Multiple Exciton Generation in Si and Ge Nanoparticles with High Pressure Core Structures

Transcription

1 Multiple Exciton Generation in Si and Ge Nanoparticles with High Pressure Core Structures S. Wippermann, M. Vörös, D. Rocca, A. Gali, G. Zimanyi, G. Galli NanoMatFutur DPG-214, 4/3/214

2 Multiple Exciton Generation (MEG) MEG: hot exciton relaxes by exciting another exciton Promising pathway to exceed Shockley- Queisser limit in 3rd generation solar cells [1] A. Nozik, Physica E 14, 115 (22)

3 Multiple Exciton Generation (MEG) M. Beard, JPCL 2, 1282 (211) MEG: hot exciton relaxes by exciting another exciton Promising pathway to exceed Shockley- Queisser limit in 3rd generation solar cells enhanced MEG in quantum-confined nanostructures, e. g. nanocrystals [1] A. Nozik, Physica E 14, 115 (22)

4 Multiple Exciton Generation (MEG) M. Beard, JPCL 2, 1282 (211) MEG: hot exciton relaxes by exciting another exciton Promising pathway to exceed Shockley- Queisser limit in 3rd generation solar cells enhanced MEG in quantum-confined nanostructures, e. g. nanocrystals Key problem Quantum confinement required for efficient MEG, but pushes electronic gap beyond solar spectrum MEG observed in Si and Ge: gap/wavefunction engineering possible? => investigate nanocrystals with high pressure core structures [1] A. Nozik, Physica E 14, 115 (22)

5 Allotropes of Si and Ge cubic diamond, semiconductor X35H36 (X=Si,Ge)

6 Allotropes of Si and Ge cubic diamond, semiconductor BC8, semimetal Si X35H36 (X=Si,Ge) X34H38 [2] S. Ganguly et al., J. Am. Chem. Soc. 136, 1296 (214)

7 Allotropes of Si and Ge cubic diamond, semiconductor BC8, semimetal Si X35H36 (X=Si,Ge) Ge ST12 X34H38 X46H52 [2] S. Ganguly et al., J. Am. Chem. Soc. 136, 1296 (214) [3] S. J. Kim et al., J. Mater. Chem. 2, 331 (21)

8 Reduced gaps for BC8 and ST12 nanocrystals Silicon Si-I BC8

9 Reduced gaps for BC8 and ST12 nanocrystals Silicon Si-I BC8 BC8 is semimetal in the bulk, quantum confinement opens small gap in BC8 nanocrystals => significantly reduced gaps compared to Si-I

10 Reduced gaps for BC8 and ST12 nanocrystals Silicon Germanium Si-I BC8 BC8 is semimetal in the bulk, quantum confinement opens small gap in BC8 nanocrystals => significantly reduced gaps compared to Si-I

11 Reduced gaps for BC8 and ST12 nanocrystals Silicon Germanium Si-I BC8 BC8 is semimetal in the bulk, quantum confinement opens small gap in BC8 nanocrystals => significantly reduced gaps compared to Si-I Ge ST12 features reduced gaps for d < 2.5nm, significantly increased electronic density of states at band edges

12 Reduced gaps for BC8 and ST12 nanocrystals Silicon Germanium Si-I BC8 BC8 is semimetal in the bulk, quantum confinement opens small gap in BC8 nanocrystals => significantly reduced gaps compared to Si-I Ge ST12 features reduced gaps for d < 2.5nm, significantly increased electronic density of states at band edges Get realistic estimate of electronic gaps from quasiparticle calculations in GW approximation

13 GW for large systems Calculation, storage & inversion of dielectric matrix ε is major computational bottleneck => spectral representation of ε (RPA) = N ṽ i i ṽ H i = N i=1 i=1 ṽ i ( i 1)ṽ H i + I [4] H.-V. Nguyen, T. A. Pham, D. Rocca, G. Galli, PRB (212), PRB 87, (213)

14 GW for large systems Calculation, storage & inversion of dielectric matrix ε is major computational bottleneck => spectral representation of ε (RPA) = N i=1 ṽ i i ṽ H i = N i=1 n ṽ i ( i 1)ṽ H i + I Calculating eigenvectors/-values does NOT require explicit knowledge of the matrix itself; knowing the action of ε on an arbitrary vector is sufficient In linear response: ( I) VSCF = v c n V SCF Charge density response to perturbation of self-consistent field can be evaluated from density functional perturbation theory [4] H.-V. Nguyen, T. A. Pham, D. Rocca, G. Galli, PRB (212), PRB 87, (213)

