Joule Heating Induced Spin Seebeck Effect

Transcription

1 Bachelor Thesis Joule Heating Induced Spin Seebeck Effect Erich Dobler Date: 23 August 213

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3 Contents 1 Introduction 1 2 The Spin Seebeck Effect Spin Currents Metal-Ferromagnet Interface The Inverse Spin Hall Effect The Spin Seebeck Voltage Experimental Method Temperature Control Computation of the thermal Gradient Measurement of the Platinum Temperature T h Magnetisation Control by an External Field Measuring the Spin Seebeck Voltage V SSE 13.1 Reduction of the Offset Voltage Electric Effects and Data Reduction Experimental Results Raw Data and Field Sweeps Spin Seebeck Voltage as a Function of the Thermal Gradient Spin Seebeck Effect at low Temperatures T c Angular Dependence of the SSE Discussion Consistency Check Quantitative Data Analysis Change of Magnetisation Conclusion and Outlook 32 Acknowledgements 35 - i -

4 1 Introduction The spin Seebeck effect (SSE) was measured for the first time in 2 by Uchida et al. [1]. The SSE enables the generation of a pure spin current by applying a thermal gradient across a metal-ferromagnet interface. Thus, it is very important in the field of spintronics and spin calorimetry, which deal with the generation, the measurement and the manipulation of spin polarized and pure spin currents. It allows understanding of spin driven thermal coupling between magnetic and metallic matter. In the frame of this thesis, the SSE appearing at a yttrium iron garnet (YIG)/platinum (Pt) interface is investigated. The key aspect in measuring the SSE is the creation of the thermal gradient across the sample. So far, there are diverse methods, for example heating by laser (Rev. [2]) or by Peltier elements (Rev. [1]). For this thesis another method is applied. Here, the thermal gradient is generated using the Pt itself as a resistive heater. This is not without its own challenges though. The main quantity in observing the SSE is the so called spin Seebeck voltage, which is measured on the Pt film. While measuring this voltage, the Pt is concurrently used as a resistive heater and, therefore, a DC heating current is sourced into the Pt film. This current, in turn, causes electric effects that occur, additional to the SSE voltage, in the measured signal. These purely electrical effects are of the order of a few mv, whereas the SSE voltage is of the order of a few µv. Although the measured signal is three orders larger than the SSE voltage, it can be obtained by means of a very simple method, which is based on the symmetrical properties of electrical and thermal effects. The method is presented in this thesis. It finally enables measuring the SSE by the resistive heating of the Pt layer. In comparison to the other heating methods, the resistive heating does not only simplify the setup and the procedure of the experiment, especially for measurements at low temperatures in a cryostat, but it also allows a simple determination of the temperature profile of the measured sample and an uncomplicated quantitative discussion of the data. However, since this method is applied the first time in measuring SSE, measurements were done to check out, whether the observed effect is the SSE or not. This thesis is structured as follows: First, the basic physics of the SSE are presented. The third chapter contains details about the experimental setup and method. The results are shown in the fifth chapter, followed by a justification of the resistive heating method and a quantitative discussion

5 2 The Spin Seebeck Effect Since the first measurement of the SSE has been made in 2 [1], a number of theories have been proposed, but none of them is generally accepted or is able to deliver a complete theoretical description of the experiments. Nevertheless, there seems to be a general consensus about the basic physics which will briefly be covered in this chapter. 2.1 Spin Currents Spin currents are essential in the understanding of the SSE processes. A spin current contains two informations, the spin polarisation σ and the direction of flow I s (Ref. [3]). In (non-magnetic) metals, spin currents are carried by electrons, whereas in ferromagnets they are carried by magnons as well. For the experiments in this thesis the insulating ferrimagnet 1 YIG is used, where the spin current is exclusively carried by the magnons. When the spin current is carried by electrons, it is important to clarify the difference between spin and charge currents, since electrons not only carry spin but also charge. A pure charge current I el implies the same density and direction for spin up and spin down electron flow, whereas a pure spin current I s is defined by equal spin up and spin down electron current density but in opposite directions (see Fig. 1). 1 The antiferromagnetic component of YIG is small (Ref. []) and it will be disregarded in this thesis. YIG is treated as a ferromagnet in the following

6 2 The Spin Seebeck Effect e - e - e - e - e - e - e - e - e - e - e - e - I el e - e - I s e - e - e - e - e - e - e - e - (a) (b) Figure 1 Sample structure: Currents in a metal: (a) pure charge current I el and (b) pure spin current I s. They differ in the direction of spin up and spin down electron flow. 2.2 Metal-Ferromagnet Interface The main feature of the SSE is the appearance of a spin current I s at a normal metal (NM) - ferromagnet (FM) interface induced by an applied thermal gradient. The spin current is a consequence of heat and spin transfer from electrons in the NM to magnetic moments in the FM or vice versa. Following Ref. [5,, 6], the spin current I s is antiparallel to the thermal gradient T FM and the polarisation σ is antiparallel to the magnetisation M of the FM 2 : I s T FM (1) σ M (2) According to Ref. [5,, 6], the magnitude of I s is proportional to the temperature difference between magnons and electrons at the interface. This, in turn, is proportional to the gradient at the interface, which will be approximated by the gradient in the FM in the following (Ref. [6]) I s T FM (3) 2.3 The Inverse Spin Hall Effect In the NM, the spin current is converted into a charge current via inverse spin Hall effect (ishe). The ishe is based on the coupling of charge and spin currents transmitted by spin-orbit interaction. It appears in materials with strong spin orbit interaction, where the scattering of electrons is dependent on their spin polarisation (Ref. [7]). In a pure spin current, spin up and spin down electrons move in opposite directions and hence are scattered in the same direction. This produces an electron accumulation on 2 In fact, there are two kinds of SSE, the longitudinal and the transversal SSE. The geometric properties hold for the longitudinal SSE which is measured in this thesis. For detailed information see []

