Solar Cells as Light Emitters:The Key to Record Efficiencies and

Approaching the Shockley-Queiser Limits

OSA FiO 2013 / LS XXIX

Owen Miller*: Post-doc, MIT MathEli Yablonovitch: UC Berkeley, LBNL

*Currently working with Steven Johnson @MIT

Slides/Codes/Relevant Papers: math.mit.edu/~odmiller/publications

From Allen BarnettUniv. of Delaware

Value of High Photovoltaic Efficiency

Efficiency and Cost of Electricity (COE) inversely related:

OperatingFinanceEfficiencyInsolation

)Cost(CostCOE BOSmodule

Factors that impact COE: Location: decides the

available input energy Efficiency: decides the

portion that can be converted to electricity

5% 8% 12%

18%

27%

40%

0

5

10

15

20

25

30

35

40

0 100 200 300 400 500 600 700 800 900 1000

Cost of Photovoltaics ($/m2)

Cos

t of E

letr

icity

(¢/k

Wh)

Which PV technology? Ask Shockley-Queisser• Canonical method for fundamental limits

– solar cell = absorber– absorber = emitter– internals = black-box (equilibrate rates through surface)– constant quasi-Fermi level separation (thermalization)

Shockley & QueisserJ. Appl. Phys. 32, 510 (1961)

Single-Junction Intermediate-Band Multi-Carrier Multi-Junction

< 33.5% < 65% < 45% < 50%

Many more: concentrating, hot-carrier, spectrum-splitting, nanowires, etc.

Fundamental limits should be…

General(few parameters)

Robust

Shockley-Queisser

Absolutely.In simplest case,

bandgap only

A real system will never be perfect. But small imperfections in material/optics should yield small losses in performance.

No!By hiding internals, important

photon dynamics are obscured

Two consequences: (a) For technology selection, need a “modified” SQ (b) To approach SQ: explicitly design for photon extraction

Fundamental limits should be…

General(few parameters)

Robust

Shockley-Queisser

Absolutely.In simplest case,

bandgap only

A real system will never be perfect. But small imperfections in material/optics should yield small losses in performance.

No!By hiding internals, important

photon dynamics are obscured

Two consequences: (a) For technology selection, need a “modified” SQ (b) To approach SQ: explicitly design for photon extraction

To extract current, voltage at contacts must be slightly lower than Voc

But, operating voltage linked directly to Voc

kT

qV

q

kTVV OCOCOP ln

We only need to understand the open-circuit voltage

Open-Circuit Voltage Determines Operating-Point Voltage

Any bad things that can happen, will happen. Can’t extract the charges before, say, non-radiative recombination

Photon Extraction VOC

• Basic solar cell definition: absorbs sunlight• Thermodynamics: absorber = emitter

– Equilibrium:

– Non-eq:

𝑅𝑒𝑚 (𝜃 ,𝜔 )=𝑅𝑎𝑏𝑠(𝜃 ,𝜔)

∫𝑅𝑒𝑚 (𝜃 ,𝜔 )𝑑𝜔𝑑Ω=∫𝑅𝑎𝑏𝑠 (𝜃 ,𝜔 )𝑑𝜔𝑑Ω

detailed balance(Kirchhoff's Law)

open-circuit, steady-state

Emission (through front) is not a loss mechanism!

Any alternate path for photons:(a) represents loss(b) reduces effective carrier lifetimes

𝑞𝑉 𝑂𝐶=𝑞𝑉 𝑂𝐶−𝐼𝑑𝑒𝑎𝑙−𝑘𝑇|ln𝜂𝑒𝑥𝑡|

The Voltage Penalty: kT |ln hext|• Mathematically formulated by Ross, 1967

• So what?

𝑞𝑉 𝑂𝐶=𝑞𝑉 𝑂𝐶−𝐼𝑑𝑒𝑎𝑙−𝑘𝑇|ln𝜂𝑒𝑥𝑡|Generalized: Rao, PRB 76, 085303 (2007)Kirchartz & Rau, PSSA 205, 2737 (2008)

depends very non-linearly on internal parameters!𝜂𝑒𝑥𝑡

Why do many solar cells have small VOC?Photon extraction is hard! (Ask LED designers)

Heat?

Outside escape cone?

solarradiation

outgoingluminescenceAbsorbed by

contact?

Non-ideal reflectivity?

For a typical high-index material:

∼50∼50

re-absorption/re-emission events

bounces off the rear mirror

…before escape

a 99% internal radiative efficiency

or a 99% reflective back mirror

Consequently

Only half of the photons escape!

𝜼𝒆𝒙𝒕 ≈𝟓𝟎%

The Fragility of Shockley-Queisser

𝜂𝑖𝑛𝑡=100 %

1 %

Many material systems have fundamental limits to

𝜂𝑖𝑛𝑡<99.7 %𝜂𝑖𝑛𝑡<20 %𝜂𝑖𝑛𝑡<10−4

GaAs:c-Si:a-Si:

Single-Junction Efficiency Records: 1990-2013

24

26

28

30

32

34

0.8 1 1.2 1.4 1.6

GaAs

33.5%

26.4%

Theory

Previous Record (2010) Voc=1.03V25.1%Record (1990- 2007)

Bandgap (eV)

Effici

ency

(%)

25.0%Best Si (1999- )