15 GW for large systems Calculation, storage & inversion of dielectric matrix ε is major computational bottleneck => spectral representation of ε (RPA) = N i=1 ṽ i i ṽ H i = N i=1 ṽ i ( i 1)ṽ H i + I Calculating eigenvectors/-values does NOT require explicit knowledge of the matrix itself; knowing the action of ε on an arbitrary vector is sufficient In linear response: ( I) VSCF = v c n Charge density response to perturbation of self-consistent field can be evaluated from density functional perturbation theory Orthogonal iteration procedure to obtain eigenvector/-value pairs, using as trial potentials In RPA fast monotonous decay of dielectric eigenvalue spectrum Single parameter N eig to control numerical accuracy No summation over empty states, no inversion n V SCF V SCF [4] H.-V. Nguyen, T. A. Pham, D. Rocca, G. Galli, PRB (212), PRB 87, (213)

16 GW for large systems Calculation, storage & inversion of dielectric matrix ε is major computational bottleneck => spectral representation of ε (RPA) N N D. Rocca, H.-V. Nguyen, T. A. Pham (UCD) = i=1 ṽ i i ṽ H i = i=1 ṽ i ( i 1)ṽ H i + I Calculating eigenvectors/-values does NOT require explicit knowledge of the matrix itself; knowing the action of ε on an arbitrary vector is sufficient In linear response: ( I) VSCF = v c n Charge density response to perturbation of self-consistent field can be evaluated from density functional perturbation theory Orthogonal iteration procedure to obtain eigenvector/-value pairs, using as trial potentials In RPA fast monotonous decay of dielectric eigenvalue spectrum Single parameter N eig to control numerical accuracy No summation over empty states, no inversion n V SCF V SCF [4] H.-V. Nguyen, T. A. Pham, D. Rocca, G. Galli, PRB (212), PRB 87, (213)

17 Optimum gap for MEG in 4-8nm BC8 NPs Si123H1 2.5 ev Si144H ev 1.1 ev.5 ev GW calculations up to d=1.6nm (Si144H114) confirm trends observed in LDA Optimum gap for PV cells employing MEG (Eg =.5-1. ev) [5] found for BC8 NPs within typical experimental size range of d = 4-8 nm (extrapolation of GW gaps) [5] M. Hanna, A. Nozik, J. Appl. Phys. 1, 7451 (26)

18 Calculating Multiple Exciton Generation (MEG) Rates Impact Ionization (II) is dominating contribution to MEG [6] => approximate MEG rates with II rates [7] Calculate II rates from Fermi s Golden Rule: II i = 2 ~ X f hx i W XX f i 2 (E i E f ) Approximate initial exciton (Xi) and final biexciton states (XXf) as singly and doubly excited Slater determinants, built up from DFT orbitals Screened Coulomb interaction W calculated using same technique as during GW calculations [6] A. Piryatinski et el., J. Chem. Phys. 133, 8458 (21) K. Velizhanin et al., Phys. Rev. Lett. 16, 2741 (211) [7] M. Vörös et al., Phys. Rev. B 87, (213)

19 Calculating Multiple Exciton Generation (MEG) Rates Impact Ionization (II) is dominating contribution to MEG [6] => approximate MEG rates with II rates [7] Calculate II rates from Fermi s Golden Rule: M. Vörös (UCD) II i = 2 ~ X f hx i W XX f i 2 (E i E f ) Approximate initial exciton (Xi) and final biexciton states (XXf) as singly and doubly excited Slater determinants, built up from DFT orbitals Screened Coulomb interaction W calculated using same technique as during GW calculations [6] A. Piryatinski et el., J. Chem. Phys. 133, 8458 (21) K. Velizhanin et al., Phys. Rev. Lett. 16, 2741 (211) [7] M. Vörös et al., Phys. Rev. B 87, (213)

20 Optimum gap for MEG in 4-8nm BC8 NPs

21 Optimum gap for MEG in 4-8nm BC8 NPs BC8 NPs feature lower activation threshold on absolute energy scale & order of magnitude higher impact ionization rate at same energies and same NP size

22 Multiple Exciton Generation in Ge allotrope nanocrystals relative energy scale

23 Multiple Exciton Generation in Ge allotrope nanocrystals relative energy scale ST12 II rates size-independent Increasing EDOS at band edges counterbalances loss of confinement

24 Multiple Exciton Generation in Ge allotrope nanocrystals relative energy scale absolute energy scale ST12 II rates size-independent Increasing EDOS at band edges counterbalances loss of confinement Simultaneously lower electronic gaps and higher relative II efficiency translate to significantly improved MEG on absolute energy scale

