The document provides information about a summer school on modeling and simulation of photovoltaic devices and systems being held in July 2011. It outlines the course, which will cover objectives of PV modeling and simulation, device modeling, fundamental limits, system modeling of multijunction devices, and detailed numerical simulation. The instructor is Prof. Jeffery L. Gray of Purdue University and the material is provided under a Creative Commons license.

1. NCN Summer School: July 2011
Modeling an Simulation of
Photovoltaic Devices and Systems
Prof. Jeffery L. Gray
grayj@purdue.edu
Electrical and Computer Engineering
Purdue University
West Lafayette, Indiana USA

2. copyright 2011
This material is copyrighted by Jeffery L. Gray under
the following Creative Commons license.
Conditions for using these materials is described at
http://creativecommons.org/licenses/by-nc-sa/2.5/
Lundstrom 2011
2

3. Outline
1. Objectives of PV Modeling & Simulation
2. PV Device Modeling
3. Fundamental Limits
4. PV System Modeling (multijunction)
5. Detailed Numerical Simulation:
“Under the Hood”
3

4. Objectives of PV Modeling &
Simulation
1. Understanding of measured device operation
• dependence of terminal characteristics (Voc, Jsc, FF, η) on
◦ Device structure (dimensions, choice of materials, doping,
etc.)
◦ Material parameters (mobility, lifetimes, etc.)
2. Predictions of performance
• Different operation conditions
◦ Temperature, illumination conditions, etc.
Leads to improved
designs
4

5. Compact Models
• based on measured terminal characteristics, lumped
element equivalent circuit models, and semi-analytical
models
Bulk and Surface
Recombination
Dominated
ln J
Space Charge
Recombination
Dominated
q/2kT
lnJ02
q/kT
ln J01
Voltage V
5

6. Compact Models
• useful for representing
overall device operation (in
SPICE, for example)
• provides some physical
insight into device
performance
I = I SC − I o1e q (V + IRS ) kT − I o 2 e q (V + IRS ) 2 kT − (V + IRS ) Rsh
6

7. Analytic Models
• based on relevant device physics (minority
carrier diffusion equation)
• provides deeper insight into device operation and
design dependencies
• device and material characterization methods
typically based on analytic models
• limited by simplifying assumptions
7

8. Minority Carrier Diffusion
Equation: D ∂ m m−m 2
− =−G ( x) o
M
∂x 2 τm
Boundary Conditions:
BSF
Law of the Junction
ni2 qV
p N (− x N ) = e kT +
P
ND
ni2 qV
nP ( xP ) = e kT
.
NA
Contacts
d∆p S F,eff d∆n S
= ∆p (−W N ) ∆n(WP ) =
0 or = ∆n(WP )
− BSF
dx Dp dx Dn
8

9. It is worth noting that the effective front
surface recombination velocity is not
independent of the operating condition…
 W 
 D cosh N 
 WN  Lp
(1 − s) S F G N τ p  cosh − 1 + po (e − 1)  s + SF 
qV Ao kT p
 Lp   Lp W 
   sinh N 
 Lp 
S F,eff =
  WN 
(1 − s )  po (e qV Ao kT
− 1) + G N τ p  cosh − 1 
  Lp 
  
9

10. Special cases:
• No grid (s=0): S F,eff = S F
• Full metal (s=1) S F,eff → ∞
S F + s D p WN
• Dark S F,eff =
1− s
• Short-Circuit S F,eff = S F
S F + s D p WN
• V large (~Open-Circuit) S F,eff =
1− s
10

11. But, I digress…
MCDE D ∂ ∆m ∆m 2
− =( x)
−G M
∂x 2 τm
Boundary Conditions:
BSF
Law of the Junction
ni2 qV
p N (− x N ) = e kT +
P
ND
ni2 qV
nP ( xP ) = e kT
.
NA
Contacts
d∆p S F,eff d∆n S
= ∆p (−W N ) ∆n(WP ) =
0 or = ∆n(WP )
− BSF
dx Dp dx Dn
11

