Influence of resonances on the noise performance of SQUID susceptometers
AArticle
Influence of Resonances on the Noise Performance ofSQUID Susceptometers
Samantha I. Davis ∗ ,†,§ , John R. Kirtley and Kathryn A. Moler Department of Physics, Stanford University, Stanford, CA 94305-4045, USA; [email protected] Geballe Laboratory for Advanced Materials, Stanford University, Stanford, CA 94305, USA; Department of Applied Physics, Stanford University, Stanford, CA 94305, USA; [email protected] Stanford Institute for Materials and Energy Sciences, SLAC National Accelerator Laboratory, 2575 Sand HillRoad, Menlo Park, CA 94025, USA; [email protected] (K.A.M.) * Correspondence: [email protected]† These authors contributed equally to this work.§ Current address: California Institute of Technology – The Division of Physics, Mathematics and Astronomy 1200E California Blvd, Pasadena CA 91125, USAReceived: 04 November 2019; Accepted: 27 December 2019; Published: date
Abstract:
Scanning Superconducting Quantum Interference Device (SQUID) Susceptometrysimultaneously images the local magnetic fields and susceptibilities above a sample with sub-micronspatial resolution. Further development of this technique requires a thorough understanding of thecurrent, voltage, and flux ( IV Φ ) characteristics of scanning SQUID susceptometers. These sensorsoften have striking anomalies in their current–voltage characteristics, which we believe to be due toelectromagnetic resonances. The effect of these resonances on the performance of these SQUIDs isunknown. To explore the origin and impact of the resonances, we develop a model that qualitativelyreproduces the experimentally-determined IV Φ characteristics of our scanning SQUID susceptometers.We use this model to calculate the noise characteristics of SQUIDs of different designs. We find thatthe calculated ultimate flux noise is better in susceptometers with damping resistors that diminishthe resonances than in susceptometers without damping resistors. Such calculations will enable theoptimization of the signal-to-noise characteristics of scanning SQUID susceptometers. Keywords:
SQUID; susceptometers; noise; scanning
1. Introduction
Superconducting Quantum Interference Devices (SQUIDs) are superconducting loops interruptedby one or more Josephson weak links [1]. SQUIDs are used to achieve high-precision magnetic sensingfor diverse applications, including gravitational-wave astrophysics [2,3], magnetoencephalography [4],quantum information [5], and scanning SQUID microscopy [6]. In scanning SQUID microscopy (SSM),SQUIDs are used to image the local magnetic fields above samples. Enhanced spatial resolution isachieved in SSM by either making very small SQUID loops [7–9] or integrating a small “pickup loop"into the body of a larger SQUID through well-shielded superconducting coaxial leads [10,11]. Anextension of SSM is scanning SQUID susceptometry, in which susceptibility measurements are made bysurrounding the pickup loop in the latter type of SQUID by a co-planar, co-axial single-turn field coil[12], often in a gradiometric configuration (see Figure 1). As sensitive techniques for probing mesoscopicmaterials, scanning SQUID magnetometry and susceptometry are paving the way for essential advancesin superconductor physics [13]. a r X i v : . [ c ond - m a t . s up r- c on ] D ec of 13 (c) (d) xx I M (b)
F.C. xx I M (a)
F.C. Shunt resistors Shunt resistors Damping resistor
Figure 1.
