Lecture VII

Why Solids?

most elements are solid at room temperature

atoms in ~fixed position

“simple” case - crystalline solid

Crystal Structure

Why study crystal structures?

description of solid

comparison with other similar materials - classification

correlation with physical properties

Metals, insulators & semiconductors?

At low temperatures all materials are insulators or metals.

Semiconductors: resistivity decreases rapidly with increasing temperature. Semiconductors have resistivities intermediate between metals and insulators at room temperature.

Pure metals: resistivity increases rapidly with increasing temperature.

1020-

1010-

100 -

10-10-R

esis

tivi

ty (

Ωm

)

100 200 3000Temperature (K)

Diamond

Germanium

Copper

Measured resistivities range over more than 30 orders of magnitude

Material Resistivity (Ωm) (295K)

Resistivity (Ωm) (4K)

10-12

“Pure”Metals

Copper

10-5

 

 

Semi-Conductors

Ge (pure) 5 102 1012

 

 

Insulators Diamond 1014  

Polytetrafluoroethylene (P.T.F.E)

1020  

1014

1020

Potassium

2 10-6 10-10

Metals and insulators

Early ideas• Crystals are solid - but solids are not

necessarily crystalline• Crystals have symmetry (Kepler) and long

range order• Spheres and small shapes can be packed to

produce regular shapes (Hooke, Hauy)

?

Kepler wondered why snowflakes have 6 corners, never 5 or 7. By considering the packing of polygons in 2 dimensions, demonstrate why pentagons and heptagons shouldn’t occur.

Empty space not allowed

CRYSTAL TYPES

Three types of solids, classified according to atomic arrangement: (a) crystalline and (b) amorphous materials are illustrated by microscopic views of the atoms, whereas (c) polycrystalline structure is illustrated by a more macroscopic view of adjacent single-crystalline regions, such as (a).

quartz

Crystal structure

Amorphous structure

Definitions1. The unit cell

“The smallest repeat unit of a crystal structure, in 3D, which shows the full symmetry of the structure”

The unit cell is a box with:

• 3 sides - a, b, c

• 3 angles - , ,

14 possible crystal structures (Bravais lattices)

3D crystal lattice

cubica = b = c = =

tetragonala = b c = = = 90o

monoclinica b c = = 90o

90o

orthorhombica b c = = = 90o

hexagonala = b c = = 90o; = 120o

triclinica b c 90o

trigonal (rhombohedral)a = b = c = = 90o

Chemical bonding

Types:

Ionic bonding

Covalent bonding

Metallic bonding

Van der Walls bonding + -+ -

Bonding in SolidsMelting point(K)

Molecularcrystals

Metals Ioniccrystals

Covalentcrystals

3000

2000

1000

0

organiccrystals

W(3683)

Mo(2883)

Pt(2034)Fe(1808)

Cu(1336)Al(933)Pb(600)Na(371)Hg(234)

LiF(1143)KCl(1063)

C(<3500)

BN(3270)

SiO2(2001)

Si(1683)Ge(1240)

Electrons in metals

P. Drude: 1900 kinetic gas theory of electrons, classicalMaxwell-Boltzmann distributionindependent electronsfree electronsscattering from ion cores (relaxation time approx.)

A. Sommerfeld: 1928Fermi-Dirac statistics

F. Bloch’s theorem: 1928Bloch electrons

L.D. Landau: 1957Interacting electrons (Fermi liquid theory)

Metallic bond

Atoms in group IA-IIB let electrons to roam ina crystal. Free electrons glue the crystal

Na+ Na+

e-

e-

Attract

Attract

Attract

AttractRepelRepel

Additional binding due to interaction of partially filled d – electron shells takes place in transitional metals: IIIB - VIIIB

Core and Valence Electrons

Simple picture. Metal have CORE electrons that are bound to the nuclei, and VALENCE electrons that can move through the metal.

Most metals are formed from atoms with partially filled atomic orbitals.

e.g. Na, and Cu which have the electronic structure

Na 1s2 2s2 2p6 3s1

Cu 1s2 2s2 2p6 3s23p63d104s1

Insulators are formed from atoms with closed (totally filled) shells e.g. Solid inert gases

He 1s2 Ne 1s2 2s2 2p6

Or form close shells by covalent bonding i.e. Diamond

Note orbital filling in Cu does not follow normal rule

Why are metals good conductors?Consider a metallic Sodium crystal to comprise of a lattice of Na+ ions, containing the 10 electrons which occupy the 1s, 2s and 2p shells, while the 3s valence electrons move throughout the crystal.

