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Franklin Liou, Hsin‐Zon Tsai, Zachary A. H. Goodwin, Andrew S. Aikawa, Ethan Ha, Michael Hu, Yiming Yang, [Kenji Watanabe](https://orcid.org/0000-0003-3701-8119), [Takashi Taniguchi](https://orcid.org/0000-0002-1467-3105), Alex Zettl, Johannes Lischner, Michael F. Crommie

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Imaging Field‐Driven Melting of a Molecular Solid at the Atomic ScaleRESEARCH ARTICLEwww.advmat.deImaging Field-Driven Melting of a Molecular Solid at theAtomic ScaleFranklin Liou, Hsin-Zon Tsai, Zachary A. H. Goodwin, Andrew S. Aikawa, Ethan Ha,Michael Hu, Yiming Yang, Kenji Watanabe, Takashi Taniguchi, Alex Zettl,*Johannes Lischner,* and Michael F. Crommie*Solid–liquid phase transitions are basic physical processes,but atomically resolved microscopy has yet to capture their full dynamics.A new technique is developed for controlling the melting and freezingof self-assembled molecular structures on a graphene field-effect transistor(FET) that allows phase-transition behavior to be imaged using atomicallyresolved scanning tunneling microscopy. This is achieved by applying electricfields to 2,3,5,6-tetrafluoro-7,7,8,8-tetracyanoquinodimethane-decorated FETsto induce reversible transitions between molecular solid and liquid phasesat the FET surface. Nonequilibrium melting dynamics are visualized by rapidlyheating the graphene substrate with an electrical current and imaging theresulting evolution toward new 2D equilibrium states. An analytical model isdeveloped that explains observed mixed-state phases based on spectroscopicmeasurement of solid and liquid molecular energy levels. The observednonequilibrium melting dynamics are consistent with Monte Carlo simulations.1. IntroductionPhase transitions reflect the collective behavior of large num-bers of particles but originate from rapid reconfigurations atF. Liou, H.-Z. Tsai, A. S. Aikawa, E. Ha, M. Hu, Y. Yang, A. Zettl,M. F. CrommieDepartment of PhysicsUniversity of California at BerkeleyBerkeley, CA 94720, USAE-mail: azettl@berkeley.edu; crommie@berkeley.eduF. Liou, H.-Z. Tsai, A. S. Aikawa, A. Zettl, M. F. CrommieMaterials Sciences DivisionLawrence Berkeley National LaboratoryBerkeley, CA 94720, USAF. Liou, A. Zettl, M. F. CrommieKavli Energy NanoSciences Institute at the University ofCalifornia at BerkeleyBerkeley, CA 94720, USAZ. A. H. Goodwin, J. LischnerDepartment of MaterialsImperial College LondonPrince Consort Rd, London SW7 2BB, UKE-mail: j.lischner@imperial.ac.ukThe ORCID identification number(s) for the author(s) of this articlecan be found under https://doi.org/10.1002/adma.202300542DOI: 10.1002/adma.202300542the single-particle scale. Great progress hasbeen made at controlling structural, elec-tronic, and magnetic phase transitions indifferent materials by varying macroscopicparameters such as strain,[1–3] density,[4,5]and electromagnetic fields,[6–9] but imagingthe atomic-scale dynamics of such transi-tions has proved difficult. The reason forthis is the difficulty of combining atomic-scale microscopy with the high bandwidthrequired to capture fast dynamics. Herewe describe a different approach for ex-ploring dynamical processes that involverapidly quenching the thermally-inducedkinetics of 2D phase transitions and imag-ing their evolution toward equilibriumframe-by-frame using atomically resolvedscanning tunneling microscopy (STM).We have used this technique to image theelectrostatically driven solid–liquid phasetransition of 2,3,5,6-tetrafluoro-7,7,8,8-tetracyanoquinodimethane (F4TCNQ) molecules at the surfaceof a graphene field-effect transistor (FET) held at cryogenicZ. A. H. GoodwinNational Graphene InstituteUniversity of ManchesterBooth St. E. Manchester M13 9PL, Manchester UKZ. A. H. GoodwinSchool of Physics and AstronomyUniversity of ManchesterOxford Road, Manchester M13 9PL, UKK. WatanabeResearch Center for Electronic and Optical MaterialsNational Institute for Materials Science1-1 Namiki, Tsukuba 305-0044, JapanT. TaniguchiResearch Center for Materials NanoarchitectonicsNational Institute for Materials Science1-1 Namiki, Tsukuba 305-0044, JapanAdv. Mater. 2023, 35, 2300542 © 2023 Wiley-VCH GmbH2300542 (1 of 8)http://crossmark.crossref.org/dialog/?doi=10.1002%2Fadma.202300542&domain=pdf&date_stamp=2023-07-13www.advancedsciencenews.com www.advmat.detemperature in ultrahigh vacuum. Voltages applied to the backgate of such a device induce reversible freezing and melting ofmolecular structures at the surface, but the evolution of thesestructures toward new equilibrium states is quenched by thedevice’s low temperature (T ≈ 4 K). Equilibration can only beachieved by passing current through the graphene substrate,thereby transiently raising the temperature and speeding themolecular kinetics (i.e., the diffusive motion). Rapid cooling afterthe current is stopped allows “snapshots” of the surface molecu-lar configuration to be taken without the need for ultrafast imag-ing techniques. Stop-motion movies of field-induced molecularphase transitions can be made this way that have atomic spatialresolution and a time resolution set by the electronic and thermalrelaxation time constants of the device.