Effect of Doping with Shallow Donors on Radiative and Nonradiative

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Effect of Doping with Shallow Donors on Radiative and Nonradiative Relaxation in Silicon Nanocrystals: Ab-Initio Study Natalia Derbenyova, and Vladimir Anatoly Burdov J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.7b09882 • Publication Date (Web): 05 Dec 2017 Downloaded from http://pubs.acs.org on December 10, 2017

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The Journal of Physical Chemistry C is published by the American Chemical Society. 1155 Sixteenth Street N.W., Washington, DC 20036 Published by American Chemical Society. Copyright © American Chemical Society. However, no copyright claim is made to original U.S. Government works, or works produced by employees of any Commonwealth realm Crown government in the course of their duties.

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The Journal of Physical Chemistry

Effect of Doping with Shallow Donors on Radiative and Nonradiative Relaxation in Silicon Nanocrystals: ab-initio Study Natalia V. Derbenyova and Vladimir A. Burdov∗ Lobachevsky State University, 603950 Nizhny Novgorod, Russian Federation Abstract: Electronic structure of 1 – 2 nm in diameter hydrogen-passivated silicon nanocrystals, and rates of the radiative interband transitions and Auger recombination were calculated on the basis of the first-principles (DFT/TDDFT) methods for the nanocrystals doped with a single centrally located phosphorus or lithium atom. We have found a significant increase of the radiative recombination rates caused by the nanocrystals’ doping at room temperature. For the P-doped crystallites this effect takes place at zero temperature as well, while the Li-doped crystallites (at least, some of them) demonstrate strong temperature dependence of the recombination rates which drastically drop as the temperature decreases. The rates of the Auger recombination in the nanocrystals with Li, in the whole, turn out to be of the same order of magnitude as in the undoped nanocrystals. On the contrary, in the P-doped nanocrystals the Auger process becomes considerably slower.

I.

INTRODUCTION

As known, silicon is the most widespread material in microelectronics. Its low cost, high purity, and nontoxicity have made it a basic semiconductor used in microelectronic devices. However, its use in various optoelectronic applications is impeded because of indirect band gap of bulk silicon. At the same time, incorporation of silicon into the elemental base of optoelectronics, photovoltaics, bio- and medical photonics would be extremely desirable, and, therefore, remains a challenge for modern physical science and technology. Great research activity and efforts were directed to overcoming this obstacle. One of the ways of modifying the silicon band structure is nanostructuring silicon, which partially allows to straighten out the gap and obtain efficient room-temperature luminescence1,2 and optical gain.3 Sufficiently high values of recombination rates (∼ 107 s−1 ) for radiative interband transitions4 have been reached in Si nanocrystals with diameters supposedly less than 2 nm. Such a way seems to be sufficiently perspective—certain progress was achieved in methods of preparation of Si nanocrystals and their applications in photonics, photovoltaics,5,6 and biology.7 An additional reconstruction of the electronic structure of Si crystallites can be obtained with various methods such as: formation of crystallites in different dielectric matrices;8–10 surface chemistry;11–16 introduction of shallow impurities;17–21 etc. As shown in a number of theoretical12,15 and experimental16,22 works, some kinds of the nanocrystals’ covering can significantly intensify the electron-hole radiative transitions, and enhance the photoluminescence intensity. Traditionally, doping is considered as an effective means for modifying basic properties of bulk semiconductors, in particular, bulk Si. It was also revealed experimentally for silicon crystallites that, at certain conditions, their doping with shallow donors (P, Li) improves the luminescent properties.23–26 This effect was theoretically explained for Si nanocrystals doped with phosphorus by increasing the rate of the radiative transitions caused by the efficient Γ-X mixing.24,27

