# Short Barriers for Lowering Current-Density in Terahertz Quantum Cascade Lasers

^{1}

^{2}

^{*}

## Abstract

**:**

_{0.10}Ga

_{0.90}As heterostructures with short-barriers, the effect of IR scattering is mitigated, leading to low operating current-densities. A series of resonant-phonon terahertz QCLs developed over time, achieving some of the lowest threshold and peak current-densities among published terahertz QCLs with maximum operating temperatures above 100 K. The best result is obtained for a three-well 3.1 THz QCL with threshold and peak current-densities of 134 A/cm

^{2}and 208 A/cm

^{2}respectively at 53 K, and a maximum lasing temperature of 135 K. Another three-well QCL designed for broadband bidirectional operation achieved lasing in a combined frequency range of 3.1–3.7 THz operating under both positive and negative polarities, with an operating current-density range of 167–322 A/cm

^{2}at 53 K and maximum lasing temperature of 141 K or 121 K depending on the polarity of the applied bias. By showing results from QCLs developed over a period of time, here we show conclusively that short-barrier terahertz QCLs are effective in achieving low current-density operation at the cost of a reduction in peak temperature performance.

## 1. Introduction

^{2}) operating current-densities due to strong parasitic current channels in the superlattice. Despite a large number of different types of terahertz QCLs that have been published, there only a few that simultaneously achieved a large (>100 K) maximum operating temperature and low (≲300 A/cm

^{2}) operating (threshold and peak) current-densities [15,16,17,18,19]. In this paper, we show that shorter (and hence thicker) barriers could be utilized for such terahertz QCLs to mitigate the role of IR scattering, which leads to significant lowering of the threshold and peak current-densities, albeit in exchange of worsened maximum lasing temperatures. Experimental results from two different QCLs based on the three-well resonant-phonon design scheme are presented, both of which show low threshold and maximum current-densities. The best result is achieved for a 3.1 THz QCL with a threshold and maximum current-density of 134 A/cm

^{2}and 208 A/cm

^{2}respectively when operating in pulsed mode at 53 K, and a maximum operating temperature of ∼135 K at a current-density of ∼280 A/cm

^{2}. These are significantly lower current-densities for any three-well terahertz QCL published in the literature with uniform barriers. The results are comparable in performance to a step-well terahertz QCL [18], which is more challenging to grow, and a four-well resonant-phonon terahertz QCL [17], which is typically more challenging to design and optimize.

## 2. Role of IR Scattering in Three-Well QCLs with Short and Tall Barriers

_{0.15}Ga

_{0.85}As material-system that has shown maximum operating temperatures in the range of 175–200 K [5,20]. The QCL was redesigned with shorter (and hence, thicker) Al

_{0.10}Ga

_{0.90}As barriers. This modified design, named as RTRP3W198, maintains approximately similar injection/collection coupling strengths for resonant-tunneling and subband energy-separations compared to the reference design of Ref. [5]. However, the oscillator strength of the lasing transition was reduced from 0.56 to 0.42 to make the design more diagonal, which serves to suppress parasitic current channels [3]. To better understand how shorter barriers lead to lowering of the operating current-density, a simplified density-matrix transport model as reported in Ref. [21] is utilized to compute the current-density, and subband populations, and gain spectrum as a function of frequency in both the designs. Figure 1a shows the computed electron wavefunctions in a tight-binding formalism for bias corresponding to peak-gain in the typical three-well resonant-phonon QCL bandstructure that are used to compute the transport characteristics. The tight-binding wavefunctions are useful to model the inter-sub-module electron transport via resonant-tunneling. A somewhat arbitrary phenomenological dephasing time of ${T}_{2}^{\ast}$∼0.25 ps is used [3] that leads to current-densities that are similar to experimentally measured values. For simplicity, the electron temperature is kept the same as the lattice temperature for computation of subband lifetimes (which primarily affects LO-phonon scattering times). The intersubband transition rates for IR scattering are calculated as in Ref. [22] with an assumed mean height of roughness of $\Delta =0.4$ nm and a correlation length of $\mathsf{\Lambda}=7.5$ nm [23] which are also phenomenological parameters. Alloy disorder scattering and impurity scattering that are relatively weaker are neglected in the model.