[10] This has allowed us toresolve the melting and freezing processes of F4TCNQ moleculesat the single-molecule level in both liquid and solid phases, some-thing not possible via other microscopy techniques[11–13] due tothe non-crystallinity and fast dynamics of molecular liquids.[11,12]Scanning tunneling spectroscopy (STS) measurements revealthat a solid phase of F4TCNQ is favored when the graphene Fermilevel (EF) is lowered to a point where the molecules becomecharge-neutral. Raising EF sufficiently with a back-gate causesthe solid F4TCNQ phase to melt into a liquid phase in a processwhere each molecule converted to the liquid phase accepts a sin-gle electron. The F4TCNQ liquid phase is thus populated by neg-ative ions.[14] We have developed a simple theoretical frameworkthat explains the equilibrium energetics of this first-order solid–liquid phase transition as a function of gate voltage, and have per-formed Monte Carlo simulations that capture its nonequilibriummelting dynamics.2. ResultsFigure 1 shows the reversible melting/freezing of a partial mono-layer of F4TCNQ on a graphene FET as it transitions throughdifferent equilibrium states in response to the applied back-gate voltage VG. For VG = −30 V (Figure 1a) the moleculesall lie in a solid “chain” phase after flowing source–drain cur-rent ISD = 1 mA through the device for 180 s (all STM images andspectroscopy are acquired only after setting ISD to zero to quenchmolecular motion by reestablishing the device base temperatureof T = 4.5 K). Figure 1j shows close-up STM and AFM images ofthe F4TCNQ solid phase revealing two quasi-1D chain morpholo-gies that we call “linear” and “zigzag” (more detailed structuralcharacterization can be found in Figures S7 and S8, SupportingInformation). The structure factor of the chain phase shows clearperiodicity, thus indicating that it is a quasi-1D crystalline solid(see additional details in Section 8, Figure S7, Supporting Infor-mation).Subsequent raising of the gate voltage to VG = 0 V followed byapplication of “diffusive” conditions (i.e., by setting ISD = 1 mA)for 180 s causes the molecular solid to partially melt. This canbe seen in Figure 1b which shows isolated F4TCNQ moleculesdotting the surface near the edge of the solid phase in the samearea as Figure 1a (the isolated molecules belong to a 2D liquidphase as described below). This is an equilibrium configurationin the sense that the average concentrations of the liquid andsolid phases have stopped changing with time under diffusiveconditions. The images in Figure 1c,d show the equilibrium con-figurations of the same region after incrementally raising the gatevoltage first to VG = 6 V and then to VG = 30 V under diffu-sive conditions. For every step increase in VG, the solid is seento melt a little more until it is completely liquefied at VG = 30 V.Figure 1e–h shows the same surface region as VG is decreasedback to −30 V under identical diffusive conditions. The liquid–solid phase transition is completely reversible (movies of thefreezing/melting processes are shown in Movies S1 and S2, Sup-porting Information).Justification for calling the phase containing isolatedmolecules (Figure 1d) a liquid comes from an analysis ofthe molecular radial distribution function, g(r), and structurefactor, S(q). Figure 1k shows g(r) extracted from a large-areaimage containing isolated molecules prepared under equilib-rium conditions (see Section 1, Supporting Information foradditional details). g(r) shows evenly spaced peaks with a spacingof a = 3.84 nm, as expected for the shell structure of an isotropicliquid.[15] The structure factor seen in the Figure 1k inset (for thesame STM image) is also indicative of an isotropic liquid andshows no evidence of crystal or gas behavior.[15]Understanding the cause of the observed molecular phasetransition requires understanding how charge transfers betweenmolecules and graphene under different gating conditions. STSmeasurements were used to gain insight into this process by sep-arately measuring the local electronic structure of the solid andliquid phases. Figure 2a shows dI/dV spectra measured on anF4TCNQ chain (solid phase) compared to an isolated F4TCNQmolecule (liquid phase) for VG = −60 V (this is the hole-dopedgraphene regime as shown by the inset electronic structure dia-gram in Figure 2a). The bare graphene spectrum for this surface(taken 10 nm away from any molecules) is shown in the inset forreference. A dip in the bare graphene local density of states nearV = 0.34 V marks the location of the graphene Dirac point (ED),thus verifying that the graphene is in the hole-doped regime forthis gate voltage. The gap-like feature at V = 0 (EF) arises from awell-known phonon-assisted inelastic tunneling effect.