Tom MarkvartSolar Energy Laboratory

School of Engineering SciencesUniversity of Southampton, UK

Classical thermodynamics

(Carnot cycle;Ts≈6000K To ≈300K)

Detailed balance in luminescence (Einstein, Kennard, Stepanov, van

Roosbroeck & Shockley)

Detailed balance in photovoltaics / photosynthesis (Shockley & Queisser, Ross &

Calvin, …)

Thermodynamics of solar energy conversion (Duysens,

Landsberg, photothermal, endoreversible, …)

O3

H2O

H2O

H2O, CO2

H2O, CO2

UV IR

ARC

top contacts

p-n junction

back contact

Voltage V =

1

2

q

Electrical load

n

p

q

Electrical load

n

p

(Antenna & P)*P+ B-

P+ H-

P+ Q-

work

Antenna & P

red

ox

E

(Antenna & P)*P+ B-

P+ H-

P+ Q-

work

Antenna & P

red

ox

E E

“Forward” rate: photogeneration g

“Reverse” (dark) rate = recombination of e- and h+

1/ Tko

BeKKK

oo

oo

he

hehehe

][][

]][[][][]][[

1

1/ TkqVo

BeIII

Compare with Shockley solar cell equation

Shockley & Queisser, J. Appl. Phys. 1961; Ross & Calvin, Biophys J. 1967.

h he

]ln[ eTkBe etc…

oBTkEo eK /1

gK

where

1/ TkBe TkE Be /1

Reactions rate / current

Ene

rgy

prod

uced

per

pho

ton

Photogeneration rate

1/ Tko

BeKKK

Free energy per e-h pair

Current / reaction rate

Pow

er

Pmax

Maximum power is extracted at Vmax or Imax: need for control of the operating point !

1/ oBTko eKKK

At “open circuit” (K = 0):

gTkEK

KTk oB

ooB ln1ln

From detailed balance (Einstein, 1917)

sBTkEeg /

S

ooc T

TE 1

Rose, J. Appl. Phys. 1960; Baruch et al, J. Appl. Phys. 1985.

)0(,, sin TE

)(,, wToout E

i

work (w)

absorptionemission

),( inin su),( outout su

An cos2E

Etendue - a geometric characteristic of light beams …

dAdn cos2E

A

(e.g. for isotropic incidence)An2E

dydxdkdk

k

nyx

2

E

… or a volume element in the phase space, an invariant, and a measure of the number of photon states:

Ts

To

uin (sin)

q (q/To)

wi

ioS

oin T

T

Tuw

1

ioS

os T

T

Tuw

1

radnkinexpci

Entropy generation

by: Cooling of photon gas TS To

Etendue expansion Es Eout

Finite “turnover rate” of the conversion “engine”

Non-radiative recombination

Markvart, Appl. Phys. Lett. 2007

o

oBkin II

IIIk

ln

o

SB

o

SBc T

Tk

T

Tk ln1

out

inBexp k

EE

ln

in

oino

in

oin T

TkTh

T

Tu 11max

Make use of hot carriers ?

= kB ln(/s) for a planar solar cell

heat rejection into To reservoir (Carnot cycle)

photon cooling (= thermalisation)

étendue expansion

photon emission (finite “turnover rate”) kinetic losses

pow

er p

er p

hot

on (

a.u

.)

normalised current / reaction rate

LH LH

RC

coll

LH LH

RC

inj

1

2

oBconvoBLH TkEinj

TkEcol ee //

gTkE

gTkE

K

KTk

oBLH

col

injoBconv

ooB

ln

lnln

reaction rate / current

ideal

observed

K

There are no shortcuts round the basic principles of PV/ photochemical conversion

Static (energy) and kinetic (current) losses are (to some approximation) independent

Thermodynamics can be used to describe the basic energy conversion processes in photovoltaics and photosynthesis

Parallels with kinetic theory but the origins of losses are elucidated in detail, in terms of entropy generation

A fundamental similarity between PV and photosynthetic conversion but differences in

Reciprocity:

• Electricity v. electricity + chemical energy

• Nano/molecular v. macroscale

• Expression of microscopic reversibility which extends the link between kinetics & thermodynamics to realistic transport processes

• Provides a description of constraints on the conversion process on account of the 2nd law of thermodynamics

Jonathan Swift: Gulliver’s Travels (1726): the Academy of Lagado. With special thanks to Peter Landsberg

…. The first Man I saw was of a meager Aspect, with sooty Hands and Face, his Hair and Beard long, ragged and singed in several Places. His Cloathes, Shirt, and Skin were all of the same Colour. He had been Eight Years upon a Project for extracting Sun-Beams out of Cucumbers, which were to be put into Vials hermetically sealed, and let out to warm the Air in

raw inclement Summers. He told me he did not doubt in Eight Years more he should be able to supply the Governors Gardens with Sun-shine at a reasonable Rate; but he complained that his stock was low, and intreated me to give him something as an Encouragement to Ingenuity, especially since this had been a very dear Season for Cucumbers. …