25 Summary (Metastable) high pressure phases of elemental semiconductors allow for gap engineering of nanoparticles, while retaining efficient MEG (Si-BC8 and Ge-ST12) Nanoparticles with high pressure core structures can attain optimum gap range for MEG-based solar energy conversion High pressure nanoparticles can be formed via the high pressure route or directly at ambient pressure in solution by chemical bottom-up synthesis from a precursor Talk on Si NPs embedded in charge transport matrix today at 12:3pm, Tre Ma

26 Acknowledgements Giulia Galli and her group at UChicago, especially Marton Vörös and Tuan Anh Pham Francois Gygi (UC Davis), Gergely Zimanyi (UC Davis), Adam Gali (Budapest Univ.), Dario Rocca (Univ. Lorraine) NanoMatFutur 13N12972 Wi3879/1-1 NISE-project NSF/Solar DMR Max-Planck-Institute for Iron Research, Düsseldorf

27 Additional Slides

28 Motivation Structure & phase transition Electron transport Summary 24/25 Optical properties (TD-DFT RPA) red-shifted optical absorption onset for high density phases (BC8, R8, Ibam) less pronounced for ST12 and low density phases (bct, hd)

29 Motivation Structure & phase transition Electron transport Summary 25/25 cubic diamond vs. ST12 II i = 2 ~ X f hx i W XX f i 2 (E i E f ) = 2 ~ W i eff 2 TDOSi cubic diamond: NP size increase reduces Coulomb interaction Weff, trion DOS almost constant => impact ionization rate drops ST12: Weff reduced as for cubic diamond, but TDOS increases => impact ionization rate remains almost constant with increasing size

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31 Density functional theory (DFT): Hohenberg-Kohn theorem: E e [n] =T [n]+ 1 2 [Walter Kohn, Nobel Prize for chemistry in 1998] E XC [n] n(r)n(r ) drdr + r r E LDA XC [n] = n(r) hom XC (n(r))dr n(r)v (r)dr + E XC [n] atomic positions

32 Density functional theory (DFT): Hohenberg-Kohn theorem: 1 Ee [n] = T [n] + 2 EXC [n] n(r)n(r ) drdr + r r LDA EXC [n] = + 2 hom XC (n(r))dr n(r)v (r)dr + EXC [n] atomic positions [Walter Kohn, Nobel Prize for chemistry in 1998] 2 n(r) Kohn-Sham self-consistent electron structure n(r ) dr + V (r) + r r LDA EXC n(r) j (r) = j j (r) occ. n(r) = j 2 (r) j

33 Density functional theory (DFT): Hohenberg-Kohn theorem: 1 Ee [n] = T [n] + 2 EXC [n] n(r)n(r ) drdr + r r LDA EXC [n] = n(r) hom XC (n(r))dr n(r)v (r)dr + EXC [n] atomic positions [Walter Kohn, Nobel Prize for chemistry in 1998] Starting point: initial geometry external potential interatomic forces n(r ) dr + V (r) + r r Kohn-Sham self-consistent electron structure LDA EXC n(r) structurally relaxed ground-state j (r) = j j (r) occ. n(r) = j 2 (r) j

34 Density functional theory (DFT): Hohenberg-Kohn theorem: 1 Ee [n] = T [n] + 2 EXC [n] n(r)n(r ) drdr + r r LDA EXC [n] = n(r) hom XC (n(r))dr n(r)v (r)dr + EXC [n] atomic positions [Walter Kohn, Nobel Prize for chemistry in 1998] Starting point: initial geometry external potential interatomic forces n(r ) dr + V (r) + r r Kohn-Sham self-consistent electron structure LDA EXC n(r) structurally relaxed ground-state vibrational and thermal properties electronic properties j (r) = j j (r) occ. n(r) = j 2 (r) j spectroscopic and transport properties

35 Excited states (GW/BSE) DFT disregards screened e - -e - interaction and e - -h interaction for excited states => band gap underestimation, wrong distribution of spectral weights

36 Excited states (GW/BSE) DFT disregards screened e - -e - interaction and e - -h interaction for excited states => band gap underestimation, wrong distribution of spectral weights Perturbative approaches for including screening (GW) and e - -h interaction (Bethe-Salpeter), starting from Quantum Liouville equation i dˆ (t) dt Z = hĥ(t), ˆ (t) i (r, r,t)= X v v(r,t) v (r,t) single particle occ. orbitals 1 Ĥ(r, r,t) (r,t)dr = 2 r2 + v H (r,t)+v ext (r,t) Z + (r, r,t) (r,t)dr (r,t) time-dep. perturbation, i. e. electromagn. field

