Charge-Transfer Excitons in Coupled Atomically Thin Polar Nitride Quantum Wells

G roup III nitrides, including aluminum nitride (AlN), gallium nitride (GaN), and indium nitride (InN), are well-established commercial semiconductors that are widely recognized for their robust applications across various fields. These materials have been instrumental in the development of high-efficiency light-emitting diodes, 1,2 lasers, 3,4 and highelectron mobility transistors, 5,6 contributing significantly to advancements in lighting, communication, and power electronics. Despite their extensive commercial use and research, the potential for exploring excitonic quantum phenomena within these materials has remained largely unexamined, presenting significant opportunities for new advancements.

Recently, we have demonstrated [7][8][9] that GaN quantum wells can be reduced to a single-atom, monolayer scale to achieve extreme quantum confinement. This results in a significant increase in the electronic bandgap and efficient deep-ultraviolet emission from GaN without the need for alloying with AlN. The confinement also suppresses Coulombic screening, which yields strongly bound excitons even at room temperature, thereby enhancing radiative efficiency. With binding energies of >200 meV, these excitons could exhibit novel quantum phenomena such as spontaneous coherence, condensation, spin currents, and superfluidity. [10][11][12][13][14][15][16][17][18] Furthermore, they offer quantum technology-based alternatives for conventional electronic devices, such as excitonic switches, interconnects, and transistors. [19][20][21][22][23][24] To fully leverage excitonic properties, it would be desirable to achieve both a strong exciton binding energy to prevent dissociation into free electron-hole pairs and a long exciton lifetime for versatile exciton-based processing. Spatially indirect excitons (IXs), also known as charge-transfer excitons, meet these criteria by having electrons and holes located in different regions yet remaining bound by strong Coulomb attraction. Such IXs can be formed in structures like wide single quantum wells (QWs), 25,26 coupled QWs, 27,28 or type II interfaces, 29,30 and their lifetimes are longer than those of spatially direct excitons (DXs) due to reduced wave function overlap. Additionally, their dipolar nature allows for easy manipulation with the external electric fields. Most previous IX experiments have used either GaAs QWs to form IXs at low temperatures, [25][26][27][28] to counter the relatively low (∼10 meV) exciton binding energy, 31 or van der Waals 2D heterostructures, 29,30,[32][33][34][35] which are not as scalable as commercial semiconductors.

We propose the creation of strongly bound IXs suitable for room-temperature excitonic applications on a commercial semiconductor platform. This is achieved by combining the polarization properties of polar nitride heterostructures with enhanced Coulomb interactions in atomically thin quantum wells (QWs). We theoretically demonstrate that the intrinsic lack of inversion symmetry and resulting polarization fields in nitride heterostructures, 36,37 used to control carrier confinement and separation in quantum wells and superlattices, [38][39][40][41][42] yield strongly bound IXs in atomically thin GaN QWs without the need for external electric fields. We investigate the properties of IXs and DXs in pairs of atomically thin GaN QWs separated by AlN barriers of varying thicknesses using first-principles calculations and photoluminescence (PL) experiments. Our results show that by adjusting the AlN barrier thickness, we can control the overlap of electron and hole wave functions, the strength of electron-hole interactions, and the character (IX or DX) of the lowest-energy exciton. Additionally, we demonstrate that this atomically thin coupled GaN quantum well can be actually grown using molecular-beam epitaxy (MBE) and characterized through PL measurements.

Our first-principles calculations are based on density functional theory (DFT) and many-body perturbation theory. This approach yields an accurate description of excitonic properties even for atomically thin layers, where continuum methods such as the Schrodinger-Poisson approach encounter difficulties. 43 As shown in Figure 1, our supercells consist of two atomically thin GaN quantum wells [1 or 2 monolayers (MLs)] separated by AlN barriers of varying thicknesses (1-7 MLs). To minimize interactions with the periodic images of the QWs, a thick AlN barrier (19 MLs) is included, resulting in a supercell structure of (AlN) 19

We perform DFT structural relaxation and band calculations with Quantum Espresso 44 and set the in-plane lattice constant of GaN QWs to that of the AlN template to mimic pseudomorphic growth. We then apply many-body perturbation theory calculations with the BerkeleyGW package [45][46][47] to obtain the quasi-particle band structure with the G 0 W 0 method and the exciton properties with the Bethe-Salpeter equation (BSE). Computational details are listed in the Supporting Information.

