Thermal Transport in NanoStructures A review of Quantized Heat Transfer

1. Thermal Transport in NanoStructuresA 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

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.

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.

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.

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.” http://pr.caltech.edu/media/Press_Releases/PR12040.html • 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.

7. Thermal Conductance Theory • Landauer Formula used as a starting point General Landauer Formula • Landauer derived to describe limiting value of energy transport.

8. Fundamental Relation • Dependent only on temperature • Represents the maximum possible value of energy transported per Phonon mode Units of W/K

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

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

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.

12. Bibliography • Rego and Kirczenow; Quantized Thermal Conductance of Dielectric Quantum Wires; Physical Review Letters Vol 81, Num 1, 232-235, 1998. • Schwab et al; Thermal Conductance through discrete quantum channels; Physica E; 60-68; (2001). • Kouwenhoven and Venema; Heat Flow through nanobridges; Nature Vol 404, 943-944, 27, April 2000. • Fresley; Conductance ---the Landauer Formula; http://www.utdallas.edu/~frensley/technical/qtrans/node9.html; 23, July 1995. • Schwab et al; Measurement of the quantum of thermal conductance; Nature Vol 404, 974-977, 27, April 2000. • Roukes; Physicists observe the quantum of heat flow; http://pr.caltech.edu/media/Press_Releases/PR12040.html; 4/26/2000. • Phonons and the Debye Specific Heat; http://hyperphysics.phy-astr.gsu.edu/hbase/solids/phonon.html. • Collins; The Quantum of Heat Flow; Physical Review Focus; 9, July 1998; http://focus.aps.org/story/v2/st2 • 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, 205308, 2005. • Wang and Yi; Quantized phononic thermal conductance for one-dimensional ballistic transport; Chinese Journal of Physics, Vol 41, No. 1, 92-99, 2005.