Nonuniform density scaling in the quasi-two-dimensional (quasi-2D) regime is an important and challenging aspect of the density functional theory. Semilocal exchange-correlation energy functionals, developed by solving the dimensional crossover criterion in the quasi-2D regime, have great theoretical and practical importance. However, the only semilocal generalized gradient approximation (GGA) that has been designed to satisfy this criterion is the Q2D-GGA [L. Chiodo et al., Phys. Rev. Lett. 108, 126402 (2012)]. Here, we establish the applicability, broadness, and accuracy of the Q2D-GGA functional by performing an extensive assessment of this functional for transition metal surface energies. The important characteristic of the surface density localization and oscillation due to the rearrangement of the d electrons is also shown for different metal surfaces.

The possibility to design and synthesize two-dimensional (2D) materials, starting from graphene and including most complex 2D crystals, as hybrid perovskites, is driving exciting advances in technological applications.1–3 Unexpected physico-chemical effects have also been reported in 2D-materials, well beyond the expected physical changes induced by the low dimensionality and the related quantum confinement.4,5 In this framework, the need emerges for a proper description of 2D and quasi-2D electronic distributions, whose properties can determine in peculiar ways the overall behavior of the isolated sheets and layered hetero-materials. In the realm of Kohn–Sham (KS) density functional theory (DFT),6,7 which is the best-assessed method for the electronic structure description of atoms, molecules, and solids, the key quantity is the exchange-correlation (XC) energy functional Exc[n] describing all the electron–electron interaction beyond the Hartree contribution.8–17 Hence, a refined description of the electronic density is based on the quality of the approximation used for the XC functional. Powerful exact conditions of the XC functional have been found using the scaling transformations for the density,18,19 which can be written in the general form

nM,a(r)=det(M)n(Mr+a),
(1)

where M is a real invertible 3 × 3 matrix, aR3, such that the scaled density preserves the particle number ∫nM,a(r)dr = ∫n(r)dr = N. The most important density scalings from Eq. (1) are the ones given by a = 0 and diagonal matrix M with positive elements Mij = λjδij, λj > 0. Thus, the uniform density scaling, defined by λ1 = λ2 = λ3 = λ ≥ 0 [i.e., nλ(r) = λ3n(λr)], is very useful for the XC functional development. The exchange energy functional scales as Ex[nλ] = λEx[n], implying that the exchange energy per particle behaves as ϵxλ. We recall that Ex[n] = ∫drn(r)ϵx(r). On the other hand, in the high-density limit of finite systems, the correlation energy must scale to a finite constant limλEc[nλ]=EcGL2, where EcGL2 is the Görling–Levy second-order perturbation energy,20–22 while in the low-density limit,20 the correlation behaves as exchange (i.e., limλ→0Ec[nλ] = λEc[n], implying that ϵcλ).

The density scalings of Eq. (1) can also conduct the system into highly anisotropic states, and the most important examples are the two-dimensional nonuniform scaling of the density, defined by λ1 = 1 and λ2 = λ3 = λ ≥ 0 [i.e., nλzy(x,y,z)=λ2n(x,λy,λz)], under which the system approaches the 1D limit when λ and the one-dimensional nonuniform scaling of the density, defined by λ1 = λ2 = 1 and λ3 = λ ≥ 0 [i.e., nλz(x,y,z)=λn(x,y,λz)], under which the system approaches the 2D limit when λ.

Both nonuniform scalings nλzy(x,y,z) and nλz(x,y,z) are of utmost importance for practical applications because at large λ, they confine the electrons in quasi-1D and quasi-2D regimes, respectively. Under such confinements, new phenomena can appear or/and specific electronic properties can be enhanced. Nevertheless, the two-dimensional nonuniform scaling of the density nλzy(x,y,z) was less investigated because the true Coulomb interaction diverges in 1D, and there are no known exact conditions for the dimensional crossover of XC energy from 3D to 1D. On the other hand, a simple analysis of the one-dimensional nonuniform scaling of the density nλz(x,y,z) proved that the exchange energy per particle must behave as23 

