Effective work function of metals interfaced with dielectrics: A first-principles study of the Pt-HfO${}_2$ interface

A parameter-free methodology to determine the effective work function, defined as the work function of the metal on the dielectric side of a metal-dielectric interface, is presented. This method relies on the direct determination of the effective work function through the interfacial dipole moment from standard density functional theory calculations devoid of scaling the band gap and band offset. For the case of a Pt-HfO${}_2$ interface, this strategy is combined with statistical thermodynamic principles to predict the most probable effective work function at given combinations of temperature and ${\mathrm{O}}_2$ pressure (through determination of the most favored interfacial O coverage). The predicted results are in excellent agreement with observations.

Figure 1

Schematic of (a) the macroscopic band structure of a metal slab and (b) the band alignment in a metal-oxide stack. $E_{\text{vac},m}$ and $E_F$ are, respectively, the vacuum level and Fermi level of the metal. $E_{\text{vac},o}$, CBM, VBM, $\chi$, and $E_g$ are, respectively, the vacuum level, conduction band minimum, valence band maximum, electron affinity, and band gap of the oxide. VBO stands for valence band offset, and represents the energy difference between the metal Fermi level and the oxide VBM. $D$${}_m$ and $D$${}_o$ are the surface dipole moments of the metal free surface and the oxide free surface, respectively. $D$${}_x$ is excess interfacial dipole moment, i.e., the total interfacial dipole moment minus the metal free surface dipole moment.

Figure 2

(a) Atomic structures of Pt:${\theta }_{\mathrm{O}}$:HfO${}_2$ interfaces with ${\theta }_{\mathrm{O}}$ = 0, 1, and 2 ML; (b) The interface phase diagram for Pt:${\theta }_{\mathrm{O}}$:HfO${}_2$. The boundaries between two different stable interface structures are represented by the black curves. The shaded region represents the regime of expected processing conditions.

Figure 3

${\mathrm{\varphi ̄}}_{\text{eff}}$ as a function of temperature ($T$) and ${\mathrm{O}}_2$ pressure ($P_{{\mathrm{O}}_2}$). The shaded region represents the regime of expected processing conditions.