A thermodynamic cycle for the solar cell

Alejandro Jenkins

with Robert Alicki (Gdansk) &

David Gelbwaser-Klimovsky (Harvard)

arXiv:1606.03819 [cond-mat.stat-mech], to appear in Annals of Physics

16 Aug. 2016 CELEQ, U. de Costa Rica

1

Textbook picture

n-type p-typeinterface

1

2

3

I

band gap

E

conductionband

valenceband

2

“We frequently read that it is just the electric field of a pn-junction which supplies the driving force for the currents flowing during illumination […] In fact something must be wrong in our physical education, if we think that that a DC current can at all be driven in a closed circuit by a purely electrical potential difference. The word potential alone should tell us that no energy can be gained by moving a charge along any closed path.”

– Würfel & Würfel, Physics of Solar Cells, 3rd ed. (Wiley-VCH, 2016)

3

Perpetual motion

– Gould, Oddities (1944), after a design by Marquess of Worcester, ca. 1640

W =

I

Cds · F

ext

=

Z

0

dt (v · Fext

)

• W > 0 calls for periodic component of Fext in phase with v

• i.e., that leads position by τ/4 (phase of 90º)

Fext

= r ) W = 0

4

Capacitor

W = eV = e

Z A

Bds ·E

A

BR I

–

–

–

–

+

–

5

Van de Graaff

4.9 Electromotive force and the voltaic cell 209

4.9 Electromotive force and the voltaic cellThe origin of the electromotive force in a direct-current circuit is somemechanism that transports charge carriers in a direction opposite to that

I1

I2 I2I3

I4

I4

I5

V3

V1 V2

I6

I6

Figure 4.18.Currents and potentials at the nodes of anetwork.

in which the electric field is trying to move them. A Van de Graaff elec-trostatic generator (Fig. 4.19) is an example on a large scale. With every-thing running steadily, we find current in the external resistance flow-ing in the direction of the electric field E, and energy being dissipatedthere (appearing as heat) at the rate IV0, or I2R. Inside the column of themachine, too, there is a downward-directed electric field. Here chargecarriers can be moved against the field if they are stuck to a nonconduct-ing belt. They are stuck so tightly that they can’t slide backward along thebelt in the generally downward electric field. (They can still be removedfrom the belt by a much stronger field localized at the brush in the termi-nal. We need not consider here the means for putting charge on and offthe belt near the pulleys.) The energy needed to pull the belt is suppliedfrom elsewhere – usually by an electric motor connected to a power line,but it could be a gasoline engine, or even a person turning a crank. ThisVan de Graaff generator is in effect a battery with an electromotive force,under these conditions, of V0 volts.

In ordinary batteries it is chemical energy that makes the charge car-riers move through a region where the electric field opposes their motion.That is, a positive charge carrier may move to a place of higher electricpotential if by so doing it can engage in a chemical reaction that will

++

++

+

+ + + + + + ++

++

+

++++–

–––––––––

++++++++++

V = V0

V = 0

R

I

EEEE

Figure 4.19.In the Van de Graaff generator, charge carriersare mechanically transported in a directionopposite to that in which the electric field wouldmove them.

yield more energy than it costs to climb the electrical hill.To see how this works, let us examine one particular voltaic cell.

Voltaic cell is the generic name for a chemical source of electromotiveforce. In the experiments of Galvani around 1790 the famous twitchingfrogs’ legs had signaled the chemical production of electric current. Itwas Volta who proved that the source was not “animal electricity,” asGalvani maintained, but the contact of dissimilar metals in the circuit.Volta went on to construct the first battery, a stack of elementary cells,each of which consisted of a zinc disk and a silver disk separated bycardboard moistened with brine. The battery that powers your flashlightcomes in a tidier package, but the principle of operation is the same.Several kinds of voltaic cells are in use, differing in their chemistry buthaving common features: two electrodes of different material immersedin an ionized fluid, or electrolyte.

As an example, we’ll describe the lead–sulfuric acid cell which isthe basic element of the automobile battery. This cell has the importantproperty that its operation is readily reversible. With a storage batterymade of such cells, which can be charged and discharged repeatedly,energy can be stored and recovered electrically.

