1. Introduction

The continuous-wave operation of quantum cascade (QC) lasers at room temperature has been demonstrated for a wide wavelength range between 3.8 and 10.6 µm. The high laser performance has been achieved by the adoption of bound-to-continuum (BTC) [1], double phonon resonance (DPR) [2] or single phonon resonance-continuum (SPC) [3] depopulation scheme, each which results in fast electron extraction (~0.1 ps) from lower laser states and low thermal electron population at the lower laser states, due to high energy differences (> 130 meV) between the lower laser states and ground ones in the injector. On the other hand, since the first demonstration of QC laser operation [4], for pump process, only the direct pump scheme in which upper laser states are pumped directly by electron tunneling from ground states in injectors has continued to be utilized except for an accidental case [5] at cryogenic temperature, ~10 K and an intentional one [6]. In other words, no much attention has been paid for electron transport relevant to the pump process. In this work, we examine an alternative pump scheme to reveal its excellent advantages over the conventional direct pump scheme and demonstrate high performances of 8 µm QC lasers based on the alternative pump scheme: a low threshold current density of 2.7 kA/cm² and a high maximum output power of 362 mW, both at room temperature, and, in particular, a high T₀-value of 303 K around room temperature, which is the highest ever reported with QC lasers.

2. Indirect pump scheme

In the alternative pump scheme named indirect pump (IDP) one, shown in Fig. 1, electrons are injected into the intermediate state, level 4 (not level 3) via electron tunneling from the ground state, level 1′ in the previous injector and then, are relaxed quickly down to the upper laser state, level 3 by LO-phonon emissions. In this circumstance, the electron population n4 at level 4 is expected to be low, n₄=(τ₄₃/τ₃+τ₄₃/τ₃₄)n₃<n₃ where 1/τ₃=1/τ₃₂+1/τ₃₁. With the density matrix analysis [7], the (net) tunneling current density is represented by

where E ₄₁′ and τ_deph are the energy separation between levels 4 and 1′ and the dephasing time, and the tunneling time is defined as τ_tunn=2/Ω_1′4 2τ_deph with the Rabi oscillation frequency Ω1′4.

 

Fig. 1. Schematic illustration of the proposed IDP scheme.

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The current consists of the forward (∝n₁) and backward (∝n₄) components. In the IDP case, the backward tunneling current, represented by the (red-colored) dashed arrow in Fig. 1 is kept to be low even when the electron population n₃ at level 3 approaches a high value comparable to the injector doping density n_inj. In an approach based on rate equations with fixed relaxation times for the relaxation processes with the help of Eq. (1) and the particle conservation (charge neutrality in each cascade stage), n_inj=n₁+n₂+n₃+n₄, the tunneling current density in a below-threshold case reads,

where the global relaxation time is given by τ _relax=η _pump τ ₃[2–1/n_sp+(2+τ ₂₁/τ ₁₂)(τ ₄₃/τ ₃₄+τ ₄₃/τ ₃)] and the pump efficiency for level 3 is defined as η _pump=(n₄/τ ₄₃-n₃/τ ₃₄)/(j/e)=η₀/[1+(τ ₃/τ ₃₄)(1-η ₀)] with η ₀=(1/τ ₄₃)/[1/τ ₄₃+1/τ ₄₂+1/τ ₄₁]. The population-inversion parameter (or spontaneous emission factor) is also defined as n_sp=1/{1-τ ₂₁[1/τ ₃₂+(1/τ ₄₂)(1/η_pump-1)/(τ ₃/τ ₄₂+τ ₃/τ ₄₁)]} (for the definition of n _sp, see Eq. (5) in reference [8]). From the principle of detail balance, the life-time ratios are, regardless of details of systems, represented simply by τ ₂₁/τ ₁₂=exp(-E ₂₁/kT) and τ ₄₃/τ ₃₄=exp(-E ₄₃/kT). The electron populations at levels 1~4 read:

The electron population n ₂/n_inj at the lower laser state, level 2 consists of the thermal back filling (τ ₂₁/τ ₁₂)(n ₁/n_inj)=n_2therm/ninj and the excess term (1-1/n _sp)(η _pump jτ ₃/en_inj). Similarly, for a conventional direct pump (DP) case without level 4, the electron populations are represented by putting τ ₄₃/τ ₃₄+τ ₄₃/τ ₃=0, η_pump=1 and n ₄=0 in Eq. (3) and the current density is done by replacing E ₄₁′ with E ₃₁′ and by putting τ _tunn=2/Ω1′3 ² τ _deph, τ _relax=τ ₃(3-1/n _sp+τ ₂₁/τ ₁₂), and n _sp=1/[1-(τ ₂₁/τ ₃₂)] in Eq. (2).

In an ideal IDP situation: n _sp~1, τ ₄₃/τ ₃₄+τ ₄₃/τ ₃≪1 and τ _tunn≪τ_relax under near resonance, (E ₄₁′/ħ)² τ _deph ²<1, the maximum current density, η _pump j_max, actually available for pumping of level 3 is given by η _pump j_max=en_inj/τ ₃, that is, regardless of the value of η _pump, twice as high as that, en_inj/2τ ₃, of the DP case [5] for the same doping level of the injector, n_inj. Namely, in the IDP case, under the extreme condition, electrons originally existing in the injector are perfectly transferred to the upper laser state, level 3, n ₃~n_inj and, in turn, the population n ₁ in the injector is completely depleted, i.e., ultimately, n ₁~n ₄≪n ₃~n_inj. The IDP scheme presents the obvious advantages of higher available population at the upper laser level and lower population in the injector over the conventional DP one, i.e., n ₃~n ₁~n_inj/2 in the strong pump limit.

In a realistic device, however, the population-inversion parameter n _sp may be larger than unity and the sum of relaxation-time ratios τ ₄₃/τ ₃₄+τ ₄₃/τ ₃ may take a value, nominally smaller than unity at high temperatures, T~300 K. By using realistic values for the important parameters, i.e. n _sp=1.4 and τ ₄₃/τ ₃₄+τ ₄₃/τ ₃=0.4 for the IDP case (see the caption of Fig. 2), the normalized electron populations are computed with Eqs. (2) and (3), and the electron populations in the DP case are, similarly, computed, both which are shown in Figs. 2(a) and 2(b), respectively. In the IDP case, the upper level population n ₃ is able to be, still, higher than the ground state one n ₁ near the maximum current, namely global population-inversion. Although the population difference n ₃–n ₂ of the IDP case is only slightly higher than that of the DP one, the ground state population n ₁ of the IDP one decreases strongly with increasing current, compared with that of the DP one, implying stronger suppressions of the thermal backfilling at the lower laser level (level 2) and optical loss due to off-resonant intersubband transitions [9] in the injector. The quality factor for lasing, defined here as (n ₃–n ₂)/n ₁, of the IDP case is twice higher than that of the DP one around the maximum current. Thus, the IDP scheme retains the advantages even in the realistic situation as will be actually demonstrated later. The current-induced injector-population changes which are more visualized in the IDP case have never been recognized so far in analyses of laser performances (even in the DP case). In fact, while electron tunneling into high energy states of the second miniband was utilized in a superlattice QC laser [6], the characteristic temperature T₀ was reported to be low, 120 K. This is because its threshold current density,~18 kA/cm² at room temperature was much lower than the maximum current density ~80 kA/cm² estimated with a high injector doping of ~5×10¹¹ 1/cm² and a life-time of ~1 ps and, in turn, the electron population in the injector was retained to be still quite high at the threshold condition.

