Time-domain Thermoreflectance - Modeling Materials

Modeling Materials

The surface temperature of a single layer

The frequency domain solution for a semi-infinite solid which is heated by the source (frequency w) can be expressed by the following equation.

where (1)

(Λ: thermal conductivity of the solid, D: thermal diffusivity of the solid, r: radial coordinate) Hankel Transform(The Hankel transform is an integral transform equivalent to a two-dimensional Fourier transform with a radially symmetric integral kernel, here g(r) is a radially symmetric.) will be effective, because the laser beam can be assumed as cylindrical shape. By definition of Hankel transform and using Eq. (1), (2)

Also, the pump and probe beams used here have Gaussian distribution. So, the 1/e2 radius of the pump and probe beam are w0 and w1 respectively. (3)

(4)

(5)

(6)

(p(r): Gaussian distribution factor of pump beam, P(k): Hankel transform of p(r), Θ(r): The distribution of temperature oscillations at the surface (Inverse transform of P(k)G(k)), ΔT: a weighted average of the temperature distribution Θ(r))

The surface temperature of a layered structure

In the similar way, frequency domain solution for the surface temperature of a layered structure can be acquired. Instead of Eq. (2), Eq. (7) will be used for a layered structure.

(7)

\left( \begin{array}{c} B^+ \\ B^- \end{array} \right)_{n} = \frac {1} {2 \gamma_n} \left( \begin{array}{cc} exp(-u_n L_n) & 0 \\ 0 & exp(u_n L_n) \end{array} \right) \left( \begin{array}{cc} \gamma_n + \gamma_{n+1} & \gamma_n - \gamma_{n+1} \\ \gamma_n - \gamma_{n+1} & \gamma_n + \gamma_{n+1}\end{array} \right)\left( \begin{array}{c} B^+ \\ B^- \end{array} \right)_{n+1}

(Λn: thermal conductivity of nth layer, Dn: thermal diffusivity of nth layer, Ln: thickness of nth layer) Using Eqs. (6) and (7), we can calculate the changes of temperature of a layered structure.

Modeling of data acquired in TDTR

The acquired data from TDTR experiments are required to be compared with the model.

(8) (9)

(10)

(Q: quality factor of the resonant circuit) This calculated Vf/V0 would be compared with the measured one.

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