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Coulomb glass Computer 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 Marburg, V 2007

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2/80 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

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3/80 Realisation of disorder: the impurity band in semiconductor Conduction band Impurity band (donors) EE Valence band Acceptors

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4/80 Realisation of disorder: the impurity band in semiconductor Conduction band Impurity band (donors) EE Valence band Acceptors

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5/80 Realisation of disorder: the impurity band in semiconductor Conduction band Impurity band (donors) EE Valence band Acceptors Occupied donor (electron) q = 0 Empty donor (hole) q = + Occupied (electron) q = acceptor _ __ _ + + +

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6/80 Realisation of disorder: the impurity band in semiconductor Impurity band (donors) Acceptors Occupied donor (electron) q = 0 Empty donor (hole) q = + Occupied (electron) q = acceptor _ __ _ + + +

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7/80 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. N = 10 K = 0.5 Occupied donor Empty donor Occupied acceptor Since the sites positions are random - site potential are random too (disorder) + + + + + _ _ _ _ _

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8/80 Realisation of disorder: the impurity band in semiconductor System of randomly distributed sites with Coulomb interaction: Site potential: isolated sites are identical Total energy (classical electrostatic interactions): dimensionless units

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9/80 What is the Coulomb glass? System of randomly distributed sites with Coulomb interaction If the system is so sparse that the distances between sites are larger than the localisation length (n < n C ) disorder => electronic wavefunctions are localised the chemical potential is localised in “localised” part of DOS => the quantum overlap may be neglected (no tunnelling) => classical system (electrons move via incoherent hops) 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 => disorder isolator

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10/80 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

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11/80 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 = 10 K = 0.5 Occupied donor Empty donor Occupied acceptor The calculating procedure isn’t a simulation of the relaxation process. (no transition rates)

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12/80 Simulation procedure (T = 0K) System of randomly distributed sites with Coulomb interaction: Site potential: isolated sites are identical Total energy (classical electrostatic interactions): Single electron transfer: dimensionless units

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13/80 Total energy change during single electron transition A (all acceptors) DiDi DjDj Site energies + + 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 _

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14/80 Simulation procedure (T = 0K) System of randomly distributed sites with Coulomb interaction: Single electron transfer: Site potential: isolated sites are identical Total energy (classical electrostatic interactions): Salesman says: transitions for which H < 0 leads to pseudo-ground state hole-electron interaction

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15/80 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 (all occupied) 3. Distribute K·N electrons over donors 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)

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16/80 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 j 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

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17/80 The new hole appears in the neighbourhood... The origin of Coulomb gap in the ground state Other holes don’t like it - they move away...

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18/80 The new hole appears in the neighbourhood... The origin of Coulomb gap in the ground state Other holes don’t like it - they move away... Distances between sites with the same (different) occupancy raise (lessen) Holes’ escape increase the distance between them and therefore lessen the total energy: Occupied sites are closer to new empty site Empty sites are farther from the new hole

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19/80 Coulomb gap in density of states for T = 0K Coulomb gap created due to Coulomb interaction in the system Single-particle DOS Si:P 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 -4-2 0 2 4 N=500 1000 real., PBC

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20/80 “Dimensionless” units The temperature is measured in energy units The length unit The unit of energy Thus => the Coulomb interaction energy reads For example: n D =69% of n C, n C = 3.52·10 18 cm -3 => [d.u.] = 1 T[K] ≈ 200K n D = 8% of n C, n C = 3.52·10 18 cm -3 => [d.u.] = 1 T[K] ≈ 100K

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21/80 Shape of Coulomb gap for T = 0K 0 0.1 0.2 0.3 0.4 0.5 0.6 -0.2 0 0.2 0.4 0.6 0.8 1 numerical simulation result fitting of ax 2 (soft gap) fitting of (Efros) fitting of (BSE) Single-particle-DOS

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22/80 Shape of Coulomb gap for T = 0K Single-particle-DOS hard gap numerical simulation result fitting of (Efros) fitting of (BSE) 0 0.04 0.08 0.12 0.16 0 0.1 0.2 0.3 0.4 0.5 N=500 1000 real., PBC

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23/80 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 sure Else move the electron from i to j with prob. Call -sub (step 1). Repeat step 3 thousands times (Monte Carlo) Repeat steps 0-3 several thousand times (averaging) Step 2 may be omitted

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24/80 Smearing of the Coulomb gap for T > 0K Single-particle DOS T = 0.0 0.1 0.2 0.3 0.4 1 0 0.1 0.2 0.3 0.4 -4-2 0 2 4 N=500, MC=10 5 1000 real., PBC

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25/80 Smearing of the Coulomb gap for T > 0K Single-particle DOS T = 0K 22K 44K 66K 88K 222K 0 0.1 0.2 0.3 0.4 -4-2 0 2 4 N=500, MC=10 5 1000 real., PBC n C = 3.52·10 18 cm -3 = 11.4 n/n C =100% =20Å=0.3 d.u.

