7/29/2019 Coherent trajectory through the normal-to-superconducting transition reveals ultrafast vortex dynamics in a super

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Coherent trajectory through the normal-to-superconducting transition reveals

ultrafast vortex dynamics in a superconductor.

P. Kusar1, I.Madan1, M.Lu-Dac1, T. Mertelj1, V. V. Kabanov1, S.Sugai2 and D. Mihailovic1,31Complex Matter Department, Jozef Stefan Institute, Jamova 39, 1000 Ljubljana, Slovenia

2Department of Physics, Faculty of Science, Nagoya University,

Furo-cho, Chikusa-ku, Nagoya 464-8602, Japan and3CENN Nanocentre, Jamova 39, 1000 Ljubljana, Slovenia

(Dated: August 13, 2012)We report on measurements of the trajectory of the superconducting order parameter (t) through

the normal-to-superconductor transition in La1.9Sr0.1CuO4 under ultrafast quench conditions up to4 1014 K/s. Using a 3-pulse technique, we isolate the trajectory of(t) from single-particle pro-cesses on short timescales. Detailed modeling of the system trajectory using hot electron energyrelaxation, time-dependent Ginzburg-Landau theory and vortex creation according to the Kibble-Zurek mechanism describes the experimental data in quantitative detail, significantly advancing ourunderstanding of the behaviour of the normal-to-superconductor transition under highly nonequi-librium conditions. The recovery of takes place on a timescale of 20 to 60 ps, depending on theexcitation energy density, while the vortex dynamics is apparent on a timescale of 10 30 ps andis visible only for the fastest quench rates.

What happens when a system goes through a

symmetry-breaking transition depends crucially on howfast the transition occurs[1, 2]. If the quench process issufficiently fast large numbers of topological defects arecreated and the nature of the final state can be substan-tially different than if the transition takes place slowly.The study of the time evolution of complex systemsthrough symmetry breaking transitions (SBT) is thus ofgreat fundamental interest in different areas of physics[24] and beyond. The diversity of microscopic vacua incondensed matter systems makes them excellent candi-dates for the experimental study of new phenomena[5],also posing new challenges for the general theory ofSBTs. Of particular general interest is the normal-

to-superconducting (NS) state transition in which aLorentz non-invariant system breaks gauge invariance[6].Under near-equilibrium conditions the NS transitionoccurs when the thermal fluctuation timescale F be-comes comparable to the Ginzburg-Landau time whichfor typical high temperature superconductors is GL =/(T Tc) 1012 1013s[2, 5, 7, 8]. The intrinsiccollective system relaxation time GL defines the relevanttimescale and experimental resolution necessary to in-vestigate system trajectories through the transition. Tillnow, the shortest quench times through the transitionwere q 108 s[9], which were relatively long comparedto GL, so observations were focused on statistical countsof vortices in the aftermath of the transition as a func-tion of the quench rate q = 1/q. Femtosecond laserspectroscopy offers sufficient time resolution for ultra-fast quench studies and pump-probe transient reflectiv-ity experiments have recently elucidated the destructionof superconductivity (i.e. the SN transition)[1012].However, standard pump-probe techniques cannot dis-tinguish the trajectory of the order parameter (OP) (t)from photo-excited hot carrier relaxation, quasiparticle

(QP) recombination dynamics across the superconduct-

ing gap and the relaxation of pseudogap (PG) excita-tions, all of which appear simultaneously in the time-resolved response[1318].

To solve this problem we devised a 3 pulse tech-nique shown schematically in Fig. 1 which allows usto measure in real time the superconducting order pa-rameter (t) through the transition in a La1.9Sr0.1CuO4(LSCO) single crystal with ultrahigh quench rates of upto q = 1/q = (dT/dt) T

1c 4 1014 s1. Further-

more, we can vary q by changing the laser excitationenergy. We compare the data with predictions of theKibble-Zurek (K-Z) mechanism of vortex creation, find-ing evidence for ultrafast vortex dynamics in the rapidquench regime when q GL[1, 20]. La1.9Sr0.1CuO4 ischosen because it is one of the best studied superconduc-tors particularly by femto-optical methods.[10, 14, 27]

With our technique the superconducting state is firstrapidly nonthermaly[10] destroyed in 1 ps by an intensedestruction (D) laser pulse creating a pancake-like nor-mal region defined by the laser beam profile with theFWHMdiameter d = 60 m and the optical penetrationlength p = 150 nm (at 800nm). We then monitor therecovery of the OP by means of the transient reflectiv-ity R(t23)/R excited with a weaker pump (P) pulseand probed with even weaker probe (p) pulse as a func-

tion of time delay t12 between the D and P pulse (seeFig. 1).[19] The D pulse fluence is adjusted to achievedifferent quench rates.

