Phonon wavefunctions and electron phonon interactions in semiconductors

Transcription

1 Phonon wavefunctions and electron phonon interactions in semiconductors Bartomeu Monserrat University of Cambridge Quantum Monte Carlo in the Apuan Alps VII QMC in the Apuan Alps VII Bartomeu Monserrat 1

2 Outline Introduction Motivation Theoretical background Born Oppenheimer and harmonic approximations Vibrational self consistent field Phonon expectation values Results Diamond Lithium Hydride Conclusions QMC in the Apuan Alps VII Bartomeu Monserrat 2

3 Table of Contents Introduction Motivation Theoretical background Born Oppenheimer and harmonic approximations Vibrational self consistent field Phonon expectation values Results Diamond Lithium Hydride Conclusions QMC in the Apuan Alps VII Bartomeu Monserrat 3

4 Motivation Atomic motion plays an essential role: Total energy (ZPM, high temperatures,... ) Thermal expansion (phonon pressure) Structural phase transitions (unstable phonons) Superconductivity (electron phonon interactions)... Anharmonicity is important for: High temperatures Light elements QMC in the Apuan Alps VII Bartomeu Monserrat bm418@cam.ac.uk 4

5 Example: diamond band gap renormalisation Diamond direct band gap: Experiment: 7.1 ev 1 DFT: 5.6 ev QMC: 7.2 ev 2 GW: 7.7 ev 3 Differences.1 ev ε nk (ev) Γ X L W Γ 1 S. Logothetidis, J. Petalas, H. M. Polatoglou, and D. Fuchs, Phys. Rev. B 46, 4483 (1992) 2 M. D. Towler, R. Q. Hood, and R. J. Needs, Phys. Rev. B 62, 233 (2) 3 F. Giustino, S. G. Louie, and M. L. Cohen, Phys. Rev. Lett. 15, (21) QMC in the Apuan Alps VII Bartomeu Monserrat bm418@cam.ac.uk 5

6 Example: diamond band gap renormalisation Diamond direct band gap: Experiment: 7.1 ev 1 DFT: 5.6 ev QMC: 7.2 ev 2 GW: 7.7 ev 3 Differences.1 ev Electron phonon interaction: renormalisation of.6 ev ε nk (ev) Γ X L q W Γ k k + q k 1 S. Logothetidis, J. Petalas, H. M. Polatoglou, and D. Fuchs, Phys. Rev. B 46, 4483 (1992) 2 M. D. Towler, R. Q. Hood, and R. J. Needs, Phys. Rev. B 62, 233 (2) 3 F. Giustino, S. G. Louie, and M. L. Cohen, Phys. Rev. Lett. 15, (21) QMC in the Apuan Alps VII Bartomeu Monserrat bm418@cam.ac.uk 6

7 Table of Contents Introduction Motivation Theoretical background Born Oppenheimer and harmonic approximations Vibrational self consistent field Phonon expectation values Results Diamond Lithium Hydride Conclusions QMC in the Apuan Alps VII Bartomeu Monserrat 7

8 Born Oppenheimer approximation Trial wavefunction: Ψ m ({r i }, {r α }) = ψ m ({r i }; {r α })Φ({r α }) Equations of motion: Ĥ e ψ m ({r i }; {r α }) = ɛ m ({r α })ψ m ({r i }; {r α }) Ĥ n Φ({r α }) = EΦ({r α }) Hamiltonians: Ĥ e = 1 2 i r i i i r j j i i + 1 Z α Z β 2 r α r β Ĥ n = 1 2 α α β α 1 m α 2 α + ɛ m ({r α }) α Z α r i r α QMC in the Apuan Alps VII Bartomeu Monserrat bm418@cam.ac.uk 8

9 Harmonic approximation Vibrational Hamiltonian in {r α } (or {u α }): Ĥ n = α + u α D αβ u β 2 m α α αβ Normal mode analysis: {r α } {q i } Vibrational Hamiltonian in {q i }: Ĥ n = ( 1 2 i 2 q 2 i ω2 i q 2 i ) QMC in the Apuan Alps VII Bartomeu Monserrat bm418@cam.ac.uk 9

10 Vibrational self consistent field equations Phonon Schrödinger equation: ( 3N ) V ({q i }) Φ({q i }) = EΦ({q i }) 2 i=1 q 2 i Ground state ansatz: Φ({q i }) = i φ i(q i ) Self consistent equations: ( 1 2 ) 2 qi 2 + V i (q i ) φ i (q i ) = ɛ i φ i (q i ) 3N V i (q i ) = φ j (q j ) V ({q 3N k}) φ j (q j ) j i j i QMC in the Apuan Alps VII Bartomeu Monserrat bm418@cam.ac.uk 1

