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E. E. Vdovin, K. Kapralov, Yu. N. Khanin, A. Margaryan, [K. Watanabe](https://orcid.org/0000-0003-3701-8119), [T. Taniguchi](https://orcid.org/0000-0002-1467-3105), C. Yang, S. V. Morozov, D. A. Svintsov, K. S. Novoselov, D. A. Ghazaryan

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[Inelastic resonant tunnelling through adjacent localised electronic states in van der Waals heterostructures](https://mdr.nims.go.jp/datasets/45db4c08-eaa2-4542-b45c-23ce6723bd46)

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Inelastic resonant tunnelling through adjacent localised electronic states in van der Waals heterostructuresnpj | 2D materials and applications ArticlePublished in partnership with FCT NOVA with the support of E-MRShttps://doi.org/10.1038/s41699-025-00528-6Inelastic resonant tunnelling throughadjacent localised electronic states in vander Waals heterostructuresCheck for updatesE. E. Vdovin1, K. Kapralov2, Yu. N. Khanin1, A. Margaryan3, K. Watanabe4, T. Taniguchi4, C. Yang5,S. V. Morozov1, D. A. Svintsov2, K. S. Novoselov6 & D. A. Ghazaryan3,6Van der Waals heterostructures offer unprecedented opportunities to design next stage functionalelectronic 2D devices. Most architectures of those devices incorporate large bandgap insulator – hBNas an encapsulating or tunnel barrier layers. Here, we use an architecture of gated vertical tunnellingtransistors to study a generic phenomenon of electron resonant tunnelling through adjacent localisedelectronic states in hBN barriers. We demonstrate that in the case of two localised electronic states,the tunnelling can be of inelastic nature giving rise to explicitly strong resonant features. It allowsaccurate tunnelling spectroscopy of delicate features of emitting and collecting layer electronicdensity of states, such as second neutrality point bandgap of moiré monolayer and electric fieldinduced bandgap of Bernal bilayer graphene. Our findings enrich the perception of interactionmechanisms among the localised electronic states in hBN barriers paving the way for futureexplorations into their applications.The advent of 2D crystals and van derWaals heterostructures1,2 has boostedthe development of the field of nanoelectronics offering unprecedentedopportunities for creation of functional devices based on novel physicalprinciples or enhanced performance characteristics. Graphene and hBN arethepivotal components required ina varietyofdevice architecturedesigns3,4.Vertical tunnelling transistor devices present a special architecture5 with anoperational principle based on tunnel effect. The resonant mechanisms invertical tunnelling transistor devices can be broadly categorized into twotypes – twist-controlled elastic6–8 and a variety of assisted inelastic events.The latter involves an energy loss during tunnelling process due to theinteractions with quasiparticles9,10 or, for instance, with moiré potential11.The palette of assisted inelastic mechanisms can be further expanded toinclude excitons, magnons, photons and others when replacing the con-stituting graphene or hBN layers with 2D transition metal dichalcogenidesor other representatives of 2D family12–15. Notably, replacing the emitter andcollector layers with the latter allows observation of twist-controlled andspin-conserved elastic tunnelling events16. Resonant tunnelling can alsomanifest without third quasiparticle’s involvement. Those resonances,however, are of an elastic nature and arise due to the sequential tunnellingthrough atomic defect based individual localised electronic states within thetunnel barrier17–19.The elastic sequential resonant tunnelling through a singular localisedelectronic state in hBN emerges when the chemical potential of collector oremitter layer aligns with it energetically. This results in the opening of aconductive channel with step (peak) resonant feature in current-voltage(conductance-voltage) characteristics of the device17,18 and can be of great usefor infrared photodetection applications20. Notably, introducing severallocalised electronic states usually results in the emergence of parallel con-ductive channels19 exhibiting similar features. Nevertheless, late 20th centurystudies of tunnelling through two localised electronic states highlighted theroleofmorecomplexdynamicsatplay showing that thosemay interactwithinthe tunnel barriers21 and lead to the rise of correlated current behaviours22.Furthermore, more recent studies demonstrate that the sequential resonanttunnelling through localised electronic state in hBNmay alsomanifest aswithan assisted mechanism as an inelastic event when the corresponding atomicdefects within the barrier are coupled to the lattice phonons23.It is worth noting, that atomic defects in hBN can serve as effectivevisible light single-photon emitters24,25 offering numerous advantages, such1Institute of Microelectronics Technology RAS, Chernogolovka, 142432, Russia. 