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Solid State Materials - Coggle Diagram

- - - - Mechanisms
        
        Dexter mechanism
        
        Energy + Electron transfer
        
        Forster resonance energy transfer(FRET)
        
        Rate depends on d**-6, d= +-2-5 nm
        
        Via coulomb (dipole-dipole) interactions
        
        Happens via FRET if the emission spectra of the donor and the absorption spectrum of the acceptor overlap. More overlap = higher rate. Otherwise via dexter
      - Excited state electron in donor atom goes to ground state and transfers its energy to an electron of the accepter atom. D + A --> D + A
      - Examples
        
        Migration of exciton occurs via FRET or Dexter in organic solar cells, because of their localized nature in organic materials
        
        Photodynamic therapy
        
        Photosensitiser is cancer tissue excites electron to triplet state and reacts thereafter with oxygen, to form a reactive oxygen specie
    - - Absorber types
        
        Conjugated organic molecules
        
        Absorb light because Pi-->Pi* transitions are allowed
        
        Have a d(homo-lumo) if the conjugated part increases in size --> same as for a particle in the box.
        
        Semiconductors
        
        Absorbs everything above the bandgap
        
        Bandgap > 3.1 ev = white
        
        Bandgap <1.3 ev = black
        
        Absorbtivity is much higher for direct bandgap sc
        
        Loss mechanisms
        
        Thermalisation
        
        After excitation, the electron quickly falls back to the bottom of the excited state, this results in heat generation
        
        Recombination
        
        Photons below the bandgap
        
        Bandgap tuning
        
        Bandgap decreases if larger atoms are used --> less orbital overlap --.> less splitting
        
        Bandgap increases if multiple atoms are used with a large electronnegativity difference
      - Absorption strength
        
        Lambert beer law. I = I0*exp(-ecl)
        
        e = molar extinction coefficient
        
        For solids, a = optical absorbitivity
        
        Increases if there is a large transition dipole moment and thus the transition is allowed. dS=0, dl=+-1
      - Photons are a oscillating electric field E = e0*cos(wt)
    - - Radiative return to the ground state
        
        Phosperescence
        
        dS = 1
        
        Intersystem crossing converts excited state to triplet state, then phosporescence to the groundstate occurs
        
        Timescale = 10-2, 10-6
        
        Fluorescence
        
        The excited electron falls back to the groundstate, hereby releasing a photon
        
        Timescale = 10-8, 10-11
        
        dS = 0
        
        LED = fluorescence via hole recombination
        
        Frank-condon principle
        
        Absorption is much quicker than nuclei movement: only vertical transition in energy diagram. When electronic+vibration energy levels are coupled
      - Non-radiative recombination
        
        Thermal quenching
        
        Increase with higher temperature, dE is easier reached (overlap groundstate/ excited state) via boltzmann kinetics.
        
        Increases if the difference (dQ) in mean bond length between the ground state and excited state increases. --> Barrier dE decreases
        
        Increases if the bonding of the excited state is less stiff
        
        Transition to ground state via nuclear motion
        
        T1/2 = temperature in wich half of the excitons are quenched
        
        Recombination without the release of a photon, but energy release via the interactions with phonons
      - Components of phospor
        
        Medium
        
        Medium that hold sensitiser and actuator
        
        Sensitiser
        
        Absorbing part
        
        Actuator
        
        Part that emisses a larger wavelength photon
        
        Stokes shift = Eab - Eem
        
        Difference excitation energy and energy emitted photon
      - Kinetics
        
        dN/dt = -kflN - knrN
        
        Lifetimes
        
        Fluorescence = 1/kfl
        
        Non-radiative = 1/knr
        
        Exciton = 1/ktot
        
        N = N0exp(-ktott)
      - Quantum yield
        
        Emmised photons / Number of initial excitons
        
        n(t-->inf)/N(0)
        
        kfl/ktot
        
        Tnr/ (Tfl+Tnr)
        
        Quantum yield decreases if knr increases
  - - - Current density
        
        Ohms law is exctrinsic, like to go to intrinsic formula
        
        I = U/R --> J = I/A --> J = ELA/ApL --> J = 1/p E = sigmaE
        
        p = resistivity
        
        sigma = conductivity S/m = ohm*m
      - Classical approach
        
        Drudes model
        
        Drift velocity vc = mu*E
        
        Scattering of valence electrons on ionic lattice (like kgt)
        
