Surface Plasmon Assisted Laser Cooling of Solids
Jacob Khurgin
PRL 98, 177401 (2007)
URL: http://link.aps.org/abstract/PRL/v98/e177401
Laser cooling of atoms is based on the Doppler effect. Laser frequencies are tuned below an atomic transition so that atoms moving toward the laser source will be more likely to absorb a photon than those moving away. The atom will then emit a photon in a random direction as it relaxes back down to its ground state, but the radiated photon will be of higher energy than the absorbed photon. As a result, the atom has radiated away some of its kinetic energy, so it has been cooled by a laser.
In solids the basic idea is similar. The solid absorbs one frequency of light and emits at a higher frequency, losing energy as a result. The mechanism is different, and it called anti-Stokes photoluminescence. Light is absorbed at a specfic frequency, then this excited state comes to thermal equilibrium with the system. Later, light is emitted at a higher frequency. The shift in frequency would be on the order of kT, where T is the temperature of the solid, and the system would gradually cool down.
Khurgin points out that there are several difficulties in the case of semiconductors. There are a lot of nonradiative decay channels, and the high index of refraction leads to low efficiencies --- i.e., even if you generate a photon of the right frequency, it's hard for it to get out of the semiconductor. One way to improve the efficiency is to make the density of absorbing states small and the density of emitting states large. When a photon is absorbed, the energy is more likely to be transferred to a higher-energy emitting state than to remain in the absorbing state for a time, then be re-emitted at the same frequency.
Khurgin's approach to the problem is to exploit surface plasmon polaritons. These occur at the interface between a dielectric and a metal at frequencies where the dielectric constants of the two media are equal in magnitude but opposite in sign. The density of plasmon states has a sharp resonance, which leads to an increase of spontaneous emission at the resonant frequency. (Apparently, Purcell worked this out back in 1946.)
Khurgin notes that the plasmon modes still have to couple to radiative modes before they give up their energy, so it might seem that nothing has been gained. However, he goes on to demonstrate that the plasmons can couple to the phonons of the metal. The metal will heat up, but the goal was never to cool the metal and the dielectric together --- only to cool the dielectric medium.
Based on this observation, Khurgin proposes placing a layer of silver on top of a gallium arsenide layer with a gap between them of a couple nanometers. The gap is a thermal insulator between the silver and gallium arsenide. The only coupling between the two systems are the plasmons. A laser will produce excitations in the gallium arsenide layer, and many of these will relax into the many available plasmon modes. The plasmon modes will couple to the phonon modes in silver, but not in gallium arsenide, so they will gradually transfer energy from the semiconductor to the metal. This four-step process leads to laser cooling of the semiconductor:
laser ---> semiconductor excitations ---> plasmons ---> phonons in metal
Khurgin estimates that the silver and gallium arsenide system could have a cooling efficiency of 2 percent or more.
Thursday, May 17, 2007
Light to Atom to Light Again
Reversible State Transfer between Light and a Single Trapped Atom
A.D. Boozer, A. Boca, R. Miller, T.E. Northup, and H.J. Kimble
PRL 98, 193601 (2007)
URL: http://link.aps.org/abstract/PRL/v98/e193601
This group from Cal Tech claims the first demonstration of the transfer of a coherent state between a photon and an atom, then from the atom back to a photon. This is a necessity for a quantum network.
Much work in quantum computing has gone into developing and manipulating qbits. It's one thing to have a single working qbit in isolation, or a pair, or 16 of them. But if you want to develop a quantum computer with a large number of qbits, or if you want to transfer the output state of your computation somewhere else, what do you do? The authors suggest that coherent light would be able to transfer superpositions of quantum states over optical fibers. Transmitting light over optical fibers is not so hard. The difficult step is turning an atomic state into a photon, transmitting the photon, then turning the photon into the same atomic state somewhere else.
In this letter, the Cal Tech group demonstrates "the reversible mapping of a coherent optical field to and from the hyperfine ground states of a single trapped cesium atom."
The prototype for their experiment is a 3-level atom. The atom has two ground states |a> and |b>, and an excited state |e>. The atom is in an optical cavity that couples |b> and |e>, and an external field couples |a> and |e>. If the external field is turned on slowly, the state |a,n> is transformed into |b,n+1> --- i.e., there is a transition between atomic states and a single photon is generated in the cavity. Slowly turning the field off reverses the transition.
