Origin of the metal-insulator transition in TaS2

A signature feature of correlated electron materials is the occurrence of metallic and insulating phases, and of transitions between them. These metal-insulator transitions (MIT), can be caused by temperature, pressure, doping, or by other external influences. Two possible mechanisms for such an MIT are a) the Peierls instability and b) the Mott-Hubbard transition.

The Peierls instability

The Peierls instability [14] causes a metal to become an insulator by the action of a lattice distor tion which doubles the crystal unit cell (see Fig. V.10). The energy cost incurred by the elastic distortion is more than compensated by a lowering of the electronic energy, due to the opening of a Peierls gap at the electronic Fermi level. Note that the situation is similar to that for the spin Peierls effect, discussed in Chapter I, where it is the lowered magnetic energy for spin dimers which drives the lattice distortion. Both the charge Peierls instability and the spin Peierls effect are intimately connected to the motion of lattice atoms, hence the relevant time scale will be that of lattice vibrations (i.e., 10–100 fs).

The Mott-Hubbard transition

A purely electronic mechanism for producing a metalinsulator transition results from the correlation physics described by the Hubbard model (see Infobox). In this model, the motion of electrons among lattice sites is governed by the two parameters U and W. The on-site Coulomb repulsion U is the energy cost incurred when a lattice site is simultaneously occupied by two electrons. The bandwidth W measures the tendency of electrons to minimize their kinetic energy by delocalizing among the lattice sites. As shown in the Infobox, a large U/W ratio favors electron localization and hence the insulating state, while a small value for this ratio causes the electrons to become itinerant and the material to become a conductor. Therefore, at some intermediate value of U/W, an originally half-filled conduction band will split into two bands, by the creation of a Hubbard correlation gap (of order U) in the electron density of states (see Fig. V.11). Since no atomic motion is involved, the relevant time scale of the Mott-Hubbard transition is that of the electronic motion, i.e., 10 fs and faster. The characteristic time scales of the “slow” Peierls instability and the “fast” Mott-Hubbard transition are compared with electronic and lattice motions in Figure V.12. A particularly interesting material in which to investigate the nature of MIT is the 1T phase of the dichalcogenide tantalum disulfide [16]. 1T-TaS2 consists of S-Ta-S layers which are weakly coupled to one another and which, at room temperature, show an incommensurate chargedensity wave (CDW) modulation which serves to split the Ta d-electron states into three bands. Since the uppermost of these is half-filled, the material is metallic. Below 180 K, the CDW locks to the lattice, and the resistivity increases by a factor 10. In order to investigate the nature of the MIT in this material, Per fetti et al. per formed a pump-probe experiment on an insulating sample at T = 30 K. A 1.5 eV laser pump pulse, with a duration of 80 fs, excites hot electrons in the material, and at a variable time later, the time-resolved band structure is probed with angle-resolved photoemision, using 6 eV incoming photons. Without a pump signal, or with a long (4.5 ps) pump-probe delay, the photoelectron spectra (see Fig. V.13 a), show a pronounced “lower Hubbard band” (LHB), corresponding to the insulating phase. But shor tly after the pump, the LHB intensity collapses, and a tail appears, extending far above the Fermi level, demonstrating shor t-lived metallic behavior. The time-dependent LHB peak height (Fig. V.13 b) shows the ultrafast (fs) nature of the collapse and a continuous recovery of the insulating state. Both these observations argue strongly for a predominantly Mott-Hubbard nature of the MIT in 1T-TaS2. Pump-probe photoelectron spectroscopy is not a technique which is par ticularly well-suited to the SwissFEL, due to the degraded energy resolution from the spacecharge felt among the many low-energy photoelectrons which are simultaneously emitted from the sample. But other power ful X-ray methods, in par ticular photon-inphoton- out techniques, such as X-ray absorption nearedge spectroscopy (XANES) (see Chapter II) and resonant inelastic X-ray scattering (RIXS), can provide similar information pertinent to electronic band structure effects. These can be per formed in a pump-probe arrangement, perhaps even in a single-shot mode (see Infobox), at the SwissFEL.

%STARTCONTAINER%

The Hubbard Model

The Hubbard Model [32] is the simplest model of interacting particles in a lattice and the star ting point of many descriptions of correlated electron systems. It is based on the “Hubbard Hamiltonian”:
H = -W \sum_{\langle ij \rangle \sigma} c^{
\dagger}_{j \sigma} c_{i \sigma} + U \sum_i n_{i\uparrow} n_{i
\downarrow}
Where the operators c and c are electron creation and annihilation operators, n = cc is the number operator, and the sums run over the spin directions σ = ↑ and ↓ and the N lattice sites of the model. implies neighboring lattice sites. The principal parameters of the model are U and W, the on-site Coulomb repulsion and the electron bandwidth (or hopping rate), respectively. Let us consider two limiting cases [33]: a) Static electrons (W = 0) It is now enough to take into account a single site, and the possible states are {|0〉,|↑〉,|↓〉,|↑↓〉}, i.e., empty, a single spin up, a single spin down, or doubly-occupied. We calculate the partition function Z and the thermally averaged site occupancy :
Z = \sum_{\alpha} \langle \alpha | e^{-\beta (H \mu n)} | \alpha
\rangle = 1 + e^{\beta \mu} + e^{\beta \mu} + e^{-\beta U + 2 \beta
\mu}
\langle n \rangle \frac{1}{Z} \sum_{\alpha} \langle | (n_{\uparrow}
+ n_{\downarrow}) e^{- \beta (H - \mu n)} | \alpha \rangle =
\frac{1}{Z} [0 + e^{\beta \mu} + e^{\beta \mu} + 2e^{- \beta U + 2
\beta \mu}] = \frac{2 (e^{\beta \mu} + e^{-\beta U + 2 \beta \mu}}{1
+ 2 e^{\beta \mu} + e^{-\beta U + 2 \beta \mu}}
where β = 1/kBT, and \mu \equiv \frac{\partial E}{\partial n}is the chemical potential. Plotting as a function of μ (Fig. V.i6) we find that at the condition for half-filling, = 1, the energy required to add an electron to the system changes by U. It is this “Hubbard gap” which implies an insulating behavior. b) Non-interacting electrons (U = 0) In this case, it is convenient to use the reciprocal-space representation of the electron operators:
c^{\dagger}_{k \sigma} = \frac{1}{\sqrt{N}} \sum_l e^{ik \cdot l} c^{\dagger}_{l \sigma}
where k takes the values kn = 2pn/N assuming periodic boundary conditions in one dimension. The Hubbard Hamiltonian now has the form:
H = \frac{-W}{N} \sum_{k,k'} \sum_{\langle jl \rangle \sigma} e^{ikj} e^{-ik' l} c^{\dagger}_{k \sigma} c_{k' \sigma} = -2W \sum_{k \sigma} n_{k \sigma} \cos k
where the last expression follows from per forming the lattice sums. The energy levels of this Hamiltonian show a “band” behavior (see Fig. V.i7). As N goes to infinity, we obtain a (gapless) continuum of states, with bandwidth 4W, which, at half-filling, implies metallic behavior. We thus find that the Hubbard Hamiltonian describes an insulator, in the case W = 0, and a metal, for U = 0. Between these two limits, i.e., for intermediate U/W, there must occur a metal-insulator transition: the Mott transition. %ENDCONTAINER%