7 2 The Spin Seebeck Effect V SSE E ishe NM σ T FM I s z M y x Figure 2 Ferromagnet-metal interface. The thermal gradient is applied in z-direction, the spin current I s flows in negative z-direction, the polarisation σ is antiparallel to the magnetisation M. The electric field E ishe caused by the ishe is perpendicular to I s and σ one side of the material (see Fig. 3). The electron accumulation yields a transverse electric field E ishe (Ref. []): E ishe = θ SH A ρ ( 2 e h ) (I s σ) () The quantities θ SH, e, ρ and A are the spin Hall angle of the NM, the absolute value of the electron charge, the electric resistivity of the NM and the area through which the spin current flows. In this case A is the area of the M/FM interface. Eq. () is only valid, if the spin diffusion length is much larger than the thickness of the metal layer, so that spin diffusion and spin backflow can be neglected. The data analysis in Sec. 6.2 will assume this to be the case. - -

8 2 The Spin Seebeck Effect E ishe NM e - I s e - FM Figure 3 Sketch of the ishe mechanism: a pure spin current I s is generates an electric field due to spin dependent scattering. 2. The Spin Seebeck Voltage In our measurement setup, the structure of the metal is a Hall bar (see Fig. 5a). Due to the electric field E ishe, a potential difference can be measured between two opposite contact pads A, B (see Fig. ). It is called the SSE voltage: V SSE = B A E ishe ds = w HB E ishe cos(α) (5) Here, w HB is the width of the Hall bar and α is the angle between the electric field E ishe and the y-axis (Fig. ). In these measurements, the magnetisation M of the FM layer is in plane (M ẑ = ) and the thermal gradient is perpendicular to the plane ( T FM ẑ). Equations (1) and (2) yield: T FM M I s σ (6) As a consequence, the angle α between E ishe and the y-axis is equal to the angle between M and the x-axis (see Fig. ). So equation (5) can be written as: V SSE = w HB θsh A ρ ( 2 e h ) I s cos(α) (7) In summary, the following dependence of V SSE on the thermal gradient T FM and the angle α is expected: V SSE T FM cos(α) () - 5 -

9 2 The Spin Seebeck Effect A V SSE E ishe T FM α y B α M x w HB Figure Closeup of the Hall bar. The SSE voltage V SSE is measured between two transverse (y-axis) contact pads A and B. The longitudinal pads (x-axis) are used to feed the Hall bar with the heating current (see Sec.3.1). The magnitude of V SSE depends on the angle α

10 3 Experimental Method The samples investigated in the frame of this thesis were taken out of a set, that was fabricated by Matthias Althammer and Sibylle Meyer for spin Hall magnetoresistance (SMR) measurements (Ref. []). Their structure is displayed schematically in Fig. 5. The YIG was grown on either a gadolinium gallium garnet (GGG) substrate or on a yttrium aluminium garnet (YAG) substrate via laser-mbe (molecular-beam epitaxy) (Ref. []). The Pt layer was deposited in-situ, without breaking the vacuum, on the YIG via electron beam evaporation (Ref. []). A Hall bar structure with the dimensions width w HB = µm and length l HB = 1µm (compare Fig. 5a) was afterwards patterned in the Pt layer via UV-lithography and argon-ion-milling. The properties of each sample used are tabulated in Tab. 1. More details concerning the fabrication parameters are listed in Ref. []. Finally the sample is glued onto a copper heat sink. Cu V YIG l HB P h =R I h 2 y Ih Pt w HB T Cu Pt YIG GGG / YAG T h T c V x (a) top view (b) side view Figure 5 Sample structure: the Pt Hall bar is electrically conected as shown in (a). A heating current I h is sourced into the Hall bar. The measured quantities are longitudinal voltage V and transversal voltage V, the latter is measured over three tabs with a potentiometer. The heating power P h = R Ih 2 dissipated in the Pt-Hall bar creates the thermal gradient T (b). The copper heat sink keeps the bottom of the sample on the constant temperature T c. The thicknesses are not to scale. The whole structure is then mounted in a custom-built sample carrier that allows the samples to be interchanged easily. The Pt Hall bar is connected to the sample carrier - 7 -

11 3 Experimental Method sample d sub (µm) d YIG (nm) d Pt (nm) GGG/YIG(53.7)/Pt(6.1) GGG/YIG(6.7)/Pt(11.2) GGG/YIG(53)/Pt(.5) YAG/YIG(5.)/Pt(6.) Table 1 Values for the thicknesses d sub, d YIG and d Pt of the substrate, the YIG layer and the Pt layer for each sample, respectively. contacts via thin aluminium bonds. The sample preparation was completely done by Matthias Althammer and Sibylle Meyer. 3.1 Temperature Control To create the thermal gradient across the layers, the top of the sample has to be heated up to the temperature T h. This is achieved by using the Pt Hall bar itself as a resistive heater. The heating method is very simple: a Keithley 2 source meter sources current into the Hall bar with a constant DC heating current I h, which yields a power deposit P h = R Ih 2 in the Pt. One part of the measurements is made at room temperature. The temperature T c of the copper heat 1 sink is assumed to stay at room temperature when the Pt film is heated. The other part of the measurements is done in a liquid helium magnet cryostat. Here, the temperature T c is adjustable in the range from 25K down to 1K by means of a so-called variable temperature insert. A quantitative determination of the gradient in the YIG T YIG is, however, not straightforward and the specific geometry (see Fig. 6a) of the sample has to be taken into account. 3.2 Computation of the thermal Gradient Due to unequal heat conductances K YIG and K Sub (see Tab. 2), the thermal gradients T YIG and T sub in the YIG and the substrate, respectively, differ in their value. A schematic temperature profile is shown in Fig. 6b to exemplify the situation. Because the SSE is proportional to the gradient T YIG (see Eq. ()), it is an important quantity to be determined. However, since the measurable quantities are the temperatures T h and T c, a proper calculation of T YIG is indispensable. This is made in this section with the following simplifications. 1 In this thesis, the temperature of the heat sink is abbreviated with T c (cold temperature). The reader must not mistake with the Curie temperature, which is named in most of the literature with the same abbreviation. - -

