Exciton-Polarons in Doped Semiconductors in a Strong Magnetic Field

PHYSICAL REVIEW B 97, 235432 (2018)

Exciton-polarons in doped semiconductors in a strong magnetic ﬁeld

Dmitry K. Eﬁmkin and Allan H. MacDonald Center for Complex Quantum Systems, University of Texas at Austin, Austin, Texas 78712, USA

(Received 19 July 2017; revised manuscript received 9 April 2018; published 21 June 2018)

In previous work, we have argued that the optical properties of moderately doped two-dimensional semiconductors can be described in terms of excitons dressed by their interactions with a degenerate Fermi sea of additional charge carriers. These interactions split the bare exciton into attractive and repulsive exciton-polaron branches. The collective excitations of the coupled system are many-body generalizations of the bound trion and unbound states of a single electron interacting with an exciton. In this paper, we consider exciton-polarons in the presence of an external magnetic field that quantizes the kinetic energy of the electrons in the Fermi sea. Our theoretical approach is based on a transformation to a new basis that respects the underlaying symmetry of magnetic translations. We find that the attractive exciton-polaron branch is only weakly influenced by the magnetic field, whereas the repulsive branch exhibits magnetic oscillations and splits into discrete peaks that reflect combined exciton-cyclotron resonance.

DOI: 10.1103/PhysRevB.97.235432

I. INTRODUCTION dress excitons into exciton-polarons. Exciton-polarons have attractive and repulsive spectral branches that evolve from The monolayer transition metal dicholagenides (TMDCs), and generalize to degenerate carrier densities, the separate MoS , MoSe ,WS, and WSe , are widely studied 2 2 2 2 absorption processes associated with bound and unbound trion in beyond-graphene [1–3] two-dimensional semiconductors states. [4–8]. Strong Coulomb and spin-orbit interactions, tunable Here we consider exciton-polarons in the presence of optical properties, and valley selection via circularly polar- magnetic field of the moderate strengthhω ¯ B ∼ T, wherehω ¯ B ized light combine to make them promising materials for is the spacing between Landau levels for excess charge carriers. optoelectronics and valleytronics [9,10]. TMDC monolayers In this regime, the magnetic field is too weak to alter the and heterostructures have therefore emerged as an interesting structure of excitons [44,45]. On the other hand, the excess platform for the exploration of exciton physics (see Ref. [11] charge carriers are in the regime of the integer and fractional and references therein). quantum Hall effect, and the polaron dressing of excitons can Recent experiments have demonstrated that the absorption be seriously influenced. In TMDC materials with the binding spectra of TMDC monolayers is dominated by exciton features energy T ≈ 30 meV [19] the polaronic dressing of excitons with large binding energies X ≈ 0.5eV[12,13]. When charge is strongly modified at magnetic field of order B ∼ 85 T, carriers are present, the exciton absorption feature splits into while magnetic oscillations of the repulsive exciton-polaron two separate peaks [6,7,14–21] that are usually attributed to branch we present below could be observable at smaller fields. excitons (X) and to trions (T), charged weakly coupled three- Optical studies of TMDC at magnetic field B ≈ 10 T, not particle complexes formed by binding two electrons to one hole enough strong to influence the polaron dressing, have been or two holes to one electron. The difference between resonant presented [21,46–49]. Studies of undoped TMDC samples at frequencies for the peaks is interpreted as the binding energy much stronger magnetic fields B ≈ 65 T have been reported T of a trion. This phenomenon has been observed previously very [50] recently, and we anticipate that experiments with in conventional quantum well systems based on GaAs and doped samples at such fields will be undertaken soon. CdTe [22–26], although the binding energy of trions there, In our theory, the optical conductivity is proportional to the T ≈ 2 meV, is more than an order of magnitude smaller than spectral function of excitons with zero momentum AX(ω,0). in TMDC T ≈ 30 meV [19,20]. In our previous work [27], Our main results for AX(ω,0) are summarized in Fig. 1. we have argued that the three-particle picture of the additional The splitting of excitons into attractive and repulsive exciton- peak is appropriate only at low doping F T, where F is polaron branches in the absence of magnetic field is illustrated the Fermi energy of excess charge carriers. The three-particle in Fig. 1(a). The spectral weight AX(ω,0) at magnetic field picture cannot explain the competition for spectral weight strengthshω ¯ B/T = 0.5 and 1 are presented in Figs. 1(b) and between two peaks that is observed in experiments at higher 1(c). The attractive exciton-polaron branch is only weakly in- carrier densities (see also related earlier work [28–34]). In the fluenced by the magnetic field. The repulsive branch splits into wide doping range where F T, that in TMDC monolayers a number of peaks separated by the Landau quantization energy 12 −2 corresponds to concentrations n 510 cm , the appropri- hω¯ B, and experiences magnetic oscillations as the Fermi ate picture is instead one of renormalized excitons interacting level passes through different Landau levels. The absorption with the degenerate Fermi sea of excess charge carriers [27,35]. peaks that are blueshifted from the primary repulsive exciton- As in the Fermi-polaron problem [36–43], the interactions polaron peak can be interpreted as combined exciton-cyclotron

TMDCs. In Sec. III, some pertinent aspects of electronic structure theory in a magnetic ﬁeld are reviewed. In Sec. IV, some features of the two-body exciton-electron problem are discussed. Section V is devoted to the many-body exciton- polaron problem. In Sec. VI, we present and discuss results. Finally, in Sec. VII, we conclude compare our results with experimental work at conventional semiconductors, and com- ment on limitations of our theory.

II. THE MODEL The minimal model for the optical properties of single-layer TMDCs accounts for two valleys with conventional parabolic energy spectra [51]. Due to strong Coulomb interactions, absorption is dominated by excitation of direct excitons with an resonant energy ωX well below the single particle contin- uum threshold. Importantly, the two valleys can be probed independently by using circularly polarized light to achieve valley-selective absorption. Our polaronic theory is based on the difference between trion and exciton binding energy scales, i.e., on the property that T X. This separation allows us to account for the effect of excess charge carriers and magnetic fields by adding a polaronic dressing self-energy to the exciton propagator. Provided that F X, the formation of photoexcited excitons is weakly disturbed, and they can still be treated as composite bosons with bare resonant energy ωX,massmX, and optical transitions matrix element D. We assume that the bare excitons interact with the excess charge carriers via a phenomenological FIG. 1. The spectral function of excitons at zero momentum contact pseudopotential that accounts approximately for their AX(ω,0) in the absence of a magnetic field (a). The excitonic feature short-range monopole-dipole Coulomb interactions. Due to the splits into attractive and repulsive exciton-polaron branches, with presence of two valleys, excitons are disturbed by two Fermi ∗ =− − dispersions that can be approximated by T 3T/2 F/2and seas. In our model, we neglect interactions between the exciton ∗ = X F/2. The approximate dispersions are plotted as long-dashed and carriers in the same valley, which are weakened by the lines. The spectral functions AX(ω,0) are illustrated at magnetic fields exchange physics related to the possibility that electrons can = of strengthhω ¯ B/T 0.5 (b) and 1 (c). The horizontal short-dashed be part of the exciton and in the Fermi sea at the same time lines denote integer filling factors νe. The repulsive exciton-polaron [52]. Interactions with the Fermi sea from the other valley branch splits into a number of peaks, which can be interpreted as are responsible for the polaron dressing of excitons and for combined exciton-cyclotron resonances. splitting the spectral function into attractive and repulsive exciton-polaron branches. resonances and correspond to the simultaneous creation of an As we discuss in details below, renormalization of the exci- exciton and an electronic transition between Landau levels. ton binding energy and polaronic dressing can be distinguished With decreasing magnetic field, the number of additional peaks in experiments. We concentrate on the latter by letting ωX, mX, increases and the absorption pattern smoothly evolves into the and D be free parameters of the theory, which we allow to be dispersive zero-field repulsive exciton-polaron branch. weakly dependent on carrier density. With these assumptions, Our paper is organized as follows. In Sec. II, we introduce the polaronic effect on photo-excited excitons can be described the model we employ to describe the optical properties of by the following Hamiltonian:

