lecture -- solid state electromagnetics...periodic lattices. we have tomake generalizations to do...
TRANSCRIPT
9/2/2019
1
Advanced Electromagnetics:
21st Century Electromagnetics
Solid State Electromagnetics
Lecture Outline
•Periodic devices•Math describing periodic structures• Electromagnetic waves in periodic structures• Electromagnetic bands• Isofrequency contours•Appendix• Hexagonal lattices• Brillouin zones
Slide 2
1
2
9/2/2019
2
Periodic Devices
Slide 3
Examples of Periodic Electromagnetic Devices
Slide 4
Diffraction Gratings
Antennas
Waveguides Band Gap Materials
Frequency Selective Surfaces
Metamaterials
Slow Wave Devices
3
4
9/2/2019
3
What is a Periodic Structure?
Slide 5
Periodicity at the Atomic Scale Larger‐Scale Periodicity
Materials are periodic at the atomic scale. Metamaterials are periodic at a much larger scale, but smaller than a wavelength.The math describing how things are periodic is the same for both atomic scale and larger scale.
Describing Periodic Structures
• There is an infinite number of ways that structures can be periodic.• Despite this, we need a way to describe and classify periodic lattices. We have to make generalizations to do this.•We classify periodic structures into:
• 230 space groups• 32 crystal classes• 14 Bravais lattices• 7 crystal systems
Slide 6
Less specific.More generalizations.
5
6
9/2/2019
4
Symmetry Operations (Animated)
Slide 7
Infinite crystals are invariant under certain symmetry operations that involve:
PureTranslation
PureRotation
PureReflection
Combinations
Symmetry Operations (Static)
Slide 8
Infinite crystals are invariant under certain symmetry operations that involve:
Pure Translation Pure Rotation
Pure Reflections Combinations
7
8
9/2/2019
5
Definition of Symmetry Categories
• Space Groups• Set of all possible combinations of symmetry operations that restore the crystal to itself.
• 230 space groups• Bravais Lattices
• Set of all possible ways a lattice can be periodic if composed of identical spheres placed at the lattice points.
• 14 Bravais lattices• Crystal Systems
• Set of all Bravais lattices that have the same holohedry (shape of the conventional unit cell)
• 7 crystal systems
Slide 9
The 14 Bravais Lattices and the Seven Crystal Systems
Slide 10
9
10
9/2/2019
6
Two‐Dimensional Bravais Lattices
Slide 11Axis vectors = translation vectors for 2D lattices
1t
2t
1t
2t
1t
2t
1t
2t
1t
2t
1 2 120t t
1 2 90t t
1 2 90t t
Rhombic (Centered Rectangular)Rectangular Oblique
SquareHexagonal
1 22cos 90t t
1 2 90t t
Hybrid Symmetries
Slide 12
FCC Symmetry Diamond = FCC + FCC Zinc Blend
* GaAs
11
12
9/2/2019
7
Hexagonal Symmetry Has Optimal Packing Density in 2D
Slide 13
2
sq
rf
a
2
hex
2
3
rf
a
hex
sq
21.1547
3
f
f
Electromagnetic behavior from hexagonal arrays tends to be at lower frequencies than compared to square arrays. This means feature sizes can be larger than they would be for square arrays. This is important at high frequencies and photonics where small features may be more difficult to realize.
Math Describing Periodic
Structures
Slide 14
13
14
9/2/2019
8
Primitive Lattice Vectors
Slide 15
Axis vectors most intuitively define the shape and orientation of the unit cell. They cannot uniquely describe all 14 Bravaislattices, but they do uniquely identify the 7 crystal systems.
Primitive lattice vectors are the smallest possible vectors that still describe the unit cell.
Translation vectors connect adjacent points in the lattice and can uniquely describe all 14 Bravais lattices. They are less intuitive to interpret.
