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Landauer's formula with finite-time relaxation: Kramers' crossover in electronic transport.

Gruss D, Velizhanin KA, Zwolak M - Sci Rep (2016)

Bottom Line: Landauer's formula is the standard theoretical tool to examine ballistic transport in nano- and meso-scale junctions, but it necessitates that any variation of the junction with time must be slow compared to characteristic times of the system, e.g., the relaxation time of local excitations.Here, we calculate the steady-state current when relaxation of electrons in the reservoirs is present and demonstrate that it gives rise to three regimes of behavior: weak relaxation gives a contact-limited current; strong relaxation localizes electrons, distorting their natural dynamics and reducing the current; and in an intermediate regime the Landauer view of the system only is recovered.We also demonstrate that a simple equation of motion emerges, which is suitable for efficiently simulating time-dependent transport.

View Article: PubMed Central - PubMed

Affiliation: Center for Nanoscale Science and Technology, National Institute of Standards and Technology, Gaithersburg, MD 20899, USA.

ABSTRACT
Landauer's formula is the standard theoretical tool to examine ballistic transport in nano- and meso-scale junctions, but it necessitates that any variation of the junction with time must be slow compared to characteristic times of the system, e.g., the relaxation time of local excitations. Transport through structurally dynamic junctions is, however, increasingly of interest for sensing, harnessing fluctuations, and real-time control. Here, we calculate the steady-state current when relaxation of electrons in the reservoirs is present and demonstrate that it gives rise to three regimes of behavior: weak relaxation gives a contact-limited current; strong relaxation localizes electrons, distorting their natural dynamics and reducing the current; and in an intermediate regime the Landauer view of the system only is recovered. We also demonstrate that a simple equation of motion emerges, which is suitable for efficiently simulating time-dependent transport.

No MeSH data available.


Related in: MedlinePlus

Schematic representation of the model.(a) System-reservoir-environment model, with yellow and blue representing the junction region (e.g., two leads connected by a junction)—the system of interest —and  indicating the extended reservoirs. The presence of electron sources, sinks, and interactions (electron-electron, electron-phonon, etc.), here subsumed into the environments , causes the reservoirs to relax toward their respective equilibrium distributions, which, when an external bias is applied, will be at different chemical potentials. (b) Each reservoir state exchanges electrons with an environment (i.e., an external reservoir at some chemical potential), which gives rise to a non-zero relaxation rate γ. The imbalance of occupied states will drive a current through , where explicit (or implicit) relaxation mechanisms may or may not be present.
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f1: Schematic representation of the model.(a) System-reservoir-environment model, with yellow and blue representing the junction region (e.g., two leads connected by a junction)—the system of interest —and indicating the extended reservoirs. The presence of electron sources, sinks, and interactions (electron-electron, electron-phonon, etc.), here subsumed into the environments , causes the reservoirs to relax toward their respective equilibrium distributions, which, when an external bias is applied, will be at different chemical potentials. (b) Each reservoir state exchanges electrons with an environment (i.e., an external reservoir at some chemical potential), which gives rise to a non-zero relaxation rate γ. The imbalance of occupied states will drive a current through , where explicit (or implicit) relaxation mechanisms may or may not be present.

Mentions: The Hamiltonian is , where the explicit degrees of freedom are divided into three parts: the left extended reservoir (), the right extended reservoir (), and the system of interest (). The extended reservoir regions have a finite electron lifetime that pushes them towards equilibrium by allowing for the exchange of electrons with the external degrees of freedom in the implicit reservoir. In some sense, we can say this is a grand canonical approach to transport when compared to the microcanonical approach of ref. 13. In other words, and are open to some larger environment (shown as and in Fig. 1), where the latter will be composed of degrees of freedom that are treated implicitly. Finally, describes the interaction between and the left () and right () extended reservoirs. Figure 1 shows a schematic of this setup.


