The electronic density of states (DOS) highlights fundamental properties of materials that oftentimes dictate their properties, such as the band gap and Van Hove singularities. In this short note, we discuss how sharp features of the density of states can be obscured by smearing methods (such as the Gaussian and Fermi smearing methods) when calculating the DOS. While the common approach to reach a "converged" density of states of a material is to increase the discrete k-point mesh density, we show that the DOS calculated by smearing methods can appear to converge but not to the correct DOS. Employing the tetrahedron method for Brillouin zone integration resolves key features of the density of states far better than smearing methods

Anand, S

Brod, MK

Dylla, M

Ganose, AM

Jain, A

Munro, J

Park, J

Persson, KA

Snyder, GJ

Toriyama, MY

English

arXiv

Spiral - Imperial College Digital Repository

Comparison of the Tetrahedron Method to Smearing Methods for the Electronic Density of StatesMichael Y. Toriyama1, Alex M. Ganose2, Max Dylla1, Shashwat Anand1, Junsoo Park2,Madison K. Brod1, Jason Munro3, Kristin A. Persson4,5, Anubhav Jain2, and G. Jeffrey Snyder11Department of Materials Science and Engineering, Northwestern University, Evanston, IL 60208, USA.2Energy Technologies Area, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA.3The Materials Science Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA.4Department of Materials Science and Engineering, University of California, Berkeley, Berkeley, CA 94720, USA. and5The Molecular Foundry, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA.(Dated: March 8, 2021)The electronic density of states (DOS) highlights fundamental properties of materials that oftentimes dictatetheir properties, such as the band gap and Van Hove singularities. In this short note, we discuss how sharpfeatures of the density of states can be obscured by smearing methods (such as the Gaussian and Fermi smearingmethods) when calculating the DOS. While the common approach to reach a “converged” density of statesof a material is to increase the discrete k-point mesh density, we show that the DOS calculated by smearingmethods can appear to converge but not to the correct DOS. Employing the tetrahedron method for Brillouinzone integration resolves key features of the density of states far better than smearing methods.Perhaps the simplest descriptor of the electronic structure ofa material is the density of states, which condenses many fun-damental electrical and optical properties into a single infor-mative diagram. Features such as the band gap and the effec-tive mass of carriers are directly related to transport propertiesof the material, whereas sharp peaks (i.e. Van Hove singulari-ties) provide critical information about optical properties suchas the dielectric constant.Oftentimes, computational methods based on first-principles are used to calculate the density of states of mate-rials. However, computational approaches require a trade-offbetween computational cost and accuracy, contextualizing thenotion of “convergence” in which the property of interest iscalculated with increasing computational cost until the prop-erty no longer appears to change within the desired accuracy.For the density of states, convergence is especially impor-tant for generating salient features of the electronic structure.Such numerical artefacts may arise from the technique used tocalculate the density of states, which can broadly be dividedinto two categories: “smearing methods” and the “tetrahedronmethod”.Smearing methods (as described mathematically in the Ap-pendix) fix a continuous function at each band and k-point toapproximate the density of states. The tetrahedron method onthe other hand divides the Brillouin zone into tetrahedra, cal-culates the eigenenergies at the corners of each tetrahedron,and linearly interpolates the eigenenergies inside of each tetra-hedron to perform the integration. The tetrahedron methodis reminiscent of the trapezoidal rule for approximating theintegral of single-variable functions. As is the case withsingle-variable functions where a linear interpolation overes-timates regions of positive curvature and underestimates re-gions of negative curvature, the linear-tetrahedron method isprone to similar errors. In response, a correction to the in-tegration weights was introduced by Blöchl,[1] and the cor-rected Brillouin zone integration method is called the tetrahe-dron method with Blöchl corrections.In this short note, we compare the density of states thatare calculated using smearing methods and the tetrahedronmethod. We use first-principles Density Functional Theory(DFT) to calculate the density of states of the half-Heuslercompound TiNiSn (Materials Project ID: mp-924130).[2] Wecompare the density of states calculated using the tetrahe-dron method with Blöchl corrections[1] against that calculatedusing smearing methods. DFT calculations were performedusing the Vienna ab-initio simulation package (VASP).[3–5]Exchange-correlation effects were treated using the Perdew-Burke-Ernzerhof functional[6] for all calculations. We usethe recommended Ti_pv, Ni_pv, and Sn_d pseudopotentialsdistributed by VASP. Each structure was relaxed using a 520eV energy cutoff and a Γ-centered grid to sample k-points inthe Brillouin zone.