Estimation of the spatial resolution limits

Due to the very strong agreement between experimental data and the simulation results, the model introduced may be used to analyze the limitations when estimating the particle position in respect to the sensor. Figure 5.11 shows

again a top view of a TMR-map for an external field H_y=16 kA/m. The response map shows a maximum value TMRmax close to the sensor position and two local minima TMRmin in some distance from it. The response is symme-tric to two axes which are obtained by a rotation of the ellipse axes of an angle α. This rotation originates from the coupling of the two magnetic layers. Sign and size of α depend on the interplay of the two coupling effects mentioned above: While Néel-coupling favours parallel alignment, the stray field interaction between the two layers lead to an antiparallel configuration. For sensors of an area smaller than 1 µm², stray field interaction is commonly dominating [DMey06].

In a first step, we analyze the influence of the external magnetic field on the sensor response.

Similar calculations as presented in Figure 5.11 are carried out for field values of Hy = 8 kA/m, 24 kA/m, 40 kA/m and 56 kA/m. The particle magnetization is chosen according to Figure 5.8(d) and given by 0.31, 0.57, 0.69 and 0.76 times the saturation value, respectively. To ease the comparison between different maps, we employ again the relative ∆TMR-ratio

part stack

stack

TMR TMR

TMR TMR 1

∆ = −

+ (5.10)

The results are shown in Figure 5.12.

We find that an increasing external field value leads to an increasing effect change at the centre of the sensor. However at the same time, the measurable TMRpart-values decrease as shown in Figure 5.12. In particular, they will rapidly drop below the threshold of noise that can always be found in such devices [PHed09]. Therefore, high fields increase the resolution of the sensor, but decrease the area in which a particle can be detected. This is similar to the behaviour of optical lenses. If a critical field value is exceeded, the response of the sensor

Figure 5.11: TMR-map for a homogeneous magnetic field of 16 kA/m along the y-axis. Crossings of black lines correspond to grid nodes; the origin of the coordinate system coincides with the centre of the sensor (white ellipse). The pattern shows symmetries, though the symmetry axes do not coincide with the semiaxes of the ellipse but are turned by an angle α due to the coupling (stray field/Néel-coupling) of the electrodes.

Figure 5.12: ∆TMR-maps for z_part = 0.562 µm and external field strengths of H = 8, 24, 40 and 56 kA/m applied parallel to the y-axis. The degree of particle saturation is given by 0.31, 0.57, 0.69 and 0.76. The inset shows crosscuts through the centre along the y-axis.

changes (Figure 5.12, Hy = 56 kA/m); the maximum ∆TMR-value does not increase any further but decreases together with TMRpart.

An analysis of the angle values shows that this point coincides with α = 0 (Figure 5.13). This behaviour is due to two factors: while the external magnetic field increases linearly, the particle moment reaches saturation and therefore, the external field will overcome the particle influence at a certain point.

On the other hand, the sensing free top layer reaches saturation and becomes less sensitive. Thus, it should be pointed out that ideal detection conditions can be obtained by adjusting the external magnetic field to the particles to be detected and the sensor chosen for the detection.

The obtained TMR-maps can be used to estimate the position of the magnetic particle. Here, we will focus on the distance d

2 2

part part

d = x +y for z_part =0.562 mµ (5.11)

between the particle and the ellipse centre of the top electrode. As is already apparent from Figure 5.11, a single TMR-value corresponds to several particle positions and can therefore only give an upper and a lower bound for the distance. To analyze these bounds, we divide the interval [TMRmin, TMRmax] into N equally sized sub-intervals {∆MR_{i i}}_=1,...,N. For our analysis, the size of ∆MR_i is basically arbitrary; it only needs to introduce a proper class division of the data points along the discrete grid nodes. For the data analysis we choose N = 100. In the experimental situation however, the minimal size of ∆MR_i will correspond to the achievable exactness of the measurement. The map is divided according to the assignment of each map point to the corresponding interval ∆MR_i. This defines a relation between ∆MR_i and the distance d of the regarded map point to the centre of the sensor. A ∆MR_i− −d plot is presented in Figure 5.14. Due to the construction each, ∆MR_i−interval corresponds to a set of d-values. A single measurement can therefore only give an upper and a lower bound for the distance between particle and sensor; the two lines of data points show these bounds. At a given ∆TMR-value, each distance value in between is possible. However if additional measuring directions are taken into account, further information can be obtained.

Figure 5.13: Sensor characteristics for different external fields applied parallel to the y-axis. For high fields the maximum and minimum measurable TMRpart-values decrease. The angle α increases and reaches zero close to H_y = 56 kA/m

Referring to the data of our calculations, we can estimate the degree of accuracy to which the distance d between particle and sensor centre can be determined by combining different measurement directions. As a simplification, we will assume that the particle is stationary during all measurements. As can be seen from Figure 5.12, employing different field values along the y-axis will cause several difficulties: the ∆TMR-response changes at every position in the same manner, therefore, additional information on the particle position can only originate from a varying angle α. This effect, however, is difficult to exploit as significant angle perturbations require fields lying in the range of particle and sensor saturation. This reduces the visibility field as already explained above by a large amount which leads to a decreased spatial resolution.

Instead, we combine bound estimations for applying external magnetic fields parallel to x- and z-axis. Thus, we obtain upper and lower bounds

x Hy z

H H

up , up , up

d d d and d_down^Hx ,d_down^Hy ,d_down^Hz ,

respectively, according to Figure 5.15. The best estimation for the distance d is given by the interval

x y z

x y z

H H

down up H ,H ,H down H ,H ,H up

[ , ]: max , min

I d d  d d 

= =  (5.12)

where the minimum and the maximum need to be taken over all three measuring directions. For the evaluation, only ∆TMR-values are taken into account which cannot be found along the boundary of the grid. Based on our calculation, no estimation is possible for these; values are given in Figure 5.15. The distance estimates for the combined detection d_up and d_down are shown in Figure 5.16(a). Upper and lower bound almost coincide near to the sensor, while the interval size increases with increasing particle-sensor distance d. Certain features (local maxima in the lower bound-map, highlighted in Figure 5.16) can be found due to the combined measurement which increases the accuracy of the position detection. The maximum absolute error

up down

{max |, } |

d d

d d

δ δ

∆ = = − (5.13)

estimation limit

Figure 5.14: Upper and lower distance bounds obtained from the map shown in Figure 5.11.

According to a measured ∆TMR-value, the distance can be estimated by the highlighted area.

The estimation limit results from the finite grid size of 3 µm. For

∆TMR-values found along the grid boundary no evaluation is possible.

is presented in Figure 5.16(b). If a certain threshold of minimum exactness is defined, we obtain guidelines on how to construct sensor arrays, which ensure such measurement resolution along all the x-y-plane. Figure 5.16(c) shows the top view to the total error surface, the grey area corresponds to estimation bounds below an error of 0.2 µm and the resulting sensor assembly.

Figure 5.16: Error estimates according to the simulation data of Figure 5.15. (a) shows upper and lower bound for each grid point, a particle at a given coordinate leads to a corresponding distance estimation.

Different features can be found in the lower bound ddown which originate from the combination of several estimations. (b) presents the maximum error ∆d from the actual distance. The grey level coincides with

additional features

Figure 5.15: Plots for upper and lower d-bounds obtained form the TMR-maps for external magnetic fields of 16 kA/m along the positive coordinate axes. The intervals given, the estimations gaps, indicate the range of ∆TMR where there is no estimation possible because not all corresponding values con be found on the grid chosen. Jumps in the ∆TMR surface can be attributed to a finite mesh resolution.