12 3 Experimental Method heater T c T' T h Κ YIG d YIG T YIG Κ Sub J h d sub T sub heat sink (a) z (b) Figure 6 A schematic of the sample is displayed in (a). Due to the resistive heating, a heat flux J h from the Pt heater to the Cu heat sink arises. As the heat conductances K YIG and K Sub of the substrate and the YIG are unequal (see Tab. 2), the thermal gradients in each layer differ (b). The heat flux is assumed to be one dimensional and perpendicular to the surface. The thermal links between the layers are premised to be ideal. The heat conductances are supposed to be spatially constant and independent of the temperature. The later is valid as long as the induced temperature changes are small. A simplified heat flux scheme is illustrated in Fig. 7. Furthermore, the temperature T is defined as the temperature at the YIG/substrate interface. P h =R I h 2 T h Κ YIG T' J h Κ Sub T c Figure 7 Heat flux scheme: the heating power P h is deposited in the Pt Hall bar. The layers are replaced by thermal resistances with conductivity K YIG and K sub. T is the temperature at the YIG/substrate interface. At steady state, a constant heat flux J h flows through the system. It carries the resistive heating power P h = R Ih 2 to the Cu heat sink: J h = P h A (9) - 9 -

13 3 Experimental Method The quantity A = l HB w HB is the area of the Hall bar (see Fig. 5a). The connection between heat flux, heat conductances and the thermal gradients is given by (Ref. [9]): J h = K YIG T YIG T YIG = T h T d YIG (1a) J h = K sub T sub T sub = T T c d sub (1b) The quantities d YIG and d sub are the thicknesses of the YIG layer and the substrate, respectively. Because the heat conductances are assumed to be spatially constant, the thermal gradients in each layer are constant, too. Combining Eqs. (9), (1a) and (1b) the temperatures T and T h are calculated as: T h = R ( A dyig d ) sub Ih 2 K YIG K + T c (11a) sub T = R A dsub K sub I 2 h + T c (11b) Finally, the gradient in YIG is given by the following equation: T YIG = 1 d YIG + K YIG K sub d sub (T h T c ) K sub K YIG Th T c d sub (12) The approximation holds since d YIG d sub (see Tab. 1). With this equation T YIG can be computed, if the temperatures T c and T h as well as the heat conductances are known. The latter are tabulated in Tab. 2 for room temperature. The temperature of the heat sink T c is controlled as described in Sec T h could be computed using equation (11b), which, however, is derived with the assumption that the total heating power flows off through the sample. Cooling by air was not taken into account. More precise results are obtained by measuring T h directly and, afterwards, calculating the thermal gradient with Eq. (12). K YIG (W/Km) K YAG (W/Km) K GGG (W/Km) 6 9 Ref. [6] Ref. [6] Ref. [1] Table 2 Heat conductivities - 1 -

14 3 Experimental Method G G G /Y IG (5 3.7 )/P t(6.1 ) T e m p e ra tu re T h (K ) H e a tin g c u rre n t I h (m A ) G ra d ie n t Τ Y IG (1 3 K /m ) Figure Calibration measurement of the GGG/YIG(53.7)/Pt(6.1) sample. The temperature T h (left axis) is plotted versus the heating current I h. Additionally the thermal gradient is calculated (Eq. (12)) and scaled on the right axis. 3.3 Measurement of the Platinum Temperature T h Calibration measurements were made with each sample in order to replace equation (11b) by an empirical correlation T h (I h ). For that purpose, the platinum is used as a resistive temperature sensor. The temperature of the Pt Hall bar is assumed to be equal to the temperature T h on the upper boundary of YIG. If the resistance R of the Pt Hall bar is known, the temperature can be easily calculated using the following equation 2 : T h = 26.69K R R RT + 2.1K (13) This formula does not contain the absolute value of R, but the change in R relative to the resistance R RT at room temperature. This is useful, since the absolute resistance varies between the different measured samples due to different thicknesses of the Pt film, whereas the relative change in temperature can be assumed to be the same. In order to measure the resistance R, the longitudinal voltage V (Fig. 5a) was recorded while I h is gradually increased. The used measurement device is the Keithley 2 SourceMeter, that also provides I h. The four point measurement eliminates contributions from parasitic contact resistances. R (I h ) is calculated using Ohm s law. The time and the increment of the steps of gradual increase of I h must be chosen adequately to make sure that the temperatures have stabilized after each step into a steady 2 The formula was obtained by fitting data from Thermocouple Instruments Ltd., Pt1 Resistance Table, Technical Data Sheet TD-TV/PT1A, (1999). The data contain values of the Pt1 resistance at different temperatures

15 3 Experimental Method state 3. Applying Eq. (13), the data get converted into T (I h ) and T YIG. A typical result is shown in Fig.. It shows the expected quadratic dependency (see Eq. (11b)). Now, the thermal gradient T YIG can be computed according to equation (12). 3. Magnetisation Control by an External Field The magnetisation M of the YIG layer is controlled by an external field H ext. For the measurements at room temperature, the sample is placed between two pairs of coils (Fig. 9), which generate the in plane external field. This setup was provided by Michael Schreier, Niklas Roschewsky and Akashdeep Kamra. On the one hand, this 2D vector magnet arrangement allows sweeps in magnitude in the range H ext = ±1mT. The range provides a complete sweep of the magnetic hysteresis of the YIG, since the coercive field of YIG at room temperature is about µ H c = 3mT. On the other hand 36 degree in plane rotations are possible by superposition of the vertical and the horizontal magnetic field. The magnitude of the vertical and the horizontal field components are separately regulated by LakeShore Gaussmeters. For the measurements in the cryostat a magnetic field up to 2T is provided by a single pair of superconducting coils. A rotatable sample mount enables 36 degree in plane rotations. The measurements in the cryostat were supervised by Sibylle Meyer. Coils Sample Figure 9 2D vector magnet. The superposition of the fields generated by the vertical and the horizontal pairs of coils allows field rotations. 3 To check if time and increment is chosen adequately,v is not only measured while increasing the heating current I h but also while decreasing it. If both give the same results, the heating must have been quasi-static. The value is referred from []

