A general method for computing thermal magnetic noise arising from thin conducting objects a, a a a a a Joonas Iivanainen ¤, Antti J. Mäkinen , Rasmus Zetter , Koos C.J. Zevenhoven , Risto J. Ilmoniemi , Lauri Parkkonen aDepartment 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 magnetic noise from arbitrarily-shaped thin conducting objects. We model divergence-free currents on a conducting surface using a stream function and calculate the magnetically independent noise-current modes in the quasi-static regime. We obtain the power spectral density of the thermal magnetic noise as well as its spatial correlations and frequency dependence. We describe a numerical implementation of the method; we model the conducting surface using a triangle mesh and discretize the stream function. The numerical magnetic noise computation agrees with analytical formulas. We provide the implementation as a part of the free and open-source software package bfieldtools. 1. Introduction Here, we outline a direct approach to compute the quasi- static frequency-dependent magnetic noise from a conduct- Thermal agitation of charge carriers in a conductor causes ing object which can be considered as a surface with a small a fluctuating voltage and a current referred to as Johnson– but possibly non-constant thickness. We examine the inter- Nyquist noise [1, 2]. The thermal current fluctuations in nal coupling phenomena associated with the surface currents the conductor are associated with a magnetic field that in- in order to determine the independent modes of the John- terferes with nearby magnetically sensitive equipment and son current [15]. We use a stream-function formalism similar measurements. Thermal magnetic noise can, e.g., limit the to a previous analytical calculation on an infinite conducting performance of sensitive magnetometers operating in con- plane [10] and to a semi-analytical computation on a layered ducting shields (e.g., [3, 4, 5]), impose constraints on fun- grid of square conducting patches [16]. The cross-spectral damental physics experiments [6, 7] and cause decoherence density of the magnetic noise can be computed based on in atoms trapped near conducting materials [8] as well as in the current fluctuations of the individual modes described high-resolution transmission electron microscopy [9]. It is by a set of Langevin equations; the fluctuation amplitudes therefore important to estimate the magnetic noise contribu- are given by the equipartition theorem [17]. Examination of tion from nearby conductors when designing sensitive exper- the individual modes gives an intuitive picture on the physics iments and devices. that determine the field noise characteristics. Thermal magnetic noise from conductors can generally We present a numerical implementation of the approach be calculated either using direct approaches where the field which uses a discretization of the stream function on a tri- noise is computed from the modeled noise currents and their angle mesh representing the surface. The implementation statistics (e.g., [3, 4, 6, 10]) or with reciprocal approaches is applicable for any conducting surface, including curved where the noise is obtained by computing the power loss in- ones. We demonstrate computations in example geome- curred in the material by a known driving magnetic field (e.g., tries and, where possible, compare the results with analyt- [5, 11]). In simple geometries analytical expressions for the ical formulas for verification. The implementation is freely magnetic noise can be obtained using either of the two ap- available as a part of the open-source Python software pack- proaches (e.g., [3, 4, 5, 6, 10]). In more complicated geome- age bfieldtools (https://bfieldtools.github.io; [18, arXiv:2007.08963v1 [physics.comp-ph] 17 Jul 2020 tries, the noise has to be computed numerically. Numeri- 19]). cal methods using the reciprocal approach have been used to compute the frequency-dependent magnetic noise (e.g., [9, 12, 13]), while a method using the direct approach has 2. Theory been suggested to compute the low-frequency noise arising We consider the magnetic noise in a frequency range from thin conductors [14]. where the macroscopic Johnson thermal noise current is divergence-free ( ~J 0). In other words, the macroscopic r ¢ Æ charge density does not fluctuate, but the current fluctua- ¤Corresponding author Email address: [email protected] (Joonas Iivanainen ) tions are due to microscopic thermal motion of charge [10]. Preprint submitted to arXiv July 20, 2020 This allows us to use stream-function expression for the sur- above, and e(t) gives