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13

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 ...


13

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 ...


8

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} \\ ...


8

Because its structure displays translational symmetry in 2D. Atoms themselves are 3D as in other materials, but they are placed on a 2D flat plane. Compare to 1D Fullerenes.


7

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 ...


5

As far as I understand, electrons in graphene are not relativistic, although quasiparticles in graphene are indeed described by the massless Dirac equation. However, for graphene, the speed velocity in this equation is replaced by the Fermi velocity, which is much smaller.


4

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 ...


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 ...


4

Thermodynamic relation $N=-\frac{\partial J}{\partial \mu}$ exactly gives you the particle number equation, wherein $J$ is the macroscopic thermodynamic potential, i.e., the quantity $F$ in your question. In thermodynamics, the relation $dJ=-SdT+Ydy-Nd\mu$ tells you why the partial derivative equation is valid. In statistical mechanics, grand canonical ...


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

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 ...


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 ...


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

$\text{m}\Omega ^{-1}$, means milli-S, that means the resistivity is in the range of kilo-Ohm. What's the problem? Apparently, the curve in your post shows very low conductivity compared to Cu.


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

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

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

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 ...


2

All three questions can be answered by first artificially separating the graphene sheet into two sheets: (a) first sheet with only spin up electrons, and (b) second sheet with only spin down electrons. This statement alone should partially answer your third question; for the sake of organization, however, I will repeat a summary of this paragraph (in the ...


2

Well, Landau's statements were not as definitive as you appear to think. His views are summarized in Statistical Physics (Landau and Lifshitz). From my copy of the 3rd edition, part one, they are found in sections 137 and 138. The discussion is on thermal fluctuations as a function of temperature and size of the 2D film. The following quotes will get you ...


2

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 ...


2

Partial answer for the first part of your question: It is written : "As each term commutes with the reflection operator, the full Hamiltonian must commute with the reflection operator, and thus, the eigenstates of H in the symmetry-adapted basis are either symmetric or anti- symmetric about the line defect." "Antisymmetric states have a node at the line ...


1

There are indeed two honeycomb structures in the picture, but as suggested by the inset in the left bottom corner, the smaller one corresponds to the atoms. The graphene in the scanning tunneling microscopy (STM) picture you show is adsorbed on an Iridium surface with Miller indices (111). Since the lattice of the Iridium atoms at the surface and the ...


1

According to this article, the molar heat capacities of graphite and graphene should be identical above roughly 100 Kelvin. The heat capacity of graphene is dominated by phonon contributions above roughly 1 Kelvin (below that, free electron heat capacity becomes a significant contributor). The molar heat capacity of graphite at room temperature is ...


1

Your value is within the range of literature values. Hydrocarbon Lithography on Graphene Membranes states "the Fermi wavelength of the electrons in graphene of 0.74 nm". Many references cite this value. Another reference says ~0.14nm.


1

Electrons near the band edge are a little interesting in graphene, because their dispersion relation goes like $\omega \sim k$ instead of $\omega \sim k^2$ as they do for massive particles. The bands actually touch in graphene, and so are linear at the band edge. Graphene electrons propagate like massless particles with a fixed velocity $v_0$ regardless of ...



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