Design optimization of flux concentrators for magnetic tunnel junctions-based sensors

Introduction to the show: ident: Quantum Radio. Generated commentary on the latest quantum physics and condensed matter papers.

Kai: Hi, I'm Kai.

Mira: And I'm Mira, here with Lev.

Kai: We're looking at the paper: "Design optimization of flux concentrators for magnetic tunnel junctions-based sensors".

Mira: Here's the gist of the paper. The gist: This paper explores a design optimization scheme to find the best trade-off between high flux concentrator gain and low magnetic noise for miniaturized, ultra-sensitive sensors based on magnetic tunnel junctions.

Kai: Sensor Requirements and Challenges Miniaturized, ultra-sensitive and easily integrable magnetometers are needed for many applications like space exploration or health monitoring<ref:2604.21500#pg2> Achieving this goal requires a magnetic sensor with high sensitivity and low noise<ref:2604.21500#pg2> High sensitivity (>1000 %/mT) can be obtained by integrating high gain permalloy flux concentrators (FC)<ref:2604.21500#pg2> And reducing the magnetic 1/f noise can be realized by increasing the number of magnetic tunnel junctions (MTJs) in the air-gap of the FC<ref:2604.21500#pg2> However, this is obtained at the expense of a wider air-gap and consequently a decrease of the magnetic gain and thus of the sensitivity<ref:2604.21500#pg2> Measuring ultralow magnetic fields (picotesla), required for biomedical applications or for space exploration, is challenging<ref:2604.21500#pg3> Magnetoresistive sensors

1,2: are interesting candidates thanks to their compacity, low energy consumption and relatively low fabrication costs<ref:2604.21500#pg4> Optimizing the detectivity of a magnetic sensor based on tunnel magnetoresistance (TMR) can be done by following two complementary options: i) improving the sensor sensitivity; ii) reducing the noise<ref:2604.21500#pg4> Achieving these two objectives is not straightforward because there is a direct interplay between the sensitivity of a magnetic tunnel junction and its noise

3: <ref:2604.21500#pg4> Indeed, the most prominent noise in magnetic tunnel junctions is nowadays the noise due to thermal fluctuations of the magnetization, which increases at low frequency and is called 1/f magnetic noise

1,4,5: <ref:2604.21500#pg4> A given thermal fluctuation of the magnetization induces a fluctuation of the resistance that contributes to the noise, and that is also directly proportional to the sensitivity<ref:2604.21500#pg4> It is also possible to increase the sensor sensitivity by using a flux concentrator (FC) that amplifies the magnetic field at the junction site; this gain of sensitivity is obtained without changing the junction response, so the magnetic noise remains the same<ref:2604.21500#pg3> Hence to aim at the picotesla range at low frequency, these sensors have been combined with magnetic FC of various sizes and shapes

5–10: that can increase the magnetic field at the sensor location by a factor up to 440 for single staged

11: and 520 for double staged FC

10: <ref:2604.21500#pg3> Modeling Flux Concentrator Gain The paper proposes two complementary approaches to model the gain of the flux concentrator<ref:2604.21500#pg2> One is based on finite elements simulations of the FC gain where the influence of geometrical parameters of the air-gap is investigated<ref:2604.21500#pg2> Then, in a second step, we propose an analytical formula consistent with all our simulations results and based on magnetic reluctance<ref:2604.21500#pg2> The flux-concentrator is made of NiFe with a relative magnetic permeability µr = 1500 and a thickness t = 8 µm<ref:2604.21500#pg4> The flux-concentrator gain is calculated as the ratio of the magnetic field in the middle of the air-gap with respect to the external magnetic field (1 µT) applied along the x direction<ref:2604.21500#pg4> To properly describe how the flux concentrator traps the flux lines, a large volume is simulated<ref:2604.21500#pg4> The mesh is much finer close to the air-gap and particularly fine inside the air-gap in which each element size is between one and two µm<ref:2604.21500#pg4> The varying geometrical parameters of the concentrator are the width w and length l of its air-gap, which will accommodate a certain number N of junctions as illustrated in Fig.

