COOS: a wave-function based Schrödinger–Poisson solver for ballistic nanotube transistors

Abstract

This paper gives an in depth overview on a wave-function based simulation framework (called coos) for modeling ballistic nanotube transistors by solving the effective-mass Schrödinger equation. The framework considers non-parabolic electronic band structure effects, band-to-band tunneling as well as a heterojunction-like model for extended contacts to describe the injection and reception of charge carriers into and from the channel. Special emphasis is put on an efficient and reliable numerical implementation. The applicability of the simulation framework and the necessity to include the aforementioned phenomena are shown by comparing simulation results with experimental data of a $50\hbox { nm}$ long carbon nanotube transistor (cntfet). The intrinsic transit frequencies and the output characteristics for higher drain-source voltages are predicted and analyzed.

Appendices

Boundary conditions for the Poisson equation

The boundary condition for the electrostatic potential $\psi$ along metal contacts reads

$$\begin{aligned} q\psi =\frac{1}{2}E_\mathrm{g,1}-\varPhi _\mathrm{sb}+qV_\mathrm{c} \end{aligned}$$

(19)

where $V_\mathrm{c}$ is the applied voltage at contact c. Let $V_\mathrm{g}$ be the gate potential. For the gate metal, an effective model is implemented comprising the work function difference $qV_\mathrm{wd}=\varPhi _\mathrm{mg}-\varPhi _\mathrm{cnt}$ between the gate metal and the cnt as well as an effective potential $V_\mathrm{ox}$ due to charges within the gate oxide yielding the boundary condition

$$\begin{aligned} \psi =-V_\mathrm{wd}-V_\mathrm{ox}+V_\mathrm{g} \end{aligned}$$

(20)

which is set along gate contacts. Neumann boundary conditions

$$\begin{aligned} \mathrel {\mathrm {grad}}\psi =0 \end{aligned}$$

(21)

are set at the outer boundaries of the simulation domain.

Absorbing boundary conditions

Resonant states are solutions of the Schrödinger equation if absorbing boundary conditions are imposed. For these states, only the flow out of the channel is considered. While hard-wall or periodic boundary conditions will lead to unacceptable artefacts due to boundary reflections, properly chosen absorbing boundary conditions absorb the waves without reflecting them. This requires an absorbing boundary layer to be placed adjacent to the channel (i. e. the uncoated cnt portion) (see Fig. 14) replacing the heterojunction contact used for the current calculation. Thus, for the pre-detection of the resonance peaks (and in contrast to the qtbm boundary conditions, which are used for calculating the internal device quantities) the simulation domain is artificially extended. When a wave enters the absorbing layer, it is attenuated by the absorption and decays exponentially. This allows to impose closed boundary conditions at the absorbing layer boundaries $(x_{\text{ a }},\,x_{\text{ b }})$ without affecting the simulation results. A perfectly matched layer (pml) is one approach to define an absorbing layer which ensures such a reflection-free transition across the boundary [51]. This method has been applied for band structure calculations [52] and for the calculation of quasi-bound states in cmos devices [53]. A pml can be derived by analytical continuation of the wave equation into the complex plane (complex-valued coordinate stretching) leading to a non-Hermitian Hamiltonian. The real parts of the eigenenergies correspond to the energies (i. e. the positions) of the resonant states while the imaginary parts correspond to their lifetimes (i. e. the widths of the peaks).

