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1 DOI: /NNANO Quantum Hall effect in black phosphorus two-dimensional electron system Likai Li, Fangyuan Yang, Guo Jun Ye, Zuocheng Zhang, Zengwei Zhu, Wen-Kai Lou, Xiaoying Zhou, Liang Li, Kenji Watanabe, Takashi Taniguchi, Kai Chang, Yayu Wang, Xian Hui Chen * and Yuanbo Zhang * * zhyb@fudan.edu.cn, chenxh@ustc.edu.cn Content 1. Carrier density induced by the graphite back gate. Field effect mobility 3. Measurement of the quantum lifetime of holes 4. SdH oscillations at electron doping 5. Contact resistance of black phosphorus FET 6. Measurement of QH energy gaps 7. Reported values of hole effective mass 8. Transport measurement in pulsed magnetic field 9. Calculation of the spin splitting 10. Additional analysis of the coincidence condition in a tilted magnetic field 11. References NATURE NANOTECHNOLOGY 1

2 DOI: /NNANO Carrier density induced by the graphite back gate Supplementary Figure 1 Carrier density nn HH measured as a function of the gate voltage VV gg. Data were obtained from sample BP77. nn HH values were obtained from Hall measurement. Line fit of the data (black line) yields a gate capacitance of CC gg = ee cm VV 1. Such a gate capacitance implies an hbn thickness of dd ~ 5 nm, if the parallel-capacitor model, CC gg = εε 0 εε rr /dd, is assumed. Here εε 0 is vacuum permittivity, and εε rr ~ 4 is the relative permittivity of hbn. NATURE NANOTECHNOLOGY

3 DOI: /NNANO SUPPLEMENTARY INFORMATION. Field effect mobility Supplementary Figure Sample transfer characteristics at VV gg < 00 at room temperature and low temperature. Data were obtained from sample BP77. Field effect mobility μμ FFFF of the hole carriers was extracted from the slope of the line fit to the transfer characteristics (broken lines). μμ FFFF was estimated to be 90 cm V 1 s 1 and 7100 cm V 1 s 1 at TT = 300 K and TT = 4 K, respectively. These values are higher than the Hall mobilities measured under similar conditions. We attribute the discrepancy to the artificial inflation of μμ FFFF with respect to μμ HH as a result of μμ HH ss dependence on gate doping (see the Supplementary Information of ref. for a detailed discussion). NATURE NANOTECHNOLOGY 3

4 DOI: /NNANO Measurement of the quantum lifetime of holes Supplementary Figure 3 Measurement of the quantum lifetime of holes. a, Magnetoresistance (subtracted by a smooth background) as a function of magnetic field recorded at varying hole doping levels (VV gg < 0). Curves are shifted vertically for clarity. Data were obtained from sample BP77 at TT = 0.3 K. SdH oscillations start to emerge at a critical magnetic field of BB cc ~ T. b, Dingle plots of log [ RR xxxx sinhλλ(tt)/(λλ(tt)rr 0 )] versus 1/BB, where λλ(tt) = ππ kk BB TTmm /ħeeee, and RR 0 is the sample resistance at B = 0. The SdH oscillation amplitude RR xxxx is extracted from a. Line fits of the Dingle plots (solid lines) yield an average quantum lifetime of 360 fs for the three hole doping levels under investigation. y-intercepts close to 4 were obtained for all three sets of data, indicating the good quality of the Dingle plots S1. Meanwhile, a transport lifetime t 1 ps can be obtained from the Hall mobility, which gives a t / q ratio of ~ 3. Similar t / q ratio has been reported in graphene, where it was found that the charged impurities close to the sample were the main source of scattering S. Similar scattering mechanism may be at play in black phosphorus DHG. 4 NATURE NANOTECHNOLOGY

5 DOI: /NNANO SUPPLEMENTARY INFORMATION 4. SdH oscillations at electron doping Supplementary Figure 4 SdH oscillations at electron doping. Magnetoresistance (subtracted by a smooth background) is shown here as a function of magnetic field recorded at varying electron doping levels (VV gg > 0). Curves are shifted vertically for clarity. Data were obtained from sample BP77 at TT = 1.5 K. The critical magnetic field, BB cc ~ 9 T, marks the onset of SdH oscillations. Such a critical field is much lower than the value reported previously (15 T; ref. ), and indicates a significant improvement on the electron mobility in our sample. NATURE NANOTECHNOLOGY 5