Mira: 1.b<ref:2604.21500#pg4> Optimization Trade-offs The study aims at optimizing the FC design in conjunction with the integration of the magnetic tunnel junctions, seeking a balance between the gain achieved by the design of the FC and the total volume of MTJs<ref:2604.21500#pg4> Since an increase of the diameter of the MTJs directly results in a wider air-gap between the concentrators, we have first studied the variation of the gain for different air-gap widths, with an arbitrary fixed length of 200 µm<ref:2604.21500#pg4> As expected, increasing the gap width drastically reduces the gain<ref:2604.21500#pg4> For example, doubling it from 5 to 10 µm reduces the gain by almost half (from 495 to 258), and an increase beyond 15 µm reduces the gain below 200<ref:2604.21500#pg4> It appears therefore that increasing the magnetic volume of the sensor does not have to be achieved solely by increasing the junction diameter, but rather by using more MTJs in the gap, arranged along y-axis and interconnected<ref:2604.21500#pg4> This approach would prevent an excessive increase of the air-gap width, which is detrimental to the gain, as previously observed<ref:2604.21500#pg4> Nonetheless, elongation reduces the concentration of field lines in the gap, thereby also reducing the gain<ref:2604.21500#pg4> Analytical Model and Optimal Design The magnetic reluctance of a magnetic core of length L and crosssection S is R = 1/µ L/S<ref:2604.21500#pg4> The total reluctance to be calculated is R = R1 + R2 + R3, where R1is the reluctance of the rectangular part of the magnetic arm, R2 is the reluctance of the wedged region and R3 the half air-gap<ref:2604.21500#pg4> The dimensionless gain G is expected to be inversely proportional to Rl<ref:2604.21500#pg4> Finally, we obtain: 1/G ∝ l t µr

L1 W + L2 W − l ln (W l): + w2t<ref:2604.21500#pg4> This formula fits quite well the simulated gain as a function of the gap width<ref:2604.21500#pg4> We can easily conclude from Fig.

Kai: 3 that it is more efficient to add MTJs in the length of the air-gap compared to the width<ref:2604.21500#pg4> The optimization of the geometrical parameters requires to minimize the function F(d, N) defined as 1/ (d√N G(d + 2b, N(d + a)))<ref:2604.21500#pg4> The overall optimum is obtained when this extremum is reached for a number of junctions such as the extremum coincides with the maximal diameter allowed<ref:2604.21500#pg4> For the optimal design found here, the flux-concentrator is a rectangular FC, without wedge, with an air-gap width of about d + 2b = 19 µm and a gain of 80<ref:2604.21500#pg4> This value has to be compared with the detectivity reference αH (µm2) s (%/mT) √alphaH/(10s) (10-6 T.µm)<ref:2604.21500#pg4> Thus, the use of a flux concentrator and of a large number of junctions with the optimum design allows to gain 3 orders of magnitude on the sensor detectivity<ref:2604.21500#pg4> The authors conclude that using finite element simulations and fitting a reluctance model, they modelled the gain of the flux concentrator with the air-gap dimensions that welcome a number of series/parallel magnetic tunnel junctions<ref:2604.21500#pg4> Their model shows that the better detectivity is obtained for relatively small junctions and using a flux concentrator with no wedged region<ref:2604.21500#pg4> This is explained by a larger reduction of noise with the magnetic volume that largely compensates for the loss of gain of the FC while increasing its length<ref:

Title and authors: Kai: So, to recap, this paper is titled "Design optimization of flux concentrators for magnetic tunnel junctions-based sensors," and the authors are Thomas Brun, Javier Rial, Lucia Risoli, Johanna Fischer, Philippe Sabon, Guillaume Jannet, Matthieu Kretzschmar, Hélène Béa and Claire Baraduc.

Mira: They’re focusing on designing those flux concentrators to work best with magnetic tunnel junctions for sensors that need high sensitivity and low noise.

Lev: I mean, it's a lot of people involved in the modeling and simulation side, which tells you this is a pretty detailed piece of work.

Kai: Right, so the core idea is optimizing the design of those concentrators to balance magnetic field amplification against noise reduction from the tunnel junctions.

Mira: They are looking at how to improve sensitivity and reduce noise in TMR sensors by following two complementary options: improving sensitivity and reducing noise.

Lev: That interplay between sensitivity and noise is what makes it hard; you can't just improve one without affecting the other significantly.

Kai: Exactly, because the most prominent noise in these tunnel junctions is that thermal fluctuation of magnetization, which is called one/f magnetic noise <ref:2604.21500#pg1>.

