Coulomb glass Computer simulations Jacek Matulewski, Sergei Baranovski, Peter Thomas Departament of Physics Phillips-Universitat Marburg, Germany Faculty

Coulomb glassComputer simulations

Jacek Matulewski, Sergei Baranovski, Peter Thomas

Departament of PhysicsPhillips-Universitat Marburg, Germany

Faculty of Physics, Astronomy and InformaticsNicolaus Copernicus University in Toruń, Poland

Marburg, V 2007

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

3/80

Realisation of disorder: the impurity band in semiconductor

Conduction band

Impurity band (donors)

E

Valence band

Acceptors

4/80


Conduction band


E

Valence band

Acceptors

5/80


Conduction band


E

Valence band

Acceptors

Occupied donor (electron) q = 0

Empty donor (hole) q = +

Occupied (electron) q =acceptor

_

_ __

+ + +

6/80



Acceptors

Occupied donor (electron) q = 0

Empty donor (hole) q = +

Occupied (electron) q =acceptor

_

_ __

++ +

7/80

Realisation of disorder: the impurity band in semiconductorThe electrostatic potential at every donor site is due to Coulomb interactionwith every acceptor (-) and every other empty site (+) in the system.

N = 10K = 0.5

Occupied donor

Empty donor

Occupied acceptor

Since the sites positions are random - site potential are random too (disorder)

++

+

+

+

_

_

_

_

_

8/80


i ji ij

ji

i

i

r

nn

rn

rH

)1)(1(

2111

21

System of randomly distributed sites with Coulomb interaction:

ij ij

j

ii r

n

rE

11Site potential: isolated sites are identical

Total energy (classical electrostatic interactions):

dimensionless units

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 < nC)

disorder => electronic wavefunctions are localisedthe 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 semiconductorsamorphous semiconductors and alloyshopping behaviour of quasicrystalsgranular filmssilicon MOSFET’s heterostructureselectrically conducting polymers and stannic oxides nanowires

=> disorder isolator

10/80

Outline



a) Searching for the pseudo-ground state (T = 0K). Coulomb gap

b) Monte Carlo simulations (T > 0K)




11/80

Simulation procedure (T = 0K)Metropolis algorithm: the same as used to solve the salesman 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

The calculating procedureisn’t a simulation of therelaxation process.(no transition rates)

12/80

Simulation procedure (T = 0K)

i ji ij

ji

i

i

r

nn

rn

rH

)1)(1(

2111

21


ij ij

j

ii r

n

rE

11Site potential: isolated sites are identical


Single electron transfer:

H

dimensionless units

13/80

ij

beforei

afterj rr

EEH11)()(

Total energy change during single electron transition

A (all acceptors)

Di

Dj

ir jr

ij rrr

rrE

i

beforei

11)( j

beforej r

E1)(

Site energies

+

i

afteri r

E1

)( rr

Ej

afterj

11 )(

+Total energy of the system:

j

before

rH

1)(

i

after

rH

1)(

ij

beforeafter

rrHHH

11)()(

In order to make the calculation possible we need to express the energy difference using sites energy values before the transition

rEE

rrEEH before

ibefore

jij

beforei

afterj

111 )()()()(

_

14/80

Simulation procedure (T = 0K)

i ji ij

ji

i

i

r

nn

rn

rH

)1)(1(

2111

21

ij ij

j

ii r

n

rE

11


Single electron transfer:

Site potential: isolated sites are identical


ij

beforei

beforej

beforei

afterj r

EEEE1)()()()(

Salesman says: transitions for which H < 0 leads to pseudo-ground state

H

hole-electron interaction

15/80

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 (-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)

16/80

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)

01 ij

ijij rEEE

Simulation procedure (T = 0K)Metropolis algorithm for searching the pseudo-ground state of system

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 ...

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)

i ji ij

ji

r

nnH

)1)(1(

21

...

Holes’ escape increase the distance between them and therefore lessen the total energy:

kjk jk

kj E

rn

E ,1

...

Occupied sites are closer to new empty site

iij ij

ji E

r

nE ,

1...

