Electrical spin injection and detection in molybdenum disulfide multilayer channel

Abstract

Molybdenum disulfide has recently emerged as a promising two-dimensional semiconducting material for nano-electronic, opto-electronic and spintronic applications. However, the demonstration of an electron spin transport through a semiconducting MoS₂ channel remains challenging. Here we show the evidence of the electrical spin injection and detection in the conduction band of a multilayer MoS₂ semiconducting channel using a two-terminal spin-valve configuration geometry. A magnetoresistance around 1% has been observed through a 450 nm long, 6 monolayer thick MoS₂ channel with a Co/MgO tunnelling spin injector and detector. It is found that keeping a good balance between the interface resistance and channel resistance is mandatory for the observation of the two-terminal magnetoresistance. Moreover, the electron spin-relaxation is found to be greatly suppressed in the multilayer MoS₂ channel with an in-plane spin polarization. The long spin diffusion length (approximately ∼235 nm) could open a new avenue for spintronic applications using multilayer transition metal dichalcogenides.

Introduction

Transition metal dichalcogenides (TMDs) have emerged as a promising 2D crystal family, demonstrating solutions for several novel nano-electronic and opto-electronic applications^{1,2,3,4,5,6,7}. In contrast to graphene and boron nitride, which are respectively a metal and a wide-gap semiconductor, TMDs family displays a large variety of electronic properties ranging from semiconductivity to superconductivity⁸. As a representative of TMDs, molybdenum disulfide (MoS₂) has a tunable bandgap that changes from an indirect gap of 1.2 eV in the bulk to a direct gap of 1.8 eV for one monolayer (ML)¹. The ML MoS₂ is characterized by a large spin-orbit splitting of ∼0.15 eV in the valence band^3,4 together with a small value of ∼3 meV for the conduction band⁹. The lack of inversion symmetry combined with the spin-orbit interaction leads to a unique coupling of the spin and valley degrees of freedom, yielding robust spin and valley polarizations^4,5,6,7.

To realize electron spin transport in the vicinity of the conduction band of MoS₂ channel, one of the prerequisites is the investigation of the electron spin-relaxation mechanism within the host material. For ML MoS₂, both the intrinsic spin splitting of the valence band and the Rashba-like spin-orbit coupling (SOC) due to the breaking of the inversion symmetry along the growth direction favour the spin transport through MoS₂ with an out-of-plane spin polarization¹⁰. This is because that the SOC creates an equivalent perpendicular k-dependent magnetic field due to the Dresselhaus effective interactions and it is associated to the D’yakonov-Perel (DP) spin relaxation mechanism¹¹. If electrons with in-plane spin polarization are injected into ML MoS₂, the effective magnetic field can induce an efficient in-plane spin precession along the field¹² as well as the spin-dephasing. Consequently, this yields a predicted short spin lifetime (10–200 ps)¹³ together with a small spin diffusion length (∼20 nm)¹⁴. Recently, a hole spin injection into ML TMDs has been demonstrated either with perpendicular magnetized GaMnAs injector¹⁵ or NiFe injector at large perpendicular magnetic field¹⁶ by electrical injection and optical detection method. This particularly emphasizes on the difficult issue to electrically inject and detect electron with in-plane spin polarization in a lateral ML MoS₂ device. To avoid the DP spin relaxation, one solution is to recover the inversion symmetry with thicker multilayer MoS₂. The recent measurement of the second-harmonic generation (SHG) efficiency as a function of the number of MLs is a good probe of the TMDs material symmetry¹⁷. For one ML, a strong SHG is detected due to the lack of inversion symmetry¹⁸. However, in the case of bilayers and 4ML MoS₂, the magnitude of SHG signal decreases by three orders of magnitude due to the recovery of inversion symmetry. Longer spin relaxation time can be expected for such structures. Thus, we consider only multilayer MoS₂ to demonstrate the in-plane electrical spin injection and detection.

Here, we provide a clear demonstration of a robust spin-valve magnetoresistance (MR) (1.1%) through the conduction band of 6ML thick MoS₂ channel with ferromagnetic (FM) Co/MgO tunnel injector at low temperature. This occurs in the optimal experimental situation of impedance matching between the interface resistance and the channel resistance. The clear spin-injection signals in MoS₂ demonstrate a spin-transport in MoS₂ with a relative long spin-diffusion length larger than 200 nm. This could open future avenues to use multilayer TMDs as an in-plane spin transport template and the electron spin can be manipulated by SOC for a well-defined MoS₂ thickness controlled by plasma etching technique¹⁹.

