Nonradiative Electron–Hole Recombination Rate Is Greatly Reduced

Letter pubs.acs.org/JPCL

Nonradiative Electron−Hole Recombination Rate Is Greatly Reduced by Defects in Monolayer Black Phosphorus: Ab Initio Time Domain Study Run Long,*,† Weihai Fang,† and Alexey V. Akimov‡ †

College of Chemistry, Key Laboratory of Theoretical & Computational Photochemistry of Ministry of Education, Beijing Normal University, Beijing, 100875, P. R. China ‡ Department of Chemistry, Natural Sciences Complex, State University of New York at Buffalo, Buffalo, New York 14260-3000, United States ABSTRACT: We report ab initio time-domain simulations of nonradiative electron−hole recombination and electronic dephasing in ideal and defect-containing monolayer black phosphorus (MBP). Our calculations predict that the presence of phosphorus divacancy in MBP (MBP-DV) substantially reduces the nonradiative recombination rate, with time scales on the order of 1.57 ns. The luminescence line width in ideal MBP of 150 meV is 2.5 times larger than MBP-DV at room temperature, and is in excellent agreement with experiment. We find that the electron−hole recombination in ideal MBP is driven by the 450 cm−1 vibrational mode, whereas the recombination in the MBP-DV system is driven by a broad range of vibrational modes. The reduced electron−phonon coupling and increased bandgap in MBP-DV rationalize slower recombination in this material, suggesting that electron−phonon energy losses in MBP can be minimized by creating suitable defects in semiconductor device material.

T

Recent experiments reported pump−probe ultrafast spectroscopy measurements of the black phosphorus flakes carrier dynamics.16−19 The angle-resolved pump−probe transient reflection spectroscopy revealed anisotropic properties of black phosphorus.17 Mu et al. suggested that a fast electron− electron scattering in black phosphorus occurs on the time scale of 25 fs and a slow electron decay occurs much slower on the 1.9 ps time scale.19 The attribution of the slow relaxation to either nonradiative electron relaxation or electron−hole recombination has not been presented yet. The diversity of experimentally measured time scales is large: He and coauthors have reported a lifetime for excited electronic states at long as 100 ps.16 Suess et al. reported two relaxation time scales in black phosphorus with several layers of material of 180 ps and 1.3 ns, respectively.18 Although the experimental measurements provide important information about the charge carrier dynamics and excited electron lifetimes, they lack a mechanistic understanding of these processes and have difficulty in attributing time scales to distinct processes. First-principles atomistic simulations are required to fully understand electron−hole recombination in black phosphorus, in order to provide valuable guidelines for materials design and further improvement. Many ground state density functional theory calculations have been performed in the past few years to explore the electronic properties of black phosphorus, MBP, and their derivates. Liu and coauthors predicted that few-layer black

wo-dimensional black phosphorus, also known as phosphorene, has been attracting intense attention due to its tunable direct bandgap1−5 and high charge mobility.6 The first field-effect transistors with high drain current and carrier mobility have been fabricated recently.7 The measured high room-temperature charge mobility of 104 cm2 V−1 S−1 is far above that in other two-dimensional semiconductors.6,8 This large magnitude plays a vital role in applications of twodimensional electron gases in high-speed field-effect transistor devices.9 Other experiments reported that black phosphorus has a high photoresponse10 and broadband response,11,12 making it an appealing material for optoelectronic and gas sensor applications.13,14 A monolayer black phosphorus (MBP) can be obtained via mechanical1 and liquid15 exfoliation of the bulk material. The former is even more useful than the latter due to a strong resonant absorption of MBP in the nearinfrared wavelength range. The direct bandgap varies from 0.3 eV in bulk black phosphorus,5 to 0.9−1.3 eV in the systems with less than five layers, and up to 2.0 eV in MBP.3 However, the precise bandgap magnitude of MBP remains a matter of debate varying in the range from 1.0 eV, as determined in the field-effect measurements,2 to 1.45 eV in photoluminescence detection experiments,1 and to 2.0 eV as inferred from the scanning tunnelling microscopy data.3 Although a suitable bandgap of MBP makes the material a promising candidate for optoelectronic and solar energy conversion applications, the overall suitability is controlled by other factors. The nonradiative electron−hole recombination constitutes one of the major energy loss mechanisms and can compromise the efficiency of various devices based on MBP, despite its advantageous bandgap. © XXXX American Chemical Society

