Why are insulators insulating and metals conducting?

Europhysics News, Jul 2018

Raffaele Resta

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Why are insulators insulating and metals conducting?

insulating and metals conducting? 0 Raffaele Resta, University ofTrieste , Italy A n insulator is distinguished from a metal by its vanishing dc .1'1. conductivity at low temperature. In contrast to what happens in metals, the electronic charge in insulators (and quite generally nonmetals) cannot flow freely under an applied dc field, but instead undergoes static polarization. Within classical physics, this qualitative difference is attributed to the nature of the electronic charge, as sketched in Fig. 1: either "bound" (Lorentz model for insulators) or"free" (Drude model for metals). In other words, electrons are localized in insulators and delocalized in metals. Switching to quantum physics, this dearcut distinction is apparently lost. In most textbooks [1], the insulating/metallic behaviour is explained by means of band structure theory, focussing on the position of the Fermi level of the given material: either in a band gap (insulators), or across a band (metals), as in Fig. 2. . Why do we need a theory of the insulating state different-and formally more complex-than the familiar one sketched in Fig. 2? The point is that such a picture applies only to a crystalline material, within the independent-electron approximation [1]: a very limited dass of insulators indeed. In some materials the insulating behaviour is dominated by disorder (Anderson insulators), in some it is dominated by electron correlation (Mott insulators): therefore, for a large number of insulators, the band picture is grossly inadequate. The present theory of the insulating state [2,3, 4] deals with all kinds of insulators on the same basis: either crystalline or disordered, either independent-electron or correlated. - The insulating/metallic state of matter is characterized by the excitation spectrum, but the qualitative difference in dc conduc­ tivity must also reflect a qualitative difference in the organization of the electrons in their ground state: a concept first emphasized byW Kohn in a milestone 1964 paper [ 5 ]. Its outstanding message is that even within quantum mechanics the cause for insulating behaviour is electron localization. Such localization, however, manifests itself in a very subtle way: in fact the electrons in a con­ densed system appear, from several viewpoints, about equally delocalized in nonmetals and metals. For instance, the Bloch orbitals in eit.\cr crystalline silicon or crystalline alu.-mnum are similarly delocalized, and do not reveal any sharp difference. The challenge is to show how electron localization can be detected and measured in the ground wavefunction of a condensedmany-elec­ tron system. The difference between localized and delocalized must be, in the thermodynamic limit, a sharp one. A solution to this problem was provided by Kohn in his original 1964 paper. In 1999 the problem was reconsidered and a solution different from Kohn's-and in many respects simpler-was found [ 3 ]. There is an outstanding phenomenologicallink between macroscopic polarization and the insulating state of matter. Sup­ pose we expose a finite macroscopic sample to an electric field, say .. Fig. 1: Schematic view of insulators and metals in classical physics. Top sketch: Lorentz model for insulators, where each electron is tied (by an harmonic force) to a particular center. Bottom sketch: Drude model for metals. where electrons roam freely over macroscopic distances, hindered only by atomic scattering potentia Is. I i i I i I ~~ ....._J Insulator Metal I I i i 1 ••.• - •••• Ji. Fig. 2: Traditional textbook view of the qualitative difference between insulators and metals. The plots show the energy band structures of crystalline (Le. ordered) materials, chosen one­ dimensional for the sake of simplicity. The insulating/metallic behaviour depends on the position of the Fermi level, which in turn is determined by the number of electrons per cell. Afilled band results in insulating behaviour, whi:le an half-filled one results in metallic behaviour. For many insulating materials (e.g. disordered and/or correlated) such a band picture is inappropriate. The present theory ofthe insulating state is based on quite different concepts, and applies on the same grounds to any insulator. inserting it in a capacitor and applying a voltage. Then the induced macroscopic polarization is qualitatively different in metals and insulators. In the former materials polarization is triv­ ial: universal, material-independent, due to surface phenomena only (screening by free carriers). Therefore polarization in met­ als is not a bulk phenomenon. The opposite is true for insulators: macroscopic polarization is a nontrivial, material-dependent, bulk phenomenon. On the theoretical side, the concept itself of macroscopic polar­ ization in quantum physics has long evaded even a correct definition: most textbooks contain incorrect statements [I]. The modern theory of polarization [ 2,6 ], based on a Berry phase [7], was developed a decade ago: it revolutionizes both the very defi­ nition of the relevant bulk observable, and