Metal-Mott insulator transitions

Transcription

1 University of Ljubljana Faculty of Mathematics and Physics Seminar I b Metal-Mott insulator transitions Author: Alen Horvat Advisor: doc. dr. Tomaž Rejec Co-advisor: dr. Jernej Mravlje Ljubljana, September 10, 2013 Abstract In this seminar, we discuss the metal-insulator tranisition that occurs as a consequence of electronic interactions. The basic experimental phenomenology and simple theoretical explanations are presented. We start with the band picture of metals and insulators. Next we present the Slater picture of magnetically ordered insulators and continue with the Mott theory of magnetically disordered insulators. In the end the Brinkman-Rice theory of metals and the dynamical mean-field theory are presented.

2 Seminar I b Metal-Mott insulator transition 2 CONTENTS I. Introduction 2 II. Theoretical pictures 3 A. Band picture 3 B. Slater picture 4 C. Mott picture 6 D. Brinkman-Rice picture 8 E. Dynamical mean-field theory (DMFT) 11 III. Conclusion 13 References 14 I. INTRODUCTION Electrical insulators or poor electric conductors are transparent due to the existence of an energy gap, E g. Because energy of the visible light ( ev) is less than the energy gap E g 5 10 ev in a typical insulator, photon absorption does not occur, and the insulator is transparent. A diamond, which is good electrical insulator, is transparent and has electrical resistivity ρ Ωm. Metals or good electric conductors partially absorb and reflect light. Reflection and absorption occur because metals have partially filled band and behave as ideal dielectrics. A silver, which is good electrical conductor, has electrical resistivity ρ 10 8 Ωm and its reflection coefficient is R = 0.90, for the visible light. When a material undergoes a metal-insulator phase transition (MIT), its electrical resistivity changes for several orders of magnitude. An example is V 2 O 3, for which a pressure/electrical resistivity diagram is presented in fig. 2a. At particular external pressure first order phase transition occurs, where electrical resistivity of V 2 O 3 changes for 9 orders of magnitude. Two parameters that control the MIT are electronic correlation strength and electronic band filling. Both are manipulated via external electric/magnetic fields, pressure, and carrier doping. In materials, that are explicable well in the noninteracting band theory, manipulating conductivity is difficult, e.g., introducing carriers to the diamond. Another set of materials exist where electronic correlation is dominant. An example is V 2 O 3. Its insulating state is called a Mott insulator and these are the the subject of the present seminar.

3 Seminar I b Metal-Mott insulator transition 3 To get insight into these phenomenon we present theoretical pictures of metallic and insulating phases and mechanisms that cause phase transitions. We present the band, Slater, Mott, and Brinkman-Rice theoretical pictures. In the end we depict the dynamical mean-field theory (DMFT). II. THEORETICAL PICTURES A. Band picture Consider a Fermi gas placed into a periodic lattice potential. If the lattice potential is constant in space and time, electrons are equally distributed in the space, and obey the Fermi-Dirac distribution in the momentum space. When a weak periodic lattice potential V (r) = V (r + R) is introduced, where R is the lattice vector, an energy gap E g = 2 V in vicinity of the Fermi energy occurs that separates the electronic band into a lower and upper band. If the lowest unoccupied state is separated from highest occupied state by the energy gap, the system is insulating. In this case, a weak external electric field does not modify the electron distribution and therefore cannot induce flow of electrons. At zero temperature fields exceeding the size of a gap per a small length scale cause Zener breakdown to occur. When the number of electrons per lattice site is even (odd), the lower band is always fully (partially) occupied. If E g > 0 and the Fermi level intersects one of the bands, the system is a metal. A simple schematic band structure is presented in fig. 1. Typical FIG. 1: Schematic band picture. E F is Fermi energy, E g is energy gap, k is reciprocal lattice vector. If E F intersects electronic band, system is metal. If Fermi level intersects energy gap E g, system is insulator [1].