Two types of susceptometer layouts: ( a ) That of Huber et al. [14,15], without a damping resistor,and ( b ) that of Gardner et al., [12] with a damping resistor. I labels the current leads, M the modulationcoil leads, and F . C . the field coil leads. The Josephson junctions are indicated by X s. The semi-transparentregions indicate superconducting shields. Superconducting coaxial leads connect the central regions withjunctions and modulation coils to the pickup loop/field coil pairs to the left and right. ( c ) Current–voltage(IV) characteristic for an undamped susceptometer at various magnetic fluxes, and ( d ) IVs for a dampedsusceptometer. One necessary step to advancing scanning SQUID technologies is understanding scanning SQUIDbehaviors. In this paper, we analyze the current–voltage–flux ( IV Φ ) properties of scanning SQUIDsusceptometers. Typically, sensitive SQUID magnetic flux measurements are made using a flux-lockedloop [1]. Here, we calculate the behaviors of our SQUID susceptometers when current-biased: The voltageacross the SQUID at constant current is held fixed by feeding back on the flux through the SQUID usinga modulation coil (see Figure 1a,b). The flux through the modulation coil compensates for changes inthe flux through the pickup loop. If there is sufficient feedback through the flux-locked loop, the currentthrough the modulation coil is proportional to the flux through the pickup loop. The sensitivity of theSQUID is due to the fact that near the critical current, small changes in flux result in large changes involtage.Scanning SQUID susceptometers also measure magnetic susceptibility by applying a localizedmagnetic field to the sample through the field coils. SQUID susceptometers are laid out in a gradiometricconfiguration so that they are insensitive to both uniform magnetic fields and currents that pass throughboth field coils (Figure 1a,b). With this layout, SQUID susceptometers image the local magnetic flux andmagnetic susceptibility of materials directly below one of the pickup loop/field coil pairs simultaneously.To optimize the performance of scanning SQUID sensors, it is important to understand their IV Φ characteristics in the presence of noise. Although the noise properties of ideal SQUIDs are well understood[16,17], the noise properties of new-generation SQUIDs, such as SQUID susceptometers, are not. Onepuzzling phenomenon is the presence of anomalies in the IV Φ characteristics of SQUID susceptometers of 13 (Figure 1c). These anomalies take the form of “steps" in voltage (peaks in the dynamic resistance dV / dI ),which occur at currents that disperse strongly with applied magnetic flux Φ and have a period of onesuperconducting flux quantum ( Φ ). Previous studies have shown that the current–voltage and alternatingcurrent (a.c.) characteristics of SQUIDs can be affected by parasitic capacitances in their input circuitry[18–20] and that the SQUIDs’ performances can be improved by resistive damping of the resultant input coilresonances [21]. We believe our anomalies are of similar origin and therefore refer to them as “resonances."To explore the origin and impact of resonances on the IV Φ characteristics of SQUID susceptometers,we perform an analysis of two-junction, direct current (d.c.) SQUID susceptometers that seeks the answersto two queries: 1) What causes the resonances? and 2) Do the resonances enhance or diminish thesensitivity of SQUID susceptometers?To address the first query, we develop models of susceptometers that reproduce the resonancesin simulations of their IV Φ characteristics. We hypothesize that the resonances occur due to parasiticcapacitances and inductances that arise from complex features of the SQUIDs, such as the field coils,the gradiometric layout, and pickup loops that are integrated into the bodies of the SQUIDs throughsuperconducting coaxial leads (Figure 1). These parasitic inductances and capacitances introduceinductor-capacitor (LC) resonances that are driven by the a.c. Josephson oscillations of the junctionsin the voltage state. Consequently, when the LC resonance frequency matches the Josephson frequency,there are voltage steps in the IV Φ characteristics, which translate to peaks in the IR Φ characteristics of thesusceptometers.Our hypothesis is supported by basic estimates of the voltage steps. The resonances in oursusceptometers have a characteristic voltage of roughly 10 µ V (see Figure 1c). Combining the Josephsonrelations [22], I s = I sin ϕ V = π d ϕ dt , (1)where I s is the supercurrent through the junction, I is the junction critical current, V is the voltage, and ϕ is the quantum mechanical phase drop across the junction, with the resonance frequency for an LC circuit, ω = (cid:112) L p , eq C p , eq , (2)we expect voltage steps to occur at V LC = Φ π (cid:112) L p , eq C p , eq , (3)where L p , eq and C p , eq are the equivalent lumped parasitic inductance and capacitance of the circuit,respectively. To estimate L p , eq and C p , eq , we use the FASTCAP, FASTHENRY, and INDUCT softwarepackages from the Whiteley Research web site [23]. We estimate that the susceptometer of Figure 1a has L p , eq = ±
20 pH and C p , eq = ± µ V < V LC < µ V . Using this intuition, we are able to successfully simulate the complex resonant behavior of thesusceptometers. We also reproduce the behavior of susceptometers with damping resistors, which greatlyreduce the amplitude of the resonances.In what follows, we first demonstrate in Section 2.1 that the addition of a parasitic capacitance tothe standard model for a SQUID produces peaks in the IR Φ characteristics similar to those observedexperimentally (see Figure 2). We then show that we can qualitatively reproduce the highly complex IR Φ characteristics of an undamped SQUID using a relatively simple model with distributed parasiticinductances and capacitances, as well as the much simpler IR Φ characteristics that result when a damping of 13 resistor is introduced (see Figure 3). We proceed to calculate SQUID noise in Section 2.2, first demonstratingthat we can reproduce previous work on basic SQUID layouts. After confirming our procedure, wecalculate the noise in our more complicated undamped and damped models at selected positions in the IR Φ plane. We conclude that the lowest intrinsic noise in the damped layout is significantly lower thanthat in the undamped layout for susceptometers for parameters that give similar critical curves.
