Effects of Vacancy Defects on the Electronic and Optical Properties of

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Energy Conversion and Storage; Plasmonics and Optoelectronics

Effects of Vacancy Defects on the Electronic and Optical Properties of Monolayer PbSe Chinedu E Ekuma J. Phys. Chem. Lett., Just Accepted Manuscript • DOI: 10.1021/acs.jpclett.8b01585 • Publication Date (Web): 20 Jun 2018 Downloaded from http://pubs.acs.org on June 21, 2018

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The Journal of Physical Chemistry Letters

Effects of Vacancy Defects on the Electronic and Optical Properties of Monolayer PbSe C. E. Ekuma∗,†,‡ U.S. Army Research Laboratory, Aberdeen Proving Ground, MD 21005-5069 E-mail: [email protected]

∗ To

whom correspondence should be addressed Army Research Laboratory, Aberdeen Proving Ground, MD 21005-5069 ‡ George F. Adams Research Fellow † U.S.

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Abstract Defect-engineering is promising for tailoring the properties of atomically thin materials. By creating defects via Se vacancies, we study the optoelectronic properties of monolayer PbSe. We obtain the single-particle properties using the density functional theory plus a firstprinciples-based typical medium approximation. The absorption spectra is explored by solving the Bethe-Salpeter equation. Our results reveal that monolayer PbSe is defect sensitive but defect-tolerant. The latter fingerprint is due to the absence of defect-induced in-gap, localized states. Our results predict that Se vacancies are the dominant defect-type in disordered PbSe monolayer. We observe that increasing Se vacancy concentrations δ renormalize the energy bandgap Eg increases from 0.21 eV for the pristine to as high as 0.45 eV at high δ . The high tunability of the optoelectronic properties of monolayer PbSe using defect-engineering makes this material a candidate for exploring flexible electronic with potential technological applications in nanoelectronics.

TOC Graphic

Among the rapidly evolving class of two-dimensional (2D) materials, monolayer lead chalcogenides and their alloys, such as PbSe, are intensively being studied for next-generation nanoelectronics applications. Their unusual and unique properties relative to other semiconductors make them attractive candidate materials for both potential device applications and for exploring fundamental physical phenomena. Some of their unique properties are high mobility, high dielectric 2


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constant, low-cost, and a narrow fundamental bandgap. Because of these attractive properties, they have been extensively studied for diverse applications such as thermoelectric energy converters and photovoltaics, 1–3 spintronics and optoelectronic devices, 4–6 infrared lasers and diodes. 7,8 Recently, the nanostructure of PbSe and related materials have been explored for creating new materials with novel and tunable properties. 9–12 There is by now a general consensus on the origin of free carriers, e.g., n- (p-) type semiconductor originates from Se (Pb) vacancies in bulk PbSe. 13,14 However, in the case of PbSe monolayer, the nature and stability of the defect states have yet to be systematically studied. Defects inevitably exist in crystals, which could be unintentionally introduced during experimental sample preparation or material characterization. In the case of atomically thin 2D materials, due to their high susceptibility to disorder, the impurity states could dramatically affect their properties for device applications. Defects in materials could be studied with the density functional theory (DFT) 16,17 using the supercell. However, the DFT supercell approach can only describe ordered defect structures. Even more importantly, the supercell approach is prohibitively expensive for exploring low defect concentrations. Also, supercell model is not adequate for calculating the energy of isolated defect states especially “shallow” impurity states in semiconductors 18 and the finite-size effects of the supercell have been reported to adversely diminish the dispersion of the deep-level in-gap states. 19 Depending on the location of the defect-induced states with respect to the bandgap, exciton recombination processes could be radiative or nonradiative. 20–22 For instance, defect-induced localized, deep-level, in-gap states could be detrimental to device performance as they act as local scattering sites, which facilitates nonradiative recombination of photoexcited excitons vis-à-vis, the reduction of the mobility of charge carriers and quantum efficiency. This is one of the major limiting factors of materials’ applications in optoelectronics devices such as field-effect transistors, solar cells, photodetectors, and light emitters. Hence, a proper understanding of the nature of the defect-induced states: deep-level localized states or shallow impurity, resonant levels within the bandgap is of great importance. The former occurs if the defect-induced states lie inside the

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Figure 1: (A) The top-view of the atomic structure and (B) the first Brillouin zone of monolayer PbSe along the [001]-direction. (C). The density functional theory band structure (black dashed bands) reproduced by the downfolded Hamiltonian H0 (solid red bands) for the pristine monolayer of PbSe. A direct single-particle bandgap 0.21 eV is predicted at the given X-point of the high symmetry zones of the Brillouin zone. The side plot is the associated density of states of the Pb-p, Se-p states, and the quasiparticle total density of states obtained from the self-consistent quasiparticle Green function and screened Coulomb interaction calculations using VASP. 15 The horizontal black dash line is the Fermi level.

