The carrier recombination rate (*R*) is defined as the number of carriers recombined per unit time and unit volume. Based on the ABC efficiency model [31–33], the *R* is mainly composed of Shockley-Read-Hall (SRH) recombination, radiative recombination, and Auger recombination, which are proportional to the first, second, and third power of the carrier concentration, respectively [34–36]. The carrier generation rate (*G*) is defined as the number of carriers generated per unit time and unit volume. Under equilibrium conditions, the carrier generation rate in the active region of a µ-LED is approximately equal to the recombination rate in it, expressed by the following formula [35, 37]:

$$G=R=An+Bnp+C\left( {{n^2}p+n{p^2}} \right)$$

1

where *A*, *B*, and *C* represent SRH recombination coefficient, radiative recombination coefficient, and Auger recombination coefficient, respectively, *n* represents electron concentration, and *p* represents hole concentration.

Under high current density, the excess carriers dominate, while the excess electron concentration balances with the excess hole concentration [24, 38–41]. When a high-frequency small-amplitude signal is injected at high current density, the increase in electron concentration is equal to that of hole concentration, and the increase in carrier concentration is much smaller than the carrier concentration at direct current (DC) bias. Therefore, the relationship between the increment of the carrier generation rate (*ΔG*) and the increment of the carrier concentration is (*Δn*) as follows:

$$\begin{gathered} \Delta G=A\left( {n+\Delta n} \right)+B\left( {n+\Delta n} \right)\left( {p+\Delta n} \right) \\ +C\left( {{{\left( {n+\Delta n} \right)}^2}\left( {p+\Delta n} \right)+\left( {n+\Delta n} \right){{\left( {p+\Delta n} \right)}^2}} \right) \\ - \left( {An+Bnp+C\left( {{n^2}p+n{p^2}} \right)} \right) \\ \approx A\Delta n+B\left( {n+p} \right)\Delta n+C\left( {{n^2}+{p^2}+4np} \right)\Delta n \\ \end{gathered}$$

2

The differential carrier lifetime (τ) can be obtained from the derivative of the carrier generation rate with respect to the carrier concentration, expressed by the following formula [12, 42]:

$$\frac{1}{\tau }=\frac{{\Delta G}}{{\Delta n}}=A+B\left( {n+p} \right)+C\left( {{n^2}+{p^2}+4np} \right)$$

3

In the frequency response of µ-LEDs, the − 3 dB modulation bandwidth (*f**− 3dB*) is defined as the corresponding frequency when the normalized power drops to half of the maximum value. Generally, the differential carrier lifetime has a relationship with the 3dB modulation bandwidth of the LED as follows [12, 25]:

$${f_{ - 3dB}}=\frac{1}{{2\pi \tau }}$$

4

In the physical model of simulation, band offset, internal loss, and the SRH recombination lifetimes are set to 70:30, 2000 m− 1, and 200 ns, respectively. Besides, the Auger recombination coefficient is set to 3 × 10− 31 cm− 6/s [34, 43, 44]. Built-in polarizations ranging from 20–80% of theoretical predictions have been reported, and 50% are chosen for simulation in this study [34, 45, 46]. Other physical parameters can be found in references [47].

The structures of the µ-LEDs in this work are shown in Fig. 1. There is a layer of 10 µm thick sapphire substrate at the bottom, followed by a 3 µm thick GaN layer with an n-type doping concentration of 5 × 1018 cm− 3 and a three-period GaN/InGaN multiple quantum well (MQW) layer. The thickness of QBs is 10 nm, where the n-type doping concentration is 3 × 1017 cm− 3. The indium content in QWs is set to 20% to ensure a blue light emission. On the top of the active area, there is a 20 nm thick Al0.23Ga0.77N as an electron blocking layer (EBL) and a 50 nm thick GaN as a cladding layer. The hole concentration levels for the p-EBL and the p-GaN are set to 1.2 × 1018 cm− 3 [34]. The ohmic contact on the cladding layer is defined as the p-electrode of the µ-LED and that on the n-type GaN layer is defined as the n-electrode of the µ-LED.

To study the effect of QWs on the modulation bandwidth, µ-LEDs with two different QW structures have been designed, represented by µ-LED A and µ-LED B in Fig. 1. The size of the µ-LED is defined as 20 µm × 20 µm, making the influence of the RC time constant negligible. For LED A, the thickness of one QW was 3 nm with an indium composition of 0.2. For LED B, the thicknesses of the falling side, bottom, and rising side of one QW are 0.5 nm, 2 nm, and 0.5 nm, respectively, with an indium composition ranging from 0 to 0.2. This design enables the two µ-LEDs with the same QW thickness, supporting the subsequent comparative analysis.