6 DOI: /NNANO Contact resistance of black phosphorus FET Supplementary Figure 5 Contact resistance of black phosphorus FET. Two-terminal (red) and four-terminal resistance (black) is measured as a function of VV gg on sample BP77. The difference between the two resistances gives an order-of-magnitude estimate of the contact resistance. Data were obtained at TT = 1.5 K. The large contact resistance at low hole doping levels (> 10 kω for V < VV gg < 0 V) makes it difficult to precisely measure the quantized RR xxxx at νν = 1 and in high magnetic fields (Fig. b). Meanwhile, the contact resistance stays above 0 kω at electron doping (VV gg > 0 V). Such a large contact resistance is probably due to the high Schottky barrier for electrons 1, and may have contributed to the fact that no quantum Hall effect was observed on the electron side. 6 NATURE NANOTECHNOLOGY

7 DOI: /NNANO SUPPLEMENTARY INFORMATION 6. Measurement of QH energy gaps Supplementary Figure 6 Measurement of QH energy gaps and an estimation of the effective g-factor at νν = 33 and 44. Magnetoresistance measured as a function of magnetic field at various temperatures. Gate voltage is fixed at VV gg =.5 V, which corresponds to a hole carrier density of cm during measurement. Data were obtained from mmmmmm sample BP77. Inset: Arrhenius plot of RR xxxx at νν = 3 (black) and 4 (red). Linear fits of the data (solid lines) yield activation gaps of 3.4 K for νν = 3 at 7.5 T and 30.9 K for νν = 4 at 0.6 T. Because ΔΔΔΔ oooooo = gg SS μμ BB BB ΓΓ νν and ΔΔΔΔ eeeeeeee = EE CC gg LL μμ BB BB ΓΓ vv (gg LL ; see discussion in the main text), we can obtain a rough estimation of EE CC and gg SS μμ BB BB, pending a separate estimation of the LL broadening ΓΓ νν. One order-of-magnitude estimation of ΓΓ νν comes from ΓΓ νν ~ħ/ττ qq, where ττ qq = 0.36 ps is the carrier quantum lifetime that we extracted from the Dingle plot of the low-field SdH oscillations (Supplementary Section 3). We therefore estimate that gg SS μμ BB BB and EE CC are approximately.7bb K and 4.9BB K at νν = 3 and νν = 4, respectively. These values imply an enhanced gg SS of ~ 4 and an EE SS /EE CC ratio of ~ 0.6 at νν = 3 and 4. NATURE NANOTECHNOLOGY 7

8 DOI: /NNANO R xx (k) a T (K) = 1 b = 1 c = 1 d = R xy (h/e ) = = = = V g (V) V g (V) V g (V) V g (V) 0.0 Supplementary Figure 7 Measurement of QH energy gaps at νν = 11 and. RR xxxx and RR xxxx as functions of gate voltage at various temperatures. Measurements were performed on sample BP4 (the same sample discussed in Fig. b and 3). Panels a-d display data obtained in fixed magnetic fields 33 T, 31 T, 9 T and 7 T, respectively. Vertical lines mmmmmm mark the filling factors νν = 1 and. The Arrhenius plots of RR xxxx (shown in Fig. 3b) yields the excitation gap as a function of magnetic field at νν = 1 and (Fig. 3c). 8 NATURE NANOTECHNOLOGY

9 DOI: /NNANO SUPPLEMENTARY INFORMATION 7. Reported values of hole effective mass Hole Effective Mass mm /mm 0 Magnetic Field BB (T) Carrier Density nn HH (10 1 cccc ) Source 0.34± ref * ref ± ref. 6 Supplementary Table 1 Reported values of the effective mass of holes in black phosphorus DHG. * No carrier density value was provided in ref. 5. We estimate the carrier density from the frequency of the SdH oscillations shown in the paper. NATURE NANOTECHNOLOGY 9

