Alloy Quantum Dots

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Optical Transitions and Excitonic Properties of Ge Sn Alloy Quantum Dots Denis Olexander Demchenko, Venkatesham Tallapally, Richard J Alan Esteves, Shopan din Ahmad Hafiz, Tanner A. Nakagawara, Indika U. Arachchige, and Umit Ozgur J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.7b06458 • Publication Date (Web): 31 Jul 2017 Downloaded from http://pubs.acs.org on August 2, 2017

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The Journal of Physical Chemistry

Optical Transitions and Excitonic Properties of Ge1-xSnx Alloy Quantum Dots Denis O. Demchenko,1* Venkatesham Tallapally,2 Richard J Alan Esteves,2 Shopan Hafiz,3 Tanner A. Nakagawara,3 Indika U. Arachchige,2 and Ümit Özgür3 1

Department of Physics, Virginia Commonwealth University, Richmond, Virginia 23284, United States 2

Department of Chemistry, Virginia Commonwealth University, Richmond, Virginia 23284, United States

3

Department of Electrical and Computer Engineering, Virginia Commonwealth University, Richmond, Virginia 23284, United States

*Corresponding Author Email: [email protected]

ABSTRACT Using hybrid functional calculations and experimental characterization we analyze optical properties of 2-3 nm Ge1-xSnx alloy quantum dots, synthesized by colloidal chemistry methods. Hybrid functional theory, tuned to yield experimental bulk band structure of germanium, reproduces directly measured properties of Ge1-xSnx quantum dots, such as lattice constants, energy gaps, and absorption spectra. Time-dependent hybrid functional calculations yield optical absorption in good agreement with experiments, and allow probing the nature of the dark

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excitons in quantum dots. Calculations suggest a spin-forbidden dark exciton ground state, which is supported by the changes in the photoluminescence lifetimes with temperature and tin concentrations. The synthesis and theoretical understanding of Ge1-xSnx alloy quantum dots will add to the overall tool box of low to non-toxic, silicon compatible Group IV semiconductors with potential application in visible to near infrared optoelectronics.

INTRODUCTION Ge1-xSnx alloys are a class of environmentally benign semiconductors that exhibit composition tunable energy gaps in the mid to near-infrared (IR) spectrum. With increasing concentration of Sn in the alloy, significant decrease in the energy gaps occurs, inducing an indirect-to-direct band structure crossover. Such behavior has led to a noteworthy interest in these materials for the fabrication of Si-compatible electronic and photonic devices, field effect transistors, and novel charge storage device applications. The synthesis of Ge1-xSnx, thin film alloys has been widely studied via molecular beam epitaxy (MBE) and chemical vapor deposition (CVD) methods.1,2 Nonetheless, significant challenges have been reported in the growth of homogeneous alloys. Attempts to produce Ge1-xSnx alloy films often resulted in phase segregation owing to large discrepancy in lattice constants (~14%) of the constituents and high reaction and growth temperatures (>400°C) employed with MBE and CVD.3 Successful results have been achieved with low-temperature (~200°C) nonequilibrium growth techniques due to kinetic suppression of Sn segregation and Ge1-xSnx thin film alloys with Sn composition as high as ~34% have been reported.4 However, the resulting narrow energy gaps, as evidenced from a handful of available reports on Ge1-xSnx alloys, have become another obstacle for their efficient use in optical devices.5 For instance, with Ge1-xSnx, an


