SUPPLMNTARY INFORMATION. Dtmin th gat inducd bgap cai concntation. Th fild inducd bgap cai concntation in bilay gaphn a indpndntly vaid by contolling th both th top bottom displacmnt lctical filds D t D b with th cosponding gats. Th lation btwn th displacmnt lctical fild gat voltags is dscibd by D = ( ) ε ( V - V )/ d, bt () bt () bt () bt () bt () wh th top gat dilctic lay has a dilctic constant εt = 7.5 fo amophous Al O 3 a thicnss d t = 8 nm, wh th bottom gat dilctic lay has a dilctic constant ε b = 3.9 fo SiO a thicnss d b = 85 nm. V bt () is th ffctiv bottom (top) offst voltag du to initial nvionmnt inducd cai doping, which is dtmind though lctical tanspot chaactization following Rf.. Th cai concntation in bilay gaphn is st by th discontinuity of th fild D=D b -D t. Following Maxwll s quation, w hav cai concntation of n= ε D, which cosponds to 5.6 cm fo D of V/nm. Th tunabl bgap Δ is dtmind by th avag displacmnt lctical fild D = (D b D t )/. Slf-consistnt tight binding calculation givs an accuat dsciption of th bgap ngy as a function of D.,3 Fo D <3 V/nm, it can b wll appoximatd by th xpssion Δ= 84 D 34.7 sin( D ), with Δ xpssd in mv D in V/nm.. Dtaild calculations of th many body Fano systm in bilay gaphn. W will focus on th tunabl Fano sonanc in bilay gaphn at diffnt inducd bgap but with zo cai doping. Th Hamiltonian dscibing low ngy xcitations in this many body systm with on dominant activ phonon mod is natu nanotchnology www.natu.com/natunanotchnology 9 Macmillan Publishs Limitd. All ights svd.
H= aa d d N M ( ad ad ), / v v v v v v α α v α α α α, α wh th lmntay xcitations a dscibd by th phonon cation opato a lcton-hol pai cation opato d c c c, v, v = v v, is th ba phonon ngy, α dnots th lcton spin vally dgs of fdom, = c, v v v, is th conduction valnc lcton ngy diffnc at wavvcto v, N is th numb of lctonic stats, M v α is th lcton-phonon coupling matix lmnt. W considd h only tansitions with zo momntum chang, sinc ths a th stats that coupl to infad photons which hav ngligibl momntum. Th whol systm is tan to b in a finit box of volum V so that th valus fo v a disct; th matix lmnt M v α is valuatd with th lcton phonon wavfunctions nomalizd to on in a singl bilay gaphn unit cll. Th ngy dispsion v c, v v, 4-6 th lcton-phonon coupling matix M v 7 α a calculatd using slf-consistnt tight-binding modl. In bilay gaphn two G-mod optical phonons, th symmtic antisymmtic mods, a psnt. Only th symmtic mod (Fig. a in th main txt) coupls to lcton-hol xcitations whn n (du to lcton-hol symmty) nds to b considd in ou Fano systm 7. Sinc th is only on phonon which is lvant in th poblm, to dal with th lag numb of dgnat lcton-hol pai stats, w intoduc a paticula stat composd of supposition of lcton-hol xcitations of ngy as d = N V M d /, α α α, (S) natu nanotchnology www.natu.com/natunanotchnology 9 Macmillan Publishs Limitd. All ights svd.
wh V = N, M α α is an avag coupling stngth N, α = is th numb of lctonic stats with ngy. Only this supposition of lcton-hol pai stats coupls to th phonon vibation, any oth lcton-hol pai stat othogonal to it has zo phonon coupling. With this constuction, th many-body Hamiltonian in bilay gaphn can b wittn in th fom / H = a a dd N% V( a d da) d d d d α α α. (S) v α H, N ~ is th numb of diffnt d stats dfind in q. (S), i.., having diffnt ngis. Th fist pat of th Hamiltonian dscibs th phonon th spcific lcton- hol xcitations d that coupl to th phonon. It is sponsibl fo all th Fano bhavio in bilay gaphn. Th scond pat dscibs all th oth lcton-hol pai xcitations which a dcoupld to th phonon. Ths lcton-hol pai stats can still b IR activ contibut to th optical absoption. But thy poduc only a lativly boad continuous absoption bacgound which dos not intf with th disct phonon sonanc. To gain physical insights of th coupld phonon lcton-hol pais systm, i.., th tms in th fist panthsis in q. (S), w us scond-od ptubation thoy. [W hav confimd that th sults fom scond-od ptubation thoy a in quantitativ agmnt with thos obtaind fom a full diagonalization of th Hamiltonian shown in q. (S).] Und a canonical tansfomation, th Fano sonanc pat of th many-body Hamiltonian can b diagonalizd to H = h h to scond od in V with th intoduction of th hybid phonon-xciton xcitation cation opato natu nanotchnology www.natu.com/natunanotchnology 3 9 Macmillan Publishs Limitd. All ights svd.
