Tunable Feshbach resonances and their spectral signatures in bilayer semiconductors

Feshbach resonances are an invaluable tool in atomic physics, enabling precise control of interactions and the preparation of complex quantum phases of matter. Here, we theoretically analyze a solid-state analogue of a Feshbach resonance in two dimensional semiconductor heterostructures. In the presence of inter-layer electron tunneling, the scattering of excitons and electrons occupying different layers can be resonantly enhanced by tuning an applied electric field. The emergence of an inter-layer Feshbach molecule modifies the optical excitation spectrum, and can be understood in terms of Fermi polaron formation. We discuss potential implications for the realization of correlated Bose-Fermi mixtures in bilayer semiconductors.

Feshbach resonances are an invaluable tool in atomic physics, enabling precise control of interactions and the preparation of complex quantum phases of matter. Here, we theoretically analyze a solid-state analogue of a Feshbach resonance in two dimensional semiconductor heterostructures.
In the presence of inter-layer electron tunneling, the scattering of excitons and electrons occupying different layers can be resonantly enhanced by tuning an applied electric field. The emergence of an inter-layer Feshbach molecule modifies the optical excitation spectrum, and can be understood in terms of Fermi polaron formation. We discuss potential implications for the realization of correlated Bose-Fermi mixtures in bilayer semiconductors.
Recently bilayer structures of two-dimensional materials have emerged as fascinating platforms for realizing exotic phases of electronic matter [1,2]. Much of their success is driven by a new level of control, arising from twisting the two layers with respect to each other during stacking. Such twisted bilayers generate a moiré potential for electrons or holes, which quenches the kinetic energy and therefore enhances correlations. Most notably this has lead to the discovery of unconventional superconductivity [3,4], correlated insulators, and charge density waves [5][6][7] in bilayer graphene and transition metal dichalcogenides (TMDs). In addition to electronic phases, semiconductors such as TMDs can host excitons, which are strongly bound electron-hole pairs. They act as mobile composite bosons and remain rigid due to their large binding energies. Moreover, excitons interact with free electrons or holes and can form charged molecules, termed trions. This renders bilayer TMDs promising candidates to study complex Bose-Fermi mixtures. Such mixtures have been recently investigated in dilute quantum gases [8][9][10], where Feshbach resonances are routinely used to control interactions between the atomic species [11][12][13][14][15]. By contrast, in solid state structures the molecular binding energies, and correspondingly the interaction strength among particles, are generically fixed by material properties, limiting the experimentally accessible regimes.
Here, we address this challenge by introducing a solidstate analogue of a Feshbach resonance. Using the layer degree of freedom as a pseudo-spin, we demonstrate that the energy of a closed-channel bound state can be tuned with respect to scattering states in an open channel, simply by applying an external electric field E z . The counterpart of hyperfine interactions in atomic systems, is provided by coherent inter-layer electron or hole tunneling. We demonstrate the impact of such Feshbach resonances on the spectrum of a single optically-injected exciton immersed in a Fermi sea of charge carriers, tak- Illustration of exciton-carrier scattering in a bilayer TMD. The electrostatic potential energy is different in the two layers and can be tuned by a perpendicular electric field Ez. Scattering between excitons and electrons is enhanced when the intra-layer trion energy is tuned into resonance with the energy of an electron and an exciton in seperate layers.
ing into account the radiative exciton decay. Close to the Feshbach resonance we find a striking modification of the exciton spectrum. In particular, we show that the spectral shape is sensitive to the finite range of the effective interactions relative to the Fermi wavelength.
Our work is motivated by a recent experimental observation of an electrically tunable Feshbach resonance in a twisted bilayer TMD heterostructure [16]. We theoretically analyze a more generic scenario with vanishing twist angles and discuss how resonantly enhanced polaron formation can be observed in reflection measurements. Our findings demonstrate the potential for bilayer TMDs to control valley-selective interactions between itinerant carriers and establish a novel platform for exploring correlated quantum dynamics of degenerate Bose-Fermi mixtures.
Effective bilayer Hamiltonian.-We consider a bilayer semiconductor setup as depicted in Fig. 1. Electrons (or holes) can tunnel from one layer to the other with coupling constant t, which can be adjusted by incorporating tunnel barriers [7]. For concreteness we focus on TMD bilayers spatially separated by a distance d. Generically, the exciton resonances in the top and bottom layers have different energies, either due to difference in material properties or strain, enabling layer-selective exciton creation. Furthermore, hybridization between inter-and intra-layer excitons is small, due to the sizable difference in binding energies. This allows us to focus only on intra-layer excitons [17]. For simplicity we assume that excitons are injected optically and are present only in the top layer. The system is then described by the effective Hamiltonian where x † k creates an exciton of mass M in the top layer, and c † k,T and c † k,B create fermions of mass m in the top and bottom layer, respectively. From now on we refer to itinerant charges as electrons, although all conclusions apply equally to holes. We omit the valley and spin degree of freedom and assume that electrons and excitons reside in different valleys, since only this scattering channel will be resonantly enhanced. As the exciton's Bohr radius is small, excitons and electrons experience sizable attractive contact interactions U , only when both particles are in the same layer and opposite valleys. We also neglect the composite nature of the exciton and treat it as a structureless boson [18]. The potential energy difference ∆ = qdE z between the two layers, can be tuned by changing E z , as illustrated in Fig. 1. We consider the scenario where ∆ is chosen such that electrons reside predominantly in the bottom layer.
