Resonant Surface Scattering 1 on Nanowires -Introduction BY DAVID - - PowerPoint PPT Presentation

Resonant Surface Scattering

• n Nanowires

BY DAVID ROFFMAN

• Introduction
• Model
• Mass Disorder vs Skutterudites
• One Dimensional Models
• Sprayed Skutterudites
• Scaling in Clean Lattices, Ordered

Sprays and Disordered Sprays

• Conclusion

Thanks to:

• Dr. Selman Hershfield
• Dr. Hai Ping Cheng
• Dr. Khandker Muttalib
• Dr. Simon Phillpot
• Dr. Gregory Stewart

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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)

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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?

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Motivation

 Electricity can be generated by a temperature difference

(thermoelectric devices). This thesis studies the heat transfer aspect

• f 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

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Sketch of Geometry Being Solved

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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

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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

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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)

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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

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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

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Define Mean Free Path

 The momentum relaxation length  Where Nm 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

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Mass Disorder vs. Skutterudites

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3-D Harmonic Lattices Part II

 Notice that mass disorder suppresses the transmission function at moderate to high

frequencies, but has virtually no impact at low energy

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Skutterudites of the Form MX3

 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

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Clean vs Disorder vs Skutterudite

 The assumption here is square X geometry

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Changing the Skutterudite Parameters

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1-D “Off” Atoms: Motivating Sprays

 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

• f 1-D models are tried first

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Model I: The clean sample

 Simplest Model  What effects transmission is easily understood:

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Model II: Changing one spring

 What happens if springs vary?  This will eventually result in the motivation for using spring constant

disorder for resonant surface scattering on wires

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Model III: A cage in 1-D

 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:

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Model IV

 Another model for a spray to better understand if it is an effective

idea

 Once again there is a dip in the transmission:

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Controlling the Dips in Transmission

 By altering the springs that connect the sample to mP (or mP itself), the location

• f 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?

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Models V and VI: A Better Analog

 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

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Models V and VI Plot

 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

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Sprayed Skutterudites

 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

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Sprayed Skutterudites Results

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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

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T(Length) for Various Cross Sections

 Wiggles in T are due to different resonances

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Normalized T(Area) for Various Lengths

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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

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T(Length) for Various Cross Sections

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Normalized T(Area) for Various Lengths

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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

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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

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Transmission (Disorder)

 More disorder means less transmission  Randomness is accounted for by averaging

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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

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Thermal Conductance of Disordered Sprays

 This is for 100% disorder  Note how increasing length reduces thermal conductance

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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

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