The valence electrons form a very dense ‘electron gas’.

_

_

_

_

_ _ __

_

_

__

_

_

__

+ +

+

+

+

+ +

+

+ + +

+

+

+

+ +

+

+

+ + + + + +

Na+ ions:Nucleus plus 10 core electrons

We might expect the negatively charged electrons to interact very strongly with the lattice of negative ions and with each other.

In fact the valence electrons interact weakly with each other & electrons in a perfect lattice are not scattered by the positive ions.

Free classical electrons:AssumptionsWe will first consider a gas of free classical electrons subject to external electric and magnetic fields. Expressions obtained will be useful when considering real conductors

(i) FREE ELECTRONS: The valance electrons are not affected by the electron-ion interaction. That is their dynamical behaviour is as if they are not acted on by any forces internal to the conductor. (ii) NON-INTERACTING ELECTRONS: The valence electrons from a `gas' of non-interacting electrons. They behave as INDEPENDENT ELECTRONS; they do not show any `collective' behaviour. (iii) ELECTRONS ARE CLASSICAL PARTICLES:  (iv) ELECTRONS ARE SCATTERED BY DEFECTS IN THE LATTICE: ‘Collisions’ with defects limit the electrical conductivity. This is considered in the relaxation time approximation.

Ohms law and electron drift

V = E/L = IR (Volts)

Resistance R = rL/A (Ohms)

Resistivity r = AR/L (Ohm m)

E = V/L = / rI A = rj (Volts m-1)

Conductivity = 1/ s r (low magnetic field)

j = sE (Amps m-2)

I = dQ/dt (Coulomb s-1)

Area A

dx

vd

denvdt

dxen

dt

dQ

A

1j

L

Area A

Electric field E

Force on electron F

Drift velocity vd

Current density j = I/A

n free electrons per m3 with charge –e ( e = +1.6x10-19 Coulombs )

Force on electrons F = -eE results in a constant electron drift velocity, vd.

Charge in volume element dQ = -enAdx

Electrical ConductivityIn the absence of collisions, the average momentum of free electrons subject to an electric field E would be given by  

 The rate of change of the momentum due to collisions is  

 At equilibrium  

 Now j = -nevd = -nep/me = (ne2tp /me) E So the conductivity is s = j/E = ne2tp /me

Ep

eFdt

d

Field

p/dt

d

Collisions

p

p

Εppp

e- So0dt

d

dt

d

CollisionsField

The electron mobility, , m is defined as the drift velocity per unit applied electric field  

m = vd / E = etp /me (units m2V-1s-1)

sodium ion (Na+)

Examples of ionic bonding• Metal atoms with 1 electron to lose can form

ionic bonds with non-metal atoms which need to gain 1 electron:– Eg. sodium reacts with fluorine to form sodium

fluoride:

sodium atom

(Na)

fluoride ion (F-)

fluorine atom

(F)

So the formula for

sodium fluoride is

NaF

CsCl Crystal Structure

• Chloride ions form simple cubes with cesium ions in the center

Examples of ionic bonding

Examples of ionic bonding:NaCl• Each sodium atom is surrounded by

its six nearest neighbor chlorine atoms (and vice versa)

• Electronically – sodium has one electron in its outer shell: [Ne]3s1 and Chlorine has 7 (out of 8 “available” electron positions filled in its outer shell) [Ne]3s23p5

• Sodium “gives up” one of its electrons to the chlorine atom to fill the shells resulting in [Ne] [Ar] cores with Na+ and Cl- ions

• Coulombic attraction with tightly bound electron cores

NaCl

mr

B

r

eU

2

04

1

• Potential energy:

a - Madelung constant, m – integer number

for ro, the equilibrium position between the ions:

mr

eU

11

4 0

2

00

U0 is the cohesive energy, i.e. the energy per ion to remove the ion out of the crystal.

