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We study the impact of interface roughness on the operation of mid-IR and THz quantum cascade lasers. Particular emphasis is given towards the differences between the Gaussian and exponential roughness distribution functions, for which we present results from simulation packages based on nonequilibrium Green’s functions and density matrices. The Gaussian distribution suppresses scattering at high momentum transfer which enhances the lifetime of the upper laser level in mid-IR lasers. For THz lasers, a broader range of scattering transitions is of relevance, which is sensitive to the entire profile of the interface fluctuations. Furthermore we discuss the implementation of interface roughness within a two band model.
© 2015 Optical Society of America
1. Introduction
Quantum cascade lasers (QCLs) [1] have become an important source of infrared radiation for spectroscopy applications [2]. They consist of a vast number of specifically designed semiconductor layers. As the interfaces of these layers are never entirely perfect, the lateral translational invariance is broken and interface roughness scattering becomes inevitable. Several studies have focused on its relevance for the lifetime of the upper laser level [3–7], which is a key element for the lasing performance. Furthermore, interface roughness is relevant for the broadening of tunneling transitions [8, 9] in QCLs. The deviations from an ideal interface are treated statistically, where the spatial correlation function of the height fluctuations contains the relevant information to evaluate the scattering matrix elements. (Correlations between different interfaces are of minor importance as they are washed out under typical growth conditions unless barriers are very thin [4].) It is common to model this correlation function by a Gaussian with two fit parameters “although there is no definite physical ground“ [10]. In this paper such a Gaussian correlation function is compared to an exponential fit and calculations for different QCLs are presented under both assumptions. While there are some specific differences, the calculated current-voltage characteristics and gain spectra are comparable if the respective fit parameters are correctly transformed.
2. Interface roughness models
The central assumption for modeling interface roughness is that the position of the interface between two materials is fluctuating by η (r), where r is a two-dimensional vector in the x − y plane of the heterostructure layers. Averaging over a large area, the statistical properties of η (r) become important. In order to quantify roughness scattering, the square of the matrix element for momentum transfer q between different subband states is required. Next to prefactors, it contains the integral
where A is the sample area. This is the Fourier transformation of the spatial correlation function 〈 η (r)η (0)〉 for the fluctuations, which we denote by f(q). It is common to assume a Gaussian distribution [10] with Alternatively, the idea, that there is a constant likelihood to be at the rim of a roughness plateau, suggests an exponential distribution: The average fluctuation height Δ and the spatial correlation length Λ are not directly measurable, but should be seen as fit parameters. Thus it is meaningless to compare the Gaussian and an exponential distribution with the same set of parameters, i.e. setting Δ̃ = Δ and Λ̃ = Λ. In this case the exponential distribution would result in twice the scattering of the Gaussian one at q ≈ 0. Rather, Λ and Δ should be changed as to achieve similar scattering rates in a wide range of q, in order to quantify the difference using either distribution. Provided the dominating scattering events have small q, both distributions are expected to provide similar results with the so that the functions f (q) as well as their second derivatives coincide at the maximum |q| = q = 0. The corresponding functions f (q) are displayed in Fig. 1 for different roughness parameters. We find, that the exponential distribution provides much stronger scattering for large q with this translation. A second natural translation is given by requiring an identical average fluctuation height 〈η (r = 0)η (0) 〉 and identical f (q = 0) for the Gaussian and exponential distribution. This provides which better reproduces the fall-off at larger q-values but agrees less well for small q as can be seen in Fig. 1.
Fig. 1 Fourier transforms f (q) of the correlation functions for different roughness distribution functions. In panel (a), the Gaussian distribution, Eq. (2), has the parameters Λ = 9 nm and Δ = 0.1 nm. The parameters for the exponential distributions, Eq. (3), Expon 1 (Λ̃ = 3.6 nm, Δ̃ = 0.17 nm) and Expon 2 (Λ̃ = 6.3 nm, Δ̃ = 0.1 nm), are transformed via Eq. (4) and (5), respectively. These distributions are used for the InGaAs/InAlGaAs based IR QCL. Panel (b) shows the exponential distribution with Λ̃ = 10 and Δ̃ = 0.2 nm, and the parameters for the two Gaussian distributions, Gauss 1 (Λ̃ = 24.5 and Δ̃ = 0.115 nm) and Gauss 2 (Λ̃ = 14.1 and Δ̃ = 0.2 nm), which are transformed via Eqs. (4) and (5), respectively. These distributions are used for the GaAs/AlGaAs based THz-QCLs.
