ME 432 Fundamentals of Modern Photovoltaics

Discussion 22: PN Junctions

13 October 2020

Fundamental concepts underlying PV conversion

solar spectrum

light absorption

carrier excitation & thermalization

charge transport

charge separation

charge collection

Courtesy: Yosuke Kanai, University of North Carolina

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After Light Absorption & Thermalization:en

• We are left with electrons at the conduction band edge, and holes in the valence band edge.

• These are free carriers that contribute to conduction• Effectively, there are two possible outcomes from here:

direction of electron flow

direction of hole flow

Outcome 1: Electrons and holes are physically separated in space, and extracted into an external circuit to do useful work

• The electron and hole are extracted into opposite terminals of an external circuit, and used to do electrical work

• This is the desired outcome for a solar cell

• Recombination takes place in the interior of the solar cell• The stored energy is lost, as a photon of light is emitted and

the electron drops back to the valence band• Undesirable for a solar cell (whereas this is the desired

outcome for an LED)

Outcome 2: Electrons and holes recombine before they can be physically separated and collected

Learning Objectives: PN Junction

1. Draw pictorially, with fixed and mobile charges, how the built-in electric field of a PN junction is formed.

2. Describe what the depletion approximation is. Describe how to solve for the electric field, potential, and band energy diagrams starting from the depletion approximation.

3. Describe how the built in field in a PN junction can separate charge carriers

4. Explain conceptually the nature of band bending and the drift, diffusion, and illumination current in a PN junction under zero, forward, and reverse bias.

Some Suggested Readings: Green, Chapter 4 (posted online)Luque & Hegedus, Chapter 3 (available for download through UIUC online library)www.pveducation.org/pvcdrom, Chapters 3 & 4

A PN junction is born

Let’s imagine the n- and p- type materials in contact, but with an imaginary barrier in between them.

When the imaginary boundary is removed, electrons and holes diffuse into the other side.

electronsholes

Diffusion current: governed by electrons on the n-type side and holes on the p-type side. Hence we call it a majority carrier current.

Eventually, an electric field builds up. The electric field induced drift current cancels out the diffusion current, and steady state is reached.

Free-carrier charge distribution across a PN junction at steady state

electronsholes

Drift current: governed by holes on the n-type side and electrons on the p-type side. Hence we call it a minority carrier current.

At equilibrium, the minority current is equal and opposite to the majority current. This is true for holes and for electrons separately: Jh,drift=-Jh,diff and Je,drift=-Je,diff. The total current (the sum of all four contributions) is zero.

Determining the Built-in Potential at Equilibrium

qψo = Eg − EF −EV( )− EC −EF( )

ln NAND

yo = built-in potential across pn junction

Determining the Built-in Potential at Equilibrium

Derivation:

n = ND = Nc exp −EC −EF

p = NA = Nv exp −EF −EVkT

Note: we used our carrier statistics equations for doped semiconductors in the derivation:

Free-carrier charge distribution across a PN junction at steady state

Jh,drift + Jh,diff = 0

qµh pξ = qDhdpdx

ξ =kTq1pdpdx

ψo = −kTqln p

−qψo

kT= ln

p b( )p a( )

p b( ) = p a( )exp − qψo

Einstein relations

integrate

Free-carrier charge distribution across a PN junction at equilibrium

To analyze the current flow in a PN junction, we invoke the depletion approximation, in which we divide the PN junction into three regions.

quasi-neutral region quasi-neutral region

• This region is “depleted” of free carriers. Hence it is called the “depletion region.”• Because the net charges here do not cancel out (in contrast to the quasi-neutral

region), it is also sometimes referred to as the “space charge region”.

The Depletion Approximation

To analyze the current flow in a PN junction, we invoke the depletion approximation, in which we divide the PN junction into three regions.

Dashed line is the actual charge distribution

Solid line is the depletion approximation

Solving for electric field, voltage, and energy

Start with depletion approximation ρ x( )

ξ x( )

Start with depletion approximation

Integrate the Poisson Equation:

ρ x( )

dξ x( )dx

=ρ x( )ε

ξ x( )

ρ x( )

ψ x( ) = built-in potential

dξ x( )dx

=ρ x( )ε

Integrate the E-field to get the potential:

dψ x( )dx

= −ξ x( )

ψ x( )

ξ x( )

ρ x( )

E x( )

= built-in potential

band energy EC, EV

dξ x( )dx

=ρ x( )ε

Integrate the E-field to get the potential:

dψ x( )dx

= −ξ x( )

E x( ) = −qψ x( )

Solving for electric field, voltage, and energyCarrying out the analysis (that is, integrating successively and applying charge neutrality constraints), one can obtain expressions for the maximum value of the electric field, the width of the depletion region, the built in potential, and the amount of band bending. The built-in potential matches the expression we derived earlier.

