Model-based design and optimization of solar energy

Model-based design andoptimization of solar energy

technologies

Raymond A. Adomaitis

Lecture 7

1/56

Lecture 7

Main topic: Cell efficiency

1 Internal and external quantum efficiency

2 Spectral responsivity

2/56

Cell currents

At this point we have

I = −Iph + Io

[exp

(qV − IRs

βkBT

)− 1

]+

V − IRs

Rsh

Physically, we understand Io , Rs , and Rsh . . .

=⇒ How to we relate Iph to AM1.5 data and spectralirradiance in general?

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Power as a function of X

We consider a system that allows irradiance to vary by a factor X .

For standard conditions, X = 1 and Iph = I stdph .

I = −XI stdph + Io

[exp

(qV

βkBT

)− 1

]As before, differentiating the power equation P = IV with respectto V and setting the result equal to zero results in a nonlinearequation for Vmp

XI stdph + Io

[1− exp

(qVmp

βkBT

)]− VmpIo exp

(qVmp

βkBT

)q

βkBT= 0

which must be solved numerically for Vmp.

4/56

Power as a function of X

Again, we can simplify equation g using

XI stdph

Io= exp

(qVoc

βkBT

)−1 therefore Voc =

βkBT

qln

(XI stdph

Io+ 1

)

to find the same iterated equation for Vmp

V i+1mp = Voc −

βkBT

qln

[1 + V i

mp

q

βkBT

]but with Voc corresponding to Voc(X ).

5/56

A preview of concentrating PV systems

Computing the maximum power point for Iph = 2.25 A andIo = 8.97× 10−9 A:

X 0.2 1 2

Vmp (V) 0.387 0.426 0.443Imp (A) -0.422 -2.121 -4.252

|Pmp| (W) 0.163 0.904 1.884|Pmp|/X 0.816 0.904 0.942

As expected, the power increases with solar concentrating ratio,but we also observe the efficiency likewise increases.

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This is one motivating factor for concentrating PV systems

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This is one motivating factor for concentrating PV systems

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Cell currents

To start the discussion, with our example PV cell

Iph =2.25 A

120 cm2

(100 cm

1 m

)2

= 187.5A

m2

or

187.5A

m2

(1

q

)= 1.17× 1021

charge carriers

m2 s

Using our class software and peak irradiance of 818 W,

818 W

1000 W

∫ 4000

280.5

EEλ(AM1.5)

hc/λdλ = 0.818

(4.31× 1021

photons

m2 s

)= 3.53× 1021

photons

m2 s

=⇒ ≈ 1/3 quantum efficiency

8/56

Cell currents

To start the discussion, with our example PV cell

Iph =2.25 A

120 cm2

(100 cm

1 m

)2

= 187.5A

m2

or

187.5A

m2

(1

q

)= 1.17× 1021

charge carriers

m2 s

Using our class software and peak irradiance of 818 W,

818 W

1000 W

∫ 4000

280.5

EEλ(AM1.5)

hc/λdλ = 0.818

(4.31× 1021

photons

m2 s

)= 3.53× 1021

photons

m2 s

=⇒ ≈ 1/3 quantum efficiency

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Solar spectral irradiance

500 1000 1500 2000 2500 (nm)

0.00

0.25

0.50

0.75

1.00

1.25

1.50

1.75

2.00

E E W

/(m2 n

m)

AM0AM1.5AM1.5 directAM1.5 diffuse

Figure: Spectral irradiance spectrum of the sun.9/56

Photon energy

We recall the silicon the band gap of Ebg = 1.12 eV and that theenergy of a single photon is

Ep = hν

where h is Planck’s constant and c is the speed of light

h = 4.136× 10−15 eV·sc = 2.998× 1017 nm/s

and ν =c

λ

10/56

Photon energy

Converting Si band gap to the corresponding photon wavelength:

ν =E

h, λ =

c

ν=

hc

Ebg= 1107 nm

which gives the cutoff point in the spectrum

above which photons have insufficient energy to promote anelectron from the valance to conduction band.

This is a characteristic common to all semiconductors.

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Photon energy

Figure: Solar spectral irradiance. Only photons with wavelength shorterthan λ = 1107 nm are absorbed by the Si solar cell.

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Photon energy

CB

VB

1.12 eV

Ep = h!

Ep - 1.12 eV

Figure: A closer look at howelectrons are promoted fromthe valence to conductionband in Si, illustrating theenergy loss (hν − 1.12 eV)by photons with wavelengthless than 1107 nm.