If the cavity is empty, the process can be used to generate a single photon. The atom is prepared in state |a,0> and the field is slowly turned on. The resulting state is |b,1>. This is interesting, but transitions between single atomic states are not the building blocks of quantum computing. Entanglement and coherent superpositions of states are the tools of the trade. The useful thing about the process just described is that is works on a superposition of states:
(A |a> + B |b> ) |0> <---> |b> ( A |0> + B |1> )
This represents the transfer of a superposition of atomic states to a superposition of photon states, and all that was required is the turning off of a classical field.
The authors never mention the word "entangled" when discussing their experiments. I don't know what the exact definition of entangled states is, but I recall something about the impossibility of writing such a state as a product of states. If this is true, then the relation above does not describe the transfer of an entangled state from atom to photon. Still, it's a good first step!
A.D. Boozer, A. Boca, R. Miller, T.E. Northup, and H.J. Kimble
PRL 98, 193601 (2007)
URL: http://link.aps.org/abstract/PRL/v98/e193601
This group from Cal Tech claims the first demonstration of the transfer of a coherent state between a photon and an atom, then from the atom back to a photon. This is a necessity for a quantum network.
Much work in quantum computing has gone into developing and manipulating qbits. It's one thing to have a single working qbit in isolation, or a pair, or 16 of them. But if you want to develop a quantum computer with a large number of qbits, or if you want to transfer the output state of your computation somewhere else, what do you do? The authors suggest that coherent light would be able to transfer superpositions of quantum states over optical fibers. Transmitting light over optical fibers is not so hard. The difficult step is turning an atomic state into a photon, transmitting the photon, then turning the photon into the same atomic state somewhere else.
In this letter, the Cal Tech group demonstrates "the reversible mapping of a coherent optical field to and from the hyperfine ground states of a single trapped cesium atom."
The prototype for their experiment is a 3-level atom. The atom has two ground states |a> and |b>, and an excited state |e>. The atom is in an optical cavity that couples |b> and |e>, and an external field couples |a> and |e>. If the external field is turned on slowly, the state |a,n> is transformed into |b,n+1> --- i.e., there is a transition between atomic states and a single photon is generated in the cavity. Slowly turning the field off reverses the transition.
If the cavity is empty, the process can be used to generate a single photon. The atom is prepared in state |a,0> and the field is slowly turned on. The resulting state is |b,1>. This is interesting, but transitions between single atomic states are not the building blocks of quantum computing. Entanglement and coherent superpositions of states are the tools of the trade. The useful thing about the process just described is that is works on a superposition of states:
(A |a> + B |b> ) |0> <---> |b> ( A |0> + B |1> )
This represents the transfer of a superposition of atomic states to a superposition of photon states, and all that was required is the turning off of a classical field.
The authors never mention the word "entangled" when discussing their experiments. I don't know what the exact definition of entangled states is, but I recall something about the impossibility of writing such a state as a product of states. If this is true, then the relation above does not describe the transfer of an entangled state from atom to photon. Still, it's a good first step!
Tuesday, May 15, 2007
Phonon Effects in Graphene
Electron-Phonon Coupling Mechanism in Two-Dimensional Graphite and Single-Walled Carbon Nanotubes
G.G. Samsonidze, E.B. Barros, R. Saito, J.Jiang, G. and M.S. Dresselhaus
PRB 75, 155420 (2007)
URL: http://link.aps.org/abstract/PRB/v75/e155420
The authors have analyzed the effects of phonons on the fermi energies and wave vectors of graphene. Their analysis is based on the group theory of the wave vector and demonstrates both the power of the technique and my lack of understanding.
They observe two phenomena associated with the phonons. First, there is a Pierls instability, which means the phonon modes open up a frequency-dependent band gap. The second is a Kohn anomaly, which is electron screening of a particular phonon mode.
Section 2 and Appendix A are very useful, as they show the general procedure for introducing phonon effects into the tight-binding model. When the phonon mode breaks the symmetry of the lattice, the unit cell of graphene must be enlarged to become a supercell of six atoms. This leads to a 6 by 6 Hamiltonian instead of the more familiar 2 by 2 version. However, the larger Hamiltonian is what Wigner calls a supermatrix --- a matrix composed of smaller matrices. There are two diagonal 3 by 3 matrices for the on-site terms, H[AA] and H[BB]. The hopping terms are described by two 3 by 3 matrices, H[AB] and H[BA], with every entry equal to t (for k=0). The latter is surprising, as it suggests that every A site is connected to every B site --- a fact that is not obvious from the diagrams provided by the authors.
