Photodetectors and Solar Cells - Cornell University · Photodetector Bandwidth: Small Signal Model...

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ECE 407 – Spring 2009 – Farhan Rana – Cornell University

Photodetectors and Solar Cells

In this lecture you will learn:

• Photodiodes• Avalanche Photodiodes• Solar Cells• Fundamentals Limits on Solar Energy Conversion • Practical Solar Cells


Photodiodes

x-xpxn

1 (p doped) 2 (n doped)+ ++ ++ +

- -- -- -

V+ -

Wn-Wp

I

1KT

qV

o eII

Consider the standard pn diode structure:

The pn diode can be used as a photodiode

2


Photogeneration in the Quasineutral Region

x-xpxn

1 (p doped) 2 (n doped)+ ++ ++ +

- -- -- -

Wn-Wp

Light

IL

Consider an unbiased photodiode with light shining on the p quasineutral region:

We need to compute the current IL in the external circuit

Start from:

xGxRxGxJxqt

nLeee

1

Generation term due to light0

LL

e

poee

ee

GxG

nxnxRxG

xxn

qDxJ

constant

only diffusion by current carrier minority

1

1

Assumptions:


Shockley’s Equations for Photodiodes: Electron Density

x-xp xn

1 (p doped) 2 (n doped)+ ++ ++ +

- -- -- -

Wn-Wp

Light

IL

We get for the minority carrier density:

1

21

2

2

e

L

e

po

DG

L

nxn

x

xn

Boundary conditions:

pop

poKTqV

pop

nWxn

nenxxn

At depletion region edge

At the left metal contact

Solution:

1

1111

sinh

sinhsinh

e

pp

e

p

e

p

eLeLpo

L

xW

L

xx

L

xW

GGnxn

Solution in the short base limit is:

ppe xWL 1

2

1

2

1 2222

pp

e

Lpp

e

Lpo

xWx

DGxW

DG

nxn

3


Shockley’s Equations for Photodiodes: Electron and Hole Currents

x

-xpxn Wn-Wp

npo

n(x)

1

111

1

sinh

sinhsinh

e

pp

e

p

e

p

eL

eLpo

L

xW

L

xx

L

xW

G

Gnxn

Electron Current:Electron (minority carrier) current flows by diffusion:

limit base Short 2

sinh

coshcosh

1

1111

ppL

e

pp

e

p

e

p

eLee

xWxqG

L

xW

L

xx

L

xW

LqGxn

DqxJ


Shockley’s Equations for Photodiodes: Electron and Hole Currents

Total Current:Total current is due to the electrons that reach the depletion region edge

limit base Short 2

sinh

1cosh

1

11

ppL

e

pp

e

pp

eLpeT

xWqG

L

xW

L

xW

LqGxJJ x-xp xn Wn-Wp

npo

n(x)

Hole Current:

xJJxJ eTh

x-xpxn Wn-Wp

Jh(x)

Je(x)

JT

Je(x)=JT

Jh(x)=00

4


Shockley’s Equations for Photodiodes: Hole Currents

Quasineutrality implies:

xnxp

x-xp xn Wn-Wp

npo

n(x)

x-xp xn Wn-Wp

ppo

p(x)

Hole Diffusion Current:

xJDD

xn

Dq

xp

DqxJ

ee

hh

hdiffh

1

11

1,

Hole Drift Current:

xJDD

JxJ

xJxJJxJ

xJJxJxJ

xJJxJ

ee

hTdrifth

diffheTdrifth

eTdiffhdrifth

eTh

1

1,

,,

,,

1


External Circuit Current

x-xpxn

1 (p doped) 2 (n doped)+ ++ ++ +

- -- -- -

Wn-Wp

Light

IL

limit base Short 2

sinh

1cosh

1

11

ppL

e

pp

e

pp

eLL

TL

xWAqG

L

xW

L

xW

ALqGI

AJI

Even if one ignores recombination inside the quasineutral region (short base limit), the circuit current corresponds to only half the total number of electron-hole pairs generated by light inside the detector (why?)

