band structures calculations - fzu

1

Title page

Band structures calculations

• reciprocal space of k-vectors, Brillouin zone

• secular equation, variational method

• band structure, periodic potential

• density of states, Fermi energy

• almost free electrons method

• tight binding method, MO-LCAO, Bloch function

2

Literature Band structures

• T. A. Albright, J. K. Burdett, M.-H. Whangbo, Wiley (2013)Orbital Interactions In Chemistry.

• J. K. Burdett, Progress in Solid State Chemistry 15 (1984) 173-255From Bonds to Bands and Molecules to Solids.

• E. Canadell , M.-H. Whangbo, Chem. Rev. 91 (1991) 965-1034Conceptual aspects of structure-property correlations and electronic instabilities, with applications to low-dimensional transition-metal oxides.

• R. Hoffmann, Angew. Chem. Int. Ed. Engl. 26 (1987) 846-878How chemistry and physics meet in the solid state.

• G. L. Miessler, P. J. Fischer, D. A. Tarr: Inorganic chemistry 5th ed., chap.5Molecular Orbitals.

• S. Cottenier (2013)Density Functional Theory and the Family of (L)APW-methods: a step-by-step introduction.

http://lom.fzu.cz/main/lom/krystalochemie/index.htmlhttp://lom.fzu.cz/main/lom/chapl/index.html

3

Reciprocal space – k-vectors space Band structures

space of k-vectors - reciprocal space

321 321321 aaaRaaar nnnzyx

321321 bbbGbbbg lkhwvu

rrr VVV

213

132

321

aab

aab

aab

222

r

c

c

r

VV

V

V 38

321

321

bbb

aaa

direct lattice: Vr crystal lattice

reciprocal space: Vc reciprocal lattice

-/a 0 /a

-X X

/b Y

-/b -Y4

Brillouin zones Band structures

holds: - E(k) = E(- k)

- One value of E for each k within one band

- E(k) is a periodical function of k, it is sufficient to be displayed

within the interval (-/a ; /a) – the first Brillouin zone

the first Brillouin zone – Wigner-Seitz cell in the reciprocal latticeWigner-Seitz cell is always primitive and it always has the same symmetry as the lattice

(primitive crystalographic cell may have lower symmetry than the lattice)

Construction: the planes normal to b1, b2, b3 through the points ±b1, ±b2, ±b3

5


6


bccbcc in direct space

corresponds to fcc in

reciprocal space

XM

R

sc

simple

cubic

fccfcc in direct space

corresponds to bcc in

reciprocal space

7


Brillouin zones of higher order:• the same volume as the 1. Brillouin zone.• the same symmetry as the 1. Brillouin zone.• translation by a reciprocal lattice vector shifts them to the 1. Brillouin zone.

1. Brillouin zone

2. Brillouin zone

3. Brillouin zone


Triclinic 1/2 Trigonal 1/6

Monoclinic 2/m 1/4 1/12

Orthorhombic mmm 1/8 Hexagonal 6/m 1/12

Tetragonal 4/m 1/8 6/mmm 1/24

4/mmm 1/16 Cubic 1/24

1/48

1 3

3m

3m m


=000 X=100 Y=010 Z=001

S=110 T=011 U=101 R=111

10

Schrödinger equation Crystal field

Schrödinger equation

Hydrogen atom:

in spherical coordinates:

re

oV 4

2ˆ

m: electron mass

o: permitivity of vacuum

: wave functions

e: electron charge

E: energy

ħ: Planck’s constant

R: radial function

Y: angular function),()( ,,,, mllnmln YrR

mlnnmln EH ,,,,ˆ

mlml YllYL ,

2

,

2 )1(ˆ

mllmlz YmYL ,,ˆ

2

2

2

2

2

2

zyx

)()()(ˆ)(

E. ípotenciálnE. kinetická

2

2

rErrVrm

2

2

2

2

2

2

r

VTH ˆˆˆ

n: principal quantum number

l: orbital quantum number

determine the orbital angular momentum

l = 0 ... n-1

ml: magnetic quantum number

projection of the angular momentum into z-axis

m l = -l … l

kinetic energy potential energy

11

Nearly free electrons | Tight binding model Band structures

Nearly free electrons :

Kinetic energy predominates over potential energy

Basis = plane waves

metallic bond, electron gas

]exp[)( k

k xkicx

Tight binding:

Potential energy predominates over kinetic energy

Basis = atomic orbitals

Covalent and ionic bonds

: exact wave function

: approximate wave function expressed in the basis

= for N

: e.g. atomic orbitals, plane waves, ...

