Magnon-drag thermopile I. DEVICE FABRICATION AND CHARACTERIZATION Our devices consist of a large number of pairs of permalloy (NiFe) wires (30 nm wide, 20 nm thick and 5 µm long) connected in a zigzag configuration by short silver (Ag) wires (40 nm thick). Each NiFe wire in a pair is identical to the other except for their geometry that results in different coercive fields that allow us to control the relative orientation of their magnetizations. A scanning electron microscope image (SEM) of a typical device is shown in Fig. S1b. Every other wire has a nucleation pad (200 nm wide and 400 nm long) that results in a reduced coercive field as shown below (Figs. S2 and S3). The main elements for the device fabrication involve two electron-beam lithography steps and a multi-angle shadow evaporation as described in our previous work [1, 2]. A heater consisting of a 50 nm thin film of Pt with a 5 nm Ti adhesion layer is defined in the first lithographic step. Next, a suspended shadow mask is fabricated as described below and the thermopile is formed by depositing, sequentially, the NiFe and the Ag wires, where the latter connect the NiFe wires electrically in series. The shadow evaporation allows for material deposition without breaking the vacuum, therefore producing clean contacts between NiFe and Ag. The suspended shadow mask (Fig. S1a) is fabricated on a highly-doped Si (100) wafer with thermally grown oxide [1, 2]. We use a methyl-methacrylate (MMA)/poly(methylmethacrylate) (PMMA) bilayer in combination with selective electron-beam exposure. The base resist (MMA) has a sensitivity that is about 5 times larger than the top resist (PMMA), which allows us to generate a controlled undercut by exposing the bilayer with a dose that is sufficient to expose the MMA layer, but insufficient to expose the PMMA layer. The exposed bilayer is developed in an isopropanol:methyl-isobutyl-ketone (3:1) solution and placed in a high-vacuum electron-beam evaporator (base pressure 10 8 Torr). The material evaporation sequence is shown in Fig. S1a. First, we evaporate NiFe under an angle of 45 o relative to the substrate normal, which creates the NiFe wires. Next, the substrate is rotated 90 o around an axis normal to the substrate and sequentially tilted in opposite directions (+/- 45 o ) to create continuous electrodes by Ag evaporation. With the 45 o tilt only the mask features in the deposition plane that contains the substrate normal and the evaporation source project onto the substrate. The rest of the mask features result in material deposition onto the side-wall of the top PMMA resist, which later on is removed NATURE MATERIALS www.nature.com/naturematerials 1
by lift-off. After fabrication the devices are kept in vacuum. This avoids oxidation of the interface between NiFe and Ag [3], which could result in a variation of the thermopile response over time. a 2 Ag 45 o b Py 45 o 1 Magnon-drag thermopile Pt heater Ag 45 o 3 FIG. 1. Sample fabrication. a, A partial view of the design of the shadow mask made with e-beam lithography using a PMMA/MMA bilayer on a SiO 2 /Si substrate. The metal deposition sequence is shown. b, Scanning electron microscope image of a device. The central part of the device is formed by N Py wires pairs connected at both ends with Ag wires. The dc voltage generated between the end electrodes is measured as a function of a temperature gradient created with an on-chip Pt heater (bright vertical feature) in close proximity to the Py wires. The two vertical arrows indicate the positions of the resistive Pt thermometers used for quantifying the temperature gradient. Generation and quantification of the temperature gradient was done following methods similar to those in Refs. [4 6]. The temperature gradient was produced by the local Pt heater through Joule heating and quantified using two resistive Pt thermometers situated at the position of the Ag transverse wires (Fig. S1b) in calibration samples. We monitored the resistance of both thermometers, R 1 and R 2, with a standard four probe technique as a function of the heater current I h and obtained R 1 and R 2 as a function of the power dissipated. Each thermometer was calibrated prior to the experiments by sweeping the temperature of the cryostat. A linear relationship between temperature and resistance was observed in the temperature range of interest with a coefficient β = R/R 0 T where R = R 1,2 and R 0 is the resistance at 295 K; β =1 10 3 K 1, in agreement with previous results [5, 7]. 2 NATURE MATERIALS www.nature.com/naturematerials