^{2}in the tall-barrier design to 360 A/cm

^{2}in the short-barrier design. A reduction of peak-gain by ∼25% from 51 cm

^{−1}to 38 cm

^{−1}also occurs, which is primarily due to an almost similar fractional reduction in the radiative oscillator strength of the respective designs. It can therefore be concluded that lowering of current-density does not negatively impact gain, and hence, a high current-density is not a necessary requirement for high-gain. The lifetime of the upper radiative subband due to $4\to 3$ IR scattering (${\tau}_{\mathrm{IR}},4\to 3$) is ∼2.6 times longer in the short-barrier QCL RTRP3W198 compared to the reference design. This is directly reflected in experimental results as shown subsequently, and is the primary benefit of using shorter and thicker barriers that suppress IR scattering. The density-matrix model predicts a relatively large (∼31 cm

^{−1}) gain even at high-temperatures for the short-barrier QCL as seen from the computation results at 150 K in Table 1. It may be noted, however, that presumably increased thermal leakage into continuum at higher operating temperatures should lead to rapid degradation in gain with temperature, thereby lowering its the maximum lasing temperature considerably in such a design. This is indeed what is observed experimentally.

## 3. Experimental Results from Two Different QCLs

^{18}cm

^{−3}contact layers grown above (60 nm thick) and below (50 nm thick) the 10-μm thick active region. The exposed top contact layer adds extra loss and hence, for ease of fabrication, it is removed entirely. However, for negative polarity operation as shown in the following text, the contact layer must be kept under the metal for biasing. A 200 nm Al

_{0.55}Ga

_{0.45}As etch-stop layer underlies the entire growth. Details regarding epitaxy of teraherz QCLs could be found in Refs [24,25]. A metal-metal waveguide was fabricated by a standard Cu-Cu wafer bonding and processed using the method as shown in Ref. [26]. Ta/Cu/Au layers were deposited as both top (18/200/150 nm) and bottom (18/200/150 nm) contact to lower the waveguide losses [27]. Ridges were processed by wet-etching using 1:8:80::H

_{2}SO

_{4}:H

_{2}O

_{2}:H

_{2}O etchant. The fabricated devices were cleaved and indium soldered on copper mount, wire bonded and mounted on the cold-stage of a Stirling cooler.

^{2}and 208 A/cm

^{2}respectively. The corresponding peak optical power as detected by the power meter was 19 mW. Its maximum lasing temperature was 135 K. In comparison to the QCLs in Refs. [5,20] a loss in the maximum lasing temperature by ∼40–65 K is compensated by a significant reduction (by a factor of ∼7) in the operating current-densities, and hence, the electrical power dissipation. The latter is an important performance enhancement if the QCL was to be operated below 100 K in a miniature cryocooler, notwithstanding its lower maximum operating temperature. This result also indicates that while IR scattering plays an important role in electron transport at low-temperatures, it is not the primary mechanism limiting maximum operating temperatures of resonant-phonon terahertz QCLs.

^{2}and 217 A/cm

^{2}respectively at 53 K, which is lower than the values in Ref. [28] owing to lower waveguide losses and also possibly due to growth including possibly doping variations. Note that the maximum lasing temperatures of BIDR3W198 are lower than that in Ref. [28] in both polarities. We note the I-Vs near peak bias of BIDR3W198 in both polarities are very rough which may indicate slight growth problems. Nevertheless, the experimental results show that bidirectional terahertz QCLs based on short-barriers are effective in providing broadband gain from the same superlattice structure with low operating current-densities and reasonably high maximum operating temperatures.

