Morphology and band structure of orthorhombic PbS nanoplatelets: an indirect band gap material

PbS quantum dots and nanoplatelets (NPLs) are of enormous interest for the development of optoelectronic devices. However, some important aspects of their nature remain unclear. Recent studies have revealed that colloidal PbS NPLs may depart from the rock-salt crystal structure of bulk


Introduction
PbS quantum dots have attracted much interest due to the possibility of tuning their bandgap in the NIR region through control of the size, 1 thus extending light absorption beyond the visible spectrum. Consequently, PbS dots have been extensively employed in the development of optoelectronic devices as solar cells, 2 LEDs 3 or photodetectors. 4,5 Recently, the synergistic combination with halide perovskites has been also explored, showing influence of PbS dots on the morphology of the halide perovskite, producing an increase of solar cell performance 6,7 and a significant increase on the long term device stability, 8 as well as advanced LEDs 9 and photodetectors. 10,11 The use of NPLs instead of dots offers additional possibilities for optoelectronic applications, which follow from their quasi-two-dimensional structure. These include large exciton binding energies, which are convenient for room temperature devices, giant oscillator strength enabling extremely bright emission, and extremely narrow emission linewidth, resulting from the precise control of the NPL thickness. [12][13][14] Specifically, PbS NPLs have been shown to combine a relatively narrow band gap, which gives access to near infrared photodetection and emission, [15][16][17] with other attractive properties, such as low-threshold carrier multiplication, 18 gate-controlled spin-orbit interaction, 19 strong multiquantum-well coupling, 20,21 stable dielectric response, 22 high surface homogeneity and efficient cross-linking, 23 and facile realization of topological and valleytronic effects. 24 Very recently, we have also shown that the addition of PbS NPLs increases the long term stability of unencapsulated perovskite solar cells as well as the performance and reproducibility of perovskite solar cells prepared at ambient conditions. 25 Despite the high interest in these systems, there are some aspects of PbS NPLs that remain unclear.
While early works on PbS NPLs assumed or inferred rock-salt crystal structure (as in bulk and nanocrystals), later experimental studies have evidenced that orthorhombic (P nma space group) modification may be also formed. 15,16 Orthorhombic NPLs seem to present a distinct optical response, namely weak luminescence at 650 − 800 nm, 15,16 which is much shorter than the emission wavelength often reported for rock-salt NPLs (1200 − 1500 nm) despite having similar thickness. Short wavelength emission was also observed in earlier studies of PbS NPLs, whose crystal structure was not unambiguously defined. 17,20 The nature of such a different optical response is still an open question. Previous theoretical studies about the electronic properties of PbS NPLs have mostly focused on the cubic modification, using either density function theory (DFT) 19,23,24,26 or k·p models. 27 Recently, Akkerman and co-workers calculated the band structure of orthorhombic PbS NPLs, within DFT schemes. 16 Their results suggest that the band structure is very similar to that of rock-salt NPLs, both showing a direct gap of similar energy. This result is in contrast with that reported for SnS and GeS NPLs, which crystallize in the same type of lattice, but exhibit indirect bandgaps. 28 It also leaves the seemingly distinct optical features of the orthorhombic PbS NPLs unexplained.
The band structure of bulk PbS in P nma phase has been calculated using DFT by Zagorac et al. 29 A direct band edge gap of 0.7 eV, wider than that in rock-salt PbS (0.4 eV), was obtained. While the increase in band gap (∆E g = 0.3 eV) is qualitatively consistent with the shorter wavelength of orthorhombic NPLs, it is much smaller than the empirical value, ∆E g ≈ 0.8 eV. Obviously, for comparison with actual NPLs, the role of quantum confinement must be addressed, as it is a major energetic contribution to the optical gap. 26,30,31 Besides, the calculations missed spin-orbit interactions, which are certainly important in lead, and were performed in the local density approximation (LDA), which is particularly unsuited to estimate band gaps. 32 To gain further insight into the photophysics of orthorhombic PbS NPLs, in this work we synthesize PbS NPLs, analyze the crystallographic structure, and use DFT and k·p modelling to analyze the associated electronic structure and optical transitions. From transmission electron microscopy (TEM) and selected area electron diffraction (SAED) measurements we determine the crystallographic phase and unit cell parameters. From DFT, we calculate the associated bulk band structure -including spin-orbit interaction-, and obtain effective masses in the valleys of interest. These masses are then plugged into effective mass Hamiltonians to account for quantum confinement and to quantify excitonic properties.
We find that the bulk band structure shows a band gap of 0.5 eV, similar to that of rocksalt PbS, but of indirect nature. From the effective masses, we deduce that the indirect band edge character is reinforced by quantum confinement along the a axis of the lattice (short axis (x) of the NPL), but it can switch to direct if strong confinement takes place in orthogonal directions. The exciton binding energy and Bohr radius are found to be comparable to those in cubic NPLs, which translates into weak influence of lateral confinement beyond ∼ 15 nm in-plane sides (longitudinal (y) and lateral (z) dimensions). The potential connection of the above results with spectroscopic experiments is discussed.