Therefore, photons with wavelengthλ = 1107 nm have precisely the rightamount of energy to create anelectron/hole pair in the Si solar cell.

Photons corresponding to shorterwavelengths have more than sufficientenergy to do this.

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Photon spectral flux

Thinking in terms of individual photons, we know that

photon spectral flux = FEλ =spectral irradiance

energy/photon=

EEλ

hc/λ

Total photons between λ0 and λ1

photons =

∫ λ1

λ0

EEλ

hc/λdλ

14/56

IQE

We define internal quantum efficiency IQE as

IQE =number of electron-hole pairs produced

number of photons absorbed by the PV semiconductor

The IQE (λ) is wavelength dependent:

0 ≤ IQE (λ) ≤ 1 for λ ≤ λbg

IQE (λ) = 0 λ > λbg

For the case where IQE = 1 for values of λ < λbg , we canrepresent the IQE (λ) by adjusting the upper limit of integrationover the

photon spectral flux =spectral irradiance

energy/photon

to correspond to λ = λbg .15/56

Cell currents

Recall

n-type emitter

p-type base

+ + + + - - - -

+

-

Ih,sc

Ie,sc

hυ

h+ e-

h+ e-

h+ e-

h+ e-

e- n0 n1 n2

Figure: PV cell currents under typical operating conditions.

16/56

EQE

We define external quantum efficiency EQE (λ) as amodification to IQE (λ):

EQE = (1− s) [1− a(λ)] [1− r(λ)] IQE (λ)electron-hole pairs

photon

that takes into account

1 the photons reflected from the PV cell top surfacer(λ) ∈ [0, 1]

2 absorption of photons by any coating or cover above the PVcell a(λ) ∈ [0, 1]

3 as well as the shading effect of the front surface contactss ∈ [0, 1].

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EQE



photon





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EQE



photon





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EQE



photon





17/56

Spectral responsivity

The photon energy in excess of 1.12 eV is absorbed by the Si solarcell in the form of waste heat - only the 1.12 eV of photon energygoes into the useful work of creating the electron/hole pair.

To take into account the fraction of photon energy converted touseful electrical work, we now scale the spectral irradiance EEλ(λ)spectrum in the following, wavelength-dependent manner to obtainthe maximum useable spectral irradiance EEλ,u for this solar cell.

Key concept

To find a relationship between the incident global irradiance EG ofthe solar cell and the photoelectrical current it produces Iph.

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Key concept


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Key concept


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To accomplish this, we define the PV cell spectral response(responsively) SR.

photon spectral flux =EEλ

hc/λ

photons

m2 nm s

charge carrier spectral distribution = EQE (λ)EEλ

hc/λ

electrons

m2 nm s

Maximum usable power is obtained integrating over λ:

EG ,u =

∫ ∞λ=0

Ebg × EQE (λ)×[EEλ(λ)

hc/λ

]dλ

W

m2

=

∫ λbg

λ=0

[Ebg

hc/λ

]EEλ(λ)dλ under ideal conditions.

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To accomplish this, we define the PV cell spectral response(responsively) SR.

photon spectral flux =EEλ

hc/λ

photons

m2 nm s

charge carrier spectral distribution = EQE (λ)EEλ

hc/λ

electrons

m2 nm s

Maximum usable power is obtained integrating over λ:

EG ,u =

∫ ∞λ=0

Ebg × EQE (λ)×[EEλ(λ)

hc/λ

]dλ

W

m2

=

∫ λbg

λ=0

[Ebg

hc/λ

]EEλ(λ)dλ under ideal conditions.

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Another way to look at this is to compute the photocurrent Iphdirectly

Iph =

∞∫λ=0

q × EQE (λ)× EEλ(λ)

hc/λdλ

=

[C

electron

] [1

e-h pair

photon

]W/(m2nm)

(J·s)(nm/s)/(nm photon)(nm)

=A

m2

where h has units J·s. Note the conversion factor

Iph =

∞∫λ=0

qEQE (λ)

hc/λEEλ dλ

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The spectral responsivity SR(λ) is defined by

SR(λ) =qEQE (λ)

hc/λwith units

A

W.

