P-N Junction Diodes Current Flowing through a Diode I-V Characteristics Quantitative Analysis (Math, math and more math)

P-N Junction Diodes Current Flowing through a Diode

I-V Characteristics

Quantitative Analysis

(Math, math and more math)

p-n Junction I-V Characteristics

In Equilibrium (no bias) Total current balances due to the sum of the individual components

Electron Drift Current

Electron DiffusionCurrent

Hole Drift CurrentHole Diffusion

Current

Diode under no Bias.mov

no net current!

EC

EV

EF

Ei

p-Type Material n-Type Material- qVBI

+ +++++ +++ ++ ++ ++++++

0 nDqnEqJJJ nnDiffusionnDriftnn

no net current!


EC

EV

EF

Ei

n vs. E

p vs. E

In Equilibrium (no bias)Total current balances due to the sum of the individual components

0 pDqpEqJJJ ppDiffusionpDriftpp


Forward Bias (VA > 0)

I

Hole Drift Current


Electron DiffusionCurrent

Hole Diffusion Current IP

IN

Diode under Forward Bias.mov

Current flow is dominated by majority carriers flowing across the junction and becoming minority carriers

VA

Current flow is proportional to e(Va/Vref) due to the exponential decay of carriers into the majority carrier bands

Lowering of potential hill

by VA

surmount potential barrier

PN III

Hole Diffusion Current negligible due to large energy barrier

Hole Drift Current


Electron Diffusion Current negligible due to large energy barrier

Reverse Bias (VA < 0)

Diode under Reverse Bias.mov


Current flow is constant due to thermally generated carriers swept out by E fields in the depletion region

Current flow is dominated by minority carriers flowing across the junction and becoming majority carriers

Increase of potential hill b

y VA

Where does the Reverse Bias Current come from?

Generation near the depletion region edges “replenishes” the current source.


Putting it all together


for Ideal diodeVref = kT/q

-I0

10

kT

qVexpII

: Diode Ideality Factor


Diode Equation

Quantitative p-n Diode Solution

Assumptions:1) Steady state conditions

2) Non- degenerate doping

3) One- dimensional analysis

4) Low- level injection

5) No light (GL = 0)

Current equations:

)x(J)x(JJ npp

dx

dpqDpEqJ ppp

dx

dnqDnEqJ nnn

Application of the Minority Carrier Diffusion Equation

Since electric fields exist in the

depletion region, the minority carrier diffusion equation

does not apply here.

Quisineutral Region Quisineutral Region

00 0 0


minority carrier diffusion eq. minority carrier diffusion eq.



kT)FF(i

PNennp 2

quasi-Fermi levels formalism

kTEE

i

kTEE

i

fi

if

enp

enn

)(

0

)(

0

Equilibrium

kTFEi

kTEFi

Pi

iN

enp

enn

)(

)(

Non-Equilibrium

The Fermi level is meaningful only when the system is in thermal equilibrium.

The non-equilibrium carrier concentration can be expressed by defining Quasi-Fermi levels Fn and Fp .

Equilibrium Non-Equilibrium

Quasi - Fermi Levels



kT)FF(i

PNennp 2

quasi-Fermi levels formalism

?



dx

dnDnEqJ nnn

dx

nndqD

p

n

0

dx

ndqD

p

n

dx

dpDpEqJ ppp

dx

ppdqD n

p

0

dx

pdqD n

p

0 0

Approach:

Solve minority carrier diffusion equation in quasineutral regions.

Determine minority carrier currents from continuity equation.

Evaluate currents at the depletion region edges.

Add these together and multiply by area to determine the total current through the device.

Use translated axes, x x’ and -x x’’ in our solution.


x”=0 x’=0



x”=0 x’=0


Holes on the n-side



x”=0 x’=0 Holes on the n-side


Similarly for electrons on the p-side…


x”=0 x’=0


Depletion Region

Negligible thermal R-G implies Jn and Jp are constant throughout the depletion region. Thus, the total current can be define in terms of only the current at the depletion region edges.


0 0

0 0 NPppP

pNppN

xJxxxJ

xJxxxJ

)x(J)x(JJ pPpN

Continuity equation

...light as suchprocesses other All

GR Thermal

P

...light as suchprocesses other All

GR Thermal

N

t

p

t

pJ

qt

p

t

n

t

nJ

qt

n

1

1

Continuity Equations



x”=0 x’=0


Total on current is constant throughout the device. Thus, we can characterize the current flow components as…

-xp xn

J

pn-junction diode structure used in the discussion of currents. The sketch shows the dimensions and the bias convention. The cross-sectional area A is assumed to be uniform.

Hole current (solid line) and recombining electron current (dashed line) in the quasi-neutral n-region of the long-base diode of Figure 5.5. The sum of the two currents J (dot-dash line) is constant.

Hole density in the quasi-neutral n-region of an ideal short-base diode under forward bias of Va volts.

The ratio of generation-region width xi to space-charge-region width xd as a function of reverse voltage for several donor concentrations in a one-sided step junction.

The current components in the quasi-neutral regions of a long-base diode under moderate forward bias: J(1) injected minority-carrier current, J(2) majority-carrier current recombining with J(1), J(3) majority-carrier current injected across the junction. J(4) space-charge-region recombination current.

(d) Adapted from [8]. Current-voltage characteristic for a diode near the boundary between (a) and (c), showing diffusion current at lower voltages and a transition to thermionic-emission current at higher biases.

Current in a Heterojunction

(a) Transient increase of excess stored holes in a long-base ideal diode for a constant current drive applied at time zero with the diode initially unbiased. Note the constant gradient at x = xn as time increases from (1) through (5), which indicates a constant injected hole current. (Circuit shown in inset.) (b) Diode voltage VD versus time.

(a) Transient decay of excess stored holes in a long-base ideal diode. In the case shown, the initial forward bias applied through the series resistor is abruptly changed to a negative bias at time t = 0. (Circuit shown in inset.) (b) Diode current ID versus time.

Junction and Free-Carrier Storage

Jelec

x

n-region

J = Jelec + Jhole

SCL

Minority carrier diffusioncurrent

Majority carrier diffusionand drift current

Total current

Jhole

Wn–Wp

p-region

J

The total current anywhere in the device is constant.

Just outside the depletion region it is due to the diffusion of minority carriers.



Thus, evaluating the current components at the depletion region edges, we have…

Note: Vref from our previous qualitative analysis equation is the thermal voltage, kT/q

J = Jn (x”=0) +Jp (x’=0) = Jn (x’=0) +Jn (x”=0) = Jn (x’=0) +Jp (x’=0)

Ideal Diode Equation Shockley Equation

Current-Voltage Characteristics

of a Typical Silicon p-n Junction


ExamplesDiode in a circuit

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P-N Junction Diodes Current Flowing through a Diode I-V Characteristics Quantitative Analysis (Math, math and more math)