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1 Sampling the Brillouin-zone: Andreas EICHLER Institut für Materialphysik and Center for Computational Materials Science Universität Wien, Sensengasse 8, A-090 Wien, Austria b-initio ienna ackage imulation A. EICHLER, SAMPLING THE BRILLOUIN-ZONE Page

2 Overview introduction k-point meshes Smearing methods What to do in practice A. EICHLER, SAMPLING THE BRILLOUIN-ZONE Page 2

3 Introduction For many properties (e.g.: density of states, charge density, matrix elements, response functions,... ) integrals (I) over the Brillouin-zone are necessary: I ε Ω BZBZ F ε δ ε nk ε dk To evaluate computationally integrals weighted sum over special k-points Ω BZBZ ω ki k A. EICHLER, SAMPLING THE BRILLOUIN-ZONE Page 3

4 k k function f k k-points meshes - The idea of special points Chadi, Cohen, PRB 8 (973) with complete lattice symmetry introduce symmetrized plane-waves (SPW): A m k R Cm eıkr sum over symmetry-equivalent R C m Cm develope f SPW shell of lattice vectors in Fourier-series (in SPW) f f0 k m f m A m A. EICHLER, SAMPLING THE BRILLOUIN-ZONE Page 4

5 k 0 m k evaluate integral (=average) over Brillouin-zone f 2π Ω 3 BZ f dk with: Ω 2π 3 BZ A m dk f f0 2 taking n k-points with weighting factors ω k so that n i ω ki A m ki 0 m N f = weighted sum over k-points for variations of f that can be described within the shell corresponding to C N. A. EICHLER, SAMPLING THE BRILLOUIN-ZONE Page 5

6 Monkhorst and Pack (976): Idea: equally spaced mesh in Brillouin-zone. Construction-rule: k prs u p b u r b 2 usb3 u r 2rq r 2q r r 2 qr b i q r reciprocal lattice-vectors determines number of k-points in r-direction A. EICHLER, SAMPLING THE BRILLOUIN-ZONE Page 6

7 k k3 k Example: quadratic 2-dimensional lattice q q2 4 6 k-points only 3 inequivalent k-points ( IBZ) 4k 4k 2 8k ω ω2 ω ½ 0 b k k k b IBZ BZ Ω BZ BZ F dk 4 F k 4 F k2 2 F A. EICHLER, SAMPLING THE BRILLOUIN-ZONE Page 7

8 k 2πnk more Fourier-components Interpretation: representation of functionf E on a discrete equally-spaced mesh N n 0 a n cos -½ density of mesh 0 ½ k higher accuracy Common meshes : Two choices for the center of the mesh centered on Γ ( Γbelongs to mesh). centered around Γ. (can break symmetry!!) A. EICHLER, SAMPLING THE BRILLOUIN-ZONE Page 8

9 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. EICHLER, SAMPLING THE BRILLOUIN-ZONE Page 9

10 " Smearing methods Problem: in metallic systems Brillouin-zone integrals over functions that are discontinuous at the Fermi-level. Fourier-components high dense grid is necessary. Solution: replace step function by a smoother function. Example: bandstructure energy with: Θ ω k ε nk Θ nk x ε nk µ x 0 0 x0 E ω k ε nk f nk ε nkµ! -½ 0 ½ k necessary: appropriate function f f equivalent to partial occupancies. A. EICHLER, SAMPLING THE BRILLOUIN-ZONE Page 0

11 $ % Fermi-Dirac function f ε nk µ exp ε nkµ!# consequence: energy is no longer variational with respect to the partial occupacies f. F E n S f n 2 S f f ln f F free energy. new variational functional - defined by (). 3 kb T f ln f S f entropy of a system of non-interacting electrons at a finite temperature T. smearing parameter. can be interpreted as finite temperature via (3). calculations at finite temperature are possible (Mermin 965) A. EICHLER, SAMPLING THE BRILLOUIN-ZONE Page

12 , ( )( * ( )( 4 2 $ & ' $ Consistency: 9 (fn % F E n S fn S f kbt f ln f f ln f fn E fn S f E fn εn exp F µ n fn N S εn fn ln εnµ exp µ ln f fn fn ε nk-µ. + µ 0 ln 0 f n 0 f % ln f f A. EICHLER, SAMPLING THE BRILLOUIN-ZONE Page 2

13 / & ' Gaussian smearing broadening of energy-levels with Gaussian function. f becomes an integral of the Gaussian function: f ε nk µ 2 erf ε nk µ no analytical inversion of the error-function erf exists entropy and free energy cannot be written in terms of f. S µ ε 2 π exp 0 µ ε 2 has no physical interpretation. variational functional F differs from E forces are calculated as derivatives of the variational quantity (F not necessarily equal to forces at E ). A. EICHLER, SAMPLING THE BRILLOUIN-ZONE Page 3

14 2 4 ( Improvement: extrapolation to (0. F 32E 0 γ2 F E S S F 2γ E E 2E 2Ê 0 0 γ 2 2 F E A. EICHLER, SAMPLING THE BRILLOUIN-ZONE Page 4

15 Method of Methfessel and Paxton (989) Idea: expansion of stepfunction in a complete set of orthogonal functions term of order 0 = integral over Gaussians generalization of Gaussian broadening with functions of higher order. A. EICHLER, SAMPLING THE BRILLOUIN-ZONE Page 5

