1. Introduction

The design of a high optical-confinement and low optical-loss structure to guide light of long wavelengths is one of the key challenges in the development of terahertz (THz) quantum cascade lasers (QCLs). Proposed by Kohler et al., a semi-insulating surface-plasmon (SISP) waveguide was instrumental in demonstrating the first THz QCL at 4.4 THz in 2002 [1]. In 2012, a THz QCL based on a metal-metal (MM) waveguide structure, first proposed by Williams et al. [2], achieved a record high lasing temperature (T_max) of 199.5 K [3]. Compared to SISP waveguides, MM waveguides provide high mode confinement (near unity) and relatively low loss for THz waves. However, on the negative side they often support multiple high-order transverse modes [4], which significantly degrade emission beam quality [5] and lower collection efficiency of desirable output optical power from the fundamental transverse mode. To overcome this shortcoming, one commonly-deployed fabrication technique is formation of narrow strips of exposed highly-doped n⁺ GaAs along both sides of the otherwise metalized top ridge waveguide edges [6].

The n⁺ top contact layer has a dramatic effect on the resonator transverse modes. When the heavily-doped layer between the top metal and the semiconductor active region along the ridge edges is exposed, the confined high-order optical modes are guided by the surface plasmons at the active region/n⁺ top contact layer interface, leading to very high loss and strong mode suppression [7]. As the fundament mode profile is concentrated mostly in the center of the laser ridge away from the lossy edges, this mode dominates the output, which at the same time improves the beam quality significantly. Fan et al. demonstrated in their design that the maximum peak output power from devices with exposed side strips was more than three times higher than the power from devices without such side strips [8]. On the other hand, improper strip width in relation to the total ridge width in devices with a narrow waveguide ridge may diminish the fundamental mode severely and thus degrade the QCL temperature performance. However, the exact impact of the ridge width and the side strip dimensions on THz QCL performance remains elusive to date.

This paper shows, theoretically and experimentally, that the optical loss due to the exposed side strips in the highly-doped GaAs contact layer has a stronger influence on laser performance in devices with narrower waveguide ridge widths. Three groups of THz QCLs based on metal-metal waveguides with two strips of highly-doped, top contact layer left uncovered by Ti/Au top metallization were fabricated and characterized and their light-current-voltage (LIV) curves were measured. Higher threshold current densities and lower T_max are observed in QCLs with narrower ridge widths, which is in good agreement with optical gain and loss simulation results. The inferior performance of the devices with a narrower ridge is attributed to higher optical loss of the fundamental mode due to the exposed side strips. In addition, through two-dimensional (2D) finite-element waveguide mode simulation and analysis using COMSOL PHYSICS, the effect of side strip width on the waveguide loss and high-order transverse mode suppression is investigated. The simulation results point toward the existence of an optimum exposed strip width at which the higher-order modes are effectively suppressed without generating excessive losses of the fundamental mode at the same time.

2. Device fabrication

To illustrate the effects of the exposed side strips and ridge width on device performance, we processed and characterized three series of MM waveguide THz QCLs (Groups A-C) with various waveguide ridge widths in each group. The active regions of all three groups of THz QCLs are based on phonon-photon-phonon transition [9,10], and were grown using molecular beam epitaxy. The fabricated devices consist of 276 repeats of a module with nominal layer thicknesses of 22 + δ + 22/62.5/10.9/66.5/22.8/84.8/9.1/61 Å (the barriers are indicated in bold fonts, the δ-doping positions are shown as well) for Group A, 44/32.51 + δ + 30/10.91/66.48/22.77/84.79/9.11/61 Å for Group B, and 244 repeats a module with layer thicknesses of 22 + δ + 22/66.85/12.55/88.95/22.55/73.4/3.45/94 Å for Group C. In these designs, the δ-doping with Si atoms to a sheet doping density of 3.25 × 10¹⁰ cm⁻² for Groups A and B, and 3.68 × 10¹⁰ cm⁻² for Group C, which translate into 3D doping concentration for the ~10 μm thick active regions of 9 × 10¹⁵ cm⁻³ for all three groups of devices. To achieve ohmic contacts, the active region is sandwiched between a bottom stack consisting of 100 nm of n⁺ GaAs (3 × 10¹⁸ cm⁻³) followed by 20 nm of intrinsic GaAs spacer and a top stack consisting of 40-50 nm of n⁺ GaAs (8 × 10¹⁷-5 × 10¹⁸ cm⁻³) followed by a 10 nm layer doped at 5 × 10¹⁹ cm⁻³, capped with 3 nm of low temperature grown (LT) GaAs. The peak lasing frequencies are 3.23 THz (T = 8K) and 3.19 THz (110K) for group A, 2.95 (8K) and 2.97 THz (50K) for group B, 3.91 THz (13 K) and 3.87 THz (110 K) for group C (measured by FTIR, not shown in this paper).

