Spin polarised quantised transport via one-dimensional nanowire-graphene contacts

Introduction

Spin-based electronics (spintronics) offers a promising route to achieving low-power and high-speed computation beyond CMOS technology^1,3,4,5,6, and potentially to energy-efficient spin qubits for applications in quantum information processing^7,8,9,10. Graphene, with its high carrier mobility and gate tuneable charge carrier density^11,12, is an ideal choice of material for studying spin transport^13,14. There have been numerous experimental demonstrations of graphene-based quantum devices built upon various architectures, including quantum dots^9,15,16,17 and quantum point contacts^{18,19,20,21,22}, which can enable further development of quantum technology. Yet, such architectures have not been employed within graphene spintronic devices.

Modern graphene spintronic devices typically use hexagonal boron nitride (hBN) to support and encapsulate the graphene channel, providing it with an atomically flat substrate, enhancing its charge and spin transport capabilities^{23,24,25,26,27,28,29,30,31,32} relative to earlier devices on Si/SiO₂ substrates^33,34,35. As the quality of graphene devices is improved, the electronic mean free path can reach up to μm scales, approaching or exceeding device dimensions and giving rise to ballistic transport^{18,19,22,36,37,38}. This is of particular interest in the field of low-power spintronics due to suppressed scattering in the ballistic regime, and the potential for coherent control over spin transport^39,40. However, at present there are very few experimental studies regarding ballistic spintronics in graphene, despite this being the focus of studies in low dimensional nanostructures based on III–V semiconductors^41,42,43,44, carbon nanotubes^45,46,47, and evidence of spin-polarised edge states in graphene nanoribbons⁴⁸.

In the case of an encapsulated channel, metal electrodes can be placed over the hBN/graphene/hBN stack to form a one-dimensional (1D) contact^49,50,51 where the metal bonds with the exposed graphene edge. 1D contacts have been shown to be far less invasive than direct, and tunnel barrier, top contacts, where a large area of the graphene surface bonds to the electrode^{50,52,53,54,55,56,57,58}. We recently demonstrated fully encapsulated graphene channels with nanoscale magnetic 1D contacts that combine high charge mobility and mean free paths in excess of ~1 μm, with efficient spin injection and spin diffusion lengths up to ~20 μm⁵⁸. The separation between contacts in these devices was on the order of ≥10 μm, while the channel widths were typically 1–2 μm, implying spin transport through the channel occurs in the diffusive regime².

In this work, we present observations on the injection of spin-polarised carriers via 1D magnetic nanowire–graphene interfaces with widths on the order of ~100 nm (Fig. 1a–d). We find that carriers are confined on a length scale below their mean free path upon injection to the channel, causing the emergence of quantised energy subbands in their band structure, evidenced by quantised conductance through the 1D contacts in the absence of a magnetic field. Contact conductance is analysed with the Landauer equation to quantitatively evaluate the effective constriction width, defined by the magnetic nanowire, and the energy scale of the observed subbands. Both extracted values agree with ballistic injection with transmission probability in the range T = 0.08–0.30, orders of magnitude higher than tunnel barrier contacts and comparable to values found in studies regarding edge-contacted graphene^59,60,61. Further evidence of the ballistic nature of transport via the 1D magnetic contacts is explored by probing the transition into the quantum Hall regime, with magnetic fields as low as ∣B∣_⊥ ≤ 1 T, yielding a consistent analysis of the 1D interface characteristics. Crucially, we obtain these results without the need to engineer any physical constriction within the graphene itself, overcoming nanofabrication challenges associated with defining quantum point contacts^18,19,22,36. The demonstration of quantised transport via spin-polarised injectors to a graphene channel represents the first step towards the realisation of ballistic graphene spintronic devices.