Landauer's formula with finite-time relaxation: Kramers' crossover in electronic transport.

Gruss D, Velizhanin KA, Zwolak M - Sci Rep (2016)

Schematic representation of the model.(a) System-reservoir-environment model, with yellow and blue representing the junction region (e.g., two leads connected by a junction)—the system of interest —and  indicating the extended reservoirs. The presence of electron sources, sinks, and interactions (electron-electron, electron-phonon, etc.), here subsumed into the environments , causes the reservoirs to relax toward their respective equilibrium distributions, which, when an external bias is applied, will be at different chemical potentials. (b) Each reservoir state exchanges electrons with an environment (i.e., an external reservoir at some chemical potential), which gives rise to a non-zero relaxation rate γ. The imbalance of occupied states will drive a current through , where explicit (or implicit) relaxation mechanisms may or may not be present.
© Copyright Policy - open-access
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC4837356&req=5

f1: Schematic representation of the model.(a) System-reservoir-environment model, with yellow and blue representing the junction region (e.g., two leads connected by a junction)—the system of interest —and indicating the extended reservoirs. The presence of electron sources, sinks, and interactions (electron-electron, electron-phonon, etc.), here subsumed into the environments , causes the reservoirs to relax toward their respective equilibrium distributions, which, when an external bias is applied, will be at different chemical potentials. (b) Each reservoir state exchanges electrons with an environment (i.e., an external reservoir at some chemical potential), which gives rise to a non-zero relaxation rate γ. The imbalance of occupied states will drive a current through , where explicit (or implicit) relaxation mechanisms may or may not be present.
Mentions: The Hamiltonian is , where the explicit degrees of freedom are divided into three parts: the left extended reservoir (), the right extended reservoir (), and the system of interest (). The extended reservoir regions have a finite electron lifetime that pushes them towards equilibrium by allowing for the exchange of electrons with the external degrees of freedom in the implicit reservoir. In some sense, we can say this is a grand canonical approach to transport when compared to the microcanonical approach of ref. 13. In other words, and are open to some larger environment (shown as and in Fig. 1), where the latter will be composed of degrees of freedom that are treated implicitly. Finally, describes the interaction between and the left () and right () extended reservoirs. Figure 1 shows a schematic of this setup.

Bottom Line: Landauer's formula is the standard theoretical tool to examine ballistic transport in nano- and meso-scale junctions, but it necessitates that any variation of the junction with time must be slow compared to characteristic times of the system, e.g., the relaxation time of local excitations.Here, we calculate the steady-state current when relaxation of electrons in the reservoirs is present and demonstrate that it gives rise to three regimes of behavior: weak relaxation gives a contact-limited current; strong relaxation localizes electrons, distorting their natural dynamics and reducing the current; and in an intermediate regime the Landauer view of the system only is recovered.We also demonstrate that a simple equation of motion emerges, which is suitable for efficiently simulating time-dependent transport.

View Article: PubMed Central - PubMed

Affiliation: Center for Nanoscale Science and Technology, National Institute of Standards and Technology, Gaithersburg, MD 20899, USA.

ABSTRACT
Landauer's formula is the standard theoretical tool to examine ballistic transport in nano- and meso-scale junctions, but it necessitates that any variation of the junction with time must be slow compared to characteristic times of the system, e.g., the relaxation time of local excitations. Transport through structurally dynamic junctions is, however, increasingly of interest for sensing, harnessing fluctuations, and real-time control. Here, we calculate the steady-state current when relaxation of electrons in the reservoirs is present and demonstrate that it gives rise to three regimes of behavior: weak relaxation gives a contact-limited current; strong relaxation localizes electrons, distorting their natural dynamics and reducing the current; and in an intermediate regime the Landauer view of the system only is recovered. We also demonstrate that a simple equation of motion emerges, which is suitable for efficiently simulating time-dependent transport.

No MeSH data available.


Related in: MedlinePlus