Letter pubs.acs.org/NanoLett

Temperature and Magnetic-Field Dependence of Radiative Decay in Colloidal Germanium Quantum Dots István Robel,† Andrew Shabaev,§ Doh C. Lee,† Richard D. Schaller,† Jeffrey M. Pietryga,† Scott A. Crooker,‡ Alexander L. Efros,⊥ and Victor I. Klimov*,† †

Chemistry Division and ‡National High Magnetic Field Laboratory, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, United States § School of Physics, Astronomy, and Computational Sciences, George Mason University, Fairfax, Virginia 22030, United States ⊥ Naval Research Laboratory, Washington, D.C. 20375, United States ABSTRACT: We conduct spectroscopic and theoretical studies of photoluminescence (PL) from Ge quantum dots (QDs) fabricated via colloidal synthesis. The dynamics of latetime PL exhibit a pronounced dependence on temperature and applied magnetic field, which can be explained by radiative decay involving two closely spaced, slowly emitting exciton states. In 3.5 nm QDs, these states are separated by ∼1 meV and are characterized by ∼82 μs and ∼18 μs lifetimes. By using a four-band formalism, we calculate the fine structure of the indirect band-edge exciton arising from the electron−hole exchange interaction and the Coulomb interaction of the Γpoint hole with the anisotropic charge density of the L-point electron. The calculations suggest that the observed PL dynamics can be explained by phonon-assisted recombination of excitons thermally distributed between the lower-energy “dark” state with the momentum projection J = ± 2 and a higher energy “bright” state with J = ± 1. A fairly small difference between lifetimes of these states is due to their mixing induced by the exchange term unique to crystals with a highly symmetric cubic lattice such as Ge. KEYWORDS: Germanium, nanocrystal, quantum dot, dark and bright exciton, electron−hole exchange interaction, photoluminescence, magnetic field

B

the interpretation of spectroscopic data owing to the fact that CdSe QDs are among the most heavily studied and wellunderstood nanomaterials. Specifically, variations in behavior between CdSe and Ge QDs, particularly at low temperatures, can be expected to yield valuable insights into the effects of quantum confinement on the phonon-assisted optical transitions of indirect-gap materials like Ge. In this work, we investigate the nature of emitting states in Ge QDs by applying time-resolved PL spectroscopy to highquality samples prepared via colloidal synthesis.14 On the basis of observed temperature (T) and magnetic-field (B) dependences, we establish that late-time low-temperature (T = 1.75− 100 K) PL dynamics are dominated by radiative processes involving two finely separated excitonic states. Specifically, our measurements reveal that either an increase in sample temperature at zero field or application of a magnetic field at a fixed T progressively shortens the radiative lifetime, with remarkably similar trends. Our magneto-optical measurements are consistent with field-induced mixing of a lower-energy spin-

and-edge photon emission and absorption in indirect-gap bulk semiconductors occur with the participation of momentum-conserving phonons, which significantly reduce the strengths of these transitions compared to direct-gap semiconductors. If indirect-gap materials are prepared as nanosized particles,1−4 that is, as quantum dots (QDs), both of these processes are expected to be enhanced, as crystal momentum conservation is relaxed, and mixing of states at different points of the Brillouin zone occurs.5−8 Indeed, confinement-induced emergence of photoluminescence (PL) has been observed in porous and nanocrystalline Si2,3,9−12 and Ge,13,14 materials that are essentially nonemissive in their bulk forms;15 although, whether such emission is intrinsic or surfacerelated is still a subject of debate.16−24 In some respects, Ge QDs are more suitable than Si QDs for studying the effects of quantum confinement on radiative recombination. The separation between the indirect (L-point) and the direct (Γ-point) conduction band minima in Ge is fairly small (140 meV vs 1.5 eV in Si); therefore, these minima are expected to be strongly mixed by spatial confinement even in particles of relatively large size. Moreover, the direct band edge of Ge at the Γ-point of the Brillouin zone is similar to that of CdSe, and the band-edge transitions in both materials are well described by the Luttinger model.25 This similarity can aid in © 2015 American Chemical Society

Received: January 27, 2015 Revised: March 12, 2015 Published: March 20, 2015 2685

