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Philipp Werner (Fribourg)
in collaboration with
Martin Eckstein (Hamburg)Karsten Held (Vienna)
Cargese, September 2016
IMPACT ionization and thermalization in photo-doped Mott insulators
Photo-doping: nonequilibrium phase transition from a correlation induced insulator to a non-thermal conducting state
Thermalization of large-gap insulators
Impact ionization in small-gap insulators
Cooling by magnon scattering
Mott insulating solar cells
Motivation
S. Iwai et al. (2003), H. Okamoto et al. (2007), ...
U
Hubbard model: simplified model for a correlated electron material
Sign problem / exponential scaling: lattice model not solvable use approximate description
Model and method
Gutzwiller, Kanamori, Hubbard (1963)
Ut
Model and method
Dynamical mean field theory DMFT: mapping to an impurity problem
Formalism can be extended to nonequilibrium systems
Impurity solver: computes the dynamics on the correlated site
t
�latt � �imp
Glatt � Gimp
Schmidt & Monien (2002); Freericks et al. (2006)
Metzner & Vollhardt (1989); Georges & Kotliar (1992)
kt
Strong-coupling perturbation theory: Eckstein & Werner (2009)
lattice model impurity model
Equilibrium DMFT phase diagram (half-filling)
Paramagnetic calculation: Metal - Mott insulator transition at low T
Smooth crossover at high T
Model and method
“Mott” insulatormetal
U
T
“Photo-excitation” of carriers across the Mott gap
Question: How quickly does the electronic system thermalize?
Pulse excited Mott insulator
Eckstein & Werner (2011)
metal
U
T
“Mott” insulator
“Photo-excitation” of carriers across the Mott gap
Question: How quickly does the electronic system thermalize?
Pulse excited Mott insulator
Eckstein & Werner (2011)
-4-3-2-1 0 1 2 3 4
14121086420
E(t)
t
1 5 gap1 5 2 x gap
-0.06
-0.05
-0.04
-0.03
-0.02
-0.01
0
0.01
14121086420
ener
gy(t)
t
pulse form total energy Te↵
“Photo-excitation” of carriers across the Mott gap
Question: How quickly does the electronic system thermalize?
Pulse excited Mott insulator
Eckstein & Werner (2011)
thermal value
0.01 0.011 0.012 0.013 0.014 0.015 0.016 0.017 0.018
14121086420
d(t)
t
1 5 gap1 5 2 x gap
“Photo-excitation” of carriers across the Mott gap
Question: How quickly does the electronic system thermalize?
Pulse excited Mott insulator
Eckstein & Werner (2011)
U=5
-5
-4
-3
-2
-1
0 5 10 15 20
log 1
0 |d
(t)-d
(Tef
f)|
t
U=3
U=2.5
U=2 U=1.5
T
U
“Photo-excitation” of carriers across the Mott gap
Strong correlation regime: Relaxation time depends exponentially on U
Pulse excited Mott insulator
Eckstein & Werner (2011)
U=5
-5
-4
-3
-2
-1
0 5 10 15 20
log 1
0 |d
(t)-d
(Tef
f)|
t
U=3
U=2.5
U=2 U=1.5 0
1
2
3
2 3 4 5lo
g 10 o r
elax
U
Pulse energy dependence of the relaxation rate
Small-gap Mott insulator
Werner, Held & Eckstein (2014) thermal valueextrapolated value
0
0.05
0.1
0.15
0.2
0.25
0.3
-6 -4 -2 0 2 4 6
A(t
)
t
U=2U=2.5
U=3U=3.5
U=4U=4.5
0
1
2
0 10 20 30 40 50 60
D(t)
/D(t=
12)
t
U=2.5
1=3//21=2.5//21=2//2
1=1.5//2
0
0.05
0.1
0.15
0.2
0.25
0.3
-6 -4 -2 0 2 4 6
A(t
)
t
U=2U=2.5
U=3U=3.5
U=4U=4.5
Pulse energy dependence of the relaxation rate
Evidence for fast and slow relaxation time
Small-gap Mott insulator
Werner, Held & Eckstein (2014)
0
1
2
0 10 20 30 40 50 60
D(t)
/D(t=
12)
t
U=3.5
1=3.5//21=3//2
1=2.5//21=2//2
thermal valueextrapolated value
Pulse energy dependence of the relaxation rate
Evidence for fast and slow relaxation time
Small-gap Mott insulator
Werner, Held & Eckstein (2014)
0
50
100
150
200
250
2 2.5 3 3.5 4
rela
xatio
n tim
e
1/(//2)
U=4U=3.5
U=3U=2.5
U=2
0
1
2
0 10 20 30 40 50 60
D(t)
/D(t=
12)
t
U=3.5
1=3.5//21=3//2
1=2.5//21=2//2
thermal valueextrapolated value
“Impact ionization”
Fast doublon-hole production by the scattering process
Small-gap Mott insulator
Werner, Held & Eckstein (2014)
doublon
high
! doublon
low
+ doublon
low
+ hole
low
“Impact ionization”
Fast doublon-hole production by the scattering process
Small-gap Mott insulator
Werner, Held & Eckstein (2014)
doublon
high
! doublon
low
+ doublon
low
+ hole
low
hole
high
! hole
low
+ doublon
low
+ hole
low
“Impact ionization”
Fast doublon-hole production by the scattering process
Consider only upper Hubbard band:
Small-gap Mott insulator
Werner, Held & Eckstein (2014)
hole
high
! hole
low
+ doublon
low
+ hole
low
doublon
high
! doublon
low
+ doublon
low
+ hole
low
doublon
high
! 3⇥ doublon
low
“Impact ionization”
Fast doublon-hole production by the scattering process
Consider only upper Hubbard band:
fast time-scale associated with these processes?
