Ignition of quantum cascade lasers in a state of oscillating electric field domains

Quantum cascade lasers (QCLs) are generally designed to avoid negative differential conductivity (NDC) in the vicinity of the operation point in order to prevent instabilities. We demonstrate that the threshold condition is possible under an inhomogeneous distribution of the electric field (domains) and leads to lasing at an operation point with a voltage bias normally attributed to the NDC region. For our example, a terahertz QCL operating up to the current maximum temperature of 199 K, the theoretical findings agree well with the experimental observations. In particular, we experimentally observe self-sustained oscillations with GHz frequency before and after threshold. These are attributed to traveling domains by our simulations. Overcoming the design paradigm to avoid NDC may allow for the further optimization of QCLs with less dissipation from stabilizing background currents.

Figure 1

Operating in the NDC region enhances the ratio between the lasing driven current and the background current: (a) Calculated current versus bias assuming a homogeneous field distribution for the sample [21]. The green circle depicts the nominal operation point (NOP). (b) Corresponding result for the sample V812 [17], which is considered here. (c) Conduction band profile and relevant electronic states (sequence between 0 and 50 meV as in legend and periodically shifted with different linestyles) at the NOP of 54 mV/module for the sample V812 where two modules are displayed. Starting from the extraction barrier, the layer sequence of a module is 46/158/46/86/31.7/83 Å where the boldface font stands for ${\mathrm{Al}}_{0.15}{\mathrm{Ga}}_{0.85}\mathrm{As}$ barriers and roman font indicates the GaAs wells. The center 5 nm of the 158 Å injector well is Si doped at $6\times {10}^{16}/{\mathrm{cm}}^3$. The entire device contains 222 modules.

Figure 2

Current-voltage relations with domain formation: (a) Quantitative comparison of experimental and simulated current-voltage characteristics. Data taken at 9 K from three devices fabricated with different contacts and waveguide sizes are shown. (See Appendix pp2 for the derivation of experimental bias per module.) The three thicker sections in the electrical characteristic of the PdGe mesa (purple solid) are proportional to the amplitude of voltage self-oscillations, or instabilities, as measured with 1-GHz bandwidth oscilloscope. The theoretical domain solution (red line) without (solid) and with (dashed) the lasing field is compared to the corresponding result for a homogeneous bias drop, as shown by the black line. The dash-dotted dark gray line denotes the load line in Eq. (A3). (b) Zoom in on the experimental current-bias relation and lasing intensity around threshold for different temperatures for the TiAu laser. The signification of “raw” QCL voltage, the vertical axis label, is explained in Appendix pp2. (The temperatures are ordered from 9 K (left) to 70 K (right) at the bottom of the figure.) (c) Simulated results comparing the effects of temperature and total losses. In panel (a) and elsewhere in the paper, losses of 20 ${\mathrm{cm}}^{-1}$ and a temperature of 77 K are applied to the simulations. The vertical dashed lines in panels (b) and (c) indicate the boundaries between the regions I–V at 9 and 77 K, respectively. The small arrows in panel (c) mark the operation points shown in Fig. 4.

Figure 3

Oscillatory behavior detected experimentally: Panel (a) shows the same data as presented in Fig. 2 but plotted against external bias $U_0$. (The temperatures are ordered from 9 K (left) to 80 K (right) at the bottom of the figure.) The shaded regions indicate voltage oscillations monitored by an oscilloscope with 1-GHz bandwidth, which are observed in two different ranges of bias for temperatures below 70 K. For 9 K, we show the QCL voltage, as measured with a 15-GHz bandwidth real-time oscilloscope, during the 2-$\mu \mathrm{s}$ pulse at $U_0=54.68\mathrm{V}$ and $57.36\mathrm{V}$ in panels (b)–(e), respectively, which are representative for the two different regions of oscillation. The amplitude spectrum shown in panels (b) and (d) are taken for the time interval $1.08\mu \mathrm{s}\le t\le 1.28\mu \mathrm{s}$ of the pulses displayed in panels (c) and (e). In panel (d), up to 20 harmonics can be observed, hence demonstrating that the sharp and small features in the time-resolved voltage in panel (e) are highly periodic. Because of the incomplete RF design, details of the oscillations on the subnanosecond timescale in panels (c) and (e) may be improperly recorded; see Appendixes pp5 and pp2.

