Question

E(k)t Er K. 4. (Advanced Level) Graphene is a special type of material which is two- dimensional and has a linear E-k diagram. The conduction and valence bands touch each other at a singularity, known as the Dirac point. In the unexcited state, the valence band is completely filled, while the conduction band is completely empty From the E-k diagram shown above, deduce the equation of the E-k curve. Hence, from the equation you deduced, speculate the special properties of the electron (Hint: your deduction may need to involve Einstein's theory of Special Relativity. It is the case where electrons are moving too fast and very close to the speed of light, and thus the classical kinetic energy of the electron, E-2mv, is no longer valid. You would thus need to use the relativistic energy of the electron) a. b. Comment if you are able find the effective electron mass from the E-k diagram. Why or why not? If not, describe the quantity you are able to obtain from the slope E-k curve. (Hint: what happens when you take the second derivative of the linear E-k curve?) Hence, derive the 2-D Density of States function for graphene. c. (Hint: you may use the already derived 2-D DOS function for parabolic materials as a template for your derivation) d. From the E-k diagram, deduce if graphene is an insulator, semiconductor, or metal? (Hint: use Heisenberg Uncertainty Principle to aid in your deduction.)

Answer 1

Grophene Betore going to wrte Hhe us dee Rous rlokviy qahin tor rophune dvant hexe Emey -momenlunm rdn. ivh by, Jimde2 He know, vedocit Pc we Considt mass le Porkde , we kay ond E = bc a, The Emeny iven diapron momendum Aeve Unto disprsion qedaien ; So, we Jee hot we Siwilerty ond xfet diagron rphnt, Con see Lhe E- Is The txpein quontum txprciononaloy Con be denv ed wrikng eahvi w en down me chani colly , but hr LA Ught Con in to E K) t Scanned with CS CamScanner

Nou, a we ov een in Lhe inea cohriy tor massporkd (mz0) dispersiun Here velakon clechns porkiedes NASSLE SS PARTI CLES Hhe NHICH BEHAVE ar e AS Some O Elehons in Jhe a u , bchove ai ing mass lu Porku4 , ConjMassl Diroc firmions Jump insonto neous Con duukn ba nd decrons Vclone bond o band gof Eleden mbiiy Can a nup ttramdy high al so nonli nea The jiven by Jhe echyc (b) mass d an hedcond di vehye Vanish u, fnite , Henu fechve cde tron nass dound his be torny in Con be found Velouib Con be vedokon Scanned with CS CamScanner

Densil Ses (Dos) Jor ephene E(K) Ky two dimunsin) Ju= Yule deqenray u2) Coms om n'^oUs Le us Consicde^ a K Ky Jhe dimansions f Lx, Jfhene bonds N(K)dk Kdk N(K)dk = Kdlk arr La have E(K)- K EdE k dk E dE N(K) dk A CAup2 216 Hh.r - jtihane a insulotor mt or neih d. rte Semi conductos h tb Semi. md Cs Scanned with CamScanner