Question

Problem 4. Low-dimensional materials play an important role in nanotechnology. Consider a two-dimensional Fermi gas of N non-interacting electrons confined to a plane of area A. Find the Fermi energy &r (in terms of A and N) and the average electron energy. Find (analytically) the chemical potential p as a function of er and T.

Answer 1

eOTTespanding Ferm momentum S Er nh2 where, s the dlensrty and ms the A patele Chem: cal pojent al mass EP and T as a anon inte racting Ferm: ga evalualed us'no Chem: cas potental at low dempe Yo tute the aue Sommeyetd method When the demperotue intreses the chemical potental decreses belos Ferm energy ar d mens ons equal to The chem cal potental 1s o undamental quantty whch chauacterises mony mechanics in stast.cal 0 thermal equ ll brum patele system Wle can nterpred theormula o Ferm gas the non in FD Space val d in the igh temperaf uwe Tange The auerage Pautcle is exp CEk-u)/kBT) + 1 number N- E K o here, Tadernal energy s Eexp (Ek-)/keT)4 1 U Ahen Ket FF EP H=EF 1i8

S)Constdes too dimens onal Ferme gas elechrons conyocd N non-interadeg to a plane auea A. aFind the Ferm encrq4 Ep Ltn-lerm R& NJ Roswer Ferme Two fmporlan chauacter:stes Ferm gases Fp and the Ferm ase the energy momentum kP The se properhes to the drspersion Yelatbn aue connected he paaticle For ee paatc le we have and FF can be 2ero temperatuue delermnedn the m where ttrepresents the energy the hghest 1ying occupied pau ticles occupy quantum stale States wth energy Such tet all othey E <EF and s alcalated N =J9 (E)nFD CE) de as where Ntotal alom numbeT Ferm energu and 2ero elsewhere Rs NFo unty below the at 2ero demperatue, N-J gce)de & momentum aie then, the FErm energ t2 3m2)/3 Ein) =N-dens:ty the gas where, V u rel by subst tutag Ferm: moment oas Then the Ferm: enevgy R&N is deduceol as arh N EP 1 Am

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