2 p + e A p2 = + e c e − + + X + − + + H dr e (r) μe e(r) X (r) ωX X(r) Ue (r)X (r)X(r)e(r) , (1) 2me 2mX

where e, μe, and me are the magnitude of charge for electrons valleys [53]. For the sake of definiteness, we assume that the (e), their chemical potential and mass. Ae = (−By/2,Bx/2,0) excess charge carriers are electrons; the generalization to holes is the vector potential for the magnetic field of strength B is straightforward. Note that we have neglected interactions in the symmetric gauge. In Eq. (1), U is the short-range between the excess electrons, Since the two valleys can be attractive exciton-electron interactions and e(r) and X(r) probed independently by using circularly polarized light, we are field operators for electrons and excitons in different do not introduce a valley index.

235432-2 EXCITON-POLARONS IN DOPED SEMICONDUCTORS IN A … PHYSICAL REVIEW B 97, 235432 (2018)

= (a) (b) νe ne/NB is the total filling factor, while [νe] denotes the integer part of νe and gives the total number of fully occupied ω GX Landau levels. DD The states |nl can be constructed√ with the help of ladder√ op- = x − y = x + y Γ erators a lB( e i e )/ 2¯h and b lB(Ke iKe )/ 2¯h as follows: (c) (a+)n(b+)l |nl= √ |00, (4) Γ = + n! l! Γ where the ground state is given by r2 | = 1 − e FIG. 2. (a) Schematic representation of Eq. (2), which specifies re 00 exp 2 . (5) 2 4lB our approximation for the excitonic contribution to the optical 2πlB conductivity. The triangle vertices correspond to the optical matrix K elements D. The paired solid and dashed lines represent an exciton as The kinematic, e, and and magnetic, e, momenta introduced above are related to the canonical momentum operator a bound state of an electron and a hole and stand for the exciton’s free = + = − bosonic Green function. (b) Exciton self-energy due to interactions pe by e pe eAe/c, and Ke pe eAe/c. Note that x y =− x y = + = with Fermi sea fluctuations. (c) Bethe-Salpeter equation for the [ e , e ] ih/l¯ B and [Ke ,Ke ] ih/l¯ B, so that [a,a ] + = = + = exciton/Fermi-sea interaction vertex . 1, [b,b ] 1, [a,b] 0, and [a,b ] 0. Our polaronic calculations require manipulations with the l l −ipr density form-factor of electrons F (p) =n l |e e |nl.Its Valley-resolved optical conductivity is illustrated schemat- n n explicit form and the derivation of the following helpful ically by the Feynman diagram depicted in Fig. 2(a) and is identities: given by 2 2 2 p l e 2 l l1 − l1l = l l n B σ (ω) = |D| ¯ A (ω,0). (2) Fn n ( p)Fn n(p) δn nn , (6) h X X 1 1 p 1 2 l1 2 2 = 4 2 2 Here ¯X me /κ h¯ is the bare binding energy of excitons l l ll n p lB F (−p )F (p) = δ N , (7) in the absence of doping and magnetic field with κ to be the n n nn pp B n 2 l l effective dielectric constant of the TMDC material. AX(ω,p) = ··· −2Im[GX(ω,p)] is the spectral function of the excitons. In the are presented in Appendix A.Here p denotes averaging n absence of electrons, over direction of momenta p and the form-factor n (x)is given by 2γX = AX(ω,0) , (3) γ 2 + ω2 n |n −n| min[n ,n]! |n −n| 2 −x X (x) = x L [x] e . (8) n max[n ,n]! min[n ,n] where γX is a phenomenological exciton radiative life- m time/scattering rate and ω = ω − ωX is the offset frequency. Here Ln [x] is the generalized Laguerre polynomial. (In the following, we omit the use ofh ¯ to distinguish wave The origin of the macroscopic degeneracy of Landau levels vector/momentum and frequency/energy, but preserve it else- is magnetic translation symmetry. In the presence of a field, where). Our main goal is to calculate AX(ω,0) in the presence translations are generated by the magnetic momentum Ke of the Fermi sea of excess electrons that experience an external and translational invariance implies that [He,Ke] = 0. This magnetic field. Before proceeding to the many-body polaronic symmetry is noncommutative since different components of problem, we briefly review some aspects of the theories of the magnetic momentum do not commute with each other. two-dimensional electrons in a magnetic field (Sec. III) and of Physically, the motion of an electron in a magnetic field is two-particle exciton-electron systems (Sec. IV). along a Larmor orbit, and the components of Ke determine its center. The macroscopic degeneracy of Landau levels is with III. LANDAU LEVELS respect to the position of the orbit center.

In the presence of an external magnetic field, the kinetic- IV. TWO-PARTICLE EXCITON-ELECTRON PROBLEM energy spectrum of electrons is quenched to a set of degenerate Landau levels [54,55]. In the convenient symmetric gauge, In the absence of magnetic field, the polaronic exciton- electronic states are labeled by a pair of quantum numbers |n,l. electron problem is simplified by the decoupling of the cen- e = + ter of mass and relative motion degrees of freedom. The They have energy n hω¯ B(n 1/2) and definite orbital mo- z = − = separation is guaranteed by translational symmetry. In the mentum Mnl h¯ (n l), where ωB eB/mec is the Larmor frequency. Energy does not depend on l and Landau levels have presence of magnetic field, translational symmetry is replaced = 2 macroscopic√ degeneracy NB 1/2πl per unit area, where by noncommutative magnetic translation symmetry, defined B = lB = hc/eB¯ is a magnetic length. When Landau levels are by [H,K] 0, and the separation is not possible anymore. all either completely filled or completely empty, electrons form Here K = Ke + pX is the total magnetic momentum of the an incompressible quantum system. At zero temperature, the charged electron and the neutral exciton. The components = dependence of their chemical potential μe on concentration ne of K do not commute [Kx ,Ky ] ih/l¯ B, and the symmetry√ + is discontinuous and is given by μe = hω¯ B(1/2 + [νe]). Here implies that [H,B B] = 0, where B = lB (Kx + iKy )/ 2¯h

