1. Thermal Transport in NanoStructures A review of Quantized Heat Transfer
Patrick Miller
April 12, 2006

2. Outline Thermal Conductivity Phonon Quantum of Thermal Conductance
Thermal Conductance Theory
Fundamental Relation
Conditions for Quantum Thermal Conductance
Future issues
12 April 2006

3. Thermal Conductivity What is Thermal Conductivity?
Measure of how well a material transfers heat
Usually discussed as a macroscopic parameter
Apply Heat to one side and will flow to another.
Mesoscopic Scale
refers to the length scale at which one can reasonably discuss the properties of a material or phenomenon without having to discuss the behavior of individual atoms. For solids and liquids this is typically a few to ten nanometers, and involves averaging over a few thousand atoms or molecules. Hence, the mesoscopic scale is roughly identical to the nanoscopic or nanotechnology scale for most solids.
The mesoscopic scale thus lies between the macroscopic scale of the world we live in, and the atomic scale in which each atom is considered separated resolved.
It is well documented that the thermal transport in nanomaterials (nanoscale or nanostructured) can be very different from that in bulk materials
12 April 2006

4. Thermal Conductivity (cont)
Heat transfer involves Electrons (non-insulators) and/or phonons.
For technologically important semiconductors acoustic phonons are the dominant carriers.
Presentation will focus on mesoscopic scale, acoustic phonons, at low temperature.
12 April 2006

5. Phonon What is a Phonon? Quanta of lattice vibrations
Can’t vibrate independently
Wavelike motion characterized by mass spring model (however phonons are massless)
Small structures can only support one Phonon mode and have a fundamental limit for thermal conductivity
Quantum of Thermal Conductance.
Atoms in materials always vibrate at the frequencies that are particular to the materials (molecules and crystals). The atomic vibration is severe when the material is hot, and moderate when cold. In fact, temperature is the severeness of the atomic vibration.
The period of the thermal vibration in molecules and crystals is generally between 10 femtoseconds (10-14s) and 1 picoseconds (10-12s). Thermal vibration is random. That means, it is not plausible that all the atoms vibrate toward the same direction and at the same speed.
Discussing the lattice vibration
Lattice waves
Due to the connections between atoms, the displacement of one or more atoms from their equilibrium positions will give rise to a set of vibration waves propagating through the lattice. One such wave is shown in the figure below. The amplitude of the wave is given by the displacements of the atoms from their equilibrium positions. The wavelength λ is marked.
The quantum of thermal conductance is best understood by beginning with a simple explanation of heat flow. In the everyday world, the amount of heat carried by an object can vary in a smooth and continuous way. Heat actually flows by means of collective, wavelike vibrations of the atoms that make up a solid material. Usually immense numbers of such waves, each inducing a unique type of synchronous motion of the atoms, act simultaneously to carry heat along a material.
12 April 2006

6. Quantum of Thermal Conductance (QTC)
What is Quantum of Thermal Conductance?
“When an object becomes extremely small, only a limited number of phonons remain active and play a significant role in heat flow within it.”
As devices become smaller a strict limit exists for heat conduction
Maximum Value is a Fundamental Law of Nature.
Only way to increase thermal conductance is to increase the size.
12 April 2006

7. Thermal Conductance Theory
Landauer Formula used as a starting point
General Landauer Formula
Landauer derived to describe limiting value of energy transport.
Evaluation of this integral provides two terms. The first is the conductance of the massless modes and the second is the contribution of the higher energy modes. The massless mode
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8. Fundamental Relation Dependent only on temperature
Represents the maximum possible value of energy transported per Phonon mode
Units of W/K
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9. Criteria for QTC Ballistic phonon transport in 1D waveguide required
Transmission coefficient must be close to unity
Temperature bounded
Low temp by transmission coefficient going to zero.
Upper temp by onset of higher-energy modes
Need to be Close to 1
12 April 2006

10. Discretized transport.
Strain map in structure. Interesting to note the blue areas on the bridges. Indicates discrete flow otherwise strain map would be a gradient
Picture is from Index 16.
An Alternative Explanation of Ballistic Heat Transfer
Ok sometimes, things get hot. It's inevitable, really.  When those things get hot, they get really angry until they go ballistic and throw their heat away from them onto something else nearby.  So instead of calmly letting the heat seep away, until they are even, they throw it away all at once so that they are cooler than the other things. That's ballistic heat transfer
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11. Applications Importance for future of NanoScale devices
represents max energy transfer per channel I.e there is a temp rise of one kelvin when a thousandth of a billionth of a watt is applied.
From index 20
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12. Bibliography
Rego and Kirczenow; Quantized Thermal Conductance of Dielectric Quantum Wires; Physical Review Letters Vol 81, Num 1, ,
Schwab et al; Thermal Conductance through discrete quantum channels; Physica E; 60-68; (2001).
Kouwenhoven and Venema; Heat Flow through nanobridges; Nature Vol 404, , 27, April 2000.
Fresley; Conductance ---the Landauer Formula; 23, July 1995.
Schwab et al; Measurement of the quantum of thermal conductance; Nature Vol 404, , 27, April 2000.
Roukes; Physicists observe the quantum of heat flow; 4/26/2000.
Phonons and the Debye Specific Heat;
Collins; The Quantum of Heat Flow; Physical Review Focus; 9, July 1998;
Balandin; Nanophononics: Phonon Engineering in Nanostructures and Nanodevices; Journal of Nanoscience and Nanotechnology Vol 5, 1-8, 2005.
Tanaka et al; Lattice thermal conductance in nanowires at low temperatures; Physical Review B 71, , 2005.
Wang and Yi; Quantized phononic thermal conductance for one-dimensional ballistic transport; Chinese Journal of Physics, Vol 41, No. 1, 92-99, 2005.
12 April 2006