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Vibrational Quantum Transport PhD Defence June 2010
- 1. Vibrational Quantum Transport PhD defence June 4th, 2010 Mads Engelund Supervisors: Antti-Pekka Jauho Mads Brandbyge
- 2. •Introduction •Basic approach •Results DTU Nanotech, Technical University of Denmark 2
- 3. Moore, Electronics, 38, 3 (1965) 3 DTU Nanotech, Technical University of Denmark
- 4. http://commons.wikimedia.org/wiki/User:Wgsimon 3 DTU Nanotech, Technical University of Denmark
- 5. Courtesy of Ray Kurzweil 3 DTU Nanotech, Technical University of Denmark
- 6. New transistor approaches Song and Lee, Nature, 462, 1039 (2009) http://www.sciencedaily.com/releases/2009/12/091223133343.htm 4 DTU Nanotech, Technical University of Denmark
- 7. New transistor approaches Lin et al., Science, 327, 662 (2010) Song and Lee, Nature, 462, 1039 (2009) http://www.sciencedaily.com/releases/2009/12/091223133343.htm 4 DTU Nanotech, Technical University of Denmark
- 8. New transistor approaches Lin et al., Science, 327, 662 (2010) Song and Lee, Nature, 462, 1039 (2009) http://www.sciencedaily.com/releases/2009/12/091223133343.htm 4 DTU Nanotech, Technical University of Denmark
- 9. Heat dissipation 7 DTU Nanotech, Technical University of Denmark
- 10. Doing work Heat dissipation Flickr.com/mikebaird 7 DTU Nanotech, Technical University of Denmark
- 11. Doing work Heat dissipation Overheating Flickr.com/mikebaird Flickr.com/mikebaird 7 DTU Nanotech, Technical University of Denmark
- 12. Ralph Group, Cornell University, http://people.ccmr.cornell.edu/~ralph/projects/emig_movies/ 8 DTU Nanotech, Technical University of Denmark
- 13. Ralph Group, Cornell University, http://people.ccmr.cornell.edu/~ralph/projects/emig_movies/ 8 DTU Nanotech, Technical University of Denmark
- 14. Heat 5 DTU Nanotech, Technical University of Denmark
- 15. Heat =Energy we’ve lost track of 5 DTU Nanotech, Technical University of Denmark
- 16. Heat =Energy we’ve lost track of Public domain 5 DTU Nanotech, Technical University of Denmark
- 17. Vibrations =main storage of heat Macroscopic suspended beam Microscopic graphene edge Wikimedia commons 6 DTU Nanotech, Technical University of Denmark
- 18. Heat flow on different scales 9 DTU Nanotech, Technical University of Denmark
- 19. Heat flow on different scales Sierra Pacific Innovations: ww.x20.org 9 DTU Nanotech, Technical University of Denmark
- 20. Heat flow on different scales IEEE Spectrum: Carbon Nanotubes Takes the Heat of Chips, Bryan Christie Design Sierra Pacific Innovations: ww.x20.org 9 DTU Nanotech, Technical University of Denmark
- 21. Heat flow on different scales Wikimedia Commons IEEE Spectrum: Carbon Nanotubes Takes the Heat of Chips, Bryan Christie Design Sierra Pacific Innovations: ww.x20.org 9 DTU Nanotech, Technical University of Denmark
- 22. Heat flow on different scales Wikimedia Commons Wikimedia Commons IEEE Spectrum: Carbon Nanotubes Takes the Heat of Chips, Bryan Christie Design Sierra Pacific Innovations: ww.x20.org 9 DTU Nanotech, Technical University of Denmark
- 23. Electronics Photonics Vibronics? 10 DTU Nanotech, Technical University of Denmark
- 24. Electronics Photonics Saser Vibronics? Physical Review Letters, 104, 083901(2010) 10 DTU Nanotech, Technical University of Denmark
- 25. Electronics Photonics Saser Vibronics? Physical Review Letters, 104, 083901(2010) Thermal wave guide and rectifier Science, 314, 5802, 1121-1124(2009) 10 DTU Nanotech, Technical University of Denmark
- 26. Measurements on vibrations 11 DTU Nanotech, Technical University of Denmark
- 27. Measurements on vibrations Raman Spectroscopy Nano Lett., 8-3, 919-924(2008) 11 DTU Nanotech, Technical University of Denmark
- 28. Measurements on vibrations Thermal probing J. of App. Phys., 105, 104306 (2009) Raman Spectroscopy Nano Lett., 8-3, 919-924(2008) 11 DTU Nanotech, Technical University of Denmark
- 29. •Introduction •Basic approach •Results 12 DTU Nanotech, Technical University of Denmark
- 30. Ab-initio • Data-> Basic principles -> model Empirical Data -> model 18 DTU Nanotech, Technical University of Denmark
- 31. Ab-initio • Data-> Basic principles -> model Empirical Data -> model 18 DTU Nanotech, Technical University of Denmark
- 32. Ab-initio • Data-> Basic principles -> model DFT Empirical Accurate tool but only app. 1000 atoms Data -> model 18 DTU Nanotech, Technical University of Denmark
- 33. Gold chains 13 DTU Nanotech, Technical University of Denmark
- 34. Gold chains Simple structure Eur. Phys. J. D, 16, 395(2001) 13 DTU Nanotech, Technical University of Denmark
- 35. Gold chains Simple structure Clear inelastic signal Phys. Rev. Lett., 88, 216803(2002) Eur. Phys. J. D, 16, 395(2001) 13 DTU Nanotech, Technical University of Denmark
- 36. 14 DTU Nanotech, Technical University of Denmark
- 37. The dynamical matrix 14 DTU Nanotech, Technical University of Denmark
- 38. The dynamical matrix Equation of motion (Newton’s 2.) 14 DTU Nanotech, Technical University of Denmark
- 39. The dynamical matrix Equation of motion (Newton’s 2.) 14 DTU Nanotech, Technical University of Denmark
- 40. DOS Green’s function DTU Nanotech, Technical University of Denmark
- 41. Leads DOS Green’s function DTU Nanotech, Technical University of Denmark
- 42. Calculating the Self-Energy 17 DTU Nanotech, Technical University of Denmark
- 43. The Surface Green’s function k 0 0 18 DTU Nanotech, Technical University of Denmark
- 44. Discrete modes vs. broadened modes
- 45. Discrete modes vs. broadened modes Density of States Frequency/Energy
- 46. Discrete modes vs. broadened modes Density of States Frequency/Energy Amplitude Time
- 47. Discrete modes vs. broadened modes Density of States Frequency/Energy Amplitude Time
- 48. Discrete modes vs. broadened modes Density of States Frequency/Energy Amplitude Time
- 49. Q-factors 17 DTU Nanotech, Technical University of Denmark
- 50. Q-factors 17 DTU Nanotech, Technical University of Denmark
- 51. Q-factors Amplitude High Q-factor 0 Low Q-factor Time 17 DTU Nanotech, Technical University of Denmark
- 52. Programming 8000 lines 19 DTU Nanotech, Technical University of Denmark
- 53. getGreensFunction(system) Programming getDynamicalMatrix(system) getSelfEnergy(system) Read getFCMatrix(system) SIESTA Result getSelfEnergy(surface) Read getFCMatrix(surface) SIESTA getUnpertubedDynamicalMatrix(surface) Result Read getlFCMatrix(crystal) SIESTA getFCMatrix(surface) Result 8000 lines getSurfaceGreensFunctionPrDiffVector(surface) getDynamicalMatrix(surface) 19 DTU Nanotech, Technical University of Denmark
- 54. •Introduction •Basic approach •Results –Gold Chains –Graphene edges 20 DTU Nanotech, Technical University of Denmark
- 55. Study of gold chains 21 DTU Nanotech, Technical University of Denmark
- 56. Study of gold chains 21 DTU Nanotech, Technical University of Denmark
- 57. Study of gold chains 21 DTU Nanotech, Technical University of Denmark
- 58. Study of gold chains 21 DTU Nanotech, Technical University of Denmark
- 59. Study of gold chains 21 DTU Nanotech, Technical University of Denmark
- 60. Study of gold chains 21 DTU Nanotech, Technical University of Denmark
- 61. Study of gold chains 21 DTU Nanotech, Technical University of Denmark
- 62. Study of gold chains Eur. Phys. J. D, 16, 395(2001) 21 DTU Nanotech, Technical University of Denmark
- 63. 24
- 64. Amplitude in chain High High Q-factor Low 24
- 65. Amplitude in chain High Low High Q-factor Low 24
- 66. Amplitude in chain High Low High Q-factor Low 24
- 67. 22 DTU Nanotech, Technical University of Denmark
- 68. 22 DTU Nanotech, Technical University of Denmark
- 69. 22 DTU Nanotech, Technical University of Denmark
- 70. 22 DTU Nanotech, Technical University of Denmark
- 71. Length 26 DTU Nanotech, Technical University of Denmark
- 72. Type of lead 27 DTU Nanotech, Technical University of Denmark
- 73. Gold Chains DTU Nanotech, Technical University of Denmark
- 74. Gold Chains •Instances of very low damping DTU Nanotech, Technical University of Denmark
- 75. Gold Chains •Instances of very low damping Frederiksen et al., Phys. Rev. B, 75, 205413(2007) DTU Nanotech, Technical University of Denmark
- 76. Gold Chains •Instances of very low damping •Sensitivity Frederiksen et al., Phys. Rev. B, 75, 205413(2007) DTU Nanotech, Technical University of Denmark
- 77. Gold Chains •Instances of very low damping •Sensitivity quantrans.org/introduction DTU Nanotech, Technical University of Denmark
- 78. •Introduction •Basic approach •Results –Gold Chains –Graphene edges 20 DTU Nanotech, Technical University of Denmark
- 79. 24 DTU Nanotech, Technical University of Denmark
- 80. www.als.lbl.gov/pics/154graphene01.png 24 DTU Nanotech, Technical University of Denmark
- 81. www.als.lbl.gov/pics/154graphene01.png www.technologyreview.com/files/ 11636/graphene_x220.jpg 24 DTU Nanotech, Technical University of Denmark
- 82. www.als.lbl.gov/pics/154graphene01.png Lin et al., Science, 327, 662 (2010) www.technologyreview.com/files/ 11636/graphene_x220.jpg 24 DTU Nanotech, Technical University of Denmark
- 83. IEEE Spectrum: Carbon Nanotubes Takes the Heat of Chips, Bryan Christie Design www.als.lbl.gov/pics/154graphene01.png Lin et al., Science, 327, 662 (2010) www.technologyreview.com/files/ 11636/graphene_x220.jpg 24 DTU Nanotech, Technical University of Denmark
- 84. An interesting experiment *Jia et al., Science, 2009, 323, 1701 25 DTU Nanotech, Technical University of Denmark
- 85. An interesting experiment electronic current *Jia et al., Science, 2009, 323, 1701 25 DTU Nanotech, Technical University of Denmark
- 86. An interesting experiment electronic current *Jia et al., Science, 2009, 323, 1701 25 DTU Nanotech, Technical University of Denmark
- 87. An interesting experiment electronic current *Jia et al., Science, 2009, 323, 1701 25 DTU Nanotech, Technical University of Denmark
- 88. An interesting experiment electronic current *Jia et al., Science, 2009, 323, 1701 25 DTU Nanotech, Technical University of Denmark
- 89. An interesting experiment electronic current *Jia et al., Science, 2009, 323, 1701 25 DTU Nanotech, Technical University of Denmark
- 90. 26 DTU Nanotech, Technical University of Denmark
- 91. 26 DTU Nanotech, Technical University of Denmark
- 92. 26 DTU Nanotech, Technical University of Denmark
- 93. The quick description 33 DTU Nanotech, Technical University of Denmark Armchair edges in graphene 02/28/2010
- 94. Vibrations 27 DTU Nanotech, Technical University of Denmark
- 95. Electrons Vibrations 27 DTU Nanotech, Technical University of Denmark
- 96. Electrons Vibrations 27 DTU Nanotech, Technical University of Denmark
- 97. Electrons Vibrations 27 DTU Nanotech, Technical University of Denmark
- 98. Electrons ? Vibrations 27 DTU Nanotech, Technical University of Denmark
- 100. e
- 101. e
- 102. e
- 103. Potential energy of electrons e Bias
- 104. Potential energy of electrons e Bias
- 105. Potential energy of electrons e Bias
- 106. Potential energy of electrons e Bias
- 107. Potential energy of electrons e Bias
- 108. Potential energy of electrons e Bias
- 109. Potential energy of electrons e Bias
- 112. <1.6
- 113. <1.6
- 114. Graphene Edges 37 DTU Nanotech, Technical University of Denmark Armchair edges in graphene 02/28/2010
- 115. Graphene Edges *Jia et al., Science, 2009, 323, 37 DTU Nanotech, Technical University of Denmark Armchair edges in graphene 02/28/2010
- 116. Graphene Edges 37 DTU Nanotech, Technical University of Denmark Armchair edges in graphene 02/28/2010
- 117. Graphene Edges 37 DTU Nanotech, Technical University of Denmark Armchair edges in graphene 02/28/2010
- 118. Graphene Edges Bye, bye armchair dimer 37 DTU Nanotech, Technical University of Denmark Armchair edges in graphene 02/28/2010
- 119. Conclusions 30 DTU Nanotech, Technical University of Denmark
- 120. Conclusions • Vibrations sensitive to everything Phys. Rev. Lett., 88, 216803(2002) 30 DTU Nanotech, Technical University of Denmark
- 121. Conclusions • Vibrations sensitive to everything • Importance of ab-initio Phys. Rev. Lett., 88, 216803(2002) 30 DTU Nanotech, Technical University of Denmark
- 122. Conclusions • Vibrations sensitive to everything • Importance of ab-initio Science, 2009, 323, 1701 • Control of matter 30 DTU Nanotech, Technical University of Denmark
- 123. Conclusions • Vibrations sensitive to everything • Importance of ab-initio Science, 2009, 323, 1701 • Control of matter Phys. Rev. Lett., 104, 30 DTU Nanotech, Technical University of Denmark
- 124. Background-Theoretical Modeling of Vibrational Properties 39 DTU Nanotech, Technical University of Denmark
- 125. Background-Theoretical Modeling of Vibrational Properties Non-conservative Forces Dundas et al., Nat. Nano., 4, 99-103(2009) 39 DTU Nanotech, Technical University of Denmark
- 126. Background-Theoretical Modeling of Vibrational Properties Non-conservative Forces Dundas et al., Nat. Nano., 4, 99-103(2009) Dynamics McEni r y et al., Phys. Rev. B, 035446(2008) 39 DTU Nanotech, Technical University of Denmark
- 127. Background-Theoretical Modeling of Vibrational Properties Non-conservative Forces Dundas et al., Nat. Nano., 4, 99-103(2009) Anharmonic Transport Mingo , Phys. Rev. B, 125402(2006) Dynamics McEni r y et al., Phys. Rev. B, 035446(2008) 39 DTU Nanotech, Technical University of Denmark
- 128. Vibrational Quantum Transport PhD defence June 4th, 2010 Mads Engelund Supervisors: Antti-Pekka Jauho Mads Brandbyge
Editor's Notes
- First of all I would like to welcome everyone in the audience, the evaluation commity, my collegues, my friends and family...all who have all taken the time to be here today and hear about the work I&#x2019;ve done these last three years.