A fully charged lead–sulfuric acid cell has positive plates that holdlead dioxide, PbO2, as a porous powder, and negative plates that holdpure lead of a spongy texture. The mechanical framework, or grid, ismade of a lead alloy. All the positive plates are connected together and

– Purcell & Morin, Electricity and Magnetism, 3rd ed., (Cambridge U. P., 2013), sec. 4.9

6

Battery

Pb + HSO4 ! PbSO4 +H+ + 2e

PbO2 +HSO4 + 3H+ + 2e ! PbSO4 + 2H2O

7

• Battery is not cyclic device

• May be recharged, but only by externally reversing current

RI

–

–

–

–

––

–

–

–

– –

+

+

++

+

+

+

– +

PbO2

Pb

I

H2SO4 (aq)

–

–

+

: electron: H+ ion : HSO4

– ion

Key

+

–

Self-oscillation• Two heat baths, working

fluid, piston

• Piston both modulates external power and extracts net work W

• Positive feedback between piston and valve action allows W > 0

• See: AJ, Phys. Rep. 525, 167 (2013) [arXiv:1109.6640 [physics.class-ph]]

8

Andronov, Vitt & Khaikin, Theory of Oscillators (Dover, 1987 [1966]), ch. VIII, sec. 10

Rayleigh-Eddington• For a cycle:

• Temperature of working substance:

• Clausius theorem:

U =

I(Q W ) = 0 ) W

IW =

IQ

Td T T ; T = T

1 +

Td

T

= I

Q

T=

IQ

T1 + Td/T

0 )I

Q

1 + Td/T 0

9

W I

Q

1 1

1 + Td/T

=

IQ · Td

T + Td

– Eddington, MNRAS 79, 177 (1919)

Rayleigh-Eddington, cont.

• “In any problem which may present itself of the maintenance of a vibration by heat, the principal question to be considered is the phase of the communication of heat relatively to that of the vibration.”

– Rayleigh, Theory of Sound, 2nd ed., vol. II (Dover, 1945 [1896]), sec. 322g

• “We must make the [Cepheid variable] star more heat-tight when compressed than when expanded.”

– Eddington, Internal constitution of the stars (Cambridge U. P., 1988 [1926]), sec. 137

10

W I

Q · Td

T + Td' 1

T

IQ · Td

Quantum thermodynamics• Heat engine as open quantum system, governed

by master eq.

• Absorbs heat in one state of working substance, rejects it in another state

• Net heat absorbed available as work

• See: Alicki, J. Phys. A: Math. Gen. 12, L103 (1979); Kosloff, J. Chem. Phys. 80, 1625 (1984); Gelbwaser-Klimovsky, Alicki & Kuricki, PRE 87, 012140 (2013) [arXiv:1209.1190 [quant-ph]]

11

Quantum thermodynamics, cont.

• Modulation of state and extraction of work performed by semi-classical, oscillating “piston”

• Described by time-dependent Hamiltonian:

• H = H0 + Hpiston(t)

• Master eq. taken in Markovian approximation (Lindblad eq.)

• Periodicity needed to define power output

12

“Piston-less” engines• Circulation of working substance

in external temp. gradient may perform piston functions

• e.g.: Feynman’s rubber-band engine, hurricanes, etc.

• Reversing circulation turns engine into refrigerator

• Interrupting circulation stops work output

• Can’t apply to solar cell: cycle must be within semiconductor

13

Feynman Lectures, vol. I, no. 42

Hurricane“Attempts to regard the condensation heat source as external lead to the oft-repeated statement that hurricanes are driven by condensation of water vapor, a view rather analogous to that of an engineer who proclaims that elevators are driven upward by the downward acceleration of counterweights. Such a view, though energetically correct, is conceptually awkward; it is far more natural to consider the elevator and its counterweight as a single system driven by a motor.”

– Emanuel, Annu. Rev. Fluid Mech. 23, 178 (1991)

14

Solar cell• baths at phonon temperature (~ 300 K) + incident photons at

high effective temperature (~ 1000 K)

• Plasma oscillation at p-n junction acts as piston, with frequency Ω/2π ~ 1 THz

• Lindblad eq., with H = H0 + Hpiston(t)

15

H0 =X

k

Ec(k)c

†kck + Ev(k)v

†kvk

Hpiston

= g sin (t) (Nc hNci0)

d

dt(t) = i

~ [H(t), (t)] + L(t)(t)

3

electron - holecreation

conductionband

valenceband

intrabandthermalization

k

gap

!ω0

photon

!ωk

recombinationphoton

!ωk′

intrabandthermalization

FIG. 1. Schematic picture of leading processes involving elec-trons, photons and phonons in a semiconductor with a directband.

