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high temperature superconductivityMSci Project
Benjamin Horvath2 March 2015
The University of BirminghamThe School of Physics and Astronomy
overview
Structure and phase diagram
Finding the Hamiltonian
Modelling our system
1
structure and phase diagram
crystalline structure
∙ Cuprate superconductors have highest known Tc (138K)∙ Layered structure:
S. Tanaka (2006)
∙ Superconductivity confined within the CuO2 layers∙ Neighbouring layers stabilise structure, increase oxygen contentand dope
3
phase diagram
∙ C. Chen (2006)
∙ The parent compound, La3+2 Cu2+O2−4 is anti-ferromagnetic
∙ AFM region reduces more rapidly on the hole doped side∙ SC region is much wider on the hole doped side
4
electron doping
∙ Doped electrons fill up the Cu shells: Cu2+ → Cu+
∙ Spins start to disappear∙ Anti-ferromagnetic coupling gets diluted, eventually disappear
5
hole doping
∙ A basic energy diagram: Disturbed AFM lattice:
∙ Oxygen sites take on holes∙ As they move around in the lattice, anti-ferromagnetism isquickly destroyed
6
finding the hamiltonian
degenerate perturbation theory
∙ A large number of possible superconducting ground states
V.J. Emery (1987)
∙ Use degenerate perturbation theory:
H = H0 +H1 +H2 = H0 + VH1 + V2H2
∙ One hop → Moving away from ground state∙ Two hops → Possible return to ground state∙ Need to eliminate terms of O(V)
8
second quantisation & canonical transformation
∙ Propose Hamiltonian:
H0 = −∆∑iσ
d†iσdiσ + U
∑i
d†iσdiσd
†iσ̄diσ̄
H1 = V∑⟨ij⟩σ
(d†iσpjσ + p†
jσdiσ
)∙ Eliminate O(V) by transformation into a new basis and find H2
∙ Rotation in Hilbert space |ψ⟩ → eS |ψ⟩, S to be determined
9
zhang-rice singlet
∙ Once H2 is found, restrict it to the ground state∙ We find:
H2 =V2
∆
∑⟨ij⟩σ
∑⟨im⟩
{(p†jσpmσ
)+
U2(∆− U)
((d†iσp
†jσ̄−d†
iσ̄p†jσ)(pmσ̄diσ−pmσdiσ̄
))}
∙ Singlet term is called the Zhang-Rice singlet
F.C. Zhang & T.M. Rice(1988)
10
hubbard model
∙ Let us now consider H for electron doping∙ There are no holes on px and py shells of the oxygen∙ Allows greater simplification of H2
∙ We find: H2 = − V2∆
∑⟨il⟩σ
d†iσdlσ
P.A. Lee (2006)
11
modelling our system
1d hubbard model
∙ 1D Hubbard model as a linear chain of atoms:
∙ Keep system in ground state configuration∙ Spin degeneracy
13
hole doping with u ≈ ∆ in 1d
∙ 1D linear chain representation:
∙ Oxygen sites with holes → singlet formation∙ Applying H2 to state |n⟩ we find:
H2 |n⟩ = − UV2∆(∆− U)
(4 |n⟩ − |n+ 1⟩ − |n− 1⟩
)∙ Singlet hopping → spin degeneracy
14
hole doping with u ≈ ∆ in 2d
∙ Consider a triangular closed loop
∙ Spins get permuted by passing hole∙ Full cycle in 6 hops → Z is 6th roots of unity∙ Z3 = ±1∙ |ψ1 ⟩, |ψ2 ⟩ & |ψ3⟩ are either singlets or triplets∙ We find Z = 1 in G.S. → triplet → ferromagnetic G.S.∙ Nagaoka’s Theorem (1966)
15
hole doping with u≫ ∆ in 1d
∙ Currently working on the U≫ ∆ limit∙ Oxygen hole is incorporated into AFM arrangement → destroyslong range AFM ordering
∙ Apply Hamiltonian to get:
16
conclusion
∙ Goal was to explain the asymmetry of the phase diagram∙ Found the Hamiltonian of the ground state∙ Built models of linear chains and closed loops → isolate linearmotion and loop motion
∙ Hopping in the lattice described by both of these types ofmotion
∙ In the limit U ≫ ∆ only the 1D case was considered∙ Hubbard model and U ≈ ∆ limit are similar and cannot deducedifference in the phase diagram
∙ The U≫ ∆ limit is completely different from former two andcould cause the asymmetry
17
next steps
∙ Turn the Hamiltonian into a pure spin problem∙ Recognise that the Hamiltonian is related to the Heisenbergmodel:
H2 = −J∑i,j
S⃗i · S⃗j
∙ Find the lowest energy state of U≫ ∆ model
18
Questions?
19
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