1. Solids - types
MOLECULAR. Set of single atoms or molecules bound to adjacent due to weak electric force between neutral objects (van der Waals). Strength depends on electric dipole moment
No free electrons  poor conductors
easily deformed, low freezing temperature
freezing boiling bonding energy
He K K
H K K
Ar K K eV/molecule H2O K K eV/mol CH K K eV/mol
correlates with bonding energy

2. Ionic Solids
Positive and negative ions. Strong bond and high melting point. no free electrons  poor conductor
similar potential as molecule. ~5 eV molecules and ~6 eV solid (NaCl)
each Cl- has 6 adjacent Na+, 12 “next” Cl-, etc
energy levels similar to molecules except no rotations….electronic in UV and vibrational in IR. Often transparent in visible R
Potential vs sep
distance R

3. COVALENT. Share valence electrons (C, H, etc)
strong bonds (5-10 eV), rigid solids, high melting point
no free electrons  insulators
usually absorb in both visible and UV
METALLIC. s-p shell covalent bonds. But d shell electrons “leftover” (smaller value of n  lower energy but larger <r>)
can also be metallic even if no d shell if there is an unfilled band
1-3 eV bonds, so weaker, more ductile, medium melting temp
“free” electrons not associated with a specific nuclei. Wavelength large enough so wavefunctions overlap and obey Fermi-Dirac statistics  conductors  EM field of photon interacts with free electrons and so absorb photons at all l

4. Bands in Solids E  large kinetic energy  large p, small l
lowest energy levels very similar to free atoms
 large kinetic energy  large p, small l
little overlap with electrons in other atoms and so narrow energy band
higher energy levels: larger l wavefunctions of electrons from different atoms overlap
need to use Fermi-Dirac statistics
many different closely spaced levels: Band: “valence” vs “conduction” depends on whether band is filled or not
4s,4p,3d
3s,3p
2s,2p 1s E

5. Multielectron energy levels
the need for totally antisymmetric wave functions causes the energies to split when the separation distance R < wavelength
if far apart  N degenerate(equal) states
overlap  still N states but different energy (different spins and different spatial states, mixed symmetries)
N based on how many electrons overlap  large for the outer shell
small DE between different levels  an almost continuous energy band
nature of the energy bands determines properties of solid filled bands empty bands partially filled bands energy “width” of band energy gaps between bands density of states in bands
6 electrons E R

6. Conduction vs valence E
energy levels in 4s/4p/3d bands overlap and will have conduction as long as there isn’t a large DE to available energy states (and so can readily change states)
x00000x0
xxxxxxxx T= xxx0xx0x T>0
xxxxxxxx xxxxxxxx
x=electron 0=empty state (“hole”)
sometime current is due to holes and not electrons
good conductors have 1 or more conduction/free electrons/holes per atom
conduction
valence E

7. Li and Be Bands But solids have energy bands which can overlap E
Atoms: Li Z=3 1s22s1 unfilled “conductor” Be Z=4 1s22s2 filled “insulator”
But solids have energy bands which can overlap
there is then just a single 2p2s band
Be fills the band more than Li but the “top” (the Fermi Energy) is still in the middle of the band. So unfilled band and both are metals 2p 2s 1s E
Atom solid

8. Magnesium Bands
Atoms: Z=12 1s22s22p63s2 filled “insulator” like Be
the 3p level becomes a band with 6N energies. The 3s becomes a band with 2N energies
They overlap becoming 1 band with 8N energy levels and no gaps and is a metal as unfilled levels above Fermi energy
BUT, if R becomes smaller, the bands split (bonds) giving an energy gap for C, Si, Ge 6N 3p 3s 8N E 2N
Atomic separation R

9. C,Si,Ge Bands similar valence C:2s22p2 Si:3s23p2 Ge:4s24p2
8N overlapping energy levels for larger R
R becomes smaller, the bands split into 4N “bond” and 4N “antibond”. an energy gap for C (7eV) and Si, Ge (~1 eV) 4N 6N
2p,3p,4p
2s,3s4s E 4N 2N
Atomic separation R
empty T=0
E(gap)
filled T=0

10. Fermi Gas Model Quantum Stats:
What are the number of conduction electrons excited to E > EF for given T? and so available for conduction
T=0
T>0
n*D EF

11. 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
Fermi energy on center of gap for undoped. Always where n(E)=0.5
D(E) typically goes as sqrt(E) at top of valence band and at bottom of conduction band n D
valence
conduction EF
D*n
If T>0

12. Semiconductors II Distribution function is
so probability factor depends on gap energy
estimate #electrons in conduction band of semiconductor. Integrate over n*D factors at bottom of conduction band