Empty sites are farther from the new hole

19/80

Coulomb gap in density of states for T = 0K

Coulomb gap created due to Coulomb interaction in the system

μEi

Sin

gle-

part

icle

DO

S

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=5001000 real., PBC

20/80

“Dimensionless” units

1Bk

14

3/12

Dne

The temperature is measured in energy units

The length unit

The unit of energy

11

3/1 Dn

14

2

e

Thus => the Coulomb interaction energy readsij

ji

r

nn )1)(1(

For example: nD=69% of nC, nC = 3.52·1018 cm-3 => [d.u.] = 1 <=> T[K] ≈ 200K

nD= 8% of nC, nC = 3.52·1018 cm-3 => [d.u.] = 1 <=> T[K] ≈ 100K

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 ax2 (soft gap)

fitting of (Efros)xE

ae/0

fitting of (BSE) 47

0

0

ln

/

xE

xE

ae

Sin

gle-

part

icle

-DO

S

iEx

22/80

Shape of Coulomb gap for T = 0K S

ingl

e-pa

rtic

le-D

OS

hard gap

numerical simulation result

fitting of (Efros)xE

ae/0

fitting of (BSE) 47

0

0

ln

/

xE

xE

ae

iEx

0

0.04

0.08

0.12

0.16

0 0.1 0.2 0.3 0.4 0.5

N=5001000 real., PBC

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

01 ij

ijij rEEE

kT

Eij

eTp

)(

24/80

Smearing of the Coulomb gap for T > 0K

μEi

Sin

gle-

part

icle

DO

S

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=105

1000 real., PBC

25/80


μEi

Sin

gle-

part

icle

DO

S

T = 0K

22K

44K

66K

88K

222K

0

0.1

0.2

0.3

0.4

-4 -2 0 2 4

N=500, MC=105

1000 real., PBC

nC = 3.52·1018 cm-3

= 11.4

n/nC=100%=20Å=0.3 d.u.

26/80


μEi

Sin

gle-

part

icle

DO

S

T = 0K

10K

20K

30K

40K

100K

0

0.1

0.2

0.3

0.4

-4 -2 0 2 4

N=500, MC=105

1000 real., PBC

nC = 3.52·1018 cm-3

= 11.4

n/nC=8%=20Å=0.13 d.u.

27/80

Pair distribution (T = 0K)

ij

occupiedi

emptyj r

EE1

ijr

N=400, T=0K, a=0.3, MC=103

28/80

Pair distribution (T > 0K)

ijr

ij

occupiedi

emptyj r

EE1

N=400, T=1/8 (28K for n/nC=1), =0.3, MC=103

29/80

Pair distribution (T > 0K)

ijr

ij

occupiedi

emptyj r

EE1

N=400, T=1 (222K for n/nC=1), =0.3, MC=103

30/80

Outline






31/80

The dynamics of the Coulomb gap: the time scales

Conduction band


Acceptors Valence band

Averaged thermal activation time (T = 7K, E=31.27 meV): 104 s

)T/exp(10 BkE

s121

00 10 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?

E

32/80


Conduction band


)ln(2 00 tvR

ij

B

ijij r

k

Er 2exp

T

2exp 1

01

0

Miller-Abrahams transfer rate for VRH:

E Averaged thermal activation time (T = 7K, E=31.27 meV): 104 s

Question 2: How long does it takes to transfer electron from donor i to empty donor j?

ijr

33/80


Conduction band


)ln(2 00 tvR

E Averaged thermal activation time (T = 7K, E=31.27 meV): 104 s

Conclusion:For Si:P (n = 69% of nC) during 103s electron travel only 0.03A

ijr

< size of atom

One need to decrease the n/nC and/or wait very longThe Coulomb glass is an isolator (n/nC < 1)

34/80

The dynamics of the Coulomb gap: the gap evolution

Sin

gle-

part

icle

DO

S

0

0.1

0.2

0.3

0.4

-4 -2 0 2 4

R0 = 0.1

1.0

1.2

1.4

2.0

5.0

μEi

T = 0K

35/80


Sin

gle-

part

icle

DO

S

0

0.1

0.2

0.3

0.4

5 2.51 0.5

Energy:

0-0.5

-1-2

)ln(2 00 tvR

36/80


Sin

gle-

part

icle

DO

S

0

0.1

0.2

0.3

0.4

5 2.51 0.5

Energy:

0-0.5

-1-2

)ln(2 00 tvR

)( ln/1),0( 00 tttg R0(t0) = 1.25

Fitting: 1.26b

37/80


Sin

gle-

part

icle

DO

S

0

0.1

0.2

0.3

0.4

5 2.51 0.5

Energy:

0

)ln(2 00 tvR

)( ln/1),0( 00 tttg R0(t0) = 1.25

Fitting: 1.26

Yu (SCE+numeric, 1999): b = 1

Malik and Kumar (analytical., 2004): b = 2

38/80

The dynamics of the Coulomb gap: the experiment proposal

Conduction band


)ln(2 00 tvRrij

T = 300K

R0 = 0.1

] )/(exp[)( 0 TTT 1/4

Mott’s formula for DC conductivity(constant DOS near the Fermi level):

Random occupations of sites

T = 7K

n/nC = 8%

39/80


Conduction band


)ln(2 00 tvRrij

R0 = 1.0

Relaxing (1st hour) ...

T = 7K

Random occupation of sites n/nC = 8%

40/80


Conduction band


)ln(2 00 tvRrij

R0 = 1.2

Relaxing (2nd hours) ...

T = 7K

n/nC = 8%

41/80


Conduction band


)ln(2 00 tvRrij

R0 = 1.4

Relaxing (3rd hour) ...

T = 7K

n/nC = 8%

42/80


Conduction band


)ln(2 00 tvRrij

R0 = 5.0

] )/(exp[)( 0 TTT 1/2

SE’s formula for DC conductivity(gap in g(E ) around the Fermi level):

Pseudo-grand state reached

T = 7K

n/nC = 8%

43/80


Conduction band


)ln(2 00 tvRrij

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 104 s (3h).

n/nC = 8%

44/80


Sin

gle-

part

icle

DO

S

0

0.1

0.2

0.3

0.4

-4 -2 0 2 4

R0 = 0.1

1.0

1.2

1.4

2.0

5.0

μEi

T = 0K

45/80


Sin

gle-

part

icle

DO

S

μEi

T = 0.1 d.u.

0

0.1

0.2

0.3

0.4

-4 -2 0 2 4

R0 = 0.1

1.0

1.2

1.4

2.0

5.0

46/80


Sin

gle-

part

icle

DO

S

μEi 0

0.1

0.2

0.3

0.4

-4 -2 0 2 4

T = 0.2 d.u. R0 = 0.1

1.0

1.2

1.4

2.0

5.0

47/80

Outline






48/80

Edwards-Anderson order parameter (EAOP)T = 0K - no transitions in the pseudo-ground state

N = 10K = 0.5

Occupied donor

Empty donor

Occupied acceptor

T = 6K - some transitions (VRH)T = 100K - a lot of transitions (NNH)

T

exp2

exp0

B

ijijijij k

Er

49/80

T = 6K - some transitions (VRH)

timeni

Edwards-Anderson order parameter (EAOP)

T = 0K - no transitions in the pseudo-ground state

timeni

T = 100K - a lot of transitions (NNH)

time

ni

time

time

time

nsrealisatiodonorsCarloMonteinq2

12 Order parameter

(per analogy to spin glass)

q = 1.0

q = 0.8

q = 0.1

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 {ni}

a

51/80

Glass transition


Yu: glass transition in Coulomb glass is the phase transition of second order

52/80

Glass transition



53/80

Glass transition



54/80

Glass transition



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 higherenergy than surrounding andpreserve its occupation unchangedeven for high temperatures

a

The lattice model well describesremote donors’ interaction

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

ord

er p

aram

eter

[d.u.]