Figure 4

Simulated oscillations for four different operation points as indicated in Fig. 2. For each panel, the space and time dependence of the electric field is displayed in color scale. The bottom module ($m=222$) is next to the positive contact. The upper part of each panel shows the bias along the QCL structure (blue) and the total lasing intensity (black), which is also resolved for the individual modes with frequency marked by color. Data for further operation points and Fourier transforms of the voltage signals are displayed in Appendix pp3.

Figure 5

Lasing spectra demonstrating a significant red-shift after threshold: (a) Experimental spectra for the sample V812 at different temperatures and currents. Close to threshold, a high-frequency signal can be seen at 3.84 THz, while lower frequencies around 3.3 THz are observed for higher current. (b) Simulated spectra at different current densities plotted with a linewidth of 0.2 GHz. Lasing starts in a high-field domain, see, e.g., Fig. 4, and is shifted toward the lower frequencies as current increases.

Figure 6

Electron density calculated by the NEGF approach for different biases per module together with the Wannier-Stark levels.

Figure 7

(a) Circuit including a parasitic capacitance $C_p$ in parallel to the QCL as well as a voltage probe of resistance $R_p$, a load resistance $R_L$, and a Schottky voltage barrier $V_B$ as well as the pulser at an external bias of $U_0$. In addition to this model, a 50-$\mathrm{\Omega }$ coaxial cable connects the pulsed bias source to the load which is neglected here. The voltage probe resistance $R_p$ includes the 50-$\mathrm{\Omega }$ input impedance of the oscilloscope. (b) Numbering of electron densities $n_m$, fields $F_m$, and current densities $J_{m\to m+1}$.

Figure 8

In panels (a) and (b), we compare the NEGF results (shown as dashed lines with symbols) with the fitted effective expressions for current, Eq. (A17) [full lines in panel (a)], and gain, Eq. (A15) [dash-dotted lines for linear response, full lines for nonlinear response in panel (b)], respectively. The nonlinear response, i.e., simulations using a finite field strength of $e{F}_{\mathrm{ac}}d=5$ meV, is shown by dashed lines with circles, whereas linear response calculations, only panel (b), are shown with dashed lines with squares. Panel (c) shows a map of the gain as a function of frequency and bias per module as obtained from $G^{\mathrm{fit}}$ for low $F_{ac}$.

Figure 9

Cold finger inside the 10 K closed-cycle refrigerator: (a) Image of the TiAu metal-metal waveguide laser bar, $1049\mu \mathrm{m}$ long, soldered onto a high-resistive Si carrier. The fan-out electrodes deposited on carrier can be seen, but with a low contrast due to their small thickness; this is why they showed a residual resistivity. (b) The laser bar mounted on the cold finger. The blue wire is a 18-GHz SMA cable used for voltage detection and on the other side, it is connected to a 50-$\mathrm{\Omega }$ input channel of a high-bandwidth oscilloscope. About 25 mm separate the laser bar from the SMA cable with, in between, a circuitry that is not optimized for RF.

Figure 10

Traveling domains simulations for external biases 54.2–54.8 V as indicated in the panels to complement the data in Fig. 4. The color bars in the top and bottom panels of the left section apply also to the rest of the column, where they have been removed for increased clarity. Fast Fourier transforms (FT) of the simulated voltage oscillations $U_{\mathrm{QCL}}\left(t\right)$ are shown with matching external bias in the right column. The voltage oscillations for the FT were recorded over a sample time of 200 ns. In each such panel, the frequency of the strongest harmonic is indicated.

Figure 11

Traveling domains simulations for external biases 55.0–56.2 V as indicated in the panels to complement the data in Fig. 4. The color bars in the top and bottom panels of the left section apply also to the rest of the column, where they have been removed for increased clarity. Fast Fourier transforms (FT) of the simulated voltage oscillations $U_{\mathrm{QCL}}\left(t\right)$ are shown with matching external bias in the right column. The voltage oscillations for the FT were recorded over a sample time of 200 ns. In each such panel, the frequency of the strongest harmonic is indicated.

Figure 12

Traveling domains simulations for external biases 56.4–57.0 V as indicated in the panels to complement the data in Fig. 4. The color bars in the top and bottom panels of the left section apply also to the rest of the column, where they have been removed for increased clarity. Fast Fourier transforms (FT) of the simulated voltage oscillations $U_{\mathrm{QCL}}\left(t\right)$ are shown with matching external bias in the right column. The voltage oscillations for the FT were recorded over a sample time of 200 ns. In each such panel, the frequency of the strongest harmonic is indicated. If the fundamental frequency differs, its value is added inside brackets.

