Quick and Dirty Introduction to Mott Insulators Branislav K. Nikoli ć Department of Physics and Astronomy, University of Delaware, U.S.A.
PHYS 624: Introduction to Solid State Physics http://www.physics.udel.edu/~bnikolic/teaching/phys624/phys624.html Weakly correlated electron liquid: Coulomb interaction effects
When local perturbation δU(r ) potential is switched on, some electrons will leave this region in order to ensure µε ≃ constant (chemical potential is a F thermodynamic potential; therefore, in equilibrium it must be homogeneous throughout the crystal).
ε δδ = n()r eD ()F U () r
εδ ≪ assume:e U (r ) F θ ε ε ε → = − f( , T 0) (F )
PHYS 624: Quick and Dirty Introduction to Mott Insulators Thomas –Fermi Screening
Except in the immediate vicinity of the perturbation charge, assume that is causeδU(r ) d eδ n (r ) by the induced space charge → Poisson equation: ∇2δU(r ) = − ε − 0 1 ∂ ∂ αe r/ r TF ∇2 = r 2 ⇒ δ U (r ) = 2 −1/ 6 ∂ ∂ 2 rrr r 1 n 4π εℏ ≃ = 0 rTF , a 0 ε 2 a3 me 2 r = 0 0 TF 2 ε = ⋅23− 3 Cu = e D (F ) nCu8.510 cm , r TF 0.55Å q in vacuum: α D δ (ε )= 0, U (r ) = = F πε 4 0 2 1/3 312n m 2/3ℏ 2/3− 4 1/3 n π π ε π D()ε = =() 3,32 n = () 222 nr⇒ = () 3 π F π ε 2ℏ 2 F TF 2F 2 2 m a 0
PHYS 624: Quick and Dirty Introduction to Mott Insulators Mott Metal Insulator Transition
1 a r2≃0 ≫ a 2 TF 4 n1/3 0 −1/3 ≫ n4 a 0
Below the critical electron concentration, the potential well of the screened field extends far enough for a bound state to be formed → screening length increases so that free electrons become localized → Mott Insulators Examples: transition metal oxides, glasses, amorphous semiconductors
PHYS 624: Quick and Dirty Introduction to Mott Insulators Metal vs. Insulator: Theory
ω ω σ ω = Ohmlaw: β αβ jα (,)q∑ (,) q E (,) q β
Theoretical Definition of a Metal: τ Re ωσ αβ (T= 0, → 0) = () D c αβ ωπ τ(1+ 2 2 ) π 2 ne −1 Drude:()D δ ω= τ ω σ αβδ αβ ,Re ( T =→→= 0, 0, 0) () D () c αβ m* c αβ
Theoretical Definition of an Insulator:
limlimlimRe ωσ α β (q , ) = 0 T →0ω → 0||q → 0
PHYS 624: Quick and Dirty Introduction to Mott Insulators Metal vs. Insulator: Experiment
Fundamental requirements for electron transport in Fermi systems: 1) Quantum mechanical states for electron hole excitations must be available at energies immediately above the ground state ( no gap! ) since ρ the external field provides ρ vanishingly small energy. 2) These excitations must describe delocalized charges ( no wave function localization! ) that can contribute to transport over the macroscopic T T sample sizes.
