Analytical models of electron leakage currents in gallium nitride-based laser diodes and light-emitting diodes: supplement

The electrical-to-optical power conversion efficiencies of the light-emitting devices based on gallium nitride (GaN) are seriously limited by electron leakage currents due to the relatively low mobility and activation ratio of holes. However, there have been few theoretical models on the behavior of the leakage current with an increasing total current. We develop an Ohmic-law-like method to describe the transport behaviors of the systems with electron and hole currents simultaneously. Based on reasonable assumptions, the ratio of the leakage current to the total current is related to the differential resistances of the devices. Through the method, we develop analytical models of the leakage currents in GaN-based laser diodes (LDs) and light-emitting diodes (LEDs). The ratios of the leakage currents with total currents in LDs and LEDs are shown to increase, which explains the sublinear behaviors of the luminescence-current (LI) curves of the devices. The theory agrees well with the numerical simulation and experimental results in larger current ranges in comparison to the traditional ABC model. The above analytical model can be used to fast evaluate the leakage currents in GaN-based LDs and LEDs.

This is due to a limited length of the p area, compared to the diffusion length of electrons [S1], as well 28 as the exponential nature of the Boltzmann distribution. The former determines that the carrier 29 concentrations do not change violently within a single layer due to recombination while the latter 30 determines that the Fermi levels change gently with the carrier concentrations.

The transmittance of the EBL 52
According to the thermal emission theory [S2], when the electrons flow to a heterojunction barrier, only those with an energy higher than the barrier can cross. From the theory of semiconductors [S2], the 54 electron concentration at the well side of the heterojunction is exp And by the thermal 55 emission theory, the concentration at the barrier side is exp as if the Fermi level is the 56 same as the well side [S3,S4], as illustrated in Fig. S3. Here c N is the effective density of states and

57
we have ignored the difference in it between the barrier and the well for simplicity. Hence, the 58 transmittance of the electrons through an EBL with a conductive band difference Notice that a constant transmittance also guarantees a proportion between the electron concentrations 62 of the waveguide layer and the cladding layer in the LED case, since in some cases there are also p-63 AlGaN layers in LEDs. Thus, in the LED case, our method of treating the electron resistance of the whole 64 p area inversely proportional to the electron concentrations from the beginning of the p area is reasonable.

Drift-diffusion model in LDs 70
As is mentioned above, the low-energy electrons driven by the forward electric field in the upper 71 waveguide layer will be blocked and bounced back by the EBL, gaining a reverse velocity against the 72 electric field as illustrated in Fig. S4. Eventually, they will slow down and accumulate near the 73 heterojunction and form a spatially descending concentration distribution, macroscopically causing a 74 reverse diffusion current against the electric field and increasing the effective electron resistance of the 75 waveguide layer. Let the coordinate origin be at the heterojunction between the waveguide layer and the 76 EBL, x be the distance from the origin and the increasing direction be the direction of the current. Notice that n j must be equal everywhere in the whole p area to satisfy a current continuity when the 93 recombination is negligible outside the active region. Thus, at a given n j , Eq. (S5) and (S6) are enough 94 for a problem of boundary condition, whose solution is: The exponential behavior of Eq. (S7) explains the simulation results in Fig. 5(c) in the main text. The 98 carrier concentration is proportional to the leakage current. Therefore, the reverse diffusion current is 99 proportional to the leakage current assuming an exponential form of the electron distribution. Larger T P 100 will lead to a larger reverse diffusion current as well. The characteristic length λ /

Recombination in the upper waveguide layer 108
One of the possible problems of placing the EBL after the waveguide layer is that a relatively high 109 concentration of electrons may recombine with holes in the waveguide layer. Fig. S5 Shows (a) the 110 radiative recombination rate and (b) the electron current density of the p area of a typical LD structure in 111 the main text at a current density of 20kA/cm 2 . It is obvious that the recombination in the waveguide 112 layer is negligible compared to that in the quantum well.

Energy levels in the quantum well 118
In the numerical simulation, the Poisson-Schrodinger equation is solved self-consistently for the 119 quantum well. Fig. S6 shows a band diagram and the wave function of several sub-band energy levels in 120 the quantum well near the turn-on voltage. Although there are two electron sub-bands in the quantum 121 well, only the lower one dominates in luminescence due to the thermal distribution. The selection rule 122 decides that the main inter-band transition occurs between the first conductive sub-band and the first valance sub-band. It is shown in Fig. S6 that at the turn-on voltage, fn is about a hundred meV 124 and will be larger when the current increases, much larger than B k T  26meV as is mentioned in the 125 main text. Thus, we apply the approximation (also in the main text) since the exponential term is much 126 larger than one: The approximation is used in the main text.

Self-heating effect in the LEDs 144
To study the leakage current effect on the efficiency droop phenomenon of LEDs, the self-heating 145 effect must be excluded. We apply up-to-500mA (2.5kA/cm 2 ) pulsed current with a 10μs pulse width 146 and a 30Hz frequency on the LED sample and measure its luminescence intensity. As a reference, a 147 continuous current is applied afterward. The duty ratios of the pulses are 0.03% and 100% 148 respectively. The results are shown in Fig. S8. The results show no significant difference between 149 the pulse condition and the continuous condition. Therefore, the self-heating effect in our 150 experiments is negligible. This may be due to the small sizes of the LEDs.