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[PhysRevLett.133.026501.pdf](https://mdr.nims.go.jp/filesets/fa2252b8-a3ee-4e99-8246-d3f16d02ab2d/download)

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[Nadine Leisgang](https://orcid.org/0000-0002-6612-866X), [Dmitry Miserev](https://orcid.org/0000-0001-7838-7274), Hinrich Mattiat, [Lukas Schneider](https://orcid.org/0009-0000-9212-6644), [Lukas Sponfeldner](https://orcid.org/0000-0002-4634-2722), [Kenji Watanabe](https://orcid.org/0000-0003-3701-8119), [Takashi Taniguchi](https://orcid.org/0000-0002-1467-3105), [Martino Poggio](https://orcid.org/0000-0002-5327-051X), [Richard J. Warburton](https://orcid.org/0000-0002-3095-3596)

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[Creative Commons BY Attribution 4.0 International](https://creativecommons.org/licenses/by/4.0/)

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[Exchange Energy of the Ferromagnetic Electronic Ground State in a Monolayer Semiconductor](https://mdr.nims.go.jp/datasets/d7376f89-bb0b-405a-bc46-9c6440c61bd9)

## Fulltext

Exchange Energy of the Ferromagnetic Electronic Ground State in a Monolayer SemiconductorExchange Energy of the Ferromagnetic Electronic Ground Statein a Monolayer SemiconductorNadine Leisgang ,1,2,* Dmitry Miserev ,1 Hinrich Mattiat,1 Lukas Schneider ,1 Lukas Sponfeldner ,1Kenji Watanabe ,3 Takashi Taniguchi ,4 Martino Poggio ,1 and Richard J. Warburton 11Department of Physics, University of Basel, Klingelbergstrasse 82, 4056 Basel, Switzerland2Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA3Research Center for Electronic and Optical Materials, National Institute for Materials Science,1-1 Namiki, Tsukuba 305-0044, Japan4Research Center for Materials Nanoarchitectonics, National Institute for Materials Science,1-1 Namiki, Tsukuba 305-0044, Japan(Received 31 October 2023; accepted 28 May 2024; published 8 July 2024)Mobile electrons in the semiconductor monolayer MoS2 form a ferromagnetic state at low temperature.The Fermi sea consists of two circles: one at the K point, the other at the K̃ point, both with the same spin.Here, we present an optical experiment on gated MoS2 at low electron density in which excitons areinjected with known spin and valley quantum numbers. The resulting trions are identified using a modelwhich accounts for the injection process, the formation of antisymmetrized trion states, electron-holescattering from one valley to the other, and recombination. The results are consistent with a complete spinpolarization. From the splittings between different trion states, we measure the exchange energy Σ, theenergy required to flip a single spin within the ferromagnetic state, as well as the intervalley Coulombexchange energy J. We determine Σ ¼ 11.2 meV and J ¼ 5 meV at n ¼ 1.5 × 1012 cm−2 and find thatJ depends strongly on the electron density n.DOI: 10.1103/PhysRevLett.133.026501Ferromagnetism represents a state of matter in whichspontaneous alignment of electron spins leads to a netmagnetization. A key metric of a ferromagnet is theexchange energy Σ, the energy required to flip one spin.Σ determines the Curie temperature separating the ferro-magnetic (magnetically ordered) and the paramagnetic(magnetically disordered) ground states. For metallic fer-romagnets, e.g., iron, Σ is large, ∼100 meV, resulting inenormous Curie temperatures, ∼1000 K. The phase tran-sition is second order.Ferromagnetic ordering of mobile electrons has beenobserved in various two-dimensional (2D) systems, e.g.,monolayer MoS2 [1], an AlAs quantum well [2], mono-layer WSe2 [3], and twisted bilayer graphene [4]. As theMermin-Wagner theorem precludes magnetic order in 2Dfor isotropic spins [5], magnetic anisotropy induced by aspin-orbit interaction or a small Zeeman splitting of theFermi surfaces is required to stabilize the ferromagneticorder of a 2D electron gas (2DEG). The