Resonant Surface Scattering 1 on Nanowires -Introduction BY DAVID ROFFMAN -Model Thanks to: -Mass Disorder vs Skutterudites -Dr. Selman Hershfield -One Dimensional Models -Dr. Hai Ping Cheng -Sprayed Skutterudites -Dr. Khandker Muttalib -Scaling in Clean Lattices, Ordered Sprays and Disordered Sprays -Dr. Simon Phillpot -Conclusion -Dr. Gregory Stewart
2 New Content 1- D Model for “off” atoms (motivation for sprays) More detailed model for skutterudites Effect of ordered sprays on 3-D lattices and skutterudites Scaling of transmission in ordered sprayed samples as a function of sample cross section and length Disordered spraying of cubic lattices and the effects on mean free path, transmission, and thermal conductance (the focus of the thesis)
3 Questions Asked How do transmission and conductance vary due to different sample masses and spring constants in 1-D chains and cubic lattices? How to modify the properties of skutterudites? How to consistently lower transmission, thermal conductance, and mean free path via sprays?
4 Motivation Electricity can be generated by a temperature difference (thermoelectric devices). This thesis studies the heat transfer aspect of such devices. Making thermoelectric devices more efficient by modifying the sample region Understanding what affects transmission through lattices, to set the framework to develop more efficient materials (eventually) Learn how to create materials with low lattice conductivity
5 Sketch of Geometry Being Solved
6 Model Basics As I am using a classical model, in order to obtain transmission, mean free path and conductivity, the positional amplitudes of the atoms are required In my model by using the harmonic approximation of a generic potential, one will arrive at the Rosenstock-Newell Model: the EOMs of each spatial coordinate are independent. Example: the X EOM doesn’t contain Y or Z Nearest neighbor only interactions
7 3-D Harmonic Lattices Part I Similar to previous models, except now in higher dimensions There are multiple allowed incident modes for a given frequency Mode transitions can occur only if there is mass disorder
8 Scattering Boundary Condition Expressing the first layers of the sample in terms of the reflection and transmission coefficients This is accomplished by using EOMs for the first layers of the baths to solve for the first layers of the sample Necessary if the sample has disorder or structure (see thesis Numerical Methods Chapter for a detailed example)
9 Define Transmission In this model, transmission can be greater than unity in the 2-D and 3-D cases: It’s maximum value is the number of incident modes However, for each incident mode the transmission and reflection still sum to one
10 Define Conductance This equation is based off the Landauer Formula for phonons Thermal conductance has units of [Energy]/([Time][Temperature]) Classical equations lead to quantized modes
11 Define Mean Free Path The momentum relaxation length Where N m is the number of modes excited in the bath, l is the mean free path, and L is the sample length: solve this for l
Mass Disorder vs. Skutterudites 12
3-D Harmonic Lattices Part II 13 Notice that mass disorder suppresses the transmission function at moderate to high frequencies, but has virtually no impact at low energy
Skutterudites of the Form MX 3 14 A substance with caged atoms, low thermal conductivity, and is easy to modify Caged atoms are rectangular structures sealed in a cubic cage Out of all the springs in the LK model (k1-k6), it appears that k1 (Cubic-Cage) has the most drastic effect on reducing transmission Adding these caged atoms decreases transmission at lower energies, whereas mass disorder doesn’t really have any effect in that regime
Clean vs Disorder vs Skutterudite 15 The assumption here is square X geometry
Changing the Skutterudite Parameters 16
1- D “Off” Atoms: Motivating Sprays 17 Skutterudites have lower transmission when compared to a cubic lattice because of caged atoms What happens if “extra” atoms are added to a system on the surface instead of inside the sample? This is not easy to understand analytically in higher dimensions, so a set of 1-D models are tried first
18 Model I: The clean sample Simplest Model What effects transmission is easily understood:
Model II: Changing one spring 19 What happens if springs vary? This will eventually result in the motivation for using spring constant disorder for resonant surface scattering on wires
Model III: A cage in 1-D 20 An attempt to understand why caged atoms reduce transmission in a skutterudite mP can also be viewed as a spray This extra atom actually causes a dip in the transmission function:
Model IV 21 Another model for a spray to better understand if it is an effective idea Once again there is a dip in the transmission:
22 Controlling the Dips in Transmission By altering the springs that connect the sample to mP (or mP itself), the location of the dip can be changed or even eliminated altogether There is only one dip, so the next question is: would additional sprayed masses create more zeros in the transmission function?
Models V and VI: A Better Analog 23 Testing the effects of two sprayed - on atoms Model VI adds an interaction in between the sprayed atoms Model V has two dips unless the masses and interactions are the same: These models motivate the use of multiple sprayed atoms in 3-D
Models V and VI Plot 24 In Model VI there can be two dips even if the pairs of sprayed masses and interactions are identical In Model VI the number of dips can be reduced to one for certain values of kP3
Sprayed Skutterudites 25 The previous 1-D models motivated a use of a surface spray A surface spray on a skutterudite was tested, and it did indeed reduce the transmission and conductance The geometry is: As a reminder, the 1-D analog is below
26 Sprayed Skutterudites Results
27 Scaling of Transmission in Cubic Lattices The issue with all of the previous results is that increasing the sample length doesn’t further reduce transmission To find out why this is, the effects of increasing sample length and cross section in a clean cubic lattice are studied
28 T(Length) for Various Cross Sections Wiggles in T are due to different resonances
29 Normalized T(Area) for Various Lengths
30 Ordered Spray Geometries Scaling results are similar to the clean lattice The transmission is lower if there is a spray, but the length of the sample is still irrelevant Two geometries are tested: Geometry 1 and Geometry 2
31 T(Length) for Various Cross Sections
32 Normalized T(Area) for Various Lengths
33 Spring Constant Disordered Spray The previous slides indicated that the reason for decreased transmission for a spray is the mismatch between heat bath and sample cross sections Based on the results of cubic mass disorder and Model II from the 1-D cases, it was decided to try a spray with disordered springs This was successful, as the transmission was lower than for a clean cubic lattice and a uniform spray Most importantly the transmission will decrease as the sample length increases
34 Geometry of a Disordered Spray Sprayed atoms can be connected to the inner sample with springs that vary randomly in between 0 and 1 The amount of disorder is defined as the percent of springs that are attached to the inner sample
35 Transmission (Disorder) More disorder means less transmission Randomness is accounted for by averaging
36 Mean Free Path As a reminder, the mean free path is obtained from: This will result in a linear Length vs 1/Transmission plot (on the left) If plotted over the spectrum of allowed frequencies (and showing various amounts of disorder), the mean free path goes down
37 Thermal Conductance of Disordered Sprays This is for 100% disorder Note how increasing length reduces thermal conductance
38 Conclusion The thermal properties of cubic lattices, sprayed cubic lattices, skutterudites, sprayed skutterudites, and spring disordered sprays were examined Skutterudites have low energy dips in the transmission function due to the caged atoms Ordered sprays reduce transmission due to mismatching between the bath and sample, however this effect doesn’t depend on sample length To decrease transmission, conductance, and the mean free path as function of length, a disordered surface spray is effective
Recommend
More recommend
