Temperature sensitivity of the electro-optical characteristics for mid-infrared (λ  =  3–16 μm)-emitting quantum cascade lasers

The threshold-current density, J_th, is the sum of J_th in the absence of backfilling and electron leakage, J_0,th, and the current densities required at threshold to compensate for backfilling, ${{J}_{\text{bf},\text{th}}}$, and electron leakage, ${{J}_{\text{leak},\text{th}}}$, [11]:

$$\begin{eqnarray}&&{{J}_{\text{th} ~}}={{J}_{0,\text{th}~}}+{{J}_{\text{bf},\text{th}~}}+{{J}_{\text{leak},\text{th}~}}\end{eqnarray} \tag{ 9a }$$

which, considering the pumping-efficiency term [17, 20, 41] ${{\eta}_{\text{p}}}=1-{{J}_{\text{leak},\text{th}}}/{{J}_{\text{th}}}$, can be rewritten:

$$\begin{eqnarray}&&{{J}_{\text{th}}}=\frac{{{J}_{0,\text{th}~}}+{{J}_{\text{bf},\text{th}~}}}{{{\eta}_{\text{p}}}}\cong \frac{{{J}_{0,\text{th}~}}+{{J}_{\text{bf},\text{th}~}}}{\eta _{\text{inj}}^{\text{tot}}};\,\,\eta _{\text{inj}}^{\text{tun}}\cong 1.\end{eqnarray} \tag{ 9b }$$

Note that η_p, as defined here, is the efficiency with which electrons injected into the upper laser level are used for lasing at threshold, when taking the tunnel-injection efficiency $\eta _{\text{inj}}^{\text{tun}}$ to be unity. We also can define a total injection efficiency $\eta _{\text{inj}}^{\text{tot}}=\eta _{\text{inj}}^{\text{tun}}{{\eta}_{\text{p}}}$ for which, to a very good approximation, $\eta _{\text{inj}}^{\text{tun}}$ is taken to be a factor in the denominators of J_0,th [6] and J_bf,th, when the actual $\eta _{\text{inj}}^{\text{tun}}$ is close to unity. It just happens that η_p, as used in equation (9b), is equal to the differential pumping efficiency in lasing, at and slightly above threshold [11, 17, 20, 49], as is the case for leakage currents in interband lasers [34].

In the case of 4.5–5.5 μm emitting lasers of DPR lower-level depopulation scheme, as that shown for a conventional QCL in figure 4, ${{J}_{\text{leak},\text{th}}}$ is given by equation (3), while the other components are defined as follows [11]:

$$\begin{eqnarray}&&{{J}_{0,\text{th}}}=q\frac{{{\alpha}_{\text{tot}}}}{{{\tau}_{\text{up,g}}}{{g}_{\text{c}}}},\end{eqnarray} \tag{ 9c }$$

$$\begin{eqnarray}&&{{J}_{\text{bf},\text{th}}}=\frac{q}{{{\tau}_{\text{up}.\text{g}}}}\left[{{n}_{\text{s}}}\exp \left(-\frac{{{\Delta }_{\text{inj}}}+\hslash {{\omega}_{\text{LO}}}\left[\left({{T}_{\text{eg}}}/{{T}_{\text{l}}}\right)-1\right]}{k{{T}_{\text{eg}}}}\right)\right. \\ &&\left. \qquad \quad +\left(\frac{{{n}_{5}}}{{{\tau}_{53}}}+\frac{{{n}_{6}}}{{{\tau}_{63}}}\right){{\tau}_{3}}\right],\end{eqnarray} \tag{ 9d }$$

where ${{\tau}_{\text{up,g}}}={{\tau}_{4\text{g}}}\left(1-{{\tau}_{4\text{g}}}/{{\tau}_{43}}\right)$ is the global 'effective' upper-state lifetime [17] (due to both inelastic and elastic scattering) in the absence of carrier leakage, and taking $\eta _{\text{inj}}^{\text{tun}}$ to be unity, with τ_4g and τ_3g being the global lifetimes in the upper and lower levels; α_tot is the sum of the mirror losses (α_m) and actual waveguide losses (α_w); g_c is the gain cross-section [58]; n_s is the electron sheet density in the injector; Δ_inj is the energy difference between the lower laser level and the ground state in the next injector region; T_eg is the electronic temperature in the injector miniband [44];$~{{\tau}_{53}}/{{\tau}_{63}}$ is the lifetime corresponding to electron relaxation from state 5/6 to state 3; ${{\tau}_{3}}$ is the lifetime in the lower laser level (state 3). τ_4g and τ_3g are global lifetimes, in that they correspond to transitions to both lower AR states (e.g. to states 3, 2, 1 from state 4) and to extractor states penetrating into the AR (e.g. the three blue-coloured extractor states penetrating the AR in figure 4). That is, the global upper-level lifetime can be expressed as:

$$\begin{eqnarray}&&\frac{1}{{{\tau}_{4\text{g}}}}={{\frac{1}{{{\tau}_{4\text{g}}}}}_{\text{lowerAR} ~\text{states}}}+{{\frac{1}{{{\tau}_{4\text{g}}}}}_{\text{extractor} ~\text{states} ~\text{into} ~\text{AR}~}}.\end{eqnarray} \tag{ 10 }$$

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The expression for ${{J}_{\text{bf},\text{th}}}$ is derived [16] by writing a rate equation for the lower laser state, which is populated (backfilled) via LO-phonon reabsorption from states 1 and 2 (assumed to be strongly coupled to the injector states and thus sharing a common electronic temperature T_eg) and by LO-phonon emission from states 5 and 6. Taking T_eg  =  (1  +  δ_eg) T_l one obtains for equation (9d) the following expression:

$$\begin{eqnarray}&&{{J}_{\text{bf},\text{th}}}=\frac{q}{{{\tau}_{\text{up}.\text{g}}}}\left[{{n}_{\text{s}}}\text{exp}\left(-\frac{{{\Delta }_{\text{inj}}}+{{\delta}_{\text{eg}}}\hslash {{\omega}_{\text{LO}}}\left[\left({{T}_{\text{eg}}}/{{T}_{\text{l}}}\right)-1\right]}{kT\left(1+{{\delta}_{\text{eg}}}\right)}\right)\right. \\ &&\left. \qquad \quad +\left(\frac{{{n}_{5}}}{{{\tau}_{53}}}+\frac{{{n}_{6}}}{{{\tau}_{63}}}\right){{\tau}_{3}}\right].\end{eqnarray} \tag{ 11 }$$

For 4.5–5.5 μm emitting QCLs, around room temperature, the δ_eg quantity takes values in the 0.15–0.20 range depending on the J_th value considered. Then, as seen from equation (11), the numerator in the exponential part of the expression hardly changes, and what mostly affects the backfilling current is the denominator. The non-exponential term turns out to be negligible compared to the exponential term [11]. Thus, equation (11) can be well approximated by:

$$\begin{eqnarray}&&{{J}_{\text{bf}~}}\approx \frac{q{{n}_{\text{s}}}}{{{\tau}_{\text{up,g}}}}\exp \left(-\frac{{{\Delta }_{\text{inj}}}}{k{{T}_{\text{eg}}}}\right).\end{eqnarray} \tag{ 12a }$$

For a device with J_th and Δ_inj values of 1.5 kA cm⁻² and 150 meV one obtains δ_eg  =  0.18, and then, from equation (11), the effect of having hot electrons in the injector basically doubles the J_bf value by comparison to the case when they are not considered. That is, hot electrons strongly affect backfilling. This may explain why, for devices designed with Δ_inj  =  130–150 meV, the derived J_bf value was found to be significantly higher than the calculated value [59] when considering that T_eg is equal to with lattice temperature (i.e. equation (12a) with T_l instead of T_eg). For short-cavity (1–2 mm) devices [46, 60] or for devices of low Δ_inj (20 meV) [61] the backfilling current at RT can be as high as 70% of the J_th value. While for the latter case it is clear why the RT J_bf value is large, the former can be understood from the fact that for short-cavity devices the RT J_th value is relatively high, which will increase the T_eg value and, in turn, both J_bf and J_leak will increase, leading to high so-called 'transparency' currents. Revin et al [60] have shown that, if the lower level is not depopulated via resonant LO-phonon transitions to lower AR energy states, the backfilling current becomes  >  70% the RT J_th value, since most electrons in the injector reside in the lower level. As a result T₀ decreases from a typical value of 137 K for moderately high doped, conventional 5.5 μm emitting devices, to a value of only 106 K.

