast as
Fourier series for t ∈]0, t0 [
X
f (t, r) =
f (n, r) einω0 t ,
(5)
0.1
B (u)
z
n∈Z
where ω0 t0 = 2π and f (t, r) , f (n, r) hold for Bz (t, r),
Hz (t, r), Eθ (t, r), jθ (t, r) and Bz (n, r), Hz (n, r),
Eθ (n, r), jθ (n, r), respectively. Replacing Eθ , jθ , Bz , Hz
in Eqs.(2,3,4) by their expression in Eqs.(5), while taking
into account
Bz (n, r) = µ (nω0 ) Hz (n, r)
u
e
0.01
0.001
0.0001
,
where µ (nω0 ) = µ0 (1 + χs (nω0 )) and χs (ω) is the magnetic susceptibility of superconducting electrons at frequency ω, yields for n 6= 0
1+inω0 τ
σ
jθ (n, r)
(n,r)
inω0 Bz (n, r) = − Eθ (n,r)
+ ∂Eθ∂r
r
∂Bz (n,r)
= −µ (nω0 ) (2jθ (n, r) + inω0 ǫ0 Eθ
∂r
Eθ (n, r) =
10
-5
10
-6
0
5
(6)
(n, r))
u=
r
10
15
λ
Fig. 2: Semi-logarithmic plots of Bz (u), eu .
Eliminating Eθ (n, r) from Eqs.(6) gives
∂ 2 Bz (n, r)
Bz (n, r) ∂Bz (n, r)
= 2
−
∂r2
δ (nω0 )
r∂r
λL
δ(ω) = r
2iωτ
−
(1 + χs (ω)) 1+iωτ
,
ω2
ωp2
.
ωp =
(7)
s
ρe2
ǫ0 m
refer to skin depth [6, 7] and plasma frequency [1, 24], respectively. As Eqs.(6) make up a system of 3 linear equations in terms of 3 unknowns jθ , Eθ , Bz , there is a single
solution, embodied by Eq.(7), which proves Eq.(1)to be
superfluous. It should be replaced by the following n ∈ Z
sequence, inferred from Eqs.(6)
where λn
Bz (n, r) + µ0 λ2n curljθ (n, r) = 0 ,
q
= λL 1 − nωi0 τ . Practical values (ω0 <
105 Hz, τ ≤ 10−11 s) ensure ω0 τ << 1 ⇒ |λn | >> λL .
The solution of Eq.(7), which has been integrated over
dBz
r ∈ [0, r0 ] with the initial condition
(r = 0) = 0, is
dr
a Bessel function, having the property Bz (r) ≈ er/δ(nω0 )
if r >> |δ(nω0 )|, as illustrated in Fig.2. Finally, Eqs.(6)
entail that Eθ (n = 0, r) = 0 ⇒ jθ (n = 0, r) = 0, which in
turn results into Hz (n = 0, r) = Hz (n = 0, r0 ), ∀r ∈ [0, r0 ].
II-permanent regime. – Because of Eθ (t > t0 , r) =
0, the friction force ∝ − jτθ is no longer at work in Eq.(2)
for t > t0 . This conveys the very property, which superconductivity owes its name to. Hence Eq.(2) is recast into
that both Eqs.(2,8) are most helpful in so far as they describe very accurately all experimental observations but
Eq.(8) claims by no means to be a physical explanation
for the existence of persistent currents, an issue which lies
anyhow far beyond the purview of this work.
Eqs.(5) thence yield
X
jθ (t > t0 , r) =
jθ (n, r) .
(9)
n∈Z
The second Maxwell equation reads now
−
Comparing
Hz (t0+ , r).
as
∂Hz
(t > t0 , r) = jθ (t0 , r)
∂r
.
(10)
Eqs.(4,10) reveals that Hz (t0− , r)
6=
The penetration depth λM is defined
∂LogHz (t0+ , r0 )
1
=
λM
∂r
.
