A general method for computing thermal magnetic noise arising from thin conducting objects
Joonas Iivanainen, Antti J. Mäkinen, Rasmus Zetter, Koos C.J. Zevenhoven, Risto J. Ilmoniemi, Lauri Parkkonen
AA general method for computing thermal magnetic noise arising from thin conductingobjects
Joonas Iivanainen a, ∗ , Antti J. Mäkinen a , Rasmus Zetter a , Koos C.J. Zevenhoven a , Risto J. Ilmoniemi a , Lauri Parkkonen a a Department of Neuroscience and Biomedical Engineering, Aalto University School of Science, FI-00076 Aalto, Finland
Abstract
Thermal motion of charge carriers in a conducting object causes magnetic field noise that interferes with sensitive measure-ments nearby the conductor. In this paper, we describe a method to compute the spectral properties of the thermal magneticnoise from arbitrarily-shaped thin conducting objects. We model divergence-free currents on a conducting surface using astream function and calculate the magnetically independent noise-current modes in the quasi-static regime. We obtain thepower spectral density of the thermal magnetic noise as well as its spatial correlations and frequency dependence. We describea numerical implementation of the method; we model the conducting surface using a triangle mesh and discretize the streamfunction. The numerical magnetic noise computation agrees with analytical formulas. We provide the implementation as apart of the free and open-source software package bfieldtools .
1. Introduction
Thermal agitation of charge carriers in a conductor causesa fluctuating voltage and a current referred to as Johnson–Nyquist noise [1, 2]. The thermal current fluctuations inthe conductor are associated with a magnetic field that in-terferes with nearby magnetically sensitive equipment andmeasurements. Thermal magnetic noise can, e.g., limit theperformance of sensitive magnetometers operating in con-ducting shields (e.g., [3, 4, 5]), impose constraints on fun-damental physics experiments [6, 7] and cause decoherencein atoms trapped near conducting materials [8] as well as inhigh-resolution transmission electron microscopy [9]. It istherefore important to estimate the magnetic noise contribu-tion from nearby conductors when designing sensitive exper-iments and devices.Thermal magnetic noise from conductors can generallybe calculated either using direct approaches where the fieldnoise is computed from the modeled noise currents and theirstatistics (e.g., [3, 4, 6, 10]) or with reciprocal approacheswhere the noise is obtained by computing the power loss in-curred in the material by a known driving magnetic field (e.g.,[5, 11]). In simple geometries analytical expressions for themagnetic noise can be obtained using either of the two ap-proaches (e.g., [3, 4, 5, 6, 10]). In more complicated geome-tries, the noise has to be computed numerically. Numeri-cal methods using the reciprocal approach have been usedto compute the frequency-dependent magnetic noise (e.g.,[9, 12, 13]), while a method using the direct approach hasbeen suggested to compute the low-frequency noise arisingfrom thin conductors [14]. ∗ Corresponding author
Email address: [email protected] (Joonas Iivanainen )
Here, we outline a direct approach to compute the quasi-static frequency-dependent magnetic noise from a conduct-ing object which can be considered as a surface with a smallbut possibly non-constant thickness. We examine the inter-nal coupling phenomena associated with the surface currentsin order to determine the independent modes of the John-son current [15]. We use a stream-function formalism similarto a previous analytical calculation on an infinite conductingplane [10] and to a semi-analytical computation on a layeredgrid of square conducting patches [16]. The cross-spectraldensity of the magnetic noise can be computed based onthe current fluctuations of the individual modes describedby a set of Langevin equations; the fluctuation amplitudesare given by the equipartition theorem [17]. Examination ofthe individual modes gives an intuitive picture on the physicsthat determine the field noise characteristics.We present a numerical implementation of the approachwhich uses a discretization of the stream function on a tri-angle mesh representing the surface. The implementationis applicable for any conducting surface, including curvedones. We demonstrate computations in example geome-tries and, where possible, compare the results with analyt-ical formulas for verification. The implementation is freelyavailable as a part of the open-source Python software pack-age bfieldtools ( https://bfieldtools.github.io ; [18,19]).