[16]The blue curve in Figure 2a shows the dI/dV spectrum fora single, isolated F4TCNQ molecule (SM) in this hole-dopedregime. The leading edge of the first peak marks the lowest un-occupied molecular orbital (LUMO) energy as discussed in pre-vious work[14,17] (ESML = 0.2 eV and is marked by a dashed blueline), while the second peak (Vb ≈ 0.4 V) is a phonon satellitearising from intramolecular vibrations.[17] The F4TCNQ LUMOlevel is unoccupied for this value of VG. The second curve (red)shows the dI/dV spectrum measured with the STM tip held overthe end molecule of an F4TCNQ solid chain (the chain end (CE)as shown in Figure 1j). The CE spectrum is nearly identical to thesingle molecule spectrum except that EL is shifted up by 0.06 eV.The third curve (orange) shows the spectrum for a molecule inthe middle of a chain (CM) (as shown in Figure 1j). Here ELis pushed up even further by an additional 0.05 eV. The overallenergy-level structure is schematically represented by the insetsketch which shows the energy level alignment of the SM LUMO,the CE LUMO, and the CM LUMO relative to ED and EF (the ex-perimental energy levels of the zigzag and linear chains are iden-tical, as shown in Figure S9, Supporting Information).This energy-level structure has important consequences forF4TCNQ/graphene solid–liquid phase transitions. For example,suppose that VG was first set to VG = −60 V (the case shownAdv. Mater. 2023, 35, 2300542 © 2023 Wiley-VCH GmbH2300542 (2 of 8) 15214095, 2023, 39, Downloaded from https://onlinelibrary.wiley.com/doi/10.1002/adma.202300542 by Cochrane Japan, Wiley Online Library on [21/10/2023]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons Licensewww.advancedsciencenews.com www.advmat.deFigure 1. Gate-tunable solid–liquid molecular phase transition. a–d) STM images show the melting of self-assembled chains of F4TCNQ molecules(solid phase) into isolated molecules (liquid phase) as VG is increased through −30 V (a), 0 V (b), 6 V (c), 30 V (d). e–h) The reverse phase transition(liquid to solid) is observed at the same spot on the surface with molecules coalescing from the liquid phase into self-assembled chains as VG isdecreased through 6 V (e), 0 V (f), −6 V (g), −30 V (h). i) Schematic of the experimental setup shows F4TCNQ molecules adsorbed onto the surfaceof a graphene FET device. j) Closeup STM images of the solid molecular chain phase (with structural overlays) show two observed geometries (linearand zig-zag), both having a center-to-center molecular distance of 8.5 Å. A bond-resolved nc-AFM image in the bottom row (obtained with a CO tip[21])reveals the linear geometry in greater detail. k) The radial distribution function g(r) of molecular positions in the liquid phase shows shell-like structurehaving an average shell spacing of 3.84 nm (the corresponding STM image can be seen in Figure S1b, Supporting Information). The correspondingstructure factor S(q) shown in the inset indicates that the liquid is isotropic. STM images were obtained at T = 4.5 K.in Figure 2a) and then slowly increased under diffusive condi-tions. This would cause EF to slide to the right and eventuallyintersect ESML . The first molecules to fill with charge due to theincreasing VG would thus be isolated F4TCNQ molecules. Asshown previously,[14] under these conditions EF becomes pinnedclose to ESML and so never reaches the chain orbitals (ECEL or ECML )which therefore remain charge-neutral (i.e., unoccupied) for awide range of VG values. Increasing VG while EF is pinned inthis way causes molecules to melt from the neutral solid and tofill with charge, thereby increasing the molecular density of thecharged liquid phase (separation between the isolated moleculesis explained by Coulomb repulsion).A useful thermodynamic variable to characterize this processis the total charge density in the molecule-decorated graphenesystem, −ΔQ (this counts the excess density of electrons). Whenthe molecular chains begin to melt in response to increasedVG, −ΔQ exhibits a discontinuous jump when plotted as afunction of EF as shown in Figure 2b. Here −ΔQ is obtainedfrom the relationship −ΔQ = CVG where C is the capacitanceper area between the graphene and the gate electrode. EF andED are measured as a function of VG from STM spectroscopy(by fitting dI/dV spectra such as that shown in the inset toFigure 2a), and the discontinuity in −ΔQ is observed to occurat EF − ED ≈ −0.125 eV. For EF − ED < −0.125 eV the moleculesAdv. Mater. 