1

Carrier Action: Motion, Recombination and Generation.

What happens after we figure out how many electrons and holes are

in the semiconductor?

2

Carrier Motion I

Described by 2 concepts:• Conductivity:

(or resistivity: )• Mobility:

Zero Field movement:

Random – over all e-

Thermal – Energy Distribution.

Motion

Electrons are scattered by impurities, defects etc.

What happens when you apply a force?

3

Carrier Motion II

Apply a force: Electrons accelerate:

-n0qEx=dpx/dt {from F=ma=d(mv)/dt}Electrons decelerate too.

• Approximated as a viscous damping force(much like wind on your hand when driving)dpx = -px dt/ {dt = time since last “randomizing

collision” and = mean free time between randomizing collisions.}

Net result: deceration = dpx/dt = -px/

xqEEqF x ˆ

4

Carrier Motion III

Acceleration=Deceleration in steady state. • dpx/dt(accel) + dpx/dt(decel) = 0

• -n0qEx - px/ = 0.

Algebra: • px/n0 = -qEx = <px>

• But

• <px> = mn*<vx> Therefore: xnxm

qx EEv

n

*

Mobility!

** ,pn m

qpm

qn

5

Currents

“Current density” (J) is just the amount of charge passing through a unit area per unit time.

Jx = (-q)(n0)<vx> in C/(s m2) or A/m2

= +(qn0n)Ex for e-’s acting alone.

= n Ex (defining e- conductivity) If both electrons and holes are present:

6

Current, Resistance

How do we find: • current (I)? We integrate J.

• resistance (R)?

• Provided , w, t are all constants along the x-axis.

ttozwtoydydzJI

00

t

w

L

V

E

x

wtL

wtL

L

xtxwdxxR

1

0)()(

)(

7

Mobility changes …

Although it is far too simplistic we use:

n = q/mn* depends upon:

• # of scatter centers (impurities, defects etc.)More doping => lower mobility (see Fig. in books)More defects (worse crystal) => smaller mobility too.

• The lattice temperature (vibrations)Increased temp => more lattice movement => more

scattering => smaller and smaller .

is the “mean free time.”mn* is the “effective mass.” (depends on material)

IncreasingDoping

8

Mobility Changes II

Mobility is also a function of the electric field strength (Ex) when Ex becomes large. (This leads to an effect called “velocity saturation.”)

<vx>

Ex (V/cm)

105 cm/s

106 cm/s

107 cm/s

102 103 104 105 106

elec

trons

hole

s

Vsat

At ~107 cm/s, the carrier KE becomesthe same order of magnitude as kBT.Therefore: added energy tends to warmup the lattice rather than speed up thecarrier from here on out. The velocity becomes constant, it “saturates.”

Here is constant (low fields). Note constant => linear plot.

9

What does Ex do to our Energy Band Diagram?

Drift currents depend upon the electric field. What does an electric field do to our energy band diagrams?

It “bends” them or causes slope in EC, EV and Ei. We can show this.

• Note:Eelectron = Total E

= PE + KEHow much is PE vs. KE???

Eelectron

EC

EV

Eg

e-

h+

10

Energy Band Diagrams in electric fields

EC is the lower edge for potential energy (the energy required to break an electron out of a bonding state.)

Everything above EC is KE then. PE always has to have a

reference! We’ll choose

one arbitrarily for the

moment. (EREF = Constant)

Then PE = EC-EREF

We also know: PE=-qV

Eelectron

EC = PE

EV = PEEg

e-

h+

KE

KE

EREF

PE

11

Energy Band Diagrams in electric fields II

Electric fields and voltages are related by:

E = -V (or in 1-D E=-dV/dx)• So: PE = EC-EREF = -qV or V = -(EC-EREF)/q

• Ex = -dV/dx = -d/dx{-(EC-EREF)/q} or

Ex = +(1/q) dEC/dx

12

Energy Band Diagrams in electric fields III

The Electric Field always points into the rise in the Conduction Band, EC.

What about the Fermi level? What happens to it due to the Electric Field?

Eelectron

EC

EVEg

EREF

Ei

Ex

13

Another Fermi-Level Definition

The Fermi level is a measure of the average energy or “electro-chemical potential energy” of the particles in the semiconductor. THEREFORE:

The FERMI ENERGY has to be a constant value at equilibrium. It can not have any slope (gradients) or discontinuities at all.

The Fermi level is our real-life EREF!

14

Let’s examine this constant EF

Note: If current flows => it is not equilibrium and EF must be changing.

In this picture, we have no connections. Therefore I=0 and it is still equilibrium!

Brings us to a good question:• If electrons and holes are moved

by Ex, how can there be NO CURRENT here??? Won’t Ex move the electrons => current?

The answer lies in the concept of “Diffusion”. Next…

Eelectron

EC

EV

EF

Semiconductor

Ei

Ex

+ V -

LooksN-type

LooksP-type

Ex

15

Diffusion I

Examples: • Perfume, • Heater in the corner (neglecting convection), • blue dye in the toilet bowl.

What causes the motion of these particles?• Random thermal motion coupled with a density

gradient. ( Slope in concentration.)

