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Non-local effects in charge and energy transport with dissipative electrodes

Rodolfo A. Jalabert

Abstract

Recent advances in nano-thermometry motivate the extension of the Landauer-Büttiker scattering theory as to include the non-local dissipation associated with charge transport. Such a program is implemented by describing the inelastic scattering in the connecting electrodes within an electrostatically self-consistent scheme. The restriction to quasi-one-dimensional geometries, weak excitation and low temperature allows to obtain general expressions of the current density and the dissipated power, valid in different regimes, for the cases of an energy-independent mean-free-path or an energy-independent relaxation-rate. In particular, the dissipation asymmetry at both sides of a nano-device and the conditions for observing heating spots with a local maximum of the dissipated power are formulated in terms of the key parameters that define the nano-device and its environment.

Non-local effects in charge and energy transport with dissipative electrodes

Abstract

Recent advances in nano-thermometry motivate the extension of the Landauer-Büttiker scattering theory as to include the non-local dissipation associated with charge transport. Such a program is implemented by describing the inelastic scattering in the connecting electrodes within an electrostatically self-consistent scheme. The restriction to quasi-one-dimensional geometries, weak excitation and low temperature allows to obtain general expressions of the current density and the dissipated power, valid in different regimes, for the cases of an energy-independent mean-free-path or an energy-independent relaxation-rate. In particular, the dissipation asymmetry at both sides of a nano-device and the conditions for observing heating spots with a local maximum of the dissipated power are formulated in terms of the key parameters that define the nano-device and its environment.
Paper Structure (17 sections, 126 equations, 1 figure)

This paper contains 17 sections, 126 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Quasi-one dimensional model
  3. Velocity-independent mean-free-path $\ell$
  4. Weak-excitation, low-temperature regime
  5. Velocity-independent relaxation-time $\tau$ in the weak-excitation, low-temperature regime
  6. Dissipated power
  7. Power dissipation with a velocity-independent mean-free-path $\ell$
  8. Spatial asymmetry and thermal spots in the weak-excitation, low-temperature regime
  9. Power dissipation with a velocity-independent relaxation-time $\tau$ in the weak-excitation, low-temperature regime
  10. Conclusions
  11. Self-consistent treatment of the electrostatic potential for a velocity-independent mean-free-path
  12. Weak-excitation, low-temperature limit for a velocity-independent mean-free-path
  13. Strong Coulomb limit for a velocity-independent mean-free-path
  14. Weak-excitation, low-temperature regime
  15. Power dissipation
  16. ...and 2 more sections

Figures (1)

  • Figure 1: Energy diagram (upper panel) and carrier density (lower panel) for the model-system of a scatterer (shadowed region of length $L$ defined by $z_{-} \le z \le z_{+}$) connected to two semi-infinite wires. The scatterer (assimilated to a tunnel barrier) is characterized by an energy-dependent transmission coefficient $\mathcal{T}(\varepsilon)$ and a permittivity $\epsilon_{\rm B}$, the wires are characterized by a velocity-independent mean-free-path $\ell$ and permittivity $\epsilon$. The parameters $\nu_{\rm L,R}$ are defined in Eq. \ref{['eq:nuLR']}, $\mu_0$ is the equilibrium chemical potential found in the wires far away from the scatterer, $n_0$ is the equilibrium carrier density, $e$ is the charge of the carriers, and $V$ is the potential drop due to the presence of the scatterer. The electro-chemical potential $\mu_{\rm L,R}(z)$, the electrostatic potential energy $e\phi_{\rm L,R}(z)$, and the carrier density $n_{\rm L,R}(z)$ for the left (right) wire are labeled by arrows and represented by the orange, blue, and green curves, respectively. The linear drop of the electrostatic potential across the scatterer follows from the assumption that the scatterer is electrically neutral. The linear behavior of the $\mu_{\rm L,R}(z)$ in the wires is a consequence of the assumption of a velocity-independent mean-free-path. For the sake of presentation, the non-equilibrium features are exaggerated.