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bandgap. They behave as dominant electron or hole traps as well as recombination centers while the latter is when the defect-induced states are located outside the bandgap. They generally led to a small perturbation to the lattice potential reminiscent of substitutional dopant. Understanding the defect morphology could greatly aid in exploiting the full technological potentials, e.g., topological states, optoelectronic, and other diverse promising applications of monolayer PbSe. Despite the need, no computational data exploring the nature of the defect states in monolayer PbSe have been published so far to our knowledge. In this letter, we use the recently developed first-principles based typical medium dynamical cluster approximation (TMDCA) 23,24 that takes into account the intrinsic, random nature of impurity states in crystals to systemically study the optoelectronic properties of disordered monolayer PbSe. We determine the stability of the defect states in the disordered crystal by calculating the defect formation energy ∆E f , which is the bond energy cost due to the creation of a defect. The calculated ∆E f profiles show that Se vacancies are energetically more favorable to form in monolayer PbSe. Our results reveal that randomly distributed Se vacancies could impact the optoelectronic properties of monolayer PbSe in diverse ways. We show that defect-engineering could be used to tune the electronic properties of monolayer PbSe. Our data show that monolayer PbSe is sensitive to defects, but it is, however, defect-tolerant 25–27 due to the absence of localized, deep-trapped levels within the bandgap. The optoelectronic fingerprint of monolayer PbSe makes it a potential candidate material to explore flexible nanoelectronics and for solar cells applications. To study disorder induced by vacancy in monolayer PbSe, we use Anderson Hamiltonian of noninteracting electrons subjected to random vacancy configurations

H = H0 + ∑ Viσα nαiσ ,

(1)

iασ

where H0 is the pristine, single-particle Hamiltonian, Viσα is a disorder potential, and nαiσ is the number operator; i, α, and σ are site, orbital, and spin indices, respectively. The terms in Eq. 1 denote the single-particle and disorder in that order.

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The single-particle Hamiltonian is obtained from DFT 16,17 calculations using the linearized augmented planewave (LAPW) method with a modified Becke-Johnson (mBJ) potential, 28 as implemented in WIEN2K. 29 The mBJ potential yields energy bandgap of semiconductors in reasonable agreement with experiment in contrast to standard density functionals and at a fraction of the computational cost of the many-body perturbation theory, e.g., the Green function plus screened Coulomb approximations. 6,28,30,31 We first carried out structural optimization using the PerdewBurke-Ernzerhof 32 exchange-correlation functional and then self-consistent electronic structure calculations of the periodically repeated monolayers of PbSe separated ≈ ∼20 Å vacuum using the optimized lattice constant to obtain the Kohn-Sham eigenvalues and eigenfunctions. The DFT electronic structure calculation used a well-converged basis set with a dense Brillouin zone sampling grid of 12 ×12×1 ~k-points and an Rmin Kmax = 9, where Rmin is the smallest sphere radius in the system and Kmax is the plane wave momentum cutoff. A LAPW sphere radius of 2.50 Bohr was used for Pb and Se, respectively. All calculations were performed relativistically including spinorbit coupling, which could be important for the band structures. 33 Using downfolding technique as implemented in the WANNIER90, 34 we obtain H0 from the converged Kohn-Sham eigenstates by generating a set of Wannier functions. The H0 included the Pb-p and Se-p states, which are the dominant states around the Fermi level. This observation is in agreement with our recent bulk structure calculations. 6 We show in Fig. 1(B) the first Brillouin zone of PbSe [001] monolayer. The 12-bands that make up H0 reproduce the DFT band structure, with a direct single-particle energy bandgap Eg ≈ 0.21 eV at the X-point of the k-space [Fig. 1(C)]. We model the disorder using a bimodal site potential Viσα = Vi ∈ {0,Wb } randomly generated using the probability mass functions P(Vi = W ) = δ and P(Vi = W ) = 0, where 0 (W ) is for the ordered (disordered) site. The vacant site potential energy is set to be far greater than the pristine bandwidth. Taking the Se vacancies as an example, the average Se vacancy concentration δ satisfies the stoichiometry PbSe1−δ of the disordered material. The TMDCA, through a set of self-consistent field (SCF) equations, maps the disordered lattice onto a periodically finite cluster of Nc primitive cells embedded in a typical medium characterized

