# G stefanucci s kurth a rubio eku gross phys rev b 77

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G. Stefanucci, S. Kurth, A. Rubio, E.K.U. Gross, Phys. Rev. B 77, 075339 (2008)
Bound state oscillations and memory effects Analytical : G. Stefanucci, Phys. Rev. B, 195115 (2007)) Numerical : E. Khosravi, S. Kurth, G. Stefanucci, E.K.U.G., Appl. Phys. A 93 , 355 (2008), and Phys. Chem. Chem. Phys. 11 , 4535 (2009) If Hamiltonian of a (non-interacting) biased system in the long-time limit supports two or more bound states then current has steady, I (S), and dynamical, I (D) , parts: ) t ( I I ) t ( I ) D ( ) S ( (D) bb' b b' b,b' I (t) sin[( )t]    Note : - bb’ depends on history of TD Hamiltonian (memory!) Questions : -- How large is I (D) vs I (S) ? -- How pronounced is history dependence? Sum over bound states of biased Hamiltonian
1-D model: start with flat potential, switch on constant bias, wait until transients die out, switch on gate potential with different switching times to create two bound states note: amplitude of bound-state oscillations may not be small compared to steady-state current History dependence of undamped oscillations
Model system U Time-dependent picture of Coulomb blockade
QD T bias L,R ˆ ˆ ˆ ˆ ˆ H t H H H H t  T link 1 , 0 , i 1 ˆ ˆ ˆ H V c c h.c.       i 1 , i , i 1 ˆ ˆ ˆ H t Vc c h.c.        bias i , , i 1 ˆ ˆ H t U t n       QD ext 0 0 0 ˆ ˆ ˆ ˆ H v n Un n
KS QD,KS T bias L,R ˆ ˆ ˆ ˆ ˆ H t H t H H H t  QD,KS KS 0 0 ˆ ˆ H t v n t n 0 0 n t n t KS 0 ext 0 xc 0 1 v n t v Un t v n t 2 Solve TDKS equations (instead of fully interacting problem): LDA functional for v xc is available from exact Bethe-ansatz solution of the 1D Hubbard model. N.A. Lima, M.F. Silva, L.N. Oliveira, K. Capelle, PRL 90, 146402 (2003)
S. Kurth, G. Stefanucci, E. Khosravi, C. Verdozzi, E.K.U.G., Phys. Rev. Lett. 104, 236801 (2010)
Steady-state density as function of applied bias for KS potential with smoothened discontinuity U L /V Fingerprint of Coulomb blockade S. Kurth, G. Stefanucci, E. Khosravi, C. Verdozzi, E.K.U.G., Phys. Rev. Lett. 104, 236801 (2010)
Most commonly used approximation for t r ρ v xc Adiabatic Approximation adiab approx xc xc,stat n (r t) v r t : v n  hom stat , xc v = xc potential of static homogeneous e-gas How restrictive is the adiabatic approximation, i.e. the neglect of memory in the functional v xc [ρ(r’,t’)](r,t) ? Can we assess the quality of the exact adiabatic approximation? e.g. ALDA hom xc xc,stat v r t : v r t
M. Thiele, E.K.U.G., S. Kuemmel, Phys. Rev. Lett. 100, 153004 (2008) 4-cycle pulse with λ = 780 nm, I 1 = 4x10 14 W/cm 2 , I 2 =7x10 14 W/cm 2 Solid line: exact Dashed line: exact adiabatic

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