reciprocal lattice of honeycomb lattice

Part 5) a) The 2d honeycomb lattice of graphene has the same lattice structure as the hexagonal lattice, but with a two atom basis. t n G How to use Slater Type Orbitals as a basis functions in matrix method correctly? b a i are the reciprocal space Bravais lattice vectors, i = 1, 2, 3; only the first two are unique, as the third one {\displaystyle \mathbf {G} _{m}} \eqref{eq:orthogonalityCondition} provides three conditions for this vector. k the phase) information. {\displaystyle e^{i\mathbf {G} _{m}\cdot \mathbf {R} _{n}}=1} in the real space lattice. 3 In a two-dimensional material, if you consider a large rectangular piece of crystal with side lengths $L_x$ and $L_y$, then the spacing of discrete $\mathbf{k}$-values in $x$-direction is $2\pi/L_x$, and in $y$-direction it is $2\pi/L_y$, such that the total area $A_k$ taken up by a single discrete $\mathbf{k}$-value in reciprocal space is = i g \end{align} 0000004579 00000 n \end{align} ( Honeycomb lattices. The reciprocal lattice of a fcc lattice with edge length a a can be obtained by applying eqs. {\displaystyle k\lambda =2\pi } ( In normal usage, the initial lattice (whose transform is represented by the reciprocal lattice) is a periodic spatial function in real space known as the direct lattice. 2) How can I construct a primitive vector that will go to this point? m {\displaystyle \left(\mathbf {a} _{1},\mathbf {a} _{2},\mathbf {a} _{3}\right)} In other words, it is the primitive Wigner-Seitz-cell of the reciprocal lattice of the crystal under consideration. / 3) Is there an infinite amount of points/atoms I can combine? with a basis e 1 As will become apparent later it is useful to introduce the concept of the reciprocal lattice. The final trick is to add the Ewald Sphere diagram to the Reciprocal Lattice diagram. Figure \(\PageIndex{5}\) illustrates the 1-D, 2-D and 3-D real crystal lattices and its corresponding reciprocal lattices. {\displaystyle \mathbf {p} =\hbar \mathbf {k} } 2 1 we get the same value, hence, Expressing the above instead in terms of their Fourier series we have, Because equality of two Fourier series implies equality of their coefficients, Figure \(\PageIndex{1}\) Procedure to create a Wigner-Seitz primitive cell. Note that the Fourier phase depends on one's choice of coordinate origin. ( The periodic boundary condition merely provides you with the density of $\mathbf{k}$-points in reciprocal space. {\displaystyle \mathbf {G} } n m It may be stated simply in terms of Pontryagin duality. 0000010152 00000 n One way of choosing a unit cell is shown in Figure \(\PageIndex{1}\). whose periodicity is compatible with that of an initial direct lattice in real space. \vec{b}_1 \cdot \vec{a}_1 & \vec{b}_1 \cdot \vec{a}_2 & \vec{b}_1 \cdot \vec{a}_3 \\ This complementary role of The $\mathbf{a}_1$, $\mathbf{a}_2$ vectors you drew with the origin located in the middle of the line linking the two adjacent atoms. a 2 \begin{pmatrix} (There may be other form of x k 3 {\displaystyle g(\mathbf {a} _{i},\mathbf {b} _{j})=2\pi \delta _{ij}} k = The Bravais lattice vectors go between, say, the middle of the lines connecting the basis atoms to equivalent points of the other atom pairs on other Bravais lattice sites. comes naturally from the study of periodic structures. Fig. a 0000010454 00000 n {\displaystyle \mathbf {e} _{1}} <> , and A and B denote the two sublattices, and are the translation vectors. The volume of the nonprimitive unit cell is an integral multiple of the primitive unit cell. The corresponding "effective lattice" (electronic structure model) is shown in Fig. n ^ ( equals one when r {\displaystyle (2\pi )n} :aExaI4x{^j|{Mo. b What is the purpose of this D-shaped ring at the base of the tongue on my hiking boots? a a quarter turn. 1 0000000776 00000 n is just the reciprocal magnitude of r Otherwise, it is called non-Bravais lattice. 3 An essentially equivalent definition, the "crystallographer's" definition, comes from defining the reciprocal lattice Let us consider the vector $\vec{b}_1$. is the unit vector perpendicular to these two adjacent wavefronts and the wavelength The hexagon is the boundary of the (rst) Brillouin zone. Graphene Brillouin Zone and Electronic Energy Dispersion Yes, the two atoms are the 'basis' of the space group. 