Electron motion in high temperature superconductors, MSci project

1. high temperature superconductivity
MSci Project
Benjamin Horvath
2 March 2015
The University of Birmingham
The School of Physics and Astronomy

2. overview
Structure and phase diagram
Finding the Hamiltonian
Modelling our system
1

3. structure and phase diagram

4. 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 content
and dope
3

5. 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

6. electron doping
∙ Doped electrons fill up the Cu shells: Cu2+
→ Cu+
∙ Spins start to disappear
∙ Anti-ferromagnetic coupling gets diluted, eventually disappear
5

7. hole doping
∙ A basic energy diagram: Disturbed AFM lattice:
∙ Oxygen sites take on holes
∙ As they move around in the lattice, anti-ferromagnetism is
quickly destroyed
6

8. finding the hamiltonian

9. 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 + V2
H2
∙ One hop → Moving away from ground state
∙ Two hops → Possible return to ground state
∙ Need to eliminate terms of O(V)
8

10. second quantisation & canonical transformation
∙ Propose Hamiltonian:
H0 = −∆
∑
iσ
d†
iσdiσ + U
∑
i
d†
iσdiσd†
i¯σdi¯σ
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

11. zhang-rice singlet
∙ Once H2 is found, restrict it to the ground state
∙ We find:
H2 =
V2
∆
∑
⟨ij⟩σ
∑
⟨im⟩
{
(
p†
jσpmσ
)
+
U
2(∆ − U)
((
d†
iσp†
j¯σ−d†
i¯σp†
jσ
)(
pm¯σdiσ−pmσdi¯σ
))
}
∙ Singlet term is called the Zhang-Rice singlet
F.C. Zhang & T.M. Rice(1988)
10

12. 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

13. modelling our system

14. 1d hubbard model
∙ 1D Hubbard model as a linear chain of atoms:
∙ Keep system in ground state configuration
∙ Spin degeneracy
13

15. 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

16. 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

17. hole doping with u≫ ∆ in 1d
∙ Currently working on the U≫ ∆ limit
∙ Oxygen hole is incorporated into AFM arrangement → destroys
long range AFM ordering
∙ Apply Hamiltonian to get:
16

18. 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 linear
motion and loop motion
∙ Hopping in the lattice described by both of these types of
motion
∙ In the limit U ≫ ∆ only the 1D case was considered
∙ Hubbard model and U ≈ ∆ limit are similar and cannot deduce
difference in the phase diagram
∙ The U≫ ∆ limit is completely different from former two and
could cause the asymmetry
17

19. next steps
∙ Turn the Hamiltonian into a pure spin problem
∙ Recognise that the Hamiltonian is related to the Heisenberg
model:
H2 = −J
∑
i,j
⃗Si · ⃗Sj
∙ Find the lowest energy state of U≫ ∆ model
18

20. Questions?
19