A Double Quantum Dot Spin Valve

A most fundamental and longstanding goal in spintronics is to electrically tune highly efficient spin injectors and detectors, preferably compatible with nanoscale electronics. Here, we demonstrate all these points using semiconductor quantum dots (QDs), individually spin-polarized by ferromagnetic split-gates (FSGs). As a proof of principle, we fabricated a double QD spin valve consisting of two weakly coupled semiconducting QDs in an InAs nanowire (NW), each with independent FSGs that can be magnetized in parallel or anti-parallel. In tunneling magnetoresistance (TMR) experiments at zero external magnetic field, we find a strongly reduced spin valve conductance for the two anti-parallel configurations, with a single QD polarization of $\sim 27\%$. The TMR can be significantly improved by a small external field and optimized gate voltages, which results in a continuously electrically tunable TMR between $+80\%$ and $-90\%$. A simple model quantitatively reproduces all our findings, suggesting a gate tunable QD polarization of $\pm 80\%$. Such versatile spin-polarized QDs are suitable for various applications, for example in spin projection and correlation experiments in a large variety of nanoelectronics experiments.

gesting a gate tunable QD polarization of ±80%. Such versatile spin-polarized QDs are suitable for various applications, for example in spin projection and correlation experiments in a large variety of nanoelectronics experiments.
Spin injection and detection are two of the most fundamental processes in semiconductor spintronics, [1][2][3][4][5] for example to exploit the electron spin for information storage, logic and sensing, [6][7][8] or to determine and control spin states in quantum physics. [9][10][11] Significant efforts are dedicated to improve the efficiencies of these processes in a variety of material platforms and physical phenomena. [12][13][14][15][16][17] Most of these concepts rely on electrical contacts to ferromagnetic reservoirs, 1 or on magnetic tunnel barriers, 18 with significant obstacles 19 like a low polarization (20 − 40%), 20 the magneto-Coulomb effect, 21,22 the conductivity mismatch at the metallic ferromagnet-semiconductor interface 23 and uncontrolled stray field effects. 4 All these effects are particularly challenging in sub-micrometer scaled electronic devices.
Here we provide an alternative route for spin injection and detection in semiconductor devices using quantum dots (QDs) without ferromagnetic contacts. As illustrated in figure 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 with D σ the QD transmission density of states (t-DoS) for spin state σ ∈ {↑, ↓} at E F . This spindependent 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 2 a ferromagnetic material, which we term ferromagnetic split-gate (FSG). The FSG generates a stray field B str at the QD position in the direction given by its magnetization, either parallel or antiparallel to its long axis 24 and can also be used for electrical gating. The FSG magnetization, and with it B str , can be inverted at a characteristic external switching field B sw , determined by the FSG width in the device design. 25 To demonstrate spin injection and detection, we combine two QD-FSG elements in series in a double QD-spin valve (DQD-SV), in which one element acts as spin-injector (polarizer) and the other as spin detector (analyzer). This concept is illustrated in figure 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 σ ∝ σ . Following typical tunneling magnetoresistance (TMR) experiments, 1 we show that in such nano structures both mutually parallel (p) and both anti-parallel (ap) magnetization states of the two FSGs can be accessed at zero external magnetic field, B = 0, and reoriented by cycling A schematic of a DQD-SV and a scanning electron microscopy (SEM) image of the investigated InAs nanowire (NW) device are shown in figures 1b and 1c, 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 nm and 230 nm, respectively, determining the corresponding switching and stray fields, which can be extracted from independent experiments as demonstrated in Supplementary Information S1 and S2. 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 4 connected accidentally and are tuned in unison, which we refer to as "gate 1" (G1) and "gate 2" (G2), while the other FSGs are labelled individually (see figure 1c). The DC current I resulting from a bias voltage V SD and the differential conductance G = dI/dV SD , were measured simultaneously using standard DC and lock-in techniques (V ac = 10 µV), at a base temperature of ∼ 50 mK.
In figure 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. 37 This allows us to independently extract most of the single QD parameters used for modelling later, e.g. the lever arms of each gate to each QD (see Supplementary S4). We now discuss various types of TMR experiments for two resonances, in figures 2 and 3, respectively, while data for a third resonance are discussed in Supplementary Information S7.
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 figure 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 parallel to the FSG axes, which results in relatively abrupt switchings of the FSG magnetizations (details in Supplementary Information S2). Such a map for the trace in figure 2a is shown in figure 2b for decreasing and increasing magnetic fields, as indicated by the blue and red arrows, respectively, 5 each starting at fields much higher (+0.5 T), or lower (−0.5 T) than shown, to ensure the formation   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 which results in TMR ≈ 6% at V SD = 500 µV and B = −55 mT.

