1. Chernogolovka, October 2009 Nolinear nonequilibrium phenomena in stacked junctions Vladimir Krasnov Experimental Condensed Matter Physics Fysikum, AlbaNova, Stockholm University Phonon lasing in stacked intrinsic Josephson junctions Motivation: Motivation: Non-equilibrium phenomena are central in many superconducting detectors, but may be detrimental for in superconducting electronics. Non-equilibrium phenomena are central in many superconducting detectors, but may be detrimental for in superconducting electronics. “Heat” (Energy) conduction at low T in the absence of thermal conductivity “Heat” (Energy) conduction at low T in the absence of thermal conductivity Extreme non-equilibrium states in stacked JJ – new nonlinear phenomena. Superconducting Cascade Laser in THz frequency range Extreme non-equilibrium states in stacked JJ – new nonlinear phenomena. Superconducting Cascade Laser in THz frequency range

2. E E   B  eV-2  N(E)N(E) eV   R  2  22 Bremsstr phonon Recomb. Relaxation of non-equilibrium Quasi-Particles in Josephson junctions 1-stage: QP Relaxation – Bremsstrahlung phonons QP Recombination – Recombination phonons 2-stage: Phonon down conversion (luminescence) Reabsorption of non-eq. Phonons: QP excitation Pair breaking -> Secondary QP 3-stage: Relaxation of secondary QP And so on… Phonon heating / radiative cooling Electron heating /cooling

3. Intrinsic stacked Josephson junctions in layered HTSC Bi 2 Sr 2 CaCu 2 O 8+x : anisotropy  c /  ab ~10 6 c-axis

4. Factors enhancing nonequilibrium effects in IJJs Very rough estimation: Al/AlO x /AlNb/AlO x /NbIJJs Bi-2212 J c (A/cm 2 ) ~1010 2 -10 3 3x10 2 -3x10 3  (meV) 0.41.430-40 J(V g ) (A/cm 2 ) ~1010 2 -10 3 ~ 10 3 -10 4 dos(1/eVcm 3 ) ~2x10 22 ~10 22 d (nm) ~100~200~0.4  (ns) ~1000~0.2~2x10 -3 (opt.)  f (V g ) (a.u.) ~500.05-0.55-50 Bosons CascadingN = 10-10 3 QPs Confinementno leakage Additional effects of stacking

5. Quantum cascade laser Operation principle: Coupled quantum wells Population inversion by resonant tunneling Cascade amplification of light intensity J.Faist, et al., Science 264 (1994) 553

6. P.Offermans et al., Appl.Phys.Lett. 83 (2003) 4131 Cross-sectional STM of InAlAs/InGaAs quantum cascade laser J.Faist, et al., Science 264 (1994) 553

7. From Z.I.Alferov, Nobel lecture Rev.Mod.Phys. 73, 767 (2001) Effect of stacking in semiconducting heterostructure lasers

8. E E   B  eV-2  N(E)N(E) eV   R  2  22 Junction-1 Junction-2 Braking phonon Recomb. Non-equilibrium effects in stacked Josephson junctions Cascade amplification of Non-equilibrium phonons

9. Kinetic balance equations Tunnel QP injection rate (bias dependent) QP escape rate (via tunneling) Phonon escape rate Phonon injection rate (bias independent)

10. Quasiparticle relaxation rate Spontaneous emissionAbsorptionStimulated emission Recombination – pair breaking absorption-emission Relaxation: emission-absorption

11. Phonon relaxation rate Spontaneous emissionAbsorptionStimulated emission

12. f(E) = F(E) +  f(E) g(  ) = G(  ) +  g(  ) Expansion of the quasiparticle relaxation rate Spontaneous emissionAbsorptionStimulated emission No equilibrium terms here

13. Expansion of the phonon relaxation rate Spontaneous emissionAbsorptionStimulated emission No equilibrium terms here Recombination – pair breaking Relaxation: absorption-emission

14. Self-consistency equation: Numerical solution for non-equilibrium   : Equilibrium    : QP’s at the bottom of the band are most important

15. RelaxationEscapeInjection Numerical procedure: obtain obtain  n from the self-consistency Eq. calculate Proceed with itteration (n+1) QP balance Phonon balance Itteration (n): Solve the system of 2K linear Eqs. Dayem & Wiegand PRB 5, 4390 (1972) Chang & Scalapino PRB 15, 2651 (1977)

16. Linear scheme (1-st itteration): Relaxation time is independent of non-equilibrium part of distribution E E N(E)N(E) 22 Bremsstr phonon Recomb. Electron Cooling What to expect: Electron Heating Phonon Heating / Cooling

17. Linear scheme (1-st itteration): Relaxation time is independent of non-equilibrium part of distribution

18. RESULTS: relaxation of tunnel QP’s Linear scheme Full: Non-linear & self-consistent Electron heating by current injection But no thermal conductance: Non-equilibrium heat transfer

19. Nonlinear solution for a double stacked junctions ordinary ”absorptive solution” Nonlinearity appears when  f > F. QP relaxation is always nonlinear at low enough T or high enough E where F(T,E)→0. Nonlinearity stimulates QP relaxation  f n.l. <  f lin. Net accumulation of QPs at E’=0 and absorption of bosons with  =0. Slow QP relaxation due to reabsorption of bosons.

