Observation of excited states in a graphene double quantum dot F. Molitor,∗ H. Knowles, S. Dro¨scher, U. Gasser, T. Choi, P. Roulleau, J. Gu¨ttinger, A. Jacobsen, C. Stampfer,† K. Ensslin, and T. Ihn Solid State Physics Laboratory - ETH Zurich, Switzerland (Dated: January 26, 2010) We study a graphene double quantum dot in different coupling regimes. Despite the strong capacitivecouplingbetweenthedots,thetunnelcouplingisbelowtheexperimentalresolution. We observe additional structures inside the finite-bias triangles, part of which can be attributed to 0 electronic excited dot states, while others are probably due to modulations of the transmission of 1 thetunnelbarriers connecting the system to source and drain leads. 0 2 PACSnumbers: 73.63.Kv,73.23.Hk,73.22.Pr n a J I. INTRODUCTION a) b) 6 100 nm 0.2 I (A) 2 GC 10-9 Double quantum dot structures are promising candi- GL 0.1 ] datesfortheimplementationofsolidstatespinqubits1,2. GR 10-10 hall Driaolubsylestdemotss,haasvfeorbeeexnamrepalleiziendGinaAmsanhyetderiffoestrreuncttumraetse3-, CL L R V (V)GL0 10-11 semiconductornanowires4–6 andcarbonnanotubes7,and - CR -0.1 s the control of individual electrons and spins has been e achieved8–12. Graphene has been predicted to be par- S D 10-12 m -0.2 ticularly well-suited for spin-basedquantum information -0.2 -0.1 0 0.1 0.2 . V (V) t processing, because spin-orbit interaction and hyperfine GR a interaction are expected to be much weaker than in the m material systems mentioned above, leading potentially FIG. 1: (a) AFM image of the sample topology. The edges - to much longer spin coherence times13,14. Significant of thegraphene regions are highlighted by dashed lines. The d twodots,labelled byLand R,haveadiameterof90nmand n progress has been made recently in the fabrication and are seperated by a 30 nm wide constriction. The dots are o the understanding of graphene-based nanostructures, as connected by 20 nm wide constrictions to source and drain c for example constrictions and quantum dots15–25. Also contacts (S and D). The global back gate and five in-plane [ graphene double quantum dots have been demonstrated graphene gates (CL, GL, GC, GR, CR) allow to tune the 1 recently26,27. For example, Liu et al.28 showed the pres- sample. (b) Current through the double dot measured as v ence of excited states in a double dot created in a top- a function of VGR and VGL at T = 1.4 K, VBG = VGC = 0 gated graphene nanoribbon. In this work, we demon- 0 V,VCR =−0.1 V,VCL =−3.25 V and Vbias=−0.5 mV. 0 strate the presence of excited states in a side-gated 7 graphenedouble dotstructure withadifferentgeometry, 4 formedbyetchingtheislandsoutofagrapheneflake. We oxygen. . 1 studyindetailthecouplingbetweenthe twodotsfordif- The transport experiments were carried out in a vari- 0 ferent gate voltages. We show that anin-plane magnetic able temperature insert at 1.4 K, and at 120 mK base 0 field changes the excited states spectrum. temperatureofastandard3He/4Hedilutionrefrigerator. 1 : In total, measurements of three different cool-downs are v presented. Even if some details changed from one cool- i X down to the other, the main features presented in this II. EXPERIMENTAL DETAILS work were present in every cool-down. r a The sample consists of a double dot structure carved out of a graphene flake. Mechanical exfoliation of natu- III. RESULTS AND DISCUSSION ral graphite flakes, followed by deposition onto a highly doped siliconwafercoveredby 285nm of silicondioxide, is used to produce the graphene flakes. Thin flakes are Fig. 1(b) shows a measurement of the chargestability identified with an optical microscope, and Raman spec- diagram, recorded at T = 1.4 K. The hexagon pattern troscopyisusedtomakesurethattheflakeconsistsonly characteristic for double dots is clearly visible, and uni- of one single graphene layer30,31. The selected flake is form over many double dot charge configurations. The contacted using electron beam lithography (EBL) and current is maximal at the triple points, where the elec- metal evaporation (Cr/Au). Finally it is patterned into trochemicalpotentialsinbothdotsarealignedwitheach the double dot structure shown in Fig. 1(a) by a second other and with the Fermi energy in the leads. These EBL step and reactive ion etching based on argon and triplepoints areconnectedby faintlinesofmuchsmaller 2 a) b) V =-12 V a) I (A) b) 3 BG -10 I (nA) 10-11 I (nA) 12 V (V)GL0.1 12 V (mV)GL-20 1100--1132 V(cid:54)GL (cid:124)tA B (cid:124)t 0 0 0 -30 10-14 -20 -10 0 10 20 0 0.1 V (V) V (V) 10-15 BG GR 140 150 160 (cid:54)V V (mV) GR GR c) VBG=-13 V I (nA)d) VBG=-16 V I (nA)e) VBG=-19 V I (nA) c)-10 I (A10)-11 d)10-11 ttt===111426(cid:43)(cid:43)(cid:43)eeeVVV 0.1 3 0.1 0.4 0.1 0.8 V) measurement V (V)GL 12 V (V)GL 0.2 (V)VGL 0.4 V (mGL-20 1100--1132 I (A)max1100--1132 0 0 0 0 0 0 -30 10-14 0 0.1 0 0.1 0 0.1 VGR (V) VGR (V) VGR (V) 140 150 160 10-15 10-14-20 -15 -10 V (mV) V (mV) GR GL FIG. 2: T=1.4 K, Vbias = 0.5 mV≈ 4kBT (a) Current as a functionofVBG,alltheothergatesat0V.