4 Seminar I b Metal-Mott insulator transition 4 energy of a gap is few ev and the thermal energy k B T 1/40 ev is not enough to induce a flow of electrons. MIT occurs if a material is doped by a charge carriers or if the lattice structure is rearranged by an external pressure. In an insulating state, where the Fermi level lies in the energy gap, adequate doping of electrons causes the Fermi level E F to rise above the lowest occupied electronic band, and the system becomes metallic, or vice versa. External pressure exerted on the system changes the effective lattice potential and consequently width of the energy gap. When an energy gap in vicinity of the Fermi level disappears or arises, a MIT occurs [2]. The band theory fails to predict behavior of the transition-metal oxides (V 2 O 3, NiO, Fe 3 O 4, etc.) and Mott insulators. Some transition-metal oxides are insulators although they have odd number of electrons per lattice site. This is inconsistent with the band theory. Examples are Fe 3 O 4 and Fe 2 O 3, which have same lattice structure and 5 1/2 and 5 electrons per lattice site, respectively. First is conducting times better than the second one. The band theory cannot explain the insulating phase for a partially filled band, neither the occurrence of non-integer band filling [3]. Another example is V 2 O 3, which should have been a metal throughout the phase diagram, while it displays insulating regime, fig. 2 b), at low temperatures. B. Slater picture Some metal-oxides, like V 2 O 3, with partially filled d or f orbitals have anti-ferromagnetic ground state, narrow bands and are not explicable within non-interacting theories due to strong electron-electron correlations. Materials corresponding to this description are correlated materials and are partially explicable by the Hubbard model. In metal-oxides the overlap of the two adjacent d orbitals is small, consequently the electronic bands are narrow. Degeneracy, of the formed bands, is lifted under the strong influence of an anisotropic crystal field. The ligand p orbitals in vicinity of the Fermi level are strongly hybridized with the corresponding d orbitals or are far from the Fermi level. Hence, a single narrow electronic band remains in vicinity of the Fermi level and the single band Hubbard model can be used for the description of correlated materials. The Hubbard model in second quantization reads H = t ij (c iσ c jσ + h.c.) + U i n i n i. (1)

5 Seminar I b Metal-Mott insulator transition 5 (a) (b) FIG. 2: a) Resistivity versus pressure at different doping of V 2 O 3. Sudden change in resistivity is a consequence of metal-insulator transition [4]. b) Phase diagram for V 2 O 3. Band theory predicts metallic phase all over the phase diagram, however experimental data proof the existence of metallic phase. Pressure and doping concentrations are varied. First order MIT occurs and ends in critical (Mott) point [4]. t is the overlap of two neighboring Wannier orbitals or the tunneling amplitude, which depends of the lattice constant a. U is the on-site Coulomb repulsion. c iσ, c iσ are the creation and annihilation operators for electrons on the site i with a spin σ. n iσ = c iσ c iσ is number operator. First term, the tight-binding approximation for independent electrons, comes from assumption that electrons can tunnel between neighboring atoms. If only this term is considered, it results in cosine dispersion relation. D = Zt, where Z is a coordination number, determines the width of a band. Second term describes a local electron-electron interactions under assumption, that the electron-electron interactions are local and that the electrons dominantly sit on and not between the lattice sites. Due to strong crystal-field splitting, multiband effects can be neglected. Intersite Coulomb interactions are also omitted due to strong screening effects, that cause exponential decay