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indirect-to-direct crossover has been observed at x = 6-11%, at which point the fundamental bandgap becomes significantly narrower (Eg < 0.35 eV) than that of bulk Ge (0.67 eV).6,7 The bandgap lowering of such alloy films is often accompanied by a decrease in photo-response, making them ill-suited for visible to near IR optical applications. In an effort to expand the spectral range and to improve the efficiency of the optical transitions, recently there has been increased interest in the synthesis of alloy nanostructures to exploit the size confinement effects. Taking advantage of the low-temperature colloidal synthesis and nucleation and growth control, we have reported the synthesis of three distinct size regimes of homogeneous Ge1-xSnx nanoalloys. The larger Ge1-xSnx nanocrystals (NCs, 15-23 nm) exhibit weak confinement, with energy gaps (0.2-0.4 eV) that are significantly redshifted from bulk Ge, and non-linear expansion of Ge lattice with increasing Sn composition. In contrast, intermediate size (3.4-4.6 nm) alloy quantum dots (QDs) exhibit moderate size confinement effects and composition tunable energy gaps (0.75-1.47 eV) in the visible to near IR spectrum.8 Very recently, the synthesis of ultra-small Ge1-xSnx alloy QDs (1.8-2.2 nm) with composition tunable absorption and orange to red color emission has also been reported.9,10 Temperature dependent, time resolved photoluminescence (PL) spectroscopy has been utilized to study the carrier dynamics of Ge1-xSnx QDs, which suggest slow decay (27 μs) of PL at 15 K, likely due to slow recombination of dark excitons and carriers trapped at surface states, and roughly one order of magnitude faster recombination with increasing Sn concentration to 23.6%. Increasing temperature to 295 K led to three orders of magnitude faster decay (9-28 ns) owing to the thermal activation of bright excitons and carrier de-trapping from surface states. Herein, we perform detailed theoretical analysis of the optical properties of ultra-small (2-3 nm) Ge1-xSnx QDs using accurate first principles methods, and compare calculations with experimental data.


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Theoretical calculations of optical properties of Ge QDs, which included excitonic effects, have been performed using the k⋅p effective mass theory.11 This approach has also been used extensively to analyze the effects of Ge-Sn alloying in nanocrystals, to develop a theory of bright and dark excitons in semiconductor QDs, and to derive effective mass parameters for bulk Ge1x

Snx alloys.12-14

Recently, an accurate ab initio theory based on modified Becke-Johnson

potential within local density approximation (LDA) was used for theoretical description of Ge1x

Snx bulk alloys.15 However no ab initio theoretical study has been attempted for Ge1-xSnx alloys

on the nanoscale. Furthermore, current state of the art electronic structure methods beyond semi/local approximations to the density functional theory (DFT) are rarely applied to QDs, due to their higher computational costs. For example, hybrid density functionals, which yield accurate electronic structure, are employed in only a handful of theoretical studies of QDs.16-18 In addition, in order to capture excitonic properties using ab initio approach, such as dark-bright exciton splittings, Bethe-Salpeter equation (BSE) calculations or some versions of timedependent DFT are needed.19,20 In this work, we are combining these current state of the art theoretical approaches to describe structural, electronic, and optical properties of Ge1-xSnx alloy QDs. We use local DFT theory to obtain equilibrium lattice structures, tuned HSE hybrid functional to obtain electronic properties, and time-dependent hybrid functional calculations (TD-HSE) to calculate optical absorption and dark/bright excitonic states of Ge1-xSnx QDs.

THEORETICAL AND EXPERIMENTAL METHODS Theoretical Calculations. we use an accurate approach based on the tuned Heyd-ScuseriaErnzerhof (HSE) hybrid functional.21,22 In a hybrid functional calculation the (semi)local exchange correlation part of the density functional is mixed with a Fock-type exchange part in


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varying proportions. In HSE hybrid functional the Fock exchange interactions are separated into long- and short-range parts. The non-local Fock exchange potential is screened at long distances (similar to a screened exchange approach), and the long-ranged part is replaced with an approximate semi-local expression.23 In certain cases it is also useful to tune the hybrid functional. In the literature, either the fraction of exact exchange or the effective screening length of the exact exchange are tuned for a particular material in order to obtain the best agreement of computed band structure with experiment. In this work, we use a tuned HSE functional, where we keep the fraction of exact exchange at a standard 0.25, while the screening parameter is set to 0.29. Since hybrid functionals are significantly more computationally expensive than common semi/local density functional approximations, only relatively small nanocrystals, containing a few hundred atoms, can be addressed with this method. We perform HSE hybrid functional calculations for Ge1-xSnx alloy nanocrystals of 1.4 – 2.7 nm in diameter. The results are compared to experimental measurements of nanocrystals with diameters of 2 – 3 nm, synthesized using colloidal chemistry methods. All calculations are performed using VASP code, and projector augmented wave (PAW) potentials, treating Ge(Sn) semicore 3d(4d)-states as core electrons.24,25 The energy cutoff for the plane-wave basis set is 174 eV in all calculations. Surface dangling bonds are passivated by hydrogen atoms. In an alloy nanocrystal atomic relaxations are essential, however, full HSE relaxation for a nanocrystal with at least several hundred atoms is computationally expensive. Therefore relaxation with a less expensive semi/local approximation to the DFT is desirable, provided it can correctly reproduce the structure for the subsequent static HSE calculation. Relaxation with semilocal functionals, such as PBE, leads to significantly overestimated lattice constants for both Ge and Sn.26,27 This leads to HSE calculation using PBE relaxed lattice of bulk Ge erroneously


