= Solution
Exchange-split bands in a <ferromagnet> generally give a different spin-resolved <density of states> for each <spin> branch, $D_\uparrow(E_F)$ and $D_\downarrow(E_F)$. If <spin> is approximately conserved across a multilayer, its two <spin> channels conduct in parallel. A <spin valve> combines a comparatively fixed reference-layer <magnetization> with a soft free layer that rotates in a small applied field. Its readout depends on the relative angle, not simply the absolute <spin> population. Biasing the layers near a perpendicular alignment gives a useful <linear response> instead of operating at the flat parallel/antiparallel extrema.
In <giant magnetoresistance>, a metallic spacer separates ferromagnetic layers. Spin-dependent bulk and interface scattering make their channel resistances unequal. The density of available final $d$ states influences scattering of more mobile states; however, a large total <density of states> does not by itself guarantee a large conductivity. The <scattering-weighted spin-channel asymmetry> also involves velocity and scattering matrix elements. In the elementary identical-layer <two-current resistor model>, let each magnetic layer have resistances $a,b$ in the two <spin> channels and neglect spacer resistance. Parallel moments give channels $2a$ and $2b$, while antiparallel moments give $a+b$ in each channel. Therefore
$$
R_P=\frac{2ab}{a+b},\qquad R_{AP}=\frac{a+b}{2},\qquad
\boxed{\frac{R_{AP}-R_P}{R_P}=\frac{(a-b)^2}{4ab}.}
$$
One favorable <spin> channel traverses both aligned layers efficiently; reversal removes that common easy path. A conducting spacer adds resistance, and <spin> mixing weakens the separate-channel approximation. The channel asymmetry is material/interface dependent, so positive magnetoresistance is not a theorem for every pair of different <ferromagnets>.<Spin dependent transport: GMR & TMR by Alain Schuhl and Daniel Lacour>[§§2.1–2.2, pp.947–948, spin-dependent scattering and two-current construction]{ref}
In <tunnelling magnetoresistance>, the spacer is an ultrathin insulator and current flows across a <magnetic tunnel junction>. In the simple spin-conserving <density-of-states tunnelling magnetoresistance model>, assume spin-independent matrix elements and low <voltage>. Available initial and final states give
$$
G_P=C(D_{1\uparrow}D_{2\uparrow}+D_{1\downarrow}D_{2\downarrow}),\qquad
G_{AP}=C(D_{1\uparrow}D_{2\downarrow}+D_{1\downarrow}D_{2\uparrow}).
$$
Writing $P_i=(D_{i\uparrow}-D_{i\downarrow})/(D_{i\uparrow}+D_{i\downarrow})$ and $G_0=CD_1D_2/2$, these become $G_P=G_0(1+P_1P_2)$ and $G_{AP}=G_0(1-P_1P_2)$. Thus
$$
\boxed{\mathrm{TMR}\equiv\frac{R_{AP}-R_P}{R_P}=\frac{2P_1P_2}{1-P_1P_2}.}
$$
The effective tunnelling polarization can depend on the barrier and orbital symmetry, not just a bulk density-of-states sketch. Crystalline MgO junctions exploit symmetry-selective transmission, so the elementary product model is not a quantitative universal formula for them.<Giant room-temperature magnetoresistance in Fe-MgO-Fe junctions by Shinji Yuasa and others>[2004 publisher abstract, crystalline MgO junctions and large room-temperature response]{ref}
Both devices convert a free-layer rotation into an electrical change. A useful approximate GMR interpolation is $R(\theta)=R_P+\Delta R(1-\cos\theta)/2$; the elementary tunnelling model instead gives $G(\theta)=G_0(1+P_1P_2\cos\theta)$. The <spin-valve small-field sensitivity> is a chain rule,
$$
\boxed{\frac{dR}{dB}=\frac{dR}{d\theta}\frac{d\theta}{dB},\qquad
\delta V=I_{\rm bias}\frac{dR}{dB}\,\delta B.}
$$
<magnetic anisotropy>, biasing and pinning set the rotation susceptibility. Conducting-spacer GMR usually offers low resistance and supports current-in-plane or current-perpendicular geometries. <TMR> uses perpendicular tunnelling, often offers a larger fractional signal and a larger resistance-area product, but demands uniform, stable barriers and has bias dependence and breakdown limits. Neither is automatically the better small-field sensor: heating, Johnson and low-frequency noise, domain switching, hysteresis and stability set the detectable field. Reverse sweep direction, vary current and <temperature>, and independently test reference-layer pinning to distinguish a reproducible magnetic signal from drift or heating.
\Image[/past-exam-of-the-physics-of-nanoelectronic-systems-course-of-the-university-of-cambridge/2012/paper-1-spin-sensors.png]
{title=Spin-dependent sensor response}
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