2. Modeling Φ Characteristics
We use commercial software to model our devices: XIC, a layout tool, and WRSPICE, a simulationtool, both developed by Whiteley Research [23]. WRSPICE is based on the JSPICE [24] simulation toolfor electronic circuits and includes Josephson junctions. The layout tool XIC produces a list of nodes thatspecify connections between devices from a schematic. In our case, the devices are resistors R , capacitors C ,inductors L , mutual inductances M , and Josephson junctions J J . Each device has a constitutive equation: V = IR for the resistors, Q = CV for the capacitors, V = LdI / dt for the inductors, and the Josephsonrelations (Equation 1) for the Josephson junctions.The nodes, devices, and constitutive equations are combined by WRSPICE into a matrix equationof the form AX = B , where the elements of the vector X are the device responses and the elements of B are the excitations (e.g., voltage and current sources). In general, the matrix equation is non-linear andis solved by LU (lower, upper) decomposition iteratively with Newton’s method. In our case, we do atransient analysis that produces the time dependence of the circuit response in the presence of d.c. biases,magnetic flux, and noise.We assume that the pairs of critical currents I and shunt resistances R J for the two Josephsonjunctions are identical for each SQUID. To calculate the IV characteristics at each flux Φ , we ramp thecurrent through the SQUID at a rate of 1 µ A/nsec and average the resulting voltage time trace (which haslarge Josephson oscillations) in bin widths of 1 µ A. Figure 2b displays typical results for an “ideal" SQUIDwith parasitic inductance L p but no parasitic capacitance C p . of 13 Mod coil SQUID JJ JJ R J R J C J C J L p (a) (b) (c) (d) I ( μ A ) Mod coil SQUID C J R J JJ C J R J C p L p Figure 2.
Adding a parasitic capacitance to an ideal Superconducting Quantum Interference Device (SQUID)produces a resonance. ( a ) Ideal SQUID schematic, and ( b ) calculated dV/dI characteristic for an idealSQUID with no parasitic capacitance at T = 4.2 K. In this instance, the upper inductances L p = 30 pH, thelower inductances L p = 1 pH, the Josephson critical currents I = 22 µ A , the shunt resistors R J = 2 Ω , andthe junction capacitances C J = 10 fF. ( c ) Schematic with a parasitic capacitance, and ( d ) calculated dV/dIcharacteristic at T = 4.2 K. Here I = 22 µ A, R s = 2 Ω , C j = 10 fF, upper L p =30 pH, lower L p = 1 pH, upper C p =10 pF, and lower C p = 1 pF. Figure 2c displays the schematic and Figure 2d displays the dV/dI characteristic for the same circuitas in Figure 2a, but with parasitic capacitances C p added in parallel with the parasitic inductances. Thesecapacitances could result from, e.g., the overlapping superconducting layers between the junctions inFigure 1a. In this case, there are single resonances at half-integer multiples of Φ and voltages of ≈ µ V ( ≈ µ A ), but no resonances at higher voltages. The resonances occur at junction voltages in goodagreement with Equation (3), taking L p , eq = pH and C p , eq = pH . The simulations also show strongpeaks in the variance of the current through the parasitic inductors at 19 µ V, supporting the hypothesis thatthe resonances arise when the Josephson oscillations drive the parasitic LC s at their resonance frequency.The more complicated schematic of Figure 3a qualitatively reproduces the complex behavior ofthe resonances seen experimentally for an undamped susceptometer (see Figure 3b,c). In this case,the resonances are generated in a “ladder" of paired L p s and C p s, which physically correspond to thedistributed inductances and capacitances of the superconducting coaxes leading to the pickup loops. Wefind that there is not a one-to-one correspondence between the number of resonances and the number of L p C p pairs, but rather that the fine details of the resonances depend on the number and values of L p C p pairs included in the simulation. The details of the model (listed in the caption of Figure 3) are chosen tofit the experiment by tweaking the various parameters. The quality of the fit is measured by calculatingthe mean variance between the model and calculated IR Φ characteristic χ = ∑ n , m ( R exp. ( I n , Φ m ) − R model ( I n , Φ m )) / N , where N is the total number of calculated points in the I Φ plane. The model results of 13 displayed in Figure 3c correspond to χ = Ω . We are not able to find a set of I , C p , L m , and L p parameters that cause the modeled peaks in IR Φ to perfectly overlap with the experimental peaks.Nevertheless, we find the qualitative agreement between experiment and modeling exhibited in Figure3b,c supports the hypothesis that the structure in the IV characteristics is due to LC resonances driven byJosephson oscillations. Figure 3.