10 DOI: /NNANO Transport measurement in pulsed magnetic field Supplementary Figure 8 Mageto-transport of holes in black phosphorus DHG in pulsed magnetic field. Hall resistance (upper panel) and magnetoresistance (lower panel) are plotted as a function of magnetic field measured at varying hole doping levels (VV gg < 0). Data were obtained in a pulsed magnetic field at TT = 1.5 K. Horizontal lines and the fractions 1/νν in the upper panel mark the quantized values in unit of h/ee. Developing quantum Hall plateaus are observed at νν = 1 and. Magnetoresistance curves are shifted vertically by multiples of 3 kω for clarity, and broken lines indicate zero-resistance level at each gate voltage. Measurements were performed on sample BP108, which has a hole mobility of ~ 000 cm V 1 s 1 and an electron mobility of ~ 1000 cm V 1 s 1 at TT = 1.5 K. 10 NATURE NANOTECHNOLOGY

11 DOI: /NNANO SUPPLEMENTARY INFORMATION Supplementary Figure 9 Mageto-transport of electrons in black phosphorus DEG in pulsed magnetic field. Hall resistance (upper panel) and magnetoresistance (lower panel) are plotted as a function of magnetic field measured at two different electron doping levels ( VV gg > 0 ). Data were obtained in a pulsed magnetic field at TT = 1.5 K. Magnetoresistance at VV gg = 10 V is shifted vertically by kω for clarity. Vertical lines mark the filling factor νν. Measurements were performed on sample BP108. NATURE NANOTECHNOLOGY 11

12 DOI: /NNANO Calculation of the spin splitting 9.1 Electronic structure of black phosphorus D quantum well In this section we calculate the electronic structure of the gate-induced D quantum well (QW) at the surface of black phosphorus. The k p Hamiltonian of the black phosphorus QW near Γ point can be expressed as S3 where EE cc + ηη cc kk xx + νν cc kk yy + ħ kk mm zz HH = [ cccc γγkk xx γγkk xx EE vv ηη vv kk xx νν vv kk yy ħ mm vvvv kk zz ] + eeee zzzz + VV(zz), V z describes the potential profile in the out-of-plane direction. V z consists of the hard-wall confining potential at the sample surfaces and the internal electrostatic potential V z caused by charge distribution in the QW. The subband dispersion and the in corresponding eigenstates can be obtained numerically from the Schrödinger equation where s is the subband index, and H E, s s s s the envelope function. To solve the Schrödinger equation, we adopt the hard-wall boundary condition and m expand s as s sin z, where L is the width of the quantum well. The L m L internal electrostatic potential V z is determined by the Poisson equation where nz and and is the dielectric constant. in dv z n z p z in, dz p z are the densities of electrons and holes in z direction, respectively, nz and p z can be obtained from S4 i i i i c i v i n z n z p z p z, where c and v refer to the conduction and valence bands, respectively, and 1 NATURE NANOTECHNOLOGY

13 DOI: /NNANO SUPPLEMENTARY INFORMATION c Ef Ei c kt B kbt ni mi e0 ln e 1. v Ei Ef v kt B kbt pi mi e0 ln e 1 * * Here E is the Fermi energy, and m* refers to the effective mass given by f mx m y (ref. S5). We obtain the eigenstates and eigenenergies of the QW numerically by solving the Schrödinger and Poisson equation self-consistently. Because the samples under investigation are p-type, we only need to consider the electric static potential induced by the hole states in numerical calculations. Supplementary Figure 10 Electronic structure of gate-induced black phosphorus QW. a, Band structure of a 10-nm-thick pristine black phosphorus without gate electric field. The pink dashed line indicates the Fermi level. b, Band structure of black phosphorus QW induced by a gate electric field of EE = 1.1 V/nm. System geometry is the same as in a. c, Spatial distribution of hole density at the surface of black phosphorus under a gate electric field of EE = 1.1 V/nm. d, Spatial distributions of the wavefunction of the highest hole subband (red) and the lowest electron subband (green). Blue curve denotes the electric potential induced by the gate electric field. NATURE NANOTECHNOLOGY 13

