2.8.2 SCHOTTKY Barrier Model of CNT-FET Operation

In general a charge transfer will take place at the metal-CNT interface leading to band-bending and the creation of a SCHOTTKY barrier. For example, a CNT-FET with titanium-carbide contacts shows equal hole and electron currents depending on the sign of the applied gate bias, so called ambipolar conduction [64]. This suggests the existence of two barriers, one for electrons and one for holes, of approximately equal height, implying that each must be about half the band-gap ( $E_\mathrm{g}/2\approx\mathrm{300~meV}$). Applying conventional semiconductor analysis, which assumes that thermionic emission contributes mostly to the total current through a SCHOTTKY barrier, indeed yields similar thermal activation barriers for electrons and holes, however, on the order of 10 meV [64]. This finding suggested that the thermionic contribution alone cannot account for the observed current levels, which is supported by modeling results showing that SCHOTTKY barriers in one-dimension are much thinner than their planar analogues [72,73]. Consequently, carrier tunneling through these thin barriers becomes the dominant conduction mechanism and cannot be neglected when quantifying the barrier height [74].

Similar conclusions can be drawn from the sub-threshold behavior of CNT-FETs, in particular when plotted as a function of gate oxide thickness. The switching behavior of a MOSFET is described by the inverse sub-threshold slope, $S\simeq (k_\mathrm{B}T/\ensuremath {\mathrm{q}})\mathrm{ln(10)}(1+C_\mathrm{D}/C_\mathrm{g})$ where $C_\mathrm{D}$ and $C_\mathrm{g}$ are the depletion and gate capacitance, respectively. In the case of a fully depleted device, $C_\mathrm{D}$ is zero and, therefore, $S$ depends only on the temperature, having a value of $65~\mathrm{mV/decade}$ at room temperature. The original CNT-FETs with thick gate oxides in back-gated geometry had unexpectedly high $S$ values of approximately $1~\mathrm{V/decade}$. On the other hand, when devices are fabricated using thinner oxides, such as the top-gated CNT-FET in [68], the value of $S$ dropped significantly into the range of $100-150~\mathrm{mV/decade}$ [68], $80~\mathrm{mV/decade}$ [75], and $67-70~\mathrm{mV/decade}$ [76]. Such a dependence of $S$ is not consistent with the bulk switching mechanism which should give $\mathrm{65~mV/decade}$ in the long channel limit. Instead, this kind of scaling of the sub-threshold slope with oxide thickness is compatible with the existence of sizeable SCHOTTKY barriers at the metal-CNT interfaces, and theoretical modeling showed that the gate field impact on this interface is responsible for the observed improvement in $S$ [77,78], see Fig. 2.17.

Figure 2.17: Inverse sub-threshold slope $S$ as a function of gate capacitance. Symbols are experimental data. Lines represent calculations for MOS-FETs with (dashed line) and without (solid line) SCHOTTKY barriers of $0.3~\mathrm{eV}$ [77].

Further evidence of the presence of SCHOTTKY barriers in CNT-FET devices is found in local gating experiments, where the on-current is shown to increase significantly by application of a local potential from a metal coated scanning probe tip only at the positions above the metal-CNT interface [79]. Similarly, the impact of SCHOTTKY barriers in the sub-threshold characteristics of the CNT-FET is clearly observed in transistors with multiple top-gates [80]. In this case, local gates over the metal-CNT interface are used to electrostatically thin the SCHOTTKY barriers and reduce the value of $S$ closer to that of the bulk switching device [80].

Hole (electron) injection into the CNT depends on the line-up of the metal FERMI level and the valence (conduction) band of the CNT, which is defined here as the SCHOTTKY barrier height. In this picture, other details of the contacts such as any changes in the metal-CNT coupling as a function of the curvature of the CNT are incorporated in an effective SCHOTTKY tunneling barrier height. This barrier height depends on a number of material parameters such as the band-gap of the CNT, work-function difference, as well as the interface quality. The CNT band-gap is inversely proportional to the diameter of the CNT, according to (2.10). Figure 2.18-a shows qualitative band diagrams for CNT-FETs with different diameters. Assuming a constant work function for all CNTs^2.4, the SCHOTTKY barrier increases linearly with increasing band-gap. On a log scale, current injection through the SCHOTTKY barrier is inversely proportional to the barrier height. Therefore, the CNT-FET with a small diameter delivers low on-current. The choice of the metal contacts also affects the device performance. Figure 2.18-b depicts the band diagrams for CNT-FETs using different source and drain contact materials. Identical energy band-gaps are drawn here to represent CNTs of the same diameter. CNT-FETs with Pd contacts deliver the highest on-current (Fig. 2.18-c), since Pd has the highest work function (5.1 eV), which forms a low SCHOTTKY barrier height to the valence band of the CNT. The trend shown follows that of the clean metal work functions: 4.3 eV for Ti and 4.1 eV for Al.

Figure 2.18: a) Schematic band diagram showing the different SCHOTTKY barrier heights for b) CNT-FETs with the same contact, but with different CNT diameters and b) CNT-FETs with the same diameter, but using Pd, Ti, and Al contacts, respectively. c) Plot of the on-current as a function of the CNT diameter [81].

M. Pourfath: Numerical Study of Quantum Transport in Carbon Nanotube-Based Transistors