^{2,a)}, E. Dupont

^{1,b)}, S. Fathololoumi

^{1,2}, C. W. I. Chan

^{3}, M. Lindskog

^{4}, Z. R. Wasilewski

^{1,2}, G. Aers

^{1}, S. R. Laframboise

^{1}, A. Wacker

^{4}, Q. Hu

^{3}, D. Ban

^{2}and H. C. Liu

^{5,c)}

### Abstract

We designed and demonstrated a terahertz quantum cascade laser based on indirect pump injection to the upper lasing state and phonon scattering extraction from the lower lasing state. By employing a rate equation formalism and a genetic algorithm, an optimized active region design with four-well cascade module was obtained and epitaxially grown. A figure of merit which is defined as the ratio of modal gain versus injection current was maximized at 150 K. A fabricated device with a Au metal-metal waveguide and a top GaAs contact layer lased at 2.4 THz up to 128.5 K, while another one without the top GaAs lased up to 152.5 K ( ). The experimental results have been analyzed with rate equation and nonequilibrium Green's function models. A high population inversion is achieved at high temperature using a small oscillator strength of 0.28, while its combination with the low injection coupling strength of 0.85 meV results in a low current. The carefully engineered wavefunctions enhance the quantum efficiency of the device and therefore improve the output optical power even with an unusually low injection coupling strength.

The authors thank Dr. Marek Korkusinski from NRC for providing the genetic algorithm and Pietro Patimisco from the Università and Politecnico di Bari for helpful stimulating discussions. They also would like to acknowledge the financial supports from Natural Science and Engineering Research Council (NSERC) of Canada, Canadian Foundation of Innovation (CFI), the CMC Microsystems, and Ontario Research Fund (ORF). H.C.L. was supported in part by the National Major Basic Research Project (2011CB925603) and the Natural Science Foundation of China (91221201 and 61234005).

I. INTRODUCTION

II. DEVICE DESIGN

A. Wavefunction engineering challenges

B. Figure of merit

C. Rate equation modeling assumptions

D. Evaluation of selected design

III. EXPERIMENTAL RESULTS

IV. RATE EQUATION ANALYSIS

A. Electrical characteristics

B. Maximum current density versus temperature

C. Intermediate resonances

D. Differential resistance at threshold

E. Cavity loss estimation

V. CONCLUSION

### Key Topics

- Tunneling
- 44.0
- Electric fields
- 26.0
- Current density
- 20.0
- Quantum cascade lasers
- 18.0
- Negative resistance
- 17.0

## Figures

Schematic diagram of a scattering-assisted QCL active region based on a phonon-photon-phonon configuration. Throughout this paper and whatever the electric field, the states within a module are labeled in energy ascending order e, 1, 2, i, and 5. The solid lines show the forward scatterings, while the dashed lines indicate the back scatterings. Δ and Ω are the detuning and the coupling between states i and e, respectively. The green lines indicate the correct injection and extraction, while the red lines show the wrong injection and extraction in each module. 2 and 1 are the ULS and LLS, respectively.

Schematic diagram of a scattering-assisted QCL active region based on a phonon-photon-phonon configuration. Throughout this paper and whatever the electric field, the states within a module are labeled in energy ascending order e, 1, 2, i, and 5. The solid lines show the forward scatterings, while the dashed lines indicate the back scatterings. Δ and Ω are the detuning and the coupling between states i and e, respectively. The green lines indicate the correct injection and extraction, while the red lines show the wrong injection and extraction in each module. 2 and 1 are the ULS and LLS, respectively.

Conduction band diagram and the moduli squared of wavefunctions of the THz 3P-QCL, V845, at 21 kV/cm. The “+” signs denote the position of Si doping in each module. The intersubband lifetimes by LO-phonon emission are given at the resonant in-plane kinetic energy.