It is known, however, that an emittance of a nanocrystals’ ensemble depends not only on the intensity of the radiative transitions but rather on the interplay of both radiative and nonradiative processes. Among latter, one can single out capture on surface defects, tunnel migration, and Förster exciton transfer.28 Also, the Auger recombination becomes possible in doped nanocrystals even if only one electron-hole pair is created by the laser pumping. F¨ orster transitions in Si crystallites are slow enough in comparison with the radiative transitions.29,30 Tunnel migration is more efficient process31 (with maximum rates ∼ 1013 s−1 ) capable of influencing the emission spectrum in dense ensembles.32 On the other hand, in low-density arrays this process becomes ineffective, because its rates drastically depend on the inter-crystallite distance,31 and exponentially tend to zero as the distance increases. Capture on surface dangling bonds can be fast—its rate varies in a wide range with varying the crystallite size, and reaches maximum values ∼ 1011 s−1 .33 In order to completely exclude or, at least, minimize undesirable effect of dangling bonds, the nanocrystal surface should be passivated by atoms with suitable valence. In the following, we investigate electronic and optical properties of individual (isolated) silicon nanocrystals (12 nm in diameter) doped with a single P or Li atom, and suppose the surface dangling bonds to be fully saturated by hydrogen. As a rule, surface hydrogen does not crucially influence the properties of Si crystallites, which are mainly governed by the quantum confinement effect in this case. Thus, we do not consider further the capture on dangling bonds as well as the tunnel and Förster migration. Meanwhile, it is difficult to completely exclude the Auger recombination (eeh-process in the case of doping with donors) having high efficiency, as, at least, expected by analogy with pure crystallites.34–36 Various aspects of physics of doped silicon nanocrystals have been already studied, especially for the case of doping with boron or phosphorus,17,21,23–25,37–40 or codoping with boron and phosphorus.21,39,41 In particular, the formation energy,39,41 electronic structure,10,39–41 radiative

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recombination rates,24,27,42 absorption spectra,10,39 etc. have been calculated by different methods. At the same time, Li-doped nanocrystals were substantially less investigated: there were reported on the calculations of their formation energies43 and measurements of the photoluminescence spectra.26 Besides, the Auger recombination in doped Si crystallites remains till now, in fact, beyond the viewing field of researchers despite its importance for the process of light emission. The present paper directly concerns these problems. Here, within the framework of ab-initio methods, we theoretically explore an influence of doping with shallow donors (Li and P) on the interband radiative transitions and Auger recombination in silicon nanocrystals.

II.

COMPUTATIONAL DETAILS

Here, we consider silicon nanocrystals built around the impurity atom (P or Li) by successive addition of Sifilled coordination spheres. After filling the last sphere with Si, all dangling bonds are saturated by hydrogen in order to exclude an appearance of deep surface levels. Initial positions of Si atoms in the nanocrystals correspond to those of bulk Si. Initial lengths and orientation of the Si-H bonds coincide with those of SiH4 molecule. The phosphorus atom is a substitutional dopant, while the lithium atom is an interstitial one. Before the relaxation, both donors have always central positions inside the nanocrystal but the former is introduced instead of the silicon atom while the latter is placed at the tetrahedral interstitial site. Such a central position is, presumably, more preferable energetically.43 As a result of this procedure, we have obtained the unrelaxed nanocrystals with, mainly, spherical-like shape. Meanwhile, some of the nanocrystals were faceted. For example, two of the Li-doped nanocrystals, Si42 H64 Li and Si148 H120 Li, have the shape that is rather tetrahedral-like. The difference in the two types of the nanocrystal shape is clearly seen from Fig.1, where Si42 H64 Li and Si66 H64 Li nanocrystals are presented. Optimization of the nanocrystals’ geometry was carried out in two stages. Initially, the undoped SiH nanocrystals (centered with respect to the Si atom or the tetrahedral interstitial) were relaxed by means of the Universal Force Field method44 and the program Avogadro45 which yield the first approximation to the optimized nanocrystals’ geometry. Further correction of the structure was achieved with the algorithm Basin Hopping46,47 using the BFGS method48 based on the density functional theory (DFT) which is realized in the program complex Octopus49 combined with the Atomic Simulation Environment (ASE).50 The relaxation procedure continued until maximum (over the nanocrystal) Hellmann-Feynman forces became less than 10−4 eV/˚ A for the nanocrystals whose diameters are smaller than 1.4 nm, and less than 10−2 eV/˚ A for the greater nanocrystals. At the second stage, the P or Li atom was in-

Figure 1: Si42 H64 Li (left) and Si66 H64 Li (right) nanocrystals before (top) and after (bottom) relaxation. Big grey balls represent the Si atoms, small light balls represent the H atoms, and the central violet ball represents the atom of Li.