a b 1.0000 R/R B=0 0.9998 - B c1-0.2 0.0 0.2 B(T) -0.2 0.0 0.2 B(T) FIG. 2. Coercive fields, B c1 and. a, Normalized resistance R of a single wire without a nucleation pad as a function of the magnetic field B at 295 K. A linear dependence of R is observed, which is interrupted at the switching fields ±, as expected from magnon-induced magnetoresistance (MR). The B swept direction is from B > 0 to B < 0 (red) and from B < 0 to B>0 (blue). b, Comparison of the normalized MR for wires with a nucleation pad (open symbols) and without it (solid symbols). The switching of the magnetization of the wire with a nucleation pad is abrupt and occurs at B c1 <. In a and b the resistance is normalized to its value at B = 0, R B=0. The insets in b show SEM images of the devices used for these measurements. We have also verified that the magnetization switching of the electrodes occurs in a narrow range of magnetic fields and that we obtain an antiparallel configuration of the magnetizations as indicated in Figs. 2 and 3 of the main text. Such verification was done by measuring the magnetoresistance (MR) of individual wires as a function of magnetic field along the wires as in the measurements of Fig. 2. Typical results are presented in Fig. S2. Figure S2a shows the resistance of a single wire without a nucleation pad. We observe a linear dependence of the MR for positive and negative fields up to the switching field, ±. This is in agreement with the results reported using FePt films [12]. The same behaviour is observed when the nucleation pad is included (Fig. S2b) but with a switching field B c1 <. Similar to the previous case, the switching is abrupt and there are no features in the magnetoresistance that would indicate a multistep switching process. Moreover, the normalized slopes of the linear MR before and after switching are identical for the wires with and without the nucleation pad, which shows that the starting and final 3 NATURE MATERIALS www.nature.com/naturematerials 3
15 R ( ) B c1 B c1-0.2 0.0 0.2 B(T) FIG. 3. Magnetoresistence in a magnon-drag thermopile. At T = 295 K (top) we observe a step-like switching due to magnons. The broadening of the step is due to the dispersion in the magnitude of for the wires that form the thermopile. At low temperatures (4.2 K, bottom), the progressive switching of the wires result in clear anisotropic magnetoresistance minima. The B swept direction is from B>0 to B<0 (red) and from B<0 to B>0 (blue). The measurements were displaced vertically for clarity. states of both wires are equivalent. This also demonstrates that the MR of the wires with the nucleation pad is dominated by the narrow wires and that the nucleation pads do not play any significant role. The measurements in Figs. S2a and S2b were performed at room temperature (295 K). If the temperature decreases, the switching occurs more gradually but still in a narrow field range. In this situation, standard anisotropic magnetoresistance (AMR) can be observed, where the resistance drops when the magnetization reverses. This process becomes evident below 200 K for the wires with a nucleation pad and below 100 K in the wires without it. Fig. S3 shows the AMR for a magnon-drag thermopile, which has both types of wires connected in series. At 295 K the switching at B c1 is barely observed but becomes obvious at low temperatures. The narrow minima in the resistance, which appear at about B c1 and as extracted in Fig. S2b, are well resolved. No additional features are observed. These measurements demonstrate a narrow distribution in the switching fields of the NiFe wires that form the thermopile and a well-developed antiparallel configuration of the magnetizations in between the switching fields. 4 4 NATURE MATERIALS www.nature.com/naturematerials
