Synopsis: GNRphonon(dimer)

GNRphonon is the class for the armchair nanoribbon used for the computation of the fullband electron and phonon spectra and of the momentum relaxation rates (within the Deformation Potential Approximation (DPA)) corresponding to absorption and emission of both acoustic and optical phonons.

The attributes of the classes are the following:

**N**:**(int)**the number of grid points along the longitudinal direction (kx) in the K-space. Note that the number of points to be used in the computation of scattering rates, in order to reduce*noise*, strongly depends on the value of the acoustic deformation potential Dac, which is an attribute of the class GNRphonon. For example, if Dac=16 eV, N=16000 can be sufficient, whereas by using Dac=4.5 eV, N ranging from 140000 to 250000 could be required. Attention has to be posed also to the*memory storage*, which increases with increasing the GNR width W and it poses severe limit to the maximum possible value of N. Finally let us observe that in “phonon_GNR.c” a grid is also defined in the qy direction in order to compute the GNR phonon subbranches (# of points: Ny). Here Ny (> dimer) is imposed equal to 140, but for some GNR widths it is sufficient Ny smaller than 140.

In the following table we give advice for the values of N and Ny required for different GNR widths W (Dac=4.5 eV) and compatible to the available RAM, if the RAM for each core is 2 GigaBytes (and assuming that both LA and TA modes are taken into account by means of an effective acoustic deformation potential):

# of dimer lines |
10(W=1.12 nm) | 22(W=2.62 nm) | 40(W=4.86 nm) | 82(W=10.10 nm) |

N |
250000 | 250000 | 250000 | 140000 |

Ny |
80 | 140 | 80 | 100 |

**dimer : (int)**the number of dimer lines of the armchair GNR**rank : (int)**the rank of the process**phi : (double)**the channel potential, also known as the midgap energy

**numberAC : (int)**the number of acoustic (AC) modes of different simmetry (LA,TA) which we want to simulate. Formally, the DPA leads to a zero coupling to the TA modes. Electron-phonon coupling is different from zero only for LA, LO and TO modes within DPA. Anyway their effects on electron transport can be introduced by using an effective deformation potential Dac. If**numberAC=2**both LA and TA modes are taken into account in the computation of the momentum relaxation rates. Instead if**numberAC=1**only LA modes are considered

