Sampling the Brillouinzone:


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1 Sampling the Brillouinzone: Andreas EICHLER Institut für Materialphysik and Center for Computational Materials Science Universität Wien, Sensengasse 8, A090 Wien, Austria binitio ienna ackage imulation A. EICHLER, SAMPLING THE BRILLOUINZONE Page
2 Overview introduction kpoint meshes Smearing methods What to do in practice A. EICHLER, SAMPLING THE BRILLOUINZONE Page 2
3 Introduction For many properties (e.g.: density of states, charge density, matrix elements, response functions,... ) integrals (I) over the Brillouinzone are necessary: I ε Ω BZBZ F ε δ ε nk ε dk To evaluate computationally integrals weighted sum over special kpoints Ω BZBZ ω ki k A. EICHLER, SAMPLING THE BRILLOUINZONE Page 3
4 k k function f k kpoints meshes  The idea of special points Chadi, Cohen, PRB 8 (973) with complete lattice symmetry introduce symmetrized planewaves (SPW): A m k R Cm eıkr sum over symmetryequivalent R C m Cm develope f SPW shell of lattice vectors in Fourierseries (in SPW) f f0 k m f m A m A. EICHLER, SAMPLING THE BRILLOUINZONE Page 4
5 k 0 m k evaluate integral (=average) over Brillouinzone f 2π Ω 3 BZ f dk with: Ω 2π 3 BZ A m dk f f0 2 taking n kpoints with weighting factors ω k so that n i ω ki A m ki 0 m N f = weighted sum over kpoints for variations of f that can be described within the shell corresponding to C N. A. EICHLER, SAMPLING THE BRILLOUINZONE Page 5
6 Monkhorst and Pack (976): Idea: equally spaced mesh in Brillouinzone. Constructionrule: k prs u p b u r b 2 usb3 u r 2rq r 2q r r 2 qr b i q r reciprocal latticevectors determines number of kpoints in rdirection A. EICHLER, SAMPLING THE BRILLOUINZONE Page 6
7 k k3 k Example: quadratic 2dimensional lattice q q2 4 6 kpoints only 3 inequivalent kpoints ( 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 BRILLOUINZONE Page 7
8 k 2πnk more Fouriercomponents Interpretation: representation of functionf E on a discrete equallyspaced 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 BRILLOUINZONE Page 8
9 Algorithm: calculate equally spacedmesh shift the mesh if desired apply all symmetry operations of Bravaislattice to all kpoints extract the irreducible kpoints (IBZ) calculate the proper weighting A. EICHLER, SAMPLING THE BRILLOUINZONE Page 9
10 " Smearing methods Problem: in metallic systems Brillouinzone integrals over functions that are discontinuous at the Fermilevel. Fouriercomponents 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 BRILLOUINZONE Page 0
11 $ % FermiDirac 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 noninteracting 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 BRILLOUINZONE 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 nkµ. + µ 0 ln 0 f n 0 f % ln f f A. EICHLER, SAMPLING THE BRILLOUINZONE Page 2
13 / & ' Gaussian smearing broadening of energylevels 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 errorfunction 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 BRILLOUINZONE 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 BRILLOUINZONE 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 BRILLOUINZONE 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 Hermitepolynomial 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 BRILLOUINZONE 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 BRILLOUINZONE Page 7
18 Linear tetrahedron method Idea:. dividing up the Brillouinzone 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 BRILLOUINZONE 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 BRILLOUINZONE Page 9
20 k j ad 2. interpolation Seite von X n j k c j k X n j... kpoints A. EICHLER, SAMPLING THE BRILLOUINZONE Page 20
21 k k k j ad 3. kspace integration: simplification by Blöchl (993) remapping of the tetrahedra onto the kpoints ω n j Ω BZΩBZ dkc j f ε n effective weights ωn j for kpoints. kspace summation: ω n j X n n j A. EICHLER, SAMPLING THE BRILLOUINZONE Page 2
22 Drawbacks: tetrahedra can break the symmetry of the Bravaislattice at least 4 kpoints are necessary Γ must be included linear interpolation under or overestimates the real curve A. EICHLER, SAMPLING THE BRILLOUINZONE Page 22
23 µ Corrections by Blöchl (993) Idea: linear interpolation under or overestimates the real curve for fullbands 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 (kpoint) of the tetrahedront DOS for the tetrahedron T at ε F. A. EICHLER, SAMPLING THE BRILLOUINZONE Page 23
24 Result: best kpoint 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 USPP and PAW practically impossible A. EICHLER, SAMPLING THE BRILLOUINZONE 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. kpoint spacing : A. EICHLER, SAMPLING THE BRILLOUINZONE 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 : MethfesselPaxton (N= or 2) always: test for energy with LT+Blöchlcorr. in any case: careful checks for kpoint convergence are necessary A. EICHLER, SAMPLING THE BRILLOUINZONE Page 26
27 The KPOINTS  file: > kpoints 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 BRILLOUINZONE Page 27
28 mesh parameter determine the number of intersections in each direction longer axes in realspace shorter axes in kspace less intersections necessary for equally spaced mesh Consequences: molecules, atoms (large supercells) is enough. Γ surfaces (one long direction 2D Brillouinzone) x for the direction corresponding to the long direction. y typical values (never trust them!): metals: semiconductors: /atom /atom A. EICHLER, SAMPLING THE BRILLOUINZONE Page 28
29 b ; b G Example  realspace/ 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 BRILLOUINZONE 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 kpoints Gamma point centered mesh preserves symmetry A. EICHLER, SAMPLING THE BRILLOUINZONE Page 30
31 Convergence tests with respect to and number of kpoints in the IBZ G.Kresse, J. Furthmüller, Computat. Mat. Sci. 6, 5 (996). A. EICHLER, SAMPLING THE BRILLOUINZONE Page 3
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