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NANO266 - Lecture 7 - QM Modeling of Periodic Structures

UCSD NANO 266 Quantum Mechanical Modelling of Materials and Nanostructures is a graduate class that provides students with a highly practical introduction to the application of first principles quantum mechanical simulations to model, understand and predict the properties of materials and nano-structures. The syllabus includes: a brief introduction to quantum mechanics and the Hartree-Fock and density functional theory (DFT) formulations; practical simulation considerations such as convergence, selection of the appropriate functional and parameters; interpretation of the results from simulations, including the limits of accuracy of each method. Several lab sessions provide students with hands-on experience in the conduct of simulations. A key aspect of the course is in the use of programming to facilitate calculations and analysis.

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Quantum Mechanical
Modeling of Periodic
Structures
Shyue Ping Ong
The MaterialsWorld
NANO266
2
Molecules
Isolated gas
phase
Typically use
localized basis
functions, e.g.,
Gaussians
Everything else
(liquids,
amorphous solids,
etc.)
Too complex
for direct QM!
(at the
moment)
But can work
reasonable
models
sometimes
Crystalline solids
Periodic
infinite solid
Plane-wave
approaches
What is a crystal?
A crystal is a time-invariant, 3D arrangement of
atoms or molecules on a lattice.
NANO266
Perovskite SrTiO3
The “motif”
repeated on each point in the cubic
lattice below…
3
Translational symmetry
All crystals are characterized by translational
symmetry
NANO266
4
t = ua + vb + wc, u,v,w ∈ Z
1D 2D (single layer MoS2)
3D
The 14 3D Bravais Lattices
NANO 106 - Crystallography of Materials by
Shyue Ping Ong - Lecture 2
P: primitive
C: C-centered
I: body-centered
F: face-centered
(upper case for 3D)
a: triclinic (anorthic)
m: monoclinic
o: orthorhombic
t: tetragonal
h: hexagonal
c: cubic
3D unit cells
Infinite number of unit cells for all 3D lattices
Always possible to define primitive unit cells for non-primitive lattices,
though the full symmetry may not be retained.
NANO 106 - Crystallography of Materials by
Shyue Ping Ong - Lecture 2
Conventional cF cell Primitive unit cell
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NANO266 - Lecture 7 - QM Modeling of Periodic Structures