After destruction, an OP front propagates fromthe bulk towards the surface on a timescale p/v pGL/ 103GL, where v is the OP frontvelocity[32] and is the out-of-plane SC correlationlength. is much longer than both, the quench timeq and the Zurek time Z qGL. As a consequencethe normal region is supercooled and we can expect vor-

a

rXiv:1207.2879v2

[cond-mat.supr-con]10Aug2012

7/29/2019 Coherent trajectory through the normal-to-superconducting transition reveals ultrafast vortex dynamics in a super

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2

t1-2t2-3

t1-3

z

SCN

sample

Te

Tc

detector

vTv

RapidQuench

t 1ps

t 3 ps

t 10-80 ps

time

SlowQuench

A

B

Figure 1: a) A schematic diagram of the pulse sequence. Thetime delays t12,t13 and t23 refer to delays betweenthe D pulse (blue) and the P pulse (red), the D pulse and theprobe pulse (green) and between the P pulse and the probepulse, respectively. The temperature front where Te = Tcand the S/N phase boundary move with velocities vT and vrespectively. Vortices are created in the wake of the temper-ature front. b) The system trajectory (depicted by the silverball) in a temporally evolving potential. In the rapid quenchscenario (A), the potential changes faster than the system canfollow. The opposite is true in the slow quench scenario (B).

tex formation according to the K-Z scenario.[1, 5, 9, 2022] Depending on the quench conditions, the propagat-ing velocity of the temperature T = Tc front vT canbe either faster or slower than the critical velocity[20]vcrit v 4

GL/q 10

5 cm/s. In the ultrafast quenchlimit when vT > vcrit, we expect vortex formation, whilein the slow quench limit (vT < vcrit), vortex formationbecomes suppressed [20, 22].

The time-evolution of the electronic and lattice tem-peratures Te(t) and TL(t) respectively, which govern thequench process, are calculated using the standard two-temperature model[23, 26]: eTe

dTe

dt

=

L(Te

TL) +

E(t) and CL(T)dTLdt = L(TeTL), where E(t) is a func-

tion (Gaussian) describing the energy density per unittime supplied by the laser pulse [33]. For maximumaccuracy we used the temperature-dependent thermalconstants e(T) and CL(T) from experimental data[25]and the measured energy loss rate L 340 K/ps forLa1.9Sr0.1CuO4 [24]. The dependence of Te(t) and TL(t)on the D pulse fluence, FD, is shown in Fig. 2a) for thefirst 2.5 picoseconds. The red contour lines show wheneither Te or TL crosses Tc, which is the critical point ofthe quench.

For low

FD

15J/cm2, we have a rapid quench sce-

nario as Te rapidly crosses Tc within a few hundred fem-toseconds, the quench rate being governed by the elec-tronic energy relaxation rate q e 41014 K/s. ForFD >15 J/cm2 however, the lattice is heated above Tcby the D pulse, so the cooling rate through Tc is gov-erned by diffusion processes on timescales longer than 30 ps. Thus by changing FD we can effectively varythe regime from rapid to slow quench, over a very widerange of quench times from q = 100 fs to hundreds ofpicoseconds.

Pulse

flue

nce,FD

[J/c

m2]

Figure 2: The time evolution of the electronic and latticetemperatures Te and TL respectively. The ball signifies thetrajectory of Te(t) for a D pulse fluence

FD = 38 J/cm

2.

The blue lines are the trajectories of Te(t) corresponding tothe D pulse fluences used in the experiments. The red linessignify the time when Te and TL cross the critical temperatureTc = 30 K. Rapid quench occurs for FD < 15 J/cm2.

The transient reflectivity amplitude in the supercon-ducting state is proportional to the square of the orderparameter As = R/R |(t12)|2[19], so to detect thetrajectory (t) through Tc, we measure the transient re-flectivity by the Pump-probe technique at different timedelays after the D pulse. The experimental data on thetransient reflectivity R(t23)/R at different delayst12 during the system recovery is shown in Fig. 3a).Two distinct and easily identifiable contributions are ob-served: one has a peak around 0.2 ps which has beenpreviously identified to be due to pseudogap excitations,and the other is the QP recombination signal of the su-perconducting condensate with a peak near 2 ps, extend-ing to tens of ps[14, 27]. We observe a clear suppressionof the QP peak at short delay t12 and a gradual emer-gence of the SC state with increasing delay. A distinctfeature in Fig. 3a) is the convergence of all QP curvesafter a delay t23 = th 25 ps which corresponds tothe complete thermalization of all degrees of freedom.

