Solid state physics - v karpus

Point groupsA set of point symmetry operations of a body constitutes its point group Examples of point groups: A set of elements A, B, C, ..... Examples of crystal structures: Ionic crys

Solid State Physics • Kietųjų kūnų fizika

Crystal structures

The crystalline state = the equilibrium state of solids

Experimental fact or theorem?

classification of crystals

V Karpus, Solid State Physics: Lecture 1 (2004)

Translation symmetry

Wigner–Seitz cell primitive cell

atoms & equivalent points

K , 2 , 1 , 0 3

3 2

2 1

Point groups

A set of point symmetry operations of a body constitutes its point group

Examples of point groups:

A set of elements A, B, C, is called a group G

if a law of their “multiplication” is defined,

and the following conditions are satisfied:

• if A and B ∈ G, then AB ∈ G

• multiplication is associative, (AB)C = A(BC)

• the set contains an element E such that

AE = EA = A

• for all A ∈ G there exist B such that

AB = BA = E

Group

Classification of crystals

System Parameters of

crystallographic cell

Lattice type Point groupTriclinic (T) a ≠ b ≠ c

h w v

1:

1:

[[uvw]]

3 1 0

2 2 _ _

Examples of crystal structures: Ionic crystals

Cs+

Cl

-Bravais lattice: sc σ= 2

KCl, AgBr, MgO, MnO

V Karpus, Solid State Physics: Lecture 1 (2004)

Examples of crystal structures: Metallic crystals

a

c b a

a

−+

== − +

++

c b a

1 2 2

1 1

coordination number = 12

Al, Cu, Ag, Ne

Examples of crystal structures: Covalent crystals

coordination number = 4

aZnS = 5.41 Å GaAs, AlAs, ΙnP, CdTe

zinc blende structure

V Karpus, Solid State Physics: Lecture 1 (2004)

The point symmetry of crystals determines the symmetry of macroscopic physical quantities

Crystallophysics

Neumann, P Curie, Voigt 19th c

[J F Nye, Physical properties of Crystals (Clarendon Press, 1964)]

Tensors

E j

isotropic system

3 33 2

32 1

31 3

3 23 2

22 1

21 2

3 13 2

12 1

11 1

E E

E j

E E

E j

E E

E j

σσ

sum

Let the vector components in the old reference system are A = (A1, A2, A3)

What will be the A-components in the new system?

)'cos(

)'cos(

'

3

2

3 1 3

2 1 2

1 1 1

1

A

A

x x A

x x A

x x A

A1' 1

or =

old) via (new

' ij j

i a A

new) via (old

'

j ji

)'cos(

)'cos(

)'cos(

)'cos(

)'cos(

)'cos(

)'cos(

)'cos(

3 3 2

3 1

3

3 2 2

2 1

2

3 1 2

1 1

1

x x x

x x

x

x x x

x x

x

x x x

x x

x

a ij

transformation matrix

il jl

ija

old) via (new

' mi lj ij

ml a a T

new) via (old

'

ij jl im

ml a a T

Tensor components transform as multiplications

or r-vector components: x m'x l'= a mi a lj x i x j

V Karpus, Solid State Physics: Lecture 1 (2004)

Tensors and symmetry

Symmetry operations as transformations

z z

y y

x x

y y

x x

0

0 1 0

0 0 1

0 1 0

0 0 1

1 0 0

0 1 0

a

z

y x

C3

z z

y y

x x

→

−

→

z z

y y

x x

0

0 1 0

0 0 1

0 1 0

0 0 1

1 0 0

0 1 0

a

x

y z

Math expression of the Neuman principle

ion ransformat symmetry t

the is

if ij a ij

lj mi

ij

2 1 S A

S 4

S 2

S 5

S 6

S 1

S 33

S 23

S 22

S 13

S 12

S 11 S

T

T T

T T

T T

T T

T T

T

Tensors of the nth rank

i ij

x

U x

piezoelectric tensor of the 3 rd rank

kl ijkl

m

l x x a a a x x x

ijkl rl pk nj mi mnpr a a a a T

V Karpus, Solid State Physics: Lecture 1 (2004)

• 2.7.5: Trečiojo rango tenzoriai Simetrija (1 pt)

• 2.7.6: Hooke’o dėsnis C tenzorius kubinėje sistemoje Izotropinės aplinkos (1 pt)

[E R Cohen and B N Taylor, Phys Today, 52(8), BG5 (1999)] – physical constants

Reciprocal lattice (Gibbs 19 th c.)

