1. 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

2. Starting from first-principles, can we efficiently and accurately  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

3. Direct application of Standard QM !!  In Standard QM, Ψ which is the solution of the many-body Schrödinger Eq. is the basic variable  Main problem: Ψ is a function of 3N variables, and N is of order of 10 24 for a realistic condensed matter sample.  Thus, direct application of Standard QM is simply impossible.  Remark: In Eq. (1) the nuclei are assumed to be at fixed positions adiabatic or Born-Oppenhiemer approximation. adiabatic or Born-Oppenhiemer approximation....... (1)

4. Density Functional Theory (DFT) Hohenberg and Kohn, PRB 136, 864 (1964) {about 500 citations per year}  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 Nobel Prize in Chemistry in 1998, for his development of DFT.

5. n(r) as a basic variable V(r) n(r) Ψ(r 1, … r N ) 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] =. So, {unkown functional of n}

6. Kohn-Sham formalism of DFT Kohn and Sham, PRA 140, 1133 (1965)  KS have introduced the following separation of F[n] where, and E XC is called exchange correlation energy E XC E XC =E X +E C +(T-T o ) {the only unknown or difficult to calculate terms == to be approximated} K.E. of non-interacting e-system. Classical e-e interaction energy.

7. Exact self-consistent single-particle equations  Varying E[n] with respect to n(r) under the constraint of constant number of electrons  Now, suppose that we have a non-interacting electronic system with the same density n(r), sustained by an effective potential V eff. Then,  Eqs. (2) and (3) are mathematically equivalent, and

8.  This leads to exact (no approximation is used so far for E XC ) transform of to  Therefore, E GS and n GS (r) can be obtained by solving a set of N single- particle Schrödinger like equations (known as KS equations):  Note that  Thus, equations 3 to 5 have to solved self-consistently.

9. Periodic Boundary Conditions and Bloch’s Theorem  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  Bloch’s Theorem: The wave-functions of the electrons moving in a periodic potential are given as u nk (r) have the same periodicity as the potential. n is the band index k is a wave-vector inside the 1 st BZ.

10.  This transforms the problem into calculating few wavefunctions for, in principle, infinite number of k points.  The great simplification comes from the fact that Ψ nk are weakly varying functions with respect to k … only few carefully chosen k- points (known as special k-points) are required. No. special E (H) Lattice Bulk No. special E (H) Lattice Bulk MeshK-points (a=10.4 Bohr)constant (Å) Mesh K-points (a=10.4 Bohr) constant (Å) modulus (Mbar) 2x2x2 2 -7.930764 5.392 0.959 4x4x4 10 -7.936765 5.384 0.956 8x8x8 60 -7.936879 5.384 0.954  Convergence test: Si in the diamond structure Expt. 5.431 0.99 Example: 2x2 mesh For 2D square lattice

11. Approximations to E XC  Local density approximation (LDA) –Assumption: E XC 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.

12.  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.

13.  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.

14. Main problem with LDA, GGA and MGGA –They allow for spurious self-interaction (SI). –Exact DFT is SI free:

15.  Theory of Exact-exchange (EXX) –Total energy –Single-particle equations [Stadele et al. PRB 59, 10 031 (1999)]

16.  Hybrid DFT/HF functionals –Adiabatic connection formula –Three empirical parameters hybrid fucntionals –One empirical parameter hybrid fucntionals –Parameter free hybrid fucntionals Example: B3LYB (Becke exchange and Lee-Yang-Parr Corr.) Example: B1LYB Examples: B0LYB PBE0

17. Single-particle energies  Whence a certain approximation for E XC is adopted, one has to solve self- consistently the Schrödinger like single-particle equations  What is the physical meaning of ε nk ?  Answer: two points of view -According to the optimized effective potential (OEP) approach: V KS is the best local approximation to the non-local energy dependent electron self-energy operator (in many-body quasi-particle theory) -- ε nk are approximate quasi-particle energies --- can be used to interpret band structure data. - According to the KS derivation of the single-particle equations: ε nk are mathematical construct {Lagrange multipliers} -- no physical meaning. ε nk are mathematical construct {Lagrange multipliers} -- no physical meaning.

18.  Si band structure

19. Computational 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 Ψ nk interstitial core

20. Some results

21. I.Phase stability and structural properties {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 most stable phase of ZnS, in agreement with experiment Rocksalt (cubic – 2 atom unit cell) [Qteish and Parrinello, PRB 61, 6521 (2000)]

22. Structural parameter ZB RS cinnabar SC16 a0 ~Å! 5.352,a 5.410b 5.017,a 5.060c 3.765a 6.555a B0 ~GPa! 83.1,a 76.9b 104.4,a 103.6c 89.3a 78.4a B0 8 4.43,a 4.9b 4.28a 4.51a 4.73a aPP-PW calculations. bExperimental data ~Ref. 18!. cExperimental data, obtained by using a fixed value of B0 8 of 4.0 ~Ref. 19!. Structural Properties: ZnS  Zinc-blende structure (equilibrium phase) Structural Parameter Theory Expt. Error (%) Structural Parameter Theory Expt. Error (%) Lattice constant (Å) 5.352 5.401 0.9 Lattice constant (Å) 5.352 5.401 0.9 Bulk modulus (GPa) 83.4 76.9 8.5 Bulk modulus (GPa) 83.4 76.9 8.5  Rocksalt structure (high pressure phase) Structural Parameter Theory Expt. Error (%) Structural Parameter Theory Expt. Error (%) Lattice constant (Å) 5.017 5.060 0.8 Lattice constant (Å) 5.017 5.060 0.8 Bulk modulus (GPa) 104.4 103.6 0.7 Bulk modulus (GPa) 104.4 103.6 0.7 The theoretical values are obtained by fitting the calculated E to Murnaghan’s EOS.

23. II. Structural phase transformation under high pressure Enthalpy (H) vs Pressure for ZnS Transition pressure (GPa) Transition Theory Expt. ZB to RS 14.5 15 ZB to SC16 12.5 --- ZB to cinnabar 16.4 --- SC16 to RS 16.2 ---

24. III. Phonons: inter-planer force constant approach IPFC’s are calculated by displacing the atoms of one layer by small amount F i = -k i u IPFC’s are then used to calculate the phonon spectra along some high-symmetry direction. Ben Amar, Qteish and Meskini, PRB 53, 5372 (1996)

25. 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) Hamdi, Aouissi, Qteish and Meskini, PRB 73, 174114 (2006) Elastic constant Of ZnSe a DFPT b Direct method

26. 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)

27.  DFT is a very powerful tool in theoretical/computational condensed matter physics.  It has wide applications in physics, chemistry, material science, geophysics, … etc.  Exciting and continuous progress on the level of theory, algorithms and applications.  Highly suitable for scientists working in developing countries – workstations are enough. workstations are enough. Conclusions

28. End