2. Theory
We consider the magnetic noise in a frequency rangewhere the macroscopic Johnson thermal noise current isdivergence-free ( ∇ · (cid:126) J = Preprint submitted to arXiv July 20, 2020 a r X i v : . [ phy s i c s . c o m p - ph ] J u l his allows us to use stream-function expression for the sur-face current. We shortly introduce stream function expression of the sur-face current and describe how it relates to physical quantitiessuch as power dissipation and inductive energy. Specifically,we assume a thin surface S with conductivity σ ( (cid:126) r ) and thick-ness d ( (cid:126) r ). A divergence-free surface-current density on S canbe expressed with a stream function Ψ (units A/m) as (e.g.,[18, 20, 21]) (cid:126) J ( (cid:126) r , t ) = ∇ (cid:107) Ψ ( (cid:126) r , t ) × (cid:126) n ( (cid:126) r ), (1)where (cid:126) n ( (cid:126) r ) is the unit surface normal and ∇ (cid:107) is the tangentialgradient on the surface. We further express the stream func-tion as a linear combination Ψ ( (cid:126) r , t ) = (cid:80) i s i ( t ) ψ i ( (cid:126) r ), resultingin the current density (cid:126) J ( (cid:126) r , t ) = (cid:88) i s i ( t ) ∇ (cid:107) ψ i ( (cid:126) r ) × (cid:126) n ( (cid:126) r ) = (cid:88) i s i ( t ) (cid:126) k i ( (cid:126) r ), (2)where (cid:126) k i ( (cid:126) r ) = ∇ (cid:107) ψ i ( (cid:126) r ) × (cid:126) n ( (cid:126) r ) represent spatial patterns ofsurface-current density (units 1/m) and s i ( t ) their time-dependent amplitudes (units A). The magnetic field can becomputed from the patterns using the Biot–Savart law (cid:126) B ( (cid:126) r , t ) = µ π (cid:90) S (cid:126) J ( (cid:126) r (cid:48) , t ) × (cid:126) r − (cid:126) r (cid:48) | (cid:126) r − (cid:126) r (cid:48) | dS (cid:48) = (cid:88) i s i ( t ) µ π (cid:90) S (cid:126) k i ( (cid:126) r (cid:48) ) × (cid:126) r − (cid:126) r (cid:48) | (cid:126) r − (cid:126) r (cid:48) | dS (cid:48) = (cid:88) i s i ( t ) (cid:126) b i ( (cid:126) r ),(3)where µ is the vacuum permeability and (cid:126) b i ( (cid:126) r ) is the mag-netic field from the pattern (cid:126) k i with a unit amplitude.The instantaneous power dissipation between patterns (cid:126) k i and (cid:126) k j is [18, 21] P i j ( t ) = s i ( t ) s j ( t ) (cid:90) S σ ( (cid:126) r ) d ( (cid:126) r ) (cid:126) k i ( (cid:126) r ) · (cid:126) k j ( (cid:126) r ) dS = s i ( t ) s j ( t ) R i j ,(4)where R i j is the mutual resistance between the patterns. Sim-ilarly, the instantaneous inductive energy between the pat-terns is given by the mutual inductance M i j [18, 21] E i j ( t ) = s i ( t ) s j ( t ) µ π (cid:90) S (cid:90) S (cid:126) k i ( (cid:126) r ) · (cid:126) k j ( (cid:126) r (cid:48) ) | (cid:126) r − (cid:126) r (cid:48) | dSdS (cid:48) = s i ( t ) s j ( t ) M i j . (5)The amplitudes of the patterns evolve according to a cou-pled system of equations ([21]; Appendix A) M dd t s ( t ) + Rs ( t ) − e ( t ) =
0, (6)where s is a vector containing the pattern amplitudes s [ i ]( t ) = s i ( t ), M and R are the mutual inductance and resistance ma-trices with elements M [ i , j ] = M i j and R [ i , j ] = R i j defined above, and e ( t ) gives the electromotive force (emf) that iscoupled to the patterns. Equation system (6) is analogous tothat of coupled RL-circuits, where s contains the circuit cur-rents. However, we note that circuit quantities such as M and R depend on the normalization of the circuit basis functions (cid:126) k i , whereas energy quantities such as power dissipation andinductive energy are free of this ambiguity [21]. Next, we investigate how to model the magnetic Johnson–Nyquist noise using the stream-function approach. Thethermal