Model

equa-7 Excitation Intensity Dependence of the Photoluminescence Properties of ZnO

tions,

dn

dt =G−cnpnp−cnanN_a⁰−cndnN_d⁺= 0 (7.3) dN_a⁰

dt = cnanN_a⁰−capN_a⁻p=0 (7.4)

dN_d⁺

dt = cndnN_d⁺−cd pN_d⁰p= 0 (7.5) whereN_a⁰andN_a⁻are the neutral and ionized deep-acceptor concentrations, andN_d⁰andN_d⁺are the neutral and ionized deep-donor concentrations. Their sums are equal to the total concen-tration of the corresponding deep centers, respectively,

N_a⁰+N_a⁻= Na (7.6)

N_d⁺+N_d⁰ = Nd (7.7)

The charge conservation gives,

n+N_a⁻= p+N_d⁺+Nds (7.8)

The photoluminescence intensities are related to the recombination rates as

IN BE ∝cnpnp (7.9)

IDLE ∝cnanN_a⁰ or cndnN_d⁺ (7.10) The equation system 7.3–7.8 can be solved numerically. There are several constant parameters, including the concentrations of defects Na, Nd, Nds and the recombination coefficients ci j, which are dependent on the specific growth method and conditions of the ZnO materials.

These parameters are carefully adjusted during the simulation until good agreement with the experimental results is obtained.

Figure 7.8 shows the final fitting results of the excitation intensity dependence of the lumines-cence intensities as a function of electron-hole pair generation rate for ZnO nanowires (a) and wafer (b), respectively. The calculated results based on the above model are represented by solid lines. With regard to the photoluminescence experiments, some facts have to be kept in mind. Due to the refractive index dispersion of light in optical lenses, the NBE and DLE with different wavelengths cannot be accurately co-focused and equally coupled into the spectrom-eter. Thus, the absolute intensities of the NBE and DLE bands in the recorded spectra cannot exactly represent the real emission of the samples. Due to this consideration, the calculated and experimental results shown in Fig. 7.8 are normalized to the values at the crossing points, respectively. The laser power intensity used in experiments is converted into electron-hole generation rate assuming that every absorbed photon excites one electron-hole pair in ZnO.

The effective thickness of the photo-excited layer beneath the surface is estimated to be 100 nm according to the absorption coefficient of∼ 1.5×10⁵cm⁻¹of ZnO at 325 nm [197]. The parameters used for calculations are listed below, In order to obtain a good fitting to the exper-imental results of ZnO nanowires (Fig. 7.8(a)), relatively higher defect concentrations (Ndand Nds) have to be used in comparison to those used for fitting to the results of the wafer. This is consistent with their lower crystal quality compared to the commercial ZnO wafers. The shal-low donor concentrations (Nds) used for both cases are consistent in order of the magnitude 96

7.4 Model

Figure 7.8: Comparison between calculated and measured luminescence intensities of ZnO nanowires (a) and ZnO wafers (b) as a function of electron-hole pair generation rate. The red lines in both figures are the simulated evolution of the ratio of the hole density to the electron density. The calculated and experimental results are normalized to the values at 10²⁵ cm⁻³s⁻¹ in (a) and 10²⁴ cm⁻³s⁻¹in (b), respectively. The parameters used for calculation are listed in Table 7.1.

Table 7.1: Parameters used for the calculations of the excitation-intensity-dependent lumines-cence intensities of ZnO nanowires and wafer shown in Fig. 7.8.

Sample Na

(cm⁻³)

Nd

(cm⁻³)

Nds

(cm⁻³)

cnp

(cm³s⁻¹)

cna

(cm³s⁻¹)

cap

(cm³s⁻¹)

cnd

(cm³s⁻¹)

cd p

(cm³s⁻¹) NW 10¹⁵ 3.5×10¹⁶ 10¹⁷ 10⁻¹¹ 10⁻¹¹ 10⁻⁷ 10⁻⁹ 10⁻⁸

wafer 10¹⁵ 10¹⁵ 10¹⁶ 10⁻¹¹ 10⁻⁹ 10⁻⁶ 10⁻¹² 10⁻⁷

with the commonly observed free electron density in ZnO [198–201]. The same bimolecular coefficient cnp = 10⁻¹¹ cm³s⁻¹ for band-edge recombination processes is used. The recom-bination coefficients for deep level transitions used for the simulations are different for the nanowire and wafer samples, which is reasonable due to the different luminescence origins of their DLE bands.

