How Solar Cells Work - OrgaNextGenerationorganext.org/userfiles/talks-conference/peter_w-rfel.pdf · How Solar Cells Work Basic theory of photovoltaic energy conversion Peter Würfel

How Solar Cells Work

Basic theory of photovoltaic energy conversion

Peter WürfelUniversity of Karlsruhe, Germany

Three Messages

Sun is a heat source, solar cells must be heat engines

Important conversion stepsolar heat chemical energy of electron-hole pairslimited by thermodynamics

Conversion of chemical energy electrical energycan be 100% efficient and needs more than a pn-junction

Important: generation of (entropy-) Free Energy by cooling

F = E - TS

Carnot process

P

V

isothermal

isothermal adiabatic

adiabatic

1

2

3

4

isothermal

isothermal

adiabaticadiabatic

1 2

34

T

S

TA

T0

Solar cells are heat engines

Principle

I = T IE,in S S,in

IS,in

T I0 S,outI IS,out S,in≥

I I (1 - T /T )E,out E,in 0 S£

The solar cell as a heat engine?

Questions:

What is the working medium (the gas)?

What kind of free energy is produced?

Conversion of solar heat into chemical energy of electrons and holes

Energy per photon ħω→ chemical energy µeh per e-h pair by thermalisation

εC εFCεF

εVεFV

εe

gdn /dh eε

dn /de eε

10 s-14 10 s-12

> meh

energy gap is necessary

Maximum chemical energyReduce entropy generation during thermalisation by

reducing the energy range, for which Fermi-distribution is established

ideal: isoenergetic thermalisation in narrow energy ranges is isentropic

DeC eFCeF

DeVeFV

ee

ω

dn /dh ee

dn /de ee

10 s-14 10 s-12

μeh

e

h

tandem cells

Tandem cells2 cells → 4 Fermi-energies for 4 energy ranges

4-level system3 Fermi-energies for connection in series

eF1

eF2

eF3

eF4

Recombinationrecombination

at the surface in the materialnon-radiative non-radiative

radiative

surface recombination bulk lifetimevelocity

effective lifetime

RecombinationDirect optical transitions in 2-level system

drup drspont drstim

[ ]212 1 2

d ( )d ( ) ( ) 1 ( ) d( )

d( )up

jr M D f f γ ω

ω ε ε ωω

= −

e1

e2

[ ]212 2 1

d ( )d ( ) ( ) 1 ( ) d( )

d( )stim

jr M D f f γ ω

ω ε ε ωω

= −

[ ]2 012 2 1

cd ( ) ( ) ( ) 1 ( ) d( )nspontr M D D f fγω ω ε ε ω= −

( )3

23 3 3

0

( )4

nDcγ ω ω

πΩ

=

density of states for photons

2 1ε ε ω− =

⎫⎪⎪⎬⎪⎪⎭

upwards

downwards

Absorption coefficient

[ ]212 1 2

12

d ( ) d ( ) d ( )

( ) ( ) d ( )( ) d ( )

abs up stimr r r

M D f f jj

γ

γ

ω ω ω

ε ε ω

α ω ω

= −

= −

=

absorption coefficient [ ]212 12 1 2( ) ( ) ( )M D f fα ω ε ε= −

12 2 1( ) 0 for ( ) ( )f fα ω ε ε< >

( ) ( )12( , ) ( ,0) 1 ( ) exp ( )j x j r xγ γω ω ω α ω= − −

amplificationj xg( )

x

Spontaneous emissionreplace in spontaneous emission rate

[ ][ ]

1 2012

1 2

1 ( ) ( )d ( ) ( ) ( ) d

( ) ( )spont

f fcr Dn f fγ

ε εω α ω ω ω

ε ε−

=−

[ ]2 12

121 2

( )( ) ( )

M Df fα ωε ε

=−

1 21 2

1 1( ) and ( )exp 1 exp 1FV FC

f f

kT kT

ε εε ε ε ε

= =− −⎛ ⎞ ⎛ ⎞+ +⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠

2 1ε ε ω− =with and

0 d( )d ( ) ( ) ( )( )exp 1

spontFC FV

cr Dn

kT

γωω α ω ω

ω ε ε=

− −⎛ ⎞ −⎜ ⎟⎝ ⎠

Production of chemical energy

2

3 3 2

( )( ) ( )( )4 exp 1FC FV

dj a dc

kT

γωω ω ω

ω ε επΩ

=− −⎡ ⎤ −⎢ ⎥⎣ ⎦

djg,abs

djg,emit

djeh = djg,abs – djg,emit

only radiative recombination, monochromatic

generalized Planck law

Sun: T = TS εFC – εFV = 0Semiconductor: T = T0 εFC – εFV ≠ 0

eFC – eFV = meh

absorptance ( ) ( ){ }( ) 1 ( ) 1 exp ( )a R dω ω α ω= − − −

djeh = dgeh – dreh

Characteristic for production of chemical energyby monochromatic lightdjeh(meh) = djg,abs – djg,emit(meh)

