6.1 Quantum Ballistic Transport Calculations

Throughout this section we consider GaAs wells and barriers composed of Al_xGa_1-xAs.

6.1.1 Tunneling Current Density

On the basis of the Robin boundary conditions (3.19) and (3.20) the current density is numerically evaluated for periodic QCL structures as well for quasi-periodic systems. The transport of charge through the structure arises as a property of the wave function, and the current density is expressed as  [1]

(6.1)

where A is the cross-sectional area of the quantum well structure. The work of P. Harrison  [95] suggests that the electron distributions in both the active region and the injector subbands are thermalized and thus the energy distribution of electrons in each subband can be approximated by a Fermi-Dirac distribution function

(6.2)

where E_F,α is the quasi Fermi energy of the α-th subband and E_α(k_∥) = E_α + ℏ²k_∥²∕2m^⋆.

The electron states are evaluated using a selfconsistent Schrödinger-Poisson solver, where the electron concentration is related to the electronic wave functions and the electron sheet densities in the corresponding subbands which can be calculated according to the Fermi-Dirac distribution. For a given total sheet density the quasi Fermi level is obtained iteratively, where the initial value of the quasi Fermi level is taken to be  [96]

while the band gap in eV is

with

The conduction band edge is given by

with

6.1.1.1 Comparison with the Tsu-Esaki Model

The Tsu-Esaki Model describes the tunneling current density. The corresponding formula is usually written as an integral over the product of two independent parts which only depend on the energy component perpendicular to the interface  [97].

(6.3)

T_c (E) is the transmission coefficient which characterizes the penetrability of the considered energy barrier. The supply function N(E) describes the supply of carriers for tunneling.

Figure 6.2 compares the simulated current density using the Robin boundary condition approach with the current density calculated by the Tsu-Esaki model for a GaAs/Al_0.3Ga_0.7As Fibonacci superlattice (FSL)  [98] which is a quasi-periodic multibarrier system. The generalized FSL is generated by an iterative process according to the Fibonacci sequence  [99]

where A and B are the blocks of the well and barrier. A special kind of FSLs with a higher quasi-periodicity follows the iteration rule  [100]

Table 6.1 presents a few initial generations of FSLs of the two different kinds.

The FSL type considered is S₅^′ which has the sequence BBABBABBBABBABBBA, where A and B are the elementary blocks corresponding to the GaAs quantum well and the Al_0.3Ga_0.7As barrier, respectively. The width of the well block is taken to be 5 unit cells of GaAs monolayers, whereas the number of the unit cells belonging to Al_0.3Ga_0.7As monolayers for the barrier block equals to 3. The lattice constants for the well and barrier materials are 5.6533 and 5.65564 , respectively. The appearance of resonance-type peaks in the current density curves is typical for quasi-periodic systems, and the results obtained are in good agreement with the Tsu-Esaki model.

6.1.1.2 Comparison with the NEGF Approach

The ability of the Robin boundary conditions to produce satisfactory current carrying states is also verified by comparing our results for the tunneling current density with calculations based on nonequilibrium Green’s functions (NEGF)  [1]. For this purpose a typical example of a midinfrared quantum cascade laser is considered  [9].

Figure 6.3 illustrates the conduction band profile of this device. The layer sequence of one period belonging to the GaAs/Al_0.33Ga_0.67As structure, in nanometers, starting from the injection barrier is: 5.8, 1.5, 2.0, 4.9, 1.7, 4.0, 3.4, 3.2, 2.0, 2.8, 2.3, 2.3, 2.5, 2.3, 2.5, and 2.1, where normal scripts represent the wells, bold the barriers.

The comparison of the obtained current-voltage characteristics with the simulation employing the nonequilibrium Green’s functions method is illustrated in Figure 6.4. The simulation is performed with the number of periods to be 30 and the temperature is taken to be 77 K. As in the case of quasi-periodic superlattices, the application of our method to calculate current carrying states proves to be very promising for periodic QCL structures as well.

To illustrate the convergence of the solution to the Robin problem and the Dirichlet problem as E_α → ∞, we investigated the transformations of the wave functions as a function of the applied electric field, because the energy levels increase as the electric field decreases. For this purpose the real part of the wave functions belonging to the Robin problem are compared to the solutions of the Dirichlet problem. The imaginary part, indicating the development of the current carrying states, are studied analogously.

The order of magnitude we use for the electric field is 10³ kV/cm on the one hand and 10^-6 kV/cm for the weak field calculation. The investigated structure is the same three-level QCL design as above. The real components of the wave functions are plotted in Figure 6.5 for the electric fields F = 10³ kV/cm and F = 10^-6 kV/cm. For F = 10³ kV/cm the calculations show obvious deviations in the spatial dependence between the solutions of the Robin and Dirichlet boundary value problems, unlike the weak field case F = 10^-6 kV/cm, where a convergence between these solutions is observable.

