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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