ϵxϵxLDAbs1/2,   when   s,
(2)

in order to be finite in the 2D limit. Here, ϵxLDA=3(3π2n)1/3/(4π) is the 3D Local Density Approximation (LDA) exchange energy per particle, s=|n|/[2(3π2)1/3n4/3] is the reduced gradient of the density, and b is a parameter that can be found from the 2D limit of the quasi-2D infinite barrier model (IBM) quantum well. We recall that the quasi-2D IBM quantum well of thickness L in the z-direction23–27 is defined by the KS effective potential

vKS(x,y,z)=0,z[0,L],otherwise
(3)

such that by shrinking the z-coordinate, keeping fixed the total number of electrons per unit area (n2D) is equivalent with a nonuniform scaling in one dimension [i.e., nλz(x,y,z)=λn(x,y,λz), with λL−1]. When L → 0, then the true 2D uniform electron gas limit is recovered, and for b = 0.5217, Eq. (2) correctly gives the 2D LDA exchange energy per particle. Such a powerful condition was incorporated into the Q2D generalized gradient approximation (GGA) XC functional of Ref. 25. Moreover, the Q2D-GGA fulfills not only the 2D LDA exchange (ϵx2DLDA) but also the 2D LDA correlation (ϵc2DLDA) using the accurate parametrization of Quantum Monte Carlo data proposed in Ref. 28. To our knowledge, the Q2D-GGA is the only functional accurate for both the uniform scaling nλ(r) and the one-dimensional nonuniform scaling of the density nλz(x,y,z). Here, we recall that the recent Made-Very-Simple (MVS)29 and Strongly-Constrained-Appropriately-Normed (SCAN)9 meta-GGA functionals keep the condition of Eq. (2), relaxing the value of the parameter b such that bMVS = 1.878 and bSCAN = 5.808. Consequently, they are performing modestly for the quasi-IBM quantum well, as shown in Fig. 11 of Ref. 30. Moreover, the MVS and SCAN meta-GGAs do not contain any quasi-2D conditions for their correlation functionals.

The Q2D-GGA was reformulated from the popular Perdew–Burke–Ernzerhof (PBE) GGA8 to constrain Ex/N and Ec/N to a finite limit under nonuniform scaling in one dimension to be accurate for the quasi-2D IBM quantum well and exact in its 2D limit. Besides, other important 3D exact constraints are also respected by the Q2D-GGA,25 such as the fulfillment of the second-order gradient expansions (GE2) of exchange31 and correlation (only in high-density limit),32 and the Lieb–Oxford bound.33,34 In particular, we mention that the GE2 exchange energy has high relevance for solid-state physics and surfaces, being the key element in the construction of PBEsol GGA.35 Note that the GGA functional of Armiento and Mattsson (AM05)36 is also a good functional for solids and surfaces, constructed from the airy gas and jellium surfaces. An article by Ropo et al.37 suggests that the accuracy of PBEsol and AM05 functionals is in same level for surface calculations. Hence, in the calculation, we do not include the AM05 for the comparison.

The Q2D-GGA has been successfully applied to close-packed (111) surfaces of several transition metals. It has been proved that, due to a rearrangement of the d electrons at the (111) surface, a localized plane of d electrons can be formed with mild quasi-2D character.25 Consequently, the Q2D-GGA and LDA become accurate for such systems, providing realistic surface energies. We recall that LDA performs reasonably for the mild quasi-2D regime, being better than popular GGA functionals.23 To put the Q2D-GGA functional performance into a broader prospect, in this article, we study various surface energies of transition metals: (100), (110), and (111) surfaces of face-centered and body-centered cubic (fcc and bcc) metals, as well as (0001), (101¯0), and (112¯0) surfaces of hexagonal closed packed (hcp) metals. We compare the Q2D-GGA surface energy results with LDA, PBE, and PBEsol ones.

All the calculations are performed with the Vienna Ab initio Simulation Package (VASP) code.39–42 The equilibrium lattice constants of all the transition metals are computed using LDA, PBE, PBEsol, and Q2D-GGA functionals. These lattice constants are used to form corresponding surface slabs for every functional.

