Solution for Examination WS 0/0 Materials and Devices in Electrical Engineering Monday 17 th of March, 9:00 11:00, International Department, SR. 0 Notice 1. It is allowed to use any kind of aids (books, scripts, notes,...) during the examination.. Any type of communication with other people is forbidden.. Please write down your name and registration number on every page that contains a part of your results. 4. Please use blue or black pen or ball pen, don t use pencil. 5. The examination duration is hours. Information on the exercises The examination consists of two parts: a) Questions (0* = 60) This part contains 0 questions. For multiple choice questions, notice that there could be one or more correct answer(s) for each question. You should decide, which one is correct, then cross (X) the corresponding letter. You will get points for each question, only when all the correct answer(s) is/are chosen. For simple answer questions or fill in blanks questions, you should give a short, accurate answer to the corresponding question, or fill a certain number/value/word in the blank to make the sentence complete and correct. The total points for one such question is. The maximum points you can achieve is 60. b) Calculations (60) The second part contains 4 calculations, each consisting of several tasks a), b), c),... If you are not able to solve one task, go on with the next one using the given values. The problems should be solved on the sheet below the task. If there is not enough space you can ask for additional sheets. Please use an extra sheet for each problem. You can get maximum 60 points for the calculations. The number of points for each task is given in the text. If you solve only a part of a calculation or of a task, or if your results are incorrect, you will get a lower value of points. Questions Calculations 1 4 Σ Result - 1 -
Questions (0 questions, each points, S = 60 points) 1. For Bohr atom model, it is true that (b, d) a) electrons revolve around the atomic nucleus in continuous orbitals. b) the position of the electron is well defined in terms of its orbital. c) the position of the electron is described by a probability distribution. d) the energies of electrons are quantized. e) electrons exhibit both wavelike and particle-like characteristics.. The atomic number for element Calcium is 0 ; its electron configuration is _ 1s s p 6 s p 6 4s. The valence electrons for this element are 4s.. When the interatomic separation for two isolated atoms reaches equilibrium spacing r 0, a) the corresponding potential energy is the bonding energy for these two atoms. b) the net potential energy E N = E 0 = 0. c) the net force F N = F A + F R = 0. (a,c) d) the net force F N = F A + F R = minimum. 4. What is correct for atomic bonding in solids? (a,c) a) The sp hybrid bonds in CH 4 molecules are covalent, there is a sharing of valence electrons between C and H, and the bonding angle is about 9. b) The larger the difference in electronegativity, the smaller the degree of ionicity. c) Metallic bonding leads to high conductivity of the material, because the valence electrons form a sea of electrons. d) The chemical bonds in element semiconductors are highly ionic. 5. What is FCC crystal structure? face-centered cubic. A total of atoms can be assigned to a given BCC unit cell. For the BCC structure, the coordination number and the APF are 8 and 0,68. 6. When concerning the crystal systems, it is true that (d) a) for FCC and BCC crystal structure, the (001), (0) and (001) planes all belong to the {0} family, and the (1) atomic planes have the same atomic packing. b) the monoclinic system has the least symmetry. c) in all crystal systems, crystallographic directions are specified by indices as [uvw]. d) an assumed unit cell, a=b=0,406nm, c=0,94nm; α =β =γ =90, has a tetragonal crystal structure. 7. For point defects in crystal lattice of solids: (b) a) Schottky defects only occur in ionic solids. b) Their number increases exponentially with increasing temperature. c) They do not change the density of the material. d) At room temperature, their concentration is the same in different crystals. 8. About imperfections in solids, which statement is correct? (a,b) a) The concentrations of vacancies are significantly higher than that of self-interstitials. b) Linear defects include edge- and screw- dislocations, which can be expressed in terms of Burgers vectors c) Interfacial defects include grain boundaries, twin boundaries, pores, and inclusions. - -