At low
√ frequency such that ωτ << 1, there is |δ| ≈
λL / 2ωτ . Then given that λL < 10−7 m, the inequality
r0 >> |δ(nω0 )| holds for any n under typical experimental conditions r0 ≈ 1mm, ω0 < 105 Hz. Thus we take
r−r0
advantage of Eq.(9) and jθ (n, r → r0 ) ≈ jθ (n, r0 )e δ(nω0 )
to integrate Eq.(10) in order to obtain
P
1
n jθ (n, r0 )
,
(11)
≈ P
δ(nω
)j
(n,
r0 ) − Hz (n = 0, r0 )
λM
0
θ
n
where the sum is performed for n 6= 0 and |n|ω0 < ωp .
Thanks to Eq.(11) and the inequality |δ(nω0 )| >> λL ,
valid for n such that |n|ω0 τ << 1, |λM | is likely to be
so that the transient current jθ (t < t0 ) turns to persistent, much larger than λL , which is well documented in experthat is jθ (t > t0 , r) = jθ (t0 , r), ∀r. It must be emphasized imental data [6–12]. More importantly, λM depends on
∂jθ
=0
∂t
,
(8)
p-3
Jacob Szeftel1 Nicolas Sandeau2
experimental conditions via ω0 and the jθ (n, r0 )’s, so that the property that a superconducting state carries no enλM , unlike λL , is no intrinsic property of a superconduc- tropy [1,2] entails that F = EK . Equating this expression
K
of ∂E
tor.
∂M with that inferred from Eq.(12) yields finally
It results from the mainstream analysis [1, 2, 4, 5] based
2
λL
on Eqs.(1,10) that
.
χs (ω) = −
2|δ(ω)|
0
Hz (t0 , r0 ) r−r
e λL
.
jθ (t > t0 , r) =
As expected, χs is found diamagnetic (χs < 0) and
λL
|χs (ω)| << 1 for ω << 1/τ . The calculation of χs (0) proConsequently there should be a one to one correspondence ceeds along the same lines, except for the second Maxwell
between the applied magnetic field Hz (t0 , r0 ) and the per- equation reading ∂Hz = −jθ and λM showing up instead
∂r
sistent current distribution jθ (t > t0 , r ≤ r0 ). However of δ(ω), whence
this statement appears to be of questionable validity for
2
two reasons
λL
χs (0) = −
.
|λM |
• Hz (t0 , r0 ) is not well-defined because the second
Maxwell equation for t < t0 , as expressed in Eq.(4),
Note that our definition of χs = µ0M(r)
Hz (r) , where
differs from its expression for t > t0 , given in
at r,
Eq.(10), which causes the discontinuity Hz (t0− , r) 6= Hz (r), M (r) refer to local field and magnetization
M
differs
from
the
usual
[1,
2,
4,
5]
one
χ
=
s
µ0 Hz (r0 ) with
Hz (t0+ , r), ∀r ∈ [0, r0 ];
Hz (r0 ), M being external field and total magnetization.
While the sample is in its normal state at T > Tc , the
• the persistent current distribution jθ (t > t0 , r ∈
[0, r0 ]) has been shown to depend actually not only applied magnetic field Hz penetrates fully into bulk matter
on Hz (t0+ , r0 ) but rather on the whole transient be- and induces a magnetic induction
havior Hz (t ∈ [0, t0 [, r0 ).