- As an introduction I would like to talk about why effective modelling of vibrations is so important. I would say that one of the principal motivations comes from the design of computers- a technology that has a huge impact on society.
- In the sixties Gordon Moore, who later went on to co-found Intel, noted a remarkable trend. The number of transistors on new microchips seemed to double every two years. He predicted this trend to last for approximately ten years. But the trend proved much more resilient that $- since it has now continued for over forty years! And in an alternative formulation, tracking the computational power that can be bought $ the trend can be traced back over a 100 years- even as the basic building blocks that make up computers have change immensely. What I would like to emphasize here, is that the relentless expansion of technological capabilities is constantly pushing the need to investigate smaller and smaller structures. Structures where the positions of individual atoms can become important
- In the sixties Gordon Moore, who later went on to co-found Intel, noted a remarkable trend. The number of transistors on new microchips seemed to double every two years. He predicted this trend to last for approximately ten years. But the trend proved much more resilient that $- since it has now continued for over forty years! And in an alternative formulation, tracking the computational power that can be bought $ the trend can be traced back over a 100 years- even as the basic building blocks that make up computers have change immensely. What I would like to emphasize here, is that the relentless expansion of technological capabilities is constantly pushing the need to investigate smaller and smaller structures. Structures where the positions of individual atoms can become important
- In the sixties Gordon Moore, who later went on to co-found Intel, noted a remarkable trend. The number of transistors on new microchips seemed to double every two years. He predicted this trend to last for approximately ten years. But the trend proved much more resilient that $- since it has now continued for over forty years! And in an alternative formulation, tracking the computational power that can be bought $ the trend can be traced back over a 100 years- even as the basic building blocks that make up computers have change immensely. What I would like to emphasize here, is that the relentless expansion of technological capabilities is constantly pushing the need to investigate smaller and smaller structures. Structures where the positions of individual atoms can become important
- In the sixties Gordon Moore, who later went on to co-found Intel, noted a remarkable trend. The number of transistors on new microchips seemed to double every two years. He predicted this trend to last for approximately ten years. But the trend proved much more resilient that $- since it has now continued for over forty years! And in an alternative formulation, tracking the computational power that can be bought $ the trend can be traced back over a 100 years- even as the basic building blocks that make up computers have change immensely. What I would like to emphasize here, is that the relentless expansion of technological capabilities is constantly pushing the need to investigate smaller and smaller structures. Structures where the positions of individual atoms can become important
- In one example, researchers have created a transistor where the current-carrying medium consists of a single atomic layer of carbon, also known as graphene. $ And in this example- even more extreme, a single benzene ring acts as the transistor.
- In one example, researchers have created a transistor where the current-carrying medium consists of a single atomic layer of carbon, also known as graphene. $ And in this example- even more extreme, a single benzene ring acts as the transistor.
- When designing circuits, the subject of heat, specifically how to get rid of heat, is very important. $ In practise, any purposefull activity that uses energy will also generate heat. And for any system, the ability to get rid of heat can act as a barrier to this activity. $ Fortunately, humans have the ability to realize this in most cases...
- When designing circuits, the subject of heat, specifically how to get rid of heat, is very important. $ In practise, any purposefull activity that uses energy will also generate heat. And for any system, the ability to get rid of heat can act as a barrier to this activity. $ Fortunately, humans have the ability to realize this in most cases...
- &#x2026;because overheating will cause a collapse. In this movie we see an example of how an exessive current destroys a gold structure (start movie). Surprisingly the structure does not break at the narrowest part of of the structure but at a short distance thereafter. This goes to show that at short length scales, heat generation and heat dissipation exhibit new phenomena, unseen at larger scales.
- Before we move any further let me take a step back to explain what heat actually is . This is my explanaition $w Let me illustrate this $ In the beginning of this movie we have an atom moving with a certain kinetic energy. In the end all this energy is redistributed among all the other atoms. In principle, we could keep track of exactly where the energy goes but in practise this is an impossible task. In stead we say that energy well distributed over all degrees of freedom is &#x201C;heat&#x201D;.
- Before we move any further let me take a step back to explain what heat actually is . This is my explanaition $w Let me illustrate this $ In the beginning of this movie we have an atom moving with a certain kinetic energy. In the end all this energy is redistributed among all the other atoms. In principle, we could keep track of exactly where the energy goes but in practise this is an impossible task. In stead we say that energy well distributed over all degrees of freedom is &#x201C;heat&#x201D;.
- Before we move any further let me take a step back to explain what heat actually is . This is my explanaition $w Let me illustrate this $ In the beginning of this movie we have an atom moving with a certain kinetic energy. In the end all this energy is redistributed among all the other atoms. In principle, we could keep track of exactly where the energy goes but in practise this is an impossible task. In stead we say that energy well distributed over all degrees of freedom is &#x201C;heat&#x201D;.
- Most of the heat in solids is stored in vibrations, so a proper understanding of vibrations is crucial in understanding heat dissipation. And as devices are made smaller and smaller we also need to understand vibrations on a smaller and smaller scale. Note that the vibrations on the smallest scales are fundamentally different from the large scale vibrations. At this small scale the solid can no longer be considered continous since it clearly consists of atoms at specific positions.
- Effective dissipation of heat is an increasingly important design consideration- The use of multiple-core processors in stead of single-core is primarily due to heat considerations. Switching and passing currents generates heat at the smallest scale. This heat must be transported all the way from the devices on the chip $...$$$ to the surroundings of the computer. It is important that this flow of heat is uninterupted across several scales. A bottleneck, can occur at any scale. And in a recent state-of-the-art development carbon nanotubes were used to avoid the bottleneck between the chip and the heat sink.
- Effective dissipation of heat is an increasingly important design consideration- The use of multiple-core processors in stead of single-core is primarily due to heat considerations. Switching and passing currents generates heat at the smallest scale. This heat must be transported all the way from the devices on the chip $...$$$ to the surroundings of the computer. It is important that this flow of heat is uninterupted across several scales. A bottleneck, can occur at any scale. And in a recent state-of-the-art development carbon nanotubes were used to avoid the bottleneck between the chip and the heat sink.
- Effective dissipation of heat is an increasingly important design consideration- The use of multiple-core processors in stead of single-core is primarily due to heat considerations. Switching and passing currents generates heat at the smallest scale. This heat must be transported all the way from the devices on the chip $...$$$ to the surroundings of the computer. It is important that this flow of heat is uninterupted across several scales. A bottleneck, can occur at any scale. And in a recent state-of-the-art development carbon nanotubes were used to avoid the bottleneck between the chip and the heat sink.
- Effective dissipation of heat is an increasingly important design consideration- The use of multiple-core processors in stead of single-core is primarily due to heat considerations. Switching and passing currents generates heat at the smallest scale. This heat must be transported all the way from the devices on the chip $...$$$ to the surroundings of the computer. It is important that this flow of heat is uninterupted across several scales. A bottleneck, can occur at any scale. And in a recent state-of-the-art development carbon nanotubes were used to avoid the bottleneck between the chip and the heat sink.
- Although heat dissipation is my primary reason to take a keen interest in vibrations, something new has emerged in recent years- detailed control over vibrations. Earlier this year a really exciting piece of news was announced by two independent groups- the creation of a SASER $ (Sound Amplification through Simulated Emission of Radiation), the vibrational analog of the optical laser. And considering the amazing progress in the field of optics since the invention of the laser this was enough to give me goose-bumps. $ Coupled with new effective methods for controlling vibrations once they are created- perhaps what we are seeing is the emergence of an entirely new field- design of devices based on vibrations.
- Although heat dissipation is my primary reason to take a keen interest in vibrations, something new has emerged in recent years- detailed control over vibrations. Earlier this year a really exciting piece of news was announced by two independent groups- the creation of a SASER $ (Sound Amplification through Simulated Emission of Radiation), the vibrational analog of the optical laser. And considering the amazing progress in the field of optics since the invention of the laser this was enough to give me goose-bumps. $ Coupled with new effective methods for controlling vibrations once they are created- perhaps what we are seeing is the emergence of an entirely new field- design of devices based on vibrations.
- Although heat dissipation is my primary reason to take a keen interest in vibrations, something new has emerged in recent years- detailed control over vibrations. Earlier this year a really exciting piece of news was announced by two independent groups- the creation of a SASER $ (Sound Amplification through Simulated Emission of Radiation), the vibrational analog of the optical laser. And considering the amazing progress in the field of optics since the invention of the laser this was enough to give me goose-bumps. $ Coupled with new effective methods for controlling vibrations once they are created- perhaps what we are seeing is the emergence of an entirely new field- design of devices based on vibrations.
- Although heat dissipation is my primary reason to take a keen interest in vibrations, something new has emerged in recent years- detailed control over vibrations. Earlier this year a really exciting piece of news was announced by two independent groups- the creation of a SASER $ (Sound Amplification through Simulated Emission of Radiation), the vibrational analog of the optical laser. And considering the amazing progress in the field of optics since the invention of the laser this was enough to give me goose-bumps. $ Coupled with new effective methods for controlling vibrations once they are created- perhaps what we are seeing is the emergence of an entirely new field- design of devices based on vibrations.
- New and exciting measurements have also appeared in recent years. $ Raman spectroscopy for example, Raman spectroscopy has reached increadible sophistication. It is now possible to perform spetroscopy on a single molecule suspended between electrodes- and to obtain a current-voltage characteristic at the same time. $ And using the method of thermal probing it is possible to find the temperature distrbution along a single carbon nanotube. These extrordinary measurement allow us to see what is actually going on on the nano-scale.
