Semiconductor Devices 14

8/7/2019 Semiconductor Devices 14

1/19

Semiconductor Devices - Hour 14 Minority Carriers, Generation and Recombination, "Continuity Equations"

Grand recap of recent relevant lectures:

1) DRIFT = Movement of carriers in an electric field,

v t( ) v t( )

=> => v

average

=t t

q

meff= = average interval between scattering = "Mobility" = velocity / electric field

a) Conventional "Low Field" regime: Mobility vs. Ion Concentration plots

n-Si For "cool" carriers:

Book Figure 5.3

1350

cm

2

volt sec

p-Si400 Link to my better web figure

Total Ion Concentration (1/cm3)

1014

1016

1018

=============================== Test Warning! =======================================

Semiconductor_Devices_14.mcd 1 7/19/2010


2/19

From figure, mobilities are not constant!!!

This terribly misleading table gives "Typical mobility values" = values to left of curves above

They are not "typical" (!$#!@$!)

They are the values you get if and only if the ion concentration (doping) is low!

Book'sTable 5.1

============================= (End Test Warning) ====================================

Curves above identify NO dependence on electric field: is constant = v / So expect v() always

Forever? Will velocity really continue to increase with no matter how strong becomes?

No: At very, very large , carrier scattering gets larger and velocity begins to saturate

Why? Carriers become so energetic that their collisions transfer so much energy to crystal that its atoms

vibrate (dance around) more, getting in the way more => Increased scattering, decreased and

Eventual fall off in mobility (and related phenomena) called "Hot Carrier Effects"



3/19

b) Actual v() plot extending to high fields and hot carriers:

v (cm/sec)

High Field

107

105

(Book Figure 5.7)Low Field10

3

volts

cm

1 102 104 106

Which will we use?

- We will use "low field" vs. ion concentration plots

- Velocity saturation (at high fields) is an issue only in newest, smallest, state-of-the-art devices

So we will continue to use following expressions for field driven DRIFT currents:

Jdrift_p q p p = Jdrift_n q n n =



4/19

2) Second type of current: DIFFUSION current due to spontaneous rearrangement of concentration gradients

n or p concentration

=> Flux Dx

Concentration x( )d

d=

x

D = "Diffusivity" or "diffusion constant" =L2

2 L = average distance traveled between scattering

= average time interval between scattering

Multiply fluxes (numbers per area per time) by charge to get current densities (charge per area per time)

Yields "diffusion current densities:" Jdiffusion_p q Dpx

pd

d= Jdiffusion_n q Dn

xn

d

d=

3) Relationship between band diagrams and electric field intensity

Ec

if is non-zero, energy changes with position:Ei

x( )1

q xEi

d

d= Ev

Ei ~ mid bandgap energy



5/19

4) "Einstein Relationship" between D and for a each carrier

Isolated bar of semiconductor w/ doping gradient: Inferred must have compensating drift & diffusion currents

Led to:D

kB T

q= or more accurately:

Dp

p

kB T

q= and

Dn

n

kB T

q=

=============================== Test Warning! =======================================

Mobilities weren't constant! D's 's So they can't be constant either!!

This (2nd) terribly misleading table gives "Typical diffusivitivity values"

= values based on erroneous "Typical Mobility" values of Table 5.1

So these D's are not typical either (!$#!@$!) but are instead a special low doping case

Book'sTable 5.2

============================= (End Test Warning) ====================================

Need one more critical piece of knowledge before can put it all together to explain devices!



6/19

Minority Carrier Thermal Generation and Recombination

a) Intrinsic Material at Equilibrium (undoped: ~ no acceptors or donors / no voltages or fields applied)

Using heat energy, electrons continuously jump up from valence band to conduction band:

ConductionBand

"Thermal Generation"

creates a pair of n & p=>

ValenceBand

"Generation Rate" = Gi = Number created / volume / time "i" denotes this intrinsic case

Gi = function of temperature / thermal energy available

+ function of light intensity (if light energy is larger than bandgap) + f (other energy sources)

NOT function of doping: Total electrons in valence band >> changes induced by doping



7/19

Reverse process is also continuous:

=> "Recombination" of n / p pair

"Recombination Rate" = R = number recombining / volume / time

But recombination rate should be: i) Proportional to the number of electrons trying to recombine

ii) Proportional to the number of holes available to be recombined with

Ri r n p=

But we are still discussing an intrinsic semiconductor where n = n i and p = ni so:

Ri r ni2= "recombination rate" in intrinsic material r to be determined or measured

BUT, in equilibrium generation and recombination rates must be equal(or concentrations would be changing!)



