🔬 mesoscale physics

Zandpack: A General Tool for Time-dependent Transport Simulation of Nanoelectronics

This paper introduces Zandpack, an open-source tool that implements a refined auxiliary mode approach for efficient time-dependent transport simulations of large nanoelectronic systems far from equilibrium, demonstrating its capabilities through applications to spin currents in hydrogenated graphene, electron excitation in graphene nanoribbons, and non-adiabatic behavior in gold break-junctions.

Original authors: Aleksander Bach Lorentzen, Alexander Croy, Antti-Pekka Jauho, Mads Brandbyge

Published 2026-02-25

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CC BY 4.0

Original authors: Aleksander Bach Lorentzen, Alexander Croy, Antti-Pekka Jauho, Mads Brandbyge

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to predict how a crowd of people (electrons) will move through a tiny, complex building (a nanodevice) when a sudden, powerful windstorm (a THz pulse) hits it.

In the world of nanoelectronics, this is a massive challenge. The building is so small that the people don't just walk; they behave like waves, jumping, interfering with each other, and reacting instantly to the wind. If you try to simulate this with old methods, your computer would need to be the size of a planet to do the math in a reasonable time.

Enter "Zandpack."

Think of Zandpack as a brand-new, super-smart navigation app for these electron crowds. It's a software tool created by researchers to simulate exactly how these tiny devices behave when hit by fast-changing electrical signals.

Here is how it works, broken down into simple concepts:

1. The "Lego" Problem (The Fitting)

To simulate the electrons, the computer needs to know how the "walls" of the building (the electrodes) interact with the "people" inside. In physics, this interaction is described by a complex, jagged mountain range of data.

The Old Way: Trying to calculate every single jagged peak of that mountain in real-time is impossible.
The Zandpack Way: The software takes that jagged mountain and approximates it using a stack of smooth, round Lego bricks (mathematically called "Lorentzians"). Instead of tracking the jagged peaks, it tracks how these smooth bricks stack up. This turns a nightmare of calculus into a manageable set of rules.

2. The "Ghost" Assistants (Auxiliary Modes)

This is the magic trick. Usually, to know where the electrons are right now, you have to remember where they were a long time ago and how the wind changed since then. This requires a massive amount of memory.

The Analogy: Imagine you are trying to predict the path of a ball rolling down a hill. Instead of remembering the entire history of the wind, you hire a team of ghost assistants.
How it works: Zandpack creates these "ghost" variables (called auxiliary modes). Each ghost carries a tiny piece of the history. As the simulation runs, the main computer just asks the ghosts, "What's the wind doing?" and the ghosts update their positions instantly.
The Result: Instead of needing a supercomputer to remember the past, the software just solves a standard set of moving equations. It's like turning a heavy, slow-moving truck into a fleet of fast, agile motorcycles.

3. The Three Test Drives

The researchers tested their new app on three very different scenarios to prove it works:

The Spin Switch (Hydrogen on Graphene):
Imagine a single hydrogen atom sitting on a graphene sheet (a material like a chicken wire made of carbon). When a THz pulse hits it, the hydrogen atom flips its magnetic "spin" (like a tiny compass needle). Zandpack simulated this flip, showing how the "spin current" (a flow of magnetic direction) moves. It even calculated how much "information" was shared between the hydrogen and the graphene, like two friends whispering secrets.
The STM Tip (Graphene Ribbon):
Imagine a tiny metal needle (like a pin) hovering over a narrow strip of graphene. The researchers simulated what happens when a strong pulse hits the needle. They found that if the pulse pushes electrons out of the strip, it's smooth. But if it tries to pull electrons in against the flow, the electrons start vibrating wildly (oscillating). Zandpack caught these high-speed vibrations that older, "lazy" simulations would have missed.
The Gold Gap (The Break-Junction):
Imagine two gold wires almost touching, with a tiny gap between them. As you pull them apart, the gap widens. Zandpack showed that as the gap gets bigger, the electrons stop behaving like they are in a steady state and start reacting to the speed of the change. It proved that for very fast pulses, you can't just assume things are static; you have to watch the movie frame-by-frame.

Why Does This Matter?

We are moving toward a future where computers and sensors are made of single atoms. To design these, engineers need to know how they react to light and electricity in picoseconds (trillionths of a second).

Old tools were like trying to watch a movie by looking at one frozen frame at a time. Zandpack is like having a high-speed camera that can capture the entire action, even when the scene is chaotic and changing faster than the eye can see.

In a nutshell: Zandpack is a new, open-source tool that makes it possible to simulate the chaotic, high-speed dance of electrons in future nanodevices, using a clever trick of "ghost assistants" to do the heavy lifting. It turns an impossible math problem into a solvable one, helping us design the electronics of tomorrow.

1. Problem Statement

The paper addresses the computational challenge of simulating time-dependent quantum transport in open nanoscale systems driven far from equilibrium.

The Core Difficulty: Accurately modeling the evolution of electrons in a device region connected to electrodes under strong, time-dependent biases (e.g., THz or optical pulses).
Limitations of Existing Methods:
- Non-Equilibrium Green's Function (NEGF): While established, standard time-dependent NEGF is computationally heavy, often requiring the calculation of the full density matrix at every time step or complex embedding self-energies.
- Alternative Approaches: Methods like the Driven Liouville-von Neumann (DLvN) approach are phenomenological and not strictly based on NEGF. Other codes (e.g., NESSi, Tkwant) often struggle with scaling for large systems or require specific approximations that limit their applicability to first-principles Density Functional Theory (DFT).
Specific Needs: A method that can handle arbitrary electrode electronic structures, incorporate mean-field electron-electron interactions (via DFT), and remain computationally efficient enough to simulate large systems with time-dependent biases.

