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Yuxin Zhang, Haidong Tian, Huaixuan Li, Chiho Yoon, Ryan A. Nelson, Ziling Li, [Kenji Watanabe](https://orcid.org/0000-0003-3701-8119), [Takashi Taniguchi](https://orcid.org/0000-0002-1467-3105), Dmitry Smirnov, Roland K. Kawakami, Joshua E. Goldberger, Fan Zhang, Chun Ning Lau

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[Quantum octets in high mobility pentagonal two-dimensional PdSe2](https://mdr.nims.go.jp/datasets/002a17fa-cac0-4ef7-9c0f-f1ac0835c04d)

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Quantum octets in high mobility pentagonal two-dimensional PdSe2Article https://doi.org/10.1038/s41467-024-44972-2Quantum octets in highmobility pentagonaltwo-dimensional PdSe2Yuxin Zhang 1, Haidong Tian1, Huaixuan Li2,3, Chiho Yoon2, Ryan A. Nelson4,Ziling Li1, Kenji Watanabe 5, Takashi Taniguchi 6, Dmitry Smirnov 7,Roland K. Kawakami 1, Joshua E. Goldberger4, Fan Zhang 2 &Chun Ning Lau 1Two-dimensional (2D) materials have drawn immense interests in scientificand technological communities, owing to their extraordinary properties andtheir tunability by gating, proximity, strain and external fields. For electronicapplications, an ideal 2D material would have high mobility, air stability, siz-able band gap, and be compatible with large scale synthesis. Here wedemonstrate air stable field effect transistors using atomically thin few-layerPdSe2 sheets that are sandwiched between hexagonal BN (hBN), with largesaturation current > 350 μA/μm, and high field effect mobilities of ~ 700 and10,000 cm2/Vs at 300K and 2K, respectively. At low temperatures, magne-totransport studies reveal unique octets in quantumoscillations that persist atall densities, arising from 2-fold spin and 4-fold valley degeneracies, which canbe broken by in-plane and out-of-plane magnetic fields toward quantum Hallspin and orbital ferromagnetism.Recently, a class of 2D materials with puckered pentagonal latticestructures1–4 has arrived at scene. These materials have broken sub-lattice symmetry, and have been predicted to host exotic new prop-erties partly due to the in-plane and cross-plane anisotropy5–7,including larger band gap variation, axis-dependent conductionpolarity, enhanced spin−orbit coupling8, and nonsymmorphicsymmetry-enforced band topology in the monolayer limit9. One suchexample is PdSe210–17. Notably, bulk PdSe2 are reported to display airstability18, a widely tunable band gap that varies from 0.5 eV in bulk to1.3 eV for monolayers1,18–20, ambipolar transport21, superconductivityupon transforming to a cubic polymorph under high pressure22, andsuperior optical and thermoelectric properties23–25. Wafer-scalesynthesis of few-layer PdSe2 has already been developed17,26,27. Mobi-lities of up to 200 cm2/Vs have been reported, albeit only under a largesource-drain bias of 1–2 V1,18,28. Thus, as a recent addition to the familyof 2D materials, PdSe2 holds great promise for digital electronic,thermoelectric, and optoelectronic applications.Despite the increasing interest in PdSe2 in the past few years,transport measurements so far have been performed only under highbias and at room temperature, with little optimization with regard tocontact resistance or systematic investigation of device behavior.Here, we report transport studies of atomically thin PdSe2 field-effecttransistors that are stable in ambient conditions for >1 month, highsaturation current among atomically thin semiconductors, as well ashigh field-effect mobility. This ultra-high mobility enabled the experi-mental observation of Shubnikov-de Hass oscillation and the quantumhall effect in this pentagonal 2Dmaterial. Interestingly, the Landau fanreveals an eightfold degeneracy, arising from twofold spin and four-fold valley degeneracies29; increasing in-plane (B||) and out-of-plane(B⊥) magnetic