zero-temperatureferromagnetic phase transition controlled by the electrondensity is predicted to be first order [6], an idea supportedexperimentally [7].Here, we present photoluminescence (PL) with quasire-sonant excitation on gated monolayer MoS2 in all fourpolarization channels. The novelty with respect to previousexperiments [1,7] lies in an interpretation of the energies ofthe emission lines. We first identify the emission lines oneby one. We then argue that the splitting between emissionlines provides a direct measurement of the ferromagneticexchange energy Σ, as well as the intervalley Coulombexchange energy J.Monolayer MoS2 is a semiconductor with direct bandgaps at theK and K̃ points of theBrillouin zone [8] [Fig. 1(a)].The spin-orbit splitting is large in the valence band(∼150 meV [9]) and small in the conduction band (a fewmeV [9–11]). Resonant σþ polarized (σ− polarized) lightcreates a bright exciton at the K point (K̃ point). (We notethat the lowest-energy excitons are dark [12,13], irrelevanthere as we inject and detect bright excitons.) Recently, apronounced optical dichroism of a 2DEG in monolayerMoS2 was interpreted as ferromagnetic ordering. TheFermi surface consists of a circle at the K point and acircle at the K̃ point [1]. If the spins point down, theK↓ andK̃↓ bands are occupied up to the Fermi energy; conversely,theK↑ and K̃↑ bands are pushed above the Fermi energy bythe Coulomb interactions and are unoccupied [Fig. 1(b)].Published by the American Physical Society under the terms ofthe Creative Commons Attribution 4.0 International license.Further distribution of this work must maintain attribution tothe author(s) and the published article’s title, journal citation,and DOI.PHYSICAL REVIEW LETTERS 133, 026501 (2024)Editors' Suggestion0031-9007=24=133(2)=026501(6) 026501-1 Published by the American Physical Societyhttps://orcid.org/0000-0002-6612-866Xhttps://orcid.org/0000-0001-7838-7274https://orcid.org/0009-0000-9212-6644https://orcid.org/0000-0002-4634-2722https://orcid.org/0000-0003-3701-8119https://orcid.org/0000-0002-1467-3105https://orcid.org/0000-0002-5327-051Xhttps://orcid.org/0000-0002-3095-3596https://ror.org/02s6k3f65https://ror.org/03vek6s52https://ror.org/026v1ze26https://ror.org/026v1ze26https://crossmark.crossref.org/dialog/?doi=10.1103/PhysRevLett.133.026501&domain=pdf&date_stamp=2024-07-08https://doi.org/10.1103/PhysRevLett.133.026501https://doi.org/10.1103/PhysRevLett.133.026501https://doi.org/10.1103/PhysRevLett.133.026501https://doi.org/10.1103/PhysRevLett.133.026501https://creativecommons.org/licenses/by/4.0/https://creativecommons.org/licenses/by/4.0/The energy separation between the ↑ and ↓ bands is Σ,the exchange energy. The close-to-complete spin polariza-tion implies that Σ is larger than the Fermi energy. Thedichroism disappears rather abruptly at electron densityn ≃ 3 × 1012 cm−2, evidence of a first-order transition to aparamagnetic state [1]. These experimental observations areconsistent with theory which predicts both spin ordering(but not valley ordering) and a first-order phase transitiondriven by subtle corrections to Fermi-liquid theory [6]. Thegoal here is to determine Σ at low n.The sample consists of a MoS2 monolayer sandwichedbetween two hBN layers [14,17,18] [Fig. 1(c)]. Electronsare injected into the monolayer via a gate electrode; theelectron density n is proportional to the applied voltage,with a capacitance calculated from the device geometry. Weperform a quasiresonant, quasilocal PL experiment: Thelaser photon energy is 1.96 eV, just above the excitonenergy, 1.94 eV; the PL is collected from a region withdiameter 500 nm. The laser intensity is chosen such that thesteady-state exciton density is at least 2 orders of magnitudesmaller than n. At these laser powers, photodoping (theinfluence of the laser on n via charge trapping in theenvironment) is both weak and slow. The remainingphotodoping