Minority Carrier Diffusion Length Is, and Calculate Its

Toward a 1D Device Model Part 2: Material Fundamentals Lecture 8 – 10/4/2011 MIT Fundamentals of Photovoltaics 2.626/2.627 – Fall 2011 Prof. Tonio Buonassisi

1 Buonassisi (MIT) 2011 2.626/2.627 Roadmap

You Are Here

2 Buonassisi (MIT) 2011 2.626/2.627: Fundamentals

Every photovoltaic device must obey: Output Energy Conversion Efficiency    Input Energy

For most solar cells, this breaks down into: Inputs Outputs

Charge Light Charge Charge Charge Solar Spectrum Drift/Diff Absorption Excitation Separation Collection usion

total absorption excitation drift/diffusion separation collection

3 Buonassisi (MIT) 2011  Liebig’s Law of the Minimum

S. Glunz, Advances in Optoelectronics 97370 (2007)

Image by S. W. Glunz. License: CC-BY. Source: "High-Efficiency Crystalline Silicon Solar Cells." Advances in OptoElectronics (2007).

total absorption excitation drift/diffusion separation collection

4 Buonassisi (MIT) 2011

 Rough Depiction of Interrelated Materials & Device Effects

Devices Solar Cell Efficiency ()

J V  FF   sc oc 

Open‐Circuit Short‐Circuit Fill Factor Current (J ) Voltage (Voc) sc (FF)

Jsc  qIQE p d

Internal Quantum Shunt Series

Efficiency (IQE) Resistance (Rsh) Resistance (Rs)

1 Materials  1  IQE  1  to NA,D,i  Leff 

Built‐In Voltage Surface Fermi‐ Absorption Diffusion

(Vo) Level Pinning Coefficient () Length (Leff)

  kT NAND Vo  ln  L  D  2 to Rs diff bulk q  ni 

Carrier Concent‐ Surface Density Im[Dielectric Lifetime () Mobility (µ) ration (NA, ND, ni) Of States Constant] (2)

Source: T. Buonassisi, unpublished. Buonassisi (MIT) 2011

5 Learning Objectives: Toward a 1D Device Model

1. Describe what minority carrier diffusion length is, and calculate its

impact on Jsc, Voc. Describe how minority carrier diffusion length is affected by minority carrier lifetime and minority carrier mobility. 2. Describe how minority carrier diffusion length is measured. 3. Lifetime: • Describe basic recombination mechanisms in semiconductor materials. • Calculate excess carrier concentration as a function of carrier lifetime and generation rate. Compare to background (intrinsic + dopant) carrier concentrations. 4. Mobility: • Describe common mobility-limiting mechanisms (dopants, temperature, ionic semiconductors).

6 Buonassisi (MIT) 2011 Minority Carrier Diffusion Length

Definition: Minority carrier diffusion length is the average distance a minority carrier moves before recombining.

Importance to a Solar Cell: Photoexcited carriers must be able to move from their point of generation to where they can be collected. Longer diffusion lengths generally result in better performance.

7 Buonassisi (MIT) 2011 Minority Carrier Diffusion Length

Definition: The average distance a minority carrier moves before recombining.

Importance to a Solar Cell: Carriers must be able to move from their point of generation to where they can be collected.

Cross section of solar cell made of high-quality material

Minority carrier diffusion length (Ldiff) is LARGE. Solar cell current output (Jsc) is large.

Most electrons diffuse through the solar cell uninhibited, contributing to high photon-to- electron (quantum) efficiencies.

8 Buonassisi (MIT) 2011 Minority Carrier Diffusion Length

Definition: The average distance a minority carrier moves before recombining.

Importance to a Solar Cell: Carriers must be able to move from their point of generation to where they can be collected.

Cross section of solar cell made of defect-ridden material

Minority carrier diffusion length (Ldiff) is small. Solar cell current output (Jsc) is small.

Electrons generated closer to the surface make it to the contacts, but those in the bulk are likely to “recombine” (lose their energy, e.g., at bulk defects, and not contribute to the solar cell output current).

9 Buonassisi (MIT) 2011 Mathematical Formalism

Recall that the current produced in an illumianted pn-junction device, is limited by the minority carrier flux at the edge of the space-charge region.