Feshbach resonance in exciton-electron scattering.-To understand scattering properties in such a heterostructure, we focus on the two-particle subspace of the system. In the center of mass frame, Eq. 1 can then be expressed in first quantization as: where µ = 1/(m −1 + M −1 ) and m tot = m + M are the reduced mass and the total mass respectively. The wave function carries the layer degree of freedom and the part describing the relative motion can be expressed as ψ(r) = (ψ T (r), ψ B (r)) T / √ 2. Asymptotic eigenstates with large spatial separation between the two particles define the open and closed channel. We consider E z for which ∆ |E 0 B |, where E 0 B is the binding energy of the intra-layer trion. Although both channels are hybridized between the layers, only the open channel is energetically accessible and electrons reside predominantly in the bottom layer (Fig. 2). The scattering threshold for the open (ε O ) and closed (ε C ) channel is The outgoing scattering states |ψ + α , in channel α with energy E can be found as solutions of the Lippmann-Schwinger equation: where r|φ α ∼ e ikx is an incoming plane wave. We can reformulate the problem by introducing the T-matrix T R |φ α =Û |ψ + α , which connects the incoming plane waves with the full outgoing scattering state. Eq. 3 translates to an equation for the off-shell T-matrixT R (E): We solve Eq. 4 analytically in a plane-wave basis which diagonalizesĤ 0 : where E is the scattering energy, and k is the total incoming momentum. The 2 × 2 matrix structure ofT R (E, k), andÛ , due to the two channels, is implicitly assumed. Scattering can be resonantly enhanced if E z is tuned such that the closed channel bound state is in proximity of the open channel threshold ε O , see Fig. 2 (b) for an illustration. Similar to cold atomic systems, we are interested in two-particle collisions with small incoming momenta. In this case, scattering is accurately described by a finite-range expansion, which is performed by expanding the denominator of the T-matrix in powers of E − ε O . In two dimensions the finite range expansion of the on-shell T-matrix takes the universal form: which is characterized by the scattering length a and effective range r 0 [19]. We relate this expansion to our effective description by integrating Eq. 5 and matching the open channel scattering amplitude T R OO (k 2 /2µ, 0) to Eq. 6. This way we obtain the open channel scattering length a O and effective range r 0 : where a = 2µE 0 B is the scattering length of the closed channel in absence of tunnel coupling. Analyzing Eq. 5-7 we find that the open channel T-matrix has a pole at energies below the scattering threshold ε O . This is the signature of a Feshbach molecule which forms in interlayer scattering [20]. Eq. 6 demonstrates that the energy of the molecule depends on both the scattering length a O and range r 0 . We plot the energy of the Feshbach molecule as a function of detuning in Fig. 3 for three different t. As the detuning becomes large and positive, the scattering length starts to diverge while the molecular energy approaches the scattering threshold. For large detunings the binding energy is then approximately given by 1/2µa 2 O . In the case E 0 B , ∆ t we obtain simple expressions for the binding energy of the Feshbach molecule and the effective range close to resonance, which read This demonstrates the power of a Feshbach resonance: complete control over the energy of the Feshbach molecule can be achieved simply by changing E z . Thus the system can be electrically tuned to arbitrarily large scattering lengths [21,22]. While the binding energy of the Feshbach molecule changes exponentially, the effective range r 0 depends only linearly on E z . We find that weakly coupled layers lead to large values of r 0 , and the resulting physics is reminiscent of narrow Feshbach resonances in three dimensions. In contrast to the three dimensional case however, the bound state does not dissolve for any value of E z , as long as there is no repulsive background scattering. The purely two dimensional geometry of the system also distinguishes the proposed resonance from realizations in cold atom systems, where scattering always remains effectively three dimensional due to the finite transverse confinement [14]. Optical impurities strongly coupled to a Fermi sea.-Resonantly enhanced two-particle scattering affects correlations in electron-exciton mixtures. We consider a low concentration of excitons injected into a Fermi sea of electrons in the open channel. The excitons in such a system are mobile impurities and form collective excitations known as Fermi polarons [23][24][25]. Here we analyze the polaron spectrum as E z is tuned over the Feshbach resonance.
