Phase equilibria at ${\text{Si-HfO}}_{2}$ and ${\text{Pt-HfO}}_{2}$ interfaces from first principles thermodynamics

Phase equilibria at ${Si-HfO}_{2}$ and ${Pt-HfO}_{2}$ interfaces from first principles thermodynamics

H. Zhu, C. Tang, and R. Ramprasad

Phys. Rev. B 82, 235413 – Published 9 December 2010

Abstract

The phase diagrams of ${Si-HfO}_{2}$ and ${Pt-HfO}_{2}$ interfaces as a function of temperature and oxygen pressure have been determined using first principles thermodynamics (FPTs), i.e., by combining conventional density-functional theory results with thermodynamics. The vibrational and configurational entropic contributions to the free energies of the condensed phases are explicitly included in this treatment. We demonstrate that the predictions of the FPT approach are in quantitative agreement with experiments for the classes of interfaces considered here. In particular, under ultrahigh vacuum (UHV) conditions, we show that FPT methods predict the correct ${Si-HfO}_{2}$ silicalike interface configurations. Likewise, we also show that an interfacial oxygen coverage of 0.5–1.0 monolayer is favored under UHV conditions at the ${Pt-HfO}_{2}$ interface before rapid oxidation of Pt may be expected (for higher oxygen pressures). These results have important implications both for the applicability of FPT methods for the considered classes of interfaces as well as for “high- $K$ ” dielectrics-based electronic devices in which such interfaces are expected.

Received 9 June 2010

DOI:https://doi.org/10.1103/PhysRevB.82.235413

Authors & Affiliations

H. Zhu, C. Tang^*, and R. Ramprasad^†

Department of Chemical, Materials and Biomolecular Engineering, Institute of Materials Science, University of Connecticut, 97 North Eagleville Road, Storrs, Connecticut 06269, USA

^*Present address: Fritz-Haber-Institut der Max-Planck-Gesellschaft, Faradayweg 4-6, D-14195 Berlin, Germany.
^†rampi@ims.uconn.edu

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Vol. 82, Iss. 23 — 15 December 2010

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Figure 1
(Color online) The representative atomic structures of the $X {-HfO}_{2}$ heterostructures with $X$ at the bottom: (a) ${Si-HfO}_{2}$ interface and (b) ${Pt-HfO}_{2}$ interface. The Hf and O atoms of the top ${HfO}_{2}$ parts in both (a) and (b) are represented by light blue (gray) and red (dark) spheres, respectively.Reuse & Permissions
Figure 2
(Color online) DFT energy difference between $X : θ_{O} : {HfO}_{2}$ and $X : 0.5 : {HfO}_{2} + (2 θ_{O} - 1) O_{2}$ , $Δ E_{θ_{O}}^{DFT}$ . Open and solid squares stand for $Δ E_{θ_{O}}^{DFT}$ of ${Si-HfO}_{2}$ with ${HfO}_{2}$ under strain and Si under strain, respectively. Open and solid circles are the corresponding data for ${Pt-HfO}_{2}$ with Pt under strain and ${HfO}_{2}$ under strain, respectively.Reuse & Permissions
Figure 3
(Color online) The relative interface free energy as a function of $Δ μ_{O_{2}}$ for the (a) ${Si-HfO}_{2}$ and (b) ${Pt-HfO}_{2}$ interfaces. The inner boundaries represent the lowest interface energy at each $Δ μ_{O_{2}}$ value. The corresponding $θ_{O}$ value is indicated in the figure. The vertical dashed-dotted lines are boundaries between two different stable interface structures. The onset $Δ μ_{O_{2}}$ to form bulk ${SiO}_{2}$ and $α {-PtO}_{2}$ is indicated by the vertical dotted line in (a) and (b), respectively.Reuse & Permissions
Figure 4
(Color online) (a) The interface phase diagrams for the ${Si-HfO}_{2}$ interface, determined using Eq. (10), i.e., after neglecting the vibrational and configurational entropic contributions of the condensed phases to the relative interface free energy. (b) Same as (a), but with the vibrational entropic contribution $(Δ F_{θ_{O}}^{v})$ included. (c) Same as (a), but with the configurational entropic contribution $(Δ F_{θ_{O}}^{c})$ included. (d) The interface phase diagrams determined using Eq. (9); this is the same as (a), but with both the vibrational and configurational entropic contributions included. The solid curves indicate the interface phase boundaries and the dotted curves represent the onset of formation of bulk ${SiO}_{2}$ . The open and solid circles stand for the condition to form ${SiO}_{2}$ and SiO at the ${Si-HfO}_{2}$ interfaces in experiments, respectively (Refs. 9, 10). The gray-filled circle stands for the critical point for interfacial ${SiO}_{2}$ to decompose to SiO (Ref. 10).Reuse & Permissions
Figure 5
(Color online) (a) The interface phase diagrams for the ${Pt-HfO}_{2}$ interface, determined using Eq. (10), i.e., after neglecting the vibrational and configurational entropic contributions of the condensed phases to the relative interface free energy. (b) Same as (a), but with the vibrational entropic contribution $(Δ F_{θ_{O}}^{v})$ included. (c) Same as (a), but with the configurational entropic contribution $(Δ F_{θ_{O}}^{c})$ included. (d) The interface phase diagrams determined using Eq. (9); this is the same as (a), but with both the vibrational and configurational entropic contributions included. The solid curves indicate the interface phase boundaries and the dotted curves represent the onset of formation of bulk ${PtO}_{2}$ . The open and solid squares are the $T$ and $P_{O_{2}}$ when a 0.25 ML O-adsorbed (111) Pt surface and a clean (111) Pt surface were observed, respectively (Ref. 34).Reuse & Permissions
Figure 6
(Color online) The vibrational and configurational contributions (i.e., $Δ F_{θ_{O}}^{v} / A$ and $Δ F_{θ_{O}}^{c} / A$ , respectively) to the relative interface free energy, $Δ γ_{θ_{O}}$ [defined in Eq. (9)], as a function of temperature for the ${Si-HfO}_{2}$ and ${Pt-HfO}_{2}$ interfaces. Results pertaining to $θ_{O}$ values of 0 and 2 are shown for the vibrational contribution, and those for 0, 1, and 2 for the configurational contribution. By definition, the $Δ F_{θ_{O}}^{c} / A$ values are identical for the three coverages. The slight difference in the $Δ F_{θ_{O}}^{c} / A$ values for the ${Si-HfO}_{2}$ and ${Pt-HfO}_{2}$ interfaces are due to the difference in the interface area $A$ for these two cases.Reuse & Permissions

Physical Review B

covering condensed matter and materials physics

Phase equilibria at ${Si-HfO}_{2}$ and ${Pt-HfO}_{2}$ interfaces from first principles thermodynamics

H. Zhu, C. Tang, and R. Ramprasad

Phys. Rev. B 82, 235413 – Published 9 December 2010

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Physical Review B

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Phase equilibria at Si-HfO2 and Pt-HfO2 interfaces from first principles thermodynamics

H. Zhu, C. Tang, and R. Ramprasad

Phys. Rev. B 82, 235413 – Published 9 December 2010

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Phase equilibria at ${Si-HfO}_{2}$ and ${Pt-HfO}_{2}$ interfaces from first principles thermodynamics