fields leads to Landau level crossings and broken spinand valley symmetries. Those observations indicate a unique bandstructure and spin-valley interplay in the pentagonal PdSe2.Each unit cell of PdSe2 consists of two inversion-symmetric planesof d8 Pd2+ ions bonded in square-planar-like coordination withReceived: 4 October 2023Accepted: 11 January 2024Check for updates1Department of Physics, The Ohio State University, Columbus, OH 43210, USA. 2Department of Physics, The University of Texas at Dallas, 800West CampbellRoad, Richardson, TX 75080-3021, USA. 3Department of Physics, Carnegie Mellon University, Pittsburgh, PA 15213, USA. 4Department of Chemistry andBiochemistry, The Ohio State University, Columbus, OH 43210, USA. 5Research Center for Electronic and Optical Materials, National Institute for MaterialsScience, 1-1 Namiki, Tsukuba 305-0044, Japan. 6Research Center for Materials Nanoarchitectonics, National Institute for Materials Science, 1-1 Namiki,Tsukuba 305-0044, Japan. 7National High Magnetic Field Laboratory, Tallahassee, FL 32310, USA. e-mail: lau.232@osu.eduNature Communications |          (2024) 15:761 11234567890():,;1234567890():,;http://orcid.org/0009-0004-5106-2604http://orcid.org/0009-0004-5106-2604http://orcid.org/0009-0004-5106-2604http://orcid.org/0009-0004-5106-2604http://orcid.org/0009-0004-5106-2604http://orcid.org/0000-0003-3701-8119http://orcid.org/0000-0003-3701-8119http://orcid.org/0000-0003-3701-8119http://orcid.org/0000-0003-3701-8119http://orcid.org/0000-0003-3701-8119http://orcid.org/0000-0002-1467-3105http://orcid.org/0000-0002-1467-3105http://orcid.org/0000-0002-1467-3105http://orcid.org/0000-0002-1467-3105http://orcid.org/0000-0002-1467-3105http://orcid.org/0000-0001-6358-3221http://orcid.org/0000-0001-6358-3221http://orcid.org/0000-0001-6358-3221http://orcid.org/0000-0001-6358-3221http://orcid.org/0000-0001-6358-3221http://orcid.org/0000-0003-0245-9192http://orcid.org/0000-0003-0245-9192http://orcid.org/0000-0003-0245-9192http://orcid.org/0000-0003-0245-9192http://orcid.org/0000-0003-0245-9192http://orcid.org/0000-0003-4623-4200http://orcid.org/0000-0003-4623-4200http://orcid.org/0000-0003-4623-4200http://orcid.org/0000-0003-4623-4200http://orcid.org/0000-0003-4623-4200http://orcid.org/0000-0003-2159-6723http://orcid.org/0000-0003-2159-6723http://orcid.org/0000-0003-2159-6723http://orcid.org/0000-0003-2159-6723http://orcid.org/0000-0003-2159-6723http://crossmark.crossref.org/dialog/?doi=10.1038/s41467-024-44972-2&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41467-024-44972-2&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41467-024-44972-2&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41467-024-44972-2&domain=pdfmailto:lau.232@osu.edudiselenide dianions (Fig. 1a, b), yet the two adjacent layers are relatedby glide-mirror symmetry. The lattice constants are a = 5.74 Å,b = 5.86Å, and c = 7.69Å as measured with x-ray diffraction (see Sup-plementary Information (SI)). Note that the unit cell of the bulk con-sists of two PdSe2 layers separatedby a van derWaals gap; fromatomicforce microscope (AFM) measurements, each monolayer step has aheight of 5.2 Å (see SI). When projected onto a plane, each layer con-sists of a puckered network of pentagonal rings. Our First-principlescalculations show that PdSe2 few-layers are indirect-gap semi-conductors, with a 1.06 eV band gap for monolayer and a <0.1 eV bandgap for heptalayer (Fig. 1c, d, and see SI). Their conduction bandminima are located close to theM point in the Brillouin zone and thusform four symmetry-related valleys. This valley quartet is universal inour few-layer calculations, although the band gap decreases withincreasing the thickness as approaching the bulk limit (see SI). (Notethat the GWmethod yields larger band gaps for quasiparticle bands29.)Results and discussionSample