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PV Performance Limits by Shockley-Queisser (SQ) Triangle
M. A. Alam
alam@purdue.edu
Electrical and Computer Engineering
Purdue University
West Lafayette, IN USA
Theory and Practice of Solar Cells: A Cell to System Perspective
1
M. A. Alam, PV Lecture Notes
Outline
3
1) Motivation
2) Derivation of the SQ-Triangle
3) Applications of SQ-Triangle
4) Conclusions
5) Appendix: Self-test questions
M. A. Alam, PV Lecture Notes
An Inefficient Machine!
4
𝜂𝐴 ∼ 1/2
𝜂𝑁 = 2/𝜋
𝜂𝑆𝑄 = 1/3
𝜂 = 𝜂𝑁 × 𝜂𝑆𝑄 × 𝜂𝑀 × 𝜂𝐴=2
𝜋×1
3×5
6×1
2∼
1
10
Tandem Bifacial
𝜂𝑀 = 4/5
M. A. Alam, PV Lecture Notes
Power and efficiency
5
𝐸1
𝐸2
Incre
asing
ban
dgap
Incre
asing b
andgap
𝑞𝑉𝑚𝑝 = 0.95𝐸𝑔 − 0.3
𝐽𝑚𝑝 = 82(1 − 0.428𝐸𝑔)
𝐽𝑆𝑈𝑁∗
𝐽𝑚𝑝
𝑉𝑚𝑝 (𝐸𝑔)
𝐽𝑚𝑝,1
𝑉𝑚𝑝,1
𝜂
𝐸𝑔,1
Counting the photons – a roadmap
6
Shockley-QueisserYablonovitch
Partition,
Cell-Module
Radiative
Angle Entropy
Carnot
Below bandgap
Above bandgap
LED
CPV
Tandem
MEG
>4n2
Striping
c
c c
M. A. Alam, PV Lecture Notes
Outline
7
1) Motivation
2) Derivation of SQ-Triangle
3) Application of SQ-Triangles
4) Conclusions
M. A. Alam, PV Lecture Notes
Defining the SQ-Triangle
8
𝐽𝑚𝑝 = 82(1 − 0.428𝐸𝑔)
𝑞𝑉𝑚𝑝 = 0.95𝐸𝑔 − 0.3
𝐽𝑚𝑝 = 72 1 − 0.52 𝑉𝑚𝑝
1. Calculate 𝑉𝑚𝑝 and 𝐼𝑚𝑝
2. Calculate efficiency:
𝜂 = 𝑉𝑚𝑝 𝐼𝑚𝑝 /𝑃𝑖𝑛
3. Locate in the triangle
M. A. Alam, PV Lecture Notes
Step 1: Calculate 𝐽𝑚𝑝
9
𝐽𝑚𝑝 = 83.75(1 − 0.428𝐸𝑔)
M. A. Alam, PV Lecture Notes
Step 2: Derive 𝑉𝑚𝑝 by a 2-Level Atom
E1
E2
m1
m2
nph(Ts)
TDSome deep sea Plackton live
in a similar diffused light environment
Single frequency filtered light
Isotropic illumination
Atom with only two levels
10
M. A. Alam, PV Lecture Notes
Detailed balance between Photons and Electrons
E1
E2
m1
m2
nph(TS)
TD
1 2(1 )( 1) phf nf
2 2(1 ) phf f nUpward transition
Downward transition
1 2 2 2(1 )( 1) (1 ) ph phf n f ff n
At steady state up and down transitions are equal …
1
2
1 /
2 /
1
1
1
1
D
D
T
T
f
f
e
e
1 1 1 2 2 2 m m B Bk E k E
11M. A. Alam, PV Lecture Notes
Detailed balance of photons and electrons
1 2 2 2(1 )( 1) (1 )ph phf f fn nf
2 1
1 2 2 1
/ /
/ / / /
1 1( 1) (
1)
1 1 1
D D
D D D Dph
T T
T T pT T hne
e en
e
e e
E1
E2
m1m2
TD
/
/ /
11
1 1
S
S S
B
B Bph p Th
k T
k T kn n
e
e e
Bose Einstein
distribution
12M. A. Alam, PV Lecture Notes
PV Efficiency of 2-level System
nph(Ts)
1 2 1 2 1( ) ( )m m
D
S
ET
ET
Single molecule-single frequency gives Carnot efficiency!
