CH676 Physical Chemistry: Principles and Applications. CH676 Physical Chemistry: Principles and Applications

Transcription

1 CH676 Physical Chemistry: Principles and Applications

2 Band Theory Fermi-Dirac Function f(e) = 1/[1 + e (E-E F)/kT ] Where the Fermi Energy, E F, is defined as the energy where f(e) = 1/2. That is to say one half of the available states are occupied. T is the temperature (in K) and k is the Boltzman constant (k = ev/k) f(e) for a metal f(e) for a semiconductor

3 Band Theory Resistivity and Band Structure The resistivities of real materials span nearly 25 orders of magnitude. This is due to differences in carrier concentration (n) and mobility (μ). Compound Resistivity (Ω cm) Compound Resistivity (Ω cm) Ca Si ~ 0.1 Ti Ge ~ 0.05 Mn ReO Zn Fe3O Cu TiO Ag ZrO Pb Al 2 O

4 Band Theory Resistivity and Band Structure σ = ne eτ/m* =ne 2 τ/m* The resistivities of real materials span nearly 25 orders of magnitude. This is due to differences in carrier concentration (n) and mobility (μ). Carrier concentration The carrier concentration only includes electrons which can easily be excited from occupied states into empty states. The remaining electrons are localized. In the absence of external excitations (light, voltage, etc.) the excitation is thermal, this is on the order of kt (~ 0.03 ev at RT) Only electrons whose energies are within a few kt of EF can contribute to the electrical conductivity. Carrier Mobility μ =eτ/m* (where, e = electronic charge; m* = the effective mass; τ = the relaxation time between scattering events What entities scatter the carriers and reduce the mobility? A defect or impurity (τ increases as purity increases) Lattice vibrations, phonons (τ decreases as temperature increases) What factors determine the effective mass? m* depends upon the band width, which in turn depends upon orbital overlap.

5 Band Theory Resistivity and Band Structure: Metal EF σ = ne 2 τ/m* cuts the very wide (disperse) s band, giving rise to a large carrier concentration, along with high mobility. This combination gives rise to high conductivity. The carrier concentration, n, increases very slowly with temperature. τ is inversely proportional to temperature (τ α 1/T), due to scattering by lattice vibrations (phonons). τ is inversely proportional impurity concentration. Conductivity goes down as temperature and impurity increases.

6 Band Theory Resistivity and Band Structure: Semiconductors Doping Semiconductors The Fermi-Dirac function shows that a pure semiconductor with a band gap of more than a few tenths of an ev would have a very small concentration of carriers. Therefore, impurities are added to introduce carriers.

7 Band Theory Resistivity and Band Structure: Semiconductors When a p-type and an n-type semiconductor are brought into contact electrons flow from the n-doped semiconductor into the p-doped semiconductor until the Fermi levels equalize (like two reservoirs of water coming into equilibrium). This causes the conduction and valence bands to bend as shown above.

8 Band Theory Band to Band Transitions Optical Properties of Semiconductors We can examine the relationship between bonding (spatial and energetic overlap) and optical properties by considering the band gaps of those compounds. Since electronic transitions from the valence to conduction band span a fairly large range of energies, semiconductors act as sort of a long pass filter. This can give rise to only certain colors. UV nm ev Violet nm ev Blue nm ev Green nm ev Yellow nm ev Orange nm ev Red nm ev Near IR 10, nm ev E = hc/λ = {( x ev-s)(2.998 x 108 m/s)}/λ E (ev) = 1240/λ(nm)

9 Band Theory Band to Band Transitions Optical Properties of Semiconductors Spatial Overlap and Band Gap (Eg) Ionicity and Band Gap (Eg) What are the effects of increasing the spatial overlap? Primary Effect: Increases the overall energy level of the conduction band (more highly antibonding) [Eg ] (Van-Vechten&Phillipsassumedspatial overlap is proportional to d -2.5 (where d is the bond distance)). What are the effects of increasing the electronegativity difference between the elements? Primary Effect: Increases the separation of the valence and conduction bands (the bonds become more ionic) [Eg ]