235432-3 DMITRY K. EFIMKIN AND ALLAN H. MACDONALD PHYSICAL REVIEW B 97, 235432 (2018) is a new ladder operator. The presence of magnetic translation the underlying magnetic translational symmetry, but works in symmetry drastically simplifies the exciton-polaron problem, a different way. In this section, we present how the transfor- however, its exploitation is not as simple as in the absence of mation works for the two-particle exciton-electron problem. a magnetic field. Before presenting the two-particle basis we employ, it is For the case of few-particle charged complexes in a useful to discuss the more obvious direct product representa- magnetic field, a successful approach has been developed tion of the electron plus exciton Hilbert space [58]: by A. Dzyubenko and collaborators [56,57]. It involves a r r |nlp=r |nl eiprX . (9) set of Bogoliubov transformations to a basis respecting the e X e symmetry of magnetic translations. The approach requires that These wave functions are normalized over the unit area. all involved particles be charged and can be applied only for For every exciton momentum p, the product state |nlp can be two- and three-particle systems. As a result, it cannot be used obtained from the corresponding ground state |00p with the for the many-body exciton-electron polaronic problem. Here help of ladder operators a and b. The two-particle Hamiltonian we have developed a related approach which also involves a corresponding to Eq. (1) in this basis can be decomposed as transformation to an appropriately chosen basis that respects follows:

2 e p l l H = ξ + + ω |nlpnlp|− UF (p − p)|n l p nlp|. (10) n 2m X n n nlp X nlp n l p

e = + − Here ξn hω¯ B(n 1/2) μe is the electron energy and The new states, which will be distinguished by an overbar − l l = | ipre | n l Fn n(p) n l e nl is the density form-factor. The prod- accent, are labeled by the same quantum numbers, and uct basis is the one naturally employed in diagrammatic and p, as the product basis. Importantly, excited states |nlp perturbation theory because it is the representation in which for each momentum p can be constructed from the corre- + the kinetic energy, and hence the bare electron and exciton sponding ground√ state |00p with the help of A = UaU√ = x y + x y propagators, are diagonal. On the other hand, the form-factor lB ( − i )/ 2¯h and B = UbU = lB (K + iK )/ 2¯h. l l − = + p K = K + p total Fn n(p p ) couples both Landau levels n and orbits l in a Here e X and e X are kinematic complicated manner. The simpliﬁcations afforded by magnetic and magnetic momenta we introduced above. As a result, + + translation symmetry are not explicit. new states Eq. (11) are eigenstates of B B as B B|nlp= Now we construct a new basis. We notice that the magnetic l|nlp and magnetic translation symmetry is respected. The momentum of electrons Ke, respected in the usual basis, can overlap between the basis states is the density form-factor

+ | = l l be transformed to the total one K as UKeU = K with the n l p nlp δp pFn n(p), and the transformation between the − unitary operator U = e ipXre . This observation motivates the two representations can be alternatively presented as introduction of a new basis as follows: | = l l | nlp Fn n(p) n l p . (12) n l In the new representation, the Hamiltonian of two-particle ip(rX −re) rerX|nlp≡rerX|U|nlp=re|nl e . (11) exciton-electron problem can be decomposed as

± Here we introduced compact notations p = px ± ipy, and V. EXCITON-POLARON PROBLEM μ = m m /(m + m ) is the reduced mass of exciton and T e X e X We now return to the many-body exciton-electron problem. electron. The main advantage of new basis is that Landau orbits The approximation we employ for the dressed exciton prop- with different intra-Landau-level label, l, are decoupled from agator is illustrated diagrammatically in Figs. 2(b) and 2(c). each other. Moreover, interaction matrix elements between l l the electron and the exciton are also diagonal with respect The many-body scattering amplitude n n(ω,p ,p)isusedto construct the exciton self-energy, illustrated in Fig. 2(b), that to the Landau level kinetic energy index, n. As we see below, l l these properties make the basis convenient for analysis of the describes the the polaronic dressing. n n(ω,p ,p) accounts exciton-electron scattering problem. for the scattering amplitude between the exciton-electron

235432-4 EXCITON-POLARONS IN DOPED SEMICONDUCTORS IN A … PHYSICAL REVIEW B 97, 235432 (2018) two-particle product states |nlp and |n l p via repeated electron-exciton interactions in a ladder diagram approximation. After summation over Matsubara frequencies and analytical continuation, the Bethe-Salpeter equation for the −vertex is e ν¯ l l l l l l1 n1 l1l (ω,p ,p) =−UF (p − p) − UF (p − p1) (ω,p1,p). (13) n n n n n n1 − e − X + n1n ω ξn p iγX n1l1 p1 1 1 X = 2 e = − e Here p p /2mX is the energy of excitons. The factorν ¯n 1 νn accounts for Pauli blocking by excluding ﬁlled states from e the scattering, while νn is the ﬁlling factor of Landau level n. Because magnetic translation symmetry is not respected in this usual basis, all two-particle states |nlp and |n l p are coupled with each other in a complicated way that makes further manipulations problematic. Now we transform the scattering amplitude to the more convenient basis ¯ using

l l l l1 l1l2 l2l (ω,p ,p) = F (p )¯ (ω,p ,p)F (−p). n n n n1 n1n2 n2n

l1l2 n1n2 Transformation of Eq. (13) to this representation leads to

e ν¯ l l l l l l1 n1 l1l2 l2l ¯ (ω,p ,p) =−Uδ − UF (−p1) F (p1)¯ (ω,p1,p). (14) n n n n n n1 − e − X + n1n2 n2n ω ξn p iγX n1l1 n2l2 p1 1 1

Because the interaction matrix elements in the new basis do transformation to the modiﬁed representation: − not depend on transferred momentum p p, the scattering X = e ll ¯ + e l l − ¯ l l p p(ω) νnFnn (p )n ω ξn Fn n( p). (18) amplitude is momentum independent n n(ω). Summation nl over l1 and averaging over the direction of momentum p1 with the help of the identity Eq. (6) decouples all scattering n l

