1. Introduction

InGaN/GaN multi-quantum well (MQW) light-emitting diodes (LEDs) exhibit surprisingly high quantum efficiencies at room temperature (RT), despite the presence of a high density of threading dislocations in the films prepared by hetero-epitaxial growth, making them perfect candidates for general solid-state lighting applications^[1]. Unfortunately, GaN-based LEDs normally only hit their peak efficiency at relatively low current densities ($\sim $10 A/cm$^{2})$ and suffer from an emerging efficiency reduction problem at higher injection levels: the so-called efficiency droop phenomenon. Negative efficiency droop occurs not only in blue LEDs, but also in green and ultraviolet devices, which severely deteriorates the performance of the diodes in critical high-current applications^[2]. It is therefore essential to reveal the physical origin of this droop behavior. Many researchers from all over the world have performed intensive investigations into this critical issue, and several competing mechanisms have been proposed, including enhanced Auger recombination at high current densities, polarization field induced electron leakage, carrier delocalization from radiative recombination centers, and a shortage of holes due to inefficient ionization of the p-type dopant^[3-7]. However, to date, none of these models has been generally accepted, and many confusing and sometimes contradictory observations and explanations of droop behavior still exist^[8].

In this work, based on the fact that most of the structure and electronic defects in (In)GaN are generally optically deactivated at low temperatures, we present a study on the low-temperature light emission efficiency behaviors of the GaN-based blue LEDs^[9]. We qualitatively explain the observed high-current efficiency reduction behavior, and theoretically derive the inverse square root dependence between the light emission efficiency and the drive current. We suggest that the excess electron leakage into the p-type side of the LEDs may be the major cause for the efficiency droop behavior.

2. Experimental

The LED samples used in this study have an InGaN/GaN MQW structure grown on $c$-plane sapphire substrates using metal-organic chemical vapor deposition. Figure 1 shows the cross section of an InGaN/GaN MQW blue LED. The epistructure consists of a 2 $\mu $m GaN:Si n-contact layer ($N_{\rm D}$ $\approx$ --5 $\times $ 10$^{18}$ cm$^{-3})$, a ten-period of 3 nm undoped In$_{0.2}$Ga$_{0.8}$N and 7 nm GaN:Si MQW layer, a 70 nm p-AlGaN electron blocking layer (EBL), and a 0.2 $\mu $m GaN:Mg ($N_{\rm A}$ > 1 $\times $ 10$^{19}$ cm$^{-3})$ p-contact layer. LED chips with a mesa size of 300 $\times $ 300 $\mu $m$^{2}$ were fabricated using standard photolithography and dry etching processes. Annealed Ti/Al/Ti/Au and Ni/Au (2.5 nm/2.5 nm) multi-layers deposited by e-beam evaporation were employed as the ohmic contact and semi-transparent p-contact, respectively.

The current-dependent emission intensity (proportional to the radiant flux) was measured under pulsed operation, with the injection current ranging from $\sim $10$^{-6}$ to 0.1 A. With this pulsed driving test, the self-heating effect in the LEDs can be substantially suppressed. The peak emission wavelength of the LEDs is about 466 nm with a full-width-at-half-maximum of 36 nm. The emission efficiency is approximately denoted by the ratio of EL intensity $P$ to current $I$ in this work.

3. Results and discussion

Figure 2 shows the experimental light emission efficiencies as a function of drive current measured at different temperatures. At low temperatures, the emission efficiency of the LEDs displays a monotonous decrease with current, while with a further increase in temperature, the efficiency curve exhibits a typical roll-off behavior, with a maximum value at $\sim $10 A/cm$^{2}$. Low emission efficiency on the low-current side of the maxima can be due to enhanced nonradiative recombination at the extended crystal defects in the active layer. Its dependence on current (and thus carrier density) can be described by the Shockley--Read--Hall model $\sim Bn^{2}$/$(Bn^{2}+An)$, where $Bn^{2}$ and $An$ are the radiative and nonradiative recombination rates, respectively, and $n$ is the free carrier concentration^[10]. With increasing injection current, the emission efficiency of the LEDs is expected to quickly approach unity^[11]. The movement of the current at the efficiency maxima towards the high-current direction at elevated temperatures can be attributed to the increased concentration of positively charged holes in the p-side of the LEDs^[6]. The droop origin on the high-current side of the efficiency maxima has been a controversial subject. In the following, based on the idea that most of the structure and electronic defects in â…¢-nitride materials are generally optically deactivated at low temperatures, we study the low-temperature efficiency reduction behavior of the LEDs.

We consider first the forward current transport characteristics of the GaN-based LEDs, since the major recombination processes of the electron--hole pairs are closely associated with these charge injection processes. Figure 3 shows the $I$--$V$ curves measured at 60, 80 and 100 K, respectively. As can be seen, these curves have identical bias dependences but are almost insensitive to temperature, which indicates that the current is mainly due to tunneling rather than a recombination process. An empirical equation for an experimental $I$--$V$ relationship of a pn junction is usually given by

$ $$\begin{equation} J_{\rm F} \propto \exp \frac{qV}{nkT}, \end{equation}$$ $

(1)

where the ideality factor $n$ $=$ 2 when the recombination current dominates and $n$ $=$ 1 when the diffusion current dominates. However, when the tunneling current is significant, $n$ has a value much larger than 2. Depending on the bias level, two successive linearly dependent segments with different slopes can be observed in the low-and medium-bias regions. The corresponding ideality factors determined from the current slopes are 19.5 and 7.5 at 100 K, respectively. According to our previous work, the current in the low-bias region (segment a) might be dominated by electron tunneling through defect-induced intermediate states, while in the medium-bias region (segment b), the heavy holes can be the dominant tunneling entities^[12]. At higher current levels ($>$ 0.1 mA), the $I$--$V$ curves gradually bend downwards, which is caused by an increasing voltage drop on the series resistance of the diodes. Visible light emission can be detected even at temperatures as low as 10 K (not shown here), indicating that a complete carrier freeze-out does not occur in the diodes. Since the ionization energy of the p-type dopant: Mg ($\sim $150--200 meV) is considerably larger than that of the n-type dopant: Si ($\sim $20 meV) in GaN, the electrons are the excess carriers and the holes are the minority. Therefore, under high forward bias, where the conduction band on the n-side becomes relatively higher than the conduction band on the p-side, many electrons can have sufficient kinetic energy to fly over the quantum well and the EBL barriers into the p-side of the LEDs^[13].

