PHYSICAL REVIEW B 84, 214502 (2011)

Hybrid superconducting quantum magnetometer

F. Giazotto* and F. Taddei†

NEST Istituto Nanoscienze-CNR and Scuola Normale Superiore, I-56127 Pisa, Italy(Received 17 May 2011; revised manuscript received 15 November 2011; published 2 December 2011)

A superconducting quantum magnetometer based on magnetic flux-driven modulation of the density of statesof a proximized metallic nanowire is theoretically analyzed. With optimized geometrical and material parameterstransfer functions up to a few mV/�0 and intrinsic flux noise ∼10−9�0/

√Hz below 1 K are achievable. The

opportunity to access single-spin detection joined with limited dissipation (of the order of ∼10−14 W) make thismagnetometer interesting for the investigation of the switching dynamics of molecules or individual magneticnanoparticles.

DOI: 10.1103/PhysRevB.84.214502 PACS number(s): 72.25.−b, 85.75.−d, 74.50.+r

I. INTRODUCTION

The superconducting quantum interference device1

(SQUID) is recognized as the most sensitive magnetic fluxdetector ever realized, and combines the physical phenom-ena of Josephson effect2 and flux quantization3 to operate.SQUID’s are nowadays exploited in a variety of physi-cal measurements4,5 with applications ranging, for instance,from pure science to medicine and biology.1,6 Recently, theinterest in the development of nanoscale SQUID’s7–9 hasbeen motivated by the opportunity to exploit these sensorsfor the investigation of the magnetic properties of isolateddipoles,10–14 with the ultimate goal to detect one single atomicspin, i.e., one Bohr magneton.

Here we theoretically analyze a hybrid superconductinginterferometer that exploits the phase dependence of thedensity of states (DOS) of a proximized metallic nanowireto achieve high sensitivity to magnetic flux. The operation of aprototype structure based on this principle, the SQUIPT,15 hasbeen recently reported. We show that with a careful design,transfer functions as large as a few mV/�0 and intrinsic fluxnoise ∼10−9�0/

√Hz can be achieved below 1 K. Limited

dissipation joined with the opportunity to access single-spindetection make this structure attractive for the investigation ofthe switching dynamics of individual magnetic nanoparticles.

The paper is organized as follows. The model of thehybrid superconducting magnetometer is presented in Sec. II.The Josephson and quasiparticle currents are calculated inSecs. III and IV, respectively. The flux resolution and deviceperformance are finally presented in Sec. V, where we addressbriefly the feasibility of this structure as a single-spin detector.Sec. VI is devoted to conclusions.

II. MODEL

The interferometer [sketched in Fig. 1(a)] consists ofa diffusive normal metal (N) wire of length L in goodelectric contact (i.e., ideal interface transmissivity) with twosuperconducting electrodes (S1) that define a ring. We assumethe wire transverse dimensions to be much smaller than L

so that it can be considered as quasi-one-dimensional. Thecontact with S1 induces superconducting correlations in Nthrough the proximity effect,16–20 which is responsible for themodification of the wire DOS.21 For lower-transparency NS1

interfaces the proximity effect in the wire will be reduced,

thus weakening the effects described below. In addition, asuperconducting junction (S2) of width w and normal-stateresistance Rt is tunnel-coupled to the middle of the N region.The loop geometry allows the change of the phase difference(ϕ) across the NS1 boundaries through the application ofa magnetic field which modifies the wire DOS22 and thetransport through the tunnel junction.15,23

The proximity effect in the wire can be described withthe quasiclassical Usadel equations.16 The short-junction limit(i.e., for �1 � hD/L2 = ETh, where �1 is the order parameterin S1, D is the wire diffusion constant, and ETh is the Thoulessenergy) will be considered in the following, since in sucha regime the Usadel equations allow an analytic expressionfor the wire DOS,24 thus simplifying the device analysis. Inaddition, the interferometer performance is optimized in thislimit as the proximity effect in the wire is maximized.22,24

Assuming a step-function form for the order parameter �1,25

i.e., constant in S1 and zero in the N wire, the wire DOSnormalized to the DOS at the Fermi level in the absence ofproximity effect is given by24

NN (x,ε,T ,ϕ) = Re{cosh[θ (x,ε,T ,ϕ)]}, (1)

where

θ = arcosh(α(ε,ϕ,T )cosh{2xarcosh[β(ε,ϕ,T )]}), (2)

