8.4 μm-emitting quantum cascade lasers (QCLs) have been designed to have, right from threshold, both carrier-leakage suppression and miniband-like carrier extraction. The slope-efficiency characteristic temperature T1, the signature of carrier-leakage suppression, is found to be 665 K. Resonant-tunneling carrier extraction from both the lower laser level (ll) and the level below it, coupled with highly effective ll-depopulation provide a very short ll lifetime (~0.12 ps). As a result the laser-transition differential efficiency reaches 89%, and the internal differential efficiency ηid, derived from a variable mirror-loss study, is found to be 86%, in good agreement with theory. A study of 8.8 μm-emitting QCLs also provides an ηid value of 86%. A corrected equation for the external differential efficiency is derived which leads to a fundamental limit of ~90% for the ηid values of mid-infrared QCLs. In turn, the fundamental wallplug-efficiency limits become ~34% higher than previously predicted.
© 2016 Optical Society of America
2 November 2016: A correction was made to Eq. (2).
Quantum cascade lasers (QCLs) emitting in the 8-10 μm range, with low enough threshold-current density, Jth, values to operate continuous-wave (CW) at room temperature (RT), have generally been grown with core regions composed of InGaAs/AlInAs superlattices lattice-matched to InP. Typical RT threshold-current density Jth and slope efficiency (ηsl)/period values have been reported to be [1,2] ~1.7 kA/cm2 and ~ 25 mW/A, respectively. Although in the active region (AR) the energy difference, E54, between the upper laser level, energy state 4, and the next higher energy level, state 5, is higher for 8-10 μm- than for 4-5 μm-emitting conventional QCLs (i.e., ~60 meV vs. ~45 meV), carrier leakage can be significant, with the main path being intra-AR shunt leakage through the state 5 , just like in the case of 4.5-5.5 μm-emitting QCLs . Furthermore, a recent highly sophisticated theoretical analysis  predicts that, in RT operation, electrons in the upper laser level of 8.5 μm-emitting conventional QCLs are quite hot (i.e., electronic temperatures 120-200 K higher than the lattice temperature). This is consistent with experimental findings [6,7] that electrons are hot in 8-9 μm-emitting QCLs, which explains why carrier leakage is still significant despite larger E54 values . This is evident from the relatively low threshold-current temperature coefficient T0 values (~170 K) reported for 8.4 μm-emitting conventional QCLs  of moderately high injector sheet-doping density (~1.2 x1011 cm−2), required for high-power operation. (T0 is defined as: Jth(Tref + ΔT) = Jth(Tref) exp(ΔT/T0), where Tref + ΔT is the heatsink temperature and Tref is the reference heatsink temperature). Similarly, relatively low slope-efficiency temperature coefficient, T1, values (~260 K) were obtained from the same QCLs  with T1 defined as: ηsl(Tref + ΔT) = ηsl(Tref) exp(-ΔT/T1)); confirming strong carrier leakage [4,7].
We have recently published  that by stepwise tapering both the barrier heights and well depths in the ARs of 8.8 μm-emitting, low-injector-doping (~0.7 x1011 cm−2) QCLs, the carrier leakage was substantially suppressed. This leakage suppression was achieved by decreasing the scattering rate from the upper level to AR-level 5, as a result of increasing the E54 value from 57 meV to 76 meV, and virtually doubling the τ54 lifetime. This so-called step-taper active-region (STA) QCL exhibited very high T0 and T1 values: ~277 K and ~551 K, respectively, by comparison to values for conventional, low-doped 8-9 μm QCLs . Resonant-tunneling extraction from the lower level occurred as well; however, only at drive levels ≥ 1.4 x threshold. The observed ηsl/period values (35 mW/A) were significantly higher than for conventional devices, due to efficient, miniband-like carrier extraction from the AR, indicating significantly higher internal differential efficiency ηid values. However, at the time, we were unable to accurately estimate those values. Now we have performed a variable mirror-loss study on those QCLs which reveal a high ηid value of 86%, in good agreement with theoretical estimates. Here, we present a new-design STA QCL, in that miniband-like carrier extraction is ensured from below threshold. The devices are moderately high-doped and lase at 8.4 μm. Based on a mirror-loss study, a similar high ηid value: 86%, is obtained by using constant ηsl values from threshold to ~1.7 x threshold. The recorded T0 and T1 values: ~219 K and ~665 K, are significantly higher than the above-mentioned values for moderately high-doped devices . Based on these results, the fundamental upper limit for the ηid value of mid-infrared (IR)-emitting devices is assessed. Then the previously published fundamental limits for mid-IR QCL wallplug efficiency  need to be increased by ~34%.
