Seamless transition from the single-electron regime to field-effect transistor operation of nanoscale Schottky-barrier FETs

One of the major challenges for the simulation of nanoscale field-effect transistors (FET) consists in an adequate description of the Coulomb interaction within the transistor channel: a proper simulation approach has to account for the Coulomb interaction of a few fluctuating electrons and at the same time has to be able to describe non-equilibrium transport in an open nanosystem. Present device simulators, however, only deal with one of the two aspects: For instance, in the limit of a quasi-isolated quantum dot system, the orthodox theory of many-body Coulomb interaction [1] correctly describes single-electron charging effects such as Coulomb blockade [2,3] but does not account for renormalization and dissipation tefms which are important in open transistor systems. On the other hand, the non-equilibrium Green's function formalism (NEGF) [4] is the most appropriate approach for the simulation of quantum transport in realistic device systems. However, a Hartree approximation is commonly employed (selfconsistent potential), rendering the approach unable to describe the Coulomb interaction of a few fluctuating electrons. Recently, we have presented a novel multi-configurational selfconsistent Green's function approach (MCSCG) [5] which allows for the inclusion of few-electron Coulomb interaction effects within the framework of the NEGF. In this paper, we present for the first time a direct comparison of a conventional Hartree NEGF calculation with the results of the MCSCG approach. It will shown that the MCSCG is able to describe Coulomb blockade effects in the low temperature limit, while for the case of strong nonequilibrium and room temperature conditions, the Hartree approximation is retained. Hence, the MCSCG approach covers the single-electron transport regime as well as the transistor operation at room temperature. Deviations from a mean-field approximation become most apparent in a system with quasi-bound states, exhibiting single-electron charging effects as a function of external electrode potentials. As a typical example, we will therefore consider a one-dimensional (ID) coaxially gated nanowire transistor with Schottky-barrier source and drain contacts [5] as sketched in Fig. 1. Here, we assume a channel length of L = 20nm with a diameter of dnt = 4nm, surrounded by a gate oxide with do, = 10nm. For such a system, the Coulomb interaction within the channel becomes equivalent to an effective 1D interaction [6]. As a key element, our algorithm identifies trapped single-electron states which are subject to occupation fluctuations. To a good approximation, the system Green's functions can then be written as a weighted average over many-body configurations, which are defined as eigenstates of a projected many-body Hamiltonian within the Fock-subspace of quasi-trapped single-particle states. Fig.2 visualizes the simulated drain current ID for the single-electron transport regime (T= 77K) as a grayscale plot. In contrast to the Hartree-only calculation (Fig.2a), the MCSCG approach (Fig.2b) correctly reveals diamond-like shaped patterns due to the quantized Coulomb interaction (as predicted by the orthodox theory and observed in experiments). While the MCSCG treatment is able to cope with the mixture of manybody configurations, the Hartree theory only provides a mean interaction potential for the description of the Coulomb interaction. In addition, Fig.3 shows ID(VGs) curves for different drain voltages VDS. In the MCSCG case (Fig.3b), single-electron transport can be identified in terms of Coulomb oscillations for the two lowest VDS, whereas the Hartree-only simulation (Fig.3a) lacks these features; the Hartree-only case exhibits broader peaks solely due to the single-particle levels of the system. However, with increasing VDS, both approaches become equivalent. (Note that the sub-threshold regime shows the regular behavior and has been omitted here.) Finally, Fig.4 shows the room temperature (T= 300K) characteristics. Apart from the slight modulation in the MCSCG calculation (Fig.4b), which is a remnant of the Coulomb oscillation, the Hartree (Fig.4a) and MCSCG (Fig.4b) results are in good agreement. Effects beyond a mean-field picture of the system will obviously have a significant impact on application-relevant device properties such as the system capacitance and the transconductance. In summary, we have compared for the first time the conventional Hartree NEGF with the MCSCG and have shown that the multi-configurational approach is able to describe single-electron charging effects in the low temperature limit. In case of strong nonequilibrium (with an almost depleted channel) and room temperature conditions, the MCSCG and the well-established Hartree approximation lead to equivalent results. As such, the MCSCG yields a seamless transition from the single-electron transport regime to transistor operation at room temperature.