Yoseph Imry - Physics@Technion

Slow relaxations and aging in
electron glasses
Yoseph Imry, work with
Yuval Oreg and Ariel Amir

Outline
• What are glasses?
• Slow relaxations and aging
• Electron glass model
• statics (Coulomb gap)
• conductance (Variable Range Hopping)
Dynamics:
• mapping to a new class of Random Matrix Theory (RMT)
• Solution of this RMT,
distribution
• Implications of
distribution on aging.
• Connection to 1/f noise
• Conclusions

Common features of glassy models
• Quenched disorder.
• Rugged energy landscape:
Many states close to ground state.
• Aging and memory effects:
relaxation depends on system ‘age’
Judean desert
General questions: structure of states, out-of-equilibrium dynamics?

Common features of glassy models
• Quenched disorder.
• Rugged energy landscape:
Many states close to ground state.
• Aging and memory effects:
relaxation depends on system ‘age’
Physical examples
Judean desert
• Structural glass (window glass)
• Various magnetic materials.
• Packing of hard spheres.
• Electron glass!
(long-ranged interactions, but not infinite!)
General questions: structure of states, out-of-equilibrium dynamics?

Common features of glassy models
• Quenched disorder.
• Rugged energy landscape:
Many states close to ground state.
• Aging and memory effects:
relaxation depends on system ‘age’
Physical examples
Judean desert
• Structural glass (window glass)
• Various magnetic materials.
• Packing of hard spheres.
• Electron glass!
(long-ranged interactions, but not infinite!)
General questions: structure of states, out-of-equilibrium dynamics?

Slow relaxations in nature
string
W. Weber, Ann. Phys. (1835)
D. S. Thompson, J. Exp. Bot. (2001)
mass

Slow relaxations in nature
string
W. Weber, Ann. Phys. (1835)
D. S. Thompson, J. Exp. Bot. (2001)
mass
Electron glass- Experimental system
Logarithmic relaxations for 5 days!

Slow relaxations in nature
string
W. Weber, Ann. Phys. (1835)
D. S. Thompson, J. Exp. Bot. (2001)
mass
Electron glass- Experimental system
Logarithmic relaxations for 5 days!
What are the ingredients leading
to slow relaxations?

An Aging Protocol (Ludwig, Osheroff)
5 V/µm
E
t w
E
15 µm thick Mylar (slide)
(amorphous polyester)
Or BK7-window glass
t=0 t=hours
t

An Aging Protocol (Ludwig, Osheroff)
• Step I: Cool the system and let it relax for a long time (days)
5 V/µm
E
t w
E
15 µm thick Mylar (slide)
(amorphous polyester)
Or BK7-window glass
t=0 t=hours
t

An Aging Protocol (Ludwig, Osheroff)
• Step I: Cool the system and let it relax for a long time (days)
• Step II: Switch on “the perturbation”
5 V/µm
E
t w
E
15 µm thick Mylar (slide)
(amorphous polyester)
Or BK7-window glass
t=0 t=hours
t

An Aging Protocol (Ludwig, Osheroff)
• Step I: Cool the system and let it relax for a long time (days)
• Step II: Switch on “the perturbation”
• Step III: After aging time t w – Switch the perturbation off
5 V/µm
E
t w
E
15 µm thick Mylar (slide)
(amorphous polyester)
Or BK7-window glass
t=0 t=hours
t

An Aging Protocol (Ludwig, Osheroff)
• Step I: Cool the system and let it relax for a long time (days)
• Step II: Switch on “the perturbation”
• Step III: After aging time t w – Switch the perturbation off
• Throughout the experiment a physical property (dielectric constant) is measured
as a function of time
5 V/µm
E
t w
E
15 µm thick Mylar (slide)
(amorphous polyester)
Or BK7-window glass
t=0 t=hours
t

