Metal-to-Insilator Phase Transition. Hubbard Model.

Transcription

1 Master M2 Sciences de la Matière ENS de Lyon Phase Transitions and Critical Phenomena Metal-to-Insilator Phase Transition. Hubbard Model. Vitaly Gorelov January 13, 2017 Abstract In this essay about metal-insulator transition several models describing metals, insulators and transition between these two states was studied. Starting from a simple band theory, which can predict the transition for several materials, the study goes to more sophisticated models, like Hubbard model, where there was taken into account the electron correlation. Next, the scope of this essay touches an importance of disorder, which can also drive the phase transition. And as the final fascinating result, using t-j model, the magnetic correlations were discussed, particular the appearance of anti-ferromagnetic ordering of the local moments at the low-temperature phase of the Mott insulator. Also, in the end of the essay there is given a short overview of several possible algorithms for numerical study of MIT problem.

2 Figure 1: From [1]. Metal-insulator phase diagram based on the Hubbard model in the plane of the Coulomb interaction scale to the bandwidth (tunnelling) (U/t) and filling of the charge carriers n. 1 Introduction The metal-insulator transition (MIT) is the problem that fascinates scientists minds since the very old times, and still it remains one of the most difficult one and it has not been understood completely yet. There is a limited set of materials being well understood and there are simple theories describing them. The first theory is based on noninteracting or weakly interacting electron systems. The theory makes a general distinction between metals and insulators at zero temperature based on the filling of the electronic bands: for insulators the highest filled band is completely filled; for metals, it is partially filled. In the noninteracting electron theory, the formation of band structure is totally due to the periodic lattice structure of atoms in crystals. Later it has been noted by Peierls (1937) that electron-electron correlation plays an important role in MIT restricting electrons movement and therefore making, what by the band theory is predicted to be a metal, insulator in reality. The easiest way to add correlations was proposed by Hubbard in 1963 [3]. In his work he included only the Coulomb interaction and neglected the exchange one, which lead to significant improvement of the theory and extending the number of materials whose metal-insulator transition is well understood. Schematic phase diagram of metal-insulator transition by Hubbard is shown on Fig. 1. Following the development in the understanding of phase transitions, it become clear that by introducing disorder one can enhance correlation effects and completely suppress a first-order transition. That discovery has led to development of a new theory called Anderson localisation (Anderson 1958). The Anderson localisation requires writing another essay and will be studied in the nearest future. To be able to discuss all aspects of MIT one might need the whole proper course and as I do not poses that amount of time I will try to describe theories that will give a clear picture of the physics of the MIT. 1

3 Figure 2: From [2]. Metal-insulator transition of band-crossing type. E(k) is plotted against k in the insulating (a) and metallic (b) states. 2 Band-crossing transitions The first and simplest theory that one can imagine, but which gives the insight into the metal-insulator transition is Band-crossing theory. Here I will follow the book of Mott [2] where the reader who knows little about MIT can get one s first knowledge about it. According to this model one can define insulators as the materials with all energy bands either completely full or completely empty. On the other hand metal can be defined as a material in which one or more energy bands (valence bands) are partly full. In other words in metals valence states are full up to the limiting energy E F (Fermi energy) and states with higher energies are empty (conduction bands). In that way the density of states at fermi level N(E F ) must vanish for a crystalline insulator. It is necessary to state that this formulation remains valid for crystalline materials at absolute zero. Using the definitions of metal and insulator stated above it is rather obvious that transition will occur when two bands cease to overlap (see Fig. 2). That can happen by modifying different parameters for instance: specific volume, ratio of carriers concentration to a lattice spacing or the composition in an alloy. Interestingly, taking into consideration only the model of non-interacting electrons one can find that the transition is second-order, in the sense that there is no discontinuity in the value of the number of carriers n. One can modify the band gap by varying the lattice parameter a as a a 0, where a 0 is the value of lattice parameter at the transition. Following the same notation, the number of carriers will vary as a a 0 3/2 and the energy as a a 0 5/2. What is interesting now is that if one takes into account electron-electron interaction the transition becomes discontinuous, the number of carriers will jump from 0 to a critical value n c at the transition. The 2

