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

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

2 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

Transport Equation Generation and Recombination Drift-Diffusion-Equation/Electron Mobility Semiconductor Transport

3 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

in general many types of materials possible: monocrystalline, polycrystalline, amorphous, organic consists

4 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

ressources (electrical efficiency) limiting processes/properties ideal material? highest electrical efficiency?

5 Power from the Solar Spectrum Spectral Intensity / Wm 2 nm Solar Spectrum AM1.5 blackbody 58K Wavelength / nm highest incoming photon rate between 1-4 ev Efficiency Shockley Queisser Limit 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 (1961) 5 / 24

5 blackbody Shockley-Queisser-Limit 1 : not material specific, except bandgap, absorbtion α(e) = θ(e E g ), cell blackbody above bandgap highest efficiency

6 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 e 22.1 g 21.7 a e 14.9 g 14.2 b b b b b b b b b b b b d e 6.18 g e 1.41 g f.95 g e 15.9 g e 9.16 g f 1.4 h :9 3.5 i 3:9 : f 4.8 i 4., e 2.88 g e 7.14 g 6: e 3.82 g e 1.85 g e 2.9 g e 4.15 g e 2.87 g f 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) ScN GaAs Si Ge LDA MBJLDA HSE G W GW CdS AlP FeO SiC NiO GaN MnO ZnO ZnS C BN AlN MgO Experimental band gap (ev) 2 F. Tran, P. Blaha, Phys. Rev. Lett. 12, (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

sp-semiconductors 11.6 Ref. [32] and references therein). For comparison better band gaps with mbj-lda Potential 2 puricated in parenthesis.

Silicon diamond structure (fcc with diatomic basis) a = 5.431 Å Eg exp Eg exp = 1.17 ev @K = 1.

7 Silicon diamond structure (fcc with diatomic basis) a = Å Eg exp Eg exp = 1.17 = 1.12 DOS / states/ev Si DOS, mbjlda Eg GGA Eg LDA Eg mbj =.5 ev =.48 ev = 1.16 ev total 7 / 24

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

9 Gallium arsenide zinkblende structure (fcc with diatomic basis, different) a = Å Eg exp Eg exp = 1.52 = 1.47 DOS / states/ev GaAs DOS, mbjlda total Eg GGA Eg LDA Eg mbj =.5 ev =.29 ev = 1.64 ev 9 / 24

47 ev @3K DOS / states/ev GaAs DOS, mbjlda 5 4.5 total 4 3.

10 Gallium arsenide GaAs, mbjlda GaAs Absorption Coefficient 5 α / cm W L Γ X W K k-points 1 2 DFT E F 1 1 exp. T = 297K 1 1 DOS / states/ev GaAs JDOS, mbjlda total 1 / 24

11 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

mechanical v = ke n(k) df dt = v rf + 1 F kf + f t = f t coll collision term includes: intraband

12 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

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

13 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

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

14 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

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

15 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 xp wp wn xn 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

16 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

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

17 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

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

18 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

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

19 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 m p m p Γ L m n -.1 m p m p strong dependence on number of k-points included in fit 19 / 24

m n.784.73 m p.218.76 m p.131.235 Γ L m n -.1 m p.99.87 m p.666.

20 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 wp 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 Ec EF Ev 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 wp n p wn wp wn Ec.2 EF Ev / 24

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.

21 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 wn 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 Ec EF Ev 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 wpwn n p wpwn Ec EFp EFn Ev / 24

22 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 wp wn n p wp wn EFn Ec EFp 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 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 wp wn n p wp wn Ec EFp EFn Ev / 24

23 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

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

FYS3410 - Vår 2015 (Kondenserte fasers fysikk) http://www.uio.no/studier/emner/matnat/fys/fys3410/v15/index.html

FYS3410 - Vår 2015 (Kondenserte fasers fysikk) http://www.uio.no/studier/emner/matnat/fys/fys3410/v15/index.html FYS3410 - Vår 015 (Kondenserte fasers fysikk) http://www.uio.no/studier/emner/matnat/fys/fys3410/v15/index.html Pensum: Introduction to Solid State Physics by Charles Kittel (Chapters 1-9 and 17, 18, 0,