Optical Saturation Produces Spurious Evidence for Photoinduced Superconductivity in K 3C60 J. Steven Dodge Leya Lopez and Derek G. Sahota

2025-05-02 0 0 309.26KB 12 页 10玖币

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Optical Saturation Produces Spurious Evidence for

Photoinduced Superconductivity in K3C60

J. Steven Dodge, Leya Lopez, and Derek G. Sahota

Department of Physics, Simon Fraser University, Burnaby, British Columbia V5A 1S6, Canada

(Dated: March 8, 2023)

We discuss a systematic error in time-resolved optical conductivity measurements that becomes important at

high pump intensities. We show that common optical nonlinearities can distort the photoconductivity depth pro-

ﬁle, and by extension distort the photoconductivity spectrum. We show evidence that this distortion is present

in existing measurements on K3C60, and describe how it may create the appearance of photoinduced supercon-

ductivity where none exists. Similar errors may emerge in other pump-probe spectroscopy measurements, and

we discuss how to correct for them.

A series of experiments over the last decade suggests that

intense laser pulses may induce superconductivity in several

materials [1, 2]. Time-resolved terahertz spectroscopy has

supplied the main evidence for this eﬀect, since it has the

electrodynamic sensitivity and the subpicosecond time reso-

lution necessary to observe its evolution [3, 4]. These mea-

surements are commonly reported in terms of the complex

photoexcited surface conductivity σs=σs1 +iσs2, which is

derived from experiment as a function of frequency ωand

pump-probe time delay ∆tusing a standard analysis proce-

dure [3–5]. Here, we show that this procedure distorts σs(ω)

when the photoinduced response has a nonlinear dependence

on pump ﬂuence—precisely the regime in which photoin-

duced superconductivity has been reported. As an example,

we describe how the evidence for photoinduced superconduc-

tivity in K3C60 [6–9] is susceptible to these distortions, and

we present an alternative explanation for the results that does

not involve superconductivity.

At equilibrium, the electrodynamic response of K3C60 ex-

hibits the characteristic features of a superconductor in the

dirty limit, as shown in Fig. 1. The equilibrium complex

conductivity ¯σ=¯σ1+i¯σ2above the critical temperature Tc

can be described by a semiclassical Drude-Lorentz model [9],

where the Lorentz oscillators account for the broad mid-

infrared conductivity at ~ω&10 meV and the Drude response

dominates at lower frequencies. (We use an overbar to distin-

guish static, equilibrium quantities from their time-dependent,

nonequilibrium counterparts.) Below Tca gap opens in ¯σ1at

~ω.6 meV, as spectral weight condenses into the supercon-

ducting δfunction at ω=0. Over the same frequency range,

the equilibrium reﬂectance is lossless, with ¯

R=1, and the in-

ertial response of the superﬂuid causes ¯σ2to diverge as 1/ω.

The optical properties reported for the photoexcited state

with T>Tcat ∆t=1 ps are qualitatively similar to those

of the equilibrium superconducting state. At low frequen-

cies, photoexcitation suppresses σs1, enhances σs2, and, af-

ter an adjustment that we discuss below, causes the reported

reﬂectance Radj to approach unity. The evidence for photoin-

duced superconductivity in K3C60 hinges on these similari-

ties [6–9].

But there is a crucial diﬀerence between the two sets of

measurements. The equilibrium conductivity is spatially uni-

form, so for a given background relative permittivity ∞there

is a unique mapping from the measured complex reﬂection

amplitude ¯rto the quantity of interest, ¯σ. This is not the

case for the photoinduced response, since the photoconduc-

tivity ∆σis not uniform. To determine σsuniquely from the

photoexcited reﬂection amplitude r, we must also specify the

conductivity proﬁle Pas a function of the depth zfrom the

surface. If Pis not known independently, we must assume a

model for it. Any error in this model will be passed on to σs.