- New and exciting measurements have also appeared in recent years. $ Raman spectroscopy for example, Raman spectroscopy has reached increadible sophistication. It is now possible to perform spetroscopy on a single molecule suspended between electrodes- and to obtain a current-voltage characteristic at the same time. $ And using the method of thermal probing it is possible to find the temperature distrbution along a single carbon nanotube. These extrordinary measurement allow us to see what is actually going on on the nano-scale.
- New and exciting measurements have also appeared in recent years. $ Raman spectroscopy for example, Raman spectroscopy has reached increadible sophistication. It is now possible to perform spetroscopy on a single molecule suspended between electrodes- and to obtain a current-voltage characteristic at the same time. $ And using the method of thermal probing it is possible to find the temperature distrbution along a single carbon nanotube. These extrordinary measurement allow us to see what is actually going on on the nano-scale.
- New and exciting measurements have also appeared in recent years. $ Raman spectroscopy for example, Raman spectroscopy has reached increadible sophistication. It is now possible to perform spetroscopy on a single molecule suspended between electrodes- and to obtain a current-voltage characteristic at the same time. $ And using the method of thermal probing it is possible to find the temperature distrbution along a single carbon nanotube. These extrordinary measurement allow us to see what is actually going on on the nano-scale.
- New and exciting measurements have also appeared in recent years. $ Raman spectroscopy for example, Raman spectroscopy has reached increadible sophistication. It is now possible to perform spetroscopy on a single molecule suspended between electrodes- and to obtain a current-voltage characteristic at the same time. $ And using the method of thermal probing it is possible to find the temperature distrbution along a single carbon nanotube. These extrordinary measurement allow us to see what is actually going on on the nano-scale.
- New and exciting measurements have also appeared in recent years. $ Raman spectroscopy for example, Raman spectroscopy has reached increadible sophistication. It is now possible to perform spetroscopy on a single molecule suspended between electrodes- and to obtain a current-voltage characteristic at the same time. $ And using the method of thermal probing it is possible to find the temperature distrbution along a single carbon nanotube. These extrordinary measurement allow us to see what is actually going on on the nano-scale.
- Now that I&#x2019;ve hopefully convinced you all that the study of vibrations is worthwhile, I will now move on to describe what I have done in this field. (long 10)
- What me and my group have worked on is on making effective computer simulation of vibrations in real structures. Two basic approaches exist to do this problem. In the empirical approach you take experimental data and create a model based on a statistical fit to a model. With this approach it is possible to model huge structures, but it is difficult to know when it will fail. It is dependent upon having data that represent any situation you might encounter. $ In our group we therefore primarily work with ab-initio models. Here you try to find the basic principles underlying the experimental data and then create a model. $ We base our work on the DFT method, which is a stable workhorse of materials physics giving among many other things- very reliable vibrational energies.
- What me and my group have worked on is on making effective computer simulation of vibrations in real structures. Two basic approaches exist to do this problem. In the empirical approach you take experimental data and create a model based on a statistical fit to a model. With this approach it is possible to model huge structures, but it is difficult to know when it will fail. It is dependent upon having data that represent any situation you might encounter. $ In our group we therefore primarily work with ab-initio models. Here you try to find the basic principles underlying the experimental data and then create a model. $ We base our work on the DFT method, which is a stable workhorse of materials physics giving among many other things- very reliable vibrational energies.
- What me and my group have worked on is on making effective computer simulation of vibrations in real structures. Two basic approaches exist to do this problem. In the empirical approach you take experimental data and create a model based on a statistical fit to a model. With this approach it is possible to model huge structures, but it is difficult to know when it will fail. It is dependent upon having data that represent any situation you might encounter. $ In our group we therefore primarily work with ab-initio models. Here you try to find the basic principles underlying the experimental data and then create a model. $ We base our work on the DFT method, which is a stable workhorse of materials physics giving among many other things- very reliable vibrational energies.
- What me and my group have worked on is on making effective computer simulation of vibrations in real structures. Two basic approaches exist to do this problem. In the empirical approach you take experimental data and create a model based on a statistical fit to a model. With this approach it is possible to model huge structures, but it is difficult to know when it will fail. It is dependent upon having data that represent any situation you might encounter. $ In our group we therefore primarily work with ab-initio models. Here you try to find the basic principles underlying the experimental data and then create a model. $ We base our work on the DFT method, which is a stable workhorse of materials physics giving among many other things- very reliable vibrational energies.
- Mono-atomic gold chains are excellent structures to test and illustrate calculational schemes. $ First of all because they have been extensively studied. $ And secondly, it is clear from experiments that vibrations in the chains are heated when a current flows. And the heat dissipation was found in to be surprisingly low. Which means there is something for theorists to explain.
- Mono-atomic gold chains are excellent structures to test and illustrate calculational schemes. $ First of all because they have been extensively studied. $ And secondly, it is clear from experiments that vibrations in the chains are heated when a current flows. And the heat dissipation was found in to be surprisingly low. Which means there is something for theorists to explain.
- Vibrations in equilibrium and at low temperature are governed by two relatively simple equations. $ First, the dynamical matrix, K, is proportional to the second derivative of the total energy with respect to the position of the atoms. In a system with one degree of freedom this would simply be the spring constant. $ Secondly, there is the equation of motion which is simply a matrix version of Newtons second Law. If we have a finite structures and a way of calculating the total energy these two equations are all we need. But when we add the macroscopic crystals $ then the equation of motion has an infinite number of variables. This we cannot solve straigth-forwardly. (short 11)
- Vibrations in equilibrium and at low temperature are governed by two relatively simple equations. $ First, the dynamical matrix, K, is proportional to the second derivative of the total energy with respect to the position of the atoms. In a system with one degree of freedom this would simply be the spring constant. $ Secondly, there is the equation of motion which is simply a matrix version of Newtons second Law. If we have a finite structures and a way of calculating the total energy these two equations are all we need. But when we add the macroscopic crystals $ then the equation of motion has an infinite number of variables. This we cannot solve straigth-forwardly. (short 11)
- Vibrations in equilibrium and at low temperature are governed by two relatively simple equations. $ First, the dynamical matrix, K, is proportional to the second derivative of the total energy with respect to the position of the atoms. In a system with one degree of freedom this would simply be the spring constant. $ Secondly, there is the equation of motion which is simply a matrix version of Newtons second Law. If we have a finite structures and a way of calculating the total energy these two equations are all we need. But when we add the macroscopic crystals $ then the equation of motion has an infinite number of variables. This we cannot solve straigth-forwardly. (short 11)
- Vibrations in equilibrium and at low temperature are governed by two relatively simple equations. $ First, the dynamical matrix, K, is proportional to the second derivative of the total energy with respect to the position of the atoms. In a system with one degree of freedom this would simply be the spring constant. $ Secondly, there is the equation of motion which is simply a matrix version of Newtons second Law. If we have a finite structures and a way of calculating the total energy these two equations are all we need. But when we add the macroscopic crystals $ then the equation of motion has an infinite number of variables. This we cannot solve straigth-forwardly. (short 11)
- To handle infinite systems we define the retarded Green&#x2019;s function, D. The Green&#x2019;s function contains all the really interesting information about the vibrations, such as the frequency and the amplitude at a specific place in the structure. -the density of states The clever thing about the Green&#x2019;s function is that it is possible to reduce the problem to a much smaller one. $ A boundary term, called the self-energy, can account for the effect of the macroscopic crystals. The self-energy term is a property of the leads so different systems with the same leads could be investigated wit the same self-energy. Although constructing the self-energy is quite involved, I wont elaborate further due to time-constraints.
- To handle infinite systems we define the retarded Green&#x2019;s function, D. The Green&#x2019;s function contains all the really interesting information about the vibrations, such as the frequency and the amplitude at a specific place in the structure. -the density of states The clever thing about the Green&#x2019;s function is that it is possible to reduce the problem to a much smaller one. $ A boundary term, called the self-energy, can account for the effect of the macroscopic crystals. The self-energy term is a property of the leads so different systems with the same leads could be investigated wit the same self-energy. Although constructing the self-energy is quite involved, I wont elaborate further due to time-constraints.
- To handle infinite systems we define the retarded Green&#x2019;s function, D. The Green&#x2019;s function contains all the really interesting information about the vibrations, such as the frequency and the amplitude at a specific place in the structure. -the density of states The clever thing about the Green&#x2019;s function is that it is possible to reduce the problem to a much smaller one. $ A boundary term, called the self-energy, can account for the effect of the macroscopic crystals. The self-energy term is a property of the leads so different systems with the same leads could be investigated wit the same self-energy. Although constructing the self-energy is quite involved, I wont elaborate further due to time-constraints.
- To handle infinite systems we define the retarded Green&#x2019;s function, D. The Green&#x2019;s function contains all the really interesting information about the vibrations, such as the frequency and the amplitude at a specific place in the structure. -the density of states The clever thing about the Green&#x2019;s function is that it is possible to reduce the problem to a much smaller one. $ A boundary term, called the self-energy, can account for the effect of the macroscopic crystals. The self-energy term is a property of the leads so different systems with the same leads could be investigated wit the same self-energy. Although constructing the self-energy is quite involved, I wont elaborate further due to time-constraints.
- In two regions 1 and 2(13 atoms each), K is considered to be perturbed by A compared to the perfect surface. The self-energy can be calculated through the Green&#x2019;s function of a perfect surface. The size of the 1 and 2 regions are convergence parameters
- Periodicity of the surface. Periodic in directions parallel->fourier transform Direction perpendicular-periodicity beyond first layers-recursive method
- Dealing with infinite structures is a bit more complicated than dealing with finite ones. $ Let us look at a finite system of three atoms. The dynamics of this system can be analysed in terms of eigenmodes- distinct types of motion each with a specific frequency. $ If these modes are set in motion they will occilate at their respective frequencies indefinately. If we now couple this system to macroscopic leads $ we might still be able to define modes of the three-atom system- but some of these modes will be damped modes. $ The amplitude of the vibrations will decrease with time as energy is transferred to the leads.
- Dealing with infinite structures is a bit more complicated than dealing with finite ones. $ Let us look at a finite system of three atoms. The dynamics of this system can be analysed in terms of eigenmodes- distinct types of motion each with a specific frequency. $ If these modes are set in motion they will occilate at their respective frequencies indefinately. If we now couple this system to macroscopic leads $ we might still be able to define modes of the three-atom system- but some of these modes will be damped modes. $ The amplitude of the vibrations will decrease with time as energy is transferred to the leads.
- Dealing with infinite structures is a bit more complicated than dealing with finite ones. $ Let us look at a finite system of three atoms. The dynamics of this system can be analysed in terms of eigenmodes- distinct types of motion each with a specific frequency. $ If these modes are set in motion they will occilate at their respective frequencies indefinately. If we now couple this system to macroscopic leads $ we might still be able to define modes of the three-atom system- but some of these modes will be damped modes. $ The amplitude of the vibrations will decrease with time as energy is transferred to the leads.
- Dealing with infinite structures is a bit more complicated than dealing with finite ones. $ Let us look at a finite system of three atoms. The dynamics of this system can be analysed in terms of eigenmodes- distinct types of motion each with a specific frequency. $ If these modes are set in motion they will occilate at their respective frequencies indefinately. If we now couple this system to macroscopic leads $ we might still be able to define modes of the three-atom system- but some of these modes will be damped modes. $ The amplitude of the vibrations will decrease with time as energy is transferred to the leads.
- Dealing with infinite structures is a bit more complicated than dealing with finite ones. $ Let us look at a finite system of three atoms. The dynamics of this system can be analysed in terms of eigenmodes- distinct types of motion each with a specific frequency. $ If these modes are set in motion they will occilate at their respective frequencies indefinately. If we now couple this system to macroscopic leads $ we might still be able to define modes of the three-atom system- but some of these modes will be damped modes. $ The amplitude of the vibrations will decrease with time as energy is transferred to the leads.
- Dealing with infinite structures is a bit more complicated than dealing with finite ones. $ Let us look at a finite system of three atoms. The dynamics of this system can be analysed in terms of eigenmodes- distinct types of motion each with a specific frequency. $ If these modes are set in motion they will occilate at their respective frequencies indefinately. If we now couple this system to macroscopic leads $ we might still be able to define modes of the three-atom system- but some of these modes will be damped modes. $ The amplitude of the vibrations will decrease with time as energy is transferred to the leads.