16 Measuring the Spin Seebeck Voltage V SSE The simultaneous usage of the Pt Hall bar as heater and SSE detector via ishe is not without its own challenges though. The transversal signal V, however, does not only include the SSE voltage V SSE, but also electrical effects V el and an offset V: V = V SSE + V el + V (1) The origin and the handling of these additional components is discussed in the following. V is measured by a Keithley 212 nanovoltmeter..1 Reduction of the Offset Voltage Since, for the resistive DC heating method, the heating current I h has to be sourced into the platinum to generate the thermal gradient, the longitudinal voltage V drops across the Hall bar. However, since there is a small unintended displacement between opposite contact pads, the pads are not on the same longitudinal potential (see Fig. 1). This potential difference causes an offset V in the transversally measured voltage V of the order of a few 1mV at a heating current of a few 1mA. A precise detection of the SSE voltage V SSE µv, however, requires the lowest range (1mV) of the nanovoltmeter 1 to be used. Because the offset V is generally out of this range, it has to be compensated. 1 The Keithley 212 nanovoltmeter (7.5 digits) provides a resolution of 1nm in the range 1mV. This yields a precise determination of the SSE voltage V SSE µv

17 Measuring the Spin Seebeck Voltage V SSE ΔV I h Figure 1 Closeup of the Hall bar. The displacement between opposite tabs is displayed disproportionately. Due to the heating current I h the offset V appears between them. R f R p R' p V I h R ll R' ll V SSE V el R Figure 11 Equivalent circuit. The longitudinal resistance is decomposed into the resistances R and R. V SSE and V el are represented by ideal voltage sources. The transversal resistance R represents the internal resistance of those voltage sources. However, no current flows through the transversal measurement circuit, because the internal resistance of the voltage measurement device (V ) is quasi infinite, so R is not of interest. The offset reduction is implemented by 3-point contacting and a compensation circuit consisting of two ten-turn potentiometers with the resistances (R p + R p ) and R f. One of them is for raw (R p + R p = 1KΩ) and the other for fine (R f 1kΩ) adjustment. The total resistance of both potentiometers is chosen to be much larger than the resistance of the Hall bar R HB 2Ω, in order to avoid a current leak that would reduce the heating power. An equivalent circuit diagram is displayed in Fig. 11, where the Hall bar is locally replaced by the resistances R and R

18 Measuring the Spin Seebeck Voltage V SSE According to Kirchhoff s laws the transversal voltage V in this configuration is given by with ( Rp + R f V = V SSE + V el + C R ) I h C = R p R } {{ } = V R R p R f + R p + R p + R + R (15) (16) The offset vanishes if the ratios of the resistances are equal: V = R p + R f R p = R R The manual adjustment is not precise enough to comply with this compensation condition, but the circuit allows a sufficient reduction up to the point where the transversal voltage is within the required range..2 Electric Effects and Data Reduction Besides the thermal SSE, there are additional electric effects that arise in a currentcarrying metal. However, in this constellation thermal and electric effects differ in one essential property: V SSE ( I h ) = V SSE (+I h ) (17a) V el ( I h ) = V el (+I h ) (17b) If the direction of the current I h gets inverted, the SSE remains unchanged, whereas the electric effects change their signs. Equation (17a) can easily be understood, because the SSE depends on the direction of the thermal gradient, which does not change if the heating current is inverted. The temperature T h is not affected by a change in the sign of I h as it goes quadratically as a function of I h. Equations (11b) and (12) yield: V SSE (I h ) T SSE Ih 2 The electric effects show another symmetry (Eq. (17b)). They are based on the direction of the conduction electrons movement and therefore they depend on the direction of the current I h. V el I h To exemplify, let us consider the conventional Hall effect. Due to the manual alignment of the sample in the magnetic field and due to possible magnetic stray field of

19 Measuring the Spin Seebeck Voltage V SSE the magnetised YIG, there is a component H of the magnetic field perpendicular to the Hall bar. As a consequence, the transversal Hall voltage [9] occurs: V el,h = µ R H d Pt H I h (1) The quantities R H and d Pt are the Hall resistance and the thickness of the Pt Hall bar, respectively. The Hall voltage shows the symmetry of equation (17b). In the Pt/YIG system the major electric effect affecting the transversal voltage is the spin Hall magnetoresistance (SMR). A presentation of the mechanism of SMR would go beyond the scope of this thesis (for more detailed information see Refs. [, 3, 11]). However, SMR shows also the symmetry of equation (17b): V el,smr I h (19) The symmetrical properties of V SSE and V el can be used as follows to separate the components of the transversal voltage V. All measurements have to be made with both directions of the DC heating current +I h and I h separately. Equation (17b) then yields: V (+I h ) = V SSE (+I h ) + V el (+I h ) + V(+I h ) V ( I h ) = V SSE ( I h ) + V el ( I h ) + V( I h ) = V SSE (+I h ) V el (+I h ) V(+I h ) Afterwards, the data are summed, which reduces them to the thermal SSE voltage: V SSE (±I h ) = 1 2 (V (+I h ) + V ( I h )) (2) Due to the addition, the thermal SSE voltage V SSE gets isolated, because the electric components are cancelled out. The offset cancels too, since it is also an electric effect that is directly proportional to the heating current (see Eq. (15)). Nevertheless, the compensation of V is necessary to get V into the proper range. Of course, the electric effect can also be isolated out of data by subtraction: V el (+I h ) + V(+I h ) = 1 2 (V (+I h ) V ( I h )) (21) Altogether, the measurement V SSE with the DC resistive heating method requires to perform all measurements twice. All the magnetic field hysteresis sweeps and all the magnetic field rotations have to be done with both +I h and I h. The reduction of the raw data from the measured transversal voltage V to the SSE voltage V SSE is exemplified at the beginning of the next chapter

