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12

I am still not sure what you precisely want to be a Klein Bottle, but let me make some comments that might help you clarify what exactly you want to know. (Warning: I am writing this while being very tired, people are invited to correct me.) First of all one must be careful to distinguish band structure of the bulk from band structure of a semi-infinite ...


7

In calculating the electron dispersion you probably obtained the diagonalized Hamiltonian in the momentum space $$ H=\sum_\mathbf{k}\left[c^{\dagger}_{\mathbf{k}A},c^{\dagger}_{\mathbf{k}B}\right]\left[\begin{array}{cc}0 & \Delta(\mathbf{k})\\ \Delta^{\dagger}(\mathbf{k}) &0\end{array}\right]\left[\begin{array}{c}c_{\mathbf{k}A} \\ ...


6

Although it's not strictly what happens, you can think of the bonds around a carbon atom as repelling each other because the electrons localised into those bonds want to get as far away from each oither as possible. That's why when a carbon atom forms three bonds you get the bonds separated by 120º. When you have four bonds they arrange themselves into a ...


5

As an ex-physicist who now works as a quant in power markets I think it's safe to say the physics of the matter will be swamped by the economics in commodities and how power markets work. Two things to note: power prices are set by markets and not by the viability of the technology (prime mover) solar is hard to make money with w/o a long term Power ...


4

A decent terrestrial space elevator could be built with a material with a tensile strength of 50 Gigapascals (including a decent safety factor), so this material may suffice. Note that there is no prospect of having one 100,000 km nanotube - they would actually be much shorter (maybe 10 cm) and held together by the much weaker inter-tube molecular bonds (if ...


3

The dimensionality of a system in practice means the number of dimensions in which objects confined to that system are free to move. For graphene we are generally talking about the motion of electrons within it (though I guess we could be talking about phonons). Anyhow, the thickness of the sheet is around one atom, which means that in the direction normal ...


3

There is such a material where each carbon atom binds to four other atoms. It's not a square lattice (due to the character of the so-called sp3-hybridization: the energetically most stable configuration is in 3D, not 2D). There are several standard bondings for carbon (and many other materials): the sp2-hybridization is in 2D and has three bonds (like ...


3

In the atomic ground state a carbon atom has the electronic configuration $1s^22s^22p^2$. In the sp$^2$ hybridization the $2s$, $2p_x$, and $2p_y$ participate in the formation of the three $\sigma$ bonds and the $2p_z$ orbital forms a $\pi$ bond. According to molecular orbital theory this $2p_z$ state would form the bonding ($\pi$) and anti-bonding orbitals ...


3

According to this article: http://physics.aps.org/articles/v5/24: The statement that in graphene the "conduction electrons are massless" is because the energy levels (bands) are proportional to their momenta. So the $E = \sqrt{p^2+m^2}$ relation of a free electron becomes $E\propto p$ in graphene. Massless particles travel all at the same speed because ...


2

The answer you'll get from most high-energy physicists is that there are no implications whatsoever. Lorentz invariance is extraordinarily well-tested: see, e.g., http://arxiv.org/abs/0801.0287. In particular, there are many relevant operators in the Standard Model that one would expect to be generated if physics at a high scale is not Lorentz-invariant. ...


2

The group velocity $v_g$ of a wave packet (that's the speed of the maximum of the wave packet) is given by $v_g=\frac{\partial\omega}{\partial k}$. In this case, $\frac{\partial\omega}{\partial k}=\frac 1 \hbar\frac{\partial E}{\partial k}$, which easily evaluates to $v_g=\frac{3ta}{2}=:v_f$ for $k=0$. That's actually the definition of $v_f$: it is the group ...


2

When silicene is buckled on the substrate it has a substantial band gap or in other words it can be turned on or off thus making it appropriate for digital applications. Graphene doesn't have a band gap so it isn't so good for digital circuits. Although techniques have been developed to produce a band gap and transistors have been made, they say that the ...


2

I think you are looking for something like this: We measured the elastic properties and intrinsic breaking strength of free-standing monolayer graphene membranes by nanoindentation in an atomic force microscope. The force-displacement behavior is interpreted within a framework of nonlinear elastic stress-strain response, and yields second- and ...


