Contributions from the Depletion Region

Contributions from the Depletion Region

at reverse bias, E 

E_Fp

E_Fn

ideal reverse current

added reverse generation current due to generation of carriers from trap level in the depletion region

E_T

E_V E_C

I₀

Space-charge-region (SCR) currents

at small forward bias,

E_Fp E_Fn

E_T

ideal diffusion current

ideal diffusion current recombination current

E_C

E_V

E 

I_SCR

I_SCR

0 SCR

I I I

ideal diffusion SCR

I  I  I

Space-charge-region (SCR) currents Inside the depletion region

( ) /

(EC EV) /kT EFn EFp kT 2 qV kT/

C V i

pn N N e 

e

 n e

The recombination rate is the largest where n   p n e

.

( )

(

1)

dep

Net recombination generation rate per unit volume n e

  

/ 2

: / .

:

( 1) : 0 :

:

dep

qV kT

the generation recombination lifetime in the depletion layer net generation

e equilibrium

net recombination









/ 2 / 2

( 1) ( 1)

N

P

x i qV kT i dep qV kT

SCR x

dep dep

n qn W

I qA e dx A e

 

  

  

/ / 2

,

(

1)

(

1)

dep

Total diode current I I I I e A qn W e

       

Total diode current can be written as,

/ / 2

0

/ 0

( 1) ( 1)

( 1), 1 2 :

i dep

qV kT qV kT

dep qV kT

I I e A qn W e

I e

idealty factor





   

    

Junction leakage current is a very important issue in DRAM technology and generate noise in major devices. Manufacturing these devices requires special care to make the generation/recombination lifetime long with super-clean and nearly crystal-defects free processing to minimize the density of

recombination traps.

Under reverse bias,

(

)

leakage

dep

I I A qn W

   

leakage dep r

I  W  V

Stored excess minority carrier charge in neutral region (Q)

negligible

( , )

xP

Q qA



Charge Storage

For forward biased one sided N

P junction,

2 2

( , )

1

n

n

n

n

n x t d n n

t D dx

dJ n

q dx





 

  



  

, since E 0 in neutral region .

n n

J qD dn dx

   

By multiplying (-qA ) at both sides of the continuity equation and integrating,

( )

( )

1

n

P n P P

J

x J x n x

n

d qA n dx A dJ qA n dx

dt 

  

     

                

: Continuity equation

( ) ( ) ( )

n n P n P nP diffusion diffusion

AJ AJ x AJ x AJ i

     

diffusion

n

dQ Q

dt i 

   : Charge control equation

(continuity equation for excess electron)

include both ac and DC

( )

n

dQ Q

dt ^ i t ^ 

In steady state (DC),

0

dQ and i I I

dt    for one-sided junction

n s

Q Q

I  

   

is called the charge-storage time.

• In a one-sided junction, is the recombination lifetime ( ) on the lighter-doping side.





s n

  

• For N

P one-sided junction,

In general, is an average of the recombination lifetime on the N side and the P side.



Similarly, for forward biased one sided P

N junction,

( )

p

dQ Q

dt ^ i t ^ 

For one sided N

P junction, i

 i t ( ) : total current

Small-Signal Model of the Diode

a.c. small signal,

V

sin , where v

^kT

V v  t  q

  I  i

The small-signal equivalent circuit of a PN diode.

 V

RS

intrinsic part I i

R

/ /

0

(

1)

I  I e   I e

I

-I₀ V

V

I

Slope = 1/R V₀

0

/ 0

/ /

0 0

1 ( ) ( 1)

/

qV kT V V

qV kT qV kT

DC

dI d

G I e

R dV dV

d q kT

I e I e I

dV kT q

 

 

  

R is called diffusion resistance

( )

/

d V V S S S DC

dQ dI kT

C G I

dV

 dV   q

   

Small signal diffusion capacitance

Diffusion capacitance:

dominant in forward bias

Junction capacitance:

dominant in reverse bias

Transient and A-C Conditions

• Most solid state devices are used for switching or for processing a-c signals.

• We investigate the important influence of excess carriers in transient and a-c problems.