37 Excited states (GW/BSE) DFT disregards screened e--e- interaction and e--h interaction for excited states => band gap underestimation, wrong distribution of spectral weights Perturbative approaches for including screening (GW) and e--h interaction (Bethe-Salpeter), starting from Quantum Liouville equation h dˆ (t) i = H (t), ˆ(t) dt Z i (r, r, t) = H (r, r, t) (r, t)dr = + Z 1 2 r + vh (r, t) + vext (r, t) 2 v (r, t) time-dep. perturbation, (r, r, t) (r, t)dr i. e. electromagn. field 1 COH (r, r ) = (r r )Wp (r, r ) 2 X SEX (r, r, t) = v (r, t) v (r, t)w (r, r ) v (r, t) v (r, t) v single particle occ. orbitals self-energy X statically screened Bethe-Salpeter equation (BSE)

38 Excited states (GW/BSE) DFT disregards screened e--e- interaction and e--h interaction for excited states => band gap underestimation, wrong distribution of spectral weights Perturbative approaches for including screening (GW) and e--h interaction (Bethe-Salpeter), starting from Quantum Liouville equation h dˆ (t) i = H (t), ˆ(t) dt Z i (r, r, t) = v (r, t) v (r, t) v single particle occ. orbitals H (r, r, t) (r, t)dr = + self-energy X Z 1 2 r + vh (r, t) + vext (r, t) 2 (r, t) time-dep. perturbation, (r, r, t) (r, t)dr i. e. electromagn. field statically screened 1 Bethe-Salpeter COH (r, r ) = (r r )Wp (r, r ) 2 X equation (BSE) SEX (r, r, t) = v (r, t) v (r, t)w (r, r ) screened Coulomb interaction v

39 Excited states (GW/BSE) To correct for DFT s band gap underestimation, quasiparticle energies can be obtained in GW approximation from GW (r, r ; i) = 1 2 Z G(r, r ; i( ))W (r, r ; i )d

40 Excited states (GW/BSE) To correct for DFT s band gap underestimation, quasiparticle energies can be obtained in GW approximation from GW (r, r ; i) = 1 2 W (r, r )= Z Z G(r, r ; i( ))W (r, r ; i )d Screened Coulomb interaction required (in random phase (RPA) approx.) 1 (r, r )v c (r, r )dr

41 Excited states (GW/BSE) To correct for DFT s band gap underestimation, quasiparticle energies can be obtained in GW approximation from GW (r, r ; i) = 1 2 W (r, r )= Z Z G(r, r ; i( ))W (r, r ; i )d Screened Coulomb interaction required (in random phase (RPA) approx.) 1 (r, r )v c (r, r )dr Bottleneck: calculation, storage & inversion of dielectric matrix is very computationally demanding, involves large sums over empty states and is hard to converge

42 Excited states (GW/BSE) To correct for DFT s band gap underestimation, quasiparticle energies can be obtained in GW approximation from GW (r, r ; i) = 1 2 W (r, r )= Z Z G(r, r ; i( ))W (r, r ; i )d Screened Coulomb interaction required (in random phase (RPA) approx.) 1 (r, r )v c (r, r )dr Bottleneck: calculation, storage & inversion of dielectric matrix is very computationally demanding, involves large sums over empty states and is hard to converge Solution: spectral representation of RPA dielectric matrix; obtain matrix from directly calculating eigenvectors and eigenvalues = NX i=1 ṽ i i ṽ H i = NX i=1 ṽ i ( i 1)ṽ H i + I => no summation over empty states, no inversion, storage of eigenvector/-value pairs only

43 How to calculate the screening Obtaining the eigenvectors/-values does NOT require explicit knowledge of the matrix; knowledge of the action of the matrix on an arbitrary vector is sufficient in linear response: charge density response to perturbation of self-consist. field V SCF can be evaluated from density functional perturbation theory orthogonal iteration procedure to obtain eigenvectors/-values, using as trial potentials V SCF in RPA fast monotonous decay of dielectric eigenvalue spectrum single parameter numerical accuracy ( I) V SCF = v c n N eig n to control = NX i=1 ṽ i i ṽ H i = NX i=1 ṽ i ( i 1)ṽ H i + I [H. Wilson et al., PRB 79, (29); D. Rocca et al., J. Chem. Phys. 133, (21); H.-V. Nguyen et al., PRB 85, 8111 (212)]