Modern Computational condensed Matter Modern Computational condensed Matter Physics:Physics:

Basic theory and Basic theory and applicationsapplications

Prof. Abdallah QteishDepartment of Physics, Yarmouk University,

21163-Irbid, Jordan

Chemistry Dept, YU, 14 May 2007

Starting from first-principles, can we efficiently and accuratelyStarting from first-principles, can we efficiently and accurately

Calculate the various properties (structural, electronic structure, vibrational, Calculate the various properties (structural, electronic structure, vibrational, thermal, elastic, magnetic, …, etc) of bulk solids;thermal, elastic, magnetic, …, etc) of bulk solids;

Investigate the surface and interface properties of solids;Investigate the surface and interface properties of solids; Study defects;Study defects; Construct the phase diagrams of alloys;Construct the phase diagrams of alloys; Study the properties of liquids and amorphous materials;Study the properties of liquids and amorphous materials; Investigate the material properties under extreme condition (very high Investigate the material properties under extreme condition (very high

temperature and pressure);temperature and pressure); Deal with biological systems;Deal with biological systems; Others ?? Others ??

Answer: YES

Direct application of Standard QM !!Direct application of Standard QM !! In Standard QM, In Standard QM, ΨΨ which is the solution of the many-body Schrödinger Eq. which is the solution of the many-body Schrödinger Eq.

),...,,(),...,,(|

1

2

1)(

2

12121

11

2NN

N

ji ji

N

iiext

N

ii EV rrrrrr

rrr

is the basic variable quantity physical observableany is ];[ XXX

Main problem: Main problem: ΨΨ is a function of is a function of 3N3N variables variables, and , and NN is of order of 10 is of order of 102424 for a for a realistic condensed matter sample.realistic condensed matter sample.

Thus, direct application of Standard QM is simply impossible.Thus, direct application of Standard QM is simply impossible.

Remark: In Eq. (1) the Remark: In Eq. (1) the nuclei are assumed to be at fixed positionsnuclei are assumed to be at fixed positions

adiabatic or Born-Oppenhiemer approximation.adiabatic or Born-Oppenhiemer approximation.

...... (1)

Density Functional Theory (DFT)Density Functional Theory (DFT)Hohenberg and Kohn, PRB 136, 864 (1964) Hohenberg and Kohn, PRB 136, 864 (1964)

{about 500 citations per year}{about 500 citations per year}

DFT is based on two theorems:DFT is based on two theorems:

– The charge density, The charge density, n(n(rr)) is a basic variable is a basic variable E=E[n].E=E[n].– Variational principle: Variational principle: E[n]E[n] has a minimum at the ground has a minimum at the ground

state state n(n(rr),), nnGSGS((rr), or), or

Nobel Prize in Chemistry in1998, for his development of DFT.

][ ][)(][ nEnFdnVnE GSGSGSGSext rr

n(r) as a basic variablen(r) as a basic variable

NN ddn rrrrr ...|),...,(|)( 22

11

V(r) n(r)

Ψ(r1, … rN)

DFT: one-to-one correspondence

Standard QM

solve M.B. Schr.Eq.

• Since n determines V (to an additive constant), Ψ and hence the K.E. (T) and the e-e interaction energy (U) are functionals of n.

• One can then define a universal energy function ≡ F[n] = <Ψ| T + U| Ψ>. So,

][)(][ nFdrrnVnE ext {unkown functional of n}

Kohn-Sham formalism of DFTKohn-Sham formalism of DFT

Kohn and Sham, PRA 140, 1133 (1965)Kohn and Sham, PRA 140, 1133 (1965)

KS have introduced the following separation of KS have introduced the following separation of F[n]F[n]

][][][ nEnTnF Ho

where,

r,rr dT i

N

iio )(

2

1)(

1

2*

'|'|

)'()(

2

1][ rr

rrrr

ddnn

nEH

and

EXC is called exchange correlation energy

EXC

EXC=EX+EC+(T-To){the only unknown or difficult to calculate terms == to be approximated}

K.E. of non-interacting e-system.

Classical e-e interaction energy.

