Lecture X
Solid state
dr hab. Ewa Popko
Metals and insulators
Measured resistivities range over more than 30 orders of magnitude
Material
Resistivity
(Ωm) (295K)
Resistivity
(Ωm) (4K)
10-5
10-12
Copper
2  10-6
10-10
SemiConductors
Ge (pure)
5  102
1012
Insulators
Diamond
1014
1014
1020
1020
Potassium
“Pure”
Metals
Polytetrafluoroethylene
(P.T.F.E)
Metals, insulators & semiconductors?
Pure metals: resistivity
increases rapidly with
increasing temperature.
Diamond
Resistivity (Ωm)
At low temperatures all
materials are insulators
or metals.
1020-
1010Germanium
100 Copper
10-100
100
200
Temperature (K)
300
Semiconductors: resistivity decreases rapidly with increasing
temperature.
Semiconductors have resistivities intermediate between metals
and insulators at room temperature.
Bound States in atoms
Electrons in isolated
atoms occupy discrete
allowed energy levels
E0, E1, E2 etc. .
The potential energy of
an electron a distance r
from a positively charge
nucleus of charge q is
F6
F7
F8
F9
00
-1
V(r)
E2
Increasing
Binding
E1
Energy
-2
E0
-3
-4
 qe
V( r ) =
4  o r
2
-5
-8
-6
-4
-2
0
r
r
2
4
6
8
Bound and “free” states in solids
The 1D potential energy
of an electron due to an 0
0
array of nuclei of charge
q separated by a distance
-1
R is
2
V(r ) = 
n
 qe
4 o r  nR
V(r)
E2
-2
E1
E0
Where n = 0, +/-1, +/-2 etc. -3
V(r)
Solid
-3
This is shown as the
black line in the figure.
V(r) lower in solid (work
function).
-4
-5
-8
+
-6
-4
-2
+
Nuclear positions
00
r2
r
+r
4
+
R
6
8
+
Energy Levels and Bands
In solids the electron states of tightly bound (high binding
energy) electrons are very similar to those of the isolated atoms.
Lower binding electron states become bands of allowed states.
We will find that only partially filled bands conduct
Band of allowed energy states.
E
+
position
+
+
+
Electron level similar to
that of an isolated atom
+
Band Theory
The calculation of the allowed electron states in a solid is
referred to as band theory or band structure theory.
Free electron model
Band Theory
The calculation of the allowed electron states in a solid is
referred to as band theory or band structure theory.
U(r)
U(r)
Free electron model:
Neglect periodic potential & scattering (Pauli)
Reasonable for “simple metals” (Alkali Li,Na,K,Cs,Rb)
Energy band
theory
2 atoms
6 atoms
Solid state
N~1023 atoms/cm3
Metal – energy band theory
The effects of
temperature
At a temperature T the
probability that a state is
occupied is given by the
Fermi-Dirac function
where μ is the chemical
potential. For kBT << EF μ is
almost exactly equal to EF.
The finite temperature only
changes the occupation of
available electron states in
a range ~kBT about EF.
Fermi-Dirac function for a Fermi temperature
TF =50,000K, about right for Copper.
N(E) dE
n(E)dE


 E - 
f(E) =  exp 
 + 1 
 kB T 


-1
kBT
T=0
T>0
EF
E
Band theory ctd.
To obtain the full band structure, we need to solve
Schrödinger’s equation for the full lattice potential. This
cannot be done exactly and various approximation schemes
are used. We will introduce two very different models, the
nearly free electron and tight binding models.
We will continue to treat the electrons as independent, i.e.
neglect the electron-electron interaction.
Influence of the lattice periodicity
In the free electron model, the allowed energy states are
where for periodic boundary conditions
2n y
2nx
2nz
kx 
; ky 
; kz 
L
L
L
nx , ny and ny positive or negative integers.
E
2 2
E
(k x  k y2  k z2 )
2m
0 k
L- crystal dimension
0
Periodic potential
Exact form of potential is complicated
-1
Has property V(r+ R) = V(r) where
-2
R = m1a + m2b + m3c
where m1, m2, m3 are integers and a ,b ,c
are the primitive lattice vectors.
E
-3
-4
-5
r
Tight Binding Approximation
Tight Binding Model: construct wavefunction as a linear combination
of atomic orbitals of the atoms comprising the crystal.
 (r) =  c j f (r - r
j
j
)
Where f(r) is a wavefunction of the isolated atom
rj are the positions of the atom in the crystal.
The tight binding approximation for s states
Solution leads to the E(k) dependence!!
1D:
E (k) = -  -  (ei k xa  e- i k xa)  - - 2 cos(k xa)
+
+
Nuclear positions
+
+
a
+
E(k) for a 3D lattice
(a,0,0); (0,a,0); (0,0,a)
Simple cubic: nearest neighbour atoms at
So
E(k) =   2(coskxa + coskya + coskza)
Minimum E(k) =   6
for kx=ky=kz=0
Maximum E(k) =   6
for kx=ky=kz=+/-/2
0
-2
 10
-4
1
-6
E(k)
-8
Bandwidth = Emav- Emin = 12
-10
For k << /a
cos(kxx) ~ 1- (kxx)2/2
-12
etc.
E(k) ~ constant + (ak)2/2
c.f. E = (k)2/me
F1
-14
-16
-18
-4
-2
/a
/a
0
2
4
k [111] direction
Behave like free electrons with “effective mass” /a2
Each atomic orbital leads to a band
of allowed states in the solid
Band of allowed states
Gap: no allowed states
Band of allowed states
Gap: no allowed states
Band of allowed states
Reduced Brillouin zone scheme
The only independent values of k are those in the first Brillouin zone.
Discard for
|k| > /a
Results of tight binding calculation
The number of states in a band
Independent k-states in the first Brillouin zone, i.e. kx < /a etc.
Finite crystal: only discrete k-states allowed k x  
2n x
, n x  0,1,2,.... etc.
L
Monatomic simple cubic crystal, lattice constant a, and volume V.
One allowed k state per volume (2)3/V in k-space.
Volume of first BZ is (2/a)3
Total number of allowed k-states in a band is therefore
 2 