This Hamiltonian gives a six-band spectrum, with the middle four bands degenerate. The K-point phonon mode breaks the degeneracy from fourfold to twofold and opens a bandgap that depends on the phonon coupling strength and amplitude.
The appendix gives the corresponding Hamiltonian for phonons with wave vectors not at a highly symmetric point of the Brillioun zone. It is not as symmetric. Although the authors do not analyze the energy bands of the general tight-binding Hamiltonian with phonon interactions, I assume that a phonon that does not respect any symmetries of the underlying lattice would lift all the degeneracies. The only exception might be the Kramers degeneracy imposed by time-reversal symmetry.
One aspect of this work I don't understand is why the phonon modes considered by the authors are more important than others. Are they the modes of lowest energy? Do the phonon bands cross the electron bands at the fermi energy? It seems that they are simply the easiest to analyze, but that does not mean their physical effects are the most important.
G.G. Samsonidze, E.B. Barros, R. Saito, J.Jiang, G. and M.S. Dresselhaus
PRB 75, 155420 (2007)
URL: http://link.aps.org/abstract/PRB/v75/e155420
The authors have analyzed the effects of phonons on the fermi energies and wave vectors of graphene. Their analysis is based on the group theory of the wave vector and demonstrates both the power of the technique and my lack of understanding.
They observe two phenomena associated with the phonons. First, there is a Pierls instability, which means the phonon modes open up a frequency-dependent band gap. The second is a Kohn anomaly, which is electron screening of a particular phonon mode.
Section 2 and Appendix A are very useful, as they show the general procedure for introducing phonon effects into the tight-binding model. When the phonon mode breaks the symmetry of the lattice, the unit cell of graphene must be enlarged to become a supercell of six atoms. This leads to a 6 by 6 Hamiltonian instead of the more familiar 2 by 2 version. However, the larger Hamiltonian is what Wigner calls a supermatrix --- a matrix composed of smaller matrices. There are two diagonal 3 by 3 matrices for the on-site terms, H[AA] and H[BB]. The hopping terms are described by two 3 by 3 matrices, H[AB] and H[BA], with every entry equal to t (for k=0). The latter is surprising, as it suggests that every A site is connected to every B site --- a fact that is not obvious from the diagrams provided by the authors.
This Hamiltonian gives a six-band spectrum, with the middle four bands degenerate. The K-point phonon mode breaks the degeneracy from fourfold to twofold and opens a bandgap that depends on the phonon coupling strength and amplitude.
The appendix gives the corresponding Hamiltonian for phonons with wave vectors not at a highly symmetric point of the Brillioun zone. It is not as symmetric. Although the authors do not analyze the energy bands of the general tight-binding Hamiltonian with phonon interactions, I assume that a phonon that does not respect any symmetries of the underlying lattice would lift all the degeneracies. The only exception might be the Kramers degeneracy imposed by time-reversal symmetry.
One aspect of this work I don't understand is why the phonon modes considered by the authors are more important than others. Are they the modes of lowest energy? Do the phonon bands cross the electron bands at the fermi energy? It seems that they are simply the easiest to analyze, but that does not mean their physical effects are the most important.
Nonlinear Resistance in Nanotubes
Scaling of Resistance and Electron Mean Free Path of Single-Walled Carbon Nanotubes
M.S. Purewal, B.H. Hong, A. Ravi, B. Chandra, J. Hone, and Philip Kim
PRL 98, 186808 (2007)
URL: http://link.aps.org/abstract/PRL/v98/e186808
This group from Columbia University has attached a series of electrodes to individual carbon nanotubes and measured the current-voltage characteristics of carbon nanotubes as a function of temperature and length between electrodes. This allowed them to analyze the scaling behavior of the resistance in single nanotubes.
Using the Landauer-Buttiker model of resitance, the group extracted the mean free path of electrons in the nanotube and studied this as a function of temperature. They find crossover behavior between two types of scaling. Below a critical temperature, the mean free path approaches a constant that differs from one nanotube to the next --- i.e., it depends on the diameter and chiral angle. Above the critical temperature, the authors find universal scaling: the mean free path is inversely proportional to the temperature.