¯ l l = l l ¯ channels n n(ω) δn nn n(ω) and makes the Bethe-Saltpeter Summation over Landau orbits l and l with the help of identity equation algebraic ¯ n(ω) =−U − U n(ω)¯ n(ω). Here the Eq. (7) yields a self-energy that is diagonal in momentum space kernel n(ω)isgivenby given by 2 2 2 2 p l e n p lB e n B e X(ω,p) = NB ν ¯ n ω + ξ . (19) ν¯nn 2 n n n (ω) = , (15) 2 n ω − ξ e − X + iγ nn pn n p X n Here NB is the Landau level degeneracy factor and n (x) n is the form factor introduced in Eq. (8). Only excitons with and n (x) is the form factor introduced in Eq. (8). The kernel = (ω) has the ultraviolet logarithmic divergence that is also zero momentum p 0 are optically active and the self-energy n expression reduces further to present in the absence of a magnetic field [27]. It is instructive to introduce the energy cutoff and rewrite the scattering (ω,0) = N νe¯ ω + ξ e . (20) ¯ X B n n n amplitude n as follows: n 1 The self-energy modifies the spectral function of excitons ¯ =− = − + −1 n . (16) AX(ω,0) [ ω X(ω,0) iγX] that is proportional to NT ln + n(ω,) T the optical conductivity, given by Eq. (2) and probed in absorption experiments. = − Here T exp[ 1/NTU] is the binding energy of the two- The closed set of Eqs. (15), (16), and (20) describes the po- body state of electron and exciton, that is the simplified model laronic dressing of excitons. In the limit of vanishing magnetic = 2 of trion, in the absence of magnetic field, while NT μT/2πh¯ fieldhω ¯ , these equations reduce to the ones derived ¯ B T is their reduced density of states. Because n does not depend in our previous work [27], which are briefly summarized in explicitly on and U, we treat T as an independent input Appendix B. A formal proof of this statement, constructed by parameter [59]. employing the momentum space coarsening picture of Landau ¯ The many-body scattering amplitude n defines the self- level formation, is presented in Appendix C. In the opposite energy of excitons (ω,p) depicted in Fig. 2(b). After summa- regime of strong magnetic fields T hω¯ B, the assumption of tion over Matsubara frequencies and analytical continuation, our polaronic theoryhω ¯ B X are not well satisfied. In this the self-energy is regime, which we discuss in Appendix D, only a few Landau levels are relevant. X = e ll + e p p(ω) νnnn ω ξn ,p ,p , (17) nl VI. RESULTS where is expressed in the product state representation. For T T, the exciton spectral function depends on four To take advantage of translational symmetry, we make a dimensionless control parameters: offset frequency ω/T,

235432-5 DMITRY K. EFIMKIN AND ALLAN H. MACDONALD PHYSICAL REVIEW B 97, 235432 (2018) magnetic fieldhω ¯ B/T, exciton decay/scattering rate γX/T, and doping F/T.HereF is the Fermi energy of electrons in the absence of magnetic field, which is related to the total filling factor by νe = ne/NB = F/hω¯ B.WefixγX/T = 0.1 and consider two values for the dimensionless magnetic field hω¯ B/T = 0.5; 1, which we refer to as to the weak and moderate magnetic field cases. In TMDC materials, the band masses for conduction and valence bands are almost the same and we used mX = 2me in numerical calculations. In Fig. 1(a), we present our results for the exciton spectral function AX(ω,0) in the absence of a magnetic field. Due to interactions with the Fermi sea of electrons, the excitonic feature splits into a redshifted attractive exciton-polaron branch, normally identified as a trion branch, and a bluesifted repulsive exciton-polaron branch, normally identified as an exciton branch. These branches are the many-body generalizations of the zero center of mass momentum bound and unbound exciton-electron states and smoothly evolve from them. With increasing doping, the attractive branch peak evolves between two approximately linear behaviors as a function of Fermi ∗ =− − ∗ =− − energy T T F and T 3T/2 F/2, whereas the ∗ = repulsive branch peak varies linearly as X F/2. These approximate dependences on carrier-density are represented in Fig. 1 by dashed lines. The exciton spectral function AX(ω,0) is plotted in FIG. 3. Frequency dependence of the excitonic spectral function = = Figs. 1(b) and 1(c) at two different values of magnetic field. The AX(ω,p 0) at different filling factors ranging from νe 0.15 (a) to = = attractive exciton-polaron branch is only weakly influenced νe 0.9 (f). The strength of the magnetic field ishω ¯ B/T 1. The by the magnetic field. For T = 0, the spectral function has partial weight of the repulsive exciton-polaron branch near ω = 0 weak cusps at integer filling factors because of the singular disappears when Landau level n = 0 becomes fully occupied at = dependence on field of the filling factors of particular Landau νe 1, where the weight of the first exciton-cyclotron resonance at = levels when they are first occupied and when they are empty. ω hω¯ B achieves its maximum. The attractive mode is the many-body generalization of the a result, it achieves a maximum when Landau level n is fully bound state of an exciton and an electron and is not strongly sensitive to the discretization of the electronic state energies filled and level n is fully empty. This doping dependence of the due to Landau level formation. Even for the magnetic field spectral function AX(ω,0) is captured in Figs. 3(a)–(f) for the = hω¯ B ∼ T, the attractive branch can still be very well approx- case of the strong magnetic fieldhω ¯ B/T 2 and the filling ∗ =− − factor range from νe = 0.15 to νe = 0.9. imated by the curve T 3T/2 F/2. As seen in Fig. 1, we find that the repulsive exciton-polaron With decreasing magnetic field, the number of additional branch is more strongly influenced by the magnetic field blueshifted exciton-cyclotron resonances grows dramatically. and experiences oscillatory behavior. It is the many-body They compete with each other and finally merge into the broad generalization of the unbound exciton-electron state which dispersive resonance representing repulsive exciton-polaron experiences Landau quantization. The main dispersionless branch. exciton-polaron feature near ω = 0 originates from exciton creation processes that are not accompanied by inter-Landau- VII. DISCUSSION AND CONCLUSIONS level electronic transitions. The set of blueshifted peaks separated by the energyhω ¯ B can be attributed to combined exciton- We have investigated the polaronic dressing of excitons by cyclotron resonances, in which excitons are created and their excess electrons using a partially phenomenological approach dressing results in electronic transitions to unoccupied higher that treats the exciton creation energy ωX, the exciton mass mX, Landau levels. Our approximations involve the summation of and the optical matrix element for photon to exciton coversion all ladder exciton-electron scattering diagrams; nevertheless, D as parameters. The presence of excess electrons makes the doping dependence of these features can be explained based these parameters doping dependent. However, the only role on the second-order perturbation theory, which we present in of X is to shift the absorption spectrum rigidly in frequency, Appendix C. In the perturbation theory, the contribution of and the only role of D is to rigidly scale the absorption intralevel transitions, responsible for the feature at ω = 0, is strength, which is proportional to |D|2. The goal of our e e proportional to νnν¯n. Due to the Pauli blocking effect, the fea- theory is to explain the frequency splitting between different ture smoothly disappears at integer filling factors and achieves features, and their relative amplitudes/spectral weight, which maximum when the corresponding Landau level is half-filled. are independent of D and ωX. The dependence of absorption The contribution of electronic transitions between different on exciton mass is much more complicated, but fortunately Landau levels n and n , corresponding to the exciton-cyclotron we believe that exciton mass renormalization is weak. Our = − e e resonance at ω hω¯ B(n n), is proportional to νnν¯n .As earlier zero-magnetic-field calculations [27], which combined

