Quick and Dirty Introduction to Mott Insulators

1 Quick and Dirty Introduction to mott InsulatorsBranislav K. Nikoli Department of Physics and Astronomy, University of Delaware, 624: Introduction to Solid State Physics ~bnikolic/teaching/phys624 624: Quick and Dirty Introduction to mott InsulatorsWeakly correlated electron liquid: Coulomb interaction effects( )( )( )FneDU =rrassume:( )( ,0)()FFe UfT = r When local perturbation potential is switched on, some electrons will leave this region in order to ensure constant (chemical potential is a thermodynamic potential; therefore, in equilibrium it must be homogeneous throughout the crystal).( )U rF PHYS 624: Quick and Dirty Introduction to mott InsulatorsThomas Fermi Screening Except in the immediate vicinity of the perturbation charge, assume that is caused by the induced space charge Poisson equation:20( )( )e nU = rr/2220201( )()in vacuum: () 0,( )4 TFr rTFFFerUrrrrre DqDU = = ====rr()()()21/ 32/ 32/ 31/ 3222222031 24( )3,33222 FFTFFnmnDnnrma === = 1/ 620032023341, 10, TFCuCuTFnraamencmr = = = ( )U rPHYS 624: Quick and Dirty Introduction to mott InsulatorsMott Metal6 Insulator Transition Below the critical electron concentration, the potential well of the screened fieldextends far enough for a bound state to be formed screening length increases so that free electrons become localized mott Insulators Examples.

2 Transition metal oxides, glasses, amorphous semiconductors22001/ 31/ 30144 TFaranna PHYS 624: Quick and Dirty Introduction to mott InsulatorsMetal vs. Insulator: Theory00 | |0lim lim lim R e( ,)0T = qq Theoretical Definition of an Insulator: Theoretical Definition of a Metal:Ohm law :( , )( , )( , )jE = qqq()2 2Re(0,0)(1)cTD = = +()()21*Drude:, Re(0,0,0)( )ccneDTDm == = PHYS 624: Quick and Dirty Introduction to mott InsulatorsMetal vs. Insulator: ExperimentT T Fundamental requirements for electron transport in Fermi systems:1) Quantum6mechanical states for electron6hole excitations must be available at energies immediately above the ground state (no gap!) since the external field provides vanishingly small ) These excitations must describe delocalized charges (no wave function localization!

3 That can contribute to transport over the macroscopic sample 624: Quick and Dirty Introduction to mott InsulatorsSingle6 Particle vs. Many6 Body InsulatorsInsulators due to electron-ion interaction (single-particle physics): Band Insulators (electron interacts with a periodic potential of the ions gap in the single particle spectrum) Peierls Insulators (electron interacts with static lattice deformations gap) Anderson Insulators (electron interacts with the disorder=such as impurities and lattice imperfections) mott Insulators due to electron-electron interaction (many-body physics leads to the gap in the charge excitation spectrum): Mott6 Heisenberg (antiferromagnetic order of the pre6formed local magnetic moments below N el temperature) Mott6 Hubbard (no long6range order of local magnetic moments) Mott6 Anderson (disorder + correlations) Wigner Crystal (Coulomb interaction dominates at low density of charge, rs(2D)=Ee6e/EF=ns1/2/ns=33 or rs(3D)=67, thereby localizing electrons into a Wigner lattice)PHYS 624: Quick and Dirty Introduction to mott InsulatorsEnergy Band TheoryElectron in a periodic potential (crystal) energy band ( : 16D tight6binding band)N= 1N= 2N= 4N= 8N= 16N= EFkinetic energy gain( )2 cos()ktka = PHYS 624: Quick and Dirty Introduction to mott InsulatorsBand (Bloch6 Wilson) InsulatorWilson s rule 1931: partially filled energy band metalotherwise insulatormetalinsulatorsemimetalCounter example.

4 Transition6metal oxides, halides, chalcogenidesFe: metal with 3d6(4sp)2 FeO: insulator with 3d6 PHYS 624: Quick and Dirty Introduction to mott InsulatorsAnderson Insulator Ht =+ mmnmm,nm mm nW=B disorder:,22W W mPHYS 624: Quick and Dirty Introduction to mott InsulatorsMetal6 Insulator TransitionsFrom weakly correlated Fermi liquid to strongly correlated mott insulatorsnc2ncnSTRONG CORRELATIONWEAK CORRELATIONINSULATORSTRANGE METALF. L. METALMott Insulator:A solid in which strong repulsionbetween 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 624: Quick and Dirty Introduction to mott InsulatorsMott Gedanken Experiment (1949)energy cost Uelectron transfer integral ttCompetition between W(=2zt) and U Metal-Insulator : V2O3, Ni(S,Se)2datomic distanced (atomic limit: no kinetic energy gain): insulatord 0 : possible metal as seen in alkali metalsPHYS 624: Quick and Dirty Introduction to mott InsulatorsMott vs.

