Two-terminal transport in biased lattices: transition from ballistic to diffusive current

This paper demonstrates that quantum transport in a biased tight-binding lattice with weak decoherence transitions from a ballistic Landauer regime to a diffusive Esaki-Tsu regime as the tilt increases, with the crossover occurring when the Wannier-Stark localization length matches the lattice size.

The Gist

Imagine a long, narrow hallway lined with a series of stepping stones. This hallway represents a crystal lattice (a grid of atoms), and the people walking on it are electrons (charged particles).

At either end of this hallway, there are two large, crowded rooms (the reservoirs or "leads"). One room is packed with people, and the other is less crowded. Naturally, people want to move from the crowded room to the empty one. This difference in crowd density creates a "pressure" that pushes the people down the hallway. In physics, this pressure is like an electric field.

This paper explores what happens to the flow of people (the electric current) when we tilt the hallway and introduce some "noise" or "distractions."

The Two Main Scenarios

The author, Andrey Kolovsky, looks at two very different ways people move through this hallway, depending on how steep the tilt is.

1. The "Super-Highway" (Ballistic Transport)

When the tilt is gentle:
Imagine the hallway is slightly sloped. The people (electrons) start running. Because the hallway is perfectly smooth and there are no obstacles, they run straight through without stopping. They don't bump into the walls or each other.

• The Physics: This is called Ballistic Transport (or Landauer transport). The current is fast and efficient. It doesn't matter how long the hallway is; if the people can run without hitting anything, they get to the other end just as fast as if the hallway were short.
• The Catch: In a perfect, frictionless world, if you tilt the hallway too much, something weird happens. The people get stuck in a loop. They run forward, hit a "wall" created by the tilt, bounce back, run forward again, and bounce back. They end up vibrating in place (this is called Bloch Oscillation) and make zero progress down the hallway. The current stops completely. This is known as Wannier-Stark Localization.

2. The "Crowded Market" (Diffusive Transport)

When the tilt is steep (and there is noise):
Now, imagine the hallway is tilted so steeply that, in a perfect world, everyone would get stuck vibrating in place. But, in the real world, there is noise. Maybe there are people shuffling their feet, or a slight breeze blowing them sideways. In physics, we call this decoherence or relaxation.

• The Magic Trick: This "noise" actually helps! Instead of getting stuck in a perfect loop, the noise bumps the people out of their trapped spots. It breaks the perfect rhythm of the bouncing.
• The Result: Now, instead of running super-fast or getting stuck, the people move like a crowd in a busy market. They shuffle forward, bump into each other, get distracted, and shuffle forward again. This is Diffusive Transport (or Esaki-Tsu transport).
• The Twist: Here is the surprising part: The steeper you tilt the hallway, the slower the crowd moves. Why? Because the steeper the tilt, the harder it is to break the "trapped" vibration. The "noise" has to work harder to push them forward. This is called Negative Differential Conductivity: pushing harder (more voltage) actually results in less flow.

The "Critical Tipping Point"

The paper's main discovery is finding the exact moment where the system switches from the "Super-Highway" mode to the "Crowded Market" mode.

• The Rule: The switch happens when the "stuck zone" (the distance a particle vibrates before getting trapped) becomes smaller than the length of the hallway itself.
• The Analogy: Imagine the hallway is 100 meters long.
  • If the "stuck zone" is 200 meters long, the people can run the whole way without getting trapped. (Ballistic).
  • If the "stuck zone" shrinks to 50 meters, the people get stuck before they reach the end. (Localization).
  • BUT, if you add a little bit of "noise" (decoherence), you can rescue them from the 50-meter trap and get them moving again, just slowly. (Diffusive).

Why Does This Matter?

For a long time, scientists thought about these two scenarios separately:

1. Theoretical Physics: "If we have a perfect crystal, electrons will oscillate and stop."
2. Real Experiments: "In the lab, we always see current flowing, even with high voltage."

This paper bridges the gap. It shows that real-world imperfections (noise) are actually the reason we see current flowing in high-voltage situations. Without that tiny bit of "messiness," the current would vanish completely.

Summary in a Nutshell

• Low Voltage (Gentle Tilt): Electrons zoom through like race cars on a smooth track. (Ballistic).
• High Voltage (Steep Tilt) + Perfect World: Electrons get stuck vibrating in place. No current flows. (Localization).
• High Voltage (Steep Tilt) + Real World (Noise): The noise bumps the electrons loose, allowing them to shuffle through slowly. The harder you push, the slower they shuffle. (Diffusive/Esaki-Tsu).

The paper provides a mathematical map to predict exactly when this switch happens, helping engineers design better electronic devices that can handle high voltages without shutting down.

Technical Summary

1. Problem Statement

The paper addresses the fundamental problem of quantum transport in finite-sized tight-binding lattices connecting two particle reservoirs (leads) with different chemical potentials.

• The Conflict: There is a historical dichotomy in transport theory.
  • Infinite Lattices (Esaki-Tsu): In the presence of an electric field and inelastic scattering, electrons exhibit Bloch oscillations that decay into a directed current. This is described by the Esaki-Tsu equation, which predicts a diffusive current regime and negative differential conductivity at high fields.
  • Finite Lattices (Landauer): Standard Landauer theory describes transport in finite systems as ballistic, where current depends on transmission probability and is independent of lattice length. Crucially, standard Landauer theory assumes no internal lattice tilt.
• The Gap: The paper seeks to unify these views by modeling a finite lattice that is physically "tilted" due to the potential difference between leads, while accounting for weak relaxation and decoherence processes within the lattice itself. The goal is to understand how the transport regime transitions from ballistic to diffusive as the electric field (tilt) increases.

