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Dynamics of the Meissner Effect: How Superconductors Expel Magnetic Fields
J. E. Hirsch
Department of Physics, University of California, San Diego, La Jolla, CA 92093-0319, USA
The Meissner effect presents us with a fundamental puzzle that has surprisingly not been noticed before: how is the mechanical momentum of the supercurrent that expels the magnetic field compensated, so that momentum conservation is not violated? The only possible answer is, the body as a whole has to acquire equal and opposite momentum to the one developed by the supercurrent. In a cylindrical geometry, the supercurrent has mechanical angular momentum parallel to the applied magnetic field, hence the body has to acquire angular momentum antiparallel to the applied field. How does that happen? The Faraday electric field that develops in the process of magnetic field expulsion transmits angular momentum to the body in the wrong direction, parallel to the magnetic field, of magnitude that is many orders of magnitude too large. How does the body manage to ignore this enormous Faraday torque and rotate in the opposite direction? Any momentum transfer between electrons and the body as a whole has to occur without entropy generation since the transition is thermodynamically reversible. This excludes scattering processes involving impurities or phonons, that generate entropy. The theory of hole superconductivity [1] explains this puzzle [2]. The explanation relies on the facts that within this theory (a) normal metals becoming superconducting expel electrons from the interior to the surface [3], and (b) the normal state charge carriers are necessarily holes [4]. The conventional theory of superconductivity does not have those physical ingredients, hence we argue that it cannot explain this puzzle. Therefore we argue that superconducting materials described by the conventional theory of superconductivity would either (i) not expel magnetic fields or (ii) violate momentum conservation. Consequently, they don’t exist. The alternative theory of hole superconductivity explains superconductivity as arising through pairing of hole carriers driven by lowering of kinetic energy [5], predicts that superconductors have inhomogeneous macroscopic charge distribution with more negative charge near the surface and more positive charge in the interior [3], and that a spin current flows near the surface in the absence of applied fields [6]. It also provides guidelines for the search for new and better superconducting materials. [1] References in https://jorge.physics.ucsd.edu/hole.html. [2] J. E. Hirsch, Europhys. Lett. 114, 57001 (2016); Annals of Physics 373, 230 (2016); Phys. Rev. B 95, 014503 (2017
); IJMPB 32, 1850158 (2018).
[3] J. E. Hirsch, Phys. Rev. B 68, 184502 (2003). [4] J. E. Hirsch, Phys. Lett. A134, 451 (1989). [5] J. E. Hirsch and F. Marsiglio, Phys. Rev. B 39, 11515 (1989); B 62, 15131 (2000). [6] J. E. Hirsch, Ann. Phys. (Berlin) 17, 380 (2008).
https://jorge.physics.ucsd.edu/hole.htmlhttp://iopscience.iop.org/article/10.1209/0295-5075/114/57001https://www.sciencedirect.com/science/article/pii/S0003491616301038https://journals.aps.org/prb/abstract/10.1103/PhysRevB.62.15131https://journals.aps.org/prb/abstract/10.1103/PhysRevB.95.014503https://www.worldscientific.com/doi/abs/10.1142/S0217979218501588https://journals.aps.org/prb/abstract/10.1103/PhysRevB.68.184502https://www.sciencedirect.com/science/article/pii/0375960189903708https://journals.aps.org/prb/abstract/10.1103/PhysRevB.39.11515https://journals.aps.org/prb/abstract/10.1103/PhysRevB.62.15131http://onlinelibrary.wiley.com/doi/10.1002/andp.200810298/abstract
Dynamics of the Meissner effect: how superconductors expel magnetic fields
J.E. Hirsch, UCSD M2S-2018 Beijing
YBaCuO, LiFeAs, FeSe, MgB2, UPt3, Sr2RuO4, V3Si, K3C60, Pb, Al, Nb, all exhibit a Meissner effect
How does the supercurrent that expels the magnetic field start and stop, withoutviolating momentum conservation?
By understanding the Meissner effect, we will learnsomething that is relevant to ALL superconductors
twistedgraphene
Dynamics of the Meissner effect: how superconductors expel magnetic fields
J.E. Hirsch, UCSD M2S-2018 Beijing
YBaCuO, LiFeAs, FeSe, MgB2, UPt3, Sr2RuO4, V3Si, K3C60, Pb, Al, Nb, all exhibit a Meissner effect
How does the supercurrent that expels the magnetic field start and stop, withoutviolating momentum conservation?