characterization at room temperatureBulk PdSe2 crystals were purchased commercially, or grown via ver-tical Bridgman inSe flux (see SI), and exfoliated into atomically thinlayers onto insulating substrates. Two types of devices are fabricated.We first fabricate “bare” devices on Si/SiO2 substrates, by transferringfew-layer graphene on few-layer PdSe2 sheets as electrodes, anddepositing Cr/Au contacts on graphene. The blue curve in Fig. 1e dis-plays the four-terminal resistanceRxx as a function of back gate voltageVbg for an as-fabricated 4 nm-thick device (device B1). The deviceappears to be ambipolar, with maximum Rxx ~ 600 kΩ located atVbg = 7.8 V. In the linear response regime, its electron and hole fieldeffect mobilities μFE= (1/e)(dσ/dn) are ~24 and 11 cm2/Vs, respectively(here σ is the 2D conductivity, n the charge density, and e the electroncharge). To test the device’s stability in ambient condition, we leavethe device in the air and monitor the transfer curve as a function oftime. Surprisingly, despite previous claims of PdSe2’s air stability1,20,30,the device deteriorates steadily with time—μFE decreases, while theresistancemaximumshifts to the right; thisp-doping aging effect likelyarises from oxidation into PdSe2Ox31. By day 27, the device loses allresponse to gate, and Rxx = 350kΩ. We find that thermal annealing at200 °C in vacuum restores the mobility of the electron-doped regime,though the resistancemaximum remains atVbg < −40V, indicating thatthe device is electron-doped, which could arise from the formation ofSe vacancies. Exposing the annealed device to air for a few hours, weagain observe p-doping (Fig. 1e, green curve).To fabricate air-stable, high-performance field effect transistordevices, we take advantage of hexagonal BN (hBN) layers, which havebeen shown to provide protection for air-sensitive materials such asphosphorene32, InSe33, and CrI334. PdSe2 sheets that are 3–8 layers thickare contacted by few-layer graphene leads and sandwiched betweenhBN layers (Fig. 1b). The global Si/SiO2 back gate tunes both the chargedensity n of PdSe2 layer and the contact between graphene and PdSe2,while the top gate covers only the channel region and tunes only ntherein. Graphene contacts are advantageous over conventionalmetallic (e.g., Cr/Au or Ti/Au) contacts, since both the work functionand surface potential of graphene are gate-tunable, thus the Schottkybarrier between graphene and the semiconductor can be effectivelylowered. As shown in Fig. 1f, for Vbg ≤0, as-fabricated hBN-encapsu-lated devices have high resistance, >~ MΩ (see Fig. S3a); as Vbgincreases from 0, four-terminal resistance drops rapidly to a few kΩupon electron doping. The high resistance and low mobility in the p-doped regime in these pristine hBN-encapsulated devices suggest thatthe hole conduction in unencapsulated PdSe2 devices is not intrinsic,but arises from oxidation of PdSe2 that can be partially reversed byvacuumannealing. In contrast to “bare” devices, the hBN-encapsulateddevices are very stable in air, with no degradation inmobility and onlyslight hole doping after 29 days (Fig. 1f).We now focus exclusively on hBN-sandwiched PdSe2 devices.Figure 1g, h displays the two-terminal current-voltage (I–V) char-acteristics of a PdSe2 device A2 that is ~2.5 nm thickor ~5 layers at room102461002461000600 Vbg (V) Day 0 Day 6 Day 2910μmefgh3400150 V (V)Vbg=70VT=290K10V 8V 5V 3V 0VVtg1600100 V (V) 70V 50V 30V 20V 10V 0V -30VVtg=0VVbg800400060-30 Vbg (V) Day 0Day 27 Day 27annealed4 hrs after annealingTop gateh-BNh-BNGraphene PdSe2cabcdΓ X Y ΓMHeptalayer without SOC-1.5-1-0.500.5Energy (eV)11.5Γ X Y ΓM-1.5-1-0.500.5Energy (eV)11.5Heptalayer with SOCFig. 