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Downloaded by guest on September 26, 2021 lcrnbn fetv as(nuiso h reeeto ), free the of units ε (in mass a effective that For band electron so challenging. have 2DES extremely we clean is 2DES, single- and however, and dilute disorder localization, by very particle hindered not a are phenomena Achieving see, interaction calculations; 7.) Carlo Monte ref. other for e.g., by corroborated state all crystal not Wigner are a to full system sition to electron transition when a 2D predict a (6) calculations For Carlo (4). Monte transition (2DES), ferromagnetic the three-dimensional for a In , energies). Fermi (3D) (equivalently, to Coulomb radius the Bohr of effective ratio the of the units con- with in is system it electron parameter description, interacting dimensionless quantitative the a characterize ordered For to an , venient (2). Wigner a of into elec- condense densities, array state to lower ground is predicted even ferromagnetic At are densities ferromagnet. trons a Bloch electron to a transition namely, low (1), a sufficiently make to at itin- expected disorder-free, system A interaction electron energy. the kinetic erant densities the low over At dominates (1–7). energy many-body in est T – system electron 2D dilute into realistic taken a be of system. phases also electron different must the disorder understanding in of that account role implying the ferromagnetism, spontaneous interaction, the besides of onset at the state insulating ing an when to transition solid, a makes sample Wigner ferro- of pinned a suggestive characteristics, a to current-voltage transition nonlinear spontaneous phases, strongly when a of state the densities, sequence magnetic lower large paramagnetic a a we at find scenario: As above we the energies. bias, with Fermi consistent gate qualitatively to via parameter reach Coulomb density to interaction the electron us the of allows ratio of system the values this large in mass The very well. effective quantum electron AlAs modulation-doped large a interact- electron density to two-dimensional of confined clean, low states system exceptionally and an ground in disorder) the electrons electron (low ing explore an we quality supports Here high that simultaneously. because platform very limited material with Experimen- been a system of occur. has however, absence should regimes, the electrons, of these (Wigner) to of a access array to tal ordered transition another an densities, solid, lower tran- even ferromagnetic At (Bloch) in a sition. enhancement to gained leading the energy, energy exceed (Fermi) kinetic should exchange the spins the electron lowered, the aligning is that by density expected Sarachik) P. been electron Myriam long and the has Kivelson, it as disorder, A. of Steven Ceperley, absence electron David low-density the by interacting, In reviewed an system? of 2020; states 28, ground August the review are for What (sent 2020 3, November Pfeiffer, N. L. by Contributed a Shayegan M. and .S Hossain S. M. system electron a two-dimensional in ferromagnetism spontaneous of Observation www.pnas.org/cgi/doi/10.1073/pnas.2018248117 eateto lcrclEgneig rneo nvriy rneo,N 08544 NJ Princeton, University, Princeton Engineering, Electrical of Department stedeeti osat and constant, the is t antcpoete aeln entpc fgetinter- great of topics been and long system have electron properties dilute magnetic interacting, its an of state ground he r s a ausaetpclywl eo hs required those below well typically are values | r .K Ma K. M. , ferromagnetism s a,1 (me = r s | r s inrsolid Wigner xed 6 olwdb nte tran- another by followed 26, exceeds upse 5 n hnapaewith phase a then and 35, surpasses 2 r /4π s h vrg neeeto distance interelectron average the , a .A ilgsRosales Villegas A. K. , ~ r 2 | s εε magnetotransport exceeds n 0 )/(π r ste2E est.For density. 2DES the is s > n Teepredictions (These 35. ) 1/2 ' 8 oee,our However, 38. where , r | s r s ' r endas defined , s 7 preced- 27, sas the also is a m .J Chung J. Y. , sthe is doi:10.1073/pnas.2018248117/-/DCSupplemental at online information supporting contains article This 1 ste2E ostruhteMTat MIT the finite through remains goes and 2DES (6) the results as calculations’ Carlo Monte the lows of density a (m to 2DES responds GaAs a in example, hwdta h pnssetblt osntdvreu othe to up diverge not does achieved susceptibility experimentally data highest AlAs the The (17). that well showed quantum AlAs to (4.5-nm-wide) confined 2DES a narrow thin) a on However, very measurements isotropic, 16). (single-valley, in (15, ideal corroborated nearly at MIT not the were diverges conclusions with susceptibility these coincides depen- spin divergence temperature the the the that that Si/SiO in In claims role 11–22). also a (9, conductivity plays of systems dence carrier systems. 2D carrier polarization spin/valley in 2D the that enigmatic dilute revealed the experiments in Numerous of of (MIT) context subject the transition a in metal–insulator became controversy and 2DESs interest interacting intense of polarization spin the large where not wells, Si/SiO is a holes at 2D of inter- polarization spin–orbit spin strong of (11–13). the straightforward extraction of the because near (m action, formation partly systems solid However, hole Wigner 2D (10). a GaAs of In hints 9). (8, large attain 0.4), to difficult very ulse ne the under Published interest.y York. competing New no of declare University authors City The University; the Stanford of S.A.K., College City Urbana-Champaign; The at M.P.S., and Illinois of University M.S.H. D.C., and Reviewers: measurements; the paper. with y the helped wrote K.A.V.R. M.S. and and M.K.M., M.K.M. tools; research; reagents/analytic data; new analyzed performed contributed M.S.H. M.S. M.S.H. and K.W.B., research; K.W.W., L.N.P., Y.J.C., designed K.A.V.R., M.S.H. contributions: Author owo orsodnemyb drse.Eal [email protected] rshayegan@ or [email protected] princeton.edu.y Email: addressed. be may correspondence whom To a ecnrle ytnn h lcrn’vle ereof degree valley electrons’ the ferromagnetism tuning of by freedom. onset controlled the be that (ferromagnetic) find a can possibly also phase, We solid solid. pinned sug- Wigner a characteristics is whose it state that insulating gest strongly of and a state signs to ground observe ferromagnetic then we a density, to transition the spontaneous lower a further inter- an and we disorder to As by high state action. stabilized metallic from is that a Starting phase from insulating lowered. transition exotic a is observe sequence density we a the densities, reveal as system transitions on electron of data two-dimensional experimental clean Our a electron– phenomena. understanding in interaction interest electron fundamental of sys- are electron tems dilute interacting, low-disorder, of states Ground Significance hr r te eiodco DS ihlarge with 2DESs other are There a .N Pfeiffer N. L. , r 2 s rMZOZOitraeo nAA quantum AlAs in or interface MgZnO/ZnO or auscnb ece oeesl,adi fact, in and easily, more reached be can values NSlicense.y PNAS r s a,1 auscnb ece 1–3.Indeed, (14–23). reached be can values .W West W. K. , n 4 = 0 = .6 r s × .067 . (' y 10 a .W Baldwin W. K. , https://www.pnas.org/lookup/suppl/ .Ised tcoeyfol- closely it Instead, 10). r 8 s and NSLts Articles Latest PNAS cm ' r s 8 2 −2 ε ' 1) nvr recent very In (17). DS,teewere there 2DESs, 13), = y 35 hc sindeed is which , eereported were r s r s a ' , ' m 26 | 9 e.g., , f7 of 1 cor- and '