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Nano Letters forbidden (“dark”) exciton state with a higher-lying (∼1 meV separation) spin-allowed (“bright”) exciton and provide the first clear evidence of intrinsic radiative recombination in QDs of indirect-gap Ge. We also find that the thermal activation of PL in Ge QDs is explained by two separate mechanisms: in addition to an increased population of the “bright” state with rising temperature, as seen in direct-gap QDs, there is also an enhancement of phonon-assisted transitions owing to the increasing population of phonon modes. Finally, a theoretical model of the fine structure of the indirect exciton and its phonon-assisted radiative recombination closely describes the observed temperature and magnetic-field dependences of the measured PL decay and specifically demonstrates an important role of an anisotropic contribution to the electron−hole exchange splitting. Ge QD samples used in this study were synthesized by a colloidal route based on the chemical reduction of GeI2 in a hexadecylamine/1-octadecene solvent/ligand mixture.14 This synthesis produces octadecene-passivated particles with mean diameters from 3.2−6.4 nm and size dispersion of about 8− 10%. Only the smallest QDs with mean diameters of 3−4 nm (corresponding PL energies of 1.0−1.3 eV) show PL intensities sufficient for time-resolved measurements. The room-temperature PL quantum yields of these samples are around 1%. A rapid reduction of the PL efficinecy with increasing QD size, that is, decreasing band gap (Eg), is a trend general to infraredemitting QDs. For example, the absence of measurable steadystate PL from Ge QDs with radii larger than 2.3 nm was pointed out in ref 26. Further, the studies of narrow-gap PbSe and HgTe QDs showed a very significant decrease in their PL quantum yield with decreasing band gap, which was around two orders of magnitude for a two-fold reduction in Eg.27 A possible explanation to this trend, proposed in ref 27, invoked PL quenching due to energy transfer to vibrational modes of the passivating ligands, which had increasing efficiency for QDs with smaller band gaps. For low-temperature measurements, Ge QDs were incorporated into poly(methyl methacrylate) with sufficient dilution to avoid interparticle energy transfer. The samples were excited at 3.06 eV using 70 ps pulses from a diode laser operating at 10 kHz repetition rate. Time-resolved, spectrally integrated PL was collected using a silicon avalanche photodiode coupled to a system for time-correlated single-photon counting. The temporal resolution was ∼1 ns. The short-time dynamics were evaluated with a transient absorption (TA) pump−probe experiment in which the temporal evolution of photoinduced absorption at 1.1 eV was monitored following excitation with 100 fs, 3.1 eV pump pulses, as previously described.29 The band gap of the QDs, determined from both optical absorption and PL measurements (symbols in Figure 1),14,28−32 shows a size-dependent trend, which is roughly consistent with a particle-in-a-box model (solid black line in Figure 1). In our calculations, we used an infinitely high potential barrier, which allowed for a fairly close description of Eg for larger QD sizes. However, this model leads to an overestimation of confinement energies for smaller particle sizes where leakage of electronic wave functions outside of the QD is more significant. For these sizes, a more accurate description of the QD band gap is provided by tight-binding calculations (dashed red line in Figure 1).32 A close correlation between positions of the absorption edge and the PL peak, as well as a good agreement of calculated and measured band gap energies, suggests that the observed PL is intrinsic to the QDs and is not related to surface

Figure 1. Band gap of Ge QDs determined from PL (blue triangles are data from ref 14, and red open circles are from ref 28) and optical absorption (black squares are from ref 14, orange circles are from ref 29, and green inverted triangles are from ref 30). The black solid line is the size dependence of the band gap calculated using a particle-in-abox effective mass approximation with bulk-Ge parameters,31 while the red dashed line shows tight-binding calculations from ref 32.

states. The measured band gap energies and their size dependence13,14 are distinct from those of earlier reports on visible emission from Ge QDs,33 which was later ascribed to GeO2.34 On the other hand, our data are consistent with the measurements of infrared-emitting plasma-synthesized Ge QDs reported by Kortshagen and co-workers28 (Figure 1) and glassembedded Ge QDs studied by Takeoka et al.13 The TA measurements conducted at low excitation fluences when the average number of photons absorbed per QD per pulse () is less than 0.1 reveal a fast nonexponential decay (Figure 2a, inset), which indicates that during the first nanosecond, at least 70% of photoexcited carriers escape from intrinsic QD states via nonradiative processes such as trapping at surface defects. Since electrons and holes provide additive contributions to the TA signal, the observed decay corresponds to the lifetime of the carrier, whether electron or hole, that exhibits the longer trapping time constant. On the other hand, decay of an exciton (an electron−hole pair) monitored by PL spectroscopy is defined by the relaxation of the carrier with the shorter time constant. Therefore, lifetimes obtained from PL decay traces might be even faster than those observed in TA measurements. A distribution in the type and number of surface defects across a QD ensemble leads to the nonexponential carrier dynamics typically seen for poorly passivated, low-PL-quantumyield samples. In QDs containing surface defects, carrier trapping is usually significantly faster than radiative recombination even in the case of CdSe where radiative lifetimes (τr) are fairly short (τr ≈ 20 ns35). This disparity leads to a distinct two-component carrier population decay, with the initial fast component due to nonradiative processes in the subensemble of almost nonemissive dots followed by slower decay due to radiative processes in the subensemble of well-passivated QDs. The distinction between the two components is expected to be even stronger in Ge QDs as they are characterized by extremely slow radiative decay with τr on the order of tens of microseconds (see the following). On the basis of these considerations, a derivation of radiative time constants in QDs typically relies on an analysis of late-time PL dynamics.35,36 Because of their limited ∼1 ns temporal resolution, our PL 2686