235432-6 EXCITON-POLARONS IN DOPED SEMICONDUCTORS IN A … PHYSICAL REVIEW B 97, 235432 (2018) static Random Phase Approximation (RPA) screening with the valid at very low densities F T, while the interplay at Hartree-Fock approximation, suggest that the carrier-density elevated doping F ∼ T in the polaronic regime is still not dependent change in exciton mass m 0.07me. To the best well understood. Singlet and triplet trions in conventional semi- of our knowledge, measurements of the renormalization of conductors correspond partially to intervalley and intravalley electron and hole masses in TMDCs, which could substantiate excitions in TMDC materials. The physics of intravalley these estimates, have not yet been reported. exciton-polarons cannot be specifically addressed using our In TMDC monolayers, the trion binding energy T ≈ approach, and its consideration is postponed to future work. 30 meV, implying that the exciton-polaron regime F T is We have neglected correlations between excess electrons. 12 −2 active at carrier densities smaller than ne 5 × 10 cm . The latter are likely to be especially important and interesting in 12 −2 At carrier densities ne 10 cm , the influences of carriers the fractional quantum Hall effect regime. For example, we can can be safely estimated from the three-particle trion physics. At anticipate that trion binding energies will jump upon passing 13 −2 carrier densities ne 10 cm , the Fermi energy of excess from below to above a filling factor at which an incompressible electrons becomes comparable even to the exciton binding state appears. Recently, additional blueshifted peaks have been energy, which is renormalized to substantially smaller values, observed at some special filling factors [76,77]. We have absorption is then governed by the interplay of Mahan and postponed an attempt to account for the role of Coulomb Mott exciton physics [11,14,60], which has been extensively correlations in the integer and fractional quantum Hall regimes studied in semiconductor quantum wells [61–66]. The zero to future work. temperature limit we used for numerical calculations implies To conclude, we have developed an approach for the T T [67], that corresponds to wide temperature range T many-body exciton-polaron problem in magnetic field. The 330 K. The criterionhω ¯ B ∼ T ≈ 85 T can be used to identify approach involves a transformation to a basis that respects the the scale of magnetic field that is required to strongly alter noncommutative symmetry of magnetic translations. We have the polaronic dressing of excitons. The magnetic oscillations shown that the attractive exciton-polaron branch is very weakly we have predicted for the repulsive exciton-polaron branch influenced by the magnetic field. The repulsive branch is can be observed at smaller magnetic fields if γX ∼ T hω¯ B. strongly modified by the magnetic field, however, and exhibits For radiative lifetime/scattering rate γX = 5 meV reported in oscillatory behavior and spectral splitting related to Landau doped TMDC samples [18] and temperature T ≈ 50 K, this quantization. The additional peaks correspond to combined condition is well satisfied for the magnetic field B ≈ 65 T exciton-cyclotron resonances and we expect that their future employed in recent experiments [50]. experimental study will be revealing. The optical properties of the conventional semiconductor quantum well systems GaAs and CdTe in a magnetic field have been extensively studied [61–66] (see Ref. [68]for ACKNOWLEDGMENTS a review). For CdTe quantum wells with trion binding energy T = 2.1 meV, the exciton-polaron regime F ∼ T This material is based upon work supported by the Army 11 −2 corresponds ne ∼ 10 cm , and the magnetic field required Research Office under Award No. W911NF-15-1-0466 and to influence the polaronic dressing is equal to B ∼ 2T. Robust by the Welch Foundation under Grant No. F1473. The au- attractive exciton-polarons, usually identified as singlet thors acknowledge helpful interactions with Atac Imamoglu, trions, and combined exciton-cyclotron resonance peaks that Fengcheng Wu, Scott Crooker, and Jesper Levinsen. are blueshifted from the repuslive exciton-polarons, usually identified as excitons, have both been reported. The resonances have been explained by using second-order perturbation theory APPENDIX A: THE DENSITY FORM-FACTOR to account for Coulomb interactions [69–71], as we briefly discuss in Appendix D. It should be noted that combined Here we present a set of useful relations for the density − l l = | ipre | exciton-cyclotron resonances have been observed only in form-factor Fn n(p) n l e nl that is given by clean samples and even then only at low temperatures [69,72]. In our theory, the crossover to the zero-field repulsive-polaron 2 2 absorption features has a complex oscillatory behavior which, − p lB − + l l = 2 n l to the best of our knowledge, has not yet been observed. Fn n(p) e Gn (p )Gl (p ). (A1) We expect that the improving quality of two-dimensional materials will make it possible to resolve the complex Here we have introduced the compact notation p± = px ± magnetic-field dependent structure of the repulsive-polaron y n ± absorption feature in the future. ip , the function Gn (p )isgivenby At enough strong magnetic fields, the absorption spectra of the conventional semiconductor GaAs and CdTe quantum ⎧ ± n −n 2 2 ⎨ p √lB n! n −n p lB wells acquires an additional feature between the attractive and L ,n n n ± i2 2 n ! n 2 G (p ) = , repulsive exciton-polaron branches [68], which is attributed to n ∓ n−n − 2 2 ⎩ p√lB n ! n n p lB L ,n

235432-7 DMITRY K. EFIMKIN AND ALLAN H. MACDONALD PHYSICAL REVIEW B 97, 235432 (2018) of useful identities [55] involving Gn2 : n1

n G (0) = δn n, (A2) n n ± = 2 ≡ Gn(p ) (2π) NBδ(p) NBδp,0, (A3) n ∓ ∓ 2 − p2 p1 lB ± ± ± ± 2 n n − = n − e Gn (p2 )Gn ( p1 ) Gn (p2 p1 ). (A4) n The relation for the products of form factors we use in the main part of the paper: 2 2 2 2 2 2 p lB p lB p l l l1 l1l − n − n1 − l l − n − n1 − l l n B F (−p)F (p) = δl le 2 G (p )G (−p ) = δ e 2 G (p )G (−p ) ≡ δ , n n1 n1n p n1 n p n n n1 n n n n1 2 l1 and p2l2 2 2 l l ll − B n − n − n p lB F (−p )F (p) = δp pNBe 2 G (p )G (−p ) ≡ δp pNB , n n nn p n n n1 2 l l

were derived using these identies. Here · · ·p denotes an relative momenta p and p. For contact interactions between n average over the direction of the momentum p and n (x)is electrons and excitons, it becomes independent of p and p given by and the Bethe-Salpeter equation reduces to the algebraic one (ω,P) =−U − U (ω,P)(ω,P). Here the kernel (ω,P)

min[n ,n]! | − | is given by n = |n −n| n n 2 −x n (x) x Lmin[n ,n][x] e , (A5) max[n ,n]! 1 − n ( + ) (ω,P) = F p P/3 . (B1) + − − P2 − p2 + m p ω X 2M 2μ F where Ln [x] is the generalized Laguerre polynomial. T T Here MT = 3me and μT = 2me/3 are the total and reduced masses of the exciton-electron system in the considered case APPENDIX B: EXCITON-POLARON DRESSING IN THE mX = 2me. Direct calculations result in ABSENCE OF A MAGNETIC FIELD 2πh¯ 2 1 Here we briefly present the main steps for the exciton (ω,P) = , (B2) μ log T + iπ dressing into exciton-polarons without magnetic field. Without T magnetic field the total momentum of exciton and electron where T = exp[−1/NTU] is the binding energy of the two- P = pe + pX is a good quantum number and the -vertex body state of electron and exciton, and is the ultraviolet (ω,P,p ,p) denotes the amplitude of scattering between cutoff energy. is given by