the electromotive force (emf) that is face current. coupled to the patterns. Equation system (6) is analogous to that of coupled RL-circuits, where s contains the circuit cur- 2.1. Stream function and surface current rents. However, we note that circuit quantities such as M and We shortly introduce stream function expression of the sur- R depend on the normalization of the circuit basis functions ~ face current and describe how it relates to physical quantities ki , whereas energy quantities such as power dissipation and such as power dissipation and inductive energy. Specifically, inductive energy are free of this ambiguity [21]. we assume a thin surface S with conductivity σ(~r ) and thick- ness d(~r ). A divergence-free surface-current density on S can 2.2. Magnetic Johnson–Nyquist noise be expressed with a stream function ª (units A/m) as (e.g., Next, we investigate how to model the magnetic Johnson– [18, 20, 21]) Nyquist noise using the stream-function approach. The ~J(~r ,t) ª(~r ,t) ~n(~r ), (1) thermal current fluctuations are driven by the Johnson emf, Æ rk £ which is proportional to a zero-mean Gaussian white noise where ~n(~r ) is the unit surface normal and is the tangential rk process [17]. In this context, equations (6) are coupled gradient on the surface. We further express the stream func- Langevin equations. tion as a linear combination ª(~r ,t) P s (t)Ã (~r ), resulting Æ i i i To determine the statistics of the current fluctuations, we in the current density apply the equipartition theorem to the system [17]. Accord- X X ing to the theorem, in a thermal bath with temperature T ~J(~r ,t) si (t) Ãi (~r ) ~n(~r ) si (t)~ki (~r ), (2) Æ i rk £ Æ i each independent degree of freedom of the system has an average energy of kBT /2, with kB being the Boltzmann con- where ~ki (~r ) Ãi (~r ) ~n(~r ) represent spatial patterns of stant. The independent degrees of freedom of the system are Æ rk £ surface-current density (units 1/m) and si (t) their time- given by the eigenvectors of M as they diagonalize the energy dependent amplitudes (units A). The magnetic field can be (5). computed from the patterns using the Biot–Savart law We thus look for independent patterns~·i (~r ) with diagonal M as linear combinations of~k (~r ). We further require that the Z j ¹ ~r ~r 0 ~ 0 ~ ¡ patterns ~·i (~r ) diagonalize R so that also the Langevin equa- B(~r ,t) J(~r 0,t) 3 dS0 Æ 4¼ S £ ~r ~r 0 tions (6) decouple. As the inductance and resistance matri- Z j ¡ j X ¹0 ~r ~r 0 X ces are symmetric positive-definite for an ordinary conduc- ~ ¡ ~ si (t) ki (~r 0) 3 dS0 si (t)bi (~r ), Æ 4¼ S £ ~r ~r Æ tor [21], these independent patterns can be found, for exam- i j ¡ 0j i (3) ple, by solving a generalized eigenvalue equation [20, 22], i.e., finding an invertible matrix V such that ~ where ¹0 is the vacuum permeability and bi (~r ) is the mag- ½ T ~ V RV diag(ri ,...,rN ) netic field from the pattern ki with a unit amplitude. RV MV¤ T Æ (7) Æ , V MV diag(li ,...,lN ), The instantaneous power dissipation between patterns ~ki Æ ~ and k j is [18, 21] where ¤ diag(¸ ,...,¸ ) is a diagonal matrix with ¸ Æ 1 N i Æ r /l . The independent patterns are given by the columns of Z 1 i i Pi j (t) si (t)s j (t) ~ki (~r ) ~k j (~r )dS si (t)s j (t)Ri j , the invertible but generally non-unitary matrix V as ~·i (~r ) Æ σ(~r )d(~r ) ¢ Æ Æ S P V ~k (~r ). (4) j ji j We can transform Eq. (6) to the new basis: where Ri j is the mutual resistance between the patterns. Sim- d T 1 T 1 T ilarly, the instantaneous inductive energy between the pat- V MVV¡ s(t) V RVV¡ s(t) V e(t) 0. (8) dt Å ¡ Æ terns is given by the mutual inductance Mi j [18, 21] 1 T By defining s˜(t) V¡ s(t) and e˜(t) V e(t), we obtain a set of Z Z ~ ~ Æ Æ 1 ¹0 ki (~r ) k j (~r 0) decoupled Langevin equations Ei j (t) si (t)s j (t) ¢ dSdS0 Æ 2 4¼ S S ~r ~r j ¡ 0j d 1 s˜i (t) ¸i s˜i (t) e˜i (t)/li 0. (9) si (t)s j (t)Mi j . (5) dt Å ¡ Æ Æ 2 Effectively, we now have a number of independent RL- The amplitudes of the patterns evolve according to a cou- circuits with time constants ¿i li /ri 1/¸i driven by emfs pled system of equations ([21]; Appendix A) Æ Æ e˜i (t). d The Johnson emf has a white noise (frequency- M s(t) Rs(t) e(t) 0, (6) independent) power spectral density (PSD) S that can dt Å ¡ Æ e˜i be used to solve the PSD of s˜i from the decoupled Langevin where s is a vector containing the pattern amplitudes s[i](t) equation [17] Æ si (t), M and R are the mutual inductance and resistance ma- Se˜i 1 Ss˜i (!) 2 2 , (10) trices with elements M[i, j] Mi j and R[i, j] Ri j defined Æ r 1 (!/¸ ) Æ Æ i Å i 2 where ! is the angular frequency.