Fig. 14

Extending the channel (uncoated tube portion) by adding perfectly matched layers (absorption layers). Additionally, the shape of the stretching function $s(x)$ is shown

Let $s(x)$ be a properly chosen complex-valued coordinate stretching function. Applying the coordinate stretching, the real-valued coordinate variable $x$ then changes to the complex-valued coordinate $\xi$ by

$$\begin{aligned} \xi =\int \limits _{0}^{x}s(\tau )\text{ d }\tau . \end{aligned}$$

(22)

However, since complex-valued coordinates are inconvenient, they are transferred back to real coordinates. In the new real coordinates the differential operator $\partial /\partial \xi$ reads

$$\begin{aligned} \frac{\partial }{\partial \xi }\rightarrow \frac{1}{s( x)}\frac{\partial }{\partial x}. \end{aligned}$$

(23)

Thus, the SE can be written as

$$\begin{aligned} -\frac{\hbar ^{2}}{2}\frac{1}{s( x)}\frac{\partial }{\partial x}\left( \frac{1}{m^{*}(E,x)}\frac{1}{s( x)}\frac{\partial \phi }{\partial x}\right) +V( x)\phi ( x)=E\phi ( x). \end{aligned}$$

(24)

After discretization one gets

$$\begin{aligned}&-\frac{\hbar ^{2}}{2} \frac{1}{s_i}\frac{2}{\Delta x_i + \Delta x_{i-1}} \left( \frac{1}{s_{i+1/2}} \frac{1}{m^{*}_{i+1/2}} \frac{\phi _{i+1}-\phi _{i}}{\Delta x_i} -\right. \nonumber \\&\qquad \left. \frac{1}{s_{i-1/2}} \frac{1}{m^{*}_{i-1/2}} \frac{\phi _{i}-\phi _{i-1}}{\Delta x_{i-1}} \right) +V_i\phi _i=E\phi _i \; , \end{aligned}$$

(25)

where

$$\begin{aligned} \frac{1}{s_{i+1/2}}=\frac{1}{2}\left( \frac{1}{s_{i+1}}+\frac{1}{s_{i}}\right) \; , \end{aligned}$$

(26)

and

$$\begin{aligned} m^{*}_{i+1/2}=\frac{1}{2} \left( m^{*}_{i+1} + m^{*}_{i}\right) . \end{aligned}$$

(27)

The parameters given by the hetero-junction contact model (see Table 1) are used for the potential $V$ and the effective mass in the absorption layers.

The problem is to find a stretching function suitable to decay oscillating as well as evanescent solutions. For oscillating solutions such as $\exp (\hbox {i}kx)$, $s(x)$ should be imaginary and positive with a high absolute value as one can check by replacing $x$ by (22). For evanescent solutions such as $\exp (\kappa x)$, $s(x)$ should be purely real with a small negative value. An imaginary part of $\sigma _{x}$ would add oscillations to the evanescent solution. To the author’s experience,

$$\begin{aligned} s(x)=\left\{ \! \begin{array}{ll} 1-(\alpha +\mathrm {i}\beta )\left( (x-x_{1})/(x_{\text{ a }}-x_{1}) \right) ^{2}\!,&{}x<x_{1}\\ 1-(\alpha +\mathrm {i}\beta )\left( (x-x_{n})/(x_{\text{ e }}-x_{n}) \right) ^{2}\!,&{}x>x_{n}\\ 1,&{} x_{1}\le x \le x_{n} \end{array}\right. \end{aligned}$$

(28)

is a good compromise for the calculation of resonant states. The parameters $\alpha$ and $\beta$ equal $1.0$ and $1.4$, respectively.

By imposing closed boundary conditions at the new boundaries $(x_{\text{ a }},\,x_{\text{ e }})$, the calculation of the resonant states reduces to the solution of a linear complex-valued eigenvalue problem in case of a parabolic band structure approximation. Employing an energy-dependent effective mass demands the evaluation of a non-linear complex-valued eigenvalue problem which is more expensive in terms of numerical effort [14].

However, pml is only reflection-less if the exact wave equation is solved. Discretization only allows for an approximate solution, and the analytical perfection of the pml method is no longer preserved. Reflections, however, can be made arbitrarily small as long as the absorption increases slowly with the distance from the channel. The quadratic stretching function usually turns on the absorption slowly enough with negligible reflections for a pml layer of only half a wavelength or thinner [51].