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Phononless AC conductivity in Coulomb glass
Monte-Carlo simulations
Jacek Matulewski, Sergei Baranovski, Peter Thomas
Departament of PhysicsPhillips-Universitat Marburg, Germany
Faculty of Physics, Astronomy and InformaticsNicolaus Copernicus University in Toruń, Poland
Ráckeve, 30 VIII 2004
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
3/38
Experimental resultsM. 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)
4/38
Experimental resultsM. 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)
5/38
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
6/38
Shklovskii and Efros’s model of zero-phonon AC hopping conductivity of disordered system
i ij ij
ji
iii r
nnnEH
21
ji ij
j
ji ij
ji r
n
r
nEE 0
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 interactionsare taken into account!
7/38
Shklovskii and Efros’s modelPair of sites
)ˆˆˆˆ)((ˆˆ21
ˆˆˆ122112
12
21221112 aaaarI
rnn
nEnEH
Hamiltonian of a pair of sites:
2,1 1
1 j j
j
r
nE
Site energy is determined by Coulomb interaction with surrounding pairs
Overlap of site’s wavefunction
)exp()( 1212012 ararIrI
21,2112 ,,ˆ21
nnWnnH nn Notice that because of overlap I(r) “intuitive” states can be not good eigenstates
mmm WH 12ˆ
,
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
0,0
1,1
8/38
0
0
2
1
2
1
m
m
WEI
IWE
Shklovskii and Efros’s modelPair of sites
Only pairs with one electron are interesting in context of conductivity:
mmm WH 12ˆ
1,00,1
1,00,1)ˆˆˆˆ)((ˆˆ21
ˆˆ
21
21122112
212211
mW
aaaarIrnn
nEnE
1,00,1 21 m
The isolated sites base1
2
2
2
1 Normalisation
2122
1211
m
m
WIE
WIE
2121
EEE 2212 4IEE where
9/38
Energy which pair much absorb or emit to move the electron between split-states(from to ):
Shklovskii and Efros’s modelPair of sites
2212 4IEEWWW
Source of energy: photons
20
2)(
i
iQ
And finally the conductivity: Shklovskii and Efros formula for conductivity in Coulomb glasses
r
r1
)( 4
02
lnI
ar
Numerical calculation (esp. for T > 0)
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 · )(4
2 2
re
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
11/38
Additional Coulomb energy in transition
2212 4IEEWWW
Correction to sites energy difference
12 EEE
12/38
Additional Coulomb energy in transition
A (all acceptors)
Di
Dj
ir jr
ij rrr +
rrE
i
beforei
11)( j
beforej r
E1)(
Site energies Energy of the system:
i
afteri r
E1
)( rr
Ej
afterj
11 )(
+
ij
beforei
afterj rr
EEE11)()(
In order to make the calculation possible we need to express the energy difference using sites energy values before the transition
rEE
rrEEE before
ibefore
jij
beforei
afterj
111 )()()()(
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
rEE
rrEEE before
jbefore
jij
beforej
afterj
111 )()()()(
2
2
12 41
Ir
EEWWW
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
15/38
Simulation procedure (T = 0K)Metropolis algorithm: the same as used to solve the milkman problemGeneral: Searching for the configuration which minimise some parameterIn our case: searching for electron arrangement which minimise total energy
N=10K=0.5
Occupied donor
Empty donor
Occupied acceptor
16/38
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
Step 1 (-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)
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)
01 ij
ijij rEEE
Simulation procedure (T = 0K)Metropolis algorithm for searching the pseudo-ground state of system
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)
01 ij
ijij rEEE
kT
Eij
eTp
)(
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
20/38
Coulomb gap in density of states for T = 0K
Coulomb gap created due to Coulomb interaction in the system
0
0.2
0.4
0.6
0.8
1
-4 -2 0 2 4
01 ij
ijij rEEE
Si:P
μEi
Nor
mal
ized
sin
gle-
part
icle
DO
S
Coulomb term, but not only ...
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)
μEi
Nor
mal
ized
sin
gle-
part
icle
DO
S
22/38
Pair distribution (T = 0K)
ij
occupiedi
emptyj r
EE1
ijr
N=400, T=0K, NMonte-Carlo=1000, a=0.27
23/38
Pair distribution (T > 0K)
ijr
ij
occupiedi
emptyj r
EE1
N=400, T=1/8 (300K), NMonte-Carlo=1000, a=0.27
24/38
Pair distribution (T > 0K)
ijr
ij
occupiedi
emptyj r
EE1
N=400, T=1 (2415K), NMonte-Carlo=1000, a=0.27
25/38
Pairs mean spatial distance (T = 0K)
2.4
2.6
2.8
3
3.2
3.4
3.6
3.8
4
0 0.05 0.1 0.15 0.2
pair
mea
n sp
atia
l dis
tanc
e
Mott’s formula
simulations
Distribution of pairs’ distances is very wide in contradiction to Mott’s assumption
N=1000, K=0.5, 2500 realisationsperiodic boundary conditions, AOER
02ln
Iar
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.N
umbe
r of
pai
rs
27/38
Conductivity (T=0K)
1e-010
1e-009
1e-008
1e-007
1e-006
1e-005
0.001 0.01 0.1
Con
duct
ivit
y (a
rb. u
n.)
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 nC means a = 0.27 [l69%]
(in units of n-1/3)
fixed parameters for Si:P: a = 20Å, and nC = 3.52·1024 m-3 (lC = 65.7Å)
There is no crossover in numerical results!
28/38
Conductivity (T=0K)
Con
duct
ivit
y (a
rb. u
n.)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
29/38
Number of pairs for T > 0K
Num
ber
of p
airs
N=500, T=0, K=0.5, aver. over 100 real.
T = 0T = 0.125T = 0.3T = 0.5T = 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
30/38
Conductivity for T > 0K
Con
duct
ivit
y (a
rb. u
n.)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 = 0.0T = 0.1T = 0.5T = 1.0
0
5e-005
0.0001
0.0002
0.0003
0.0004
0.0005
31/38
1e-008
1e-007
1e-006
1e-005
0.0001
0.001
0.001 0.01 0.1 1
Conductivity for T > 0KC
ondu
ctiv
ity
(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 (0.5·1013 Hz)
temperature (dimensionless units)
Si:P 69%
32/38
Shape of Coulomb gap for T = 0K (corresponding to conductivity in low frequency)
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
0 0.1 0.2 0.3 0.4 0.5
Nor
mal
ized
sin
gle
part
icle
-DO
S
this hump probably isonly a model artefact
hard gap
numerical simulations result
fitting of (Efros)xE
ae/0
fitting of (Baranovskii et al.) 47
0
0
ln
/
xE
xE
ae
iEx
Efros: many particle-hole excitations in whichsurrounding electrons were allowed to relax