2023, 35, 2300542 © 2023 Wiley-VCH GmbH2300542 (3 of 8) 15214095, 2023, 39, Downloaded from https://onlinelibrary.wiley.com/doi/10.1002/adma.202300542 by Cochrane Japan, Wiley Online Library on [21/10/2023]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons Licensewww.advancedsciencenews.com www.advmat.deFigure 2. Electronic energy level alignment and charge accumulation under electrostatic gating. a) dI/dV spectra taken at VG = −60 V for F4TCNQmolecules at a chain middle (CM), a chain end (CE), and for single, isolated molecules (SM) (images shown in Figure 1). The inset plot shows the dI/dVspectrum measured on bare graphene for VG =−60 V. The inset sketch shows the relative energy alignments of the CM LUMO state, the CE LUMO state,the SM LUMO state, the Fermi energy (EF), and the Dirac point (ED). b) Total charge density accumulated in the molecule/graphene surface (measuredcapacitively and plotted in terms of electron density) as a function of EF − ED (as determined by STS). A discontinuity is seen at EF − ED = −0.125 eV.STS spectra obtained at T = 4.5 K.are in the charge-neutral chain phase where increases in −ΔQreflect the filling of the graphene Dirac band (thus leading to awell-known parabolic dependence of −ΔQ on EF in graphene).[18]When EF reaches the critical value of EF − ED = −0.125 eV, how-ever, charge begins to flow into the F4TCNQ LUMO states as thechain phase melts to accommodate added charge. The moleculeshave a high quantum capacitance at this energy and so devicecharge accumulates rapidly with increasing EF, thus resulting indiscontinuous behavior as shown in Figure 2b. While the discon-tinuity in −ΔQ reflects the electronic part of the phase transition,concurrent imaging of molecular chain dissociation (i.e., molec-ular melting) shows that this electronic change accompaniesthe structural phase transition. The EF dependence of −ΔQ inFigure 2b is reminiscent of the temperature dependence of trans-ferred heat in a standard temperature-driven solid–liquid meltingtransition (such as ice to water), where latent heat must be pro-vided to increase entropy as the solid converts to a liquid. HereEF is analogous to temperature and the number of excess elec-trons (−ΔQ) is analogous to entropy, so one can think of “latentcharge” as being necessary to induce 2D molecular melting inour devices (see Section 10, Supporting Information for more de-tailed discussions of this analogy). Similar thermodynamic mod-els of electrostatically driven phase transitions have been utilizedto explain solid–solid phase transitions in 2D materials.[19]These insights enable us to develop a theoretical model forquantitatively understanding the microscopic energetics of theF4TCNQ/graphene solid–liquid phase transition. We first notethat the F4TCNQ molecules and graphene both exchange elec-trons with the gate which acts as a reservoir. The thermodynam-ics of such an open system for electrons is described by the grandpotential (see Section 10, Supporting Information for a more de-tailed discussion). Under our low-temperature experimental con-ditions (which rise to ≈25 K when ISD ≠ 0) the entropy contribu-tion TS to the grand potential is expected to be small, and so wemodel the grand potential as follows:Φ = U − EFNe (1)Here U is the total energy of the graphene plus molecules and Neis the total number of electrons in the graphene/molecule systemrelative to a reference state. The reference here is the configura-tion where all electrons occupy graphene band states with energyE < ELSM and the molecules are uncharged. Since the LUMO en-ergy of the chains is higher than that of isolated molecules, weignore the possibility of the chains becoming charged and as-sume that electrons occupy either single-molecule LUMO statesor graphene Dirac band states. The graphene contribution to thetotal energy relative to the reference state is denoted by Ug(EF) =∫ EFEL𝜀g(𝜀) d𝜀, where g(𝜀) is found from the well-known linear bandmodel[20] to be g(𝜀) = 2A(ED − 𝜀)∕𝜋ℏ2v2F (here ED is the Diracpoint energy, A is the area of graphene, and vF is the Fermi ve-locity). If we assume that our system has a total of N moleculesthat are all in the neutral chain phase, then the molecular energycan be approximated as Us(N) ≈ −𝛼N where −𝛼 corresponds tothe energy per bond between adjacent molecules. We denote thenumber of electrons in this pure solid phase as Ne,s, in which casethe grand potential isΦs = Us (N) + Ug(EF)− EFNe,s (2)On the other hand, if the N molecules are all in the charged liq-uid phase then the molecules are each charged by one electronin the LUMO and the molecular contribution to the total energybecomes Ul(N) = ELN (for simplicity we have dropped the super-script “SM” from EL). We denote the number of electrons in thispure liquid phase as Ne,l, in which case the grand potential isΦl = Ul (N) + Ug(EF)− EFNe,l (3)The critical Fermi level (EcF) at which the phase transition occursis determined by settingΦs = Φl . At this Fermi level Ne,l − Ne,s =N since N electrons are needed to charge the molecules, therebyyielding EcF = EL + 𝛼. For EF < EcF all of the electrons reside ingraphene band states and all of the molecules are condensed intosolid chains due to the energy gain of bond formation. For EF >EcF, on the other hand, all of the molecules are in the charged liq-uid state. The transition from the solid phase to the liquid phaseAdv. Mater. 