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Erik S.
Skibinsky-Gitlin
^{a},
Salvador
Rodríguez-Bolívar
^{ab},
Marco
Califano
*^{c} and
Francisco M.
Gómez-Campos
^{ab}
^{a}Departamento de Electrónica y Tecnología de Computadores, Facultad de Ciencias, Universidad de Granada, 18071 Granada, Spain
^{b}CITIC-UGR, C/Periodista Rafael Gómez Montero, n 2, Granada, Spain
^{c}Pollard Institute, School of Electronic and Electrical Engineering, Bragg Centre for Materials Research, University of Leeds, Leeds LS2 9JT, UK. E-mail: m.califano@leeds.ac.uk

Received
11th October 2019
, Accepted 24th November 2019

First published on 25th November 2019

We present a theoretical study at the atomistic level of the optical properties of semiconductor nanocrystal films. We investigate the dependence of the absorption coefficient on size, inter-dot separation, surface stoichiometry and morphology, temperature, position of the Fermi level and light polarization. Our results show that, counter-intuitively, huge blue shifts are expected in some intra-band transitions for strongly coupled arrays, in contrast with the predicted and observed red shift of the band gap absorption in such systems. Furthermore, we find that the energies of such transitions can be tuned within a range of several hundreds of meV, just by engineering the inter-dot separation in the film through the choice of appropriately sized capping ligands. Finally we discuss the application of this effect to nanocrystal-based intermediate-band solar cells.

We showed in the past^{3} that both stoichiometry and surface morphology of the single dots have a strong influence on the film's miniband structure. However, how these features translate into measurable differences in the optical properties remains to be investigated.

Another interesting question concerns the effect of the Fermi level's position on the optical properties of a NC film, considering that in some situations the samples may be doped, either optically (photodoping) or electrochemically.

In order to address these issues and to investigate the direction and origin of the energy shift(s) of the different intra-conduction-band transitions, here we carry out a comprehensive theoretical study of the optical properties of nanocrystal films, using LDA-quality wave functions for the isolated dot (obtained within the atomistic semiempirical pseudopotential method^{13}), as a basis set for a tight-binding calculation of the film's band structure and optical spectra. We investigate the dependence of the latter on (i) size, (ii) inter-dot separation, (iii) surface stoichiometry and (iv) morphology, (v) temperature, (vi) Fermi level's position, and (vii) light polarization. We find that the key properties determining the shift in any transition energy are specifically the width and shape of the minibands involved (which can differ in films made of NCs of the same size and material, but different stoichiometries), rather than simply the degree of deconfinement of the carriers in general: indeed such shifts can be either to the red or to the blue, depending on the relative positions (with respect to each other) of the miniband minima in the Brillouin zone, with larger shifts occurring, in decreasing order of importance, (a) for smaller dots, (b) for shorter inter-dot separations, (c) in films made of dots with anion-rich surfaces, and (d) for structured surface morphologies that allow some degree of dot interlocking. These effects are more clearly seen when the Fermi level is situated at or below the CBM-derived miniband minimum and at low temperatures.

We also show that, in the strong coupling regime, the film's optical spectrum can be very different from that of the isolated dot and therefore cannot be simply deduced from single-dot spectroscopy. Finally, based on our results we propose a new scheme for the realisation of NC-based intermediate band solar cells.

Fig. 1 Miniband structure for the lowest conduction band states in structures A–D (type A, B and D NCs are separated by one bond length, whereas type C dots are 0.25 bond lengths apart, as in this structure the dots are interlocked through their surface As atoms^{3}). The energies are referred to the vacuum level. The insets show the atomistic structure of the different systems, where yellow and green spheres represent As and In atoms, respectively. |

The absorption coefficient is calculated within the electric dipole approximation using Fermi's Golden Rule, as

(1) |

Film | Isolated dot | |||||||
---|---|---|---|---|---|---|---|---|

A | B | C | D | A | B | C | D | |

Δ_{1→2} |
930 | 779 | 545 | 483 | 648 | 706 | 470 | 475 |

Δ_{1→3} |
969 | 838 | 576 | 558 | 669 | 726 | 502 | 506 |

Δ_{1→4} |
1019 | 843 | 577 | 569 | 669 | 726 | 502 | 506 |

Δ_{1→5} |
1060 | 961 | 892 | 828 | 815 | 880 | 815 | 814 |

System A exhibits a larger miniband intermixing, owing to its small size and the specific nature (anion-rich) of its surface. Interestingly system B, which is obtained from system A by exchanging anions and cations, exhibits a dramatically different miniband structure because of the significantly reduced wave function overlap between neighbouring dots originating from the different type of surface atoms. Such overlap is similarly small in systems C and D, due to their large size,^{3} leading to flatter minibands (M4 in particular). In all systems, with the only exception of system C, M2 features a flat region close to the Γ point, especially pronounced in system D.

