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ABSTRACT
By virtue of substructure technique, the atomistic Green’s function method can be applied to efficiently evaluate the coherent phonon transport in a large device. Comprised of several substructures without inner atoms, the device can be described by a small system of algebraic equations, which can give a small but crucial submatrix of the retarded Green’s function of the open device without loss in accuracy. While avoiding solving the entire device problem, the method can basically give all the information to describe the quantum transport within the ballistic limit, such as the transmission function and local density of states. The validity and efficiency of the proposed method is demonstrated by the one- and three-dimensional cases. The multilevel substructure technique is also feasible.
Nomenclature
| A | = | spectral function matrix |
| a | = | lattice constant |
| D | = | effective stiffness matrix |
| G | = | Green’s function matrix |
| GD1 | = | first column of the device Green’s function block matrix, defined in Eq. (10) |
| GD3 | = | third column of the device Green’s function block matrix, defined in Eq. (10) |
| g | = | uncoupled Green’s function of contacts or the substructure’s inner part |
| I | = | identity matrix |
| i | = | unit of imaginary number |
| K | = | harmonic matrix, defined in Eq. (1) |
| K(j, j′) | = | coupling matrix between substructure j and j′ |
| K(LC,LD) | = | coupling matrix between left contact and left device region |
| K(RD,RC) | = | coupling matrix between right contact and right device region |
| k | = | element of the harmonic matrix |
| m | = | mass |
| N | = | number of substructures |
| n | = | number of the degrees of freedom |
| q | = | wave number |
| qmax | = | maximum wave number |
| U | = | bonding energy |
| u | = | displacement |
| V | = | coupling harmonic matrix |
| 0+ | = | broadening constant, which is a positive infinitesimal number |
| β | = | spring constant |
| Γ | = | broadening function, defined in Eq. (7) |
| Ξ | = | transmission function |
| ρ | = | local density of states |
| Σ | = | self-energy matrix |
| τ | = | nonzero submatrix of self-energy matrix, defined in Eq. (5) |
| ω | = | angular frequency of phonons |
| = | complex angular frequency of phonons |
| Subscripts | = | |
| C | = | contact region |
| D | = | device region |
| ID | = | intermediate device region not directly bonding with either contact |
| I, E | = | the inner part and external part |
| i, j | = | index of a degree of freedom |
| LC | = | left contact region |
| LC0 | = | 0th layer in left contact region |
| LD | = | left device region bonding with the left contact |
| RC | = | right contact region |
| RC0 | = | 0th layer in right contact region |
| RD | = | right device region bonding with the right contact |
| Superscripts | = | |
| T | = | matrix transpose |
| R | = | reduced matrix |
| (j) | = | substructure number |
| * | = | conjugate transpose of a matrix |
Nomenclature
| A | = | spectral function matrix |
| a | = | lattice constant |
| D | = | effective stiffness matrix |
| G | = | Green’s function matrix |
| GD1 | = | first column of the device Green’s function block matrix, defined in Eq. (10) |
| GD3 | = | third column of the device Green’s function block matrix, defined in Eq. (10) |
| g | = | uncoupled Green’s function of contacts or the substructure’s inner part |
| I | = | identity matrix |
| i | = | unit of imaginary number |
| K | = | harmonic matrix, defined in Eq. (1) |
| K(j, j′) | = | coupling matrix between substructure j and j′ |
| K(LC,LD) | = | coupling matrix between left contact and left device region |
| K(RD,RC) | = | coupling matrix between right contact and right device region |
| k | = | element of the harmonic matrix |
| m | = | mass |
| N | = | number of substructures |
| n | = | number of the degrees of freedom |
| q | = | wave number |
| qmax | = | maximum wave number |
| U | = | bonding energy |
| u | = | displacement |
| V | = | coupling harmonic matrix |
| 0+ | = | broadening constant, which is a positive infinitesimal number |
| β | = | spring constant |
| Γ | = | broadening function, defined in Eq. (7) |
| Ξ | = | transmission function |
| ρ | = | local density of states |
| Σ | = | self-energy matrix |
| τ | = | nonzero submatrix of self-energy matrix, defined in Eq. (5) |
| ω | = | angular frequency of phonons |
| = | complex angular frequency of phonons |
| Subscripts | = | |
| C | = | contact region |
| D | = | device region |
| ID | = | intermediate device region not directly bonding with either contact |
| I, E | = | the inner part and external part |
| i, j | = | index of a degree of freedom |
| LC | = | left contact region |
| LC0 | = | 0th layer in left contact region |
| LD | = | left device region bonding with the left contact |
| RC | = | right contact region |
| RC0 | = | 0th layer in right contact region |
| RD | = | right device region bonding with the right contact |
| Superscripts | = | |
| T | = | matrix transpose |
| R | = | reduced matrix |
| (j) | = | substructure number |
| * | = | conjugate transpose of a matrix |