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The edge emission from AlGaInP light-emitting diodes showed a red-shifted peak in addition to the peak of surface emission. This shift resulted from the quantum-well absorption of the guided wave. Although the shift degrades the color quality and the extraction efficiency of the device, it helps elucidate many important optical properties of the material and the dynamics of carrier recombination, including the electron-hole recombination lifetime, the optical joint density of state, the spontaneous emission spectrum and the absorption spectrum. A simple concept of the bimolecular recombination is established. The corresponding coefficient can be expressed by a simple formula and was therefore determined.
©2008 Optical Society of America
1. Introduction
Since the wavelength of the direct band-to-band transition of (AlxGa1-x) 0.5In0.5P can be easily varied from 550 nm to 650 nm by varying x, AlGaInP alloys are the most important materials for fabricating visible light-emitting diodes (LEDs) from the yellow-green to the red spectral range [1–4]. Efforts are ongoing to increase the efficiency and product lifetime and to reduce the manufacturing cost of AlGaInP LEDs [5–11]. The current maximum achieved luminous efficiency is 100 lm/W [12], a value which exceeds not only that of light bulbs (~15 lm/W) but also that of fluorescent lamps (~90 lm/W) [13,14]. The typical full width at half maximum of the emission spectrum is less than 20 nm. Both of these advantages have supported the successful commercialization of AlGaInP LEDs in applications that depend on color light sources, including decorative lighting, traffic lights, LED displays and various electronics. Despite these important achievements, some yellow-green AlGaInP LED lamps were found to emit yellow at a particular angle, which represents a falling short of the generally recognized high color quality and single color of these lamps. The origin of this unexpected emission is readily identified by inspecting a working bare chip under a microscope. Figures 1(a) and 1(b) present an example. The edge emission that escaped from the GaP current spreading layer had the same yellow-green color as the surface emission, but the edge emission from the AlGaInP active layer was yellow. Other AlGaInP LEDs that operate at longer wavelengths do not exhibit a similar phenomenon that can be easily resolved by the naked eye, but they do produce extra emissions that can be identified by using spectral analysis. Figure 1(c) presents the surface and edge emissions from yellow-green, amber and red chips. All of the edge emissions comprised not only the surface emission peaks but also other peaks with longer wavelengths. Thus, the existence of red-shifted edge emission is independent of the operating wavelength. In fact, based on our examination, the red-shifted edge emissions are not accidental and are exhibited by all AlGaInP LEDs that are based on a multiple-quantum-well (MQW) design. When this red-shifted emission is collected along with the surface emission to increase optical output power, the color quality of the device is degraded. Therefore, a compromise must be made between the efficiency and the color quality of the devices when packaging.
Fig. 1. (a). Edge image of a working yellow-green chip. (b). Surface image of a working yellow-green chip. (c). Surface and edge emission spectra from yellow-green, amber and red chips at 20mA.
Despite the above considerations, the presence of this extra red-shifted peak has rarely been discussed in the literature. The authors recently determined that the red-shift was caused by the quantum-well absorption of the guided wave. Although the shift degrades the color quality and the extraction efficiency of the device, it helps to elucidate many important optical properties of the material and the dynamics of carrier recombination, including the electronhole (e-h) recombination lifetime, the optical joint density of state, the spontaneous emission spectrum and the absorption spectrum. A simple concept of the bimolecular recombination of electrons and holes is developed. The corresponding coefficient can be expressed by a simple formula and its value was therefore determined.
2. Surface and transverse electric mode edge emission spectra
Although all AlGaInP LEDs that are based on the MQW design exhibit a similar red-shift, a wavelength shift in the yellow-green spectral range is associated with the largest color contrast, and so a device that operates at this wavelength range is used herein as an example. The wafer structure has an n-type GaAs substrate, a 390 nm-thick n-type Al0.5In0.5P hole-blocking layer, 65 pairs of MQWs, each of which consists of a 3.3 nm (Al0.3Ga0.7)0.5In0.5P well/2.9 nm (Al0.58Ga0.42)0.5In0.5P barrier, a 517 nm-thick p-type Al0.5In0.5P electron-blocking layer and a 9.38 µm-thick p-type GaP current spreading layer. Chips made from this wafer have an area of 225×225 µm2 and a height of 180 µm.