8/19

Gi Ri= r ni2= (equation 1)

b) Non-Intrinsic Non-Equilibrium (far more interesting) Material

n(t) and p(t) are actual non-equilibrium carrier concentrations

no and po are equilibrium carrier concentrations in same material

n and p are how much concentrations have changed from equilibrium

n t( ) no n t( )+=

p t( ) po

p t( )+=

Define:

Another way of defining:

n and p are deviation from equilibrium values

Deviations can be either plus (more) or minus (less)

Called "excess carrier concentrations" - even though "excess" can be < 0

n t( ) n t( ) no=

p t( ) p t( ) po=

Definition will allow us tofocus on the interestingchangesfrom equilibriumthat occur in activeelectronic devices

Present

Value

Equilibrium

Value

"Excess"

Value

Consider effect of G and R (alone) on the value of n(t)

tn t( )

d

d= [ generation rate ] - [recombination rate ] = [ generation rate ] -

rn t( ) p t( )



9/19

However, generation rate (valence band to conduction band jumps) depends only on available thermal energy

SHOULD be same here (doped / non-intrinsic case) as was above (undoped / intrinsic case)

G Gi= r ni2= (using equation 1 above)

So:

tn t( )d

dr ni

2 r n t( ) p t( )= Sorta makes sense: change in n deviation from intrinsic values

Now plug in alternate expressions for n(t) and p(t):

t no n t( )+( )d

d r ni

2

r no n t( )+( ) po p t( )+( )= on left tnod

d 0= plus some algebra:

tn t( )

d

dr ni

2no po po n t( ) no p t( ) n t( ) p t( )=

At equilibrium, know that no x po = ni2 so 1st and 2nd terms on right will cancel:

tn t( )

d

dr po n t( ) no p t( ) n t( ) p t( )( )=

Now assume special case of "LOW LEVEL INJECTION"



10/19

LOW LEVEL INJECTION = change in MAJORITY carrier is small compared to its equilibrium value

If N-type material: n no

With these, take look at equation:From assumption of "Low Level" injection know that po >> p

tn t( )

d

dr po n t( ) no p t( ) n t( ) p t( )( )=

FIRST term on right will be largest: - Changes p and n tend to be comparable

- Then po in 1st term >> no in 2nd term

- Both parts of 3rd term are small

So for our minority n carriers in this piece of P-type material, expect

tn t( )

d

dr po n t( )=

r and po are assumed to be constants so has easy solution



11/19

n t( ) n 0( ) er po t= n 0( ) e

tno=

no

1

r po

= "excess minority carrier lifetime"

n t( )

n t( ) n 0( ) e

t

no=

n 0( )

not

Perturbation in minority carrier concentration (away from equilibrium) will die out exponentially

Typical survival lifetime of an excess minority carrier = no



12/19

Can also loose minority carriers because they flow out of the volume we are considering (escape rather than die!)

Flow in Flow out

x x x+

Inside Box: 1) Drift of carriers 2) Diffusion of carriers 3) Generation - Recombination of carriers

Taking all into account, expect:

tn

d

d= Net flux into volume + Net generation rate - Net recombination rate (equation 2)

First term: Net flux into volume = (flow in from left - flow out from right) /x or

Fn x( ) Fn x x+( )

x x

Fn x( )d

d

= where Fn(x) is flux (flow) of electrons at x



13/19

Equation 2 then becomes (now expressing both x and t dependences):