2Center for Photonics and 2D Materials, Moscow Institute of Physics andTechnology, Dolgoprudny, 141701, Russia. 3Laboratory of Advanced Functional Materials, Yerevan State University, Yerevan, 0025, Republic of Armenia.4National Institute for Materials Science, Namiki 1-1, Tsukuba, 305-0044 Ibaraki, Japan. 5State Key Laboratory of Physical Chemistry of Solid Surfaces, Colla-borative InnovationCenter of Chemistry for EnergyMaterials (iChEM), College of Chemistry andChemical Engineering, XiamenUniversity, Xiamen, 361005, China.6Institute for Functional Intelligent Materials, National University of Singapore, Singapore, 117575, Republic of Singapore. e-mail: kostya@nus.edu.sg;davitghazaryan@ysu.amnpj 2D Materials and Applications |             (2025) 9:7 11234567890():,;1234567890():,;http://crossmark.crossref.org/dialog/?doi=10.1038/s41699-025-00528-6&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41699-025-00528-6&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41699-025-00528-6&domain=pdfmailto:kostya@nus.edu.sgmailto:davitghazaryan@ysu.amwww.nature.com/npj2dmaterialsas room-temperature operation and potential compatibility with photonicdevices. Recent studies demonstrate that hBN can host a variety26,27 ofoptically active atomic defects including carbon or oxygen impurities orboron and nitrogen vacancies. The former can be controllably introducedinto the hBN crystal. Those defects exhibit strong photo- and electro-luminescent narrowemission lines28,29,whichare essential characteristics fordesigningnext generationquantum light sources30,31 basedon single-photonemission mechanism. Despite this promising development, challengesremain in understanding the underlying mechanisms governing the emis-sion properties of a variety of atomic defects in hBN.In this work, we use dual gated vertical tunnelling transistors to studyresonant electron tunnelling through two adjacent localised electronic statesin few atomic layer hBN barriers sandwiched in-between moiré monolayerand AB Bernal bilayer graphene collector and emitter layers. We demon-strate that the resonant tunnelling through such two adjacent localisedelectronic states canbeof an inelastic nature as a generic phenomenon in thecase of wide tunnel barriers and short localization lengths. We also displaythat those inelastic tunnelling events dominate all over the other channels ofelastic events through individual localised electronic states giving rise toexplicitly strong resonant features. It allows their utilization in accurateresonant tunnelling spectroscopy of delicate features in electronic density ofstates of emitter andcollector layers, suchas secondneutrality point valence-band bandgap of moiré monolayer and electric field induced bandgap ofBernal bilayer graphene. Our work enriches the understanding of interac-tionmechanisms among localised electronic states in hBN facilitating futureinvestigations into their potential applications.Inelastic resonant tunnelling through adjacent localised elec-tronic states in hBNThe tunnelling current resulting from inelastic resonant events between twoadjacent localised electronic states can, under certain conditions, surpass theelastic resonant current associated with each state. As illustrated in Fig. 1a, aschematic band diagram depicts the distribution of localised electronicstatesA and B within the hBN tunnel barrier. Notably, the resonant featurein the tunnelling conductance as a function of bias voltage—correspondingto inelastic transitions between statesA andB in series (denoted asA!B inFig. 