 

Fig. 2. (a). Normalized electron populations in an IDP laser, computed with Eq. (3) using the parameters: n _sp=1.4, τ ₂₁/τ ₁₂=0.2, τ ₄₃/τ ₃₄+τ ₄₃/τ ₃=0.4, and normalized maximum current density computed with Eq. (2) using τ _tunn/τ ₃=0.18 and η _pump=0.8. These parameter values are very close to room temperature ones actually used in the computation of the threshold current density of the test devices. (b) Normalized electron populations in a DP laser, computed by using the corresponding parameters: n _sp=1.3, τ ₂₁/τ ₁₂=0.2 and τ _tunn/τ ₃=0.18. The quality factor, (n ₃–n ₂)/n ₁ in the IDP case is also plotted. The vertical dashed lines indicate maximum normalized currents, commonly in Figs. (a) and (b).

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3. Design of active region and laser performances

The BTC active region was designed, as shown in Fig. 3, to incorporate the intermediate state, level 4 for the IDP process and importantly so as to avoid couplings of injector states with adjacent upper laser states, level 3 under any bias fields. As shown in Fig. 3(b), the injector states, level 1′ is already located to be higher than the upper laser states, level 3 even under the low bias field, 18 kV/cm which results in a low applied voltage, ~4 V, around current turn-on. As a consequence of localization of the wavefunction of level 4, the pump efficiency can be designed to be sufficiently high, η _pump≥0.8 at room temperature. Note that the designed energy separation, E ₂₁~60 meV, between the lower laser states and injector ground state is substantially lower than those (>130 meV) of the high performance QC lasers reported previously [1–3, 9]. The relaxation times at zero phonon temperature and the Rabi oscillation frequency are estimated: τ₄₃₀=0.27 ps, τ₄₂₀=3 ps (≪τ₄₁₀), τ ₃₀~τ ₃₂₀=1.5 ps, and ħ Ω_1′4=7 meV (>ħ Ω _1′3≤3 meV), which will be actually used in the computation of the temperature dependencies of the threshold current density jth and slope dj_th/d(1/L) of the threshold current density-versus-the inverse of cavity length. The stronger coupling, ħ Ω 1′4=7 meV between levels 1′ and 4 results from the wavefunction localization of level 4. The transition dipole moment is, also, estimated to be enough large, Z₃₂=1.92 nm.

 

Fig. 3. Conduction band diagram and moduli squared of the relevant wavefunctions in the designed active region of the IDP QC laser. The lattice-matched In_0.53Ga_0.47As/In_0.52Al_0.48As layer sequence of one period of the active layers, in angstroms, starting from the injection barrier (toward the right side) is as follows: 45/24/21/69/10/60/17/46/21/40/20/37/22/34/24/31/30/29/33/27/35/25 where In0.52Al0.48As barrier layers are in bold, In_0.53Ga_0.47As QW layers in roman, and doped layers (Si, ~10¹⁷ cm^-3) are red-colored. (a) The bias field is assumed to be strong, 34.5 kV/cm enough to align the ground state of the injector to the level 4. (b) The bias field is assumed to be weak, 18 kV/cm, corresponding to a current turn-on voltage. The injector state 1′ is already located to be higher than the upper laser states, level 3. The structure is designed to avoid couplings of the injector states with the adjacent upper laser states, level 3 under any bias fields.

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All of the lattice-matched In_0.53Ga_0.47As/In_0.52Al_0.48As layer structures, shown in Fig. 3, with a relatively low injector Si-doping of n_inj~5×10¹⁰ cm^-2 were grown on an n-InP substrate by MBE technique. The active/injector stages with 33 repetitions were used as the emitting region and sandwiched between two 0.4 µm thick n-In_0.53Ga_0.47As layers (Si, ~10¹⁷ cm^-3). The upper cladding layer consists of a thick n-InP (Si, ~10¹⁷ cm^-3) followed by a thin n+- In_0.53Ga_0.47As (Si, ~10¹⁸ cm^-3) cap layer, both the layers which were grown by MOVPE technique. After the growth, the wafer was processed into a 12 µm-width ridge-structure. Finally, the Ti/Au films were evaporated on top of the ridge. The mirror facets of Fabry-Perot cavities were formed simply by cleaving. No high reflection coatings were employed for the mirror facets. The laser output from one facet was measured by using a calibrated HgCdTe detector (HPK: P4361-10 (specially designed for the 8 µm band)) in pulsed mode: 50 ns pulse width at a repetition rate of 1 kHz.