APPLIED PHYSICAL SCIENCES studies which revisited the MIT problem in Si/SiO2 2DESs (21, occupy two in-plane conduction-band valleys with longitudinal 22), it was also concluded that there is no divergence of the spin and transverse effective ml = 1.1 and mt = 0.20, lead- 1/2 susceptibility, even at densities lower than those reached in refs. ing to an effective in-plane band mass m equal to (ml mt ) = 15, 16. Therefore, there has been no experimental evidence for 0.46 (25). We refer to these valleys, whose electron occu- a transition to full in a 2DES at zero magnetic pancy we can control via the application of in-plane, uniaxial field prior to this work. strain, by the direction of their major axis in the plane, [100] and [010] (SI Appendix, section I). The large effective mass, Experimental Results combined with the exceptionally high purity of the samples, We report here the observation of a spontaneous transition to allows us to achieve very large values of rs while maintaining a a fully magnetized ground state and experimentally construct a strongly interacting system. This is evinced by our observation comprehensive ground-state phase diagram for the interacting of signatures of fractional quantum Hall states in a perpen- 2DES as a function of electron density (and rs ), as summa- dicular magnetic field a at density as low as n = 1.20 (rs ' rized in Fig. 1A. Our material platform is a very low disorder, 45) (see, e.g., Fig. 2 for data at n = 1.70 and relevant discus- dilute 2DES confined to a modulation-doped, 21-nm-wide AlAs sion); throughout the manuscript, we give the 2DES densities in quantum well (ε = 10) (24). The 2D electrons in our sample units of 1010 cm−2.

