1. Introduction

The wide band gap in ZnO is beneficial for UV-optoelectric devices, and ZnO nanorods are used for various applications in nanotechnology such as optical resonators, solar cells, piezo-electric energy generators, and in bio-medical applications [1]. Although nanorods are generally considered to be one-dimensional (1D) nanostructures, their 1D nature is determined by the degree of lateral confinement. As the exciton binding energy in ZnO is large (∼ 60meV), the lateral size of a ZnO nanorod needs to be comparable to the exciton diameter (∼ 3.6nm) to show this 1D nature. Otherwise, large-diameter nanorods are likely to produce bulk-like excitons or show weak confinement. When confinement dimensionality is determined by comparing the confinement size to the exciton diameter, it is difficult to distinguish the limits of weak and no confinement [2]. Alternatively, quantum confinement can also be induced by applying a strong external magnetic field. For example, if a high magnetic field is applied perpendicular to the lateral plane of a quantum well, quantum dot-like three dimensional confinement can be achieved. Because these magnetically-induced quantum dots have very uniform size-homogeneity when compared to inhomogeneous self-assembled quantum dots, long-range spatial coherence can be achieved. This enables collective coherent emission such as super-fluorescence and super-radiance [3, 4]. Likewise, 1D nanostructures can be induced from bulk by applying a large magnetic field, taking the system into the strong-field regime, where the magnetic cyclotron energy (h̄ω_c) is larger than the exciton binding energy ( $R_y^{*}$). On the other hand, for a lower magnetic field, the so-called weak-field regime ( $\overline{h}{\omega }_{\text{c}}\ll R_y^{*}$), it is difficult to define the dimensionality of magnetically-induced confinement [5].

As an alternative method for evaluating the confinement dimensionality, the temperature dependence of the radiative decay time can be used. Because the intra-relaxation of the thermally excited exciton towards the bottleneck range of radiative decay for the center-of-mass exciton dispersion depends on the density of states, confinement dimensionality affects the temperature dependence of the radiative decay time. The validity of this method was confirmed in quantum wires (1D), quantum wells (2D), and bulk (3D) [6, 7]. Although this method is rarely used for magneto-photoluminescence, it is useful to evaluate magnetically-induced confinement. Under a sufficiently large magnetic field ( $\overline{h}{\omega }_{\text{c}}\gg R_y^{*}$), the induced one-dimensional nature from bulk can be confirmed in terms of the temperature dependence of radiative decay time similar to the case of quantum wires ( $~\sqrt{T}$) [7]. However, because of the large exciton binding energy of ZnO, the intermediate-field regime ( $\overline{h}{\omega }_{\text{c}}~R_y^{*}$) requires ∼ 81T. In the case of the weak-field regime ( $\overline{h}{\omega }_{\text{c}}\ll R_y^{*}$), the Coulomb energy still dominates and the external magnetic field can be treated as a perturbation [5]. Recently, enhancement of the effective mass anisotropy was claimed as a consequence of magnetic perturbation [8]. Although a 1D nature cannot be achieved by a weak-field, the increase of the effective mass perpendicular to the external magnetic field may affect the degree of the induced-confinement.

In this work, we have evaluated the confinement dimensionality of our ZnO nanorods in terms of the temperature dependence of the radiative decay time (τ_r) with and without an external magnetic field. For large diameter ZnO nanorods (250nm) in the absence of a magnetic field, a bulk nature was confirmed (τ_r ∼ T^1.5). On the other hand, when 6T is applied (the weak-field regime), we conclude that the perturbation effect of the weak magnetic field results in a quasi-dimensionality between 2D and 3D according to the power coefficient of the radiative decay time with temperature (τ_r ∼ T^1.3).