A chip was attached to the margin of a two-pin TO-46 lead frame using silver paste without an encapsulant. The lead frame enables the spectra of various orientations to be easily obtained by bending the pins of the lead frame without any adjustment of the measurement system. The emissions of the intended orientations were then collected by a microscope on top of the package and analyzed by a monochromator and a silicon photodetector. Since the wavelength-dependent parameters, including the responsivity of the silicon photodetector and the reflectivity of the grating, were taken into account, the raw spectra were the relative wavelength-dependent energy spectra P(λ). Therefore, the relative frequency-dependent energy spectra P(hν) were derived using the transformation P(hν)=(λ 2 h/C 0)P(λ), where C 0 is the speed of light in a vacuum and h is Planck’s constant [15]; and the relative frequency-dependent photon spectra of interest, N(hν), described in the following paragraphs, were then obtained by N(hν)=P(hν)/hν.
Figure 2(a) shows the spectrum of the surface emission on a semi-log scale. However, since the radiation of a transition in a quantum-well structure has a preferred polarization [16,17], and the purpose of this study is to understand why the edge emission has a red-shifted peak in addition to the peak of surface emission, care should be taken to ensure that the measured edge emission is produced by the same electronic transitions as the surface emission, to ensure comparability between these two orientations. Thus, the relationship between the electronic transition and the radiation is considered. For a spontaneous transition of an electron in the conduction band with a wave function ϕ c(r⃗)=u c(r⃗)exp(ik⃗c·r⃗)/√V to the valence band with a wave function ϕ v(r⃗)=u v(r⃗)exp(ik⃗v·r⃗)/√V via the emission of a photon with a vector potential where u c,v(r⃗) and
exp (ik⃗c,v·r⃗)/√V are the Block lattice function and the envelope function, respectively, of the conduction band and the valence band; k⃗c, k⃗v and k⃗ℓ are the wave vectors of the electron in the conduction band, the electron in the valence band and the radiation, respectively; V is the volume; ε̑ℓ is a unit vector in the direction of the electric field of the radiation; ε 0 is the dielectric constant of free space; n r is the refractive index, and the other terms have their usual meanings, the transition rate according to Fermi’s golden rule is,
where the superscript ℓ refers to the optical mode of the emission; E ci and E vf are the energies of the initial and final states; p⃗op is the operator -iħ∇⃗, and m 0 is the free electron mass [17–19]. This equation describes not only the k-selection rule and energy conservation for the transition, but also the fact that the transition rate is dominated by the polarization of the radiation ε̑ℓ rather than the transmission direction of the radiation k⃗ℓ. Since the surface emission was collected on top of the chip, the electric field vector of that emission is parallel to the quantum wells, and the emission was dominated by the electron-heavy hole recombination [17]. The transverse electric (TE) mode of the edge emission has the same direction as the electric field vector of the surface emisson. The surface and the TE mode edge emissions are therefore associated with an equal probability of electronic transition with the emission of a photon, and should have the same spontaneous emission spectra. Hence, a polarizer was inserted into the optical path to measure the TE mode of the edge emission, yielding Fig. 2(b) with a semi-log scale. As stated above, the edge emission has an extra red-shifted peak, which is 47.9 meV lower than the main peak.
Fig. 2. (a). Solid lines represent the surface emission spectrum of the sample at 20 mA. The dashed line represents the linear regression of the high-energy tail and the dotted line represents the obtained broadened step function. (b). Transverse electric mode edge emission spectrum of the device operated at 20 mA.