The presence of the intermediate states, level 4 was, in fact, verified by EL spectral measurements, indicating an energy separation of E ₄₃~40 meV between levels 4 and 3. The results of I–V measurements over the wide current range, 0~8 kA/cm², shown in Fig. 4(a), convince us of stable electron injection from the injector to intermediate states, level 4, being continuous over the current range for lasing, 0~5 kA/cm². The higher current density (>5 kA/cm ²) is understood to be brought about by the alignment of injector states with the higher spurious states, level 5. The EL peak height ratio of 4–2 and 3–2 transitions in the current range, 0~5 kA/cm² was observed to be relatively insensitive to temperature change. For instance, the peak height ratio increased only by a factor of ~1.6 with increasing temperature from 77 K to 300 K. This fact supports more directly the electron transport from the injector to intermediate states, level 4 in the same current range, since from the excitation-relaxation balance n ₄/τ ₄₃-n ₃/τ ₃₄=n ₃/τ3 at the upper laser states, level 3 under the IDP condition, the temperature dependence of the peak height ratio is represented by n ₄/n ₃=(τ ₄₃/τ ₃+τ ₄₃/τ ₃₄) which has a weak temperature-dependence, by a theoretical factor of 2.3 in the present case. Note that if electrons should transport directly from the injector to the upper laser states, level 3, because of 1/τ ₄₃≫1/τ ₄₂, 1/τ ₄₁, the peak height ratio is to be proportional to τ ₄₃/τ ₃₄=exp(-E ₄₃/kT), being strongly dependent on temperature by a factor of 80.

 

Fig. 4. (a). Current-light output characteristics of an IDP QC laser with a cavity length of 4 mm at different temperatures, 77 k~380 K. The current-voltage characteristics as well as the lasing spectrum at 300 K are also shown. (b) Current-light output and current-voltage characteristics of another IDP QC laser with a cavity length of 1.5 mm at different temperatures, 77 k~330 K, together with lasing spectrum at 300 K.

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Ridge-structured lasers with the active regions were driven in pulsed current mode, 50 ns pulse width and 1 kHz repetition rate at different heat sink temperatures, 77 to 400 K. An 8 µm IDP laser with a cavity length of 4 mm has demonstrated the high device performance: a low threshold current density of 2.7 kA/cm² and a high maximum output power of 362 mW (from one facet) both at room temperature and a maximum lasing temperature of 390 K, despite the relatively narrow ridge width of 12 µm, cascade stage repetition of 33 and no HR coatings, as shown in Fig. 4(a). Furthermore, despite the low energy separation, E ₂₁~60 meV, the temperature-dependence of the threshold current of the 4 mm-long laser has been observed to be weak, T ₀~243 K around room temperature in marked contrast to T ₀~74 K in the low temperature range, as shown in Fig. 5(a), where the characteristic temperature T ₀ is defined by the empirical relation: j _th=j₀ exp(T/T₀). The obtained T ₀-value of 243 K is slightly higher than a reported T ₀-value of 231 K in an HR-coated 1.5-mm long (approximately same mirror loss as that of the present 4-mm laser) 9.8 µm DP-BTC-QC laser with a threshold current density of 3.1 kA/cm² at room temperature in pulsed mode operation [10]. The energy separation E ₂₁ of the cited laser was designed to be high, E ₂₁~150 meV, enough to suppress the thermal backfilling at the lower laser states. This implies that the obtained high T ₀-value of the present IDP laser with the low energy separation, E ₂₁~60 meV, is brought about by the substantial suppression of thermal backfilling due to the current-induced decrease in the injector population. Another 8 µm IDP laser with a shorter cavity length of 1.5 mm has, as shown in the upper panel of Fig. 5(a), exhibited further weaker temperature-dependence: a surprisingly high T₀-value of 303 K around room temperature associated with a room-temperature threshold current density of 3.3 kA/cm².