Spin Spatial r A CDs n = 1.30 r = 42.8 50 38 30 26 5 s 1.6 Paramagnet Metal 1.60 38.6 4 r 34.5 1.2 sc ) 26 2.00 -2

cm 3 Insulator 2.10 10 33.7 (T) 30 s 0.8 r M B

n (10 2 34 Ferromagnet 38 nc Wigner solid 32.2 0.4 2.30 1 50

2.60 30.3 0.0 0 (a. u.) 2.75 29.5 B [100] E 20 B < B B = B B > B R || M || M || M 28.7 E E E 2.90 16 rsc E F EF

E 26.9 /gm F E > E * Z F 3.30 EZ = EF 12 m

E < E * Z F g(E) 25.1 3.80 = g =

χ 8 0.2 / *

) 24.1

I 4.10 χ Ω B M n (M B|| 4 c [100] R V 0.1 0 0 12-3 -2 -1 0 1 2 3 01234 10 -2 B|| (T) B|| (T) n (10 cm )

Fig. 1. Highlights of our experimental findings. (A) An experimental phase diagram, suggested by our study, for the ground states of an interacting 2DES at zero magnetic field as it is made progressively more dilute. The sample is paramagnetic at high densities, and its resistance vs. temperature dependence switches from metallic to insulating below a density of 3.2 (in units of 1010 cm−2). As the density is further lowered, the 2DES makes an abrupt transition to a fully magnetized state below 2.00; this is signaled by a vanishing of BM (B–D). At even lower densities, when n . 1.70, the sample turns highly insulating and exhibits nonlinear I-V, suggestive of a pinned Wigner solid. (B) Our experimental technique to measure spin polarization of the 2DES, demonstrated using data taken at n = 2.63 and density-of-states diagrams. The sample resistance R[100], measured along the [100] direction (Inset), increases as the magnetic field Bk applied parallel to the 2DES plane polarizes the electron spins and then saturates once the system is fully magnetized at the field BM. Above BM, the Zeeman energy (EZ ) exceeds the Fermi energy (EF ); g(E) is the . (C) Resistance (R[100]) vs. Bk data taken at different 2DES densities (R[010] data show a similar behavior). Traces are shown for both up- and down-sweeps of Bk and are offset vertically. The vertical arrows mark the positions of BM above which the resistance saturates. They are placed symmetrically with respect to Bk = 0 and are based on the average of four values of BM (for up- and down-sweeps and +Bk and −Bk). For n ≤ 2.00, the traces are flat, and there is no hint of a BM.(D) BM and (E) the deduced spin susceptibility plotted as a function of density (lower axis) and rs (top axis). All measurements were performed at T = 0.30 K.

