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# Lecture 14

## on Jul 11, 2007

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## Lecture 14Presentation Transcript

• Today’s objectives-Electrical Conduction
• What are the forms of Ohm’s law?
• What are conductance and resistance and some standard values for materials?
• Know the equation which accounts for scattering.
• Describe the contribution of thermal vibrations, atomic defects, and more macroscopic defects on resistivity/conductivity?
• How do available electron energy states vary for atoms, molecules, large molecules, and solids?
• Sketch the 3 possible metallic band diagrams.
• Sketch a simple metallic, semiconducting, and insulating band diagram.
• How is conductivity described based on metal, semiconductor, and insulating band diagrams?
Reading for today Semiconductors and Integrated Circuits Chapter sections: 18.1-9
• Voltages and Electric Fields
• An electric field exists whenever there is a voltage difference between two points.
+ - + - Often, we consider the positive bias to simply be at ‘V’ and the negative pole to be at 0 Volts, or ground. V
• ELECTRICAL CONDUCTION 3 • Ohm's Law:  V = I R voltage drop (volts) resistance (Ohms) current (amps) • Resistivity,  and Conductivity,  : --geometry-independent forms of Ohm's Law E: electric field intensity resistivity (Ohm-m) J: current density conductivity • Resistance:
• Ohm’s Law
• Voltage source, current meter, resistor.
• Note: current is measured in Amperes.
V I I = V / R voltage drop (volts) resistance (Ohms) current (amps) Energy Specimen (R) + -
• R depends on specimen shape—but we need a term that is independent of geometry.
R = V / I = Volts/Amps = Ohms Resistivity: current density (Ohm*meters)
• Other geometry independent relationships.
• Rewriting Ohm’s law (V=IR) using these terms yields a second, more materials friendly version:
• Resistivity or conductivity
• Electric field
• Current density (Amps/m)
General rules Conductivity is largest for metals & smallest for insulators. Resistivity is smallest for metals and largest for insulators.
• CONDUCTIVITY: COMPARISON Remember that Resistivity is the inverse =1/conductivity, and is thus huge for insulators and very small for conductors. way out there Pluto’s orbital diameter 1/2 mile atom 1E+06 1E+03 1E-07 1E-20 good metal metal Semi-conductor insulator
• 10 million light years (10 23 m). The distant galaxy is the Milky Way.
• 1 million light years (10 22 m) The disc becomes visible.
• 1 light year (10 16 m), within the Milky Way. This is our sun.
• 1 trillion km (10 15 m) The sun even bigger.
• 100 billion km (10 14 m) Our solar system…
• 10 billion Km (10 13 m) Our solar system.
• 1 billion Km (10 12 m) The orbits of Venus, Earth, Mars.
• 10 million Km (10 10 m) Orbit of Earth...
• 1 million Km (10 9 m) Earth and the orbit of Moon.
• 100.000 Km (10 8 m) Third rock from the sun…
• 10.000 Km(10 7 m) The northern hemisphere of Earth. As usual, another wonderful day in CT…
• 1.000 Km (10 6 m) FLA
• 10 Km (10 4 m) You start to differentiate buildings.
• 1 Km (10 3 m)
• 100 m (10 2 m) An ordinary view from an helicopter.
• 10 m (10 1 m)
• 1 m (10 0 m)
• 10 cm (10 -1 m)
• 1 cm (10 -2 m) You can see the structure of a leaf.
• 1 mm (10 -3 m) Even closer.
• 100 micron (10 -4 m) you can see the cells.
• 10 micron (10 -5 m)
• 1 micron (10 -6 m). The cell itself.
• 1.000 angstrom (10 -7 m) You can see the chromosomes.
• 100 angstrom (10 -8 m) You can see the DNA chain.
• 1 nm (10 -9 m) The chromosomes.
• 1 angstrom. (10 -10 m) The carbon atom.
• 1 Pico metre (10 -12 m) The orbit of electrons, if we could see it....
• 100 Fermi (10 -13 m) Inside an atom.
• 10 Fermi (10 -14 m), protons and neutrons… Too bad we can’t see them…
• EX: CONDUCTIVITY PROBLEM • Question 18.2, p. 649, Callister 6e : a) What is the minimum diameter (D) of the wire so that  V < 1.5V, given that sigma=6.07*10 7 /(Ohm*m)? b) What is R? Solve to get D > 1.88 mm. With D, R can be determined: 0.59 Ohms or less. < 1.5V 2.5A 6.07 x 10 (Ohm-m) 7 -1 100m
• Reasons for resistivity/conductivity
• Why does resistivity vary from one material to the next?
• Electronic Structure (metallic, semiconducting, insulating)
• Scattering Events (characterized by ‘ μ ’)
Note that by convention, electrons move opposite to the field direction. n=number of electrons per volume e=charge on electron (or hole) μ =mobility of electrons in a given material
• Scattering: origin of resistivity/conductivity
• Primary Scattering Events
• Thermal defects (k b T at room temperature is about 25 meV).
• Atomic Defects (impurities/dopants)
• 2d and/or 3d defects (grain boundaries, particles, dislocations)
• Oxygen free high conductivity copper is usually employed when very low resistivity is required. Al is an adequate substitute. Ag or Au are great, but too expensive.
• Resistivity Components • Resistivity increases with: --temperature --wt% impurity --% deformation
• Strengthening Mechanisms in Metals
• Grain Boundary Strengthening: Smaller grains result in stronger materials. The GB acts as an obstacle to dislocation motion.
• Work-hardening: As a metal is plastically deformed, the dislocation density increases. Dislocation-dislocation interactions result in reduced mobility of dislocations.