PC1D Simulation for 300um thick Si Solar Cell Jsc  qGLdiff 10 9

8 Jsc  Ldiff 7

6 From Eq. 4.44 in Green. 5 Jsc = illuminated current = JL G = carrier generation rate 4 3

ShortCirctui Current [mA/cm2] 2  1 0 0.00 0.10 0.20 0.30 0.40 0.50 Diffusion Length [microns]

10 Buonassisi (MIT) 2011 Mathematical Formalism

Note that the voltage is also affected by Ldiff.

  kBT Jsc From Eq. 4.45 in Green. Voc  ln 1 Voc = open-circuit voltage q Jo  Jo = saturation current density

2 From Eq. 4.37 in Green. qDni D = minority carrier diffusivity Jo  N = majority carrier dopant concentration LdiffN ni = intrinsic carrier concentration

2 Voc  lnLdiff  2lnLdiff  lnLdiff 

11 Buonassisi (MIT) 2011  Mathematical Formalism

Assuming a weak dependence of FF on Ldiff, we have the following relationship:

  JscVoc  Ldiff lnLdiff 

12 Buonassisi (MIT) 2011 Mathematical Formalism

Assuming a weak dependence of FF on Ldiff, we have the following relationship:

  JscVoc  Ldiff lnLdiff 

Ldiff >> device thickness.

Ldiff << device thickness.

13 Buonassisi (MIT) 2011 Minority Carrier Diffusion Length

When your material has short minority carrier diffusion length relative to absorber thickness, two engineering options: Reduce absorber layer thickness (if light management permits) or increase diffusion length.

14 Buonassisi (MIT) 2011 Learning Objectives: Toward a 1D Device Model

1. Describe what minority carrier diffusion length is, and calculate its

15 Buonassisi (MIT) 2011 Collection Probability

Courtesy of Christiana Honsberg. Used with permission.

16 Buonassisi (MIT) 2011 Collection Probability

n-type

p-type

Courtesy of Christiana Honsberg. Used with permission.

From PV CDROM 17 Buonassisi (MIT) 2011 Collection Probability

Courtesy of Christiana Honsberg. Used with permission.

From PV CDROM 18 Buonassisi (MIT) 2011 Spectral Response (Quantum Efficiency)

Courtesy of PVCDROM. Used with permission.

Standards: IEC 60904-3 and 60904-8 from PVCDROM 19 Buonassisi (MIT) 2011 Minority Carrier Diffusion Length

At each point… Mapped over an entire sample…

Graph and photo © source unknown. All rights reserved. This content is excluded from our Creative Commons license. For more information, see http://ocw.mit.edu/fairuse. See: P. A. Basore, IEEE Trans. Electron. Dev. 37, 337 (1990).

20 Buonassisi (MIT) 2011 What Limits Diffusion Length?

21 Buonassisi (MIT) 2011 What Limits the Minority Carrier Diffusion Length?

Quick answer: Recombination-active defects, intrinsic mobility limitations, or absence of percolation pathways.

Thin films: Effect of bulk defects (GBs) on . Nanostructured: Effect of morphology on .

Please see lecture video for visuals.

R.B. Bergmann, Appl. Phys. A 69, 187 (1999) F. Yang et al., ACS Nano 2, 1022 (2008)

22 Buonassisi (MIT) 2011 Mathematical Formalism Diffusion length is governed by “lifetime” and “mobility”

Ldiff = bulk diffusion length Ldiff  Dtbulk D = diffusivity tbulk = bulk lifetime

kBT For carriers in an electric field D   kB = Boltzmann coefficient q T = temperature q = charge  µ = mobility Definition of “bulk lifetime”: The average time an excited carrier exists before recombining. (I.e., temporal analogy to diffusion length.) Units = time. tbulk of µs to ms is typical for indirect bandgap semiconductors, while tbulk of ns to µs is typical of direct bandgap semiconductors. Definition of “mobility”: How easily a carrier moves under an applied field.(i.e., ratio of drift velocity to the electric field magnitude.) Expressed in units of cm2/(V*s). Mobilities of 10-100’s cm2/Vs typical for most crystalline semiconductors. Can be orders of magnitude lower for organic, amorphous, and ionic materials. 23 Buonassisi (MIT) 2011 Mathematical Formalism Diffusion length is governed by “lifetime” and “mobility”

Ldiff = bulk diffusion length Ldiff  Dtbulk D = diffusivity tbulk = bulk lifetime

