Quasiparticle electronic structure and optical absorption of diamond nanoparticles from ab initio many-body perturbation theory Huabing Yin, Yuchen Ma, Xiaotao Hao, Jinglin Mu, Chengbu Liu, and Zhijun Yi Citation: The Journal of Chemical Physics 140, 214315 (2014); doi: 10.1063/1.4880695 View online: http://dx.doi.org/10.1063/1.4880695 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/140/21?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Enhanced many-body effects in 2- and 1-dimensional ZnO structures: A Green's function perturbation theory study J. Chem. Phys. 139, 144703 (2013); 10.1063/1.4824078 Communication: Electronic band gaps of semiconducting zig-zag carbon nanotubes from many-body perturbation theory calculations J. Chem. Phys. 136, 181101 (2012); 10.1063/1.4716178 Ab initio many-body study of the electronic and optical properties of MgAl2O4 spinel J. Appl. Phys. 111, 043516 (2012); 10.1063/1.3686727 Ab initio calculations of optical absorption spectra: Solution of the Bethe–Salpeter equation within density matrix perturbation theory J. Chem. Phys. 133, 164109 (2010); 10.1063/1.3494540 Many-body effects in nonadiabatic molecular theory for simultaneous determination of nuclear and electronic wave functions: Ab initio NOMO/MBPT and CC methods J. Chem. Phys. 118, 1119 (2003); 10.1063/1.1528951

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THE JOURNAL OF CHEMICAL PHYSICS 140, 214315 (2014)

Quasiparticle electronic structure and optical absorption of diamond nanoparticles from ab initio many-body perturbation theory Huabing Yin,1 Yuchen Ma,1,a) Xiaotao Hao,2 Jinglin Mu,1 Chengbu Liu,1,b) and Zhijun Yi3 1

School of Chemistry and Chemical Engineering, Shandong University, Jinan 250100, China School of Physics, State Key Laboratory of Crystal Materials, Shandong University, Jinan 250100, China 3 Department of Physics, China University of Mining and Technology, Xuzhou 221116, China 2

(Received 29 January 2014; accepted 16 May 2014; published online 5 June 2014) The excited states of small-diameter diamond nanoparticles in the gas phase are studied using the GW method and Bethe-Salpeter equation (BSE) within the ab initio many-body perturbation theory. The calculated ionization potentials and optical gaps are in agreement with experimental results, with the average error about 0.2 eV. The electron affinity is negative and the lowest unoccupied molecular orbital is rather delocalized. Precise determination of the electron affinity requires one to take the off-diagonal matrix elements of the self-energy operator into account in the GW calculation. BSE calculations predict a large exciton binding energy which is an order of magnitude larger than that in the bulk diamond. © 2014 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4880695] I. INTRODUCTION

Owing to the highly unique physical and chemical properties, hydrogen-terminated diamond nanoparticles (DNPs), also named diamondoids, have generated great excitement in regards to their potential applications as electron emitters, photovoltaics, and other nanoscale devices.1–3 In recent years, solar cell devices based on conjugated polymers and DNPs have been studied by experiments, and it has been shown that DNPs can enhance the optical and electrochemical properties of polymers.4, 5 Many experiments and theoretical calculations have been conducted to investigate the electronic and optical properties of isolated DNPs. Diamondoids are composed entirely of sp3 -hybridized carbon. The smallest diamondoid is adamantane (C10 H16 ). In contrast to other semiconductor nanocrystals, such as Si and Ge, which exhibit clear quantum confinement (QC) effects, the optical properties of DNPs, e.g., optical gaps and oscillator strength, are determined by not only the size but also the shape and symmetry of the nanoparticles.6–9 It has also been found that the electron affinity and optical gap of DNPs are determined by Rydberg states.10, 11 Experimentally, Bostedt et al.9, 12, 13 measured the ionization potentials (IP) and optical gaps of a series of smalldiameter DNPs in the gas phase which contain 10–26 carbon atoms. Their works provide good references for theoretical calculations. Various theoretical methods have been applied to study the electronic and optical properties of DNPs,10, 14–16 including density-functional theory (DFT), time-dependent DFT (TDDFT), quantum Monte Carlo (QMC), and GW method within the many-body perturbation theory (MBPT). QMC predicts correctly the negative electron affinity property of diamondoids. However, QMC overestimates the optical gaps by 0.8−1.2 eV,14, 15 and overestimates the ionization potential of adamantane by about 1.0 eV.16 DFT a) Electronic mail: [email protected] b) Electronic mail: [email protected]