3. Ray tracing and optical mode calculation
In our LED, all photons were generated in the AlGaInP active layer. However, as presented in Fig. 1(a), while the edge emission from the GaP current spreading layer yielded a similar spectral shape to that of the surface emission, the edge emission from the AlGaInP active layer was red-shifted. Hence, the path along which the photon passed before it escaped from the chip is critical to the red-shift and was considered herein. The calculation was based on the refractive indices obtained by the modified single-effective-oscillator method with parameters that were proposed by Kaneko and Kishino at 572.1 nm [20]. Since the thickness of each well and barrier was much less than this wavelength, the MQWs were considered with a thickness-weighted ed index n MQWs=3.179. As indicated by the solid arrows in Figs. 3(a) and 3(b), since the refractive index of the MQWs differs greatly from that of air, the photons that can escape from the edge of the chip were limited to the edge escape cone with an angle that is determined by Snell’s law, sin-1 (n MQWs -1)=18.3°. As the photons inside the cone escaped from the GaP current spreading layer, their spectrum has no reason to be substantially differed from that of the surface emission. However, the refractive indices of the MQWs and the carrier-blocking layers also differ. As indicated by the dashed arrows in Figs. 3(a) and 3(b), when the angles of the generated photons were less than 90°-sin-1(n cladding/n MQWs)=16.6°, successive total internal reflections kept the photons within the absorbing AlGaInP active layer, and the spectrum of these photons was modified by the absorption effect and the photons emerged from the AlGaInP active layer. This fact qualitatively explains why the edge emission from the GaP current spreading layer has a similar spectral shape as the surface emission, but the edge emission from the AlGaInP active layer was red-shifted.
Fig. 3. (a). Active layer and ray traces. Solid arrows indicate the edge escape cone while dashed arrows indicate the angle of total internal reflection. (b). When the angles of the photons are inside the edge escape cone and exceed 16.6°, the photons can emerge from the GaP layer. As the angles are less than 16.6°, the photons are confined and emerge from the AlGaInP active layer.
To see how the active layer confines and transports the red-shifted light, the guided modes were numerically solved based on the assumption that the carrier-blocking layers were infinitely thick. Figure 4 plots the results, and with the real thicknesses of the carrier-blocking layers as dashed lines. The TE mode has two solutions- TE0 and TE1. The tails of the power distributions outside the real thicknesses of the carrier-blocking layers are extremely small in these two modes, indicating that these modes are effectively confined. The proportions of the power confined in the MQWs (ΓMQWs) are 92.9%, 62.7% for TE0 and TE1, respectively. However, only the wells in the active layer can absorb light. The proportions of the power confined in the wells (Γw) are also presented, and their values are 46.6% and 26.6% in modes TE0 and TE1, respectively.
Fig. 4. Numerical power distributions of guided transverse electric modes in the sample. ΓMQWs and Γw are the fractions of the power that is confined in the MQWs and wells, respectively.
4. Spontaneous emission, absorption and red-shift
Since the red-shifted edge emission was probably a spontaneous emission but modified by the absorption, an examination of the spontaneous emission and absorption spectra is essential to any thorough understanding of the underlying mechanisms. The relative spontaneous emission spectrum can be approximated as the surface emission spectrum, presented in Fig. 2(a), since the total thickness of the quantum wells is only 214.5 nm. However, the absorption spectrum can not be directly measured from the device. A derivation of the absorption spectrum from the spontaneous emission spectrum was carried out.