- Dealing with infinite structures is a bit more complicated than dealing with finite ones. $ Let us look at a finite system of three atoms. The dynamics of this system can be analysed in terms of eigenmodes- distinct types of motion each with a specific frequency. $ If these modes are set in motion they will occilate at their respective frequencies indefinately. If we now couple this system to macroscopic leads $ we might still be able to define modes of the three-atom system- but some of these modes will be damped modes. $ The amplitude of the vibrations will decrease with time as energy is transferred to the leads.
- The damping of modes with time can be measured by the socalled Q-factor. $ The Q-factor measures how well a specific mode retains energy, so the higher Q-factor the lower the heat dissipation for this mode. $ Here we see the amplitude of two modes with the same frequency but with vastly different Q-factors. The initial motion of the Low-Q-factor mode quickly disappears while the high-Q-factor mode retains a large amplitude for a much longer period.
- The damping of modes with time can be measured by the socalled Q-factor. $ The Q-factor measures how well a specific mode retains energy, so the higher Q-factor the lower the heat dissipation for this mode. $ Here we see the amplitude of two modes with the same frequency but with vastly different Q-factors. The initial motion of the Low-Q-factor mode quickly disappears while the high-Q-factor mode retains a large amplitude for a much longer period.
- Now, all these methods that I have discussed actually had to be implemented. And this implementation was no a simple task. This diagram $ represents the core part of program I&#x2019;ve developed during my Ph.D studies. It shows how to get the Green&#x2019;s function... Self-energy much more complicated than I let on. Parts of the program uses work by Thomas Frederiksen and Magnus Paulsson, but an additional 8000 lines of code has been written during the Phd. project which makes this a quite extensive programming project.
- Let us now move on to examine some of the results I have found using these methods. First of all, let us take a closer at the gold chains that was used as an example in the method section of this talk.
- One of the questions we wanted to answer is: How sensitive is the heat dissipation is to the exact configuration of the gold chains? Plenty of studies use very crude models of heat dissipation so we basically wanted to see if the work I&#x2019;d done was really nessecary. Don&#x2019;t let me hold you in suspense...we found that it was indeed nescesary:-) We made an extensive study of chains with different lengths, $3, $4, $5, $6, and $7 atom chains $ at different strain- pulling and pushing the chains as far as they would go without breaking in each direction $ And finally we investigated chains between differently oriented crystals.
- One of the questions we wanted to answer is: How sensitive is the heat dissipation is to the exact configuration of the gold chains? Plenty of studies use very crude models of heat dissipation so we basically wanted to see if the work I&#x2019;d done was really nessecary. Don&#x2019;t let me hold you in suspense...we found that it was indeed nescesary:-) We made an extensive study of chains with different lengths, $3, $4, $5, $6, and $7 atom chains $ at different strain- pulling and pushing the chains as far as they would go without breaking in each direction $ And finally we investigated chains between differently oriented crystals.
- One of the questions we wanted to answer is: How sensitive is the heat dissipation is to the exact configuration of the gold chains? Plenty of studies use very crude models of heat dissipation so we basically wanted to see if the work I&#x2019;d done was really nessecary. Don&#x2019;t let me hold you in suspense...we found that it was indeed nescesary:-) We made an extensive study of chains with different lengths, $3, $4, $5, $6, and $7 atom chains $ at different strain- pulling and pushing the chains as far as they would go without breaking in each direction $ And finally we investigated chains between differently oriented crystals.
- One of the questions we wanted to answer is: How sensitive is the heat dissipation is to the exact configuration of the gold chains? Plenty of studies use very crude models of heat dissipation so we basically wanted to see if the work I&#x2019;d done was really nessecary. Don&#x2019;t let me hold you in suspense...we found that it was indeed nescesary:-) We made an extensive study of chains with different lengths, $3, $4, $5, $6, and $7 atom chains $ at different strain- pulling and pushing the chains as far as they would go without breaking in each direction $ And finally we investigated chains between differently oriented crystals.
- One of the questions we wanted to answer is: How sensitive is the heat dissipation is to the exact configuration of the gold chains? Plenty of studies use very crude models of heat dissipation so we basically wanted to see if the work I&#x2019;d done was really nessecary. Don&#x2019;t let me hold you in suspense...we found that it was indeed nescesary:-) We made an extensive study of chains with different lengths, $3, $4, $5, $6, and $7 atom chains $ at different strain- pulling and pushing the chains as far as they would go without breaking in each direction $ And finally we investigated chains between differently oriented crystals.
- One of the questions we wanted to answer is: How sensitive is the heat dissipation is to the exact configuration of the gold chains? Plenty of studies use very crude models of heat dissipation so we basically wanted to see if the work I&#x2019;d done was really nessecary. Don&#x2019;t let me hold you in suspense...we found that it was indeed nescesary:-) We made an extensive study of chains with different lengths, $3, $4, $5, $6, and $7 atom chains $ at different strain- pulling and pushing the chains as far as they would go without breaking in each direction $ And finally we investigated chains between differently oriented crystals.
- One of the questions we wanted to answer is: How sensitive is the heat dissipation is to the exact configuration of the gold chains? Plenty of studies use very crude models of heat dissipation so we basically wanted to see if the work I&#x2019;d done was really nessecary. Don&#x2019;t let me hold you in suspense...we found that it was indeed nescesary:-) We made an extensive study of chains with different lengths, $3, $4, $5, $6, and $7 atom chains $ at different strain- pulling and pushing the chains as far as they would go without breaking in each direction $ And finally we investigated chains between differently oriented crystals.
- One of the questions we wanted to answer is: How sensitive is the heat dissipation is to the exact configuration of the gold chains? Plenty of studies use very crude models of heat dissipation so we basically wanted to see if the work I&#x2019;d done was really nessecary. Don&#x2019;t let me hold you in suspense...we found that it was indeed nescesary:-) We made an extensive study of chains with different lengths, $3, $4, $5, $6, and $7 atom chains $ at different strain- pulling and pushing the chains as far as they would go without breaking in each direction $ And finally we investigated chains between differently oriented crystals.
- One of the questions we wanted to answer is: How sensitive is the heat dissipation is to the exact configuration of the gold chains? Plenty of studies use very crude models of heat dissipation so we basically wanted to see if the work I&#x2019;d done was really nessecary. Don&#x2019;t let me hold you in suspense...we found that it was indeed nescesary:-) We made an extensive study of chains with different lengths, $3, $4, $5, $6, and $7 atom chains $ at different strain- pulling and pushing the chains as far as they would go without breaking in each direction $ And finally we investigated chains between differently oriented crystals.
- One of the questions we wanted to answer is: How sensitive is the heat dissipation is to the exact configuration of the gold chains? Plenty of studies use very crude models of heat dissipation so we basically wanted to see if the work I&#x2019;d done was really nessecary. Don&#x2019;t let me hold you in suspense...we found that it was indeed nescesary:-) We made an extensive study of chains with different lengths, $3, $4, $5, $6, and $7 atom chains $ at different strain- pulling and pushing the chains as far as they would go without breaking in each direction $ And finally we investigated chains between differently oriented crystals.
- One of the questions we wanted to answer is: How sensitive is the heat dissipation is to the exact configuration of the gold chains? Plenty of studies use very crude models of heat dissipation so we basically wanted to see if the work I&#x2019;d done was really nessecary. Don&#x2019;t let me hold you in suspense...we found that it was indeed nescesary:-) We made an extensive study of chains with different lengths, $3, $4, $5, $6, and $7 atom chains $ at different strain- pulling and pushing the chains as far as they would go without breaking in each direction $ And finally we investigated chains between differently oriented crystals.
- One of the questions we wanted to answer is: How sensitive is the heat dissipation is to the exact configuration of the gold chains? Plenty of studies use very crude models of heat dissipation so we basically wanted to see if the work I&#x2019;d done was really nessecary. Don&#x2019;t let me hold you in suspense...we found that it was indeed nescesary:-) We made an extensive study of chains with different lengths, $3, $4, $5, $6, and $7 atom chains $ at different strain- pulling and pushing the chains as far as they would go without breaking in each direction $ And finally we investigated chains between differently oriented crystals.
- One of the questions we wanted to answer is: How sensitive is the heat dissipation is to the exact configuration of the gold chains? Plenty of studies use very crude models of heat dissipation so we basically wanted to see if the work I&#x2019;d done was really nessecary. Don&#x2019;t let me hold you in suspense...we found that it was indeed nescesary:-) We made an extensive study of chains with different lengths, $3, $4, $5, $6, and $7 atom chains $ at different strain- pulling and pushing the chains as far as they would go without breaking in each direction $ And finally we investigated chains between differently oriented crystals.
- One of the questions we wanted to answer is: How sensitive is the heat dissipation is to the exact configuration of the gold chains? Plenty of studies use very crude models of heat dissipation so we basically wanted to see if the work I&#x2019;d done was really nessecary. Don&#x2019;t let me hold you in suspense...we found that it was indeed nescesary:-) We made an extensive study of chains with different lengths, $3, $4, $5, $6, and $7 atom chains $ at different strain- pulling and pushing the chains as far as they would go without breaking in each direction $ And finally we investigated chains between differently oriented crystals.
- One of the questions we wanted to answer is: How sensitive is the heat dissipation is to the exact configuration of the gold chains? Plenty of studies use very crude models of heat dissipation so we basically wanted to see if the work I&#x2019;d done was really nessecary. Don&#x2019;t let me hold you in suspense...we found that it was indeed nescesary:-) We made an extensive study of chains with different lengths, $3, $4, $5, $6, and $7 atom chains $ at different strain- pulling and pushing the chains as far as they would go without breaking in each direction $ And finally we investigated chains between differently oriented crystals.
- Needless to say the information on all the modes of all the variations of gold chains is huge. So some effort is needed to condense all this information. To do this we represent each mode of the system as a dot. $ The modes with a large amplitude inside to chain are represented as black dots The modes with large Q-factor $ are represented as large dots. These modes dissipate heat very slowly. As I&#x2019;ve illustrated here the modes with high Q-factor are the ones that do not cause the leads to osccilate. $ And finally in the lower right corner we see the type of mode that is far most numorous in the total system since all modes in the crystals are of this type, low Q-factor and low amplitude in the chain.
- Needless to say the information on all the modes of all the variations of gold chains is huge. So some effort is needed to condense all this information. To do this we represent each mode of the system as a dot. $ The modes with a large amplitude inside to chain are represented as black dots The modes with large Q-factor $ are represented as large dots. These modes dissipate heat very slowly. As I&#x2019;ve illustrated here the modes with high Q-factor are the ones that do not cause the leads to osccilate. $ And finally in the lower right corner we see the type of mode that is far most numorous in the total system since all modes in the crystals are of this type, low Q-factor and low amplitude in the chain.
- Needless to say the information on all the modes of all the variations of gold chains is huge. So some effort is needed to condense all this information. To do this we represent each mode of the system as a dot. $ The modes with a large amplitude inside to chain are represented as black dots The modes with large Q-factor $ are represented as large dots. These modes dissipate heat very slowly. As I&#x2019;ve illustrated here the modes with high Q-factor are the ones that do not cause the leads to osccilate. $ And finally in the lower right corner we see the type of mode that is far most numorous in the total system since all modes in the crystals are of this type, low Q-factor and low amplitude in the chain.
- Needless to say the information on all the modes of all the variations of gold chains is huge. So some effort is needed to condense all this information. To do this we represent each mode of the system as a dot. $ The modes with a large amplitude inside to chain are represented as black dots The modes with large Q-factor $ are represented as large dots. These modes dissipate heat very slowly. As I&#x2019;ve illustrated here the modes with high Q-factor are the ones that do not cause the leads to osccilate. $ And finally in the lower right corner we see the type of mode that is far most numorous in the total system since all modes in the crystals are of this type, low Q-factor and low amplitude in the chain.
- Needless to say the information on all the modes of all the variations of gold chains is huge. So some effort is needed to condense all this information. To do this we represent each mode of the system as a dot. $ The modes with a large amplitude inside to chain are represented as black dots The modes with large Q-factor $ are represented as large dots. These modes dissipate heat very slowly. As I&#x2019;ve illustrated here the modes with high Q-factor are the ones that do not cause the leads to osccilate. $ And finally in the lower right corner we see the type of mode that is far most numorous in the total system since all modes in the crystals are of this type, low Q-factor and low amplitude in the chain.