2. The internal differential efficiency in quantum cascade lasers
To address what influences the ηid value let us first look at the definition of the external differential efficiency for a QCL of Np number of periods :10], encompasses all differential photon-transition and carrier-usage efficiencies. We have defined it for QCLs  as the product of the differential pumping efficiency [4,12], ηp = 1- Jleak/Jth, where Jleak is the leakage-current density, and the differential transition efficiency  ηtr, when the injection efficiency into the upper laser level ηinj is taken to be unity. However, by solving the rate equations for a 3-level active region and taking into account ηinj, while assuming that a fraction of 1- ηinj carriers are scattered into the lower laser level, Hamadou et al.  derived a total transition efficiency, ηtr,tot:14]. Note that, while typically ηinj ≥ 0.95, there are cases when ηinj is lower than 0.95, even if leakage to the continuum is negligible; for example due to electron scattering and/or injection from the injector ground state(s) to AR energy state(s) above the upper laser level [15,16] or to scattering from the injector ground state(s) to the lower laser level . Even for those cases, the second term in the numerator of Eq. (2) is basically ≤ 10% of the first term; thus, Eq. (3) remains a good approximation for ηid. The net effect is that ηinj should be considered a factor in the ηid expression, a fact that has not been generally taken into account in QCL analyses. Note that the ηinjηp product can be considered to be a “total” injection efficiency ηinj,tot since in QCLs the carriers employed in downward transitions from the upper laser level are the result of two sequential mechanisms: carrier tunneling injection into the upper laser level and carrier relaxation from the upper laser level to the lower energy states in the active region. That is, ηinj is the efficiency of the first mechanism while ηp is the efficiency of the second mechanism (i.e., the fraction of carriers leaking from the upper laser level is 1- ηp). The ηinj,tot term is analogous to the internal-differential-efficiency term in interband-transition lasers . Note also that the ηinj,tot term is the same as the total-injection-efficiency term used in the denominator of the Jth expression when the ηinj value is close to unity [7,18].
The STA-type QCL structure has been redesigned for resonant-tunneling extraction from the lower level, state 3, for biases below threshold, while maintaining strong suppression of carrier leakage from the upper level, state 4. The conduction band diagram and relevant wavefunctions are shown in Fig. 1, and relevant lower AR and extractor states are shown in an inset. The barrier heights increase stepwise in the AR: x = 0.51, 0.51, 0.58 and 0.58 in AlxIn1-xAs, and the wells depths increase stepwise: x = 0.56, 0.56, 0.61 and 0.62 in InxGa1-xAs (i.e., the 3rd and 4th wells are so-called deep wells , having lower-energy bottoms than wells in the injector). As before , the two deep wells on the right side of the AR enhance the step-tapering of the AR-barrier heights which, in turn, cause  significantly larger E54 values than in conventional AR structures composed of wells and barriers of fixed alloy composition, respectively. More specifically the achieved E54 value (75 meV) is higher than typical values of 56-60 meV in conventional devices, which together with a relatively long (0.65 ps) τ54 lifetime  ensure a fourfold reduction in carrier leakage through energy state 5. (As before , the lifetimes are calculated using an 8-band k•p code whose parameters have been chosen to agree well with experimental results from conventional, deep-well and tapered-active QCLs). That is, ηp increases from ~88% to ~97%. However, the two deep wells have been made deeper than in the prior STA design  which, together with another design change: the two barriers following the exit barrier are taller (Al0.56In0.44As) than the rest of the barriers in the extractor/injector region (Al0.54In0.46As), ensure, as described below, resonant-tunneling extraction from the lower laser level, for electric field values below lasing threshold.
