Resonator element, resonator, electronic device, electronic apparatus, and moving object
A resonator element includes a vibrating portion that vibrates in a thickness shear vibration and includes a first main surface and a second main surface which are in a front and back relationship to each other, a first excitation electrode that is provided on the first main surface, and a second excitation electrode that is provided on the second main surface, and an energy trapping coefficient M satisfies a relationship of 33.6≦M≦65.1.
1. A resonator element comprising:
a vibrating portion that vibrates in a thickness shear vibration and includes a first main surface and a second main surface which are in a front and back relationship to each other;
a first excitation electrode that is provided on the first main surface; and
a second excitation electrode that is provided on the second main surface,
wherein an energy trapping coefficient M, a frequency decrease amount Δ, a cut-off frequency fs, and a frequency fe satisfy relationships of
33.6 ≦M≦ 65.1,
M=K ×( hx/ 2 ×ts )×√Δ,
Δ=( fs−fe )/ fs,
fs=R/ts , and
fe=R/[ts+te ×(ρ e/ρx )],
where M is an energy trapping coefficient, K is an anisotropy coefficient of the vibrating portion, hx is a length along the thickness shear vibration direction of the first excitation electrode and the second excitation electrode, is is a thickness of the vibrating portion, Δ is a frequency decrease amount, fs is a cut-off frequency of the vibrating portion, fe is a frequency when the first excitation electrode and the second excitation electrode are disposed in the vibrating portion, R is a frequency integer of the vibrating portion, te is a sum of the thicknesses of the first excitation electrode and the second excitation electrode, ρe is densities of the first excitation electrode and the second excitation electrode, and ρx is a density of the vibrating portion.
2. The resonator element according to claim 1 ,
wherein the energy trapping coefficient M satisfies a relationship of 33.6 M 46.7.
3. The resonator element according to claim 1 ,
wherein the vibrating portion is a quartz crystal substrate.
4. The resonator element according to claim 3 ,
wherein the quartz crystal substrate is an AT cut quartz crystal substrate.
5. The resonator element according to claim 1 ,
wherein when the length along the direction orthogonal to the thickness shear vibration direction of the first excitation electrode and the second excitation electrode is set to hz,
a relationship of 1.25≦hx/hz≦1.31 is satisfied.
6. The resonator element according to claim 1 ,
wherein the first excitation electrode and the second excitation electrode are respectively formed of a two-layer structure in which a first layer and a second layer are laminated in this order from the vibrating portion side,
wherein the first layer includes nickel (Ni), and
wherein the second layer includes gold (Au).
7. A resonator comprising:
the resonator element according to claim 1 ; and
a package that accommodates the resonator element.
8. An electronic device comprising:
the resonator element according to claim 1 ; and
a circuit.
9. An electronic apparatus comprising:
the resonator element according to claim 1 .
10. A moving object comprising:
the resonator element according to claim 1 .