Electronic Transport in Solar Cells and DFT Calculations for Si and GaAs

Electronic Transport in Solar Cells and DFT Calculations for Si and GaAs Jean Diehl Institut für Theoretische Physik Goethe-Universität, Frankfurt June 24th, 211 1 / 24

Contents 1 Introduction to Solar Cells 2 Results of Calculations for Si and GaAs DFT Calculations with wien2k Bandstructure, DOS, Absorption Coefficient 3 Electronic Transport in Semiconductors Semiclassical Transport/Boltzmann Transport Equation Generation and Recombination Drift-Diffusion-Equation/Electron Mobility Semiconductor Transport Equations Depletion Approximation Solution of Transport Equations Effective Masses Carrier Concentrations 4 Todo Current Densities 2 / 24

What is a Solar Cell? converts energy of photons to energy of electrons directly (photovoltaic cell), no photothermal converters in general many types of materials possible: monocrystalline, polycrystalline, amorphous, organic consists of two parts: 1. generation of electrons n and holes p 2. separation of charge carriers pn-junction build of seminconducting material incident light p-doped n-doped substrate x p w p w n x n 3 / 24

Why to consider Solar Cell Physics? power supply, alternative way to nuclear power or fossil fuels important to know how they work: efficient use of ressources (electrical efficiency) limiting processes/properties ideal material? highest electrical efficiency? η max = J maxu max P s with total incident solar power ( rs P s = de Eb s (E), b s (E) = and maximum power of the solar cell r SE ) 2 2π h 3 c 2 E 2 d(j(u)u) du Current-Voltage-Characteristic J(U) needed e E 1 = U=Umax 4 / 24

Power from the Solar Spectrum Spectral Intensity / Wm 2 nm 1 2 1.8 1.6 1.4 1.2 1.8.6.4.2 Solar Spectrum AM1.5 blackbody 58K 5 1 15 2 25 3 35 4 Wavelength / nm highest incoming photon rate between 1-4 ev Efficiency.35.3.25.2.15.1.5 Shockley Queisser Limit.5 1 1.5 2 2.5 3 Eg / ev AM1.5 blackbody Shockley-Queisser-Limit 1 : not material specific, except bandgap, absorbtion α(e) = θ(e E g ), cell blackbody above bandgap highest efficiency η max 3 33% between E g 1 1.5eV Si and GaAs good candidates material dependent properties/defects reduce efficiency 1 W. Shockley, H. J. Queisser, J. Apl. Phys. 32 pp. 51-519 (1961) 5 / 24

gap of 23 solids. We used the WIEN2K package [31] which LDA, GGA (PBE) underestimate band gaps for is based on the full-potential (linearized) augmented planewave and local orbitals [FP-ðLÞAPW þ lo] method (see sp-semiconductors 11.6 Ref. [32] and references therein). For comparison better band gaps with mbj-lda Potential 2 puricated in parenthesis. For ture which were obtained SE3, HSE6, G W, and es were taken from Hartree-Fock potential [22], Eq. (1) can be seen as a kind of hybrid potential whose amount of exact exchange DFT Calculations is given by c. with wien2k G W GW Expt. 19.59 e 22.1 g 21.7 a 13.28 e 14.9 g 14.2 b b b b b b b b b b b b d 9.8 5.5 e 6.18 g 5.48 1.12 e 1.41 g 1.17.66 f.95 g.74 13.27 e 15.9 g 14.2 9.4 7.25 e 9.16 g 7.83.95 f 1.4 h :9 3.5 i 3:9 :4 2.4 1.1 f 4.8 i 4., 4.3 2.27 e 2.88 g 2.4 6.1 e 7.14 g 6:25 2.8 e 3.82 g 3.2 1.3 e 1.85 g 1.52 2.44 e 2.9 g 2.45 3.29 e 4.15 g 3.91 2.6 e 2.87 g 2.42 5.83 f 6.28 2.51 f 3.8 g 3.44 ef. [4]. ce [17]. f Reference [15]. eference [14]. Table I and Fig. 1 show the results obtained with the LDA and MBJLDA potentials for the fundamental band Theoretical band gap (ev) 16 14 12 1 8 6 4 2 ScN GaAs Si Ge LDA MBJLDA HSE G W GW CdS AlP FeO SiC NiO GaN MnO ZnO ZnS C BN AlN MgO 2 4 6 8 1 12 14 16 Experimental band gap (ev) 2 F. Tran, P. Blaha, Phys. Rev. Lett. 12, 22641 (29) FIG. 1 (color online). Theoretical versus experimental band gaps. The values are given in Table I (Ne is omitted). LiCl Xe Kr LiF Ar 6 / 24

