Download Mobility (cont.)

CHAPTER 3: CARRIER
CONCENTRATION PHENOMENA
Part I
SUB-TOPICS IN CHAPTER 3:







Carrier Drift
Carrier Diffusion
Generation & Recombination
Process
Continuity Equation
Thermionic Emission Process
Tunneling Process
High-Field Effect
Part I



Carrier Drift
Carrier Diffusion
Generation & Recombination
Process
CARRIER DRIFT
Mobility

The electron in s/c have 3 degree of freedom – they
can move in a 3-D space. The kinetic energy K.E of
electron is given by
1
3
2
K e  mn vth  kT
2
2
(1)
From the theorem for equipartition of energy, ½ kT unit
energy per degree of freedom.


mn – effective mass of electron,
vth – average thermal velocity (~ 107cm/s at
T=300K)
Mobility (cont.)




Electron in s/c is moving rapidly in all direction, where
thermal motion of an individual electron may be
visualized as a succession of random scattering from
collisions with lattice atoms, impurity atoms, and other
scattering centers, as shown in Fig. 3.1(a).
Average distance between collisions – mean free path.
Average time between collisions – mean free time C.
For typical mean free path ~ 10-5cm,
C = 10-15/vth~10-12s (or in 1ps).
Mobility (cont.)




When small electric field, E, is applied to s/c sample,
each electron will experience a force (–qE) from the
field and accelerated along the field (in opposite
direction) during the time between collisions –
additional thermal velocity component.
This additional component called drift velocity.
Combination displacement of an electron (due to
random thermal motion) & drift component illustrated
in Fig. 3.1(b).
Note that: net displacement of the electron is in the
opposite direction of applied field.
Mobility (cont.)
Without electric field
hole
Figure 3.1. Schematic path of an electron in a semiconductor. (a) Random
thermal motion. (b) Combined motion due to random thermal motion and an
applied electric field.
Mobility (cont.)

The momentum is applied to an electron is given by qEC, and momentum gained is mnvn. Thus, using
physics conservation of energy, electron drift velocity:
vn    n E


(2)
Note that: vn is proportional to E
The proportionality factor may be written as
n
q C

mn
(3)
• The proportionality factor also called electron mobility.
• A similar expression may be written for holes in valence
band may be written as: vp = p E
• Mobility is very important parameter for carrier transport – it
describes how strongly the motion of an electron is influenced
by an applied electric field.
Mobility (cont.)





From eq. (3), mobility is related directly to mean free
time between collisions determined by the various
scattering mechanism.
Two MOST important mechanisms: lattice scattering
and impurity scattering.
Lattice scattering – results from thermal vibrations of
the lattice atoms at any temperature, T>0K (it
becomes dominant at high temp. – mobility decreases
with increasing temp.) – theoretically mobility due to
lattice scattering L decrease in proportion to T-3/2
Impurity scattering – results when charge carrier
travels past an ionized doping impurity (donor or
acceptor). It depends on Coulomb force interaction.
Impurity scattering depends on total concentration of
ionization impurities (sum of +ve and –ve charge
ions). It becomes less significant at higher
temperatures.
Mobility (cont.)

The probability of a collision taking place in unit
time, 1/C, - the sum of the probabilities of collision
due to the various scattering mechanism:
or
1
C
1



1
 C , lattice
1
L


1
 C , impurity
(4)
1
I
L – lattice scattering mobility
I – impurity scattering mobility
(4a)
Mobility (cont.)




Electron mobility as a function of temp. for Si with 5
different donor concentration is given by Fig. 3.2.
For lightly doping (i.e. 1014 cm-3) – lattice scattering
dominates and mobility decreases as the temp.
increases.
For heavily doping (i.e. 1019cm-3) – at low temp.
impurity scattering is most pronounced. Mobility is
increases as temp. increases.
For a given temp., mobility decreases with increasing
impurity concentration (due to enhanced impurity
scattering).
Lightly doped
Heavily doped
Figure 3.2.
Electron mobility in silicon versus temperature for various donor
concentrations. Insert shows the theoretical temperature
dependence of electron mobility.
Mobility (cont.)



