Superconductivity · 624 CE, Brahmagupta in his book "Brahmasphutasiddhanta": "Make a wheel of light timber, with uniformly hollow spokes at equal intervals. Fill each spoke up to

Superconductivity

Lecture 1: Superfluidity, superconductivity, materials -Superfluidity and superconductivity

-Materials, types of SC: conventional, heavy fermion, high Tc

Lecture 2: Microscopic properties, BCS theory

-Pairing mechanisms: phonons, spin-fluctuations -BCS theory

-Experimental Properties

Dirk van der Marel Université de Genève 0

624 CE, Brahmagupta in his book "Brahmasphutasiddhanta": "Make a wheel of light timber, with uniformly hollow spokes at equal intervals. Fill each spoke up to half with mercury and seal its opening situated in the rim. Set up the wheel so that its axle rests horizontally on two upright supports. Then the mercury runs upwards in some hollow spaces and downwards in some others, as a result of which the wheel rotates automatically forever."

The perpetuum mobile

1

1932:Discoveryofaspecificheatanomalyinliquid4HeWillemHendrikKeesomandAnnaPetronellaKeesom

T 0

c(T)

0

2.3 K

“λ-transition”

The two phases of liquid 4He

2

The phase diagram of liquid 4He

3

1937: Discovery that liquid He-II is a superfluid

PyotrKapitsa,JohnF.Allen,andDonMisener 4

In 1924 Albert Einstein and Satyendranath Bose predicted that cooling bosonic atoms to a very low temperature would cause them to fall

(or "condense") into the lowest accessible quantum state, resulting in a new form of matter.

There exists no classical explanation for the complete absence of friction in a superfluid

In 1938 Fritz London proposed BEC as a mechanism for superfluidity

in 4He and superconductivity.

5

4He atoms are Bosons

6

Bose-Einstein statistics

Numberofbosonsinstate k : Nk =

1e εk−µ( )/kBT −1

N bosons in a recipient of volume V Density : n=N/V

μ = chemical potential

μ self-adjusts such that

Nkk∑ =N

7

Fraction of bosons having εk ≠ 0 :

=ζ 3/2( )n!3

mkBT2π

⎛

⎝⎜⎞

⎠⎟

3/2

Remaining fraction N0/N = 1−N+/N : BEC

Bose-Einstein condensation

The fraction N+/N is proportional to T3/2 !

N+

N= 1

eεk/kBT −1k≠0∑

For k=0: Nk = ∞ Interpretation: N0 is a macroscopic fraction of N

At low temperatures : μ = 0 ⇒ Nk =1

eεk/kBT −1

8

T/T0

N0(T)/N

Bose-Einstein condensation

Condensationtemperature:kBT0=3.3125!2n2/3

m

N0N

=1− N+

N=1− T

T0

⎛

⎝⎜⎞

⎠⎟

3/2

Liquidhelium:n=2.2⋅1022cm−3⇒T0 =3.1K

9

Experimental λ point: Tc=2.3 K

315314 1010 −− << cmncm1995: Dilute cold atom gases. Wolfgang Ketterle ; Eric Cornell; Karl Wiemann

BECof87Rb

(web-siteNobelprize2001)

Bose-Einstein condensation

⇒0.5µK <T0 <2µK

10

Back to 1911

H. Kamerling Onnes�

Mercury

The discovery of superconductivity

11

Single-element superconductors

12

Thermodynamic properties: Specific heat jump at Tc

ZrB12

13

Magnetic properties: Perfect Conductor

⇒!∇×!E = m

ne2∂!∇×!j

∂t

⎫⎬⎪

⎭⎪⇒ ∂

!B∂t

= mne2c

∂!∇×!j

∂tFaraday:

1⎡⎣ ⎤⎦ !E = m

e2n1τ− iω

⎡

⎣⎢

⎤

⎦⎥!j

⇒σ ω( )≡!j!E= e

2nτ /m1− iωτ (opticalconductivity)