7
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 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, 38 which yields for a constant weak inter-dot coupling T 12 and the magnetization orientations i, j ∈ {+, −} along the FSG axes, 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 elec-8 trochemical 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 which yields, using the definition of the QD polarizations in equation (1), 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.
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 figure 1d, for which we again plot I as a function of B and V G1 at V SD = 10µV. These curves show qualitatively similar characteristics as discussed for figure 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 optmize the TMR signal. First, we apply a small homogenous 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, for example with many superconducting circuit elements. We measure I along cross section C 2 indicated in figure 1d, which is chosen on the resonance maximum along the base of the bias triangle (see Supplementary Information S6) so that a shift in the resonance energies is negligible. We expect that the QD polarizations are also gate tunable to large values, but since an external field is applied, the above symmetry argument cannot be used for a simple estimate. We therefore resort to numerically evaluating the model introduced above. To do so, we define the total magnetic fields B  In the model it is straight forward to extract the spin polarizations, e.g. P 1 for QD1 as a function of V G1 at B = 40 mT, which is plotted in figure 4b, with P 2 ≈ 27% for QD2, being independent of V G1 . P 1 can be gate tuned over a large range, with a maximum absolute value of P 1 ≈ 80%, and a zero-field value of ≈ 59%. This analysis demonstrates that the DQD-SV † we also include cross lever arms in the model without stating this explicitly for simplicity 13 is a highly tunable spin valve with one QD acting as a gate-tunable spin injector and the other as a detector, such that transport through the DQD can be electrically tuned from predominantly spin down electrons to spin up electrons, depending on the orientation of B str and B.

METHODS
The InAs NWs were grown using 30 nm gold (Au) colloid assisted chemical beam epitaxy 46     As a control experiment to the DQD-SV, we fabricate a device with a single pair of FSG as shown in figure S1a. The device is a 3-terminal InAs NW contacted with Ti/Au normal metal contacts S, D 1 and D 2 . Adjacent to the NW segment between S and D 1 , we placed a 170 nm wide Permalloy FSG pair in a split gate geometry at a distance of 35 nm from the NW. We apply a dc (V SD ) and ac bias to the source (S) contact and simultaneously measure the differential conductance G (1,2) = dI1,2 dV at contacts D 1 and D 2 using standard lock-in techniques. Figures S1b and S1c show colorscale plots of G (1) and G (2) , respectively as a function of V SD and back gate voltage V BG . We observe a regular pattern of Coulomb Blockade (CB) diamonds, suggesting the formation of QDs in both NW segments. We then apply an external magnetic field B along the FSG long axes and measure G (1),(2) as a function of B for the QD resonances in the S − D 1 and S − D 2 segment, similar to experiments in the main text.

A Double Quantum Dot Spin Valve -supplementary information
The maximum conductance G (1) max over one CB resonance (V BG = 1.2 V) as a function of B is shown in figure S1d. It is clearly hysteretic for the up (red) and down (blue) sweeps and mirror symmetric around B = 0. We also observe a sharp switching in this magnetoconductance (MC) at B sw = 25 mT for both sweeps, suggesting a reversal in the magnetization of the FSG. We note that we only observe a single switching, suggesting that the two parts of the FSG switch in unison. Furthermore, for both sweeps in the NW segment S − D 2 , we observe a small hysteresis and no switching, as shown in figure S1f. We note that we do not observe any hysteresis in the magnetoconductance for QD devices without any FSGs nearby (not shown).
For a two terminal device with B str = 0, the MC G 0 (B) is necessarily an even function in B This relation allows us to determine the stray field of the FSGs assuming B str || B.      figure 2c of the main text, measured at V SD = 500 µV. We assign an average zero level of the measured data, shown as black dashed line and define a lower and upper current limit for a significant deviation of ∆I from the average zero. We use the B values at which the upper horizontal line meets ∆I as the two switching fields, B sw1 140 mT and B sw2 5 mT, respectively. We use a similar analysis for the lower horizontal line. We point out that a similar analysis of Figure 3b, i.e. on a different resonance, results in the same switching fields, as shown in figure S2b.

Sweep Sequence
The determination of the switching fields enables us to define the exact procedure to obtain the four magnetization states at B = 0. The measurements in the main text were all done in the following orders:   figure S3. The capacitance between the QD and the respective gate is given by: C G = e/∆V G . We find C G1 = 6.4 aF and C G4 = 5.94 aF to the respective QD. The total capacitances of the two QDs are C 1 = 64 aF and C 2 = 65.2 aF, while the mutual capacitance is C m = 7 aF. The addition energy of the QDs are E add,1 ≈ 2.5 meV and E add,2 ≈ 2.7 meV, while the level spacings are δE 1 ≈ 0.7 meV and δE 2 ≈ 0.81 meV, respectively. The lever arms for both dots are found as: a 11 ≈ 0.1, a 12 ≈ 0.015, a 21 ≈ 0.0 and a 22 ≈ 0.09, i.e. the cross lever arms are one order of magnitude smaller than the direct lever arms.       Table S1: Summary of the parameters extracted from the resonant tunneling model for the three resonances measured (g-factors and energy offset E (0) of the two QDs).