EH

N

i

iic EH

12

Plane wave

Plane wave:

• constant frequency

• spreads like infinite parallel planes

normal to vector of motion.

]exp[)( k

k xkicx

13

Secular equation Band structures

nnnnnnnnnn

nnnn

nnnn

EcEcEccHcHcH

EcEcEccHcHcH

EcEcEccHcHcH

22112211

22221122222112

12211111221111

ˆˆˆ

ˆˆˆ

ˆˆˆ

System of equations has a non-trivial solution, only if the matrix determinant = 0:

nnnnn

n

n

nnnnn

n

n

c

c

c

EEE

EEE

EEE

c

c

c

HHH

HHH

HHH

2

1

21

22212

12111

2

1

21

22212

12111

ˆˆˆ

ˆˆˆ

ˆˆˆ

0

0

0

ˆˆˆ

ˆˆˆ

ˆˆˆ

2

1

21

22212

12111

2

1

21

22212

12111

nnnnn

n

n

nnnnn

n

n

c

c

c

EEE

EEE

EEE

c

c

c

HHH

HHH

HHH

0

0

0

ˆˆˆ

ˆˆˆ

ˆˆˆ

2

1

2211

2222221212

1121211111

nnnnnnnnn

nn

nn

c

c

c

EHEHEH

EHEHEH

EHEHEH

0

0

0

1ˆ

2

1

2211

22222121

11121211

nnnnnnn

nn

nn

iiijjiijji

c

c

c

EHESHESH

ESHEHESH

ESHESHEH

SHH

0det

2211

22222121

11121211

EHESHESH

ESHEHESH

ESHESHEH

ESH

nnnnnn

nn

nn

ijij

Multiply equation (3) from left subsequently by functions 1, 2,..., n, and create system of equations:

Convert to matrix form, for the constant E it holds iEj = Eij:

Convert all on one 1 side and join into 1 matrix:

Matrix eigen vectors:

Symmetry – axis vector:

𝐴 Ԧ𝑣 = 1 Ԧ𝑣

Eigen-functions:

𝐻Φ = 𝐸Φ

(1) 𝐻Φ = 𝐸Φ, (2) Φ = σ𝑖=1𝑛 𝑐𝑖𝜑𝑖

By substitution of (2) into (1) (3)

(3) 𝐻 𝑐1𝜑1 + 𝑐2𝜑2 + … + 𝑐𝑛𝜑𝑛 =

𝐸 𝑐1𝜑1 + 𝑐2𝜑2 + … + 𝑐𝑛𝜑𝑛

Unknowns: 𝐸, 𝑐𝑖

14

Retrieval of eigen-values and eigen-vectors Band structures

1 1 1

1

1

1

22

2

2

1

2

2

2

2

1

21

22221

11211

2

1

21

inii

nj

j

j

nnnn

n

n

n

n

ccc

c

c

c

ccc

ccc

ccc

E

E

E

RI-

IR IR,

Complex matrix:

(1) 𝐻Φ = 𝐸Φ, (2) Φ = σ𝑖=1𝑛 𝑐𝑖𝜑𝑖

By substitution of (2) into (1) (3)

