1. Introduction

Semiconductor heterostructure-based Terahertz (THz) quantum cascade (QC) lasers represent today an efficient coherent source spanning the Far-IR region of the electromagnetic spectrum extending to the sub-THz region if we include magnetically assisted devices [1, 2]. The main limitation for such kind of devices is the operating temperature, still well below the level reachable with Peltier coolers even in pulsed operation. The development of THz QCLs has followed a path which has substantially simplified the active region designs. The number of wells per period has been drastically reduced from the 7-wells, superlattice-based structure employed for the first demonstration [3] to the last, high performance resonant-phonon three well diagonal design [4].

In this paper we report on a new design of a two-well based THz laser based on a photon-phonon cascade. This results to be the simplest configuration for an intersubband laser: in a simple picture, only three states are involved in the transport and light emission processes (at low temperatures). Intersubband lasing with a three-level, two well system was reported for optically pumped structures at energies above the LO phonon [5] and only recently it has been realised experimentally in an electrically pumped, THz emitting structure [6, 7].

2. Active region design

The bandstructure for two periods of such a structure is reported in Fig. 1. The injection of electrons in the ground state ∣3〉 of a thin well via resonant tunneling is followed by a photon-assisted tunneling transition to state ∣2〉, which is the first excited state of a large quantum well; the first two levels of this well are tuned to be resonant with the GaAs LO phonon energy (E ₂₁ ≃ h̄ω_LO).

The design strategy is simple: we try to exploit the subpicosecond scattering time of the electrons in the lower state of the lasing transition (τ ₂₁ ≃ 0.22ps at 50 K when resonant with LO phonon) in order to obtain a robust population inversion, which does not rely on a resonant tunneling extraction of the carriers as in other phonon-depopulation based structures [4,2]. The resonant tunneling injection into the ground state of the first quantum well should ensure an high injection efficiency. The optical transition is kept strongly diagonal by using a thick (3.8 nm) barrier, in order to enhance the upper state lifetime reducing the wavefunction’s overlap. This leads to a low value (z ₃₂ = 3.0 nm) for the dipole matrix element for the ∣3〉 → ∣2〉 transition. The right vertical axis of the figure represents the doping level and the lightly shaded pink area is the doped region The injection coupling is kept strong and the calculated value at the anticrossing for the energy splitting between the injector and the upper state is 2h̄Ω_31′ = 3 meV This arrangement of the levels is expected to produce a broad gain curve due to the presence of two diagonal transitions with a relevant dipole matrix element on the lower state ∣2〉 (z _31′ = 2.5 nm).

 

Fig. 1. Calculated bandstructure for two periods of sample EV1183 for an applied electric field of 14 kV/cm. The layer sequence is (nm, starting with the injection barrier): 4.5/8.3/3.8/17.9. The figures in bold face represent the Al_0.15Ga_0.85As barriers. The central part of the 17.9 nm GaAs well is doped in order to obtain a sheet carrier density of 1.5 × 10¹⁰ cm ^-2, measured with C-V technique. The location of the doping in the well is chosen to minimize the overlap of the state ∣2〉 wavefunction with the doped region.

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The structure has been grown by molecular beam epitaxy, repeating 220 times the active module and embedding it in a double-side, 50 nm thick, heavily doped (3 × 10¹⁸ cm^-3) GaAs cladding. The presence of a 200 nm thick Al_0.5Ga_0.5As etch stop layer allows the selective removal of the substrate and the processing in double-metal configuration [1, 2].

3. Laser results: direct bias

The lasers are processed in standard double metal waveguide configuration [8], by wet etching ridges of various widths ranging from 70 to 200 μm. For this first set of results, the top metal cladding completely covers the laser ridge (see inset of Fig. 2) Laser bars of different lengths are then cleaved and In-soldered on copper submounts. The data collected from a 1.7 mm long, 100 μm wide laser ridge operating in pulsed-mode are reported in Fig. 2. Laser action is observed up to an heatsink temperature of 125 K, with a threshold current density of 250 A/cm² at 10 K: the dynamic range of the laser at the same temperature is in excess of 60%. The temperature dependence of the lasing threshold can be fitted, starting at a temperature of 70 K, with the usual phenomenological relation J(T) = J₀ · e^T/T₀ which yields a value of T₀=64 K, rather low if compared with structures based on a resonant-phonon design (T₀=160 K for the structure of Ref. [4]). The strong photon driven transport is evident from the pronounced change of slope of the I-V curve at laser threshold.

