II crystal structure

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Lucas Harmon
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1 II crstal structure 2-1 basic concept > Crstal structure = lattice structure + basis > Lattice point: positions (points) in the structure which are identical. > Lattice translation vector > Lattice plane > Unit cell > Primitive unit cell 1 lattice point/unit cell Several crstal structures: CsCl crstal structure = simple cubic (s.c.) lattice structure + basis s.c. lattice structure is primitive 2 ions at each lattice point basis= Cs + + Cl -

2 Fe (ferrite), Cr, Mo, W bod-centered cubic (bcc) bcc crstal structure =bcc lattice structure + basis basis= 1 atom/lattice point 2 lattice points /bcc b.c.c is not primitive Al, Au, Ag, Pt face-centered cubic (fcc) fcc crstal structure =fcc lattice structure + basis basis= 1 atom/lattice point 4 lattice points /fcc f.c.c is not primitive GaAs, AlP, InP, ZnS, CdTe, HgTe Zinc blende crstal structure = Ga (fcc) + As (fcc) (for GaAs) fcc lattice structure fcc lattice structure + basis basis = Ga 2+ + As 2- (for GaAs)

3 Si, Ge, diamond diamond crstal structure = fcc lattice structure + basis basis = 2 atoms/lattice point NaCl = Na (fcc) + Cl (fcc) NaCl crstal structure = fcc lattice structure + basis basis = Na + + Cl -

hcp (for CdS) wurtzite structure = simple

4 Mg, Zn, heagonal close packed (hcp) hcp crstal structure = simple heagonal lattice + basis basis = 2 atoms/lattice point CdS, ZnO, ZnS Wurtzite structure => Cd 2+ hcp + S 2- hcp (for CdS) wurtzite structure = simple heagonal lattice + basis basis = 2 Cd S 2- (for CdS)

BaTiO 3, CaTiO 3 Perovskite crstal structure perovskite crstal

5 CaF 2 Fluorite crstal structure fluorite crstal structure = fcc lattice structure + basis basis = Ca F - (at tetrahedral sites) BaTiO 3, CaTiO 3 Perovskite crstal structure perovskite crstal structure = simple cubic lattice structure + basis basis =1 Ba Ti O 2-

6 2-2 Miller Indices in a crstal direction The direction [u v w] is epressed as a vector The direction <u v w>are all the [u v w] tpes of direction, which are crstallographic equivalent. z plane The plane (h k l) is is the Miller inde of the plane. z c c / l (hkl) b a a / h b / k a / h b / k z c / l 1 {h k l} are the (h k l) tpes of planes which are crstllographic equivalent.

7 2-2-3 meaning of miller indices (100) (200) d d >Low inde planes are widel spaced. >Low inde directions correspond to short lattice translation vectors (110) (120) >Low inde directions and planes are important for slip, and cross slip electron mobilit.

8 2-3 Miller-Bravais indices in cubic sstem (1) Direction [h k l] is pepredicular to ( h k l ) plane in the cubic sstem, but not true for other crstal sstems. (110) (110) In heagonal sstem using Miller - Bravais indeing sstem:( hkil ) and [hkil] Reason (i) Tpe [110] does not equal to [010], but these directions are crstallographic equivalent. Reason (ii) z ais is [001], crstallographicall distinct from [100] and [010].

9 2-3-3 Miller-Bravais indices for the heagonal sstem (a) direction The direction [h k i l] is epressed as a vector u a Note : 2110 plane. 3 is the shortest translation vector on the basal (b) planes (h k i l) ; h + k + i = 0 Plane (hkl) (hkil) u (100) (100) (1010) ( hkil)

10 u (010) (0110) ( hkil) u (2110) Proof: u (hkl); (hkil) For plane (hkl), the intersection with the basal plane (001) is a line that is epressed as

11 Where we set the lattice constant a =b=1 in the heagonal lattice for simplicit. Therefore the line equation becomes h+k =1 The line along the ais can be epressed as = The intersection point of the two lines occurs at the point [1/(h+k), 1/(h+k)] The vector from origin to the point can be epressed along the ais as In other words, according to the definition (c) Transformation from Miller [z] to Miller-Bravais inde [hkil] 2 h 3 u 2 k rule : 3 ( ) i i 3 z l m Proof: h i The same vector is epressed as [z] in miller indices and as [hkil] in Miller-Bravais indices [,, z] [h k i l]

12 Therefore, = h - i = k - i z = l Moreover, h + k = - i We can obtain = h i = h +h + k = 2h +k = k i = k + h + k = h +2k 2 h 3 2 k 3 ( ) i 3 z l

13 2-4 Stereographic projections direction hkl N uvw horizonetal plane uvw S hkl uvw uvw

14 2-4-2 plane Great circle: the plane passing through the center of the sphere. N S Small circle: the plane not passing through the center of the sphere. N S

15 2-4-3 Stereographic projection of different Bravais sstems Cubic

16 Trigonal and Heagonal

17 Orthorhombic and Monoclinic

18 2-5 Two convections used in stereographic projection (1) plot directions as poles and planes as great circles (2) plot planes as poles and directions as great circles find angle between two directions (a) find a great circle going through them (b) measure angle b Wulff net (i) If two poles up angle (ii) If one pole up, one pole down angle

19 2-5-2 measuring the angle between planes This is equivalent to measuring angle between poles > use of stereographic projections (i) plot directions as poles ---- used to measure angle between directions ---- use to establish if direction lie in a particular plane (ii) plot planes as poles ---- used to measure angles between planes ---- used to find if planes lies in the same zone

Bravais Lattice + Basis = Crystal Structure

Bravais Lattice + Basis = Crystal Structure A crystal structure is obtained when identical copies of a basis are located at all of the points of a Bravais lattice. Consider the structure of Cr, a I-cubic