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Coulomb glass

Coulomb glass. 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 Marburg, V 2007. Computer simulations. Outline

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Coulomb glass

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  1. Coulomb glass 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 Marburg, V 2007 Computer simulations

  2. Outline 0. Example of disordered system with long-range interactions 1. Simulation procedure for computation of single particle DOS 2. The dynamics of the Coulomb gap 3. The Coulomb glass and the glass transition 4. Phononless AC conductivity in Coulomb glass

  3. Realisation of disorder: the impurity band in semiconductor Conduction band DE Impurity band (donors) Acceptors Valence band

  4. Realisation of disorder: the impurity band in semiconductor Conduction band DE Impurity band (donors) Acceptors Valence band

  5. Realisation of disorder: the impurity band in semiconductor Conduction band DE + + + Impurity band (donors) Occupied donor (electron) q = 0 Empty donor (hole) q = + _ Occupied (electron) q =acceptor _ _ _ Acceptors Valence band

  6. Realisation of disorder: the impurity band in semiconductor + + + Impurity band (donors) Occupied donor (electron) q = 0 Empty donor (hole) q = + _ Occupied (electron) q =acceptor _ _ _ Acceptors

  7. Realisation of disorder: the impurity band in semiconductor The electrostatic potential at every donor site is due to Coulomb interaction with every acceptor (-) and every other empty site (+) in the system. Since the sites positions are random - site potential are random too (disorder) N = 10K = 0.5 + _ + + _ Occupied donor Empty donor _ Occupied acceptor + _ + _

  8. Site potential: isolated sites are identical Realisation of disorder: the impurity band in semiconductor System of randomly distributed sites with Coulomb interaction: Total energy (classical electrostatic interactions): dimensionless units

  9. What is the Coulomb glass? System of randomly distributed sites with Coulomb interaction disorder => electronic wavefunctions are localised the chemical potential is localised in “localised” part of DOS If the system is so sparse that the distances between sites are larger than the localisation length (n < nC) => the quantum overlap may be neglected (no tunnelling) => classical system (electrons move via incoherent hops) => disorder isolator Examples: compensated lightly doped semiconductors amorphous semiconductors and alloys hopping behaviour of quasicrystals granular films silicon MOSFET’s heterostructures electrically conducting polymers and stannic oxides nanowires

  10. Outline 0. Example of disordered system with long-range interactions 1.Simulation procedure for computation of single particle DOS a) Searching for the pseudo-ground state (T = 0K). Coulomb gap b) Monte Carlo simulations (T > 0K) 2. The dynamics of the Coulomb gap 3. The Coulomb glass and the glass transition 4. Phononless AC conductivity in Coulomb glass

  11. Simulation procedure (T = 0K) Metropolis algorithm: the same as used to solve the salesman problem General: Searching for the configuration which minimise some parameter In our case: searching for electron arrangement which minimise total energy N = 10K = 0.5 Occupied donor Empty donor Occupied acceptor The calculating procedure isn’t a simulation of the relaxation process. (no transition rates)

  12. Site potential: isolated sites are identical Single electron transfer: Simulation procedure (T = 0K) System of randomly distributed sites with Coulomb interaction: Total energy (classical electrostatic interactions): dimensionless units

  13. + Total energy of the system: + In order to make the calculation possible we need to express the energy difference using sites energy values before the transition Total energy change during single electron transition Dj Site energies Di _ A (all acceptors)

  14. hole-electron interaction Simulation procedure (T = 0K) System of randomly distributed sites with Coulomb interaction: Total energy (classical electrostatic interactions): Site potential: isolated sites are identical Single electron transfer: Salesman says: transitions for which DH < 0 leads to pseudo-ground state

  15. Simulation procedure (T = 0K) Metropolis algorithm for searching the pseudo-ground state of system Step 01. Place N randomly distributed donors in the box2. Add K·N randomly distributed acceptors (all occupied)3. Distribute K·N electrons over donors Step 1 (m -sub)3. Calculate site energies of donors4. 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)

  16. Simulation procedure (T = 0K) Metropolis algorithm for searching the pseudo-ground state of system Step 2 (Coulomb term)6. Searching the pairs checking for occupied site i and empty j If there is such a pair then move electron from i to jand call m -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)

  17. The new hole appears in the neighbourhood ... Other holes don’t like it - they move away ... The origin of Coulomb gap in the ground state