According to Eq. (1), the radiative recombination lifetime τ r of an e-h pair that satisfies the k-selection rule is the inverse of the total transition rate summed over all optical modes:
where M(hv) is the optical mode density that equals 8πn r 3 v 2/hC 3 0; is the average of over all polarizations of the optical modes, and the anisotropy of the transition rate caused by the quantum-well structure was temporary ignored and was considered later. As a result, the spontaneous emission rate is
where ρ(hν) is the optical joint density of state, which is the state density of optically permissible transitions that satisfy the k selection rule; f c is the probability that an electron is in the initial state and 1-f v is the probability that a hole is in the final state [21]. Since LEDs are operated under low-level injection, both f c and 1-f v can be approximated using the Boltzmann distribution. Therefore, f c(1-f v)≈exp⌊(E fn-E fp)/KT⌋exp[-hν/KT] where E fn-E fp is the separation between the quasi-Fermi levels of the electron and the hole, and Eq. (3) becomes
Additionally, since the energies of the carriers in the wells are quantized in the direction of crystal growth, both the densities of states of the electron and the hole in a perfect quantum well are proportional to step functions, and the corresponding optical joint density of state is ρ(hν)=m r u(hν-E g)/πħ 2 d where u(hν-E g) is the step function; m r is the reduced mass of an e-h pair; d is the quantum-well thickness, and E g is the effective band gap, including the quantum-confined zero-point energies of electron and hole. As a result, the equation for the spontaneous emission rate of the quantum wells is
Following the determination of the theoretical spontaneous emission rate, the information in the surface emission spectrum can be studied. In Eq. (5), when τ r is assumed to be independent of the energy hν of the photons, only the term exp(-hν/KT) depends on hν when hν≫Eg, and the high energy tail of the spontaneous emission spectrum versus hν, plotted on a semi-log scale, should be linear with a slope of 1/KT. This relationship was experimentally demonstrated by the high energy tail of the measured surface emission spectrum, presented as the dashed line in Fig. 2(a), and the gradient corresponds to a junction temperature of 350K. Additionally, if the surface emission spectrum is divided by exp(-hν/KT), then, according to Eq. (5), the resulting function should be proportional to the step function u(hν-E g) and the coefficient of proportionality can be obtained from the resulting function at hν≫E g. The previously obtained function was divided by the coefficient of proportionality, and the step function was then experimentally determined from the surface emission spectrum, represented by the dotted line in Fig. 2(a). The step function thus obtained, unlike the theoretical step function that suddenly changes from zero to unity, has a transition region from ~2.00 to ~2.16 eV over which the value changes, indicating broadening of the edge of the corresponding joint density of state ρ(hν)=m r u(hν-E g)/πħ 2 d and the effective band gap of the wells. This broadening is commonly encountered in semiconductors and has various causes such as carrier collision [22], well width fluctuation [23–24], composition fluctuation [25], low-energy tail [26] and variations among wells. Although the reasons for the broadening are not discussed further herein, the broadening has a great impact on the absorption spectrum and the red-shift.
In contrast to spontaneous emission, the absorption spectrum of the wells can not be measured from the device. However, since the mechanisms of spontaneous emission and absorption are highly correlated, the absorption spectrum can be derived from the spontaneous emission spectrum using the relationship between Einstein’s A and B coefficients for spontaneous and stimulated emissions, respectively. The result is [27–31]
Until now, the above information has sufficed for quantitatively determining the red-shift. Consider a spontaneous transition at a distance of x from the edge of the chip; the resulting radiation r sp(hν) is guided by the active layer with a confinement factor Γw; the emerging emission should be attenuated by the absorption and becomes r sp(hν)exp(-Γw α(hν)x. As a result, the red-shifted edge emission r edge(hν) should be the integral of the attenuated emission over the area of the quantum well:
where L is the edge length of chip and L 2 in the denominator is the area, which is used for normalization. After r sp(hν) from the measured surface emission and the previously simulated Γw are substituted into Eq. (7), and u(hν-E g) from the obtained broadened step function and m r=0.096m 0 (from electron effective mass m*e=0.113m 0 [32] and heavy hole effective m*hh=0.635m 0 [33]) are substituted into Eq. (6), τ r is the only undetermined parameter that remains in these two equations and is set as a fitting parameter.
Figure 5(a) presents the absorption spectra derived from Eq. (6) with τ r=0.03 ns, 0.3 ns and 3.0 ns, respectively. The absorption coefficients increased as the radiative recombination lifetime decreased. At hν<2.16eV, the absorption coefficients increased exponentially with hν, because of the broadening of the effective band gap. Figure 5(b) presents the edge emission spectra obtained using Eq. (7) with the TE0 and TE1 guided modes denoted as dashed and dotted lines, respectively. However, spontaneous emission should radiate into TE0 and TE1 modes with equal probability; the averages of these two modes are favorable to correspond to the measured data, and are represented as solid lines. A comparison of the surface emission spectrum represented as a dashed-dotted line (which is also identical to the red-shifted simulation curve with τ r→∞) indicates that all of the solid curves were attenuated and red-shifted, because of the increase in the absorption coefficient with hν. The attenuation and red-shift increased as the radiative recombination lifetime decreased. Figure 5 (c) plots the relationship between the shift and radiative recombination lifetime. The red-shift of 47.9 meV, measured and presented in Fig. 2(b), revealed that the radiative recombination lifetime is 0.3 ns.