Single-Junction Efficiency Records: 1990-2013

24

26

28

30

32

34

0.8 1 1.2 1.4 1.6

GaAs

33.5%

26.4%

Theory

Previous Record (2010) Voc=1.03V25.1%Record (1990- 2007)

~30%

Alta Devices28.8% Voc=1.12V (2012)

Bandgap (eV)

Effici

ency

(%)

25.0%Best Si (1999- )

0 0.2 0.4 0.6 0.8 130.5

Reflectivity

Cell

Effici

ency

(%)

0 0.2 0.4 0.6 0.8 1

1.16

Reflectivity

Voc

(Vol

ts) 1.14

1.12

1.10

1.08

1.06

Reflectivity0 0.2 0.4 0.6 0.8 1

Jsc

(mA/

cm2 )

32

32.2

32.2%

33.2%

31.9%

1.1041.115

1.145

32.4332.46

32.50

31.5

32.5

33.5

32.4

32.6

90% Rear

ReflectivityIs Not

Enough!

GaAs 3mm

Efficiency vs. Rear Mirror Reflectivity

e-

h+

h h h(<25%)

h(>25%)

hhhg hg

e-

h+

GaAs Alta (x1)

GaAs ISE (x20)

M. A. Green, “Radiative Efficiency of state-of-the-art photovoltaic cells”

2010 GaAs (ISE)

2013 GaAs (Alta)

𝐽𝑆𝐶𝑉 𝑂𝐶

𝐹𝐹

𝐸𝑓𝑓

29.8𝑚𝐴𝑐𝑚2 29. 7

𝑚𝐴𝑐𝑚2

1 .030𝑉

8 6.0 %

2 6.4 %

Courtesy of Alta Devices

1 .122𝑉

8 6.5 %

2 8.8 %

Clarification: enhanced extraction photon recycling

Photon recycling is one way to achieve light extraction, and thereby high Voc.

However, there are ways to improve light extraction without enhancing photon recycling. For example: surface texturing. Voc increases.

Light extraction is the general parameter that determines Voc, not photon recycling.

cf. also Lush & Lundstrom, Solar Cells 30, 337 (1991)

Fundamental limits should be…

General(few parameters)

Robust

Shockley-Queisser

Absolutely.In simplest case,

bandgap only

A real system will never be perfect. But small imperfections in material/optics should yield small losses in performance.

No!By hiding internals, important

photon dynamics are obscured

Two consequences: (a) For technology selection, need a “modified” SQ (b) To approach SQ: explicitly design for photon extraction

Large bandgap

Small bandgap Eg1

Eg2

Multi-Junction

Photon down-conversion

Eg0

2Eg

3Eg

Photon up-conversion

GaAs Single-Junction

Application to many solar technologies

Hanna & Nozik J. Appl. Phys. 100, 074510 (2006)

Eg

0

2Eg

Multiple-exciton generation: Shockley-Queisser

3Eg

• Depends on exact assumptions – geometry, refractive index• Illustrates how important it is to look at robustness, photon dynamics

Robustness analysis: multi-exciton generation

Δ𝑉 𝑂𝐶=𝑘𝑇 ln𝜂𝑒𝑥𝑡The absolute voltage penalty is independent of bandgap

Δ𝑉 𝑂𝐶

𝑉 𝑂𝐶

∼Δ𝑉 𝑂𝐶

𝐸𝑔

The relative voltage penalty is much worse for small

Sub-wavelength solar cells: a new wrinkle

Atwater et. al., “Design Considerations

for Plasmonic Photovoltaics,” 2010

Pillai et. al. “Surface plasmon enhanced silicon solar cells,” 2007

Zhu et. al. “Nanodome Solar Cells with Efficiency Light Management…,” 2009

New physics: emission partially de-coupled from absorption emission can occur into near-field (plasmons, quenching, etc.)

Modeling emission: fluctuation-dissipation theorem

Joint work led by Xiang Zhang groupPhys. Rev. Lett. 109, 138701 (2012)

𝑞𝑉 𝑂𝐶=𝑘𝑇 ln( 𝑅𝑎𝑏𝑠 ,𝑆𝑢𝑛

𝑅𝑒𝑚 , 300𝐾) 𝑅𝑒𝑚 ,300𝐾=∫

0

𝑛𝑑𝜔ℏ𝜔

⟨𝑺(𝒓 ,𝜔 )⟩ ⋅�̂�

⟨𝑷 (𝒓 ,𝜔 )⋅𝑷 (𝒓 ′ ,𝜔 ) ⟩𝑆=ℏ

𝑒ℏ𝜔 /𝑘𝑇−1𝜖𝐼 (𝒓 ,𝜔 ) 𝛿(𝒓−𝒓 ′)

Example system: air-Si-Au thin film

Emission into plasmon modes: - reduces extraction through the front - reduces carrier lifetimes - reduces

ray opticsnear field

(FDT)

Here, plane waves don’t couple to plasmons no absorption effect (JSC unaffected)

This is ONLY an emission/VOC effect!

• Photon extraction: driver behind record 28.8% efficiency– Power output = current * voltage– voltage = extraction (difficult!)– There are significant returns to:

>90% material quality>90% rear mirror reflectivity

– New considerations at nano-scale

• Technology selection: Shockley-Queisser is fragile– Adding a single parameter, , enables robustness analysis– Can be applied everywhere SQ can be applied

Slides/Codes/Relevant Papers: math.mit.edu/~odmiller/publications