Axis Translation Vectors
Slide 16
1 1
2 2
3 3
1 0 0
0 1 0
0 0 1
t a
t a
t a
1 1
2 2
3 3
1 2 1 2 1 2
1 2 1 2 1 2
1 2 1 2 1 2
t a
t a
t a
1 1
2 2
3 3
0 1 2 1 2
1 2 0 1 2
1 2 1 2 0
t a
t a
t a
1 1
2 2
3 3
1 2 1 2 0
1 2 1 2 0
0 0 1 2
t a
t a
t a
1 1
2 2
3 3
1 1 3 1 3
1 3 1 3 1 3
1 3 1 1 3
t a
t a
t a
Simple
Body‐Centered
Face‐Centered
Base‐Centered
Trigonal
15
16
9/2/2019
9
Non‐Primitive Lattice Vectors
Slide 17
Almost always, the label “lattice vector” refers to the translation vectors, not the axis vectors.
A translation vector is any vector that connects two points in a lattice. They must be an integer combination of the primitive translation vectors.
1 2 3pqrt pt qt rt
, 2, 1,0,1, 2,
, 2, 1,0,1, 2,
, 2, 1,0,1, 2,
p
q
r
1t
2tPrimitive
translation vectors
The Reciprocal Lattice (1 of 10)
Slide 18
Vectors , , and define the direct lattice.
1t
2t
3t
17
18
9/2/2019
10
The Reciprocal Lattice (2 of 10)
Slide 19
and define a plane.
This plane repeats itself through the lattice with a spacing of 3.
is defined to be perpendicular to these planes with magnitude
1t
2t
3T
33
2T
The Reciprocal Lattice (3 of 10)
Slide 20
and define a plane.
This plane repeats itself through the lattice with a spacing of 2.
is defined to be perpendicular to these planes with magnitude
1t
3t
2T
22
2T
19
20
9/2/2019
11
The Reciprocal Lattice (4 of 10)
Slide 21
and define a plane.
This plane repeats itself through the lattice with a spacing of 1.
is defined to be perpendicular to these planes with magnitude
2t
3t
1T
11
2T
The Reciprocal Lattice (5 of 10)
Slide 22
Vectors , , and define the reciprocal lattice.
1T
2T
3T
21
22
9/2/2019
12
The Reciprocal Lattice (6 of 10)
Slide 23
and define a plane.
This plane repeats itself through the lattice with a spacing of a3.
is defined to be perpendicular to these planes with magnitude
1T
2T
3t
33
2t
a
The Reciprocal Lattice (7 of 10)
Slide 24
and define a plane.
This plane repeats itself through the lattice with a spacing of a2.
is defined to be perpendicular to these planes with magnitude
1T
3T
2t
22
2t
a
23
24
9/2/2019
13
The Reciprocal Lattice (8 of 10)
Slide 25
and define a plane.
This plane repeats itself through the lattice with a spacing of a1.
is defined to be perpendicular to these planes with magnitude
2T
3T
1t
11
2t
a
The Reciprocal Lattice (9 of 10)
Slide 26
Vectors , , and define the direct lattice.
1t
2t
3t
25
26
9/2/2019
14
The Reciprocal Lattice (10 of 10)
Slide 27
Direct Lattice Reciprocal Lattice
Each direct lattice has a unique reciprocal lattice so knowledge of one implies knowledge of the other.
Reciprocal Lattices (2 of 2)
Slide 28
Direct Lattice Reciprocal Lattice
Simple Cubic Simple Cubic
Body‐Centered Cubic Face‐Centered Cubic
Face‐Centered Cubic Body‐Centered Cubic
Hexagonal Hexagonal
27
28
9/2/2019
15
Calculating the Reciprocal Lattice Vectors (1 of 4)
Slide 29
The area of this parallelogram is calculated from the magnitude of the cross product…
2 3A t t
Calculating the Reciprocal Lattice Vectors (2 of 4)
Slide 30
1 2 3V t t t
The volume of this parallelpiped is calculated using the scalar triple product.
