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VIEWS: 325 PAGES: 32

 Electrical conductivity
    Energy bands in solids
    Band structure and conductivity
 Semiconductors
    Intrinsic semiconductors
    Doped semiconductors
         o n-type materials
         o 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

 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),
    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: materials with zero resistivity     2
     at very low temperature
some representative resistivities ():
   R = L/A, R = resistance, L = length, A = cross
    section area; resistivity at 20o C
  resistance(in )         (L=1m, 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      3
 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:
    o 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;
    o 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
    o 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;
    o semiconductors: gap is small - about 0.1 to 1 eV;
    o conductors: valence band only partially filled, or (if it is filled), the
      next allowed empty band overlaps with it
Band structure and conductivity

 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
 at T > 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”.     6
Intrinsic silicon:

  o “doped semiconductor”: (also “impure”, “extrinsic”) =
    semiconductor with small admixture of trivalent or
         pentavalent atoms;

n-type material

  donor (n-type) impurities:
    o dopant with 5 valence electrons (e.g. P, As, Sb)
    o 4 electrons used for covalent bonds with surrounding Si atoms,
      one electron “left over”;
    o left over electron is only loosely bound  only small amount of
      energy needed to lift it into conduction band (0.05 eV in Si)
    o  “n-type semiconductor” has conduction electrons, very few
      holes (just the few intrinsic holes)
    o 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.

 p-type material
 acceptor (p-type)
  o 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
  o “p-type semiconductor”
    has mobile holes, very
    few mobile electrons
    (only the intrinsic ones).

 advantages of doped semiconductors:
   o can”tune” conductivity by choice of doping fraction
   o can choose “majority carrier” (electron or hole)
   o can vary doping fraction and/or majority carrier within piece
     of semiconductor
   o can make “p-n junctions” (diodes) and “transistors”             9
n – type material   p– type material

   Majority and Minority Carriers

 n-type material:
   o majority carrier: electrons
   o minority carrier: holes
 p-type material:
   o majority carrier: holes
   o minority carrier: electrons    11
   o p-n junction = semiconductor in which impurity changes abruptly from
     p-type to n-type ;
   o “diffusion” = movement due to difference in concentration, from
     higher to lower concentration;
   o in absence of electric field across the junction, holes “diffuse”
     towards and across boundary into n-type and capture electrons;
   o electrons diffuse across boundary, fall into holes (“recombination of
     majority carriers”);  formation of a “depletion region” (= region
     without free charge carriers) around the boundary;
   o 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.

                   p-n junction
 Formation of depletion region in p-n junction:


   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:
     o the direction of the electric field is from p-side
       towards n-side
     o  p-type charge carriers (positive holes) in p-side are
       pushed towards and across the p-n boundary,

     o n-type carriers (negative electrons) in n-side are
       pushed towards and across n-p boundary
                      current flows across p-n boundary         14
           Forward biased pn-junction
 Depletion region and potential barrier reduced


     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-
     diodes used in “rectifiers”, to convert ac
      voltage to dc.                                 16
           Reverse biased diode

   Depletion region becomes wider,
     barrier potential higher

 (bipolar) transistor =
  combination of two
  diodes that share
  middle portion, called
  “base” of transistor;
  other two sections:
  “emitter'' and
 usually, base is very
  thin and lightly doped.

        two kinds of bipolar transistors: pnp and npn
        “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;
       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
           Transistor operation
 Number of holes captured depends on the number of n-
  type carriers in the base
   o 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;
   o base current is usually very small;
   o 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
 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.

      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.

 “Forward biased” pn junction – diode:
    “forward biased”, i.e. p-side +, n-side –
    current flows  charge-carriers (electrons and electron
     holes) flow into the junction from the electrodes
    electron meets a hole  falls into a lower energy level 
     releases energy in the form of a photon (light)
    wavelength of emitted light depends on the materials
     used to make the diode (h = gap energy)


    Red: aluminum gallium arsenide (AlGaAs).
    Blue: indium gallium nitride (InGaN)
    green: aluminum gallium phosphide (AlGaP)
    "White" light by combining light from red,
    green, and blue (RGB) LEDs, or by coating a
     blue LED with yellow phosphor.

 mobile electrons in conductor move through
  lattice of atoms or ions that vibrate (thermal
 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       24
  temperatures below 4.12 K.
 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:
   o can cool with (common and cheap) liquid nitrogen rather
     than with (rare and expensive) liquid helium;
   o much easier to reach and maintain LN temperatures (77
     K) than liquid Helium temperatures (few K).

      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
 examples: In, Sn, V, Mo, Nb-Zr, Nb-Ge, Nb-Ti
  alloys                                                26
          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
   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):
       o create images of human body to detect tumors, etc.;
       o need uniform magnetic field over area big enough to cover
       o can be done with conventional magnets, but s.c. magnets
         better suited - hundreds in use
    magnetic levitation - high speed trains??                       27
    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”.                   28
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
 (explanation given by John Bardeen, Leon N.
  Cooper, J. Robert Schrieffer, 1957)             29
    Hall Effect
         Edwin Hall (1879):
           magnetic field
            perpendicular to current
             potential difference
            perpendicular to current
            and magnetic field
         Hall effect measurements
         allow determination of
         charge carrier density in
         metals and semiconductors

    magnetic field exerts                  Hall effect explanation
    force on moving charge
    carrier of charge q (Lorentz
    force) in the lateral F  qvB

   Lateral displacement of
    charges  accumulation of
    charges  electric field                                               I
    (Hall field) perpendicular                        t
    to current and magnetic                                            w
    field direction            FE  qEH
    force due to Hall field
    opposite to Lorentz force
   Equilibrium reached when                                      EH
                                          qvB  qEH          v
    magnitude of force due to                                     B
    Hall field = mag. of
    Lorentz force  get drift             J  nqv 
    speed v                                             B
   Current density J, density
    of charge carriers n, Hall                   EH   1
                                          RH       
    coefficient RH                               JB nq                         31
               Hall effect measurements
 In the lab, we measure current I,
  B-field, Hall voltage VH, size
  (width w, height t) of sample

    VH  EH w       I  JA  Jwt
 calculate RH from measurements,
  and assume |q| = e  find n.
 sign of VH and thus RH tells us
  the sign of q                               I
          EH   V /w     V t  1
   RH        H        H 
          JB I / wt B IB nq


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