We first analyze the positions of electrons and holes in our investigated heterostructures to identify the presence of charge-transfer exciton states. In Figure 1, we show the wave functions of the lowest unoccupied and highest occupied states for a typical simulated heterostructure, along with the atomic layers. We average wave functions Ψ e,h over the plane perpendicular to the c-axis

where A is the normalization area along the x-y plane of the supercell, and obtain their envelope functions ψ ̅ e,h (z) by interpolating the wave function local maxima with splines. The shapes of the full wave functions and the envelope functions for all simulated structures are illustrated in Figure S1. Both AlN and GaN inherently exhibit spontaneous polarization due to their lack of inversion symmetry within their wurtzite crystal structures, leading to the emergence of a built-in electric field even in the absence of an external electric field. Moreover, the formation of a heterostructure between AlN and GaN induces strain in the crystal lattice due to lattice mismatch, resulting in piezoelectric polarization. The discontinuity in the macro-scopic polarization across the AlN/GaN interface causes charge accumulation and a potential drop at the interface, 36,37 as illustrated by the black solid lines in Figure 1. These characteristics create an energy differences of ∼0.2 and ∼0.8 eV (depending on the specific geometry) between equivalent quasiparticle states of the two QWs. As a result, the lowestenergy electrons and holes are separately confined in different QWs, forming a spatially indirect electron-hole pair, as shown in Figure 1. This promising result suggests that these structures offer exciting opportunities to spatially separate electrons and Nano Letters To verify the formation of IX, we visualize the electron part of the exciton wave function of monolayer QWs in real space:

where c and v denote the indices for the electron and hole states, respectively (i.e., electronic vacancy in the v band), and k represents the momentum. The BSE computation produces electron-hole amplitude A vck S of exciton state S. For visualization, we select the lowest-energy exciton state for S, while ψ c and ψ v correspond to electron and hole wave functions, respectively. We fix hole position r h to the left GaN QW to plot eq 2 as a function of electron position r e . Interestingly, we observe that most of the electron charge is localized on the right QW for 1 ML AlN barriers, forming an IX for both monolayer and bilayer QWs (Figure 1a,c). However, as we increase the barrier thickness from 1 to 7 MLs, the electroncharge distribution shifts to the left QW in the monolayer case, indicating the transition from IX to DX (Figure 1b). Meanwhile, excitons in bilayer QWs remain indirect even for 7 ML barriers, with its electron-charge distribution localized in the right QW (Figure 1d). The electronic parts of the exciton wave functions for all simulated structures are illustrated in Figure S2. We denote that throughout the following discussion, IXs refer to excitons primarily originating from electrons in the right quantum well (e1 state in Figure 1) and holes in the left quantum well (h1 state in Figure 1). Conversely, DXs refer to excitons primarily originating from electrons in the left quantum well (e2 state in Figure 1) and holes in the left quantum well (h1 state in Figure 1).

To determine which factor governs the formation of IX or DX, we first analyze how the energy levels of exciton states change relative to those of unbound electron-hole pairs as we vary the separation distance between two atomically thin QWs. Panels a and b of Figure 2 compare the energy levels of the unbound indirect and direct electron-hole pair states (E e1h1 = E e1 -E h1 , and E e2h1 = E e2 -E h1 ), the DX state (E DX ), and the energetically lowest exciton state (E LX ) as a function of this separation. We observe that the energy differences between the spatially indirect (E e1h1 ) and spatially direct (E e2h1 ) electronhole pair states are smaller in the monolayer QWs, indicating the higher probability of the hybridization between direct and indirect states in monolayer QWs. In addition, the difference between the energy of the DX state and the lowest electronhole pair state is much smaller in monolayer QWs than in bilayer QWs. Most importantly, the crossover between E DX and E e1h1 (★ in Figure 2a) indicates that the DX state eventually becomes bound below the unbound electron-hole states as the AlN barrier thickness increases. On the basis of this observation, we conclude that the transition from IX to DX emerges when E DX becomes lower than E e1h1 . The energy levels of the conduction and valence band states of all simulated structures as well as the polarization fields in the AlN barrier are summarized in Table S2. The electrostatic potential of the coupled atomically thin GaN QWs, plotted for each supercell structure, is presented in Figure S3.

To determine whether the lowest-energy excitons are IXs or DXs, we analyze the ratio R LX = ϕ indirect /ϕ total between the oscillator strength of the indirect e-h pair [

and the total oscillator strength [ϕ total = ∑ cv |ϕ cv (r = 0)| 2 ] for the lowest-energy singlet exciton integrated over the Brillouin zone. We evaluated this ratio directly from our BSE calculations as a function of AlN barrier thickness, as shown by the solid lines in panels c and d of Figure 2. A ratio approaching 1 signifies that the lowest-energy exciton primarily comprises the indirect electron-hole pair, classifying it as an IX. Conversely, a ratio that approaches 0 indicates the lowest-energy exciton is mainly composed of the direct electron-hole pair, thereby identifying it as a DX. For monolayer QWs, we observe a sharp decrease in the ratio as the number of AlN separation layer increases from 5 to 7 MLs. This indicates that the IX-to-DX transition occurs beyond 5 ML, consistent with our observations depicted in Figure 2a. For bilayer QWs, the ratio remains close to 1 for all simulated heterostructures, indicating that the lowest-energy exciton predominantly consists of spatially indirect electron-hole pairs.

