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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

Regimes of the electronic current.The steady-state current, Eq. (5) (or Eq. (3)), of the single-state system connected to two 1D extended reservoirs of size Nr = 64. The potential difference is V = 0.5Jħ and the temperature is given by β = 40(Jħ)−1. The dashed lines show the approximations in the small and large γ regimes, and the dotted line is the Landauer calculation of the closed system, , with infinite  and  without relaxation. The small γ regime has a current increasing linearly with γ, as it dominates the rate at which electrons flow through the whole setup. In the large γ regime, the fast relaxation localizes electrons in the extended reservoir, causing the current to decay as 1/γ. In the intermediate relaxation regime, the current matches that from a Landauer calculation.
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f2: Regimes of the electronic current.The steady-state current, Eq. (5) (or Eq. (3)), of the single-state system connected to two 1D extended reservoirs of size Nr = 64. The potential difference is V = 0.5Jħ and the temperature is given by β = 40(Jħ)−1. The dashed lines show the approximations in the small and large γ regimes, and the dotted line is the Landauer calculation of the closed system, , with infinite and without relaxation. The small γ regime has a current increasing linearly with γ, as it dominates the rate at which electrons flow through the whole setup. In the large γ regime, the fast relaxation localizes electrons in the extended reservoir, causing the current to decay as 1/γ. In the intermediate relaxation regime, the current matches that from a Landauer calculation.

Mentions: Figure 2 shows the calculation of the current I from Eq. (5) (or Eq. (3)) as a function of the relaxation rate γ for a reservoir size Nr = 64. There are three regimes visible: (1) a small γ regime with current I1, (2) an intermediate regime with I2, and (3) a large γ regime with I3. We first discuss the intermediate regime.


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

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

Regimes of the electronic current.The steady-state current, Eq. (5) (or Eq. (3)), of the single-state system connected to two 1D extended reservoirs of size Nr = 64. The potential difference is V = 0.5Jħ and the temperature is given by β = 40(Jħ)−1. The dashed lines show the approximations in the small and large γ regimes, and the dotted line is the Landauer calculation of the closed system, , with infinite  and  without relaxation. The small γ regime has a current increasing linearly with γ, as it dominates the rate at which electrons flow through the whole setup. In the large γ regime, the fast relaxation localizes electrons in the extended reservoir, causing the current to decay as 1/γ. In the intermediate relaxation regime, the current matches that from a Landauer calculation.
© Copyright Policy - open-access
Related In: Results  -  Collection

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

f2: Regimes of the electronic current.The steady-state current, Eq. (5) (or Eq. (3)), of the single-state system connected to two 1D extended reservoirs of size Nr = 64. The potential difference is V = 0.5Jħ and the temperature is given by β = 40(Jħ)−1. The dashed lines show the approximations in the small and large γ regimes, and the dotted line is the Landauer calculation of the closed system, , with infinite and without relaxation. The small γ regime has a current increasing linearly with γ, as it dominates the rate at which electrons flow through the whole setup. In the large γ regime, the fast relaxation localizes electrons in the extended reservoir, causing the current to decay as 1/γ. In the intermediate relaxation regime, the current matches that from a Landauer calculation.
Mentions: Figure 2 shows the calculation of the current I from Eq. (5) (or Eq. (3)) as a function of the relaxation rate γ for a reservoir size Nr = 64. There are three regimes visible: (1) a small γ regime with current I1, (2) an intermediate regime with I2, and (3) a large γ regime with I3. We first discuss the intermediate regime.

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