[7] Spin-orbit coupling effects were not in-cluded in the calculations. The density of states is calculatedusing 5001 energy bins. We employ the Gaussian and Fermismearing methods, where we use a smearing width of 0.05 eV.As shown in Figure 1, key analytical features representa-tive of the electronic structure of a material are clearly evi-dent when the tetrahedron method with Blöchl corrections isemployed. Sharp Van Hove peaks such as those at 0.8 eVand 2 eV below the valence band maximum are visible fromthe density of states calculated using the tetrahedron method,whereas the peaks are obscured by Gaussian and Fermi smear-ings. Gaps in the density of states, such as the one at 1.6 eVabove the valence band maximum observed by the tetrahedronmethod, close when smearing methods are employed.Smearing methods should in principle reach the true densityof states by increasing the computational cost by e.g. increas-ing the k-point mesh density. However, as shown in Figure1, stark features such as Van Hove singularities and the bandedge shape are obscure even at higher k-point densities usingsmearing methods without carefully adjusting the smearingparameters, whereas they appear clearly with the tetrahedronmethod. Although the density of states of the Gaussian andFermi smearing methods separately converge to a stable dis-tribution as the k-point density increases, they do not approachthe density of states calculated using the tetrahedron method.Moreover, the width parameter of the smearing methods mustbe carefully chosen to reach the density of states calculatedusing the tetrahedron method. As shown in Figure 2, using acomparatively large smearing width of 50 meV obscures thearXiv:2103.03469v1  [cond-mat.mtrl-sci]  5 Mar 20212−5 −4 −3 −2 −1 0 1 2E - EVBM (eV)Density of States(a) Tetrahedron7×7×711×11×1115×15×1517×17×1721×21×21−5 −4 −3 −2 −1 0 1 2E - EVBM (eV)Density of States(b) GaussianTetrahedon7×7×711×11×1115×15×1517×17×1721×21×21−5 −4 −3 −2 −1 0 1 2E - EVBM (eV)Density of States(c) FermiTetrahedron7×7×711×11×1115×15×1517×17×1721×21×21FIG. 1: Density of states of TiNiSn using different k-point samplingmethods. The width parameter is fixed to 50 meV for the smearingmethods. The tetrahedron method (a) shows a clear band gap, vanHove singularities and flat DOS regions even with a 7×7×7 k-pointmesh. The Gaussian (b) and Fermi (c) smearing methods appear toconverge the density of states above a mesh size of 15, but even theconverged density of states blurs the features seen in the tetrahedronmethod and can produce artificial features even at a much denser k-point mesh.peaks at ∼ 2 eV below the valence band maximum, whereasusing a small width of 10 meV introduces spurious noise. Itis therefore clear from this comparison that the tetrahedronmethod is the preferred computational method for evaluatingthe electronic structure of a semiconductor, since many fea-5 4 3 2 1 0 1 2E - EVBM (eV)Density of StatesTetrahedronWidth = 10 meVWidth = 20 meVWidth = 50 meVFIG. 2: The density of states calculated using the Gaussian smearingmethod with different smearing parameters, using a dense k-pointmesh of 21×21×21.tures of the electronic structure are markedly expressed.We stress the importance of the Brillouin zone integrationmethod over the k-point mesh density in this note. Specif-ically, the tetrahedron method for Brillouin zone integrationis preferred over smearing methods to generate sharp quali-ties of the density of states. Although many in the commu-nity already follow this recommendation, they are importantto re-emphasize as the user community of electronic structuremethods grows.AppendixThe density of states g(E) is calculated asg(E) =1V ∑n∫Vδ (E− εn,k)dk (1)where εn,k is the energy of band n at k-point k, and V is thevolume of the reciprocal primitive cell. The Gaussian andFermi smearing methods approximate the δ (E − εn,k) func-tion in the following ways:Gaussian smearing:δ (E− εn,k)≈1σ√πe−(E−εn,kσ)2(2)Fermi smearing:δ (E− εn,k)≈e−E−εn,kσσ(1+ e−E−εn,kσ)2 (3)AcknowledgementsWe acknowledge support from NSF DMREF award#1729487. MYT acknowledges support from the United3States Department of Energy through the Computational Sci-ence Graduate Fellowship (DOE CSGF) under Grant NumberDE-SC0020347. This research was supported in part throughthe computational resources and staff contributions providedfor the Quest high performance computing facility at North-western University which is jointly supported by the Office ofthe Provost, the Office for Research, and Northwestern Uni-versity Information Technology. Work by AG, JP, and AJwas supported by the U.S Department of Energy, Office ofBasic Energy Sciences, and the Early Career Research Pro-gram. JM and KAP acknowledge support from the U.S. De-partment of Energy, Office of Science, Office of Basic EnergySciences, Materials Sciences and Engineering Division undercontract no. DE-AC02-05-CH11231 (Materials Project pro-gram KC23MP). This research used resources of the NationalEnergy Research Scientific Computing Center (NERSC), aU.S. Department of Energy Office of Science User Facilityoperated under Contract No. DE-AC02-05CH11231.[1] P. E. Blöchl, O. Jepsen, and O. K. Andersen, Physical Review B49, 16223 (1994).[2] A. Jain, S. P. Ong, G. Hautier, W. Chen, W. D. Richards,S. Dacek, S. Cholia, D. Gunter, D. Skinner, G. Ceder, et al., APLMaterials 1, 011002 (2013).[3] G. Kresse, Journal of Non-Crystalline Solids 192, 222 (1995).[4] G. Kresse and J. Furthmüller, Computational materials science6, 15 (1996).[5] G. Kresse and J. Furthmüller, Physical review B 54, 11169(1996).[6] J. P. Perdew, K. Burke, and M. Ernzerhof, Physical review letters77, 3865 (1996).[7] H. J. Monkhorst and J. D. Pack, Physical review B 13, 5188(1976).

Comparison of the tetrahedron method to smearing methods for the electronic density of states

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Comparison of the tetrahedron method to smearing methods for the electronic density of states

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