Moreover, we analyze the square overlap of the lowestenergy electron and hole envelope functions for all simulated heterostructures, as shown by the dashed lines in panels c and d of Figure 2. For both monolayer and bilayer QWs, we observe that the square overlap of the electron-hole envelope functions decreases exponentially with an increase in separation distance between electrons and holes. Bilayer QWs show a more rapid decrease in the overlap with an increasing barrier thickness compared to those of monolayer QWs, which can be attributed to the better confinement of carriers in bilayer QWs. This behavior is consistent with previously reported electronic bandgaps of 5.44 eV for monolayer GaN and 4.69 eV for bilayer GaN in the AlN barrier. 7,8 The lower bandgap in bilayer QWs results in a higher potential barrier, leading to a reduced level of wave function leakage into the AlN barrier and the neighboring GaN QW. This reduction in the level of leakage prevents the coupling between spatially direct and indirect states in bilayer QWs, effectively inhibiting the transition from IX to DX.

Furthermore, we focus on how excitonic properties such as the exciton binding energy and the radiative decay time vary with an increase in barrier thickness. To realize excitonic devices that are operational at room temperature, a strong exciton binding energy that prevents thermal dissociation of excitons is necessary. Figure 3a shows the binding energies of the lowest-energy excitons in the simulated QWs. For both monolayer and bilayer QWs, extreme quantum confinement effects lead to significantly enhanced exciton binding energies (75-240 meV) compared to that of bulk GaN (20 meV) 48 that prevent the thermal dissociation of excitons at room temperature. [7][8][9] For the bilayer QWs, the binding energy monotonically decreases in proportion with 1/d, with an increasing AlN barrier thickness (d). We attribute the reduction of the binding energy to the larger spatial separation between the electron and hole wave functions, similar to what has been found for the IXs in van der Waals heterostructures. 49,50 Meanwhile, the exciton binding energy of the monolayer QWs deviates from the 1/d trend as we increase the barrier thickness from 5 to 7 MLs, which indicates that the spatial separation of the electron and hole decreases and the lowest-energy exciton gradually changes its character from IX to DX.

We also characterize the binding energy of the lowest-energy triplet exciton for the limiting cases of monolayer QWs with a 1 ML AlN barrier and bilayer QWs with a 7 ML AlN barrier. The singlet-triplet exchange splitting energies were estimated to be 6 and 0.05 meV, respectively. The small splitting energies indicate that dark triplet excitons can be easily converted into singlet excitons by thermal excitation. Also, the larger spatial separation of electrons and holes results in much smaller exchange splitting, which agrees with previous reports. 51,52 Next, we analyze the radiative properties of excitons in this coupled QW system. In planar structures, the light-matter interaction conserves momentum q between the in-plane momentum of photons and the center-of-mass momentum of excitons. For light, the in-plane momentum is strictly limited by the wavelength of the photon, |q| ≤ 2π/λ, where the wavelength is ∼250 nm for the GaN QWs studied here. In contrast, exciton wave vector q is ∼100 times larger than the exciton size, which is ∼2 nm according to Figure 1. Thus, momentum conservation implies that only a small fraction of bright excitons with |q| ≤ 2π/λ can undergo radiative decay. Radiative decay constant γ S , which characterizes how the population of bright excitons in state S decays as e -γ S t , has been microscopically derived using methods such as BSE 53,54 or semiconductor luminescence equations, 31 leading to the following expression (in SI units):

where q S ≡ E S /ℏc is the photon wave vector matching the energy of exciton E S and

A vck S is the excitonenhanced dipole moment. Here d k cv is the microscopic dipole moment between the conduction and valence bands, and A vck S is the electron-hole amplitude, as also defined in eq 2. Finally, the radiative decay time is given as the inverse of the radiative decay rate (τ S = 1/γ S ).

Figure 3b shows the radiative decay time of the lowestenergy 1s excitons (τ 1s ) as a function of barrier thickness. The bilayer QWs exhibit a roughly exponentially increasing τ 1s over 2 orders of magnitude (from 1 to 100 ps) for an increasing barrier thickness from 1 to 7 MLs. This result confirms that bilayer QWs can serve as an excellent platform for realizing long-lived IXs. At the same time, the monolayer QWs exhibit a much faster τ 1s (from 300 fs to 1 ps) that first increases and then decreases with an increase in well separation. The τ 1s peak matches the IX-to-DX transition (Figure 2c) for the lowestenergy exciton state because DXs have stronger electron-hole overlap and lower τ 1s (larger γ S ) than do IXs.