**Ecutoff : (double)**the maximum electron energy used for the computation of scattering rates, i.e. rates are only computed for electron energies smaller than Ecutoff (in eV)**delta : (int)**the sampling along the kx grid used in the computation of scattering rates, i.e. rates are calculated only every delta points along the kx direction**deltak : (double)**(the discretization step of the kx grid) x (delta)**kyE: (numpy array of length dimer)**the quantized transverse electron wavevector**qy : (numpy array of length dimer)**the quantized transverse phonon wavevector. Note that the same boundary conditions are imposed to both the phonon and electron wavefunctions**kx : (numpy array of length N)**the longitudinal electron wavevector**qx : (numpy array of length N)**the longitudinal phonon wavevector**qx0 : (double)**the longitudinal phonon wavevector at which the graphene branches are computed**qy0 : (double)**the transverse phonon wavevector at which the graphene branches are computed**kxup : (double)**the maximum phonon wavevector associated to a rank when computing the scattering rates**kxdown: (double)**the minimum phonon wavevector associated to a rank when computing the scattering rates**dim1 : (int)**the number of entries in the first index (dimension) of a matrix double** which represents the phonon energy (energyP2D). The first index of energyP2D runs over each qx value and therefore dim1=N**dim2 : (int)**the number of phonon subbands which arise from a particular graphene phonon branch, and therefore dim2=dimer**dim3 : (int)**the number of graphene phonon branches (dim3=6). Note that one half of the GNR phonon subbranches is acoustic and the other one is optical**mmin : (int)**the minimum quantum electron index considered in the computation of the scattering rates. Its value is univocally defined by Ecutoff**mmax : (int)**the maximum quantum electron index considered in the computation of the scattering rates. Its value is univocally defined by Ecutoff**kxmin : (double)**the minimum longitudinal electron wavevector kx considered for the computation of the scattering rates. Its value is univocally defined by Ecutoff**kxmax : (double)**the maximum longitudinal electron wavevector qy considered for the computation of the scattering rates. Its value is univocally defined by Ecutoff**Phi_r1, Phi_ti1, Phi_to1 : (double)**force constant parameters for the 1th-nearest neighbors extracted from DFT calculations (L. Wirtz and A. Rubio, Solid State Communications vol. 131, pp. 141-152 (2004)**Phi_r2, Phi_ti2, Phi_to2 : (double)**force constant parameters for the 2th-nearest neighbors extracted from DFT calculations (L. Wirtz and A. Rubio, Solid State Communications vol. 131, pp. 141-152 (2004)**Phi_r3, Phi_ti3, Phi_to3 : (double)**force constant parameters for the 3th-nearest neighbors extracted from DFT calculations (L. Wirtz and A. Rubio, Solid State Communications vol. 131, pp. 141-152 (2004)**Phi_r4, Phi_ti4, Phi_to4 : (double)**force constant parameters for the 4th-nearest neighbors extracted from DFT calculations (L. Wirtz and A. Rubio, Solid State Communications vol. 131, pp. 141-152 (2004)**energyE : (numpy matrix of length Nx(2*dimer))**the electron energy matrix where the first index runs over the qx value, whereas the second index runs over the quantum electron subband (conduction, the first dimer index and valence, the last ones) index**energyP2D :****(numpy matrix of length Nx(dimer*6))**the phonon energy matrix where the first index runs over the qx value, whereas the second index runs over the quantum phonon subband index corresponding to acoustic and optical phonons. The order is the following: ZA(0)-TA(0)-LA(0)-ZO(0)-TO(0)-LO(0)-ZA(1)-TA(1)-……., where the number between round brackets is the order of the phonon mode.**minAC :****(numpy matrix of length dimerx3)**the energy matrix which includes the minimum phonon energy for each acoustic phonon subbranches. The second index runs over all acoustic graphene branches (ZA-TA-LA) from which the GNR phonon subbranches arise**Egraphene :****(numpy array of length 6)**the energy array which includes the graphene phonon energies (ZA-TA-LA-ZO-TO-LO) in correspondence of the (kx,ky) points**rateAA : (numpy matrix of length Nxdimer)**the matrix corresponding to the momentum relaxation rates for ABSORPTION of ACOUSTIC phonons. The first index runs over the qx points, whereas the second index runs over the GNR conduction electron subbands

**rateAE : (numpy matrix of length Nxdimer)**the matrix corresponding to the momentum relaxation rates for EMISSION of ACOUSTIC phonons. The first index runs over the qx points, whereas the second index runs over the GNR conduction electron subbands**rateOA : (numpy matrix of length Nxdimer)**the matrix corresponding to the momentum relaxation rates for ABSORPTION of OPTICAL phonons. The first index runs over the qx points, whereas the second index runs over the GNR conduction electron subbands**rateOE : (numpy matrix of length Nxdimer)**the matrix corresponding to the momentum relaxation rates for EMISSION of OPTICAL phonons. The first index runs over the qx points, whereas the second index runs over the GNR conduction electron subbands**electron_GNR : (function)**function which computes the GNR electron (conduction and valence) subbands**phonon_GNR : (function)**it computes the GNR phonon subbranches using the fourth-nearest neighbor force constant (4NNFC) approach**phonon_graphene : (function)**it computes the graphene branches in correspondence of the (kx,ky) point**rateACABS : (function)**it computes the momentum relaxation rates for scattering involving the ABSORPTION of ACOUSTIC phonons**rateACEM : (function)**it computes the momentum relaxation rates for scattering involving the EMISSION of ACOUSTIC phonons**rateOPTABS : (function)**it computes the momentum relaxation rates for scattering involving the ABSORPTION of OPTICAL phonons**rateOPTEM : (function)**it computes the momentum relaxation rates for scattering involving the EMISSION of OPTICAL phonons**Dac : (double)**deformation potential value (in eV)**temp : (double)**temperature (in kelvin)**thop : (double)**hopping parameter (in eV)**aCC : (double)**lattice constant (in meter)