Quantitative Model of Unilluminated Diode – part I

G.R. Tynan UC San Diego MAE 119

Lecture Notes

A Solar PV Cell is just a p-n junction (“diode”) illuminated by light….

P-type n-type

Photon Flux with E>Egap

Examine This p-n Junction…

Diode current-voltage characteristics

3 http://www.electronics-tutorials.ws/diode

Key elements: Quasineutral regions Quasi-neutral region

Net charge density ~ 0 Quasi-neutral region

Net charge density ~ 0

Key elements: Depletion region Depletion region With net charge,

finite E-field

What is needed for quantitative model of this diode?

•  What are the electron and hole densities? •  How do they move within the device?

– Diffusion due to concentration gradients – Motion under action of electric fields

•  Given these items, what is the current in the device as a function of voltage across device?

What is needed for quantitative model of this diode?

•  What are the electron and hole densities? •  How do they move within the device?

– Diffusion due to concentration gradients – Motion under action of electric fields

•  Given these items, what is the current in the device as a function of voltage across device?

Determining Charge Carrier Densities

•  Find number of mobile electrons/volume in conduction band from – Density of allowed states, N(E) and – Energy distribution of e and h, f(E):

Charge Carrier Energy Distribution

•  Charge carrier particle energies follow Fermi-Dirac probability distribution

EF = “Fermi Energy” characteristic upper limit for energy T = Temperature of particles f(E)dE = probability of finding particle between (E, E+dE)

Density of allowed charge carrier states

•  Conduction Band Allowed State Density (states/vol-energy)

•  Valence Band Allowed State Density (states/vol-energy:

NC E( ) = 8 2π me

*( )h3

3/2

E − EC( )1/2; E ≥ EC

NC E( ) = 8 2π me

*( )h3

3/2

E − EC( )1/2; E ≥ EC

Mobile Charge Carriers found from product of N(E)f(E):

•  Find number of mobile electrons/volume in conduction band from N(E) and f(E):

Mobile Charge Carrier Density:

•  Number of mobile electrons/volume in conduction band:

•  Result:

CE E≥( ) ( )maxc

C

E

En f E N E dE= ∫

n = NC exp EF − EC( ) kT⎡⎣ ⎤⎦

NC = 2 2πme*kT

h2⎛⎝⎜

⎞⎠⎟

3/2

Mobile Charge Carrier Densities

•  Number of mobile holes/volume in valence band:

•  Result: p = NV exp EV − EF( ) / kT⎡⎣ ⎤⎦

p = NV (E) f (E)dE0

EV

∫

NV = 2 2πmh*kT

h2⎛⎝⎜

⎞⎠⎟

3/2

A few useful definitions:

•  Useful to define “intrinsic concentration, ni :

•  Since n=p in a pure semiconductor can write

•  Which gives:

ni2 = np = NCNV exp EV − EC( ) / kBT⎡⎣ ⎤⎦

NV exp EV − EF( ) / kT⎡⎣ ⎤⎦ = NC exp EF − EC( ) / kT⎡⎣ ⎤⎦

ln2 2

c V VF

C

E E NkTEN

⎛ ⎞+= + ⎜ ⎟⎝ ⎠

Charge Carrier Densities – n-type doped semiconductors

•  Charge neutrality tells us

•  For n-type doped material

•  And therefore in n-type material

p + n + ND+ − NA

− = 0

n = ND+ and ND

+ ≈ ND

p = n − ND+ << n

Charge Carrier Densities – p-type doped semiconductors

•  Charge neutrality tells us

•  For p-type doped material

•  And therefore in p-type material

p + n + ND+ − NA

− = 0

p = NA− and NA

− ≈ NA

n = p − NA− << p

What is needed for quantitative model of this diode?

•  What are the electron and hole densities? •  How do they move within the device?

– Diffusion due to concentration gradients – Motion under action of electric fields

•  Given these items, what is the current in the device as a function of voltage across device?

