Range Determination of the Influence of Carrier Concentration on Lattice Thermal Conductivity for Bulk Si and Nanowires

Mathematical modeling has been extended to simulate some physical systems to calculate some parameters that may need a sophisticated cost or may have some obstacles to be measured directly with an experimental method. In this study, the Modified Callaway Model has been used to calculate size dependence lattice thermal conductivity (LTC), and the influence of carrier concentration for bulk Si and its nanowires (NWs) with diameters of 22, 37, 56, and 115 nm has been investigated. Calculations were performed from 3K to 1600K for all cases. The effects of carrier concentration on LTC has found to begin from (10 cm) for the bulk state and that increased to (10 cm) for the NW with a diameter of 22 nm. The temperature that the maximum effect of carrier concentration can occur, has found to be at (10 K) for the bulk, and that increased to (340 K) for the (22 nm) Si NW.

It is well known that the scattering process is responsible for calculating LTC in solids, mainly phonon-phonon at high temperature, defects including electron-phonon scattering at the intermediate and at the lower temperature the boundary is controlling. However, the electronphonon scattering is the list been investigated [7][8][9]. Li et al. [10] and Hochbaum et al. [11] experimentally investigated LTC for Si NWs and found a reduction of its values for smaller size diameters. For small size diameters, similar results have also been reported for GaN and Si NWs by Mamand et al. [12] and Qader et al. [2]. On the other hand, Liao et al. [13] proved a significant effect of carrier concentration as the electron-phonon scattering process in LTC.
In this work, we try to investigate the electron concentration-effect and electron-phonon scattering interaction at a low and intermediate temperature range of LTC in Si NWs. The modified mass density for a nanoscale is also used. The method of modified Callaway model calculations will be used for bulk Si and then applied to the nanosize wires as stated in the following method of calculations:

