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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