Results

MoS₂ contact and channel resistance

A key issue for electrical spin injection is the conductivity mismatch between the FM injector and the semiconducting MoS₂ channel, which generally results in a vanishing MR^20,21 due to the spin-backflow processes²¹. In FM/MoS₂ contacts, a Schottky barrier (SB) height (Φ_b) of 100–180 meV is generally created at the interface with an extended depletion region^22,23. However, it has been recently demonstrated that an efficient reduction of Φ_b down to ∼10 meV at zero back-gate voltage can be achieved by inserting a 1–2 nm layer of MgO (ref. 23), Al₂O₃ (ref. 24) or TiO₂ (ref. 25) as a thin tunnel barrier between the FM and MoS₂. A careful design of the interface structure consisting of an oxide tunnel barrier injector (MgO) on top of an unavoidable Schottky contact thus appears mandatory to get efficient electrical spin injection and spin-detection²⁶.

In our devices, MoS₂ flakes were mechanically exfoliated onto a SiO₂/Si (n++) substrate. Four FM contacts composed of MgO (2 nm)/Co (10 nm)/Au (10 nm) were deposited on one MoS₂ flake (see details in Methods). The four electrodes have almost identical width around 300 nm with channel distances varying from 450 to 2,800 nm (Fig. 1a). The thickness of the flake is determined by atomic force microscopy characterization to be about 4.3 nm (Fig. 1b,c). Considering 0.72 nm for one ML MoS₂ (ref. 27), the thickness of the flake corresponds to 6ML MoS₂. Figure 1d shows schematics of the device. A drain-source bias (V_ds) between two top contacts was applied to inject a current I_ds. Meanwhile, a back-gate voltage (V_g) was applied between the substrate and one top contact to modulate the carrier density in the MoS₂ channel.

Figure 1: Multilayer MoS₂-based lateral spin-valve device.

(a) Optical image of the device with the multilayer MoS₂ flake on 100 nm SiO₂/Si(n++) substrate, the E1, E2, E3 and E4 indicate the four Au/Co/MgO electrodes. (b,c) AFM measurement (in tapping mode) focused on the MoS₂ channel between E1 and E2 electrodes. The thickness of MoS₂ is determined by the Gaussian distribution of pixel height in the square region in b. (d) Schematics of the lateral spin-valve device. The multilayer MoS₂ serves as a spin transport channel, and two Au/Co/MgO electrodes are used to inject spin (V_ds) and measure the current (I_ds). A back-gate voltage (V_g) between the substrate and one top contact is used to modulate the carrier density in the MoS₂ channel.

Let us first focus on the two-terminal I_ds–V_ds characteristics at 12 K between electrodes E1 and E2 with different V_g (Fig. 2a). At V_g=0 V, the current level is rather low (I_ds∼−100 nA at V_ds=−1 V). By applying a back-gate voltage, a large increase of the current is observed with positive V_g, while the current density is greatly suppressed at negative V_g. The quasi-symmetric nonlinearity of I_ds–V_ds is attributed to the back-to-back Schottky diode structures of the device (inset of Fig. 2a), thus indicating the important role played by the Co/MgO/MoS₂ Schottky contacts. The contact region characterized by the contact resistance (R_C) is constituted by the MgO tunnel barrier and the depletion zone of MoS₂ underneath. To extract the respective contribution of the contact resistance and MoS₂ channel resistance, we have acquired I_ds–V_ds curves at V_g=+10 V with different channel distances (Fig. 2b). Those emphasize on a larger contribution of R_C in the total resistance (R_total) with a shorter channel distance. Since the resistance of the MoS₂ channel (R_MS) is proportional to the channel distance, R_C can be extracted from the intercept of linear fitting of the total resistance as a function of channel distance if one assumes a constant contact resistance (see details in Supplementary Note 1). To ensure an identical voltage drop on the contact region, we have plotted in Fig. 2c R_total versus the channel distance (normalized by its width) at different I_ds. In Fig. 2d, the extracted value of R_C rapidly drops off on increasing |V_ds| whereas it dominates the total resistance at low V_ds. The large variation of R_C is ascribed to the change of the Schottky profile versus V_ds. This is a major issue of our devices as discussed in the following. The extracted MoS₂ channel resistance (R_MS) displays a smaller variation on increasing |V_ds|. The ratio of R_MS/R_total becomes saturated to be about 35% when |V_ds|>0.35 V (inset of Fig. 2d). If one assumes that this ratio is still valid at |V_ds|=1 V, R_MS can be estimated to be 102 kΩ for V_g=+10 V and 32.8 kΩ for V_g=+20 V (sheet resistance R_sq∼1 × 10⁵Ω). At |V_ds|=1 V and V_g=+20 V, R_C approaches 2R_MgO=63.7 kΩ corresponding to two intrinsic MgO barriers in series free of depletion tunnelling zone, thus giving R_MgO=31.8 kΩ.

