p/n junction

FE

AE

iE

FE

VE

CEDE

vacE

Isolated p, n regions: no electric contact, not in equilibrium

p n

iE

VE

CE

p/n junctionIn equilibrium, the Fermi level must be constant. Shift the energy levels in p and n regions up/down to align EF:

AE

iE

FE

VE

CE

DE

vacE

VE

iE

vacE

CE

p nelectronenergy

Electrons tend to lowest available energy:The junction must have an electric field pointing from n to p.

p

n

p/n junction

vac F vac C C F

F V C V

V

Aln

p p

p g

E E E E E E

E E E ENkT EN

vac F vac C C F

Dln

n n

Cn

E E E E E E

NkTN

C V

A D

2

A D

A D2

ln

eln

ln

g

p n g

E kTi

g

i

N NqV E kTN N

nE kTN N

N NqV kTn

2C V

2C V

ee

g

g

E kTi

E kTi

n N NN N n

Find potential difference (voltage):

//p-side work function//n-side work function

vac vac F Fp n p np np nqV E E E E

Equilibrium: F F Fp nE E E

Built-in voltage: A D2lnbi

i

kT N NVq n

using:

Depletion approximation: space-charge region

xnwpw

Assume the regions directly adjacent to the junction on either sideare completely depleted of free carriers.

metallurgicaljunction

p nionized

acceptorsionizeddonors

depletion regionor

space-charge region

electronsholes

0

neutral region neutral region

This is called the depletion region or space-charge region (SCR).

Depletion approximation: charge density p x n x

D AN x N x

D Ax p x n x N x N x

np

---

+ ++

+ ++++ +

--- ---

---

+ ++

netcharge density

ionizeddopants

freecarriers

A

D

0,, 0, 0

0,

p

p

n

n

x wN w x

xN x w

w x

Depletion approximation orabrupt space-charge approx.

AN

DN

DN

AN

We are left with ionized acceptors and donors in the SCR.

DN

AN

Depletion approximation: electric field

A

D

0,

, 0Ε

, 0

0,

p

p p

xn n

n

x wqN x w w xq x dxqN w x x w

w x

Ε q xddx

Ε x

np

x

A Dp nN w N w

No surface charge:

0 0E Ex x Ε

maxΕ

maxΕ a p d nqN w qN w

//Poisson equation

No net charge: Ε 0 outside of SCR

Depletion approximation: potential

2A

22 2A D

2 2A D

0,

, 02Ε

, 02

,2

p

p p

xp n n n

p n n

x wqN x w w x

V x x dx q N w N w w x x w

q N w N w w x

ΕdVdx

V x np

biV

pw nwx

0

2 2A D2bi p n

qV N w N w

So:

Depletion approximation: depletion width

2 2A D

A D A D

1 1 1 12 2bi p nq qV N w N wN N N N

D

Ap n

Nw wN A

Dn p

Nw wN

D

A D

1 121 1

bin Vw N qN N

A

A D

1 121 1

bip Vw N qN N

A D

2 1 1bip n

Vw w w q N N The depletion width is:

Depletion approximation: energy diagram

AEFEDE

vacE

VE

iE

CE

p nelectron

energy

biVp

n

p/n junction: Non-equilibrium (I)External voltage (e.g., applied bias or photovoltage):

2 2A DΕ

2

n

p

w

p n bix w

qx dx N w N w V V

A D

2 1 1bip n

V Vw w w

q N N

A

D

e

e

i Fp

F in

E E kT

i

E E kTi

p N n

n N n

In the neutral regions:

F F2 2A D e ei in p bi

E E E E kT q V V kTi iN N n n

F Fn p biE E q V V A D

2lni

N NkTn

qV

The quasi-Fermi level splitting in the SCR equals the external voltage.