5


Photogeneration in the Depletion Region

x-xpxn

1 (p doped) 2 (n doped)+ ++ ++ +

- -- -- -

Wn-Wp

Light

IL

Electron Current:

pnLxx Lphnh

Lh

pnLxx Lnepe

Le

xxqGdxGqxJxJ

Gx

xJq

xxqGdxGqxJxJ

Gx

xJq

np

np

1

1

Hole Current:


Electron, Hole, and Total Currents

x-xpxn

1 (p doped) 2 (n doped)+ ++ ++ +

- -- -- -

Wn-Wp

Light

IL

Electron and Hole Current:The electric field inside the depletion region sweeps the electrons towards the n-side and the holes towards the p-side. Consequently, it must be that:

0

0

nh

pe

xJ

xJ

pnLneph xxqGxJxJ Therefore,

Total Current:

pnL

phpenhneT

xxqG

xJxJxJxJJ

6


x-xpxn Wn-Wp

Je(x)=0

JT Je(x)=JT

Jh(x)=0

Jh(x)=JT

Electron, Hole, and Total Currents

0

x-xpxn

1 (p doped) 2 (n doped)+ ++ ++ +

- -- -- -

Wn-Wp

Light

IL

External Circuit Current

pnLTL xxqGAJI

The circuit current corresponds to the total number of electron-hole pairs generated by light inside the detector!


Photodetector Circuits: Electrical Characteristics

x-xpxn

1 (p doped) 2 (n doped)+ ++ ++ +

- -- -- -

VR

+-

Wn-Wp

I

+ -V

R

Photodetectors are operated in reverse bias (why?)

The circuit current I has two components:

i) The current due to the biased pn junction given as:

ii) The current IL due to photogeneration

These two components can be added together (why?) to give the total current:

1KTqVo eI

LKTqV

o IeII 1

7


Photodetector Circuits: Electrical Characteristics

x-xp xn

1 (p doped) 2 (n doped)+ ++ ++ +

- -- -- -

VR+-

Wn-Wp

I

+ -V

R

We had:

LKTqV

o IeII 1 (1)

Kirchhoff’s Voltage Law gives:

0 VVIR R (2)

V

I

Isc = -( Io + IL )

Increasing R

Voc-VR

The solution of these two equations gives the circuit current and the junction voltage

Open circuit voltage (R = ∞):

1ln

o

Loc I

Iq

KTVV

Short circuit current (R = 0): :

Losc IIII

Current due to photogeneration

Current due to thermal generation

(1)(2)


Common Photodetector Structures

n doped substrate

p+ doped

Light

x

0

AR coatingPassivation

Metal contacts

Front side illuminated homojunction photodetector

n doped substrate

p+ doped

Light

x

0

AR coating

intrinsic

Passivation

Metal contacts

Front side illuminated homojunction pin photodetector

x

o

L

eI

xI

RRxG

xoeIxI

8


Common Photodetector Structures

n doped substrate (InP)

p+ doped (InP)

Light

x

0

AR coating

Intrinsic In0.53Ga0.47As

Passivation

Metal contacts

Front side illuminated heterojunction pin photodetector(used for 1550 nm fiber optic communications)

Photogeneration only in the intrinsic region!


Figures of Merit of Photodetectors

Responsivity (units: Amps/Watt):

inc

L

PI

R

External Quantum Efficiency:

RqP

IqP

qI

inc

L

inc

Lext

W

External quantum efficiency measures the fraction of photons incident on the photodetector that ended up producing an electron in the external circuit

Internal Quantum Efficiency:

abs

L

abs

L

PI

qPqI

int

Internal quantum efficiency measures the fraction of photons that caused photogeneration and also ended up producing an electron in the external circuit

9


Figures of Merit of Photodetectors

Dark Current:

The dark current of a photodetector is the current present even if there no light and limits the detector sensitivity

Lo III

When VR=0:

Current due to thermal generation → dark current

Device leakage current also contributes to dark current

kT

E

vciio

g

eNNnAnI 222 where

Recall that:

Small bandgaps and high temperatures increase dark current


Photodetector Bandwidth

We assume that the optical power incident of the photodiode is time dependent:

ftjoinc efPPtP 2Re

Suppose the current in the external circuit is:

tI

ftjo efIItI 2Re

We know that: oo PRI

But how do we relate I(f ) to P(f )?

tPinc

tI

10


Photodetector Bandwidth: Small Signal Model

RdRext

Cj R h(f ) P(f )

I(f)

Cd

dR Differential resistance of the pn junction KTqV

o

KTqV

o

ReqIkT

eqIkT

jC Diode junction capacitance

dC Diode depletion capacitance due to charge storage in the quasineutral regions and can be ignored in a reverse biased pn junction

fh A frequency dependent function that describes the intrinsic frequency response of the photogenerated carriers inside the photodiode

extjextdjextd

d

RCfjfhfRP

RRCfjRR

RfhfRPfI

211

2

The RC limit of the detector bandwidth

0

10

fh

fhLimiting behavior of h(f ):


