Solid State Devices. Section 20 PN Diode I-V Characteristics

Klimeck – Solid State Devices

Section 20

PN Diode I-V Characteristics

Gerhard Klimeck

[email protected]

Solid State Devices

Section 20

PN Diode I-V Characteristics

• 20.1 Band diagram with applied bias

• 20.2 Derivation of the forward bias formula • 20.3 Forward Bias - Non-linear Regime

»Resistive drop »Ambipolar regime • 20.4 Non-ideal effects: »Junction recombination »Tunneling »Impact ionization status V V V V ( ) 2 − β 2 β

=

n p

=

qV_A i i F F

n e

n e

n

p

Klimeck – Solid State Devices (4,6) Junction Recombination ( ) 2 − β 2 β

=

n p

=

qV_A i i F F

n e

n e

n

p

(4,6) Junction Recombination 0 W R n I qA dx t ∂ = − ∂

∫

2 1 1 ( ) ( [ ] [ ] ) ] ) [ ( ( ) τ τ − ∂ = − ∂ + + + i p n p x p x n x n n x n n t p p n

τ

τ

=

E

_i

=

E

_T

n

₁

=

p

₁

=

n

_i ] 2 ) ( ) ( [ ) 1 ( / 2 i kT qV i n x p x n e n t n A + + − − = ∂ ∂ τ Assume

What is the recombination current?

Shockley-Reed Hall

Follows from assuming midgap traps ( ) 2 − β 2 β

=

n p

=

qV_A i i F F

n e

n e

n

p

Klimeck – Solid State Devices

Electron/Hole Concentrations at Junction

) ( ) (x E qV x E_i = _iL − ( ) [ ( ] ) ( ) / / ( ) N i N iL kT F E x F E qV x i kT i n e x n e n − − + = = / 2 [ ]/ [ ( )]/ / ( ) ( ) N A N iL iL A qV kT i kT i F E F E qV qV x kT qV kT i x n e x e n p n e− − + + − + = = position ] 2 ) ( ) ( [ ) 1 ( / 2 i kT qV i n x p x n e n t n A + + − − = ∂ ∂ τ

Junction Recombination kT E F U_FN = N − iL q kT V U_A A / = ] [ ) 1 ( A FN FN A U U U U U U i e e e n t n + − − + ₊ − − = ∂ ∂ τ ] [ ) ( 2 / 2 / 2 / 2 / 2 / 2 / A FN A FN A A A A U U U U U U U U U U i e e e e e e n t n + − − − + − + − − = ∂ ∂ ⇒ τ ( / 2) [ / 2] i A FN A n sinh U n t τ cosh U U U ∂ ⇒ = − ∂ + −     = − _{ }ni × _UA _×

∫

W dx I qA sinh 0 W R n I qA dx t ∂ = − ∂

∫

( ) [ ( ] ) ( ) / / ( ) N i N iL kT F E x F E qV x i kT i n e x n e n − − + = = / 2 [ ]/ [ ( )]/ / ( ) ( ) N A N iL iL A qV kT i kT i F E F E qV qV x kT qV kT i x n e x e n p n e− − + + − + = = ] 2 ) ( ) ( [ ) 1 ( / 2 i kT qV i n x p x n e n t n A + + − − = ∂ ∂ τ

Klimeck – Solid State Devices

Junction Recombination in Forward Bias

( / 2) 0 ( ) ( ) 2 FN U A W i A R U U n U dx I qA sinh e τ + − ⇒ ≈ −

∫

max ( ) /( / 2 ) 0 ( ) ( ) 2 W i A R _{x kT q} n U dx I qA sinh e τ − ⇒ ≈ − ×

∫

_E / 2 max 2 A qV kT i Dep n kT I qA e q τ  _{  } ⇒ = − _{   }    E  V_A 2 4 5 1 3 6,7 ln(I) E_max

Effective spatial width of recombination

Excess Carrier at mid-junction Factor ½ indicates

recombination / traps in junction Traps hidden in τ 0 2 [ / 2] τ     = − _{ }× _{ }× + −    

∫

W i A R FN A n U dx I qA sinh cosh U U U

Keep positive branch

Approximate electric field as constant

Junction Leakage in Practice n p d r_{j rj} Insulating Layer

Junction Design Considerations Electric field stronger at corners, sharp edges.  increased recombination!