2023, 35, 2300542 © 2023 Wiley-VCH GmbH2300542 (4 of 8) 15214095, 2023, 39, Downloaded from https://onlinelibrary.wiley.com/doi/10.1002/adma.202300542 by Cochrane Japan, Wiley Online Library on [21/10/2023]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons Licensewww.advancedsciencenews.com www.advmat.dedoes not occur when EF = EL because melting the chains requiresextra energy to break the bond between a chain end moleculeand its neighbor (i.e., the latent heat of melting). The processof adding a charged, isolated molecule to the liquid phase onlybecomes energetically favorable when the Fermi level reaches avalue equal to EL plus the energy required to break one bond (𝛼).This insight allows us to, in principle, experimentally obtain 𝛼 bycomparing the measured value of EcF at which the phase transi-tion occurs (which is marked by Fermi level pinning) to spectro-scopic measurements of EL. Experimentally we observe EcF to be120 ± 20 meV below the Dirac point energy and EL to be 140 ±5 meV below the Dirac point (EL was determined previously[14]).The difference between these quantities is on the order of our ex-perimental uncertainty, and so we are not yet able to extract anaccurate value of 𝛼 from our data. We are, however, able to placean upper limit on 𝛼: 𝛼 ≤ 40 meV (which is consistent with a DFT-based estimate of 𝛼, see Section 5b, Supporting Information).While the grand potential is continuous at the phase transition,its first derivative with respect to EF is not. From Equations (2)and (3) we see that 𝜕Φl𝜕EFand 𝜕Φs𝜕EFdiffer by N at EF = EcF , confirm-ing that this is a first-order phase transition. In a heat-driven first-order phase transition, such as the transformation of ice to liquidwater, latent heat is required to convert the phases at the tran-sition temperature. Our phase transition, however, is not heat-driven but is rather driven by electrostatic gating. There is thus alatent charge of N electrons required for complete conversion ofN molecules in the solid phase to the liquid phase rather than alatent heat. This is consistent with the experimental discontinuityin −ΔQ seen in Figure 2b which reflects the charge transferred tomelt F4TCNQ while EF is pinned at the critical value, analogousto how latent heat is transferred to melt a solid while the temper-ature is pinned at the melting point in a heat-driven solid–liquidphase transition.The preceding discussion is relevant for equilibrium condi-tions of the pure liquid phase (EF > EcF) vs the pure solid phase(EF < EcF), but we are also able to characterize the nonequilibriumsolid–liquid (mixed phase) coexistence regime (i.e., unstable ex-cursions from EF = EcF ) where the proportion of molecules inthe chain and liquid phases can be adjusted from one equilib-rium state to another (Figure 3). Figure 3a shows a plot of theexperimental liquid phase molecular density (Nl∕A, where A isthe graphene area) vs VG − V0 where V0 = −10 V is the gate volt-age at which isolated molecules first appear in STM images. Theyellow dots in Figure 3a shows that the experimental equilibriumvalues for Nl∕A exhibit a linear dependence on gate voltage. Themagenta dots, on the other hand, show experimental nonequi-librium data obtained by changing VG and ISD in such a way thatdiffusive conditions do not last long enough for the system tofully equilibrate. Figure 3b–g shows a full cycle of the system(measured at a single location on the device) as it evolves fromone equilibrium configuration to a different one (yellow dots) andthen back again by transitioning through a series of intermediatenonequilibrium states (magenta dots).To understand this experimental process, we start withFigure 3b which shows a patch of the surface that was initially inan equilibrium state at VG − V0 = 60 V. At this gate voltage a rela-tively high liquid phase density (Nl∕A = 4.1 × 1012 molecules percm2) coexists with a much lower concentration of the solid phase.The gate voltage was then changed to VG − V0 = 50 V under non-diffusive conditions (i.e., ISD = 0) to set a new equilibrium tar-get, but