Following previous practice [6, 7, 10, 11], Budden et al. [8]

use a proﬁle that we denote by Pexp, which they express in

terms of the refractive index as

n(ω, z;Pexp)=¯n(ω)+ ∆ns(ω)e−αz,(1)

where ¯nis the equilibrium refractive index, αis the pump at-

tenuation coeﬃcient, and ∆nsis the photoinduced change in

the refractive index at the surface. We include the label Pexp

explicitly to emphasize its role in inferring n(ω, z;Pexp) from

the measured r(ω). In terms of the conductivity, the proﬁle is

σ(ω, z;Pexp)=¯σ(ω)+ ∆σ(ω, z;Pexp),

=−iω0{[n(ω, z;Pexp)]2−∞}.(2)

Now it is possible to determine σs(ω;Pexp)=σ(ω, 0; Pexp)

from r(ω) by solving the Maxwell equations with Pexp and

matching the usual electromagnetic boundary conditions at

the surface [12].

The problem with this procedure is that Pexp implicitly re-

lies on two assumptions that are both unreliable. First, it as-

sumes that the pump absorption remains linear in the pump

intensity, so that the energy density Eabsorbed by the pump

decays as E ∝ e−αz. Second, it assumes that nis linear in E.

Jointly, these assumptions imply that Eq. (2) is independent of

the pump intensity. But none of these assumptions are sound

at the high pump intensities used in the experiments. Indeed,

the measured photoresponse consistently shows a nonlinear

dependence on the incident ﬂuence F[6, 10, 13–21], so ana-

lyzing them in terms of the proﬁle Pexp is not self-consistent.

And as we demonstrate here, neglecting nonlinearity can in-

troduce errors in σs(ω;Pexp) that are profoundly misleading.

The pump attenuation length Λ = 1/α is less than a third of a

typical probe attenuation length in K3C60 (see inset to Fig. 1),

so the pump excites only a fraction of the probe volume and

arXiv:2210.01114v4 [cond-mat.supr-con] 7 Mar 2023

FIG. 1. Conductivity (a,b) and reﬂectance (c) of K3C60, in equi-

librium and after photoexcitation with ﬂuence F=3.0 mJ/cm2and

pump photon energy ~ω≈170 meV, adapted from Budden et al. [8].

Equilibrium results are shown above and below Tc=20 K, at

100 K and 10 K, respectively [7]. Photoexcited results are shown

at 100 K for ∆t=1 ps. The photoexcited surface conductivity

σs(open circles) is inferred by assuming the proﬁle Pexp with ∞=5

and Λ = 220 nm [6, 8]. The distinction between the adjusted re-

ﬂectance (open squares) and the raw reﬂectance (open diamonds)

is described in the text. The inset shows E(z) for the pump and the

probe in the linear optical regime at ~ω=6.46 meV, each normal-

ized to their surface value.

the photoinduced change in ris much weaker than it would be

with uniform excitation. The change ∆ris then mainly sen-

sitive to the sheet photoconductance, ∆G= ∆σsdeﬀ, where

deﬀ=Rdz ∆σ(z)/∆σsis the eﬀective perturbation thickness.

For Pexp, we get deﬀ= Λ, independent of ﬂuence. But this is

no longer true if the photoconductivity is nonlinear, and fail-

ing to account for this will introduce error in deﬀ. Any error in

deﬀwill introduce a compensating error in ∆σs, distorting σs.

The diﬀerence between the raw and adjusted reﬂectance in

Fig. 1(c) reveals the scope for such an error. Budden et al. [8]

do not report raw measurements of the photoexcited re-

FIG. 2. Local photoconductivity ∆σas a function of depth from the

surface (a) and pump ﬂuence (b),(c) for two models of nonlinearity.

(a) The proﬁles Psat (orange, solid lines) and PTPA (purple, dashed

lines) are shown for the same four values of the normalized pump

ﬂuence f, indicated by markers of the corresponding color in (b) and

used by Budden et al. [8] is shown for ~ω=6.46 meV. Markers in

(a) indicate the 1/edepth for each curve.