the ways to compute it in real solids. This theory has been the subject of a previous arti­ cle in this journal [ 8 ], and starts making its way in elementary textbooks [ 9 ]. The recent advances about the insulating state of matter [ 3,4 ] are deeply rooted in the modern theory of polariza­ tion: in fact polarization and localization can be regarded as two aspects of the same phenomenon, and stem from the same for­ malism. In order to provide an oversimplified treatment, here I only deal with a system of N one-dimensional electrons, chosen spin­ less (or parallel-spin) for the sake of simplicity. The many-body ground wavefunction is then q'(Xl,x2,... Xj, ... XN), and all the elec­ trons are confined to a segment of length 1. Eventually, we will be interested in the thermodynamic limit, defined as the limit N -., co and L -., co, while the density N/L is kept constant. For practical purposes, this limit is well approximated when L is much larger than a typical atomic dimension. A crucial role in our treatment is played by the boundary conditions chosen for the wavefunction: we adopt here-as almost mandatory in condensed matter physics [I]-Born-von Karman periodic boundary conditions, which amount to imposing that the wavefunction q' is periodic, with period L, over each electronic variable Xj separately. Equiva­ lently, one can imagine the electrons to be confined in a circular rail of length L: the coordinates Xj are then proportional to the angles 2 'TtXj IL, defined modulo 2 1t. Following Refs. [ 2,3 ], the key quantity needed to deal with both polarization and localization is the ground-state expectation value ZN = (wlUlw) = f dx\ ... foL dXNlw(x\, ... XNWU(Xt> ... XN), where the unitary operator U, called the "many-body phase operator" or "twist operator", is defined as and clearly obeys periodic boundary conditions. The expecta­ tion value ZN is a dimensionless complex number, whose modulus is no larger than one. The electronic contribution to the macroscopic polarization of the system can be expressed in the very compact form [ 2,4 ]: P,I = 2e7r }~.~ IIII log z-"" where e is the electron charge. Notice that, for a one-dimensional system, the polarization has the dimensions of a charge (dipole per unit length). The essential ingredient in Eq. (3) is Im 10gzN, i.e. the phase of the complex number ZN. This phase, which is a rather peculiar kind of Berry phase [ 7 ], is ill defined whenever ZN van­ ishes. And here comes the key message [ 3, 4 ]: what differentiates very sharply metals from insulators is the behaviour of the mod­ ulus of ZN in the thermodynamic limit: in the former materials it goes to zero, while in the latter it goes to one. We find therefore, in agreement with the above phenomenological considerations, that macroscopic polarization is well defined in insulators and ill defined in metals. The modulus of ZN can be used to measure the localization of the many-body wavefunction, thus providing a quantitative Insulator Metal o o o o o o 0 o o 0 0 0 o o o 0 o o (1) (2) (3) Ji. Fig. 3: Same energy band structure as in fig. 2, for a finite one­ dimensional system with periodic boundary conditions. In drawing the figure, the period Lhas been taken as 14 times the lattice constant. In the msulating case the band is filled, cmd tile ground wavefunction '¥ is the antisymmetrized product (Slater determinant) of 14 Bloch orbitals, whose k-vectors and corresponding energies are indicated by dots. Notice that only one of the states at the BriIJouin-zone boundaries must be occupied to avoid double counting. In the metallic case the band is half-filled, and the ground wavefunction is the antisymmetrized product of the 7 Bloch orbitals whose energy is below the Fermi level, indicated by dots. assessment of Kohn's [5) main idea. To this aim, we have intro­ duced [3) the intensive quantity (x2)e = - Nh-.t-m001V1" (L-21)r2 log IZN!2 , (4) having the dimensions of a squared length. It can be proved [4) that, in insulators, the modulus of ZN differs from one by the order of 1/N, hence (x?)c is finite, while in metals it is divergent: this finding clearly vindicates the classical viewpoint of Fig. 1. I stress that (x?)c is an intensive property characterizing the localization of the many-body wavefunction qt as a whole, which applies on the same grounds to ordered/disordered and correlat­ ed/uncorrelated many-electron systems. It is now expedient to focus on a special case: ordered and uncorrelated, i.e. a crys­ talline system of independent electrons, as in the band-structure picture of Fig. 2, where a single band is considered. In this case the ground many-bodywavefunction qt(Xl,x2,•.