IMPACT ionization and thermalization in photo-doped Mott insulators

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IMPACT ionization and thermalization in photo-doped Mott insulators Philipp Werner (Fribourg) in collaboration with Martin Eckstein (Hamburg) Karsten Held (Vienna) Cargese, September 16

Motivation Photo-doping: nonequilibrium phase transition from a correlation induced insulator to a non-thermal conducting state S. Iwai et al. (3), H. Okamoto et al. (7),... Thermalization of large-gap insulators Impact ionization in small-gap insulators Cooling by magnon scattering U Mott insulating solar cells

Model and method Hubbard model: simplified model for a correlated electron material Gutzwiller, Kanamori, Hubbard (1963) t U Sign problem / exponential scaling: lattice model not solvable use approximate description

Model and method Dynamical mean field theory DMFT: mapping to an impurity problem Metzner & Vollhardt (1989); Georges & Kotliar (199) lattice model impurity model t G latt G imp t k latt imp Formalism can be extended to nonequilibrium systems Schmidt & Monien (); Freericks et al. (6) Impurity solver: computes the dynamics on the correlated site Strong-coupling perturbation theory: Eckstein & Werner (9)

Model and method Equilibrium DMFT phase diagram (half-filling) Paramagnetic calculation: Metal - Mott insulator transition at low T Smooth crossover at high T T metal Mott insulator U

Pulse excited Mott insulator Photo-excitation of carriers across the Mott gap Eckstein & Werner (11) Question: How quickly does the electronic system thermalize? T metal Mott insulator U

Pulse excited Mott insulator Photo-excitation of carriers across the Mott gap Eckstein & Werner (11) Question: How quickly does the electronic system thermalize? pulse form total energy T e 4.1 E(t) 3 1-1 - -3-4 4 6 gap x gap 8 1 1 14 energy(t) -.1 -. -.3 -.4 -.5 -.6 4 6 8 1 1 14 t t

Pulse excited Mott insulator Photo-excitation of carriers across the Mott gap Eckstein & Werner (11) Question: How quickly does the electronic system thermalize?.18.17.16 thermal value.15 d(t).14.13.1.11.1 4 gap x gap 6 8 1 1 14 t

Pulse excited Mott insulator Photo-excitation of carriers across the Mott gap Eckstein & Werner (11) Question: How quickly does the electronic system thermalize? -1 log 1 d(t)-d(t eff ) - -3-4 U=5 U=3 U=.5 T -5 U=1.5 U= 5 1 15 t U

Pulse excited Mott insulator Photo-excitation of carriers across the Mott gap Eckstein & Werner (11) Strong correlation regime: Relaxation time depends exponentially on U -1 3 log 1 d(t)-d(t eff ) - -3-4 U=5 U=3 U=.5 log 1 relax 1-5 U=1.5 U= 5 1 15 t 3 4 5 U

Small-gap Mott insulator Pulse energy dependence of the relaxation rate Werner, Held & Eckstein (14) thermal value extrapolated value A( ).3.5..15.1 U= U=.5 U=3 U=3.5 U=4 U=4.5 D(t)/D(t=1) 1 U=.5.5-6 -4-4 6 =3 / =.5 / = / =1.5 / 1 3 4 5 6 t

Small-gap Mott insulator Pulse energy dependence of the relaxation rate Werner, Held & Eckstein (14) thermal value extrapolated value A( ).3.5..15.1 U= U=.5 U=3 U=3.5 U=4 U=4.5 D(t)/D(t=1) 1 U=3.5.5-6 -4-4 6 =3.5 / =3 / =.5 / = / 1 3 4 5 6 t Evidence for fast and slow relaxation time

Small-gap Mott insulator Pulse energy dependence of the relaxation rate Werner, Held & Eckstein (14) thermal value extrapolated value 5 U=4 U=3.5 U=3 U=.5 U= relaxation time 15 1 D(t)/D(t=1) 1 U=3.5 5.5 3 3.5 4 /( /) =3.5 / =3 / =.5 / = / 1 3 4 5 6 t Evidence for fast and slow relaxation time

Small-gap Mott insulator Impact ionization Werner, Held & Eckstein (14) Fast doublon-hole production by the scattering process doublon high! doublon low + doublon low + hole low

Small-gap Mott insulator Impact ionization Werner, Held & Eckstein (14) Fast doublon-hole production by the scattering process doublon high! doublon low + doublon low + hole low hole high! hole low + doublon low + hole low

Small-gap Mott insulator Impact ionization Werner, Held & Eckstein (14) Fast doublon-hole production by the scattering process doublon high! doublon low + doublon low + hole low hole high! hole low + doublon low + hole low Consider only upper Hubbard band: doublon high! 3 doublon low

Small-gap Mott insulator Impact ionization Werner, Held & Eckstein (14) Fast doublon-hole production by the scattering process doublon high! doublon low + doublon low + hole low hole high! hole low + doublon low + hole low Consider only upper Hubbard band: doublon high! 3 doublon low fast time-scale associated with these processes? Slow time scale related to multi-particle scattering processes

Small-gap Mott insulator Impact ionization Werner, Held & Eckstein (14) Time evolution of the spectral function.1 U=3.5 t=18 t=4 t=3 t=36 =4 / t=4 I(, t) -4-3 - -1 1 3 4