For devices with Δ_inj  ≈  100 meV [38, 40] equation (12a) with T_l instead of T_eg was found to overestimate the J_bf value. Then, a model involving backfilling from all subbands in the injector has been proposed by Maulini et al [38], but, while it provides lower J_bf values than when using the conventional formula, it implies much higher injector-doping values than indicated by the maximum current-density J_max value. For instance, in [40] the thus estimated n_s value was 5.7  ×  10¹¹ cm⁻², while the J_max value of ~10 kA cm⁻² indicates an actual n_s value of ~2.4  ×  10¹¹ cm⁻², in agreement with typical values for 8–9 μm emitting QCLs [50]. We find that Maulini et al's J_bf formula predicts well the J_bf value only if one assumes a T_eg value ~100 K higher than the lattice temperature at RT, which agrees with the findings that electrons are hot in 8–9 μm QCL as well [46, 47]. That is, reasonably accurate J_bf values are obtained with Maulini's derived expression, only when considering hot electrons:

$$\begin{eqnarray}&&{{J}_{\text{bf}~}}\cong \frac{q{{n}_{\text{s}}}}{{{\tau}_{\text{up,g}}}}\exp \left(-\frac{{{\Delta }_{\text{inj}}}}{2k{{T}_{\text{eg}}}}\right)\frac{\sinh \left[{{\Delta }_{\text{inj}}}\text{/}\left(2{{N}_{\text{inj}}}k{{T}_{\text{eg}}}\right)\right]}{\sinh \left[(N+1){{\Delta }_{\text{inj}}}\text{/}\left(2{{N}_{\text{inj}}}k{{T}_{\text{eg}}}\right)\right]};\,\\ &&{{\Delta }_{\text{inj}}}\approx 100\text{meV,}\end{eqnarray} \tag{ 12b }$$

where N_inj stands for the number of subbands in the injector lower miniband. In either case, the electron temperature in the injector lower miniband has a strong effect on the J_bf value; thus, lowering the T_eg value, for instance by lowering the overall J_th value, is likely to significantly decrease the backfilling current.

Currently, for characterizing device performance, the following J_th expression is used:

$$\begin{eqnarray}&&{{J}_{\text{th}}}=\frac{{{\alpha}_{\text{m}}}~+~{{\alpha}_{\text{w},\text{eff}}}}{\Gamma g},\end{eqnarray} \tag{ 13 }$$

where α_w,eff is an effective waveguide loss, Γ is the transverse optical-confinement factor to the core region, and g is the differential gain. Using equations (9b) and (9c) and comparing to equation (13), the following relationships can be written:

$$\begin{eqnarray}&&\Gamma g={{\eta}_{\text{p}}}\left(~{{\tau}_{\text{up,g}}}{{g}_{\text{c}}}/q\right),\end{eqnarray} \tag{ 14a }$$

$$\begin{eqnarray}&&{{\alpha}_{\text{w,eff}}}={{\alpha}_{\text{w}}}+\left({{\tau}_{\text{up,g}}}{{g}_{\text{c}}}/q\right){{J}_{\text{bf,th}}}={{\alpha}_{\text{w}}}+~{{\alpha}_{\text{bf}}}.\end{eqnarray} \tag{ 14b }$$

That is, from a J_th versus 1/L plot, where L is the cavity length, the derived value for the modal differential gain, Γg, reflects not only the transverse optical-mode confinement (i.e. Γ), but also the degree of carrier leakage, as expressed by the pumping-efficiency term η_p, which for conventional mid-IR QCLs is ~0.85 at 300 K and ~0.77 at 360 K [11]. In effect, η_p represents a carrier-confinement term, as expected given the shunt-current nature of J_leak. By defining, from (14a), a modified effective upper-state lifetime τ'_up,g  =  η_pτ_up,g, and using equations (9b) and (12a), the J_th equation becomes:

$$\begin{eqnarray}&&{{J}_{\text{th}}}=\frac{q}{\tau _{\text{up,g}}^{'}}\frac{{{\alpha}_{\text{m}}}+~{{\alpha}_{\text{w}}}}{{{g}_{\text{c}}}}+\frac{{{J}_{\text{bf}}}}{{{\eta}_{\text{p}}}}\\ &&\approx \frac{q}{{{\tau}^{'}}_{\text{up,g}}}\left[\frac{{{\alpha}_{\text{m}}}+~{{\alpha}_{\text{w}}}}{{{g}_{\text{c}}}}+{{n}_{\text{s}}}\exp \left(-\frac{{{\Delta }_{\text{inj}}}}{k{{T}_{\text{eg}}}}\right)\right]\end{eqnarray} \tag{ 15 }$$

in a form that includes all optical losses, the backfilling term as well as takes into account the carrier leakage through the use of $\tau _{\text{up,g}}^{'}$ rather than τ_up,g. The difference is significant, especially for conventional devices in which case carrier leakage increases [11] the J_th value by ~18% at 300 K and ~30% at 360 K. As for the experimentally derived value of the effective waveguide loss (equation (14b)), it contains the actual waveguide loss plus a term reflecting backfilling, α_bf, sometimes called 'resonant' waveguide loss [59]. Due to backfilling, α_w,eff is a strongly temperature-dependent quantity, which represents the actual waveguide loss α_w only at cryogenic temperatures. In order to extract α_w values at room temperature one has to compare the slope efficiencies of devices of different α_m values [40, 59].

As mentioned above, we have found that for E₅₄  ⩾  50 meV the primary leakage path is through state 5 [11, 16]; thus, carrier leakage can be assumed to occur only via state 5 (figure 3). Then, from the rate equations [58] with J  =  J₄  +  J_5,leak, where J₄  =  J₀  +  J_bf  =  qn₄/τ_4g and J_5,leak   =  qn₅/τ_5,leak   =  qn₄τ_5,tot/τ₄₅τ_5,leak, an expression for η_p is derived (equation (8)) which, in turn, provides the following expression for effective upper-state lifetime $\tau _{\text{up,g}}^{'}$:

$$\begin{eqnarray}&&\frac{1}{{{\tau}^{'}}_{\text{up,g}}}=\frac{1}{{{\tau}_{\text{up,g}}}{{\eta}_{\text{p}}}}=\frac{1}{\left(1-{{\tau}_{3\text{g}}}/{{\tau}_{43}}\right)}\left(\frac{1}{{{\tau}_{4\text{g}}}}+\frac{1}{{{\tau}_{45}}}\frac{{{\tau}_{5,\text{tot}}}}{{{\tau}_{5,\text{leak}}}}\right).\end{eqnarray} \tag{ 16 }$$

That is, due to carrier leakage, the required value for the upper-state carrier-sheet-density n₄ at threshold is $1/\left[1+\left({{\tau}_{4}}_{\text{g}}{{\tau}_{5,\text{tot}}}/{{\tau}_{45}}{{\tau}_{5}}_{,\text{leak}}\right)\right]$ higher than in the case of no leakage. Note that for a short-barrier case (i.e. no level 5; thus, leakage to the continuum) τ₄₅ becomes τ_4,leak and the equation reduces to equation (2b).

Finally, in the J_th equation some authors [62, 63] call the term which is added to the J_0,th term, transparency-current density J_tr, since backfilling is thought of as corresponding to 'resonant' intersubband (ISB) absorption [59]. However, the leakage-current density at and above room temperature is by no means a negligible term for QCLs in general [11, 41] and the shunt current through the high-energy AR states contributes to the backfilling current (equation (9d)). Thus, considering the J_th  −  J_0,th quantity to be a transparency-current density is incorrect, since carrier leakage and carrier-leakage-induced backfilling have nothing to do with absorption.

Conventional high-performance QCLs emitting in the 4.5–5.5 μm range have suffered from severe carrier leakage, which manifested itself as low characteristic temperature T₀ values (130–150 K) [11, 45, 60, 65]. Such devices had moderately high-doped injector regions, in order to obtain the large dynamic range necessary for high-power operation. If the devices have low- doped (i.e. n_s  =  (0.5–0.7)  ×  10¹¹ cm⁻²) injectors, the T₀ value is ~200 K, as the backfilling current is significantly reduced [42, 66–68]; thus, such devices are useful mostly for low-power-dissipation, low-output-power applications.

By suppressing the carrier leakage via multidimensional CB engineering [17], in moderately high-doped devices, has led to T₀ values in the 230–285 K range [10, 18–20]. More specifically, the carrier leakage was suppressed by increasing the ${{E}_{\text{ul}+1,\text{ul}}}$ value, from ~44 meV for conventional devices to values in the 60–90 meV range, and the ${{\tau}_{\text{ul}+1,\text{ul}}}$ value, from ~0.22 ps for conventional devices to values in the 0.32–0.72 ps range, by means outlined in section 5. For low-doped devices with suppressed carrier leakage a T₀ value of 363 K was recorded [13] due to reduced backfilling and, due to a strongly diagonal transition, a relatively large, highly temperature-insensitive J_th,0 term. Good agreement with experiment was obtained for calculated T₀ values for both deep-well (DW) and conventional moderately high-doped, 4.6–4.8 μm emitting devices, of DPR lower-laser-level depopulation scheme, by using the above outlined definitions for J_th and an iterative process outlined in [11, 16] (table 1). For calculating T_ei values we used α_e–l  =  35 K cm² kA⁻¹, as measured for T_eg in 4.8 µm emitting QCLs [44]. Since state g is strongly coupled to the upper level, state 4, it was assumed that at threshold T_e4  ≈  T_eg, which was recently validated to be accurate from Monte-Carlo based calculations of GaAs QCLs [32]. We also used T_e5  ≈  T_e6  ≈  T_e4, although such assumptions may well not be accurate. In any event, since leakage primarily occurs through state 5, the T_e5 and T_e6 values are of little relevance.

Similar good agreement between experimental and theoretical T₀ values (i.e. 143 K versus 147 K, over the 260–300 K temperature range) has been obtained for 4.6 μm emitting devices of single-phonon-resonance (SPR) lower-level depopulation scheme [45], by assuming hot electrons in the upper level. The T₀ values are lower than for DPR-type devices of ~70 meV ${{E}_{\text{ul}+1,\text{ul}}}$ value, most likely due to higher contribution from backfilling currents, since Δ_inj is significantly smaller than for DPR devices (i.e. ~125 meV [58] versus ~150 meV).