20. Phonon intensity Energy eV-2  22 Bremsstrahlung phonons Recombination phonons eV < 4  Phonon intensity Energy eV-2  22 Bremsstrahlung phonons eV = 4  Recombination phonons Enhanced depairing Secondary QP-band 0<E-  < eV-4  New bands appear at eV=2n  Nonlinear effects at even-gap bias: Secondary nonequilibrium QP and bosons Stimulated emission?

21. Bias-dependence of the nonlinear absorptive solution for a double stacked junctions

22. Phonon generation-detection experiment R.C.Dynes and V.Narayanamurti, Phys.Rev.B 6 (1972) 143 Time of flight experiments 0.4 cm (Ge) 1.5 cm (Al 2 O 3 )

23. Nonequilibrium I-V characteristics Note, that I-V curves are very similar for both solutions. Therefore, power dissipation P=IV is also the same. However, suppression of  is much smaller in the radiative state. This is due to radiative cooling = ballistic boson emission from the stack. Radiative cooling is the only heat transport mechanism considered here,  =0. The stack effectively (100% efficiency) converts electric power into boson emission without ac-Josephson effect.

24. Overdoped Bi-2212 V.M.Krasnov, Phys.Rev.Lett. 97,257003 (2006) Observation of even-gap peculiarities in Bi-2212 intrinsic tunneling characteristics

25. Changing the QP injection rate: Non-linear

26. P.Berberich, R.Buemann and H.Kinder, Phys.Rev.Lett. 49 (1982) 1500 Sn-SnO x -SnAl-AlO x -Al Monochromatic phonon generation by Josephson junctions

27. Height of the mesa 4a 4b I+I+ I-I- V-V- V+V+ Tripple-mesa with common junctions for injection-detection experiments: Three and Four-probe measurements N=52 N=28 N=52 N=28 V.M.Krasnov, Phys.Rev.Lett. 97,257003 (2006)

28. AEBC D Bi 2 Sr 2 CaCu 2 O 8+  I V V.M.Krasnov, Phys.Rev.Lett. 97,257003 (2006) Detection of recombination radiation

29. Appearance of a second ”radiative solution” at large bias No net accumulation of QPs at E’=0 – fast QP relaxation due to stimulated emission of low  bosons. Eistence of two solutions is a result of nonlinerity

30. From O.Heikkilä et al., J.Appl.Phys. 105, 093119 (2009) Semiconducting Light Emitting Diode Absorptive and Radiative states in stacked IJJs bare some similarity with light emitting and lasing states in heterostructure injection diodes. Population inversion by electron injection in a superlattice. Note that in LED J th =10-100 A/cm 2 at 300K, J th ~exp(  T). For IJJs J = 10 4 A/cm 2 at 4K. Mesa itself acts as a Fabry-Perot resonator, selecting cavity (Fiske) modes.

31. Conclusions: Linear approximation fails already at relatively small disequilibrium: the nonequilibrium part has to be small compared to thermal population. Linear approximation fails already at relatively small disequilibrium: the nonequilibrium part has to be small compared to thermal population. Nonequilibrium effects are always nonlinear at low enough effects T. This has to be taken into account in analysis of superconducting devices at low T. Nonequilibrium effects are always nonlinear at low enough effects T. This has to be taken into account in analysis of superconducting devices at low T. In stacked IJJ extreme nonequilibrium state can be achieved. The obtained radiative state indicates a possibility of realization of a new type of Superconducting Cascade Laser (SCL). Unlike existing Josephson oscillators which utilize the ac-Josephson effect for conversion of electric power into radiation, the SCL is based on direct conversion of electric power into boson emission via nonequilibrium QP relaxation upon sequential tunneling in stacked junctions. The mechanism is similar to lasing in semiconducting heterostructures and allows very high radiation efficiency. In stacked IJJ extreme nonequilibrium state can be achieved. The obtained radiative state indicates a possibility of realization of a new type of Superconducting Cascade Laser (SCL). Unlike existing Josephson oscillators which utilize the ac-Josephson effect for conversion of electric power into radiation, the SCL is based on direct conversion of electric power into boson emission via nonequilibrium QP relaxation upon sequential tunneling in stacked junctions. The mechanism is similar to lasing in semiconducting heterostructures and allows very high radiation efficiency. Emitted are bosons that participate in pairing. Therefore, nonequilibrium intrinsic tunneling spectroscopy may provide a direct probe for HTSC coupling mechanism. Emitted are bosons that participate in pairing. Therefore, nonequilibrium intrinsic tunneling spectroscopy may provide a direct probe for HTSC coupling mechanism.