### Device and characterization

As illustrated in Fig. 1a, the spin degeneracy of a QD state can be lifted by a magnetic field, resulting in a spin polarization at the Fermi energy *E*_{F} of

$$P=frac{{D}_{uparrow }({E}_{{rm{F}}})-{D}_{downarrow }({E}_{{rm{F}}})}{{D}_{uparrow }({E}_{{rm{F}}})+{D}_{downarrow }({E}_{{rm{F}}})},$$

(1)

with *D*_{σ} the QD transmission density of states (t-DoS) for spin state *σ* ∈ {*↑*, *↓*} at *E*_{F}. This spin-dependent transmission directly results in a spin-polarized current through the QD. In practice, a single QD can be spin-polarized individually by placing it in the narrow gap in a long strip of a ferromagnetic material, which we term FSG. The FSG generates a stray field ({B}_{{rm{str}}}) at the QD position in the direction given by its magnetization^{28,29}, either parallel (p) or antiparallel (ap) to its long axis^{30}, and can also be used for electrical gating. The FSG magnetization, and with it ({B}_{{rm{str}}}), can be inverted at a characteristic external switching field *B*_{sw}, determined by the FSG width in the device design^{31,32}.

Similar to conventional spin valves with two ferromagnetic contacts with Stoner split bands, we combine two QD-FSG elements with Zeeman split QD states in series to form a DQD-SV. In the first QD-FSG unit (spin injector), a spin-polarized tunnel current is generated, which is then detected at a different position by a second QD-FSG unit (spin detector). This concept is illustrated in Fig. 1a: electrons in state *σ* from the unpolarized electrical contacts tunnel sequentially through the two QDs with a probability that depends on the FSG states of both QDs, to first order resulting in the respective current ({I}_{sigma }propto {D}_{sigma }^{(1)}{D}_{sigma }^{(2)}). Following typical TMR experiments^{1}, we show that in such nano structures both mutually p and both ap magnetization states of the two FSGs can be accessed at zero external magnetic field, *B* = 0, and reoriented by cycling *B*. The individual QD polarizations and TMR signals can be continuously electrically tuned up to values close to the theoretical limits. In contrast to previously employed, very large polarizing external magnetic fields^{27,33,34}, the stray and external magnetic fields required for such optimizations are small enough and decay over short enough length scales, to be compatible with various spin injection and detection experiments, e.g., with superconducting components in Cooper pair splitters^{18,19} for electron spin correlation measurements^{35}, or to demonstrate equal spin Andreev reflection^{20} at Majorana-type superconducting bound states^{36,37,38}.

A schematic of a DQD-SV and a scanning electron microscopy image of the investigated InAs NW device are shown in Fig. 1b, c, respectively. The FSGs are long Permalloy (Py) strips fabricated by electron beam lithography with a narrow gap at the NW position, forming the split-gate geometry. The strip widths are 120 and 230 nm, respectively, determining the corresponding switching and stray fields, which can be extracted from independent experiments as demonstrated in Supplementary Notes 1 and 2. The electrical contacts at the NW ends are made of titanium/gold with a split central gate (CG) to electrically form the two QDs fabricated in the same step. One part of the narrower FSG and the CG gate are electrically connected accidentally and are tuned in unison, which we refer to as “gate 1” (G1) and “gate 2” (G2), whereas the other FSGs are labeled individually (see Fig. 1c). The DC current *I *resulting from a bias voltage *V*_{SD} and the differential conductance *G* = d*I*/d*V*_{SD}, were measured simultaneously using standard DC and lock-in techniques (*V*_{ac} = 10 μV), at a base temperature of ~50 mK.

In Fig. 1d, we plot *I* flowing through the DQD-SV at *V*_{SD} = 1 mV, as a function of *V*_{G1} and *V*_{G4}. This map shows several bias triangles characteristic for a weakly coupled DQD. These triangles originate from one resonance of each QD aligning in energy within the bias transport window^{39}. This allows us to independently extract most of the single QD parameters used for modeling later, e.g., the lever arms of each gate to each QD (see Supplementary Note 3). We now discuss various types of TMR experiments for two resonances, in Figs. 2 and 3, respectively.

