Phonons and the Isotopically Induced Mott transition - Physics

Phonons and the Isotopically Induced Mott transition - Physics

This energy difference U H is known as the Hubbard (on site) energy and is typically

large (several eV). In the condensed state, this onsite e-e repulsion was expressed

by Mott as: 3 2


φ ( ) φ ( )



0 12

2 2


= ∫∫ r1 r2 dr1d






r is the inter-electron separation at a site, and ( r )

φ is the electron

wavefunction. In other words, in this model the e-e repulsion is neglected except for

two electrons on the same site. Conventional band theory, which incorporates e-e

interactions within a suitably chosen periodic potential within the single particle

Schrödinger equation, cannot take the effects of this Hubbard energy into account.

Figure 2.2

The electronic density of states N(E) in a cubic crystalline material where E F denotes the Fermi

2.2.3 The Hubbard energy

For particular classes of crystalline materials where electron concentrations are low

and hence screening ineffective, 1 as well as in materials with filled d and f orbitals,

the e-e interactions cannot be neglected. In these materials the band picture breaks

down dramatically with the occurrence of a first order phase transition between

metallic and insulating states. To understand this transition within the framework

developed by Hubbard, 14 and later refined by Mott, 3 it is instructive to first consider

the situation regarding electronic excitations in the extreme limit of isolated alkali

atoms. Instead of there being, as in the band case, unoccupied electron energy

levels immediately above E F allowing electronic excitation and metallic behaviour,

electron hopping between neutral isolated atoms requires a large energy. In order to

remove an electron from an atom, an ionization energy I el must be supplied and,

although some energy is recovered as the electron affinity χ when this electron is

added to another neutral atom to give a negatively charged ion, the difference

between these two energies and therefore the net energy required to place two

U = I − χ.


electrons in a given orbital, is given by: 4



For strongly correlated materials the energy cost in placing a second electron on a

site within this model can (for


≥ W

) open a gap in the otherwise continuous band

and therefore make the material electrically insulating. This gap is generally not

equal to the Hubbard energy because of bandwidth effects. If we consider the


example of two Ni

+ ions 4 in the limit of infinite separation, the energy cost in placing

an extra electron on a



+ is equal to U H given by Eq. (2.1). As the interatomic

separation is reduced Hubbard bands form, with the lower one corresponding to the

motion of ‘holes’ among the ion sites and the upper band to the motion of electrons.

This is illustrated in Figure 2.3

As the bandwidth of these bands increases it becomes easier for electrons and holes

to hop between sites and the eventual overlap of the two bands marks the Mott

transition from the insulating to the metallic state.

2.3 Phonons

In the same way that the energy of an electromagnetic wave is quantized with the

photon as the quantum of energy, the quantum of energy associated with collective

modes of vibration on a lattice is called a phonon. Two types of phonon can be

distinguished depending on the nature of the mode: ‘acoustic’ and ‘optical’. An

acoustic mode is one in which all ions in a primitive cell move essentially as a unit in

phase, and the dynamics are dominated by interactions between cells: an optical

energy for (a) a normal metal and (b) an insulator. 3 12


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