The semiconductor wafers were processed into double Ti/Au (50/3000 Å) metal-metal ridge waveguides structures. In-Au wafer bonding technology and standard photolithography was used [11] to fabricate QCL devices with target ridge widths of ~90, ~120 and ~150 μm, respectively for each group of lasers A, B, C. The waveguide ridges were fabricated by using reactive ion etching, minimizing lateral current spreading across the active region. Exposed side strips of highly-doped top contact layers were created along both longitudinal sides of the ridge edges in all QCL waveguides, as shown in Fig. 1(a). The actually-fabricated ridge width was measured by using scanning electron microscopy (SEM) to be ~81, ~112, and ~141 μm, respectively, slightly reduced from the design values. The SEM images confirmed uncovered strips with a uniform width of 6.5 μm on the top of the MM ridge waveguides in the three groups of THz QCLs, as shown in Fig. 1(b). The fabricated samples were cleaved into THz QCL bars with ~1 mm Fabry-Perot cavity length, then indium soldered on oxygen-free copper packages and mounted on a cold finger in a Janis Helium closed-cycle cryostat for LIV measurements in pulse mode (pulse width = 250 ns, repetition rate = 1 kHz).

 

Fig. 1 (a) Schematic diagram of the cross section of fabricated THz QCLs with a metal-metal waveguide from three groups of samples (Group A, B, and C). The active regions are based on four wells phonon-photon-phonon design, with thicknesses of ~10 μm. To achieve ohmic contacts, the active regions are sandwiched between a top stack of 100 nm of 3 × 10¹⁸ cm⁻³ n⁺ GaAs followed by 20 nm of unintentionally doped GaAs spacer for Group A, B, and C and a bottom stack of 8 × 10¹⁷/5 × 10¹⁸/5 × 10¹⁹ cm⁻³ n⁺ GaAs layers with thicknesses of 40/50/10 nm for Group A and B, and of 5.2 × 10¹⁷/5 × 10¹⁸/5 × 10¹⁹ cm⁻³ n⁺ GaAs layers with thicknesses of 40/50/10 nm for Group C. Metal cladding layers are deposited on both sides of the semiconductor region to form electric contacts. Two side strips of exposed highly doped top contact layer at the longitudinal edges of the top metal cladding are fabricated to suppress higher order transverse modes. (b) Scanning electronic microscope graph focuses at the edge of one THz QCL metal-metal waveguide (top view) shows a side strip with a width of 6.5 μm.

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3. Results and discussion

Figures 2(a)–2(c) show the measured threshold current density (J_th) of the three groups of devices (Group A-C) as a function of waveguide ridge width. Within the experimental uncertainty, one can observe a clear trend – J_th decreases as the laser ridge waveguide becomes wider for all three groups of THz QCLs. In particular, as the waveguide ridge width increases from 81 to 139 μm, the threshold current density at 10 K in Group C devices drops from over 1.83 to below 1.19 kA/cm² – a decrease by ~42%. This could be attributed to the higher optical loss in a narrower waveguide. In a device with a wider ridge waveguide, the waveguide loss is lower and a lower threshold current density is needed to satisfy the lasing threshold condition [9,10]. The LIV measurement results of Group A, B, and C are shown in Figs. 2(d)–2(f). The smooth LI curves of all samples indicate that only fundamental mode exist in waveguides.