 

Fig. 2. (a) Pulsed L-I-V measurements (150 ns wide double-pulses square-wave modulated at 400 Hz) for a 1.7 mm long, 100 μm wide ridge processed as a standard double metal resonator (see scheme in the inset). The detector used is a calibrated He-cooled Si-bolometer. (b): Threhsold current density, maximum current density and dynamic range as a function of the heatsink temperature.

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The spectral emission as a function of the injected current is reported in Fig. 3(a,b) for pulsed and CW operation respectively. The emission, as expected, is extremely broadband and extends over more than 1.2 THz, from the lowest bias above threshold to the breaking point of the I-V curve. It presents a clear and wide gap centered around 14.8 meV (3.55 THz): the energy separation between the center of gravity of the two family of modes below and above the gap is ΔE_spec ≃ 2.7 meV. This characteristic spectral feature is independent from the cavity dimensions (many devices with different lengths and widths were tested) and it is not an artefact of the pulsed operation since very similar results are obtained in continuous wave (CW) mode (see Fig. 3(b)).

 

Fig. 3. (a) Spectral emission as a function of increasing injected current density in pulsed operation for a 1.7 mm long, 100 μm wide double metal cavity. (b): Spectral emission as a function of the injected current density in CW operation for a 1.5 mm long, 110 μm wide double metal cavity

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We attribute this feature to the coherent coupling of the upper state of the lasing transition (state ∣3〉) with the ground state of the “phonon” well (labeled ∣1′〉 in Fig. 1): the calculated coupling 2h̄Ω_31′ = 3 meV compares well with the mode separation ΔE_spec ≃ 2.7 meV reported from the spectra. The fact that we can observe the strong coupling within this two states naturally poses some constraints on their individual broadening which must not exceed the coupling strength. We will extensively discuss about the gain broadening by strong coupling in Sec.4 where simulation data are also presented.

The same epilayer has been processed in a slightly modified version of the double metal waveguide. The top of the laser ridge has been partly left uncovered from the metal, as done in other works in order to suppress lateral high-order modes [9]. In contrary to Ref.[9] we remove the doped contact layer from the top of the laser ridge and we leave a wider portion of exposed semiconductor (17.5 μm on each side) [10] (see inset of Fig. 4). This results in slightly higher radiative losses from the double metal structure. A series of L-I-V curves as a function of the temperature is reported in Fig. 4. Peak powers of more than 30 mW at a duty cycle of 1.5 % are recorded at 10 K and 25 mW are still available at liquid nitrogen temperature. As in Ref.[9] the temperature performance is degraded (T_max = 110 K) but we observe an increase in the threshold current density and in the maximum current density and a small decrease of the dynamic range. It is worth to note that in this configuration the J_max feature is quite insensitive and hardly moves as function of the heatsink temperature. If we compare with the standard double metal waveguide (see data of Fig. 2), in that case the J_max ^10K = 620 A/cm² and J_max ^125K = 790 A/cm².

 

Fig. 4. Pulsed measurements in direct bias as a function of heatsink temperature for a 1.36 mm long, 180 μm wide double metal laser ridge with the top partially covered with metal, as schematized in the inset. The ridge width is taken as the width of the metal. The power has been measured with a broad area, calibrated THz power meter.

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The laser efficiency can be estimated by comparing the transport of a lasing cavity and a non-lasing structure (a ridge waveguide with highly absorbing ends), reported in Fig. 5. Adopting the same method of Ref. [11] we can quantify the radiative contribution to the current circulating in the device by simply subtracting the value of the current for the laser from the one of the non-lasing structure at the maximum bias for the laser to be operational (see black dashed lines in Fig. 5(a)). The ratio between the radiative contribution to the total current will give the quantum efficiency of the laser that in the case of the two well structure is $\eta =\frac{J_{\mathrm{rad}}}{J_{\mathrm{tot}}}\simeq 0.43$. The photon driven transport accounts then for nearly one half of the current circulating in a lasing device.