We have done a surface energy convergence test by varying the thickness of the slab. In Fig. 1, we show the variation of the surface energy with the number of layers (N) for the Q2D-GGA functional. The upper panel contains surfaces of fcc platinum (Pt) and the lower panel contains bcc tantalum (Ta). We observe that (i) the quantum size effects are small and the quantum oscillations are dumped with respect to the case of simple metals.43 However, the oscillations are more pronounced for the (110) surfaces, (ii) for all the surfaces of Pt and Ta, the results are well converged when N ≥ 10, and (iii) our analysis agrees well with the one shown in Fig. 3 of Ref. 44.

FIG. 1.

Surface energies of (111), (110), and (100) surfaces of fcc Pt (upper panel) and bcc Ta (lower panel) are shown by varying number of layers.

FIG. 1.

Surface energies of (111), (110), and (100) surfaces of fcc Pt (upper panel) and bcc Ta (lower panel) are shown by varying number of layers.

Close modal

We consider thirteen layers-thick surface slabs with (1 × 1) surface unit cells and more than 20 Å vacuum. All the atomic positions are fully relaxed. In all our calculations, we employ the conjugate gradient algorithm and the Gaussian smearing with width 0.05 eV. 16 × 16 × 1 Γ-centered Monkhorst–Pack k-points are used to sample the Brillouin zone (BZ) integration with an energy cutoff of 800 eV for the planewave basis set and an energy convergence criterion of 1.0 × 10−6 eV. Except iron and cobalt, all the bulk and surface calculations are done without considering the spin-polarization.

To investigate the performance of density functional methods, we consider the surface energies of the above mentioned real metallic fcc, bcc, and hcp structures. The surface energy is defined as the energy required to construct the surface from the bulk structure,

σ=12αϵNslabNϵatombulk,
(4)

where ϵNslab is the total energy of the relaxed surface slab with N atoms, ϵatombulk is energy of the bulk per atom, and α is the slab surface area. The 1/2 factor accounts for the two surfaces in the slab unit cell.

In Table I, we report all the calculated surface energy results and observe the tendency as PBE > PBEsol > LDA > Q2D-GGA. For the reference, 0 K extrapolated experimental data38 are provided against all the cases. In the case of the fcc transition metals, the most stable surface is the close-packed one and the (111) Q2D-GGA surface energies of Au, Pt, and Pd are in excellent agreement with the experimental values, while LDA is the most accurate for Ag, Cu, Ir, and Rh. The fcc results are in agreement with the ones of Ref. 25, and for all fcc metals, the (111) Q2D-GGA surface energies are remarkably close to the experimental values. For the bcc transition metals, the most stable surface is the (110) facet. The Q2D-GGA is the most accurate for Nb and Ta. At the same time, for the other transition metals (Cr, Fe, Mo, V, and W), it shows a significant overestimation of the (110) surface energy and the PBEsol and LDA are more accurate. The remarkable performance of Q2D-GGA is also more evident in the case of the most stable (0001) surface energies of the hcp transition metals, as four cases out of total seven considered metals are more close to the experimental energies. However, for the hcp metals, the (101¯0) and (112¯0) facets are also quite stable, contributing significantly to the experimental value.

TABLE I.

Transition metal surface energies (J/m2). The experimental reference values are supplied for the most stable surface of each metal. The closest result to the experiment for the most stable surface is highlighted in bold.

