ch676 physical chemistry: principles and applications · pdf file · 2011-10-11and...

CH676 Physical Chemistry Principles and Applications

CH676 Physical ChemistryPrinciples and Applications


Band Theory

f(E) for a metal f(E) for a semiconductor

Fermi-Dirac Function

f(E) = 1[1 + e(E-EF)kT]

Where the Fermi Energy EF is defined as the energy where f(E) = 12 That is to say onehalf of the available states are occupied T is the temperature (in K) and k is the Boltzmanconstant (k = 862 times 10-5 eVK)


Band TheoryResistivity and Band StructureThe resistivities of real materials span nearly 25 orders of magnitudeThis is due to differences in carrier concentration (n) and mobility (μ)

Compound Resistivity (Ωcm) Compound Resistivity (Ω cm)Ca 39 times 10-6 Si ~ 01Ti 42 times 10-6 Ge ~ 005Mn 185 times 10-6 ReO3 36 times 10-6

Zn 59 times 10-6 Fe3O4 52 times 10-6

Cu 17 times 10-6 TiO2 9 times 104

Ag 16 times 10-6 ZrO2 1 times 109

Pb 21 times 10-6 Al2O3 1 times 1019


Band Theory

Carrier concentrationbull The carrier concentration only includes electrons which can easily be excited from occupied states

into empty states The remaining electrons are localizedbull In the absence of external excitations (light voltage etc) the excitation is thermal this is on the

order of kT (~ 003 eV at RT)bull Only electrons whose energies are within a few kT of EF can contribute to the electrical conductivity

Resistivity and Band StructureThe resistivities of real materials span nearly 25 orders of magnitudeThis is due to differences in carrier concentration (n) and mobility (μ)

Carrier Mobilitybull μ = eτm (where e = electronic charge m = the effective mass τ = the relaxation time

between scattering events

bull What entities scatter the carriers and reduce the mobilitybull A defect or impurity (τ increases as purity increases)bull Lattice vibrations phonons (τ decreases as temperature increases)

bull What factors determine the effective massbull m depends upon the band width which in turn depends upon orbital overlap

σ = nemiddoteτm =ne2τm


Band TheoryResistivity and Band Structure Metal

Conductivity goes down as temperature and impurity increases

σ = ne2τm

bull EF cuts the very wide (disperse) s band giving rise to a large carrierconcentration along with high mobility This combination gives rise to highconductivity

bull The carrier concentration n increases very slowly with temperature

bull τ is inversely proportional to temperature (τ α 1T) due to scattering by latticevibrations (phonons)

bull τ is inversely proportional impurity concentration


Band Theory

Doping SemiconductorsThe Fermi-Dirac function shows that a pure semiconductor with a band gap ofmore than a few tenths of an eV would have a very small concentration ofcarriers Therefore impurities are added to introduce carriers

Resistivity and Band Structure Semiconductors


Band Theory

When a p-type and an n-typesemiconductor are brought into contactelectrons flow from the n-dopedsemiconductor into the p-dopedsemiconductor until the Fermi levelsequalize (like two reservoirs of watercoming into equilibrium) This causesthe conduction and valence bands tobend as shown above



Band TheoryBand to Band Transitions Optical Properties of Semiconductors

bull We can examine the relationshipbetween bonding (spatial andenergetic overlap) and opticalproperties by considering the bandgaps of those compounds

bull Since electronic transitions from thevalence to conduction band span afairly large range of energiessemiconductors act as sort of a longpass filter This can give rise to onlycertain colors

E = hcλ = (41357 x 10-15 eV-s)(2998 x 108 ms)λ

E (eV) = 1240λ(nm)

UV 100-400 nm 124 - 310 eVViolet 400-425 nm 310 - 292 eVBlue 425-492 nm 292 - 252 eVGreen 492-575 nm 252 - 215 eVYellow 575-585 nm 215 - 212 eVOrange 585-647 nm 212 - 192 eVRed 647-700 nm 192 - 177 eVNear IR 10000-700 nm 177 - 012 eV



Spatial Overlap and Band Gap (Eg)

What are the effects of increasing thespatial overlap

Primary Effect Increases the overallenergy level of the conduction band(more highly antibonding) [Eg uarr](Van-Vechten amp Phillips assumed spatialoverlap is proportional to d-25 (where d isthe bond distance))

Ionicity and Band Gap (Eg)

What are the effects of increasing theelectronegativity difference between theelements

Primary Effect Increases the separationof the valence and conduction bands (thebonds become more ionic) [Eg uarr]


Band TheoryColored Semiconductors

CdS (Eg=242 eV) CdTe (Eg=150 eV) ZnS (Eg=36 eV) ZnSe (Eg=258 eV)

Light Emitting Diodes

GaAs (Eg=143 eV) rarr Near IR

GaPN (Eg = 225 eV) rarr Yellow

GaPZnO (Eg = 225 eV) rarr Red

GaN SiC rarr Blue



Energetic amp Spatial OverlapIII IV V VIII


Crystal Structures and Magnetic PropertiesMagnetic Ordering


Crystal Structures and Magnetic PropertiesMagnetic Moment of Atoms Ions and ElectronsMagnetism in solids originates in the magnetic properties of an electronμS = g [S(S+1)]12 [(eh(4πme)]μB = (eh(4πme)μS = g [S(S+1)]12 μB(S = frac12 the spin quantum number g ~ 2 the gyromagnetic ratio μB = 92742 times 10-24