6 Seminar I b Metal-Mott insulator transition 6 of the Coulomb interactions. In spite of these tremendous simplifications, the Hubbard model can reproduce the Mott insulating phase and the transition between Mott insulators and metals [5]. The Slater theory explains insulating states that arise from an antiferromagnetic ground state and can be described by the single-band Hubbard model. Consider a bipartite lattice constituted of A and B sublatices, with one s-electron per site. Nearest neighbor coupling is antiferromagnetic and nearest neighbors of electron from A are from B, and vice versa. Because the, spins mutually repel due to the Coulomb interaction (potential term in eq. (1)), they arrange themselves that, e.g., spins on A are and spins on B are. Hence spins form a spin-density wave which is self-stabilizing. The system is not translational invariant, therefore we have gain in the potential energy and equivalent loss in the kinetic energy, due to the localization of electrons. Therefore, more favorable is if doubling of the lattice cells occurs (the Brillouin zone is halved), where the energy of the system is lower. At the new Brillouin zone boundary a band splitting and a energy gap for charge excitations occur, therefore the system is insulating [6]. Analogically an attractive local electron-electron interactions form a charge-density waves, which also lead to an insulating state. MIT is closely connected with disappearance of a magnetic ordering at Néel temperature T N. As temperature rises, thermal fluctuations affect the ordering and the energy gap is narrowed. When the energy of thermal fluctuations becomes comparable to the coupling energy, the ordering is destroyed and first order phase transition occurs. Some antiferromagnetic oxides, e.g., V 2 O 3, remain insulating well above the T N, as can be noticed in phase diagram, fig. 2 b). In the Slater theory width of an energy gap is comparable to the k B T N, what is much less than width of the energy gap in realistic materials [2]. C. Mott picture Magnetically disordered insulators are known as Mott insulators. Consider a system, with half filling and narrow bands, described by the Hubbard model (1). Whether the system is a metal or a insulator depends on ratio between the energy gap and the strength of electron-electron correlations, 2t/U. When electron-electron correlations are negligible, t/u 1, the electrons tunnel between the atoms, and the system is metallic. In the atomic limit, t/u 1, where electronelectron correlations prevail, the electrons are well localized, and the system is an insulator [7]. A schematic Mott transition is presented in fig. 3 a). Let us consider a system with half filling

7 Seminar I b Metal-Mott insulator transition 7 E F UHB U-D Intensity (arb. unit) U=5.0eV, T=1160K U=5.0eV, T=300K hν=700ev, T=175K Schramme et al. [14] hν=60ev, T=300K 2D LHB Decreasing U (a) E-E F (ev) (b) FIG. 3: a) Schematic Mott transition, where due to decreasing electron correlations lower (LHB) and upper (UHB) Hubbard band start to overlap and system becomes metallic. b) Photoemission spectra and the dynamical mean-field theory (DMFT) calculations for V 2 O 3. The quasi-particle peak (E E F = 0) and the LHB are clearly seen, however the quasi-particle peak is much wider than the dynamical mean-field theory simulation predicted [9]. and small overlap of the atomic orbitals, where 2D U. Energy µ + = E 0 (N + 1) E 0 (N) necessary to add an electron is given by µ + (N = L) = U D, since charge excitations with energy U are mobile, such that they form a band with width D, as in case of the non-interacting electrons. This band is the upper Hubbard band, which is generally not a one-electron band and it describes the spectrum of charge excitations for an extra electron added to the ground state. Energy required for removal of an electron is µ (N = L) = E 0 (N) E 0 (N 1) = U + D. The corresponding spectrum forms the lower Hubbard band. Both bands are presented in fig. 3 a). When D U we expect that the chemical potential is not continuous and a gap for charge excitations occurs µ(n = L) U D > 0. Hence the system is an insulator. If the overlap of the atomic orbitals is increased U D, the bandwidth of the upper and the lower Hubbard band D and tendency of electrons to delocalize increase. As U D the bands start to overlap, as presented in fig. 3 a), the gap for charge excitations vanishes and the system is a metal [8]. For the intermediate values U D a metal-insulator transition occurs. Energy gap in insulators is of the order of 1eV 10 4 K, which exceeds the energy scale (the Neel temperature) predicted by Slater, E g k B T N. If on-site repulsion in a insulator is large compared to kinetic energy, stability of the insulator is guaranteed [2].

