The Nernst-Einstein equation indicates that the ratio β /D for a given material varies only with temperature. Calculate β/d for oxygen ions in Zr 0.

Transcription

1 The Nernst-Einstein equation indicates that the ratio β /D for a given material varies only with temperature. Calculate β/d for oxygen ions in 0.8 Y at 800 C. 1

2 The Nernst-Einstein equation indicates that the ratio β /D for a given material varies only with temperature. Calculate β/d for oxygen ions in 0.8 Y at 800 C. 2

3 Simpson and Carter (J. Am. Ceram. Soc. 49 (1966) 139) measured the self diffusion coefficient for oxygen in 0.85 Ca and found it to be D = cm 2 /s at 1100 C. Calculate the electrical mobility and conductivity of oxygen ions based on this. Assume density of 0.85 Ca g/cm 3 and molecular weight g/mole. 3

4 4 Ca Ca Ca 0.85 Ca SI

5 Simpson and Carter (J. Am. Ceram. Soc. 49 (1966) 139) measured the self diffusion coefficient for oxygen in 0.85 Ca and found it to be D = cm 2 /s at 1100 C. Calculate the electrical mobility and conductivity of oxygen ions based on this. Assume density of 0.85 Ca g/cm 3. 5

6 Electrical conductivity Zahl der 0.85 Ca Einheiten per m 3 6

7 For intrinsic silicon, the room-temperature electrical conductivity is 4apple10-4 Ω -1 m -1 ; the electron and hole mobilities are, respectively, 0.14 and m 2 V -1 s -1. Compute the electron and hole concentrations at room temperature. 7

8 For intrinsic silicon, the room-temperature electrical conductivity is 4apple10-4 Ω -1 m -1 ; the electron and hole mobilities are, respectively, 0.14 and m 2 V -1 s -1. Compute the electron and hole concentrations at room temperature. Solution: 8

9 Calculate concentration of the charge carriers in intrinsic Si in a function of temperature. Temperature dependence: E g =1.14 ev energy gap, k= ev/k (mole fractions) Temperature (K) K g N e =N h

10 Intrinsic Silicon (mole fractions) Temperature dependence: E g =1.14 ev energy gap, k= ev/k Temperature (K) K g N e =N h

11 What is the number of the oxygen vacancies in the unit cell of 0.8 Y ? Assuming the lattice parameter of (cubic) YSZ is 0.54 nm, calculate a concentration of the oxygen vacancies (number per m 3 ). (Y) Fluoritstruktur (CaF 2 -Typ) 11

12 In 0.8 Y , how many oxygen vacancies are there per unit cell? If the lattice parameter of (cubic) YSZ is 0.54 nm, calculate the density of vacancies (number per m 3 ) (Y) Formula V per unit cell V c = m 3 Fluoritstruktur (CaF 2 -Typ) 12

13 Defektkonzentration n/n 0 bei verschiedenen Temperaturen Temperatur [ o C] Aktivierungsenergie ev

14 Write the Kröger-Vink notation for the following fully charged species in Mg: Cation and anion on their normal sites xygen vacancy Magnesium vacancy Interstitial magnesium ion 14

15 Write the Kröger-Vink notation for the following species in 2 : Cation and anion on their normal sites xygen vacancy Zirkonium vacancy Yttrium dopant substituting Nitrogen ion (N 3- ) substituting for oxygen ion Write the Kröger-Vink notation for the following fully charged species in CaTi 3 : Calcium vacancies Titanium vacances xygen vacances Ti ions on Ca sites and vice versa Ti interstitials 15

16 Write the Kröger-Vink notation for the following species in 2 : Cation and anion on their normal sites xygen vacancy Zirkonium vacancy Yttrium dopant substituting Nitrogen ion (N 3- ) sobstituting for oxygen ion Write the Kröger-Vink notation for the following fully charged species in CaTi 3 : Calcium vacancies Titanium vacances xygen vacances Ti ions on Ca sites and vice versa Ti interstitials 16

17 Write the electroneutrality condition for defects in silicon : pure boron-doped phosphorous-doped 17

18 Write the electroneutrality condition for M 1-x Write the electroneutrality condition for M 1+x (oxygen interstitial sites) Write the electroneutrality condition for M 1-x Write the electroneutrality condition for M 1+x (metal interstitial sites) 18

19 Write the electroneutrality condition for M 1-x 19

20 Write the electroneutrality condition for M 1+x (oxygen interstitial sites) 20

21 Write the electroneutrality condition for M 1-x 21

22 Write the electroneutrality condition for M 1+x (metal interstitial sites) 22

23 Metal oxide Me 2 is doped with Mf 2 3 at the doping level At a certain temperature T and oxygen partial pressure 10-9 atm, concentration of oxygen vacancies is Make a plot showing dependence of point defects concentration (, and ) on oxygen partial pressure at T. Identify the charge carriers and regions of intrinsic and extrinsic conductivity. 23