For 3.9 μm emitting SPR devices [41], suppression of carrier leakage, by increasing by design the ${{E}_{\text{ul}+1,\text{ul}}}$ value to ~120 meV, has resulted in a relatively high T₀ value (i.e. 175 K over the 150–300 K temperature range), not as high as for optimized ~3.8 μm emitting SPR devices [69]; possibly due to a relatively low Δ_inj value (135 meV) and/or higher injector doping level.

Many of the T₀ values mentioned above are from 3 mm long, uncoated-facets devices. However, T₀ is a function of mirror loss, generally decreasing slowly with increasing cavity length [41, 60]. This can be understood from equations (9b) and (9c); in that, as ${{\alpha}_{\text{m}}}$ decreases the relatively temperature-insensitive portion of J_th (i.e. J_th,0) decreases; thus, J_th becomes more temperature sensitive for longer-cavity devices.

As expected, another factor that affects the T₀ value is the electronic temperature in the upper level, T_eul. Since at threshold and a lattice temperature, T_l, T_eul  =  T_l  +  α_e–l  ⋅  J_th, it follows that, if J_th significantly decreases, T_eul will decrease which will lead to lower backfilling and leakage currents; thus, to higher T₀ values. A good example is what happened in the case of 4.6 μm emitting devices of non-resonant-extraction (NRE) lower-level depopulation scheme when going from 3 mm long, uncoated devices of α_w ~ 1.2 cm⁻¹ [70] to 3 mm long, high-reflectivity (HR)-coated devices of α_w ~ 0.5 cm⁻¹ [63]. Then, the RT J_th value decreased from ~1.5 kA cm⁻² to ~0.95 kA cm⁻². In turn, the T₀ value increased from 140 K [70] to 168 K over the 290–350 K temperature range [55]. However, it is worth noting that although the ${{E}_{\text{ul}+1,\text{ul}}}$ value was relatively high (~63 meV); thus, comparable to those for DW and TA devices [17], the T₀ value for 3 mm long, uncoated devices, with α_w ~1.2 cm⁻¹ is much smaller than for DW/TA devices (i.e. 140 K versus 230–255 K). We attribute this difference in large part to the fact that, unlike in the case of DW and TA devices, the increase in ${{E}_{\text{ul}+1,\text{ul}}}$ value is not accompanied by an increase in the ${{\tau}_{\text{ul}+1,\text{ul}}}$ value, since there is strong overlap between the wavefunctions of the upper level and the next higher AR energy state [71]; thus, the leakage current is similar to that in conventional QCLs. Further evidence that carrier leakage is significant in NRE-type QCLs comes, in part, from the fact that the experimentally [63] measured value for the internal differential efficiency per period [17] $\eta _{i}^{d}$ is significantly smaller (i.e. ~50%) than that (i.e. ~70%) for 4.9 μm emitting 'shallow-well' QCLs of suppressed carrier leakage [19], which have been shown [17] to, de facto, be TA-SPR-type QCLs. (In fact, the NRE-QCL $\eta _{i}^{d}~$ value is smaller than that for conventional 4.65 μm emitting DPR devices (i.e. ~60%) [72]). Another reason for the lower $\eta _{i}^{d}$ value in NRE versus TA-SPR devices is a lower laser-transition efficiency [73] η_tr value (i.e. 0.81 versus 0.89), as calculated from device modelling including elastic scattering [58].

Similarly, 4.0 μm emitting NRE-type QCLs with In_0.73Ga_0.27As QWs [55] most likely owe their relatively low T₀ and T₁ values (135 K and 168 K, respectively) to carrier leakage through state 5, since significantly higher J_th(300 K) values translate into higher T_eul values than for NRE-type 4.6 μm emitting QCLs. The suggested reason: leakage to indirect valleys, is highly unlikely, given that the most recent measurements of the L-valley position in strained In_0.73Ga_0.27As QWs [74] found it to be 40 meV higher than expected above the Γ valley; thus, for the QCLs in [55], the L valley lies above state 5, similar to what Flores et al found for their 3.9 μm emitting QCLs [41].

Another example of the effect of the T_eul value on the T₀ value is what happened for SPR-type 4.7 μm emitting devices of strong coupling [75]. Although the ${{E}_{\text{ul}+1,\text{ul}}}$ value was high (71 meV) [58], T₀ was only moderately high (188 K) since the RT J_th value was high (~2.5 kA cm⁻²) which, in turn, increased the T_eul value, and thus the leakage current. Evidence of carrier leakage similar to that in conventional DPR 4.7 μm QCLs is that the measured T₁ had a rather small value of 149 K [76].

The external differential quantum efficiency η_d for a QCL having N_p periods can be written:

$$\begin{eqnarray}&&{{\eta}_{\text{d}}}=\eta {{_{\text{i}}^{\text{d}}}_{{}}}\frac{{{\alpha}_{\text{m}}}}{{{\alpha}_{\text{m}}}+{{\alpha}_{\text{w}}}}{{N}_{\text{p}}}=\eta _{\text{inj}}^{\text{tot}}\eta {{_{\text{tr}}^{{}}}_{{}}}_{{}}\frac{{{\alpha}_{\text{m}}}}{{{\alpha}_{\text{m}}}+{{\alpha}_{\text{w}}}}{{N}_{\text{p}}}\cong {{\eta}_{\text{p}}}\eta {{_{\text{tr}}^{{}}}_{{}}}\frac{{{\alpha}_{\text{m}}}}{{{\alpha}_{\text{m}}}+{{\alpha}_{\text{w}}}}{{N}_{\text{p}}},\end{eqnarray} \tag{ 17a }$$

where η_p, as derived in [49], is the differential pumping efficiency at and close to threshold [11, 49] and is given by:

$$\begin{eqnarray}&&{{\eta}_{\text{p}}}=1-\frac{{{J}_{\text{leak,th}}}}{{{J}_{\text{th}}}},\end{eqnarray} \tag{ 17b }$$

η_tr is the laser-transition differential efficiency [73]; α_m and α_w are the mirror loss and actual waveguide loss (due to free-carrier absorption (FCA) and ISB absorption), respectively. The product η_pη_tr can be thought of as the internal differential efficiency per period $\eta _{i}^{d}~$ [16], when the tunnel-injection efficiency (into the upper level) $\eta _{\text{inj}}^{\text{tun}}~$ is taken to be unity. However, in the case $\eta _{\text{inj}}^{\text{tun}}~$ is less than unity, due for example to electron scattering from the injector ground state in the next higher energy state(s) above the upper level [42, 43], it becomes a multiplying factor in equation (17a), since it accounts for another leakage path [34]. Then, the total injection efficiency, $\eta _{\text{inj}}^{\text{tot}}~$ in equation (17a), is equal to $\eta _{\text{inj}}^{\text{tun}}$η_p, basically as used in equation (9b). For QCLs, the validity of using the η_p term in the equation (17a) was confirmed by the interpretation [10] of Yang et al's experimental data from QCLs of tall exit barrier [33], and by Flores et al's experimental analysis of carrier leakage via IFR [53].

Severe carrier leakage in conventional high-performance QCLs emitting in the 4.5–5.0 μm range has also manifested itself as low characteristic-temperature T₁ values (140–170 K) for the slope efficiency over wide heatsink-temperature ranges above RT [11, 42, 65]. Since η_tr is basically temperature independent, it is obvious from equation (17a) that the T₁ parameter is primarily determined by the variations with temperature of both η_p and α_w [11]. Note that, just as in the case of interband-transition lasers [14], carrier leakage strongly affects the T₁ value. That is, T₁ is a much better indicator of carrier leakage than the T₀ parameter, since T₀ reflects the temperature dependence of the backfilling current as well.

While T₀ values have been successfully predicted theoretically for both conventional [11, 45] and CB-engineered 4.6–4.8 μm QCLs [11], that has not been the case for T₁ values. As shown in [11], the use of the calculated J_leak values that worked in predicting well the experimental T₀ values, resulted in T₁ values significantly higher than the experimental ones. We concluded that an increase with temperature of the α_w value had to account for this difference; a reasonable assumption since α_w of state-of-the-art 4.6–4.8 μm emitting devices have been reported [70] to be primarily (~90%) due to ISB absorption. Since the ISB-absorption portion of α_w, α_ISB, increases with temperature [77, 78], α_w should also increase with temperature. While, for devices emitting in the 4.5–5.5 μm wavelength range, there are no experimental data as far as the temperature dependence of α_w, such data have been published for 8.4 μm emitting QCLs [79]. We plot in figure 5(a) the α_w and α_w,eff variations with increasing temperature, extracted from figure 4.8 of [79], for DPR- and bound-to-continuum (BC)-type devices [50]. Since the free-carrier-absorption part of α_w is weakly temperature dependent, and α_ISB can be as much as 73% of α_w [80], it becomes clear that the increase of α_w with temperature (e.g. from 7.6 to 11.3 cm⁻¹ over the 243–400 K range, for DPR devices) is primarily due to ISB absorption for both device types. The much stronger increase with temperature of α_w,eff is due to increased backfilling, and it is quite similar to the behaviour of α_w,eff for 8.2 µm emitting DPR-type devices [78].