The low temperature behavior is consistent with models based on impurity scattering while the high temperature behavior is consistent with electron-phonon scattering.
At extremely long length scales and low temperatures, the authors find that the resistance no longer scales linearly with the distance between the source and drain. The critical length at which nonlinear behavior starts to dominate is much larger than the electron mean free path, which suggests it is not the result of Anderson localization or a similar type of quantum interference. The phase coherence length is the same order of magnitude as the mean free path, and therefore much shorter than the critical length scale. The authors imply there is no satisfactory theoretical explanation of the observed behavior.
This was the first paper on nanotubes with a recommendation from the editors. It is well-written and reports an interesting result, so the endorsement is well-deserved.
M.S. Purewal, B.H. Hong, A. Ravi, B. Chandra, J. Hone, and Philip Kim
PRL 98, 186808 (2007)
URL: http://link.aps.org/abstract/PRL/v98/e186808
This group from Columbia University has attached a series of electrodes to individual carbon nanotubes and measured the current-voltage characteristics of carbon nanotubes as a function of temperature and length between electrodes. This allowed them to analyze the scaling behavior of the resistance in single nanotubes.
Using the Landauer-Buttiker model of resitance, the group extracted the mean free path of electrons in the nanotube and studied this as a function of temperature. They find crossover behavior between two types of scaling. Below a critical temperature, the mean free path approaches a constant that differs from one nanotube to the next --- i.e., it depends on the diameter and chiral angle. Above the critical temperature, the authors find universal scaling: the mean free path is inversely proportional to the temperature.
The low temperature behavior is consistent with models based on impurity scattering while the high temperature behavior is consistent with electron-phonon scattering.
At extremely long length scales and low temperatures, the authors find that the resistance no longer scales linearly with the distance between the source and drain. The critical length at which nonlinear behavior starts to dominate is much larger than the electron mean free path, which suggests it is not the result of Anderson localization or a similar type of quantum interference. The phase coherence length is the same order of magnitude as the mean free path, and therefore much shorter than the critical length scale. The authors imply there is no satisfactory theoretical explanation of the observed behavior.
This was the first paper on nanotubes with a recommendation from the editors. It is well-written and reports an interesting result, so the endorsement is well-deserved.
Friday, May 11, 2007
Transfer of Spin Polarization
Spin Transfer from an Optically Pumped Alkali Vapor to a Solid
K. Ishikawa, B. Patton, Y.Y. Jau, and W. Happer
PRL 98, 183004 (2007)
URL: http://link.aps.org/abstract/PRL/v98/e183004
This editor recommendation was a very well-written paper.
As the title indicates, the authors demonstrated the transfer of spin from optically pumped cesium vapor to cesium salt on the surface of a sealed glass cylinder. The authors deposited a layer of CsH salt on the walls of the container about 10 microns thick. Next, they put in pure cesium and nitrogen at various pressures and sealed the cylinders.
They used a laser tuned to a specific atomic transition in the cesium vapor to pump the system using both right and left handed circularly polarized light. To see the effects of pumping the vapor on the salt layers, the authors measured the free induction decay of the sample --- a common technique in NMR. They were able to discriminate between the vapor and the salt because the resonance of the vapor is at 52.39 MHz while the salt is at 53.15 MHz. They found that in optically pumped samples, the polarization of the salt was 4 times its value in the unpumped samples. They take this as demonstration of transfer of spin polarization from the vapor to the salt.
Charles Marcus spoke at Penn a couple months ago and discussed his work on one and two electron quantum dots. He makes artificial atoms on solid state chips and inserts electrons, makes them interact, and reads out their final states. It's one of the first approaches to quantum computing that I would consider feasible. Anyway, Marcus and his group observed that when they sent spin polarized electrons through the system, they could polarize the nuclear spins in the dot. I thought he called it a "nuclear zamboni," but I can't track down that reference right now. The idea is that you can use an electron beam to smooth out the nuclear environment and make it less likely that nuclear disorder will destroy your carefully prepared electron qbits.
The authors of the current paper discuss a similar phenomenon, and they give a nice theoretical description of the process. They argue that nuclear and electronic spins obey a set of coupled diffusion equations (and give very clear descriptions of where all the terms in the equation come from). They give approximate solutions to this set of equations, then show that these approximate and easy-to-understand solutions reproduce the major features of a much more detailed numerical model. Their simple model explains why the electron current scales inversely with pressure, and why the nuclear current is proportional to the pressure for small pressure, and constant at large pressures.