(b)-(e): Currentas FIG.3: (a)Measurementofachargestabilitydiagramaround a function of the voltage applied to GR and GL for different one pair of triple points at small bias voltage Vbias = 15 µV negative values of back gate voltage. and VBG = −8 V, VCL = VCR = 0 V, VCG = −0.5 V and T = 120 mK. (b) Schematic drawing of a charge stability diagram for two tunnel coupled quantum dots around two triple points labelled A and B. The corners of the hexagons, current,originatingfrominelastic cotunneling processes. atthetriplepoints,areroundedduetotunnelcoupling,with Along these lines, the energy level in one dot is aligned a deviation from the straight, dotted lines proportional to with the electrochemical potential in the corresponding the tunnel coupling strength t. (c) Simulation of a pair of lead. Such well controlled double dot behavior, some- triple points for t = 14 µeV, and the energy independent times with less symmetric barriers, could be observed in part of the tunneling rates to the leads γL = 1.26 GHz and the whole accessible range of positive back gate voltages γR =1.69 GHz. (d) Maximum current along thecotunneling (0-30 V) for all three cool-downs,as long as the barriers lines in the range of the upper triple point for each value of were not too closed to allow current detection. VGL for the measurement (black crosses) and simulations for For negative back gate voltages, however, the situa- t=12 µeV (green), t=14 µeV (blue) and t=16 µeV (red). tion can be quite different. This difference can be ob- served in Fig. 2(a), displaying the current through the double dot as a function of back gate voltage at 1.4 K. dot, and therefore transport is dominated by the left While the current is completely suppressed for positive dot. For V = −13 V, the opposite situation is real- BG values of V , resonances can be observed at negative ized, with strong coupling between the left dot and the BG gate voltages. Charge stability diagram measurements left lead. Fig 2(d), recorded at VBG = −16 V, corre- give a better understanding of this region. Fig. 2(b)-(e) spondstoamoresymmetricsituation,wherethecurrent represent a selection of such measurements for four dif- along the cotunneling lines in both directions is almost ferentvaluesofV between-10Vand-20V, wherethe equally strong. The coupling between both dots Em is BG C resonancesarestrongest. Theycorrespondtocompletely very strong compared to the charging energies of the in- differentsituations: atV =−12V,representedinFig. dividual dots ER and EL (Em ≈ 0.5·ER ≈ 0.5·EL), BG C C C C C 2(b), the current is high along the horizontal cotunnel- leadingtoalmostdiagonallines,whichwouldcorrespond ing lines and almost continuous across the triple points. toonelargedotdelocalizedoverbothislands. Finally,at Fig. 2(c) represents the opposite situation, occurring at VBG = −19 V, the situation corresponds again to a well V =−13V, with high currentat the triple points and defined double dot, with all three tunnel barriers well BG along the vertical cotunneling lines. At V = −16 V closed (Em ≈0.2ER ≈0.2EL). BG C C C [Fig. 2(d)], the regions of finite current describe diago- Fig. 3(a) shows a measurement for a charge stability nal,wavylines,whileatVBG =−19V[Fig. 2(e)]aclean diagraminthevicinityofonepairoftriplepoints. Ithas double dot chargestability diagramwith currentonly at been recorded at low temperature T ≈ 120 mK and at the triple points, is recovered. low bias voltage V = 15 µV to prevent an expansion bias These different cases represent different coupling of the triple points to triangles. A negative back gate regimesbetweenthe twodots, andbetweenthe dots and voltage V = −8 V has been chosen, because only in BG the leads. In the case of V = −12 V, the current is this regime the cotunneling lines are visible even at this BG high whenever the energy level in the left dot is aligned low bias voltage. A corresponding schematic drawing of with the chemical potential in the left lead. This can a charge stability diagram for two tunnel coupled quan- be understood assuming the coupling between the right tum dots can be seen in Fig. 3(b). The tunnel coupling dot and the right lead is very strong compared to the leads