- Needless to say the information on all the modes of all the variations of gold chains is huge. So some effort is needed to condense all this information. To do this we represent each mode of the system as a dot. $ The modes with a large amplitude inside to chain are represented as black dots The modes with large Q-factor $ are represented as large dots. These modes dissipate heat very slowly. As I&#x2019;ve illustrated here the modes with high Q-factor are the ones that do not cause the leads to osccilate. $ And finally in the lower right corner we see the type of mode that is far most numorous in the total system since all modes in the crystals are of this type, low Q-factor and low amplitude in the chain.
- Needless to say the information on all the modes of all the variations of gold chains is huge. So some effort is needed to condense all this information. To do this we represent each mode of the system as a dot. $ The modes with a large amplitude inside to chain are represented as black dots The modes with large Q-factor $ are represented as large dots. These modes dissipate heat very slowly. As I&#x2019;ve illustrated here the modes with high Q-factor are the ones that do not cause the leads to osccilate. $ And finally in the lower right corner we see the type of mode that is far most numorous in the total system since all modes in the crystals are of this type, low Q-factor and low amplitude in the chain.
- Needless to say the information on all the modes of all the variations of gold chains is huge. So some effort is needed to condense all this information. To do this we represent each mode of the system as a dot. $ The modes with a large amplitude inside to chain are represented as black dots The modes with large Q-factor $ are represented as large dots. These modes dissipate heat very slowly. As I&#x2019;ve illustrated here the modes with high Q-factor are the ones that do not cause the leads to osccilate. $ And finally in the lower right corner we see the type of mode that is far most numorous in the total system since all modes in the crystals are of this type, low Q-factor and low amplitude in the chain.
- Needless to say the information on all the modes of all the variations of gold chains is huge. So some effort is needed to condense all this information. To do this we represent each mode of the system as a dot. $ The modes with a large amplitude inside to chain are represented as black dots The modes with large Q-factor $ are represented as large dots. These modes dissipate heat very slowly. As I&#x2019;ve illustrated here the modes with high Q-factor are the ones that do not cause the leads to osccilate. $ And finally in the lower right corner we see the type of mode that is far most numorous in the total system since all modes in the crystals are of this type, low Q-factor and low amplitude in the chain.
- Needless to say the information on all the modes of all the variations of gold chains is huge. So some effort is needed to condense all this information. To do this we represent each mode of the system as a dot. $ The modes with a large amplitude inside to chain are represented as black dots The modes with large Q-factor $ are represented as large dots. These modes dissipate heat very slowly. As I&#x2019;ve illustrated here the modes with high Q-factor are the ones that do not cause the leads to osccilate. $ And finally in the lower right corner we see the type of mode that is far most numorous in the total system since all modes in the crystals are of this type, low Q-factor and low amplitude in the chain.
- Needless to say the information on all the modes of all the variations of gold chains is huge. So some effort is needed to condense all this information. To do this we represent each mode of the system as a dot. $ The modes with a large amplitude inside to chain are represented as black dots The modes with large Q-factor $ are represented as large dots. These modes dissipate heat very slowly. As I&#x2019;ve illustrated here the modes with high Q-factor are the ones that do not cause the leads to osccilate. $ And finally in the lower right corner we see the type of mode that is far most numorous in the total system since all modes in the crystals are of this type, low Q-factor and low amplitude in the chain.
- Needless to say the information on all the modes of all the variations of gold chains is huge. So some effort is needed to condense all this information. To do this we represent each mode of the system as a dot. $ The modes with a large amplitude inside to chain are represented as black dots The modes with large Q-factor $ are represented as large dots. These modes dissipate heat very slowly. As I&#x2019;ve illustrated here the modes with high Q-factor are the ones that do not cause the leads to osccilate. $ And finally in the lower right corner we see the type of mode that is far most numorous in the total system since all modes in the crystals are of this type, low Q-factor and low amplitude in the chain.
- Needless to say the information on all the modes of all the variations of gold chains is huge. So some effort is needed to condense all this information. To do this we represent each mode of the system as a dot. $ The modes with a large amplitude inside to chain are represented as black dots The modes with large Q-factor $ are represented as large dots. These modes dissipate heat very slowly. As I&#x2019;ve illustrated here the modes with high Q-factor are the ones that do not cause the leads to osccilate. $ And finally in the lower right corner we see the type of mode that is far most numorous in the total system since all modes in the crystals are of this type, low Q-factor and low amplitude in the chain.
- Needless to say the information on all the modes of all the variations of gold chains is huge. So some effort is needed to condense all this information. To do this we represent each mode of the system as a dot. $ The modes with a large amplitude inside to chain are represented as black dots The modes with large Q-factor $ are represented as large dots. These modes dissipate heat very slowly. As I&#x2019;ve illustrated here the modes with high Q-factor are the ones that do not cause the leads to osccilate. $ And finally in the lower right corner we see the type of mode that is far most numorous in the total system since all modes in the crystals are of this type, low Q-factor and low amplitude in the chain.
- Let us look at the chain with 7 atoms at different strains $ In this plot the horisontal axis is the average distance between the atoms in the chain. Each vertical line of dots represent all the modes from one calculation. I would like to direct your attention to the diagonal line of modes with high Q-factor in this plot. $ These modes represent a similar type of motion inside the chain and yet we see a huge variation in the Q-factor- from 40 to 1500 with only a minor increase in the average distance between the atoms in the chain. $ This is the calculated mode osccilation corresponding to the high Q-factor mode. (10+11)
- Let us look at the chain with 7 atoms at different strains $ In this plot the horisontal axis is the average distance between the atoms in the chain. Each vertical line of dots represent all the modes from one calculation. I would like to direct your attention to the diagonal line of modes with high Q-factor in this plot. $ These modes represent a similar type of motion inside the chain and yet we see a huge variation in the Q-factor- from 40 to 1500 with only a minor increase in the average distance between the atoms in the chain. $ This is the calculated mode osccilation corresponding to the high Q-factor mode. (10+11)
- Let us look at the chain with 7 atoms at different strains $ In this plot the horisontal axis is the average distance between the atoms in the chain. Each vertical line of dots represent all the modes from one calculation. I would like to direct your attention to the diagonal line of modes with high Q-factor in this plot. $ These modes represent a similar type of motion inside the chain and yet we see a huge variation in the Q-factor- from 40 to 1500 with only a minor increase in the average distance between the atoms in the chain. $ This is the calculated mode osccilation corresponding to the high Q-factor mode. (10+11)
- Let us look at the chain with 7 atoms at different strains $ In this plot the horisontal axis is the average distance between the atoms in the chain. Each vertical line of dots represent all the modes from one calculation. I would like to direct your attention to the diagonal line of modes with high Q-factor in this plot. $ These modes represent a similar type of motion inside the chain and yet we see a huge variation in the Q-factor- from 40 to 1500 with only a minor increase in the average distance between the atoms in the chain. $ This is the calculated mode osccilation corresponding to the high Q-factor mode. (10+11)
- Short chains->long chains Larger Q-factors Greater variation in damping (variation an order of magnitude larger)&#x200F;
- Similar peak energies Generally stronger localisation for (111) than (100)&#x200F; Different type of variation Little difference when breaking symmetry(not shown)&#x200F;
- I will not go through the entire study with 3-7 length chains and different crystal directions, but only mention the main conclusions. $First of all, we found that the damping can fall off to extremely low values, as low as 5 micro -electron-volts- a value 1000s of times lower than the peak energy. $This minimal value also matches that of a previous study based on experimental data. $Secondly, the damping is very sensitive -to the atomic structure of the system, to the crystal direction, to the length of the chain and to the precise value of the average distance between the atoms. The precise atomic configuration around these chains must be taken accurately into account to predict heat dissipation. $ This is perhaps the most important conclusion, since similar narrow junctions between gold electrodes are often investigated -both experimentally and theoretically.
- I will not go through the entire study with 3-7 length chains and different crystal directions, but only mention the main conclusions. $First of all, we found that the damping can fall off to extremely low values, as low as 5 micro -electron-volts- a value 1000s of times lower than the peak energy. $This minimal value also matches that of a previous study based on experimental data. $Secondly, the damping is very sensitive -to the atomic structure of the system, to the crystal direction, to the length of the chain and to the precise value of the average distance between the atoms. The precise atomic configuration around these chains must be taken accurately into account to predict heat dissipation. $ This is perhaps the most important conclusion, since similar narrow junctions between gold electrodes are often investigated -both experimentally and theoretically.
- I will not go through the entire study with 3-7 length chains and different crystal directions, but only mention the main conclusions. $First of all, we found that the damping can fall off to extremely low values, as low as 5 micro -electron-volts- a value 1000s of times lower than the peak energy. $This minimal value also matches that of a previous study based on experimental data. $Secondly, the damping is very sensitive -to the atomic structure of the system, to the crystal direction, to the length of the chain and to the precise value of the average distance between the atoms. The precise atomic configuration around these chains must be taken accurately into account to predict heat dissipation. $ This is perhaps the most important conclusion, since similar narrow junctions between gold electrodes are often investigated -both experimentally and theoretically.
- I will not go through the entire study with 3-7 length chains and different crystal directions, but only mention the main conclusions. $First of all, we found that the damping can fall off to extremely low values, as low as 5 micro -electron-volts- a value 1000s of times lower than the peak energy. $This minimal value also matches that of a previous study based on experimental data. $Secondly, the damping is very sensitive -to the atomic structure of the system, to the crystal direction, to the length of the chain and to the precise value of the average distance between the atoms. The precise atomic configuration around these chains must be taken accurately into account to predict heat dissipation. $ This is perhaps the most important conclusion, since similar narrow junctions between gold electrodes are often investigated -both experimentally and theoretically.
- I will not go through the entire study with 3-7 length chains and different crystal directions, but only mention the main conclusions. $First of all, we found that the damping can fall off to extremely low values, as low as 5 micro -electron-volts- a value 1000s of times lower than the peak energy. $This minimal value also matches that of a previous study based on experimental data. $Secondly, the damping is very sensitive -to the atomic structure of the system, to the crystal direction, to the length of the chain and to the precise value of the average distance between the atoms. The precise atomic configuration around these chains must be taken accurately into account to predict heat dissipation. $ This is perhaps the most important conclusion, since similar narrow junctions between gold electrodes are often investigated -both experimentally and theoretically.
- I&#x2019;ve mentioned graphene a few times already but now we move on to section of the talk entirely devoted to this remarkable material.