The carrier-extraction scheme is schematically shown in the inset of Fig. 1. Extraction consists of resonant extraction from both the lower laser level, state 3, and AR state 2. That is, state 3 is coupled to extractor state 3′, and state 2 is coupled to extractor state 2’. More specifically, while threshold corresponds to a field strength of ~53 kV/cm, states 3 and 3′ have a splitting at resonance of ~7 meV at a field of 51 kV/cm, while states 2 and 2’ have a splitting at resonance of ~9 meV at a field of 49 kV/cm. Is should be stressed that this is not a double-phonon-resonance (DPR)-type device since electrons are not extracted directly from state 1, but due to a relatively small E21 value (33 meV) they are in large part excited to state 2 via phonon absorption, and then extracted out of the AR. Thus, we call this device a STA-Resonant Extraction (RE) QCL.
The device lower-levels depopulation scheme is schematically shown in Fig. 2. Both states 3 and 3′ are depopulated to states 2, 2’ and 1. State 1 plays a significant role since it helps to further depopulate states 3 and 3′, in that it causes lowering of the τ3g value from 0.16 ps, in the case of depopulation only to states 2 and 2’, to 0.12 ps.
Thus, the net effect of resonant extraction from the lower level coupled with efficient depopulation is a very short global lower-level lifetime of 0.12 ps; much shorter than in DPR and/or bound-to-continuum devices (~0.2 ps) [1,19] or nonresonant-extraction (~0.25 ps)  devices. One has to go to three-phonon-resonance (TPR) devices  to get comparably short lifetimes (0.14 ps), but then a significant price has to be paid, in that carrier leakage through AR state 6 dramatically increases, as the energy difference between the upper level, state 5 in that case, and the next higher energy level, state 6, decreases from ~60 meV to ~40 meV . Therefore, we have miniband-like carrier extraction, just like when using lower minibands in superlattice-type QCLs [21,22] as well as an improvement over the shallow-well QCLs  which, due to only two lower AR states, have longer (0.16 ps) lower-level lifetimes . The upper-level lifetime τup,g is found to be 0.97 ps which, together with the low τ3g value, provides a differential transition efficiency ηtr value of 0.89. In contrast, for conventional 8-9 μm-emitting QCLs [1,20] the ηtr values are only ~0.81 .
The QCL structures were grown via metal-organic chemical vapor deposition (MOCVD) with core-layer thicknesses (in Å) for one period, starting right after the exit barrier: 37, (15), 36, (16), 33, 18, 33, 20, 32, 21, 30, 23, 29, 37, 21, 9, 57, 11, 49, , 42 , where bold italic script are In0.56Ga0.44As quantum wells (QWs), bold normal script are the 3rd and 4th QWs in the AR: In0.61Ga0.39As and In0.62Ga0.38As, normal script are Al0.54In0.46As barriers, italic script are Al0.51In0.49As barriers, bracketed italic script are Al0.58In0.42As barriers, italic script in parentheses are Al0.56In0.44As barriers, and underlining indicates a nominal doping of 1.1x1017 cm−3. Below a 35-period core, on a (1-2) x 1017-doped InP substrate, the following layers were grown: 2 μm-thick 2x1016-doped InP layer and a 0.2 μm-thick 5x1016cm−3-doped In0.53Ga0.47As layer. Above the core, the grown layers were: 0.2 μm-thick 5x1016cm−3-doped In0.53Ga0.47As layer, 2 μm-thick 2x1016cm−3-doped InP layer, 2.0 μm-thick 1017 cm−3-doped InP layer, 0.5 μm-thick 5x1018 cm−3-doped plasmon InP layer, and 0.15 μm-thick 2x1019 cm−3-doped InP cap layer. Note that the InGaAs light-confining layers have been changed from 0.5 μm-thick 1017cm−3-doped  to 0.2 μm-thick 5x1016cm−3-doped, in order to decrease the waveguide loss by ~0.4 cm−1. Devices were fabricated into ~21 μm-wide wet-etched ridges with current confinement provided by 400 nm-thick Si3N4 and with Ti/Au episide metallization. After wafer lapping and metallization, 3 mm-long bars were cleaved, high-reflectivity (HR)-coated on one facet and separated into chips.