!∇×!E − c−1∂

!B /∂t =0

14

Perfect conductor: τ = ∞

⎫

⎬⎪⎪

⎭⎪⎪

⇒!E = m

ne2∂!j

∂t

Ampère'slaw:!j = c

4π!∇×!B⇒∇2 ∂

!B∂t

= 1λL2∂!B∂t

⇒∂!B x ,t( )∂t

=∂!B 0,t( )∂t

e− x/λL

⇒Deepinsidetheperfectconductor:∂!B/∂t=0

Magnetic properties: Meisner-Ochsenfeld effect (1933)

15

Perfectconductor:

!E = e−2mn−1∂

!j /∂t

∂!B /∂t =0

⎧⎨⎪

⎩⎪

Magnetic properties: Meisner-Ochsenfeld effect (1933)

Meisner&Ochsenfeld,deepinsidesuperconductor:!B =0

FritzLondon'shypothesis 1935( ):!j = − c4πλL2

!A

λL =mc2

4πnse2;ns = "superfluiddensity"

!E = −c−1∂

!A/∂t⇒ !E = 4πλL2∂

!j /∂t 1t Londonequation( )

!B =!∇×!A⇒

!B = − 4πc−1λL2

!∇×!j 2d Londonequation( )

16

Magnetic properties: Fritz London (1935)

!j = − c

4πλL2!A

2d Londoneqn: !B = − 4πc−1λL2!∇×!j

Ampère'slaw:!j = c4π!∇×!B

⎫⎬⎪

⎭⎪⇒

⎧

⎨⎪

⎩⎪

⇔ Superconductivity

⇓!B x ,t( ) = !B0 e− x/λL

Deepinsideasuperconductor

!B x ,t( ) =0

Ampère'slaw⇒!j x ,t( ) =0

0th Londonequation⇒ !A x ,t( ) =0 1t Londonequation⇒ !E x ,t( ) =0

∇2 !B = λL

−2 !B

17

Magnetic and AC properties of a superconductor:

LondonEquation:!js = −

nse2

m!A

A supercurrent can coexist with a normal current

Normalcurrent:!jn =

e2nn /mτ −1 − iω

!E ;nn = "normalelectrondensity"

Opticalconductivity:σ ω( ) =

!js +!jn!

E= c

2

λL2 πδ ω( )+ i

ω⎡

⎣⎢

⎤

⎦⎥+

τe2nn /m1− iωτ

Important consequences •  An external magnetic field penetrates only within a surface layer of thickness λL •  The supercurrent is confined to a surface layer of thickness λL •  The magnetic flux through a superconductor is quantised in units of ϕ0=h/(2e)

ns(T) vanishes at Tc, and is maximal at T=0

18

Field- and current distribution inside a superconductor

E. F. Talantsev, A. E. Pantoj, W. P. Crum & J. L. Tallon,

Scientific Reports 8, 1716 (2018)

19

Magnetic properties: Types I and II superconductivity

20

Type I superconductivity

Type II superconductivity

Bc1 < B < Bc2 : Magnetic Flux Penetrates as Vortices Vortex: Angular momentum = ħ & Magnetic Flux = h/2e

Vortices interact with each other à Abrikosov vortex lattice

Magnetic properties: Type II superconductivity

21

Heavy Fermion Superconductors

Tc m*/m (from γ) CeIn3 0.2 K ~50 CeCu2Si2 0.7 K ~1000 CePt3Si 0.75 K ~400 CeCoIn5 2.3 K ~250 UPt3 0.48 K ~200 UBe13 0.85 K ~300 URu2Si2 1.3 K ~25 PuCoGa5 18.5 K ~80

F Steglich et al, PRL 43 (1979) 1892 A de Visser, A Menovsky, JJM Franse, Physica B+C 147, 81 (1987) MB Maple et al, PRL 56, 185 (1986) MJ Rice, PRL 20, 1439 (1968) K Kadowaki and SB Woods, Solid State Communications 58, 507 (�1986)

