Chernogolovka, October 2009Chernogolovka, October 2009

Nolinear nonequilibrium phenomena in stacked Nolinear nonequilibrium phenomena in stacked

junctionsjunctions Vladimir KrasnovVladimir Krasnov

Experimental Condensed Matter PhysicsExperimental Condensed Matter Physics Fysikum, AlbaNova, Stockholm UniversityFysikum, AlbaNova, Stockholm University

Phonon lasing in stacked Phonon lasing in stacked intrinsic Josephson junctionsintrinsic Josephson junctions

Motivation:Motivation:

• Non-equilibrium phenomena are central in many superconducting Non-equilibrium phenomena are central in many superconducting detectors, but may be detrimental for in superconducting electronics. detectors, but may be detrimental for in superconducting electronics.

• ““Heat” (Energy) conduction at low Heat” (Energy) conduction at low TT in the absence of thermal in the absence of thermal conductivityconductivity

• Extreme non-equilibrium states in stacked JJ – new Extreme non-equilibrium states in stacked JJ – new nonlinear nonlinear phenomena. Superconducting Cascade Laser in THz frequency rangephenomena. Superconducting Cascade Laser in THz frequency range

E

E

B eV-2

N(E)

eVR 2

2

Bremsstrphonon

Recomb.

Relaxation of non-equilibrium Quasi-ParticlesRelaxation of non-equilibrium Quasi-Particlesin Josephson junctionsin Josephson junctions

1-stage:QP Relaxation – Bremsstrahlung phononsQP 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

Intrinsic stacked Josephson junctions in layered HTSC Intrinsic stacked Josephson junctions in layered HTSC

BiBi22SrSr22CaCuCaCu22OO8+x 8+x : anisotropy : anisotropy cc//abab ~10 ~1066

c-axisc-axis

Factors enhancing nonequilibrium effects in IJJsFactors enhancing nonequilibrium effects in IJJs

IfDoS ~Very rough estimation:)(

~3cmdosd

Jf

Al/AlOx/Al Nb/AlOx/Nb IJJs Bi-2212

Jc (A/cm2) ~10 102-103 3x102-3x103

(meV) 0.4 1.4 30-40

J(Vg) (A/cm2) ~10 102-103 ~ 103-104

dos(1/eVcm3) ~2x1022 ~2x1022 ~1022

d (nm) ~100 ~200 ~0.4

(ns) ~1000 ~0.2 ~2x10-3 (opt.)

f (Vg) (a.u.) ~50 0.05-0.5 5-50

Bosons Cascading N = 10-103

QPs Confinement no leakage

Additional effects of stackingAdditional effects of stacking

Quantum cascade laserQuantum 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

P.Offermans et al., Appl.Phys.Lett. 83 (2003) 4131

Cross-sectional STM of InAlAs/InGaAs quantum cascade laserJ.Faist, et al., Science 264 (1994) 553

From Z.I.Alferov, Nobel lecture Rev.Mod.Phys. 73, 767 (2001)

Effect of stacking in semiconducting heterostructure lasersEffect of stacking in semiconducting heterostructure lasers

Kinetic balance equations

)()()(

),(

escinjrel t

N

t

N

t

N

t

EN

Tunnel QP injection rate (bias dependent)

)()()()()(

2)(

EfeVEfeVEEdE

Ret

EN

ninj

QP escape rate (via tunneling)

)()()()()(

2)(

eVEfEfeVEEdE

Ret

EN

nesc

Phonon escape rate dvN

t

N s

esc

)(

)(

Phonon injection rate (bias independent)

)()(

)()(

escJ

inj t

NN

t

N

Quasiparticle relaxation rate

EPh

E

Ph

Ph

QP

rel

gEfEfgEfEfEE

EEDd

gEfEfgEfEfEE

EEDd

gEfEfgEfEfEE

EEDd

dEDV

tEN

)()(1)(1)(1)()()(

1

)()(1)()(1)(1)()(

1

)(1)(1)()()(1)()(

1

04

22

0

22

0

22

)(

Spontaneous emissionAbsorption Stimulated emission

Recombination – pair breaking

absorption-emissionRelaxation: emission-absorption

Phonon relaxation rate

)(1)()()()(1)(1)(

121

)(1)(1)()()(1)()(

1

08

2

2

2

)(

gEfEfgEfEfEE

EEdE

gEfEfgEfEfEE

EEdE

dDDV

tN PhQP

rel

Spontaneous emissionAbsorption Stimulated emission

E

E

QP

rel

EFEFggEFEfEfgEfEE

EEd

EFEFggEFEfEfgEfEE

EEd

EFEFggEFEfEfgEfEE

EEd

EdbDV

tEN

1)()()()()()()()()(

1

)()()()()()()(1)()(

1

)()()()(1)()()()()(

1

)(04

2

2

2

0

2

2

2

0

2

2

2

4

)(

f(E) = F(E) + f(E)g() = G() + g()

Expansion of the quasiparticle relaxation rate

Spontaneous emissionAbsorption Stimulated emission

No equilibrium terms here

Expansion of the phonon relaxation rate

1)()()()()()()()()(

121

)()()()(1)()()()()(

1

08

2

2

2

24

)(

EFEFggEFEfEfgEfEE

EEdE

EFEFggEFEfEfgEfEE

EEdE

dbDV

tN QP

rel

Spontaneous emissionAbsorption Stimulated emission

No equilibrium terms here

Recombination – pair breaking

Relaxation: absorption-emission

DDD

dEE

EfdE

E

kTEdE

E

Ef222222

)(2

)2/tanh()(211

Self-consistency equation:

dE

E

Efd

TT

TkTT

Tkc

c

c

c

221 2

0

0

0

)(2

1

)2

tanh()2

tanh(

Numerical solution for non-equilibrium :

