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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)
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