• Alloying: Strain fields of alloying elements (substitutional and interstitial) stop dislocation motion. Plus, they may form secondary phases and the GB between phases may reduce dislocation mobility.
How do these mechanisms affect electrical conduction?
• Strengthening Mechanisms in Metals
• Any defect structure will reduce electron mobility .
• Ideal metallic conductor should be:
• Single-crystal (no grain boundaries, twin boundaries, anti-phase boundaries, vacancies [impossible!!!]… )
• No defects (dislocations).
• As pure as possible (no impurities, no secondary phasesand phase boundaries).
They ALL decrease electrical conductivity!!!
• To understand the electronic structure of a crystal, we have to consider several steps of increasing complication:
• Atoms
• Molecules
• Crystals
• Metals
• Semiconductors
• Insulators
• Electrons in atoms
• Electrons in an atom have particular energies (quantized energy states ) depending on which orbital they are in.
Pauli exclusion principle H 1s He 1s B 1s 1p Energy
• Electrons in solids
• In a solid, there are so many electrons with energies very near each other that ‘bands’ of states develop.
Isolated atoms Solid All we draw is the “band diagram”
• Energy band structures for various metals
• Partially filled or empty bands are called ‘conduction bands.’
• Any band that is totally filled is considered to be a “valence band.”
• We usually ignore ‘deep’ valence bands.
Metal (Cu) partially filled 4s (conduction) filled 3d, 3p, 3s, 2p, 2s, 1p, 1s (valence) Empty 4p (conduction) Band gap Band gap Energy Filled 3d (valence) Deep valence-only an issue for optical properties
• Energy band structures for various metals
• Energy bands overlap differently depending on material and esp. valence electrons.
conducting conducting conducting Filled 1p, 1s (deep valence) E f Metal (Be) Filled 2s (valence) Empty 2p (conduction) Empty 3s (conduction) Band gap Band gap Energy E f , Fermi level Metal (Cu) partially filled 4s (conduction) filled 3d, 3p, 3s, 2p, 2s, 1p, 1s (valence) Empty 4p (conduction) Band gap Band gap Metal (Mg, 3p and 3s overlap) E f Filled [mostly] 3s (valence) Empty [mostly] 3p (conduction) filled (valence) Empty 4s (conduction) Band gap Band gap
• Definition of Conductivity
• The free-est electron (the electron with the highest energy) defines the position of the “Fermi level.”
• Above E f , all available electron states in the energy bands are empty
• Below E f , they are all filled.
• If there is no gap between filled and empty states, the material is conductive .
• If there is a gap, the material is a semiconductor or insulator.
Metal (Cu) partially filled 4s (conduction) Empty 4p (conduction) Band gap Band gap Energy Filled (valence) E f , Fermi level
• Band structures for semiconductors and insulators
• Semiconductors and Insulators have totally full valence bands and empty conduction bands with a bandgap between them. E f exists in the bandgap .
• The distinction between semiconducting and insulating materials is arbitrarily set to a bandgap of < or > 2 eV, respectively.
Energy E f , Fermi level Metal (Cu) partially filled 4s (conduction) filled 3p, 2p, 2s, 1p, 1s (valence) Empty 4p (conduction) Band gap Band gap Filled (deep valence) E f Semiconductor (Si) Filled (valence) Empty (conduction) Band gap Band gap Filled (deep valence) E f Insulator (Al 2 O 3 ) Filled (valence) Empty (conduction) Band gap Band gap
• CONDUCTION & ELECTRON TRANSPORT
• At room temperature, atoms have kinetic energy = kT, which is approximately 25 meV.
• This is sufficient to jump from a filled state to an empty state in a metal since the Fermi level (topmost electron) has empty states nearby.
• Once in the empty state, an electron can be swept away by an electric field.
• In a metal, nearly any electric field is sufficient to pull a substantial number of electrons into these nearby, empty, conducting states.
• Conduction in insulators/semiconductors • Insulators: --Higher energy states not accessible due to gap. • Semiconductors: --Higher energy states possibly accessible due to smaller gap.
• Note: Conductivity can sometimes be enhanced by adding:
• Dopants to generate excess charges
• Impurities or defects that create states or bands within the gap.
• Metallic conduction
• Once an electron has jumped across the Fermi level, or whenever there is not an electron in a site beneath the Fermi level, this is considered a ‘hole.’
• A hole behaves similar to an electron, but has the opposite charge. It can contribute to conductivity.
• SUMMARY Reading for next class Semiconductors and Integrated Circuits, Chapter sections: 18.10-15
• How are conductance and resistance characterized?
• What are the forms of Ohm’s law?
• How does conductivity vary for conductors, semiconductors, and insulators?
• How do available electron energy states vary for atoms, molecules, large molecules, and solids?
• Sketch the 3 possible metallic band diagrams.
• Sketch a simple metallic, semiconducting, and insulating band diagram.
• How is conductivity described based on metal, semiconductor, and insulating band diagrams?
• What are, and what are the equations, for the drift velocity and conductivity with respect to the mobility?
• Describe and be able to calculate the thermal and impurity components of conductivity?
• What kinds of scattering sites contribute to the deformation term of conductivity?