Anomalous Coulomb Drag between InAs Nanowire and Graphene Heterostructures
Richa Mitra, Manas Ranjan Sahu, Kenji Watanabe, Takashi Taniguchi, Hadas Shtrikman, A.K Sood, Anindya Das
AAnomalous Coulomb Drag between InAs Nanowire and Graphene Heterostructures
Richa Mitra , Manas Ranjan Sahu , Kenji Watanabe , TakashiTaniguchi , Hadas Shtrikman , A.K Sood and Anindya Das Department of Physics,Indian Institute of Science, Bangalore, 560012, India. National Institute for Materials Science, Namiki 1-1, Ibaraki 305-0044, Japan. Department of Physics, Weizmann Institute of Technology, Israel. (Dated: February 25, 2020)Correlated charge inhomogeneity breaks the electron-hole symmetry in two-dimensional (2D) bi-layer heterostructures which is responsible for non-zero drag appearing at the charge neutrality point.Here we report Coulomb drag in novel drag systems consisting of a two-dimensional graphene and aone dimensional (1D) InAs nanowire (NW) heterostructure exhibiting distinct results from 2D-2Dheterostructures. For monolayer graphene (MLG)-NW heterostructures, we observe an unconven-tional drag resistance peak near the Dirac point due to the correlated inter-layer charge puddles.The drag signal decreases monotonically with temperature ( ∼ T − ) and with the carrier densityof NW ( ∼ n − N ), but increases rapidly with magnetic field ( ∼ B ). These anomalous responses,together with the mismatched thermal conductivities of graphene and NWs, establish the energydrag as the responsible mechanism of Coulomb drag in MLG-NW devices. In contrast, for bilayergraphene (BLG)-NW devices the drag resistance reverses sign across the Dirac point and the mag-nitude of the drag signal decreases with the carrier density of the NW ( ∼ n − . N ), consistent with themomentum drag but remains almost constant with magnetic field and temperature. This deviationfrom the expected T arises due to the shift of the drag maximum on graphene carrier density. Wealso show that the Onsager reciprocity relation is observed for the BLG-NW devices but not forthe MLG-NW devices. These Coulomb drag measurements in dimensionally mismatched (2D-1D)systems, hitherto not reported, will pave the future realization of correlated condensate states innovel systems. Correlated electronic states continue to be the focusof the condensed matter community, thanks to their richcomplexity in physics and fascinating technological po-tential in the near future. Over the years the search forrealizing highly correlated states have led to the discov-ery of novel many-body states like excitonic condensatestates [1–4], fractional quantum Hall states [5, 6], Lut-tinger liquid phase[7–10] etc. Coulomb drag has provento be the quintessential tool for probing the electron-electron interaction in correlated systems and studied indiverse set of systems like 2D electron gas (2DEG) based(AlGaAs/GaAs) heterostructures [1, 2, 4, 11–14] to quan-tum wires [7–10] . In Coulomb drag, current ( I D ) passingin one of the layers produces an open circuit voltage ( V D )in the other layer without any particle exchange. Veryrecently, graphene based heterostructures [15–19] haverevealed intriguing feature of the drag signal at the Diracpoint [15, 17, 18]; namely that it can have both positive[15] and negative [18] amplitudes. A puzzling feature isits temperature dependence which shows monotonic be-havior with a maximum at the lowest temperature inBLG [18] whereas non-monotonic variation with a maxi-mum at an intermediate temperature ( ∼ n ), temperature ( T ) and magneticfield ( B ). The MLG-NW devices show a drag resistance( R D = V D /I D ) maximum around the Dirac point and itsdependence on n , T and B establish the Energy drag asthe dominant mechanism. In comparison, absence of thedrag signal at the Dirac point for the BLG-NW devicesand flipping sign across the Dirac point with negligibledependence on T and B suggest the dominance of mo-mentum drag mechanism.The device and measurement configuration areschematically presented in Fig. 1a. All the devicescomprise of heterostructures of hexagonal boron nitride(hBN) encapsulated graphene stack and InAs NW withdiameter between 50 to 70 nm. The heterostructureswere assembled by the standard hot pick up technique[29, 30], where the ∼
10 nm thick top hBN of thegraphene stack separates the graphene channel and theNW (SI-1 of Supplemental Material [24]). The inhomo-geneity ( δn ) of graphene is ∼ . × /cm , which cor- a r X i v : . [ c ond - m a t . m e s - h a ll ] F e b (a)(b)I D V GR V BG V DR D ( Ω ) (c)(d) B(T) R D ( Ω ) T (K) R D ( Ω ) R D T -2 (e)(f)(g) R D B R D -4 n N -2 -1 0 1 2 n G (/cm ) × 10 -0.200.20.40.6 R D ( Ω ) -4 -2 0 2 4 n G (/cm ) × 10 R D ( Ω ) -10 -5 0 5 10 I D ( µ A) -2-1012 n G ( / c m ) × 10
11 -0.5 0 0.5 R D ( Ω ) n N (/cm) × 10 FIG. 1. (a) Device schematic : The heterostructure consists of a InAs nanowire on top of a hBN encapsulated graphenestack assembled on
Si/SiO substrate. (b) 2D colormap of R D at T =1.5K plotted against the I D and n G for the D1 device(MLG-NW). (c) Response of the R D at different temperatures. (d) R D versus n G plot for different magnetic fields at T =1.5K.(e) Peak values of the R D (blue circles) plotted with temperature. R D decreases with temperature and fits (red solid line) wellwith R D ∝ T − . (f) The pink open circles are the peak value of R D plotted with magnetic field at T =1.5K. The black solidline shows data upto 200mT fits well with B . (g) R D at the Dirac point as a function of n N at T =1.5K. The solid line is anoverlay of n − N with the data. The errorbars in n N have been estimated from different sweeps of measurements of device shownin Fig S-3B (a) of the Supplemental Material [24]. responds to a Fermi energy broadening of ∆ ∼ meV and ∼ . meV for MLG and BLG, respectively. TheNWs could only be electron doped due to Fermi energypinning near the conduction band. The 1D nature ofthe NW used is ascertained by measuring the electricalconductance as a function of the V BG for shorter chan-nel length showing participation of 3-5 sub-bands (seeSI-1E of Supplemental Material [24]). The charge in-homogeneity in the NW was investigated by measuringthe temperature-dependent conductance as shown in Fig.S-1F of Supplemental Material [24], which suggests thelocalization length of ∼ I D was passed through the grapheneand V D was measured on the NW as shown in Fig. 1a orvice versa. The carrier density of the graphene ( n G ) andNWs ( n N ) were tuned by the SiO back gate ( V BG ) andby a voltage ( V GR ) between the graphene and the NW(SI-2 of Supplemental Material [24]). In our DC measure-ments, the drag signal contains a predominant flippingcomponent (sign reversal of the drag voltage with I D )together with a small non-flipping component. Here, wepresent the extracted flipping part (in the linear regime)as mentioned in section SI-2B of Supplemental Material [24], which is consistent with the drag signal measured bythe low-frequency AC (at 7Hz) technique (SI-2 of Sup-plemental Material [24]). The tunneling resistance of the ∼ − G Ω in all the devices. We have used twoMLG-NW (D1, D2) and two BLG-NW (D3, D4) devicesfor the drag measurements.Figure 1b shows the 2D colormap for the MLG-NWdevice (D1) at T =1.5K and n N ∼ × cm − , where R D is plotted with I D varying from -10 µA to +10 µA and n G varying from 0 to 2 × /cm for both electron andhole doping. The drag signal peaks near the Dirac pointand subsequently