MetalsSurfaceLDAPBEPBEsolQ2D-GGAExpt.38 
Face-centered cubic (fcc) 
 (100) 1.29 0.80 1.09 1.48  
Ag (110) 1.35 0.86 1.17 1.58  
 (111) 1.17 0.73 0.98 1.34 1.25 
 (100) 1.35 0.86 1.19 1.60  
Au (110) 1.40 0.90 1.23 1.67  
 (111) 1.14 0.71 0.98 1.36 1.50 
 (100) 1.98 1.45 1.78 2.20  
Cu (110) 2.05 1.51 1.88 2.34  
 (111) 1.77 1.30 1.58 1.98 1.83 
 (100) 3.47 2.88 3.34 3.79  
Ir (110) 3.54 2.91 3.35 3.84  
 (111) 2.90 2.36 2.77 3.19 3.00 
 (100) 2.13 1.53 1.92 2.36  
Pd (110) 2.23 1.60 2.00 2.48  
 (111) 1.87 1.33 1.69 2.12 2.05 
 (100) 2.44 1.87 2.26 2.70  
Pt (110) 2.48 1.87 2.31 2.79  
 (111) 2.00 1.49 1.85 2.27 2.48 
 (100) 3.04 2.40 2.85 3.30  
Rh (110) 3.11 2.43 2.88 3.37  
 (111) 2.65 2.08 2.47 2.89 2.70 
Body-centered cubic (bcc) 
Cr (100) 4.19 3.62 4.03 4.24  
 (110) 3.76 3.18 3.59 3.62 2.30 
 (111) 4.08 3.48 3.86 4.21  
Fe (100) 3.44 2.50 3.04 3.55  
 (110) 3.17 2.43 2.92 3.29 2.48 
 (111) 3.47 2.74 3.20 3.70  
 (100) 3.73 3.26 3.60 3.99  
Mo (110) 3.24 2.82 3.12 3.48 3.00 
 (111) 3.52 3.06 3.36 3.77  
 (100) 2.71 2.31 2.60 2.93  
Nb (110) 2.38 2.07 2.30 2.60 2.70 
 (111) 2.71 2.37 2.61 2.95  
 (100) 2.84 2.46 2.76 3.07  
Ta (110) 2.66 2.36 2.60 2.90 3.15 
 (111) 3.03 2.70 2.99 3.33  
 (100) 2.83 2.35 2.67 3.09  
(110) 2.78 2.38 2.66 3.04 2.55 
 (111) 3.07 2.64 2.95 3.38  
 (100) 4.42 4.01 4.36 4.67  
(110) 3.64 3.25 3.57 3.82 3.68 
 (111) 4.01 3.61 3.91 4.27  
Hexagonal close packed (hcp) 
Co (0001) 2.76 2.09 2.53 2.86 2.55 
 (101¯0) 2.95 2.25 2.71 3.17  
 (112¯0) 3.25 2.48 3.03 3.47  
Cd (0001) 0.37 0.21 0.41 0.56 0.69 
 (101¯0) 0.74 0.46 0.72 0.90  
 (112¯0) 0.93 0.57 0.86 1.08  
 (0001) 1.68 1.49 1.90 2.14 2.15 
Hf (101¯0) 2.14 1.93 2.10 2.36  
 (112¯0) 2.10 1.84 2.06 2.32  
 (0001) 2.74 2.26 3.01 3.43 3.05 
Ru (101¯0) 3.59 2.98 3.42 3.87  
 (112¯0) 4.07 3.40 3.87 4.35  
 (0001) 2.21 1.97 2.16 2.25 2.10 
Ti (101¯0) 2.00 2.03 2.22 2.37  
 (112¯0) 2.17 1.89 2.08 2.33  
 (0001) 0.48 0.28 0.47 0.74 0.91 
Zn (101¯0) 0.83 0.55 0.76 1.05  
 (112¯0) 1.20 0.86 1.22 1.55  
 (0001) 1.57 1.38 1.74 1.98 2.00 
Zr (101¯0) 1.90 1.68 1.84 2.09  
 (112¯0) 1.93 1.66 1.83 2.11  
