Lecture 5: Charge carrier dynamics

1
Lecture 5 Charge carrier dynamics

• Carrier diffusion
• Einsteins relation
• Carrier generation and recombination
• Introduction to optical processes in
  semiconductors

2
Charge carrier diffusion

• Whenever there is a gradient in the concentration
  of a species of mobile particle, the particles
  diffuse from the regions of high concentration to
  the regions of low concentration.
• For electrons (holes) the collision process can
  be described by the mean free path l (the average
  distance the carrier moves between successive
  collisions) and the mean collision time tsc.
• Consider a concentration profile n(x,t) of
  electrons at time t. The electron flux f(x,t)
  across a plane xx0 is
• nL and nR are the average carrier densities in
  the two regions.

3
Charge carrier diffusion
Since the two regions L and R are separated by l,
write
4
Charge carrier diffusion

• The net flux is
• Dn is called the diffusion coefficient, it
  depends upon the scattering processes that
  control l and tsc and the temperature.
• The hole diffusion coefficient gives the hole
  flux to be
• The electron and hole flux can result in a
  current flow

Current flows in opposite directions
5
Einsteins relation

• The combined influence of an external E-field and
  carrier diffusion gives the current density
• We want to establish a relationship between the
  mobility and diffusion coefficients.
• Let us consider the effect of electric fields on
  the energy bands in a semiconductor.
• In the case where a uniform E-field is applied
  there will be a potential energy gradient, with a
  positive potential on the left hand side in
  relation to the right hand side.

6
The effect of E-fields in semiconductors
For a uniform E-field the potential energy
is The applied force is related to the
potential energy by Since electron charge is
e, the bands obey The electrons drift
downhill.
7
Einsteins relation

• At equilibrium, the total electron and hole
  currents are individually zero the E-field can be
  written
• If we write n(x) in terms of the intrinsic Fermi
  level, Efi, and the Fermi level in the
  semiconductor, EF(x), we can obtain the
  derivative of the carrier concentration using the
  Boltzmann approximation

8
Einsteins relation

• At equilibrium, the Fermi level cannot vary
  spatially, otherwise the probability of finding
  electrons along a constant energy position
  position will vary along the semiconductor.
• Otherwise, electrons at a given energy in a
  region where the probability is low to move to
  the same energy in a region where the probability
  is high.
• This is not allowed, no current is flowing, the
  Fermi level has to be uniform in space at
  equilibrium
• The Einstein relation for electrons is hence
  written

9
Einsteins relation Example

• Obtain the diffusion coefficient of electrons in
  Silicon at an electric field of 1kV/cm and
  10kV/cm at 300K.
• From v-F relations, the velocity of electrons in
  silicon is 1.4?106cm/s and 7?106cm/s at 1kV/cm
  and 10kV/cm.
• Using the Einstein relation, the diffusion
  coefficient
• Giving

10
Diffusion length

• Excess electrons (holes) injected will recombine
  with the holes (electrons).
• They travel a distance Ln(Lp) before recombining
  in the absence of electric fields.
• Rate of particle flow Particle flow due to
  current- Particle loss due to recombinationPartic
  le gain due to generation.
• dn(dp) is the excess electron (hole) density.

11
Diffusion length
Carrier injection
How does the excess density vary with position?
The general solution of the second-order
differential equation is
12
Diffusion length

• If LgtgtLn and dn(L)0
• The semiconductor sample is much longer than Ln.
  This happens in the case of a long p-n diode. The
  carriers are injected at the origin and the
  excess density decays to zero deep in the
  semiconductor.
• If LltltLn
• Very important for bipolar transistors and narrow
  p-n diodes. The carrier density goes linearly
  from one boundary value to the other

13
Diffusion length Example

• A p-type GaAs sample has electrons injected from
  a contact. The minority carrier mobility is
  4000cm2/V s at 300K. What is the diffusion length
  given the recombination time is 0.6ns?
• The diffusion time constant (use Einstein
  relation)
• The diffusion length is
• Using the recombination time

14
Carrier generation and recombination

• As a sample of pure silicon has its temperature
  increased from 0K to 300K, the electron density n
  will increase from 0 to 1.5?1010cm-3.
• Thermal equilibrium is achieved by electron
  excitation and recombination RGRR.
• The absorption and emission of light is a second
  very important carrier generation process.
• Depending on the nature of the recombination
  process, the released energy that results from
  the recombination process can be emitted as a
  photon or dissipated as heat to the lattice.
• The process involving the emission of a photon is
  called radiative recombination, otherwise it is
  called nonradiative recombination.

15
Optical processes

• The most important optoelectronic interaction in
  semiconductors is the band-to-band transition.
• In the photon absorption process, a photon
  scatters an electron in the valence band, causing
  the electron to go into the valence band.
• In the reverse process the electron in the
  conduction band recombines with a hole in the
  valence band to generate a photon.
• Conservation of energy
• Conservation of crystal momentum

16
Optical processes

• A 1eV photon corresponds to a wavelength of
  1.24um. The k-values relevant are 10fm, which is
  essentially zero compared to the k-values for
  electrons.
• k-conservation ensures that the initial and final
  electrons have the same k-value.
• Only vertical k transitions are allowed.
• Direct bandgap semiconductors have a strong
  interaction with light.
• Indirect bandgap semiconductors have a weak
  interaction with light.
• Photon attenuation through a semiconductor is
  described simply as

Absorption coefficient
17
Optical processes

• Each photon absorbed will create an electron-hole
  pair. With knowledge of the optical power density
  P of light impinging on a semiconductor, the
  photon flux is
• Hence the electron-hole pair generation rate is
• Electrons in the conduction band can then
  recombine with holes in the valence band to
  generate a photon.

18
Radiative recombination

• The electron-hole pair generation rate in optical
  processes was defined as
• Electrons in the conduction band will then
  recombine with holes in the valence band.
• Consider an n-type semiconductor with ngtgtp. If an
  excess hole density dp is injected, these excess
  minority carriers will recombine with the
  majority carriers with a rate given by
• tn and tp are the electron and hole radiative
  lifetimes (minority carrier lifetimes).
• Typically 1ns in heavily doped semiconductors.

Excess holes in n-type
Excess electrons in p-type
19
Nonradiative recombination

• In real semiconductors the forbidden bandgap
  region always has intentional or unintentional
  impurities that produce electronic levels that
  are in the bandgap.
• These regions can arise from chemical impurities
  or from native defects such as a lattice vacancy.
• Bandgap levels are states in which the electron
  is localised in a finite space near the defect
  not like free states.
• As the electrons move in the allowed bands they
  can get trapped by these defects.
• Such defects can allow the recombination of an
  electron (hole) without he emission of a photon.
• This non-radiative process competes with
  radiative recombination.

20
Nonradiative recombination

• A mid-bandgap level with density Nt is referred
  to as a lattice trap.
• A capture cross-section s can be assigned to a
  lattice trap. For carrier velocity vth, the rate
  at which the carrier encounters a trap is
• The trap lifetime tnr is the time it takes for a
  carrier to be captured by a trap.
• This process is referred to as Shockley-Read-Hall
  SRH recombination.
• If the assumptions
• The trap levels are midgap
• npgtgtni2 under injection conditions

21
Summary of lecture 5

• Carrier diffusion
• Einsteins relation
• Carrier generation and recombination
• Introduction to optical processes in
  semiconductors