Silicon diamond structure (fcc with diatomic basis) a = 5.431 Å Eg exp Eg exp = 1.17 ev @K = 1.12 ev @3K DOS / states/ev 2.5 2 1.5 1.5 Si DOS, mbjlda -1-5 5 1 Eg GGA Eg LDA Eg mbj =.5 ev =.48 ev = 1.16 ev total 7 / 24

Silicon Si, mbjlda 5-5 -1 W L Γ X W K k-points α / cm 1 1 7 1 6 1 5 1 4 1 3 1 2 Si Absorption Coefficient DFT 1 1 DFT broad. E F 1 exp. T = 3K 1 1 JDOS / states/ev 8 7 6 5 4 3 2 1 Si JDOS, mbjlda 2 4 6 8 1 total 8 / 24

Gallium arsenide zinkblende structure (fcc with diatomic basis, different) a = 5.653 Å Eg exp Eg exp = 1.52 ev @K = 1.47 ev @3K DOS / states/ev GaAs DOS, mbjlda 5 4.5 total 4 3.5 3 2.5 2 1.5 1.5-1 -5 5 1 Eg GGA Eg LDA Eg mbj =.5 ev =.29 ev = 1.64 ev 9 / 24

Gallium arsenide GaAs, mbjlda 1 7 1 6 GaAs Absorption Coefficient 5 α / cm 1 1 5 1 4 1 3-5 -1 W L Γ X W K k-points 1 2 DFT E F 1 1 exp. T = 297K 1 1 DOS / states/ev 45 4 35 3 25 2 15 1 5 GaAs JDOS, mbjlda 2 4 6 8 1 total 1 / 24

Boltzmann Transport Equation appliend when λ db λ c electrons distributed with classical phase-space-distribution function dw = f (r, k, t)d 3 pd 3 r force F = φ and velocity defined quantum mechanical v = ke n(k) df dt = v rf + 1 F kf + f t = f t coll collision term includes: intraband contributions (collisions with impurities/lattice, electron mobility) interband contributions (generation/recombination of electron-hole-pairs) 11 / 24

Generation and Recombination generation by incident light: G = (1 r)αb s e α(x+xp) recombination processes: a) radiative, b) Auger, c) Shockley-Reed-Hall/Trap a) b) c) E c E c E c E t E v E v E v have in common: R n n n τ n, R p p p τ p surface recombination: J n (x s ) = qs n (n(x s ) n ), J p (x s ) = qs p (p(x s ) p ) 12 / 24

Drift-Diffusion-Equations/Electron Mobility assume distribution function is close to equilibrium (f 1 f ): f c (r, k, t) f (E, E Fn, T ) + f 1 (r, k, t) approximate BTE (LHS f f, E c = φ, stationary case): k E c (k) r f 1 φ kf = f c f τ J n 2q V k vf c (r, k, t) = 2q V f c = f + f E τv E Fn k v 2 τ f E E F for non-degenerate semiconductor f E = βf define mean-value: A k Af / k f with n 1/V k f J n = qn v2 τ E Fn = nµ n E Fn µ n q v2 τ = q v2 τ = qτ m 1 2 m n v 2 = 1 2 k BT 13 / 24

