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Kronig-Penney model

Solutions can be found in region I and region IIMatch boundary conditions

I II

2 22 ( )2

d V x Em dx

Linear differential equations with periodic coefficients

or solutions of the form eikxuk(x)

Have exponentially decaying solutions,

T ( )( ) ( ) ( )ikx ik x a ika ikx ika

k k kTe u x e u x a e e u x e ikae

Kronig-Penney model

Eigenfunctions of the translation operator can be found in terms of any two linearly independent solutions. A convenient choice is:

Solutions can be found that are simultaneous eigenfunctions of the Hamiltonian and the translation operator.

1 21 2(0) 1, (0) 0, (0) 0, (0) 1.d ddx dx

Kronig-Penney model

11 1 21

sin( )( ) cos( ), ( ) k xx k x x k for 0 < x < b

for b < x < a1 2 11 2 1

22 1 2 12

1 2

sin( ( ))sin( )( ) cos( ( ))cos( ) ,cos( ( ))sin( ) sin( ( ))cos( )( ) .

k k x b k bx k x b k b kk x b k b k x b k bx k k

Except for the coefficients, these are the same solutions as we found for light in a layered material.

Kronig-Penney modelat x = a

1 2 11 2 12

2 1 2 121 2

sin( ( ))sin( )( ) cos( ( ))cos( ) ,cos( ( ))sin( ) sin( ( ))cos( )( ) .

k k a b k ba k a b k b kk a b k b k a b k ba k k

1 111 122 221 22

.x a xT Tx a xT T

The translation operator translates the function a distance a.

The elements of the translation operator can be evaluated at x = a.

Kronig-Penney model

111 1

2 222

da ax a xdxx a xda adx

2 1 22 1

2 ( ) 1( ) ( ), ,( ) 2( )ax x xd a adx

The eigen functions and eigen values are

2 4 2 2 11 2 1 2 1

1 2

( )( ) 2cos cos sin sin . d a k ka k a b k b k a b k bdx k k If > 2, the potential acts like a mirror for electrons

Kronig-Penney model1, 2E k k

1, 2 1 2( , , , , )k k V V a b

12

2 4

ikae 211 4tan k a

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 = 12.5 eV, a = 2 × 10-10 m, b = 5 × 10-11 m, and an electron density of n = 3 electrons/primitive cell.

A separable potential

Y is the product of the solutions to the Kronig-Penney model.

I II

2 2 ( ( ) ( ) ( ))2 V x V y V z Em Y Y Y

( , , ) ( ) ( ) ( )KP KP KPx y z x y z Y

A separable potential I II

http://lampx.tugraz.at/~hadley/ss1/separablecrystals/thermo.html

Band structure in 1-D

Bloch Theorem

( )( ) ( )i k G r ik r iG r ik rk k G k G kG G

r C e e C e e u r

( ) ik rkk

r C e

1

( ) i k G rk GGk Bz

r C e

Eigen function solutions of the Schrödinger equation have Bloch form.

Bloch form 1 2 3 1 2 3( ) ( )

1 2 3( ) ( ) ( )ik r ma na la ik ma na lamnl k k kT r e u r ma na la e r

These k's label the symmetries

( ) ( )ik rk kr e u r

periodic function

Any wave function that satisfies periodic boundary conditions

Bloch waves in 1-D

Plane wave method

( ) ( )iG r ik rMO G kG k

U r U e r C e

2 2 ( )2 MOU r Em

Write U and as Fourier series.

2 22

0 0

1( ) 4iG r

MOj Gj

Ze Ze eU r V Gr r

For the molecular orbital Hamiltonian

volume of a unit cell

Plane wave method

( ) ( )iG r ik rMO G kG k

U r U e r C e

2 2 ( )2 U r Em

2 2 ( )2

ik r i G k r ik rGk k kGk k k

k C e U C e E C em

2 2 02 Gk k GGk E C U Cm

Central equations (one for every k in the first Brillouin zone)

Must hold for each Fourier coefficient. k G k k k G

Plane wave method

2

20

iji j

ZeM V G G

2 22ii iM k Gm

MC EC

Off-diagonal elements:

The central equations can be written as a matrix equation.

Diagonal elements:

Central equations - one dimension

Central equations couple coefficients k to other coefficients that differ by a reciprocal lattice wavevector G.

