Transcript Solids - types • MOLECULAR. Set of single atoms or molecules bound to adjacent due to weak electric force between neutral objects (van.

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
He
H
Ar
H2O
CH4
freezing
boiling
1K
4K
14 K
20 K
84 K
87 K
273 K
373 K
90 K
111 K
bonding energy
.08 eV/molecule
0.5 eV/mol
0.1 eV/mol
• correlates with bonding energy
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Ionic Solids
• Positive and negative ions. Strong bond and high
melting point. no free electrons  poor conductor
R
Potential vs sep
distance R
• similar potential as molecule. ~5 eV molecules and
~6 eV solid (NaCl)
• each Cl- has 6 adjacent Na+,
12 “next” Cl-, etc2
2
V (6 Na  )  
6e
4 0 R
V (12Cl  )  
12e
4 0 2 R
1.7e 2
  all  
4 0 R
• energy levels similar to molecules except no
rotations….electronic in UV and vibrational in IR.
Often transparent in visible
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• 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
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Bands in Diatomic Molecules-Reminder
• assume all valence electrons are shared
• if both atoms are the same then |y|2 same if switch
atom(1) and atom(2) --- electron densities around
each atom are the same (even sort of holds if
different atoms like CO)
H(1s)
<-- very far apart --->
H(1s)
close together H(“1s”)H(“1s”)
electron wavefunctions overlap -“shared”
• two energy levels (S=0,1) (spatial
symmetric and antisymmetric)which have
| y (1,2) | | y (2,1) |
2
2
E
bands
R=infinity
(atoms)
1s*1s
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Vib and rot
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Bands in Solids
• lowest energy levels very similar to free atoms
•  large kinetic energy  large p, small l
Z2
K  2 13.6eV
n
h
l
p
• 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
4s,4p,3d
3s,3p
2s,2p
E
1s
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Multielectron energy levels
•
3 electrons: 12  12  12  32  12  12
• 3/2 symmetric spin. Each 1/2 has different mixed
symmetry  3 different spacial wavefunctions and
(usually) 3 different 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
6 electrons
E
R
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y (1,2,3,4...N )  N! terms
•
N different combinations of spatial wavefunctions
gives N energy levels
• 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
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Bands in Solids 2
•
as their atomic radii are larger (and wavelength
larger) there is more overlap for outer electrons
• larger N (# shared)  wider bands (we’ll see later
when discuss Fermi gas)
2
n a0
re 
atoms
Zeff (n)
• valence electrons will also share though band width
is narrower
• “valence” vs “conduction” depends on whether
band if filled or not
conduction
valence
E
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Conduction vs valence
•

•
•
•
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)
00000000
x00000x0
xxxxxxxx T=0
xxx0xx0x T>0
xxxxxxxx
xxxxxxxx
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
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Li and Be Bands
• Atoms: Li Z=3 1s22s1 unfilled “conductor”
Be Z=4 1s22s2 filled “insulator”
• But solids have energy bands which can
overlap
2p
2s
E
1s
Atom
solid
• 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
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Magnesium Bands
• Atoms: Z=12 1s22s22p63s2 filled “insulator” like
Be
6N
3p
8N
3s
E
2N
Atomic separation R
• 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
BUT, if R becomes smaller, the bands split (bonds)
giving an energy gap for C, Si, Ge
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C,Si,Ge Bands
• similar valence C:2s22p2 Si:3s23p2 Ge:4s24p2
4N
6N
2p,3p,4p
E
2s,3s4s
4N
2N
Atomic separation R
• 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)
empty
T=0
E(gap)
filled
T=0
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Properties of Energy Bands
• Band width, gaps, density of states depend on the
properties of the lattice (spacing, structure)
• First approximation is Fermi Gas. But if
wavelength becomes too small, not overlapping
 sets width of band
• next approximation adds in periodic structure of
potential
• can cause interference of “traveling” waves
(reflection/transmission). Essentially vibrational
modes of the solid
• destructive interference causes energy gaps which
are related to dimensions of lattice
• Note often the “band energy” is measured from the
bottom of the band (which is the electronic energy
level)
• “real” calculation needs to use 3D structure of solid
 complicated D(E), need measurements
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Fermi Gas Model
• Quantum Stats:
probabilty/ energy  n( E ) 
1
e ( E  EF ) / kT  1
8V (2m3 )1/ 2 1/ 2
density of states  D( E ) 
E
3
h
h 2  3 
EF (T  0) 
 