1 | Electronic and room temperature transport properties of few-layerPdSe2 field effect transistors. a Top and side views of the lattice structure of 2Dpuckered PdSe2. Blue and yellow spheres represent the Pd and Se atoms, respec-tively. The vertical lattice constant, c = 7.69 Å. b The device schematic. c, d Bandstructures of 7-layer PdSe2 without and with spin-orbit coupling included in thecalculation, respectively. Our first-principles calculations based on density func-tional theory were performed using the Vienna ab initio simulation package (seeSupplementary Note 3). e, f Transport characteristics Rxx(Vbg) of PdSe2 devices onSiO2 (device B1) and encapsulated within hBN (device A1), respectively, after peri-ods of time in air. Rxx is the four-probe longitudinal resistance, and Vbg is the backgate voltage. Insert of f: optical image of a hBN/PdSe2/hBN stack. Yellow and whitedotted lines outline graphene and PdSe2, respectively. g, h I–V characteristics of a5-layer device A2 at different gate voltages, showing saturation current of ~350 μA/μmwhen highly doped. g: varying top gate voltage Vtg at Vbg = 70V. h: varying Vbgat Vtg =0V.Article https://doi.org/10.1038/s41467-024-44972-2Nature Communications |          (2024) 15:761 2temperature. When Vbg <0V, the device is intrinsic the zero-biasresistance is ~350MΩ, and saturation current Isat is a few nA/μm. Withincreasing doping, the two-terminal resistance R2T decreases and Isatincreases. At Vbg = 70V and Vtg = 10V, Isat ~ 350 μA/μm, which is veryhigh for atomically thin 2D semiconductors. Though Isat per layer islower than the state-of-the-art value of, e.g. MoS235, we note that thesaturation current does not necessarily scale with the number of layersin the few-layer limit. Moreover, the channel lengths of our devices arerather long (~1μm),while the contact resistance is still quite prominent(e.g. when highly doped, R2T and Rxx are ~35 kΩ and 7 kΩ, respectively),thus we expect that Isat can be improved by an order of magnitude byminimizing contact resistance of the future generation of devices, forexample, via fabricating gates that independently tune the workfunction of the graphene contacts.Observation of quantum octetsTo examine the dominant scattering mechanism, we plot the aver-age field effect mobility μFE between 4.5 and 6.5 × 1012cm−2 as afunction of temperature T for two separate devices. As shown byFig. 2a, the room temperature μFE is ~150 and 780 cm2/Vs for the5-layer (A2) and 7-layer (A3) devices, respectively. At high tem-peratures, μFE increases with decreasing temperature, with a power-law behavior T-α. Fitting the data yields α = 1.3 and 1.6 for ~5-layer and7-layer device, respectively, indicating that the main scatteringmechanism is phonon scattering, most likely longitudinal acousticand longitudinal optical phonons36. For T < 30 K, μFE saturates to2000 and 10,000 cm2/Vs, respectively, indicating that the deviceshave reached the regime where the mobility bottleneck is scatteringby intrinsic defects and/or impurities. These mobility values, whichare 1-3 orders of magnitude higher than prior reports on PdSe21,19,23,37and close to state-of-the-art MoS2 devices38, suggest that hBNencapsulation significantly reduces the formation of defects andscattering by charged impurities in the substrates.We now turn to the magnetotransport measurements at thecryogenic temperatures. Here, we focus on a sample that is ~3.5 nmthick or ~7 layers (device A4). Figure 2b shows the background sub-tracted longitudinal resistance ΔRxx plotted versus Vtg and the per-pendicularmagnetic fieldB⊥, while the backgate voltage ismaintainedat Vbg = 65 V. Clear Shubnikov-de Hass (SdH) oscillations from PdSe2start to be resolved at ~4 T, indicating quantum mobility of 2500 cm2/Vs. Interestingly, the Landau fan features resistance minima at fillingfactors ν = nh/eB⊥ that are integer multiples of 8 (Fig. 2b, d); suchquantum Hall octets that persist at all density ranges have not beenobserved in other 2Dmaterials. To