22

C=γT

ρ=AT2 γ=c’Ν(0) A=cΝ(0)2

1986: Possible High Tc Superconductivity in the Ba-La-Cu-O System

J.G. Bednorz and K.A. Müller Z. Phys. B - Condensed Matter 64, 189-193 (1986)

Karl Alex Mueller

J. Georg Bednorz

23

How to make a room temperature superconductor

24

RoomtemperatureSCexistsalready:NEUTRONSTARS

CassiopeiaAShterninetal:Tc>2’000’000’000K

Pageetal:Tc=500’000’000K

Toward superconductivity at room temperature

25

1987: Superconductivity at 93 K in a New Mixed-Phase Y-Ba-Cu-0 Compound System at Ambient Pressure

M. K. Wu, J. R. Ashburn, C. J. Torng, P. H. Hor, R. L. Meng, L. Gao, Z. J. Huang, Y. Q. Wang & C. W. Chu

PRL 58, 908-910 (1987)

1993: Superconductivity above 130 K in the Hg-Ba-Ca-Cu-O system A. Schilling, M. Cantoni, J. D. Guo & H. R. Ott

Nature 363, 56-58 (1993)

1993: Superconductivity above 150 K in HgBa2Ca2Cu3O8+δ at high pressures

C. W. Chu, L. Gao, F. Chen, Z. J. Huang, R. L. Meng & Y. Y. Xue Nature 365, 323-325 (1993)

Toward superconductivity at room temperature

26

Ba0.6K0.4BiO3:Tc=30KR.J.Cava,B.Batlogg,J.J.Krajewski.R.Farrow,L.W.Rupp,A.E.White,K.Short,

W.F.Peck&T.Kometai,Nature332,222(1988).

RbCs2C6O:Tc=33KK.Tanigawa,T.W.Ebbesen,S.Saito,J.Mizuki,J.S.Tsai,Y.Kubo,S.Kuroshima,

Nature352,222(1991).

MgB2:Tc=39KJ.Nagamatsu,N.Nakagawa,T.Muranaka,Y.Zenitani&J.Akimitsu,Nature410,63(2001)

SmFeAsO:Tc=55K

RENZhi-An,LUWei,YANGJie,YIWei,SHENXiao-Li,LIZheng-Cai,CHEGuang-Can,DONGXiao-Li,SUNLi-Ling,ZHOUFang,ZHAOZhong-Xian,Chin.Phys,Lett.25,2215(2008)

FeSeonSrTiO3:Tc>100KJian-FengGe,Zhi-LongLiu,CanhuaLiu,Chun-LeiGao,DongQian,Qi-KunXue,

YingLiuandJin-FengJia,NatureMat.14,285(2015)

Toward superconductivity at room temperature

27

2015: Superconductivity at 203 kelvin at high pressures (140 GPa) in sulfur hydride

AP Drozdov, MI Eremets, IA Troyan, VK Senofontov & SI.Shylin, Nature 525,73

Toward superconductivity at room temperature

28

Toward superconductivity at room temperature December 2018: SC at 250 K in LaH10 /170 Gpa �

AP Drozdov, PP Kong, VS Minkov, SP Besedin, MA Kuzovnikov, S Mozaffari, L Balicas, �F Balakirev, D Graf, VB Prakapenka, E Greenberg, DA Knyazev, M Tkacz & MI Eremets;

ArXiv:1812.01561

August 2018: SC at 280 K in LaH10 /202 GPa M Somayazulu, M Ahart, AK Mishra, ZM Geballe, M

Baldini, Y Meng, VV Struzhkin & RJ Hemley; �arXiv:1808.07695

29

220

240

260 LaH10 (170 GPa)

Toward superconductivity at room temperature LaH10 (202 GPa)

(140 Gpa)

30

Summary

31

•  Superfluidity and superconductivity: Realisations of the perpetuum mobile

•  Basic principles of Bose-Einstein condensation

•  The discovery of superconductivity

•  Meisner effect, type I superconductivity

•  Fritz Londons’ interpretation of the Meisner effect

•  Type II superconductivity and vortices

•  Heavy fermion superconductors •  High Tc and the quest for room temperature superconductors