        Current density is J = vden = sigma*E
        
        Sigma = nemu, n = charger carrier concentration
        
        Mobility of electron in E is mu = e*t/m
        
        tau t = scattering time = l/Vrms
        
        Falsely assumes that every electron can contribute
        
        Conductivity decreases with increased scattering
        
        Band orbitals are needed + filling
      - Quantum approach
        
        Free electron model
        
        For bands with a parabolic band dispersion
        
        Metals usually have a non-parabolic shape
        
        effective mass m*
        
        h2/d2E/dk2
        
        Smaller than me for wide bands
        
        Or m = h2/2a2*H12
        
        Energy electron E = E0 + h2k2/2m =+- E0+ h2k2/2m*
        
        Sigma depends lineairly on n if E follow k2, for sc
        
        Occupation of states
        
        product f(E)*N(E)
        
        Density of states
        
        Fermi -dirac distribution function
        
        At T>0 the fermi state is always halve filled
        
        Contributing electrons
        
        Velocity
        
        Fermi velocity
        
        Vf = root(2Ef/m*)
        
        The velocity of the electrons near the fermi energy
        
        v = 1/h * dE/dK
        
        Increases for wide bands
        
        Only the electrons that can be excited near de fermi energy, contribute to the conductivity of the material
        
        Electric field changes causes more negative/positive k levels filled --> uncompensated momentum electrons -> origin nett electron flux
        
        Scattering types
        
        Phonon scattering
        
        Increases with T
        
        Defect scattering
        
        Independent of T
        
        Dominant for small T
        
        Causes residual resistivity
    - - Semiconductors
        
        metals > sigma > insolators
        
        Conductivity increasers for higher temperatures due to the formation of more charge carriers
        
        Drudes result can be used because follows maxwell distribution for sc. Because dE/kbt >> 1.
        
        Ef = 1/2Eg
      - Metals
        
        sigma > sigma sc
        
        Drudes result cannot be used because does not follow maxwell dist
        
        Conductivity lowers for higher temperatures due to scattering
        
        The movement of nuclear motion will produce a rough energy profile, that slows down the electrons
      - Insolators
        
        Sc > sigma
  - - - In common
        
        Electrochemistry
        
        Nernst equation
        
        For each half reaction
        
        E = E0 + RT/nF * ln( [O]/[R])
        
        dG = -nFE
        
        Involves ion transport over long distances
        
        Production or storage of energy
      - Batteries
        
        Primary
        
        Bases on irreversible chemical reactions
        
        Non-rechargeble
        
        Alkeline batteries
        
        Secondary
        
        Based on reversible chemical reactions
        
        Rechargable
        
        Lead acid / Li-ion batteries
        
        Performace terms
        
        Capacity (Ah/kg)
        
        Voltage
        
        Energy density
      - Fuel cells
        
        The fuel is externally supplied
        
        Ion conducting material is needed
      - Electrochemical cells
    - - Proton conductors
        
        Nafion
        
        Only works when hydrated, so higher more efficient T is not possible
        
        Works by forming close packed hydrogen bonds that lower transport energy
        
        Only works below 90 degrees
      - Oxide conductors
        
        Oxygen sensors
        
        Generates a potential for different oxygen pressures at each side
        
        Can conduct O2- via oxide vacencies
      - Ionic compounds
        
        Superionic conductors
        
        Cation superconductor
        
        AgI
        
        Conducts after phase change
        
        Anion superconductor
        
        PbF2
        
        Requirements
        
        Lot of interstituals (n)
        
        Polarizable static counterions
        
        Low charge of mobile ions to lower activation energy
        
        Low energy migration routes
        
        Small mobile ions
        
        Absence of trapping mechanisms
        
        Solid conductor that has conductivities close to the molten state
        
        Conduction mechanism
        
        Sigma = nqmu
        
        Interstitual hopping
        
        Direct hopping
        
        Interstitialcy hopping
        
        Jumps to a filled state and kick it away
        
        Vacency hopping
        
        Schotsky defects
        
        Dominates at large T
        
        Sigma depends on D0, number of schotsky defects (dH/2kT) and act E of migration
        
        Extrinsic defects
        
        Dominates at low T
        
        Sigma depends on D0, number of defects and act E of migration
      - May not have electrical conductivity
    - - The reversible insertion of guest species into vacent sites of a host structure
      - Uses
        
        Graphite
        
        K
        
        N dopes the graphite
        
        Br2
        
        P dopes the graphite
        
        Intercalation increases its electrical conductivity
        
        Li-ion batteries
        
        Electrode material
        
        Cathode
        
        LixCoO2
        
        Unstable if x<0.5
        
        Disadvantages
        
        3 more items...
        