account for this 8-fold degeneracy,we note that the conduction band bottom occurs near the M point inthe rectangular Brillouin zone, and that the glide and screw symme-tries dictate a 4-fold valley degeneracy (Fig. 2c). Additionally, near theconductionband bottom, the spin-orbit coupling is negligibly small, ascan be seen in Fig. 1c, d and Fig. S6–7, the bands have a 2-fold spindegeneracy even for even-layer systems in which inversion symmetryis broken. Consequently, the charge carriers in PdSe2 are endowedwith the observed 8-fold degeneracy. In addition, Hall measurementswas performed on a 3-L device A5, as shown in Fig. 2d, quantized Hallresistance Rxy plateaus are resolved at Rxx minimum, attesting to thequantumHall nature of the observed octets. Additional transport dataon A5 can be found in Supplementary Fig. 3b–d.Fig. 2 | Transport data of PdSe2 transistors at low temperatures. a Field effectmobility as a function of temperature of two PdSe2 transistors that are~3.5 nm(green) and 2.5 nm (red) in thickness. Dashed lines are fits to T-γ. b SdHoscillations of device A4 versus Vtg and the perpendicular magnetic field B⊥. Tobetter display the oscillations, a smooth polynomial background is subtracted.c Energy contours of conduction band of 7-layer PdSe2 in the kx-ky plane, showingthe four valleys. To better showcase the valleys, segments of the axes are omitted(indicated by dotted lines). The MATLAB function interp2 was used to interpolatethe first-principles results and generate the equal-energy contours at specificenergies (see Supplementary Note 3). d Longitudinal resistance Rxx (red curve) andHall resistance Rxy (blue curve) of device A5 plotted versus B⊥ at Vbg = 80V andT = 50mK. Inserted values indicate the quantized Hall resistance. e Rxx(Vtg) ofdevice A4 at B⊥ = 7.5 T and different temperatures. f Normalized amplitude ofoscillations in d plotted versus temperature, fitted to the Lifshitz-Kosevichequation.Article https://doi.org/10.1038/s41467-024-44972-2Nature Communications |          (2024) 15:761 3The effective mass of the charge carriers is extracted by evaluat-ing the amplitude of the SdH oscillations with temperature. Figure 2eplots R(Vtg) at T ranging from 1.7 K to 7 K at B⊥ = 7.5 T, and Fig. 2f theoscillation amplitude as a function of temperature. Fitting the data tothe Lifshitz–Kosevich formula ΔR / χsinhðχÞ, where χ = 2π2kTm*_eB?, k is theBoltzman constant, _ is the reduced Planck constant, and T is thetemperature, we obtain an effective mass m*=0:29me, here me is thebare electron mass in vacuum. This result is in excellent agreementwith previous theoretical calculations29.Magnetotransport measurements in tilted fieldsAt higher fields, degeneracy of the quantum Hall octets is partially lif-ted (Fig. 3d and Fig. S3b). Major resistance valleys occur at filling factorsν=8N, where N= 1, 2, 3… is an integer denoting the Landau level (LL)index; between the major valleys, minor resistance dips are visible atν=4Nodd, whereNodd= 1, 3, 5… is an odd integer. To gain insight into thenature of this quantumHall ferromagnetism,weperformmeasurementsin tilted magnetic fields, since the valley (orbital) degrees of freedomdepend only on the perpendicular component of B, whereas the spincouples to the total magnetic field Bt. Figure 3a–c plots the R(Bt, ν) fordata acquired at angles θ=0, 35.6° and 48.2°, respectively, and linetraces of R(Vtg) are shown in Fig. 3d. At θ=0 (i.e., Bt=B⊥ and B|| =0), theQH states at ν=8, 16, 24, and 32 are resolved at lower fields, and those atν=4, 12, 20 and 28 are resolved at higher fields with shallower minima,indicating that the gaps for the former (latter) are larger (smaller). Atθ=35.6°, the adjacent states are approximately equally resolved, sug-gesting that the gaps at ν=8N (ν=4Nodd) decrease (increase) withincreasingB∣∣. Upon further