Electron transport in ultra-thin films and ballistic electron emission microscopy

Abstract

Abstract. We have developed a calculation scheme for the elastic electron current in ultra-thin epitaxial heterostructures. Our model uses a Keldysh's non-equilibrium Green's function formalism and a layer-by-layer construction of the epitaxial film. Such an approach is appropriate to describe the current in a Ballistic Electron Emission Microscope (BEEM) where the metal base layer is ultra-thin and generalizes a previous one based on a decimation technique appropriated for thick slabs. This formalism allows a full quantum mechanical description of the transmission across the epitaxial heterostructure interface, including multiple scattering via the Dyson equation, which is deemed a crucial ingredient to describe interfaces of ultra-thin layers properly in the future.We introduce a theoretical formulation needed for ultra-thin layers and we compare with results obtained for thick Au(111) metal layers. An interesting effect takes place for a width of about ten layers: a BEEM current can propagate via the center of the reciprocal space (Γ) along the Au(111) direction. We associate this current to a coherent interference finite-width effect that cannot be found using a decimation technique. Finally, we have tested the validity of the handy semiclassical formalism to describe the BEEM current.

Accurate ab initio determination of ballistic electron emission spectroscopy: Application to Au/Ge

“…26 The latter are computed iteratively solving Dyson's equation as explained in what follows. 7 The Green's functions, g l,l ′ , for a system composed of l superlayers are obtained from the Green's functions for a system composed of l − 1 superlayers and the interaction with a single non-interacting superlayer, t l−1,l :…”

“…Furthermore, since it only involves N-order calculations, it allows the study of large enough threedimensional atomistic models that can be realistically compared to experiments. 7 Therefore, we combine our own ultra-high vacuum (UHV), low-temperature BEEM experiments for the Au(001)/Ge(001) interface with the ab initio theory presented below to extract accurate values for the Schottky barrier.…”

Macroscopic Versus Microscopic Schottky Barrier Determination at (Au/Pt)/Ge(100): Interfacial Local Modulation

“…Using this formalism, it is possible to compute Green's functions of order N + 1 starting from uncoupled Green's functions of order N , therefore making the calculation an efficient N -order procedure that allows to tackle thick layers, while still solving the problem from first-principles. 45 A best-fit of Equation 5 to Equation 3 determines a parameter-free value for µ, which is found to depend mostly on the metal-semiconductor combination. To get a consistent value for µ it is necessary to restrict the fitting procedure to an interval where the main hypotheses behind Equation 3 apply; namely, (i) only a single minimum in the conduction band adds to the current (other minima are expected to work as new channels with the same value of µ)…”

“…Using this formalism, it is possible to compute Green’s functions of order N + 1 starting from uncoupled Green’s functions of order N , therefore making the calculation an efficient N -order procedure that allows to tackle thick layers, while still solving the problem from first-principles …”

Phase-space ab-initio direct and reverse ballistic-electron emission spectroscopy: Schottky barriers determination for Au/Ge(100)

“…A serious description of BEES must consider four basic steps: (i) tunnelling from the STM tip to the metal base, (ii) transport through the metal base, (iii) transmission and reflection at the interface and (iv) injection into the semiconductor conduction (V > 0, electrons) or valence (V < 0, holes) bands. The Keldysh non-equilibrium Greens functions formalism developed in (Claveau et al, 2017) performs such tasks. Such theory has been complemented with an ab-initio LCAO procedure (Lewis et al, 2011) to build the relevant Hamiltonians making the calculations free from a particular parametrization and more general .…”