Transcript Continuity Equation

CHAPTER 3: CARRIER
CONCENTRATION PHENOMENA
Part 2
CHAPTER 3: Part 2
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Continuity Equation
Thermionic Emission Process
Tunneling Process
High-Field Effect
CONTINUITY EQUATION
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Continuity Equation – to consider the overall effect when
drift, diffusion, and recombination occur simultaneously in s/c
material.
Fig. 3.15 – infinitesimal slice with a thickness dx located at x –
it may derive by one dimensional continuity equation.
The number of electron in the slice – increase due to net
current flow into the slice & the net carrier generation in the
slice.
The overall rate of electron increase is algebraic sum of 4
components:
A–B+C–D
where; A = number of electrons flowing into the slice at x
B = number of electrons flowing out at x + dx
C = rate which electrons are generated
D = rate at which they are recombined with holes in
the slice
Figure 3.15. Current flow and generation-recombination processes in an
infinitesimal slice of thickness dx.
CONTINUITY EQUATION (cont…)

The overall rate of change in the number of electrons
in the slice:
 J n ( x) A J n ( x  dx) A 
n
Adx  

 (Gn  Rn ) Adx

t
q
 q


(1)
For 1-D, under low injection condition, the continuity eq. for
minority carriers:
yk
yk
 2 yk
( yk  yko )
E
 zyk  k
 k E
 Dk
 Gk 
2
t
x
x
x
k
(2)
where, y = n, k = p, and y = p, k = n, (i.e np – in p-type s/c,
and pn – n-type s/c.
CONTINUITY EQUATION (cont…)
• In addition to continuity equations, Poisson’s equation:
dE  s

dx
s
(3)
must be satisfied, and s – s/c dielectric permittivity, s – space charge
density, where s = q(p – n + ND+ - NA-)
Continuity Equation (cont…)
Solve the Continuity Equation
Steady-state injection from
one side
Minority carriers at the surface
The Haynes-Shockley Experiment
Steady-State Injection From One Side
n-type semiconductor
• Assume that light is negligibly small, and
assumption of zero field & zero generation at
x > 0.
• At steady state there is a concentration
gradient near surface.
From (2) the diff. equation for minority carriers
inside s/c is
pn
 2 pn ( pn  pno )
 0  Dp

2
t
x
p
Pn: Holes in n-type s/c
(4)
Figure 3.16. Steady-state
carrier injection from one side.
(a) Semi-infinite sample. (b)
Sample with thickness W.
Steady-State Injection From One Side
(cont..)

The solution of pn(x) by considering the boundary conditions, pn(x = 0)
= pn(0) = constant value, and pn (x  ) = pno, thus
pn ( x)  pno   pn (0)  pno exp(  x / L p )
L p  D p p 
(5)
1/ 2
Where,
is called diffusion length.
• With thickness x = W, thus
pn ( x)  pno   pn (0)  pno 

Where,   sinh (W  x ) / L p
(6)

sinh( W / L p )
Current density at x = W is given by
J p  q pn (0)  pno 
Dp
L p sinh( W / L p )
(7)
Minority Carriers at the
Surface
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When surface recombination is introduced (Fig. 3.17), the hole
current density flowing into the surface from the bulk of the s/c. It’s
given by qUs.
Assume that the sample is uniformly illuminated with uniform
generation of carriers.
Surface recombination leads to a lower carrier concentration at the
surface.
The solution of the continuity equation based on boundary condition
(x = 0, and x = ), is
  S exp(  x / L (8)
)
pn ( x)  pno   p GL 1 


p
lr
( L p   p Slr )
p



Graph pn(x) versus x in Fig. 3.17 for a finite Slr. When Slr  , thus,
pn ( x)  pno   p GL 1  exp(  x / L p )
Slr: low injection surface recombination velocity
(9)
Minority Carriers at the Surface (cont.)
Figure 3.17. Surface recombination at x = 0. The minority carrier distribution
near the surface is affected by the surface recombination velocity.
The Haynes-Shockley Experiment
-One of the classic experiments in semiconductor physics to demonstrate drift
and diffusion of minority carriers.
Phys. Rev. Vol 81 pg. 835 (1951)
• The voltage source V1 establishes
an electric field in the +x direction in
the n-type semiconductor bar.
Excess carriers are produced and
effectively injected into the
semiconductor bar at contact (1)
by a pulse.
Without applied field
• Contact (2) may collect a fraction
of the excess carriers as they drift
through the s/c.
Carrier distributions
With applied field
Figure 3.18.
The Hayes-Shockley experiment.
(a) Experimental setup. (b) Carrier
distributions without an applied field. (c)
Carrier distributions with an applied field.
The Haynes-Shockley Experiment
(cont…)

