Definitive Band Gaps for Single-Wall Carbon Nanotubes

We report ab initio quantum mechanical calculations of band structures of single-walled carbon nanotubes (SWNTs) using the B3LYP flavor of density functional theory. In particular, we find excellent agreement with the small band gaps in “metallic” zigzag SWNTs observed by Lieber et al. [0.079 vs 0.080 eV for (9,0), 0.041 vs 0.042 eV for (12,0), and 0.036 eV vs 0.029 eV for (15,0)]. This contrastswith the results fromLDAandPBE,which lead to band gaps 70-100% too small, and with those from the GW correction to LDA, which leads to a gap two times too large. Interestingly we find that the (5,0) system, expected to be a large gap semiconductor, is metallic. These results show that B3LYP leads to very accurate band gaps for CNTs, suggesting its use in designing CNT devices. We find that the effective mass of the CNT (significant in designing CNT devices) scales inversely proportional to the square of the diameter. SECTION Electron Transport, Optical and Electronic Devices, Hard Matter C arbon nanotubes (CNTs) provide a number of unique and special properties that suggest great promise for nanoelectronics applications. In particular, the high electrical conductivity of quantum wires provides a potential solution for on-chip interconnect metals and transistors of future integrated circuits. One crucial obstacle to overcome in fabrication is controlling whether the CNT is metallic or semiconducting. The critical parameter determining the electronic properties of CNTs is the chiral vector, Ch = (na1 þ ma2) (n,m), where n andmare integers anda1 and a2 are the real spaceunit vectors of thegraphene sheet.Ck specifies theway the graphene sheet is wrapped. When n-m is a multiple of 3, the simple theory leads to a crossing of bands at the Fermi energy, implying that CNT ismetallic; otherwise, it is expected to be a semiconductor. Thus, armchair (n,n) CNTs are expected to always be metallic, whereas zigzag (n,0) CNTs are expected to be metallic only when n is a multiple of 3. However, on the basis of measurements under ultrahigh vacuum conditions at 5 K on a Au(111) substrate, Lieber et al. showed that some (3m,0) zigzag single-walled carbon nanotubes (SWNTs) have finite band gaps [0.080 ( 0.005 eV for (9,0), 0.042( 0.004 eV for (12,0), and 0.029( 0.004 eV for (15,0)]. Previous quantummechanical (QM) calculations were not able to account for the observed band gaps. The local density approximation (LDA) functional in density functional theory (DFT) led to gaps of 0.024 eV for (9,0), 0.002 eV for (12,0), and 0 eV for (15,0), which are 70, 95, and 100% too small. Of course, it is well-known that LDA leads to band gaps that are too small. A common approach to correcting these LDA band gaps is the GWapproximation, which calculates the poles of the Green's functions explicitly. For (9,0), GW leads to a band gap that is too large by 213%. The generalized gradient approximation (GGA) functional leads tomuchmore accurate cohesive energies than LDA, but the Perdew-BurkeErnzerhof (PBE) flavor leads to band gaps of 0.030 eV for (9,0), 0.010 eV for (12,0), and 0 eV for (15,0), which are 63, 86, and 100% too small. The Perdew-Wang 91 (PW91) flavor of GGA corrected with an empirical uniform scale factor (1.20) leads to band gaps of 0.20 eV for (9,0), 0.08 eV for (12,0), and 0.14 eV for (15,0), which are to large by 250, 190, and 483% of the experimental values, respectively, following no consistent trend. Since the band gap is the most significant property in designing CNTs for electronics applications, it is essential to find a way of predicting accurate band gaps. We report such an approach here. The problem with bad band gaps from DFT calculations has been encountered before. A spectacular case is for the undoped parent compound, La2CuO4, of cuprate superconductors, where LDA and GGA lead to highly overlapping bands at the Fermi energy and hence a metal, whereas this system has an experimental band gap of 2.0 eV. Perry et al. solved this problem by showing that the Becke-Lee-YangParr (B3LYP) flavor of DFT leads to an accurate band gap of 2.0 eV. Indeed, similar results have now been reported for many other semiconductor and insulator systems. The B3LYP functional combines the Becke GGA exchange potential based with Hartree-Fock (HF) exact exchange plus the Lee-Yang-Parr correlation functional. B3LYP has been shown to provide the most accurate cohesive energies, ionization potentials, and electron affinities for a range of finite molecules. The inclusion of exact HF exchange helps to correct for the self-energy problem with standard DFT formulations. Received Date: June 30, 2010 Accepted Date: September 14, 2010