Superfluidity and 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 The perpetuum mobile 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."

1 The two phases of liquid 4He

1932: Discovery of a specific heat anomaly in liquid 4He Willem Hendrik Keesom and Anna Petronella Keesom

“λ-transition”

c(T)

0 2.3 K T 2 The phase diagram of liquid 4He

3 1937: Discovery that liquid He-II is a superfluid

Pyotr Kapitsa, John F. Allen, and Don Misener 4 There exists no classical explanation for the complete absence of friction in a superfluid

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.

In 1938 Fritz London proposed BEC as a mechanism for superfluidity in 4He and superconductivity.

5 4He atoms are Bosons

6 Bose-Einstein statistics

N bosons in a recipient of volume V

Density : n=N/V

1 Number of bosons in state k : N k = /k T (εk −µ) B e −1

μ = chemical potential

μ self-adjusts such that N N ∑ k = k

7 Bose-Einstein condensation

1 At low temperatures : μ = 0 N ⇒ k = ε /k T e k B −1

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

Fraction of bosons having εk ≠ 0 : 3/2 N 1 ζ 3/2 ⎛ mk T ⎞ + = = ( ) B ∑ εk /kBT 3 ⎜ ⎟ N k≠0 e −1 n! ⎝ 2π ⎠

3/2 The fraction N+/N is proportional to T !

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

8 Bose-Einstein condensation

3/2 N N ⎛ ⎞ 0 + T = 1− = 1− ⎜ ⎟

)/N N N ⎝ T0 ⎠ T

( 2 2/3 0 ! n N Condensation temperature: k T = 3.3125 B 0 m

T/T0

Liquid helium: n = 2.2⋅1022cm−3 ⇒T = 3.1 K 0

Experimental λ point: Tc=2.3 K

9 Bose-Einstein condensation

1995: Dilute cold atom gases. Wolfgang Ketterle ; Eric Cornell; Karl Wiemann 1014 cm −3 < n < 1015cm −3 ⇒0.5µK

BEC of 87Rb

(web-site Nobel prize 2001)

10 The discovery of superconductivity Back to 1911 Mercury

H. Kamerling Onnes

11 Single-element superconductors

12 Thermodynamic properties: Specific heat jump at Tc

ZrB12

13 Magnetic properties: Perfect Conductor ⎫ ⎪ ! ⎪ ! m ∂ j ⎬⇒ E = 2 ⎪ ne ∂t Perfect conductor: τ = ∞ ⎪ ⎭ ! ! ! ! m ∂∇ × j ! ! ! ⇒ ∇ × E = 2 ⎪⎫ ∂B m ∂∇ × j ne ∂t ! ! ! ⎬ ⇒ = 2 Faraday: ∇ × E − c−1 ∂B /∂t = 0 ∂t ne c ∂t ⎭⎪ ! ! ! ! ! c ! ! B 1 B 2 ∂ ∂ ∂B x,t ∂B 0,t x/ Ampère's law: j B ( ) ( ) − λL = ∇ × ⇒ ∇ = 2 e 4π ∂t λ ∂t ⇒ = L ∂t ∂t ! Deep inside the perfect conductor: B/ t=0 ⇒ ∂ ∂

! m ⎡1 ⎤ ! ⎡1⎤ E = ⎢ − iω ⎥ j ⎣ ⎦ e2n τ ! ⎣ ⎦ j e2nτ /m ⇒σ ω ≡ ! = (optical conductivity) ( ) E 1− iωτ 14 Magnetic properties: Meisner-Ochsenfeld effect (1933)

15 Magnetic properties: Meisner-Ochsenfeld effect (1933) ! Meisner & Ochsenfeld, deep inside superconductor: B = 0 ! ! ⎪⎧ E = e−2mn−1 ∂ j /∂t Perfect conductor: ⎨ ! B / t 0 ⎩⎪ ∂ ∂ = ! c ! Fritz London's hypothesis 1935 : j A ( ) = − 2 4πλL mc2 λ = ; n = "superfluid density" L 4πn e2 s s ! ! ! ! E = −c−1 ∂A/∂t ⇒ E = 4πλ 2 ∂ j /∂t 1t London equation L ( ) ! ! ! ! ! ! B = ∇ × A ⇒ B = − 4πc−1λ 2∇ × j 2d London equation L ( )

