Monte Carlo Study of Thermal Transport of Direction and Frequency Dependent Boundaries in High Kn Systems N.A. Roberts and D.G. Walker Department of Mechanical Engineering Vanderbilt University May 30, 2008 Sponsor: DARPA 1/12
Introduction Thermal rectification is a phenomenon where transport through a device is dependent on direction T q 1 Forward Conduction q 2 Reverse Conduction x q = ka dt dx q 1 > q 2 k 1 > k 2 Widespread applications in thermal management problems Cooling of micro/nano electronics Improvement of macroscale refrigeration and energy saving buildings 2/12
Analog to pn-junction (Diode) Thermal Engineering Laboratory, Vanderbilt University In equilibrium electrons diffuse into p-type semiconductor and holes diffuse into n-type semiconductor, this creates the space charge region and built-in potential (voltage) A forward bias decreases the potential across the junction which enhances transport A reverse bias increases the potential across the junction reduces transport resulting in minimal current flow Thermal rectifying behaviour should be observed if a device could be created in which phonon transport can be manipulated P q,i Space Charge Region Carrier Concentration (log) (Phonon Concentration) Internal Electric Field V Potential Distribution (Temperature Distribution) N T, V 3/12
Boltzmann Transport Equation Phonons treated as particles with distribution function f = f( r, p, t) ( ) f t + v rf + F f p f = f 0 f t τ relax scat It is very difficult to obtain a closed-form solution In the current work we solve it using a probabilistic simulation known as Monte Carlo Method 4/12
Simulation Method Initialization Number of phonons initially prescribed (1,000,000) Randomly distributed throughout the device Polarization and frequency obtained based on initial temperature Momentum calculated from analytic dispersion relation Three-phonon and impurity scattering were not considered to investigate the system in the ballistic regime Cross sections of 100 100 nm to 1000 1000 nm and lengths of 100 nm to 1000 nm 1 cos ka Analytic phonon dispersion ω(k) = ω max,b 2 3-D Density of States D(k) = k2 2π 2 V g Phonon group velocity V g = k ω 1 Bose-Einstein distribution n = exp k hω B T 1 5/12
Simulation Method Cont. The device is composed of an isotropic material with Silicon-like properties Max. longitudinal acoustic phonon frequency 1.23 10 13 Hz Max. transverse acoustic phonon frequency 4.5 10 12 Hz Normalized density function F 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 F(ω) = b 0 0 2 4 6 8 10 12 14 Phonon Frequency (THz) T = 10K T = 20K T = 80K T = 640K ω0 n D(ω)dω ω max,b 0 n D(ω)dω The normalized density function is a key element in this study by shown the concentration of high frequency (energy) phonons at a specified temperature 6/12
Definition of Direction and Frequency Dependent Boundaries Boundaries parallel to x-axis have direction and frequency dependence Boundaries are flat in simulation Directional dependence comes from asymmetric sawtooth geometry Frequency dependence comes from specified surface roughness Both dependencies come from difference roughnesses of the surfaces If a phonon with negative x-momentum strikes a boundary parallel to the x-axis a parameter, p(ω, η), is calculated based on the phonon frequency and characteristic roughness if p 1, the phonon has a high probability of a diffuse reflection [ ] p(ω) = exp 64π5 η 2 ω 2 V 2 g α α Specular Reflection Diffuse Reflection 7/12
Definition of Direction and Frequency Dependent Boundaries Cont. w Smooth t Rough l and w are length and width of the device, respectively t is the sawtooth depth (of order or smaller than dominate phonon wavelength) l phonons moving to the right see smooth surfaces, phonons moving to the left see rough surfaces 8/12
Simulation Results - Temperature distribution Thermal Engineering Laboratory, Vanderbilt University α = 1 α = 10 T 10K 640K Temperature (K) Temperature (K) 15 14 13 12 11 10 9 8 7 30 25 20 15 10 5 0 η = 1e-9 m η = 1e-10 m η = 1e-11 m η = 1e-12 m η = 0 m 0 0.25 0.5 0.75 1 Distance along device (x/l) η = 1e-9 m η = 1e-10 m η = 1e-11 m η = 1e-12 m η = 0 m 0 0.25 0.5 0.75 1 Distance along device (x/l) Temperature (K) Temperature (K) 1600 1400 1200 1000 800 600 400 200 7000 6000 5000 4000 3000 2000 1000 0 η = 1e-9 m η = 1e-10 m η = 1e-11 m η = 1e-12 m η = 0 m 0 0.25 0.5 0.75 1 Distance along device (x/l) η = 1e-9 m η = 1e-10 m η = 1e-11 m η = 1e-12 m η = 0 m 0 0.25 0.5 0.75 1 Distance along device (x/l) 9/12
Simulation Results - Temperature difference at T = 10 K Temperature Difference (K) 25 20 15 10 5 0 1e-12 1e-11 1e-10 1e-09 1e-08 η (m) α = 0.1 α = 0.2 α = 0.5 α = 1.0 α = 2.0 α = 5.0 α = 10.0 Smooth α = l w Increased biasing with increased surface roughness of rough surface Increased biasing with increased aspect ratio (greater percentage of direction and frequency dependent surface area) w t Rough l 10/12
Simulation Results - Normalized temperature difference for η = 5 10 12 m T/T 10 8 6 4 2 0 T = 10K T = 20K T = 40K T = 50K T = 80K T = 160K T = 320K T = 480K T = 640K 0 1 2 3 4 5 6 7 8 9 10 Normalized temperature difference increases with increasing temperature and aspect ratio This increase with temperature is explained by the normalized density function α 11/12
Conclusions Self-biasing devices can be achieved with the use of asymmetric geometries and surface roughnesses (anisotropic behavior in an isotropic material) With the addition of thermalizing boundaries we should see thermal rectification in the ballistic transport regime The impact of the boundaries will be reduced at higher temperatures when scattering is included and when the device surface area to volume ratio is decreased Fabrication Atomically smooth η = 0 One unit cell variation η 0.5a 12/12