2 2 2 1 1 1m m
D S D
E E E E
T T T
E1
E2
m1
m2
TD
300
6000 1 0.95
1 2
1 2
1m m
D
S
R T
E E R T
1 2( ) 0
if
m m
SDT T
Can not extract
energy at same
temperature
13
M. A. Alam, PV Lecture Notes
Problem of Angular Anisotropy
The angle mismatch leads to Vmp loss
12 1 2 2 2(1 )( 1) (1 ) ph phf f n f f n
negative number
12
SUN
1 2 2 1 2 1 1
2
or, ln
B D B S B D
E E E E
k T k T k T
m
m
2
11 2 1 ln
m m
Dmp g B D
S
TV E k T
T
14
Step 1: Derivation of Vmp complete
5
1 06 1
2 4
2 2
Without mirror
With mirror
1 2 1
1 2 1 2 2
1 lnD B D
S
T k T
E E T E E
m m
Consistent with all experimental data !
(1.10
0.706)
(1.45,
1.111)
(1.01
0.736)Green,
Prog. In PV, 2011
1 2 2
1
2
0.95 0.31
,
Set, 6000 , 300 ,
2 :
1 ln
m
S D
g
Dg B D
S
g
T K T K
E E E
TE k TqV E
T
12
SUN
15
Concentrator cells fool the cells
, S C D
1 1ln
S
D
Dmp g B D
S
Dg
S
TE
T
TV E k T
T
Number of suns … 2 / 104720SN
16
M. A. Alam, PV Lecture Notes
Step 3: Creating the SQ Triangle
𝐸1
𝐸2
𝑞𝑉𝑚𝑝 = 𝐸𝑔 1 −𝑇𝐷𝑇𝑆
– 𝑘𝐵 𝑇𝐷 ln𝜃𝐷𝑐 𝜃𝑆
= 𝑐𝑓𝐸𝑔 – Δ
𝐽𝑚𝑝 𝐸𝑔 = 𝑐𝐼𝑠𝑢𝑛 1 − 𝛽′𝐸𝑔
= 𝐽0 1 − 𝛽 𝑉𝑚𝑝
𝑉𝑚𝑝
𝐼𝑚𝑝
Hirst, PIP, 2011
Khan & Alam, APL, 2015,
Also, see AJP, 2012
𝛽 ≡ 𝛽’ /𝑐𝑓
𝑉0 ≡ (1 − 𝛽Δ Τ) 𝛽
)𝐽0 ≡ 𝑐𝐼𝑠𝑢𝑛 (1 − 𝛽Δ
17
Outline
18
1) Motivation
2) Derivation of SQ-Triangle
3) Applications of SQ-Triangle
4) Conclusions
M. A. Alam, PV Lecture Notes
𝑆1 = 𝑥 1 − 𝑥
𝑑𝑆1𝑑𝑥
= 1 − 2𝑥 = 0 → 𝑥 =1
2
Maximum Single Junction Efficiency
𝐽𝑠𝑢𝑛∗
𝐽𝑚𝑝
𝑉𝑚𝑝 (𝐸𝑔)
𝐽𝑠𝑢𝑛∗ /2
𝑉𝑚𝑝,1
1.0
