高温超导理论研究总结

2 ξ2 k + ∆k
• gap ∆k ≡ ∆ (s-wave)
[Edegger, Muthukumar, Gros ‘06] 4
d -wave superconductivity
gap-function ∆k = ∆(cos kx − cos ky )
2
- 1.5
N(ω )
⊲ nodes along (1, 1) direction
i U iS −iS i2 2 [S, [S, H ]]
Td U
≈ H + i[S , H ] +
+ ...
W/2 0 −W/2
T+
T−
diagonal in double occupancy ⇒ T+ − T− + O(t 2 /U 2 )
Te
projected Hilbert-space Ht −J = He f f + O(t 3 /U 2 ) = Te + Td + U ∑i ni,↑ ni,↓ + J ∑<i, j> Si · S j J = 4t 2 /U
= 4.3
7
doped Mott-Hubbard insulators
H = −t
<i, j>,σ
∑
† c + c c† jσ ciσ + U ∑ ni,↑ ni,↓ iσ jσ i
strong correlation U ≫ t : reduced double occupancy † c† has energy U c i↑ i↓ |0 low-energy states c† singly-occupied iσ |0 |0 empty sites low-energy Hamiltonian canonical transformation Ht −J ≈ He f f = eiS H e−iS He f f block-diagonal in double-occupancy
novelty superconductivity in a doped oxide (bad metal) doped Mott-Hubbard insulator do AF and SC compete or cooperate?
3
BCS superconductivity
Bardeen-Cooper-Schrieffer ⊲ standard theory for ‘conventional’ superconductors. ⊲ superconductivity = ˆ pair condensation ξk
upper Hubbard band U
W/2 0 −W/2 lower Hubbard band
8
canonical transformation
kinetic energy terms T = Te + Td + T+ + T− H = Te + Td + U ∑i ni,↑ ni,↓ + T+ + T− perturbation expansion He f f = e H e S =
nitrogen
O LaSrCuO O LaBaCuO O Nb3Ga O Nb3S
Year
1986: Discovery by of high-temperature superconductivity by Bednorz & Muller ¨
2
high-temperature superconductors
2 ξ2 + ∆ k k
|ΨBCS =
∏
k
† uk + vk c† c k ↑ −k ↓
|0
h ¯ 2 k2 2m
2 u2 / v k k
⊲ one-particle dispersion ξk =
Bogoliubov quasiparticles
1 1± = 2 ∆=0

−µ ∆>0
• dispersion Ek =
+ +
+ +
1
0.5
[Fang. et. al. (2005)]
- 0 0 0.5
ω /∆
1
1.5
2
• predicted by theory
⊲ correlation-induced superconductivity
• experimental verification
⊲ phase-sensitive interference ⊲ tunneling, ARPES
?? 140 120 100 80 60 40 Liquid 20 0 hydrogen O Pb O Hg 1910 O NbC O Nb 1930 1950
111111111111111111111111111111111111111 000000000000000000000000000000000000000 111111111111111111111111111111111111111 000000000000000000000000000000000000000 111111111111111111111111111111111111111 000000000000000000000000000000000000000 111111111111111111111111111111111111111 000000000000000000000000000000000000000 O HgCaBaCuO 111111111111111111111111111111111111111 000000000000000000000000000000000000000 111111111111111111111111111111111111111 000000000000000000000000000000000000000 111111111111111111111111111111111111111 000000000000000000000000000000000000000 O TlCaBaCuO 111111111111111111111111111111111111111 000000000000000000000000000000000000000 111111111111111111111111111111111111111 000000000000000000000000000000000000000 111111111111111111111111111111111111111 000000000000000000000000000000000000000 111111111111111111111111111111111111111 000000000000000000000000000000000000000 111111111111111111111111111111111111111 000000000000000000000000000000000000000 O BiCaSrCuO 111111111111111111111111111111111111111 000000000000000000000000000000000000000 111111111111111111111111111111111111111 000000000000000000000000000000000000000 111111111111111111111111111111111111111 000000000000000000000000000000000000000 111111111111111111111111111111111111111 000000000000000000000000000000000000000 O YBaCuO 111111111111111111111111111111111111111 000000000000000000000000000000000000000 111111111111111111111111111111111111111 000000000000000000000000000000000000000 111111111111111111111111111111111111111 000000000000000000000000000000000000000 111111111111111111111111111111111111111 000000000000000000000000000000000000000 111111111111111111111111111111111111111 000000000000000000000000000000000000000 111111111111111111111111111111111111111 000000000000000000000000000000000000000 111111111111111111111111111111111111111 000000000000000000000000000000000000000 Liquid 111111111111111111111111111111111111111 000000000000000000000000000000000000000 111111111111111111111111111111111111111 000000000000000000000000000000000000000
K 277
111111111111111111111111111111111111111 000000000000000000000000000000000000000 111111111111111111111111111111111111111 000000000000000000000000000000000000000 111111111111111111111111111111111111111 000000000000000000000000000000000000000 Room temperature 111111111111111111111111111111111111111 000000000000000000000000000000000000000 111111111111111111111111111111111111111 000000000000000000000000000000000000000 111111111111111111111111111111111111111 000000000000000000000000000000000000000 111111111111111111111111111111111111111 000000000000000000000000000000000000000
Physics of Doped Mott-Hubbard Insulators and the High-Temperature Superconductors
Claudius Gros
J.-W. Goethe University Frankfurt
1
transition temperatures
⊲ transport, ARPES ⊲ pseudo ∆k ≈ ∆(cos kx − cos ky )
0 kx 1
spectral gap
ky
0
45 Fermi surface angle
90
0
• Fermi arc
6
BCS-ratio
• universal for weak-coupling
2∆ = kB Tc 3.52 s-wave 4.3 d-wave
5
key issue: the pseudogap
what happens for T > Tc ?
T
ps
T*(x)
eu do ga p
∆(Τ)
∆ ∆ sc
Tc(x) AF d−wave SC
doping x
Τc T*
1
SC PG NS PG SC
• one-particle excitations
[Kugler, Fischer, Renner, Ono, Ando ‘01]
8 7
STM
6
T*/TC
5 4 3 2 1 0 0 5 10 15 20 25 30 35
2 ∆ p/k T*= 4. 3
B
2∆ p/ kBTC
• high-temperature superconductors
2∆ kB T ∗