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by a hybridization function. This effective hybridization function is calculated self-consistently, employing the TMDCA condition. 23,35–42 The TMDCA SCF equations were iteratively solved to obtain the cluster density of states ρ c = − π1 ℑGc , which is averaged over a large number of configurations to calculate the wave-vector-resolved, non-self-averaged cluster typical density of states (for details, see Ref. 23 ). To calculate the two-particle Green functions, we used the converged single-particle Green functions obtained from the TMDCA self-consistency calculations to solve the Bethe-Salpeter equations as described in Ref. 24 The TMDCA allows for the study of very small impurity concentrations as low as 0.10% since it uses the same primitive cell of the host material without the need for supercell. While this is advantageous since the computation cost does not scale with the system size as in supercell approaches, it, however, neglects possible structural relaxation processes around the vacancies. To verify the effects of this neglect, we performed relaxed supercell calculations of different sizes: 2×2×1 [Fig. 2(A)] and 4×4×1 (not shown) corresponding to 25% and 6.25% Se vacancy concentrations. The DFT supercell calculations used the mBJ functional as implemented in the Vienna Ab Initio Simulation Package (VASP). 15 We show results obtained with the TMDCA using typical (indigo dashed line) and algebraic (green solid line) averaging methods [Fig. 2(A)]. Both the DFT and TMDCA calculations show single-particle bandgap enhancement. Most importantly, the DFT result shows no defect-induced deep-level states. As the answer was negative since the defect-induced states were all shallow impurity levels, one may expect that the main conclusions of this work, related to PbSe monolayer being defect-tolerant material, do not depend on crystal structure relaxation or spin-orbit interaction. Before proceeding to the discussion of our main results, we will first consider the properties of the pristine PbSe monolayer. The band structure and the associated partial density of states are shown in Fig. 1(C). Our results show that monolayer PbSe is a direct and narrow bandgap semiconductor with a single-particle energy bandgap Eg ≈ 0.21 eV at the X-point of the k-space. The calculated Eg of monolayer PbSe is about 5.0% smaller than the bulk Eg ≈ 0.28 eV. 6 Based on the crystal symmetry, we believe that the smaller Eg of the PbSe monolayer is justified. Aside from the difference in the crystal symmetry of the monolayer (tetragonal symmetry) and the bulk

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(cubic symmetry), the monolayer structure could have a markedly stronger crystal field effects. 43 Also, due to quantum confinement, smaller Eg of the monolayer PbSe is consistent with increased pressure that could decrease the bandgap since the lattice constant of the monolayer PbSe a0 = 4.25 Å [Fig. 1(A)] is smaller than the bulk lattice constant a0 = 6.12 Å. 44

(PbSe)1-δ PbSe1-δ Pb1-δSe

DFT TMDCA

0.4

-1

∆fE (eV)

4

(B)

(A)

D(E) (eV )

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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2

0

0.2

0 E (eV)

0

5

10 15 δ (%)

20

0

Figure 2: (A) A comparison of the density of states obtained using the supercell approach with the TMDCA@DFT for Se vacancy concentration δ = 25%. The DFT calculations used the mBJ potential. The indigo dashed line correspond to the typical density of states. Observe that in all cases, the defect-induced states are all resonant levels with no localized deep-level states within the single-particle bandgap. (B) The vacancy formation energy as a function of Pb, Se, and Pb-Se vacancy concentrations. To gain some insight on the energetics of the vacancy formation processes and the stability of the disordered monolayer PbSe crystal, we calculated the defect formation energy ∆E f = Ev − E0 , where Ev and E0 are the total energies of the vacancy and pristine structure, respectively. We checked the stability of the disordered crystal for the Se-rich, Pb-rich, and when we have vacancies on both the Pb- and Se-sublattices. As shown in Fig. 2(B), ∆E f is positive and increases with 8


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δ = 2%

δ = 1%

δ = 5%

δ = 10%

3 -1

D(E) (eV )

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2 1 0

-2

0

2

-2

0

2 -2 E (eV)