29
30
9/2/2019
16
Calculating the Reciprocal Lattice Vectors (3 of 4)
Slide 31
V
A
The distance between the planes is
1 2 3
2 3
t t t
t t
Calculating the Reciprocal Lattice Vectors (4 of 4)
Slide 32
1
2T
The reciprocal lattice vector has a magnitude of
2 31
2 31
t tT
t tT
and direction of
2 31 1
2 3
t tT T
t t
2 3
2 3
2 t t
t t
2 3
1 2 3 2 3
2 3
2 t t
t t t t t
t t
2 3
1 2 3
2t t
t t t
31
32
9/2/2019
17
Reciprocal Lattice Vectorsfor 3D Lattices
Slide 33
The reciprocal lattice vectors can be calculated from the direct lattice vectors (and the other way around) as follows:
2 3 3 1 1 21 2 3
1 2 3 1 2 3 1 2 3
2 3 3 1 1 21 2 3
1 2 3 1 2 3 1 2 3
2 2 2
2 2 2
t t t t t tT T T
t t t t t t t t t
T T T T T Tt t t
T T T T T T T T T
There also exists primitive reciprocal lattice vectors. All reciprocal lattice vectors must be an integer combination of the primitive reciprocal lattice vectors.
1 2 3PQRT PT QT RT , 2, 1,0,1, 2,
, 2, 1,0,1,2,
, 2, 1,0,1, 2,
P
Q
R
Reciprocal Lattice Vectorsfor 2D Lattices
Slide 34
The reciprocal lattice vectors can be calculated from the direct lattice vectors (and the other way around) as follows:
2,
2,1 21
2 y
x
t
tt tT
1,
1,2
1 2
2 y
x
t
tT
tt
2,
2,1 2
1
2 y
x
T
TT Tt
1,
1,2
1 2
2 y
x
T
Tt
TT
33
34
9/2/2019
18
Grating Vector 𝐾
Slide 35
A grating vector 𝐾 is very much like a wave vector 𝑘 in that its direction is normal to planes and its magnitude is 2 divided by the spacing between the planes.
avg cosr r K r
ˆ ˆ ˆr xx yy zz
position vector
ˆ ˆ ˆx y zK K x K y K z
Reciprocal Lattice Vectors are Grating Vectors
Slide 36
Reciprocal lattice vectors are grating vectors!!
2a a
a
T K
There is a close and elegant relationship between the reciprocal lattice vectors and wave vectors. For this reason, periodic structures are often analyzed in reciprocal (grating vector) space.
35
36
9/2/2019
19
Miller Indices
Slide 37
Miller indices identify repeating planes within periodic structures like crystals.
Recall the definition of a reciprocal lattice vector…
1 2 3PQRT PT QT RT
P, Q, and R are the Miller indices of the planes
described by the reciprocal lattice vector 𝑇 .
PQR
Primitive Unit Cells
Slide 38
Primitive unit cells are the smallest volume of space that can be stacked onto itself (with no voids and no overlaps) to correctly reproduce the entire lattice.
The Wigner‐Seitz cell is one method of constructing a primitive unit cell. It is defined as the volume of space around a single point in the lattice that is closer to that point than any other point in the lattice. A proper unit cell contains only one “atom.”
These illustrations show the relationship between the conventional unit cell (shown as frames) and the Wigner‐Seitz cell (shown as volumes) for the three cubic lattices.
37
38
9/2/2019
20
Constructing the Wigner‐SeitzPrimitive Unit Cell (1 of 2)
Slide 39
BCC Conventional Unit Cell Wigner‐Seitz Unit Cell
How to we construct this?
Constructing the Wigner‐SeitzPrimitive Unit Cell (2 of 2)
Slide 40
Step 1 – Pick a point in the lattice to build the unit cell around.Step 2 – Construct planes that bisect the region between all adjacent points.Step 3 – The unit cell is the region enclosed by all of the planes.
39
40
9/2/2019
21
Brillouin Zones
Slide 41
The Brillouin zone is constructed in the same manner as the Wigner‐Seitz unit cell, but it is constructed from the reciprocal lattice.
The Brillouin zone is closely related to wave vectors and diffraction so analysis of periodic structures is often performed in “reciprocal space.”
The Brillouin zone for a face‐centered‐cubic lattice is a “truncated” octahedron with 14 sides.
This is the most “spherical” of all the Brillouinzones so the FCC lattice is said to have the highest symmetry of the Bravais lattices.
Of the FCC lattices, diamond has the highest symmetry.
Different Brillouin Zones
Slide 42
1st
2nd
3rd4th
http://www.eelvex.net/physics/finding‐brillouin‐zones‐a‐visual‐guide/
41
42
9/2/2019
22
Degree of Symmetry
Slide 43
The degree of symmetry refers to how spherical the Brillouin zone is.