(3) 𝐻 𝑐1𝜑1 + 𝑐2𝜑2 + … + 𝑐𝑛𝜑𝑛 =

𝐸 𝑐1𝜑1 + 𝑐2𝜑2 + … + 𝑐𝑛𝜑𝑛

Unknowns: 𝐸, 𝑐𝑖

𝐻 Ԧ𝑐𝑘𝐻 = 𝐸𝑘

𝐻 Ԧ𝑐𝑘𝐻, 𝑘 = 1 … 𝑛

𝐵 = 𝑃−1𝐻𝑃: 𝐸𝑘𝐵 = 𝐸𝑘

𝐻 Ԧ𝑐𝑘𝐵 = 𝑃−1 Ԧ𝑐𝑘

𝐻 Ԧ𝑐𝑘𝐻 = 𝑃 Ԧ𝑐𝑘

𝐵

Jacobi’s method, Givens’ matrices P

𝐵 =

𝐸1 00 𝐸2

⋯ 0⋯ 0

⋮ ⋮0 0

⋱ ⋮⋯ 𝐸

𝐸1 − 𝐸𝑘 00 𝐸2 − 𝐸𝑘

⋯ 0⋯ 0

⋮ ⋮0 0

⋱ ⋮⋯ 𝐸 − 𝐸𝑘

𝑐𝑘1𝐵

𝑐𝑘2𝐵

⋮𝑐𝑘𝑛

𝐵

=

00⋮0

Ԧ𝑐1𝐵= 1,0, … , 0 , Ԧ𝑐1

𝐵 = 0,1, … , 0 , …

𝐻 Ԧ𝑐𝑘𝐻 = 𝐸𝑘

𝐻 Ԧ𝑐𝑘𝐻

𝐻 Ԧ𝑐𝑘𝐻 − 𝐸𝑘

𝐻 Ԧ𝑐𝑘𝐻, = 0

𝐻 − 𝐼𝐸𝑘𝐻 Ԧ𝑐𝑘

𝐻, = 0

𝐼:unitary matrix

15

Variational method – secular equation Band structures

System of equations for i = 1, 2, ..., N

0][][][

0][][][

0][][][

222111

2222221211

1112122111

EHcESHcESHc

ESHcEHcESHc

ESHcESHcEHc

nnnnnnn

nnn

nnn

Calculation of the determinant

secular equation of the N.order.

The solution is N eigen-values Ei

(energy). For each Ei we get N

coefficients cij (eigen-vectors) by

solving the system of equations.

Ei: energy of the function

i = j cij i

0det

2211

22222121

11121211

EHESHESH

ESHEHESH

ESHESHEH

ESH

nnnnnn

nn

nn

ijij

N

i

ijijj ESHc1

0][ Sii = 1

System of equations has solution, only if determinant Hij – ESij = 0:

If the potential depends on the

wave functions i, i.e. on the

searched coefficients cij, the secular

equation must be solved iteratively,

by the so-called SCF method (self-

consistent field).

dHH ijijˆ*

dS ijij

*

Hij: exchange integral

Hii (i=j): ”on-site” energy of individual base states.

Sij: overlap integral. Sii (i=j) = 1, Sij (ij) 0.

16

Variational method Band structures

: exact wave function

: approximate wave function expressed in the basis

= for N

: e.g. atomic orbitals, plane waves, ...

EH

N

i

iic

0

ˆˆ

ˆˆˆ

,

*

,

*

,

*

,

*

**

**

*

*

****

iji

N

ji jij

N

ji ij

iji

N

ji j

ij

N

ji ij

N

j ii

N

j jj

N

j ii

N

j jj

SccEHcc

Scc

Hcc

dcc

dcHc

d

dHE

dEdHEHEH

dHH ijijˆ*

dS ijij

*

Hij: exchange integral

Hii (i=j): ”on-site” energy of individual base states.

Sij: overlap integral. Sii (i=j) = 1, Sij (ij) 0.

EH

17

Nearly free electrons Band structures

]exp[V]exp[VV ]exp[V)( 42

210 xixixGixV

aa

G

G jG

a

2

Exact potential: a huge attraction force near

the core.

If we are interested in the potential in which

the electrons (especially the valence) move,

we can neglect the vicinity of the nucleus.

Function: ]exp[)()( k

k xkicLaxx

lkLa

2

Potential is repeated after period a, function is repeated after period La.

The 1.Brillouin zone is 2/a in the reciprocal space, the function is calculated in 2/La.

:j

Lattice vectors. For 1D j = 0, 1, 2, ...

a

La2

a

aL

a

La2

La2

:*

GG VV Potential is real

For 1D l = 0, 1, 2, ..., L/2

a 0a

18


Master equation: system of L equations, formulation

of secular equation for the plane wave basis.

Various solutions ck within 1. Brillouin zone.

-G/2 k G/2 (- /a l(2/La) /a)

]exp[)( k

k xkicx

EVm

ˆ2

2

]exp[V)( G

G xGixV

k

xki

k

k G

xGki

Gk

k

xki

kmk ecEeVcec

)(

2

22

, kkeVcGkkk G

xki

GGk

0 2

22

xki

k G

GGkkkmk eVccE

0 2

22

G

GGkkkmk VccE

In order the sum to be =0,

each term in [] must be =0.