Klimeck – Solid State Devices

Junction Recombination in Reverse Bias

2

i

n

n

t

∂

_{= −}

∂

τ

W=x_n+x_p (Recombination in depletion region n=p=0) Depletion width 2 1 1 ( ) ( [ ] [ ] ) ] ) [ ( ( ) τ τ − ∂ = − ∂ + + + i p n p x p x n x n n x n n t p p n

τ

τ

=

E

_i

=

E

_T

n

₁

=

p

₁

=

n

_i ] 2 ) ( ) ( [ ) 1 ( / 2 i kT qV i n x p x n e n t n A + + − − = ∂ ∂ τ Assume Shockley-Reed Hall

Follows from assuming midgap traps ( ) 2 − β 2 β

=

n p

=

qV_A i i F F

n e

n e

n

p

Junction Recombination in Reverse Bias 0

₂

R W i

n

I

qA

dx

τ





≈ −

_

_





∫

2

i

n

n

t

∂

_{= −}

∂

τ

V_A 2 4 5 1 3 6,7 ln(I

)

2

i bi A

W

n

q

A

V

V

τ

∝

−

= −

W=x_n+x_p (Recombination in depletion region n=p=0) Integrate… Depletion width

Ideally the flat – now it depends as a square root of voltage => another probe mechanism for traps in junction

Klimeck – Solid State Devices

Section 20

PN Diode I-V Characteristics

• 20.1 Band diagram with applied bias

• 20.2 Derivation of the forward bias formula • 20.3 Forward Bias - Non-linear Regime

»Resistive drop »Ambipolar regime • 20.4 Non-ideal effects: »Junction recombination »Tunneling »Impact ionization status V V V V

Section 20

PN Diode I-V Characteristics

• 20.1 Band diagram with applied bias

• 20.2 Derivation of the forward bias formula • 20.3 Forward Bias - Non-linear Regime

»Resistive drop »Ambipolar regime • 20.4 Non-ideal effects: »Junction recombination »Tunneling »Impact ionization status V V V V

Klimeck – Solid State Devices

Forward Bias Nonlinearity (7): Esaki Diode

I V_A F_{n F} p F_n F_n F_p F_p X

V

_A 2 1 3 6,7

ln(I)

Heavy doping 1 empty No states!

Esaki-Diode: Heavily doped diode

Tunneling in diodes. Nobel Prize (Esaki)

Reverse Bias (5): Zener Tunneling 2 2 4 4 cosh sinh (p.49 ADF) = υ = α   α +_ − _ α α   I qpT T k d d k F_n F_p 4 5 +V_A empty empty Tunneling (triangular barrier)

Remember: Tunneling through a triangular barrier Zener tunneling occurs in every diode. (reverse bias)

Klimeck – Solid State Devices

Various Regions of I-V Characteristics

1. Diffusion limited 2. Ambipolar transport 3. High injection 4. R-G in depletion 5. Breakdown 6. Trap-assisted R-G 7. Esaki Tunneling

V

_A 2 4 5 1 3 6,7

ln(I)

Section 20

PN Diode I-V Characteristics

• 20.1 Band diagram with applied bias

• 20.2 Derivation of the forward bias formula • 20.3 Forward Bias - Non-linear Regime

»Resistive drop

»Ambipolar regime

• 20.4 Non-ideal effects: »Junction recombination »Tunneling

»Impact ionization status

V

V

V

Klimeck – Solid State Devices Avalanche Breakdown 1. Diffusion limited 2. Ambipolar transport 3. High injection 4. R-G in depletion 5. Breakdown 6. Trap-assisted R-G 7. Esaki Tunneling

V

_A 2 4 5 1 3 6,7

ln(I)

Nonlinearity due to Impact-Ionization

Klimeck – Solid State Devices

Nonlinearity due to Impact-Ionization

Exponential current growth

(Impact Ionization or Inverse Auger process)

High Reverse Bias > bandgap (typically 3/2 bandgap)

