Carrier Action: Motion, Recombination and Generation.

1
Carrier Action Motion, Recombination and
Generation.

• What happens after we figure out how many
  electrons and holes are in the semiconductor?

2
Carrier Motion I

• Described by 2 concepts
• Conductivity s
• (or resistivity r 1/s)
• Mobility m
• Zero Field movement
• Random over all e-
• Thermal Energy Distribution.
• Motion

• Electrons are scattered by impurities, defects
  etc.

What happens when you apply a force?
3
Carrier Motion II

• Apply a force
• Electrons accelerate
• -n0qExdpx/dt from Fmad(mv)/dt
• Electrons decelerate too.
• Approximated as a viscous damping force
• (much like wind on your hand when driving)
• dpx -px dt/t dt time since last
  randomizing collision and t mean free time
  between randomizing collisions.
• Net result deceration dpx/dt -px/t

4
Carrier Motion III

• AccelerationDeceleration in steady state.
• dpx/dt(accel) dpx/dt(decel) 0
• -n0qEx - px/t 0.
• Algebra
• px/n0 -qtEx ltpxgt
• But
• ltpxgt mnltvxgt Therefore

Mobility!
5
Currents

• Current density (J) is just the amount of
  charge passing through a unit area per unit time.
  
• Jx (-q)(n0)ltvxgt in C/(s m2) or A/m2
• (qn0mn)Ex for e-s acting alone.
• sn Ex (defining e- conductivity)
• If both electrons and holes are present

6
Current, Resistance

• How do we find
• current (I)? We integrate J.
• resistance (R)?
• Provided r, w, t are all constants along the
  x-axis.

E
x
t
L
V
w
7
Mobility changes

• Although it is far too simplistic we use
• mn qt/mn
• t depends upon
• of scatter centers (impurities, defects etc.)
• More doping gt lower mobility (see Fig. in books)
• More defects (worse crystal) gt smaller mobility
  too.
• The lattice temperature (vibrations)
• Increased temp gt more lattice movement gt more
  scattering gt
• smaller t and smaller m.

t is the mean free time. mn is the effective
mass. (depends on material)
m
Increasing Doping
8
Mobility Changes II

• Mobility is also a function of the electric field
  strength (Ex) when Ex becomes large. (This leads
  to an effect called velocity saturation.)

Here m is constant (low fields). Note constant m
gt linear plot.
ltvxgt
Vsat
107 cm/s
At 107 cm/s, the carrier KE becomes the same
order of magnitude as kBT. Therefore added
energy tends to warm up the lattice rather than
speed up the carrier from here on out. The
velocity becomes constant, it saturates.
106 cm/s
electrons
holes
105 cm/s
Ex (V/cm)
102
103
104
105
106
9
What does Ex do to our Energy Band Diagram?

• Drift currents depend upon the electric field.
  What does an electric field do to our energy band
  diagrams?
• It bends them or causes slope in EC, EV and Ei.
  We can show this.
• Note
• Eelectron Total E
• PE KE
• How much is PE vs. KE???

Eelectron
e-
EC
Eg
EV
h
10
Energy Band Diagrams in electric fields

• EC is the lower edge for potential energy (the
  energy required to break an electron out of a
  bonding state.)
• Everything above EC is KE then.
• PE always has to have a
• reference! Well choose
• one arbitrarily for the
• moment. (EREF Constant)
• Then PE EC-EREF
• We also know PE-qV

Eelectron
e-
KE
EC PE
Eg
EV PE
PE
KE
h
EREF
11
Energy Band Diagrams in electric fields II

• Electric fields and voltages are related by
• E -ÑV (or in 1-D E-dV/dx)
• So PE EC-EREF -qV or V -(EC-EREF)/q
• Ex -dV/dx -d/dx-(EC-EREF)/q or
• Ex (1/q) dEC/dx

12
Energy Band Diagrams in electric fields III

• The Electric Field always points into the rise in
  the Conduction Band, EC.
• What about the Fermi level? What happens to it
  due to the Electric Field?

Eelectron
Ex
EC
Ei
EV
Eg
EREF
13
Another Fermi-Level Definition

• The Fermi level is a measure of the average
  energy or electro-chemical potential energy of
  the particles in the semiconductor. THEREFORE
• The FERMI ENERGY has to be a constant value at
  equilibrium. It can not have any slope
  (gradients) or discontinuities at all.
• The Fermi level is our real-life EREF!

14
Lets examine this constant EF
V -

• Note If current flows gt it is not equilibrium
  and EF must be changing.
• In this picture, we have no connections.
  Therefore I0 and it is still equilibrium!
• Brings us to a good question
• If electrons and holes are moved by Ex, how can
  there be NO CURRENT here??? Wont Ex move the
  electrons gt current?
• The answer lies in the concept of Diffusion.
  Next

Semiconductor
Ex
Eelectron
Ex
EC
Ei
EF
EV
Looks P-type
Looks N-type
15
Diffusion I

• Examples
• Perfume,
• Heater in the corner (neglecting convection),
• blue dye in the toilet bowl.
• What causes the motion of these particles?
• Random thermal motion coupled with a density
  gradient. ( Slope in concentration.)

