Solids From Bonds to Bands Atom E Molecule
Solids: From Bonds to Bands Atom E Molecule Levels Bond Band 1 -D Solid 1
Real Materials more complex 1. Many orbitals per atom 2. Multiple dimensions (3 -D) Let us first recap the 1 -D bandstructure, so we can see how to generalize it in 3 -D. Being systematic helps !! 2
Summary of 1 D bandstructure STEP 1: Find period in real space R =R a= a On occasion, this may need you to choose a multiatom or multiorbital basis STEP 2: Find k-space periodicity (connecting equivalents points in k-space) K. a = 2 p K x x x -2 p/a 0 STEP 3: Find BZ by bisecting nearest neighbor connectors. This gives the smallest zone in k-space for a non-repeating band. In this case, it’s between –p/a and p/a. x 2 p/a x x 3
Summary of 1 D bandstructure STEP 4: Choose N allowed k-points by imposing periodic boundary conditions after N unit cells. For complex solids, we may need to choose specific directions. k = (n/N)K, where n=0, 1, 2, …, N-1, and K=2 p/a STEP 5: Identify nearest neighbors and find Fourier transform of H terms over this range for each allowed k. Hn, n-1 Hnn n Hn, n+1 Hk = [Hnn] + [Hn, n+1]eika + [Hn, n-1]e-ika b bands n+1 Each [Hnn] has size bxb (b: # basis sets) STEP 6: Find eigenvalues E(k). This gives b bands for each k within the BZ. x x. . x -p/a 0 p/a x x 4
Summary of 1 D bandstructure STEP 6: Use this bandstructure E-k to calculate DOS D(E), fit parabolas to extract effective mass m*, etc. These are then used for calculating electronic properties like transmission, I-V, etc. 5
General prescription in 3 -D 1 2 0 4 [H]nm 3 • • • Identify real and k-space lattice vectors Identify Brillouin zone Choose grid points along suitable directions in k-space Find H(k) by summing over nearest neighbor H terms with Fourier phases Find eigenvalues to get E-k, which we then use as needed 6
Finding real-space periodicity (Lattice vectors) 7
Lattice Vectors Simple cubic lattice a = (1, 0, 0)a b = (0, 1, 0)a c = (0, 0, 1)a Three primitive vectors are ‘coordinates’ in terms of which all lattice coordinates R can be expressed R = ma + nb + pc (m, n, p: integers) 8
Face-centered cube a = a(0, ½, ½) b = a(½, 0, ½) c = a(½, ½, 0) 6 face center atoms shared by 2 cubes each, 8 corners shared by 8 cubes each, giving a total of 8 x 1/8 + 6 x 1/2 = 4 atoms/cell 9
Body-centered cube a = a(½, ½, ½ ) b = a(-½, ½ ) c = a(½, -½ ) 8 x 1/8 corner atom + 1 center atom gives 2 atoms per cell 10
Finding k-space periodicity (Reciprocal Lattice vectors) 11
First let’s Fourier transform the lattice R = ma + nb + pc (m, n, p: integers) Coordinates of periodic lattice (ie atoms) Not to be confused with coordinate r of electron which is spread out everywhere Need to find reciprocal basis sets K 1, K 2, K 3 such that reciprocal lattice coordinates can be written as K = MK 1 + NK 2 + PK 3 (M, N, P: integers) 12
What are lattice vectors in reciprocal space? R = ma + nb + pc (m, n, p: integers) K = MK 1 + NK 2 + PK 3 (M, N, P: integers) To create analogy with 1 -D, want K 1 to be orthogonal to b and c, and have a 2 p overlap with a so that exp(i. K. R) = 1 K 1 = 2 p(b x c)/[a. (b x c)] Pts in k-spaced by integer # of K’s represent same electronic state 13
States spaced by K are identical 14
Choosing k values M, N, P unit cells along Reciprocal lattice vectors k = (m/M)K 1 + (n/N)K 2 + (p/P)K 3 m=0, 1, 2, …(M-1) n=0, 1, 2, …(N-1) p=0, 1, 2, …(P-1) 15
A simple 2 -D Example K 2 b a K 1 In 3 -D Reciprocal lattice of BCC FCC !! 16
Now to find smallest unit cell volume in k-space (Brillouin zone) First let’s do the same in real space (Wigner-Seitz cell) 17
How to pick a “Primitive” Unit cell? Two atoms in cell. Want to take a smaller ‘primitive’ cell which only contains 1 atom Many ways to do this 18
How to pick a “Primitive” Unit cell? 19
How to pick a “Primitive” Unit cell? 20
How to pick a “Primitive” Unit cell? The special ‘Wigner. Seitz Cell’ is the one that preserves the translational/rotational symmetry of the lattice 1. Join nearest neighbors 2. Bisect these lines 3. Join the bisectors 21
3 -D Wigner-Seitz Cell • Complicated, but it’s the smallest volume in real-space with 1 atom in it • Join nearest neighbors with center atom, draw bisecting planes, and identify volume enclosed by them • To get smallest range of unique k-values for E-k, just need to Fourier transform this! 22 • ie, Wigner-Seitz in Fourier space Brillouin Zone
Brillouin Zone of BCC WS of FCC 23
Choose specific directions within the BZ to plot E-k along 24
Real Materials more complex 1. Many orbitals per atom (many bands) 2. Multiple dimensions (3 -D) 3. Let us look at bands for silicon E K along X direction (100, 010, 001 etc) 25
Real Materials more complex 1. Many orbitals per atom (many bands) 2. Multiple dimensions (3 -D) 3. Let us look at bands for silicon E L G X L valley along (111) 26
Real Materials more complex 1. Many orbitals per atom (many bands) 2. Multiple dimensions (3 -D) 3. Let us look at bands for silicon E L G X Combine these into 1 figure 27
Now that we’ve identified k directions, let’s choose a grid of k points, and find the E-k 28
General prescriptions b basis sets per atom Equation for mth atom S m[H]nm{fm} = E{fm} 1 2 0 [H]nm (bx 1) Try {fm} = {f 0}eik. dm (Bloch’s Theorem) 4 3 Remember these are coeffs of f in atomic {um} basis Solution E{f 0} = [h(k)]{f 0} where h(k) = S m[H]nmeik. (dm-dn) n: any unit cell atom, m: all its nearest neighbors Eigenvalues of h(k) E(k) (bandstructure) 29
Some important materials Cu: Metal 30
Some important materials Ga. As: Direct Bandgap 31
Some important materials E DOS Si: Indirect Bandgap 32
More on silicon 33
2 FCC lattices interpenetrating, displaced by ¼ the body diag 3 3 1 2 Where are the dimers? ? Along body diagonal (111 dirn), atoms unequally placed But along x-axis (100) atoms equally placed 34
• With s orbitals alone, dimerization takes care of gap in (111) direction, not (100) • To get gap in (100), need s and p orbitals, since projection of p orbitals unequal • To get indirect bandgap, need s, p, s* orbitals 35
Approximations to bandstructure Properties important near band tops/bottoms Described through Constant Energy ellipsoids 36
Approximations to bandstructure In contrast, valence bands are more warped and are hard to write as parabolas. One uses 6 band k. p for instance. . hh lh 37
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