Modeling of dV/dI vs. I and Φ ( IR Φ ) for two types of SQUID susceptometers: ( a ) Undampedschematic, ( b ) experimental dV/dI characteristic, and ( c ) calculated dV/dI characteristic at T = 4.2 K fora SQUID with the layout of Figure 1a [14,15]. In this model, I = 25 µ A, R J = 2 Ω , C J = 10 fF, L m = 30pH, L p = 4 pH, and C p = 8 pF. There are a total of five L p , C p pairs in each arm to the left and right of theschematic, representing the coaxial leads to the pickup loops. ( d ) Damped schematic, ( e ) experimentaldV/dI characteristic, and ( f ) calculated dV/dI characteristic at T = 4.2 K for a SQUID with the layout ofFigure 1b [12]. In this model, I = 12 µ A, R J = 2 Ω , R D = 4 Ω , C j = 10fF, L m = 30 pH, L p = 1 pH, and C p = 8pF. There are five L p , C p pairs in each arm of the center of the schematic, representing the coaxial leads tothe pickup loops. We also find that in both the modeling and experiment, the resonances can be greatly reduced withthe addition of a damping resistor. Figure 3d displays the schematic of a SQUID susceptometer with adamping resistor (see Figure 1b, [12]), with parameters adjusted to fit the experimental IV Φ characteristics. Table 1.
Dimensionless parameters.Parameter Symbol Conversion formulaVoltage v V / I R J Magnetic flux φ Φ / Φ Thermal noise parameter Γ π k b T / I Φ Voltage noise power S v π S V / I R J Φ Flux noise ζ φ S Φ ( π I R J / Γ ) / Φ Hysteresis parameter β LI / Φ of 13 The characteristic time step for the transient analysis in JSPICE is a fraction of the inverse Josephsonfrequency—typically several GHz. Since we are interested in the noise at frequencies of several hundredHz or below, such calculations can be very time consuming (see the discussion in Ref. [16]). Noise isintroduced into our simulations as Johnson noise from the resistors with a Gaussian distributed voltagein series with the resistors with standard deviation V n = √ k b TR / dt , or equivalently, current sources inparallel with the resistors with standard deviation I n = √ k b T / dtR , where dt is the time interval. Forthe noise results we report here, we fix the flux and current through the SQUID and solve for the voltageas a function of time, typically recording the voltage V ( t ) in 1 ps intervals over 300 ns. We then Fouriertransform V ( t ) to get the power spectral density S V ( f ) , fit the results below the frequency (cid:104) V ( t ) (cid:105) /10 Φ (where (cid:104) V ( t ) (cid:105) is the average voltage over the full time trace) to a straight line, and extrapolate to zerofrequency to obtain S V . The data from any currents that have fewer than 100 points in this frequencyinterval or have a negative intercept from the linear fit are rejected. The transfer function dV / d Φ isobtained by subtracting two runs separated by 0.02 Φ centered on the flux of interest, and the flux noise is (cid:113) S Φ = (cid:113) S V / ( dV / d Φ ) . We repeat this procedure ten times. Following Tesche and Clarke [16], we reportour results using reduced units. Table 1 lists these units and conversion formulas to obtain them from S.I.units. In this table, k b is Boltzman’s constant, Φ = h /2 e is the superconducting flux quantum, I is thesingle junction critical current, R J is the single junction shunt resistance, and T is the temperature.We first verify that we can reproduce previous work. Figure 4a displays the schematic, and Figure 4b–ddisplays the dimensionless voltage noise power S v /2 Γ , the dimensionless transfer function | dv / d φ | , andthe dimensionless flux noise ζ φ respectively for the models used by Tesche and Clarke [16] and Bruineset al. [17]. The results have several qualitative