JT the Bohr magneton)

So that for a free electron μS = 173 μB

Almost all atoms have multiple electrons but most of the electrons are paired upin orbitals with another electron of the opposite spin When all of the electrons onan atom are paired the atom is said to be diamagnetic Atomsions with unpairedelectrons are paramagnetic

Diamagnetic = There is a very small magnetic moment associated with an electrontraveling in a closed path around the nucleusParamagnetic = The moment of an atom with unpaired electrons is given by the spin Sand orbital angular L and total momentum J quantum numbers


Crystal Structures and Magnetic PropertiesMagnetic Moments of Atoms amp Ions

Unpaired electrons and paramagnetism are usually associated with thepresence of either transition metal or lanthanide (actinide) ions In manytransition metal compounds the surrounding anionsligands quench the orbitalangular momentum and one needs only to take into account the spin onlymoment Consider the following examples

Ion e- Config S μS(μB) μS+L(μB) μobs (μB)

Ti4+ d1 frac12 173 301 17-18V2+ d2 1 283 449 28-31Cr3+ d3 32 387 521 37-39Fe3+ d5 (HS) 52 592 592 57-60Ni2+ d8 (HS) 1 283 449 29-39Cu2+ d9 12 173 301 19-21




Crystal Structures and Magnetic PropertiesSuperexchangeIn order for a material to be magnetically orderedthe spins on one atom must couple with the spinson neighboring atoms The most commonmechanism for this coupling (particularly ininsulators) is through the semicovalentsuperexchange interaction The spin informationis transferred through covalent interactions withthe intervening ligand


Crystal Structures and Magnetic PropertiesSuperexchange


Crystal Structures and Solid ElectrolytesIonic vs Electronic ElectrolyteA substance that conducts electricity through the movement of ions Most electrolytes aresolutions or molten salts but some electrolytes are solids and some of those arecrystalline solids Different names are given to such materials

bull Solid Electrolytebull Fast Ion Conductorbull Superionic Conductor

Ionic vs Electronic Conductivity

MetalsElectrons carry the currentConductivity Range = 10 Scm lt σ lt 105 ScmConductivity Increases linearly as temperature decreases

Solid ElectrolytesIons carry the currentConductivity Range = 10-3 Scm lt σ lt 10 ScmConductivity decreases exponentially as temperature decreases (activated transport)


Solid Electrolytes

Defects

In order for an ion to move through a crystal it must hop froman occupied site to a vacant site Thus ionic conductivity canonly occur if defects are present The two simplest types ofpoint defects are Schottky and Frenkel defects

Ion Migration

Consider the movement of Na+ ions in NaCl via vacancies originating from Schottky defects Note thatthe Na+ ion must squeeze through the lattice inducing significant local distortionrelaxation This is onefactor that limits the mobility of ions A second factor that contributes is the relatively high probabilitythat the ion will jump back to itrsquos original position leading to no net ionic migration


Solid ElectrolytesSolid Electrolyte Materials

bullAg+ Ion ConductorsbullAgI amp RbAg4I5

bullNa+ Ion ConductorsbullSodium β-Alumina(ie NaAl11O17 Na2Al16O25)bullNASICON (Na3Zr2PSi2O12)

bullLi+ Ion ConductorsbullLiCoO2 LiNiO2bullLiMnO2

bullO2- Ion ConductorsbullCubic stabilized ZrO2 (YxZr1-xO2-x2 CaxZr1-xO2-x)bullδ-Bi2O3bullDefect Perovskites (Ba2In2O5 La1-xCaxMnO3-y )

bullF- Ion ConductorsbullPbF2 amp AF2 (A = Ba Sr Ca)


Solid Electrolytes

Applications of Ionic ConductorsThere are numerous practical applications all based on electochemical cells where ionic conductivity is needed and it is advantageousnecessary to use solids for all components

bull Batteriesbull Fuel Cells

In such cells ionic conductors are needed for either the electrodes the electrolyte or bothElectrolyte (Material needs to be an electrical insulator to prevent short circuit)Electrode (Mixed ionic and electronic conductivity is needed to avoid open circuit)


Solid Electrolytes

Schematic of Rechargable Li BatteryLi-ion batteries are among the best battery systems in terms of energy density (W-hkg amp W-hL) This makes them very attractive for hybrid automobiles amp portable electronics

The cathode half-reaction (charging) is

The anode half-reaction is


Solid ElectrolytesCathode Materials Considerations1 The transition metal ion should have a large work function (highly oxidizing) to

maximize voltage

2 The cathode material should allow an insertionextraction of a large amount of lithiumto maximize the capacity High cell capacity + high cell voltage = high energydensity

3 The lithium insertionextraction process should be reversible and should induce little orno structural changes This prolongs the lifetime of the electrode

4 The cathode material should have good electronic and Li+ ionic conductivities Thisenhances the speed with which the battery can be discharged