The imaginary components of the wave functions corresponding to the Robin boundary value problem are plotted in Figure 6.6. At the very small electric field F = 10^-6 kV/cm the imaginary part is almost zero everywhere except some very small peaks whose magnitudes are insignificant compared to the imaginary part of the wave function at F = 10³ kV/cm.

Since the imaginary part of the wave function becomes zero with increasing energy, our results indicate that the tunneling current vanishes in this case. This behavior can be understood in terms of the transmission coefficient which expresses the probability of tunneling and decays rapidly with energy according to  [101]

(6.4)

where T_c denotes the transmission coefficient for an electron in a heterostructure.

As mentioned above, the energy levels increase as the electric field decreases. Due to equation (6.4) the transmission coefficient decreases significantly with decreasing electric field, which is illustrated in Table 6.2. The three energy levels correspond to the upper and lower laser levels. Thus, by reducing the bias the conduction decreases and only minimal current flows. In this low field region the open boundary condition approach yields no significant deviation from the Dirichlet boundary value problem.

6.1.2 Optical Gain

Here, we focus on the calculation of optical gain under consideration of the proposed Robin boundary conditions. Due to the strong dependence of the dipole matrix element on the wave function the determination of optical gain qualifies for verification of boundary conditions.

In QCLs intersubband transitions contribute to the gain profile, especially the transitions between the upper and the lower laser level. In general, the standard expression for optical gain in semiconductors can be written as  [102]

(6.5)

Here, f(E) denotes the distribution function for the electrons. |M_u,l|² represents the transition matrix element, ρ is the density of states, and Λ is the lineshape function. In a quantum well, the density of states can be taken as ρ = m^⋆∕πℏ²L  [103] and the transition matrix element is approximated by the dipole matix element according to M_u,l = m₀²ω²|z_ul|². Substituting these approximations for the density of states and the dipole matrix element, and using the line shape as proposed in  [104]

(6.6)

the optical gain can be estimated as  [105]

(6.7)

where z_ul is the dipole matrix element, n_r is the refractive index, ϵ₀ is the vacuum permittivity and c is the speed of light. The dipole matrix element depends strongly on the wave functions of the relevant states, and so does the optical gain.

The validity of equation (6.7) is not restricted to any particular scattering mechanism. Certain assumptions have to be made about γ. We assume that the homogeneous broadening γ is dominated by the interaction with optical phonons  [106]. Thus

The top line describes optical phonon absorption and the bottom line optical phonon emission, where γ₀ = (πe² ∕2ℏ)[1∕ε_∞- 1∕ε₀]q_ph, q_ph = (2m_eω_ph∕ℏ)^1∕2, and the phonon occupation number is given by N_ph = 1∕(exp (ℏω_ph∕k_BT) - 1). Furthermore, ε_s and ε_∞ are the low and high frequency dielectric constants respectively.

6.1.2.1 Calculation Results

We consider a laser consisting of 90 periods of a GaAs/Al_0.15Ga_0.85As heterostructure at a temperature of 70 K  [107]. The GaAs/Al_0.15Ga_0.85As layer sequence in nanometers is 3.8, 14.0, 0.6, 9.0, 0.6, 15.8, 1.5, 12.8, 1.8, 12.2, 2.0, 12.0, 2.0, 11.4, 2.7, 11.3, 3.5, and 11.6, where the AlGaAs layers are in bold. Figure 6.7 shows our simulation results to be in good agreement with measurements that are performed on this QCL design  [108].

The dipole matrix element between states Ψ_α and Ψ_β along the well growth direction z is given by  [109]

(6.8)

In order to obtain numerical illustrations for z_αβ = e^{i2πk_∥∕k_∥max} ∫ ₀^LzΦ_α^⋆(z)Φ_β(z)dz, the product |x _∥ | cos(x_∥,k_∥) is set to 2π∕k_∥^max, and k_∥^max = 0.1nm^-1. The calculations of the dipole matrix elements, which performed for the two different electric field strengths 10³ kV/cm and 10^-6 kV/cm, are illustrated in Figure 6.8 as functions of the wave vector k_∥. The dashed curves represent the results determined taking into account Robin boundary conditions and the solid curves result from Dirichlet boundary conditions. A comparison of the results belonging to the different electric fields reveals that the solutions of the Robin and Dirichlet boundary value problems converge when lowering the electric field. Thus, our calculations demonstrate the accuracy of the proposed Robin boundary conditions for QCL simulations. However, for the unbiased case Dirichlet boundary conditions yield nearly comparable results.