without allowing the system to evolve toward the newtarget (since the kinetics are quenched by keeping ISD = 0).The resulting nonequilibrium configuration is denoted t = 0(Figure 3b) and is visually identical to the equilibrium state atVG − V0 = 60 V. Figure 3c shows the same region after subjectingit to diffusive conditions (by setting ISD = 1.1 mA) for Δt = 50 mswhile holding the gate voltage constant at VG − V0 = 50 V. Thesolid phase density is seen to increase, but equilibrium is notyet established. Figure 3d shows the same region after allowingit to evolve for an additional 50 ms under diffusive conditionswhile maintaining VG − V0 = 50 V. The system is now in equi-librium with Nl∕A reduced to 3.5 × 1012 molecules per cm2 andthe solid density correspondingly increased. Figure 3e–g showthe same process in reverse as VG is reset to the original value ofVG − V0 = 60 V. The system is observed to evolve back to its orig-inal equilibrium configuration after passing through a nonequi-librium state (Figure 3f) regime.The mixed-phase solid/liquid configurations observed inFigure 3 can be understood within our theoretical framework ina straightforward way. To do this we consider the total energy ofa mixed phase state containing Nl molecules in the liquid phaseand N − Nl molecules in the chain phase given byU(Nl, EF)= Ul(Nl)+ Us(N − Nl)+ Ug(EF)(4)where Ul, Us, Ug, and N are defined the same as for Equations (2)and (3). Here N is constant, and EF is determined by VG and Nl.Only Nl remains variable, and its value at equilibrium Neql is ob-tained by minimizing Equation (4) with respect to Nl (see Sec-tion 4, Supporting Information for details). The resulting expres-sion for Neql per unit area isNeqlA= CVG +|ED −(EL + 𝛼) |2𝜋ℏ2v2F(5)where A is the area of the graphene capacitor, EL is the LUMOenergy, ED is the Dirac point energy, and vF is the Fermi velocitynear the Dirac point (1.1 × 106 m s−1). This expression is similarto an expression derived in ref. [14] using a different approach,but the new expression differs in the last term of Equation (5)which arises due to the energy required to break a bond (𝛼), a fac-tor not considered in ref. [14]. Equation (5) is plotted in Figure 3a(white dashed line) and is seen to match the equilibrium data(yellow dots) quite well. The nonequilibrium behavior (magentadots) can be explained by plotting U from Equation (4) as a colormap depending on both VG and Nl in Figure 3a. The low-energyregion of U(Nl, VG) is seen to correspond precisely to the equilib-rium density defined by Equation (5) (as expected). Excursionsfrom equilibrium, as shown by the magenta dots, thus push thesystem to higher energy. The energy landscape of Figure 3a isconsistent with the experimentally observed tendency of the sys-tem to relax back down in energy to the equilibrium configura-tion.A more dramatic example of nonequilibrium behavior isshown in Figure 4 which exhibits the time evolution of anonequilibrium melting process at a molecular solid–liquid in-terface. The STM image in Figure 4a shows the equilibriumconfiguration of this area at VG = −20 V after sufficiently longAdv. Mater. 2023, 35, 2300542 © 2023 Wiley-VCH GmbH2300542 (5 of 8) 15214095, 2023, 39, Downloaded from https://onlinelibrary.wiley.com/doi/10.1002/adma.202300542 by Cochrane Japan, Wiley Online Library on [21/10/2023]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons Licensewww.advancedsciencenews.com www.advmat.deFigure 3. F4TCNQ chain freezing and melting under nonequilibrium conditions. a) Experimental values of the equilibrium liquid phase molecule den-sity (Nl∕A) are plotted as yellow dots and nonequilibrium values as magenta dots. The theoretical total energy of the equilibrium mixed phase ofF4TCNQ/graphene is also shown (color scale) as a function of liquid phase surface density and gate voltage (V0 is the gate voltage at which melting firstbegins). The minimum energy configuration corresponds to the dashed white line (obtained from Equation (5)). b) STM image of the nonequilibriummolecular state obtained by switching VG − V0 to 50 V starting from the equilibrium state at VG − V0 = 60 V and not allowing the system to evolve underdiffusive conditions (t = 0). c) Molecular chains condense into a nonequilibrium state after allowing the system to evolve for 50 ms under diffusiveconditions (ISD = 1.1 mA, VG − V0 = 50 V). d) Molecular chain condensation advances to this equilibrium state after waiting an additional 50 ms underdiffusive conditions (ISD = 1.15 mA, VG − V0 = 50 V). e) STM image of the nonequilibrium state obtained by switching VG − V0 to 60 V and not allowingthe system to evolve under diffusive conditions (t = 0). f) Molecular chains have partially melted in this nonequilibrium state obtained after allowing thesystem