Figure 2: Transport characterization of MoS₂ field-effect transistor.

(a) Current (I_ds)–voltage (V_ds) characteristics between the E1 and E2 electrodes, measured at 12 K with applying different back-gate voltages V_g. Inset: band diagram of the back-to-back diode structure of the MoS₂ device with a two-terminal bias V_ds. (b) I_ds–V_ds characteristics measured between different electrodes with a back-gate voltage V_g=+10 V at 12 K. Inset: schematics of connection between different electrodes. (c) The total resistance (R) between the two electrodes versus the channel distance (normalized by the width) with different I_ds. (d) V_ds dependence of the total resistance (E1–E2) and the extracted contact resistance R_C and the MoS₂ channel resistance R_MS (E1–E2). Inset: The contribution of R_MS in the total resistance as a function of V_ds. (e)Transfer characteristic I_ds–V_g between E1 and E2 electrodes, measured at 12 K with applying different V_ds. (f) Extracted effective mobility μ versus V_g with different V_ds.

In Fig. 2e, I_ds as a function of V_g is plotted for different V_ds. The transistor ON/OFF ratio can be estimated from the current ratio between V_g=±20 V, which is around 2 × 10³. The lower ON/OFF ratio compared to the reported values² is due to the influence of leakage current on I_ds (∼1.5 nA at V_g=±20 V) in the OFF state because of the slight damage of contacts during wire bonding (see Supplementary Note 2). The effective field-effect mobility μ can be estimated by extracting the slope dI_ds/dV_g from the I_ds−V_g curves (Fig. 2e):

where L is the channel length, w is the channel width and C_i is the gate capacitance². It is found that μ increases with V_g as well as V_ds (Fig. 2f). At V_g=+20 V and V_ds=−1 V, the mobility equals μ∼6 cm² V⁻¹ s⁻¹ in close agreement with previously reported value (7 cm² V⁻¹ s⁻¹) at 10 K for ML MoS₂ on SiO₂/Si substrate²⁸. In this low temperature range the transport in MoS₂ is dominated by scatterings on charged impurities²⁸ or hopping through localized states²⁹.

Magnetoresistance measurements

We now focus on the key results of this paper about the MR experiments on MoS₂. These are performed at relatively low bias (|V_ds|<0.15 V) in order to avoid inherent hot-electrons spin depolarization mechanisms. Figure 3a displays the recorded magneto-current curve at 12 K with V_ds=−0.1 V and V_g=+20 V. A clear spin-valve signal is observed characterized by a larger current flowing in the parallel (P) state at high field and a smaller current in the quasi-antiparallel (AP) magnetic state at low field. The MR ratio can be calculated from (I_P−I_AP)/I_AP × 100% to be about 1.1%. This result constitutes a clear demonstration of an electron spin transport through the conduction band of MoS₂ in a lateral geometry. We have carefully checked the angle dependence of MR, leakage current and electrode resistance to rule out any possible spurious effects on MR such as charged impurities in MoS₂, spin transport in the Si substrate or anisotropic MR effect of Co electrodes (see Supplementary Note 3). In addition, micromagnetic simulations prove a comparable coercivity field as observed in our experimental configuration (see Supplementary Note 4). Reproducible results have been obtained on several samples and no spin signal can be found in the control non-magnetic samples (see Supplementary Note 3). All of these confirm that the MR results originate from spin transport from MoS₂.

Figure 3: Back-gate voltage-dependent MR characterization of the device.