Fig. 5. (a). Absorption spectra obtained at various τ r. (b). Dashed-dotted line represents surface emission while other lines represent edge emissions deduced from surface emission for various τ r. Dashed lines: TE0; Dotted lines: TE1; Solid lines: average of TE0 and TE1. (c). Relationship between peak shift and τ r (solid line). The peak intensity (long dashed line) and integrated intensity (short dashed line) decrease with τ r as well.
Further taking the anisotropy of the transition rate caused by the quantum-well structure into account, though the quantum-well structure do not alter the total transition rate as well as the radiative recombination lifetime due to the conservation rule of momentum matrix elements, the transition rate of an electron-heavy hole pair is enhanced by 50% for the radiation that polarized along the quantum well, as the emissions we measured herein, at the expense of the reduced rate for the radiation that polarized vertical to the quantum well [17]. Therefore, the total transition rate 1/τ r is 50% overestimated in our case, and τ r should be modified to be 0.45 ns.
5. Radiative recombination lifetime, carrier lifetime and bimolecular recombination coefficient
Although the aforementioned red-shift mechanism certainly occurs in the sample since the wave-guiding effect was present and the emission was absorbed, the shift deduced from Eq. (7) and shown in Fig. 5(c) depends strongly on the fitting parameter τ r, and so whether all of the measured shift is attributable to this mechanism depends on the correctness of the obtained τ r. The momentum matrix element in Eq. (2) can be obtained via Kane’s model for band structure:
where P is the interband matrix element used by Kane and Δ is the spin-orbit splitting [17,34–36]; and the total transition rate is
With E g=2.16eV and Δ=0.073eV [37], τ r is theoretically estimated to be 0.59 ns, which is in agreement with our experimental result, τ r=0.45 ns.
Another evidence for this fact is found from the absorption spectra in Fig. 5(a), where the curve of τ r=0.3 ns (that corresponds to the modified τ r=0.45 ns) has an absorption coefficient of 1.8×104 cm -1 at an energy of 2.26 eV (≈E g+100 meV). Replacing ρ(hν) in Eq. (6) with a three-dimensional optical joint density of state and with τ r=0.45 ns yields the calculated absorption coefficients of the bulk material at an energy of E g+100meV of 0.70×104 cm -1 for electron-heavy hole transition and 0.34×104 cm -1 (with light hole effective mass m*ħ=0.124m 0 [37]) for electron-light hole transition and a total absorption coefficient of 1.04×104 cm -1. Both of the quantum well and the bulk absorption coefficients are typical, by comparison with those of other direct band-gap semiconductors with a zinc-blend structure such as Al0.5In0.5As (2.3×104 cm -1), GaAs (1.4×104 cm -1), InP (1.5×104 cm -1) and Ga0.5In0.5As (0.7×104 cm -1) at an energy of E g+100 meV [38], and are direct evidence of the radiative recombination lifetime and the underlying red-shift mechanism.