Mobility reaches a
maximum value at low
impurity concentrations
corresponds to the lattice
scattering limitation.
Both electron & hole
mobility decrease with
increasing impurity
concentration.
Mobility of electrons is
greater than holes due to
the smaller effective
mass of electrons.
Figure 3.3.
Mobility and diffusivities in Si and
GaAs at 300 K as a function of impurity concentration.
Resistivity





Refer to Fig. 3.4.
3.4(a) – n-type s/c & its band diagram at thermal
equilibrium.
3.4(b) – when biasing voltage is applied at right-handterminal.
Assume that contact at both terminals are ohmic (there
is negligible voltage drop at each of the contacts).
When E (electric field) is applied to s/c, each electron
may experience a force of –qE. Thus, the force is equal
to the negative gradient of the potential energy:
dEC
 qE  (gradient of electron potential energy, U)  
dx
EC – conduction band energy
(5)
Resistivity (cont.)
Figure 3.4. Conduction process in n-type semiconductor (a) at thermal
equilibrium and (b) under biasing condition.

In the gradient of U, any part of the band diagram
that is parallel to EC (e.g EF, Ei, and EV) may be used.
But it’s convenient to use intrinsic Fermi level Ei
(when consider p-n junction in Chapter 4). From (5):
d i
E
dx
(6)
where  - electrostatic potential, and defined as
 
Ei
q
which represents the relationship between electrostatic
potential and potential energy, U.
(7)




For homogenous s/c (Fig. 3.4(b)) – U and Ei
decrease linearly with distance, thus electric field
constant –ve x-direction.
Electrons in conduction band move to the right –
electron undergoes a collision, loses some or all
of its K.E to the lattice & drops toward its thermal
equilibrium position – this process will be
repeated many times.
Hole behaves in the same manner but in the
opposite direction.
Transport of carriers under applied electric field –
drift current.
• From Fig. 3.5, with application of electric field, current density for
both electron and hole, Jk may be written as
k
Ik
Jk 
  ( yqvi )  yqkvk  qk k E
A i 1
where for electron, k = n, y= -1; and hole, k=p, y=1
Figure 3.5. Current conduction in a uniformly doped semiconductor bar with
length L and cross-sectional area A.
(8)

Total current following in s/c sample is sum of the
electron and hole components, which is
J  Jn  J p
 q(n n  p p ) E

From (9), conductivity  = q(nn + pp). Thus,
resistivity of semiconductor is given by


(9)
1
q(n n  p p )
(10)
For extrinsic s/c, generally may be written as
1

qk k
(11)
For n-type (n>>p), k=n, and p-type (p>>n), k=p
EXAMPLE 1

The hole concentration is given by
  x  -3
p( x)  1015 exp 
cm
 L 
(x  0)
where L = 12m. The hole diffusion coefficient is
Dp= 12cm2/s. Calculate the hole diffusion current
density versus x
In practical, to measure resistivity –
commonly used the four-point probe
method (Fig. 3.6)

With thickness, W <<
d, thus
resistivity is given by


VW (CF )
  cm
I
(12)
where CF ~ ‘correction factor’ and it
depends on the ratio of d/s,
s – probe spacing.
Figure 3.6. Measurement of
resistivity
using a four-point probe.
Room
temperature
Figure 3.7. Resistivity versus impurity concentration3 for Si and GaAs.
EXAMPLE 2

Find the resistivity of intrinsic Si doped with n =
1450, p = 505, and n = p = ni = 9.65109 at T =
300K.
THE HALL EFFECT