djg,abs

djeh

djg,emitdjg

wmeh,mp

mehmeh,ocmeh,sc

h =

Production of chemical energy frommonochromatic radiation

0 1 2 3 4 50

102030405060708090

100

ef

ficie

ncy

/ %

photon energy / eV

maximum concentration

no concentration

Infinite tandem: η = 86% max. concentration

Production of chemical energyin wide band semiconductor with total solar spectrum(Shockley-Queisser)

0 1 2 30

0.1

0.2

0.3

0.4

η

εG / eV

max. concentration

no concentration

AM0 spectrum

Optimal materials

full absorption above energy threshold (in a thin film)

minimum recombination for given difference of Fermi-energies:

radiative recombination

good materials for solar cells should be luminescingadvantage for organic materials?

difference of the Fermi-energies (chemical energy per eh-pair) is obtained from luminescence intensity

Chemical energy → electrical energy

Charge current

-ej

eFV

eFC

eV

eC

ee

0 xjQ

G R µehjQ = - e (G - R)

Separation of electrons and holes withsemi-permeable membranes← H2 , O2 →

Voltage: eV = eF,right - eF,left = μeh

O2

O2

O2

O2 O2

O2

O2

O2

O2

O2

H2

H2 H2

H2 H2H2

H2

H2

H2

← e , h →

ee

eF,C eF,V

eV

eC

x

-ej0

absorbern-type p-type

eF,lefteF,rightµeh

Transport properties, drift current

acceleration ii

i

z e Eam∗=

,mobility C ii

i

eb

mτ∗=

drift current

Diffusion currentdiffusion current

Einstein relation

chemical potential of particles i,0 ln ii i

i

nkTN

μ μ⎛ ⎞

= + ⎜ ⎟⎝ ⎠

Fick‘s law of diffusion

total charge current

i i iz eη μ ϕ= +

for electrons (zi = -1) and holes (zi = +1)

electrochemical potential

e e FC

h h FV

ee

η μ ϕ εη μ ϕ ε

= − == + = −

field and diffusion currents do not exist

Dye Solar Cell

Problem: e and h bound in exciton

e

h

Problems with excitons in organicsemiconductors

�e

�exciton

�exciton

�C

�V

electron bound to free hole hole bound to free electron

lumo

homo

large exciton binding energy

1 2

0

exciton dissociation

Requirements for solar cell structures

+-

Le

n+

p+

absorbern+

p+

Sufficient condition:Le, Lh >> ta >> 1/a

rules out low mobility absorbers

ta

Necessary condition:Le, Lh >> distance betweenmembranes

1/α

Conditions for optical and electrical properties of absorberssplitting of Fermi-energies selective transport

ta >> 1/a

bulk heterojunction

Advantage of nano-structures in conventional solar cells

distance between membranes on nm-scale

absorbers can have poor transport properties

Problem: large interface area may increaserecombination

luminescence as a tool to proveenergy conversion efficiencyspectral intensity of luminescence

( )( ) ( )

2

3 3 2

0

2

, 3 3 2

0

( )( )4

exp 1

spectral emission of photons through surface of homogeneous system( )( )

4 ( )exp 1

(

emissioneh

FC FV

emissionemit

FC FV

dR x dc x x

kT

dj a dc

kT

a

γ

ωα ω ωπ ω ε ε

ωω ωπ ω ε ε

Ω=

⎛ ⎞− −⎡ ⎤⎣ ⎦ −⎜ ⎟⎜ ⎟⎝ ⎠

Ω=

⎛ ⎞− −−⎜ ⎟

⎝ ⎠

[ ] ( )) 1 ( ) 1 exp ( ) eR Lω ω α ω= − −⎡ ⎤⎣ ⎦

Luminescence as a characterization tool

16000

14000

12000

10000

8000

6000

4000

2000

16000

14000

12000

10000

8000

6000

4000

2000

l < 1000 nmElectroluminescence

counts per pixel

Physics of Solar Cells