Thermoelectric spin voltage in graphene

In recent years, new spin-dependent thermal effects have been discovered in ferromagnets, stimulating a growing interest in spin caloritronics, a field that exploits the interaction between spin and heat currents. Amongst the most intriguing phenomena is the spin Seebeck effect, in which a thermal gradient gives rise to spin currents that are detected through the inverse spin Hall effect. Non-magnetic materials such as graphene are also relevant for spin caloritronics, thanks to efficient spin transport, energy-dependent carrier mobility and unique density of states. Here, we propose and demonstrate that a carrier thermal gradient in a graphene lateral spin valve can lead to a large increase of the spin voltage near to the graphene charge neutrality point. Such an increase results from a thermoelectric spin voltage, which is analogous to the voltage in a thermocouple and that can be enhanced by the presence of hot carriers generated by an applied current. These results could prove crucial to drive graphene spintronic devices and, in particular, to sustain pure spin signals with thermal gradients and to tune the remote spin accumulation by varying the spin-injection bias.

To introduce the notion of the thermoelectric spin voltage (TSV), it is useful to adapt the analogy with conventional thermocouples from the original scenario that was proposed to explain the spin-Seebeck effect [3]. In graphene, thermoelectric effects are sensitive to changes in the carrier density n. The Seebeck coefficient S(n), schematically shown in Fig. 1a, changes sign across the charge neutrality point (CNP), when the dominant carriers change from electrons to holes or vice versa [18,19]. Therefore, a thermocouple can be fabricated using two sheets of this material with unequal n (Fig. 1b). For carrier densities n 1 and n 2 , the thermoelectric voltage is V S = −∆S∆T , where ∆T is the temperature difference between the two ends of the sheets and ∆S = [S 2 (n 2 ) − S 1 (n 1 )].
In the spin-transport case, carriers can be considered as flowing in independent spin channels characterized by two spin sub-bands whenever the relevant length scale is smaller than the spin diffusion length λ s . In graphene, λ s can exceed a few micrometres, while spin injection from a ferromagnet can result in a significant difference ∆n = n ↑ − n ↓ between the carrier densities for spin-down (n ↓ ) and spin-up (n ↑ ). This difference is reflected in the spin accumulation ∆µ 0 = µ ↑ − µ ↓ , where µ ↓,↑ are the spin-dependent electrochemical potentials associated to n ↓,↑ (Fig. 1c). Because n ↓ = n ↑ , the Seebeck coefficients for spindown and spin-up carriers S ↓,↑ = S(n ↓,↑ ) can also be different. By replacing n 1,2 with n ↓,↑ and S 1,2 with S ↓,↑ , it becomes apparent that, within λ s , the scenario sketched in Fig. 1b is intrinsically present in a single graphene sheet. The change in ∆µ 0 , labelled δµ, results in a thermoelectric spin voltage δµ/e ∼ −(S ↑ − S ↓ )∆T with e the electron charge. This analogy suggests that when S ↑ < S ↓ the spin accumulation ∆µ = ∆µ 0 + δµ at a remote detector can be enhanced by the presence of a temperature gradient.
In order to observe the TSV, we implement multi-terminal graphene devices comprising normal metal and spin sensitive electrodes as shown in Fig. 1d (see Methods, Supplementary   Fig. 1). Spin injection and detection is achieved with ferromagnetic electrodes 3 and 4, respectively, which delimit the spin channel [20,21]. Metal electrodes 1 and 2, and the graphene in between, define a heater that generates the temperature gradient in the spin channel and induces the TSV in electrode 4. A key aspect of the device geometry is that it allows independent control of the heat and spin sources and, therefore, separate thermal and spin injection effects. As discussed below, it can also discriminate between the spin thermoelectric effect discussed here from the thermal spin injection, which is due to a thermal gradient in the injector and the spin-dependent Seebeck effect [22,23].
We start by estimating the graphene Seebeck coefficient S from the square resistance R vs n using the Mott relation [18,19] (room temperature, RT) and 2b (77 K), is known to be a good approximation of S, both in magnitude and temperature dependence [17,18], and allows us to determine ∆T from the thermoelectric voltage.
