1. Phononless AC conductivity in Coulomb glass Monte-Carlo simulations Jacek Matulewski, Sergei Baranovski, Peter Thomas Departament of Physics Phillips-Universitat Marburg, Germany Faculty of Physics, Astronomy and Informatics Nicolaus Copernicus University in Toruń, Poland Ráckeve, 30 VIII 2004

2. 2/38 Outline 1. Experimental results of AC conductivity measurements 2. Shklovskii and Efros’s model of zero-phonon AC hopping conductivity in the disordered system 3. “Coulomb term” 4. Simulation procedure 5. Results a) Coulomb gap in states distribution b) Pairs distribution c) Conductivity

3. 3/38 Experimental results M. Lee and M.L. Stutzmann, Phys. Rev. Lett. 87, 056402 (2001) E. Helgren, N.P. Armitage and G. Gru:ner, Phys. Rev. Lett. 89, 246601 (2002)

4. 4/38 Experimental results M. Lee and M.L. Stutzmann, Phys. Rev. Lett. 87, 056402 (2001) E. Helgren, N.P. Armitage and G. Gru:ner, Phys. Rev. Lett. 89, 246601 (2002)

5. 5/38 Outline 1. Experimental results of AC conductivity measurements 2. Shklovskii and Efros’s model of zero-phonon AC hopping conductivity in the disordered system 3. “Coulomb term” 4. Simulation procedure 5. Results a) Coulomb gap in states distribution b) Pair distribution c) Conductivity

6. 6/38 Shklovskii and Efros’s model of zero-phonon AC hopping conductivity of disordered system System of randomly distributed sites with Coulomb interaction: When T  0K, the Fermi level is present If frequency of external AC electric field is small, only pairs near the Fermi level contribute to conductivity (one site below and one over) Pair approximation Site energy: (sites are identical) Electron-electron interactions are taken into account!

7. 7/38 Shklovskii and Efros’s model Pair of sites Hamiltonian of a pair of sites: Site energy is determined by Coulomb interaction with surrounding pairs Overlap of site’s wavefunction Notice that because of overlap I(r) “intuitive” states can be not good eigenstates Anyway four states are possible a priori: there is no electron, so no interaction and energy is equal to 0 there is one electron at the pair (two states) there are two electrons at the pair

8. 8/38 Shklovskii and Efros’s model Pair of sites Only pairs with one electron are interesting in context of conductivity: The isolated sites base Normalisationwhere

9. 9/38 Energy which pair much absorb or emit to move the electron between split-states (from to ): Shklovskii and Efros’s model Pair of sites Source of energy: photons And finally the conductivity: Shklovskii and Efros formula for conductivity in Coulomb glasses Numerical calculation (esp. for T > 0) Two limits Energy which must be absorbed by pairs in unit volume due to change states Q = QM transition prob. (Fermi Golden Rule) prob. of finding “proper” pair ·· prob. of finding photon with energy equals to ·

10. 10/38 Outline 1. Experimental results of AC conductivity measurements 2. Shklovskii and Efros’s model of zero-phonon AC hopping conductivity in the disordered system 3. “Coulomb term” 4. Simulation procedure 5. Results a) Coulomb gap in states distribution b) Pair distribution c) Conductivity

11. 11/38 Additional Coulomb energy in transition Correction to sites energy difference

12. 12/38 Additional Coulomb energy in transition A (all acceptors) DiDi DjDj + Site energiesEnergy of the system: + In order to make the calculation possible we need to express the energy difference using sites energy values before the transition

13. 13/38 Additional Coulomb energy in transition Physical cause of correction: changes in the Coulomb net configuration Pair approximation: only one pair changes the state at the time Unfortunately to obtain this term we need to forget about the overlap for a moment In order to make the calculation possible we need to express the energy difference using sites energy values before the transition

14. 14/38 Outline 1. Experimental results of AC conductivity measurements 2. Shklovskii and Efros’s model of zero-phonon AC hopping conductivity of disordered system 3. “Coulomb term” 4. Simulation procedure a) T = 0K (Metropolis algorithm) b) T > 0K (Monte-Carlo simulation) 5. Results a) Coulomb gap in states distribution b) Pair distribution c) Conductivity

15. 15/38 Simulation procedure (T = 0K) Metropolis algorithm: the same as used to solve the milkman problem General: Searching for the configuration which minimise some parameter In our case: searching for electron arrangement which minimise total energy N=10 K=0.5 Occupied donor Empty donor Occupied acceptor

16. 16/38 Simulation procedure (T = 0K) Metropolis algorithm for searching the pseudo-ground state of system Step 0 1. Place N randomly distributed donors in the box 2. Add K·N randomly distributed acceptors Step 1 (  -sub) 3. Calculate site energies of donors 4. Move electron from the highest occupied site to the lowest empty one 5. Repeat points 3 and 4 until there will be no occupied empty sites below any occupied (Fermi level appears)

17. 17/38 Step 2 (Coulomb term) 6. Searching the pairs checking for occupied site j and empty i If there is such a pair then move electron from j to i and call  -sub (step 1) and go back to 6. Effect: the pseudo-ground state (the state with the lowest energy in the pair approximation) Energy can be further lowered by moving two and more electrons at the same step (few percent) Simulation procedure (T = 0K) Metropolis algorithm for searching the pseudo-ground state of system Dlaczego nie gamma?????