V Karpus, Solid State Physics: Lecture 2 (2004)

without 2π in crystallography

]) [

(

] [

2

, ]) [

(

] [

2

, ]) [

(

] [

2

3 2 1

2 1 3

3 2 1

1 3 2

3 2 1

3 2 1

a a a

a a b

a a a

a a b

a a a

a a

K , 2 , 1 , 0 ,

3 3 2

2 1

• the reciprocal lattice (an arrangment of its sites) does not depend

on a choice of the primitive cell of the direct lattice

•

ij j

gl

) ( )

( if , ) i exp(

Properties of reciprocal lattice

• Each vector g of the reciprocal lattice is perpendicular

to a set of the direct lattice planes.

Miller indices of the planes coincide with the g = mibi vector indices

m1/m, m2/m, m3/m (where m is the common multiplier of mi).

The distance between the planes is

• Volume of the reciprocal cell is

• The direct lattice is the reciprocal of its own reciprocal lattice

.

2

m g

cell

3 rc

V

Examples of the reciprocal lattices

• sc (lattice constant = a)

k b

j b

i

b

a a

a

ππ

,

2,

2

3 2

• sc (lattice constant = 2π/a)

k a

j a

a

c b

a

c b a

a

−+

== − +

++

k j i b

−+

3 2 1

a a

a

πππ

• fcc (lattice constant = 4π/a)

Brillouin zone (BZ) = the Wigner-Seitz cell of the reciprocal lattice

N

P D

bcc

X

Z W

X

U S K

X-ray diffraction

1912 von Laue, Knipping, and Friedrich

the 1st experiment on X-ray diffraction

• el.-m nature of X-rays

• the inner structure of crystals

(Nobel prize, 1914)

1913 W H Bragg and W L Bragg

determined the KCl, NaCl, KBr, KI crystal structures

• X-ray crystallography

(Nobel prize, 1915)

Diffraction from lattice

~ interference of the secondary waves,emitted by atoms in a response to the incident wave

λ α

α

3 3

3 3

2 2

2 2

1 1

1 1

cos '

cos

cos '

cos

cos '

cos

m a

m a

m a

2D

3D

λ α

α

λ α

α

3 3

3 3

2 2

2 2

1 1

1 1

cos '

cos

cos '

cos

cos '

cos

m a

m a

2 2

1 1

2 '

,

2 '

,

2 '

,

m m

m

π π

π

k k a

k k a

k k a

k´ = k

(elastic scattering)

α' α

k

k´

k´ = k + g

What should be the incident wave vector,

that the way would diffract?

k k'

d sin θ θ

g k

Derivation of the Bragg law

from the Laue equations

2θ

V Karpus, Solid State Physics: Lecture 2 (2004)

incident wave:

) i exp(i

) exp(i

0

d

the secondary wave emitted by r-oscillator:

) i '

exp(i '

'

sin

0 2

2 )

(

t kr

d r

oscillator

) (r exp - i k '- k r

C E

E

Primitive crystal lattice

Model:

• The single atom corresponds to each lattice site

• Positions of oscillators coincide with those of atoms

(electrons are concentrated on nuclei) ⇒ r = ln

number of electrons on atom

C

E

l

l k k'-

i - exp

A

A

scattering amplitude A ≠ 0 for k´- k = g

A ,'

cell Z N

• Another proof of the Laue formula k´= k + g

• Intensity of diffraction reflexes ∝ ZA2

• Diffraction peaks are infinitely narrow

(the peak width is determined by sample volume)

• Intensity of diffraction reflexes (for unpolarized X-beam)

1

~ N

δ

Atomic factor

e

r l

-e

-e

How does the electron distribution in atoms

affect the diffraction?

e n r C

E

l

r l

k k

g g k

N C

(200)

(400) (420)(422)

(220) (222)

λ = 0.709 Å

Al

[B W Batterman et al., Phys Rev 122., , 68 (1961)]

Hartree-Fock calculations (for isolated atoms)

The electron distribution

• does not change the positions

of diffraction reflexes,

• does not smear them,

• affects intensity of the reflexes(enables investigations

of the electron distribution)

Structure factor

Let us consider the diffraction from the crystal lattice with basis

e

r ρ

E

l

r ρ

l k k

N C

2

1 2

1 2

1 2

1 2 1

=

α

ρ

] e

e e

oddoreveneither

areall

Quantum mechanical description of diffraction

Diffraction as a scattering of particles/waves by crystal

Initial state

(incident particle):

) / i

) / i

2

'

2 '

• Another proof of the Laue formula k´= k + g

• Justification of the diffraction of particles (electrons, neutrons, )

• Mk'k ∝ Vg possibility to analyse the interaction

k ' = +

of the de Broglie principle of duality.

