Solids
Eisberg & Resnick Ch 13 & 14
RNave:
http://hyperphysics.phy-astr.gsu.edu/hbase/solcon.html#solcon
Alison Baski:
http://www.courses.vcu.edu/PHYS661/pdf/01SolidState041.ppt
Carl Hepburn, “Britney Spear’s Guide to Semiconductor Physics”.
http://britneyspears.ac/lasers.htm
http://hyperphysics.phy-astr.gsu.edu/hbase/solcon.html#solcon
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)
Ionic Bonds
RNave, GSU at http://hyperphysics.phy-astr.gsu.edu/hbase/chemical/bond.html#c4
Ionic Bonds
Ionic Bonding
RNave, Georgia State Univ at hyperphysics.phy-astr.gsu.edu/hbase/molecule
Covalent Bonds
RNave, GSU at http://hyperphysics.phy-astr.gsu.edu/hbase/chemical/bond.html#c4
Covalent Bonding
SYM
spatial
ASYM
spin
ASYM
spatial
SYM
spin
space-symmetric tend to be closer
Covalent Bonding
not really parallel, but spin-symmetric
Stot = 1
Stot = 0
not really anti, but spin-asym
space-symmetric tend to be closer
TYPES OF SOLIDS (ER 13.2)
CRYSTALINE BINDING
•
•
•
•
molecular
ionic
covalent
metallic
Molecular Solids
•
•
•
•
most organics
inert gases
O2 N2 H2
orderly collection of molecules held together by v. d. Waals
gases solidify only at low Temps
easy to deform & compress
poor conductors
Ionic Solids
NaCl
NaI
KCl
• 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
Ge
Covalent Solids
Si
diamond
• 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
Fe Ni Co
Metallic Solids
config dhalf full
• (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
BAND STRUCTURE
Isolated Atoms
Diatomic Molecule
Four Closely Spaced Atoms
Six Closely Spaced Atoms
as fn(R)
the level of interest
has the same nrg in
each separated atom
Two atoms
Six atoms
Solid of N atoms
ref: A.Baski, VCU 01SolidState041.ppt
www.courses.vcu.edu/PHYS661/pdf/01SolidState041.ppt
Four Closely Spaced Atoms
conduction band
valence band
Solid composed of ~NA Na Atoms
as fn(R)
1s22s22p63s1
Sodium Bands vs Separation
Rohlf Fig 14-4 and Slater Phys Rev 45, 794 (1934)
Copper Bands vs Separation
Rohlf Fig 14-6 and Kutter Phys Rev 48, 664 (1935)
Differences down a column in the Periodic Table: IVA Elements
same valence
config
Sandin
The 4A Elements
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
How to choose eF
and
Behavior of the Fermi function at
band gaps
Fermi Distribution for a selected eF
Probability of an energy occuring
(not normalized)
1.5
1
T=0
1000
5000
0.5
0
0
1
2
3
4
Energy
n(e ) 
1
e (e e F ) / kT  1
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.
Fermions
T=0
RNave: http://hyperphysics.phy-astr.gsu.edu/hbase/solcon.html#solcon
Fermions T > 0
Number of Electrons at an Energy e
In QStat, we were doing

Tot KE 

e ne  N e  de
0
distrib fn
Number of ways
to have a particular
energy
Number of electrons
at energy e
# states
probability
of this nrg
occurring
# electrons
at a given nrg
Semiconductors
ER13-9, -10
Semiconductors
~1/40 eV
•
•
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
Intrinsic Semiconductors
Silicon
Germanium
RNave: http://hyperphysics.phy-astr.gsu.edu/hbase/solcon.html#solcon
Doped Semiconductors
lattice
p-type dopants
n-type dopants
5A doping in a 4A
lattice
5A in 4A lattice
3A in 4A lattice
5A in 4A lattice
3A in 4A lattice
‘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
• .


*********************************************************
•
Free-Electron Model
–
–
–
–
–
•
•
Spatial Wavefunctions
Energy of the Electrons
Fermi Energy
Density of States dN/dE
Number of States as fn NRG
E&R 13.5
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
Free-Electron Model (ER13-5)


classical description
e

p2
2m

2 K 2
2m
Quantum Mechanical Viewpoint
In a 3D slab of metal, e’s are free to move
but must remain on the inside


2 2

   0   E
2m
Solutions are of the form:
  xyz 
8
L3
sin k x x sin k y y sin k z z
nz 
L
With nrg’s:
e

h2
2
2
2

n

n

n
x
y
z
8mL2

At T = 0, all states are filled
up to the Fermi nrg
e fermi

h2
2
2
2

n

n

n
x
y
z
8mL2

max
A useful way to keep track of the states that are filled is:
nx2  ny2  nz2
 n2 max
e fermi
h2