 

Fig. 5. (a). Plots of the threshold current densities of the 4-mm and 1.5-mm long lasers as functions of device temperature. Note that the vertical scales for two devices are shifted for a clear view. The straight lines represent fits, showing empirical T ₀-values. The dashed curves labeled “K_s” are theoretical ones taking account of the suppression of the electron population in the injector. The dashed curves labeled “K_s=0” are obtained by assuming K_s=0 and ones labeled “K_s/2” are done with the half values for K_s. Note that the current range of both the curves is reasonably limited to be below the maximum current density. In Eq. (2), the uses of the injector doping n_inj=5×10¹⁰ 1/cm² together with the estimated value of τ_relax=1.73 ps (given by τ₃~τ₃₂=τ₃₂₀/[(1+δ)_Nphonon+1]=0.823 ps, n _sp=1.42, τ₂₁/τ₁₂=0.65, τ₄₃/τ₃₄=0.295, and τ₄₃/τ₃=0.212) at T=380 K (weak-lasing temperature), and the tunneling time τ_tunn=0.175 ps (given by ħ Ω 1′4=7 meV and τ_deph=100 fs) lead to a maximum current density of 3.96 kA/cm², shown by the horizontal arrows, which is fairly close to the observed current densities, ~4.0 kA/cm², for maximum output powers at high temperatures (Figs. 4(a) and 4(b)), caused by the complete alignment of the injector states with the intermediate state, level 4. (b) Plots of the slope d_jth/d(1/L) of the threshold current-density jth-versus-the inverse of cavity length 1/L as a function of device temperature. The experimental data of the jth-(1/L) characteristics for different temperatures, 100 K~330 K are shown in the inset. The curve labeled “K_s” are theoretical one taking account of the suppression of the electron population in the injector. The curve labeled “K_s=0” are obtained by assuming K_s=0.

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The differential quantum yield, η_ext=(e/ħω_opt)dp_out/dI₀, of the 4-mm (1.5-mm) long laser has been maximized at 260 K (240 K): η _ext,max=539 % (516 %) (See Figs. 4(a) and 4(b)). In addition, the slope, dj _th/d(1/L) of the threshold current density-versus-the inverse of cavity length tends to saturate and even to decrease slightly in the high temperature range, T≥250K, as shown in Fig. 5(b), which is very specific to the IDP lasers. It should be stressed that rather than further lowering of the threshold current density, the obtained high stability of the laser performances for temperature changes is quite beneficial to real device-applications. The remarkably weak temperature-dependencies of the threshold current density and slope in the high temperature range will be, below, interpreted in terms of substantial lowering of thermal backfilling populations at the lower laser states and of optical absorptions in the injectors, both associated with the anticipated suppressions of electron populations in the injectors with increasing current.

4. Calculations and discussion

In a rate equation approach [8] taking account of current-dependencies of the thermal backfilling populations n _2therm(=(τ₂₁/τ₁₂)n ₁) and optical absorptions α_inj(∝n ₁) in the injectors based on (the first one of) Eq. (3), the threshold current density reads,

where α_c, (1/L)ln(1/R) and α(0)_inj are the passive waveguide loss, mirror loss and modal optical loss in the injector with n ₁=n _inj, respectively, and M(=33) and g_c are the stage repetition and the (modal) gain cross section per stage. In the denominator of Eq. (4), the key parameter K_s representing the influence of suppressions of the thermal backfilling and injector loss on the threshold current is defined as