Davisson and G P Thomson –Nobel prize (1937) for discovery

of electron diffraction

V Karpus, Solid State Physics: Lecture 2 (2004)

Neutron diffraction

[D P Mitchell and P N Powers, Phys Rev., 50, 486 (1936)]

[C G Shull, Rev Mod Phys., 67, 753 (1995)]

Shull (Oak Ridge reactor, Brookhaven): Nobel prize (1994)

V

, n

2

2 )

• Investigations of the H-compounds

(discovery of the crystal structure of ice:

[E O Wollan et al., Phys Rev., 92, 1082 (1953)])

• Investigations of magnetic structures

(discovery of the antiferomagnetic structure:

[C G Shull et al., Phys Rev., 83, 333 (1951)])

NB:

• reciprocal lattice:

•

• Fourier transform of the periodic function:

• distance between crystallographic planes versus g-vector:

K , 2 , 1 , 0 ,

3 3 2 2 1

(

] [ 2

3 2 1

3 2

a a

b = π

K , 2 , 1 , 0 ,

= π p p

gl

) ( ) ( if , ) i exp(

)

g g

f f

f

m g

d = 2 π

• Brillouin zone of the fcc lattice:

centre: Γ [100]direction: ∆, X (BZ face) [111] direction: Λ, L (BZ face)

Reciprocal lattice

Properties of reciprocal lattice

Examples of reciprocal lattices

Classical description of X-ray diffraction

• Primitive crystal lattice

• 3.6.1: Difrakcinių refleksų plotis (0.75 pt)

V Karpus, Solid State Physics: Lecture 2 (2004)

) (

)

; (

) ( r , R ψ r1, r2, K , ri, K R1, R2, K , Rk, K

ion - el ion

el ˆ ˆ

=

i

e m

V T

H

) ( ,

2 0

2 el

el el

-el

2

1 2

ˆ

ˆ

r r

h

= +

=

k

k k

k

, , , ,

V M

V T

2 ion

ion ion

-el V , , K , , K , , K , , K

Adiabatic approximation I

or the Born–Oppenheimer (1927) approximation

is based on the fact, that

M

m0 << (e.g MSi = 14mp + 14mn ≈ 28mp → m0/M ≈ 2 10-5)

mp = 1836 m0

Ions are rather heavy and slow as compared to electrons

They do not follow all details of electron motion

and experience some averaged electron field

Electrons, on the contrary, feel istantaneous positions of atoms

Static lattice

) , ( )

( ˆ

ˆ

ion - el ion

ion

H

el el el

ion - el

el ˆ ( , )]

ˆ

) (

ion - ion

el E V R

) ( )

Adiabatic approximation II

ψ ψ

H ˆ = [ ˆion + ˆel + el-ion] =

) ( )

, ( )

, ( r R ψel r R ψion R

ion el ion

el el

ion

2 ion

el ion

2

)]

(ˆ

k

k k

k k

E M

<<

M

m E

E

ion ion

el

2

* el

3 ion

el ion

2

d)]

(ˆ

M

r E

H

k

k k

E~

ion ion

el ion ( )]

ˆ

el ion

ion ion

Chemical bonds

L Pauling, The Nature of Chemical Bond (Cornell University Press, 1939)

J C Pillips, Bonds and Bands in Semiconductors (Academic Press, 1973)

The binding energy, alias the cohesion energy,

εc is the energy required to disassemble a solid

into its constituent parts (atoms, ions or molecules)

εc is usually measured in electronvolts per atom / molecule

or in kilocalories per mole

mole

kcal05

23atom

eV

The main types of chemical bonds

Ar Ar

Ar

Ar Ar

-ionic bonds covalent bonds

metallic bonds van der Waals bonds

Si Si

Si

V Karpus, Solid State Physics: Lecture 3 (2004)

Electronic configuration of atoms

Electron states in isolated atoms are classified by 4 quantum numbers:

• n = 1, 2, 3, principal quantum number

• l = 0, 1, , (n-1) orbital quantum number

• m = -l, , -1, 0, 1, , l magnetic quantum number

• s = -½, +½ spin

3p 3s

2s

1s

2p 3d

K L

Ionic bond

Model of the ionic crystal:

a set of charged + and – spheresE.g.:

3Li: 1s2 2s

9F: 1s2 2s22p5

Li+ : 1s2

F–: 1s2 2s22p6

The valence electrons in ionic crystals

redistribute between atoms

to form + and – ions

,

) 0 ( )

0 ( 2

1

M

R R

a

2 M

C U

a core =

m = 6–10

(Pauling 1960)

van der Waals bond

Fluctuating dipole interaction

Consider two neutral atoms (in free space) without dipole momenta, d = 0

r

r d

r dr

6

2 d

d

2

5 )

(

r

w r

(alias van der Waals or London potential):

Lenard-Jones potential

6 12

LJ( )

r

A r

B r

LJ( ) 4

r r

r

0.014 0.02

3.98 3.65

0.0104 3.4

a 12

2

r

A r

A

Potential energy of molecular crystals

( ) N U u

V

j i

0 ( LJ 2

energy per atom

12

2 6 min

theor c

-0.089 -0.08

εctheor (eV) -0.027

εcexper (eV) -0.02

fcc structure:

A6 = 14.45392, A12= 12.13188

[J E Lennard-Jones and A E Ingham,

Proc Roy Soc (Lodon), A107, 636 (1925)]

V Karpus, Solid State Physics: Lecture 3 (2004)

( ns r

ns ψ

ψ r =

)()

ns np

C, Si, Ge: (core) + ns2np2

van der Waals bond

Fluctuating dipole potential

2

4 B

6

One-electron approximation

Schrödinger equation of the electron subsystem:

el el el

ion el

ˆ

el el el

) (

2 ion

el 0

-2

2

1 )

∆

−

≠ r r r

h

?many-particle

self-consistent field

techniques

самосогласованное полеsuderintinis laukas

construct the functional and vary ϕito minimize it

H H

( )

( )

( d

)

(

2 2

3 ion

el 0

2

r

r r

r

r

i j

j

'

e '

( )

( )

(

2

r r

r'

U

r r

r

d )

2

i j

j ' e

ρ

Hartree equation

• Hartree approximation reduces the many-body problem of interacting electrons to the one-particle one.

• The system of interacting electrons is changed to that of non-interacting “quasiparticles”.

• ϕi-functions describe “quasielectrons”, electrons dressed with e-e interaction.

– the one-particle equation of the electron which moves in the ion field Uel-ion

and in he effective field UH

of interaction with other electrons

V Karpus, Solid State Physics: Lecture 4 (2005)

Hartree – Fock approximation (В Фок, 1930)

)()

()( 1 2 2

1

el el el

) (

2 ion

el 0

-2

2

1)

()

(

)()

()

(

)()

()

(

!1

2 1

2 2

2 1

2

1 1

1 1

1 HF

N N N

N

N N

q q

q

q q

q

q q

q

N

ϕϕ

ϕ

ϕϕ

ϕ

ϕϕ

ϕψ

K

KK

KK

()

()(

d

) (

2

*

i i i i

j

i j j i

j i j j

(d

)

(

2 2

ion el 0

2

i i i

j j j i

∆

r r r

( ϕ χ σ

ϕ q = r

) ( )

( )

( )

(

2

r r

e r'

r r

r r

r

d )

HF,

ρ ρ

∑

=

||) spin

*

* exch,

) ( ) (

) ( ) ( ) ( )

( )

(

j i

i j

i

'

' e

'

,

r r

r r

r

r r

r

ϕ ϕ

ϕ ϕ

ϕ

ϕ ρ

• Exchange interaction depends on the state under consideration ϕi

• The exchange charge density ρexch,i

depends on r Electron interacts with

the charge, which depends on the electron position.

