Physical properties of low-dimensional ${sp}^2$-based carbon nanostructures

The last two decades have witnessed a tremendous growth in the development and understanding of ${sp}^2$ carbon-based nanostructures. The impact of this research has led to a number of fundamental discoveries that have played a central role in the understanding of many aspects of materials physics and their applications. Much of this progress has been enabled by the development of new techniques to prepare, modify, and assemble low-dimensional materials into devices. The field has also benefited greatly from much progress in theoretical and computational modeling, as well as from advances in characterization techniques developed to probe and manipulate single atomic layers, nanoribbons, and nanotubes. Some of the most fundamental physical properties of ${sp}^2$ carbon-based nanostructures are reviewed and their role as model systems for solid-state physics in one and two dimensions is highlighted. The objective of this review is to provide a thorough account on current understanding of how the details of the atomic structure affect phonons, electrons, and transport in these nanomaterials. The review starts with a description of the behavior of single-layer and few-layer graphene and then expands into the analysis of nanoribbons and nanotubes in terms of their reduced dimensionality and curvature. How the properties can be modified and tailored for specific applications is then discussed. The review concludes with a historical perspective and considers some open questions concerning future directions in the physics of low-dimensional systems and their impact on continued advances in solid-state physics, and also looks beyond carbon nanosystems.

Figure 1

Graphene (a) real space and (b) reciprocal space lattices. The unit cells in real space and reciprocal space are highlighted, and the unit vectors in real space and reciprocal space are indicated.

Figure 2

Structural models for three carbon nanotubes (top), three graphene nanoribbons (middle), compared to their mother structure graphene (bottom). The chiral vector ${\mathbf{C}}_h$ and in-plane vectors ${\mathbf{a}}_1$, ${\mathbf{a}}_2$ are shown.

Figure 3

Structures of (a) a N-AGNR and (b) a N-ZGNR. In each case, one dimer is highlighted. The unit cells are emphasized by a gray box. $W_{\mathrm{AGNR}}$ and $W_{\mathrm{ZGNR}}$ are the armchair and zigzag nanoribbon widths, respectively, while $c_{\mathrm{AGNR}}$ and $c_{\mathrm{ZGNR}}$ show the length of the translational vectors for the ribbon unit cells. Adapted from [153].

Figure 4

(a) Electronic band structure calculated using an orthogonal tight-binding model and considering only nearest-neighbor atom interactions. The inset on the lower left highlights the conical energy band dispersion near the Fermi energy. (b) The corresponding energy dependence of the electronic density of states (DOS). For a neutral graphene sample, the Fermi level $E_F$ is located at the Dirac point corresponding to zero energy. From [379].

Figure 5

(a) Electronic band structure of graphene showing the cutting lines corresponding to a nanotube’s allowed states in black. Electronic band structure of (b) an (8,8) armchair and (c) a (9,0) zigzag carbon nanotube obtained from both zone-folding (solid circles) and ab initio calculations (solid lines).

Figure 6

Electronic band structure of (a) an armchair-edged (8-AGNR) and (b) a zigzag-edged graphene nanoribbon (8-ZGNR) calculated using an orthogonal simple tight-binding model (symbols) compared to the zone-folded bands (solid lines) for the corresponding carbon nanotubes [(9,0) and (8,8), respectively]. (c) Comparison between the electronic bands of a 8-ZGNR obtained by tight-binding (symbols) and by density function theory calculations (solid line), showing the effects of edges on the calculated electronic energy band structure.

Figure 7

DFT calculated phonon dispersion relations for monolayer graphene. The insets on the left show the phonon displacements associated with the optically active LO (top) and TO (middle) phonon modes at the $\mathrm{\Gamma }$ point and the $K$ point TO phonon mode (bottom), corresponding to the modes highlighted by red circles. Adapted from [473].

Figure 8

(a) Phonon dispersion of a (10,0) carbon nanotube obtained by DFT calculations (left panel) and zone folding (middle panel) together with the vibrational density of states (VDOS) for both approaches (right panel). Solid and dashed lines are for data obtained with DFT calculations and the zone-folding scheme, respectively. Dependence of the frequencies of the (b) Raman-active and (c) IR active phonon modes as a function of the nanotube index $n$ for selected $(n,0)$ zigzag nanotubes. Adapted from [113].