to evolve for 10 ms under diffusive conditions (ISD = 1.11 mA, VG − V0 = 60 V). g) Molecular chains have melted even further in this equilibriumstate obtained after waiting an additional 90 ms under diffusive conditions (ISD = 1.11 mA, VG − V0 = 60 V), thus returning the molecular density to itsinitial configuration in (b). STM images were obtained at T = 4.5 K.diffusive conditions. A region of high solid phase density can beseen in the upper left and zero liquid phase density through-out. The surface was then put into a nonequilibrium state byrapidly changing the gate voltage to VG = 60 V (correspondingto a high liquid phase density equilibrium target). The systemwas then allowed to evolve under diffusive conditions for onlyΔt = 500 μs before being quenched and imaged as shown inFigure 4b. This nonequilibrium snapshot shows a “wave” of liq-uid phase molecules emanating from the molecular solid like wa-ter from a melting glacier. The width of the liquid layer extendsoutward from the solid by ≈80 nm and exhibits an interparticlespacing that is mostly constant. Figure 4c shows the same areaafter allowing it to evolve under diffusive conditions for another700 μs. The layer of liquid now extends outward from the solidby more than 160 nm. A full video of this process can be foundin Movie S3 (Supporting Information).The theoretical framework discussed up to now is inade-quate to model this type of nonequilibrium dynamics. To bet-ter understand this melting process we have generalized ouroverall model to account for: i) multiple chains, ii) isolated un-charged molecules, and iii) screened Coulomb interactions be-tween charged molecules. We have numerically simulated thismore complete model using the Monte Carlo method (see Section6, Supporting Information) to explain the dynamics shown inFigure 4a–c. An initial configuration was chosen with moleculesarranged into chains (Figure 4d), similar to the F4TCNQ solidswe observe experimentally. All model parameters were con-strained by the experiment except for 𝛼 (for which we only havean upper bound), but our results do not strongly depend on theprecise value of 𝛼. A fixed number of electrons was added tothe system at the start of the calculation to simulate the gat-ing process, and the resulting liquid phase density and EF valuewere subsequently determined. Overall, the simulation producedresults quite similar to the experiment. For example, isolatedmolecules were observed to dissociate from chains after onlya few Monte Carlo steps and to move toward empty grapheneAdv. Mater. 2023, 35, 2300542 © 2023 Wiley-VCH GmbH2300542 (6 of 8) 15214095, 2023, 39, Downloaded from https://onlinelibrary.wiley.com/doi/10.1002/adma.202300542 by Cochrane Japan, Wiley Online Library on [21/10/2023]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons Licensewww.advancedsciencenews.com www.advmat.deFigure 4. Nonequilibrium melting of the F4TCNQ solid. a) STM image of an equilibrium F4TCNQ solid formed under diffusive conditions on a grapheneFET at VG = −20 V. VG was stepped up to VG = 60 V before imaging, but the system was not allowed to evolve under diffusive conditions (t = 0). b)Same region of the surface after allowing it to evolve under diffusive conditions for Δt = 500 μs (ISD = 1.3 mA, VG = 60 V). A “wave” of charged liquidphase molecules can be seen emanating from the solid interface. c) Same region after allowing the system to evolve for an additional Δt = 700 μs underdiffusive conditions (ISD = 1.3 mA, VG = 60 V). The flow of charged molecular liquid has extended even further from the condensed phase interface.d–f) Monte Carlo simulations of F4TCNQ molecules disassociating from chains to model the behavior shown in (a–c). Molecules colored in blue arecharged and can be seen flowing outward from the charge-neutral condensed phase interface. STM images were obtained at T = 4.5 K.regions (Figure 4e,f), similar to the flow of molecules observedexperimentally in Figure 4b,c.3. ConclusionWe have observed a gate-tunable first-order solid–liquid phasetransition for F4TCNQ molecules adsorbed onto the surface of agraphene FET. We are able to control and image the relative abun-dances of liquid and solid phases for different equilibrium con-ditions and to directly visualize nonequilibrium processes with asingle-molecule resolution for both the solid and liquid phases.We have developed an analytical model that explains the gate-dependent equilibrium properties of this system with the onlyunknown parameter being the energy of cohesion of the molec-ular solid. The techniques described here provide a new methodfor experimentally extracting this parameter, and our results putan experimental upper bound on it of 40 meV per molecule.Monte Carlo simulations show reasonable agreement with