There are actually experimental indications for an exponential distribution for several material systems such as Si/SiO2 [11]; InAs/GaSb [12], InAs/GaInSb [13]; GaAs/InGaAs [14]. For GaAs/AlGaAs or InGaAs/InAlAs interfaces, relevant for QCL structures, less information is available. In [15], Offermans et al. report lateral fluctuations of 10 nm for an InGaAs/InAlAs QCL but do not provide a distribution function, while [16] reported values of 10–20 nm for a GaAs/AlGaAs superlattice. In [5], Leuliet et al. used a Gaussian distribution with Λ = 6 nm and Δ = 0.15 nm, to fit their data for a GaAs/Al0.33Ga0.67As QCL. Recently, 〈η (r)η(0) 〉 has been measured for a GaAs/InGaAs QCL, where three different Gaussians were required to fit the data [17].
Considering the case of an IR QCL, the out-scattering from the upper laser level requires a particular large momentum transfer q, as the large energy mismatch between initial and final states (typically the lasing energy) must be transferred into in-plane kinetic energy. For typical infrared QCLs with a lasing energy of h̄ω = 150 meV, this implies q ≈ 0.4 nm−1 (using the in-plane effective mass for InGaAs lattice matched to InP). For a typical value of Λ̃ 10 nm, the Gaussian thus provides a strong suppression of the scattering matrix element. In contrast, the exponential distribution of Eq. (3) shows a power law suppression which allows for some out-scattering, in particular with the Translation 1, see Fig. 1(a). For THz lasers, on the other hand, h̄ ω ≈ 10 meV corresponds to q ≈ 0.1 nm−1 (using the in-plane effective mass for GaAs) and scattering at intermediate q values decreases the lifetime. In this case the level spacings for the extraction process (as well as the injection for a scattering assisted design) match the optical phonon energy (∼36 meV) with q ≈ 0.25 nm−1. Thus, large q scattering might even improve performance in these structures. Similar interface engineering has been proposed in [18], where barriers are inserted to decrease the lifetime of the lower laser state.
The power law for large q can actually be related to the behavior of g(r) = 〈η (r)η (0) 〉 for r → 0. Standard rules of Fourier transformation provide the second derivative
Assuming, that g(r) = g(r) and f (q) = f (q) are rotational invariant, the asymptotic behavior f (q) ∼ 1/q3 is thus related to δ2g(r)/δ r2 → ∞ for r → 0, which means, that the gradient of g(r) is discontinuous at the origin (this corresponds to the Fourier transform asymptotic behavior theorem in the one dimensional case discussed in [19]). Such a discontinuity naturally occurs, if g(r) has a finite slope at r = 0 (which is the case for the exponential distribution in contrast to the Gaussian).A (negative) slope of g(r) at r = 0 can be motivated by the following argument: We consider the product η (r)η (r0) for a fixed reference point r0. This product is positive for r = r0 and maintains its value as long as r is on the same plateau as r0. Crossing the rim of the plateau, η (r)η (r0) changes, more likely to a negative value, as the average elongation 〈η (r) 〉 = 0. As one averages over all reference points r0, there are some points, which are precisely on the rim of a plateau and thus 〈η (r)η (0) 〉 is expected to have a finite negative slope in the direction of r for small r. Note, that this argument requires a sharp drop of the scattering potential for the conduction band electrons at the rims between the plateaus. On the other hand, if the rims of the plateaus result in a smooth change of the potential landscape, there would be no such negative slope for small r, and the Gaussian distribution would be a viable choice. To determine which behavior is the most accurate in a real situation would therefore require the precise measurement and interpretation of the actual potential landscape.
In order to demonstrate the relevance of the roughness distribution, we provide simulation results for a mid-IR QCL, a THz QCL with scattering injection, and a THz QCL with tunneling injection. We apply the interface roughness parametrizations shown in Figs. 1(a) and (b) for the mid-IR QCL and THz QCLs, respectively. In all cases we apply identical distributions for all interfaces.
3. Two-band model: including the valence band offset
For IR QCLs non-parabolicity in the conduction band is relevant as the electronic states cover a large range of energies. This can be effectively implemented by mixing the conduction band wave-function with at least one component from the valence band [20]. As the valence band offset (VBO) differs from the conduction band offset, roughness may act differently, whether one restricts to the conduction band or implements a two-band model. In a phenomenological approach, where little is known about the actual roughness distribution, appropriate choices of the roughness parametrization can to a large extent compensate for the difference between these concepts. Nevertheless the magnitude of the difference is of interest for a priori calculations.