### Tunneling magnetoresistance at zero external magnetic field

We first demonstrate the principle of a TMR experiment and show that all FSG magnetization states can be accessed at *B* = 0. Figure 2a shows a high-resolution bias triangle of a resonance (not shown in Fig. 1d) at *V*_{SD} = 500 μV. Our typical TMR experiment consists of first choosing a specific trace for the two gate voltages here by sweeping *V*_{G1} and keeping *V*_{G4} constant, as indicated by the red arrow, such that no excited states are involved in the transport process. We then measure *I* as a function of *V*_{G1} at a series of external magnetic fields, *B*, applied in p to the FSG axes, which results in relatively abrupt switchings of the FSG magnetizations. Such a map for the trace in Fig. 2a is shown in Fig. 2b for decreasing and increasing magnetic fields, as indicated by the blue and red arrows, respectively, each starting at fields much higher (+0.5 T), or lower (−0.5 T) than shown, to ensure the formation of only a single magnetic domain along the FSG axes. These maps show a clear hysteresis with a strong dependence on *B* and the sweep direction. To demonstrate this more explicitly, we extract the position, width (both shown in Supplementary Fig. 4) and the maximum current ({I}_{max }) at each *B* value. ({I}_{max }) extracted from Fig. 2b is plotted in Fig. 2c for decreasing (blue) and increasing *B* (red).

In the up-sweep, ({I}_{max }) first increases roughly linearly with increasing *B*, followed by a maximum at *B* ≈ −55 mT and a decrease around *B* = 0. At small positive *B*, ({I}_{max }) becomes flatter, followed by a small maximum at *B* ≈ 85 mT, and a roughly linear decrease towards more positive *B*. The down-sweep can be described similarly as the up-sweep, but mirrored at *B* ≈ 0 leading to a clear hysteresis. This hysteresis can be understood qualitatively by considering a smooth non-monotonous MR of the DQD that changes abruptly with the reorientation of the FSG magnetizations. In the up-sweep, at *B* > *B*_{sw2} ≈ 5 mT the wider FSG is reoriented p to the now positive *B* and the two FSG magnetizations become ap. The FSGs become magnetized in p again for *B* > *B*_{sw1} ≈ 140 mT, when the narrower FSG is also inverted (details are given in Supplementary Note 2). These configurations are shown schematically at the bottom of Fig. 2c for the down (blue) and the up-sweep (red).

As a first quantitative measure for the TMR effect, we use the maximum current values at *B* = −55 mT, using the maximum value of ({I}_{max }) in the p state, and the value in the opposite sweep direction at the same field in the ap state. We define TMR as

$${rm{TMR}}=frac{{I}_{{rm{p}}}-{I}_{{rm{ap}}}}{{I}_{{rm{p}}}+{I}_{{rm{ap}}}}$$

(2)

which results in TMR ≈ 6% at *V*_{SD} = 500 μV and *B* = −55 mT.

To explicitly demonstrate that all four magnetization states (two p and two ap) are accessible at *B* = 0, we measure the differential conductance *G* at *V*_{SD} = 0 as a function of *V*_{G1} for each FSG magnetization state. The direction of the stray fields *B*_{str1} and *B*_{str2} can be reversed individually by sweeping *B* beyond the characteristic FSG switching fields. For example, we sweep to *B* = −500 mT and back to *B* = 0 to obtain the (−, −) state, followed by sweeping to *B* = +40 mT and back to *B* = 0 to obtain the (−, +) state (see Supplementary Note 4 for more details). We note that in the used sequence, p is followed by ap and vice versa. The gate sweeps for the four magnetization states at *B* = 0 are plotted in Fig. 2d. All curves show a maximum at the same gate voltage, which corresponds to a weakly spin split energy level of each QD (*Γ* > *g**μ*_{B}*B*) being aligned with the Fermi energy. The conductance is gradually reduced to zero if the QD levels are detuned by *V*_{G1}. We find very similar maximum conductances for the same relative magnetization orientations and a clear suppression in *G* for both ap states with respect to the two p states, yielding ({rm{TMR}}=frac{Delta G}{{G}_{{rm{P}}}+{G}_{{rm{AP}}}}approx 7 %), similar to the value obtained at a larger bias and a small finite *B*.

The DQD-SV experiment can be reproduced quantitatively using a very simple model, which also allows us to estimate the QD polarizations: we assume that the current is given by elastic tunneling in two independent spin channels^{40}, which yields for a constant weak inter-dot coupling *T*_{12} and the magnetization orientations *i*, *j* ∈ {+, −} along the FSG axes,

$$begin{array}{ll}{I}^{(ij)}={I}_{uparrow }^{(ij)}+{I}_{downarrow }^{(ij)}=&frac{e}{h}{mathop{sum }limits_{sigma }}mathop{int}nolimits_{-infty }^{infty }{T}_{12}{D}_{1sigma }^{(i)}(E){D}_{2sigma }^{(j)}(E)\ &[f(E-{mu }_{{rm{S}}})-f(E-{mu }_{{rm{D}}})]dE,end{array}$$

