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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+O24 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 = i

    didi + Ui

    didididi

    H1 = Vij

    (dipj + p

    jdi

    ) Eliminate O(V) by transformation into a new basis and find H2 Rotation in Hilbert space | eS |, S to be determined

    9

  • zhang-rice singlet

    Once H2 is found, restrict it to the ground state We find:

    H2 =V2

    ij

    im

    {(pjpm

    )+

    U2( U)

    ((dip

    jd

    ip

    j)(pmdipmdi

    ))}

    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 = V

    2

    il

    didl

    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 |1 , |2 & |3 are either singlets or triplets We find Z = 1 in G.S. triplet ferromagnetic G.S. Nagaokas 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 = Ji,j

    Si Sj

    Find the lowest energy state of U model

    18

  • Questions?

    19

    Structure and phase diagramFinding the HamiltonianModelling our system

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