Despite the qualitative agreement between the diffusion model and experimental data, the authors imply the theory of spin transfer is not understood all that well. It's quite an interesting theoretical problem --- one I'm interested in myself. Usually in the analysis of collisions (as in particle physics calculations), one averages over the initial and final spin polarizations to compute the cross section. For a spin polarized sample, this is not the right approach.
In addition, atomic physics offers possibilities not allowed in particle physics. Suppose an atom in a singlet state scatters off a magnetic impurity. There is some probability for an interaction that would leave the atom in a triplet state and flip a spin in the magnetic impurity.
Another interesting situation is Coulomb drag. The Coulomb interaction does not allow for spin flips. This results in scattering processes that conserve charge current but reduce the spin current. I have an article on the phenomenon somewhere. I should write that up.
K. Ishikawa, B. Patton, Y.Y. Jau, and W. Happer
PRL 98, 183004 (2007)
URL: http://link.aps.org/abstract/PRL/v98/e183004
This editor recommendation was a very well-written paper.
As the title indicates, the authors demonstrated the transfer of spin from optically pumped cesium vapor to cesium salt on the surface of a sealed glass cylinder. The authors deposited a layer of CsH salt on the walls of the container about 10 microns thick. Next, they put in pure cesium and nitrogen at various pressures and sealed the cylinders.
They used a laser tuned to a specific atomic transition in the cesium vapor to pump the system using both right and left handed circularly polarized light. To see the effects of pumping the vapor on the salt layers, the authors measured the free induction decay of the sample --- a common technique in NMR. They were able to discriminate between the vapor and the salt because the resonance of the vapor is at 52.39 MHz while the salt is at 53.15 MHz. They found that in optically pumped samples, the polarization of the salt was 4 times its value in the unpumped samples. They take this as demonstration of transfer of spin polarization from the vapor to the salt.
Charles Marcus spoke at Penn a couple months ago and discussed his work on one and two electron quantum dots. He makes artificial atoms on solid state chips and inserts electrons, makes them interact, and reads out their final states. It's one of the first approaches to quantum computing that I would consider feasible. Anyway, Marcus and his group observed that when they sent spin polarized electrons through the system, they could polarize the nuclear spins in the dot. I thought he called it a "nuclear zamboni," but I can't track down that reference right now. The idea is that you can use an electron beam to smooth out the nuclear environment and make it less likely that nuclear disorder will destroy your carefully prepared electron qbits.
The authors of the current paper discuss a similar phenomenon, and they give a nice theoretical description of the process. They argue that nuclear and electronic spins obey a set of coupled diffusion equations (and give very clear descriptions of where all the terms in the equation come from). They give approximate solutions to this set of equations, then show that these approximate and easy-to-understand solutions reproduce the major features of a much more detailed numerical model. Their simple model explains why the electron current scales inversely with pressure, and why the nuclear current is proportional to the pressure for small pressure, and constant at large pressures.
Despite the qualitative agreement between the diffusion model and experimental data, the authors imply the theory of spin transfer is not understood all that well. It's quite an interesting theoretical problem --- one I'm interested in myself. Usually in the analysis of collisions (as in particle physics calculations), one averages over the initial and final spin polarizations to compute the cross section. For a spin polarized sample, this is not the right approach.
In addition, atomic physics offers possibilities not allowed in particle physics. Suppose an atom in a singlet state scatters off a magnetic impurity. There is some probability for an interaction that would leave the atom in a triplet state and flip a spin in the magnetic impurity.
Another interesting situation is Coulomb drag. The Coulomb interaction does not allow for spin flips. This results in scattering processes that conserve charge current but reduce the spin current. I have an article on the phenomenon somewhere. I should write that up.
Thursday, May 10, 2007
(No) e/2 States in Graphene
Electron Fractionalization in Two-Dimensional Graphenelike Structures
C.Y. Hou, Claudio Channon, and Christopher Mudry
PRL 98, 186809 (2007)
URL: http://link.aps.org/abstract/PRL/v98/e186809
This is an interesting paper that probably has no "practical application" whatsoever. The authors show how the order parameter of a Kekule phonon interaction in graphene can lead to fractionally charged states.