2. Experiment

The ZnO nanorods were grown by metalorganic chemical vapor deposition (MOCVD) on c-plane sapphire in a horizontal reactor operating at a reduced pressure of 50 Torr. Diethylzinc (DEZn) and nitrous oxide (N₂O) were used as the zinc and oxygen sources with helium gas. The concentration of DEZn was 40 μmol/mn. Vertical ZnO nanorods with high aspect ratio and hexagonal section were grown at 875 °C under moderate oxygen/zinc molar ratio (R_VI/II) of 1150 in order to favor 1D growth (high R_VI/II above 14000 leads to two dimensional layer growth). The deposition time is 40 minutes [9]. Top and side views of SEM images of the nanorods are shown in Fig. 1(a), where a single rod is of 5 μm length and hexagonal type and its diameter is approximately 250 nm. Third harmonic emission (266nm) generated from a Ti:sapphire laser (∼ 800nm) emitting pulses of 80fs duration operating at a repetition frequency of 80MHz was used for excitation. In order to avoid many-body effects, a weak excitation intensity (10pJ per pulse) was used. The beam was focused onto the sample mounted in a temperature-controlled cryostat housing a superconducting magnet, where an external magnetic flux density of 6T was applied parallel to the nanorod direction. Time-resolved photoluminescence (TR-PL) was measured by using a streak camera.

 

Fig. 1 SEM images of ZnO nanorods, where the average length is ∼ 5μm (left image) and the size of a hexagonal crosssection is ∼ 250nm (right image) (a). PL spectrum at 10 K, 20 K, and 30 K in absence/presence of magnetic field (b)/(c), where the individual PL spectrum of D⁰X (dotted) and FX (shadow) are extracted by Gaussian fitting, and the central emission energy and the linewidth in eV are shown in the parentheses

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3. Results and discussion

Although several papers have already reported TR-PL in bulk ZnO and ZnO nanorods in the context of the temperature dependence of the radiative decay time, the physics underpinning this temperature dependence for the exciton was often misunderstood [10, 11], where the following aspects were overlooked. First of all, although the photoluminescence (PL) spectrum is dominated by donor bound excitons (D⁰Xs), free excitons (FXs) need to be resolved from the PL spectrum in order to evaluate the dimensional nature of the recombination. Secondly, the excitation intensity should be low enough to avoid many-body effects. As our excitation intensity (0.1μJcm⁻²) is lower by two-orders of magnitude compared to that of reported work (40μJcm⁻²), bandgap renormalization can be ruled out. Thirdly, the temperatures used should be low, otherwise the extraction of a pure radiative decay time becomes difficult as thermal non-radiative processes become significant, such as thermal dissociation, excitation, and quenching.

As shown in Figs. 1(b) and 1(c), the individual PL spectra of both the D⁰Xs and FXs with and without external magnetic field were resolved for various temperatures by double Gaussian fitting [12]. Although various peaks are seen at energies below that of the D⁰X, which are possibly other kinds of donor-bound excitons, acceptor-bound excitons, and exciton LO-phonon replicas, we include the D⁰Xs to improve the fitting accuracy of the FXs. Given the fitted linewidths of FXs (∼ 15meV), it was not possible to resolve the two peaks expected from the small Zeeman splitting (∼ 0.9meV) under an external magnetic flux density of 6T, which was estimated by using the g-factor of the electron (1.957) and the hole (2.45) in ZnO [13]. Rather the overall magnetic effect causes a redshift of ∼ 5meV. Recently, a similar result was observed in a quantum well; when the effect of an external magnetic field applied along the in-plane is considered as a perturbation, the weak-field gives rise to a redshift in the absorption spectrum. They also found that the diamagnetic effect eventually gave rise to a blueshift beyond the threshold magnetic field (∼ 30T) [8]. Therefore, the PL redshift under 6T in Fig. 1(c) can be attributed to a weak-field perturbation effect.