𝐽𝑚𝑝/𝐽𝑠𝑢𝑛∗
𝛽𝑉𝑚𝑝
1 − 𝑥
𝑥 1.0
0.52 ∗ 𝑉𝑚𝑝 =1
2, 𝑉𝑚𝑝 = 0.96, 0.95 × 𝐸𝑔 − 0.3 = 𝑉𝑚𝑝 = 0.96, 𝐸𝑔 = 1.33𝑉
𝐽𝑚𝑝 = 72 1 − 0.52 𝑉𝑚𝑝
M. A. Alam, PV Lecture Notes
19
Tandem Solar Cell
20
𝑬𝟎
𝑬𝟏+
𝑬𝒊+
𝑬𝑷
TCO
𝑬𝟏−
𝑬𝒊−
𝑬𝑸
TCO
1-Sun
R-Sun
M. A. Alam, PV Lecture Notes
N-Junction Tandem Cell
𝐸𝑔,1, 𝑉𝑚𝑝,1
𝐸𝑔,2, 𝑉𝑚𝑝,2
𝐸𝑔,3, 𝑉𝑚𝑝,3
𝑉𝑚𝑝𝑖= Τ𝑖𝑉0 𝑁 + 1
𝐼𝑚𝑝𝑖= Τ𝐼0 (𝑁 + 1)
𝑉𝑚𝑝 = 𝑐𝑓 𝐸𝑔 – 0.31 – 𝑘𝐵 𝑇𝐷 ln 𝑐
M. A. Alam, PV Lecture Notes
21
Efficiency of N-junction Tandem
𝜂𝑁 𝑐 = (𝐼0 𝑉0 Τ/4𝑐) × 2𝑁 (𝑁 + 1)
)𝐼0 ≡ 𝑐𝐼𝑠𝑢𝑛 (1 − 𝛽Δ
𝑉0 ≡ (1 − 𝛽Δ Τ) 𝛽𝛽 = 𝛽 Τ’ 𝑐𝑓
M. A. Alam, PV Lecture Notes
22
Tandem Efficiency Limits
𝐽𝑆𝑈𝑁∗
𝐽𝑚𝑝
𝑉𝑚𝑝 (𝐸𝑔)
𝐽𝑆𝑈𝑁∗
2
𝑉𝑚𝑝,1
𝐽𝑆𝑈𝑁∗
𝐽𝑚𝑝
𝑉𝑚𝑝(𝑋𝑔)
1
3𝐽𝑆𝑈𝑁∗
2
3𝐽𝑆𝑈𝑁∗
𝐽𝑆𝑈𝑁∗
𝐽𝑚𝑝
𝑉𝑚𝑝(𝑋𝑔)
1
𝑁𝐽𝑆𝑈𝑁∗
𝑁 − 1
𝑁𝐽𝑆𝑈𝑁∗
𝐽𝑚𝑝 = 70 1 − 0.52 𝑉𝑚𝑝 𝑃𝑜𝑢𝑡 = 𝐽𝑚𝑝 × 𝑉𝑚𝑝
𝜂 = 33% 44% 66%
M. A. Alam, PV Lecture Notes
23
Bifacial Solar Cell
24
𝜂 = 𝜂𝑁 × 𝜂𝑆𝑄 × 𝜂𝑀 × 𝜂𝐴=2
𝜋×1
3×5
6×1
2∼
1
10
𝜂𝐴 ∼ 1/2
M. A. Alam, PV Lecture Notes
𝐽𝑚𝑝
𝐽0= 1
𝑉𝑚𝑝 /𝑉01 = 1
𝑥 2𝑥
𝐸1+
𝐸2+
For inverted bifacial design
𝑆4 = 𝑥(1 − 𝑥) + 𝑥 (1 − 2𝑥) + 𝑥 1 − 2𝑥 − 𝛾𝑥 + 𝑥 1 −𝑥
𝑅
𝐽𝑚𝑝
𝐽0= 𝑅
2𝑥+𝛾𝑥
𝑥+(1
−𝛾)𝑥
𝑥
𝐸0𝐸0
𝐸2−1
𝛾 = 1/(1 + 𝑅)
1 − 2𝑥 − 𝛾𝑥
1 −𝑥
𝑅
M. A. Alam, PV Lecture Notes
25
SQ Triangle for Bifacial PV
𝑆𝑁𝑆1
=2 1 + 𝑅 𝑁2
𝑁 𝑁 + 1 𝑅 + 1 − 2𝑅
𝑆𝑁𝑆1
=8𝑅 1 + 𝑅 𝑁2
2𝑅 2𝑁2 + 4𝑁 − 1 – 𝑅2 − 1
𝑁𝑐𝑟𝑖𝑡≤
1+
𝑅−1
M. A. Alam, PV Lecture Notes
26
Advantages of Bifacial PV
𝑆𝑁𝑆1
=2 1 + 𝑅 𝑁2
𝑁 𝑁 + 1 𝑅 + 1 − 2𝑅
𝑆𝑁𝑆1
=8𝑅 1 + 𝑅 𝑁2
2𝑅 2𝑁2 + 4𝑁 − 1 – 𝑅2 − 1
𝑁𝑐𝑟𝑖𝑡 ≤ 1 + 𝑅−1
M. A. Alam, PV Lecture Notes
