Charging graphene nanoribbon quantum dots

We describe charging a quantum dot induced electrostatically within a semiconducting graphene nanoribbon by electrons or holes. The applied model is based on a tight-binding approach with the electron-electron interaction introduced by a mean-field local-spin-density approximation. The numerical approach accounts for the charge of all the $p_z$ electrons and screening of external potentials by states near the charge-neutrality point. Both a homogeneous ribbon and a graphene flake embedded within the ribbon are discussed. The formation of transport gaps as functions of the external confinement potential (top-gate potential) and the Fermi energy (back-gate potential) are described in a qualitative agreement with the experimental data. For a fixed number of excess electrons, we find that the excess charge added to the system is, - depending on the voltages defining the work point of the device, (i) delocalized outside the quantum dot, - in the transport gap due to the top-gate potential; (ii) localized inside the quantum dot, - in the transport gap due to the back-gate potential; or (iii) extended over both the quantum dot area and the ribbon connections, - outside the transport gaps. The applicability of the frozen valence-band approximation to describe charging the quantum dot by excess electrons is also discussed.

Figure 1

(a) Schematics of the ribbons considered in this work with and without the central flake. The considered nanoribbons have armchair edges, and the number of atoms across the channel (18 and 36) corresponds to a semiconducting dispersion relation of a $p_z$ tight-binding Hamiltonian. The length of the ribbon is $L_y=61$ nm. The hexagonal flake of width $L_{x_2}=15.7$ nm is embedded inside the ribbon with the width $L_{x_1}=2.15$ nm (left plot). We consider also homogeneous ribbons of width $L_{x_1}=4.3$ nm (left plot). (b) External potential defining the quantum dot as given by Eq. (12) for $W=0.5$ eV.

Figure 2

Single-electron spectra for a quantum dot of length $2R=20$ nm defined within a semiconducting armchair nanoribbon of length $L=60.1$ nm, and 18 (a) or 36 (b) atoms across the channel. The colors of the lines indicate the extent of the electron localization within the dot defined as the probability to find the electron within the segment $[-1.2R,1.2R]$ (see the color scale on right-hand side of the plots). In (c) the results for the hexagonal flake inside the nanoribbon with 18 atoms are given. The black insets symbolically denote the studied system.

Figure 3

(a) Chemical potentials for $N_e=-24,\cdots ,24$ excess electrons in an 18-atom-wide nanoribbon as a function of the QD potential $W$. (b) Charge contained within the quantum dot (segment $[-1.2R,1.2R]$) defined within the 18-atom-wide nanoribbon as a function of $W$ for $N_e=-24,\cdots ,24$ excess electrons. (c) and (d) Same as (a) and (b) for a nanoribbon with 36 atoms across the channel. (e) and (f) Same as (c) and (d) but for the hexagonal flake with 44 atoms along the side of the hexagon. In (a)–(f) and elsewhere in the paper, the length of the system is $L=60.1$ nm. (g) and (h) Same as (e) and (f) but with longer ribbons, with $L$ increased to 90 nm. Insets in the figures shows an enlarged fragment from the graph. The orange lines in (a) mark the transport gap due to the Fermi energy (solid lines) and (b) shows the external potential (dotted lines); see text. The thicker line in (a) shows the chemical potential of eight electron confined in the dots ${\mu }_8$, which is discussed in detail in the text.

Figure 4

(a) and (b) Charge contained within a segment of $[-1.2R_d,1.2R_d]$ for $N_e=1,2,\cdots$ excess electrons for the quantum dot defined within the 36-atom-wide ribbon width for $W=0.5$ eV (a) and $W=0.8$ eV (b). For the energy spectrum and the charge contained within the quantum dot, see Figs. 3 and 3 for $W=0.5$ eV (a) and $W=0.8$ eV (b). (c) and (d) Same as (a) and (b) only for a semiconducting 18-atom ribbon containing a hexagonal flake across the channel.

Figure 5

Excess electron charge [(a), (d), and (g)] and total potential [(b), (e), (h)] for the nanoribbon of Figs. 3 and 3 at $W=0.5$ eV for $N_e=1$, 5, and 9, respectively. Parts (c), (f), and (i) present the cross sections of (a) and (b), (d) and (e), and (g) and (h) along the axis of the ribbon, respectively, compared the external potential of Eq. (12).

Figure 6

The charge localized in the quantum dot $Q_d$ of a 36-atom-wide nanoribbon (a) and the hexagonal flake embedded within an 18-atom-wide ribbon (b) for $W=0$, 0.5, and 1 eV as a function of the electrochemical potential of the external electron reservoir ${\mu }_r$. The region of the transport gap is marked with vertical lines. In (b), the lines (dots) denote the results obtained for a nanoribbon of length $L=60.1$ nm (90 nm). For the chemical potential spectra, see Fig. 3 and Figs. 3 and 3 for the system without (with) the flake inside the nanoribbon, respectively.

Figure 7

The charge density of the eighth excess electron added to the system calculated as a difference of the total charge for $N_e=8$ and 7 for the ribbon of Fig. 3. The ${\mu }_8$ line is marked by the thick solid line in Fig. 3.

Figure 8

(a) Chemical potentials for $N_e=-24,\cdots ,24$ excess electrons for the ribbon with 18 atoms across the channel as calculated by the standard method of the present work (red lines for $N_e=1,\cdots ,24$ and gray lines for $N_e=-24,\cdots ,-1$) and with the basis of conduction band only (black lines), with all the single-electron states below the energy gap considered as frozen. (b) Charge confined within the quantum dot for both approaches as a function of $W$.

Figure 9

Chemical potentials for $N_e=-24,\cdots ,24$ excess electrons for the ribbon with 18 atoms across the channel for the confinement potential $-Wexp(-{\left(\frac{y}{R}\right)}^2)$ instead of Eq. (12). For the results with Eq. (12), see Fig. 3.

Figure 10

Same as Figs. 3 and 3 but with the disordered edge of the flake containing both armchair and zigzag fragments as displayed in (a); see the text.