**Fig. 1: Device and 1D contact architecture.**

Results and discussion

Charge transport

Results presented in this work are taken from four devices: D1–D4. All devices comprise fully encapsulated graphene channels employing nanoscale 1D contacts (a summary of the fabrication process and measurement set-up can be found in “Methods”, while the physical properties of each device can be found in Supplementary Table SI). All data are recorded at a temperature of 20 K, unless stated otherwise. Figure 1e shows the conductivity, σ, against n, where the neutrality (Dirac) point, n = 0, is defined by the back gate voltage at which σ is a minimum. Applying a linear fit to this data at low carrier density, we extract the field-effect mobility, μ = 1/e(dσ/dn). Around n ~2 × 10¹¹ cm⁻², the extracted values are μ = 43,000 ± 2000 cm² V⁻¹ s⁻¹ at 300 K and μ = 105,000 ± 5000 cm² V⁻¹ s⁻¹ at 20 K. These values are significantly higher than those extracted from graphene spintronic devices of other architectures^28,34,50, but consistent with the highest mobilities found in devices employing full encapsulation and 1D contacts⁵⁸. Lower mobility at room temperature (300 K) indicates transport is dominated by electron-phonon scattering, with contributions from intrinsic graphene phonons and the surface optical mode of the hBN substrate^62,63, while at low temperature (20 K) the narrow Dirac peak evidences a low impurity density in the channel of <5 × 10¹⁰ cm⁻² (see ref. ²⁵). Conductivity is related to the mean free path of carriers, ℓ_mfp, via the Einstein relation, D = σ/eN, where D = v_Fℓ_mfp/2 is the diffusion coefficient, N is the density of states of graphene and v_F is the Fermi velocity (see “Methods”). At T = 20 K, we extract a value of ℓ_mfp = 0.74 ± 0.07 μm at a carrier density ∣n∣ = 5 × 10¹¹ cm⁻² (Supplementary Fig. S2c). The value of ℓ_mfp is far below the separation between contacts, L_ch = 12.2 μm, indicating that charge transport measured along the channel is diffusive.

The resistance of the 1D contacts as a function of V_BG is measured by employing the 3-terminal configuration (Fig. 2c). All contacts display an electron–hole asymmetry, with the higher transmission in the electron regime. We also observe a small shift between the contact conductance minimum, used to define the contact carrier density, n_c, and the graphene channel neutrality point, with the contact conductance minimum shifted towards negative carrier density by ~1 × 10¹¹ cm⁻² (Supplementary Fig. S2d). This implies n-type doping of graphene adjacent to the 1D interface, arising due to charge transfer from the cobalt nanowire. The n-doped region at the nanowire–graphene interface forms a tuneable potential profile that extends into the channel over a length scale, l_pn≤100 nm (Fig. 1b)^{53,54,64,65,66,67}. The extracted value of ℓ_mfp is 2 − 3 × below the physical width of the channel, W_ch = 2 μm, but is much greater than l_pn. Considering this, contact resistance consists of dominant contributions from scattering at the rough nanowire–graphene interface, and from the surrounding potential profile.

**Fig. 2: Zero-field quantised conductance via e-QPCs (device D2).**

The contacts display plateaus in their resistance as a function of V_BG (Fig. 2c and Supplementary Fig. S2d). As the value of ℓ_mfp is greater than the widths the 1D contacts (W_c = 150–350 nm), the observed plateaus in R_c are explained by the formation of subbands in the energy spectrum of injected charge carriers, due to confinement through the 1D nanowire–graphene interface. The combination of confinement from the 1D constriction defined by the magnetic nanowire–graphene interface, and the surrounding tuneable potential profile, forms an effective tuneable quantum point contact through which transport occurs ballistically. From this point on, we will refer to these as edge quantum point contacts (e-QPCs).

