Physics 215
Winter 2002
Introduction to Modern Physics
Prof. Ioan Kosztin
Lecture #23
Solid State Physics
• Bonding in solids (metals, isolators, semiconductors)
• Classical free electron theory of metals
• Quantum theory of metals
• Band theory of solids
• Semiconductors
• Lasers
Classification of solids
• Phases of matter:
• solid (well defined shape and volume)
• liquid (only well defined volume)
• gas (no defined shape or volume)
• plasma (an overall neutral collection of charged and
neutral particles)
• Solids
• crystalline (atoms form a regular periodic structure)
• amorphous (atoms have irregular spatial distribution)
• Solids
• metals (good electrical/heat conductors)
• semiconductors
• insulators (poor electrical/heat conductors)
Bonding in solids: Ionic solids
Ionic solid crystals (e.g. NaCl) are
held together by the Coulomb
attractive interaction between ions
with opposite sign (ionic bonding)
e2 b
U = −αk
+ m
r r
(m ~ 10)
k = 1 / 4πε 0
Madelung constant
(α = 1.7476 for Na +Cl − )
Ionic cohesive energy:
11

U0 = min U (r )  = −αk  1 − 
m  r0

 mb 
r0 = 

 αk 
1
m −1
Bonding in solids: Ionic solids
Properties of ionic solid crystals:
• relatively stable and hard
• poor electrical/heat conductors
• high melting/boiling temperatures
• transparent to visible light
• strong IR absorption
• soluble in polar solvents (e.g., water)
Bonding in solids: Covalent solids
Atoms in the crystal are held together
by covalent bonding
C atoms in diamond form a tetragonal
crystal structure
Properties of covalent crystals:
• very hard and stable
• high melting point
• good insulators
• do not absorb light
• larger cohesive energies (~10 eV)
than in ionic crystals
Bonding in solids: Metallic solids
Atoms in a metallic crystal are
held together by the effective
attractive electrostatic
interaction mediated by the
conduction (valence) electron
gas (metallic bonding)
Metal
ion
Conduction
electron gas
Properties of metallic crystals:
• smaller cohesive energies
(~1 eV) than in covalent/ionic
crystals
• sufficiently hard and stable
• good electrical/heat conductors
• strong interaction with light
• form solid solutions
Bonding in solids: Molecular crystals
Molecules in the crystal are held together by:
• weak Van der Waals bonds
exp: solid methane (Ec=0.10 eV/molecule)
solid argon (Ec=0.076 eV/molecule)
• relatively strong hydrogen bonds
exp: solid ice (Ec=0.53 eV/molecule)
Amorphous solids
• Ideal solid crystals exhibits structural long range order (LRO)
• Real crystals contain imperfections, i.e., defects and impurities ,
which spoil the LRO
• Amorphous solids lack any LRO [though may exhibit short range
order (SRO)]
Crystal
Glass
(amorphous)
Gas
Degree of (dis)ordering in a solid
can be quantified by the two particle correlation
(radial distribution) function
g2(r) = probability of finding a 2nd atom at a distance r
from a given atom;
g2(r) can be measured experimentally and calculated
theoretically/numerically.
Classical free electron theory of metals
• Free electron model of metals:
metal = an ideal gas of conduction electrons moving through the
fixed lattice of positive ion cores
• Features of the free electron model:

• explains the high electrical (σ) and σ ~ 106 (Ωm) −1


−
~
10
100
W/mK
K

thermal (K) conductivity of metals 
!
!
• explains the functional form of Ohm’s law J = σE 
• explains the relationship between σ and K
[K / σT = const ]
(Wiedemann-Franz law)
• fails to predict accurately the experimental
values of σ and K
Electrical conduction: Ohm’s Law
!
!
J = envd ,
current
density
!
!
vd = v (t )
electron
density
!
!
vd = − µE ,
mobility
!
E =0
!
!
vd = v (t ) = 0
v rms =
!
3kBT
v (t )2 =
m
L = v rms τ (mean free path)
!
!
!
vd = v (t ) = − µE
eτ 2
s = vd τ =
E
m
drift velocity
eτ
µ=
m
!
!
J = σE
mean free
time
(Ohm’s law)
2
τ
ne
−1
σ=ρ =
y
t
i
m
ity
tiv
v
c
i
u
t
nd
sis
co
re
ne 2L
ne 2L
=
=
mv rms
3kBTm
experimentally ρ~T and not T1/2 !!!
Heat conduction: Wiedemann-Franz law
Heat current density:
dQ
dT
= −K
JT ≡
Adt
dx
thermal
conductivity
1
1
2
τ
K =
C"v v rms L = kB nv rms
" 2
3 3
v τ
2
kB n
rms
K mkB 2
3kB2
v = 2T
=
2 rms
2e
σ 2e
⇓
K
3kB2
−8
2
1.1
10
W
Ω/K
= const =
=
×
⋅
2e 2 $%$$$$$$$
σT #$$$$$$
&
= Lorentz number
is different from the
experimental value !!!