Deprtment of Electricl nd Computer Engineering, Cornell University ECE 4070: Physics of Semiconductors nd Nnostructures Spring 2014 Exm 2 ` April 17, 2014 INSTRUCTIONS: Every problem must be done in the exm booklet Only work done on the exm booklets will be grded do not ttch your own sheets to the exm booklets under ny circumstnces To get prtil credit you must show ll the relevnt work Correct nswers with wrong resoning will not get points All questions do not crry equl points All questions do not hve the sme level of difficulty DO NOT WRITE IN THIS SPACE 1
Problem 1 (Bnd electrons) 40 points ) Consider 2D crystl in the xy plne. The crystl hs single vlence bnd with prbolic nonisotropic dispersion reltion. A constnt nd uniform mgnetic field B Boz ˆ is pplied in the z- direction. Assume tht the crystl hs lots of holes in the vlence bnd nd the holes fill the vlence bnd down to n energy E F below the vlence bnd mximum E v. The rest of the vlence bnd is full of electrons. Consider n electron in the vlnce bnd on the hole Fermi surfce whose position in k-spce t certin point in time is s shown in the figure below. /b = 2 Electron b Hole Fermi surfce Sketch the trjectory of the electron motion in rel spce, indicte the direction of electron motion with rrows, nd indicte where the electron is on tht trjectory when in k-spce it is s shown bove. (10 points) b) Consider 2D crystl with squre Brvis lttice in the xy plne. A constnt nd uniform mgnetic field B Boz ˆ is pplied in the z-direction. Fermi surfce Electron FBZ The conduction bnd dispersion is show bove. At lrge electron number the Fermi surfce in k-spce is s shown bove (solid lines) in the First Brillouin Zone. Now consider n electron on the Fermi surfce whose position in k-spce t certin point in time is s shown in the figure bove. Sketch the trjectory of the electron motion in rel spce, indicte the direction of electron motion with rrows, nd indicte where the electron is on tht trjectory when in k-spce it is s shown bove. (10 points) c) Consider 3D mteril with prbolic energy dispersion for the conduction bnd nd the effective mss tensor for the conduction bnd electrons in the xyz coordinte system is known to be digonl. A uniform nd constnt mgnetic field of 0.1 Tesl is pplied first in the x-direction nd the cyclotron frequency is mesured to be 4.66 GHz. The sme mgnetic field is then pplied in the y-direction nd then in the z-direction nd in ech cse the cyclotron frequency is mesured to be 9.32 GHz nd 13.99 GHz, respectively. Find the effective mss tensor of the electrons. (10 points) d) Consider 3D crystl. The conduction bnd energy dispersion reltion is not prbolic, nd is given by the reltion: 4 Ek Ec k 2
Assume ner-zero temperture. The Fermi energy is E F (mesured from the bnd bottom), nd the Fermi wvevector is k F. Assume tht the electron scttering time is. Energy Find the conductivity tensor of the mteril such tht the current density cn be written s sufficiently smll pplied electric fields. (10 points) J E for 3
Problem 2 (Miscellneous) 20 points ) In the limit q 0 (i.e. the long wvelength limit) the coustic phonon frequencies go to zero wheres the opticl phonon frequencies pproch non-zero vlue. Explin physiclly why in the q 0 limit the coustic phonon frequencies go to zero but not the opticl phonon frequencies. The points wrded will depend on the qulity nd clrity of the explntion provided. (5 points) b) How do you think the phonon dispersion reltion for negtive wvevectors is relted to the phonon dispersion for positive wvevectors More specificlly, does the reltion q q hold If so, give forml proof. If not, explin why not. The points wrded will depend on the qulity nd clrity of the explntion provided. (5 points) c) Consider 2D crystl (not grphene) with hexgonl symmetry nd with the first Brillouin zone shown below. k K y K k x FBZ The mteril hs conduction bnd minim t the three K nd the three K points. Ner the minim, the 2 2 conduction bnd energy dispersion is prbolic but not isotropic. At the K-point, the 3 3 inverse effective mss mtrix is known to be: 1 1 1 3 1 m1 m2 m1 2 M 1 1 3 1 m2 m1 2 m2 4 Find the inverse effective mss mtrix for the conduction bnd minimum t the K -point 0, nd 3 lso for the minimum t the K-point 2 2,. (10 points) 3 3 4
Problem 3 (Phonons in 1D) 30 points Consider the following 1D lttice of toms consisting of two different kinds of toms (blck nd gry). M 1 M 2 ) Assume solution form nd then use it to write the equtions for lttice wve in the stndrd form: 2 D q Obtin ll the elements of the dynmicl mtrix D q nd the other column vectors nd mtrices indicted by question mrks in the expression bove. Show your work. (15 points) b) If M 1 =M 2, would you expect zero bndgp between the coustic bnd nd the next higher bnd t the zone boundry Give physicl explntion. Correct nswer with wrong physics will not get you ny points. (5 points) c) If M 1 =M 2, nd =, would you expect zero bndgp between the coustic bnd nd the next higher bnd t the zone boundry Give physicl explntion. Correct nswer with wrong physics will not get you ny points. (5 points) d) For If M 1 =M 2, nd =, find the frequencies of ll the phonon bnds t the zone center without solving the mtrix eqution obtined in prt (). Justify your pproch/nswer by physicl explntion. (5points) 5