Lecture 3 Semiconductor Physics (II) Carrier Transport

Outline • Thermal •Carrier Drift •Carrier

Reading Assignment: Howe and Sodini; Chapter 2, Sect. 2.4-2.6

6.012 Spring 2007 Lecture 3 1 1. Thermal Motion

In thermal equilibrium, carriers are not sitting still: • Undergo collisions with vibrating Si (Brownian motion) • Electrostatically interact with each other and with ionized (charged) dopants

Characteristic time constant of thermal motion: ⇒ mean free time between collisions

τc ≡ collison time [s] In between collisions, carriers acquire high velocity:

−1 vth ≡ thermal velocity [cms ] …. but get nowhere!

6.012 Spring 2007 Lecture 3 2 Characteristic of thermal motion:

λ≡mean free path [cm]

λ = vthτc

Put numbers for Si at room temperature: −13 τc ≈ 10 s 7 −1 vth ≈ 10 cms ⇒ λ ≈ 0.01 µm

For reference, state-of-the-art production MOSFET:

Lg ≈ 0.1 µm ⇒ Carriers undergo many collisions as they travel through devices

6.012 Spring 2007 Lecture 3 3 2. Carrier Drift

Apply to semiconductor:

E ≡ electric field [V cm-1]

⇒ net force on carrier

F = ±qE

E

Between collisions, carriers accelerate in the direction of the electrostatic field:

qE v(t) = a •t =± t mn,p

6.012 Spring 2007 Lecture 3 4 But there is (on the average) a collision every τc and the velocity is randomized:

net velocity τc in direction of field

time The average net velocity in direction of the field:

qE qτc v = vd =± τ c =± E 2mn,p 2mn,p

This is called drift velocity [cm s-1]

Define: qτ c 2 −1 −1 µn,p = ≡ mobility[cm V s ] 2mn,p

Then, for :

vdn = −µnE and for holes: vdp = µpE

6.012 Spring 2007 Lecture 3 5 Mobility - is a measure of ease of carrier drift

• If τc ↑, longer time between collisions ⇒ µ ↑ • If m ↓, “lighter” particle ⇒ µ ↑ At room temperature, mobility in Si depends on doping:

1400

1200

electrons 1000 Vs) / 2 800

600 mobility (cm holes 400

200

0 1013 1014 1015 1016 1017 1018 1019 1020 + −3 Nd Na total dopant concentration (cm )

• For low doping level, µ is limited by collisions with lattice. As Temp ->INCREASES; µ-> DECREASES • For medium doping and high doping level, µ limited by collisions with ionized impurities • Holes “ heavier” than electrons

– For same doping level, µn > µp

6.012 Spring 2007 Lecture 3 6

Net velocity of charged particles ⇒ :

Drift current ∝ carrier drift velocity ∝ carrier concentration ∝ carrier charge

Drift current : drift Jn =−qnvdn = qnµ nE drift J p = qpvdp = qpµ p E

Check signs: E E

vdn vdp - +

drift drift Jn Jp

x x

6.012 Spring 2007 Lecture 3 7 Total Drift :

drift drift drift J = Jn + Jp = qn( µ n + pµ p )E

Has the form of Ohm’s Law E J = σ E = ρ

Where:

σ ≡ conductivity [Ω-1 • cm-1] ρ≡resistivity [Ω • cm]

Then: 1 σ = = qnµ + pµ ρ ( n p )

6.012 Spring 2007 Lecture 3 8 Resistivity is commonly used to specify the doping level

• In n-type semiconductor: 1 ρn ≈ qNdµn • In p-type semiconductor: 1 ρp ≈ qNaµ p

1E+4

1E+3

1E+2

1E+1 p-Si

1E+0 n-Si 1E-1 Resistivity (ohm.cm) 1E-2

1E-3

1E-4 1E+12 1E+13 1E+14 1E+15 1E+16 1E+17 1E+18 1E+19 1E+20 1E+21 Doping (cm-3)

6.012 Spring 2007 Lecture 3 9 Numerical Example:

16 -3 Si with Nd = 3 x 10 cm at room temperature 2 µn ≈ 1000 cm / V • s ρn ≈ 0.21Ω•cm n ≈ 3X1016 cm−3 Apply E = 1 kV/cm 6 vdn ≈10 cm / s << vth E J drift ≈ qnv = qnµ E = σ E = n dn n ρ drift 3 2 J n ≈ 4.8×10 A/ cm

Time to drift through L = 0.1 µm L td = = 10 ps vdn fast!

6.012 Spring 2007 Lecture 3 10 3. Carrier Diffusion

Diffusion = particle movement (flux) in response to concentration gradient n

x Elements of diffusion:

• A medium (Si Crystal) • A gradient of particles (electrons and holes) inside the medium • Collisions between particles and medium send particles off in random directions – Overall result is to erase gradient

6.012 Spring 2007 Lecture 3 11 Fick’s first law- Key diffusion relationship

Diffusion flux ∝ - concentration gradient

Flux ≡ number of particles crossing a unit area per unit time [cm-2 • s-1]

For Electrons: dn F =−D n n dx

For Holes: dp F =−D p p dx

2 -1 Dn ≡ diffusion coefficient [cm s ] 2 -1 Dp ≡ hole diffusion coefficient [cm s ]

D measures the ease of carrier diffusion in response to a concentration gradient: D ↑⇒Fdiff ↑ D limited by vibration of lattice atoms and ionized dopants.

6.012 Spring 2007 Lecture 3 12 Diffusion Current

Diffusion current density =charge × carrier flux

diff dn Jn = qDn dx dp J diff =−qD p p dx

Check signs:

n p Fn Fp

J Jndiff pdiff x x

6.012 Spring 2007 Lecture 3 13 Einstein relation

At the core of drift and diffusion is same physics: collisions among particles and medium atoms ⇒ there should be a relationship between D and µ

Einstein relation [will not derive in 6.012] D kT = µ q

In semiconductors: D kT D n = = p µn q µ p

kT/q ≡ thermal At room temperature: kT ≈ 25mV q

16 -3 For example: for Nd = 3 x 10 cm 2 2 µn ≈ 1000 cm / V • s ⇒ Dn ≈ 25cm / s 2 2 µ p ≈ 400 cm / V • s ⇒ Dp ≈ 10 cm / s

6.012 Spring 2007 Lecture 3 14 Total Current Density

In general, total current can flow by drift and diffusion separately. Total current density:

dn J = Jdrift + Jdiff = qnµ E + qD n n n n n dx dp J = Jdrift + Jdiff = qpµ E − qD p p p p p dx

Jtotal = Jn + Jp

6.012 Spring 2007 Lecture 3 15 What did we learn today?

Summary of Key Concepts

• Electrons and holes in semiconductors are mobile and charged – ⇒ Carriers of electrical current! • Drift current: produced by electric field dφ Jdrift ∝ EJdrift ∝ dx • Diffusion current: produced by concentration gradient dn dp Jdiffusion ∝ , dx dx • Diffusion and drift currents are sizeable in modern devices • Carriers move fast in response to fields and gradients

6.012 Spring 2007 Lecture 3 16