Tài liệu Chapter XXIV Crystalline Solids doc

We will describe the wavefunctions and energy levels of electrons insuch periodical atomic structures.. Discrete atomic states  band of crystal states  Fill according to Pauli Princip

GENERAL PHYSICS III

Optics

&

Quantum Physics

Chapter XXIV

Crystalline Solids

§ 1 Wavefunctions and energy band of electrons

§ 2 Electronic conduction in metals

§ 3 Semiconductors

To have a quantum-mechanical treatment we model a crystalline solid asmatter in which the atoms have long-range order, that is a recurring

(periodical) pattern of atomic positions that extends over many atoms

We will describe the wavefunctions and energy levels of electrons insuch periodical atomic structures

We want to answer the question: Why do some solids conduct currentand others don’t?

§1 Wavefunctions and energy band of electrons:

1.1 Potential energy of electrons:

r ( U

e r

r

e )

r ( U

Discrete atomic states  band of crystal states  Fill according to Pauli Principle

• A simple (1-D) model of the recurring structure is shown below

• An electron interacts not with one, but with many nuclei  every

energy level splits into a band of energies

1.2 Wavefunctions and Energies:

• Atomic ground state:

r ( U

e ) r (  

What do these crystal states look like?

approximately linear combinations of atomic orbitals.

Again start with

simple atomic state:

+e

r

n = 1

A

Bring N atoms together together forming a 1-D crystal (a periodic lattice)

(N atomic states N crystal states):

Energy band

• 1-D periodic lattice:

• Simple model of a crystal with covalent bonding:

N

Highest energy orbital (N-1 nodes)

1

Lowest energy orbital (zero nodes)

“in between” states

• The “in between" states:

Bloch Wavefunction for

electron in a solid:

The wavevector k has N possible values from k = /L to k = /a.

Real Envelope: eikx

Length of crystal, L

Lattice spacing, a

n

u(x)e (x) ik x

 u(x)atomic states involved:depends on the

L/a

1,2,

n L

• An example of Bloch Wavefunctions and the Energy Band:

u(x)e (x) ik x

Lowest energy wavefunction

Highest energy wavefunction

Closely spaced energy levels in this “1s-band”

For N = 6 there are six differentsuperpositions of the atomic statesthat form the crystal states

1.3 Bloch Wavefunctions and the Free Electron Model:

 Bloch wavefunction acts almost like free electron wavefunction:

 In a perfectly periodic lattice, an

electron moves freely without

scattering from the atomic cores !!

It’s in a packet of stationary states

–-the Bloch wavefunctions

Free electron:

Envelope: Re eikx

u(x)e (x) ik x

p m

2

k E

Energy Ae

(x)

2 2

2 x



mass" effective

"

m*

* m 2

k E

Energy u(x)e

(x)

2 n

2 x

ik n



e

-• Electron in a periodic potential has a well defined wavevectorand momentum:

even though it is traveling in a complicated potential

 Also lattice vibrations break the periodicity – electrons in metalsscatter more at higher temperatures Here the electrons see atime-varying potential

) h/

k

e

- If there is a defect in the crystal,

the electron may scatter to another Bloch state

I.e a Bloch state is not stationary in the full potential, with

defect

1.4 Insulators, Semiconductors, Metals:

 Energy bands and the gaps between them determine the conductivityand other properties of solids

E

conductors

 Insulators have a valence band which is

full and a large energy gap (few eV)

 apply an electric field - no states of

higher energy available for electron

 Metals have an upper band which is only

partly full

 apply an electric field - lots of states

of higher energy available for electron

 Semiconductors are insulators at T = 0

They have a small energy gap (~1 eV)

between valence and conduction bands,

so they become conducting at higher T

Why do some solids conduct current and others don’t?

2.1 Semi-classical picture of conduction:

ne E

E m

ne J

2 2

2.2 Quantum-mechanical description of conduction:

Z = 11 1s 2 2s 2 2p 6 3s 1

N states 4N states

N states

1s

2s, 2p

3s

The 2s and 2p bands overlap.

Since there is only one electron in the n = 3 shell,

we don’t need to consider the 3p or 3d bands of states, which partially overlap the 3s band.

N = total number of

atoms Fill the crystal stateswith 11N electrons

• Example: Na

Z = 11 1s 2 2s 2 2p 6 3s 1

N states 4N states

Total number of atoms = N Total number of electrons

= 11N

Fill the Bloch states

according to Pauli Principle:

The 3s band is only half

filled (N orbital states

 EE kT

F

e

f(E) Fermi-Dirac distribution

(EF: the Fermi level )

Z = 14 1s 2 2s 2 2p 6 3s 2 3p 2

4N states 4N states

Total number of atoms = N Total number of electrons =14N

Fill the Bloch states

according to Pauli Principle

It appears that, like Na,

Si will also have a half

filled band: The 3s3p

band has 4N orbital

states and 4N electrons.

But something special happens for Group IV elements.

By this analysis, Si should be

a good metal, just like Na.

Consider the structure of energy band for Si :

Z = 14 1s 2 2s 2 2p 6 3s 2 3p 2

4N states 4N states

Total number of atoms = N Total number of electrons = 14N

Fill the Bloch states

according to Pauli Principle

In Si, the eigenstates

are made of hybrid

Antibonding

Bonding

The s-p band splits into two:

Antibonding states

Bonding states

• The dependence of electronic conductivity of a semiconductor on

temperature: The example of Si Z = 14 1s22s22p63s23p2

The electrons in a filled band cannot contribute to conduction, because with

reasonable E fields they cannot be promoted to a higher kinetic energy.

Therefore, at T = 0, Si is an insulator At higher temperatures, however,

electrons are thermally promoted into the conduction band:

Metal: scattering time  gets shorter with increasing T

m

1 

 Semiconductor: number n of free

electrons increases rapidly with T (much faster than decreases)

Energy Gap

Egap ≈1 eV

Filled band at T = 0 (Valence band)





where J = current density and E = applied electric field

 Resistivity depends on the scattering time for electrons

 Resistivity depends on the number of free electrons n (carriers)

Example properties at room temperature:

 The structure as well as the bonding type determine the properties of asolid

Material Resistivity Carrier Type