Exciton energy spectra in polyyne chains

Recently, we have experimentally observed signatures of sharp exciton peaks in the photoluminescence spectra of bundles of monoatomic carbon chains stabilized by gold nanoparticles and deposited on a glass substrate. Here, we estimate the characteristic energies of excitonic transitions in this complex quasi-one-dimensional nano-system with use of the variational method. We show that the characteristic energy scale for the experimentally observed excitonic fine structure is governed by the interplay between the hopping energy in a Van der Waals quasicrystal formed by parallel carbon chains, the neutral-charged exciton splitting and the positive-negative trion splitting. These three characteristic energies are an order of magnitude lower than the direct exciton binding energy.

Recently, we have experimentally observed signatures of sharp exciton peaks in the photoluminescence spectra of bundles of monoatomic carbon chains stabilized by gold nanoparticles and deposited on a glass substrate [1]. Here, we estimate the characteristic energies of excitonic transitions in this complex quasi-one-dimensional nano-system with use of the variational method. We show that the characteristic energy scale for the experimentally observed excitonic fine structure is governed by the interplay between the hopping energy in a Van der Waals quasicrystal formed by parallel carbon chains, the neutral-charged exciton splitting and the positive-negative trion splitting. These three characteristic energies are an order of magnitude lower than the direct exciton binding energy Introduction.-Being monoatomic chains of carbon atoms carbynes represent ultimate one-dimensional crystals. Carbynes are linear chains of sp 1 -hybridized carbon atoms. Their two known allotropes are polyyne, characterized by alternating single and triple electronic bonds between carbon atoms, and cumulene, characterized by double bonds between atoms. Theoretical and recent experimental studies indicate the semiconducting behavior for polyyne chains and quasicrystals, while infinite cumulene chains are expected to be metallic [2,3]. The strain in finite size chains of polyyne results in the enhancement of the direct band gap, so that experimentally achievable polyyne structures are expected to be emitting visible light in a wide spectral range that is dependent on the specific geometry of the structure [4,5]. This makes polyyne based nanostructures highly promising for the realization of light-emitting diodes and nano-lasers. In order to predict the quantum efficiency of carbon-based optical nano-emitters one should learn more about excitons in linear carbon chains. The ex-citon binding energy in sp 1 -carbon may be quite large due to the strong two-dimensional quantum confinement. Recently, we detected the strong excitonic features in low-temperature photoluminescence spectra of polyyne bundles stabilized by gold nano-particles and deposited on a glass substrate. The focus of our study was on elongated polyynic molecules, containing straight parts of even numbers of atoms (from 8 to 24) regularly separated with kinks. The synthesised sp 1 -hybridization chains were packed in hexagon bundles characterised by the distance between neighboring parallel chains of 5.35 A [6]. The chains were hold together by the Van der Waals force (see the schematic in Fig. 1(a)). They were grown by the laser ablation in liquid (LAL) and deposited on a fused quartz substrate for the photoluminescence (PL) study. The chains were stabilized by gold anchors attached to their ends [6]. In low temperature PL spectra, we have observed the characteristic triplet structure ( Fig. 1(b,c)). The triplet is invariably composed of a sharp intense peak accompanied by two broader satellites shifted by about 15 and 40 meV [1] to the lower energy side of the main peak, respectively. Very interestingly, the triplet structure is found to be nearly identical in carbon chains of different lengths. It moved as a whole with the band-gap variation as the length of the chain changed. We assign the observed sharp peaks to the optical transitions associated with neutral and charged excitons in polyyne bundles. Indeed, as any direct band-gap semiconductor, polyyne is expected to sustain exci-tons. Sharp resonances that emerge at low temperatures are clear signatures of the excitonic emission. Our time-resolved photoluminescence (TRPL) measurements confirm this assumption [1]. In Ref. [1] we attributed the main peak of every triplet to the neutral exciton transition and two lower energy satellites to charged exciton (trion) transitions, respectively. The main argument supporting this interpretation was the detected dipole polarization of about a half of carbon-metal nanocomplexes that was revealed by their alignment in the presence of the external electric field [6].