1b)—is approximately an order of magnitude greater than elastic tun-nelling features observed through the individual electronic statesA or B. Inthis context, an electron in stateA has the option to either directly traverse tothe drain or take a shorter route to stateB. The latter pathwaymay incur theadditional cost of emitting a phonon9 or a plasmon10 or emerge due toAuger-like process.The inelastic tunnelling event’s rate contains extra small couplingconstant g responsible for electron-phonon and electron-electron interac-tions. It exhibits a weak (non-exponential) dependence on the distancebetween the localised electronic states. When separating the elastic andinelastic components of the total (from source to drain) resonant tunnellingcurrent in the balance equation (see Supplementary Note 1 for furtherdetails), it becomes evident that the inelastic component predominates ifjgj2>e�2κLSA þ e�2κLBD , where κ represents the localization length of theelectronic state, LSA (LBD) denotes the distance from the source (B) tolocalised state A (drain). Notably, inelastic resonant events consistentlydominate when the tunnelling probability is low, such as in scenariosinvolving wide barriers or short localization lengths. The evaluated plotspresenting the inelastic and elastic components of the tunnelling current as afunction of the total attenuation between the source and drain layers, e�2κd ,are illustrated inFig. 1c.Our calculations incorporate theprecise positionsoflocalised electronic stateswithin the hBNbarrier (seeMethods formodelingelectrostatic parameters), which were determined through fitting resonanttunnelling features and will be elaborated upon in next section. Here, theinelastic current significantly surpasses the elastic current as the tunnellingweakens (� 10�7 for jgj2 ¼ 0:1). This intriguing behaviour, where higher-order inelastic tunnelling events prevail overfirst-order elastic events, can beattributed to the selectionof the easiest tunnellingpath for the electron. If thecoupling is weak, i.e. the barrier is wide, and is also high, the electron wouldalways “choose” an inelastic path.Electric field fine-tuning of resonant features arisen frominelastic tunnelling events through adjacent statesTo investigate the resonant features arising from our inelastic tunnellingevents, we conducted low-temperature differential tunnelling conductancemeasurements for our dual gatedDevice 1withmoirémonolayer andBernalbilayer graphene layers used as bottomand top layers correspondingly (referto Methods for further details and Supplementary Note 2 for the deviceschematics). The results are presented as a function of backgate and biasvoltages for several fixed topgate voltages of V tg ¼ 2; 3; 4V in Figs. 2a, d,and g, respectively. The maps reveal that certain resonant features, mani-fested as peaks in the conductance, stand out prominently against a back-drop of various weaker features. This prominence can be attributed to thepredominance of inelastic tunnelling events occurring through adjacentlocalised electronic statesA and B within hBN barrier. Additionally, a rangeof other resonant features emerge from the tunnelling through individuallocalised electronic states, which are of a different origin and give rise toelastic resonant tunnelling. These include tunnelling features associatedwithFig. 1 | Inelastic resonant tunnelling through twoadjacent localised states in hBN barrier, con-ceptualization. a Schematic band diagrams illus-trating elastic (top) and inelastic (bottom) resonanttunnelling events through localised states A and B.b Tunnelling current (J) and differential tunnellingconductance curves at T ¼ 2 K as a function of bias(Vb) voltage at fixed backgate and topgate voltagesofVbg ¼ 0 V andV tg ¼ 2 V. The tunnel current andconductance values are normalized over the cross-sectional area of theDevice 1 (S � 20 μm2). The peak(step) in the conductance (current) bias voltagedependence, which is corresponding to inelastictunnelling through states A and B in series (markedas A-> B) is about an order of magnitude greaterthan the one corresponding to elastic tunnellingthrough state A (marked as A). c Evaluation ofinelastic tunnelling current through A to B for twocoupling constants jgj2, and elastic current forseparate events of tunnelling through states A and Bvs the transparency of the barrier, where κ�1 is thelength of the localization.https://doi.org/10.1038/s41699-025-00528-6 Articlenpj 2D Materials and Applications |             (2025) 9:7 2www.nature.com/npj2dmaterialslocalised electronic states A or B, as