# Tuning quantum cascade laser wavelength by the injector doping

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## Abstract

Tuning the wavelength of emitted radiation is a tremendous feature of quantum cascade lasers which enables their use in various applications. Usually, this tuning is executed by the change of the bias current or the temperature. In this paper, it is demonstrated, both experimentally and theoretically, that yet another possibility of tuning laser wavelength offers the change of doping density. For the experimental demonstration, a set of GaAs/AlGaAs devices emitting in the range 9.3–9.7 µ\({\rm {m}}\) was MBE grown and processed. For the theoretical analysis, the simulations that employ nonequilibrium Green’s function formalism, applied to the single-band effective mass Hamiltonian, are used. The analysis shows that the physical mechanism responsible for wavelength-doping correlation is a linear Stark effect. The range of tuning is limited on both low and high doping side. Both these limits are established and discussed.

## 1 Introduction

The well-known feature of a quantum cascade laser (QCL), which enables this device being used in gas tracing systems, is the possibility of tuning the wavelength of the emitted radiation. The wavelength of a QCL is tuned by changing either the laser temperature and/or the bias current [1]. Temperature tuning provides a broad tuning range; however, it is slow as the whole submount and laser needs to be temperature-controlled. Through the use of a buried heater element, the active region temperature can be modified without changing the submount one. This method has been successfully applied to buried heterostructure lasers, becoming an attractive solution for molecular spectroscopy [2]. The short period length and the diagonal nature of the laser transitions in QCLs guarantee an additional tuning mechanism of the emission due to the linear Stark effect [3].

The influence of doping density on the performance of QCLs, working in mid-infrared (MIR) range, has been studied mainly in the context of dynamic working range and efficiency [4, 5, 6, 7, 8], with only a limited number of experimental investigations, undertaken to clarify its influence on emission characteristics [9]. While, for the obvious reasons, this method cannot be used for tuning devices mounted onboard the sensing systems, it seems attractive for the preselection of the spectral range when designing devices dedicated for specific applications. In this paper, this possibility is demonstrated both experimentally and theoretically. The basic theoretical tool employed in the analysis is numerical modeling. It is shown that the Stark shift is the physical mechanism responsible for the observed phenomenon, which otherwise is limited to the certain range of doping densities.

## 2 Experiment

_{0.45}Ga

_{0.55}As devices that utilize resonance phonon depopulation scheme. The 3-well active region design of [10, 11] was adopted: the layer sequence in the single QCL module was:

**4.6**, 1.9,

**1.1**, 5.4,

**1.1**, 4.8,

**2.8**, 3.4,

**1.7**, 3.0, \(\underline{\mathbf{1.8}}\), \(\underline{2.8}\), \(\underline{\mathbf{2.0}}\), \(\underline{3.0}\),

**2.6**, 3.0 nm, starting from the injection barrier. The AlGaAs layers are denoted in bold. The underlined layers are n-doped. The injector doping was in the range \(3.4 \times 10^{17}\) to \(8.0 \times 10^{17}\,\text {cm}^{-3}\). Only two-barrier quantum well pairs in the central part of each injector were doped. The structure containing 36 modules was grown by MBE. TEM image and schematics of the grown structure are shown in Fig. 1.

The structure used a double-plasmon Al-free waveguide for planar optical confinement. The core of the structure was embedded in the lightly doped waveguide composed of 3.5 µm thick n-GaAs layers on each side (\(n = 4.0\times 10^{16}\,\text {cm}^{-3}\)), terminated by 1 µ\({\rm {m}}\) thick highly Si-doped (\(n = 1.0\times 10^{19}\,\text {cm}^{-3}\)) GaAs layers. For such a waveguide, the optical losses and the confinement factor were estimated in the range \(\alpha _\mathrm{{w}}=\)16–20 \(\; \text {cm}^{-1}\) and \({\Gamma }=0.31\), respectively [12]. The double trench lasers with 3 mm \(\times \;25\) µm current windows were fabricated using standard processing technology [13]. The facets were uncoated, so the mirror losses can be estimated as \(\alpha _\mathrm{{m}}\cong 5\,\text {cm}^{-1}\).

In the experiment, only the injector doping was subjected to the variation: the devices doped to different density \(N_{\text {dop}}\) in the range 3.4–8.0 \(\times 10^{17} \; \text {cm}^{-3}\) were fabricated and characterized. Current–voltage–light characteristics and lasing spectra were measured. For the latter, the experimental strategy was to keep the parameters, which are known to influence the wavelength, constant. In QCLs, the lasing takes place for the current density *J* above the threshold value \(J_{\text {th}}\). The threshold current is known to depend quite a bit on doping density at high temperatures [4, 6], and much more weakly at low temperatures [5, 6, 8]. This is because the major gain-deteriorating mechanism is thermal backfilling of lower laser state [14]. At high temperatures, the distribution of electrons that occupy injector states has a long energy tail reaching the lower laser state. Then, the increasing doping results in the increasing filling of this state and decreased population inversion. At low temperatures, the redistribution of the electrons, due to the increasing doping, is limited to the injector states, so the population of the lower state is not so much affected [13, 14]. Data in Fig. 2, collected for our devices, are in full correlation with this scenario: the threshold current remains practically unchanged when \(T < 100\) K, while at higher temperatures, it depends strongly on doping density. Making the planned experiment above \(T> 100\) K may then result in a disappearance of the lasing when the threshold current exceeds the measurement current. Therefore, our experimental temperature was fixed at 77 K, at which the threshold current hardly depends on the doping density. The devices were also chosen to have similar characteristic temperature \(T_{0}\), regardless of doping (see Fig. 2).