Time Variation of Stored Charge

• Any change in current  a change of charge stored in the carrier distributions.

• Since time is required in building up or depleting a charge distribution, the stored charge must inevitably lag behind the current in a time-dependent problem. This is inherently a capacitive effect.

• For a proper solution of a transient problem, we must use the time-

dependent continuity equations

For P

-N junction; two charge storage effects:

(1) The usual recombination term Q

/ 

in which the excess carrier distribution is replaced every 

seconds

(2) A charge buildup (or depletion) term dQ

/dt, which allows for the fact that the distribution of excess carriers can be increasing or decreasing in a time- dependent problem

• Solving the equation with Laplace transform, with I (t > 0)=0 and Q

(0)=I 

, we obtain

   

   

0 1 (0)

0 1

p p p

p

p p p

p

Q s sQ s Q

Q s sQ s I

  



   



     

,

,

P diffusion

p

Q t dQ t

i total diode current i t

   dt



Turn-off transient

 

i t

I

t=0

t

       

0 ( )

p p p

p p

Q t dQ t Q t

i t I for t at steady state

  dt   

 

( ) ( ) 1 f t e

f s a



  L

• The stored charge dies out exponentially from its initial value I 

with a time constant equal to the hole lifetime in the n material.

  1

( )

p p

p t

p p

Q s I

s

Q t I e

 

 

 

P^{+ N}

i(t)

v(t)

N

xN

xP



becomes non-exponential as the transient proceeds.

• Even though the current is suddenly terminated, the voltage across the junction persists until Q

disappears.

• At any time during the transient, the excess hole concentration at x

is

  



'

1 p t  p e 

• The gradient of the hole distribution at x

= 0 (zero current implies zero gradient).

• An approximate solution for v(t) can be obtained by assuming an exponential distribution for at every instant during the decay

 quasi-steady state approximation

' p

x

xN

Transient Junction Voltage, v(t)?

' p

'

p

 

' , '( )

p x t  p t e

• For the stored charge at any instant

  ^'

^'  

N

x x L

p p

Q t  qA 

p e

dx  qAL p t

  

 ^{ }

'( )

1

p

Q t p t p e

   qAL

  ^ln

¹

p n

kT I

v t e

q qAL p

 



          

During turn-off, v(t) cannot be changed instantaneously

Quasi-Steady State Approximation 

( )

p p

Q t   I e

hole distribution in the N-region as a function of time during the transient

'( , ) p x t

Reverse Recovery Transient

• For t < 0 (positive bias  steady I = I_fE/R)

• After the generator voltage is reversed (t > 0), the current must initially reverse to I = I_r - E/R (temporarily).

The reason for this unusually large reverse current through the diode is that the stored charge (and hence the junction voltage) cannot be changed instantaneously.

xN

Switching voltage

Diode current

• As the current is reversed, the junction voltage remains at the small forward-bias it had before t = 0.

• While the current is negative through the junction, the slope of the distribution must be positive at x_N.

• As long as p_nis positive, the junction voltage v(t) is positive and small.

• When the stored charge is depleted and becomes negative, the junction exhibits a negative voltage.

• As time proceeds, the magnitude of the reverse current becomes smaller as

more of –E appears across the reverse-biased junction, until finally the only current is the small reverse saturation current.

• The time t_sd( ) required for the stored charge (and therefore the junction voltage) to become zero is called the storage delay time.

The critical parameter determining t_sd( ) is the carrier lifetime (_p for the P⁺-N junction)







Part II: Application to Optoelectronic Devices

Solar Cells

Photonic Devices:

Solar Cells, Light Emitting Diode, Laser Diode, Photodiode

Solar Cell Basics

• Commonly made of silicon, solar cells, also known as photovoltaic cells, can convert sunlight to electricity with 15 to 30 % energy efficiency.

• The structure of solar cell is identical to a PN junction diode but with finger-shaped or transparent

electrodes so that light can strike the semiconductor.