Triclinic Pseudo‐periodic
HighestSymmetry
LowestSymmetry
SC BCC FCC Diamond(FCC)
Exploiting Additional Symmetry
Slide 44
If the field is known at every point inside a single unit cell, then it is also known at any point in an infinite lattice because the field takes on the same symmetry as the lattice so it just repeats itself.
Since the reciprocal lattice uniquely defines a direct lattice, knowing the solutions to the wave equation at each point inside the reciprocal lattice unit cell also defines the solution everywhere in the infinite reciprocal lattice.
Many times, there is still additional symmetry to exploit. So, the smallest volume of space that completely describes the electromagnetic wave can be smaller than the unit cell itself.
The field in each of these squares is a mirror image of each other.
Due to the symmetry in this example, the field at any point in the entire lattice can be mapped to an equivalent point in this triangle.
43
44
9/2/2019
23
The Irreducible Brillouin Zone
Slide 45
The smallest volume of space within the Brillouin zone that completely characterizes the periodic structure is called the irreducible Brillouin zone (IBZ). It is smaller than the Brillouin zone when there is additional symmetry to exploit.
Electromagnetic Waves in Periodic
StructuresSlide 46
45
46
9/2/2019
24
Waves are Perturbed by Objects
Slide 47
k
When a portion of a wave propagates through an object, that portion is delayed.
Thus, a wave gets perturbed by the object.
Fields in Periodic Structures
Slide 48
Waves in periodic structures take on the same symmetry and periodicity as their host.
k
47
48
9/2/2019
25
The Bloch Theorem
Slide 49
Waves inside of a periodic structure are like plane waves, but they are modulated by an envelope function. It is the envelope function that takes on the same symmetry and periodicity as the structure.
j rA r eE r
Overall field is the combination of the envelope and plane wave term.
Envelope function has the same symmetry and periodicity as the periodic structure.
Plane‐wave like phase “tilt” term.
Bloch wave vector
Example Waves in a Periodic Lattice
Slide 50
Wave normally incident onto a periodic structure.
Wave incident at 45° onto the same periodic structure.
49
50
9/2/2019
26
Mathematical Description of Periodicity
Slide 51
A structure is periodic if its material properties repeat. Given the lattice vectors, the periodicity is expressed as
1 2 3 pqr pqrr t r t pt qt rt
Recall that it is the amplitude of the Bloch wave that has the same periodicity as the structure the wave is in. Therefore,
1 2 3 pqr pqrA r t A r t pt qt rt
Example – 1D Periodicity
Slide 52
For a device that is periodic only along one direction, these relations reduce to
, , , ,
, , , ,
x
x
x p y z x y z
A x p y z A x y z
x
x
x p x
A x p A x
more compact
notation
Many devices are periodic along just one dimension.
x
yz
x
, , 2, 1,0,1,2, ,p
51
52
9/2/2019
27
Electromagnetic Bands
Slide 53
Band Diagrams (1 of 2)
Slide 54
Band diagrams are a compact, but incomplete, means of characterizing the electromagnetic properties of a periodic structure. It is essentially a map of the frequencies of the eigen‐
modes as a function of the Bloch wave vector 𝛽.
53
54
9/2/2019
28
Band Diagrams (2 of 2)
Slide 55
To construct a band diagram, we make small steps around the perimeter of the irreducible Brillouin zone (IBZ) and compute the eigen‐values at each step. When we plot all these eigen‐values as a function of , the points line up to form continuous “bands.”
Norm
alized
Frequen
cy
Animation of the Construction of a Band Diagram
Slide 56
55
56
9/2/2019
29
2D Animation of Bands and Bloch Waves
Slide 57
Reading Band Diagrams
• Band gaps• Absence of any bands within a range of frequencies indicates a band gap.• A COMPLETE BAND GAP is one that exists over all possible Bloch wave vectors.
• Transmission/reflection spectra• Band gaps lead to suppressed transmission and enhanced reflection
• Phase velocity• The slope of the line connecting to the point on the band corresponds to phase velocity.• From this, we get the effective phase refractive index.
• Group velocity• The slope of the band at the point of interest corresponds to the group velocity.• From this, we get the effective group refractive index.
