The scope of solid state physics Solid state physics studies

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 nanocluster … 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 (simple) 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 vectors (原始向量) and n1, n2, and n3 are integers.

• a lattice + a basis → crystal structure

lattice

basis (基元) • conversely, a crystal structure can be decomposed as a lattice + a basis

crystal structure

It’s important to distinguish 2 different types of lattice : a simple lattice v.s. a lattice with a basis • Triangular (or hexagonal) lattice

• Honeycomb lattice

Primitive vectors a2

basis a1

A simple lattice

A triangular lattice + 2-point basis (i.e. superposition of 2 triangular lattices)

• Unit cell

primitive (unit) cell (原始晶胞) nonprimitive (unit) cell

• a primitive cell contains a lattice point • a non-primitive cell contains 2 or more lattice points (sometimes it’s more convenient to use this one)

Crystal structures of elements

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

lattice constant

a

conventional unit cell (慣用晶胞, nonprimitive)

G a a1 = ( xˆ + yˆ − zˆ ) , 2 G a a2 = ( − xˆ + yˆ + zˆ ) , 2 G a a3 = ( xˆ − yˆ + zˆ ) . 2

• A bcc lattice is a simple lattice. • For convenience of description, we can treat it as a cubic lattice with a 2-point basis.

2). fcc lattice (Ne, Ar, Kr, Xe, Al, Cu, Ag, Au… etc)

One possible choice of primitive vectors

a

A primitive unit cell

G a a1 = ( xˆ + yˆ ) , 2 G a a2 = ( yˆ + zˆ ) , 2 G a a3 = ( zˆ + xˆ ) . 2 a conventional unit cell

• A fcc lattice is also a simple lattice. • It can be seen as a cubic lattice with a 4-point basis.

3). hcp structure (Be, Mg… etc.) 2 overlapping “simple hexagonal lattices”

2-point basis

• hcp structure = simple hexagonal lattice + a 2-point basis • Primitive vectors: a1, a2, c [ c=2a√ (2/3) for hcp] • The 2 atoms of the basis are located at d1=0, d2 = (2/3) a1+ (1/3) a2+(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 ≦ π/√18 (the value of fcc and hcp)

Nature, 3 July 2003

4). Diamond structure (C, Si, Ge… etc) = 2 overlapping fcc lattices (one is displaced along the main diagonal by 1/4)

= fcc lattice + a 2-point basis, d1=0, d2=(a/4)(x+y+z)

• 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), which is a familiar structure with an unfamiliar name.

One more example of crystal structure 鈦酸鈣結構

Index system for crystal planes

An Indexed PbSO4 Crystal

The Miller index (h,k,l) rules:

no need to be primitive vectors

z=1

1. 取截距 (以a1, a2, a3為單位) 得 (x, y, z)

a3 a1

2. 取倒數 (1/x,1/y,1/z) 3. 通分成互質整數 (h,k,l)

a2

x=2 (h,k,l)=(3,2,6)

y=3

For example, cubic crystals (including bcc, fcc… etc)

[1,1,1]

• 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

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]-direction + those equivalent to it by crystal symmetry