2 of 7 | www.pnas.org/cgi/doi/10.1073/pnas.2018248117 Hossain et al. Downloaded by guest on September 26, 2021 Downloaded by guest on September 26, 2021 osi tal. et Hossain µ Since rises. longer no E When resistance the 27). and (26, tized, (E electrons energy screening 2D Fermi the the the in by reduction potential B a disorder of the because of resistance higher resistance (E has the energy whose that Zeeman 27) of is addition as 26, 1B, increases Fig. 12–22, 2DES in the (9, of depicted a technique as of principle, well-established function working a a is as This resistance 2DES field the magnetic measuring via the 1B, ization to (Fig. axis transferred direction see vector are [010] wave details, Fermi electrons the longer along the whose valley is of the all i.e., valley, where [010] case the for Ferromagnetism. Spontaneous of Signatures as low of as dependence Temperature densities at states Hall tum in plateaus quantized nearly by to close (B field magnetic dicular where 2. Fig. A B F k R[100] (2π = ece ufiinl ag au (B value large sufficiently a reaches steBh antn(g magneton Bohr the is 0 0 0 0 0 anttasotdt near data Magnetotransport g n ÷2 ÷2 ÷5 3- 10123 2 1 0 -1 -2 -3 c 1/3 stebn au fteefcieLand effective the of value band the is = ~ 1/3 2 1/3 0adcniusat continues and 2.00 V n I .W rb h pnpolar- spin the probe We I). section Appendix, SI 1/3 )/m B B k ┴ ehave we , ple aallt h apepae(i.1 (Fig. plane sample the to parallel applied 1 1 1 1 F ⊥ B 1 B n =2.50 ,te2 lcrn eoeflymagne- fully become electrons 2D the ), o v este sidce i nt f10 of units (in indicted as densities five for ) ┴ M (T) 1.70 2.10 2.00 2.20 R B xy ν = =1 ae at taken , k R  xy oaie h lcrn hog the through electrons the polarizes n 1 2π tteepce aus inln eldvlpdqatmHl ttsa hs lig.Tepeec ffatoa quan- fractional of presence The fillings. these at states Hall quantum well-developed signaling values, expected the at 1 Z 1 1 µ 2 = < n n .Teprilyplrzd2DES polarized partially The ). B ~ = = n 2 c n 0dmntae h xrml ihqaiyo h DSadtepeec feeto–lcrnitrcin ( interaction. electron–electron of presence the and 2DES the of quality high extremely the demonstrates 1.70 h ogtdnlrssac rcsso epmnm tLna ee ligfactors filling level Landau at minima deep show traces resistance longitudinal The .  n norAA DS.Note 2DES). AlAs our in c 1/3 ogtdnl(R Longitudinal (A) . = gm 1/3 n Bottom, 1/3 0 hwn httehseei aihsa ihtmeaue (T temperatures high at vanishes the that showing 1.70, 1/3 , M is,w rsn data present we First, othat so ) 0.1 MΩ frmore (for Inset) E B Z ´ e R = xy g 0 0 0 0 0 g E atrand factor [100] µ 3- 10123 2 1 0 -1 -2 -3 Z B 1/3 1/3 1/3 equals B al(R Hall (B) and ) 1/3 and 10 [1] B). -h/e cm -3h/e −2 2 1 1 1 nFg 1C finding Fig. as principal that in The is 17). experiments our (6, in increases susceptibility as quickly n h eue ucpiiiya ucinof when function 2DES, interacting a dilute, as a for susceptibility deduced sus- the spin and B), 1 the 1 deduce to Fig. (Fig. us In allows ceptibility. above and 2DES discussed the of magnetization As 1C. Fig. field density-dependent a For Inset). show we field. magnetic for magnetized fully is 2DES measuring susceptibility the from for factor enhancement by the values enhanced the ing susceptibility, spin the 2DES, to related directly that sity enhancing density, 2DES the lower .A h est slwrd rnucdhseei near hysteresis pronounced a lowered, is density the As ). 1 B 2 1 ┴ i.1 Fig. (T) xy n =2.50 n 1.70 2.10 2.00 2.20 B ν eitne,maue at measured resistances, ) c =1 M g Fg 1 (Fig. , C–E R 1 m sivreypootoa oteproduct the to proportional inversely is 1 3h/e 1 1 χ [100] eoecmltl needn of independent completely become and , n h/e ∗ n hnsdel aihsblwaciia den- critical a below vanishes suddenly then and /χ, ≥ atr h vlto ecie bv.I i.1C Fig. In above. described evolution the capture 2 .Ti bevto togysget htthe that suggests strongly observation This C–E). 2 vs. 1/3 2.10, χ B 1/3 1/3 B M 1/3 r yial nacdb neato.Denot- interaction. by enhanced typically are D k noreprmns efidta swe as that find we experiments, our In . R ple ln h 10 ieto Fg 1B, (Fig. direction [100] the along applied and 2 [100] h/e B E M C T B g ' ie with rises = eso lt ftemeasured the of plots show we n n M ∗ erae ail,sgaigafast- a signaling rapidly, decreases 0K). 0.70

, R 0K hw safnto fperpen- of function a as shown K, 0.30 ≤

akdb etclarw in arrows vertical by marked , xy slwrdblw21,tetraces the 2.10, below lowered is 0 0 0 0 m n n =1.70 ∗ c and , nteasneo napplied an of absence the in n 1 1 1 1 11 -1 slowered, is B ν NSLts Articles Latest PNAS χ k = -h/e ∗ B ,accompanied 1/3, and 1 n auae above saturates and na interacting an In χ. n a determine can one , M B 2 χ ┴ 0.30 K ∗ 0.50 K 0.40 K 0.70 K (T) 0 B inl h full the signals /χ n k B sexpected As . B h/e ⊥ gm ihn sign no with = M = 2 g ν decreases =1 ∗ emerges 0 hc is which m | 1 1 1 ∗ f7 of 3 /gm B M C 2