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PL burst have been used to analyze heat transfer between the QDs and the surrounding medium.37 In the case of Ge QDs, early time fast PL decay likely also corresponds to the initial transient stage of QD cooling, while later-time dynamics reflect exciton decay after the establishment of thermal equilibrium between the QDs and the surrounding medium, that is, after the temperature of the excitonic system becomes equal to the nominal sample temperature. On the basis of these considerations, in our analysis of temperature-dependent exciton dynamics, we use the late-time slower PL component. As shown in Figure 2, panel b, the lifetime of this long component changes from 81 to 8 μs when the temperature is raised from 1.75 to 137 K, while the relative PL QY (proportional to the normalized PL intensity of Figure 2b) stays nearly constant (the observed variation is within 15%) suggesting that shortening of PL decay is not due to nonradiative effects but rather activation of a faster radiative decay channel. A similar T-dependent PL behavior was observed previously in QDs of CdSe,35,36 CdTe,38 InAs,38 Si,39 and PbSe40,41 and attributed to the presence of an exciton fine structure that comprises closely spaced states with different oscillator strengths. These effects are especially prominent in CdSe QDs,35 in which low-temperature PL decay is dominated by slow recombination of a dipole-forbidden “dark” exciton (the projection of the total angular momentum along the hexagonal c-axis |J| = 2), while an increase in the temperature leads to thermal activation of a dipole-allowed “bright” exciton (|J| = 1), resulting in a dramatic, orders-of-magnitude increase in the decay rate. In our measurements of Ge QDs, however, we observe only a moderate change in the PL lifetime (by a factor of ca. 10), which is more consistent with observations for PbSe QDs,40 where it was ascribed to the interplay of two closely spaced, slowly emitting exciton states with different but comparable decay rates. We continued our studies of the PL mechanism in Ge QDs by performing magneto-PL measurements. First, we study PL dynamics as a function of magnetic field at a fixed temperature T = 1.6 K (Figure 3a). We observe that the decay time of the long-lived PL component becomes faster with increasing magnetic field and reaches saturation at B higher than ∼7 T (Figure 3b). In contrast to CdSe QDs where magnetic field has a very profound effect on PL relaxation,36,42 in the case of Ge QDs, it causes a rather small (∼20%) but clearly discernible change in the PL lifetime (Figure 3b). Just as in the case of Tdependent studies, this again suggests that the two emitting states mixed by magnetic field do have distinct oscillator strengths, but the difference between them is not large. Next, we compare the lifetimes extracted from late-time PL dynamics for T from 1.75−20 K measured either at zero field or at B = 7 T (Figure 3c). At T below 10 K, application of a magnetic field shortens the PL decay time, while it does not have any noticeable effect on PL dynamics at higher temperatures. Interestingly, applying a magnetic field to the QDs seems to be equivalent to raising the sample temperature, and the decay trace measured at 1.75 K in 7 T field matches almost exactly the trace recorded at zero field and 8 K (Figure 3d). This again is qualitatively similar to previous observations for CdSe QDs (see Figure 5 in ref 42). The results of the above measurements along with similarities between our T- and B-dependences and those reported previously for CdSe QDs point toward the radiative character of late-time PL decay observed at low temperatures in

Figure 2. (a) Temperature dependence of PL decay in Ge QDs with mean radius a = 1.75 nm. The inset shows short-time dynamics (time window of ∼1 ns) measured using a TA experiment at low excitation fluences, = 0.03 (black) and 0.06 (red). The similarity between the traces confirms that they describe carrier dynamics in QDs excited with no more than one exciton. (b) The measured (black squares) and calculated (black solid line, fit excludes shaded area where PL intensity drops significantly) PL lifetimes as a function of temperature along with the measured PL intensity (red circles; normalized to the maximum signal).

