Field effect doping of graphene in metal$|$dielectric$|$graphene heterostructures: A model based upon first-principles calculations

We study how the Fermi energy of a graphene monolayer separated from a conducting substrate by a dielectric spacer depends on the properties of the substrate and on an applied voltage. An analytical model is developed that describes the Fermi level shift as a function of the gate voltage, of the substrate work function, and of the type and thickness of the dielectric spacer. The parameters of this model, that should describe the effect of gate electrodes in field effect devices, can be obtained from density functional theory (DFT) calculations on single layers or interfaces. The doping of graphene in metal$|$dielectric$|$graphene structures is found to be determined not only by the difference in work function between the metal and graphene and the dielectric properties of the spacer but potential steps that result from details of the microscopic bonding at the interfaces also play an important role. The doping levels predicted by the model agree very well with the results obtained from first-principles DFT calculations on metal$|$dielectric$|$graphene structures with the metals Al, Co, Ni, Cu, Pd, Ag, Pt, or Au, and a $h$-BN or vacuum dielectric spacer.

Figure 1

Schematic drawing of the metal$|$dielectric$|$graphene (M$|$D$|$Gr) structure. The electrostatic potential profile in the structure is indicated by the wide dashed line at the top. $\Delta V$ is the shift of the work function of the structure with respect to that of the clean metal $W_{\mathrm{M}}$; ${\Delta }_{\mathrm{M}}|\mathrm{D},{\Delta }_{\mathrm{D}}|\mathrm{Gr}$ are the potential steps at the interfaces; and $\Delta E_{\mathrm{F}}$ is the Fermi level shift in graphene caused by the electrostatic doping.

Figure 2

The internal field $E_{\mathrm{int}}$ (solid lines) inside the dielectric spacer of Fig. 1, as a function of the applied external field $E_{\mathrm{ext}}$, for two values of $\kappa$. The black line corresponds to a Cu$|h$-BN$|$graphene structure with five layers of $h$-BN and $\kappa =2.8$. The red line would be the response of the same system, but with $\kappa =1.0$ instead of 2.8. The curves cross at $E_{\mathrm{ext}}=E_{\mathrm{i}0}$, see Eq. (9), where $E_{\mathrm{int}}=0$. At fields $E_{\mathrm{ext}}=-E_0$, see Eq. (7), the curves cross the lines $E_{\mathrm{ext}}/\kappa$ (dashed lines).

Figure 3

Top views of the energetically favorable configurations of $h$-BN on (a) Cu, Ni(111), Co(0001), and (b) Pt, Pd, Au, Ag, Al(111) in 1$\times$1 and $\sqrt{3}\times \sqrt{3}$ surface unit cells, respectively.

Figure 4

Side views of the energetically favorable configurations of metal$|h$-BN$|$graphene structures, (a) Cu, Ni, Co and (b) Au, Pt, Ag, Pd, Al.

Figure 5

Left panel: The band structure of a Au(111)$|h$-BN monolayer|graphene stack. The eigenstates of graphene and $h$-BN with strong $\pi$ character are highlighted in black and green (gray), respectively, where the symbol sizes represent the amount of $p_z$ character. The plotting order is total, graphene $p_z$, $h$-BN $p_z$. Right panel: The superimposed band structures of free-standing graphene and a $h$-BN monolayer. The graphene ($h$-BN) bands have been shifted up (down) by 0.08 (1.28) eV.

Figure 6

The Fermi level shift versus external electric field strength for a Cu(111)$|h$-BN$|$graphene structure with two layers of $h$-BN (black) and for the same structure with the $h$-BN layers replaced by vacuum (red). The lines represent the model described by Eq. (6) with $\kappa =2.80$ and 1.0, respectively. The symbols are the values calculated with DFT.

Figure 7

Plane-averaged electron density difference $\Delta n_{\mathrm{M}|\mathrm{BN}}(z,0)$ for Ni, Pd, Cu, and Au(111) surfaces covered with a monolayer of $h$-BN. The vertical dashed lines represent the positions of the atomic layers at the interfaces. The position $z$ is measured from the top metal atom.

