# Tag Info

18

Graphene is only transparent because it is very thin (one atom thick). If it absorbs 2% per layer then just a few hundred layers would absorb almost all light and that would still be a very thin sheet of graphite. The question should be why does graphene absorb so much light compared to diamond which really is transparent? A simplified answer is that ...

14

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 ...

14

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

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 ...

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} \\ c_{\mathbf{k}B}\... 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. 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 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 ... 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 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 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 ... 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 ... 4 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 ... 4 This answer won't be very long, because there's not all too much to say: It's a typo. This is clear from the context: The sentence describes graphene, as witnessed by the words "single layer", which is the characteristic property of graphene. The sentence occurs in a paper on graphene. The 'n' is found next to the 'm' on most keyboards. 4 I've also stucked with such problem. Since this is old question, but I didn't find the full answer on it, I'll write down my attemption. It doesn't represent the full solution; however, I think that it almost gives the answer. The density of states \rho (E) is the imaginary part of the self-energy \Sigma (\mathbf r , \mathbf r, E+i\epsilon), where \... 4 The Berry phase in one dimension is usually called the Zak phase . Viewing the parameter space as a 1-D Brillouin zone, then for a two band Hamiltonian:$$ H = h_x \sigma_x + h_y \sigma_y + h_z \sigma_z,$$the Zak phase is half the solid angle of the winding path of the unit vector$$ \hat{n} = (h_x, h_y, h_z)/ \sqrt{h_x^2+h_y^2+h_z^2}$$on the Bloch ... 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 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 Graphene is already commercially used in printable, conducting inks (see e.g. here). This application works with small graphene flakes in a liquid solution. For most electronic applications, high-quality, large-area graphene placed on an insulating substrate is needed. The most promising method (in my opinion) for this end is chemical vapor deposition (CVD) ... 3 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 ... 3 There are similar topological invariants for band structures in one dimension, but an important difference is that these invariants always require some symmetry in the band Hamiltonians, for example particle-hole symmetry. In such cases, typically the invariant is given by$$ C=\int\frac{dk}{2\pi} a_k \text{ mod }1 

3

It is a very beautiful paper! But as all the old Physical Review Letters a bit cryptic, the supplementary material in the arXiv (http://arxiv.org/pdf/cond-mat/0604594v3.pdf) version is helping a bit. In the full $8\times8$ Hamiltonian, electrons from valley K are coupled with holes from valley K' via the proximized superconducting coupling $\Delta$, the ...

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

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 ...

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 ...

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$\text{m}\Omega ^{-1}$, 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.

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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 ...

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