• Dispersion• Any time the band deviates from the “light line” there is dispersion. • The phase and group velocity are the same except when there is dispersion.
Slide 58
At least five electromagnetic properties can be estimated from a band diagram.
57
58
9/2/2019
30
Reading Band Diagrams
Slide 59
The Band Diagram is Missing Information
Slide 60
x
y
y
x
M
X
Direct lattice: We have an array of air holes in a dielectric with n=3.0.
Reciprocal lattice: We construct the band diagram
by marching around the perimeter of the irreducible
Brillouin zone.
The band extremes “almost” always occur at the key points of symmetry.
But we are missing information from inside the Brillouin zone.
59
60
9/2/2019
31
x y
0
a
The Complete Band Diagram
Slide 61
The Full Brillouin Zone…
y
a
a
0
a 0 a
xThere is an infinite set of eigen‐frequencies associated with each point in the Brillouin zone. These form “sheets” as shown at right.
ΓX
M
ΓX
M
Animation of Complete Photonic Band Diagram
Slide 62
61
62
9/2/2019
32
Relation Between the Complete Band Diagram and the Band Diagram (1 of 4)
Slide 63
We start with the full band diagram.
M
XΓ
x y
02
a
c
IBZ
Relation Between the Complete Band Diagram and the Band Diagram (2 of 4)
Slide 64
We raise walls around the perimeter of the irreducible Brillouinzone.
These walls slice through the bands.
x y
02
a
c
M
XΓ
IBZ
63
64
9/2/2019
33
Relation Between the Complete Band Diagram and the Band Diagram (3 of 4)
Slide 65
We focus only on the intersections of the walls and the bands.
02
a
c
M
XΓ
IBZ
x y
Relation Between the Complete Band Diagram and the Band Diagram (4 of 4)
Slide 66
We unfold the walls to reveal the ordinary band diagram.
02
a
c
ΓX
M
Γ
x y
65
66
9/2/2019
34
Calculating Isofrequency
Contours(i.e. Index Ellipsoids for
Periodic Structures)
Slide 67
Recall Phase Vs. Power Flow
Slide 68
Isotropic Materials
k
P
x
y
Anisotropic Materials
k
x
y
Power propagates in the direction of P which is always normal to the surface of the index ellipsoid. From this, we can define a group velocity and a group refractive index.
P
Phase propagates in the direction of k. Therefore, the refractive index derived from |k| is best described as the phase refractive index. Velocity here is the phase velocity.
67
68
9/2/2019
35
Isofrequency Contours From First‐Order Band
Slide 69
02
a
c
aa
a
a0
0
y x
02
a
c
aa
a
a0
0
y x
Isofrequency contours are mostly circular. Not much interesting here.
Isofrequency From Second‐Order Band
Slide 70Index ellipsoids are “isofrequency contours” in k‐space.
69
70
9/2/2019
36
Standard View of Isofrequency Contours
Slide 71
Index ellipsoids in periodic structures are very interesting and useful because they can be things other than ellipsoids. They tend to resemble the shape of the Brillouin zone.
xa
ya
Example Applications
Slide 72
Self‐Collimation Negative Refraction
71
72
9/2/2019
37
APPENDIX
Slide 73
Geometry of a Hexagonal Lattice (1 of 3)
Slide 74
73
74
9/2/2019
38
Geometry of a Hexagonal Lattice (2 of 3)
Slide 75
1
2
3
3ˆ ˆ
2 2
3ˆ ˆ
2 2ˆ
a at x y
a at x y
t cz
1
2
3
2 2ˆ ˆ
32 2
ˆ ˆ3
ˆ2
T x ya a
T x ya a
T c z
Direct Lattice Reciprocal Lattice
x
y
z
1t
2t
a
c
x
y
z
1T
2T
2 a
2
c
conventional unit cell
Geometry of a Hexagonal Lattice (3 of 3)
Slide 76
Brillouin Zone and Irreducible Brillouin Zone
x
y
z conventional reciprocal unit cell
Brillouin zone (BZ)
Irreducible Brillouin zone (IBZ)
H L
MK
A 122
1 11 23 3
132
1 12 32 2
1 1 11 2 33 3 2
0
T
T T
T
T T
T T T
Γ
M
K
A
L
H
75
76