decays at higher n G . Figure 1c shows R D at different temperatures. The peak magnitude de-creases rapidly with temperature as shown by open cir-cles in Fig. 1e. Figure 1d shows the dependence of R D on magnetic field upto 400 mT at T =1.5K. Notably, weobserve a giant increase (by one order of magnitude) ofthe drag peak with increasing magnetic field as shown byopen circles in Fig. 1f. The dependence of the drag peakon n N measured by varying V GR (SI-3C of SupplementalMaterial [24]) is shown in Fig. 1g.Figure 2a shows the 2D colormap for the BLG-NW de-vice (D3), where R D is plotted as a function of I D and n G at T =1.5K for n N ∼ . × cm − . In contrast toMLG-NW devices, the drag signal flips sign from positive -4 -2 0 2 4 n G (/cm ) × 10 -0.6-0.4-0.2 R D ( Ω ) -10 -5 0 5 10 I D (µA) -3-2-10123 n G ( / c m ) × 10 -0.2-0.100.10.2 (a) (b) (c)(d) (e) (f) n N (/cm) × | R D | ( Ω ) R D n N-1.5 n G (/cm ) × R D n G-1.6 | R D | ( Ω ) n N = 8.7 x 10 n N = 5.9 n N = 2.8 n N = 2.3 n N = 1.7 n N = 1.1 /cm n G * R D ( Ω ) B(mT) R D ( Ω ) T(K) n G * ( x / c m ) × 10 T(K) R D ( Ω ) FIG. 2. (a) 2D colormap of R D with I D and n G at T =1.5K, V GR = 1 V for a BLG-NW device. The horizontal dashed lineis the Dirac point of the graphene. (b) R D versus n G plot at T =1.5K for different n N tuned by the V GR from 0.9 to 5V. (c)The red circles are the plot for dip value of R D at different n N . The variation of the drag signal with n G at V GR = 0 . V isindicated by the blue open circles. The solid lines are the fitting to ∼ n − . N and n − . G . (d) The variation of R D with magneticfield at T =1.5K. (e) The position of the dip ( n ∗ G ) of R D as a function of temperature (raw data in Fig. S-4B of SupplementalMaterial [24]). (f) The dip value of R D plotted as a function of temperature. The dashed lines in d, e and f are the guidinglines. to negative as n G shifts from holes to electrons across theDirac point with distinct peak and dip at finite densitiesof holes and electrons. At the Dirac point the R D is neg-ligible unlike the MLG-NW device. Figure 2b shows R D as a function of n G for different NW densities ( n N ∼ × cm − ) tuned by V GR . The blue circles in Fig.2c quantify how the magnitude of R D decreases with n G (for electron side in Fig. 2b) for n N = 1 . × cm − ,whereas the red circles show the magnitude of the dipof R D at n ∗ G (marked in Fig. 2b) as a function of n N .Figure 2d shows that R D at n ∗ G for the BLG-NW deviceremains almost constant with magnetic field (raw datain SI-4A of Supplemental Material [24]), in contrast withthe MLG-NW device. Figure 2e and 2f demonstrate thetemperature dependence of the drag signal for the BLG-NW device. It can be seen from Supplemental Material[24] Fig. S-4B (raw data) that the peak (hole side) ordip (electron side) position of R D shifts towards highercarrier density in graphene with increasing temperaturefor a fixed carrier density of the NW ( n N ∼ × cm − ). Figure 2e shows the position ( n ∗ G ) and the cor-responding value of R D in Fig. 2f as a function of tem-perature. Unlike the MLG-NW device, the drag signal inthe BLG-NW device clearly displays much less variationwith temperature.The observations of monotonic decrease of drag signalof the MLG-NW device as well as weak dependence of thedrag signal of the BLG-NW device on increasing temper-ature are anomalous as compared to the conventional mo-mentum drag which predicts T [31–34] increase as seen in double-layer MLG heterostructures [15]. Anomalyin temperature-dependence, specifically, drag signal in-creasing with lowering temperature has been observed in2DEG (GaAs-AlGaAs) [11] or 2DEG-graphene [19] het-erostructures. The anomalous upturn of the drag signalwith lowering temperature at low temperature regime in-dicated the presence of interlayer excitonic condensationin 2DEG-graphene system [19] or the Luttinger liquidstate in quantum wire systems [7]. The possibility of ex-citonic condensation in our MLG-NW devices is ruled outas the drag peak appears at the Dirac point of graphenewith the NW having a finite density.To explain our results, we first recall the three mainmechanisms of the Coulomb drag: (i) homogeneous mo-mentum drag (HMD) - momentum transfers via Coulombmediated scattering, (ii) inhomogeneous momentum drag(IMD) - momentum transfer in presence of correlatedinter-layer charge puddles and (iii) energy drag (ED) -vertical energy transfer in presence of correlated inter-layer charge puddles. Since the HMD signal should bezero at the Dirac point and increases as T in a Fermiliquid [31–34], it can be ruled out as the possible mecha-nism for our MLG-NW devices. Now, both IMD and En-ergy drag mechanisms predict a maximum of R D at theDirac point due to the presence of correlated inter-layercharge puddles, although the underlying physics is dif-ferent. The effective momentum theory (EMT) of IMD[23] suggests an increase of the drag signal with tem-perature in low temperature regime and should decreasewhen k B T > . . Further, the EMT does not explainthe effect of magnetic field on the drag signal. Thus theanomalous decrease with temperature and enhancementof R D with magnetic field in our MLG-NW devices is notconsistent with the predictions of the IMD.Coming now to the Energy drag mechanism, a positivecorrelation of charge inhomogeneities in MLG and NWgives rise to a positive drag peak around the Dirac pointdue to the combined effect of Coulomb mediated verti-cal energy transfer and thermoelectric Peltier effect [20].The Energy drag is expected to increase [22] with mag-netic field as B and display a non-monotonic behaviorwith temeprature [20]. Fig. 1f for the MLG-NW deviceclearly shows B dependence of R D at lower magneticfield which is consistent with the Energy drag mechanism[20]. To explain the temperature dependence, a quanti-tative theory of ED in 2D-1D system is required. In theabsence of such theory, we appeal to Song et.al for 2D-2Dsystem which shows [20] R D ∝ T κ ∂Q∂µ G ∂Q∂µ N , where ∂Q∂µ is the partial derivatives of the Peltier coefficient Q withrespect to the chemical potentials of drive ( µ G ) and draglayers ( µ N ). The quantity κ is the sum of the thermalconductivities ( κ G + κ N ) of the two layers. For double-layer graphene heterostructures, the Energy drag mech-anism [20] generates a non-monotonic temperature be-havior where the drag signal increases as T upto a tem-perature equivalent to ∼ ∆ and subsequently decreasesas T − . For the MLG-NW devices, the typical value of∆ is ∼ meV (equivalent to 150K). Hence, accordingto the Energy drag mechanism, the drag signal shouldhave increased monotonically upto ∼ ∼ e /h and thus poor electronicthermal conductivity ( κ e ) as compared to graphene, mak-ing phonon contributions to the thermal transport ( κ ph )dominant [35]. Hence, κ = κ G + κ N = κ e + κ ph . Since κ e ∝ T and κ ph (electron-phonon contribution) ∝ T ([36, 37]) , κ = ( aT + bT ), where a and b are the rela-tive contributions from the electronic and the phononicparts. The contribution of the interlayer dielectric hBNto κ ph is expected to be much smaller than that of theNW and hence is not expected to affect the temperaturedependence. Using ∂Q∂µ G ∝ T ∆ at the Dirac point and ∂Q∂µ N ∝ T µ N at µ N (cid:54) = 0 (SI-5 of Supplemental Material[24] for details), the temperature dependence of R D is R D ∝ T µ N ∆ ( aT + bT ) . Noticeably, the R D still has thenon-monotonic dependence on temperature, dependingon