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CRYSTALLIZATION MODELING FOR SOLAR CELLS
GERMAN OLIVEROS ENERGY SYSTEMS INITIATIVE MEETING CENTER FOR ADVANCED PROCESS DECISION MAKING
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MOTIVATION
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THE DEMAND FOR SOLAR ENERGY
q Urgent need to rely less on tradiConal energy sources • Unsustainable • Contaminant
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SILICON SOLAR CELL SUPPLY CHAIN
Silica Quartz (
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HORIZONTAL RIBBON GROWTH: THE MISSING LINK?
• First conCnuous design patented by William Shockley in 1959:
Germanium ribbons Ice ribbons
• First experimental work published by Carl Bleil in 1968:
• Improvements of the process reported by B. Kudo in 1979:
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RESEARCH OBJECTIVE
To design and analyze a con;nuous wafering system using the concepts of Systems Engineering and Materials Science to find desirable design parameters, relevant process variables and op;mal opera;ng condi;ons
…under the hypothesis that a smooth and uniform film is formed on top of the melt.
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CONCEPTUAL DESIGN
Challenges:
• CrystallizaCon and stability (micro-‐scale) to achieve single crystalline Silicon film • Complex fluid flow and heat transfer interacCon • Impurity modeling • Lack of experimental data and proof of concept
Vpull
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• ConCnuous process to produce 150 – 250 micron wafer • Silicon floats on top of its melt • No sawing, hence no material losses VcrystallizaCon
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FLUID FLOW AND HEAT TRANSFER
( )
( )( )
( ) TTV
AVqBBF
FPVVV
V
2
3
2
2
PrRe1
1Re1
0
∇=∇⋅
+
−=
+∇−∇=∇⋅
=⋅∇Continuity:
2D Navier-Stokes:
Source Term:
Energy Transport:
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CRYSTALLIZATION DYNAMICS
"↓$ &↓'$ ()↓$ /(, = -↓$ (↓↑2 )↓$ /(/↑2
"↓0 &↓'0 ()↓0 /(, = -↓0 (↓↑2 )↓0 /(/↑2 (&↓0 /(, =1(↓↑2 &↓0 /(/↑2
&↓$ =2&↓0
-↓$ ()↓$ /(/ − -↓0 ()↓0 /(/ =ρ∆345/4, −1(&↓0 /(/ = 45/4, (&↓0∗ − &↓$∗ ) )↓0 = )↓8 + 4)↓9 /4& &↓0
Solid
Interface CondiCons
-()↓$ /(, =σε()↑4 − )↓↑4 ↓∞ )
(&↓$ /(: =0
Top
Vy(y,t)
Qs(y,t)
Ql(y,t)
ΔH
Ny(y,t)
Liquid
Closed Domain composed of two moving subdomains
Qrad
“Hot”
(&↓$ /(: =0
)= )↓$ Booom
ASSUMPTIONS
a) UnidirecConal solidificaCon b) Neglect convecCon in the melt
c) No mass diffusion in solid, limited mass diffusion in liquid: Ds = 0, 0 < DL < ∞ 9
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CRYSTALLIZATION DYNAMICS
• Slow CrystallizaCon VelociCes avoids formaCon of Impurity Boundary Layer
"↓$ &↓'$ ()↓$ /(, = -↓$ (↓↑2 )↓$ /(/↑2
"↓0 &↓'0 ()↓0 /(, = -↓0 (↓↑2 )↓0 /(/↑2 (&↓0 /(, =1(↓↑2 &↓0 /(/↑2
&↓$ =2&↓0
-↓$ ()↓$ /(/ − -↓0 ()↓0 /(/ =ρ∆345/4, −1(&↓0 /(/ = 45/4, (&↓0∗ − &↓$∗ ) )↓0 = )↓8 + 4)↓9 /4& &↓0
Solid
-()↓$ /(, =σε()↑4 − )↓↑4 ↓∞ )
(&↓$ /(: =0
Top
Liquid
(&↓$ /(, =0
)= )↓$ Booom
Interface CondiCons
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CRYSTALLIZATION DYNAMICS • Contrast this with faster interface velociCes
"↓$ &↓'$ ()↓$ /(, = -↓$ (↓↑2 )↓$ /(/↑2
"↓0 &↓'0 ()↓0 /(, = -↓0 (↓↑2 )↓0 /(/↑2 (&↓0 /(, =1(↓↑2 &↓0 /(/↑2
&↓$ =2&↓0
-↓$ ()↓$ /(/ − -↓0 ()↓0 /(/ =ρ∆345/4, −1(&↓0 /(/ = 45/4, (&↓0∗ − &↓$∗ ) )↓0 = )↓8 + 4)↓9 /4& &↓0
Solid
)= )↓";
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CRYSTALLIZATION DYNAMICS
• Impurity effect on thermal profile is seen in undercooling effect at the interface
• High levels of undercooling lead to interfacial instability!