27
Plant-PV Tandem cells
𝜆
𝜆𝜆
M. A. Alam, PV Lecture Notes
28
Rescaled Tandem System
𝐽𝑆𝑈𝑁
𝐽𝑆𝐶
𝐸𝑔
For plants
𝐽𝑆𝑈𝑁𝑛𝑒𝑤
𝐽𝑆𝐶
𝑋𝑔
𝐽𝑆𝑈𝑁 83.75 mA/cm2
𝛽 0.428 eV−1
𝐽𝑆𝑈𝑁𝑛𝑒𝑤 63.2095 mA/cm2
𝛽𝑛𝑒𝑤 0.5666 eV−1
With the new parameters
we can now redesign any
N-J tandem
𝐽𝑆𝐶 𝑋𝑔 = 𝐽𝑆𝑈𝑁(1 − 𝛽𝑋𝑔)
𝐽𝑆𝐶𝑛𝑒𝑤 𝑋𝑔 = 𝐽𝑆𝑈𝑁 1 − 𝛽𝑋𝑔 − 𝐽𝑆𝐶
𝑐𝑢𝑡
= 𝐽𝑆𝑈𝑁𝑛𝑒𝑤 1 − 𝛽𝑛𝑒𝑤𝑋𝑔
M. A. Alam, PV Lecture Notes
29
Conclusions
• Solar cells are inefficient machines. A thermodynamic analysis
shows how we can improve the performance.
• A simple SQ Triangle anticipates thermodynamic limits for
tandem PV, bifacial PV, and concentrator PV cells.
• Tandem efficiency scales as: 𝜂𝑁 𝑐 = 𝐼0 𝑉0 Τ𝑁 2 𝑁 + 1 𝑐
• Bifacial PV has phase transition at 𝑁𝑐𝑟𝑖𝑡 ≤ 1 + 𝑅−1
• Thermodynamic limit serves as a beacon and a guardrail for
next generation PV.
M. A. Alam, PV Lecture Notes
30
31
Single (SJ) or multi-junction (MJ)SJ: J-V or Eg-sweepMJ: J-V or N-sweep
Input spectrum: AM1.5G or ideal BlackbodyDistance from sun (for Blackbody input)Spectral low-pass filter (for MJ calculations)Reflectance of the ground (for MJ calc.)
Simulation setup
Simulation specific input
Spectral input
These set of inputs change based on the choice of simulation setup
(a) (b) (c)
nanohub.org/resources/pvlimitshttp://arxiv.org/abs/1606.01176
M. A. Alam, PV Lecture Notes
Self-assessment
1. How does the average bandgap of tandem cell compare to that of a single
junction solar cell?
