Transport in a Two-Channel Nanotransistor Device with Lateral Resonant Tunneling

Abstract

We study field effect nanotransistor devices in the Si/SiO₂ material system which are based on lateral resonant tunneling between two parallel conduction channels. After introducing a simple piecewise linear potential model, we calculate the quantum transport properties in the R-matrix approach. In the transfer characteristics, we find a narrow resonant tunneling peak around zero control voltage. Such a narrow resonant tunneling peak allows one to switch the drain current with small control voltages, thus opening the way to low-energy applications. In contrast to similar double electron layer tunneling transistors that have been studied previously in III-V material systems with much larger channel lengths, the resonant tunneling peak in the drain current is found to persist at room temperature. We employ the R-matrix method in an effective approximation for planar systems and compare the analytical results with full numerical calculations. This provides a basic understanding of the inner processes pertaining to lateral tunneling transport.

1. Introduction

In this study, we consider field effect nanotransistor devices based on lateral resonant tunneling between two parallel conduction channels: One channel is selectively coupled to the source contact and the other channel is selectively coupled to the drain contact. A source-drain current thus can only flow when there exists tunneling transport between the two channels. Because of strong level quantization in each channel, tunneling only occurs under special resonance conditions which can be controlled with very small voltages applied to two control gates. This opens the way to low operating voltages around 100 mV suitable for low energy applications.

A similar resonant tunneling mechanism was employed in the 1990s in double electron layer tunneling transistors (DELTTs) in the GaAs/AlGaAs material system. Here, two selectively coupled electron layers in GaAs are separated by an AlGaAs tunneling barrier. Experimental solutions to this central technological problem are demonstrated in Refs. [1,2,3,4,5,6,7,8,9,10,11,12,13,14]. In a DELTT, both lateral dimensions of the electron layers lie typically in the range of tens of micrometers while the widths of the electron layers and the tunneling barrier in the perpendicular direction lie in the range of tens of nanometers. The electron layers thus assume the character of a two-dimensional (2D) electron gas which is loaded from dopants in layers above and below the electron layers and transport was discussed in terms of 2D–2D tunneling [5,15,16,17]. Under these conditions, resonant tunneling peaks were found in the transfer characteristics at low temperatures, typically in the Kelvin range.

With the substantial progress in semiconductor technology, it is now possible to fabricate double quantum wells in the technologically relevant Si/SiO₂-material system in double silicon-on-insulator (SOI) substrates ([18], see Figure 1). Furthermore, it has become possible to shrink the device lengths in the transport direction to the size of tens of nanometers, i.e., by a factor of a thousand compared to the DELTTs. Transport is then described by coherent wave functions ranging from source to drain [19]. Because of the global coherence in the transport direction from source to drain, tunneling between the electron layers assumes the character of lateral resonant tunneling rather than 2D–2D tunneling. Because of this essential difference in the device operation we call the described Si-based devices two-channel tunneling FETs (2CTFETs).

We calculate quantum transport in a 2CTFET using the R-matrix approach developed in Refs. [19,20,21]. Our approach has been confirmed in applications to nano-MOSFETs [22] and SOI transistors [23]. For the R-matrix method, we first employ the effective approximation for planar systems (EAPS, see Appendix E) which provides a physical understanding of the basic inner processes pertaining to lateral resonant tunneling transport. We then compare with the results of fully numerical calculations (2D model). One can now identify a number of features of the 2CTFET which are favorable for applications: First, while MOSFETs rely on thermal activation, 2CTFETs work with resonant tunneling which is independent of the temperature. Second, in comparison with DELTTs, the conduction channels are loaded from the lateral source and drain where the chemical potential can be increased up to the regime of electron volts in the limit of heavy doping and narrow contacts (see Appendix B), much more than the thermal energy at room temperature of 25 meV. It is therefore conceivable that thermal effects play a minor role in the 2CTFET and, indeed, we see that the resonant tunneling peak in the transfer characteristic persists at room temperature. Moreover, the extremely efficient injection from the heavily doped narrow source/drain contacts and the much lesser scattering in the short, nearly ballistic conduction channels favor relatively high drain currents.