        MX2 metal dichalgonides can also allow intercalation
        
        Anode
        
        LiC6
        
        Stable
        
        Lower Ef and 10* less energy
        
        Li metal
        
        Is unstable because can form dendrites
        
        Favourable because light and very electropositive
        
        Performance requirements
        
        Stability
        
        Stables over many cycles of (de)charging
        
        Large charge/energy density
        
        Low density material
        
        As much lithium as possible
        
        High constant Ecell
        
        Large difference between fermi level anode and cathode
        
        Low internal resistance
        
        Can charge faster
        
        Less losses
        
        Electrolyte stability window
        
        The window in wich the electrolyte is not reduced or oxidized
        
        Limits potential if reduced
        
        Lumo > Efr and Homo < Efa
        
        Ethylene carbonate is is stabilized by the solid-electrolyte interface
- - - - Solid state diffussion
        
        Random hopping
        
        Covered distance follows gaussian distribution. sigma = jump distance(lambda) * root(number of steps)
        
        Hopping under a driving force
        
        Force F
        
        Energy in/decrease = +-lambda*F/2, depended on direction
        
        Add + the activation energy
        
        Rnet = r0*Flambda/kT if Flambda <<dG
        
        Point defect movements
        
        Rate depends on
        
        Crystal structure
        
        Binding energy
        
        Temperature
        
        Rprogression = fcPdirPavailPbv
        
        Pdir = Chance it goes into the wanted direction, 1/2 or 1/6
        
        Pavail, available vacencies = xvNc(fraction vacenciesnearestneighbours), interstitual holes =1
        
        fc = correlation factor (1-2/N), chance atom goes back
        
        Pb, probability that energy overcomes binding energy
        
        v, vibrational frequency, calculate using E=kT
        
        Vacency jumps 1/xv more often
        
        Flux due to diffussion and migration
        
        From electrochemical potential the diffusion and electromigration flux can be determined
        
        Self diffusion coefficient = r0*lambda^2
        
        Mobility = Dq/kT = sigma/(q carrier density)
        
        Temperature dependence
        
        Mostly for number of vancies and an probability increase
        
        Dm = D0*exp(-Ea/RT)
        
        Atoms is solid can diffuse via point defects (vacencies, interstituals)
      - Point defects
        
        Extrinsic
        
        Doping
        
        Aliovalent substitution
        
        Charge inbalance must thus be compensated
        
        Redox compensation
        
        3 more items...
        
        Vacency formation
        
        Cation/anion addition
        
        Uses
        
        Change of electronical properties
        
        2 more items...
        
        Substituted atom has other oxidation number
        
        Isovalent substitution
        
        Maintains charge balance
        
        Vegards law
        
        Lattice constant or bandgap will change linearly with lattice constant substituted atoms
        
        May show some unlinear behaviour.
        
        Intentional addition of vancencies, interstituals, substitutionals
        
        Can be done up to solubility limit = phase seperation
        
        Are impurity defects
        
        Alloying(high conc)
        
        Intrinsic
        
        Thermodynamics of vacency formation
        
        Thermodynamic equilibrium of vacencies
        
        Schotttsky defects = N0exp(s/2k)exp(-dH/2kT, N0 is number of NaCl pairs
        
        Frekel defects, now also depends on Ni
        
        Vacencies = N0exp(s/k)exp(-dH/kT
        
        When dG = 0, ddG/dn = 0 for minimum
        
        dG = dH - TdS
        
        Bond loss will result in increase of enthalpy (linear)
        
        Bond loss will result in increase of vibration entropy (linear)
        
        Disorderd state will increase entropy via kln(possible states)
        
        Possible states = (n+N0)!/(n!*N0!)
        
