Lead iodide (PbI2) has gained much interest due to its direct electronic gap in the visible range and layered crystal structure. It has thereby been considered as a promising material for applications in atomically thin optoelectronic devices. In this work, we present a detailed investigation of the effect of spin–orbit coupling (SOC) that arises from the presence of heavy atoms on the electronic and optical properties of PbI2 using first-principles calculations based on density-functional theory and many-body perturbation theory. We find that SOC not only alters the bandgap but also induces the mixing of orbital characters, resulting in a significant change in the overall band structure and charge carrier effective masses. Moreover, the band orbital mixing caused by SOC results in the dramatic change in optical transition matrix elements and, correspondingly, the absorption spectrum. Our experimentally measured absorption spectra validate the calculation results and demonstrate the importance of SOC in the optical processes of PbI2. Our findings provide insights that are important for the potential use of PbI2 as a material platform for visible optoelectronic devices.

Lead iodide (PbI2) is a layered semiconductor, where covalently bonded triatomic layers (I–Pb–I) are stacked by van der Waals (vdW) interactions.1 PbI2 is known to exhibit a direct bandgap in the visible range,2 allowing its wide applications in high-performance optoelectronic devices, such as lasers and photodetectors.3,4 Also, due to its ability to react with amine iodides, PbI2 has gained immense popularity as a precursor for halide perovskites, which has widely been explored for high-efficiency light-emitting diodes and photovoltaic devices.5,6 Furthermore, the nature of vdW interaction allows PbI2 to form various types of heterostructures with other vdW materials, such as graphene and transition metal dichalcogenides (TMDCs).7–9 These heterostructures allow us to exploit a broader range of physical phenomena, such as interlayer excitons and high spin-polarizations.10,11

Over the past few decades, both theoretical and experimental studies have been playing an important role in deepening our understanding of the material properties of PbI2. Early studies on PbI2 revealed a strong spin–orbit coupling (SOC) in this material, which is attributed to the presence of heavy elements Pb and I.12–14 Later, this observation is supported by the recently developed first-principles calculation methods, which pointed out that the inclusion of the SOC is crucial for the precise calculation of the electronic properties for PbI2, indicating the strong SOC effect in PbI2.15–17 

Although the prior research has improved our understanding of some of the unique features of PbI2, there is still a lack of a theoretical approach that explicitly investigates the effect of SOC on the optical properties of PbI2. SOC is known to be critically important in optical processes of various materials and systems, since it defines the orbital character of bands and the fine structure of excitons. For instance, spin-polarization is a key to understanding the unique optical properties of TMDCs, which are important for spintronic and valleytronic applications.18,19 Moreover, a precise estimation of the bright-dark exciton splitting is imperative for the design of high-efficiency optoelectronic devices based on lead halide perovskites.20,21 These observations firmly demonstrate the necessity of rigorous investigation of the effect of SOC in PbI2, which also has wide potential applications in the areas of photonic devices.

In this work, we investigate the electronic, optical, and excitonic properties of PbI2 by fully incorporating the effect of SOC. Among many polytypes of PbI2, we focus on 2H-PbI2, which is the most common structure.1 We perform first-principles calculations using density functional theory (DFT) and many-body perturbation theory (MBPT). Our quasi-particle band structure calculation and band character analysis indicate that SOC mixes electron states with different orbital character and causes a huge modification in the band structure and carrier effective masses. In addition, the optical matrix elements calculation and the absorption calculation results support the pivotal role of SOC in the optical process of PbI2. Finally, we compare our simulated absorption spectrum with our experimentally measured spectrum and confirm that our simulation results are in good agreements with the experiments.