measurements discard a significant fraction of nonemissive QDs with fast nonradiative decay and automatically select a subensemble of highly emissive dots with slower recombination. In fact, as we show in the following, the T- and B-field dependent trends we observe in PL dynamics at low temperatures (T < 100 K) are consistent with exciton decay dominated almost entirely by radiative recombination. In Figure 2, we present time-resolved PL of Ge QDs measured as a function of sample temperature. Low-temperature PL traces exhibit a prominent fast initial decay followed by a slower mostly single-exponential component. A qualitatively similar behavior has been previously observed for CdSe QDs35 where both the initial PL burst as well as latertime slower dynamics have been ascribed to radiative decay with a varied emission rate controlled by the local temperature of the QD. Excitation with a laser pulse leads to almost instantaneous heating of the QD, which increases the contribution to emission from higher-energy, higher-oscillator-strength fine-structure exciton states. As the QD cools down, the emission rate decreases due to an increasing role of recombination via the lowest-energy, nominally forbidden “dark” state. Recently, the temporal dynamics of the initial 2687

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Figure 3. (a) PL decay measured at 1.6 K as a function of magnetic field (0 ≤ B ≤ 15 T). (b) The time constant extracted from late-time PL decay as a function of magnetic field. Inset: the calculated dependence of lifetime versus magnetic field (see text for details). (c) PL lifetime (derived from late-time dynamics) as a function of temperature in the presence (red circles) or absence (black squares) of a 7-T magnetic field; lines are theoretical fits (see text for details). (d) PL decay measured at 7 T and 1.75 K (red dashed line) closely matches that recorded at 0 T and 8 K (green solid line). Both are distinct from the trace measured at 1.75 K and 0 T (black line).

electron at the L-point of the Brillouin zone. This difference arises from a strong anisotropy of the effective mass of the Lpoint electron, which leads to a spatially anisotropic electron wave function. The resulting anisotropy of the electron Coulomb potential splits the four-fold degenerate ground state of the Γ-hole into two sublevels with angular momentum projections along the ⟨111⟩ axis M = ±3/2 and ±1/2 (referred to below as hole “spins”). In QDs with radius a, which is smaller than the bulk Bohr exciton radius (aex = 24.3 nm in Ge), this splitting becomes size-dependent and can be described as48 Δlh= Ec(1/2) − Ec(3/2), where Ec(|M|) is the energy of the electron Coulomb interaction with the spin-M hole. This energy can be calculated from

Ge QDs. However, despite these qualitative similarities, the absolute values of the measured radiative time constants (80− 90 μs at ∼2K) are about two orders of magnitude longer than in CdSe QDs (∼1 μs), which suggests that the emitting transition in Ge QDs is still largely indirect. On the other hand, the fact that the transition lifetime is four orders of magnitude shorter than the nominal radiative time constant of bulk Ge (0.61 s)15 indicates a significant effect of quantum confinement on radiative dynamics due to processes such as confinementinduced mixing between direct- and indirect-band minima and enhancement of electron−phonon coupling at small QD sizes43 that result in increased radiative phonon-assisted decay rates. According to tight binding calculations,32 if these effects are taken into account, the intrinsic exciton lifetime in Ge QDs with diameters between 3.75 and 4 nm is ∼100 μs, which is in remarkable agreement with time constants measured in the present work. As was pointed out earlier, the T- and B-field trends observed in the PL decay suggest that radiative recombination in Ge QDs occurs via two closely separated exciton states with slightly different oscillator strengths. Further, the 10 K activation temperature (kBT of 0.86 meV) indicated by the data in Figure 3 suggests that the two states are split by an energy of ∼1 meV. To clarify the origin of this splitting, we conduct a theoretical analysis of band-edge exciton states in Ge QDs. A splitting of the indirect band-edge exciton by about 1 meV was previously observed in bulk Ge crystals44−46 where it was explained by the difference in energies of the Coulomb interaction of the light and heavy holes47 with the band-edge

2

Ec(|M |) = −

∫ dr1dr2 ε|r e− r | ψe2(ρ1, z1)ψM2 (r2) 1

2

(1)

where z1 and ρ1 are, respectively, the projections of r1 along the ⟨111⟩ direction and perpendicular to it, ψe(ρ1, z1) is the anisotropic electron wave function (which can be found using the adiabatic approximation48), and ψM is the wave function of the spin-M hole calculated according to ref 49. For computing Δlh, we use the longitudinal and transverse effective masses of an electron at the L-minimum (ml = 1.588m0, and mt = 0.0815m0, respectively50) and light and heavy hole effective masses at the Γ-point (mlh = 0.045m0, and mhh = 0.35m0, respectively51). By using ε = 8, we obtain that in QDs with a = 1.75 nm, the light−heavy hole splitting is 3 meV. This is more than three times larger than in bulk Ge, which is a direct consequence of quantum confinement. The calculated value, 2688