4. Transport analysis and simulations

In Fig. 5(a) we report the CW experimental transport characteristics for a lasing device (red curves) and a non-lasing ridge with highly absorbing ends (blue curves) together with the respective differential resistances. It is interesting to observe that the breaking point of the I-V curve has strongly different characteristics in the lasing device in respect to the non lasing cavity. The position of the J_max in terms of current density is different (J_max ^las=690 A/cm² and J_max^no-las =462 A/cm²) and also the corresponding electric field is different (E_max^las=12.7 kV/cm and E_max^no-las=14.1 kV/cm). As already discussed in Ref.[12, 11], the strong laser field and, as a consequence, the optical cavity characteristics will influence the dynamic range of a lasing device. The laser transport shows a pronounced feature at about 11.1 kV/cm: it is the laser laser threshold to which corresponds a steep increase in differential conductance due to the reduction of the upper state lifetime caused by stimulated emission. In the case of the non-lasing device, this feature is not present and the I-V curve proceeds monotonically. From the differential analysis of the curves we can identify also other resonances in the transport, highlighted by arrows in Fig. 5(a), corresponding to stationary points of the conductance. These features indicate coupling of states belonging to different periods. This is not unexpected since the structure length is short (34.5 nm) and we already observe coupling effects through the injection barrier which is then not limiting the coherence length of the electron. The first feature at 4.1 kV/cm is attributed to the resonance between the ground state of the phonon well ∣1′〉 and the ground state of the narrow well (∣3〉) of the second following period. The feature at 7.4 kV/cm is attributed to the expected coupling between state ∣1′〉 and state ∣2〉 that constitutes the principal leakage current path of this design. The third observed resonance at 9.4 kV/cm is quite narrow and can be attributed to the parasitic coupling between state ∣3〉 and the second excited state of the phonon well of the second following period. This leakage path can represent an efficient gain degradation channel, as proposed also for a similar structure in Ref. [7].

 

Fig. 5. (a): Continuous wave measurements and differential analysis for a laser ridge (red) and a non-lasing device (blue ) at T=10 K. The arrows highlight the electric field values corresponding to the resonance features. The region between the dashed lines is the fraction of radiative current due stimulated emission. (b): Simulation of the gain curve for a lasing device for an electric field of 12.5 kV/cm for a lattice temperature T=50 K. (c): Simulated transport curves in delocalized basis (black curve), tight-binding basis (blue curve) for a lasing device and in tight binding basis for a non-lasing device (light-blue, dashed curve), all for a lattice temperature T=50 K. The red curve is an experimental, CW curve for a lasing device. The green curve is a simulation of the emitted laser signal.

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The transport and gain characteristics of the structure have been computed using a code based on a single electron density matrix equation of motion [13]. We can select a basis by cutting the structure typically at the injection barrier or otherwise leave it over the whole period, as illustrated below. The structure can therefore be split into sub-modules. The equations of motion are then implemented on the eigenbasis computed into each sub-module independently. We consider coherence with sequential resonant tunneling for doublet of states across the barriers defining sub-modules. In the sub-modules the dynamics is given by rate equations. Each scattering time is compute in the Fermi golden rule approximation. The model does not account fully for the k-space. The scattering times are resolved in k, but before they enter the equations of motion, we average them with a Fermi-Dirac distribution (introducing the electronic temperature in the model) If we compare our model with recent papers, we are very close to the model described in [20]. Our simulation tool includes alloy disorder, LO phonon scattering and interface roughness scattering: electron-electron scattering is not explicitely introduced and the electronic temperature is considered uniform in all the subbands and assumed 80 K higher than the lattice one [14]. The free parameters used are the waveguide losses (α_tot = 15 cm^-1) and the structural parameters of the interface roughness scattering [15, 16] (we used uncorrelated interfaces in the z direction with ∧= 60 Å and Δ = 1.5 Å). The choice of the basis for the modelling of the structure is not straightforward due to the peculiar nature of the gain curve, which presents two distinct maxima corresponding to the two upper lasing states constituted by the anticrossed doublet ∣3〉 and ∣1′〉. If we want to correctly model the gain curve, the simulation has to include the coherent coupling of the electronic states across the injection barrier, as visible in Fig. 1. In this way, we obtain a gain profile which accounts for the observed two-color behaviour (see Fig. 5). When simulating the transport, this approach reveals its limitation because it overestimates the parasitic coupling between the states of the same period and of different periods, producing current spikes in correspondence of the various resonances. Such features are also present in the experimental curve, as discussed before, but their magnitude is much smaller and the values of the electric field present a small discrepancy. On the other side, if we model the structure with an inter-period tight-binding approach we obtain a much smoother J-V curve which does not present spikes and matches with fair agreement the experimental data, both for the lasing and non-lasing structures, in the high-electric field regime. For low applied electric fields, however, the simulations underestimate the current density circulating in the structure by roughly a factor of two. The model accounts for the photon assisted transport effect as clearly shown by the different behavior of the two simulated curves (see Fig. 5(c), dashed light blue curve for the non-lasing cavity and continuous dark blue curve for the laser case).