a


3
2 3  V
V
a
3
N
Precisely N allowed k-states i.e. 2N electron states (Pauli) per band
This result is true for any lattice:
each primitive unit cell contributes exactly one k-state to each band.
Metals and insulators
In full band containing 2N electrons all states within the first B. Z. are
occupied. The sum of all the k-vectors in the band = 0.
A partially filled band can carry current, a filled band cannot
Insulators have an even integer number
of electrons per primitive unit cell.
With an even number of electrons per
unit cell can still have metallic behaviour
due to band overlap.
Overlap in energy need not occur
in the same k direction
EF
Metal due to
overlapping bands
EF
Empty Band
EF
Energy Gap
Partially
Filled Band
Full Band
Part Filled Band
Part Filled Band
Energy Gap
Full Band
INSULATOR
or SEMICONDUCTOR
METAL
METAL
or SEMI-METAL
Insulator -energy band theory
Covalent bonding
Atoms in group III, IV,V,&VI tend to form
covalent bond
Filling factor
T. :0.34
F.C.C :0.74
Covalent bonding
Crystals: C, Si, Ge
Covalent bond is formed by two electrons, one
from each atom, localised in the region
between the atoms (spins of electrons are
anti-parallel )
Example: Carbon 1S2 2S2 2p2
C
C
3D
Diamond:
tetrahedron,
cohesive energy 7.3eV
2D
Covalent Bonding in Silicon
•Silicon [Ne]3s23p2 has four
electrons in its outermost shell
•Outer electrons are shared with
the surrounding nearest neighbor
atoms in a silicon crystalline lattice
•Sharing results from quantum
mechanical bonding – same QM
state except for paired, opposite
spins
(+/- ½ ħ)
diamond
semiconductors
Intrinsic conductivity
ln()
1/T
 s   0s e
 Eg / 2kT
Extrinsic conductivity – n – type semiconductor
ln()
 d   0d e
1/T
 Ed / kT
Extrinsic conductivity – p – type semiconductor
Conductivity vs temperature
ln()
 s   0s e
 Eg / 2kT
 d   0d e  Ed / kT
1/T
Actinium
Aluminium (Aluminum)
Americium
Antimony
Argon
Arsenic
Astatine
Barium
Berkelium
Beryllium
Bismuth
Bohrium
Boron
Bromine
Cadmium
Caesium (Cesium)
Calcium
Californium
Carbon
Cerium
Chlorine
Chromium
Cobalt
Copper
Curium
Darmstadtium
Dubnium
Dysprosium
Einsteinium
Erbium
Europium
Fermium
Fluorine
Francium
Gadolinium
Gallium
Germanium
Gold
Hafnium
Hassium
Helium
Holmium
Hydrogen
Indium
Iodine
Iridium
Iron
Krypton
Lanthanum
Lawrencium
Lead
Lithium
Lutetium
Magnesium
Manganese
Meitnerium
Mendelevium
Mercury
Molybdenum
Neodymium
Neon
Neptunium
Nickel
Niobium
Nitrogen
Nobelium
Osmium
Oxygen
Palladium
Phosphorus
Platinum
Plutonium
Polonium
Potassium
Praseodymium
Promethium
Protactinium
Radium
Radon
Rhenium
Rhodium
Rubidium
Ruthenium
Rutherfordium
Samarium
Scandium
Seaborgium
Selenium
Silicon
Silver
Sodium
Strontium
Sulfur (Sulphur)
Tantalum
Technetium
Tellurium
Terbium
Thallium
Thorium
Thulium
Tin
Titanium
Tungsten
Ununbium
Ununhexium
Ununoctium
Ununpentium
Ununquadium
Ununseptium
Ununtrium
Uranium
Vanadium
Xenon
Ytterbium
Yttrium
Zinc
Zirconium