The temperature-dependencies of nonradiative relaxation rates may be represented by the following relations: for nearly resonant relaxations, for instance, 1/τ₄₃=(N _phonon+1)/τ₄₃₀ together with τ₄₃/τ₃₄=exp(-E ₄₃/kT), (similarly, for 1/τ₂₁ and τ₂₁/τ₁₂) and for off-resonant relaxations, for instance, 1/τ₃~1/τ₃₂=[(1+δ)N _phonon+1]/τ₃₂₀ (similarly, for 1/τ₄₂ and 1/τ₄₁) where N _phonon is the Bose-Einstein distribution function for the LO-phonon system and the difference in transition matrix elements between the phonon-absorption and emission processes is accounted by the parameter, δ~0.5. From Eq. (4), one obtains the simple relation which includes only the loss term, α_c+[α ⁽⁰⁾ _inj+(τ₂₁/τ₁₂)n_injMgc]/(1+τ₂₁/τ₁₂) and, as a result, is useful for the identification of α ⁽⁰⁾ _inj and g_c as will be demonstrated below,

where j _th0 is the threshold current density for 1/L=0.

The threshold current density jth and slope dj _th/d(1/L) are computed numerically, based on Eqs. (4) and (4a), with four steps including the empirical determination of g_c and τ₂₁/τ₁₂. (I) The following values for the physical parameters were used to estimate n _sp, η _pump, τ₃, τ₄₃/τ₃₄, and τ₄₃/τ ₃, as functions of temperature, in Eqs. (3), (4) and (4a): τ₄₃₀=0.27 ps, τ₄₂₀=3 ps (≪τ₄₁₀), τ₃₀~τ₃₂₀=1.5 ps, τ₂₁₀=0.3 ps, and E ₄₃=40 meV. (II) By neglecting the terms, (τ₂₁/τ₁₂)n_injMgc and τ₂₁/τ₁₂ in Eq. (5), the experimental value for j _th0/dj _th/d(1/L) at the lowest temperature, T~77 K together with R=0.291 leads to α_c+α ⁽⁰⁾ _inj~α⁽⁰⁾ _inj=8.92 cm^-1 (α_c<1 cm^-1 in the present devices), which is very close to the reported off-resonant intersubband absorption [9], Γα_ISB~10 cm^-1. (III) Because of the difficulty involved in the identification of linewidth of each single transition in the inhomogeneous system, the BTC one, we take the following way for the estimation of the gain cross section gc. In the low temperature range 100 K~200 K where the lasing thresholds take place at voltages relatively close to the turn-on voltages as shown in Fig. 4, the energy separation between the lower laser and ground states is safely determined to be E ₂₁~60 meV by using experimental values, V _appl, E ₄₃=40 meV and E ₃₂=photon energy in the relation: V _appl/M=E ₄₃-E ₄₁′+E ₃₂+E ₂₁ and by neglecting the voltage drop in the series resistance. The energy separation, E ₄₁′ is estimated by using the experimental threshold current density as well as ħΩ_1′4=7 meV and τ_deph=100 fs in Eq. (2). The uses of E ₂₁=60 meV (for τ₂₁/τ₁₂=exp(-E ₂₁/kT)), α⁽⁰⁾ _inj=8.92 cm^-1 and n_inj=5×10¹⁰ 1/cm² (for n_inj, see the caption of Fig. 5(a)), and of experimental values of j _th0/dj _th/d(1/L) in the low temperature range 100 K~200 K (shown in the inset of Fig. 5(b)) again together with R=0.291 in Eq. (5) lead to the estimation of the gain cross section gc. The obtained values of the gain cross section are also fitted by the curve to extrapolate the gain cross section in the higher temperature range, T>200 K as shown in Fig. 6(a). This manner for the evaluation of gain cross section is quite reliable since the gain cross section of the present BTC QC lasers at 300 K is extrapolated to be 3.55×10^-11 cm, reasonably smaller than the gain cross section, 4.3×10^-11 cm computed directly for a double phonon (bound-to-bound: homogeneous system) QC laser [11]. On the other hand, the term [α⁽⁰⁾ _inj +(τ₂₁/τ₁₂)n_inj Mgc]/(1+τ₂₁/τ₁₂) in the higher temperature range (400 K>T>200 K) is identified by substituting the experimental values of j _th0/dj _th/d(1/L) over the same temperature range into Eq. (5), followed, without knowledge of E ₂₁, by the determination of τ₂₁/τ₁₂ with uses of the extrapolated values of g_c. Note that for j _th0/djth/d(1/L) in the highest temperature range (T>350 K), its extrapolated values are used. (IV) The key parameter K_s is now easily identified, which is linked to the (already identified) terms, [α ⁽⁰⁾ _inj +(τ₂₁/τ₁₂)n_injMgc]/(1+τ₂₁/τ₁₂) and n_injMgc, in the frame work of the present model, as indicated by Eq. (4a). Finally, the threshold current density jth and the slope dj _th/d(1/L) are computed as functions of device temperature. The ratio of resonant loss to injector one (τ₂₁/τ₁₂)Mg_c/(α⁽⁰⁾inj/n_inj), the electron population n _1th in the injector and the population-inversion (n ₃′n ₂)_th both at threshold condition are also obtained, which are shown in Fig. 6(b).