Bn = µ0 (1 + χn ) Hz ,
(13)
III-field cooled sample. – The expression of χs is
needed for Eqs.(6) to be self-contained and because the where χn designates the magnetic susceptibility of conducsusceptibility not being continuous at Tc will turn out to tion electrons. It comprises [1] the sum of a paramagnetic
be solely responsible for Meissner effect to occur in a su- (Pauli) component and a diamagnetic (Landau) one and
perconductor, cooled inside a magnetic field. Since no χn > 0 in general. Moreover the magnetic induction reads
paramagnetic contribution has ever been observed in the for T < Tc (Hz )
superconducting state [1,2], it has been concluded that the
Bs = µ0 (1 + χs (0)) Hz ,
(14)
latter is always in a macroscopic singlet spin state. Consequently the only contribution to χs is of macroscopic with χ (0) < 0. Because of χ (0) 6= χ , the magnetic
s
s
n
origin and can thence be calculated using Maxwell’s equa- induction undergoes a finite step while crossing T (H )
c
z
tions. We begin with writing down the t-averaged density
of kinetic energy
δB
χs (0) − χn
Bs − Bn
=
= µ0
Hz ,
(15)
δt
δt
δt
2
m jθ (r)
,
EK (r) =
where δt refers to the time needed in the experimental
2ρ
e
procedure for T to cross Tc (Hz ). Due to the first Maxwell
associated with the current jθ (r)eiωt , flowing along the equation (see Eq.(3)), the finite δB/δt induces an electric
δB
ortho-radial axis (this latter induces in turn a magnetic field Eθ such that curlEθ = − δt , giving rise eventually to
iωt
field Hz (r)e , parallel to the z axis). The second the persistent, Hz screening current, typical of Meissner
z
effect, as detailed hereabove.
Maxwell equation simplifies into ∂H
∂r = −2jθ because the
The mainstream explanation [1, 2, 4, 5] for the magterm ∝ Eθ in the third equation in Eqs.(6) shows up negligible with respect to that ∝ jθ for practical ω << ωp . netic field Hz being driven out of the superconductor at
As this discussion is limited to the case r → r0 , both Tc (Hz ) in a field cooled sample, will be reviewed now.
For T > Tc (Hz ), i.e. in the normal state, Hz (r) =
Hz (r), jθ (r) are ∝ er/δ(ω) , so that EK (r) is recast into
Hz (r0 ), jθ (r) = 0, ∀r ≤ r0 . At T = Tc (Hz ), a finite
r−r0
2
λL
µ0
momentum density ∝ jθ (r) = jθ (r0 )e λL , is inferred to
Hz (r)
.
(12)
EK (r) =
build up from Eqs.(1,10). However since no external force
8 |δ(ω)|
has been exerted on the sample during the cooling proK
=
−H
,
where
M
=
Moreover there is the identity ∂E
cess, that statement is tantamount to violating Newton’s
z
∂M
µ0 χs (ω)Hz is the magnetization of superconducting elec- law. This inconsistency stems from failing to realize that,
∂F
trons. Actually this identity reads in general ∂M
= −Hz , though Hz remains unaltered during the cooling process,
where F represents the Helmholz free energy [25]; however the magnetic induction B is indeed modified at Tc , as
p-4
Revisiting Meissner effect
V-Hall effect. – For t > t0 , the magnetic induction
shown by Eq.(15). This B variation arouses the driving
force, giving rise to the screening current jθ , and ulti- Bz (t0 , r) exerts on each electron of the persistent current
mately to Hz expulsion, in accordance with Newton and jθ (t0 , r), a radial Lorentz force Bzρjθ . Equilibrium is se 2
Maxwell’s law, as shown by Eq.(2) and Eq.(10), respecm jθ
cured
by
the
radial
centrifugal
force
being counr
ρe
tively.
terbalanced by the sum of the Lorentz force and an elecIV-test experiment. – An experiment, enabling one trostatic one eEr (r), with the radial electric field given
to check the validity of this work, will be presented now. by
It consists of inserting the superconducting sample into
m
jθ
B
+
.
j
E
=
−
z
θ
r
a cylindrical coil of radius r0 , flown through by a timeρe
ρe2 r
dI
dependent current I(t) such that dt 6= 0, only if t ∈]0, t0 [.
∂Bz
, Er
The coil is made up of a wire of length l and radius rc (see Owing to the second Maxwell equation jθ = − µ0 ∂r
can be recast as
Fig.1). Applying Ohm’s law to the coil yields
−l (Ea (t) + Eθ (t, r0 )) = RI(t) ⇒
Eθ (t, r0 ) = Us (t)−RI(t)
l
λ2
,
(16)
where Ea , Us , R are the applied electric field, parallel to
the ortho-radial axis, the voltage drop throughout the coil
and its resistance, respectively. Besides Eθ (t, r0 ) is obtained from Eqs.(5,6) as
Eθ (t, r0 ) = −
X
n∈Z
an Bz (n, r0 )einω0 t
,
z
∂Bz Bz − rL ∂B
∂r
Er =
∂r
µ0 ρe
Because of
∂Bz
∂r
≈
Bz
λM
Z
r0
.