2. Model

We consider the realization of a 2CTFET on a double SOI substrate which is shown schematically in Figure 1. The 2CTFET can be derived from an SOI transistor on a standard SOI wafer as follows: As is well known, in an SOI transistor one adds to the structure elements of a standard SOI wafer (Si substrate, buried oxide (BOX) and Si film) the further elements source, drain, back gate, top gate oxide and top gate (see Figure 1d). Then, in a double SOI substrate an extra buried oxide BOX_top is added to the standard SOI wafer (see Figure 1a, [18]). In consequence, the Si film is split into two separate layers, SOI_top and SOI_bottom, which can be as narrow as 2 nm. The two layers thus assume the character of quantum wells which are coupled by the buried oxide BOX_top acting as a tunneling barrier. In the shown 2CTFET, the quantum well in SOI_bottom is selectively contacted to the source and the quantum well in SOI_top is selectively contacted to the drain. In the strong barrier limit, a drain current from source to drain only occurs under lateral resonant tunneling conditions leading to the resonant tunneling peak in the transfer characteristics as shown in Figure 2.

We consider a simple model potential $V\left(\overrightarrow{r}\right)$ for the 2CTFET:

• In the grounded source contact ${\Omega }_1$, we set $V(\overrightarrow{r}\in {\Omega }_1)=0$
• In the drain contact ${\Omega }_2$, we set $V(\overrightarrow{r}\in {\Omega }_2)=-e{U}_D$.
• In the device kernel ${\Omega }_0$ (‘scattering area’), we cut off the wave functions in the top-gate oxide and the BOX_bottom. In the remaining tunneling coupled quantum wells, we choose a piecewise linear potential of the separable form

  $$V(\overrightarrow{r}\in {\Omega }_0)=V_T\left(y\right)+V_L\left(x\right).$$

  (1)

  In the transport direction (x-direction), there is a linear potential drop of the applied drain voltage $U_D$

  $$V_L\left(x\right)=-{\displaystyle \frac{x}{L}}e{U}_D,$$

  (2)

  where L is the channel length. We write the transverse potential in the device kernel in the form

  $$V_T(0\le y\le D)=V_B\Theta (y-D_C)\Theta (D_C+D_B-y)+V_g+(V_{bg}-V_g){\displaystyle \frac{y}{D}}$$

  (3)

(See Figure 1b). Here, $V_B$ and $D_B$ are the height and thickness of the tunnel barrier (BOX_top) along the y-direction. Furthermore, $D_C$ is the thickness of the two quantum wells (SOI_top and SOI_bottom) formed between the tunneling barrier and the top gate oxide and the BOX_bottom. Finally, $V_g=V_T(y=0)$ is the transverse potential on the upper edge of the silicon film and $V_{bg}=V_T(y=D)$ that on the lower edge. For $V_g=V_{bg}$, the transverse potential $V_T\left(y\right)$ is symmetrical and the quantum levels in the isolated quantum wells coincide. One obtains in the interacting system a series of pairs of resonantly interacting quantum wells. As described in detail in Appendix D, these two interacting quantum levels lead to two wave functions (one symmetrical and one anti-symmetrical) which are localized at the same time in both quantum wells, thus allowing the electron to interchange between the two quantum wells (see Figure A4a). At $V_g\ne V_{bg}$, out of resonance, this interaction is broken. The quantum wells become isolated and the current from source to drain becomes very small. The results are the narrow resonant tunneling peaks shown in Figure 2.

In our calculations, we use the parameter values listed in

Table 1. Notations and parameter values, above fixed values, below varied values.

Quantity	Notation	Values
Thickness of SOI_top and SOI_bottom	$D_C$	3 nm
Isotropic effective mass	$m^{*}$	$0.32\times m_0$
Valley degeneracy	$N_V$	6
Channel length	L	30 nm
Doping in source/drain contact	$N_D$	5 × 1020 cm−3
Potential at interface BOX_bottom/SOI_bottom	$V_{bg}$	0
Gate insulator thickness	$D_G$	8 nm
Width of the 2CTFET in the z-direction	W	up to ∞
Height of tunnel barrier	$V_B$	1 eV or 2 eV
Potential at interface between top gate oxide/SOI_top	$V_g$	−0.2 eV ≤$V_g$≤ 0
Drain voltage	$U_D$	0 or 0.2 V
Device temperature	T	300 K or 10 K
Tunnel barrier thickness (BOX_top, SiO2)	$D_B$	1 nm to 3 nm, mainly 2 nm

.