troduced into the relaxed crystallite, and the described above optimization procedure was repeated. The relaxation of the spherical-like nanocrystals results in their isotropic compression so that their shape remains spherical, as shown in Fig.1 (right column) where unrelaxed (upper image) and relaxed (lower image) sphericallike Si66 H64 Li nanocrystals are presented. Such a behavior is typical for all spherical-like nanocrystals we consider here. The nanocrystals with the tetrahedrallike facetting relax more complicatedly. They undergo a centrally-directed compression in the domain of the facets’ centers, which leads to an ejection of the vertices’ atoms away from the nanocrystal center, as seen in Fig.1 (left column). In the following we shall characterize each the nanocrystal by its effective mean radius R calculated as an average distance from the nanocrystal center to the surface hydrogen atoms.51 For the sphericallike nanocrystals R always decreases owing to the relaxation, while for the tetrahedral-like nanocrystals R remains nearly constant. For example, for the unrelaxed and relaxed Si66 H64 Li crystallites we have obtained the values: R = 6.97˚ A and R = 6.77˚ A, respectively, while for the unrelaxed and relaxed Si42 H64 Li crystallites one finds R = 6.01˚ A and R = 5.98˚ A, respectively. Radii of the optimized P- and Li-doped crystallites are presented in Table 1. For comparison, we have also adduced in the table radii of the undoped nanocrystals being the prototypes for the P-doped crystallites. As seen, introduction of the P atom leads, as a rule, to a very weak compression of the nanocrystal related to its state in case of no doping. Evidently, insertion of Li into the central interstitial should slightly increase the nanocrystal size.


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The Journal of Physical Chemistry 3

Indeed, e.g. for the relaxed Si66 H64 and Si42 H64 crystallites without Li we found the values R = 6.75˚ A and R = 5.96˚ A, respectively, which are smaller by 0.02˚ A than those of the corresponding Li-doped crystallites. Thus, these two types of donors act oppositely. Table 1. Radii of the undoped and doped with P or Li relaxed nanocrystals. R (˚ A) R (˚ A) R (˚ A) Si35 H36 5.54 Si34 H36 P 5.53 Si30 H40 Li 5.28 Si47 H60 6.18 Si46 H60 P 6.17 Si42 H64 Li 5.98 Si71 H84 6.97 Si70 H84 P 6.97 Si66 H64 Li 6.77 Si87 H76 7.33 Si86 H76 P 7.33 Si82 H72 Li 7.28 Si123 H100 8.31 Si122 H100 P 8.31 Si106 H120 Li 8.10 Si147 H100 8.79 Si146 H100 P 8.78 Si148 H120 Li 8.77 Si167 H124 9.11 Si166 H124 P 9.10 Si172 H120 Li 9.19 Si239 H196 10.31 Si238 H196 P 10.30 Si220 H144 Li 9.97 After the geometry was optimized, the electronic structure of Si crystallites was determined by DFT method employing the program complex Octopus49 in which the Kohn-Sham equations52 are solved on a grid with the use of pseudopotentials. Optimal values of the grid spacing and the supercell size were selected under the requirement that the change of the total crystallite energy be less than 0.01 eV when varying these parameters. As the boundary condition we accept equality to zero of the wave function at the supercell surface. The generalized gradient approximation (GGA-PBE)53 was used for the exchange-correlation functional. Excited states have been calculated within the Casida’s approach54 developed on the basis of the time-dependent density functional theory (TDDFT) for finite-size systems. To exclude electron self-interaction, the Perdew-Zunger correction55 has been taken into account.

III.

RESULTS AND DISCUSSION

The described above method was applied to the nanocrystals doped with phosphorus and lithium in order to determine their electronic structure. Figure 2 demonstrates calculated interband energy gaps for these nanocrystals. For comparison, the gaps of the undoped (P-doped prototypical) nanocrystals are as well shown in the figure. Here, when talking about the energy gap, we imply the gap εg between the lowest energy level in the conduction band and the highest energy level in the valence band. As follows from Fig.2, phosphorus, although insignificantly affects elastic properties of the crystallite, strongly modifies its electronic structure. It is seen that, the energy gap of the P-doped crystallites turns out to be essentially smaller (by several tenths of eV) than that of undoped ones. Such a great reduction of the energy gap is mainly caused by the strong Coulomb attraction of the conduction electron and phosphorus ion. Besides, in the P-doped crystallites, an efficient valley-orbit in-

4 Li-doped Energy HeVL

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

3 Undoped P-doped

2 1 0 5

Splitting 6

7

8 9 Radius HÞL

10

11

Figure 2: Interband energy gaps of the undoped, and doped with phosphorus and lithium nanocrystals. Dots: energies of the singlet-triplet splitting in P-doped nanocrystals.