the relative magnitudes of the parameters a and b .The calculated R D for different values of a/b is shown in T(K) R D ( a r b . un it ) T=1.5K (a) (b) R D ( Ω ) R D T -2 -4 -2 0 2 4n G (/cm ) × 10 -6-4-20 - ∂ Q G R / ∂ µ G ( - / e ) -10123 R D ( Ω ) FIG. 3. (a) Theoretically calculated R D as a function oftemperature for several values of a/b ranging from 0.01 to10. The inset shows the experimental R D (open circles) as afunction of temperature for D2 device with T − fitting (solidred line). (b) Similarities between ∂Q∂µ (red solid line) and R D (blue dashed line). Fig. 3a, where one can see that the crossover happens attemperatures near ∼ K (below our temperature range)and decreases as T − consistent with our experimentaldata (the solid lines in Fig. 1e for D1, and in the in-set of Fig. 3a for D2). Furthermore, Fig. 3b shows thesimilarities between the dependence of R D and ∂Q/∂µ G on n G (SI-5 of Supplemental Material [24]), which fur-ther strengthens the Energy drag to be the dominantmechanism in MLG-NW devices. Moreover, the effectof carrier density of the NW on the drag peak showing n − N dependence (Fig. 1g) is compatible with the En-ergy drag mechanism as the ∂Q∂µ N ∝ T µ N = T n N (SI-5 ofSupplemental Material [24]) .We will now discuss the possible drag mechanism forthe BLG-NW devices. Drag being almost zero near theDirac point (Fig. 2a and 2b) rules out Energy drag andIMD, in favour of HMD as a possible mechanism, where R D ∝ ( k B T ) n . G n . N is consistent with our result as shown inFig. 2c (solid lines). However, we do not observe thepredicted T increase of the drag signal (Fig. 2f). Thiscan be due to that the drag signal not only slowly varieswith increasing temperature but also the shift of the R D maximum and minimum position ( n ∗ G ) towards higher n G (Fig. 2e and 2f). This happens due to the temperature-induced Fermi energy broadening, over and above theintrinsic disorder limited ∆ ( ∼ (a) (b) -1 -0.5 0 0.5 1 n G (/cm ) × 10 -0.500.5 R DG R ( Ω ) -505 R DN W ( Ω ) -2 -1 0 1 2 n G (/cm ) × 10 -202 R DG R ( Ω ) -202 R DN W ( Ω ) FIG. 4. (a) Onsager in MLG-NW device at T =1.5K. Thered line corresponds to the R D measured on NW whereasthe blue line corresponds to the R D measured on MLG. (b)Similar data for the BLG-NW device. erostructures hitherto not reported. We observe very dif-ferent drag signals for the MLG-NW and the BLG-NWdevices. The MLG-NW devices show a maximum of R D at the Dirac point and the peak value decreases withincreasing temperature as well as with the carrier den-sity of the NW. Further, the drag increases by one orderof magnitude with magnetic field. These results showthat the Energy drag mechanism is dominant for theCoulomb drag in the MLG-NW heterostructures, wherethe phononic thermal conductivity of the NWs plays asignificant role in reduced drag signal with increasingtemperature. In contrast, for the BLG-NW devices, thedrag reverses sign across the Dirac point as expected fromthe momentum drag mechanism, with slow variation withtemperature and magnetic field. Our results are promis-ing for realizing the correlated states in dimensionallymismatched novel devices, with different mechanisms atplay.Authors thank Dr. Derek Ho and Prof. ShaffiqueAdam for useful discussions on in-homogeneous momen-tum drag. AD thanks DST (DSTO-2051, DSTO-1597)and AKS thanks Year of Science Fellowship from DST forthe financial support. K.W. and T.T. acknowledge sup-port from the Elemental Strategy Initiative conductedby the MEXT, Japan and the CREST (JPMJCR15F3),JST. We acknowledge Michael Fourmansky for his profes-sional assistance in NWs MBE growth. HS acknowledgespartial funding by Israeli Science Foundation (grant No.532/12 and grant No. 3-6799), BSF grant No. 2014098and IMOS-Tashtiot grant No. 0321-4801. HS is an in-cumbent of the Henry and Gertrude F. Rothschild Re-search Fellow Chair. [1] M. Kellogg, I. Spielman, J. Eisenstein, L. Pfeiffer, andK. West, Physical review letters , 126804 (2002).[2] J. Seamons, C. Morath, J. Reno, and M. Lilly, Physicalreview letters , 026804 (2009).[3] J. Li, T. Taniguchi, K. Watanabe, J. Hone, and C. Dean,Nature Physics , 751 (2017).[4] D. Nandi, A. Finck, J. Eisenstein, L. Pfeiffer, andK. West, Nature , 481 (2012). [5] J. Eisenstein, G. Boebinger, L. Pfeiffer, K. West, andS. He, Physical review letters , 1383 (1992).[6] Y. Suen, L. Engel, M. Santos, M. Shayegan, and D. Tsui,Physical review letters , 1379 (1992).[7] D. Laroche, G. Gervais, M. Lilly, and J. Reno, Science , 631 (2014).[8] P. Debray, V. 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Department of Physics, Weizmann Institute of Technology, Israel.
SI-1. Coulomb drag devices
In this section we furnish all the details about the Coulomb drag devices. The section has been divided intoseveral sub-sections as mentioned below.
SI-1A. Monolayer Graphene (MLG) /Bilayer Graphene (BLG) -InAs NanowireHeterostructures
To prepare the hBN-graphene-hBN stacks, at first the graphene and hexagonal boron nitride (hBN) areexfoliated from their respective crystals using the standard scotch tape method and transferred on top of Si substrate caped with thermally grown SiO (thickness ∼
300 nm). The monolayer and bilayer grapheneflakes are initially identified using the optical microscope. The top and bottom hBN thickness is choosen tobe ∼ − nm and ∼ − nm , respectively. After exfoliation, the flakes are picked up in appropriateorder at ◦ C from the substrate using the PDMS/PC stamp and a homemade heater stage. Using a micro-manipulator equipped with high accuracy x,y, and z axis movement the flakes are carefully stacked under themicroscope. After completing the pick-up sequence, the stacks are dropped on a fresh
Si/SiO substrateat ◦ C along with polymer. The polymer is dissolved in solvents (eg. chloroform) and the substrate withthe stack is further cleaned with acetone/IPA.InAs nanowires (NW) are grown on (111)B substrate by the Molecular Beam Epitaxy (MBE) method
3, 4 . The NWs have typical diameter ranging from 50 nm to 80 nm. The NWs are first dispersed into ethanol ∗ [email protected] µl of the ethanol solution on another clean Si/SiO substrate withpre-defined alignment marks. As the ethanol evaporates, the NWs sit on the substrate due to the van der-Waals attraction force. NWs with diameter around 70-80 nm (identified by SEM imaging) are picked upfrom the substrate by the same dry pick-up technique using a PDMS/PPC stamp at room temperature andsubsequently dropped over the clean hBN/Graphene/hBN stack at around − ◦ C after careful alignmentsuch that the nanowire is parallel to the graphene edge. The polymer (PPC) is dissolved in solvents and theheterostructures are further cleaned by IPA/Acetone thoroughly.
SI-1B. Details of contact fabrication
After assembling the heterostructures, the samples are spin coated with bilayer (495A4/950A4) PMMA (e-beam resist) and baked at 180 ◦ C at the hotplate. The contacts for graphene and NW are made separately.Each time we follow the similar process of spin-coating, baking followed by contact patterning with e-beamlithography. To establish 1D contacts to the encapsulated graphene we follow the well-known reactive-ionetching technique and subsequent thermal evaporation of Cr (5 nm)/Pd (13 nm)/Au (70 nm). The InAsNWs have a thin native oxide layer on their surface which is removed in order to establish ohmic contact.We use the standard technique of chemical etching using .