2. Methodology

The author employs a Master Equation approach for the carrier's single-particle density matrix ($\rho$), offering a distinct alternative to the Green's function formalism used in previous related works (e.g., Ref [6]).

• Model Setup:
  • A finite tight-binding chain of length $L$ connects two reservoirs (modeled as tight-binding rings of size $M$).
  • A chemical potential difference $\Delta\mu$ is applied between the leads. This creates an electrostatic potential drop, effectively tilting the lattice with an electric field magnitude $F = \Delta\mu / (CL)$, where $C$ is the lead capacity.
  • The total Hamiltonian includes the lattice, the leads, and coupling terms ($\epsilon$).
• Dynamics:
  • The evolution is governed by the Lindblad master equation: $\frac{d\rho}{dt} = -i[H, \rho] - \gamma \mathcal{L}(\rho)$.
  • Relaxation in Leads: The leads act as thermal baths, relaxing the system to a steady state even without internal lattice decoherence.
  • Relaxation in Lattice: To study diffusive transport, the author introduces a specific Lindblad operator (Eq. 13) that causes the decay of off-diagonal elements of the density matrix (decoherence) at a rate $\tilde{\gamma}$.
• Numerical & Analytical Approach:
  • Numerical: The steady-state density matrix is found by solving the algebraic equation $\dot{\rho} = 0$ for the total system. The limit $M \to \infty$ is taken to ensure convergence to Landauer results for unbiased cases.
  • Analytical: A semi-analytical derivation is performed for the diffusive regime by approximating the equations of motion for the density matrix elements in the central part of the lattice.

3. Key Contributions

1. Unification of Transport Regimes: The paper demonstrates a continuous transition between the Landauer ballistic regime (weak tilt) and the Esaki-Tsu diffusive regime (strong tilt) within a single finite-system framework.
2. Critical Tilt Condition: It identifies that the crossover between regimes is determined by the condition where the Wannier-Stark localization length ($L_{WS}$) coincides with the lattice length ($L$).
   • $L_{WS} \approx 2J/F$.
   • When $L_{WS} > L$, transport is ballistic.
   • When $L_{WS} < L$, Wannier-Stark localization suppresses current unless decoherence is present.
3. Role of Weak Decoherence: The study reveals that in the strong-field regime ($F > F_{cr}$), where Wannier-Stark localization would otherwise suppress current to zero, a weak decoherence rate ($\tilde{\gamma}$) destroys the localization and restores a non-zero, diffusive current.
4. Derivation of Finite-Lattice Esaki-Tsu Law: The author derives a specific functional dependence for the current in the diffusive regime of a finite lattice, distinct from the infinite lattice case.

4. Results

• Ballistic Regime ($F < F_{cr}$):
  • The stationary current is ballistic and depends on the chemical potential difference $\Delta\mu$ but is independent of the lattice length $L$.
  • The current increases linearly with the field initially, consistent with Landauer theory.
  • The density matrix $\rho$ exhibits a band-matrix structure.
• Transition Point:
  • As the field $F$ increases (by decreasing lead capacity $C$), the current is suppressed when $L_{WS} \approx L$.
  • For $L=40$, the critical point occurs at $1/C \approx 40$; for $L=120$, it shifts to a lower field strength, confirming the scaling $F_{cr} \propto 1/L$.
• Diffusive Regime ($F > F_{cr}$ with $\tilde{\gamma} > 0$):
  • In the absence of decoherence ($\tilde{\gamma}=0$), the current vanishes due to Wannier-Stark localization.
  • With weak decoherence ($\tilde{\gamma} > 0$), the current reappears.
  • Scaling Law: The numerical and analytical results yield the current dependence:
    $$\bar{j} \sim \frac{\tilde{\gamma} J}{F^2 L} \sim \tilde{\gamma} L J \left(\frac{1}{C}\right)^{-2}$$
  • This represents a negative differential conductivity regime where current decreases as the field (or $1/C$) increases.
  • The density matrix in this regime becomes tridiagonal in the bulk, indicating diffusive transport.
• Population Inversion:
  • In the ballistic regime, site populations increase with the site index.
  • In the strong-field diffusive regime, this trend inverts, with populations decreasing along the lattice.

5. Significance

• Theoretical Bridge: The work successfully bridges the gap between the microscopic Bloch oscillation physics (infinite lattice) and mesoscopic transport (finite lattice), showing they are limits of the same physical system under different conditions.
• Experimental Relevance: The paper argues that "weak decoherence" is inevitable in real laboratory experiments. Therefore, the predicted transition from ballistic to diffusive current (and the associated negative differential conductivity) should be observable in real physical systems, such as cold atoms in optical lattices or semiconductor superlattices.
• Methodological Advancement: By using the master equation approach, the study provides a robust framework for incorporating finite thermalization rates and internal relaxation processes, which are difficult to handle in pure scattering (Landauer) or infinite-lattice (Bloch) theories.

In conclusion, Kolovsky demonstrates that the nature of transport in biased finite lattices is not fixed but is a tunable parameter dependent on the interplay between the electric field (tilt), system size, and decoherence rates. This provides a comprehensive guide for interpreting transport experiments in mesoscopic systems where both localization and dissipation play roles.