11 Approximating the Born Oppenheimer energy surface (I) V ({q i }) = V () + 3N V i (q i ) + 3N 3N i=1 i=1 j>i V ij (q i, q j ) + Static lattice DFT total energy DFT total energy along frozen independent phonon DFT total energy along frozen coupled phonons QMC in the Apuan Alps VII Bartomeu Monserrat bm418@cam.ac.uk 11

12 Approximating the Born Oppenheimer energy surface (II) V ({q i }) = V () + 3N V i (q i ) + 3N 3N i=1 i=1 j>i V ij (q i, q j ) + Energy (ev) Diamond Anharmonic Harmonic q (a.u.) QMC in the Apuan Alps VII Bartomeu Monserrat bm418@cam.ac.uk 12

13 Approximating the Born Oppenheimer energy surface (III) V ({q i }) = V () + 3N V i (q i ) + 3N 3N i=1 i=1 j>i V ij (q i, q j ) LiH Anharmonic Harmonic Energy (ev) q (a.u.) QMC in the Apuan Alps VII Bartomeu Monserrat bm418@cam.ac.uk 13

14 Methodology 1. Supercell with N atoms 3N normal modes 2. Solve for harmonic U har and anharmonic U anh energies 3. Calculate anharmonic correction U anh = U anh U har 4. Supercell size convergence for U anh 5. Add converged U anh to accurate harmonic energy QMC in the Apuan Alps VII Bartomeu Monserrat bm418@cam.ac.uk 14

15 Phonon expectation values Ground state (T = K) Â({q i}) Φ = Φ({q i }) Â({q i}) Φ({q i }) = dq 1 dq 3N Φ ({q i })Â({q i})φ({q i }) Finite temperature (T > K) Â({q i}) Φ,β = 1 Φ s ({q i Z }) Â({q i}) Φ s ({q i }) e βes s Â({q i }) can be electronic band structure, stress tensor,... QMC in the Apuan Alps VII Bartomeu Monserrat bm418@cam.ac.uk 15

16 Table of Contents Introduction Motivation Theoretical background Born Oppenheimer and harmonic approximations Vibrational self consistent field Phonon expectation values Results Diamond Lithium Hydride Conclusions QMC in the Apuan Alps VII Bartomeu Monserrat 16

17 Diamond potential and phonon wavefunction 1 Anharmonic Harmonic.2 Anharmonic Harmonic Energy (ev) φ(q) q (a.u.) q (a.u.) QMC in the Apuan Alps VII Bartomeu Monserrat bm418@cam.ac.uk 17

18 Diamond anharmonic ZPE (I) SC size U har (ev) U anh (ev) U anh (ev) Unit cell Accurate.368 xc-functional: LDA QMC in the Apuan Alps VII Bartomeu Monserrat bm418@cam.ac.uk 18

19 Diamond anharmonic ZPE (II) 5 4 U anh (mev) xc-functional: LDA Number of normal modes QMC in the Apuan Alps VII Bartomeu Monserrat bm418@cam.ac.uk 19

20 Diamond Γ point el ph renormalisation -.6 E Γ (ev) Harmonic Anharmonic AHPT 1 MBPT 2 PI MD (1 K) Number of normal modes 1 F. Giustino, S. G. Louie, and M. L. Cohen, Phys. Rev. Lett. 15, (21) 2 E. Cannuccia and A. Marini, Europ. Phys. J. B (212), accepted 3 R. Ramírez, C. P. Herrero, and E. R. Hernández, Phys. Rev. B 73, (26) QMC in the Apuan Alps VII Bartomeu Monserrat bm418@cam.ac.uk 2

21 LiH potential and phonon wavefunction Anharmonic Harmonic.2.15 Anharmonic Harmonic Energy (ev) φ(q) q (a.u.) q (a.u.) QMC in the Apuan Alps VII Bartomeu Monserrat bm418@cam.ac.uk 21

22 LiH anharmonic ZPE (I) E xc H Li U har (ev) U anh (ev) U anh (ev) LDA PBE Supercell size: 48 modes LDA lattice parameter: bohr stiffer phonons PBE lattice parameter: 3.78 bohr softer phonons QMC in the Apuan Alps VII Bartomeu Monserrat bm418@cam.ac.uk 22

23 LiH anharmonic ZPE (I) E xc H Li U har (ev) U anh (ev) U anh (ev) LDA PBE Supercell size: 48 modes LDA lattice parameter: bohr stiffer phonons PBE lattice parameter: 3.78 bohr softer phonons QMC in the Apuan Alps VII Bartomeu Monserrat bm418@cam.ac.uk 23

24 LiH anharmonic ZPE (I) E xc H Li U har (ev) U anh (ev) U anh (ev) LDA PBE Supercell size: 48 modes LDA lattice parameter: bohr stiffer phonons PBE lattice parameter: 3.78 bohr softer phonons QMC in the Apuan Alps VII Bartomeu Monserrat bm418@cam.ac.uk 24