) h/e

APPLIED PHYSICAL SCIENCES of a BM whatsoever. A most logical interpretation of this behav- We believe it is the exceptionally high sample quality that ior is that for n ≤ 2.00, the 2DES is fully magnetized at Bk = 0, allows us to explore the interaction phenomena even in the expected for spontaneous ferromagnetism. The critical density extremely dilute case and observe a spontaneous magnetization. nc = 2.00 corresponds to rs = 34.5. The high quality of our 2DES can be inferred from Fig. 2 A It is noteworthy in Fig. 1C that the trace taken at a slightly and B traces. At Landau level filling factor ν = 1 and 1/3 (ν = larger density, n = 2.10, exhibits a pronounced hysteresis near nh/eB⊥), we observe clear indications of integer and fractional Bk = 0. In Fig. 2 A and B we show R[100] and Rxy (Hall quantum Hall states, signaled by resistance minima in R[100] resistance) traces taken as a function of perpendicular mag- minima and reasonably well-quantized plateaus in Rxy even at netic field, B⊥. Hysteretic features near zero magnetic field n = 1.70 (rs ' 38). (We observe R[100] minima at ν = 1 and are clearly visible here too and are most pronounced when 1/3 down to n = 1.20 [rs ' 45].) The observation of fractional n is close to nc . These features suggest a possible forma- quantum Hall states down to such small densities is particularly tion of magnetic domains, as expected near a magnetic tran- noteworthy as it attests to the presence of interaction even when sition. Notably, as seen in Fig. 2C, the hysteresis vanishes at the 2DES is extremely dilute. high temperatures; e.g., at n = 1.70, the hysteresis disappears We would like to highlight two points before closing this at T ' 0.70 K. section. First, note that Eq. 1 assumes that gm (i.e., the spin sus- We note that the technique we use to probe the enhance- ceptibility, χ) is a constant, independent of Bk. This assumption ment and divergence of the spin susceptibility is similar to what is consistent with the conclusions of previous studies (17). While was used in many previous studies of the spin polarization of our determination of χ∗ relies on this assumption, we emphasize dilute 2D carriers (9, 12–22, 26, 27). None of the previous that our main conclusion, namely, that the 2DES undergoes a studies, however, reported a sudden and complete disappear- spontaneous spin polarization below a critical density, is based ance of the magnetoresistance as a function of Bk (implying a on the vanishing of BM and does not depend on such an assump- vanishing BM and full magnetization) as the 2DES density is tion. Second, as shown in SI Appendix, section IV, the value of reduced below a critical value. In some of these studies, e.g., R[100] (which is constant and independent of Bk) at n = 2.00 is in the work of ref. 16 on Si/SiO2, nonzero magnetoresistance close to the saturated value of R[100] at large Bk at n = 2.10. was observed down to the lowest achievable densities, and a This observation confirms our conclusion that the 2DES is fully plot of BM (deduced from fitting the magnetoresistance to an spin polarized at n = 2.00 at Bk = 0, just as it is at n = 2.10 at empirical expression) vs. n was extrapolated to lower densities large Bk. to conclude that BM vanishes at a finite density (corresponding to an rs ' 9). As shown in ref. 17, such extrapolation can be mis- Is There a Link between the Spontaneous Ferromagnetism and the leading, and recent new measurements (21, 22) indeed conclude MIT? As mentioned in the opening paragraphs, a possible link that there is no magnetic instability or complete spin polarization between the spin polarization and MIT in dilute 2D carrier sys- at zero magnetic field in the Si/SiO2 system even at densities tems has been debated heavily (9, 11–22). In Fig. 3A we show the lower than those achieved in ref. 16 (see, e.g., point 7 on p. 10 temperature dependence of the sample resistance measured at of ref. 21). B = 0. At the highest n, the behavior is metallic, with resistance

107 n = AB 1.2 1.10 C rs 1.20 1.5 107 1.30 1.50 Paramagnetic metal 6 1.70 n = 3.2 27 10 2.0 1.80 MIT 1.90