effects (a slight decrease in n on an hour-long timescale) are eliminated by integrating for 1200 s atone n, resetting n to zero, repeating this cycle to cover all n;see Supplemental to Ref. [7]. This is important, as other-wise photodoping can lead to a mismatch between PL andabsorption [19]. The excitation is either σþ or σ− polarized,thereby injecting an exciton with spin ↑ at the K point orspin ↓ at the K̃ point, respectively. The PL is detected withσþ or σ− polarization. Note that, in absorption, only theeigenstates of the system are probed. A magnetic field(perpendicular to the 2DEG) of þ9.00 T is applied with adirection such that only spin-↓ bands are occupied. Theoptical response is plotted as a matrix (Fig. 2): σþ/σ− refersto excitation with σþ, collection with σ−, etc.We focus initially on σþ excitation. At n ¼ 0, there isone PL line in both σþ=σþ and σþ=σ− corresponding to theneutral exciton X0. The dichroism D ¼ ½IðσþÞ − Iðσ−Þ�=½IðσþÞ þ Iðσ−Þ� is 42%. On increasing n, X0 weakens.In σþ=σþ, several trions are observed, yet, in σþ=σ−, thePL is very weak such that D increases to D ≃ 64%at n ¼ 1.5 × 1012 cm−2.We propose that the increase ofD at small n [up to about2 × 1012 cm−2, Fig. 3(c)] is a consequence of a Bir-Aronov-Pikus electron-hole exchange. At n ¼ 0, an exciton injectedinto theK valley can be scattered within its lifetime to the K̃valley by the electron-hole exchange [20–22] [Fig. 1(a)].This reduces D from the high value expected from theselection rules alone. Assuming an exciton lifetime of∼4 ps [23,24] and that the dynamics can be described witha rate equation, the measuredD implies aK → K̃ scatteringtime of ∼6 ps (see Supplemental Material [14]), consistentwith experiments in the time domain [25]. Once a Fermisurface is formed, the spin-↓ electron states at the K̃ valleyare occupied such that the scattering process is inhibited bythe Pauli principle andD increases. This is evidence that therelevant K ↔ K̃ scattering mechanism is electron-holeexchange and that the K̃↓ states become occupied.At low n, three trions are observed in σþ=σþ, labeled T1,T2, and T3 [Figs. 2 and 3(a)]. T1 and T2 are linked: Theyhave similar intensities and linewidths. In σþ=σ−, there isvery weak PL from a trion, labeled T4 [Figs. 2 and 3(a)].The energy of T4 is close to that of T3. However, the ndependence of the T3 and T4 linewidths are quite different[Fig. 3(b)], indicating that T3 and T4 arise from differenttrion species.We turn to σ− excitation. Using again the trion energiesand n-dependent linewidths to identify the trions, in σ−=σþ,T1, T2, and T3 are observed; in σ−=σ−, T4 is observed.Hence, the collection channel and not the excitationchannel determines which trions appear.To proceed, we describe the trions T1…T4 microscopi-cally (see SupplementalMaterial [14]). Themodel applies inthe limit of low density where the Fermi wavelength is muchlarger than the trion size, ∼2 nm [26–28]. (At higher n,the eigenstates are exciton-Fermi sea polarons [1,29–32].)The low-density limit applies to the lowest n used in theexperiment. The electrons have two degrees of freedom: spinSz ¼ � 12and valley τz ¼ � 12(þ 12for K and − 12for K̃).According to the Pauli exclusion principle, the total wavefunction of a trion must be antisymmetric with respect toparticle exchange [33,34]. The two electronswithin the trion(a)(c)(b)FIG. 1. (a) Band structure of monolayer MoS2 showing excitonformation at the K and K̃ points and the intervalley scattering viaelectron-hole exchange. (b) Schematic of the reconstructed bandstructure containing ferromagnetically ordered itinerant electronswith spin ↓. (c) Schematic of the sample design. FLG stands forfew-layer graphene.PHYSICAL REVIEW LETTERS 133, 026501 (2024)026501-2have, therefore, six eigenstates jS; Sz; τ; τzi characterized bythe total spinS, its projectionSz, the valley pseudospin τ, andits projection τz. Four are