The scope of solid state physics Solid state

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The scope of solid state physics Solid state physics studies physical properties of materials

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

Dept of Phys M. C. Chang Crystal structure

A lattice = a set of points in which every point has exactly the

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

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

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

Crystal structures of elements

1). bcc lattice (Li, Na, K, Rb, Cs… etc) One possible choice of primitive

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

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.

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!

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

Viewing from different angles • coordination number (配位數) = 12, packing fraction 74% (Cf:

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

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

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

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

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

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,

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)

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

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