) 2.00

) 2.5 6 Ω 10

Ω 2.10

( Paramagnetic insulator 2.50 105 n = 2.0 35 [010] 3.0 c dV/dI ( R 3.2

3.5 5 10 Ferromagnetic insulator 104 4.0 nWS = 1.7 38 4.5 5.0 103 104 0.3 0.5 0.7 -40 -20 0 20 40 Ferromagnetic Wigner solid 10 -2 T (K) IDC (nA) n (10 cm )

Fig. 3. Summary of temperature-dependent and nonlinear I-V data and the phases that we observe as a function of n (or rs). (A) R[010], resistance measured along [010], as a function of temperature exhibiting metallic transport for n ≥ 3.2, and a switch over to an insulating behavior for smaller n. R[100] data show a qualitatively similar behavior. (B) Differential resistance (dV/dI) measured along [100] at T = 0.30 K, plotted against the DC bias current, showing increasing nonlinearity at small biases when n . 1.70. (C) Illustration of different phases observed in our 2DES. Starting as a paramagnetic metal at large n (small rs), the 2DES undergoes a cascade of transitions to a paramagnetic insulator, ferromagnetic insulator, and finally to a ferromagnetic solid as the density is lowered.

4 of 7 | www.pnas.org/cgi/doi/10.1073/pnas.2018248117 Hossain et al. Downloaded by guest on September 26, 2021 Downloaded by guest on September 26, 2021 rc n a ese narneo este between densities of range a in with seen this be to rise can unique linear not of and is unusual trace function behavior an This a . by as before fields followed rise higher fields, rapid low very very a at shows resistance the MIT, traces (as magnetoresistance the 2DES in near anomalies the certain of observe do of magnetization we disappearance full sudden the the from and inferred MIT the between hole 2D GaAs clean very for (10). except systems MIT, the than for larger systems much carrier is that the MIT noteworthy with the also disagrees observe is but we It 17 16. ref. ref. of of data conclusions the with consistent again is n est,wietevle-eeeaecs eurstesals est.(E density. smallest the requires case valley-degenerate the while density, of positions the mark arrows vertical The degenerate. are valleys occupied to two respect the when densities 2DES different 4. Fig. at density The is seen. switches is behavior behavior the insulating which opposite, den- lowest an the at sities, while temperature, decreasing with decreasing osi tal. et Hossain case. (50%) valley-degenerate the toward 100%) and (0 cases single-valley from move smallest the has 2DES B the on density, occupancy fixed valley a the At of effect the demonstrating densities C AB M ) c ewudlk oepaieta hl ed o e link a see not do we while that emphasize to like would We B 2 = nrae n ece t aiu au hntevlesaeeulyocpe.Tesnl-alycs ece h ermgei rniina h largest the at transition ferromagnetic the reaches case single-valley The occupied. equally are valleys the when value maximum its reaches and increases 3- -1 -2 -3 R (a. u.) M [100] 1.73 1.63 1.50 1.87 1.40 2.20 2.00 3.79 4.00 2.90 2.75 n =1.18 3.32 n eitnevs. Resistance (D) density. of function a as plotted 2DES valley-degenerate the of susceptibility spin deduced the and h ne ftetasto ofl antzto.This magnetization. full to transition the of onset the .0, MIT otoln h pnplrzto n ermgei rniinb uigtevle cuac.( occupancy. valley the tuning by transition ferromagnetic and polarization spin the Controlling B k ' n r ae nteaeaeo w ausof values two of average the on based are and at 1C, Fig. in seen As 3.2 B || 0123 (T) n MIT ' B 3. M n r 2 hnol n aly(ihr[0]o 00)i cuid seetostase rmoevle oteother, the to valley one from transfer electrons As occupied. is [010]) or [100] (either valley one only when s 3 = (r eotdi te 2D other in reported B s M utaoethe above just .30, ' nFg 1C Fig. in r ,wl above well 27), s C