α =√

ε2/[ε2 − �2

1(T )cos2(ϕ/2)], (3)

and

β =√[

ε2 − �21(T )cos2(ϕ/2)

]/[ε2 − �2

1(T )]. (4)

In the above expressions, ε is the energy relative to the chemi-cal potential of the superconductors, T is the temperature, andx ∈ [−L/2,L/2] is the spatial coordinate along the wire. NN

exhibits a minigap (εg)

εg(ϕ) = �1(T )|cos(ϕ/2)| (5)

for |ε| � εg whose amplitude depends on ϕ and is constantalong the wire. In particular, εg = �1 for ϕ = 0 and decreasesby increasing ϕ, vanishing at ϕ = π . Finally, by neglecting thering inductance the phase difference becomes ϕ = 2π�/�0,where � is the total flux through the loop area and �0 =2.067 × 10−15 Wb is the flux quantum.

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F. GIAZOTTO AND F. TADDEI PHYSICAL REVIEW B 84, 214502 (2011)

Φ

N

Rt S2

ΦL ϕ /2 -ϕ /2

w

S1

x 0

Ibias

FIG. 1. (Color online) (a) Scheme of the device. L is the wirelength whereas w is the width of the superconducting tunnel junction(S2) coupled to the middle of the N region. ϕ is the macroscopicquantum phase difference in S1, while � is the magnetic fluxthreading the loop. Furthermore, Rt is the tunnel junction normal-stateresistance and Ibias is the current flowing through the structure.

III. JOSEPHSON CURRENT

At equilibrium, a current through the system Ieq can flowthanks to a direct Josephson coupling between the supercon-ducting electrode S2 [with order parameter equal to �2(T )eiχS

where χS is the macroscopic phase] and the proximized Nwire. We assume the BCS temperature dependence for �1,2(T )with critical temperature Tc1,2 = �0

1,2/(1.764kB ) where �01,2

is the zero-temperature order parameter in S1,2 and kB is theBoltzmann constant. Since the junction NS2 is extended in thex direction, one can calculate, in the tunneling limit, Ieq usingthe quasiclassical approach16,27,28 as the following integral:

Ieq(ϕ) = − 1

8ewRt

∫ w/2

−w/2dx

×∫ +∞

−∞dεTr

[σ3

[GN

R (ε,x,ϕ),GS2R (ε)

]

× tanh

(ε

2kBT

)], (6)

where σ3 is the third Pauli matrix, [·,·] represent the commu-tator, and e is electron charge. Furthermore, GN

R (ε,x,ϕ) is theposition-dependent retarded Green’s function on the N wireand G

S2R (ε) is the retarded Green’s function of the electrode

S2. They are defined as follows:

GNR (ε,x,ϕ) =

(cosh θ sinh θeiχ

− sinh θe−iχ − cosh θ

)(7)

and

GS2R (ε) = 1√

ε2 − �22

(ε �2e

iχS

−�2e−iχS −ε

), (8)

0.0 0.5 1.00.0

0.5

1.0

1.5

2.0

2.5

3.0 0.10 0.50 0.70 0.90 0.95 0.98

I eq [Δ

0 2/ eR

t]

Φ/Φ0

T [Τc2]

FIG. 2. (Color online) Josephson current vs flux (Ieq − �) char-acteristics calculated for a few values of temperature T assuming�0

1 = 0.1ETh and �01 = 4�0

2. Thick lines represent the supercurrentcalculated using the Ambegaokar-Baratoff formula Eq. (11), whereasthe thin lines are the supercurrent calculated for an extended tunneljunction with width w = L/2.

where θ is defined in Eq. (2) and

χ = −arctan(γ (ε,ϕ,T )tanh{2xarcosh[β(ε,ϕ,T )]}), (9)

with

γ =√[

ε2 − �21(T )cos2(ϕ/2)

]/[�1(T )cos(ϕ/2)

]. (10)

Note that the supercurrent Ieq depends on the phase χS of theorder parameter in the electrode S2. We are interested in thecritical current that we determine by fixing χS such that itgives the maximum supercurrent for a given value of ϕ. InFig. 2 the equilibrium critical current is plotted, for differentvalues of temperature, as a function of ϕ in units of �0

2/(eRt ),assuming �0

1 = 0.1ETh, �01 = 4�0

2, and w = L/2. For thesake of comparison we also plot the supercurrent calculatedthrough the Ambegaokar-Baratoff29 formula, relative to apointlike NS2 junction:

IABeq = πεg(ϕ)�2(T )kBT

eRt

×∑

l=0,±1,...

