Thermal and efficiency droop in InGaN/GaN light-emitting diodes: decoupling multiphysics effects using temperature-dependent RF measurements

Multiphysics processes such as recombination dynamics in the active region, carrier injection and transport, and internal heating may contribute to thermal and efficiency droop in InGaN/GaN light-emitting diodes (LEDs). However, an unambiguous methodology and characterization technique to decouple these processes under electrical injection and determine their individual roles in droop phenomena is lacking. In this work, we investigate thermal and efficiency droop in electrically injected single-quantum-well InGaN/GaN LEDs by decoupling the inherent radiative efficiency, injection efficiency, carrier transport, and thermal effects using a comprehensive rate equation approach and a temperature-dependent pulsed-RF measurement technique. Determination of the inherent recombination rates in the quantum well confirms efficiency droop at high current densities is caused by a combination of strong non-radiative recombination (with temperature dependence consistent with indirect Auger) and saturation of the radiative rate. The overall reduction of efficiency at elevated temperatures (thermal droop) results from carriers shifting from the radiative process to the non-radiative processes. The rate equation approach and temperature-dependent pulsed-RF measurement technique unambiguously gives access to the true recombination dynamics in the QW and is a useful methodology to study efficiency issues in III-nitride LEDs.


Injection efficiency of LEDs grown on different orientations of GaN
We have observed similar injection efficiency trends for MQW c-plane (0001) and m-plane (101 ̅ 0) LEDs, suggesting that the higher injection efficiency with increasing current density is not strongly orientation dependent. Figure S1 shows the injection efficiency as a function of current density for polar, semipolar, and nonpolar LEDs. The injection efficiency at low current densities Figure S1 The injection efficiency for MQW polar, semipolar, and nonpolar LEDs as a function of current density.
depends on epitaxial structure and material properties of the LED. For instance, the injection efficiency of the semipolar LED is the highest due to the presence of an electron blocking layer (EBL), which reduces the electron leakage. The injection efficiency of the nonpolar LED is lower than that of the polar LED because of the lower acceptor doping in the p-GaN layer of the nonpolar LED, leading to lower hole injection and consequently higher electron leakage.

Calculation of the injection efficiency and carrier density
To calculate the injection efficiency, Eq. (1) and (2) are considered in steady-state form. The injection efficiency is the current associated with recombination in the quantum-well (QW) ( = ) divided by the total current injected into the device. Therefore, the total injection efficiency is However, since the small-signal measurement yields the differential carrier lifetimes, the differential forms of Eq. (1) and (2) should be used instead to find the injection efficiency. The differential forms of Eq. (1) and (2) are: By direct analogy to Eq. (S1), and using Eq. (S2) and (S3) in steady-state, the differential injection efficiency is derived as The differential and total injection efficiencies are related through Eq. (S5).
The extracted lifetimes from the fittings yield the total and differential injection efficiencies.
To calculate the total carrier number in the QW, Eq. (S2) and (S3) are considered in steadystate. Solving Eq. (S2) and (S3) for leads to: The carrier density ( ) is then calculated by knowing the total carrier number using = , where and are the QW thickness and LED area, respectively.

Example of fitting
Equation (S7) Figure S4 shows an example of simultaneous fitting to the measured data of expressions for real part of the impedance, imaginary part of the impedance, and 20log of the modulation response. Table S1 shows an example of the estimated fitting parameters and the 95% confidence intervals calculated for those parameters. All the estimated parameters fall within their confidence intervals, ensuring robustness of the fittings. Figure S4 Simultaneous fitting to the measured data of expressions for real part of the impedance, imaginary part of the impedance, and 20log of the modulation response. Confidence intervals on the predicted data are very close to the predicted data, such that they are indistinguishable in the plots.
The impedance reduces with higher current densities, but in this case it is still relatively large due to the small area of the micro-LEDs, ensuring robust fittings as it is shown in Fig. S4 and Table   S1.  Figure S5 shows the net carrier escape time as a function of current density for different stage temperatures. The carrier escape time initially decreases with increasing current density but starts to increase at high injection levels and eventually becomes negative around a current density of 3-4 kA/cm 2 . The initial behavior of the carrier escape time is attributed to the carrier leakage from the QW. At high carrier injection levels, the population of carriers in the QW and cladding layers become similar, resulting in Coulomb-enhanced capture which leads to a negative escape rate.

Carrier escape time
Carrier escape time generally reduces with increasing stage temperature due to increase of thermionic emission. The effect of carrier escape time is folded into the injection efficiency of Fig.   3(b). Figure S5 The net carrier escape time as a function of current density for different stage temperatures. Closed symbols when carrier escape time is positive ( ∆ > ∆ ) and open symbols when carrier escape time is negative ( ∆ < ∆ ).

Circuit parameters
The fitting procedure in section 4 resulted in the extraction of the parameters in equations S7 and S8. Figure S6 shows , resistance associated with carriers in the cladding region, , resistance associated with carriers in the QW, , parasitic series resistance, , , resistance associated with carriers that recombine in the cladding region, , capacitance associated with carriers in the cladding region, and , capacitance associated with carriers in the QW as a function of current density. Detailed discussion of the behavior of the circuit parameters can be found in our previous works. 1,2 Figure S6 (a) , resistance associated with carriers in the cladding region, (b) , resistance associated with carriers in the QW, (c) , parasitic series resistance, (d) , , resistance associated with carriers that recombine in the cladding region, and (e) , capacitance associated with carriers in the cladding region, and (f) , capacitance associated with carriers in the QW.