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26/80 Smearing of the Coulomb gap for T > 0K Single-particle DOS T = 0K 10K 20K 30K 40K 100K 0 0.1 0.2 0.3 0.4 -4-2 0 2 4 N=500, MC=10 5 1000 real., PBC n C = 3.52·10 18 cm -3 = 11.4 n/n C =8% =20Å=0.13 d.u.

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27/80 Pair distribution (T = 0K) N=400, T=0K, a=0.3, MC=10 3

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28/80 Pair distribution (T > 0K) N=400, T=1/8 (28K for n/n C =1), =0.3, MC=10 3

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29/80 Pair distribution (T > 0K) N=400, T=1 (222K for n/n C =1), =0.3, MC=10 3

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30/80 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

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31/80 The dynamics of the Coulomb gap: the time scales Conduction band Impurity band (donors) AcceptorsValence band Averaged thermal activation time (T = 7K, E=31.27 meV): 10 4 s the microscopic time Lifetime of donor (inverted transfer rate up to conduction band): Question 1: What must be the temperature to keep the electron in the imputity band? EE

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32/80 The dynamics of the Coulomb gap: the time scales Conduction band Impurity band (donors) Miller-Abrahams transfer rate for VRH: EE Averaged thermal activation time (T = 7K, E=31.27 meV): 10 4 s Question 2: How long does it takes to transfer electron from donor i to empty donor j?

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33/80 The dynamics of the Coulomb gap: the time scales Conduction band Impurity band (donors) EE Averaged thermal activation time (T = 7K, E=31.27 meV): 10 4 s Conclusion: For Si:P (n = 69% of n C ) during 10 3 s electron travel only 0.03A < size of atom One need to decrease the n/n C and/or wait very long The Coulomb glass is an isolator (n/n C < 1)

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34/80 The dynamics of the Coulomb gap: the gap evolution Single-particle DOS 0 0.1 0.2 0.3 0.4 -4-2 0 2 4 R 0 = 0.1 1.0 1.2 1.4 2.0 5.0 T = 0K

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35/80 The dynamics of the Coulomb gap: the gap evolution Single-particle DOS 0 0.1 0.2 0.3 0.4 5 2.51 0.5 Energy: 0 -0.5 -2

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36/80 The dynamics of the Coulomb gap: the gap evolution Single-particle DOS 0 0.1 0.2 0.3 0.4 5 2.51 0.5 Energy: 0 -0.5 -2 R 0 (t 0 ) = 1.25 Fitting: 1.26b

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37/80 The dynamics of the Coulomb gap: the gap evolution Single-particle DOS 0 0.1 0.2 0.3 0.4 5 2.51 0.5 Energy: 0 R 0 (t 0 ) = 1.25 Fitting: 1.26 Yu (SCE+numeric, 1999): b = 1 Malik and Kumar (analytical., 2004): b = 2

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38/80 The dynamics of the Coulomb gap: the experiment proposal Conduction band Impurity band (donors) T = 300K R 0 = 0.1 1/4 Mott’s formula for DC conductivity (constant DOS near the Fermi level): Random occupations of sites T = 7K n/n C = 8%

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39/80 The dynamics of the Coulomb gap: the experiment proposal Conduction band Impurity band (donors) R 0 = 1.0 Relaxing (1 st hour)... T = 7K Random occupation of sitesn/n C = 8%

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40/80 The dynamics of the Coulomb gap: the experiment proposal Conduction band Impurity band (donors) R 0 = 1.2 Relaxing (2 nd hours)... T = 7K n/n C = 8%

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41/80 The dynamics of the Coulomb gap: the experiment proposal Conduction band Impurity band (donors) R 0 = 1.4 Relaxing (3 rd hour)... T = 7K n/n C = 8%

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42/80 The dynamics of the Coulomb gap: the experiment proposal Conduction band Impurity band (donors) R 0 = 5.0 1/2 SE’s formula for DC conductivity (gap in g(E ) around the Fermi level): Pseudo-grand state reached T = 7K n/n C = 8%

