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Abstract
Most optoelectronic devices share the same basic epitaxial structure – a stack of quantum wells (QWs) sandwiched between p- and n-doped layers. In nitride semiconductors, where holes have 20-times lower mobility than electrons, the holes are able to populate only the topmost 1–2 QWs. The inability to distribute the holes in a large-enough number of QWs is a cause of high Auger recombination in nitride LEDs. Lateral carrier injection is an alternative design, in which the doped regions are situated at the sides of the QW stack and the carriers diffuse horizontally into the QWs. Given that the carriers are injected into all available QWs, it finally makes sense to grow structures with a large number of QWs. We report the results of our computer simulations, which explore the advantages of LCI-based LEDs in terms of energy efficiency.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Nitride semiconductors have become one of today’s most important materials in optoelectronics. The emitted wavelength can be varied within a wide spectrum, ranging from visible light into the ultraviolet, which other semiconductors cannot address in an energy-efficient way. Today, nitride-based optoelectronic devices, such as light-emitting diodes (LEDs) and laser diodes (LDs), are most typically based on a common epitaxial structure, schematized in Fig. 1(a), which consists of a stack of alternating quantum wells (QWs) and barrier layers, itself sandwiched between a $p$-doped layer and an $n$-doped layer to form a $p$–$n$ junction.
Fig. 1. Schematics of the epitaxial structures of nitride optoelectronic devices with the conventional design (a) and the LCI design (b).
This structure has a flaw, which happens to be particularly serious for nitride semiconductors. The carriers of each type (electrons and holes) must diffuse from one QW to the next sequentially, yet the barrier layers between QWs obstruct diffusion with their wider bandgap. This is not a problem for electrons as their high mobility lets them pass across many barriers with little effort. In nitride semiconductors, however, the mobility of the holes is about 20 times lower than that of the electrons. For this reason, most holes recombine in the QW that is closest to the $p$-doped layer before they have the chance of diffusing any further. According to experimental evidence, only about 20% of the holes make it to the second-closest QW, and virtually none reaches the third [1]. This severely limits the number of QWs that makes sense having in the device’s epitaxial structure. If further QWs cannot be reached by the holes, no recombination can take place in them.
However, increasing the number of QWs can be useful. The efficiency of nitride LEDs at high currents is often limited by the onset of Auger recombination. To reduce the concentration of carriers in the QWs, and thus minimize Auger recombination, the cumulative volume of the QWs (also known as active volume) needs to be increased. A large active volume can be useful in LD devices, too. Nitride LDs often have the problem that they cannot achieve lasing because the gain saturates at a smaller value than is needed to compensate optical losses. In theory, one can increase the active volume in different ways, but each way poses problems. Increasing the thickness of the QWs is problematic because of the small critical thickness [2] and the high piezoelectric fields caused by mechanical stress in strained nitride layers [3]. Increasing the QW size laterally requires bigger chip sizes, which might be unacceptable due to other requirements, such as cost, yield, étendue or resonator geometry constraints. Finally, increasing the number of QWs is ineffective due to the limited hole mobility in nitride semiconductors, as discussed above.
The limitation in the number of QWs arises from the fact that the conventional structure in Fig. 1(a) strives to be one-dimensional, in the sense that all layers are stacked on top of each other. This simplifies the growth process because it can be done in a single epitaxial run. However, more elaborate structures can be devised to circumvent the problem. In this article, we analyse a two-dimensional structure where a vertical stack of QWs is placed in a horizontally oriented $p$–$n$ junction, so that the carriers are injected into the QWs from the sides (Fig. 1(b)). This concept, known as lateral carrier injection (LCI), allows a large numbers of QWs to be populated with both holes and electrons.
