Sampling the Brillouin-zone: Andreas EICHLER Institut f¨ ur Materialphysik and Center for Computational Materials Science Universit¨ at Wien, Sensengasse 8, A-1090 Wien, Austria b-initio ackage imulation ienna A. E ICHLER , S AMPLING THE B RILLOUIN - ZONE Page 1
� � � � Overview introduction k-point meshes Smearing methods What to do in practice A. E ICHLER , S AMPLING THE B RILLOUIN - ZONE Page 2
✂ ✁ ✂ ✆ ✁ ✁ ✂ ✄ ☎ ✆ Introduction For many properties (e.g.: density of states, charge density, matrix elements, response functions, .. .) integrals ( I ) over the Brillouin-zone are necessary: 1 ε ε δ ε n k ε d k I F Ω BZBZ To evaluate computationally integrals weighted sum over special k-points 1 ∑ ω k i Ω BZBZ k A. E ICHLER , S AMPLING THE B RILLOUIN - ZONE Page 3
✁ ✁ ✂ ☛ ✂ ✝ � ✄ ✂ ✁ ✁ ✠ ✞ � ✂ ✂ ☞ ✁ ✄ � ✡ k-points meshes - The idea of special points Chadi, Cohen, PRB 8 (1973) 5747. function f k with complete lattice symmetry introduce symmetrized plane-waves (SPW): e ı kR k A m ∑ R C m ✝✟✞ sum over symmetry-equivalent R C m C m 1 SPW ”shell” of lattice vectors develope f k in Fourier-series (in SPW) ∞ ∑ f k f 0 f m A m k m 1 A. E ICHLER , S AMPLING THE B RILLOUIN - ZONE Page 4
� ✄ ✎ ✑ ✑ ✁ ✂ ✄ ✄ ✏ ✍ ✄ ✑ ✑ ✆ ✂ ✁ ✄ � ✏ ✆ ✁ ✄ ✌ ✍ ✎ ✌ ✞ ✂ evaluate integral (=average) over Brillouin-zone Ω ¯ k d k f f 3 2 π BZ Ω with: k d k 0 1 2 A m m 3 2 π ✏✒✑ BZ ¯ f f 0 taking n k-points with weighting factors ω k so that n ∑ ω k i A m k i 0 m 1 N ✏✒✑ i 1 ¯ f = weighted sum over k-points for variations of f that can be described within the ”shell” corresponding to C N . A. E ICHLER , S AMPLING THE B RILLOUIN - ZONE Page 5
✑ ✄ ✑ ✏ ✄ ✓ ✏ ✓ ☞ ☞ ✄ Monkhorst and Pack (1976): Idea: equally spaced mesh in Brillouin-zone. Construction-rule: k prs u p b 1 u r b 2 u s b 3 2 r q r 1 u r r 1 2 q r 2 q r ✏✒✑ b i reciprocal lattice-vectors q r determines number of k-points in r-direction A. E ICHLER , S AMPLING THE B RILLOUIN - ZONE Page 6
☞ ✄ ✏ ✂ ✁ ✄ ✔ ✄ ✁ ✚ ✏ � ✂ ✆ ✙ ✘ ✄ ✗ ✎ ✁ ✂ ✆ ✖ ✁ ✂ ☞ ✕ ✁ ✂ ✂ ✁ ✆ ✔ ✂ � ✄ ✄ ✆ � ✆ ✔ ✁ ✏ ✄ ✆ ✄ � b Example: k k ½ quadratic 2-dimensional lattice k q 1 q 2 4 16 k-points k � b only 3 inequivalent k-points ( IBZ) 0 1 1 1 ω 1 – 4 k 1 IBZ 8 8 4 3 3 1 ω 2 – 4 k 2 8 8 4 BZ 3 1 1 ω 3 – 8 k 3 8 8 2 1 1 1 1 F k d k 4 F k 1 4 F k 2 2 F k 3 Ω BZ BZ A. E ICHLER , S AMPLING THE B RILLOUIN - ZONE Page 7
