Wave-Function Mapping of Graphene Quantum Dots with Soft Confinement

Using low-temperature scanning tunneling spectroscopy, we map the local density of states of graphene quantum dots supported on Ir(111). Because of a band gap in the projected Ir band structure around the graphene $K$ point, the electronic properties of the QDs are dominantly graphenelike. Indeed, we compare the results favorably with tight binding calculations on the honeycomb lattice based on parameters derived from density functional theory. We find that the interaction with the substrate near the edge of the island gradually opens a gap in the Dirac cone, which implies soft-wall confinement. Interestingly, this confinement results in highly symmetric wave functions. Further influences of the substrate are given by the known moiré potential and a 10% penetration of an Ir surface resonance into the graphene layer.

Figure 1

(a) $(100\times 100){\mathrm{nm}}^2$ STM image of Ir(111) covered by monolayer graphene islands; $U=-0.3\mathrm{V}$, $I=0.3\mathrm{nA}$; (b) atomically resolved $(12\times 12){\mathrm{nm}}^2$ image of graphene island; (c) magnified view of zigzag edge with graphene lattice overlaid; $U=0.7\mathrm{V}$, $I=20\mathrm{nA}$.

Figure 2

(a) black line: $dI/dU(U)$ curve spatially averaged over the graphene QD shown to the right; $U_{\mathrm{stab}}=0.5\mathrm{V}$, $I_{\mathrm{stab}}=0.5\mathrm{nA}$, $U_{\mathrm{mod}}=10\mathrm{mV}$; gray line: DOS(E) of the same island as obtained by TB calculation (see text); vertical bars mark the calculated eigenstate energies with degeneracies indicated as numbers; (b)–(d) $dI/dU$ images recorded at energies $E=Ue$ as marked; $I=0.2\mathrm{nA}$; $U_{\mathrm{mod}}=10\mathrm{mV}$. (e)–(g) LDOS maps calculated with soft edge potential at energies indicated; (h),(i): LDOS of an individual state calculated without (h) and with (i) soft edge potential.

Figure 3

(a) Energy gap $\Delta E$ versus graphene-Ir distance $D$ as deduced from DFT calculations; insets: band structure around $E_{\mathrm{D}}$ for two different $D$ as marked by arrows with $\Delta E$ indicated; grey area: projected bulk bands of Ir; Thick black lines: graphene states; (b)-(g) Calculated LDOS ($=|\Psi {|}^2$) for individual confined states with energies marked: (b)-(d) with soft edge potential; (e)-(g) without soft edge potential; (h) experimental $k_{n}r=E_n/(\hslash v_{\mathrm{D}})r$ for the two peaks closest to $E_D$ at different average island radius $r$; circles: $n=0$, squares: $n=1$; dotted lines: zeros of the first two Bessel functions (see text).

Figure 4

(a) STM image and (b) $dI/dU$ map of a large graphene QD; $30\times 30{\mathrm{nm}}^2$, $U=-0.65\mathrm{V}$, $I=0.5\mathrm{nA}$, $U_{\mathrm{mod}}=10\mathrm{mV}$; (c) calculated LDOS of the same QD at $E=-0.65\mathrm{eV}$; (d)-(f) $dI/dU$ maps of a graphene QD recorded at the energies marked; $27\times 30{\mathrm{nm}}^2$, $I=0.5\mathrm{nA}$, $U_{\mathrm{mod}}=10\mathrm{mV}$; deduced wave lengths ${\lambda }_{\mathrm{out}}$ (${\lambda }_{\mathrm{in}}$) outside (inside) the QD are marked in (e); (g) resulting dispersion relations $E(\Delta k=\pi /{\lambda }_{\mathrm{in}/\mathrm{out}})$ inside (stars) and outside (triangles) of the QD as well as from standing waves scattered at Ir(111) step edges (circles); full lines are linear fits with resulting $v_{\mathrm{D}}$ indicated; energy offset is marked; dashed line is deduced from photoemission on clean Ir(111) [27]; (h) relative intensity $R$ of $S_0$ and $S_2$ in graphene as deduced from STS data (squares) and from DFT calculations ($S_0$: circles, $S_2$: triangles); inset: calculated LDOS of $S_0$ at $E=-0.4\mathrm{eV}$ along the direction perpendicular to the surface; $I_{\mathrm{Ir}}$ and $I_{\mathrm{C}}$ as used for determination of $R$ are marked.