Determination of recombination coefficients in InGaN quantum-well light-emitting diodes by small-signal time-resolved photoluminescence

While InGaN-based quantum well (QW) light emitting diodes (LEDs) can be considered a mature technology in the blue spectral region, they suffer from the "Droop", i.e., the reduction of efficiency with increasing drive current,^1,2) which has been discussed widely in literature and has been attributed to Auger recombination^3–5) experimentally. In addition, an extension of the emission wavelength into the "Green Gap"^1,6) has proven to be challenging. The origins of both phenomena are continuously debated and their detailed understanding requires deeper insight into the dominating loss mechanisms. For this purpose, the determination of the recombination coefficients corresponding to the principal channels, i.e., Shockley–Read–Hall (SRH), radiative, and Auger recombination is of utmost importance. Naturally, also a detailed analysis of their dependences on the LED structure design, InGaN composition, and temperature is quite desirable. First studies carried out in this direction^7–10) have reported on the recombination coefficients considered in terms of the commonly used ABC-model^2,11) and obtained by measuring the differential carrier lifetime (DLT) of non-equilibrium carriers by modulation of the LED operating current. Unfortunately, the recombination coefficients reported in the above studies poorly correlate with each other, which can be partly attributed to different experimental setups and to difficulties in the data interpretation for time-resolved current-modulation experiments (see Sect. 3.3).

In this paper, we suggest a novel technique for the determination of the recombination coefficients combining both, electrical and optical excitation of the LED active region. Our technique has certain advantages if compared to previously applied experimental approaches, since it avoids transient modifications of the LED band diagram, which interfere with the DLT measurements.

The so-called ABC-model is commonly used for interpreting the LED efficiency dependence upon variation of the operating current.^2,11) In this model, assuming an injection efficiency of 100% (i.e., in the absence of electron and hole leakage from the active region), the external quantum efficiency (EQE) of an LED is given by

$$\begin{equation} \mathrm{EQE} = \eta \frac{Bn}{A + Bn + Cn^{2}}, \end{equation} \tag{ 2.1 }$$

where n is the non-equilibrium carrier density assumed to be equal for electrons and holes, η is the light extraction efficiency, and A, B, and C are the SRH, radiative, and Auger recombination coefficients, respectively. Since EQE is usually derived from electroluminescence (EL) as a function of injection current, the number of degrees of freedom in Eq. (2.1) is too large for a meaningful determination of the recombination coefficients from the EL data.

The number of variables can be substantially reduced, if the equation is rewritten in terms of the normalized optical output power p, being the ratio between the LED output power P_out and the power P_m corresponding to EQE_max, the maximum of the measured EQE. In this case,

$$\begin{equation} \mathrm{EQE} = \eta \frac{Q}{Q + p^{1/2} + p^{-1/2}}. \end{equation} \tag{ 2.2 }$$

Here, Q = B/(AC)^1/2 is the so-called quality factor¹¹⁾ which is a dimensionless combination of the recombination coefficients relevant to the maximum value of internal quantum efficiency IQE_max = EQE_max/η = Q/(Q + 2). It is important that both values, EQE_max and P_m, can be derived directly from the measured EQE dependence on the LED operating current.

In order to evaluate Q, the EQE does not need to be measured in absolute units (e.g., in an Ulbricht sphere), which simplifies the measurements considerably. One can consider the ratio EQE_max/EQE depending on the normalized optical power p:

$$\begin{equation} \frac{\mathrm{EQE}_{\text{max}}}{\mathrm{EQE}} = \frac{Q + p^{1/2} + p^{-1/2}}{Q + 2}, \end{equation} \tag{ 2.3 }$$

which allows finding the Q-factor and, hence, the IQE_max value.¹²⁾ Additionally, the light extraction efficiency η could be obtained from EQE_max/IQE_max, if the EQE measurement was performed in absolute units instead.

In addition to the quality factor Q, the parameter P_m can also be expressed in terms of the recombination parameters: P_m = E_phηV_rAB/C, where E_ph is the energy of photons averaged over the LED emission spectrum and V_r is the recombination volume, which is the product of the active region area S and the effective active region width d. As P_m and E_ph are known from the EL measurements, η is evaluated by the procedure discussed above, and V_r is estimated from structural parameters. Therefore, the combination of the recombination coefficients AB/C = P_m/(E_phηV_r) can also be found from the EL data.