- Graphene is a single layer or few layers of carbon atoms arranged in a hexagonal grid and it has some very exotic material properties. In the picture you see the electronic energy dispersion of graphene. We see that the two seperate surfaces are just touching in six distinct points at the Fermi surface- This is a highly unusual feature indeed. But it&#x2019;s actually not what makes graphene so interesting from an applications point of view. instead it&#x2019;s the $ increadible strength and flexibility of graphene $ the extrordinary mobility of electrons in graphene that makes very fast electronics possible. $ and finally the hexagonal structure, common to graphene and carbon nanotubes, is a very effective heat conductor. And, carbon is literally dirt cheap. (short 11+10)
- Graphene is a single layer or few layers of carbon atoms arranged in a hexagonal grid and it has some very exotic material properties. In the picture you see the electronic energy dispersion of graphene. We see that the two seperate surfaces are just touching in six distinct points at the Fermi surface- This is a highly unusual feature indeed. But it&#x2019;s actually not what makes graphene so interesting from an applications point of view. instead it&#x2019;s the $ increadible strength and flexibility of graphene $ the extrordinary mobility of electrons in graphene that makes very fast electronics possible. $ and finally the hexagonal structure, common to graphene and carbon nanotubes, is a very effective heat conductor. And, carbon is literally dirt cheap. (short 11+10)
- Graphene is a single layer or few layers of carbon atoms arranged in a hexagonal grid and it has some very exotic material properties. In the picture you see the electronic energy dispersion of graphene. We see that the two seperate surfaces are just touching in six distinct points at the Fermi surface- This is a highly unusual feature indeed. But it&#x2019;s actually not what makes graphene so interesting from an applications point of view. instead it&#x2019;s the $ increadible strength and flexibility of graphene $ the extrordinary mobility of electrons in graphene that makes very fast electronics possible. $ and finally the hexagonal structure, common to graphene and carbon nanotubes, is a very effective heat conductor. And, carbon is literally dirt cheap. (short 11+10)
- Graphene is a single layer or few layers of carbon atoms arranged in a hexagonal grid and it has some very exotic material properties. In the picture you see the electronic energy dispersion of graphene. We see that the two seperate surfaces are just touching in six distinct points at the Fermi surface- This is a highly unusual feature indeed. But it&#x2019;s actually not what makes graphene so interesting from an applications point of view. instead it&#x2019;s the $ increadible strength and flexibility of graphene $ the extrordinary mobility of electrons in graphene that makes very fast electronics possible. $ and finally the hexagonal structure, common to graphene and carbon nanotubes, is a very effective heat conductor. And, carbon is literally dirt cheap. (short 11+10)
- Graphene is a single layer or few layers of carbon atoms arranged in a hexagonal grid and it has some very exotic material properties. In the picture you see the electronic energy dispersion of graphene. We see that the two seperate surfaces are just touching in six distinct points at the Fermi surface- This is a highly unusual feature indeed. But it&#x2019;s actually not what makes graphene so interesting from an applications point of view. instead it&#x2019;s the $ increadible strength and flexibility of graphene $ the extrordinary mobility of electrons in graphene that makes very fast electronics possible. $ and finally the hexagonal structure, common to graphene and carbon nanotubes, is a very effective heat conductor. And, carbon is literally dirt cheap. (short 11+10)
- Graphene is a single layer or few layers of carbon atoms arranged in a hexagonal grid and it has some very exotic material properties. In the picture you see the electronic energy dispersion of graphene. We see that the two seperate surfaces are just touching in six distinct points at the Fermi surface- This is a highly unusual feature indeed. But it&#x2019;s actually not what makes graphene so interesting from an applications point of view. instead it&#x2019;s the $ increadible strength and flexibility of graphene $ the extrordinary mobility of electrons in graphene that makes very fast electronics possible. $ and finally the hexagonal structure, common to graphene and carbon nanotubes, is a very effective heat conductor. And, carbon is literally dirt cheap. (short 11+10)
- Graphene is a single layer or few layers of carbon atoms arranged in a hexagonal grid and it has some very exotic material properties. In the picture you see the electronic energy dispersion of graphene. We see that the two seperate surfaces are just touching in six distinct points at the Fermi surface- This is a highly unusual feature indeed. But it&#x2019;s actually not what makes graphene so interesting from an applications point of view. instead it&#x2019;s the $ increadible strength and flexibility of graphene $ the extrordinary mobility of electrons in graphene that makes very fast electronics possible. $ and finally the hexagonal structure, common to graphene and carbon nanotubes, is a very effective heat conductor. And, carbon is literally dirt cheap. (short 11+10)
- But what more specifically motivated us to study graphene was this experiment. In this frame we see a structure of disordered graphene flakes overlapping. I think it is even possible to see individual atoms if you strain your eyes. $ As the experimenters passed an electronic current through the sample atoms started evaporating. $ If you follow the red arrow we see one type of edge growing while another is retreating. $ $ Finally one of the edges have completely disappeared. The net effect is that fewer distinct edges exist which reduces the roughness of the flake edges. $ What we see here..... The really interesting part is that the effect depends on the direction of the electronic current compared to the direction of the edges. The sample is not simply heated up- it is a more precise manipulation of the edges. And who knows what level of control we could acheive if we could understand this process
- But what more specifically motivated us to study graphene was this experiment. In this frame we see a structure of disordered graphene flakes overlapping. I think it is even possible to see individual atoms if you strain your eyes. $ As the experimenters passed an electronic current through the sample atoms started evaporating. $ If you follow the red arrow we see one type of edge growing while another is retreating. $ $ Finally one of the edges have completely disappeared. The net effect is that fewer distinct edges exist which reduces the roughness of the flake edges. $ What we see here..... The really interesting part is that the effect depends on the direction of the electronic current compared to the direction of the edges. The sample is not simply heated up- it is a more precise manipulation of the edges. And who knows what level of control we could acheive if we could understand this process
- But what more specifically motivated us to study graphene was this experiment. In this frame we see a structure of disordered graphene flakes overlapping. I think it is even possible to see individual atoms if you strain your eyes. $ As the experimenters passed an electronic current through the sample atoms started evaporating. $ If you follow the red arrow we see one type of edge growing while another is retreating. $ $ Finally one of the edges have completely disappeared. The net effect is that fewer distinct edges exist which reduces the roughness of the flake edges. $ What we see here..... The really interesting part is that the effect depends on the direction of the electronic current compared to the direction of the edges. The sample is not simply heated up- it is a more precise manipulation of the edges. And who knows what level of control we could acheive if we could understand this process
- But what more specifically motivated us to study graphene was this experiment. In this frame we see a structure of disordered graphene flakes overlapping. I think it is even possible to see individual atoms if you strain your eyes. $ As the experimenters passed an electronic current through the sample atoms started evaporating. $ If you follow the red arrow we see one type of edge growing while another is retreating. $ $ Finally one of the edges have completely disappeared. The net effect is that fewer distinct edges exist which reduces the roughness of the flake edges. $ What we see here..... The really interesting part is that the effect depends on the direction of the electronic current compared to the direction of the edges. The sample is not simply heated up- it is a more precise manipulation of the edges. And who knows what level of control we could acheive if we could understand this process
- But what more specifically motivated us to study graphene was this experiment. In this frame we see a structure of disordered graphene flakes overlapping. I think it is even possible to see individual atoms if you strain your eyes. $ As the experimenters passed an electronic current through the sample atoms started evaporating. $ If you follow the red arrow we see one type of edge growing while another is retreating. $ $ Finally one of the edges have completely disappeared. The net effect is that fewer distinct edges exist which reduces the roughness of the flake edges. $ What we see here..... The really interesting part is that the effect depends on the direction of the electronic current compared to the direction of the edges. The sample is not simply heated up- it is a more precise manipulation of the edges. And who knows what level of control we could acheive if we could understand this process
- We made an investigation of the vibrations in structures that mixes edges of different type. And we believe to have found a very good candidate for the process behind the evaporation of the edges. For these system we noticed a class of modes that all share some characteristics. $ The modes are similar to modes of finite structures with a definite frequency and no damping and the modes are combinations of simple types of motion at a specific edge- the socalled armchair edge. $ This basic type of motion is not damped by the vibrations in the graphene sheet because the bond between the two outer carbon atoms is very strong- even stronger than the bond inside the graphene sheet which is one of the strongest bonds known. This mismatch makes the vibration incompatible with the vibrations in the sheet.
- We made an investigation of the vibrations in structures that mixes edges of different type. And we believe to have found a very good candidate for the process behind the evaporation of the edges. For these system we noticed a class of modes that all share some characteristics. $ The modes are similar to modes of finite structures with a definite frequency and no damping and the modes are combinations of simple types of motion at a specific edge- the socalled armchair edge. $ This basic type of motion is not damped by the vibrations in the graphene sheet because the bond between the two outer carbon atoms is very strong- even stronger than the bond inside the graphene sheet which is one of the strongest bonds known. This mismatch makes the vibration incompatible with the vibrations in the sheet.
- We made an investigation of the vibrations in structures that mixes edges of different type. And we believe to have found a very good candidate for the process behind the evaporation of the edges. For these system we noticed a class of modes that all share some characteristics. $ The modes are similar to modes of finite structures with a definite frequency and no damping and the modes are combinations of simple types of motion at a specific edge- the socalled armchair edge. $ This basic type of motion is not damped by the vibrations in the graphene sheet because the bond between the two outer carbon atoms is very strong- even stronger than the bond inside the graphene sheet which is one of the strongest bonds known. This mismatch makes the vibration incompatible with the vibrations in the sheet.
- When a mode is coupled to large reservoir of vibrations then the temperature of the mode is the same as that of the reservoir. It could actually just as well be considered as part of the reservoir. It doesn&#x2019;t really matter that electrons also couple to the mode $ because this coupling will mostly be feable in comparison. But if the coupling to the vibrations isen&#x2019;t there $ then it is a different matter. Then the mode will only exchange energy with electrons, and even if this happens rarely- the mode will reach an equilibrium with the electrons. $ If no electronic current flows the mode will simply have the same temperature as the electronic system. But if a current flows however something much more violent- and complicated $ can happen.
- When a mode is coupled to large reservoir of vibrations then the temperature of the mode is the same as that of the reservoir. It could actually just as well be considered as part of the reservoir. It doesn&#x2019;t really matter that electrons also couple to the mode $ because this coupling will mostly be feable in comparison. But if the coupling to the vibrations isen&#x2019;t there $ then it is a different matter. Then the mode will only exchange energy with electrons, and even if this happens rarely- the mode will reach an equilibrium with the electrons. $ If no electronic current flows the mode will simply have the same temperature as the electronic system. But if a current flows however something much more violent- and complicated $ can happen.
- When a mode is coupled to large reservoir of vibrations then the temperature of the mode is the same as that of the reservoir. It could actually just as well be considered as part of the reservoir. It doesn&#x2019;t really matter that electrons also couple to the mode $ because this coupling will mostly be feable in comparison. But if the coupling to the vibrations isen&#x2019;t there $ then it is a different matter. Then the mode will only exchange energy with electrons, and even if this happens rarely- the mode will reach an equilibrium with the electrons. $ If no electronic current flows the mode will simply have the same temperature as the electronic system. But if a current flows however something much more violent- and complicated $ can happen.
- When a mode is coupled to large reservoir of vibrations then the temperature of the mode is the same as that of the reservoir. It could actually just as well be considered as part of the reservoir. It doesn&#x2019;t really matter that electrons also couple to the mode $ because this coupling will mostly be feable in comparison. But if the coupling to the vibrations isen&#x2019;t there $ then it is a different matter. Then the mode will only exchange energy with electrons, and even if this happens rarely- the mode will reach an equilibrium with the electrons. $ If no electronic current flows the mode will simply have the same temperature as the electronic system. But if a current flows however something much more violent- and complicated $ can happen.
- When a mode is coupled to large reservoir of vibrations then the temperature of the mode is the same as that of the reservoir. It could actually just as well be considered as part of the reservoir. It doesn&#x2019;t really matter that electrons also couple to the mode $ because this coupling will mostly be feable in comparison. But if the coupling to the vibrations isen&#x2019;t there $ then it is a different matter. Then the mode will only exchange energy with electrons, and even if this happens rarely- the mode will reach an equilibrium with the electrons. $ If no electronic current flows the mode will simply have the same temperature as the electronic system. But if a current flows however something much more violent- and complicated $ can happen.
