Spring 2010 MSE 111. Midterm Exam. Prof. Eugene E. Haller. University of California at Berkeley Department of Materials Science and Engineering

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1 Spring 00 MS Midterm xam Prof. ugene. Haller University of California at Berkeley Department of Materials Science and ngineering 3/6/0, 9:40 am 80 minutes, 74 points total, 0 pages ame: SID: Problem OAL Points Possible Points SHOW ALL OF YOUR WORK!!! Answers given without supporting calculations will be marked wrong, even if they are numerically correct. of

2 . (5 pts.) rue/false a) rue or False he number of free electrons in a metal increases according to the Boltzmann distribution. b) rue or False he position and momentum of an electron can be known precisely at the same time. c) rue or False SiC has purely covalent bonds. d) rue or False Bonding electrons in Cr have localized character. e) rue or False Deep level dopants are more delocalized in k-space compared to shallow level dopants. of

3 . (6 points) Phonons a) (4 pts.) eatly sketch the dispersion relation for phonons in a -D chain of two different types of atoms (i.e., K-Br-K-Br ) on the axes below..o. L.O. L.A..A. 0 A B -/a 0 /a k b) (4 pts.) Given the following three expressions, label all endpoints on your sketch in a for all longitudinal phonon frequencies (i.e., both optical and acoustic if both are present). A M C lighter _ atom C B o C M heavier _ atom M M c) (3 pts.) Assuming a lattice constant of a, label the minimum and maximum values of k on your sketch. Why did you choose these values (one short sentence)? ±/a because this is the edge of the Brillouin Zone. Outside of these values indicates waves shorter than the distance between reciprocal lattice points which mean nothing A 0 d) (5 points) ow, given that the optical frequency at k/k max is 4x0 Hz for a KBr chain, calculate the optical frequency at k0 for KBr. he atomic masses are 39 for K and 80 for Br. 4x0 C 80 Hz C C 3.x x0 39 Hz 6 (Don t ding them twice if they labeled the above graph wrong. You can assume math is ok.) 3 of

4 3. ( points) Kronig Penney A quantum well structure is grown with Å GaAs layers sandwiched between 5 Å Al -x Ga x As layers. his means that the well width, a, is Å while the well spacing, w, is 5Å. he following expressions and plot are provided to answer parts a-d. You may assume that the effective mass of an electron in this structure is 9.x0-3 kg. Make your procedure obvious since you will be reading numbers off of a graph. P m* a h V w * m 0 h sin( a) F( a) P cos( a) a / a a a 3 a) (4 pts.) Calculate the height of the well, V 0, in ev using the plot. Show your work. F( / ) ( fromgraph).6 P 0 P / Ph V0 3.eV mawe 4 of

5 b) (4 pts.) Shade the allowed minibands on the plot above. See plot c) (4 pts.) What is the width of the first allowed energy band and first disallowed energy gap in ev? h me ( ) where a / a a.5 -> 4.8eV a3.->9.7ev a >39.3 W allowed eV W disallowed 0.4eV xtra credit ( pts.): Suggest a quantitative cutoff for the minimum energy gap to be observable at 300K eV k 5 of

6 4. Hall ffect (0 points) he setup shown below is used to measure the electrical characteristics of a piece of Si. he esla magnetic field points IO the page (-z direction) and the ma applied current flows in the x direction. a) ( pts.) In which direction would holes be deflected due to the Hall effect? In which direction would electrons be deflected? Direction deflected Holes Up or y lectrons Up ory b) ( pts.) ow, considering the voltmeter reading of 5 mv, is the Si sample n-type or p- type? n-type 6 of

