Prof. Jeffery L. Gray

grayj@purdue.eduElectrical 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

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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”

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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

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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

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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

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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=

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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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