 

Fig. 2 Calculated threshold gain of TM₀₀ modes in metal-metal waveguides of THz QCLs with different ridge widths for Group A (a), B (b), and C (c). The TM₀₀ mode threshold gain of waveguides with a fixed side strip width of 6.5 μm increase while shrinking the waveguide ridge widths for all three groups (red curves). However, the calculated threshold gain for waveguides without side strips fabricated are almost independent on waveguide ridge width (blue dashed curves). Measured threshold current densities from light-current experiments for Group A, B, and C are shown as circles in the (a), (b), and (c). The elevated threshold current densities in waveguides with narrower ridge widths for all three group of devices are in good agreement with the trend of the curve of calculated threshold gain vs. waveguide ridge width. Measured LIV curves, with the waveguide ridge widths and cold finger temperatures, for Groups A, B, and C are shown in (d), (e), and (f), respectively.

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To confirm the experimental observation and verify this hypothesis, we calculated the threshold gain $(g_{th})$of the fundamental mode (TM₀₀) in THz QCL waveguides of Group A, B, and C by using COMSOL Multiphysics. The threshold gain is defined as $g_{th}=({\alpha }_{\omega }+{\alpha }_m)/\Gamma$, where ${\alpha }_{\omega }$ is the waveguide loss; ${\alpha }_m$, the mirror loss, (2.87 cm⁻¹, 2.61 cm⁻¹, and 3.71 cm⁻¹ for Group A, B, and C, respectively) is taken from [12]; $\Gamma$is the confinement factor, which is close to unity in metal-metal waveguides. The device model in the simulation consists of a 1 mm Fabry-Perot resonant cavity and a metal-metal waveguide with variable width ranging from 60 to 160 μm. Two exposed side strips (6.5 μm wide) lie longitudinally along the ridge edges. The waveguide losses ${\alpha }_{\omega }$ were calculated by simulating the complete 2D structure of devices shown schematically in Fig. 1(a) at lasing frequency of a lasing frequency of 3.2, 2.9, and 3.9 THz for QCLs Group A, B, and C, respectively. The red curves in Fig. 2 show the calculated TM₀₀ threshold gain of the three groups of devices as a function of ridge width. For comparison, the threshold gain of the TM₀₀ mode in similar structures without exposed side strips were also calculated and are plotted as blue dotted curves in Fig. 2. The temperature dependent permittivity of the active region and highly-doped contact layers were calculated using Drude-Lorentz approximation as

In contrast to almost ridge-width-independent threshold gain in metal-metal ridge waveguides without exposed side strips (which can explain a 25 μm ridge width THz QCL without side exposed strips can still lase at a high temperature of 97 K at cw operation in [18]), the threshold gain in devices with exposed side stripes rise by about ~25 cm⁻¹ (Group A), ~35 cm⁻¹ (Group B) and ~20 cm⁻¹ (Group C) when the waveguide ridge width decreases from 160 to 60 μm. This strongly supports the experimental observation of increased threshold current density in narrower ridge devices. It is clear that the optical loss due to the uncovered side strips region becomes dominant in narrow devices (60 μm or less) and thus significantly downgrades their performance.

Another important impact of variable-width waveguides with exposed side strips on device performance is reflected by the maximum lasing temperature (T_max). The measured T_max data of Group A, B, and C are presented in Fig. 3(a). The experimental results show T_max of 117 K (132K/135 K) for devices of Group A, 102 K (115 K/128 K) for devices of Group B, and 101 K (115 K/119 K) for devices of Group C with ridge widths of ~81 μm (~112 μm/~141μm) for each group. T_max clearly exhibits a monotonic dependence on ridge waveguide width in all three groups of the THz QCLs. Maximum lasing temperature enhancement of up to 26 K was observed by just increasing the ridge width from 81 to 141 μm in group B.