The “Hall effect” was discovered
in 1879 by the American
physicist, Edwin Hall (1855 –
1938). He discovered the "Hall
effect" while working on his
doctoral (PhD) thesis in Physics.
In 1880, full details of Hall's
experimentation with this
phenomenon formed his doctoral
thesis and was published in the
American Journal of Science and
in the Philosophical Magazine.
THE HALL EFFECT




Hall effect is used to measure the carrier concentration.
It is also one of the most convincing methods to show the
existence of holes as charge carriers – measurement can give
directly the carrier type.
Fig. 3.8 show the Hall effect set-up (consider a p-type
sample). Using Lorentz force F = qv x B = qvxBz. (B: magnetic
field)
There is no net current flow along y-direction (in steady-state),
thus Ey exactly balances the Lorentz force:
Hall field
 Jp 
Ey  
 qp 
 Bz  RH J p Bz


Hall coefficient
1
RH 
qp
(13)
(14)
Figure 3.8. Basic setup to measure carrier concentration using the Hall effect.

The measurement of the Hall voltage for a known
current and magnetic field yields
J p Bz
1
IB zW
p


qRH
qE y
qVH A
All quantities in RHS can be measured, thus carrier
concentration and carrier type can be obtained directly
from Hall measurement.
RHS: right-hand-side
(15)
EXAMPLE 3


Given Si sample of unknown doping. Hall
measurement provides the following
information: W = 0.05cm, A = 1.6 x 10-3 cm2, I
= 2.5mA, and B = 30 nT (1T = 10-4 Wb/cm2). If
Hall voltage of +10mV is measured, find the Hall
coefficient
Bonus for who solve the three examples
CARRIER DIFFUSION
Diffusion Process


Carriers move from a high concentration region to
low concentration region ~ called diffusion current.
From Fig. 3.9, current density may explain by
mathematical formalism below:
n( l )vth
2
n(l )vth
RHS: F2 
2
LHS: F1 
1 
dn  
dn  
n
(
0
)

l

n
(
0
)

l





2 
dx  
dx  
dn
dn
  vthl
  Dn
(16)
dx
dx
F  F1  F2 
F ~ average electron flow
dn
per unit area.

J

qD
n
n
l ~ mean free path
dx
Dn ~ diffusion coefficient
(17)
Figure 3.9. Electron concentration versus distance; l is the mean free path.
The directions of electron and current flows are indicated by arrows.
EINSTEIN RELATION

Rewrite Eq. (17) using
theorem for equipartition of
energy:
1
1
2
mn vth  kT
2
2
(18)
• Using (3), (16), & (18), Einstein relation
may be written as
 kT 

Dn  vthl   n 
 q 
(relation of diffusivity & mobility)
(19)
DENSITY EQUATIONS
Total current density at any point is the sum of
the drift & diffusion components:

dk
J k  q k kE  yqDk
dx
where k = n, with y=1, and k=p, with y= -1.
•
Total conduction current density is given by
Jcond = Jn + Jp
(20)
GENERATION & RECOMBINATION
TYPES
1. Direct Recombination





For the direct-band gap s/c in thermal equilibrium –
the continuous thermal vibration of lattice atoms –
cause bonds between neighboring atoms to be broken.
Bonds broken cause electron-hole pair.
Carrier generation – electron to make upward
transition to conduction band & leaving a hole in
valence band. It is represented by the generation rate
Gth (number of electron-hole pair generated/cm3/s) –
Fig. 3.10(a).
Recombination – electron makes transition downward
from cond. band. It is represented by recombination
rate Rth (Fig. 3.10(a)).
At thermal equilibrium conduction : Gth = Rth for pn =
ni2 to be maintained.
• The rate of generation & recombination in n-type is
Gth  Rth  nno pno
(21)
• When we shine a light, it produced electron-hole pair at a
rate GL, recombination and generation rate
R   (nno  n)( pno  p)
G  GL  Gth
(22)
(23)
n  nn  nno ; p  pn  pno
nno & pno – electron and hole
densities
 - proportionality constant
Figure 3.10. Direct generation and
recombination of electron-hole pairs: (a)
at thermal equilibrium and (b) under
illumination.
• The net change of hole concentration is given by
dpn
 G  R  GL  Gth  R
dt
(24)
At steady-state, dpn/dt = 0;
GL  R  Gth  U
(25)
And at low level injection, pno << nno, the net recombination is
U 
pn  pno
p
; p
1