PHYS 624: Quick and Dirty Introduction to Mott Insulators Single Particle vs. Many Body Insulators
Mott Insulators due to electron-electron interaction (many-body physics leads to the Insulators due to electron-ion gap in the charge excitation spectrum ): interaction ( single-particle physics ): Mott Heisenberg (antiferromagnetic order of Band Insulators (electron interacts the pre formed local magnetic moments below → with a periodic potential of the ions Néel temperature) gap in the single particle spectrum ) Mott Hubbard (no long range order of local Peierls Insulators (electron interacts magnetic moments) with static lattice deformations → gap ) Mott Anderson (disorder + correlations) Wigner Crystal ( Coulomb interaction dominates Anderson Insulators (electron at low density of charge , rs (2D)=Ee e/E F interacts with the disorder=such as 1/2 =n s /n s=33 or rs (3D)=67 , thereby localizing impurities and lattice imperfections) electrons into a Wigner lattice )
PHYS 624: Quick and Dirty Introduction to Mott Insulators Energy Band Theory
Electron in a periodic potential (crystal) → energy band ( ε ()k= − 2cos( t ka ) : 1 D tight binding band)
N = 1 N = 2 N = 4 N = 8 N = 16 N = ∞
EF
kinetic energy gain
PHYS 624: Quick and Dirty Introduction to Mott Insulators Band (Bloch Wilson) Insulator
Wilson’s rule 1931: partially filled energy band → metal otherwise → insulator metal insulator semimetal
Counter example: transition metal oxides, halides, chalcogenides Fe: metal with 3d 6(4sp) 2 FeO: insulator with 3d 6
PHYS 624: Quick and Dirty Introduction to Mott Insulators Anderson Insulator
ˆ =ε + ε ∈ − W W H∑mmm ∑ t mn mn disorder:m , m m,n 2 2
δ=W B
PHYS 624: Quick and Dirty Introduction to Mott Insulators Metal Insulator Transitions
Mott Insulator: A solid in which strong repulsion between the particles impedes their flow → simplest cartoon is a system with a classical ground state in which there is one particle on each site of a crystalline lattice and such a large repulsion between two particles on the same site that fluctuations involving the motion of a particle from one site to the next are suppressed.
From weakly correlated Fermi liquid to strongly correlated Mott insulators INSULATOR STRANGE METAL n 2n c F. L. METAL c n
STRONG CORRELATION WEAK CORRELATION
PHYS 624: Quick and Dirty Introduction to Mott Insulators Mott Gedanken Experiment (1949)
electron transfer integral t
energy cost U
d atomic distance d → ∞ (atomic limit: no kinetic energy gain): insulator d → 0 : possible metal as seen in alkali metals Competition between W(=2zt) and U → Metal-Insulator Transition
e.g.: V 2O3, Ni(S,Se) 2
PHYS 624: Quick and Dirty Introduction to Mott Insulators Mott vs. Bloch Wilson insulators
Band insulator, including familiar semiconductors, is state produced by a subtle quantum interference effects which arise from the fact that electrons are fermions . Nevertheless one generally accounts band insulators to be “simple” because the band theory of solids successfully accounts for their properties. Generally speaking, states with charge gaps (including both Mott and Bloch Wilson insulators) occur in crystalline systems at isolated “occupation numbers”* where is the number of particles per unit celν * l. ν ν= Although the physical origin of a Mott insulator is understandable to * any child, other properties, especially the response to doping ν ν δ→ − are only partially understood. Mott state, in addition to being insulating, can be characterized by: presence or absence of spontaneously broken symmetry (e.g., spin antiferromagnetism); gapped or gapless low energy neutral particle excitations; and presence or absence of topological order and charge fractionalization.
PHYS 624: Quick and Dirty Introduction to Mott Insulators Trend in the Periodic Table
U ↑ U ↓
PHYS 624: Quick and Dirty Introduction to Mott Insulators Theoretical modeling: Hubbard Hamiltonian
Hubbard Hamiltonian 1960s: on-site Coulomb interaction is most dominant
band structure correlation e.g.: U ~ 5 eV, W ~ 3 eV for most 3 d transition metal oxide such as MnO, FeO, CoO, NiO : Mott insulator
♠ Hubbard’s solution by the Green’s function decoupling method → insulator for all finite U value
♦ Lieb and Wu’s exact solution for the ground state of the 1 D Hubbard model (PRL 68) → insulator for all finite U value
PHYS 624: Quick and Dirty Introduction to Mott Insulators Solving Hubbard Model in Dimensions∞
PHYS 624: Quick and Dirty Introduction to Mott Insulators Dynamical Mean Field Theory in Pictures
Dynamical mean-field theory (DMFT) of correlated electron solids replaces the full lattice of atoms and electrons with a single impurity atom imagined to exist in a bath of electrons. The approximation captures the dynamics of electrons on a central atom (in orange) as it fluctuates among different atomic configurations, shown here as snapshots in time. In the simplest case of an s orbital occupying an atom, fluctuations ↑ ↓ ↓ ↑ could vary among |0 〉, | 〉, | 〉, or | 〉, which refer to an unoccupied state, a state with a single electron of spin up, one with spin down, and a doubly occupied state with In ∞ D, spatial fluctuation can be neglected. opposite spins. In this illustration of one → mean field solution becomes exact. possible sequence involving two transitions, Hubbard model → single impurity Anderson an atom in an empty state absorbs an electron from the surrounding reservoir in model in a mean field bath. each transition. The hybridization Vν is the Solve exactly in the time domain quantum mechanical amplitude that specifies → “dynamical ” mean field theory how likely a state flips between two different configurations.