well as contributions from numerousother states located in different layers of hBN barrier (see SupplementaryNote 3 for more details). Notably, recent studies23 display that some ofweaker features observed can be also attributed to the defect-phonon cou-pling assisted tunnelling throughhBNbarriers.Here,we limit our discussionto the generic phenomenon describing conceptually the inelastic resonanttunnelling through two adjacent localised electronic states.To leverage our findings, we model the electrostatic parameters of ourdevices (see Methods for details) to elucidate the characteristics associatedwith inelastic tunnelling events. In line with31, our approach employs gatedconfiguration featuring graphene electrodes being gated by metallic gateswith an additional Au topgate (see Supplementary Note 2). To characterizethe energetic spectrum of Bernal bilayer graphene, we adopt a four-bandmodel with a self-consistent bandgap, as introduced in ref.32, alongside aneffective screening approach for evaluating the electric field between thegraphene layers. Next, we apply modified Landauer-Buttiker model toderive an expression for inelastic resonant tunnelling conductance (current)through our localised electronic states A and B (see Methods for furtherdetails). Our best fit to the experimental data for Device 1 indicates that thehBN tunnel barrier consists of six monolayers (d) with localised states of Aand B positioned at zA ¼ d=6, and zB ¼ 2 d=3. Their zero electric fieldenergies correspond to E0A ¼ 83 meV,E0B ¼ 73 meV. As a result, whenintroducing the resonant conditions of μGr>EA>EB>μBGr andμGr<EA<EB<μBGr, where μGr (μBGr) denote the chemical potentials ofmoirémonolayer (Bernal bilayer) graphene, and EA (EB) represent the in-fieldenergies of localised electronic state A (BÞ, we generate differential tunnel-ling conductance maps. These maps, plotted as a function of backgate andbias voltages, are presented in Figs. 2b, e, h for fixed topgate voltages ofV tg ¼ 2, 3, 4 V, respectively. Here, dashed white lines indicate deeps in thedifferential tunnelling conductance illustrating the alignment of chemicalpotential of moiré monolayer graphene with its neutrality point.When we take a close look at the differential tunnelling conductancemaps, we can see that inelastic resonant tunnelling events between twolocalised electronic statesA and B are influenced significantly by the electricfield, specifically, the topgatevoltage applied to thedevice. Figure 2cpresentsthe band diagram of an arbitrary point of the inelastic resonant tunnellingconductive channel indicated by white dot in Fig. 2b. Here, the changes inelectrostatic field (due to the application of bias and backgate voltage) alongthe conductive channel shift the energy levels of localised electronic statesAand B fulfilling the resonant conditions all over the channel. Figure 2 f, ipresent the band diagrams at more specific points also indicated by whitedots in Fig. 2 e, h. At these points, the energy levels of states A and B aligncausing the tunnelling process to shift from inelastic to elastic. Interestingly,when topgate voltage of 4 V is applied, the four conductive channels con-verge at a single point.Notably, topgate voltage canbefinely adjusted to tuneA andB states in such away to fail the resonant conditions along the inelasticsequential tunnelling conductive channel (see Supplementary Note 4 foradditional details). Those regions are highlighted as a transparent shadedarea in Fig. 2b, e. The fact that resonant inelastic tunnelling can take placealready at very small bias voltages (see Fig. 2g, h) implies the absence ofenergy threshold for this process. Obviously, it does not necessarily excludethe resonant inelastic tunnelling associated with phonon emission fromphysical candidates as it alsomay take place for higher bias voltage cases, butpredominantly displays that the origin of these inelastic transitions can beassociated with Auger-like processes or, for instance, an emission of 2Dplasmons possessing gapless energy spectrum.In contrast, within the weak coupling regime and at high barrierconditions, the tunnelling current resulting from inelastic resonant eventsFig. 