5 Bloch6 Wilson Insulators Band insulator, including familiar semiconductors, is state produced by a subtle quantum interference effects which arise from the fact thatelectrons 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 Bloch6 Wilson Insulators ) occur in crystalline systems at isolated occupation numbers where is the number of particles per unit cell. Although the physical origin of a mott insulator is understandable to any child, other properties, especially the response to doping are only partially understood. mott state, in addition to being insulating, can be characterized by: presence or absence of spontaneously broken symmetry( , spin antiferromagnetism); gapped or gapless low energy neutralparticle excitations; and presence or absence of topological order and charge fractionalization.

6 * =* * PHYS 624: Quick and Dirty Introduction to mott InsulatorsTrend in the Periodic TableU U PHYS 624: Quick and Dirty Introduction to mott InsulatorsTheoretical modeling: Hubbard HamiltonianHubbard Hamiltonian 1960s: on-site Coulomb interaction is most dominant Hubbard s solution by the Green s function decoupling method insulator for all finite Uvalue Lieb and Wu s exact solution for the groundstate of the 16D Hubbard model (PRL 68) insulator for all finite : U~ 5 eV, W~ 3 eV for most 3dtransition6metal oxide such asMnO, FeO, CoO, NiO : mott insulatorband structurecorrelationPHYS 624: Quick and Dirty Introduction to mott InsulatorsSolving Hubbard Model in Dimensions PHYS 624: Quick and Dirty Introduction to mott InsulatorsDynamical Mean6 Field Theory in Pictures In 6D, spatial fluctuation can be neglected. mean6field solution becomes exact.

7 Hubbard model single6impurity Anderson model in a mean6field bath. Solve exactly in the time domain dynamical mean6field theoryDynamical mean-field theory (DMFT)of correlated6electron 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 spin6up, one with spin6down, and a doubly occupied state with opposite spins. In this illustration of one possible sequence involving two transitions, an atom in an empty state absorbs an electron from the surrounding reservoir in each transition.

8 The hybridization V is the quantum mechanical amplitude that specifies how likely a state flips between two different configurations. PHYS 624: Quick and Dirty Introduction to mott InsulatorsStatic vs. Dynamic Mean6 Field Theory Static = Hartree6 Fock or Density Functional Theory: Dynamic = Dynamical Mean6 Field Theory:3233[ ( )][ ( )]( ) ( )( )( )( )1( ) ( )2[ ( )]2||kineticextKSiexchangeEVdVmd dE =+ + = ++ rrrrrrrrrrr rrr r 23[ ( )]( )[ ( )]( ), ( )( ) | ( ) |||( )exchangeKSextiiiEVVdf =++= rrrrrrrr rr[]3133[ ( ), ][ ( ), ]( ) ( )[ ( )][ ( )]1( ) ( )[ ( ), ][ ( )]( ) 1 / [ ( )]2||kineticextexchangeGEGVdGtd dEGG =+ = ++ + kkrrrrrrrr rrr rPHYS 624: Quick and Dirty Introduction to mott InsulatorsTransition from non6 Fermi Liquid Metal to mott InsulatorModel:Mobile spin6 electrons interact with frozen spin6.

9 DOS well6defined even though there are no fermionic 624: Quick and Dirty Introduction to mott InsulatorsExperiment: Photoemission Spectroscopyh (K, ) > We-(Ek,k, )N-particle(N 1)-particleP(| i | f )Sudden approximationEinstein s photoelectric effectPhotoemission current is given by:EiNEfN 1 +> <= fiNiNfrTkEEEiTfeZABNi,12/)(||1)( PHYS 624: Quick and Dirty Introduction to mott InsulatorsMott Insulating Material: V2O3 a= c= (1012) cleavage planeVanadiumOxygensurface-layer thickness =side top viewPHYS 624: Quick and Dirty Introduction to mott InsulatorsTheory vs. Experiment:Photoemission SpectroscopyPhotoemission spectrumof metallic vanadium oxide V2O3near the metal insulatortransition. The dynamical mean6field 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.

10 Higher6energy 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 InsulatorsPhase Diagram of V2O3 PHYS 624: Quick and Dirty Introduction to mott InsulatorsWigner CrystalSince the mid61930s, theorists have predicted the crystallization of electrons. If a small number of electrons are restricted to a plane, put into a liquid6like state, and squeezed, they arrange themselves into the lowest energy configuration possible66a series of concentric rings. Each electron inhabits only a small region of a ring, and this bull's6eye 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.