A typical RT light vs. current-density curve is shown in Fig. 3(a). The Jth value is 1.88 kA/cm2 and Jmax is ~7.5 kA/cm2, indicating that the injector sheet-density ns is ~1.65x1011 cm−2 rather than the 1.1x1011cm−2 target value. The ns value was obtained from the expression for the maximum operating-current density: Jmax = nsq/τtransit, where τtransit is the total time the electron spends in a period at resonance , and employing τtransit ~3.5 ps, for MOCVD-grown devices at λ = 8.4 μm, as estimated from the data presented by Chiu et al. . The large dynamic range and a high slope efficiency (1.15 W/A) result in 3.3 W peak pulsed, single-facet power. These values correct for the 92% measurement efficiency of our test setup, which takes into account a 96% collection efficiency and a combined 96% transmission efficiency through two antireflection-coated lenses. The measurement setup consists of a calibrated thermopile and two high numerical-aperture, plano-convex lenses. Compared to conventional moderately-high-doped (1.2x1011 cm−2), 8.4 μm-emitting 35-period devices  of same geometry, the Jth value is only 10% higher, although the doping level is ~40% higher. We attribute this relatively low Jth value primarily to carrier-leakage suppression for the STA device; that is, ηp ~97% vs. ~88% for that conventional QCL . The ηp values are calculated by using the expression for shunt-type carrier leakage through state 5  and considering an electron-lattice coupling constant  αe-l value of 35 K cm2/kA, taken as the result of compensation of the high αe-l value found for 8.5 μm-emitting, low-doped QCLs (~70 K cm2/kA)  by the fact that αe-l is found to be almost inversely proportional with the injector sheet-doping density . (We have also calculated ηp for αe-l values in the 25-70 K cm2/kA range and find that it varies by ± 0.01 with respect to the value when considering 35 K cm2/kA). The high ηp value together with a high ηtr value (89%) by and large explain a RT ηsl value ~30% higher than for same-geometry conventional QCLs .
Figure 3(b) shows the T0 and T1 values. Over the 20-60 °C range the T0 and T1 values are 219 K and 665 K, respectively. Since the doping more than doubled vs. the previous STA device  the T0 value has decreased from ~277 K to ~219 K, primarily due to more backfilling. Yet, because of carrier-leakage suppression, the 219 K figure is higher than 170 K for conventional 8.4 μm-emitting devices  even though those were lower doped (1.2 x1011cm2). Still the best evidence of strong carrier-leakage suppression is, as expected, the T1 value: 665 K, which is much higher than that for conventional 8.4 μm QCLs (~260 K) .
We show in Fig. 4(a) the inverse slope efficiency vs. inverse mirror loss for moderately-high-doped STA-RE devices. The derived internal differential efficiency is 86% and αw is 6.6 cm−1. The high value for ηid reflects that STA devices achieve both significant carrier-leakage suppression and highly efficient carrier extraction. Using calculated values of 97% ± 1% for ηinj (field-strength variations of ± 2 kV/cm are considered around the 53 kV/cm threshold field), 97 ± 1% for ηp, and 89% for ηtr, Eq. (3) gives an ηid value of: 84 ± 2%. It should be noted that for the ηtr calculations we considered only inelastic scattering. However, in STA-type structures, elastic scattering and in particular interface roughness (IFR), is likely to affect more the τ3g value than the τup,g value. This is because the IFR effect on lifetimes is proportional with the square of the conduction-band offsets . Since the transitions from state 4 mostly occur in the vicinity of the shallow AR barriers, and transitions from state 3, to states 2 and 1, mostly occur in the vicinity of the tall AR barriers, τ3g is likely to decrease more due to IFR than τup,g. This is similar to previously proposed IFR-engineering methods for reducing the τ3g /τup,g ratio [27,28]. In turn, ηid values considering both elastic and inelastic scattering are likely to have higher values than when considering only inelastic scattering.