Zero-field ballistic transport through nanoscale 1D contacts

Figure 2c presents contact conductance, G, extracted from an e-QPC, in units of G₀ = 2e²/h and plotted as a function of Fermi wave vector, ({k}_{{{{rm{F}}}}}=sqrt{pi {n}_{{{{rm{c}}}}}}), where n_c is the e-QPC carrier density. The data shows a series of periodic plateau-like features, which emerge at low temperature. Such features are caused by the quantisation of available energy states in the e-QPC into discrete 1D subbands^68,69. This is a fingerprint of ballistic conduction, indicating that carriers are confined by the 1D magnetic nanowire–graphene interface upon injection to the channel. Conductance in the ballistic regime is governed by the Landauer equation, which, for graphene, can be approximated as¹⁸,

$$G=frac{4{e}^{2}}{h}MT=frac{4{e}^{2}}{h}frac{{k}_{{{{rm{F}}}}}{W}_{{{{rm{c}}}}}}{pi }T,$$

(1)

where e²/h is the conductance quantum, T is the transmission probability, and M is the mode number, defined as the number of half Fermi wavelengths, λ_F/2 = π/k_F, that fit within the constriction width, W_c^68,69. The factor of 4 is unique to graphene and reflects its four-fold degeneracy, coming from the 2 spin and 2 valley states that its carriers can occupy. T takes a value between 0 and 1, and modifies the magnitude of conductance to account for carriers that are back-scattered, for example, off rough edges. The value of T can be estimated from the spacing of consecutive 1D subbands, relative to the ideal case, when T = 1 and plateaus are spaced by 4e²/h. Subbands emerging in Fig. 2c are separated by ~e²/h, a factor of ~4 reduction from the ideal case. Values of T extracted by this method, from all devices, are in the range T = 0.08 − 0.30 (see Supplementary Table SI).

As the transmission probability is crucial to the Landauer description of ballistic conduction, we verify our estimation of T by performing finite bias spectroscopy on the e-QPCs, where conductance at a fixed carrier density is measured as a function of a bias voltage applied between the electrode and channel (see “Methods”)⁷⁰. Conducting bias sweeps at different back gate voltages allows us to generate bias maps for the hole (Fig. 2a) and electron (Fig. 2b) transport regimes, which help to visualise the energy subbands. These can be seen at points along zero bias where the lines are closely spaced, or overlapping, with their positions agreeing with the plateaus observed in Fig. 2c.

Transmission across the magnetic nanowire–graphene interface is determined by the coupling of graphene p-orbitals to the surface d-orbitals of the ferromagnetic electrodes. Hence, there is a mismatch in work functions on either side of the interface, and it is expected that T <1^59,65,71. In fact, T across the e-QPCs is somewhat tuneable via the back gate voltage, owing to the tuneable potential profile surrounding the 1D interface. Close to the interface, graphene is n-type doped due to charge transfer from the magnetic nanowire. However, the Fermi energy of the graphene channel is free to change as the back gate voltage is varied^{53,54,64,66,67}. Hence, in the electron doping regime, the e-QPC has an n–n potential profile, scattering is reduced and T is increased. Meanwhile, in the hole doping regime, the e-QPC has an n–p profile, which increases scattering and reduces T in this region^18,65. Consequently, the plateaus in the hole doping regime appear smeared and occur at increasingly smaller spacings as hole density increases and T decreases. This behaviour is responsible for the observed divergence of conductance between electrons and holes for k_F >150 × 10⁶ m⁻¹ (Fig. 2c).

With the above considerations in mind, the following analysis is performed in the mid carrier density range k_F ≈50–150 × 10⁶ m⁻¹ or ∣n_c∣ ≈ 0.5 − 7.0 × 10¹¹ cm⁻², where conductance is comparable between the two carrier regimes. Averaging the spacing between plateaus within this carrier density range, across the two datasets (Fig. 2a–c), yields a transmission of T(e) = 0.26 ± 0.04 for electrons, and T(h) = 0.23 ± 0.05 for holes. The asymmetry in these results is consistent with the interpretation of a p–n junction in the adjacent graphene. With these estimations of T, Eq. (1) can be used to evaluate the effective constriction width defined by the magnetic nanowire–graphene interface, which is W_c(e) = 240 ± 20 nm for electrons, and W_c(h) = 220 ± 20 nm for holes. The width of the electrode used for this measurement is 300 nm. Hence, the extracted width for both carrier polarities agree within their uncertainty and fall within 20% of the nominal physical width of the magnetic nanowire.