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exhibits large errors in both the electronic and optical gaps, however, TDDFT calculations with hybrid functionals for the exchange-correlation kernel, such as PBE0 and B3LYP, achieve high accuracy in optical gaps.10, 15, 16 Most surprisingly, in a recent work by DFT and importance sampling Monte Carlo method, Patrick and Giustino11 propose that the nuclear quantum motion of carbon atoms play a remarkable role in the optical gaps of diamondoids, e.g., 650 meV for adamantane, i.e., the electron-phonon coupling should be considered to get a good agreement between theoretical calculations and experiments for the optical gap. Apparently, there is still controversy on the excited state of DNPs. In this work, we reexamine the electronic and optical properties of DNPs by the GW method and Bethe-Salpeter equation (BSE) within the ab initio MBPT which is based on a set of Green’s function equations.17, 18 The GW method has achieved great success in quasiparticle (QP) calculations and its combination with BSE has also become a standard procedure for optical excitations of many systems of different dimensions. We will show that the ionization energy and optical gap of DNPs could be predicted acceptably (about 0.2 eV) by MBPT. In the conventional one-shot perturbative GW calculation, the QP wave functions are usually approximated by the Kohn-Sham eigenfunctions from DFT, and the off-diagonal matrix elements of the self-energy operator can thus be neglected. However, we find that this simplification can lead to an error of 0.2−0.3 eV in the electron affinity of DNPs.

II. MODELS AND METHODS

Fig. 1 shows the nine kinds of DNPs studied in this work. Their structures are extracted from bulk diamond and the boundary carbon atoms are terminated by hydrogen atoms. These DNPs have been well studied in recent experiments.9, 13 Ground-state geometries of the DNPs are optimized within DFT using the SIESTA code,19 employing the PBE generalized gradient approximation (GGA)20 for the

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J. Chem. Phys. 140, 214315 (2014) TABLE I. The quasiparticle (QP) gap, optical gap (E), and exciton binding energy (EB ) of the lowest excited state for adamantane calculated by Gaussian basis sets with or without the extra decay constant 0.02. “diag.” refers to calculations within the first-order perturbation evaluation, i.e., only the diagonal matrix elements of  − Vxc are considered. “full” refers to calculations that include both the diagonal and off-diagonal matrix elements of  − Vxc . The energy is in unit of eV. Plane wavea

Gaussian basis α

exchange and correlation energy and norm-conserving Troullier-Martins pseudopotentials21 to describe the interaction between the ion cores and the valence electrons. The double-ζ plus a single shell of polarization function is used as the basis set for geometry optimization. Based on the optimized ground-state geometries, the electronic and optical properties are then calculated by a set of Gaussian-orbital based codes.22–24 First, DFT calculations are carried out to get the Kohn-Sham eigenvalues and eigenfunctions that will be used in the GW+BSE procedure to construct physical quantities, such as self-energy operator, electronhole interaction kernel, exciton wave functions, etc. Previous studies have shown that the lowest unoccupied molecular orbital (LUMO) of the DNP is a Rydberg state which is rather delocalized in space.1, 10, 11, 14 Drummond et al.14 found that diffuse basis functions must be added to the Gaussian basis in order to describe the diffuse character of LUMO. In the work of Wang et al.1 where SIESTA code is used to calculate the electronic properties of DNPs, a set of ghost atoms are introduced in the vacuum region to reproduce the results from plane-wave based DFT. In our work, the basis sets are composed by two parts: (i) atom-centered Gaussian orbitals with decay constants (in atomic unit) of 0.2, 0.71, and 2.5 for carbon and 0.15 and 0.6 for hydrogen; (ii) Gaussian orbitals with a much smaller decay constant of 0.02 whose centers are positioned at the center of the DNP. Gaussian orbitals with s, p, d, and s* symmetry are included for each decay constant. The 0.02 decay constant is supplemented in order to describe the diffuse character of Rydberg states. This strategy has been applied in previous CASPT2 calculations by Roos et al.25–27 The decay constants have been well adjusted so that the orbital energies calculated by the Gaussian-orbital based DFT reproduce (within 0.1 eV) those obtained by a well-converged plane-wave based DFT, such as VASP.28, 29 Here, it is worthwhile to note that the energies of all the occupied orbitals obtained by our Gaussian basis sets are in agreement with the plane-wave results, and for the lowest 3−4 unoccupied or-

With 0.02

 − Vxc

full

full

diag.

diag.

