Abstract

We analyze the electromagnetic properties of InP/InAs nanowire solar cells for different geometries. We address both eigenvalue calculations to determine the wave propagation as well as source problems to simulate direct perpendicular illumination by three-dimensional finite element calculations. We demonstrate the validity of a 2D waveguide modal analysis as a method of estimating the results of the computationally far more demanding 3D analysis. The resulting data is employed in a detailed balance analysis in order to determine the optimum set of bandgap energies for a single-junction and dual-junction cell as well as the corresponding efficiency limit. The efficiency of the nanowire design can approach the efficiency of conventional thin-film designs despite the low volume fill-factor.

© 2009 Optical Society of America

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  1. L. Tsakalakos, "Nanostructures for photovoltaics," Mater. Sci. Eng. R 62(6), 175-189 (2008).
    [CrossRef]
  2. W. Shockley and H. Queisser, "Detailed Balance Limit of Efficiency of p-n Junction Solar Cells," J. Appl. Phys. 32, 510 (1961).
    [CrossRef]
  3. B. Tian, T. Kempa, and C. Lieber, "Single nanowire photovoltaics," Chem. Soc. Rev. 38(1), 16-24 (2009).
    [CrossRef]
  4. E. Palik, Handbook of optical constants of solids (Academic Press, 1985).
  5. J. Jianming, "The finite element method in electromagnetics,"Wiley & Sons (1993).
  6. Q1. J. Nedelec, "Mixed finite elements in R 3," Numerische Mathematik 35(3), 315-341 (1980).
    [CrossRef]
  7. F. R¨omer, B. Witzigmann, O. Chinellato, and P. Arbenz, "Investigation of the Purcell effect in photonic crystal cavities with a 3D finite element Maxwell solver," Opt. Quantum Electron. 39(4), 341-352 (2007).
    [CrossRef]
  8. M. Hano, "Finite-element analysis of dielectric-loaded waveguides," IEEE Trans. Microwave Theory Tech. 32(10), 1275-1279 (1984).
    [CrossRef]
  9. Q2. R. Orobtchouk, "On Chip Optical Waveguide Interconnect: the Problem of the In/Out Coupling," Springer Series in Optical Sciences 119, 263-290 (2006).
    [CrossRef]
  10. Q3. G. L’etay and A. Bett,"EtaOpt-a program for calculating limiting efficiency and optimum bandgap structure for multi-bandgap solar cells and TPV cells," Spectrum 20 (2001).
  11. M. Schubert, F. Mont, S. Chhajed, D. Poxson, J. Kim, and E. Schubert, "Design of multilayer antireflection coatings made from co-sputtered and low-refractive-index materials by genetic algorithm," Opt. Express 16(8), 5290-5298 (2008).
    [CrossRef]
  12. S. Choi, C. Palmstrøm, Y. Kim, D. Aspnes, H. Kim, and Y. Chang, "Dielectric functions and electronic structure of InAsP films on InP," Appl. Phys. Lett. 91, 041,917 (2007).
    [CrossRef]

2009 (1)

B. Tian, T. Kempa, and C. Lieber, "Single nanowire photovoltaics," Chem. Soc. Rev. 38(1), 16-24 (2009).
[CrossRef]

2008 (2)

2007 (2)

S. Choi, C. Palmstrøm, Y. Kim, D. Aspnes, H. Kim, and Y. Chang, "Dielectric functions and electronic structure of InAsP films on InP," Appl. Phys. Lett. 91, 041,917 (2007).
[CrossRef]

F. R¨omer, B. Witzigmann, O. Chinellato, and P. Arbenz, "Investigation of the Purcell effect in photonic crystal cavities with a 3D finite element Maxwell solver," Opt. Quantum Electron. 39(4), 341-352 (2007).
[CrossRef]

2006 (1)

Q2. R. Orobtchouk, "On Chip Optical Waveguide Interconnect: the Problem of the In/Out Coupling," Springer Series in Optical Sciences 119, 263-290 (2006).
[CrossRef]

2001 (1)

Q3. G. L’etay and A. Bett,"EtaOpt-a program for calculating limiting efficiency and optimum bandgap structure for multi-bandgap solar cells and TPV cells," Spectrum 20 (2001).

