Spring 2007

PHYS 5830: Condensed Matter PhysicsCourse Code 2536

Instructor: Dr. Tom N. Oder

• Dr. Tom N. Oder

• WBSH 1016, E-mail: tnoder@ysu.edu,

• Phone (330) 941-7111

• Website: http://www.ysu.edu/physics/tnoder

• Class Website: www.ysu.edu/physics/tnoder/S07-PHYS2536/index.html

• Office Hours: M, W, F 2:00 pm – 3:00 pm.

• Research: (Wide Band Gap) Semiconductors.

Physics 5830: Condensed Matter Physics

Course Code 2536

Required Texts:

1. R.F. Pierret: Advanced Semiconductor Fundamentals, (Second Edition) - Modular Series on Solid-State Devices, Volume VI, Addison-Wesley, 1988. By.

2. R. C. Jeager: Introduction to Microelectronics Fabrication (2nd Edition) - Modular Series on Solid-State Devices, Volume V, Addison-Wesley.

3. Supplemental reference materials will come from archival journal papers selected by the instructor.

Prerequisite: Phys. 3704 Modern Physics. [May be waived by instructor].

Course Structure: Lecture sessions and hands-on laboratory activities. Course Objectives:1.      To develop a background knowledge of semiconductor theory sufficient to understand modern semiconductor devices.2.      To provide students with practical experience in cutting-edge technology related to electronic device fabrication including lithography, thin film deposition and device characterization.

• Adhere to given instructions for safe handling of processing tools and chemicals.• Safe handling to avoid equipment damage or bodily harm.• Carelessness will lead to removal from class.• Repair/replacement of a damaged equipment carelessly handled.

Grading Policy:     Homework/Quizzes 20%.     Midterm Exam (1) 20%.Week right after Spring break     Laboratory Project 20%.     Final exam 40%. Homework problems will be assigned throughout the semester to reinforce the class material.

Final Grade:

90% - 100% = A

80% - 89% = B

70% - 79% = C

60% - 69% = D

0% - 59% = F

Cell Phones: Cell phones must be turned off during class and exam sessions.

1. Semiconductor Physics2. Device Physics3. Processing and Characterization

1 + 2: Mondays, Wednesdays3: Fridays

Major Areas to be covered

Relevant References:

http://ece-www.colorado.edu/~bart/book/book/contents.htm

• Crystals: Structure and Growth.• Energy Bands• Carriers in Semiconductors • Phonon Spectra and Optical Properties

of Semiconductors• Basic Equations for Semiconductor

Device Operation

Semiconductor Physics

1. Crystal Structure of Solids

What is “Crystal” to the man on the street?

• Computers, palm pilots, laptops, Silicon (Si) MOSFETs, ICs, CMOS, anything “intelligent”

• Cell phones, pagers Si ICs, GaAs FETs, BJTs

• CD players AlGaAs and InGaP laser diodes, Si photodiodes

• TV remotes, mobile terminals Light emitting diodes

• Satellite dishes InGaAs MMICs

• Fiber networks InGaAsP laser diodes, pin photodiodes

• Traffic signals, car GaN LEDs (green, blue)

Taillights InGaAsP LEDs (red, amber)

Significance of Semiconductors

Fundamental Properties of Matter

States of Matter

1. Solids – Definite volume, definite shape.

2. Liquids – Definite volume, no fixed shape. Flows.

3. Gases – No definite volume, no definite shape. Takes the volume and shape of its container.

Matter: - Has mass, occupies spaceMass – measure of inertia - from Newton’s first law of motion. It is one of the fundamental physical properties.

•Regarded as fourth state of matter. No definite volume, no definite shape. Composed of electrically charged particles.•Fully ionized gas at low density with equal amount of positive and negative charges – net electrically neutral.•Affected by electric and magnetic fields.•Plasma is the main state of matter in planetary objects such as stars.

Plasma:

•Regarded as fifth state of matter obtained when atoms/molecules are at very low temperature and their motion is halted. •They lose their individual identity and become a different entity.•Bose-Einstein condensates – Formed by bosons.•Fermionic condensates – By fermions.