DFT calculations were performed within the Quantum Espresso code.22 We used fully relativistic pseudopotentials and local density approximations (LDA) for the exchange–correlation functional.23,24 The 5d, 6s, and 6p electrons of Pb were included in the valence states. Two-components spinor wavefunctions were calculated on 8×8×4 Monkhorst-Pack mesh. A plane cutoff energy of 90 Ry converged the total energy to within 1mRy/atom. Using spinor wavefunctions, the quasi-particle band energy was obtained from the G0W0 method, as implemented in the BerkeleyGW code.25 We used the screening cutoff energy of 15 Ry and included a number of bands up to the half of the screening cutoff energy (7.5 Ry). We adopted the generalized plasmon-pole model for the calculation of frequency-dependent dielectric effects26 and the static-remainder approach to increase the computation speed and achieve the better convergence of the self-energy corrections.27 Next, the quasi-particle band structure was interpolated with the maximally localized Wannier function method as implemented in the Wannier90 code.28,29 Finally, using the modified dielectric matrix and quasi-particle band structure, we solved a fully relativistic form of the Bethe–Salpeter equation (BSE) to obtain excitonic properties including spin–orbit interactions.30,31 We interpolated the Coulomb matrix elements calculated on a coarse 8×8×4 k-grid into a finer 24×24×12 k-grid using the dual-grid method, as implemented in the BerkeleyGW code.25 The top four valence bands and the bottom four conduction bands were included in the BSE calculation. In addition, the scissor-shift method was applied to fix the direct bandgap at the k=A point to 2.52 eV.

Flakes of PbI2 were mechanically exfoliated from single crystal PbI2 (HQ Graphene) onto quartz substrates pretreated by oxygen plasma.32 The absorption was measured using a white light source (HL-2000-LL, Ocean Insight). The white light was focused using a microscope objective and the transmitted light was collected and analyzed using a high-resolution spectrometer (IsoPlane-320 from Princeton Instruments, 300 grooves/mm grating, Pixis 400 CCD camera).

First, we calculated the quasiparticle band structure of bulk PbI2 with and without SOC effects (Fig. 1). Since the s orbitals are not affected by SOC due to their zero angular momentum, we used the energies of the deep-energy bands originating from the 6s orbital of Pb atoms as a common reference to align the band structures with and without SOC effects and demonstrate the absolute effects of SOC on the higher-energy valence and conduction bands (Fig. S1). Without SOC, our calculation overestimated the direct bandgap by almost 25% Eg=3.16eV, as pointed out by previous studies.15 However, we observed that accounting for SOC reduced the bandgap by almost 1 eV to a value of 2.23 eV. We also found that including the 5s25p6 core electrons of Pb atoms as valence states lead to an increase in the bandgap to 2.52 eV, which matched well with the experimentally reported electronic gap33 (Table I). However, due to the extremely high computation cost of excitonic calculations with the semicore pseudopotential of Pb (5s25p65d106s26p3), we instead used the valence pseudopotential of Pb (5d106s26p3) and rigidly increased the bandgap by 0.29 eV to match the gap obtained with semicore pseudopotential in subsequent excitonic and optical calculations with the BSE method. Details on the comparison between the calculation results with the semicore Pb pseudopotential and the valence Pb pseudopotential is included in the supplementary material (Fig. S2).

TABLE I.

Electronic gap of bulk PbI2 obtained from valence (5d6s6p) and semicore (5s5p5d6s6p) Pb pseudopotentials.

w/o SOC (valence) w/ SOC (valence) w/ SOC (semicore)
EGDFT  1.98eV  1.24eV  1.22eV 
EGGW  3.16eV  2.23eV  2.52eV 
ΔEGGW  1.18eV  0.99eV  1.30eV 
w/o SOC (valence) w/ SOC (valence) w/ SOC (semicore)
EGDFT  1.98eV  1.24eV  1.22eV 
EGGW  3.16eV  2.23eV  2.52eV 
ΔEGGW  1.18eV  0.99eV  1.30eV 
Next, we summarize the effective mass of electrons and holes along different directions in Table II. The effective masses of electrons and holes were obtained by fitting the energy-momentum curves near the band extrema k=A to the parabolic equation as follows:
Ek=E0 ±2k22me,h,
where Ek is the energy of the band at the reciprocal point k, E0 is the conduction band minimum for an electron and the valence band maximum for a hole, and me,h is an effective mass of an electron or a hole. Overall, the electron effective mass along the out-of-plane direction increases while the hole effective mass along the in-plane direction decreases with the inclusion of SOC. This change can be attributed to the mixing of band characters induced by SOC. The electron effective mass parallel and perpendicular to c-axis of PbI2 are reported to be me=2.1m0 and me=0.48m0.34 Meanwhile, the hole effective mass was estimated to be isotropic (mh=0.195m0).35 Our calculation that include SOC also show highly anisotropic effective masses for electrons and nearly isotropic effective masses for holes. At the same time, the predicted values are in good agreements with previous reported data.15 We assume that the deviation of the experimental values from the simulation results can be attributed to the limit of a rough analytical approximation used in experimental data analysis. In particular, we reckon the experimental value for hole effective mass to be underestimated, due to the poor fit with the absorption peak energies.36,37
TABLE II.

