Elastic constants of single-crystal $\mathrm{Ti}{\mathrm{N}}_x\left(001\right)(0.67⩽x⩽\mathrm{1.0})$ determined as a function of $x$ by picosecond ultrasonic measurements

The elastic constants $c_{11}$ and $c_{44}$ of single-crystal NaCl-structure $\delta \text{−}\mathrm{Ti}{\mathrm{N}}_x\left(001\right)$ layers, with $x$ ranging from 0.67 to 1.0, were determined using sound velocity measurements. Picosecond ultrasonic optical pump/probe techniques were employed to generate and detect longitudinal sound waves and surface acoustic waves (SAW) in order to obtain $c_{11}\left(x\right)$ and $c_{44}\left(x\right)$, respectively. SAW generation was achieved by depositing a periodic series of Al bars on the $\mathrm{Ti}{\mathrm{N}}_x\left(001\right)$ layers to spatially modulate the surface reflectivity. $c_{11}$ and $c_{44}$ were found to decrease continuously from 626 and $156\mathrm{GPa}$ with $x=1$ to 439 and $92\mathrm{GPa}$ with $x=0.67$. The Voit–Reuss–Hill average aggregate elastic moduli $E_{\mathrm{VRH}}\left(x\right)$ obtained from our measured $c_{11}\left(x\right)$ and $c_{44}\left(x\right)$ values are in good agreement with previous $\mathrm{Ti}{\mathrm{N}}_x\left(001\right)$ nanoindentation results.

Figure 1

Mass density $\rho$ of epitaxial $\mathrm{Ti}{\mathrm{N}}_x$ layers grown on MgO(001) as a function of $x$.

Figure 2

AFM image of an Al grating formed on $\mathrm{Ti}{\mathrm{N}}_x\left(001\right)$ layers. The grating consists of 150 parallel $15\text{−}\mathrm{nm}$-thick Al bars of width $220\mathrm{nm}$ and separated by $180\mathrm{nm}$ (the wavelength $\Lambda =400\mathrm{nm}$). The inset is a schematic illustration of the sample geometry.

Figure 3

(a) A typical plot of the change in reflectivity $\Delta R$ as a function of the time delay $t$ between laser pump and probe pulses incident on a blanket-Al-coated stoichiometric TiN(001) sample. The solid triangles indicate $\Delta R$ peaks corresponding to acoustic pulses reflected from the $\mathrm{Al}∕\mathrm{Ti}{\mathrm{N}}_x$ interface. The open triangles mark $\Delta R$ peaks corresponding to additional reflections from $\mathrm{Ti}{\mathrm{N}}_x∕\mathrm{Mg}\mathrm{O}$ $\left(▿\right)$ and $\mathrm{Al}∕\mathrm{Ti}{\mathrm{N}}_x$ $\left(▵\right)$ interfaces as shown schematically in (b). (c) A plot of $\Delta R$ as a function of the time delay $t$ for a SAW propagating perpendicular to a ⟨100⟩-oriented Al grating on $\mathrm{Ti}{\mathrm{N}}_{0.67}\left(001\right)$.

Figure 4

(a) Measured longitudinal sound velocity $v_1$ and (b) surface acoustic wave (SAW) velocities $v_{\mathrm{SAW}}^{100}$ and $v_{\mathrm{SAW}}^{110}$ along ⟨100⟩ and ⟨110⟩ directions in epitaxial $\mathrm{Ti}{\mathrm{N}}_x\left(001\right)$ as a function of $x$.

Figure 5

$\mathrm{Ti}{\mathrm{N}}_x\left(001\right)$ elastic constants (a) $c_{11}$ and (b) $c_{44}$ plotted as a function of $x$. (c) The Voigt–Reuss–Hill elastic moduli $E_{\mathrm{VRH}}$, calculated from our $c_{11}\left(x\right)$ and $c_{44}\left(x\right)$ results, and nanoindentation $\left(E_n\right)$ elastic moduli (Ref. 20) vs $x$. The dashed lines in all three panels were calculated using Bruggeman effective medium theory (EMT).