ELECTRONIC AND OPTICAL PROPERTIES OF GRAPHENE. J. González Instituto de Estructura de la Materia, CSIC, Spain

Transcription

1 ELECTRONIC AND OPTICAL PROPERTIES O GRAPHENE J. Gonzálz Instituto d Estructura d la Matria, CSIC, Spain

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3 Graphn has opnd th way to invstigat th bhavior of a gnuin two dimnsional matrial: vry larg mchanical strngth, flxibility, transparncy and conductivity lctrons show rmarkabl proprtis from th thortical point of viw (rlativistic lik bhavior) (J. C. Myr t al., Nano Ltt. 8, 358 (008)) But som challngs hav to b facd: sampls hav significant corrugation th intraction with th substrat and boundary conditions modify significantly th transport proprtis rom E. Stolyarova t al., Proc. Natl. Acad. Sci.. 104, 909 (007)

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5 Th lctronic proprtis can b undrstood from a simpl tight binding modl H tb t + + ψ r ) ψ ( r) t ψ ( r o ( ) ψ ( r) o H t j 0 1,,3 ik v j j 1,,3 0 ik v j ε ( K) ± t 1+ 4cos ( ak / ) + 4cos( ak y y / ) cos( 3aK x / ) Expanding around ach cornr of th Brillouin zon, w obtain th hamiltonian for a two componnt frmion (Dirac hamiltonian) H 3 ta k x 0 + ik y k x ik 0 y W hav to introduc a Dirac frmion for ach indpndnt rmi point, at which ε ( k) ± v k, D( ε ) ε

6 Th scattring from disordr is vry unconvntional in graphn,, du to th chirality of lctrons. Whn a quasiparticl ncircls a closd path in momntum spac, it picks up a Brry phas of π H v k 0 iφ k 0 iφ ψ 1 i ± φ / iφ / ψ i π ( σ / ) z ψ In th absnc of scattrrs that may induc a larg momntum transfr, transfr, backscattring is thn supprssd (H. Suzuura and T. Ando, Phys. Rv. Ltt.. 89,66603 (00)). w ~ * A + A A + A + [ A A + A A * ] A A A A * iπ ( σz /) A A < 0

7 Anothr way of xplaining th supprssion of backscattring is by considring that, for th masslss Dirac frmions, th psudospin givs ris to th consrvd quantity σ k k which commuts with th Dirac hamiltonian H v σÿk. This also xplains th pculiar proprtis of lctrons whn tunnling nling across potntial barrirs: th transmission probability is qual to 1 at normal incidnc, and 0 for backscattring M. I. Katsnlson,, K. S. Novoslov,, and A. K. Gim,, Natur Physics,, 60 (006)

8 Th first xprimntal obsrvations and masurmnt of unusual transport proprtis pointd at th xistnc of a conical disprsion of lctron quasiparticls in graphn th lctric fild ffct shows that, moving th rmi lvl with th gat voltag, a substantial concntration of lctrons (or hols) can b inducd n V g rom K. S. Novoslov t al., Natur 438, 197 (005) th rspons to a magntic fild is also unusual, as obsrvd in particular in th quantum Hall ffct σ xy 4 n h + 1 rom K. S. Novoslov t al., Natur 438, 197 (005)

9 Coming back to th conductivity, this can b computd in Boltzmann transport thory in trms of th dnsity of stats D as σ ( ε ) v D( ε ) τ ( ε ) v ε τ ( ε ) Th linar dpndnc obsrvd xprimntally on gat voltag implis that σ should b proportional to th lctron dnsity, that is, σ ε This rquirs a particular typ of scattrrs for which τ ε in th cas of short rang rang scattrrs, τ 1/k σ const. in th cas of chargd Coulomb scattrrs, τ k σ ε ²

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11 Th nonvanishing conductivity at th Dirac point can b also undrstood starting from th linar quasiparticl disprsion ε ± v k. Th conductivity can b computd from th Kubo formula σ ( i lim ω q 0 j( q, j( q, which, using th continuity quation t ρ ÿ j, can b convrtd into ω σ ( i lim q 0 q ρ( q, ρ( q, Th dnsity dnsity corrlator corrsponds to th charg suscptibility χ( q, 4h q v q ω and lads to a nonvanishing DC conductivity σ ( ω 0) 4h

12 Howvr, th qustion of th minimum conductivity of undopd graphn is not so simpl, and thr has bn a long controvrsy rgarding whthr it should b givn by a univrsal valu. Th xprimntal masurs wr pointing at a valu qual to four tims th quantum of conductanc σ 44 /h, in disagrmnt with th thortical prdiction for clan graphn π σ ( ω 0) h K. S. Novoslov t al., Natur 438, 197 (005) and abov th mor ralistic rsult obtaind aftr including th ffct of disordr (th mystry of th missing π). 4 σ ( ω 0) π h Nowadays w know that th intraction with som substrats may giv ris to a minimum valu of σ. Thus, chargd impuritis in SiO lad to th formation of lctron hol puddls in graphn,, and prcolation of th lctron currnt is what producs a valu σ 44 /h. J. Martin t al., Nat. Phys. 4,, 144 (008)