to rounded edges of the hexagons, with the point coupling of the left dot to the leads and to the right of charge balance shifted from the original triple point 3 proportionaltothestrengthofthetunnelcoupling. Con- 0 (cid:98) (meV) 6 trary to the situation depicted in Fig. 3(b), no rounding a) c) I of the corners is visible in the measurement. This allows ustoestimateanupperboundforthetunnelcouplingt≤ b)120 I (p3A) 120 d(aI/.duV.)GL 1 20 µeV. This energyscale is comparableto the tempera- V) tapunardceitabivbreooaucdtoeutnwpilnoingogordfeentrhesregocyfomtEuCamngnn≈eilti1un.dg3elmisnmeeVsa.l(lkTeBrhTteh≈laenv1et0rhµaerecmVas)- V (mV)GL10800 (cid:98) 12 V (mGL10800 0 necessary for the determination of these energy scales were extracted from a measurement of the same pair of -20 0 20 0 -20 0 20 -1 triple points at Vbias = 1 mV. Fig. 3(c) shows the re- VGR (mV) VGR (mV) sult of a numerical calculation of the current based on the rate equation using the lever arms and charging en- d) e) I (pA) dI/dV ergies deduced from the measurement. Best agreement GL 120 120 (a.u.) is found for γ = 1.26 GHz and γ = 1.69 GHz for 1 the energy-indLependent part of theRtunneling rates to mV) 1 V) ttwheeelneabdost,hadnodtst32=,331.4AµmeVorefodretthaeiletdundneseclrcipotuiponlinogf tbhee- V (GL100 0.5V (mGL100 0 calculation can be found in Ref.32. Measurement and 80 80 simulation are quite similar, except for the finite current 0 -1 -40 -20 0 20 -40 -20 0 20 measured along the line connecting both triple points, V (mV) GR V (mV) GR whichisunexpectedandcannotbereproducedwiththis simple model. Fig. 3(d) shows a closer comparison be- FIG.4: CloserlookatonepairoftriplepointsatT =120mK. tween the measurement and the calculation. For each (a)Currentalongthedetuningaxis. Thearrows indicatethe value of VGL in the range of the upper triple point, the position of excited states. (b), (d): Current as a function of maximum current in the region of the cotunneling line VGR and VGL for Vbias=6 mV (b) and Vbias=-6 mV (d). The is plotted for the measurement and for calculations with dashed linein (b) representsthedetuninglinechosen forthe t=12µeV,t=14µeVandt=16µeV.Thetunnelcou- measurement in (a). (c),(e): Corresponding representations pling determines how fast the current drops as one goes ofthecurrent,numerically derivatedbyVGL aftersmoothing away from the triple point along the cotunneling lines. overthree data points. Bestagreementisfoundfort=14µeV.However,due to the uncertainty in the lever arms in this regime and in the electronic temperature, the tunnel coupling strength TABLEI:Overviewofthemainleverarmsandenergyscales. canonlybe determineduptoafactoroftwo. The differ- The lever arms and the mutual charging energy ECm are de- encebetweenmeasurementandcalculationinthetailsof terminedfromFig. 4(b). Thesingledotchargingenergiesare determined from Fig. 1(a) using thelever arms of thistable. the peak far away from the triple point arises from the fact that the peak current from the measurement does not take values lower than the noise level. αGR,R αGL,L αGR,L αGL,R 0.18 0.20 0.062 0.066 Despite the strong capacitive coupling between the dots, the tunnel coupling is low. For the observation ECR ECL ECm level spacing of Coulomb blockade a resistance of the order of h/e2 is 13.0 meV 14.0 meV 2.4 meV ≈0.5 meV required. This resistance may arise from a tunnel bar- rier, as it is usually the case for GaAs-based quantum dot systems. In graphene, this tunnel coupling may be weak if there is a narrow but high barrierseparating the Table I gives an overview of the main lever arms and dots. Suchasituationcouldgiverisetostrongcapacitive energy scales. The symmetry of the structure is remark- coupling (see Fig. 2(d)) while the tunnel coupling itself able: the lever arms of both dots, as well as both single remains below the experimental resolution. Additional dot charging energies, are very similar. The charging resonances in the central constriction17–22 and interac- energies are considerably higher than those reported in tions might lead to an even more complicated situation. Ref.28 (single dot charging energies ≈ 3 meV, mutual Fig. 4displaysacloserlookatonepairoftriplepoints coupling energy ≈ 0.4 meV) despite the fact that these for a finite applied bias voltage of V = ±6 mV. This dots are slightly smaller. This is consistentwith the fact bias measurement is recorded at V =25 V, a region where thatinthe caseofRef.28 the sample is partlycoveredby BG the dot-lead coupling strengths are weak, and therefore top gates, leading to increased screening. no cotunneling lines arevisible. Becauseofthe highbias Inside the triangles of finite current, additional paral- voltage, the triple points evolved into triangles3. The lel lines can be seen. These lines are even clearer when extent of these triangles gives the lever arms needed to plotting the derivativeofthe currentalongthe V -axis, GL determine the energy scales of the double dot system. taken numerically after smoothening over 3 data points. 