The Hamiltonian for interface roughness scattering is written in second quantization as
where zj denote the interface positions. In the two-band model, where c and v denote the conduction and valence band components, respectively, and ΔEc/v are the respective band offsets. This gives terms in the matrix elements squared as where the wave functions are taken at the respective zj. If only the conduction band offset is taken into account, then the resulting expression is instead Since for type I heterostructures ΔEc and ΔEv have opposite signs, the former case will exhibit lower interface roughness scattering.Using our Nonequilibrium Green’s function (NEGF) simulation scheme [21, 22] we performed simulations for different QCLs in order to compare the expressions (8) and (9). For the THz QCL of [23], Fig. 2 shows that the neglect of the valence band offset in the roughness scattering term neither changes the current density nor the gain. In contrast, a slight increase in the current density at the peak and a slight reduction of the gain is visible for the IR QCL of [24] in Fig. 3. This follows the expected trend, that nonparabolicity becomes more relevant with reduced band gap and increased state energies. Consequently, the difference between both approaches may become more prominent for QCLs at shorter wavelengths.
Fig. 2 (a) Current-field simulations of the THz QCL [23] using the NEGF model with and without inclusion of the valence band offset (VBO) in the roughness scattering. The simulation temperature is 140 K and the dashed line shows the experimental data at 150 K. Here, the exponential distribution function from Fig. 1(b) is used. (b) Gain simulations for the same sample at a bias of 74 mV/period.
Fig. 3 (a) Current-field simulations of the mid-IR QCL [24] using the NEGF model with and without inclusion of the valence band offset (VBO) in the roughness scattering. The Gaussian interface roughness from Fig. 1(a) is applied. (b) Results for the gain at an electric field of 50 kV/cm and a lattice temperature of 300 K.
In both cases the simulations agree quantitatively with experimental data. As the simulations shown do not include the lasing field they cannot reproduce the experimental currents above the threshold current density of 1.5 kA/cm2 in Fig. 3.
In the rest of this work, all simulations of the mid-IR QCL are performed using Eq. (9), whereas all simulations for the THz QCLs employ Eq. (8). In the calculations of the energy levels as well as the matrix elements of all other scattering mechanisms, i.e. acoustical and optical phonons, alloy and impurity scattering which do not depend explicitly on the band offsets, we use the two-component wavefunctions within the effective two-band model [20].
4. Results
4.1. Mid-IR laser
In order to quantify the impact of different interface roughness distributions on the QCL performance, we simulate the IR device [24] shown in Fig. 4 with the DM model from [25] and the NEGF model. A detailed discussion of the model differences is given in [26], where we applied the Gaussian roughness model with Λ = 9 nm and Δ = 0.1 nm. Here we compare these results with the two exponential models applying different translations of the parameters as shown in Fig. 1(a).
Fig. 4 Band structure with the square moduli of the wavefunctions, together with the carrier density from the NEGF simulation, for the mid-IR structure of [24]. The arrow indicates the main laser transition.
A comparison of the current-field characteristics is shown in Fig. 5(a). We find that the Expon 1 roughness distribution provides generally higher currents than Expon 2 and Gauss. This can be attributed to the shorter lifetime of the upper laser state, as shown in Table 1, due to enhanced roughness scattering with large momentum transfer, which facilitates the transfer of carriers through the device. The trend is the same in both simulation schemes, albeit the currents from the NEGF model are generally smaller than the DM results. Note that the field for the experimental data does not take into account any possible bias drop in contacting regions, which would reduce the field slightly. Thus, comparison with experiment cannot clearly support a certain model.
Fig. 5 (a) Current-field characteristics of the QCL in [24] for the DM (dashed lines) and NEGF (full lines) simulation schemes and different roughness distributions given in Fig. 1(a). (b) Peak gain vs. electric field. The dotted line denotes the gain required to compensate the losses. The red crosses show the experimental threshold data in both panels.
Table 1. Scattering times in the DM model for the upper and lower laser state at a bias of 50 kV/cm. The NEGF simulations show the same trend but due to the intricate treatment of coherences, it is less straightforward to extract a single time.
