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MSE 141 ELECTRONIC MATERIALSElectrical Conduction in Materials
Dr. Benjamin O. Chan
Associate Professor
January 2012
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ELECTRONS IN A CRYSTALHow are the electrons distributed
among the available energy levels?Probabilistic distribution expected
We can never specify the location and kinetic energy of each individual electron (out of ~ 1022 cm-3)
Fermi Distribution Applies to fermions
Pauli Exclusion Principle applies Bosons require Bose-Einstein distribution
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FERMI DISTRIBUTION FUNCTION
kB = Boltzmann’s constant = 8.616 x 10-5 eV/K
EF = Fermi energy Metals: Highest occupied energy at T = 0 K Semiconductors: energy level with a 0.5 probability
of occupation chemical potential
T = absolute temperature
1exp
1)(
TkEE
EF
B
F
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FERMI DISTRIBUTION FUNCTION (T=0 K)
Ground StateT = 0 KE = EF
Satisfactory for MetalsFor semiconductors,
EF is usually in the energy gap: these energies have no chance of being occupied!0
1
1
1)(
e
EEF F
110
1
1
1)(
e
EEF F
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FERMI DISTRIBUTION FUNCTION (T>0 K)
At room temperature, DE~1%
For E >> EF
Boltzmann tail
2
1
11
1
1
1)(
0
e
EF F
Tk
EEEF
B
Fexp)(
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DENSITY OF STATES Distribution of energy levels over a band Consider lower part of valence band (e.g. 4s
for Cu) Electrons essentially free
Assume electrons are confined in a square potential well
where nx, ny and nz are the principal quantum numbers and a is now the length of the crystal
2222
22
2 zyxn nnnma
E
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QUANTUM NUMBER SPACE
nx, ny and nz specify an energy state En
If we take n to be the radius of a sphere containing quantum numbers
The surface of the sphere will contain states yielding the same En
2222zyx nnnn
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NUMBER OF QUANTUM STATESAll points inside sphere represent
states with E < En
Number of states with E En is proportional to the volume of the sphere with radius n
Only the positive octant and integral values of nx, ny and nz are allowed
23
23
22
23 2
63
4
8
1E
man
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DOS Z(E) WITHIN A BAND
Density of states (DOS) = number of energy states per unit energy
where V = a3 is the volume occupied by the electrons
2/1
2/3
22
22
4)( E
ma
dE
dEZ
2/12/3
22
2
4)( E
mVEZ
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SOME NOTES ON DOSE vs Z(E) graph
Not the usual Z vs E !Parabolic graphLower end of the band has fewer
available E values compared to higher energies
Area within the curve = number of states with E En Element area d = Z(E)dE
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POPULATION DENSITYPauli exclusion principle
2 electrons of opposite spin per stateNumber of electrons per unit energy
N(E)
Where F(E) is the Fermi distribution function
N(E) = population density
)()(2)( EFEZEN
1
2
2)( /
2/12/3
22
TkEE BFe
EmVEN
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POPULATION DENSITY N(E)
As T0 and E<EF, N(E)=2Z(E)
T0 and E<EF, N(E) is smeared out
Area within the curve = number of electrons, N*, that have an energy E<En
For an energy interval E to E + dE dEENdN )(*
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CALCULATING EF
For T0 and E<EF
Defining N’= number of electrons per unit volume and solving for EF, we get
Note: N* does not change with T
FF EE
dEEmV
dEENN0
2/12/3
220
* 2
2)(
m
NmV
NEF 2
32
32
3/2223/2*
2
2/32/3
22* 2
3 FEmV
N
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CRYSTAL DOS Z(E) WITHIN A BAND
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CRYSTAL Z(E) WITHIN FBZ
For low E, constant energy curves are circlesFree-electron-likeZ(E) is parabolic
For larger energies, Z(E)0 as FBZ boundary is reached
Largest number of energy states available is near the center of a band
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SIMPLIFIED ENERGY BANDS
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CONSEQUENCES OF BAND MODEL
Solids in which the highest filled band is completely occupied by electrons are insulators Including s-band?
Alkali metals (monovalent) have half-filled valence band
Bivalent metals should be insulatorsClosed s-shell
Semiconductor4 valence electrons: valence band full!