* * * ' χ / χ = g m /gm BM (T) B 27 0.0 0.4 0.8 1.2 1.6 B ' 12 16 M 0 4 8 M 3.60 twhich at (for 01234 mre ihvria ros.Tevle cuac sgahclyidctdo h right. the on indicated graphically is occupancy valley The arrows). vertical with (marked data), B +B k and B at k n n k ) n (10 and c(2V) c(2V) B M lte gis h alyocpny hwn la nraein increase clear a showing occupancy, valley the against plotted n ttelws este,i ssrnl olna.A demon- As nonlinear. strongly is it in densities, strated lowest the at below and lowered densities is density At the data. additional and I rpryo u DSa eylwdniis i.3B Fig. densities. low ( very resistance at ferential 2DES our of property Solid. Wigner of Hints in data Transition. the Ferromagnetic of the interpretation our in features lous Si/SiO for recently below erates of slope the in n change a is feature thy ' iu tde neteeydlt Dcrirssesat systems carrier 2D dilute extremely Pre- on temperatures. studies high at vious vanishes and temperature of function −B DC 10 MIT .Aohrnotewor- Another S4). Fig. Appendix, (SI MIT the near 3.05, k cm .For ). see ; tapasta the that appears it : -2 ) n o esrmn details measurement for V–VII, sections Appendix, SI ≤ i.S11 Fig. Appendix, SI 0 h rcsaeflt n hr sn ito a of hint no is there and flat, are traces the 1.40, n MIT ulttvl iia etr a reported was feature similar qualitatively A . 2 D DS 2) ewl eunt hs anoma- these to return will We (22). 2DESs dV E eadesaohrfundamental another address we 3B, Fig. In

eitnevs. Resistance A) B (T) M -1 R [100] (a. u.) 0.0 0.2 0.4 0.6 0.8 /dI B n =2.00 M safnto fapidD current DC applied of function a as ) rpwt oeigtedniyaccel- density the lowering with drop 01 0 06 02 0 20 40 60 80 100 n 020406080100 B c , M I hyaepae ymtial with symmetrically placed are They . hsnniert sastrong a is nonlinearity this , -V n -1 B ≥ n =1.63 B lwybcmsnonlinear, becomes slowly k B n M B aatknat taken data || NSLts Articles Latest PNAS k c % [010] 01 (T) % [100] , vs. aatkna he 2DES three at taken data dV n /dI lti i.1D Fig. in plot -1 nepeainof Interpretation n =1.40 slna.As linear. is T ipasdif- displays = 01 n =2.90 B 0Kfor K 0.30 M B M (B . | B 2.00 1.63 1.40 2.20 swe as f7 of 5 0 = and at

APPLIED PHYSICAL SCIENCES (10), or at very small Landau level fillings (28), have concluded interaction is playing a dominant role. An alternative interpreta- that the nonlinear I -V suggests the depinning of a Wigner tion of our data is then as follows. It has been shown theoretically solid which is pinned by the ubiquitous disorder potential. This that a direct first-order transition between a metallic Fermi conclusion has been corroborated by numerous experimental state and a Wigner solid is forbidden in 2D and that there should studies, including microwave resonance and other measurements be one or more exotic microemulsion phases in an intermediate (29–32). It is possible that in our 2DES, too, the nonlinear I - density range (20, 35). It might be that the ferromagnetic tran- V signals the formation of a pinned Wigner solid at rs & 38 sition we observe is occurring in such a microemulsion phase. (n . 1.70). The anomalous features seen in the resistance vs. Bk data near An intriguing observation in our experiments is that the frac- the MIT (Fig. 1C and SI Appendix, Fig. S4) could be mani- tional quantum Hall states still show up in a density regime festations of such exotic intermediate phases. Also, given the where the longitudinal resistance at B = 0 is extremely large anisotropy of the effective mass in our system and that the resis- 2  h/e and the 2DES exhibits a strongly insulating temperature tance we measure in this phase is anisotropic (R[100] < R[010]), dependence. The resistance is also extremely large at ν = 1/3, it is possible that the intermediate phase is nematic or smec- but remarkably, its value at ν = 1/3 decreases as we lower the tic as proposed theoretically (20, 35). Moreover, the theory of temperature, as expected for a fractional quantum Hall liquid refs. 35, 36 concludes that the Wigner solid phase stabilized at state. The notion that a pinned Wigner solid at B = 0 would the lowest densities has ferromagnetic order. This scenario is make a transition to a correlated liquid state in a perpendicular consistent with our data: we see BM = 0 even below n ' 1.70 field might sound counterintuitive but in fact is consistent with where we infer a transition to a pinned Wigner solid phase in conclusions implied by the theoretical work of Zhao et al. (33). our 2DES.