2. Methodology

The authors present Zandpack, an open-source Python code implementing a refined Auxiliary Mode Expansion (AME) approach. The methodology relies on transforming the integro-differential equations of NEGF into a system of coupled Ordinary Differential Equations (ODEs).

Key Theoretical Components:

Pole Expansion of Level-Width Functions:
- The electrode level-width functions ( $\Gamma_\alpha$ ), which describe the coupling between the device and electrodes, are fitted to a sum of Lorentzian functions.
- This allows the energy integrals in the self-energies to be solved analytically using the residue theorem, converting the history-dependent memory terms into auxiliary variables.
Auxiliary Mode Expansion (AME):
- The method introduces auxiliary wave vectors ( $\vec{\Psi}_{\alpha xc}$ ) and coupling matrices ( $\Omega$ ) that evolve according to a set of coupled ODEs.
- This avoids the need for matrix-matrix products at every time step, significantly reducing computational cost.
- The system is solved using an explicit Runge-Kutta adaptive solver.
Handling Non-Orthogonal Bases:
- Since DFT codes (like SIESTA and DFTB+) often use non-orthogonal basis sets, the authors implement a transformation (based on Löwdin orthogonalization) to map the non-orthogonal device-electrode overlap into an orthogonal basis compatible with the AME theory.
Linearization of the Hamiltonian:
- To avoid the high cost of recalculating the full DFT Hamiltonian at every time step, the authors employ a linearization scheme.
- The Hamiltonian dependence on the density matrix is approximated via a Taylor expansion around the equilibrium state, utilizing the Mulliken charge matrix. This allows for fast updates of the Hamiltonian based on the instantaneous charge distribution.
Steady-State Initialization:
- The code includes a solver to find the initial steady-state density matrix and auxiliary vectors self-consistently before time propagation begins, ensuring the system starts in a physical equilibrium state.

3. Key Contributions

Zandpack Software: A modular, open-source Python package that interfaces with external DFT codes (specifically SIESTA and DFTB+) via a generic Python wrapper.
Refined AME Algorithm: A novel decomposition of the level-width function (Eq. 16 in the paper) that ensures the method produces no spurious currents under equilibrium conditions, a common issue in previous implementations.
General Linearization Scheme: A robust method to incorporate mean-field density dependence (TDDFT-like effects) into time-propagation without the full cost of iterative DFT calculations at every step.
Broad Applicability: The framework is designed to handle various system types, including normal metals, semiconductors, and superconducting junctions (via Bogoliubov-de Gennes extension).

4. Results and Demonstrations

The paper validates Zandpack using three distinct examples of increasing complexity:

Hydrogen on Graphene (Hubbard Model):
- Setup: A tight-binding model with a mean-field Hubbard $U$ term simulating a hydrogen atom on graphene.
- Findings: The code successfully simulates spin-polarized currents, spin transitions, and mutual information. It demonstrates a pump-probe setup where an IR pulse triggers a transition from a spin-polarized to a spin-unpolarized state, observable via rectified charge transfer.
STM on Armchair Graphene Nanoribbon (AGNR) (DFTB+):
- Setup: A metallic STM tip probing a semiconducting AGNR using the DFTB+ code.
- Findings: The simulation captures the rectified current and rapid oscillations when the bias drives electrons from the valence band of the AGNR into the tip. Crucially, it highlights the importance of density feedback: a static Hamiltonian fails to capture high-frequency oscillations and overestimates current, whereas the dynamic (linearized) Hamiltonian correctly models the electrostatic repulsion and band-gap transitions.
Gold Break-Junction (SIESTA):
- Setup: A gold atomic junction with varying gap distances, modeled using SIESTA with the linearized Hamiltonian.
- Findings: The results show significant deviations between adiabatic (instantaneous) approximations and full time-dependent simulations, particularly for wider gaps where coupling is weak. This underscores the necessity of time-dependent methods for sub-picosecond dynamics.

5. Significance

Bridging Theory and Experiment: Zandpack provides a practical tool for interpreting emerging experimental techniques like THz-STM and pump-probe spectroscopy, where non-equilibrium dynamics are central.
Computational Efficiency: By converting the problem into a set of ODEs and utilizing linearization, the method scales favorably (roughly $N^2$ with orbital count for intermediate sizes), making it feasible to simulate experimentally relevant device sizes that were previously too large for time-dependent NEGF.
First-Principles Integration: The ability to interface with standard DFT codes (SIESTA, DFTB+) allows researchers to move beyond simplified tight-binding models to realistic material descriptions while maintaining computational tractability.
Open Science: As an open-source tool with Jupyter notebook demonstrations, it lowers the barrier to entry for researchers wishing to perform complex time-dependent quantum transport simulations.

In conclusion, Zandpack represents a significant step forward in the simulation of nanoelectronics, offering a general, efficient, and physically rigorous framework for studying electron dynamics in open quantum systems under strong time-dependent perturbations.