relevant here:j0; 0; 1; 1i≡ jSdi;j0; 0; 1; 0i≡ jSii;j1; 0; 0; 0i≡ jT0i;j1;−1; 0; 0i≡ jT−i ð1Þand are shownpictorially in Fig. 4. jSdi is the intravalley spinsinglet at the K point; jSii the intervalley spin singlet; andjT0i and jT−i are two spin components of the intervalleyspin triplet.Consider σþ excitation which creates a bright exciton attheK point. The injected electron state is jK↑i. This electronbinds with a second electron to form a trion. Binding to asecondK↑ electron is forbidden by the Pauli principle. If thesecond electron is K↓, the electrons form the intravalleyspin-singlet state jSdi [Eq. (1)]. The second spin can residein the opposite valley, but only spin-↓ electrons are availablein the ferromagnetic state. The antisymmetrized stateformed is ð1= ffiffiffi2p Þ½jK1↑1;K̃2↓2i− jK2↑2;K̃1↓1i�. This stateis not an eigenstate: It decomposes to ð1= ffiffiffi2p Þ½jT0i þ jSii�and gives rise to two lines in the spectrum: one at the jT0ienergy, the other at the jSii energy. Under σþ=σþ, thelowest-energy trion T3 is thereby identified as jSdi; thehigher-energy pair,T1 andT2, are identified as jSii and jT0i.The model explains the observation that T1 and T2 arelinked: The lines arise from recombination of the same state.Switching to σ− excitation, a bright exciton is created atthe K̃ point. The injected electron state is now K̃↓. In thepresence of only spin-↓ electrons, the only trion that can beformed is jT−i. Under σ−=σ−, only T4 is observed. T4 isthereby identified as jT−i.Finally, we analyze the cross-channels. Under σþ=σ−,the bright exciton at the K point is scattered to the K̃ pointby electron-hole exchange. Only spin-↓ electrons areFIG. 2. PL for quasiresonant excitation on gated monolayer MoS2 at þ9.00 T and 4.2 K shown as a matrix: excitation in σþ or σ−,collection in σþ or σ−. Fitting of each spectrum determines the energies of all the lines.PHYSICAL REVIEW LETTERS 133, 026501 (2024)026501-3available such that the only possible trion is jT−i. This isconsistent with the observation of T4 in the spectrum.Under σ−=σþ, the bright exciton at the K̃ point is scatteredto the K point, making a spin-↑ electron available, leadingto the formation of jSdi and ð1= ffiffiffi2p Þ½jT0i þ jSii�, such thatlines T1, T2, and T3 appear in the spectrum, exactly asobserved.Themodel gives a consistent description of the lines in thePL matrix and is consistent with a two-band, spin-↓ferromagnetism. If spin-↑ states were occupied in theFermi sea, then a jSdi-like trion (specifically, j0; 0; 1;−1i)would be observed under σ−=σ−. This is not the case.Furthermore, a doublet corresponding to ð1= ffiffiffi2p Þ½jT0i þjSii� would be observed under σ−=σ−—this is also not thecase. Thus, only the spin-↓ bands in eachvalley are occupied.We now consider the energies of the states (seeSupplemental Material [14]), first, states jT0i and jT−i.In a single-particle interpretation, these two states would besplit by a small Zeeman energy. (Using the spin and valleyg factors [10,35], the single-particle splitting between jT0iand jT−i is −1.03 meV.) This is not the case: jT0i and jT−iare split by a much larger energy, ≃10 meV; see Fig. 3(d).The explanation is that Σ contributes to jT0i but not to jT−i.Subtracting the Zeeman splitting, we find Σ ≈ 11.2�1.4 meV at n ¼ 1.5 × 1012 cm−2. At this density, theFermi energy is 2.6 meV (taking an electron mass of0.7m0 [36]), much smaller than Σ, as required for theconsistency of Fig. 1(b).Second, the splitting between jT0i and jSii arises from anintervalley Coulomb exchange interaction J, which lowersthe energy of the spin-triplet jT0i with respect to the spin-singlet jSii, similar to Hund’s rule in atoms. The splittingbetween T1 ¼ jSii and T2 ¼ jT0i provides us with Jas a function of n [Fig. 3(d)]. We extract J ≈ 5 meV atn ¼ 1.5 × 1012 cm−2, indicating the importance of interval-ley Coulomb exchange scattering, as pointed out in Ref. [6].We