1√[ω2

l + ε2g(ϕ)

][ω2

l + �22(T )

] , (11)

where ωl = πkBT (2l + 1). Interestingly, for our choice ofparameters the difference between Ieq and IAB

eq is hardlynoticeable: the lateral spatial extension of the NS2 junctionalong x plays a marginal role. As a matter of fact, thesupercurrent turns out to be negligible in the experimentsreported in Refs. 15 and 37. The reason for such Josephsoncurrent suppression is currently unclear. One possibility is thatthe system is brought out of equilibrium by voltage fluctuationsthat might originate from the measuring circuit. Such voltagefluctuations will drop mostly across the tunneling barrier, beingthe most resistive component of the system. As a result, theJosephson current will oscillate at high frequency, hinderingthe possibility of detection. This fact is fortunate, since asupercurrent might prevent a correct voltage readout of thedevice, which is not observed experimentally.

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HYBRID SUPERCONDUCTING QUANTUM MAGNETOMETER PHYSICAL REVIEW B 84, 214502 (2011)

IV. QUASIPARTICLE CURRENT

The current through the tunnel junction biased at voltageV is therefore dominated by the quasiparticles, and can bewritten as3

I = 1

ewRt

∫ w/2

−w/2dx

∫dεNN (x,ε,ϕ)NS2(ε)F (ε,ε), (12)

where

NS2(ε,T ) = |ε|√ε2 − �2(T )2

�[ε2 − �2(T )2] (13)

is the normalized DOS of the S2 electrode, ε = ε − eV,�(y) is the Heaviside step function, F (ε,ε) = [f0(ε) −f0(ε)], and f0(ε) = [1 + exp(ε/kBT )]−1 is the Fermi-Diracenergy distribution. In the following we set �0

2 = 200 μeVand �0

1 = 4�02 = 800 μeV as representative values for a

structure exploiting aluminum (Al) and vanadium (V) assuperconductors,30,31 respectively, w = L/2 and Rt = 5 M�.

Figure 3 shows the interferometer current vs voltage (I -V )characteristics calculated at T = 0.1Tc2 for different values ofthe applied flux �.32 In particular, for � = 0 the I -V charac-teristic resembles that typical of a superconductor-insulator-superconductor junction (i.e., S1IS2 where I denotes an insu-lator) where the minigap in the wire is maximized [i.e., εg =�1(T )], and the onset for large quasiparticle current occursat3 V = [�1(T ) + �2(T )]/e. For � = �0/2 the characteristicis similar to that of a normal metal-insulator-superconductorjunction (i.e., NIS2) with εg suppressed. The curves show apeak at V = |�1(T ) − �2(T )|/e, which corresponds to thesingularity appearing in the tunneling characteristic betweendifferent superconductors.3 In a current-biased setup the inter-ferometer operates as a flux-to-voltage transducer providing avoltage response V (�) that depends on the bias current Ibias

fed through the tunnel junction (see Fig. 3). For any Ibias, V (�)is determined by solving the equation Ibias − I = 0.

Figure 4(a) shows V (�) at T = 0.1Tc2 calculated forseveral Ibias values. V (�) turns out to be maximized at thelower bias currents where the voltage swing obtains values aslarge as 4�0

2/e, whereas it is gradually reduced by increasing

0 2 4 610-12

10-10

10-8

10-6

10-4

10-2

100

0 0.4 0.1 0.45 0.2 0.5 0.3

I [Δ0 2

/eR

t]

V [Δ0

2/e]

Φ [Φ0]

V (Φ)

Ibias

FIG. 3. (Color online) Interferometer quasiparticle current vsvoltage (I − V ) characteristics calculated for a few values of �

at T = 0.1Tc2. Tc2 is the critical temperature of S2, Ibias is thecurrent flowing through the device, and V (�) is the resulting voltagemodulation.

