展开查看详情
1 . Lecture #5
ANNOUNCEMENT
• Discussion Section 102 (Th 10-11AM) moved to 105 Latimer
OUTLINE
– Mobility dependence on temperature
– Diffusion current
– Relationship between band diagrams & V,
– Non-uniformly doped semiconductor
– Einstein relationship
– Quasi-neutrality approximation
Read: Chapter 3.2
Spring 2003 EE130 Lecture 5, Slide 1
Mechanisms of Carrier Scattering
Dominant scattering mechanisms:
1. Phonon scattering (lattice scattering)
2. Impurity (dopant) ion scattering
Phonon scattering mobility decreases when T increases:
1 1
µ phonon ∝ τ phonon ∝ ∝ ∝ T −3 / 2
phonon density × carrier thermal velocity T × T 1/ 2
µ = qτ / m v th ∝ T
Spring 2003 EE130 Lecture 5, Slide 2
1
2 . Impurity Ion Scattering
Boron Ion Electron
_
- -
Electron +
Arsenic
Ion
There is less change in the electron’s direction of travel
if the electron zips by the ion at a higher speed.
v th3 T 3/2
µ impurity ∝ ∝
NA + ND NA + ND
Spring 2003 EE130 Lecture 5, Slide 3
Temperature Effect on Mobility
1 1 1
= +
τ τ phonon τ impurity
1 1 1
= +
µ µ phonon µ impurity
Spring 2003 EE130 Lecture 5, Slide 4
2
3 .Example: Temperature Dependence of ρ
Consider a Si sample doped with 1017cm-3 As.
How will its resistivity change when the temperature is
increased from T=300K to T=400K?
Solution:
The temperature dependent factor in σ (and therefore
ρ) is µn. From the mobility vs. temperature curve for
1017cm-3, we find that µn decreases from 770 at 300K to
400 at 400K. As a result, ρ increases by
770
= 1.93
400
Spring 2003 EE130 Lecture 5, Slide 5
Diffusion
Particles diffuse from regions of higher concentration
to regions of lower concentration region, due to
random thermal motion.
Spring 2003 EE130 Lecture 5, Slide 6
3
4 . Diffusion Current
dn dp
J N,diff = qDN J P,diff = − qDP
dx dx
x x
D is the diffusion constant, or diffusivity.
Spring 2003 EE130 Lecture 5, Slide 7
Total Current
J = JN + JP
dn
JN = JN,drift + JN,diff = qnµn + qDN
dx
dp
JP = JP,drift + JP,diff = qpµp – qDP
dx
Spring 2003 EE130 Lecture 5, Slide 8
4
5 .Band Diagram: Potential vs. Kinetic Energy
increasing electron energy
electron kinetic energy
increasing hole energy
Ec
Ev
hole kinetic energy
Ec represents the electron potential energy:
P.E. = Ec − Ereference
Spring 2003 EE130 Lecture 5, Slide 9
Electrostatic Potential V
0.7V V(x)
E
0.7V
+ – x
N-Si
0
• Potential energy of a –q charged particle is related to
the electrostatic potential V(x):
P.E. = − qV
1
V= ( Ereference − Ec )
q
Spring 2003 EE130 Lecture 5, Slide 10
5
6 . Electric Field
0.7V
V(x)
E 0.7V
+ – x
N-Si 0
dV 1 dEc
=− =
dx q dx
• Variation of Ec with position is called “band bending.”
Spring 2003 EE130 Lecture 5, Slide 11
Non-Uniformly-Doped Semiconductor
• The position of EF relative to the band edges is
determined by the carrier concentrations, which is
determined by the dopant concentrations.
• In equilibrium, EF is constant; therefore, the band
energies vary with position:
Ec(x)
EF
Ev(x)
Spring 2003 EE130 Lecture 5, Slide 12
6
7 .• In equilibrium, there is no net flow of electrons or holes
JN = 0 and JP = 0
Î The drift and diffusion current components must
balance each other exactly. (A built-in electric field
exists, such that the drift current exactly cancels out the
diffusion current due to the concentration gradient.)
dn
J N = qnµ n + qDN =0
dx
Spring 2003 EE130 Lecture 5, Slide 13
Consider a piece of a non-uniformly doped semiconductor:
n = N c e − ( Ec − EF ) / kT
n-type semiconductor
dn N dE
= − c e −( Ec − EF ) / kT c
Decreasing donor concentration dx kT dx
Ec(x) n dEc
=−
EF kT dx
n
=− q
Ev(x) kT
Spring 2003 EE130 Lecture 5, Slide 14
7
8 . Einstein Relationship between D and µ
Under equilibrium conditions, JN = 0 and JP = 0
dn
J N = qnµ n + qDN =0
dx
qDN kT
0 = qnµ n − qn DN = µn
kT q
kT
Similarly, DP = µp
q
Note: The Einstein relationship is valid for a non-degenerate
semiconductor, even under non-equilibrium conditions
Spring 2003 EE130 Lecture 5, Slide 15
Example: Diffusion Constant
What is the hole diffusion constant in a sample of silicon
with µp = 410 cm2 / V s ?
Solution:
kT
DP = µ p = (26 mV) ⋅ 410 cm 2 V −1s −1 = 11 cm 2 /s
q
Remember: kT/q = 26 mV at room temperature.
Spring 2003 EE130 Lecture 5, Slide 16
8
9 . Potential Difference due to n(x), p(x)
• The ratio of carrier densities (n, p) at two points depends
exponentially on the potential difference between these points:
n n
EF − Ei1 = kT ln 1 => Ei1 = EF − kT ln 1
ni ni
n
Similarly, Ei2 = EF − kT ln 2
ni
n n n
Therefore Ei1 − Ei2 = kT ln 2 − ln 1 = kT ln 2
ni ni n1
1
V2 − V1 = (Ei1 − Ei2 ) = kT ln n2
q q n1
Spring 2003 EE130 Lecture 5, Slide 17
Quasi-Neutrality Approximation
• If the dopant concentration profile varies gradually
with position, then the majority-carrier concentration
distribution does not differ much from the dopant
concentration distribution.
Spring 2003 EE130 Lecture 5, Slide 18
9
10 . Summary
• Carrier mobility varies with temperature
– decreases w/ increasing T if lattice scattering dominant
– decreases w/ decreasing T if impurity scattering dominant
• Electron/hole concentration gradient Æ diffusion
dn dp
J N,diff = qDN J P,diff = −qDP
dx dx
• Current flowing in a semiconductor is comprised of
drift & diffusion components for electrons & holes
J = JN,drift + JN,diff + JP,drift + JP,diff
– In equilibrium, JN = JN,drift + JN,diff = 0
Spring 2003 EE130 Lecture 5, Slide 19
• The characteristic constants of drift and diffusion
are related: D kT
=
µ q
• In thermal equilibrium, EF is constant
• Ec represents the electron potential energy
Variation in Ec(x) Æ variation in electric potential V
dEc dEv
Electric field = =
dx dx
• E - Ec = electron kinetic energy
Spring 2003 EE130 Lecture 5, Slide 20
10
热门城市