14 DOI: /NNANO The band structure of black phosphorus QW without and with gate electric field is shown in Figs. S10a and S10b, respectively. Here the thickness of the black phosphorus sample is assumed to be 10 nm and the gate electric field at the sample surface is EE = 1.1 V/nm. Both values are close to those used in the experiment. Other parameters used in the calculation are given in Table S. Our results indicate that only the highest hole subband is occupied and the energy gap in the QW remains ~ 0.3 ev. Our calculation further shows that free holes are mostly confined within ~ atomic layers at the surface, as shown in Fig. S10c. EE cc (ev) EE vv (ev) ηη cc (ev nm ) νν cc (ev nm ) γγ (ev nm) mm cccc (mm 0 ) mm cccc (mm 0 ) ηη vv (ev nm ) νν vv (ev nm ) εε (εε 0 ) Supplementary Table Parameters used in the calculation. εε 0 is vacuum permittivity. 9. Single-particle g-factor of the DHG in black phosphorus quantum well Armed with the electronic structure of the QW, we now calculate the single-particle g-factor of the DHG based on a multiband k p theory. Within the k p framework, the Schrödinger equation can be expressed as S6 D kk E 0 nn nn, n n Here the Hamiltonian matrix element coefficients are given by DD nnnn αααα = ħ δδ mm nnnn δδ αααα + ħ nn PP αα ll ll PP ββ nn 0 mm 0 EE (0) (0) nn EE ll ll with PP = pp + so the secular equation can be written as ħ 4mm 0 cc σσ VV, 1 S A D k, k D k, k E 0 nn nn nn n n (1) 14 NATURE NANOTECHNOLOGY

15 DOI: /NNANO SUPPLEMENTARY INFORMATION where D S nn' { S Dnn and A Dnn are the symmetric and antisymmetric part of D (i.e., nn' A D D ) / and D D D ) / ), respectively. k, k ] and ( nn' nn' nn' ( nn' nn' k, k } are commutator and anticommutator, respectively. S Dnn [ can be written as SS DD nnnn αααα = ħ δδ mm nnnn δδ αααα + ħ nn PP αα ll ll PP ββ nn + nn PP ββ ll ll PP αα nn 0 mm 0 ll (0) (0), EE nn EEll and the antisymmetric part AA DD nnnn αααα S Dnn = ħ is given by ll (0) (0). EE nn EEll mm 0 nn PP αα ll ll PP ββ nn nn PP ββ ll ll PP αα nn When the system is subjected to a magnetic field, the secular equation becomes 1 S A D k, k D k, k B B E 0 nn nn nn n n () where Bohr magneton μμ BB = ee ħ/(mm 0 ), [kk αα, kk ββ ] = εε αααααα eebb γγ /(iiħ), and is the Levi- Civita symbol. We consider an external magnetic field applied along z axis. ħkk is now A B yx,,0 / is the replaced by the canonical momentum, ħkk ħkk + eeee, where vector potential adopting the symmetry gauge. In this case, So the secular equation () becomes m0 x, k y B Bz k. i 1 S D k, k nn 1 np l lp n np l lp n B mb E nn (3) B z B s z n n im0 l En El From equation (3) we obtain the analytical form of the effective magnetic moment of the electron in a crystal: so the g-factor is give by 1 np l lp n np l lp n B 0 0 im0 l En E, l z NATURE NANOTECHNOLOGY 15

16 DOI: /NNANO g / B g where gg 0 = is the electron g-factor. 1 np l lp n np l lp n. 0 im l En El It has been shown that the dominant components of electrons states near Γ point are p orbitals S7,S8. lp, is therefore very small, and can be safely neglected. So z and gg xxxx = gg 0 [1 + 1 gg yyyy = gg 0 [1 + 1 z v iimm 0 Γ vv ll iimm 0 Γ vv ll +, PP yy ll l PP zz Γ + vv, Γ + vv, PP zz ll l PP yy Γ + vv, (0) (0) EE Γvv + EE, l +, PP zz ll l PP xx Γ + vv, Γ + vv, PP xx ll l PP zz Γ + vv, (0) (0) EE Γvv + EE, l ] gg 0 ] gg 0 g zz g 1 1, Pl lp,, P l lp, v x y v v y x v im0 l E E v, l g c, 4c,, P,, P, v x 4c 4c y v 0 0 E E,, v 4c, P,, P, v y 4c 4c x v 0 0 E E,, v 4c im 0 v Px 1v 1v Py v 1v, 0 0 E E v, 1v,,,,,, P,, P v y 1v 1v x v 1v, 0 0 E E v, 1v,,. In the QW, the total wave function Φ is the product of envelope function and band-edge Bloch function, so i i i. The above expression for g zz can then be written as 16 NATURE NANOTECHNOLOGY

