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6.730 Physics for Solid State Applications Lecture 6: Periodic Structures Tuesday February 17, 2004

Outline • Point Lattices •Crystal Structure= Lattice + 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 6.730

PSSA

-a

0

a

2a

3a

Spring Term 2004

1

2D Bravais Lattices hexagonal rectangular

square

oblique

centered rectangular 6.730

PSSA

Spring Term 2004

3D: 14 Bravais Lattices

6.730

PSSA

Spring Term 2004

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

PSSA

Spring Term 2004

Cubic

6.730

PSSA

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

3

Body Centered Cubic

6.730

PSSA

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

FACE CENTERED CUBIC

6.730

PSSA

Spring Term 2004 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 unit cell.

6.730

PSSA

Spring Term 2004

Wigner-Sietz Cell

6.730

PSSA

Spring Term 2004

5

Wigner-Sietz Cell

FCC 6.730

PSSA

BCC Spring Term 2004

Crystal Structure = Lattice + Basis

A two dimensional lattice with a basis of three atoms 6.730

PSSA

Spring Term 2004

6

Cubic Structure with atomic basis

6.730

PSSA

Spring Term 2004

Diamond Crystal Structure

6.730

PSSA

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

7

Zincblende Crystal Structure

http://cst-www.nrl.navy.mil/lattice 6.730

Spring Term 2004

PSSA

Diamond Crystal Structure: Silicon

Bond angle = 109.5º

• 2 atom basis to a FCC Lattice • Tetrahedral bond arrangement • Each atom has 4 nearest neighbors and 6.730

PSSA

12 next nearest neighbors Spring Term 2004

8

Perovskite and Related Structures

6.730

YBa2Cu3O7-x High-Tc Structure

Spring Term 2004

PSSA

Fourier Transform Review EE-convention for Fourier Transforms

S(f)

S(t) (1)

(1)

(1)

(1)

(1)

f

t -2T

6.730

PSSA

-T

0

T

2T

2 T

1 T

0

1 T

2 T

Spring Term 2004

9

Physics Convention for Fourier Transforms

In general,

6.730

Spring Term 2004

PSSA

1D Periodic Crystal Structures

Sa(q)

Sa(x) (1)

(1)

(1)

(1)

(1)

x -2a

6.730

PSSA

-a

0

a

2a

q 4π a

2π a

0

2π a

4π a

Spring Term 2004

10

1D Periodic Crystal Structures M(x) (1) (1)

-2a

-a

0

a

2a

x

=

-2a

0

0

a

(1)

2a

x

-a

0

4π a

2π a

6.730

2π a

a

2a

x

Ma(q)

Sa(q)

= 2π a

(1) (1)

-a

M(q)

4π a

Ma(x)

Sa(x)

0 2π a

4π a

q 4π a

2π a

0 2π a

Spring Term 2004

PSSA

Reciprocal Lattice Vectors 1. The Fourier transform in q-space is also a lattice 2. This lattice is called the reciprocal lattice 3. The lattice constant is 2 π / a 4. The Reciprocal Lattice Vectors are

q K-2 6.730

PSSA

K-1

K0

K1

K2

Spring Term 2004

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

6.730

Spring Term 2004

PSSA

1D Periodic Crystal Structures with a basis M(x) (1) (1)

-2a

-a

0

a

2a

x

=

-2a

0

0

a

(1)

2a

x

d1

-a

d2 0

2π a

4π a

2π a

0 2π a

a

2a

x

Ma(q)

Sa(q)

= 2π a

(1) (1)

-a

M(q)

4π a

Ma(x)

Sa(x)

4π a

q 4π a

2π a

0 2π a

Delta functions still only occur where the reciprocal lattice is, independent of the basis. But the basis determines the weights.

