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Jeffrey A Cloninger, Raine Harris, Kristine L Haley, Randy M Sterbentz, [Takashi Taniguchi](https://orcid.org/0000-0002-1467-3105), [Kenji Watanabe](https://orcid.org/0000-0003-3701-8119), [Joshua O Island](https://orcid.org/0000-0002-6074-9414)

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[A back-to-back diode model applied to van der Waals Schottky diodes](https://mdr.nims.go.jp/datasets/4c3ddf49-effe-42d3-a1cf-b63ff7ed932c)

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A back-to-back diode model applied to van der Waals Schottky diodesJournal of Physics: CondensedMatter     PAPER • OPEN ACCESSA back-to-back diode model applied to van derWaals Schottky diodesTo cite this article: Jeffrey A Cloninger et al 2024 J. Phys.: Condens. Matter 36 455301 View the article online for updates and enhancements.You may also likeEfficiency-loss analysis of monolithicperovskite/silicon tandem solar cells byidentifying the patterns of a dual two-diodemodel’s current-voltage curvesYuheng Zeng, Zetao Ding, Zunke Liu et al.-A novel analytical method fordetermination of diode parameters fromthe dark forward I–V characteristics of asilicon solar cellAbhishek Kumar, S N Singh, Jyoti et al.-Electrical characteristics and trapsignatures for Schottky barrier diodes on4H-SiC, GaN-on-GaN, AlGaN/GaNepitaxial substratesShikha Kumari, Rashmi Singh, ShivamKumar et al.-This content was downloaded from IP address 144.213.253.16 on 15/08/2024 at 02:28https://doi.org/10.1088/1361-648X/ad69ef/article/10.1088/1674-4926/44/8/082702/article/10.1088/1674-4926/44/8/082702/article/10.1088/1674-4926/44/8/082702/article/10.1088/1674-4926/44/8/082702/article/10.1088/1402-4896/ace55d/article/10.1088/1402-4896/ace55d/article/10.1088/1402-4896/ace55d/article/10.1088/1402-4896/ace55d/article/10.1088/1361-6641/ad4a65/article/10.1088/1361-6641/ad4a65/article/10.1088/1361-6641/ad4a65/article/10.1088/1361-6641/ad4a65Journal of Physics: Condensed MatterJ. Phys.: Condens. Matter 36 (2024) 455301 (5pp) https://doi.org/10.1088/1361-648X/ad69efA back-to-back diode model applied tovan der Waals Schottky diodesJeffrey A Cloninger1, Raine Harris1, Kristine L Haley1, Randy M Sterbentz1,Takashi Taniguchi2, Kenji Watanabe3 and Joshua O Island1,∗1 Department of Physics and Astronomy, University of Nevada Las Vegas, Las Vegas, NV 89154, UnitedStates of America2 Research Center for Materials Nanoarchitectonics, National Institute for Materials Science, 1-1 Namiki,Tsukuba 305-0044, Japan3 Research Center for Electronic and Optical Materials, National Institute for Materials Science, 1-1Namiki, Tsukuba 305-0044, JapanE-mail: joshua.island@unlv.eduReceived 11 June 2024, revised 16 July 2024Accepted for publication 31 July 2024Published 9 August 2024AbstractThe use of metal and semimetal van der Waals contacts for 2D semiconducting devices has ledto remarkable device optimizations. In comparison with conventional thin-film metaldeposition, a reduction in Fermi level pinning at the contact interface for van der Waals contactsresults in, generally, lower contact resistances and higher mobilities. Van der Waals contactsalso lead to Schottky barriers that follow the Schottky–Mott rule, allowing barrier estimates onmaterial properties alone. In this study, we present a double Schottky barrier model and apply itto a barrier tunable all van der Waals transistor. In a molybdenum disulfide (MoS2) transistorwith graphene and few-layer graphene contacts, we find that the model can be applied to extractSchottky barrier heights that agree with the Schottky–Mott rule from simple two-terminalcurrent–voltage measurements at room temperature. Furthermore, we show tunability of theSchottky barrier in-situ using a regional contact gate. Our results highlight the utility of a basicback-to-back diode model in extracting device characteristics in all van der Waals transistors.Keywords: diodes, back-to-back, model, applied, MoS2, van der Waals1. IntroductionIn metal-semiconductor-metal devices, diode models havebeen used to calculate and extract device characteristics, suchas the Schottky barriers and diode ideality factors, from simpleelectrical measurements [1–4]. Due to strong Fermi level pin-ning (FLP) though, these characteristics typically do not agreewith the Schottky–Mott theory, and it is challenging to sep-arate model inadequacies and intrinsic device characteristics.