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Figure 5(b) shows, for the 8.4 μm emitting DPR QCLs [50], the relative variations with temperature of the slope efficiency η_sl and of the quantity α_m/(α_m  +  α_w), with α_w values taken from figure 4.8 of [79], with respect to their 20 °C values. While η_sl decreases by ~33% over the 20–103 °C range; thus, giving a T₁ value of 263 K, the α_m/(α_m  +  α_w) quantity decreases by only 18%. The difference; that is, an additional 18% decrease, is due to the temperature variation of the η_pη_tr product. Since η_tr is basically temperature insensitive, the ~18% decrease reflects the increase in relative carrier leakage (i.e. J_leak/J_th) with increasing temperature, similar to what happens in 4.5–5.0 μm QCLs [11]. Such a relatively high increase in J_leak/J_th is expected, in view of the fact that electrons have been found to be 'hot' in the upper level of 8–9 μm emitting QCLs as well [46, 47]. Such leakage, at and above RT, explains the relatively low values for T₀ and T₁: ~160 K [50] and ~263 K, respectively. Further proof of carrier leakage is the relatively low RT $\eta _{\text{inj}}^{\text{tot}}$ value (~0.86) obtained from more advanced modelling of the same devices [81]. We calculate for the 8.4 μm DPR devices [50] values of ~0.98 and ~0.89 for $\eta _{\text{inj}}^{\text{tun}}$ and η_p which give an $\eta _{\text{inj}}^{\text{tot}}$ value of ~0.87; in very good agreement with the 0.86 value deduced in [81]. As we shall see in section 6.2.3, the carrier leakage can be suppressed by employing a step-taper TA (STA)-type AR design quite similar to that proposed for high-performance 4.5–5.0 μm QCLs [17].

Carrier-leakage suppression via conduction-band engineering of 3.9–5.0 μm QCL [17, 41], has led to T₁ values in the 285–800 K range [13, 18–20, 41, 49]. For 4.9 μm, TA-STR-type ridge-guide devices [13, 19], increasing the average doping per period and the period number from 30 to 40 led to an increase in α_w from ~0.5 cm⁻¹ to ~0.75 cm⁻¹, consistent with the observation that for state-of-art-devices α_w is mostly due to ISB losses in the core [70] and with that fact that α_ISB is proportional with the number of carriers in the injector (i.e. out of the 50% increase only 14% is due to the higher Γ value, with the rest being mostly due to the ~25% increase in doping). Since, as mentioned above, α_ISB increases with temperature, an increase in the α_w value with temperature will, in turn, lead to a decrease in the T₁ value. Indeed, for the same mirror-loss value, T₁ decreased from 645 to 343 K; thus, confirming, for 4.5–5.0 μm QCLs as well, that the α_w increase with temperature is, in a significant way, partly responsible for the T₁ value. Note also that, if α_w is relatively cavity-length (L)-independent, the T₁ value will increase with decreasing L, as long as the carrier leakage does not significantly increase.

As mentioned above in the threshold-current section, significantly lowering the J_th value causes carrier-leakage suppression due reduced electronic temperature in the upper level, as evidenced by an increase in the T₀ value of NRE-type devices as J_th was lowered. Similarly, for those same NRE devices, significantly J_th lowering led to an increase in the T₁ value from 140 to 295 K [55]. Since for those devices the ratio of α_w to α_m is basically the same, it can be clearly seen from equations (3), (7) and (17a) that the increase in T₁ value can be primarily attributed to better carrier-leakage suppression through reduced electronic temperature. Nonetheless, as discussed in section 3.3, there is still a fair amount of carrier leakage, as evidenced from lower $~\eta _{i}^{d}~$ values than in conventional 4.6 μm emitting. DPR-type QCLs (i.e. 50% versus 60%).

The expression for the pulsed maximum wallplug efficiency η_wp,max can be written as [11]:

$$\begin{eqnarray}&&{{\eta}_{\text{wp,max}}}={{\eta}_{\text{s}}}{{\eta}_{\text{p}}}{{\eta}_{\text{tr}}}\frac{{{\alpha}_{\text{m,opt}}}}{{{\alpha}_{\text{m,opt}}}+{{\alpha}_{\text{w}}}}\left(1-\frac{{{J}_{\text{th}}}}{{{J}_{\text{wpm}}}}\right)\frac{{{N}_{\text{p}}}h\nu}{q{{V}_{\text{wpm}}}},\end{eqnarray} \tag{ 18 }$$

where η_s reflects the deviation from a straight line of the pulsed L–I curve at the η_wp,max point [11], α_m,opt is the optimal mirror loss [63], J_wpm is the current density at the η_wp,max point, hν is the photon energy and V_wpm is the voltage at J_wpm. The value for α_m,opt has been found both theoretically [63] and experimentally [19, 63] to be ~2.2 cm⁻¹. Analysis of published data reveals that the J_wpm value, in both pulsed and CW operation, is an approximate fixed multiple of the pulsed J_th value at RT, J_th (300 K); that is: J_wpm  ≅  BJ_th (300 K). For instance, for devices with significant carrier leakage [64, 71] we found the B value to be ~2.5 [11]. However, for devices of moderately high doping and suppressed carrier leakage [19] the B value increases to ~3.0. As for η_s, its value is found to be ~0.90 at 2.5  ×  threshold for devices with typical carrier leakage [63]. For devices with suppressed carrier leakage [19] η_s is ~0.95 at 2.5  ×  threshold and ~0.92 at 3  ×  threshold. The higher η_s value at the same drive level most likely reflects carrier-leakage suppression. That is, J_leak continues to increase about threshold, since it is proportional with n₄, which increases above threshold, and, as observed by Pflugl et al [42], at high drives above threshold there is leakage due to carrier thermally activation from the injector ground state to high-energy AR states. Thus, the higher the energy difference between the upper level ul and/or state g and the state ul  +  1, the less droop is expected for the L–I curve at high drives above threshold.

At a heatsink temperature T_h above room temperature (i.e. above 300 K) η_wp,max can be written as such:

$$\begin{eqnarray}&&{{\eta}_{\text{wp,max}}}\left({{T}_{\text{h}}}\right)\cong {{\eta}_{\text{s}}}{{\eta}_{\text{d}}}\left(300~~\text{K}\right)\exp \left(-\frac{{{T}_{\text{h}}}-300~\,\text{K}}{{{T}_{1}}}\right)\,\,\,\,\\ &&\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\times \left[1-\frac{1}{B}\exp \left(\frac{{{T}_{\text{h}}}-300\,~\text{K}}{{{T}_{0}}}\right)\right]\frac{h\nu}{q{{V}_{\text{wpm}}}}.\end{eqnarray} \tag{ 19 }$$

With increasing heatsink temperature the pulsed η_wp,max is clearly affected by the T₁ and T₀ values. We define a characteristic temperature T₂ for η_wp.max from the relation:

$$\begin{eqnarray}&&{{\eta}_{\text{wp,max}}}\left({{T}_{\text{h}}}\right)={{\eta}_{\text{wp,max}}}\left(300~~\text{K}\right)\exp \left(-\frac{{{T}_{\text{h}}}-300~K}{{{T}_{2}}}\right).\end{eqnarray} \tag{ 20 }$$

Taking both the T₀ and T₁ values for conventional, 3 mm long, back-facet HR-coated high-performance QCLs [11, 42, 82] to be 143 K over the 300–360 K temperature range, and B  =  2.5, we obtain T₂  =  71 K. In sharp contrast, for TA-type devices [20] with T₀  =  231 K, T₁  ≅  750 K and B  =  3, the calculated T₂ value is 249 K. This means that over the 300–360 K temperature range η_wp.max would vary 2.66 times slower than for conventional QCLs. For TA-type devices there are experimental results for the temperature dependence of η_wp,max for the so-called shallow-well QCLs [19]. Given T₀  =  244 K, T₁  =  343 K and B  =  3 in that case, the calculated T₂ value is 186 K in very good agreement with the 195 K value that can be extracted from the experimental data [19]; thus, confirming the accuracy of the approximation in equation (19). Then, for 'shallow-well' TA-type devices the η_wp,max value varies 2.15 times slower than in conventional devices; reflecting the beneficial effect that carrier-leakage suppression brings about as far as low temperature sensitivity of the η_wp,max value. Furthermore, carrier-leakage suppression is the major factor leading to the almost doubling of the single-facet, pulsed η_wp,max value at ~300 K; that is, from 15% in 4.6 μm emitting NRE-type [71] and SPR-type devices [45, 75] to 27% in 4.9 μm emitting TA-SPR devices [19]. More specifically, out of the 1.8 factor improvement in η_wp,max over the NRE-QCL value, 1.55 is primarily due to carrier-leakage suppression and higher η_tr value (i.e. 1.40 from higher value and 1.11 from the higher drive level at which η_wp,max occurs) with the rest of 1.16 being mostly due to a lower V_wpm value. Furthermore, the 27% η_wp,max value at λ = 4.9 μm corresponds to an ~29% value at λ = 4.6 μm [73]; thus, there is virtual doubling of the η_wp,max value at a given wavelength.