Conduction band diagram and the moduli squared of wavefunctions of the THz 3P-QCL, V845, at 21 kV/cm. The “+” signs denote the position of Si doping in each module. The intersubband lifetimes by LO-phonon emission are given at the resonant in-plane kinetic energy.

The 4-level RE simulation results of the structure presented in Fig. 2 . (a) Different characteristic times at 20 K ( , thick blue lines). The scattering time presented in figure are defined as follows: is tunneling time (solid line), (dotted line), and (dashed dotted line) are the transit times—excluding the tunneling time—across the four wells before and after threshold, respectively; is injection state lifetime (dashed line); and is the modified effective lifetime (dashed dotted dotted line). (b)Normalized populations of the four states at 20 K (thick blue lines) and 150 K (thin red lines) lattice temperatures and the population inversion ( ) at 20 K (blue solid circles) and 150 K (red solid circles), (c)Current density, lasing frequency (dashed line), and optical gain-bandwidth product vs electric field at 20 and 150 K lattice temperatures.

The 4-level RE simulation results of the structure presented in Fig. 2 . (a) Different characteristic times at 20 K ( , thick blue lines). The scattering time presented in figure are defined as follows: is tunneling time (solid line), (dotted line), and (dashed dotted line) are the transit times—excluding the tunneling time—across the four wells before and after threshold, respectively; is injection state lifetime (dashed line); and is the modified effective lifetime (dashed dotted dotted line). (b)Normalized populations of the four states at 20 K (thick blue lines) and 150 K (thin red lines) lattice temperatures and the population inversion ( ) at 20 K (blue solid circles) and 150 K (red solid circles), (c)Current density, lasing frequency (dashed line), and optical gain-bandwidth product vs electric field at 20 and 150 K lattice temperatures.

Left axis: The bias voltage of THz 3 P-QCL V845 versus the current density, (a) device A (b) device B. The short vertical arrows show the change in the slope of the V-J curves at laser threshold and the lowest temperature (10 K for device A or 7.8 K for device B). Right axis: Collected THz light (optical output power) versus current density at different heat sink temperatures. Since the measurement set-up and the waveguide properties are different, the collected light, the maximum current density, and the threshold current are different in plots (a) and (b). Drop voltage on device B is higher than on device A, the latter having the top 100 nm contact GaAs layer hence, a top Schottky contact with a short depleted region ( ).

Left axis: The bias voltage of THz 3 P-QCL V845 versus the current density, (a) device A (b) device B. The short vertical arrows show the change in the slope of the V-J curves at laser threshold and the lowest temperature (10 K for device A or 7.8 K for device B). Right axis: Collected THz light (optical output power) versus current density at different heat sink temperatures. Since the measurement set-up and the waveguide properties are different, the collected light, the maximum current density, and the threshold current are different in plots (a) and (b). Drop voltage on device B is higher than on device A, the latter having the top 100 nm contact GaAs layer hence, a top Schottky contact with a short depleted region ( ).

Maximum current density and threshold current density as functions of heat sink temperature for devices A and B. The dashed line shows the result of a 5-level rate equation simulation assuming a constant product .

Maximum current density and threshold current density as functions of heat sink temperature for devices A and B. The dashed line shows the result of a 5-level rate equation simulation assuming a constant product .

THz spectra recorded for different biases and temperatures. The current density, the applied voltage bias, and voltage drop per module are reported in the figure. Spectrum at 150 K was collected from device B while all other spectra were measured from device A.

THz spectra recorded for different biases and temperatures. The current density, the applied voltage bias, and voltage drop per module are reported in the figure. Spectrum at 150 K was collected from device B while all other spectra were measured from device A.

The current density vs electric field were calculated by using a 5-level first-order rate equation formalism at 10 K for lasing (red) and non-lasing (blue) devices. The green, pink, and cyan lines represent the leakage currents from the wrong extraction 2-e, and the wrong injections i-1, i-e, respectively. The vertical dashed lines were plotted to determine the first NDR and threshold voltage of the device at 10 K. The black dashed line shows the current density by using the second-order model of tunneling. The experimental curve of device A, shown as an orange dotted line, was measured at 10 K for comparison.

