Modern Computational condensed Matter Physics Basic theory and
Modern Computational condensed Matter Physics: Basic theory and applications Prof. Abdallah Qteish Department of Physics, Yarmouk University, 21163 -Irbid, Jordan Chemistry Dept, YU, 14 May 2007
Starting from first-principles, can we efficiently and accurately u u u u Calculate the various properties (structural, electronic structure, vibrational, thermal, elastic, magnetic, …, etc) of bulk solids; Investigate the surface and interface properties of solids; Study defects; Construct the phase diagrams of alloys; Study the properties of liquids and amorphous materials; Investigate the material properties under extreme condition (very high temperature and pressure); Deal with biological systems; Others ? ? Answer: YES
Direct application of Standard QM !! In Standard QM, Ψ which is the solution of the many-body Schrödinger Eq. u . . . (1) is the basic variable u Main problem: Ψ is a function of 3 N variables, and N is of order of 1024 for a realistic condensed matter sample. u Thus, direct application of Standard QM is simply impossible. u Remark: In Eq. (1) the nuclei are assumed to be at fixed positions adiabatic or Born-Oppenhiemer approximation.
Density Functional Theory (DFT) Hohenberg and Kohn, PRB 136, 864 (1964) {about 500 citations per year} Nobel Prize in Chemistry in 1998, for his development 1999 of DFT. u DFT is based on two theorems: – The charge density, n(r) is a basic variable E=E[n]. – Variational principle: E[n] has a minimum at the ground state n(r), n. GS(r), or
n(r) as a basic variable Standard QM Ψ(r 1, … r. N) solve M. B. Schr. Eq. n(r) V(r) DFT: one-to-one correspondence • 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, {unkown functional of n}
Kohn-Sham formalism of DFT Kohn and Sham, PRA 140, 1133 (1965) u KS have introduced the following separation of F[n] EXC where, K. E. of non-interacting e-system. Classical e-e interaction energy. and EXC is called exchange correlation energy EXC=EX+EC+(T-To) {the only unknown or difficult to calculate terms == to be approximated}
Exact self-consistent single-particle equations u Varying E[n] with respect to n(r) under the constraint of constant number of electrons u Now, suppose that we have a non-interacting electronic system with the same density n(r), sustained by an effective potential Veff. Then, u Eqs. (2) and (3) are mathematically equivalent, and
This leads to exact (no approximation is used so far for E XC) transform of u to u Therefore, EGS and n. GS(r) can be obtained by solving a set of N singleparticle Schrödinger like equations (known as KS equations): u Note that u Thus, equations 3 to 5 have to solved self-consistently.
Periodic Boundary Conditions and Bloch’s Theorem u Periodic Boundary Conditions: Finite systems are assumed to be periodically repeated to fill the whole space An efficient recipe to study atoms, molecules, surfaces, Interfaces, … etc u Bloch’s Theorem: The wave-functions of the electrons moving in a periodic potential are given as unk(r) have the same periodicity as the potential. n is the band index k is a wave-vector inside the 1 st BZ.
u This transforms the problem into calculating few wavefunctions for, in principle, infinite number of k points. u The great simplification comes from the fact that Ψnk are weakly varying functions with respect to k … only few carefully chosen kpoints (known as special k-points) are required. u Convergence test: Si in the diamond structure Mesh 2 x 2 x 2 4 x 4 x 4 8 x 8 x 8 No. special K-points 2 10 60 Example: 2 x 2 mesh For 2 D square lattice E (H) (a=10. 4 Bohr) -7. 930764 -7. 936765 -7. 936879 Expt. Lattice constant (Å) 5. 392 5. 384 5. 431 Bulk modulus (Mbar) 0. 959 0. 956 0. 954 0. 99
Approximations to EXC u Local density approximation (LDA) – Assumption: EXC depends locally on ρ( r ) – Recommended LDA functional: Perdew-Wang (PRB 45, 13244, 1992) – LDA is currently being used to study fundamental problems in physics, chemistry, geology, material science and pharmacy.
u Generalized gradient approximation (GGA) – Assumption: – Recommended GGA functional: Perdew. Burke-Ernzerhof (PBE) [PRL 77, 3865 (1996)]. – GGA is found to improve the binding energies, but not the band gaps.
u Meta-GGA (MGGA) – Assumption: – Here, τ is the kinetic energy density – Recommended MGGA functional: Toa. Perdew-Staroverov-Scuseria (TPSS) [PRL 91, 146401 (2003)] – Self-interaction free correlation. Not well tested yet.