V VV γ h a G d θ. H tanθ ' = sin θ ( R ' ' ) cos d γ ' γ G ' = ' iγ is th ba lcton-pai stat Gn s function 8. Th fist tm in h cosponds to th nomalizd disct tansition. It dscibs th phonon dssd by its couplings to th off-sonanc lcton-hol pai xcitations. This vitual lcton-hol cloud givs th phonon its infad activity. Th nomalizd phonon cnt fquncy boadning γ is givn by = V ' R G ' γ = γ V ' Im G. ' ' With th hybid phonon-xciton stats bing ignstats of bilay gaphn xcitations, th infad absoption spctum can b obtaind by valuating th optical tansition matix lmnt of this hybid phonon-xciton xcitations. Th intfnc btwn th phonon (fist tm of h ) th xciton (scond tm of h ) componnts poducs th obsvd Fano linshaps. This calculation quantitativly poducs th obsvd gat-tunabl Fano sonancs in bilay gaphn, th compaison of th xpimntal thotical sults a shown in Fig. 3 in th main txt. Anoth impotant ingdint in dscibing th Fano sonanc is th paamt ' q = π D ( - h ) V - p h μ μ ph -h (s th main txt) calculatd, in ou cas, by q = R Im i γ i γ α α,, M M * α * α d d α α T ˆ T ˆ wh Tˆ is th opato fo lcton-hol pai gnation though photon absoption. Th al imaginay pat of th ba 4 natu nanotchnology www.natu.com/natunanotchnology 9 Macmillan Publishs Limitd. All ights svd.
lcton-pai stat Gn s function giv th acquid lcton-hol chaact of th phonon th joint lcton-hol pai dnsity of stats, spctivly. It is notwothy that q dos not dpnd on th absolut valu of th lcton-phonon coupling stngth bcaus th gaind optical activity of th dssd phonon is popotional to th lctonphonon coupling stngth, cancling th on in th dnominato. To account fo th finit liftim inhomognous boadning of th lcton-hol pai xcitations, w hav st th lcton-hol tansition width γ to b 4 mv.. Oostinga, J. B., Hsch, H. B., Liu, X. L., Mopugo, A. F. & Vsypn, L. M. K. Gat-inducd insulating stat in bilay gaphn dvics. Nat Mat 7, 5-57 (8).. Zhang, Y. t al. Dict Obsvation of a Widly Tunabl Bgap in Bilay Gaphn. Natu 459, 8-83 (9). 3. Min, H. K., Sahu, B., Banj, S. K. & MacDonald, A. H. Ab initio thoy of gat inducd gaps in gaphn bilays. Phys Rv B 75, 555 (7). 4. McCann,. Asymmty gap in th lctonic b stuctu of bilay gaphn. Phys Rv B 74, 643 (6). 5. Abgl, D. S. L. & Fal'o, V. I. Optical magnto-optical fa-infad poptis of bilay gaphn. Phys Rv B 75, 5543 (7). 6. Wang, F. t al. Gat-vaiabl optical tansitions in gaphn. Scinc 3, 6-9 (8). 7. Ando, T. & Koshino, M. Fild ffcts on Optical Phonons in Bilay Gaphn. J Phys Soc Jpn 78, 3479 (9). 8. Fano, U. ffcts of Configuation Intaction on Intnsitis Phas Shifts. Phys Rv 4, 866 (96). natu nanotchnology www.natu.com/natunanotchnology 5 9 Macmillan Publishs Limitd. All ights svd.