8 Seminar I b Metal-Mott insulator transition 8 Although the Mott-Hubbard theory gives good description of insulating states, it fails to describe metallic states accurately. In correlated metals a quasi-particle peak located at the Fermi energy, in addition to the lower Hubbard band, is observed in photoemission experiments. As an example a spectra of V 2 O 3 is shown in fig. 3 b) [9]. Note that the photoemission experiments resolve only the occupied states and hence cannot resolve the upper Hubbard band (the inverse photoemission experiments, which probe the unoccupied states are able to resolve it). The quasiparticle peak corresponds to itinerant excitations with similar properties than the non-interacting electrons. In contrast to the Mott-Hubbard picture, the quasi-particle peak is thus observed. Its width diminishes on approaching the insulating phase continuously. Simultaneously, the specific heat coefficient increases. This can be thought of as the consequence of the increase of the quasi-particle density of states. A better theory that describes itinerant quasi-particle states is thus needed. D. Brinkman-Rice picture The simplest theory that describes the quasi-particle states correctly is based on the Gutzwiller method. We describe this theory, developed by Brinkman and Rice below. Valence electrons in the transition metals are from the d-orbitals, where electron-electron correlations are strong [8], and sites with one electron per site are favorable. In uncorrelated systems most states are doubly occupied, where the double occupancy occurs as a consequence of minimization of kinetic energy. In strongly correlated systems double occupancies are too costly due to the strong Coulomb repulsion. Consider a Hubbard model, where degeneracy of the d-orbitals is lifted by a strong, anisotropic crystal field, so a single d band remains near the Fermi level. Furthermore, let ligand p orbitals be far from the Fermi level or be strongly hybridized with the relevant d orbitals [5]. Due to strong screening of the ionic potential we neglect intersite Coulomb interactions. Solution, for this model, is equivalent to the solution for narrow, half filled, s bands. We construct Gutzwiller s wave function (GWF) Ψ; let G, Γ denote the set of lattice sites occupied by, spins, respectively. Φ 0 is the vacuum state, and particular spin configuration is ψ GΓ = c + g c + γ Φ 0. (2) g G γ Γ

9 Seminar I b Metal-Mott insulator transition 9 Any strongly correlated state Ψ is expressed as Ψ = g [1 (1 η) ν n g n g ]Φ, (3) where Φ is wave function of the uncorrelated system [10]. In strongly correlated systems sites with one electron per site are favorable, while in uncorrelated system most states are doubly occupied. The exponent, ν, tells us the number of identical lattice sites among the sets G, Γ. Parameter η is a weight for particular configuration, and depends on the occupancy of states. In uncorrelated systems η = 1, while in correlated systems η < 1. Pair correlation function ρ 2 (g, g ; g, g ), when number of electrons N and lattice dimensions L limit toward infinity, is ρ 2 (g, g ; g, g ) = ν, η = ν(l 2N + ν) N ν = 2 ν 1 2 ν. (4) Weight parameter η depends on the average double occupancy ν. We express the effective quasiparticle mass as m = m/z. Z q is discontinuity of single particle density function at Fermi level. In terms of average double occupancy, ν, discontinuity reads q = 32 ν(1 2 ν). Discontinuity q in terms of weight η, q = 4η/(1 + η) 2, indicates, that for strong interactions, η 0, discontinuity disappears and electrons are well localized. Therefore, insulators can be envisaged as systems with one electron per site, while metals can be treated as a sea of doubly occupied sites. Let us estimate the ground state energy. Potential contribution from the Coulomb interaction is Ψ H U Ψ = DU, where D ν. To estimate the kinetic part, we neglect spin and charge short-range correlations, and obtain ɛ η = q( ɛ + ɛ ) + Ud, where d = D/N, ɛ σ = Φ H t Φ, q = 1 (U/U 0 ) 2. When we minimize the energy in terms of d, we get ( ɛ η (U < U c ) = ɛ + ɛ 1 U ) 2, U c = 8 ɛ + ɛ. (5) U c For U > U c system is insulator and η = 0 and ɛ η = 0. When U U 0, the effective mass, m /m = 1/q, diverges as q approaches 0. We express the static electric susceptibility as χ 1 s = 1 (U/U 0) 2 ρ(ɛ F ) [ 1 ρ(ɛ F )U 1 + U/2U ] 0, (6) (1 + U/U 0 ) 2 which also diverges as U U 0. Beside the susceptibility and the mass enhancement, the ratio χ/m is constant, as we approach U 0 and that was observed in the experiments, fig. 4. Mass,