Each individual PL spectrum of the FX was obtained as a function of temperature by using Gaussian fitting as shown in Figs. 1(b) and 1(c), whereby the peak PL energy was estimated and a corresponding TR-PL trace was also measured up to a temperature of ∼ 100K within the spectral resolution of the TR system ∼ 10meV. As shown in Figs. 2(a) and 2(b), both the TR-PL with and without magnetic field show a monotonic decay on a log plot, where the PL decay is more rapid at low temperature. Although the increase in the PL decay time for increasing temperature contains the essential physics associated with the density of states, it is crucial to distinguish the radiative decay time from the PL decay time. As shown schematically in Fig. 3(b), FXs decay radiatively only in the small k-range where the center-of-mass FX wave vector is small when compared to that of the photon (k < k₀) at finite temperature, where k₀ is determined by the intersection of the parabolic center-of-mass FX dispersion (E = h̄²k²/2M) and the linear photon dispersion ( $E=\overline{h}ck/\sqrt{\epsilon }$). Given the total FX mass (M = 0.69m₀) and the dielectric constant ε₀(ε_∞) = 8.656(3.67), where m₀ is the electron rest mass, the maximum kinetic energy for radiative decay of the FXs ( $\mathrm{\Delta }={\overline{h}}^2{k}_0^2/2M~0.14$ meV(0.059 meV)) could be determined [14, 15]. On the other hand, for the FXs with large center-of-mass wave vector (k > k₀) the decay is dominated by non-radiative processes during their intra-relaxation along the dispersion via various inelastic scattering processes to conserve their momentum. As a result, the PL decay time (τ) represents the population decay as a consequence of both radiative and non-radiative processes. Provided that high quality samples are measured at very low temperature (< 10K), the non-radiative decay can be negligible [6]. Otherwise the non-radiative decay time (τ_nr) needs to be considered in the PL decay.

 

Fig. 2 TR-PL intensity of the FX for various temperatures in the absence/presence ((a)/(b)) of magnetic field. Temperature dependence of the PL decay time (τ(T)) (c) and ratio of the IQE with temperature to the IQE at 4K (ζ(T) = η(T)/η(4K)) (d) are shown with (open triangles) and without (filled triangles) external magnetic field up to ∼ 40K, beyond which non-radiative decay dominates, resulting in significant increase in τ(T) and a decrease in ζ(T) as shown in insets.

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Fig. 3 (a) Temperature dependence of the radiative decay time with and without magnetic field. (b) Excitons in the small k-range (k_ex < k₀) decay radiatively, whilst the population ratio with all excited excitons determines the temperature dependence of the radiative decay time (τ_r(T)). In the case of the weak-field regime, the perturbation energy of external magnetic field (B = 6T) gives rise to a redshift, and the effective mass perpendicular to the c-axis in a ZnO nanorod becomes heavier relatively ( $M_{\perp }^{*}>M_{\perp }$).

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For the monotonic decay of the time-resolved PL intensity, the population of FXs (N) after transient excitation (g) is given by

Theoretically, τ_r(T) is given by

Figure 2(c) shows τ(T) obtained from the monotonic decay of the TR-PL in the absence and presence of a magnetic field (Figs. 2(a) and 2(b)). The temperature dependence of the calibration factor ζ(T) was also obtained by normalizing I(T) to I(4K) as shown in Fig. 2(d). While τ(T) increases with temperature, a critical change in the temperature dependence is seen beyond 40K, where I(T) also becomes significantly suppressed (insets of Figs. 2(c) and 2(d)). These results indicate that the PL decay is dominated by non-radiative decay due to thermal effects when the temperature increases beyond 40K. It should also be noted that the accuracy of τ(T) deteriorates when the temperature is higher than 40K. Although Figs. 2(a) and 2(b) are normalized, the raw intensity weakens with a small signal-to-noise ratio as the temperature rises. As a result, the error bar for τ(T) becomes large. Since the dimensional nature of the nanorod is associated with radiative decay processes, α should be obtained by considering lifetime data below the critical temperature (< 40K) only. The enhancement of η(T) up to ∼ 40K can be attributed to suppression of FX trapping to donors [18]. Zhang et al. measured a quadratic dependence for τ(T), but calibration was overlooked [10]. Although Jen et al. considered the temperature dependence of the integrated PL intensity in order to calibrate τ(T), a linear temperature dependence of τ_r(T) was obtained in a ZnO film, which is in contrast to the 3D case (τ_r(T) ∼ T^1.5). This disagreement possibly results from the inappropriate temperature range. Because the temperature dependence of the FXs was considered from 40K up to 180K [11], the linear temperature dependence is likely to arise from thermal linewidth broadening. On the other hand, as we have divided τ(T) by the calibration factor (ζ(T)) for the temperature dependence of the radiative decay time according to Eq. (3), the power-order of τ_r(T) decreases when compared to that of τ(T). As shown in Fig. 3(a), we have confirmed that the temperature dependence of τ_r(T) in our ZnO nanorods follows that expected for the bulk case (∼ T^1.5). Given that the unknown η(4K) is less than 1, τ_r(T) seems somewhat long compared to the tens of picoseconds expected for the intrinsic radiative decay time in a semiconductor due to the long range spatial coherence of the small k. This can be explained by the large PL linewidth of the FXs in the nanorods; when the linewidth broadening is large, the uncertainty of the wave vector becomes comparable to k₀. This limits the spatial coherence by localization, resulting in a relatively long radiative decay time.