ﬂectance R=|r|2, so we have deduced it from their reported

Pexp and σs. What Budden et al. [8] do report is Radj, which

they compute for an interface between a diamond window

(used in the measurements) and a ﬁctitious medium with uni-

form σ(ω) that they set equal to σs(ω;Pexp). While the raw

reﬂectance Rexceeds ¯

Rby at most 3.4%, Radj exceeds it by as

much as 15%, a discrepancy of more than a factor of 4. Note

that Radj(ω) is derived from σs(ω;Pexp), not the other way

around, so any error in σswill also appear in Radj. If we over-

estimate deﬀ, we will underestimate both |∆σs|and |Radj −¯

R|,

and if we underestimate deﬀwe will overestimate them.

And as Fig. 2 makes clear, nonlinearity can cause deﬀto

change by an order of magnitude or more as the ﬂuence

increases. We show proﬁles for two common nonlineari-

ties, which we discuss in more detail in the Supplemental

Material [22]. In one, which we label as Psat, we assume

that E ∝ e−αzand that the local photoconductivity ∆σsatu-

rates with E. Deﬁning the dimensionless ﬂuence parameter

f=F/Fsat, where Fsat is the characteristic scale for satura-

tion, we express σas [23, 24]

σ(ω, z,f;Psat)=¯σ(ω)+ ∆σsat(ω)f e−αz

1+f e−αz,(3)

which yields

∆G(ω, f;Psat)= ∆σsat(ω)Λln(1 +f).(4)

Note that ∆G(ω, f;Psat) continues to increase with feven

as ∆σs(ω, f;Psat)= ∆σsat(ω)f/(1 +f) saturates. This is be-

cause ∆σgrows more slowly at the surface than it does in the

interior as fincreases, which causes deﬀto increase also. The

logarithmic growth of G(ω, f;Psat) with fdoes not depend

on the detailed form of the saturation in Eq. (3), since it fol-

lows from the assumption that E ∝ e−αz.

For the second proﬁle, PTPA, we assume that ∆σremains

proportional to Ebut that the absorption is nonlinear, with a

two-photon absorption (TPA) coeﬃcient β[22]. For simplic-

ity, we further assume that the pump intensity has a rectangu-

lar temporal proﬁle with duration τpand that the pump reﬂec-

tion coeﬃcient Rpremains constant. This allows us to express

σanalytically as

σ(ω, z,f;PTPA)=¯σ(ω)+ ∆σTPA(ω)f(1 +f)e−αz

1+f(1 −e−αz)2,(5)

where now f=F/FTPA with FTPA =(ατp/β)/(1 −Rp).

As Fig. 2(b) shows, the surface photoconductivity

∆σs(ω, f;PTPA)=f(1 +f)∆σTPA(ω) increases quadrati-

cally with fwhen f1, where TPA dominates. At the same

time, deﬀ= Λ/(1 +f) decreases with f, which compensates

for the superlinear growth of ∆σsand causes the sheet

photoconductance,

∆G(ω, f;PTPA)= ∆σTPA(ω)Λf,(6)

to remain strictly proportional to f.

Now consider the systematic error that we introduce if we

assume the wrong proﬁle. If the true proﬁle is Psat but we as-

sume it is Pexp, for example, then we would infer the surface

conductivity to be σs(ω, f;Psat 7→ Pexp), where the notation

Psat 7→ Pexp indicates that we use the Psat proﬁle to compute

r(ω) with a source spectrum σs(ω, f;Psat), then use the Pexp

proﬁle to infer an image spectrum σs(ω;Psat 7→ Pexp) from

r(ω). The requirement that the source and image proﬁles yield

the same r(ω) is roughly equivalent to holding ∆G= ∆σsdeﬀ

constant for K3C60, so the image transformation eﬀectively

rescales the source ∆σsby deﬀ(Psource)/deﬀ(Pimage). Since

deﬀ= Λ for Pexp, we divide Eq. (4) by Λto get

∆σs(ω, f;Psat 7→ Pexp)≈∆σsat(ω) ln(1 +f),(7)

which overestimates ∆σs(ω, f;Psat) by (1 +f) ln(1 +f)/f.