Effective mass of electrons and holes of bulk PbI2 in units of the free electron mass. Inclusion of SOC increases the electron effective mass parallel to the c axis me and reduces the hole effective mass perpendicular to the c axis mh.

This work Previous theory15  Experiments
With SOC Without SOC
me  1.78  AΓ  1.30  AΓ  1.05  2.134  
me  0.26  AH  0.26  AH  0.21  0.4834  
  0.27  AL  0.26  AL     
mh  0.72  AΓ  0.68  AΓ  0.56  0.19535  
mh  0.60  AH  1.06  AH  0.59  0.19535  
  0.62  AL  1.11  AL     
This work Previous theory15  Experiments
With SOC Without SOC
me  1.78  AΓ  1.30  AΓ  1.05  2.134  
me  0.26  AH  0.26  AH  0.21  0.4834  
  0.27  AL  0.26  AL     
mh  0.72  AΓ  0.68  AΓ  0.56  0.19535  
mh  0.60  AH  1.06  AH  0.59  0.19535  
  0.62  AL  1.11  AL     
Based on the band structure calculations, we further investigate how the SOC mixes the band character. To analyze the band character, we calculated the projected density of states (PDOS), gijψnk, where gij is the atomic orbital i of atom j, and ψnk is the Bloch wave function for band index n and electron wave vector k. We normalized this value by the total PDOS as follows:
gij|ψnk2j=Pb,Ii=s,p,dgij|ψnk2.
(1)
This normalized PDOS value ranges from 0 to 1, indicating the orbital origin of the conduction and valence bands at each k point. In Fig. 2, we analyzed the normalized PDOS value for the pz orbital of Pb and I atoms, respectively, excluding SOC corrections. The figure shows that the normalized PDOS value is close to 1 only near the band extrema where k=A. This result indicates that the 6pz orbitals of Pb contribute to the bands near the conduction band minimum while the 6s orbitals of Pb and the 5pz orbitals of I contribute to the bands near the valence band maximum. The next two conduction bands are derived from the 6px and 6py orbitals of Pb, while deeper valence bands are formed by the 5px and 5py orbitals of I.
FIG. 1.

Quasiparticle band structure of bulk PbI2 obtained from G0W0 method using scalar-relativistic pseudopotentials (without SOC, red dashed curve) and fully relativistic pseudopotentials (with SOC, blue curve).

FIG. 1.

Quasiparticle band structure of bulk PbI2 obtained from G0W0 method using scalar-relativistic pseudopotentials (without SOC, red dashed curve) and fully relativistic pseudopotentials (with SOC, blue curve).

Close modal
FIG. 2.

State-projected band structure obtained from the scalar-relativistic pseudopotentials. Bloch wavefunctions are projected on the (a) pz orbital of Pb atom, (b) pz orbital of I atoms, (c) pxy orbital of Pb atom, (d) pxy orbital of I atoms, (e) s orbital of Pb atom, and (f) s orbital of I atoms. The values are normalized by the total projected density of states. The normalized PDOS indicate that the pz orbitals of lead contribute to the bands near the CBM, while the bands near VBM mainly originate from the pz orbitals of iodine.

FIG. 2.

State-projected band structure obtained from the scalar-relativistic pseudopotentials. Bloch wavefunctions are projected on the (a) pz orbital of Pb atom, (b) pz orbital of I atoms, (c) pxy orbital of Pb atom, (d) pxy orbital of I atoms, (e) s orbital of Pb atom, and (f) s orbital of I atoms. The values are normalized by the total projected density of states. The normalized PDOS indicate that the pz orbitals of lead contribute to the bands near the CBM, while the bands near VBM mainly originate from the pz orbitals of iodine.