24 Brouwer (Patterson)-Diagramm T=const -1/6-1/4 0 intrinsic extrinsic 24

25 Cobalt oxide: The electronic conductivity of Co 1-y at 1350 C and p 2 = 0.1 atm is 25 S/cm. Thermogravimetric measurements show that y = under the same conditions. It is assumed that singly charged cobalt vacancies are the dominating point defects. Identify the charge carriers responsible for the conductivity and calculate their charge mobility. (Assume that the density of Co at 1350 C equals that at room temperature, 6.4 g/cm 3. Atomic weights M Co = 58.93; M = 16.00; q= C) Platzverhältnis Die Anzahl an Kationenplätzen (K) einer Verbindung K x A y muss immer im richtigen Verhältnis zur Anzahl der Anionenplätze (A) stehen 25

26 Holes mobility 26

27 Nickel oxide: Assume that doubly charged nickel vacancies and holes are the dominating defects in Ni 1-y under oxidising conditions. At 1245 C and p 2 = 1 atm we know the following for the compound: The self diffusion coefficient for nickel: D Ni = cm 2 /s Electrical conductivity: σ = 1.4 S/cm (Data from M.L. Volpe and J. Reddy, J. Chem. Phys., 53 (1970) 1117) Nickel vacancy concentration, in site or mole fraction: [V Ni ] = 2.5apple10-4 (Data from W.C. Tripp and N.M. Tallan, J. Am. Ceram. Soc., 53 (1970) 531) (Atomic weights M Ni = 58.71, M = 16.00, density of Ni = 6.67 g/cm 3 ) a) Calculate the charge mobility of the nickel vacancies and the ionic conductivity under the conditions referred to above (Nernst-Einstein Gleichung) b) Calculate the concentration of electron holes under the given conditions, given as site fraction and as volume concentration ( number/m 3 ). c) Calculate the charge mobility of the holes. q= C k= J/K 27

28 Nernst-Einstein Point a 28

29 a) nickel vacancies Compare the obtained value with σ = 1.4 S/cm =140 S/m are not dominating carriers 29

30 b) holes site fraction Volume concentration 30

31 c) holes σ for nickel vacances 31

32 1. Calculate EMF (EMK) at 500 and 1100K for fuel cells in which Methane (CH 4 ) or Hydrogen is used as a fuel. Assume that the partial pressures of all the gaseous reactants are equal 1 bar (pure oxygen at the cathode!). 2. Calculate what will be change of EMF at 1100K in the case of CH 4 fuel, assuming total pressure of the gases at both the electrodes 1 bar (pure oxygen at the cathode!) and composition at anode 50%H 2, 25%C 2 and 25%CH 4. I. Barin,. Knacke, Thermochemical properties of inorganic substances, Springer- Verlag,

33 ½ 2 + H 2 apple H 2 Kathode: ½ 2 + 2e - apple 2- Anode: 2- + H 2 apple H 2 + 2e - 500K appleg= ( )-0.5 ( )= kcal/mol = J/mol E=-( )/( )=1.139V 1100K kcal/mol J/mol 0.973V 33

34 2 2 + CH 4 apple 2H 2 +C 2 Kathode: e - apple 4 2- Anode: CH 4 apple 2H 2 + C 2 +8e K appleg = J/mol E o =-( )/( )=1.039V V 34

35 n the diagram show the doping regions for intrinsic and doped silicon at room temperature. (mole fractions) n n-type p 35

36 Doped silicon 1. Phosphorus is added to high-purity silicon to give a concentration of m -3 of charge carriers at room temperature. a) Is the material n-type or p-type? b) Calculate the room-temperature conductivity of this material, assuming that electron and hole mobilities (respectively, 0.14 and m 2 V -1 s -1 ) are the same as for the intrinsic material Density of Si 2.33 g/cm 3 ; molecular weight g/mol (mole fractions) q= C 36

37 1. a) Phosphorus- V group, will act as a donor in silicon b) m -3 electron concentration is greater than that for the intrinsic case (mole fractions) 37

38 Doped silicon 2. The room-temperature conductivity of intrinsic silicon is 4apple10-4 Ω -1 m -1. An extrinsic n-type silicon material is desired having a room-temperature conductivity of 150 Ω -1 m -1. a) Specify a donor element type that may be used and its concentration in atom percent. b) Calculate the equilibrium hole concentration Assume that electron and hole mobilities (respectively, 0.14 and m 2 V -1 s -1 ) are the same as for the intrinsic material, and that at room temperature the donor atoms are already ionized. Density of Si 2.33 g/cm 3, molecular weight g/mol. (mole fractions) E g =1.14 ev, k= ev/k 38

39 2. a) P, As, Sb 39

40 2. b) 40