∗Author to whom any correspondence should be addressed.Original Content from this work may be used under theterms of the Creative Commons Attribution 4.0 licence. Anyfurther distribution of this work must maintain attribution to the author(s) andthe title of the work, journal citation and DOI.For instance, no change in the extracted Schottky barriers froma back-to-backmodel for tin oxide nanobelt junctions has beenreported, even for different contact metals [4].FLP at the interface between a metal and a semicon-ductor can be detrimental to device performance, ultimatelyincreasing contact resistance and reducing device mobility. Intwo-dimensional (2D) material devices with evaporated metalcontacts, FLP has been identified as a significant contribu-tion to inhibiting device performance [5, 6]. They also leadto Schottky barriers that deviate greatly from the Schottky–Mott rule and typically, a pinning factor is incorporated inSchottky–Mott theory to account for FLP [6, 7]. FLP inthese devices results from both intrinsic and extrinsic effects,complicating estimates of Schottky barriers based on con-tact metals alone. Intrinsic effects leading to FLP includethe formation of metal-induced gap states (MIGS) from1 © 2024 The Author(s). Published by IOP Publishing Ltdhttps://doi.org/10.1088/1361-648X/ad69efhttps://orcid.org/0000-0003-3701-8119https://orcid.org/0000-0002-6074-9414mailto:joshua.island@unlv.eduhttp://crossmark.crossref.org/dialog/?doi=10.1088/1361-648X/ad69ef&domain=pdf&date_stamp=2024-8-9https://creativecommons.org/licenses/by/4.0/J. Phys.: Condens. Matter 36 (2024) 455301 J A Cloninger et almetal d-orbital and chalcogen p-orbital overlap [8], metalizedinterfaces for some contact metals [9, 10], and interfacialcharge redistribution [11]. Extrinsic effects further promotingFLP in these devices include defect induced gap states (DIGS)from intrinsic vacancies and surface vacancies introduced dur-ing metal deposition [12].Several strategies have been implemented to reduce FLPin 2D material devices [13]. Some successful achievementshave been using metallic 2D materials as contacts [14–16],inserting a buffer layer between the metal and the channelmaterial [17], phase engineering of the 2D material itself [18,19], mechanical transfer of metals on top of 2D materials[20], and edge contacts [21]. Generally, these methods act tostrongly reduce MIGS and further contributions from DIGS tode-pin the Fermi level at the interface. Using 2D materials asmetal contacts is particularly attractive because, in the case ofgraphene, this allows tunability of the Schottky barrier throughelectrostatic modulation of the Fermi energy in the contactmetal. The graphene barristor showed that the Schottky barriercan be directly tuned at the interface between graphene andsilicon [22]. Using van der Waals metals as a contact is alsotheoretically expected to reduce MIGS and thereby lower FLP[14]. This can then lead to devices that approach the Schottky–Mott limit and Schottky barriers that can be estimated based onmaterial characteristics alone. For example, accurate Schottkybarriers can be extracted for van der Waals materials con-nected to tungsten disulfide (WS2) [23], tungsten diselenide(WSe2) [24] and indium selenide (InSe) [25]. This allows foroptimized devices for next generation computing, sensing, anddigital-analog chips [26, 27].A diode model that shows consistency with Schottky–Mott theory for van der Waals contacted 2D semiconduct-ors is desirable to efficiently extract device characteristicsfrom simple electrical measurements without the need ofmodel additions such as pinning factors. Here, we show thata straightforward back-to-back diode model can be appliedto metal van der Waals contacted MoS2 junctions. Themodel extracts barriers that are in general agreement with theSchottky–Mott rule. In few-layer graphene contacted MoS2devices, we extract barriers of 0.46 eV−0.58 eV, varyingaccording to the gating conditions. For a graphene contact weextract barriers of 0.37 eV−0.74 eV for different contact gatevoltages. Our results here show that the simple model is cap-able of capturing reasonable barrier values for both grapheneand few-layer graphene contacts to few-layer MoS2 flakes.2. ResultsA back-to-back diode model is implemented to extractdevice characteristics from simple two-terminal electricalmeasurements [1–4]. Our open source python script of themodel is available for others to use [28]. The current througheach contact of the device can be written from