Mira: And that one/f magnetic noise actually increases at low frequency, and it’s directly proportional to the sensitivity you are trying to achieve <ref:2604.21500#pg1>.

Lev: That means you have to be careful because increasing the sensitivity by using a flux concentrator doesn't change the inherent magnetic noise of the junction response itself.

Kai: But that FC does amplify the magnetic field right at the junction site, which gives you that extra gain without adding new sources of noise.

The paper's summary: Kai: The paper summarizes their approach by using two complementary modeling methods: first, finite element simulations to investigate how geometrical parameters of the air gap affect the FC gain.

Mira: Then they take those simulation results and propose an analytical formula consistent with all their simulations, which is based on magnetic reluctance.

Lev: So they aren't just guessing; they're using simulation to guide the creation of a more accurate mathematical model based on physical principles like reluctance.

Kai: They are modeling the flux concentrator itself as being made of NiFe with a relative magnetic permeability mu r equal to one thousand five hundred and a thickness t equal to eight microns.

Mira: The flux concentrator gain is calculated as the ratio of the magnetic field in the middle of the air gap with respect to an external magnetic field of one microtesla applied along the x direction.

Lev: They simulate this using a large volume, where each element size is between one and two microns, especially fine inside that air gap area.

Kai: The main geometrical parameters they are testing are the width w and length l of the air gap, which determines how many magnetic tunnel junctions N can fit there.

Mira: They explore how increasing the diameter of MTJs directly widens the air gap between concentrators, which drastically reduces the gain.

Lev: That's when they realized that simply making junctions bigger isn't enough; you need to use more MTJs in the gap, arranged along the y-axis and interconnected.

Kai: This suggests a design where you trade off junction size against air gap width, trying to avoid that excessive increase in width that hurts the gain.

The paper's improvements: Mira: The paper proposes this optimization trade-off by studying the variation of the gain for different air-gap widths, with an arbitrary fixed length of two hundred microns.

Lev: They showed that increasing that gap width drastically reduces the gain, and they gave a concrete example: doubling it from five to ten microns cuts the gain from four hundred ninety-five to two fifty eight.

Kai: And when you go beyond fifteen microns, the gain drops below two hundred, so it really shows how sensitive the design is.

Mira: The key improvement is that they moved away from just increasing junction diameter as a way to increase magnetic volume.

Lev: Instead, they suggested using more MTJs in the gap along the y-axis and interconnected; this prevents that excessive air gap width while still building up magnetic volume.

Kai: So, even though elongating those lines reduces the concentration of field lines in the gap, which also lowers the gain.

Mira: The analytical model they derived for one/G is inversely proportional to l t mu r

L1 W + L2 W − l ln (W l): + w2t <ref:2604.21500#pg1>.

Lev: And then they minimize that function F defined as one / (d√N G(d + 2b, N(d + a))).

Kai: The ultimate finding is that the best design is found when this minimization function reaches an extremum for a number of junctions such that the extremum coincides with the maximal diameter allowed.

Mira: And for this optimal design, they got a rectangular FC without a wedge region, with an air gap width of about d + 2b equals nineteen microns and a gain of eighty.

Lev: And that result is compared against the detectivity reference alphaH which is measured in (µm2) s (% per milliTesla) root alphaH over (10s) times (ten to the negative six Tesla micrometers) <ref:2604.21500#pg2>.

Kai: So, they found that this optimal design allows them to get three orders of magnitude on the sensor detectivity compared to a single junction.

Conclusion: Mira: To wrap up, they concluded that using finite element simulations and fitting a reluctance model for the flux concentrator gain with air gap dimensions accommodating series or parallel magnetic tunnel junctions gives them their best results.

Lev: They found that the better detectivity comes from relatively small junctions and using a flux concentrator without a wedge region.

Kai: The reason they say this is because the larger reduction of noise with the magnetic volume largely compensates for the loss of gain from the FC while increasing its length.

Mira: So, to put it simply, they found that by optimizing these parameters, you can achieve better performance than before by finding a balance between gain and noise.

Lev: It’s about using that extra magnetic volume to lower the noise enough that it makes up for the lost gain from the concentrator structure.

Kai: So, this paper is a solid piece of work showing how you can design these systems to handle ultralow fields while keeping them sensitive and low noise.

Mira: It’s a useful framework for anyone looking to design similar designs in this field.