Even if LCI improves the distribution of the holes among the QWs, it still introduces a new problem, in that a uniform distribution of the holes within each individual QW is not guaranteed. The holes injected from the $p$-doped region diffuse along the QW until they meet the electrons coming from the $n$-doped region, whereupon they recombine. Given that the electrons are much more mobile that the holes, recombination will take place preferably next to the $p$-doped region. So it seems that the LCI design is unable to take advantage of the entire volume provided the QW stack, just like the conventional design. However, as we will demonstrate shortly, the holes are able to diffuse a few micrometres along the QWs of the LCI structure before they eventually recombine. If this diffusion length is compared to the 5–10 nm (corresponding to the cumulative thickness of 2–3 typical InGaN QWs) of the conventional structure shown in Fig. 1(a), the comparison is merciless. On top of this, at least in the case of LEDs, the LCI design allows the $p$- and $n$-doped regions to be contacted by a metal line on the surface of the wafer, and therefore multiple $p$–$n$ junctions can easily be placed side-by-side (as shown in Fig. 2) to form stripes filling the whole chip surface. LCI can therefore provide a large active volume on a chip of limited surface area, in a way that the conventional design could only match by making extensive use of tunnel junctions [4] or resorting to optical pumping [5].
Fig. 2. Scheme of an LED based on the LCI design, seen from above and through a section. The LCI structure is wound into meanders to cover the whole chip surface.
LCI structures have been employed since 1987 in GaAs and GaInAsP VCSELs that could easily be integrated into optoelectronic integrated circuits [6–10], proving that carriers could be injected laterally into thin QWs. Lateral injection of electrons (but not holes) was applied to nitride LEDs in 2009 with the purpose of improving current spreading [11]. In 2012, LCI was then proposed as a solution to the hole distribution problem in a stack of several nitride QWs [12]. In this article, we finally present the results of computer simulations aimed at proving that LCI can indeed be a viable, theoretically valid solution.
2. Simulator setup
In this article we present the results of finite-element simulations on LED structures based on the drift–diffusion model that prove that the LCI design can indeed theoretically improve LED efficiency. The computer software used is DDCC-2D, which is being developed by the Optoelectronic Device Simulation Laboratory of the National Taiwan University [13]. The LCI structure used in the simulations (shown in Fig. 2) contains a variable number $N$ of 2 nm-thick green-emitting In$_{0.24}$Ga$_{0.76}$N QWs separated by 5 nm GaN barriers. It was chosen to simulate green-emitting QWs so that the advantages of the LCI design will be more evident. The QW stack has variable lateral extent $\ell$, defined as the distance between the $p$- and $n$-doped regions. Above the QW stack, a 25 nm GaN layer is used as a cap. Beneath the QWs, a 95-nm In$_{0.04}$Ga$_{0.96}$N underlayer is included, which is believed necessary to minimize the density of point defects in the above InGaN layers when the structure is grown by metal-organic vapour-phase epitaxy (MOVPE) [14]. The $p$- and $n$-doped regions have a lateral extent of 0.5 ${\mathrm{\mu}}$m each and a dopant concentration of $5\times {10}^{18}$ cm$^{-3}$ (Mg and Si are assumed as dopant species). Conventional LED structures, which were simulated for comparison purposes, include exactly the same QW stack, except that it is sandwiched between $p$- and $n$-doped layers. The first and last barriers of the QW stack are doped as well, as this proved more effective at reducing carrier overflow in our simulations than the addition of any type of electron-blocking layer (EBL).
The full set of simulations parameters are listed in Table 1. Most of the parameters are the simulator’s defaults, with the exception of the carriers’ mobility and the recombination-rate coefficients $A$, $B$ and $C$. The $A$, $B$ and $C$ coefficients were chosen in such a way that the simulation of a conventional non-LCI structure with one single QW would reproduce similar efficiency and differential-lifetime curves as in the real-life green ($\lambda =500$ nm) LEDs presented in [15]. This choice guarantees that their values are at least credible, whereas the simulator’s defaults would not (as they do not even depend on the indium content).
Table 1. The material parameters for each material used our simulations. Most parameters are the simulator’s defaults with the exception of the carriers’ mobility and the recombination-rate coefficients $A$, $B$ and $C$.