Carrier LifetimeFor carriers in an electric field kBT D   kB = Boltzmann coefficient q T = temperature q = charge µ = mobility Carrier Mobility  Definition of “bulk lifetime”: The average time an excited carrier exists before recombining. (I.e., temporal analogy to diffusion length.) Units = time. tbulk of µs to ms is typical for indirect bandgap semiconductors, while tbulk of ns to µs is typical of direct bandgap semiconductors. Definition of “mobility”: How easily a carrier moves under an applied field.(i.e., ratio of drift velocity to the electric field magnitude.) Expressed in units of cm2/(V*s). Mobilities of 10-100’s cm2/Vs typical for most crystalline semiconductors. Can be orders of magnitude lower for organic, amorphous, and ionic materials. 24 Buonassisi (MIT) 2011 Learning Objectives: Toward a 1D Device Model

1. Describe what minority carrier diffusion length is, and calculate its

25 Buonassisi (MIT) 2011 Excess Electron Carrier Concentration

Generally equal to doping concentration

n  n0  n n  Gt

Excess Electron Carrier Generation rate Density carriers  Excited Carrier

cm 3  sec Lifetime carriers   3   cm  [sec] 26 Buonassisi (MIT) 2011

  Minority, Majority Carriers in Silicon Under AM1.5G n  Gt 1011cm 3

Excited Carrier AM1.5G ~ 1016 Lifetime carriers  ~10μs cm 3  sec if : 11 3 N 1016cm 3 n  p 10 cm D then :  10 3 ni 10 cm n 2 p  p  i 104 cm 3 0 N n  p  ni D ND  n 27 Buonassisi (MIT) 2011

  Carrier Lifetime and Recombination

Bulk Minority Carrier Lifetime:

n n = Excess minority t  carrier concentration R R = Recombination rate Minorit Majority y Carrier Carrier First approximation of minority carrier lifetime, for low injection (e.g., illumination) conditions.

h+ h+ e- More Detailed Calculation: h+ h+ 1 1  tbulk trad

Radiative p-type silicon recombination, can be derived from thermodynamics: [ε=α] 28 Buonassisi (MIT) 2011 Carrier Lifetime and Recombination

Bulk Minority Carrier Lifetime:

h+ h+ e- More Detailed Calculation: h+ h+ 1 1 1   tbulk trad t Auger

1 Dominant under t  p-type silicon very high injection Auger 2 CNA conditions

29 Buonassisi (MIT) 2011  Carrier Lifetime and Recombination

Bulk Minority Carrier Lifetime:

h+ h+ e- More Detailed Calculation: h+ h+ 1 1 1 1    tbulk trad t Auger tSRH

Defect level- p-type silicon mediated recombination

30 Buonassisi (MIT) 2011 Radiative Recombination

2 R  G  Bnp  Bn i Under equilibrium dark conditions

R  B np  n2 Net Recombination i  non-equilibrium

= R-Gequilibrium

hν n  n0  n

p  p0  p

1 trad  n-type ( where n0=ND) BNA  n  1 trad  p-type ( where p0=NA) BNDp 

31 Buonassisi (MIT) 2011

 Radiative Recombination

1 hν t  100ms rad, Si 2 1015 1016 

B n

Radiative recombination is very slow in silicon, and is rarely the limiting lifetime in silicon-based solar cells. However, radiative recombination is often the lifetime- limiting recombination pathway for high-quality “thin- film” materials, including GaAs. 32 Buonassisi (MIT) 2011 Defects and Carrier Recombination: τSRH

Defects can form in semiconductors. Defects can introduce midgap stages.

Defect formation energy (EA) determines Midgap states are characterized by their equilibrium concentration at a given energy level(s) (ED) and capture cross temperature. sections for electrons and holes (se, sh) ) i Calculated midgap states for Fe complexes in Si Interstitial Iron (Fe Iron Interstitial ) s Substitutional Iron (Fe Iron Substitutional

S.K. Estreicher et al., Phys. Rev. B 77, 125214 (2008). © APS. All rights reserved. This content is excluded from our Creative Commons license. For more information, see http://ocw.mit.edu/fairuse. Buonassisi (MIT) 2011 33 Defects and Carrier Recombination: τSRH

Trap State (introduced by defect) tn0 n0  p1  n t p0 p0  n1  n  tSRH  n0  p0  n E E   T c   kT  n1  Nc  e

Ev ET  Ec    kT  p1  Nv  e ET 1 Ev t p 0  Nvths p 1 t  n 0 Nv s

capture capture th n capture capture - + emission emission e emission emission - h + e

h N  trap density s  capture cross- section

34 Buonassisi (MIT) 2011

 Defects and Carrier Recombination: τSRH

Trap State (introduced by defect)