, r0 >> λL and |λM | >> λL , the
∂B 2
approximation Er (r) ≈ ∂rz / (2µ0 ρe) can be used for significant r >> λL . For Hall effect to be observed, a sample
in shape of a cylindrical crown of inner and outer radius
r1 , r0 , respectively, is needed (see Fig.1). Finally the Hall
(17) voltage reads, for r0 − r1 >> |λM |
B 2 t+ , r0
Er (r)dr ≈ − z 0
2µ0 ρe
r−r0
UH = −
,
where Eθ (n, r → r0 ) ≈ Eθ (n, r0 )e δ(nω0 ) and an =
r1
inω0 δ(nω0 ). Working out Bz (n, r0 ) in Eq.(17) requires
2µ0
to solve the second Maxwell equation for Bz (t, r) inside a with Bz t+
0 , r0 = − πrc I(t0 ), as inferred from Eq.(19)
+
dI
cross-section of the coil wire
and dt (t0 ) = 0. As in normal metals [1], measuring UH
gives access to ρ. Note that UH is independent of r1 .
∂Ec
∂Bz
,
(18)
= −µ0 2jc + ǫ0
VI-measurement of λL . – Most experiments [6–11]
∂r
∂t
have consisted of measuring complex impedances at frequencies ω ∈ [10M Hz, 30GHz], which is tantamount to
where jc = I(t)
πrc2 and Ec = Ea (t) + Eθ (t, r0 ) are both
assessing
δ(ω). Because of
√ ωτ << 1√in that frequency
assumed to be r-independent.
Moreover integrating
2ωτ = 1/ 2µ0 σω. However
range,
there
is
|δ|
≈
λ
/
L
Eq.(18) for r ∈ [r0 + 2rc , r0 ] with the boundary condition
whereas
σ
can
be
measured
by several methods, there is
Bz (t, r0 + 2rc ) = 0, ∀t (see Fig.1), while taking advantage
no
experimental
way
to
determine
τ , so that the exact
of Eq.(16), yields
value of λL is not known and thence nor that of ρ/m.
Therefore it is suggested to work at higher frequencies,
ǫ0 rc R dI
2I(t)
√
Bz (t, r0 ) = 2µ0
−
.
(19) such that ω >> 1/τ, ω << ωp , because δ(ω) = λL / 2
l dt
πrc
is independent from τ in that range. Typical values
τ ≈ 10−11 s, ωp ≈ 1016 Hz would imply to measure light
Combining Eqs.(16,17,19) leads finally to
absorption in the IR range. Then for an incoming beam
Z t0 being shone at normal incidence on a superconductor of
Us (t) − RI(t)
refractive index ñ ∈ C, the absorption and reflection coef+ an Bz (t, r0 ) e−inω0 t dt = 0 .
l
0
ficients A, R read [24]
Inserting the measured value of Us (t) − RI(t) into that
(1 − ñ)2 A
=
1
−
R
=
1
−
equation and checking that it is fulfilled for significant, low
(1 + ñ)2 .
|n|’s, would eventually ensure the validity of this analysis.
√
(t)−RI(t)
= r, where The refractive index ñ and the complex dielectric constant
In addition Eq.(17) predicts that UUns (t)−RI(t)
Un (t), r are respectively the voltage drop measured in the ǫ = ǫR + iǫI , conveying the contribution of conduction
normal state and the conductivity ratio [8–11] in the su- electrons, are related [24] by
perconducting and normal state. A qualitative agreement
2
is likely to have been overlooked in the discovery experiǫ
(ωp /ω)
2
ñ
=
=
1
−
.
ment [3].
