Lecture on materials properties calculated from first-principles / density functional theory (DFT)

(1)

Lecture on materials properties calculated from first-principles / density functional theory (DFT)

TOHOKU

UNIVERSITY

2019.05.22:

10:30 – 11:30 = fundamentals of DFT

11:30 – 12:00 = guide of software installation 2019.05.29 and 06.05:

electronic structure of silicon and graphene phonon dispersion, etc…

Evaluation of this class: 1 quiz (today) and 1 report (2019.07.26) Class webpage: http://ccmp.home.blog

Emails: [email protected] or [email protected]

(2)

What is DFT?

• An approach to calculate the (ground state) properties of many-electron systems from first-principles

• Features:

- Ab initio / first-principles / no empirical information - Quantum mechanical and numerical

- Predictive capability

- Very accurate for most materials (errors typically < few %)

Quantum ESPRESSO will be used in this class

ESPRESSO = opEn-Source Package for Research in

Electronic Structure, Simulation, and Optimization

(3)

G. Hautier, C.C. Fischer, A. Jain, T. Mueller, and G. Ceder, “ Finding Nature ’ s Missing Ternary Oxide Compounds Using Machine Learning and Density Functional Theory, ”

Chem. Mater. 22, 3762-3767 (2010)

DFT for discovery of new materials

(4)

DFT for establishing materials database

A. Jain, S.P. Ong, G. Hautier, W. Chen, W.D. Richards, S.

Dacek, S. Cholia, D. Gunter, D. Skinner, G. Ceder, K.A.

Persson, Applied Physics Letters Materials 1, 011002 (2013)

https://www.materialsproject.org/

(5)

DFT for your publications ☺

Example research subject: Optical properties

(6)

DFT for your publications ☺

Example research subject: Thermoelectrics

(7)

Outline

Formalism

– Hydrogen Atom

– Density Functional Theory

• Exchange-Correlation Potentials

• Pseudopotentials and Related Approaches

Practical Issues

– Implementation

• Periodic boundary conditions

• k-Points

• Plane-wave basis sets

– Parameters controlling numerical precision

(8)

Introduction

The Hydrogen Atom

Proton with mass M 1 , coordinate R 1 Electron with mass m 1 , coordinate r 1

r  r 1  r 2 , R  M 1 R 1  m 2 r 2

M 1  m 2 , m  M 1 m 2

M 1  m 2 , M  M 1  m 2

Y( R , r)   cm (R)  r ( r)

(9)

Hydrogen Atom

Switch to Spherical Coordinates



 



 q

q q 

q

 q







 E

r e r

r r r

r r

m   

 ö

 

æ  



 ö



 æ

 

 ö



 æ

2 2

2

sin sin 1

sin 1 1

2



 ( r, q ,  )  R nl (r ) Y l m ( q ,  )

  2 2m

1 r 2

d

dr r 2 d dr æ

  ö

   l( l  1)

r 2  e 2 r æ

 

ö

  R nl ( r)  E n R nl (r)

E n   me 4 2 2

1

n 2   13.6

n 2 eV

Commonly used units in calculation:

1 Ry ~ 13.6 eV, 1 Ha = 2 Ry,

1 a.u. = Bohr’s radius ~ 0.53 angstrom

ℏ

(10)

Many-Electron Problem

• collection of

– N ions

– n electrons

• total energy

computed as a function of ion positions

– Use approximation in quantum mechanics!

electrons

ions

(11)

Born-Oppenheimer Approximation

• Mass of nuclei exceeds that of the electrons by a factor of 1000 or more

– we can neglect the kinetic energy of the nuclei – treat the ion-ion interaction classically

– significantly simplifies the Hamiltonian for the electrons

• Consider Hamiltonian for n electrons in potential of N nuclei with atomic numbers Z i

External electron-ion potential

 V ext   r j

(12)

Density Functional Theory

Hohenberg and Kohn (1964), Kohn and Sham (1965)

