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  (  

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  

FLUID  FLOW  AND  HEAT  TRANSFER  

( )

( )( )

( ) TTV

AVqBBF

FPVVV

V

2

3

2

2

PrRe1

1Re1

0

∇=∇⋅

+

−=

+∇−∇=∇⋅

=⋅∇Continuity:

2D Navier-Stokes:

Source Term:

Energy Transport:

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

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  

)= )↓";

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

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