Exp.b

QP gap E EB

11.49 7.49 4.00

9.73 6.67 3.06

9.97 ... ...

10.04 ... ...

... 6.49

a

FIG. 1. Structures of the diamond nanoparticles studied in this work: (a) adamantane (C10 H16 ), (b) diamantane (C14 H20 ), (c) triamantane (C18 H24 ), (d) [1(2)3]tetramantane (C22 H28 ), (e) [123]tetramantane (C22 H28 ), (f) [121]tetramantane (C22 H28 ), (g) [1(2, 3)4]pentamantane (C26 H32 ), (h) [12312]hexamantane (C26 H30 ), and (i) [1212]pentamantane (C26 H32 ). Carbon and hydrogen atoms are represented in cyan and gray, respectively.

Without 0.02

b

Calculated by the ABINIT package. Reference 9.

bitals which are most relevant to the optical gaps, the error is also within 0.1 eV. Therefore, the excited states in the lowenergy side of the spectra calculated by our GW+BSE calculations are reliable. Inclusion of the extra diffuse functions with the decay constant of 0.02 is mandatory. For example, they influence the QP gap between the highest occupied molecular orbital (HOMO) and LUMO, the optical gap, and the exciton binding energy of the lowest excited state by 1.76, 0.82, and 0.94 eV, respectively, for adamantane (Table I).

III. RESULTS AND DISCUSSION

First of all, in order to verify the reliability of the use of diffuse orbitals (i.e., the 0.02 decay constant) in this work, a GW calculation, which is based on a plane-wave basis set with a cutoff energy of 25 Ry, is carried out using the ABINIT package30, 31 for the smallest diamondoids, adamantane. In the plane-wave GW calculation, norm-conserving pseudopotentials are used to describe the electron-ion interactions. When we only consider the diagonal terms of  − Vxc , the QP gap of adamantane obtained from the plane-wave GW calculation is 10.04 eV, which is consistent with our results (9.97 eV) calculated by the Gaussian basis sets with the 0.02 decay constant (see Table I). Recently, Marsili et al.32 performed GW calculations on the electronic gaps of Ge nanoparticles by the plane-wave-based code Yambo, which also show the Rydberg characters of LUMOs. In their work, the QP gap for Ge10 H16 is 8.3 eV. We also study Ge10 H16 by our GW scheme. If the decay constant 0.02 is included in the Gaussian basis set, its QP gap is 8.24 eV, if not, it is 9.68 eV. GW calculations are usually carried out within firstorder perturbation theory by assuming that the QP wave function can be approximated by the Kohn-Sham eigenfunction. Within this approximation, off-diagonal matrix elements of the QP correction term  − Vxc are neglected and only its diagonal terms are considered. Here,  is the non-Hermitian electron self-energy operator and Vxc is the exchange-correlation potential within DFT. This approximation is valid for the occupied orbitals. Table II gives the ionization potentials of the DNPs calculated by the GW method,

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TABLE II. Ionization potentials (in eV) for the diamondoids calculated by the GW method and their comparison with experimental data. Diamondoids

Formula

GW

Exp.a

Adamantane Diamantane Triamantane [1(2)3]tetramantane [123]tetramantane [121]tetramantane [1(2, 3)4]pentamantane [12312]hexamantane [1212]pentamantane

C10 H16 C14 H20 C18 H24 C22 H28 C22 H28 C22 H28 C26 H32 C26 H30 C26 H32

9.17 8.66 8.34 8.10 8.11 8.04 8.00 7.98 7.86

9.23(5) 8.80(5) 8.44(5) ... ... 8.23(5) 8.18(5) ... ...

a

Reference 13.

and the maximal error is about 0.2 eV. With the increase of DNP diameter, the energy of HOMO rises gradually (see Fig. 2), exhibiting a strong quantum confinement effect. In DFT, the LUMO energies of DNPs are negative, e.g., −0.80 eV for adamantane, and LUMOs are thus bound states. This is different from the experiments which have found that DNPs are negative electron affinity materials, i.e., LUMO should be an unbound state. By the GW method, within the first-order perturbation theory, the energies of LUMOs are lifted up to about 0.60 eV (Fig. 2). This is consistent with previous GW results from Nguyen et al.16 Since QP LUMO is now above the vacuum level and is an unbound state, its wave function should be more delocalized than its corresponding DFT LUMO. The difference between QP LUMO and DFT LUMO is controlled by the off-diagonal matrix elements of  − Vxc . If this difference is large, the first-order perturbation theory becomes invalid.24 We took the full matrix elements of  − Vxc , both diagonal and off-diagonal, into account and calculated the QP energies by diagonalizing the QP Hamiltonian. We found that the QP energy of LUMO is reduced by 0.2−0.3 eV compared to that from first-order perturbation theory (Fig. 2). The QP LUMO is a mixture of the bound DFT LUMO and higher unbound DFT empty orbitals. For the nine

