President University Erwin Sitompul SDP 4/1
Lecture 4Semiconductor Device Physics
Dr.-Ing. Erwin SitompulPresident University
http://zitompul.wordpress.com
2 0 1 3
President University Erwin Sitompul SDP 4/2
Ev
Ec
Ec
EvGaAs, GaN(direct semiconductors)
Si, Ge(indirect
semiconductors)
Photon PhotonPhonon
Direct and Indirect SemiconductorsChapter 3 Carrier Action
E-k Diagrams
• Little change in momentum is required for recombination
• Momentum is conserved by photon (light) emission
• Large change in momentum is required for recombination
• Momentum is conserved by mainly phonon (vibration) emission + photon emission
President University Erwin Sitompul SDP 4/3
0nnn
0ppp
, 0n p
Equilibrium valuesDeviation from
equilibrium values
Excess Carrier ConcentrationsChapter 3 Carrier Action
Positive deviation corresponds to a carrier excess, while negative deviation corresponds to a carrier deficit.
Values under arbitrary conditions
pn Charge neutrality condition:
, 0n p
President University Erwin Sitompul SDP 4/4
Often, the disturbance from equilibrium is small, such that the majority carrier concentration is not affected significantly.
However, the minority carrier concentration can be significantly affected.
For an n-type material
For a p-type material
0 0, p n n n
0 0, n p p p
“Low-Level Injection”Chapter 3 Carrier Action
This condition is called “low-level injection condition”.The workhorse of the diffusion in low-level injection condition is
the minority carrier (which number increases significantly) while the majority carrier is practically undisturbed.
0p p
0n n
President University Erwin Sitompul SDP 4/5
p TR
p c N pt
G G-equilibrium
p pt t
NT : number of R–G centers/cm3
Cp : hole capture coefficient
Indirect Recombination RateChapter 3 Carrier Action
Suppose excess carriers are introduced into an n-type Si sample by shining light onto it. At time t = 0, the light is turned off. How does p vary with time t > 0?
Consider the rate of hole recombination:
In the midst of relaxing back to the equilibrium condition, the hole generation rate is small and is taken to be approximately equal to its equilibrium value:
R-equilibrium
pt
p T 0c N p
President University Erwin Sitompul SDP 4/6
R G R G
p p pt t t
p TR G p
p pc N pt
n TR G n
n nc N nt
pp T
1c N
where
where nn T
1 c N
Indirect Recombination RateChapter 3 Carrier Action
The net rate of change in p is therefore:
p T p T 0c N p c N p p T 0c N p p
• For holes in n-type material
• For electrons in p-type material
Similarly,
President University Erwin Sitompul SDP 4/7
pp T n T
1 1 nc N c N
Minority Carrier LifetimeChapter 3 Carrier Action
The minority carrier lifetime τ is the average time for excess minority carriers to “survive” in a sea of majority carriers.
The value of τ ranges from 1 ns to 1 ms in Si and depends on the density of metallic impurities and the density of crystalline defects.
The deep traps originated from impurity and defects capture electrons or holes to facilitate recombination and are called recombination-generation centers.
President University Erwin Sitompul SDP 4/8
PhotoconductorChapter 3 Carrier Action
Photoconductivity is an optical and electrical phenomenon in which a material becomes more electrically conductive due to the absorption of electro-magnetic radiation such as visible light, ultraviolet light, infrared light, or gamma radiation.
When light is absorbed by a material like semiconductor, the number of free electrons and holes changes and raises the electrical conductivity of the semiconductor.
To cause excitation, the light that strikes the semiconductor must have enough energy to raise electrons across the band gap.
President University Erwin Sitompul SDP 4/9
16 30 10 cmp
p n
Example: PhotoconductorChapter 3 Carrier Action
Consider a sample of Si at 300 K doped with 1016 cm–3 Boron, with recombination lifetime 1 μs. It is exposed continuously to light, such that electron-hole pairs are generated throughout the sample at the rate of 1020 per cm3 per second, i.e. the generation rate GL = 1020/cm3/s.
a) What are p0 and n0?
