Superconductivity

“door meten tot weten”－reaching a truth via the measurements－

Kameringh Onnes (1911)

g

R()

HHg

Temperature (K)Zero resistance → no loss of energy

persistent current is forever

Discovery of S d ti it i H

Small magnetic field S d ti itSuperconductivity in Hg suppresses Superconductivity

TResistance Tc Critical Magnetic field

Normal State

Superconductivity

TemperatureTemperature

Electron Gas or Liquid in Metals

Ions form a lattice in metals

Free electrons model in metal

energy

potential energy in metal (b) simple model

In model (b) where cubic box has a volume V=L3, Schrödinger equation is given by

Here,

otherwiseU i i di b d ditiUsing a periodic boundary condition

We obtain a following plane-wave function :

With eigen energy :

where wave numbers are quantized as follow:

n-dependence of electron’s wave length in cubic boxWave length Momentum

positive integer

Density of states, D( )per unit volume(cm3) is d fi d b fdefined as number of states in between and +d.

kz=0 surface +d.

Since two electrons with spin-up and-down exist in k-

l (2 / )3 b

surface

kspace volume (2/L)3, number of states per unit volume in k-space is 2 (L/2)3

kx

k space is 2 (L/2)

ky

Using = h2k2/2m , g / ,the above equation is rewritten as the f ti ffunction of

Mechanism of formation of electron pairs due to lattice vibrationto lattice vibration

5 ~ 10 eV

: Fermi energy: Fermi energy

Electrons density :

Concept for Superconductivity

Fermi energy

Fermions

BBosons

Energy gap

Cooper pairs: |k , - k >Fermi-Dirac Statistics Bose-Einstein Statistics

Cooper pairs: |k , k

Provided that attractive interaction works between electrons near the Fermi level, electrons are always bounded making

i C i I d hi hpairs - Cooper pairs -. In order to prove this theorem, we deal with a simple case where two electrons are added on the Fermi sea as illustrated belowFermi sea as illustrated below.

We deal with a following Schrödinger equation for two electrons with attractive potential V (r1, r2) ;

In case of V (r1, r2) = 0 , the wave function with a lowest energy at zero total momentum is described by the following formula;

a wave function is expressed as follow;Then, for

Note that since this wave function is symmetric in orbital sector, the spin function is in anti symmetric spin singlet statefunction is in anti-symmetric spin-singlet state.

Inserting (11.11) into (11.9) and using,

We can derive this eigen equation.

When the eigen energy in the eq. (11.12) has a solution for E < 2F , two- electron bounded state (Cooper pair) is formed. P id d h i i d f llProvided that V (r1, r2) is approximated as follows;

otherwiseHere, if the constant attractive interaction V is assumed to be effective only among electrons within energies in the range between the Fermi energy F and the Debye energy which is the highest one of lattice vibration, we obtain the following equation;

When taking , form we have

Since , we obtain the following relation is considered and as a result

we obtain the eigen energy as

It was proved that electrons near the Fermi surface are bounded making pairs of (k, ) and (-k, ) -C i i th tt ti i t ti di t dCooper pair- via the attractive interaction mediated by lattice vibration with highest energy -Debye energy - . Here the Cooper pair is in the zero total momentum and the spin singlet statetotal momentum and the spin-singlet state.

Since Cooper pairs are formed by many body of electrons near the Fermi level these areelectrons near the Fermi level, these are condensed into a macroscopic quantum statewhich is regarded as a Bose condensation. This outstanding aspect of superconductivity wasoutstanding aspect of superconductivity was theoretically clarified by Bardeen, Cooper and Schriefer, and hence this theory is called as BCS theory which is epoch-making event in condensedtheory which is epoch making event in condensed matter physics in the 20th century.

I thi BCS t t i t i In this BCS state, an isotropic energy gap opens on the Fermi level, yielding a perfect diamagnetism called Meissner effect and zero-resistance effectresistance effect.