features that are common to all the models studied:The dynamic resistance dV / dI (not shown), the voltage noise power S v (Figure 4b), and the transferfunction | dv / d φ | (Figure 4c) have peaks, and the flux noise ζ φ (Figure 4d) has a broad minimum, atsimilar flux -ependent currents I . The error bars for S v /2 Γ , dv / d φ , and ζ φ are calculated through errorpropagation. Using the standard error propagation formula, we find that the statistical uncertainty for S v /2 Γ is proportional to the standard deviation of the fit to the voltage periodogram at low frequencies.Since the amplitude of the voltage noise is greater at large dV / dI , the statistical uncertainty in S v /2 Γ isgreater in the vicinity of the critical curve and resonance. For dv / d φ , we find that the uncertainty in dv / d φ is proportional to the sum in quadrature of the uncertainties of the voltages at the two fluxes, so the errorbars are also greater near the critical curve and resonance. For ζ φ = V noise / ( dv / d φ ) , the contributionfrom the error of dv / d φ is proportional to ( dv / d φ ) − , so the uncertainty in ζ Φ is large where dv / d φ isclose to zero. The results from Bruines et al. [17] differ from those by Tesche and Clarke [16] in the transferfunction | dv / d φ | and flux noise ζ φ because of a numerical error in Tesche and Clarke. Our results agreewith the results of Bruines et al. [17] to within statistical uncertainty. of 13 b) d)c) Τ | d 𝑣 d 𝜙 | 𝜁 𝜙 Τ a) Mod. Coil
SQUID C J R J C J R J JJ JJL m L m Τ S 𝑣 Γ 𝐼/𝐼 𝐼/𝐼 𝐼/𝐼 Figure 4.
Comparison with previous work: ( a ) Schematic of the model used. In this case, the junctioncritical current I = 17.2 µ A, junction capacitance C j = 0 pF, modulation inductance L m = 30 pH, shuntresistance R J = 2 Ω , Φ = Φ , and T = 20.56 K. This choice of parameters leads to β = Γ = b ) is inferred from the curves labelled Bruines in ( c ) and( d ). After confirming our procedure by reproducing previous work, we then proceed to calculate thenoise for the more complicated undamped susceptometer, with conceptual layout given by Figure 1a, andschematic given by Figure 3a. It would take prohibitively long to calculate the noise for all currents andfluxes. Instead, we choose four values for flux at currents along the “critical curve", at which the junctionis just entering the voltage state. The symbols superimposed on the dV / dI plot in Figure 5a show thevalues of current I and flux φ used for each calculation, with paired fluxes (required to calculate | dv / d φ | )separated by 0.02 Φ and centered on each flux value plotted in Figure 5 b–d. of 13 ΤdV dI (Ω) Τ Φ Φ C u rr e n t ( μ A ) a) d)c) Τ | d 𝑣 d 𝜙 | 𝜁 𝜙 Τ Τ S 𝑣 Γ b) Figure 5.
Noise calculations for an undamped SQUID along the critical curve: ( a ) Plot of dV/dI vs. current(I) and flux ( Φ ). The crosses correspond to the values of I and Φ for which noise was calculated. There aretwo sets of crosses, separated by 0.02 Φ , to enable the calculation of the derivative dv / d φ at each flux value.( b ) Plots of the dimensionless low-frequency voltage noise power S v /2 Γ vs. current I for four different fluxvalues. ( c ) Plots of the dimensionless transfer junction | d ν / d φ | . ( d ) Plots of the dimensionless flux noise ζ φ . The schematic used for these calculations was that of Figure 3a, with I = 25 µ A, L m = 30 pH, R J = 2 Ω , L p = 4 pH, C J = 10 fF, C p = 8 pF, and T = Figure 6 displays similar calculations for the same model and parameters as Figure 5, but for currentand flux values along the first resonance in the IR Φ characteristic. ΤdV dI (Ω)
ΤΦ Φ C u rr e n t ( μ A ) a) d)c) Τ | d 𝑣 d 𝜙 | 𝜁 𝜙 Τ b) Τ S 𝑣 Γ Figure 6.