Received: January 1, 2016 Accepted: January 28, 2016

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DOI: 10.1021/acs.jpclett.6b00001 J. Phys. Chem. Lett. 2016, 7, 653−659

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The Journal of Physical Chemistry Letters

study electron transfer and relaxation dynamics in a wide variety of systems.31−43 The approach satisfies the detailed balance condition, making the transitions upward in energy less probable than those downward.44 Many other methods for simulating quantum nonadiabatic dynamics are available nowadays,45 including nonequilibrium Green’s (Keldysh) functions,46,47 wavepacket-based propagation48 path integral approaches,49−51 and even fully quantal methods to nuclear-electronic dynamics.52,53 The applicability of these methods to our problem is limited due to their large computational costs or the lack of commonly available implementation. Other, more computationally efficient methods based on the Fermi golden-rule formula54 lack the atomistic details of the electron−nuclear dynamics. Finally, the methods such as the reduced density matrix55,56 or the atomistic Ehrenfest dynamics57,58 are on par with the presently utilized approach, especially when the latter is used with suitable decoherence corrections.59 The present method selection is therefore motivated by the availability of the suitable tools and by the capability of such tools to provide atomistic resolution dynamics at moderate computational costs, yet capture essential physics of the recombination process. The geometry optimization, adiabatic MD, and NA coupling calculations are performed using the Vienna Ab initio Simulation Package (VASP),60 employing projected-augmented potential61 to describe electron−ion interactions and Perdew− Burke−Ernzerhof (PBE)62 functional based on generalized gradient approximation (GGA) to treat nonlocal electron exchange-correlation interactions. The NA-MD simulations for electron−hole recombination with decoherence correction are carried out by the PYXAID code.29,30 Decoherence effects enter the NA-MD simulation via a semiclassical scheme, as described elsewhere.30,38 The present study focuses on periodic structures of ideal MBP. In our calculations, a (4 × 4 × 1) supercell is used with and without P divacancy. The representative geometries taken from the MD trajectory at ambient temperature are shown in Figure 1. A 20 Å vacuum layer in the z-direction is used to eliminate spurious interactions between the periodic images. The ideal MBP is represented by a puckered layer with the puckering along the armchair direction. A zigzag bonding direction is shown in Figure 1a. The defect model MBP-DV is