Given this definition of SR we can write the usable portion of thespectral irradiance as

EEλ,u(λ) =Ebg

qSR(λ)× EEλ(λ)

= EQE (λ)×[Ebg

hcλ

]× EEλ(λ) (see equation for EG ,u)

= EQE (λ)× SR∗(λ)× EEλ(λ)

with a “dimensionless spectral responsivity” defined by

SR∗(λ) =Ebg

hc/λ=

λ

λbg

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The spectral responsivity SR(λ) is defined by

SR(λ) =qEQE (λ)

hc/λwith units

A

W.

Given this definition of SR we can write the usable portion of thespectral irradiance as

EEλ,u(λ) =Ebg

qSR(λ)× EEλ(λ)

= EQE (λ)×[Ebg

hcλ

]× EEλ(λ) (see equation for EG ,u)

= EQE (λ)× SR∗(λ)× EEλ(λ)

with a “dimensionless spectral responsivity” defined by

SR∗(λ) =Ebg

hc/λ=

λ

λbg

21/56

Photon energy

CB

VB

1.12 eV

Ep = h!

Ep - 1.12 eV

Energy loss (hν − 1.12 eV) by photons with wavelength greaterthan 1107 nm (left).

Resulting usable spectral irradiance EEλ,u and EEλ(AM1.5) (right).

Examine, for example, the curves at λ = 1107/2 nm.

22/56

Photon energy

CB

VB

1.12 eV

Ep = h!

Ep - 1.12 eV

Energy loss (hν − 1.12 eV) by photons with wavelength greaterthan 1107 nm (left).

Resulting usable spectral irradiance EEλ,u and EEλ(AM1.5) (right).

Examine, for example, the curves at λ = 1107/2 nm.

22/56


We summarize our results as follows

Iph =

∞∫λ=0

SR(λ)EEλ(λ) dλ (A/m2)

EG ,u =

∞∫λ=0

Ebg

qSR(λ)EEλ(λ) dλ (W/m2)

=

∞∫λ=0

SR∗(λ)EQE (λ)EEλ(λ) dλ

=

∞∫λ=0

EEλ,u(λ) dλ (W/m2)

23/56


We compute the true theoretical maximum power EG ,u that canbe generated by a 1.12 eV band gap Si solar cell by integratingEEλ,u over the usable portion of the spectrum:

EG ,u =

∫ 1107

λ=0EEλ,u(λ) dλ = 490.7 W/m2

which gives the theoretical maximum efficiency of a Si solar cell as

ηSi ,max =490.7 W/m2

1000 W/m2= 49.1%

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Maximum cell efficiency

An immediate practical question we can answer is: what is theoptimal band gap for a PV cell?

Approach: integrate the product of our “dimensionless” spectralresponsivity SR∗ function evaluated for s = r = 0 and ideal IQEand the spectral irradiance EEλ (AM1.5 scaled to 1kW/m2)...

...to obtain the maximum useable solar flux, and divide that by theglobal irradiance:

ηmax(Ebg ) =EG ,u

EG=

1

1000 W/m2

∫ hc/Ebg

λ=0SR∗(λ)EEλ dλ

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ηmax(Ebg ) =EG ,u

EG=

1

1000 W/m2

∫ hc/Ebg


25/56





ηmax(Ebg ) =EG ,u

EG=

1

1000 W/m2

∫ hc/Ebg


25/56

Cell efficiency, continued

These results are plotted in the next slide.

Plotting some commonly used semiconductors for PV and othersolar energy applications (such as for photoelectrochemical watersplitting reactions), we obtain a good idea of the potential relativeeffectiveness of each.

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Summary: PV maximum efficiency

Figure: Maximum theoretical efficiency as a function of semiconductorband gap.

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Lecture 7, continued


1 Multijunction cells

2 Splitting the spectrum

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Multijunction devices

Because of the linear drop in λ of the spectral responsivity, thehigh-frequency portion of the spectral irradiance is underused.

=⇒ create a PV cell by stacking multiple junctions where thetop-most junction (the first to interact with incident light)corresponds to the junction with the highest band gap.

29/56


Because of the linear drop in λ of the spectral responsivity, thehigh-frequency portion of the spectral irradiance is underused.

=⇒ create a PV cell by stacking multiple junctions where thetop-most junction (the first to interact with incident light)corresponds to the junction with the highest band gap.

29/56


Tandem device

glass

TCO (SnO2)

p a-Si:H

i a-Si:H

n a-Si:H

contact-

+

n/p tunnel junction

i a-SiGe:H

Multi (tandem) junction a-Si cell design (left)and spectral irradiance (right). Abscissa is λ innm.