(a) Magneto-resistance response of the multilayer MoS₂-based lateral spin-valve device measured at 12 K with V_g=+20 V and V_ds=−0.1 V. (b) Back-gate voltage dependence of MR measured at 23 K with V_ds=−0.1 V. The error bars have been calculated by taking account of the signal noise and the contribution of leakage current. (c) Back-gate voltage-dependent Schottky barrier height (Φ_b). The deviation from the linear response at low V_g (blue solid line) defines the flat band voltage (V_F) and the real Φ_b of Co/MgO on MoS₂. Insets: schematics of MoS₂ band structure with different V_g. (d) Variation of the total resistance (R) as a function of V_g with different V_ds at 23 K. The error bars in b have been calculated by taking account of the signal noise and the contribution of leakage current.

A very interesting feature is the particular back-gate voltage dependence of MR near the optimal condition for spin-injection/detection. Figure 3b displays MR versus V_g showing a characteristic maximum MR signal at a gate voltage of V_g=+20 V. In order to clarify this point, one should first understand the effect of V_g on the transport properties. As shown in the inset of Fig. 3c, the back-gate mainly plays two roles. One is to modulate the Fermi level (E_F) inside the MoS₂ bandgap yielding a change of the carrier density in the channel². The second role is to modify the SB profile and width. This scenario can be supported by the measurement of the back-gate dependence of the SB height (Φ_b) of Co/MgO on MoS₂, as shown in Fig. 3c (see Supplementary Note 5). It is noted that all Φ_b values are extracted from the measurements between 180 K to 240 K and with V_ds from −0.4 V to −1 V. Here, we can identify two regions for the variation of Φ_b versus V_g. For V_g<+2.6 V, that is when the depletion layer is thick, the thermionic emission dominates and this results in a large linear increase of Φ_b on the negative V_g. However, for V_g>+2.6 V, the tunnel current through the thin SB impinges on the linearity of Φ_b. The real value of Φ_b for Co/MgO on MoS₂ is obtained at the point of the onset of the deviation (+2.6 V) equalling thus 5.3 meV in good agreement with the value reported for Co/Al₂O₃ (2.5 nm) on multilayer MoS₂ (ref. 24). This analysis would provide the SB height only when the thermal activation is dominant, that is, below +2.6 V. Above this value, the analysis can still be performed, but it does not lead to the interpretation of SB height, since tunnelling is dominant. In Fig. 3d, we have plotted the total resistance versus V_g for different V_ds. As mentioned above, the resistance is mainly attributed to the contact resistance R_C at V_ds=−0.04 V. At V_ds=−1 V, the contribution from R_MS can reach 35% in R_total when the depletion region is much reduced. At large negative V_g when E_F is far away from the bottom of the conduction band, the resistance is rather high and does not vary with V_ds certainly preventing any spin transport in MoS₂. For positive V_g, the channel and contact resistance both decrease rapidly with V_g when E_F moves close to the conduction band. From −20 V to +20 V, the MoS₂ channel resistance decreases much faster than the contact resistance (5 × 10⁴ times for V_ds=−1 V versus 1.2 × 10³ times for V_ds=−0.04 V).

The bias dependence of MR measured with V_g=+20 V at 12 K is displayed in Fig. 4a. It is found that the MR ratio decreases with the increase of bias |V_ds|. When |V_ds| is larger than 0.15 V, MR almost disappears. We note that the total resistance also decreases rapidly with the increase of |V_ds| (Fig. 4b), especially in the range |V_ds|<0.15 V where R_C is considered to be dominant as mentioned above. This indicates that the observation of MR could be related to the large contact resistance introduced to overcome the impedance mismatch at Co/MgO/MoS₂ interface. In Fig. 4c, we display the temperature (T) dependence of MR acquired with V_g=+20 V and V_ds=−0.1 V: the MR rapidly decreases with T. When T>60 K, MR almost disappears. This thermal behaviour could be also associated to the variation of R_C versus T if one assumes a constant spin polarization at Co/MgO interface³⁰. In Fig. 4d, the total resistance as a function of T is plotted for different V_ds conditions. It appears obviously that the resistance at V_ds=−0.04 V (R_C dominant) decreases rapidly when T>60K, while the resistance with V_ds=−1 V (R_MS+2R_MgO) decreases more slowly with T. Therefore, a strong correlation between the contact resistance and MR is also revealed from the temperature dependence of MR. In Fig. 4d, we also show the temperature variation of mobility derived at V_g=+20 V and V_ds=−1 V. It is noted that the mobility of MoS₂ channel shows a slight increase with temperature up to 200 K, which could be due to the scattering from the charged impurities²⁸. Supplementary data for MR measurements (temperature, bias and back-gate dependence) can be found in Supplementary Note 6.

