PN Diode I-V Characteristics

Current-Voltage Characteristics Qualitative Analysis: Thermal Equilibrium

Diffusion - Electron diffusion from n-side to p-side - Most will be reflected - Few electrons will have enough energy and overcome the barrier height

Drift - Electron drift from p-side to n-side - Few electrons will be drifted (e- is minority carriers in p-side) - At thermal equilibrium, diffusion and drift are balanced Net electron current = 0 Similarly, net hole current=0

Thus, the total current=0

Electrons EC

EF Ei

EV

Equilibrium (VA = 0)

Particle Flow Current

Electron Diffusion

Electron Drift

Hole Drift

Hole Diffusion

1

2

Current-Voltage Characteristics Qualitative Analysis: Forward Bias (VA>0)

More electron diffusion from n-side to p-side due to lowering of barrier height

Electrons drift from p-side to n-side remains same (minority concentration unchanged)

Electron diffusion dominates

Diffusion current increases exponentially with VA

Potential barrier (Vbi-VA) linearly decrease with increasing VA

Plus electron concentration (in n-side, majority carrier) above the barrier height increases exponentially when the energy level moves upward

Electrons

IN

IP

P N

VA>0

Forward Bias

EC EFn Ei

EV

Current-Voltage Characteristics Qualitative Analysis: Forward Bias (VA>0)

Similar analysis applied to holes

The total current is dominated by both electron and hole diffusion and given as

JF=JN, F+JP, F Forward current increases

exponentially with VA

Will VA be larger than Vbi?

IN

IP


Electron Diffusion

Electron Drift

Hole Drift

Hole Diffusion

EC EFn Ei

EV

3

Current-Voltage Characteristics Qualitative Analysis: Reverse Bias (VA<0)

- Very few electron diffusion from n-side to p-side due to increased barrier height

- Electrons drift from p-side to n-side remains same (minority concentration unchanged)

- Electron current is dominated by drift

-Thus electron current is very small Similar analysis applied to holes So, the total reverse current dominated by both electron and hole drift currents

Reverse currents are very small and nearly independent with VA value

Electrons IN

IP

EC EFn Ei

EV


Electron Diffusion

Electron Drift

Hole Drift

Hole Diffusion

Reverse Bias 4

Current-Voltage Characteristics Qualitative Analysis: I-V Characteristics

Constant

Exponential

VA

I

5

Current-Voltage Characteristics Quantitative Analysis: I-V Characteristics

Assumptions (1) The diode is being operated under steady state conditions (2) The doping profile: a non-degenerately doped step junction models (3) The diode is one dimensional (4) Low level injection prevails in the quasi-neutral regions (5) There are no processes other than drift, diffusion, and thermal R-G (with negligible) taking place inside the diode.

Thus, the current can be calculated as

AJI =

)()( xJxJJ PN +=

dxdnqDnEqxJ NnN += µ)(

dxdpqDpEqxJ ppp −= µ)(

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Quasi-neutral regions - The minority carrier diffusion equations are

E≈0 E≈0

-xp xn

Depletion region

Quasi-neutral regions

P Region N Region

E≠0

np

nnP

pn

ppN

xxpdx

pdD

xxn

dxnd

D

≥∆

−∆

=

−≤∆

−∆

=

τ

τ

2

2

2

2

0

0

From the above two equations (plus boundary condition, will be discussed later), both and can be obtained pn∆ np∆

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- Since low level injection (E≈0) and

000 ==dx

dpdx

dn

Thus, current density in these two quasi-neutral region can be calculated as

dxnd

qDxJ pNN

∆=)(

dxpdqDxJ n

PP∆

−=)(

For x ≤ -xp

For x ≥ xn

Recall: pqDpEqJJJ ppdiffpdriftpp ∇−=+= µ,,

nqDnEqJJJ nndiffndriftnn ∇+=+= µ,,

Thus, we are able to calculate the current density (diffusion current) at quasi-neutral regions!!!

Since low-level injection, the majority carrier concentration remains the same within quasi-neutral regions, thus the E-field is almost zero.

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Current-Voltage Characteristics

Quantitative Analysis: I-V Characteristics Depletion region

GRP

GRN

tp

dxdJ

q

tn

dxdJ

q

−

−

∂∂

+−=

∂∂

+=

|10

|10- Recall

- From assumption (5), we have 0|

0|

=∂∂

=∂∂

−

−

GR

GR

tp

tn

- Then, we have

0=dx

dJ N

0=dx

dJ P

- Which means that JN and JP are constant within the depletion region

)()(

)()(

nPnpP

pNnpN

xJxxxJxJxxxJ

=≤≤−

−=≤≤−

E≈0 E≈0

-xp xn

Depletion region


P Region N Region

E≠0

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Depletion region

- Since JN and JP are constant within depletion region

)()(

)()(

nPnpP

pNnpN

xJxxxJxJxxxJ

=≤≤−

−=≤≤−

- The total current density can be given as below

)()()()( nPpNPN xJxJxJxJJ +−=+=

Thus, the total current density can be calculated as the sum of diffusion currents at both depletion edges JN(-xp) and JP(xn)!!!