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predicting a direct fundamental bandgap at the Γ-point, instead of an indirect Γ-L bandgap. On the other hand, LDA underestimates lattice constants only slightly, and for bulk Ge HSE calculation using LDA relaxed lattice leads to a correct band order, and an indirect Γ-L bandgap.28 Table 1 shows the band structure parameters of bulk Ge calculated using the above parametrization of HSE hybrid functional, with lattice structure relaxed using LDA. (The band structure parameters are illustrated in Ref. 29). The table also provides comparison of results, with spin-orbit (SO) coupling included in the calculations, since it is needed to capture the spinforbidden dark excitons in calculations of Ge1-xSnx alloy nanocrystals. Although some band energies are slightly underestimated, overall HSE yields band energies in good agreement with experiment. Therefore, in all calculations of Ge1-xSnx nanocrystals, the crystal structure was relaxed using LDA, minimizing forces to 0.05 eV/Å or less, and electronic structure was computed using the above parametrization of HSE hybrid functional.

Table 1. Band structure parameters (at 300 K) of bulk Ge calculated using tuned HSE hybrid functional, with and without spin-orbit (SO) coupling. The standard bulk Ge band diagram, illustrating band energy labels can be found in Ref. 29. Experimental data is taken from Ref. 30. Experiment (eV) Tuned HSE (eV) Tuned HSE+SO (eV) Eg

0.66

0.69

0.60

EX

1.2

1.12

1.02

E1

0.8

0.80

0.70

E2 Γ

3.22

3.06

3.02

ΔE

0.85

0.92

0.82

Γ

ESO 0.29

0.31


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In order to include excitonic effects, time-dependent hybrid functional calculations (TD-HSE) were performed, following Ref. 31, which are based on tuned hybrid HSE functional. In the TDHSE calculations, the excitonic effects are approximately described by replacing the electronhole ladder diagrams with the screened exchange. The dielectric function is obtained by solving the Cassida’s equation.32 This procedure is equivalent to the Bethe-Salpeter equation (BSE), with the screened interaction, W, replaced with one-quarter of the nonlocal screened exchange term.33 The TD-HSE calculations (including SO coupling) of smaller QDs (1.4 nm) were used to analyze the nature of spin-forbidden dark and bright bound excitons, and trace their evolution with admixture of Sn into the nanocrystal. These calculations included SO coupling selfconsistently with non-collinear spins according to Ref. 34.

Synthesis of ~ 2nm Ge1-xSnx Quantum Dots Materials. Germanium diiodide (99.99+ %) and tin dichloride (99.9985 %), were purchased from Strem Chemicals and Alfa Aesar, respectively. N-butyllithium (BuLi) 1.6 M in hexane was purchased from Sigma Aldrich and stored in a N2 glove box at < 5°C. 1- octadecene (ODE, 90%) was purchased from Fisher Scientific. Oleylamine (OLA, >98% primary amine) was purchased from Sigma Aldrich. Toluene, chloroform, carbon tetrachloride, and methanol of ACS grade were purchased from Acros. OLA and ODE were dried by heating at 120 °C under vacuum for one hour prior to storage in a N2 glovebox. Methanol and toluene were dried over molecular sieves and Na, respectively and distilled prior to use. Caution: n-butyllithium is highly pyrophoric and ignite in air so must be handled in air free conditions by properly trained personnel. Carbon tetrachloride is highly toxic and its use should be minimized to limit exposure.