2 2 + 2 − 2 1 P pF (pF P ) (pF P ) = ω + iγX − − + s ω + iγX − ω + iγX − , 2 4MT 4me 4me 4me

= − 2 − 2 APPENDIX C: RECOVERING THE where s sign( ω pF/4me P /4me). The many-body scattering amplitude does not depend separately on U and , ZERO-MAGNETIC-FIELD LIMIT but only on their combination T. As a result, it is convenient to Here we prove that expressions for the kernel Eq. (15) and treat T as independent parameter of the model instead of them. self-energy Eq. (20) reduce to their counterparts Eqs. (B1) and Electron-exciton complex represents a simplified model for a (B3) in the limit of vanishing magnetic field. In this limit, trion and T is the corresponding binding energy. Excitonic as well as for high Landau levels n, the reconstruction of self-energy is connected with the -vertex as follows: the electronic spectrum into Landau levels can be presented as a coarsing of momentum space. All states from pn to 2 = e e pn+1 quench to a single Landau level n, where pn/2me (ω,0) = nF ξ ω + ξ ,P . (B3) P P hω¯ B(n + 1/2). The number of states inside the corresponding P ring is equal to the Landau level degeneracy NB.Theset of wave functions can be approximated by plane waves | → iper = Equations (B1) and (B3) are the counterparts of Eqs. (15) n,l e with fixed absolute value of momentum pe and (20). The latter reduce to them in the limit of vanishing pn. The summation over Landau orbits l corresponds to the magnetic field, as we prove in the next section. integration over states in the ring, while all rings span the full

235432-8 EXCITON-POLARONS IN DOPED SEMICONDUCTORS IN A … PHYSICAL REVIEW B 97, 235432 (2018) momentum space ··· = ··· ··· → ··· ( ) NB pe , ( ) ( ). (C1)

l nl pe ··· Here pe denotes averaging over direction of momenta pe. To prove the reduction n(ω) → (ω,Pn) resulting in n(ω) → (ω,Pn) we at first rewrite Eq. (B1) as follows: (1 − n ( ))(P,p ,p ) = F pe e X (ω,P) 2π 2 2 . (C2) + pe pX p p ω − − + F e X 2me 2mX Here we have introduced the form factor as follows: = − − (P,pe,pX) 2π δ(P pe pX) pe,pX (C3) In the limit of vanishing magnetic field reduction of Eq. (15)to(C2) requires 2 2 2 n pXlB l → ¯ (P ,p ,p ). (C4) B n 2 n e,n X n To prove this relation we recall the definition of the form factor n (x): 2 2 l l n pXlB l l ll 2 2 δ = F (−p )F (p ) → (2πh¯ ) δ(P − P )(2πh¯ ) N δ(P − p − p ) . (C5) n n n 2 n n X nn X pX n n B n e,n X pX,pe l

l l → 2 − The ﬁrst multiplier in the last term corresponds to δn n (2πh¯ ) δ(Pn Pn ). The second multiplier ensures the connection Eq. (C4). The connection between the self-energies Eqs. (20) and (B3) in the limit of vanishing magnetic ﬁeld can be proven as follows: (ω,0) = N νe¯ ω + e → N n e ω + e ,P → n e ω + e ,P . (C6) B n n n B F Pn Pn n F P P n n P

APPENDIX D: STRONG MAGNETIC FIELD LIMIT appears instead of the reduced density of states of electron and 2 exciton NT = μT/2πh¯ since kinetic energy of electrons is In the limit of the strong magnetic field T hω¯ B, only a quenched. The function I [z] is defined as follows: few Landau levels are important and need to be taken into ac- nn count. Their number can be estimated as NL ≈ X/hω¯ B, where the binding energy of excitons X is the energy scale at which ∞ n n (x) I z = dx . excitons cannot be considered as structureless quasiparticles nn [ ] − + (D2) anymore. Taking into account that in experiments exciton and 0 z x iδ trion binding energies differ by an order of magnitude X/T ∼ 10, we can estimate the number of relevant Landau levels as =− n Its imaginary part is given by Inn [z] π(z)n (z), while NL ≈ X/hω¯ B ∼ 5. In that case, the self-energy (ω,p = 0) the real part Inn [z] for the first three Landau levels is presented of excitons at zero momentum can be rewritten as in the table below. Here Ei(z) is the exponential integral

function. The expression of I [z] for higher Landau level hω¯ α νe nn =− B X n , (D1) indices n and n has the same functional form. e n ω 2 1 + αX ν¯ I 2 + n − n n n n n hω¯ B In this spectral function of excitons at zero momentum, AX(ω,0) depends on dimensionless frequency ω/hω¯ B and where αX = NX|U| is the dimensionless coupling constant doping νe = F/hω¯ B, as well as on dimensionless coupling 2 with NX = mX/2πh¯ to be the excitonic density of states. It constant αX and number of involved Landau levels NL.For

Inn [z]0 1 2

−z 2 0 e−zEi(z), e−zzEi(z) − 1, e z Ei(z)−z−1 − 2 −z − −z − 2 − + e z(z−2)2zEi(z)−z2+z+2 1 e zEi(z) 1 e (z 1) Ei(z) z 1 2 e−zz2Ei(z)−z−1 e−z(z−2)2zEi(z)−z2+z+2 (z2−4z+2)2e−zEi(z)−z3+7z2−14z+6 2 2 2 4

235432-9 DMITRY K. EFIMKIN AND ALLAN H. MACDONALD PHYSICAL REVIEW B 97, 235432 (2018)

αX = 0.4 and NL = 5, the spectral function AX(ω,0) is presented in Fig. 4(a). Compared to results described in the main text, the few Landau level approximation very well captures both splitting of the exciton level into attractive and repulsive exciton-polaron branches. It also captures the appearance of features corresponding to exciton-cyclotron resonances. It correctly predicts frequencies of all resonance and doping dependencies of their amplitudes. To investigate the capability of the perturbation theory we expand the self-energy Eq. (D1) in the coupling constant (1) αX up to the second order. The ﬁrst-order term (ω) = −αXνehω¯ B/2 represents the shift of the energy of excitons due to their interactions with electrons. The second-order term is given by

(2) 2 hω¯ B e e ω (ω) = α ν ν¯ I 2 + n − n , X 2 n n nn hω¯ nn B

and represents the effect of electronic transitions between Lan- dau levels. The corresponding frequency and doping depen- FIG. 4. Analytical results for the spectral function of excitons dence of the spectral function of excitons AX(ω,0) is presented = = = AX(ω,p 0) at αX 0.4andNL 5. Full ladder approximation (a) in Fig. 4(b). The perturbation theory very well captures the and the second-order perturbation theory (b). The perturbation theory appearance and doping dependence of the exciton-cyclotron does not capture the splitting of redshifted attractive exciton-polaron resonances. Nevertheless, it is not sufﬁcient to describe the branch, but well describes the appearance and doping dependence of splitting of redshifted attractive exciton-polaron branch. exciton-cyclotron resonances.