The effective constriction width can also be evaluated from the energy spacing between 1D subbands, which can be directly determined from spectroscopy data. Here, the non-linear conductance response is studied by identifying the bias value at which plateaus from two adjacent subbands merge⁷⁰. To more easily visualise this effect, Fig. 2d presents the bias spectroscopy data as a 2D colour map of the transconductance (dG/dV_BG). When plotted in this way, a distinct diamond pattern is visible, formed from plateaus at zero bias (purple squares) and the finite bias point at which they merge (green triangles). Fitting a local minima to the crossing point (black boxes Fig. 2d) allows us to quantify the average energy spacing of the subbands, which gives ΔE = 9.5 ± 1.4 meV.

By combining the graphene dispersion relation, E = ℏv_Fk_F, where v_F = 1 × 10⁶ ms⁻¹ is the Fermi velocity of graphene, with the number of modes, M = k_FW_c/π, the energy of a given subband is given by,

$$E=frac{hslash {v}_{{{{rm{F}}}}}pi }{{W}_{{{{rm{c}}}}}}M.$$

(2)

Employing Eq. (2), along with the estimate of ΔE, the effective constriction width is calculated as W_c(e) = 220 ± 30 nm. This value agrees within the experimental error with that extracted by applying the Landauer equation to Fig. 2c. Thus, using two different methods, applied to separate data sets, we demonstrate a consistent quantitative picture of ballistic conduction through an e-QPC.

Spin transport

We demonstrate efficient spin injection through the e-QPCs, via non-local (inset Fig. 3b) spin valve and spin precession (Hanle) signals, which are shown respectively in Fig. 3a, b. The spin valve signal shows two anti-parallel states, one for negative B field and one for positive, which occur due to an abrupt reversal in the magnetisation direction of a magnetic electrode, relative to its adjacent electrode^2,33,72,73. These switches signify the injection and detection of spin currents via the e-QPCs, with a spin signal magnitude of ΔR_NL = 180 ± 10 mΩ. Spin precession data shown in Fig. 3b is fitted with the standard spin precession equation^2,74,75,76, which yields a spin diffusion coefficient D_s = 0.37 ± 0.04 m² s⁻¹, a spin lifetime τ_s = 170 ± 14 ps, a contact polarisation P = 4.8 ± 0.5%, and a spin diffusion length λ_s = 7.9 ± 0.7 μm. These values compare favourably with those found across several of our previous devices⁵⁸, consistent with the high mobility found for this spintronic device.

We perform spin valve measurements over a gate voltage range where the clearest plateaus are observed in contact conductance (−2.5 to 20 V), extracting values of ΔR_NL at each V_BG value (Fig. 3c). The spin signal dependence is plotted alongside conductance, G, through the spin-injecting contact, which displays several plateaus, arising from confinement, spaced by ~e²/h in agreement with the previous measurement; plateaus are highlighted by dashed red lines, placed according to minima in dG/dV_BG (Supplementary Fig. S5b). Note that the first three of the plateaus seen in Fig. 3c (V_BG < 10 V), are also visible in a separate measurement of contact conductance taken from the same contact, during a subsequent cool down (Fig. 4b). Hence, the observed low field confinement plateaus are reproducible across several thermal cycles. The first of these subbands occurs in a region of rapid increase in ΔR_NL, explained in part by a similar decrease in mismatch parameters close to neutrality (Supplementary Fig. S5a). Subsequent plateaus are located close to weak features in the spin signal, but more devices are needed to determine a connection between the presence of quantised transport and modulations of spin signal, as was found for carbon nanotube quantum dots with ferromagnetic contacts^45,46,47.

