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**PHYS 5830: Condensed Matter Physics Instructor: Dr. Tom N. Oder**

Spring 2007 PHYS 5830: Condensed Matter Physics Course Code 2536 Instructor: Dr. Tom N. Oder

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**Physics 5830: Condensed Matter Physics**

Course Code 2536 Dr. Tom N. Oder WBSH 1016, Phone (330) Website: Class Website: Office Hours: M, W, F 2:00 pm – 3:00 pm. Research: (Wide Band Gap) Semiconductors.

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Required Texts: R.F. Pierret: Advanced Semiconductor Fundamentals, (Second Edition) - Modular Series on Solid-State Devices, Volume VI, Addison-Wesley, By. R. C. Jeager: Introduction to Microelectronics Fabrication (2nd Edition) - Modular Series on Solid-State Devices, Volume V, Addison-Wesley. Supplemental reference materials will come from archival journal papers selected by the instructor. Prerequisite: Phys Modern Physics. [May be waived by instructor].

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**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.

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**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.

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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.

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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.

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**Major Areas to be covered**

Semiconductor Physics Device Physics Processing and Characterization 1 + 2: Mondays, Wednesdays 3: Fridays

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Relevant References:

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**Semiconductor Physics**

Crystals: Structure and Growth. Energy Bands Carriers in Semiconductors Phonon Spectra and Optical Properties of Semiconductors Basic Equations for Semiconductor Device Operation

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**Crystal Structure of Solids**

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

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**Significance of Semiconductors**

• 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)

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**Fundamental Properties of Matter**

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

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Plasma: 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.

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Condensate: 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.

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Spin Occupancy Examples Fermions Half integral Only one per state electrons, protons, neutrons, quarks, neutrinos Bosons Integral spin Many allowed photons, 4He atoms, gluons

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**Element – one type of atoms**

Compound – 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.

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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.

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**Under what categories could this class be grouped?**

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Amorphous Solids 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.

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**Continuous random network structure of atoms in an amorphous solid**

Amorphous Solids Continuous random network structure of atoms in an amorphous solid

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**Polycrystalline 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.

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**Polycrystalline Solids**

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

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**Single- Vs Poly- Crystal**

• 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

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**Single- Vs Poly- Crystal**

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**Single- Vs Poly- Crystal**

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

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**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.

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**Hard Sphere Model of Crystals**

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 (, , ).

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Lattice Parameters a b c

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Atoms in a Crystal

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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.

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**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.

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Unit cell

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**Equivalent points of the lattice is defined by translation vector. **

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.

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**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

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Crystal System 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.

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**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.

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**1. Cubic (Isometric) System**

3 Bravais lattices Symmetry elements: Four 3-fold rotation axes along cube diagonals a = b = c = = = 90o a b c

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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.

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

• Rare due to poor packing (only Po has this structure) • Close-packed directions are cube edges. Coordination # = 6 (# nearest neighbors) 1 atom/unit cell

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**Coordination Number = Number of**

nearest neighbors

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One atom per unit cell 1/8 x 8 = 1

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**Atomic Packing Factor • APF for a simple cubic structure = 0.52 a**

Adapted from Fig. 3.19, Callister 6e.

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

• Exhibited by Al, Cu, Au, Ag, Ni, Pt Close packed directions are face diagonals. Coordination number = 12 4 atoms/unit cell All atoms are identical Adapted from Fig. 3.1(a), Callister 6e. 6 x (1/2 face) + 8 x 1/8 (corner) = 4 atoms/unit cell

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**3 mutually perpendicular planes. **

FCC Coordination number = 12 3 mutually perpendicular planes. 4 nearest neighbors on each of the three planes.

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**All atoms are identical**

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

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

• Exhibited by Cr, Fe, Mo, Ta, W Close packed directions are cube diagonals. Coordination number = 8 All atoms are identical 2 atoms/unit cell

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**All atoms are identical**

How is a and R related for an BCC? [a= unit cell dimension, R = atomic radius]. All atoms are identical 2 atoms/unit cell

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**Which one has most packing ?**

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**Which one has most packing ?**

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

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**Only one Bravais lattice**

2. Hexagonal System Only one Bravais lattice Symmetry element: One 6-fold rotation axis a = b c = 120o = = 90o

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**Hexagonal Closed Packed Structure (HCP)**

2D Projection Exhibited by …. ABAB... Stacking Sequence Coordination # = 12 APF = 0.74 3D Projection Adapted from Fig. 3.3, Callister 6e.

