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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