Computational Solid State Physics

計算物性学特論　第４回

4. Electronic structure of crystals

Single electron Schroedinger equation 　

)()(

)(2

22

rr

r

H

Vm

H

nmmn

nnn

Sd

a

rrr

rr

)()(*

)()(

Expansion by base functions Φn

: overlap integral

m: electron mass

V(r): potential energy

h: Planck constant

2h

mnnm

nmn

nnmn

n

nnnm

nnnm

nnn

nnn

HdH

dadHa

dadaH

aaH

r

rr

rr

*

**

**

n

nmnn

nmn aSaH

:matrix element of Hamiltonian

:algebraic equation

2

1

a

a

a

2221

1211

HH

HH

H

2221

1211

SS

SS

S

aa SH

mnnm HdH r *

mnnm Sdrrr )()(*

:expression of algebraic equation by matrixes

and vectors

aa SH

nmnmS

0||det

IH

H

aa

nmnmI

: ortho-normalized bases

: unit matrix

eigenvalue equation

condition of existence of inverse matrix of

secular equation

)( IH

Solution (1)

Solution (2)

nmnmS

aa HS 1

0||det 1 IHS

aa SH

Potential energy in crystals

mln

VV

cbaR

rRr

)()(

a,b,c: primitive vectors of the crystaln.l.m: integers

G

G rGr ivV exp)( G: reciprocal lattice vectors

:periodic potential

Fourier transform of the periodic potential energy

Primitive reciprocal lattice vectors

baG

acG

cbG

c

b

a

2

2

2

0

2

cGbG

aG

aa

a

cba

3)2( cbaG GGGV

: volume of a unit cell

Volume of 1st Brilloluin zone

Properties of primitive reciprocal lattice vectors

Bloch’s theorem for wavefunctions in crystal

)()exp()( rRkRr kk ii i

)(exp()( rr)ukr kk ii i

)()( rRr ikik uu mln cbaR

(1)

(2)

k is wave vectors in the 1st Brillouin zone.

Equations (1) and (2) are equivalent.

Plane wave expansion of Bloch functions

)(]exp[)( rrkr kk ii ui

G

Gk,k rGr ]exp[)( icu ii

G

Gk, rGkr )(exp)( iciik

)()( rRr ikik uu

G : reciprocal lattice vectors

mln cbaR

Normalized plane wave basis set

)()exp()(

)()(*

])(exp[1

)(

''

rRkRr

rrr

rGkr

kGkG

GGkGkG

kG

i

d

iV

kG :satisfies the Bloch’s theorem

V : volume of crystal

mln cbaR

Schroedinger equation for single electron in crystals

'

02

2

)(

)(2

)(

GGvH

vm

H

GG'

GG

k

Gkk

2

1

G

G

a

a

a

0|)()(|det

IkkH

H

aa

: Bragg reflection

G

G rGr ivV exp)( : potential energy in crystal

: secular equation to obtain the energy eigenvalue at k.

Energy band structure of metals

Zincblende structure

a

bc

Brillouin zone for the zincblende lattice

Empirical pseudopotential method

Energy band of Si, Ge and Sn

Empirical pseudopotential method

Si Ge Sn

Tight-binding approximation

nlmiii

ijii

j

mlnmlniN

c

)()](exp[1

)(

)()(

cbarcbakr

rr

k

kk

)()exp()( rakar kk ii i

)( ii mln cbar i-th atomic wavefunction at (n,l,m)-lattice sites

Linear Combination of Atomic Orbits (LCAO)

satisfies the Bloch theorem.)(rik

1-dimensional lattice (1)

a

'

'

,

'

)()exp(

)(])'()(exp[1

)()(*)'exp(1

)()(*

)()exp(1

)(

kk

lkk

mmn

nm crystal

ss

crystal

kk

sk

lSikal

mnSmkkimnikaN

dmnamikikanN

d

nikanN

rararrrr

arr

S(n-m)

rrr

rrr

rr

d

dHk

kH

kk

kk

kk

)()(*

)()(*)(

)()()(

)exp()exp(

)()(*)exp(

)'()(*)]'(exp[1

)()(*

101000

'

ikaHikaHH

dHnikan

dnHnnnikaN

dH

n

nnkk

rrar

rararrrr

:Schroedinger equation

1-dimensional lattice (2)

1-dimensional lattice (3)

)cos(2)( 0 katk

ε0=H00: site energy

t=H10=H-10: transfer energy

ka

ε(k)/-t

ak

a

t < 0

Energy dispersion relation

1st Brillouin zone

Valence orbits for III-V compounds

4 bonds

;4/]111[

;4/]111[

;4/]111[

;4/]111[

4

3

2

1

ad

ad

ad

ad

Matrix elements of Hamiltonian between atomic orbits

ssV

spV

ssV

ssV

Matrix element of Hamiltonian between atomic orbit Bloch functions

ksksss acac HkH ||)(

nlm

cnlmsnlmksi

Nc )()exp(

1)( τRrRkr

mlnnlm cbaR

nlm

anlmsnlmksi

Na )()exp(

1)( τRrRkr

)()( 0 kgVkH ssss ac

Calculation of Hamiltonian matrix element

i

dikssss

iac eVkH )(

][ 4/)(4/)(4/)(4/)( akkkiakkkiakkkiakkkiss

zyxzyxzyxzyx eeeeV

4321

4321

4321

4321

)(

)(

)(

)(

3

2

1

0

dikdikdikdik

dikdikdikdik

dikdikdikdik

dikdikdikdik

eeeekg

eeeekg

eeeekg

eeeekg

Matrix element between atomic orbits

ppppxy

ppppxx

spsp

ssss

VVE

VVE

VE

VE

3/13/1

3/23/1

3/ 2/1

Hamiltonian matrix for the zincblende structure

1-fold

Bottom of conduction band: s-orbit

Top of valence band: p-orbit

0)0()0()0(

4)0(

321

0

ggg

g

Energy at Gamma point (k=0)

3-fold

;422

;422

2

2

2

2

xx

ap

cp

ap

cp

ss

ax

cs

as

cs

EE

EE

Energy band of Germanium

Energy band of GaAs, ZnSe, InSb, CdTe

Spin-orbit splitting at band edge

Efficiency and color of LED

Periodic table

B C N

Al Si P

Ga Ge As

In Sn Sb

PL energy is determined by the energy gap of direct gap semiconductors.

Bond picture (1): sp3 hybridization

]|||[|2

1|

]|||[|2

1|

]|||[|2

1|

]|||[|2

1|

4

3

2

1

zyx

zyx

zyx

zyx

pppsh

pppsh

pppsh

pppsh

ijjihh |

[111]

[-1-1-1]

[-11-1]

[-1-11]

Bond picture (2)

32

23

VV

VVH

3V

Hamiltonian for two hybridized orbits

2V

22

23 VV

: hybridized orbit energy

: transfer energy

bonding and anti-bonding states

Successive transformations of linear Combinations of atomic orbitals, beginning with atomic s and p orbitals and proceeding to Sp3 hybrids, to bond orbitals, and finally to band states. The band states represent exact solution of the LCAO problem.

Problems 4

Calculate the free electron dispersion relation within the 1st Brillouin zone for diamond structure.

Calculate the energy dispersion relation for a graphen sheet, using a tight-binding approximation.

Calculate the dispersion relation for a graphen sheet, using pane wave bases.