20 5 Experimental Results In this chapter the experimental results are presented and described. The discussion follows in Ch Raw Data and Field Sweeps I G G G /Y IG (5 3.7 )/P t(6.1 ) h = 1 5 m A M a g n e tic fie ld µ H e x t (m T ) R = 3 6 Ω ,2 +,2 u p s w e e p +,1 5 d o w n s w e e p +,1 5 T ra n s v e rs a l V o lta g e V (m V ) S S E V o lta g e V S S E (µv ) +,1 +, 5 +, -, -, 5 -,1 -,1 5 -,2 -, = +,1 V (+ I h ) +, 5 +, -, -, 5 -,1 -,1 5 V -,2 (-I h ) -, V S S E = (V 2 (+ I h ) + V (-I h )) M a g n e tic fie ld µ H e x t (m T ) Figure 12 Raw data V (+I h ) and V ( I h ) of a H ext -sweep are plotted in the upper diagram. The lower diagram shows the reduced data V SSE. The transversal voltage is of the order of V mv, whereas the SSE voltage is of the order of V SSE µv. Figure 12 shows a H ext -sweep. The field is oriented along the x-axis (see Fig. 5a), the longitudinal direction of the Hall bar. The measurement was done with the GGG/YIG(53.7)/Pt(6.1) sample at room temperature. In the upper panel the raw data V (+I h ) and V ( I h )

21 5 Experimental Results are plotted. First the different components of V (see Eq.(15)) shall be outlined. At the maximal negative field µ H ext = 1mT, there is the offset of about V(±I h ) ±.1mV for both current directions, which remains after reducing it with the compensation circuit. The field is swept up (black symbols) and in the region of the coercive field, a peak occurs due to the SMR. For large positive field the transverse voltage is constant again. The down sweep shows the same behaviour with the peak at the negative coercive field. The peaks are of the order of V el 1µV. If one compares V (+I h ) and V ( I h ) the SMR shows the symmetry of an electric effect (see Eq. (17b)). After the reduction of the data according to Eq. (2), the electric components cancel. The remaining thermal SSE voltage is plotted in the lower panel of Fig. 12. As a function of the external magnetic field, V SSE is on a constant negative level V SSE µv for high negative H ext, then in the region of the coercive field it switches sign to a positive constant level V SSE µv for large positive H ext. This behaviour is to be expected, since the magnetisation M is inverted by α = 1 during the field sweep and as a consequence the SSE voltage changes its sign (compare Eq. (7)). Because the SSE component V SSE is much smaller than the other components V and V el, it is hardly visible in the raw data V. In Fig. 13 the electric components V el + V of the transversal voltage V are plotted. The values are obtained by subtracting the raw data V (+I h ) and V ( I h ) (see Eq. (21)), whereby the thermal SSE voltage cancels. Obviously, there is hardly a difference between the raw data and the reduced electric components. G G G /Y IG (5 3.7 )/P t(6.1 ) I h = 1 5 m A R = 3 6 Ω V e l + V (m V ),1,1 6,1,1 2,1,, 6, u p s w e e p d o w n s w e e p 1 V e l + V = (V 2 (+ I h ) V (-I h )) M a g n e tic fie ld µ H e x t (m T ),1,1 6,1,1 2,1,, 6, Figure 13 The diagram shows the electric components V and V el of V from the same H ext-sweep as displayed in the upper panel of Fig. 12. They are obtained by subtracting the the measured voltages for both current directions V (+I h ) and V ( I h ). Figure 1 shows such H ext -sweeps in different samples. Here the data are already reduced to the SSE voltage. All samples show the same behaviour for high fields. First, the value of V SSE at high fields where the magnetisation of the YIG layer is saturated is determined. To this end the curves are fitted with a hyperbolic tangent: ( ) µ H ext ± C V SSE (µ H ext ) = V SSE tanh (22) D While this fit function may not neccessarily reproduce the exact shape of V SSE, it allows to quantify the magnitude of the signal from computer based analysis. Another - 1 -

22 5 Experimental Results possibility to determine the magnitude of V SSE is to read the value at large magnetic field H ext directly out of the data. But this method implies a large error, because the measurements show noise of the order of about 2µV. That is why the fit given above is used. It averages the values and makes the data analysis uniform and more precise. However, it also involves a systematic error: for some samples a peak occurs during the change of magnetisation (see H ext -sweeps in the samples GGG/YIG(53)/Pt/(.5) (green symbols) and GGG/YIG(6.7)/Pt(11.2) (blue symbols) in Fig. 1). This peak is also taken into account by the fit function and, consequently, the values at high fields are not reproduced exactly. The systematic error is clearly visible at the measurement of GGG/YIG(6.7)/Pt(11.2) (blue symbols) in Fig. 1. The fitted curve lies at large fields about.5µv above the measured level. So, for further investigations, it is indispensable to elaborate another way of data analysis to avoid the systematic error. The data shown in Fig. 1 are all recorded with the same heating current I h = 1mA at room temperature T c = 293K, but apparently they all differ in the magnitude of V SSE. This can be explained with the different resistances R of the Pt Hall bar, which directly affect the heating power and the thermal gradient. M a g n e tic fie ld µ H e x t (m T ) I h = 1 m A T c = K G G G /Y IG (5 3 )/P t(.5 ) d o w n s w e e p - u p s w e e p G G G /Y IG (6.7 )/P t(1 1.2 ) R = 9 Ω - S S E V o lta g e V S S E (µv ) - G G G /Y IG (5 3.7 )/P t(6.1 ) R = Ω - - Y A G /Y IG (5. )/P t(6. ) R = Ω - R = Ω M a g n e tic fie ld µ H e x t (m T ) Figure 1 Field sweeps with different samples. The data were recorded at room temperature T c = 293K using the same heating current magnitude I h = 1mA in all cases. The solid lines represent a fit to the data via. Eq. (22)