2 | The dominance of inelastic resonant tunnelling events and their electricfield tuneability. a, d, and g, Experimental differential tunnelling conductancemapsat T ¼ 2 K as a function of bias (Vb) and backgate (Vbg) voltages at fixed topgatevoltages of V tg ¼ 2V(a), V tg ¼ 3V(d) and V tg ¼ 4V(g). The tunnelling con-ductance values are normalized over the cross-sectional area of Device 1(S � 20μm2). b, e, and h, Evaluated differential tunnelling conductancecounterplots corresponding to inelastic tunnelling events through states A and B (BandA) in series as a function of backgate and bias voltages at fixed topgate voltages ofV tg ¼ 2V (b), V tg ¼ 3V(e) and V tg ¼ 4V(h). Dashed white lines in (b, e, h)illustrate the alignment of the chemical potential of moiré monolayer graphene withits neutrality point. e, f, and i, illustrate band diagrams corresponding to white dotsin (b, e, h) maps, respectively.https://doi.org/10.1038/s41699-025-00528-6 Articlenpj 2D Materials and Applications |             (2025) 9:7 3www.nature.com/npj2dmaterialsbetween adjacent localised electronic states does not consistently prevailover elastic events. For Device 2, which comprises nine monolayers of hBNthat form the tunnel barrier, that features several localised states distributedthroughout the barrier, we observe only elastic resonant events undersequential tunnelling mechanisms through two localised states (refer toDevice 2 data in Supplementary Note 5). This event manifests as a pro-nounced negative differential conductance as a function of bias voltage andis comparable in magnitude to the resonant events associated with indivi-dual states. Moreover, the tuning of the electric field can indeed shift therelative positions of these elastic resonant features in relation to the backgateand topgate, however, this adjustment does not result in the emergence of aforbidden tunnelling regions observed in Device 1.Inelastic resonant tunnelling for the spectroscopy of delicateelectronic density of state featuresStudies of low-temperature differential tunnelling conductance at elevatedbackgate voltages reveal that our inelastic tunnelling events can serve as apowerful tool for probing subtle features in the electronic density of states ofemitter or collector layers (see Fig. 3). This technique further boosts thedefect-assisted tunnelling spectroscopy33 enabling the visualization34 ofelectronic bandgap at the second neutrality point of the bottom moirémonolayer graphene, which is aligned with underlying encapsulating hBNlayer. Additionally, it facilitates the measurement of the electric field-tuneable bandgap of top Bernal bilayer graphene with an exceptional pre-cision.The experimental differential tunnelling conductance contour plot, asa functionofbias andbackgate voltages atfixed topgate voltageofV tg ¼ 4V,is presented in Fig. 3a. The evaluated differential tunnelling conductancemap for our inelastic events is shown in Fig. 3b. In both cases, resonantfeatures associated with inelastic events through two adjacent localisedelectronic statesA and Bmanifest as a series of kinks and forbidden regionswithin a limitedwindows of bias and backgate voltages applied to the device.These regions are indicated by transparent squares in the experimental (seeFig. 3a) and theoretical (see Fig. 3b)maps and are further detailed in Figs. 3c,e (experimental) and Figs. 3d, f (theoretical) as zoomed-inmaps. Figures 3c,d present the case of a reconstruction of energetic band structure at thevalence band second neutrality point of moirémonolayer graphene. Figures3e, f, on the other hand, illustrate the case of electric field induced bandgapopening of Bernal bilayer graphene. Notably, our experimental results are inan excellent agreement with theoretical evaluations.The bias and backgate voltage positions of these resonant featuresenable the determination of the twist angle in moiré monolayer graphene35measured as the distance between the two parallel dashed lines in Fig. 3b.Here, similar to the approach used for the extraction of second neutralitypoint energetic values from the gate dependence of field-effect moirémonolayer transport devices36, we used Eq. (1) system of electrostaticequations to obtain the corresponding lines when the chemical potential ofmoiré monolayer nests at its first neutrality point and valence band secondneutrality point (see white dashed lines in Fig. 2). Our evaluations showedthat the latter corresponds to the energies of 220meV. Next, it was directlyconverted to the twist angle of approximately 1° using the approach pre-sented in ref.37,38. This twist angle induces a bandgap at the valence band ofthe energetic spectrum37–39, which starts to reconstruct at 220meV. Here,the resonant tunnelling from electronic state A into the collector moirégraphene (see the inset in Fig. 3a) is effectively prohibited within a narrowFig. 