Figure 4: Drain-source bias and temperature-dependent MR characterization of the device.

(a) V_ds dependence of MR measured at 23 K with V_g=+20 V. (b) The total resistance (R) of the device versus V_ds. The area with orange colour indicates the V_ds range where the total resistance is dominated by the contact resistance. (c) Temperature dependence of MR measured with V_g=+20 V and V_ds=−0.1 V. (d) Temperature dependence of the total resistance (R) with different V_ds and the MoS₂ channel mobility μ (V_g=+20 V, V_ds=−1 V). The error bars in a,c have been calculated by taking account of the signal noise and the contribution of leakage current.

Spin diffusion length

In order to explain our experimental results, we have calculated the MR from the standard theory of spin-injection adapted to lateral devices with tunnelling injectors^20,31,32. The expected MR is calculated versus the characteristic resistances which are the tunnelling interface resistance (R_I) and the channel spin-resistance (R_N*=R_sq˙l_sf/w) (ref. 31) for a lateral spin-valve in the so-called ‘open’ geometry (that is, the injected spin may diffuse freely in the channel outwards both contact regions)³². The generic formula is given by:

where γ is the tunnel spin polarization (injector and detector), chosen to be 0.5. (The spin polarization γ is deduced from the tunnel magnetoresistance (TMR) of a magnetic tunnel junction with amorphous Al₂O₃ barrier by γ=[TMR/(2+TMR)]^1/2=0.5 with TMR=70%. We assume that the MgO tunnel barrier is amorphous on MoS₂). L is the channel length (450 nm) and l_sf is the spin diffusion length (SDL), found to be close to 235 nm from refined analyses (detailed hereafter). The channel resistance R_N possesses a certain relationship with its spin-resistance R_N* according to R_N=R_N*·L/l_sf, and this makes R_N close to R_N* when L≈l_sf. As shown in Fig. 5a, one can observe a theoretical maximum value of MR of about 1% in a narrow window when the spin-dependent tunnel resistance R_I is almost equal to the channel spin-resistance R_N*. This condition corresponds to a perfect balance between the spin-injection rate (1/R_I) and the spin relaxation rate (1/R_N*) giving a same maximum of MR for the whole range of R_I and R_N*. To better demonstrate the relationship between R_I and R_N*, we have plotted in Fig. 5b the MR as a function of the ratio R_I/R_N* for different l_sf. The maximum MR increases with l_sf due to the evanescent exponential prefactor describing the spin-memory loss scaling with exp(−L/l_sf), however always being localized around R_I/R_N*=1. From this optimal condition, a larger R_I would reduce the rate of spin-injection compared to the spin-flip rate and consequently reduce the spin-accumulation in MoS₂. On the contrary, a larger R_N* gives rise to a spin-backflow process by which the spin would relax in the ferromagnet (Co) and resulting in a reduced injected spin-current and MR from the optimum condition of equality between the tunnel contact resistance and channel spin-resistance (see ref. 32 and Supplementary Note 7).

Figure 5: Calculation of MR with spin diffusion theory.

(a) Calculated MR of FM/I/MoS₂/I/FM structure as a function of the interface tunneling resistance (R_I) and the channel spin-resistance (R_N*). (b) Calculated MR as a function of the ratio R_I/R_N* with different spin diffusion length (l_sf). Inset: theoretical calculated MR versus channel length (blue line) and experimentally observed MR between different electrodes (E1–E2, E2–E3 and E1–E3). The error bars have been calculated by taking account of the signal noise and the contribution of leakage current.

