Coherence of resonant-tunneling transport in terahertz quantum-cascade lasers

Sushil Kumar and Qing Hu

Phys. Rev. B 80, 245316 – Published 15 December 2009

Abstract

We develop simple density-matrix models to describe the role of coherence in resonant-tunneling (RT) transport of quantum-cascade lasers (QCLs). Specifically, we investigate the effects of coherent coupling between the lasing levels with other levels on the transport properties and gain spectra. In the first part of the paper, we use a three-level density-matrix model to obtain useful analytical expressions for current transport through the injector barrier in a QCL. An expression for the slope discontinuity in the current-voltage characteristics at the lasing threshold is derived. This value is shown to be a direct measure of the population inversion at threshold and contradicts the previously held belief of it being indicative of ratio of the laser level lifetimes. In the second part of the paper, we use density matrices to compute the gain spectrum for a resonant-phonon terahertz QCL design. The large anticrossing of the doublet of lower radiative levels is reflected in a broad gain linewidth due to a coherent RT assisted depopulation process. At certain bias conditions, the gain spectrum exhibits double peaks which is supported by experimental observations.

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Received 23 July 2009

DOI:https://doi.org/10.1103/PhysRevB.80.245316

Authors & Affiliations

Sushil Kumar and Qing Hu

Department of Electrical Engineering and Computer Science and Research Laboratory of Electronics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA

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Vol. 80, Iss. 24 — 15 December 2009