When a strong magnetic field is applied, a one-dimensional structure can be induced from the bulk through the Landau levels, where the quantized levels in the plane perpendicular to an external magnetic field are analogous to the confined levels in quantum wires, and the 1D-excitons remain free to move along the direction of the quantum wires and external magnetic field, respectively [19, 20]. Alternatively, the reduction in confinement dimension can be understood in terms of the density of states. In the case of bulk, the states are equally spaced (Δk) in k-space for the various coordinate configurations (k_x, k_y, k_z). On the other hand, an external magnetic field gives rise to quantized Landau levels in the magnetically-induced parabolic potential for the rotating radius. Schematically, the equally spaced (Δk) points in the k-plane (k_x, k_y), which represent the states in the absence of magnetic field, turn into the rings, which correspond to the Landau levels in the presence of an external magnetic field. The lateral distance between the rings in the k-plane (Δk_m) decreases as the Landau levels are equidistant in energy. As a result, the density of states is reduced in the presence of an external magnetic field [19, 20].

As the level spacing versus the thermal energy can be a criterion for confinement instead of a comparison of the confinement size to the exciton diameter, the magnetically-induced confinement can be evaluated in terms of the Landau level spacing (h̄ω_c) compared to the exciton binding energy ( $R_y^{*}$), where the cyclotron resonance frequency (ω_c = eB/μ) can be estimated by the reduced mass of the exciton (μ) and the applied magnetic flux density (Bẑ). Therefore, a 1D nature can be claimed in the so-called strong-field regime ( $\overline{h}{\omega }_{\text{c}}\gg R_y^{*}$). It would be challenging to measure the gradual change of this quasi-dimension towards the ideal 1D by increasing the magnetic field to this strong-field regime. However, at a flux density of 6T we are still in the weak-field regime ( $\overline{h}{\omega }_{\text{c}}\ll R_y^{*}$) as the Landau level spacing (h̄ω_c ∼ 4.4meV) is still less than the exciton binding energy ( $R_y^{*}~60\text{meV}$) [5, 21]. This is somewhat similar to the center-of-mass exciton confinement when the exciton binding energy is larger than the confinement energy. For a diameter of 250nm in our ZnO nanorod, Δk = 2.5 × 10⁵ cm⁻¹ was estimated. Δk_m = 7.0×10⁵ cm⁻¹ for 6T was also obtained by using the level spacing between the ground and the first excited state of the Landau levels. However, this value can be further reduced when Coulomb interactions are considered. In the case of the weak-field regime, the magnetically-induced confinement effect is likely to be suppressed due to thermal effects, but a reduction in the density of states might be expected as the arrayed points in k-space merge into the rings.

