Graphene Nanoribbon bandstructure

One of the biggest surprises in graphene nanoribbons is the absence of chirality, as evidenced by the experiments from the Phil Kim and Hongjie Dai groups. This contrasts sharply with nanotubes -- in fact, a simple tight-binding theory of nanoribbons gives three chiralities, but the data cluster around the largest energy set. We believe the reason is three-fold: the transverse wavefunctions do not experience a precise hard-walled boundary condition and diffuse outside, as confirmed by our simulations. This alone removes any metallicity -- Extended Huckel theory (black curve on top right) agrees. Add in a 3.5 per-cent compressive edge strain as the unpassivated edge pi-bonds start to resemble benzene with a partial double bond structure, and the gaps become larger. Finally, edge roughness (classified as modulation or dislocation) tends to favor the segments with the largest band-gaps, as long as the segment widths (correlation lengths) are larger than the tunneling decay constants.

Our calculations under-score the importance of a proper band-structure method. We have found Extended Huckel Theory (EHT) to be quite accurate -- capturing the edge strains and roughnesses as well as dangling bond chemistry as well as DFT (with much less computational cost).

One can see the creation and passivation of edge-states at the GNR edges, simulated using EHT.

Nanoribbon Electronics

Narrowing down a GNR to less than 10 nm opens a modest band-gap that can be used to gate the channel. In our models, we took a really narrow (7,0) strip with a better band-gap. While such strips are difficult to make, the results underscored a few important points. Even for such narrow strips, the current jumped up beyond saturation due to band-to-band tunneling. Furthermore, the role of edge roughness was higher on the OFF current than the ON, and for smaller correlation lengths than larger. We thus quantified the role of edge roughness on GNR performance.

Wide-narrow-wide all graphene devices

Since wide ribbons are metallic and narrow ones are semiconducting, one can imagine a wide-narrow-wide strip monolithically patterned out of a single template. The I-Vs of such a device (below) are quite attractive. Due to the 2-D electrostatics from the planar contacts vs the 3-D electrostatics from the gate, the potential profile is well controlled by the gate (top right) -- the drain pulls it down minimally and the gate capacitance is much larger than the drain even for a small sheet shown here (the ratio is 34 for the high-k geometry shown, but drops down to about 24 if 3D contacts are used). A gate in the same plane as the channel will not work as well, as the gate will need to turn the channel on with fringing fields from its edge capacitance. In our geometry with the gate running across the channel on top of it with a high-k below, the currents saturate very well due to the elimination of short-channel effects by the 2D contact electrostatics. Furthermore, the C-C bonds at the wide-narrow interface promote Ohmic contacts that with delocalized gap states that have minimal pinning effect on the channel.

Low-bias mobility of graphite derivatives

All graphite derivatives (graphene, nanoribbons, bilayer) have an asymptotic constraint on their band-structure, namely, that the high energy dispersion must be photon-like (linear). This leads to a fundamental trade-off, so that opening a band-gap EG increases the mass of the electrons by the lost energy, EG = mv₀². The trade-off means that we can estimate the highest mobilty at room temperature and the inversion relation with bandgap based on bandstructure considerations alone on a 3-parameter E_G-mu-lambda plot. The largest predicted mobility is 400,000 cm²/Vs. The trade-off depends on the scattering length, and shows that graphene does not necessarily have strong advantages over other semiconductors when it comes to digital logic (which needs both a high switching speed or mobility and a high ON-OFF ratio or bandgap).

High-bias saturation in bulk graphene

The I-V of graphene shows a tendency to saturate followed by a rise, due to the low density of states near the Gamma point and the subsequent band-to-band tunneling (blue dashed lines). Elastic impurity scattering makes the saturation better -- for large gate voltages (higher curves) the scattering reduces the current, while for lower voltages near the Dirac point the lack of density of states means that the scattering helps the current (red curves, left). Optical phonon scattering also contributes to saturation (right red curves), creating a pronounced inflection point at a source-drain voltage set by the phonon frequency and unmodified by the gate voltage -- in other words, the `kink' arises as a gate-independent feature in the I-V.

Graphene Circuits

Scalability is a problem with nanoribbons, as the gaps require ultra-thin widths below 10 nm. Nanotubes have problems with scaling (providing adequate current for high speed operation). While graphene seemed scalable, the reliance on quantization for opening a band-gap works against that advantage. A series of notches cut into a graphene sheet may allow some scalability, but the quantitative performance advantages remain to be seen.

• "Graphene Devices, Interconnect and Circuits -- Challenges and Opportunities", Mircea R. Stan, Dincer Unluer, Avik Ghosh and Frank Tseng, submitted (ISCAS 2009).
  
  
• "Diluted chirality dependence in edge rough graphene nanoribbon field-effect transistors", F. Tseng, D. Unluer, K. Holcomb, M. Stan and A. W. Ghosh, Appl. Phys. Lett. Vol. 94, 2231 12 (2009).
  
  
• "Performance Advantages of Monolithically Patterned Wide-Narrow-Wide All-Graphene on Insulator Devices", Dincer Unluer, Frank Tseng, Avik W. Ghosh, Mircea R. Stan, cond-mat/arXiv:0809.3756
  
  
• "Extended Huckel theory for bandstructure, chemistry and transport. Part I: Carbon Nanotube", D. Kienle, J-I. Cerda and A. W. Ghosh, J. Appl. Phys. Vol. 100, 043714 (2006).