k-points

Sampling the Brillouin-zone:Andreas EICHLER Institut fur Materialphysik and Center for Computational Materials Science Universit t Wien, Sensengasse 8, A-1090 Wien, Austria a

b-initio

ackage ienna imulation

A. E ICHLER , S AMPLING

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Overviewintroduction k-point meshes Smearing methods What to do in practice


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IntroductionFor many properties (e.g.: density of states, charge density, matrix elements, response functions, . . . ) integrals (I) over the Brillouin-zone are necessary: I

1 BZ

F nk

dk

To evaluate computationally weighted sum over special k-points integrals

1 BZ


BZ

kik

BZ

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k-points meshes - The idea of special pointsChadi, Cohen, PRB 8 (1973) 5747.

function f k with complete lattice symmetry introduce symmetrized plane-waves (SPW): Am k

R

ekRCm

sum over symmetry-equivalent R Cm SPW

shell of lattice vectors

develope f k in Fourier-series (in SPW) f k

Cm

f0

m 1

1

f m Am k


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evaluate integral (=average) over Brillouin-zone f with: 2 3

2 3

BZ

f k dk

BZ

Am k dk

0

m

12

taking n k-points with weighting factors k so thati 1

f = weighted sum over k-points for variations of f that can be described within the shell corresponding to CN .

f

f0

ki A m

n

ki

0

m

1

N


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Monkhorst and Pack (1976):Idea: equally spaced mesh in Brillouin-zone. Construction-rule: k prs ur bi qr u p b1

ur b2 r

us b3 qr

2r qr 1 2qr

12

reciprocal lattice-vectors determines number of k-points in r-direction


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Example: quadratic 2-dimensional lattice q1 q2 4 16 k-points IBZ)

r b"

only 3 inequivalent k-points ( 4 4 81 BZ

0

k k k r b#$ & % !

k1 k2 k3

1 8 3 8 3 8

1 8 3 8 1 8

1 2 3 k1

1 4 1 4 1 2 1 4F

IBZ BZk21 2F

BZ

F k dk

1 4F

k3


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Interpretation: representation of functionF k on a discrete equally-spaced mesh

En 0

an cos 2nk

N

-

0

kmore Fourier-components higher accuracy

density of mesh

Common meshes : Two choices for the center of the mesh centered on ( belongs to mesh).

centered around . (can break symmetry !!)


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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 ( calculate the proper weighting IBZ)'


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Smearing methodsProblem: 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 energynk

k nk nk (

with: x

1 x 0 x 2

0 0

E e4

nk

k nk f

) 0

nk

1

-

0

k

necessary: appropriate function f


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f equivalent to partial occupancies.B RILLOUIN - ZONE Page 10

Fermi-Dirac function

f

nk

exp

nk

1

1

consequence: energy is no longer variational with respect to the partial occupacies f . 1 2 3 F S f

53

2

E

S fnn

F free energy. new variational functional - dened by (1). S f

entropy of a system of non-interacting electrons at a nite temperature T.

f ln f

1

f ln 1

f

6

7

kB T

smearing parameter. can be interpreted as nite temperature via (3). calculations at nite temperature are possible (Mermin 1965)A. E ICHLER , S AMPLINGTHE

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Consistency: 1 2 3 4 F E S fnn

S f

f ln f

1

f ln 1

f

kB T F8

6

7

fn E fn S f E fn

fnn

N9

0 ln 1 f f

1

4

5

S fn

0 ln 1

@ A

5

2

6

ln f

1

f

1

7

@

7

8

n 1 fn

6

7

n

ln 1 fnfn

0

8

@ A

9

exp fn

n

11

@

B

C

9

exp

nk E

1

@


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Gaussian smearingbroadening of energy-levels with Gaussian function. f becomes an integral of the Gaussian function: f nk

1 1 28

nk erf

no analytical inversion of the error-function erf exists entropy and free energy cannot be written in terms of f . S

9

1 2 G

exp

2

has no physical interpretation. variational functional F differs from E 0 . forces are calculated as derivatives of the variational quantity (F ). not necessarily equal to forces at E 0 . A. E ICHLER , S AMPLING

H

I

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Improvement: extrapolation to

0.

@

1 2 3

F F S

E 0

2 S

Q P

E

1 1

3

4

E

F

2

P

E 0

21 2

P

@

4

E 0

E

F

E

P


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Method of Methfessel and Paxton (1989)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.