current fluctuations are driven by the Johnson emf,which is proportional to a zero-mean Gaussian white noiseprocess [17]. In this context, equations (6) are coupledLangevin equations.To determine the statistics of the current fluctuations, weapply the equipartition theorem to the system [17]. Accord-ing to the theorem, in a thermal bath with temperature T each independent degree of freedom of the system has anaverage energy of k B T /2, with k B being the Boltzmann con-stant. The independent degrees of freedom of the system aregiven by the eigenvectors of M as they diagonalize the energy(5).We thus look for independent patterns (cid:126) κ i ( (cid:126) r ) with diagonal M as linear combinations of (cid:126) k j ( (cid:126) r ). We further require that thepatterns (cid:126) κ i ( (cid:126) r ) diagonalize R so that also the Langevin equa-tions (6) decouple. As the inductance and resistance matri-ces are symmetric positive-definite for an ordinary conduc-tor [21], these independent patterns can be found, for exam-ple, by solving a generalized eigenvalue equation [20, 22], i.e.,finding an invertible matrix V such that RV = MV Λ ⇔ (cid:189) V T RV = diag( r i ,..., r N ) V T MV = diag( l i ,..., l N ), (7)where Λ = diag( λ ,..., λ N ) is a diagonal matrix with λ i = r i / l i . The independent patterns are given by the columns ofthe invertible but generally non-unitary matrix V as (cid:126) κ i ( (cid:126) r ) = (cid:80) j V ji (cid:126) k j ( (cid:126) r ).We can transform Eq. (6) to the new basis: dd t V T MVV − s ( t ) + V T RVV − s ( t ) − V T e ( t ) =
0. (8)By defining ˜ s ( t ) = V − s ( t ) and ˜ e ( t ) = V T e ( t ), we obtain a set ofdecoupled Langevin equations dd t ˜ s i ( t ) + λ i ˜ s i ( t ) − ˜ e i ( t )/ l i =
0. (9)Effectively, we now have a number of independent RL-circuits with time constants τ i = l i / r i = λ i driven by emfs˜ e i ( t ).The Johnson emf has a white noise (frequency-independent) power spectral density (PSD) S ˜ e i that canbe used to solve the PSD of ˜ s i from the decoupled Langevinequation [17] S ˜ s i ( ω ) = S ˜ e i r i + ( ω / λ i ) , (10)2here ω is the angular frequency. The average energy (5) ofthe i th independent degree of freedom is: 〈 E i ( t ) 〉 = = l i 〈 ˜ s i ( t ) 〉 = l i π (cid:90) ∞ S ˜ s i ( ω ) d ω = l i S ˜ e i r i λ i = S ˜ e i r i , (11)where the brackets 〈·〉 denote the ensemble average. On theother hand, according to the equipartition theorem the aver-age energy is 〈 E i 〉 = k B T , which can be used together withequation (11) to solve the Nyquist formula for the PSD of theJohnson emf: S ˜ e i ( ω ) = k B Tr i , (12)where r i is associated with the average power dissipation 〈 P i ( t ) 〉 = r i 〈 ˜ j i ( t ) 〉 .To compute the cross-spectral density (CSD) of the mag-netic noise due to the Johnson current, we note that theFourier transform of the field from the independent patternsis obtained as F { (cid:126) B ( (cid:126) r )}( ω ) = F (cid:169)(cid:88) i s i ( t ) (cid:126) b i ( (cid:126) r ) (cid:170) = F (cid:169)(cid:88) i ˜ s i ( t ) (cid:126) β i ( (cid:126) r ) (cid:170) = (cid:88) i F { ˜ s i }( ω ) (cid:126) β i ( (cid:126) r ), (13)where (cid:126) β i ( (cid:126) r ) denotes the magnetic field from (cid:126) κ i . The CSD be-tween magnetic field components at (cid:126) r and (cid:126) r (cid:48) along unit vec-tors (cid:126) n and (cid:126) n (cid:48) is given by (cid:68) (cid:126) n · F { (cid:126) B ( (cid:126) r )} ∗ F { (cid:126) B ( (cid:126) r (cid:48) )} · (cid:126) n (cid:48) (cid:69) = (cid:68) (cid:126) n · (cid:161)(cid:88) i F { ˜ s i } ∗ (cid:126) β i ( (cid:126) r ) (cid:162)(cid:161)(cid:88) k F { ˜ s k } (cid:126) β k ( (cid:126) r (cid:48) ) (cid:162) · (cid:126) n (cid:48) (cid:69) = (cid:126) n · (cid:179)(cid:88) i (cid:88) k (cid:126) β i ( (cid:126) r ) (cid:68) F { ˜ s i } ∗ F { ˜ s k } (cid:69) (cid:126) β k ( (cid:126) r (cid:48) ) (cid:180) · (cid:126) n (cid:48) = (cid:126) n · CSD (cid:126) B ( (cid:126) r , (cid:126) r (cid:48) , ω ) · (cid:126) n (cid:48) , (14)where we defined CSD (cid:126) B = (cid:80) i (cid:80) k (cid:126) β i ( (cid:126) r ) 〈 F { ˜ s i } ∗ F { ˜ s k } 〉 (cid:126) β k ( (cid:126) r (cid:48) ) asthe CSD tensor of the magnetic field.The CSD tensor can be simplified by noting that the am-plitudes ˜ s i are independent: their temporal cross-correlationis (cid:82) ˜ s i ( t ) ˜ s k ( t + t (cid:48) ) d t = i (cid:44) k . For i = k , the auto-correlation with exponential decay is given as the Fouriertransform of the PSD of Eq. (10). The CSD of ˜ s i and ˜ s k isthereby 〈 F { ˜ s i } ∗ F { ˜ s k } 〉 = S ˜ s i ( ω ) δ ik and the CSD tensor of themagnetic noise is CSD (cid:126) B ( (cid:126) r , (cid:126) r (cid:48) , ω ) = (cid:88) i (cid:126) β i ( (cid:126) r ) S ˜ s i ( ω ) (cid:126) β i ( (cid:126) r (cid:48) ). (15)We next describe how to compute the CSD between fieldmeasurements by an array of sensors. We approximate themeasurement of the i th sensor y i ( t ) as a weighted sum of themagnetic field over the spatial extent of the sensor y i ( t ) = (cid:90) (cid:126) w i ( (cid:126) r ) · (cid:126) B ( (cid:126) r , t ) dV ≈ N i (cid:88) l = (cid:126) w i ( (cid:126) r l ) · (cid:126) B ( (cid:126) r l , t ), (16) where (cid:126) r l are the N i integration points of the sensor i and (cid:126) w i ( (cid:126) r l ) are their vector weights. The CSD between measure-ments y i and y k is thenCSD y i , y k ( ω ) = (cid:173) F { y i } ∗ F { y k } (cid:174) = N i (cid:88) l = N k (cid:88) h = (cid:126) w i ( (cid:126) r l ) ∗ · CSD (cid:126) B ( (cid:126) r l , (cid:126) r h , ω ) · (cid:126) w k ( (cid:126) r h ). (17)
3. Implementation
In this Section, we briefly outline the numerical implemen-tation of the magnetic noise computation. The implemen-tation is a part of the bfieldtools
Python software pack-age [19] and uses its stream-function discretization as well asnumerical integrals and functions to compute the resistanceand inductance matrices. The theoretical and computationalaspects of the software are presented in detail in Ref. [18].In bfieldtools , the conducting surface is representedby a triangle mesh and the stream-function basis Ψ ( (cid:126) r ) = (cid:80) i s i h i ( (cid:126) r ) consists of piecewise linear functions (or "hat func-tions") h i ( (cid:126) r ). The hat function value is one at the vertex i andzero at other vertices with linear interpolation on the trianglefaces. Each of these basis functions represents an elemen-tary current pattern which circulates around the correspond-ing vertex i . The magnetic field is obtained from the streamfunction s i with a linear map (Eq. (3)). For example, the z -component of the field at N evaluation points is b z = Cs , (18)where C is an N × M matrix mapping the M vertex-circulatingcurrents ( s [ i ] = s i ) to field component amplitudes at the eval-uation points.The resistance matrix R (with surface conductivity σ ( (cid:126) r ) d ( (cid:126) r ) discretized as constant on the triangles) and induc-tance matrix M of the elementary current patterns can becomputed using the software. In the case of an open mesh,the boundary conditions of the stream function are set asdescribed in Ref. [18]. Multiple separate conductors canbe handled by computing the inductances between all thepatterns and by forming a block resistance matrix comprisingthe resistance matrices of