18. 18/38 Simulation procedure (T > 0K) Monte-Carlo simulations Step 3 (Coulomb term) 7. Searching the pairs checking for occupied site j and empty i If there is such a pair Then move electron from j to i for sure Else move the electron from j to i with prob. Call  -sub (step 1). Repeat step 3 thousands times Repeat steps 0-3 several thousand times (parallel)

19. 19/38 Outline 1. Experimental results of AC conductivity measurements 2. Shklovskii and Efros’s model of zero-phonon AC hopping conductivity in the disordered system 3. “Coulomb term” 4. Simulation procedure 5. Results a) Coulomb gap in states distribution b) Pair distribution c) Conductivity

20. 20/38 Coulomb gap in density of states for T = 0K Coulomb gap created due to Coulomb interaction in the system Si:P Normalized single-particle DOS Coulomb term, but not only...

21. 21/38 Smearing of the Coulomb gap for T > 0K 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 -4-2 0 2 4 T = 0 (0K) T = 0.3 (725K) T=1 (2415K) Normalized single-particle DOS

22. 22/38 Pair distribution (T = 0K) N=400, T=0K, N Monte-Carlo =1000, a=0.27

23. 23/38 Pair distribution (T > 0K) N=400, T=1/8 (300K), N Monte-Carlo =1000, a=0.27

24. 24/38 Pair distribution (T > 0K) N=400, T=1 (2415K), N Monte-Carlo =1000, a=0.27

25. 25/38 Pairs mean spatial distance (T = 0K) pair mean spatial distance Mott’s formula simulations Distribution of pairs’ distances is very wide in contradiction to Mott’s assumption N=1000, K=0.5, 2500 realisations periodic boundary conditions, AOER

26. 26/38 Pair energy distribution (T = 0K) 0 50000 100000 150000 200000 250000 0 1 2 3 4 5 6 7 8 9 10 We work here!!! N=500, T=0, K=0.5, aver. over 100 real. Number of pairs

27. 27/38 Conductivity (T=0K) Conductivity (arb. un.) Helgren et al. (T=2.8K) n = 69% simulations N=500, T=0, K=0.5, aver. over 25k real. Δ(hw)=0.001 (blue), Δ(hw)=0.01 (green) n = 69% of n C means a = 0.27 [l 69% ] (in units of n -1/3 ) fixed parameters for Si:P: a = 20Å, and n C = 3.52·10 24 m -3 (l C = 65.7Å) There is no crossover in numerical results!

28. 28/38 Conductivity (T=0K) Conductivity (arb. un.) N=500, T=0, K=0.5, aver. over 25k real. Δ(hw)=0.001 (blue), Δ(hw)=0.01 (green) 1e-008 1e-007 1e-006 1e-005 0.0001 0.001 0.01 0.1 simulations Helgren 69% Si:P clearly visible crossover a = 0.36

29. 29/38 Number of pairs for T > 0K Number of pairs N=500, T=0, K=0.5, aver. over 100 real. T = 0 T = 0.125 T = 0.3 T = 0.5 T = 1 0 0.5 1 1.5 2 2.5 3 0 50000 100000 150000 200000 250000 0 0.5 1 1.5 2 2.5 3

30. 30/38 Conductivity for T > 0K Conductivity (arb. un.) N=500, T=0, K=0.5, aver. over 1000 real. Δ(hw)=0.01, T>0 0.00015 0.00025 0.00035 0.00045 0 0.05 0.1 0.15 0.2 0.25 0.3 T = 0.0 T = 0.1 T = 0.5 T = 1.0 0 5e-005 0.0001 0.0002 0.0003 0.0004 0.0005

31. 31/38 Conductivity for T > 0K Conductivity (arb. un.) N=500, T=0, K=0.5, aver. over 1000 real. Δ(hw)=0.01, T>0  = 0.001  = 0.01  = 0.1 (0.5 · 10 13 Hz) temperature (dimensionless units) Si:P 69%

32. 32/38 Shape of Coulomb gap for T = 0K (corresponding to conductivity in low frequency) Normalized single particle-DOS this hump probably is only a model artefact hard gap numerical simulations result fitting of (Efros) fitting of (Baranovskii et al.) Efros: many particle-hole excitations in which surrounding electrons were allowed to relax

33. 33/38 The End