## 3 Model

*k*. The full non-interacting effective mass Hamiltonian reads:

*U*, and the mean field term \(V_{sc}(z)\). Although Eq. (1) is strictly one-band Hamiltonian, it accounts for mixing with remote (valence) bands through energy-dependent effective mass, \(m(E,z)=m^{*}(z)\left\{ 1+[E-E_{c}(z)]/E_{g}(z)\right\}\). The in-plane dynamics included by kinetic energy terms uses the same mass,

*m*(

*E*,

*z*). It was shown that such a formulation preserves the in-plane non-parabolicity, comparable to the results predicted by the 8-band \(\text {k}\cdot \text {p}\) method [16]. The Hamiltonian of Eq. (1) was used with nonequilibrium Green’s function (NEGF) formalism to get the reliable results which account for both quantum coherence and scattering. The use of this method is a must as our conclusions (to be presented further on) relay on the estimation of the gain peak magnitude, which in this method is calculated without any simplifying assumptions. In other methods, this value may depend on somewhat arbitrary assumed broadening [17].

*h*and average diameter \({\Lambda }\). These are hardly measurable quantities and usually are evaluated by adjusting experimental and calculated characteristics [24]. Such a procedure applied to our devices gives \(h=0.19\) nm and \({\Lambda }=9\) nm. Similar values are reported in the literature, e.g., in [25]. Reasonable estimation of the

*h*and \({\Lambda }\) values is further confirmed by the calculations of optical gain presented in Figs. 4 and 5. The gain was calculated with the use of theory presented in [26] in the first order approximation, i.e., the terms \(\delta {\Sigma }\) were ignored.

## 4 Results and discussion

*J*constant, this increase must be compensated by the increased detuning from the resonance \({\Delta }\). This mechanism is illustrated in Fig. 5.

The I–V characteristics plotted according to Eq. (3) were compared to the simulations. While drawing the lines, a linear relation between \(n_{\text {g3}}\) and \(N_{\text {dop}}\), and \({\Delta }=V_{\text {res}}-V\), with \(V_{\text {res}}=10\) V, was assumed. The horizontal line is pinned to the value of the constant current density \(J \approx 1.5 J_{\text {th}}=5.5 \; \text {kAcm}^{-2}\). As can be seen, both Eq. (3) and the results of simulation predict a decrease of the bias voltage *V* with increasing doping density. This was also observed in the experiment (see Fig. 3a). The decreased *V* transfers then to the decreased separation \(E_{3}-E_{2}\) of laser levels and so the energy \(h\nu =E_{3}-E_{2}\) of emitted photons through the linear *Stark effect*.

*U*must decrease to keep the current constant. At low doping densities, there is little band bending, and so the local field in active wells, \(F_{\text {a-w}}\), equals approximately the mean field \(\left\langle F \right\rangle =U/L\), where

*L*is the length of the structure. Then, the energetic separation of laser levels, \(E_{3}-E_{2}\), decreases due to the linear Stark effect. With the increasing doping, the bands bend upward in the injector wells, where most of the charge is accumulated. Therefore, the field in the remaining (active) wells must be higher than the average field, \(F_{\text {a-w}}> \left\langle F \right\rangle\). This means that when doping increases (bias voltage decreases), \(F_{\text {a-w}}\) drops less than \(\left\langle F \right\rangle\) (see Fig. 6a). Consequently, the Stark shift is less sensitive to the increasing doping when the latter goes to large values. Our numerical simulations show that, with further increase of the doping, the field in the active wells, \(F_{\text {a-w}}\), stops decreasing at all, but instead saturates at certain minimum value. This causes saturation of the Stark shift for the highest doping, observed both in Fig. 4b (experiment) and Fig. 7 (simulations), and simultaneously defines the lower limit of the frequencies that can be reached by changing doping density. The upper limit of frequency tuning is more trivial: for the doping, as low as \(1 \times 10^{17} \; \text {cm}^{-3}\), there are not enough carriers to reach the experimental current (see Fig. 5), or even the threshold current required for lasing.

## 5 Conclusions

In this paper, we have characterized in detail QCL wavelength tuning by the injector doping. A rigorous model of the observed effect, based on nonequilibrium Green’s function formalism, has been presented. The physical mechanism responsible for this phenomenon is the linear Stark effect. The range of tuning is limited on both low and high doping sides. On the low doping side, the limitation is caused by the insufficient free carrier density to reach the threshold current/gain. On the high doping side, the tuning is limited by: (i) the saturation of the Stark shift caused by band bending and (ii) the decreased efficiency of carrier injection due to their redistribution caused by strong attraction of the dopants. The phenomenon can be utilized for the rough tuning of devices wavelength, dedicated to operate in the specific frequency bands.

^{17}to \(2.07\,\times 10^{17}\,\text {cm}^{-3}\)). According to Eq. (3), this difference must have been compensated by the simultaneous decrease of detuning from resonance \({\Delta }\), that is an increase of the bias voltage (see Fig. 5). Then, the associated Stark effect leads to the observed blueshift of the lasing frequency. Unfortunately, in [9], the I–V characteristics were not provided, so drawing definite conclusions on the physical mechanism, responsible for tuning the laser wavelength by the injector doping in InP-based devices, must be left for future studies.

## Notes

### Acknowledgements

This work was supported by the National Centre for Research and Development (NCBR) grant no. TECHMATSTRATEG1/347510/15/NCBR/2018 (SENSE). The authors would like to thank Dr Piotr Karbownik for fabrication of the investigated QCLs.

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