Original Atomistic Green’s Function Methods

• The Atomistic Green’s Function methods is based on the lattice dynamics of a finite body coupled to a semi-infinite thermal reservoir.
• It used the force constant matrices to compute the atomistic Green’s function of the finite body
• In the original form of the AGF, the transmission function is a summation of the transmittances of all the phonon modes for a given frequency
• The original AGF method solves the phonon transmission problem in terms of the surface Green’s functions of the lead as well as the coupling between the channel and the leads.
• The phonon transmission is calculated from the retarded Green function of the center
  • Self-energy terms from coupling to the left and right lead
  • The transmission is a summation of all the phonon modes for a given frequency

• Caroli formula: transmission function

• Given the harmonic matrix $H$ of a close system, the following eigenvalue equation is satisfied: \([ \omega ^ 2 I -H] \phi = 0\)

• When the reservoirs or contacts are held at a constant temperature by an external heat source or sink (open system), the eigenvalue equation became to: \([ (\omega^2 + i 0^+)I - H] \phi = s\) + here $0^+$ is a small positive infinitesimal factor that introduces damping in the open system

• The total harmonic matrix of the coupled system is: \(H_{\mathrm{tot}} = \begin{bmatrix} H_1 & \tau_1^{+} & 0 \\ \tau_1 & H_d & \tau_2 \\ 0 & \tau_2^{+} & H_2\\ \end{bmatrix}\)
  • The size of different matrix is related to the degree-of-freedom of different region

    Externded Atomistic Green’s Function Method

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What and How:

• The extended method is based on the concept of the Bloch matric
• It allow us to determine the wavelength and polarizatoin dependence of the phonon transmission

• Bloch matrix: describe the phase change of the eigenmodes as they propagate from one rincipal layer to another and is associated with the translational symmetry of the leads.

• Lack of knowledge of phonon transmission coefficients between specific modes on each side of the interface and how the coefficients are affected by atomic scale structure.
• Precise transmission and reflection processes that lead to thermal resistance and how they depend on phonon frequency as well as incident angle.
  • The process can be partly explained with familiar concepts such as conservation of transverse momentum.
• Mode-resolved formalism of the AGF
  • transmission coefficient from a specific mode in a material to a specific mode in the other material.