3 | Resonant inelastic tunnelling spectroscopy of delicate features in elec-tronic density of states of emitter and collector layers. a Experimental differentialtunnelling conductance at T ¼ 2 K as a function of bias (Vb) and backgate (Vbg)voltages at fixed topgate voltage of V tg ¼ 4V. Insert illustrates the schematic banddiagram corresponding to the forbidden tunnelling event presented in a zoomedmap in (c) and evaluated in (d). The tunnelling conductance values are normalizedover the cross-sectional area of Device 1 (S � 20 μm2). b Evaluated differentialtunnelling conductance maps corresponding to inelastic tunnelling as a function ofbias (Vb) and backgate (Vbg) voltages at fixed topgate voltage of V tg ¼ 4V. Thewhite dashed lines correspond to the alignments of chemical potentials of moirémonolayer and Bernal bilayer graphene electrodes with their neutrality points, andthe alignment of the chemical potential of moiré monolayer graphene with valenceband second neutrality point arising from its alignment with encapsulating hBNlayer. Insert illustrates the schematic band diagram corresponding to the forbiddentunnelling event presented in a zoomedmap (e) and evaluatedmap (f). c–fZoomed-in regions of experimental (a) and theoretical (b) differential tunnelling con-ductancemaps corresponding to resonant features associatedwith the twist-angle inmoiré monolayer graphene of approximately 1° with a valence band bandgap ofΔ2NP�Gr ¼ 7 meV (c–d), and with the electric-field induced bandgap of ΔBGr ¼42meV of Bernal bilayer graphene (e–f).https://doi.org/10.1038/s41699-025-00528-6 Articlenpj 2D Materials and Applications |             (2025) 9:7 4www.nature.com/npj2dmaterialsrange of bias and backgate voltages, particularly when its chemical potentialnests at secondneutrality point bandgap (seeFig. 3c),whichweobserve tobeof Δ2NP-Gr ¼ 7 meV. The same twist angle of moiré graphene layer alsoinfluences the conduction band. However, in this case, it instead manifestsas a subtle, smeared feature at the background of inelastic and elasticresonant features (see the loci distortions indicated by black arrows inSupplementary Fig. 4), which we attribute to the absence of conductionband second neutrality point bandgap. Instead, it represents a deep in thedependence of differential tunnelling conductance from bias or backgatevoltages.Utilizing a similar approach, one may enable the determination of theelectric field-tuneable electronic bandgap of Bernal bilayer graphene. Here,the emergence of the bandgap results in a distinct window of bias and gatevoltages (at a fixed topgate voltage) characterized by forbidden states fortunnelling events from state B to the collector layer. This occurs when thechemical potential of bilayer graphene resides within its bandgap (see theinset in Fig. 3b). For topgate voltage ofV tg ¼ 4V, as illustrated in Fig. 3, weobserve strong correlation between experimental and theoretical results,yielding to the bandgap of ΔBGr ¼ 42meV (see Figs. 3e, f). Our techniquefacilitates high precision extraction of the bandgap of Bernal bilayer gra-phene across various topgate voltages (see Supplementary Fig. 7). Fur-thermore, under perpendicular magnetic fields, the studies of differentialtunnelling conductance displays that our resonant inelastic tunnellingevents can also serve as a valuable tool for magneto-spectroscopy, likequantum-dot assisted spectroscopy40, revealing subtle features, such as spinand valley associated gaps in quantized spectra of emitter/collector layers atmoderate magnetic fields (see Supplementary Fig. 8).In conclusion, ourwork highlights the dominance of inelastic resonanttunnelling events over elastic tunnelling events for two localised electronicstates closely positioned in wide hBN barriers and exhibiting short locali-zation lengths. Our findings reveal that inelastic tunnelling current reso-nances, which may be driven by interactions like electron-phonon, canexceed the elastic current resonances by an order of magnitude under theconditions of small