Despite the importance of the e-h recombination lifetime τ r, τ r is easily mistaken for the carrier lifetime τ c since these two terms are both used to describe e-h recombination. Hence, the differences and the correlations between these terms are worthy of note. According to Eq. (2), τ r is a quantum mechanical parameter that specifies the transition rate between two energy states. Consequently, τ r depends only on the wave functions of the initial and the final states but not on the populations of carriers. In contrast, the carrier lifetime τ c is defined using the bimolecular recombination equation
where B is the bimolecular recombination coefficient; R is the recombination rate; n and p denote the concentrations of electrons and holes, respectively; subscript 0 denotes the values at thermal equilibrium; Δ denotes excess concentrations, and the contribution of non-radiative recombination is ignored. Therefore, τ c strongly depends on the carrier population. Nevertheless, these two time constants are highly correlated since the recombination rate R can be derived from Eq. (4)
where N r is similar to the electron (hole) effective density of state N c=2(2πm*e KT/h 2)1.5 (N v=2(2πm*hh KT/h 2)1.5 but with the effective mass replaced by the reduced mass of an e-h pair [39]. Notably, the term N r exp{-⌊-E g-(E fn-E fp)⌋/KT} in Eq. (11) strongly resembles that in the expression for electron concentration n=N c exp⌊-(E c-E fn)/KT⌋ where E c is the bottom of the conduction band. This similarity results mathematically from the resemblance between the densities of states v.s. and occupation probabilities (exp{-⌊hν-(E fn-E fp)⌋/KT} v.s. exp⌊-(E-E fn)/KT⌋) of the e-h pair and electron. Therefore, N r can be called as the e-h pair effective density of state and n e-hpair=N rexp{-⌊E g-(E fn-E fp⌋/KT} is the concentration of e-h pairs. Accordingly, Eq. (11) is R=n e-h pair/τ r, which describes a simple concept in which the e-h pair recombination rate corresponds to the concentration of the e-h pair divided by the lifetime of the e-h pair. Additionally, since exp{-⌊E g-(E fn-E fp⌋/KT}np/N c N v, the recombination rate R=(1/τ r)(N r/N c N v)np, and the bimolecular recombination coefficient B can be expressed as a simple formula:
If both of the heavy-hole and the light-hole bands are considered, based on the hole distributions in these two bands, the bimolecular recombination coefficient is
With τ r=0.45 ns and the effective masses previously used, the bimolecular recombination coefficients are B e→hh=1.37×10-10 cm 3 sec-1 and B e→hh=7.68×10-10 cm 3 sec-1 for electron-heavy hole and electron-light hole transitions, respectively, at room temperature. Therefore, the total bimolecular recombination coefficient is Be→hh=1.87×10-10 cm 3 sec-1, which is very close to the corresponding values for other direct-band gap semiconductors (2.0×10-10 cm 3 sec-1 for GaAs [26,40], 1.2×10-10 cm 3 sec-1 for InP [40], 2.2×10-10 cm 3 sec-1 for GaN [40], and 0.92×10-10 cm 3 sec-1 for compressively strained Ga0.45In0.55P [41]). This value is thus further supports the radiative recombination lifetime and the underlying red-shift mechanism.
6. Dependence of red-shift on temperature and bias
Since similar red-shifts have been observed in nanowire lasers [42] and edge emitting lasers [26], this work is extended to address how the red-shift depends on the population distribution of carriers by simulation since both the optical joint density of state and the radiative recombination lifetime had been determined. However, the previously used Boltzmann distribution can fail in extreme cases, and so the Fermi-Dirac distribution was used in the simulation. Since only the effects of population distribution are of interest, the temperature-dependence of the band gap, Joule-heating and current-crowding caused by high current injection were all neglected to ensure that the results concerning the carrier distribution would be clear.
Figure 6 shows the peak positions of the emissions at various temperatures with a carrier concentration on of 1017 cm-3 in the wells. The peak of the edge emission only weakly depends on the temperature. According to Eq. (7) and Fig. 5(b), the intensity of the edge emission was substantially suppressed by absorption when Γw α(hν)L≫1 but was the same as that of spontaneous emission when Γw α(hν)L≫1. Therefore, the peak of the edge emission should be around a characteristic energy hν c=2.11eV, obtained from Fig. 5(a), satisfying α(hν c)(Γw L)-1≈164cm -1 and should not depend strongly on temperature. Since the absorption coefficient strongly increases with the energy in the band tail and equals the gain when the population of the corresponding states is fully inverted, this characteristic energy should also denote the position of the lasing peak if the similar quantum wells are used as the gain material in a edge emitting laser, and should also be red-shifted from the spontaneous emission. In contrast, the peak position of the spontaneous emission is more sensitive to the temperature. When the temperature is low, most of the carriers are in the band tail. Since the density of states in the tail strongly increases with the energy, the effect of increasing the temperature is to raise the overall distribution of carriers to higher energies, and the peak of the corresponding spontaneous emission is blue-shifted. However, as the temperature is high, many carriers are already at high energies where the density of states is a constant, and the peak position only weakly depends on the temperature. Therefore, the red-shift, presented in the inset in Fig. 6, increases with temperature for T<200K, and then gradually becomes saturated at high temperature.