Exact self-consistent single-particle equationsExact self-consistent single-particle equations

Varying Varying E[n]E[n] with respect to with respect to n(r)n(r) under the constraint of under the constraint of constant number of electronsconstant number of electrons

0)(][][)()(

drrnnEnEdrrnVTrn XCHexto

Now, suppose that we have a non-interacting electronic system with the Now, suppose that we have a non-interacting electronic system with the same density same density n(rn(r), sustained by an effective potential ), sustained by an effective potential VVeffeff. Then, . Then,

(2) )()(

][

XCHexto VVV

n

T

n

nE

rr

0)()()(

drrndrrnVTrn effo

(3) )()(

][

effo V

n

T

n

nE

rr

Eqs. (2) and (3) are mathematically equivalent, andEqs. (2) and (3) are mathematically equivalent, and

(4) ... XCHextKS

eff VVVVV

This leads to exact (no approximation is used so far for EThis leads to exact (no approximation is used so far for EXCXC) transform of ) transform of

(5) ......... )()(][2

1 2 rr iiiKS nV

to

Therefore, Therefore, EEGSGS and and nnGSGS(r)(r) can be obtained by solving a set of can be obtained by solving a set of NN single-particle Schrödinger like single-particle Schrödinger like equationsequations (known as KS equations): (known as KS equations):

Note thatNote that

. state of no. occupation the

is and states, occupiedpartially or occupied

allover runs e wher,|)(|)( 2

i

f

ifn

i

ii

i rr

Thus, equations 3 to 5 have to solved self-consistently. Thus, equations 3 to 5 have to solved self-consistently.

Periodic Boundary Conditions and Bloch’s TheoremPeriodic Boundary Conditions and Bloch’s Theorem

Periodic Boundary ConditionsPeriodic Boundary Conditions: Finite systems are assumed to be : Finite systems are assumed to be periodically repeated to fill the whole space periodically repeated to fill the whole space

An efficient recipe to study atoms, molecules, surfaces, Interfaces, … etc

Bloch’s TheoremBloch’s Theorem: The wave-functions of the electrons moving in a : The wave-functions of the electrons moving in a periodic potential are given asperiodic potential are given as

rkkk rr .)()( i

nn eu

unk(r) have the same periodicity as the potential.n is the band indexk is a wave-vector inside the 1st BZ.

This transforms the problem into calculating few wavefunctions for, This transforms the problem into calculating few wavefunctions for, in principle, infinite number of in principle, infinite number of k k points.points.

The The great simplificationgreat simplification comes from the fact that comes from the fact that ΨΨnnkk are weakly varying functions with are weakly varying functions with respect to respect to kk … only few carefully chosen … only few carefully chosen kk-points (known as special -points (known as special kk-points) are required. -points) are required.

No. special E (H) Lattice Bulk No. special E (H) Lattice Bulk MeshMesh K-points K-points (a=10.4 Bohr(a=10.4 Bohr)) constant (Å)constant (Å) modulus (Mbar) 2x2x2 2 -7.930764 5.392 0.959 2x2x2 2 -7.930764 5.392 0.959 4x4x4 10 -7.936765 5.384 0.9564x4x4 10 -7.936765 5.384 0.9568x8x8 60 -7.936879 5.384 0.954 8x8x8 60 -7.936879 5.384 0.954

Convergence test: Si in the diamond structureConvergence test: Si in the diamond structure

Expt. 5.431 0.99Example: 2x2 meshFor 2D square lattice

Approximations to EApproximations to EXCXC

Local density approximation (LDA)Local density approximation (LDA)

– AssumptionAssumption: E: EXCXC depends locally on depends locally on ρρ( ( rr ) )

– Recommended LDA functional: Perdew-Wang (PRB Recommended LDA functional: Perdew-Wang (PRB 4545, 13244, 1992), 13244, 1992)

– LDALDA is currently being used to study fundamental is currently being used to study fundamental problems in physics, chemistry, geology, material problems in physics, chemistry, geology, material science and pharmacyscience and pharmacy. .

rrr dE XCLDAXC )()( hom

Generalized gradient approximation Generalized gradient approximation (GGA)(GGA)

– Assumption:Assumption:

– Recommended GGA functional: Perdew-Recommended GGA functional: Perdew-Burke-Ernzerhof (PBE) [PRL Burke-Ernzerhof (PBE) [PRL 7777, 3865 , 3865 (1996)].(1996)].

