Semiconductor-Semimetal Transition in van der Waals Carbyne Crystals

TL;DR

This paper investigates the semiconductor-semimetal transition (SST) in freestanding van der Waals crystals of carbynes (carbyne). It develops a fully three-dimensional tight-binding model, validated against density functional theory, to capture intra-chain Peierls distortion and inter-chain coupling in AA and AB stackings. The study predicts SST to occur for chains longer than about 42 atoms and identifies a hyperbolic van Hove singularity leading to unique hyperbolic exciton states in the semiconductor phase. Experimental photoluminescence measurements on finite chains corroborate the predicted chain-length dependence of the optical oscillator strength and support the SST picture, highlighting chain-length engineering as a route to band-gap control in one-dimensional carbon crystals.

Abstract

Freestanding van der Waals crystals made of single-atom carbon chains (carbynes) have been recently realized technologically. Here we investigate their electronic and optical properties experimentally, by continuous-wave and time-resolved photoluminescence spectroscopy, and theoretically. Employing a fully three-dimensional tight-binding formalism benchmarked against density functional theory calculations we predict the semimetal-semiconductor transition to occur in van der Waals carbyne crystals composed by the chains of about 42 atoms long. The semiconductor phase is characterized by a hyperbolic van Hove singularity which gives rise to unconventional hyperbolic exciton states. Experimentally, we access the semiconductor phase, where resonant features associated with hyperbolic excitons are clearly visible. The exciton oscillator strength is found to be strongly sensitive to the length of carbon chains: our experiments show that it decreases with the increasing chain length. This tendency, confirmed by the theoretical modeling, manifests the evolution of the hyperbolic exciton state on the way to the semiconductor-semimetal crossover. Our approach accounts for the actual crystalline geometry, including alternating intra-chain hoppings and inter-chain couplings. By fitting tight-binding dispersions to density functional theory data we extract consistent parameters and establish a comprehensive framework for the physics of carbyne crystals. This study paves the way towards efficient band-gap engineering in ultimate one-dimensional carbon crystals.

Figures (3)

• Figure 1: Schematic representation of the van der Waals carbyne crystal in the two stacking geometries. a) Three-dimensional view of the AA stacking, with intra-chain hoppings $t_1$, $t_2$ and inter-chain coupling $\gamma_1$. b) Three-dimensional view of the AB stacking, where the nonequivalent couplings $\gamma_2$ and $\gamma_3$ are included. c) Two-dimensional schematic of bond alternation, with atoms $a_i,b_i$ (first chain) and $\tilde{a}_i,\tilde{b}_i$ (second chain) labeled explicitly.
• Figure 2: Electronic band structures of van der Waals carbyne crystals. (a) Comparison of DFT and TB band dispersions along the $k_z$ direction for the AA stacking. (b) Three-dimensional representation of the electronic band structure for the AA stacking obtained from the tight-binding model. (c) Comparison of DFT and TB band dispersions along the $k_z$ direction for the AB stacking. (d) Three-dimensional representation of the electronic band structure for the AB stacking obtained from the tight-binding model. The fitted TB parameters are summarized in Table \ref{['tab:TBparams']}.
• Figure 3: Finite-size band gap $E_{\mathrm{gap}}(N)$ as a function of the chain length $N$, obtained from Eq. \ref{['eq:Eg_finite']} (red curve, left scale). The change of sign of $E_{\mathrm{gap}}$ indicates the semiconductor-to-semimetal transition, which occurs at $N \approx 42$. Points show the experimental PL data from Refs. Kutrovskaya2020Kutrovskaya2021 with black curve serving as a guide for the eye. The violet curve and points show the predictions of a tight-binding model for a single finite polyyne chain confined with gold atoms. The data are taken from Refs. Portnoi. The blue curve (right scale) shows the oscillator strength $f_{cv}(N)$ of the hyperbolic exciton plotted as a function of chain length $N$ (normalized units). The left inset shows the saddle point $(0,0,\pi/c_0)$ of the conduction band associated with the considered exciton resonance. The right inset shows the zoom on theoretical dependence of the exciton oscillator strength on the length of the chains compared to the experimental data obtained by time-resolved photoluminescence spectroscopy at 4K. The vertical line indicates the critical chain length corresponding to SST.