Supplementary information. Coulomb oscillations in a gate-controlled few-layer graphene quantum dot

Transcription

1 Supplementary nformaton Coulomb oscllatons n a gate-controlled few-layer graphene quantum dot Ypu Song * Haonan Xong Wentao Jang Hongy Zhang Xao Xue Cheng Ma Yuln Ma Luyan Sun Hayan Wang and Lumng Duan * Center for Quantum Informaton IIIS Tsnghua Unversty Bejng Chna Department of Physcs Tsnghua Unversty Bejng Chna Department of Physcs Unversty of Mchgan Ann Arbor Mchgan USA * Correspondng Author. E-mal: lmduan@umch.edu ypsong@mal.tsnghua.edu.cn S1

2 Devce Type of devce Devce geometry Sngle quantum dot; Number of graphene layers Resstance at 300K D1 Quantum dot Dameter 506 2nm Trlayer 1.10 kω D2 Quantum dot Double quantum dots; QD dameter 95 2nm Trlayer 3.97 kω D3 Quantum dot Double quantum dots; QD dameter 102 2nm Blayer 3.28 kω D4 Quantum dot Double quantum dots; QD dameter 120 2nm Blayer kω D5 Feld-electrontransstor (FET) Source-dran length: 1.6um channel Blayer 5.98 kω D6 Feld-electrontransstor (FET) Source-dran length: 1.7um channel Blayer 4.98 kω Table S1: Descrpton of the measured devces shown n Fgure 2 n the man text S2

3 Fgure S1: Graphene thckness and stackng order are characterzed by a combnaton of optcal contrast atomc force mcroscopy (AFM) and Raman spectroscopy. 1-3 (a) AFM lne scan taken over a sngle-layer step-edge of a graphene sheet. (b) The 2D Raman band of a trlayer graphene wth ABC stackng n the dot area of devce D1. The expermental data s shown n black dot lne. The red short dashed lne s a ft to sx Lorentzan components. S3

4 I (na) I (na) Peak spacng (V) Peak spacng (V) 4 (a) (c) V ' (V) t1 4 (b) (d) V ' (V) t3 Fgure S2 (a) (b): Source-dran oscllaton current as a functon of the top gate voltage V t1 and V t3 at a base temperature along a vertcal cuttng lne n Fg.3(d) at V t3 = 2.13V and horzontal cuttng lne at V t1 = 1.94 V respectvely. A seres resstance of the RC flter 22.4 kω has not been subtracted; (c) and (d) the correspondng peak spacng vares wth changng the top gate voltage. The x-axs label ndcates the sequence number of peak spacng countng from the left n (a) and (b). Error bars ndcate the measurement uncertanty. Dash lnes represent the lnear ft. S4

5 Current (na) Peak spacng (V) 0.8 (a) B=0.275 T (b) B=0.275 T B=0 T B=0 T V ' b (V) Fgure S3 (a): Source-dran oscllaton current as a functon of the back gate voltage V b at a base temperature along two horzontal cuttng lnes n Fg.4(a) at B=0T (black) and 0.275T (red) respectvely. The red curve s shfted up 0.45 na for clarty. A seres resstance of the RC flter has not been subtracted. (b) The correspondng peak spacng vares wth a change n the back gate voltage. The x-axs label ndcates the sequence number of peak spacng countng from the left n (a). The red dots are shfted up 2.5 V for clarty. Black and red lnes are to gude the eyes. Smulaton method: 1. Band structure smulatons We consder a system consstng of a top gate back gate and three layers of grapheme between the top and back gates. We apply a parallel-plate capactor model to calculate the densty of electrons on each graphene layer and the Ferm energy of the system. 45 The top (back) gate has an electron excess densty n t(b) nduced by V t(b) = en t(b) d t(b) /ε 0 ε r where d t(b) s the S5

6 dstance from top (back) gate to a closest graphene layer (d t = 100nm d b = 300nm) ε 0 s vacuum permttvty and ε r = 3.9 s the delectrc constant of SO 2. Consderng the lmted wdth of the top gates Posson equaton s solved by COMSOL to calculate the electrc feld dstrbuton upon electrcal gatng. The total excess densty of electrons s consdered to be zero.e. 3 n = n t + n b + =1 n = 0 (S1) where = s the layer number n 1(3) denotes the excess electron densty on the closest layer to the top(back) gate. To determne the excess densty on each layer we follow a self-consstent Hartree approxmaton by Avetsyan et al.. 4 The layer asymmetres between layer 1 2 and layer 2 3 are determned by Δ 12 = α(n 2 + n 3 + n b ) (S2) Δ 23 = α(n 3 + n b ) (S3) where α = e 2 c 0 /ε 0 ε r. c 0 = 0.35nm s the nterlayer dstance. The tght-bndng Hamltonan for ABC-stacked TLG n the presence of the top and back gates s ( Δ 12 γ 0 f γ 0 f Δ 12 γ γ 1 0 γ 0 f γ 0 f 0 γ γ 1 Δ 23 γ 0 f γ 0 f Δ 23 ) (S4) S6

