Surface-bound states in nanodiamonds

We show via ab initio calculations and an electrostatic model that the notoriously low, but positive, electron affinity of bulk diamond becomes negative for hydrogen passivated nanodiamonds and argue that this peculiar situation (type-II offset with a vacuum level at nearly midgap) and the three further conditions: (i) a surface dipole with positive charge on the outside layer, (ii) a spherical symmetry, and (iii) a dielectric mismatch at the surface, results in the emergence of a peculiar type of surface state localized just outside the nanodiamond. These states are referred to as “surface-bound states” and have consequently a strong environmental sensitivity. These type of states should exist in any nanostructure with negative electron affinity. We further quantify the band offsets of different type of nanostructures as well as the exciton binding energy and contrast the results with results for “conventional” silicon quantum dots.

Figure 1

(a) Single-particle HOMO-LUMO gaps from DFT-LDA calculation with (blue squares) and without (black circles) exciton-binding energy versus radius for hydrogen passivated NDs. The left panel shows the bare LDA results while the right panel shows LDA-corrected (see Sec. 4) results. (b) Schematic showing the hydrogen passivated NDs with negative electron affinity. The $E_{\mathrm{LUMO}}$ represents the energetic position of the surface-bound states.

Figure 2

Schematic picture showing the alignment procedure used to calculate IP and EA. The value for ${\varepsilon }_{\mathrm{vac}}$ is taken from the potential output. The figure shows a negative electron affinity (NEA). The figure is not to scale; the average potentials are typically 50 eV from the eigenvalues in our case.

Figure 3

Band alignment of H-terminated nanodiamond [ND-H], H-terminated diamond (110) surface [C-H(110)], bare diamond (110) surface [C(110)], H-terminated diamond (111) surface [C-H(111)], bared diamond (111) surface [C(111)], H-terminated Si(110) surface [Si-H(110)] and bare Si(110) surface [Si(110)], and two H-terminated Si QDs (Si-H QD1 and Si-H QD2). Results from our DFT-LDA calculations are indicated by “This work,” $GW{\mathrm{\Gamma }}^1$ calculations from Ref. [23] are indicated by “$GW{\mathrm{\Gamma }}^1$,” while experimental results from Ref. [25] are indicated by “Exp.” with references. The values of the IPs are given in black above the bottom line of the figure. The values of the EA are given in black close to the vacuum level. A semiquantitative correction of our DFT-LDA calculated IP and EA are given in bold blue and bold red numbers.

Figure 4

Schematic showing the electron affinity of bulk diamond, hydrogen-terminated diamond surface, and hydrogen-terminated nanodiamonds and the surface contribution to the band offset ${\mathrm{BO}}^{\mathrm{surf}}$ along with the surface state in H-terminated diamond (110) surface and the surface-bound states in H-terminated nanodiamond. The IP and EA values are given in blue and red, respectively.

Figure 5

Local electrostatic potential averaged over each atom (a) and (c) and bond length distribution (b) as a function of the distance to the center of the NDs. The bond length of bulk diamond is shown as dashed lines.

Figure 6

Wave functions of (a)–(g) LUMO ($e_0$) and excited unoccupied states ($e_1$ to $e_{15}$) along with (h) the HOMO wave function of ${\mathrm{C}}_{281}{\mathrm{H}}_{172}$ with isosurfaces corresponding to 75% of the maximum (red isosurface) and 75% of the minimum (blue isosurface) value. The carbon and passivating hydrogen atoms are represented as cyan and white spheres.

Figure 7

The distributions of the charge density for a surface-bound state in H passivated nanodiamond (a) and a surface state of H passivated diamond surface (b).

Figure 8

Atomic charges of (a) ${\mathrm{C}}_{147}{\mathrm{H}}_{100}$, (b) ${\mathrm{C}}_{281}{\mathrm{H}}_{172}$, (c) ${\mathrm{C}}_{465}{\mathrm{H}}_{228}$, (d) ${\mathrm{C}}_{633}{\mathrm{H}}_{300}$ NDs, and (e) ${\mathrm{Si}}_{281}{\mathrm{H}}_{172}$ QD obtained from DFT calculations and plotted as a function of distance to the center of the NDs.

Figure 9

Surface charge density (a) for different size NDs (black, red, green) and for a 2D surface (blue) that can be seen as the infinite radius limit of the NDs. Dielectric constant as a function of position (b) and resulting electronic potential (c). Electronic potential as a function of the distance to the center of the ND for different dielectric properties of the outside region. (d) Electronic potential as a function of the distance to the center for a ND with radius = 8.0 Å embedded in a material with varying dielectric properties.

Figure 10

(a) Electrostatic potential $V\left(r\right)$ (black solid line) induced by the surface dipole along with the radial part of the ground state wave function $R_{00}\left(r\right)$ of the model calculation (red solid line) and from the LUMO state $e_0$ obtained via DFT (green dashed line). (b) Electrostatic potential obtained for a surface dipole pointing in the reverse direction [negative shell outside, shown in (c)] as would be the case in silicon QDs [see Fig. 8].

Figure 11

Exciton binding energy for NDs and for Si QDs as a function of radius. For the NDs, the dielectric environment ${\varepsilon }_{\mathrm{out}}$ has been varied from vacuum with ${\varepsilon }_{\mathrm{out}}=1$ to a rather high dielectric environment with ${\varepsilon }_{\mathrm{out}}=5.7$. Results at the single-configuration level (circles, squares, triangles) and at the CI level (diamonds) are shown.

Figure 12

Schematic view of the electronic potential considering only the surface contribution ${\mathrm{BO}}^{\mathrm{surf}}$ (dotted line), adding to this potential the chemical offset ${\mathrm{BO}}^{\mathrm{chem}}$(solid line). The upper part shows the nanodiamond results and the lower part the silicon quantum dot results. The surface trap disappears in the case of silicon.