0

2

-2

0

2

Figure 3: TMDCA@DFT results of monolayer PbSe showing the typical density of states at four different Se vacancy concentrations δ . For reference, we also includes the density of states of pristine monolayer PbSe obtained from the mBJ potential. The Fermi level has been shifted to zero at the top of the valence bands. The energy bandgap systematically increased from 0.21 eV for the pristine crystal to 0.45 eV for the disordered crystal at δ = 10%. increasing vacancy concentration. Our data reveal that disorder due to Se vacancies is the most probable to form in monolayer PbSe crystal since the defect formation energy for vacancies on Se-sites are greater than that of Pb-sites and for the case where the Pb- and Se-sublattices are both disordered. Since our calculations show that Se vacancy is the most stable defect structure, unless otherwise stated, the reported results below are for the effects of Se vacancies on the optoelectronic properties of monolayer PbSe. Next, we investigate the effects of Se vacancies on the electronic properties of monolayer PbSe. Figure 3 shows the typical density of states at [low (1 and 2%) and high (5 and 10%)] Se vacancy concentrations δ . As δ increases (Fig. 3), the single-particle bandgap systematically increases as 0.21, 0.24, 0.27, 0.31, and 0.45 for the pristine, 1, 2, 5, and 10% Se vacancy concentrations, respectively. The Fermi level also shifts to higher energy defined with respect to δ = 0, implying n-type doping. The data for Pb vacancies (not shown) show p-doping as expected. In the case where the vacancies are introduced in both Pb and Se sublattices, it was dominant n-doping at low concentrations but evolves to a behavior reminiscent of p − n doping character at high concentra9



tions. This could be significant as one could engineer a p − n junction by creating vacancies in both sublattices. Experimentally, this could be achieved by modulating the threshold energy for knock-on damage, which may be markedly different for Pb and Se. Our data show significant redistribution and the enhancement of states around the Fermi level. The defect-induced redistribution and enhancement of the states around the Fermi level without the formation of isolated impurity states 45 have been shown to lead to increased thermoelectric figure of merit, e.g., in PbTe, SnTe, and Bi2 Te3 alloys. 46–51 Observe that for all the Se vacancy concentrations considered herein, aside from the formation of some resonant (virtual bound) states, there are no localized, deep-level defect states within the single-particle bandgap. To properly discern the exact band characters and their contribution to the states in the proximity of the Fermi level, we count the number of states in the band extrema. The character of the states in the proximity of the band extrema [valence band maximum (VBM) and conduction band minimum (CBM)] have been shown to be crucial to whether the defect induced impurity states will be shallow or localized, deep-trapped states. 25–27 As shown in Fig. 4, the single-particle bandgap of monolayer PbSe is predominantly derived from Pb-p and Se-p states. The conduction band minimum is formed by a strong hybridization between Pb-pz and Se-p, which contributed ∼ 64% and 36%, respectively of the total CBM states. The valence band maximum is composed mainly of Se-p, which contributed ∼ 92% of the total VBM states. When disorder due to Se vacancies is introduced in the crystal, while the above picture is generally maintained, the states were renormalization and the lattice symmetry-induced degenerate px+y orbital characters are lifted to form distinct px and py states. The electronic signature of the states that form the single-particle bandgap, namely, the antibonding character of the conduction and bonding character of the valence bands extrema will favor the bonds broken due to Se vacancies to mostly reside inside the bands as resonant states leaving the single-particle bandgap free of defect-induced deep-trapped localized states as schematically shown in Fig. 4 (A). This could be attributed to a symmetry-protected property already observed in some semiconductors and such materials are said to be defect-tolerant. 25–27 This also suggests that

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the defect-induced states are reminiscent of shallow, intrinsic defects with minimal or absence of surface states. 14,26 This picture is different from what is observed in some semiconductors where the defect-induced impurity states appear as deep-trapped levels inside the bandgap [Fig. 4(B)], which could adversely diminish the quantum efficiency of the material for device applications.

Figure 4: Schematics of the orbital character of the valence band maximum and conduction band minimum in a pristine and disordered crystal for (A) monolayer PbSe and (B) a conventional semiconductor of orbital characters α and β , respectively, where α (β ) could be a combination or any of the orbitals: s, p, and d. The absorption spectroscopy provides a detailed and direct information about the electronic signature of materials and has the advantage of being a true “bulk” probe. 6 While there have been a number of optical (experimental and theoretical) studies of both pristine and disordered bulk PbSe, 52–61 there are yet no comprehensive computational studies of the optical properties of disordered monolayer PbSe. Herein, we provide a detailed first-principles based calculations of the absorption spectra of disordered monolayer PbSe. We calculate the absorption spectra, which includes the effects of electron-hole interactions by solving the Bethe-Salpeter equation in the particle-hole channel to obtain the complex dielectric function ε(ω). The two particle calculations used the single-particle Green functions obtained from the converged, self-consistent TMDCA calculations.