[1] K. S. Novoselov, A. K. Geim, S. V.Morozov, D. Jiang, Y.Zhang, [12] A. Chernikov, T. C. Berkelbach, H. M. Hill, A. Rigosi, Y. Li, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, Science 306, O. B. Aslan, D. R. Reichman, M. S. Hybertsen, and T. F. Heinz, 666 (2004). Phys. Rev. Lett. 113, 076802 (2014). [2] K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, M. I. [13] K. He, N. Kumar, L. Zhao, Z. Wang, K. F. Mak, H. Zhao, and Katsnelson, I. V. Grigorieva, S. V. Dubonos, and A. A. Firsov, J. Shan, Phys.Rev.Lett.113, 026803 (2014). Nature 438, 197 (2005). [14] A. Chernikov, A. M. van der Zande, H. M. Hill, A. F. Rigosi, A. [3] A. K. Geim and A. H. MacDonald, Phys. Today 60(8), 35 Velauthapillai, J. Hone, and T. F. Heinz, Phys.Rev.Lett.115, (2007). 126802 (2015). [4] K. F. Mak, C. Lee, J. Hone, J. Shan, and T. F. Heinz, Phys. Rev. [15] F. Cadiz, S. Tricard, M. Gay, D. Lagarde, G. Wang, C. Robert, Lett. 105, 136805 (2010). P. Renucci, B. Urbaszek, and X. Marie, Appl. Phys. Lett. 108, [5] D. Jariwala, V. K. Sangwan, L. J. Lauhon, T. J. Marks, and 251106 (2016). M. C. Hersam, ACS Nano 8, 1102 (2014). [16] B. Zhu, X. Chen, and X. Cui, Sci. Rep. 5, 9218 (2015). [6] J. S. Ross, S. Wu, H. Yu,N. J. Ghimire, A. M. Jones, G. Aivazian, [17] C. Zhang, H. Wang, W. Chan, C. Manolatou, and F. Rana, Phys. J. Yan, D. G. Mandrus, D. Xiao, W. Yao, and X. Xu, Nat. Rev. B 89, 205436 (2014). Commun. 4, 1474 (2013). [18] E. Courtade, M. Semina, M. Manca, M. M. Glazov, C. Robert, [7] K. F. Mak, K. He, C. Lee, G. H. Lee, J. Hone, T. F. Heinz, and F. Cadiz, G. Wang, T. Taniguchi, K. Watanabe, M. Pierre, J. Shan, Nat. Mater. 12, 207 (2013). W. Escofﬁer, E. L. Ivchenko, P. Renucci, X. Marie, T. Amand, [8] K. F. Mak, K. He, J. Shan, and T. F. Heinz, Nat. Nano. 7, 494 and B. Urbaszek, Phys.Rev.B96, 085302 (2017). (2012). [19] H. S. Lee, M. S. Kim, H. Kim, and Y. H. Lee, Phys. Rev. B 93, [9] Q. H. Wang, K. Kalantar-Zadeh, A. Kis, J. N. Coleman, and 140409 (2016). M. S. Strano, Nat. Nano. 7, 699 (2012). [20] A. M. Jones, H. Yu, J. R. Schaibley, J. Yan, D. G. Mandrus, T. [10] X. Duan, C. Wang, A. Pan, R. Yu, and X. Duan, Chem.Soc.Rev. Taniguchi, K. Watanabe, H. Dery, W. Yao, and X. Xu, Nat. Phys. 44, 8859 (2015). 12, 323 (2016). [11] G. Wang, A. Chernikov, T. F. Glazov, Mikhail M. Heinz, X. [21] G. Plechinger, P. Nagler, A. Arora, R. Schmidt, A. Chernikov, Marie, T. Amand, and B. Urbaszek, Rev. Mod. Phys. 90, 021001 A. G. del Aguila, P. C. Christianen, R. Bratschitsch, C. Schuller, (2018). and T. Korn, Nat. Commun. 7, 12715 (2014).

235432-10 EXCITON-POLARONS IN DOPED SEMICONDUCTORS IN A … PHYSICAL REVIEW B 97, 235432 (2018)

[22] G. V. Astakhov, V. P. Kochereshko, D. R. Yakovlev, W. Ossau, of TMDC semiconductors has been discussed in several recent J. Nürnberger, W. Faschinger, and G. Landwehr, Phys. Rev. B papers [78–83]. They are important only for excited excitonic 62, 10345 (2000). states which are excluded from considerations here. [23] G. Yusa, H. Shtrikman, and I. Bar-Joseph, Phys.Rev.B62, 15390 [52] The low-energy model describes excitonic transitions between (2000). low-energy bands of the TMDC four band electronic structure, [24] V. Ciulin, P. Kossacki, S. Haacke, J.-D. Ganière, B. Deveaud, which are referred as A excitons. Excitonic transitions between A. Esser, M. Kutrowski, and T. Wojtowicz, Phys. Rev. B 62, high-energy bands, referred as B excitons, can be dressed R16310 (2000). by Fermi seas from two valley since low-energy bands are [25] K. Kheng, R. T. Cox, Merle Y. d’ Aubigné, F. Bassani, K. ﬁlled by electrons and exchange interactions are absent. The Saminadayar, and S. Tatarenko, Phys.Rev.Lett.71, 1752 (1993). corresponding valley splitting of the attractive exciton-polaron [26] V. Huard, R. T. Cox, K. Saminadayar, A. Arnoult, and S. branch have been recently reported [20,21]. Tatarenko, Phys.Rev.Lett.84, 187 (2000). [53] We consider here only exciton-polarons formed from the ground [27] D. K. Eﬁmkin and A. H. MacDonald, Phys.Rev.B95, 035417 excitonic state. Excited states and delocalized electron-hole ones (2017). are well energy separated from it and are excluded from the [28] R. Suris, V. Kochereshko, G. Astakhov, D. Yakovlev, W. Ossau, consideration. That is why excitons have only the center of mass J. Nurnberger, W. Faschinger, G. Landwehr, T. Wojtowicz, G. degree of freedom that is the argument of their annihilation

Karczewski, and J. Kossut, Phys. Stat. Solid. (b) 227, 343 (2001). operator X(r) in Hamiltonian (1) and in wave functions ((9) [29] R. Suris, in Optical Properties of 2D Systems with Interacting and (11)). Electrons,editedbyW.OssauandR.A.Suris(NATOScientiﬁc [54] Z. F. Ezawa, Qunatum Hall Eeffects (World Scientiﬁc, 2000). Series, Kluwe, 2000). [55] A. H. MacDonald, arXiv:cond-mat/9410047. [30] M. Baeten and M. Wouters, Phys. Rev. B 89, 245301 (2014). [56] A. B. Dzyubenko, Phys. Rev. B 65, 035318 (2001). [31] M. Baeten and M. Wouters, Phys. Rev. B 91, 115313 (2015). [57] A. B. Dzyubenko and A. Y. Sivachenko, Phys.Rev.Lett.84, [32] M. Combescot, J. Tribollet, G. Karczewski, F. Bernardot, C. 4429 (2000). Testelin, and M. Chamarro, Europhys. Lett. 71, 431 (2005). [58] We consider here only exciton-polarons formed from the ground [33] S.-Y. Shiau, M. Combescot, and Y.-C. Chang, Phys.Rev.B86, excitonic state. Excited states and delocalized electron-hole ones 115210 (2012). are well energy separated from it and are excluded from the [34] M. Combescot, O. Betbeder-Matibet, and F. Dubin, Eur. Phys. consideration. That is why excitons have only the center of mass J. B 42, 63 (2004). degree of freedom that is the argument of their annihilation

[35] M. Sidler, P. Back, O. Cotlet, A. Srivastava, T. Fink, M. Kroner, operator X(r) in Hamiltonian (1) and in wave functions ((9) E. Demler, and A. Imamoglu, Nat. Phys. 13, 255 (2016). and (11)).

