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