4 Figure 3: From [4]. Phase diagram for a half-filled Hubbard model calculated from DMFT theory. straightforward proof of this statement is given in the book of Mott [2]. Usually, when the gap is small, the discontinuity will be small as well and can be observed only at low temperatures. Band theory works very good for quite a few systems when the kinetic energy of the electrons is dominant over the other energy scales in the problem, but when it comes to materials close to metal-to-insulator transition the fermi energy is typically small and comparable to the potential energy, and electrons become bound or localised and the band theory stops being correct. Class of materials that by conventional band theory are predicted to be metal but are insulators when measured experimentally are called Mott insulators. The next step will be including electron-electron interactions, which was done by Hubbard in 1963 [3]. 3 Mott-Hubbard transitions A prototype of theoretical understanding for the transition between the Mott insulator and metals was achieved by using Hubbard model. Let us now consider the model which from the first sight is very simplistic but gives a lot of insight and plays important role in the understanding of metal-insulator transition. The Hubbard Hamiltonian for the narrow energy band approximation written in a second-quantized form takes into account tunnelling of electrons between weakly hybridized atomic orbitals and Coulomb interaction U. 3

5 Ĥ = t <ij>σ(a iσ a jσ + h.c.) + U i ˆn i ˆn i. (1) Here a iσ (a iσ) describes creation(annihilation) of a single band electron on the site i. In this way the first term describes tunnelling of electron from site i to the nearestneighbor site j and t the hopping integral, is equal to B/2z, where B is the band width and z is the coordination number. U is the Coulomb interaction term, where ˆn iσ being number (or density) operator ˆn iσ a iσ a iσ and N = iσ ˆn iσ. This picture is appropriate for the narrow-band systems and therefore one can neglect all the exchange interactions and to be left with only Coulomb repulsion. Lowenergy and low-temperature properties are often well described after this simplification since only a small number of bands (sometimes just one band) are crossing the Fermi level and have to do with low-energy excitations. The parameters of the simplified models in this case should be taken as effective values derived from renormalized bands near the Fermi level. In a simple phenomenological way Hubbard model can be understood as the following: electrons tunnel between atomic orbitals localized on individual lattice sites while double occupancy of a lattice site incurs an energetic penalty associated with the mutual Coulomb interaction [5]. The phase behaviour of the model (Fig. 3) depends on three dimensionless parameters as it is defined in [5]: the ratio of the Coulomb interaction scale to the bandwidth (tunnelling) U/t, the particle density or filling fraction n (i.e. the average number of electrons per site), and the (dimensionless) temperature, T/t. Firstly, the simplest dilute limit n 1 will be considered. In this limit the typical electron wavelength is greater than the site separation and the dynamics are free. One can expect weakly interacting nearly free electron system. While the interaction remains weak U/t 1 and the lattice has integer filling per unit cell one expects a metallic behavior to persist. When, on the other hand, the interaction is very strong U/t 1 electrons do not have enough energy to jump on the neighboring site, and a gap opens, leading to Mott insulating behavior. When the tunneling energy and the Coulomb interaction are comparable U/t 1, the system finds itself in the vicinity of the Mott transition. One can conclude that the band-width can be controlled by modifying the tunnelling energy. Experimentally that can be achieved by pressurising the system. An interesting result in understanding the Mott-Hubbard transition from the metallic side is that close to the transition the strong number of charge carriers n (or effective mass) enhancement is predicted [4]. n (U c U) 1. (2) As the result one can expect a large resistivity increase with the following crossover to insulating behaviour. This important result was obtained using dynamical mean-field theory (Fig. 4). The most crucial simplification in the Hubbard model is to consider only electrons in a single s orbit, when experimentally is has been found that d electrons play important role in MIT. In transition metals with narrow energy bands the electron charge density in a d-band is concentrated near nuclei of the solid, making electrons being on a particular atom. In this way it is possible to talk about d-electrons exhibiting both types of behaviour the ordinary band model and the atomic model. Only by taking correlation effects into account it is possible to understand how d-electrons exhibit 4

