Nonlinear Meissner Effect and Enhancement of the Field Onset of Penetration of Vortices in Dirty Nb

Films Under RF and DC Fields.

Alex Gurevich

National High Magnetic Field Laboratory, Tallahassee, FL 32310

6-th Thin Film SRF Workshop, Padova, Italy, Oct. 4-6, 2010

N. Groll, AG and I. Chiorescu, Phys. Rev. B81, 020504(R) (2010)

Outline

• Linear and nonlinear Meissner effect. Crucial effect of impurities.

• Thin film strip resonator. Frequency shift due to field dependent superfluid density.

• Effect of dc parallel magnetic field

• Theory: Usadel equations

• Comparison of theory with experiment

• Nonlinearity of the surface resistance

• Implications for Nb cavities and multilayer coating

Meissner Effect

Meissner current density Js(y):

For H = 100 mT the Meissner current density in Nb:

λλ

λ/0/

0 )(,)( ys

y eH

yJeHyH −− ==

)(

)(

T

THJ cd λ

≅ Jd(0) ≈ 4 MA/mm2

for pure Nb

Js(0) ≈ 2 108 A/cm2

Maximum (pairbreaking) current density:1

3 mm

100 nm

London equations (1935)

Two-fluid model: coexisting SC and N “liquids” with the densities ns(T) + nn(T) = n.

Second Newton law yields the first London equation:

dJs/dt = (e2ns/m)E J = σE (ballistic electron flow in SC) (viscous N flow in metals)

TTc

ns

nn

Maxwell equations, ∇× E = -µ0∂tH and ∇× H = Js give the second London equation:

λ2∇H - H = 0

London penetration depth:

2/1

02 )(

=

µλ

Tne

m

s

Superconducting current

ABne

mAJ

s

×∇=

=−= ,,

2/1

20

20 µ

λλµ

Superfluid current density ns is independent of current density at J << Js: linear Meissner effect

Any dependence of ns(J) on current density results in the nonlinear Meissner effect

GL current pairbreaking near Tc

Crucial effect of nonmagnetic impurities on current pairbreaking at T << Tc

Clean limit: ns is equal to the total electron density: no NLME?

Dependence of the SC gap on current

• Clean limit (l >> ξ 0) • Dirty limit (l << ξ 0)

In the clean limit the gap ∆ (J) is independent of RF field at low T J. Bardeen, Rev. Mod. Phys. 34, 667 (1962).

∆2(J)/∆2(0)

J/Jd(0)1

T << Tc

1

0.54

T ≈ Tc

2/3

∆2(J)/∆2(0)

J/Jd(0)1

T << Tc1

0.54

T ≈ Tc

2/3

Intermodulation caused by NLME

• NLME in oscillating rf field

• Higher order harmonics in current density

λωλλµ

/2

2

20

)(,sin)(),(,1 xa

c

eBxBtxBtxAA

AC

AJ −==

−−=

220

3

32

2

20

1

31

8

)(,

8

)(31

)(

3sin)(sin)(),(

cc B

xCBJ

B

xCBxBJ

txJtxJtxJ

λµλµ

ωω

=

+−=

+=

Intermodulation measurements is one of the ways of revealing the NLME

T. Dahm and D.J. Scalapino, J. Appl. Phys. 81, 2002 (1997); Phys. Rev. B 60, 13125 (1999); D.E. Oates, J. Supercond. Novel Magn. 20, 3 (2007).

Effect of NLME on kinetic inductance

• NLME be probed by measuring the reacive part of the surface impedance: Z = Rs + iXs

• Kinetic energy of SC currents in a film of

width w, length s and thickness d < λ defines the kinetic inductance Lk: m

eAv

VvmnILs

ssk 2,

22

22

==

wd

BTJsLk

),,(20 λµ=

• NLME manifests itself in the dependence of Lk on H• Change of the resonance frequency of a strip line

Thin film resonator stripline

• 65 nm thick Nb thin film stripline w = 100 µm, s = 3mm

• resonance frequency f = 19.2 GHz

• dirty Nb with ρn(Tc) = 26.3 µΩcm Tc = 7.25K, λ = 226 nm, ξ = 9 nm

• 3 coils enabled us to vary the orientation of the dc field ( < 2T)

• NLME produced by dc Meissner currents is probed by very weak rf currents ( Prf < 10 pW)

Measure the change of the resonance frequency f = 1/2π (CL)1/2 as a function of the parallel dc magnetic field: 02

)()0(

L

HLL

f

f kk −=δ

T = 80 mK

Effect of dc filed

y

1,2

,1

00

20

2

20

≅=

−−=

CA

A

AC

AJ

πξφ

λµ

Field B inclined by angle ϕ with respect to the x-axis induces vector potentials:

Ay = -zBcos ϕ and Ax = zBcosϕ + δAω: only Ax is coupled linearly with weak rf field

Total rf current defines λ (H)

2

2

0

22

0

)2cos2(2

31)(,

)(λϕ

φξπλ

λµδδ ω

ω

−

+=−= dBC

BB

AdwI

The NLME correction to is quadratic in B and has 2 – cos2ϕ angular dependence

d

w

z

x

H

ϕδAω

Field range of the Meissner state is greatly increased in a thin film with d < λ :

• NLME can only exist in the Meissner state below the lower critical field B < Bc1

• NLME correction becomes significant for B ∼ Bc so for type-II superconductors with κ = λ /ξ >> 1, the NLME correction is very small because Bc1 << Bc

• Main problem with the observation of NLME in bulk superconductors

• Enhancement of parallel Bc1 in a thin film with d << λ : (Abrikosov, 1964):

+== 5.0ln4

,ln2

20

120

1 ξλ

πλφ

ξπφ bulk

cfilmc B

d

dB

NLME can also reveal the enhancement of the vortex penetration field by thin film coating

Calculation of λ(H) in the dirty limit

• Solve perturbatively in A and caclulate the first quadratic NLME correction at T = 0:

∑∑>

Ω

>

−==∆

∆−=

−∇

0

2

0

20

222

,sin2,sin2

,cossin2sin2

2

ωωαπαπγ

ααωαφ

πα

QeNDTJT

ADn

ππ

ππ

λϕξφ

ξπλ

6

1

24

7

32,

18

1

96

)2cos2)((2

1)(

2

21

222

21

2

0

2

++=+=

−+

+=

CC

CdCB

B

Solve Usadel equations to calculate λ (B) for any T

D = lvF/3, ωn = πT(2n+1)

Detecting the Meissner state by the behavior of ∆f

• Cusp in ∆ f if B is parallel to the film

• Q does not change much up to B = 1T

• Drop of Q(H) for B > 1T

• Meissner state up to 1T

• Consistent with Bc1

Field and angular dependence of ∆f

• Quadratic field dependence of ∆ f up to 1T

• 2 – cos2ϕ dependence is in good agreement with experiment

• Quantitative agreement between theory and experiment

RF dissipation

• Thermal activation of normal electrons

• Scattering mechanisms and normal state conductivity: σ n = e2n0l/pF, pF = ħ(3π 2n0)1/3

- Normal skin effect ( l << λ ): multiple impurity scattering in the λ - belt:

Rs ∼ (µ 02ω 2λ 3σn∆ /T)exp(-∆ /T)

- Anomalous skin effect in the clean limit ( l >> λ ): scattering by the gradient of the rf field:

k

E

∆

H(t)E(z,t)

∆−

+

∆∆∝

TkC

Tpk

nR

BFBs expln 0

0422

0

ωλωµ

Effect of nonmagnetic impurities on low field RBCS

• Effect on the linear surface resistance

- No suppression of the superconducting gap (Anderson theorem)

- Increase of the London penetration depth

- Increase of the BCS surface resistance

- Decrease the lower critical field (the onset of vortex penetration)

Nonmagnetic impurities appear to be not too bad for RBCS, but are they benign at high rf fields?

E. Palmieri

Thermal activation in rf field in the clean limit

2∆ - 2vpF2∆ + 2vpF

Rocking “tilted” electron spectrum in the current-carrying rf state J = J0cosω t

Superfluid velocity vs(t) = J/nse

Reduction of the gap ∆ (vs) = ∆ - pF|vs| in the electron spectrum increases the density

of thermally-activated normal electrons nr(J), thus increasing Rs (Kulik and Palmieri; Gurevich)

)()2/()( 222 tvpEmppE sFF

±−+∆±=

Critical pairbreaking velocity:

Fc pv

∆= Clean limit

Nonmagnetic impurities are pairbreakers at high fields

• NLME experiment and theory show that nonmagnetic impurities are pairbreakers at high fields, reducing the superconducting gap and the superfluid density

• Increase of Rs as the field increases

Rs ∝ λ 3(H)∆ (H)exp[-∆(H)/T] ∝ exp [C∆(H/Hc)2 ]∆ /T]

Here λ (H) = [ 1 + Cλ(H/Hc)2 ]λ and ∆ (H) = [ 1 - C∆(H/Hc)2 ]∆ and Cλ and C∆ are ∼ 1

• Exponential increase of Rs (H) with H

Conclusions

• Dirty Nb surface can result in the nonlinear Meissner effect

• Intermodulation effects: generation of higher harmonics of rf current

• NLME causes pairbreaking at high fields – possible mechanism for high-field Q slope

• Measurements of the nonlinear Meissner effect revealed the 5-fold increase of the vortex penetration field in a thin Nb film