Shape bulk surface interface nano-cluster … etc

1. Solid state physics = {A} × {B} × {C} × {D} Always try to understand a physical phenomenon from the microscopic point of view (atoms plus electrons)! The scope of solid state physics Solid state physics studies physical properties of materials Material metalsemiconductor insulator superconductor magnetic … etc Structure crystal amorphous … etc Shape bulk surface interface nano-cluster … etc Properties electrical optical thermal mechanical … etc

2. Dept of Phys M.C. Chang Crystal structure

3. A lattice = a set of points in which every point has exactly the same environment ! • A lattice vector can be expanded as r = n1a1+n2a2+n3a3, • where a1, a2, and a3 are called primitive (translation) vectors 原始向量andn1, n2,andn3are integers For example, in 2-dim, • primitive (unit) cell (原始晶胞) nonprimitive (unit) cell one primitive unit cell contains one lattice point

4. A crystal structure = a lattice + a basis lattice basis, 基元 crystal structure

5. 2-atom basis • An example: graphite (honeycomb structure) • Is it a simple lattice (i.e. the basis consists of only 1 atom)? • Find out the primitive vectors and basis. honeycomb structure = triangular lattice + 2-atom basis

6. Crystal structures of elements

7. One possible choice of primitive vectors A conventional unit cell, 傳統晶胞 (nonprimitive) 1). bcc lattice (Li, Na, K, Rb, Cs… etc) lattice constant a • Note: A bcc lattice is a simple lattice. • But we can also treat it as a cubic lattice with a 2-point basis! (to take advantage of the cubic symmetry.)

8. One possible choice of primitive vectors • 2). fcc lattice (Ne, Ar, Kr, Xe, Al, Cu, Ag, Au… etc) lattice constant A primitive unit cell a conventional unit cell • A fcc lattice is also a simple lattice, but we can treat it as a cubic lattice with a 4 point basis.

9. 2-point basis • 3). hcp structure(= simple hexagonal lattice + a 2-point basis.) • e.g. Be, Mg… etc. • 2 overlapping “simple hexagonal lattices” • Primitive vectors: a1, a2, c [ c=2a(2/3) for hcp] • The 2 atoms of the basis are located at d1=0 and • at d2 = (2/3) a1+ (1/3) a2+(1/2)c

10. The tightest way to pack spheres: • ABCABC…= fcc, ABAB…= hcp!

11. Viewing from different angles • coordination number (配位數) = 12, packing fraction  74% • (Cf: bcc, coordination number = 8, packing fraction  68%) • Other close packed structures: ABABCAB… etc.

12. Kepler’s conjecture (1611): The packing fraction of spheres in 3-dim /18 (the value of fcc and hcp) Nature, 3 July 2003

13. 4). Diamond structure (C, Si, Ge… etc) • = fcc lattice + a 2-point basis, d1=0, d2=(a/4)(x+y+z) • = 2 overlapping fcc lattices • (one is displaced along the main diagonal by 1/4 distance) • Very low packing fraction ( 36% !) • If the two atoms on the basis are different, then it is called a Zincblend (閃鋅) structure (eg. GaAs, ZnS… etc). • It is a familiar structure with an unfamiliar name.

14. One example of more complicated crystal structure 鈦酸鈣結構

15. We can view a lattice as • a stack of planes, instead of a collection of atoms. • The Miller index (h,k,l) for crystal planes no need to be primitive vectors • rules: • 1. 取截距 (以a1, a2, a3為單位) 得 (x, y, z) • 2. 取倒數 (1/x,1/y,1/z) • 3. 通分成互質整數 (h,k,l)

16. For example, cubic crystals (including bcc, fcc… etc) [1,1,1] • Note: Square bracket [h,k,l] refers to • the “direction” ha1+ka2+la3, instead of a crystal plane! • For cubic crystals, [h,k,l] direction  (h,k,l) planes

17. Diamond structure (eg. C, Si or Ge) • Termination of 3 low-index surfaces {h,k,l} = (h,k,l)-plane + those equivalent to it by crystal symmetry <h,k,l>= [h,k,l]-direction + those equivalent to it by crystal symmetry

18. Surface reconstruction (表面重構) • Actual Si(001) surface under STM (Kariotis and Lagally, 1991)

19. Theoretical model Different surface reconstructions on different lattice planes Si(111) surface