If we analyze the calculated gain spectrum (Fig. 5(b)) and we compare it to the measured data from a laser operating at its maximum current density, we can see that there is a discrepancy in the energies. The experimental value for the high energy modes is higher than the calculated one: this is due to the fact that the model does not account for a two-color lasing with two upper states which are coherently coupled. In order to fully take into account the effect of the laser action we should include the optical field in the coherences of the density matrix and not only in the rate equations for the populations. We believe the pyhsical process responsible for the two color lasing [17, 18, 19] to be the following: as the bias is increased state ∣1′〉 starts to overlap with state ∣2〉 and the 1’–2 transition displays gain and hits the threshold condition starting laser action. The laser emits then in the range 11.5 – 13.5 meV As more current flows into the structure, population of the state ∣3〉 grows and also the second state of the doublet reaches threshold. The clamping of the electric field at threshold allows for the observation of laser action from a pair of states which are anticrossed. A similar, weaker broadening of the gain spectrum due to the coherent coupling of the lower states of the laser transition has been reported for a 4-level resonant-phonon depopulation design [20]. The temperature peformance of the laser structure are not as good as expected from a structure where the lower state is almost empty and its lifetime is very weakly temperature-dependent. The simulations predict gain for temperatures higher than 200 K. The fact that both upper lasing states are localized also where all the charge resides (n_s = 1.7 × 10¹⁰ cm ^-2) could point towards a relevant role played by carrier-carrier scattering which is not taken into account by our model. The presence of a very hot electron distribution in the upper state could explain the bad thermal behaviour of such a structure. The leakage paths identified in the transport analysis will also be activated thermally contributing to the gain reduction.

 

Fig. 6. (a): Pulsed measurements in reverse bias for a standard double metal laser ridge of 1.7 mm length and 160 μm wide. The detector used is an He-cooled Si-bolometer. (b): Complete spectral emission in pulsed mode at T=10K for both direct (red curves) and reverse (black curves) bias as a function of the injected current density.

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5. Laser results: reverse bias

Due to the presence of a one-well injector and the intrinsic quasi-symmetry of the two-well structure it is very likely that the structure behaves as a bi-directional laser [21, 19]. Laser action in reverse bias is observed up to a temperature of 90 K in pulsed operation: the L-I-V curves together are reported in Fig. 6. Threshold current density at 10 K is 340 A/cm² and operation in CW is observed only up to 20 K with very low efficiency. The transport shows some pronounced features but there is no evidence of a strong, photon-driven transport as in the direct bias case. The spectral emission (see Fig. 6 (b)) does not show the same behavior as in direct bias: it spans a frequency range of about 0.4 THz and displays a pronounced Stark shift but it does not present two distincts lasing regions with a wide gap as observed in the other polarity. The overall worse performance of the laser is consistent with the lower dipole matrix elements for the two transitions (z^rev ₃₂ = 2.0 nm and z^rev _1′2 =1.8 nm), and the ratio of the thresholds in reverse and direct bias for the same device is $\frac{J_{\mathrm{thresh}}^{\mathrm{rev}}}{J_{\mathrm{thresh}}^{\mathrm{dir}}}\simeq 1.5$, revealing a ratio of 3/2 for the gains.

In Fig. 6 (b) are reported the spectral emission of the devices in both polarities over the whole dynamic range of the lasers. The reverse bias spectra fill partially the gap created by the splitting of the two upper states in the direct bias polarity.

6. Conclusions

A THz laser based on a photon-phonon cascade was demonstrated. Broad spectral emission spanning over 1.2 THz of bandwidth is reported together with peak optical powers in excess of 30 mW from standard double-metal waveguides. The broadband lasing is attributed to a coherent coupling of two upper lasing states achieving simultaneously enough gain to overcome the losses. The observed gap in the laser emission spectrum fits well with the predicted gain profile. The fact that the ground state ∣1′〉 of the phonon well is also the upper state of a lasing transition constitutes an example of quantum cascade laser where the population of the upper state is not achieved via resonant tunneling. The quasi-symmetric nature of the design makes bi-directional operation possible. The limitation in the maximum operating temperature to 125 K is attributed to the reduction of the upper state lifetime. The broadband spectrum of the laser could be potentially interesting for THz amplifiers as well as for mode-locked sources.