Tham khảo tài liệu 'heat transfer mathematical modelling numerical methods and information technology part 9', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | Introduction to Nanoscale Thermal Conduction 309 To understand the effects of a periodic interatomic potential acting on the electron waves consider a simple yet effective model for the potential experienced by the electrons in a periodic lattice. This model the Kronig-Penny Model assumes there is one electron inside a square periodic potential with a period distance equal to the interatomic distance a mathematically expressed as i 0 for 0 z b I Vq for c z 0 11 subjected to the periodicity requirement given by v z b c V z where a b c. Solutions of Eq. 10 subjected to Eq. 11 are i D1exp iMz D2exp iMz for 0 z b 1- y I D3exp iLz D4exp iLz for c z 0 where D1 D2 D3 and D4 are constants determined from boundary conditions ti2 M2 13 and . h2L2 v 2m 14 with M and L related to the electron energy. Although the full mathematical derivation of the predicted allowed electron energies will not be pursued here see for example Griffiths 2000 one important part of this formalism is recognizing that the periodicity in the lattice gives rise to a periodic boundary condition of the wavefunction given by y z b c y z exp iz b c y z exp ika 15 where k is called the wavevector. Equation 15 is an example of the Bloch Theorem. The wavevector is defined by the periodicity of the potential . the lattice and therefore the goal is to determine the allowed energies defined in Eq. 13 as a function of the wavevector. The relationship between energy and wavevector e k known as the dispersion relation is the fundamental relationship needed to determine all thermal properties of interest in nanoscale thermal conduction. After incorporating the Bloch Theorem and continuity equations for boundary conditions of Eq. 12 and making certain simplifying assumptions Chen 2005 the following dispersion relation is derived for an electron subjected to a periodic potential in a one-dimensional lattice A. sin Mc cos Mc cos kc . K 16 Here A is related to the electron energy and atomic potential V and from Eq. .