By understanding the Meissner effect, we will learnsomething that is relevant to ALL superconductors
Only materials that have hole carriers can expel magnetic fields
twistedgraphene
< >
magnetsuperconductor
I
reversible
Meissnereffect(1933)
magnetic field
NàS
SàN
Tc(H)
H(H)(H)
(H) (H)
H H
H
no entropyis generated
II=0
I=0I
Meissner effect kinetics
B
I
JEH, Annals of Physics 362, 1 (2015)
super
normal
expands and inthe process expels the magnetic field
superconducting region
λ
expansion something about the process of
expels the magnetic field
Meissner effect kinetics
B
I
JEH, Annals of Physics 362, 1 (2015)
super
normal
expands and inthe process expels the magnetic field
superconducting region
λ
expansion something about the process of
expels the magnetic field
FL
vr
B=0
!
" =#
B=0
(a) Growth of superconducting phase
I
(b) Outward flow of perfectly conducting fluid
r0
E I
E I
I
Faraday’s law/Lorentz force
What does BCS explain?* That in the superconducting state, there cannot be a magnetic field in the interior (phase coherence)* That the energy is lower in the superconducting state with no B than in the normal state with B
ψBCS =|ψ | eiθ (r ) !
i"∇ ="p =m"vs +
ec"A
!p = 0!vs = −
emc!A==> ==>
!J = − c
4πλL
!A ==>
!∇×!J = − c
4πλL
!B ∇2
!B = 1
λL2
!B==>
* BCS says nothing about HOW the magnetic field gets expelled to reach the final state with .
!p = 0
WHAT BCS EXPLAINS
, B=0 B=0,
WHAT BCS EXPLAINS
What the really isWHAT THE MEISSNER EFFECT IS
BCS has not AND cannot explain the process
BB=0
B=0
, B=0 B=0,
WHAT BCS EXPLAINS
bodyrotation
Faraday field
How is the momentum of the electrons (Le) and body (Li) generated?Faraday field pushes in the wrong direction (Lenz’s law)
Faraday force on ions
Faraday force on electrons
‘Meissner force’
‘Meissner force’
Normal to superconductor transition in a magnetic field
EF = −1c∂φ∂t
bodyrotation
Faraday field
Faraday force on ions
Faraday force on electrons
‘Meissner force’
‘Meissner force’
Normal to superconductor transition in a magnetic field
normal H=Hc
Hc λLsuperconducting
H=0
EF
cool
Hc
View fromthe top
vs = 70,000cm / s vbody = 0.000001cm / s
How is the momentum of the electrons and body generated?
bodyrotation
Faraday field
Faraday force on ions
Faraday force on electrons
‘Meissner force’
‘Meissner force’
Superconductor to normal transition in a magnetic field
normal H=Hc
Hc λLsuperconducting
H=0
EF
heat
Hc
View fromthe top
vs = 70,000cm / s vbody = 0.000001cm / s
How is the momentum of the electrons transferred to the body?
λLλL
normal region
supercond region
Hc Hc
λL
supercond region
EF EF
normal region
EF slows down electrons inside superconducting (S) regionEF imparts counterclockwise momentum to body in S region
Expansion of superconducting phase
???body rotation
???
WHAT IS NEEDED TO EXPLAIN THE PROCESS
What the really isWHAT THE MEISSNER EFFECT IS
BCS cannot explain the process of field expulsion
1) Electrons flowing outward
3) Lorentz force2) Ions flowing outward
FL = q!vc×!B
BB=0
B=0
rWHAT IS NEEDED TO EXPLAIN THE PROCESS
What the really isWHAT THE MEISSNER EFFECT REALLY IS
BCS cannot explain the process of field expulsion
Hc
λLEF
normal region
supercond region
1) Electrons flowing outward
3) Lorentz force2) Ions flowing outward
FL = q!vc×!B
radial flowof charge
EF
rWHAT IS NEEDED TO EXPLAIN THE PROCESS
What the really isWHAT THE MEISSNER EFFECT REALLY IS
BCS cannot explain the process of field expulsion
Hc
λLEF
normal region
supercond region
1) Electrons flowing outward
3) Lorentz force2) Ions flowing outward
FL = q!vc×!B
EF
2) Electrons backflowing inwardradial flowof charge
nnormal superfluid
hot cold
4HeFlow and backflow in superfluid 4He
Fountain effect
nnormal superfluid
hot cold
4HeFlow and backflow in superfluid 4He
λL
normal region
supercond region
Hc
vs
vnvnvs
vs vn
cold hot
body rotation
EF
Lorentz force:
FL = e!vc×!B
HOW?
