Quantum Mechanical Modeling of Periodic Structures

Shyue Ping Ong

The Materials World

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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.

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Perovskite SrTiO3

The “motif”

repeated on each point in the cubic lattice below…

3

Translational symmetry

All crystals are characterized by translational symmetry

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t = ua+ vb+wc, u,v,w ∈ Z1D 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

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*a j = 2πδij

Reciprocal lattice

Translation vectors in the reciprocal lattice is given by:

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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

Periodic Boundary Conditions

Repeat unit cell infinitely in all directions.

What does this mean for our external potential (from the nuclei)?

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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

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H = −12∇+V (x)

where V (x) =V (x +ma).

V (x) = Vnei2πanx

n=−∞

∞

∑

Bloch’s Theorem

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

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ψn,k (r) = eik.run,k (r)

Plane wave

ψn,k+K (r) =ψn,k (r)

Brillouin Zones for common lattices

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simple cubic

fcc

bcc hexagonal

Bloch waves

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From wikipedia

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.

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ψn,k (r) = eik.run,k (r)

f (x) = cnei2πanx

n=−∞

∞

∑

Reciprocal lattice vector in 1D

Plane waves as a basis

Let us now write u(r) as an expansion

Our wave function then becomes

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un,k (r) = cGn,keiGr

G∑

ψn,k (r) = cGn,kei(k+G).r

G∑

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

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ψn,k (r) = cGn,kei(k+G).r

G∑

Infinite sum over reciprocal space

ψ(r) = eik.r and the corresponding energy is E = !2

2mk2

E = !2

2mk+G 2

Energy cutoff

Solutions with lower energy are more physically important than solutions with higher energies

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Ecut =!2

2mGcut2

ψn,k (r) = cGn,kei(k+G).r

k+G<Gcut

∑

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)

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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) = cGn,kei(k+G).r

G∑

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.

Comparison of different PPs

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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)

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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:

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N1t1, N2t2, N3t3

ψ(r+ N1t1) =ψ(r+ N2t2 ) =ψ(r+ N3t3) =ψ(r)

eikN1t1 = eikN2t2 = eikN3t3 =1

k = m1N1g1 +

m2

N2

g2 +m3

N3

g3

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.

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f = Vcell(2π )3

f (k)dkBZ∫

f = 1Nk

f (k)k∑

Choice of k-points

1.  Sampling at one point (Baldereschi point, or Gamma point)

2.  Monkhorst-Pack – Sampling at regular meshes

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Monkhorst-Pack mesh

Regular equi-spaced mesh in BZ

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Unshifted Shifted

Convergence with respect to k-points

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Similar exercise in lab 2!

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)

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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.

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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)

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.

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