our modelDavies, Lee, Rice

0.2

0.4

0.6

0.8

1

0.001 0.01 0.1 1 10 100

57/80

Glass transition

EA

ord

er p

aram

eter

our modelDavies, Lee, Rice (B = 2)

0

0.1

0.2

0.3

0.4

1 10

[d.u.] 3


58/80

Glass transition

EA

ord

er p

aram

eter


0

0.1

0.2

0.3

0.4

100K 1000K

[K] 300K

Si:Pn/nC = 8%


59/80

Glass transition

EA

ord

er p

aram

eter


0

0.1

0.2

0.3

0.4

197K 1965K

[K] 590K

Si:Pn/nC = 69%


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

nsrealisatiodonorsCarloMonteiCarloMonteim nnq

2

0

1212

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

61/80

Goes to zero faster than the EA order parameter.

Modified Edwards-Anderson order parameter

mod

ifie

d E

A o

rder

par

amet

er our model (N = 100)

Davies, Lee, Rice (B = 2)

0

0.2

0.4

0.6

0.8

1

1.2

0.01 0.1 1 10

[d.u.]

our model (N = 500)

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 transitionNeed for the parameter which rapidly goes to zero. What mechanism behind it?

N = 10K = 0.5

Occupied donor

Empty donor

Occupied acceptor

Our idea is totrace the electroninstead of the site’s occupation!

63/80


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

64/80


0

0.2

0.4

0.6

0.8

1

0.001 0.01 0.1 1 10 100

EA order parameterthe new order parameter

EA

and

new

ord

er p

aram

eter

s

[d.u.]

about 1%of electronsgot stacked

65/80

Glass transitions versus Coulomb gap smearing

Sin

gle-

part

icle

DO

S f

or 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

ord

er p

aram

eter

0.0

0.5

1.0

EA order parameter

the new order parameter

[d.u.]

66/80

Glass transitions vs Coulomb gap evolution

Sin

gle-

part

icle

DO

S f

or 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

ord

er p

aram

eter

0.0

0.5

1.0

EA order parameter

the new order parameter

electrons leavethe initial sites

almost noneelectron rests

completerandomness

gap starts to form

gap is formed

67/80

The gap transition with time limitation

EA

ord

er p

aram

eter

s

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 = 1.0No limit

[d.u.]

The gap starts to form for R > 1.2, while the order is established for R < 1.5

68/80


new

ord

er p

aram

eter

s

[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 = 1.0No limit

69/80

0.05

0.1

0.15

0.01 0.1 1 10 100


0.6

0.8

1

T = 0.1T = 0.2

0

0.5

1

EA

ord

er p

ar.

new

ord

er p

ar.

T = 0.1T = 0.2

R [d.u.]

T = 0.1T = 0.2

From random

From ground

From ground

70/80

Outline






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

71/80

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)

72/80

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)

73/80

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

74/80

0

0

2

1

2

1

m

m

WEI

IWE


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

75/80

Energy which pair much absorb or emit to move the electron between split-states(from to ):


2

2

1212 4

1I

rEEWWW

Source of energy: photons

20

2)(

iiQ

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 el. transition

Q = QM transition prob.(Fermi Golden Rule)

prob. of finding“proper” pair· · prob. of finding photon

with energy equals to · )(4

2 2

re

76/80

Pair distribution (T = 0K)

ij

occupiedi

emptyj r

EE1

ijr

N=400, T=0K, NMonte-Carlo=1000, a=0.27

77/80

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

In contradiction to the Mott’s assumption the distribution of pairs’ distances is very wide

N=1000, K=0.5, 2500 realisationsperiodic boundary conditions, AOER

02ln

Iar

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.N

umbe

r of

pai

rs

79/80

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!

80/80

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

simulationsHelgren 69% Si:P

crossover (?)

a = 0.36

81/80

The End

http://www.phys.uni.torun.pl/~jaceke-mail: [email protected]

Documents

Coulomb glass Computer simulations Jacek Matulewski, Sergei Baranovski, Peter Thomas Departament of Physics Phillips-Universitat Marburg, Germany Faculty