IMPACT ionization and thermalization in photo-doped Mott insulators

Transcription

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

2 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

3 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

4 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)

5 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

6 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

7 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) gap x gap energy(t) t t

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

9 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 ) U=5 U=3 U=.5 T -5 U=1.5 U= t U

10 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 ) U=5 U=3 U=.5 log 1 relax 1-5 U=1.5 U= t U

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

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

13 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 / =3 / =.5 / = / t Evidence for fast and slow relaxation time

14 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

15 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

16 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

17 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

18 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)

19 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

20 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

21 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 ( ) initial high-energy populations impact ionization thermalization very small D1: single-exponential relaxation

22 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 ( ) initial high-energy populations impact ionization thermalization

23 Simple model Impact ionization Werner, Held & Eckstein (14) Two-step relaxation predicted by the model. thermal normalized doublon population 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 ) t-t s

24 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 amplitude > : doublon-conserving scattering processes start to deplete the high-energy population

25 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.] U=3.5 I(, t=36)-i(, t=4) [a. u.] U=

26 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?

27 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

28 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)

29 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] z=6 z=5 z=4 b c density ( E: magnetic for each hopping: energy gain Ea, but kinetic energy is bounded..1 E F z=3 z= z= T=1/8 1/5, 1 1 3

30 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] z=6 z=5 z=4 b c density ( E: magnetic..1 E F z=3 z= z= T=1/8 1/5, 1 1 3

31 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