After a pulse, by setting Gp = 0, and E/x = 0 (applied electric is
constant across the conduction bar), thus transport equation is given
by:
p n
p n
 2 p n ( p n  p no )
(10)
And the solution may be written as
  p E
 Dp

2
t
x
p
x
p n ( , t ) 

2
t 
exp 

  p no
4D p t
 4 D p t  p 
N
(11)
For no electric field applied along the sample,  = x, and with electric
field,  = x - pEt . N = number of electrons or holes generated per unit area.
Illustrated by Fig. 3.18(b).
For E = 0, carriers diffuse away from the point of injection and recombine.
For E  0, all excess carriers move with drift velocity pE, and diffuse
outward and recombine as in the field-free case.
Thermionic Emission Process
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At the s/c surface, carriers may recombine with recombination
centers due to the dangling bonds of the surface region.
Thermionic Emission Process – condition where the carriers
have sufficient energy to ‘thermionically’ emitted into the
vacuum.
Fig. 3.19(a) – band diagram of an isolated n-type s/c. q is the
energy difference between the condition band edge & the
vacuum level in the s/c.
qs – work function (energy between Fermi level & vacuum level
in the s/c).
If energy > q - electron can be thermionically emitted into the
vacuum.
• Electron density with energies > q
may be written as
(a)

 q (   Vn ) 
nth   n( E )dE  N C exp  
 (12)
kT 

q
NC – effective density of states in cond. band.
Vn – is the difference between bottom of
cond. band & Fermi level.
Figure 3.19.
(b)
(a) The band diagram of an isolated ntype semi-conductor.
(b) The thermionic emission process.
TUNNELING PROCESS
• Fig. 3.20a – the energy diagram when two
isolated s/c samples are brought close together.
•qV
qVo =two
q. s/c sample and
Distance
between
o respectively.
potential barrier height represents by
d and
• If d<<<, electron at left-side s/c may transport
across the barrier & and move to the other side
(even if electron is << barrier height.)
– called Quantum Tunneling Phenomena.
Figure 3.20.
(a) The band diagram of two isolated
semiconductors with a distance d.
(b) One-dimensional potential barrier.
(c) Schematic representation of the wave
function across the potential barrier.
TUNNELING PROCESS (cont…)
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Classic case: particle is always reflected (if E < qVo).
Quantum case: particle has finite probability to transmit or ‘tunnel’
through the potential barrier.
As usual, the behavior of particle (conduction electron) in the region
with qV(x) = 0 can be described by Schrödinger equation:
 2 d 2

 E
2
2mn dx
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or
2m n
d 2
  2 E
2
dx

(13)
mn – effective mass, ħ - reduced Planck constant, E – kinetic energy,
and  - wave function of the particle.
The solution of (13) are
 ( x)  A exp( jkx)  B exp(  jkx)
 ( x)  C exp( jkx)
Where k = (2mnE/ ħ2)1/2.
for x  0
for x  d
(14)
Schrodinger Equation
Additional
Fact!!
The Schrodinger equation plays the role of Newton’s Law and
conservation of energy in classical mechanics - i.e., it predicts the
future behavior of a dynamic system. It is a wave equation in
terms of the wave function which predicts analytically and precisely
the probability of events or outcome. The detailed outcome is not
strictly determined, but give a large number of events, the
Schrodinger equation will predict the distribution of results.
The kinetic and potential
energies are transformed
into the Hamiltonian which
acts upon the wave
function to generate the
evolution of the wave
function in time and space.
The Schrodinger equation
gives the quantized
energies of the system and
gives the form of the wave
function so that other
properties may be
calculated
TUNNELING PROCESS (cont…)
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For x  0 : incident-particle wave function with amplitude A.
: reflected wave function with amplitude B.
For x  d : transmitted wave function with amplitude C.
Inside the potential barrier, wave function may be written as
 d 2

 qV0  E
2
2mn dx
or
d 2 2mn (qV0  E )