8m   
2/3
• Ex 13-2. What are the number of conduction
electrons excited to E > EF for given T?(done
earlier)
E
N  total 
F
 D( E )dE 
0
V 8m
3
(
h2
) 3 / 2 E F3 / 2
3 N 1/ 2
at T  0  D( E ) 
E
3/ 2
2 EF
DN  n( E F ) D( E F )DE
1
 D ( E F )(2  3)kT
2
DN 3kT .025eV


N
2 EF
4eV
n*D
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T=0
T>0
EF 14
Fermi Gas Model II
• Solids have energy bands and gaps
ideal
D( E )  0 in gaps
D
D( E ) varies in band
 often  E1/ 2 at bottom
real
E
• Can calculate density of states D(E) from lattice
using Fourier Transform like techniques (going
from position to wavelength space)
• can change D(E) by changing lattice
- adding additional atoms during fabrication
- pressure/temperature changes
 PHYS 566, 480, 690A techniques
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Fermi Gas Model III
• 1D model. N levels and min/max energy
2
2
Na=L
2 2
p
h
hn
E


2
2m 2ml 8m L2
n  0  min E  0 (band bottom)
a
h2
n  N  max E 
(band width)
2
8m a
• For 2D/3D look at density of states. Grows as E.5
until circle in k-space “fills up” then density falls
(can’t have wavelength ~ smaller than spacing)
ky
2D/3D
D
kx
EF
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Interactions with Lattice
• Study electron wavefunction interactions with the
lattice by assuming a model for the potential
• Kronig-Penney has semi-square well and barrier
penetration
• will sort of look at 1D  really 3D and dependent
on type of crystal which gives inter-atom separation
which can vary in different directions
• solve assuming periodic solutions
• Bragg conditions give destructive interference but
different “sine” or “cosine” due to actual potential
variation. will have different points where
wavefunction=0
V0
y  eikx  eix / a  right traveling wave
y  e ikx  e ix / a  left traveling wave
  y 1   aiy i
all reflected
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Interactions with Lattice
• Get destructive interference at
2

k

l
a
• leads to gaps near those wavenumbers
• once have energy bands, can relate to conductivity
• materials science often uses the concept of effective
mass. Electron mass not changing but “inertia”
(ability to be accelerated/move) is. So high m* like
being in viscous fluid  larger m* means larger
interaction with lattice, poor conductor
• m* ~ m in middle of unfilled band
m* > m near top of almost filled band
m* < m near bottom of unfilled band
• always dealing with highest energy electron
(usually near Fermi energy)
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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
 65
E g  6eV  n  10
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 =

DN   n( E ) D( E  Ebot )dE
Ebot
Ebot  EF  Egap / 2  bottom conduction
D( E  Ebot )  AE0.5  sam e as valence
DN  e
 E g / 2 kT
 A( E  Ebot ) DE  e
0.5
 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  AE F
0
3
• the fraction in the conduction band is then
DN
kT 3 / 2  Eg / 2 kT
 3( ) e
N
EF
kT .025
DN
for

, E g  1eV 
 1014
EF
4
N
DN kT
m etal

 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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Doped semiconductors
•
Si(14) 3s23p2 P(15) 3s23p3 Al(13) 3s23p1
Si
||
4 covalent bonds. Fill all valence
Si= Si =Si
energy levels (use all electrons)
||
1 eV gap
Si
Si
||
single electron loosely bound to P
Si= P =Si
(~looks like Na)
||
0.05 eV  conduction band
Si
e
Si =Si
|| ||
0.06 eV can break one of the Si=Si
Si= Al -Si
bonds. That electron  Al. The “hole”
|| || ||
moves to the Si atom
Si=Si
“hole”
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Doped semiconductors II
conduction band
.05 eV
E
donor electrons
acceptor holes
.06 eV
valence band
P-doped n-type
Al-doped p-type
“extra” e
“missin” e= (hole)
.05 eV to move from
donor to conduction band
.06 eV to move from
valence to conduction band
• The Fermi Energy is still where n(EF) = ½.
 doping moves EF
• Complex compounds shift Fermi Energy and D(E)
(AlxOyYz...)
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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
n-type
D
conduction
valence
EF
D(E)
p-type
EF
• adding Al (p-type). some electrons move from
valence to acceptor band. n(E)= ½ now in that
band
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Doped Semiconductors IV
• undoped semicon. have Ncond ~ 10-11Nvalence
• doping typically 10-7 increases conductivity
• but if raise T then the probability to move from
valence band to conduction band increases e-E/kT.
Can see this as a change in Fermi Energy with
temperature  at some point all the donor
electrons are in the conduction band and many
valence move to conduction  EF back in middle
of gap
donor
E
EF
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