teraction56 takes place, which is due to the short-range potential57 of the phosphorus ion, resulting in a creation of the so-called “fine structure” of the energy spectrum in the conduction band. In particular, according to the tetrahedral symmetry of the system, the six-fold degenerate energy level of the conduction ground state splits into the singlet (unit irreducible representation A1 of the point group Td ), doublet (two-dimensional irreducible representation E), and triplet (three-dimensional irreducible representation T2 ). The arrangement of the singlet, doublet, and triplet is as follows: the doublet and triplet have close energies, and the doublet level lies a little higher than the triplet one, while the singlet level is strongly split off deep into the forbidden band. Such a “fine structure” is typical also for the P-doped Si crystallites with greater sizes58 (> ∼ 2 nm in diameter) and for the P levels in bulk Si where, however, the splitting is small enough: ∼ 45 meV.59 Quantum confinement substantially strengthens the splitting that becomes order of magnitude greater as shown by dots in Fig.2. Strong singlet-triplet splitting gives extra rise to the energy gap reduction in the P-doped crystallites. In contrast to the case of the phosphorus doping, effect of the lithium doping on the nanocrystal energy gap εg is, in fact, negligible. It is seen in Fig.2 that, the dependencies of εg on the crystallite radius for the undoped and Li-doped crystallites are very resembling, and differ insignificantly. The short-range field of the lithium ion mixes six equivalent valleys in the conduction band of silicon similarly to the ion of phosphorus, and also produces the valley-orbit splitting of the ground state into the singlet, doublet, and triplet. It is known, however, that in bulk Si, this splitting has an inverse structure: the singlet splits off up, while the doublet and triplet have a little lower (∼ 1 – 2 meV), almost identical, energies.59 The order of the singlet, doublet, and triplet levels in Li-doped nanocrystals can change depending on the nanocrystal size as shown in Fig.3. The bulk-like structure, when the singlet level is the highest one, takes place for Si30 H40 Li, Si42 H64 Li, and Si220 H144 Li nanocrystals among all the



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Doublet

Si42 H64 Li 0

5

Wave Function

10 5 0 -5 -10 -15 -10 -5

10 5 0 -5 -10 -15

Triplet Si66 H64 Li -10 -5

10

Figure 3: Schematic representation of the ground state “fine structure” in the conduction band of Li-doped crystallites listed in Table 1. The circles under the energy levels symbolize Li-doped crystallites with rising sizes (from left to right) as they listed in Table 1. The numbers denote corresponding energies of the splitting in meV.

10 5 0 -5 -10 -15

Singlet Si106 H120 Li -10 -5

0 x( )

considered here. Si106 H120 Li and Si148 H120 Li nanocrystals have the standard structure of levels with the lowest singlet. The three remaining nanocrystals have the structure of levels with the singlet situated between the doublet and triplet. We see again that the confinement strongly increases the splitting compared to the case of bulk Si—typical values of the splitting energies vary within ∼ 0.2 eV. Nevertheless, these values are appreciably smaller than the ones in the P-doped crystallites. The energy level of the ground (highest) single-particle state in the valence band is not, in fact, qualitatively affected by the donor atom. It is triply degenerate in undoped crystallites, and this degeneracy remains invariable in both P- and Li-doped crystallites. Such a degeneracy in silicon nanocrystals is a consequence of the tetrahedral symmetry that does not change after the donor introduction into the nanocrystal center. x-dependence of the ground-state single-particle wave functions in the conduction (dashed lines) and valence (solid lines) bands are shown in Fig.4 for the Li- and P-doped nanocrystals. The x-axis was chosen along the crystallographic direction [100] of a face-centered lattice. Among the Li-doped nanocrystals we present in Fig.4 the three ones with different types of the ground state (singlet, doublet, and triplet) in the conduction band. For the P-doped crystallites, the singlet is always the ground state in the conduction band. Therefore, anyone of the eight crystallites considered here can be taken as an illustrative example. It is seen that, independently of the donor kind, the singlet wave functions are qualitatively resembling (see lower images in Fig.4). There is, however, some quantitative distinction between them at the donor site. In the P-doped crystallite, the singlet wave function has some finite value at x = 0 in contrast to the Li-doped crystallite where the singlet wave function tends to zero. Obviously, this distinctive feature causes the substantially different character of the ground-state splitting in the conduction band for the P- and Li-doped crystallites. The doublet and triplet wave functions in the Li-doped crystallites behave differently from each other, as well as from the singlet wave function. They exactly equal zero at the lithium site.

0

5

10

x( ) Wave Function

x( ) Wave Function

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

Wave Function

4

5

10

10 5 0 -5 -10 -15

Si86 H76 P

Singlet -10 -5

0

5

10

x( )

Figure 4: Kohn-Sham wave functions of the lowest conduction (dashed lines) and the highest valence (solid lines) singleparticle states in Li- and P-doped crystallites. The three Lidoped crystallites were chosen so that the lowest conduction states be the doublet (upper left image), triplet (upper right image), and singlet (lower left image). Typical situation for the P-doped crystallites, with the singlet being the ground conduction single-particle state, is presented in the lower right image.

