Eastern Illinois University The Keep Faculty Research and Creative Activity Physics January 2008 Calculation of phonon dispersion in semiconductor nanostructures: An undergraduate computational project Jie Zou Eastern Illinois University, [email protected] Follow this and additional works at: http://thekeep.eiu.edu/physics_fac Part of the Physics Commons Recommended Citation Zou, Jie, "Calculation of phonon dispersion in semiconductor nanostructures: An undergraduate computational project" (2008). Faculty Research and Creative Activity. 1. http://thekeep.eiu.edu/physics_fac/1 This Article is brought to you for free and open access by the Physics at The Keep. It has been accepted for inclusion in Faculty Research and Creative Activity by an authorized administrator of The Keep. For more information, please contact [email protected]. Calculation of phonon dispersion in semiconductor nanostructures: An undergraduate computational project ͒ Jie Zoua Department of Physics, Eastern Illinois University, Charleston, Illinois 61920 ͑Received 14 August 2007; accepted 24 November 2007͒ This paper discusses an undergraduate research project that involves the numerical calculation of phonon dispersion in semiconductor nanostructures. Phonon dispersion is calculated in the elastic continuum approximation by applying a finite-difference method to solving the elastic wave equation with boundary conditions. Students did simulations for two nanostructures: a free-standing GaN thin film and a free-standing AlN/GaN/AlN heterostructure. A comparison of these two structures helps students understand the effect of boundary conditions on the nature of the phonon dispersion. The project is related to current research in phonon thermal transport in semiconductor nanostructures and nanodevices. © 2008 American Association of Physics Teachers. ͓DOI: 10.1119/1.2825396͔ I. INTRODUCTION three-layer AlN/GaN/AlN heterostructure. The choice of GaN and its heterostructure is due to their potential use in Current technology in solid-state devices has reached the high-power and high-temperature device applications. The nanometer scale. Development of solid-state devices is goal of the project is to introduce students to current research driven by understanding of the novel properties and new topics in physics and engineering and to help them develop physical phenomena in the underlying materials. Most solid- computational skills to solve more complicated problems. state devices, such as transistors, are based on semiconduc- The project also helps students better understand the effect of tors and their nanostructures. The advances in technology boundary conditions on the characteristics of the phonon dis- have stimulated much research interest among physicists and persion. Although the project focuses on phonons, the con- engineers in the properties of semiconductor nanostructures, cept of quantization induced by boundary conditions applies such as quantum wells, quantum wires, and quantum dots.1 equally well to an electron confined in a quantum well. Although there has much work on the electronic properties The rest of the paper is organized as follows. The next of semiconductor nanostructures, research in phonon proper- section presents the general theory of lattice thermal conduc- ties has attracted significant attention in recent years.2 Pho- tivity in order to illustrate the importance of phonon disper- non engineering, an application analogous to electronic sion. Section III presents the continuum elastic wave equa- band-gap engineering,3 has recently been proposed, and pro- tions, followed by a discussion in Sec. IV on the numerical vides new opportunities for controlling phonon transport in algorithm for solving phonon dispersion. The simulation re- semiconductor nanostructures by tuning the phonon disper- sults are discussed in Sec. V. sion. In a semiconductor, heat is conducted mainly via lattice waves or phonons. In analogy to electronic band structure, phonon dispersion is an important property that directly af- II. LATTICE THERMAL CONDUCTIVITY fects the lattice thermal conductivity of the semiconductor. AND PHONON DISPERSION One of the main factors that drives the current interest in phonon heat conduction in semiconductor nanostructures is The law of thermal conduction can be found in most in- troductory texts.8 In the one-dimensional case, it takes the the increased heat dissipation associated with an increase in ␬͉ / ͉ 8 the transistor packing density. form, J= dT dx , where J is the heat flux or the thermal Students in introductory solid-state physics often solve energy transmitted per unit cross-sectional area per unit time, and ͉dT/dx͉ is the magnitude of the temperature gradient. phonon dispersion problems in simple one-dimensional ␬ cases.4 Two typical examples are the monatomic and di- The proportionality constant is the thermal conductivity. More generally, dT/dx should be replaced by the tempera- ١ atomic linear chain