**Drift-Diffusion Mode-Space Approach to Subband Transport in
Devices with Transverse Quantum Confinement**

**Introduction**

The Drift-Diffusion Mode-Space model (DDMS) is a semi-classical approach to transport in devices with strong transverse confinement. The solution for carrier density and current is decoupled into 1D, 2D or cylindrical Schrödinger equation in transverse direction and 1D classical drift-diffusion equation to account for carrier transport in each subband. Thus, the model rigorously captures quantum effects in transverse direction and yet inherits all familiar ATLAS models for mobility, recombination, impact ionization, and band-to band tunneling.

**Simulation**

Usage of DDMS is quite analogous to that of Schrödinger or mode-space NEGF models. The DDMS models is activated by an option DD_MS along with SCHRO (for electrons) and/or P.SCHRO (for holes) on the MODELS statement. Variable CARR on the METHOD statement should be set to zero, since multidimensional drift-diffusion solvers are not used. Due to a variation of electron density and potential in transverse direction, Dirichlet boundary conditions in contacts are not the best option. Instead, a quasi Fermi level is fixed, while electrostatic potential is subjected to von Neumann (zero electric field) boundary conditions, which are set by specifying REFLECT on the CONTACT statement.

Schrödinger equation is solved in each transverse slice to find electron and/or hole eigen energies and wave functions. The minimum required number of eigen states is determined automatically, but may also be set by parameter EIGEN on the MODELS statement. The DDMS model is compatible with all ATLAS Schrödinger solvers: 1D and cylindrical in ATLAS2D and 2D in ATLAS3D. Position dependent eigen energies play a role of conduction (valence) band edge and are used to solve 1D drift-diffusion transport equations in each subband. The equations are discretized using Scharfetter-Gummel scheme. ATLAS mobility models are fully integrated with the method and used in the discretization. DDMS always employs a rigorous mobility-diffusion relation, which is dimensionality dependent in case of Fermi-Dirac statistics.

Generation-recombination (G-R) mechanisms such as Shockley-Read-Hall, Auger and optical recombination, band-to-band tunneling and impact ionization are available in the method. When generation-recombination mechanisms are present, it is possible to iterate between carrier density and G-R rates before solving the Poisson equation. The self-consistency between carrier density and G-R rates, achieved in this inner iteration procedure, will result in a more stable Poisson convergence. The number of inner iterations is controlled by the RGITER.DDMS=N and the RGCONV.DDMS=X parameters on the METHOD statement. Here, N is the maximum number of inner iterations (default is 0) and X is minimum error between 0 and 1 (default is 1e-5). It is recommended, to first run a simulation without G-R mechanisms to get a feeling of device behavior and then increase G-R parameters.

In addition to a regular structure file, a detailed information on each subband can be stored in an extra log file by using option DDMS.LOG on SAVE or SOLVE statements. The file will contain subband-resolved quantities, such as eigen energies, carrier densities, current densities, quasi Fermi levels and generation-recombination rates.

Unlike the mode-space NEGF model, the DDMS model can handle only devices with uniform cross-section, because quantum mechanical coupling between electron subbands is neglected. Below we give several examples of DDMS model applied to various devices with transverse quantum confinement.

**A. Single gate SOI transistor
**Performance and scaling limits of SOI or double gate transistors is significantly affected by short channel and floating body effects. The major mechanisms behind these effects are electrostatics in the channel with quantum confinement, band-to-band tunneling, impact ionization and recombination. While some applications may suffer from floating body effects, others such
as zero-capacitor DRAM cells (Z-RAM), may employ them. In either case, it is important to take into account all the major mechanisms on equal footing in a single simulation. The DDMS model is an excellent tool to handle such cases.

We consider a single gate SOI transistor with 10 nm channel thickness, 60 nm gate length and 1 nm gate oxide, schematically shown in Figure 1. Source and drain extension regions are n-doped to Nd=1e20 cm^{-3}, while channel is p-doped to Na=1e18 cm^{-3}. In order to estimate device characteristics, we start with a simple calculation where we switch off generation-recombination term. We use a velocity saturation mobility model with saturation velocity of 1.1e7 cm/s. Figure 2 shows self-consistently computed carrier and current density profile under drain bias of 1 V and gate bias of 1.5 V. In this example, both electron and holes are treated quantum mechanically within isotropic effective mass approximation. The wave functions and eigen energies are shown in Figure 3. In Figure 4(left) and 5(left), we can see that the device shows excellent Id-Vd and Id-Vg characteristics, with no short-channel effects, steep subthreshold slope and no DIBL.