Condensate:

Spin Occupancy Examples

Fermions Half integral

Only one per state

electrons, protons, neutrons, quarks, neutrinos

Bosons Integral spin

Many allowed

photons, 4He atoms, gluons

Element – one type of atomsCompound – Two or more different atoms chemically joined. Constituent atoms (fixed ratios) can be separated only by chemical means.Mixture - Two or more different atoms combined. Constituent atoms (variable ratios) can be separated by physical means.

Solid-State Physics – branch of physics dealing with solids.Now replaced by a more general terminology - Condensed Matter Physics. To include fluids which in many cases share same concepts and analytical techniques.

STRUCTURE OF SOLIDS

•Can be classified under several criteria based on atomic arrangements, electrical properties, thermal properties, chemical bonds etc.•Using electrical criterion: Conductors, Insulators, Semiconductors•Using atomic arrangements: Amorphous, Polycrystalline, Crystalline.

Under what categories could this class be grouped?

•No regular long range order of arrangement in the atoms.•Eg. Polymers, cotton candy, common window glass, ceramic. •Can be prepared by rapidly cooling molten material. •Rapid – minimizes time for atoms to pack into a more thermodynamically favorable crystalline state. •Two sub-states of amorphous solids: Rubbery and Glassy states. Glass transition temperature Tg = temperature above which the solid transforms from glassy to rubbery state, becoming more viscous.

Amorphous Solids

Continuous random network structure of atoms in an amorphous solid

Amorphous Solids

•Atomic order present in sections (grains) of the solid.•Different order of arrangement from grain to grain. Grain sizes = hundreds of m.•An aggregate of a large number of small crystals or grains in which the structure is regular, but the crystals or grains are arranged in a random fashion.

Polycrystalline Solids

Polycrystalline Solids

Atoms arranged in a 3-D long range order. “Single crystals” emphasizes one type of crystal order that exists as opposed to polycrystals.

Crystalline Solids

• Properties of single crystalline materials vary with direction, ie anisotropic.

•Properties of polycrystalline materials may or may not vary with direction.If the polycrystal grains are randomly oriented, properties will not vary with direction ie isotropic.

•If the polycrystal grains are textured, properties will vary with direction ie anisotropic

Single- Vs Poly- Crystal

E (diagonal) = 273 GPa

E (edge) = 125 GPa

Single- Vs Poly- Crystal

-Properties may/may not vary with direction.-If grains are randomly oriented: isotropic. (Epoly iron = 210 GPa)-If grains are textured, anisotropic.

200 m

Single- Vs Poly- Crystal

Solid state devices employ semiconductor materials in all of the above forms.

Examples:

Amorphous silicon (a-Si) used to make thin film transistors (TFTs) used as switching elements in LCDs.

Ploycrystalline Si – Gate materials in MOSFETS.

Active regions of most solid state devices are made of crystalline semiconductors.

• Assumes atoms are hard spheres with well defined diameters that touch.

•Atoms are arranged on periodic array – or lattice

•Repetitive pattern – unit cell defined by lattice parameters comprising lengths of the 3 sides (a, b, c) and angles between the sides (, , ).

Hard Sphere Model of Crystals

a

b

c

Lattice Parameters

Atoms in a Crystal

The Unit Cell Concept

•The simplest repeating unit in a crystal is called a unit cell. •Opposite faces of a unit cell are parallel.•The edge of the unit cell connects equivalent points.•Not unique. There can be several unit cells of a crystal.•The smallest possible unit cell is called primitive unit cell of a particular crystal structure. •A primitive unit cell whose symmetry matches the lattice symmetry is called Wigner-Seitz cell.

• Each unit cell is defined in terms of lattice points. •Lattice point not necessarily at an atomic site.• For each crystal structure, a conventional unit cell, is chosen to make the lattice as symmetric as possible. However, the conventional unit cell is not always the primitive unit cell. •A crystal's structure and symmetry play a role in determining many of its properties, such as cleavage (tendency to split along certain planes with smooth surfaces), electronic band structure and optical properties.