the individual conductors.We decouple the elementary circuits by solving the gener-alized eigenvalue equation (7) for eigenvalues Λ and eigen-vectors V using SciPy [23]. We then evaluate the CSD matrix Σ b of the magnetic field component at ω using equation (18)as Σ b = (cid:173) b z b T z (cid:174) = CV (cid:173) ˜ s ˜ s T (cid:174) V T C T = CV Σ ˜ s V T C T , (19)where s = V ˜ s and Σ ˜ s is a diagonal matrix with elements Σ ˜ s [ i , i ] = S ˜ s i ( ω ) (Eq. (10)).We model the measurement y i in Eq. (16) as y i = w T i b i ,where w T i is a row vector comprising the sensor weights and b i = C i s is a column vector of the magnetic noise along thedirections of the vector weights at the integration points. Theelements of the measurement CSD matrix can then be com-puted as follows Σ y [ i , k ] = w T i (cid:173) b i b T k (cid:174) w k = w T i C i V Σ ˜ s V T C T k , w k . (20)3 R e l a t i v e e rr o r ( % ) z/R = 0.05 z/R = 0.1010 Distance ( z/R )10 R e l a t i v e e rr o r ( % ) N = 630 N = 1844 N = 541810 Distance ( z/R )10 B z n o i s e ( f T / r H z ) Analytic N = 541810 Mode index10 𝜏 ( m s ) N = 630 N = 1844 N = 5418 Mesh Noise-current patterns N = 630 N = 1844 N = 5418 AB C D E
Figure 1: Comparison of the numerical solution of magnetic thermal noise of a circular conducting disk with radius R = xy -plane) toan analytical formula. A: Three meshes with different numbers of triangles ( N ) representing the disk and exemplary contours of the numerically solved noise-current patterns. Blue and red contours depict current flows in opposite directions. B: The time constants τ of the modes computed using the meshes. C–D:
Comparison between the numerical and analytical solution of the low-frequency magnetic noise ( B z ) on the z -axis. C: Comparison between the numericalsolution obtained using the densest mesh and the analytical formula. D: The relative errors of the numerical solutions to the analytical formula. E: Relativeerror as a function of number of current modes for the densest mesh.
In practice, we compute the cross-spectral densities usingmultidimensional NumPy-arrays [24] and by summing overthe relevant dimensions of the arrays. This way, we can, e.g.,compute the cross-spectral density of the magnetic field in300 observation points at 100 frequencies and store the resultin an array with dimensions of 300 × × × ×
4. Validation and numerical results
We first computed special case examples that allowed com-paring our numerical computation of the magnetic noise toanalytical formulas at the low-frequency limit. Specifically,we investigated the following:• B z noise along the z -axis due to a uniform conductingdisk centered on the x y -plane • B noise at the center of a spherical conducting surfaceas a function of the sphere radius• B noise at the center of a cylindrical conducting surfacealong the long axis of the cylinder.The analytical formulas for these three cases can be foundin the paper by Lee and Romalis [5]. Besides validation, wepresent other example computations. Unless stated other-wise, we used d = σ = × Ω − m − , corre-sponding to aluminium at room temperature T =
293 K.
Figure 1 presents the computation of the low-frequency B z noise along the z -axis due to a disk with a radius R = x y -plane. The disk was modeled with three dif-ferent meshes with 630, 1 844 and 5 418 triangles. Fig. 1A4 R z R (m)10 B z n o i s e ( f T / r H z ) AnalyticNumerical B z n o i s e ( f T / r H z ) z (m)2030405060 AnalyticNumerical Figure 2: Numerical computation of low-frequency magnetic noise in-side spherical and cylindrical conducting surfaces (represented as trianglemeshes) and comparison to analytical formulas.
