Chapter 8

Microwave Applications

AS-Chap. 8 - 2

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• passive microwave devices – filters – resonators

based on the very small losses due to the small microwave surface resistance of superconducting materials improved performance and/or smaller device size

• active microwave devices – microwave sources – mixers

Josephson junctions as voltage controlled oscillators: 𝑓

𝑉= 483 597.9 GHz/V

Josephson junctions as nonlinear elements

wide application field

8 Microwave Applications

AS-Chap. 8 - 3

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8.1 High Frequency Properties of SCs

• we make use of the two-fluid model of superconductivity total carrier density 𝑛 = 𝑛𝑛 + 𝑛𝑠/2 (factor ½ due to Cooper pairs)

• transport properties of normal fluid:

Ohm‘s law 𝐽𝑛 = 𝜎𝑛𝐸 =1

𝜌𝑛𝐸

𝜎𝑛 =𝑛𝑛𝑒2𝜏

𝑚 (normal conductivity)

• transport properties of superfluid: 1. London equation (linearized)

(London coefficient)

(London penetration depth)

AS-Chap. 8 - 4

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8.1.1 AC Conductivity

• we assume a harmonic current with angular frequency 𝜔: with 𝐉𝑛 = 𝑛𝑛𝑒𝐯𝑛 and we obtain

complex normal conductivity:

with 𝜎0 =𝑛𝑒2𝜏

𝑚 (normal state conductivity: 𝑛𝑛 = 𝑛)

normal fluid component

AS-Chap. 8 - 5

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8.1.1 AC Conductivity

• we assume a harmonic supercurrent with angular frequency 𝜔: from we obtain

superfluid component

purely complex conductivity of superfluid:

• note that the conductivities 𝜎𝑛 and 𝜎𝑠 are strongly temperature dependent

AS-Chap. 8 - 6

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8.1.1 AC Conductivity

no resistive part (no loss)

purely inductive part (inertia) 𝐿𝑠(𝑇)

Two-fluid picture in electrotechnical language

~ V Ls(T)

Ln(T)

Rn(T)

I

superfluid channel normal channel

superconductor

both resistive (loss)

and inductive part (inertia)

𝐿𝑛 𝑇 , 𝑅𝑛(𝑇)

two parallel conduction channels

AS-Chap. 8 - 7

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8.1.1 AC Conductivity

Two-fluid picture in electrotechnical language

• 𝜔 = 0: all current flows through superfluid channel

• 𝜔 > 0: 𝐽𝑠 decreases and 𝐽𝑛 increases

• 𝜔 = 𝜔𝑛𝑠 = 𝑅𝑛/𝐿𝑠: 𝐽𝑠 ≃ 𝐽𝑛 crossover frequency

normal channel dominates at high frequencies

• 𝜔𝑛𝑠 < 𝜔 < 𝜔𝜏 =𝑅𝑛

𝐿𝑛=

1

𝜏 : resistive contribution dominates in normal channel

• 𝜔𝜏 < 𝜔 < 𝜔Δ : inductive contribution dominates in normal channel

• 𝜔Δ < 𝜔 : above the gap frequency Cooper pairs can be broken up complicated behavior

superfluid channel dominates at low frequencies

AS-Chap. 8 - 8

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8.1.1 AC Conductivity

𝝎 𝝎𝚫 𝝎𝝉 𝝎ns

𝝎 𝝎𝚫 𝝎𝝉 𝝎ns

superfluid channel

dominates

normal channel dominates

ohmic inductive

superfluid channel dominates

normal channel dominates inductive

0

0

T close to Tc

T << Tc

crossover frequencies depend on temperature since 𝑛𝑛 and 𝑛𝑠 depend on T

AS-Chap. 8 - 9

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8.1.1 AC Conductivity

total conductivity

• normal component

• superfluid component

• 𝜔𝜏 ≪ 1 (low-frequency approximation):

resistive losses in normal component

inductive response due to inertia of Cooper pairs

AS-Chap. 8 - 10

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8.1.2 Surface Impedance

• surface impedance – characteristic impedance seen by a plane wave incident perpendicular upon a

flat surface of a conductor – given by the ratio of the electric and the magnetic field at the surface

𝑯𝒚 𝑬𝒙 x

y z

(skin depth)

with ∇ × 𝐸 = 𝜕𝐵/𝜕𝑡

AS-Chap. 8 - 11

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8.1.2 Surface Impedance

• surface impedance of a normal metal

for 𝜔𝜏 ≪ 1, we have 𝜎 𝜔 ≃𝑛𝑒2𝜏

𝑚= 𝜎0 real number

𝑅𝑠 and 𝑋𝑠 are proportional to 𝜔 !! example: Au or Cu @t 100 GHz and room temperature 𝛿 ≃ 0.25 𝜇𝑚 and 𝑍𝑠 ≃ 0.1 1 + 𝑖 Ω/⊡

AS-Chap. 8 - 12

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8.1.2 Surface Impedance

• surface impedance of a superconductor

use

for 𝜔𝜏 ≪ 1, and using 𝜎 = 𝜎1 − 𝑖𝜎2 with 𝜎1 = 𝜎𝑛 and 𝜎2 = 1/𝜔𝜇0𝜆𝐿2

2𝑖 = (1 + 𝑖)

AS-Chap. 8 - 13

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8.1.2 Surface Impedance

further simplification at 𝑇 ≪ 𝑇𝑐: 𝜎𝑠 ≪ 𝜎2 approximation 1 + 𝑥 −1/2 ≃ 1 −1

2𝑥

with 𝜎1 = 𝑛𝑛𝑒2𝜏/𝑚𝑛 and 𝜎2 = 1/𝜔𝜇0𝜆𝐿2

𝑅𝑠 ∝ 𝜔2𝜆𝐿3 𝑛𝑛

𝑛 and 𝑋𝑠 ∝ 𝜔𝜆𝐿!! different from normal metals: 𝑅𝑠 ∝ 𝜔

example: Nb @ 100 GHz and 𝑇 ≪ 𝑇𝑐 , 𝑍𝑠 ≃ 𝑖𝜔𝜇0𝜆𝐿 𝜆𝐿 ≃ 0.1 𝜇𝑚 and 𝑍𝑠 ≃ 𝑖 0.08 Ω/⊡

AS-Chap. 8 - 14

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108

109

1010

1011

10-6

10-4

10-2

surf

ace

resi

stan

ce (

/ÿ)

frequency (Hz)

8.1.2 Surface Impedance

AS-Chap. 8 - 15

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8.1.2 Surface Impedance

for 𝜔𝜏 ≪ 1 and 𝑇 ≪ 𝑇𝑐:

kinetic inductance

• surface reactance 𝑋𝑠 is purely inductive the equivalent inductance, 𝑋𝑠 = 𝑖𝜔𝐿𝑘 , is denoted as kinetic inductance

𝐿𝑘 reflects the kinetic energy of the superfluid (acceleration of superfluid requires energy)