Electronic structure

Quantum confinement adds one important constraint to the electronic structure of the 2DEL. As a consequence, the electronic structure of the LaAlO₃/SrTiO₃ interface has been shown to be distinct from the one of bulk SrTiO3. This difference has stimulated considerable research interest and has been studied intensively. Before discussing the electronic structure, it is worth mentioning the study on the thickness of the conducting layer, which provides an estimate of the quantum confinement. Basletic et al.[46] "imaged" the conducting layer by using conductive atomic force microscopy (C-AFM) on the cross-section of a sample. From their work, the confinement of the electrons and the role of the oxygen post-annealing were clearly revealed: at room temperature the 2DEL was found to be confined within a few nanometers in fully oxidised samples while the conducting region extends to hundreds of nanometers in samples prepared without oxygen annealing. Following this work, Copieet al.[47]

measured the temperature dependence of the thickness of the 2DEL, which evolves from∼4 nm at room temperature (an estimation probably limited by the tip resolution) to∼12 nm at 8 K, as shown in Fig.1.9(a). Using an electric field dependent dielectric permittivity (E), they also modelled the confining potential as well as the charge distribution by solving Poisson equation self-consistently. As they are transferred to the interface, electrons induce an electric field that lowers the dielectric constant . The consequence is to reduce the screening efficiency of SrTiO3and confines the electrons within a few nanometers next to the interface, as observed experimentally.

Figure 1.9 (b) illustrates the electric field and confinement length calculated as a function of the carrier density n2D. Forn2D ≥ n^lim_2D, the electric field increases dramatically, resulting in an enhanced confinement. Inversely, for n_2D < n^lim_2D the electron liquid extends a few hundred nanometers inside the SrTiO3 crystal. This first study highlights the relationship between the confinement and the carrier density; we will come back to this in Chapter2. Infrared ellipsometry experiments [50] performed at 10 K confirmed that the 2DEL extends for about 11 nm. At room temperature, different experiments including hard X-ray photoelectron spectroscopy [67], soft X-ray angle resolved photoemission spectroscopy (ARPES) [68] suggest that the 2DEL is extremely confined within 1-3 u.c.. This change in confinement with temperature is due to the giant temperature dependence of the dielectric constant of SrTiO3.

The effect of the quantum confinement is to modify the orbital order of the bands and to generate sub-bands due to the quantisation of the out of plane component of the momentum. In a quantum well, the energy levels (E_ψ) associated with states ψ having an effective mass m^∗_⊥ along the confinement direction are quantised as Eψ ∝1/m^∗_⊥(ⁿ_L)², beingLthe quantum well width andnthe quantisation number.

As a consequence, for a confinement alongz,dxy orbitals have a lower energy than d_xz/d_yzorbitals since their effective mass alongzis large [(m^∗_xy)⊥∼7-10, (m^∗_xy)_k ∼ 0.7-2].Ab initiocalculations without considering spin-orbit effects support this orbital reconstruction [69]. In the case of a strong confinement (thickness of a few u.c.), they show thatd_xy orbitals appear lowest in energy (Fig.1.10). However, these orbitals are probably filled with charges subject to in-plane localisation, which consequently do not contribute to transport [71]. At higher energies sitdxz/dyzorbitals that extend over several layers and contribute markedly to transport [71,72]. This effect will

(a)

(b)

Figure 1.9 – The thickness of the LaAlO3/SrTiO3interface:(a) Measured by C-AFM at 300 K and 8 K showing a conducting thickness of 4 nm and 12 nm, respectively; (b) Calculated confinement length and electric fieldEas a function of the 2D carrier density n2D. Then^lim_2D marks the criticaln2Dabove which the electrons are confined within a few nanometers close to the interface. Adapted from [47].

(a) (b)

Figure 1.10 – Band structure for the LaAlO3/SrTiO3 interface (1): (a) Ab initio calculation of the band structure for a sheet carrier densitynsof 0.5 /2D u.c. [69]; (b) Fermi surface measured by soft X-ray resonant ARPES compared with DFT calculations [70].

The O2psurface band refers to the hole pocket predicted in the Zener breakdown scenario forming at the LaAlO3surface.

Figure 1.11 – Band structure for the LaAlO₃/SrTiO₃interface (2): A sketch of the electronic structure according to experimental observations.

further be discussed in Chapter2. When the thickness of the 2DEL becomes large (for instance, due to a low carrier density), the confinement is less effective. In this case, the electronic structure is rather similar to that of bulk SrTiO3,dxz/dyzbeing lowest in energy [73]. The first measurements on the electronic structure of the LaAlO3/SrTiO3

interface was performed by Salluzzoet al.[74] using X-ray absorption spectroscopy.

They found that the degeneracy of the Ti 3d t2g orbitals is lifted and an electronic reconstruction occurs at the interface. The Ti 3dxy orbitals become the first available states in the system. This orbital reconstruction is sketched in Fig.1.4. The electronic structure of the system has also been investigated by soft X-ray resonant ARPES measurements [68,70,75], suggesting a reconstructed Fermi surface, as compared to bulk SrTiO₃. The experimental picture of the electronic structure is sketched in Fig.1.11. No signature of further sub-bands splitting (d_xy1,d_xy2, ...,d_xz,yz1,d_xz,yz2, ..., etc.), as suggested byab initiocalculations, has so far been observed. This discrepancy may originate from the difference between the thickness measured experimentally at low temperature and the thickness imposed by the calculations. Rashba spin-orbit coupling also modifies the electronic structure [76,77], for instance, by mixing different orbital states and creating features at crossings between bands of different orbitals.

However, these features are currently below the resolution of the ARPES technique, thus difficult to be revealed.