comment on the behavior at higher n. First, J: Jdecreases with n [Fig. 4(d)]. The spin-down states belowthe Fermi energy are occupied in the ferromagneticallyordered phase such that they are excluded from the spin-down component of the trion. Conversely, the spin-upstates remain unoccupied such that the spin-up componentof the trion does not depend on n. The overlap between thespin-up and spin-down densities within the trion decreaseswith n and tends to zero at kF ≫ 1=atr, where atr is the trionsize and kF ¼ ffiffiffiffiffiffiffiffi2πnpthe Fermi momentum in the ferro-magnetic phase. This allows us to estimate the trion sizeatr ≈ 1=ffiffiffiffiffiffiffiffiπn0p ≈ 3 nm, a value consistent with previousresearch [26–28]. Here, n0 ≈ 3.5 × 1012 cm−2 is the densitywhere J ≈ 0 meV in Fig. 3(d). Second, Σ: Σ also decreaseswith increasing n [Fig. 3(d)]. The exchange energy of theFIG. 4. Schematic of the trion eigenstates showing in each casethe two electron states and the hole state from which the trion isconstructed. The interpretation of the PL spectra leads to theassignment Si ≡ T1, Sd ≡ T3, T0 ≡ T2, and T− ≡ T4.(a) (b) (c)(d)FIG. 3. (a) PL spectra at n ¼ 1.5 × 1012 cm−2 (atþ9.00 T and 4.2 K) for σþ/σþ excitation/collection and σ−/σ− excitation/collection.(b) Trion linewidths versus n. (c) n dependence of the optical dichroism D for σþ and σ− excitation. (d) Energy splittingΔE versus n.PHYSICAL REVIEW LETTERS 133, 026501 (2024)026501-4ferromagnetic state should decrease with increasing n as theinteractions stabilizing the ferromagnetism become weaker.However, our description of the trions [Eq. (1)] is valid onlyat low n. At higher n, two effects potentially reduce the T2,T4 splitting: the dependence of exchange on density and themany-body effects on the trion energies, such that therelationship between Σ and the T2, T4 splitting is no longerclear. Our determination of Σ applies instead at low nwhereEq. (1) is valid. Finally, D: D with σþ excitation increasesmonotonically with n, and D with σ− excitation has a diparound n ¼ 3.0 × 1012 cm−2 [Fig. 3(c)]. At the highest n,the difference inD probably reflects the paramagnetism. Atintermediate n, the behavior is not understood, but we notethat it is determined from PL intensities which depend onmany factors.A key component of this analysis is the observation ofthe T1 ≡ jSii, T2 ≡ jT0i “doublet” [Fig. 3(a)] not resolvedin previous experiments [1,7]. Here, smaller linewidthsallowed us to resolve the doublet. The doublet is notobserved at every location on the sample. Dividing a region6 × 6 μm2 into pixels, the doublet is observed with aprobability of 20% (see Supplemental Material [14]).There is no obvious correlation, doublet versus no doublet,with the energies, for instance, the X0 energy. It is likelythat inhomogeneities result in these statistical properties.In conclusion, we identify all the PL lines from gatedmonolayer MoS2. We find that only spin-↓ bands at eachvalley are occupied, signaling ferromagnetic order. Atn ¼ 1.5 × 1012 cm−2, we extract from the PL spectra theferromagnetic exchange energy Σ ≈ 11.2� 1.4 meV andthe intervalley Coulomb exchange energy J ≈ 5 meV. Thelarge exchange energy suggests that ferromagnetic orderingshould survive up to tens of degrees Kelvin. This isconsistent with the observation of a pronounced dichroismeven at 30 K [7]. However, at elevated temperatures, theoptical probe is no longer useful on account of phononbroadening of the optical lines—this motivates an inves-tigation with a sensitive magnetometer [37,38].The work was supported by the Ph.D. School QuantumComputing and Quantum Technology (QCQT) of theUniversity of Basel, the Georg H. Endress Foundation, andthe Swiss Nanoscience Institute (SNI). N. L. acknowledgessupport from the Swiss National Science Foundation (SNF)(Project No. P500PT_206917). K.W. and T. 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