Size effect on the phonon heat conduction in semiconductor nanostructures

Lattice thermal conductivity for silicon nanowires and quantum well are theoretically investigated in the temperature range from 2K to 300K. The modified Gallaway method for bulk crystal is used for calculating lattice thermal conductivity. All important phonon relaxation mechanism such as Umklapp scattering, Mass-difference scattering and boundary scattering are calculated at 300K. The result show that the modification of the acoustic phonon modes and phonon group velocities due to spatial confinement of phonons lead to significant increase in the all phonon relaxation rate. From our numerical results, we predicate a significant decrease of the lattice thermal conductivity in cylindrical Size effect on the phonon heat conduction in semiconductor nanostructures. ٦٥ nanowires with diameter (D=10-nm), and quantum well with thickness of the same size, results compared to that of the reported experimental as well as theoretical values. Introduction The physical properties of nanostructure have been investigated extensively both theoretically and experimentally due to their scientific and industrial importance. Single crystal silicon thin films and layers with thickness in the range of 1nm-100μm are widely used in modern applications such as SOI (silicon-on-insulter) device, integrated circuit transistor, and nanowires applications in the area of IR detectors and IR night vision, and thermal sensor. (Makdadi et al., 2005; Feng et al., 2003). As the size of low-dimensional materials decreases to the nanometers size range, the thermal, electronic, magnetic, optical, and thermodynamical properties of the material are significantly altered from those of either the bulk or a single molecule. (Feng et al., 2003) Additionally a large degree of solid-state material behavior depends on phonon dynamics, and these dynamics are substantially altered as nanostructure dimensions approach the phonon mean free-path length. The continuous scaling down of feature sizes in Micro-electronic and Micro-mechanical devices to nanometer sizes leads to increased power dissipation per unit area. This makes it important to understand heat conduction in various kinds of nanostructure, such as: quantum well (two dimension (2D), nanowires and nanotube(1D) and quantum dot(0D), and in particular to understand the effects of the confinement of phonons as the nanostructures size approaches the phonon mean-free-path length. Indeed, in bulk materials internal scattering dominates heat transfer processes. (Alassafee., 2005; Yang et al., 2006; Makdadi et al., 2005) For crystalline nanowires, though, as its size decreases the frequency of phonon-boundary collisions increases. In addition the ratio of the surface area to volume increases, the result is that the thermal conductivity of nanostructures differs significantly from that of bulk materials this has been shown experimentally by many author's. (Zou et al., 2000; Liang et al., 2006). In this work the dependence of thermal conductivity on the diameter of free standing nanowires and on the thickness of free standing quantum well is investigated theoretically taking into account spatial confinement of phonon induced by the boundaries which leads to altering phonon spectra. Our recent work has focused on silicon because:1) the material properties are well known. 2) The relatively high sound velocity makes it easier to observe size effect at large diameter and higher temperatures, 3Experimental measurements are currently available for silicon nanowires and quantum well. Abdurrahman Khaleel Suleiman & Abdul-Ghefar Kamil Faiq ٦٦ THEORY 1phonon Boltzmann equation:A phonon of energy ћωS(q) and velocity VS(q) in the direction of q contributes ћωS(q)VS(q) to the heat current where q represent phonon wave vector. The net phonon heat current with a small temperature gradient ∇T is given by:JQ =-ΣÑq.sћωS(q)VS(q) (1) Where subscript s refers to a particular phonon polarization type, VS(q) is the phonon group velocity, and Ñq.s =Nq.s –Nq.s is the deviation of the phonon distribution, Nq.s, from its equilibrium value, Nq.s. The equilibrium phonon distribution, Nq.is given by the Bose-Einstein distribution:1 1 ) / ) ( ( , − = ° T K q s q b s e N ω h (2) By definition JQ =-K∇T (3) Thus the problem of determining the lattice thermal conductivity is essentially that of obtaining Ñq.s. In order to do this we need to solve the Boltzmann equation for Ñq.s, (Zou et al., 2000; Callawy., 1958). In steady state, the phonon Boltzmann equation can be written as:0 ) ( ) ( , , = ∂ ∂ + ∂ ∂