For further analysis, we subtract the PG response fromthe data[34] and plot As as a function t12, thus obtain-ing the trajectory of |(t)|2 shown by the square blackdots in Fig. 3b). From the exponential fits of the decay ofR/R(t23) we also obtain the QP relaxation time QPas a function oft12 which is shown in Fig 3b), plottedas a relaxation rate 1/QP. We observe that 1/QP showsa similar trajectory as the order parameter; indeed su-perimposing |(t12)| on the graph of1/QP(t12) (solidline in Fig. 3b)), we see that the QP recombination

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3

0.1 1 10 1000

1

2

t2-3

[ps]

R/R

[10-4] t1-2=100 ps

10 ps

1ps

th

Increasingt1

-2

SC

PG

0.00

0.05

0.10

0.15

1 10 100

t1-2

[ps]

1/

QP

1 10 100

0.0

0.5

1.0

As[x10-4]

t1-3

[ps]

a) b)

Figure 3: The transient reflectivity data R/R forLa1.9Sr0.1CuO4 at 4 K as a function of t23, for differentt12. The D pulse fluence is FD = 12 J/cm2, which isapproximately three times above the destruction threshold(FD 4 J/cm2)[10]. The red line indicates the pseudogapsignal measured above Tc. The recovery time of the SC staterec is indicated. b) The same data as in a) after subtrac-tion of the PG signal plotted as a function of t13 . The

black squares are the fitted maximum amplitudes of As andgive |(t)|2 for each t12. The blue squares are the QP re-laxation rate superimposed on the measured order parametertrajectory |(t)| As as a function of delay t12.

rate closely follows the order parameter as a function oft12, consistent with the fact that 1/QP(t) is propor-tional to the superconducting gap s(t), which is in turnproportional to |(t)| [13].

Tuning FD significantly changes the trajectory of thesystem through the transition, and the SC state recoversmuch more slowly with increasing FD. Fig. 4a) showsthe trajectory of ||

2

for a number of different FD.[35]For the lowest two values ofFD, which correspond to thefastest quench, we see that destruction of the condensateis incomplete throughout the entire excitation volume,since ||2 is non-zero in the first 2 ps after the D pulse.With increasing FD the recovery becomes slower. Theinitial recovery rate 1/i can be obtained from the ini-tial slope of |(t12)|2 for small t12 and is shownin Fig. 4b). Plotting the recovery time versus FD inFig 4c), we find a phenomenological i which increasesapproximately linearly with FD before eventually satu-rating at high FD. For comparison, we see that the QPrecombination time QP (shown on the same plot) first

dramatically decreases below the destruction thresholdfluence for FD < 4J/cm2, and then slightly increaseswith increasing FD. This impressively demonstrates howwell we can separate the order parameter recovery fromQP in these experiments, and more importantly, showsthat the QP relaxation has very different systematics andplays no significant role in determining the recovery dy-namics.

To analyze the trajectory of the system through thetransition as accurately as possible, we use the full set

100

101

102

104

106

0.0

0.5

1.0

FD[J/cm

2]

3.7

7

12

14

18

24

34

||2

[norm.]

t1-2

[ps]

a)

4 10 20 300.2

0.4

0.6

0.8

0 10 20 30 40 500

20

40

60

80

0

20

40

60

80

4K

c)

QP

[ps]F

T

rec

FD[J/cm

2]

`

0 1 2 3 4 5

0.0

0.1

0.2

0.3

24

34

44

FD[J/cm

2]

7

12

14

18

||2[

norm.]

t1-2

[ps]

b)

Figure 4: a) The trajectories of ||2 as a function of t12for different D pulse fluences FD. The lines are calculatedtrajectories from the TDGL theory. The insert shows thepredicted |(t)|2 with (black) and without (orange) vorticesat 7J/cm2. The vertical scales are normalized to the equilib-rium |0|2 in the absence of the D pulse. b) The relaxation of||2 in the first 5 ps for different fluences. c) The dependenceof the recovery time rec on pulse fluence (red dots). Forcomparison, the QP recombination time QP is also shownfor different FD (diamonds). The dashed line is the predictedQP from the Rothwarf-Taylor model[28, 29] which is validonly below the destruction threshold FD < 4J/cm2.

of TDGL equations where in the absence of an externalfield, the dimensionless order parameter (t) is describedby[30]:

u

t+ i

= r(t, z) ||2 +2 + , (1)

2 = i

2( )

, (2)

where temporal and spatial coordinates are measured inunits of GL and correlation length , respectively. Theonly parameter in the equation is the ratio u = GL be-tween the longitudinal relaxation time and the transverserelaxation time. For our LSCO sample, 1 ps, u 5[8] and 2 nm. To initiate vortex formation arisingfrom microscopic fluctuations, the customary Langevinnoise term is added [31]. We also take into account theinhomogeneous nature of light penetration into the sam-ple and thermal diffusion processes, which take place on

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4 6 8 10 12 14

FD

[J/cm2]

-0.070

-0.028

0.015

0.057

0.090

4 6 8 10 12 14

10

20

30

40

50

FD

[J/cm2]

D-P[

ps]

-0.040

-0.016

0.0080.032

0.056

0.080

b) c)

a)

Figure 5: a) The predicted vortex structure of ||2 at dif-ferent delay times corresponding to rapid quench conditions(FD = 12 J/cm2) showing the |(r, t)|2 in the time intervalt12 = 0 25 ps in an area of 200 200 2. b) The dif-ference between the experimental trajectory and the fits cal-culated without Langevin noise: ||2 = ||2novortex ||2expshows a clear suppression of ||2 around 10-20 ps for 5