Figure 8

Influence of an external electric field on the Cu(111)$|$BN interface dipole layer. The plane-averaged electron density difference $\Delta n_{\mathrm{Cu}|\mathrm{BN}}(z,E_{\mathrm{ext}})$ is shown for three different electric fields (perpendicular to the interface). The vertical dashed lines represent the positions of the atomic layers at the interface.

Figure 9

Influence of an external electric field on the BN$|$Gr interface dipole layer. The plane-averaged electron density difference $\Delta n_{\mathrm{BN}|\mathrm{Gr}}(z,E_{\mathrm{ext}})$ is shown for different electric field strengths (solid lines). The dotted and dashed lines show the polarization of isolated $h$-BN and Gr sheets, respectively, in these external fields. The vertical dashed lines represent the positions of the atomic layers at the interfaces.

Figure 10

Plane-averaged electron density difference $\Delta n_{\mathrm{tot}}(z,E_{\mathrm{ext}})$ for a Cu(111)$|$five-layers-of-$h$-BN$|$graphene structure for external electric fields $E_{\mathrm{ext}}=-E_0,0,E_{i0}$ V/Å. The vertical dashed lines represent the positions of the atomic layers at the interfaces.

Figure 11

The Fermi level shift versus the thickness $d$ of the $h$-BN film, plotted as a function of $1/\sqrt{d}$ and (inset) as a function of $d$. The symbols are the results of DFT calculations on Cu(111)$|h$-BN$|$graphene structures with 1–6 layers of $h$-BN. The dashed line represents the model calculated with Eq. (6), the dotted line represents the asymptotic $1/\sqrt{d}$ dependence.

Figure 12

The value of the intrinsic electric field $E_0$ [Eq. (7)] as a function of $1/d$, where $d$ is the thickness of the $h$-BN film. An external electric field $E_{\mathrm{ext}}^0=-E_0$ is required to restore graphene to charge neutrality. The symbols are the results of DFT calculations for Cu(111)$|h$-BN$|$graphene structures with 1–6 layers of $h$-BN.

Figure 13

The Fermi level shift versus external electric field strength for a Cu(111)$|h$-BN$|$graphene structure with 2–5 layers of $h$-BN. The lines represent the model of Eq. (6) for different thicknesses of dielectric. The symbols are values of the doping level calculated with DFT.

Figure 14

The Fermi level shift for metal$|h$-BN$|$graphene structures with two layers of $h$-BN, plotted against $V_0$, according to Eqs. (8) and (12) with $V_{\mathrm{g}}=0$ and parameters from Table . The line is for a fixed $d$, and the circles are for the equilibrium metal$|h$-BN bonding distances.

Figure 15

(a) Irreducible Brillouin zone (IBZ) of the graphene unit cell sampled with a $6\times 6$ $\mathbf{k}$-point grid. (b) The eigenvalues of free-standing graphene in the vicinity of the Dirac ($K$) point sampled with a $36\times 36$ $\mathbf{k}$-point grid. The dashed contour about the $K$ point in (a) and (b) represents a cut through the Dirac “cone” at $-$1.0 eV from the Dirac point while the solid contour in (b) represents a cut at $-$0.5 eV.

Figure 16

DoS of graphene close to the Dirac point obtained by successively doubling the linear sampling density of the $\mathbf{k}$-point grid used in the tetrahedron scheme. With increasing grid density the DoS converges to a smooth line.

Figure 17

DoS of free-standing graphene (red/gray) and of graphene on $h$-BN (black), close to the Dirac point. The solid and dashed curves were calculated using $36\times 36$ and $288\times 288$ grids, respectively.

Figure 18

Cross sections of the graphene Dirac cone at energies $-0.1$,$-0.5$, and $-1.0$ eV with respect to the charge neutrality point, as calculated using a 288$\times$288 $\mathbf{k}$-point grid. The black contours correspond to graphene on $h$-BN and the red (gray) contours to free-standing graphene. The axes are in units of $2\pi /a$.