Depletion of fossil-fuel deposits and recent history and projection of world energy consumption

assuming 3% annual growth. (From [3]. © 1992 IEEE.)

pnJunction Si solar cells at work. Honda„s two seated Dream car is powered by photovoltaics. The Honda Dream was first to finish 3,010 km in four days in the 1996 World Solar Challenge.

SOURCE: Courtesy of Centre for Photovoltaic Engineering, University of New South Wales, Sydney, Australia.

SOURCE: Courtesy of NASA, Dryden Flight Center

Solar cell inventors at Bell Labs (left to right) Gerald Pearson, Daryl Chapin and Calvin Fuller are checking a Si solar cell sample for the amount of

voltage produced (1954).

SOURCE: Courtesy of Bell Labs, Lucent Technologies

The principle of operation of the solar cell (exaggerated features to highlight principles)

Neutral

n-region Neutral

p-region

W E_o

Voc Medium

Long

Depletion region

Diffusion Drift

Finger electrode

Back electrode

n 

p

Lh

Short

Le

• Photogenerated carriers within the volume L

+ W + L

give rise to a photocurrent I

.

• The variation in the photogenerated EHP concentration with distance is also shown where α is the absorption coefficient at the wavelength of interest.

L

L

W

I

x EHPs

exp(  x)

I

 

(a) The solar cell connected to an external load R and the convention for the definitions of positive voltage and positive current. (b) The solar cell in short circuit. The current is the photocurrent, Iph. (c) The solar cell driving an external load R. There is a voltage V and current I in the circuit.

R

I

(a) Light

I

V = 0

I

= I

ph

(b)

I

I = I

d

I

ph

V

I

R (c)

V

Dry Battery + -

I

V

V I (mA)

Dark

Light

Twice the light

0.4 0.6 0.2

20

-20 0

V_oc

I_ph

Short circuit solar cell current in light

K I I

I _sc   _ph  

Constant that depends on the particular device Light intensity Photocurrent generated by light

Solar cell I-V

 

 



  



 



 



 _ph exp 1

kT I eV

I

I _o



where I_ois the reverse saturation current and is the ideality factor: 1 - 2

short circuit current (I_sc)

open circuit voltage

Typical I-V characteristics of a Si solar cell. The I-V curves for positive current requires an external bias voltage. Photovoltaic operation is always In the negative current region.

V

I (mA)

0.6 0.2 0.4

0

Voc

100

I_sc = I_ph The load line for

R = 3 

(I-V for the load)

I-V for a solar cell under an illumination of 700 W m

Operating point Slope = - 1/R

P I

I

I

R V

I

(a) (b)

0.5 0.1 0.3

V

200

(a) When a solar cell drives a load R. R has the same voltage as the solar cell but the current through it is in the opposite direction to the convention that current flows from high to low potential.

(b) The current I and voltage V in the circuit of (a) can be found from a load line construction. Point P is the operating point .The load line is for R = 3 Ω.( ',I V')

Load line

R

I   V

Fill factor

oc sc

FF I V V I



• Electron-hole pair (EHP) generation by light illumination.

EHPs generated in SCR move toward to their respective majority-carrier regions, due to electric field in SCR and form generation current called, - I

• The power dissipated by the diode is negative (i.e., the product (V×I) is negative)

 power generator (Solar power  electrical power)



light illumination

EHP generation

(a) Light can produce a current in PN junction at V = 0.

(b) Solar cell IV product is negative, indicating power generation.

/ 0

,

(

1)

Total diode current I I  I e   I

V

Light Penetration Depth-Direct-Gap and Indirect-Gap Semiconductors

( ) hc 1.24 ( )

Photon energy eV  m

 

 

Photons with energy less than E_g are not absorbed by the semiconductor. Photons with energy larger than E_g are absorbed but some photons may travel a considerable distance in the semiconductor

before being absorbed.

Light intensity( ) x  e

: 1 :

absorption coefficient penetration depth





The thickness of solar cell 1 :

  

• The Si or Ge solar cell must be thick ( > 50 μm) : due to low α

• The GaAs or InP solar cell is thin (~ 1 μm) : due to high α

low  high 

Related to the specific energy band structure