Note: We won’t do it here. See Green Chapter 4 for the full development & exposition.

ξmax = −

ψo( )1NA

Extreme Value of the Electric Field

lp =WND

NA + ND

%& ln =W

NA + ND

W = ln + lp =2εq

ψo( ) 1NA

Width of the Depletion Region

Summary of current understanding

1. The built-in electric field is established at a pn-junction because of the balance of electron and hole drift and diffusion currents.

2. When light creates an electron-hole pair, a pn-junction can separate the positive and negative charges because of the built in electric field

How the built-in electric field can separate charges for solar cell operation.

electric field+-

• Say we hook up our pn junction to a wire (still with no applied bias).• If the illumination happens to create an electron-hole pair close to the transition

region on the p-type side, the built-in electric field quickly sweeps the electron to the n-type side. Charge separation is achieved!

• Note that this is a minority current. Key concept: Current that is generated by solar cells arises from the minority carriers.

How the built-in electric field can separate charges during solar cell operation.

electric field+-

• Say we hook up our pn junction to a wire (still with no applied bias).• If instead the illumination happens to create an electron-hole pair close to the

transition region but now on the n-type side, the built-in electric field quickly sweeps the hole to the p-type side. Charge separation is achieved!

• Note that this is a still minority current. Key concept: The current that is generated by a solar cell arises from the minority carriers.

How the built-in electric field can separate charges for solar cell operation.

electric field+-

• Key concept: Current that is generated by solar cells arises from the minority carriers.

• Note: If an electron-hole pair is created by light excitation far away from the transition region, it is more likely that the minority carrier will recombine with a majority carrier before it reaches the electric-field region and can be separated.

• Implication: to make the best solar cells, we want to maximize the minority carrier diffusion length (average distance a minority carrier will travel before recombining)

Application of a Bias Voltage to pn junction

P N P N

Forward bias: Va > 0 Reverse bias: Va < 0

electric fieldelectric field+ - +-

+ - +-

• Opposes the built-in potential

• Aligns with the built-in potential

The application of a bias (forward or reverse) perturbs the net balance between majority and minority current. Consequently, current now flows in through the pn junction.

Application of a Bias Voltage to pn junction

P N P N

Forward bias: Va > 0 Reverse bias: Va < 0

electric fieldelectric field+ - +-

+ - +-

• Opposes the built-in potential

• Reduces band-bending

• Aligns with the built-in potential

• Increases band-bending

The application of a bias (forward or reverse) perturbs the net balance between diffusion (majority) and drift (minority) current. Consequently, current now flows in through the pn junction.

First, the punchline …• The total current in a pn junction can be written as the sum of the

response in the dark and the response from the illumination • The application of a bias (forward or reverse) to a pn junction solar cell

affects the current flow through the device exponentially• The total current in a pn junction is given by the following expression:

Jtot = Jdark + Jill

= Jo exp qVAkT

%&−1

,-+ Jill

response in the dark

Response to illuminationDiffusion

currentDrift current

pn junction: in the dark

Jtot = Jdark + Jill

= Jo exp qVAkT

%&−1

,-+ Jill

Consider first the pn junction in the dark: Jill = 0

• This is known as the “ideal diode equation”• Note that if VA = 0 (no applied bias), then the total current is

zero because the diffusion and drift components cancel out• This is consistent with what we found before

pn junction: in the dark, with VA=0

Jdrift = 0, Jdiff=0,Jtot = Jdrift + Jdrift = 0

In the quasi-neutral regions:

pn junction: in the dark, with VA=0

Jdrift = -Jdiff ≠ 0,Jtot = Jdrift + Jdiff = 0

In the transition region:

pn junction: in the dark, VA ≠ 0This figure shows what a typical J-V curve for a pn junction in the dark looks like as a function of applied bias VA.