3% (
N H ) S solution for 30 minutes at ◦ C and quickly load the samples in the evaporation chamber for thermal evaporation of Ti (5 nm)/Al (100 nm).Before the chemical etching, we perform O plasma for few seconds which helps to achieve better contactsto the NWs. The optical and the SEM images of the Coulomb drag devices are shown in Fig. S-1A (a) and(b). The typical channel length of the graphene and the nanowire are respectively ∼ µm and ∼ µm .After the fabrication, the samples were cut and mounted on a chip carrier for wire bonding. All the sampleswere carefully checked at first at room temperature and then at 1.5K for further characterization.For the Coulomb drag study, the thickness of the top hBN is very important, as it determines theseparation between the drive (graphene) and the drag (nanowire) layer. Too thick hBN will reduce the inter-layer Coulomb scattering and may lead to smaller or negligible drag signal which is difficult to measure,whereas very thin layer of hBN will increase the possibility of interlayer leakage current. Significant amountof inter-layer leakage current is unwanted in this kind of scenario as it can mask the actual signal and leadto other effects . For this reason, we limit the top hBN thickness in our devices to be ∼ − nm (Fig.S-1A (c)). 2 Distance ( µ m) -5051015 H e i gh t ( n m ) (a) (b) (c) ~10 nm Fig. S-1A: Device fabrication. (a) Optical image of a conventional Coulomb drag device. The black dashedline represents the InAs NW. The outer and inner metal contacts are for graphene and NW respectively. Thescale bar is 5 µm . (b) Scanning electron microscope image of a Coulomb drag device. The scale is 2 µm .The pink dashed line outlines the area of the graphene flake, whereas the yellow dashed line borders theposition of the NW which sits on top of the encapsulated graphene stack. (c) AFM height thickness profileof the top hBN of the encapsulated graphene stack showing ∼ nm . SI-1C. Device characterization of Graphene
In this sub-section we will discuss about the extraction of the device parameters such as the density inhomo-geneities ( δn ), mobilities for the MLG and BLG. We will also discuss how the carrier density of graphene( n G ) have been converted from the gate voltage V BG . Fig. S-1B and S-1C represent the device charac-teristics of two MLG-NW and two BLG-NW devices, respectively. Fig. S-1B(a) and (d) show backgateresponses of two MLG devices (D1 and D2). Similar plots for BLG devices (D3 and D4) are shown in Fig.S-1C(a) and (d). All four plots have inset images where 2-probe graphene resistance R is plotted against n G and the plots are fitted with the formula R = 2 R C + L/Weµ
F E q n G + δn to extract the field-effect mobility µ F E , where R C , L , W are contact resistance, length and width of the graphene channel, respectively. µ F E is found to be ∼ cm /V S for D1 and D2, whereas 65,000 and 53,000 cm /V S for D3,D4, respectively. To find the density inhomogeneities in graphene, the graphene conductance G is plottedagainst n G in the log-log fashion as shown in Fig. S-1B (b), (e), and S-1C (b),(e). We obtain δn to be around − × /cm which is standard for encapsulated graphene devices
8, 9 .The graphene density ( n G ) is tuned by the back gate voltage V BG . We apply the backgate voltage V BG in the p++ doped Si substrate across the 300 nm thick thermal oxide. Applying V BG would dope3
20 -10 0 10 20 V BG (V) R ( Ω ) -20 -10 0 10 20 V BG (V) G ( e / h )
14 14.5 15 log(n G ) -3.2-3-2.8-2.6 l og ( G ) HoleSideElectronSide -20 -10 0 10 20 V BG (V) R ( Ω ) log(n G ) -4-3.5-3-2.5-2 l og ( G ) HoleSideElectronSide (1/cm ) × R ( k Ω ) datafitted curve (1/cm ) × R ( k Ω ) datafitted curve -10 0 10 20 30 40 V BG (V) G ( e / h ) V GR (V) G ( e / h ) (a) (b) (c)(d) (e) (f) G n G n Fig. S-1B: Monolayer Graphene-NW (MLG-NW) device characterization. (a) and (d) 2-probe resis-tance of MLG-NW devices (D1 and D2) is plotted against backgate voltage V BG at T=1.5K. Inset showsthe gate response of the respective devices fitted with the mobility formula with mobility µ F E and contactresistance R C as fitting parameters. The extracted mobilities are found to be ∼ , cm /V S and ∼ , cm /V S , respectively for D1 and D2. In (b) and (e) the intrinsic inhomogeneity ( δn ) have beenextracted by plotting log G vs. log n G for the D1 and D2 devices, respectively for electron as well as holeside. The extracted δn values are . . × /cm and . . × /cm for electron (hole) sidein D1 and D2, respectively. (c) and (f) are backgate responses of 2-probe conductance of the InAs NW at1.5K while graphene contacts are in floating condition for D1 and D2 devices, respectively. Inset panel in(f) is the NW conductance plotted with V GR .the graphene. The carrier density per unit area accumulated on the graphene can be expressed as: n G =1 e C G ( V BG − V DP ) , where V DP is the backgate voltage where the graphene becomes charge neutral. Thequantity C G is the effective capacitance per unit area between the graphene and the doped Si , given by C G = (cid:15) (cid:15) r d , where (cid:15) = 8 . × − F m − is the permittivity of free space, (cid:15) r = 3 . is the relativepermittvity of SiO and d = 300 nm is the thickness of the oxide layer. By putting all the parameters, weobtain C G = 115 aF µm − .Apart from using optical microscope, we have employed Quantum hall (QH) plateaus to identifymonolayer and bilayer graphene in the heterostructures. Fig. S-1D (a) and (b) shows the the characteristic4
30 -20 -10 0 10 20 V BG (V) R ( Ω ) l og ( G ) HoleSideElectronSide -20 -10 0 10 20 30 V BG (V) G ( e / h ) V GR (V) G ( e / h ) -20 0 20 V BG (V) R ( Ω ) (1/cm ) × 10 R ( k Ω ) datafitted curve
14 14.5 15 15.5 log(n G ) -3.8-3.6-3.4-3.2-3-2.8-2.6 l og ( G ) log(n G ) -20 0 20 V BG (V) G ( e / h ) (1/cm ) × R ( k Ω ) datafitted curve (a) (b) (c)(d) (e) (f) HoleSideElectronSide G n n G Fig. S-1C : Bilayer graphene-NW (BLG-NW) device characterisation. (a) and (d) 2-probe resistanceof BLG devices D3-BLG and D4-BLG plotted against backgate voltage V BG at T=1.5K. Inset shows thegate response fitted with the mobility formula with mobility µ F E and contact resistance R C being the fittingparameters. D3 and D4 have mobilities ∼ cm /V S and ∼ cm /V S , respectively. (b) and(e) the intrinsic inhomogeneities ( δn ) of the BLG devices are calculated by plotting log G vs. log n G . δn ∼ × /cm for electron (hole) side for D3 and D4, respectively. (c) and (f) arebackgate response of 2-probe conductance of the InAs NWs for D3 and D4 devices respectively at 1.5Kwhile graphene contacts are in floating condition. Inset panel is the NW conductance plotted with V GR .QH plateaus for MLG and BLG. SI-1D. Device characterization of InAs Nanowires