ǫ0
1 − i/ (ωτ )
p-5
Jacob Szeftel1 Nicolas Sandeau2
At last we get
λL =
2
µ0 cσA
,
τ = µ0 σλ2L
[10]
[11]
[12]
[13]
,
where c refers to light velocity in vacuum. The same
procedure could be applied in normal metals too; however due to τ ≈ 10−14 s, the available
frequency
14
range
16
would
be
much
narrower
:
10
Hz,
10
Hz
versus
11
10 Hz, 1016Hz in a superconductor.
conclusion. – This comprehensive explanation of
Meissner effect resorts solely to macroscopic arguments.
The applied, time-dependent magnetic field excites transient eddy currents according to Newton and Maxwell’s
equations, which turn to persistent ones, after the magnetic field stops varying and the induced electric field
thereby vanishes. Those eddy currents thwart the magnetic field penetration. Were the same experiment to
be carried out in a normal metal, eddy currents would
have built up the same way. However, once the electric
field vanishes, they would have been destroyed quickly by
Joule dissipation and the magnetic field would have subsequently penetrated into bulk matter. As a matter of
fact, Meissner effect shows up as a mere outcome of persistent currents, the very signature of superconductivity.
It is thence unrelated to any microscopic property of the
superconducting wave-function [2, 6, 7, 13, 17, 18, 26]. The
common physical significance of the Meissner and skin effects, both stemming from ǫR (ω) < 0 for ω < ωp , has been
unveiled too. Hall effect can be observed. At last it is
worth pondering upon the physical significance of Eq.(1):
it postulates that the transient regime (0 < t < t0 ) is
completely irrelevant to the permanent one (t > t0 ), while
this work demonstrates quite the opposite, i.e. the permanent regime, characterizing Meissner effect, is entirely
determined by the transient one.
[14]
[15]
[16]
[17]
[18]
[19]
[20]
[21]
[22]
[23]
[24]
[25]
[26]
∗∗∗
We thank M. Caffarel, E. Ilisca and G. Waysand for
encouragement.
REFERENCES
[1] Ashcroft N.W. and Mermin N. D., Solid State Physics,
edited by Saunders College (PA) 1976
[2] Parks R.D., Superconductivity, edited by CRC Press
1969
[3] Meissner W. and Ochsenfeld R., Naturwiss., 21
(1933) 787
[4] London F., Superfluids, edited by Wiley, Vol. 1 (NewYork) 1950
[5] de Gennes P.G., Superconductivity of Metals and Alloys,
edited by Addison-Wesley (Reading, MA) 1989
[6] Pippard A.B., Proc.Roy.Soc., A203 (1950) 98
[7] Pippard A.B., Proc.Roy.Soc., A216 (1953) 547
[8] Hashimoto K. et al., , Phys.Rev.B, 81 (2010) 220501R
[9] Hashimoto K. et al., Science, 336 (2012) 1554
p-6
Gordon R. T. et al., Phys.Rev.B, 82 (2010) 054507
Hardy W. N. et al., Phys. Rev. Lett., 70 (1993) 3999
Sonier J. E. et al. ,Phys. Rev. Lett., 72 (1994) 744
Ginzburg V.L. and Landau L.D., Zh. Eksperim. i. Teor.
Fiz., 20 (1950) 1064
Edwards W.F., Phys. Rev. Lett., 47 (1981) 1863
Essen H. and Fiolhais M., Am.J.Phys., 80 (2012) 164
Yoshioka D., arXiv:1203.2227 (2012)
Bardeen J., Cooper L.N. and Schrieffer J.R., Phys.
Rev., 108 (1957) 1175
Cooper L.N., Phys. Rev., 104 (1956) 1189
Henyey F. S., Phys. Rev. Lett., 49 (1982) 416
Szeftel J. and Sandeau N., arXiv:1407.1231
Reuter G. E. H. and Sondheimer E. H., Proc. Roy.
Soc., A195 (1948) 336
Chambers R. G., Proc. Roy. Soc., A215 (I952) 481
Josephson B. D., Phys. Letters, 1 (1962) 251
Born M. and Wolf E., Principles of optics, edited by
Cambridge University Press 1999
Landau L.D. and Lifshitz E.M., Statistical Physics,
edited by Pergamon Press (London) 1959
Gor’kov L.P., Soviet Phys. JETP, 9 (1959) 1364