Seven layers, like a network stack

In a network (digital track D1) every layer has one job and talks only to its neighbours: the physical layer moves bits, the data-link layer builds frames, and so on up to the application. A receiver unpacks the data from the bottom up. A sensor is a receiver for the physical world: the field enters at the bottom and climbs to the application on top. What each layer hands up is its "protocol data unit" (PDU), here a physical quantity.

The analogy goes beyond the picture. As in a network, a layer can be replaced without touching the others: put a Hall plate in place of layers 2–4 (see Compare) and layers 5–7 stay the same, just as Wi-Fi can replace Ethernet underneath IP.

Move the magnet or the field and follow the "packet" up the stack. The right panel decodes it layer by layer, like a network analyser. Click a layer (or its button) for a closer look.

Where the field comes from, and how big it is

The input of the sensor is the magnetic flux density B, measured in tesla (T). In air it is tied to the field strength H by B = μ₀H, so 1 mT corresponds to 796 A/m (and to 10 gauss). For scale: the earth's field is about 50 μT, a fridge magnet gives a few mT, the surface of a neodymium magnet about 0.5 T. Sensors meet three kinds of sources (chapter 3 and chapter 2):

• a permanent magnet on a rotating shaft (angle sensors). Far from it the field is that of a dipole and falls with the cube of the distance;
• a current in a wire (current sensors). Ampère's law gives circles around the wire, falling as 1/r;
• the earth (electronic compasses). Only its horizontal part, about 22 μT in central Europe, points north.

The magnet is a 6 mm × 2.5 mm NdFeB disc (B_r = 1.3 T, magnetic moment m = 0.073 A·m²) treated as a point dipole. That is accurate once the distance exceeds the magnet's size; very close to it the real field is weaker than the dipole formula.

A tiny compass needle with a preferred axis

The sensing element is a ferromagnetic film only about 2 nm thick, the free layer. All its electron spins are aligned (chapter 3), so the whole film behaves like one compass needle: a magnetization vector of fixed length that can only rotate. It prefers to lie along an easy axis, set by the element's shape or crystal; the strength of that preference is the anisotropy field H_K. An outside field H pulls the needle away from the easy axis. The Stoner–Wohlfarth model writes the energy, in units of the anisotropy energy, as a function of the magnetization angle θ:

θ_K is the direction of the easy axis, θ_H that of the field. The magnetization sits in a valley of e(θ), like a ball. The same film can be used in three ways:

• Field across the easy axis (linear sensor). One valley that moves smoothly: the component along the field is exactly cos θ = h for |h| ≤ 1. The response is linear, with no hysteresis.
• Field along the easy axis (memory). Two valleys. The ball stays in its valley until that valley disappears at |h| = 1, then it flips. The result is a square hysteresis loop, which is how MRAM stores a bit.
• Strong rotating field (angle sensor). The needle follows the field direction. A small lag remains, at most about 1/(2h) radians.

In memory mode the field is tilted 0.1° from the easy axis, as in any real experiment; otherwise the ball could balance forever on the top of the hill.

Electrons keep their spin while they tunnel

Every electron carries a spin, a built-in magnetic moment that points "up" or "down" along any chosen axis. In a ferromagnet the energy bands of the two spin directions are shifted against each other by the exchange splitting (about 1–2 eV in iron). At the Fermi energy, where conduction happens, there are therefore more states for one spin direction (the majority spins) than for the other (the minority spins). The spin polarization P measures the imbalance in the density of states D:

A magnetic tunnel junction is two ferromagnets separated by an insulator about 1 nm thick. Electrons cross it by quantum tunneling (chapter 9) and keep their spin on the way. Each electron needs a free state of the same spin on the other side, so each spin channel conducts in proportion to the product of the densities of states on both sides (Jullière, 1975). If the two magnetizations are parallel (P), majority meets majority and the current is large. If they are antiparallel (AP), majority meets minority and the current is small:

At an angle θ between the magnetizations, a spin that is "up" for the first magnet is, for the second, a quantum superposition: majority with probability cos²(θ/2), minority with probability sin²(θ/2). Adding the four channels gives a simple law. Note that it is the conductance G = 1/R, not the resistance, that varies with cos θ:

With P = 0.6 Jullière's formula gives 112.5 %. Real junctions do better than this simple model. Crystalline MgO lets mainly one symmetry of wave function through (called Δ₁), and in Fe and CoFe(B) that symmetry is present at the Fermi energy almost only for majority spins. That raised room-temperature TMR from about 70 % with amorphous Al₂O₃ barriers to several hundred percent (record about 600 % in 2008). Sensors typically use 100–200 %.