1984 (1)

M. Hano, "Finite-element analysis of dielectric-loaded waveguides," IEEE Trans. Microwave Theory Tech. 32(10), 1275-1279 (1984).
[CrossRef]

1980 (1)

Q1. J. Nedelec, "Mixed finite elements in R 3," Numerische Mathematik 35(3), 315-341 (1980).
[CrossRef]

1961 (1)

W. Shockley and H. Queisser, "Detailed Balance Limit of Efficiency of p-n Junction Solar Cells," J. Appl. Phys. 32, 510 (1961).
[CrossRef]

Arbenz, P.

F. R¨omer, B. Witzigmann, O. Chinellato, and P. Arbenz, "Investigation of the Purcell effect in photonic crystal cavities with a 3D finite element Maxwell solver," Opt. Quantum Electron. 39(4), 341-352 (2007).
[CrossRef]

Aspnes, D.

S. Choi, C. Palmstrøm, Y. Kim, D. Aspnes, H. Kim, and Y. Chang, "Dielectric functions and electronic structure of InAsP films on InP," Appl. Phys. Lett. 91, 041,917 (2007).
[CrossRef]

Chang, Y.

S. Choi, C. Palmstrøm, Y. Kim, D. Aspnes, H. Kim, and Y. Chang, "Dielectric functions and electronic structure of InAsP films on InP," Appl. Phys. Lett. 91, 041,917 (2007).
[CrossRef]

Chhajed, S.

Chinellato, O.

F. R¨omer, B. Witzigmann, O. Chinellato, and P. Arbenz, "Investigation of the Purcell effect in photonic crystal cavities with a 3D finite element Maxwell solver," Opt. Quantum Electron. 39(4), 341-352 (2007).
[CrossRef]

Choi, S.

S. Choi, C. Palmstrøm, Y. Kim, D. Aspnes, H. Kim, and Y. Chang, "Dielectric functions and electronic structure of InAsP films on InP," Appl. Phys. Lett. 91, 041,917 (2007).
[CrossRef]

Hano, M.

M. Hano, "Finite-element analysis of dielectric-loaded waveguides," IEEE Trans. Microwave Theory Tech. 32(10), 1275-1279 (1984).
[CrossRef]

Kempa, T.

B. Tian, T. Kempa, and C. Lieber, "Single nanowire photovoltaics," Chem. Soc. Rev. 38(1), 16-24 (2009).
[CrossRef]

Kim, H.

S. Choi, C. Palmstrøm, Y. Kim, D. Aspnes, H. Kim, and Y. Chang, "Dielectric functions and electronic structure of InAsP films on InP," Appl. Phys. Lett. 91, 041,917 (2007).
[CrossRef]

Kim, J.

Kim, Y.

S. Choi, C. Palmstrøm, Y. Kim, D. Aspnes, H. Kim, and Y. Chang, "Dielectric functions and electronic structure of InAsP films on InP," Appl. Phys. Lett. 91, 041,917 (2007).
[CrossRef]

Lieber, C.

B. Tian, T. Kempa, and C. Lieber, "Single nanowire photovoltaics," Chem. Soc. Rev. 38(1), 16-24 (2009).
[CrossRef]

Mont, F.

Nedelec, J.

Q1. J. Nedelec, "Mixed finite elements in R 3," Numerische Mathematik 35(3), 315-341 (1980).
[CrossRef]

Orobtchouk, R.

Q2. R. Orobtchouk, "On Chip Optical Waveguide Interconnect: the Problem of the In/Out Coupling," Springer Series in Optical Sciences 119, 263-290 (2006).
[CrossRef]

Palmstrøm, C.

