What have we so far

1
What have we so far?

• electric field from charge distributions
• Coulombs law
• relation between flux and charge
• Guasss law
• work and potential from charge distributions
• relation between potential and field

2
Whats next?

• Id still like to understand how signals move
  (propagate) through a long telegraph cable
• to do that we need a better understanding of
  materials
• conductors
• dielectrics

3
Conductors

• 1600 William Gilbert (1544-1603), after 18 years
  of experiments with loadstones, magnets and
  electrical materials, finishes his book De
  Magnete.
• coined the modern Latin word electricus from
  ??e?t??? (elektron), the Greek word for amber,
  which soon gave rise to the English words
  electric and electricity
• the work included the first major classification
  of electric and non-electric materials the
  relation of moisture and electrification showing
  that electrification affects metals, liquids and
  smoke noting that electrics were the attractive
  agents (as opposed to the air between objects)
  that heating dispelled the attractive power of
  electrics and showing the earth to be a magnet.
• 1729 Stephen Gray classified materials as
  conductors and insulators

4
Charged particles in materials

• charge transport in conductors
• real particles are the carrier of current
• in a normal metal, the carrier is the electron
• electrons and holes in a solid respond to an
  electric field almost as if they were free
  particles in a vacuum, but with a different mass
• ordinary mass of an electron me 9.1110-31 kg
• effective mass (m) helps capture the fact that
  the carrier is not really in free space
• consider a point charge in an external electric
  field
• electrons accelerate in response to the force
• velocity would continue to increase the longer
  the charge stays in the field
• just like falling in a gravitational field

5
Friction and terminal velocity

• in real materials the particle does not continue
  to increase its velocity without bound
• again, just like falling in a gravitational field
  when there is air
• if you jump out of an airplane youll reach a
  terminal velocity due to friction
• in the spread eagle position a sky diver goes a
  max of a little over 50 m/sec (110 miles/hour)
• terminal velocity calculator

6
Scattering and drift velocity

• for us, lets assume that throughout the material
  there are scattering centers
• on average the electron will travel for a time t
  before it scatters
• electrons collide with scatterers, randomizing
  their velocities
• this leads to an overall average velocity that
  does NOT increase without bound in a constant
  force field (as a free particle would)
• we can calculate the drift velocity from the
  characteristic scattering time t
• consider an electron starting at rest in a
  constant electric field E
• the force on the electron is qeE
• acceleration is constant (Newtons Second Law
  Fma)

7
Drift velocity and the scattering time

• assume that during the time t the particle is in
  free-flight in the electric force field E, so at
  the end of time t the velocity is
• assume that after the time interval t the
  scattering event randomizes things so that we
  restart at zero velocity again
• hence, the drift velocity is just
• note that the drift velocity is (approximately)
  linearly proportional to the magnitude of the
  electric field E
• m is the mobility (units of velocity/electric
  field cm2 / Vsec)

8
Current density

• how big is the current density due to this
  electric field?
• this is a flux concept, very similar to what we
  discussed before
• the vector current density is set by the density
  of carriers rv and their vector (drift) velocity
  v
• assume there are n free electrons per unit volume
  (number per unit volume), i.e., the electron
  density n
• the electron charge is qe
• then the charge density rv is just
• and the current density is
• this suggests that current density is linearly
  proportional to the applied electric field!!!
• Ohms Law!

9
Conductivity and Ohms Law

• so far we have
• where we have used the symbol s, the conductivity
  of the material
• in field form Ohms Law is given by
• where
• or using and
• higher conductivity is the result of
• higher mobility (longer scattering time and/or
  lower effective mass)
• larger carrier density

10
Summary page

• so far we have
• field properties
• material properties
• what varies amongst materials?
• somewhat mobility (or equivalently, effective
  mass and scattering time)
• a lot carrier density n!!!
• all the way from approximately zero to 1023 per
  cm3

11
Band structure of solids

• a potential energy picture of materials that is
  fundamentally connected to the periodicity of a
  crystal and the quantum mechanics
• there are states (or bands) at various
  energies that are allowed, i.e., carriers can
  occupy a state only at certain energies
• there are energies that are NOT allowed band
  gaps
• electrons in a solid tend to settle into lowest
  available energy states
• loosely speaking, highest occupied energy at zero
  temperature is called the Fermi energy
• in semiconductors and insulators, the Fermi
  energy is inside one of the disallowed regions.
  This means that the electrons fill up to the top
  of one of the bands, and no electrons enter the
  next highest band. The highest filled band is
  called the valence band, the next highest band is
  called the conduction band, and the energy
  difference between the two is called the band
  gap. Electrons in the valence band cannot
  accelerate in response to an electric field,
  because there are no states available where the
  electrons would be moving any faster. Hence there
  is no conduction at zero temperature.
• at FINITE temperature, due to thermal energy,
  some of the electrons from the valence band will
  be thermally excited into the conduction band
• the number of electrons depends on how big the
  band gap is