10 Band Theory Colored Semiconductors CdS (Eg=2.42 ev) CdTe (Eg=1.50 ev) ZnS (Eg=3.6 ev) ZnSe (Eg=2.58 ev) Light Emitting Diodes GaAs (Eg=1.43 ev) Near IR GaP:N (Eg = 2.25 ev) Yellow GaP:ZnO (Eg = 2.25 ev) Red GaN, SiC Blue

11 Band Theory Band to Band Transitions Optical Properties of Semiconductors Energetic & Spatial Overlap II III IV V VI

12 Crystal Structures and Magnetic Properties Magnetic Ordering

13 Crystal Structures and Magnetic Properties Magnetic Moment of Atoms, Ions, and Electrons Magnetism in solids originates in the magnetic properties of an electron. μ S = g [S(S+1)] 1/2 [(eh/(4πm e )] μ B = (eh/(4πm e ) μ S = g [S(S+1)] 1/2 μ B (S = ½, the spin quantum number; g ~ 2, the gyromagnetic ratio; μb = J/T, the Bohr magneton) So that for a free electron: μs = 1.73 μb Almost all atoms have multiple electrons, but most of the electrons are paired up in orbitals with another electron of the opposite spin. When all of the electrons on an atom are paired the atom is said to be diamagnetic. Atoms/ions with unpaired electrons are paramagnetic. Diamagnetic = There is a very small magnetic moment associated with an electron traveling in a closed path around the nucleus. Paramagnetic = The moment of an atom with unpaired electrons is given by the spin, S, and orbital angular, L and total momentum, J, quantum numbers.

14 Crystal Structures and Magnetic Properties Magnetic Moments of Atoms & Ions Unpaired electrons and paramagnetism are usually associated with the presence of either transition metal or lanthanide (actinide) ions. In many transition metal compounds the surrounding anions/ligands quench the orbital angular momentum and one needs only to take into account the spin only moment. Consider the following examples: Ion e- Config. S μ S (μ B ) μ S +L(μ B ) μ obs (μ B ) Ti 4+ d1 ½ V 2+ d Cr 3+ d3 3/ Fe 3+ d5 (HS) 5/ Ni 2+ d8 (HS) Cu 2+ d9 1/

15 Crystal Structures and Magnetic Properties Magnetic Ordering

16 Crystal Structures and Magnetic Properties Superexchange In order for a material to be magnetically ordered, the spins on one atom must couple with the spins on neighboring atoms. The most common mechanism for this coupling (particularly in insulators) is through the semicovalent superexchange interaction. The spin information is transferred through covalent interactions with the intervening ligand.

17 Crystal Structures and Magnetic Properties Superexchange

18 Crystal Structures and Solid Electrolytes Ionic vs. Electronic Electrolyte A substance that conducts electricity through the movement of ions. Most electrolytes are solutions or molten salts, but some electrolytes are solids and some of those are crystalline solids. Different names are given to such materials: Solid Electrolyte Fast Ion Conductor Superionic Conductor Ionic vs. Electronic Conductivity Metals Electrons carry the current Conductivity Range = 10 S/cm < σ <10 5 S/cm Conductivity Increases linearly as temperature decreases Solid Electrolytes Ions carry the current Conductivity Range = 10-3 S/cm < σ <10S/cm Conductivity decreases exponentially as temperature decreases (activated transport)

19 Solid Electrolytes Defects In order for an ion to move through a crystal it must hop from an occupied site to a vacant site. Thus ionic conductivity can only occur if defects are present. The two simplest types of point defects are Schottky and Frenkel defects. Ion Migration Consider the movement of Na + ions in NaCl via vacancies originating from Schottky defects. Note that the Na + ion must squeeze through the lattice, inducing significant local distortion/relaxation. This is one factor that limits the mobility of ions. A second factor that contributes is the relatively high probability that the ion will jump back to it s original position, leading to no net ionic migration.