E≈0 E≈0

-xp xn

Depletion region


P Region N Region

E≠0

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Boundary Conditions

At the Ohmic contacts: - Consider long “wide” base diode, which means the diode has contacts which are larger than the minority carrier diffusion lengths. The contacts can be viewed at - So, the boundary conditions are

±∞=x

0)(

0)(

=∞=∆

=−∞=∆

xnp

xpnEC EFn Ei

EV

FN

FP EFp

∆np(p-contact) ∆pn(n-contact) ∆pn(xn) ∆np(-xp)

-xp xn

(a)

(b) 11


Boundary Conditions

At Depletion Region Edges

- Introducing “quasi-Fermi level” FN and FP (FN-FP=VA). Inside depletion region, we have - So, the boundary conditions are

EC EFn

Ei

EV

FN

FP EFP

-xp xn

(b)

∆np(p-contact) ∆pn(n-contact) ∆pn(xn) ∆np(-xp)

(a)

at x = -xp

at x = xn

App Nxp =− )(

kTqVi

Aennp /2=

kTqV

A

ipp

AeNnxn /

2

)( =−

)1()( /2

−=−∆ kTqV

A

ipp

AeNnxn

Dnn Nxn =)(

)1()( /2

−=∆ kTqV

D

inn

AeNnxp

kTqV

D

inn

AeNnxp /

2

)( =

kTEFi

iNenn /)( −=kTFE

ipienp /)( −=

Recall:

P N

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Calculation of Current JP(xn)

-xp

x

x’ X’=0

(1) Shift the origin of coordinates to xn

(2) The equation and boundary conditions are

(3) Solving the above equations, we have

LpxkTqV

Dp

pnpP ee

NLnD

qdx

pdqDxJ Ai /'/2

)1('

)'( −−=∆

−=

(4) So, JP(xn) can be given as

)1()0'()( /2

−==== kTqV

Dp

pPnP

Ai eNLnD

qxJxxJ

p

nnp

pdx

pdDτ∆

−∆

= 2'

2

0 X’ ≥ 0

0)'( =∞=∆ xpn

)1()0'( /2

−==∆ kTqV

D

in

AeNnxp

Continuity Equation

Ohmic Contact

Law of Injection

Subject to the boundary conditions

xn

13


Calculation of Current

(1) Similarly, JN(-xp) can be obtained as follow:

)1()( /2

−=−= kTqV

An

npN

Ai eNLnD

qxxJ

(3) A useful and important equation:

)1)(( /22

−+== kTqV

AN

n

DP

p Aii eNLnD

NLnD

qAAJI

(2) So, the total current is

)(

)1(22

0

/0

AN

n

DP

p

kTqV

NLnD

NLnD

qAI

eII

ii

A

+≡

−=

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Current-Voltage Characteristics Some Useful Discussions

I-V Characteristics (Ideal)

(1) For reverse biases greater than a few kT/q, (2) For forward biasing larger than a few kT/q, (3) In forward bias conditions, I-V is always plotted as semi-log scale since (4) Saturation current - depends on intrinsic concentration - depends on doping concentration

0II −=

)/exp(0 kTqVII A→

AVkTqII += )ln()ln( 0

)(22

0AN

n

DP

p

NLnD

NLnD

qAI ii +≡

I0

I

)/0

kTqVAeIqkTVA />>

VA

ln(I)

ln(I0)

Slope=q/kT

VA

15


Carrier Current Components

Think Again: How do we calculate the current?

JP

JN

JN

JP

0

J

-xp xn x

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Carrier Concentrations (Forward bias)

- Carriers are injected into other side by diffusion and become minority carriers. - The excess minority carriers are eliminated by recombination during diffusion deeper into the regions - Thus, there is a build-up of minority carrier in the quasi-netural regions immediately adjacent to the edges of the depletion regions.

A

i

Nn2

D

i

Nn2

-xp xn

pp

nn

np

pn

x

NA ND

n or p (Log-scale)

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Current-Voltage Characteristics

Some Useful Discussions

Carrier Concentrations (Reverse bias)

- The depletion region acts like a “sink” for minority carriers, draining the carriers from the adjacent quas-ineutral region. - A reverse bias of a few kT/q effectively reduces the minority carrier concentrations to zero at the edges of the depletion region. - Larger reverse biases have little effect on the carrier concentration (consistent with the saturation current in reverse bias).

D

i

Nn2

Liner Scale

n or p

A

i

Nn2

-xp xn x

np

pn

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PN Diode I-V Characteristics

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