Assuming the balance condition R_I=R_N*, the spin diffusion length (l_sf) estimated from formula (2) for a 1% MR is close to 235 nm. This appears as a lower bound because of the relative high spin-polarization γ (0.5) chosen for Co/MgO. Other tunnelling processes than direct tunnelling would give rise to small MR by several orders of magnitude. In particular, sequential tunnelling processes via interface states (IS) between MgO and MoS₂, like observed in Co/Al₂O₃/GaAs structures³³ and responsible for spin-accumulation amplification, would be detrimental for MR. (In a spin injection/transport/detection experiments, the level of spin accumulation generated in MoS₂ through spin-injection is generally reduced by the presence of intermediate states³³. This would result in an overall reduction factor of (R_MgO)²·R_MS*/(R_SC)³ even in the case of infinite spin-relaxation time on these IS. Here, R_MgO is the MgO barrier resistance (about 30 kΩ), R_MS* is the spin-resistance of MoS₂ (R_sq˙l_sf/w=20–60 kΩ) and R_SC is the Schottky resistance (MΩ).) This excludes a sequential tunnelling to form the interface resistance and emphasizes that a direct tunnelling process through one MgO and Schottky composite barrier should be taken into account. Another important conclusion is that an l_sf of 235 in multilayer MoS₂ at low temperature is already ten times larger than the value predicted in ML MoS₂ taking into account the DP spin-depolarization mechanism¹⁴. To strengthen that point, the inset of Fig. 5b displays the characteristic MR versus channel distance acquired between different electrodes (E1–2, E2–3 and E1–3) together with the simulation curve obtained for l_sf=235 nm. It fingerprints the distance-dependence of the spin-injection/detection process beyond other spurious effects. The exponential decay of MR is in good agreement with the simulated MR versus the channel distance. This validates the estimated long spin diffusion length at least larger than 200 nm in our 6ML MoS₂. From the extracted mobility of MoS₂ channel (μ=6 cm² V⁻¹ s⁻¹) at 12 K (V_g=+20 V, V_ds=−1 V), one can estimate the spin lifetime τ_sf to be 46 ns from τ_sf=l_sf²/(2D)=l_sf²e/(2μk_BT), where D is the diffusion constant. Remarkably, this spin lifetime is one order of magnitude larger than the electron spin relaxation time recently measured in ML MoS₂ by optical Kerr spectroscopy³⁴. Since the multilayer MoS₂ possesses an indirect band gap restricting the ability to observe spin relaxation by optical means, the electric spin-injection/detection method provides an efficient alternate method to probe quantitatively the spin relaxation mechanisms in such compound.

Discussion

As mentioned above, an optimized MR occurs at the balance condition of a tunnel contact resistance approaching the channel spin-resistance. It seems however impossible to perfectly fulfil such condition in our device at low V_ds bias. As shown in Fig. 5a, in the best MR situation, the MoS₂ channel resistance (R_MS) is estimated to be about 150 kΩ. The contact resistance in the investigated range of bias and temperature lies in the MΩ range, well beyond the characteristic threshold, and should then exclude any MR. How may one then reconcile with the standard spin-injection model? To clarify that point, let us focus on the T-dependence of the conductance as a fingerprint of the electronic hopping process involved in the transport. Figure 6a displays the Arrhenius plot of the T-dependence conductance at different V_ds. It becomes obvious that the charge transport may be described by two distinct mechanisms in the respective high and low temperature regimes, with a characteristic threshold at T*∼70 K. For T>T*, the transport is dominated by the nearest-neighbour hopping (NNH) with a conductivity varying like G∼exp(−T₀/T). For T<T*, the conductance can be fitted by a 2D variable-range hopping (VRH) equation according to G∼exp[−(T₁/T)^0.33]. Such characteristic T-dependence has been observed in many low-dimensional systems and is a signature of hopping transport via localized states^29,35,36. In MoS₂ system, it is reported that the sulphur vacancies can introduce localized donor states inside the bandgap²⁹. The two temperature regions for the different hopping regimes are even more pronounced at small |V_ds| (0.04 V) when the contact resistance dominates the total resistance. This highlights a transport dominated by hopping in the contact region more than in the MoS₂ channel by itself.

Figure 6: Evidence of hopping transport in the contact region.

(a) Arrhenius plot of the temperature dependent conductance (symbols) at different V_ds from Fig. 4d and the fitting results by different hopping models (grey and pink lines). Two hopping regimes are clearly separated by T* (vertical line). (b) Band diagram of the Schottky contact region of the MoS₂ device. The device can be divided into three regions. The direct tunnelling region consists of the MgO tunnelling barrier (R_MgO) and one part of Schottky contact (R_SC1) taken as a whole. The second region is in the tail part of depletion layer where electrons transport in a hopping behaviour (R_SC2). The third region is the region where electrons either transport in the MoS₂ channel by a hopping behaviour or transport in the MoS₂ conduction band (R_MS), depending on the carrier density.

In this scenario, we propose that the contact region may be constituted by three different zones (Fig. 6b): (i) the chemical MgO tunnel barrier, (ii) the strongly depleted zone underneath MgO (SC1) playing the role of an additional composite tunnel barrier and (iii) the tail of the MoS₂ depletion zone where the electronic conduction is ensured by hopping mechanism (SC2). When one considers an inhomogeneous spin-channel made of two parts: a depletion part ‘D’ of length t and a semi-infinite channel part ‘B’, the effective channel spin-resistance reads where is the ‘bulk’ spin-resistance of the channel and ( is the spin diffusion length in the depletion region). For thin depletion case (), like considered here, , leads to .