The wave functions of the highest valence state (solid lines in Fig.4) in all the Li-doped crystallites behave almost identically. All they are antisymmetric with respect to the donor site and oscillate with increasing amplitude towards the crystallite surface. This looks like an ejection of the oscillations (and the wave function itself) from the area where the lithium atom is located. In the P-doped crystallites, the highest valence state has the wave function being also antisymmetric related to the donor site. However, in contrast to the Li-doped crystallites, no suppression of the wave function oscillations is observed in the vicinity of the phosphorus atom. As has been already mentioned in the Introduction, the no-phonon radiative interband transitions in bulk silicon are forbidden by the momentum conservation law. In nanocrystals, momentum is no longer a good quantum number (as evident from the explicit form of the wave functions plotted in Fig.4), and its conservation is not required. Therefore, the no-phonon radiative electronhole transitions become possible. Here, we calculate their rates using the Fermi’s golden rule adapted54 to manyelectron systems within the first-order perturbation theory. The decay rate for the transition between the initial (i) and the final (f ) states can be written in the following standard form: −1 τR,if =

2π ∑ |Wif |2 δ(εif − ¯hωα (q)). ¯h q,α

(1)

Here, ωα (q) stands for the frequency of the emitted photon with the wave vector q and polarization α, εif is the energy of the electron-hole excitation, and Wif is the


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108 HaL (2)

q ,µ

ˆ is the elecdescribing the electron-photon interaction. p tron momentum operator, eq′ µ is the polarization vector, the operator a ˆq′ µ annihilates and a ˆ†q′ µ creates a photon ′ with the wave vector q and polarization µ, and V stands for the volume of the electromagnetic resonator that is usually chosen equal to the supercell volume.40 The initial state in Eq.(1) corresponds to some excited state of the electron subsystem plus an ensemble of photons whose distribution over q is described by the BoseEinstein statistics. In the final state, the valence band is totally populated by electrons, and the number of photons increases by one. The parameter κ represents the correction factor in the electric field magnitude inside the nanocrystal due to the difference in refraction indices of the nanocrystal and its surrounding. Within the frames of a macroscopic approach, it is possible to treat the nanocrystal as a sphere with radius R and permittivity ϵN C embedded in a homogeneous media with bulk permittivity ϵ. In this case, the parameter κ is defined as59 ( )2 3ϵ κ= . (3) 2ϵ + ϵN C Since all the above calculations of the electronic structure were carried out for the nanocrystals in vacuum, we set hereafter ϵ = 1. Parameter κ varies within a wide range depending on ϵN C . For instance, in a static case, if the sphere size is great enough, one can ascribe to ϵN C its bulk value (12 for bulk Si), which yields κ = 0.046. However, for small nanocrystals ϵN C should be less.61,62 In the limiting case of extremely small crystallites it is possible to expect that ϵN C → ϵ = 1, and κ → 1. We should also take into account a non-stationary character of the electromagnetic field. This means that ϵN C is a function of ω and should be taken at ω = εif /¯ h. Unfortunately, we do not really know the dependence of ϵN C on both the photon frequency and nanocrystal size. As a consequence, an exact value of κ is unknown as well. In this situation, it −1 is more logical to calculate not τR,if itself but rather the −1 −1 ratio τR,if /κ or, equivalently, τR,if at κ = 1. After some algebra, the recombination rate at κ = ϵ = 1 acquires the form: −1 τR,if =

4e2 εif |pif |2 , 3m20 ¯ h2 c3

(4)

where pif = ⟨ψi |ˆ p|ψf ⟩ with ψi,f being the initial and final electron orbitals. In order to compute ψi,f and the momentum matrix element, we use here the Casida’s approach,54 as has been pointed out above. Results of our calculations are presented in Fig.5(a) for the basic electron-hole radiative transition occurring

107 Rate H1sL

matrix element of the operator √ ) ∑ 2π¯ he2 κ ( ˆ = ˆ W a ˆq′ µ + a ˆ†q′ µ eq′ µ p 2 ′ m0 ωµ (q )ϵV ′

T =0

106 105 104 5

6

7

8

9

10

11

Radius HÞL 108 HbL Room Temperature

107 Rate H1sL

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

106 105 104 5

6

7

8

9

10

11

Radius HÞL Figure 5: Rates of the radiative decay at κ = 1 for undoped (circles), P-doped (squares), and Li-doped (rings) crystallites listed in Table 1. (a) Rates of the basic interband transitions realized at T = 0: the electron-hole pair has minimum energy. (b) Rates of the radiative recombination at room temperature—the electron-hole excitations with all possible energies are involved into the process.