Impact-ionization and Flux Conservation ( ) _n( ) _n( ) _{p p}( ) n n I x + dx = I x + α I x dx + α I x dx 𝑑𝑑𝐼𝐼𝑛𝑛(𝑥𝑥) 𝑑𝑑𝑥𝑥 = 𝛼𝛼𝑝𝑝 𝐼𝐼𝑇𝑇 − 𝐼𝐼𝑛𝑛(𝑥𝑥) + 𝛼𝛼𝑛𝑛𝐼𝐼𝑛𝑛(𝑥𝑥) x _{W 0}

Impact Ionization probabilities

Steady state: Define I_T= I_N+I_P (total current) Constant in steady state

Differential equation 𝑑𝑑𝐼𝐼_𝑛𝑛(𝑥𝑥) 𝑑𝑑𝑥𝑥 − 𝛼𝛼𝑛𝑛 − 𝛼𝛼𝑝𝑝 𝐼𝐼𝑛𝑛(𝑥𝑥) = 𝛼𝛼𝑝𝑝𝐼𝐼𝑇𝑇 𝐼𝐼_𝑝𝑝 = 𝐼𝐼_𝑇𝑇 − 𝐼𝐼_𝑛𝑛(𝑥𝑥) ( ) ( ) ( ) ( ) ( ) ( ) ( ) n n n n p p n n n p p I x dx I x I x I x dx dI x I x I x dx + − = α + α ⇒ = α + α

Klimeck – Solid State Devices

Impact-ionization and Flux Conservation

Differential equation 𝑑𝑑𝐼𝐼_𝑛𝑛(𝑥𝑥) 𝑑𝑑𝑥𝑥 − 𝛼𝛼𝑛𝑛 − 𝛼𝛼𝑝𝑝 𝐼𝐼𝑛𝑛(𝑥𝑥) = 𝛼𝛼𝑝𝑝𝐼𝐼𝑇𝑇 𝑦𝑦𝑦 − 𝑎𝑎𝑦𝑦 = 𝑏𝑏 𝑦𝑦 = 1_{𝑎𝑎 𝑒𝑒}𝑎𝑎𝑎𝑎+𝑐𝑐𝑎𝑎 − _𝑎𝑎𝑏𝑏 x _{W 0} I_p(W)~0 I_n(0)~0 ( )

(

)

( ) 0 0 ' 0 ' 0 (0) 1 ( ) x n p x n p W dx p T n W dx T p n n e dx I I W I e dx I − α −α − α −α ∫ α + = ∫ + α − α

∫

∫

Solution form of differential equation _{Reverse diffusion} current

Impact-ionization ( ) ( (0) 1 ( ) _{n T} ) _T p T n n p I W I I W I I M I W I + = ⇒ ≈ ≡ � 𝑊𝑊 𝛼𝛼_𝑛𝑛𝑒𝑒− ∫0𝑥𝑥 𝛼𝛼𝑛𝑛−𝛼𝛼𝑝𝑝 𝑑𝑑𝑎𝑎𝑑𝑑𝑑𝑥𝑥 = 1 − 1 𝑀𝑀_𝑝𝑝 ( )

(

)

( ) 0 0 ' 0 ' 0 (0) 1 ( ) x n p x n p W dx p T n W dx T p n n e dx I I W I e dx I − α −α − α −α ∫ α + = ∫ + α − α