Assuming that bright excitons are thermally distributed at temperature T, the total radiative decay constant of bright excitons follows from a thermal average (at q = 0)

for dilute excitations that follow the Boltzmann distribution.

The corresponding bright exciton (BX) lifetime [τ(T) = 1/ γ(T)] is plotted as a function of temperature in Figure 4. For monolayer QWs, the BX lifetime depends only weakly on temperature, indicating that the radiative decay constants of the lowest-energy S states are similar in magnitude. For bilayer QWs (Figure 4b), the lifetimes for 1 and 3 ML AlN barriers are similarly insensitive to temperature, whereas 5 and 7 ML AlN barriers produce a strong temperature dependence. Specifically, the 7 ML AlN barrier results in a maximum exciton lifetime at ∼130 K, after which the BX lifetime decreases with further increases in temperature. We attribute this nonmonotonic behavior to the diminishing energy difference between the IX and DX, as shown in Figure 2b. For higher temperatures, the reduced energy gap leads to a significant occupation of the direct exciton states, as illustrated in Figure S4. This increased presence of DX states at higher temperatures explains the observed decrease in the exciton lifetime. Additionally, the BX exciton lifetime can approach 100 ps at room temperature for the 5 ML AlN barrier, which is much longer than that typically expected for BXs.

To estimate the exciton lifetime under operational conditions, we need to consider the momentum distribution of the entire thermalized exciton population. A fraction, denoted as F bright , of bright excitons decays radiatively with a rate γ(T) via the s-polarized modes. The total exciton density decays exponentially with a time constant given by T T F ( ) ( ) life 4 3 bright , where the factor of 4 / 3 accounts for the emission into both s-and p-polarized light modes. 55 This results in an exciton lifetime [τ life (T) = 1/γ life (T)], and the fraction of bright excitons at thermal equilibrium [F bright(TD) ] is given by 56

= F T ( ) 1 e q Mk T bright(TD) /2 2 1s 2 B

where M is the total electron-hole mass. Using eq 5 often overestimates the decay constant and underestimates the lifetime because momentum-selective recombination depletes bright excitons, creating a "hole" in the momentum distribution of excitons. 57,58 The correct F bright changes with excitation conditions and over time, as predicted by fully dynamic computations [57][58][59] that include the nonequilibrium dynamics among radiative hole burning, bright-dark exciton thermalization, and exciton formation. Experiments have already demonstrated a reduction in F bright by a factor of 20 compared to F bright(TD) at low temperatures, 60,61 including weak plasma-dominated PL. However, F bright will approach its thermal limit F bright(TD) whenever the relaxation process replenishes the momentum distribution faster than the recombination process depletes it. At 300 K, relaxation time τ rlx can be on the order of subpicoseconds, 62,63 implying that in cases with 3, 5, and 7 ML AlN barriers (as shown in Figure 4b), the condition τ(T) ≫ τ rlx is satisfied. Then, eq 5 predicts that at 300 K (room temperature), F bright approaches F bright(TD) = 5 × 10 -4 , which enhances the exciton lifetime {τ life (T) = 1/ γ life (T) = 3/[4γ(T)F bright(TD) ]} to ∼100 ns for the 5 ML AlN barrier structure. This demonstrates the significant potential of these bilayer QWs as platforms for long-lived excitons at room temperature.

In addition to theoretical predictions, we used molecularbeam epitaxy (MBE) to grow, to the best of our knowledge, the first set of samples with atomically precise GaN QW pairs, separated by varying AlN barrier thicknesses as the control parameter. Using Veeco GEN II plasma-assisted MBE, we grow these heterostructures on 3 in. Si(111) substrates. Our previous study shows that we can accurately grow III-nitride monolayers, [7][8][9]64 whereas the direct growth of Al(Ga)N on Si substrates results in highly coalesced nanowires and/or quasifilm structures, 65 complicating the realization of the desired IX heterostructures. To overcome this challenge, we adopt a multisegment growth strategy to achieve the required precision. The process starts with 100 nm GaN nanowires as a template, followed by 120 nm Si-doped AlN nanowires and a subsequent 20 nm intrinsic AlN segment. The critical heterostructure layers consist of two ML GaN QWs separated by AlN barriers of varying thicknesses. Detailed experimental information is available in the Supporting Information.

In panels a and b of Figure 5, we show the structural characterization of coupled GaN quantum wells separated by a thin AlN barrier. One can see that a continuous thin AlN barrier and GaN quantum wells are formed in the AlN nanowire. These atomically thin GaN/AlN heterostructures provide us with a material platform for exploring the presence of IX signatures. However, direct absorption measurements are