To gain better understanding of the observed resonances, one needs to compare the anticipated exciton and trion energies in the system, estimate the thermal hopping energy and the splitting between positively and negatively charged trions. Here we attempt analysing the energy spectra of excitons in carbyne-based nano- systems with use of the variational method. We rely on the effective mass approximation. Using this formalism we were able to shed light on the interplay between several characteristic energies that may be responsible for the fine structures observed experimentally, namely, the positive-negative trion energy splitting, the neutralcharged exciton energy splittings and the hopping energy that splits spatially direct and indirect exciton states.
The exciton modelling.-We recall that a Wannier-Mott exciton is a neutral quasiparticle [7], whose optical features strongly depend on the dimensionality of a semiconductor crystal [8]. Much attention has been focused on studies of excitons in strongly confined quantum systems [9,10]. Stronger confinements typically result in larger binding energies of excitons [11]. In addition to the exciton features, similar spectral resonances corre- sponding to negatively (X -) and positively (X+) charged exciton complexes, trions, consisting of two electrons and one hole and two holes and an electron, respectively, have been studies in doped semiconductor structures [12,13]. From the theoretical point of view, the excitonic problem is usually treated in the framework of the effective mass approximation. Resolving the Schroedinger equation for the relative motion of electron and hole [14], one finds the fine structure of excitonic transition that varies from hydrogen-like, in bulk crystals [7], to 3D-quantum box spectra in small nanocrystals [10]. More challenging is the description of trion states, where a variety of many-body effects may come into play [15,16]. In the present study, we rely on the simplest quasi-analytical approach, that seems to be the best adapted at the stage where very little is known about the electronic and optical properties of the relatively complex hybrid nanostructure under study. We extend the approaches developed in Refs. [14,17] for the variational calculations of exciton states and Ref. [18] for the analysis of trion states. We assume that electron and hole localization radii in the plane normal to the axis of the cylinder are given, respectively, by the parameters Re and Rh that are much smaller than the exciton Bohr radius. In this case, the electron and hole confinement problem in the frozen out at the liquid Helium temperature. From this ensemble of data, one can estimate the thermal hopping energy as 10-15 meV.
In conclusion, the developed variational model allowed us to confirm the origin of characteristic energy splittings experimentally observed in low-temperature PL spectra of polyyne chains. We predict the exciton and trion binding energies to be of the order of 250-300 meV. The triplet fine structure that has repeatedly been observed in polyyne chains of different lengths is likely to be due to the spatially direct neutral exciton, position and negative trion peaks, respectively. The non-radiative exciton decay channel observed in the time-resolved PL spectra at the room temperature is most probably associated with the thermal hopping of one of the carriers forming the ex-citon between parallel polyyne chains. Excitons are not destroyed by this hopping process but they become radiatively inactive or dark. We realise that the proposed model has several shortcomings. First, we assumed the carbon chains to be infinitely long. The finite sizes of the chains can be incorporated to the model at the cost of one or two supplementary adjustable parameters. We opted for keeping the model as simple as possible having in mind that, experimentally, the exciton-trion triplet appears to be essentially independent of the length of the chain. Another limitation of the validity of our approach comes from the limited accuracy of the effective mass approximation in nano-systems of such a small size as bundles of polyyne chains composed by 10-20 atoms each. Still, we are confident that the method predicted a correct order of magnitude for the exciton and trion binding and hopping energies, as the comparison with available experimental data certifies. Further experimental and theoretical studies are needed to reveal the spin structure and transport properties of quasiparticles in linear carbon chains.
Acknowledgement. The work is supported by the Westlake University, project 041020100118 and the Program 2018R01002 funded by Leading Innovative and Entrepreneur Team Introduction Program of Zhejiang, by RFBR grant 18-32-20006 and by MSHE within the State assignment VlSU 0635-2020-0013. A.K. acknowledges Saint-Petersburg State University for the research grant ID 40847559.