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43/80 The dynamics of the Coulomb gap: the experiment proposal Conduction band Impurity band (donors) Pseudo-grand state reached T = 7K Change from 1/4-law (Mott) to 1/2-law (SE) not because of the cooling of the sample, but because it relaxed for 10 4 s (3h). n/n C = 8%

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44/80 The dynamics of the Coulomb gap: the gap evolution Single-particle DOS 0 0.1 0.2 0.3 0.4 -4-2 0 2 4 R 0 = 0.1 1.0 1.2 1.4 2.0 5.0 T = 0K

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45/80 The dynamics of the Coulomb gap: the gap evolution Single-particle DOS T = 0.1 d.u. 0 0.1 0.2 0.3 0.4 -4-2 0 2 4 R 0 = 0.1 1.0 1.2 1.4 2.0 5.0

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46/80 The dynamics of the Coulomb gap: the gap evolution Single-particle DOS 0 0.1 0.2 0.3 0.4 -4-2 0 2 4 T = 0.2 d.u. R 0 = 0.1 1.0 1.2 1.4 2.0 5.0

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47/80 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

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48/80 Edwards-Anderson order parameter (EAOP) T = 0K - no transitions in the pseudo-ground state N = 10 K = 0.5 Occupied donor Empty donor Occupied acceptor T = 6K - some transitions (VRH) T = 100K - a lot of transitions (NNH)

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49/80 T = 6K - some transitions (VRH) time nini Edwards-Anderson order parameter (EAOP) T = 0K - no transitions in the pseudo-ground state time nini T = 100K - a lot of transitions (NNH) time nini Order parameter (per analogy to spin glass) q = 1.0 q = 0.8 q = 0.1

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50/80 Glass transition Davies, Lee, Rice (lattice model, no actual acceptors): glass transition from random (T > 0.3K) to ordered (T = 0K) system of {n i } a

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51/80 Glass transition Davies, Lee, Rice (lattice model, no actual acceptors): glass transition from random (T > 0.3K) to ordered (T = 0K) system of {n i } Yu: glass transition in Coulomb glass is the phase transition of second order

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52/80 Glass transition Davies, Lee, Rice (lattice model, no actual acceptors): glass transition from random (T > 0.3K) to ordered (T = 0K) system of {n i } Yu: glass transition in Coulomb glass is the phase transition of second order

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53/80 Glass transition Davies, Lee, Rice (lattice model, no actual acceptors): glass transition from random (T > 0.3K) to ordered (T = 0K) system of {n i } Yu: glass transition in Coulomb glass is the phase transition of second order

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54/80 Glass transition Davies, Lee, Rice (lattice model, no actual acceptors): glass transition from random (T > 0.3K) to ordered (T = 0K) system of {n i } Yu: glass transition in Coulomb glass is the phase transition of second order

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55/80 Glass transition Our model: random positions of sites, actual acceptors present a But closely locates groups are not present in the lattice model The crowded group has higher energy than surrounding and preserve its occupation unchanged even for high temperatures a The lattice model well describes remote donors’ interaction

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56/80 Glass transition Our model: EAOP has the same value for N=100 and N=500, glass transition has long exponential tail (nonzero values even for T > 300K) EA order parameter [d.u.] our model Davies, Lee, Rice 0.2 0.4 0.6 0.8 1 0.001 0.01 0.1 1 10 100

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57/80 Glass transition EA order parameter our model Davies, Lee, Rice (B = 2) 0 0.1 0.2 0.3 0.4 1 10 [d.u.] 3 Our model: EAOP has the same value for N=100 and N=500, glass transition has long exponential tail (nonzero values even for T > 300K)

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58/80 Glass transition EA order parameter our model Davies, Lee, Rice (B = 2) 0 0.1 0.2 0.3 0.4 100K 1000K [K] 300K Si:P n/n C = 8% Our model: EAOP has the same value for N=100 and N=500, glass transition has long exponential tail (nonzero values even for T > 300K)

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59/80 Glass transition EA order parameter our model Davies, Lee, Rice (B = 2) 0 0.1 0.2 0.3 0.4 197K 1965K [K] 590K Si:P n/n C = 69% Our model: EAOP has the same value for N=100 and N=500, glass transition has long exponential tail (nonzero values even for T > 300K)

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60/80 - introduced to demonstrate that the Coulomb fields induce sine ordering at low temperatures - in our (random site) model it depends on N! Modified Edwards-Anderson order parameter normal “spin”“spin” in system with no interactions the difference report the contribution to EA order parameter related to presence of the Coulomb interaction within the system