increaseofB∣∣, atθ=48.2°, only theQHstatesat ν=4Nodd= 12, 20, 28, and 36 are resolved, i.e., the cyclotron andZeemangapsbecomeequal inmagnitude, and theLLswith indices (N,↑)and (N+ 1,↓) cross each other. This is the so-called coincidence angle, atwhich LLs with opposite spin indices cross.These data enable us to determine the schematic of LL resolution.At low field, the 8-fold degenerate LLs are separated by cyclotron gapsEc = _ωcðN + 12Þ, whereωc =eB?m* is the cyclotron frequency. Increasing B⊥first lifts the spin degeneracy, giving rise to Zeeman-split LLs withenergy difference EZ = gμBBt, where μB is the Bohr magneton and g theeffective electron g-factor, including the Coulomb interaction correc-tion. In titled field measurements, increasing B|| further enhances theZeeman splitting, giving rise to LL crossing at large B|| (Fig. 3e). Usingthis schematic, we can extract the effective g-factor from the coin-cidence angle at which the cyclotron and Zeeman gaps are equal inmagnitude (θ = 48.2°). This condition yields g = _eμBm*B?Bt= 2cosðθ2Þm*=me= 4.60,respectively, using m* =0.29me obtained earlier. This enhancementfrom the bare value of g = 2 in a nearly spin-orbit-coupling-free systemsubstantiates the presence of electron–electron interaction effect andthe resulting quantum Hall spin ferromagnetism.Lifting of all degeneracyFinally, we perform measurements at even higher B⊥ fields up to 29 Tand observe that, after the lifting of the spin degeneracy, the fourfoldvalley degeneracy is also broken (Fig. 4a). This quantum Hall orbitalferromagnetism is clearly illustrated by the emergence of additionalresistance minima between filling factor 4, 8, and 12 (Fig. 4b). Futurestudy is required to showwhether these quantumHall states are valleycoherent phases breaking translational symmetry or valley polarizedphases exhibiting in-plane ferroelectricity39. Notably, quantum halleffects down to the lowest LL are resolved, attesting to the excellentquality of our sample. The entire sequence of degeneracy breaking isschematically illustrated in Fig. 4c.In conclusion, we demonstrate that few-layer pentagonal PdSe2,when sandwiched between hBN layers, is an excellent 2D semi-conductor, boasting air stability, high saturation current, exceedinglyhigh field effectmobility, and quantumHall octets, and ferromagnetismFig. 3 | Magnetotransport data in tilted magnetic fields for device A4 atT = 50mK.a–cRxx in kΩ versus totalmagneticfieldBt andfilling factor ν at differenttilting angles. d Line traces of R(Vtg) at B⊥ = 12 T and different tilting angles. Thetraces are offset for clarity. Black numbers indicate filling factors. e Schematics ofthe evolution of Landau levels with in-plane magnetic field while B⊥ is kept con-stant. Ec = _ωc, is the cyclotron gap. EZ = gμBBt, is the Zeeman gap. θ1 indicates theangle at which all adjacent states are approximately equally resolved. θ2 is thecoincidence angle at which LLs with opposite spin indices cross.Article https://doi.org/10.1038/s41467-024-44972-2Nature Communications |          (2024) 15:761 4in magnetic fields. Our work paves the way for future electronic, pho-tonic, and topological applications of this pentagonal material.MethodsPdSe2 synthesisBulk PdSe2 crystals were purchased from Sixcarbon Technology orgrown via vertical Bridgman-type growth. Elemental Pd powder ismixed with a large excess of elemental Se powder (molar ratio of3:97 Pd:Se). The typical 7-gram charge is loaded into a fused-silicaampoule with 2–3mm thick walls and a tapered end. It is important toensure that even if all of the Se was volatilized, the pressure generatedwould be much less than the critical hoop stress of fused-silica(<50MPa). The ampoule is evacuated and sealed under a typicalvacuum pressure of ~60mtorr. After sealing, a small fused-silica hookis welded to the ampoule opposite the tapered end. The