thermionic the-ory as:I1 = Is1(eqV1kT − 1), I2 =−Is2(e−qV2kT − 1), (1)whereIs1,s2 = S1,2A∗T2(e−ϕB01,B02kT), (2)are the reverse saturation currents, V1,2 are the voltage dropsacross each contact, k is the Boltzmann constant, T is the tem-perature, S1,2 are the contact areas, A∗ is the Richardson con-stant, and ϕB01,B02 are the Schottky barriers. For simplicity, wemake the assumption that the voltage drops are equal acrosseach contact, V1 = V2 = V/2. The total current through thedevice must be equal to the currents through each junction(Itot = I1 = I2) and the total voltage drop is equal to the sum ofthe voltage drops across each junction (V= V1 +V2). Througha little algebra, the total current can be expressed as:Itot =2Is1Is2sinh( qV2kT)(Is1eqV2kT + Is2e−qV2kT) . (3)Figures 1(a) and (c) shows the calculated current for two idealcases (n1,n2 = 1) with equal and unequal junction barriers.The exponential increase in current around zero bias is present.The current saturates at higher biases set by the reverse sat-uration current of the reversed-biased junction. For non-idealcases in nanoscale junctions, the intrinsic Schottky barriers(ϕB01,B02) can be replaced by effected barriers that include thevoltage dependence (image charge lowering) and an idealityconstant, n, to account for defects and interface oxides. Theeffective barriers can be written as:ϕB1,B2 = ϕB01,B02 ± eV1,2(1− 1n1,2). (4)Two examples of non-ideal diode curves are shown infigures 1(b) and (d) for equal and unequal barrier strengths.A more gradual nonlinearity is observed that is reminiscent ofexperimental current–voltage curves for nanoscale devices.This model is used to extract ideality constants (n1,n2) andSchottky barriers from simple two terminal current–voltagemeasurements in an all van der Waals transistor composedof MoS2 with graphene and few-layer graphene contacts. Wenote that the model could also be applied to metal contacteddevices but the extracted Schottky barriers typically will notagree with the Schottky–Mott rule due to strong Fermi levelpinning. Figure 2(a) shows an optical image of the device. Tofabricate the device, prepatterned gold electrodes, accompan-ied by a chromium layer for adhesion, are deposited onto aheavily doped silicon wafer with 285 nm of SiO2. The dopedsubstrate is used as a global back gate to gate the MoS2 chan-nel. A van der Waals heterostructure is stacked and transferredon top of electrodes using dry stacking and heated transfertechniques [29, 30]. The heterostructure is composed of, instack order, an encapsulating top boron nitride (BN), two few-layer graphene flakes, an MoS2 flake, a graphene flake, and abottom BN flake used as a gate dielectric for the contact gate.On the left side (figure 2(a)), theMoS2 flake (labeled 2) is con-nected to a gold electrode by a graphene flake (labeled 1, con-tact area 15×10−8 cm−2). At the junction of the MoS2 flakeand the graphene, there is a gold finger gate. On the right side,2J. Phys.: Condens. Matter 36 (2024) 455301 J A Cloninger et alFigure 1. Calculated current–voltage curves for a back-to-backdiode model. (a) An ideal case in which the ideality constants are 1and the Schottky barriers are equal. (b) A non-ideal case in whichthe ideality constants are 1.3 and the barriers are equal. (c) An idealcase in which the ideality constants are 1 and the barriers are notequal. (d) A non-ideal case in which the ideality constants are 1.3and the barriers are unequal.the same MoS2 flake is connected to two few-layer grapheneflakes (labeled 3, contact area 100×10−8 cm−2 and 4, con-tact area 15×10−8 cm−2). From contrast analysis [31], the fewlayer graphene electrodes are both determined to be 8 layersor 2.7 nm thick. For all model calculations, we have used aRichardson constant of 80.3 A cm−2 K−2 for MoS2 [32].We first investigate the right side of the device, using thetwo few-layer graphene contacts as source and drain elec-trodes and the highly doped silicon substrate as a back gate.Figure 2(b) shows a 2D color plot of the drain current as a func-tion of bias and gate voltage. The typical transistor responseis observed, with the left side of the plot indicating the OFFstate and the right side of the plot the ON state. The 1D gatesweeps for increasing bias voltage are shown in figure 2(c).Figure 2(d) shows I−V curves in the OFF (black) and ON(red) states. Using the two diode model, we extract idealityconstants close to 1 (n1 = 1.09, n2 = 1.17 in the OFF state andn1 = 