Work function of single-walled and multiwalled carbon nanotubes: First-principles study

Carbon nanotubes can be viewed as rolled-up graphene sheets. As such, their work functions should be closely related to those of graphene due to geometric and structural similarities. In this paper, we have systematically investigated the work functions of single-walled and multiwalled carbon nanotubes by density functional calculations. The work functions of single-walled carbon nanotubes (SWCNTs) are very close to those of graphene in the armchair conformation, while for the zigzag and chiral conformations, the work functions are close to those of graphene as the diameter is larger than a certain threshold. When the diameter of the tube is smaller than $10\mathrm{\AA }$, the work functions of zigzag and chiral tubes increase dramatically as the diameter decreases. The deviation in the work function from that of graphene for small tubes can be explained and qualitatively estimated by the downshift of the Fermi level due to the curvature effect. For multiwalled carbon nanotubes (MWCNTs), we only consider zigzag and armchair MWCNTs. We find that the work functions of all armchair and zigzag MWCNTs with inner tube diameters larger than $10\mathrm{\AA }$ are very close to those of graphene. The work functions of zigzag MWCNTs with inner tube diameters smaller than $10\mathrm{\AA }$ exhibit significant variations depending on the diameters of the inner and outer tubes. Using a very simple model, we find that the work functions of MWCNTs can be successfully estimated from the work functions and electronic structures of the constituent SWCNTs.

Figure 1

Work functions of SWCNTs of different diameters (squares, armchair tubes; diamonds, chiral tubes; triangles, zigzag tubes). Lines serve as visual guides. The dashed line indicates the work function of graphene. The inset shows the work function of armchair (3,3), (4,4), (5,5), and (10,10) nanotubes calculated using the structure of Shan and Cho. The dashed line of the inset indicates the work function of graphene calculated using the structure of Shan and Cho.

Figure 2

(Color online) Band structures of SWCNTs of different diameters. Dashed line: results using a zone-folding approach. Solid line: LDA results.

Figure 3

(a) The lowest eigenvalues relative to the vacuum level at the $\Gamma$ point of SWCNTs of different diameters (squares, armchair SWCNTs; diamonds, chiral SWCNTs; triangles, zigzag SWCNTs). The solid line is a linear fit to the data of various SWCNTs. (b) Same plot as (a) but shown with eigenvalues relative to the Fermi level. The solid line is a linear fit to the data of armchair SWCNTs.

Figure 4

Work functions of SWCNTs of different diameters [squares, armchair tubes; diamonds, chiral tubes; triangles, zigzag tubes; crosses, calculated by Eq. (1)]. Lines serve as visual guides.

Figure 5

(Color online) Contour plots of the redistributed charge density of (a) (4,0)@(13,0) DWCNTs and (b) (5,0)@(14,0) DWCNTs. Plots are on the plane perpendicular to the tube axis, with black circles denoting the position of carbon atoms. The calculated averages of the redistributed charge density in (4,0)@(13,0) and (5,0)@(14,0) are also shown in (c) and (d), respectively.

Figure 6

(Color online) The band structures calculated by LDA for an individual (4,0) nanotube, an individual (13,0) nanotube, and (4,0)@(13,0) DWCNTs. The vacuum level is set to zero. The work functions of the (4,0), (13,0), and (4,0)@(13,0) tubes are 5.93, 4.54, and $4.99\mathrm{eV}$, respectively. The dashed line is the Fermi level of the system. The dotted lines serve as visual guides.

Figure 7

A schematic diagram of a cylindrical capacitor for the charge transfer model of DWCNTs. The effective radius is $r_1+\Delta$ and $r_2-\Delta$ for the inner and outer tubes, respectively, where $r_1$ and $r_2$ are the radius of inner and outer tubes and $\Delta$ is equal to $1.0\mathrm{\AA }$.

Figure 8

(Color online) The band structures calculated by LDA for an individual (8,0) nanotube, an individual (17,0) nanotube, and (8,0)@(17,0) DWCNTs. The vacuum level is set to zero. The work functions of the (8,0), (17,0), and (8,0)@(17,0) tubes are 4.80, 4.52, and $4.72\mathrm{eV}$, respectively. The dashed line is the Fermi level of the system. The dotted lines serve as visual guides.

Figure 9

(Color online) The band structures of a (4,0)@(13,0) tube calculated by (a) LDA and (b) charge transfer model. The dashed line is the Fermi level of the system.

Figure 10

(Color online) Work functions of various zigzag $N$-walled nanotubes with $(n,0)$ inner tube, calculated by the charge transfer model (circles) for $\Delta =1.0\mathrm{\AA }$ and real LDA calculation (crosses). Lines serve as visual guides.