 

Fig. 6. (a). Gain cross section g_c and key parameter K_s as functions of device temperature. The gain cross sections identified at three temperatures, 100 K, 150 K and 200 K are fitted by the curve represented by g_c=G_c/(aT-bT2-1) with G_c=1.19×10^-10 cm, a=0.021 1/K and b=2.16×10^-5 1/K² for the temperature range, 100 K~400 K. (b) Computed electron population n_1th/n_inj in the injector and population-inversion (n ₃–n ₂)th/n_inj of the 4-mm long QC lasers at threshold conditions, and ratio M_gc(τ₂₁/τ₁₂)/(α ⁽⁰⁾ inj/n_inj) as functions of device temperature.

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The latter ones, n _1th and (n ₃–n ₂)_th are counted by substituting the experimental values of j _th of the lasers into j in Eq. (3).

The experimental data of the threshold current and slope are fitted consistently, without any intentional adjustments of the absolute values, by the theoretical curves labeled “K_s”, as shown in Figs. 5(a) and 5(b). The fair agreement between them indicates the validity of the present model where both the thermal backfilling populations n _2therm=(τ₂₁/τ₁₂)n ₁ at the lower laser states and optical absorptions α _inj=(α ⁽⁰⁾ inj/ninj)n ₁ in the injectors are viewed as being quenched with increasing current, as a consequence of suppressions of electron populations in the injectors. The agreement also implies that spurious leakage currents do not play any significant roles in the elevation of the T ₀-values. Since the ratio Mg_c(τ₂₁/τ₁₂)/(α⁽⁰⁾ _inj/ninj) is larger than unity in the high temperature range, T≥ 300 K, as shown in Fig. 6(b), the observed high T ₀-values are ascribed primarily to the suppressions of the thermal backfilling at the lower laser states in the present devices. The electron population n _1th in the injector at the lasing threshold decreases down to n _1th/n_inj~0.25 with increasing temperature, i.e. increasing threshold current density. Notifying the loss-gain balance at thresholds, the decrease in the required population-inversion (n ₃–n ₂)_th with increasing temperature (T>250 K) (Fig. 6(b)) is really consistent to the quenching of the injector-loss α_inj caused by the decreasing injector-population n _1th. From the numerical results on n ₁ (shown in Figs. 2(a) and 2(b)), the key parameter K_s of a corresponding DP laser is supposed to be about half that of the present IDP laser. The theoretical curves, labeled “K_s/2” in Fig. 5(a), obtained with the half values for K_s, predict substantially lower T ₀-values of 139 K and 127 K in L=4 mm and 1.5 mm DP-devices (with E ₂₁~60 meV) in the temperature ranges: 270~330 K and 250~290 K. Moreover, in addition to the high stability of the laser performances for temperature changes, the IDP scheme may, also, make the threshold current density less sensitive to the detuning between peak gain and resonant wavelengths in a DFB or external cavity system since the key parameter K_s represented by Eq. (4a) increases with decreasing gain-cross section g_c associated with the detuning. This is an additional advantage of the IDP scheme.

