Charge transport in topological graphene nanoribbons and nanoribbon heterostructures

Although it is generally accepted that structural parameters like width, shape, and edge structure crucially affect the electronic characteristics of graphene nanoribbons (GNRs), the exact relationship between geometry and charge transport remains largely unexplored. In this paper, we present in situ through-transport measurements of various topological GNRs and GNR heterostructures by lifting the ribbon with the tip of a scanning tunneling microscope. At the same time, we develop a comprehensive transport model that enables us to understand various features, such as obscuring of localized states in through transport, the effect of topology on transport, as well as negative differential conductance in heterostructures with localized electronic modes. The combined experimental and theoretical efforts described in this paper serve to elucidate general charge transport phenomena in GNRs and GNR heterostructures.

Figure 1

Relationship between eigenstate localization, density of states (DOS), and transport probed in a 7/14/7-armchair graphene nanoribbon (AGNR). (a) Scanning tunneling microscope (STM) topographic scan ($V=-1.2\mathrm{V}$, $I=20\mathrm{pA}$) of a 7/14/7-AGNR on Au(111), next to a NaCl island. The arrow denotes where the STM tip was attached for conductance measurements. (b) Map of the differential conductance $dI/dV\left(V,z\right)$ for the ribbon in (a), lifted on the NaCl island. (c) Calculated zero-bias transport from tip to surface (teal), employing an extra Lorentzian broadening of 2 meV, and DOS (gray). The highest occupied molecular orbital (HOMO)–lowest unoccupied molecular orbital (LUMO) gap and transport gap are indicated. (d) Calculated molecular orbitals of frontier states 191 to 201. Orbitals 195 and 198 are reversed for the spin-down eigenstates (blue labels) compared with the spin-up eigenstates (red labels), as indicated; all other states are the same for the two spin channels (purple labels).

Figure 2

Transport in open- and closed-shell 5-armchair graphene nanoribbons (AGNRs). (a) Transport in a long 5-AGNR as a function of tip height and bias voltage. Reproduced with permission from Lawrence et al. [37]. (b) Calculated frontier orbitals of the $5_{14}\text{−}\mathrm{AGNR}$ and $5_8\text{−}\mathrm{AGNR}$, respectively, revealing an open-shell ground state in the former and closed-shell ground state in the latter. (c) Calculated $I$($V$) (solid) and $dI/dV$ (dashed) transport for the $5_8\text{−}\mathrm{AGNR}$ (red) and $5_{14}\text{−}\mathrm{AGNR}$ (green). The $dI/dV$ was Gaussian-broadened by 20 meV. (d) Transport $dI/dV$ and resistance $R$ measured in three different 5-AGNRs, labeled I, II, and III. Spectra are offset by 100 nS (200 MΩ). (e) Scanning tunneling microscope (STM) topographic scans ($V=-1.2\mathrm{V}$, $I=20\mathrm{pA}$) of the three 5-AGNRs whose transport is shown in (c). The arrows indicate the position where the tip was contacted to the GNR end. All scale bars are 10 nm.

Figure 3

Relationship between graphene nanoribbon (GNR) topology, ground state, end states, and zero-bias transmission. (a) Table showing calculated frontier states for the $N_2\text{−}\mathrm{armchair}$ GNR (AGNR) and $N_6\text{−}\mathrm{AGNR}$ (first column), calculated zero-bias transmission for the $N_2\text{−}\mathrm{AGNR}$ (red) and $N_6\text{−}\mathrm{AGNR}$ (blue) as a function of energy (second column), zero-bias transmission for $N_n\text{−}\mathrm{AGNRs}$ as a function of length $n$ (third column), and Kekulé resonance structures corresponding to the closed- and open-shell configurations of $N$-AGNRs. The rows correspond to the $N=5\text{−}\mathrm{AGNR}$, 7-AGNR, 9-AGNR, and 11-AGNR, respectively. (b) Tight-binding calculation employing $U=0$ for the $9_2\text{−}\mathrm{AGNR}$ and $9_6\text{−}\mathrm{AGNR}$. The frontier states are shown, with a schematic energy diagram. (c) Calculated transmission of the frontier states of $5_n\text{−}$, $7_n\text{−}$, $9_n\text{−}$, and ${11}_n\text{−}\mathrm{AGNR}$ as a function of length $n$. (d) Doubly open-shell Kekulé resonance structure for the 11-AGNR.

Figure 4

Negative differential resistance (NDR) in a 5/7/5-armchair graphene nanoribbon (AGNR) tunnel junction. (a) Calculated frontier orbitals of the $5_4{7}_8{5}_4^t$-AGNR. (b) Calculated zero-bias transmission. (c) Calculated transmission after application of a 1 V internal bias drop. (d) Charge distribution isosurface (top), and charge per atomic row, and potential energy per atomic row in the $5_4{7}_8{5}_4^t$-AGNR. (e) Charge distribution isosurface (top), charge per atomic row, and potential energy per atomic row in the $5_4{7}_8{5}_4^t$-AGNR after application of a 1 V internal bias drop. (f) Calculated transmission as a function of energy and internal bias drop. (g) Calculated total current as a function of energy and bias drop. The diagonals represent different fractions of internal to external bias drop. (h) Calculated $I$($V$) curves for different values of α, corresponding to the cross-sections along the diagonals indicated in (g). The gray curve is the experimental result, reproduced from Jacobse et al. [65]. (i) The 5/7/5-AGNR tunnel junction lifted off the gold surface by the scanning tunneling microscope (STM) tip with frontier orbitals shown, schematically coarse grained into an effective two-site model.

Figure 5

Transport through a notched 7-armchair graphene nanoribbon (AGNR) on NaCl, showing pronounced negative differential resistance (NDR). (a) Transport for the $7_2{5}_2{7}_8^c$-AGNR on NaCl as a function of lifting height $z$ and bias voltage $V$. (b) Measured $I$($V$) spectra at specific heights. The curves are offset by 2 nA for clarity. (c) Scanning tunneling microscope (STM) scans ($I=30\mathrm{pA}$, bias voltage as indicated) of the $7_2{5}_2{7}_8^c$-AGNR on NaCl. (d) Calculated frontier orbitals of the $7_2{5}_2{7}_8^c$-AGNR. (e) Calculated $I$($V$) curves for the $7_2{5}_2{7}_8^c$-AGNR. (f) Calculated $I$($V$) curves for the $7_{12}\text{−}\mathrm{AGNR}$. (g) Calculated transmission as a function of energy and internal bias drop for the $7_2{5}_2{7}_8^c$-AGNR on NaCl. (h) Calculated transmission as a function of energy and internal bias drop of the $7_{12}\text{−}\mathrm{AGNR}$ on NaCl.