Electron and hole mobility are parameters worth discussing in detail, because the results presented in this article depend heavily upon their values. The carrier mobility $\mu$ is linked to the carriers’ effective mass $m^{*}$ and the average scattering time $\tau$ by the formula $\mu =q\tau /m^{*}$ [16]. The scattering time depends heavily on ionized-impurity scattering caused by dopant species, as electrically charged impurities have a large scattering cross section, but it also depends on alloy scattering in the case of ternary alloys such as InGaN. The dependence of both carriers’ mobility upon the doping level in GaN has already been researched in depth by others. In particular, Mnatsakonov et al. [17] have shown that electron mobility can be as high as 1000 cm$^{2}$ V$^{-1}$ s$^{-1}$ for $\textrm {[Si]}<10^{16}$ cm$^{-3}$, but it decreases rapidly at higher doping levels, reaching 100 cm$^{2}$ V$^{-1}$ s$^{-1}$ for $\textrm {[Si]}=5\times 10^{18}$ cm$^{-3}$. Hole mobility follows a similar trend, starting at almost 200 cm$^{2}$ V$^{-1}$ s$^{-1}$ for $\textrm {[Mg]}<10^{16}$ cm$^{-3}$, and decreasing to 4 cm$^{2}$ V$^{-1}$ s$^{-1}$ for $\textrm {[Mg]}=5\times 10^{18}$ cm$^{-3}$. Our own data are similar, in that we measure an electron mobility around 150 cm$^{2}$ V$^{-1}$ s$^{-1}$ for $\textrm {[Si]}=5\times 10^{18}$ cm$^{-3}$, and a hole mobility around 8 cm$^{2}$ V$^{-1}$ s$^{-1}$ for $\textrm {[Mg]}=5\times 10^{18}$ cm$^{-3}$. We ended up using our values in the simulations.
In ternary alloys, however, alloy scattering can also have a significant effect on the scattering time. Sohi et al. [18] have researched electron mobility in high-electron-mobility transistors (HEMTs) with In$_x$Ga$_{1-x}$N channel layers of variable indium content $x$, and found that the electron mobility decreases to 600 cm$^{2}$ V$^{-1}$ s$^{-1}$ for $x=3$%, and decreases further to about 150 cm$^{2}$ V$^{-1}$ s$^{-1}$ for $x=20$%. However, given the large variation in alloy homogeneity reported for InGaN layers and our scepticism for such high values, we decided to stay on the safe side and select the much smaller value of 20 cm$^{2}$ V$^{-1}$ s$^{-1}$ for our In$_{0.24}$Ga$_{0.76}$N QWs. As for the holes, presuming that electrically neutral scattering centres such as indium have a comparable scattering cross section for both electrons and holes, we assumed their loss of mobility to be about the same as for the electrons. Therefore, we kept the constant ratio of 20 between electron and hole mobility for all the materials used in our simulations.
3. Discussion: optimal lateral extent
A first series of simulations was carried out to determine the optimal lateral extent $\ell$ of the QWs. The simulated structures have $N=20$ QWs with $\ell$ varying from 0.5 to 6 $\mathrm {mu}$m. To obtain results that we can meaningfully compare, the voltage bias applied in the simulation was set so as to get a current equal to 35 A cm$^{-2}\,(\ell +1\,\mathrm{\mu}\textrm {m})$, which corresponds to 0.35 A for a 1 mm$^{2}$ LED chip as in Fig. 2. The larger the QW lateral extent is, the higher the applied current must be to compensate the fact that we can fit fewer stripes on the chip’s surface. Fig. 3 shows the density plot of the radiative-recombination rate (which is linked to the density of carriers) in the QW stack for $\ell =1$, 3 and 5 $\mathrm{\mu}$m. The density of holes is always higher next to the $p$-doped region because the holes have a shorter diffusion length than the electrons. For this reason, the radiative-recombination rate decreases as we move towards the $n$-doped region. Moreover, it can be noticed that radiative recombination can sometimes be stronger in the topmost QW, because this QW is supplied with carriers not only from the side but also from the cap layer above. The same would be true for the bottommost QW if the InGaN underlayer did not trap most of the carriers coming from beneath.
Fig. 3. Density plot of the radiative-recombination rate (or, equivalently, the photon-generation rate) in LCI-based LEDs with 20 QWs of lateral extent $\ell =1$, 3 and 5 $\mathrm{\mu}$m, at the equivalent current density of 35 A cm$^{2}$. Each visible ’slice’ corresponds to one of the QWs.
Table 2 shows a summary of the simulation results for each value of $\ell$. If we define the distribution homogeneity as the ratio between the radiative-recombination rate next to the $p$- and $n$-doped layer, we obtain the values in Table 2’s last column. We consider any value above 50%, such as in the case of $\ell =3$ $\mathrm{\mu}$m , to be good for our purposes. The lateral extent of 3 $\mathrm{\mu}$m is also definitely within the capabilities of most photolithographic setups used in optoelectronics laboratories.