assumed in our derivations to be much smaller than thefrequency ω0 corresponding to the energy gap Eg = !ω0.On the other hand, it should be much larger than therecombination rate in order to justify the approxima-tions leading to (10). The only phenomenon which sat-isfies all these requirements is plasma oscillation visiblefor p-n junctions. Their appearance is due to the factthat a p-n junction creates an inter-phase between re-gions of different electron concentrations which can os-cillate in space producing collective macroscopic electricfield oscillations. In several experiments such oscilla-tions have been observed [14, 15], with typical frequenciesΩ/2π ≃ 1T Hz, much lower than ω0 ≃ 1400T Hz corre-sponding to the energy gap ∼ 1eV . In the final step ofcell operation the THz plasma oscillations must be con-verted into a direct current. Here again a p-n junctionplays a crucial role acting as a diode which rectifies thehigh frequency periodic current. A qualitative pictureof this mechanism is shown on Fig. 2. The plasmonicdegree of freedom is represented by the quantum levelsin the asymmetric potential which is harmonic for lowerenergies. Asymmetry is due to the p-n junction whichdefines an “easy” direction for the carrier flow (to theleft). The work supplied to the oscillator drives the uni-directional electric current.

Hamiltonians, generators and stationary states Theelectrons in a semiconductor occupying the conductionand valence bands are described by the annihilation andcreation operators ck, c†

kand vk, v†

k, respectively, sub-

ject to canonical anticommutation relations. The elec-trons are treated as non-interacting particles moving ina self-consistent potential. The single-particle states are

ħΩ

FIG. 2. Schematic picture of rectification of plasma os-cillations. Plasma oscillations caused by the collective mo-tion of free carriers are described by the quantum harmonicoscillator coherently pumped by the feed-back mechanism.Due to the diode-type potential at the edges, oscillations areconverted into a DC current.

labeled by k and the unperturbed Hamiltonian reads

H0 =!

k

"

Ec(k)c†kck + Ev(k)v†

kvk

#

. (11)

In the p-n junction a non-homogeneous free carrier distri-bution created in a self-consistent build-in potential Ubi

can be perturbed producing collective plasma oscillationswith the frequency Ω. The associated Hamiltonian per-turbation which should be added to the electronic Hamil-tonian (11) has form

ξM = ξ1√N

Eg

!

k

$

c†kck + vkv†

k

%

, (12)

where ξ is a small dimensionless parameter describing theamplitude of deformation, Eg is the relevant energy scale

(typically eUbi ≃ Eg), and c†kck, vkv†

kare number opera-

tors of free electrons and holes, respectively. The mean-field type scaling (N - number of atoms in the sample)reflects the fact, that only carriers which are close to thejunction feel the potential distortion. Among the basicirreversible processes the intraband thermal relaxationis the fastest (thermalization time ∼ 10−12s) and there-fore, the stationary state of the electronic systems withthe total Hamiltonian H0 + ξM is , within a reasonableapproximation, a product of grand canonical ensemblesfor electrons in conduction and valence band with thesame temperature T of the device and different chem-ical potentials µc and µv, respectively. The associated

Solar cell, cont.

16

d

dt(t) = i

~ [H(t), (t)] + L(t)(t)• Stationary state ρ(t) and

generator L(t) depend on piston’s g sin (Ωt)

• Two-temperature ρ (hot photons & cold phonons)

• Net extracted power, explicitly limited by Carnot’s theorem times kinematic factor (“ultimate efficiency”)

• Current rectified by asymmetry of plasmon potential

ħΩ

Alicki, Gelbwaser-Klimovsky & Szczygielski, J. Phys. A: Math. Theor. 49 (2016) 015002; [arXiv:1501.00701 [cond-mat.stat-mech]]

Self-oscillating pistonhot bath

cold bath

x

workingsubstance

valve 1

valve 2

piston

17

Putt-putt steamboat

18

https://www.youtube.com/watch?v=2BTqbf8ZZSo

Putt-putt, cont.• Positive feedback between rate of

condensation and level of liquid water in tank

• Water level acts as self-oscillating piston

• Momentum transferred away from boat only during expulsion phase

• Finnie & Curl, Proc. Internat. Union of Theor. & Appl. Mech. 3, 486 (1963); Am. J. Phys. 31, 289 (1963)

• AJ, Eur. J. Phys. 32, 1213 (2011) [arXiv:0908.3190 [physics.flu-dyn]]

19

https://www.youtube.com/watch?v=zUTEJo2Bxw0

Hydrodynamical model

x+ x+ !