SuperconductivityConventional superconductivity:

Cooper pairCooper pair attractive interaction: electron-phonon coupling

s-wave spin singlet

order parameter: �� r

r ei r

pairing channel: angular momentum l=0 and spin s=0

broken symmetry: U(1) gauge

p r r e

• Meissner-Ochsenfeld-effect (Higgs)• persistent currentsro en symmetry ( ) gaug p• flux quantization

§2 どんな超伝導体があるのだろうどんな元素が超伝導になるのだろう？どんな元素が超伝導になるのだろう？

H

Li Be B C N O F

He

Ne20 K

0.026K9 11 2K 0.6K34K

超伝導を示さない元素

Na

K

Mg

C S Ti V C M F C Ni C Z

Al

G

Si

G

P

A

S

S

Cl

B

Ar

K

20 K 9K 11.2K

8.5K 17K5.8K1.17K

3.6K

34K

超伝導を示す元素

圧力下もしくは薄膜で超伝導を示す元素

K

Rb

Ca

Sr

Sc

Y

Ti

Zr

V

Nb

Cr

Mo

Mn

Tc

Fe

Ru

Co

Rh

Ni

Pd

Cu

Ag

Zn

Cd

Ga

In

Ge

Sn

As

Sb

Se

Te

Br

I

Kr

Xe

15K

4K

0.3K

2.8K

2K 5.4K 2.7K 7K 1.4K

3.6K 7.4K 1.2K

0.4K

0.6K

5.4K17.2K

9.25K9.7K 0.92K 6.2K 0.5K 0.035mK

0.85K

0.52K

7.9K8.4K

3.4K4 2K

3.7K4.7K

Cs

Fr

Ba

Ra

Hf Ta W Re Os Ir Pt Au Hg Tl Pb Bi Po At Rn

5K1.5K 8.7K0.16K 4.4K

0.92K

0.01K5.5K

6.2K

1.7K

0.5K

0.7K 0.1K

0.52K

4.15K

4.2K

2.4K 7.2K

Fr RaLa Ce Pr Nd Pm Sm Eu Gd Tb Dy Ho Er Tm Yb Lu

Ac Th Pa U Np Pu Am Cm Bk Cf Es Em Md No Lr

1.7K6K

12K

0 7K

0.1K

AoyamaAoyama--Gakuin Gakuin UniversityUniversity

1.4K 1.4K0.7K2.2K

1K

超伝導はどのようにしておこるか？S理論による予測－BCS理論による予測－

左から、左から、

John Bardeen、 Leon N. Cooper、

J Robert SchriefferJ. Robert Schrieffer

)1(

expDcT

格子 周波数が高 方が

映像：日立サイエンスシリーズ 超

格子の周波数が高い方が

高いTcに有利

AoyamaAoyama--Gakuin Gakuin UniversityUniversity

映像：日立サイエンスシリーズ, 超伝導 よりPictures from Nobelprize.org

超伝導現象の解明-BCS理論

1957 Bardeen, Cooper, Schrieffer (BCS) 理論1957  Bardeen, Cooper, Schrieffer (BCS) 理論

Nobel Prize (1972)Bardeen           Cooper         Schrieffer

電子２個が対をつくって運動電子２個が対をつくって運動（クーパー対を形成）

電子－格子相互作用を

媒介とした電子間引力 － － －+－媒介とした電子間引力

「BCSの壁」を越えた超伝導の発見「室 超伝導

H C B C O -100℃1986年以前の超伝導転移温度

1993年「室温超伝導」

人類の「夢」の「実現」へ

HgCaBaCuO（高圧下）

HgCaBaCuO（高圧下）

140

160

-100℃超伝導転移温度の最高記録 24Ｋ（Ｎｂ３Ｇｅ） 銅酸化物

BiCaSrCuO

TlCaBaCuOHgCaBaCuOTlCaBaCuO

（高圧下）

（Ｋ

） 120

140

理論家は、、、、

液体窒素

YBaCuO

BiCaSrCuO

界温

度（

80

100

超伝導転移温度は高くても

LaSrCuO

Nb G

臨界

40

6030~40K程度まで

「BCSの壁」金属系超伝導

Hg

LaBaCuO

PbNbNbC

NbN

Nb3Sn

V3Si Nb3Ge

Nb-Al-Ge

0

20

-273℃

「BCSの壁」

年1910 1930 1950 1970 19900 273℃

1986年

高温超伝導の発見

物質科学における最大の発見のひとつ

1987ノーベル物理科学賞“Possible High Tc Superconductivity

in the Ba-La-Cu-O System”

Bednorz Müller

酸化物高温超伝導体の発見

J.G.Bednorz and K.A.Muller, Z.Physik B64,189 (1986)