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In a typical synthesis of 1.8-2.3 nm Ge1-xSnx QDs, appropriate amounts of GeI2 and SnCl2, 0.6 mmol of metal total, were combined with 20 mL of OLA in a 50 mL three neck flask inside a glovebox. The sealed set up was transferred to a Schlenk line, and degassed under vacuum at 120 °C to produce a homogeneous orange color solution. The reaction temperature was then raised to 230 °C under nitrogen and 0.5-0.9 mL of BuLi in 3.0 mL of ODE was swiftly injected. The temperature dropped to 209-213 °C and the mixture was reheated to 300 °C within 10-14 min to produce Ge1-xSnx alloy QDs. The flask was then rapidly cooled with compressed air to ~100 °C and 10 mL of freshly distilled toluene was added. Then, 60-90 mL of freshly distilled methanol was added, followed by centrifugation at 4000g for 5-10 min to precipitate the alloy QDs. The supernatant was discarded and the precipitate was purified by dispersing in toluene and subsequent precipitation with methanol. Physical Characterization of QDs. A Cary 6000i spectrophotometer (Agilent Technologies) was used for solid-state diffuse reflectance (DRA) with an internal DRA 2500 attachment. Solid sample measurements were performed by mixing the dry QDs in a BaSO4 matrix prior to analysis. Elemental compositions were recorded by energy dispersive spectroscopy (EDS). EDS data were obtained in a Hitachi FESEM Su-70 scanning electron microscope (SEM) operating at 20 KeV with an in-situ EDAX detector. Dried QDs were adhered to an aluminum stub with double sided carbon tape prior to analysis. The elemental compositions were determined by averaging the atomic percentages of Ge and Sn acquired from 5 individual spots per sample. Photoluminescence (PL) studies were performed using a frequency doubled Ti:sapphire laser (385 nm wavelength, 150 fs pulse width, 160 kHz to 80 MHz repetition rate) as the excitation source. The detector was a liquid N2 cooled charge coupled device (CCD) camera connected to a


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spectrometer. Samples were drop cast onto a clean Si substrate and dried and stored under nitrogen. RESULTS AND DISCUSSION Crystal structure. When Ge1-xSnx alloy QDs are experimentally synthesized with Sn concentrations below 22%, Sn atoms are homogeneously distributed within the Ge QD host lattice, with no evidence of clustering, as recently shown in Ref. [9]. Transmission electron microscopy (TEM) images of 2-3 nm Ge1-xSnx QDs are available in Supporting Information (Figure S1). Here we obtain experimental lattice constants using the powder X-ray diffraction patterns of alloy QDs. Diffraction peak positions of (111), (220), and (311) Bragg reflections were accurately determined and the lattice constants were calculated by applying the Bragg’s equation to all peaks. Fig. 1 shows experimental lattice constants for somewhat larger 4-6 nm QDs, because there are large errors in the measured data for small 2-3 nm Ge1-xSnx QDs, due to broad diffusive XRD peaks.9 In contrast, larger 4-6 nm QDs (Fig. 1) show more consistent data. It has been shown that admixing Sn to Ge in the bulk leads to the increase of the lattice constant, which deviates from the Vegard’s rule.5,15,35 Unlike the bulk lattice constants, which show significant bowing, averaged lattice constants calculated using LDA relaxed structures of Ge1x

Snx QD alloys, show essentially linear Vegard’s behavior (see Figure 1), for Sn contents below

~30%.15 It is expected that LDA lattice constants are slightly underestimated compared to experiment (for instance in the bulk, LDA lattice constants are 0.5% below the measured values).27 In addition, the calculated Ge1-xSnx QD lattice constants are lower than those expected from the LDA derived Vegard’s rule in the bulk (shown in Figure 1 as a dashed line). Figure 1 also shows that for small QDs, i.e. 6 nm and below in diameter, the average lattice constants are


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practically independent of the QD size. Overall behavior of lattice constants with admixture of Sn reproduces experimentally measured average lattice constants,

Figure 1. Average lattice constant of the 1.4, 2.1 and 2.7 nm diameter Ge1-xSnx QDs calculated using the LDA relaxed atomic structures compared with experimental measurements of 4-6 nm Ge1-xSnx QDs. Dashed line shows the Vegard’s rule for the bulk Ge1-xSnx alloy, obtained from LDA.