Resulting carrier density distribution

EXAMINE Depletion Region IN MORE DETAIL…

DEPLETION REGION”

Charge Distribution Across Unbiased p-n diode

Can Find Electric Potential in Depletion Region:

Charge Distribution From Semiconductor Material Theory E-field from Poisson’s Equation Potential Profile by Integrating E-field

Idealized Model of Charge Distribution Across Depletion Region:

Poisson’s eq’n relates charge density to E field:

Poisson’s Equation:

DD NN ≈+AA NN ≈−

Donor/Acceptor Ion Density

p, n distribution given by step functions seen in previous viewgraph

Integrate Poisson’s Equation to Find E(x); Integrate Again to find potential distribution

dEdx

= qε(p − n + ND

+ − NA− )

Idealized Model of Depletion Region:

( )2/1

0max112

⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛+−−=

DAext NNVqE ψ

ε

RESULT:

( )2/1

0112

⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛+−−=+=

DAextnp NNV

qllw ψε

DA

Dp NN

NWl

+=

DA

An NN

NWl+

=

Carrier concentration at edges of un-biased junction

With NO Ext. Voltage, Vext=0:

D

ipnnb N

nkToqeppp

2

)(00

≈−== ψ

A

inpb N

nkToqennn

a

2

00≈⎟

⎠⎞⎜

⎝⎛ −== ψ

What is needed for quantitative model of this diode?

•  What are the electron and hole densities? •  How do they move within the device?

– Diffusion due to concentration gradients – Motion under action of electric fields

•  Given these items, what is the current in the device as a function of voltage across device?

Question: What happens when we apply an external voltage to this diode? In order to answer, need to first define current density, J (Amps/m2)…

Need to define current density, J: Consider a collection of n (p) electrons/unit volume (holes/unit volume) that are moving towards the right

Q: How many charges pass thru the surface per unit area/unit time? A: This is defined as the electron (hole) current density, Je (Jh )

Can be caused by E-field and/or by diffusion

Basic Equations of Semiconductor Device Physics

Poisson’s Equation: )( −+ −+−= AD NNnpqdxdE

DD NN ≈+AA NN ≈−

Donor/Acceptor Ion Density

dxdpqDpEqJ

dxdnqDnEqJ

eheh

eee

−=

+=

µ

µCurrent

Transport

S.S. Charge Conservation: )(1

1

GUdxdJ

q

GUdxdJ

q

h

e

−−=

−=U~Source G~Sink

Question: What happens when we apply an external voltage to this diode? A: Currents can begin to flow…

How does charge distribution respond to Vbias?

Jh = qpµhE − qDh

∂p∂x

In general current density depends on E and gradient, i.e.

We will simplify analysis by assuming that

qpµhE ≈ qDh

dpdx even if Jh ≠ 0

Now…

E ≈ kTq

1p

dpdx

And since we can integrate to find potential drop across depletion region…

E = −∂φ ∂x

e e

h h

kTDqkTDq

µ

µ

=

=

How does charge distribution respond to Vbias?

And since we can integrate to find potential drop across depletion region…

E = −∂φ ∂x

φ! −Va = + kT

qln

ppa

pnb

Which can be re-arranged to give:

pnb

= ppaexp

qφ!kT

⎡

⎣⎢

⎤

⎦⎥exp

qVa

kT⎡

⎣⎢

⎤

⎦⎥

How does charge distribution respond to Vbias?

And since we can integrate to find potential drop across depletion region…

E = −∂φ ∂x

φ! −Va = + kT

qln

ppa

pnb

Which can be re-arranged to give:

pnb

= ppaexp

qφ!kT

⎡

⎣⎢

⎤

⎦⎥exp

qVa

kT⎡

⎣⎢

⎤

⎦⎥

Get similar result for electrons…

How does charge distribution respond to Vbias?

And since we can integrate to find potential drop across depletion region…

E = −∂φ ∂x

φ! −Va = + kT

qln

ppa

pnb

Which can be re-arranged to give:

pnb

= ppaexp

qφ!kT

⎡

⎣⎢

⎤

⎦⎥exp

qVa

kT⎡

⎣⎢

⎤

⎦⎥

pn0

Minority carrier density w/o ext. bias

Get similar result for electrons… THIS IS A KEY RESULT!

Exponential Increase w/ Va>0 !

Forward Bias Increases Minority Charge Carrier Density at Edge

of Quasineutral region:

kTqV

D

ikTqV

nn

appapp

be

nn

epp2

0==

kTqV

A

ikTqV

pp

appapp

ae

Nn

enn2

0==

Concentration Of MINORITY CARRIERS at Edge of Depletion Region INCREASES EXPONENTIALLY W/ Ext. Bias è KNOWN AS MINORITY è CARRIER INJECTION

Minority Carrier Density at edge of quasineutral region increases EXPONENTIALLY forward bias

pnb= pn0

expqVa

kT⎡

⎣⎢

⎤

⎦⎥

npa= np0

expqVa

kT⎡

⎣⎢

⎤

⎦⎥