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

220

2 3 4 522

0 22 3 42 2

2 2 322

03 2 2

220

4 3 2

22

22

22

2

G G G G G

kG G G G G

G G G G G

G G G G G

G G G G G

k G E U U U U Umk G CU E U U U Um

kU U E U U Umk GU U U E U Um

k GU U U U E Um

0

0

0

02

0

G

k G

kk G

k G

CC

CC

2 2 02 k G k GG

k E C U Cm

Central equations - one dimension

Central equations 3d - simple cubic

22

unit cell 0G

ZeU V G

( ) iG rGG

V r U e

Molecular orbital Hamiltonian

2 22 ik Gm

2

2unit cell 0 i j

ZeV G G

2 2 02 Gk k GGk E C U Cm

Central equations:

diagonal elements:

off-diagonal elements:

22

2 2 2 2 2 2 2 2 2 2 2 22 2 2 2 2 2

unit cell 0 unit cell 0 unit cell 0 unit cell 0 unit cell 0 unit cell 02

22 2 2 2

2unit cell 0 unit

2 ˆ2 8 8 4 8 8 16

2 ˆ8 2 8

z

y

k k Ze a Ze a Ze a Ze a Ze a Ze aam V V V V V V

k kZe a Ze aaV m V

2 2 2 2 2 2 2 2

2 2 2 2 2 cell 0 unit cell 0 unit cell 0 unit cell 0 unit cell 0

22

2 2 2 2 2 2 2 22 2 2 2

unit cell 0 unit cell 0 unit cell 0 unit cell 0

4 8 16 82 ˆ

8 8 2 4 16x

Ze a Ze a Ze a Ze aV V V V

k kZe a Ze a Ze a Ze aaV V m V V

2 2 2 22 2

unit cell 0 unit cell 02 2 2 2 2 2 2 2 2 2 2 2 2 2

2 2 2 2 2 2unit cell 0 unit cell 0 unit cell 0 unit cell 0 unit cell 0 unit cell 0

2 2 2 22

unit cell 0

8 8

4 4 4 2 4 4 4

8

Ze a Ze aV V

Ze a Ze a Ze a k Ze a Ze a Ze aV V V m V V V

Ze a Ze aV

22

2 2 2 2 2 2 2 22 2 2 2 2

unit cell 0 unit cell 0 unit cell 0 unit cell 0 unit cell 0

2 2 2 2 2 2 2 22 2 2

unit cell 0 unit cell 0 unit cell 0 unit cel

2 ˆ8 16 4 2 8 8

8 16 8 4

xk kZe a Ze a Ze a Ze aaV V V m V V

Ze a Ze a Ze a Ze aV V V V

22

2 2 2 22 2 2

l 0 unit cell 0 unit cell 0

22 2 2 2 2 2 2 2 2 2 2 2

2 2 2 2 2 2unit cell 0 unit cell 0 unit cell 0 unit cell 0 unit cell 0 unit cell 0

2 ˆ8 2 8

216 2 8 4 8 8

yk kZe a Ze aaV m V

kZe a Ze a Ze a Ze a Ze a Ze aV V V V V V

2ˆ2

zkam

Central equations - simple cubic

Central equations - simple cubic

E [aJ]

G X M RG

Central equations - simple cubic

G X G X

empty lattice

Plane wave method

fcc Z= 0

empty lattice

Plane wave method

fcc hydrogen

Plane wave method

Approximate solution near the Bz boundary2 2 02 k G k G

Gk E C U Cm

For just 2 terms

2 2

222 0

2

kk G

k E U CmCk GU Em

Near the Brillouin zone boundary k ~ G/222

222 2 0

2 2

kk G

G E U CmCGU Em

E

G/222

2 2GE Um

2U

Review: MoleculesStart with the full Hamiltonian

2 22 2 22 2, 0 0 02 2 4 4 4

A A Bi Ai A i A i j A Be A iA ij AB

Z e Z Z eeH m m r r r

Use the Born-Oppenheimer approximation

Neglect the electron-electron interactions. Helec is then a sum of HMO.22 2

10 12 4

AMOAe A

Z eH m r r

The molecular orbital Hamiltonian can be solved numerically or by the Linear Combinations of Atomic Orbitals (LCAO)

2 22 22, 0 0 02 4 4 4

A A Belec ii i A i j A Be iA ij AB

Z e Z Z eeH m r r r