液体窒素温度を超える高温超伝導体

金属の超伝導を解明したBCS理論

電圧計

電流源金属の超伝導を解明したBCS理論では、高温超伝導を説明できない。

それに代わる理論も、発見後、25年経った現在も、決定的なものは現れていなかった。

銅酸化物 YB C O銅酸化物 YBa2Cu3O7‐y

高温超伝導物質の構造と物性

Hg系 (高温超伝導の世界記録)Tc~ 130K

T 90KTc~ 90K

Tc~ 30K

Why T is so high？

200

Why Tc is so high？

200

Hg-Ba-Ca-Cu-O

High-Tc cupratemetalIron-based system

K) Superconductivity was

1911

163at a suitable pressure

150

Hg Ba Ca Cu O（ ）

Hg-Ba-Ca-Cu-O

Tl-Ba-Ca-Cu-Oature

(K Superconductivity was

discovered163

100

a Ca Cu O

Bi-Sr-Ca-Cu-O

Y-Ba-Cu-Otem

pera 1986

High-Tc cuprate was discovered77

50SmO

0.9F

0.11FeAs

nsi

tion t discovered

2006

nitrogen temperature77

0

LaO0.89

F0.11

FeAs

LaOFeP

MgB2

NbGeNbNNbC

NbPb

Tra

Hg

La-Ba-Cu-O Iron-based system was discovered

01900 1920 1940 1960 1980 2000 2020

Year

g

Quasi-particle DOS in SC states wave with isotropic gaps-wave with isotropic gap

Ns(E)=N0E/(E2 - 2)1/2

p-wave with point–node gap

p-wave with line–node gap

(point node)

(li d )(line nodes)

Spin susceptibilitySpin polarization in superconducting phase

Spin singlet pairing:• breaking up of Cooper pairs• decrease of spin susceptibility• vanishing susceptibility at T=0

p

TTc

Spin triplet pairing:

• polarization without pair breaking

• no reduction of spin susceptibilityfor equal-spin pairing

p

const. ford (k )

H 0

TTc

Spin susceptibilitySpin polarization in superconducting phase

Spin triplet pairing: 17O K i ht hift I S R OSpin triplet pairing:

• polarization without pair breaking• no reduction of spin susceptibility

for equal spin pairing

17O-Knight shift In Sr2RuO4

Hcfor equal-spin pairing

const. ford (k )

H 0

H c

measured( )

Yosida-behaviorspin-singlet17O-Knight shift

measured

g(High-Tc oxides)

Ishida et al., Nature 396, 242 (1998)

inplane equal-spin pairing d || ˆ z

High-Tc SC : 63Cu-NMRSummary1 : Knight shift

ds-wave

d-wave

Ks ~ T

(line nodes)

s

27Al Knight shift p-wave

Heavy-Fermion SC

UPt 195Pt NMRUPt3 : 195Pt-NMR

Summary2 : pairing state in unconventional superconductors

Spin-singlet Spin-triplet

CeCu2Si2UPt3

Spin-singlet Spin-triplet

2 2

UPd Al UNi AlUPd2Al3UNi2Al3

CeCoIn5Sr2RuO4

Line-nodes gap SC in correlated electrons SC

p-wave

AFM-SC

UPd2Al31/T ∝T3 !!

d-wave

s-wave2 31/T1 ∝T !!

Strong-coupling effect

Possible SC order parameters and their spin-state

BCS SC High-T oxides UPtBCS SC High-Tc oxidesCeCu2Si2 UPd2Al3, CeRIn5

UPt3 Sr2RuO4

Spin-fluctuations mediated d-wave superconductivity

When the eigen energy in the eq. (11.12) has a solution for E < 2F , two- electron bounded state (Cooper pair) is formed. P id d h i i d f llProvided that V (r1, r2) is approximated as follows;

otherwiseHere, if the constant attractive interaction V is assumed to be effective only among electrons within energies in the range between the Fermi energy F and the Debye energy which is the highest one of lattice vibration, we obtain the following equation;

When taking , form we have

Since , we obtain the following relation

A general SC gap equation

d-wavesuperconductivity due to the on site Coulombto the on-site Coulomb

repulsive interaction

Possible SC order parameters and their spin-state

BCS SC High-T oxides UPtBCS SC High-Tc oxidesCeCu2Si2 UPd2Al3, CeRIn5

UPt3 Sr2RuO4

NMR/NQR probes of emergent properties in p g p pcorrelated-electron superconductors

・ Symmetry of the Cooper pair of either spin-singlet or spin-triplet

・ SC gap with either isotropic or nodal structure

・ Characters of spin fluctuationsCharacters of spin fluctuations