The scope of solid state physics Solid state
- Slides: 19
The scope of solid state physics Solid state physics studies physical properties of materials Material Structure Shape Properties metal semiconductor insulator superconductor magnetic … etc crystal amorphous … etc bulk surface interface nano-cluster … etc electrical optical thermal mechanical … etc Solid state physics = {A} × {B} × {C} × {D} Always try to understand a physical phenomenon from the microscopic point of view (atoms plus electrons)!
Dept of Phys M. C. Chang Crystal structure
A lattice = a set of points in which every point has exactly the same environment ! A lattice vector can be expanded as r = n 1 a 1+n 2 a 2+n 3 a 3, where a 1, a 2, and a 3 are called primitive (translation) vectors 原始向量 and n 1, n 2, and n 3 are integers For example, in 2 -dim, primitive (unit) cell (原始晶胞) nonprimitive (unit) cell one primitive unit cell contains one lattice point
A crystal structure = a lattice + a basis lattice basis, 基元 crystal structure
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. 2 -atom basis honeycomb structure = triangular lattice + 2 -atom basis
Crystal structures of elements
1). bcc lattice (Li, Na, K, Rb, Cs… etc) One possible choice of primitive vectors lattice constant a A conventional unit cell, 傳統晶胞 (nonprimitive) 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. )
2). fcc lattice (Ne, Ar, Kr, Xe, Al, Cu, Ag, Au… etc) One possible choice of primitive vectors 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.
3). hcp structure (= simple hexagonal lattice + a 2 -point basis. ) e. g. Be, Mg… etc. 2 overlapping “simple hexagonal lattices” 2 -point basis Ø Primitive vectors: a 1, a 2, c [ c=2 a (2/3) for hcp] Ø The 2 atoms of the basis are located at d 1=0 and at d 2 = (2/3) a 1+ (1/3) a 2+(1/2)c
The tightest way to pack spheres: ABCABC…= fcc, ABAB…= hcp!
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.
Kepler’s conjecture (1611): The packing fraction of spheres in 3 -dim / 18 (the value of fcc and hcp) Nature, 3 July 2003
4). Diamond structure (C, Si, Ge… etc) = fcc lattice + a 2 -point basis, d 1=0, d 2=(a/4)(x+y+z) = 2 overlapping fcc lattices (one is displaced along the main diagonal by 1/4 distance) Ø Very low packing fraction ( 36% !) Ø If the two atoms on the basis are different, then it is called a Zincblend (閃鋅) structure (eg. Ga. As, Zn. S… etc). It is a familiar structure with an unfamiliar name.
One example of more complicated crystal structure 鈦酸鈣結構
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a 1, a 2, a 3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)
For example, cubic crystals (including bcc, fcc… etc) [1, 1, 1] Note: Square bracket [h, k, l] refers to the “direction” ha 1+ka 2+la 3, instead of a crystal plane! For cubic crystals, [h, k, l] direction (h, k, l) planes
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
Actual Si(001) surface under STM (Kariotis and Lagally, 1991) Surface reconstruction (表面重構)
Different surface reconstructions on different lattice planes Si(111) surface Theoretical model
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