MetalsSurfaceLDAPBEPBEsolQ2D-GGAExpt.38 
Face-centered cubic (fcc) 
 (100) 1.29 0.80 1.09 1.48  
Ag (110) 1.35 0.86 1.17 1.58  
 (111) 1.17 0.73 0.98 1.34 1.25 
 (100) 1.35 0.86 1.19 1.60  
Au (110) 1.40 0.90 1.23 1.67  
 (111) 1.14 0.71 0.98 1.36 1.50 
 (100) 1.98 1.45 1.78 2.20  
Cu (110) 2.05 1.51 1.88 2.34  
 (111) 1.77 1.30 1.58 1.98 1.83 
 (100) 3.47 2.88 3.34 3.79  
Ir (110) 3.54 2.91 3.35 3.84  
 (111) 2.90 2.36 2.77 3.19 3.00 
 (100) 2.13 1.53 1.92 2.36  
Pd (110) 2.23 1.60 2.00 2.48  
 (111) 1.87 1.33 1.69 2.12 2.05 
 (100) 2.44 1.87 2.26 2.70  
Pt (110) 2.48 1.87 2.31 2.79  
 (111) 2.00 1.49 1.85 2.27 2.48 
 (100) 3.04 2.40 2.85 3.30  
Rh (110) 3.11 2.43 2.88 3.37  
 (111) 2.65 2.08 2.47 2.89 2.70 
Body-centered cubic (bcc) 
Cr (100) 4.19 3.62 4.03 4.24  
 (110) 3.76 3.18 3.59 3.62 2.30 
 (111) 4.08 3.48 3.86 4.21  
Fe (100) 3.44 2.50 3.04 3.55  
 (110) 3.17 2.43 2.92 3.29 2.48 
 (111) 3.47 2.74 3.20 3.70  
 (100) 3.73 3.26 3.60 3.99  
Mo (110) 3.24 2.82 3.12 3.48 3.00 
 (111) 3.52 3.06 3.36 3.77  
 (100) 2.71 2.31 2.60 2.93  
Nb (110) 2.38 2.07 2.30 2.60 2.70 
 (111) 2.71 2.37 2.61 2.95  
 (100) 2.84 2.46 2.76 3.07  
Ta (110) 2.66 2.36 2.60 2.90 3.15 
 (111) 3.03 2.70 2.99 3.33  
 (100) 2.83 2.35 2.67 3.09  
(110) 2.78 2.38 2.66 3.04 2.55 
 (111) 3.07 2.64 2.95 3.38  
 (100) 4.42 4.01 4.36 4.67  
(110) 3.64 3.25 3.57 3.82 3.68 
 (111) 4.01 3.61 3.91 4.27  
Hexagonal close packed (hcp) 
Co (0001) 2.76 2.09 2.53 2.86 2.55 
 (101¯0) 2.95 2.25 2.71 3.17  
 (112¯0) 3.25 2.48 3.03 3.47  
Cd (0001) 0.37 0.21 0.41 0.56 0.69 
 (101¯0) 0.74 0.46 0.72 0.90  
 (112¯0) 0.93 0.57 0.86 1.08  
 (0001) 1.68 1.49 1.90 2.14 2.15 
Hf (101¯0) 2.14 1.93 2.10 2.36  
 (112¯0) 2.10 1.84 2.06 2.32  
 (0001) 2.74 2.26 3.01 3.43 3.05 
Ru (101¯0) 3.59 2.98 3.42 3.87  
 (112¯0) 4.07 3.40 3.87 4.35  
 (0001) 2.21 1.97 2.16 2.25 2.10 
Ti (101¯0) 2.00 2.03 2.22 2.37  
 (112¯0) 2.17 1.89 2.08 2.33  
 (0001) 0.48 0.28 0.47 0.74 0.91 
Zn (101¯0) 0.83 0.55 0.76 1.05  
 (112¯0) 1.20 0.86 1.22 1.55  
 (0001) 1.57 1.38 1.74 1.98 2.00 
Zr (101¯0) 1.90 1.68 1.84 2.09  
 (112¯0) 1.93 1.66 1.83 2.11  