Unit cell

•Unit cell: Simplest portion of the structure which is repeated and shows its full symmetry.•Basis vectors a and b defines relationship between a unit cell and (Bravais) lattice points of a crystal.•Equivalent points of the lattice is defined by translation vector.r = ha + kb where h and k are integers. This constructs the entire lattice.

a

b

•By repeated duplication, a unit cell should reproduce the whole crystal.•A Bravias lattice (unit cells) - a set of points constructed by translating a single point in discrete steps by a set of basis vectors. •In 3-D, there are 14 unique Bravais lattices. All crystalline materials fit in one of these arrangements.

In 3-D, the translation vector is

r = ha + kb + lc

•The crystal system: Set of rotation and reflection symmetries which leave a lattice point fixed. •There are seven unique crystal systems: the cubic (isometric), hexagonal, tetragonal, rhombohedral (trigonal), orthorhombic, monoclinic and triclinic.

Crystal System

Bravais Lattice and Crystal System

Crystal structure: contains atoms at every lattice point.

•The symmetry of the crystal can be more complicated than the symmetry of the lattice.

•Bravais lattice points do not necessarily correspond to real atomic sites in a crystal. A Bravais lattice point may be used to represent a group of many atoms of a real crystal. This means more ways of arranging atoms in a crystal lattice.

1. Cubic (Isometric) System

Symmetry elements: Four 3-fold rotation axes along cube diagonalsa = b = c = = = 90o

3 Bravais lattices

ab

c

By convention, the edge of a unit cell always connects equivalent points. Each of the eight corners of the unit cell therefore must contain an identical particle.

• Rare due to poor packing (only Po has this structure)• Close-packed directions are cube edges.

Coordination # = 6 (# nearest neighbors)

1 atom/unit cell

(1-a): Simple Cubic Structure (SC)

Coordination Number = Number of nearest neighbors

One atom per unit cell

1/8 x 8 = 1

APF = Volume of atoms in unit cell*

Volume of unit cell

*assume hard spheres

• APF for a simple cubic structure = 0.52

APF = a3

4

3(0.5a)31

atoms

unit cellatom

volume

unit cellvolume

a

R=0.5a

Adapted from Fig. 3.19, Callister 6e.

Atomic Packing Factor

Adapted from Fig. 3.1(a), Callister 6e.

• Exhibited by Al, Cu, Au, Ag, Ni, Pt• Close packed directions are face diagonals.• Coordination number = 12• 4 atoms/unit cell

6 x (1/2 face) + 8 x 1/8 (corner) = 4 atoms/unit cell

(1-b): Face Centered Cubic Structure (FCC)

All atoms are identical

FCCCoordination number = 12

3 mutually perpendicular planes.4 nearest neighbors on each of the three planes.

How is a and R related for an FCC?[a= unit cell dimension, R = atomic radius].

All atoms are identical

• Exhibited by Cr, Fe, Mo, Ta, W• Close packed directions are cube diagonals.• Coordination number = 8

2 atoms/unit cell

(1-c): Body Centered Cubic Structure (BCC)

All atoms are identical

2 atoms/unit cell

All atoms are identical

How is a and R related for an BCC?[a= unit cell dimension, R = atomic radius].

Which one has most packing ?

Which one has most packing ?

For that reason, FCC is also referred to as cubic closed packed (CCP)

Symmetry element: One 6-fold rotation axisa = b c= 120o

= = 90o

2. Hexagonal System

Only one Bravais lattice

• Exhibited by …. • ABAB... Stacking Sequence• Coordination # = 12• APF = 0.74

3D Projection

2D Projection

A sites

B sites

A sites

Bottom layer

Middle layer

Top layer

Adapted from Fig. 3.3, Callister 6e.

