From transition metal oxides to cosmic strings (and how electronic structure computations can help) Nicola Spaldin Materials Theory Department of Materials, ETH Zürich
Important things that happened as the universe cooled Today Low Time Temperature Very long time ago High
Important things that happened as the universe cooled Today Time Very expensive } ß Hadron Colliders ß Cosmic Microwave Background Very long time ago
Important things that happened as the universe cooled Today Low Time Temperature Inaccessible? } Very long time ago High
Important things that happened as the universe cooled Today Low Time Inaccessible? } Cosmologists are often wrong but never in doubt Lev Landau Temperature Very long time ago High
Transition metal oxides Compounds containing transition metals (d electrons) and oxygen (p electrons) Many interactions at similar energy scale Coupled and competing interactions! Energy scale ~ 1eV Thursday, September 20, 2012 D-MATL / Materials Theory 6
Where do transition metal oxides fit in the universe? MATERIALS THEORY Today ß Complex Oxides Low Time Temperature Very long time ago High
We can use transition metal oxides to study the GUT MATERIALS THEORY Today ß Complex Oxides Low Time Temperature Inaccessible? } Very long time ago High
MATERIALS THEORY We will identify a transition metal oxide with a phase transition described by the same mathematics as proposed for the GUT spontaneous symmetry breaking described by a non-trivial homotopy group Results in formation of topological defects (Kibble) P63/mmc Early Universe TM oxide YMnO3 10 µm 1 µm 0 2 4 6 8 10 P6 cm Relative PFM contrast (%) 3 SU(3)xSU(2)xU(1) Then we will use the TM oxide to test in the lab the scaling laws (number of defects as a function of cooling rate) proposed for the GUT! Kibble-Zurek scenario Thursday, September 20, 2012 D-MATL / Materials Theory 9
Plan Show how first-principles electronic structure calculations using supercomputers have helped with understanding the structural phase transition in YMnO 3 In particular, our computations have allowed us to show that the phase transition in YMnO 3 results in formally topologically protected defects (Kibble symmetry requirement) Test whether the defects follow so-called Kibble-Zurek scaling laws which are predicted for early-universe-like phase transitions Thursday, September 20, 2012 D-MATL / Materials Theory 10
Difficulties: Fast things are relativistic Many-body interactions lead to computational complexity MATERIALS THEORY How do we calculate the structural and electronic properties of transition metal oxides? Schrödinger Equation OR: (-h 2 /2m)d 2 ψ/dx 2 + V(x)ψ(x) = E ψ(x) H ψ = E ψ H Ψ(r1,r2 )= E Ψ(r1,r2 ) Pauli principle: Total Ψ must be antisymmetric with respect to the interchange of any two electrons CORRELATIONS
Our tool: Density Functional Theory interacting many-electron system Kohn-Sham Equations {T+V ei (r)+v H (r)+v xc (r)} φ i (r) = ε i φ i (r) system of non-interacting electrons V xc (r) incorporates the beyond-mean-field correlations; standard approximations inadequate for magnetic insulators Allows us in principle to calculate all ground state properties: charge densities and energies; crystal structures; magnetic ordering; phonon frequencies; ferroelectric polarizations; dielectric/piezoelectric response; magnetoelectric coupling with various tricks (finite E or H fields, linear response, etc.) AT ZERO KELVIN!