The mean error (ME), mean absolute error (MAE), and mean absolute relative error (MARE) for fcc, bcc, and hcp sets along with the overall errors are tabulated in Table II. The upper panel represents the errors of most-stable surfaces from the experimental surface energies. The MAEs of both fcc and hcp for Q2D-GGA are very accurate in comparison to all other methods, whereas for bcc, PBEsol has the least MAE of 0.43 J/m2, and the MAEs of bcc for all the methods lies between 0.4 J/m2 and 0.5 J/m2. Overall, Q2D-GGA is the best functional in describing the surface energies of most-stable surfaces of transition metals with the least MAE of 0.27 J/m2.

TABLE II.

LDA, PBE, PBEsol, and Q2D-GGA error statistics [mean errors (MEs), mean absolute errors (MAEs), and mean absolute relative errors (MAREs)] of the most stable surface energies (γcalc) and the Wulff weighted average surface energies (γWulffcalc) with respect to experimental values.

ErrorLDAPBEPBEsolQ2D-GGA
γcalc 
Fcc 
ME (J/m2−0.19 −0.69 −0.36 0.05 
MAE (J/m20.19 0.69 0.36 0.15 
MARE (%) 9.57 34.65 18.44 7.14 
Bcc 
ME (J/m20.25 −0.20 0.13 0.41 
MAE (J/m20.50 0.45 0.43 0.50 
MARE (%) 19.54 16.15 16.77 20.10 
Hcp 
ME (J/m2−0.23 −0.54 −0.18 0.07 
MAE (J/m20.33 0.54 0.19 0.17 
MARE (%) 22.95 35.80 19.93 10.11 
Overall 
ME (J/m2−0.05 −0.47 −0.13 0.17 
MAE (J/m20.33 0.55 0.32 0.27 
MARE (%) 17.35 28.86 17.38 12.44 
γWulffcalc 
Fcc 
ME (J/m2−0.11 −0.62 −0.28 0.14 
MAE (J/m20.14 0.62 0.28 0.20 
MARE (%) 7.08 31.78 14.84 9.40 
Bcc 
ME (J/m20.32 −0.15 0.19 0.47 
MAE (J/m20.52 0.42 0.44 0.54 
MARE (%) 20.58 15.50 17.39 21.20 
Hcp 
ME −0.03 −0.37 −0.05 0.24 
MAE 0.20 0.37 0.16 0.24 
MARE 12.68 25.72 11.03 9.92 
Overall 
ME (J/m20.05 −0.38 −0.04 0.28 
MAE (J/m20.28 0.47 0.29 0.32 
MARE (%) 13.44 24.33 14.42 13.50 
ErrorLDAPBEPBEsolQ2D-GGA
γcalc 
Fcc 
ME (J/m2−0.19 −0.69 −0.36 0.05 
MAE (J/m20.19 0.69 0.36 0.15 
MARE (%) 9.57 34.65 18.44 7.14 
Bcc 
ME (J/m20.25 −0.20 0.13 0.41 
MAE (J/m20.50 0.45 0.43 0.50 
MARE (%) 19.54 16.15 16.77 20.10 
Hcp 
ME (J/m2−0.23 −0.54 −0.18 0.07 
MAE (J/m20.33 0.54 0.19 0.17 
MARE (%) 22.95 35.80 19.93 10.11 
Overall 
ME (J/m2−0.05 −0.47 −0.13 0.17 
MAE (J/m20.33 0.55 0.32 0.27 
MARE (%) 17.35 28.86 17.38 12.44 
γWulffcalc 
Fcc 
ME (J/m2−0.11 −0.62 −0.28 0.14 
MAE (J/m20.14 0.62 0.28 0.20 
MARE (%) 7.08 31.78 14.84 9.40 
Bcc 
ME (J/m20.32 −0.15 0.19 0.47 
MAE (J/m20.52 0.42 0.44 0.54 
MARE (%) 20.58 15.50 17.39 21.20 
Hcp 
ME −0.03 −0.37 −0.05 0.24 
MAE 0.20 0.37 0.16 0.24 
MARE 12.68 25.72 11.03 9.92 
Overall 
ME (J/m20.05 −0.38 −0.04 0.28 
MAE (J/m20.28 0.47 0.29 0.32 
MARE (%) 13.44 24.33 14.42 13.50 

The experimental surface energies have been obtained from liquid metal surface tension measurements or from polycrystalline samples with multiple facets such that they can be seen as weighted averages over the surface energies of various crystal faces. The weight of a given facet should be proportional to its stability, in the sense that the more stable the facet is, the more significant contribution the surface energy must have. We consider the weights given by the percentage of the area exposed by a given surface, using the Wulff representation of crystal equilibrium shapes.38,45,46 The details of the percentage of exposed facets from the Wulff construction are given in the supplementary material. This approach has been recently discussed in Refs. 38 and 45. The surface energy of each facet is used to construct the Wulff structure, and the Wulff contributions at 0 K are shown in the supplementary material. Using Wulff percentage of each facet, we calculate the equilibrium surface energy of every transition metal corresponding to every exchange-correlation functionals. Different errors of such surface energies in accordance with the experimental values are tabulated in the lower panel of Table II. Due to the comparatively large surface energies of facets other than the most stable surfaces, the overall MAE of Q2D-GGA is larger than the PBEsol functional. The principal contribution is from the magnetic calculation of iron and cobalt. The LDA method has the lowest MAE of 0.28 J/m2. It is observed that the Q2D-GGA functional is the best when we compare the surface energies of the most stable surface of transition metals with the 0 K extrapolated experimental data, but the Q2D-GGA is not up to the mark in the case of Wulff construction. However, such deterioration in the case of the Wulff constructed averaged value should not be taken critically, as it is not clear whether the 0 K extrapolated values would be the same as the most stable surfaces or Wulff averaged surfaces.