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BIVALENT METAL
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BIVALENT METAL Is it insulating or conducting?
p band overlaps with s band Weak binding forces of valence electrons on their
atomic nuclei Upper s band electrons empty into low levels of p band Partially filled energy bands: metal!
Semi-metal Are alkali metals more conducting than bivalent
metals?
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EFFECTIVE MASS Empirical evidence shows that mass of
electron in a crystal can be larger or smaller compared to the mass of a free electron We’ll call this the effective mass m* of the
electron m*/mo describes deviation from mo
Usually bigger than 1 (so what?) Determine an expression for m*
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VELOCITY OF AN ELECTRON IN AN ENERGY BAND
Wave packet velocity given angular velocity w and wavenumber k
Acceleration a
For a free electron
This gives us
dk
dE
dk
hEd
dk
d
dk
dg
1)/2()2(v
dt
dk
dk
Ed
dt
da g
2
21v
kp
dt
dk
dt
dp
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EFFECTIVE MASS Substituting,
Applying Newton’s Second Law
Comparing the two previous equations
m* is inversely proportional to the curvature of the energy band!!
*m
Fa
Fdk
Ed
dt
dp
dk
Eda
2
2
22
2
2
11
1
2
22*
dk
Edm
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ENERGY BAND STRUCTURES REVISITED High curvature = low mass Low curvature = high mass Usually at the center or boundary of a BZ
m*/mo can be as low as 1% Multiple curves for same k leads to multiple
values for effective mass More than one type of carrier available!
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AL BAND STRUCTURE
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CU BAND STRUCTURE
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SI BAND STRUCTURE
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GAAS BAND STRUCTURE
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MORE NOTES ON ENERGY BAND DIAGRAMS
Ideal electron band within FBZ
Calculate first derivative and reciprocal of second derivative
m* is small and positive near center of FBZ and increases for larger values of k
Electrons in the upper part have a negative mass Defect electron/ electronhole Particle travels opposite to an
electron under an applied electric field
Important in semiconductors Effective mass is a tensor!
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PROBLEM SET 2Hummel, 3rd editionChapter 6
5, 8, 12Chapter 7
1, 3, 8, 9
Due Jan. 26, 2012
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ELECTRICAL CONDUCTION IN METALS AND ALLOYS
Electrical Conductivity The ability of materials to let charge carriers
move through it Units: Siemens/m = 1/Wm Resistivity is the inverse (Wm)
Insulators/Dielectrics Unable to conduct electricity High resistivity
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ROOM TEMPERATURE CONDUCTIVITY
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CURRENT DENSITY AND ELECTRIC FIELD Current Density
Flow of charge per unit time per unit area Units: A/m2
Electric Field Region of influence of a charge carrier Units: V/m
In general, the relationship between the current density j across a material to the
applied electric field e across it is
Where s is called the conductivity of the material This equation does not assume s to be constant!
j
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OHM’S LAW If s is constant, we get the general form of
Ohm’s Law
Hey, nothing changed!! It’s true the equation did not change externally,
but s suffered a transformation, from being a function to becoming constant
j
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OHM’S LAW Consider the material to be a cylinder of
length l and cross-sectional area A
Substituting into Ohm’s law equation,
Solving for I,
A
l
lVAIj ab //
l
V
A
I ab
abVl
AI
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CONDUCTANCE GDefining the conductance G to be
Ohm’s Law becomes
If we set G=1/R, R being the resistance of the material, we get
Or, the more familiar form is
l
AG
abGVI
abGVI
IRVab
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RESISTANCE AND RESISTIVITYThe resistance R of the material is
then defined as
with r=1/s, the resistivity of the material R is due to collisions with lattice atoms
Crystal defects tend to increase R More collisions take place!
Particle Picture
A
l
A
lR
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WAVE PICTURE OF RR is due to scattering of charge
carriers by atomsAtoms absorb energy of incoming wave
and re-emit them in the form of spherical waves
Treat atoms as oscillators Periodic structures: waves propagating in the
forward direction are “in-phase” (constructive interference)
electron wave passes without hindrance through crystal (coherent scattering)
Non-periodic structures: incoherent scattering
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CAUTION!R, G, r and s (while being constant) are
temperature dependent!