Figure 9

Raman spectra of (a) graphene, (b) an isolated single-wall carbon nanotube, and (c) a $N_r=7$ AGNR. Adapted from [285], [205], and [63].

Figure 10

(a) Raman spectra of 5 Å PTCDA on graphene (colored line) and on a blank ${\mathrm{SiO}}_2/\mathrm{Si}$ substrate (black line) with a 532 nm excitation laser wavelength ($E_{\text{laser}}=2.33\mathrm{eV}$). The inset shows the structure of the PTCDA molecule. From [184]. (b) Calculated Raman intensity as a function of laser excitation energy for $E_F=E_H+\hslash {\omega }_q$. A homogeneous broadening of $\gamma =0.03\mathrm{eV}$ is applied. Adapted from [28].

Figure 11

(a) A uniaxial and (b) shear strain applied to graphene nanoribbons in real space. The red shaded rectangles denote the unit cell of the strained structures. (c) The displacement of the $K$ point in reciprocal space due to the tensile strain (along the armchair and zigzag directions) and the shear strain is shown. The effect of strain is to displace the Dirac cone from the $K$ point to another position of the Brillouin zone (d). (e) The allowed states for low-dimensional nanostructures, such as nanoribbons and nanotubes, are illustrated by the cutting lines. The effect of strain on those structures can be interpreted based on the displacement of the Fermi points by a vector $\mathrm{\Delta }{\mathbf{k}}_F$. Adapted from [251].

Figure 12

Band gap opening $E_{\text{gap}}$ as a function of strain for (a) graphene and (b) armchair nanoribbons. The inset to (a) shows the variation of the Poisson ratio $\nu$ along the armchair and zigzag directions as a function of strain. Adapted from [251].

Figure 13

Electronic band structure considering spin-polarization effects for a zigzag nanoribbon as a function of (a) uniaxial and (b) shear strain. (c) Band gap opening as a function of uniaxial strain for the band structure shown in (a). Adapted from [251].

Figure 14

(a) $G$-band evolution as a function of the amount of strain applied [numbers on the right side of (a)] in arbitrary units to a graphene layer immersed in a polymer matrix, as illustrated in (b). (c) Eigenvectors for the $G^{-}$ and $G^{+}$ modes obtained by DFT based calculations in strained graphene. Adapted from [307].

Figure 15

Effects of uniaxial and torsional strain on the band gap of several $(n,m)$ nanotubes. Lines are for the analytical model and points are simulation data using the tight-binding method. Adapted from [476].

Figure 16

(a) Torsional ${\epsilon }_{\tau }$, (b) axial ${\epsilon }_T$, (c) radial strains ${\epsilon }_R$, and (d) the resulting injected charge $q$ as a function of the Fermi energy ($\mu$) for the (8,7) semiconducting SWNT. The charge is given in units of the elementary charge $e$ per carbon atom in the SWNT structure. The vertical lines show, as a guide to the eye, the approximate energy values for the valence subband maxima ($E_1^v$, $E_2^v$, and $E_3^v$) and the conduction subband minima ($E_1^c$, $E_2^c$, and $E_3^c$). From [440].

Figure 17

Scanning tunneling microscopy (STM) measurements on graphene nanoribbons obtained by unzipping carbon nanotubes. (a) STM topographical image ($V_S=0.3\mathrm{V}$, $I=60\mathrm{pA}$, $T=7\mathrm{K}$) of the edge of an (8,1) graphene nanoribbon over an Au(111) surface and (b) its corresponding structural model. STS measurements of the graphene nanoribbon measured along a line (c) perpendicular (black points) and (d) parallel (red points) to the ribbon edge. The inset to (c) shows a high-resolution $dI/dV$ spectrum for another nanoribbon with a (5,2) geometry, thus showing the energy splitting $\mathrm{\Delta }$ of the edge states. From [421].

Figure 18

(a) Electronic band structure and (b) the corresponding electronic density of states for the (8,1) graphene nanoribbon, calculated using a Hubbard-like model without (dashed blue lines) and with (solid red lines) electron-electron interactions. The calculated energy splitting of $\mathrm{\Delta }=29\mathrm{meV}$ in (b) is comparable to the experimentally observed value of $23.8\pm 3.2\mathrm{meV}$. From [421].