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Nano Letters however, is also significantly greater than the separation between the two emitting excitonic levels derived from our measurements of Ge QDs, which suggests that the latter is not due to light−heavy hole splitting. Next, we consider the effect of the electron−hole exchange interaction, which as we show below is the most likely reason for the experimentally observed splitting. The energy spectrum of the “exchange-correlated” indirect exciton in Ge QDs can be calculated by adopting the approach from ref 49 for the situation of an anisotropic electron wave function.48 This anisotropy can be accounted for by introducing an additional parameter ξ = εaexch/ εsexch, where εs(a)exch is the symmetric (antisymmetric) exchange constant. First, we consider the case of large Δlh, when Δlh is much greater than δ0 = 3ηh (1 + ξ). Here, ηh is the heavy-hole exchange constant defined as ηh = (a30/3π)εsexch ∫ Fh (r,θ) ψ2e (r cos θ, r sin θ) sin θ dθr2 dr, where Fh = 0.5[R20 (r) + R0 (r) R2 (r) (1−3 cos2θ) + R22 (r) cos4θ] is expressed via radial components R0(r) and R2(r) of the hole wave function ψM (ref 48) and the electron wave function ψe(r cosθ, r sinθ) with cylindrical symmetry (see ref 49). Calculation of the integral gives ηh = 0.57(a0/a)3εsexch. In the approximation when Δlh ≫ δ0, we can construct two excitons with the total momentum projections J = ±2 and J = ±1 by combing the ±1/2 electron spins and the ±3/2 spin projections of the heavy hole. In zero magnetic field, these excitons are split by the exchange interactions into two states with energies48 ε2 = −

3η (1 + ξ) 3η (1 + ξ) Δlh Δ − h and ε1L = − lh + h 2 2 2 2 (2)

On the basis of these expressions, the energy separation between the dark (ε2) and bright (εL1 ) states, Δ12 = εL1 − ε2, is simply equal to δ0, that is, Δ12 = δ0 = 3ηh(1 + ξ)

(3)

Next, we consider exact solutions for the five level fine structure of the band-edge exciton obtained numerically. In Figure 4, we show examples of calculated energies as a function of δ0 for two values of the parameter of anisotropy ξ = 0 (panel a) and ξ = 3 (panel b). The fine structure exciton states form two manifolds: one associated with a heavy hole and the other with a light hole. The lower-energy manifold comprises dark and bright exciton states (J = ±2 and ±1, respectively) discussed previously. The higher-energy manifold contains states obtained by combing the spins of the electron (±1/2) and the light hole (±1/2), which yields three exciton levels with distinct energies εU1 (J = ±1), εU0 (J = 0), and εL0 (J = 0). The upper εU0 and lower εL0 exciton states with J = 0 are symmetric and antisymmetric superpositions of electron (±1/2) and light hole (±1/2) states that are split by the electron−hole exchange interaction. As can be seen in Figure 4, panel a, in the isotropic case (ξ = 0), the energy separation Δ12 between the two lowest-energy states is always smaller than the experimentally measured dark− bright exciton splitting of ∼1 meV independent of δ0. Our calculations show that Δ12 can reach values of ∼1 eV or greater only if the anisotropy parameter exceeds 0.35. This indicates that anisotropy of electronic wave functions has a significant effect on the fine-structure of band-edge exciton states in Ge QDs. Since the experimental value of the exchange splitting satisfies the condition Δ12 ≪ Δlh, we can use eq 3 to estimate

Figure 4. Energy diagram of the exciton fine structure in the absence of magnetic field as a function of δ0 for (a) Δlh = 3 meV; ξ = 0 and (b) Δlh = 3 meV; ξ = 3. Shaded areas mark the region of energy splitting derived from the experiment. (c) Energy diagram of the two lowestenergy excitonic states (dark and bright states) in the presence of magnetic field that further splits these levels due to the Zeeman effect. Solid and dashed arrows indicate radiative recombination pathways that produce left and right circularly polarized light, respectively.