23 5 Experimental Results 5.2 Spin Seebeck Voltage as a Function of the Thermal Gradient To obtain the spin Seebeck voltage V SSE as a function of the thermal gradient T YIG, the H ext sweeps are recorded for different heating currents I h. The temperature T h and the thermal gradient are determined with the calibration measurements as described in Sec. 3.2 and 3.3. A typical measurement is displayed in Fig. 15. As expected, V SSE increases with higher T h. Repeating the procedure and fitting the data to Eq. (22) then gives V SSE ( T YIG ) for each sample. The results are plotted in Fig. 16. Obviously, one sample (YAG/YIG(5.)/Pt(6.)) shows an enormous spin Seebeck effect, which will be discussed later. G G G /Y IG (5 3.7 )/P t(6.1 ) M a g n e tic fie ld µ H e x t (m T ) R R T = Ω T c = K I h = 1 5 m A T h = 3 6. K - - S S E V o lta g e V S S E (µv ) - I h = 9 m A I h = m A T h = K T h = K M a g n e tic fie ld µ H e x t (m T ) Figure 15 H ext -sweeps with different heating currents. The SSE voltage V SSE increases with the platinum temperature T h and hence the thermal gradient in the YIG. 5.3 Spin Seebeck Effect at low Temperatures T c For the sample YAG/YIG(5.)/Pt(6.) field sweeps were also recorded in the magnet cryostat at different heat sink temperatures T c. The measurements were made with the support of Sibylle Meyer. Some of them are plotted in Fig. 17. All measurements were done with I h = ±1mA. Obviously, the magnitude of V SSE changes as a function of T c. But here a quantitative discussion is not possible, because the thermal gradient could not be computed due to unknown parameter values. The temperature dependence of K YIG and K sub are not known and, additionally, the correlation T h (R) of the Pt resistance given in Eq. (13) fails for low temperatures. The correlation yields T h < T c, which is physically not possible, since the Pt layer is heated not cooled

24 5 Experimental Results S S E V o lta g e V S S E (µv ) ,,5 1, 1,5 2, 2,5 3, 3,5 G G G /Y IG (5 3.7 )/P t(6.1 ) G G G /Y IG (6.7 )/P t(1 1.2 ) G G G /Y IG (5 3 )/P t(.5 ) Y A G /Y IG (5. )/P t(6. ) T Y IG (µk /Å ) T Y IG (1 3 K /m ) Figure 16 SSE voltage at different thermal gradients. Y A G /Y IG (5. )/P t(6. ) I h = 1 m A M a g n e tic fie ld µ H e x t (m T ) S S E V o lta g e V S S E (µv ) d o w n s w e e p T c = 1 K u p s w e e p - - T c = 5 K - - T c = 1 K - - T c = 1 5 K M a g n e tic fie ld µ H e x t (m T ) Figure 17 Field sweeps at different base temperatures T c in the cryostat. It was shown in Sec. 3.2 that that V SSE is directly proportional to the dissipated electrical power, which, in turn, is a function of the sample resistance R. Since R itself is a function of temperature, it is reasonable to normalize the measured data to the dissipated heat. This is plotted in Fig. 1. Obviously, V SSE decreases strongly for temperatures below 15K. However, a quantitative discussion is not possible, because a change in the thermal conductivities can not be qualitatively taken into account as already mentioned above

25 5 Experimental Results, , 1 2 I h = 1 m A, 1, 1 V S S E /P h (V /W ),, 6,,, 6,, 2 Y A G /Y IG (5. )/P t(6. ), 2,, H e a t s in k te m p e ra tu re T c (K ) Figure 1 Values for V SSE /P h measured at different heat sink temperatures T c. 5. Angular Dependence of the SSE So far, all measurements presented were field sweeps. Figure 19 shows a in plane rotation of the external magnetic field in the cryostat at T c = 1K. The magnitude of the field is µ H ext = 2T and not 1mT as in the sweeps above to check out whether V SSE is depending on the magnitude of the external magnetic field or not. The rotation was done in both directions, clockwise and anti-clockwise. V is plotted for both directions of the heating current in the upper panel. It is dominated by the SMR, which has the functionality of sin 2 (α) (Ref. [, 3, 11]). After the raw data V (+I h ) and V ( I h ) have been summed up and divided by two, the thermal SSE voltage remains. The arithmetic average of both directions of rotation is plotted in the lower panel. The SSE voltage V SSE shows the expected cos(α) dependence (see Eq. ). On this specific sample, V SSE apparently does depend on the magnitude of the external field H ext, the value of V SSE determined by the H ext -sweep (α = ) is about 2µV at 1K and 1mA (see Fig. 17), whereas, at the same conditions, the maximum of the cos in the field rotation (α = ) is about 3µV. Further investigations have to be done, to examine, if this effect is sample specific or not

26 5 Experimental Results Y A G /Y IG (5. )/P t(6. ) I h = 1 m A T c = 1 K µ H e x t = 1 T A n g le α , + 2, T ra n s v e rs a l V o lta g e V (m V ) S S E V o lta g e V S S E (µv ) + 2, 7 + 2, 6 + 2, 5-2, 5-2, 6-2, 7-2, V (+ I h ) V (-I h ) V S S E = (V (+ I h ) + V (-I h )) + = A n g le α , 7 + 2, 6 + 2, 5-2, 5-2, 6-2, 7-2, Figure 19 Field rotations. The upper diagram shows the transversal voltage with a sin 2 (α) angle dependence. The lower diagram shows the SSE voltage, which is obtained after data reduction. It shows the expected cos(α) dependence