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**3. Tetragonal System Symmetry element: One 4-fold rotation axis**

Two Bravais lattices Symmetry element: One 4-fold rotation axis a = b c = = = 90o

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**4. Trigonal (Rhombohedral) System**

One Bravais lattice Symmetry element: One 3-fold rotation axis a = b c = 120o = = 90o

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5. Orthorhombic System Four Bravais lattices Symmetry element: Three mutually perpendicular 2-fold rotation axes a b c = = = 90o

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**6. Monoclinic System Symmetry element: One 2-fold rotation axis**

Two Bravais lattices Symmetry element: One 2-fold rotation axis a b c = = 90o, 90o

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**7. Triclinic System Symmetry element: None a b c 90o**

One Bravais lattice Symmetry element: None a b c 90o

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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.

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**Layer Stacking Sequence**

HCP = ABAB… = ABCABC.. FCC

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**FCC: Coordination number**

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

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**Diamond Lattice Structure**

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.

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**Diamond Lattice Structure**

a = lattice constant

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**Diamond Lattice Structure**

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**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

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

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Zinc Blende 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.

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**Zinc Blende Each Zn bonded to 4 Sulfur - tetrahedral**

Equivalent if Zn and S are reversed Bonding often highly covalent

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

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**Red = Ga-atoms, Blue = As-atoms**

GaAs Red = Ga-atoms, Blue = As-atoms Equal numbers of Ga and As ions distributed on a diamond lattice. Each atom has 4 of the opposite kind as nearest neighbors.

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**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.

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WURTZITE

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**Wurtzite Gallium Nitride (GaN)**

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**Step 1 : Identify the intercepts on the x- , y- and z- axes. **

Miller Index Step 1 : Identify the intercepts on the x- , y- and z- axes. Step 2 : Specify the intercepts in fractional co-ordinates Step 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

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**Miller Index For Cubic Structures**

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}.

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**Miller Indices: Directions**

z y x x y z [1] Draw a vector and take components a a [2] Reduce to simplest integers [3] Enclose the number in square brackets [0 1 1]

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**Negative Directions z y x x y z [1] Draw a vector and take components**

a a [2] Reduce to simplest integers [3] Enclose the number in square brackets

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**Miller Indices: Equivalent Directions**

Equivalent directions due to crystal symmetry: z y x 1 2 3 1: [100] 2: [010] 3: [001] Notation <100> used to denote all directions equivalent to [100]

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Directions

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Directions _ [1 0 1] _ [1 2 1]

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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: Identify the intersections with the axis, namely 2, -3 and 4. Calculate the inverse of each of those intercepts, resulting in 1/2, -1/3 and 1/4. 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. These are obtained in this case by multiplying each fraction by 12. Resulting Miller indices is Negative index indicated by a bar on top.

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**Miller Indices of Planes**

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 /∞ /∞ [3] Convert to the smallest integers [4] Enclose the number in round brackets (1 0 0)

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**Miller Indices of Planes**

z y x Miller Indices of Planes x y z [1] Determine intercept of plane with each axis 2a a a [2] Invert the intercept values 1/2a /2a /2a [3] Convert to the smallest integers [4] Enclose the number in round brackets (1 1 1)

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**Planes with Negative Indices**

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 /a /a [3] Convert to the smallest integers [4] Enclose the number in round brackets

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z y x (100) plane (010) plane (001) plane Equivalent Planes 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.

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(0 1 1)

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The (110) surface Assignment 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).

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**Crystallographic Planes**

Miller Indices (hkl) reciprocals

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**Crystallographic Planes**

_ (1 1 1)

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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

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Family of Planes All planes that are identical are included in 1 family denoted by { hkl} Example {111} (111) and (111) have equivalent atoms

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The (111) surface Assignment Intercepts : a , a , a Fractional intercepts : 1 , 1 , 1 Miller Indices : (111) The (210) surface Assignment Intercepts : ½ a , a , ¥ Fractional intercepts : ½ , 1 , ¥ Miller Indices : (210)

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**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.

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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

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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.

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QUESTIONS 1.

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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-k where 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/S i is redundant and not necessary.

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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.

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

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**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

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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:

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CRYSTAL GROWTH Seed Ingot Buole

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**Miller Indices – Silicon Wafers**

(100) plane (surface) Flat edge [011] direction Flats are cut out of wafers to indicate crystal orientation of wafer surface Essential to know surface orientation of Si wafer if you want to fabricate circuits on it

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**ATOMIC PACKING FACTOR: BCC**

• APF for a body-centered cubic structure = 0.68 Close-packed direction - diagonal 4R 3 a = Adapted from Fig. 3.2, Callister 6e.

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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.

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**ATOMIC PACKING FACTOR: FCC**

• APF for a face-centered cubic structure = 0.74 Close-packed directions (diagonal): length = 4R a = 2R 2 Unit cell contains 4 atoms/unit cell Adapted from Fig. 3.1(a), Callister 6e. 16 R3 APF = 4 3 p atoms unit cell atom volume R3 2 unit cell volume

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