9. For steady-state diffusion, which description is correct? (b,c) a) The diffusion flux (J) is proportional to the concentration gradient (dc/dx), and is timedependent. dc b) It can be represented as Fick s 1 st law: J = D. dx c) The diffusion coefficient D is temperature- and activation energy-dependent. Qd 1 d) There is a relationship of: ln D = ln D0 + ( ). R T. Regarding thermal properties of the materials: (b,c) a) Heat capacity is not associated with the contribution of vibrational energy. b) The larger the interatomic bonding energy, the deeper and more narrow the potential energy trough, as a result, the smaller the coefficient of thermal expansion α l. c) Thermal shock is the fracture of a body resulting from thermal stresses induced by rapid temperature changes. 11. There are four types of stress-strain tests: tension, compression, torsion, and shear_ For a material that is stressed by tensile load, the correct sequence is l,i,j,k. i. experiences the gradual elastic-plastic transition where yielding occurs; j. continues plastic deformation up to the maximum point of stress where tensile strength correspondingly reaches; k. results in fracture if stress is continuously applied; l. undergoes elastic deformation, i.e. stress and strain are proportional; 1. Amorphous solids (b) a) can be manufactured using the same procedure as that for single crystal materials. b) show only short range order but no long range order. c) exhibit anisotropic behavior. d) have discontinuous volume change with the change of temperature. 1. About polymers, which statement is correct? (b,d) a) Organic molecules that have single or double covalent bonds are termed saturated. b) Semi-crystalline polymer consists of crystalline and amorphous regions. c) Polyethylene (PE) is a kind of copolymer, the monomer of this polymer is H H. d) Linear, branched, crosslinked, and network molecular structures are possible for polymers. 14. The electrical conduction (b,d) a) takes place in metal via phonons. b) in semiconductor arises from electrons and mobile holes. c) is exponentially related to the number of free electrons and the electron mobility. d) is contributed by both electrons and ions in ionic-bonded materials. e) doesn t happen in any polymeric materials. 15. For applications of metals and alloys: Precision-resistor should possess high resistivity and very low temperature coefficient. Thermocouples should have a high Seebeck-coefficient. Resistive strain gauges can be used as force and strain sensors. C C H H - -
16. The temperature dependence of the electrical conductivity of a highly pure silicon crystal is predominantly determined (a) a) by the temperature dependence of the charge carrier concentration. b) by the temperature dependence of the charge carrier mobility. c) by the ionization energy of Si. d) by the relatively narrow band gap E g. 17. For insulators and intrinsic semiconductors, (b,c,d) a) increasing the temperature leads to an enhancement of resistivity. b) E f lies near the band gap center. c) at a given temperature, the larger the band gap, the lower the electrical conductivity. d) electrons can be excited into conduction band only when the excitation energy is at least equal to the band gap. 18. What is Fermi energy E f? _the energy corresponding to the highest filled state at 0K_. What is band gap energy? E g = E C- E V. 19. For semiconductors, it is correct that (b,d) a) Si and GaAs are two element semiconductors. b) The energy level of an acceptor state is just above the top of the valence band. c) n = p is not valid for an intrinsic semiconductor. d) The Hall effect experiment can be conducted for determination of majority charge carrier type, concentration and mobility of a certain material. e) transistors are electronic devices that allow the current to flow only in one direction. 0. In dielectrics, the orientation polarization (b,c) a) arises from the displacement of cations and anions in an ionic crystal. b) offers a decreasing contribution to the total polarization with increasing temperature. c) results from orientation of the permanent dipoles in the direction of the applied field. d) results from the displacement of the center of electron cloud relative to the nucleus. 1. Two parallel metal plates are separated from each other for a certain distance, and are connected with constant voltage. A ceramic disk is then introduced to the gap. Thus, the capacitance, the electric field and the dielectric displacement of the capacitor: (b) a) C, E become smaller, D remains the same b) C, D become larger, E remains the same. c) all become larger. d) all remain the same. e) all become smaller.. When electric field present, the electric susceptibility of dielectric materials χ Σ can be represented as: In C, Si and Ge: χ Σ = χ e. in NaCl and MgO: χ Σ = χ e + χ ion. in BaTiO (T > 600 K): χ Σ = χ e + χ ion. in Sb + and Fe + doped BaTiO barrier layer capacitor: χ Σ = χ e + χ ion + χ or + χ RL. - 4 -