Energy Gaps. Since in this work we compare the results of ab initio calculations to the experimental measurements, a brief discussion of theory/experiment comparison for Ge QDs is in order. Experimental energy gaps of QDs are often determined from diffuse reflectance spectroscopy, using Kubelka-Munk remission function which converts reflectance to absorption


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edge measurements.36,37 The nature of electronic transitions is typically deduced from the Tauc equation fits into reflectance data.8 The energy gaps obtained in these measurements usually follow expected trends. However, when compared to theoretical calculations, energy gaps obtained from the absorption onsets often appear to be systematically lower. Figure 2 illustrates a comparison of measured and calculated energy gaps obtained by several different experimental and theoretical methods for a series of pure Ge QDs. Ab initio calculations using HSE hybrid functional, performed in this work, are in good agreement with the empirical pseudopotential (EPM) calculations, where pseudopotential parameters were fit to reproduce experimental bulk Ge band structure.38 Power law fit (Eg=4.27·x-0.72 eV) into both theoretical calculations reproduces correct asymptotic of large QDs, where for QD sizes above 12 nm there is essentially no quantum confinement. However, experimental measurements using absorption onsets in Ge QDs have consistently yielded lower gap energies by 0.2-0.4 eV, with the difference increasing for smaller QDs, as shown in Figure 2.39-41 This could be due to surface states lowering absorption onset due to imperfect surface passivation in experiment. Recently, scanning tunneling spectroscopy (STS) was used to directly probe single-particle energy gaps in Ge QDs as a function of QD size.42 As evident from Figure 2, the results of these measurements are in a significantly better agreement with theoretical predictions. Since in this work the measured optical properties of ultra-small Ge1-xSnx alloy QDs are compared with theory, this underestimation of energy gaps from absorption onset analysis should be kept in mind. In addition, in this work energy gaps are also extracted from the photoluminescence (PL) maxima and compared to HSE hybrid functional calculations. However, there is an expected difference between absorption onsets and the PL maxima, i.e. the Stokes shift, as discussed below.


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Figure 2. Comparison of energy gaps for Ge QDs obtained from HSE calculation (this work), EPM calculation,38 and experimental data measured from the absorption onsets (ABS) Refs. [26,27,28], and using scanning tunneling spectroscopy (STS) Ref.[42]. Dashed curve represents a power law fit (Eg=4.27·x-0.72 eV) to HSE and EPM calculations.


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Figure 3. Energy gaps of Ge1-xSnx alloy QDs obtained from HSE calculation as a function of Sn concentration for 1.4, 2.1, and 2.7 nm diameter QDs (solid lines show linear regression of the data points). Experimental data are obtained from room temperature absorption onset (ABS) and photoluminescence spectra peaks (PLmax).

Figure 3 shows comparison of the calculated energy gaps to experimental measurements. Calculations are performed on LDA relaxed Ge1-xSnx QD crystal structures, and single particle gaps are calculated with tuned HSE hybrid functional (at this point spin-orbit coupling is not included in the calculations). The experimentally measured gaps are obtained from the KubelkaMunk analysis24 of the diffuse reflectance spectra for 2-3 nm Ge1-xSnx QDs.37 The absorption


