Question

1. All students: (a) By combining kinetic theory with classical mechanics, Drude was able to derive a sensible equation for the electrical conductivity of metals. In the case of the thermal conductivity however, Drude's theory fails. Explain why (in bullet-point form). (b) The Fermi energy of copper is ~8 eV. What fraction of electrons are above the Fermi-level at room temperature? (c) Write down a general expression for the electron density in terms of the density of states and the Fermi-Dirac distribution function for a two-dimensional material such as graphene. (d) A friend asks if electrical resistance is caused by electrons colliding with the nu- clei in the crystal lattice. Provide an answer to your friend's question and include what you would expect for the resistance of a perfect crystal, where the nuclei are arranged in a perfect array at low temperature (ignore superconductivity). (e) Provide a brief microscopic explanation of why most materials expand when heated. How might you design a material with a very low coefficient of thermal expansion? (15 marks)

Answer 1

ude theoy elong Theany Says hat, exa a paxticu has a Kinetie enorg y f E T For a system fuymoviny U=NKThence Heat JE also implies That metals, Which have many fh tectons, should have a much langer heat capacity with classical Kinetic freedom g of movement That contains N-Higid parieus in Thuee dine ctions The total enrgy is capacity C-3Ne 7han insulators. However 7his is nat Obseved The abore failur of the Duude - modul du to tro easons: moinly Come Daude model is based On classical Maxwell boltzmann Stahistits Which follow The La parition f energy which in turn say5 thal all the electnons inside a material can absonb thermal energy (which is nat The neality) case in (ii DAude model only in The model, but completely ignohes The effects jans cansiders the elacthons The matix osihvely changed

Anser Fermi Energy at finite demperatur calculated as Can be KOT - Flo 12 F Whn EFCO8ev Fermi energy at T-OK. (briven -23 Boltzmans constant T 300K (ROom tempenature E7 20h 8ev ne Kno far elkethons, Now 3 EF () 8mc2 m Wheu h 6.634XL0 Plancks Constant m- o.31 Max densiy electuns n no. EF8ev fnding puting we get valuus nom (2 in (1) , and then 22 n 1.02770727 Xl6elactrony/m3 These ane tn electrns above the Fermi uvel

Ne can te the average number atces Whue aca Densiy 9 as, aCE) PCE) dE N = States rFermi-Dinai Statistics PCE)=PAobability disbuibution We have 1 -e)-f) PCE ensity states foor cdimensin is The general foumula enby Wert 2m L. Lengh Scole l 2 2-dimen sion, co a CE C. (3) 9.Mytiha puHing ) )in 4), we get )N- dE n aCE) flE) dE L2

Whore, L Anea of 2 D matorial N - number density elecbron L2 f(E) pCE) Fermi- Dihat distnibutin functiom aCE 12 m deni states C. Hence EF acf)E =u 2 (d) Answet electrons electrical hesistance is causecl Lattice olliding ystael with the nucei in The Then, it follows 7he Dudu Thy esishwity is ren by" a Theasy Frm Dude When T- Aetaxation time be hveen tmo collissions Nor, if The cuystal is perfect honce Then Thure would be no vibhatin, hence no collisian,Then Aelaxation time be comes in finite, Then Aesistivity S-0 (42Ttemp

Answer: in a chemical bond against Consider energy drstance between The bound atoms euive in dinection ar stnetching oppoite compression band ungth > Expansion > intornucuar 6eparation strngth Bond fig(D uom above in uft hand side ,hence it is 6ay tough to 0, As energy cuive is steeper a bond thn to expand because hand sid Compne ss in ight 7 , eny as in The uft hand side cure is nat as steep So, easier to stnetchthan So, when we add dtoms co mokcules do indeeo We can say That chemical bonds ane are to CompheSs heat enorg ,The thy mone oscillate mone vigonously in The latice. And The bonds boTh Comphess and stnetch, with the oscillation But because The estouing Fonce is stiffer against compuesion but weaker against extension,7h shaking is asymmetrical The Sthutch distance is Langer Than sauee ze distance

Considur, two cures Enenory intemucuc dittance lon Thermal as it is steepu Than Black'cuhve CmateniaD will have expansion Curve J (Material will have tow thigh Thurmal expansjon Than oTher cuive Cuire as it is less Steeper