Hexagonal Closed Packed Structure (HCP)

Symmetry element: One 4-fold rotation axisa = b c= = = 90o

3. Tetragonal System

Two Bravais lattices

Symmetry element: One 3-fold rotation axisa = b c= 120o

= = 90o

4. Trigonal (Rhombohedral) System

One Bravais lattice

5. Orthorhombic System

Symmetry element: Three mutually perpendicular 2-fold rotation axesa b c = = = 90o

Four Bravais lattices

6. Monoclinic System

Symmetry element: One 2-fold rotation axisa b c = = 90o, 90o

Two Bravais lattices

7. Triclinic System

Symmetry element: Nonea b c 90o

One Bravais lattice

•The crystal system: Set of symmetries which leave a lattice point fixed. There are seven unique crystal systems.

• Some symmetries are identified by special name such as zincblende, wurtzite, zinc sulfide etc.

A sites

B sites

A sites

HCP

FCC

Layer Stacking Sequence

= ABCABC..

= ABAB…

FCCCoordination number = 12

3 mutually perpendicular planes.4 nearest neighbors on each of the three planes.

FCC: Coordination number

•Exhibited by Carbon (C), Silicon (Si) and Germanium (Ge).•Consists of two interpenetrating FCC lattices, displaced along the body diagonal of the cubic cell by 1/4 the length of the diagonal. • Also regarded as an FCC lattice with two atoms per lattice site: one  centered on the lattice site, and the other at a distance of a/4 along all axes, ie an FCC lattice with the two-point basis.

Diamond Lattice Structure

a = lattice constant

Diamond Lattice Structure

Diamond Lattice Structure

Two merged FCC cells offset by a/4 in x, y and z.

Basic FCC Cell Merged FCC Cells

Omit atoms outside Cell Bonding of Atoms

8 atoms at each corner, 6 atoms on each face, 4 atoms entirely inside the cell

•Similar to the diamond cubic structure except that the two atoms at each lattice site are different.      • Exhibited by many semiconductors including ZnS, GaAs, ZnTe and CdTe.•GaN and SiC can also crystallize in this structure.

Zinc Blende

Zinc Blende

Each Zn bonded to 4 Sulfur - tetrahedral

Equivalent if Zn and S are reversedBonding often highly covalent

Zinc sulfide crystallizes in two different forms: Wurtzite and Zinc Blende.

GaAs

•Equal numbers of Ga and As ions distributed on a diamond lattice.• Each atom has 4 of the opposite kind as nearest neighbors.

Red = Ga-atoms, Blue = As-atoms

Wurtzite (Hexagonal) Structure

•This is the hexagonal analog of the zinc-blende lattice.• Can be considered as two interpenetrating close-packed lattices with half of the tetrahedral sites occupied by another kind of atoms. • Four equidistant nearest neighbors, similar to a zinc-blende structure. •Certain compound semiconductors (ZnS, CdS, SiC) can crystallize in both zinc-blende (cubic) and wurtzite (hexagonal) structure.

A sites

B sites

A sites

WURTZITE

Wurtzite Gallium Nitride (GaN)

Miller Index

Step 1 :  Identify the intercepts on the x- , y- and z- axes.Step 2 :  Specify the intercepts in fractional co-ordinatesStep 3 :  Take the reciprocals of the fractional intercepts

(i)  in some instances the Miller indices are best multiplied or divided through by a common number in order to simplify them by, for example, removing a common factor. This operation of multiplication simply generates a parallel plane which is at a different distance from the origin of the particular unit cell being considered. e.g.   (200) is transformed to (100) by dividing through by 2 . (ii)  if any of the intercepts are at negative values on the axes then the negative sign will carry through into the Miller indices; in such cases the negative sign is actually denoted by overstriking the relevant number.e.g.   (00 -1) is instead denoted by       

100

•Miller index is used to describe directions and planes in a crystal.•Directions - written as [u v w] where u, v, w.• Integers u, v, w represent coordinates of the vector in real space.•A family of directions which are equivalent due to symmetry operations is written as <u v w>•Planes: Written as (h k l).•Integers h, k, and l represent the intercept of the plane with x-, y-, and z- axes, respectively.• Equivalent planes represented by {h k l}.