Repulsive potential 1/rm

Attractive potential -1/r

Total potential

Ionic bonding

Properties of the ionic crystals

• medium cohesive energy (2-4 eV/ atom).– low melting and boiling temp. .

• Low electrical conductivity.– (the lack of the free electrons).

• Transparent for VIS light– ( energy separation between neighbouring levels > 3 eV)

• Easily dissolved in water.– Electrical dipoles of water molecules attract the ions

Covalent bonding: molecular orbitalsConsider an electron in the ground, 1s, state of a hydrogen atom

The Hamiltonian is

The expectation value of the electron energy is

This gives <E> = E1s = -13.6eV

o

2

4e = where

RadiusBohr theis a where a 1

= (r) i.e. oo e ar/-3/2 o

r

- 2m

- = H

22

(r)dV H (r) = > E <

+

E1s

V(r)

F(r)

Hydrogen Molecular Ion

Consider the H2+ molecular ion in which

one electron experiences the potential

of two protons. The Hamiltonian is

We approximate the electron wavefunctions as

and

|R - r|-

r -

2m

- = )rU( +

2m

- = H

2222

] + A[ |)] R - r(| + )r([ A = )r( 21

] B[ |)]R - r(| )r([ B = )r( 21

p+ p+

e-

R

r

Bonding andanti-bonding states

Expectation values of the energy are:

E = E1s – g(R) for

E = E1s + g(R) for

g(R) - a positive function

Two atoms: original 1s stateleads to two allowed electron states in molecule.

Find for N atoms in a solid have N allowed energy states

)r(

)r(

)r(

-6 -4 -2 0 2 4 6

-1.4

-1.2

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

r

-6 -4 -2 0 2 4 6

-1.4

-1.2

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

r

V(r)

2)r(

)r(

1s

2s

bonding

Anti-bonding

Anti-bonding

bonding

covalent bonding – H2 molecule

• 8

6

4

2

0

-2

-4

-6

R00.1 0.2 0.3 0.4

nuclear separation (nm)

ener

gy(e

V)

parallel spin

antiparallel spin

system energy (H2)

Covalent bonding

Crystals: C, Si, Ge

Covalent bond is formed by two electrons, one from each atom, localised in the region between the atoms (spins of electrons areanti-parallel )

Example: Carbon 1S2 2S2 2p2

C C

Diamond: tetrahedron, cohesive energy 7.3eV

3D 2D

Covalent Bonding in Silicon

• Silicon [Ne]3s23p2 has four electrons in its outermost shell

• Outer electrons are shared with the surrounding nearest neighbor atoms in a silicon crystalline lattice

• Sharing results from quantum mechanical bonding – same QM state except for paired, opposite spins (+/- ½ ħ)

Covalent bond

Atoms in group III, IV,V,&VI tend to form covalent bond

Filling factor

T. :0.34 F.C.C :0.74

ionic – covalent mixed

diamond lattice

zinc blend crystals (ZnS, GaAs)

As

Properties of the covalent crystals

• Strong, localized bonding.

• High cohesive energy (4-7 eV/atom).– High melting and boiling temperature.

• Low conductivity.

The Hall Effect

An electric field Ex causes a current jx to flow.

Ex, jxEyBz

vd = vx

The Hall coefficient is RH = Ey/jxBz = -1/ne

The Hall resistivity is rH = Ey/jx = -B/ne

jx = -nevx so Ey = -jxBz/ne

Therefore Ey = +vxBz

F = -e (E + v B). In equilibrium jy = 0 so Fy = -e (Ey - vxBz) = 0

A magnetic field Bz produces a Lorentz force in the y-direction on

the electrons. Electrons accumulate on one face and positive charge on the other producing a field Ey .

j

For a general vx.

vx+ve or -ve

The Hall Effect

The Hall coefficient RH = Ey/jxBz = -1/ne

The Hall angle is given by tan f = Ey/Ex = rH/r

For many metals RH is quiet well described by this expression which is useful for obtaining the electron density, in some cases.

However, the value of n obtained differs from the number of valence electrons in most cases and in some cases the Hall coefficient of ordinary metals, like Pb and Zn, is positive seeming to indicate conduction by positive particles!