17 DOI: /NNANO SUPPLEMENTARY INFORMATION g zz g 0 1 4c, 1v, 1v,, P,, P, v x 4c 4c y v 0 0 v, 4c,, P,, P, v y 4c 4c x v 4c, E E v, 4c, im0 v Px 1 v 1 v Py v E E,,,, E E 0 0 v, 1v,, P,, P v y 1v 1v x 0 0 E E v, 1v, v, P g0 1 E g 4 x1 v 4c where P, P,, x1 4c x v i V,, c, c v 4mc 0 x From our calculation shown in Fig. S10d, one clearly sees that the gate electrical field pushes electrons and holes in opposite directions, so the envelope wavefunctions of the electron and hole states (red and green in Fig. S10d, respectively) are spatially separated. The overlap integral v 4c therefore becomes negligibly small, so gg zzzz gg 0 In summary, the single-particle g-factor of the hole carriers in the gate-induced DHG in black phosphorus is gg. In addition, the anisotropy of the g-factor is negligible (i.e., gg xxxx gg yyyy gg zzzz ) as a result of the weak intrinsic spin-orbit interaction in black phosphorus. 9.3 The effect of exchange interaction on spin splitting of the Landau levels Previous experimental S9,S10 and theoretical studies S11,S1 indicate that the LLs in black phosphorus D electron system can be described by a decoupled Hamiltonian S1 in low-energy regime. Taking the Landau gauge AA = ( BBBB, 0,0) for a perpendicular magnetic field B, the kk pp Hamiltonian for the valence band of the QW can be written as S11 HH vv = EE vv (aa aa + 1 ) ħωω vv + 1 σσσσμμ BBBB ħ kk zz + eeee mm zz zz + VV(zz) vvvv NATURE NANOTECHNOLOGY 17

18 DOI: /NNANO with the creation and annihilation operators aa = mm vvvvωω vv ħ (yy yy 0 + iipp yy mm vvvv ωω vv ), aa = mm vvvvωω vv ħ (yy yy 0 iipp yy mm vvvv ωω vv ) where ωω vv = eeee mm vvvv mm vvvv is the cyclotron frequency with mm vvvv = ħ /ηη vv, mm vvvv = ħ / vv vv. ηη vv = ηη vv + γγ EE gg, yy 0 = kk xx ll BB is the cyclotron center, ll BB = ħ/eeee is the magnetic length, σ = ±1 is the spin index, and g =.0 is the free electron g-factor. Here we only consider the valence band of a black phosphorus DHG. Next we calculate the spin splitting of the LLs caused by the exchange interaction in the QW. In the presence of a magnetic field, the electron wavefunction Ψ nnzz,nn,kk xx (r, zz) can be written as (σσ) Ψ nnzz,nn,kk xx (r, zz) = ψψ (σσ) nnzz (zz) nn, kk xx = ψ (σσ) nnzz (zz) eeiikk xxxx φφ nn [κκ(yy yy 0 )] LL xx where the envelope function along the z direction is obtained from the Schrödinger and Poisson equations described in Section 8.1, and nn zz is the index of the electronic subbands. κκ = mm vvvv ωω vv /ħ, φφ nn (yy) = ee yy / HH nn (yy) ππ nn nn! are the harmonic oscillator wave functions. The corresponding LL energy EE nn (ii) can be obtained analytically and numerically S11. Because the black phosphorus DHG has only one occupied subband S9, the index nn zz can be omitted. We assume a density of states (DOS) given by D(EE) = 1 ππll BB ππγ ee where Γ is the LL broadening due to disorder S1, when calculating the filling factor. The expression for the exchange correction to the LL energy, calculated from the (σσ) wavefunctions Ψ nnzz,nn,kk xx (r, zz), has the following form S13,S14 + dddd (σσ) Σ nn = ddzz nn,σσ kk xx kk xx + EE Γ + ddr + ddr Ψ (σσ) (σσ nn,kkxx (r, zz)ψ ) nn,kk (r, zz )VV(r r, zz, zz (σσ )Ψ ) xx nn,kk (r, zz )Ψ (σσ) xx nn,kkxx (r, zz) where VV(r r, zz, zz ) = VV( r r, zz, zz ) is the Coulomb Green s function describing the interaction between two point charges located at (r, zz) and (r, zz ), with r r the distance between the two point charges in the DHG plane. The Fourier transform for the Coulomb interaction is VV( r r, zz, zz ) = dd q (ππ) DD (qq, zz, zz ) ee iiq (r r ) (σσ) 18 NATURE NANOTECHNOLOGY

