Download ELECTRICAL CONDUCTIVITY - FSU Physics Department

Conductivity
 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
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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: materials with 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 )
aluminum
brass
copper
platinum
silver
carbon
germanium
silicon
porcelain
teflon
blood
fat
(L=1m, diam =1mm)
2.8x10-8
8x10-8
1.7x10-8
10x10-8
1.6x10-8
3.5x10-5
0.45
 640
1010 - 1012
1014
1.5
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resistivity
in  m
3.6x10-2
10.1x10-2
2.2x10-2
12.7x10-2
2.1x10-2
44.5
5.7x105
 6x108
1016 - 1018
1020
1.9x106
3
3x107
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:
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
structure:
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
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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 > 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:
o “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:
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.
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p-type material
 acceptor (p-type)
impurities:
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”
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n – type material
p– type material
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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
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DIODES AND TRANSISTORS
 p-n JUNCTION:
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.
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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:
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
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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 nside 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 nside
 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 ntype 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
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 ptype 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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LEDs
 “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)
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LEDs




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.
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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:
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).
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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):
o create images of human body to detect tumors, etc.;
o need uniform magnetic field over area big enough to cover
person;
o 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
I
 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
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




magnetic field exerts
force on moving charge
carrier of charge q (Lorentz
force) in the lateral F  qvB
B
direction:
Lateral displacement of
charges  accumulation of
charges  electric field
(Hall field) perpendicular
to current and magnetic
FE  qEH
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
Hall effect explanation
I
t
qvB  qEH
J  nqv 
RH 
w
v
EH
B
nqEH
B
EH
1

JB nq
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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
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
t
RH 
w
EH
V /w
V t 1
 H
 H 
JB I / wt B IB nq
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