Due to its importance in absorption, we report in Table 1 the energy difference between bands M1 and Mn at the Γ point, and compare it with the energy separation between the same levels in isolated dots, where it corresponds to the position of the lowermost inter-band absorption peak (black arrows in Fig. 2, 3, S1, and S2 ESI†).

We find that all values of in the films are blue-shifted compared with the same energy separations in isolated dots. The shifts range from about 8 meV, for in system D, to as much as 350 meV for in system A. As a consequence, if an intra-miniband transition in the film originates from the Γ point, we expect its energy to be blue-shifted: in the case of the (intra-miniband) absorption edge, Table 1 would predict blue shifts of up to 282 meV (system A).

When considering larger inter-dot separations, we will keep the absolute position of E_{F1} and the relative position of E_{F2} fixed. This means that, as displayed in Fig. 4, E_{F1} will be found progressively deeper into the band gap, while E_{F2} will remain in the middle of M1, as the energy of the bottom of M1 increases (i.e., moves upwards), due to the decrease in inter-dot coupling (Fig. 5(a)) – hence mininband width (Fig. 5(b)) – occurring with increasing separation. This will allow us to investigate the effects of both a constant and a varying Fermi level.

Fig. 5 Effect of interdot separation on the film's band structure: (a) coupling energy (i.e., overlap integral V_{11} = |〈ψ_{1}(r)|V(r)|ψ_{1}(r − r′)〉|; here ψ_{1}(r) is the single dot wave function for state 1 – the CBM – and V(r) is the potential), and (b) miniband width, calculated for M1 as a function of separation (indicated in both Å, lower x axis, and bond lengths, upper x axis), for all structures considered here. These quantities were calculated for d = 1 bl in films made of different materials (InAs, InSb, CdSe, and PbSe) in ref. 3. |

3.2.1 Effect of light polarization.
We find that, for each system, the positions of absorption thresholds and absorption peaks are rather independent of light polarization for each particular Fermi level and temperature, whereas their amplitude is more sensitive to it, although remaining approximately within the same order of magnitude for the different polarization directions. The order of magnitude of the absorption coefficient for all the systems considered is around 10^{5} to 10^{6} m^{−1}, similar to what is predicted for InAs/GaAs epitaxial dot arrays,^{15} suggesting an efficient absorption.

Interestingly we find that the same light polarization affects each system in a different fashion, depending on the symmetry of the wave functions in the various cases: while 110-polarized light yields the strongest absorption in A, it has the opposite effect in B and D; similarly, polarization along 100 leads to the highest absorption C but not in B and D, where the absorption is strongest for light polarized along 111 and 001, respectively.

3.2.2 Effect of T.
For E_{F} = E_{F1} and T = 77 K, M1 is mainly occupied around the Γ point, i.e., the lowest energy in the miniband: the calculated absorption peaks occur at the values of Δ_{1→m} (see Table 1), indicating that the absorption threshold is closely related to the 1 → 2 transitions taking place around Γ. At higher temperatures (T = 300 K) the absorption peaks remain at the same positions, but they are broadened to the red, due to the increased (thermal) occupation of higher energy states within M1 (which are closer to M2), leading to an enhanced photon absorption at lower energies.

In the case of systems B and C this broadening is also accompanied by the appearance of additional low-energy peaks leading to a substantial red shift in the absorption threshold. This is due to the small width of M1 in these systems (136 meV and 81 meV, respectively – see Fig. 5(b)) that leads to a more uniform occupation, and hence to a greater reduction of the absorption threshold energy due to the efficient photon absorption from M1 to M2 from every -state. If M1 is wide, however, only the states around the Γ point are occupied, and therefore the absorbed photons are mainly those with energies close to Δ_{1→2} (as in the case of systems A and D).

For E_{F} = E_{F2} the spectra exhibit little dependence on temperature. As the Fermi level is placed in the middle of M1 all the states around the Γ point are completely occupied, and so are also states higher in energy. The 1 → n transitions for different cover a wide range of energies, most of which are lower than Δ_{1→n}. Therefore the absorption threshold energy is significantly reduced. In this case the flatter M1, the smaller the temperature-induced energy red shift of the absorption threshold, when compared to the case E_{F} = E_{F1}.