Aging and universality
Amir, Oreg and Imry, to be published

The model
• Strong localization due to disorder
randomly positioned sites, on-site disorder.
• Coulomb interactions are included
• “Phonons” induce transitions between configurations.
• Interference (quantum) effects neglected.
e.g:
Pollak (1970)
Shklovskii and Efros (1975)
Ovadyahu and Pollak (2003)
Muller and Ioffe (2004)

The model
• Strong localization due to disorder
randomly positioned sites, on-site disorder.
• Coulomb interactions are included
• “Phonons” induce transitions between configurations.
• Interference (quantum) effects neglected.
e.g:
Pollak (1970)
Shklovskii and Efros (1975)
Ovadyahu and Pollak (2003)
Muller and Ioffe (2004)

“Local mean-field” approximation - Dynamics

“Local mean-field” approximation - Dynamics

“Local mean-field” approximation - Dynamics

“Local mean-field” approximation - Dynamics
• includes the interactions
• N is the Bose-Einstein distribution
• ξ - the localization length

“Local mean-field” approximation - Dynamics
• includes the interactions
• N is the Bose-Einstein distribution
• ξ - the localization length
At long times:
• The system reaches a locally stable point (metastable state).
• Many metastable states
(“Pseudo-ground-states”, Baranovski et al., J. Phys. C, 1979)

“Local mean-field” approximation - Equilibrium
• Detailed balance leads to Fermi-Dirac statistics (n j f j ).
• Self-consistent set of equations for the energies:
(assuming half filling).

“Local mean-field” approximation - Equilibrium
• Detailed balance leads to Fermi-Dirac statistics (n j f j ).
• Self-consistent set of equations for the energies:
(assuming half filling).
Yields linear Coulomb gap
(+ temperature dependence)
M. Pollak, Discuss. Faraday Soc (1970)
A.L. Efros and B.I. Shklovskii,
J. Phys. C (1975)
A. L. Efros, J. Phys. C (1976).
M. Grunewald et al., J. Phys. C. (1982)
AA et al., PRB (2008)
Surer et al., PRL (2009)
Goethe et al., PRL (2009)

“Local mean-field” approximation - Equilibrium
• Detailed balance leads to Fermi-Dirac statistics (n j f j ).
• Self-consistent set of equations for the energies:
(assuming half filling).
Yields linear Coulomb gap
(+ temperature dependence)
M. Pollak, Discuss. Faraday Soc (1970)
A.L. Efros and B.I. Shklovskii,
J. Phys. C (1975)
A. L. Efros, J. Phys. C (1976).
M. Grunewald et al., J. Phys. C. (1982)
AA et al., PRB (2008)
Surer et al., PRL (2009)
Goethe et al., PRL (2009)

“Local mean-field” approximation - Equilibrium
• Detailed balance leads to Fermi-Dirac statistics (n j f j ).
• Self-consistent set of equations for the energies:
(assuming half filling).
AA et al., PRB (2009)
Yields linear Coulomb gap
(+ temperature dependence)
M. Pollak, Discuss. Faraday Soc (1970)
A.L. Efros and B.I. Shklovskii,
J. Phys. C (1975)
A. L. Efros, J. Phys. C (1976).
M. Grunewald et al., J. Phys. C. (1982)
AA et al., PRB (2008)
Surer et al., PRL (2009)
Goethe et al., PRL (2009)

“Local mean-field” approximation – Steady State
Miller-Abrahams resistance network (no interactions)
Equilibrium rates,
obeying detailed balance
A. Miller and E. Abrahams, (Phys. Rev. 1960)
Generalization
1) Find n i and E i such that the system is in steady
state.
2) Construct resistance network.