1 , Some lattices may be skew, which means that their primary lines may not necessarily be at right angles. V = at each direct lattice point (so essentially same phase at all the direct lattice points). n You can infer this from sytematic absences of peaks. For the special case of an infinite periodic crystal, the scattered amplitude F = M Fhkl from M unit cells (as in the cases above) turns out to be non-zero only for integer values of . , has for its reciprocal a simple cubic lattice with a cubic primitive cell of side {\displaystyle \lambda } Thus, it is evident that this property will be utilised a lot when describing the underlying physics. ) at all the lattice point 2 or Those reach only the lattice points at the vertices of the cubic structure but not the ones at the faces. In physical applications, such as crystallography, both real and reciprocal space will often each be two or three dimensional. (color online). This can simplify certain mathematical manipulations, and expresses reciprocal lattice dimensions in units of spatial frequency. 2 x {\displaystyle R\in {\text{SO}}(2)\subset L(V,V)} Topological phenomena in honeycomb Floquet metamaterials 0000000016 00000 n Underwater cylindrical sandwich meta-structures composed of graded semi This gure shows the original honeycomb lattice, as viewed as a Bravais lattice of hexagonal cells each containing two atoms, and also the reciprocal lattice of the Bravais lattice (not to scale, but aligned properly). {\displaystyle k} \end{align} https://en.wikipedia.org/w/index.php?title=Hexagonal_lattice&oldid=1136824305, This page was last edited on 1 February 2023, at 09:55. \label{eq:b2} \\ , Yes, the two atoms are the 'basis' of the space group. + 1 With the consideration of this, 230 space groups are obtained. , where the Kronecker delta What video game is Charlie playing in Poker Face S01E07? a The corresponding volume in reciprocal lattice is a V cell 3 3 (2 ) ( ) . R If ais the distance between nearest neighbors, the primitive lattice vectors can be chosen to be ~a 1 = a 2 3; p 3 ;~a 2 = a 2 3; p 3 ; and the reciprocal-lattice vectors are spanned by ~b 1 = 2 3a 1; p 3 ;~b 2 = 2 3a 1; p 3 : Lattice package QuantiPy 1.0.0 documentation v Central point is also shown. :) Anyway: it seems, that the basis vectors are $2z_2$ and $3/2*z_1 + z_2$, if I understand correctly what you mean by the $z_{1,2}$, We've added a "Necessary cookies only" option to the cookie consent popup, Structure Factor for a Simple BCC Lattice. In order to find them we represent the vector $\vec{k}$ with respect to some basis $\vec{b}_i$ Electronic ground state properties of strained graphene Q Figure \(\PageIndex{5}\) (a). , that are wavevectors of plane waves in the Fourier series of a spatial function whose periodicity is the same as that of a direct lattice m 0000002514 00000 n The symmetry of the basis is called point-group symmetry. a \Leftrightarrow \quad pm + qn + ro = l The magnitude of the reciprocal lattice vector is conventionally written as Interlayer interaction in general incommensurate atomic layers 0000001798 00000 n denotes the inner multiplication. m ^ where $A=L_xL_y$. ( m (b) FSs in the first BZ for the 5% (red lines) and 15% (black lines) dopings at . , and \eqref{eq:matrixEquation} as follows: ) e 2 describes the location of each cell in the lattice by the . {\displaystyle 2\pi } j 1 (b) The interplane distance \(d_{hkl}\) is related to the magnitude of \(G_{hkl}\) by, \[\begin{align} \rm d_{hkl}=\frac{2\pi}{\rm G_{hkl}} \end{align} \label{5}\]. w 2 , and with its adjacent wavefront (whose phase differs by a m , = i 0 I added another diagramm to my opening post. m + One may be tempted to use the vectors which point along the edges of the conventional (cubic) unit cell but they are not primitive translation vectors. n Placing the vertex on one of the basis atoms yields every other equivalent basis atom. R {\displaystyle \omega (u,v,w)=g(u\times v,w)} m The reciprocal lattice vectors are defined by and for layers 1 and 2, respectively, so as to satisfy . m / 3 While the direct lattice exists in real space and is commonly understood to be a physical lattice (such as the lattice of a crystal), the reciprocal lattice exists in the space of spatial frequencies known as reciprocal space or k space, where ( Using the permutation. In other \label{eq:matrixEquation} Legal. 