5 The cathode should be chemically stable over the entire voltage range and not reactwith the electrolyte

6 The cathode material should be inexpensive environmentally friendly andlightweight


Crystal Structures and Solid ElectrolytesCathode Materials Considerations


Solid ElectrolytesSolid Oxide Fuel Cells amp Proton Exchange Membrane Fuel CellsTypical Fuels

bullAnode Reaction 2H2 rarr 4H+ + 4e-

bullCathode Reaction O2 + 4H+ + 4e- rarr 2H2ObullOverall Cell Reaction 2H2 + O2 rarr 2H2O

A fuel cell generates electricity and heat by electrochemically combining a gaseous fueland an oxidizing gas via an ion conducting electrolyte typically at elevated temperatures(eg 800-1000 ordmC)

Advantages vs Conventional Power Generation MethodsbullHigher conversion efficiencybullLower CO2 emissions

Cathode amp AnodebullHigh electronic conductivitybullChemical and mechanical stabilitybullThermal expansion coefficient that matches electrolytebullSufficient porosity to facilitate transport of O2 from the

gas phase to the electrolyte

ElectrolytebullHigh oxygen ion conductivitybullVery low electronic conductivity

bullAnode Reaction 2H2 + 2O2ndash rarr 2H2O + 4endash

bullCathode Reaction O2 + 4endash rarr 2O2ndash

bullOverall Cell Reaction 2H2 + O2 rarr 2H2O


Solid ElectrolytesDesign Principles O2- Conductors

bull High concentration of anion vacancies

bull necessary for O2- hopping to occur

bull High Symmetry

bull provides equivalent potentials between occupied and vacant sites

bull High Specific Free Volume (Free VolumeTotal Volume)

bull void spacevacancies provide diffusion pathways for O2- ions

bull Polarizable cations (including cations with stereoactive lonepairs)

bull polarizable cations can deform during hopping which lowers theactivation energy

bull Favorable chemical stability cost and thermal expansioncharacteristics

bull for commercial applications


Band Theory

f(E) for a metal f(E) for a semiconductor

Fermi-Dirac Function

f(E) = 1[1 + e(E-EF)kT]

Where the Fermi Energy EF is defined as the energy where f(E) = 12 That is to say onehalf of the available states are occupied T is the temperature (in K) and k is the Boltzmanconstant (k = 862 times 10-5 eVK)









Band Theory













σ = ne2τm






Band Theory




Band Theory







E = hcλ = (41357 x 10-15 eV-s)(2998 x 108 ms)λ

E (eV) = 1240λ(nm)

















GaN SiC rarr Blue






























Solid Electrolytes

Defects


Ion Migration










Solid Electrolytes





Solid Electrolytes






maximize voltage
























bull High Symmetry
















Band Theory













σ = ne2τm






Band Theory




Band Theory







E = hcλ = (41357 x 10-15 eV-s)(2998 x 108 ms)λ

E (eV) = 1240λ(nm)

















GaN SiC rarr Blue






























Solid Electrolytes

Defects


Ion Migration










Solid Electrolytes





Solid Electrolytes






maximize voltage
























bull High Symmetry











σ = ne2τm






Band Theory




Band Theory







E = hcλ = (41357 x 10-15 eV-s)(2998 x 108 ms)λ

E (eV) = 1240λ(nm)

















GaN SiC rarr Blue






























Solid Electrolytes

Defects


Ion Migration










Solid Electrolytes





Solid Electrolytes






maximize voltage
























bull High Symmetry









Band Theory




Band Theory







E = hcλ = (41357 x 10-15 eV-s)(2998 x 108 ms)λ

E (eV) = 1240λ(nm)

















GaN SiC rarr Blue






























Solid Electrolytes

Defects


Ion Migration










Solid Electrolytes





Solid Electrolytes






maximize voltage
























bull High Symmetry









Band Theory







E = hcλ = (41357 x 10-15 eV-s)(2998 x 108 ms)λ

E (eV) = 1240λ(nm)

















GaN SiC rarr Blue






























Solid Electrolytes

Defects


Ion Migration










Solid Electrolytes





Solid Electrolytes






maximize voltage
























bull High Symmetry












E = hcλ = (41357 x 10-15 eV-s)(2998 x 108 ms)λ

E (eV) = 1240λ(nm)

















GaN SiC rarr Blue






























Solid Electrolytes

Defects


Ion Migration










Solid Electrolytes





Solid Electrolytes






maximize voltage
























bull High Symmetry























GaN SiC rarr Blue






























Solid Electrolytes

Defects


Ion Migration










Solid Electrolytes





Solid Electrolytes






maximize voltage
























bull High Symmetry





































Solid Electrolytes

Defects


Ion Migration










Solid Electrolytes





Solid Electrolytes






maximize voltage
























bull High Symmetry
















Solid Electrolytes





Solid Electrolytes






maximize voltage
























bull High Symmetry









Solid Electrolytes






maximize voltage
























bull High Symmetry










maximize voltage
























bull High Symmetry


























bull High Symmetry








ch676 physical chemistry: principles and applications · pdf file · 2011-10-11and...

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