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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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