19 DOI: /NNANO SUPPLEMENTARY INFORMATION where DD (qq, zz, zz ) = πee exp( qq z zz ) εεεε. Here we can omit the z dependence because the electrons are strongly confined in the QW (with a thickness of about 1 nm) according to our self-consistent calculations in Section 9.1. Following above equations, the calculation of the matrix elements of the Coulomb potential via the wavefunctions (σσ) (r, zz) is given by Ψ nnzz,nn,kk xx mm, kk 1 ee iiq r nn, kk = ee ii κκ kk 1 +kk qq yy ll BB ee κκκκ nn! mm! [ κκ (qq yy + iiqq xx )ll BB ] mm nn LL nn mm (κκκκ)δδ kk1 kk,qq xx for m n, where u = qq ll BB, LL mm nn (xx) are associated polynomials, and q = qq xx + qq yy. Using those matrix elements, we find the exchange correction to the screening Σ nn (σσ) = vv nn nn,σσ LL nn,nn (σσ ) + qqqqqq ee κκκκ (κκκκ) nn nn ππ + dddd (σσ,σσ ) DD (qq, zz, zz ) (σσ (κκκκ, zz, zz),σσ) εε(qq) LL nn,nn (κκκκ, zz, zz ) + ddzz (σσ) (σσ,σσ where vv nn is the filling factor, ) LL nn,nn (uu, nn zz, zz) = nn1! nn! LL nn 1 nn1 (uu)ψψ (σσ) (σσ nnzz (zz)ψψ ) nnzz (zz) with nn 1 = min(nn, nn ) and nn = max(nn, nn ). The dielectric function of the DHG is given by ε(qq) = 1 + DD(EE FF EE nn (σσ) ) nn,σσ + dddd + ddzz The energy of the LLs therefore has the form (σσ,σσ ) LL nn,nn (0, zz, zz)dd (qq, zz, zz (σσ,σσ) )LL nn,nn (0, zz, zz ) (σσ) (σσ) (σσ) EE nn = EEnn + Σnn with the exchange interaction taken into account. The effective g-factor can be extracted from ( ) gg = EE nnff ( ) EE nnff = gg μμ BB BB 0 + Σ ( ) ( ) nn Σnn μμ BB BB Fig. S11 displays the gg calculated as a function of hole density at a fixed magnetic field of 31 T. The exchange interaction induces sizable enhancement of gg when the Fermi level lies between two spin-split states from the same LL (i.e., at odd νν), whereas no enhancement is found when the Fermi level lies between states from adjacent LLs (i.e., at even νν). The exchange enhancement strongly depends on the LL broadening, which was assumed independent of νν in our calculation. Finally, the maximum gg at odd filling factors (denoted as gg SS in the main text) is found to slightly decrease as hole density NATURE NANOTECHNOLOGY 19

20 DOI: /NNANO increases. This trend is opposite to our experimental observations in Fig. 4b. The discrepancy may be a result of possible filling-factor-dependent LL broadening, which was not considered in our calculation Filling factor = mev = 3 mev.6 g* Hole density (10 1 cm - ) Supplementary Figure 11 Exchange enhancement of the effective g-factor in black phosphorus DHG. gg was calculated as a function of hole density. Sample thickness was taken to be 10 nm. The magnetic field and temperature was fixed at 31 T and 300 mk, respectively. Other parameters are kept the same as those used in Fig. S10b. 0 NATURE NANOTECHNOLOGY