Inserting in the Schrödinger equation the z-independent potential $V\left(\overrightarrow{r}\right)=V(x,y)$, one obtains with the ansatz

$$\Psi \left(\overrightarrow{r}\right)=\psi (x,y)\sqrt{{\displaystyle \frac{2}{W}}}sin\left({\displaystyle \frac{n_z\pi }{W}}z\right)$$

(4)

the two-dimensional Schrödinger equation

$$\left[-{\displaystyle \frac{{\hslash }^2}{2m^{*}}}\left({\displaystyle \frac{{\partial }^2}{\partial x^2}}+{\displaystyle \frac{{\partial }^2}{\partial y^2}}\right)+V(x,y)-E^{xy}\right)\psi (x,y)=0$$

(5)

with the energy of the motion in the x-y-direction $E^{xy}=E-({\hslash }^2/2m^{*}){(n_z\pi /W)}^2$ where E is the total energy. For the calculation of I-V-traces of the 2CTFET, we use the Landauer–Büttiker formula for the drain current

$$I_D={\displaystyle \frac{2e}{h}}{N}_V{\int }_0^{\infty }d{E}^{xy}\left[S(E^{xy}-\mu )-S(E^{xy}-\mu +e{U}_D)\right]T^{2d}\left(E^{xy}\right)$$

(6)

with the valley degeneracy $N_V$ [19]. In the subsequent sections, we describe the further quantities in this expression in detail:

• The chemical potential $\mu$ in the source in Appendix B
• The supply function $S\left(\alpha \right)$ in Appendix C
• For the evaluation of the two-dimensional current transmission $T^{2d}$, we apply in Section 3.1 the EAPS and in Section 3.2 the fully numerical 2D model.

Our 2D model is based on either the full R-matrix method [24] or Kwant [25]. The Kwant simulation tool is an open source Python library designed for transparent quantum transport simulations based on the tight binding formalism and a numerical method known as wave function matching (WFM) [26,27]. Wave function matching is a technique employed to compute the scattering matrix of a mesoscopic system that can also be described by a Hamiltonian in tight binding form and which is also suitable for ab initio calculations. In a continuous description, the method is based on slicing the Hamiltonian on the transport direction and imposing matching conditions for the wave function. The WFM is closely related to other methods, like the R-matrix method and Green’s function techniques. However, in the R-matrix method, the Hamiltonian of the central system is diagonalized only once, independent of the total energy and the matching conditions are imposed between the scattering region and the leads. On the other hand, Green’s function techniques are more general and one may also include dissipative effects. For non-interacting systems, these methods provide consistent results as is shown for R-matrix and WFM implemented in Kwant (see Figure A9).

The scattering potential is determined by solving the Poisson equation with boundary conditions set by $(U_D,U_G)$, while the system is considered invariant in the z-direction (see Figure 1f). The Dirichlet type boundary conditions for the potential are set on a rectangular shaped region, defined by the intersection of the leads with the scattering region, ${\Omega }_s\cap {\Omega }_0$, with additional 8 nm buffers of oxide. The remaining segments are located at the positions of the top and back gates.

3. Results

3.1. Effective Approximation for Planar Systems (EAPS)

In the following plots, we evaluate in the limit $W\to \infty$ the quantity $J_D=I_D/W$ which is usually given in experimental work. Here, Equation (A10) is used for the supply function which is the only quantity depending on the width making sure that $I_D$ scales linearly with W. In Figure 2, we show the calculated transfer characteristics $J_D$ vs. $V_g$ of the 2CTFET. A central narrow resonant tunneling peak around zero control voltage $U_G=U_{BG}=0$ V is found with a small full width at half maximum of ∼100 mV. At higher control voltages, there are much smaller satellite peaks. As expected from the quantum mechanical nature of the resonant tunneling mechanism, the tunneling peak is nearly unchanged with varying device temperature, growing very weakly with decreasing temperature. This is opposite to what is found in a MOSFET. In Figure 3, it is demonstrated that the maximum of the tunnelling peak decreases strongly, nearly exponentially, with increasing barrier height and barrier thickness.