models, which can be solved analytically using lattice dynamics.4 Students are usually not familiar ture gradient T. The heat flux J can be expressed in terms of phonons as a sum over contributions from all phonon with three-dimensional problems and are not exposed to 9 problems that require numerical solutions. Additionally, the modes: results discussed in a typical undergraduate textbook apply ប␻ ͑ ͒ ͑ ͒ ͑ ͒ ١T. 1 J = ͚ nk,␣ ␣ k v␣ k only in bulk materials. Modifications are expected in a con- k,␣ fined structure, such as a thin film ͑quantum well͒, a nano- wire ͑quantum wire͒, and a heterostructure. In a semiconduc- A phonon mode is specified by the wave vector k and the ␣ tor, such as silicon, heat conduction is due mainly to long- phonon branch . nk,␣ is the number of phonons of type wavelength acoustic phonons.5 An alternative approach to ͑k,␣͒ per unit volume of the crystal. The phonon angular the calculation of phonon dispersion is the elastic continuum frequency is given by ␻, and the energy of a phonon of type ͑ ␣͒ ប␻ ͑ ͒ ͑ ͒ ١T is the component of the approximation, in which the equations of motion of the at- k, is given by ␣ k . v␣ k oms are reduced to the continuum elastic wave equation.6,7 phonon group velocity in the direction of the temperature This paper discusses an undergraduate project in which gradient. Equation ͑1͒ for the heat flux is analogous to that students apply a finite-difference method to determine the for the electric current density, in which electron density, phonon dispersion in a free-standing GaN thin film and a charge, and drift velocity are the counterparts of the phonon 460 Am. J. Phys. 76 ͑4&5͒, April/May 2008 http://aapt.org/ajp © 2008 American Association of Physics Teachers 460 This article is copyrighted as indicated in the article. Reuse of AAPT content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 139.67.65.234 On: Thu, 19 Dec 2013 20:55:09 tion. According to Hooke’s law, if the deformations are small, the stress components are linearly related to the strain components; the coefficients are called the elastic stiffness constants. In a hexagonal crystal, such as wurtzite GaN, the independent elastic stiffness constants are C11, C12, C13, C33, 14 C44, and C66. Fig. 1. Schematics of a free-standing AlN/GaN/AlN heterostructure ͑left͒ We look for solutions to Eq. ͑2͒ of the form of a sinusoidal and a free-standing GaN thin film ͑right͒. The coordinate system is shown in wave, the middle. ͑ ͒ ͓ ͑␻ ͔͒ ͑ ͒ Uy = uy z exp i t − kx , 3 where u is the amplitude, which depends on the z coordi- distribution, energy, and group velocity. An important term in y nate, and k is the x component of the wave vector. If we ͑ ͒ ␻ ͑k͒ Eq. 1 is the phonon dispersion relation ␣ , which de- substitute Eq. ͑3͒ into Eq. ͑2͒ and use the relations between scribes the dependence of the phonon angular frequency on the stress, strain, and displacement components, as described the wave vector. in Ref. 4, we derive the elastic wave equations for a GaN We model the thermal conductivity in a semiconductor / / 9–11 ͑ ͒ thin film and an AlN GaN AlN heterostructure. The deriva- nanostructure as follows: 1 Find the phonon distribution tion is straightforward, and we leave it as an exercise. The by solving the phonon Boltzmann transport equation in the ͑ ͒ ͑ ͒ 9 ͑ ͒ results are given in Eqs. 4 and 7 for a GaN thin film and relaxation-time approximation; 2 derive the phonon dis- an AlN/GaN/AlN heterostructure, respectively. persion by solving the continuum elastic wave equation with ͑ ͒ ͑ ͒ For a GaN thin film, Eq. 2 can be simplified to the the appropriate boundary conditions; 3 calculate the pho- second-order ordinary differential equation non group velocity V=d␻/dk; and ͑4͒ replace the summa- tion in Eq. ͑1͒ by an integral over the phonon density of d2u ͑z͒ 1 y + ͑␳ ␻2 − C k2͒u ͑z͒ =0. ͑4͒ states. 2 GaN 66,GaN y dz C44,GaN A generalized expression for the lattice thermal conductiv- ١T͉. The phonon The free-standing boundary conditions require that the stress͉/ity can be derived using the relation ␬=J dispersion plays an important role in determining the lattice and strain are zero at the surfaces, that is, thermal conductivity through its effects on other phonon du ͑z͒ properties, such as the group velocity and density of states. ͯ y ͯ =0, ͑5͒ In the following we discuss the numerical calculation of dz z=Ϯt/2 phonon dispersion in two types of nanostructures: a free- where t is the thickness of the thin film. The boundary-value standing thin film and a three-layer heterostructure. We fol- problem represented by Eq. ͑4͒ with the free-standing bound- low a general approach outlined by Balandin and 12 ary conditions can be solved analytically. co-workers. The derivation and calculation of the lattice A three-layer heterostructure is different because it is in- thermal conductivity in a thin film and heterostructure can be homogeneous along the z direction, and therefore the mate- found in Ref.