msci presentation
TRANSCRIPT
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
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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
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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
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electron doping
∙ Doped electrons fill up the Cu shells: Cu2+ → Cu+
∙ Spins start to disappear∙ Anti-ferromagnetic coupling gets diluted, eventually disappear
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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
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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)
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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
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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)
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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)
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modelling our system
1d hubbard model
∙ 1D Hubbard model as a linear chain of atoms:
∙ Keep system in ground state configuration∙ Spin degeneracy
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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
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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)
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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:
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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
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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
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Questions?
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