impact ionization and thermalization in photo-doped mott

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





-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↵





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





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


Strong correlation regime: Relaxation time depends exponentially on U



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


Evidence for fast and slow relaxation time


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


Evidence for fast and slow relaxation time



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



doublon

high

! doublon

low

+ doublon

low

+ hole

low





doublon

high

! doublon

low

+ doublon

low

+ hole

low

hole

high

! hole

low

+ doublon

low

+ hole

low



Consider only upper Hubbard band:



hole

high

! hole

low

+ doublon

low

+ hole

low

doublon

high

! doublon

low

+ doublon

low

+ hole

low

doublon

high

! 3⇥ doublon

low



Consider only upper Hubbard band:

fast time-scale associated with these processes?

Slow time scale related to multi-particle scattering processes



hole

high

! hole

low

+ doublon

low

+ hole

low

doublon

high

! 3⇥ doublon

low

doublon

high

! doublon

low

+ doublon

low

+ hole

low


Time evolution of the spectral function



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


Time evolution of the spectral function



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


High (low) energy population

Two exponential relaxations

Obtain by fitting

Simple model


⇣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]



Simple model


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



Simple model


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]


Two-step relaxation predicted by the model

Simple model


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


Fluence dependence: impact ionization timescale shows little fluence dependencethermalization timescale shows larger fluence dependence

Simple model


�

⌧

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


Fluence dependence: increasing role of doublon-conserving scattering processes

Competing effects


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


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


(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


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



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



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

impact ionization and thermalization in photo-doped mott

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