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The application of translation symmetries in quantum mechanics, specifically in the context of one-body Hamiltonians commonly used in mean-field approaches. the concept of Bloch states, the Brillouin zone, and the orthogonality of Bloch states. It also explains the importance of these concepts in the context of periodic potentials and the Hamiltonian's block diagonalization.
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International summer School in electronic structure Theory: electron correlation in Physics and Chemistry
Centre Paul Langevin, Aussois, Savoie, France
Xavier Blase
Institut N´eel, CNRS, Grenoble, France.
June 12, 2015
I (^) Introduction to Solid State Physics, 8th Edition, Charles Kittel, John Wiley and Sons Eds. I (^) Solid State Physics, N.W. Ashcroft and N.D. Mermin, Saunders College Publishing. I (^) P. Yu and M. Cardona, Fundamentals of semiconductors, Springer. I (^) Ibach and L¨uth, Solid-state physics, Springer (with sections devoted to experimental techniques).
More advanced: I (^) Theoretical Solid State Physics, W. Jones and March, Dover Eds. I (^) Quantum Theory of Solids, Charles Kittel, John Wiley and Sons.
The unit cell is a portion of space that repeated periodically can reconstruct the entire crystal. A unit cell can contain several atoms (the motif). The lattice vectors: ~Rijk = i~a 1 + j~a 2 + k~a 3 allow to reconstruct the crystal from the atoms in the unit-cell with (~a 1 ,~a 2 ,~a 3 ) the basis vectors. The minimum volume cell is a primitive cell.
Figure: (Left) 2D square lattice with one atom per cell. A unit-cell is shaded in blue. (Right) 2D hexagonal cell with one atom per cell. Two different unit cells are represented. For the blue cell, 1/4th of each connected atom belong to this cell. The yellow one is called the Wigner-Seitz cell that is invariant with respect to the crystal symmetry point group.
Depending on the shape of the unit-cell, one can categorize 3D crystal under 7 different ”lattice systems”, that yield 14 Bravais lattices depending on the disposition of atoms in the unit cell (the motif). For example, a cubic lattice can be ”simple”, body-centered (BCC) or face-centered (FCC). Silicon and diamond are FCC lattices with 2 atoms per primitive cell.
Courtesy: http : //chemwiki.ucdavis.edu/Wikitexts/UC Davis/UCD Chem 2 B/UCD Chem 2 B
We consider now the case of a crystal with discrete (not infinitesimal) translation properties.
Assume that the potential is periodic: V(x+R)=V(x) (with R=na) and call (TR ) the translation operator. Then:
TR [V (x)ψ(x)] = V (x − R)ψ(x − R) = V (x)TR ψ(x)
which means that the potential, and thus the Hamiltonian, commute with the translation operator: [TR , H] = 0. Then quantum mechanics says that one can find a common eigenbasis for the two operators.
H|ψk > = Ek |ψk > TR |ψk > = Ck (R)|ψk >
We can find the expression of the Ck by simple considerations. The translation operator should preserve the normalisation of ψ: ∫ dx|ψ(x − R)|^2 =
dx|TR ψ(x)|^2 =
dx|C (R)|^2 |ψ(x)|^2 =
dx|ψ(x)|^2
so |C (R)|^2 = 1 and C (R) = eiθ(R). Further:
TaTaψ(x) = ψ(x − 2 a) = T 2 aψ(x) ⇒ C (a)C (a) = C (2a)
The only mathematical function satisfying such conditions is:
C (a) = e−ika^ ⇒ C (2a) = C (a)C (a) and C (R = na) = e−ikR^.
The quantum number (k) is associated with the translation operator.
The ei~k~r^ phase term can be regarded as an ”envelope function” that modulates the periodic function u~k (~r ). In the 1D example here below, assume that each atom has one (pz ) orbital. One can create different Bloch states by changing the magnitude of the k-vector (we represent e.g. the real part of the wavefunctions).
For the first/second Bloch state, (k = π/ 2 a) and (k = π/ 4 a).
The ~k-vectors are homogeneous to the inverse of a distance and lives in the ”reciprocal space”. If (~a 1 ,~a 2 ,~a 3 ) are the periodic vector of the crystal, we choose to represent the ~k-vectors as a function of the reciprocal space basis: (~b 1 , ~b 2 , ~b 3 ) vectors such that:
~bi = 2π ~aj^ ×^ ~ak ~ai · (~aj × ~ak )
⇒ ~ai · ~bj = 2πδij
Defining the reciprocal space vectors: G~ = l 1 ~b 1 + l 2 ~b 2 + l 3 ~b 3 , then the ei^ ~G^ ·~r^ vectors form a basis for periodic functions since for any lattice vector in real space ~R = n 1 ~a 1 + n 2 ~a 2 + n 3 ~a 3 ,
ei^ G~ ·(~r +R~) = ei^ ~G ·~r +i P α nαlα~aα·~bα = ei^ ~G ·~r +i P α nαlα 2 π = ei^ ~G ·~r
For example, u~k (~r ) =
~G u~k (G~^ )e i G~ ·~r (^) , the planewave expansion of u ~k (~r^ ).