30/56


Tandem device

glass

TCO (SnO2)

p a-Si:H

i a-Si:H

n a-Si:H

contact-

+

n/p tunnel junction

i a-SiGe:H

Multi (tandem) junction a-Si cell design (left)and spectral irradiance (right). Abscissa is λ innm.

30/56


In this cell, where an amorphous Si layer is deposited before (andso above with respect to the incident light) an a-SiGe layer in thissuperstrate design.

The band gap of each layer is as follows:

Ebg ,aSi = 1.75 eV Ebg ,aSiGe = 1.3 eV

λbg ,aSi = 708 nm λbg ,aSiGe = 954 nm

Note that a transparent conducting oxide (TCO) layer of tin oxide(SnO2) has been deposited above (before) the a-Si layer:

Ebg ,TCO = 3.5 eV, λbg ,TCO = 354 nm

31/56


In this cell, where an amorphous Si layer is deposited before (andso above with respect to the incident light) an a-SiGe layer in thissuperstrate design.

The band gap of each layer is as follows:

Ebg ,aSi = 1.75 eV Ebg ,aSiGe = 1.3 eV

λbg ,aSi = 708 nm λbg ,aSiGe = 954 nm

Note that a transparent conducting oxide (TCO) layer of tin oxide(SnO2) has been deposited above (before) the a-Si layer:

Ebg ,TCO = 3.5 eV, λbg ,TCO = 354 nm

31/56


Given the AM1.5 spectral irradiance EEλ we compute the fractionof the spectrum absorbed by each of the three layers.

If we assume the TCO to be transparent to all wavelengths belowλbg ,TCO

ETCO =

∫ λbg,TCO

λ=0EEλ dλ

= 16.1 W/m2

therefore, less than 2% of the total incident power is absorbed bythe TCO film.

32/56


Given the AM1.5 spectral irradiance EEλ we compute the fractionof the spectrum absorbed by each of the three layers.

If we assume the TCO to be transparent to all wavelengths belowλbg ,TCO

ETCO =

∫ λbg,TCO

λ=0EEλ dλ

= 16.1 W/m2

therefore, less than 2% of the total incident power is absorbed bythe TCO film.

32/56


Compute the fraction of light remaining that can be effectivelyconverted to electron-hole pairs:

ηmax ,aSi =1

1000 W/m2

∫ λbg,aSi

λbg,TCO

SR∗aSi (λ)EEλ dλ

= 0.361

ηmax ,aSiGe =1

1000 W/m2

∫ λbg,aSiGe

λbg,aSi

SR∗aSiGe(λ)EEλ dλ

= 0.192

and so the maximum total potential efficiency of this tandemjunction cell is

ηmax = 36.1 + 19.2 = 55.3%

33/56


Tandem device

glass

TCO (SnO2)

p a-Si:H

i a-Si:H

n a-Si:H

contact-

+

n/p tunnel junction

i a-SiGe:H

200 400 600 800 1000 1200 1400

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

EE

W/(

m2

nm

)

c-Sia-Si a-SiGeSnO2

200 400 600 800 1000 1200 1400 (nm)

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

EE

,EE

,u W

/(m

2nm

) EE

EE ,u,top

EE ,u,bot

The usable spectral irradiance (lower).

34/56


ηmax = 36.1 + 19.2 = 55.3%

a substantial improvement over the single-junction a-Si and a-SiGecells of

ηaSi ,max = 36.1% ηaSiGe,max = 46.0%

respectively.

Improvement is partially offset by increased resistance across thetunnel junction and through the cell itself.

Recall most efficient PV cells that currently exist are multijunctiondevices (next page).

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36/56


We take note of two important features in this diagram

1 c-Si solar cells have improved from 13 to 25% over 25 years,but are a long way from the 49.1% potential efficiency wecomputed (note that mc-Si cells are limited to slightly over20%);

2 Both in multijunction and c-Si solar cells, significantimprovement is observed when concentrated solar radiation isused.