from the lowest excited (initial) state to the ground (final) one. In fact, only those transitions are realized in the limiting case of zero temperature (T = 0), when the nonradiatively relaxed (after the laser pumping) electronhole excitation occupies exclusively the lowest energy level before the radiative interband annihilation, and may not populate higher energy levels. As seen, in the whole, the crystallites doped with phosphorus demonstrate faster transitions (with the rates varying within an order of magnitude around ∼ 106 s−1 ) than the Li-doped or undoped crystallites. Some exception is Si86 H76 P crystallite, in which the transition turns out to be very slow. At the same time, crystallites Si35 H36 , and Si66 H64 Li and Si82 H72 Li exhibit high enough rates equal, or close, to 5×106 s−1 . It is also seen that the distributions of the rates for the P-doped and undoped crystallites are sufficiently narrow (the rates vary, in fact, within an order of magnitude), especially if we exclude the “particular” crystallites Si86 H76 P and Si35 H36 from the consideration. On the contrary, the Li-doped nanocrystals are characterized by a wider spread of the rate values (within ∼ 3 orders of magnitude), which can be resulted from the reconstruction of the ground state in the conduction band with increasing the nanocrystal



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Figure 6: Schematic representation of the Auger eeh recombination. Initially, the hole in the valence band and both electrons in the conduction band are in their ground states.

size as shown in Fig.3. It is also interesting to trace an influence of the finite (room) temperature on the radiative interband transitions. Evidently, in this case the transitions with the energies greater than εg may efficiently contribute to the luminescence. In order to estimate the radiative decay rate at finite temperature T ̸= 0 we have to calculate the overall contribution from all the transitions with taking into account occupation probabilities of the excited electron-hole states: ∑ −1 τR,if exp [−(εif − εg )/kB T ] −1 ∑ τR (T ) = , (5) exp [−(εif − εg )/kB T ] where kB stands for the Boltzmann constant. Results of our calculations for this case are presented in Fig.5(b). First of all, it should be noted that the radiative recombination rates for the P-doped crystallites remain almost invariable—the statistic averaging does not, in fact, change their values. This means that the radiative transitions with the energies slightly greater than the band gap εg have low efficiency and weakly contribute to τR−1 (T ). The nanocrystals doped with lithium turn out to be sensitive to the temperature. At least for three nanocrystals (two smallest and the greatest ones), the rates drastically increase—by 1 – 3 orders of magnitude. As a result, for six of the eight nanocrystals the rate values range from ∼ 5 × 105 to 5 × 107 s−1 , and four smallest Lidoped nanocrystals demonstrate the fastest radiative decay compared to the corresponding, close in size, P-doped nanocrystals. Radiative rates of the undoped crystallites do not strongly change. For some of them (two smallest and Si147 H100 crystallites) the rates became several times (less than an order) greater, while for Si87 H76 crystallite the rate became even several times less. Thus, the temperature finiteness is almost unimportant for the P-doped crystallites and weakly, in the whole, influence the radiative decay in the undoped crystallites. Meanwhile, the radiative decay in some of the Li-doped crystallites substantially quickens. This is, obviously, a

consequence of high decay rates of the transitions from the nearest vicinity of the interband energy gap, for which ¯hω − εg < ∼ kB T . In order to estimate the photon generation efficiency in the nanocrystals, not only radiative but also nonradiative processes have to be taken into account, as has been already pointed out above. In particular, in donordoped nanocrystals, the Auger eeh process is expected to be highly efficient. Below we focus on the calculations of the Auger-recombination rates for the P-, Li-, and undoped (for comparison) silicon nanocrystals at low level of excitation: we suppose only one electron-hole pair to be excited in the doped crystallites, as shown schematically in Fig.6. In the undoped crystallites, to “make” the Auger recombination possible, we have to formally assume that two electron-hole pairs exist inside the crystallite. The rate of the Auger eeh transition can be calculated with the Fermi’s golden rule as was similarly done for the rates of the radiative recombination [Eq.(1)]: −1 τA,if =