InAs is a n-type semiconductor with direct bandgap of 0.35 eV. Due to fermi level pinning within theconduction band, the nanowires are always n-type. The 2-probe conductance ( G ) versus V BG (while thegraphene is at floating condition) corresponding to devices D1, D2, D3 and D4 are shown in Fig. S-1B(c),(f)and S-1C(c),(f), respectively. All the nanowires start conduction at a threshold voltage ( V T H ). Below V T H the conductance remains negligible which gradually increases and become constant (determined by5
30 -20 -10 V BG (V) G ( e / h ) -30 -25 -20 -15 -10 -5 V BG (V) G ( e / h ) B= 5 T B= 8 T(a) (b) BLGMLG
Fig. S-1D: Quantum Hall plateaus in MLG and BLG . (a) and (b) Quantum Hall response of D1-MLGand D3-BLG devices at 1.5K and captured at 5T and 8T magnetic field, respectively. (a) shows character-istic conductance plateaus of the MLG which appears at conductance of 2, 6, 10 G , whereas (b) showsconductance plateaus of BLG with 4, 8 G conductance plateaus.the contact resistance) at higher gate voltages. Since for most of the devices we obtain V T H < , thenanowires show significant conduction even at V BG = 0 due to electron doping. During the Coulomb dragmeasurement one of the graphene contacts is kept grounded. To change the carrier density ( n N ) of thenanowire in this situation, we apply a voltage V GR between the graphene and the nanowire through the tophBN, as discussed in the measurement technique section. Fig. S-1B(f) and S-1C(c) inset show the nanowireconductance as a function of V GR . For calculating n N from V GR , we use n N = 1 e C N ( V GR − V T H ) ,where C N is the capacitance per unit length. In order to evaluate C N , we use the cylinder on a infiniteplate capacitance model
10, 11 , where C N = 2 π(cid:15) (cid:15) r cosh − ( t/r ) , where t is the distance between the center of thenanowire to the graphene, and r is the radius of the NW. For our devices, the top hBN thickness ∼ nm and r ∼ nm and thus t ∼ nm . Putting all the parameters, we obtain C N = 320 aF µm − . Mobility calculation of the Nanowire:
We calculate the field-effect mobility of the InAs nanowiresusing the analytical expression : µ = LC N dGd ( V GR − V T H ) (1)Where, L is the channel length, C N , V GR and V T H are the capacitance per length, gate voltage and thethreshold voltage respectively as mentioned earlier. The term dGd ( V GR − V TH ) is calculated from the slope ofthe gate response of the nanowire 2-probe conductance. For most of the nanowires used in the Coulomb6 n N (cm -1 ) × 10 G ( e / h ) n N (cm -1 ) × 10 G ( e / h ) (a) (b) Fig. S-1E : 1D nature of the InAs nanowires. (a)Conductance of InAs nanowires as a function of carrierconcentration for 100 nm and (b) 1 µm channel lengths.drag experiments, we obtain mobilities to be ∼ − cm /V.s . SI-1E. The 1D nature of the InAs nanowires
In order to establish that the InAs nanowires used in our studies are indeed 1D systems, we have measuredthe conductance as a function of gate voltage for devices having two different channel lengths i.e. ∼ nm and ∼ µm , as shown in Fig SI-1E. The 100 nm channel device ( Fig. SI-1E a) clearly shows thesignature of different 1D sub-bands by exhibiting conductance plateaus at 2 e h and 4 e h . It can be seen thatwithin the density range of ∼ × cm − (range accessed in our experiments), maximum of five sub-bands are populated. In comparison, the 1 µm channel length shows monotonic increase of the conductance(Fig. SI-1E b) with gate voltage and saturates around 2 e h . This suggests that the transport in ∼ µm nanowire channel (used in our drag experiments) is not in the ballistic regime but rather in diffusive regimewith mobility around ∼ cm /V.s as mentioned in the previous section. The above arguments alsojustifies that the InAs nanowires used in our experiments are not heavily doped, rather only few sub bandsare populated. The conductance value for 1 µm channel length is limited due to its diffusive nature.7 V GR (V) G ( e / h ) T -1/2 (K -1/2 ) -4-2024 l n ( σ T ) ( S K ) n N (/cm) × 10 T ( K ) n N (/cm) × 10 ξ ( n m ) (a) (b)(c) (d) /cm Fig. S-1F : Localization length ( ξ ) extraction. (a) 2-probe conductance of the NW (D2-MLG) plotted with V GR at different temperatures from 1.5K to 30K. The conductance oscillations present at low temperaturedisappear at the high temperature regime. (b) Localization length ( ξ ) is extracted by fitting σ at different Taccording to Mott formula of variable range hopping. log ( σT ) with T − / for different V GR gives T . Theopen circles and the red solid lines are respectively data and fitted lines for multiple n N . (c) Extracted T plot with 1D NW densities n N . (d) Localization length ( ξ ) plotted with 1D NW density n N .8 I-1F. Localization length ( ξ ) in InAs Nanowires The localisation length ( ξ ) in the InAs NWs has been calculated using the Mott formula from the temperaturedependence of the conductivity σ at different nanowire densities. By calculating ξ we can estimate the lengthscale of the puddles present in the NW. The Fig. S-1F (a) shows the gate response of InAs NW (belongs toD2 device) at different temperatures. Here we have modeled the NW transport by variable range hopping(VRH), where the conductivity σ of the d dimensional system at temperature T is expressed as : σ = σ ( T ) exp [ − ( T /T ) d + 1 ] (2)where T and d are correlation energy scale and dimensionality of the system respectively, and σ = AT m , m ≈ . − . We have extracted the T by plotting ln ( σT ) vs. T − / which corresponds to d=1of equation (2), and measuring the intercept while linearly fitting the data (red lines in Fig. S-1F (b). InFig. S-1F (c), the T is plotted with the 1D NW densities n N which shows the energy dependence of T . Toextract ξ from T , we use ξ = 13 . k B T D ( E ) , where D ( E ) ∼ × eV − cm − is the typical surfacedensity of charged trap at oxide substrate. In the VRH model, we consider the electron transport occurs viaband of localized disordered states . Figure S-1F(d) shows ξ plotted against n N where ξ ∼ − nm . SI-2: Measurement technique
In Coulomb drag, a constant current ( I D ) is passed through the drive layer and as a result of inter-layermomentum and energy transfer, an open circuit voltage ( V D ) is generated in the drag layer without anyexchange of particle. Now, Coulomb drag for the MLG/BLG-NW devices can be measured in two con-figurations: (a) driving current in graphene and measuring voltage in NW, (b) driving current in NW andmeasuring voltage in graphene. SI-2A. DC vs. AC measurement
In Coulomb drag measurements, both DC and AC techniques have been utilized to measure the drag voltage( V D ). The measurement schematic for the DC and AC technique are presented in Fig. S-2A(a) and S-2A(b),9 a) V BG GrapheneInAs Nanowire V GR V D DMM V DC V BG GrapheneInAs NanowireLock-in AmplifierV AC V GR V D (b) Fig. S-2A: DC and AC measurement schematic.
