Coherent tunneling of mixed state hole wave packets in coupled quantum well structures
Vasu Sankaran, Jasprit Singh
Applied Physics Letters
The time-dependent Schrodinger equation is solved numerically to study the coherent tunneling of hole wave packets in asymmetric coupled quantum wells. The importance of selection rules and band mixing is evident in the extremely low rates of the wave-packet leakage from heavy-hole state to a resonant light-hole state at zero in-plane wave vector (k"). But these rates increase dramatically away from kll = 0, when the hole states acquire mixed character, and rapidly become comparable to the
... -hole to heavy-hole resonant tunneling rates. The effect of inhomogeneous level broadening arising from well size fluctuations in multicoupled quantum well systems is shown to greatly reduce the effective tunneling rates near resonance. Tunneling in quantum well structures has recently gained considerable interest.td This is especially true for cases where the representation of electronic states involves bases states of differing symmetry, and the character of the electron or hole wave function changes across the heterostructure. This is the case, for instance, when a direct bandgap electron (r type) escapes through an indirect bandgap barrier (X or L type). Similar considerations arise in hole tunneling when the hole state is described using a degenerate k-p formalism, as is appropriate in III-V materials like GaAs where the top of the valence band is fourfold degenerate. This letter, focuses on the tunneling of a hole wave packet in an asymmetric coupled quantum well (CQW) shown in Fig. 1 (a) . Such structures are often chosen by experimentalists because of the ease with which a hole wave packet can be selectively introduced in one well and the coherent tunneling studied with application of an electric field. Figure 1 (b) shows the calculated subband energies over a range of electric field strengths in a 56/ 48/88 A GaAs/Al,.,Ga",As double quantum well calculated at kll = 0. As seen from this figure, the heavy-hole (HH) state HHI in the wide well (WW) can be resonantly tuned to the light-hole (LH) state LHl in the narrow well (NW) at 31 kV/cm and to HH2(NW) at 52 kV/cm. A wave packet can be injected by pulsed laser near the HH 1 (WW) state and the luminescence decay can then study the tunneling escape of injected carriers. Some controversy exists at present on whether or not the HH wave packet can tunnel through the LH state when they are resonantly lined up by the electric field. Leo et al. ' have observed essentially no tunneling when the HH state is lined up with the LH state, while Norris6 and Liu et ~1.' report rapid tunneling under these conditions. The tunneling results are obtained from time-resolved photoluminescence experiments and since these techniques measure the decay of carriers in a particular well they depend upon coherent tunneling and scattering processes. Since scattering is somewhat sample dependent, it is important to study the intrinsic process of coherent tunneling alone. We will briefly describe our approach for studying the time evolution of multiband hole wave packets, which is an extension of a well-known method used to study the prop-agation of electron wave packets whose time evolution is determined by a scalar Schriidinger equation.' We choose to describe the hole states using the multiband effective mass theory of Luttinger and Kahn.' In the envelope function approximation resulting from a four-band description, the time-dependent hole state is given by where uti are the zone center Bloch functions of the valence band, f" the corresponding envelop functions, k" the in-plane wave vector, rll the in-plane coordinate, and z the %-40 J5 8 20 0 I -20 -40 -60 NARROW WIDE WELL WELL DISTANCE 20 30 40 ELECTFX nElD &klm) 60 FIG. 1. (a) Schematic band diagram and hole subbands in an asymmetric double quantum well structure, and (b) subband levels as a function of electric field in a 56/48/88 A G~As/A~,-,~G+.,As double quantum well calculated at k" = 0. Center of the wide well is set to zero energy.