thehighly nonequilibrium kinetics observed in our experiment. Thephenomenology observed here should be generalizable to otheradsorbate/surface systems that are similarly gate-tunable.4. Experimental SectionGraphene Transistor Fabrication: Graphene/hexagonal boron nitride(hBN) FETs were fabricated on highly doped SiO2/Si by mechanical exfo-liation. Electrical source and drain contacts were fabricated by depositing3 nm thickness of Cr and 10 nm thickness of Au through a stencil mask.The doped silicon substrate was used as the back gate. After placing inUHV, the graphene surface was cleaned by high-temperature annealing ina vacuum at 400 °C for 12 h.Molecule Deposition: F4TCNQ molecules were loaded into a Knudsencell evaporator and heated to 120 °C under ultrahigh-vacuum (UHV) con-ditions for deposition onto a graphene FET held at room temperature.Sub-monolayer molecular coverage was achieved by keeping the deposi-tion time under 15 s.STM/STS Measurements: STM/STS measurements were performedunder UHV conditions at T = 4.5 K using a commercial Omicron LT STMwith Pt/Ir tips. STM topography was obtained in constant-current mode.STM tips were calibrated on an Au(111) surface by measuring the Au(111)Shockley surface state before all STS measurements. STS was performedunder open feedback conditions by lock-in detection of the tunnel currentdriven by a wiggle voltage having a magnitude of 6–16 V rms at 401 Hzadded to the tunneling bias. WSxM software was used to process all STMand AFM images.Supporting InformationSupporting Information is available from the Wiley Online Library or fromthe author.AcknowledgementsThis work was supported by the U.S. Department of Energy, Office of Sci-ence, Office of Basic Energy Sciences, Materials Sciences and Engineer-ing Division (DE-AC02-05-CH11231), within the Nanomachine program(KC1203 which provided for STM imaging, spectroscopy, and analysis).Support was also provided by the Molecular Foundary at LBNL funded bythe U.S. Department of Energy, Office of Science, Office of Basic EnergyAdv. Mater. 2023, 35, 2300542 © 2023 Wiley-VCH GmbH2300542 (7 of 8) 15214095, 2023, 39, Downloaded from https://onlinelibrary.wiley.com/doi/10.1002/adma.202300542 by Cochrane Japan, Wiley Online Library on [21/10/2023]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons Licensewww.advancedsciencenews.com www.advmat.deSciences, Scientific User Facilities Division (DE-AC02-05CH11231), whichprovided for graphene device fabrication; by the National Science Foun-dation Award CHE-2204252 (molecular deposition and characterization);by the EPSRC grant EP/S025324/1 (calculation of molecular vdW bond-ing energy from DFT); by the Thomas Young Centre under grant numberTYC-101 (Monte Carlo simulations); by the Imperial College London Re-search Computing Service (DOI: 10.14469/hpc/2232) (DFT calculation ofmolecular chain electronic structure); and by JSPS KAKENHI Grant Num-ber 20H00354, 21H05233, and 23H02052 and World Premier InternationalResearch Center Initiative (WPI), MEXT, Japan. (growth of hBN crystals).F.L. acknowledges support from a Kavli Ensi Philomathia Graduate Stu-dent Fellowship. Z.G. was supported through a studentship in the Centrefor Doctoral Training on Theory and Simulation of Materials at ImperialCollege London funded by the EPSRC (EP/L015579/1). We thank J.M. Kahkfor useful discussions.Note: The marking for the three corresponding authors, as Alex Zettl,Johannes Lischner, and Michael Crommie, was clarified on September 27,2023, after initial publication online, following a misunderstanding and atechnical issue in the proofing process.Conflict of InterestThe authors declare no conflict of interest.Author ContributionsF.L. and H.-Z.T. contributed equally to this work. Conceptualization wasdone by F.L., H.-Z.T., A.A., and M.C.; Methodology was done by F.L., H.-Z.T., A.A., Z.G., and J.L.; Investigation was done by F.L., H.-Z.T., A.A., E.H.,M.H., K.W., T.T., and M.C.; Visualization was done by F.L., H.-Z.T., A.A., andY.Y.; Funding acquisition was done by J.L. and M.C.; Project administrationwas done by J.L. and M.C.; Supervision was done by J.L. and M.C.; Writingwas done by F.L., H.-Z.T., Z.G., J.L., and M.C.; Review and editing werewritten by F.L., H.-Z.T., Z.G., J.L., and M.C.Data Availability StatementThe data that support the findings of this study are available from the cor-responding author upon reasonable request.Keywordsfield-driven phase transitions, graphene field-effect transistor, molecularsolids, nonequilibrium dynamics, solid–liquid phase coexistenceReceived: January 17, 2023Revised: June 6, 2023Published online: July 13, 2023[1] H. Guo, K. Chen, Y. Oh, K. Wang, C. Dejoie, S. A. Syed Asif, O.L. Warren, Z. W. Shan, J. Wu, A. M. 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