But if minority carriers typically live for (shortening no)

then recombination rate, Rn, should be n /tn x t,( )d

d xFn x t,( )

d

d Gn+ Rn=

tn x t,( )d

d xFn x t,( )

d

d Gn+

n x t,( )

=

But flux (# / area / time) is just current density (charge / area / time) divided by charge: Fn x t,( )Jn x t,( )

q=

Given that Jn q n n q Dnx

nd

d+= get by plugging in above

tn x t,( )d

d xn n x t,( ) x t,( ) Dn

xn x t,( )d

d

d

d Gn+

n x t,( )

=

= n x t,( )

x

n x t,( )d

d

n x t,( )

x

x t,( )d

d

+

Dn 2x

n x t,( )d

d

2+ Gn+

n x t,( )

In regions where the electric field is ~ constant, this simplifies to (dropping explicit arguments)

tn

d

d

n

xn

d

d D

n 2xn

d

d

2+ G

n+

n

= Substitute in n = n

o

+ n remembering that no

is a constant



14/19

tn

d

d n xn

d

d Dn 2x n

d

d

2

+ Gn+

no

n

=

Gn, the total generation rate, can be divided into two parts

Gn Gnon_thermal Gthermal+=Gnon_thermal = Generation due to light, electron bombardment ...

Gthermal = What we discussed above

But the thermal generation rate = equilibrium generation rate = equilibrium recombination rate

Gthermal = Ro = no /

So Gthermal - no / = 0

Get upon substituting into equation above:

tn

d

dn

xn

d

d Dn 2

xn

d

d

2+ Gnon_thermal+

n

=

Can go through same arguments to count minority holes in N-type material. Then get pair of equations



15/19

(equation 3)

tn

d

dn

xn

d

d Dn 2

xn

d

d

2+

n

n Gnon_thermal+=

tp

d

dp

xp

d

d Dp 2

xp

d

d

2+

p

p Gnon_thermal+=

"Minority Carrier Continuity Equations"

(equation 4)

left term 1 term 2 term 3 term 4

Left = Total rate at which the "excess" minority carrier concentration will change with time

Term 1 = Change due to DRIFT ( pushing carriers) in / out of volume

Term 2 = Change due to DIFFUSION (spontaneous rearrangement of gradients) from volume

Term 3 = Net loss of carriers due to RECOMBINATION unbalanced by thermal generation

Term 4 = Generation due to things other than heat: absorbing light, gamma rays . . .

ASSUMPTIONS:

1) is approximately constant (so d/dx term could be thrown out)

2) Low Level injection (implicit in assumption that recombination proportional to 1 /)



16/19

APPLIES ONLY TO MINORITY CARRIERS:

When we are dealing with P-type material, can use electron equation # 3

When we are dealing with N-type material, can use hole equation # 4

BOOK does not limit to ~ constant special caseThus each of its "Ambipolar Transport Equations" retains an additional term

But will never use those more complicated equations - so stick with my versions!!

Recounting, each equation has the general form:

tminority_carrier

d

d= +/-

xminority_carrier

d

d Drift

+ D2x

minority_carrierd

d

2 Diffusion

-minority_carrier

minority_carrierNet Recombination

+ Gnon_thermal

Non-thermal Generation



17/19

The important part, understanding what these terms really mean:

Drift term requires:x

minority_carrierd

d

If no gradient in carrier concentraion, as many drift out of box at right as drifted in at left => no change

Diffusion term requires: D2

xminority_carrier

d

d

2

Diffusion current starts by being proportional to gradient

So to get more diffusion in to box than out of box must have a gradient of a gradient (a 2nd derivative)

Equations deal only with offsets from equilibrium ("excess carrier" populations)

"DC" components (no and po) were all knocked out by the derivatives!



18/19

And the very good news:

We will virtually never have to deal with full equations. Instead will treat special cases of:

1) Gnon-thermal = 0 - Good for ~ all but solar cells

AND/OR:

2) "Steady-state" - Voltages are constant so time derivatives => 0

AND/OR:

3) = 0 - Many regions of devices are essentially free of electric field

Next Time!



19/19


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Semiconductor Devices 14