Now we consider the simulated peak value of the gain as a function of applied electric field, displayed in Fig. 5(b). Here we find the highest gain for the Gaussian roughness, somewhat lower values for Expon 2 and relatively low gain for Expon 1. Again this can be directly attributed to the lifetime of the upper laser state, which is central for the inversion. Quantitatively, the NEGF model provides an inversion Δn of 2.54, 1.14, and 2.33 × 109cm−2 and a linewidth Γ of 13.8, 13.1, and 11.6 meV for the Gauss, Expon 1 and Expon 2 distribution, respectively, at an electric field of 50 kV/cm. Thus the key contribution Δn/Γ suggests a reduction in gain by 53% (Expon 1) and an increase by 9% (Expon 2) of the peak gain compared to the Gaussian distribution. This reflects the trend in the full NEGF calculations at 50 kV/cm, where the corresponding relative changes are a reduction by 42% (Expon 1) and an increase by 10% (Expon 2), respectively. Γ is dominated by intra-subband scattering with low q, hence the similar Γ for the Gauss and Expon 1 distributions. Expon 2 has a lower Γ, as expected from the lower f (q) at low q. From this reasoning, the Expon 1 distribution is expected to have the largest Γ, however we find that Gauss results in a somewhat higher value, for which we currently have no clear explanation.
A gain of ∼9 cm−1 is required in order to overcome the total losses of the experimental sample [26], and this is observed at the experimental threshold field of 48 kV/cm. All three roughness distributions agree reasonably with the experimental threshold current density, the Gauss and Expon 2 requiring slightly higher losses and the Expon 1 slightly lower, and both the DM and NEGF models provide the same threshold field when the same distribution is employed. Finally, we note that the DM model provides significantly larger gain than the NEGF model at higher fields. The reasons are not yet fully understood, however similar output powers are found if gain saturation is considered [26].
4.2. Scattering assisted injection THz QCL
For the THz structures studied below, the exponential distribution function with Λ̃ = 10 nm and Δ̃ = 0.2 nm provide results in close agreement with experimental data [23,27]. Employing translations (4) and (5) thus provides the corresponding Gaussian distributions displayed in Fig. 1(b). Here Gauss 1 agrees with the exponential at low q while Gauss 2 agrees better at higher q.
First, we study the scattering assisted injection design presented in [23]. For this structure, see Fig. 6, the laser level separation is ∼ 14 meV, which corresponds to a momentum transfer of q ∼ 0.16 nm−1. One would therefore expect that in this case the intermediate-q is of high relevance. Furthermore, the electron transport through the device is also relying on two different optical phonon resonances (with an energy difference matching q ≈ 0.25 nm−1) and a tunneling resonance at the main operation point. This explains the differences in the current densities shown in Fig. 7(a), where the current essentially increases with the size of the scattering matrix elements in the range of 0 < q ≲ 0.25 nm−1 as determined by f (q) displayed in Fig. 1(b).
Fig. 6 Electron density and square of the wavefunctions moduli at a bias of 74 mV/period, for the structure of [23], based on resonant phonon injection and extraction. The upper and lower laser levels are labeled by u and l, respectively, i label the injector level and e the extractor level. The simulations are carried out for a lattice temperature of 140 K.
Fig. 7 (a) Current-field simulations of the THz QCL [23] using the NEGF model with different roughness distributions given in Fig. 1(b). The simulations are carried out for a lattice temperature of 140 K. (b) Gain at 74 mV/period for the THz QCL [23], using different roughness models. The linewidths Λ of the gain peaks are without interface roughness (IFR): 2.8 meV; for Gauss 1: 3.6 meV; for Expon: 3.9 meV; and for Gauss 2: 4.3 meV.
These effects are more pronounced at lower temperature, as phonon scattering becomes weaker while roughness scattering is less temperature dependent. However, at low temperature, the NEGF simulations provide an extremely non-thermalized carrier distribution in the upper laser state, which appears to be an artifact due to the neglect of electron-electron scattering. Thus we consider our results as not reliable under these conditions.
Figure 7(b) shows the gain evaluated near the current peak. As expected, the peak gain decreases with the scattering strength around q = 0.16 nm−1, which dominates the scattering from upper to the lower laser level. Furthermore, the linewidth is affected, which enhances the impact of the different roughness models. Here, the difference in linewidth between the Gauss 1 and Expon roughness distribution is actually the least, as low q-scattering dominates the width. Neglecting interface roughness altogether (dashed lines) gives a large and sharp gain peak, showing that roughness plays a vital role, deteriorating the performance of the QCL at 140 K for this design.