0.0 0.5 1.0 1.5 2.0-15

-10

-5

0

5

10

15

[Δ0 2

/eΦ 0]

Φ [Φ0]

0.0 0.5 1.0 1.5 2.01

2

3

4

5

6

0.2 1.0 1.5 2.0 2.5 3.0 4.0

V [ Δ

0 2/e

]

Φ [Φ0]

Ibias

[Δ0

2/eR

t ]

(a)

(b)

FIG. 4. (Color online) (a) V vs � calculated for several biascurrents Ibias at T = 0.1Tc2. (b) F vs � calculated for the same Ibias

values and T as in (a).

Ibias. The interferometer performance is thus improved at lowIbias.

An important figure of merit of the interferometer isrepresented by the flux-to-voltage transfer function1

F(�) = ∂V

∂�, (14)

which is shown in Fig. 4(b) for the same Ibias values as inpanel (a). In particular, F as large as �12.5�0

2(e�0)−1 canbe obtained at the lowest currents, whereas it is graduallysuppressed at higher Ibias.

The role of the temperature is shown in Fig. 5(a) whichdisplays V (�) calculated for several T values at I =1.0�0

2/(eRt ). An increase in T leads to a reduction of V (�)as well as to a suppression and smearing of the voltage swing.This directly reflects on the transfer function, as displayed inFig. 5(b). We note that even at T = Tc,2, i.e., when S2 is driveninto the normal state, F as large as �8.6�0

2(e�0)−1 can beachieved. It follows that voltage swings up to 0.8 mV and Fas large as 2.5 mV/�0 can be achieved with the suggestedmaterials combination for T � 1 K.

V. NOISE AND DEVICE PERFORMANCE

We now turn to the discussion of the noise properties of theinterferometer. In the actual current-biased setup an importantquantity is represented by the voltage noise spectral density(SV ) defined as

SV = R2dSI , (15)

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F. GIAZOTTO AND F. TADDEI PHYSICAL REVIEW B 84, 214502 (2011)

FIG. 5. (Color online) (a) V vs � calculated for a few temper-atures at Ibias = 1.0�0

2/eRt . (b) F vs � calculated for the same T

values and Ibias as in (a).

where Rd = ∂V/∂I is the tunnel junction dynamic resistance,and SI is the current noise spectral density (shot noise) givenby34

SI = 2

wRt

∫ w/2

−w/2dx

∫dεNN (x,ε,�)NS2(ε)M(ε,ε), (16)

where

M(ε,ε) = f0(ε)[1 − f0(ε)] + f0(ε)[1 − f0(ε)]. (17)

The intrinsic flux noise per unit bandwidth of the interfer-ometer (�ns) is related to the voltage noise spectral densityas1

�ns =√

SV

|F(�)| . (18)

Note that �ns ∝ √Rt , as SV ∝ Rt and F(�) is independent

of tunnel junction resistance.Figure 6(a) shows �ns versus Ibias for several flux values

at T = 300 mK. �ns is a nonmonotonic function of Ibias witha minimum that depends, for each �, on the bias current. Inparticular, an increase in � leads to a general reduction of�ns at low Ibias, while its minimum moves toward lower biascurrent. We stress that �ns as low as 10−9 �0/

√Hz or better

can be achieved at this temperature in the ∼10–80 pA rangefor suitable values of �. This good flux sensitivity stems fromthe low shot noise SI (which is peculiar to all-superconductingtunnel junctions) together with a small Rd at the biasing pointand large F(�).

10 100

1

10

100

0.1 0.2 0.3 0.4 0.45

Φ ns [n

Φ 0/Hz1/

2 ]

Ibias

[pA]

Φ [Φ0]

25 50 75 1001

10

100

0.1 0.2 0.3 0.4 0.45

Φ [Φ0]

P [f

W]

Ibias

[pA](a)

(b)

(c)

40 60 80 100

100

101

102

103

104

Φ ns [n

Φ 0/Hz1/

2 ]

Ibias

[pA]

0.3 0.6 0.9 1.2 1.5

T [K]

FIG. 6. (Color online) (a) �ns vs Ibias calculated for different� values at T = 0.3 K. (b) �ns vs Ibias for � = 0.3�0 calculatedat different temperatures. (c) P vs Ibias calculated for different �

values at T = 0.3 K. In all calculations we set �02 = 200 μeV, �0

1 =800 μeV, and Rt = 5 M�.