- Let us examine what happens when a current is flowing from left to right in this system. $ If and electron comes in from the left there are several thing that can happen. $ First of all, there is a chance that the electron reaches the right without exchanging energy with the mode $ Secondly, the electron can exchange energy with the mode and scatter back $ This is a diagram of the potential energy of electrons, the boxes represent occupied electronic state. And in this picture the bias that allows the current to flow is simply the difference in height between the occupied levels. $The electron comes in from one of the occupied levels in the left and exchanges energy with the vibration. $ If the electron scatteres back then the vibration cannot gain energy because the electron cannot go down in energy since these states are already occupied. $ The vibration can however loose energy because the electron can go up in energy. So backward scattering can only cause the vibration to loose energy. $ If the electron exchanges energy with the mode and scatters forward however then the vibration can both $loose -and $gain energy All these three events happen randomly- sometimes the vibration gets a little energy and sometimes it is taken away. The average energy contained in the mode we can interpret as an effective temperature. How hot the mode gets is determined by the relative probability of the forward and backwards scattering which is something that varies from mode to mode. The experts will realize I&#x2019;ve assumed that the system is conducting, which we have verified that this is. (11+10+6)
- Let us examine what happens when a current is flowing from left to right in this system. $ If and electron comes in from the left there are several thing that can happen. $ First of all, there is a chance that the electron reaches the right without exchanging energy with the mode $ Secondly, the electron can exchange energy with the mode and scatter back $ This is a diagram of the potential energy of electrons, the boxes represent occupied electronic state. And in this picture the bias that allows the current to flow is simply the difference in height between the occupied levels. $The electron comes in from one of the occupied levels in the left and exchanges energy with the vibration. $ If the electron scatteres back then the vibration cannot gain energy because the electron cannot go down in energy since these states are already occupied. $ The vibration can however loose energy because the electron can go up in energy. So backward scattering can only cause the vibration to loose energy. $ If the electron exchanges energy with the mode and scatters forward however then the vibration can both $loose -and $gain energy All these three events happen randomly- sometimes the vibration gets a little energy and sometimes it is taken away. The average energy contained in the mode we can interpret as an effective temperature. How hot the mode gets is determined by the relative probability of the forward and backwards scattering which is something that varies from mode to mode. The experts will realize I&#x2019;ve assumed that the system is conducting, which we have verified that this is. (11+10+6)
- Let us examine what happens when a current is flowing from left to right in this system. $ If and electron comes in from the left there are several thing that can happen. $ First of all, there is a chance that the electron reaches the right without exchanging energy with the mode $ Secondly, the electron can exchange energy with the mode and scatter back $ This is a diagram of the potential energy of electrons, the boxes represent occupied electronic state. And in this picture the bias that allows the current to flow is simply the difference in height between the occupied levels. $The electron comes in from one of the occupied levels in the left and exchanges energy with the vibration. $ If the electron scatteres back then the vibration cannot gain energy because the electron cannot go down in energy since these states are already occupied. $ The vibration can however loose energy because the electron can go up in energy. So backward scattering can only cause the vibration to loose energy. $ If the electron exchanges energy with the mode and scatters forward however then the vibration can both $loose -and $gain energy All these three events happen randomly- sometimes the vibration gets a little energy and sometimes it is taken away. The average energy contained in the mode we can interpret as an effective temperature. How hot the mode gets is determined by the relative probability of the forward and backwards scattering which is something that varies from mode to mode. The experts will realize I&#x2019;ve assumed that the system is conducting, which we have verified that this is. (11+10+6)
- Let us examine what happens when a current is flowing from left to right in this system. $ If and electron comes in from the left there are several thing that can happen. $ First of all, there is a chance that the electron reaches the right without exchanging energy with the mode $ Secondly, the electron can exchange energy with the mode and scatter back $ This is a diagram of the potential energy of electrons, the boxes represent occupied electronic state. And in this picture the bias that allows the current to flow is simply the difference in height between the occupied levels. $The electron comes in from one of the occupied levels in the left and exchanges energy with the vibration. $ If the electron scatteres back then the vibration cannot gain energy because the electron cannot go down in energy since these states are already occupied. $ The vibration can however loose energy because the electron can go up in energy. So backward scattering can only cause the vibration to loose energy. $ If the electron exchanges energy with the mode and scatters forward however then the vibration can both $loose -and $gain energy All these three events happen randomly- sometimes the vibration gets a little energy and sometimes it is taken away. The average energy contained in the mode we can interpret as an effective temperature. How hot the mode gets is determined by the relative probability of the forward and backwards scattering which is something that varies from mode to mode. The experts will realize I&#x2019;ve assumed that the system is conducting, which we have verified that this is. (11+10+6)
- Let us examine what happens when a current is flowing from left to right in this system. $ If and electron comes in from the left there are several thing that can happen. $ First of all, there is a chance that the electron reaches the right without exchanging energy with the mode $ Secondly, the electron can exchange energy with the mode and scatter back $ This is a diagram of the potential energy of electrons, the boxes represent occupied electronic state. And in this picture the bias that allows the current to flow is simply the difference in height between the occupied levels. $The electron comes in from one of the occupied levels in the left and exchanges energy with the vibration. $ If the electron scatteres back then the vibration cannot gain energy because the electron cannot go down in energy since these states are already occupied. $ The vibration can however loose energy because the electron can go up in energy. So backward scattering can only cause the vibration to loose energy. $ If the electron exchanges energy with the mode and scatters forward however then the vibration can both $loose -and $gain energy All these three events happen randomly- sometimes the vibration gets a little energy and sometimes it is taken away. The average energy contained in the mode we can interpret as an effective temperature. How hot the mode gets is determined by the relative probability of the forward and backwards scattering which is something that varies from mode to mode. The experts will realize I&#x2019;ve assumed that the system is conducting, which we have verified that this is. (11+10+6)
- Let us examine what happens when a current is flowing from left to right in this system. $ If and electron comes in from the left there are several thing that can happen. $ First of all, there is a chance that the electron reaches the right without exchanging energy with the mode $ Secondly, the electron can exchange energy with the mode and scatter back $ This is a diagram of the potential energy of electrons, the boxes represent occupied electronic state. And in this picture the bias that allows the current to flow is simply the difference in height between the occupied levels. $The electron comes in from one of the occupied levels in the left and exchanges energy with the vibration. $ If the electron scatteres back then the vibration cannot gain energy because the electron cannot go down in energy since these states are already occupied. $ The vibration can however loose energy because the electron can go up in energy. So backward scattering can only cause the vibration to loose energy. $ If the electron exchanges energy with the mode and scatters forward however then the vibration can both $loose -and $gain energy All these three events happen randomly- sometimes the vibration gets a little energy and sometimes it is taken away. The average energy contained in the mode we can interpret as an effective temperature. How hot the mode gets is determined by the relative probability of the forward and backwards scattering which is something that varies from mode to mode. The experts will realize I&#x2019;ve assumed that the system is conducting, which we have verified that this is. (11+10+6)
- Let us examine what happens when a current is flowing from left to right in this system. $ If and electron comes in from the left there are several thing that can happen. $ First of all, there is a chance that the electron reaches the right without exchanging energy with the mode $ Secondly, the electron can exchange energy with the mode and scatter back $ This is a diagram of the potential energy of electrons, the boxes represent occupied electronic state. And in this picture the bias that allows the current to flow is simply the difference in height between the occupied levels. $The electron comes in from one of the occupied levels in the left and exchanges energy with the vibration. $ If the electron scatteres back then the vibration cannot gain energy because the electron cannot go down in energy since these states are already occupied. $ The vibration can however loose energy because the electron can go up in energy. So backward scattering can only cause the vibration to loose energy. $ If the electron exchanges energy with the mode and scatters forward however then the vibration can both $loose -and $gain energy All these three events happen randomly- sometimes the vibration gets a little energy and sometimes it is taken away. The average energy contained in the mode we can interpret as an effective temperature. How hot the mode gets is determined by the relative probability of the forward and backwards scattering which is something that varies from mode to mode. The experts will realize I&#x2019;ve assumed that the system is conducting, which we have verified that this is. (11+10+6)
- Let us examine what happens when a current is flowing from left to right in this system. $ If and electron comes in from the left there are several thing that can happen. $ First of all, there is a chance that the electron reaches the right without exchanging energy with the mode $ Secondly, the electron can exchange energy with the mode and scatter back $ This is a diagram of the potential energy of electrons, the boxes represent occupied electronic state. And in this picture the bias that allows the current to flow is simply the difference in height between the occupied levels. $The electron comes in from one of the occupied levels in the left and exchanges energy with the vibration. $ If the electron scatteres back then the vibration cannot gain energy because the electron cannot go down in energy since these states are already occupied. $ The vibration can however loose energy because the electron can go up in energy. So backward scattering can only cause the vibration to loose energy. $ If the electron exchanges energy with the mode and scatters forward however then the vibration can both $loose -and $gain energy All these three events happen randomly- sometimes the vibration gets a little energy and sometimes it is taken away. The average energy contained in the mode we can interpret as an effective temperature. How hot the mode gets is determined by the relative probability of the forward and backwards scattering which is something that varies from mode to mode. The experts will realize I&#x2019;ve assumed that the system is conducting, which we have verified that this is. (11+10+6)
- Let us examine what happens when a current is flowing from left to right in this system. $ If and electron comes in from the left there are several thing that can happen. $ First of all, there is a chance that the electron reaches the right without exchanging energy with the mode $ Secondly, the electron can exchange energy with the mode and scatter back $ This is a diagram of the potential energy of electrons, the boxes represent occupied electronic state. And in this picture the bias that allows the current to flow is simply the difference in height between the occupied levels. $The electron comes in from one of the occupied levels in the left and exchanges energy with the vibration. $ If the electron scatteres back then the vibration cannot gain energy because the electron cannot go down in energy since these states are already occupied. $ The vibration can however loose energy because the electron can go up in energy. So backward scattering can only cause the vibration to loose energy. $ If the electron exchanges energy with the mode and scatters forward however then the vibration can both $loose -and $gain energy All these three events happen randomly- sometimes the vibration gets a little energy and sometimes it is taken away. The average energy contained in the mode we can interpret as an effective temperature. How hot the mode gets is determined by the relative probability of the forward and backwards scattering which is something that varies from mode to mode. The experts will realize I&#x2019;ve assumed that the system is conducting, which we have verified that this is. (11+10+6)
- Let us examine what happens when a current is flowing from left to right in this system. $ If and electron comes in from the left there are several thing that can happen. $ First of all, there is a chance that the electron reaches the right without exchanging energy with the mode $ Secondly, the electron can exchange energy with the mode and scatter back $ This is a diagram of the potential energy of electrons, the boxes represent occupied electronic state. And in this picture the bias that allows the current to flow is simply the difference in height between the occupied levels. $The electron comes in from one of the occupied levels in the left and exchanges energy with the vibration. $ If the electron scatteres back then the vibration cannot gain energy because the electron cannot go down in energy since these states are already occupied. $ The vibration can however loose energy because the electron can go up in energy. So backward scattering can only cause the vibration to loose energy. $ If the electron exchanges energy with the mode and scatters forward however then the vibration can both $loose -and $gain energy All these three events happen randomly- sometimes the vibration gets a little energy and sometimes it is taken away. The average energy contained in the mode we can interpret as an effective temperature. How hot the mode gets is determined by the relative probability of the forward and backwards scattering which is something that varies from mode to mode. The experts will realize I&#x2019;ve assumed that the system is conducting, which we have verified that this is. (11+10+6)
- Let us examine what happens when a current is flowing from left to right in this system. $ If and electron comes in from the left there are several thing that can happen. $ First of all, there is a chance that the electron reaches the right without exchanging energy with the mode $ Secondly, the electron can exchange energy with the mode and scatter back $ This is a diagram of the potential energy of electrons, the boxes represent occupied electronic state. And in this picture the bias that allows the current to flow is simply the difference in height between the occupied levels. $The electron comes in from one of the occupied levels in the left and exchanges energy with the vibration. $ If the electron scatteres back then the vibration cannot gain energy because the electron cannot go down in energy since these states are already occupied. $ The vibration can however loose energy because the electron can go up in energy. So backward scattering can only cause the vibration to loose energy. $ If the electron exchanges energy with the mode and scatters forward however then the vibration can both $loose -and $gain energy All these three events happen randomly- sometimes the vibration gets a little energy and sometimes it is taken away. The average energy contained in the mode we can interpret as an effective temperature. How hot the mode gets is determined by the relative probability of the forward and backwards scattering which is something that varies from mode to mode. The experts will realize I&#x2019;ve assumed that the system is conducting, which we have verified that this is. (11+10+6)
- Let us examine what happens when a current is flowing from left to right in this system. $ If and electron comes in from the left there are several thing that can happen. $ First of all, there is a chance that the electron reaches the right without exchanging energy with the mode $ Secondly, the electron can exchange energy with the mode and scatter back $ This is a diagram of the potential energy of electrons, the boxes represent occupied electronic state. And in this picture the bias that allows the current to flow is simply the difference in height between the occupied levels. $The electron comes in from one of the occupied levels in the left and exchanges energy with the vibration. $ If the electron scatteres back then the vibration cannot gain energy because the electron cannot go down in energy since these states are already occupied. $ The vibration can however loose energy because the electron can go up in energy. So backward scattering can only cause the vibration to loose energy. $ If the electron exchanges energy with the mode and scatters forward however then the vibration can both $loose -and $gain energy All these three events happen randomly- sometimes the vibration gets a little energy and sometimes it is taken away. The average energy contained in the mode we can interpret as an effective temperature. How hot the mode gets is determined by the relative probability of the forward and backwards scattering which is something that varies from mode to mode. The experts will realize I&#x2019;ve assumed that the system is conducting, which we have verified that this is. (11+10+6)
- Let us examine what happens when a current is flowing from left to right in this system. $ If and electron comes in from the left there are several thing that can happen. $ First of all, there is a chance that the electron reaches the right without exchanging energy with the mode $ Secondly, the electron can exchange energy with the mode and scatter back $ This is a diagram of the potential energy of electrons, the boxes represent occupied electronic state. And in this picture the bias that allows the current to flow is simply the difference in height between the occupied levels. $The electron comes in from one of the occupied levels in the left and exchanges energy with the vibration. $ If the electron scatteres back then the vibration cannot gain energy because the electron cannot go down in energy since these states are already occupied. $ The vibration can however loose energy because the electron can go up in energy. So backward scattering can only cause the vibration to loose energy. $ If the electron exchanges energy with the mode and scatters forward however then the vibration can both $loose -and $gain energy All these three events happen randomly- sometimes the vibration gets a little energy and sometimes it is taken away. The average energy contained in the mode we can interpret as an effective temperature. How hot the mode gets is determined by the relative probability of the forward and backwards scattering which is something that varies from mode to mode. The experts will realize I&#x2019;ve assumed that the system is conducting, which we have verified that this is. (11+10+6)
- Here we see the calculated effective temperature vs. bias for each of the 5 modes in the two systems we&#x2019;ve investigated. $ This vertical line shows the temperature where armchair graphene edges becomes unstable. The effective temperature of the selected modes reach this temperature at a bias much smaller than the $1.6 V used in the experiment. I don&#x2019;t want to oversell this point because, first of all, the heating of one mode, is not the same as the heating all modes. Secondly, we have omitted an effect that would tend to redistribute the energy among the modes, the effect of anharmonicity. What I do want to note is that these modes do stick out considerably when comparing with the other modes in the system that are damped by vibrations. $ The heating of those modes fall below this line. So it is difficult to see how general heating could account for evaporation seen in experiment.
