Preprint IFLC11 Dr.Paul.Kinsler@physics.org http://www.kinsler.org/physics/ Interface and confined phonons in stepped quantum wells. P. Kinsler, R. W. Kelsall, and P. Harrison † Institute of Microwaves and Photonics, School of Electronic and Electrical Engineering, University of Leeds, Leeds LS2 9JT, United Kingdom. The simple symmetric and antisymmetric interface phonon modes of a square quantum well do not occur in an asymmetric quantum well. The presence of extra materials and interfaces affects the potentials of the interface (IF) phonons in addition to the widely recognised effect on the electronic wavefunctions. Our “stepped” quantum well structures have phonon modes (potentials) localised at the well/step interface, so these structures have 8 confined-LO (LC) phonon modes and 10 IF-phonons. We calculate the non-trivial dispersion of the IF-phonons for increasing asymmetry (step height); classify the resulting phonon potentials into “even-like”, “odd-like”, and “centered”; and explain the strongest electron–IF-phonon scattering processes in terms of their matrix elements. The result is that the LC scattering rates increase with increasing asymmetry, whilst the IF rates decrease. This is a draft of a paper published as Physica B (263-264) , 507 (1999), with references and email addresses updated. Layered GaAs/Ga x Al 1 x As heterostructures confine not only the quantum states of the electrons, but also those of the phonons. Stepped (asymmetric) quantum wells have been proposed as terahertz emitters [1, 2]. The utility of a given de- sign is determined by non-radiative scattering, including both electron-electron and electron-phonon processes. So far, anal- ysis has been restricted to using bulk phonon modes [3]. Here we vary the asymmetry of simplified stepped quantum wells in order to study the resulting changes in the behaviour of the phonon dispersion and potentials, as well as the intra-subband scattering rates. The dielectric continuum model (DCM) is a continuum theory based on electro-magnetic boundary conditions[4]. The dominant electron–phonon scattering processes involve confined-LO phonons (LC) and interface phonons (IF) – TO- phonon modes do not couple to electrons, and can be ignored. The total scattering rates obtained using the DCM are known to be reasonably accurate in the types of structures we con- sider. The dispersionless LC-phonons have sinusoidal inter- action potentials that are completely confined to each layer. In contrast, the IF-phonon potentials extend across the struc- ture while decaying exponentially away from the interfaces. We consider four-layer AlGaAs/GaAs semiconductor mi- crostructures (Fig. 1). The thick AlAs barrier layers have (bulk) LO and TO phonon frequencies: the LO frequency leads to a semi-infinite set of LC modes confined by the barrier/well (A) interface and the left hand edge of the mi- crostructure; and similarly for the right hand barrier layer (in- terface C and edge). Finally, the LO and TO frequencies for the barrier layers contribute two IF modes. For our numerical calculations, these barriers are taken to be 100 ˚ A thick, which is easily sufficient to confine the wavefunctions of the electron subbands. There are two inner layers of equal thickness (100 ˚ A), the well layer and the step layer. The average composition of the Electronic address: Dr.Paul.Kinsler@physics.org † Electronic address: p.harrison@physics.org C 4 3 2 1 9 8 10 5 7 6 B A Fig. 1: Asymmetric Well and IF-phonon potentials Barrier Well Step Barrier A B C A B C A B C FIG. 1: Diagram of our simple four-layer asymmetric quantum structure, where the barriers are AlAs and the well and step are Al x Ga 1 x As as described in the text. The interfaces are labeled A, B, C, and correspond to the peaks of the IF-phonon potentials drawn schematically. The absolute amplitude of the potential peaks varies depending on the particular phonon considered as well as both k and h. well and step combined is Al 0 080 Ga 0 920 As, but we vary the asymmetry by decreasing the Al composition of the well by a small fraction (to Al 0 080 h Ga 0 920 h As), while increasing that of the step by an equal amount (to Al 0 080 h Ga 0 920 h As). AlGaAs has AlAs-like LO and TO frequencies and GaAs-like LO and TO frequencies, so the well and step layers each con- tribute 2 sets of LC modes and 4 IF-phonon modes. This means the four-layer structure contains 8 LC modes, and its three distinct materials contribute to 10 IF modes. Fig. 2 compares the IF-phonon dispersion curves for our nearly symmetric structure (h 0 005) with those for our most asymmetric structure (h 0 060). The 10 branches, numbered upwards with increasing energy, are visible, al- though modes 5, 6, and 7 are compressed into only 0.5meV. For a symmetric structure (h 0), we would only see branches similar to 2, 3, 6, 7, 9, and 10 from our nearly symmetric structure. Branches 1, 4, 5, and 8 which are associated with the B interface (not present in the symmetric case), interest- ingly have little dispersion. There is no continuous transi- 1