1.12, n2 = 1.14 in the ON state) and Schottky barriers ofϕ1 = 0.58 eV, ϕ2 = 0.58 eV in the OFF state and ϕ1 = 0.49eV, ϕ2 = 0.46 eV in the ON state. The most significant dif-ference between the model and data is a low bias nonlinear-ity, highlighted by the black arrow in figure 2(d). This can beattributed to the absence of the series resistance of the MoS2flake itself from the model [2]. This series resistance leads tosoftening of the low bias nonlinearity in devices with similarbarriers for source and drain.The observed barrier lowering for increasingly positivegate voltage for both the source and drain due to the imagecharge effect is consistent with previous results showingFigure 2. An MoS2 transistor with graphene and few-layergraphene contacts. (a) An optical image of the device. The dottedlines outline the various flakes in the van der Waals heterostructure.1 is the graphene flake on the left side of the image, 2 is the MoS2flake, and 3 and 4 are both few-layer graphene (8 layers, 2.7 nmthick) flakes on the right side. (b) Color plot of the current throughthe few-layer graphene contacts (3 source and 4 drain) as a functionof bias voltage (V) and gate voltage (Vg). (c) I−Vg curves forincreasing positive bias voltage. (d) I−V curves for gate voltagesof −10 V (black) and 10 V (red). The solid lines indicate fits to theI−V curves using the two diode model in the main text.similar behavior [33–35]. As a result of van derWaals contact,there is minimal Fermi level pinning and the Schottky–Mottrelationship, ϕ1,2 = ϕm −χS where ϕm is the work functionof the metal and χS is the electron affinity of the semicon-ductor, can be directly applied. Given that the electron affin-ity of MoS2 is 4.0 eV [36] and the work function of few-layer graphene (graphite) is 4.5 eV [37], we would expect aSchottky barrier of 0.5 eV. This is consistent with our resultsand highlights the ease of extracting barrier information fromsimple two-terminal measurements. From the Schottky–Mottrule, any change in the work function of the contact mater-ial, including thickness dependent changes, would reflect in achange in the Schottky barrier height.Using the contact gate, the Schottky barrier for thegraphene to MoS2 contact can be tuned independently.Figure 3(a) shows the current–voltage sweeps at Vg = 10 Vfor different contact gate voltages. For negative contact gatevoltages, the I−V curves show a clear diode-like response,with the forward current completely suppressed. For positivecontact gate voltages, this suppression is lifted and finite for-ward currents are observed. By employing the double diodemodel once again, we can extract the Schottky barriers andideality factors for different contact gate voltages. Figure 3(b)plots the curve in figure 3(a) for contact gate of −0.5 V(red circles). The diode model fit (black) produces Schottky3J. Phys.: Condens. Matter 36 (2024) 455301 J A Cloninger et alFigure 3. Tunable Schottky barriers in graphene contacted MoS2.(a) Current as a function of bias voltage for different contact gatevoltages. (b) Current as a function of bias voltage for a contactvoltage of −0.5 V (red circles). The black curve shows the fit fromthe diode model. The inset shows a calculation of the Fermi energyin graphene as a function of contact gate voltage. (c) ExtractedSchottky barriers for the graphene (ϕ1) and few-layer graphenecontact (ϕ2) for the data shown in panel (a). The error bars showone-sigma uncertainty in the fitting parameters. (d) Extractedideality factors for the graphene (n1) and few-layer graphene (n2)contact. The error bars show one-sigma uncertainty in the fittingparameters. All measurements made with source contact 1(graphene) and drain contact 3 (few-layer graphene)..barriers of ϕ1 = 0.44 eV (few-layer graphene) and ϕ2 = 0.73eV (graphene), and ideality factors of n1 = 1.19 and n2 = 2.42(respectively). The negative contact voltage pulls the Fermienergy of graphene into the valence band and increases theSchottky barrier for the corresponding contact. The contactgate completely shields the graphene-MoS2 contact from theglobal back-gate.The Schottky barriers and ideality factors from fits for allthe data curves in figure 3 are shown in panels (c) and (d).Error bars in these figures indicate the standard deviations foreach parameter from the fit. While the Schottky barrier for thefew-layer graphene contact stays relatively constant over thechange in contact gate voltage, the barrier at the graphene