6.730

PSSA

Spring Term 2004

12

Atomic Form Factors & Geometrical Structure Factors Ma(x) d1

-a

Different Atoms (1) and (2)

d2 0

a

2a

x

Ma(x) d1

-a

6.730

PSSA

Same atoms at d1 and d2

d2

0

a

2a

x

Spring Term 2004

13

View more...
Outline • Point Lattices •Crystal Structure= Lattice + 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 6.730

PSSA

-a

0

a

2a

3a

Spring Term 2004

1

2D Bravais Lattices hexagonal rectangular

square

oblique

centered rectangular 6.730

PSSA

Spring Term 2004

3D: 14 Bravais Lattices

6.730

PSSA

Spring Term 2004

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

PSSA

Spring Term 2004

Cubic

6.730

PSSA

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

3

Body Centered Cubic

6.730

PSSA

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

FACE CENTERED CUBIC

6.730

PSSA

Spring Term 2004 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 unit cell.

6.730

PSSA

Spring Term 2004

Wigner-Sietz Cell

6.730

PSSA

Spring Term 2004

5

Wigner-Sietz Cell

FCC 6.730

PSSA

BCC Spring Term 2004

Crystal Structure = Lattice + Basis

A two dimensional lattice with a basis of three atoms 6.730

PSSA

Spring Term 2004

6

Cubic Structure with atomic basis

6.730

PSSA

Spring Term 2004

Diamond Crystal Structure

6.730

PSSA

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

7

Zincblende Crystal Structure

http://cst-www.nrl.navy.mil/lattice 6.730

Spring Term 2004

PSSA

Diamond Crystal Structure: Silicon

Bond angle = 109.5º

• 2 atom basis to a FCC Lattice • Tetrahedral bond arrangement • Each atom has 4 nearest neighbors and 6.730

PSSA

12 next nearest neighbors Spring Term 2004

8

Perovskite and Related Structures

6.730

YBa2Cu3O7-x High-Tc Structure

Spring Term 2004

PSSA

Fourier Transform Review EE-convention for Fourier Transforms

S(f)

S(t) (1)

(1)

(1)

(1)

(1)

f

t -2T

6.730

PSSA

-T

0

T

2T

2 T

1 T

0

1 T

2 T

Spring Term 2004

9

Physics Convention for Fourier Transforms

In general,

6.730

Spring Term 2004

PSSA

1D Periodic Crystal Structures

Sa(q)

Sa(x) (1)

(1)

(1)

(1)

(1)

x -2a

6.730

PSSA

-a

0

a

2a

q 4π a

2π a

0

2π a

4π a

Spring Term 2004

10

1D Periodic Crystal Structures M(x) (1) (1)

-2a

-a

0

a

2a

x

=

-2a

0

0

a

(1)

2a

x

-a

0

4π a

2π a

6.730

2π a

a

2a

x

Ma(q)

Sa(q)

= 2π a

(1) (1)

-a

M(q)

4π a

Ma(x)

Sa(x)

0 2π a

4π a

q 4π a

2π a

0 2π a

Spring Term 2004

PSSA

Reciprocal Lattice Vectors 1. The Fourier transform in q-space is also a lattice 2. This lattice is called the reciprocal lattice 3. The lattice constant is 2 π / a 4. The Reciprocal Lattice Vectors are

q K-2 6.730

PSSA

K-1

K0

K1

K2

Spring Term 2004

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

6.730

Spring Term 2004

PSSA

1D Periodic Crystal Structures with a basis M(x) (1) (1)

-2a

-a

0

a

2a

x

=

-2a

0

0

a

(1)

2a

x

d1

-a

d2 0

2π a

4π a

2π a

0 2π a

a

2a

x

Ma(q)

Sa(q)

= 2π a

(1) (1)

-a

M(q)

4π a

Ma(x)

Sa(x)

4π a

q 4π a

2π a

0 2π a

Delta functions still only occur where the reciprocal lattice is, independent of the basis. But the basis determines the weights.

6.730

PSSA

Spring Term 2004

12

Atomic Form Factors & Geometrical Structure Factors Ma(x) d1

-a

Different Atoms (1) and (2)

d2 0

a

2a

x

Ma(x) d1

-a

6.730

PSSA

Same atoms at d1 and d2

d2

0

a

2a

x

Spring Term 2004

13

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