✁ ✞ ✁ � ✂ ✆ � ✆ ☛ ✂ Interpretation: representation of function F k on a discrete equally-spaced mesh E N 2 π nk ∑ a n cos n 0 k -½ 0 ½ density of mesh more Fourier-components higher accuracy Common meshes : Two choices for the center of the mesh centered on Γ ( Γ belongs to mesh). centered around Γ . (can break symmetry !!) A. E ICHLER , S AMPLING THE B RILLOUIN - ZONE Page 8
� � � � ✛ � Algorithm: calculate equally spaced-mesh shift the mesh if desired apply all symmetry operations of Bravaislattice to all k-points extract the irreducible k-points ( IBZ) calculate the proper weighting A. E ICHLER , S AMPLING THE B RILLOUIN - ZONE Page 9
✥ ☎ ✤ ✓ ✦ ✠ ✣ ✢ ✜ ✄ ✂ ✁ ✧ ✆ ✆ ✂ ✁ ✆ ✆ Smearing methods Problem: in metallic systems Brillouin-zone integrals over functions that are discontinuous at the Fermi-level. high Fourier-components dense grid is necessary. Solution: replace step function by a smoother function. Example: bandstructure energy ω k ε n k ¯ Θ ε n k ∑ µ E n k � 1 x 0 with: ¯ Θ x 0 x 0 ε n k µ ω k ε n k f ∑ k -½ 0 ½ σ n k necessary: appropriate function f f equivalent to partial occupancies. A. E ICHLER , S AMPLING THE B RILLOUIN - ZONE Page 10
✄ ☎ ☎ ✂ ✩ ✁ ✂ ☞ ✁ ✁ ☎ ☎ ✂ ✄ ✁ ✂ ✁ ✂ ✂ ☎ ✆ ✄ ✂ ✥ ✁ ✓ ✄ ✂ ✁ ✪ ✁ ε n k µ 1 Fermi-Dirac function f ε n k µ σ exp 1 σ ✦★☞ consequence: energy is no longer variational with respect to the partial occupacies f . n σ S 1 F E ∑ f n 2 f ln f 1 ln 1 S f f f σ 3 k B T F free energy . new variational functional - defined by (1). S f entropy of a system of non-interacting electrons at a finite temperature T. σ smearing parameter . can be interpreted as finite temperature via (3). calculations at finite temperature are possible (Mermin 1965) A. E ICHLER , S AMPLING THE B RILLOUIN - ZONE Page 11
✭ ✄ ✓ ✄ ✪ ☎ ✂ ☎ ✁ ☎ ☞ ✩ ☎ ✁ ✁ ✂ ✁ ✭ ✂ ✁ ✄ ☎ ☎ ✂ ✁ ✁ ✏ ✡ ✳ ✁ ✂ ✁ ✁ ✄ ✂ ☎ ✯ ✓ ✓ ✰ ☎ ✄ ☞ ✂ ✁ ✁ ✂ ✁ ☎ ✂ ✭ ✁ ✄ ✄ ✱ ✲ ✂ ✂ ✪ ☎ ✁ ✁ ✄ ✂ ✂ ☎ ✁ ✂ ✫ ☎ ✁ ☞ ✩ ✄ ☎ ✄ ✂ ✁ ✂ ✁ ✂ ✁ ☎ ☎ ✄ ✬ ✂ ✂ Consistency : n σ S 1 F E ∑ f n 2 S f f ln f 1 f ln 1 f σ 3 k B T ∂ 4 F µ ∑ n f n N 0 ∂ f n ∂ E σ ∂ S 1 4 5 µ 0 ∂ f n ∂ f n ✂✮✭ f ∂ S ln 1 2 6 ln f 1 ln 1 f 1 ∂ f f ∂ E ε n 7 ∂ f n σ ln 1 f n ε n 5 7 8 µ 0 f n ✂✮✭ ε n µ 1 8 9 exp 1 σ f n 1 9 f n µ ε n k exp 1 σ A. E ICHLER , S AMPLING THE B RILLOUIN - ZONE Page 12
✂ ✆ ✄ ✁ ✁ ✆ ✴ ✬ ✂ ✵ ☎ ☎ ☎ ☎ ✂ ✶ ✫ � ✂ ✄ � ✁ ☎ ✁ � ✂ ✆ ☎ Gaussian smearing broadening of energy-levels with Gaussian function. f becomes an integral of the Gaussian function: ε n k ε n k µ 1 µ f 1 erf σ σ 2 no analytical inversion of the error-function erf exists entropy and free energy cannot be written in terms of f . 2 ε ε µ 1 µ S π exp σ σ 2 σ has no physical interpretation. σ variational functional F differs from E 0 . σ forces are calculated as derivatives of the variational quantity ( F ). not necessarily equal to forces at E 0 . A. E ICHLER , S AMPLING THE B RILLOUIN - ZONE Page 13