The above two combinations are insufficient for the unambiguous determination of the recombination coefficients. However, their determination becomes possible, if the measured DLT of non-equilibrium carriers is additionally involved. Starting from the rate equation for the unsteady carrier density N = n + δn(t)

$$\begin{equation} \frac{dN}{dt} = j - AN - BN^{2} - CN^{3}, \end{equation} \tag{ 2.4 }$$

and considering δn(t) to be a small perturbation of the steady-state carrier density n, one can show that the perturbation decays mono-exponentially with the DLT τ = (A + 2Bn + 3Cn²)⁻¹. In terms of the normalized optical power p the DLT can be expressed by

$$\begin{equation} \tau = \frac{A^{-1}}{1 + 2Qp^{1/2} + 3p}. \end{equation} \tag{ 2.5 }$$

Equation (2.5) shows that the measurement of the DLT corresponding to a certain steady-state non-equilibrium carrier density n (equivalent to a certain p) enables the direct evaluation of the SRH recombination coefficient A.

With Q and A known, the coefficients B and C could be calculated, assuming that the active volume V_r is known. Generally this is not the case, because of the uncertainty in the estimation of the effective active region width d for multiple quantum wells¹¹⁾ (MQW). Hence, we will use the corresponding sheet parameters: B_2D = B/d and C_2D = C/d² instead. For the sheet recombination coefficients, the quality factor has the same form: Q = B_2D/(AC_2D)^1/2. Another combination of the recombination coefficients, AB_2D/C_2D = P_m/(E_phηS) is now dependent on the active region area S which can be estimated unambiguously and is nearly equal to the LED chip area. Using both expressions, one can express the coefficients B_2D and C_2D via coefficient A and other experimentally determined parameters as follows:

$$\begin{align} B_{\text{2D}} &= A^{2}Q(2 + Q)\left(\frac{qS}{I_{\text{max}}}\right), \\ C_{\text{2D}} &= A^{3}(2 + Q)^{2}\left(\frac{qS}{I_{\text{max}}}\right)^{2}. \end{align} \tag{ 2.6 }$$

Here, q is the elementary charge and I_max the LED operating current corresponding to EQE_max, which is also found from the EL data without requiring any measurement in absolute units.

As described above, only two spectroscopic measurements are necessary in order to obtain the recombination coefficients. Firstly, an EL measurement as a function of drive current in arbitrary units and secondly, one, preferably more, differential lifetime measurement(s) within this current range.

3.1. Samples

3.2. Electroluminescence

3.3. Differential carrier lifetime

3.4. Small-signal time-resolved photoluminescence

3.5. Evaluation

The EL measurement is used in two ways. First, it proves that the sample under investigation can be treated by the ABC-model and that the currents applied in the DLT measurement do not cause significant internal heating, which would otherwise result in a reduction of the efficiency observed at high currents. Second, as described in Sect. 2, the measurement allows to determine the dimensionless quality factor Q. Figure 1 shows the result of this evaluation for a single quantum well LED at a temperature of 350 K. Applying Eq. (2.3) yields a quality factor of 4 corresponding to a peak IQE of 67%. With this Q-factor, the ABC-model reproduces the LED efficiency behaviour over the entire current range.

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In order to determine the entire set of recombination coefficients of the ABC-model, SSTRPL measurements at various drive currents above the current of peak efficiency were performed. This choice has been made on the basis of signal-to-noise ratio requirements, but generally the measurement can be performed at any current, which is approved by previous EL measurements to be within the ABC-model envelope. Once the raw SSTRPL transient decays have been treated in the manner described in Sect. 3.4, they can be fitted by a mono-exponential decay as detailed in Sect. 2. Generally, if I₀ is the peak intensity of a mono-exponential decay, the transient decay should have the form $I(t) = I_{0}e^{ - \tfrac{t}{\tau }}$, where τ is the decay constant. In order to account for the remaining carriers from the previous excitation pulses, which are separated by the repetition time T from each other, this equation becomes

$$\begin{equation*} I(t) = I_{0}\sum_{n = 0}^{\infty}\exp\left(-\frac{t + nT}{\tau}\right) = I_{0}\cfrac{\exp(-t/\tau)}{1-\exp(-T/\tau)}. \end{equation*}$$

Adding the constant EL background originating from the current injection leads to the final expression for the SSTRPL fitting function:

$$\begin{equation} I(t) = EL + I_{0}\frac{\exp(-t/\tau)}{1 -\exp(-T/\tau)}, \end{equation} \tag{ 3.1 }$$

which is used to determine the DLT (τ). Figure 2 (right) shows an exemplary transient decay recorded for a SQW at a temperature of 350 K with an injection current of 148 mA, fitted with Eq. (3.1). While in principle such a measurement allows the extraction of the recombination coefficient A, the accuracy of the estimate can be improved significantly by varying the drive current and repeating this measurement. This leads to the dependence depicted by data points in Fig. 2 (left). The solid line is a fit of these data points according to Eq. (2.5) and yields a SRH recombination coefficient of A = (4.46 ± 0.04) × 10⁶ s⁻¹.