II. MAGNON-DRAG MODEL. MAGNETIC FIELD EFFECTS In a ferromagnetic metal, electrons are scattered by magnons as demonstrated by longitudinal magnetoresistance measurements. In analogy to phonon scattering, which results in a phonon-drag effect, the electron-magnon interaction produces a magnon-drag effect. In a magnetic material a magnon current flows down the temperature gradient, which causes a thermoelectric voltage because of its interaction with the electron system. It was suggested first by Bailyn [8] that the theory of magnon-drag should be analogous to that of phonondrag [9]. Following this work the magnon-drag component, S MD, of the thermopower has been calculated by Grannemann and Berger [10] who considered a magnon gas in addition to a free-electron gas and obtained the low temperature limit of equation (1) introduced in the main text. At low temperatures, where the magnon-electron interaction is dominant, S MD is proportional to T 3/2, following the same dependence of the magnon gas specific heat. In addition, the magnon-drag effect has a magnetic field contribution given by the so-called quenching function, F (y), which is given by: F (y) = 0 x 3/2 exp(x + y) (x + y) dx, (1) [exp(x + y) 1] 2 where x = Dq 2 /k B T and y = gµ B B int /k B T with B int the external magnetic field induction B plus the ferromagnet magnetization µ 0 M (about 1 T for Py). A quadratic dispersion law is assumed [10 13] which does not lead to great error because of the integration in the entire momentum space [14]. There, the states with larger q will make the greatest contribution [10, 14]. The considered approximation is thus valid for thermal magnons, except at very low temperatures. The exchange interaction plays the main role, while dipole-dipole interaction makes a relatively small contribution to the dispersion relation [14, 15]. We numerically integrate F (y) and plot F (y)/f (0) versus y in Fig. S4a, where F (0) = 4.45. In Fig. S4b, F (y)/f (0) is shown for different values of the temperature in a range relevant to our experiments (see Fig. 4, main text). The arrow corresponds to B int = µ 0 M = 1 T. For small variations B 0.1 TofB int about µ 0 M, F (y)/f (0) presents a markedly linear response, which is in a good agrement with our measurements (Figs. 3a and b, main text). NATURE MATERIALS www.nature.com/naturematerials 5
a 1.0 b 1.00 0.9 0.99 F(y)/F(0) 0.8 0.7 F(y)/F(0) 0.98 0.97 T (K) 50 100 150 200 250 300 0.0 0.2 0.4 g B B int /k B T 0.96 0 50 100 g B B int ( ev) FIG. 4. Normalized quenching function. a, F (y)/f (0), calculated by numerical integration. F (y) encodes the influence of the magnetic field on the magnon drag effect. b, F (y)/f (0) versus gµ B B int for different temperatures. The arrow corresponds to B int = µ 0 M = 1 T, while the bar indicates the magnetic field excursion during data acquisition. The calculations predict that the magnon drag effect should have a linear dependence on the magnetic field for our experimental conditions, which is in excellent agrement with our observations. [1] Valenzuela, S. O., & Tinkham, M. Spin current induced Hall effect. J. Appl. Phys. 101, 09B103 (2007). [2] Costache, M. V., & Valenzuela, S. O. Experimental spin ratchet. Science 330, 1645-1648 (2010). [3] Mihajlović, G., et al. Enhanced spin signal due to native oxide formation in Ni 80 Fe 20 lateral spin valves. Appl. Phys. Lett. 97, 112502 (2010). [4] Small, J. P., Perez, K. M. & Kim, P. Modulation of thermoelectric power of individual carbon nanotubes. Phys. Rev. Lett 91, 256801 (2003). [5] Shapira, E., Tsukernik, A., & Selzer, Y. Thermopower measurements on individual 30 nm nickel nanowires. Nanotechnology 18, 485703 (2003). [6] Boukai, A. I. et al., Silicon nanowires as efficient thermoelectric materials. Nature 451, 168-171 (2008). [7] Zhang, X. et al. Thermal and electrical conductivity of a suspended platinum nanofilm. Appl. Phys. Lett. 86, 171912 (2005). 6 6 NATURE MATERIALS www.nature.com/naturematerials
[8] Bailyn, M. Maximum variational principle for conduction problems in a magnetic field, and the theory of magnon drag. Phys. Rev. 126, 2040-2054 (1962). [9] Ziman, J. M. Electrons and Phonons (Oxford University Press, London, 1960). [10] Grannemann, G. N. & Berger, L. Magnon-drag Peltier effect in a Ni-Cu alloy. Phys. Rev. B 13, 2072-2079 (1976). [11] Raquet, B., Viret, M., Sondergard, E., Cespedes, O., & Mamy, R. Electron-magnon scattering and magnetic resistivity in 3d ferromagnets. Phys. Rev. B 66, 024433 (2002). [12] Mihai, A. P., Attan, J. P., Marty, A., Warin, P. & Samson, Y. Electron-magnon diffusion and magnetization reversal detection in FePt thin films. Phys. Rev. B 77, 060401(R) (2008). [13] Tulapurkar, A. A. & Suzuki, Y. Contribution of electron-magnon scattering to the spindependent Seebeck effect in a ferromagnet. Solid State Commun. 150, 466 (2010). [14] Gurevich, A. G. & Melkov, G. A. Magnetic Oscillations and Waves (CRC Press, New York, 1996). [15] Misra, A. & Victora, R. H. Ferromagnetic relaxation by magnon-induced currents. Phys. Rev. B 73, 172414 (2006). NATURE MATERIALS www.nature.com/naturematerials 7