ampoule isthen suspended by this hookwithin a vertical single-zone tube furnaceusing nichromewire to suspend the ampoule. Togrow the PdSe2 singlecrystals, the ampoule is suspended in the vertical furnace which isramped up to 850° C in 24 h. The temperature is held at 850 °C for 50 hbefore the ampoule is dropped through the natural thermal gradientof the vertical furnace at a rate of 1mmh−1 over the course of ~100 h.X-ray diffraction was used to confirm the lattice parameters of thebulk, giving a = 5.7423(5) Å, b = 5.8646(5) Å, c = 7.6924(2) Å.Sample characterization and device fabricationThin layer flakes were exfoliated on PDMS using the standard Scotchtape method. The thickness of thin flakes was measured using anatomic force microscope (AFM). For hBN-encapsulated devices andbare devices, hBN/graphene/PdSe2/hBN or graphene/PdSe2/hBNstacks were assembled using a dry transfer technique with a PCstamp40, then released onto 290 nm SiO2/Si substrates at 170°C ~ 180°C. Afterwards, the stackswere shaped to thedesiredHall bar structurethrough electron-beam lithography and reactive ion etching (RIE) withSF6 and Ar, followed by the deposition of top gate dielectric (Al2O350~70 nm). Finally, top gate andmetal leads thatmake edge-contact tographene were patterned by electron-beam lithography and sub-sequent deposition of metals (Cr 5 nm /Au 100nm).Electrical and magnetotransport measurementsTwo-terminal transport characteristics were measured by applying ana.c. voltage bias of 50–100μV (a standard lock-in amplifier SR830) tothe source and recording the drain current. For four-terminal mea-surements, the SR830was used to apply a constant a.c current of 10 to100nA andmeasure the voltage drop across the channel. Top gate andback gate d.c bias were applied using Keithley 2400.Magnetotransport measurements were performed in a Janis cryostatand two other He3 cryostats at the National High Magnetic FieldLaboratory (NHMFL).Data availabilityThe data that support the findings of this study are available within themain text and Supplementary Information. Any other relevant data areavailable from the corresponding authors upon request. Source dataare provided with this paper.Code availabilityThe codes for band structure calculations are available upon requestfrom F.Z. and H.L.References1. Oyedele, A. D. et al. PdSe2: pentagonal two-dimensional layers withhigh air stability for electronics. J. Am. Chem. Soc. 139, 14090 (2017).2. M. Yagmurcukardes et al. Pentagonalmonolayer crystals of carbon,boron nitride, and silver azide. J. Appl. Phys. 118, 104303 (2015).3. Zhang, S. et al. Penta-graphene: a new carbon allotrope. Proc. Natl.Acad. Sci. USA 112, 2372 (2015).4. Shen, Y. & Wang, Q. Pentagon-based 2D materials: classification,properties and applications. Phys. Rep. 964, 1 (2022).5. Zhuang, H. L. From pentagonal geometries to two-dimensionalmaterials. Comput. Mater. Sci. 159, 448 (2019).6. Qin, D. et al. Monolayer PdSe2: a promising two-dimensional ther-moelectric material. Sci. Rep. 8, 1 (2018).7. Bravo, S., Correa, J., Chico, L. & Pacheco, M. Symmetry-protectedmetallic and topological phases in penta-materials. Sci. Rep. 9,1 (2019).8. R. A. Nelson et al. Axis dependent conduction polarity in the air-stable semiconductor, PdSe2. Mater. Horiz. 10, 3740–3748 (2023).9. Bravo, S., Pacheco,M., Correa, J. D. &Chico, L. Topological bands inthe PdSe2 pentagonal monolayer. Phys. Chem. Chem. Phys. 24,15749 (2022).10. Liu, X. et al. Temperature-sensitive spatial distribution of defects inPd Se 2. Phys. Rev. Mater. 5, L041001 (2021).11. Chen, X. et al. Broadband nonlinear photoresponse and ultrafastcarrier dynamics of 2D PdSe2. Adv. Opt.l Mater. 10, 1 (2022).12. 