[36] J. R. Engelbrecht and M. Randeria, Phys. Rev. Lett. 65, 1032 [59] The logarithmic ultraviolet divergence of the kernel n(ω), given (1990). by (15), comes from Landau levels with high kinetic energy [37] J. R. Engelbrecht and M. Randeria, Phys. Rev. B 45, 12419 index n. In Appendix C we use the momentum space coarsening (1992). picture, that is valid for their formation, to show to that their [38] R. Schmidt and T. Enss, Phys.Rev.A83, 063620 (2011). contribution can be approximated by the related expressions [39] F. Chevy, Phys. Rev. A 74, 063628 (2006). derived by us in the absence of magnetic ﬁeld. In the latter case, [40] J. Levinsen and S. K. Baur, Phys.Rev.A86, 041602 (2012). brieﬂy overviewed in Appendix B, -vertex does not depend

[41] P. Massignan, M. Zaccanti, and G. M. Bruun, Rep. Prog. Phys. explicitly on and U, but only on their combination T. 77, 034401 (2014). [60] A. Steinhoff, M. Florian, M. Rosner, G. Schonhoff, T. O. [42] F. Chevy and C. Mora, Rep. Prog. Phys. 73, 112401 (2010). Wehling, and F. Jahnke, Nat. Commun. 8, 1166 (2017). [43] J. Levinsen and M. M. Parish, Strongly interacting two- [61] C. Schüller, K.-B. Broocks, C. Heyn, and D. Heitmann, Phys. dimensional fermi gases, in Annual Review of Cold Atoms and Rev. B 65, 081301 (2002). Molecules (World Scientiﬁc, 2015), Chap. 1, pp. 1–75. [62] D. Andronikov, V. Kochereshko, A. Platonov, T. Barrick, S. A. [44] I. V. Lerner and L. Yu. E., Zh. Eksp. Teor. Fiz 78, 1167 (1980) Crooker, and G. Karczewski, Phys.Rev.B72, 165339 (2005). [JETP 51, 588 (1980]. [63] D. Sanvitto, D. M. Whittaker, A. J. Shields, M. Y. Simmons, [45] C. Kallin and B. I. Halperin, Phys. Rev. B 30, 5655 (1984). D. A. Ritchie, and M. Pepper, Phys.Rev.Lett.89, 246805 (2002). [46] A. Srivastava, M. Sidler, A. V.Allain, D. S. Lembke, A. Kis, and [64] I. Bar-Joseph, G. Yusa, and H. Shtrikman, Solid State Commun. A. Imamoglu, Nat. Phys. 11, 141 (2015). 127, 765 (2003). [47] G. Aivazian, Z. Gong, A. M. Jones, R.-l. Chu, J. Yan, D. G. [65] A. J. Shields, M. Pepper, M. Y. Simmons, and D. A. Ritchie, Mandrus, C. Zhang, D. Cobden, W. Yao, and X. Xu, Nat. Phys. Phys. Rev. B 52, 7841 (1995). 11, 148 (2015). [66] G. Yusa, H. Shtrikman, and I. Bar-Joseph, Phys. Rev. Lett. 87, [48] D. MacNeill, C. Heikes, K. F. Mak, Z. Anderson, A. Kormányos, 216402 (2001). V. Zólyomi, J. Park, and D. C. Ralph, Phys.Rev.Lett.114, [67] H. Hu, B. C. Mulkerin, J. Wang, and X.-J. Liu, arXiv:1708.03410 037401 (2015). [cond-mat.quant-gas]. [49] Z. Wang, J. Shan, and K. F. Mak, Nat. Nanotech. 12, 144 (2016). [68] I. Bar-Joseph, Semicond. Sci. Technol. 20, R29 (2005). [50] A. V. Stier, N. P. Wilson, G. Clark, X. Xu, and S. A. Crooker, [69] D. R. Yakovlev, V. P. Kochereshko, R. A. Suris, H. Schenk, W. Nano Lett. 16, 7054 (2016). Ossau, A. Waag, G. Landwehr, P. C. M. Christianen, and J. C. [51] The model we consider is the non-relatvistic limit of a four Maan, Phys.Rev.Lett.79, 3974 (1997). band model for massive Dirac particles with strong spin-valley [70] S. A. Moskalenko, M. A. Liberman, and I. V. Podlesny, Phys. coupling. The role of Berry phases and the Dirac-like spectrum Rev. B 79, 125425 (2009).

235432-11 DMITRY K. EFIMKIN AND ALLAN H. MACDONALD PHYSICAL REVIEW B 97, 235432 (2018)

[71] A. B. Dzyubenko, Phys.Rev.B64, 241101 (2001). [78]F.Wu,F.Qu,andA.H.MacDonald,Phys. Rev. B 91, 075310 [72] G. Finkelstein, H. Shtrikman, and I. Bar-Joseph, Phys. Rev. Lett. (2015). 74, 976 (1995). [79] D. K. Eﬁmkin and Y. E. Lozovik, Phys. Rev. B 87, 245416 [73] J. J. Palacios, D. Yoshioka, and A. H. MacDonald, Phys. Rev. B (2013). 54, R2296 (1996). [80] I. Garate and M. Franz, Phys.Rev.B84, 045403 [74] A. Wójs and J. J. Quinn, Phys.Rev.B75, 085318 (2007). (2011). [75] A. Wójs, J. J. Quinn, and P. Hawrylak, Phys.Rev.B62, 4630 [81] M. Trushin, M. O. Goerbig, and W. Belzig, Phys. Rev. B 94, (2000). 041301 (2016). [76] S. Smolka, W. Wuester, F. Haupt, S. Faelt, W. Wegscheider, and [82] J. Zhou, W.-Y. Shan, W. Yao, and D. Xiao, Phys. Rev. Lett. 115, A. Imamoglu, Science 346, 332 (2014). 166803 (2015). [77] S. Ravets, P. Knüppel, S. Faelt, O. Cotlet, M. Kroner, W. [83] A. Srivastava and A. Imamoglu, Phys. Rev. Lett. 115, 166802 Wegscheider, and A. Imamoglu, Phys.Rev.Lett.120, 057401 (2015). (2018).

235432-12