2

When the Möbius strip is cut down the middle you don't get two cylinders. See here and here for example. Fig. 3(b) should be interpreted as two cylinders, each with an extra (and different, thus two cases, $y<0$ and $y>0$) on-site potential that accounts for the twist. After the transformation the field operators obey periodic boundary conditions so ...


2

1) The Bloch theorem comes from the fact that the group of translations is Abel, thus its representations are defined by number which is called $\mathbf{k}$. It means that when you translate (by let's say vector $\mathbf{a}$) the wavefunction with given $\mathbf{k}$ it is multiplied by exponent $e^{i\mathbf{ka}}$ (more or less by definition), which gives you ...


2

You are right with your assumption - the special behaviour at the Dirac cone allows for an application of the holographic principle. But how is this possible? It turns out that since in this region of the band structure the Fermi velocity is very large, i.e. two orders of magnitude below the speed of light, graphene behaves effectively as a relativistic ...


1

There are really two separate parts to your question: what is the difference between a supercapacitor and a battery? why does graphene make such good supercapacitors? A battery and a supercapacitor work in very different ways. A battery generates electricity using a chemical reaction. As the reaction procedes a current is generated, and once the reagents ...


1

The $sp^2$ bonds can be regarded as approximately localised because each bond only involves two carbon atoms. I suppose they will form bands in graphene, but very narrow ones. Anyhow, when two $sp^2$ orbitals interact you get a bonding and an antibonding state. Since each carbon atom contributes one electron you fill the bonding state and leave the anti ...


1

Electric resistance is low in a layer of graphene due to delocalized electrons of the benzene rings. Your chemistry students propably know about the visualisation with three double/single-bonds and wondered about that. This simple experiment with a multimeter also demonstrates graphene as an electric insulator inbetween two layers. Concluding the fact that ...


1

oh no! it appears I'm too late.. so this is a popular claim, and further popularized by Michio Kaku (youtube). Hover, graphene cannot be as thin as cling film. Why? because graphene by definition is an atomically thin substance! It's literally one layer of graphite, which is how it was discovered. Saran wrap is literally a million times thicker than a sheet ...


1

TL;DR version: you want periodic boundary conditions with an extra twist. You don't want straightforward periodic boundary conditions as this would be solving for a structure that is periodic. You can solve for an infinite sheet of graphene using Bloch's theorem. I'll give a couple of details: Find a basic finite unit that you can tile to make the ...


1

You're mixing two things here. One is what the structure of the boundary is, e.g. armchair or zigzag. The other is what the wavefunction does at the boundary. For your finite size cluster of carbon atoms, you have to decide what shape it has, which basically means deciding how many lattice sites you include and where you put them. This would decide whether ...


1

The dielectric function to which you refer describes screening. From a phenomenological point of view, you can imagine the function acting as a damper (or sometimes an enhancer) of momentum and energy transfer. The wave vector $q$ and frequency $\omega$ dependence are these quantities, momentum $\hbar q$ and energy $\hbar\omega$ transfer, respectively. They ...


1

Resistivity is the relevant parameter for three-dimensional materials. Sheet resistance (less commonly called "sheet resistivity") is the relevant parameter for two-dimensional materials, and its inverse is called "sheet conductance" or "sheet conductivity". In the Novoselov paper you cited, they talk about sheet resistance and sheet conductance. Please ...


1

There are, in fact, a wide variety of techniques for producing graphene other than the scotch-tape method. A very good review of these techniques can be found in this recent review article: http://onlinelibrary.wiley.com/doi/10.1002/adma.201202321/abstract It is extremely difficult to obtain the dimensions you require using the scotch-tape method. In my ...


1

Note that as opposed to the case of neutrinos where the Dirac-Weyl equation is unambiguous, the effective equation for electrons in graphene has some ambiguity. Specifically it depends on the orientation of the axis with respect to the graphene lattice, and on the implied basis (which is often not explicitly written). So people just redefine the helicity ...


1

The real economics will come into play via electricity. Space based solar transmitting electricity down graphene cables solves our energy crisis basically forever. Once you build the first cable, building more is an order of magnitude cheaper. Once you make that initial investment, the solar farms become trivial, although it will take years if not decades to ...



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