Paper 146 Abstract

Proceedings of the 1997 Advanced Workshop on Frontiers in Electronics
ed. by G. S. Pomrenke, et al., (IEEE, 1997) pp. 39-42.
(WOFE'97 Proc. ISBN 0-7803-4059-0)

Abstract of the invited talk at the

Advanced Workshop on Frontiers in Electronics (WOFE)
Tenerife, Canary Islands, Spain (January 6-12, 1997)

Hot Electrons in Quantum Cascade Lasers

Serge Luryi
Department of Electrical Engineering
State University of New York at Stony Brook
Stony Brook, NY 11794-2350


"Hot electrons in quantum cascade lasers"

The quantum cascade laser is a new mid-infrared laser, based on unipolar transitions of electrons between energy levels created by quantum confinement. Since the first demonstration of QCL (Faist et al. 1994), its design has continuously improved through a series of elegant innovations by the Bell Labs group culminating in their recent report of a high power room-temperature operation (Faist et al. 1996). We have carried out a theoretical analysis of the QCL operation (Gelmont et al., 1996; Gorfinkel et al., 1996), which suggests that it is dominated by hot-electron effects. These effects arise from the power dissipated in each cascade period, mainly due to nonradiative electron transitions. The energy stored in the transverse degrees of freedom, corresponding to in-plane motion of carriers, fundamentally changes both the lineshape of intersubband resonance and the spectral characteristics of gain.

Conventional two-level models do not adequately describe the intersubband gain function in experimentally realized QCL heterosystems. The reason is two-fold. Firstly, one cannot neglect the nonparabolicity: its inclusion suppresses the peak gain value by more than an order of magnitude. Secondly, the relaxation rate of intersubband resonance, which is mainly due to transverse intrasubband scattering processes, is strongly dependent on the electron kinetic energy. Hot-carrier effects are therefore indispensable for the correct physical understanding of the temperature behavior of quantum cascade lasers.

Depending on the laser design, there may be different scenarios of how the kinetic energy is distributed among various groups of electrons and dissipated into the lattice. The most attractive situation arises at low carrier concentrations, when electron-electron collisions are not fast enough to establish a common electron temperature between the two subbands. It turns out that in this regime the lower-subband electron distribution can be approximately characterized by a negative temperature, so that states near the subband bottom are mostly unoccupied. At the same time, the upper subband remains approximately in a thermal ensemble, characterized by the ambient temperature. This situation is most favorable for high gain. It is even possible to achieve high positive values of the gain at room temperature without an overall population inversion between the two subbands. Within the framework of our theory, the low concentration regime can be treated rigorously.

For higher (but still moderate) concentrations, the electron temperature approximation becomes applicable, but it is still possible (at least semi-quantitatively) to regard optical phonon scattering as the dominant phase breaking mechanism. In this range, we predict a peculiar hot electron effect, manifested in the dependence of the lasing wavelength on the pump current, including a regime where the wavelength switches "digitally" between two stable values.

Very high concentrations require a special consideration, as it becomes necessary to explicitly include electron-electron scattering in the calculation of scattering rates. Also the energy balance equation in this range should include the dependence of the electron cooling rate on the carrier density and effective temperature due to optical phonon bottleneck. Moreover, as shown by Warburton et al. (1996), at high carrier concentrations the nonparabolicity is strongly suppressed by depolarization, so that intersubband resonance becomes a collective effect. Our theory does not directly apply to the high-concentration limit.

Experimentally reported QCL heterostructures (Faist et al. 1994-1996) operate neither in the high-concentration nor in the low-concentration regime. For moderate concentrations our theory provides a semiquantitative account of all experimentally observed features, such as the existence of a sharply defined critical temperature beyond which the QCL does not reach the generation regime. This is due to the fact that the subthreshold gain is not a monotonic function of the injection current in QCL but reaches a maximum and then declines due to the hot-carrier effect. Above threshold, the dependence of gain on the carrier temperature leads to a lower than expected slope efficiency of the laser.

Given the tremendous applications potential of the QCL, we believe the device deserves a major development effort. The improvement strategies that have born most fruit so far, have dealt with the kinetics of subband population and with the laser engineering, such as improved heat sink, cavity design, etc. We have identified three new strategies. First, one should strive to reduce the effects of non-parabolicity by employing new material systems and designs to engineer subbands with approximately equal kinetic masses in the QW plane. Next, one can attempt to reduce the carrier heating effects by introducing new carrier cooling channels, e.g., via intervalley phonon emission in the doped reservoir region. Third, one can render hot-carrier effects harmless by implementing the low concentration regime of operation. Our work shows that the QCL still has an enormous reserve for improvement.


Faist, J., Capasso, F., Sivco, D. L., Sirtori, C., Hutchinson, A. L., Cho, A. Y. (1994) "Quantum Cascade Laser," Science 264, pp. 553-556.

Faist, J., Capasso, F., Sirtori, C., Sivco, D. L., Baillargeon, J. N., Hutchinson, A. L., Chu, S.-N. G., Cho, A. Y. (1996) "High power mid-infrared quantum cascade lasers operating above room temperature", Applied Physics Letters 68, pp. 3680-3682.

Gelmont, B., Gorfinkel, V., Luryi, S. (1996) "Theory of the spectral lineshape and gain in quantum wells with intersubband transitions", Applied Physics Letters 68, pp. 2171-2173.

Gorfinkel, V., Luryi, S., Gelmont, B. (1996) "Theory of gain spectra for quantum cascade lasers and temperature dependence of their characteristics at low and moderate carrier concentrations", IEEE Journal of Quantum Electronics 32, pp. 1995-2003.

Warburton, R. J., Gauer, C., Wixforth, A., Kotthaus, J. P., Brar. B., Kroemer, H. (1996) "Collective effects in the intersubband resonance of InAs/AlSb quantum wells", Superlattices and Microstructures 19, pp. 365-374.

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