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onsets shift significantly with the increasing concentration of Sn. The experimental energy gaps were also measured from the PL maxima of the same Ge1-xSnx QD samples. The error bars are obtained from measurements of multiple samples with the same composition. As shown in Figure 3, both measurements are in reasonably good agreement with each other, showing the gaps decreasing from about 2 to 1.6-1.7 eV, when the Sn content varies from zero to ~24%. HSE calculations for 2.7 nm QDs appear to agree with those obtained in experiment. However, taking into account the underestimation of the energy gap in optical measurements, as discussed above, the sizes of Ge1-xSnx QDs in experimental samples are likely between 1.8 and 2.5 nm. Overall, the calculations and experiment are in good agreement regarding the linear decrease of the energy gap with the Sn concentration. The experimental data shown in Figure 3 also demonstrates the existence of the Stokes shift, i.e. the redshift of the PL maxima relative to the absorption edge. In our samples the values of Stokes shift vary between 0.05 to 0.35 eV. The observed values of the Stokes shift could be caused by non-resonant absorption in the presence of large size distribution of QDs in the samples.9 More poly-disperse samples exhibit larger Stokes shift, for example for 2% and 5% Sn samples in Fig. 3. Another reason for the observed Stokes shift is a dark exciton ground state, with several possible mechanisms of its formation, as discussed below. Nature of dark exciton. In addition to the changes in the value of the energy gap, calculations help us trace the changes in the character of single particle electronic states with increasing Sn content in Ge QDs. Figure 4 shows HSE calculated single-particle eigenvalues for 2.7 nm Ge1x

Snx QDs with 0%, 10%, and 20% of Sn. The orbital character of wavefunctions is obtained by

calculating their projections onto spherical harmonics around each atom within a radius of 1.22 Å for Ge atoms (1.57 Å for Sn). The eigenvalues in Figure 4 are color coded to reveal the orbital


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character of the one-electron states, and their hybridization, from predominantly p-states to sstates. With increasing Sn content, the accompanying atomic relaxations lift the degeneracies of the three-fold degenerate top of the valence band state (and the adjacent three-fold degenerate state below), retaining overall p-character of the states near the top of the valence band. On the other hand, unoccupied states show significant s-p hybridization. In the absence of Sn, there is a series of states of predominantly p-character roughly 0.05 eV above the lowest conduction band derived state. Probability of optical transitions involving these states is suppressed due to the orbital angular momentum selection rules. However, the transitions between the band edge states is always orbitally allowed, with the s-p hybridized states moving away from the conduction band edge with increased Sn concentration (Figure 4), and therefore do not explain the possible existence of dark exciton ground state. Alternatively, a dark exciton ground state could be due to different electron and hole symmetries of envelope parts of the wavefunctions. This spatial symmetry induced dark exciton is characterized by the P-like envelope function of the hole ground state.43 In the past, theoretical calculations based on k·p theory suggested existence of this dark exciton ground state in relatively small CdS (Se, Te) QDs, which would become bright (S-like character of envelope function of the hole ground state) with increasing QD size.44-46 However, k·p theory can predict different energy level ordering compared to first principles methods,47 with latter predicting a bright S-like hole ground state over a wide size range of CdS QDs.43 To test this spatial dark exciton, along with single particle energy levels, Figure 4 also shows corresponding single particle wavefunctions (charge densities). First, for 0% of Sn in 2.7 nm Ge QDs, the envelope functions of both electron and hole band edge states are of S-character, indicating spatially bright states. In the valence band, the P-like envelope function hole state is about 0.05 eV below the top


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of the valence band. Admixture of Sn to the Ge QD leads to a stronger wavefunction localization, as evident from comparison of wavefunction isosurfaces (in all cases at 15% of maximal value) plotted for 0% and 20% Sn (Figure 4). In addition, with increased Sn content the orbital character of the envelope function is less distinct S- or P-type. Nevertheless, at 20% Sn, band edge states retain their predominant S-character of envelope function, while P-like state moves deeper into the valence band, by about 0.15 eV. This suggests that even in small Ge1-xSnx QDs spatial symmetry induced dark exciton does not form.