2i
00
nnp
210
16
10
10 4 310 cm
b) What are Δn and Δp?
LG 20 610 10 14 310 cm• Hint: In steady-state
(equilibrium), generation rate equals recombination rate
President University Erwin Sitompul SDP 4/10
Example: PhotoconductorChapter 3 Carrier Action
Consider a sample of Si at 300 K doped with 1016 cm–3 Boron, with recombination lifetime 1 μs. It is exposed continuously to light, such that electron-hole pairs are generated throughout the sample at the rate of 1020 per cm3 per second, i.e. the generation rate GL = 1020/cm3/s.
c) What are p and n?
d) What are np product?
• Note: The np product can be very different from ni
2 in case of perturbed/agitated semiconductor
0p p p 16 1410 10 16 310 cm
0n n n 4 1410 10 14 310 cm
16 1410 10np 30 310 cm 2in
President University Erwin Sitompul SDP 4/11
2i
R G R G p 1 n 1( ) ( )n npp n
t t n n p p
• ET : energy level of R–G center
Net Recombination Rate (General Case)Chapter 3 Carrier Action
For arbitrary injection levels and both carrier types in a non-degenerate semiconductor, the net rate of carrier recombination is:
T i( )1 i E E kTn n e
i T( )1 i
E E kTp n e
where
President University Erwin Sitompul SDP 4/12
JN(x) JN(x+dx)
dx
Area A, volume A.dx
N N1 ( ) ( )nAdx x dx x A
t q
J J
Continuity EquationChapter 3 Carrier Action
Consider carrier-flux into / out of an infinitesimal volume:
Flow of current
Flow of electron
President University Erwin Sitompul SDP 4/13
NN N
( )( ) ( ) xx dx x dxx
JJ J
N
thermal other R G processes
( )1 xn n nt q x t t
J
N ( )1 xnt q x
J
Continuity EquationChapter 3 Carrier Action
• Taylor’s Series Expansion
P
thermal other R G processes
( )1 xp p pt q x t t
J
The Continuity Equations
President University Erwin Sitompul SDP 4/14
Minority Carrier Diffusion EquationChapter 3 Carrier Action
The minority carrier diffusion equations are derived from the general continuity equations, and are applicable only for minority carriers.
Simplifying assumptions:The electric field is small, such that:
N n N Nn nq n qD qDx x
EJ
P p P Pp pq p qD qDx x
J E
• For p-type material
• For n-type material
Equilibrium minority carrier concentration n0 and p0 are independent of x (uniform doping).
Low-level injection conditions prevail.
President University Erwin Sitompul SDP 4/15
NL
n
( )1 xn n Gt q x
J
0 0N L
n
( ) ( )1 n n n n nqD Gt q x x
2
N L2n
n n nD Gt x
2
P L2p
p p pD Gt x
Minority Carrier Diffusion EquationChapter 3 Carrier Action
Starting with the continuity equation for electrons:
Therefore
Similarly
thermal nR G
n nt
Lother processes
n Gt
President University Erwin Sitompul SDP 4/16
2p p p
N L2n
n n n
D Gt x
Carrier Concentration NotationChapter 3 Carrier Action
The subscript “n” or “p” is now used to explicitly denote n-type or p-type material.
pn is the hole concentration in n-type materialnp is the electron concentration in p-type material
Thus, the minority carrier diffusion equations are:
2n n n
P L2p
p p pD Gt x
• Partial Differential Equation (PDE)!
