Topologically Protected Photovoltaics in Bi Nanoribbons

Abstract

Photovoltaic efficiency in solar cells is hindered by many unwanted effects. Radiative channels (emission of photons) sometimes mediated by nonradiative ones (emission of phonons) are principally responsible for the decrease in exciton population before charge separation can take place. One such mechanism is electron-hole recombination at surfaces or defects where the in-gap edge states serve as the nonradiative channels. In topological insulators (TIs), which are rarely explored from an optoelectronics standpoint, we show that their characteristic surface states constitute a nonradiative decay channel that can be exploited to generate a protected photovoltaic current. Focusing on two-dimensional TIs, and specifically for illustration purposes on a Bi(111) monolayer, we obtain the transition rates from the bulk excitons to the edge states. By breaking the appropriate symmetries of the system, one can induce an edge charge accumulation and edge currents under illumination, demonstrating the potential of TI nanoribbons for photovoltaics.

Keywords: Exciton; Optics; Photovoltaics; Topological insulator; Two-dimensional materials.

Figure 1

Schematic representation of the proposed mechanism. Device where an exciton wave packet is created at the bulk of the sample, where it will diffuse in any direction. Excitons entering the top ribbon present a finite momentum Q, giving rise to an out-of-equilibrium edge carrier population with nonzero momentum and velocity, thus forming a topologically protected current.

Figure 2

(a) Bi(111) zigzag nanoribbon where the dissociation process takes place. The highlighted atoms denote the unit cell, and a⃗ is the Bravais vector. The edge atoms are identified with the rectangles and labeled as L (left) or R (right). We introduce onsite energies on the left edge to split the topological edge bands. (b) Band structure of a Bi(111) zigzag ribbon for N = 20, with the edge bands highlighted in green.

Figure 3

Splitting of the edge bands. (a) For a topological insulator with inversion symmetry, the edge bands of both sides are degenerate, resulting in identical rates for intraedge and interedge transitions. (b) The introduction of an edge offset potential allows the splitting of the edge bands, producing a distinction among the different transitions. (c) For each edge e–h pair, we can determine its total velocity as v_e–h = v_e – v_h(43) to establish whether it carries current. For the pair drawn in (b), we observe that v_e > 0 and v_h < 0, meaning that v_e–h = v_e – v_h > 0. (See the Supporting Information for the definition of the velocity.)

Figure 4

Transitions at Q = 0. (a) Band structure of the Bi(111) ribbon for N = 20 and w = 0.2 eV. The edge bands are colored according to the electronic occupation at the edges of the ribbon. (b) Real-space electronic density probability of the ground-state exciton for N = 12. (c) Transition rates of the ground-state exciton to the different edge electron–hole pairs as a function of the width of the ribbon N for w = 0.2 eV. (d, f) Transition rates and edge occupation as a function of the edge offset potential w for N = 14. (e, g) Transition rates and ground-state exciton energy as a function of the dielectric constant ε for N = 14. (c, d, e, and f) share the same legend.

Figure 5

Transitions at finite Q. (a, b) Band structure of the Bi(111) ribbon for N = 20 and w = 0.2 eV. The first one shows the edge occupation of the bands, and the second one shows the average spin projection of the bands. (c) Transition rates of the ground-state exciton as a function of Q for N = 14. (d) Low-energy exciton band structure. Each color corresponds to four excitonic states in total. (e) Center-of-mass velocity of the ground-state exciton. The shadowed region denotes the fraction of excitons that do not contribute to the formation of an edge current. (f) Velocity v = v_e – v_h of the relevant electron–hole pair and of each component individually, for N = 14 and w = 0.2 eV. (g) Transition rates as a function of N for Q = 0.1 Å^–1 and w = 0.2 eV. (h) Transition rates as a function of the staggered potential V_st for N = 14 and w = 0.2 eV. The inset shows the total spin projection of the ground-state exciton, , as a function of the staggered potential.