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If, as it is the case here, the current at which the device efficiency is maximal (I_max) and the active area S is known, the sheet recombination coefficients B_2D and C_2D can now be evaluated using Eq. (2.6). This yields the values B_2D = (2.31 ± 0.04) × 10⁻⁵ cm⁻² s⁻¹ and C_2D = (7.3 ± 0.2) × 10⁻¹⁸ cm⁻⁴ s⁻¹, which are well comparable to values found in literature.¹⁰⁾

We have performed the measurements lined out in the previous section for various samples with one, three and five QWs in the active region, respectively. The results are summarized in Fig. 3, where multiple data points with the same number of QWs correspond to different chips manufactured from the same wafer. In particular, the quality factor (top left) increases drastically if the active region consists of multiple QWs. This is mirrored by a significant decrease in the A coefficient (bottom left). Both, Q-factor and A coefficient do not vary strongly if structures with three and five QWs are compared. In contrast, both the B and C sheet coefficients decrease continuously. We note that for a single QW with a nominal thickness of 3 nm the corresponding bulk coefficients B and C are of the order 10⁻¹¹ cm⁻³ s⁻¹ and 10⁻³⁰ cm⁻⁶ s⁻¹, respectively. While the sheet coefficients decrease with increased number of QWs, the corresponding bulk coefficients (assuming a nominally linearly increasing active volume) increase.

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We note that the scatter of the obtained values for nominally identical chips is rather large. This is especially notable, because individual chips were not only taken from the same wafer, but also from adjacent positions. Such large scatter is not uncommon¹⁰⁾ and suggests a relatively large inhomogeneity in the material. The emission spectrum however, is basically identical. Therefore the large difference in recombination properties arises from local differences in microscopic properties (e.g., point defect density, threading dislocation density) from chip to chip, rather than from more global properties such as QW composition and thickness, which may vary across the wafer and would cause varying EL spectra. This material inhomogeneity may originate from non-optimal growth-conditions of the LED structures studied here, which is also suggested by the relatively low quality factors compared to commercial blue LEDs, which show up to Q = 8 at 350 K.¹²⁾

The reduction of the SRH recombination parameter with QW number (M) is consistent with the assumption that all devices contain one highly defective QW¹⁶⁾ and M − 1 less defective QWs with a defect density ratio of approximately $7:1$. The QW with a higher defect density is most likely the top-most QW, which suffers from an increased point defect incorporation due to Mg back diffusion during p-GaN growth.¹⁶⁾

The fact that the radiative and Auger sheet recombination coefficients do not scale with M⁻¹ and M⁻² respectively, strongly suggests that the effective active region width does not increase linearly with the number of QWs. Therefore not all of the QWs are efficiently pumped by the applied bias.^17–19) As a result, any further addition of QWs does neither increase the quality factor (Fig. 3) nor shift the maximum of efficiency to higher currents. In this particular case, increasing the number of QWs beyond three has no positive effects on the overall efficiency of the LED.

We have demonstrated a novel technique that allows to directly measure the recombination coefficients corresponding to SRH, radiative, and Auger recombination in InGaN/GaN (multi) QW LEDs. This is achieved without requiring EQE measurements in absolute units and relies on the novel small-signal time-resolved photoluminescence (SSTRPL) approach. We have tested this method on a large quantity of SQW and MQW devices, where the determination of the recombination coefficients allows to pin the reduction in efficiency in SQW devices on an increased point defect density. In addition the lack of efficiency improvement from three to five QWs is shown to be caused by inhomogeneous pumping of the QWs. The suggested technique can be used for all devices (and in all operating regimes), which conform to the standard ABC-model. Therefore, SSTRPL will enable a deeper understanding of the underlying physical processes that limit device efficiency, providing a promising tool to analyse device shortcomings in the green spectral region.

We gratefully acknowledge the financial support of the European Union FP7-ICT Project NEWLED, No. FP7-318388, and the German Science Foundation within the Collaborative Research Center 787 (CRC 787).