Figure 4. HSE calculated single particle eigenvalues around the energy gap for 2.7 nm diameter Ge1-xSnx QDs with different Sn concentrations. Orbital character of the states is color coded, going from predominantly p-character (blue) to s-character (red). Zero energy is at the highest


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occupied state in each case, and the energy axis within the gap is broken to focus on the evolution of the eigenvalues. Single-particle wavefunctions are also shown with isosurfaces plotted at 15% of the maximal value for selected eigenstates. Dark exciton ground state can also be produced due to the electron-hole exchange interaction creating a triplet exciton ground state, which is forbidden for optical transition. 13,48 Dark exciton ground state in Ge QDs has been studied both theoretically and experimentally,33 where PL measurements from 3–4 nm QDs (larger QDs show low PL intensity) accompanied by the effective mass type calculations suggested 1 meV electron-hole exchange splitting between dark and bright exciton states. This splitting is strongly size dependent, and in small QDs (~ 1 nm diameter) can reach tens of meV.43 In order to test the spin-forbidden dark exciton in Ge1-xSnx alloy QDs, here we perform TD-HSE calculations, including spin-orbit coupling, of the smallest QDs addressed in this work, with diameter of 1.4 nm.


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Figure 5. Oscillator strengths of optical transitions for a series of D=1.4 nm Ge1-xSnx alloy QDs of varying Sn concentrations. Red arrows indicate the dark-bright exciton splitting (see text) in each QD. Zero of the energy axis is placed at the lowest unoccupied HSE single particle state.


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In order to quantify probability of optical transitions for dark/bright excitons we use TD-HSE theory to calculate oscillator strengths of optical transitions around the energy gap, taking into account electron-hole interactions. Oscillator strength defines the probability of optical transition, proportional to the radiative recombination rate. We assume an excitonic state to be bright if its oscillator strength is at least a few hundred times larger than that of a dark exciton (similar to Ref. [19]). Figure 5 shows TD-HSE (+SO) calculated oscillator strengths of optical transitions for 1.4 nm diameter Ge1-xSnx QDs, with concentrations of Sn varying from 0 to 20%. In all cases, ground states of bound excitons are dark, characterized by the low values of oscillator strengths 10-4 – 10-2. For pure Ge QDs the dark versus bright exciton transitions are very distinct, with dark ground state having oscillator strength that is five orders of magnitude lower than that of a bright state. The addition of Sn introduces mixing of excitonic states, smearing the dark-bright exciton distinction, and bright/dark excitons oscillator strength ratio decreases to a few hundred. Binding energies of ground state dark excitons are indicated by positions of the left ends of the red arrows in Figure 5. For a pure 1.4 nm Ge QD the ground state dark exciton binding energy is 97 meV, with the bright state split by electron-hole exchange interaction by 81 meV. The ground state dark exciton binding energy shifts towards the lowest unoccupied single particle state, from 97 meV to 79 meV with increasing Sn concentration from 0 to 20%. At the same time, the bright excitons show an opposite trend, with the lowest bright state binding energy increasing from 16 to 41 meV. As a result, dark-bright exciton splitting (indicated by red arrows in Figure 5) decreases from about 80 to 40 meV. In experiment, Ge1x

Snx QD sizes are 1.8 – 3 nm, larger than QDs for which these results are computed. Since

electron-hole exchange splitting is size dependent, we roughly estimate this splitting in experimental 1.8 – 3 nm Ge1-xSnx QDs to be 30 – 40 meV in pristine Ge QDs, lowered with


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admixture of Sn to 15 – 20 meV. These values are smaller than the observed values of Stokes shift (0.05 to 0.35 eV, as shown in Figure 3), suggesting that in Ge1-xSnx alloy QDs Stokes shift is caused by a combination of spin-forbidden dark exciton ground state and the size distribution of QDs in the samples. The above findings are in agreement with the PL lifetime measurements of 1.8 – 3 nm Ge1-xSnx QDs.10 At 15 K, PL decays were three orders of magnitude slower (2.7 - 27 μs) than those at room temperature (9 - 28 ns) due in part to slow recombination of spin-forbidden dark excitons. Increasing the temperature allowed thermal activation of much faster recombining bright excitons as well as escape of carriers from any surface-related trap states. Consequently, PL peaks originating from bright exciton recombination at room temperature were blueshifted by 34 – 112 meV, somewhat larger than the estimated dark-bright exciton splitting values. This slight discrepancy is possibly due to size distribution and effects of surface states, which are not taken into consideration in theoretical calculations. Absorption spectra. Using TD-HSE calculations we obtain optical absorption spectra of 2.7 nm Ge1-xSnx QDs (in these calculations spin-orbit coupling is not included, since it has negligible effect on the computed spectra). Figure 6 presents comparison of these calculations to the experimentally measured absorption spectra, which are obtained from solid state diffuse reflectance spectroscopy (DRA). Reflectance data of QDs were converted to pseudo-absorption using Kubelka-Munk remission function.49 The energy gap values were obtained from linear extrapolation of the absorption onsets to the intersection point of the baseline. All measured spectra were normalized to the absorption at 3 eV. Then, to isolate the absorption of QDs only, the spectrum of the capping ligands (oleylamine) was subtracted. The absorption minimum at 3.5 eV is the result of this subtraction, and not present in the raw data (Supporting Information,


