Introduction to Condensed Matter Physics

Transcription

1 Introduction to Condensed Matter Physics The Reciprocal Lattice M.P. Vaughan

2 Overview

3 Overview of the reciprocal lattice Periodic functions Reciprocal lattice vectors Bloch functions k-space Dispersion relations roup velocity Free electrons Nearly free model Band structure Kronig-Penny model Valence and conduction bands Number of orbitals in a band

4 Periodic functions

5 Periodic functions A real, periodic function U(r) defined within a crystal structure must have the same periodicity as the lattice. This means that U r R Ur, where R is a lattice vector.

6 Real functions - examples Potential energy Charge density NOTE: The electronic wave-functions will not necessarily be periodic over the primitive cell Wave-functions are complex May be differ by phase factor on boundaries

7 Periodic functions Consider a plane wave with wavevector of the form ir r Ae. This will be periodic over the primitive cell if This may be achieved if for some integer n. r R r R 2n

8 Reciprocal lattice vectors The vectors may be written in the form m, b m2b 2 m3b 3 for integers m, m 2 and m 3, where the b i are called primitive reciprocal lattice vectors.

9 Reciprocal lattice vectors matrix form The primitive reciprocal lattice vectors may be written in matrix form as b b b b x 2x 3x b b b y 2 y 3y b b b z 2z 3z. The primitive reciprocal lattice is known as the Brillouin zone. We shall consider some simple examples.

10 Reciprocal lattice vectors If b i a j 2 ij, then we will have R m n m n m n n as we need.

11 Reciprocal lattice vectors In matrix form, the condition b i a j 2 ij, becomes ba T 2.

12 Reciprocal lattice vectors This may be expressed in a more concise form as ba T 2 I, where I is the identity matrix. Multiplying this on T the right by the inverse of a b 2 T a.

13 Reciprocal lattice vectors Note that most text books normally give the b i in the form b i a a 2 j k ai j k a a, with cyclic permutation of the indices i, j and k.

14 Lattice vectors examples. a a simple cubic. 2 a a. 2 a a FCC BCC

15 Reciprocal lattice vectors FCC Using b 2 T a. we find 2 b a. i.e. the Brillouin zone has a BCC structure.

16 Reciprocal lattice vectors BCC Using b 2 T a. we find b a. i.e. the Brillouin zone has an FCC structure.

17 Reciprocal lattice vectors The vectors are known as reciprocal lattice vectors, with the b vectors being the primitive reciprocal lattice vectors. Let a real, periodic function (such as the potential) be written as U ir r Ue.

18 Reciprocal lattice vectors Now U irr r R U e U U r, e ir as required. Note that U is the Fourier transform of U.

19 The Fourier transform Thus We may use plane waves with the reciprocal lattice vectors as a basis to perform Fourier transformation of functions with the periodicity of the crystal lattice.

20 Bloch functions

21 Bloch s theorem According to Bloch s theorem the electronic eigenfunctions of the crystal lattice take the form n, k ikr r u r e, n, k where n is the band index and u n,k (r) is a function with the periodicity of the crystal lattice.

22 Bloch s theorem Using the general form for a periodic function, we may write u n,k (r) in a plane-wave representation as Thus u ir C e. r n, k n, k ik C e r n, k n, k r.

23 Bloch s theorem Now, for some lattice vector R we will have ikr r R. n e, k n, k r Let us assume that is periodic over a vector R Na, (in one dimension) where a is the lattice constant.

24 Bloch s theorem In the x-direction, for instance, we then have n, k ik xna x Na e x. The periodic boundary conditions then require that This is satisfied when e ik x Na. 2n k x. Na n, k

25 k-space The difference between the x-components of the k vector are then k x 2 n Na 2n Na, 2 Na.

26 k-space Thus a k -vector occupies a reciprocal volume in k -space of 2 Na 3 3 2, V C where V C is the crystal volume (volume of the supercell). In general, however, N is usually large enough that the k-vectors are assumed to lie in a continuum of values.

27 k-space The periodicity condition of u n,k (r) is which implies u x a u, n, k n, k x e i x a and so 2n x a.

28 k-space Therefore the shortest x translating a point in k space into the next Brillouin zone is x 2. a Typically, we allow k x to vary from /a to /a, which then mark the boundaries of the Brillouin zone in the x-direction.

29 Dispersion relations

30 Dispersion relations in k-space Each of the Bloch functions is a simultaneous eigenvector of both momentum and energy p k and Ek k. The dependence of energy on momentum (or wave-vector) is encapsulated in the dispersion relations. We may plot the dispersion relations in k-space to visualise the electronic structure of the system.