Intrinsic Frequency Limitations of Photodetectors: h(f )

x-xp xn

1 (p-doped) 2 (n-doped)

VR

+-

Wn-Wp

I

+ -V

x

Consider an electron generated at point x inside the junction at time t=0

t t

pn

e

xx

Eq

pn

h

xxEq

E

xx

e

n

E

xx

h

p

tIxe tIxh

Ramo-Shockley Theorem:As the electron and hole move inside the depletion region under the influence of the electric field, there is current flow in the external circuit due to image charges (or capacitive coupling)

qE

xx

xxE

qE

xxxx

EqdttItI

h

p

pn

h

e

n

pn

e

oxhxe

11



hhh

eee

x

xxhxe

pn

ttq

ttq

dxtItIxx

tIn

p

01

011

The actual detector current consists of photogenerated pairs at all locations inside the depletion region. So the average current waveform due to one photogeneration event is:

The above transients can be approximated as:

E

xx

e

pne

E

xx

h

pnh

Electron and hole transit times through the entire depletion region are:

022

teq

eq

tI he

t

h

t

e

t t

e

q

h

q

e h

tIxe tIxh



022

teq

eq

tI he

t

h

t

e

The average current waveform due one photogeneration event is:

This is the impulse response h(t ) of the detector: q

tIth

The Fourier transform of the impulse response will give h(f ):

he fjfj

fh 21

212121

Describes the frequency limitations due to the electron and hole transit times through the depletion region

extjhe

extj

RCfjfjfjfRP

RCfjfPfRhfI

211

2121

2121

211

Finally we have:

Describes the frequency limitations due both intrinsic and extrinsic (circuit level) factors

12


Avalanche Photodiodes (APDs): Basic Principle

Photomultiplier tubes:

Provide high sensitivity for light detection even at the single photon level Employs electron multiplication to increase the charge output per photon

Impact Ionization in Semiconductors:

he

eg mm

mEE 1electronmin

he

hg mm

mEE 1holemin

Highly energetic electrons and holes in semiconductors can create electron-hole pairs via impact ionization


Modeling Impact Ionization in Semiconductors

e

Impact Ionization Coefficients:

The electron ionization coefficient (units: 1/cm) defined as the number of electron-hole pairs created by one electron in unit distance of travel

h The hole ionization coefficient (units: 1/cm) defined as the number of electron-hole pairs created by one hole in unit distance of travel

EChh

ECee

h

e

eB

eA

Electrons and holes in high electric fields gain enough energy to cause impact ionization

cE

vE

13


Avalanche Photodiodes

xxn -xp

-qV

Avalanche pin photodiode

Light

xqGxJxJx

xJ

xqGxJxJxGxGqx

xJ

Lhheee

LhheeLee

The equation for the electron current is:

The equation for the hole current is:

xqGxJxJx

xJ

xqGxJxJxGxGqx

xJ

Lhheeh

LhheeLhh



xxn-xp

-qV

Light

The equation for the electron current becomes:

Total current is:

xJxJJ heT

xqGJxJx

xJLThehe

e

he

xx

ThLxx

pene

pnhe

pnhee

JqGexJxJ

1

Solution, assuming uniform illumination, is:

0 pe xJ 0nh xJ

Boundary conditions:

Total current is:

nenhneT xJxJxJJ

14



xxn-xp

-qV

Light

Total current is:

nenhneT xJxJxJJ

1

1

pnhe

pnhe

xxhhe

xx

LTe

eqGJ

The multiplication gain M of the photodetector is:

1

11pnhe

pnhe

xxhhe

xx

pnpnL

T

e

e

xxxxqGJ

M



n doped substrate (InP)

p+ doped (InP)

Light AR coating

Intrinsic In0.53Ga0.47As (Light absorption)

Passivation

Metal contacts

n- doped InP (Multiplication layer)

n+ doped InP

Front side illuminated heterostructure avalanche photodiode with separate photogeneration and multiplication layers

Avalanche photodiodes are good for high sensitivity but low speed applications

Most avalanche photodiodes have separate regions for light absorption and carrier multiplication.

