Bonds

1
Bonds

• Chapter 5 of Solymar

2
Introduction

• When two hydrogen atoms come close to each other
• They form a chemical bond, resulting in a
  hydrogen molecule (H2)
• When many silicon atoms come close
• They form many chemical bonds, resulting in a
  crystal
• What brings them together?
• The driving force is

To reduce the energy
3
Interactions between Atoms

• For atoms to come close and form bonds, there
  must be an attractive force
• Na gives up its 3s electron and becomes Na
• Cl receives the electron to close its n 3 shell
  and becomes Cl-
• The Coulomb attractive force is proportional to
  r-2
• In the NaCl crystal, Na and Cl- ions are 0.28 nm
  apart
• There must be a repulsive force when the ions are
  too close to each other
• When ions are very close to overlap their
  electron orbitals and become distorted, a
  repulsive force arises to push ions apart and
  restore the original orbitals
• This is a short-range force

4
Equilibrium Separation

• There is a balance point, where the two forces
  cancel out (Fig. 5.1)
• The energy goes to zero at infinite separation
• As separation decreases, the energy decreases, so
  the force is attractive
• At very small separation, the energy rises
  sharply, so the force is strongly repulsive
• The minimum energy point (Ec, or the zero force
  point) corresponds to the equilibrium separation
  ro
• The argument is true for both molecules and
  crystals

5
Mathematical

• Mathematically
• A and B are constants
• The first term represents the repulsion and the
  second attraction
• Minimum energy
• It must be negative, so m

6
Mechanical Properties

• Hookes law
• Deformation is linearly proportional to stress
  (elasticity)
• Microscopically, when the crystal is compressed,
  i.e. when the interatomic distance is reduced,
  the energy will increase. When the force is
  removed, interatomic distance will return to
  equilibrium
• This can be expressed by the energy-separation
  curve

7
Bulk Elastic Constant
T

• A cube of side a under isotropic compression
• E(r) is the energy per atom
• Naa3E(ro) is the total energy of the crystal at
  equilibrium
• Na is the number of atoms per unit volume
• If ro changes ro Dr, Taylor series of E(r) is

T
T
a
T stress
8
Bulk Elastic Constant

• The net change in energy
• The total change in crystal dimensions
• Each side is moved by
• The total work done on the cube
• Bulk elastic constant is defined by
• So

9
Bond Types

• Four types in total
• Ionic
• Covalent
• Metallic
• van der Waals

10
Ionic Bonds

• NaCl a good example
• What is the cohesive energy of an ionic crystal?
• A Na ion has 6 Cl ions as distance a
• 12 Na ions at distance
• 8 Cl ions at
• Electrostatic energy
• Its within 10 of the experimental value

11
Metallic Bonds

• Each atom in a metal donates one or more
  electrons and becomes a lattice ion
• The electrons move around and bounce back and
  forth
• They form an electron sea, whose electrostatic
  attraction holds together positive lattice ions
• The electrostatic attraction comes from all
  directions, so the bond is non-directional
• Metals are ductile and malleable

12
Covalent Bonds

• When two identical atoms come together, a
  covalent bond forms
• The hydrogen molecule
• A hydrogen atom needs two electrons to fill its
  1s shell
• When two hydrogen atoms meet, one tries to snatch
  the electron from the other and vice versa
• The compromise is they share the two electrons
• Both electrons orbit around both atoms and a
  hydrogen molecule forms
• The chlorine molecule
• A chlorine atom has five 3p electrons and is
  eager to grab one more
• Two chlorine atoms share an electron pair and
  form a chlorine atom

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Group IV

• Carbon 1s22s22p2 Si 1s22s22p63s23p2 Ge
  1s22s22p63s23p63d104s24p2
• Each atom needs four extra electrons to fill the
  p-shell
• They are tetravalent
• sp3 hybridization
• s shell and p shell hybridize to form four
  equal-energy dangling electrons
• Each of them pairs up with a dangling electron
  from a neighbor atom
• There are four neighbor atoms equally spaced
• Each atom is at the center of a tetrahedron
• Interbond angle 109.4?
• Covalent bond is directional

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Group IV

• At 0 K
• All electrons are in bonds orbiting atoms
• None can wander around to conduct electricity
• They are insulators
• At elevated temperatures
• Statistically, some electrons can have more
  enough energy to escape through thermal
  vibrations and become free electrons
• They are semiconductors
• The CC bond is very strong, making diamond the
  hardest material known (Table 5.1)
• Diamond has excellent thermal conductivity
• It burns to CO2 at 700?C

15
van der Waals Bonds

• Argon has outer shell completely filled
• When argon is cooled down to liquid helium
  temperature, it forms a solid
• The electrons are sometimes here and sometimes
  there, so the centers of the positive charge
  (nucleus) and negative charge (electrons) are not
  always coincident
• The argon atom is a fluctuating dipole
  (instantaneous dipole)
• It induces an opposite dipole moment on another
  argon atom, so they attract each other
• Such attraction is weak, so the materials have
  low melting and boiling temperatures
• They are often seen in organic crystals

16
Miscellaneous

• Mixed bonds
• In real crystals, bonds may not be any of these
  pure types
• III-V and II-VI compound semiconductors have a
  combination of ionic and covalent bonds
• GaAs and CdTe
• Carbon
• Carbon has both sp3 and sp2 hybridization
• It can crystallize into graphite and fullerenes,
  in addition to diamond

17
Coupled Mode Approach

• What happens to the energy levels of the atoms
  when they come close to each other?
• Feynmans coupled mode approach
• Schrodingers equation
• General solution is a superimposed one
• We care less about the spatial probability but
  what is the probability the electron is in state
  j at time t?

18
Getting Rid of Spatial Variation

• Multiply by yk and integrate over volume
• It can be shown that yk and yj are orthogonal
• Ckj 1 can be done by properly choosing
  constants in yk and yj
• Notation
• Hkj is a constant
• No spatial variables
• Each wj represents a state

19
Without Coupling

• Minimum number of states to have coupling two
• When H12 H21 0, the two states are not
  coupled
• The solution
• The probability of the electron being in state 1
• Its a constant
• If the electron begins in this state, it will
  always be in this state

20
Coupled States

• Physical picture of coupling (Fig. 5.5)
• The hydrogen molecular ion two protons and one
  electron
• The electron can attach to proton 1 (state 1) or
  proton 2 (state 2)
• When the two states are uncoupled, the electron
  will remain at the proton it started with
• When two protons come closer to each other, the
  electron can leave proton 1 and tunnel to proton
  2, or vice versa
• Classically, the electron can not do so because
  of the energy barrier

21
Feynmans Approach

• Simplification
• H11 H22 E0 H12 H21 -A
• Attempt solution
• Substitute to get
• The determinant
• Solution
• If no coupling, E E0, i.e. both states are at
  the same energy level
• If there is coupling, the energy level is split
  to two new levels

22
Energy Split

• Whenever there is coupling, the energy level
  splits
• A is related to the tunneling probability
• It decreases exponentially with distance
• E0 A has a minimum, which gives the equilibrium
  separation between two hydrogen atoms (Fig. 5.7)
• In the hydrogen molecule, the electrons jump from
  one proton to the other, resulting in a molecule
• Electron exchange, instead of electron sharing

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HW Assignment

• 5.3, 5.6