Handout 5 The Reciprocal Lattice In this lecture you will learn: • Fourier transforms of lattices • The reciprocal lattice • Brillouin Zones • X-ray diffraction • Fourier transforms of lattice periodic functions ECE 407 – Spring 2009 – Farhan Rana – Cornell University Fourier Transform (FT) of a 1D Lattice Consider a 1D Bravais lattice: a1 a xˆ Now consider a function consisting of a “lattice” of delta functions – in which a delta function is placed at each lattice point: f x x a1 a xˆ f x x n a n The FT of this function is (as you found in your homework): i kx x i kx n a 2 2 f k x dx x n a e e k x m n n a m a The FT of a train of delta functions is also a train of delta functions in k-space ECE 407 – Spring 2009 – Farhan Rana – Cornell University 1 Reciprocal Lattice as FT of a 1D Lattice f x 1 x a1 a xˆ FT is: f k x 2 a 2 k b xˆ x 1 a The reciprocal lattice is defined by the position of the delta-functions in the FT of the actual lattice (also called the direct lattice) Direct lattice (or the actual lattice): x a1 a xˆ Reciprocal lattice: 2 k x b xˆ 1 a ECE 407 – Spring 2009 – Farhan Rana – Cornell University Reciprocal Lattice of a 1D Lattice For the 1D Bravais lattice, a1 a xˆ The position vector R n of any lattice point is given by: Rn n a1 f x x a1 a xˆ f x x Rn n i kx x i k.Rn The FT of this function is: f k x dx x Rn e e n n The reciprocal lattice in k-space is defined by the set of all points for which the k- vector satisfies, ei k . Rn 1 for ALL R n of the direct lattice For the points in k-space belonging to the reciprocal lattice the summation e i k . R n becomes very large! n ECE 407 – Spring 2009 – Farhan Rana – Cornell University 2 Reciprocal Lattice of a 1D Lattice For the 1D Bravais lattice, a1 a xˆ The position vector R n of any lattice point is given by: Rn n a1 The reciprocal lattice in k-space is defined by the set of all points for which the k- vector satisfies, ei k . Rn 1 for ALL R n of the direct lattice i k . Rn For k to satisfy e 1 , it must be that for all R n : k . Rn 2 integer k x na 2 integer 2 k m where m is any integer x a Therefore, the reciprocal lattice is: 2 k x b xˆ 1 a ECE 407 – Spring 2009 – Farhan Rana – Cornell University Reciprocal Lattice of a 2D Lattice y Consider the 2D rectangular Bravais lattice: If we place a 2D delta function at each lattice point we get the function: a2 c yˆ x f x,y x n a y m c a1 a xˆ nm The above notation is too cumbersome, so we write it in a simpler way as: 2 The summation over “j ” is over all the lattice points f r r R j j 2 2 A 2D delta function has the property: d r r ro gr gro and it is just a product of two 1D delta functions corresponding to the x and y components of the vectors in its arguments: 2 r ro x ro.xˆ y ro.yˆ Now we Fourier transform the function f r : 2 i k . r 2 2 i k . r f k d r f r e d r r R j e j 2 i k . R j 2 2 2 e k x n ky m j ac nm a c ECE 407 – Spring 2009 – Farhan Rana – Cornell University 3 Reciprocal Lattice of a 2D Lattice 2 i k . R j 2 2 2 f k e k x n ky m j ac nm a c ky y 2 b2 yˆ Reciprocal lattice c a2 c yˆ x 2 k x a1 a xˆ b xˆ 1 a Direct lattice • Note also that the reciprocal lattice in k-space is defined by the set of all points for which the k-vector satisfies, i k . R e j 1 for all R j of the direct lattice • Reciprocal lattice as the FT of the direct lattice or as set of all points in k-space for which exp i k . R j 1 for all R j , are equivalent statements ECE 407 – Spring 2009 – Farhan Rana – Cornell University Reciprocal Lattice of a 2D Lattice y ky 2 b2 yˆ Reciprocal lattice c a2 c yˆ x 2 k x a1 a xˆ b xˆ 1 a Direct lattice • The reciprocal lattice of a Bravais lattice is always a Bravais lattice and has its own primitive lattice vectors, for example, b 1 and b 2 in the above figure • The position vector G of any point in the reciprocal lattice can be expressed in terms of the primitive