Our previous discussions focused on two-body scattering with small but finite momentum, for which the exciton is a long-lived excitation and the scattering matrix is essentially unitary [26]. Here, we focus on optically excited k = 0 excitons. In this regime excitons couple to the radiation field, which allows them to decay via electronhole recombination via emission of an optical photon. As this decay process is essentially memory-less, it can be described by a Lindblad master equatioṅ where Γ(k) is the decay rate of the exciton. In the presence of a Fermi sea Eq. 9 constitutes a complex manybody system, which can not be solved exactly. However, it was found that key properties can already be inferred purely from the scattering properties of the system [27,28] and that T-matrix approximations provide an accurate description of the ground and excited states of mobile impurities [29][30][31][32][33][34][35][36][37]. For our heterostructure setting we develop a T-matrix approximation to include dissipation as well as finiterange corrections from the Feshbach resonance: .
(10) Details on the calculation can be found in the supplemental materials [38]. Compared to Eq. 5, the momentum of the electron in the open channel is now restricted to lie above the Fermi surface due to Pauli blocking by the Fermi sea. Exciton recombination results in an imaginary part iΓ(k) of the exciton energy [39,40]. Using this T-matrix, we then determine the self-energy of the exciton as a function of frequency ω: This equation originates from the creation of a particlehole pair in the open channel, with hole momentum q < k F . The spectral function of the exciton then reads (12) As the master equation fulfills fluctuation-dissipation relations and we have treated dissipation exactly, the resulting spectral function respects the sum rule dω 2π A x (ω, k) = [x k , x † k ] = 1. We compute the spectrum as a function of detuning, by integrating Eq. 12 numerically. We show the resulting exciton spectra in Fig. 4 for three different Fermi energies (a)-(c). They are characterized by the formation of an attractive branch, with maxima at negative frequencies; and a repulsive branch, with maxima at positive frequencies. For small Fermi energies ( Fig. 4(a)) the two resonances approach the Feshbach molecule and bare exciton energy respectively: the spectrum can be understood in terms of the formation of a Fermi polaron and is clearly distinct from an avoided level crossing. We observe that the repulsive polaron abruptly loses spectral weight and blue-shifts in energy, as the Feshbach molecule becomes weakly bound. With increasing carrier density, the maximal splitting between the repulsive and attractive branch grows ( Fig. 4(b)). Surprisingly, the spectrum at high densities starts to resemble an avoided crossing (see Fig. 4(c)). This change in spectral shape cannot be explained assuming contact interactions, but rather arises from significant finite range corrections [41,42]. Since the average scattering process involves momenta on the order of k F , the non-logarithmic terms in Eq. 6 become successively more important at high densities and strongly renormalize the spectrum. The spectral asymmetry we observe is distinct from solidstate realizations based on polaritons, whose steep dispersion obscures the relevant scattering physics [43]. As the spectral function of the exciton is directly accessible in reflection measurements, the features we identified provide particularly clear experimental signatures, which result from many-body effects.
Conclusions and Outlook.-We have investigated an electrically tunable solid-state Feshbach resonance, using the layer pseudo-spin degree of freedom of semiconductor bilayers. Our scheme allows for a controlled enhancement of electron-exciton scattering in experiments. We find that much of the resulting Feshbach physics, such as Fermi-Polaron formation for a dilute concentration of excitons, may be observed in experiments as a highly asymmetric and density-dependent reflection spectrum. This makes TMD bilayers ideal systems to study two dimensional Fermi polarons in parameter regimes that have so far been inaccessible in cold atomic gases and monolayer semiconductors. By extending our setup to finite exciton densities, Feshbach resonances could enable precise control of degenerate Bose-Fermi mixtures in solid state systems. This is particularly appealing as excitations of an excitonic Bose gas can mediate superconductivity in a Fermi sea [44,45]. Since the bound state exists only for excitons and electrons with different spin/valley degree-of-freedom, the Feshbach-resonance could allow for spin selective interaction control and may induce instabilities in exotic pairing channels.
Feshbach resonances can also form in different scattering channels than the one considered here, i.e. inter-layer excitons in resonance with an intra-layer bound state, which could prove to be useful in the context of long-lived indirect exciton condensates [46]. Furthermore, our work generates the opportunity to study few-body physics in two dimensional semiconductors. The tunable scattering length can be used to explore exotic multi-particle bound states, where a single electron binds multiple excitons [47,48]. While we specifically considered resonant scattering between excitons and electrons, Feshbach physics in 2D materials could be a generic phenomenon that may also be relevant for understanding purely electronic processes [49].
Acknowledgments.-We thank I. Schwartz and Y. Shimazaki for discussions which motivated this work. The authors also thank A. Cavalleri, F. Helmrich and P.A. Murthy for fruitful discussions. This work was supported by ETH Zurich. We also acknowledge support from the Technical University of Munich -Institute for Advanced Study, funded by the German Excellence Initiative and the European Union