2π |Uif |2 δ(Ei − Ef ), ¯h

(6)

where Ei,f are the initial and final two-electron energies, and the two-electron wave functions are constructed as products of the single-electron Kohn-Sham orbitals. Uif is the matrix element of the two-particle Coulomb interaction operator having the long-range and short-range ˆ =U ˆLR + U ˆSR . Within the macroscopic model, parts: U when the nanocrystal is treated as a sphere with radius R and permittivity ϵN C situated in vacuum, the longrange Coulomb interaction includes the direct screened electron-electron interaction and the interaction of the point charges (−e) with the images of each other. If both electrons are located inside the nanocrystal, this part is usually written as follows: ULR (r1 , r2 ) =

e2 e2 (ϵN C − 1) + ϵN C |r1 − r2 | ϵN C R ∞ ∑ (l + 1)Pl (cos θ) rl rl 1 2 × , lϵN C + l + 1 R2l

(7)

l=0

where r1 and r2 stand for the position-vectors of the two conduction electrons, Pl (cos θ) is the Legendre polynomial of the argument cos θ with θ being the angle between r1 and r2 . The short-range part manifests oneself only at small distances between the point charges (about Bohr radius aB = 0.53˚ A) because of impossibility of screening at such small distances. The short-range part of the point charges’ interaction can be introduced through the dielectric function ϵ(q) obtained numerically by Walter and Cohen63 and approximated by the following expression:27 Aq 2 β 2 + (1 − A)q 2 ϵN C 1 = 2 + , ϵ(q) α + q2 (β 2 + q 2 )ϵN C

(8)

with A = 1.142, α = 0.82/aB , and β = 5/aB . Earlier, resembling expression was obtained by Pantelides


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and Sah57 who fitted ϵ(q) obtained numerically from the model of almost free electrons by Nara.64 We use here more accurate pseudopotential-based calculations of Walter and Cohen63 and find (according to Eq.(8)) the short-range part of the Coulomb interaction of the two point charges in the form: e2 USR (r) = [Ae−αr + (1 − A − 1/ϵN C )e−βr ], r

(9)

where r = |r1 − r2 |. Strictly speaking, Eqs (8) and (9) were obtained for bulk material. However, due to the short-range nature of USR , these expressions may be applied to nanocrystals as well. Among a variety of possible two-electron initial states suitable for the eeh Auger process we restrict ourselves here with only one state, that is more preferable from the energy point of view, where two electrons occupy the lowest energy level in the conduction band, and the valence band is completely populated except for the highest energy level that has one empty state, as shown in Fig.6. In the final state, one of the two conduction electrons sinks into the empty valence state while the second one climbs up (see Fig.6) and occupies some higher energy level in the conduction band. The difference of the final and initial energies εf −εi of the “second” electron should be equal to the nanocrystal energy gap εg due to the energy conservation law provided by the delta-function in Eq.(6). If we choose, for simplicity, the origin of the energy axis so that εi = 0 (as shown in Fig.6), and assume small, but finite, broadening of the energy levels, then the transition may occur with εf ̸= εg . In this case, it is natural to sum up the contributions to the Auger recombination rate from the transitions with all possible final states of the Auger electron: τA−1 =

2Γ ∑ |Uif |2 . h ¯ Γ2 + (εf − εg )2

(10)

f

Here, the delta-function was broadened with half-width Γ. In the subsequent calculations we set Γ = 10 meV. Figure 7 illustrates behavior of τA−1 as function of the nanocrystal size. The main conclusion is that doping with donors reduces, in the whole, efficiency of the Auger recombination. This effect is more pronounced for the crystallites with the phosphorus atom except for the crystallite Si166 H124 P where the rate is high enough and almost coincides with the Auger recombination rate for Si35 H36 crystallite having the highest rate among all the undoped crystallites. The Li-doped crystallites have, in the mean, some intermediate position between the undoped and P-doped crystallites. Sharp deviation from this “rule” takes place, in fact, only for Si82 H72 Li nanocrystal in which the rate turns out to be ∼ 3 times greater than the maximum rates in P-doped and undoped nanocrystals. It is worth noting that the distribution of the rate values is sufficiently wide for all kinds (doped or undoped) of the nanocrystals. This is a typical situation for the Auger

1013 1012 Rate H1sL

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

1011 1010 109 108 107 5

6

7

8 9 Radius HÞL

10

11

Figure 7: eeh Auger recombination rates for undoped (circles), P-doped (squares), and Li-doped (rings) crystallites listed in Table 1. The initial and final states correspond to the ones shown in Fig.6.