Slow time scale related to multi-particle scattering processes
Small-gap Mott insulator
Werner, Held & Eckstein (2014)
hole
high
! hole
low
+ doublon
low
+ hole
low
doublon
high
! 3⇥ doublon
low
doublon
high
! doublon
low
+ doublon
low
+ hole
low
“Impact ionization”
Time evolution of the spectral function
Small-gap Mott insulator
Werner, Held & Eckstein (2014)
0
0.010
-4 -3 -2 -1 0 1 2 3 4
I(t, t
)
t
1=4//2
U=3.5 t=18t=24t=30t=36t=42
“Impact ionization”
Time evolution of the spectral function
Small-gap Mott insulator
Werner, Held & Eckstein (2014)
0
0.001
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
I(t, t
)-I(t
, t=2
4)
t
t=30t=36t=42
gain in low-energy weight= 2.3 x loss in high-energyweight
“Impact ionization”
High (low) energy population
Two exponential relaxations
Obtain by fitting
Simple model
Werner, Held & Eckstein (2014)
⇣dD1
dt
⌘
imp= � 1
�D1
⇣dD2
dt
⌘
imp= �3
⇣dD1
dt
⌘
imp⇣ d
dtD2
⌘
therm=
1⌧
⇣Dth �D2
⌘
Dth �D(t) = 2⌧+�⌧�� D1(ts)e�
t�ts� +
⇣Dth �D(ts)� 2⌧+�
⌧�� D1(ts)⌘e�
t�ts⌧
�, ⌧, D1(ts)
D1 (D2) [D = D1 + D2]
“Impact ionization”
High (low) energy population
Simple model
Werner, Held & Eckstein (2014)
D1 (D2) [D = D1 + D2]
impact thermalizationionization
U ⌦ D1(ts)D(ts) � ⌧
2.5 3⇡2 0.0088 7.20 18.8
2.5 2.5⇡2 0.0067 7.75 19.0
2.5 2⇡2 0.0044 9.35 19.6
3 3.5⇡2 0.046 13.4 60.3
3 3⇡2 0.040 15.0 61.4
3 2.5⇡2 0.026 16.5 64.9
3.5 3.5⇡2 0.15 44.0 376
3.5 3⇡2 0.083 48.4 257
( 4 4⇡2 0.19 86.9 5990 )
very small D1:single-exponentialrelaxation
initial high-energypopulations
“Impact ionization”
High (low) energy population
Simple model
Werner, Held & Eckstein (2014)
impact thermalizationionization
U ⌦ D1(ts)D(ts) � ⌧
2.5 3⇡2 0.0088 7.20 18.8
2.5 2.5⇡2 0.0067 7.75 19.0
2.5 2⇡2 0.0044 9.35 19.6
3 3.5⇡2 0.046 13.4 60.3
3 3⇡2 0.040 15.0 61.4
3 2.5⇡2 0.026 16.5 64.9
3.5 3.5⇡2 0.15 44.0 376
3.5 3⇡2 0.083 48.4 257
( 4 4⇡2 0.19 86.9 5990 )
initial high-energypopulations
D1 (D2) [D = D1 + D2]
“Impact ionization”
Two-step relaxation predicted by the model
Simple model
Werner, Held & Eckstein (2014)
2.00
1.50
1.00
0.50
0 0 100 200 300 400 500
norm
aliz
ed d
oubl
on p
opul
atio
n
t-ts
U=3.5, 1=3.5//2
DMFT dataD1(t-ts)/D(ts)D2(t-ts)/D(ts)D(t-ts)/D(ts)
Dth/D(ts)
thermal
small high energy population contributessignificantly to doublonproduction
“Impact ionization”
Fluence dependence: impact ionization timescale shows little fluence dependencethermalization timescale shows larger fluence dependence
Simple model
Werner, Held & Eckstein (2014)
�
⌧
amplitude D(ts) Dth �th � ⌧0.25 0.000108 0.000236 4.884 19.9 2140.5 0.000429 0.000917 4.593 19.5 1941 0.00167 0.00334 3.879 18.3 1472 0.00593 0.0105 2.854 15.6 85.06 0.0165 0.0252 1.996 11.2 46.1
amplitude > 2: doublon-conserving scattering processesstart to deplete the high-energy population
“Impact ionization”
Fluence dependence: increasing role of doublon-conserving scattering processes
Competing effects
Werner, Held & Eckstein (2014)
doublon
high
+ doublon
low
! 2 doublon
intermediate
-1
-0.5
0
0.5
1
0 1 2 3 4 5
I(t, t
=36)
-I(t
, t=2
4) [a
. u.]