Jellium model

) ( )

( ) ( )

(

2

r r

|

|

~ ) (

2 ion

el

R r r

=

'

' e

r'

U

r r

r

d )

as if there is no e-e interaction at all

V Karpus, Solid State Physics: Lecture 4 (2005)

(

H r = V − + r

U

) i exp(

1 )

+

) (

jellium 3

d)

(

V

e R

V

R r

V eZN /

jellium =

+

ρ

) ( )

( ) ( )

(

2

r r

Electron gas Drude 1900Sommerfeld (1927-1928)

Boltzmann gas and Fermi gas

)2(

)2(4)

E

2 / 3

)2(

)2(4)

) exp[(

Θ

=

f

2 / 3 2

F 2

23

m n

]/

exp[

]/

|

|exp[ kBT kBT

2 / 3 2

B c

B c

2

2)

Fermi gas in metals

rs,Al = 1.1 Å, rs,Au = 1.6 Å, rs,Na = 2.1 Å

• Fermi energy and momentum

m

n

2

) 3

( 2 2 / 3 2

F = h

F ( 3 n )

Sommerfeld integration

) ( ' )

( 6

) ( d

) ( ) (

6

0 0

µ

π ε

ε ε

( d

) ( ) (

d

ε

ε ε

ε ε

K +

− +

− +

2

1 '' ( ) ( ) )

( ) ( ' )

( )

a

K +

1

d )

( d )

( '' )

( )

( ) (

d

ε

µ ε ε µ

µ ε

a

Proof

ε µ

ε

d

df

− ) ( ε

a

V Karpus, Solid State Physics: Lecture 4 (2005)

Wigner crystal

kin

e - e

ε

V

=Γ

3

m B 3

c

19

3 / 1 c

B

27

803

[E Wigner, Phys Rev., 46, 1002 (1934)]

[W M Itano et al., Science, 279, 686 (1998)]

(r = −e n r −n

ρ

local concentration of electrons

screening radius (Debye radius)

r V

n e

2 2

s

V a

rV r

)(

∂

∂

)/exp(

)(

e r

κ

V Karpus, Solid State Physics: Lecture 4 (2005)

(r = −e n r −n

ρ

local concentration of electrons

screening radius (Thomas–Fermi radius)

r V

2 2

46

1

a

k n

)(

e r

ε

∝

n

Screening and the dielectric function

4 )

exp(

), i i

exp(

), i i

exp(

), i i

ω ω

4

q q

q

= Φ

ω ω

ω

χ κ

, 2

2 ,

4 1

V s

+

=

) ,

ε q

ω

χ κ

π ω

1 1

) 0 ,

(

q a

n

B ,ω =

Linhard formula

V H

, ,

2 ) 0 ( 2

ω ω

-V e

f

k

k k

k q

)(2

dlim

3 0

,

q k k

q k k

q

f f

h

)

-(2

d 2 lim

4 1 ) ,

3 0

2 2

q k k

q k k

q e

h t/

i i 2 /

++

+

+Ψ

=

Ψ

−

+ + +

+

αω

εεαω

εε

α ω ω

α ω ω

hh

h

ei

-e1

t t i -i s*

, t

t i i s , )

0 (

q k k

kr q

q k k

kr q k

F) 2 / ( 1

) 0 ,

(

q a

k q F

0

(

ω

ω ω

m

n e

π

-2

,ee

t/

i i 2 / 3

ρ( ) = 0

• DOS:

3/2 F B

ε

V

=Γ

• Gaseous parameter:

)/exp(

2

s

a

r r

11

)0,

(

q a

Band structure

ψ ε

h

Kronig–Penney problem Mathieu problem

tight binding approximationnearly free electron approximation

Bloch theorem

φ κ

i L

i

R

i L

i

R

e e

e e

b b

a a

*

*

R

L L

a t

r

b

b t

r a

|

|

)(

cos1

i

|

|

)cos(

e

t

a t

Mathieu problem

δ

2468

z

1 d

d

2 0

2

h

∈ = ε δ / Ua

2

2

∈

i = +

) ( e

)

i i

+

3 2

) ( r l iklψ r

) ( )

( )

(

2 0

2

r r

, ) ( )

) (

) (

e

) (

e )

2 (

i i

i

r

a r a

r

a a

δ

i i

i

i i

=

+

= +

) ( e

) ( r = ikru r

( ˆ ) ( ) ( ) ( ) 2

0

r r

r k

) ( r l i lψ r

ψ ( r + l ) = eiδklψ ( r )

For any electron quantum state there exists a vector k such,

that a translation by the lattice vector l is equivalent

to multiplication of the wave function by the factor exp(ikl)

hk is called as quasimomentum

) ( )

( ,

) (

ψ