process in Si nanocrystals.34 One of the reason is that the sizes of the nanocrystals are small enough, and the wave functions strongly change with adding new atoms when filling new coordination spheres for the construction of the nanocrystals with greater sizes. This, in turn, can result in significant changes of the matrix element Uif in Eq.(10) and the recombination rates. Another reason is due to the discreteness of the energy spectrum in nanocrystals, which leads to a resonant profile of the density of states. If the energy εf falls onto the energy level (see Fig.6), the density of states and, as a consequence, the probability τA−1 [Eq.(10)] become maximal in contrast to the case when εf falls in between the levels. Accordingly, the rate of the Auger transition demonstrates as well a resonant structure36 as function of the nanocrystal size, and can be, therefore, resonant (greater) or nonresonant (smaller) for different nanocrystals considered here. Precisely this case is presented in Fig.8, where the conduction-band density of states (CB DOS) for the two, Li- and P-doped, nanocrystals is depicted. The positions of the resonant final energies εf = εg of the Auger electrons are shown by the arrows. It is seen, that in Si82 H72 Li nanocrystal, resonant value of εf coincides, in fact, with the position of one of the CB DOS peaks, while in Si86 H76 P nanocrystal, resonant εf corresponds to one of the CB DOS minima between two peaks. Consequently, the Auger recombination rates in these two, close in size, nanocrystals differ by ∼ 6 orders of magnitude, as shown in Fig.7. Presumably, the smaller values of τA−1 for the P-doped crystallites are also, at least in part, owing to this circumstance. Since the P-doped crystallites have smaller interband gaps εg , the energy εf turns out to be smaller as well because of the approximate equality εf ≈ εg required for the transition. The resonant structure of the CB DOS in nanocrystals should be, obviously, more rarefied at lower energies. This trend, in particular, is evident in Fig.8 for Si86 H76 P nanocrystal as compared it



Page 8 of 11

Si82 H72 Li

Si86 H76 P

0

CB DOS

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

CB DOS

8

1

2

3

0

εf (eV)

1

2

3

εf (eV)

Figure 8: Density of states in the conduction band of the Liand P-doped nanocrystals. Vertical arrows indicate the final energy of the Auger electron counted from the energy of the ground single-electron state, as shown in Fig.6, in the case of exact resonance: εf = εg .

to Si82 H72 Li nanocrystal. It is possible, therefore, to conclude that the energy of the Auger electron in the P-doped crystallites corresponds to the lower density of states (greater, in the mean, values of |εf −εg | in Eq.(10)) compared to that of the Li-doped and undoped crystallites. This is accompanied by a certain decrease of τA−1 in the P-doped crystallites.

IV.

CONCLUSION

The performed computation of the recombination rates for the radiative and Auger transitions allows us to estimate and briefly discuss an effect of doping on the photoluminescence intensity in Si nanocrystals. The photoluminescence intensity is defined by the quantum efficiency of photon generation—the so-called quantum yield that can be calculated as the ratio of the radiative recombination rate τR−1 and the total decay rate of the photoluminescence τP−1 L . The latter is the sum of the rates of both radiative and nonradiative processes: −1 −1 τP−1 L = τR + τN R . Since all nonradiative processes, except for the Auger recombination, were “switched off” in −1 −1 our model, the rate τN R may be replaced by τA . Taking into account also that the Auger recombination for all considered nanocrystals is much faster than the ra-

VII. 1

2

3

4

diative interband recombination, it is possible to write the quantum yield as τR−1 /τA−1 . In accordance with the obtained above results, this ratio is small enough—as a rule, much less than a per cent. Consequently, within the model suggested here, doping cannot be considered as an efficient means for improving the luminescent properties of Si nanocrystals. At the same time, it is clear that, this model is strongly idealized, and, e.g., surface dangling bonds will most likely exist after finishing the process of the crystallite fabrication. In this case, the excessive electron from the donor atom may populate the dangling bond. Thereby, both the Auger recombination and repeated capture on this dangling bond (after the electron-hole pair creation) can be strongly suppressed, while the radiative channel of the electron-hole recombination turns out to be efficient and, even, enhanced by the donors. Correspondingly, it is possible to conclude that, at some favorable conditions, doping can be capable of increasing the radiative recombination rates and the photon generation efficiency.

V.

The authors thank prof. S.K. Ignatov and Dr. A.A. Konakov for their interest to this work and fruitful discussions. The work was supported by the Russian Ministry of Education and Science (Task No 3.2637.2017/4.6, and the project RFMEFI61614X0008), and the Russian Foundation for Basic Research (project No 1632-00683). The calculations were performed with the use of the supercomputer complexes “Lobachevsky”65 and“Lomonosov”.66

VI.

AUTHOR INFORMATION

Corresponding Author ∗ E-mail (V.A. Burdov): [email protected] Notes The authors declare no competing financial interest.

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Effect of Doping with Shallow Donors on Radiative and Nonradiative

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