Synthesis of PbS Nanoplatelets
PbS NPLs were prepared according to an established procedure. 15 Briefly, potassium hydroxide and octadecanol were mixed in a 1:1 molar ratio at 150 • C under air. After cooling down to room temperature, carbon sulfide was injected swiftly. In this way, potassium octadecylxanthate (KOctdX) is formed. Then, lead octadecylxanthate (PbOctdX) was synthesized by mixing an aqueous solution of lead nitrate and KOctdX in an equimolar amount under vigorous stirring. The final product was filtered and washed with Milli-Q water for several times.
For the NPLs synthesis, 90 mg PbOctdX and 5 mL trioctylamine (TOA) were mixed in a 50 mL three-neck flask for 30 min at room temperature under vacuum. Then, the temperature of the reaction mixture was slowly increased to 80 • C under a N 2 atmosphere. It took around 15 min to reach the desired temperature. With the purpose to vary the dimensions of the NPLs, the reaction was conducted for 0.5, 1, 3 and 5 h, observing a change from orange to a darker color. After the NPLs synthesis was completed, the reaction was cooled down to room temperature, and 2 mL of toluene, 1 mL of oleic acid and 3 mL of 1,2-dichlorobenzene were added. The NPLs were next purified by adding excess of acetonitrile and centrifuged at 5000 rpm for 5 min. The NPLs were then dispersed in toluene and again precipitated by adding acetonitrile and centrifuged at 5000 rpm for 5 min. Finally, the purified NPLs were dispersed in hexane.

Morphological and optical characterization
TEM images and SAED patterns of PbS NPLs were achieved by a field emission gun TECNAI G2 F20 microscope operated at 200 kV. Average dimensions of the NPLs were obtained from the TEM images with ImageJ software. UV-Vis absorption spectra of NPLs solutions at different reaction times were acquired by using a PerkinElmer UV/VIS/NIR spectrometer Lambda 1050+. The wavelength range for the measurements was 450-1050 nm. Steady state photoluminescence (PL) measurements were conducted through a photoluminescence spectrophotometer (Fluorolog 3-11, Horiba). An excitation wavelength of 420 nm was used to perform the steady state PL in a wavelength range between 600-800 nm. For the calculation of the cell parameters from SAED patterns, we used the known Miller indixes (hkl) and the equation for the interplanar distance d in orthorhombic structures: with a, b, c the cell parameters to calculate.

Theoretical Methods
The band gap of rock-salt NPLs has been well described within DFT using a Perdew-Burke-Ernzerhof (PBE) exchange-correlation functional, with full inclusion of spin-orbit coupling (SOC). 26 To study the P nma modification, which in bulk form is stable at high pressure, 29,33 we used a PBEsol functional instead, which is known to be more accurate for solids under pressure, 34 also with SOC inclusion. By using the generalized gradient approximation (GGA), both PBE and PBEsol outperform bare LDA in estimating bulk band gaps. 35 Where necessary, to confirm the robustness of the results, the calculations at the PBEsol level were compared against PBE functional and GW approach.
Rather than calculating the band structure of few-monolayer NPLs, we calculated that of bulk PbS. In this way, we exploited the symmetry of the lattice to carry out numerically accurate estimates. Quantum confinement was introduced in a second stage using singleband (effective mass) k·p theory, which has proved successful in quantitatively describing the photophysics of thin NPLs prepared out of other materials. 27,30,31,36 This approach also circumvents possible problems of atomistic models related to uncertainties on the hybrid (inorganic NPL-organic ligand) surface details, for which experimental information is limited.
DFT calculations were performed with the Quantum Espresso code. 37 We geometrically relaxed the experimental PbS othorhombic unit cell (ICSD 648451) until forces on the individual nuclei were less than 0.001 R y /a 0 , where R y is the Rydberg constant and a 0 the Bohr radius. The first Brillouin zone was sampled with a Γ-centered Monkhorst-pack grid of 4x8x8 k points. In order to avoid possible phase transitions, we controlled cell dynamics in the relaxation process with the Wentzcovitch Extended Lagrangian. 38 We also run the Yambo code, 39,40 which implements the GW approximation. 41,42 To study optoelectronic properties we described excitons with effective mass Hamilto-