We have also performed a length study for the low-doped STA-RE devices. As seen in Fig. 4(b) we still get an ηid value of 86%, while αw drops to 5.5 cm−1. In this case, the calculated ηid value using Eq. (3) and only inelastic-scattering lifetimes is 85 ± 2%. The αw values in both cases agree very well with values derived from conventional 8-9 μm-emitting QCLs [29,30]. For instance, an estimate for the αw of low-doped devices can be obtained by comparing data from 9.0 μm-emitting low-doped, low-loss QCLs  and 8.2-9.3 μm-emitting, high-doped QCLs . By using the “total” waveguide-loss (i.e., the sum of αw and the “backfilling” loss αbf)  value of 9.0 μm devices (obtained from comparing the Jth values of the same uncoated and HR-coated chips) ; scaling αbf by employing the injector doping levels and transverse optical-confinement factor Γ values in  and ; and adding the extra free-carrier absorption loss (~0.4 cm−1), due to thicker and higher doped InGaAs optical-confinement layers in 8.8 μm-emitting STA devices  vs. the 9.0 μm-emitting devices , we obtain an αw value is ~4.9 cm−1. Then, taking into account that the so-called αempty (i.e., αw minus the losses in the core region such as intersubband absorption)  is ~2.3 cm−1  at 8.8 μm wavelength vs. 1.8 cm−1 for the low-loss 9.0 μm devices of , we estimate αw ~5.4 cm−1, a value in excellent agreement with our measured value. For moderately high-doped devices we calculate a loss increase of ~1.4 cm−1, due to higher free-carrier absorption in the core, and taking into account the above-mentioned extra 0.4 cm−1 loss for low-doped devices the estimated additional loss vs. low-doped devices is ~1 cm−1. This extra loss, added to the measured low-doped device αw value (5.5 cm−1), leads to an αw estimate of ~6.5 cm−1. Again there is excellent agreement with our measured value. These estimates give us confidence that the measured ηid values are fairly accurate as well. Significantly lower αw values can be obtained, for 8-9 μm-emitting QCLs, by using a larger number of periods (e.g., 45) and especially by using heavily-strained core regions, as Lyakh et al. did , since heavily-strained structures definitely appear to dramatically lower the losses in the core region by comparison to lattice-matched structures [29,30] (e.g., at 9 μm wavelength, core losses of only ~0.5 cm−1  vs. ~2.7 cm−1 ).
The 86% values compare well with measured and/or derived ηid values for several conventional devices emitting in the 7-11 μm wavelength range: ~63.5% for 7.1 μm-emitting TPR QCLs ; ~57% for 8.4 μm-emitting DPR QCLs [2,32]; ~67% for 9.0 μm-emitting DPR QCLs [9,33]; 58% for 9.0 μm-emitting QCLs with nonresonant extraction ; and 64% for 10.7 μm-emitting DPR QCLs . Thus, the ηid values for STA-RE devices are 30-50% higher than those for conventional devices. The achievement of ηid values significantly higher than for any type of 7-11 μm-emitting QCL demonstrates not only superior performance from STA-RE devices, but also that ηid values close to theoretical limits can be achieved, as long as there is both effective carrier-leakage suppression and fast, miniband-like carrier extraction.
6. Impact on the fundamental limits for the wall-plug efficiency of mid-IR QCLs
By looking at Eq. (3) one can estimate what is the fundamental upper limit for the ηid value. Assuming 100% injection efficiency and complete carrier-leakage suppression (i.e., ηp = 1) one obtains: ηid ≅ ηtr. As shown above, we have designed 8.4 μm-emitting QCLs with ηtr = 0.89. More recently, we have designed 4.8 μm-emitting STA-RE QCLs with ηtr = 0.90. Considering miniband-like extraction the lower-level lifetimes will assume values close to 0.1 ps, and since the τup,g calculated values, for devices emitting in the 4.5-9.0 μm range, are on the average ~1 ps [7,8,20] it follows that the upper limit for the ηid value is ~90%. That should hold true within a couple of percentage points when considering elastic scattering as well. By comparison, for interband-transition lasers the upper limit for ηid is 100% .