FIG. 2. Energies (in eV) of HOMO and LUMO for diamondoids calculated by the GW method. “diag.” refers to calculations that include only the diagonal matrix elements of  − Vxc . “full” refers to calculations that include both the diagonal and off-diagonal matrix elements of  − Vxc .

TABLE III. Optical gaps (in eV) of diamondoids calculated by GW+BSE and their comparison with experimental data. Diamondoids

Formula

GW+BSE

Exp.a

Adamantane Diamantane Triamantane [1(2)3]tetramantane [123]tetramantane [121]tetramantane [1(2, 3)4]pentamantane [12312]hexamantane [1212]pentamantane

C10 H16 C14 H20 C18 H24 C22 H28 C22 H28 C22 H28 C26 H32 C26 H30 C26 H32

6.67 6.78 6.04 5.90 5.92 6.13 5.86 6.00 5.82

6.49 6.40 6.06 5.94 5.95 6.10 5.81 5.88 5.85

a

Reference 9.

DNPs we studied, the energy of LUMO is independent of the size of DNP, exhibiting no quantum confinement effect. Optical gap is an important factor that determines the optoelectronic properties of nanoclusters, such as Gan Asm 33 and Sin Hm nanoclusters.34 Table III lists our GW+BSE optical gaps of diamondoids and those from experiments. Our calculations reproduce the experimental data acceptably with average deviations smaller than 0.2 eV except diamantane where the deviation reaches 0.4 eV. In Table III, the GW+BSE optical gap refers to the lowest dipole-allowed transition. For diamantane which is of D3d symmetry, the HOMO→LUMO transition is dipole-forbidden, and this transition forms the lowest excited state (S1 ) with zero oscillator strength at the energy of 6.26 eV in GW+BSE. The second and the third excited states (S2 and S3 ) are also dark states, which come from transitions HOMO−1→LUMO and HOMO−2→LUMO, respectively. S2 and S3 are degenerate at the energy 6.50 eV. The fourth excited state (S4 ) of diamantane, which has the energy of 6.78 eV, comes from the dipole-allowed transition HOMO→LUMO+1. Compared with the experiment by Landt et al.,9 we find that the calculated bright S4 state for diamantane coincides with the first absorption peak at 6.77 eV in Fig. 2(a) of Ref. 9, while the optical gap of 6.40 eV they give in Table I of Ref. 9 is more close to our calculated dark states. In our calculations, we did not take the temperature effect into account. In experiment,9 the optical absorption is measured at 20−220 ◦ C. So we think that the experimental optical gap of diamantane may originate from phonon-assisted electronic transitions. For [121]tetramantane and [12312]hexamantane which exhibit the C2h and D3d symmetry, respectively, the HOMO→LUMO transition, which forms the S1 state of these two clusters at the energy of 5.81 and 5.78 eV, respectively, is also dipole-forbidden. For [121]tetramantane, the GW+BSE optical gap is at the S2 state (6.13 eV) which originate from the HOMO→LUMO+2 transition. The HOMO→LUMO+1 transition is also dipole-allowed for [121]tetramantane and forms the S3 state at 6.17 eV with the oscillator strength an order of magnitude larger than S2 . It is the larger exciton binding energy in S2 than S3 that makes the HOMO→LUMO+2 exciton lower in energy than the HOMO→LUMO+1 one. However, due to the small energy difference between S2 and S3 and the vibrational effects of nuclei in finite temperature, these two absorption peaks should be indiscernible in experiment.