16 F E f 0 f N S N x x x with: 2 f0 A n H N : x erf 2 A NH 2N n!4 n x N m n 4 π x A m H 2m ex 2 x e x 2 Hermite-polynomial of order N advantages: deviation of F from E 0 only of order 2+N in extrapolation for (0 usually not necessary, but also possible: E 0 2Ê N 2 N A. EICHLER, SAMPLING THE BRILLOUIN-ZONE Page 6

17 The significance of N and MP of order N leads to a negligible error, if X degree 2N around ε F. ε is representable as a polynomial of linewidth can be increased for higher order to obtain the same accuracy entropy term (S n SN f n ) describes deviation of F from E. if S 5few mev then Ê 32F 2E 32E 0. forces correct within that limit. in practice: smearings of order N= or 2 are sufficient A. EICHLER, SAMPLING THE BRILLOUIN-ZONE Page 7

18 Linear tetrahedron method Idea:. 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. EICHLER, SAMPLING THE BRILLOUIN-ZONE Page 8

19 ad. How to select mesh for tetrahedra map out the IBZ b b use special points b b A. EICHLER, SAMPLING THE BRILLOUIN-ZONE Page 9

20 k j ad 2. interpolation Seite von X n j k c j k X n j... k-points A. EICHLER, SAMPLING THE BRILLOUIN-ZONE Page 20

21 k k k j ad 3. k-space integration: simplification by Blöchl (993) remapping of the tetrahedra onto the k-points ω n j Ω BZΩBZ dkc j f ε n effective weights ωn j for k-points. k-space summation: ω n j X n n j A. EICHLER, SAMPLING THE BRILLOUIN-ZONE Page 2

22 Drawbacks: tetrahedra can break the symmetry of the Bravaislattice at least 4 k-points are necessary Γ must be included linear interpolation under- or overestimates the real curve A. EICHLER, SAMPLING THE BRILLOUIN-ZONE Page 22

23 µ Corrections by Blöchl (993) Idea: linear interpolation under- or overestimates the real curve for full-bands or insulators these errors cancel for metals: correction of quadratic errors is possible: δω kn T 40 D T ε F 4 j ε jn ε kn j D T corners (k-point) of the tetrahedront DOS for the tetrahedron T at ε F. A. EICHLER, SAMPLING THE BRILLOUIN-ZONE Page 23

24 Result: best k-point convergence for energy forces: with Blöchl corrections the new effective partial occupancies do not minimize the groundstate total energy variation of occupancies ω nk w.r.t. the ionic positions would be necessary with US-PP and PAW practically impossible A. EICHLER, SAMPLING THE BRILLOUIN-ZONE Page 24

25 Convergence tests (from P.Blöchl, O. Jepsen, O.K. Andersen, PRB 49,6223 (994).) bandstructure energy of silicon: conventional LT -method vs. LT+Blöchl corrections bandstructure energy vs. k-point spacing : A. EICHLER, SAMPLING THE BRILLOUIN-ZONE Page 25

26 What to do in practice energy/dos calculations: linear tetrahedron method with Blöchl corrections ISMEAR=-5 calculation of forces: semiconductors: Gaussian smearing (ISMEAR=0; SIGMA=0.) metals : Methfessel-Paxton (N= or 2) always: test for energy with LT+Blöchl-corr. in any case: careful checks for k-point convergence are necessary A. EICHLER, SAMPLING THE BRILLOUIN-ZONE Page 26

27 The KPOINTS - file: > k-points for a metal 2> 0 3> Gamma point 4> > st line: comment 2nd line: 0 ( automatic generation) 3rd line: Monkhorst or Gammapoint (centered) 4th line: mesh parameter 5th line: (shift) A. EICHLER, SAMPLING THE BRILLOUIN-ZONE Page 27

28 mesh parameter determine the number of intersections in each direction longer axes in real-space shorter axes in k-space less intersections necessary for equally spaced mesh Consequences: molecules, atoms (large supercells) is enough. Γ surfaces (one long direction 2-D Brillouin-zone) x for the direction corresponding to the long direction. y typical values (never trust them!): metals: semiconductors: /atom /atom A. EICHLER, SAMPLING THE BRILLOUIN-ZONE Page 28

29 b ; b G Example - real-space/ reciprocal cell 4 4 k k k k 2 k 3 ACBED 0 IBZ BZ b : KMLEN 0 IBZ BZ b F doubling the cell in real space halves the reciprocal cell zone boundary is folded back to Γ same sampling is achieved with halved mesh parameter A. EICHLER, SAMPLING THE BRILLOUIN-ZONE Page 29

30 OQP Example - hexagonal cell RTSVU Z[]\ ^`_ before symmetrization after WTXVY shifted to atbvc in certain cell geometries (e.g. hexagonal cells) even meshes break the symmetry symmetrization results in non equally distributed k-points Gamma point centered mesh preserves symmetry A. EICHLER, SAMPLING THE BRILLOUIN-ZONE Page 30

31 Convergence tests with respect to and number of k-points in the IBZ G.Kresse, J. Furthmüller, Computat. Mat. Sci. 6, 5 (996). A. EICHLER, SAMPLING THE BRILLOUIN-ZONE Page 3

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