        Sublineair with n
        
        Does not take into account kinetics
        
        Ionic compounds
        
        Schottsky defects = vacency of cation and anion, vacency attracts each other
        
        Frenkel defects = vacency of (cat)/anion + interstitual (an)/cation
        
        Most of time interstitual is smallest atom
        
        Needs to satisfy charge balance = at least two defects needed
        
        Other defects
        
        Interchange of cation/anion
        
        Color center: electron trapped at anion site
        
        Defects that form naturally in each structure, caused by entropic behaviour
        
        1d defects: vacencies, interstituals, substitutions
        
        Kruger vink notation
        
        Ac^b
        
        B = net charge of defect
        
        Positive = dot
        
        Negative = '
        
        C = name of lattic position, v, i, Ca
        
        A = chemical name atom
      - Have great influence on the properties of materials
    - - Translational
        
        Crystal structure has translational symmetry defined by a lattice
        
        14 Bravais lattices
        
        lattice
        
        Hexagonal
        
        Orthorombic
        
        Cubic
        
        Triclinic
        
        Tetragonal
        
        Top has lowest symmetry
        
        Basis
        
        Base centered
        
        Face centered
        
        Body centered
        
        Primitive
        
        Ordered, periodic arrangement of atoms
        
        Points that have indentical surroundings
        
        Unit cell
        
        Dimensions
        
        3 angles alpha, beta, gamma
        
        3 lattice vectors a, b, c
        
        When drawing planes in unit cell
        
        Then draw other equivalent planes such that the unit cell can be structured
        
        First draw the plane indicated by the miller indices
        
        Can be translated to form the entire crystal
        
        Nomenclature
        
        Direction in lattice
        
        vector [uvw]
        
        <uvw> is a set of equivalent lattice directions <100> = [100], [010], [001]
        
        Planes in lattice
        
        Miller indices h, k, l
        
        h, k, l are the inverts of the intersections with the x, y, z axis.
        
        bar h = -h
        
        No intersection means 1/infinity=0
        
        (hkl) is one plane
        
        {hkl} is a set of equivalent planes
      - Rotational
        
        Crystallographic point groups
        
        Set of symmetry operators that leave the structure of the crystals unchanged.
      - Space groups symbols
        
        Takes together bravais lattices and rotational operations
    - - 8-N generalized rule
        
        VECa = (mvalency cation + n valency anion)/n = 8 + (m/n)*CC-AA
        
        So if Veca>8 the extra electrons will locate in cation bonding or lone pairs
        
        If VECa< 8 Anion bonding will occur
        
        Electronegative sp atoms of valency N, contain 8-N atoms in bonds. To obtain a stable configuration
        
        Shapes for each AA/CC
        
        2 = trigonal
        
        3 = prism/triple bond
        
        1 = dimer
        
        4 = graphite like
        
        Can also be used to guess oxidation number
      - Crystal chemical formula C^[N]mA^[m]n
        
        Has to follow three balances
        
        Connectivity (nM=Nm)
        
        Bond valence (c/N)=(a/M)
        
        Electroneutrality
        
        List each non-equivalent atom, with the coordination number and geometry in the subscript
        
        Coordination number+geometry
        
        6 = o
        
        4 = t
        
        [3,1] is 3 bonds to first metal, 1 to other
        
        Can be graphically shown in a bond model
    - - Filling of holes in HCP/FCC
        
        Octahedral holes
        
        1 per atom
        
        SIze r = 0.414r0,
        
        Tetrahedal holes
        
        2 per atom
        
        Size r = 0.225r0
        
        Determination filling
        
        if Y<0.225, smaller a will fill tetrahedral holes
        
        Y = (Radius smaller a/ Radius large a)
        
        If 0.225<Y<0.414 smaller a will fill octahedral holes
        
        Voor 0.414<Y<0.73 more like a BCC structure
        
        Anions are bigger and form dense packing of spheres, cations are smaller and may fit in between.
      - Most important closed packed crystals
        