In addition, the experimental results of the DLE of the ZnO nanowires can only be well fitted by the deep donor related recombination in the model with bimolecular recombination coeffi -cientscnd > cna, as shown in Table 7.1. For ZnO wafers, the situation is inverse. The DLE is fitted with the deep acceptor related recombination withcnd < cna. This difference illustrates again the different origins for the DLE of the nanowire and the wafer samples.

The calculated dependencies show good agreement with the experimental results. In the calcu-lated curves, turning points can be seen for both kinds of samples where the increasing rates of the NBE and DLE with the generation rate significantly change. For the nanowires, at around 10²⁵cm⁻³s⁻¹, the DLE transits from a linear to a square-root dependence on excitation intensity while the NBE curve increases superlinearly and then returns to a linear increase. The depen-dence of the DLE is similar to the simulated results in the GaN model in the literatures which mainly studied the defect luminescence of GaN [90, 196]. However, the dependence of the NBE (with transition from a linear dependence on the excitation intensity to a superlinear one)

7 Excitation Intensity Dependence of the Photoluminescence Properties of ZnO

and the successful fitting with the recombination model, to the best of the author’s knowledge, has not been reported yet. The trends for the ZnO wafer are similar. But the turning point shifts to a lower generation rate (between 10²² and 10²³ cm⁻³s⁻¹). The excitation-intensity-dependent photoluminescence is further discussed in the following for low and high excitation intensities, respectively.

Simple algebraic transformation of Equ. 7.3−7.8 gives the recombination rates of the deep-level transitions as

RDLEa= cnanN_a⁰ = cnacapNanp

cnan+capp (7.11)

RDLEd =cndnN_d⁺ = cndcd pNdnp

cndn+cd pp (7.12)

with

N_a⁰= capp

cnan+cappNa (7.13)

N_d⁺= cd pp

cndn+cd ppNd (7.14)

Substituting the above expressions into Equ. 7.3 yields G= cnpnp+ cnacapNa

cnan+cappnp+ cndcd pNd

cndn+cd ppnp (7.15) This equation describes the balancing between the generation rateG and the recombination processes through those three channels. The first term on the right side represents the rate of the inter-band recombination. It linearly depends on the product of the electron and hole densities np. The last two deep-level recombination terms depend on np but with variable coefficients. Increasing electron and hole densities will decrease the strength of deep-level recombinations relative to the direct inter-band recombination.

The simulated evolution of the concentration ratio of the holes to electronsp/nwith increasing generation rateG is plotted in Fig. 7.8. Taking the nanowire sample for instance, under low excitation intensities (< 10²⁴ cm⁻³s⁻¹), the photogenerated electron and hole densities are negligible compared to that of the intrinsic conduction electrons due to the ionization of the shallow donors. Thus, the total electron densitynis much higher than the hole density p. The concentration ratio p/nis almost zero. According to Equ. 7.13 and 7.14, it has N_a⁰ ≈ 0 and N_d⁺ ≈ 0, indicating that the acceptor centers are almost completely ionized while the deep donor centers keep neutral. The Equ. 7.15 can be simplified to be

G≈cnpnp+capNap+cd pNdp (7.16) Because of the large base number ofn, it can be regarded as constant whilepincreases linearly with the generation rateG. Therefore, all the three terms in Equ. 7.16 for band-edge and deep-level recombination increase near-linearly withpand hence the generation rateG.

As the excitation intensity increases, more electron-hole pairs are generated. pincreases and cannot be ignored anymore compared to the electron densityn. When the generation rate for 98