Nine films, each a few nanometres thick

The real device is a stack of ultrathin films, sputtered onto the wafer and etched into pillars. Each film has a single job. Click a film in the picture to see its job.

• An antiferromagnet (IrMn or PtMn) pins the magnetization of the film on top of it by exchange bias. Its direction is set once, by cooling the wafer in a strong field.
• The pinned layer, a 0.85 nm ruthenium spacer and the reference layer form a synthetic antiferromagnet (SAF): through the Ru the two magnets couple antiparallel (RKKY coupling). The pair is very stiff and its stray fields almost cancel.
• The MgO barrier sets the resistance. Since tunneling falls off exponentially with thickness, the resistance-area product RA grows about ten-fold for every additional 0.25 nm.
• The free layer (CoFeB, sometimes with NiFe) turns with the field (layer 2), and a cap protects the stack.

The TMR also shrinks with the voltage across the barrier: hot electrons excite magnons and the spin filtering weakens. Sensors therefore connect many junctions in series in each bridge arm, so that every junction sees only some tens of millivolts. Series strings also average out noise and the variation between junctions.

Thicknesses and RA are typical values for CoFeB/MgO junctions; the formulas are fits that show the trends, not data of a specific product.

Four junctions turn resistance into a clean voltage

A single junction is a poor sensor on its own. Its resistance also changes with temperature, and it differs from chip to chip. A full Wheatstone bridge uses four junction strings. In two of them the reference layer is pinned one way, in the other two the opposite way (push-pull), so that a field raises the conductance of one pair exactly as much as it lowers that of the other. Everything that affects all four alike (temperature drift of R̄, supply noise, slow ageing) cancels, and the signal doubles. With R = R̄/(1 ± x) and x = P²h the output is exactly linear:

With V_b = 3.3 V, P² = 0.36 and H_K = 5 mT the sensitivity is 72 mV per volt of supply per mT, and the output swings up to ±36 % of the supply. A silicon Hall plate gives about 0.1 mV/V/mT. What does not cancel: the TMR itself falls by about 0.1 % per kelvin, so the gain drifts. The electronics corrects that with a temperature sensor. Small mismatches between the four arms leave a small offset.

Amplify, digitize, and send it as bits

The bridge voltage is analog. An amplifier with gain G scales it to the input range V_FS of an analog-to-digital converter (ADC), which outputs an N-bit number. The smallest step, one LSB, corresponds to a field step ΔB. Noise from the junctions (mostly 1/f noise at low frequency) and from the amplifier blurs the signal. Averaging N_avg samples reduces random noise by √N_avg, at the cost of speed. Too little gain wastes ADC resolution; too much gain clips strong fields. Here the sensor measures the earth's field while the device turns once every two seconds:

The codes at the bottom are what leaves the chip. A serial interface such as SPI or I²C packs them into frames. From here the sensor is literally part of a network stack: the bits travel over buses (AXI4-Lite on the processor side) and possibly over Ethernet and TCP (D1).

Angle, current, compass and memory

Angle sensors (steering wheels, motors, knobs) sit under a magnet on a rotating shaft. Two bridges with reference directions 90° apart give V_cos ∝ cos φ and V_sin ∝ sin φ, and the processor computes φ = atan2(V_sin, V_cos). Since the magnitude drops out, the result does not depend on the exact distance or temperature. It is sensitive to imperfections: an offset, unequal amplitudes or bridges not exactly 90° apart each cause a characteristic angle error. In the plot the circle becomes a shifted or squeezed ellipse.

• Current sensors measure the field of a conductor, B = μ₀I/(2πr), without touching it: no shunt loss and galvanic isolation for free. Two sensors on opposite sides of the conductor see opposite fields; their difference doubles the signal and cancels homogeneous stray fields.
• Electronic compasses measure the 22 μT horizontal earth field along two or three axes and compute the heading with atan2, plus an accelerometer for tilt.
• MRAM stores a bit in the free layer: parallel = 0, antiparallel = 1 (layer 2, memory mode). It is read through the TMR and written by a current whose spin-polarized electrons exert a torque on the free layer (spin-transfer torque). It keeps its data without power like flash (chapter 11), but survives more than 10¹² writes instead of about 10⁵.
• Hard-disk read heads have used TMR since about 2005 to read magnetic bits only some tens of nanometres wide.

Hall, AMR, GMR and TMR: four ways to fill layers 2–4

The stack on one page

Where next? Chapter 11 stores data with tunneling electrons as well, as charge on a floating gate instead of in a magnet. Back to the learning path.

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