Main problem with LDA, GGA and MGGA – They allow for spurious self-interaction (SI). – Exact DFT is SI free:
u Theory of Exact-exchange (EXX) [Stadele et al. PRB 59, 10 031 (1999)] – Total energy – Single-particle equations
u Hybrid DFT/HF functionals – Adiabatic connection formula – Three empirical parameters hybrid fucntionals Example: B 3 LYB (Becke exchange and Lee-Yang-Parr Corr. ) – One empirical parameter hybrid fucntionals Example: B 1 LYB – Parameter free hybrid fucntionals Examples: B 0 LYB PBE 0
Single-particle energies u Whence a certain approximation for EXC is adopted, one has to solve selfconsistently the Schrödinger like single-particle equations u What is the physical meaning of εnk ? u Answer: two points of view - According to the KS derivation of the single-particle equations: εnk are mathematical construct {Lagrange multipliers} -- no physical meaning. - According to the optimized effective potential (OEP) approach: VKS is the best local approximation to the non-local energy dependent electron self-energy operator (in many-body quasi-particle theory) -εnk are approximate quasi-particle energies --- can be used to interpret band structure data.
u Si band structure
Computational approaches All-electron: Pseudopotential: - all the electron are explicitly included - the space is separated in core are interstitial regions. - electrons = valence+ core. - only Valence electrons are explicitly included. interstitial - Two main approaches core I- LAPW {partial waves (core) and PW (interstitial)} II- LMTO {partial waves (core) and Hankel functions (interstitial)} - effective potential (pseudopotential) due to the nucleus are the core electrons - PW basis sets to expand Ψnk
Some results
I. Phase stability and structural properties {example Zn. S} [Qteish and Parrinello, PRB 61, 6521 (2000)] E vs V curves of Zn. S The ZB structure is the most stable phase of Zn. S, in agreement with experiment Zincblende (cubic – 2 atom unit cell) Rocksalt (cubic – 2 atom unit cell) SC 16 (cubic – 16 atom unit cell) Cinnabar (hexagonal – 6 atom unit cell)
Structural Properties: Zn. S u Zinc-blende structure (equilibrium phase) Structural Parameter Lattice constant (Å) Bulk modulus (GPa) u Theory 5. 352 83. 4 Expt. 5. 401 76. 9 Error (%) 0. 9 8. 5 Expt. 5. 060 103. 6 Error (%) 0. 8 0. 7 Rocksalt structure (high pressure phase) Structural Parameter Lattice constant (Å) Bulk modulus (GPa) Theory 5. 017 104. 4 The theoretical values are obtained by fitting the calculated E to Murnaghan’s EOS.
II. Structural phase transformation under high pressure • Enthalpy (H) vs Pressure for Zn. S • Transition pressure (GPa) Transition ZB to RS ZB to SC 16 ZB to cinnabar SC 16 to RS Theory 14. 5 12. 5 16. 4 16. 2 Expt. 15 -------
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 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) DFPT b Direct method a Elastic constant Of Zn. Se Hamdi, Aouissi, Qteish and Meskini, PRB 73, 174114 (2006)
V. Thermal Properties (details are in Lecture III) • Linear thermal expansion coefficient of Zn. Se • Constant pressure heat capacity at of Zn. Se Hamdi, Aouissi, Qteish and Meskini, PRB 73, 174114 (2006)
Conclusions u DFT is a very powerful tool in theoretical/computational condensed matter physics. u It has wide applications in physics, chemistry, material science, geophysics, … etc. u Exciting and continuous progress on the level of theory, algorithms and applications. u Highly suitable for scientists working in developing countries – workstations are enough.
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