Conductivity

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Conductivity

• Electrical conductivity
• Energy bands in solids
• Band structure and conductivity
• Semiconductors
• Intrinsic semiconductors
• Doped semiconductors
• n-type materials
• p-type materials
• Diodes and transistors
• p-n junction
• depletion region
• forward biased p-n junction
• reverse biased p-n junction
• diode
• bipolar transistor
• operation of bipolar pnp transistor
• FET
• Superconductivity
• Hall effect lab experiment

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ELECTRICAL CONDUCTIVITY

• in order of conductivity superconductors,
  conductors, semiconductors, insulators
• conductors material capable of carrying electric
  current, i.e. material which has mobile charge
  carriers (e.g. electrons, ions,..) e.g. metals,
  liquids with ions (water, molten ionic
  compounds), plasma
• insulators materials with no or very few free
  charge carriers e.g. quartz, most covalent and
  ionic solids, plastics
• semiconductors materials with conductivity
  between that of conductors and insulators e.g.
  germanium Ge, silicon Si, GaAs, GaP, InP
• superconductors certain materials have zero
  resistivity at very low temperature.

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resistivities

• some representative resistivities (?)
• R ?L/A, R resistance, L length, A cross
  section area resistivity at 20o C

resistance(in ?) (L1m, diam 1mm) resistivity
in ? m aluminum 2.8x10-8 3.6x10-2 brass
?8x10-8 10.1x10-2 copper 1.7x10-8 2.2x10-
2 platinum 10x10-8 12.7x10-2 silver
1.6x10-8 2.1x10-2 carbon 3.5x10-5
44.5 germanium 0.45 5.7x105 silicon ?
640 ? 6x108 porcelain 1010 - 1012 1016 -
1018 teflon 1014 1020 blood 1.5 1.9x106
fat 24 3x107
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ENERGY BANDS IN SOLIDS

• In solid materials, electron energy levels form
  bands of allowed energies, separated by
  forbidden bands
• valence band outermost (highest) band filled
  with electrons (filled all states occupied)
• conduction band next highest band to valence
  band (empty or partly filled)
• gap energy difference between valence and
  conduction bands, width of the forbidden band
  
• Note
• electrons in a completely filled band cannot
  move, since all states occupied (Pauli
  principle) only way to move would be to jump
  into next higher band - needs energy
• electrons in partly filled band can move, since
  there are free states to move to.
• Classification of solids into three types,
  according to their band structure
• insulators gap forbidden region between
  highest filled band (valence band) and lowest
  empty or partly filled band (conduction band) is
  very wide, about 3 to 6 eV
• semiconductors gap is small - about 0.1 to 1
  eV
• conductors valence band only partially filled,
  or (if it is filled), the next allowed empty band
  overlaps with it

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Band structure and conductivity
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INTRINSIC SEMICONDUCTORS

• semiconductor material for which gap between
  valence band and conduction band is small (gap
  width in Si is 1.1 eV, in Ge 0.7 eV).
• at T 0, there are no electrons in the
  conduction band, and the semiconductor does not
  conduct (lack of free charge carriers)
• at T gt 0, some fraction of electrons have
  sufficient thermal kinetic energy to overcome the
  gap and jump to the conduction band fraction
  rises with temperature e.g. density of
  conduction electrons in Si 0.9x1010/cm3
  at 20o C (293 K) 7.4x1010/cm3 at 50o C (323
  K).
• electrons moving to conduction band leave hole
  (covalent bond with missing electron) behind
  under influence of applied electric field,
  neighboring electrons can jump into the hole,
  thus creating a new hole, etc. ? holes can
  move under the influence of an applied electric
  field, just like electrons both
  contribute to conduction.
• in pure Si and Ge nb. of holes (p-type charge
  carriers) nb. of conduction electrons
  (n-type charge carriers)
• pure semiconductors also called intrinsic
  semiconductors.

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• Intrinsic silicon
• DOPED SEMICONDUCTORS
• doped semiconductor (also impure,
  extrinsic) semiconductor with small admixture
  of trivalent or pentavalent atoms

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n-type material

• donor (n-type) impurities
• dopant with 5 valence electrons (e.g. P, As, Sb)
  
• 4 electrons used for covalent bonds with
  surrounding Si atoms, one electron left over
  
• left over electron is only loosely bound ? only
  small amount of energy needed to lift it into
  conduction band (0.05 eV in Si)
• ? n-type semiconductor has conduction
  electrons, very few holes (just the few intrinsic
  holes)
• example doping fraction of 10-8 Sb in Si yields
  about 5x1016 conduction electrons per cubic
  centimeter at room temperature, i.e. gain of
  5x106 over intrinsic Si.