tunnelling probabilities. We also demonstrate that theobserved strong resonances associatedwith inelastic tunnelling through twoadjacent localised electronic states can be of great use for the spectroscopy ofdelicate electronic density of state features in collector and emitter layers.This includesaccurate spectroscopyof secondneutralitypoint valence-bandbandgap of moiré monolayer and electric field induced bandgap of Bernalbilayer graphene. Our observations not only deepen the understanding ofunderlying mechanisms governing the electron tunnelling in quantumsystems, but also suggest potential pathways for designing unique devicearchitectures for future electronic applications.MethodsDevice preparationTunnelling devices were fabricated using the dry transfer technique41applied to the micro-mechanically cleaved layers of monolayer graphene,Bernal bilayer graphene, and hBN of various thicknesses (tunnelling layer,two encapsulating top and back gate electrode layers) from bulk graphiteand hBN crystals. Those were assembled on top of each other into van derWaals heterostructures using PC (PolyBisphenol carbonate) stamps pre-pared on commercial PDMS (poly-dimethylsiloxane) films, and eventually,were deposited on top of 285 nm thick silicon dioxide/strongly p-dopedsilicon wafers. Twist angle of about 1° was introduced in-between the bot-tomencapsulating layer ofhBNandbottommonolayer grapheneduring thefabrication procedure for Device 1. Next, Cr/Au edge contacts42 were madeon the bottom monolayer and top Bernal bilayer graphene using electron-beam lithography followed by hBN etching, metal deposition and a lift-offprocess. hBN was etched in a reactive ion etching system using CHF3chemistry. Contacts were made in such a way to have a four-probe mea-surement geometry. The top encapsulating hBN layer was additionallycovered by a Cr/Au pad at the cross-sectional area ofmonolayer and Bernalbilayer grapheneof S � 20μm2 (S � 25μm2) forDevice1 (Device 2),whichserved as topgate electrode.Measurement techniqueTunnelling current-voltage curves were measured using a 2636B sourcemeter from Keithley Instruments. The differential tunnelling conductancemeasurements were performed using anAC-DCmixing techniquewith thehelp of symmetric voltagedivider,Keithley Instruments 2636B sourcemeter(forDC voltage), and StanfordResearch SR860 lock-in amplifier (for outputAC sine voltage at fixed frequency of f ¼ 7 Hz). The measurements wereperformed in a two-probe configuration considering that the contact andlateral resistances of electrode layers were several orders of magnitudesmaller than the tunnelling resistance at high biases, backgated and topgatevoltages. To obtain high-resolution data no additional device was intro-duced to the measurement circuits. The samples were held in a closed-loopcryostat at fixed temperature of T ¼ 2 K.Theoretical simulationsAn electrostatic model of the device under the study with an hBN tunnelbarrier ofwidth of d, the gap between the two graphene layers of d0 in Bernalbilayer graphene and with distances to the top dtg and back gates dbg can begiven by the following system of equations:eVb ¼ μGr � μBGr � edFbeVbg ¼ μGr þ edbgFgeV tg ¼ μGr � edtgFt � edFbΔBGr ¼ 4πe2d0ðnGr μGr� �� n0BGrðμBGr;ΔBGrÞÞ8>>>><>>>>:; ð1ÞwhereV tg,Vbg andVb are the backgate, topgate and bias voltages, Fg is theelectric field between the monolayer graphene and Si backgate electrode, Ftis the electric field between the Bernal bilayer graphene and Au topgateelectrode,Fb is the electricfieldwithin the tunnel barrier region,μGrðμBGrÞ isthe chemical potential of monolayer (Bernal bilayer) graphene with respectto its neutrality point (or the middle of the bilayer’s bandgap, in case it isopen),nGr μGr� � ¼ μ2Gr=π_2v20 is the carrier density inmonolayer graphene,nBGrðμBGr; ΔBGrÞ ¼ m=πRe½Sqrtð2μ2BGr � Δ2BGrÞ� is the carrier density inBernal bilayer graphene, and n0BGrðμBGr;ΔBGrÞ ¼ nBGrðμBGr;ΔBGrÞ=2 þΛnSðμBGr;ΔBGrÞ is the carrier density in one particular layer of Bernalbilayer, where nSðμBGr;ΔBGrÞ is the screening term, and Λ � 1 is thedimensionless parameter describing the effectiveness of the screening in thebilayer graphene. Note, that Λ ¼ 0 describes poor screening when thedensity on each layer is equal to nBGrðμBGr;ΔBGrÞ=2.Similar to17, an inelastic sequential tunnelling current (J) can be pre-sented by phenomenological