The observation of MR at low bias goes in favour of an impedance matching at the level of the contact region, by acting with back-gate voltage, however without preventing spin injection and spin transport in the MoS₂ channel. The impedance matching is achieved when the ratio between the tunnelling injector resistance R_I (R_MgO+R_SC1) and the channel effective spin-resistance () close to R_N (R_SC2+R_MS) approaches unity from small to large values depending on which part of MoS₂ consists in the SC1 or SC2 regions. At low bias, the impedance matching in our Co/MgO/MoS₂ devices is then only possible thanks to the high resistivity region (SC2) in the channel. Since the maximum MR is expected at about equal value of R_I and , the non-linear variation of the observed MR versus V_g reflects the balance ratio between R_I and . When V_g increases from +8 V to +20 V, due to the shift of E_F in the MoS₂ band gap, the faster decay of the MoS₂ channel resistance (R_MS) compared to the tunnel injector resistance (R_I) contributes to the enhancement of MR (Fig. 3d). One can also invoke the increase of the electron mobility (Fig. 2f) and spin-flip time. For larger V_g, due to the shrinking of the Schottky depletion layer, a continuous decreasing of R_SC2 cannot fulfil anymore the balance condition between R_I and , and results in the drop of MR. When increasing the bias or the temperature, the electrical field³⁷ or thermal activation energy³⁸ can favour electron hopping via localized states, especially for the variable-range hopping process and can effectively reduce R_SC2. This explains the drop of MR with increase of V_ds and T is also due to the deviation of the impedance balance condition.

In conclusion, we have demonstrated the electrical spin injection and detection through the conduction band of a 450 nm long, 6ML thick MoS₂ channel. From the systematic studies of the bias, temperature and back-gate voltage dependence of MR, it is found that the hopping via localized states in the contact depletion region plays a key role to keep the balance condition between the interface tunnelling resistance and the channel resistance, which is mandatory for the observation of the two-terminal MR. Moreover, the electron spin-relaxation is found to be greatly suppressed in the multilayer MoS₂ channel for an in-plane spin injection geometry. The underestimated long spin diffusion length (∼235 nm) could open a new avenue for spintronic applications using multilayer TMDs.

Methods

Nano-device fabrication

The MoS₂ flakes were exfoliated from a bulk crystal (SPI Supplies), using the conventional micro-mechanical cleavage technique, onto a clean SiO₂ (100 nm)/n++-Si substrate. First e-beam lithography (Raith-150) was performed to define the four electrodes on the selected flake. Then the sample was introduced into a molecular beam epitaxy system to deposit the FM electrodes, which consists of 2 nm MgO, 10 nm Co and 10 nm Au. After deposition and lift-off, a second e-beam lithography procedure was used to define the four large pads for electrical connection. Then Ti(10 nm)/Au(190 nm) was thermally evaporated in a PLASSYS MEB400s system for the large pads. After lift-off, the device was annealed in the vacuum at 200 °C for 1 h followed by the coverage of 10 nm MgO protection layer. To check the thickness of MoS₂ flake and the distance of channel, we have performed atomic force microscopy characterization in tapping mode on the device. In order to precisely extract the flake thickness, Gaussian fitting of the distribution of height has been employed.

Magneto-transport measurements

The magneto-transport measurements have been performed in a cryostat varying temperature from 12 to 300 K with a maximum magnetic field of 4kOe. For the device presented in the main text, in order to reach a well-defined AP magnetic configuration, a magnetic field was applied at a 45° angle to the electrodes. Magnetic domains are then generated through the reservoir of the large triangle areas of the electrodes (Fig. 1a) before propagating towards the injector and detector regions above the MoS₂ flake³⁹ (see more micromagnetic simulations in Supplementary Note 4). For the back-gated two-terminal spin-valve measurement as described in Fig. 1d, we have used a Keithley 2400 to apply the drain-source bias V_ds, and used a Keithley 6487 picoamperometer to measure the drain-source current I_ds. At the same time, another Keithley 2400 was employed to apply the back-gate voltage V_g.

Data availability

The data that support the findings of this study are available from the corresponding author on request.