Electronic Squeezing by Optically Pumped Phonons: Transient Superconductivity in K 3 C 60. With: Eli Wilner Dante Kennes Andrew Millis
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1 Electronic Squeezing by Optically Pumped Phonons: Transient Superconductivity in K 3 C 60 With: Eli Wilner Dante Kennes Andrew Millis
2 Background: Mean-Field Theory of Simple Superconductivity If the effective electron-electron interaction is attractive (negative), then it is easy to show that starting from the non-interacting Fermi sea, 2 electrons with energies above the Fermi ( 1 2)/2 energy will form a bound state with a wave function ( 1 2) i q r r r e r. At q=0 this is just ik r1 ik r2 ( k) e e, namely a pairing (if made into singlet) of (k,-k ) plane wave functions k together. For real metals, the effective e-e interaction is indeed attractive for states within shell around the Fermi level (surface) due to over-screening of ions via phonons. Important to note-electron-phonon interaction taken to be linear in phonon coordinate. Together, motivates guess for the ground state wave function of the form k Can then show by minimization that this variational guess gives lower energy than noninteracting Slater determinant. ~ p g ( u k vkc ) 0. k c k
3 Consequences: Find within variational calculation that energy difference between normal metal and superconducting state is EN ES V N(0) D0 where D (0) is the energy gap 2 0 D0 2 e N V p N(0) is the density of states at the Fermi surface, V strength of e-e attraction, p is characteristic phonon frequency. Same calculations can easily be performed at finite T. Find a transition at to superconducting state kt ~ D. Gap is T-dependent-disappears at T c. b c 0 Meaning of gap-minimum energy to create single particle excitation from superconducting ground state is 2D 0 : D 0 for removing it from one state and D 0 for placing it in another. Finite energy to break up pairs-protected from scattering.
4 Re Background: Primer on Optical Conductivity Consider simple model for motion of electrons in a metal: Then dv( t) mv( t) it m ee( t), E( t) Ee. dt 2 e / m ( ne / m) v( t) E( t), j nev j ( ) E, ( ). 1i 1i This defines the frequency dependent (optical) conductivity. It can be microscopically and quantum mechanically computed from Kubo relation (current-current correlator). Real part of above is Lorentzian centered at =0.
5 Re Re Background: Primer on Optical Conductivity In superconductor, no dissipation so expect and the Lorentzian should become a delta function with 2 ne weight But this should only be for the fraction of pairs formed, not the rest of the normal electrons m which should only be visible in optical conductivity at frequencies corresponding to twice the gap and higher. Delta function with weight ne s m 2 As well by the Kramers-Kronig relation, the imaginary part ne s Im ( ) ~ m 2 1
6 Background: Superconductivity in K 3 C 60 Parent FCC lattice insulator Doping 3 electrons per C 60 metallic. Tc high: K. Generally thought to be standard superconductivity mediated by linearly-coupled phonons (albeit with strong couplings). Recent evidence-this is not quite the case (but will ignore here).
7 Feb Effect transient (non-equilibrium)- lasts ~10ps. Seen cleanly when pumping performed at temperatures ~100K! Equilibrium (T c ~ 20K) Optically active mode driven by IR laser/probed by THZ laser
8
9 Building a Model Hamiltonian The phonon modes that mediate equilibrium superconductivity in K 3 C 60 are well characterized. These modes are linearly coupled to the electronic degrees of freedom. However the modes that are IR active (the 4 triply degenerate T 1u modes) by symmetry can only couple quadratically to the electronic degrees of freedom. These are the modes pumped in the experiment. Strength of quadratic coupling order of magnitude smaller than linear. Generally ignored in equilibrium. In model with just linear coupling can remove coupling with polaron transformation. Will make band narrower and U negative (favoring pairing). We assume the band and U terms are effective parameters and already have absorbed this transformation. Take values consistent with known Tc. Have worked out full tensor structure of couplings but will ignore that and keep simple here.