Close modal

However, the inclusion of the spin–orbit coupling leads to the mixing of the band orbital characters. As illustrated in Fig. 3, the characters of px and py orbitals of Pb are significantly enhanced for the conduction bands near the CBM, while the character of pz orbital of Pb diminishes. Similarly, the characters of px and py orbitals of I are strengthened for the valence bands near the VBM, but the degree of band mixing is less significant than that of conduction bands near the CBM. Overall, these changes in band character indicate that SOC has a significant impact on the optical properties, which heavily depend on the characteristics of band edges.

FIG. 3.

State-projected band structure obtained from the fully-relativistic pseudopotentials. Bloch wavefunctions are projected on the (a) pz orbital of Pb atom, (b) pz orbital of I atoms, (c) pxy orbital of Pb atom, (d) pxy orbital of I atoms, (e) s orbital of Pb atom, and (f) s orbital of I atoms. The values are normalized by the total projected density of states. The normalized PDOS indicate that the SOC mixes the band characters and leads to the enhanced contribution from px and py orbitals for the bands near the band extrema.

FIG. 3.

State-projected band structure obtained from the fully-relativistic pseudopotentials. Bloch wavefunctions are projected on the (a) pz orbital of Pb atom, (b) pz orbital of I atoms, (c) pxy orbital of Pb atom, (d) pxy orbital of I atoms, (e) s orbital of Pb atom, and (f) s orbital of I atoms. The values are normalized by the total projected density of states. The normalized PDOS indicate that the SOC mixes the band characters and leads to the enhanced contribution from px and py orbitals for the bands near the band extrema.

Close modal

Even though the band character analysis based on the orbital angular momentum provides a better understanding of the effect of SOC, we further investigate the total angular momentum j since the orbital angular momentum l and the intrinsic spin angular momentum s are strongly coupled in PbI2. Correspondingly, we projected electron wavefunctions onto the eigenfunctions of the total angular momentum.38 We demonstrate that Pb 6p orbitals (j=0.5,mj=±0.5) mainly comprise the CBM, while I 5p orbitals (j=1.5,mj=±0.5) contribute to the VBM (Fig. S3). Due to the inversion symmetry and time-reversal symmetry, both the conduction band minimum and the valence band maximum of PbI2 exhibit spin-degeneracy.39 Moreover, we observe the splitting of the conduction bands into the bands with j=3/2 and j=1/2, yielding a SOC gap of 0.93 eV at k=A [Figs. S3(a) and S3(c)]. Such strong spin–orbit splitting is attributed to the heavy-element Pb atom and is also reported for lead halide perovskites.40,41

The mixing of the band character is directly reflected in the change in the optical matrix elements. In Fig. 4, we plot the matrix elements of the momentum operator between the lowest conduction band and the highest valence band evaluated along high-symmetry directions. At the band extrema (k=A), if the SOC effect is not considered, the momentum matrix element shows a large value for light polarization along the out-of-plane direction while the value for the in-plane direction is zero [Fig. 4(a)]. Based on the band character analysis (Figs. 2 and 3), the optical transitions at the band extrema are dominated by the transitions between pz orbitals of Pb and I atoms. However, we observe that the inclusion of SOC changes the transition probabilities due to the mixing of band characters [Fig. 4(b)]: in-plane transitions are enabled and relatively stronger than transitions along the out-of-plane direction. This should, therefore, result in strong optical absorption/emission peaks.

FIG. 4.

Interband momentum matrix elements along high-symmetry directions evaluated without and with SOC. (a) Without SOC, the magnitude of the matrix elements at the band extrema k=A is strong for light polarization perpendicular to the PbI2 plane (i.e., along the z axis) and zero within the plane (x axis), indicating anisotropic optical properties. (b) When the SOC effect is included, the matrix element becomes nonzero for both polarizations, with the matrix element within the plane becoming much stronger than along the z axis, which indicates that orbital mixing by SOC enables dipole-forbidden transitions for light polarized within the PbI2 planes to become allowed.