10 Seminar I b Metal-Mott insulator transition 10 (a) (b) FIG. 4: a) Strong mass enhancement in mono-layer 3 He films on graphite. In this system Mott insulating phase can be approached when the density is increased by hydrostatic pressure. Solid dot measurements were obtained from heat capacity and triangle points from magnetization [11]. b) Normalized Wilson ratio χ/γ between susceptibility χ (T 300K) and specific heat γ. Lower figure clearly shows specific heat and susceptibility enhancement as a function of filling x. Measurements were performed on Sr 1 x La x TiO 3 [12]. specific heat, and electric susceptibility enhancement are presented along with the Wilson ratio, which is the ratio between electric susceptibility and specific heat. Solid dots and triangles present measurements, and solid line is fit to the data [11]. Susceptibility and specific heat enhancement, that scale equivalently to mass enhancement, also appear in Sr 1 x La x TiO 3, fig. 4 b). Indication for MIT is the effective mass divergence m and disappearance of the discontinuity in the single particle density function at the Fermi level. Eq. (4) shows, that for strong interaction (η = 0) average double occupancy is zero, and therefore half filled band with well localized electrons can form an insulating state. The Gutzwiller s variational method for solving the Hubbard model (1) correctly predicts mass, and specific heat enhancement, that were observed in the experiments [10]. Major drawback of the Brinkman-Rice theory is, it fails to describe the occurrence of the lower and upper Hubbard band in an insulating state. Non of the presented theories accounts for the quantum effects, which are

11 Seminar I b Metal-Mott insulator transition 11 Material (crystalline solid) Atom Effective Medium FIG. 5: The Dynamical Mean-Field Theory (DMFT) concept. Solid is replaced by a single atom exchanging electrons with a self-consistent medium. Quantum fluctuations are taken fully into account and spatial fluctuations are suppressed [13]. important for MIT. A theory that would combine Mott and Brinkman-Rice picture is thus desired. E. Dynamical mean-field theory (DMFT) DMFT describes the bulk correlated problem in terms of an atom embedded in a self-consistent bath, which contains all the information about the intra-atomic interactions and is schematically presented in fig.5. The DMFT takes quantum fluctuations fully into account, but suppresses spatial fluctuations due to infinite-dimensional lattice. All the information about the self-consistent bath is contained in a local single-particle spectral function (ω). The DMFT reduces the problem to a quantum impurity problem, which has to be solved self-consistently, from where one determines the (ω). Consider the single impurity Anderson model or SIAM [14], which in second quantization reads as H SIAM = kσ [ ] ɛ k a kσ a kσ + V k (a kσ c 0σ + h.c.) µ(n 0 + n 0 ) + Un 0 n 0. (7) The parameters in DMFT, ɛ k, V k, are computed from the self-consistency condition. The first and second term in the sum represent non-correlated electronic levels of the effective medium, and coupling of the impurity with the effective medium, trough the hybridization parameters V k, respectively. The last two terms were already introduced in the Hubbard model (1) and describe the electron-electron interactions. Single particle content of the Anderson model can be expressed in terms of the hybridization function (ω) that represents all possible ways in which electrons tunnels of the site to the band,