6 Figure 4: From [4]. Mass enhansement evidence can be seen in experiments on mono-layer He 3 films on graphite (Casey, Patel, Nyéki, Cowan and Saunders, 2003). In this system, the solid phase (Mott insulator) can be approached when the density is increased by the application of hidrostatic pressure. both types of behaviour simultaneously. Good examples of such behaviour are describe in the original paper of Hubbard [3]. Furthermore in d-electron system, orbital degeneracy is an important source of complicated behaviour and more interestingly considering d-electron system orbital symmetry breaking will play important role in MIT transitions. However, in a narrow energy band, even a very simple approximate representation of electron interaction allow building a theoretical description of MIT for a certain number of materials. It is worth to mention that since the metal and the Mott insulator do not differ on symmetry grounds, two phases can coexist (Fig. 5) in a finite range of parameter space, which was proved by theories and several experiments. Similarly as in standard liquidgas systems, the coexistence region, and the associated first-order line, terminate at the critical end-point at T = T c. The corresponding critical behaviour was found to belong to the standard Ising universality class. 4 Weak disorder near Hubbard - Mott transitions. To illustrate this argument I will follow the chapter in the book Introduction to Metal- Insulator Transitions of V. Dobrosavljević [4]. As it has been seen, all quantities display a discontinuity across any first-order phase transition, but this jump can be reduced in presence of impurities or disorder. Following here the original Imry and Ma droplet arguments (Imry, Ma 1975), where in the presence of random field system will favour breaking into domains (droplets), which then suggests that sufficiently strong disorder can completely suppress such a first-order transition, eliminating the finite 5

7 Figure 5: From [4]. Phase diagram of the organic salt illustrating coexistence region (shaded), which displays hysteresis in transport. temperature metal-insulator coexistence region. To illustrate this argument, consider doped semiconductor, where the bandwidth can be modified by varying the donor concentration n. Assume that n is close and below its critical value n c (T ), so that the system is close to a Mott transition and a large resistivity drop. Introducing the reduced donor density δn(t ) = (n n c (T ))/n c (T ), which will drive the system through the first order phase transition, and can be understood as magnetic field in an ordinary ferromagnet. Taking into account non-idealistic periodicity of donor ions, one can introduce for given region of size L local concentration of donors n(l) n c, which in principle will be a random quantity and will favour the formation of metallic droplet. For δn L = n(l) n c > 0, the energy of metallic phase will be lower by the amount E M I (L) = ɛδn L, (3) where ɛ is a constant measuring the density-dependent (free) energy difference between the metal and the insulator. Creating such a droplet will create a domain wall, which costs surface energy E s = σl d 1, (4) where σ is proportional to the surface tension of the droplet. formed only if The droplet will be E M I (L) > σl d 1. (5) But E M I (L) is proportional to a random quantity δn L, so to calculate the probability of droplet to be formed one has to calculate probability of density fluctuation. 6

8 Figure 6: From [4]. The simplest model for disorder-induced cluster states near first-order phase tran- sitions is provided by the random-field Ising model (Imry and Ma, 1975). δn L > σl d 1 /ɛ. (6) Under assumption that the donor density fluctuations are uncorrelated on large enough scales, the probability distribution is given by the central-limit theorem P (δn L ) exp( 1 2 L d ), (7) W 2 where W is a disorder strength (constant) on the microscopic level. Therefore, the probability of the droplet of size L will be of order P (δn L ) exp( 1 σ 2 L 2d 2 2 ɛ 2 L d W 2 ) = exp( 1 σ 2 L d 2 2 ɛ 2 ). (8) W 2 Clearly, for d > 2 formation of larger drops will be exponentially suppressed, i.e. their concentration is exponentially small, and for sufficiently weak disorder (W σ/ɛ), even the very small droplets are exponentially rare. The first-order transition remains sharp, at least at low enough temperatures. However, when the temperature is approaching the critical end-point at T = T c, one expects the droplet surface tension to decrease as a power of the correlation length ξ (T c T ) ν (in mean-field theory ν = 1/2), so even weak disorder starts to have an appreciable effect. More precisely, here σ(t ) ξ 3, and ɛ(t ) ξ 1, and even small droplets start to proliferate. The transition is then smeared down to temperatures such that δn 2 L W σ(t )/ɛ(t ) ξ 2 (T c T ) 2ν, (9) and one can conclude that in the presence of weak disorder, the critical temperature is depressed by δt c (W ) = T c (0) T c (W ) W 1/2ν. (10) 7