Fundamental limits for the wall-plug efficiency of mid-IR lasers were derived about a decade ago . The curve of the maximum wall-plug efficiency vs. emission wavelength was “anchored” using parameters from a 9.0 μm-emitting QCL . For that case we derive from  an ηid value of ~67%, which reflected the state of the art at the time. Meanwhile, most of the highest pulsed wall-plug efficiencies published at 298 K or adjusted for 298 K operation by using their respective T0 and T1 values : 27% at 4.9 μm , 18.3% at 7.1 μm , 15.4% at 9 μm  and 10% at 10.7 μm  are found to agree well with the predicted maximum wall-plug-efficiency ηwp,max curve  (see Fig. 5). However, that happens either because the ηid values for those devices are close to 67% (as pointed out in the preceding section for 7-11 μm-emitting QCLs and as reported (70%) for the 4.9 μm-emitting QCL in ) or because a low loss coefficient in an uncoated-facets device compensated for a relatively low ηid value (i.e., 58% in ). Since we find that ηid for mid-IR QCLs can ultimately reach values of ~90%, that means that the fundamental limits for ηwp,max can be ~34% higher than previously predicted. We show in Fig. 5 a comparison between ηwp,max vs. wavelength curves with ηid = 67%  and with ηid = 90%. As can be seen, over the 8-9 μm wavelength range ηwp,max increases from 15.3 to 13% to 20.5-17.6%, respectively. Similarly, the predicted 29% ηwp,max value at 4.6 μm wavelength  becomes 39%.
Relevant wall-plug-efficiency experimental results at various wavelengths have been inserted, for comparison, in Fig. 5. It is interesting to note that the three data points in the 4.6-4.9 μm range (i.e., 15.4% , 22%  and 27% ) track rather well with their reported ηid values (i.e., 50%, 60% and 70%, respectively). We make a clear distinction between results from devices for which the output power was obtained from a single facet, while the back facet was high-reflectivity (HR) coated, and devices for which the output power was considered from both facets (i.e., both facets were uncoated). The both-facets ηwp,max values are generally higher than the single-facet values simply because of significantly higher mirror-loss αm values . Then to obtain single-facet operation, while maintaining the same αm value, one can adjust the cavity length and the mirror-facets’ reflectivities , but the Jth values remain relatively high compared to those for devices that have to start with an HR-coated back facet. Those high Jth values, for high pulsed ηwp,max devices [20,31,37], impair CW performance since the core-temperature rise ΔTact is directly proportional with Jth . In turn, the CW ηwp,max values are significantly lower than in pulsed operation (e.g., 10% vs. 18.9% for 7.1 μm-emitting devices , and 10% vs. 16% for 9 μm-emitting devices ), and the ΔTact value at the ηwp,max point becomes rather high (e.g., ~58 °C for the 9 μm-emitting devices  compared to typical ~30 °C values for state-of-the-art QCLs [7,35]). Such high ΔTact values raise issues of long-term reliability . Furthermore, the authors of the paper reporting the highest pulsed ηwp,max value (28.3%)  did not present CW results, apparently since a high Jth value coupled with a low T0 value (140 K) led to high CW Jth values. That is, optimization for high pulsed ηwp,max operation does not necessarily mean optimization for high CW ηwp,max operation .. For instance, when conventional 4.6-4.8 μm-emitting, HR-coated QCLs were optimized for maximum wall-plug efficiency  smaller pulsed ηwp,max values were observed for single-facet than for both-facets devices of similar αmand ηid values (i.e., 15.4% vs. 22%, as seen from Fig. 5), but the CW ηwp,max value was comparable to the pulsed one (i.e., 13% vs. 15.4%) . Similarly, for the highest single-facet ηwp,max value (i.e., 27% at λ = 4.9 μm) , achieved from devices with both carrier-leakage suppression and miniband-like extraction , the CW value (21%) was relatively close to the pulsed one. As shown in , one can get CW values as high as 25% for 4.9 μm-emitting QCLs of same ηid (i.e., 70%), if the inherently low thermal-conductance values of the QCLs in  are increased, by using other device designs, to match conventional values.