• For each external potential there is a unique groundstate electron density

• Energy can be obtained by minimizing a density functional with respect to density of electrons n(r)

E groundstate =min{E tot [n(r)]}

        int     ext     ...

tot n r  T n r  E n r   d r V r n r 

E

Kinetic Energy Electron-Electron Interactions

Other terms Electron-Ion

Interactions

(13)

Kohn-Sham Approach

 





n

i

i r e

r n

1

) 2

( )

( 

)]

( [ ' '

) ' ( ) ( 2

1

) ( ) 2 (

}]

[{

3 3

3 1

3 2

* 2

r n E

r rd r d

r

r n r n

r d r n r V

r m d

E

xc ext n

i

i i i e

i

 











  





 

Many-body electron-electron and other interactions are all put into E xc [n(r)]

“ Exchange-Correlation Energy ”

ℏ

(14)

Kohn-Sham Equations

) ( )

( )

( ' '

) ' ) (

2 (

3 2

2

r r

r V

r r d

r r r n

m e i V ext xc  i  e i  i

 





 



 

 





  

) (

)]

( ) [

( n r

r n r E

V xc xc

d

 d

ℏ

Most of the research on improvement of DFT were dedicated

to establish good exchange-correlation “functionals”

(15)

Local Density Approximation

(e.g., J. P. Perdew and A. Zunger, Phys. Rev. B 23, 5048 (1981))

r d r

n r

n

E xc [ ( )]   e xc hom ( ( )) ( ) 3

(r) Density

of Gas Electron

s Homogeneou of

Energy n

Correlatio Exchange

)) (

hom (

n r

xc n  

e

Generalized Gradient Approximation

J. P. Perdew, K. Burke and M. Ernzerhof, Phys. Rev. Lett. 77 (1996)

r d s r

F r

n r

n

E xc GGA [ ( )]   e x hom ( ( )) ( ) xc ( s ,  , ) 3

(16)

A Note on Accuracy and Ongoing Research

• LDA leads to “ overbinding ”

− Lattice constants commonly 1-3% too small, elastic constants 10-15%

too stiff, cohesive energies 5-20% too large

but, errors are largely systematic

− Energy differences tend to be more accurate

• GGA corrects for overbinding

− Sometimes “overcorrects”

• “Beyond DFT” Approaches

− For “highly correlated” systems LDA & GGA perform much worse and corrections required (DFT+U, Hybrid Hartree-Fock/DFT, …)

− Non-bonded interactions, e.g., van der Waals interactions in graphite,

require additional terms or functionals (e.g., vdW-DF)

(17)

External electron-ion interaction

• Potential due to ions is singular at ion core

• Eigenfunctions oscillate rapidly near singularity

• Eigenfunction in bonding region is smooth



) ( )

( )

( ' '

) ' ) (

2 (

3 2

2

r r

r V

r r d

r r r n

m e i V ext xc  i  e i  i

 





 



 

 





 ℏ  

valence electrons

ion core

Difficult to solve KS equation!

(18)

Pseudopotentials

• For plane-wave basis sets, rapid oscillations require large number of basis functions

– expensive – unnecessary

• these oscillations don't alter bonding properties

• Replace potential with nonsingular potential – preserve bonding tails of eigenfunction

– preserve distribution of charge between core and tail regions

– reduces number of plane waves required for accurate expansion of wavefunction

“Pseudopotential”



 pseudo

(19)

Summary of Kohn-Sham DFT

 





n

i

i r e

r n

1

) 2

( )

( 

Electron Density

Electron Wavefunctions  i (r )

Exchange-Correlation Energy E xc [n(r )]

Form depends on whether we use LDA or GGA

Potential Electrons Feel from Nuclei V ext (r )

)]

( [ ' '

) ' ( ) ( 2 1

) ( ) 2 (

}]

[{

3 3

3 1

3 2

* 2

r n E r rd r d

r r n r n

r d r n r V r

m d E

xc ext N

i

i i i e

i

e

 











  