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61/80 Goes to zero faster than the EA order parameter. Modified Edwards-Anderson order parameter modified EA order parameter our model (N = 100) Davies, Lee, Rice (B = 2) [d.u.] our model (N = 500)

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62/80 New order parameter related to the electron diffusion The quest for the phase transition parameter in the Coulomb glass (Lee, Yu) Analysis of “Binder g” suggests that the glass transition is the phase transition Need for the parameter which rapidly goes to zero. What mechanism behind it? N = 10 K = 0.5 Occupied donor Empty donor Occupied acceptor Our idea is to trace the electron instead of the site’s occupation!

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63/80 New order parameter related to the electron diffusion The quantities which may be a base for a new order parameter: - the distance of the electron from the final site to the initial one - the total hops length - the number of hops The value of the new order parameter may be the percentage of all electrons for which: - the distance between final and initial site is smaller than... - the total hops length is smaller than... - the number of hops is smaller than... We just measure the percentage of the electrons which stay for all simulation at the initial position. Thus the new order parameters = 1 only if the EA order parameter = 1. The disadvantage: its value is related to the measurement time (stronger than the EA order parameter) The advantage: its value more rapidly goes to zero => the phase transition apply to the electron diffusion in the Coulomb glass

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64/80 New order parameter related to the electron diffusion 0 0.2 0.4 0.6 0.8 1 0.001 0.01 0.1 1 10 100 EA order parameter the new order parameter EA and new order parameters [d.u.] about 1% of electrons got stacked

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65/80 Glass transitions versus Coulomb gap smearing Single-particle DOS for E = 0 0 0.1 0.2 0.3 0.01 0.1 1 10 g(0) g(0) (Grannan and Yu; lattice model) EA order parameter 0.0 0.5 1.0 EA order parameter the new order parameter [d.u.]

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66/80 Glass transitions vs Coulomb gap evolution Single-particle DOS for E = 0 0 0.1 0.2 0.3 0.01 0.1 1 10 g(0) g(0) (Grannan and Yu; lattice model) EA order parameter 0.0 0.5 1.0 EA order parameter the new order parameter electrons leave the initial sites almost none electron rests complete randomness gap starts to form gap is formed

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67/80 The gap transition with time limitation EA order parameters 0.2 0.4 0.6 0.8 1 0.001 0.01 0.1 1 10 100 R = 0.3 R = 0.5 R = 0.75 R = 1.0 No limit [d.u.] The gap starts to form for R > 1.2, while the order is established for R < 1.5

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68/80 The gap transition with time limitation new order parameters [d.u.] The gap starts to form for R > 1.2, while the order is established for R < 1.5 0.2 0.4 0.6 0.8 1 0.001 0.01 0.1 1 10 100 R = 0.3 R = 0.5 R = 0.75 R = 1.0 No limit

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69/80 0.05 0.1 0.15 0.01 0.1 1 10 100 The gap transition with time limitation 0.6 0.8 1 T = 0.1 T = 0.2 0 0.5 1 EA order par. new order par. T = 0.1 T = 0.2 R [d.u.] T = 0.1 T = 0.2 From random From ground

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70/80 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 a) Experimental results of AC conductivity measurements b) brief introduction to Shklovskii and Efros’s model of zero-phonon AC hopping conductivity of disordered system c) results of computer simulations for T = 0K

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71/80 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)

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72/80 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)

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73/80 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

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74/80 Shklovskii and Efros’s model Pair of sites Only pairs with one electron are interesting in context of conductivity: The isolated sites base Normalisationwhere

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75/80 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) Energy which must be absorbed by pairs in unit volume due to el. transition Q = QM transition prob. (Fermi Golden Rule) prob. of finding “proper” pair ·· prob. of finding photon with energy equals to ·

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76/80 Pair distribution (T = 0K) N=400, T=0K, N Monte-Carlo =1000, a=0.27

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77/80 Pairs mean spatial distance (T = 0K) pair mean spatial distance Mott’s formula simulations In contradiction to the Mott’s assumption the distribution of pairs’ distances is very wide N=1000, K=0.5, 2500 realisations periodic boundary conditions, AOER

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78/80 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

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79/80 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!

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80/80 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 crossover (?) a = 0.36

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81/80 The End http://www.phys.uni.torun.pl/~jacek e-mail: jacek@phys.uni.torun.pl

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