the upper limit of the heavy-hole exchange constant. Specifically, by using Δ12 of ∼1 meV and ξ ≥ 0.35, we obtain ηh ≤ 0.25 meV. On the basis of this value and using the QD radius a = 1.75 nm and the bulk Ge lattice constant a0 = 0.566 nm, we can further estimate the upper limit of the symmetric exchange constant from εsexch = 1.75ηh(a/a0)3, which yields εsexch ≤ 13 meV. This estimation suggests that the strength of the electron−hole exchange interaction in indirect-gap Ge is significantly smaller than that in direct-gap semiconductors such as CdSe (εsexch = 450 meV49) likely because of the separation of the electron and the hole in the momentum space. As in CdSe QDs,35 the lowest energy ε2 state corresponds to J = ±2 and is an optically forbidden “dark” exciton because an emitted photon cannot carry two units of angular momentum in the dipole approximation. Consequently, an additional spin− flip process mediated, for example, by coupling to paramagnetic dangling bonds on the QD surface or carrier−phonon 2689

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agreement with the ∼1 meV splitting estimated from the B- and T-dependent data in Figure 3. An interesting result of the above analysis is that in Ge QDs, the dark exciton lifetime is only about four times longer than the bright exciton lifetime, while in wurtzite CdSe QDs, it is longer by approximately two orders of magnitude. This difference might be linked to the distinct symmetries of Ge and CdSe crystal structures. Indeed, in the case of the cubic lattice of Ge, the difference between lifetimes of dark and bright excitons is diminished by mixing between these two states induced by a special crystal-symmetry-related term of the electron−hole exchange interaction,25 which can be expressed as αcub[σxJx3 + σyJy3 + σzJz3], where αcub is the exchange constant, and σi and Ji (i = x, y, and z) are components of the Pauli matrix and the matrix of spin 3/2, respectively. Because of the reduced symmetry of a wurtzite lattice, this term is not present in the case of CdSe QDs, which eliminates the zero-Bfield mixing between the bright and dark states and thus increases the disparity between their lifetimes. Let us now consider the effect of a magnetic field on bandedge exciton recombination. The energy spectrum of the bandedge exciton in each valley depends on the projections of the magnetic field along (Bz) and perpendicular to (B⊥) the Γ−L direction. The first projection is responsible for splitting of the ±2 dark and ±1 bright excitons, while the second for the admixture of these states. By using the approach developed in ref 49, we can find the energy levels of the split-off band-edge excitons:

interaction in the presence of a spin−orbit interaction, is required to explain a fairly efficient PL seen at low temperatures when the upper ε1 “bright” exciton state (corresponding J = ±1) is not thermally populated. The involvement of this additional process should make decay of the dark exciton much slower than that of the bright exciton. While this is indeed true in CdSe QDs, the distinction between “dark” and “bright” exciton lifetimes in Ge QDs is much less pronounced. As we discuss later in the article, this might be linked to the difference in crystal structures of these two materials. As the Ge QD sample temperature is raised and the occupancy of the higher-energy bright state is increased, the PL decay becomes faster. However, it still remains much slower than in CdSe QDs, which suggests that the radiative transition stays largely indirect and requires the participation of momentum-conserving phonons.50 This leads to an additional source of temperature dependence in the PL decay as rates of both phonon emission and phonon absorption are controlled by thermal occupancies of the phonon mode (Nq) and scale, respectively, as (Nq + 1) and Nq. While activation of the bright exciton can explain acceleration of PL relaxation in the temperature range up to 10 K, a further enhancement in the PL decay rate at higher temperatures (10−100 K; Figure 2b) is likely attributed to increased rates of phonon emission/ absorption due to increased occupation numbers of phonon modes. In the temperature range below 100 K, where the observed PL decay is presumably dominated by radiative recombination, we expect primarily contributions from the two lowest energy acoustic phonon modes, one transverse (TA) with energy ETA = 7.8 meV and the other longitudinal (LA) with ELA = 27.7 meV.52,53 The coupling between L- and Γ-band minima via the TA mode is nominally forbidden in bulk Ge by selection rules dictated by translational momentum conservation,53 but it becomes allowed in the QDs due relaxation of momentum conservation. To capture the effect of phonons on the temperature dependence of radiative decay, we describe lifetimes of the dark (τ2) and the bright (τ1) excitons as follows:

ε±L1 =

ε±2 =

τ02

P(T ) 0 τ2,1

−Δlh ± 3μB ghBz 2

(4)