7 c) (6 pts.) Assuming a carrier effective mass of 0. and resistivity of 0.0 -cm, what do you expect the mobility and lifetime of a carrier in this sample to be? (Hint: check out the equation sheet and clearly show your work.) n JB 0.00A wherej 5A/ m and e 0.0mx0.m H n 4.7x0 9 m 3 μ en 0.000m 4.7x0 μm x0 9 e.6x0 3 9 m 3 8.5x0 H 749m e V 0.0m / Vs secor0.85ns 0.75V / m 5. (7 pts.) Density of States Derive the density of states, Z(), for electrons in a very thin metal film. You should treat the system as two dimensional. his is not just an academic exercise, as there are materials that have two dimensional properties. Quantum wells and graphene (a single sheet of graphite) are two examples, although the physics is a bit more complicated. In the free electron model, the D system can be represented as a thin film with sides of length a. hese a define the length of a D infinite potential well. For a D infinite potential well, the energy is given by: h 8m a e ( n n ) 7 of

8 Only positive n and n are allowed. ach n and n combination is an orbital state. We can define a new variable n as: n n n herefore h 8m e a n Let us consider how many states there are with energies less than. corresponds to n n. h n n 8a m e 8m e a h ach state, or electron wavefunction in the crystal, can be represented by a box at n, n. n n n n n n' n 0 n n, n From the previous figure, we know that all states within the quarter arc defined by n have <. he area of this quarter arc is the total number of orbital states. he total number of states, S, including spin is twice as many, S n a m e h S 4a m e h he density of states g is defined as the number of states per unit area per unit energy. herefore, ds 4 a me 4 me g a d a h h From the previous result we can conclude that, for a two dimensional system, the density of states is constant. n 8 of

9 9 of 6. (8 pts.) Semiconductor Statistics a) ( pts.) Write down the form of the Fermi distribution function for holes () k xp f b F holes b) ( pts.) Simplify the above equation in the case of F >> k. () () k xp f k xp f b F holes b F holes c) (4 pts.) At what temperature is there a 5 percent probability that a state with an energy 0. ev below the Fermi energy will be occupied by a hole? () K xp k xp f b F Alternatively, () K xp k xp f b F holes (6 pts.) Semiconductors a) (4 pts.) In intrinsic semiconductors the concentration of free carriers is a strong function of temperature. he intrinsic carrier concentration in Ge ( g 0.66 ev) at 300 K (room

10 temperature) is 30 3 cm -3. Ignore the temperature dependencies of g, C, and V. At what temperature will the concentration be two orders of magnitude lower? n i g exp 30 c v exp 5 kb c v c v cm exp K b) (6 pts.) Semiconductor Doping Fill in the tables below by specifying the impurity type in Si and in AlAs. For AlAs, give an answer for both atomic sites. Use acceptor A, double acceptor A, donor D, double donor D, or neutral. Mark an X for dopants that would be triple donors or acceptors. Dopant Al site As site Ge in AlAs Ga in AlAs e in AlAs Mg in AlAs D X A A A D X Dopant Sb in Si Al in Si Be in Si P in Si ype D A A D c) ( pts.) Briefly explain the difference between extrinsic and intrinsic semiconductors. An intrinsic semiconductor is a material with no impurity atoms present in the crystal. An extrinsic semiconductor is defined as a semiconductor in which dopants or impurity atoms have been added. d) ( pts.) In one sentence, what is a phonon? A phonon is a quantized lattice vibration e) ( pts.) Within the framework of the hydrogenic model, can you get the carrier effective mass 0 of

11 from a plot of vs. k of a given semiconductor? If so, how? If not, why not? Yes, you can. he effective mass is proportional to the inverse of the second derivative of the /k parabola. of

12 Constants and quations: x0 kg h.054x0 J s h 6.65x0 J s m e k b 8.6x0 ev / K e.6x0 C c 3x0 m / s ev 9 3.6x0 J A 6.0x0 atoms / mole sin x p h / F m a h h mv p k n d sin p m h c n h 8mL mv h n L R A e μ m e ( n e μe nh h μ ) J evd e μ e H RH JB R H e time-independent Schrödinger equation: h V m x g n i cv exp kb g F n c exp kb v * 3 / m hkb h p exp F v kb c m h * e k b 3 / 4C M / ka sin of