phosphorus can undergo a continuous transformation from a normal insulator to a topological insulator and finally to metal phases, when an electric field is applied.20 Han et al. reported that the bandgap and effective masses of charge carriers in black phosphorus nanoribbons depend strongly on the external strain and on the orientation of nanoribbons.21 Other authors suggested that a phosphorus divacancy is much more stable than a monovacancy.22,23 The former can be formed in high concentrations, and does not create gap states in the electronic structure of MBP.14 To date, no quantum nonadiabatic dynamics studies of electron−hole recombination in MBP have been reported. The influence, negative or positive, of divacancy defects on the nonradiative electron−hole recombination rates remains unclear. Motivated by the recent experimental works,16−18 we report a first time-domain density functional theory (TD-DFT) study of the electron−hole recombination dynamics in MBP. We employ the nonadiabatic molecular dynamics (NA-MD) technique to explore in detail the mechanism of the electron−hole recombination at the MBP with and without divacancy defects (MBP-DV) and to provide guidelines for minimizing energy loss. We study the phonon-induced pure dephasing of the highest occupied (HOMO) and the lowest unoccupied (LUMO) molecular orbitals in ideal MBP and MBP-DV at ambient temperature. The obtained nonradiative electron−hole recombination rate in ideal MBP sheets is in excellent agreement with the experimentally measured ones.16,18 Our computations predict that the relaxation slows down significantly in MBP-DV as opposed to the ideal MBP, suggesting that the nonraditative energy loss of MBP can be minimized via creating suitable defects. The computed time scales are consistent with those measured experimentally.18 Our calculations reveal that slow electron−hole recombination in MBP-DV can be attributed to small magnitudes of NA coupling and large bandgap. In addition to weak electron− phonon coupling and consequently small energy fluctuations, our simulations indicate that the decrease in atomic fluctuations in MBP-DV results in increased coherence times in MBP-DV. The homogeneous luminescence line width is predicted to be 151 meV in ideal MBP at ambient temperaturein good agreement with the experimental data.24 It is 2.5 times larger than that in MBP-DV. The electronic subsystem of ideal MBP exhibits exclusive coupling to the 450 cm−1 vibrational mode, while the MBP-DV couples to a broad spectrum of modes. Our simulations employ the quantum-classical fewestswitches surface hopping (FSSH)25 technique implemented within the time-dependent Kohn−Sham theory.26,27 The lighter and faster electrons are treated quantum mechanically, whereas the heavier and slower nuclei are described classically. The current simulation uses a simplified and more efficient version of FSSH, employing the classical path approximation (CPA), which greatly reduces computational cost while still correctly describing thermal equilibrium in quantum-classical dynamics.28−30 CPA assumes that the classical trajectory does not depend on electronic dynamics, while the electronic dynamics still relies on the classical coordinates. CPA is valid when no significant structural changes are involved, which is the case in the present study. The geometric structure of the black phosphorus and the material with divacancies are relatively rigid. They are slightly affected by electronic dynamics, providing us with solid ground for applying CPA to study quantum dynamics in these materials using the chosen methodology. The method had been applied successfully to

Figure 1. Top and side views of monolayer black phosphorus (MBP) in the stable configuration at 300 K. (a) Ideal MBP and (b) MBP-DV obtained by removing the atoms labeled 1 and 2 on the panel (a). The defects in MBP-DV only induce local distortion around the divancancy and minimize strain to form a stable 5−8−5 ring without creating dangling bonds. 654

DOI: 10.1021/acs.jpclett.6b00001 J. Phys. Chem. Lett. 2016, 7, 653−659

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The Journal of Physical Chemistry Letters

LUMO (Figure 2b). The symmetry breaking between the two sublayers of the MBP-DV relaxes selection rules. The HOMO and LUMO are the main participants in the electron−hole recombination in the two systems under investigation. Thus, their time evolution is important for understanding the dynamics of recombination and the factors that affect this process. Atomic motions induce fluctuations of the electronic orbital energy levels. The extent of the fluctuation reflects the strength of electron−phonon coupling. The canonically averaged bandgaps of MBP and MBP-DV computed at the PBE level are 0.77 and 0.99 eV, respectively. Figure 3 shows the evolution of the LUMO and HOMO energy levels of ideal MBP (black) and MBP-DV (red) systems whose bandgaps scaled to the HSE06 values. The HOMO energies fluctuate little and the fluctuations are similar in the two systems. This can be rationalized by the delocalized nature of the orbitals, which are spread nearly uniformly on the nanocrystalline phosphorus sheets, Figure 2a and 2b. The defects have relatively small impact on this delocalization, and hence do not affect energy level fluctuations strongly. The fluctuations of the LUMO energy levels are much larger in the ideal MBP than in the MBP-DV (Figure 3a). Generally, defects increase electron−phonon coupling due to a more localized nature of defect states, as was observed in carbon nanotubes.38 On the contrary, black phosphorus relaxes the strain and does not create gap states and weakens the coupling, similarly to graphanes41 and graphene nanoribbons.38 The defects increase the excitation energy in the MBP-DV system, as measured by the difference of the HOMO and LUMO energies. The LUMO energy of ideal MBP exhibits fluctuation with a well-defined frequency, Figure 3a. The defects create a broad range of vibrational modes that couple to the electronic degrees of freedom. Furthermore, the symmetry breaking in both the HOMO and LUMO of the MBP-DV helps it couple to more phonons (Figure 2b). To characterize the phonon modes that couple to the electronic states participating in electron−hole recombination, we compute the spectral density (Figure 3b). The spectral density is computed by the Fourier transform (FT) of the HOMO−LUMO energy gap fluctuation, as detailed elsewhere.42,68 The electronic degrees of freedom in the ideal MBP couple exclusively to the 450 cm−1 vibrations. The frequency approximates to either the 470 cm−1 P−P stretching mode, with the P−P stretching taking place along the armchair direction, or to the 440 cm−1 mode, with P−P stretching along the zigzag direction in black phorphorus.4,24,69 The existing experimental data and computational studies,4,24,69,70 suggest that the vibrational mode along the armchair direction is more probable, because it contributes strongly to the electronvibrational coupling. In contrast to the ideal MBP, a broad range of phonon modes couples to the electronic degrees of freedom in the MBP-DV system. To characterize the strength of electronic transitions in MBP and MBP-DV, we have computed the averaged values of the magnitude of NA coupling (Table 1). Our calculations show that the NA coupling in MBP is 1.5 times larger than that in MBP-DV. This can be rationalized by large atomic fluctuations in ideal MBP relative to those in MBP-DV. The coupling is proportional to derivative coupling,⟨φĩ |∇R |φj̃ ⟩, and to velocity of nuclei, Ṙ . Both quantities are increase as the magnitude of atomic fluctuations increases. The discussion of atomic motion in MBP and MBP-DV is presented below.