The temperature dependence is displayed in Fig. 6(b) where�ns vs Ibias is plotted for different T values at � = 0.3�0.Notably, the minimum of �ns turns out to be quite insensitiveto the temperature up to �900 mK. Then, higher T yieldsto a reduction of the current window suitable for high fluxsensitivity and to an overall enhancement of �ns . Furthermore,for T � Tc2 [see the line corresponding to T = 1.5 K inFig. 6(b)], �ns is significantly degraded in the whole Ibias

range. This emphasizes the effectiveness of a superconductingtunnel probe for a drastic suppression of �ns .

The impact of dissipation P = V I is displayed in Fig. 6(c),which shows P vs Ibias for different � values at T = 0.3 K.P can largely change by varying � and Ibias as well. Inparticular, in the ∼10–80 pA current range, P can vary from afew femtowatts to some tens of femtowatts. Such a small powerhas the additional advantage to prevent substantial electronheating in the N wire.35 By contrast, in conventional SQUID’sdissipation is typically from two to five orders of magnitudelarger.1,6 As P ∝ R−1

t , dissipation can be tailored by choosinga proper value of the tunnel junction resistance.

For a correct operation of the interferometer the twofollowing conditions should be fulfilled: (i) 2πI 0

c LG/�0 � 13

(where I 0c is the zero temperature critical current of the S1NS1

Josephson junction and LG is the loop geometric inductance),and (ii) LS1

k � LNk

15,23 [where LS1,Nk � hRS1,N/π�0

1 is thekinetic inductance3 and RS1,N is the normal-state resistance ofS1(N)]. Condition (i), where I 0

c = 0.66π�01/eRN ,24 ensures

the avoidance of magnetic hysteresis whereas (ii), which isequivalent to RS1 � RN , ensures that the phase differenceset by � drops entirely at the wire ends thus allowinga full modulation of its DOS. As an additional set ofparameters we choose a silver (Ag) wire with L = 80 nm, cross

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HYBRID SUPERCONDUCTING QUANTUM MAGNETOMETER PHYSICAL REVIEW B 84, 214502 (2011)

section A = 30 × 10 nm2, and D = 0.02 m2s−1 which yieldsRN = L/(AνF e2D) � 5.2 �, where νF = 1 × 1047 J−1m−3

is the DOS at the Fermi level in Ag, �1/ETh � 0.3, andI 0c � 318 μA. By choosing, for instance, a circular washer

geometry1 with 2r = 150 nm as the internal diameter andexternal radius R r , we get LG = 2μ0r ≈ 0.19 pH, whereμ0 is the vacuum permeability, so that 2πI 0

c LG/�0 ≈ 0.18.Condition (ii) can be fulfilled as well by choosing a suitablewasher thickness and R.

As LG has to be kept small to satisfy condition (i) itfollows that the present structure could be suitable for themeasurement of the magnetic properties of small isolatedsamples. In this context, the magnetometer sensitivity (Sn)to an isolated magnetic dipole placed at the center of the loopis approximately given by10,12

Sn = 2r�ns

μ0μB

, (19)

where μB is the Bohr magneton. With our choice for r andby coupling the device to a cryogenic voltage preamplifier36

(which we assume dominates the voltage noise) with√

SpreV �

0.1 nV/√

Hz yields a total flux noise �totns =

√S

preV

max|F(�)| �40 n�0/

√Hz, leading to Sn ≈ 1 atomic spin/

√Hz below 1 K.

Furthermore, the best achievable energy resolution1 would be

E = (�totns )2

2LG� 170h.

VI. CONCLUSIONS

In summary, we have theoretically investigated a hybridsuperconducting magnetometer whose operation is based onmagnetic flux-driven modulation of the density of states of aproximized metallic nanowire. In particular, we have shownthat with suitable geometrical and material parameters theinterferometer can provide large transfer functions (i.e., of theorder of a few mV/�0) and intrinsic flux noise down to a fewn�0/

√Hz below 1 K. Furthermore, joined with limited power

dissipation, the structure has the potential for the realization ofsensitive magnetometers for the investigation of the switchingdynamics of small spin populations.

ACKNOWLEDGMENTS

We acknowledge L. Faoro, R. Fazio, P. Helisto, L. B.Ioffe, M. Kiviranta, M. Meschke, Yu. V. Nazarov, J. P.Pekola, and S. Pugnetti for fruitful discussions. The FP7program “MICROKELVIN” and the EU project “SOLID” areacknowledged for partial financial support.

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