The Kekule texture, as the authors call it, is a periodic variation in the hopping amplitudes of the tight-binding model. One of my colleagues has done some work on this type of perturbation. It expands the unit cell in real space to include 4 lattice sites, and reduces the Brillouin zone accordingly. The order parameter that describes the Kekule phonons can be a complex number. However, the authors point out that a constant phase can be removed by an appropriate chiral transformation of the Hamiltonian.
Interesting things happen when the phase becomes a local parameter. Randy, one of my profs who taught a great course on liquid crystals, emphasized this point repeatedly. The central idea behind Goldstone modes is that gapless excitations arise when a global symmetry operation is applied locally. Phonons were the first example. In a lattice, the crystal is the same if the entire lattice is translated by any amount. However, if we make the translation a local operation --- i.e., each lattice site is translated by a different amount --- it leads to a gapless excitation in the system: acoustic phonons.
The work of the authors is similar. They introduce vortices in the Kekule parameter. The Hamiltonian can be solved to give a single-valued normalizable wave function for a zero mode --- a mode that occurs precisely in the middle of the band gap. (The Kekule parameter introduces a gap into the graphene energy surface.)
Next, they calculate the charge bound to a vertex by taking the difference of the local density of states when there is one vortex and no vortices. Conservation of the number of electrons says that 2 times the number of electrons associated with a vortex plus the integral of the zero mode --- i.e. 1 --- must vanish. Ergo, there is a charge of -e/2 associated with each vortex. This would seem to violate charge conservation --- adding one vortex changes the charge of the system by half an electron charge. The authors point out, however, that vortices come in pairs, so there is no contradiction.
The disappointed disclaimer of this work is that when you factor in the degeneracy associated with spin, the fractionally charged states disappear. Only scalar electrons permit fractionally charged excitations. Too bad. The authors go on to discuss how one might introduce local variations in the Kekule parameter, but the point seems moot.
The authors made a few statements I would like to know more about. They claim that the time-reversal symmetry of the tight-binding Hamiltonian follows from the fact that the hopping parameters are real. Moreover, they say that states come in pairs of energy +E and -E because there exists a gauge transformation that can flip the sign of the ladder operators on the A sublattice without affecting those on the B sublattices. This sounds like a 2 pi rotation of a spin-1/2 particle, but I don't understand how it works for a tight-binding model.
In a four page paper, the authors managed to explain quite a lot!
C.Y. Hou, Claudio Channon, and Christopher Mudry
PRL 98, 186809 (2007)
URL: http://link.aps.org/abstract/PRL/v98/e186809
This is an interesting paper that probably has no "practical application" whatsoever. The authors show how the order parameter of a Kekule phonon interaction in graphene can lead to fractionally charged states.
The Kekule texture, as the authors call it, is a periodic variation in the hopping amplitudes of the tight-binding model. One of my colleagues has done some work on this type of perturbation. It expands the unit cell in real space to include 4 lattice sites, and reduces the Brillouin zone accordingly. The order parameter that describes the Kekule phonons can be a complex number. However, the authors point out that a constant phase can be removed by an appropriate chiral transformation of the Hamiltonian.
Interesting things happen when the phase becomes a local parameter. Randy, one of my profs who taught a great course on liquid crystals, emphasized this point repeatedly. The central idea behind Goldstone modes is that gapless excitations arise when a global symmetry operation is applied locally. Phonons were the first example. In a lattice, the crystal is the same if the entire lattice is translated by any amount. However, if we make the translation a local operation --- i.e., each lattice site is translated by a different amount --- it leads to a gapless excitation in the system: acoustic phonons.
The work of the authors is similar. They introduce vortices in the Kekule parameter. The Hamiltonian can be solved to give a single-valued normalizable wave function for a zero mode --- a mode that occurs precisely in the middle of the band gap. (The Kekule parameter introduces a gap into the graphene energy surface.)
Next, they calculate the charge bound to a vertex by taking the difference of the local density of states when there is one vortex and no vortices. Conservation of the number of electrons says that 2 times the number of electrons associated with a vortex plus the integral of the zero mode --- i.e. 1 --- must vanish. Ergo, there is a charge of -e/2 associated with each vortex. This would seem to violate charge conservation --- adding one vortex changes the charge of the system by half an electron charge. The authors point out, however, that vortices come in pairs, so there is no contradiction.