✂ ✂ ✂ ✁ ✏ ✂ ✂ ✁ ✁ ✂ ✄ ☎ ☎ ✌ ✍ ✁ ✁ ✷ ☎ ✷ ✁ ✂ ✭ ☎ ✁ ✁ ✂ ✭ ✁ ✂ ✂ ☞ ✂ ✁ ✂ ✂ ✁ ✁ ☞ ✂ ✁ ✁ ✁ ✂ ✄ ☎ ✁ ✂ ✂ ✁ ✁ ✷ ✂ ✄ ✂ ✁ ✁ ✂ ✁ Improvement: extrapolation to σ 0. γσ 2 σ 1 F E 0 ✂✸✷ σ σ σ S σ 2 F E ∂ F σ σ 2 γσ 3 S ∂σ γσ 2 σ 1 3 4 E E 0 ˆ 1 σ σ σ 1 4 E 0 E F E 2 A. E ICHLER , S AMPLING THE B RILLOUIN - ZONE Page 14
✆ ✆ Method of Methfessel and Paxton (1989) Idea: expansion of stepfunction in a complete set of or- thogonal functions term of order 0 = integral over Gaussians generalization of Gaussian broadening with functions of higher order. A. E ICHLER , S AMPLING THE B RILLOUIN - ZONE Page 15
✓ ✂ ✁ ✓ ✷ ✁ ✂ ✂ ✁ ✓ ✭ � ✁ ✂ ✞ ✄ ✁ ✂ ✁ ✁ ✂ � ✓ ✹ ✍ ✁ ✂ ✄ ✌ ✂ ✂ ✂ ✄ ✂ ✁ ✁ ☎ ☞ ✁ ✂ ✂ ✡ ✁ ✁ ✂ ✂ ✄ ☞ ✁ ✁ ✂ ☞ ✁ ✄ 1 f 0 x 1 erf x 2 N x 2 f N x f 0 x ∑ A m H 2 m x e 1 m 1 x 2 1 S N x 2 A N H 2 N x e n 1 with: A n n !4 n π H N : Hermite-polynomial of order N advantages: σ only of order 2+N in σ deviation of F from E 0 extrapolation for σ 0 usually not necessary, but also possible: ˆ 1 σ σ σ E 0 E N 1 F E N 2 A. E ICHLER , S AMPLING THE B RILLOUIN - ZONE Page 16
✁ ✂ ✷ ✁ ✁ ✁ ✂ ✆ � ✺ � ✆ ✂ ✁ ✂ ✁ ✂ ✄ ✁ ✂ ✁ � � The significance of N and σ ε MP of order N leads to a negligible error, if X is representable as a polynomial of degree 2 N around ε F . linewidth σ can be increased for higher order to obtain the same accuracy σ∑ n S N σ σ ”entropy term” ( S f n ) describes deviation of F from E . if S few meV then ˆ σ σ σ E F E E 0 . ✂✸✷ ✂✸✷ forces correct within that limit. in practice: smearings of order N=1 or 2 are sufficient A. E ICHLER , S AMPLING THE B RILLOUIN - ZONE Page 17
Linear tetrahedron method Idea: 1. dividing up the Brillouin-zone into tetrahedra 2. Linear interpolation of the function to be integrated X n within these tetrahedra 3. integration of the interpolated function ¯ X n A. E ICHLER , S AMPLING THE B RILLOUIN - ZONE Page 18
✻ ✾ ✽ ✼ ad 1. How to select mesh for tetrahedra map out the IBZ use special points � � b b 1 1 3 4 3 4 � � b b 1 1 1 1 3 4 A. E ICHLER , S AMPLING THE B RILLOUIN - ZONE Page 19
✁ ✂ ✂ ✁ ✁ ✂ ✄ ad 2. interpolation ¯ X n k ∑ c j k X n k j j j .......... k-points A. E ICHLER , S AMPLING THE B RILLOUIN - ZONE Page 20
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