created by removing two adjacent phosphorus atoms, labeled 1 and 2 (Figure 1a), which has the smallest formation energy, as suggested by Li et al.14 Figure 1b shows that the divacancy is self-healing and forms one 8-fold ring and two adjacent 5-fold rings (top view of Figure 1b). The side view of Figure 1b further confirms that the divacancy leads to only local geometry distortion around the defects, not affecting the overall crystal structure. The ideal MBP has a direct bandgap at the Γ point. The value of 0.87 eV is calculated at the PBE level of theory,62 whereas the hybrid HSE06 functional predicts the value of 1.56 eV.63−65 The HSE06 value agrees well with previous theoretical22,66 and experimental2,67 findings. The presence of the divacancy increases the bandgap to 1.16 and 1.86 eV according to PBE and HSE06 functionals, respectively. The absence of gap states may arise from the self-healing of divacancy eliminating dangling bonds. The charge densities of the HOMO and the LUMO of the two systems are shown in Figure 2. In ideal MBP, the σ bonds

Figure 2. Charge densities of the HOMO and LUMO molecular orbitals. (a) For ideal MBP, the orbitals are distributed uniformly on the sheet. (b) For MBP-DV, the LUMO is localized mainly around the phosphorus defects, whereas the HOMO density is depleted around the 8-atom ring.

remain intact at the ambient temperature. Photoexcitation promotes an electron from the bonding σ orbital distributed uniformly in the two sublayers along the P−P bonds to the antibonding σ orbitals of the in-plane P atoms. This is clearly depicted in the side views of the HOMO and LUMO charge densities in Figure 2a. The introduction of the P divacancy into the ideal MBP induces significant changes in the charge densities of both HOMO and LUMO. The LUMO becomes localized in the vicinity of the 5−8−5 ring (Figure 2b). The HOMO remains largely delocalized, although a significant depletion of the charge density in the vicinity of the 5−8−5 ring is observed. Thus, upon the photoexcitation of the MBPDV system, the electron density shifts from the HOMO to the 655

DOI: 10.1021/acs.jpclett.6b00001 J. Phys. Chem. Lett. 2016, 7, 653−659

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The Journal of Physical Chemistry Letters