Coulomb drag measurement schematic for (a) DC and(b) AC technique. In DC technique, (a) a DC voltage is applied across the drive layer through a large( ∼ M Ω ) resistance and the open circuit voltage appearing in drag layer is measured by a digital multimeter(DMM) having the input impedance of 100M Ω . Backagte voltage V BG is applied in Si through SiO forchanging graphene density n G , while NW is maintained at a constant conductance. An additional voltage V GR is applied in graphene which shifts the fermi energy of the NW with respect to graphene. For the ACtechnique, (b) a small frequency ( ∼ V GR is connected with the circuit using a isolation transformer as shown.10 GR Open Open DMMOpen V S1 GrapheneInAs Nanowire R1R2V S2 + - V BG (c) Fig. S-2A: Double Relay circuit. (c) We have performed DC measurement using this circuit to confirmthe Onsager reciprocity relation. Each relay R1 and R2 are connected to the two layers separately as shown.The circuit is connected such that one can independently interchange between the graphene and NW layersas drive layers without changing the V GR (NW density). The blue (red) color lines indicate where graphene(NW) is the drive layer and the NW (graphene) is the drag layer. V S and V S is the DC source voltageconnected to the graphene and NW, respectively. 11espectively. In the DC technique, a DC voltage from a voltage source (Keithley 2400 or Yokogawa GS200)is applied across the drive layer (Graphene) through a high resistance ( ∼ − M Ω ) path, and the opencircuit output voltage is measured across the drag layer (nanowire) by a Digital multimeter (Agilent 34401A6 / DMM). For the AC technique, a low frequency ( ∼ Hz ) AC signal from the output of SR 830 lock-inamplifier is applied across the drive layer whereas the output drag signal is fed into the lock-in amplifier.The input voltage is changed such that I D varies from + 10 µA to - 10 µA . In order to change the nanowiredensity, we apply V GR through graphene. For the AC circuit (Fig. S-2A (b)), an isolation transformer circuitis used for applying the DC voltage V GR . Since drag voltage is very sensitive to the carrier density of theNW, we have kept V GR fixed at a certain value while measuring the V D verses V BG . We have applied acircuit (Fig. S-2A(c)) using two relays such that the drag can be measured in both the configurations. Byswitching the relays in appropriate manner, we are able to measure the V D for multiple NW densities n N inboth the configurations. It has helped us to investigate the validity of the Onsager principle appropriately. SI-2B. Flipping and non-flipping part extraction
Although both DC and AC techniques (section
SI-2A ) yield same drag features (Fig. SI-2B (d)), the rawsignals in DC measurement are interpreted in a different way. We have observed that, the raw DC signalcontain predominantly a drag signal superposed with a small non-flipping signal. The drag signal (flippingpart) flips sign when the current direction is flipped whereas the sign of the non-drag signal (non-flippingpart) remains unchanged. In order to extract the actual drag signal from the raw data, we use the followingprotocol: V D → − V D , as I D → − I D for drag signals, but the non-flipping part which originates fromheating effect ( ∝ I R ) doesn’t changes its sign. So, we can write in equations that: V + raw = V F P + V NF P V − raw = − V F P + V NF P (3)where V + raw and V − raw are raw drag signals when I D is positive and negative, respectively. V F P and V NF P are contributing flipping and non-flipping part of the drag signal, respectively. Combining thetwo equations, we get: V F P = 12 ( V + raw − V − raw ) (4)12 (µV) D V (µV) D V (µV) D I D ( µ A) V D ( µ V ) (a) (b) (c)(d) (e) (f) -100 -50 0 50 100 V GR (mV) -100-50050100 I O u t ( p A ) R Leak Ω ~-1 -0.5 0 0.5 1 n G (/cm ) × 10 V D ( µ V ) flip part: DC method7 Hz AC method -10 -5 0 5 10 I D ( µ A) -50510 V B G ( V ) -50 0 50 -10 -5 0 5 10 I D ( µ A) -50510 V B G ( V ) -50 0 50 -10 -5 0 5 10 I D ( µ A) -50510 V B G ( V ) -50 0 50 Fig. S-2B: Flipping and non-flipping part extraction. (a) 2D colormap (raw data) of open circuit dragvoltage plotted against drive current I D along the x-axis and backgate voltage V BG along the y-axis for aMLG-NW device at 1.5K. (b) and (c) are similar 2D colormaps of the extracted flipping and non-flippingpart of the drag signal as mentioned in the text. In (b), the drag is negative for positive I D and flips signas the current direction is flipped unlike (c). Color-bar ranges are same for plots of (a)-(c) and the blackdashed line indicates the Dirac point of the system. (d) Comparison between drag voltage acquired in DCand low frequency AC technique at 1.5K for a MLG-NW device. (e) V D plotted against the I D extractedfrom the 2D plot. V D varies quite linearly with I D . (f) Inter-layer leakage current ( I Out ) is plotted againstthe inter-layer voltage ( V GR ) applied across the graphene and the NW. The black dashed line corresponds tothe leakage resistance R Leak ∼ G Ω . Our data implies R Leak in our device is much larger than G Ω .13ll the data presented in the main text are flipping part and extracted using equation (4). Fig S-2B(a)is the raw data collected by DC technique for a MLG-NW device, and S-2B(b) and S-2B(c) are the flippingand non-flipping part extracted by the above mentioned process. We can observe that, the non-flipping partis much smaller in magnitude compared to the drag signal. To cross-check the validation of this extractionprocess, we compare the extracted flipping DC data with the raw AC drag signal and we have found thatthey match remarkably well with each other as shown in Fig S-2B(d). Figure S-2B(e) shows the drag signalas a function of drive current, which varies linearly beyond µA which also ensures that all the data hasbeen recorded while the system is in the linear regime. Figure S-2B(f) shows the leakage current throughthe top hBN as a function of V GR . One can clearly see that the leakage resistance is much larger than G Ω .As seen from Fig. S-2B(b) and Fig. 1(b), 2(a) of the main manuscript, the drag signals (flippingpart) in our samples have magnitude comparatively smaller ( ∼ ) than the well-studied 2D-2D systemsreported so far. In a dimensional mismatched system, smaller drag is expected due to the limited phase spaceinvolved in scattering as compared to the 2D-2D systems. In hybrid systems like ours, only a fraction of thedrive current can interact with the carriers in the other layer to produce the Coulomb drag. In a simplifiedpicture, the drag resistance will be proportional to the ratio of width of the drag layer to the width of thedrive layer ( W Drag W Drive ). In 2D-2D system this ratio is unity, whereas in 2D-1D hybrid the ratio is two orderssmaller. Therefore, the observed drag resistance in our case ( ∼ ) roughly scales to ∼ −
200 Ω forthe 2D-2D devices, close to the observed values .Although measuring small drag signal ( ∼ ) was challenging but we could measure it accurately asthe drag voltage was few tens of micro volts, which was much higher than the resolution of 100 nV. SI-3A. Tuning n G and n N In the Coulomb drag measurements, the drag resistance R D = V D I D has been captured as a function of both n G and n N . Applying V BG in doped Si through the SiO tunes the n G . In order to change n N , we apply V GR to the graphene layer as shown Fig. S-2A. The following two equations demonstrate how the carrierdensities change with the gate voltages: C G ( V BG − V GR ) = n G e (5) C N V GR = n N e (6)14here C G and C N are the capacitance per unit area and capacitance per unit length, respectivelybetween graphene sheet and the p-doped Si with SiO as the dielectric medium, and between the cylindricalshaped NW and the graphene sheet where the top hBN of the heterostructure acts like a dielectric medium.The quantities n G and n N are 2D and 1D carrier densities of the graphene and the nanowire, respectively.In the equations above, we have not taken into account the effect of quantum capacitance of the layers, asthat doesn’t affect the qualitative outcome of our results. SI-3B. Shifting of Dirac point with application of V GR The 2-probe resistance versus backgate voltage V BG response of the MLG of the D2 device at T=1.5K fordifferent V GR values is shown in Fig. S-3A. The shift of the Dirac point towards more electron-side isgoverned by equations (4) and (5). We have taken into account this effect for cases where V GR = 0 whilepresenting n G . -20 -10 0 10 20 V BG (V) R ( Ω ) V GR =0VV GR =1VV GR =2VV GR =3VV GR =4V Fig. S-3A: Dirac point shift.