20 Solid Electrolytes Solid Electrolyte Materials Ag + Ion Conductors AgI & RbAg 4 I 5 Na + Ion Conductors Sodium β-alumina (i.e. NaAl 11 O 17, Na 2 Al 16 O 25 ) NASICON (Na 3 Zr 2 PSi 2 O 12 ) Li + Ion Conductors LiCoO 2, LiNiO 2 LiMnO 2 O 2- Ion Conductors Cubic stabilized ZrO 2 (Y x Zr 1-x O 2-x/2, Ca x Zr 1-x O 2-x ) δ-bi 2 O 3 Defect Perovskites (Ba 2 In 2 O 5, La 1-x Ca x MnO 3-y,...) F - Ion Conductors PbF 2 & AF 2 (A = Ba, Sr, Ca)

21 Solid Electrolytes Applications of Ionic Conductors There are numerous practical applications, all based on electochemical cells, where ionic conductivity is needed and it is advantageous/necessary to use solids for all components. Batteries Fuel Cells In such cells ionic conductors are needed for either the electrodes, the electrolyte or both. Electrolyte (Material needs to be an electrical insulator to prevent short circuit) Electrode (Mixed ionic and electronic conductivity is needed to avoid open circuit)

22 Solid Electrolytes Schematic of Rechargable Li Battery Li-ion batteries are among the best battery systems in terms of energy density (W-h/kg & W-h/L). This makes them very attractive for hybrid automobiles & portable electronics. The cathode half-reaction (charging) is: The anode half-reaction is:

23 Solid Electrolytes Cathode Materials Considerations 1. The transition metal ion should have a large work function (highly oxidizing) to maximize voltage. 2. The cathode material should allow an insertion/extraction of a large amount of lithium to maximize the capacity. High cell capacity + high cell voltage = high energy density 3. The lithium insertion/extraction process should be reversible and should induce little or no structural changes. This prolongs the lifetime of the electrode. 4. The cathode material should have good electronic and Li + ionic conductivities. This enhances the speed with which the battery can be discharged. 5. The cathode should be chemically stable over the entire voltage range and not react with the electrolyte. 6. The cathode material should be inexpensive, environmentally friendly and lightweight.

24 Crystal Structures and Solid Electrolytes Cathode Materials Considerations

25 Solid Electrolytes Solid Oxide Fuel Cells & Proton Exchange Membrane Fuel Cells Typical Fuels: Anode Reaction: 2H 2 4H + +4e - Cathode Reaction: O 2 + 4H + +4e - 2H 2 O Overall Cell Reaction: 2H 2 +O 2 2H 2 O A fuel cell generates electricity and heat by electrochemically combining a gaseous fuel and an oxidizing gas, via an ion conducting electrolyte, typically at elevated temperatures (eg ºC) Advantages vs. Conventional Power Generation Methods Higher conversion efficiency Lower CO 2 emissions Cathode & Anode High electronic conductivity Chemical and mechanical stability Thermal expansion coefficient that matches electrolyte Sufficient porosity to facilitate transport of O 2 from the gas phase to the electrolyte Electrolyte High oxygen ion conductivity Very low electronic conductivity Anode Reaction: 2H 2 + 2O 2 2H 2 O + 4e Cathode Reaction: O 2 + 4e 2O 2 Overall Cell Reaction: 2H 2 + O 2 2H 2 O

26 Solid Electrolytes Design Principles: O 2- Conductors High concentration of anion vacancies necessary for O 2- hopping to occur High Symmetry provides equivalent potentials between occupied and vacant sites High Specific Free Volume (Free Volume/Total Volume) void space/vacancies provide diffusion pathways for O 2- ions Polarizable cations (including cations with stereoactive lone pairs) polarizable cations can deform during hopping, which lowers the activation energy Favorable chemical stability, cost and thermal expansion characteristics for commercial applications