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f0 x fN x SN x

1 2

1

erf xN

with:

advantages: deviation of F from E 0 only of order 2+N in extrapolation for

f0 x

An

1 2 AN H2N 1n n!4n R

m 1

Am H2m xe

1

xe

x2

x2

HN : Hermite-polynomial of order N

0 usually not necessary, but also possible: E

@

E 0

N 2

1

N

1F

E

P


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The signicance of N and MP of order N leads to a negligible error, if X is representable as a polynomial of degree 2N around F . linewidth can be increased for higher order to obtain the same accuracy entropy term (S S

n SN fn ) describes deviation of F from E .

if S few meV F then E P Q

forces correct within that limit.

in practice: smearings of order N=1 or 2 are sufcient

E

E 0.

P

P Q


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Linear tetrahedron method

Idea: 1. dividing up the Brillouin-zone into tetrahedra 2. Linear interpolation of the function to be integrated Xn within these tetrahedra 3. integration of the interpolated function Xn


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ad 1. How to select mesh for tetrahedra map out the IBZ use special points

r bU

r bW

3 1 3

1 3 r 1 b

4 1 4

1 4 r 1 b

V

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ad 2. interpolation

Seite 1 von 1

Xn k

c j k Xn k jj

j .......... k-points


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ad 3. k-space integration: simplication by Blochl (1993)

remapping of the tetrahedra onto the k-points n j 1 BZ dkc j k f n k

BZ

effective weights n j for k-points. k-space summation:

n j Xnnj

kj


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Drawbacks: tetrahedra can break the symmetry of the Bravaislattice at least 4 k-points are necessary must be included linear interpolation under- or overestimates the real curve


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Corrections by Bl chl (1993) oIdea: linear interpolation under- or overestimates the real curve for full-bands or insulators these errors cancel for metals: correction of quadratic errors is possible: kn j DT

T

1 40 DT

F jnj 1

4

kn

corners (k-point) of the tetrahedronT DOS for the tetrahedron T at F .


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Result: best k-point convergence for energy forces: with Bl chl corrections the new effective partial occupancies do not minimize the o groundstate total energy variation of occupancies nk w.r.t. the ionic positions would be necessary with US-PP and PAW practically impossible


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Convergence tests

(from P.Blochl, O. Jepsen, O.K. Andersen, PRB 49,16223 (1994).)

bandstructure energy of silicon: conventional LT -method vs. LT+Bl chl corrections o bandstructure energy vs. k-point spacing :


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What to do in practiceenergy/DOS calculations: linear tetrahedron method with Bl chl corrections o ISMEAR=-5 calculation of forces: semiconductors: Gaussian smearing (ISMEAR=0; SIGMA=0.1) metals : Methfessel-Paxton (N=1 or 2) always: test for energy with LT+Bl chl-corr. o

in any case: careful checks for k-point convergence are necessary


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The KPOINTS - le:1> 2> 3> 4> 5> k-points for a metal 0 Gamma point 9 9 9 0 0 0

1st line: comment 2nd line: 0 ( automatic generation)

3rd line: Monkhorst or Gammapoint (centered) 4th line: mesh parameter 5th line: 0 0 0 (shift)


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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) 1 1 1 is enough.'

2-D Brillouin-zone) surfaces (one long direction x y 1 for the direction corresponding to the long direction.

typical values (never trust them!):

metals: semiconductors:

9

9 4

9 /atom

4

4 /atom


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Example - real-space/ reciprocal cellr bY

r bq

44ige h f

0c a db`

k1 k2 k3 r bX

24

k1 k2

0t s dbr

r bp

x v iwu

IBZ BZ

IBZ BZ

doubling the cell in real space halves the reciprocal cell zone boundary is folded back to same sampling is achieved with halved mesh parameter


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Example - hexagonal cellbefore after symmetrization

shifted to G

r y

r

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 symmetryA. E ICHLER , S AMPLINGTHE

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Convergence testswith respect to and number of k-points in the IBZ

A. E ICHLER , S AMPLINGTHE

G.Kresse, J. Furthm ller, Computat. Mat. Sci. 6, 15 (1996). uB RILLOUIN - ZONE Page 31

k-points

Documents

k1 k2 k3

bl chl corrections

equally spaced

linear tetrahedron

linear interpolation

nite temperature

point convergence

variational