outflowing superelectrons generate Meissner current
vs
backflowing normal electronstransmit their momentum to the body.
vn
nnormal superfluid
hot cold
4He4
λλL
normal region
supercond region
Hc
vv
v
cold hot
EF
FL = e!vc×!B
HOW?
vvn
vnvn
vn
Jx
H
ionEy
FAmp
!FAmp =
Ic
!L×!H
What is the difference between electrons and holes?
Jx
H
!FAmp =
Ic
!L×!H
ionEy
FAmp
RH0
FAmp= EyFAmp= Ey
xzy
Ampere force on the body iscaused by electric field Ey
L
Jx
H
EyFAmp
!FAmp =
Ic
!L×!H
What is the difference between electrons and holes?
Jx
H
!FAmp =
Ic
!L×!H
ionEy
FAmp
RH0
FAmp= EyFAmp= Ey
xzy
Ampere force on the body iscaused by electric field Ey
L
Ampere force on the body isin direction opposite to electric field
m*0
Hole current transfers momentum to the body=FAmp
Flatt
Fon-latt
FEFH
FEFH
FE
Jx
H
ionEy
FAmp
!FAmp =
Ic
!L×!H
RH>0
FAmp= Ey
m*
λL
normal region
supercond region
Hc
vs
vn
vs
vs
cold hotvn
vn
There is an outflowing hole currentcausing an Ampere torque on body,in opposite direction to EF torque
EF
orbit expansion:N S
to preserve charge neutrality
Orbit expansion è kinetic energy lowering:
€
K0 =12mev
2 =L2
2mer02 =
!2
2mer02
r0=kF-1 r1=2λL
€
K1 =!2
2mer12
2λL orbits in superconductors
SλL
vs R
€
L = (mevsR)ns(2πRλLh)angular momentum of supercurrent:
€
=
€
=
N S
2λL
€
L = (mevs2λL )ns(πR2h)
S
vs
Ground state of a superconductor (no magnetic field applied)
r=2λL orbits r=2λL orbits
Electron spin into screen Electron spin out of screen
Currents in the interior cancel out, near the surface survive==> there is a spontaneous spin current in the ground state of superconductors near the surface
€
vφ =!
4meλL
€
L = mevφ ⋅ (2λL ) =!2
Macroscopic zero point motion in the ground state of superconductors
phasecoherence
There is a spontaneous spin current in the ground state ofsuperconductors, flowing within λL of the surface
€
! µ =
e"2mec
! σ
For λL=400A, vσ0=72,395cm/s# of carriers in the spin current: ns
µ
µnvσ0
€
! v σ 0 = −"
4meλL
! σ × ˆ n
When a magnetic field is applied:
€
! v σ =! v σ 0 −
emec
λL! B × ˆ n
vσ
B
no external fields applied
The slowed-down spin component stops when
€
B = meceλL
vσ 0
€
=!c4eλL
2
€
=φ04πλL
2 ~ Hc1!
Electronic orbits have radius (to explain Meissner effect)
Angular momentum: ==>
€
L = ! /2
€
L = mevσ 0(2λL )
€
2λL
!Jσ = ens
!vσ =!Jσ 0 −
c4πλL
!B× n̂
Ann. der Phys.17, 380 (2008)
There is a spontaneous spin current in the ground state ofsuperconductors, flowing within λL of the surface
€
! µ =
e"2mec
! σ
For λL=400A, vσ0=72,395cm/s# of carriers in the spin current: ns
µ
µnvσ0
€
! v σ 0 = −"
4meλL
! σ × ˆ n
When a magnetic field is applied:
€
! v σ =! v σ 0 −
emec
λL! B × ˆ n
vσ
B
no external fields applied
The slowed-down spin component stops when
€
B = meceλL
vσ 0
€
=!c4eλL
2
€
=φ04πλL
2 ~ Hc1!
