Real-space methods for ab initio modelling of surfaces and interfaces under external potential bias

Abstract: Accurate ab initio modelling of surfaces and interfaces, especially under an applied external potential bias, is important for describing and characterizing various phenomena that occur in electronic, catalytic, and energy storage devices. Leveraging the ability of real-space density functional theory (DFT) codes to accommodate generic boundary conditions, we introduce two methods for applying an external potential bias that can be suitable for modelling surfaces and interfaces. In the first method, an external constant electric field is applied by modifying the DFT Hamiltonian via the introduction of an auxiliary linear potential while solving the electrostatic potential arising in DFT using a Poisson equation with zero-Neumann boundary conditions. The second method directly enforces the desired external potential bias by imposing constraints on the electrostatic potential, thereby naturally mimicking experimental conditions. We describe the underlying DFT governing equations for the two setups within the real-space formalism employing finite-element discretization. First, we validate the constant electric field setup within real-space finite-element DFT (DFT-FE) with an equivalent approach using plane-wave DFT (i.e., using periodic boundary conditions) on three representative benchmark systems, namely La-terminated Li$7$La$_3$Zr$_2$O${12}$, GaAs (111), and Al FCC (111) slabs. Subsequently, we present a comprehensive evaluation of the two setups in terms of the average ground-state properties, such as surface and adsorption energies. The methods developed in our work provide an attractive alternative to plane-wave DFT approaches in applying external potential bias that usually suffer from the periodic boundary conditions restrictions and poor scalability on parallel computing architectures.

Real-Space Methods for Ab Initio Modeling of Surfaces and Interfaces Under External Potential Bias

This paper describes innovative advancements in the ab initio modeling of surfaces and interfaces, addressing critical computational bottlenecks associated with the application of external potential biases. The authors introduce two computational frameworks within the real-space density functional theory (DFT) paradigm that accommodate generic boundary conditions and provide scalable solutions for large systems.

Methodological Advances

The authors propose two distinct methods for introducing external potential biases: the Constant Electric Field (CEF) setup and the Applied Potential Difference (APD) setup. Both methods are implemented within a real-space finite-element DFT framework (DFT-FE), enabling efficient and accurate large-scale simulations:

1. Constant Electric Field (CEF) Setup: In this method, an external constant electric field is applied by introducing an auxiliary linear potential into the DFT Hamiltonian. This is achieved by modifying the electrostatic potential with a Poisson equation, utilizing zero-Neumann boundary conditions to simulate neutral slabs. Validation against plane-wave DFT approaches demonstrates the equivalency and reliability of the CEF method for systems such as Li$_7$La$_3$Zr$_2$O$_{12}$, GaAs, and Al surfaces.
2. Applied Potential Difference (APD) Setup: This method enforces the external potential bias directly by imposing constraints on the electrostatic potential. It mimics experimental conditions more naturally and provides a more accurate control over the potential difference across the slab, relative to the controlled parameters in the CEF setup.

Numerical Results and Practical Implications

Through benchmarking with plane-wave DFT and extensive validation, the authors demonstrate the robustness of their approaches in handling systems with varying electronic properties, from insulators to metals. The paper reports on the effectiveness of both setups in capturing dielectric responses, surface energies, and adsorption properties.

• Dielectric Response: The results show close agreement between the CEF setup in DFT-FE and plane-wave DFT setups, validating the capability of the DFT-FE tool to produce accurate dielectric responses across a range of systems.
• Surface and Adsorption Energies: The variations in surface energies and adsorption properties across different potential biases indicate distinct behaviors between the CEF and APD setups, showing the latter's superior ability in reflecting experimental conditions in computational models.

Future Directions and Speculative Outlook

The development of these real-space methods significantly enhances the computational toolkit available for surface science and interface modeling, offering promising directions for future applications in electronics, catalysis, and energy storage. The ability to efficiently simulate large systems under controlled potential biases facilitates the exploration of properties related to electrochemical interfaces, surface reactivity under bias, and other critical phenomena in condensed matter physics. Furthermore, the scalability of the finite-element discretization opens avenues for incorporating more sophisticated electronic structure methods, such as hybrid functionals and beyond-DFT approaches, in future research.

In conclusion, this paper presents significant advancements in the real-space modeling of surfaces and interfaces, presenting flexible and scalable methodologies that align closely with experimental setups and offer novel insights into material behaviors under external biases.