The Brillouin zone is usually taken to be the highest-symmetry primitive cell of the reciprocal lattice, namely the ”Wigner-Seitz” primitive cell. Plot the planes normal to reciprocal lattice vectors cutting them ”in the middle”. The volume that such planes will define is the Brillouin zone. High symmetry directions and k-points have ”standard” names.
Figure: (Left) 2D square and hexagonal BZ. (Middle) The BZ of a face-centered cubic (FCC) lattice is a truncated octaedron. (Right) The BZ of a body-centered cubic lattice is a rhombic dodecahedron.
For illustration, let’s go back to 1D. The values of (k) are governed by the boundary conditions. Solid-state physicists adopt usually the Born and von Karman periodic boundary conditions where the solid ”closes” onto itself. This means that with N cells, one has the condition:
ψ(x + Na) = ψ(x) ⇒ eikNa^ = 1 ⇒ k = integer × (2π/Na)
The first BZ is: − π a < k ≤ π a.
There are thus N k-vectors in the BZ, as many as unit cells.
Sum rule:
~R ei
n=0 e
ikna (^) = 1 −eikNa 1 −eika^ = 0 for^ k^6 = 0 in the BZ.
Since the Hamiltonian is periodic, then the arguments developed here above hold and:
〈ψn′~k′ | Hˆ|ψn~k 〉 = 〈ψn′~k | Hˆ|ψn~k 〉δ(~k − ~k′)
where the indices (n,n’) serve to distinguish Bloch states with the same ~k-vector (not necessarily eigenstates of Hˆ, e.g. basis vectors).
The Hamiltonian does not couple Bloch states with different Bloch vectors.
This is the central result.
Since there are as many k-point in the BZ as there are unit cells in the crystal, in the absence of orthogonality and block diagonalization of the Hamiltonian, the Bloch representation would not have helped much.
We start from Bloch theorem: ψn~k (~r ) = ei~k~r^ u~k (~r ), where un~k (~r ) is periodic. Then:
[ −ℏ^2 ∇^2 2 m
ψn~k (~r ) = En~k ψn~k (~r )
yields straighforwardly:
[ (~p + ℏ~k)^2 2 m
un~k (~r ) = En~k un~k (~r ), with ~p = −iℏ∇.
This is a ~k-specific Hamiltonian: one has to set-up and diagonalize a different Hamiltonian for each ~k-point of interest in the BZ.
Compare e.g. to the hydrogen case where we had a specific radial equation for each l-quantum number, with the (L^2 / 2 mr 2 ) centrifugal term coming from the kinetic operator in spherical coordinates.
We will provide in the lecture ”DFT for solids” a more detailed account of DFT calculations with planewaves. The question: ”which is the best basis ?” (planewaves, Gaussians, real-space grid, wavelets, etc.) has probably no answer besides ”it depends on the system you study !”.
Courtesy: http://www.iue.tuwien.ac.at/phd/osintsev/disserch4.html
Atomic-like orbitals are extremely compact and allow a natural description of the variations of wavefunctions close to the atoms.
Plenewave basis are on the contrary not very good for describing strong variations of the density, but are more systematic and allow to sample the density far away from the atoms (e.g. diffuse orbitals, interstitial sites, etc.)
Besides the translations, the crystal is invariant under various operations {g } containing rotations, reflexions, inversion, etc. with respect to a point, an axis, or a plane, and combinations of such operations with translations: (g |a)~r = g ?~r + ~τ , with ~τ not necessarily a lattice vector.
ψ~k (g ?~r + ~τ ) = eik·(g^ ?~r^ +~τ^ )uk(g ?~r + ~τ ) = eig^
− (^1) k·~r ˜u(~r ), with ˜u(~r ) periodic.
The eigenstates ψ~k (g ?~r + ~a) are also the eigenstates of Hˆg − (^1) k. In particular, the eigenvalues of ˆHg − (^1) k and ˆHk are the same.
The irreducible Brillouin zone defines the set of non-equivalent ~k-points.
Irreducible Brillouin zone (in red) for an hexagonal lattice. The irreducible BZ represents here 1/24 of the full Brillouin zone. Letters specify the name of the ”high symmetry” ~k-vector and directions. All other ~k-vector in the BZ can be obtained by symmetry.