  18. Holes’ escape increase the distance between them and therefore lessen the total energy: Empty sites are farther from the new hole Occupied sites are closer to new empty site The origin of Coulomb gap in the ground state The new hole appears in the neighbourhood ... Other holes don’t like it - they move away ... Distances between sites with the same (different) occupancy raise (lessen)

  19. 0.4 Si:P 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 -4 -2 0 2 4 Coulomb gap in density of states for T = 0K Coulomb gap created due to Coulomb interaction in the system N=5001000 real., PBC Single-particle DOS

  20. Thus => the Coulomb interaction energy reads “Dimensionless” units The unit of energy The temperature is measured in energy units The length unit For example: nD=69% of nC, nC = 3.52·1018 cm-3 => t[d.u.] = 1 <=> T[K] ≈ 200K nD= 8% of nC, nC = 3.52·1018 cm-3 => t[d.u.] = 1 <=> T[K] ≈ 100K

  21. numerical simulation result fitting of (Efros) fitting of (BSE) fitting of ax2 (soft gap) Shape of Coulomb gap for T = 0K 0.6 0.5 0.4 Single-particle-DOS 0.3 0.2 0.1 0 -0.2 0 0.2 0.4 0.6 0.8 1

  22. numerical simulation result fitting of (Efros) fitting of (BSE) Shape of Coulomb gap for T = 0K 0.16 0.12 Single-particle-DOS 0.08 hard gap 0.04 N=5001000 real., PBC 0 0 0.1 0.2 0.3 0.4 0.5

  23. 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 i to j for sureElse move the electron from i to j with prob. Callm -sub (step 1). Repeat step 3 thousands times (Monte Carlo) Repeat steps 0-3 several thousand times (averaging) Step 2 may be omitted

  24. 0.4 T = 0.0 0.1 0.2 0.3 0.3 0.4 1 0.2 0.1 0 -4 -2 0 2 4 Smearing of the Coulomb gap for T > 0K N=500, MC=1051000 real., PBC Single-particle DOS

  25. 0.4 0.3 0.2 0.1 0 -4 -2 0 2 4 Smearing of the Coulomb gap for T > 0K N=500, MC=1051000 real., PBC nC = 3.52·1018 cm-3e = 11.4 n/nC=100%a=20Å=0.3 d.u. T = 0K 22K 44K 66K 88K Single-particle DOS 222K

  26. 0.4 0.3 0.2 0.1 0 -4 -2 0 2 4 Smearing of the Coulomb gap for T > 0K N=500, MC=1051000 real., PBC nC = 3.52·1018 cm-3e = 11.4 n/nC=8%a=20Å=0.13 d.u. T = 0K 10K 20K 30K 40K Single-particle DOS 100K

  27. Pair distribution (T = 0K) N=400, T=0K, a=0.3, MC=103

  28. Pair distribution (T > 0K) N=400, T=1/8 (28K for n/nC=1), a=0.3, MC=103

  29. Pair distribution (T > 0K) N=400, T=1 (222K for n/nC=1), a=0.3, MC=103

  30. Outline 0. Example of disordered system with long-range interactions 1. Simulation procedure for computation of single particle DOS 2.The dynamics of the Coulomb gap 3. The Coulomb glass and the glass transition 4. Phononless AC conductivity in Coulomb glass

  31. DE Question 1: What must be the temperature to keep the electron in the imputity band? Lifetime of donor (inverted transfer rate up to conduction band): Acceptors Valence band the microscopic time The dynamics of the Coulomb gap: the time scales Conduction band Averaged thermal activation time (T = 7K, DE=31.27 meV): 104 s Impurity band (donors)

  32. Question 2: How long does it takes to transfer electron from donor i to empty donor j? Miller-Abrahams transfer rate for VRH: The dynamics of the Coulomb gap: the time scales Conduction band DE Averaged thermal activation time (T = 7K, DE=31.27 meV): 104 s Impurity band (donors)

  33. The dynamics of the Coulomb gap: the time scales Conduction band DE Averaged thermal activation time (T = 7K, DE=31.27 meV): 104 s Impurity band (donors) Conclusion: For Si:P (n = 69% of nC) during 103s electron travel only 0.03A < size of atom One need to decrease the n/nC and/or wait very long The Coulomb glass is an isolator (n/nC < 1)