        Zinc blende
        
        0.5 th holes filled + fcc
        
        Diamond
        
        0.5 th holes filled with same atoms + fcc
        
        Wurtzite
        
        0.5 th holes filled + hcp
        
        Rocksalt
        
        All oh holes filled + FCC
        
        HCP
        
        Hexagonal close packing = a-b-a
        
        Coordination = 12
        
        unit cell has ground surface area of (root(2)/2)*a^2
        
        Volume = abc*sin(g)
        
        FCC
        
        Face centered cubic = close packing = a b c
        
        Coordination = 12
        
        Number of atoms/unit cell increases with holes filled
      - Interactions between atoms in solids
        
        Mettalic bonding
        
        See of electrons, maximises for densest structures
        
        Covalent bonding
        
        Overlap of specific orbitals
        
        Ionic interactions
        
        Coulomb interactions, maximises for densest structures
        
        VDW kracht
        
        If no densest structure is formed, other non-isotropic interaction play a role, for example for polonium
        
        Densest packing = isotropic interactions
      - Packing of hard spheres
        
        Larger atom/ion forms densest packing HCP/FCC
  - - - Classical description of ionic bonding = coulomb point particle interactions
      - Total lattice energy = Ua+Ur
        
        Attractive
        
        (z1e)(z2e)A/(4Pied)
        
        A = madelung constant
        
        Summation of all coulomb interactions for on ion in the lattice with its surroundings
        
        6 opposite charges at distance d
        
        12 same charges at distance root(2)d
        
        8 opposite charges at distance root(3)d
        
        Can differ for various crystal structures
        
        Repulsive
        
        Repulsion overlap orbitals
        
        B*e^-d/p
        
        p is measure for size orbitals +- 0.365 Ang
        
        Total = at equilibrium distance do
        
        dE/d0 = 0
        
        ((z1e)(z2e)/(4Pied))*(A-p/d0)
        
        VDW (-1%), +polarisation(-1%), + zero point energy (+1%)
    - - QM description of atomic orbitals
        
        Quantum number
        
        l = angular quantum number, (shape)
        
        ml = magnetic quantum number, (direction)
        
        n = principal quantum number (size)
        
        s = spin quantum
        
        The more charge density close to the nucleus, the lower the energy of the orbital
        
        To determine radial distribution, multiply by 4PiRdr
        
        Determine energy with SE = kinetic + potential energy (ionic interaction)
        
        Phi*HPhi
        
        Orbital has n-1 nodes, n-1-l radial nodes and l nodal planes
      - MO's=linear combination of atomic orbitals (LCAO)
        
        phi = C1phi1+C2phi2
        
        MO
        
        Symmetry
        
        Pi symmetry
        
        Sigma symmetry
        
        Symmetry with rotation over z-axix
        
        ungerade
        
        No inversion center
        
        Gerade
        
        Inversion center
        
        Only atomic orbitals with same symmetry have overlap
        
        s-p overlap
        
        s12 = smax*cos(t)
        
        s-d overlap
        
        si2 = 1/root3smaxcost*sint
        
        Bonding type
        
        Bonding E = H11+H12/(1+S12)
        
        H11 approximately energy AO
        
        H12 energy gained with overlap orbitals
        
        High electron density between nuclei
        
        Hij can be written as integral(phiiHphij)/normalisation
        
        Antibonding
        
        Is more antibonding than bonding mo bonding, because of s12 term
        
        E = H11-H12(1-S12)
        
        Node between nuclei
        
        Higher kinetic energy and less coulom interactions
        
        Bond order
        
        0.5*(number of bonding electrons - number of antibonding electrons)
        
        Higher bond order
        
        Stronger bond
        
        Shorter bond distance
        
        MO energy diagrams
        
        Polyatomic
        
        SALC's
        
        Symmetry adapted linaeir combinations
        
        For each geometry, different overlap of atomic orbitals occur
        
        2 more items...
        