To compare S Mott with the actual thermoelectric response of the device, the voltage in the spin channel, V dc , is measured vs the dc current, I dc , in the heater. Figures 2c and   2d show the results at RT and at 77 K, respectively. In both cases, we observe a change from an upward (V dc > 0) to a downward (V dc < 0) response with n, as expected from a thermoelectric effect. Since the graphene is used as part of the heater, both lattice and carriers are simultaneously heated. However, the carrier temperature T can differ from, and be much larger than, the lattice temperature [14][15][16]. Hot carriers propagate away from the heater region, but at RT they thermalize with the lattice before reaching the voltage electrodes [17]. This leads to a parabolic dependence of V dc = −S∆T = −ΣI 2 dc , owing to the classical relationship between temperature increase and Joule heating, ∆T = αI 2 dc . Here α is a constant that depends on the heater resistance, the geometry, and the thermal conductivity of the device components and substrate. The coefficient Σ is extracted from At 77 K, the presence of hot carriers leads to a much larger temperature gradient than at RT and to the break-down of the classical relationship between Joule heating and ∆T [17]. Despite that S increases monotonically with T [18,19], V dc at 77 K is larger than at RT by an order of magnitude, while the parabolic dependence of V dc vs I dc transforms into a characteristic V-shape. The hot carriers obey a thermal distribution [14] but ∆T is difficult to model and depends strongly on n [16,17]. However, noting that S ∝ T , we can estimate ∆T from the ratio between S Mott and V dc in Fig. 2d. At the CNP a more precise estimation can be obtained from the slopes of S M ott and V dc vs n; we find that ∆T ∼ 60 K (CNP) and ∆T ∼ 30 K (n = 10 12 cm −2 ), which is significantly larger than ∆T ≈ 1 K at RT. These large ∆T in the hot-carrier regime are highly favorable to observe the TSV.
Next, we investigate the graphene spin transport properties when an ac current i ac is applied in the non-local configuration, using electrodes 3 and 4 as spin injector and detector, respectively (Fig. 1d). Figure 2e shows the non-local resistance, R N L = V ac /i ac , as a magnetic field B along the long axis of the electrodes is swept to generate parallel (↑↑) and antiparallel (↓↑) magnetization alignments. The non-local spin signal, defined as ∆R N L = Figure 2f shows spin precession measurements with out-of-plane B, from which we determine the spin relaxation length λ s , the spin relaxation time τ s , as well as the effective polarization P of the electrodes ( Supplementary Fig. 2).
Having characterized the spin transport and thermoelectric properties, the device is now biased in the configuration shown in Fig. 3a, where I dc and i ac are simultaneously applied.
The current I dc generates a thermal gradient in graphene that induces the TSV, which is quantified from the change δR S N L = P δµ eiac in the non-local signal ∆R N L . Note that I dc also generates a thermal gradient in the spin injector. Since i ac passes through the heater such gradient is modulated by it, therefore, an additional change δR th N L , associated with thermal spin injection, should be expected in ∆R N L . Nevertheless, as explained in the Supplementary Information, δR S N L and δR th N L are even and odd functions of I dc , respectively, providing a straightforward procedure to disentangle them: δR S N L is obtained from the average between the measurements for ±I dc , while δR th N L is obtained by calculating half of the difference between the same measurements. An increase in ∆R N L must originate from δR S N L and/or δR th N L . Because I dc is injected in non-magnetic electrodes and the detection is non-local, the increase is of thermal origin, however, it cannot be explained by a simple increase in the graphene temperature since ∆R N L weakly decreases with temperature ( Supplementary Fig. 2). Similar conclusions are drawn from Fig. 3c, which also shows that the overall thermal effect is largest about the CNP. Figure 3d shows δR S N L vs n, as extracted from the measurements acquired with |I dc | = 50 µA; δR th N L from the same measurements is shown in Supplementary Fig. 4. It is observed that δR S N L displays a maximum value of ≈ 0.3 Ω at the CNP, which represents a significant ∼ 20% increase in ∆R N L ; δR S N L also presents an electron-hole asymmetry, it decreases away from the CNP and clearly changes sign in the hole side for |n| = 7 × 10 11 cm −2 . To understand these results, we first consider the scheme in Fig. 4. As depicted in Fig. 4a, the spin accumulation, leading to n ↓ = n ↑ , effectively shifts S ↓ and S ↑ . As explained in the introduction, δµ ∼ −e(S ↑ − S ↓ )∆T . Therefore, for small spin accumulation (typically ∆n is below 10 11 cm −2 ), δµ can be approximated by δµ ∼ −e ∂(S∆T ) ∂n ∆n, where we assume that, when hot carriers are involved, both S and ∆T can depend on n. If ∆T was weakly dependent on n, then δµ ∝ ∂S ∂n , which qualitatively describes δR S N L ∝ δµ and suggests that the change in sign in δR S N L roughly occurs when S reaches an extreme (compare Figs. 3d and 4b).
