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Home , Bravais lattice

6.730 Physics for Solid State Applications

Lecture 6: Periodic Structures

Tuesday February 17, 2004 Outline

• Point Lattices • Structure= + Basis •Fourier Transform Review •1D Periodic Crystal Structures: Mathematics

6.730 Spring Term 2004 PSSA

Point Lattices: Bravais Lattices

Bravais lattices are point lattices that are classified topologically according to the symmetry properties under rotation and reflection, without regard to the absolute length of the unit vectors.

A more intuitive definition: At every point in a Bravais lattice the “world” looks the same.

1D: Only one Bravais Lattice

-2a -a0 a3a 2a

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1 2D Bravais Lattices

hexagonal

rectangular

square

oblique

centered rectangular

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3D: 14 Bravais Lattices

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2 Lattice and Primitive Lattice Vectors

A Lattice is a regular array of points {Rl } in space which must satisfy (in three dimensions)

The vectors ai are know as the primitive lattice vectors.

A two dimensional lattice with different possible choices of primitive lattice vectors. 6.730 Spring Term 2004 PSSA

Cubic

6.730 Spring Term 2004 PSSA http://cst-www.nrl.navy.mil/lattice/struk.picts/a_h.s.png

3 Body Centered Cubic

6.730 Spring Term 2004 PSSA http://cst-www.nrl.navy.mil/lattice/struk.picts/a2.s.png

FACE CENTERED CUBIC

6.730 Spring Term 2004 PSSA http://cst-www.nrl.navy.mil/lattice/struk.picts/a1.s.png

4 Wigner-Sietz Cell

1. Choose one point as the origin and draw lines from the origin to each of the other lattice points.

2. Bisect each of the drawn lines with planes normal to the line.

3. Mark the space about the origin bounded by the first set of planes that are encountered. The bounded space is the Wigner-Sietz .

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Wigner-Sietz Cell

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5 Wigner-Sietz Cell

FCC BCC

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Crystal Structure = Lattice + Basis

A two dimensional lattice with a basis of three

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6 Cubic Structure with atomic basis

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Diamond

http://cst-www.nrl.navy.mil/lattice 6.730 Spring Term 2004 PSSA

7 Zincblende Crystal Structure

http://cst-www.nrl.navy.mil/lattice

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Diamond Crystal Structure: Silicon

Bond angle = 109.5º

• 2 basis to a FCC Lattice • Tetrahedral bond arrangement • Each atom has 4 nearest neighbors and 12 next nearest neighbors 6.730 Spring Term 2004 PSSA

8 Perovskite and Related Structures

YBa2Cu3O7-x High-Tc Structure 6.730 Spring Term 2004 PSSA

Fourier Transform Review EE-convention for Fourier Transforms

S(t) S(f) (1) (1) (1) (1) (1)

t f 2 -2T-T 0 T2T 1 0 1 2 T T T T

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9 Physics Convention for Fourier Transforms

In general,

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1D Periodic Crystal Structures

Sa(q) Sa(x) (1) (1) (1) (1) (1)

x q 4π 2π -2a-a 0 a2a 2π 0 4π a a a a

6.730 Spring Term 2004 PSSA

10 1D Periodic Crystal Structures M(x) Sa(x) Ma(x) (1) (1) (1) (1) (1)

x x x -2a-a 0 a2a = -2a-a 0 a2a -a 0 a2a

Ma(q) M(q) Sa(q)

= q 4π 2π 2π 4π 4π 2π 2π 0 4π 2π 2π 0 a a a a 0 a a a a a a

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Reciprocal Lattice Vectors

1. The Fourier transform in q-space is also a lattice

2. This lattice is called the

3. The is 2 π / a

4. The Reciprocal Lattice Vectors are

q

K-2 K-1 K0 K1 K2

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11 Periodic Function as a Fourier Series

Recall that a periodic function and its transform are

Define then the above is a Fourier Series:

and the equivalent Fourier transform is

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1D Periodic Crystal Structures with a basis M(x) Sa(x) Ma(x) d (1) (1) (1) (1) (1) 1 d2

x x x -2a-a 0 a2a = -2a-a 0 a2a -a 0 a2a

Ma(q) M(q) Sa(q)

= q 4π 2π 2π 4π 4π 2π 2π 0 4π 2π 2π 0 a a a a 0 a a a a a a

Delta functions still only occur where the reciprocal lattice is, independent of the basis. But the basis determines the weights. 6.730 Spring Term 2004 PSSA

12 Atomic Form Factors & Geometrical Structure Factors

Ma(x) d 1 Different Atoms (1) and (2) d2

x -a 0 a2a

Ma(x) d Same atoms at d and d 1 d2 1 2

x -a 0 a2a

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