By assuming the injector population to be clamped, n ₁=n _1th above threshold, the quantum yield, defined as the ratio, (p_out/ħω_opt)/(I₀-I_th)/e, of output photon flux to the input electron flow, of a single-mode laser is represented by [8],

The uses of the estimated values of n _1th/n_inj (for the 4-mm long laser, Fig. 6(b)) of the 4-mm (1.5-mm) long laser as well as the appropriate values for the remaining parameters in Eq. (6), lead to quantum yields of 328 % (614 %) at 77 K and 394 % (692 %) at 260 K (240 K), respectively. These theoretical values correspond to the experimental values: 460 % (478 %) at 77 K and 539 % (516 %) at 260 K (240 K) (see Figs. 4(a) and 4(b)), indicating qualitatively that the higher yield at the higher temperature is attributable to the stronger quenching of the injector loss. Further theoretical work including additional effects such as an unclamped n ₁ above threshold is, however, required for quantitative understanding of the observed higher yields of the 4-mm long laser.

5. Conclusions

The alternative pump scheme named indirect pump one has been proposed to clarify its own advantages over the conventional direct pump scheme. The high performance of the ridge-structured 8-µm QC laser with a cavity length of 4 mm based on the indirect pump scheme has been demonstrated: a low threshold current density of 2.7 kA/cm² and a high maximum output power of 362 mW both at room temperature, and a maximum lasing temperature of 390 K. In particular, despite the low energy separation between the lower laser and ground states, ~60 meV, the lasers with cavity lengths of 4 mm and 1.5 mm have exhibited high T ₀- values of 243 K and 303 K around room temperature, respectively. The latter T ₀-value, 303 K, associated with a room-temperature threshold current density of 3.3 kA/cm², is the highest record ever reported with QC lasers and, in more general, with semiconductor lasers except for quantum dot lasers (T ₀~400 K). The differential quantum yield has been observed to be maximized at temperatures, 240 K~260 K. For temperature changes, the observed high stabilities of the threshold current and slope of the threshold current-versus-the inverse of cavity length have been interpreted in terms of suppressions of electron populations in the injectors caused by electron-transport dynamics, which are specifically visualized in the indirect pump scheme. The obtained robustness of the lasing characteristics for temperature changes has a potential impact on practical applications of QC lasers. The room-temperature continuous-wave operation of the IDP QC lasers with threshold current densities of ~3 kA/cm² would be possible by the incorporation of appropriate heat-dissipation structures such as thick top contact layers and/or buried hetero-structures. Moreover, the threshold current density would be further lowered by a slight increase in the energy separation between the lower laser and ground states, holding the high T ₀-value. The higher T ₀-values of the IDP lasers may result in smaller differences in temperature-dependencies of laser performances between pulsed and CW modes.

The indirect pump scheme may open up a new opportunity for longer wavelength (>10 µm) mid-infrared QC lasers including DFB and external cavity lasers since the scheme promises higher electron populations at upper laser states and, simultaneously, stronger suppressions of electron populations in injectors, the latter which may result in substantial absorption quenching, being particularly beneficial to such long wavelength QC lasers. Furthermore, the scheme would be a powerful harness for high temperature (≥250 K) operation of THz QC lasers, which has not been achieved yet with the conventional direct pump scheme [12].