12. We can learn a lot from solving
the MCDE…
∂ 2 ∆m ∆m
DM − =( x)
−G
∂x 2
τm
∆mM ( x) = ∆mM ogeneous ( x) + ∆mM
hom particular
( x)
= AM sinh[( x − xM ) Lm ] + BM cosh[( x − xM ) Lm ]
+ ∆mM
particular
( x)
12

13. Effects of Base Lifetime on
Solar Cell Figures of Merit …
13

14. Effects of BSF on Solar Cell
Figures of Merit …
14

15. Spectral Response
15

16. What makes a good solar cell?
The key is the open-circuit voltage…
Consider a solar cell with a perfect BSF and very thin
emitter, then
• All recombination occurs in the base (minority carrier
lifetime is τm)
• At open-circuit, minority carrier concentration in the
base (width W) is constant wrt position and total
recombination must equal total generation
∆m
W W
q ∫ R( x)dx = q ∫ G ( x)dx → q W = JL
0 0
τm
16

17. What makes a good solar cell?
Combining the “law of the junction” at open-circuit
ni2 qVOC
=∆m
NB
e ( kT
−1)
J Lτ m
with the ∆m = from the previous slide, yields
qW
17

18. What makes a good solar cell?
N Bτ m J L
VOC = kT ln
qni2W
kT
VOC − ln[q VOC kT + 0.72]
FF =
q J SC = J L
VOC + kT q
VOC FFJ SC
η=
Pin
FF expression from: M. A. Green, Solar Cells: Operating Principles, Technology, and System
18
Applications, Prentice Hall, 1982.

19. What makes a good solar cell?
High VOC yields high FF and JSC, hence efficiency
N Bτ m J L
VOC = kT ln
qni2W
• Optically thick (light trapping)
• Mechanically thin
• High doping (trade-off with lifetime and ni {bandgap
narrowing})
• Wide bandgap [low ni] (trade-off with JL)
• Plus, assumptions of perfect BSF and thin emitter
• Slight modifications for high-injection conditions and for other
dominant recombination mechanisms (Auger, radiative)
19

20. What makes a good solar cell?
20

21. What makes a good solar cell?
21

22. Fundamental Limits
“Ultimate” Efficiency1
But a single junction solar cell
does not use all the photons
efficiently.
JSC=JL
FF=1
qVOC=EG
1W. Shockley, W. and H. J. Queisser, “Detailed Balance Limit of Efficiency of p-n Junction Solar Cells,” J. 22
of
Appl. Phys., 32(3), 1961, pp. 510-519.

23. Carnot Limit (thermodynamic)
Tsolar cell
η=
1− =
94.8%
TSun (~ 5800 K )
• More detailed calculations put the limit at ~87% as the
number of junctions approaches infinity (~300K)
• Efficiency actually peaks for a finite number of junctions
and approaches zero as the number of junctions
approaches infinity
23

24. Fundamental Limits
Gray, J.L.;et. al., "Peak efficiency of multijunction photovoltaic systems," Photovoltaic Specialists Conference
24
(PVSC), 2010 35th IEEE , pp.002919-002923, 20-25 June 2010

25. System Modeling LIGHT
Modeling and analysis of
multijunction PV systems can
benefit from a different view of
the efficiency.
1
η=
Pin
∑
junctons
VOC , j FFj J S Cj
,
25

26. System Efficiency
ηsys = ηultimate η photon ηic ∑ β i FFi ηV ,i ηC,i
1
ηphoton: efficiency of photon absorption EG,i Igen,i
βi = q
ηic: electrical interconnect efficiency ∑ 1
q
EG,i Igen,i
ηV,i: voltage efficiency (qVOC/EG)
ηC,i: collection efficiency
Achievement of a PV system efficiency of greater than 50%
requires that the geometric average of these six terms
(excluding β) must exceed ( 0.5 ) = 0.891
1
6
Gray, J. L.; et.al. , "Efficiency of multijunction photovoltaic systems," Photovoltaic Specialists Conference,
26
2008. PVSC '08. 33rd IEEE , pp.1-6, 11-16 May 2008.