• The so called “Heat Conduction Equation”
President University Erwin Sitompul SDP 4/17
p n0, 0n pt t
2 2p n
N P2 20, 0n pD Dx x
p n
n p
0, 0n p
L 0G
Simplifications (Special Cases)Chapter 3 Carrier Action
Steady state:
No diffusion current:
No thermal R–G:
No other processes:
• Solutions for these common special-case diffusion equation are provided in the textbook
President University Erwin Sitompul SDP 4/18
2n n n
2 2P p P
p p px D L
P P pL D
2n n
P 2p
0 p pDx
n n0 (0)p p
N N nL D Similarly,
Minority Carrier Diffusion LengthChapter 3 Carrier Action
Consider the special case:Constant minority-carrier (hole) injection at x = 0Steady state, no light absorption for x > 0
L 0 for 0G x
The hole diffusion length LP is defined to be:
President University Erwin Sitompul SDP 4/19
2n n
2 2P
p px L
P P
n ( ) x L x Lp x Ae Be
n ( ) 0p
n n0(0)p p
Pn n0( ) x Lp x p e
Minority Carrier Diffusion LengthChapter 3 Carrier Action
The general solution to the equation is:
A and B are constants determined by boundary conditions:
Therefore, the solution is:
0B
n0 A p
• Physically, LP and LN represent the average distance that a minority carrier can diffuse before it recombines with a majority carrier.
President University Erwin Sitompul SDP 4/20
2p 437 cm V s
P pkTDq
P P pL D
Example: Minority Carrier Diffusion LengthChapter 3 Carrier Action
Given ND=1016 cm–3, τp = 10–6 s. Calculate LP.
225.86 mV 437 cm V s 211.3cm s
2 611.3cm s 10 s 33.361 10 cm
= 33.61 m
From the plot,
President University Erwin Sitompul SDP 4/21
F i i F( ) ( ) i i, E E kT E E kTn n e p n e
N i( )i
F E kTn n e i P( )i
E F kTp n e
P ii
ln pF E kTn
N i
i
ln nF E kTn
Quasi-Fermi LevelsChapter 3 Carrier Action
Whenever Δn = Δp ≠ 0 then np ≠ ni2 and we are at non-
equilibrium conditions. In this situation, now we would like to preserve and use the
relations:
On the other hand, both equations imply np = ni2, which does
not apply anymore.The solution is to introduce to quasi-Fermi levels FN and FP
such that:
• The quasi-Fermi levels is useful to describe the carrier concentrations under non-equilibrium conditions
President University Erwin Sitompul SDP 4/22
17 30 D 10 cm ,n N
23 3i
00
10 cmnpn
17 14 17 30 10 +10 10 cmn n n
3 14 14 30 10 +10 10 cmp p p
17 14 31 310 10 =10 cmnp
Example: Quasi-Fermi LevelsChapter 3 Carrier Action
Consider a Si sample at 300 K with ND = 1017 cm–3 and Δn = Δp = 1014 cm–3.
• The sample is an n-typea) What are p and n?
b) What is the np product?
President University Erwin Sitompul SDP 4/23
Example: Quasi-Fermi LevelsChapter 3 Carrier Action
Consider a Si sample at 300 K with ND = 1017 cm–3 and Δn = Δp = 1014 cm–3.
c) Find FN and FP?
N i ilnF E kT n n
5 17 10N i 8.62 10 300 ln 10 10F E
0.417 eV
P i ilnF E kT p n
5 14 10i P 8.62 10 300 ln 10 10E F
0.238 eV
N i i Pi i
F E kT E F kTnp n e n e 0.417 0.238
10 100.02586 0.0258610 10e e 311.000257 10
31 310 cm
Ec
Ev
Ei
FP
0.238 eV
FN
0.417 eV
President University Erwin Sitompul SDP 4/24
1.(6.17)A certain semiconductor sample has the following properties:
DN = 25 cm2/s τn0 = 10–6 sDP = 10 cm2/s τp0 = 10–7 s
It is a homogeneous, p-type (NA = 1017 cm–3) material in thermal equilibrium for t ≤ 0. At t = 0, an external light source is turned on which produces excess carriers uniformly at the rate GL = 1020 cm–3 s–1. At t = 2×10–6 s, the external light source is turned off. (a) Derive the expression for the excess-electron concentration as a function of time for 0 ≤ t ≤ ∞.(b) Determine the value of the excess-electron concentration at
(i) t = 0, (ii) t = 2×10–6 s, and (iii) t = ∞.(c) Plot the excess electron concentration as a function of time.
Chapter 3 Carrier Action
Homework 3
2.(4.38)Problem 3.24Pierret’s “Semiconductor Device Fundamentals”.
Due date: 14.10.2013.
Top Related