R tends to increase (G decreases) with increasing T
What exceptions can you think of?
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CLASSICAL ELECTRICAL CONDUCTION THEORYPre-QM periodDrude: consider the valence electrons
(in a metal) as a gas or plasmaMonovalent atoms contribute one electron
per atom Na=number of atoms=number of charge
carriers=N
Where No is Avogadro’s number, d the density of the material and M is the atomic mass (monatomic crystal)
M
NNN o
a
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FREE ELECTRON MOTION
Electrons are accelerated by an applied electric field
For constant e, acceleration is constant v builds up to infinity!
When e =0, v = constant This can happen only in superconductors!
This is not what we observe…
edt
dm v
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ELECTRON PATH THROUGH THE CONDUCTOR
Electrons suffer intermittent collisions with lattice ions
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EFFECT OF COLLISIONSDrag: let g = damping coefficient
Steady state case dv/dt = 0
where vf = drift velocity
edt
dm γvv
γvv
ef
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ELECTRON VELOCITY
Modified Differential Equation
Solution
Average time between collisions t
e
e
dt
dm
f
vv
v
fmte
f e v1vv
em fvτ
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CONDUCTIVITY AND T
Current density
Conductivity
Conductivity is large for large N and t Mean free path l
vNeJ
m
Ne 2σ σv eN f
fvl
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EFFECT OF TEMPERATURE AND LATTICE DEFECTS
As T increases, atoms vibrate more about their equilibrium positionst decreases and s
decreasesMore collision sites
(specially for point and line defects)t decreases and s
decreases
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QUANTUM MECHANICAL CONSIDERATIONS
Plot velocities in velocity spaceMomentum spacek-space
Since there is no preferential velocity, the range of velocities form a disk with radius vFvF = Fermi velocity, the maximum velocity
the electrons can assume In 3-D, the velocity distribution forms a
sphere, the surface of which we call a Fermi surface
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VELOCITY DISTRIBUTION OF ELECTRONS
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MORE ON VELOCITY…At equilibrium, all points inside the
Fermi sphere are occupiedVelocities cancel each other out resulting
in zero net velocity for the whole groupUnder an applied electric field, the
Fermi sphere is displaced opposite the field directionNet velocity gain resultsGreat majority of electrons still cancel
each other pair-wiseUncompensated electrons cause the
observed current
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SO?Not all conduction electrons contribute
to currentTheir drift velocity is very close to the
Fermi velocity (which is relatively high!)
Classical Mechanics assumes that all conduction electrons contribute to current and move with a modest velocity
How do you compare vf and vF?
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POPULATION DENSITY
There are more electrons near EF
Only a little energy DE is needed to raise a substantial number of electrons into slightly higher states
Velocity of excited electrons only slightly higher than vF Mean velocity remains vF
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QM CONDUCTIVITY Taking v=vF and N=N’=the number of
displaced electrons as shown by the previous figure, Ohm’s law gives
OrEEeNNej FFF )(vv
kdk
dEEeNj FF )(v
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QM CONDUCTIVITY For free electrons
Substituting into equation for j
We now calculate the displacement Dk of the Fermi sphere in k-space. We note that
edt
dk
dt
dp
dt
dmF v
kEeNj F )(v2F
Fm
pk
mdk
dEv
2
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QM CONDUCTIVITYThus,
And
Where t is the relaxation time. Thus,
)(v 22F FENej
e
te
k
dte
dk
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QM CONDUCTIVITY Assume electric field vector points in the –
v(k)x direction
v(k)y components cancel out pairwise
Only projections of v(k) on the +v(k)x axis contribute to the current
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2D VELOCITY SPACE
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QM CONDUCTIVITY Summing the velocities in the first and fourth
quadrants
d
ENej FF22/
2/
2 )cosv()(
2/
2/
22
2 cos)(
dv
ENe FF
22
v)(2 FFENe
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QM CONDUCTIVITY For a spherical Fermi surface
Since s = j/e
22
v)(3 FFENe
j
22
v)(3 FFENe
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SOME NOTESConductivity depends on the square of
the Fermi velocity, the relaxation time and the population density at EF
Not all free electrons participate in the conduction processOnly those near EF
Monovalent vs. bivalent metal Insulators and semiconductors
N(EF) effectively zero!