A quantitative comparison with the experiment can be made from δR S N L = P δµ eiac ∼ −P ∂(S∆T ) ∂n ∆n iac . Here, ∂(S∆T ) ∂n is directly extracted from Fig. 2b, and ∆n estimated from the spin injection rate P iac e , τ s , λ s and the width w of the graphene as ∆n ∼ P τs 2eλsw i ac . Using values at the CNP: P = 6 %, τ s = 250 ps, and λ s = 1 µm, we find ∆n iac ∼ 3.6 × 10 9 cm −2 µA −1 . Combining with ∂(S∆T ) ∂n ∼ 2.4 × 10 −9 µV cm −2 , we obtain δR S N L ∼ 0.5 Ω, which is in reasonable agreement with the experimentally found value. Moreover, the modelled δR S N L vs n, obtained by differentiating S∆T (solid line in Fig. 3d), successfully reproduces all of the trends in the data, including the electron-hole asymmetry and the change in sign at |n| = 7 × 10 11 cm −2 . Such an agreement conveys confidence to the interpretation of δR S N L as arising from the TSV. The somewhat smaller experimental δR S N L at the CNP could be a consequence of an overestimation of ∆n at the injector when the temperature gradient is applied. Note that because the carrier cooling times are dependent on n [17,24], the temperature for spin-up and spin-down carriers can be different, as observed in a nanopillar spin valve subjected to a heat current [25]. This is considered in the model, as it estimates δR S N L from the product between S(n) and ∆T (n). The spin splitting induced by the thermoelectric spin voltage in graphene is at least two orders of magnitude larger than that deriving from the spin-dependent Seebeck effect in metals [22]. This striking result is a consequence of a relatively large Seebeck coefficient, its strong variation at the Fermi level, and the low density of states of graphene. It also stems from the temperature gradient induced by hot carriers; the lattice temperature gradient is about an order of magnitude smaller and cannot explain the magnitude of the observed effect.
Because the peak Seebeck coefficient scales as 1/ √ n r , the thermoelectric spin voltage can be enhanced by decreasing the residual carrier concentration n r . The temperature gradient can also increase if the spin injector was part of the heater, resulting in a strong bias dependence of the device performance even at room temperature; however, other effects, such as a bias dependence of the polarization, could play a role in this case [26][27][28]. The demonstrated phenomenon is analogous to the enhancement, away from the CNP, generated with a drift current [29] but based on a thermal drift in combination with an n-dependent Seebeck coefficient. These observations are not exclusive to graphene and could be relevant for other materials presenting an energy-dependent conductance, for instance common semiconductors [30], or topological insulators, which exhibit exotic spin properties and strong thermoelectric effects. They can therefore lead to advances in spin caloritronics, where spin currents are controlled or sustained over long distances using heat currents.

Data availability
The data that support the findings of this study are available from the corresponding authors upon reasonable request. comprising two graphene sheets, 1 and 2, with carrier densities n 1 and n 2 , and thus different ∆T is builtup due to the temperature difference ∆T between the cold and the hot side of the sheets, which derives from the temperature gradient ∇T . For the case drawn, n 1 = −n 2 and thus V S = δµ/e.
The flow direction of the majority carriers in 1 (holes) and 2 (electrons) is shown with green and blue arrows, respectively. c, At length scales smaller than the spin relaxation length, carriers with opposite spins belong to two independent transport channels. When the spin accumulation ∆µ 0 = 0, S becomes spin dependent. A thermoelectric effect analogous to that in b leads to a TSV and a remote increase (decrease) of the spin accumulation ∆µ = ∆µ 0 + δµ at the cold end.
For simplicity, the case n ↓ = −n ↑ is drawn. The flow direction of the carriers for spin-down and spin-up sub-bands is shown with green and blue arrows, respectively. d, Device configuration to detect the TSV proposed in c. A heating current between electrodes 1 and 2 generates ∇T . Spin injection and detection is achieved with electrodes 3 and 4, which define the spin channel. The temperature difference in the spin channel generates the TSV that leads to a change in the spin signal measured in 4. The color of the substrate in b-d represents the temperature of the carriers in graphene.  c, R N L vs out-of-plane magnetic field B ⊥ at different n for I dc = 0 (black) and 50 µA (red). The magnetization of the electrodes are in parallel configuration. In b and c, an increase of R N L is observed when I dc is applied; for clarity, a small spin-independent background has been subtracted for I dc = 50 µA. d, Change in R N L induced by the TSV, δR S N L , as a function of n (open symbols and left axis). The comparison to ∂V dc ∂n at I dc = 50 µA (line and right axis) is suggested by theoretical modelling (Fig. 4). The data in d are obtained at fixed gate from the difference in spin signal for the parallel and antiparallel magnetization configuration of the electrodes at B = 0; the error bars reflect the uncertainty in R N L . Note that in c the positive contribution due to the thermal spin injection ( Supplementary Fig. 4) suppresses the overall change induced by I dc at n = ±1.5 × 10 12 cm −2 . This leads to a TSV δµ/e ∝ − dS dn ∆T when a temperature difference ∆T is built up between spin injector and detector; here it is assumed that ∆T is independent of n.