12
Band diagrams for materials
band gap (forbidden states)

• Fermi level indicates how states are actually
  occupied
• a flat fermi level indicates that no external
  voltages are applied
• most above the Fermi level are empty of electrons
• most below are full of electrons
• a small bandgap usually gives a larger number of
  carriers for current
• metals zero bandgap, n 1023 cm-3
• semiconductors medium bandgap, n 1010-1021
  cm-3
• insulators large bandgap, n 0 cm-3

13
Metals

• the Fermi energy is in the middle of one of the
  bands
• electrons in this band can easily accelerate,
  since there is no energy gap to available
  conduction states
• the number of available carriers in a metal is
  very much higher than in an insulator
• typically about 1022 / cm3
• typical scattering time in metals is about 10-14
  sec
• typical mobility is a few hundred cm2 / Vsec
• conductivity (units 1/(ohmslength) is high
• equivalent material parameter is the
  resistivity r (units ohmslength)

14
Calculating dc resistance

• consider a block of uniform conducting material
  with perfect electrical contacts on each end
• the current density from Ohms law J s E
• but the normal form of Ohms law is I R V
• here the total current I is just the current
  density integrated across a cross section
• here the field is uniform and hence so is the
  current density
• so I (cross sectional area) J w t s E
• the voltage difference V between one end and the
  other is just the line integral of the electric
  field
• again, since everything is uniform this is easy
• V E l
• so we should have (w t s E) R E l
• or

15
Sheet resistance

• consider a block of uniform conducting material

• if the width and length are the same (i.e., its
  a square)

or

• Rs is the sheet resistance of the material
• example for a uniformly doped piece of
  semiconductor

16
Common conducting materials

• silver
• highest conductivity of all metals
• conductivity s 63x106 / mohm
• 1.6 ????cm, 0.02 ????? _at_ 1?? m
• copper
• primarily used as an interconnect/wiring material
• conductivity s 59.6x106 / mohm
• 1.7 ????cm, 0.02 ????? _at_ 1?? m
• gold
• very inert adheres poorly
• conductivity s 45.2x106 / mohm
• 2.5 ??? cm, 0.025 ????? _at_ 1? m
• aluminum
• common interconnect/wiring material
• conductivity s 37.7x106 / mohm
• 3 ????cm, 0.03 ????? _at_ 1?? m
• excellent adherence to oxides
• good ohmic contacts to Si

17
Other conducting materials

• polysilicon
• used mainly for gates in MOSFETs
• 300 ????cm, ?????????_at_ 1??m
• high temperature stability
• refractory metals
• chromium, palladium, tungsten
• very high temperature stability
• used mainly as reaction barriers in ICs
• refractory silicides
• moderate resistivities
• WSi2 70 ??cm
• Pd2Si 30 ???cm
• good high temperature stability
• used with poly as gate metallization in MOSFETs

18
Semiconductors

• for a typical band gap of about 1eV only about
  1010 electrons per cubic centimeter are thermally
  excited into the conduction band at room
  temperature
• this relatively small number of carriers is
  responsible for conduction in pure (aka
  intrinsic) semiconductors
• exciting electrons into the conduction band
  leaves behind holes in the valence band, which
  may also conduct electricity
• impurities can be added to the material to
  significantly alter the number of carriers, and
  hence the conductivity (or equivalently, the
  resistivity)
• donor (n-type) impurities introduce states near
  the conduction band, allowing electrons from
  normally filled sites easy access to the
  conduction band
• acceptor (p-type) impurities introduce states
  just above the valence band, allowing the easy
  formation of holes.
• even one impurity atom out of every billion
  host atoms can have a significant effect on
  conductivity

19
Electrical Conduction in Semiconductors

• Semiconductors
• depending on what kind of impurities are
  incorporated, the charge carriers in
  semiconductors may be either electrons (called
  n-type material) or holes (called p-type
  material) compared to metals (which have only
  electrons), semiconductor have fairly high
  resistance
• Electrons
• negative charge, flow downhill
• Holes
• positive charge, flow uphill