15


Solar Cell Basics


Solar Cell Basics

x-xpxn

1 (p doped) 2 (n doped)+ ++ ++ +

- -- -- -

Wn-Wp

Light Pinc

I

R

V+ -

LKT

qV

o IeII

1

0VIR

V

I

Isc = -( Io + IL )

Increasing R

Voc

LKTqV

o IeII

1

RV

I

How to deliver the maximum power to the resistor R?

01

VIeI

dVd

LKT

qV

o

VIeIP LKTqV

oout

1

16


Maximizing Output Power of Solar Cells

V

I

Isc = -( Io + IL )

Increasing R

VocVm

Im

LKTqV

o IeII

1

RV

I

Area of rectangle = -ImVm = Pout = Power delivered to the resistor R

Vm = Voltage at maximum output power Im = Current at maximum output power

11

01

o

LmKTqV

LKT

qV

o

II

KTqV

e

VIeIdVd

m

ocscmmout VIVIP

LKT

qV

om IeIIm

1

x-xpxn

1 (p doped) 2 (n doped)+ ++ ++ +

- -- -- -

Wn-Wp

Light Pinc

I

R

V+ -


Resistance Matching in Solar Cells

V

I

Isc = -( Io + IL )

Increasing R

VocVm

Im

LKTqV

o IeII

1

RV

I

What value of the resistor R lets one achieve the maximum output power?

dm

KT

qVo

Re

KT

qI

R

m 11

ocsc

mm

VIVI

FF

Solar cell Fill Factor (FF):

Solar Efficiency:

inc

mm

PVI

x-xpxn

1 (p doped) 2 (n doped)+ ++ ++ +

- -- -- -

Wn-Wp

Light Pinc

I

R

V+ -

Solar cells are always operated in forward bias!

17


Black Body Radiation

Solar radiation spectrum near Earth can be very well approximated as that of a black body at temperature Ts = 5760K

r

z T P

Consider a point P at a distance r from a black body at temperature T

We need to calculate the photon flux and the radiation power at the point P

Each mode of radiation emitted from the black body with wavevectior will have photon occupancy given by:

1

1

KTqeqn

q

The total photon flux per unit area at P can then be written as an integral:

ngdcqnc

dqqdF p

oN

o

0

2

0 03

2

4

sin

2

2cossin2

32

2

cgp

Where the photon density of states in free space is:


Black Body Radiation

r

z T P

The energy flux at the point P is:

4232

422

0

2sin

60sin

4sin

Tc

KTngdcF oop

oE

428

K-m

Watts1067.5constant BoltzmannStephan

s

degrees267.0s 2Watts/m1350EF

Sun

Radiation at the surface of the Earth:

Ts

18


Fundamental Limits to Solar Energy Conversion Efficiency

Carnot engine

Sun

Device

Earth

A device at temperature Tc absorbs solar energy, radiates a part of it away, and then converts the rest into useful work W by an ideal heat engine operating between the temperature Tc and the ambient temperature on Earth Ta = 300K

Net radiative energy absorbed by the device (and which is available for useful work):

442sin css TT

Energy conversion efficiency of an ideal heat engine (Carnot engine):

ca TT1Energy conversion efficiency of the device:

42

442

sin

1sin

ss

c

acss

T

T

TTT

But the above expression does not consider the fact that solar energy can be concentrated!


Solar Energy Concentration and Energy Conversion Efficiency

Suns

Device

Sun

actual

Lens

Device

One can increase the efficiency by focusing or concentrating sunlight

Focusing can increase the effective half-angle s of the Sun from 0.267 degrees to the maximum possible value of 90 degrees

s

actualX

2

2

sin

sin

Light concentration is measured in units of the parameter X:

42

106.4sin

11

s

X

c

a

s

c

ss

c

acss

T

T

T

T

T

T

TTT

11sin

1sin

4

4

42

442

With full concentration, the maximum efficiency for solar energy conversion is:

A maximum efficiency of close to 85% is achieved for Tc = 2450K

Ts Tc

Ts

Tc

19


Fundamental Energy Conversion Efficiency of a PhotodiodeConsider a solar cell made from a semiconductor with a bandgap Egat temperature Ta = 300K

Assumption: The solar cells absorbs all photons whose energies are larger than the bandgap

We know from Chapter 3 that in the spontaneously emitted radiation by a forward bias junction the occupancy of a radiation mode of frequency is:

1

1

aKTqVe

n

Sunlight

Ta

SPE

1

14

,,min

min

KTqVpNe

gdc

VTF

1

14

,,min

min

KTqVpEe

gdc

VTF

General expressions for emitted photon number and energy fluxes:

Cell


Fundamental Energy Conversion Efficiency of a PhotodiodeSunlight

Ta

SPESun

s

Ta

Photodiode

Ts

The radiation power per unit area incident on the cell from the Sun is:

Cell

0,0,sin2sEs TFA

The photon flux per unit area incident on the cell from the Sun and absorbed by the cell is: gsNs ETFA ,0,sin2

A

The photon flux per unit area incident on the cell from the surrounding ambient and absorbed by the cell is:

gaNs ETFA ,0,sin1 2

The photon flux per unit area emitted by the cell is:

1

14

sin2

sg

KTpE

se

gdc

A

1

14

sin2

sg

KTpE

se

gdc

A

1

14

sin1 2

ag

KTpE

se

gdc

A

gaN EVTAF ,, 1

14

ag

KTqVpE e

gdc

A

20


Fundamental Energy Conversion Efficiency of a PhotodiodeSunlight

Ta

SPESun

s

Ta

Photodiode

Ts

Cell

AAssumption: The internal quantum efficiency of the cell is 100% and each photon absorbed by the cell results in one electron flow in the external circuit:

I

gaNgaNsgsNs EVTAFETFAETFqAI ,,,0,sin1,0,sin 22

0,0,sin

,,,0,sin1,0,sin

0,0,sin

2

22

2

sEs

gaNgaNsgsNs

sEs

TFA

VEVTFETFETFqA

TFA

IV

The energy conversion efficiency of the cell is:

To find the maximum value of one must maximize the above w.r.t. to the voltage V

V+ -

R


Shockley–Queisser LimitSunlight

Ta

SPE

Cell

AI

V+ -

R

No Concentration (s = 0.26o): A maximum efficiency of ~31% is reached at a bandgap of ~1.27 eV

Full Concentration (s = 90o): A maximum efficiency of ~41% is reached at a bandgap of ~1.1 eV.

Shockley–Queisser result

Bandgap too small Output voltage too small

goc EqVqV

Bandgap too large Low energy photons do not get absorbed

Why is there an optimal bandgap?

21


Material Voc (V) Jsc (Amp/cm2) FF (%) Efficiency (%)Crystalline Si 0.705 42.7 82.8 25.0Crystalline GaAs 1.045 29.7 84.7 26.1Poly-Si 0.664 38.0 80.9 20.4a-Si 0.859 17.5 63.0 9.5CuInGaSe2 (CIGS) 0.716 33.7 80.3 19.4CdTe 0.845 26.1 75.5 16.7

Typical Performances of Semiconductor Photocells (Green at al., Prog. Photovolt: Res. Appl., 17, 85 (2009))

Energy Conversion Efficiencies of Some Common Solar Cells

Shockley–Queisser result


Design of Silicon Solar Cells

The PERL Si solar cell (Green et al., 1994)Efficiency ~25%

Light trapping by a top lambertiansurface and bottom reflector can increase the optical path length by ~4n2

22


Design of Solar Cell Modules

The IV characteristics of pn junctions connected in series must identical (or nearly so)


Improving Efficiencies of Photodiode Solar Cells

Eg

Ec

Ev

qV

Efe

Efh

Energy loss

Light

X

0 1 2 3 4 50

0.2

0.4

0.6

0.8

1

1.2

Energy (eV)

So

lar

Sp

ectr

um

Ts = 5760K

Energy Loss Pathways:

i) Photons with energies smaller than the bandgap are not absorbed

ii) Photons with energies much larger than the bandgap generate electrons (holes) with energies much above (below) the band edge. This extra energy is lost to phonons and does not contribute to output electrical power. Recall that:

goc EqVqV

23


Improving Efficiencies of Solar Cells: Tandem Cells

Eg1

LightEg2

Eg3

Eg1 > Eg2 > Eg3

A three junction tandem cell

Eg1

Light

Eg2

Eg3

A three junction tandem cell with spectral filtering


Improving Efficiencies of Solar Cells: Tandem Cells

InGaP/GaAs two-junction tandem cell with a reversed biased Esaki tunnel junction and an efficiency of 30.3% (w/o concentration) (Takamoto et al., 1997).

Photodetectors and Solar Cells - Cornell University · Photodetector Bandwidth: Small Signal Model...

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