Origin of Enhanced Electromechanical Coupling in (Yb,Al)N Nitride Alloys

Our experiments demonstrate that alloying the cubic-phase $\mathrm{Yb}\mathrm{N}$ into the wurtzite-phase $\mathrm{Al}\mathrm{N}$ results in clear mechanical softening and enhanced electromechanical coupling of $\mathrm{Al}\mathrm{N}$. First-principle calculations reproduce experimental results well, and predict a maximum 270% increase in electromechanical coupling coefficient caused by (1) an enhanced piezoelectric response induced by the local strain of $\mathrm{Yb}$ ions and (2) a structural flexibility of the $(\mathrm{Yb},\mathrm{Al})\mathrm{N}$ alloy. Extensive calculations suggest that the substitutional neighbor $\mathrm{Yb}$-$\mathrm{Yb}$ pairs in wurtzite $\mathrm{Al}\mathrm{N}$ are energetically stable along the $c$ axis, and avoid forming on the basal plane of the wurtzite structure due to the repulsion between them, which explains why $(\mathrm{Yb},\mathrm{Al})\mathrm{N}$ films with high $\mathrm{Yb}$ concentrations are difficult to fabricate in our sputtering experiments. Moreover, the neighbor $\mathrm{Yb}$-$\mathrm{Yb}$ pair interactions also promote structural flexibility of $(\mathrm{Yb},\mathrm{Al})\mathrm{N}$, and are considered a cause for mechanical softening of $(\mathrm{Yb},\mathrm{Al})\mathrm{N}$.

Figure 1

The $c$-lattice spacing (a) and density (b) of $(\mathrm{Yb},\mathrm{Al})\mathrm{N}$ films from experimental measurement and theoretical calculation. The solid squares represent the measurement values, and the solid circles show the calculated values. Note that $(\mathrm{Yb},\mathrm{Al})\mathrm{N}$ films with high $\mathrm{Yb}$ concentrations ($x>0.25$) are difficult to fabricate by sputtering.

Figure 2

(a) The six pair configurations in the $3\times 3\times 2$ supercell (purple, $\mathrm{Al}$ atoms; brown, $\mathrm{N}$ atoms). A $\mathrm{Yb}$-$\mathrm{Yb}$ pair ($n$) is defined as the two $n\mathrm{th}$ neighbor substitutional $\mathrm{Yb}$ atoms relative to a $\mathrm{Yb}$ atom located at the origin “0.” (b) The pair interaction energies $\mathrm{\Delta }{E}^{(n)}$ for $n=1,2,3,4,5,6$ neighbor $\mathrm{Yb}$ pairs. For comparison, $\mathrm{\Delta }{E}^{(n)}$ in Sc-doped $\mathrm{Al}\mathrm{N}$ is also calculated.

Figure 3

(a) The calculated $k_{33}^2$ values are compared with the experimentally determined $k_t^2$ for $(\mathrm{Yb},\mathrm{Al})\mathrm{N}$ films with different rocking curve FWHM values. The smaller the FWHM value, the stronger the $c$-axis orientation. (b) The calculated $V_{33}$ and $V_{44}$ values are compared with the experimentally measured $V_{33}^D$ and $V_{44}^E$ for $(\mathrm{Yb},\mathrm{Al})\mathrm{N}$ films with $\mathrm{FWHM}<3^{\circ }$, respectively. Connected lines represent the calculated values, whereas the scattered points show the experimentally measured ones. In (a), the open square symbol represents the measurement value for the general pure $\mathrm{Al}\mathrm{N}$ FBAR with $\mathrm{Mo}$ electrodes, and the asterisk shows the $k_t^2$ value for $(\mathrm{Sc},\mathrm{Al})\mathrm{N}$ at $x=43\mathrm{\%}$ [9].

Figure 4

Calculated contributions of $e_{33}$ introduced in Eq. (3). (a) The clamped-ion contribution and (b) the ionic contribution, which is related to the strain effect and the Born charges $Z_{33}$ calculated from Eq. (3).

Figure 5

Compositional weighted and site-resolved internal strain sensitivity and Born effective charges of $(\mathrm{Yb},\mathrm{Al})\mathrm{N}$.

Figure 6

Energy landscape of wurtzite $(\mathrm{Yb},\mathrm{Al})\mathrm{N}$ ($x=0.33$).

Figure 7

Local structures of $\mathrm{Yb}$ and $\mathrm{Al}$ atoms in the relaxed $(\mathrm{Yb},\mathrm{Al})\mathrm{N}$ alloy system ($x=0.33$ and $c/a=1.52$). The index of nitrogen atoms in the local structure is given. The deformed bipyramidal structure in (b) is shown as a visual guide.

Figure 8

Charge density differences of $(\mathrm{Yb},\mathrm{Al})\mathrm{N}$ with $x=0.33$ under the compressed state of $c/a=1.40$. (a) An $a$-$b$ basal plane is selected, and (b) an $a$-$c$ plane is selected to show charge density difference along the $c$ axis. The loss of electrons is indicated in light blue and electron accumulation is indicated in yellow. Each isovalue is set to 10% of the maximum. (purple, $\mathrm{Al}$ atoms; blue, $\mathrm{Yb}$ atoms; brown, $\mathrm{N}$ atoms).

Figure 9

Calculated pair interaction energies $\mathrm{\Delta }{E}^{(n)}$ at different $c/a$ ratios: (a) the lattice spacing along the $a$ axis is fixed, and (b) the lattice spacing along the $c$ axis is fixed. Here $\mathrm{\Delta }{E}^{(3)}$ is the pair interaction energy due to the neighbor pair along the $c$ axis, and $\mathrm{\Delta }{E}^{(2)}$ and $\mathrm{\Delta }{E}^{(6)}$ are the pair interaction energies due to the neighbor pairs on the $a$-$b$ basal plane in wurtzite $(\mathrm{Yb},\mathrm{Al})\mathrm{N}$. A $3\times 3\times 2$ $\mathrm{Al}\mathrm{N}$ structure with the lattice parameters of $(\mathrm{Yb},\mathrm{Al})\mathrm{N}$ ($x=0.33$) is selected as the reference, and a $3\times 3\times 3k$ mesh is used for all the calculations.