22 Semiconductor Nanoparticles-3

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Semiconductors

  • Semiconductors are materials with a (relatively) small band gap (typically 1eV) between a filled valence band and an empty conduction band.
  • Chemical potential, μ (often called Fermi energy) lies in the band gap.
  •  Insulators at T=0, with a small density of electrons excited at finite temperatures.
  • Typical semiconductors are Silicon and Germanium or III-V compounds such as GaAs 2 atoms in primitive basis have 4 electrons each (or 3 + 5); 8 electrons fill 4 bands made of s and p orbitals.

 

Band structure of Semiconductors

Graph of Energy (E) vs. wave vector (k). EF separates filled and empty states.

 

Energy levels of electrons and holes

 

Close to the band edge minima and maxima we can write:

Band gap determines the optical properties – strong absorption when hν > EG

 

Optical absorption

  • Excitation promotes an electron from the valence band to conduction band.
  • An empty state left in valence band is known as a hole.
  •  If band minima and maxima are at different points then we have an Indirect semiconductor.
  • The classic example is Silicon.
  • This affects the optical properties such as absorption where ∆k ≈ 0

Hole picture

 

Remove one electron from a filled band and electricity can be conducted by the movement of all of the electrons present.

 

The sum of this motion is equivalent to one positively charged particle: a hole

 

Holes and their properties

Effective masses
Force accelerates the electron or hole:
Meaning of the effective mass
  • Effective mass changes as we move through a band in k-space positive (electron-like) at the bottom, becoming negative (hole-like) at the top

Typical values in semiconductors are in the range 0.01 to 0.5 me

 

Concentrations of Electrons and Holes

 

Calculate carrier density from density of states and distribution function:

For most semiconductors the chemical potential, ?, (often also called the Fermi Energy (EF)), lies in the band gap so:           

 

LAW OF MASS ACTION

 

Densities of holes and electrons depend on

Intrinsic Semiconductors

 

A semiconductor is said to be intrinsic if it is undoped, and the only source of electrons and holes is by its thermal excitation from the valence band to the conduction band.

We can use this relation to measure the Band Gap, by measuring the carrier densities from the Hall Effect At low temperatures ni 0 and impurities are important

 

Doping Semiconductors

 

We can control the numbers of electrons and holes in a semiconductor by adding impurities which dope the material.

 

Donors donate an electron to make the material more n-type. A typical example is by adding a group V element (such as As or P) to a group IV semiconductor such as silicon.

 

Four of the electrons participate in the sp3 bonds as if they were from silicon, but the fifth electron is left over. Extra charge on P nucleus creates a +ive core, and the fifth electron can be bound to this, but the binding is weak.

 

Shallow Donors

 

Impurity binding looks like a hydrogen atom Binding energy

 

Binding energy is very small because:

(i)   The effective mass is small (typically m* = 0.1 me)

(ii)   The wave function is large (much more than the crystal unit cell), so we must include the relative

dielectric constant of the medium εr, – typically ≈10.

∴  ∗ ≈ 10−3  R0 = 13.6 meV (155K)

 

Acceptors

 

Dope semiconductors with holes by adding group III elements to a group IV material. e.g. put Ga into silicon.

 

One valence electron is missing. This creates a vacant state, a hole, which binds to the ion core of the Ga which is negatively charged. Binding energy is:

Where is the energy level?

 

Ionized Acceptor or Donor is a free hole or electron at the top or the bottom of the band

∴ Acceptor is R* above the valence band edge

Donor is R* below the conduction band edge

 

Extrinsic Carrier Densities

 

Density of impurities (e.g. Donors) usually much less than NC, NV. Impurities can be ionized Nd+ , or neutral Nd0 so:

Nd= Nd++Nd0

 

Using charge neutrality we have:

n=p+Nd+

 

Simple argument:

 

At high (e.g. Room) Temperature kT > R* therefore all donors will be ionized but the density of holes created by excitation across the band gap is still small

∴ n ≈Nd+=Nd

 

Chemical Potential μ

 

At high (room) temperature most impurities are ionized

∴ n ≈Nd+=Nd

 

μ  lies below the Donor level so that most impurities are empty (ionized), but is still close to conduction band.

 

Minority Carriers

 

By Law of Mass action the (very small) density of holes is:

p=ni(T)2⁄ Nd

 

Temperature dependent density and chemical potential

 

  • Density is constant in region around room temperature.
  • High temperature gives Intrinsic behaviour.

 

Conductivity of Semiconductors

μ, is the mobility, which is defined by v = μ E, the drift velocity per unit electric field.

 

Conductivity is dominated by variation in densities.

 

Mobility is determined by scattering rate:

 

Low T: impurity scattering gives μ  ~ T3/2

 

High T: phonon scattering gives μ  ~ T3/2

 

p-n junction

 

p-n junctions

 

 

What is current flow across junction?             J=nev

What is v? – due to diffusion of carriers with diffusion coefficient D and lifetime         .

 

Low Dimensional Structures and Materials

 

•  Anisotropic Materials

• Artificial layered structures – Quantum Wells and Superlattices

•  Electric or Magnetic Fields applied in one direction.

Heterojunctions

Reduced Dimensionality

Quantum Well removes 1 Dimension by quantization

 

 

Electron is bound in well and can only move in plane

 

2-D system – motion in x, y plane

 

Quantum Well – Type I

 

Typical Materials:

1: GaAs

2: (Al0.35Ga0.65)As

(Eg = 1.5 eV)           (Eg = 2.0 eV)

Energy levels are quantized in Z-direction with values En for both electrons and holes

 

Infinite well – Particle in a box

 

Density of States 

g(k)dk

g(ε)dε

 

Optical Properties

 

3-D Absorption coefficient is proportional to the density of states:

∴α  ~  ε1/2

Modified close to the band gap due to ‘excitons’

 

2-D Big Changes

 

Multiple Band gaps –

Band gap shift –

Sharper edge-

 

For wide wells the sum of many 2-D absorptions becomes equivalent to the 3-D absorption shape (ε1/2)

 

•  Sharp peaks due to excitons

•  peaks doubled due to heavy and light holes

 

Semiconducting Lasers

 

Forward biased p-n junction

  • Quantum Well laser
  • Fibre Optic Communications,
  • CD players, laser pointers

How do we achieve low dimensionality?

 

Naturally anisotropic crystals

Controlled growth of layers and/or apply external potential Deposit thin layers of single crystals to create

‘heterostructures’

 

Two Main techniques:

 

I) Molecular Beam Epitaxy (MBE)

II) Metal Organic Vapour Phase Epitaxy (MOVPE)

 

Molecular Beam Epitaxy (MBE)

  • Ultra High Vacuum molecular (Molecular Beam) evaporation of species of elements
  • Epitaxy-   maintaining crystal structure of the ‘substrate’ – which is a single crystal
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