Figure 13

Traveling domains simulations for external biases 57.2–57.8 V as indicated in the panels to complement the data in Fig. 4. The color bars in the top and bottom panels of the left section apply also to the rest of the column, where they have been removed for increased clarity. Fast Fourier transforms (FT) of the simulated voltage oscillations $U_{\mathrm{QCL}}\left(t\right)$ are shown with matching external bias in the right column. The voltage oscillations for the FT were recorded over a sample time of 200 ns. In each such panel, the frequency of the strongest harmonic is indicated.

Figure 14

Time-resolved oscillations at 9 K over the 2-$\mu \mathrm{s}$-long pulse with a residual positive slope as measured with a 2-GHz bandwidth oscilloscope. Panels (a) and (b) are screen captures of the oscilloscope for $U_0=55.24$ V (i.e., around lasing threshold) and $U_0=57.9$ V respectively. As also indicated in panel (a), the magenta trace represents the current ($I_{\mathrm{tot}}$) pulse, the green trace is the raw QCL voltage, and the orange trace shows the QCL voltage channel with a 312.5-MHz low-pass filter. The sharp step down on the filtered voltage in panel (b) illustrates the right side of the merlon, region IV, previously recorded in average mode [see Fig. 2]. Panels (c) and (d) provide a detailed analysis of the data from panels (a) and (b), respectively. The purple curve (labeled Frequency) represents the instantaneous first harmonic frequency computed within a moving integration window, which is represented by a filled red curve (labeled Gaussian window) at the bottom. The dashed black line represents the voltage averaged within the Gaussian window. The inset in panel (c) focuses on the identification of threshold, the boundary between regions I and II, at $t\approx 0.65\mu \mathrm{s}$: (i) The average slope of the QCL voltage decreases; (iii) the average slope of the QCL current increases slightly; and (iii) the voltage exhibits a tiny spike of 10 mV at $t\approx 0.65\mu \mathrm{s}$, similar to Fig. 2 at threshold (where 20 mV is observed in average mode). The vertical dashed lines represent region boundaries, which we could clearly identify. The slow decline of current and increase of average voltage for $t\gtrsim 1.3\mu \mathrm{s}$ are attributed to sample heating.

Figure 15

Time-resolved oscillations recorded with the 2 GHz oscilloscope at 9 K and $U_0=57.76$ V for two different pulses. All curves correspond to Fig. 14. The deviations between panels (a), (c) and (b), (d) show typical pulse-to-pulse fluctuations. In both cases a multi-segment region III and an instantaneous oscillation frequency evolving like a staircase are observed. The $\sim 0.1\mathrm{V}$ drop of average voltage (dashed black line) marks the region IV, the right side of merlon in Fig. 2.

Figure 16

Time-resolved oscillations recorded with the 15 GHz oscilloscope at 9 K and $U_0=57.54\mathrm{V}$ for a pulse with good stability. Panel (a), left axis, shows the top of the 2-$\mu \mathrm{s}$ long pulse of the raw QCL voltage. The right axis is the instantaneous frequency as computed within a 50-ns wide Gaussian window with 8.5-ns standard deviation displayed around $t=1\mu \mathrm{s}$. Panel (b) shows details of the recorded bias signal in a 20-ns wide window. The dominant frequency of 0.9 GHz corresponds to the period indicated. However, by looking into more detail, only one of three periods are identical, as it can be seen by the wider peaks marked with a black bar. Panel (c) shows the same 20-ns wide window after filtering out the ringing effects. Panel (d) shows that the $f/3$ subharmonic regime is observed when the average raw voltage is stabilized at $\approx 12.65\mathrm{V}$. Panel (e) shows the instantaneous frequency spectrum.

Figure 17

Exemplary deconvolution of the time-resolved oscillations at 9 K and $U_0=57.65\mathrm{V}$: (a) The 2-$\mu \mathrm{s}$-long green trace is the raw signal from the voltage probe recorded with a 15-GHz bandwidth oscilloscope and the magenta trace with lower amplitude oscillations is the result from deconvolution. The dashed black trace is the average voltage computed within a moving 50-ns-wide Gaussian average window (with 8.5 ns standard deviation). A multisegment region III of oscillations can be observed. Panels (b)–(d) represent three 10-ns-wide enlarged areas. Panel (b) corresponds to the $f/3$ subharmonic regime, which becomes more distinct after deconvolution.