27 6 Discussion 6.1 Consistency Check The primary purpose of all these measurements is to check out, whether the signal, that is supposed to be the SSE voltage, does show the properties of the SSE or not. This is essential, because the experimental method (current heating) is applied for the first time in measuring the SSE, and the assumption, that the voltage remaining after data reduction is the SSE voltage, could be wrong. Since spin currents can hardly be measured directly, a verification is not possible, but other effects can be falsified and excluded. First of all, the fact, that the measurement works with all four samples (see Fig. 1), shows, that the effect is not a sample specific occurrence, but rather a general effect. Any resistive electric effect can be excluded, since it would cancel due to its symmetry after the data reduction. If the signal was an artefact from the data reduction due to differences between V el (+I h ) and V el ( I h ) caused by measuring inaccuracies, the measurements would not be reproducible. As a consequence, the measured signal must be a thermal effect. A thermal effect only appears, if there is an applied thermal gradient. So the effect must vanish, when T YIG goes to zero, as confirmed by the measurements (see Fig. 16). Since the voltage depends on the magnetisation, it must be a magnetic effect. The conventional Seebeck effect for example can therefore be excluded. To restrict the possible thermal effects even further, a measurement with an additional sample was done. This sample also stems from the set of Matthias Althammer and Sibylle Meyer and it has an additional aluminium oxide (AlOx) layer between the YIG and the Pt layer. Figure 2 shows the H ext -sweeps that were done with this sample. In comparison to the measurements with the other samples (see Fig. 1) there is no sizeable SSE voltage V SSE at all. As a consequence it can be concluded that the signal measured with the other samples must be due to the spin interaction of the NM with the FM. The insulating AlOx layer prevents the spin transfer. The measured signal also shows the expected cos(α) dependence on the direction of the magnetisation (see Fig. 19). Consequently, it is assumed that the measured voltages indeed stem from SSE. A summary of the argumentation is displayed in Tab

28 6 Discussion T c = K Y A G /Y IG /A lo x /P t M a g n e tic fie ld µ H e x t (m T ) R = 3 1 Ω S S E V o lta g e V S S E (µv ) I h = 1 m A 2 1 d o w n s w e e p u p s w e e p -2 I h = 1 2 m A I h = 6 m A M a g n e tic fie ld µ H e x t (m T ) Figure 2 Field sweeps with the (YAG/YIG/AlOx/Pt) sample at different heating currents. Obviously, the insulating AlOx layer suppresses the interface mechanism. The SSE voltage is zero. Effect Argument Measurement sample specific same results at different samples Fig. 1 electric effect exclusion by data reduction - magnetic effect dependence on the magnetisation Fig. 1 thermal effect dependence on thermal gradient Fig. 16 FM/NM interface effect AlOx layer suppresses the signal Fig. 2 angle dependence of SSE cos(α) dependence (compare Eq. ()) Fig. 19 Table 3 Summary of falsified ( ) and verified ( ) aspects. 6.2 Quantitative Data Analysis This chapter is focused on the data containing V SSE as a function of the gradient T YIG (see Fig. 16). So far, the measured value was always the SSE voltage V SSE. However, the effect that has to be evaluated is the interface process and V SSE is not a quantity that is directly correlated with the interface process but indirectly via inverse spin Hall effect. Due to the ishe there are further temperature dependent quantities that affect the SSE voltage, for example the specific resistance ρ = ρ(t h ) (see Eq. (7)). So, to investigate the dependence of the interface process on the thermal gradient in the YIG, it is more insightful to compute the spin current density J s, that causes V SSE

29 2 6 Discussion via the ishe. The absolute value of J h per unit area can be obtained by converting equation (7): J s = I h A = h ( 2 1 e w HB θ SH ) VSSE ρ The term in parenthesis is assumed to be constant, the temperature dependence (Ref. [12]) of the spin Hall angle is neglected, instead the value θ SH =.37 (Ref. [13]) at room temperature is used in the calculations. The specific resistance of Pt is a function of the temperature. It can be calculated as: (23) ρ(t h ) = ρ R(T h) R RT (2) ρ is the specific resistance at room temperature. It is useful to express it as a function of the thermal gradient T YIG, which can easily be achieved by combining Eqs. (12) and (13): K YIG d ρ( T YIG ) = ρ K sub T YIG + T c 2, 1K sub 26, 69K (25) Using this equation and Eq. (23) J s is calculated from the measured data (Fig. 16). The result is plotted in Fig. 21. The dimension of a spin current density is J/m 2, it is the flow of h/2 per unit time unit area. T Y IG (µk /Å ) ) S p in c u rre n t J s (1 - J /m,,5 1, 1,5 2, 2,5 3, 3,5 2, 2, G G G /Y IG (5 3.7 )/P t(6.1 ) 1, G G G /Y IG (6.7 )/P t(1 1.2 ) 1, 1,6 G G G /Y IG (5 3 )/P t(.5 ) Y A G /Y IG (5. )/P t(6. ) 1,6 1, 1, 1,2 1,,,6,,2 1,2 1,,,6,,2,, T Y IG (1 3 K /m ) Figure 21 Spin current in each sample as a function of the thermal gradient T YIG : the data were obtained by calculating J s with Eqs. (23) and (25) and the measured data of the V SSE (see Fig. 16). Evidently, all samples except YAG/YIG(5.)/Pt(6.) (orange symbols) show at T YIG = K/m a spin current density of about J s.2 1 J/m 2. In order to get closer to the interface process, one has to think about the mechanism that generates the spin current. There is a spin and energy transfer from the platinum