8mL2
n 2 max
total number of states up to an energy efermi:
N  2
e fermi
h 3N 



8m   V 
2
1
8
 volume
  21
of 
8
 sphere 
3
4 nmax
3
2/3
# states/volume ~
# free e’s / volume
Sample Numerical Values for Copper slab
N
V
e fermi
= 8.96 gm/cm3
1/63.6 amu
h 3N 



8m   V 
2
6e23 = 8.5e22 #/cm3 = 8.5e28 #/m3
2/3
efermi = 7 eV
nmax = 4.3 e 7
so we can easily pretend that there’s a smooth distrib of nxnynz-states

Density of States
Tot KE 

0
How many combinations of are there
within an energy interval e to e + de ?
e fermi
N
dN
dN
dE
h 3N 



8m   V 
2
 V   8mE 
 
 2 
 3  h 
 V 
 

 3 
2/3
3/ 2
1/ 2
3  8mE 
 2 
2 h 
8 V
3

2
m
3
h


1/ 2
8m
h2
E 1/ 2
dE
e ne  N e  de
At T
≠ 0 the electrons will be spread out among the allowed states
How many electrons are contained in a particular energy range?
 number of ways to have


 a particular energy 
8 V
3
2m
3
h
 
1/ 2
E
 probabilit y of



 this energy occuring 
1
1/ 2
e
( E e f ) / kT
1
this assumes there are no other issues
Distribution of States:
Simple Free-Electron Model vs Reality
Problems with Free Electron Model
(ER13-6, -7)