16
Green dye in a fishbowl

• If you placed green dye in a fishbowl, right in
  the center, then let it diffuse, you would see it
  spread out in time until it was evenly spread
  throughout the whole bowl. This can be modeled
  using the simple-minded motion described in the
  figure below. L-bar is the mean (average) free
  path between collisions and t the mean free
  time. Each time a particle collides, its new
  direction is randomly determined. Consequently,
  half continue going forward and half go
  backwards.

32
Dye Concentration
16
16
8
8
8
8
4
4
8
8
4
4
x
-3 -2 -1
0 1 2 3
17
Diffusion II

• Over a large scale, this would look more like

t0
t1
Lets look more in depth at this section of the
curve.
t2
t3
tequilibrium
18
Diffusion III

• What kind of a particle movement does Random
  Thermal motion (and a concentration gradient)
  cause?

n(x)
It causes net motion from large concentration
regions to small concentration regions.
nb0
nb1
nb2
Bin (1)
Bin (0)
Bin (2)
Line with slope
Half of e- go left half go right.
x-axis
19
Diffusion IV

• Net number of electrons crossing x0 is
• Number going right 0.5nb1lA
• Minus Number going left 0.5nb2lA
• Net is 0.5lA(nb1-nb2)
• (note lAvolume of a bin.)
• Flux of particles crossing a plane per unit
  time and unit area. Symbol is f
• f 0.5lA(nb1-nb2) (t mean free
  time.)
• tA
• Or f 0.5l (nb1-nb2)
• t

20
Diffusion V

• Using the fact that slope (dn/dx) -(nb1-nb2)/l
  gives
• f - 0.5l2 dn or f -Dndn/dx
  (electrons)
• t dx
• or f -Dpdp/dx (holes)
• Now When charges move we get current.
  Consequently, the current density is directly
  related to the particle flux. The equations are
  
• (electrons) (holes)

21
Diffusion VI

• Lets look at an example

dn/dx 0 here
n(x)
x
J(x)
x
The electrons are diffusing out of the center and
toward the edges.
22
Currents round-up

• So now we know that our total currents have 2
  components
• DRIFT due to any electric field we apply
• DIFFUSION due to any (dp/dx, dn/dx) we apply
  and thermal motion.

23
Answering that old question
V -

• How can we have an electric
• Field and still have no current?
• (Still have J 0?)
• Diffusion must balance Drift!
• Example

Semiconductor
Ex
Eelectron
Ex
EC
Ei
EF
EV
Looks P-type
Looks N-type
24
Einstein Relationship

• We next remember pniexp((Ei-EF)/kBT)
• Plugging this into our equation for the electric
  field and noting that dEF/dx 0 we get
• The Einstein Relationships.
• These are very useful. You will never find a
  table for both Dp and mp as a result of these.
  Once you have m, you have D too, by this
  relationship.

25
A sanity check

• Pretend we have

• What will be the fluxes and currents?

x
Holes Mechanism Electrons
Ex
Diffusion Flux (f) Current Density (J) Drift Flux
(f) Current Density (J)
n(x)
p(x)
26
Recombination Generation I

• Generation (G) How e- and h are produced or
  created.
• Recombination (R) How e- and h are destroyed or
  removed
• At equilibrium r g and
• since the generation rate is set by the
  temperature, we write it as r gthermal

• The concepts are visually seen in the energy band
  diagram below.

Ee
R
G
EC
hv
hv
EV
x
27
Recombination Generation II

• Recombination must depend upon
• the of electrons no
• the of holes po
• (If no e- or h, nothing can recombine!)
• From the chemical reaction
• e- h ? Nothing
• we can know that
• r arnopo arni2 gthermal

• When the temperature is raised
• gthermal increases
• Therefore
• ni must increase too!

The recombination rate coefficient
28
Recombination Generation III

• A variety of recombination mechanisms exist

Ee
Direct, Band to Band
Auger
R
G
EC
Ee
R
G
hv
hv
EC
EV
x
Ee
EV
Indirect via R-G centers
x
R
G
EC
R-G Center Energy Level
EV
x
29
GaAs band structure produced by J. R. Chelikowsky
and M. L. Cohen, Phys. Rev. B 14, 556
(1976)using an empirical Pseudo-potential method
see also Cohen and Bergstrasser, Phys. Rev. 141,
789 (1966).
GaAs is a Direct Band Gap Semiconductor
Eg The Band Gap Energy
Direct recombination of electrons with holes
occurs. The electrons fall from the bottom of
the CB to the VB by giving off a photon!
30
GaAs band structure produced by W. R. Frensley,
Professor of EE _at_ UTD using an empirical
Pseudo-potential method see also Cohen and
Bergstrasser, Phys. Rev. 141, 789 (1966).
31
Silicon band structure produced by J. R.
Chelikowsky and M. L. Cohen, Phys. Rev. B 14, 556
(1976)using an empirical Pseudo-potential method
see also Cohen and Bergstrasser, Phys. Rev. 141,
789 (1966).
Si is an Indirect Band Gap Semiconductor
Eg The Band Gap Energy
Only indirect recombination of electrons with
holes occurs. The electrons fall from the
bottom of the CB into an R-G center and from
the R-G center to the VB. No photon!
32
Silicon band structure produced by W. R.
Frensley, Professor of EE _at_ UTD using an
empirical Pseudo-potential method see also Cohen
and Bergstrasser, Phys. Rev. 141, 789 (1966).