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QM VS CLASSICALNf constantN(E) changes very little with T t decreases with increasing T
Large rate of collisions between drifting electrons and vibrating lattice atoms
s decreases!Both QM and classical models
accurately describe the temperature dependence of resistivity
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RESISTIVITY AND TEMPERATURE For metals, r decreases linearly with T
according to
Where a is the linear coefficient of resistivity Oscillating atoms increase incoherent
scattering of electron waves (electron-atom collisions)
Residual resistivity is due to crystal imperfections Impurities, vacancies, GB’s, dislocations Essentially temperature independent
)](1[ 1212 TT
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RESISTIVITY VS. TEMPERATURE
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MATTHIESSEN’S RULEThe resistivity arises from independent
scattering sources that are additive
rth is the ideal resistivityrres is due to defects and impurities
Number of impurities usually constant Number of vacancies and GB’s can be changed
via heat treatment
resthdefimpth
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RESISTIVITY AND TEMPERATUREAnnealing exercise
Annealing near melting temperature + rapid quenching to room temperature increases r due to frozen in vacancies
Room temperature aging or annealing at slightly elevated temperature annihilates some vacancies, decreasing r
Recrystallization and grain growth can also change r
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RESISTIVITY OF ALLOYS r increases with increasing amount of
solute (Cu-Ni graph)Slope of r vs T graph essentially
constantSmall additions of solute cause linear
shift of r vs T graph to higher rDifference in size alter lattice parameter,
increasing electron scatteringDifference in valence produces local
charge difference, increasing scatteringDifference in electron distribution alters EF
This changes N(E) and thus s
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RESISTIVITY VS. SOLUTE TYPE
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VS. SOLUTE TYPELinde’s Rule
r of dilute single phase alloys increases with the square of the valence difference between solute and solvent constituents
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RESISTIVITY FOR ORDERED OR DISORDERED ALLOYS
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ORDERING Solute atoms are usually
randomly distributed in the solvent resulting in incoherent scattering of electron waves
If they are periodically arranged (e.g., alternately with host atoms for a 50/50 alloy) coherent scattering can occur Dramatic decrease in r results
(increase in mean free path) Only selected alloys show
long-range ordering Cu3Au, CuAu, Au3Mn, …
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HOW TO ACHIEVE ORDER AND DISORDER
OrderAnnealing slightly below order-disorder
transition (395 °C for Cu3Au) followed by moderate cooling rate
Slow cooling from above the transition temperature
DisorderQuench rapidly in ice brine from slightly
above transition temperatureAnneal above transition temperature
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r FOR ISOMORPHOUS ALLOYSMaxima recorded at around 50%
compositionNordheim’s Rule
Residual resistivity depends on the fractional atomic composition (XA and XB) of the constituents
Where C is a material constantHolds strictly only for a few binary systems
Does not take into account changes in DOS arising from composition (alloys containing transition metal)
BABBAA XCXXX
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r FOR TWO-PHASE ALLOYS Mixture rule
Sum of the resistivities of each component taking the volume fractions of each phase into consideration
Crystal structure and the kind of distribution of the phases must also be considered
No maxima observed with respect to composition More like a linear interpolation of the individual
phases
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RESISTORSFor limiting current flow in an electrical
circuit I = V/R
Usual materialCarbon compositesWire-wound (around a ceramic body)
Alloys of high resistivity (e.g., nichrome) Needs to withstand corrosion and high
temperaturesMetal films on glass or ceramic substrates
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JOULE HEATINGHeat dissipation in resistors
Expressed in Watts
Variable resistor = varistorUsually with sliding contact
RIP 2
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SUPERCONDUCTIVITYResistivity becomes immeasurably
small (zero!) below a critical temperature Tc
H.K. Onnes (1911)Hg below 4.15 K
27 elements, numerous alloys, ceramics containing CuO, organic compounds based on Se or S
Tc ranges from 0.01 to 130 KHigh Tc superconductors are hot!