16

Green dye in a fishbowl … If you placed green dye in a fishbowl, right in the center,

then let it diffuse, you would see it spread out in time until it was evenly spread throughout the whole bowl. This can be modeled using the simple-minded motion described in the figure below. L-bar is the “mean (average) free path between collisions” and the mean free time. Each time a particle collides, it’s new direction is randomly determined. Consequently, half continue going forward and half go backwards.

x

Dye Concentration

0 1 2 3-3 -2 -1l

32

16 16

8 8 8 8

4 4 8 8 4 4

17

Diffusion II

Over a large scale, this would look more like:

t=0

t1

t2

t3

tequilibrium

Let’s look more in depthat this section of the curve.

18

Diffusion III

What kind of a particle movement does Random Thermal motion (and a concentration gradient) cause?

n(x)

x-axis

Bin(1)

Bin(2)

Bin(0)

lx 0 lx 00x

nb1

nb2

nb0It causes net motion from large concentration regions to small concentration regions.

Line with slope:

Half of e- go lefthalf goright.

0

12xdx

dnl

nn bb

19

Diffusion IV

Net number of electrons crossing x0 is:

• Number going right: 0.5*nb1*ℓ*A

• Minus Number going left: 0.5*nb2*ℓ*A

• Net is = 0.5*ℓ*A*(nb1-nb2)

• (note ℓ*A=volume of a bin.)

Flux = # of particles crossing a plane per unit time and unit area. Symbol is:

= 0.5*ℓ*A*(nb1-nb2) ( = mean free time.)

*A

Or = 0.5*ℓ (nb1-nb2)

20

Diffusion V

Using the fact that slope (dn/dx) = -(nb1-nb2)/ℓ gives:

= - 0.5*ℓ2 dn or = -Dn*dn/dx (electrons) dx

or = -Dp*dp/dx (holes)

Now: When charges move we get current. Consequently, the current density is directly related to the particle flux. The equations are: • (electrons) (holes)

)3(

)1(

DnqD

DqD

qJ

n

dxdn

n

nn

)3(

)1(

DpqD

DqD

qJ

p

dxdp

p

pp

21

Diffusion VI

Let’s look at an example: n(x)

J(x)

x

x

dn/dx = 0 here

The electrons are diffusingout of the center and toward the edges.

22

Currents round-up

So now we know that our total currents have 2 components: • DRIFT – due to any electric field we apply• DIFFUSION – due to any (dp/dx, dn/dx) we

apply and thermal motion.

nptotal JJJ

23

Answering that old question

How can we have an electric

Field and still have no current?

(Still have J = 0?)

Diffusion must balance Drift!Example:

Eelectron

EC

EV

EF

Semiconductor

Ei

Ex

+ V -

LooksN-type

LooksP-type

Ex

24

Einstein Relationship

We next remember: p=niexp((Ei-EF)/kBT)Plugging this into our equation for the electric

field and noting that dEF/dx = 0 … we get The Einstein Relationships.

These are very useful. You will never find a table for both Dp and p as a result of these. Once you have , you have D too, by this relationship.

qTkD

qTkD

B

n

nB

p

p and

25

A sanity check

Pretend we have: What will be the fluxes and currents?

x

Ex

n(x)

p(x)

Holes Mechanism Electrons

DiffusionFlux ()

Current Density (J)

DriftFlux ()

Current Density (J)

26

Recombination – Generation I

Generation (G): How e- and h+ are produced or created.

Recombination (R): How e- and h+ are destroyed or removed

At equilibrium: r = g andsince the generation

rate is set by the temperature, we write it as: r = gthermal

The concepts are visually seen in the energy band diagram below.

EC

EV

Ee

x

G R

hv hv

27

Recombination – Generation II

Recombination must depend upon• the # of electrons: no

• the # of holes: po

(If no e- or h+, nothing can recombine!)

From the chemical reaction• e- + h+ → Nothing

we can know that• r = αrnopo = αrni

2 = gthermal

When the temperature is raised• gthermal increases

Therefore

• ni must increase too!

The recombination “rate coefficient”

28

Recombination – Generation III

A variety of recombination mechanisms exist:

EC

EV

x

G R

hv hvEC

EV

x

G R

EC

EV

x

G R

Ee

Ee

Ee

Direct, Band to Band Auger

Indirect via R-G centers

R-G Center Energy Level

29GaAs band structure produced by J. R. Chelikowsky and M. L. Cohen, Phys. Rev. B 14, 556 (1976)

using an empirical Pseudo-potential method see also: Cohen and Bergstrasser, Phys. Rev. 141, 789 (1966).

Eg – TheBand Gap

Energy

GaAs is a Direct

Band GapSemiconductor

Directrecombinationof electronswith holes occurs. The electrons fallfrom the bottomof the CB to theVB by givingoff a photon!

30GaAs band structure produced by W. R. Frensley, Professor of EE @ UTD

using an empirical Pseudo-potential method see also: Cohen and Bergstrasser, Phys. Rev. 141, 789 (1966).

31Silicon band structure produced by J. R. Chelikowsky and M. L. Cohen, Phys. Rev. B 14, 556 (1976)

using an empirical Pseudo-potential method see also: Cohen and Bergstrasser, Phys. Rev. 141, 789 (1966).

Eg – TheBand Gap

Energy

Si is an Indirect

Band GapSemiconductor

Only indirectrecombinationof electronswith holes occurs. The electrons fallfrom the bottomof the CB into an R-G centerand from theR-G center to theVB. No photon!

32Silicon band structure produced by W. R. Frensley, Professor of EE @ UTD

using an empirical Pseudo-potential method see also: Cohen and Bergstrasser, Phys. Rev. 141, 789 (1966).