Top:
Magnetic noise in thecenter of the sphere with different sphere radii R . Bottom:
Noise in the fieldcomponent along the long axis ( z ) of the cylinder. The analytical formulaonly applies in the center of the cylinder ( z = shows examples of the stream-function contours of the nu-merically computed patterns of the noise current, whileFig. 1B shows their time constants. At a relative distance of z = R , the relative error of the numerical solution of B z noise to the analytical formula is 2.7% when the densest meshis used; with a larger distance, the relative error is smaller.Higher-order modes contribute to the noise when the dis-tance is small (Fig. 1E).Figure 2 shows the numerical results for the low-frequencymagnetic noise inside a closed sphere (2 562 vertices; 5 120triangles) and a cylinder (3 842; 7 680). The computation andanalytical formula agree in both cases, with relative errors of0.06% and 0.03% in the case of the sphere and the cylinder,respectively. Next, we examined the magnetic noise and its frequency de-pendence using a simple conductor. We computed the B z noise on the z -axis as well as the magnetic noise CSD alongthe x -axis due to a circular conducting disk centered on the x y -plane ( R = B z noise due to the disk is shown inFig. 3. The same figure also shows the estimated frequencyat which the PSD is reduced by three decibels from the zero-frequency value. The 3-dB cutoff frequency (4 µ σ d z ) − for Frequency (Hz)10 B z n o i s e ( f T / r H z ) Distance ( z/R )10 - d B c u t o ff f r e q u e n c y ( H z ) Infinite planeDisc
Figure 3: The spectral density of the thermal magnetic noise B z on the z -axis due to a circular conducting disk with radius R = xy -plane. Left:
Spectral density a function of frequency and distance. Thecurves with different colors present the noise with different relative distancesfrom the mesh (ranging from 0.05 R to R ). Right:
The frequency at whichthe noise power has decreased by 3 dB from its zero-frequency value. Thesolid line gives the cutoff frequencies for an infinite plane calculated usingan analytical formula. x (m)0200400600800 50 Hz100 Hz500 HzNear DC 1 0 1 x (m)1000100 B x B y B x B z B y B z x (m)0200400600800 B x B y B z x (m)0200400600800 P S D ( f T / H z ) B x B y B z C S D t o x = m ( f T / H z ) A BDC C S D ( f T / H z ) C S D t o x = m ( f T / H z ) Figure 4: Magnetic noise cross-spectral density (CSD) along the x -axis ( z = R ) due to a a circular conducting disk with radius R = xy -plane. A: Low frequency noise power spectral density along the x -axis. B: Low-frequency noise CSD to x = C: Noise cross-spectral densityof B z to x = D: Noise CSD between different compo-nents of the magnetic field. an infinite planar conductor [3] is also shown. At small rela-tive distances to the disk ( z < R ), the numerical 3-dB fre-quencies scale as those for an infinite plane. At distancescomparable to the radius z ≈ R , the 3-dB frequency is con-stant suggesting contribution from a single mode with thelargest time constant. Fig. 4 shows examples of cross-spectraldensity of magnetic noise calculated on the x -axis.We then investigated the magnetic noise due to a planar5onductor with a star shape (1 442 vertices, 2 702 triangles).Fig. 5 illustrates the noise-current patterns on the conduc-tor and the magnetic noise spectral density at different per-pendicular distances from the conductor. At small relativedistances, the magnetic noise spectral density has a spatialstructure that resembles the shape of the conductor. At largerdistances, the magnetic noise loses the structural detail, re-flecting different fall-off distances of the field noise compo-nents that correspond to the noise-current modes with dif-ferent levels of detail.Last, as a practical example, we computed the low-frequency magnetic noise CSD seen by a superconductingquantum interference device (SQUID) array (102 magne-tometers; MEGIN Oy, Helsinki, Finland). We investigated twogeometries where the magnetometer array was either nearan aluminium plate or inside a closed cylindrical aluminiumshield (Fig. 6). The aluminium had a thickness of 5 mm andwas at room temperature in both cases. The magnetometerswere modelled as point-like for simplicity.The computed low-frequency noise spectral CSD in theSQUID array is presented in Fig. 6. Compared to the intrinsicnoise level of these commercial SQUID sensors ( ∼ (cid:112) Hz),the magnetic noise is significant in both cases.
5. Conclusion and outlook
We presented a method to compute the (cross) spectraldensity of magnetic thermal noise due to an arbitrarilyshaped conductor that can be considered a surface, i.e., thincompared to the distance to the observation points. Wehave made the implementation openly available (and con-tributable) as a part of the open-source Python softwarepackage bfieldtools . The numerical approach allows vi-sualization of the noise-current patterns, providing an intu-itive perspective on the physics. We validated the numericalimplementation by comparing the results to analytical for-mulas, and found agreement within ∼ Acknowledgments
This work has received funding from the European Union’sHorizon 2020 research and innovation programme undergrant agreement No. 820393 (macQsimal). The content issolely the responsibility of the authors and does not necessar-ily represent the official views of the funding organizations.