Figure 4 schematically illustrates the individual current component inside the LEDs under the high forward-bias condition. From the illustration, we can readily deduced that, (1) an increase in drive bias will cause a greater fraction of electrons to escape from the MQW region into the p-side of the LEDs, where they will recombine non-radiatively with the holes, and (2) the radiative recombination current within the MQW layer equals the hole current $I_{\rm hole}$ injected from the p-side into the active layer. At high forward bias, $I_{\rm hole}$ is not as important as electron leakage current $I_{\rm Leak}$, and the emission efficiency of the LEDs can be represented by

$ $$\begin{equation} \eta = \frac{I_{\rm hole} }{I_{\rm Leak}}, \end{equation}$$ $

(2)

where $I_{\rm Leak}$ is the electron leakage current component. In addition, the relationship between $I_{\rm hole}$ and $I_{\rm Leak}$ can be empirically approximated by

$ $$\begin{equation} I_{\rm Leak} =CI_{\rm QW}^K, \end{equation}$$ $

(3)

where both $C$ and $K$ are the fitting parameters^[14]. It is believed that Equation (3) not only covers the carrier leakage by TE from the quantum wells, but can also be used for the fly-over carriers not captured by the quantum wells, or for the trap-assisted tunneling leakage^[15]. Then, substituting Eq. (3) into Eq. (2), yields

$ $$\begin{equation} \eta=\alpha I_{\rm Leak}^m, \end{equation}$$ $

(4)

where $\alpha =C^{-1/K}$, $m=1/K$ --1. This equation predicts that at a given low temperature, the emission efficiency and the electron leakage current of the LEDs should follow a power-law dependent relationship. Figure 4 shows the experimental result measured at 80 K. As observed, these data points can be nicely fitted with $m=$ $-0.5$ in the measured drive current range, which is consistent with the form of Eq. (4).

We shall now briefly address the low-temperature efficiency curve with $m =$ $-$0.5. The generation rate of minority carriers is extremely slow due to the wide bandgap nature of GaN, so the generation process may be discarded in the analysis. With low temperature and high forward bias conditions, the electron leakage current into the p-type region of the LEDs should approximately equal the recombination current $I_{\rm R}$ in the p-type side:

$ $$\begin{equation} I_{\rm Leak} =I_{\rm R} \propto Rn_{\rm e} p_{\rm h}, \end{equation}$$ $

(5)

where $R$ is the non-radiative recombination coefficient, $n_{\rm e}$ the concentration of electrons captured by the various traps and acceptors, and $p_{\rm h}$ the concentration of holes induced by the injected electrons. To satisfy the electro-neutrality condition in the p-type region, we have

$ $$\begin{equation} p_{\rm h} \approx n_{\rm e} . \end{equation}$$ $

(6)

Substituting Eq. (6) into Eq. (5) yields

$ $$\begin{equation} I_{\rm Leak} \propto p_{\rm h}^2 . \end{equation}$$ $

(7)

At high injection levels

$ $$\begin{equation} I_{\rm P} \propto p_{\rm h} \ll I_{\rm Leak} \approx I, \end{equation}$$ $

(8)

and combining with Eq. (7) gives

$ $$\begin{equation} \frac{P}{I}\propto I^{-0.5}. \end{equation}$$ $

(9)

Obviously, Equation (9) predicts an inverse square root dependent relationship between the emission efficiency and the drive current, which is in good agreement with the experimental result. This consistency supports the light emission efficiency droop behavior of the GaN-based LEDs at low-temperature being well explained by the electron leakage model, and electron leakage might be the essential cause of a significant efficiency reduction at high current levels.

Finally, it is worth noting that although the strong polarization field can effectively reduce the barrier to electron leakage from the quantum well and over the EBL, and Auger recombination can produce high-energy electrons far above the EBL conduction band-edge, enhancing the electron leakage behavior of the LEDs, neither of them can be the major cause of efficiency droop because obvious efficiency collapse behavior has been detected in the nonpolar $m$-plane samples and at extremely low current density (Auger recombination becomes significant only at high carrier densities), respectively. Furthermore, there is no obvious improvement in the droop performance of the LEDs on bulk GaN substrates, with the low-density defects not greatly improved compared with those of conventional $c$-plane chips, indicating that the carrier delocalization regime may not be the main cause of the droop either^[16].

4. Summary

The emission efficiency of GaN-based blue LEDs was measured and studied. The low-temperature efficiency curves show an inverse square root dependent relationship as a function of drive current. The forward-bias current characteristics suggest that the large electron injection into the p-side of the LEDs can suppress the effective injection of holes into the quantum wells, resulting in decreased emission efficiency. Based on the electron leakage model, the inverse square root dependent relationship between the emission efficiency and drive current is derived, and is in good agreement with the experimental observation. Electron leakage may be the real cause of efficiency droop in LEDs at low temperature.