For several years now it has been recognized that the effective-mass model is insufficient to accurately describe tunneling through indirect gap barriers [1,2,3,4]. The interaction between quasi-bound X states in the barrier with continuum 
states results in resonance-antiresonance features known as Fano resonances [5]. Such features have been observed experimentally as negative differential resistance in 
single barrier heterostructures [6]. In order to describe tunneling in these structures one must use models which take material bandstructures into account. In this work we choose to describe bandstructure using tight binding models. The Quantum Transmitting Boundary Method (QTBM) is implemented to obtain open system boundary conditions for the tight-binding Hamiltonian. Using this method one obtains scattering states by simply solving a sparse system of linear equations.
In order to accurately model the electronic structure of the material we have employed the 
empirical tight-binding model [7]. In this basis the Schrödinger equation is expressed as:
is a subvector containing the atomic orbital coefficients for the jth layer. The Hamiltonian matrix elements are contained in the submatrices 
and 
. The tight-binding parameters used in this work are taken from Boykin [8]. In our formulation each layer corresponds to a lattice unit cell. Therefore these subvectors and submatrices are of order ten for the 
model ( 5 anion and 5 cation orbitals ). At the GaAs/AlAs interface the tight binding parameters are taken to be the average of those in the bulk materials on either side.
This Hamiltonian is coupled to the semi-infinite contact regions by adapting the Quantum Transmitting Boundary Method (QTBM) [9,10] to the tight binding basis. The QTBM operator is equivalent to the inverse of the propogator (
) for the system. We have developed efficient numerical techniques to locate the poles and zeros of 
Our approach is to determine the position and width of the resonance lineshape by numerically locating the poles of the propogator 
in the complex energy plane. We have developed a shift and invert non-symmetric Lanczos algorithm which allows us to rapidly determine the location of these poles [11]. The complex energies associated with the poles provides the location and width of the sharp features in the transmission coefficient. With this information we may obtain an analytic fit to the transmission characteristic. In order to obtain the fit, we assume a rational form for the transmission lineshape and expand the denominator as a partial fraction.
Here, 
and 
represent the location of the poles and zeros of 
. The partial fraction expansion coefficients (
) are treated as fitting parameters for the transmission lineshape. Once the fits are obtained, integration over the resonances becomes a trivial numerical task.
We apply our method to calculate current density versus voltage in a GaAs/AlAs resonant tunneling diode. In this structure, resonances due to states confined by the X-point conduction band profile significantly contribute to the current flowing through the device. In addition, non-parabolicity of the imaginary bands in the AlAs barrier result in increased tunneling current in the device. Thus, a full bandstructure model is necessary to accurately simulate current in this structure. In Figure 1, single band and multi-band simulations of the I-V characteristic of a GaAs/AlAs RTD are compared with experiment. Figure 2 shows the analytic fit to the multi-band transmission probability. In Figure 3 and 4, the electron density of states arising from the single band and multi-band models are compared.
Figure 1. Current density vs. voltage for a GaAs/AlAs double barrier resonant tunneling diode. Solid line is experimental data, dashed line is a full bandstructure calculation, and dotted line is a single band effective mass calculation.
Figure 2. Analytic fit for several resonances in the GaAs/AlAs double barrier resonant tunneling diode. Solid line is the actual transmission characteristic, dashed line is the analytic fit.
Figure 3. Single band density of states for the GaAs/AlAs RTD.
Figure 4. Multi-band density of states for the GaAs/AlAs RTD. The full bandstructure model predicts additional resonances due to states confined by the X-point profile. This results in the additional current predicted in Figure 1.