4.3. Resonant Tunneling Injection THz QCL
As the resonant-tunneling designs have received much attention and have also repeatedly broken the temperature record, it is of interest to expand this study to cover those designs as well. In this work we focus on the three well structure shown in Fig. 8 investigated by Kumar [27] which was later improved to reach operation temperatures of ∼ 200 K [28].
Fig. 8 Electron density and square of the wavefunctions moduli at a bias of 56 mV/period, for the structure of [27], based on resonant tunneling injection from the injector level (i) into the upper laser level (u) and phonon extraction from the extraction level (e). The lower laser level is labeled by l.
A collection of simulation results is presented in Fig. 9. Let us first consider the validity of the simulations by comparing the ab initio calculations to the experimental results of [27]. Threshold current is reported to increase from 410 A/cm2 at 5 K heatsink temperature to 800 A/cm2 at 180 K. At the design bias of 56 mV/period the low temperature simulations provide a high gain of the order of 100 cm−1and a current density above the experimental threshold (except for Gauss 1), consistent with the onset of gain at a lower current. While the current peak without laser operation, only reaches ≈ 550 A/cm2, a significantly higher current is found employing simulations under laser operation [29], shown by crosses in Fig. 9(a). This is consistent with experimentally measured currents for this sample. For higher temperatures gain drops and currents around ≈ 800 A/cm2 match well the experimental threshold current.
Fig. 9 (a) Current field characteristics of the resonant-tunneling design demonstrated in [27] at both high and low temperatures for different roughness distributions given in Fig. 1(b). Crosses show current densities under laser operation, assuming total losses of 30 cm−1. (b) Gain simulations at the same temperatures where in addition the case of no interface roughness is also included.
The results for different roughness distributions follow the same trend as for the scattering-assisted design. Comparing the low temperature threshold current indicates that roughness scattering in accordance with the Expon or Gauss 2 distribution is adequate. In general the differences between the roughness distributions are slightly less significant than for the scattering-assisted design addressed above, as there are fewer interfaces per length present in this design. For low temperatures, switching on the scattering given by the exponential correlation function leads to a decrease of peak gain with 20%, which is also consistent with Monte-Carlo studies for resonant-tunneling designs [30].
Below threshold, the current-field characteristics in Fig. 9(a) show a pre-peak around 37 mV/period. This is due to the intermediate resonance between the injector state and the extraction state as shown in Fig. 8, in accordance with the discussion in [27]. At low temperatures the experiment observes a plateau region for a bias range of ≈ 1.6 V with a current of ≈ 400 A/cm2 after which lasing sets in. This is consistent with our calculated current-field relation, if one takes into account domain formation for biases surpassing the pre-peak [31]: In this case a high-field domain forms with a field around 50 mV/period, while the current and the low-field domain is locked at the pre-peak. Within the high-field domain we calculate a gain of 50/cm and thus lasing sets in if approximately half of the device (222 periods in total) are covered by the high-field domain. The difference between the fields in the low- and high-field domain, thus suggests a bias range of 1.4 V for the current plateau before lasing sets in.
5. Conclusions
We have studied the relevance of different distribution functions for the interface roughness in Quantum Cascade Lasers. In principle, exponential distribution functions provide a slower decay of scattering with increasing transition wave-vector q, compared to Gaussians, which can be related to a finite slope of the spatial correlation function at the origin. We find that for IR QCLs, the scattering at large q values is most important and that the results for Gaussian and exponential distribution functions are comparable, if they provide similar matrix elements in this region. These findings are recovered by different simulation schemes, which demonstrates that they hold beyond specific approximations in the respective models. In contrast, for THz QCLs scattering at intermediate q is more relevant and we showed that the width of the gain spectrum is most sensitive to low q scattering. For all structures studied here, increased roughness scattering enhances current and deteriorates gain. However, scattering matrix elements with different ranges of q are of relevance for IR and THz structures, which has to be taken into account, when choosing a model distribution functions.
Furthermore we have shown that corrections due to the valence band components used for the treatment of non-parabolicity are negligible for THz QCLs and are of minor importance for IR-QCLs in the 8 μ m region.
Acknowledgments
The research leading to these results has received funding from the European Union Seventh Framework Programme (FP7/2007–2013) under grant agreement n° 317884, the collaborative Integrated Project MIRIFISENS and the Swedish Research Council (VR).
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