This is totally inexplicable within the free electron model.

j=jx

Ey

Bzvd = vx

Ex

E

The (Quantum)Free Electron model: Assumptions

(i) FREE ELECTRONS: The valance electrons are not affected by the electron-ion interaction. That is their dynamical behaviour is as if they are not acted on by any forces internal to the conductor. (ii) NON-INTERACTING ELECTRONS: The valence electron from a `gas' of non-interacting electrons. That is they behave as INDEPENDENT ELECTRONS that do not show any `collective' behaviour. (iii) ELECTRONS ARE FERMIONS: The electrons obey Fermi-Dirac statistics. (iv) ‘Collisions’ with imperfections in the lattice limit the electrical conductivity. This is considered in the relaxation time approximation.

Free electron approximation

U(r)U(r)

Neglect periodic potential & scattering (Pauli)

Reasonable for “simple metals” (Alkali Li,Na,K,Cs,Rb)

Eigenstates & energies

dtd

iUm2

22

)L/n,L/n,L/n(2k

km2

E

e)t,r(

zzyyxx

22

k

)rkt(i0k

Ek

|k|

Unit volume in k-space: 1/(2p3)

Density of states

G(E) - this is the number of allowed states within a band:

g(E)dE=2g(k)dk

with g(k) equal to the density of states within the k-space. 2 is due to two possible spin values.

dkdE

kgEg

/

)(2)(

g(k) is equal to the number of states within the space between two spheras of radii k and k+dk, which is equal to the number of states per unit volume (1/(2p)3) multiplied by the volume between the spheras (4pk2dk). Thus :

2

22

3 2)(4

)2(

1)(

kkgdkkdkkg

m

kE

2

22

2

2

2)(

k

kg

mEmmk

k

mk

dkdE

kEg 2

2

2

/

1)(

322222

2

2

2

For:

dkdE

k

dkdE

kgEg

/

1

/

)(2)(

2

2

g(E)

E

Statistics & DOS

Density of states:

mEm

dE

dkkgEg 2)(2)(

32

Fermi-Dirac statistics:

1e

1)E(f

kT

EFD

T=0.1m

0.5 1 2

1

m=EF

g(E)

E

2kBT

Occupation of states

|kF|

FermiSurface

D(E)

E

2kBT

EF

Bound States in atoms

r4

qe = )r(V

o

2

Electrons in isolated atoms occupy discrete allowed energy levels E0, E1, E2 etc. .

The potential energy of an electron a distance r from a positively charge nucleus of charge q is

-8 -6 -4 -2 0 2 4 6 8-5

-4

-3

-2

-1

0

F6 F7 F8 F9

r

V(r)E2

E1

E0

r

0

Increasing Binding Energy

Bound and “free” states in solids

-8 -6 -4 -2 0 2 4 6 8-5

-4

-3

-2

-1

0

F6 F7 F8 F9

r

-8 -6 -4 -2 0 2 4 6 8-5

-4

-3

-2

-1

0

F6 F7 F8 F9

r

-8 -6 -4 -2 0 2 4 6 8-5

-4

-3

-2

-1

0

F6 F7 F8 F9

r

V(r)E2

E1

E0

The 1D potential energy of an electron due to an array of nuclei of charge q separated by a distance R is

Where n = 0, +/-1, +/-2 etc.

This is shown as the black line in the figure.

n o

2

nRr4

qe = )r(V

r

0

0

+ + + + +RNuclear positions

V(r) lower in solid (work function).

Naive picture: lowest binding energy states can become free to move throughout crystal

V(r)Solid

Energy Levels and Bands

+E

+ + + +position Electron level similar to

that of an isolated atom

Band of allowed energy states.

In solids the electron states of tightly bound (high binding energy) electrons are very similar to those of the isolated atoms.

Lower binding electron states become bands of allowed states.

We will find that only partial filled band conduct

Solid stateN~1023 atoms/cm32 atoms 6 atoms

Energy band theory

Metal – energy band theory

Insulator -energy band theory

diamond

semiconductors

Intrinsic conductivity

kTEss

ge2/

0

ln(s)

1/T

1/T

ln(s)

kTEdd

de /0

Extrinsic conductivity – n – type semiconductor

Extrinsic conductivity – p – type semiconductor

Conductivity vs temperature

kTEss

ge2/

0

ln(s)

kTEdd

de /0

1/T