4 a) B=0 T b) B=4 T differentvalues ofthe magnetic field, orientedparallelto dI/dVGL dI/dVGL the graphene plane. The figure displays the current dif- (a.u.) (a.u.) 1 1 ferentiated by VGL, measured directly by applying an ac V) V) modulationtoV ,andrecordingtheaccurrent. Again, m80 m80 GL V (GL 0 V (GL 0 sthtaeteuspppaerraelldegletoofththeebtarsiealningeleaasrweevlilsaibsleli.neTshpeapraolslietliotno 60 60 ofthetrianglesinthegatevoltageplanealmostdoesnot 60 80 −1 60 80 -1 change at all up to B = 12 T. This is in contrast to the V (mV) V (mV) caseofaperpendicularmagneticfield,wheretheposition GR GR ofthetriplepointsandtheintensitieschangesignificantly c) B=8 T d) B=12 T on a magnetic field scale ∆B ≈ 250 mT due to the ef- dI/dV dI/dV fects of the field onthe orbitalpartof the wavefunctions GL GL (a.1u.) (a.1u.) (not shown). Two effects of the parallel magnetic field mV)80 mV)80 onthe triple points canbe observed. First, with increas- V (GL 0 V (GL 0 itnogthmeabgnaseetilcinfieeilndc,rethaesesn,uwmhbiecrhoifsvmisoisbtlepsrtoantoeusnpcaerdalfloerl B = 12 T [Fig. 5(d)]. This effect was observable for 60 60 bothpairs oftriple points whichwerestudied, andcould 60 80 -1 60 80 -1 originate from Zeeman splitting. However, it was not V (mV) GR VGR (mV) possible to analyze in detail the appearance of these ad- ditionallines because their broadeningis similar to their FIG. 5: dI/dVGL as a function of VGR and VGL for T = spacing. The second effect is the appearance of a line 120 mK and Vbias = −4 mV for different values of in-plane parallel to the left edge of the triangle at high magnetic magnetic field. dI/dVGL is measured with a lock-in amplifier fields. This line originates from modulated transmission by adding an ac modulation of 200 µV to VGL and detecting betweentherightdotandtherightlead. Theappearance theac component of the current. of this line with high magnetic field is surprising, as one would not expect the in-plane magnetic field to localize states. Wespeculatethatsomeareasofthestructureare The most prominent lines run parallel to the baseline of exposed to a finite component of the out-of plane field, thetriangles. Alongsuchaline,thedetuningbetweenthe owing to ripples always present in graphene flakes34. energy levels in both dots is kept fixed. These lines are usuallyattributedtoexcitedstatesintheright(left)dot forpositive(negative)biasvoltage. Thelinescanalsobe clearlyseeninacutalongthedetuningline(arrowinFig. IV. CONCLUSION 4(a)). Theyhaveatypicallevelspacingof≈0.5meVfor excited states in the right dot, and 0.4-0.8 meV in the We have studied a graphene double quantum dot in left dot, and are much broader than k T ≈10 µeV as a B different coupling regimes. Despite the strong capaci- result of inelastic tunneling processes. tive coupling between both dots, the tunnel coupling is However, at closer inspection additional lines parallel below the experimental resolution and no roundening of to the lower edge of the triangle for V = 6 mV and bias the hexagones at the triple points can be resolved. A to the upper edge for V = −6 mV are visible. Along bias numerical calculation of the current based on the rate these lines, the alignment between the energy level of equation leads to an estimation for the tunnel coupling the left dot and the Fermi energy in the left lead is kept of t≈14 µeV. Inside the finite-bias triangles, additional constant. Inthe caseofnegativebiasvoltage,theselines structurescanbeobserved,whichweattributetoexcited can not originate from an excited state in the left dot, dot states, but partly also to imperfections in the tunnel assuming the number of carriers in both dots to stay barriers. With the application of an in-plane magnetic constant. These lines are probably due to modulations field, additional states become visible within the finite- ofthetunnelingcouplingbetweentheleftdotandtheleft bias triangles. lead, because of resonances in this constriction18. These lines are broader than the lines parallel to the baseline, and only occur parallel to the nearly horizontal edge of Acknowledgments the triangle, which corresponds to the direction of the stronger cotunneling lines. 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