 

Fig. 3 Measured maximum lasing temperatures of Group A, B, and C of THz QCLs with different waveguide ridge widths as denoted by discrete symbols, respectively (red squares: Group A, blue circles: Group B, and green triangles: Group C). The experimental results show a significant degradation of temperature performance (T_max) of THz QCLs caused by shrinking waveguide ridge width: T_max drops 18/26/18 K with waveguide ridge width decreasing from ~141 μm to ~81 μm in devices of Group A/B/C. Calculated T_max of three groups of QCLs are presented in colored curves (red: Group A, blue: Group B, and green: Group C). The calculated T_max of THz QCLs various ridge widths shows relatively good agreement with the experimental trend despite discrepancies in the exact values of T_max, which might be attributed to the inaccuracy of the models used for optical gain and waveguide loss, as well as waveguide imperfections in the fabricated devices.

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To better understand this trend, we calculated optical gain and optical loss of the three groups of THz QCLs as a function of ridge width at different temperatures. The optical gain decreases while the optical loss increases with increasing temperature. When they cross over, the device reaches its maximum lasing temperature. The simulated maximum lasing temperatures as a function of ridge width are also presented in Fig. 3. The calculation of the optical gain was performed using an improved rate equation model presented in [10], with a bandwidth of 1.5 THz. In this model, except doping concentration in the active region, all parameters are independent from which used in loss calculation. Intersubband LO phonons, ionized impurity and the interface roughness scattering mechanism were taken into consideration. Key simulation parameters such as electron heating temperature and the pure dephasing time constant are the same as those reported in [10]. Figure 3 shows that the theoretical T_max is enhanced quickly when increasing the ridge width from 75 to 150 μm, gradually approaching an upper limit – the T_max of a THz QCL with infinite ridge width. Despite of discrepancy in absolute value, the calculated T_max of THz QCLs with various ridge widths can fairly predict the trend of the observed experimental results for all three groups. The quantitative discrepancy in T_max could be attributed to the inaccuracy of the modeling used for optical gain and waveguide loss calculation, as well as imperfections in real device fabrication and characterizations.

Evidently, as demonstrated from both simulation results and the experimental data, THz QCLs with narrower ridge width display relatively higher optical loss (according to Fig. 2) and hence are expected to have lower T_max. The deteriorated temperature performance (T_max) in narrow-ridge devices could be attributed mainly to the increased optical loss contribution from the exposed side strips, which will be discussed in more detail later. It is worth mentioning that further extending the waveguide ridge width from 150 μm to infinity would result in the T_max of THz QCLs rising to its upper limit. However, broad ridge waveguides will cause a negative effect in heat dissipation [19] which may prevent continuous-wave operation of the lasers as an example. The theoretical curves in Fig. 3 show that a QCL device with a ridge width of 150 μm almost reaches the predicted T_max saturation value. This ridge width might be an optimum point as the Joule heat dissipation is still manageable in such a case [20].

The exposed side strips on the MM waveguide can also play a critical role in suppressing higher-order transverse modes. This is because of the different mode profile distribution between the fundamental and the higher-order modes. The waveguide losses of TM₀₀, TM₀₁, and TM₀₂ lateral modes in a metal-metal waveguide structure with variable-width exposed side strip width (0 to 12 μm) are calculated and plotted in Fig. 4(a). For simplicity, the thickness and doping concentration of each layer were taken from devices of Group C, and the ridge width of the waveguides was fixed to 80 μm. The waveguide losses of the TM₀₀, TM₀₁, and TM₀₂ mode are almost identical when there is no exposed side strips on the top metallization. With the exposed side strips becoming wider from 0 to 2.5 μm, the waveguide losses of the three modes all start to increase, while those of the higher-order modes (TM₀₁ and TM₀₂) increase much faster than that of the fundamental mode (TM₀₀). Beyond 2.5 μm, the waveguides losses of the three modes moderately first decrease slightly due to the mode re-distribution inside the waveguide then increase monotonously with the exposed side strips widths. It clearly reveals that the uncovered side strips impose much larger optical losses to higher-order transverse modes than to the fundamental mode. The elevated waveguide losses observed in Fig. 4(a) can be explained by the special model profile distribution resulting from the exposed side strips. Figure 4(b) shows the 2D TM₀₁ mode distribution schemes inside the waveguides with two specific strip widths: 1 μm, and 2.5 μm. The two mode distribution profiles look quite similar, but the greatest difference is observed around the exposed side strips region, as shown in enlarged insets. At the strip width of 2.5 μm, the mode distribution is much more pronounced beneath the side metal trenches (exposed side strips) than that at the strip widths of 1 μm. This mode re-distribution is because of the surface plasmon effect that exists at the interface between the highly-doped (n⁺) top contact layer (with a negative real part of the dielectric constant) and the lightly doped active region (with a positive real part of the dielectric constant) at the side strips region, (a similar effect in SISP waveguide was discussed in [21]). The optical mode is guided along this interface and leaks out through the highly doped exposed side strips region, leading to the much higher waveguide losses of the TM₀₁ mode.