PbS Orthorhombic Structure
Our first goal was to confirm the orthorhombic phase of PbS NPLs. We synthesized NPLs following Khan and co-workers. 15  The presence of a cubic phase is attributed to the formation of some PbS nanoparticles together with NPLs. Nevertheless, since the capping ligands used during the synthesis promote the laminar growth of NPLs from the attachment of monomers or small PbS nanoparticles, 47 the orthorhombic phase is the main crystalline structure of the materials. This fact can be confirmed through UV-Vis absorption and photoluminescence (PL) measurements obtained from NPLs samples (see Fig. S5). An absorption shoulder at 720 nm was distinguished, and then red-shifted to 736 nm by increasing the time of NPLs synthesis. Furthermore, the typical PL peak of 2D PbS NPLs was reached between 736-741 nm, similar to previous works. 15 We next attempted to model the same orthorhombic lattice (ICSD 648451) using DFT.
The first attempts to relax the crystal structures inevitably led to a phase change from orthorhombic to rock-salt. This fact reflects that rock-salt is more stable and that the orthorhombic phase is achieved by external factors in the environment. Previous studies preserved the orthorhombic phase by keeping the cell parameters fixed. 16 We opted to carry out the relaxation process using the Wentzcovitch Extended Lagrangian instead, which is useful for the cell parameters optimization and improves numerical simulations within a single phase, allowing a slight freedom on the cell parameters. 38 The cell parameters obtained using PBE and PBEsol functionals approximately match the experimental ones, and are listed in Table 1.

Band Structure
We started by calculating the energy bands of bulk (P nma) PbS with PBEsol functional following the paths between high symmetry points in the first Brillouin zone, that we depict in Fig. 2d. As usual in DFT, the absolute values of the band gaps may be offset from experimental ones, but the predicted trends are usually reliable. 23,24,26 In Fig. 3 we compare the energy bands with and without SOC. The figure shows that SOC, which was missing in some earlier works, 29 has a non-negligible influence due to the presence of a heavy metal, lead in this case. It gives rise to a sizable narrowing of the band gap between valence and conduction bands. Similar results were obtained with PBE functional (See Fig. S2).
where m * is the effective mass. Diagonal values of the mass, m * i were obtained by taking five points E(k s ) along the k i axis (separated from each other by 200/(a,b,c)Å −1 ), centered around the band edge wave vector, and fitting the effective mass value. We checked that quantitatively similar results were obtained by fitting the second derivative of the band edge with finite difference derivations. Table 3 summarizes the effective masses obtained for the valleys of interest along different crystallographic directions. Similar effective masses are found with the PBE functional as given in Table S1 (SI). As in the case of rock-salt structures, 59 masses are highly anisotropic, with masses in one direction (x, in our case) being very different from the other two. The two latter present small differences, indicating that the reduced symmetry in the y, z directions has only moderate impact on the masses.  Table 3. For example, the heavier masses of c (1) and v (2) (HOMO and LUMO in the system) along the crystallographic axis a (m x ), as compared to the other bands, imply that confining the NPL in that crystallographic direction reinforce the indirect character of the band edge transition. By contrast, confinement along the in-plane b or c axis would favors direct transitions instead.