2
2
dx

(15)
The solution of E < qVo:
 ( x)  F exp( x)  G exp(  x)
Where  = {2mn(qVo – E)/ħ2}1/2. (x) illustrated at Fig. 3.20(c).
(16)
TUNNELING PROCESS (cont…)
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Transmission coefficient may be written as

(qV0 sinh d ) 
C 
   1 

4
E
(
qV

E
)
 A
0


2
2
1
(17)
Transmission coefficient decreases as E decreases.
When d >> 1, (C/A) <<< and varies as

C 
  ~ exp( 2d )  exp  2d
 A

2
2mn (qV0  E ) 



* Used for tunneling diodes in Chapter 8
(18)
HIGH-FIELD EFFECT
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At low electric field, drift velocity is proportional to the applied field,
and assume that time interval between collision c is independent to
applied field.
Fig. 3.21 shows the measured drift velocity of electrons and holes in
Si as a function of the electric field.
Drift velocity increases less rapidly when electric field increased. At
large field, the drift velocity approaches a saturation velocity.
Thus, from experimental investigation, it may be expressed by
Drift velocity:
vn , v p 
vs
1  E / E  
 1/ 
(19)
0
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Where vs – saturation velocity (107cm/s for Si at 300K).
E0 – constant field which is 7 x 103 V/cm for electrons and
E0 = 2 x 104V/cm for holes in high-purity Si materials.
 - 2 for electrons and 1 for holes.
vs at high field is particularly likely for FET – discuss more in Chapter
6.
HIGH-FIELD EFFECT (cont…)
Figure 3.21. Drift velocity versus electric field in Si.
HIGH-FIELD EFFECT (cont…)
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Fig 3.22 shows the differentiation between high-field
transport in n-type GaAs and Si.
For n-type GaAs – vs reached maximum level, then decreases
when the field increases. This phenomenon is due to
energy bands structure of GaAs that allows the transfer
of conduction electrons from high mobility energy
minimum (called valley) to low mobility.
Means that, electron transfer from the central valley to the
satellite valleys along [111] direction (discussed in Chapter
2).
HIGH-FIELD EFFECT (cont…)
Figure 3.22. Drift velocity versus electric field in Si and GaAs. Note that for n-type
GaAs, there is a region of negative differential mobility.
HIGH-FIELD EFFECT (cont…)
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Fig. 3.23 gives a clear view of the phenomena in Fig. 3.22,
where it considers the simple two-valley model of n-type
GaAs at various conditions of electric fields.
Energy separated between two-valleys is E = 0.31eV. The
lower valley’s electron effective mass, electron mobility, and
electron density are represented by m1, 1, and n1
respectively. The upper level represents by the same
expression with subscript 2.
HIGH-FIELD EFFECT (cont…)
Figure 3.23. Electron distributions under various conditions of electric fields for a twovalley semiconductor.
HIGH-FIELD EFFECT (cont…)

Total electron concentration is given by n = n1 + n2. The steadystate conductivity of n-type GaAs may be written as
  q(1n1   2 n2 )  qn

The average mobility is
 

Drift velocity may be written as
v s  E



1 n1   2 n2
n1  n2
(20)
(21)
(22)
At Fig. 2.32(a), E << and all electrons remain in the lower valley.
Fig. 2.32(b), E is higher and some electrons gain sufficient
energies from the field to move to the higher valley.
Fig. 2.32(c), E >>, it may transfer all electrons to the higher
valley.
HIGH-FIELD EFFECT (cont…)
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In mathematical view;
For 0 < E < Ea
n1  n
Ea < E < E b n  n  n
1
2
E > Eb
n1  0

