Valley Zeeman effect in elementary optical excitations of monolayerwse 2 Ajit Srivastava 1, Meinrad Sidler 1, Adrien V. Allain 2, Dominik S. Lembke 2, Andras Kis 2, and A. Imamoğlu 1 1 Institute of Quantum Electronics, ETH Zurich, CH-8093 Zurich, Switzerland and 2 Electrical Engineering Institute, Ecole Polytechnique Federale de Lausanne (EPFL), CH-1015 Zurich, Switzerland. S1. Intercellular orbital magnetic moment in three-band tight binding model of WSe 2 including excitonic effects As mentioned in the main text, the intercellular contribution to the orbital magnetic moment in a two-band model is identical for conduction and valence band. However, this particle-hole symmetry is broken in TMDs and one can expect a finite difference between the intercellular orbital magnetic moment of the conduction and the valence band. We use the third-nearest neighbour (TNN) three-band tight binding model of Liu et al. [1] which reproduces the lowest energy band structure obtained from first-principles calculations throughout the Brillouin zone. In order to compute the momentum-resolved (k-resolved) orbital magnetic moment of the three bands, we use the following formula [2, 3] - L n (k) =i m [ un,k H k x u i,k u i,k H ] k y u n,k c.c, (1) E i E n j =n where E i and u i are the energy and the wave function of the i-th band, respectively. The orbital magnetic moment for the valence and conduction bands are shown in Fig. S1 (a) and (b) and their difference in Fig. S1 (c), all in units of µ B. In order to include the effect of strong excitonic effects on the measured orbital magnetic moment, we assume that the excitonic wave function in real space to be a superposition of electronhole pair states with a momentum spread of π/a B where a B 1-1.5 nm is the exciton Bohr radius. In particular, we assume the 2-dimensional exciton wave function of the form [4] ψ(r) 1 a B exp( r/a B ) and compute its Fourier-Bessel transform a B /(1+a 2 k 2 ) 3/2 as the weight function of such a superposition, as shown in Fig. S1(d) on a log scale. This weight function is large only NATURE PHYSICS www.nature.com/naturephysics 1
around the K-points but decays sufficiently slowly and is finite at values of k close to ±K where the difference in orbital magnetic moment of the two bands is large. This region is where the conduction band starts to change its mass drastically along the Γ K line in the Brillouin zone. Finally, we average the difference of orbital magnetic moments of the valence and conduction bands using the weight function to obtain the intercellular contribution to the exciton orbital magnetic moment. A magnetic field induced splitting of 4.3µ B B z, corresponds to a B 1.3 nm which is in excellent agreement with the predicted value of exciton Bohr radius in TMDs [5]. S2. Orbital magnetic moment of trion in WSe 2 due to finite Berry curvature According to Yu et al. [6], the trion (X ) dispersion near ±K-point has an exchange-induced gap of δ ex which results in a large Berry curvature, Ω X (k). If we treat X as a charged quasiparticle described by a two-band model of Yu et al. near ±K-points, the finite Berry curvature is accompanied by an orbital angular momentum L(k) which is identical for both bands. Using Eq. 18 of S1, we get L(k) as, [ L(k) =i m u1,k H k x u 2,k u 2,k H ] k y u 1,k c.c. (2) E g (k) ( ) 1/2 4J where E g (k) =δ ex 1+ 2 k 4 K 2 δex 2 (k+k TF) is the k-dependent gap. Expressing in terms of 2 ΩX (k), L(k) = m E g(k)ω X (k). (3) The trion magnetic moment due to the e-h exchange is then given by µ X = e 2m L(k), µ X (k) = ej 2 k 2 ( (k +2k TF ) 4k 4 J 2 ) 1 K 2 δ ex (k + k TF ) 3 1+ (k + k TF ) 2 K 2 δex 2 (4) where J is the electron-hole exchange coupling strength and k TF is the Thomas-Fermi wavevector corresponding to carrier screening. Fig. S2(a) shows µ(k) in units of µ B for k up to 0.01K and three different values of k TF.Ak TF of 0.1 ω 0 /c corresponds to a charge density of 10 9 cm 2. The average (intercellular) magnetic moment due to electron-hole exchange depends on the range of k-wavevectors involved in the radiative recombination and the carrier doping density through k TF. It can be calculated by averaging over certain wave vector range δk, say from 0 to k lim. This is shown in Fig. S2(b) for different values of k TF and different values of k lim (from k ph = ω 0 /c to 0.03K). 