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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

Expansion of the plateau.The steady-state current as a function of the relaxation rate γ for the  systems (dashed lines) and the Nr → ∞ limit, I23 (solid line). The parameters of the system are the same as in Fig. 2. In the limit that Nr → ∞ and then γ → 0, we recover the standard Landauer current. The vertical dotted line demarcates the regions where the Markovian master equation is valid and not valid. Since the size of the plateau grows linearly with Nr, the plateau will eventually extend into the region where the Markovian equation is valid, allowing for transport in this intermediate plateau regime to be simulated with the much simpler Markovian approach.
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f3: Expansion of the plateau.The steady-state current as a function of the relaxation rate γ for the systems (dashed lines) and the Nr → ∞ limit, I23 (solid line). The parameters of the system are the same as in Fig. 2. In the limit that Nr → ∞ and then γ → 0, we recover the standard Landauer current. The vertical dotted line demarcates the regions where the Markovian master equation is valid and not valid. Since the size of the plateau grows linearly with Nr, the plateau will eventually extend into the region where the Markovian equation is valid, allowing for transport in this intermediate plateau regime to be simulated with the much simpler Markovian approach.

Mentions: Figure 2 shows that there is an intermediate range of γ for which the current is approximately flat. That is, in this crossover region between small and large values of γ, a plateau forms and subsequently elongates as the size of the extended reservoir increases (see Fig. 3). The current in this regime is the same as that predicted by a Landauer calculation for alone. That calculation gives the current as . In linear response, this yields


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

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

Expansion of the plateau.The steady-state current as a function of the relaxation rate γ for the  systems (dashed lines) and the Nr → ∞ limit, I23 (solid line). The parameters of the system are the same as in Fig. 2. In the limit that Nr → ∞ and then γ → 0, we recover the standard Landauer current. The vertical dotted line demarcates the regions where the Markovian master equation is valid and not valid. Since the size of the plateau grows linearly with Nr, the plateau will eventually extend into the region where the Markovian equation is valid, allowing for transport in this intermediate plateau regime to be simulated with the much simpler Markovian approach.
© Copyright Policy - open-access
Related In: Results  -  Collection

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

f3: Expansion of the plateau.The steady-state current as a function of the relaxation rate γ for the systems (dashed lines) and the Nr → ∞ limit, I23 (solid line). The parameters of the system are the same as in Fig. 2. In the limit that Nr → ∞ and then γ → 0, we recover the standard Landauer current. The vertical dotted line demarcates the regions where the Markovian master equation is valid and not valid. Since the size of the plateau grows linearly with Nr, the plateau will eventually extend into the region where the Markovian equation is valid, allowing for transport in this intermediate plateau regime to be simulated with the much simpler Markovian approach.
Mentions: Figure 2 shows that there is an intermediate range of γ for which the current is approximately flat. That is, in this crossover region between small and large values of γ, a plateau forms and subsequently elongates as the size of the extended reservoir increases (see Fig. 3). The current in this regime is the same as that predicted by a Landauer calculation for alone. That calculation gives the current as . In linear response, this yields

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