PDF Publication Title:
Text from PDF Page: 035
Tight-Binding Model for Electrons on the Honeycomb Lattice 31 Eq. (2.24) is satisfied when γkD = 0, i.e. when ReγkD = 1 + cos(kD · a2) + cos(kD · a3) (2.25) = 1 + cos "√3a(kxD + √3kyD)# + cos "√3a(−kxD + √3kyD)# = 0 and, equally, ImγkD = sin "√3a(kxD + √3kyD)# + sin "√3a(−kxD + √3kyD)# = 0. (2.26) 22 The last equation may be satisfied by the choice kyD = 0, and Eq. (2.25), thus, when √3aD ! D 4π 1 + 2 cos 2 kx = 0 ⇒ kx = ± 3√3a . Comparison with Eq. (1.6) shows that there are, thus, two5 inequivalent Dirac points D and D′, which are situated at the points K and K′, respec- 22 tively, D 4π k = ±K = ±3√3aex. (2.27) Although situated at the same position in the first BZ, it is useful to make a clear conceptual distinction between the Dirac points, which are defined as the points where the two bands π and π∗ touch each other, and the purely crystallographic points K and K′, which are defined as the corners of the first BZ. There are, indeed, situations where the Dirac points move away from the points K and K′, e.g. when the nn hopping amplitudes are no longer the same in the directions δ1, δ2, and δ3 [19]. In the following chapters, we will, however, consider the natural situation for graphene where the Dirac points are situated at the BZ corners and use the notation K and K′ for the Dirac points. Notice that because of the symmetry ǫ−k = ǫk, which is a consequence of time-reversal symmetry, Dirac points occur necessarily in pairs – if kD is a 5We remind the reader that there are only two inequivalent points, and not six as Fig. 2.2 (b) might suggest. As pointed out in Sec. 1.2.1, there are pairs of three points that may be connected to one another by a reciprocal lattice vector and that are, thus, crystallographically equivalent.PDF Image | Physical Properties of Graphene
PDF Search Title:
Physical Properties of GrapheneOriginal File Name Searched:
CoursGraphene2008.pdfDIY PDF Search: Google It | Yahoo | Bing
Salgenx Redox Flow Battery Technology: Power up your energy storage game with Salgenx Salt Water Battery. With its advanced technology, the flow battery provides reliable, scalable, and sustainable energy storage for utility-scale projects. Upgrade to a Salgenx flow battery today and take control of your energy future.
| CONTACT TEL: 608-238-6001 Email: greg@infinityturbine.com | RSS | AMP |