****************************
1)
2)
3)
Bragg reflection
.
.
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
• .
• .
• .
Nearly-Free Electron Model
version 1 – SP221
k  /2
a /2
2
/2
k
2
a /2
/2
k
Nearly-Free Electron Model
version 2 – SP324
This treatment assumes that when
a reflection occurs, it is 100%.
•
•
•
Bloch Theorem
Special Phase Conditions, k = +/- m /a
the Special Phase Condition k = +/- /a


~~~~~~~~~~
(x) ~ u e i(kx-wt)
amplitude
In reality, lower energy waves are sensitive to the lattice:
Bloch’s
Theorem
(x) ~ u(x) e i(kx-wt)
Amplitude varies with location
u(x) = u(x+a) = u(x+2a) = ….
(x) ~ u(x) e i(kx-wt)
u(x+a) = u(x)
(x+a) e -i(kx+ka-wt)  (x) e -i(kx-wt)
(x+a)  e ika
(x)
Something special happens with the phase when
e ika = 1
ka = +/ m 
k
 

a
, 2
What it is ?
m = 0 not a surprise
m = 1, 2, 3, …

a
, ...
Consider a set of waves with +/ k-pairs, e.g.
k = + /a moves 
k
 
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

a
Kittel
+ ~ 2 cos kx
 ~ 2i sin kx
|+|2 ~ 4 cos2 kx
||2 ~ 4 sin2 kx
Free-electron
Nearly Free-electron
Kittel
Discontinuities occur because the lattice is impacting the movement of electrons.
Effective Mass m*
A method to force the free electron
model to work in the situations where
there are complications
free electron KE functional form
e

2 k 2
2 m*
ER Ch 13 p461 starting w/ eqn (13-19b)
Effective Mass m*
-- describing the balance between applied ext-E and lattice site reflections
1

m*
1  2e
 2 k 2
m* a = S Fext
q Eext
2)
greater curvature, 1/m* > 1/m > 0,  m* < m 
net effect of ext-E and lattice interaction
provides additional acceleration of electrons
m = m*
greater |curvature| but negative,
At inflection pt
net effect of ext-E and lattice interaction
de-accelerates electrons
1)
No distinction between m & m*,
m = m*, “free electron”, lattice structure does
not apply additional restrictions on motion.
Another way to look at the discontinuities
e

2 k 2
2m
apply perturbation from lattice

2 k 2
2 m*
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
From earlier:
Even when above barrier,
reflection and transmission coefficients can
increase and decrease depending upon the energy.
change in motion
due to applied field
enhanced by change in reflection coefficients
change in motion
due to reflections
is more significant
than change in motion
due to applied field
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.
A reminder:
Waves from the left behave like:
 from  eiKx  r e  iKx
the
left
 from  t eiKx
the
left
e

2 K 2
2m
Waves from the right behave like:
 from
the
right
 t e  iKx
 from  e  iKx  r eiKx
the
right
e

2 K 2
2m
sum

A left  B right
unknown weights
Bloch’s Theorem defines periodicity of the wavefunctions:
sum x  a   eika sum x 
 x  a   eika sum
 x 
sum
Related to
Lattice spacing
Applying the matching conditions at x   a/2
 x  a   eika sum
 x 
sum
sum x  a   eika sum x 
A + B
left
right
A + B
left
right
A + B
left
A + B
right
left
right
And eliminating the unknown constants A & B leaves:
t 2  r 2 iKa 1 iKa
cos ka 
e  e
2t
2t
e

2 K 2
2m
For convenience (or tradition) set:
t  t ei
1 t  r
2
2
r   i r ei
cosKa   
 cos ka
t
Related to
Energy
e

cosKa   
 cos ka
t
Related to
possible
Lattice spacings
2 K 2
2m
allowed solution regions
allowed solution regions
Superconductivity
ER 14-1, 13-4
Temperature Dependence of Resistivity
R Nave:
http://hyperphysics.phy-astr.gsu.edu/hbase/solids/supcon.html#c1
Joe Eck:
superconductors.org
Temperature Dependence of
Resistivity
R 
L

A
• Conductors
– Resistivity  increases with increasing Temp
– Temp  t but same # conduction e-’s  
• Semiconductors & Insulators
– Resistivity  decreases with increasing Temp
– Temp  t
but more conduction e-’s  
First observed Kamerlingh Onnes 1911
Note: The best conductors & magnetic materials tend not to be superconductors (so far)
Superconductors.org
Only in nanotubes
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
Superconductor Classifications
Type I
Bardeen, Cooper, Schrieffer
1957, 1972
“Cooper Pairs”
e
Q: Stot=0 or 1? L? J?
e
Symmetry energy ~ 0.01 eV
Popular Bad Visualizations:
correlation lengths
Pairs are related by momentum ±p,
NOT position.
Sn 230 nm
Al 1600
Pb
83
Nb 38
Best conductors  best ‘free-electrons’  no e – lattice interaction
 not superconducting
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
• Experimental Support of BCS Theory
– Isotope Effects
– Measured Band Gaps corresponding to Tcrit
predictions
– Energy Gap decreases as Temp  Tcrit
– Heat Capacity Behavior
Normal Conductor
Semiconductor
or
Superconductor
Another fact about Type I:
-- Interrelationship of Bcrit and Tcrit
Type II
Yr
mixed normal/super
Q: does BCS apply ?
Composition
Tc
May
2006
InSnBa4Tm4Cu6O18+
150
2004
Hg0.8Tl0.2Ba2Ca2Cu3O8.33
138
1986 (La1.85Ba.15)CuO4
30
YBa2Cu3O7
93
actual ~ 8 mm
Sandin
Type II – mixed phases
fluxon
Q: does BCS apply ?
Y Ba2 Cu3 O7
crystalline
may control the electronic config of the conducting layer
La2-x Bax Cu O2
solid solution
Another fact about Type II:
-- Interrelationship of Bcrit and Tcrit
Applications
OR
Other Features of Superconductors
http://superconductors.org/Uses.htm
Meissner Effect
Magnetic Levitation – Meissner Effect
Kittel states this explusion effect
is not clearly directly connected
to the  = 0 effects
Q: Why ?
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
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.
Josephson Junction
~ 2 nm
Recall: Aharonov-Bohm Effect
-- from last semester
affects the phase of a wavefunction
~ e
i ( p eA ) r1 / 
~ eikx ~ eipx / 
A
Source
B
~ e
i ( p  eA ) r2 / 
SQUID
superconducting quantum interference device
o
 ~ oe
ileft
 ~ o e
i right
 ~ o ei
  fn (location )
Add up change in flux as go around loop
   dl  n 2
loop
qB
   


Aharonov Bohm
2 
B  n
q
2 
(2e)
 2.07  10 15 Telsa m 2
Typical B fields
(Tesla)
(# flux quanta)
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 will be able to locate
targets without flying close to
the surface.
Image courtesy Department of
Defence.
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.
http://www.csiro.au/science/magsafe.html
Phillip Schmidt etal. Exploration Geophysics 35, 297 (2004).
Arian Lalezari
SQUID
2 nm
1014 T SQUID threshold
Heart signals 10 10 T
Brain signals 10 13 T
•
•
•
•
•
•
•
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
•
•
•
•
•
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
Redraw LHS
Sn 230 nm
Al 1600
Pb 83
Nb 38
Best conductors  best ‘free-electrons’  no e – lattice interaction
 not superconducting