As shown in Fig. 3(a), we found that τ_r(T) in the presence of a flux density of 6T increases with temperature as τ_r(T) ∼ T^1.3. Although the power order 1.3 may not represent the presence of an intermediate dimension, the power order of the temperature-dependent radiative decay time can be used to evaluate the nature of the effective confinement. On the other hand, when the magnetic length ( $L=\sqrt{\frac{\overline{h}}{eB}}$) becomes comparable to the exciton Bohr radius (a_B), it is also known that the oscillator strength of electron-hole pair becomes enhanced as the magnetic field increases due to shrinkage of the wavefunction of the relative motion of the electron and hole in the magnetic field. As a result, the radiative decay time decreases. We have estimated that the magnetic length (L = 10.4nm) at 6T is still larger than a_B ∼ 1.8nm in ZnO, which confirms again the weak-field regime. Therefore, the relative decrease of the radiative decay time at 6T compared to that in the absence of a magnetic field may not be dominated by the wavefunction shrinkage effect [22].

As the exciton binding energy ( $R_y^{*}~60\text{meV}$) is still larger than the Landau level spacing (h̄ω_c ∼ 4.4meV), the diamagnetic perturbation energy of the center-of-mass exciton ( $\delta \epsilon (B)=\frac{e^2{B}^2{r}^2}{8M_{\perp }}$) should be considered, where M_⊥ is the total FX mass in the plane perpendicular to external field (Bẑ) corresponding to the vector potential $\overrightarrow{A}=\frac{1}{2}rB\widehat{\theta }$ in polar coordinates (r, θ). The first-order perturbation of the diamagnetic effect is always of positive value (ε⁽¹⁾ > 0), but the second-order perturbation energy for the ground state can be negative (ε⁽²⁾ < 0) in particular conditions. It is noticeable that the second-order perturbation for the excited states are also of positive value. In the case of a quantum well, the negative ε⁽²⁾ occurs only when the state filling from the ground state is extended beyond a half of the energy between the ground state and the first excited state. Therefore, the redshift under 6T in Fig. 1(c) is possibly a consequence of the two competing terms (δε(B) ≃ ε⁽¹⁾ + ε⁽²⁾ < 0), where ε⁽²⁾ is more pronounced than ε⁽¹⁾. It was known that ZnO is intrinsically n-type without intentional doping due to zinc interstitials, oxygen vacancies, and hydrogen. As the doping density is high (10¹⁷ ∼ 10¹⁸cm⁻³), the negative ε⁽²⁾ in our ZnO nanorods is possibly associated with state filling (or the Fermi level), which is similar to the case of a doped quantum well. However, further work is necessary to uncover the origin of the negative ε⁽²⁾, which is not clear at the moment. Nevertheless, when ε⁽²⁾ is negative, it is also known that the effective mass in the plane perpendicular to magnetic field becomes heavier ( $M_{\perp }^{*}>M_{\perp }$) [8]. As shown schematically in Fig. 3(b), suppose a weak magnetic field gives rise to a redshift and an enhancement of the effective mass in the plane perpendicular to the external magnetic field for the center-of-mass exciton dispersion, the range of radiative decay becomes smaller, i.e., both Δ and k₀ decrease. These effects may explain why the power order of the temperature-dependent radiative decay time has decreased (τ_r(T) ∼ T^1.3) in the presence of a weak external magnetic field (6T) when compared to the bulk nature in the absence of a magnetic field (τ_r(T) ∼ T^1.5). Intuitively, the induced anisotropy of effective mass seems to change the confinement dimensionality effectively, i.e., the heavier effective mass in the plane perpendicular to the magnetic field affects transport, by which the lateral confinement effect is enhanced although the weak magnetic field is not enough to induce a one-dimensional nature.

4. Conclusion

The dimensional nature of confinement has been investigated in ZnO nanorods by using the temperature dependence of the radiative decay, where the temperature dependence of the integrated PL intensity was also used as a calibration factor. We confirmed that 250nm-diameter ZnO nanorods behave in a similar manner to bulk material (τ_r ∼ T^1.5) at low temperatures (< 40K), whereby the alternative method of temperature-dependent decay time for determining dimensionality has been verified. In the weak magnetic field regime for ZnO (6T), a significant decrease in the power order of the temperature dependence of the radiative decay time (∼ T^1.3) was attributed to an enhancement of the effective mass perpendicular to the magnetic field and a redshift of the center-of-mass exciton as a consequence of perturbation effects in the weak-field regime.