dc SQUID

(1)

Enrico Silva - diritti riservati - Non è permessa, fra l’altro, l’inclusione anche parziale in altre opere senza il consenso scritto dell’autore

dc SQUID

Negligible screening, zero voltage

Circulating currents do not compensate the external ﬂux

I1,s= Icsin(γ1) I2,s= Icsin(γ2)

β L = 2LI ^c Φ ⁰ ≪ 1

I s,max ≃ 2I c

!

! cos

"

π Φ a

Φ ⁰

#!

!

Figures from:

R. Gross, A. Marx, Walther Meissner Institut http://www.wmi.badw.de/teaching/Lecturenotes/

Φ ≃ Φ ^a

Nonzero screening, zero voltage

Circulating currents compensate the external ﬂux.

Strong screening: towards ﬂux quantization in the loop.

Figures from:

β L = 2LI ^c Φ ⁰

Φ ^a Φ ⁰ = Φ

Φ ⁰ + β L

2 + sin

! π Φ

Φ ⁰

"

Φ != Φ ^a

Φ ^a /Φ ⁰

Thursday, November 8, 12

(2)

!"#$%&'

**()*)+(**

,

Negligible screening, nonzero voltage

β L = 2LI ^c Φ ⁰ ≪ 1

Φ ≃ Φ ^a

!V (t)" = I ^c R N

!

" I 2I ^c

# ²

− cos ²

"

π Φ ^a Φ ⁰

#

Maximum modulation of the average voltage for

I=2I c .

Figures from:

Circulating currents do not compensate the external ﬂux

!"#$%&'

**()*)+(**

_,

dc SQUID: transfer function

!V (t)" = I

^c

R

N

!

" I 2I

c

#

²

− cos

²

"

π Φ

^a

Φ

⁰

#

Figures from:

Usual operational point:

Flux-to-voltage converter.

Transfer function:

I ! 2I ^c H =

!

∂ !V (t)"

∂Φ ^a

!

Φ ^a ≃ (2n + 1) Φ ⁰

4 H ≃ I c R N

Φ ⁰ /2 ≃ R N

L One ﬁnds numerically β ! 1

dc SQUID: parameters

!V (t)" = I

c

R

N

!

" I 2I

c

#

²

− cos

²

"

π Φ

^a

Φ

⁰

#

Figures from:

Usual operational point:

Flux-to-voltage converter.

Transfer function:

I ! 2I ^c H =

!

∂ !V (t)"

∂Φ ^a

!

Φ ^a ≃ (2n + 1) Φ ⁰

4 H ≃ I ^c R ^N

Φ ⁰ /2 ≃ R ^N L One ﬁnds numerically β ! 1

Other requirements:

• large loop (large ΔΦ with small ΔB). But then large L! –> small H and small β

Thursday, November 8, 12

(3)

“washer” SQUIDs

Figures from:

junctions

“ﬁeld focussing” device: the large superconducting area expels the ﬂux, and

focusses into the small hole.

One ﬁnds:

• effective area (where ﬁeld is collected) scales with W:

• inductance scale with D: L ≃ 1.25µ ⁰ D

A _{ef f} ∼ D · W

Caveats: W too large –> ﬂuxons enters the superconducting area, and thermal activated motion leads to noise

real dc SQUID

Figures from:

Coupling to the signal.

Figures from:

Fossheim-Sudbo

Current nodes I

Signal coil

Base electrode

Counter electrode Josephson junctions

DC SQUID

I

High frequencies Low frequencies Parasite capacitance

A spiral coil brings the signal over the SQUID area

Thursday, November 8, 12

(4)

SQUIDs as magnetometers

Biomagnetism

^!"

Figures from:

Transformer conﬁgurations

Figures from:

Fossheim Sudbo

Pickup coil

→

Signal coil

SQUID B _ext

→

B _coupled

Gradiometer conﬁgurations

Sensitive to spatial derivatives

Figures from:

∂B

∂z

**!"#!$%&'!(#)+,+' -$!.!'%&'!(#)+,+'**

1

^st

!"#$" 2

^nd

!"#$" 1

^st

!"#$"%&dB

z

'#( 1

^st

!"#$"%&dB

z

'#)

Thursday, November 8, 12

dc SQUID

dc SQUID

Negligible screening, zero voltage

Circulating currents do not compensate the external ﬂux

β L = 2LI c Φ 0 ≪ 1

I s,max ≃ 2I c

!

!

!

! cos

"

π Φ a

Φ 0

#!

!

!

!

Φ ≃ Φ a

Nonzero screening, zero voltage

Circulating currents compensate the external ﬂux.

Strong screening: towards ﬂux quantization in the loop.

β L = 2LI c Φ 0

Φ a Φ 0 = Φ

Φ 0 + β L

2 + sin

! π Φ

Φ 0

"

Φ != Φ a

Φ a /Φ 0

Thursday, November 8, 12

()*)+(

Negligible screening, nonzero voltage

β L = 2LI c Φ 0 ≪ 1

Φ ≃ Φ a

!V (t)" = I c R N

!

" I 2I c

# 2

− cos 2

"

π Φ a Φ 0

#

Maximum modulation of the average voltage for

I=2I c .

Circulating currents do not compensate the external ﬂux

()*)+(

dc SQUID: transfer function

!V (t)" = I

R

!

" I 2I

#

− cos

"

π Φ

Φ

#

Usual operational point:

Flux-to-voltage converter.

Transfer function:

I ! 2I c H =

!

!

!

!

∂ !V (t)"

∂Φ a

!

!

!

!

Φ a ≃ (2n + 1) Φ 0

4 H ≃ I c R N

Φ 0 /2 ≃ R N

L One ﬁnds numerically β ! 1

dc SQUID: parameters

!V (t)" = I

R

!

β L = 2LI ^c Φ ⁰ ≪ 1

Φ ⁰

Φ ≃ Φ ^a

β L = 2LI ^c Φ ⁰

Φ ^a Φ ⁰ = Φ

Φ ⁰ + β L

Φ ⁰

Φ != Φ ^a

Φ ^a /Φ ⁰

**()*)+(**

β L = 2LI ^c Φ ⁰ ≪ 1

Φ ≃ Φ ^a

!V (t)" = I ^c R N

" I 2I ^c

# ²

− cos ²

π Φ ^a Φ ⁰

**()*)+(**

I ! 2I ^c H =

∂Φ ^a

Φ ^a ≃ (2n + 1) Φ ⁰

Φ ⁰ /2 ≃ R N

I ! 2I ^c H =

∂Φ ^a

Φ ^a ≃ (2n + 1) Φ ⁰

4 H ≃ I ^c R ^N

Φ ⁰ /2 ≃ R ^N L One ﬁnds numerically β ! 1

• inductance scale with D: L ≃ 1.25µ ⁰ D

A _{ef f} ∼ D · W

SQUID B _ext

B _coupled

**!"#!$%&'!(#)+,+' -$!.!'%&'!(#)+,+'**