Phase Diagram for an Interacting 2D Electron System. The obser- Role of the Valley Degree of Freedom in the Spin Polarization. vations of MIT, spontaneous magnetization, and nonlinear I -V Another important highlight of experiments is captured in Fig. 4 response as a function of n (or rs ) lead us to suggest an exper- where we illustrate how the valley polarization impacts the fer- imental phase diagram for the ground states of our interacting romagnetic transition. How we tune the valley occupancy is 2DES, as highlighted in Fig. 1A. The ground states include four described in SI Appendix, section I. In Fig. 4A we show a plot distinct phases as a function of decreasing n (i.e., increasing rs ), similar to Fig. 1C except that here the valleys are degenerate. as schematically represented in Fig. 3C: 1) paramagnetic metal, We also show plots of the measured BM and the deduced spin 2) paramagnetic insulator, 3) ferromagnetic insulator, and 4) susceptibility as a function of n in Fig. 4 B and C. Qualitatively ferromagnetic Wigner solid. similar to Fig. 1 C–E, when n is lowered, BM decreases rapidly as susceptibility increases. However, in the valley-degenerate case, Interpretation of the Ferromagnetic Transition. A simple but possi- the critical density below which BM vanishes is nc(2V ) = 1.40 bly na¨ıve interpretation of the observed ferromagnetic transition instead of nc(1V ) = 2.00 of the single-valley–occupied 2DES we observe is that it is a Bloch transition (1). As alluded to (Fig. 1 C–E). This finding is consistent with previous mea- in the Introduction, in 1929, suggested that itiner- surements (at much higher densities) (37, 38) and theoretical ant electrons should spontaneously magnetize at sufficiently low calculations (39) which concluded that the enhancement of the densities. Our observation of the spontaneous ferromagnetism is spin susceptibility, at a given density, is smaller for a two-valley highly reminiscent of such a phenomenon. A transition at lower system compared to a single-valley one. The phenomenon can densities to a state that we identify as a Wigner solid would also be attributed to the modification of the exchange energy in the make qualitative sense in this simple picture. Such a picture is system because of the extra (valley) degree of freedom. also qualitatively consistent with the Monte Carlo calculations The tunability of the valley occupancy allows us to investigate of ref. 6 which were done for a disorder-free 2DES, although the ferromagnetic transition as a function of partial valley occu- there are quantitative discrepancies. For example, the value of rs pancy. As shown in Fig. 4 D and E, BM is largest when the valleys (' 35) at which we observe the ferromagnetic transition is larger are degenerate. As we lift the degeneracy and move toward valley than the value (' 26) predicted by ref. 6. So is the value of rs polarization, BM starts to drop and eventually attains its smallest (' 38) above which we see signs of a possible Wigner solid for- value when the electrons are completely valley polarized. This mation (ref. 6 predicts rs ' 35). These quantitative discrepancies means that the spin susceptibility is maximum when the 2DES is might come from the fact that the calculations are performed for valley polarized and minimum when the valleys are degenerate, an ideal 2DES, with no disorder, zero electron layer thickness, for a given density. Thanks to this interplay between the spin and and an isotropic effective mass. In our 2DES, on the other hand, valley degrees of freedom, we can control the onset of the ferro- the electrons have a nonzero layer thickness and occupy a valley magnetic transition via tuning the valley occupancy, opening up with an anisotropic effective mass. Both the finite electron layer exciting avenues to integrate and valleytronics in the thickness and the mass anisotropy are known to reduce the inter- same device. action strength and the enhancement of the spin susceptibility at a given rs (34) and can move the onset of full magnetization to Data Availability. All study data are available in the main text and larger rs . SI Appendix. However, there is an important caveat. As clean as our sample ACKNOWLEDGMENTS. We acknowledge support through the US might be, there is some finite disorder because of the presence Department of Energy Basic Energy Science (Grant DEFG02-00-ER45841) of ionized . The disorder is indeed the likely cause for for measurements, the National Science Foundation (Grants DMR the 2DES exhibiting an insulating behavior below nMIT . On the 1709076, ECCS 1906253, and MRSEC DMR 1420541), and the Gordon other hand, our observation of a spontaneous ferromagnetism and Betty Moore Foundation’s Emergent Phenomena in Quantum Systems Initiative (Grant GBMF9615 to L.N.P.) for sample fabrication and and the fractional quantum Hall features that we observe at nc characterization. We also thank D. M. Ceperley, H. D. Drew, J. K. Jain, and even lower densities (Fig. 2) provide strong evidence that S. A. Kivelson, M. A. Mueed, and M. P. Sarachik for illuminating discussions.

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