PHYS 624: Quick and Dirty Introduction to Mott Insulators Static vs. Dynamic Mean Field Theory
Static = Hartree Fock or Density Functional Theory: ρ ρ Γ[()]ρ r =E [()] r + V ()() rrr d 3 kinetic∫ ext ℏ2 + Ψ=Ψε 1ρ ρ ()()r r ′ ⇒ VKS()r () r i () r +d3r d 3 r′ + E [ρ ( r )] 2m ∫ exchange 2 |r− r ′ | ρδ ρ(r′ ) E [ (r )] 2 VVd[()]ρ rrr=+ ()3 ′ +exchange ε ,()ρ r =Ψ f ()|()| r KS ext ∫ − ′ δρ ∑ i i ||r r () r i Dynamic = Dynamical Mean Field Theory:
3 ρ ρ Γ[(),]ρ rGE = [(),] r GV + ()() rrr d −1 kinetic∫ ext ∆ ω ω ω =[] −Σ∆ − G[()]∑ [()] t k 1ρ ρ ()()r r ′ ⇒ k +ddE3r 3 r′ + [ρ (), r G ] ∫ exchange Σ∆[()] ω ω ω ω ≡∆ () − 1/G [()] ∆ + 2 |r− r ′ |
PHYS 624: Quick and Dirty Introduction to Mott Insulators Transition from non Fermi Liquid Metal to Mott Insulator
NOTE: DOS well defined even though there are no fermionic quasiparticles. ↑ Model: Mobile spin electrons interact with ↓ frozen spin electrons.
PHYS 624: Quick and Dirty Introduction to Mott Insulators Experiment: Photoemission Spectroscopy
Einstein’s photoelectric effect hν (K,λ) > W - σσσ e (Ek,k, )
N-particle (N−1)-particle
Sudden approximation
N −1 N Ef Ei P(| i 〉 → | f 〉) Photoemission current is given by: 1 − N 2 − ω = Ei / kBT < > δ ω + N 1 − N A− ( ) ∑e f | Tr | i ( E f Ei ) Z i, f
PHYS 624: Quick and Dirty Introduction to Mott Insulators Mott Insulating Material: V 2O3
surface-layer thickness =
2.44Å – (1012) cleavage plane side view → c = 14.0 Å
→ a = 4.95 Å Vanadium
Oxygen top view
PHYS 624: Quick and Dirty Introduction to Mott Insulators Theory vs. Experiment: Photoemission Spectroscopy
Photoemission spectrum of
metallic vanadium oxide V 2O3 near the metal−insulator transition. The dynamical mean field theory calculation (solid curve) mimics the qualitative features of the experimental spectra. The theory resolves the sharp quasiparticle band adjacent to the Fermi level and the occupied Hubbard band, which accounts for the effect of localized d electrons in the lattice. Higher energy photons (used to create the blue spectrum) are less surface sensitive and can better resolve the quasiparticle peak.
Phys. Rev. Lett. 90 , 186403 (2003)
PHYS 624: Quick and Dirty Introduction to Mott Insulators Phase Diagram of V2O3
PHYS 624: Quick and Dirty Introduction to Mott Insulators Wigner Crystal
Since the mid 1930s, theorists have predicted the crystallization of electrons. If a small number of electrons are restricted to a plane, put into a liquid like state, and squeezed, they arrange themselves into the lowest energy configuration possible a series of concentric rings. Each electron inhabits only a small region of a ring, and this bull's eye pattern is called a Wigner crystal. Only a handful of difficult experiments have shown indirect evidence of this phenomenon → Electrons trapped on a free surface of liquid helium offer an excellent high mobility 2D electron system. Since the free surface of liquid He is extremely smooth, the mobility of electrons increases enormously at low temperatures.
PHYS 624: Quick and Dirty Introduction to Mott Insulators Beyond Solid State Physics: Bosonic Mott Insulators in Optical Lattices
EVOLUTION: Superfluid state with coherence, Mott Insulator without coherence , and superfluid state after restoring the coherence.
PHYS 624: Quick and Dirty Introduction to Mott Insulators