With deep traps (i.e. mid-gap) and low-injection conditions: 1 tSRH, n type  t p 0  Nvths p Ec 1 ET tSRH, p type  tn 0  Nvths n Ev

Deep traps and high-injection capture capture

capture capture conditions: - + emission emission e emission emission - h + e

h tSRH  tn0 t p0

35 Buonassisi (MIT) 2011 Localized Defects Create Efficient Recombination

Pathway

Direct Bandgap Indirect Bandgap Semiconductor Semiconductor

E E

hv = Eg Recombination efficient Eg Recombination inefficient (no phonon required) (phonon required) t ~ ns to µs k k t ~ ms

(a) Direct (b) Indirect

Image by MIT OpenCourseWare.

36 Buonassisi (MIT) 2011 Localized Defects Create Efficient Recombination

Pathway

Direct Bandgap Indirect Bandgap Semiconductor Semiconductor

E E

hv = Eg Recombination efficient Eg Efficient recombination (no phonon required) via defect level! t ~ ns to µs k k t < µs with high defect concentrations

(a) Direct (b) Indirect

Image by MIT OpenCourseWare.

37 Buonassisi (MIT) 2011 Defects Impact Minority Carrier Lifetime

Impurity Point Defects (c-Si) Dislocations (c-Si)

© American Institute of Physics. All rights reserved. This content Courtesy of Elsevier, Inc., http://www.sciencedirect.com. Used with permission. is excluded from our Creative Commons license. For more information, see http://ocw.mit.edu/fairuse.

A.A. Istratov, Materials Science & Engineering B, 134, 282 (2006) C. Donolato, Journal of Applied Physics 84, 2656 (1998) 38 Buonassisi (MIT) 2011 Effect of Defects on Minority Carrier t, Ldiff The distribution of defects matters as much as their total concentration!

Reprinted by permission from Macmillan Publishers Ltd: Nature Materials. Source: Buonassisi, T., et al. "Engineering Metal-Impurity Nanodefects for Low- Cost Solar Cells." Nature Materials 4, no. 9 (2005): 676-9. © 2005. T. Buonassisi et al., Nature Mater. 4, 676 (2005)

39 Buonassisi (MIT) 2011 Surfaces Introduce Many Mid-Gap States

mid-gap surface states provide additional pathway for

Surface Surface recombination.

They are often formed by dangling bonds on the surface.

Sample Si Si Si Si trap thickness 2 Si Si Si Si trap W 1 W  Ref: Horányi et al., tsurf     Appl. Surf. Sci. 63, 306 (1993) Si Si Si Si trap 2S D 

Si Si Si Si trap Surface recombination Carrier velocity diffusivity http://pvcdrom.pveducation.org/CHARACT/surface.htm 40 Buonassisi (MIT) 2011  Proper Passivation Reduces Surface Recombination Surface Surface Proper surface passivation ties up dangling bonds, reduces density of trap states.

“Perfect” passivation yields:

S 0, tsurf  Si Si Si Si H Practically, S approaching 1 cm/s have Si Si Si Si H been achieved on silicon.

Si Si Si Si H

41 Buonassisi (MIT) 2011 Measuring Surface Recombination Velocity

1. Measure lifetime of samples of varying thickness, but with same bulk and surface properties. (The lifetime will be affected by both bulk and surface

conditions, thus measurement will reveal an “effective” lifetime, or teff.)

2. Any variation in lifetime should be due to a changing tsurf, per below:

Ref: Horányi et al., Appl. Surf. Sci. 63, 306 (1993)

3. With >2 lifetime measurements (sample thicknesses), can solve for tsurf!

42 Buonassisi (MIT) 2011 Auger Recombination, τAuger

2 2 Rn type ~ pn Rp type ~ np 1 1 t  t  n type Cn 2 p type Cp2

*Above equations true at sufficiently high doping 18 -3 densities (≥10 cm in silicon) 43 Buonassisi (MIT) 2011   Auger Recombination in n-type silicon

2 Rn type ~ pn 1 tn type  2 Cn © AIP. All rights reserved. This content is excluded from our Creative Commons license. For more information, see http://ocw.mit.edu/fairuse. J. Dziewior, W. Shmid, Appl. Phys. Lett., 31, p346 (1977)