3.2.3 Effect of the position of the Fermi level E_{F}.
A change of the Fermi level position from E_{F1} (the bottom of M1) to E_{F2} (the middle of M1) affects the occupation of minibands, and therefore the absorbed photon energies, proportionally to the width of M1 (see Fig. 4). For E_{F} = E_{F2}, system A exhibits, at both 77 K and 300 K, a considerable reduction in the absorption threshold position and a change in absorption coefficient behaviour as a function of energy, which shows a linear increase from very low energies, as opposed to the exponential increase found for E_{F} = E_{F1}. This is due to the strong inter-dot coupling in this system (see Fig. 5(a)), yielding both wide minibands and large miniband intermixing (see Fig. 1), that leads to enhanced absorption for a wider range of photon energies.

The red shift of the intra-conduction-band absorption threshold, when moving from E_{F1} to E_{F2}, is also predicted to occur, albeit to a lesser extent, in all the other systems, where it will, however, be more evident at low temperatures, due to the similar way in which temperature and Fermi level position affect the lower miniband's occupation.

3.2.4 Effect of inter-dot separation.
When the distance between the dots is increased, their coupling decreases leading to a flattening of all minibands^{3} (see Fig. 5 and 4), with the electronic structure tending to the discrete levels of the isolated dot, for large separations. The effect of this flattening is the more pronounced, the wider M1 for the lowest inter-dot separation: system A, therefore, is affected more dramatically than the other systems, as can be seen from Fig. 6(a) [the variation of the full spectra with increasing distance is displayed in Fig. S1 and S2 (ESI†)], which focuses on the case E_{F} = E_{F1} and T = 77 K (i.e., panels d and h in Fig. S1 and S2 ESI†). The main effects of increasing interdot separation are: (i) the increasing red shifts of the intraband absorption edge – Fig. 6(a) – (and of all other peaks), and (ii) the decreasing magnitude of the absorption coefficient – Fig. 6(b). The former feature evidences the progressive blue shift in the intra-band absorption edge occurring when transitioning from the dilute solution limit (large inter-dot distance) to close packed arrays (minimum dot-to-dot separation), which starkly contrasts with the red shift of the band gap absorption that we predict (see Fig. 7), and that has been observed experimentally,^{4,9–12} in the same conditions.

for the M1 → M2 transition, although nearly unchanged at Γ, increases strongly with inter-dot separation at the Brillouin zone boundaries (this is particularly evident in Fig. 8 for 110 polarization) leading to stronger absorption when M1 is flatter, hence more uniformly populated for all values of . This is exactly what happens for E_{F} = E_{F2}, when the flattening of the minibands also leads to a narrowing of the range of absorption energies (as the M1–M2 separation is more uniform), compared to systems with wider minibands. As a consequence, the absorption peaks become not only higher but also narrower for greater inter-dot distances. This is also the origin of the increase, with increasing separation, in the first absorption peak amplitude predicted for system D in Fig. 6(b).

Fig. 6 Effect of increasing interdot separation – expressed both in bond lengths (upper x axis) and Å (lower x axis) – on (a) the energy shift ΔE and (b) the amplitude, of the lowermost absorption peak in films of NCs of type A–D for E_{F} = E_{F1} and T = 77 K. (Full spectra are shown in panels d and h in Fig. S1 and S2 ESI†). ΔE is calculated with respect to the position of the first transition in isolated dots. In the case of system C, the dots are closer together due to the interlocking of the surface atoms. |

As discussed above, both effects (i) and (ii) are more pronounced in system A, where the peaks can shift by more than 250 meV and the absorption coefficient decreases by as much as 22 orders of magnitude for inter-dot separations ranging from 1 to ∼2 bond lengths, in contrast to a maximum variation of about 4 orders of magnitude for systems B and C. This striking difference in absorption reduction between system A on one side, and systems B and C on the other, is due to the fact that, considering the larger width of M1 (i.e., stronger coupling) in the former system, when E_{F} = E_{F1}, the distance from M1 to E_{F} increases more rapidly with increasing dot-to-dot separation (i.e., decreasing coupling) for A than it does for the other systems, in which the band is already much flatter and the coupling much weaker for the closest inter-dot separation (see Fig. 5 and 4). In other words, given that the width of M1 in system A is 392 meV and E_{F} is at its bottom for the shortest inter-dot separation, when this separation is increased the band's bottom can shift upwards by 392 meV increasing its distance from E_{F} by as much, whereas a narrow band such as M1 in D (width = 48 meV) can only move 48 meV away from E_{F}. The reduction in absorption is proportional to the distance between M1 and E_{F}.