Method of Calculations
It is obviously known that in semiconductors acoustic phonons are responsible for transferring heat for temperature gradient between two points. This is due to the value of group velocity in semiconductors is bigger than the optical phonon velocity [14]. Callaway's phenomenological theory, which is progressed under the Boltzmann transport equation, is used for LTC of a semiconductor as follows [15][16][17]: where = ( /ℏ) 3 ( /(2 2 )), ( ) = 4 ( − 1) −2 , = ℏ / ; also, and ћ are the In Eq.(3a) each 1 and 2 are the usual Debye-Callaway terms which are given by: Similarly, in Eq.(3b) 1 and 2 for the transverse phonons can be expressed as [22]: where superscript and subscript, which denoted by T and L, are the transverse and longitudinal phonons, respectively; also is Debye temperature which is split to transverse and longitudinal value. The term of ( ) is obtained by using the following relations [23]: where ( ) is the transverse (longitudinal) acoustic phonon group velocity.
The scattering process rates, which take into account in this study, are including, phononphonon (normal) scattering, ( ) inharmonic interaction or three-photon Umklapp scattering, ( ) , phonon-impurity, ( ) , phonon-electron, ℎ− ( ) , phonon-boundary, ( ) , and phonondislocation ( ) scattering. All phonon scattering processes are represented as follows [4]: In Eq. (9), the umklapp scattering is divided into longitudinal and transverse modes as: where ( ) is the umklapp parameter strength for longitudinal (transverse) mode and expressed as: where M is the average mass of an atom in the crystal, ( ) is the Gruneisen parameter that is handled to fitting the theoretical LTC curve with corresponding experimental data and normal phonon scattering has a significant effect for determining the peak value of the LTC [9]. The following mathematical expression can be used for calculating transverse and longitudinal modes of Debye temperature of bulk semiconductor [20]: with ( ) is longitudinal (transverse) zone-boundary phonon frequency. However, it is not a resistive process [22] and mathematically can be expressed as: where ( ) is the normal parameter strength for longitudinal and transverse mode, which is equal to: The phonon-impurity scattering rate due to the present impurity or isotopes in the crystal structure, can be calculated as [9]: where ( ) is the phonon scattering due to different isotopes and for each mode is given as: likewise, ( ) is the phonon scattering distribution due to impurity, which is expressed as: here S is the scattering factor, which is equal to one [24], is the concentration of impurity, and Γ is strength of the mass-difference scattering and can be found by: where is the percentage of atomic natural abundance, is the atomic mass of the ℎ isotope, ̅ is the average atomic mass, which is equal to ̅ = ∑ and V is the lattice volume.
The phonon-boundary scattering rate for bulk is assumed self-sufficient of each frequency and temperature and expressed as [ where is the effective diameter of the specimen which is found to be 3mm from the LTC fitting at low temperature. In nanocrystals, ( ) depends on the group velocity and the effective diameter of the sample, . Also, it is independent of either temperature or frequency such that: where L is the length of the sample. For absolute temperature smaller than Debye temperature the value of approaches to cross-sectional dimensions, which is known as Casimir length ( ). Furthermore, phonons scattered from grain or surface boundary of the sample, whereby, the relaxation rate for longitudinal (transverse) mode is given by [22]: The value of 1/ is the specularity parameter, which depends on the frequency of phonon and the rate of surface roughness (ε). Moreover, ε has the value from the range (0 1), with the maximum value (1) represents phonon completely specular reflection and minimum value (0) shows the phonon is completely diffuse from the surface.
A non-linear phonon scattering is necessary for phonon-dislocation interaction. Crystal including dislocation (linear defect) undergoes another phonon scattering process which is divided into short and long-range scattering. The short-range is similar to three-phonon Umklapp scattering, that phonons scatter on the center of the dislocation lines, as well as the long-range is scattering of phonon due to elastic strain field of dislocation lines [14]: scatter phonons, so the rate of phonon-electron scattering for each longitudinal (transverse) mode is as follows [25]: here is electron concentration density, is deformation potential and is the mass density.
In this work, the modified mass density for a nanoscale is used, which is not constant as in the bulk material. Therefore, nanosize dependent mass density is calculated from the following relation [26,27]: where (∞) is the bulk mass density and ( ) its nanosize dependence. The sound group velocity ( ) is a nanoscale size-dependent parameter. It can be calculated as a function of nanowires diameter from the nanosize dependent of Debye temperature ( ), according to the following [28,29]: Hence, r denoted by the nanowire radius, (∞) and (∞) are bulk state group velocity and the Debye temperature. ( ) can be calculated from the nanosize dependence of the melting temperature ( ) as in the following relation: where (∞) is the bulk melting temperature and the ( ) can be calculated depending on the (∞) and the lattice volume nanosize dependence ( ) as [30,31]: where (∞) is the melting entropy, is gas constant, , it is a critical radius in which the materials melt at (0 K) and is equal to 3h, where h is the 1 st surface layer high and is equal to where ( ) is the lattice mean bond length in Si NWs and can be calculated from the following relation [26]: where ∆ ( ) is the increase in mean bond length as a function of ( ) and calculated by the following relation [26]: Hence ∆ ( ) is the maximum increase in the mean bond length and for Si is equal to (nm) calculation for ( ), ( ), and ( ) used in the calculation for LTC in this work.

Analysis of results
The parameters that obtained for fitting the calculated LTC curves to that of the experimental data by process of trial and error are listed in Table 1. These parameters are increases with the decrease of nanowires diameter, including impurities (Nimp), dislocation (ND), carrier concentration (ne), surface roughness (ε), longitudinal and transverse Grüneisen parameter (γL and γT), are affected by the decrease of NWs diameters but in nanosize systematic.  The obtained results including  (Table 2). Fig. 1 shows the size-dependent values for all calculated parameters and LTC values as they obtained by using the data listed in Table 1 and Table 2 for Bulk Si and its NWs of diameters 22 nm, 37 nm, 56 nm, and 115 nm. For the bulk Si, Eqs. (1) to (22) were used with the bulk parameters that are tabulated in Table   3. For nanowires, the nanosize dependence of mass density, group velocity, Debye temperature, melting point, and mean bond length values were calculated through Eqs. (23) to (27). The recalculations temperature dependence of LTC with different electron concentration is shown in Fig. 2. In general and as expected, the increase of will decrease LTC, that's since an electron-phonon scattering, which shows the decrease of LTC as a function of temperature, it looks the maximum effects have occurred at temperature 10 K in bulk Si and 91, 86, 107 and 333 K in Si nanowires with diameters of 55, 37, and 22 nm, respectively. According to the results, the effects of carrier concentration decreased by decreasing the diameter of Si nanowire.
The temperature where shows its maximum effect were also investigated. The results are shown in Fig. 3. This temperature strongly depends on the type of material and the diameter of the nanowire.