con-tact changes considerably, spanning more than 300 meV inenergy. This behavior is also reflected in the ideality constants.3. DiscussionPart of the barrier modulation can be understood from the elec-tric field-induced change to the work function of graphene.The work function of graphene is marginally lower thangraphite. Both experimental and theoretical studies reportthe work function to be approximately 4.3 eV [38–40].This would correspond to Schottky barrier of 0.3 eV fromthe Schottky–Mott rule. Self-consistent density functionalcalculations of an explicit graphene-MoS2 heterostructureestimate the Schottky barrier to be 0.37 eV [41]. This roughlyaligns with our extracted barrier at zero contact gate voltageof 0.48 eV although residual doping of the graphene layer incontact with MoS2 may be responsible for the discrepancy.The gate-induced change in the work function of graphenehas been reported using surface Kelvin probemicroscopy [42].The Fermi energy in single-layer graphene can be calculatedfrom:EF = sign(∆Vg) h̄vF (απ|∆Vg|)1/2 (5)where α is the capacitance and vF is the Fermi velocity. A cal-culation of the change in the Fermi energy with gate voltagefor this device is shown in the inset of figure 3(b). For this cal-culation, the dielectric constant of BN is taken to be 3.76 [43].The BN thickness for the graphene contact insulating layer is6 nm, estimated by optical contrast analysis [44]. Over the 1-volt range of the contact gate, an isolated graphene layer typ-ically exhibits a modulation of EF ≈ 300 meV. In the case ofa graphene in contact with MoS2, there is an enhanced tun-ability of the barrier due to the influence of the gate on bothmaterials. Calculations for this heterostructure show a signi-ficant change of 0.75 eV−0.03 eV when subjected to slightlylarger electric field strengths (0.01 V/Ang vs. 0.008 V/Ang)[41]. This agrees with the magnitude of the Schottky barriersfor negative contact gate in our measurements.The lack of modulation of the Schottky barrier beyond 100mV contact gate could be due to pinning from impurity levelsin the MoS2 itself [45]. While the influence of MIGS has beenlargely mitigated using 2D material contacts, there still maybe some influence from DIGS. Rhenium impurities are knownto be present in MoS2 samples and lead to impurity states nearthe conduction band (0.29 eV below the conduction band)[45].A similar trend in the change in Schottky barrier heights hasbeen reported in graphene and few-layer graphene contactedMoS2 devices [46, 47]. The barrier height modulation flattensfor increasingly positive gate voltages.4. ConclusionsWe have shown that a straightforward back-to-back diodemodel can be used to extract Schottky barrier heights andideality factors in MoS2 transistors with graphene and few-layer graphene contacts. The lack of strong FLP leads to bar-rier heights that reasonably agree with the Schottky–Mottrule with consideration of defect doping. In an MoS2 tran-sistor with two few-layer graphene contacts, we extract barrierheights of 0.46 eV−0.58 eV depending on gating conditions.In an MoS2 transistor with a graphene contact on one side,we extract a barrier height of 0.48 eV. Using a contact gateon the graphene contacted side of the device, we showed howthe barrier heights could be directly tuned. Our results sup-port the use of a simple back-to-back diodemodel in extractingcharacteristics in van derWaals contacted 2D semiconductors.4J. Phys.: Condens. Matter 36 (2024) 455301 J A Cloninger et alData availability statementThe data that support the findings of this study are openlyavailable at the following URL/DOI: https://github.com/islandlab-unlv/Back-to-back-diode-model.AcknowledgmentsThis work was supported by the National Science Foundationunder Grant No. (2047509) and by, or in part by, the U.S.Army Research Laboratory and the U.S. Army ResearchOffice under Contract/Grant Number (W911NF2310160). KW and T T acknowledge support from the JSPS KAKENHI(Grant Numbers 21H05233 and 23H02052) and WorldPremier International Research Center Initiative (WPI),MEXT, Japan.ORCID iDsKenji Watanabe https://orcid.org/0000-0003-3701-8119Joshua O Island https://orcid.org/0000-0002-6074-9414References[1] Wang Z et al 2020 Phys. Status Solidi a 217 1901018[2] Osvald J 2015 Phys. Status Solidi a 212 2754–8[3] Grillo A and Di Bartolomeo A 2021 Adv. Electron. 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Introduction 2. Results 3. Discussion 4. Conclusions References