2 NATURE PHYSICS www.nature.com/naturephysics
SUPPLEMENTARY INFORMATION In the following, we assume that the contribution to trion magnetic moment is primarily from the electron-hole induced orbital moment and that the confinement of the excess electron is much weaker than the exciton confinement. With the above assumptions, in order to observe a splitting of 5.5 to 6.2 µ B, the trion final state intercellular magnetic moment should be between 4.1 and 4.45 µ B since the intracellular contribution to orbital magnetic moment is µ intra =2µ B, and the intercellular magnetic moment for the initial state of an electron in the conduction band was estimated to be 3.35 µ B (see supplementary S1). This interval of µ avg is shown as white band in Fig. S2(b). It is evident that a wide of range of parameters can lead to the experimentally observed trion splitting. [1] Liu, G.-B. et al. Three-band tight-binding model for monolayers of group-vib transition metal dichalcogenides. Phys. Rev. B 88, 085433 (2013). [2] Chang, M.-C. et al. Berry phase, hyperorbits, and the Hofstadter spectrum: Semiclassical dynamics in magnetic Bloch bands. Phys. Rev. B 53, 7010 (1996). [3] Yafet, Y. Solid State Physics:Advances in Research and Applications Vol. 14 (Academic, New York, 1963). [4] Zaslow, B., Zandler, M. E. Two-Dimensional Analog to the Hydrogen Atom. Am. J. Phys. 35, 1118 (1967). [5] Ye, Z. et al. Probing excitonic dark states in single-layer tungsten disulphide.nature 513, 214-218 (2014). [6] Yu, H. et al. Dirac cones and Dirac saddle points of bright excitons in monolayer transition metal dichalcogenides. Nat. Commun. 5, 3876 (2014). NATURE PHYSICS www.nature.com/naturephysics 3
Figure S1: Orbital magnetic moment from tight binding model with excitonic effects. a, (b,) Momentum-resolved orbital magnetic moment of valence (conduction) band calculated using third-nearest neighbour tight-binding model of Ref. [1]. The dotted hexagon denotes the Brillouin zone with K-points as the vertices and Γ point as the centre. c, Difference in orbital magnetic moment of valence and conduction bands. The region where the difference is large along Γ K line is where the conduction band changes its mass drastically. d, The weight function used to average the momentum-resolved orbital magnetic moment obtained from Fourier-Bessel transform of the ground state exciton wave function with a Bohr radius of 1.3 nm, plotted on a log-scale. Figure S2: Orbital magnetic moment of trion arising from large Berry curvature. a, Momentum-resolved orbital magnetic moment of trion (X ) as a function of X centre of mass wavevector k measured from the K-point for different Thomas-Fermi wave vectors (k TF ) of carrier doping densities. b, Average magnetic moment of X for a range of wavevectors involved in radiative recombination (δk) and k TF. The white band depicts the range of magnetic moments consistent with the experimentally observed trion splitting. 4 NATURE PHYSICS www.nature.com/naturephysics
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a 25 k TF = 0.1 ω/c b 14 20 k TF = 1 ω/c k TF = 10 ω/c 8 12 10 µ (µ B ) 15 10 k TF (ω 0 /c) 6 4 8 6 4 2 5 2 0 0.000 0.005 k/k 0.010 10 δk/k (10-3 ) 20 30 6 NATURE PHYSICS www.nature.com/naturephysics