Figure 19

(a) STM image and (b) a line profile of height measurements of a CVD-grown 6.4 nm wide nanoribbon. (c) An atomic model is shown for the ribbon with the (3,1) chirality on one edge and a pentagon–heptagon 5–7 reconstructed edge on the other side. The local density of electronic states is calculated as a function of energy in (d) considering the different edges shown in (c). (e) The experimental $dI/dV$ measurements measured at the edges of the nanoribbon are shown [see the + in (a)]. From [336].

Figure 20

Schematic diagrams for the electronic band structure of a (a) half-metal, (b) spin-semiconductor, and (c) a spin gapless semiconductor nanoribbon along with their spin-polarization configurations at the edges (d). (e) The atomic model of the sawtooth nanoribbons defining the chiral index $(n_1,n_2)$ is shown for $(n_1,n_2)=(4,3)$. (f), (g) The electronic band structures $E-E_F$ vs $k$ of the graphene nanoribbons for external electric fields $E_{\mathrm{ext}}$ of 0.04 and 0.0 eV/Å are shown. The right panels in (f) and (g) denote the partial charge densities for the four levels close to the Fermi level $E_F$. From [453].

Figure 21

(a)–(e) Structural model and nomenclature for graphene nanowiggles, which are made up of successive oblique and parallel cuts in armchair ($A$) or zigzag ($Z$) graphene patches. (a) Illustration of how the widths of the parallel ($W_p$) and oblique ($W_o$) sectors are defined in terms of the number of C-C lines (red and green). $W_p$ and $W_o$ refer, respectively, to the number of parallel and oblique atomic rows that are present in a nanowiggle in the two indicated directions. The structure shown in (b) is one of the structures actually synthesized by [63]. From [91].

Figure 22

Electronic band structures $E(k)$ calculated along $\mathrm{\Gamma }-X$ using DFT (dashed lines) and tight-binding (solid lines) methods corresponding to the different magnetic states for the representative $AA$, $AZ$, $ZA$, and $ZZ$ graphene nanowiggles, shown in Figs. 21. The schematic spin distributions (red: down, black: up) are shown on top of each panel. From [91].

Figure 23

Schematic diagrams of the moiré patterns that are formed by the twisting angles $\theta =5.1°$, 7.3°, 13.2°, and 21.8°. The gray area highlights the unit cell, which decreases as the twisting angle $\theta$ increases. Adapted from [71].

Figure 24

(a) Brillouin zone for a twisted bilayer graphene (see Fig. 23) rotated by $\theta$, where the circles represent the isoenergies for Dirac cones from the bottom layer (black) and from the top layer (red). (b) The energy dispersion for each Dirac cone close to the $K$ point and in the vicinity where the two cones overlap, giving rise to van Hove singularities as shown in (c). The two Dirac cones are separated in reciprocal space by $\mathrm{\Delta }k$ which depends on $\theta$, as shown in (a). Adapted from [219].

Figure 25

(a)–(c) Electronic band structure for twisted bilayer graphene for three distinct twist angles $\theta$ (9.4°, 13.2°, and 21.8°, respectively) taken over the range of energy $-2.0$ to $+2.0\mathrm{eV}$. (d) Calculated optical absorption spectra near the absorption peak as a function of twist angle $\theta$. (e) $\theta$ dependence for the optical absorption peaks at $E_L^{\max }$, according to the results calculated in (d), and others not shown. Squares denote calculated data and circles are values replicated by symmetry. The solid line is a fit to the equation $E_L^{\max }=E_0|\sin (3\theta )|$. Adapted from [71].

Figure 26

(a) Atomic force microscope image of tBLG folded using the AFM tip as schematically shown in (b). The inset to (a) is the atomic resolution AFM image used for estimating the twist angle as 6°. (c) Raman spectra of tBLG shown in (a), using excitation laser energies at $E_{\text{laser}}=1.96$, 2.33, 2.41, and 2.54 eV. Besides the first-order allowed $G$ band, a peak centered at $1625{\mathrm{cm}}^{-1}$ is observed in (c). The absence of the disorder-induced $D$ band ($1350{\mathrm{cm}}^{-1}$) provides evidence that the folded region has a low density of defects. The Raman image obtained from the same region shown in (a) for the $G$-band [(d)] and the $R^{\prime }$ band centered at $1625{\mathrm{cm}}^{-1}$ in (e). Adapted from [70].