In the SCR:

p p

n n

F F

F F

E E

E E

p/n junction: Non-equilibrium (II)

AE

p nEelectronenergy

biV V

p n

FpE

Forward bias: + -V

2 1 1bi

a d

V Vw q N N

FnEDE

vacE

VE

iE

CE

p/n Junction: Non‐equilibrium (III)

AE

FnEFpEqV

FE

FE

pwnw

First, analyze without photogeneration:

DE

VE

iE

CE

p/n junction: non-equilibrium (IV)

2 e kTin p n In the SCR:

A

0 e

p

p qV kTp

p w N

n w n

The depletion approximation gives:

D

0 e

n

n qV kTn

n w N

p w p

qV

Far from the junction:

2

0A

2

0D

p i

n i

nn nN

np pN

0

0 e 1

pp p

p qV kT

n w n w n

n

0

0 e 1

nn n

n qV kT

p w p w p

p

where:

Excess minority-carrier concentrations at the edges of the SCR:

First, analyze without photogeneration:

Photogeneration RateAbsorption:

, , ,d b E x E x b E xdx

//separation of variables

0 0

b x x

b b x

db x dx xb

//treat each E separately

uniform

ln0

b x xb

, ,0 e E xb E x b E

Absorptioncoefficient

Carrier generation rate:

, ,

,

, 1 e

E xs

dg E x b E xdxE b E x

g E x R E E b E

,0 1 sb E R E b E

Spectral photon flux density in

material

p/n junction solar cell (I)

In steady-state:

0

p pg

n nn n

n n nG U G

g nn G

0

n ng

p pp p

p p pG U G

g pp G

n pG G G

, 1 1sg E x R E E b E g E

e 1E x no attenuation

0E

G x dE g E G

//uniform generation

//band-to-band absorption

p-side n-side

//Assume weak absorption

Photogeneration:

0

p pgn n n n

0n n

gp p p p

p/n junction solar cell (II)

FnE pFEqV

pwnw

With uniform photogeneration,there are excess minority carriers in the neutral regions,so the quasi-Fermi levels are split.

FE

DE

VE

iE

CE

AE

FE

p/n junction solar cell (III)

pp pn w n w n

0 0

0

e

e 1

n qV kT nn g

n qV kTg

p w p p p

p p

pn x n x n np x p x p

Consider generation in the quasi-neutral regions (assuming uniform generation).p-side n-side

Outside the SCR:

0 0

0

e

e 1

p pqV kTp g

p qV kTg

n w n n n

n n

0 ep qV kT

pn w n

The carrier concentrations are still related to the quasi Fermi-level spitting:

0 en qV kT

np w p

nn np w p w p

At the edges of the SCR:

So the excess minority-carrier concentration at the edges of the SCR are:

p/n junction solar cell (IV)

n pJ J x J x

Total current density:

sign convention for PV

//must be constant at any point in circuit

Positive J when device is delivering power

Calculate using: n p p pJ J w J w

n n p nJ J w J w

Depletion approx: All the potential difference occurs across the SCR.So, E=0 at the SCR edges: , p nx w x w

Εn n n ndn dnJ q q D q Ddx dx

Εp p p pdp dpJ q q D q Ddx dx

or:

p/n junction solar cell (V)

e e

e e

p n p n

p pn p

x w L x w Ln n

x w Lx w Lp p

n x A B

p x A B

e

e

p n

n p

x w Lp

x w Ln

n x n w

p x p w

In the neutral regions, we have diffusion only:p

n

x ww x

00

n

p

AB

nn n

n

pp p

p

d n q DJ x q D n xdx Ld p q DJ x q D p x

dx L

nn p p

n

pp n n

p

q DJ w n wL

q DJ w p wL

p

n

x ww x

The boundary conditions give:

0 e 1p qV kTp gn w n n

0n

0 e 1n qV kTn gp w p p

0p

0 e 1n p qV kTn p g

n

q DJ w n nL 0 e 1p n qV kT

p n gp

q DJ w p pL

Assume infinitely thick neutral regions.