. About different types of capacitors, _Foil capacitors_ may have large capacitance due to the very large surface area. Electrolyte capacitors possess a high volume capacity, because on the anode there is a very thin anodic oxidation layer which acts as dielectric. Ceramic barrier layer capacitors have conductive grains and insulating barrier layers. 4. Pb(Zr,Ti)O (a,b,c,d) a) show a linear correlation of P ~ E above 500 C. b) may exhibit different structure concerning its composition at room temperature. c) possesses a maximum coupling coefficient k at the morphotropy phase boundary. d) is a kind of ferroelectric material with spontaneous polarization, and can be used as electromechanical transducer. 5. BaTiO (a) a) possesses permanent electric dipoles. b) experiences phase transitions when temperature increases from -00 C to 00 C. c) shows a pronounced hysteresis of P(E) diagram above Curie temperature. d) exhibits P = P r at coercive E-field. 6. The ionic contribution to the total conductivity _increases_ with increasing temperature. For solid oxide fuel cell (SOFC), _Y-doped ZrO _ can be used as solid electrolyte material, Sr or Co doped LaMnO is a typical _cathode_ material, and Ni is a very common _anode_ material. 7. For magnetic properties, it is correct that (a,b,c) a) in ferromagnetic materials H << M is valid. b) With the increment of the applied external H field, the domains in ferromagnetic material are oriented in the direction nearly aligned with the H field. c) for electrons, magnetic moments are originated from two different sources. d) Fe O 4 is a kind of antiferromagnetic material which possesses no net magnetic moment as a whole. 8. The susceptibility of paramagnetic materials (a) a) decreases with increasing temperature. b) is negative. c) is temperature independent. d) increases with increasing temperature. 9. For ferromagnetic B(H) hysteresis loop, (a,c) a) a residual B exists at zero H field means that the material remains magnetized in the absence of the external H field. b) the remanent flux density is reached when all the magnetic dipoles are aligned. c) materials with rectangle hysteresis curve are ideal choices for digital magnetic storage. 0. Which statement about ferromagnetic and ferroelectric materials is correct? (c,d) a) They are oxide-ceramic material which must be prepared by a sintering process. b) They both contain iron. c) They exhibit hysteresis behavior below Curie temperature. d) Ferromagnetic materials possess permanent magnetic, while ferroelectrics possess electric dipoles. - 5 -
Calculations (S = 60 points) 1. A silver wire has a diameter of mm and a length of 6 m. (S = 14 points) a) Compute the resistance of the wire. The electrical conductivity for Ag is 6,8 7 (Ω m) -1 at room temperature. ( points) For Ag: σ = 6,8 7 (Ω m) -1 l l 6 R = ρ = = A σa 7 6,8,14 ( ) = 6 =,81 1,5 Ω b) When the potential drops across the ends of the wire is 0,000 V, compute the electric field strength and the current density across the two ends of the wire. (4 points) E = U / l = 0,000V / 6m =, -5 V/m V 0,000 J = σ E = σ = l 6 or j = I / A = U /R x A =,7 x A / m 7 6,8 =,7 A / m c) Calculate the equilibrium number of vacancies per cubic meter for silver at 800 C. The energy of vacancy formation (activation energy) is 1, ev/atom; the atomic weight and density (at 800 C) for silver are 7,9 g/mol and 9,5 g/cm, respectively. (* Boltzmann s constant k = 8,6 x -5 ev / atom K) (5 points) The number of atomic sites per cubic meter for Ag: N = N A A ρ Ag (6,0 = 6 atoms/ mol)(9,5g / cm )( cm 7,9g / mol / m ) = 5, 8 atoms/ m The number of vacancies at 800 C (7 K): N v Qv = Nexp( ) = ( 5, kt = ( 5, 8 atoms / m ) e 8 ( 1189, ) atoms / m = 5, ( 1, ev / atom ) )exp ( 8, 6 ev / atom K )( 7K ) 8 atoms / m 686, 6 =, 64 vacancies / m - 6 -