– GGAGGA is found to is found to improve the binding improve the binding energiesenergies, but , but not the band gapsnot the band gaps. .

rrrr dE XCGGAXC ))(),(()(

Meta-GGA (MGGA)Meta-GGA (MGGA)

– Assumption:Assumption:

– Here, Here, ττ is the kinetic energy density is the kinetic energy density

– Recommended MGGA functional: Toa-Perdew-Recommended MGGA functional: Toa-Perdew-Staroverov-Scuseria (TPSS) [PRL Staroverov-Scuseria (TPSS) [PRL 9191, 146401 , 146401 (2003)](2003)]

– Self-interaction free correlationSelf-interaction free correlation. . Not well tested Not well tested yetyet..

rrrrr dE XCMGGAXC ))(),(),(()(

2|)(|)( rr occ

ii

Main problem with LDA, GGA and MGGAMain problem with LDA, GGA and MGGA

– They allow for spurious They allow for spurious self-interactionself-interaction (SI).(SI).

– Exact DFT is SI free:Exact DFT is SI free:

GGA. and LDAfor case not the is This

and between n cancelatioExact

||

)()(

2

1

,For

.||

)()()()(

2

1

''

''

''

*''*

''

''''

XH

SelfH

vvselfx

vv

vvvvx

VV self

Eddrr

E

vv

ddrr

E

rrrr

kk

rrrrrr

kk

kk

kkkk

Theory of Exact-exchange (EXX)Theory of Exact-exchange (EXX)

– Total energyTotal energy

– Single-particle equationsSingle-particle equations

][||

)()()()(

2

1

||

)()(

2

1][

''

'''' ''

*''*

''

'

Cvv

vvvv

ieotot

Edd

ddETE

kk

kkkk rrrr

rrrr

rrrr

rr

)(

)(..

)(

)(

)(

][

)()(

with,][][][2

1

''

''

'

'

2

rrr

rVcc

rV

r

r

EEV

VVVV

KS

vk KS

vk

vk

XXX

iiiCXHion

[Stadele et al. PRB 59, 10 031 (1999)]

Hybrid DFT/HF functionals Hybrid DFT/HF functionals

– Adiabatic connection formulaAdiabatic connection formula

– Three empirical parameters hybrid fucntionalsThree empirical parameters hybrid fucntionals

– One empirical parameter hybrid fucntionalsOne empirical parameter hybrid fucntionals

– Parameter free hybrid fucntionalsParameter free hybrid fucntionals

|'|

)'()('

2

1||

with ,

33,

1

0

,

rr

rrrr

ddVE

EdE

eeXC

XCXC

GGAC

GGAX

LSDX

HFLSDXC

ACMXC EaEaEEaEE 321

3 )(

Example: B3LYB (Becke exchange and Lee-Yang-Parr Corr.)

)(11 GGA

XHFGGA

XCACMXC EEaEE Example: B1LYB

)(4

11 GGAX

HFGGAXC

ACMXC EEEE Examples: B0LYB

PBE0

Single-particle energiesSingle-particle energies

Whence a certain approximation for Whence a certain approximation for EEXC XC is adopted, one has to solve self- is adopted, one has to solve self-consistently the Schrödinger like single-particle equationsconsistently the Schrödinger like single-particle equations

)()(][2

1 2 rr kkk nnnKS nV

What is the physical meaning of What is the physical meaning of εεnnkk ? ?

Answer: two points of viewAnswer: two points of view

- According to the optimized effective potential (OEP) approach: According to the optimized effective potential (OEP) approach: VVKSKS is is the best local approximation to the non-local energy dependent the best local approximation to the non-local energy dependent electron self-energy operator (in many-body quasi-particle theory)electron self-energy operator (in many-body quasi-particle theory) -- -- εεnk nk

are approximate quasi-particle energies --- can be used to are approximate quasi-particle energies --- can be used to interpret band structure data. interpret band structure data.

- - According to the KS derivation of the single-particle equations:According to the KS derivation of the single-particle equations:

εεnnk k are mathematical construct {Lagrange multipliers}are mathematical construct {Lagrange multipliers} -- -- no physical meaning.no physical meaning.