Backgate response of graphene 2-probe resistance of D2-MLG at 1.5Kfor different graphene gate voltage V GR . With increasing V GR , the Dirac point of graphene shifts towardspositive backgate voltage as the graphene becomes more hole type doped for V GR > .15 I-3C. Drag peak dependence on Nanowire carrier density for MLG-NW device
In this section we present the data showing variation of R D with the n N . The conversion of n N from V GR has been mentioned before in section SI-1D where it is shown that the threshold voltage value V T H of thedevice can directly influence n N . Evidently, any error in obtaining the V T H can lead to an error estimating n N from V GR and further the density dependence of the drag signal. As shown in Fig. S-3B (a), the gateresponses of the nanowires in our devices are quite reproducible and we could determine the thresholdvoltage quite accurately from the logG versus V BG plot. However, from Fig. S-3B (a) we see that thereis small difference in the threshold voltages between two successive gate voltage sweeps. We can quantifythe error in estimating n N as δn = C N δV BG ∼ . × cm − , where δV BG is the spread between thethreshold voltages of different sweeps. Fig. S-3B(b) shows the drag resistance ( R D ) response with thegraphene density ( n G ) at different values of n N for a MLG-NW device. The drag resistance peak decreasesas the n N increases. The R D peak magnitude appearing at n G = 0 is plotted against n N in Fig. S-3B(c)(also shown in Fig. 1(g) of the main text). We have also included the error in estimating n N as horizontalerror bars shown in Fig S-3B (c). The red solid line shows the agreement of our data with R D ∼ n − N .From fig S-3B(b), we notice that for certain nanowire densities, the drag resistance becomes negativein the intermediate n G values (for n N = 1 . × cm − ). Although we don’t have a clear understanding ofthis negative R D at finite n G , We believe that the answer may lie in the dimensionality mismatched 2D-1Dsystem. However, for the temperature and magnetic field data presented in the manuscript shown in Fig1(b)-1(f), has been measured for n N ∼ × cm − where R D is positive for all values of n G . -2 -1 0 1 2 n G (/cm ) × 10 -50510 R D ( Ω ) n N =1.22n N =1.60n N =1.79n N =2.36n N =2.73n N = 6.51 n N (cm -1 ) × 10 R D ( Ω ) (a) (b) (c) -15 -10 -5 0 5 10 V BG (V) -4 -3 -2 -1 G ( e / h ) FirstSecond
Fig. S-3B: R D vs. n n plot with errorbars. (a) Nanowire conductance as a function of gate voltages fortwo successive gate voltage sweeps. (b) R D plotted against the n G at T= 1.5K for different n N in unit of cm − controlled by the V GR . (c) The R D magnitude at n G = 0 from (b) is plotted with corresponding n N with horizontal black errorbar extracted from error in estimating the threshold voltage.16
10 -5 0 5 10 I D ( µ A) -4-2024 V B G ( V ) -20 0 20 V (µV) D -1 0 1 2 3 V BG (V) -20-10010 V D ( µ V ) -2 0 2 4 V BG (V) -60-40-20020 V D ( µ V ) (a) (b)(c) B(mT) V D ( µ V ) V D B (d) Fig. S-3C: MLG-NW sample with a dip at the Dirac point. (a) 2D colormap of V D (flipping part)plotted against I D along the x-axis and V BG along the y-axis at T=1.5K. The black dashed line indicates theposition of the Dirac point of the sample. At the Dirac point, V D shows a dip instead of a peak, unlike otherMLG-NW samples. (b) Backgate response of V D at different perpendicular magnetic fields at T=1.5K. V D rapidly increases in magnitude with magnetic field. (c) Backgate response of V D at different temperatures.At T=1.5K, V D shows a central dip at the Dirac point which diminishes quickly at higher temperature. (d)Dip magnitude of V D plotted with the applied magnetic field. The red line shows B fitting of the existingdata at low magnetic field. SI-3D. MLG-NW device with a dip at the Dirac point
Although most of the MLG-NW devices shows a peak near the Dirac point, for some devices we have ob-served a dip in drag signal near the Dirac point as shown in Fig. S-2C. The dip has the same characteristics17s the peak appearing in other MLG-NW devices. As shown in Fig. S-2C (b) and (c) the V D increases inmagnitude in presence of perpendicular magnetic field and diminishes very quickly with increasing temper-ature. These data are similar to devices D1 and D2, but with a dip instead of a peak at the Dirac point. Fig.S-2C (d) shows that the dip magnitude plotted with B fits with B at smaller values off B .The possible reason that some of the monolayer graphene devices shows a dip instead of a peak, can berelated to the type of inter-layer correlation present between the charge puddles. It is known from literature that, positive (negative) inter-layer correlation i.e. δµ δµ > < leads to positive (negative)drag signal due to Energy transfer mechanism. Positive correlation is expected when disorder potential isdominated by charged impurities, whereas puddles due to layer strain often bear negative correlation . SI 4A. Drag signal at different magnetic fields for the BLG-NW device
The raw data corresponding to Fig. 2(d) of the main text showing the dip values of R D for discrete magneticvalues which belongs to the D3 device is presented in this section. As shown in Fig. S-4A, R D dip magnitudein the electron-side remains almost constant at non-zero magnetic field unlike the MLG-NW devices. -4 -2 0 2 4 n G (/cm ) × 10 -0.2-0.100.1 R D ( Ω ) Fig. S-4A: R D of BLG-NW device with magnetic field. R D vs. n G is plotted for different magneticfields for the D3 device at T=1.5K. The R D doesn’t change significantly in presence of the magnetic fieldsas compared to the MLG-NW devices. The black dashed lines indicate the zero drag magnitude and the n G = 0 . 18 ig. S-4B: R D peak position shift with T. Backgate response of drag signal for different temperaturesplotted together for D3-BLG device at V GR = 0 . V . For all temperatures, drag signal flips across the Diracpoint. However, at higher temperature the drag signal magnitude diminishes and the peak/dip position ofdrag shifts towards higher density which affects the allover temperature variation of the drag signal. Thehorizontal black dashed line is the zero drag magnitude. SI-4B: Gate response of Drag signal at different temperatures for the BLG-NW device
In this section, we discuss the anomalous temperature dependence observed for our BLG-NW devices. Forthe BLG-NW devices, drag follows all the momentum drag properties except that, R D doesn’t increase as T with the temperature. Instead the drag peak/dip magnitude reduces slowly with increasing temperature.We also observe that the n G value at which the peak/dip appears ( n ∗ G ), shifts towards the higher value, i.e. n ∗ G increases as the temperature rises (shown in Fig. S-4B). The variation of n ∗ G and drag magnitude at thepeak with temperature have been demonstrated in Fig. 2(e) and (f) of the main text. We believe that, sincein Momentum drag the peak/dip position in density is determined by the temperature induced broadeningas well as intrinsic inhomogeneities ( δn ) of the system, different temperature regime has a role to play. Atlower temperature regime when k B T < µ δn (Here µ δn is the equivalent chemical potential due to intrinsicinhomogeneity δn ), the peak/dip position is determined by the µ δn whereas at higher temperature regime,the temperature induced Fermi energy broadening determines the peak/dip position. Since, R D magnitudevaries inversely with carrier density ( n − . ), peak/dip appearing at higher densities with increasing temper-ature leads to allover slow variation with temperature rather than usual T increase.19 × -0.200.20.40.60.8 R D ( Ω ) -4 -2 0 2 4 × -10123456 R D ( Ω ) R D ( Ω ) -10 -5 0 5 10 I D ( µ A) -2-1012 n G ( / c m ) × 10
11 -0.5 0 0.5-2 -1 0 1 2 × R ( Ω ) n G (cm -2 ) n G (cm -2 )n G (cm -2 ) n N (cm ) × G ( e / h ) -1 (a)(b) -2 -1 0 1 2 n G (/cm ) × -0.200.20.40.60.8 R D ( Ω ) (c) (d)(e)(f) Fig. S-4C: Data for MLG-NW (D1). (a) NW conductance plot with the 1D nanowire density n N atT=1.5K. (b) and (c) respectively shows plot of 2-probe graphene resistance and Drag resistance R D plotwith density n G . (d) and (e) R D vs. n G plot for different temperatures and magnetic fields respectively.(f) 2D colormap of R D as a function of drive current I D and n G . Data of (c)-(f) has been obtained for n N ∼ × cm − . 20 I-4C: Density, Temperature and Magnetic field dependence of MLG-NW and BLG-NWdevices
In this section we present nanowire and graphene response with density, magnetic field and temperatureconcomitantly with the drag resistance response. Fig. S-4C and 4D respectively shows the data where R D vs. n G plot is shown concomitantly with graphene and nanowire responses for MLG-NW D1 and BLG-NW D3 devices. Fig. S-4E has the data for nanowire conductance ((a), (b)) for different temperature andmagnetic fields respectively. SI-5A: Calculation of ∂Q∂µ for Graphene and Nanowire
In this section we will discuss how we have obtained the expressions for ∂Q∂µ for the graphene and the NWand finally the total thermal conductivity of the graphene-NW system.