37/56

Combined thermal and PV

For a semiconductor with Ebg = 1.75 eV (e.g., a-Si:H) soλbg ,aSi = 708 nm

PV 368 W/m2

thermal 513 W/m2

low-grade 119 W/m2

total 1 kW/m2

500 1000 1500 2000 2500 (nm)

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

EE

W/(

m2

nm

)

AM1.5AM1.5u

h < h crit

crit

38/56

Combined thermal and PV

A potential PV-thermal concentrating system

PV 368 W/m2

thermal 513 W/m2

low-grade 119 W/m2

total 1 kW/m2

load I waste

heat

Challenges: Engineering the thin-film PV/reflector (Ebg ), systemsintegration, optimization, (static) concentrator design

Potential: Efficient power + heating, power + desalination

39/56

Lecture 7, continued


1 Anti-reflection coatings

2 A final look at ultimate efficiency

40/56

PV cell fabrication processing steps

Starting with p-doped Si wafers, the basic steps relevant to c-Si,mc-Si PV cell manufacturing are

1 Surface finishing andtexturing

2 Phosphorous diffusion tocreate the pn-junction

3 Edge isolation to increaseshunt resistance

4 AR deposition;reflectance for bare Sican be 30%; AR coatingsreduce losses < 10%;potentially to < 1%

p-type Si

Starting wafer

Texture etch

P diffusion

Edge isolation

n-type Si

AR deposition

Front contact printing

Back contact print

Firing

41/56

PV cell fabrication processing steps

5 Front contact printing

6 Back contact printing

7 Firing; this thermalprocess also canprofoundly affectother characteristicsof the cellperformance

p-type Si

Starting wafer

Texture etch

P diffusion

Edge isolation

n-type Si

AR deposition

Front contact printing

Back contact print

Firing

At this point the PV cell is operational. However, for a workablepanel, the cells, typically producing 0.5 V and 3.6 A from each10× 10 cm2 cell remain to be wired and packaged.

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Design of anti-reflection coatings

Crystalline silicon nitride chemical formula: Si3N4.

Because AR films are amorphous and contain significant hydrogen,they are denoted SiNx :H with x = 1.33 corresponding tostoichiometric N/Si ratios

n-type

p-type

+ + + + + - - - - -

+

-

Ih,ph

Ie,ph

h!

h+ e-

h+ e-

h+ e-

h+ e-

e- (a)

(b)

(c)

(b)

n0 n1

n2

43/56

Film molecular structure and typical film compositions

x = 0

SiH

NN

N

H

SiSiSi

Si

SiN

H

NH

NN

HN

NHN

N

HNN

Si

Si

Si

Si Si

Si

Si

Si

Si

SiSi

Si

Si

H H

Si

H

Si

Si

Si

H

SiSi

H

H

SiH

Si

x > 1

SiH

NN

N

H

SiSiSi

Si

SiN

H

NH

NN

HN

NHN

N

HNN

Si

Si

Si

Si Si

Si

Si

Si

Si

SiSi

Si

Si

H H

Si

H

Si

Si

Si

H

SiSi

H

H

SiH

Si

The limiting case of a pureamorphous Si:H film - thinkthin-film PV (left) - andamorphous SiNx :H (right)

n-type

p-type

+ + + + + - - - - -

+

-

Ih,ph

Ie,ph

h!

h+ e-

h+ e-

h+ e-

h+ e-

e- (a)

(b)

(c)

(b)

n0 n1

n2

Representative film compositions:

x 0 1.33

H fraction (at) 0.3 0.15N fraction (at) 0 0.5Si fraction (at) 0.7 0.35

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SiNx :H AR films

Film optical and electrical properties vary strongly with filmcomposition x .

The film refractive index n1 has been shown to display acompositional dependence proportional to 1/x = Si/N

n1 =0.65

x+ 1.3 (1)

The band gap decreases with increasing Si content from 5.3 eV fora stoichiometric film to 1.8 eV for amorphous Si.

Ebg ,1 = 1.8 + 2.63x (eV). (2)

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Reflectance

I

T

r01

r12

n0

n1

n2 w

Given a film thickness w , a range ofwavelengths λ, and n0, n1, and n2 asthe refractive indices of thesurrounding medium, AR film, andSi, respectively, we can write thesingle-interface reflectances as thesquares of

r01 =n1 − n0n1 + n0

r12 =n2 − n1n2 + n1

Consider the (idealized) case where no film exists; under theseconditions n0 = 1 and n2 = 3.85 (for Si) and so

r02 =

[n2 − n0n2 + n0

]2=

[3.85− 1

4.85

]2= 35% reflectance.

46/56

Reflectance

I

T

r01

r12

n0

n1

n2 w

Given a film thickness w , a range ofwavelengths λ, and n0, n1, and n2 asthe refractive indices of thesurrounding medium, AR film, andSi, respectively, we can write thesingle-interface reflectances as thesquares of

r01 =n1 − n0n1 + n0

r12 =n2 − n1n2 + n1

Consider the (idealized) case where no film exists; under theseconditions n0 = 1 and n2 = 3.85 (for Si) and so

r02 =

[n2 − n0n2 + n0

]2=

[3.85− 1

4.85

]2= 35% reflectance.