However, it is worth noting that the highest reported pulsed value for η_wp,max (i.e. 27% for 4.9 μm emitting devices) was obtained from devices with an $\eta _{i}^{d}$ value of only 70%; in that, with virtually suppressed carrier leakage and a calculated η_tr value of ~0.89 [58] one would expect $\eta _{i}^{d}$ values of ~86%. The 27% η_wp,max value is in good agreement with theoretical predictions by Faist [73], but only because those predictions consider $\eta _{i}^{d}$  ≈  70%. That is, we find that the solid-line prediction curve in figure 1 of [73], derived with key parameters from a 9 μm emitting QCL ([12] of that work), corresponds to $\eta _{i}^{d}$  ≈  70%. The difference between the experimentally obtained $\eta _{i}^{d}$ value and the expected ultimate value may be due to IFR-mediated carrier leakage, since the TA-SPR device uses has high CB offsets in the injector region which, in turn, are conducive to IFR-assisted leakage from state g to the high-energy AR states [43, 54]. Such leakage means that the $\eta _{\text{inj}}^{\text{tun}}~$ value is less than unity, which, in turn, decreases the $\eta _{\text{i}}^{\text{d}}~$ value in equation (17a). In sharp contrast, recently developed TA-type 8.7–8.8 μm emitting devices have provided $\eta _{i}^{d}$ values in the 85–90% range [83]; thus, approaching theoretical limits. Since for 8–9 μm QCLs the CB offsets are significantly smaller than for 4.5–5.0 μm QCLs, it is quite possible that IFR-mediated carrier leakage has a negligible impact on 8–9 μm QCLs, unless their core regions consist of heavily strain-compensated structures, in which case $\eta _{i}^{d}$ drops again to relatively low values (58%) [40].

Figure 9(b) shows the so-called moderate-taper TA design [17]; that is, a stepwise approximation of the linear-taper design shown in figure 9(a). The design was implemented for the purpose of easing fabrication. The introduced asymmetry causes antisymmetric states 2 and 5 to be 'pulled' to the right which, in turn, results in higher τ_4g and τ₅₄ values, respectively, than in DW devices (see table 3). In particular, as seen from table 4, where we compare moderate-taper to conventional QCLs, τ_4g recovers to values close to those in conventional QCLs. The net effect is that the effective upper-level lifetime τ_up,g is pretty much the same for both cases. In addition, due to barrier-height tapering, the E₅₄ value has increases to 67 meV versus 44 meV in conventional QCLs, and the τ₅₄ value is two times larger than in conventional QCLs (table 3). In turn, carrier-leakage suppression is expected to be significantly more pronounced than in DW devices. By using the same approach as in [11] for calculating the leakage-current density, and considering global values for both τ_5,leak and τ_5,tot lifetimes, we obtain a relative leakage-current density value J_leak/J_th of ∼ 4%, that is, less than half that for DW devices and almost four times less than for conventional devices. Note that the assumption of halving the 300 K τ_4g values in order to take into account elastic scattering [11, 52] is still basically valid, since for both TA and conventional QCLs the CB offsets, where the lasing transition occurs, are similar (~800 meV), and τ₄ involves diagonal transitions to only one lower energy level. In turn, since τ_up,g is basically the same, an ∼ 10% decrease in the J_th (300 K) value is expected, primarily due to carrier-leakage suppression. Thus, tapering the barrier heights in the AR significantly suppresses carrier leakage and, unlike in DW devices, it also reduces the J_th (300 K) value.

Thirty-period QCL devices of active-region design shown in figure 9(b) were grown by MOCVD. Details of the growth, the material characterization, and the structural design are provided in [20] and [96]. Initial devices [49, 97] involved inadvertently heavy-doped injector regions and resulted, due to increased backfilling, in high J_th values (~2 kA cm⁻²) and relatively low T₀ values (~180 K). However, the T₁ values were quite high (454 K); thus, confirming that TA QCLs suppress carrier leakage much more than DW QCLs. As seen below, for medium-doped TA devices the T₁ values are much higher (~750 K), as expected since the ISB part of α_w is a basically proportional with the carrier concentration in the core region and thus with the injector-doping level [78, 80]. A similar behaviour is observed for TA-SPR devices of different doping levels (i.e. the T₁ value decreases with increasing the injector-doping level) [13, 19]. In figure 10(a), we show, for medium-doped devices, the light–current (L–I) curves in pulsed operation (100 ns, 2 kHz) for 3 mm long, 19 μm wide ridge, uncoated-facets devices as the heatsink temperature is varied. An inset shows that the devices lased at ∼4.8 μm. J_th (300 K) is 1.58 kA cm⁻², a value ∼15% less than for conventional QCLs of same geometry [11]. 10% of that difference is due to carrier-leakage suppression, with the rest of 5% most likely due to lower backfilling current in these TA QCLs versus conventional QCLs, as a result of lower injector-region sheet-doping density in the former (i.e. 0.83  ×  10¹¹ cm⁻² versus 10¹¹ cm⁻²).

The carrier-leakage suppression is evidenced by high T₀ and T₁ values (figure 10(b)). The T₀ values are ∼ 230 K, typical of devices with suppressed carrier leakage and moderately high injector-doping level [11, 19], as needed for high-CW-power performance. T₁ values as high as 797 K have been measured which, to the best of our knowledge, is the highest T₁ value reported to date from any narrow-gain, mid-IR QCL. On average, the T₁ values are ∼ 750 K, that is, almost three times larger than for DW QCLs [11, 18] and more than twice the T₁ value (i.e. 343 K) for TA QCLs of SPR design and similar injector-doping level [19]. However, the latter is mainly due to a stronger influence, in the case of TA-SPR QCLs, of the temperature dependence of the ISB part of α_w on the temperature dependence of ${{\eta}_{d}}$, since α_m and the FCA-related part of α_w are significantly smaller than those for the TA DPR devices reported in [20].

We note, from table 3, that the linear-taper TA QCL has significantly higher E₅₄ and τ₅₄ values than moderate-taper TA QCL, as expected due to increased barrier-height tapering [17]. In turn, further carrier-leakage suppression will occur (J_leak/J_th of ∼ 3%) [49] and higher T₀ and T₁ values are expected. Furthermore, there is resonant extraction from both lower AR energy states 3 and 2 (i.e. miniband-type extraction [83]), which, in turn, will provide higher η_tr and slope-efficiency values than for moderate-taper TA QCL. However, miniband extraction while keeping a vertical transition (i.e. a z₄₃ value  >  15 Å) comes at a price in τ_up,g; thus, τ_up stays basically the same as for moderate-taper devices. With respect to conventional QCLs there is almost full recovery of the τ_up,g value, while achieving substantial carrier-leakage suppression; that is, one is likely to obtain high- T₀ and -T₁ QCLs with RT thresholds ~10–12% less than conventional QCLs of same injector-doping level as well as significantly higher RT slope efficiencies due to increases in both η_p and η_tr. Next we discuss the application of linear AR-barrier tapering to QCLs of SPR lower-level depopulation scheme and miniband-type extraction.

Figure 11(a) shows the conduction-band diagram, calculated at the operating point, for the so-called 'shallow-well' QCL [13]; which is a TA-type QCL, since a shallow barrier (Al_0.48In_0.52As) (adjacent to a shallow well), a normal-height barrier in the AR centre (Al_0.64In_0.36As) and an exit barrier composed of Al_0.64In_0.36As layers sandwiching a thin (6 Å) AlAs layer, form a linear-taper TA-type AR structure [17]. In addition, there are six AlAs layers inserted in the barriers of the extractor/injector region, in order to ensure a large enough minigap to prevent carrier escape from the energy state right above the upper level, state 4, to the upper Γ miniband; thus, to the continuum. Furthermore, we calculate that the six AlAs layers ensure the high value for the energy difference between the upper level and state 4, E₄₃ (i.e. ~85 meV rather than ~69 meV in the case without the AlAs layers) [98]. This 4.9–5.0 μm emitting TA-type QCL was reported [13] as having an AR structure of SPR lower-level depopulation scheme. Actually, it is not the 'classical' SPR scheme involving lower-level depopulation to one energy state and resonant extraction from that state. Rather, there is resonant extraction from both the lower laser level, state 2, and state 1; thus, extraction happens from a miniband [58] similar to the bound-to-continuum (BC) AR design. This miniband-type extraction was recently also employed for 8.7–8.8 μm emitting STA-type QCLs [83], and is a key feature of the 4.5–5.0 μm emitting, STA-type QCLs treated below in section 5.3.

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As pointed out by Faist [58], the shallow well and shallow barrier, around where the lasing transition occurs in the AR, cause a significant decrease in the negative influence of IFR on the τ_up,g value. Furthermore, the effect of sharply reduced IFR scattering appears to compensate for the increase in alloy-disorder scattering [51] associated with stronger upper-level wavefunction penetration in the barriers as well as with the presence of layers of mole-fraction compositions close to 0.50. This is reflected by the fact that when comparing the TA-SPR device to conventional-height barrier devices [58] the former has the highest (inelastic  +  elastic) τ_up,g value. In addition, by using values calculated in [58], we note that the ratio of τ_up,g values considering both inelastic and elastic scattering to τ_up,g values considering only inelastic scattering is basically the same (i.e. ~0.22) for the TA-SPR device and an SPR device of conventional-height barriers [45]. This means that while for TA-SPR devices the decrease in IFR scattering, by lowering the barriers where the transition occurs, is compensated in part by alloy-disorder scattering, for conventional-barrier SPR devices the higher IFR scattering is compensated in part by lower alloy-disorder scattering.

The structure has strong coupling between the injector ground state, g and the upper level, state 3; in that, the splitting at resonance between those levels is ~14 meV (at a field of 128 kV cm⁻¹) by comparison to typical splitting values of 4–6 meV for QCLs emitting in the 4.0–5.5 μm range. Then, the operating point is generally taken such that the energy difference between the upper level, state 3, and the injector g state, E_3g, is about one kT larger than the splitting value [45, 75]. Specifically, we chose the operating point at 87 kV cm⁻¹, since there, with our k  ⋅  p code, E_3g is 41 meV and Δ_inj is 123 meV, similar to the 124 meV value Faist considered when calculating this same device [58]. Then, as shown in figure 11(a), the energy difference between the upper level and the next higher energy level, E₄₃, is 85 meV, in agreement with the 90 meV value reported by Razeghi et al [99]. In addition, τ₄₃ has a quite high value: 0.72 ps; ensuring strong carrier-leakage suppression. Thus, this steep linear-taper TA device follows the same trends seen in table 3 for TA-type devices. However, in sharp contrast to those devices, the strong carrier-leakage suppression comes at the price of virtual halving the thermal conductance, due to the presence of the seven additional AlAs barriers in the injector region (see section 4.2).