  ℏ

) ( )

( )

( ' '

) ' ) (

2 (

3 2

2

r r

r V r r d r

r r n

m e i V ext xc  i  e i  i

 





 



 

 





 ℏ  

KS equation Total energy

V xc (r )  d E xc [ n(r)]

d n( r)

Note:  i depends on n(r) which depends on  i

 Solution of Kohn-Sham equations must be done iteratively

(20)

Self-Consistent Solution to DFT Equations

Input Positions of Atoms for a Given Unit Cell and Lattice Constant

guess charge density

compute effective potential

solve KS equation, recalculate density

compare output and input charge densities

Energy for Given Lattice Constant

different

same

1. Set up atom positions

2. Make initial guess of “input” charge density (often overlapping atomic charge densities) 3. Solve Kohn-Sham equations with this input

charge density

4. Compute “output” charge density from resulting wavefunctions

5. If energy from input and output densities differ by amount greater than a chosen

threshold, mix output and input density and go to step 2

6. Quit when energy from input and output densities agree to within prescribed

tolerance (e.g., 10 -5 eV)

(21)

Some of Widely Used DFT Codes

Free

• Quantum Espresso / PWSCF (http://www.quantum-espresso.org/)

• ABINIT (http://www.abinit.org/)

• CASTEP (http://ccpforge.cse.rl.ac.uk/gf/project/castep/)

Commercial:

• VASP (http://cms.mpi.univie.ac.at/vasp/)

• WIEN2K (http://www.wien2k.at/)

(22)

Outline

Formalism

– Hydrogen Atom

– Density Functional Theory

• Exchange-Correlation Potentials

• Pseudopotentials and Related Approaches

Practical Issues

– Implementation

• Periodic boundary conditions

• k-Points

• Plane-wave basis sets

– Parameters controlling numerical precision

(23)

Implementation of DFT for a Single Crystal

a

a a

Unit Cell Vectors a 1 = a (-1/2, 1/2 , 0) a 2 = a (-1/2, 0, 1/2) a 3 = a (0, 1/2, 1/2) Example: Diamond Cubic Structure of Si

Crystal Structure Defined by Unit Cell Vectors and Positions of Basis Atoms

Basis Atom Positions 0 0 0

¼ ¼ ¼

All atoms in the crystal can be obtained by adding integer

multiples of unit cell vectors to basis atom positions

(24)

Electron Density in Crystal Lattice

n   r  n  r  R uvw 

a

a a

Unit-Cell Vectors

a 1 = a (-1/2, 1/2 , 0) a 2 = a (-1/2, 0, 1/2) a 3 = a (0, 1/2, 1/2)

Electron density is periodic with periodicity given by R uvw

R uvw  ua 1  va 2  wa 3

Translation Vectors:

(25)

because wavefunctions in a crystal obey Bloch’s Theorem

  r  k r    k r

k

b

 b  exp i  u

For a given band b

where is periodic in real space: u k b   r  u k

b  r  R uvw 

u k b   r

R uvw  ua 1  va 2  wa 3

Translation Vectors:

Electron Density is Periodic

(26)

Representation of Electron Density

  r  k r    k r

k

b

     



 

b

b e e



BZ F

k

k f d

e n

BZ

2 3

) (

)

( r k r

In practice the integral over the Brillouin zone is replaced with a sum over a finite number of k-points (N kpt )

One parameter that needs to be checked for numerical convergence is

number of k-points

    





b 

b

b e e

 ( )

) (

1

2

F N

j

_j

kpt

j

f

w e

n r k r k

Integral over k-points in first Brillouin zone

f( e – e

_F

) is Fermi-Dirac distribution function with Fermi energy e

_F









N

e

i

e i

n

1

) 2

r ( )

r

( 

(27)

Representation of Wavefunctions

Fourier-Expansion as Series of Plane Waves

        

lmn

lmn i

u

u k b r k b G exp G r

  r  k r    k r

k

b

For a given band:

