3. OUTLINE
• Review Ionic / Covalent Molecules
• Types of Solids (ER 13.2)
• Band Theory (ER 13.3-.4)
– basic ideas
– description based upon free electrons
– descriptions based upon nearly-free electrons
• ‘Free’ Electron Models (ER 13.5-.7)
• Temperature Dependence of Resistivity (ER 14.1)
4. Ionic Bonds
RNave, GSU at http://hyperphysics.phy-astr.gsu.edu/hbase/chemical/bond.html#c4
12. Molecular Solids
• orderly collection of molecules held together by v. d. Waals
• gases solidify only at low Temps
• easy to deform & compress
• poor conductors
most organics
inert gases
O2 N2 H2
13. Ionic Solids
• individ atoms act like closed-shell, spherical, therefore binding
not so directional
• arrangement so that minimize nrg for size of atoms
• tight packed arrangement poor thermal conductors
• no free electrons poor electrical conductors
• strong forces hard & high melting points
• lattice vibrations absorb in far IR
• to excite electrons requires UV, so ~transparent visible
NaCl
NaI
KCl
14. Covalent Solids
• 3D collection of atoms bound by shared valence
electrons
• difficult to deform because bonds are directional
• high melting points (b/c diff to deform)
• no free electrons poor electrical conductors
• most solids adsorb photons in visible opaque
Ge Si
diamond
15. Metallic Solids
• (weaker version of covalent bonding)
• constructed of atoms which have very weakly
bound outer electron
• large number of vacancies in orbital (not enough
nrg available to form covalent bonds)
• electrons roam around (electron gas )
• excellent conductors of heat & electricity
• absorb IR, Vis, UV opaque
Fe Ni Co
config dhalf full
29. Band Spacings
in
Insulators & Conductors
electrons free to roam
electrons confined to small region
RNave: http://hyperphysics.phy-astr.gsu.edu/hbase/solcon.html#solcon
30. How to choose eF
and
Behavior of the Fermi function at
band gaps
31. Fermi Distribution for a selected eF
0
0.5
1
1.5
0 1 2 3 4
Energy
Probability
of
an
energy
occuring
(not
normalized)
T=0
1000
5000
1
1
)
( /
)
(
kT
F
e
n e
e
e
32. How does one choose/know eF
If in unfilled band, eF is energy of highest energy electrons at T=0.
If in filled band with gap to next band, eF is at the middle of gap.
35. Number of Electrons at an Energy e
e
e
e
e d
N
n
KE
Tot
0
distrib fn Number of ways
to have a particular
energy
In QStat, we were doing
Number of electrons
at energy e
40. Semiconductors
• Types
– Intrinsic – by thermal excitation or high nrg photon
– Photoconductive – excitation by VIS-red or IR
– Extrinsic – by doping
• n-type
• p-type
•
~1 eV
~1/40 eV
46. ‘Free-Electron’ Models
• Free Electron Model (ER 13-5)
• Nearly-Free Electron Model (ER13-6,-7)
– Version 1 – SP221
– Version 2 – SP324
– Version 3 – SP425
• .
47. • Free-Electron Model
– Spatial Wavefunctions
– Energy of the Electrons
– Fermi Energy
– Density of States dN/dE E&R 13.5
– Number of States as fn NRG E&R 13.5
• Nearly-Free Electron Model (Periodic Lattice Effects) – v2 E&R 13.6
• Nearly-Free Electron Model (Periodic Lattice Effects) – v3 E&R 13.6
*********************************************************
49.
E
m
0
2
2
2
z
k
y
k
x
k
L
xyz z
y
x sin
sin
sin
8
3
In a 3D slab of metal, e’s are free to move
but must remain on the inside
Solutions are of the form:
L
nz
2
2
2
2
2
8
z
y
x n
n
n
mL
h
e
With nrg’s:
Quantum Mechanical Viewpoint
50. At T = 0, all states are filled
up to the Fermi nrg
max
2
2
2
2
2
8
z
y
x
fermi n
n
n
mL
h
e
A useful way to keep track of the states that are filled is:
max
2
2
2
2
n
n
n
n z
y
x
51. total number of states up to an energy efermi:
3
3
max
4
8
1
8
1
2
2
n
sphere
of
volume
N
3
/
2
2
3
8
V
N
m
h
fermi
e
max
2
2
2
8
n
mL
h
fermi
e
# states/volume ~ # free e’s / volume
52. Sample Numerical Values for Copper slab
V
N
= 8.96 gm/cm3 1/63.6 amu 6e23 = 8.5e22 #/cm3 = 8.5e28 #/m3
efermi = 7 eV
3
/
2
2
3
8
V
N
m
h
fermi
e
nmax = 4.3 e 7
so we can easily pretend that there’s a smooth distrib of nxnynz-states
53. Density of States
How many combinations of are there
within an energy interval e to e + de ?