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Figure 1
(Color online) Plot of the magnitude squared envelope wave functions for a three-level QCL design with two different sets of basis functions. Plot (a) is for the “extended” scheme, where energy splitting due to the injector anticrossing $2 ℏ Ω_{1^{'} 3}$ is visible and the wave functions are calculated for a potential profile ${\hat{H}}_{ext}$ as it appears in the figure. Plot (b) is for the “tight-binding” scheme, where the potential ${\hat{H}}_{tb}$ is formed by making the barriers at the boundaries of a module sufficiently thick to confine the wave functions within the module.Reuse & Permissions
Figure 2
(Color online) (a) A plot of the factor $4 Ω_{1^{'} 3}^{2} τ_{∥} τ_{3}$ to determine the regime of RT transport through the injector barrier ( $⪢ 1$ coherent and $⪡ 1$ incoherent). A phenomenological value of $T_{2}^{*} = 0.75 ps$ is assumed. (b) A related plot to determine the maximum value of population inversion $(n_{3} - n_{2})$ that can be attained for a given $Ω_{1^{'} 3}$ , calculated using Eq. (10). The top axis converts $Δ n$ on the bottom axis to corresponding values of peak gain for the following typical parameters: $n_{tot} / volume = 5 \times 10^{15} {cm}^{- 3}$ , $f_{32} = 0.5$ , $Δ ν = 1.0 THz$ , and using the expression $g \approx 70 \frac{[Δ n / (10^{15} {cm}^{- 3})] f_{32}}{[Δ ν / (1 THz)]} [{cm}^{- 1}]$ for GaAs material (Ref. 6) that assumes a Lorentzian gain linewidth of $Δ ν$ (note that $f_{32}$ is the oscillator strength of the radiative transition).Reuse & Permissions
Figure 3
(Color online) Calculations of the current density $(I / area)$ as a function of the energy detuning $ℏ Δ_{1^{'} 3}$ [which is proportional to the external bias voltage $V$ , as in Eq. (12)] and its inverse slope $\frac{ℏ}{| e |} \frac{d Δ_{1^{'} 3}}{d I}$ [which is proportional to the differential resistance $R$ , as in Eq. (13)]. The calculations are done for a range of parameters using Eq. (5) for the three-level model. Following default values are chosen corresponding to the typical values for RPTQCL designs: $2 ℏ Ω_{1^{'} 3} = 1.5 meV$ , $T_{2}^{*} = 0.75 ps$ , $g_{th} = 40 / cm$ (Ref. 25), $τ_{21} = 0.5 ps$ , $τ_{32} = 3.0 ps$ , and $τ_{31} = 5.0 ps$ . A doping density $n_{tot} / volume = 5 \times 10^{15} {cm}^{- 3}$ is used. The radiative transition is assumed to have a Lorentzian linewidth of $Δ ν = 1.5 THz$ (the broad linewidth is based on the findings of Sec. 3) and an oscillator strength of $f_{32} = 0.6$ to determine the population inversion $Δ n_{th}$ required to attain a particular value of threshold gain $g_{th}$ . The thin (red) portion of the curves corresponds to $I < I_{th}$ while the thick (blue) region is for $I > I_{th}$ where the occurrence of threshold is marked by a discontinuity in $R$ . Each of the subplots show the variation in the $I − V$ ’s and the $R − V$ ’s when only a single parameter is changed, the others being kept at the values mentioned above. Note that additional linewidth broadening due to coherent effects (discussed in Sec. 3) is not considered for calculations of the curves above threshold.Reuse & Permissions
Figure 4
(Color online) Typical experimental $I − V$ ’s versus temperature for (a) a four-level and (b) a five-level resonant-phonon terahertz QCL, respectively. The corresponding band diagrams show tight-binding wave functions at design bias calculated by splitting the QCL module into multiple submodules across the relevant barriers. The radiative transition is from $3 \to 2$ , and the depopulation of the lower level is via $2 \to 2 a$ RT and $2 a \to 1 / 1 a$ electron-longitudinal-optical phonon scattering where $E_{21} \approx ℏ ω_{LO}$ . The $I − V$ ’s are measured in cw operation for metal-metal ridge lasers. The plot in (a) is from a 3.9 THz QCL labeled OWI222G (Ref. 10) of dimensions $60 μ m \times 310 μ m$ $(T_{max, cw} \sim 76 K)$ and that in (b) is from a 2.7 THz QCL labeled FL178C-M10 (Ref. 6) of dimensions $35 μ m \times 670 μ m$ $(T_{max, cw} \sim 108 K)$ . The onset of lasing results in a change in the slope of the $I − V$ , which is indicated by arrows for the curves recorded below the $T_{max, cw}$ .Reuse & Permissions
Figure 5
(Color online) Plots of the effective lower-level lifetime $τ_{2, eff}$ (also equivalent to the tunneling time through the collector barrier) for the resonant-phonon designs of the type shown in Fig. 4 calculated using Eq. (11). Plot (a) shows $τ_{2, eff}$ versus the energy detuning $ℏ Δ_{2, 2 a}$ between the levels 2 and $2 a$ and (b) shows $τ_{2, eff}$ versus the phenomenological dephasing time $T_{2}^{*}$ calculated at the $2 − 2 a$ resonance bias $(ℏ Δ_{2, 2 a} = 0)$ . $τ_{2 a} \sim 0.2 ps$ is assumed owing to the typical value obtained in $GaAs / {Al}_{0.15} {Ga}_{0.85} As$ quantum wells for $E_{2 a, 1} \sim ℏ ω_{LO}$ .Reuse & Permissions
Figure 6
(Color online) (a) Conduction-band diagram of a recently demonstrated two-well terahertz QCL design, labeled TW246 (Ref. 23). Only three levels (1, 2, and 3) participate in transport at low temperatures; however, at higher temperatures additional parasitic levels (4 and 5) are also likely to contribute to the current flow. (b) Experimental cw $I − V$ ’s and $R − V$ ’s at different heat-sink temperatures from a $20 μ m \times 1.56 mm$ metal-metal ridge laser with a $T_{max, cw} \sim 63 K$ . The lasing threshold is marked by a discontinuity in the $R$ curves that is shown more clearly in an expanded view in the lower inset. The upper inset shows the variation in fractional discontinuity in $R$ at threshold $\frac{Δ R_{th}}{R_{th}} (\equiv \frac{R_{th}^{-} - R_{th}^{+}}{R_{th}^{-}})$ and of $J_{max}$ with temperature.Reuse & Permissions