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Reflectance

First and second-surface reflected wave phase difference (inradians)

φd = 2π(2w)n1λ

results in maximum destructive interference when the filmthickness w corresponds to w = λ/(4n1).

Final form of total reflectance

r(w , x , λ) =r201 + r212 + 2r01r12 cosφd

1 + r201r212 + 2r01r12 cosφd

Valid in the limit w → 0?

x = 1 and w = 200 nm

n0 = 1 for air

n1 = 0.65/x + 1.3 for SiNx :H

n2 = 3.85 for Si

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Reflectance


φd = 2π(2w)n1λ



r(w , x , λ) =r201 + r212 + 2r01r12 cosφd

1 + r201r212 + 2r01r12 cosφd


x = 1 and w = 200 nm

n0 = 1 for air

n1 = 0.65/x + 1.3 for SiNx :H

n2 = 3.85 for Si

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Reflectance


φd = 2π(2w)n1λ



r(w , x , λ) =r201 + r212 + 2r01r12 cosφd

1 + r201r212 + 2r01r12 cosφd


x = 1 and w = 200 nm

n0 = 1 for air

n1 = 0.65/x + 1.3 for SiNx :H

n2 = 3.85 for Si

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Reflectance, final note

Where have we seen this before?

48/56

Reflectance, final note

Where have we seen this before?

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AR film absorbance

Light absorption in thin films generally is modeled as anexponential function of film thickness with awavelength-dependent absorption coefficient.

We will model absorption as a high-frequency filter, with thecut-off wavelength λa defined by the x-dependent band gap:

λa(x) =hc

Ebg ,1(x)

thereforea(λ) = H(λ− λa)

where H is the Heaviside (step) function.

Contact shading is proportional to covered area.

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Reflectance, contact shading, SR, ...

Consider an AR film with x = 1 and w = 200 nm and AM1.5normalized to 1 kW/m2

recall min r(λ)at λ = 310 and520 nm

SR∗ =(Ebg/hc)λ

200 400 600 800 1000 1200 1400 (nm)

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

pow

er

W/(

m2

nm

)

EE

EE ,u

(1-r)EE ,u

(1-s)(1-r)EE ,u

50/56

The cumulative effects of a, s, r , and SR∗

Film composition,

thickness, uniformity

Device model

!"

Solar - optical model

Igen

200 400 600 800 1000 1200 1400 (nm)

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

pow

er

W/(

m2

nm

)

EE

EE ,u

(1-r)EE ,u

(1-s)(1-r)EE ,u

Iph,max =

∫ ∞0

SR(λ)EEλ(λ) dλ

=q

Ebg

∫ 1107

280(1− s) [1− a(λ)] [1− r(λ)]EEλ(λ)SR∗(λ) dλ

=q

Ebg339.5 W/m2 with Ebg in J

= 303 A/m2

51/56

Final notes on AR films

Surface roughening by etching

AR film over roughened surface

Surface contact optimization

Buried and laser-etched contact designs

Back-surface roughening

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Final notes on max theoretical efficiency

Schockley and Queisser in 1961 used a thermodynamics-basedargument to compute ultimate PV cell efficiency.

Krogstrup and coworkers in 2013 presented a simpler tounderstand interpretation:

Voc = Ebg

(1− T

Tsun

)− kBT

(ln

Ωemit

Ωsun+ ln

4n22X− ln(QE )

)Ebg , kB in eV

Ωemit , Ωsun radiant energy solid angles (steradians)n, X refractive index, solar conc ratio

QE emission QE

A more detailed discussion of the model terms can be found in ourclass notes.

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found after simplifying to:

Voc = Ebg

(1− T

Tsun

)− kBT

(ln 10 + ln

4n22X

+ ln 1

)

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To summarize our computational strategy for computing themaximum theoretical efficiency, for a given Ebg compute

Voc = Ebg

(1− T

Tsun

)− kBT

(ln 10 + ln

4n22X

)

Iph =

λbg∫λ=0

qλ

hcEEλ(λ) dλ

Io = Iph

[exp

(qVoc

kBT

)− 1

]−1

Pmp : from our ideal diode equation

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Max efficiency: 34.5%

Plus a small shift to the higher Ebg materials

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Documents

Model-based design and optimization of solar energy