  34. 0.4 R0 = 0.1 1.0 1.2 0.3 1.4 2.0 5.0 0.2 0.1 0 -4 -2 0 2 4 The dynamics of the Coulomb gap: the gap evolution T = 0K Single-particle DOS

  35. 0.4 Energy: 0 -0.5 0.3 -1 -2 0.2 0.1 0 0.5 1 2.5 5 The dynamics of the Coulomb gap: the gap evolution Single-particle DOS

  36. 0.4 Energy: 0 -0.5 0.3 -1 -2 0.2 0.1 0 0.5 1 2.5 5 The dynamics of the Coulomb gap: the gap evolution Single-particle DOS Fitting: 1.26 b R0(t0) = 1.25

  37. Energy: 0 Yu (SCE+numeric, 1999): b = 1 Malik and Kumar (analytical., 2004): b = 2 The dynamics of the Coulomb gap: the gap evolution 0.4 0.3 Single-particle DOS 0.2 0.1 Fitting: 1.26 R0(t0) = 1.25 0 0.5 1 2.5 5

  38. T = 7K R0 = 0.1 Mott’s formula for DC conductivity (constant DOS near the Fermi level): 1/4 The dynamics of the Coulomb gap: the experiment proposal Conduction band T = 300K Random occupations of sites n/nC = 8% Impurity band (donors)

  39. R0 = 1.0 The dynamics of the Coulomb gap: the experiment proposal Conduction band T = 7K Random occupation of sites n/nC = 8% Impurity band (donors) Relaxing (1st hour) ...

  40. R0 = 1.2 The dynamics of the Coulomb gap: the experiment proposal Conduction band T = 7K n/nC = 8% Impurity band (donors) Relaxing (2nd hours) ...

  41. R0 = 1.4 The dynamics of the Coulomb gap: the experiment proposal Conduction band T = 7K n/nC = 8% Impurity band (donors) Relaxing (3rd hour) ...

  42. SE’s formula for DC conductivity (gap in g(E) around the Fermi level): 1/2 The dynamics of the Coulomb gap: the experiment proposal Conduction band T = 7K Pseudo-grand state reached n/nC = 8% Impurity band (donors) R0 = 5.0

  43. The dynamics of the Coulomb gap: the experiment proposal Conduction band T = 7K Pseudo-grand state reached n/nC = 8% Impurity band (donors) Change from 1/4-law (Mott) to 1/2-law (SE) not because of the cooling of the sample, but because it relaxed for 104 s (3h).

  44. 0.4 R0 = 0.1 1.0 1.2 0.3 1.4 2.0 5.0 0.2 0.1 0 -4 -2 0 2 4 The dynamics of the Coulomb gap: the gap evolution T = 0K Single-particle DOS

  45. 0.4 R0 = 0.1 1.0 1.2 0.3 1.4 2.0 5.0 0.2 0.1 0 -4 -2 0 2 4 The dynamics of the Coulomb gap: the gap evolution T = 0.1 d.u. Single-particle DOS

  46. 0.4 R0 = 0.1 1.0 1.2 0.3 1.4 2.0 5.0 0.2 0.1 0 -4 -2 0 2 4 The dynamics of the Coulomb gap: the gap evolution T = 0.2 d.u. Single-particle DOS

  47. Outline 0. Example of disordered system with long-range interactions 1. Simulation procedure for computation of single particle DOS 2. The dynamics of the Coulomb gap 3.The Coulomb glass and the glass transition 4. Phononless AC conductivity in Coulomb glass

  48. T = 6K - some transitions (VRH) T = 100K - a lot of transitions (NNH) Occupied donor Empty donor Occupied acceptor Edwards-Anderson order parameter (EAOP) T = 0K - no transitions in the pseudo-ground state N = 10K = 0.5

  49. q = 1.0 T = 6K - some transitions (VRH) q = 0.8 ni time time T = 100K - a lot of transitions (NNH) q = 0.1 ni time Order parameter(per analogy to spin glass) time time time Edwards-Anderson order parameter (EAOP) T = 0K - no transitions in the pseudo-ground state ni

  50. Glass transition Davies, Lee, Rice (lattice model, no actual acceptors): glass transition from random (T > 0.3K) to ordered (T = 0K) system of {ni} a

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