Noise calculations for an undamped SQUID along resonance at the lowest magnitude bias current("first resonance"): ( a ) Plot of dV/dI vs. current (I) and flux ( Φ ). The crosses correspond to the values of I and Φ for which noise was calculated. ( b ) Plots of the dimensionless low-frequency voltage noise power S v /2 Γ vs. current I for four different flux values. ( c ) Plots of the dimensionless transfer junction | d ν / d φ | .( d ) Plots of the dimensionless flux noise ζ φ . The schematic used for these calculations was that of Figure3a, with I = 25 µ A, L m = 30 pH, R J = 2 Ω , L p = 4 pH, C J = 10 fF, C p = 8 pF, and T = 4.2 K. Finally, Figure 7 displays results for the damped SQUID susceptometer model with layout in Figure1b and schematic in Figure 3d. For these calculations, the parameters were chosen to match those for theundamped SQUID, except for the addition of a damping resistor R d = Ω , for more direct comparisonbetween the damped and undamped cases. ΤdV dI (Ω)
ΤΦ Φ a) d)c) Τ | d 𝑣 d 𝜙 | 𝜁 𝜙 Τ b) Τ S 𝑣 Γ C u rr e n t ( μ A ) Figure 7.
Noise calculations for a damped SQUID along the critical curve: ( a ) Plot of dV/dI vs. current (I)and flux ( Φ ). The crosses correspond to the values of I and Φ for which noise was calculated. ( b ) Plots ofthe dimensionless low-frequency voltage noise power S v /2 Γ vs. current I for four different flux values.( c ) Plots of the dimensionless transfer junction | d ν / d φ | . ( d ) Plots of the dimensionless flux noise ζ φ . Theschematic used for these calculations was that of Figure 3d, with I = 22 µ A, L m = 30 pH, R J = 2 Ω , R d = 2 Ω , L p = 1 pH, C J = 10 fF, C p = 8 pF, and T = 4.2 K. A summary of the noise analysis described in Section 2.2 is reported in Figure 8. The calculatedminimum flux noise is similar for the undamped susceptometer model on the critical curve vs. on the firstresonance, but is significantly lower for the damped susceptometer model than for the undamped one forparameters that give similar critical curves.
Flux ( ) M i n . / Undamped CCUndamped ResonanceDamped
Figure 8.
Minimum flux noise for damped vs. undamped SQUIDs. The square blue symbols correspond tothe undamped SQUID along the critical curve, the diamond red symbols are for the undamped SQUIDalong the first resonance, and the triangular green symbols correspond to the damped SQUID along thecritical curve. These data were generated by varying the current ( I ) at fixed flux. The noise values that we calculate are comparable to experimentally reported noise floors for scanningSQUID susceptometers. The undamped SQUIDs presented in Figure 1c have a critical current I = 25 µ A and a shunt resistance R J =2 Ω . At T = ζ φ = S φ = µ Φ / Hz for these values of I and R J . The damped SQUIDs presented in Figure 1d have a criticalcurrent I =12.5 µ A and shunt resistance of R J =4 Ω , so ζ φ = S φ = µ Φ / Hz at 4.2 K. Gardner et al. [12] report an intrinsic noise of 3 µ Φ / Hz for damped scanning SQUIDsusceptometers at 4.2 K. Kirtley et al. [15] report an intrinsic noise of 2 µ Φ / Hz for undampedsusceptometers at 4.2 K, while Huber et al. [14] report a noise of 0.25 µ Φ / Hz at 125 mK above 10 kHzfor undamped susceptometers. Author Contributions:
Conceptualization, J.R.K.; Formal analysis, S.I.D. and J.R.K.; Funding acquisition, K.A.M.;Software, S.I.D. and J.R.K.; Writing – original draft, S.I.D. and J.R.K.; Writing – review and editing, S.I.D. and J.R.K.
Funding:
This research received no external funding.
Acknowledgments:
We would like to thank Diana Chamaki for making some of the IV measurements of dampedsusceptometers presented in this paper.
Conflicts of Interest:
The authors declare no conflict of interest.
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