-1.0 -0.5 0.5 1.0VA

100Current Density (mA/cm2)

Jtot = Jo expqVAkT

%&−1

Using: Jo =0.0001 mA/cm2

kT = 0.026 eVq=+1, and VA in Volts

As applied bias VAgets larger, the current grows exponentially

(Volts)

Jtot = Jo expqVAkT

%&−1

This figure shows what a typical J-V curve for a pn junction in the dark looks like as a function of applied bias VA.

-1.0 -0.5 0.5 1.0VA

(Volts)

As applied bias VAgets more negative, the current drops exponentially

pn junction: in the dark, VA ≠ 0

Jtot = Jo expqVAkT

%&−1

This figure shows what a typical J-V curve for a pn junction in the dark looks like as a function of applied bias VA.

-1.0 -0.5 0.5 1.0VA

(Volts)

zoomed in:-1.0 -0.8 -0.6 -0.4 -0.2

-0.0002

pn junction: in the dark, VA ≠ 0

As applied bias VAgets more negative, the current drops exponentially, leaving behind current -Jo

why the exponential dependence on VA?Recall: for VA = 0 we have no current in the quasineutral region (or anywhere)

The minority carrier concentration depends exponentially on the band bending 𝜓!

why the exponential dependence on VA?

Forward bias reduces the band bending from 𝜓! to .

It exponentially increases the minority carrier concentration at the edge of the depletion region.

And then exponentially increases the minority carrier diffusion current!

this is called minority carrier injection

P depletion n

region

minorityminority

carrier'am'¥fusiow

diffusion currentcurrent

electrons←→ holes

# midininihomacmwnent

pn junction: under illumination

Jtot = Jdark + Jill

= Jo exp qVAkT

%&−1

,-+ Jill

• The effect of illumination is to increase the minority carrier current (the drift current). Thus, Jill is negative and its magnitude depends on the intensity of the illumination.

• Additionally, Jill is also independent of the applied bias VA. • Therefore, the net effect of turning illumination on is a constant shift to the J-V

curve.

electric field+-

• Say we hook up our pn junction to a wire (still with no applied bias).• If the illumination happens to create an electron-hole pair close to the transition

region on the p-type side, the built-in electric field quickly sweeps the electron to the n-type side. Charge separation is achieved!

• Note that this is a minority current.

electric field+-

• Say we hook up our pn junction to a wire (still with no applied bias).• If instead the illumination happens to create an electron-hole pair close to the

transition region but now on the n-type side, the built-in electric field quickly sweeps the hole to the p-type side. Charge separation is achieved!

• Note that this is a still minority current

electric field+-

• Note: If an electron-hole pair is created by light excitation far away from the transition region, it is more likely that the minority carrier will recombine with a majority carrier before it reaches the electric-field region and can be separated.

• Implication: under illumination, the contribution to Jill arises from excitations that occur within a diffusion length of the transition region

• This is true, independently of VA. Hence Jill is a constant, negative offset on the J-V curves.

pn junction: under illumination, VA ≠ 0This figure shows what a typical J-V curve for a pn junction under illumination looks like as a function of applied bias VA.

Jtot = Jo expqVAkT

%&−1

,-+ Jill

kT = 0.026 eVq=+1, and VA in VoltsJill = -42.5 mA/cm2

Jo+Jill(Volts)

-1.0 -0.5 0.5 1.0

Current Density (mA/cm2)

illuminated

pn junction: under illumination, VA ≠ 0

Question: Which quadrant of the J-V curve is typically used under

solar cell operation?

Solar Cell Operation:The power density dissipated

P=JV is <0 here (i.e. we are generating power)

Jtot = Jo expqVAkT

%&−1

,-+ Jill

kT = 0.026 eVq=+1, and VA in VoltsJill = -42.5 mA/cm2

(Volts)-1.0 -0.5 0.5 1.0

Jo+Jill

illuminated

Jtot = Jo expqVAkT

%&−1

,-+ Jill

Note that the standard convention adopted by the solar industry is to flip the y-axis on the J-V curves.

Out[8]=

-1.0 -0.5 0.5 1.0Volts

100Current mA

VOut[9]=

-1.0 -0.5 0.5 1.0Volts

100Current mA

Key Concepts:• The current response of an ideal pn-junction under illumination can

be described by the “ideal diode equation” plus a contribution due to the light-generated e-h pairs.

• When light creates an e-h pair within one diffusion length of the depletion region, a pn-junction can separate the positive and negative charges because of the built in electric field

• In a PV device, minority carriers determine the current that is converted to useful energy (electricity)

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