Wave function and

potential put into

Schrödinger equation

19


master equation – system of L equations, various

solutions ck within 1. Brillouin zone (- /a k /a). 0

2

22

G

GGkkkmk VccE

G

GG GxGxxV )sin(B)cos(AV)( 0

)(

)(

)(

12

11

21

okGk

okk

okGk

VEVV

VVEV

VVVE

mk

k 2

22

-10 -8 -6 -4 -2 0 2 4 6 8 10 -10 -8 -6 -4 -2 0 2 4 6 8 10

GGGGGGGG iBAViBAVVV , :*

E

k

a

a a

a

ka

e (

ka)

m

k

m

mvE

222

2222

p

𝑉0 ≠ 0, 𝑉𝐺 → 0 for 𝐺 > 0

20

k -vectors, band width, band gap Band structures

k - quantum number

wave vector

number of allowed values k = number of elementary cells in crystal

free electrons:

band width: determined by the overlap of interacting orbitals (as for MO)

2k k

hm

vp

m

k

m

mvE

222

2222

p

-/a /a

e(k

a)

ka

ka

e (

ka)

/a-/a-2/a 2/a

e(k

a)

21

Density of states Band structures

DOS(E), g(E) - number of allowed energy levels per energy interval

holds: g(E)*dE = number of levels in the interval (E ; E+dE)

1 dimension: generally:

numerically:

1

22

k

EaEg

0

E

DOS(E)0.0

s

s

-/a /a

e(k

a)

ka

nS

knk

k

BZk E

dS

VEg

,

2

n k

EE kn

eEg

2

,

2

2

22

Density of states Band structures

R MX N(e)

e (

ka)

MX N(e)

e (

ka)

2-D 3-D

XM

R

sc

X

M

23

Fermi level Band structures

Fermi level – the highest occupied level at T=0 K

T>0: Fermi-Dirac statistic holds:

occupied states DOS(E)*f(E) 1/)(exp

1

TkEEEf

BF

Fermi plane – the set of k in k-space, for which holds E(k) = EF

ChemPot.exe

../../DOCUME~1/knizek/LOCALS~1/Temp/_tc/David_prednasky/ChemPot/ChemPot.exe

24

MO-LCAO = Molecular orbitals – linear combination of atomic orbitals Band structures

i: molecular orbital, : atomic orbital

N

ii c

BA

SSRRRRS

HRHRRHRH

HRHRH

BAAABBBBAAAB

BAAABBBBAAAB

BBAAAAAA

BA

***

****

*

)()()()(

)(ˆ)()(ˆ)(

)(ˆ)(

,

EES

ESE

EHESH

ESHEH

BBBABA

ABABAA

**

Cell containing 2

identical orbitals

N

i

ijijj ESHc1

0][

25

MO-LCAO = Molecular orbitals – linear combination of atomic orbitals Band structures

0det

22

*

ESE

EES

ESE

2112 0 ,1

EEβS

E

= : coulombic energy (energy of AO)

(<0) = t : exchange energy (degree of bonding energy)

S (0-1) : overlap integral

212

2

1 , E

0 ,0

2211

S

EE

ESE

)()(

)(ˆ)(

)(ˆ)(

*

*

*

BBAA

BBAA

AAAA

RRS

RHR

RHR

1 ,0

21

S

2121212 00 ccc cc : cE

2121211 00 ccc cc : cE

211

2

1 ,E

12

2

2

1 cc

00

0det 21

2

1

EE

E

E

26

Band structure – Bloch orbitals Band structures

BO : Bloch orbital, : atomic orbital

)exp()(),( 1 iknanarkrN

nNBO

n=0 n=1 n=2 N

BO = (r) + (r-a)eika + (r-2a)eik2a + ... + (r-na)eikNa

k=0 ()

e0 = 1

0 1 2 3-2-3 -1

a

k=/a (X)

ein = (-1)n = 1,-1,...

0 1 2 3-2-3 -1

a

0

E(1-2)

E(1+2)

X

E()

/a

ka

...

k=/2a

cos(n/2) = 1,0,-1,0, ... sin(n/2) = 0,1,0,-1, ...

A A A A

)sin()cos()exp( knaiknaikna

27

Symmetry of orbitals Band structures

0.0

s

s

-/a /a

e(k

a)

ka

0.0

p

px

-/a /a

e(k

a)

ka

0 1 2 3-2-3 -1

a

x/a

0 1 2 3-2-3 -1

a

x

0 1 2 3-2-3 -1

a

x

0 1 2 3-2-3 -1

a

x/a

28

Symmetry of orbitals Band structures

X0.0

p

dxy

-/a /a

e(k

a)

ka

X

0.0

s

s

-/a /a

e(k

a)

ka

py

x

0cos2 ak x

0cos2 ak x

29

Formation of bands – orbitals pyBand structures

30

Formation of bands – orbitals pyBand structures

Relative contribution of individual orbitals

to each energy levels is also displayed

31

Formation of bands – orbitals pxBand structures

Relative contribution of individual orbitals

to each energy levels is also displayed

32

Band width Band structures

z

Band width W

Wp > Ws

p orbitals reach closer to each other, bigger

overlap

Wz > Wx,Wy

-bonding > - bonding

valence > core

Delocalization of orbitals:

W(5d) > W(4d) > W(3d)

Small difference of orbitals energies

W(Co-O) > W(Ti-O)

33

Tight binding method (CO-LCBO) Band structures

Bloch orbitals: - basis

(BO)

n

nnjNj i )exp()(),( 1 kRRrrk

Crystal orbitals:

(CO)

cij(k) , Ei (k) = ?

