Kronig-Penney model. 2m dx. Solutions can be found in region I and region II Match boundary conditions

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1 Kronig-Penney model I II d V( x) m dx E Solutions can be found in region I and region II Match boundary conditions

2 Linear differential equations with periodic coefficients Have exponentially decaying solutions, or solutions of the form e ix u (x) T Te u x e u x a e e u x e ix ( ) ( ) i x a ( ) ia ix ( ) ia ia e

3 Kronig-Penney model Solutions can be found that are simultaneous eigenfunctions of the Hamiltonian and the translation operator. Eigenfunctions of the translation operator can be found in terms of any two linearly independent solutions. A convenient choice is: d 1 d 1() 1, (), (), () 1. dx dx

4 Kronig-Penney model for < x < b ( x) cos( x), ( x) 1 1 sin( 1x) 1 for b < x < a sin( ( x b))sin( b) 1 1 1( x) cos( ( x b))cos( 1b), cos( ( x b))sin( b) sin( ( x b))cos( b) 1 1 ( x). 1 Except for the coefficients, these are the same solutions as we found for light in a layered material.

5 Kronig-Penney model at x = a sin( ( a b))sin( b) 1 1 1( a) cos( ( a b))cos( 1b), cos( ( a b))sin( b) sin( ( a b))cos( b) 1 1 ( a). 1 The translation operator translates the function a distance a. 1 x a 1 x. x a x T11 T1 T1 T The elements of the translation operator can be evaluated at x = a.

6 Kronig-Penney model d 1 1 a a 1 x a dx 1 x x a d x a a dx The eigen functions and eigen values are ( a) 1 x 1( x) ( x),, d ( a) 1( a) dx 4 d ( a) 1 1( a) cos abcos 1b sin a bsin 1b. dx 1 If >, the potential acts lie a mirror for electrons

7 Kronig-Penney model E 1, (, V, V, a, b) 1, ia e tan a

8 Kronig-Penney model (a) The energy-wave number dispersion relation. The dashed line is the Fermi energy. (b) The density of states. (c) The internal energy density (solid line) and Helmholtz free energy density (dashed line). (d) The chemical potential (solid line) and the specific heat (dashed line). All of the plots were drawn for a square wave potential with the parameters: V = 1.5 ev, a = 1-1 m, b = m, and an electron density of n = 3 electrons/primitive cell.

9 A separable potential I II m Y ( V( x) V( y) V( z)) Y EY Y is the product of the solutions to the Kronig-Penney model. Y( x, y, z) ( x) ( y) ( z) KP KP KP

10 A separable potential I II

11 Band structure in 1-D

12 Bloch Theorem ( r ) C e i r Any wave function that satisfies periodic boundary conditions These 's label the symmetries ( r) C e 1 Bz i r ( ) ( r ) i r C e i r i r e C e e i r u ( r) Bloch form ( ) i r e r u ( r) T r e u r ma na la e r mnl 1 3 i ( r ma1 na la3 ) i ( ma1 na la3 ( ) ( ) ) ( ) Eigen function solutions of the Schrödinger equation have Bloch form. periodic function

13 Bloch waves in 1-D

14 Plane wave method m UMO ( r) E Write U and as Fourier series. ir U ( r ) U e ( r ) C e MO i r For the molecular orbital Hamiltonian U MO ( r) ir Ze 1 Ze e 4 r r V j j volume of a unit cell

15 Plane wave method U( r) E m ir U ( r ) U e ( r) C e MO i r C e U C e E C e i r i( ) r i r m Must hold for each Fourier coefficient. m E C U C Central equations (one for every in the first Brillouin zone)

16 Plane wave method The central equations can be written as a matrix equation. MC EC Diagonal elements: M ii m i Off-diagonal elements: M ij Ze V i j

17 Central equations - one dimension Central equations couple coefficients to other coefficients that differ by a reciprocal lattice wavevector E U U U U U m C U E U U U U m U U E U U U m U U U E U U m U U U U E U m C C C C E C U C m

18 Central equations - one dimension

19 Central equations 3d - simple cubic V ( r ) i r U e Molecular orbital Hamiltonian U V Ze unit cell Central equations: m E C U C diagonal elements: m i off-diagonal elements: Ze V unit cell i j

20 Central equations - simple cubic ˆ z a Ze a Ze a Ze a Ze a Ze a Ze a m 8V unit cell 8V unit cell 4V unit cell 8V unit cell 8V unit cell 16V unit cell ˆ y Ze a a Ze a Ze a Ze a Ze a Ze a 8V unit cell m 8V unit cell 4V unit cell 8V unit cell 16V unit cell 8V unit cell ˆ x Ze a Ze a a Ze a Ze a Ze a Ze a 8V unit cell 8V unit cell m 4V unit cell 16V unit cell 8V unit cell 8V unit cell Ze a Ze a Ze a Ze a Ze a Ze a 4V unit cell 4V unit cell 4V unit cell m 4V unit cell 4V unit cell 4V unit cell ˆ Ze a Ze a Ze a Ze a x a Ze a Ze a 8V unit cell 8V unit cell 16V unit cell 4V unit cell m 8V unit cell 8V unit cell ˆ Ze a Ze a Ze a Ze a Ze a y a Ze a 8V unit cell 16V unit cell 8V unit cell 4V unit cell 8V unit cell m 8V unit cell ˆ Ze a Ze a Ze a Ze a Ze a Ze a z a 16V unit cell V unit cell 8V unit cell 4V unit cell 8V unit cell 8V unit cell m

21 Central equations - simple cubic E [aj] X M R

22 Central equations - simple cubic empty lattice X X

23 Plane wave method

24 Plane wave method fcc Z= empty lattice

25 Plane wave method fcc hydrogen

26 Approximate solution near the Bz boundary E C U C m For just terms E U C m C U E m Near the Brillouin zone boundary ~ / E U C m C U E m E / E U m U

27 Review: Molecules Start with the full Hamiltonian H Z e e Z Z e A A B i A i me A ma i, A 4 ria i j 4 rij AB 4 rab Use the Born-Oppenheimer approximation H elec Z e e A A B i i me i, A 4 ria i j 4 rij AB 4 rab Z Z e Neglect the electron-electron interactions. H elec is then a sum of H MO. H MO ZAe 1 m 4 r r e A 1 A The molecular orbital Hamiltonian can be solved numerically or by the Linear Combinations of Atomic Orbitals (LCAO)