model based on a modification of the Land-auer-Buttiker’s model using the following equations:JA�>B ¼ �ξRdELA E � EA� �ΓB E � EB� �× γbγABγbþγABf BGr Eð Þ 1� f Gr EA� �� �;JB�>A ¼ �ξRdELB E � EB� �ΓA E � EA� �× γtγABγtþγABf Gr Eð Þ 1� f BGr EB� �� �;ð2Þwhere γtðγbÞ and γAB are the electronic tunnelling rates from a localised stateA(B) into the top (bottom) electrode, and in-between A and B localisedstates, respectively, f Gr;BGr ¼1=ð1þ expððE � μGr;BGrÞ=kBTÞ are the corre-sponding Fermi functionswith a temperatureT andBoltzmann constant kB,Γ is a singly peaked function with full width half maximum (FWHM) of γ,andE0 ¼ E0i þ eFbzi (E0i corresponds to theunperturbedenergiesofA andB localised states, zi, corresponds to their positions in the tunnelling barrier).Here, the tunnelling is possible only in cases when μGr>EA>EB>μBGr andμGr<EA<EB<μBGr.We used the experimental resonant features presented inFig. 2 of themain text to get the theoretical curves that are obtained using theelectrostaticmodel closest to themby varying themodel parameters, i. e., theenergies of localised states at zero electric field and their locations within thehBN barrier). For each localised state, the zero-field energy, and the locationwithin the hBN barrier at which the best match with the experimentallyobservedmaps is achievedwere successively chosen ðE0A ¼ 83 meV, E0B ¼73 meV, zA ¼ d= 6, and zB ¼ 2d=3Þ.https://doi.org/10.1038/s41699-025-00528-6 Articlenpj 2D Materials and Applications |             (2025) 9:7 5www.nature.com/npj2dmaterialsData availabilityThedata is available fromcorresponding authors upona reasonable request.Received: 6 November 2024; Accepted: 16 January 2025;References1. Geim, A. K. et al. Van der Waals heterostructures. Nature 499,419–425 (2013).2. Novoselov, K. S. et al. 2D materials and van der Waalsheterostructures. Science 353, 6298 (2016).3. Liu Y. et al. Van der Waals heterostructures and devices. Nat. Rev.Mater. 1, 16042 (2016).4. Satterthwaite, P. F. et al. Van derWaals device integration beyond thelimits of van der Waals forces using adhesive matrix transfer. Nat.Electron. 7, 17–28 (2024).5. Britnell, L. et al. 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A.M. andD.A.G. acknowledgethe support from NAS of Republic of Armenia within the framework of the“Young Scientists” support program 24YSSPS-7. E.E.V., Yu.N.K. andS.V.M. acknowledge the support from the State task No. 075-00295-25-00.Author contributionsThe project was conceived and directed by D.A.G. and K.S.N. Themeasurements were carried out by E.E.V., Yu.N.K., S.V.M. and D.A.G. Thetheoretical model was devised by K.K. and D.A.S. High quality hBN crystalswere prepared by K. W. and T. T. The devices were prepared by C. Y. Thedata were analysed by E.E.V., Yu.N.K., S.V.M., K.K., A. M. and D.A.G. Themanuscript was prepared by E.E.V., K.K., D.A.G. and K.S.N. with the inputfrom all the other authors.https://doi.org/10.1038/s41699-025-00528-6 Articlenpj 2D Materials and Applications |             (2025) 9:7 6www.nature.com/npj2dmaterialsCompeting interestsThe authors declare no competing interests.Additional informationSupplementary information The online version contains supplementarymaterial available athttps://doi.org/10.1038/s41699-025-00528-6.Correspondence and requests for materials should be addressed toK. S. Novoselov or D. A. 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To viewa copyof this licence,visit http://creativecommons.org/licenses/by-nc-nd/4.0/.© The Author(s) 2025https://doi.org/10.1038/s41699-025-00528-6 Articlenpj 2D Materials and Applications |             (2025) 9:7 7https://doi.org/10.1038/s41699-025-00528-6http://www.nature.com/reprintshttp://creativecommons.org/licenses/by-nc-nd/4.0/www.nature.com/npj2dmaterials Inelastic resonant tunnelling through adjacent localised electronic states in van der Waals heterostructures Outline placeholder Inelastic resonant tunnelling through adjacent localised electronic states in hBN Electric field fine-tuning of resonant features arisen from inelastic tunnelling events through adjacent states Inelastic resonant tunnelling for the spectroscopy of delicate electronic density of state features Methods Device preparation Measurement technique Theoretical simulations Data availability References Acknowledgements Author contributions Competing interests Additional information