2.57/2.570 Midterm Exam No. 1 March 31, :00 am -12:30 pm

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1 2.57/2.570 Midterm Exm No. 1 Mrch 31, :00 m -12:30 pm Instructions: (1) 2.57 students: try ll problems (2) students: Problem 1 plus one of two long problems. You cn lso do both long problems, nd one will be considered for bonus points. (3) Clerly stte your ssumptions. (4) Closed book, one sheet of notes llowed. Unless otherwise noted, ssume electron mss equls tht of free electron in vcuum. 1. Answer the following short questions. Briefly explin your nswer (60 Points) (1) Infrred light t 10 micron is emitted between the first two energy levels of onedimensionl quntum well with infinite potentil brrier height. Wht is the width of the quntum well? (2) Air molecules t 1 tm hve men free pth pproximtely 70 nm. Estimte the men free pth t 1 torr, i.e., 1/760 of 1 tm. (3) Light is incident from medium of refrctive index n=2 towrds n interfce with vcuum. Clculte the reflectivity t the interfce if (3) The ngle of incidence is 0 o. (3b) If the ngle of incidence is 65 o. (4) Consider blckbody rdition t 400 K in vcuum. On verge, how mny photons re there in quntum stte tht hs photon wvelength of 10 m? 1

2 (5) An electron with n energy of 1 ev encounters potentil brrier of height 0.5 ev. The electron effective mss equls tht of free electrons. Wht is the probbility tht the electron cn go over the potentil brrier? (6) The therml conductivity of ir t 300 K nd 1 tmosphere is W/m-K. Estimte its therml conductivity t 600 K nd 1 tm. (7) Copper hs n fcc lttice with lttice constnt of 3.61 Å (one Cu tom per lttice point). Ech Cu tom contributes one vlence electron. Clculte its Fermi level. (8) A mteril hs n fcc structure with two toms s the bsis. The lttice constnt of the conventionl cubic unit cell is 5.0 Å. The verge phonon velocity is 2000 m/s nd phonon men free pth is 50 nm. Answer the following two questions: (8) Wht is the phonon specific het per unit volume t high tempertures? (8b) Wht is the high temperture therml conductivity of the mteril? 2

3 Long Problems: 2. Liner Dispersion. Sometimes, the electron energy-wvevector dispersion cn be liner (n exmple is in grphene t specil points in k-spce). Consider simplified liner dispersion model, 2 2 E E vk v k k k 2 c x y z 2 4 Where k x, k y, k z,,... We ssume tht the number of electrons in the mteril is L L fixed t n per unit volume. Answer the following questions () Derive n expression for the electron density of sttes per unit volume nd per energy intervl. (b) Derive n integrl expression tht determines the chemicl potentil, using Fermi-Dirc sttistics. (c) Assuming tht Fermi-Dirc sttistics cn be well pproximted by Boltzmnn sttistics, derive n explicit expression for the electron chemicl potentil. In doing the energy integrtion, you cn ssume tht upper limit is energy pproches infinity, despite you know tht the dispersion is limited to the first Brillouin zone. 3

4 3. Phonon Reflection t One-Dimensionl Atomic Chin Interfce. Consider n infinite one-dimensionl tomic chin s shown in the following figure. One the left side, the toms hve mss m. On the right hnd side, the toms hve different mss M. The spring constnt nd interctomic spcing re identicl on the two sides, respectively. The phonon dispersion on the two sides re given by 1 K k 1 K k r sin sin 2 m 2 2 M 2 where k r or k,..., re wvevectors in the first Brillouin zone for phonons on the left nd the right hnd side. We now consider lttice wve comes in from the left hnd side, nd the tomic displcement from equilibrium position is expressed s u (x ) Aexp i t k x i i i where x i is the tomic position t equilibrium. The reflected tomic wve is u r (x i ) B exp i t k x i nd trnsmitted wve is u (x ) C exp i t k x t i r i We know tht when wve trnsmit cross the interfce, the frequency remins constnt. Answer the following questions, () Find out reltion between k r nd k. (b) Write down the equtions of motion for toms 1 nd 2 t the interfce. (c) Determine the reflection coefficient B/A, nd trnsmission coefficient C/A. (d) Consider tht power equls force times velocity, determine power reflectnce nd trnsmittnce t the interfce. K m 1 2 K M X 4

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