The connection of the quasi-2D regime with the surfaces of transition metals is analyzed in Fig. 2 by plotting the plane-averaged electron-density variation with the normalized distance. In the case of fcc(111) surfaces, presented in the left panel, we observe that Pt and Pd have two prominent peaks, which represent localized layers of d-electrons, that give a quite important quasi-2D character of these surfaces. These layers are considerably smaller for Rh(111), being closer to the Friedel oscillations of simple metals, represented here by the Al(111) curve. These observations are in agreement with the results of Table I because the Q2D-GGA surface energies are very accurate for Pt and Pd but are overestimating the Rh(111).

FIG. 2.

The electron density difference [Δn(z) = nsurf,l(z) + nsurf,r(z) − nbulk, where n(z) = ∫dx dy n(x, y, z) is the xy-averaged density and nsurf,l(z) and nsurf,r(z) are the densities of the left and right surfaces, respectively] vs normalized distance for selected fcc (left panel), bcc (middle panel), and hcp (right panel) metal surfaces. For comparison, the simple metal structures [fcc Al(111), bcc Na(110), and hcp Mg(0001)] are also considered.

FIG. 2.

The electron density difference [Δn(z) = nsurf,l(z) + nsurf,r(z) − nbulk, where n(z) = ∫dx dy n(x, y, z) is the xy-averaged density and nsurf,l(z) and nsurf,r(z) are the densities of the left and right surfaces, respectively] vs normalized distance for selected fcc (left panel), bcc (middle panel), and hcp (right panel) metal surfaces. For comparison, the simple metal structures [fcc Al(111), bcc Na(110), and hcp Mg(0001)] are also considered.

Close modal

For the (110) surfaces of bcc transition metals, as shown in the middle panel of Fig. 2, there is only one localized layer of electrons at the surface such that the quasi-2D regime is milder than the one found for the (111) surface. Consequently, Q2D-GGA is performing, in general, worse than that for close-packed surfaces, overestimating the results, as shown in Table I. We also observe that, going into the bulk region, the quantum oscillations of W(110) become similar to the Friedel oscillations of Na(110), which behave as limzΔn(z)cos(2kFz+ϕ)/[kfz]2.47 

Finally, in the right panel of Fig. 2, we report the density variation Δn for the close-packed (0001) surfaces of hcp metals. Both Ru and Hf transition metals have a localized layer of d-electrons at the surface, followed by a region where Δn is constant. Such a region does not allow the quasi-2D region of the localized layer to dissipate. This phenomenon is more evident for the Hf surface where the plateau has Δn(z) ≈ 0 as a boundary of the electron layer. Indeed, the Q2D-GGA is very accurate for the Hf(0001) surface energy.

In summary, we have assessed the performance of the Q2D-GGA for the surface energies of different fcc, bcc, and hcp transition metals. The Q2D-GGA shows its accuracy for the different facet surface energies, and its performance has been analyzed with respect to the most stable surface energy as well as the Wulff admixture of the different surfaces. The importance of the quasi-2D regime at the transition metal surfaces has been showed considering the localization of the charge density, providing a rationale for the accuracy of the Q2D-GGA.

Finally, we would like to underline the relevance of Q2D-GGA as a tool for studying systems with layered electronic charge distributions to be used in future investigations not only of transition metals surfaces but also of surfaces with adsorbates48,49 and of heterostructured materials.1–5 

See the supplementary material for the percentage of exposed facets from the Wulff construction for LDA, PBE, PBEsol, and Q2D-GGA.

A.P. acknowledges the financial support from the Department of Atomic Energy, Government of India. This work has been performed in a high-performance computing facility of NISER.

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Supplementary Material