30 6 Discussion to the YIG. The spin of an electron in the Pt flips from up to down and transfers the spin h to a magnon in the YIG. Due to this process, the density of spin up electrons at the interface decreases, whereas the the density of spin down electrons increases, as a consequence, the (diffusive) spin current I s appears. So, the number Ṅ s of spin flips per unit time and unit area can be calculated as follows: Ṅ s = J s h (26) Furthermore, with the lattice constant 1 of Pt of about a Å (Ref. [9]), the number of spin flips per unit time and per Pt atom at the interface is about Ṅ s a s for T YIG = K/m. Now, it is focused on the the heat flux J h s, that goes with the spin transfer from Pt to the YIG. It is given by the following equation: J h s = Ṅ s E (27) E is the energy transfer from a electron to a magnon. According to Ref. [1], E is given as: E = k B T h (2) k B is the Boltzmann constant. The results are plotted in Fig. 22. Here the data are fitted with lines through the origin: J h s = κ i s T YIG (29) This equation is the analogue to the normal heat flux relation through a bulk material (compare Eq. (1a)). κ i s is the heat conductance via spin transfer. The values that have been obtained by the fits are listed in Tab.. Additionally the ratio of spin heat current J h s to the total heat current J h is calculated for each data point of Fig. 22) using Eq. (1a). The average values of J h s /J h for each sample are tabulated in Tab. sample κs(w/km) i Js h /J h GGG/YIG(53.7)/Pt(6.1) GGG/YIG(6.7)/Pt(11.2) GGG/YIG(53)/Pt(.5).6.1 YAG/YIG(5.)/Pt(6.).5.65 Table Results of the quantitative data analysis. The heat conductances κ h s of each sample are obtained from the linear fit (Eq. (29)). J h s /J h is the average ratio of the spin current to the total heat flux. To better illustrate matters, the original heat flux scheme (Fig. 7) can be extended. This is displayed in Fig. 23. Here, the Pt/YIG interface is modelled with two additional branches. Those branches replace the phonon coupling by κph i and the spin 1 The lattice constant at the interface would be different to the one in the bulk, since it is grown on YIG with a lattice constant of about 12Å (Ref. [])

31 6 Discussion T Y IG (µk /Å ) ) S p in h e a t c u rre n t J h s (k W /m 2,,5 1, 1,5 2, 2,5 3, 3, G G G /Y IG (5 3.7 )/P t(6.1 ) G G G /Y IG (6.7 )/P t(1 1.2 ) 1 9 G G G /Y IG (5 3 )/P t(.5 ) 9 Y A G /Y IG (5. )/P t(6. ) T Y IG (1 3 K /m ) Figure 22 Clculated values of the spin heat current J h s for different thermal gradient in the YIG. coupling by κ i s. With the data given in Tab. it can be assumed, that about 15% of the total heat flux J h flows through the spin heat channel across the interface. The rest must go via phonon heat transport. Obviously, for the YAG/YIG(5.)/Pt(6.) sample, the value for J h s /J h is much larger. A possible reasons could be, that the phonon heat channel is very bad (κ i ph κi s). This would mean that the main part of the heat must flow through the spin heat channel across the interface. The calculations presented above were made with many simplifications: the spin diffusion (Eq. ()) and the temperature dependence of the spin Hall angle θ SH (Eq. (23)) were disregarded. Furthermore, the thermal gradient was computed assuming one dimensional heat flow and spatial constant total heat conductivities (Sec. 3.2) and the temperature T h was determined using the linear correlation (13), etcetera. However, the purpose of this section is not to produce exact quantitative results, but rather to convey a quantitative notion of the SSE. Since the original cause of the heat and spin transfer is a difference between the electron and magnon temperature (Ref. [6]), it would be more reasonable to compute the heat conductances κph i and κi s according to that difference and not according to the thermal gradient T YIG as done above using the fit function (Eq. (29)). But a determination of the electron-magnon temperature profile would go beyond the scope of this thesis. The quantities Ṅ s, J s and Js h /J h, in contrast, are more reasonable, because they were computed without involving the temperature profile

32 6 Discussion In summary, there are two accesses to the interface process. The first one is the SSE, which allows measuring the spin current I s and the spin-flip-rate Ṅ s. This is symbolized by a spin-current measuring symbol in the spin heat channel (Fig. 23). The second access is the total heat flux J h and the temperature profile. It can be calculated by measuring T h and T c. With known heat conductances K YIG and K sub a value for κs i can be calculated, by measuring the temperature T h. For example, if the spin heat channel broke down due to a complete demagnetisation of the YIG, then it must be visible in a increase of T h, since about 15% of the heat conductance across the interface drops out. A simultaneous measurement of both, the spin current and the heat flux, could lead to an experimentel determination of the energy transfer E (see Eq. 27). P h =R I h 2 Pt T h I s J ph h κ i ph J s h κ i s Κ YIG T' J h Κ Sub T c Figure 23 The heat flux scheme is extended by two additional branches with κ i ph and κi s, which replace the phonon and the spin heat conductance across the interface, respectively. Those heat channels carry the heat current J ph h and Js h. 6.3 Change of Magnetisation So far V SSE has only been discussed in the high field limit, but not during the change of the magnetisation. In Fig. 2 a H ext -sweep (α = ) at room temperature is plotted. The levels at high fields are more or less constant, but in the region of the coercive field a peak occurs. Three possible explanations for that peak are discussed in the following. The first and easiest is, that the peaks are artefacts from the data reduction. This could be caused by a difference in the magnitude of V (+I h ) and V ( I h ) due to measurement errors. Because the peaks show high symmetry and a are reproducible, a measurement error caused by noise is improbable, instead, it is rather a unknown systematic error. Investigations has to be done to find such a systematic error. The second explanation is, that the heat conductance of YIG decreases, because its