“Local mean-field” approximation – Steady State
With interactions (ES)
Without interactions (Mott, VRH)
GeAs samples Shlimak, Kaveh, Yosefin,
Lea and Fozooni (1992)
Ovadyahu (2003)

“Local mean-field” approximation – Steady State
With interactions (ES)
Without interactions (Mott, VRH)
GeAs samples Shlimak, Kaveh, Yosefin,
Lea and Fozooni (1992)
Ovadyahu (2003)

“Local mean-field” approximation – Steady State
With interactions (ES)
Without interactions (Mott, VRH)
GeAs samples Shlimak, Kaveh, Yosefin,
Lea and Fozooni (1992)
Ovadyahu (2003)

VRH (Mott) to E-S Crossover
Scaling is obeyed at crossover,
Slope is ~0.49 0.02
Slope is ~0.34 0.01
Amir, Oreg and Imry , PRB (2009)

“Local mean-field” approximation - Dynamics

“Local mean-field” approximation - Dynamics
We saw: approach works well for statics & steady-states!
Moving on to dynamics…

Solution near locally stable point
Close enough to the equilibrium (locally) stable point,
one can linearize the equations, leading to the equation:

Solution near locally stable point
Close enough to the equilibrium (locally) stable point,
one can linearize the equations, leading to the equation:
(Anderson Localization)
Sum of columns vanishes (particle conservation number)

Solution near locally stable point
Close enough to the equilibrium (locally) stable point,
one can linearize the equations, leading to the equation:
(Anderson Localization)
Sum of columns vanishes (particle conservation number)
For low temperatures, near a local minimum,
second term is negligible
All eigenvalues are real and negative

Eigenvalue Distribution
Solving numerically shows a distribution proportional to :
Eigenvalue distribution
Eigenvalues

Changing the interaction strength
• Slow relaxations occur also without interactions.
• Interactions push the distribution to (s)lower values.

Outline
• What are glasses?
• Slow relaxations and aging
• Electron glass model
• statics (Coulomb gap)
• conductance (Variable Range Hopping)
Dynamics:
• mapping to a new class of Random Matrix Theory (RMT)
• Solution of RMT
• Implications of
distribution
distribution on aging.
• Connection to 1/f noise
• Conclusions

Digression: Random Matrix Theory in a nutshell
• Define a matrix with A ij independent, Gaussian variables.
• Distribution of eigenvalues follows the “Wigner semicircle law”:

Digression: Random Matrix Theory in a nutshell
• Define a matrix with A ij independent, Gaussian variables.
• Distribution of eigenvalues follows the “Wigner semicircle law”:
• Why is this interesting? Wide applications include:
Nuclear physics Wigner and Dirac (1951) Atomic physics Camarda and
Georgopulos PRL (1983) Mesoscopics Imry, EPL (1986), Beenakker, PRL
(1993) Chaotic systems Bohigas, Giannoni and Schmit, PRL (1984) …And
many more.

Digression: Random Matrix Theory in a nutshell
• Define a matrix with A ij independent, Gaussian variables.
• Distribution of eigenvalues follows the “Wigner semicircle law”:
• Why is this interesting? Wide applications include:
Nuclear physics Wigner and Dirac (1951) Atomic physics Camarda and
Georgopulos PRL (1983) Mesoscopics Imry, EPL (1986), Beenakker, PRL
(1993) Chaotic systems Bohigas, Giannoni and Schmit, PRL (1984) …And
many more.
In the following we present a different class of RMT, also broadly applicable

Digression: What are Random Distance Matrices?
1) Choose N points randomly and uniformly in a d-dimensional cube.
Bogomolny, Bohigas, and Schmit, J. Phys. A: Math. Gen. (2003).
Mezard, Parisi and Zee, Nucl. Phys. (1999)

Digression: What are Random Distance Matrices?
1) Choose N points randomly and uniformly in a d-dimensional cube.
2) Define the off-diagonals of our matrix as:
(Euclidean distance)
r ij
Bogomolny, Bohigas, and Schmit, J. Phys. A: Math. Gen. (2003).
Mezard, Parisi and Zee, Nucl. Phys. (1999)

Digression: What are Random Distance Matrices?
1) Choose N points randomly and uniformly in a d-dimensional cube.
2) Define the off-diagonals of our matrix as:
(Euclidean distance)
3) Define diagonal as:
r ij
sum of every column vanishes
(will come from a conservation law)
Bogomolny, Bohigas, and Schmit, J. Phys. A: Math. Gen. (2003).
Mezard, Parisi and Zee, Nucl. Phys. (1999)

Digression: What are Random Distance Matrices?
1) Choose N points randomly and uniformly in a d-dimensional cube.
2) Define the off-diagonals of our matrix as:
(Euclidean distance)
3) Define diagonal as:
r ij
sum of every column vanishes
(will come from a conservation law)
Q: What is the eigenvalue distribution?
What are the eigenmodes?