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p-type material

• acceptor (p-type) impurities
• dopant with 3 valence electrons (e.g. B, Al, Ga,
  In) ? only 3 of the 4 covalent bonds filled ?
  vacancy in the fourth covalent bond ? hole
  
• p-type semiconductor has mobile holes, very few
  mobile electrons (only the intrinsic ones).

• advantages of doped semiconductors
• cantune conductivity by choice of doping
  fraction
• can choose majority carrier (electron or hole)
  
• can vary doping fraction and/or majority carrier
  within piece of semiconductor
• can make p-n junctions (diodes) and
  transistors

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n type material
p type material
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Majority and Minority Carriers

• n-type material
• majority carrier electrons
• minority carrier holes
• p-type material
• majority carrier holes
• minority carrier electrons

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DIODES AND TRANSISTORS

• p-n JUNCTION
• p-n junction semiconductor in which impurity
  changes abruptly from p-type to n-type
• diffusion movement due to difference in
  concentration, from higher to lower
  concentration
• in absence of electric field across the junction,
  holes diffuse towards and across boundary into
  n-type and capture electrons
• electrons diffuse across boundary, fall into
  holes (recombination of majority carriers) ?
  formation of a depletion region ( region
  without free charge carriers) around the
  boundary
• charged ions are left behind (cannot move)
• negative ions left on p-side ? net negative
  charge on p-side of the junction
• positive ions left on n-side ? net positive
  charge on n-side of the junction
• ? electric field across junction which prevents
  further diffusion.

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p-n junction

• Formation of depletion region in p-n junction

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DIODE

• diode biased p-n junction, i.e. p-n junction
  with voltage applied across it
• forward biased p-side more positive than
  n-side
• reverse biased n-side more positive than
  p-side
• forward biased diode
• the direction of the electric field is from
  p-side towards n-side
• ? p-type charge carriers (positive holes) in
  p-side are pushed towards and across the p-n
  boundary,
• n-type carriers (negative electrons) in n-side
  are pushed towards and across n-p boundary ?
  current flows across p-n boundary

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Forward biased pn-junction

• Depletion region and potential barrier reduced

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Reverse biased diode

• reverse biased diode applied voltage makes
  n-side more positive than p-side ? electric
  field direction is from n-side towards p-side ?
  pushes charge carriers away from the p-n boundary
  ? depletion region widens, and no current flows
• diode conducts only when positive voltage
  applied to p-side and negative voltage to n-side
• diodes used in rectifiers, to convert ac
  voltage to dc.

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Reverse biased diode

• Depletion region becomes wider, barrier
  potential higher

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TRANSISTORS

• (bipolar) transistor combination of two diodes
  that share middle portion, called base of
  transistor other two sections emitter'' and
  collector
• usually, base is very thin and lightly doped.
  

• two kinds of bipolar transistors pnp and npn
  transistors
• pnp means emitter is p-type, base is n-type,
  and collector is p-type material
• in normal operation of pnp transistor, apply
  positive voltage to emitter, negative voltage to
  collector

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operation of pnp transistor

• if emitter-base junction is forward biased,
  holes flow from battery into emitter, move into
  base
• some holes annihilate with electrons in n-type
  base, but base thin and lightly doped ? most
  holes make it through base into collector,
  
• holes move through collector into negative
  terminal of battery i.e. collector current
  flows whose size depends on how many holes have
  been captured by electrons in the base

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Transistor operation

• Number of holes captured depends on the number of
  n-type carriers in the base
• Number of n-type carriers can be controlled by
  the size of the current (the base current) that
  is allowed to flow from the base to the emitter
• base current is usually very small
• small changes in the base current can cause a big
  difference in the collector current
  
• transistor acts as amplifier of base current,
  since small changes in base current cause big
  changes in collector current.
• transistor as switch if voltage applied to base
  is such that emitter-base junction is
  reverse-biased, no current flows through
  transistor -- transistor is off
• therefore, a transistor can be used as a
  voltage-controlled switch computers use
  transistors in this way.

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Field-effect transistor (FET)

• In FETs, current through channel from source
  to drain is controlled by voltage (electric
  field) applied to the gate
• in a pnp FET, current flowing through a thin
  channel of n-type material is controlled by the
  voltage (electric field) applied to two pieces of
  p-type material (gate) on either side of the
  channel (current depends on electric field).
• Advantage of FET over bipolar transistor very
  small gate current small power consumption
• Many different kinds of FETs
• FETs are the kind of transistor most commonly
  used in computers.