7. Key factors affecting the fundamental limits for the wall-plug efficiency
Looking at the expression for the maximum (pulsed) wallplug efficiency, as given in , with the minor modification of replacing the ηs term; that is, the droop in the pulsed L-I curve at the ηwp,max point, with ηinj, since ηs is typically in the 0.90-0.92 for state-of-the-art devices  while ηinj is typically in the 0.96-0.98 range (For the ultimate limit in wallplug efficiency both terms can be considered unity). Then the ηwp,max expression is given by:Eq. (3), αm,opt is the optimal mirror loss, Jwpm is the current density at the ηwp,max point, hν is the photon energy and Vwpm is the voltage at Jwpm. Since Jwpm is generally found to be ~80% of the maximum operating current Jmax (e.g., for the 4.9 μm-emitting QCL of highest wallplug efficiency  Jwpm is ~78% of Jmax) we consider, like Faist did , that Jwpm = Jmax and Vwpm = Vmax. For Vmax Faist took Np (hν/q + Δinj) where Δinj is the energy difference between the lower laser level and the Fermi energy of the injector, at injection resonance. The term due to Joule heating [i.e., (Imax- Ith) Rs, where Rs is the series resistance] was justifiably neglected since for state-of-the-art devices it is a relatively small part of the maximum voltage (e.g., for the 4.9 μm-emitting QCL of  that voltage drop was only ~13% of Vmax). This approximation is further justified by the fact that in reality Vwpm is ~90% of Vmax . Then, we obtain the following:9] with the significant distinction that carrier leakage is taken into account via the ηp term, and that the ηinj term is considered as well. Since the 1- Jth/Jmax term varies little for high-power devices (i.e., it is typically in the range: 0.67-0.75 ), to maximize the ηwp,max value, for a given αm,opt value, one needs to primarily: (a) maximize the ηid value; (b) minimize the αw value; and (c) minimize the Δinj value. Thus, since STA-RE-type device designs have been realized with low Δinj values: 120-130 meV (for example the 4.9 μm- and 4.7 μm-emitting STA-RE-like designs in ) and ηid is not a function of the αw value, maximization of the ηid value can be performed independently of minimizing the αw and Δinj values. For instance, for the previously highest ηid value (i.e., 67%)  for QCLs emitting in 7-11 μm wavelength range the wall-plug efficiency was only 3.3% , due to rather high αw values , by comparison to the predicted 13% fundamental-limit value at λ = 9.0 μm.
For the presented STA-RE devices we have obtained a maximum, single-facet wall-plug efficiency of 8% which is only slightly above the highest single-facet value reported for devices emitting in the 7-11 μm wavelength range (Fig. 5). Although we achieved record-high ηid values, the αw value was relatively high (5.5-6.6 cm−1) and the Δinj value was relatively high as well (~200 meV). As pointed out above, for 8-9 μm-emitting QCLs one can significantly lower the αw value by using strain-compensated core regions and increasing the number of stages . Similarly the Δinj value can be reduced to values close to 120 meV by using three-quantum-well, STA-RE-like active regions [7,23]. Thus, we expect that optimized STA-RE devices can definitely come close to the fundamental wall-plug efficiency limits. In fact, a recently published 5.6 μm-emitting QCL , with both significant carrier-leakage suppression and apparently highly effective carrier extraction, has achieved both the highest ηid value (75%) for devices emitting in the 4.5-6.0 μm range as well as the highest pulsed, both-facets wall-plug efficiency (28.3%) for QCLs. Thus, as seen in Fig. 5, it approaches the fundamental limit at that wavelength (i.e., 33%), albeit, as discussed above, high Jth and low T0 values prevent CW operation.
In conclusion, by combining carrier-leakage suppression with fast, miniband-like extraction we have realized 8-9 μm-emitting QCLs of internal-differential-efficiency values close to theoretical limits. Significantly higher T0 and T1 values are obtained compared to moderately high-doped, conventional 8.4 μm-emitting QCLs. Such devices should prove useful for obtaining both high-power CW operation  and high CW wallplug-efficiency values .
US Army (W911NF-12-C-0033).
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