9 When disorder is sufficiently strong (W σ(0)/ɛ(0)), the first order jump is completely eliminated at finite temperature, and only a smooth metal-insulator crossover remains. Formation of clusters for a different value of disorder parameter using random-field Ising model is illustrated on Fig. 6. Only at T = 0 a sharp distinction between metal and insulator states occurs, where the transition reduces to the conventional quantum critical point. It is important to mention that in a clean system the behaviour is completely different, there all the conduction electrons simultaneously turn into local magnetic moments at a well defined critical concentration n c. Interestingly, by experiment observations it is often found that the low-temperature phase of the Mott insulator is accompanied by the anti-ferromagnetic ordering of the local moments. The origin of these magnetic correlations can be traced to a mechanism known as superexchange and can be understood straightforwardly within the framework of the Hubbard model system. 5 t-j Model Coming back to Hubbard model it is evident that for large lattice systems calculating the ground state is evidently infeasible. Instead, one can effect a perturbation theory which projects the insulating system onto a low-energy effective spin Hamiltonian, specifically, treating the hopping part of the Hamiltonian Ĥt as a weak perturbation of the Hubbard interaction ĤU [5]. Here it is useful to invoke a canonical transformation of the Hamiltonian, namely Ĥ Ĥ e tôĥe tô = e t[ô,] Ĥ t2 Ĥ t[ô, Ĥ] + [Ô, [Ô, Ĥ]] +..., (11) 2! where the exponentiated commutator is defined by the series expansion on the right. By choosing the operator Ô such that Ĥt + t[ĥu, Ô] = 0, all terms at first order in t can be eliminated from the transformed Hamiltonian. As a result, the effective Hamiltonian is brought to the form Ĥ = ĤU + t 2 [Ĥt, Ô] + O(t3 ). (12) Applying the ansatz, tô = [ ˆP s Ĥ t ˆPd ˆP d Ĥ t ˆPs ]/U, where ˆP s and ˆP d are operators that project onto the singly and doubly occupied subspaces respectively, the first-order cancellation is assured. Substituting this result into Eq. (12) and projecting onto the singly occupied subspace one obtains ˆP s Ĥ ˆPs = 1 U ˆP s Ĥ t ˆPd Ĥ t ˆPs = 2 t2 U ˆP ( ( s 1 + a 1σ a a 2σ 1σ 2σ) a ˆPs = J Ŝ 1 Ŝ 2 1 ), 4 (13) where J = 4t 2 /U denotes the strength of the anti-ferromagnetic exchange interaction that couples the spins on neighbouring sites. The result derived from perturbation theory shows that the neighbouring electrons tend to adopt an antiparallel or antiferromagnetic spin configuration. This observation has a simple physical interpretation. Anti-parallel spins can take advantage of the hybridization (however small) and reduce their kinetic energy by hopping to a neighbouring site (see Fig. 7). Parallel spins on the other hand are restricted from participating in this virtual process by the Pauli 8