The drift velocity:
For 0 < E < Ea

For E > Eb



v n  1 E
vn   2 E
and
and
and
n2  0
n2  n
n2  n
(23)
(24)
If 1E > 2E – there is a region which the vs decreases with an
increasing field at Ea < E < Eb shown in Fig. 3.24.
With this characteristic of n-type GaAs drift velocity – this
materials is used in microwave transferred-electron
devices (discuss in Chapter 8).
HIGH-FIELD EFFECT (cont…)
Figure 3.24. One possible velocity-field characteristic of a two-valley semiconductor.
HIGH-FIELD EFFECT (cont…)
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When E in s/c is increased above a certain value – the carriers
gain enough K.E to generate electron-hole pairs by an
avalanche process shown in Fig. 3.25 (electron in cond. band
represented by 1).
If E >>> - electron can gain K.E before it collides with the
lattice.
On impact with the lattice – electrons imparts most of its K.E to
break a bond – to ionize a valence electron from the valence
band to the cond. band & generate an electron-hole pair
(represented by 2 and 2’).
This process continued to generate another electron-hole pairs
(e.g 3 and 3’, 4 and 4’) and so on. This process called
avalanche process. This process will results in breakdown in
p-n junction (discussed in Chapter 4).
Figure 3.25.
Energy band
diagram for the avalanche
process.
HIGH-FIELD EFFECT (cont…)
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Consider the process of 2 – 2’:
Just prior to the collision, fast moving electron (no. 1) has a K.E = ½
m1vs2, and momentum, p = m1vs, (m1 – effective mass).
After collision, there are 3 carriers: the original electron + electron
hole pair (no.2 and 2’). If we assume the 3 carriers have same
effective mass, same K.E, and same p, thus the total K.E = 3/2
(m1vf2), and total p = 3m1vf.
vf - velocity after collision.
To conserve both energy and momentum before and after the
collision, thus
and
1
3
m1v s2  E g  m1v 2f
2
2
m1v s  3m1v f
respectively.
(25)
(26)
HIGH-FIELD EFFECT (cont…)

Eg – band gap corresponding to the minimum energy required to
generate an electron-hole pair. By substitute (26) into (25), thus the
required K.E for the ionization process may be written as
E0 


1
m1v s2  1.5 E g
2
(27)
E0 > Eg for the ionization process to occur. It depend on the band
structure, where for Si, electron and hole are E0 = 3.6eV (3.2Eg) and
E0 = 5.0eV(4.4Eg) respectively.
The number of electron-hole pairs generated by an electron per unit
distance traveled – ionization rate, where for the electron and hole is
represented by n and p respectively.
HIGH-FIELD EFFECT (cont…)
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Measurement of ionization rates for Si and GaAs are shown in
Fig. 2.36. (n and p are strongly dependent on the electric
field.
For large ionization rate (say 104cm-1), the corresponding
electric field is  3 x 105 V/cm for Si and  4 x 105 V/cm for
GaAs.
Electron-hole pair generation rate GA from the avalanche
process is given by
GA 


n
| J n |  p | J p |
(28)
q
Where Jn and Jp are the electron and hole current densities,
respectively. This expression may be used in the continuity
equation for devices operated under the avalanche condition.
HIGH-FIELD EFFECT (cont…)
Figure 3.26.
Measured ionization
rates versus
reciprocal field for Si
and GaAs.
CONCLUSION
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Excess carriers in s/c cause non-equilibrium condition, where
most of s/c devices operate under this circumstances.
Carriers may be generated by: forward-bias of p-n junction,
incident light, and impact ionization.
Continuity equation – the governing equation for the rate of
charge carriers.
Thermionic emission occurs when carriers in the surface
region gains enough energy to be emitted into vacuum level.
Tunneling process – based on the quantum tunneling
phenomena that results in the transport of electrons across a
potential barrier even if the electron energy is less than the
barrier height.
When the electric field become higher, drift velocity departs
from its linear relationship with the applied field & approaches a
saturation velocity. This phenomena is important in the study of
short-channel field-effect transistor (Chapter 6).
CONCLUSION (cont…)
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When the electric field exceeds a certain value, the carriers
gain enough K.E to generate electron hole-pair by colliding
with the lattice & breaking a bond. This effect particularly
important in the study of p-n junctions.
Impact ionization or avalanche process – high field
accelerates a new electron-hole pairs, which collide with the
lattice to create more electron-hole pairs.
From avalanche process, the p-n junction breaks down and
conducts a large current (you may learn more about this topic
in Chapter 4).
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