Analytical Description of Acoustic Eigenmodes in Thin Films and Membranes

In the following paragraph, the analytical description of the acoustic eigenmodes, in this context called “coupled resonances”, will be derived. The cases considered here are a single layer on a substrate, and two-layered membranes. These calculations were provided by Vitalyi Gusev and provide an alternative to the numerical modeling for a better understanding of the physics of the investigated system.

The calculations start from a general three layer system, and are later reduced to the cases of interest. Figure 2.10 illustrates the considered system and the labels used. The system consists of two (thin) layers, labeled 1 and 2, and a third layer, which could be air in the case of a membrane or a substrate, labeled 3. All layers can be of different materials. In the case of membranes, the properties of layer three are set to air/vacuum later. In the following the strain in each layer is considered in the form of plane waves

2 Theoretical Framework

x

H

₂

-H

₁

1 2 3

ε

₊⁽¹⁾

e

^-ik1x

ε

₊⁽²⁾

e

^-ik2x

ε

_-⁽¹⁾

e

^ik1x

ε

_-⁽²⁾

e

^ik2x

0

Figure 2.10: Illustration of the two layers on a substrate with the respective strain waves in each layer. The coupled resonances are calculated assuming the strain distributions in each layer and their components traveling in positive (+) or negative (-) x-direction. Further, the boundary conditions at the interfaces (-H1, 0, H) have to be considered. In case of a membrane, the layer 3 is set to air/vacuum.

·e^i(ωt−kx). The time dependence (∼ e^iωt) can be neglected, as it occurs in all terms.

The strain in both layers is therefore composed of the following four components:

⁽¹⁾₊ e^−ik1x strain originated in layer 1, traveling in positive x-direction ⁽¹⁾₋ e^ik1x strain originated in layer 1, traveling in negative x-direction ⁽²⁾₊ e^−ik2x strain originated in layer 2, traveling in positive x-direction ⁽²⁾₋ e^ik2x strain originated in layer 2, traveling in negative x-direction. denotes the strain amplitude, the subscript (+) the direction of traveling and the superscript (1/2) the layer. Next, the boundary conditions at the positions -H1, 0 and H1 have to be considered. At the free surface, x=-H1, the strain gets entirely reflected, thus:

⁽¹⁾₋ =−⁽¹⁾₊ e^2ik1H1. (2.29) At the interface to the substrate, x=H2, the amount of reflected strain is given by the reflection coefficient R23

⁽²⁾₋ =R23⁽²⁾₊ e^2ik2H2. (2.30) Finally, at the interface between layers 1 and 2, the boundary conditions yield for the amplitude in layer 1:

⁽¹⁾₋ = R12⁽¹⁾₊

| {z }

reflected part of⁽¹⁾₊

+ T21⁽²⁾₋

| {z }

transmitted part of⁽²⁾₋

, (2.31)

where T21 is the transmission coefficient from layer 2 into layer 1. Combining

Equa-30

2.5 Analytical Description of Acoustic Eigenmodes in Thin Films and Membranes

tions 2.29, 2.30 and 2.31, and rewriting, one obtains for x= 0 and layer 1:

R12e^2ik1H1

⁽¹⁾₊ +T21R23⁽²⁾₊ e^−2ik2H2 = 0. (2.32) Further, at position x= 0 in layer 2, one has:

⁽²⁾₊ =R21⁽²⁾₋ +T12⁽¹⁾₊ , (2.33) so that using Equations 2.32 and 2.33 the resulting condition is:

T12⁽¹⁾₊ + R23R21e^−2ik2H2 −1

⁽²⁾₊ = 0 (2.34)

Finally, Equations 2.32 and 2.33 can be combined into one, as both are coupled. The solution of the coupled system is given when the determinant diminishes:

det(A) =

R21+e^2ik1H1 T21R23e^−2ik2H2 T12 R23R21e^−2ik2H2 −1

| {z }

A

⁽¹⁾₊ ⁽²⁾₊

!

= 0.! (2.35)

The acoustic mismatch between two layers is defined as Z12 = ρ₁v₁

ρ₂v₂ = Z1

Z2

, and thus the reflection and transmission coefficients are

R12 = 1−Z12

1 +Z12

and T12= 2 1 +Z12

,

in accordance to Equations 2.14 and 2.15. Using further the definition R12 =−R21 =R, and calculating the determinant of Equation 2.35, it follows that:

R23Re^2ikH1 +Re^2ik−2H2 +e^{2i(k2H2+k1H1)}=R23

= 0.! (2.36)

This equation can be separated into a real and an imaginary part, so that:

R[R23cos (2k₁H1) + cos (2k₂H2)] + cos [2(k₂H2+k₁H1)] = 0^! (2.37) R[R23sin (2k₁H1) + sin (2k₂H2)] + sin [2(k₂H2+k₁H1)] = 0^! (2.38) where both equations have to be valid. The results in the following section will be deducted from these equations.