16 Magnetic properties: Fritz London (1935) ! ! ! ! ! 2d London eqn: B = − 4πc−1λ 2∇ × j ⎫ ⎧ ∇2B = λ −2B L ⎪ ⎪ L ! c ! ! ⎬⇒ ⎨ ⇓ Ampère's law: j = ∇ × B ⎪ ⎪ ! ! − x/λ 4π ⎭ B x,t = B e L ⎩ ( ) 0 Deep inside a superconductor ! B(x,t) = 0 ! Ampère's law ⇒ j x,t = 0 ( ) ! 1t London equation ⇒ E(x,t) = 0 ! 0th London equation ⇒ A x,t = 0 ( ) ! c ! j = − A ⇔ Superconductivity 4πλ 2 L 17 Magnetic and AC properties of a superconductor:

2 ! nse ! London Equation: js = − A m ns(T) vanishes at Tc, and is maximal at T=0

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)

A supercurrent can coexist with a normal current ! e2n /m ! Normal current: j n E ; n "normal electron density" n = −1 n = τ − iω ! ! j + j c2 ⎡ i ⎤ τe2n /m Optical conductivity: s n n σ ω = ! = 2 ⎢πδ ω + ⎥ + ( ) E λ ( ) ω 1− iωτ L ⎣ ⎦

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

Type I superconductivity

Type II superconductivity

20 Magnetic properties: 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

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

ρ=AT2 A=cΝ(0)2 C=γT γ=c’Ν(0)

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 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 Toward superconductivity at room temperature

Room temperature SC exists already: NEUTRON STARS

Cassiopeia A

Shternin et al: Tc > 2’000’000’000 K

Page et al: Tc = 500’000’000 K 25 Toward superconductivity at room temperature

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)

26 Toward superconductivity at room temperature

Ba0.6K0.4BiO3 : Tc = 30 K R.J. Cava, B.Batlogg, J.J. Krajewski. R.Farrow, L.W.Rupp, A.E.White, K. Short, W.F. Peck & T. Kometai, Nature 332, 222 (1988).

RbCs2C6O: Tc = 33 K K. Tanigawa, T. W. Ebbesen, S. Saito, J. Mizuki, J. S. Tsai, Y. Kubo, S.Kuroshima, Nature 352, 222 (1991).

MgB2: Tc = 39 K J. Nagamatsu, N. Nakagawa, T. Muranaka, Y. Zenitani & J. Akimitsu, Nature 410, 63 (2001)

SmFeAsO: Tc = 55K REN Zhi-An, LU Wei, YANG Jie, YI Wei, SHEN Xiao-Li, LI Zheng-Cai, CHE Guang-Can, DONG Xiao-Li, SUN Li-Ling, ZHOU Fang, ZHAO Zhong-Xian, Chin. Phys, Lett. 25, 2215 (2008)

FeSe on SrTiO3: Tc > 100K Jian-Feng Ge, Zhi-Long Liu, Canhua Liu, Chun-Lei Gao, Dong Qian, Qi-Kun Xue, Ying Liu and Jin-Feng Jia, Nature Mat. 14, 285 (2015) 27 Toward superconductivity at room temperature

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

28 Toward superconductivity at room temperature

August 2018: SC at 280 K in LaH10 /202 GPa December 2018: SC at 250 K in LaH10 /170 Gpa M Somayazulu, M Ahart, AK Mishra, ZM Geballe, M AP Drozdov, PP Kong, VS Minkov, SP Besedin, MA Kuzovnikov, S Mozaffari, L Balicas, Baldini, Y Meng, VV Struzhkin & RJ Hemley; F Balakirev, D Graf, VB Prakapenka, E Greenberg, DA Knyazev, M Tkacz & MI Eremets; arXiv:1808.07695 ArXiv:1812.01561

29 Toward superconductivity at room temperature

LaH10 (202 GPa)

260

LaH10 (170 GPa) 240

220

(140 Gpa)

30 Summary

• 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