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Acrobat Distiller, Job 4

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EE130 Lecture 5, Slide 1Spring 2003

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

EE130 Lecture 5, Slide 2Spring 2003

Dominant scattering mechanisms:

1. Phonon scattering (lattice scattering)

2. Impurity (dopant) ion scattering

2/3 2/1 1 velocityermalcarrier thdensityphonon 1 ?×?×??TTT phononphonon Phonon scattering mobility decreases when Tincreases:

μ= qτ/ m

Mechanisms of Carrier Scattering

Tv th 2

EE130 Lecture 5, Slide 3Spring 2003

_

ElectronBoron Ion

Electron

Arsenic

Ion

DADAth

impurity NNT NNv 2/33 There is less change in the electron's direction of travel if the electron zips by the ion at a higher speed.

Impurity Ion Scattering

EE130 Lecture 5, Slide 4Spring 2003

Temperature Effect on Mobility

impurityphononimpurityphonon

111111

3

EE130 Lecture 5, Slide 5Spring 2003

Consider a Si sample doped with 10

17 cm -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 μ

n . From the mobility vs.temperature curve for 10 17 cm -3 , we find that μ n decreases from 770 at 300K to

400 at 400K. As a result, ρincreases by

Example: Temperature Dependence of ρ

93.1400770=

EE130 Lecture 5, Slide 6Spring 2003

Particles diffuse from regions of higher concentration to regions of lower concentration region, due to random thermal motion.

Diffusion

4

EE130 Lecture 5, Slide 7Spring 2003

dxdnqDJ N diffN, dxdpqDJ P diffP,

D is the diffusion constant, or diffusivity.

xx

Diffusion Current

EE130 Lecture 5, Slide 8Spring 2003

J N = J

N,drift

+ J

N,diff

= qnμ n dxdnqD N J P = J

P,drift

+ J

P,diff

= qpμ p dxdpqD P J= J N + J P

Total Current

5

EE130 Lecture 5, Slide 9Spring 2003

Band Diagram: Potential vs.Kinetic Energy

electron kinetic energy increasing electron energy E c E v hole kinetic energy increasing hole energy referencec

P.E.EE-=

E c represents the electron potential energy:

EE130 Lecture 5, Slide 10Spring 2003

N- +- 0.7 V Si E V x 0.7V x 0

Electrostatic Potential V

• Potential energy of a -qcharged particle is related to the electrostatic potential V(x): )(1 creference

EEqV-=

qV-=P.E. 6

EE130 Lecture 5, Slide 11Spring 2003

N- +- 0.7 V Si E V x 0.7V x 0

Electric Field

dxdE qdxdV c 1=-= • Variation ofE c with position is called "band bending."

EE130 Lecture 5, Slide 12Spring 2003

Non-Uniformly-Doped Semiconductor

• The position of E F relative to the band edges is determined by the carrier concentrations, which is determined by the dopant concentrations. •In equilibrium, E F is constant; therefore, the band energies vary with position: E v (x) E c (x) E F 7

EE130 Lecture 5, Slide 13Spring 2003

• In equilibrium, there is no net flow of electrons or holes

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.)

0=+=dxdnqDqnJ

NnN J N = 0 and J P = 0

EE130 Lecture 5, Slide 14Spring 2003

n-type semiconductor

Decreasing donor concentration

dxdEekTN dxdn ckTEEcFc dxdE kTn c kTEE c Fc eNn Consider a piece of a non-uniformly doped semiconductor: E v (x) E c (x) E F qkTn-= 8

EE130 Lecture 5, Slide 15Spring 2003

0=+=dxdnqDqnJ

NnN

Under equilibrium conditions,J

N = 0 andJ P = 0 kTqDqnqn N n -=μ0 n qkTD N

Similarly,

Einstein Relationship between Dand μ

p qkTD P Note: The Einstein relationship is valid for a non-degenerate semiconductor, even under non-equilibrium conditions

EE130 Lecture 5, Slide 16Spring 2003

What is the hole diffusion constant in a sample of silicon with μ p = 410 cm 2 / V s ?

Solution

Remember:kT/q = 26 mVat room temperature.

/scm 11sVcm 410)mV 26( 2112
P p qkTD

Example: Diffusion Constant

9

EE130 Lecture 5, Slide 17Spring 2003

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: 12 i2i1121 2 i1 i2 i2i1i 2 Fi2i 1 Fi1 i1 i1Fquotesdbs_dbs7.pdfusesText_13