## 4. Summary

_{0.10}Ga

_{0.90}As material system with short-barriers compared to the conventional Al

_{0.15}Ga

_{0.85}As barriers used in literature. The RTRP3W197 QCL was grown several years ago and was briefly reported in Ref. [29]. The results presented in this manuscript with new designs that are grown at different time-periods now conclusively show the benefit of using short-barriers to suppress IR scattering in resonant-phonon terahertz QCLs, which, in conjunction with the use of a diagonal radiative transition [3], can lead to a significant lowering of operational current-density in such QCLs. While this benefit comes at the expense of lower maximum operating temperatures, the smaller electrical dissipation in the QCLs could be a significant advantage when such QCLs are operated below 100 K in miniature cryocoolers that have a limited cooling capacity. In the future, two QCLs based on short barriers (Al-10%) and tall barriers (Al-15%) with very similar design parameters, such as lasing frequency, oscillator strength, and subband coupling could be designed, fabricated and tested in order to give a direct conclusion about the impact of IR scattering on the performance of terahertz QCLs.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**(

**a**) Plot of magnitude-squared wavefunctions computed in a tight-binding formalism for a typical three-well resonant-phonon terahertz quantum cascade laser (QCL) design at the bias corresponding to peak-gain. A QCL module is divided into two sub-modules at barriers affecting transport of electrons via resonant-tunneling (RT). The radiative transition $4\to 3$ occurs in the sub-module consisting of quantum-wells QW 1 and QW 2, whereas the injector and extraction subbands 2 and 1 respectively are localized in the sub-module comprising of quantum-well QW 3. (

**b**) Theoretically calculated gain spectra using a simplified density-matrix transport model for the three-well resonant-phonon designs RTRP3W198 (with short Al-10% barriers) and the design from Fathololoumi et al. [5] (with taller Al-15% barriers) respectively, at the bias corresponding to peak-gain. In the model, the electron and lattice temperatures are set to 50 K and the inter-module electron transport occurs via RT whereas intra-module transport is modeled by electron-LO-phonon scattering and interface-roughness scattering.

**Figure 2.**(

**a**) One module conduction band diagram of the short-barrier three-well resonant-phonon QCL design named RTRP3W198 at the peak-gain bias. Starting from injector barrier, layer thicknesses in monolayers (ML) are (with barriers indicated in bold face)

**23**/31/

**14**/30/

**22**/61 where the center of the widest well is n-doped with sheet-density of 2.8 × 10

^{10}cm

^{−2}. The radiative transition is between $4\to 3$ where E

_{43}= 12.4 meV (∼3 THz) and the radiative oscillator strength f

_{43}= 0.41. ${\Delta}_{\mathrm{mn}}$ is the energy splitting between subbands m and n when they are aligned for optimal resonant-tunneling. ${\Delta}_{32}$ = 3.86 meV. (

**b**) Experimental light-current-voltage (L-I-V) characteristics from a 3.2 mm × 120 μm ridge laser with metal-metal cavity. A threshold current-density of 134 A/cm

^{2}and a maximum of 208 A/cm

^{2}was measured at 53 K in pulsed mode of operation with 300 ns pulses repeated at 100 kHz. Insets show the representative spectra of the QCL measured at 53 K. The QCL emits in the frequency range of 3.0–3.2 THz.

**Figure 3.**(

**a**) and (

**b**) Conduction band diagrams for a bidirectional three-well resonant-phonon terahertz QCL design named BIDR3W198 at peak-gain bias corresponding to positive and negative polarity operation respectively. In contrast to RTRP3W198 of Figure 2, the bidirectional design BIDR3W198 is characterized by injection and extraction barriers of same thickness. Starting from injector barrier, layer thicknesses in monolayers (ML) (where barriers are indicated in bold face) are

**23**/31/

**12**/30/

**23**/60 where the center of the widest well is n-doped with a sheet-density of 2.8 × 10

^{10}cm

^{−2}. Key design parameters are indicated alongside the band diagrams. (

**c**) and (

**d**) Experimental light-current-voltage (L-I-V) characteristics from a 1.6 mm × 120 μm ridge laser with metal-metal cavities biased under positive and negative polarity respectively. The threshold current-densities at positive and negative polarity bias are 167 A/cm

^{2}and 217 A/cm

^{2}respectively at 53 K, and the corresponding maximum operating temperatures are 141 K and 121 K in pulsed mode of operation. Pulsed L-I-V measurement was done with 300 ns pulses repeated at 100 kHz. The QCL radiates in the frequency range 3.1–3.4 THz under positive polarity and 3.3–3.7 THz under negative polarity bias respectively. Representative spectra measured at 53 K are shown the insets of the corresponding figures.