Semiconductor Transport Equations obtained by integrating BTE over k-space: Drift-Diffusion-Equations for spatially invariant material: n t = 1 q J n + G n R n p t = 1 q J p + G p R p J n = qd n n + qµ n ne J p = qd p n + qµ p pe with D n k BTµ n q assume stationary case, with poisson s equations, set of coupled : D n 2 n nµ n 2 φ n n τ n + (1 r)αb s e α(x+xp) = D p 2 p + pµ p 2 φ + p p τ p (1 r)αb s e α(x+xp) = 2 φ = q ɛ (n p + N a N d ) 14 / 24

Depletion Approximation potential is completely dropped for w p < x < w n neutral regions for x < w p, x > w n 1 16 incident light p-doped n-doped substrate Charge Density / cm 3 1 15 1 1 15 xp wp wn xn 1 16-2 -1.5-1 -.5.5 1 1.5 2 Solution for φ: φ(x) = qn a 2ɛ (x + w p) 2 for w n < x < φ(x) = qn d 2ɛ (x w n) 2 + U bi U for < x < w p depletion width obtained by requiring φ and φ to be continuous at x = 15 / 24

1D Transport Equations neutral regions (L n τ n D n ): d 2 n dx 2 n n L 2 + (1 r)αb se α(x+xp) = for x < w p n D n d 2 p dx 2 p p L 2 + (1 r)αb se α(x+xp) = for x > w n p D p boundary conditions: n( w p ) = n e qu p(w n ) = p e qu dn dx = S n [n( x p ) n ] x= xp D n dp dx = S p [p(x n ) p ] x=xn D p with equilibrium carrierconcentrations n = n 2 i /N a, p = n 2 i /N d space-charge-region: np = n 2 i e q(φ+u) and qu E Fn E Fp 16 / 24

Solution of 1D Transport Equations ( x + wp n(x, E) = A n cosh L n ) +B n sinh ( x + wp L n γ n = [1 r(e)] α(e)b s(e)l 2 n D n (α 2 L 2 n 1) ( ) A n = n e qu 1 + γ n e α(xp wp) ) γ n e α(x+xp) +n B n = γ [ n (β7 cosh β 5 + sinh β 5 ) e β 6 (β 7 + L n α) ] β 7 sinh β 5 + cosh β 5 ( ) n e qu 1 (β 7 cosh β 5 + sinh β 5 ) + β 7 sinh β 5 + cosh β 5 β 7 LnSn D n, β 5 xp wp L n, β 6 α(w p w p ) 17 / 24

Simplified case for Transport equations no incoming light: b s = γ n = γ p = no surface recombination: S n = S p = Carrier Concentration in the different regions: x < w p w p < x < n ( e qu n 1 ) e x+wp Ln + n N a N a e qu ni 2 N a e q(φ+u) ni < x < w 2 n N a e qu w n < x N d p ( e qu p N a e q( φ+u) 1 ) e x wn Lp + p 18 / 24

Effective Masses obtained by parabolic fit of bandstructure at VB max, CB min: E(k) = E ± 2 2m (k k ) 2 Si / m e GaAs / m e Γ X m n.784.73 m p.218.76 m p.131.235 Γ L m n -.1 m p.99.87 m p.666.661 strong dependence on number of k-points included in fit 19 / 24

Carrier Concentrations Na = 1.e+16 cm 3, Nd = 1.e+16 cm 3, Ln = e+ m, Lp = e+ m, V =. V Na = 1.e+16 cm 3, Nd = 1.e+16 cm 3, Ln = e+ m, Lp = e+ m, V =. V, Eg = 1.16 ev Carrier Concentration / cm 3 1 2 1 15 1 1 1 5 1 wp 1 5-2 -1.5-1 -.5.5 1 1.5 2 Na = 1.e+14 cm 3, Nd = 1.e+14 cm 3, Ln = e+ m, Lp = e+ m, V =. V wn n p wp wn 1.8.6.4.2 Ec EF -.2 -.4 -.6 -.8-1 Ev -1.2-2 -1.5-1 -.5.5 1 1.5 2 Na = 1.e+14 cm 3, Nd = 1.e+14 cm 3, Ln = e+ m, Lp = e+ m, V =. V, Eg = 1.16 ev Carrier Concentration / cm 3 1 2 wp 1 15 1 1 1 5 1 1 5-2 -1.5-1 -.5.5 1 1.5 2 n p wn wp wn 1.8.6.4 Ec.2 EF -.2 -.4 -.6 -.8 Ev -1-2 -1.5-1 -.5.5 1 1.5 2 2 / 24