 

Fig. 4 (a) Calculated waveguide loss of TM₀₀, TM₀₁, and TM₀₂ modes as a functions of side strip width of THz QCLs (with layer thickness and doping concentrations same as those from Group C QCLs) are presented. The waveguide ridge width is fixed to 80 μm. (b) Two dimensional lateral modes distributions of TM₀₁ modes (as a sample of all transverse modes) inside the waveguides with a fixed ridge width of 80 μm and various side strip widths of 1 μm and 2.5 μm. At a side strip of 2.5 μm, the mode leakage at side strips region is displayed. (c) Calculated contour map of TM₀₀ to TM₀₁ mode suppression of Group C device with various ridge widths and strip widths. The purple curves show the mode suppression (logarithmic ratio of threshold gain) of 1 dB, 2 dB, 3 dB, and 4 dB.

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Although processing exposed side strips is an efficient way to suppress the higher-order transverse modes, it also brings extra losses to the fundamental mode in narrow ridge width MM waveguides. In order to determine an optimum width of the exposed side strips considering the tradeoff between achieving effective higher-mode suppression and maintaining sufficiently low fundamental mode loss, we introduce higher-order mode suppression strength (L_db), which is defined as a logarithmic ratio of threshold gains of TM₀₀ and TM₀₁ mode: L_db = 10log₁₀(g_{th_TM01}/g_{th_TM00}). A contour map of this higher-mode suppression strength for various waveguide ridge widths and exposed side strip widths is presented in Fig. 4(c). It shows that higher modes are more effectively suppressed in a narrower waveguide with wider exposed side strips. Obviously, a fixed side strip width (such as 6.5 μm used in all tested QCLs in this study) might not be the best choice when the ridge width is in a wide range. For QCLs with a narrow ridge width waveguide - for example, 80 μm ridge width devices in Group C - the side strips width should be narrowed down to 2~4 μm to lower fundamental mode loss yet still uphold an efficient higher mode suppression strength of above 2 dB. However, for QCLs with a wider ridge width waveguide (i.e., 150 μm or above), the side strip width needs to be much larger (over 8 μm) in order to attain efficient higher mode suppression.

4. Conclusion

Three groups of THz QCLs with different waveguide ridge widths in each group and a fixed width of exposed side strips for higher-mode suppression were fabricated and characterized. The measured elevated J_th and downgraded T_max in narrower waveguide devices from all three groups of devices clearly reveal substantial performance degradation of THz QCLs due to shrinking the ridge width of metal-metal waveguides. Numerical simulation based on 2D finite-element waveguide mode analysis together with the temperature dependent Drude model were employed to calculate the waveguide losses and predict the T_max of the three groups of devices with various waveguide ridge widths. The simulation results are in good agreement with experimental observations. The losses induced by the exposed highly-doped top contact layer at the side strip region become relatively more dominant in THz QCLs with narrower ridge widths. An optimal width of the side uncovered strips can be obtained by calculating higher-mode suppression strength as a function of ridge width and side strip width, which should satisfy the tradeoff of achieving adequate higher-order mode suppression as well as maintaining sufficiently low fundamental mode loss.