κ ±(B , θ ) =

τ01

where and are the dark and bright exciton lifetimes, respectively, at T = 0, and Nq(T) = 1/[1 − exp(Eq/kBT)] is the temperature-dependent occupation number of the phonon mode with energy Eq (q = TA, LA), and λq are the coefficients that describe branching between emission channels involving the TA- and LA-phonon modes (λ2TA + λ2LA = 1). Further, assuming thermal equilibrium between the dark and bright exciton states, we obtain the following temperature dependence of the PL lifetime, which accounts for both thermal activation of the bright exciton and temperature-controlled occupancy of the phonon modes: P(T ) 1 + τ ̅ exp[−Δ12 /kBT ] 1 = τ (T ) τ20 1 + exp[−Δ12 /kBT ]

2

+

−

p±2 + |n|2 2 p±2 + |n|2 (6)

2

Here, p± = 3ηh (1 + ξ) ± μBge∥Bz, n = μBge⊥B⊥, μB is the Bohr magneton, gh is the hole Lande g-factor, and ge∥ and ge⊥ are, respectively, longitudinal and transverse g-factors of the L− valley electron. The strength of admixture, κ± , of the dark ±2 excitons to the bright ±1 excitons by the “transverse” magnetic field is described by

1 1 2 2 = 0 [λ TA (2NTA(T ) + 1) + λLA (2NLA(T ) + 1)] τ2,1(T ) τ2,1 =

−Δlh ± 3μB ghBz

p∓2 + |n|2 − p∓ 2 p∓2 + |n|2

(7)

where θ is the angle between the valley orientation and the applied magnetic field. As a result, the radiative decay time of the four lowest mixed exciton states can be written as 1 τ±L1(B ,

⎫ ⎧1 ⎡1 1⎤ = ⎨ 0 + ⎢ 0 − 0 ⎥κ ±(B , θ )⎬P(T ) τ2 ⎦ θ ) ⎩ τ1 ⎣ τ1 ⎭ ⎪

⎪

⎪

⎪

⎫ ⎧1 ⎡1 1 1⎤ = ⎨ 0 − ⎢ 0 − 0 ⎥κ ±(B , θ )⎬P(T ) τ±2(B , θ ) ⎩ τ2 τ2 ⎦ ⎣ τ1 ⎭

(5)

⎪

⎪

⎪

⎪

(8)

where τL± 1 and τ±2 are the decay times of the upper and lower exciton states, with energies εL± 1 and ε±2 as described by eq 6. To obtain the average radiative lifetime in the ensemble of randomly oriented QDs, we need to take into account the thermal distribution of excitons over all band-edge levels and

where τ ̅ = τ02/τ01 is the ratio of the dark and bright exciton lifetimes at T = 0 K. By using this expression for fitting the experimental data (solid line in Figure 2b), we obtain τ02 = 81.9 μs, τ01 = 18.4 μs, and Δ12 = 0.93 meV; the latter value is in 2690

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Letter

Nano Letters

and magnetic field on the radiative lifetime of intrinsic, bandedge PL in colloidal Ge QDs and have revealed similarities and crucial differences relative to QDs of direct-gap materials. In the range of temperatures 1.75−100 K, the PL lifetime shortens with increasing T from ∼80 μs to ∼10 μs, while the PL quantum yield remains almost unchanged. A qualitatively similar behavior is observed when the B-field is increased for a fixed temperature. These observations can be rationalized under the assumption that PL dynamics are controlled by radiative recombination of band-edge excitons thermally distributed between the two closely spaced states (∼0.93 meV separation) characterized by about a four-fold difference in oscillator strengths. To support a quantitative analysis of these experimental findings, we developed a four-band model of the indirect exciton, which comprises a hole at the Γ-point band minimum and an electron at the L-point. The Coulomb interaction of the hole with the anisotropic charge distribution of the electron splits the hole level into two sublevels separated by 3 meV for the 1.75 nm radius QDs studied in this work. Since this splitting is greater than the one observed experimentally, we conclude that it cannot be responsible for the observed T- and B-dependent trends in the PL decay. We further consider the effect of the exchange interaction, which splits the band-edge exciton formed by the spin ±3/2 hole and ±1/2 electron into the lower-energy “dark” exciton (J = ±2) and the higher-energy “bright” exciton (J = ±1). We show that the combination of two phenomena, thermal activation of excitons from the dark to the bright state along with Tdependent occupancies of phonon modes involved in radiative decay of indirect excitons, can explain the observed temperature dependence of the PL decay, while mixing between the dark and bright states is responsible for the increased PL decay rate with increasing B. This first detailed description of the optically active states in QDs of an indirect-gap Group IV semiconductor reveals important subtleties emerging from the structural characteristics of Ge that will contribute to the development of a universal understanding of the optical properties of quantum confined systems of arbitrary composition. The specific findings of this study may also provide enabling insights toward the optimization of Group-IV QDs for applications in, for example, environmentally benign solutionprocessed solar cells or light emitters, especially in view of recent reports of relatively efficient near-infrared emission14,55 and high carrier multiplication efficiencies56−58 in these materials.