        Central atom with coordinated ligands
        
        Pi conjugation
        
        Pix,y orbitals of benzene
        
        More nodal planes = lower energy
        
        Heteronuclear
        
        AO's have different energy
        
        Different contributions of AO's to MO
        
        Splitting is larger if AO have same energy
        
        Homonuclear
        
        AO's have same energy
        
        Same contribution of each AO
        
        s-p mixing occurs between 2s and 2pz orbitals if close in energy, 2s energy's go down, 2pz energies go up. Above Pi(2P) for N2 and below
        
        Number of AO's is number of MO's
        
        Degeneracy
        
        Number of MO's with same energy
        
        Nomenclature
        
        Born oppenheimer approximation
        
        Electrons move much faster than nuclei so there wavefunction can be decoupled
        
        Variational principle
        
        The lowest calculated energy of an orbital is the best approximation. Ecalculated is always >=Ereal. So choose C1 amd C2 such that the energy is as low as possible
  - - - Bloch function = exp(ikr)*u(r)
        
        Bloch function = wave functions that describe the orbitals in solid, superposition of AO's or MO's. N bloch functions.
        
        u(r)basis set
        
        Tight binding method = basis set are AO's
        
        Follows periodicity of the chain u(r) = u(r+Na), a = distance atoms
        
        Periodic BC
        
        Allowed k values are quantized, using exp(iKNa) = cos(kNa) + i sin(kNa). cos = 1, sin = 0
        
        k = n2Pi/(N*a), for n = 0, +-1 ...N/2
        
        Phi(r)=Phi(r+Na), same probability needed
        
        If N --> inf. K becomes continious betweet -Pi/a, Pi/a
        
        prefactor exp(ikr)
        
        Determines coefficients of the superposition
        
        for k = 0, all wavefunctions are times 1 = totally bonding
        
        for k = Pi/a, all wavefunctions are alternating 1, -1, 1 = totally antibonding
        
        Brillioun zone
        
        AOs unit cell = number of bands = number of brillioun zones
        
        All wavefunctions described by k values in the range -Pi/a, Pi/a
        
        Is a reciprocal space
      - Number of AO's = Number of MO's in solid. So for many atoms, a continious energy band will be formed. Total antibonding at the top and totally bonding at the bottom
      - Resulting wavefunction
        
        Crystal momentum (quasi momentum)= hbar*k
        
        Comibination of periodic part and real part of phase factor
        
        Wavelength = 2Pi/k
        
        2a if k = Pi/a
        
        infinity if k = 0
    - - Semiconductor
        
        Filled valence band and small bandgap
      - Insulator
        
        Filled valence band and high bandgap
      - Metal
        
        Half filled or overlapping bands
      - Bandgap type
        
        Direct bandgap
        
        VBM is directly below the CBM. In K terms
        
        Much easier to excite electron
        
        Indirect bandgap
        
        VBM is not directly below the CBM.
        
        Absorbtivity decreases, because to change the momentum of an electron. Photon absorption has to be paired with phonon interaction, because photon has 0 momentum. Less likely so less absorption. Unless the energy of the photon is high enough to go straight up.
    - - Band structure diagram
        
        Energy electrons are a denoted as a function of k. E(k) = H11+2H12cos(ka) = E0 +0.5Wcos(ka).
        
        Is symmetric around k = 0, because energy goes with k**2
        
        Band width W depends on orbital overlap. If assumed s12 = 0. W = 4*H12
        
        Velocity electrons depends on slope dE/dk
        
        More complicated for 3d heterogenic structures
      - DOS
        
        Density of states N(E) representation
        
        Number of states per unit energy and unit volume
        
        Great increase in states at the ends of the bands
        
        Bands are filled at T=0k up to the fermi level, can be excited for T>0
      - Energy of H2 chain is favoured above H chain because of favourable splitting
        
        Bandgap = H-H interactions
        
        Bandwith = H2-H2 interactions
        
        Note: energy does not always increase with k: always draw MO's
        
        Valence band and conduction band are formed
        
        Peirls distortion: Partially filled band in 1d chain results in alternating chain lengths. --> dubbele binding-enkele binding-dub
    - - 2D
        
        Reciprocal lattice vectors (2d)
        
        a* = 2Pi/a
        
        b* = 2Pi/b
        
        Are perpendicular with real lattice vectors a b = 0, ab = 0
        
        Can be used to draw brillioun zones
        
        Brillioun zone = -a/2, -b/2 --- a/2, b/2
        
        gamma = lowest energy, k = 0,0
        
        X = higher energy at boundary, k = a*/2, 0
        
        M = highest energy, k = a/2, b/2
        
        k becomes now a vector k(a,b)
        
        Can be used to describe surfaces
      - 3D
        
        Works the same, only other r lattice vectors