10 Model Hamiltonian ˆ K ˆ 1 H J c c h c U n n X Pˆ gk Xˆ n ( ˆ ˆ..) ˆ ˆ ( ) ˆ ij i j i i i i i i i, j i i 2 2M i ˆ ˆ ˆ 1u i i i K, g, M T, n c c We have not included the interaction of the vibrations with the laser field. For simplicity here-we will assume this merely leads to a time dependence in the amplitude of the phonon modes
11 In particular. We consider decoupled interaction of T 1u phonons with the laser field: ˆ boson 0 ( i i ) F( t) ( i i ), i 2 i H 1 ˆ K X 2 2 i MK i i, 0. M i i0t ( t) e if ( t), t i0 ( t) ( ) ( ), 0 i f t d e F ~ f( t) 2 i i 1
12 Physical Role of Quadratic Coupling: Squeezing Always >0! 1 K U (2, ) (0, ) 2 (1, ) ( ) eff E m E m E m U m 2 1 2g 1 1 4g 2 M Vs. Thus-high amplitude phonon excitation leads to an increasingly negative (attractive) interaction between electrons!
13 Physical Role of Quadratic Coupling: Squeezing Note-density dependent change in oscillator stiffness requires g> -1/4. A change in stiffness without change in mass: change in uncertainty ellipse for oscillator-squeezing in phase space! ( i/2) ( ˆ ˆ ˆ ˆ j X jpj Pj X j ) ˆ j S e, (1/ 4)ln[1 2 g( nˆ nˆ )] i i i ˆ ˆ ˆ ˆ ˆ 1 SHS S( J ( cˆ cˆ h. c.)) S U nˆ nˆ ( nˆ )( ) ij i j i i i i i i, j i i 1 1 K 2 ( nˆ ) (1 2 ˆ ), ˆ 2 2 i gni X i MK i i M
14 Approximation: simplify hopping (kinetic energy) and drop inelastic terms. 2 2 Remaining model interesting: ( * g ( nb 2 nb J Je 1)/8 ) Hˆ J cˆ cˆ h c g U nˆ nˆ 2 * ( i j..) (2 i i 1) i, j 2 i i i i 1 g g 2 0 ( i i ) ( ) 0 (2 i i i i 1)ˆ ni Higher occupancy of phonon From known distortion caused by optical pumping can estimate mean occupancy of boson mode ( t ) 10 Disorder weak but not so far from boundary of localization. Higher occupancy of phonon modes INCREASES site disorder and kills all conductivity! mode DECREASES U-stronger attraction
15 How good is our approximate Hamiltonian? Can test with a 2-site model which we can (numerically) solve exactly: Double Occupancy Spectral Function
16 How good is our approximate Hamiltonian? Phonon properties:
17 The Hamiltonian suggests a phase diagram of system under driving:
18 Putting in numbers: Optical Conductivity
19 Putting in numbers: transition temperatures.
20 Some Comments (I) We have some numerical evidence from exact (1D) DMRG simulations (Wilner et al, to be published) that the field-induced disorder effect is real-and the induction of collective behavior competes with this. The couplings we have used are on the high end but not unrealistic. On the other hand, we have ignored many features of the true band structure (multi-orbital character, phonon multiplicity, Hund s coupling, etc) and have used a BCS-like calculation.
21 Some Comments (II) There are at least 2 other recent explanations for these nonequilibrium effects. Demler et al. have a theory similar in spirit but focusing on different couplings, Georges et al. have a theory based on excitonic cooling that actually does not involve enhanced pairing at all. Still need to be able to explain similar effects seen in cuprates
22 Conclusions: The notion that coherent excitation may induce transient collective electronic effects is actually not as implausible as seems at first sight. The mechanism put forward here is general-applies to other systems. Predict distinct transient changes ( phase transitions ) other than superconductivity as well On other hand-see no way to make these effects long lasting-phonon dephasing, relaxation will always kill behavior at longer times.
23 Thx-E.Wilner, D.Kennes and A.Millis Kennes, Wilner, Reichman and Millis, Nature Physics, online publication (DOI /NPHYS4024) January Kennes, Wilner, Reichman and Millis, arxiv (2017) Wilner, Kennes, Reichman, Millis (to be submitted).
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