Similarly, dividing Eq. (6) by Λgives

∆σs(ω, f;PTPA 7→ Pexp)≈∆σTPA(ω)f,(8)

which underestimates ∆σs(ω, f;PTPA) by 1/(1 +f).

Figure 3 shows the ﬂuence dependence reported by

Mitrano et al. [6] for ∆σs1(ω;Pexp) in K3C60, which we use

to infer the proﬁle. The measurements reveal a clear sub-

linear ﬂuence dependence that is inconsistent with the re-

lationship expected for ∆σs1(ω, f;PTPA 7→ Pexp) given in

FIG. 3. Least-squares ﬁt with ∆σs1(ω, f;Psat 7→ Pexp) (dashed

line) to the ﬂuence dependence of ∆σs1(ω;Pexp) reported by

Mitrano et al. [6] (points with error bars). The ﬁt is constrained

to pass through the anchor point ∆σs1(ω;Pexp) (open circle) at

F=3 mJ/cm2reported by Budden et al. [8] for ~ω=6.46 meV.

We multiply the results of Mitrano et al. [6] by an overall

scale factor Ato account for systematic diﬀerences from the

results of Budden et al. [8]. Best-ﬁt parameter values are

Fsat =(1.0±0.5) mJ/cm2and A=0.65 ±0.06 (χ2=3.7, d.o.f. =5).

The solid line extrapolates the source function ∆σs1(ω, f;Psat) from

its value of at F=3 mJ/cm2(open triangle).

Eq. (8) [22]. And as we noted earlier, the deviation from

linearity is also incompatible with the assumptions that yield

the Pexp proﬁle used in the original analysis. A ﬁt with

∆σs1(ω, f;Psat 7→ Pexp), however, is nearly indistinguishable

from the experimental results—which means that the source

function ∆σs1(ω, f;Psat), shown as a solid line in Fig. 3, is

the best estimate for the true surface photoconductivity. Note

that this deviates signiﬁcantly from the originally reported re-

sults at all ﬂuences, and is nearly a factor of 2 smaller than the

result reported by Budden et al. [8] at F=3.0 mJ/cm2.

We ﬁx Fsat at the value obtained from this ﬁt and extend

our analysis as a function of frequency in Fig. 4. We derive

the alternative spectrum σs(ω;Psat) so that its image in Pexp

is equal to σs(ω;Pexp) reported by Budden et al. [8]. Both

spectra show decreases in σs1(ω) and increases in σs2(ω), but

by diﬀerent amounts. Since ∆σsis inversely related to deﬀ,

∆σs(ω;Pexp) has a smaller magnitude than ∆σs(ω, f;Psat).

These quantitative diﬀerences suggest qualitatively diﬀer-

ent physical interpretations. The spectrum with Pexp looks

like that of a superconductor [6–8]: σs1(ω;Pexp) falls to near

zero below ~ω≈10 meV, σs2(ω;Pexp) is enhanced at low fre-

quencies, and a Drude-Lorentz ﬁt yields a carrier relaxation

rate γ=0 [22]. But the spectrum with Psat looks like a nor-

mal metal with a photoenhanced mobility: σs1(ω, f;Psat) lies

well above zero at all ωand clearly increases with decreas-

ing ωbelow ~ω≈9 meV, while σs2(ω, f;Pexp) shows more

moderate enhancement at low frequencies. A Drude-Lorentz

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OpticalSaturationProducesSpuriousEvidenceforPhotoinducedSuperconductivityinK3C60J.StevenDodge,LeyaLopez,andDerekG.SahotaDepartmentofPhysics,SimonFraserUniversity,Burnaby,BritishColumbiaV5A1S6,Canada(Dated:March8,2023)Wediscussasystematicerrorintime-resolvedopticalconductivitymeasurementsthatbecomesi...

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Optical Saturation Produces Spurious Evidence for Photoinduced Superconductivity in K 3C60 J. Steven Dodge Leya Lopez and Derek G. Sahota

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