25 LiH anharmonic ZPE (I) E xc H Li U har (ev) U anh (ev) U anh (ev) LDA PBE Supercell size: 48 modes LDA lattice parameter: bohr stiffer phonons PBE lattice parameter: 3.78 bohr softer phonons QMC in the Apuan Alps VII Bartomeu Monserrat bm418@cam.ac.uk 25

26 LiH anharmonic ZPE (II) H Li U har (ev) U anh (ev) U anh (ev) Supercell size: 162 modes xc functional: PBE QMC in the Apuan Alps VII Bartomeu Monserrat bm418@cam.ac.uk 26

27 LiH anharmonic ZPE (III) 6 5 U anh (mev) H 7 Li 1 H 6 Li 1 xc-functional: PBE 2 H 7 Li 2 H 6 Li Number of normal modes QMC in the Apuan Alps VII Bartomeu Monserrat bm418@cam.ac.uk 27

28 LiH constant volume heat capacity 5 4 c V (1-4 ev/k) H 7 Li 1 H 6 Li 2 H 7 Li 1 SC size: 162 modes xc-functional: PBE 2 H 6 Li 6k B Temperature (K) QMC in the Apuan Alps VII Bartomeu Monserrat bm418@cam.ac.uk 28

29 El ph renormalised LiH band structure (I) 6 4 ε nk (ev) Γ X L W Γ QMC in the Apuan Alps VII Bartomeu Monserrat bm418@cam.ac.uk 29

30 El ph renormalised LiH band structure (II) ε nk (ev) Γ X L W Γ QMC in the Apuan Alps VII Bartomeu Monserrat bm418@cam.ac.uk 3

31 LiH el ph band gap renormalisations Supercell size: 162 modes xc functional: PBE H Li E X (ev) E X (ev) Static lattice QMC in the Apuan Alps VII Bartomeu Monserrat bm418@cam.ac.uk 31

32 Table of Contents Introduction Motivation Theoretical background Born Oppenheimer and harmonic approximations Vibrational self consistent field Phonon expectation values Results Diamond Lithium Hydride Conclusions QMC in the Apuan Alps VII Bartomeu Monserrat 32

33 Conclusions Summary: VSCF for phonon wavefunctions of bulk materials. Phonon expectation values for electronic band structure. Preliminary tests on diamond and LiH. Future work: Other expectation values: stress tensor, atomic positions,... Ice, solid hydrogen,... QMC in the Apuan Alps VII Bartomeu Monserrat 33

34 Acknowledgements Prof Richard J. Needs Dr Neil D. Drummond TCM group EPSRC QMC in the Apuan Alps VII Bartomeu Monserrat 34

35 Table of Contents Additional Material QMC in the Apuan Alps VII Bartomeu Monserrat 35

36 Additional Material: diamond phonon dispersion ω nq (ev).8.4 Γ X L W Γ QMC in the Apuan Alps VII Bartomeu Monserrat bm418@cam.ac.uk 36

37 Additional Material: LiH phonon dispersion.12 1 H 7 Li 2 H 7 Li ω nq (ev).8.4 Γ X L W Γ QMC in the Apuan Alps VII Bartomeu Monserrat bm418@cam.ac.uk 37

38 Additional Material: Diamond coupling term Coupled optical phonons at the Γ point q 2 (a.u.) q 2 (a.u.) Energy (ev) Energy (ev) q 1 (a.u.) q 1 (a.u.) QMC in the Apuan Alps VII Bartomeu Monserrat bm418@cam.ac.uk 38

39 Additional Material: LiH coupling term Coupled optical phonons at the Γ point q 2 (a.u.) q 1 (a.u.) q 2 (a.u.) Energy (ev) q 1 (a.u.) Energy (ev) QMC in the Apuan Alps VII Bartomeu Monserrat bm418@cam.ac.uk 39

40 Additional Material: Allen Heine perturbation theory (I) Expansion of el ph interaction to second order in u α : V el ph ({u α }) Ĥ1 + Ĥ2 Ĥ 1 = u α α V el ph ({u α }) α Ĥ 2 = 1 u α u β α β V el ph ({u α }) 2 αβ Second order perturbation theory: ɛ nk ({u α }) ɛ nk + nk Ĥ1 + Ĥ2 nk + n k Ĥ1 nk 2 ɛ n k nk ɛ n k QMC in the Apuan Alps VII Bartomeu Monserrat bm418@cam.ac.uk 4

41 Additional Material: Allen Heine perturbation theory (II) Thermal average within harmonic approximation (reciprocal space): 1 2m α Nω νq (2n νq + 1) n νq = 1 e βωνq + 1 (1) QMC in the Apuan Alps VII Bartomeu Monserrat bm418@cam.ac.uk 41