13. 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

14. Doped semiconductors e “hole” Si(14) 3s23p2 P(15) 3s23p3 Al(13) 3s23p1
|| covalent bonds. Fill all valence
Si= Si =Si energy levels (use all electrons)
|| eV gap
|| single electron loosely bound to P
Si= P =Si (~looks like Na)
|| eV  conduction band
Si = Si
|| || eV can break one of the Si=Si
Si= Al -Si bonds. That electron  Al. The “hole”
|| || || moves to the Si atom
Si=Si=Si e
“hole”

15. Doped semiconductors II
conduction band
.05 eV
donor electrons
acceptor holes E
.06 eV
valence band
P-doped n-type
“extra” e
.05 eV to move from donor to conduction band
Al-doped p-type
“missing” e= (hole)
.06 eV to move from valence to conduction band
The Fermi Energy is still where n(EF) = ½  doping moves EF

16. Doped Semiconductors III
Adding P (n-type). Since only .05 eV gap some electrons will be raised to conduction band  where n(E)= ½ is in donor band
adding Al (p-type). some electrons move from valence to acceptor band. n(E)= ½ now in that band
n-type D
valence
conduction EF
D(E)
p-type EF

17. Superconductivity Resistance goes to 0 below a critical temperature Tc
element Tc resistivity (T=300)
Ag mOhms/m
Cu mOhms/m
Ga K mO/m Al Sn Pb Nb
many compounds (Nb-Ti, Cu-O-Y mixtures) have Tc up to 90 K. Some are ceramics at room temp
Res. T

18. Superconductors observations
For different isotopes, the critical temperature depends on mass. ISOTOPE EFFECT
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. B field adds energy, can cause quench if abiove critical value
plasma  gas  liquid  solid  superconductor

19. 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

20. 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)

21. Conditions for superconductivity 2
2 electrons in pair have equal but opposite momentum. Maximizes 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)
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 pair size)
different times
different pairs

22. Energy levels in S.C.
electrons in Cooper pair have energy as part of the Fermi sea (E1 and E2=EF+D) plus from their binding energy into a Cooper pair (V12)
E1 and E2 are just above EF (where the action is). If the condition is met then have transition to the lower energy superconducting state
can only happen for T less than critical temperature. Lower T gives larger energy gap. At T=0 (from BCS theory)
Egap
normal
s.c.
Temperature

23. Magnetic Properties of Materials
H = magnetic field strength from macroscopic currents
M = field due to charge movement and spin in atoms - microscopic
can have residual magnetism: M not equal 0 when H=0
diamagnetic  c < 0. Currents are induced which counter applied field. Usually Superconducting c = -1 (“perfect” diamagnetic)

24. Paramagnetism
Atoms can have permanent magnetic moment which tend to line up with external fields
if J=0 (Helium, filled shells, molecular solids with covalent S=0 bonds…)  c = 0
assume unfilled levels and J> n = # unpaired magnetic moments/volume n+ = number parallel to B n- = number antiparallel to B n = n+ + n-
moments want to be parallel as

25. Paramagnetism II
Use Boltzman distribution to get number parallel and antiparallel
where M = net magnetic dipole moment per unit volume
can use this to calculate susceptibility(Curie Law)

26. Paramagnetism III
if electrons are in a Fermi Gas (like in a metal) then need to use Fermi-Dirac statistics
reduces number of electrons which can flip, reduces induced magnetism, c smaller
antiparallel
parallel EF
turn on B field. shifts by mB
antiparallel states drop to
lower energy parallel

27. Ferromagnetism
Certain materials have very large c (1000) and a non-zero B when H=0 (permanent magnet). c will go to 0 at critical temperature of about 1000 K ( non ferromagnetic)
4s2: Fe26 3d Co d Ni d s2: Gd f Dy66 4f10
All have unfilled “inner” (lower n) shells. BUT lots of elements have unfilled shells. Why are a few ferromagnetic?
Single atoms. Fe,Co,Ni D subshell L= Use Hund’s rules  maximize S (symmetric spin)  spatial is antisymmetric and electrons further apart. So S=2 for the 4 unpaired electrons in Fe
Solids. Overlap between electrons  bands but less overlap in “inner” shell. overlapping changes spin coupling (same atom or to adjacent atom) and which S has lower energy. Adjacent atoms may prefer having spins parallel. depends on geometry  internuclear separation R

28. Ferromagnetism II
R small. lots of overlap  broad band, many possible energy states and magnetic effects diluted
R large. not much overlap, energy difference small
R medium. broadening of energy band similar to magnetic shift  almost all in state
P A
P A
E(unmagnetized)-
E(magnetized)
Fe Co Ni R Mn