20
Other conducting materials Electrolytes

• electric currents in electrolytes are flows of
  electrically charged atoms (ions)
• example if an electric field is placed on a
  solution of Na and Cl, the sodium ions will
  move towards the negative electrode (anode), and
  the chlorine ions will move towards the positive
  electrode (cathode). If the conditions are right,
  redox reactions will take place, which release
  electrons from the chlorine, and allow electrons
  to be absorbed into the sodium.
• in water ice and in certain solid electrolytes,
  flowing protons constitute the electric current
• because the mass of the carrier is much larger
  than the mass of an electron, these materials are
  usually more resistive, and have very different
  behavior than metals when things change wrt time

21
Gases and plasmas

• in neutral gases, electrical conductivity is very
  low.
• act as a dielectric or insulator,
• until the electric field reaches a breakdown
  value, stripping the electrons from the atoms
  thus forming a plasma.
• plasma allows the conduction of electricity,
  forming a spark, arc or lightning
• for ordinary air below the breakdown field, the
  dominant source of electrical conduction is via
  mobile particles of water, which shuttle electric
  charge, forming a current
• a plasma is the state of matter where some of the
  electrons in a gas are stripped or "ionized" from
  their parent molecules or atoms
• plasmas can be formed by
• high temperature
• application of an electric field in excess of the
  breakdown strength
• electrical conduction in a plasma is due to the
  motion of both the electrons and the
  positively-charged ions

22
Conductors in electrostatics (dc)

• the interior of a conductor is (almost) always
  space charge neutral
• there is a large density of negatively charged
  free electrons,
• BUT there is also an equal but opposite charge
  (positive) density that is fixed in space
• the atoms in the material
• for us, well always assume that the numbers are
  exactly equal, so there is no NET charge inside a
  conductor
• consider a piece of conductor (metal) that is
  insulated from external connections, no time
  variations
• current must be zero
• current density must be zero
• by Ohms law, electric field inside must be
  zero!!!!

23
Properties of conductors in electrostatics

• no currents tells us
• zero electric field inside, otherwise there would
  be a current!
• E tangent on the outside surface of a conductor
  must be zero
• if not thered be a current
• well look at this in detail on the next slide
• the surface of a conductor is an equipotential!!
• if its not, there would have to be a field along
  the surface, and hence there would be current
  flow!
• can there be a field perpendicular to the surface
  of a conductor?
• we need to do this with a picture and Gausss law
• charge can be induced on the surface of a
  conductor
• well derive this a few slides from now
• E must be perpendicular to the surface of a
  conductor!!!

24
Tangential electric field at the surface of a
conductor

• imagine there is some E at the surface of a
  conductor
• lets find the voltage going around the path
  a-b-c-d-a

• we already know this must be zero
• E is a conservative field

• all the pieces add up to zero since V around a
  closed loop is zero

25
Tangential electric field at the surface of a
conductor

• E at the surface of a conductor
• voltage around closed loop must be zero

• lets let the path get small enough that E
  doesnt vary much along each side

• if there is zero current (purely electroSTATIC)
  then E inside 0

26
Tangential electric field at the surface of a
conductor

• E at the surface of a conductor
• weve let the path get small enough that E
  doesnt vary much along each side
• and if there is zero current (purely
  electrostatic) then E inside 0

• when no currents are flowing, the component of
  the electric field tangent to the surface of a
  conductor must be zero

27
Normal electric field at the surface of a
conductor

• image there is some D at the surface of a
  conductor
• this time lets construct a gaussian surface and
  look at flux
• Gausss law
• again, lets select a small enoughsurface that D
  doesnt vary much
• for the sides, the D?dS would pick out the
  component of D tangent to the metal surface
• but we already know Dtan is zero (since Etan
  0)!
• for the bottom, inside the conductor
• for statics (no current flow) we know the field
  is zero inside!
• all that is left is the top!!
• D?dS picks out the normal component of D

28
Normal electric field at the surface of a
conductor

• image there is some D at the surface of a
  conductor
• if there is no current flow we know the field is
  zero inside!
• Gausss law
• all that is left is the top!!
• D?dS picks out the normal component of D

29
Electrostatic boundary conditions for conductors

• in the absence of current flow we have the
  following conditions for a conductor
• the electric field (and D as well) inside is
  identically zero
• at the surface of a conductor, the field is
  everywhere normal to that surface
• the conductor is an equipotential
• at the surface of a conductor, any normal
  component of the field induces a surface charge
  that is proportional to the field strength

30
Now what? the opposite of a metal is a

• a large bandgap usually gives a smaller number of
  free charged carriers
• dielectrics large bandgap, nfree 0 cm-3