What s known about the structure of YMnO 3 from experiments? High temperature P6 3 /mmc Low temperature P6 3 cm How do we get between the high temperature and low temperature structures? Polarization (5.6 µc/cm 2 ) paraelectric ferroelectric Thursday, September 20, 2012 D-MATL / Materials Theory 13
Calculate phonons for high symmetry structure We find that 2 soft phonon modes take us to the low symmetry structure MATERIALS THEORY Trimerization (K 3 ) Y displacements (Γ 2 ) Thursday, September 20, 2012 D-MATL / Materials Theory 14
We can then map out the energy surface for these phonons DFT calculations show us that the ferroelectricity is improper: The trimerization (K3) mode is the energy-lowering primary order parameter C.J. Fennie and K.M. Rabe, Ferroelectric phase transition in YMnO3 from first principles Phys. Rev. B 72, 100103 (2005) Thursday, September 20, 2012 D-MATL / Materials Theory 15
We can see how the polarization evolves with the primary order parameter MATERIALS THEORY P z ~ Q K3 3 P z ~ Q K3 C.J. Fennie and K.M. Rabe, Ferroelectric phase transition in YMnO3 from first principles Phys. Rev. B 72, 100103 (2005) Thursday, September 20, 2012 D-MATL / Materials Theory 16
We can construct model Hamiltonians using parameters derived from DFT e.g. here is the Landau free energy S. Artyukhin, K.T. Delaney, NAS and M. Mostovoy, Landau theory of topological defects in multiferroic hexagonal manganites, arxiv:1204.4126 Using DFT we can extract a, b, c, c, g, g and χ 0 Then we can derive the potential for the phase transition: And explain the unusual domain structure: 1µm 0 2 4 6 8 10 Relative PFM contrast (%) Thursday, September 20, 2012 D-MATL / Materials Theory 17
Combining the result of our electronic structure calculations with those from homotopy theory we find that the vortex cores in the YMnO 3 domain structure are formally topologically protected defects that should obey Kibble-Zurek scaling S. Griffin, M. Lilienblum, K. Delaney, Y. Kumagai, M. Fiebig and N. A. Spaldin, From multiferroics to cosmology: Scaling behaviour and beyond in the hexagonal manganites, arxiv:1204.3785 (2012) Also, experimentally: 60 nm 100 u.c. 2 µm V 2µm So now let s use YMnO 3 to test Kibble-Zurek scaling laws! Thursday, September 20, 2012 D-MATL / Materials Theory 18
What is the physics of a Kibble-Zurek phase transition? Kibble-Zurek scenario: Predicts how many defects should form at a GUT-like transition defects form when they meet isolated low-symmetry regions nucleate Depends on time spent at the transition temperature Cool slowly: Different regions can communicate their choice of phase à Large regions of the same choice à Low density of defects Cool quickly: Not much time to communicate choice of phase à Many smaller regions with different choice of phase à High density of defects
A bit more rigorously Thursday, September 20, 2012 D-MATL / Materials Theory 20
Quantitatively, Kibble Zurek theory predicts: domain size as a function of cooling rate, defined as Tc (1400K) / cooling rate Critical exponents: ν = 0.6717 µ=1.3132 from MC simulations for 2D XY model M. Campostrini et al., Phys. Rev. B 74, 144506 (2006) zero-temperature correlation length ~ domain wall width in ferroelectrics zero-temperature relaxation time = ξ 0 / speed of sound speed of sound = 640 m/s (DFT) Thursday, September 20, 2012 D-MATL / Materials Theory 21
MATERIALS THEORY DFT calculations of domain walls in YMnO3 β+ α- Yu Kumagai and NAS, arxiv:1207.4080 Thursday, September 20, 2012 Domain wall width effectively zero! D-MATL / Materials Theory 22
Comparison of KZ theory (DFT parameters) with experiment Red line: our calculations with ξ 0 = 0.06 A Thursday, September 20, 2012 D-MATL / Materials Theory 23
Comparison of KZ theory (DFT parameters) with experiment Red line: our calculations with ξ 0 = 0.06 A Red circles: measured by Chae et al. S. C.. Chae et al., Direct observation of the proliferation of ferroelectric loop domains and vortexantivortex pairs, PRL 108, 167603 (2012) Blue circles: measured by Fiebig et al. REMARKABLE AGREEMENT! Thursday, September 20, 2012 D-MATL / Materials Theory 24
Comparison of KZ theory (DFT parameters) with experiment Red line: our calculations with ξ 0 = 0.06 A Red circles: measured by Chae et al. S. C.. Chae et al., Direct observation of the proliferation of ferroelectric loop domains and vortexantivortex pairs, PRL 108, 167603 (2012) Blue circles: measured by Fiebig et al. REMARKABLE AGREEMENT AT SLOW COOLING RATES! Thursday, September 20, 2012 D-MATL / Materials Theory 25
MATERIALS THEORY Open questions: Is this beautiful Kibble-Zurek behavior too good to be true? What is the origin of the turnaround? S. Griffin, M. Lilienblum, K. Delaney, Y. Kumagai, M. Fiebig and N. A. Spaldin, From multiferroics to cosmology: Scaling behaviour and beyond in the hexagonal manganites, arxiv:1204.3785 (2012) Thursday, September 20, 2012 D-MATL / Materials Theory 26
MATERIALS THEORY The Materials Theory group at ETH Yu Kumagai Sinead Griffin Kris Delaney, UC Santa Barbara The Multifunctional Ferroic Materials group at ETH Manfred Fiebig and Martin Lilienblum Thursday, September 20, 2012 D-MATL / Materials Theory 27
Summary YMnO 3 displays rich physics Electronic structure calculations on powerful supercomputers are helpful in unraveling some of this physics YMnO 3 seems to provide the first example of Kibble-Zurek scaling in a condensed matter system Use of condensed matter systems to explore questions in other areas of physics is a lot of fun K. Rushchanskii, NAS et al., A multiferroic material to search for the permanent electric dipole moment of the electron, Nature Materials 9, 649 (2010) Thursday, September 20, 2012 D-MATL / Materials Theory 28