),()(),( rkkrk j

j

iji c

)()()(ˆ kkk iii EH

0)()()( kkk jlijl SEH 0)()()()( kkkk jijlijl cSEH

)()()()(ˆ)()( kkkkkk ljjlljjl SHH

ljjlljjljjj HtHE ˆˆparameters:

matrix

elements:

dH ljˆ*

dlj

*lj H ˆ

lj

34-1 -0.5 0 0.5 1

Tight binding method (CO-LCBO) Band structures

akeekH x

aikaik xx cos2)(

)coscos(cos2

)(

akakak

eeeeeekH

zyx

aikaikaikaikaikaik zzyyxx

Only the interaction with the nearest neighbours are taken into account:

only the exchange integral with the nearest neighbour (E~, t~,S<<1)

(0,0)

(a,0)

(a,0)

(0,a)(0,a)

x

y

ak xcos2

aa

35

Linear crystal, 2 atoms basis Band structures

= 1 = 2 , t = t1 = t2

0 1-1

a

x

1()

0 1-1

a

x

2()

0 1-1

a

x

1(X)

0 1-1

a

x

2(X)

1

2

X X

212

2

2

1

E

211

1

2

1

E

MO ~ (k=0)

36


= 1 = 2 , t = t1 = t2

0 1-1

a

x

1()

0 1-1

a

x

2()

0 1-1

a

x

1(X)

0 1-1

a

x

2(X)

2

1

X X

212

2

2

1

E

211

1

2

1

E

MO ~ (k=0)

37

Linear crystal, 2 atoms basis – general formulas Band structures

0 1 2-2 -1

pa (1-p)a

x

t1

t2

*

122121

2121

)1(

2112

)(

)(

)1(

HeeH

eeeeeeeH

appaa

ikaikpa

ikaikpaikpaikaikpaapikikpa

38

Linear crystal, 2 atoms basis – general formulas Band structures

0 1 2-2 -1

a

x

0 1-1

a

x

0 1-1

a

x

t1

t2

= c1 1 + c2 2CO ? E, c1 , c2

E(k)=

BO

39


= 1 = 2 , t1 < t2 < 0

w = 2t2 eg = 2(t2–t1)

1 < 2 , t = t1 = t2 < 0

w = 2t eg = 2– 1

40

CuO22- plane Band structures

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

MX

E [

eV

]

X

M

d = - 1.9 p

p = - 4.3

tpd = -1.15

tpp= -0.2

-8 -6 -4 -2 0 2

0

1

2

3

n = 0.15

tot

dx

2-y

2

px

py

DO

S [eV

-1]

E [eV]

bonding

(b1g)

41


X

MX

42


M

MX

43

Formation of band gap Band structures

• ionic insulators

-10 -5 0 5 10

0

2

4

6

8NaCl

Na - 3s

Cl - 3s

Cl - 3p

DO

S(E

) [e

V-1]

E [eV]

• covalent insulators

Na-3s

Cl-3p

-20 -15 -10 -5 0 5

0.0

0.5

1.0

1.5

C - diamond

C - 2sC - 2p

DO

S(E

) [e

V-1

]

E [eV]

C-2s

C-2p

44

Some basic relations

linear operator

commuting operators

fOccfOfOfOffO ˆˆ ;ˆˆ)(ˆ 2121

0ˆˆˆˆˆ,ˆ ABBABA

0ˆ,2 xLL ipx x ˆ,ˆ zyx LiLL ˆˆ,ˆ

dHdH *

1

*

22

*

1ˆˆ

dSij

*

H is Hermitian operator

K*: Complex conjugate

Hermitian matrix

unitary matrix

ortogonal matrix

Sii = 1: normalised function

Sij = 0: orthogonal function

Sij = ij : orthonormal functionxx ip

ˆ

jijiijij

HT ibaibaKKK ;*

1 tj.,1 KKKKKK HHH

1 tj.,1 KKKKKK TTT

ijijijij ibaKibaK * ;

tFvmp

maF

vplFE

band structures calculations - fzu

Documents