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CRITICAL TEMPERATURES
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SOME NOTES Superconducting transition is reversible Can be considered a separate state
Apart from solid, liquid, gas Higher degree of order
Entropy = 0 !!
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HIGH TC SUPERCONDUCTORS
Cu oxide based ceramics exhibit large Tc (40 K)
1-2-3 (rare earth to alkaline earth to copper ratio)
High Tc>77 K (boiling point of liquid nitrogenUsual coolants: liquid H (20 K) and liquid
He (4 K)Dry Ice (195 K or -79C)
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APPLICATIONS
Strong electromagnets MRI, particle accelerators, electric power storage
devices Currently Nb-Ti or Nb3Sn alloy microns in diameter
embedded in a Cu matrix Lossless power transmission
Once a current is induced in a loop, it continues to flow without significant decay
MAGLEV trains More compact and faster computers Switches: cryotrons
Coiled Nb wire around Ta wire immersed in liquid He
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SUPERCONDUCTORS.ORG
…reports the observation of record high superconductivity near 254 Kelvin (-19C, -2F). This temperature critical (Tc) is believed accurate +/- 2 degrees, making this the first material to enter a superconductive state at temperatures commonly found in household freezers. This achievement was accomplished by combining two previously successful structure types: the upper part of a 9212/2212C and the lower part of a 1223. The chemical elements remain the same as those used in the 242K material announced in May 2009. The host compound has the formula (Tl4Ba)Ba2Ca2Cu7O13+ and is believed to attain 254K superconductivity when a 9223 structure forms (shown below left).
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TRANSITION TEMPERATURE TC
Sharp transition for pure and structurally perfect superconductorsTransition width ~10-5 K for GaTransition can be as wide as 0.1 K for
alloysCeramics display an even wider spread
Varies with atomic mass, ma
Where a is a materials constantHg: Tc varies from 4.185 K to 4.146 K as ma
changes from 199.5 amu to 293.4 amu
constantcaTm
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PURE VS. IMPURE SUPERCONDUCTORS
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ELIMINATING SUPERCONDUCTING STATE
Raise T above Tc
Raise magnetic field H above critical magnetic field strength Hc
Hc depends on the actual temperature of the superconductor T
Where Ho is the critical magnetic field strength at 0 K
Ceramics usually have smaller Hc compared to metals (more vulnerable to losing superconductivity)
2
2
1c
oc T
THH
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HIGH STRENGTH ELECTROMAGNETS
Large currents required Generates substantial resistive heating (water
cooling required) Heat dissipation occurs anyway
Superconductors are immune to resistive losses But they need to be operated below Tc
Costly cooling process Economic issue
Weigh acquisition price and operation cost
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LIMITING FACTORS
H must be below Hc
Superconducting state is destroyed by its own magnetic field! (a case of self-destruction!)
Critical current Ic above which superconductivity disappears
THI critical space where superconductivity reigns!
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SUPERCONDUCTOR LIMITS
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SUPERCONDUCTOR CLASSES
Type ISharp transition for Hc or Tc
Not used for magnet coilsType II
Gradual transition for Hc or Tc
Superconductivity extends to Hc2 which can be 100 times bigger than the initial Hc1
Mainly utilized for superconducting solenoids
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TYPE I SUPERCONDUCTOR
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TYPE II SUPERCONDUCTOR
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MORE ABOUT TYPE IISupermagnets >> 10TTransition metals and alloys
Nb, Al, Si, V, Pb, Sn, Ti, Nb3Sn, NbTi
Ceramics work too!The region between Hc1 and Hc2
represent a “mixed” state of superconducting and normal regionsNormal regions grow as we go from Hc1 to
Hc2
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OF VORTICES AND FLUXOIDSSmall circular regions in the normal
state carrying the smallest possible unit of magnetic flux (flux quantum)
Surrounded by superconducting regions
Forms a 2-D superlatticeParallel to magnetic field linesRegularly arranged in space
mutual repulsion
)(1005.22
215 mTe
ho
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ELECTRONS AND VORTICESElectrons flowing perpendicular to
fluxoids would always find an unobstructed path through the superconducting matrixUnlimited superconductivity!