Relaxation in electron glasses
Amir, Oreg and Imry, PRB 2008
Distance matrices – Motivation
Anomalous diffusion
Scher and Montroll, PRB 1975
Metzler, Barkai and Klafter, PRL 1999
Localization of phonons
Ziman, PRL 1982
Photon propagation in a gas
Akkermans, Gero and Kaiser, PRL 2008

Results – 2D

Results

Results
(no fitting parameters)

Results
(no fitting parameters)

Exponential Distance Matrices- results
(arbitrary dimension d)
C d =volume of a d-dimensional sphere
• Logarithmic corrections to
• In dimensions > 1: cutoff at
Amir, Oreg and Imry, arxiv: 1002.2123

Analytical approach in a nutshell
Part I – Moment calculation
The k’th moment:
leads to the distribution function.

Renormalization group approach

Renormalization group approach
Mechanical intuition: network of masses and springs

Renormalization group approach

Renormalization group approach

Renormalization group approach
RG procedure yields growing clusters

Renormalization group approach
3’rd 1000
3000 4000
Examples of eigenmodes of a 5000X5000 matrix

Renormalization group approach
Eigenvalue distribution
+
RG approach
Number of points in a cluster of a given eigenvalue

Renormalization group approach
Eigenvalue distribution
+
RG approach
Number of points in a cluster of a given eigenvalue
• Eigenmodes are localized clusters (“phonon localization”)
• Size of clusters diverges at low frequencies
Amir, Oreg and Imry, Localization, anomalous diffusion and slow relaxations:
a random distance matrix approach, arxiv: 1002.2123

Intermediate Summary
• Statics: Coulomb gap
• Steady-state: variable range hopping, crossover between two regimes
• Dynamics: Microscopic model reduces to RMT problem of a different class.
• Eigenvalue distribution (relaxation times) follow approximately
distribution.
• Same RMT class relevant for various other problems.

Electron glass aging– experimental protocol

Electron glass aging– experimental protocol
Step I
System equilibrates for long time

Electron glass aging– experimental protocol
Step I
System equilibrates for long time
Step II
V g is changed, for a time of t w .

Electron glass aging– experimental protocol
Step I
System equilibrates for long time
Step II
V g is changed, for a time of t w .
Experiment
Theory
Throughout the experiment
Conductance is measured as a
function of time.
Data: Ovadyahu et al.

“Full” aging
• For various physical systems:

“Full” aging
• For various physical systems:
Electron glass
Vaknin et al., PRL (2000)

“Full” aging
• For various physical systems:
Other experiments showing full aging:
•Spin glass
Rodriguez et al., PRL (2003)
• Vortices in superconductors
Electron glass
Vaknin et al., PRL (2000)
Du et al., Nature Physics (2007)
•Granular metals
N. Kurzweil et al., PRB (2007)

Aging – physical picture
Assume a parameter of the system is slightly modified (e.g: V g )
After time t w it is changed back. What is the repsonse?
Initially, system is at some local minimum

Aging – physical picture
Assume a parameter of the system is slightly modified (e.g: V g )
After time t w it is changed back. What is the repsonse?
At time t=0 the potential changes,
and the system begins to roll towards the new minimum

Aging – physical picture
Assume a parameter of the system is slightly modified (e.g: V g )
After time t w it is changed back. What is the repsonse?
At time t w the system reached some new configuration

Aging – physical picture
Assume a parameter of the system is slightly modified (e.g: V g )
After time t w it is changed back. What is the repsonse?
Now the potential is changed back to the initial formthe
particle is not in the minimum!
The longer t w , the further it got away from it.
It will begin to roll down the hill.