10 Figure 7: From [5]. Top: hybridization of spin polarized states is forbidden by Pauli exclusion. Bottom: superexchange mechanism by which two antiparallel spins can lower their energy by a virtual process in which the upper Hubbard band is occupied. principle. This mechanism, which involves a two-step process, was first formulated by Anderson14 and is known as superexchange. Extending the calculation presented above one arrives to the the quantum spin-(1/2) Heisenberg Hamiltonian Ĥ = J Ŝ m Ŝ n, (14) <mn> where < mn > denotes a sum of neighbouring spins on the lattice and the positive exchange constant J t 2 /U. When doped away from half-filling, the behaviour of the Hubbard model is difficult to resolve. Removal of electrons introduces vacancies (holes) that may propagate through the lattice. For a low concentration of holes, the strong coupling Hubbard system may be described by the effective t - J Hamiltonian, Ĥ t J = t ˆP s a mσa nσ ˆPs + J Ŝ m Ŝ n. (15) <mn> <mn> However, the path of vacancies is blocked by the antiferromagnetic spin correlations of the background. Here transport depends on the competition between the exchange energy of the spins J and the kinetic energy of the holes t. Oddly, at J = 0 (i.e. U = ), the ground state spin configuration is known to be driven ferromagnetic by a single hole while, for J > 0, it is generally accepted that a critical concentration of holes is required to destabilize antiferromagnetic order. The following interpretation of t-j model leads to a hight-temperature superconductivity and the reader may refer to the paper of N. M. Plakida [6]. 6 Numerical studies of metal-insulator transitions. The difficulties in solving Hubbard model analytically in arbitrary dimensions has led to development of several numerical algorithms which nowadays applied extensively. One is the exact diagonalization of the Hamiltonian matrix including the Lanczos method. Using this method, one can obtain physical quantities with high accuracy but only for small clusters. For instance, for the Hubbard model, the largest number of sites for the ground state is around 20. If one is interested in finite temperature 9

11 full diagonalization is possible only for even smaller-sized system than the Lanczos method for the ground state only. To relax this severe limitation for treating finitetemperature and dynamic properties, a combination of diagonalization and statistical sampling was introduced a decade ago (Imada and Takahashi, 1986). In this method, the trace summation is replaced by a sampling of randomly generated basis states, while the imaginary-time or real-time evolution, e τĥ or e iτĥ, is computed exactly as in the power method. In this quantum transfer Monte Carlo method and the quantum molecular dynamics method, the system size is extended to sizes comparable to the Lanczos method. A very similar method can also be applied to the t J model by replacing the power method with Lanczos diagonalization. The quantum Monte Carlo method provides another way of calculating strong-correlation effects for relatively larger systems. A frequently used algorithm of quantum Monte Carlo calculation is based on the path- integral formalism combined with the Stratonovich - Hubbard transformation. Here it has been only listed the most frequent algorithms related to the metal-insulator transition, for more details reader can consult review article of Imada, Fujimori and Tokura [1]. 7 Conclusion It is clear that metal-insulator transition is the problem that still requires many brilliant minds to work on. In this essay I just described the theories that were established decades ago and that lies in the foundation of current research in the field of metalinsulator transition. The reason I did so is the fact that this field is new for me and I did not have any courses of solid state physics, therefore I tried to understand and to give in this work the basic physics that drives this process and mainly it was focused of Hubbard model. Despite the simplicity of the Hubbard model it was proved experimentally that it works for many materials very well. The next logical step of this work should be taking into account the disorder which leads to the theory called Anderson localisation and Anderson transition, which is very fascinating and I will study it by myself in the future. 10

12 References [1] M. Imada, A. Fujimori, Y. Tokura, Metal-insulator transitions, Reviews of Modern Physics, 70, (1998) [2] Nevill Mott, Metal-Insulator Transitions, Taylor & Francis, (1990). [3] J. Hubbard, Electron Correlations in Narrow Energy Bands, Proc. R. Soc. Lond. A, 276, (2006). [4] V. Dobrosavljević, Introduction to Metal - Insulator Transitions. In Conductor- Insulator Quantum Phase Transitions, Oxford: Oxford University Press, (2012). [5] A. Altland, and B. Simons, Condensed matter field theory, Cambridge: Cambridge University Press (2006). [6] N. M. Plakida, Superconductivity in the t-j model, Condensed Matter Physics, (2002). 11