EE105 - Spring 2007Microelectronic Devices and Circuits

Lecture 2Semiconductor Basics

2

Periodic Table of Elements

3

Electronic Properties of Silicon

Silicon is in Group IV (atomic number 14)– Atom electronic structure: 1s22s22p63s23p2

– Crystal electronic structure: 1s22s22p63(sp)4

– Diamond lattice, with 0.235 nm bond length Very poor conductor at room temperature: why?

(1s)2

(2s)2

(2p)6 (3sp)4

Hybridized State

4

The Diamond Structure3sp Tetrahedral Bond

A43.5

A35.2

5

States of an Atom

Quantum Mechanics: The allowed energy levels for an atom are discrete (2 electrons with opposite spin can occupy a state)

When atoms are brought into close contact, these energy levels split

If there are a large number of atoms, the discrete energy levels form a “continuous” band

Ene

rgy

E1

E2

...E3

Forbidden Band Gap

AllowedEnergyLevels

Lattice ConstantAtomic Spacing

6

Silicon

Si has four valence electrons. Therefore, it can form covalent bonds with four of its neighbors.

When temperature goes up, electrons in the covalent bond can become free.

7

Electron-Hole Pair Interaction

With free electrons breaking off covalent bonds, holes are generated.

Holes can be filled by absorbing other free electrons, so effectively there is a flow of charge carriers.

8

Free Electron Density as a Function of Temperature

Eg, or bandgap energy, determines how much effort is needed to break off an electron from its covalent bond.

There exists an exponential relationship between the free-electron density and bandgap energy.

15 3/ 2 32

0 10 3

0 15 3

5.2 10 /

( 300 ) 1.08 10 /

( 600 ) 1.54 10 /

gE

kTi

i

i

n T e electrons cm

n T K electrons cm

n T K electrons cm

9

N Type Doping

If Si is doped with group-V elements such as phosphorous (P) or arsenic (As), then it has more electrons and becomes N type (electron).

Group-V impurities are called Donors

10

P Type Doping

If Si is doped with group-III elements such as boron (B), then it has more holes and becomes P type.

Group-III impurities are called Acceptors

11

Summary of Charge Carriers

12

Thermal Equilibrium (Pure Si)

Balance between generation and recombination determines no = po

Strong function of temperature: T = 300 K

2

10

( )( )

( ) ( )( ) / ( )

( ) 10 -3 cm at 300K

th opt

th

th i

i

G G T GR k n pG R

k n p G Tn p G T k n Tn T

13

Majority Carrier Conc.

= Doping Conc.

Minority Carrier Conc.

(Mass Action Law)

N-Type

P-Type

Mass Action Law

The product of electron and hole densities is ALWAYS equal to the square of intrinsic electron density, regardless of doping levels

2 10 3( 300 , 10 ) K cmo o i ip n n T n

dd NNn 0

aa NNp 0

2

0i

d

np

N

2

0i

a

nn

N

14

Compensated Doping

Si is doped with both donor and acceptor atoms:

– More donors than acceptors: Nd > Na N type

– More acceptors than donors: Na > Nd P type

2

2

i

i

o d a od a

o a d oa d

nn N N p

N N

np N N n

N N

15

First Charge Transportation Mechanism: Drift

The process in which charge particles move because of an electric field is called drift.

Charge particles will move at a velocity that is proportional to the electric field.

Ev

Ev

ne

ph

Mobility

16

Mobility vs. Doping in Silicon at 300K

Typical values 1350450

2

2

V-sec / cm V-sec / cm

n

p

17

Current Flow: General Case

Electric current is calculated as the amount of charge in v meters that passes thru a cross-section if the charge travel with a velocity of v m/s.

I v W h n qI

J v n qWh

18

( )

n n

p p

tot n p

n p

J E n qJ E p qJ E n q E p q

q n p E

Current Flow: Drift

Since velocity is equal to E, drift characteristic is obtained by substituting v with E in the general current equation.

The total current density consists of both electrons and holes.

19

Velocity Saturation

A topic treated in more advanced courses is velocity saturation. In reality, velocity does not increase linearly with electric field. It

will eventually saturate to a critical value.

0

0

0

0

1

1

sat

sat

bE

vb

v EE

v

20

Second Charge Transportation Mechanism: Diffusion

Charge particles move from a region of high concentration to a region of low concentration.

21

Current Flow: Diffusion

Diffusion current is proportional to the gradient of charge (dn/dx) along the direction of current flow.

Total diffusion current density consists of both electrons and holes.

( )

n n

p p

tot n p

dnJ qD

dxdp

J qDdxdn dp

J q D Ddx dx

Diffusion Coefficient

22

Example: Linear vs. Nonlinear Charge Density Profile

Linear charge density profile means constant diffusion current, whereas nonlinear charge density profile means varying diffusion current.

LN

qDdxdn

qDJ nnn dd

nn L

xL

NqDdxdn

qDJ exp

23

Einstein's Relation

While the underlying physics behind drift and diffusion currents are totally different, Einstein’s relation provides a link between the two.

p

qkTD

24

Resistivity of Uniformly Doped Si

1 1

n n

n

n

J E n q E

nq

nq

1

Ohm's LawV R IV E LI J tW

I V EL LJ E E

A RtW RtW RtWL L

RtW tW

25

Sheet Resistance (Rs)

IC resistors have a specified thickness – not under the control of the circuit designer

Eliminate thickness, t, by absorbing it into a new parameter: the sheet resistance (Rs)

S

L L LR R

Wt t W W

“Number of Squares”

26

Using Sheet Resistance (Rs)

Ion-implanted (or “diffused”) IC resistor

27

Idealizations

Why does current density Jn “turn”?

What is the thickness of the resistor? What is the effect of the contact regions?