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Figure S2). TD-HSE calculations show two main peaks above the band edge absorption, which are smeared out in experiment, likely due to the size distribution of QDs in the samples. Although calculated absorption spectra are somewhat redshifted compared to experiment, overall agreement between theory and experiment is satisfactory. It is worth noting that independent particle absorption calculations (not shown here) produce spectra with absorption peaks that are blue shifted by ~0.8 eV for all Sn concentrations. Bandgap energies, which are obtained from Kubelka-Munk analysis in experiment (Fig. 6(a)), and calculated from HSE (Fig. 6(b)), are shown with dashed arrows. Overall trend of redshifting absorption with increased concentration of Sn is reproduced in calculations. Calculated band edge absorption (absorption value at the energy of the gap) increases slowly from 0% to 10% Sn QDs (by 37.5%), and raises by about a factor of two for 20% Sn QD. Integrated absorption also increases for 20% Sn QD by about 20%.


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Figure 6. Comparison of absorption spectra for 2-3 nm Ge1-xSnx QDs measured experimentally (a), and calculated using TD-HSE calculations (for 2.7 nm Ge1-xSnx QDs). Dashed arrows show calculated and measured energy gaps. CONCLUSIONS In conclusion, we reported the theoretical hybrid functional calculations of electronic and optical properties of Ge1-xSnx alloy QDs with diameters ranging roughly from 1 to 3 nm, and compare them to the experimental measurements. Contrary to the bulk Ge1-xSnx alloy behavior, the average lattice constants follow the Vegard’s rule, increasing linearly with increasing Sn concentrations. Similarly, the energy gaps decrease linearly with the increased Sn content. Using TD-HSE theory we calculate spin-forbidden dark exciton ground state in all Ge1-xSnx alloy QDs, with the dark-bright exciton splitting decreasing with admixture of Sn. For the smallest QDs of


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1.4 nm we obtain the splitting of 80 meV in pure Ge QDs, decreasing to 40 meV when the Sn concentration reaches 20%. We estimate that in experimental 2-3 nm QDs this splitting is 30-40 meV in pure Ge QDs, decreasing to 15-20 meV with the addition of Sn. TD-HSE calculations of optical absorption spectra reproduce the trends found in experiment, such as the redshift of the spectra with Sn concentration, and increased absorption around the band edge. In all cases, we find HSE hybrid functional calculations in agreement with experimental data, and therefore, conclude that they are a reliable tool for the prediction of properties of semiconductor QDs.

ASSOCIATED CONTENT SUPPORTING INFORMATION TEM images of 2 – 3 nm Ge1-xSnx alloy QDs with Sn compositions varying from 4% to 24%, absorption spectra of the same samples. This material is available free of charge via the Internet at http://pubs.acs.org. AUTHOR INFORMATION Corresponding Author Denis O. Demchenko E-mail: [email protected] ACKNOWLEDGMENTS The calculations were performed at the VCU Center for High Performance Computing. We acknowledge the use of the Analytical Instrumentation Facility (AIF) at North Carolina State University, which is supported by the State of North Carolina and the National Science Foundation and Dr. Yang Liu for his assistance with HRTEM and STEM analysis. Authors


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