Calculation of ∂Q∂µ : To explain the temperature dependence of our MLG-NW devices, a quantitativetheory of ED in 2D-1D system is required. In the absence of such theory, we appeal to the Energy drag bySong et.al for 2D-2D systems. From ref , we obtain the expression for Energy drag as: ρ D ∝ T κ (cid:16) ∂Q GR ∂µ G (cid:17)(cid:16) ∂Q NW ∂µ N (cid:17) (7)Equation (7) shows that the drag resisivity is directly proportional to the ∂Q∂µ of both the graphene andNW layers and inversely related to the total thermal conductivity of both the layers κ . In the experiment,we observe the R D peak appearing at the Dirac point i.e. when µ G = 0 , while the nanowire has a finitecarrier density, i.e. µ N = 0 . This situation indicates that in our devices only ∂Q GR ∂µ G (cid:12)(cid:12)(cid:12)(cid:12) µ G =0 and ∂Q NW ∂µ N (cid:12)(cid:12)(cid:12)(cid:12) µ N =0 contributes towards the observed non-zero drag peak at the Dirac point.In Energy drag the heat current ( j q ) and the charge current ( j ) are coupled by: j q = Q j , where Q is thethe Peltier coefficient. The general expression of Q can be written in terms of the layer conductivity σ andchemical potential µ as: 21 × R ( Ω ) -6 -4 -2 0 2 4 6 × -0.6-0.4-0.200.20.4 R D ( Ω ) n N (cm -1 ) × G ( e / h ) n G (cm -2 )n G (cm -2 ) (a)(b)(c) n G (cm -2 ) × -6 -4 -2 0 2 4 6 × -0.3-0.2-0.100.10.2 R D ( Ω ) -10 -5 0 5 10 I D (µA) -3-2-10123 n G ( c m - ) × 10 -0.2-0.100.10.2 R D ( Ω ) (d)(e)(f) n G (cm -2 ) Fig. S-4D: Data for BLG-NW (D3). (a) NW conductance plot with the 1D nanowire density n N at T=1.5K.(b) and (c) respectively shows plot of 2-probe graphene resistance and Drag resistance R D plot with density n G . (d) and (e) R D vs. n G plot for different temperatures and magnetic fields respectively. (f) 2D colormapof R D as a function of drive current I D and n G . Data of (c)-(f) has been obtained for n N ∼ × cm − . Q = π k B T e ( dσ/dµ ) σ (8)22 n N (cm -1 ) × -0.100.10.20.30.40.5 G ( e / h ) n N (cm -1 ) × -0.100.10.20.30.40.5 G ( e / h ) (a) (b) Fig. S-4E (a) and (b) NW conductance vs. n N plot for different temperatures and magnetic fields respec-tively. The NW belongs to device D2.Now for graphene the conductivity can be expressed as σ = σ + n G eµ F E , where σ is the residualconductivity at the charge neutrality point, µ F E is the field-effect mobility and n G is the carrier density ingraphene. Using this expression for conductivity and the linear energy-momentum relation for the mono-layer graphene µ G = ~ v f √ πn G , equation (8) reduces to: Q GR = 2 π k B T e µ G µ G + ∆ (9)where Q GR is the Peltier coefficient for the monolayer graphene, ∆ = σ ~ v f πeµ F E , here ~ is thereduced Plank constant, v f is the Fermi velocity, k B is the Boltzmann constant, e being the electronic chargeand µ G is the graphene chemical potential respectively. The partial differentiation of equation (9) withrespect to µ G leads to ∂Q GR ∂µ G (cid:12)(cid:12)(cid:12)(cid:12) µ G =0 ∝ T ∆ . Putting appropriate parameters for graphene, i.e. σ ∼ µS , v f = 10 m/s , µ F E = 100 , cm /V s , we obtain ∆ ∼ meV .Now to calculate ∂Q NW ∂µ N , we remember that InAs has parabolic band structure, i.e. µ N = ~ π n N m ∗ ,where m ∗ is the effective mass corresponding to the InAs band structure and we take the NW conductivityto be σ = n N eµ F E which yields: 23 NW = π k B T e µ N (10)The partial differentiation of equation (10) leads to ∂Q NW ∂µ N (cid:12)(cid:12)(cid:12)(cid:12) µ N =0 ∝ T µ N . Calculation of κ : The total thermal conductivity κ is the sum of the thermal conductivities ofgraphene ( κ G ) and nanowire ( κ N ), i.e. κ = κ G + κ N . Now for graphene, the electronic contributiontowards κ is dominant at low temperatures, so κ G ∝ c T . Whereas for nanowires, the phononic contribu-tion towards the thermal conductivity is very prominent, as the nanowires are poor electrical and hence poorthermal conductors. So, κ N ∝ ( c T + bT ) , here c , c and b are constants. Combining the effects for boththe drive and the drag layer yields κ ∝ ( aT + bT ) , where a = c + c and b are the relative contributionfrom the electronic and phononic part towards the thermal conductivity. This brings the equation (7) to: ρ D ∝ T µ N ∆ ( aT + bT ) (11)In this calculation we have not taken into account the contribution of the electron-phonon couplingfrom the interlayer dielectric hBN towards the total thermal conductivity κ of the system. This is becausethe contribution from the mentioned effect will be much smaller than the corresponding contribution fromnanowire and hence may not affect the temperature dependence of R D .Now to explain the temperature dependence of our MLG-NW devices from equation (11), we seethat (shown in main text Fig 3(a)) the drag resistivity at first increases upto a certain T and monotonicallydecreases further where the ab ratio determines the value of T . According to equation (11), the value of T is below our base temperature T=1.5K for ab = 5 (shown in main text fig 3(a)), i.e. even when the phononicpart is 5 times smaller then the electronic part. SI-5B. Plot of Q vs. µ and ∂Q∂µ vs. µ from experimental data In the Fig. 3(b) of the main text, we have compared the density response of the ∂Q GR ∂µ G derived from theexperimental results to that of the measured drag resistance R D and we have found that the width of both24he plots near the Dirac point are very similar. This similarity in dependence supports our claim that thedrag in MLG-NW devices are originated from the Energy drag mechanism . In Fig. S-5 (a) the Peltiercoefficient of the graphene Q GR vs. µ G is plotted for one of the devices. We calculate Q GR from equation(8) from experimentally obtained gate response of the graphene. σ vs. µ G is obtained from the 2-proberesistance R vs. V GR using σ = LW R , where L and W are the length and width of the graphene channeland µ G = ~ v f √ πn G , n G being the graphene density; The conversion of n G from V BG has been discussedbefore in section SI-1C. Fig. S-5 (b) shows ∂Q GR ∂µ G plot with µ G which is calculated by performing derivativeof equation (8): ∂Q GR ∂µ G = π k B T e (cid:2) σ d σdµ G − (cid:0) dσ/dµ G σ (cid:1) (cid:3) (12) -0.02 -0.01 0 0.01 0.02 µ G (eV) -4-2024 Q G R ( µ V ) -0.02 -0.01 0 0.01 0.02 µ G (eV)0246 ∂ Q G R / ∂ µ G ( x10 - / e ) (a) (b) Fig. S-5: (a) Peltier coefficient ( Q GR ) obtained from experimental data plotted with the chemical potentialof graphene µ G . (b) Partial derivative of Q GR with respect to graphene chemical potential, ∂Q GR /∂µ G plotted against µ G . SI-5C: Relation between the chemical potential ( µ N ) and carrier density ( n N ) of thenanowire In Fig. 1(g) of the main text, we see that the R D for MLG-NW devices varies with the nanowire density n N as R D ∝ n N and we say that this relation follows directly from the Energy drag mechnism in MLG-NWdevices. To prove that, we recall equation (10), which says Q NW ∝ µ N . Partial differentiation of equation2510) w.r.t µ N yields ∂Q NW ∂µ N (cid:12)(cid:12)(cid:12)(cid:12) µ N =0 ∝ µ N . The relation between the chemical potential µ N and the carrierdensity n N for InAs nanowire is µ N ∝ n N , which comes from the parabolic energy-momentum relationand the 1D density of states. Since, µ N = ~ k F m ∗ and k F = n N π for one-dimensional system, where k F isthe Fermi wavevector, m ∗ is the effective mass of electron in the conduction band for InAs nanowire bandstructure, we obtain µ N = ~ π n N m ∗ . This leads to, ∂Q NW ∂µ N (cid:12)(cid:12)(cid:12)(cid:12) µ N =0 ∝ n N as obtained experimentally.26 eferences
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