The disappointed disclaimer of this work is that when you factor in the degeneracy associated with spin, the fractionally charged states disappear. Only scalar electrons permit fractionally charged excitations. Too bad. The authors go on to discuss how one might introduce local variations in the Kekule parameter, but the point seems moot.
The authors made a few statements I would like to know more about. They claim that the time-reversal symmetry of the tight-binding Hamiltonian follows from the fact that the hopping parameters are real. Moreover, they say that states come in pairs of energy +E and -E because there exists a gauge transformation that can flip the sign of the ladder operators on the A sublattice without affecting those on the B sublattices. This sounds like a 2 pi rotation of a spin-1/2 particle, but I don't understand how it works for a tight-binding model.
In a four page paper, the authors managed to explain quite a lot!
Excitons in Boron Nitride Nanotubes
Excitons and Many-Electron Effects in the Optical Response of Single-Walled Boron Nitride Nanotubes
C.H. Park, C.D. Spataru, and Steve Louie
PRL 96, 126105 (2006)
URL: http://link.aps.org/abstract/PRL/v96/e126105
I took two main points from this paper.
First, the larger band gap in boron nitride nanotubes makes particle interactions even more important. The shift of the free particle band gap due to electron interactions is larger than in nanotubes, and the exciton binding energies are also larger. The authors do not present any scaling results as they only analyzed a single nanotube.
Second, the exciton is much more localized in boron nitride nanotubes than in carbon nanotubes. In a carbon nanotube, there is little variation in the magnitude of the exciton wave function around the tube circumference. In boron nitride, there is quite a lot of variation. The electron is localized around the hole, and the probability of finding it on the opposite side of the tube is very small. The amplitude is also strongly peaked on the boron sites.
In effect, the second result implies the excitons in boron nitride are qualitatively different than in carbon. They cannot be considered one-dimensional objects. The exciton envelope function in a carbon nanotube is a smooth localized function. That of a boron nitride nanotube must describe the variations in amplitdue between the two sublattices. The boron nitride exciton is a two-dimensional wave function that happens to live on a cylinder. It is not strongly influenced by the periodic boundary conditions. (At least not in the tube studied here. In a tube with a smaller radius, the amplitude might not decay to zero over half a circumference.)
An interesting question that has probably been answered: Are the excitons in a boron nitride nanotube much different from those of a boron nitride sheet? The sheet has 3-fold rotational symmetry, which the tube does not. However, the tightly-localized excitons in the tube might not "be able to tell the difference," so to speak. There are no excitons in graphene because it is a semi-metal with no band gap. The large gap of planar boron nitride should allow for excitons, and I would expect them to be quite similar to their nanotube analogs.
C.H. Park, C.D. Spataru, and Steve Louie
PRL 96, 126105 (2006)
URL: http://link.aps.org/abstract/PRL/v96/e126105
I took two main points from this paper.
First, the larger band gap in boron nitride nanotubes makes particle interactions even more important. The shift of the free particle band gap due to electron interactions is larger than in nanotubes, and the exciton binding energies are also larger. The authors do not present any scaling results as they only analyzed a single nanotube.
Second, the exciton is much more localized in boron nitride nanotubes than in carbon nanotubes. In a carbon nanotube, there is little variation in the magnitude of the exciton wave function around the tube circumference. In boron nitride, there is quite a lot of variation. The electron is localized around the hole, and the probability of finding it on the opposite side of the tube is very small. The amplitude is also strongly peaked on the boron sites.
In effect, the second result implies the excitons in boron nitride are qualitatively different than in carbon. They cannot be considered one-dimensional objects. The exciton envelope function in a carbon nanotube is a smooth localized function. That of a boron nitride nanotube must describe the variations in amplitdue between the two sublattices. The boron nitride exciton is a two-dimensional wave function that happens to live on a cylinder. It is not strongly influenced by the periodic boundary conditions. (At least not in the tube studied here. In a tube with a smaller radius, the amplitude might not decay to zero over half a circumference.)
An interesting question that has probably been answered: Are the excitons in a boron nitride nanotube much different from those of a boron nitride sheet? The sheet has 3-fold rotational symmetry, which the tube does not. However, the tightly-localized excitons in the tube might not "be able to tell the difference," so to speak. There are no excitons in graphene because it is a semi-metal with no band gap. The large gap of planar boron nitride should allow for excitons, and I would expect them to be quite similar to their nanotube analogs.
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