"↓$ &↓'$ ()↓$ /(, = -↓$ (↓↑2 )↓$ /(/↑2
"↓0 &↓'0 ()↓0 /(, = -↓0 (↓↑2 )↓0 /(/↑2 (&↓0 /(, =1(↓↑2 &↓0 /(/↑2
&↓$ =2&↓0
-↓$ ()↓$ /(/ − -↓0 ()↓0 /(/ =ρ∆345/4, −1(&↓0 /(/ = 45/4, (&↓0∗ − &↓$∗ ) )↓0 = )↓8 + 4)↓9 /4& &↓0
Solid
-()↓$ /(, =σε()↑4 − )↓↑4 ↓∞ )
(&↓$ /(: =0
Free Boundary
Liquid
(&↓$ /(, =0
)= )↓$ Booom
Interface CondiCons
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MULLINS-‐SEKERKA STABILITY
• Morphological stability theory developed by Mullins and Sekerka proved condiCons for stability under diffusive heat and mass flow.
• Stability is determined by temperature gradients in both phases, concentraCon
gradients on the liquid side and surface tension effects.
Apply PerturbaCon Crystal
Melt I(y,t) = I(y,t) + δ(t) sin (ωx)
Crystal
Melt
Grows?
Decays?
Crystal
Melt
Crystal
Melt
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MULLINS-‐SEKERKA CONDITION
( )
( ) c
cM
mGpDVGG
DVmGp
DVGGp
DVTV
ωω
ωωωωω
δδ
2*'
*2*'*2 2
+⎥⎦
⎤⎢⎣
⎡⎟⎠
⎞⎜⎝
⎛−−
⎭⎬⎫
⎩⎨⎧
⎥⎦
⎤⎢⎣
⎡⎟⎠
⎞⎜⎝
⎛−+⎥⎦
⎤⎢⎣
⎡⎟⎠
⎞⎜⎝
⎛−+−⎥⎦
⎤⎢⎣
⎡⎟⎠
⎞⎜⎝
⎛−Γ−
=
•
INSTABILITY = CAPILLARITY + THERMAL GRADIENTS
CONCENTRATION GRADIENTS
+
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PROOF OF CONCEPT Challenges: How to make it work?
1. CrystallizaCon should begin at the center of the melt surface
2. Once film is formed, how to develop a pulling mechanism that resembles the operaCon with
silicon?
• Start seeding process inside of the bath Strategy:
• Flat non-‐conducCng solid surface to facilitate nucleaCon
• HeaCng on the sidewalls
• Water inlet
Start pulling …and hope for the best
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PROOF OF CONCEPT
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THANK YOU
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EXISTING TECHNOLOGIES
Edge-‐Defined Film Fed Growth (EFG) Molded wafer (MW)
Shaping die
Silicon melt Substrate
Silicon sheet
Crystal growth Ribbon
pulling
Ribbon-‐Growth on Substrate (RGS) String Ribbon
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EXISTING TECHNOLOGIES
Method Thickness† (µm)
Pulling speed
(cm/min)
Throughput (cm2/min)
Furnaces/ 100 MWP
EFG 300 1-‐2 136 100 SR 300 1-‐2 13 600 RGS 300 650 10140 2-‐3 MW 450 300 4680 6-‐7
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RELEVANT PROPERTIES AND SYSTEM PARAMETERS
Property Value
Density of Liquid Silicon 2293 [kg/m3]
Density of Solid Silicon 2570 [kg/m3]
Thermal Conductivity of Liquid Silicon 18 [W/m ⁰C]
Thermal Conductivity of Solid Silicon 58 [W/m ⁰C]
Heat Capacity of Liquid Silicon 968 [J/kg ⁰C]
Heat Capacity of Solid Silicon 1040 [J/kg ⁰C]
Mass Diffusivity of Aluminum in Silicon 7e-8 [m2/s]
Segregation Coefficient of Aluminum in Silicon 2.8e-3
Slope of the Liquidus Line in Al-Si Phase Diagram -0.000095 [⁰C/ppm] (or -0.95 [⁰C/%wt]
Melting Point of Pure Silicon 1414 ⁰C
Temperature of “Cold” Wall 1300 [⁰C]
Temperature of “Hot” Wall 1500 [⁰C]
Initial Temperature 1500 [⁰C]
Initial Concentration of Aluminum in Silicon Melt 5000 [ppm]
Height of System 3 [cm]
Desired Thickness 1.2 millimeters
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CURRENT TECHNOLOGY: THE CZOCHRALSKI AND WIRE SAW PROCESS
• Process discovered by Jan Czochralski in 1916 • SCll one of the most common techniques to produce the silicon wafers • High purity product with minimal amount of imperfecCons • Extensive modeling by Robert Brown at MIT
Disadvantages
û Final product has circular cross-‐secConal area û Batch Process
û Tremendous amount of material losses
Half of “technological effort” is lost 21