Figure 27

(a) Atomic force microscopy image of monolayer graphene sitting on a $\mathrm{Si}/\mathrm{S}\mathrm{i}{\mathrm{O}}_{\mathrm{2}}$ substrate. The white arrows stand for the directions over which the AFM tip was used in contact mode to scan the graphene sheet and to generate defects. (b) Three regions of folded graphene (labeled 1, 2, and 3) were obtained with the AFM tip. (c), (d) denote, respectively, the $G$-band and $R$-band map intensities for the boxed area in (b). (e) Raman spectra of tBLG obtained from regions 1, 2, and 3 defined in (b). The scale bars are $4\mu \mathrm{m}$ (a), (b) and $1\mu \mathrm{m}$ (c), (d). Adapted from [71].

Figure 28

(a) Brillouin zones for the top and bottom graphene layers, rotated from each other by a small twist angle $\theta$. (b) The intravalley double-resonance process involving elastic electron scattering by the static potential generated by the moiré pattern. (c) High energy in-plane transverse (TO) and longitudinal (LO) optical phonon branches along high symmetry directions in the first Brillouin zone of graphene. Adapted from [437]. The scattering phonon wave vectors ${\mathbf{Q}}_{\text{intra}}$ and ${\mathbf{Q}}_{\text{inter}}$ stand for scattering the electron or hole within the same valley and different valleys, respectively. The electron-phonon coupling is stronger for the LO and TO phonon branches near the $\mathrm{\Gamma }$ and $K$ points, respectively, and relevant ${\mathbf{Q}}_{\text{intra}}$ and ${\mathbf{Q}}_{\text{inter}}$ wave vectors are outlined in (d). Adapted from [70].

Figure 29

$R$ and $R^{\prime }$-band frequency [${\omega }_R(\theta )$ and ${\omega }_R^{\prime }(\theta )$] as a function of the twist angle $\theta$. The solid circles, squares, diamonds, and up triangles are experimental data. From [71], ([175], [219], and [452], respectively. Adapted from [71].

Figure 30

(a) Schematic of the measurement setup showing the STM tip and an optical microscope image of a graphene and h-BN heterostructure where the graphene layer is on the top of the h-BN layer. (b) Superlattice wavelength and rotation as a function of the angle $\varphi$ between the graphene and h-BN lattices. (c)–(e) STM topography images for three different moiré patterns with the wavelength of the periodic potential $\lambda =$ (c) 2.4, (d) 6.0, and (e) 11.5 nm. The white scale bars in all images are 5 nm. Adapted from [480].

Figure 31

(a) Side and top views of a single layer of phosphorene on top of graphene, where $d_{P/G}$ is the distance between the graphene and phosphorene layers. (b) Band structure of the phosphorene/graphene heterostructure near the Fermi energy. The energy differences between the energy of the graphene Dirac point and the band edges of the phosphorene conduction (${\mathrm{\Delta }}_{CB}$) and valence (${\mathrm{\Delta }}_{VB}$) bands are shown as a function of the wave vector from $\mathrm{\Gamma }$ to one corner of the Brillouin zone $Y$. (c) Evolution of the band edges, as a function of the field relative to the graphene Dirac point, for single-layer phosphorene. Adapted from [332].

Figure 32

Temperature ($T$) dependence of the resistivity $\rho (T)$ for different charge-carrier densities in graphene. The measured sample resistivity increases linearly with temperature $T$ in the high-temperature limit, thereby indicating that a quasiclassical phonon distribution is responsible for the electron scattering. As $T$ decreases, the resistivity decreases more rapidly, following a $T^4$ dependence. This low-temperature behavior is described by a Bloch-Grüneisen model, taking into account the quantum distribution of the two-dimensional acoustic phonons in graphene, as explained by [147]. From [122].

Figure 33

(a) Off-side view of a scanning electron microscopy (SEM) image using a suspended six-probe graphene device. (b) Atomic force microscopy image of the suspended device before the measurements, and (c) after the measurements with graphene removed by a short oxygen plasma etch. (d) Side view of the device (schematic). The doped silicon gate, partly etched ${\mathrm{SiO}}_2$, suspended single-layer graphene and Au/Cr electrodes are shown, respectively, in blue, green, pink, and orange (from bottom to top). The white rectangle denotes vacuum. From [51].