Table 5 provides key parameters of the device, and is used for comparing to other designs of ${{E}_{\text{ul}+1,\text{ul}}}$  >  80 meV and miniband-type extraction (i.e. 4.5–5.0 μm emitting STA-type QCLs), discussed below in the next subsection. Since the transition is diagonal, the dipole matrix element, z₃₂, is relatively small (10.6 Å) and the calculated τ_up,g is relatively large (2.14 ps). By comparison, for conventional [82] and TA-DPR devices [20] vertical transitions give z₄₃ ~15.6 Å and τ_up,g ~1.25 ps. In addition, the EL linewidth of vertical-transition devices is at least 1.35 times smaller than that of diagonal-transition devices [75]. Considering a conventional DPR device of same geometry [72] we calculate a J_th (300 K) value of ~1.20 kA cm⁻². Since the ratio of $~{{\eta}_{p}}~$ values between this device and conventional ones [72] is ~1.14, the estimated value of the J_th (300 K) value, minus the difference in leakage-current densities, is ~1.05 kA cm⁻². In contrast, for the TA-SPR device the J_th (300 K) value is ~25% higher (i.e. 1.31 kA cm⁻²). The difference is likely due to lower z₃₂, larger EL linewidth and backfilling current, in spite of higher τ_up,g value, even when interface roughness and alloy scattering are taken into account [58]. Nonetheless, this fits the classical tradeoff of higher threshold for higher CW power, although such a tradeoff does not necessarily apply for obtaining maximum CW wallplug efficiency [17].

The carrier-leakage suppression is evidenced by high T₀ and T₁ values (figure 11(b)). The T₀ value is 244 K and the T₁ value, measured from the published data, is 343 K. Further proof of strong carrier-leakage suppression is an $\eta _{i}^{d}$ value of 70% [19], the highest value reported to date from 4.5–5.0 μm emitting QCLs. However, as pointed above in section 3.3.2, the theoretical expected $\eta _{i}^{d}$ value is ~86%; thus, another carrier-leakage mechanism is likely at work, that decreases the injection efficiency and, in turn, lowers the $\eta _{i}^{d}$ value. Nevertheless, the above-mentioned attributes led to the achievement of record-high values for maximum pulsed and CW wallplug efficiency (27% and 21%, respectively) and maximum CW power (5.1 W) [19]. The same AR design was used by Metaferia et al [100], when employing a novel method for quick fabrication of BH-type devices, for obtaining 2.4 W single-facet CW power, in a single spatial mode with no beam steering. High T₀ and T₁ values were also obtained from those devices: 275 K and 442 K, respectively [101]. The somewhat higher values than those for the devices reported in [19] are most likely due to somewhat lower doping level, as evidenced by a lower J_max value [101]. However, as mentioned above in section 4.2, the seven highly strained layers inserted in the extractor/injector region cause a thermal resistance value more than twice that of conventional 4.5–5.0 μm QCLs, and, in turn, a rather high ΔT_act value (~50 K) is obtained at the CW η_wp,max point. Thus, long-term reliable operation, which is well known to be a strong function of ΔT_act in QCLs [102], is doubtful. Therefore, for a practical device one needs to implement TA-type or STA-type designs [17] that negligibly affect the device thermal resistance value.

We have presented [17] an STA-DPR design of high E₅₄ and τ₅₄ values (97 meV and 1.17 ps, respectively) which, in turn, resulted in negligible J_leak/J_th values (~1%) and a projected CW η_wp,max value of 27%, since the G_th value was estimated to be similar to that for conventional QCLs. However, that DPR-type design involves resonant extraction only from state 2. Although, carrier depopulation of state 1 via phonon absorption to state 2 may solve the issue (i.e. a pocket-extractor approach similar to the pocket-injector approach [87]), that is not a certainty. That is the reason why we present below an STA-DPR device with resonant extraction from both levels 2 and 1 as well as from the lower laser level; that is, miniband-type extraction, as in the cases of 4.9–5.0 μm emitting TA-SPR [13, 19] and 8.7–8.8 μm emitting STA-DPR QCLs [83].

The structure of the STA-DPR device with miniband extraction is shown on figure 12. The AR has deep wells and two 'normal' height barriers (Al_0.59In_0.41As and Al_0.65In_0.35As) followed by a tall (Al_0.93In_0.07As) third barrier. The design provides very high values for both E₅₄ and τ₅₄ (100 meV and 0.84 ps), somewhat higher than for TA-SPR devices, but, unlike in that case, without the need of additional AlAs layers inserted throughout the extractor/injector region. Key device parameters are listed in table 5. Due to Stark-effect reduction [17], the E₅₄ value is significantly higher than for the linear TA-DPR device shown in figure 9(a) (i.e. 100 meV versus 77 meV) and the τ₅₄ value is almost twice that for linear TA-DPR devices (i.e. 0.84 ps versus 0.49 ps (table 3)). Then, the calculated J_leak/J_th value is negligible (~1%). However, the constraint of resonant extraction from all three lower AR states, lowers the τ_up,g value to 1.15 ps; that is, τ_up,g is ~8% lower than for conventional QCLs. Nonetheless, due to virtual complete carrier-leakage suppression, the J_th (300 K) value is calculated to be ~7% less than that for conventional QCLs.

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The transition is basically vertical, given a z₄₃ value of 15.3 Å, basically the same as in conventional and TA-DPR QCLs (15.6 Å); thus, much higher than for TA-SPR QCLs (10.6 Å). The figure of merit; that is, $z_{\text{tr}}^{2}$  ⋅  τ_up,g, where z_tr is the lasing-transition dipole matrix element, considering only inelastic scattering, is somewhat higher than for TA-SPR QCLs (i.e. 268 Ų ps versus 239 Ų ps). When considering interface roughness, the τ_up,g value of TA-SPR devices will decrease much less than for STA-DPR devices, due to much lower CB offsets in the part of the AR where the lasing transition occurs (i.e. ~500 meV versus ~700 meV). However, the EL linewidth will be significantly narrower in STA-DPR devices, due to much less transition diagonality than in TA-SPR devices. To that effect, when calculating the IFR factor, summed over all transition energies, affecting the EL linewidth (i.e. equation (1) in [50]), we find it to be 59% that for the TA-SPR QCL. In any event, while J_th (300 K) for STA-DPR devices is expected to be ~7% less than in conventional QCLs, we have found above that for TA-SPRs devices J_th (300 K) is ~25% higher than in conventional QCLs; thus, STA-DPR devices are expected to have J_th (300 K) values ~26% less than for TA-SPR devices. The significantly lower J_th (300 K) value is likely to compensate for lower V_wpm values in SPR devices that in DPR devices, due to lower Δ_inj values (i.e. ~124 meV versus ~150 meV). Thus, the difference may well come down to the much higher G_th values, due to fewer interfaces per period and less strain, for STA-DPR devices compared to TA-SPR devices (table 5). Then, as pointed out in section 4.3, the devices are projected to reach CW η_wp,max values of at least 25%, while the core-temperature rise will be less than half that of TA-SPR devices (i.e. ΔT_act ~20 K) at the η_wp,max point. Such devices are expected to be reliable at high (3–4 W) CW powers, since, to date, the only lifetest data showing long-term QCL reliability [65] have been obtained from (low-power) QCLs with ΔT_act ~15 K.

It should be pointed out that a 4.8 μm emitting STA-like DPR QCL has been reported [44, 92]; in that, both AR barriers and QWs were step-tapered via inserting submonolayer-thick AlAs and InAs spikes in the barriers and QWs. While the thermal conductivity did improve compared to that of conventional QCLs, due to potential interface broadening [44], the STA-like AR design resulted in carrier-leakage suppression only below 0 °C [18]. That is, the inserted submonolayer-thick layers led only to modest increases in E₅₄ and τ₅₄ compared to conventional QCLs: ~51 meV versus 44 meV, and 0.30 ps versus 0.22 ps [98], because their overall effect was minor as far as increasing the effective CB offsets. In turn, while T₀ had a high value of ~240 K below 0 °C, it decreased to 154 K above 0 °C, and T₁ was only 143 K above 0 °C [18]. The T₁ value, in particular, indicates that, above 0 °C, the device had basically the same high degree of carrier leakage as conventional QCLs [11].

The STA-DPR concept with miniband extraction has been demonstrated for 8.4 μm and 8.8 μm emitting QCLs [83] and resulted in very high T₀ and T₁ values, for low-J_th 8–9 μm emitting devices, as well as in record-high $\eta _{i}^{d}$ values (85–90%) for any type of QCL. The results are discussed below in section 6.2.3.

