Bonding in Solids

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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
(α = 1.7476 for Na +Cl − )

(m ~ 10)
k = 1 / 4πε 0

Madelung constant

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) 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

Metal ion

Conduction electron gas

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   K ~ 10 − 100 W/mK  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 , ! ! vd = v (t )
drift velocity

! ! vd = − µE ,

current density

electron density

eτ µ= m
! ! J = σE

mobility

mean free time

(Ohm’s law)

! E =0
! ! vd = v (t ) = 0 ! ! ! vd = v (t ) = − µE

v rms =

L = v rms τ (mean free path)

! 3kBT v (t )2 = m

eτ 2 s = vd τ = E m

ne 2τ σ = ρ −1 = ty ivi ty m ct ivi u t...
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