Small-gap Mott insulator Impact ionization Werner, Held & Eckstein (14) Time evolution of the spectral function.1 t=3 t=36 t=4 I(, t)-i(, t=4) gain in low-energy weight =.3 x loss in high-energy weight.5 1 1.5.5 3 3.5 4 4.5

Simple model Impact ionization Werner, Held & Eckstein (14) High (low) energy population dd1 dt dd dt d dt D imp imp D 1 (D ) [D = D 1 + D ] = therm = 1 1 D1 dd1 = 3 dt imp D th D Two exponential relaxations D th D(t) = + D 1 (t s )e t ts + D th D(t s ) + D 1 (t s ) e t t s Obtain,, D 1 (t s ) by fitting

Simple model Impact ionization Werner, Held & Eckstein (14) High (low) energy population D 1 (D ) [D = D 1 + D ] D 1 (t s ) D(t s ) U 3.5.88 7. 18.8.5.5.67 7.75 19..5.44 9.35 19.6 3.5 3.46 13.4 6.3 3 3.4 15. 61.4.5 3.6 16.5 64.9 3.5 3.5.15 44. 376 3 3.5.83 48.4 57 4 (4.19 86.9 599 ) initial high-energy populations impact ionization thermalization very small D1: single-exponential relaxation

Simple model Impact ionization Werner, Held & Eckstein (14) High (low) energy population D 1 (D ) [D = D 1 + D ] D 1 (t s ) D(t s ) U 3.5.88 7. 18.8.5.5.67 7.75 19..5.44 9.35 19.6 3.5 3.46 13.4 6.3 3 3.4 15. 61.4.5 3.6 16.5 64.9 3.5 3.5.15 44. 376 3 3.5.83 48.4 57 4 (4.19 86.9 599 ) initial high-energy populations impact ionization thermalization

Simple model Impact ionization Werner, Held & Eckstein (14) Two-step relaxation predicted by the model. thermal normalized doublon population 1.5 1..5 small high energy population contributes significantly to doublon production U=3.5, =3.5 / DMFT data D 1 (t-t s )/D(t s ) D (t-t s )/D(t s ) D(t-t s )/D(t s ) D th /D(t s ) 1 3 4 5 t-t s

Simple model Impact ionization Werner, Held & Eckstein (14) Fluence dependence: impact ionization timescale shows little fluence dependence thermalization timescale shows larger fluence dependence amplitude D(t s ) D th th.5.18.36 4.884 19.9 14.5.49.917 4.593 19.5 194 1.167.334 3.879 18.3 147.593.15.854 15.6 85. 6.165.5 1.996 11. 46.1 amplitude > : doublon-conserving scattering processes start to deplete the high-energy population

Competing effects Impact ionization Werner, Held & Eckstein (14) Fluence dependence: increasing role of doublon-conserving scattering processes doublon high + doublon low! doublon intermediate 1 amplitude=5 amplitude= amplitude=.5 1 amplitude=5 amplitude= amplitude=.5 I(, t=36)-i(, t=4) [a. u.].5 -.5 U=3.5 I(, t=36)-i(, t=4) [a. u.].5 -.5 U=4-1 1 3 4 5-1 1 3 4 5

Competing effects Impact ionization Werner, Held & Eckstein (14) Scattering with external degrees of freedom (phonons, spins...): (dd 1 /dt) imp+ph/mag = ( 1/ 1/apple)D 1 (dd /dt) imp+ph/mag = (3/ +1/apple)D 1 Reduction in high-energy population decreases effect of impact ionization How effective is the cooling of photo-doped carriers by scattering with phonons/spins?

Cooling of carriers Effect of short-ranged antiferromagnetic correlations Eckstein & Werner (14) 4-site cluster calculations for D Hubbard give cooling rate =3 S NN NN spin correlations

Mobility of photo-doped carriers Mott insulating solar cells LaVO 3 on top of SrTiO 3 has suitable gap size Strong internal fields (carrier separation) Assmann, Held, Sangiovanni... (13) Strong correlations (impact ionization)

Mobility of photo-doped carriers Mott insulating solar cells LaVO 3 on top of SrTiO 3 has suitable gap size Nonequilibrium DMFT simulations show Assmann, Held, Sangiovanni... (13) Localization by strong internal fields a A(,z) [offset].7.6.5.4.3 z=6 z=5 z=4 b c 1.4 1.996 1.8 density ( E: magnetic for each hopping: energy gain Ea, but kinetic energy is bounded..1 E F z=3 z= z=1 - -1 1.6.4. T=1/8 1/5, 1 1 3

Mobility of photo-doped carriers Mott insulating solar cells LaVO 3 on top of SrTiO 3 has suitable gap size Nonequilibrium DMFT simulations show Assmann, Held, Sangiovanni... (13) Localization by strong internal fields Efficient separation of carriers in the presence of AFM order =3 vdrift a A(,z) [offset].7.6.5.4.3 z=6 z=5 z=4 b c 1.4 1.996 1.8 density ( E: magnetic..1 E F z=3 z= z=1 - -1 1.6.4. T=1/8 1/5, 1 1 3

Summary 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 1/ m Drift velocity in polar heterostructures limited by scattering with spins v drift m Contribution of impact ionization to the efficiency of Mott solar cells requires more detailed analysis

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