- Here we see the calculated effective temperature vs. bias for each of the 5 modes in the two systems we&#x2019;ve investigated. $ This vertical line shows the temperature where armchair graphene edges becomes unstable. The effective temperature of the selected modes reach this temperature at a bias much smaller than the $1.6 V used in the experiment. I don&#x2019;t want to oversell this point because, first of all, the heating of one mode, is not the same as the heating all modes. Secondly, we have omitted an effect that would tend to redistribute the energy among the modes, the effect of anharmonicity. What I do want to note is that these modes do stick out considerably when comparing with the other modes in the system that are damped by vibrations. $ The heating of those modes fall below this line. So it is difficult to see how general heating could account for evaporation seen in experiment.
- Here we see the calculated effective temperature vs. bias for each of the 5 modes in the two systems we&#x2019;ve investigated. $ This vertical line shows the temperature where armchair graphene edges becomes unstable. The effective temperature of the selected modes reach this temperature at a bias much smaller than the $1.6 V used in the experiment. I don&#x2019;t want to oversell this point because, first of all, the heating of one mode, is not the same as the heating all modes. Secondly, we have omitted an effect that would tend to redistribute the energy among the modes, the effect of anharmonicity. What I do want to note is that these modes do stick out considerably when comparing with the other modes in the system that are damped by vibrations. $ The heating of those modes fall below this line. So it is difficult to see how general heating could account for evaporation seen in experiment.
- Here we see the calculated effective temperature vs. bias for each of the 5 modes in the two systems we&#x2019;ve investigated. $ This vertical line shows the temperature where armchair graphene edges becomes unstable. The effective temperature of the selected modes reach this temperature at a bias much smaller than the $1.6 V used in the experiment. I don&#x2019;t want to oversell this point because, first of all, the heating of one mode, is not the same as the heating all modes. Secondly, we have omitted an effect that would tend to redistribute the energy among the modes, the effect of anharmonicity. What I do want to note is that these modes do stick out considerably when comparing with the other modes in the system that are damped by vibrations. $ The heating of those modes fall below this line. So it is difficult to see how general heating could account for evaporation seen in experiment.
- What we have done is by no means a realistic simulation of the experiment that motivated us. $The real system is far to big and complicated for that. But on the small test systems $ we have found a type of mode that should exist on in any size system with armchair edges. We&#x2019;ve demonstrated that these modes accumulate energy as a current flows. $ and estimated that this accumulated energy would allow C-C dimers to evaporate. $ (10+11+12)
- What we have done is by no means a realistic simulation of the experiment that motivated us. $The real system is far to big and complicated for that. But on the small test systems $ we have found a type of mode that should exist on in any size system with armchair edges. We&#x2019;ve demonstrated that these modes accumulate energy as a current flows. $ and estimated that this accumulated energy would allow C-C dimers to evaporate. $ (10+11+12)
- What we have done is by no means a realistic simulation of the experiment that motivated us. $The real system is far to big and complicated for that. But on the small test systems $ we have found a type of mode that should exist on in any size system with armchair edges. We&#x2019;ve demonstrated that these modes accumulate energy as a current flows. $ and estimated that this accumulated energy would allow C-C dimers to evaporate. $ (10+11+12)
- What we have done is by no means a realistic simulation of the experiment that motivated us. $The real system is far to big and complicated for that. But on the small test systems $ we have found a type of mode that should exist on in any size system with armchair edges. We&#x2019;ve demonstrated that these modes accumulate energy as a current flows. $ and estimated that this accumulated energy would allow C-C dimers to evaporate. $ (10+11+12)
- What we have done is by no means a realistic simulation of the experiment that motivated us. $The real system is far to big and complicated for that. But on the small test systems $ we have found a type of mode that should exist on in any size system with armchair edges. We&#x2019;ve demonstrated that these modes accumulate energy as a current flows. $ and estimated that this accumulated energy would allow C-C dimers to evaporate. $ (10+11+12)
- What we have done is by no means a realistic simulation of the experiment that motivated us. $The real system is far to big and complicated for that. But on the small test systems $ we have found a type of mode that should exist on in any size system with armchair edges. We&#x2019;ve demonstrated that these modes accumulate energy as a current flows. $ and estimated that this accumulated energy would allow C-C dimers to evaporate. $ (10+11+12)
- What we have done is by no means a realistic simulation of the experiment that motivated us. $The real system is far to big and complicated for that. But on the small test systems $ we have found a type of mode that should exist on in any size system with armchair edges. We&#x2019;ve demonstrated that these modes accumulate energy as a current flows. $ and estimated that this accumulated energy would allow C-C dimers to evaporate. $ (10+11+12)
- What we have done is by no means a realistic simulation of the experiment that motivated us. $The real system is far to big and complicated for that. But on the small test systems $ we have found a type of mode that should exist on in any size system with armchair edges. We&#x2019;ve demonstrated that these modes accumulate energy as a current flows. $ and estimated that this accumulated energy would allow C-C dimers to evaporate. $ (10+11+12)
- What we have done is by no means a realistic simulation of the experiment that motivated us. $The real system is far to big and complicated for that. But on the small test systems $ we have found a type of mode that should exist on in any size system with armchair edges. We&#x2019;ve demonstrated that these modes accumulate energy as a current flows. $ and estimated that this accumulated energy would allow C-C dimers to evaporate. $ (10+11+12)
- What we have done is by no means a realistic simulation of the experiment that motivated us. $The real system is far to big and complicated for that. But on the small test systems $ we have found a type of mode that should exist on in any size system with armchair edges. We&#x2019;ve demonstrated that these modes accumulate energy as a current flows. $ and estimated that this accumulated energy would allow C-C dimers to evaporate. $ (10+11+12)
- What we have done is by no means a realistic simulation of the experiment that motivated us. $The real system is far to big and complicated for that. But on the small test systems $ we have found a type of mode that should exist on in any size system with armchair edges. We&#x2019;ve demonstrated that these modes accumulate energy as a current flows. $ and estimated that this accumulated energy would allow C-C dimers to evaporate. $ (10+11+12)
- What we have done is by no means a realistic simulation of the experiment that motivated us. $The real system is far to big and complicated for that. But on the small test systems $ we have found a type of mode that should exist on in any size system with armchair edges. We&#x2019;ve demonstrated that these modes accumulate energy as a current flows. $ and estimated that this accumulated energy would allow C-C dimers to evaporate. $ (10+11+12)
- I will now end this talk with some concluding remarks. $ The vibrational energies are low compared to the energy in the electron system or the energy stored in the stress and strain. This means that vibrational properties are very sensitive to even small changes in the configuration of systems. $ The sensitivity of the vibrational system to me highlights the importance of ab-initio for these systems. Measurements will have difficulty in controlling all the relevant parameters and reliable simulations would be nescesary we understand what is happening. $ The study of vibrations is part of the study of the movement of atoms. If we gain a better understanding of how to manipulate atoms, for example by current as we went through or by laser or saser $ it would perhaps be a way to gain precise control over the structure of matter. (34)
- I will now end this talk with some concluding remarks. $ The vibrational energies are low compared to the energy in the electron system or the energy stored in the stress and strain. This means that vibrational properties are very sensitive to even small changes in the configuration of systems. $ The sensitivity of the vibrational system to me highlights the importance of ab-initio for these systems. Measurements will have difficulty in controlling all the relevant parameters and reliable simulations would be nescesary we understand what is happening. $ The study of vibrations is part of the study of the movement of atoms. If we gain a better understanding of how to manipulate atoms, for example by current as we went through or by laser or saser $ it would perhaps be a way to gain precise control over the structure of matter. (34)
- I will now end this talk with some concluding remarks. $ The vibrational energies are low compared to the energy in the electron system or the energy stored in the stress and strain. This means that vibrational properties are very sensitive to even small changes in the configuration of systems. $ The sensitivity of the vibrational system to me highlights the importance of ab-initio for these systems. Measurements will have difficulty in controlling all the relevant parameters and reliable simulations would be nescesary we understand what is happening. $ The study of vibrations is part of the study of the movement of atoms. If we gain a better understanding of how to manipulate atoms, for example by current as we went through or by laser or saser $ it would perhaps be a way to gain precise control over the structure of matter. (34)
- I will now end this talk with some concluding remarks. $ The vibrational energies are low compared to the energy in the electron system or the energy stored in the stress and strain. This means that vibrational properties are very sensitive to even small changes in the configuration of systems. $ The sensitivity of the vibrational system to me highlights the importance of ab-initio for these systems. Measurements will have difficulty in controlling all the relevant parameters and reliable simulations would be nescesary we understand what is happening. $ The study of vibrations is part of the study of the movement of atoms. If we gain a better understanding of how to manipulate atoms, for example by current as we went through or by laser or saser $ it would perhaps be a way to gain precise control over the structure of matter. (34)
- I will now end this talk with some concluding remarks. $ The vibrational energies are low compared to the energy in the electron system or the energy stored in the stress and strain. This means that vibrational properties are very sensitive to even small changes in the configuration of systems. $ The sensitivity of the vibrational system to me highlights the importance of ab-initio for these systems. Measurements will have difficulty in controlling all the relevant parameters and reliable simulations would be nescesary we understand what is happening. $ The study of vibrations is part of the study of the movement of atoms. If we gain a better understanding of how to manipulate atoms, for example by current as we went through or by laser or saser $ it would perhaps be a way to gain precise control over the structure of matter. (34)
- I will now end this talk with some concluding remarks. $ The vibrational energies are low compared to the energy in the electron system or the energy stored in the stress and strain. This means that vibrational properties are very sensitive to even small changes in the configuration of systems. $ The sensitivity of the vibrational system to me highlights the importance of ab-initio for these systems. Measurements will have difficulty in controlling all the relevant parameters and reliable simulations would be nescesary we understand what is happening. $ The study of vibrations is part of the study of the movement of atoms. If we gain a better understanding of how to manipulate atoms, for example by current as we went through or by laser or saser $ it would perhaps be a way to gain precise control over the structure of matter. (34)
- I will now end this talk with some concluding remarks. $ The vibrational energies are low compared to the energy in the electron system or the energy stored in the stress and strain. This means that vibrational properties are very sensitive to even small changes in the configuration of systems. $ The sensitivity of the vibrational system to me highlights the importance of ab-initio for these systems. Measurements will have difficulty in controlling all the relevant parameters and reliable simulations would be nescesary we understand what is happening. $ The study of vibrations is part of the study of the movement of atoms. If we gain a better understanding of how to manipulate atoms, for example by current as we went through or by laser or saser $ it would perhaps be a way to gain precise control over the structure of matter. (34)
- Relate to own topic
- Relate to own topic
- Relate to own topic
- Thank you for your attention. (10+11+6+3)