31 roup velocity The gradient of k with respect to k gives the group velocity v k of a wave-packet centred at k k k v k ke k.

32 Free electrons For free electrons in one dimension, the solution for a single particle is found from Schrodinger s equation with the potential energy term set to zero: 2 2 m e d dx 2 2 k x E x, k k where m e is the mass of a free electron.

33 Free electrons The solution is a plane-wave with energy eigenvalues E k 2 2 k 2 m e. These dispersion relations are plotted in the succeeding graph.

34 Free electrons

35 Free electrons We go beyond primitive the extended Brillouin zone. The primitive BZ is then known as the reduced Brillouin zone. The points of the k x axis show equivalent k- points Associated eigenvectors must be common to the reduced Brillouin zone.

36 Free electrons

37 Nearly free electron model Electrons assumed to be weakly perturbed by the positively charged ionic cores. Ionic cores assumed to lie at the corners of a cubic unit cell of side a. Along a given direction, the wave-function is reflected specularly from the atomic planes.

38 Nearly free electron model Constructive interference required between the paths of the wave-function. The occurs when the Bragg condition, is satisfied. 2a sin m where is the wavelength and m is an integer (see graph for ).

39 The Bragg condition

40 Nearly free electron model In terms of wave-vector, we have k m. a sin In one dimension, the waves are both normally incident and this result reduces to k m a.

41 Nearly free electron model Putting m =, we can construct standing waves from these wavefunctions via k k ix / a / 2cos / ix a x e e x a ix / a / 2 sin ix a x e e i x / a. These standing waves are out of phase with each other, so their squared moduli will have peaks in different places.,

42 Nearly free electron model will pile its charge density on the ions at the zone boundaries charge density of will be highest away from the ions. Potential is lower at the ionic sites, so the will have a lower energy than the at the zone boundaries. Since both have the same wave-vector, this means there will be an energy gap at these points.

43 Nearly free electron model

44 Band structure

45 The Kronig Penney Model The potential profile for the Kronig Penney Model.

46 The Kronig Penney Model Solution of the Kronig Penney model (valid solutions in blue areas where curve = cos k(a+b)).

47 Valence and conduction bands From the second and third regions (from the left) of allowed solutions to the Kronig-Penney Model, we obtain graphs of the valence and conduction bands.

48 Valence and conduction bands

49 k x Number of orbitals in a band Earlier, we saw that The total series of k x -points in the BZ will be N Na 2, k x 4, 2 Na 2.,, 2, 4,, Na Na Na Na a where we have omitted /a since it is equivalent to /a.

50 Number of orbitals in a band There are therefore N independent points in the crystal from each primitive cell. So, allowing for spin, the number of orbitals in each band is then 2N This result also holds for 3D.

51 Number of orbitals in a band Hence If a primitive cell contains just one atom with a single valence electron, a band may be half filled With two atoms of valence one (or one atom of valence two), a band may be entirely filled.

52 Additional material

53 The Fourier transform We may use the prescription for a periodic function to perform Fourier transformation. Let W be the volume of a primitive cell. If there are N C primitive cells in the crystal volume V C, then V C N C W and we can write a normalised volume integral over all space as I V C d 3 N C r W.

54 The Fourier transform With a change of variable, I can be written as r R r', I can be written as I N C W R 3 d r'. W

55 The Fourier transform Using this prescription, we can write. ' 3 ' ' ' 3 ' W W W r r r r r R R r d U e e N N d U e i i C V C i C Making the substitution, r r i e U U

56 The Fourier transform. ' 3 ' ', ' 3 ' W W W r r r r R R r d U e e N N d U e i i C V C i C we have For the time being, we will focus on the integration over the primitive cell.

57 The Fourier transform For the sake of simplicity, we shall assume that the unit cell is simple cubic with lattice constant a. We then have W 3 i' r ' d r' i U e U xx y y z e W W a where, for the simple cubic structure, i 2 a and m i m i ' is an integer. m i m i ' 3 dxdydz,

58 The Fourier transform. ' 2 i i x x m m i x a x x i a x i e i i e dx e Unless i =, the integral over any spatial direction gives a factor

59 The Fourier transform Hence. ' ', 3 ' ' r r U d U e i W W., ',, ' 3 ' R R r r r U U e N N d U e i C V C i C W Substituting this into the expression for the integration over all space, we then have