D

dEE

kTE

0 022

)2/tanh(1

Equilibrium :

QP’s at the bottom of the band are most important

,)5.0(),(

),...2,1(,)5.0(),(

dEigx

KidEiEEfx

iiKi

iii

Relaxation Escape Injection

Numerical procedure:

));();(( 111)( nnn

nij ij

EgEfRR

025.0

,7max

dE

KdEE

obtain

)}();({ EgEfx nni

obtain n from the self-consistency Eq.

calculate

YR ni

nij

)1()1( ,

Proceed

with itteration (n+1)

),221,21(,

),221,1(,

KKjKiKYxxR

KKjKiYxxR

iPiUjij

iIiYjij

QP balancePhonon balance

Itteration (n): Solve the system of 2K linear Eqs.

Dayem & Wiegand PRB 5, 4390 (1972)Chang & Scalapino PRB 15, 2651 (1977)

Nonlinear solution for a double stacked junctionsordinary ”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 fn.l.< flin.

Net accumulation of QPs at E’=0 and absorption of bosons with =0.Slow QP relaxation due to reabsorption of bosons.

Pho

non

inte

nsity

EnergyeV-2 2

Bremsstrahlungphonons Recombination

phonons

eV < 4

Pho

non

inte

nsity

EnergyeV-2 2

Bremsstrahlungphonons

eV = 4

Recombinationphonons

•Enhanced depairingSecondary QP-band0<E-< eV-4

New bands appear at eV=2n

Nonlinear effects at even-gap bias: Nonlinear effects at even-gap bias: Secondary nonequilibrium QP and bosonsSecondary nonequilibrium QP and bosons

Stimulated emission?

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

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 (Al2O3)

Nonequilibrium Nonequilibrium I-VI-V characteristics 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.

Overdoped Bi-2212-400 -200 0 200 400

0

2

4

6

S811b69.7K75.0K80.0K84.1K87.5K89.9K92.0K

- (

100K

) (m

S)

V (mV)V.M.Krasnov, Phys.Rev.Lett. 97,257003 (2006)

Observation of even-gap peculiarities in Observation of even-gap peculiarities in Bi-2212 intrinsic tunneling characteristics Bi-2212 intrinsic tunneling characteristics

0 20 40 60 800

10

20

30

40

811b 6S/e

4S/e

2S/e

Vol

tage

per

junc

tion

(m

V)

T (K)

Height of the mesa

4a 4bI+

I-

V-

V+

0

0.1

0.2

0.3

0.4

0 500 1000 1500 2000 2500

IV4b124b6a T=4.19K Three-probe

I (m

A)

V(mV)

0

0.1

0.2

0.3

0.4

0 250 500

IV4a124b6aT4_19bFour-probe

V(mV)

I (m

A)

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)

AA EEBB CCD

BiBi22SrSr22CaCuCaCu22OO8+8+II

VV

V.M.Krasnov, Phys.Rev.Lett. 97,257003 (2006)

Detection of recombination radiationDetection of recombination radiation

0 1 2

1E-3

0.01

0.1

0 1 2

0.01

0.1

1

10

IQP

T =0.5Tc ,

U=0.1,

I=0.1.

eV-2

a)

eV/0=3.0

F

f

f,

F

E'/0

Nonlinearabsorptive solution

Absorption b)

G

g

g, G

/0

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

From O.Heikkilä et al., J.Appl.Phys. 105, 093119 (2009)

Semiconducting Light Emitting Diode 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 Jth=10-100 A/cm2 at 300K, Jth~exp(T). For IJJs J = 104 A/cm2 at 4K.

Mesa itself acts as a Fabry-Perot resonator, selecting cavity (Fiske) modes.

Conclusions:Conclusions:

• Linear approximation fails already at relatively small disequilibrium: the Linear approximation fails already at relatively small disequilibrium: the nonequilibrium part has to be small compared to thermal population. nonequilibrium part has to be small compared to thermal population.

• Nonequilibrium effects are always nonlinear at low enough effects Nonequilibrium effects are always nonlinear at low enough effects TT. This . This has to be taken into account in analysis of superconducting devices at low has to be taken into account in analysis of superconducting devices at low TT..

• In stacked IJJIn stacked IJJ extreme nonequilibrium state can be achieved. The obtained extreme nonequilibrium state can be achieved. The obtained radiative state indicates a possibility of realization of a new type of radiative state indicates a possibility of realization of a new type of Superconducting Cascade LaserSuperconducting Cascade Laser (SCL). Unlike existing Josephson (SCL). Unlike existing Josephson oscillators which utilize the ac-Josephson effect for conversion of electric oscillators which utilize the ac-Josephson effect for conversion of electric power into radiation, the SCL is based on direct conversion of electric power power into radiation, the SCL is based on direct conversion of electric power into boson emission via nonequilibrium QP relaxation upon sequential into boson emission via nonequilibrium QP relaxation upon sequential tunneling in stacked junctions. The mechanism is similar to lasing in tunneling in stacked junctions. The mechanism is similar to lasing in semiconducting heterostructures and allows very high radiation efficiency. semiconducting heterostructures and allows very high radiation efficiency.

• Emitted are bosons that participate in pairing. Therefore, nonequilibrium Emitted are bosons that participate in pairing. Therefore, nonequilibrium intrinsic tunneling spectroscopy may provide a direct probe for HTSC coupling intrinsic tunneling spectroscopy may provide a direct probe for HTSC coupling mechanism. mechanism.

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