In Figure 4, we plot the output characteristics of the 2CTFET. Since here we are interested in the low-voltage properties of the 2CTFET, we examine gate and drain voltages below 0.2 V as an operating voltage. The basic structure of the output characteristics resembles that of a MOSFET. However, as can be expected from the different transport mechanisms, the drain current of the 2CTFET becomes smaller with increasing gate voltage, opposite to the MOSFET. Also, there is no clear saturation voltage. Finally, the traces for small gate voltages develop at higher drain voltages a weak negative differential conductance. A negative differential conductance in the output characteristic is typical for resonant tunneling devices [28,29] and it is also present in DELTTs.

3.2. Full Numerical Calculations on 2D Models

Figure 5 shows the behavior of the drain current as the gate voltage is varied, for several values of the drain bias. For $U_G=0$, the two quantum wells are symmetric and the electron wave functions are localized in both channels, leading to a significant current flow into the drain. In contrast, by increasing the magnitude of the gate voltage, $|U_G|$, the asymmetry of the two quantum wells becomes larger, resulting in localization of the wave functions in each of the two channels. In effect, the source-to-drain transport is diminished, similar to the analytical results show in Figure 2b. The normally ON state of the 2CTFET at $U_G=0$ V is thus turned into an OFF state. Unlike the analytic model, the solution domain of the Poisson equation included the top and bottom oxide layers (see Figure 1a), which in effect decrease the potential drop on the active region ${\Omega }_0$. For a proper comparison, we plotted in the insets of Figure 5 the drain currents against the potentials found at the upper edge of the drain channel ($x=L/2$, $y=0$, see Figure 1c). By varying the drain voltage, we observe an almost linear increase in the drain current, as depicted in Figure 6, for two barrier sizes ($V_b=1$ eV and $V_b=2$ eV) and several $U_G$ values. When $U_G$ is increasing, the slopes $d{J}_D/d{U}_d$ are decreasing, which is consistent with the characteristics presented in Figure 4, reflecting the ON–OFF transition. For a more comprehensive overview on the dependence of J-V characteristics on the bias conditions, we plotted in Figure 7 contour plots of $J_D$ vs. $(U_D,U_G)$ for $V_b=1$ eV and $V_b=2$ eV. This confirms the appearance of narrow maxima in the drain current around $(U_G,U_D)=(0,0.2)$ V.

The transmission function shown in Figure A9 presents a series of sharp peaks which decrease in amplitude with total energy E, while their widths increase. For each transverse mode, a matching condition between the maxima and minima of the quasi-stationary waves in the two channels occurs for certain values of the total energy, leading to narrow peaks in the transmission. We call this in-phase wave function-matching condition and this is exemplified for $E=0.231$ eV in Figure 8a, for the first transverse mode. A dip in the transmission function corresponds to out-of-phase quasi-stationary waves in the two channels, as indicated, e.g., for $E=0.248$ eV. One obtains a similar picture for the second transverse mode, as one can see from Figure 8b. For the in-phase condition, the large overlap of the wave function favors the tunneling between the two channels, which is minimized for the out-of-phase condition. At the same time, from Figure A9 one can see that the oscillation period in the transmission function increases with the total energy. This can be explained by observing the overlap of the two stationary waves as the longitudinal k-vector is increased. Moreover, one should note that, as the electrons are incoming from the left in the lower channel, one expects in the upper channel a transition from quasi-stationary states to purely outgoing waves in the drain lead, which is visible near the interface with the drain.

Further investigations shall be focused on the effects of the Coulomb interaction. In confined quantum dot systems in SiO_x films, weakly coupled to the leads, Coulomb blockade is manifested [30]. Although the transistor system has a much stronger coupling between the leads and the active region, the Coulomb interaction can redefine the transversal mode energies, introducing energy shifts in the resonant transmission conditions.

4. Summary

We study the transport properties of field effect nanotransistor devices which are based on lateral resonant tunneling between two parallel conduction channels. In the transfer characteristics, we find narrow resonant tunneling peaks around zero control voltage allowing one to switch the drain current with small control voltages. This opens the way to low-energy applications. Furthermore, we study in an effective approximation to the general R-matrix approach the basic inner processes pertaining to lateral resonant tunneling transport. This analysis provides an in-depth understanding of the tunneling process, revealing the complex structure of the transmission function. Analytical calculations accurately provide the positioning of the transverse modes and the distribution of transmission peaks, which follows from the matching of quasi-stationary wave functions in the two channels. The consistency of the EAPS is further confirmed by numerical calculations on 2D models, using the R-matrix method and Kwant simulation package.