3
/
2
2
3
8
V
N
m
h
fermi
e
2
/
3
2
8
3
h
mE
V
N
dE
h
m
h
mE
V
dN 2
2
/
1
2
8
8
2
3
3
2
/
1
2
/
1
3
3
2
8
E
m
h
V
dE
dN
e
e
e
e d
N
n
KE
Tot
0
54. At T ≠ 0 the electrons will be spread out among the allowed states
How many electrons are contained in a particular energy range?
occuring
energy
this
of
y
probabilit
energy
particular
a
have
to
ways
of
number
1
1
2
8
/
)
(
2
/
1
2
/
1
3
3
kT
E f
e
E
m
h
V
e
59. Other Problems with the Free Electron Model
• graphite is conductor, diamond is insulator
• variation in colors of x-A elements
• temperature dependance of resistivity
• resistivity can depend on orientation of crystal & current I direction
• frequency dependance of conductivity
• variations in Hall effect parameters
• resistance of wires effected by applied B-fields
• .
• .
• .
61. Nearly-Free Electron Model
version 2 – SP324
• Bloch Theorem
• Special Phase Conditions, k = +/- m /a
• the Special Phase Condition k = +/- /a
This treatment assumes that when
a reflection occurs, it is 100%.
62. (x) ~ u e i(kx-wt)
(x) ~ u(x) e i(kx-wt)
~~~~~~~~~~
amplitude
In reality, lower energy waves are sensitive to the lattice:
Amplitude varies with location
u(x) = u(x+a) = u(x+2a) = ….
Bloch’s
Theorem
63. u(x+a) = u(x)
(x+a) e -i(kx+ka-wt) (x) e -i(kx-wt)
(x) ~ u(x) e i(kx-wt)
(x+a) e ika (x)
Something special happens with the phase when
e ika = 1
ka = +/ m m = 0 not a surprise
m = 1, 2, 3, …
...
,
2
,
a
a
k
What it is ?
64. a
k
Consider a set of waves with +/ k-pairs, e.g.
k = + /a moves k = /a moves
This defines a pair of waves moving right & left
Two trivial ways to superpose these waves are:
+ ~ e ikx + e ikx ~ e ikx e ikx
+ ~ 2 cos kx ~ 2i sin kx
67. Effective Mass m*
A method to force the free electron
model to work in the situations where
there are complications
ER Ch 13 p461 starting w/ eqn (13-19b)
*
2
2
2
m
k
e
free electron KE functional form
68. Effective Mass m*
-- describing the balance between applied ext-E and lattice site reflections
2
2
2
1
*
1
k
m
e
m* a = S Fext
q Eext
69. No distinction between m & m*,
m = m*, “free electron”, lattice structure does
not apply additional restrictions on motion.
m = m*
greater curvature, 1/m* > 1/m > 0, m* < m
net effect of ext-E and lattice interaction
provides additional acceleration of electrons
greater |curvature| but negative,
net effect of ext-E and lattice interaction
de-accelerates electrons
At inflection pt
1)
2)
70. *
2
2
2
2
2
2 m
k
m
k lattice
from
on
perturbati
apply
e
Another way to look at the discontinuities
Shift up implies effective mass has decreased, m* < m,
allowing electrons to increase their speed and join
faster electrons in the band.
The enhanced e-lattice interaction speeds up the electron.
Shift down implies effective mass has increased, m* > m,
prohibiting electrons from increasing their speed and making
them become similar to other electrons in the band.
The enhanced e-lattice interaction slows down the electron
71. From earlier: Even when above barrier,
reflection and transmission coefficients can
increase and decrease depending upon the energy.
72. change in motion
due to reflections
is more significant
than change in motion
due to applied field
change in motion
due to applied field
enhanced by change in reflection coefficients
73. Nearly-Free Electron Model
version 3
à la Ashcroft & Mermin, Solid State Physics
This treatment recognizes
that the reflections of electron
waves off lattice sites can
be more complicated.
75. Waves from the left behave like:
iKx
iKx
left
the
from e
r
e
iKx
left
the
from e
t
m
K
2
2
2
e
76. Waves from the right behave like:
iKx
iKx
right
the
from e
r
e
iKx
right
the
from e
t
m
K
2
2
2
e
77. right
left
sum B
A
Bloch’s Theorem defines periodicity of the wavefunctions:
x
e
a
x sum
ika
sum
x
e
a
x sum
ika
sum
unknown weights
Related to
Lattice spacing
78.
x
e
a
x sum
ika
sum
x
e
a
x sum
ika
sum
Applying the matching conditions at x a/2
A + B
left right
A + B
left right
A + B
left right
A + B
left right
iKa
iKa
e
t
e
t
r
t
ka
2
1
2
cos
2
2
m
K
2
2
2
e
And eliminating the unknown constants A & B leaves:
79. For convenience (or tradition) set:
2
2
1 r
t
i
e
t
t
i
e
r
i
r
ka
t
Ka
cos
cos
87. Superconductors.org Only in nanotubes
Note: The best conductors & magnetic materials tend not to be superconductors (so far)
88. Superconductor Classifications
• Type I
– tend to be pure elements or simple alloys
– = 0 at T < Tcrit
– Internal B = 0 (Meissner Effect)
– At jinternal > jcrit, no superconductivity
– At Bext > Bcrit, no superconductivity
– Well explained by BCS theory
• Type II
– tend to be ceramic compounds
– Can carry higher current densities ~ 1010 A/m2
– Mechanically harder compounds
– Higher Bcrit critical fields
– Above Bext > Bcrit-1, some superconductivity
90. Type I
Bardeen, Cooper, Schrieffer 1957, 1972
“Cooper Pairs”
Symmetry energy ~ 0.01 eV
Q: Stot=0 or 1? L? J?
