Transcript Semiconductors

Semiconductors
• Filled valence band but small gap (~1 eV) to an
empty (at T=0) conduction band
• look at density of states D and distribution
function n
n
D
conduction
valence
EF
D*n
If T>0
• Fermi energy on center of gap for undoped. Always
where n(E)=0.5 (problem 13-26)
• D(E) typically goes as sqrt(E) at top of valence
band and at bottom of conduction band
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Semiconductors II
• Distribution function is
n( E ) 
if
1

 ( E  E F ) / kT
1
e
 1 e ( E  EF ) / kT
E  EF  E gap / 2  kT  .025eV @ T  300
 n( E g )  e
 E g / 2 kT
• so probability factor depends on gap energy
E g  1eV  n  10 11
E g  6eV  n  10
 65
Si
C
• estimate #electrons in conduction band of
semiconductor. Integrate over n*D factors at
bottom of conduction band
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• Number in conduction band using Fermi Gas
model =

N   n( E ) D( E  Ebot )dE
Ebot
Ebot  EF  E gap / 2  bottom conduction
D( E  Ebot )  AE 0.5  same as valence
N  e
 E g / 2 kT
 A( E  Ebot ) 0.5 E  e
 E g / 2 kT
A(kT ) 0.5 2kT
• integrate over the bottom of the conduction band
• the number in the valence band is about
E F  E gap / 2
2 3/ 2
0.5
N 
AE dE  AEF
0
3
• the fraction in the conduction band is then
N
kT 3 / 2  E g / 2 kT
 3( ) e
N
EF
kT .025
N
for

, E g  1eV 
 10 14
EF
4
N
N kT
metal 

 0.005
N
EF
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Conduction in
semiconductors
• INTRINSIC. Thermally excited electrons move
from valence band to conduction band. Grows with
T.
• “PHOTOELECTRIC”. If photon or charged
particle interacts with electrons in valence band.
Causes them to acquire energy and move to
conduction band. Current proportional to number
of interactions (solar cells, digital cameras, particle
detection….)
• EXTRINSIC. Dope the material replacing some of
the basic atoms (Si, Ge) in the lattice with ones of
similar size but a different number (+- 1) of valence
electrons
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Superconductivity
• Resistance goes to 0 below a critical temperature Tc
element Tc resistivity (T=300)
Ag
--.16 mOhms/m
Cu
-.17 mOhms/m
Ga 1.1 K
1.7 mO/m
Al
1.2
.28
Res.
Sn
3.7
1.2
Pb
7.2
2.2
Nb
9.2
1.3
T
• many compounds (Nb-Ti, Cu-O-Y mixtures) have
Tc up to 90 K. Some are ceramics at room temp
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Superconductors observations
• Most superconductors are poor conductors at
normal temperature. Many good conductors are
never superconductors
• --> superconductivity due to interactions with the
lattice
• practical applications (making a magnet), often
interleave S.C. with normal conductor like Cu
• if S.C. (suddenly) becomes non-superconducting
(quenches), normal conductor able to carry current
without melting or blowing up
• quenches occur at/near maximum B or E field and
at maximum current for a given material. Magnets
can be “trained” to obtain higher values
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Superconductors observations
• For different isotopes, the critical temperature
depends on mass. ISOTOPE EFFECT
M 0.5Tc  cons tan t ( Sn115,117,119 )
Evibrations 
K
M
• again shows superconductivity due to interactions
with the lattice. If M -> infinity, no vibrations, and
Tc--> 0
• spike in specific heat at Tc
• indicates phase transition; energy gap between
conducting and superconducting phases. And what
the energy difference is
• plasma -> gas -> liquid -> solid -> superconductor
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What causes
superconductivity?
• Bardeen-Cooper-Schrieffer (BCS) model
• paired electrons (cooper pairs) coupled via
interactions with the lattice
• gives net attractive potential between two electrons
• if electrons interact with each other can move from
the top of the Fermi sea (where there aren’t
interactions between electrons) to a slightly lower
energy level
• Cooper pairs are very far apart (~5,000 atoms) but
can move coherently through lattice if electric field
----> resistivity = 0 (unless kT noise overwhelms-> breaks lattice coupling)
atoms
electron
electron
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Conditions for
superconductivity
• Temperature low enough so the number of random
thermal phonons is small
• interactions between electrons and phonons large
(-> large resistivity at room T)
• number of electrons at E = Fermi energy or just
below be large. Phonon energy is small (vibrations)
and so only electrons near EF participate in making
Cooper pairs (all “action” happens at Fermi energy)
• 2 electrons in Cooper pair have antiparallel spin -->
space wave function is symmetric and so electrons
are a little closer together. Still 10,000 Angstroms
apart and only some wavefunctions overlap (low E
--> large wavelength)
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Conditions for
superconductivity 2
•
2 electrons in pair have equal but opposite
momentum. Maximises the number of pairs as
weak bonds constantly breaking and reforming. All
pairs will then be in phase (other momentum are
allowed but will be out of phase and also less
 
probability to form)
e
 
ip  r
Ppair  p1  p2  0
• if electric field applied, as wave functions of pairs
are in phase - maximizes probability -- allows
collective motion unimpeded by lattice (which is
much smaller than 2pair size)
|  total | |  1   2  .... n |2
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