13 Away from zro frquncy and ε 0, w can also tst th ac conductivity σ '( Θ h 4h ( ω ε ) which, with mor gnrality, bcoms for nonvanishing tmpratur σ '( 4h hω sinh kbt ε + hω cosh cosh kbt kbt This bhavior is consistnt with th rsults of th xprimnts carrid out with infrard spctroscopy: Z. Q. Li t al.,, Natur Phys. 4, 53 (008)

14 Th knowldg of th optical conductivity allows also to obtain th transmission of light, by solving th Maxwll quations ω c iω c ( ) E E + j E with th currnt j σ ( δ ( z) x E x Th optical transmittanc turns out to b givn by T 1 σ 1 + c 1 πα ( 0.977), α 4π hc which is in agrmnt with th xprimntal obsrvations R. R. Nair t al.,, Scinc 30, 1308 (008)

15 A dirct vidnc of th conical disprsion has bn obtaind with angl rsolvd photomission spctroscopy A. Bostwick t al.,, Natur Phys. 3, 36 (007) Ths xprimnts ar also usful to provid a masur of th intraction ffcts in graphn A. Bostwick t al.,, Natur Phys. 3, 36 (007)

16 Th quasiparticl proprtis ar anyhow significantly rnormalizd du to th strong Coulomb intraction, which has a coupling λ /4πv : + Ψ( r, t) Ψ ( r', t') ig( r r', t t') with g / 16 v G( k, ω Z v Z v σ k + iγ Z( 1 c λ log( Λ / Z v ( 1 + c' λ log( Λ / (J. G.,. Guina and M. A. H. Vozmdiano, Phys. Rv. B 59, R474 (1999)) Th ffctiv coupling λ /4πv flows to zro at low nrgis du to th nhancmnt of th rmi vlocity, lading th systm towards th non intracting rgim. This bhavior has bn obsrvd in xprimnts carrid out in graphn at vry low lctronic dnsitis (D. C. Elias t al.,, Natur Phys. 7,, 701 (011))

17 Turning now to th imaginary part Γ,, w can analyz th way in which th scattring of lctrons givs ris to a finit liftim for th lctron quasiparticls. G( k, ω Z v Z v σ k + iγ In th undopd systm, th lctron quasiparticls cannot dcay into lctron hol pairs ε ( q, 1+ q 4 v q q ω τ 1 Γ( k, ω v k ) 0 In th dopd systm, th dcay of lctron quasiparticls coms from intraband lctron hol xcitations: Th quasiparticl dcay rat bcoms τ 1 ( k k ) log ( k ) k E. H. Hwang, Bn Yu Kuang Hu and S. Das Sarma, Phys. Rv. B 76,, (007)

18 W can also comput th lctronic rspons to an xtrnal potntial tial V x (q,), xprssing th scrning ffcts in trms of th dilctric function V ( q, V x ( q, + V 1 V ε ( q, x scr ( q, ( q, Th pols in 1/ε(q, account for th xistnc of collctiv xcitations of lctrons (plasmons) ε ( q, 1+ χ( q, 0 q But plasmons can only aris in th cas of dopd graphn, with disprsion ω ( q) P ε q Th plasmon disprsion has bn masurd with grat accuracy in xprimnts with infrard nanoscopy Z. i t al., Nano Ltt. 11, 4701 (011)

19 W now turn to phonons as th rlvant sourc of scattring at low carrir dnsitis. At low tmpraturs w hav th contribution of acoustic phonons τ 1 Γ( k, v k ) g ε ( k) k B T This givs ris to a dcay rat linarly proportional to th quasiparticl nrgy. Using Boltzmann transport thory, w obtain a rsistivity that dos not dpnd on carrir dnsity and is linraly proportional to tmpratur σ ph ( ε ) 1 v D( ε ) τ ( ε ) 1 1 h g k T B S. Das Sarma and E. H. Hwang, Phys. Rv. B 87, (013)

20 Th thortical rsults hav to matchd with th xprimntal masurs of th rsistivity. It is assumd that th rsistivity has a contribution from impuritis, anothr from acoustic phonon scattring, and som xtra contributions accounting for T dpndnt nonlinar bhavior ρ V, T ) ρ ( V ) + ρ ( T ) + ρ ( V ( g imp g ph nl g, T ) J. H. Chn t al., Natur Nanotch. 3, 06 (008) Th hop is to b abl to rmov th contribution from impuritis, in ordr to rmain with th intrinsic sourc of rsistivity (phonons). In that cas th mobility would b nhancd at low carrir dnsity as J. H. Chn t al., Natur Nanotch. 3, 06 (008) σ ph ( T ) μ ( T ) n

21 Graphn sms thrfor a quit xciting matrial from th xprimntal as wll as from th thortical point of viw, with many othr aspcts bsids thos t covrd hr lctronic proprtis of bilayr and multilayr graphn,, lading in particular to th possibility of opning a gap in bilayr graphn with a transvrs lctric fild influnc of th strain in th carbon layr, and th ability to mimic th ffct of magntic filds with appropriat nginring of th applid strss rol of th adsorbats in graphn,, with th dvlopmnt of a mtal insulator transition at sufficintly larg concntration of adsorbd molculs ffct of vacancis and impuritis in th formation of local magntic momnts in th carbon layr