1ECE 407 – Spring 2009 – Farhan Rana – Cornell UniversityHandout 4Lattices in 1D, 2D, and 3DIn this lecture you will learn:• Bravais lattices• Primitive lattice vectors• Unit cells and primitive cells• Lattices with basis and basis vectorsAugust Bravais (1811-1863)ECE 407 – Spring 2009 – Farhan Rana – Cornell UniversityBravais LatticeA fundamental concept in the description of crystalline solids is that of a “Bravais lattice”. A Bravais lattice is an infinite arrangement of points (or atoms) in space that has the following property:The lattice looks exactly the same when viewed from any lattice point A 1D Bravais lattice:bA 2D Bravais lattice:bc2ECE 407 – Spring 2009 – Farhan Rana – Cornell UniversityBravais LatticeA 2D Bravais lattice:A 3D Bravais lattice:bdcECE 407 – Spring 2009 – Farhan Rana – Cornell UniversityBravais LatticeA Bravais lattice has the following property:The position vector of all points (or atoms) in the lattice can be written as follows:21amanR321apamanR1anR1D2D3DWhere n, m, p = 0, ±1, ±2, ±3, …….And the vectors,are called the “primitive lattice vectors” and are said to span the lattice. These vectors are not parallel.321 and,, aaaExample (1D): xbaˆ1Example (2D): bcxbaˆ1ycaˆ2xyb3ECE 407 – Spring 2009 – Farhan Rana – Cornell UniversityBravais LatticebdcExample (3D): xbaˆ1ycaˆ2zdaˆ3The choice of primitive vectors is NOT unique:bcxbaˆ1ycaˆ2xbaˆ1ycxbaˆˆ2All sets of primitive vectors shown will work for the 2D latticeECE 407 – Spring 2009 – Farhan Rana – Cornell UniversityBravais LatticeAll lattices are not Bravais lattices:Example (2D): The honeycomb lattice4ECE 407 – Spring 2009 – Farhan Rana – Cornell UniversityThe Primitive Cell• A primitive cell of a Bravais lattice is the smallest region which when translated by all different lattice vectors can “tile” or “cover” the entire lattice without overlappingTwo different choices of primitive cell Tiling of the lattice by the primitive cellbcxbaˆ1ycaˆ2• The volume (3D), area (2D), or length (1D) of a primitive cell can be given in terms of the primitive vectors, and is independent of the choice of the primitive vectors or of the primitive cells212aa3213. aaa11a1D2D3DExample, for the 2D lattice above:bcaa212xbaˆ1ycxbaˆˆ2bcaa 212or• The primitive cell is not uniqueECE 407 – Spring 2009 – Farhan Rana – Cornell UniversityThe Wigner-Seitz Primitive Cell• The Wigner-Seitz (WS) primitive cell of a Bravais lattice is a special kind of a primitive cell and consists of region in space around a lattice point that consists of all points in space that are closer to this lattice point than to any other lattice pointWS primitive cellTiling of the lattice by the WS primitive cellbcxbaˆ1ycaˆ2• The volume (3D), area (2D), or length (1D) of a WS primitive cell can be given in terms of the primitive vectors, and is independent of the choice of the primitive vectors212aa3213. aaa11a1D2D3DExample, for the 2D lattice above:bcaa212xbaˆ1ycxbaˆˆ2bcaa 212or• The Wigner-Seitz primitive cell is unique5ECE 407 – Spring 2009 – Farhan Rana – Cornell UniversityWigner-Seitz Primitive CellExample (2D): bbxbaˆ1ybxbaˆ2ˆ2222212baa Primitive cellExample (3D): bdcxbaˆ1ycaˆ2zdaˆ3bcdaaa 3213.Primitive cellECE 407 – Spring 2009 – Farhan Rana – Cornell UniversityLattice with a BasisConsider the following lattice:• Clearly it is not a Bravais lattice (in a Bravais lattice, the lattice must look exactly the same when viewed from any lattice point)• It can be thought of as a Bravais lattice with a basis consisting of more than just one atom per lattice point – two atoms in this case. So associated with each point of the underlying Bravais lattice there are two atoms. Consequently, each primitive cell of the underlying Bravais lattice also has two atomsbchbchPrimitive cell• The location of all the basis atoms, with respect to the underlying Bravais lattice point, within one primitive cell are given by the basis vectors: xhddˆ0216ECE 407 – Spring 2009 – Farhan Rana – Cornell UniversityLattice with a BasisConsider the Honeycomb lattice:It is not a Bravais lattice, but it can be considered a Bravais lattice with a two-atom basis I can take the “blue” atoms to be the points of the underlying Bravais lattice that has a two-atom basis - “blue” and “red” - with basis vectors:hhxhddˆ021Or I can take the small “black” points to be the underlying Bravais lattice that has a two-atom basis - “blue” and “red” - with basis vectors:Primitive cellPrimitive cellxhdxhdˆ2ˆ221Note: “red” and “blue” color coding is only for illustrative purposes. All atoms are the same.WS primitive cellWS primitive cellECE 407 – Spring 2009 – Farhan Rana – Cornell UniversityLattice with a BasisPrimitive cellaa / 2aNow consider a lattice made up of two different atoms: “red” and “black”, as shown• It is clearly not a Bravais lattice since two different types of atoms occupy lattice positions• The lattice define by the “red” atoms can be taken as the underlying Bravais lattice that has a two-atom basis: one “red” and one “black”• The lattice primitive vectors are:• The two basis vectors are:xaddˆ2021xaaˆ1yaxaaˆ2ˆ221a2aThe primitive cell has the two basis atoms: one “red” and one “black” (actually one-fourth each of four “black” atoms)7ECE 407 – Spring 2009 – Farhan Rana – Cornell UniversityBravais Lattices in 2DThere are only 5 Bravais lattices in 2DObliqueRectangular Centered RectangularHexagonal SquareECE 407 – Spring 2009 – Farhan Rana – Cornell UniversityLattices in 3D and the Unit Cellaaaxaaˆ1yaaˆ2zaaˆ3Simple Cubic Lattice:It is very cumbersome to draw entire lattices in 3D so some small portion of the lattice, having full symmetry of the lattice, is usually drawn. This small portion when repeated can generate the whole lattice and is called the