Figure 7
(Color online) Detuning energy $ℏ Δ_{1^{'} 3}$ as a function of the voltage per QCL module [ $= V / N_{p}$ in Eq. (12)] for the tight-binding wave functions of the four-level resonant-phonon design of Fig. 4a and the three-level intrawell-phonon design of Fig. 6a, respectively. Small circles represent calculated values, which are overlaid with straight line fits to the data points close to the $1^{'} − 3$ resonance. The fits indicate that $ℏ Δ_{1^{'} 3}$ is linearly related to $V / N_{p}$ close to the $1^{'} − 3$ resonance [i.e., $f_{V}$ in Eq. (12) is a constant]. Typically, lasing threshold occurs after the $1^{'} − 2$ resonance $(\sim 30 mV / module)$ beyond which the current continues to increase up to the $1^{'} − 3$ resonance when $J_{max}$ occurs.Reuse & Permissions
Figure 8
(Color online) Illustrative band diagrams for (a) double-barrier resonant-tunneling diode structure (Ref. 30) that consists of a single quantum well sandwiched between degenerately doped emitter and collector regions, and (b) a superlattice structure (Ref. 1) that consists of multiple repeated quantum wells of which only three wells are shown. In (a), the shown Fermi distributions (indicated by thin red lines) for the carrier populations apply to a quasicontinuous carrier distribution in three dimensions, whereas in (b), the shown carrier distribution applies only to the two-dimensional momentum space in the $x − y$ plane (since the carriers are confined in the $z$ direction) and is typically Maxwell-Boltzmann type (i.e., the Fermi energy $μ_{n}$ lies much below the energy of the bottom of the subband $E_{n}$ indicated by thick horizontal lines). In (a), $V$ is the voltage applied between the emitter and the collector that appears as the difference between the respective Fermi energies and in (b) $V$ represents the voltage per repeated module of the superlattice that appears as the difference in the energy of the bottom of the subbands.Reuse & Permissions
Figure 9
(Color online) Variation in the fractional change in differential resistance at threshold $\frac{Δ R_{th}}{R_{th}}$ with $τ_{31}$ calculated using the expression in Eq. (15) with same typical values of different parameters as in Fig. 3.Reuse & Permissions
Figure 10
(Color online) The four-level resonant-phonon terahertz QCL structure reproduced from Fig. 4a with a different level numbering scheme for simplicity of presentation in the following discussions. The radiative transition is from $4 \to 3$ , and depopulation of the lower level is via $3 \to 2$ RT and $2 \to 1$ e-LO phonon scattering where $E_{21} \approx ℏ ω_{LO}$ .Reuse & Permissions
Figure 11
(Color online) Computed gain spectra for the four-level structure of Fig. 10 for different injector anticrossings $2 ℏ Ω_{1^{'} 4}$ , evaluated at $1^{'} − 4$ resonance $(Δ_{1^{'} 4} = 0)$ . Levels 3 and 2 are also taken to be at resonance $(Δ_{32} = 0)$ and a small value of $2 ℏ Ω_{32} = 2 meV$ is chosen to limit additional broadening due to a coherent $3 \to 2$ RT process [as shown in Fig. 12a]. All other parameters have same equivalent values as in Fig. 3, except $τ_{21} = 0.2 ps$ that results in an effective lower-state lifetime $τ_{3, eff} (\equiv \frac{n_{3}}{n_{2}} τ_{21}) \approx 1.0 ps$ , which can also be estimated from the equivalent of Eq. (11). Also, $E_{43} = 16.5 meV$ (4 THz). The values of FWHM linewidth $Δ ν$ and population inversion $Δ n_{43} = n_{4} - n_{3}$ obtained as a result of the calculation are also indicated alongside each of the curves.Reuse & Permissions
Figure 12
(Color online) Computed gain spectra for the four-level QCL design of Fig. 10 for (a) different values of collector anticrossings $2 ℏ Ω_{32}$ at resonance $(Δ_{32} = 0)$ and (b) different values of detuning $ℏ Δ_{32}$ at $2 ℏ Ω_{32} = 5 meV$ . Levels $1^{'}$ and 4 are assumed to be at resonance $(Δ_{1^{'} 4} = 0)$ and a small value of $2 ℏ Ω_{1^{'} 4} = 1.5 meV$ is chosen to limit additional broadening due to a coherent $1^{'} \to 4$ RT process (as shown in Fig. 11). Other parameters are the same as in Fig. 11. The values of FWHM linewidth $Δ ν$ , effective lifetime of the lower state $τ_{3, eff} (\equiv \frac{n_{3}}{n_{2}} τ_{21})$ , and population inversion $Δ n_{43} = n_{4} - n_{3}$ obtained as a result of the calculation are also indicated alongside each of the curves.Reuse & Permissions
Figure 13
(Color online) (a) Computed $I − V$ and gain spectra at selected bias points for the four-level QCL in Ref. 10. These results are obtained by solving Eq. (19) in the steady state $(\frac{d}{d t} \to 0)$ , whereby, $I \equiv | e | (\frac{ρ_{44}}{τ_{41}} + \frac{ρ_{22}}{τ_{21}})$ and the gain is calculated using Eq. (22). A value of $g_{th} = 40 {cm}^{- 1}$ is assumed similar to that for Fig. 3 (Ref. 25) following which the stimulated lifetime $τ_{st}$ is determined to keep the peak gain constant as $g_{th}$ beyond threshold. Nonradiative lifetimes are calculated by including e-LO phonon scattering only whereby an electron temperature of 100 K is assumed in the upper level 4. Also, $T_{2}^{*} \sim 0.75 ps$ and $Ω_{1^{'} 3} \approx Ω_{42} \approx 0$ . (b) Experimental $I − V$ and spectra (shown as insets) at selected bias points measured at a heat-sink temperature of 20 K in cw mode. The device is a $100 μ m \times 1.39 mm$ metal-metal waveguide ridge laser from the same active region as in Ref. 10. The higher than expected bias voltages for the experimental $I − V$ are likely due to additional voltage drop at the electrical contacts.Reuse & Permissions

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