Emission Energy and dielectric effects
With the aim of studying the optical properties of orthorhombic PbS NPL, we build on the previous results of bulk, but now adding quantum confinement and electron-hole (exciton) Coulomb interaction. In this section we study the influence of such effects. Since bulk band gaps and effective masses are of the same order, it is reasonable to expect the same behavior in orthorhombic NPLs. Thus, we safely model the system using single band Hamiltonians 43 built up with the calculated gaps of Table 2 and the effective masses of In Fig. 6a and b we analyze the effect of lateral confinement (in z axis) on the exciton emission energy and binding energy, respectively, for a NPL with L x = 1.8 nm, corresponding to ∼ 7 PbS monolayers, which is the thickness of the NPLs synthesized in Ref. 15. The length in L y is fixed at 50 nm, well above the 2D exciton Bohr radius (a * B ≈ 3.6 nm, vide infra) so that lateral confinement is introduced exclusively by the variable L z .
The blue line in Fig. 6a shows the band edge (indirect, c (1) − v (2) ) transition energy. In the quantum well limit (L z > 15 nm), vertical confinement pushes the emission energy to ∼ 1 eV, i.e. about twice larger than in bulk. This is in spite of the large binding energy (∼ 0.125 eV if dielectric mismatch is taken into account, blue solid line in Fig. 6b). Sizable influence of lateral confinement is only observed for L z < 15 nm, when emission energy starts blueshifting and exciton binding energies are enhanced.
For comparison, in Fig.6a we also plot the direct c (2) − v (2) transition (red line). For large area NPLs, the direct band gap exceeds the indirect one by ∼ 0.4 eV. This splitting is about twice larger than the 0.22 eV observed for bulk in Table 3 and confirms that quantum confinement along the a axis reinforces the indirect nature of the band. Yet, c (2) electrons have heavier in-plane masses than c (1) ones (recall Table 3). One may thus expect that for strong lateral confinement the ground state switches from indirect to direct. Figure 6a shows that this can actually happen, but only for very strong lateral confinement (L z < 3 nm).
The black line in Fig.6a corresponds to rock-salt NPLs (calculated bulk band gap and masses can be found in Sec.III of SI). Clearly, the cubic lattice is an intermediate case between the direct and indirect gaps of orthorhombic PbS NPLs. Figure 6b shows that binding energies in orthorhombic NPLs with L z > 15 nm are similar to those estimated by Yang and Wise for thin 2D rock-salt nanosheets, 27 and also display a strong enhancement arising from dielectric mismatch. With regard to total exciton energy, we find such an enhancement is actually exceeded by the increase of self-energy (not shown), which leads to a moderate blueshift of the emission energy, similar to that reported for cubic PbS NPLs. 27 The binding energy of direct excitons is found to be slightly larger than that of indirect ones, because the electron and hole wave functions are more similar (cf. in-plane masses in Table 3). From our variational calculation of the exciton energy, we can also extract the effective Bohr radius, a * B . 43 Fig. 7 shows the corresponding values as a function of the NPL lateral size. For weak lateral confinement, the 2D limit is retrieved, which amounts to a 2D B = 3.6 nm for indirect excitons and to a 2D B = 2.82 nm for direct ones. The smaller radius for the direct exciton reflects the larger binding energy. When L z decreases, a * B first becomes larger, owing to the enhanced binding energy (Fig. 6b), but then decreases due to quantum confinement. This behavior is expected from the approximate scaling as 1/L z and 1/L 2 z for Coulomb and quantum confinement energies.
The effect of NPL thickness on the band edge is illustrated in Fig. 8, for a platelet with negligible lateral confinement (L y = L z = 50 nm). In thin structures, deviations of the effective mass from bulk parabolic values make single-band k·p models overestimate quantum confinement effects, 44 but these are still able to provide semi-quantitative trends. 27,31 The figure shows that both direct (red line) and indirect (blue line) gaps decrease rapidly with the number of orthorhombic PbS monolayers. The effect is however more pronounced for cubic PbS NPLs. This reflects the heavier mass m * x in orthorhombic band edges (Table 3) as compared to the cubic counterparts (see SI), especially for the valence band. Emission energy (eV)