∫

∫

x _{W 0} I_p(W)~0

I_n(0)~0 Solution form of differential equation Reverse diffusion current

At x=W, I_Nhas grown exponentially, and I_Pis now negligible. Multiplication Factor 1 = ∫0 𝑊𝑊_𝛼𝛼 𝑝𝑝𝑒𝑒− ∫0 𝑥𝑥 _𝛼𝛼 𝑛𝑛−𝛼𝛼𝑝𝑝 𝑑𝑑𝑎𝑎𝑑𝑑𝑑𝑥𝑥 + 1 𝑀𝑀_𝑝𝑝 1 + ∫₀𝑊𝑊 𝛼𝛼_𝑝𝑝 − 𝛼𝛼_𝑛𝑛 𝑒𝑒− ∫0𝑥𝑥 𝛼𝛼𝑛𝑛−𝛼𝛼𝑝𝑝 𝑑𝑑𝑎𝑎𝑑𝑑𝑑𝑥𝑥 1 + � 0 𝑊𝑊 𝛼𝛼𝑝𝑝 − 𝛼𝛼𝑛𝑛 𝑒𝑒− ∫0 𝑥𝑥 _𝛼𝛼 𝑛𝑛−𝛼𝛼𝑝𝑝 𝑑𝑑𝑎𝑎𝑑𝑑𝑑𝑥𝑥 = � 0 𝑊𝑊 𝛼𝛼𝑝𝑝𝑒𝑒− ∫0 𝑥𝑥 _𝛼𝛼 𝑛𝑛−𝛼𝛼𝑝𝑝 𝑑𝑑𝑎𝑎𝑑𝑑𝑑𝑥𝑥 + 1 𝑀𝑀𝑝𝑝 ≈ 1

Klimeck – Solid State Devices Impact-ionization � 0 𝑊𝑊 𝛼𝛼_𝑛𝑛𝑒𝑒− ∫0𝑥𝑥 𝛼𝛼𝑛𝑛−𝛼𝛼𝑝𝑝 𝑑𝑑𝑎𝑎𝑑𝑑𝑑𝑥𝑥 = 1 − 1 𝑀𝑀_𝑝𝑝 x _{W 0} ≈ 1 Assume: 𝛼𝛼_𝑛𝑛 = 𝛼𝛼_𝑝𝑝 � 0 𝑊𝑊 𝛼𝛼_𝑛𝑛 𝑥𝑥 𝑑𝑑𝑥𝑥 = 1

this cannot be quite right… n, p different masses etc

Electric Field

Position

VA<0 VA=0

VA>0

Assume: 𝛼𝛼_𝑛𝑛(𝑥𝑥) = 𝛼𝛼_𝑛𝑛 Most avalanche happens spatially _{at maximum electric field} 𝛼𝛼𝑛𝑛 𝑊𝑊 = 1

0

B

p A e

−

α = E from experiment and theory A and B material coeffients

( )

(

)

1 / 2 0 0 2 0 D n D A bi A s s D A qN x q N N V V k k N N − _{= =}  ₋   ₊    ε ε E Breakdown-Field

Impact-ionization: In Practice

Good …. Imagers

single photon can create an avalanche

Bad…. n p Insulating Layer d rj rj Photon Detector

High E-fields at junction corners  Breakdown

Klimeck – Solid State Devices

Junction Engineering

E E

Reduced field for p-i-n junction, because V_bi (area under the curve) must be the same.

n p Insulating Layer d rj rj i-region Lower E-field! Intrinsic region: E-field has to be constant, because there are no charges!

High practical relevance for MOSFET scaling Needed to reduce fields on the drain side….

Modern Considerations: Dead Space

For very small (ballistic) junctions,

electrons can cross the junction without inducing impact Ionization.

(Dead space too small)

W

Dead Space

Dead Space:

Space you need before an electron can impact ionize.

Klimeck – Solid State Devices

Zener Breakdown vs. Impact Ionization

How do you differentiate between Zener tunneling and

impact-ionization? Need bias > bandgap

Can happen for smaller reverse bias ~1V Very noisy – can be measured

Non-Ideal Effects: Conclusion

1) Junction recombination is often used as a diagnostic tool for process maturity. Defects in junction arises from misplaced donor impurities, not necessary from deep-trap impurities.

2) Impact ionization plays an important role in wide variety of devices (e.g. avalanche photo-diodes).

Klimeck – Solid State Devices

Section 20

PN Diode I-V Characteristics

• 20.1 Band diagram with applied bias

• 20.2 Derivation of the forward bias formula • 20.3 Forward Bias - Non-linear Regime

»Resistive drop

»Ambipolar regime

• 20.4 Non-ideal effects: »Junction recombination »Tunneling

»Impact ionization status

V

V

V

Section 20

PN Diode I-V Characteristics

• 20.1 Band diagram with applied bias

• 20.2 Derivation of the forward bias formula • 20.3 Forward Bias - Non-linear Regime

»Resistive drop »Ambipolar regime • 20.4 Non-ideal effects: »Junction recombination »Tunneling »Impact ionization V V V V