The increase in temperature modifies the peak profiles, broadening the absorption threshold to the red, as commented above. At room temperature, the reduction in the magnitude of the absorption coefficient in system A is not as large as at low temperatures (the maximum reduction is only of about 5 orders of magnitude).

As discussed before, when E_{F} = E_{F2} the effects of temperature are negligible. The most remarkable feature we predict for this position of the Fermi level is the slight increase in absorption with inter-dot distance. The explanation for this effect is again related to the miniband flattening for greater inter-dot separations, and the interplay between miniband occupation and oscillator strength far from Γ. Indeed, we find that the main contribution to the variations in the absorption properties of the film with separation comes from the changes in the occupation of the different minibands, f(E_{n}) in eqn (1). The magnitude of the oscillator strength

(2) |

Interestingly, although the absorption threshold is of the order of hundreds of meV, the minimum gap between these two minibands (generally located at the Brillouin zone boundaries) could be as small as 50 meV (see Fig. 1).

As we discussed previously, one of the reasons for the greater absorption around the Γ point is the electron population of the minimum of M1. However, we find that transitions from the Brillouin zone center also have a larger oscillator strength eqn (2) than transitions from any other region in -space.

In system A the highest (lowest) absorption values are, in general, observed for light polarization along the 110 (011) direction. Fig. 8 shows the oscillator strength in the whole Brillouin zone for transitions from M1 to M2 for these light polarizations and for two different values of interdot separation: As expected, the 110 direction shows higher values than 011, by about a factor of 4, reflecting the difference found in the absorption curves. More importantly, for the closest dot-to-dot separation, the oscillator strength is orders of magnitude larger around Γ than at the Brillouin zone boundaries, demonstrating that the importance of the Γ point in absorption is not only related to the electron population, but also to the larger oscillator strength of transitions originating around that point.

We find that the oscillator strengths for all 1 → n transitions with n = 2, …, 4 have a similar magnitude. The transitions from M5 upwards are much weaker, hence the reduction in amplitude of the absorption peaks at high energy. A similar behaviour is found for all polarizations. This effect is illustrated in Fig. 9 where we show the oscillator strength for transitions 1 → n (2 ≤ n ≤ 7) in system A. Here the strongest transitions are found between M1 and the miniband triplet M2, M3, M4. Higher energy minibands are split from this set, and the oscillator strength is much reduced. Indeed we find that the contribution of the M1 → Mn transitions with 4 < n ≤ 7 to the absorption coefficient in Fig. 3(a) is negligible.

Another important aspect worth discussing is the inter-dot separation realistically achievable in technologically relevant CQD films. The dot proximity (hence their coupling) in such arrays is limited by the presence of the capping ligands, routinely employed to stabilize the surface of these nanostructures, whose length may vary from ∼2 nm for oleic acid,^{25} to 0.35 nm for oxalic acid.^{6} A further reduction of the dot-to-dot distance down to 0.1 nm can be achieved by the use of inorganic ligands, such as atomic halide anions (Cl^{−}, Br^{−} and I^{−}),^{26,27} which can lead to the observation of band-like transport, high electron mobility and high photoconductivity in CQD films.^{28} Reactive self-assembly^{29–31} or laser annealing^{32} can finally lead to a facet-to-facet bonding of the dots. The range 1 ≤ d < 2 bl (0.26 ≤ d < 0.5 nm) considered in this work represents therefore a realistic estimate of the inter-dot separations achievable in experimental films.

We have also previously shown^{34} that allowed optical transitions exist between VB minibands and M1, and between M1 and higher energy conduction minibands Mn, with n ≥ 2. M1 can therefore be considered as a sort of “stepping stone”, or intermediate band (IB), for transitions from the VB minibands to CB minibands Mn with n > 1. This mechanism is similar to that at the base of the intermediate band solar cell (IBSC) concept. The difference is that, instead of exploiting sub-bandgap photon absorption, this scheme would enable the harnessing of some of the energy in excess of the NC bandgap, but lower than the threshold for carrier multiplication,^{33} which would otherwise be lost to heating of the cell, with consequent degradation of its performance. We have recently investigated its viability,^{34} and the predicted maximum achievable solar energy conversion efficiencies were far in excess of the Shockley–Queisser^{35} limit.

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## Footnote |

† Electronic supplementary information (ESI) available. See DOI: 10.1039/c9na00647h |

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