2 Theoretical Framework

2.5.1 Eigenmodes of Two-Layer Membranes

In the absence of a substrate, namely the membrane case, the reflection coefficient R23

is set to -1. After some calculations, one can derive the condition R+ 1

R−1tan(k₂H2) = tan(k₁H1). (2.39) This condition has to be fulfilled in order to fulfill Equations 2.37 and 2.38. Equation 2.39 can be reduced to

i one can derive the final equation tan

The roots of Equation 2.43 give the frequencies of the eigenmodes of the two-layered membrane.

Influence of the Acoustic Mismatch

Equation 2.43 will yield the frequencies of the eigenmodes of the two-layered membrane, i.e. the acoustic standing waves in the system. One can now consider two different

“extremes" of the acoustic mismatch between both layers, which will give good estimates of the behavior of the acoustic spectrum, i.e. no acoustic mismatch difference between both layers Z12 = 1 or a strong mismatch: Z121 and Z121.

The first case, Z12 = 1, results in the eigenmodes of a single layer membrane with thickness d = H1 +H2 in the analytical picture. In reality one still has to model the interface properties, e.g. adhesion between both layers, to consider the deviation between theory and experiment.

Open Pipe Oscillations If the top layer (1) is acoustically harder than the bottom layer, i.e. Z12 1 or Z1 Z2, Equation 2.42 can be rewritten as

tank₁H1 = 0.

32

2.5 Analytical Description of Acoustic Eigenmodes in Thin Films and Membranes

x

-H

₁

0

closed pipe

open pipe strain

strain

Figure 2.11: Illustration of the strain distribution in layer one (refers labeling of Figure 2.10) for open and closed pipe oscillations. If the layer one is acoustical harder than layer two, i.e. Z12 1, the fundamen-tal mode fits half a wavelength in the layer, similar to a standing wave with two open ends.Then the strain is zero at the interfaces. The fundamental mode fits quarter of a wavelength in the layer if Z121, or one interface (x= 0) is fixed, thus closed pipe resonances.

Here the stain is zero only at the open end.

This equation is valid fork₁H1

=! π·n, withn= 0,±1,±2, . . .. The obtained resonances are ^λ₂-resonances, and the frequencies follow the relation

fn =n· v₁ 2H1

. (2.44)

This is equal to the results of a single layer membrane of thickness H1. The resonances in the bottom layer (2) follow the conditions derived in the following paragraph, see Equation 2.45. The fundamental mode in this case fits half a wavelength in the layer, as depicted in Figure 2.11, as it can be described similar to a standing wave with two open ends, thus the name open pipe resonances (see for example organ pipes).

Closed Pipe Oscillations The last case to consider is the case where the bottom layer is the acoustically harder layer, i.e. Z12 1 or Z1 Z2. Again one can rewrite Equation 2.42, obtaining

tank₁H1 =∞.

The condition for the roots is k₁H1

=! ^π₂ ·(n+ 1), withn = 0,±1,±2, . . .. The obtained resonances are therefore ^λ₄-resonances, and follow the frequency relation

f_n =n· v₁ 4H1

. (2.45)

The frequencies in layer 1 follow in this case the condition of Equation 2.44. The fundamental mode fits quarter of a wavelength in the layer, as depicted in Figure 2.11, and can be described similar to a standing wave with one closed end, thus the name closed pipe resonances.

2 Theoretical Framework

2.5.2 Single Layer on Semi-Infinite Substrate

The second case interesting for this thesis is the case of a thin layer (∼nm) on a thick substrate (∼µm to mm). This can be achieved in the calculations by setting the layers 2 and 3 equal. The result is a single layer on a semi-infinite substrate, i.e. the case when the eigenmodes of the thin film are not notably perturbed by the "eigenmodes” of the bottom layer, as found for example in the case of a 10 nm gold film on a 3 mm thick diamond substrate.

In the case of the elastic substrate, complex frequencies have to be used in order to account for the losses to the substrate, hereω =ω⁰+iω⁰⁰. The Equations 2.37 and 2.38 in the same frequency dependence as for the membrane. In the first case, Z12 1, Equation 2.47 vanishes only for the condition

sin −H1

v₁ ω⁰

= 0,!

as the second term cannot become zero, resulting in the frequency dependance f_n =n· v₁

2H1

, (2.48)

whereas for the second case, Z12 1, results again in the condition for the modes is given by

f_n =n· v₁ 4H1

. (2.49)

Having complex frequencies, it is possible to derive the relation between reflection coef-ficient R12 and damping rate from Equation 2.46 as

|R12|=e^−2ω00Hv11,

illustrating the relation between the frequency dependent loss of acoustic energy and the acoustic mismatch of the layer and the substrate.

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Im Dokument High-Frequency Acoustic Phonons in Confined Geometries (Seite 41-47)