Semtsiv et al [95] have used a conventional 3.8 μm emitting device with tall AR barriers (AlAs) and miniband-type extraction [103] for which they replaced the first barrier within the AR with a relatively shallow barrier (i.e. Al_0.48In_0.52As) (figure 13(a)). The result was a 3.9 μm emitting device, which, because of much less IFR influence on the EL linewidth and the upper-state lifetime, resulted in significantly reduced J_th (300 K) values. Employing within the AR a shallow barrier followed by two tall barriers is equivalent to using a STA-type structure. In turn, due to strong asymmetry and Stark-effect reduction [17], the E₄₃ value was very high (i.e. ~120 meV). As a result, carrier leakage was suppressed [41] which resulted in a high T₀ value of 175 K for 3.8–4.0 μm emitting QCLs, given the relatively high injector sheet-doping density (5  ×  10¹¹ cm⁻²), and, in particular, a high T₁ value of 550 K for any type of QCL, albeit below 300 K. More recently Flores [104] found the carrier leakage in such devices to be due to both IFR and LO-phonon scattering from two injector ground states to the AR energy state situated above the upper level. Note that in that case, in equation (9b), $\eta _{\text{inj}}^{\text{tun}}<1$; thus, ${{J}_{\text{th}}}=\left({{J}_{0,\text{th}~}}+{{J}_{\text{bf,th}~}}\right)/\eta _{\text{inj}}^{\text{tot}}$.

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For emission in the 4.5–5.0 μm range, we present in figure 13(b), a STA-SPR device with miniband extraction, just as employed in TA-SPR [13] and STA-DPR devices [83]. The AR has deep wells and two 'normal'-height barriers (Al_0.58In_0.41As and Al_0.62In_0.38As) followed by a tall (Al_0.93In_0.07As) third barrier, which is also the exit barrier. The structure is strongly coupled; that is, 15.4 meV splitting at resonance (133 kV cm⁻¹) between energy levels g and 3; thus, similar to the TA-SPR design [13]. Key parameters are listed in table 5. Step tapering brings about two advantages over the TA-SPR device: (1) the key energy differential E₄₃ has a similar high value (87 meV), but without the need of AlAs additional layers inserted throughout the extractor/injector region; (b) the τ₄₃ value is significantly higher: 0.99 ps versus 0.72 ps, giving an estimated ${{\eta}_{\,p}}$ value of 0.98; and, in turn, an ultimate limit of ~87% for the $\eta _{i}^{d}~$ value. Therefore, carrier leakage is virtually completely suppressed at no price in G_th; thus, holding the potential, just like the STA-DPR design, to result in long-term reliable operation at watt-range powers.

The transition is diagonal (z₃₂  =  11 Å) with a correspondingly long, inelastic-scattering upper-state lifetime: 1.82 ps. The figure of merit, only for inelastic scattering, is somewhat smaller than that for TA-SPR QCLs (i.e. 219 Ų ps versus 239 Ų ps). However, when calculating the IFR factor, summed over all transition energies, affecting the EL linewidth (i.e. equation (1) in [50]) we find it to be 76% of that for the TA-SPR devices. Thus, a smaller EL linewidth is likely to partly compensate for less τ_up,g reduction due to higher CB offset(s), where the transition occurs in the AR (i.e. ~700 meV versus ~500 meV). Further compensation for the reduction of the IFR-scattering effect on τ_up,g (when using low CB offset(s)) comes, as discussed in section 5.2.2, from less alloy-disorder scattering when using conventional-height barriers in the AR (where the lasing transition occurs). All in all, given similar Δ_inj values, the J_th (300 K) value should basically be the same as for the moderately doped TA-SPR device (~1.3 kA cm⁻²) [19]. Then, given that the V_wpm value will be similar to that for the TA-SPR QCL and that G_th is likely to be about twice that of the TA-SPR QCL (section 4.2), the devices are very likely to reach RT CW η_wp,max values of at least 25%, with relatively low core-temperature rise (~20 K).

Conventionally QCLs emitting in the 8–10 μm range, with low enough J_th values to operate CW at room temperature, have been grown with cores composed of InGaAs/AlInAs superlattices lattice-matched to InP. Typical RT J_th and η_sl/period values have been [50, 105] ~1.7 kA cm⁻² and ~25 mW/A, respectively. Due to the smaller lasing-transition energies involved, compared to 4.5–5.5 μm emitting SC QCLs, carriers in the upper level have stronger quantum confinement; thus, resulting in relatively large E₅₄ values (56–60 meV). In turn, above 300 K, typical T₀ (160–217 K) [50, 59, 80, 105–109] and T₁ values (192–260 K) [50, 110] have been higher than for conventional 4.5–5.5 μm emitting QCLs. (A T₀ value of 231 K was reported [111], over the 250–300 K temperature range, from a 9.8 μm QCL, but from short-cavity, high-J_th (3.1 kA cm⁻² at 300 K) devices; a similar behaviour to that for short-cavity, high-J_th 4.5–5.5 μm QCLs (section 3.1.2)). Nonetheless carrier leakage can be significant, with the main path being shunt leakage through the next higher energy state above the upper level [27], just like in the case of 4.5–5.5 μm emitting QCLs [11]. A highly sophisticated analysis of 8.4 μm emitting devices [50], taking into account elastic scattering, did fairly accurately predict the T₀ value (~160 K), but the predicted η_s values were virtually temperature insensitive, in stark contrast with the experiment. (i.e. T₁ ~260 K) (figure 5(b)).

Furthermore, estimates of the electronic temperature were conflicting. On the on hand, analysis based on density-matrix modelling of 8.4 μm emitting devices [81] concluded that electrons are in equilibrium with the lattice; although, as before [50], there was good agreement with experiment only as far as T₀, not as far as T₁. That indicates that carrier leakage was not taken into account, a fact also inferred from the relatively low derived $\eta _{\text{inj}}^{\text{tot}}$ value (see section 3.2.2). On the other hand, experiments on 7.8 μm [46] and 9.0 μm emitting [94] devices led to estimates of 100–160 K differences between the electronic and lattice temperatures at RT. Recently, as already mentioned above, a comprehensive analysis of 8.5 μm QCLs [47] has found that electrons in the upper laser level can have an electronic temperature T_eul ~500 K; that is, ~200 K above the lattice temperature at RT. That analysis combined with recent experimental results from strain-compensated devices (section 6.2) clearly pointing to some degree of carrier-leakage suppression associated with increasing CB offsets, that can only be explained as a consequence of decreasing T_eul values, indicate that for 8–9 μm QCLs hot electrons are as much of a reality as for 3.9–5.0 μm QCLs. Furthermore, the fact that the recent implementation of the STA concept for 8.4 and 8.8 μm emitting QCLs [83] has resulted in dramatic increases in both the T₀ and T₁ values, at little to no penalty in J_th, has established that conventional 8–9 μm emitting QCLs have significant carrier leakage as a result of hot electrons in the upper level.

An extensive analysis by Liu et al [78] of the temperature dependence of optical gain and effective waveguide loss, α_w,eff, in 8–10 μm emitting QCLs found that for vertical-transition devices α_w,eff does increase with temperature over the 200–300 K range, and attributed the increase to increases in both free-carrier absorption and intersubband absorption. Another part of the increase in α_w,eff may well have been due to the backfilling current, as clearly evident from Wittman's study [79] of the temperature dependence of α_w and α_w,eff for 8.4 μm emitting QCLs over the 240–400 K temperature range (figure 5(a)). The increase with temperature of the α_w value, together with carrier leakage explain the relatively low T₁ value (i.e. 192 K) [110] for 8.2 μm emitting QCLs [105]. However, for diagonal-transition devices emitting around 10 μm, α_w,eff was found to decrease with temperature, a behaviour believed to be due to the complex nature of resonant absorption in those devices.

Very high T₀ and T₁ values have been reported from 8–9 μm emitting QCLs of unconventional upper-state(s) populating mechanisms [112–114]; however, at a price in J_th and/or η_s. 8.0 μm emitting, indirect-pump scheme devices [112] provided high T₀ values: 243 K and 303 K, above RT, from 4 mm and 1.5 mm long chips, respectively, due to suppression of electron populations in the injector regions. However, they had relatively high RT J_th values of 2.7 kA cm⁻² and 3.3 kA cm⁻², respectively. Dual-upper-state (DAU), broad-gain QCLs [113, 114], have displayed 306–510 K T₀ values, mainly because of weak temperature-dependence of the peak gain. However, those occur for RT J_th  =  2.1–2.6 kA cm⁻², which are relatively high values for 40-period, low-injector-doping devices. The η_s value was found to be virtually temperature independent above RT, with a typical η_s/period value of ~25 mW/A (figure 14). The DAU-design devices thus appear to be quite suitable for low-power (~100 mW), broadband-tuning applications.

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An extraction scheme involving an SPR structure with depopulation of the lower level to a miniband, so-called single phonon-continuum (SPC), has been implemented in both 7.9 μm [115] and 8.6 μm emitting [116] devices, for high-yield mass production. Devices at both wavelengths displayed lower-level lifetimes of ~0.2 ps; similar to those in DPR-scheme devices. The former showed a relatively high T₀ value (181 K) for moderately high-doped devices, as a result a carrier-leakage suppression (E₄₃ ~75 meV) balanced by relatively high backfilling (Δ_inj  =  130 meV). The latter, being somewhat lower doped, showed, for 33-period uncoated-facets devices, a T₀ value of ~200 K, and a high T₁ value of ~400 K above RT, which the authors attribute to relatively low carrier-excitation from the upper level to the next higher AR energy state. However, the RT η_s/period value was relatively modest: 22.5 mW/A.