The currents are:

Using previous results:

p/n Junction solar cell (VI)

or

,SCR

,SCR

p p p n p p

n n n p n n

J w J w J w

J w J w J wwe need either:

,SCR

,rec ,gen

n

p

wn n

x w

n n

J w q U x G x dx

J J

n pJ w p nJ wWe have and

n p p p n n p nJ J w J w J w J w To find

,SCR

,rec ,gen

n

p

wp p

x w

p p

J w q U x G x dx

J Jwhere

p/n Junction solar cell (VII)

Assume: ,rec 0pJ

,gen ,gen

n

p

wn p

x wJ J q G dx q G w

,SCR 0 e 1 n p qV kT

n n n p n n gn

q DJ w J w J w q G w n n q G wL

//no recombination in SCR

//uniform generation in SCR

,rec 0nJ

,SCR ,SCR p p n nJ w J w q G w

Just need one or the other. Let's find p n n nJ J w J w

We know: 0 e 1p n qV kTp n g

p

q DJ w p pL

,SCR n n n p n nJ w J w J w 0 e 1n p qV kTn p g

n

q DJ w n nL

,SCR n nJ w q G w

p/n Junction solar cell (VIII)

0 e 1p n qV kTp n p

p

q DJ w p q G LL

p n n nJ J w J w

n ng n n

n n

q D q Dn G q G LL L

0 e 1n p qV kTn n n

n

q DJ w n q G L wL

0 e 1n p qV kTn n g

n

q DJ w q G w n n q G wL 0 e 1p n qV kT

p n gp

q DJ w p pL

p pg p p

p n

q D q Dp G q G LL L

Use:

p/n Junction solar cell (IX)

photo n pJ q G w L L

20 00

A D

n p n pp ni

n p n p

D D D DJ q n p q nL L L N L N

//photocurrent

//dark current

photo 0 e 1qV kTJ V J J

00

photo 0

e 1

e 1

n pp n qV kTn p

n p

qV kT

q D q DJ q G w L L n pL L

J J J

SCR transit timeCan we ignore recombination in the SCR?Estimate transit time across SCR:

Ev max1E = E2 max

1 E2

v max

2E

w wv

A DmaxE

p nqN w qN w

p nw w w A

A D

11 1

pN w w

N Nmax

1E 1 1

a d

q w

N N

max

A D

E 11 1

qw

N N

A D

2 1 1 q N N

0

2

16 3A D

10

cm100 V s

10 cm

N N

112.2 10 s Short compared to typicalminority carrier lifetimes.

Assume:

Plots: depletion approx. (I)equilibrium (V=0 V, G=0):

no quasi-Fermilevel splittingin SCR

no minority-carrierconc. gradientsin neutral regions

Plots: depletion approx. (II)forward bias, no illumination (V=0.5 V, G=0):

minority-carrierconc. gradientsin neutral regions→diffusion awayfrom SCR

positivequasi-Fermi levelsplitting in SCR

Plots: depletion approx. (III)short circuit, illuminated (V=0 V, G=5e19 1/cm3/s):

minority-carrierconc. gradientsin neutral regions→diffusion towards SCR

no quasi-Fermilevel splitting

Plots: depletion approx. (IV)operating, illuminated (V=0.5 V, G=5e19 1/cm3/s):

minority-carrierconc. gradientsin neutral regions→diffusion towards SCR

positivequasi-Fermi levelsplitting in SCR

2

A

eqV kTinN

2

D

eqV kTinN

2

A

inN

2

D

inN

ANDNn

2

A

eqV kTinN

2

D

eqV kTinN

2

A

inN

2

D

inN

p

ANDNnp

pw nw

pw nw

forward bias,no generation

forward biaswith generation

Plots: carrier concentrations in depletion approx. 

electron diffusion

electron diffusion

hole diffusion

hole diffusion

2

A

ig

n nN

2

D

ig

n pN

Solving p/n homojunction characteristics without the depletion approx.

Fnn n

dEJ ndx

1 nn n

dJ G Uq dx

1 pp p

dJ G Uq dx

2

2q xd V

dx

Fpp p

dEJ pdx

//Poisson equation

//electron current

//hole current

//hole continuity

//electron continuity

Plots: no depletion approx. (I)equilibirium (V=0 V, G=0):

CE

VE

iEFn

E

tot 0p nJ J J

p n

p n

FpE

0U G

Plots: no depletion approx. (II)forward bias, no generation (V=0.7 V, G=0):

U

G-U

CE

VE

iE FnE

pJ nJtotJ

p n

p n

FpE

Plots: no depletion approx. (III)

UG-U

CE

VE

iEFn

E

FpE

pJ nJ

totJ

p n

p n

G

operating, illuminated (V=0.7 V, G=1e23 1/cm3/s):

depl. approx.: dashed linesfull calc.: solid lines