Fig. 6. Peak positions of spontaneous and edge emissions at various temperatures. Inset shows red-shift.
Figure 7 shows the peak positions of the emissions with various carrier concentrations in the wells at a temperature of 350K. The top x-axis represents the corresponding current that was produced by spontaneous radiative recombination, and the contributions from leakage current, non-radiative recombination current and stimulated radiative recombination current are not included. When the carrier concentration is less than 1018 cm -3, the peak of the edge emission is essentially unaltered. However, when the carrier concentration exceeds 1018 cm -3, many of the electronic states that correspond to the characteristic emission energy hν c=2.11eV are occupied by carriers. Accordingly, the absorption coefficient is reduced; the energy at that point where the absorption coefficient is high enough to suppress effectively the emission moves upward, and the peak of the edge emission moves to higher energy. When the carrier concentration is further increased to over 3.5×1018 cm -3, stimulated emission is activated at the characteristic energy, and the peak position moves further. In contrast, the peak position of the spontaneous emission is unaltered unless band filling occurs where the carrier concentration exceeds 1018 cm -3. Therefore, the red-shift only weakly depends on the carrier concentration when the concentration is low, decreasing as the carrier concentration increases when the concentration is sufficiently enough. However, the changeover concentration is 1018 cm -3, which corresponds to a spontaneous radiative current of 100 mA - a value that markedly exceeds the operating current, 20 mA, of the device, and not, therefore, observable in the device herein.
Fig. 7. Peak positions of spontaneous and edge emissions at various concentrations. Top x-axis represents the current that results from spontaneous radiative recombination.
7. Conclusion
The surface emission and edge emission spectra of AlGaInP LEDs were obtained and analyzed. The peaks of edge emissions from the active layers of the samples that were studied herein were all red-shifted from those of the surface emissions. This red-shifted emission was confined in the active layer because of the difference between the refractive indices of the MQWs and the carrier-blocking layers. Accordingly, the guided wave was red-shifted because of the consecutive absorption by the quantum wells before it left the device.
To analyze quantitatively this red-shift, the surface emission was compared with the theoretical spontaneous emission spectrum and the optical joint density of state of the quantum wells was obtained. This optical joint density of state was broadened at the energy near the band gap by comparison with the step function of an ideal quantum well. Based on the obtained optical joint density of state, the equations of the absorption and the red-shifted edge emission spectra were obtained but with an undetermined e-h recombination lifetime τ r. Fitting the shift in the edge emission to the measured value indicated that the e-h recombination lifetime was 0.45 ns, and the integrated and peak intensities of the red-shifted edge emission were attenuated to 3.7% and 3.2%, respectively, of the original values. Additionally, since τ r is one of the most important parameters in the dynamics of e-h recombination, not only the spontaneous emission spectrum but also the absorption spectrum were fully determined following the determination of τ r. A simple concept of bimolecular recombination between electrons and holes was presented. The bimolecular recombination coefficient can be expressed by a simple formula, and the corresponding value for τ r=0.45 ns is B=1.87×10-10 cm 3 sec-1.
In conclusion, since the red-shift was caused by absorption, both the color quality and extraction efficiency of the LED were degraded. Reducing the proportion of energy confined in wells, the broadening of the band gap, or the lateral dimensions of the device can alleviate the problems of both color quality and extraction efficiency, and thus improve device performance.
Acknowledgments
The authors would like to thank the National Science Council of the Republic of China, Taiwan, for financially supporting this research. Ted Knoy is appreciated for his editorial assistance.
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