• XN) is a Slater deter­ minant (i.e. antisymmetrized product) of N single-particle orbitals, which are usually chosen in the Bloch form both for met­ als and insulators. But there is an outstanding difference, illustrated in Fig. 3: only one half of the band is used to build the Slater determinant for the metal, while the whole band is used for the insulator. These two Slater determinants are therefore quali­ tatively very different, despite the fact that both are built of (delocalized) Bloch orbitals. Their difference is probed very sharply by the many-body phase operator U, Eq. (2): in fact it can be proved [3,4) that in the metallic case the expectation value ZN, Eq. (1), vanishes, thus leading to a formally infinite value of (x?)c, while in the insulating case (x2)c assumes a finite value. Actual values of (x?)c in nonmetallic materials, as computed e.g. for the III-V semiconductors, are of the order of a few bohr2• I stress that, in the present analysis, no use is made of the spectrum of the system: the metallic/insulating behaviours reflect a different orga­ nization of the electrons in the ground state. ~ Fig. 4: The ground wavefunction qt of an insulator can be equivalently written as an antisymmetrized product of either Bloch orbitals (Fig. 3, left panel) or Wannier orbitals. Both Bloch and Wannier orbitals obey periodic boundary co'nditions overthe period L, which we have taken as 14 times the lattice constant (same as in Fig. 3). For the sake of clarity, we have plotted (red and green) only 2 of the 14 Wannier orbitals which are needed to build 'f'.ln the thermodynamic (L -4 00) limit the Wannier orbitals are exponemially localized and the second cumulant moment oftheir distribution (x~) - (x» is finite: in fact it is equal to (x~)" Eq. (4).The situation is completely different in the metallic case (Fig. 3, right panel): one cannot write 'P as an antisymmetrized product of orbitals whose second moment is finite in the thermodynamic limit. Notice that such a sharp qualitative difference reflects a different organization of the electrons in the ground state, and makes no reference to either the excitation spectrum or conductivity properties. The intensive quantity (x?)" measuring electron localization in the many-bodywavefunction, has the meaning of a second cumu­ lant moment. Once more, it is expedient to illustrate this for the special case of a crystalline system of independent electrons, as in Figs. 2 and 3. Since a determinant is invariant under unitary transformations, we can perform any unitary transformation on the N single-particle occupied orbitals without affecting the ground N-particle wavefunction '1', and therefore leaving ZN and (x2)c invariant. Starting from orbitals of the Bloch form, hence delocalized throughout the crystal, we may look for a unitary transformation leading to orbitals which are localized around some crystalline sites. One such transformation, namely, the Wan­ nier transformation, is well known in solid-state physics: this is illustrated in Fig. 4. According to our theory, (x2)c is the mini­ mum possible value for the averaged second cumulant moment (x?) - (x?) of the electron distribution of the localized orbitals, in the N -4 00 limit. One outstanding implication is that, for insula­ tors, the many-bodywavefunction can be written as a Slater determinant oflocalized single-particle orbitals, whose distribu~ tions have finite second moments. More precisely, the averaged second moment of the single-particle orbitals can be made as small as (x?)c with a suitable choice of the unitary transforma­ tion. Suppose, instead, that we attempt a localizing transformation on the occupied Bloch orbitals of a metal. Then, since (x2)c diverges, it is impossible that all of the transformed orbitals have a finite second moment in the thermodynamic limit. In conclusion, the present theory of the insulating state sharply discriminates between an insulator and a metal without actually looking either at the excitation spectrum or at conductivity properties. Instead, it is enough to probe-with an elegant tool, the many-body phase operator ofEq. (2)-the organization of the elec­ trons in the ground state. Once our simple definition of . localization is adopted, electrons are localized in any insulator and delocalized in any metal. Localization in the ground electronic wavefullction is the key reason v.hy insulators sustain bulk dielec­ tric polarization. In the present treatment, localization and polarization appear as two aspects of the same phenomenon, and are naturally described by the same formalism. Raffaele Resta is professor of Struttura della Materia, University of Trieste, and a coordinator in the INFM DEMOCRITOS National Simulation Centre in Trieste. Previously, he was a professor at the University of Pisa and at SISSA, Trieste. He has also worked for many years at EPFL (Swiss Institute of Technology, Lausanne). [1] N. W. Ashcroft and N. D. Mermin , Solid State Physics (Saunders, Philadelphia, 1976 ); C. Kittel, Introduction to Solid State Physics, 7th . edition (Wiley, New York, 19 ~6). [2] R. Resta , Phys. Rev. Lett. 80 , 1800 ( 1998 ). [3] R. Resta and S. Sorella , Phys. Rev. Lett . 82 , 370 ( 1999 ). [4] R. Resta , J. Phys.: Condens. Matter 14 , R625 ( 2002 ). [5] W. Kohn , Phys. Rev . 133 , Al71 ( 1964 ). [6] R. Resta , Ferroelectrics 136 , 51 ( 1992 ); R.D. King-Smith , and D. Vanderbilt , Phys. Rev. B47 , 1651 ( 1993 ); R. Resta, Rev. Mod. Phys . 66 , 899 ( 1994 ). [7] R. Resta , J. Phys.: Condens. Matter 12 , R107 ( 2000 ). [8] R. Resta , Europhysics News 28 , 18 ( 1997 ). [9] M.P. Marder , Condensed Matter Physics (Wiley, New York, 2000 ), p. 609 .


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Raffaele Resta. Why are insulators insulating and metals conducting?, Europhysics News, 92-94, DOI: 10.1051/epn:2003303