Si band structureSi band structure

Computational approachesComputational approaches

All-electron:

- all the electron are explicitly included

- the space is separated in core are interstitial regions.

- Two main approaches

I- LAPW {partial waves (core) and PW (interstitial)} II- LMTO {partial waves (core) and Hankel functions (interstitial)}

Pseudopotential:

- electrons = valence+ core.

- only Valence electrons are explicitly included.

- effective potential (pseudopotential) due to the nucleus are the core electrons - PW basis sets to expand Ψnk

interstitial

core

Some results Some results

I.I. Phase stability and structural properties Phase stability and structural properties {example ZnS}{example ZnS}

E vs V curves of ZnS

Zincblende (cubic –2 atom unit cell)

SC16 (cubic –16 atom unit cell)

Cinnabar (hexagonal –6 atom unit cell)

The ZB structure is the moststable phase of ZnS, in agreement with experiment

Rocksalt (cubic –2 atom unit cell)

[Qteish and Parrinello, PRB 61, 6521 (2000)]

Structural Properties: Structural Properties: ZnSZnS

Zinc-blende structure (equilibrium phase) Zinc-blende structure (equilibrium phase)

Structural ParameterStructural Parameter TheoryTheory ExptExpt. . Error (%)Error (%)

Lattice constant (Å) 5.352 5.401 0.9Lattice constant (Å) 5.352 5.401 0.9

Bulk modulus (GPa) 83.4 76.9 8.5Bulk modulus (GPa) 83.4 76.9 8.5

Rocksalt structure (high pressure phase) Rocksalt structure (high pressure phase)

Structural ParameterStructural Parameter TheoryTheory ExptExpt. . Error (%)Error (%)

Lattice constant (Å) 5.017 5.060 0.8Lattice constant (Å) 5.017 5.060 0.8

Bulk modulus (GPa) 104.4 103.6 0.7Bulk modulus (GPa) 104.4 103.6 0.7

The theoretical values are obtained by fitting the calculated E to Murnaghan’s EOS.

.11

)/()(

'

'Cons

B

VV

B

VBVE

o

Bo

o

oo

II. Structural phase transformation under high pressure II. Structural phase transformation under high pressure

• Enthalpy (H) vs Pressure for ZnS

)()()( ppVpVEpH

• Transition pressure (GPa)

TransitionTransition TheoryTheory ExptExpt. . ZB to RS 14.5 15ZB to RS 14.5 15ZB to SC16 12.5 ---ZB to SC16 12.5 ---ZB to cinnabar 16.4 ---ZB to cinnabar 16.4 ---SC16 to RS 16.2 ---SC16 to RS 16.2 ---

III. Phonons: inter-planer force constant approach III. Phonons: inter-planer force constant approach

• IPFC’s are calculated by displacing the atoms of one layer by small amount

Fi = -kiu

• IPFC’s are then used to calculate the phonon spectra along some high-symmetry direction.

Ben Amar, Qteish and Meskini, PRB 53, 5372 (1996)

IV. Elastic constantsIV. Elastic constants

• Direct method: applying proper strain and calculate the corresponding stress [Nielson and Martin PRB 32, 3792 (1985)]

• Using density functional perturbation theory (Lec. 3)

Hamdi, Aouissi, Qteish and Meskini, PRB 73, 174114 (2006)

Elastic constantOf ZnSe

a DFPTb Direct method

V. Thermal Properties (details are in Lecture III)V. Thermal Properties (details are in Lecture III)

• Linear thermal expansion coefficient of ZnSe

• Constant pressure heat capacity at of ZnSe

Hamdi, Aouissi, Qteish and Meskini, PRB 73, 174114 (2006)

DFT is a very powerful tool in theoretical/computational DFT is a very powerful tool in theoretical/computational condensed matter physics.condensed matter physics.

It has wide applications in physics, chemistry, material science, It has wide applications in physics, chemistry, material science, geophysics, … etc.geophysics, … etc.

Exciting and continuous progress on the level of theory, Exciting and continuous progress on the level of theory, algorithms and applications.algorithms and applications.

Highly suitable for scientists working in developing countries – Highly suitable for scientists working in developing countries –

workstations are enough.workstations are enough.

ConclusionsConclusions

End