(3)

where *D*_{βσ}(*E*) denotes the spin-dependent t-DoS in dot *β* ∈ {1, 2} and *σ* ∈ {*↑*, *↓*} the spin orientation; (f(E)=1/(1+{e}^{E/({k}_{B}T)})) is the Fermi-Dirac distribution function and *μ*_{S,D} the electrochemical potential in the source and drain contacts, respectively. To start with, we assume a small bias (linear regime) to obtain the conductance, as in the experiments. Since the Zeeman shift is opposite, but of the same magnitude for opposite spins, the t-DoS of each QD obeys the identity ({D}_{sigma }^{-}(-B,{E}_{{rm{F}}})={D}_{-sigma }^{+}(+B,{E}_{{rm{F}}})) due to time-reversal symmetry. At *B* = 0, this reduces to ({D}_{sigma }^{-}({E}_{{rm{F}}})={D}_{-sigma }^{+}({E}_{{rm{F}}})), which yields, using the definition of the QD polarizations in equation (1),

$${rm{TMR}}=frac{{I}_{{rm{p}}}-{I}_{{rm{ap}}}}{{I}_{{rm{p}}}+{I}_{{rm{ap}}}}={P}_{1}{P}_{2}approx {P}^{2}.$$

(4)

In the last step we assume that both QD polarizations are identical, which results in *P* ≈ 27% on resonance at *B* = 0. We stress that this expression for the TMR signal only holds at *B* = 0 because of the non-constant QD t-DoS, in contrast to devices with ferromagnetic contacts, for which it holds also at finite external fields, limited only by the correlation energy of the band structure.

### Optimized tunneling magnetoresistance at finite fields

The non-constant t-DoS of the QDs allows us to go beyond the standard experiments, enabling us to optimize and tune the TMR signals magnetically as well as electrically. To demonstrate this, we investigate cross-section C_{1} pointed out in Fig. 1d, for which we again plot *I* as a function of *B* and *V*_{G1} at *V*_{SD} = 10 μV. Figure 3a shows the up and down sweeps, which again show a clear hysteresis, prominently visible in Fig. 3b, where we plot ({I}_{max }) as a function of *B* for the up and down sweeps (width and position are discussed in Supplementary Fig. 4). These curves show qualitatively similar characteristics as discussed for Fig. 2c. From the current maximum, we find a TMR signal of ~29% at *B* = 0 and estimate the individual QD spin polarizations as *P* ≈ 53% using equation (4). These values are larger than for the previously discussed resonance, mostly due to a smaller resonance width.

We now exploit the non-constant t-DOS to optimize the TMR signal. First, we apply a small homogeneous external field of ±40 mT, which is small enough to still access all four FSG magnetization states (*B* < *B*_{sw1}) and compatible with a wide variety of applications, e.g., with many superconducting circuit elements. We measure *I* along cross-section C_{2} indicated in Fig. 1d, which is chosen on the resonance maximum along the base of the bias triangle (see Supplementary Note 5) so that a shift in the resonance energies is negligible.

Figure 3c shows the four *I*(*V*_{G1}) curves along C_{2} for the four FSG magnetization states (*i*, *j*) (*V*_{G4} is the same for each chosen *B*). The curve for the p (−, −) [blue] and the ap configuration (+, −) [gray] were measured at *B* = −40 mT, while the ones for (+, +) [purple] and (−, +) [black] were measured at *B* = +40 mT (see Supplementary Note 4 for sweep sequence). We find that the maximum current and lineshape for both ap configurations are almost identical, while the two p ones slightly differ. Most importantly, the ap curves are reduced in amplitude by ~25% with respect to the p ones. We note that for this cross-section, the maximum occurs at the same *V*_{G1} value for both pairs of curves in Fig. 3c.

For any given *V*_{G1} and *B*, we now calculate the TMR signal using equation (2). As an example, this is plotted for the states (+, +) and (−, +) in Fig. 3d (red curve), which shows that the TMR signal is continuously gate-tunable roughly between +50% and −25%. This TMR signal can be improved significantly by exploiting the small, field-induced shifts in the resonance positions. To achieve this, we plot TMR = (*I*_{++} − *I*_{−+})/(*I*_{++} + *I*_{−+}) at *B* = 40 mT as a function of *V*_{G1} and *V*_{G4} in Fig. 4a and find the optimal cross-section labeled C_{opt}. In Fig. 3d, we plot TMR along *C*_{opt} which shows a continuously gate-tunable TMR with a well separated pronounced maximum and minimum TMR of +80% and −90%, respectively. These values are significantly larger than in most other systems.