Carrier Concentrations Na = 2.e+14 cm 3, Nd = 1.e+16 cm 3, Ln = e+ m, Lp = e+ m, V =. V Na = 2.e+14 cm 3, Nd = 1.e+16 cm 3, Ln = e+ m, Lp = e+ m, V =. V, Eg = 1.16 ev Carrier Concentration / cm 3 1 2 1 15 1 1 1 5 1 wn 1 5-2 -1.5-1 -.5.5 1 1.5 2 Na = 1.e+16 cm 3, Nd = 1.e+16 cm 3, Ln = 1e+6 m, Lp = 1e+6 m, V =.6 V n p wp wn 1.8.6.4.2 Ec EF -.2 -.4 -.6 -.8-1 Ev -1.2-2.5-2 -1.5-1 -.5.5 1 1.5 2 Na = 1.e+16 cm 3, Nd = 1.e+16 cm 3, Ln = 1e+6 m, Lp = 1e+6 m, V =.6 V, Eg = 1.16 ev Carrier Concentration / cm 3 1 2 1 15 1 1 1 5 1 wpwn n p 1.8.6.4.2 -.2 -.4 wpwn Ec EFp EFn 1 5-2 -1.5-1 -.5.5 1 1.5 2 -.6 Ev -2-1.5-1 -.5.5 1 1.5 2 21 / 24

Carrier Concentrations Na = 1.e+16 cm 3, Nd = 1.e+16 cm 3, Ln = 1e+6 m, Lp = 1e+6 m, V = -.4 V Na = 1.e+16 cm 3, Nd = 1.e+16 cm 3, Ln = 1e+6 m, Lp = 1e+6 m, V = -.4 V, Eg = 1.16 ev Carrier Concentration / cm 3 1 2 1 15 1 1 1 5 1 wp wn n p 1.5 -.5-1 wp wn EFn Ec EFp 1 5-2 -1.5-1 -.5.5 1 1.5 2 Na = 1.e+16 cm 3, Nd = 1.e+16 cm 3, Ln = 1e-6 m, Lp = 1e-6 m, V =.4 V -1.5 Ev -2-1.5-1 -.5.5 1 1.5 2 Na = 1.e+16 cm 3, Nd = 1.e+16 cm 3, Ln = 1e-6 m, Lp = 1e-6 m, V =.4 V, Eg = 1.16 ev Carrier Concentration / cm 3 1 2 1 15 1 1 1 5 1 wp wn n p 1.8.6.4.2 -.2 -.4 -.6 wp wn Ec EFp EFn 1 5-2 -1.5-1 -.5.5 1 1.5 2 -.8 Ev -2-1.5-1 -.5.5 1 1.5 2 22 / 24

Current Densities from drift-diffusion-equations without electric field: ( ) dn j n ( w p, E) = qd n Bn dx = qd n + αγ n e α(xp wp) x= wp L n J n = de j n (E) current from SCR by integrating the continuity equation: J scr = q total current densitiy: wn w p dx(r G) J = J n + J p + J scr 23 / 24

Todo effective masses from DOS (k-points problem) electron mobilities (impurity, phonon) µ n, µ p D n, D p µ T 3/2 + T 3/2 recombination times τ n, τ p L n, L p surface recombination S n, S p (maybe not possible? treatment of the surface is important) reflection coefficient (maybe same problem) r = (1 n)2 + κ 2 (1 + n) 2 + κ 2 get: quantum efficiencies (energy resultion of efficiency) QE(E) j n + j p + j scr qb s get: electrical efficiency get: contribution of different layers 24 / 24