t
U=3.5
amplitude=5amplitude=2
amplitude=0.5
-1
-0.5
0
0.5
1
0 1 2 3 4 5
I(t, t
=36)
-I(t
, t=2
4) [a
. u.]
t
U=4
amplitude=5amplitude=2
amplitude=0.5
“Impact ionization”
Scattering with “external” degrees of freedom (phonons, spins ...):
Reduction in high-energy population decreases effect of impact ionization
How effective is the cooling of photo-doped carriers by scattering with phonons/spins?
Competing effects
Werner, Held & Eckstein (2014)
(dD1/dt)imp+ph/mag = (�1/� � 1/)D1
(dD2/dt)imp+ph/mag = (3/� + 1/)D1
Effect of short-ranged antiferromagnetic correlations
4-site cluster calculations for 2D Hubbard give cooling rate
Cooling of carriers
Eckstein & Werner (2014)
NN spin correlations
=3
� S2NN
Mott insulating solar cells on top of has suitable gap sizeStrong internal fields (carrier separation)Strong correlations (impact ionization)
Mobility of photo-doped carriers
Assmann, Held, Sangiovanni ... (2013)
LaVO3 SrTiO3
Mott insulating solar cells on top of has suitable gap sizeNonequilibrium DMFT simulations show
Localization by strong internal fields
Mobility of photo-doped carriers
Assmann, Held, Sangiovanni ... (2013)
LaVO3 SrTiO3
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
-20 -10 0 10 20
A(t
,z) [
offs
et]
t
z=1
z=2
z=3
z=4
z=5
z=6
EF
a
0.996
1
1.004
6E: 0.331.5
density (T=1/8)b
1.02.0
0
0.2
0.4
0.6
0.8
1
1 2 3 4 5 6z
T=1/8
magnetic order (6E=0)c
, 1/61/5 , 1/4 , 1/3
e
d
...
...
...
...
�=-2.5�E,-1.5�E, -0.5�E, 0.5�E, 1.5�E, 2.5�E
...... ...
...
...
...
...
...
...
...
......
...�=0
...
...
...
... ...
...
...
...
...
...
...
...
�=0
t||t
U=20 U=0U=0U=0U=0
E(t)
E(t)
for each hopping: energy gain Ea, but kinetic energy is bounded
Mott insulating solar cells on top of has suitable gap sizeNonequilibrium DMFT simulations show
Localization by strong internal fieldsEfficient separation of carriers in the presence of AFM order
Mobility of photo-doped carriers
Assmann, Held, Sangiovanni ... (2013)
LaVO3 SrTiO3
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
-20 -10 0 10 20
A(t
,z) [
offs
et]
t
z=1
z=2
z=3
z=4
z=5
z=6
EF
a
0.996
1
1.004
6E: 0.331.5
density (T=1/8)b
1.02.0
0
0.2
0.4
0.6
0.8
1
1 2 3 4 5 6z
T=1/8
magnetic order (6E=0)c
, 1/61/5 , 1/4 , 1/3
e
d
...
...
...
...
�=-2.5�E,-1.5�E, -0.5�E, 0.5�E, 1.5�E, 2.5�E
...... ...
...
...
...
...
...
...
...
......
...�=0
...
...
...
... ...
...
...
...
...
...
...
...
�=0
t||t
U=20 U=0U=0U=0U=0
E(t)
E(t)
=3 v d
rift
Relaxation of photo-doped carriers - some insights from DMFT
Exponential scaling of thermalization time with gap size
If gap < width of Hubbard bands: pulse-energy dependent initial relaxation due to impact ionization
Decay of high-energy population due to scattering with spins
Drift velocity in polar heterostructures limited by scattering with spins
Contribution of impact ionization to the efficiency of Mott solar cells requires more detailed analysis
Summary
1/� � m2
vdrift � m2
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