S. Choi, C. Palmstrøm, Y. Kim, D. Aspnes, H. Kim, and Y. Chang, "Dielectric functions and electronic structure of InAsP films on InP," Appl. Phys. Lett. 91, 041,917 (2007).
[CrossRef]

Poxson, D.

Queisser, H.

W. Shockley and H. Queisser, "Detailed Balance Limit of Efficiency of p-n Junction Solar Cells," J. Appl. Phys. 32, 510 (1961).
[CrossRef]

R¨omer, F.

F. R¨omer, B. Witzigmann, O. Chinellato, and P. Arbenz, "Investigation of the Purcell effect in photonic crystal cavities with a 3D finite element Maxwell solver," Opt. Quantum Electron. 39(4), 341-352 (2007).
[CrossRef]

Schubert, E.

Schubert, M.

Shockley, W.

W. Shockley and H. Queisser, "Detailed Balance Limit of Efficiency of p-n Junction Solar Cells," J. Appl. Phys. 32, 510 (1961).
[CrossRef]

Tian, B.

B. Tian, T. Kempa, and C. Lieber, "Single nanowire photovoltaics," Chem. Soc. Rev. 38(1), 16-24 (2009).
[CrossRef]

Tsakalakos, L.

L. Tsakalakos, "Nanostructures for photovoltaics," Mater. Sci. Eng. R 62(6), 175-189 (2008).
[CrossRef]

Witzigmann, B.

F. R¨omer, B. Witzigmann, O. Chinellato, and P. Arbenz, "Investigation of the Purcell effect in photonic crystal cavities with a 3D finite element Maxwell solver," Opt. Quantum Electron. 39(4), 341-352 (2007).
[CrossRef]

Appl. Phys. Lett. (1)

S. Choi, C. Palmstrøm, Y. Kim, D. Aspnes, H. Kim, and Y. Chang, "Dielectric functions and electronic structure of InAsP films on InP," Appl. Phys. Lett. 91, 041,917 (2007).
[CrossRef]

Chem. Soc. Rev. (1)

B. Tian, T. Kempa, and C. Lieber, "Single nanowire photovoltaics," Chem. Soc. Rev. 38(1), 16-24 (2009).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (1)

M. Hano, "Finite-element analysis of dielectric-loaded waveguides," IEEE Trans. Microwave Theory Tech. 32(10), 1275-1279 (1984).
[CrossRef]

J. Appl. Phys. (1)

W. Shockley and H. Queisser, "Detailed Balance Limit of Efficiency of p-n Junction Solar Cells," J. Appl. Phys. 32, 510 (1961).
[CrossRef]

Mater. Sci. Eng. R (1)

L. Tsakalakos, "Nanostructures for photovoltaics," Mater. Sci. Eng. R 62(6), 175-189 (2008).
[CrossRef]

Numerische Mathematik (1)

Q1. J. Nedelec, "Mixed finite elements in R 3," Numerische Mathematik 35(3), 315-341 (1980).
[CrossRef]

Opt. Express (1)

Opt. Quantum Electron. (1)

F. R¨omer, B. Witzigmann, O. Chinellato, and P. Arbenz, "Investigation of the Purcell effect in photonic crystal cavities with a 3D finite element Maxwell solver," Opt. Quantum Electron. 39(4), 341-352 (2007).
[CrossRef]

Spectrum (1)

Q3. G. L’etay and A. Bett,"EtaOpt-a program for calculating limiting efficiency and optimum bandgap structure for multi-bandgap solar cells and TPV cells," Spectrum 20 (2001).

Springer Series in Optical Sciences (1)

Q2. R. Orobtchouk, "On Chip Optical Waveguide Interconnect: the Problem of the In/Out Coupling," Springer Series in Optical Sciences 119, 263-290 (2006).
[CrossRef]

Other (2)

E. Palik, Handbook of optical constants of solids (Academic Press, 1985).

J. Jianming, "The finite element method in electromagnetics,"Wiley & Sons (1993).

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Figures (6)

Fig. 1.
Fig. 1.