  • 1. Quantum Mechanical Modeling of Periodic Structures Shyue Ping Ong
  • 2. The MaterialsWorld NANO266 2 Molecules Isolated gas phase Typically use localized basis functions, e.g., Gaussians Everything else (liquids, amorphous solids, etc.) Too complex for direct QM! (at the moment) But can work reasonable models sometimes Crystalline solids Periodic infinite solid Plane-wave approaches
  • 3. What is a crystal? A crystal is a time-invariant, 3D arrangement of atoms or molecules on a lattice. NANO266 Perovskite SrTiO3 The “motif” repeated on each point in the cubic lattice below… 3
  • 4. Translational symmetry All crystals are characterized by translational symmetry NANO266 4 t = ua + vb + wc, u,v,w ∈ Z 1D 2D (single layer MoS2) 3D
  • 5. The 14 3D Bravais Lattices NANO 106 - Crystallography of Materials by Shyue Ping Ong - Lecture 2 P: primitive C: C-centered I: body-centered F: face-centered (upper case for 3D) a: triclinic (anorthic) m: monoclinic o: orthorhombic t: tetragonal h: hexagonal c: cubic
  • 6. 3D unit cells Infinite number of unit cells for all 3D lattices Always possible to define primitive unit cells for non-primitive lattices, though the full symmetry may not be retained. NANO 106 - Crystallography of Materials by Shyue Ping Ong - Lecture 2 Conventional cF cell Primitive unit cell
  • 7. The Reciprocal Lattice For a lattice given by basis vectors a1, a2 and a3, the reciprocal lattice is given basis vectors a1*, a2* and a3* where: NANO 106 - Crystallography of Materials by Shyue Ping Ong - Lecture 2 a1 * = 2π a2 × a3 a1.(a2 × a3 ) a2 * = 2π a1 × a3 a1.(a2 × a3 ) a3 * = 2π (a1 × a2 ) a1.(a2 × a3 ) ai * aj = 2πδij
  • 8. Reciprocal lattice Translation vectors in the reciprocal lattice is given by: NANO266 8 G = ha1 * + ka2 * +la3 * Direct lattice Reciprocal lattice Simple Cubic Simple Cubic Face-centered cubic (fcc) Body-centered cubic (bcc) Body-centered cubic (bcc) Face-centered cubic (fcc) Hexagonal Hexagonal
  • 9. Periodic Boundary Conditions Repeat unit cell infinitely in all directions. What does this mean for our external potential (from the nuclei)? NANO266 9
  • 10. Electron in a periodic potential For an electron in a 1D periodic potential with lattice vector a, we have For any periodic function, we may express it in terms of a Fourier series NANO266 10 H = − 1 2 ∇+V(x) where V(x) =V(x + ma). V(x) = Vne i 2π a nx n=−∞ ∞ ∑
  • 11. Bloch’sTheorem For a particle in a periodic potential, eigenstates can be written in the form of a Bloch wave Where u(r) has the same periodicity as the crystal and k is a vector of real numbers known as the crystal wave vector, n is known as the band index. For any reciprocal lattice vector K, , i.e., we only need to care about k in the first Brillouin Zone NANO266 11 ψn,k (r) = eik.r un,k (r) Plane wave ψn,k+K (r) =ψn,k (r)
  • 12. Brillouin Zones for common lattices NANO266 12 simple cubic fcc bcc hexagonal
  • 14. Plane waves as a basis Any function that is periodic in the lattice can be written as a Fourier series of the reciprocal lattice Recall that from the Bloch Theorem, our wave function is of the form Where u(r) has the same periodicity as the crystal and k is a vector of real numbers known as the crystal wave vector, n is known as the band index. NANO266 14 ψn,k (r) = eik.r un,k (r) f (x) = cne i 2π a nx n=−∞ ∞ ∑ Reciprocal lattice vector in 1D
  • 15. Plane waves as a basis Let us now write u(r) as an expansion Our wave function then becomes NANO266 15 un,k (r) = cG n,keiGr G ∑ ψn,k (r) = cG n,kei(k+G).r G ∑
  • 16. Using the plane waves as basis Plane waves offer a systematic way to improve completeness of our solution Recall that for a free electron in a box, Corresponding, each plane wave have energy NANO266 16 ψn,k (r) = cG n,kei(k+G).r G ∑ Infinite sum over reciprocal space ψ(r) = eik.r and the corresponding energy is E = !2 2m k2 E = !2 2m k+G 2
  • 17. Energy cutoff Solutions with lower energy are more physically important than solutions with higher energies NANO266 17 Ecut = !2 2m Gcut 2 ψn,k (r) = cG n,kei(k+G).r k+G<Gcut ∑
  • 18. Convergence with energy cutoff The same energy cutoff must be used if you want to compare energies between calculations, e.g., if you want to compute: Cu (s) + Pd (s) -> CuPd(s) NANO266 18
  • 19. Pseudopotentials Problem: Tightly bound electrons have wavefunctions that oscillate on very short length scales => Need a huge cutoff (and lots of plane waves). Solution: Pseudopotentials to represent core electrons with a smoothed density to match various important physical and mathematical properties of true ion core NANO266 19 ψn,k (r) = cG n,kei(k+G).r G ∑
  • 20. Types of pseudopotentials (PPs) Norm-conserving (NC) •  Enforces that inside cut-off radius, the norm of the pseudo-wavefunction is identical to all-electron wavefunction. Ultrasoft (US) •  Relax NC condition to reduce basis set size further Projector-augmented wave (PAW) •  Avoid some problems with USPP •  Generally gives similar results as USPP and all-electron in many instances. NANO266 20 Kresse, G.; Joubert, D. From ultrasoft pseudopotentials to the projector augmented-wave method, Phys. Rev. B, 1999, 59, 1758–1775.
  • 21. Comparison of different PPs NANO266 21
  • 22. How do you choose PPs? Sometimes, several PPs are available with different number of “valence” electrons, i.e., electrons not in the core. Choice depends on research problem – if you are studying problems where more (semi-core) electrons are required, choose PP with more electrons But more electrons != better results! (e.g., Rare-earth elements) NANO266 22
  • 23. Born–von Karman boundary condition Consider large, but finite crystal of volume V with edges Born-von Karman boundary condition requires Since we have Bloch wavefunctions, Therefore, possible k-vectors compatible with cyclic boundaries are given by: NANO266 23 N1t1, N2t2, N3t3 ψ(r+ N1t1 ) =ψ(r+ N2t2 ) =ψ(r+ N3t3) =ψ(r) eikN1t1 = eikN2t2 = eikN3t3 =1 k = m1 N1 g1 + m2 N2 g2 + m3 N3 g3
  • 24. Integrations in k space For counting of electrons in bands and total energies, etc., need to sum over states labeled by k Numerically, integrals are performed by evaluating function at various points in the space and summing them. NANO266 24 f = Vcell (2π)3 f (k)dk BZ ∫ f = 1 Nk f (k) k ∑
  • 25. Choice of k-points 1.  Sampling at one point (Baldereschi point, or Gamma point) 2.  Monkhorst-Pack – Sampling at regular meshes NANO266 25
  • 26. Monkhorst-Pack mesh Regular equi-spaced mesh in BZ NANO266 26 Unshifted Shifted
  • 27. Convergence with respect to k-points NANO266 27 Similar exercise in lab 2!
  • 28. Important things to note about k-point convergence Symmetry reduces integrals to be performed -> Irreducible Brillouin Zone k-point mesh is inversely related to unit cell volume (larger unit cell volume -> smaller reciprocal cell volume) NANO266 28
  • 29. k-point sampling in metals BZ in metals are divided into occupied and unoccupied regions by Fermi surface, where the integrated functions change discontinuously from non-zero to zero. => Extremely dense k-point mesh needed for integration Algorithmic solutions •  Tetrahedron method. Use k points to define a tetrahedra that fill reciprocal space and interpolate. Most widely used is Blochl’s version. •  Smearing. Force the function being integrated to be continuous by “smearing” out the discontinuity, e.g., with the Fermi-Dirac function or the Methfessel and Paxton method. NANO266 29 Fermi-surface of Copper (Cu), the color codes the inverse effective mass of the electrons, large effective masses are represented in red, from A. Weismann et al., Science 323, 1190 (2009)
  • 30. References Martin, R. M. Electronic Structure: Basic Theory and Practical Methods (Vol 1); Cambridge University Press, 2004. Grosso, G.; Parravicini, G. P. Solid State Physics: : 9780123044600: Amazon.com: Books; 1st ed.; Academic Press, 2000. NANO266 30