Recall that is periodic in real space: u k b   r u k b   r  u k b  r  R uvw 

can be written as a Fourier Series:

  r

k

u b

3 2

1 b b

b m n

lmn  l  

G

where the are primitive reciprocal lattice vectors b i ⋅ = 2

(28)

Representation of Wavefunctions

Plane-Wave Basis Set

  r  k r    k r

k

b

Another parameter that needs to be checked for convergence is the

“plane-wave cut-off energy” E cut

In practice the Fourier series is truncated to include all G for which:

  2 cut

2

2 E

m G  k 



For a given band

           

G k

k b r u b G exp i G k r



Use Fourier Expansion

ℏ

(29)

Examples of Convergence Checks

k points

To tal ene rg y ( Ry)

no offset

offset

no offset

k points

Cal c ul ati on tim e (s )

To tal ene rg y (Ry)

E cut (Ry) E cut (Ry)

Cal c ul ati on tim e (s )

(30)

Example calculation inputs

&CONTROL

calculation='bands',

restart_mode='from_scratch', prefix='si',

pseudo_dir='../pseudo/', outdir='../work/',

/

&SYSTEM ibrav=2,

celldm(1)=10.2625, nat=2,

ntyp=1,

ecutwfc=60.0, nbnd=8,

/

&ELECTRONS conv_thr=1d-8, /

ATOMIC_SPECIES

Si 28.0855 Si.pbe-rrkj.UPF ATOMIC_POSITIONS (alat) Si 0.00 0.00 0.00 Si 0.25 0.25 0.25 K_POINTS {crystal_b}

5

0.0000000000 0.0000000000 -0.5000000000 20 0.0000000000 0.0000000000 0.0000000000 30 -0.5000000000 0.0000000000 -0.5000000000 10 -0.3750000000 0.0000000000 -0.6250000000 30 0.0000000000 0.0000000000 0.0000000000 20

&CONTROL

calculation='scf',

/

&SYSTEM ibrav=2,

celldm(1)=10.2625, nat=2,

ntyp=1,

ecutwfc=60.0, /

&ELECTRONS

mixing_beta=0.7, conv_thr=1d-8, /

ATOMIC_SPECIES

Si 28.0855 Si.pbe-rrkj.UPF ATOMIC_POSITIONS (alat) Si 0.00 0.00 0.00 Si 0.25 0.25 0.25 K_POINTS automatic 4 4 4 1 1 1

By only 2 input files, we can “easily” calculate electronic band structure!

(31)

Example calculation inputs

&CONTROL

calculation='bands',

/

&SYSTEM ibrav=2,

celldm(1)=10.2625, nat=2,

ntyp=1,

ecutwfc=60.0, nbnd=8,

/

&ELECTRONS conv_thr=1d-8, /

ATOMIC_SPECIES

Si 28.0855 Si.pbe-rrkj.UPF ATOMIC_POSITIONS (alat) Si 0.00 0.00 0.00 Si 0.25 0.25 0.25 K_POINTS {crystal_b}

5

0.0000000000 0.0000000000 -0.5000000000 20 0.0000000000 0.0000000000 0.0000000000 30 -0.5000000000 0.0000000000 -0.5000000000 10 -0.3750000000 0.0000000000 -0.6250000000 30 0.0000000000 0.0000000000 0.0000000000 20

&CONTROL

calculation='scf',

/

&SYSTEM ibrav=2,

celldm(1)=10.2625, nat=2,

ntyp=1,

ecutwfc=60.0, /

&ELECTRONS

mixing_beta=0.7, conv_thr=1d-8, /

ATOMIC_SPECIES

Si 28.0855 Si.pbe-rrkj.UPF ATOMIC_POSITIONS (alat) Si 0.00 0.00 0.00 Si 0.25 0.25 0.25 K_POINTS automatic 4 4 4 1 1 1

By only 2 input files, we may be able to publish some papers!

(32)

Electronic Bandstructure

Example for Si

Brillouin Zone Energy dispersion