2x = A(N0 N)

N = (x)N +B(x)

flame

liquid waterx

air & steam

one-wayvalves

cold tankwall

lower reservoir

upper reservoir

20

Instability

x = v

v = x v n

n = fx n

21

x+ x+ !

2x = A(N0 N) N = (x)N +B(x)

! = 1 ; A = 1set

(+ ) + 1

(+ ) + f = 0Characteristic eq.:

N0 = B/ f b gN0

(x) = + gx, B(x) = B + bx, n N N0

Linearizing:

Instability, cont.

f =A (b gN0)

!2>

b ' 0

g < 0

22

Plasma oscillation• Collective oscillation of relatively dense electron fluid

in n-type layer

• in contact with dilute electron gas in p-type layer (absorber)

• Described by single degree of freedom: position x of depletion layer between phases

• Plasma frequency:

• Reported in: Sha, Smirl & Tseng, PRL 74, 4273 (1995); Kersting et al., PRL 79, 3038 (1997)

23

!p =

snee2

m

Positive feedback• Increased x compresses

working gas

• Conducting electrons “roll” down potential at interface

• Holes driven to positive terminal

• Reduces number of holes, increasing rate of photon absorption (Rayleigh-Eddington)

24

band gap

conductionband

valenceband

to positiveterminal

current pumped

to negativeterminal

equilibrium positionof interface

x

electrons

holes

p-typen-type

f =A (b gN0)

!2>

b > 0

g ' 0

Gaps• Separation of time scales

prevents working substance from thermalizing completely

• Solar cell: fast intra-band, slow inter-band transitions

• Putt-putt: fast thermalization in gas, slow interconversion between gas and liquid

• Egap = 1.1 eV

• Hvap = 0.42 eV

n-type p-typeinterface

1

2

3

I

band gap

E

conductionband

valenceband

25

Garai, Fluid Phase Equilibr. 283, 89 (2009) [arXiv:physics/0611289 [physics.chem-ph]]

“Ultimate efficiency”• Photons with E < Egap are not absorbed

• For E > Egap, excess E - Egap quickly dissipated into phonons

• ηu := Number of photons absorbed times Egap, divided by total energy of incident photons

• Under normal conditions, ηu ~ 44%

• Putt-putt may have analogous factor:

• work on liquid absorbs molecule; kinetic energy dissipated

• Only Hvap per molecule can become work

26

Effective photon temperature• Phonons at room temperature, T1 ~ 300 K

• Sunlight produced at Sun’s surface, ~ 6,000 K

• Not in thermal equilibrium when it reaches Earth

• Define local temperature Tloc(ω) by

• For

• See: Alicki & Gelbwaser-Klimovsky, New J. Phys. 17, 115012 (2015) [arXiv:1507.01660 [quant-ph]]

27

e~!/kBTloc

(!) =n(!)

1 + n(!)

n(!) =

e~!/kBT 1 = (R/R0)

2 ' 2 105

T2

' Tloc

(!0

) ' 1,000 K

cycle

T2

T1

Egap

eVoc

Qout

Maximum efficiency• Maximum work per photo-generated

conducting pair:

• Carnot’s theorem:

• Total efficiency:

• Agrees with detailed balance limit: Shockley & Queisser, J. Appl. Phys. 32, 510 (1961)

28

eVoc

Egap

· th

th = 1 T1

T2= 70%

max

= u

· th

' 31%

Predictions• Amplitude squared of plasma oscillation at pn junction

should be roughly proportional to power output

• Could be externally driven at resonance, increasing power output

• High frequency (submillimeter radiation) and noise in illuminated cell make tests challenging

• There might be small THz fluctuations in output current, if not totally smoothed out by hydrodynamics

• Compelling theoretical argument for pursuing tests

29

Summary & outlook• Thermodynamics requires self-oscillating piston in solar cell

• Offers natural hydrodynamic model for current pumping, analogous to putt-putt engine

• Simple and thermodynamically clean derivation of efficiency limit, agrees with Shockley-Queisser

• Coherent plasma oscillation of pn-interface observed, but not yet in illuminated cell

• Might impact photovoltaic cell design (new frequency ranges, etc.)

• Similar arguments should apply to thermoelectric generators, fuel cells, photosynthesis, biological proton pumping, etc.

30