Figure 3. (a) Time evolution of the HOMO and LUMO orbital energies for ideal MBP (black line) and MBP-DV (red line) at 300 K. Values are scaled to the HSE06 energy gap. The fluctuation of the LUMO energy of MBP is greater than that of MBP-DV, whereas the HOMO energies show smaller and similar fluctuations. The Fermi level is set to 0. (b) Spectral density computed by taking a Fourier transform (FT) of the HOMO− LUMO energy gap fluctuation, shown in panel a. Magnitudes for the ideal MBP are scaled down by the factor of 50. Electronic recombination in ideal MBP couples exclusively to the optical phonon mode at 450 cm−1, while MBP-DV recombination is driven by a boarder range of vibrational modes.

we compute the magnitude of atomic fluctuations in ideal and divacancy-containing systems. The magnitude of atomic fluctuation are computed as the ensemble-averaged standard

Table 1. Average NA Coupling, Pure-Dephasing Time (T2*), Nonradiatve Electron-Hole Recombination Time, and Line Width (Γ), for Ideal MBP with and without Defect (MBPDV)

MBP BP-DV

NA coupling (meV)

dephasing (fs)

recombination (ns)

line width (meV)

3.52 1.97

4.5 11.3

0.108 1.57

151 60

deviation of the position of each atom i, σi = ⟨( ri ⃗ − ⟨ ri ⃗⟩)2 ⟩ . Here, ri ⃗ represents the location of atom i at time, and angular bracket indicates ensemble and time averaging. A larger standard deviation reflects larger fluctuation of that atom. For the ideal MBP, all atoms are identical. Based on our computations, one can distinguish two types of P atoms in the MBP-DV systems: those belonging to the 5−8−5 ring and all other atoms. The computed standard deviations for each group of atoms are listed in Table 2. The results indicate that

Another factor affecting the electron−hole recombination is electronic decoherence. To characterize decoherence in the two studied systems, we compute decoherence times. Our calculations are based on the optical response theory and the second-order cumulant approximation.32,68 Pure-dephasing characterizes elastic electron−phonon scattering and can be related to luminescence line width, which can be probed experimentally. The pure-dephasing times, τ, are obtained by fitting the data in Figure 4a by the Gaussian: f(t) = A exp(−0.5(t/τ)2) + B. The dephasing times summarized in Table 1 suggest that the dephasing in ideal MBP occurs 3 times faster than in MBP-DV. Correspondingly, the fluorescence line width is 151 meV for ideal MBP at room temperature and reduces 3 times in the presence of defects. The computed line width of MBP is in excellent agreement with the experimental value of 150 meV.24 In order to understand the physical origins of weaker electron−phonon coupling and slower dephasing in MBP-DV,

Table 2. Standard Deviations in the Positions of Phosphorus Atoms in the Ideal MBP and MBP-DVa MBP MBP-DV

P

P belonging to 5-8-5 ring/rest P

0.900 0.735

0.727/0.737

a

The averaged value for the phosphorus atoms belonging to the 5-8-5 ring and the rest of atoms in MBP-DV are also listed. The units are Å.

the defects decrease the mobility of both the atoms belonging to the 5−8−5 ring and the atoms outside the ring. Particularly, the standard deviations of P atoms reduce from 0.900 Å in ideal MBP to 0.735 Å in MBP-DV.

Figure 4. (a) Dephasing functions computed using the HOMO−LUMO energy gap fluctuation. The pure dephasing times are reported in Table 1. (b) Electron−hole recombination across the HOMO−LUMO energy gap due to phonon-induced nonradiative relaxation. MBP-DV shows a much slower nonradiative decay rate than the ideal MBP, suggesting that the nonradiative electron−hole recombination can be substantially reduced via creating defects. 656

DOI: 10.1021/acs.jpclett.6b00001 J. Phys. Chem. Lett. 2016, 7, 653−659

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The Journal of Physical Chemistry Letters

are likely to be responsible for the multiple decay time scales reported in the experiment. Slow relaxation in black phosphorus induced by the defects is attributed to the weak NA electron−phonon coupling and large bandgap. The combination of these factors competes successfully with the slow phonon-induced pure-dephasing in the MBP-DV system. We rationalize the slow dephasing and smaller couplings in this system to the reduced mobility of the phosphorus atoms due to a vacancy-induced strain in the system. In the ideal MBP system, the nonradiative electron−hole recombination is promoted exclusively by the 450 cm−1 phonon mode. Although the defects create new types of vibrational modes, they decrease the mobility of phosphorus atoms, as a result, decreasing the NA coupling by the factor of 1.5 and the recombination rate by an order of magnitude. The defect-induced increase of the lifetime of an excited electron in monolayer black phosphorus ensures slow recombination of charge carriers. The simulations suggest that the vacancy defects can play an important role in the rational design of materials with longer nonradiative electron−hole recombination time scale. This knowledge can be used to minimize the recombination rate via engineering of suitable defects.