Table 2. Performance indices of the simulated LCI-based LEDs with 20 QWs of varying lateral extent $\ell$, at the equivalent current density of 35 A cm$^{-2}$.
Nonetheless, the diffusion length of the holes must be considered a function of current, so the optimal lateral extent ultimately depends on the operative conditions of the device. In the case of LDs, the optimal lateral extent is definitely smaller than in LEDs, because carrier lifetime is greatly decreased both by the high carrier density required to achieve gain (which increases the spontaneous-recombination rate), and by the high photon density (which promotes recombination by stimulated emission). However, our simulator software is not adequate for the simulation of stimulated emission, so for the moment no assessment will be made for LDs.
4. Discussion: efficiency in LEDs
In order to show that LEDs based on the LCI design can be more efficient than conventional LEDs, a new series of simulations was carried out, where the number of quantum wells $N$ was varied from 1 to 100 and the lateral extent $\ell$ was kept constant to the previously determined optimal value of 3 $\mathrm{\mu}$m. Equivalent LEDs based on the conventional design were also simulated for comparison. The results are reported in Table 3. The data clearly show that increasing the number of QWs brings minor improvements in the conventional structures, whereas the improvements are substantial in the case of the LCI structures. Quantum efficiency (IQE$\,\times \,$CIE) can be improved by 14% in the case of 5 QWs, 25% for 10 QWs, 35% for 20 QW, 50% for 50 QWs, and 58% for 100 QWs, in comparison to conventional non-LCI structures. Considering the diminishing returns of increasing the number of QWs, and the challenge involved in their growth due to pseudomorphic strain, a compromise solution would certainly have to be considered. Moreover, the voltage bias increases by about 0.30–0.36 V for each QW added to the conventional structures, due to the barrier resistances being in series, whereas the bias decreases in the case of the LCI structures due to the new QWs being inserted in parallel. In the case of only 1–2 QWs, the quantum efficiency of the LCI structure is lower than the conventional structure due to strong leakage currents. The leakage current is a function of the applied bias, so it decreases as the number of QWs increases. These results show that the efficiency could be improved by increasing the number of QWs in the case of LCI, but not with the conventional structure.
Table 3. Performance indices of the simulated LEDs, based on the (a) conventional and (b) LCI design, at the equivalent current density of 35 A cm$^{-2}$ (corresponding to 0.35 A for a 1 mm$^{2}$ chip). In the LCI structures, the lateral extent of the QWs is 3 $\mathrm{\mu}$m. The last four columns refer to the topmost QW only in the case of the conventional structures.
Figure 4(b) makes it clearer how the LCI design affects the LED efficiency. As the number of QWs is increased, the efficiency curves of the LCI-based LEDs are shifted to the right, thus moving the intersection point with the 35 A cm$^{-2}$ current line upward. This happens because increasing the QWs makes the active volume larger. A simple way to convince oneself of this is via the ABC model, which is the elementary model based on the two equations
where $\eta$ is the internal quantum efficiency (IQE), $I$ the current, and $n$ the density of free carriers (electrons or holes) in the active volume $V_a$. Therefore, the IQE depends solely on $n$, which itself depends on the $I/V_a$ ratio. So, if the active volume is doubled, the $\eta (I/V_a)$ graph is shifted rightward by a factor of two. The change of slope in the rightmost part of the curves in Fig. 4(b) is due to carrier overflow.Fig. 4. Efficiency curves (with logarithmic axes) of LEDs of the conventional design (a) and the LCI design (b), where the number of QWs is varied. The dotted vertical line shows the reference current at which LEDs are often designed to operate (35 A cm$^{-2}$).
On the other hand, the conventional LED structures behave differently at low and high currents (Fig. 4(a)). At low currents, the long carrier lifetime lets a portion of the holes diffuse into the lower QWs, thus enlarging the active volume. This only has the effect of promoting Shockley–Read–Hall recombination, lowering the IQE. At high currents, increasing the number of QWs in the conventional LED structure does not increase the active volume, and therefore the IQE curves coincide.