FIG. 4.

Interband momentum matrix elements along high-symmetry directions evaluated without and with SOC. (a) Without SOC, the magnitude of the matrix elements at the band extrema k=A is strong for light polarization perpendicular to the PbI2 plane (i.e., along the z axis) and zero within the plane (x axis), indicating anisotropic optical properties. (b) When the SOC effect is included, the matrix element becomes nonzero for both polarizations, with the matrix element within the plane becoming much stronger than along the z axis, which indicates that orbital mixing by SOC enables dipole-forbidden transitions for light polarized within the PbI2 planes to become allowed.

Close modal

The effect of SOC on the band character and transition probability is also well-illustrated in the calculated absorption spectrum (Fig. 5). To concentrate on the influence of SOC and excitonic effect, we applied a rigid shift to fix the electronic gap for all the absorption calculations to 2.52 eV. Without SOC, there are only dark exciton states below the electronic gap for the in-plane polarization [Fig. 5(a)]. This can be attributed to the pz-character of the bands near CBM and VBM. Bright exciton states and corresponding strong absorption peaks are only observed for the higher energy range above the electronic gap. These bright exciton states are mainly composed of the transitions between higher-energy conduction bands and valence bands which show px and py band characters. On the other hand, we see an absorption peak below the electronic gap in the case of out-of-plane polarization. This absorption peak originates from the bright exciton state that is mainly formed by the transitions near the band extrema (k=A). Therefore, we conclude that the strong absorption peak below the bandgap will only appear in the case of out-of-plane polarization when the SOC effect is not included.

FIG. 5.

Absorption spectrum of bulk PbI2 for (a) in-plane and (b) out-of-plane polarized light. Red and blue curves represent the absorption spectrum with and without SOC, respectively. For each case, the absorption spectrum is obtained with (BSE, solid lines) and without (RPA, dashed lines) the inclusion of excitonic effects. The electronic gap is set to 2.52 eV (vertical dashed line) for all the considered cases. Including spin–orbit significantly alters the overall absorption spectrum. Especially, the strong excitonic peak below the bandgap is observed for in-plane polarization only when SOC is considered, demonstrating the importance of SOC for the optical processes of PbI2.

FIG. 5.

Absorption spectrum of bulk PbI2 for (a) in-plane and (b) out-of-plane polarized light. Red and blue curves represent the absorption spectrum with and without SOC, respectively. For each case, the absorption spectrum is obtained with (BSE, solid lines) and without (RPA, dashed lines) the inclusion of excitonic effects. The electronic gap is set to 2.52 eV (vertical dashed line) for all the considered cases. Including spin–orbit significantly alters the overall absorption spectrum. Especially, the strong excitonic peak below the bandgap is observed for in-plane polarization only when SOC is considered, demonstrating the importance of SOC for the optical processes of PbI2.

Close modal

However, with the inclusion of SOC, we observe completely different absorption spectra for in-plane and out-of-plane polarized cases (Fig. 5). For the case of in-plane polarization, the overall absorption spectrum shifts to the lower-energy due to the band mixing, and a bright excitonic peak appears below the bandgap. Meanwhile, the intensity of the excitonic absorption peak for the out-of-plane polarized light decreased, confirming the change of the band character. The trend corresponds well with the change in dipole matrix elements (Fig. 4).

In addition, SOC also affects the exciton binding energy. Table III shows that the inclusion of SOC leads to the reduction of the binding energy, which is mainly attributed to the large reduction in the hole effective mass along the in-plane direction (Table II). Our analysis indicates that the role of SOC is critically important to understand the optical transitions of bulk PbI2.

TABLE III.