The observed changes in the nuclear dynamics can be rationalized by the increased strain in the vacancy-containing system. Although the introduced vacancies are healed and the dangling bonds are not formed in the MBP-DV, the overall structure develops a certain amount of strain, making atoms less mobile. The decreased mobility is a consequence of increased energetic costs associated with the potential displacement of the P atoms away from their new equilibrium positions. As a result, the decreased atomic mobility reduces the NA couplings, because the overall change of the wave function on the nuclear vibrations diminishes. Analogously, the decreased atomic fluctuations have a smaller potential to induce substantial fluctuations of energy levels. Consequently, the electronic coherence times increase. The computed electron−hole recombination kinetics in MBP and MBP-DV is shown in Figure 4b. The recombination time scales are obtained by fitting the computed data with the short-time linear approximation P(t) = exp(−t/τ) ≈ 1 − t/τ. The recombination times, τ, are summarized in Table 1. The obtained time scale of 108 ps in ideal MBP agrees well with the corresponding experimental data.16,18 The introduction of divacancy into the ideal MBP dramatically reduces the recombination rate, resulting in a longer excited-state lifetime of 1.57 ns. The work of Suess et al.18 reports two sets of decay times with a fast channel of 180 ps and a slow channel of 1.37 ns, respectively. Our calculations suggest that the shorter time can be attributed to the recombination in the ideal MBP phases, while the longer time corresponds to relaxation involving vacancy-containing (surface) phases. Increasing the number of layers of black phosphorus is expected to reduce quantum confinement effects and to decrease the band gap of the material. This change can increase the electron-vibrational NA couplings and accelerate decoherence (whose rate shortens with the increase of energy gap fluctuations) due to the additional interlayer interaction. The overall effect for electron−hole recombination depends on band gap, NA couplings, and decoherence. In the limit of infinite number of layers, the recombination rate of a system containing defects should coincide with that of ideal MBP, because the concentration of surface defects will decrease, and the role of this type of defect will become negligible. The magnitude of the HOMO−LUMO energy gap, the magnitude of the NA coupling, and the pure dephasing times are the three main factors affecting the dynamics of electron− hole recombination, Table 1. Based on the energy gap law,71 the smaller HOMO−LUMO energy gap in MBP system as compared to that in MBP-DV leads to faster nonradiative recombination in the former one. The NA coupling primarily determines the decay rates in both ideal and vacancy-containing systems. The electron−phonon coupling is 1.5 times larger in the ideal MBP than in the MBP-DV, Table 1. According to Fermi’s golden rule,72 the population transfer rate depends on the square of the coupling. Strong electron−phonon coupling and small HOMO−LUMO energy gap compete successfully with the effect of fast dephasing in ideal MBP, leading to the recombination being overall faster in MBP than in MBP-DV. In summary, we have conducted the time-domain ab initio atomistic simulation of electron−hole recombination in monolayer black phosphorus using nonadiabatic molecular dynamics, directly mimicking the time-resolved experiments. Our simulations indicate that a strong nonradiative electron− hole recombination channels exist in black phosphorus. The decay rate is greatly reduced by the presence of defects, which

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS R.L. is grateful to the National Science Foundation of China (Grant No. 21573022). W.H.F. thanks the National Science Foundation of China under Grant No. 21520102005 and the Science Fund for Creative Research Groups of the National Natural Science Foundation of China under Grant No. 21421003. A.V.A. acknowledges the financial support from the University at Buffalo, The State University of New York, startup package.

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