Strain compensation in 8–10 μm QCLs for the purpose of improving their performance is a relatively new development. In 2010 Leavitt et al [37] compared SC versus lattice-matched 7.7–7.8 μm QCLs and found that, while E₅₄ was basically the same (~60 meV), the strained devices had clearly superior performance; that is, lower CW RT J_th values (1.59 kA cm⁻² versus 1.77 kA cm⁻²), higher T₀ values (~222 K versus 194 K), and, most notably, higher CW slope efficiency (0.76 W/A versus 0.56 W/A). The significant slope increase, in particular, is indicative of some reduction in carrier leakage in spite of similar E₅₄ value. Now that is clearly appears that electrons can be quite hot in 8.0–9.0 μm QCLs [47], one likely explanation for such improved behaviour is that the $\Delta {{T}_{\text{eul}-\text{l}}}~$ value is reduced in these high CB-offset devices, in agreement with the observation by Spagnolo et al [30] that with increased CB offset the $\Delta {{T}_{\text{eul}-\text{l}}}~$ value decreases, since there is stronger coupling between electrons and the lattice. Another factor that may also have contributed to the increased slope efficiency is reduced ISB loss in high CB-offset devices.

Other indications of somewhat reduced carrier leakage have come from SC 7.1 μm [38], 9.0 μm [40] and 10.3–10.7 μm [117] emitting QCLs. For the 7.1 μm device, in spite of a relatively low T₀ value (159 K) due to large backfilling, the T₁ value was high (411 K); albeit from uncoated-facet devices for which the temperature variation of the low α_w value (~1.7 cm⁻¹ at 300 K) had a negligible impact on T₁. This low-loss design led to record-high, both-facets pulsed η_wp,max value (18.9%) at 7.1 μm emission wavelength, slightly above theoretically predicted limits when assuming $\eta _{i}^{d}$  ≈  70% [73]. However, the $\eta _{i}^{d}$ value was only ~63.5%; thus, the good match with predicted limits is likely due to much lower Δ_inj value: 100 meV versus 150 meV [73] as well as to a high η_tr value due to efficient extraction via the triple-phonon-resonance design [118]. The 9.1 μm device also involved a low-loss design of similar E₅₄ value (~60 meV). Similarly, although due to significant backfilling the pulsed J_th value was relatively high (2.1 kA cm⁻²) (thus, T₀ had a conventional-device value (169 K) [93]), an uncoated-facets device had a low α_w value (1.6 cm⁻¹) and a high T₁ value (331 K) [93]; indicating some carrier-leakage reduction. Just like for 7.1 μm devices, given a high mirror-loss value, the temperature dependence of the low α_w value had little impact on the T₁ value. The reduction in carrier leakage is also indicated by the fact that the differential pumping efficiency [11] η_p (equations (8) and (17a)) increased from an extracted value of ~62% from 4.6 μm emitting devices [63], considering an η_tr derived from lifetime calculations [58], to 70%. Just as for the above-mentioned SC devices, given the same E₅₄ value, the carrier-leakage reduction may well have been due to a reduction in the $\Delta {{T}_{\text{eul}-\text{l}}}~$ value with increasing CB offsets. A record-high, both-facets pulsed η_wp,max value (16%) at 9 μm wavelength was achieved, matching theoretically predicted limits with $~\eta _{i}^{d}$  ≈  70% [73]. However, the $\eta _{i}^{d}$ value was only ~58%; thus, just like for the 7.1 μm devices, the good match may be due to much lower Δ_inj value: 93 meV versus 150 meV [73]. 10.7 μm emitting, uncoated-facets QCLs [117] demonstrated a T₁ value of 290 K; which may be indicative of some degree of carrier-leakage suppression. A pulsed η_wp,max value of 10.4% at λ  =  10.7 μm was obtained, also matching theoretical predictions when assuming $\eta _{i}^{d~}$  ≈  70% [73], as expected, given that the extracted $\eta _{i}^{d}$ value is 64%.

Fujita et al [39] have reported on 8.7 μm emitting SC QCLs with anticrossed DAU laser states. Just like for the other 8–10 μm SC devices, carrier-leakage suppression occurs even though the energy difference between the upper level and the next higher energy state, E₅₄, is basically the same as for lattice-matched devices: 60 meV. The authors acknowledge that a reduced T_eul value may account for carrier-leakage suppression. Due to both optical-absorption quenching in the injector and carrier-leakage suppression, record-high T₀ values were achieved: 466 K, to heatsink temperatures as high as 400 K from low-doped devices. However, the J_th (300 K) value is somewhat high (~2.3 kA cm⁻²) for 3 mm long, HR-coated 40-period devices; thus, leading to a relatively small dynamic range (i.e. lasing up to ~1.5  ×  J_th). Due to a strong decrease in the α_w value with increasing temperature, while carrier leakage is suppressed, the slope efficiency increases with temperature; thus, resulting in negative T₁ values:  −330 K. Yet, the highest η_s/period value, obtained at 400 K heatsink temperature, is only 0.25 W/A. Both T₀ and T₁ values are better that for lattice-matched DAU lasers of same E₅₄ value (~60 meV); most likely due to significant carrier-leakage suppression in the SC structures. Such devices appear quite useful for low-power applications such as broadband tuning involving an external cavity.

The STA-DPR approach has been implemented for 8.38 μm and 8.8 μm emitting QCLs [83]. The schematic band diagram and relevant wavefunctions are shown in figure 15 for the 8.38 μm device. The design starting point was a published lattice-marched 8.2 μm emitting DPR device [105]. The AR barriers are stepwise tapered, in that the first two barriers are Al_0.51In_0.49As layers and the third and exit barriers are taller: Al_0.58In_0.42As layers. In addition, the third and fourth wells in the AR are In_0.60Ga_0.40As layers, which, having lower conduction-band energy than wells in the injector, are so-called deep wells [10]. These deep wells cause an enhancement of the step tapering of the AR-barrier heights. The results are both a higher E₅₄ value: 75–76 meV versus 57 meV, as well as a significant decrease in the spatial overlap between states 5 and 4, which results in a higher τ₅₄ value: 0.73 ps [83] versus 0.37 ps. Then, we estimate that, due to drastic carrier-leakage suppression, J_leak/J_th is at least four times less than in the conventional QCL [105], and that η_p reaches values as high as ~97%.

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The 8.8 μm emitting devices were relatively low doped (~0.7  ×  10¹¹ cm⁻²), while the 8.38 μm emitting devices were moderately high doped (~1.6  ×  10¹¹ cm⁻²). Light–current (L–I) characteristics (200 ns-wide pulses, 100 Hz repetition rate) as a function of heatsink temperature are shown in figure 16(a) for 8.8 μm devices. The L–I curves are superlinear since above ~1.3  ×  threshold there is resonant extraction from the lower laser level, which, in turn, gives a high η_tr value: ~90% [83]. For the 8.38 μm emitting devices, resonant extraction occurs right from threshold, providing a linear L–I curve (figure 16(b)) of a relatively low J_th value (1.88 kA cm⁻²) and reaching a maximum single-facet peak power of 3.3 W. The 8.38 μm devices also have a high η_tr value: ~89%. The net result of virtually complete carrier-leakage suppression and high η_tr, due to lower-level resonant extraction, are record-high extracted values for $\eta _{i}^{d}$: 85–90%, in good agreement with calculated values (86–87%) [83]. In contrast, the $\eta _{i}^{d}$ values for conventional 7–10 μm emitting devices are significantly lower: ~63.5% for 7.1 μm NRE SC devices [38]; 58% for 9.0 μm NRE SC devices [40]; and 64% for 10 μm DPR SC devices [117]. The significantly higher achieved $\eta _{i}^{d}$ values reflect the fact that this STA design takes advantage of both the good properties of the TA design [17] (significant carrier-leakage suppression) and of the miniband design with resonant extraction from the lower laser level [118] (short (~0.1 ps) lower-level lifetimes, as in superlattice-type QCLs [119, 120]).

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The drastic carrier-leakage suppression is reflected in very high T₀ and T₁ values for both low-doped (figure 17(a)) and moderately doped (figure 17(b)) devices. That is, for low-doped devices the average T₀ and T₁ values are 277 K and 551 K, respectively. By comparison, for similarly doped [110] 3.5 mm long, HR-coated QCLs, emitting at 8.2 μm, the T₀ and T₁ values were: 217 K and 192 K [110]; and for 3.0 mm long, HR-coated 8.8 μm emitting QCLs [108] the T₀ was 200 K. For moderately high doped devices the average T₀ and T₁ values are 241 K and 272 K, respectively. The decrease in average T₀ value is due to larger backfilling with increased doping, yet the value is significantly higher than for lower-doped (~1.2  ×  10¹¹ cm⁻²) 8.4 μm emitting conventional QCLs (i.e. ~160 K) [50]; thus, reflecting the carrier-leakage suppression in STA devices. The decrease in average T₁ value reflects the increase in ISB loss with increased doping. Again, the value is superior to the one from lower-doped, 8.4 μm emitting conventional QCLs (i.e. ~260 K) [50]; reflecting the carrier-leakage suppression. Further proof of carrier-leakage suppression is that, in spite of similar α_w RT values, the RT slope efficiency (1.15 W/A) is significantly higher than that (0.9 W/A) of same-geometry 8.4 μm emitting QCLs [50].

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The achievement of $\eta _{i}^{d}$ values 30–40% higher than for any type of 7–10 μm-emitting QCLs demonstrates not only the superior performance of STA-type devices, but also that $\eta _{i}^{d}$ values close to theoretical limits can be achieved, as long as carrier leakage is effectively suppressed and there is fast, miniband-type carrier extraction out of the active region.