Figure 1. Schematics of Si SOI FET with Lg=60nm, t=10nm and gate oxide of 1nm. |

Figure 2.Carrier density (top row) and current density (bottom row) for electrons (left column) and holes (right column) in SOI FET under Vd=1V and Vg=1.5V in the absence of generation-recombination mechanisms. |

Figure 3. Wave function (left column) and eigen energies (right column) for electrons (top row) and holes (bottom row) in SOI FET under Vd=1V and Vg=1.5V in the absence of generation-recombination mechanisms. |

In order to qualitatively demonstrate how various generation-recombination effects may influence the device behavior, we include a default ATLAS model for Shockley-Read-Hall recombination, Selberherr model for impact ionization with soft electric field threshold of E=2e6 V/cm and local Kane model for band-to-band tunneling. For quantitative results a more careful choice of parameters or calibration would be required. In Figure 4(right), one can see that the saturation of Id-Vd characteristics is deteriorated and the current is significant even when device is off at Vg=0.1V and Vd=1V. In Figure 5 (right), we see that the subthreshold slope decreases with drain voltage and the leakage current increases. In order to understand the physics behind this deterioration of device performance, in Figure 6 we plot electron subband, electron density and hole density profiles with and without generation-recombination. The main reason behind the change of device performance is a tremendous increase of hole concentration as seen in Figure 6(right). To maintain charge neutrality, the device electrostatics causes a lowering of source injection barrier (Figure 6 (left)) and, as a consequence, much higher drain current and electron concentration in the channel (Figure 6(center)). In Figure 7 we plot current in two lowest electron and two highest hole subbands in the presence of generation-recombination at Vd=1V and Vg=0V. One can see that electron-hole pair generation causes a small increase in electron and hole currents near channel-drain junction. Thus, this direct contribution to current increase is much smaller than the indirect effect due to electrostatics and cannot explain the change in I-V characteristics.

Figure 4. Id-Vd characteristics of SOI FET at Vg=0.1, 0.3, 0.6, 0.8, 0.9 and 1V in the absence (left) and in the presence (right) of generation- recombination. |

Figure 5. Id-Vg characteristics of SOI FET at Vd=0.1, 0.3, 0.5, 0.8 and 1V in the absence (left) and in the presence (right) of generation-recombination. |

Figure 6. Electron subbands (left), electron concentration (center) and hole concentration (right) in SOI FET under Vd=1V and Vg=0V with (blue) and without (red) generation-recombination. |

Figure 7. Current density in the two lowest electron and two highest hole subbands in SOI FET under Vd=1V and Vg=0V in the presence of generation-recombination. |

**B. Surround gate FET**

Recent progress in nanowire growth and fabrication has led to a possibility of utilizing nanowires as surround gate transistors. The DDMS model can be applied to study scaling behavior and electrostatics in such devices. In modeling of surround gate FET, a cylindrical symmetry can be employed to slash a computational cost of a 3D simulation. In this case, 1D cylindrical Schrödinger equation is solved for various orbital numbers and then 1D drift-diffusion equation is solved in each subband. Presence of generation-recombination mechanisms couples different subbands.

Figure 8 shows electron and hole density profiles in a cylindrical surround gate FET with gate length of 30nm, diameter of 4nm and the gate oxide of 1nm. Figure 9 shows I-V characteristics for similar devices with different gate lengths and diameters of 4nm and 12nm. One can see that the devices show a good scaling behavior down to Lg=5nm at D=4nm and Lg=10nm at D=12nm. All physics described in Section A is also applicable for the surround gate FET. Figures 10 and 11 show a degradation of device I-V characteristics due to floating body effects. In this case however, saturation of Id-Vd characteristics is much better then in the case of SOI, due to a better electrostatic control.

Figure 8. Electron (left) and hole (right) carrier density profiles in cylindrical surround gate FET with gate length of 30nm and diameter of 4nm. |

Figure 9. Gate length scaling of surround gate FETs with diameter of 4nm (left) and 12nm (right). |

Figure 10. Id-Vd characteristics of surround gate FETs (Lg=30nm D=4nm) in the absence (left) and in the presence (right) of generation-recombination. |

Figure 11. Id-Vg characteristics of surround gate FETs (Lg=30nm D=4nm) in the absence (red) and in the presence (green) of generation-recombination. |

**Conclusion**

This article has presented a Drift-Diffusion Mode-Space approach to modeling devices with strong transverse confinement. The new model can predict eigen energies and wave functions in transverse slices, quantum electron concentration, current density and current-voltage characteristics within drift-diffusion approximation. The model is compatible with conventional ATLAS models for mobility, recombination, impact ionization and band-to band tunneling. An additional insight can be gained by looking at various subband resolved quantities.