21 DOI: /NNANO SUPPLEMENTARY INFORMATION 10. Additional analysis of the coincidence condition in a tilted magnetic field Supplementary Figure 1 The coincidence condition in a tilted magnetic field. a, Magnetoresistance (after subtracting a smooth background) as a function of VV gg / cos(θθ) showing the amplitude of the SdH oscillations. Data were recorded in a fixed total magnetic field of 0 T while the tilt angle θθ was varied. Vertical lines mark the minima of the SdH oscillations and their corresponding filling factors νν from 5 to 7. Data were taken on the same sample as shown in Fig. 4. b, Oscillation amplitude at even filling factors (solid symbols) and odd filling factors (open symbols) as a function of cos(θθ). Data were extracted from a. Below θθ = 57, the amplitude at even filling factors decreases monotonously with increasing θθ, whereas that of odd filling factors remains constant. Such observation is unambiguous evidence that θθ cc of 46.3 corresponds to a coincidence of the lowest order (the rr = 1 coincidence discussed in ref. S15). There is indication that a rr = 1 coincidence (or the ii = 1 coincidence referred to in ref. S16) might occur between θθ = 57 and 7.8, as suggested by the appearance of a peak in the amplitude at odd filling factors. Such a coincidence was however not corroborated by a simultaneous dip in the amplitude at even filling factors. Future experiments on samples with higher mobility may be able to fully resolve the rr = 1 coincidence. NATURE NANOTECHNOLOGY 1

22 DOI: /NNANO References S1. P. T. Coleridge, Phys. Rev. B 44, (1991). S. X. Hong, K. Zou and J. Zhu, Phys. Rev. B 80, (009). S3. P. K. Li and I. Appelbaum, Phys. Rev. B 90, (014). S4. F. Stern, Phys. Rev. Lett. 18, 546 (1967). S5. F. Stern, Phys. Rev. B 5, 4891 (197). S6. R. Winkler, Spin-Orbit Coupling Effects in Two-Dimensional Electron and Hole Systems, Springer-Verlag, Berlin (003). S7. A. S. Rodin, A. Carvalho, and A. H. Castro Neto, Phys, Rev. Lett. 11, (014). S8. A. N. Rudenko and M. I. Katsnelson, Phys. Rev. B 89, 01408(R) (014). S9.L. K. Li, G. J. Ye, V. Tran, R. X.Fei, G. R. Chen, H. C. Wang, J. Wang, K.Watanabe, T. Taniguchi, L. Yang, X. H. Chen, and Yuanbo Zhang, Nature Nanotechnology10,608 (015). S10. Xiaolong Chen, Yingying Wu, Zefei Wu, Yu Han, Shuigang Xu, Lin Wang, Weiguang Ye, TianyiHan,Yuheng He, Yuan Cai and Ning Wang, Nat. Communs. 6, 7315 (015). S11. X. Y. Zhou, R. Zhang, J. P. Sun, Y. L. Zou, D. Zhang, W. K. Lou, F. Cheng, G. H. Zhou, F. Zhai and Kai Chang, Sci. Rep. 5,195 (015). S1. Yongjin Jiang, Rafael Roldan, Francisco Guinea and Tony Low, Phys. Rev. B 9, (015). S13. Tsuneya Ando and Yasutada Uemura, J. Phys. Soc. Japan 36, 959 (1974). S14. S. S. Krishtopenko, V. I. Gavrilenko and M. Goiran, J. Phys.: Condens. Matter 3, (011). S15. R. J. Nicholas, R. J. Haug, K. v Klitzing and G. Weimann, Phys. Rev. B 37, 194 (1988). S16. K. Vakili, Y. P. Shkolnikov, E. Tutuc, E. P. De Poortere and M. Shayegan, Phys. Rev. Lett. 9, 6401 (004). NATURE NANOTECHNOLOGY