solar cells lecture 3: modeling and simulation of photovoltaic devices and systems
DESCRIPTION
J. L. Gray (2011), "Solar Cells Lecture 3: Modeling and Simulation of Photovoltaic Devices and Systems," http://nanohub.org/resources/11690.TRANSCRIPT
Prof. Jeffery L. Gray
[email protected] and Computer Engineering
Purdue UniversityWest Lafayette, Indiana USA
Modeling an Simulation of Photovoltaic Devices and Systems
NCN Summer School: July 2011
Lundstrom 2011
copyright 2011
2
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/
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
Objectives of PV Modeling & Simulation1. 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
Compact Models• based on measured terminal characteristics, lumped
element equivalent circuit models, and semi-analytical models
q/2kT
q/kT
ln J
lnJ02
ln J01
Voltage V
Space Charge Recombination Dominated
Bulk and Surface Recombination Dominated
5
Compact Models
• useful for representing overall device operation (in SPICE, for example)
• provides some physical insight into device performance
( )( ) ( ) 21 2
S Sq V IR kT q V IR kTSC o o S shI I I e I e V IR R+ += − − − +
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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
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Minority Carrier Diffusion Equation: 2
2 ( )oM
m
m mmD G xx τ
−∂− = −
∂
( ) 0Pn W∆ =)(d
d effF,N
pWp
DS
xp
−∆=∆
BSF
BSFd ( )d P
n
Sn n Wx D∆
= − ∆
P+kTqVNN N
nxp e)(D
2i=−
.e)(A
2i kTqV
PP Nnxn =
Boundary Conditions:Law of the Junction
Contacts
or
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It is worth noting that the effective front surface recombination velocity is not independent of the operating condition…
−+−−
+−+
−−
=
1cosh)1e()1(
sinh
cosh)1e(1cosh)1( FF
effF,
p
NpN
kTAqVo
p
N
p
N
p
pkTAqVo
p
NpN
LW
Gps
S
LWL
W
LD
spL
WGSs
So
o
τ
τ
9
Special cases:
• No grid (s=0):
• Full metal (s=1)
• Dark
• Short-Circuit
• V large (~Open-Circuit)s
WDsSS Np
−
+=
1F
effF,
F,eff FS S=
sWDsS
S Np
−
+=
1F
effF,
F,effS →∞
F,eff FS S=
10
But, I digress…MCDE 2
2 ( )Mm
m mD G xx τ
∂ ∆ ∆− = −
∂
( ) 0Pn W∆ =)(d
d effF,N
pWp
DS
xp
−∆=∆
BSF
BSFd ( )d P
n
Sn n Wx D∆
= − ∆
P+kTqVNN N
nxp e)(D
2i=−
.e)(A
2i kTqV
PP Nnxn =
Boundary Conditions:Law of the Junction
Contacts
or
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We can learn a lot from solving the MCDE…
hom( ) ( ) ( )sinh[( ) ] cosh[( ) ]
( )
ogeneous particularM M M
M M m M M mparticularM
m x m x m xA x x L B x x L
m x
∆ = ∆ + ∆= − + −
+ ∆
2
2 ( )Mm
m mD G xx τ
∂ ∆ ∆− = −
∂
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Effects of Base Lifetime on Solar Cell Figures of Merit …
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Effects of BSF on Solar Cell Figures of Merit …
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Spectral Response
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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
0 0
( ) ( )W W
Lm
mq R x dx q G x dx q W Jτ∆
= → =∫ ∫16
What makes a good solar cell?Combining the “law of the junction” at open-circuit
L mJmqWτ
∆ =
( )2
1OCqV kTi
B
nm eN
∆ = −
with the from the previous slide, yields
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What makes a good solar cell?
2ln B m LOC
i
N JV kTqn Wτ
=
SC LJ J=OC OC
OC
ln[ 0.72]kTV q V kTqFF
V kT q
− +=
+
OC SC
in
V FFJP
η =
FF expression from: M. A. Green, Solar Cells: Operating Principles, Technology, and System Applications, Prentice Hall, 1982. 18
What makes a good solar cell?
2ln B m LOC
i
N JV kTqn Wτ
=
• 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)
High VOC yields high FF and JSC, hence efficiency
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What makes a good solar cell?
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What makes a good solar cell?
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Fundamental Limits
“Ultimate” Efficiency1
But a single junction solar cell does not use all the photons efficiently.
1 W. Shockley, W. and H. J. Queisser, “Detailed Balance Limit of Efficiency of p-n Junction Solar Cells,” J. of Appl. Phys., 32(3), 1961, pp. 510-519.
JSC=JLFF=1qVOC=EG
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Carnot Limit (thermodynamic)
58001 94.8%
(~ )solar cell
Sun K
TT
η = − =
• 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
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Fundamental Limits
Gray, J.L.;et. al., "Peak efficiency of multijunction photovoltaic systems," Photovoltaic Specialists Conference (PVSC), 2010 35th IEEE , pp.002919-002923, 20-25 June 2010 24
System Modeling
Modeling and analysis of multijunction PV systems can benefit from a different view of the efficiency.
, ,1
OC j j S Cjjunctonsin
V FF JP
η = ∑
LIGHT
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System Efficiency
η η η η β η η= ∑ , ,sys ultimate photon ic i i V i C iFF
1, ,
1, ,
G i gen iqi
G i gen iq
E I
E Iβ =
∑ηphoton: efficiency of photon absorption ηic: electrical interconnect efficiency
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 ( )
160.5 0.891=
Gray, J. L.; et.al. , "Efficiency of multijunction photovoltaic systems," Photovoltaic Specialists Conference, 2008. PVSC '08. 33rd IEEE , pp.1-6, 11-16 May 2008. 26
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
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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)
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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
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Simulation InputsThe ADEPT input file consists of a series of diktats:
*title simple examplemesh nx=500layer 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=darki-v vstart=0 vstop=.1 dv=.1solve itmax=100 delmax=1.e-6
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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]
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Sample output: terminal characteristics
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Sample output: recombination rate
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Detailed Numerical Simulation
( )p p p pJ q V V kT pµ µ= − ∇ − − ∇
( )V q p n Nε∇ ⋅ ∇ = − − +
p ppJ q G Rt
∂∂
∇ ⋅ = − −
n n
nJ q G Rt
∂∂
∇ ⋅ = − − −
( )n n n nJ q V V kT nµ µ= − ∇ + + ∇
‘Under the Hood’
Semiconductor Equations
Operating conditions, material properties, and other physics are in the B.C. and T, ε, N, G, Rp, Rn, µp, µn, Vp, and Vn.
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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
• 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
1( ) ( )k k kJ v v F v+∆ = −
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Sparseness of 1D Jacobi matrix
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Sparseness of 2D Jacobi matrix
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Questions
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