Exciton binding energy calculated for in-plane and out-of-plane polarization. Inclusion of the SOC leads to a slight decrease in the exciton binding energy due to reduction in effective mass.

w/o SOC w/ SOC
EBEc  68.9 meV  48.6 meV 
EBEc  98.5 meV  83.3 meV 
w/o SOC w/ SOC
EBEc  68.9 meV  48.6 meV 
EBEc  98.5 meV  83.3 meV 

Finally, we calculated the absorption spectrum with random-phase approximation (RPA) to verify the importance of excitonic effects in PbI2. As illustrated in Fig. 5, there are large differences between the two absorption spectra obtained without (RPA, dashed curves) and with (BSE, solid curves) the inclusion of excitonic interactions. The overall transfer of spectral weight from high energies to low energies is attributed to the characteristic effect of excitonic corrections. Especially, the bright 1s exciton absorption peak below the electronic gap appears only when the excitonic effect is considered. Meanwhile, the absorption spectra obtained without the excitonic effect (RPA, dashed curves) show onsets at the electronic gap (2.52 eV, vertical dashed line). As a consequence, we conclude that both SOC and excitonic effects play a major role in optical processes in PbI2.

To further support our calculations, we measure differential transmission (ΔTT) of bulk PbI2 at room temperature. Here, ΔTT=(Iflake+subsIsubs)/IincIsubs/Iinc and is related to absorbance (A) by A =log10(ΔTT).Iinc, Isubs, and Iflake+subs correspond to the intensities of incident light, transmitted light after substrate only, and transmitted light after both substrate and flake, respectively. The circled area in Fig. 6(a) shows the measurement area (relatively large due to the large white light spot size). The uniformity of measurement area of the mechanically exfoliated samples was confirmed by atomic force microscopy (AFM) (Fig. S4) to rule out any thickness related contributions. Please refer to additional data corresponding to various thicknesses in the supplementary material (Fig. S5). The dominant absorption peak around 2.48 eV in Fig. 6(b) is in good agreement with the simulations [2.476 eV, Fig. 5(a)] and previously reported spectra.42–44 

FIG. 6.

(a) Optical image of a mechanically exfoliated PbI2 sample. The circled region corresponds to the measurement area. (b) Experimentally measured differential transmission spectrum of bulk PbI2.

FIG. 6.

(a) Optical image of a mechanically exfoliated PbI2 sample. The circled region corresponds to the measurement area. (b) Experimentally measured differential transmission spectrum of bulk PbI2.

Close modal

In conclusion, we demonstrate that SOC is of vital importance to the electronic and optical properties of PbI2. Our first-principles calculations based on density functional theory and many-body perturbation theory illustrate how SOC modifies the band character and the related electronic parameters, such as bandgap and carrier effective masses. Following calculations on momentum matrix elements and absorption spectrum indicate that the fine structure of the exciton is strongly affected by the SOC and this change is directly reflected in the optical spectrum of PbI2. Our study provides an insight for fundamental processes of PbI2 and its wide applications as optical materials and high-efficiency optoelectronic devices.

See the supplementary material for details on the deep-energy bands originating from the 6s orbital of Pb atoms, comparison between the calculation results with the semicore Pb pseudopotential and the valence Pb pseudopotential, projected density of states (PDOS) obtained from fully relativistic pseudopotentials, optical image and atomic force microscopy (AFM) data, and differential transmission spectra of bulk PbI2.

This work was supported through the National Science Foundation (NSF) (Grant No. DMR-1904541). Computational resources were provided by the National Energy Research Scientific Computing (NERSC) Center, a Department of Energy Office of Science User Facility supported under Contract No. DEAC0205CH11231. W.L. was partially supported by the Kwanjeong Educational Foundation Scholarship.

The authors have no conflicts to disclose.

Woncheol Lee: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Software (equal); Validation (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). Zhengyang Lyu: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Validation (equal); Writing – original draft (equal); Writing – review & editing (equal). Zidong Li: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Validation (equal); Writing – review & editing (equal). Parag B. Deotare: Conceptualization (equal); Funding acquisition (equal); Project administration (equal); Resources (equal); Supervision (equal); Validation (equal); Writing – review & editing (equal). Emmanouil Kioupakis: Conceptualization (equal); Funding acquisition (equal); Project administration (equal); Resources (equal); Supervision (equal); Validation (equal); Writing – review & editing (equal).

The data that support the findings of this study are available from the corresponding author upon reasonable request.

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