Aging – Analysis
Assume a parameter of the system is slightly modified (e.g: V g )
After time t w it is changed back. What is the response?
t=t w
a
b
t=0

Aging – Analysis
Assume a parameter of the system is slightly modified (e.g: V g )
After time t w it is changed back. What is the response?
t=t w
a
Sketch of calculation
If a and b configurations are close enough in phase space:
b
t=0

Aging – Analysis
Assume a parameter of the system is slightly modified (e.g: V g )
After time t w it is changed back. What is the response?
t=t w
a
Sketch of calculation
If a and b configurations are close enough in phase space:
b
t=0

Aging – Analysis
Assume a parameter of the system is slightly modified (e.g: V g )
After time t w it is changed back. What is the response?
t=t w
a
Sketch of calculation
If a and b configurations are close enough in phase space:
b
t=0

Aging – Analysis
Assume a parameter of the system is slightly modified (e.g: V g )
After time t w it is changed back. What is the response?
t=t w
a
Sketch of calculation
If a and b configurations are close enough in phase space:
b
t=0

Aging – Analysis
Assume a parameter of the system is slightly modified (e.g: V g )
After time t w it is changed back. What is the response?
t=t w
a
Sketch of calculation
If a and b configurations are close enough in phase space:
b
t=0
Logarithmic relaxation during step II

Aging – Analysis
Assume a parameter of the system is slightly modified (e.g: V g )
After time t w it is changed back. What is the response?
t=t w
a
Sketch of calculation
If a and b configurations are close enough in phase space:
b
t=0
Time t after the perturbation is switched off:
Logarithmic relaxation during step II

Aging – Analysis
Assume a parameter of the system is slightly modified (e.g: V g )
After time t w it is changed back. What is the response?
t=t w
a
Sketch of calculation
If a and b configurations are close enough in phase space:
b
t=0
Time t after the perturbation is switched off:
Logarithmic relaxation during step II
Full aging
Only
distribution yields full aging!
See also: T. Grenet et al. Eur. Phys. J B 56, 183 (2007)

Aging Protocol - Results
Amir, Oreg and Imry, PRL 2009

Connection to conductance relaxation?
If we assume
We obtain full aging:
Assuming a linear dependence leads to:
For short times:
For long times:

Detailed fit to experimental data
• Full aging
• Deviations from logarithm start at
Amir, Oreg and Imry, PRL (2009)

Full aging and universality
Amir, Oreg and Imry, to be published

Other protocols…?
Periodic voltage
Theory vs. Experiment
(Exp.: Ovadyahu et al., PRB 2003)

Other protocols…?
Periodic voltage
Theory vs. Experiment
(Exp.: Ovadyahu et al., PRB 2003)

Connection to 1/f noise?
Amir, Oreg, and Imry
arXiv:0911.5060, Ann. Phys. 2009
Langevin Noise
Equipartition theorem: each eigenmode should get
The mean-field equations can be derived from a free energy:
From this we can find the noise correlations matrix:
The
spectrum then leads to a 1/f noise spectrum:
B.I. Shklovskiı,
Solid State Commun (1980)
K. Shtengel et al.,
PRB (2003)

Conclusions
• Statics: Coulomb gap, Steady-state (conductance): Variable Range Hopping
• Dynamics near locally stable point: many slow modes,
distribution.
How universal? We believe: a very relevant RMT class.
• One obtains full aging, with relaxation approximately of the form :
More details:
Phys. Rev. B 77, 1, 2008 (mean-field model)
Phys. Rev. Lett. 103, 126403 (2009) (aging properties)
Phys. Rev. B 80, 245214 2009 (variable-range hopping)
Ann. Phys. 18, 12, 836 (2009) (1/f noise)
arxiv: 1002.2123 (exponential matrices – solution)