Electronic orbits have radius (to explain Meissner effect)
Angular momentum: ==>
€
L = ! /2
€
L = mevσ 0(2λL )
€
2λL
!Jσ = ens
!vσ =!Jσ 0 −
c4πλL
!B× n̂
!Jc = −
c4πλL
Hc1 × n̂
Ann. der Phys.17, 380 (2008)
+ other papers
Why electronic orbits expand to radius 2λL when a metal becomessuperconducting:
ℓ = "2
From Dirac equation: Hs.o. = −e!
4me2c2"σ ⋅ ("E × "p)
H = p2
2me+Hs.o. =
12me
( !p− ec
!Aσ )
2
!Aσ =
!4mec
"σ ×!E
!p = 0 ⇒ vσ0 = −
emec!Aσ
!E = 2πρ!r = 2π | e | ns
!r!Aσ =
2π | e | ns"4mec
!σ ×!r ≡#Bσ ×!r
2⇒ !Bσ =
π | e | ns!mec
vσ0 =
eλLmec
Bσ =πe2nse
2λL!me2c2
=!
4meλL
ℓ =mevσ0 (2λL ) =me
!4meλL
(2λL ) =!2
angularmomentum
2λL
spin-orbitinteraction
F. Marsiglio
(1989)
Theory of hole superconductivity (1988-2018)S. Tang, H.Q. HongH.Q. Lin R. Teshima
G. Bach References: http://physics.ucsd.edu/~jorge/hole.html
(1992)(1990)
(2008)(2001)
(2015)(2010)
(1989-on)
(2001-2008)
(2016-18)
Materials
(1989-2000)
Prediction vs postdictionThe essential elements for explaining Meissner effect:
1) Electrons expelled radially outward (to generate Meissner current)
2) Holes are the normal charge carriers (to transfer momentum to body without dissipation)
3) Kinetic energy lowering drives superconductivity (to explain what drives electron outflow)
were part of the theory many years before the theory addressed the Meissner effect
(2001)
(1989)
(1992)
(2003-on)
body rotates
How momentum is transferred without transferring energy
Pem= momentum ofelectromagnetic field
!FH = e
"vc×!Hv
!Pem =
14πc
!E ×!H HPe E Pem
electron moves out,acquires Peem field acquires Pem
normal electron moves in,acquires from em field,transfers it to the body
Pem
E
Electromagnetic field mediates the transfer of momentum
Lem Lem
Process is reversible iff the normal state charge carriers are holes
Radial flow and counterflow is essential
S
N H
EF
body
super current
rotation
EF
λL
r0
FH
λL electrons
r0
S
N H
EF
body
super current
rotation
λL r0
λL
FH backflow
r0 FE
Lorentz
Lorentz ions
(a) (b)
FE
F
Flatt
ions EF
Fon-latt
r0 r0 r0 r0
impurity FH
backflow
electron carriers (RH
* Conventional BCS-London theory describes NONE of this* The theory of hole superconductivity describes this physics
Summary: In the Meissner transition, * Electrons need to acquire momentum in direction opposite to that dictated by the Faraday electric field.* The body needs to acquire momentum in direction opposite to that dictated by the Faraday electric field.
==> A radial outflow of superconducting electrons is needed, and a radial outflow of normal holes
* The transition is thermodynamically reversible
References: http://physics.ucsd.edu/~jorge/hole.html
* 2λL orbits describe radial outflow, give spin current, explain universal self-field critical current
Holes are necessary to transfer momentum from electrons to the body in a reversible way
* Conventional BCS-London theory describes NONE of this* The theory of hole superconductivity describes this physics
Summary: In the Meissner transition, * Electrons need to acquire momentum in direction opposite to that dictated by the Faraday electric field.* The body needs to acquire momentum in direction opposite to that dictated by the Faraday electric field.
==> A radial outflow of superconducting electrons is needed, and a radial outflow of normal holes
* The transition is thermodynamically reversible
References: http://physics.ucsd.edu/~jorge/hole.html
* 2λL orbits describe radial outflow, give spin current, explain universal self-field critical current
Holes are necessary to transfer momentum from electrons to the body in a reversible way