Miller Index For Cubic Structures

x y z

[1] Draw a vector and take components 0 2a 2a

[2] Reduce to simplest integers 0 1 1

[3] Enclose the number in square brackets [0 1 1]

z

y

x

Miller Indices: Directions

z

y

xx y z

[1] Draw a vector and take components 0 -a 2a

[2] Reduce to simplest integers 0 -1 2

[3] Enclose the number in square brackets 210

Negative Directions

Miller Indices: Equivalent Directions

z

y

x1

2

3

1: [100]2: [010]3: [001]

Equivalent directions due to crystal symmetry:

Notation <100> used to denote all directions equivalent to [100]

Directions

Directions

_[1 2 1]

_[1 0 1]

The intercepts of a crystal plane with the axis defined by a set of unit vectors are at 2a, -3b and 4c. Find the corresponding Miller indices of this and all other crystal planes parallel to this plane.

The Miller indices are obtained in the following three steps:1. Identify the intersections with the axis, namely 2, -3 and 4.2. Calculate the inverse of each of those intercepts, resulting in

1/2, -1/3 and 1/4.3. Find the smallest integers proportional to the inverse of the

intercepts. Multiplying each fraction with the product of each of the intercepts (24 = 2 x 3 x 4) does result in integers, but not always the smallest integers.

4. These are obtained in this case by multiplying each fraction by 12.

5. Resulting Miller indices is6. Negative index indicated by a bar on top.

346

z

y

x

z=

y=x=a

x y z

[1] Determine intercept of plane with each axis a ∞ ∞

[2] Invert the intercept values 1/a 1/∞ 1/∞

[3] Convert to the smallest integers 1 0 0

[4] Enclose the number in round brackets (1 0 0)

Miller Indices of Planes

z

y

x

x y z

[1] Determine intercept of plane with each axis 2a 2a 2a

[2] Invert the intercept values 1/2a 1/2a 1/2a

[3] Convert to the smallest integers 1 1 1

[4] Enclose the number in round brackets (1 1 1)

Miller Indices of Planes

z

y

x

Planes with Negative Indices

x y z

[1] Determine intercept of plane with each axis a -a a

[2] Invert the intercept values 1/a -1/a 1/a

[3] Convert to the smallest integers 1 -1 -1

[4] Enclose the number in round brackets 111

• Planes (100), (010), (001), (100), (010), (001) are equivalent planes. Denoted by {1 0 0}.

• Atomic density and arrangement as well as electrical, optical, physical properties are also equivalent.

z

yx

(100) plane

(010) plane

(001) planeEquivalent Planes

(0 1 1)

The (110) surfaceAssignment Intercepts :   a , a , Fractional intercepts :   1 , 1 , Miller Indices :   (110)

The (100), (110) and (111) surfaces considered above are the so-called low index surfaces of a cubic crystal system (the "low" refers to the Miller indices being small numbers - 0 or 1 in this case).

Miller Indices (hkl)reciprocals

Crystallographic Planes

Crystallographic Planes _(1 1 1)

in the cubic system the (hkl) plane and the vector [hkl], defined in the normal fashion with respect to the origin, are normal to one another but this characteristic is unique to the cubic crystal system and does not apply to crystal systems of lower symmetry

Family of PlanesAll planes that are identical are included in 1 family denoted by { hkl}Example {111}

(111) and (111) have equivalent atoms

The (111) surfaceAssignment Intercepts :   a , a , a Fractional intercepts :   1 , 1 , 1 Miller Indices :   (111)

The (210) surfaceAssignment Intercepts :   ½ a , a , Fractional intercepts :   ½ , 1 , Miller Indices :   (210)

Symmetry-equivalent surfaces

the three highlighted surfaces are related by the symmetry elements of the cubic crystal - they are entirely equivalent.