d) The crystal structure of silver is FCC. Calculate the atomic packing factor (APF) and determine the coordination number for this structure. ( points) Ag: FCC a = 4R a = R, totally 4 atoms in unit cell. APF = volume of atoms in unit cell / total unit cell volume = 4 x 4/ π R / a = 0,74 coordination number: 1. Semiconductivity (S = 18 points) a) Using the associated data given in Table 1, compute the electron and hole concentrations of high-purity Ge single crystal at room temperature. ( points) e = 1,6-19 C Table 1 Characteristics of intrinsic silicon at 00 K Semiconductor Band gap E g (ev) Electron mobility µ e (m² / Vs) Hole mobility µ h (m² / Vs) Electrical conductivity σ i (Ω m) -1 Ge 0,67 0,8 0,18, σ = e 0 (n µ e + p µ h ) Read from Table 1: σ =, (Ω-m) -1, µ e = 0,8 m² / Vs, µ h = 0,18 m² / Vs, and e 0 = 1,6-19 As intrinsic semiconductor n = p n = p = σ / [e 0 (µ e + µ h )] =, (Ω-m) -1 / [1,6-19 As (0,8 m² / Vs + 0,18 m² / Vs)] =,455 19 m - b) Phosphorus is now doped into this crystal to obtain a concentration of m - charge carriers at room temperature. Is the material n-type or p-type? Give a short explanation. Which kind of other elements can be used instead of this dopant? (4 points) P is a Group VA element, having 5 valence electrons, acts as a donor in germanium. Thus, the m - charge carriers will be approx. all electrons. Hence, this material is extrinsically n-type. The other Group VA elements, e.g. As and Sb can be used. - 7 -
c) How to call the energy level introduced by this type of impurity? Where is it located in the energy band? ( points) donor state, located within the band gap and just below the bottom of the conduction band. d) Calculate the conductivity of this doped germanium at room temperature (T = 00 K). Then calculate the electrical conductivity of intrinsic germanium at 00 C. (* Boltzmann s constant k = 8,6 x -5 ev / K) (8 points) 1) Read from Table 1: µ e = 0,8 m² / Vs; n = m - and e 0 = 1,6-19 As σ e 0 n µ e = (1,6-19 As) ( m - )(0,8 m² / Vs) = 6080 (Ω-m) -1 ) calculate the value of C at RT: C = lnσ + E g / kt = ln(,) + 0,67 ev / x (8,6 x -5 ev / K) x 00 K = 0,788 + 1,95 = 1,74 00 C = 47 K lnσ = C - E g / kt = 1,74 0,67 / x (8,6 x -5 ev / K) x 47 K = 1,74 8, = 5,5 σ = 49,64 (Ω-m) -1. As shown in Fig., a parallel-plate capacitor having an area of 5 mm 5 mm and a plate separation of mm. A piece of BaTiO ceramic with the corresponding dimension is positioned within the region between the plates. (S = 18 points) 000 8000 6000 ε r 4000 000 0 40 60 80 0 140 160 180 00 0 Temperature ( C) - 8 -
Fig. 1. Dielectric constant of BaTiO ceramic vs. T Fig.. Dealing process of the parallel-plate capacitor a) Firstly the capacitor is placed in an oven with a temperature of 00 C and connected to a voltage of V. Read the specific value from the given graph (Fig. 1), calculate the capacitance, the dielectric displacement and the polarization, respectively. (8 points) T = 00 C, read from Fig. 1: ε r = 000 for BaTiO 1 A A 8, 85 C = ε = ε0εr = l l Q C = Q = C V =, 1 V 0 0 a) b) 0 As/ Vm 000 5 m F V =, 1 9 C 6 m =, 1 V 1 V D = εe = ε0εr = 8, 85 As / Vm 000 = 8, 85 C / m l m 1 V 8, 85 As / Vm V P = D ε 0E = D ε0 = 8, 85 C / m = 8, 846 l m orp = ε ε E ε E = ( ε 1) ε E = 8, 846 C / m r r b) Then the voltage supply is disconnected, and the ceramic capacitor is taken out of the oven, cooled for a while until its Curie temperature is reached. Now decide which character(s) in a) will be changed, make the calculation and give your answer. (8 points) Now T = T C = C, read from Fig. 1: ε r = 000 for BaTiO. Since V is disconnected, Q remains the same value. C, V, E, P will be changed, C, Vfl, Efl, P ; while D remains the same. 1 6 A A 8, 85 As/ Vm 000 5 m 9 C = ε = ε0εr = = 15, F l l m 9 Q Q, 1 C C = V = = = V 9 V C 15, F V V E = = = V / m l m V Q Q D = εe = ε0εr = ε0εr = = 885, C/ m l C l A 1 V 885, As / Vm V P = D ε 0E = D ε0 = 8, 85 C/ m = 8849, l m F C / m C/ m - 9 -
c) Is this capacitor suitable for using in the circuit of an electrical equipment for cooking (at temperatures up to 50 C)? Why? Give a short explanation. ( points) No. Because in the temperature range of 50 C ~ 00 C, ε r of BaTiO has a very significant temperature dependency. Therefore, the capacitance value varies extremely. It can not be used into the circuit. 4. Concerning magnetic properties, (S = points) a) Giving the saturation magnetization for nickel is M S = 5,1 5 A/m. Calculate its saturation flux density. (* Vacuum permeability µ 0 = 4π x -7 H / m) ( points) B S = µ 0 M S = (4π x -7 H / m)( 5,1 5 A/m) = 0,64 tesla b) Tungsten-steel is a kind of hard magnetic material. Assuming that the material is initially unmagnetized, using the associated data given in Table 1.6 (p694), sketch the entire B(H) hysteresis curve in the given range. Scale and label both coordinate axes, denote all the characteristic points in the graph. (7 points) Assuming the saturation flux density: 1,1 tesla at 8000 A/m 0 - -