12 Seminar I b Metal-Mott insulator transition 12 ρ(ω) ω (a) (b) FIG. 6: a) Density of states ρ(ω) = Im G. Results of DMFT for d, at T = 0 and different ratios U/t = 1, 2, 2.5, 3, 4 (from top to bottom). b) Phase diagram of DMFT for Hubbard model at half filling. T/t is the temperature, U/t is the Coulomb repulsion. Magnetic ordering is suppressed by lattice frustration. At T = 0 we get continuous transition, solid line represents the first order transition and dotted lines represent the region of coexistence of metallic and insulating state [5]. propagates there and returns to the original site. (iω) = kσ V k 2 δ(ω ɛ l ). (8) At the Mott transition, (ω) plays a role of an order parameter, because in an insulating state it has a gap at the Fermi level, which disappears at the transition to a metallic state. The DMFT is exact in a limit of large coordination number [5]. The DMFT correctly describes the Fermi liquid behavior of a metal and it gives more accurate results at high temperatures, where the spatial correlations really become negligible. The self-energies Σ i (ω) describe the local correlations and define the quasi-particle mass m = 1 ( Σ i / ω) ω=0, and the inelastic scattering rate τ 1 in (ω) = ImΣ i (ω = 0) [2]. In figure 6a) results of the DMFT for the Hubbard model are presented. System with odd number of electrons per lattice site is, for relatively strong but finite Coulomb repulsion, an insulator. In a metallic state a resonance peak, that is a consequence of quasi-particle excitations, appears in the vicinity of the Fermi energy. Its height is constant, but its width becomes smaller with the increasing strength of the Coulomb repulsion and at the critical value U = U c completely disappears. A energy gap is formed. At a Mott transitions one is observing a competition of the bare bandwidth 2t with the electron-electron interaction U. The gap formation is connected with

13 Seminar I b Metal-Mott insulator transition 13 the vanishing renormalization factor Z and the diverging mass of the quasi-particles. Figure 6b) presents the phase diagram for a half-filled Hubbard model. Solid line denotes the first order transition, which occurs at higher temperatures and ends with a second order critical point (also called Mott critical point). Dotted lines denote the region of coexistence of both metallic and insulating states. At T = 0 temperature continuous transition occurs. In fig. 3b) we can compare the DMFT results and the photoemission spectra (dotted plots) of V 2 O 3. Quasi-particle peak in vicinity of the Fermi level and lower Hubbard band are observed. The DMFT predicts narrower quasi-particle peaks than observed in the experiments, but positions of the peaks match. The DMFT has given very good results in examining the vicinity of the Mott transition in clean systems, very well predicted the properties of several transition metal oxides, heavy fermion systems, and Kondo insulators. It also naturally introduces a dynamical order parameters, that characterizes the qualitative differences between the various phases [2]. On the other hand, the DMFT does not give good description of phenomena, that are dominated by spatial fluctuations. Furthermore it cannot provide description of various physical quantities near the critical point. The DMFT is very extensive theory and for more details one should see ref. [2] and [14]. III. CONCLUSION We presented simple theoretical pictures of metals and insulators. The band theory explains insulating states in terms of the filled conduction bands. Some magnetically ordered materials with half filling are insulators, due to the doubling of the lattice cell. This is correctly explained by the Slater theory. Metal-oxides, like V 2 O 3, have odd number of electrons per lattice site and are insulators, well above the Neel temperature. To explain this, electron-electron interactions have to be taken into account. Furthermore, in metals a quasi-particle peak, specific heat, and electric susceptibility enhancement were observed and explained within the Brinkman-Rice theory and the Fermi liquid theory. A theoretical approach that accounts accurately for all the mentioned phenomena by bridging the Mott and the Brinkman-Rice picture is the DMFT. The DMFT is based on a simplification that occurs in the limit of large dimensionality. This simplification correctly incorporates local charge fluctuations but neglects the long range spatial fluctuations. The

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