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