Lorentz force pushes fluxoids perpendicular to current and magnetic field directionFluxoids become obstacles to drifting
electronsCurrent decreases, resistivity increases
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PINNED VORTICESPinned down by defects
GB’s, dislocations, fine particles of alloying components
Achieved by heat treament and plastic deformationWire drawing
Used for Nb3Sn superconducting magnet
Increased resistivity is avoided!
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CERAMIC SUPERCONDUCTORS
YBa2Cu3O7-x (YBCO)Orthorhombic layered
perovskite containing periodic O vacancies
DrawbacksBrittlenessCannot carry high current
densitiesEnvironmental instability
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SUPERCONDUCTOR THEORY
BCS Theory Bardeen, Cooper, Schrieffer (1957) Works well for conventional superconductors
Key Idea Existence of Cooper pair
Pair of electrons that has lower energy than two individual electrons
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COOPER PAIR FORMATION
Consider electron in metal at 0 K Electron perturbs neighboring lattice sites Electron drifts through crystal generating temporary
perturbations along its path Displaced ion can be set oscillating by perturbation
generating a phonon Phonon interacts quickly with a second electron
lowering its energy due to lattice deformation Second electron emits phonon which interacts with
first electron Phonon is passed back and forth, coupling the
electrons together, bringing them into a lower energy state
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COOPER PAIRS AND FERMI SURFACES
All electrons on a Fermi surface having opposite momentum and spin form Cooper pairsCloud of Cooper pairs is formedCooperative drifting through the crystal
proceedsThis is an ordered state of the conduction
electrons!Ordering eliminates scattering by lattice
atoms, resulting in zero resistance!
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FERMI ENERGY EF lower in superconducting stateSuperconducting state is separated
from the normal state by an energy gap Eg
Energy gap stabilizes Cooper pair against small changes of net momentum they won’t break apart!
Eg ~ 10-4eV Observed via IR absorption measurements
at T<Tc
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DOS SUPERCONDUCTOR
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JOSEPHSON EFFECT
Another way to measure Eg
Two pieces of metal One in superconducting state, the other in normal
state Energy bands in superconductor are raised
by an appropriate voltage If the applied voltage is big enough, filled
states in superconductor are opposite empty states in normal conductor Cooper pairs can tunnel through
Eg is determined from the threshold voltage for tunneling
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JOSEPHSON JUNCTION
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COOPER PAIR BINDING
Electron-phonon binding occurs at very low temperaturesT < 40 KNoble metals = poor superconductors
(coupling is hard to achieve)Excitons (electron-hole pairs) may link
electrons to form Cooper pairsOrganic superconductors
Resonating valence bandsHigh Tc superconductors
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THERMOELECTRIC PHENOMENA
ThermocoupleTwo different wires connected togetherCharge carrier concentration gradient at
junction generates emfCharge carrier concentration gradient
usually increases with temperatureTemperature measurement is possibleUse small junctions for quick
measurement!Cu-constantan (Cu-45%Ni)
-180 C to +400 CChromel (Ni-10%Cr)-alumel (Ni-
2%Mn2%Al) Higher temperaures
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SEEBECK EFFECT
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SEEBECK EFFECTBack to back thermocouples
One cold, the other one hotPotential difference between the
junction pairs due to carrier concentration gradient
Seebeck coefficient S
after T.J. Seebeck, German physicistUsually in mV/K but can go as high as
mV/K
T
VS
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PELTIER EFFECT
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PELTIER EFFECTSeebeck Effect in reverseDC flowing through back to back
junctions cause one junction to be colder and the other one hotter
PbTe or BiTe together with metalsUp to 70 C gradients has been
achieved with n- and p-type semiconductorsThermoelectric refrigerator
Lower temperatures can be achieved by cascading refrigerators Each stage acts as heat sink for the next
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CONTACT POTENTIALTwo metals in contact with each other
Seebeck: Electrons from material with higher EF go down to material with lower EF until EF equalizes Space charge accumulated produces contact
potential Temperature dependent
Peltier: Current causes electrons with higher EF to transfer energy to material with lower EF Material with higher EF loses energy and becomes
colder!Always a concern in electrical
measurements