In fact there are a total of 6 faces related by the symmetry elements and equivalent to the (100) surface - any surface belonging to this set of symmetry related surfaces may be denoted by the more general notation {100} where the Miller indices of one of the surfaces is instead enclosed in curly-brackets.

in the hcp crystal system there are four principal axes; this leads to four Miller Indices e.g. you may see articles referring to an hcp (0001) surface. It is worth noting, however, that the intercepts on the first three axes are necessarily related and not completely independent; consequently the values of the first three Miller indices are also linked by a simple mathematical relationship

In case of a cubic structure, the Miller index of a plane, in parentheses such as (100), are also the coordinates of the direction of a plane normal. It stands for a vector perpendicular to the family of planes, with a length of d-1, where d is the inter-plane spacing.Due to the symmetries of cubic crystals, it is possible to change the place and sign of the integers and have equivalent directions and planes:

•Coordinates in angle brackets or chevrons such as <100> denote a family of directions which are equivalent due to symmetry operations. If it refers to a cubic system, this example could mean [100], [010], [001] or the negative of any of those directions.•Coordinates in curly brackets or braces such as {100} denote a family of plane normals which are equivalent due to symmetry operations, much the way angle brackets denote a family of directions.

QUESTIONS

1.

With hexagonal and rhombohedral crystal systems, it is possible to use the Bravais-Miller index which has 4 numbers (h k i l)

i = -h-kwhere h, k and l are identical to the Miller index.The (100) plane has a 3-fold symmetry, it remains unchanged by a rotation of 1/3 (2π/3 rad, 30°). The [100], [010] and the        directions are similar. If S is the intercept of the plane with the        axis, then

i = 1/Si is redundant and not necessary.

in the hcp crystal system there are four principal axes; this leads to four Miller Indices e.g. you may see articles referring to an hcp (0001) surface. It is worth noting, however, that the intercepts on the first three axes are necessarily related and not completely independent; consequently the values of the first three Miller indices are also linked by a simple mathematical relationship.

In the cubic crystal system, a plane and the direction normal to it have the same indices.[101] direction is normal to the plane (101)

Distance (d) separating adjacent planes (hkl) of a cubic crystal of lattice constant (a) is: 222 lkh

ad

Angle () between directions [h1 k1 l1] and [h2 k2 l2] of a cubic crystal is:

))(()cos(

22

22

22

21

21

21

212121

lkhlkh

llkkhh

Miller-Bravais Indices

• For hcp crystal structure• Planes (h k i l), directions [h k i l]• Sum of the first three indices h + k + i = 0

To determine the Miller-Bravais indices of the crystallographic direction indicated on the a1-a2-a3 plane of an h.c.p. unit cell (ie. c = 0), the vector must be projected onto each of the three axes to find the corresponding component, such that:

CRYSTAL GROWTH

SeedIngotBuole

(100) plane (surface)

Flat edge [011]

direction

Flats are cut out of wafers to indicate crystal orientation of wafer surfaceEssential to know surface orientation of Si wafer if you want to fabricate circuits on it

Miller Indices – Silicon Wafers

aR

• APF for a body-centered cubic structure = 0.68

Close-packed direction - diagonal

Adapted fromFig. 3.2,Callister 6e.

ATOMIC PACKING FACTOR: BCC

APF = a3

4

3( 3a/4)32

atoms

unit cell atomvolume

unit cell

volume

4R 3

a =

Calculate the maximum fraction of the volume in a simple cubic crystal occupied by the atoms. Assume that the atoms are closely packed and that they can be treated as hard spheres. This fraction is also called the packing density.

The atoms in a simple cubic crystal are located at the corners of the units cell, a cube with side a. Adjacent atoms touch each other so that the radius of each atom equals a/2. There are eight atoms occupying the corners of the cube, but only one eighth of each is within the unit cell so that the number of atoms equals one per unit cell. The packing density is then obtained from: or about half the volume of the unit cell is occupied by the atoms.The packing density of four cubic crystals is listed in the table below.

unit cell

volume

• APF for a face-centered cubic structure = 0.74

Adapted fromFig. 3.1(a),Callister 6e.

ATOMIC PACKING FACTOR: FCC

Close-packed directions (diagonal):

Unit cell contains

4 atoms/unit cell

length = 4R a = 2R 2

16 R3

APF =

4

34

atoms

unit cell atomvolume

R3

2