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SUPERCONDUCTIVITY

• mobile electrons in conductor move through
  lattice of atoms or ions that vibrate (thermal
  motion)
• cool down conductor ? less vibration ? easier
  for electrons to get through ? resistivity of
  conductors decreases (i.e. they become better
  conductors) when they are cooled down
• in some materials, resistivity goes to zero below
  a certain critical temperature TC
• these materials called superconductors --
  critical temperature TC different for different
  materials
• no electrical resistance ? electric current, once
  started, flows forever!
• superconductivity first observed by Heike
  Kamerlingh Onnes (1911) in Hg (mercury) at
  temperatures below 4.12 K.

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Superconductors

• many other superconductors with critical
  temperatures below about 20K found by 1970 --
  high TC superconductors (Karl Alex Müller and
  Johannes Georg Bednorz, 1986)
• certain ceramic oxides show superconductivity at
  much higher temperatures since then many new
  superconductors discovered, with TC up to 125K.
  
• advantage of high TC superconductors
• can cool with (common and cheap) liquid nitrogen
  rather than with (rare and expensive) liquid
  helium
• much easier to reach and maintain LN temperatures
  (77 K) than liquid Helium temperatures (few K).

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Properties of superconductors

• electrical resistivity is zero (currents flowing
  in superconductors without attenuation for more
  than a year)
• there can be no magnetic field inside a
  superconductor (superconductors expel magnetic
  field -- Meissner effect)
• transition to superconductivity is a phase
  transition (without latent heat).
• about 25 elements and many hundreds of alloys and
  compounds have been found to be superconducting
  
• examples In, Sn, V, Mo, Nb-Zr, Nb-Ge, Nb-Ti
  alloys

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applications of superconductors

• superconducting magnets
• magnetic fields stronger, the bigger the current
  - conventional magnets need lots of power and
  lots of water for cooling of the coils
  
• s.c. magnets use much less power (no power needed
  to keep current flowing, power only needed for
  cooling)
• most common coil material is NbTi alloy liquid
  He for cooling
• e.g. particle accelerator Tevatron at Fermi
  National Accelerator Laboratory (Fermilab)
  uses 990 superconducting magnets in a ring with
  circumference of 6 km, magnetic field is 4.5
  Tesla.
• magnetic resonance imaging (MRI)
• create images of human body to detect tumors,
  etc.
• need uniform magnetic field over area big enough
  to cover person
• can be done with conventional magnets, but s.c.
  magnets better suited - hundreds in use
• magnetic levitation - high speed trains??

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explanation of superconductivity -- 1

• Cooper pairs
• interaction of the electrons with the lattice
  (ions) of the material, ? small net effective
  attraction between the electrons (presence of
  one electron leads to lattice distortion, second
  electron attracted by displaced ions)
• this leads to formation of bound pairs of
  electrons (called Cooper pairs) (energy of
  pairing very weak - thermal agitation can throw
  them apart, but if temperature low enough, they
  stay paired)
• electrons making up Cooper pair have momentum and
  spin opposite to each other net spin 0 ?
  behave like bosons.

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explanation of superconductivity -- 2

• unlike electrons, bosons like to be in the same
  state when there are many of them in a given
  state, others also go to the same state
  
• nearly all of the pairs locked down in a new
  collective ground state this ground
  state is separated from excited states by an
  energy gap
• consequence is that all pairs of electrons move
  together (collectively) in the same state
  electron cannot be scattered out of the regular
  flow because of the tendency of Bose particles to
  go in the same state ? no resistance
• (explanation given by John Bardeen, Leon N.
  Cooper, J. Robert Schrieffer, 1957)

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Hall Effect

• Edwin Hall (1879)
• magnetic field perpendicular to current ?
  potential difference perpendicular to current and
  magnetic field
• allows determination of charge carrier density
  in metals and semiconductors

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Hall effect explanation

• magnetic field exerts force on moving charge
  carrier of charge q (Lorentz force) in the
  lateral direction
• Lateral displacement of charges ? accumulation of
  charges ? electric field (Hall field)
  perpendicular to current and magnetic field
  direction
• force due to Hall field opposite to Lorentz
  force
• Equilibrium reached when magnitude of force due
  to Hall field mag. of Lorentz force ? get
  drift speed v
• Current density J, density of charge carriers n,
  Hall coefficient RH

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Hall effect measurements

• In the lab, we measure current I, B-field, Hall
  voltage VH, size (width w, height t) of sample
• calculate RH from measurements, and assume q
  e ? find n.
• sign of VH and thus RH tells us the sign of q