lattice vectors: G n b1 m b2 For m and n integers So we can write the FT in a better way as: 2 2 2 2 2 2 2 f k k x n ky m k G j ac nm a c 2 j where 2 = ac is the area of the direct lattice primitive cell ECE 407 – Spring 2009 – Farhan Rana – Cornell University 4 Reciprocal Lattice of a 3D Lattice Consider a orthorhombic direct lattice: d R n a1 m a2 p a3 where n, m, and p are integers a2 a Then the corresponding delta-function lattice is: c 1 a 3 3 f r r R j j a 3 3 A 3D delta function has the property: d r r ro gr gro The reciprocal lattice in k-space is defined by the set of all points for which the k- vector satisfies: exp i k . R j 1 for all R j of the direct lattice. The above relation will hold if k equals G : 2 2 2 G n b m b p b and b xˆ b yˆ b zˆ 1 2 3 1 a 2 c 3 d Finally, the FT of the direct lattice is: 3 i k . r 3 3 i k . r f k d r f r e d r r R j e j b 2 3 3 b1 i k . R j 2 3 2 3 b3 e k G j k G j j acd j 3 j ECE 407 – Spring 2009 – Farhan Rana – Cornell University Direct Lattice Vectors and Reciprocal Lattice Vectors ky y 2 b2 yˆ Reciprocal lattice c a2 c yˆ x 2 k x a1 a xˆ b xˆ 1 a Direct lattice R n a1 m a2 G n b1 m b2 Remember that the reciprocal lattice in k-space is defined by the set of all points for which the k-vector satisfies, ei k . R 1 for all R of the direct lattice So for all direct lattice vectors R and all reciprocal lattice vectors G we must have: ei G . R 1 ECE 407 – Spring 2009 – Farhan Rana – Cornell University 5 Reciprocal Lattice of General Lattices in 1D, 2D, 3D More often that not, the direct lattice primitive vectors, a 1 , a 2 , and a 3 , are not orthogonal Question: How does one find the reciprocal lattice vectors in the general case? ID lattice: If the direct lattice primitive vector is: a1 a xˆ and length of primitive cell is: = a 2 Note: Then the reciprocal lattice primitive vector is: ˆ b1 x a a . b 2 2 1 1 f r r R j f k k G j i G . R j 1 j e p m 1 2D lattice: If the direct lattice is in the x-y plane and the primitive vectors are: a and a 1 2 and area of primitive cell is: 2 a1 a2 aˆ2 zˆ zˆ a1 Then the reciprocal lattice primitive vectors are: b1 2 b2 2 2 2 2 2 2 2 f r r R j f k k G j j 2 j i Gp . Rm Note: a j . bk 2 jk and e 1 ECE 407 – Spring 2009 – Farhan Rana – Cornell University Reciprocal Lattice of General Lattices in 1D, 2D, 3D 3D lattice: If the direct lattice primitive vectors are: a , a , and a 1 2 3 and volume of primitive cell is: 3 a1 . a2 a3 Then the reciprocal lattice primitive vectors are: Note: a . b 2 aˆ2 a3 a3 a1 a1 a2 j k jk b1 2 b2 2 b3 2 3 3 3 i G . R e p m 1 3 3 2 3 f r r R j f k k G j j 3 j Example 2D lattice: 2 b xˆ yˆ 1 b a1 b xˆ 4 b b b2 yˆ a xˆ yˆ b2 b 2 2 2 a2 b2 b b a a a 1 4 b 1 2 1 2 2 b 4 b ECE 407 – Spring 2009 – Farhan Rana – Cornell University 6 The Brillouin Zone The Wigner-Seitz primitive cell of the reciprocal lattice centered at the origin is called the Brillouin zone (or the first Brillouin zone or FBZ) Wigner-Seitz primitive cell 1D direct lattice: x a1 a xˆ Reciprocal lattice: First Brillouin zone 2 k x b1 xˆ 2D lattice: a y Wigner-Seitz ky primitive cell Reciprocal lattice 2 b2 yˆ First Brillouin zone c a2 c yˆ k a1 a xˆ x 2 x b xˆ 1 a Direct lattice ECE 407 – Spring 2009 – Farhan Rana – Cornell University Wigner-Seitz The Brillouin Zone 2D lattice: primitive cell First Brillouin zone 2 b b b1 xˆ yˆ a1 xˆ yˆ b 2 2 a1 b2 2 b b b2 xˆ yˆ a2 xˆ yˆ b 2 2 b1 a2 b2 b a a 4 b 2 1 2 2 b 4 b Direct lattice Reciprocal lattice Volume/Area/Length of the first Brillouin zone: The volume (3D), area (2D), length (1D) of the first Brillouin zone is given in the same way as the corresponding expressions for the primitive cell of a direct lattice: Note that in all dimensions (d) the following 1D b 1 1 relationship holds between the volumes, areas, 2D b b lengths of the direct and reciprocal lattice 2 1 2 primitive cells: 2 d 3D 3 b1 .