nno
p – lifetime of the excess minority carriers.
U is net recombination, defined as
U = (nno + pno + ∆p)∆p
(26)
• From (25) & (26) (in steady-state),
generation rate is given by
GL  U 
pn  pno
p
(27)
and, p  pno   p GL
(28)
• When the light is turn off, t = 0, the
boundary cond. pn(0)Eq. (28), and
pn()  pno, thus
 t 
Figure 3.11.
pn (t )  pno   p GL exp   (29)
 
Decay of photo excited carriers.
 p a) n-type sample under constant illumination.
(b) Decay of minority carriers (holes) with time.
(c) Schematic setup to measure minority carrier lifetime.
GENERATION & RECOMBINATION
2. Indirect Recombination
The derivation of the
recombination rate is more
complicated.
 Et – called the intermediatelevel states.
 There are 4 basic transitions
takes place.
 Example of the indirect-band
gap s/c – Si.
 After indirect recombination
process:
(i) Electron capture
(ii) Electron emission
(iii) Hole capture
(iv) Hole emission

Figure 3.12.
Indirect generation-recombination
processes at thermal equilibrium.

The recombination rate is given by
U 


 p  pn  ni exp C    n nn  ni exp( C )
(30)
Under low-injection condition in a n-type, so nn >> pn
, then (30) can be written as
U  vth o N t


vth n p N t pn nn  ni2
pn  pno
p  pno
 n
p
 2n 
1   i  cosh( C )
 nno 
(31)
where, vth – thermal velocity, Nt – concentration of the
recombination centre,  - capture cross section
(effectiveness of the centre to capture an electron or hole),
and
Ei  Et
C
kT
3. Surface Recombination




A large number of localize energy states (generationrecombination centers) may introduced at the surface region.
(Fig. 3.13).
It may enhance the recombination rate at the surface region
by an energy called surface-state.
The kinetics of the surface recombination are similar to those
in bulk centers.
Total number of carrier recombining at the surface per unit
area and unit time:
U s  vth p N st ( ps  pno )
(32)
And, the low-injection surface recombination velocity is defined as:
Slr  vth p N st
(33)
where, ps – concentration at surface, Nst – recombination center
density per unit area in the surface region.
Figure 3.13. Schematic diagram of bonds at a clean semiconductor surface.
The bonds are anisotropic and differ from those in the bulk.
4. Auger Recombination



Occurs by the transfer of the energy
& momentum released by the
recombination of electron-hole pair to
3rd particle (either electron or hole).
Example shown in Fig. 3.14, the 2nd
electron absorb the energy released
by direct recombination – becomes an
energetic electron.
It’s very important – carrier
concentration is very high (results
from high doping or high injection
level). The rate of this recombination
can be expressed as
RAug  Bn 2 p or RAug  Bnp 2
B – proportionality constant (strong
temperature depending)
(34)
Figure 3.14. Auger recombination.
Summary of Part 1





In part 1 of carrier transport phenomena, various
temperature process include drift, diffusion, generation,
and recombination.
Carrier drift – under influence of an electric field. At low
field, drift velocity is proportional to electric field called
Mobility.
Carrier diffusion – under influence of carrier
concentration gradient.
Total current = (drift + diffusion) components.
Four types of recombination process:
(i) Direct
(ii) Indirect
(iii) Surface
(iv) Auger
"Science is a powerful instrument.
How it is used, whether it is a blessing
or a curse to mankind, depends on
mankind and not on the instrument. A
knife is useful, but it can also kill."
Albert Einstein