First-Principles Study of the Temperature Dependence of Structural, Electronic, and Hyperfine Properties of the Cu(100) Surface

Abstract: In this work, we investigate the temperature-dependent behavior of the pure (undoped) Cu(100) surface using first-principles calculations within the Density Functional Theory framework. One of the main objectives is to determine whether the linear dependence of the predicted electric-field gradient (EFG) tensor on the outermost Cu atom on the Cu(100) surface arises from the same generation of the surface or from the reconstruction of the surface. To this end, we perform here a comprehensive $\it{ab}$ $\it{initio}$ study of the Cu(100) surface reconstruction and its associated structural, electronic, and hyperfine properties as a function of temperature, not only at the outermost atomic layer (i.e., the topmost Cu atom) but also as a function of atomic depth relative to the reconstructed surface. To study the temperature dependence of the EFG, we use experimentally determined temperature-dependent lattice parameters for bulk copper in our calculations. The anisotropic relaxation that arises when bulk symmetry is broken helps unravel the potential sources of EFG temperature dependence at the surface. Studying the electron density of conduction electrons $ρ$($\bf{r}$) at the atomic scale near the Cu nucleus and the atom-resolved partial density of states at the topmost Cu atom allows us to correlate the surface effect on the EFG with the bulk value. Finally, we correlate the temperature dependence of the EFG on the undoped Cu(100) surface with the linear behavior of the ''ionic'' contribution to the EFG.

• The paper demonstrates that intrinsic surface relaxation and reconstruction drive the linear decrease in EFG with temperature on Cu(100).
• It employs ab initio DFT calculations with a slab-supercell and full-potential APW+lo method for accurate electronic structure analysis.
• The study quantitatively matches TDPAC experimental data, validating the role of conduction-electron anisotropy in surface hyperfine properties.

First-Principles Characterization of Temperature Effects on the Cu(100) Surface: Structural, Electronic, and Hyperfine Properties

Overview and Context

This paper conducts a thorough ab initio exploration of the temperature-dependent properties of the pure Cu(100) surface using Density Functional Theory (DFT) calculations. The central focus is on the electric-field gradient (EFG) tensor at the topmost copper atoms—a property highly sensitive to local electronic anisotropy and routinely probed via time-differential perturbed angular correlations (TDPAC)—and its behavior with temperature. The study seeks to clarify whether the experimentally observed linear temperature dependence of the EFG for 111Cd probes on Cu(100) surfaces originates from intrinsic effects related to surface generation, relaxation, and reconstruction, or is dominated by impurity-induced modifications.

Computational Methodology

A slab-supercell approach with up to 11 inequivalent substrate layers (typically 7) is employed to simulate the Cu(100) surface, ensuring bulk-like conditions at deeper layers. Surface relaxation and reconstruction are determined via sequential DFT relaxations at various temperatures, taking experimentally measured lattice parameters for bulk copper as input for thermal expansion. Full-potential APW+lo calculations are implemented in WIEN2k, utilizing PBE-GGA exchange-correlation and rigorous convergence criteria for ionic forces. The EFG tensors are extracted from second derivatives of the total electrostatic potential evaluated at relaxed atomic positions.

Structural Properties Under Temperature Variation

Surface formation breaks the bulk FCC symmetry, leading to pronounced structural relaxation. The topmost interlayer separation contracts relative to the bulk, while the subsequent layer expands, consistent with medium-energy ion scattering experiments [29]. The relaxation affects only atoms coordinated outside the topmost plane, with in-plane neighbor distances unchanged regardless of temperature or reconstruction. Deep layers recover bulk symmetry and interatomic distances at elevated temperatures, demonstrating the localized nature of surface effects.

Electronic Structure and EFG Analysis

The EFG at each atomic site is determined by local charge density anisotropy. The calculations demonstrate a substantial enhancement of V33 at the topmost Cu atom relative to deeper layers, where the EFG approaches zero, as expected for bulk FCC copper. Electron density maps for conduction electrons elucidate the pronounced spatial redistribution at the surface, yielding strong axial symmetry about the [100] direction and leading to large positive EFG values. A detailed analysis of partial density of states (PDOS) confirms the dominant role of p-orbital anisotropy for the EFG at the surface. The p-contribution, due to proximity to the nucleus and favorable nodal structure, outweighs the d-contribution even when asymmetry counts are comparable.

A universal correlation between conduction electron and ionic contributions to the EFG, originally characterized for noncubic metals [31,32,33], is invoked to explain these results. Calculations reveal that the conduction-electron term is both dominant and oppositely signed to the lattice term, yielding an effective scaling factor Acu ≈ -10. This quantitative insight is consistent with known Sternheimer antishielding effects and prior EFG measurements.

Hyperfine Properties and Temperature Dependence

Ab initio predictions for V33 at the Cu(100) surface show a nearly linear decrease with increasing temperature, with reconstructed surfaces exhibiting slightly reduced slope relative to unrelaxed ones. Numerical fits yield V33(0 K) ≈ 2.35 × 10²¹ V/m² with slope m ≈ -2.2 × 10⁻⁴ per K. Experimental TDPAC data for 111Cd-doped Cu(100) reveals a similar linear trend, quantitatively matching in relative slope. This close agreement supports the assertion that surface generation and relaxation drive the linear EFG-temperature dependence, in contrast to the canonical $-aT^{3/2}$ law observed in bulk noncubic metals [19].

Charge density visualization across temperature (100 K vs 1000 K) corroborates the decrease in local anisotropy and reduction of EFG at elevated temperatures, as negative charge distribution contracts towards more symmetric (bulk-like) states.

Theoretical Implications and Future Directions

The work challenges the traditional notion that impurity-induced local electronic effects are the primary source of the linear EFG temperature dependence. Instead, it demonstrates that intrinsic surface reconstruction and thermal lattice expansion account for the majority of the effect, as captured by first-principles modeling. This insight has broader implications for EFG-based surface characterization in metals, especially in designing and interpreting hyperfine experiments.

Practically, the results bolster the interpretability of TDPAC and related measurements, providing quantitative ab initio baselines for surface-induced electronic anisotropy without impurity complications. Theoretically, the applicability of universal EFG correlations to surface states suggests new avenues for modeling hyperfine interactions in low-dimensional or nanostructured systems. Further research could extend these methods to more complex systems (e.g., doped surfaces, oxide formation, alloying) or assess dynamical effects beyond static lattice expansion.

Conclusion

This study combines DFT calculations with point-charge models and detailed PDOS analysis to elucidate the origin and temperature dependence of the EFG at the Cu(100) surface. It establishes that the linear decrease with temperature is fundamentally an intrinsic surface effect, not primarily caused by local impurity modifications. The reconstructed structural parameters are in excellent agreement with experiment, and electronic structure calculations fully account for observed hyperfine trends. These results advance the understanding of surface electronic and hyperfine properties in FCC metals, providing an authoritative baseline for future theoretical and experimental investigations on temperature-dependent surface phenomena.