Conflicts of interests
The authors declare no conflicts of interest.
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Left:
The triangle mesh representing the conductor and the numerically computedthermal current patterns on the conductor. The time constant of the pattern decreases from left to right, top to bottom. Blue and red contours depict current flows in opposite directions.
Right:
Low-frequency noise spectral density at different vertical distances to the conductor. The plot limits are the same as thesize of the conductor shown in A.
40 cm
Noise spectral density(fT/rHz)Noise CSD (fT /Hz)
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Appendix A. Matrix equation
Here, we briefly present the derivation of the matrix equation(6) using the stream-function representation of the surface-current density (2). We divide the electrical surface current (cid:126) J ( (cid:126) r , t ) to two components as (cid:126) J ( (cid:126) r , t ) = σ ( (cid:126) r ) d ( (cid:126) r ) (cid:126) E ( (cid:126) r , t ) = σ ( (cid:126) r ) d ( (cid:126) r ) (cid:179) (cid:126) E F ( (cid:126) r , t ) + (cid:126) E s ( (cid:126) r , t ) (cid:180) = σ ( (cid:126) r ) d ( (cid:126) r ) (cid:179) − ∂ (cid:126) A ( (cid:126) r , t ) ∂ t + (cid:126) E s ( (cid:126) r , t ) (cid:180) , (A.1)where (cid:126) E F is the Faraday-inductive field given as the negativetime derivative of the magnetic vector potential (cid:126) A and (cid:126) E s isthe source field. The source field represents the active com-ponents responsible for the currents in the conductor, whilethe inductive electric field is due to the magnetic field gener-ated by the currents and represents their inductive coupling.The source field can be, e.g., due to an external time-varyingmagnetic field ( (cid:126) E s = − ∂ (cid:126) A s / ∂ t ) or due to a combination of mi-croscopic thermal current fluctuations (cid:126) J f and their associ-ated macroscopic electric field ( (cid:126) E s = (cid:126) J f / σ d + (cid:126) E f ).By reordering the terms and expressing the vector potentialusing the current density, Eq. (A.1) reads ∂∂ t µ π (cid:90) S (cid:126) J ( (cid:126) r (cid:48) , t ) | (cid:126) r − (cid:126) r (cid:48) | dS (cid:48) + (cid:126) J ( (cid:126) r , t ) σ ( (cid:126) r ) d ( (cid:126) r ) − (cid:126) E s ( (cid:126) r , t ) =
0. (A.2)We consider a frequency range where the charge density doesnot fluctuate ( ∇ · (cid:126) J = ∂∂ t (cid:88) k s k ( t ) µ π (cid:90) S (cid:126) k k ( (cid:126) r (cid:48) ) | (cid:126) r − (cid:126) r (cid:48) | dS (cid:48) + (cid:88) k s k ( t ) (cid:126) k k ( (cid:126) r ) σ ( (cid:126) r ) d ( (cid:126) r ) − (cid:126) E s ( (cid:126) r ) = (cid:126) k l ( (cid:126) r ) and integrating over thesurface, we have ∂∂ t (cid:88) k s k ( t ) µ π (cid:90) S (cid:90) S (cid:126) k l ( (cid:126) r ) · (cid:126) k k ( (cid:126) r (cid:48) ) | (cid:126) r − (cid:126) r (cid:48) | dSdS (cid:48) + (cid:88) k s k ( t ) (cid:90) S (cid:126) k l ( (cid:126) r ) · (cid:126) k k ( (cid:126) r ) σ ( (cid:126) r ) d ( (cid:126) r ) dS − (cid:90) S (cid:126) k l ( (cid:126) r ) · (cid:126) E s ( (cid:126) r , t ) dS = (cid:88) k M lk ∂∂ t s k ( t ) + R lk s k ( t ) − e l ( t ) =
0, (A.5)where e l ( t ) = (cid:82) S (cid:126) k l ( (cid:126) r ) · (cid:126) E s ( (cid:126) r , t ) dS is the source emf coupled topattern ll