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30

20

ω4

Pristine 1% 2% 5% 10%

40

ε2(ω)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

ω3

Eb ω1

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ω2

10

0 0

Egq

2

4 h ω (eV) _

6

8

Figure 5: The imaginary part of the dielectric function for monolayer PbSe as a function of the excitation energy h¯ ω at various Se vacancy concentrations δ . The ω1−4 denote the main exciton peaks. We obtain exciton binding energy Eb = 0.20, 0.24, 0.27, 0.38, and 0.42 eV for the pristine, 1, 2, 5, and 10% vacancy concentrations in that order. The vertical solid black line is the quasiparticle q energy bandgap Eg ≈ 1.51 eV. Increasing the concentration initially leads to the increase and then suppression of the generation of free electron-hole pairs at higher excitation energies. We show in Fig. 5 the imaginary part of the dielectric function ε2 (ω) for various Se vacancy concentrations. The impact of the Se vacancies on the magnitude of ε2 (ω) is an initial increase due to electron-hole delocalization and then a decrease as the vacancy concentration increases. The latter effects could be attributed to spectral weight redistribution. The first peak in the absorption spectra (ω1 ∼ 1.30 eV) is related to the band edge. As Se vacancy concentration increase, ω1 is redshifted, broadened, and decreased systematically to ∼1.0 at δ = 20%. Another prominent structure in the absorption spectra occurs at ω2 with a broad shoulder observed at ω3 . The maximum of the spectra is depicted with ω4 , which occurs at ∼5.4 eV. We note that the absorption spectra of the pristine monolayer and the bulk crystal are qualitatively similar (see, e.g., Refs. 6,55 ). However, key differences are obvious, especially at lower energies where the two resonant features below 2.50 eV (ω1 and ω2 ) is just a shoulder ∼1.90 eV in the bulk structure. 6,55 The differences could be attributed to, among other things, the crystal symmetry (tetragonal versus rock-salt), dimensionality (twodimension versus three-dimension), and the inclusion of the effects of electron-hole interactions in our current data.

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q

To estimate the exciton binding energy, we calculate the quasiparticle energy bandgap Eg ≈ 1.51 eV from the self-consistent quasiparticle Green function and screened Coulomb interaction q

calculations as implemented in VASP. 15 The difference between Eg and the first excitation energy (onset of a prominent feature) in the absorption spectra is the measure of the exciton binding energy Eb . As a function of increasing Se vacancy concentrations, we obtain Eb ≈ 0.20, 0.24, 0.27, 0.38, and 0.42 eV for δ = 0, 1, 2, 5, and 10%, in that order. As expected, our calculated fundamental bandgap is lower than the quasiparticle bandgap due to the exciton binding energy. Interestingly, the calculated exciton binding energies seem high for such a narrow bandgap material. For example, the exciton binding energy in monolayer MoS2 range from 0.2–0.7 eV. 62–67 This is significant since a bound exciton is a fingerprint of a stable material properties. With such Eb , the excitons could, e.g., withstand strong electric fields and high carrier injections. Also, the calculated absorption spectra show relatively large absorption around 1.0 – 2.5 eV, which could be explored for potential solar cell applications. In summary, we have studied the effects of randomly distributed Se vacancies on the electronic and optical properties of monolayer PbSe using a first-principles-based typical medium approach. Our results show that the defect-induced states in disordered monolayer PbSe are predominantly shallow impurity resonant levels, which mostly resides inside the bandgap, a signature of a defecttolerant material. Our calculations further show that the single-particle bandgap of monolayer PbSe could be modulated by Se vacancies since increased δ led to bandgap renormalization. The calculated exciton binding energies further show that monolayer PbSe is stable even at high vacancy concentration. The research was sponsored by the Army Research Laboratory (ARL) and was accomplished under the Cooperative Agreement Number W911NF-11-2-0030 as an ARL Research [George F. Adams] Fellow. The views and conclusions contained in this document are those of the authors and should not be interpreted as representing the official policies, either expressed or implied, of ARL or the U.S. Government. The U.S. Government is authorized to reproduce and distribute reprints for Government purposes notwithstanding any copyright notation herein. This work was supported

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in part by a grant of computer time from the DOD High-Performance Computing Modernization Program at the Army Engineer Research and Development Center, Vicksburg, MS.

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