2. How does the efficiency of a triple junction solar cell compare to that of a
single junction solar cell? Hint. Efficiency ratio is given by 2N/(N+1).
3. What is the maximum short-circuit current achievable from AM1.5
illumination? Ans. 70 mA/cm^2.
4. What is the maximum short circuit current for a 4-junction solar cell? Ans.
70/(N+1) = 14 mA/cm^2
5. For R=0.3, what is efficiency of a 4-junction bifacial solar cell?
6. If R_s increases by a factor of 5, how much does the critical concentration for
a CPV decrease by? 32M. A. Alam, PV Lecture Notes
34
Wait, Wait don’t tell me …
Maximum Jsc (in mA/cm2) from AM1.5
spectrum is
a) 7000 b) 700 c) 70 d) 7
70 mA/cm2
M. A. Alam, PV Lecture Notes
35
Wait, Wait don’t tell me …
Maximum Jsc for Eg=1.7 eV perovskite
under AM1.5 illumination is
a) 100 b) 50 c) 20 d) 5
Jsc=83.75(1-0.43Eg)
=23 mA/cm2
M. A. Alam, PV Lecture Notes
36
Wait, Wait don’t tell me …
For a 3-junction tandem, maximum
Jsc in mA/cm2 is
a) 100 b) 70 c) 17 d) 4
Jsc=Jmax/(N+1)Jsc
Vmp
M. A. Alam, PV Lecture Notes
37
Wait, Wait don’t tell me …
If Eg=1eV, maximum 𝑉𝑚𝑝 is
a) 1.0 b) 0.65 c) 0.45 d) 0.25
Voc=0.95Eg-0.31
M. A. Alam, PV Lecture Notes
38
Wait, Wait don’t tell me …
In a N=3 tandem,
E1=1.9eV, E3=0.97,
what is E2?
a) 2.5 b) 1.6 c) 1.3 d) 1.1
Eg,ave=Eg,SJ=1.33
M. A. Alam, PV Lecture Notes
39
Wait, Wait don’t tell me …
In a N=3 tandem, What is Vmp?
a) 4 b) 3 c) 2 d) 1
Vmp/N=0.95Eavg-0.31
M. A. Alam, PV Lecture Notes
40
A Little Formula Sheet
𝐸𝑔,𝑆𝐽𝑜𝑝𝑡
≅ 2.55 𝑘𝑇𝑠
𝑞𝑉𝑜𝑐,𝑆𝐽 = 0.95 × 𝐸𝑔 − 0.232
𝑞𝑉𝑚𝑝,𝑆𝐽 = 0.95 × 𝐸𝑔 − 0.31
𝐽𝑠𝑐,𝑆𝐽 = 𝐽0 1 − 𝛽𝐸𝑔 (AM1.5, 𝐽0 = 83.75, 𝛽 = 0.428).
𝐹𝐹 ~ Τ𝑣𝑜𝑐 𝑣𝑜𝑐 + 4.7
𝜂𝑇.𝑆𝐽 = −26.45𝐸𝑔2 + 70.77𝐸𝑔 − 14.42 (AM1.5,
empirical)
𝐽𝑠𝑐 𝑁 =2
𝑁+1𝐽𝑠𝑐( 𝐸𝑔 )
𝑞𝑉𝑚𝑝/𝑁 = 𝐸𝑔 1 −𝑇𝐷
𝐸𝑔
𝐸𝑔,𝑚𝑎𝑥
𝑇𝑠– 𝑘𝐵 𝑇𝐷 𝑙𝑛
𝜃𝐷
𝜃𝑆
𝐸𝑔,max =𝑁 − 1
𝛽 𝑁+𝛽 1 + 𝑅 𝐸0 − 𝑅
𝛽 × 𝑁
(𝑇 − 𝑇𝑎) = 𝑃/ℎ = 1000(1 − 𝜂 − 𝑅)/ℎModule
Tandem
SJ cell
M. A. Alam, PV Lecture Notes
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