Figure 34

Differential conductance $dI/dV$ as a function of gate ($G$) voltage and source-drain ($S\text{−}D$) bias. Various quantum transport regimes of a single nanotube connected to metallic leads are illustrated depending on the coupling strength between the leads and the nanotube conduction channel. The coupling continuously changes from highly reflective (top) to highly transmissive (bottom). The diamond shapes shown on top are typical manifestations of the Coulomb blockade effect where the addition of one charge to the nanotube corresponds to each diamond. The bottom panel corresponds to a Fabry-Pérot–type interference pattern of a one-dimensional standing wave. The amount of conductance in each case is shown on the right. From [258].

Figure 35

(a) SEM micrograph of GNR devices with various channel widths fabricated on a 200 nm thick ${\mathrm{SiO}}_2$ substrate. The widths of the GNRs from top to bottom are 20, 30, 40, 50, 100, and 200 nm. Each GNR was prepared by a lithographic process. (b) AFM image of a single-layer graphene sample before lithographic thinning. (c) Cross-section measurement of the AFM image along the dashed line shown in (b) indicating that the sheet is a single layer of graphene. From [86].

Figure 36

Quantum conductance as a function of energy (center) for the PM, FeM, and AFeM states in a 12-ZGNR (solid blue curves correspond to spin-up and dashed red lines to spin-down states). For each state we present local current plots corresponding to ${\mu }_1=-0.35\mathrm{eV}$ and ${\mu }_2=-0.25\mathrm{eV}$ and to ${\mu }_1=+0.25\mathrm{eV}$ and ${\mu }_2=+0.35\mathrm{eV}$ for both the AFeM and FeM states and ${\mu }_{1/2}=\pm -0.05\mathrm{eV}$ for the PM case (chemical potential windows are marked by vertical black dotted lines), for both spin-up (blue) and spin-down (red) states. Adapted from [154].

Figure 37

Conductance measurements of a defect-free GNR obtained by the Ullmann coupling followed by the cyclo-dehydrogenation method ([63]). In this experiment, an STM tip pulls on the molecule to detach it from the gold substrate (a) and the current is recorded for various values of tip-substrate bias [inset to (a)] and tip height (b). Adapted from [221].

Figure 38

Differential conductance maps of an intramolecular junction between a pristine ($p$) and nitrogen-doped GNR heterostructures. Heterostructures are artificially outlined by white dashed lines for clarity for the reader. The $10\times 10{\mathrm{nm}}^2$ STM images recorded at $T=5\mathrm{K}$, $U=1.35\mathrm{V}$, and $I=0.15\mathrm{nA}$ are shown in (a) and (b). In (b), the $p$- and N-GNR units are indicated by gray and blue dots, respectively. (c)–(f) Selected examples of conductance maps at energy positions around the conductance band minimum. The analysis of the results shows that the heterojunction features a type-II alignment with a sharp transition region. From [64].

Figure 39

(a) Band structure of a hybrid graphene nanoribbon with both zigzag and armchair edges. The arrows point to the wave-function plot of the states around the Fermi energy, displaying the edge state associated with the zigzag edge. The lower arrow points to the junction. (b) Electronic transport plot of conductance vs energy, revealing that the nanoribbon can be used as a spin filter on an energy window just above the Fermi energy. Adapted from [52].

Figure 40

(a) DFT calculated electronic transport of boron- and nitrogen-doped graphene nanoribbons. Green arrows indicate the energy of the calculated donorlike (boron) and the acceptorlike (nitrogen) states. Local density of states plots for (b) nitrogen and (c) boron localized states at the energies indicated by the arrows in (a). Adapted from [97].

Figure 41

(a), (c) Calculated electronic band structure and (b), (d) quantum conductance of cumulene and polyyne chains. (e) The electronic densities for cumulene, polyyne, and strained polyyne chains. (f) Time-dependent electrical current measurements made during the formation of a carbon chain. The inset to (f) shows the TEM images of the formed chains, just after the sudden drop is observed in the electrical current. From [94].

Figure 42

(a) Resonance Raman spectra of linear carbon chains encapsulated in multiwall carbon nanotubes. The $D$- and $G$-band Raman modes from carbon nanotubes are shown here along with the bands from the chain. (b) Resonance window of the linear carbon chain mode. From [8] and [9].