**Table 1.**Key parameters related to the density-matrix transport model used for the three-well resonant-phonon designs RTRP3W198 (with short-barriers) and that by Fathololoumi et al. [5] (with taller barriers), computed for the bias corresponding to maximum gain. Electron temperatures in the subbands was assumed to be same as that of the lattice temperature T (as indicated) for simplicity and a general lack of consensus in terahertz QCL literature about the relation between the two.

Design | T (K) | ${\mathit{\tau}}_{\mathbf{LO},4\to 3}$ | ${\mathit{\tau}}_{\mathbf{IR},4\to 3}$ | ${\mathit{\tau}}_{\mathbf{tot},4\to 3}$ | ${\mathit{\tau}}_{2\to 1}$ | J (A/cm^{2}) | Peak Gain (cm^{−1}) |
---|---|---|---|---|---|---|---|

RTRP3W198 | 50 | 148 ps | 5.9 ps | 5.7 ps | 0.56 ps | 362 | 38 |

Fathololoumi et al. [5] | 50 | 66 ps | 2.3 ps | 2.2 ps | 0.21 ps | 942 | 51 |

RTRP3W198 | 150 | 4.0 ps | 7.3 ps | 2.6 ps | 0.28 ps | 666 | 31 |

Fathololoumi et al. [5] | 150 | 2.3 ps | 2.8 ps | 1.3 ps | 0.20 ps | 1353 | 37 |

**Table 2.**Performance summary of various three-well resonant-phonon QCLs based on GaAs/Al

_{0.10}Ga

_{0.90}As superlattices. The design RTRP3W197 was developed and grown several years ago and was briefly reported in Ref. [29]. It differs from RTRP3W198 of Figure 2 with a slightly thinner extraction barrier (by 1 ML).

Design Name | Wafer No. | Frequency | Jth (A/cm^{2}) | Jmax (A/cm^{2}) | Tmax | Doping Density |
---|---|---|---|---|---|---|

RTRP3W197 | VB0464 | 2.7 THz | 240 @77 K | 450 @77 K | 157 K | 2.8 × 10^{10} cm^{−2} |

RTRP3W198 | VB0890 | 3.1 THz | 134 @53 K | 208 @53 K | 135 K | 2.8 × 10^{10} cm^{−2} |

BIDR3W198 (pos. bias) | VB0891 | 3.2 THz | 167 @53 K | 270 @53 K | 141 K | 2.8 × 10^{10} cm^{−2} |

BIDR3W198 (neg. bias) | VB0891 | 3.5 THz | 217 @53 K | 322 @53 K | 121 K | 2.8 × 10^{10} cm^{−2} |

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**MDPI and ACS Style**

Gao, L.; Reno, J.L.; Kumar, S.
Short Barriers for Lowering Current-Density in Terahertz Quantum Cascade Lasers. *Photonics* **2020**, *7*, 7.
https://doi.org/10.3390/photonics7010007

**AMA Style**

Gao L, Reno JL, Kumar S.
Short Barriers for Lowering Current-Density in Terahertz Quantum Cascade Lasers. *Photonics*. 2020; 7(1):7.
https://doi.org/10.3390/photonics7010007

**Chicago/Turabian Style**

Gao, Liang, John L. Reno, and Sushil Kumar.
2020. "Short Barriers for Lowering Current-Density in Terahertz Quantum Cascade Lasers" *Photonics* 7, no. 1: 7.
https://doi.org/10.3390/photonics7010007