Illustration of nanowire array geometry and of the relation between 3D and waveguide simulation. Left: definition of the unit cell U, the boundary ∂U top and arbitrary volume VU. Center: nanowire geometry and coordinate reference. Right: relation between 3D and waveguide simulation.

Fig. 2.
Fig. 2.

Dispersion relation of an InAs nanowire array (a 0=600 nm, d 0=180 nm, without oxide). The distribution of the intensity of the electric field within a quarter of the unit cell of each mode is illustrated at λ 0=300 nm. Modes with ℜ{n eff,m }<5×10-5 are assumed to have reached cut-off.

Fig. 3.
Fig. 3.

Penetration depth (top) and modal transmission coefficient (bottom) of an InAs nanowire array (a 0=600 nm, without oxide) as a function of the the nanowire diameter d0 at λ 0=300 nm wavelength. The numbers assigned to the most predominant modes are referring to Fig. 2.

Fig. 4.
Fig. 4.

Comparison of spectral power absorption ratio of various InP and InAs nanowire designs (nanowire length h 0=2 µm). Solid lines: 2D waveguide calculations with impedance matching and application of Beer-Lambert law. Asterisk markers: 3D source problem calculations. Thick lines: Thin-film characteristics provided for illustrating anti-reflective properties of the array.

Fig. 5.
Fig. 5.

Shockley-Queisser efficiency limit of a single junction solar cell design under 500×AM1.5d illumination. Note the increased efficiency in the presence of a coaxial waveguide. The geometry of the device is: h 0=2 µm, a 0=600 nm, d WG=200 nm. The curve labelled as ’ideal device’ refers to the calculations presented in [10].

Fig. 6.
Fig. 6.

Shockley-Queisser efficiency limit of a dual junction solar cell for various nanowire diameters (nanowire length 2 µm, distance 600 nm) compared to a thin-film dual junction cell under 500×AM1.5d illumination. Thin-film calculation assumes perfect absorption. Numerical results not confined by the InAs/InP material system: ideal thin-film η max=0.53, 300 nm η max=0.45, 220 nmη max=0.41, 180 nm η max=0.38. Results within InAs/InP: ideal thin-film η max=0.50, 300 nm η max=0.41, 220 nm η max=0.32, 180 nm η max=0.23. The plot labeled as ’ideal device’ refers to the calculations presented in [10].

Equations (18)

Equations on this page are rendered with MathJax. Learn more.

×(μ̿1×E)k02ε̿E=jk0Z0J.
F(E)=12V(×E)μ̿1(×E)k02Eε̿EdV+jk0Z0VEJdV.
Stot(λ0)=Sinc(λ0)+Ssc(λ0)
ηp(V,λ0)=Vp(λ0)dVUtop𝓡{Sinc}·ẑda
T(λ0)=𝓗{UtopStot(x,λ0)·da}𝓗{UtopSinc(x,λ0)·da}
P(z)=x'U,z'=zStot(x,λ0)·ẑda'
E(x,y,z)=E (x,y)·ejβz
Pm(z+L)=Pm(z)·eαmLwith αm=4πλ0𝒯{neff,m}.
ηm=Einc|Em2Einc|EincEm|Emwitha|b=Utopa·b*da
tm(λ0)=4n0𝓡{neff,m}(n0+𝓡{neff,m})2.
Pm(z=0)=Pinc·tmηmmηi.
ηabs,2D(λ0)=i=1mPi·(1exp(4πλ0𝓗{neff,i}h0))
g(x,λ0)=ηeqep(λ0)hc0λ0
jsc(V)=qηeqehc0V 0λgλ p (λ)dλdV
v(j,Eg)=kBTqln(jscjj0+1)
j0=8πqn02h3c02kBT(2kB2T2+2kBTEg+Eg2)exp(EgkBT).
ηmax=maxEg,1n(jtoti=1nvi0psun(λ)dλ)withjtot=min (ji)
neff2=n02nNW2(1+2δ)n02(2δ2)n02(2+δ)+nNW2(1δ)

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