Discussion
From the point of view of the electronic structure, the main effect of the orthorhombic distortion of the PbS lattice is the formation of an indirect gap, between c (1) and v (2) bands.
This is a drastic change as compared to rock-salt, where PBEsol predicts a direct band gap instead (see Fig. S3). From Table 2, the energy splitting between the fundamental (indirect) band gap and the direct ones exceeds 0.20 eV in bulk. For NPLs confined along the a crystallographic direction, 15,16 the masses reported in Table 3 indicate that the gap will open further (e.g. 0.4 eV in Fig. 6a between c (1) -v (2) and c (2) -v (2) ). These splittings are well beyond thermal energy at room temperature. Therefore, in equilibrium most photoexcited electron-hole pairs would occupy states of the indirect gap.
One can try to draw connections between this finding and experimental observations.
Orthorhombic NPLs show weak luminescence and high energy absorption onset / emission peak (∼ 1.6 eV). 15,16 Yet, time-resolved photoluminescence shows radiative recombination is fast. 15 The weak luminescence may result from the indirect nature of the fundamental transition. After non-resonant excitations, only a small fraction of photoexcited electron-hole pairs recombines radiatively across direct gaps, while the majority decays to the indirect one.
In the direct gap recombination, giant oscillator strength permits short radiative lifetimes.
This process should be responsible for the high energy emission and absorption. By contrast, in the indirect gap recombination, optical selection rules yield long radiative lifetime, and subsequent competition with non-radiative relaxation mechanisms leads to low quantum yields.
For 7 monolayer PbS NPLs with weak lateral confinement, we estimate the direct transition energy to be ∼ 1.4 eV (Fig. 6a, red line). This value is slightly lower than the 1. Inaccuracy in the determination of the effective NPL thickness may be an issue as well. In both cubic 27 and orthorhombic (Fig. 6b) NPLs, k·p theory predicts strong excitonic effects.
The short radiative lifetimes in emission experiments are consistent with this feature, 15,66 but in the two systems measured absorption spectra show smeared peaks (see e.g. Fig.S5a).
It has been pointed out that uneven NPL thickness, leading to inhomogeneous broadening, may be responsible for this. 65 If this was the case, the different number of monolayers in the optically active region of the NPL would certainly have a drastic effect on the emission wavelength, because of the extreme quantum confinement. This can be observed in Fig. 8.
For nominal thicknesses of 7 monolayers or less, a fluctuation of a single monolayer implies a band edge energy shift exceeding 0.1 eV, i.e. comparable to exciton binding energies.
As for the effect of lateral confinement, in Ref. 15 a blueshift of ∼ 30 meV was observed when L z decreased from 12.4 to 3.5 nm, which is well below the 200 meV obtained in our calculations (Fig. 6a). This significant deviation is yet another indication that extrinsic factors, such as impurities or surface roughness, are conditioning the actual photo-physical response.
Synthetic efforts to control the NPL defects and homogeneity are then expected to be key in improving the optical properties of orthorhombic PbS NPLs and reach the maturity of their cadmium chalcogenide counterparts. Yet, even if this issues are circumvented, the indirect band gap we predict is an intrinsic factor which will ultimately determine the performance of these structures.

Conclusions
We have synthesized orthorhombic PbS NPLs and determined the lattice parameters using

Graphical TOC Entry
where σ x , σ y and σ z are the Pauli matrices, k t = (k x , k y ), σ t = (σ x , σ y ), and 1 is the 2 × 2 unit matrix. We can abbreviate the notation and write with h cv = h vc .
Our goal is to obtain effective, single-band Hamiltonians for conduction and valence bands, and the associated effective masses, from the multi-band Dimmock Hamiltonian, eq.

S2
(1). The detail procedure is explained below. Similar reasoning have been used in Ref. 3.
The eigenvalue equation from eq. (2) reads where F c and F v are the envelope function of conduction and valence band, respectively.
Equation (3) is equivalent to equation set From equation (5), replacing the last term in equation (4) we obtain an effective Hamiltonian for the conduction band Similarly, for the valence band one obtains If we write ∆ = ( Eg 2 +h Carriers in NPLs are confined within it, then if we consider the case of confinement in the z direction, E+∆ . With these considerations, and simplifying the notationh m P ≡ P , then we can write The effective masses of a single band can be obtained, but ultimately depend on the energies of the multiband calculation. Knowing that the substitution is possible, for efficiency reasons instead of substituting one by one the terms that relate the single band masses with the terms of the multiband, we will make calculations of a single band and we will find which effective masses fit better to those obtained by the multiband Dimmock calculation.

S5
In Fig. S1. we fit the effective masses, from the single band Hamiltonian, that have the smallest deviation compared to the calculation of 4-bands Hamiltonian in the L z range. In this case, the energy was calculated for a NPL with dimensions L x = 1.8 nm, L y = 50 nm.

Rock-Salt PbS Band Structure
We calculate energy bands for the rock-salt structure with the PBEsol functional and the SOC inclusion. We relaxed the structure until forces less than 0.001 R y /a 0 . The first Brillouin zone was sampled with a Γ-centered Monkhorst-pack grid of 6x6x6 k points. A direct gap E g = 0.24 e.V is found at the point of symmetry L, which is clearly underestimated with respect to the reported experimental value (0.42 e.V).