44 Buonassisi (MIT) 2011  Auger Recombination in n-type silicon

Courtesy of Daniel Macdonald. Used with permission. 1 1 1 1    t bulk t band t Auger tSRH

Daniel MacDonald thesis “Recombination and Trapping in Multicrystalline Silicon Solar Cells,” The Australian National University (May 2011) 45 Buonassisi (MIT) 2011

 Measuring Minority Carrier Lifetime Recall that 1 1 1 1    tbulk trad t Auger tSRH If defect-mitigated recombination is dominant, band-to-band radiative recombination will be suppressed: 1 1  t t rad SRH or

trad tSRH

In fact, band-to-band (radiative recombination) and defect-mitigated (non-radiative recombination) are inversely proportional.

By imaging the band-to-band (radiative) recombination using a very sensitive CCD camera, we are able to quantitatively extract the minority

carrier lifetime. 46 Buonassisi (MIT) 2011 Photoluminescence Imaging (PLI)

Experimental Setup Measurement of Wafer Chuck with d. contacts

Luminescence/heat radiation

CCD Camera

Laser

Image by MIT OpenCourseWare.

© AIP. All rights reserved. This content is excluded from our Creative Commons license. For more information, see http://ocw.mit.edu/fairuse. M. Kasemann et al., Proc. IEEE PVSC, San Diego, CA (2008). T. Trupke et al., Appl. Phys. Lett. 89, 044107 (2006).

47 Buonassisi (MIT) 2011 Temperature Changes σ0 emission emission - e

Trapped carriers are more likely to be (re-)emitted at higher thermal energies

(kbT). At lower T, trapped carriers reside in traps © Springer-Verlag. All rights reserved. This content is excluded from our Creative Commons license. For more information, see http://ocw.mit.edu/help/faq-fair-use/. longer, facilitating recombination.

S. Rein, Lifetime spectroscopy. Springer-Verlag (Berlin, 2005). p. 430 Buonassisi (MIT) 2011 48 Learning Objectives: Toward a 1D Device Model

1. Describe what minority carrier diffusion length is, and calculate its

49 Buonassisi (MIT) 2011 Effect of Reduced Mobility on Solar Cell Performance

Low carrier mobility can reduce device efficiency by several tens of percent relative (as big of an impact as lifetime!)

J. Mattheis et al., Phys. Rev. B. 77, 085203 (2008)

50 Buonassisi (MIT) 2011 What Limits Mobility?

1. Defect Scattering 2. Trapping at stretched bonds 3. Incomplete percolation pathways 4. Phonon Scattering

Example of carrier trapping Example of complex percolation pathway L. Wagner et al., PRL 101, 265501 (2008) F. Yang et al., ACS Nano 2, 1022 (2008)

Diagrams removed due to copyright restrictions. See lecture video.

51 Buonassisi (MIT) 2011 Mobility and Carrier Concentration in Semiconductor

Increased concentration of ionized dopant atoms increases conductivity, but can reduce carrier mobility (due to scattering).

52 Buonassisi (MIT) 2011 Percolation Pathways

Both µ and t play a role in determining Ldiff, and hence efficiency, for various device architectures.

Table removed due to copyright restrictions. See lecture video.

F. Yang et al., ACS Nano 2, 1022 (2008)

53 Buonassisi (MIT) 2011 Simple Example

1. Well-behaved inorganic semiconductor (no effect of trapping at stretched bonds and incomplete percolation pathways). 2. Defect scattering dominant  ionized dopants inside sample scatter carriers.

54 Buonassisi (MIT) 2011 Demo: Conductivity of Heated Intrinsic and Doped Silicon

silicon

HEAT

55 Buonassisi (MIT) 2011 Demo Explained 1 n  n  n  N  n s   qn 0 D  For intrinsic Si, n >> ND. For highly doped Si, n << ND. Conductivity is the product of µ and n.

10000  1.E+16

1.E+14 3] - 1000

1.E+12

1.E+10 100 1.E+08

"Intrinsic [cm^2/sec] Mobility 1.E+06 10

Carrier Concentration [cm^ Concentration Carrier Carriers" 1.E+04 "Carriers Doped Na=1e16" 1.E+02 1 100 250 400 550 0 100 200 300 400 500 600 Temp [K] Temp [K]

56 Buonassisi (MIT) 2011 MIT OpenCourseWare http://ocw.mit.edu

2.627 / 2.626 Fundamentals of Photovoltaics Fall 2013

For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.