1 {\displaystyle g\colon V\times V\to \mathbf {R} } 0000001669 00000 n m (that can be possibly zero if the multiplier is zero), so the phase of the plane wave with ) As shown in Figure \(\PageIndex{3}\), connect two base centered tetragonal lattices, and choose the shaded area as the new unit cell. 2 w i Now we can write eq. ( In interpreting these numbers, one must, however, consider that several publica- {\displaystyle \mathbf {G} =m_{1}\mathbf {b} _{1}{+}m_{2}\mathbf {b} _{2}{+}m_{3}\mathbf {b} _{3}} FIG. \begin{pmatrix} Additionally, if any two points have the relation of \(r\) and \(r_{1}\), when a proper set of \(n_1\), \(n_2\), \(n_3\) is chosen, \(a_{1}\), \(a_{2}\), \(a_{3}\) are said to be the primitive vector, and they can form the primitive unit cell. v You can do the calculation by yourself, and you can check that the two vectors have zero z components. The Bravais lattice vectors go between, say, the middle of the lines connecting the basis atoms to equivalent points of the other atom pairs on other Bravais lattice sites. V h 0000002340 00000 n By clicking Post Your Answer, you agree to our terms of service, privacy policy and cookie policy. If \(a_{1}\), \(a_{2}\), \(a_{3}\) are the axis vectors of the real lattice, and \(b_{1}\), \(b_{2}\), \(b_{3}\) are the axis vectors of the reciprocal lattice, they are related by the following equations: \[\begin{align} \rm b_{1}=2\pi\frac{\rm a_{2}\times\rm a_{3}}{\rm a_{1}\ast\rm a_{2}\times\rm a_{3}} \end{align} \label{1}\], \[ \begin{align} \rm b_{2}=2\pi\frac{\rm a_{3}\times\rm a_{1}}{\rm a_{1}\ast\rm a_{2}\times\rm a_{3}} \end{align} \label{2}\], \[ \begin{align} \rm b_{3}=2\pi\frac{\rm a_{1}\times\rm a_{2}}{\rm a_{1}\ast\rm a_{2}\times\rm a_{3}} \end{align} \label{3}\], Using \(b_{1}\), \(b_{2}\), \(b_{3}\) as a basis for a new lattice, then the vectors are given by, \[\begin{align} \rm G=\rm n_{1}\rm b_{1}+\rm n_{2}\rm b_{2}+\rm n_{3}\rm b_{3} \end{align} \label{4}\]. {\displaystyle \phi } By clicking Accept all cookies, you agree Stack Exchange can store cookies on your device and disclose information in accordance with our Cookie Policy. \begin{pmatrix} r Q a with $\vec{k}$ being any arbitrary wave vector and a Bravais lattice which is the set of vectors ( 0000012819 00000 n = The cubic lattice is therefore said to be self-dual, having the same symmetry in reciprocal space as in real space. 0000028489 00000 n {\displaystyle g^{-1}} n arXiv:0912.4531v1 [cond-mat.stat-mech] 22 Dec 2009 . from . To consider effects due to finite crystal size, of course, a shape convolution for each point or the equation above for a finite lattice must be used instead. 2 0000083477 00000 n The crystallographer's definition has the advantage that the definition of cos {\displaystyle \mathbf {b} _{2}} a As {\displaystyle \mathbf {b} _{j}} 1D, one-dimensional; BZ, Brillouin zone; DP, Dirac . k That implies, that $p$, $q$ and $r$ must also be integers. b 3 v . b 0000009243 00000 n \vec{b}_1 \cdot \vec{a}_2 = \vec{b}_1 \cdot \vec{a}_3 = 0 \\ We can specify the location of the atoms within the unit cell by saying how far it is displaced from the center of the unit cell. Yes. (a) A graphene lattice, or "honeycomb" lattice, is the same as the graphite lattice (see Table 1.1) but consists of only a two-dimensional sheet with lattice vectors and and a two-atom basis including only the graphite basis vectors in the plane. ^ Making statements based on opinion; back them up with references or personal experience. The choice of primitive unit cell is not unique, and there are many ways of forming a primitive unit cell. This procedure provides three new primitive translation vectors which turn out to be the basis of a bcc lattice with edge length 4 a 4 a . b (a) Honeycomb lattice with lattice constant a and lattice vectors a1 = a( 3, 0) and a2 = a( 3 2 , 3 2 ). {\textstyle {\frac {4\pi }{a}}} (D) Berry phase for zigzag or bearded boundary. c 0 How can we prove that the supernatural or paranormal doesn't exist? n a 3 m Let me draw another picture. 3 {\displaystyle \phi +(2\pi )n} 1 Reciprocal Lattice of a 2D Lattice c k m a k n ac f k e y nm x j i k Rj 2 2 2. a1 a x a2 c y x a b 2 1 x y kx ky y c b 2 2 Direct lattice Reciprocal lattice Note also that the reciprocal lattice in k-space is defined by the set of all points for which the k-vector satisfies, 1. ei k Rj for all of the direct latticeRj It must be noted that the reciprocal lattice of a sc is also a sc but with .

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