**Fig. 4: Landau fan diagram for D1, symmetrised with respect to B_⊥.**

Transition to quantum Hall regime

In order to explore quantum Hall transport, we measure conductance through an individual e-QPC, while varying its carrier density, n_c, and steadily increasing perpendicular magnetic field strength, B. Plotting transconductance against n_c and B produces a Landau fan diagram, which is shown in Fig. 4a (see further data in Supplementary Fig. S6). At low field, faint white vertical regions are visible, corresponding to 1D subbands arising from confinement (dotted lines in Fig. 4a, b); bias spectroscopy measurements were also performed on this e-QPC with the features only resolved at zero bias (Supplementary Fig. S4). The transition from quantum confinement into the quantum Hall regime, where the 1D subbands of a ballistic conductor coalesce into well-defined Landau Levels, occurs at a critical magnetic field strength, B_c, at which the cyclotron radius, l_c is equal to half the constriction width i.e. 2l_c ≈W_c⁶⁹. For a certain energy (represented by the filling factor, ν) the field strength required for the transition is given by ({W}_{{{{rm{c}}}}}approx 2{l}_{{{{rm{c}}}}}=2{l}_{{{{rm{B}}}}}sqrt{nu /2})¹⁸, where ({l}_{{{{rm{B}}}}}=sqrt{hslash /eB}) is the magnetic length³⁷. Combining the above gives the expression for the critical field,

$${B}_{{{{rm{c}}}}}=frac{h}{e{W}_{{{{rm{c}}}}}}sqrt{frac{n}{pi }},$$

(3)

with this transition indicated by the solid black line in Fig. 4a. Above this transition, the plateaus evolve into dispersive Landau levels at high magnetic field, each corresponding to a filling factor ν. The white features observed for B > 1.0 T align closely with the expected Landau level positions, indicated by dashed black lines (Fig. 4a) and consistent with the relation B = nh/eν^77,78,79.

Both quantum Hall and low field data show lower conductance for hole doping (Fig. 4), owing to the p–n junction nature of the e-QPCs. Transmission probability for electron conductance is estimated from the average spacing between plateaus, in both regimes (Fig. 4b). At low field (red trace), the average spacing gives a value of T(e) = 0.28 ± 0.04, while at high field (purple trace) the extracted value is T(e) = 0.26 ± 0.03. Finite T(e) in the quantum Hall regime, where Landau levels are expected to be spaced by 4e²/h for graphene, confirms that reduced plateau spacing arises due to scattering from disordered edges at the nanowire–graphene interface and the surrounding potential profile, which reduces the transmission probability of charge carriers injected via the e-QPCs.

Conclusion

We demonstrate quantised conductance and quantum Hall transport at 1D magnetic nanowire–graphene interfaces that form e-QPCs, capable of spin injection and detection, in a high-mobility graphene spin transistor. Consistent values of effective constriction widths are extracted using two distinct methods: (i) using the Landauer equation to analyse G vs k_F data with a transmission estimated from conductance steps, and (ii) estimating the energy spacing, ΔE, between consecutive subbands via finite bias analysis. Across all data and devices, the transmission probability falls within the range T = 0.08–0.30. Explicit evidence of spin transport, detected in a representative device, confirms that spin injection occurs ballistically, resulting in spin signals that propagate over >10 μm distances. The presence of 1D subbands in spin-injecting contacts is confirmed; weak features in the spin signal suggest that ΔR_NL depends on the position of the Fermi energy in relation to the subbands, but more evidence is needed. Applying a perpendicular magnetic field prompts a transition into the quantum Hall regime, which causes the emergence of Landau level plateaus at the expected filling factors of 4-fold degenerate graphene. These phenomena take place in a graphene spintronic device without the need to engineer a physical constriction within the graphene channel. The demonstration of ballistic spin injection via magnetic e-QPCs presents an encouraging step towards the development of low-power ballistic spintronics.