The current density vs electric field were calculated by using a 5-level first-order rate equation formalism at 10 K for lasing (red) and non-lasing (blue) devices. The green, pink, and cyan lines represent the leakage currents from the wrong extraction 2-e, and the wrong injections i-1, i-e, respectively. The vertical dashed lines were plotted to determine the first NDR and threshold voltage of the device at 10 K. The black dashed line shows the current density by using the second-order model of tunneling. The experimental curve of device A, shown as an orange dotted line, was measured at 10 K for comparison.

Conduction band diagram and the moduli squared of wavefunctions of V845 at 7.7 kV/cm. States in left module (upstream), middle module, and right module (downstream) are represented by subscripts , and n + 1, respectively. The extraction state (e) of each module is in resonance with state (1) of next module at an electric field of 7.7 kV/cm.

Conduction band diagram and the moduli squared of wavefunctions of V845 at 7.7 kV/cm. States in left module (upstream), middle module, and right module (downstream) are represented by subscripts , and n + 1, respectively. The extraction state (e) of each module is in resonance with state (1) of next module at an electric field of 7.7 kV/cm.

Left axis: The differential resistance of non-lasing (the red dashed line) and lasing (solid lines with symbols) device A versus current density at different temperatures. The L-J measurement results are also plotted (right scale) to determine the threshold current at each temperature.

Left axis: The differential resistance of non-lasing (the red dashed line) and lasing (solid lines with symbols) device A versus current density at different temperatures. The L-J measurement results are also plotted (right scale) to determine the threshold current at each temperature.

Symboled lines are the cavity loss × gain bandwidth products of device A, calculated by a 4-level RE model, at different lattice temperatures (TL ) and for two electron temperatures, Te , such as , 100 K. The solid lines are the peak gain × gain bandwidth products vs lattice temperature at 19.7 kV/cm calculated by the RE model.

Symboled lines are the cavity loss × gain bandwidth products of device A, calculated by a 4-level RE model, at different lattice temperatures (TL ) and for two electron temperatures, Te , such as , 100 K. The solid lines are the peak gain × gain bandwidth products vs lattice temperature at 19.7 kV/cm calculated by the RE model.

Current densities at different lattice temperatures, calculated with the NEGF method. is located at 19.3 kV/cm. The experimental curve of the non-lasing device, shown as an orange dotted line, was measured at 4.2 K for comparison.

Current densities at different lattice temperatures, calculated with the NEGF method. is located at 19.3 kV/cm. The experimental curve of the non-lasing device, shown as an orange dotted line, was measured at 4.2 K for comparison.

Carrier densities at (a) 18.7 kV/cm and (b) 19.3 kV/cm. Current is peaked at the bias in (b), although the tunneling resonance is greater in (a).

Carrier densities at (a) 18.7 kV/cm and (b) 19.3 kV/cm. Current is peaked at the bias in (b), although the tunneling resonance is greater in (a).

(a) The detuning energy between extraction and injection states Eei , the energy differences of the extraction , the injection and the energy spacing of the main leakage channel . (b) Population densities of the injection ni , upper laser , lower laser and extraction ne states. At electric fields where i and e are almost degenerate, the average value of ne and ni is shown.

(a) The detuning energy between extraction and injection states Eei , the energy differences of the extraction , the injection and the energy spacing of the main leakage channel . (b) Population densities of the injection ni , upper laser , lower laser and extraction ne states. At electric fields where i and e are almost degenerate, the average value of ne and ni is shown.

The gain spectrum of the structure near the NDR and for different lattice temperatures. At 50 K, the gain spectrum at lower electric field is also plotted to show the red shift in the spectrum.

The gain spectrum of the structure near the NDR and for different lattice temperatures. At 50 K, the gain spectrum at lower electric field is also plotted to show the red shift in the spectrum.

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