The elasticity and piezoelectricity of AlN containing charged vacancies

Figures & data

Figure 1. Atomic geometry of pure and native point defects in 32-atom AlN in the ideal (unrelaxed) structure. (a) Pure AlN, (b) V_N-AlN and (c) V_Al-AlN. Large blue and small gray spheres represent Al an N atoms, respectively. Vacancies are indicated by the red circles.

Figure 2. Formation energies of vacancies with different charge states in AlN as calculated by DFT. Formation energies as a function of the Fermi level is shown for the extreme growth conditions: 32-atom supercell under (a) Nitrogen rich and (b) Aluminum rich; 72-atom supercell under (c) Nitrogen rich and (d) Aluminum rich.

Figure 3. The density of states of bulk AlN (a) and the 32-atoms supercell containing an Al or N vacancy (b–e). The density of states of bulk AlN (f) and the 72-atoms supercell containing an Al or N vacancy (g–j). Fermi level is set to zero.

Figure 4. The three-dimensional (3D) charge density difference of AlN with vacancy defects. (a) AlN, (b) ${\mathrm{V}}_{\text{Al}}^{3-}$–AlN, (c) ${\mathrm{V}}_{\mathrm{N}}^{1+}$–AlN and (d) ${\mathrm{V}}_{\mathrm{N}}^{3+}$–AlN.

Figure 5. The elastic coefficients C₁₁, C₁₂, C_13, C₃₃, C₄₄ and C₆₆ of ${\mathrm{V}}_{\text{Al}}^{3-}$–, ${\mathrm{V}}_{\mathrm{N}}^{1+}$– and ${\mathrm{V}}_{\mathrm{N}}^{3+}$–AlN.

Figure 6. Young’s modulus and piezoelectric strain coefficients d₃₃ of pure AlN and different charged state defectants.

Table 1. d₃₃, d₃₃ Increase rate and E of AlN-based piezoelectric materials.

Materials	d33 (pC/N)	d33 increase rate	E (GPa)	Ref
Pure AlN	∼5.120	–	∼310.000	^[^20^,^83^,^84^]
${\mathrm{V}}_{\text{Al}}^{3-}$–AlN	34.821	6.916	237.801	–
${\mathrm{V}}_{\mathrm{N}}^{1+}$–AlN	243.692	48.400	120.535	–
${\mathrm{V}}_{\mathrm{N}}^{3+}$–AlN	100.993	20.058	222.859	–
ScxAl1-xN	24.600	∼4.731	–	^[^84^]
ScxAl1-xN	–	–	217.100	^[^85^]
(MgHf) xAl1-xN	16.417	3.205	∼208.000	^[^20^]
(MgZr) xAl1-xN	–	∼2.855	–	^[^86^]
(MgTi) xAl1-xN	–	∼2.498	–	^[^86^]
(MgHf) xAl1-xN	–	∼1.988	–	^[^86^]
Cr xAl1-xN	–	1.732	–	^[^87^]
Ta xAl1-xN	8.200	1.952	–	^[^83^]

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