e
e
91. Sn 230 nm
Al 1600
Pb 83
Nb 38
Best conductors best ‘free-electrons’ no e – lattice interaction
not superconducting
Popular Bad Visualizations:
Pairs are related by momentum ±p,
NOT position.
correlation lengths
92. More realistic 1-D billiard ball picture:
Cooper Pairs are ±k sets
Furthermore:
“Pairs should not be thought of as independent particles” -- Ashcroft & Mermin Ch 34
93. • Experimental Support of BCS Theory
– Isotope Effects
– Measured Band Gaps corresponding to Tcrit
predictions
– Energy Gap decreases as Temp Tcrit
– Heat Capacity Behavior
96. Type II
Q: does BCS apply ?
mixed normal/super
Yr Composition Tc
May
2006
InSnBa4Tm4Cu6O18+ 150
2004 Hg0.8Tl0.2Ba2Ca2Cu3O8.33 138
1986 (La1.85Ba.15)CuO4 30
YBa2Cu3O7 93
103. Magnetic Levitation – Meissner Effect
Q: Why ?
Kittel states this explusion effect
is not clearly directly connected
to the = 0 effects
104. Magnetic Levitation – Meissner Effect
MLX01 Test Vehicle
2003 581 km/h 361 mph
2005 80,000+ riders
2005 tested passing trains at relative 1026 km/h
http://www.rtri.or.jp/rd/maglev/html/english/maglev_frame_E.html
105. Maglev in Germany (sc? idi)
32 km track
550,000 km since 1984
Design speed 550 km/h
NOTE(061204): I’m not so sure this track is superconducting. The MagLev planned for the Munich area will be. France is also thinking about a sc maglev.
107. Recall: Aharonov-Bohm Effect
-- from last semester
affects the phase of a wavefunction
Source B
/
)
( 2
~ r
eA
p
i
e
/
)
( 1
~ r
eA
p
i
e
/
~
~ ipx
ikx
e
e
A
111. MAGSAFE will be able to locate
targets without flying close to
the surface.
Image courtesy Department of
Defence.
http://www.csiro.au/science/magsafe.html
Finding 'objects of interest' at sea with MAGSAFE
MAGSAFE is a new system for locating and identifying submarines.
Operators of MAGSAFE should be able to tell the range, depth and
bearing of a target, as well as where it’s heading, how fast it’s going
and if it’s diving.
Building on our extensive experience using highly sensitive magnetic
sensors known as Superconducting QUantum Interference Devices
(SQUIDs) for minerals exploration, MAGSAFE harnesses the power
of three SQUIDs to measure slight variations in the local magnetic
field.
MAGSAFE has higher sensitivity and greater immunity to external noise than conventional
Magnetic Anomaly Detector (MAD) systems. This is especially relevant to operation over shallow
seawater where the background noise may 100 times greater than the noise floor of a MAD
instrument.
114. SQUID
2 nm
1014 T SQUID threshold
Heart signals 10 10 T
Brain signals 10 13 T
115. • Fundamentals of superconductors:
– http://www.physnet.uni-hamburg.de/home/vms/reimer/htc/pt3.html
• Basic Introduction to SQUIDs:
– http://www.abdn.ac.uk/physics/case/squids.html
• Detection of Submarines
– http://www.csiro.au/science/magsafe.html
• Fancy cross-referenced site for Josephson Junctions/Josephson:
– http://en.wikipedia.org/wiki/Josephson_junction
– http://en.wikipedia.org/wiki/B._D._Josephson
• SQUID sensitivity and other ramifications of Josephson’s work:
– http://hyperphysics.phy-astr.gsu.edu/hbase/solids/squid2.html
• Understanding a SQUID magnetometer:
– http://hyperphysics.phy-astr.gsu.edu/hbase/solids/squid.html#c1
• Some exciting applications of SQUIDs:
– http://www.lanl.gov/quarterly/q_spring03/squid_text.shtml
116. • Relative strengths of pertinent magnetic fields
– http://www.physics.union.edu/newmanj/2000/SQUIDs.htm
• The 1973 Nobel Prize in physics
– http://nobelprize.org/physics/laureates/1973/
• Critical overview of SQUIDs
– http://homepages.nildram.co.uk/~phekda/richdawe/squid/popular/
• Research Applications
– http://boojum.hut.fi/triennial/neuromagnetic.html
• Technical overview of SQUIDs:
– http://www.finoag.com/fitm/squid.html
– http://www.cmp.liv.ac.uk/frink/thesis/thesis/node47.html
117. Redraw LHS
Sn 230 nm
Al 1600
Pb 83
Nb 38
Best conductors best ‘free-electrons’ no e – lattice interaction
not superconducting