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Science 342, 614 (2013).AcknowledgementsWe thank Dima Shcherbakov for the helpful discussion. This work issupported by NSF/DMR 2128945. H.T. is supported by the DOE BESDivision under grant no. DE-SC0020187. R.A.N. and J.E.G. acknowl-edge the Air ForceOffice of Scientific Research for funding fromgrantnumber FA9550-21-1-02684. Z.L. and R.K.K. acknowledge supportfrom AFOSR/MURI project 2DMagic (FA9550-19-1-0390) and the USDepartment of Energy (DE-SC0016379). A portion of this work wasperformed at the National High Magnetic Field Laboratory, which issupported by the National Science Foundation through NSF/DMR-1644779 and the State of Florida. K.W. and T.T. acknowledge supportfrom the JSPS KAKENHI (Grant Numbers 20H00354, 21H05233, and23H02052) andWorld Premier International ResearchCenter Initiative(WPI), MEXT, Japan. The theoretical work done at UT Dallas was sup-ported by NSF under Grants no. DMR-1945351, no. DMR-2105139, andno. DMR-2324033. We acknowledge the Texas Advanced ComputingCenter (TACC) for providing resources that have contributed to theresearch results reported in this work.Author contributionsC.N.L. conceived the experiment. Y.Z. fabricated the devices. Y.Z.,H.T., and D.S. performed measurements. Z.L. and R.K. performedsecond harmonic generation measurements. R.A.N. and J.E.G. grewPdSe2 bulk crystals. K.W. and T.T. provided hBN crystals. H.L. and C.Y.performed the theoretical calculations under the supervision of F.Z.C.N.L. and Y.Z. analyzed data. C.N.L, Y.Z., and F.Z. interpreted dataand wrote the manuscript. All authors read and commented on themanuscript.Competing interestsThe authors declare no competing interests.Additional informationSupplementary information The online version containssupplementary material available athttps://doi.org/10.1038/s41467-024-44972-2.Correspondence and requests for materials should be addressed toChun Ning Lau.Peer review information Nature Communications thanks the anon-ymous reviewers for their contribution to the peer review of this work. Apeer review file is available.Reprints and permissions information is available athttp://www.nature.com/reprintsPublisher’s note Springer Nature remains neutral with regard to jur-isdictional claims in published maps and institutional affiliations.Article https://doi.org/10.1038/s41467-024-44972-2Nature Communications |          (2024) 15:761 6https://arxiv.org/abs/1805.06493https://doi.org/10.1038/s41467-024-44972-2http://www.nature.com/reprintsOpen Access This article is licensed under a Creative CommonsAttribution 4.0 International License, which permits use, sharing,adaptation, distribution and reproduction in any medium or format, aslong as you give appropriate credit to the original author(s) and thesource, provide a link to the Creative Commons licence, and indicate ifchanges were made. The images or other third party material in thisarticle are included in the article’s Creative Commons licence, unlessindicated otherwise in a credit line to the material. If material is notincluded in the article’s Creative Commons licence and your intendeduse is not permitted by statutory regulation or exceeds the permitteduse, you will need to obtain permission directly from the copyrightholder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.© The Author(s) 2024Article https://doi.org/10.1038/s41467-024-44972-2Nature Communications |          (2024) 15:761 7http://creativecommons.org/licenses/by/4.0/http://creativecommons.org/licenses/by/4.0/ Quantum octets in high mobility pentagonal two-dimensional PdSe2 Results and discussion Sample characterization at room temperature Observation of quantum�octets Magnetotransport measurements in tilted�fields Lifting of all degeneracy Methods PdSe2 synthesis Sample characterization and device fabrication Electrical and magnetotransport measurements Data availability Code availability References Acknowledgements Author contributions Competing interests Additional information