5. Discussion: current leakage
One possible problem of the LCI design is that part of the carriers recombine outside of the QWs. Fig. 5 shows the horizontal component of the hole and electron currents in the LCI structure with $N=20$ QWs and $\ell =3$ $\mathrm{\mu}$m. In particular, Fig. 5(a) shows the hole current and Fig. 5(b) the free-electron current. Only the $x$ component of the current vectors is shown, as this is what actually is of interest in our analysis. The horizontal lines clearly show the high current density flowing inside the QWs. It is noteworthy that no current (neither hole nor electron) flows along the barrier layers. In fact, the polarization charges existing between QWs and barrier layers attract any electron from the barrier into the underlying QW and every hole into the overlying QW, as depicted in Fig. 6, so that no charge carrier is able to traverse a barrier layer from one side to the other without being eventually trapped into a QW. However, there is an electron current flowing along the cap layer above the QW stack, as well as a smaller electron current flowing along the top interface of the InGaN underlayer below the QW stack. The former could be minimized by reducing the thickness of the cap layer. The latter could be eliminated by removing the underlayer, but at the cost of introducing a hole current in its place. Finally, there is a hole leakage current flowing along the lower interface of the underlayer (not depicted in Fig. 5(a)), yet this is small and could be reduced by increasing the underlayer thickness.
Fig. 5. Map of the $x$ component of the hole (a) and electron (b) currents in an LCI structure with 20 QWs.
Fig. 6. The polarization charges between nitride layers push the carriers from the barriers into the QWs.
To discuss the current leakage quantitatively, we divide the quantum efficiency in two components: whereas the IQE describes the probability that an electron–hole pair inside the QWs recombines radiatively (rather than non-radiatively), the carrier-injection efficiency (CIE) indicates the fraction of electron–hole pairs that recombine (radiatively or not) inside the QWs relative to the total pairs supplied by the electrodes (i.e. $I/q$). The actual quantum efficiency is therefore the product of the two components (IQE$\,\times \,$CIE). All components are listed separately for every simulated structure shown in Tables 2 and 3. It can be noted however, that the CIE calculated from our simulations is acceptably high and in line with the CIE of the conventional structures. Even assuming that the conventional design could be improved so that it has no carrier overflow at all, our simulations still suggest that the IQE improvements made possible by the LCI design (from 45% to 62% in the case of $N=20$ QWs) can still overcompensate the CIE losses introduced by it (from 100% to 87%).
6. Final remarks
Our simulations have shown that LEDs based on the LCI design can be more energy efficient than conventional LEDs. Carrier overflow is expected to be in line with conventional LEDs, whereas the IQE is considerably increased and the operative voltage can be smaller. In the green LEDs considered in our analysis, quantum efficiency could be improved by 25% in the case of 10 QWs, 50% in the case of 50 QWs. The margin of improvement of blue LEDs is expected to be smaller, but LCI could still turn out a viable solution to the so-called green gap problem. The growth of a large number of QWs without loss of crystal quality can be challenging, but high-quality stacks of 30–40 QWs have already been reported [15,19].
As for the convenience of production, the LCI design may strike as very complicated to manufacture by means of epitaxy and lithography. At the very least, it requires three epitaxial runs (one for the QW stack and one of each of the two doped regions) and two lithographic processes. However, it should be considered that the LCI design also provides some new advantages, such as the fact that both doped regions can be contacted from the top surface. The majority of commercial LEDs are still grown on insulating sapphire substrates, which implies that additional lithographic processes are anyway necessary to access the $n$-doped layer beneath the QWs. The LCI design does not have this problem. The feasibility of the regrowth process of the doped regions should be assured by the research already carried out by other researchers on the regrowth of the source and drain region of GaN field-effect transistors [20,21]. High-quality regrowth of $p$-doped GaN regions have been reported as well [22], even in narrow trenches [23] with similar width and depth to those proposed by us.
Finally, even if this article focuses its analysis of LED applications, the LCI design may even bring greater advantages to LDs, and especially vertical-cavity surface-emitting lasers (VCSELs), where it is desirable to have a large number of active QWs.
Funding
Narodowe Centrum Badań i Rozwoju (1/POLBER-3/2018).
Disclosures
The authors declare no conflicts of interest.
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