also conduct averaging over all possible QD orientations. The resulting rate of exciton recombination can be written as 1 = 0.5 τ (T , B )

sin θdθ ∑ ∫ ∑ exp( −ε /kT ) i

i

i

exp( −εi /kT ) τi (9)

where εi(B, θ) are the energies of the four lowest exciton states described by eq 6, and τi(B, θ) are the radiative decay times of corresponding states described by eq 8. By using eq 9, we are able to closely describe the effect of magnetic field on the T-dependence of radiative lifetimes as shown in Figure 3, panel c (lines and symbols are calculations and experimental data, respectively). The theoretical dependence of the PL decay time was obtained using gh = −1.23 based on the formalism of ref 49. Since the anisotropic g-factors of the L-valley electron are not known, we have used them as fitting parameters. The best fit to experimental data yields ge|| = 0.56 and ge⊥ = 3.08. The same set of parameters also provides a qualitative description of the magnetic-field dependence of radiative lifetime measured at a fixed temperature; specifically, the model explains shortening of PL decay with increasing field (compare calculated trend in the inset of Figure 3, panel b with measurements in its main panel; T = 1.6 K). The presented description of the indirect excitons neglects intervalley interactions between the excitons in four identical L valleys. One can use this approximation if the intervalley exciton splitting, Δint, is smaller than the Coulomb splitting of the heavy and light holes, Δlh, originating from the anisotropic distribution of the electron charge. In the case when Δint > Δlh, the electron wave function has equal contributions from all four valleys, and as a result, the electron charge distribution becomes isotropic. This should eliminate the effects of both the light− heavy hole splitting and the magnetic-field-induced mixing of the ground dark exciton with the bright state. Since the experimental data indicate a significant effect of magnetic field on dark-exciton lifetime, this suggests that the situation realized in Ge QDs corresponds to the regime of a weak intervalley coupling when Δint ≪ Δlh. In this case, the four L-valley exciton states can be considered as uncoupled, and their interaction with the external magnetic field will depend on the specific orientation of a given L-valley with respect to the magnetic-field direction. Interestingly, previous measurements of Si nanocrystals formed in porous Si54 indicated the opposite trend, namely, the lengthening of the radiative decay time with increasing magnetic field. This difference, however, is not surprising given the difference between the samples investigated in the two studies. In samples used in ref 54, Si nanocrystals were elongated, and the long axis was oriented preferentially along the direction of the pores that in turn were aligned with the direction of the B-field. In this case, application of a magnetic field did produce state splitting; however, it did not appreciably mix dark and bright states. The B-field-induced splitting increased the separation between the bright and the dark exciton states, which led to a reduced population of the bright state and hence increased radiative lifetime. The situation is different in the films of Ge QDs studied in the present work. In these samples, Ge QDs are oriented randomly, and therefore the effect of field-induced state mixing is strong, which leads to the observed acceleration of PL decay with increasing B. To summarize, we have conducted the first concerted experimental and theoretical study of the effect of temperature

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Present Address

D.C.L., Department of Chemical and Biomolecular Engineering, KAIST Institute for the Nanocentury, Korea Advanced Institute of Science and Technology (KAIST), 291 Daehak-ro, Yuseong-Gu, Daejon 305−701, Korea. R.D.S., Center for Nanoscale Materials, Argonne National Laboratory, Argonne, Illinois 60439, United States; and Department of Chemistry, Northwestern University, Evanston, Illinois 60208, United States. Notes

The authors declare no competing financial interest. 2691

DOI: 10.1021/acs.nanolett.5b00344 Nano Lett. 2015, 15, 2685−2692

Letter

Nano Letters

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ACKNOWLEDGMENTS I.R., D.C.L, R.D.S., J.M.P., S.A.C., and V.I.K. were supported by the Chemical Sciences, Biosciences and Geosciences Division, Office of Basic Energy Sciences (BES), Office of Science (OS), U.S. Department of Energy (DOE). A.S. was supported by the Center for Advanced Solar Photophysics (CASP) an Energy Frontier Research Center funded by BES, OS, U.S. DOE. A.L.E. acknowledges financial support of the Office of Naval Research through the Naval Research Laboratory Basic Research Program.

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DOI: 10.1021/acs.nanolett.5b00344 Nano Lett. 2015, 15, 2685−2692