27. Detailed Numerical Simulation
• based on more rigorous device physics
• numerical solution circumvents need for simplifying
assumptions, i.e. allows spatially variable parameters
• provides predictive capability
o Terminal Characteristics (I-V, SR, C-V, etc.)
• provides diagnostic capability
o Can examine internal parameters (energy band,
recombination, etc.)
• Ability to test simplifying assumption in analytic modeling
27

28. Historical Overview of Solar Cell
Simulation at Purdue (not comprehensive)
 SCAP1D (Lundstrom/Schwartz ~1979)
 x-Si solar cells (1D)
 SCAP2D (Gray/Schwartz ~ 1981)
 x-Si solar cells (2D)
 PUPHS (Lundstrom, et. al. mid-1980s)
 III-V heterostructure solar cells (1D)
 TFSSP (Gray/Schwartz mid-1980s)
 Amorphous Si solar cells (1D)
 ADEPT (Gray, et. al. late 1980s to present)
 A Device Emulation Program and Tool(box)
 Arbitrary heterostructure solar cells (CIS, CdTe, a-Si, Si, GaAs,
AlGaAs, HgCdTe, InGaP, InGaN, …)
 Fortran version (1D, on nanoHUB )
 C versions (1D, 2D -- 3D capable, but not extensively used)
 MatLab ™ toolbox (under development – 1D, 2D, 3D)
28

29. Simulation Inputs
 solar cell structure: composition, contacts, doping,
dimensions
 material properties: dielectric constant, band gap,
electron affinity, other band parameters, absorption
coefficients, carrier mobilities, recombination
parameters, etc.
 operating conditions: operating temperature, applied
bias, illumination spectrum, small-signal frequency,
transient parameters
29

30. Simulation Inputs
The ADEPT input file consists of a series of diktats:
*title simple example
mesh nx=500
layer tm=2 nd=1.e17 eg=1.12 ks=11.9 ndx=3.42
+ nv=1.83e19 nc=3.22e19 up=400. un=800.
layer tm=200 na=1.e16 eg=1.12 ks=11.9 ndx=3.42
+ nv=1.83e19 nc=3.22e19 up=400. un=800.
genrec gen=dark
i-v vstart=0 vstop=.1 dv=.1
solve itmax=100 delmax=1.e-6
30

31. Simulation Outputs
 the numerical solution provides the value of the potential,
V, and the carrier concentrations, p and n at every point
within the device, from which one can compute and
display:
• the terminal characteristics, i.e. I-V, cell efficiency,
spectral response, etc. [predictive]
• a microscopic view of any internal parameter – for
example, recombination rate (i.e. losses) [diagnostic]
31

32. Sample output: terminal
characteristics
32

33. Sample output:
recombination rate
33

34. Detailed Numerical Simulation
‘Under the Hood’
Semiconductor Equations
∇ ⋅ ε∇V = −q ( p − n + N )
  ∂ p   ∂n
∇ ⋅= q  G − R p −
Jp  ∇ ⋅ J n = −q  G − Rn − 
 ∂t   ∂t 
 
J p = µ p ∇ (V − V p ) − kT µ p ∇p
−q J n = µn∇ (V + Vn ) + kT µn∇n
−q
Operating conditions, material properties, and other
physics are in the B.C. and T, ε, N, G, Rp, Rn, µp, µn, Vp,
and Vn.
34

35. Numerical Solution
 Transform differential equations into difference
equations on a spatial grid – yields a large set of non-
linear difference equations.
 Use a a generalized Newton method to solve – results
in a iterative sequence of matrix equations
J (v k )∆v k +1 = (v k )
−F
• v = [p n V]; F(vk) is the set of difference equations
• J(∆vk) is a sparse block tri-diagonal matrix of order 3n , where n
is the number of mesh points (1D)
• In 2D (n x m grid), J(∆vk) is a sparse block tri-diagonal matrix of
order 3nm
35

36. Sparseness of 1D Jacobi matrix
36

37. Sparseness of 2D Jacobi matrix
37

38. Questions
38