Methods

Device fabrication

Devices investigated in this work comprise Van der Waals heterostructures of single-layer graphene flakes, fully encapsulated by thin (10–30 nm) hBN flakes. Individual flakes are isolated via mechanical exfoliation, then stacked via the dry peel transfer technique²⁷. A polymer hard mask is patterned over the heterostructure using electron beam lithography (EBL), which, followed by reactive ion etching, defines the transport channels (Fig. 1a and Supplementary Fig. S1a). Due to the selectivity of the etch recipe, the hBN and graphene flakes have different etch rates, leading to a small ledge of exposed graphene (≤10 nm) forming at the edges of the channel; this is the point where the 1D contact is made (Supplementary Fig. S1b)⁵⁸. A second EBL stage is performed to pattern nanowire electrodes into a polymer coating. Ferromagnetic metal is deposited onto the sample, which adheres to the exposed pattern and forms the electrodes. For the devices in this work, the nanowires are 60 nm cobalt (Co) with a 20-nm Gold (Au) capping layer. All contacts have widths in the range W_c = 150–350 nm. Devices are stacked on 290 nm SiO₂/Si substrates, with the oxide layer acting as a dielectric between the highly doped silicon back gate and the flakes of 2D material on its surface.

Electrical characterisation

Electrical measurements are performed in a high vacuum cryostat (pressures <10⁻⁶ mbar), at temperatures between room temperature (300 K) and low temperature (20 K); temperature is lowered by a constant flow of liquid helium into the cryostat, then controlled by a thermocouple placed near the device. Devices are mounted between the poles of a rotating electromagnet, allowing us to apply both in-plane and out-of-plane magnetic fields, with a strength of up to 1.2 T. All transport measurements are performed with low frequency (<20 Hz) AC lock-in techniques, with the exception of bias spectroscopy (see below methods).

Carrier density in the graphene channel, n, is controlled by the application of a DC voltage, V_BG, to the p-doped silicon back gate, which changes the electric field between the gate and channel. In order to estimate n in the graphene, the channel and Si back gate are treated as plates of a parallel capacitor, separated by two dielectrics: (i) the SiO₂ layer, with thickness d_Si = 290 nm, and (ii) bottom hBN flake of thickness d_h, determined via AFM (Supplementary Fig. S1b). We apply the equation n = C_g(V_BG − V_D)/e, where V_D is the gate voltage at which the Dirac point occurs, and the capacitance is calculated as C_g = ϵ₀ϵ_hϵ_Si/(ϵ_hd_Si + ϵ_Sid_h), where ϵ₀ = 8.85 × 10⁻¹², ϵ_h ~3.8, and ϵ_Si = 3.9 are the permittivities of free space, hBN (value used here is for bulk hBN³¹), and SiO₂, respectively. In this interpretation, applying a positive (negative) voltage to the gate will induce negative (positive) charges in the graphene channel. Hence, by gating, we can access both the electron and hole transport regimes.

Charge transport in the graphene channel is probed by measuring resistivity, ρ, in the standard 4-terminal configuration (Supplementary Fig. S2a), while controlling channel carrier density, n, by the means explained above. The result is a Dirac curve (Supplementary Fig. S2a).

Transport at the 1D contact is measured via the 3-terminal configuration (Fig. 2c and Supplementary Fig. S2d). V_BG is again used to control the carrier density, but in this case a specific contact carrier density, n_c, is defined by the voltage at which minimum contact conductance occurs. All contacts give resistances in the range R_c = 1–10 kΩ.

The charge diffusion coefficient, D, is calculated via the Einstein relation, D = σ/eN, with N being the density of states, calculated as,

$$N=frac{sqrt{{g}_{s}{g}_{v}n}}{sqrt{pi }hslash {v}_{{{{rm{F}}}}}},$$

(4)

where g_s and g_v are the spin and valley degeneracy factors, respectively, and v_F is the Fermi velocity.

Bias spectroscopy

In the case of bias spectroscopy, a DC voltage source is used in series with an AC lock-in amplifier, such that we can measure the differential conductance of an e-QPC, G = dI/dV, while simultaneously applying a bias voltage between contact and channel, V_Bias. The amplitude of the AC signal used is chosen to be small, such that the magnitude of oscillations in V_Bias is far below the energy scale of the features being resolved. Using an independent DC source, a voltage is applied to the silicon back gate, V_BG, which controls the carrier density in the channel, n. Figure 2a, b and Supplementary Figs. S3a and S4 show the results of these measurements, where each black line represents a measurement of G as a function of V_Bias, at a fixed V_BG.