7 Where f(k x k y ) = e k xa 0 / 3 + 2e k xa 0 /2 3 cos k y a 0 /2 a 0 = 2.46 Å. The Hamltonan operates n the space of coeffcents of the tght-bndng functons c(k ) = (c A1 c B1 c A2 c B2 c A3 c B3 ). The electronc denstes on the three layers are gven by n = 2 π d2 k ( c A 2 + c B 2 ) (S5) Solvng Eqs. (S2)-(S4) self-consstently we obtan n hence obtanng the Hamltonan and the band structure n k-space. Parameters γ 0 = 3.12 ev γ 1 = 0.38 ev are used n our calculatons. Gven gate electron denstes n t(b) we can calculate the Ferm level E f valence band E v and conducton band E c of TLG based on the parallel-plate capactor model assumng the onste energy of the mddle layer graphene s the zero-pont energy. Fg.S4 shows the calculated profle of the conducton band and valence band for the QD devce D1 upon electrcal gatng V t = 5.0V and V b = 22.6V. Fgure S4: The spatal profle of the conducton band E c and valence band E v. 2. Smulatons for the magnetc feld dependence of energy-level spectrum S7

8 Egenstates can be expanded n a set of base functons to solve egenvalues of Hamltonan for ABC-stacked TLG. In our calculaton the base functons are chosen to be the egenstates correspondng to the Landau levels of a trlayer graphene sheet 6.e. T n k A1 n1 k B1 n k A2 n k B2 n1 k A3 n1 k B3 n2 k c c c c c c (S6) n k 1 n 2! n lb 2 n ky z /2 x y e e H z 1/2 n (S7) Where 2 H n s the Hermte polynomal and z x kl B l B l B eb. For n 0 nk 0. Based on nk the Hamltonan can be wrtten as H E1 BC BC1 E E BC 0 0 BC2 E E3 BC BC3 E3 (S8) In the matrx B 2vF l B γ 1 = 0.38eV E s the onste energy on dfferent layers. For n 0 C n 1 C2 n 1 C3 n 2. For n 1 C 1 0. In the case of n 2 C1 C2 C 3 0. We wll not consder ths case n later analyss as the energy s ndependent of the magnetc feld. If E vares wth a spatal coordnate for example E n E x R and y R out E else (S9) S8

9 The egenstates can be expanded n the bass of nk : a (S10) m n k n k nk For explctness rewrte the Landau egenstate as 6 n k n k n k 1 c (S11) Where represents a unt vector of the th component n (S6). Thus the egenstates subjected to a varyng potental can be expressed as a c C n k n k n k n k m n k n k n k nk n k C nk (S12) In the bass of nk the matrx element of the Hamltonan s n k ( ) nk H x y mlj H x y dxdy (S13) * m l j j An alternatve way s to solve the egenvalues and egenvectors for the Hamltonan matrx n p (S8) frst to calculate E nk x y and coeffcents c n (S11) to obtan the form of nk p nk (p s the ndex of sx egenvalues of the 6 6 matrx) and then drectly use the expanson expresson n (S10). In ths way denotng p nk as nkp the correspondng matrx element s q 6 * n k n k q m l m l m l m l 1 ( ) nkp H x y mlq nkp E x y mlq c E x y c dxdy (S14) For Landau levels energes are degenerate for varous k so the number of k should be large enough to acheve a vald perturbaton approach. In our case the calculaton s done wth S9

10 k = 400 so that most energy levels have merged nto Landau levels n the magnetc feld regon that we are nterested n. Supportng references: (1) L H.; Wu J.; Huang X.; Lu G.; Yang J.; Lu X; Xong Q.H.; Zhang H. ACS Nano (2) Hao Y. F.; Wang Y. Y.; Wang L.; N Z. H.; Wang Z.Q.; Wang R.; Koo C.K.; Shen Z.X.; Thong J.T.L. Small (3) Jhang S. H.; Cracun M. F.; Schmdmeer S.; Tokumtsu S.; Russo S.; Yamamoto M.; Skoursk Y.; Wosntza J.; Tarucha S.; Eroms J.; Strunk C. Phys. Rev. B (R). (4) Avetsyan A. A.; Partoens B.; Peeters F. M. Phys. Rev. B (5) McCann E.; Koshno M. Rep. Prog. Phys (6) Yuan S.J.; Rold an R.; Katsnelson M. I. Phys. Rev. B S10