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They may form a single peak where the S3 state dominates. For [12312]hexamantane, the S2 state, which is composed by the HOMO−1→LUMO transition, is degenerate with S1 and also dark. The optical gap is at the S3 state which comes from the HOMO→LUMO+1 transition. Recent QMC calculations overestimate the optical gaps of DNPs by about 0.8 eV compared to experiment,15 and the authors attribute this to the lack of zero-point energy correction in their calculations. Recent work by Patrick and Giustino11 seems to support this idea. They suggest that the experimental optical gaps should correspond to adiabatic excitations instead of vertical ones. It seems that our present GW+BSE calculations do not support their conclusion. It may be necessary to take into account the effects of electron-phonon coupling in the GW +BSE calculations just like the strategies proposed by Giustono et al.36, 37 and Marini et al.38, 39 in the future to further clarify this point. However, this is beyond the scope of our present work. From C10 Hm to C26 Hm , the QP gap decreases by about 1.2 eV, while the optical gap decreases by about 0.80 eV. This is induced by the decrease of exciton binding energy as the size of the cluster increases. For example, in adamantane (C10 H16 ) and [1(2,3)4]pentamantane (C26 H32 ), the exciton binding energy is 3.06 and 2.72 eV, respectively. The binding energy is close to that for hydrogenated Si nanoparticles of similar size.35 This is in contrast to the QMC calculations which predict that the exciton binding energy of C29 H36 is only 1.25 eV.14 The exciton binding energy in bulk diamond is calculated to be 0.4 eV by us, which is an order of magnitude smaller than those for the small-diameter DNPs. Fig. 3 shows the absorption spectra of the nine diamondoids (from C10 to C26 ) from GW+BSE calculations

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and experiment.9 The experimental spectra are broad, while the theoretical ones consist of many sharp peaks, especially for adamantane and diamantine which have only one and two carbon cages, respectively. The obvious sharp peaks of adamantane and diamantine originate from the existence of molecular-like excitations.9 Patrick and Giustino11 show that the non-continuous spectra are due to the neglect of carbon atom vibrations in the calculations. The spectrum also depends strongly on the symmetry of the diamondoid. In our calculations, with the size increase of DNPs, the separated sharp peaks form continuous spectra gradually. For larger DNPs (from C18 to C26 ), to a certain extent, GW +BSE calculations can reproduce the main absorption peaks of the continuous spectra in experiment. The calculated spectral intensity at high energy is lower than that from the experiment, which may also be attributed to the carbon atom vibration as demonstrated in Ref. 11. From both Table III and Fig. 3, we can see that the deviation between theoretical calculations and experiment at the low energy region is more pronounced for adamantane and diamantane. This may mean that the electron-phonon interaction is much stronger in these two small nanocrystals than that in larger ones. Electron-phonon interaction may cause a redshift of the optical absorption onset,11 leading to a more close agreement with experiment. IV. CONCLUSIONS

In summary, using GW+BSE method, we have calculated the ionization potentials and optical gaps of Hterminated diamond nanoparticles. The agreement with experiment is good, typically within 0.2 eV. However, to obtain a full agreement with experiment, especially for the small nanoparticles and for the overall shape of the optical spectrum, the role of electron-phonon interaction need to be found out. Meanwhile, it is noteworthy that the optical gaps calculated by the GW+BSE method in this work are obtained from the lowest dipole-allowed transitions, which are different from the experimental optical gaps that are obtained by integrating the oscillator strength up to a certain threshold (5×10−4 of the normalized oscillator strength). Thus, the agreement of 0.2 eV for the optical gaps is just an estimate. Our calculations also manifest the necessity to consider the off-diagonal matrix elements of the self-energy operator for an accurate prediction of electronic and optical gaps of DNPs by many-body perturbation theory. ACKNOWLEDGMENTS

FIG. 3. Absorption spectra of diamondoids. Red (dashed) line: experimental spectra from Ref. 9. Black (solid) line: calculated spectra by GW+BSE which are Gaussian-broadened by 0.03 eV.

This work was supported by the National Natural Science Foundation of China (NNSFC) (Grant Nos. 21173130, 91127014, and 11247264), the Natural Science Foundation of Shandong Province (No. BS2012CL022), and Open Fund of State Key Laboratory of Information Photonics and Optical Communications (Beijing University of Posts and Telecommunications). X.T.H. acknowledges the support from National Young 1000 Talents Program and Research Fund for the Doctoral Program of Higher Education (Grant No. 20130131110004). Computational resources have been provided by the National Supercomputing Centers in Jinan.

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Quasiparticle electronic structure and optical absorption of diamond nanoparticles from ab initio many-body perturbation theory.

The excited states of small-diameter diamond nanoparticles in the gas phase are studied using the GW method and Bethe-Salpeter equation (BSE) within t...
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