Field-dependent thermal conductivity and Lorenz number in Co/Cu multilayers

The influence of inelastic scattering mechanisms on the giant magnetoresistance (GMR) effect has been controversially discussed. This issue can be addressed, for example, by comparing charge currents with heat currents in magnetic multilayers that exhibit GMR. The interpretation of such experiments is based on the Wiedemann-Franz law. Due to the fact that this law only holds for elastic scattering, experimentally observed deviations from this law are usually attributed to the presence of inelastic scattering and vice versa. We develop two simple models to demonstrate that this interpretation can lead to wrong conclusions in the context of the GMR effect. We employ a measurement technique that is based on the so-called 3$\omega$ method to study the in-plane electrical and thermal conductivities of a Co/Cu multilayer in dependence on magnetic fields over a wide range of temperatures. Our experimental results indicate that the Wiedemann-Franz law holds over the temperature range investigated. Using the simple models developed, we conclude that the influence of electron-magnon scattering on the GMR effect in the Co/Cu multilayer is negligible. We further conclude that electron-phonon scattering is characterized by the same spin asymmetry as the predominant elastic scattering of electrons at interfaces in the Co/Cu multilayer.

Figure 1

In the presence of electron-magnon scattering, the Lorenz number $L$ of magnetic multilayers can depend on the magnetic configuration (parallel: $L_{\mathrm{P}}$, antiparallel: $L_{\mathrm{AP}}$). The curves show the ratio $L_{\mathrm{P}}/L_{\mathrm{AP}}$ according to Eq. (5) as functions of the significance of electron-magnon scattering that is quantified by the ratio ${\rho }_{\mathrm{sf}}/{\rho }_{\mathrm{M}}$.

Figure 2

In the presence of electron-phonon scattering, the Lorenz number $L$ of magnetic multilayers can depend on the magnetic configuration (parallel: $L_{\mathrm{P}}$, antiparallel: $L_{\mathrm{AP}}$). The curves show the ratio $L_{\mathrm{P}}/L_{\mathrm{AP}}$ according to Eq. (7) as functions of the significance of electron-phonon scattering that is quantified by ${\rho }_{\mathrm{M}}^{\mathrm{i}}/{\rho }_{\mathrm{M}}^0$.

Figure 3

Illustration of (a) the thin-film stack and (b) the heater geometry for 3$\omega$ measurements. The thin-film stack (magnetic multilayer, insulating layer, heater lines) is deposited on a fused silica substrate. Several heater lines are fabricated directly on the substrate, and above the Co/Cu multilayer. The lengths $l$ of the heater lines are 2 mm; the widths $2b$ of the heater lines are 5 and 10 $\mu \mathrm{m}$.

Figure 4

In-phase (circles) and out-of-phase (triangles) components of the temperature oscillations of a heater line directly fabricated on a fused silica substrate, plotted against the logarithm of the frequency of the heating current. The fit curves (solid lines) are obtained using the model explained in the main text. The measurement was performed at a temperature of 300 K.

Figure 5

In-phase (circles) and out-of-phase (triangles) components of the temperature oscillations of a heater line fabricated above a Co/Cu multilayer, plotted against the logarithm of the frequency of the heating current. The fit curves (solid lines) are obtained using the model explained in the text. The dashed lines show a simulation of the temperature oscillations without Co/Cu multilayer below the heater line. The measurement was performed at a temperature of 300 K.

Figure 6

(a) Field dependence of the 3$\omega$ voltage $U_{3\omega }\left(H\right)$ measured on a heater line, the resistance of which was 150 $\Omega$. The imposed heating current was 5 mA. (b) Field dependence of the thermal conductivity of the Co/Cu multilayer ${\kappa }_{\text{meas}}$ deduced from the data shown in (a). The thermal conductivity changes due to the giant magnetothermal resistance effect. (c) Field dependence of the electrical resistivity $\rho \left(H\right)$ of the Co/Cu multilayer. (d) Thermal conductivity $\kappa (1/\rho )$ of the Co/Cu multilayer, plotted against its electrical conductivity $\sigma =1/\rho$. The resulting curve is linear over the field-range of the giant magnetoresistance effect (between about $-$250 and 250 mT), and slightly deviates from linearity at saturation fields. The measurements were performed at 300 K.

Figure 7

Flow chart of the measurement procedure described Sec. 4.

Figure 8

Experimental results measured on a Co/Cu multilayer over the temperature range between 10 and 300 K (full black dots). (a) Electrical resistivity for the parallel magnetic configuration, ${\rho }_{\mathrm{P}}$, and for the antiparallel one, ${\rho }_{\mathrm{AP}}$. (b) Electronic thermal conductivity for the parallel magnetic configuration, ${\kappa }_{\mathrm{P}}$, and for the antiparallel one, ${\kappa }_{\mathrm{AP}}$. The data points represent the mean value averaged over the results from four heater lines. The error bars indicate the standard deviation. (c) Giant magnetothermal resistance ratio, $({\kappa }_{\mathrm{P}}-{\kappa }_{\mathrm{AP}})/{\kappa }_{\mathrm{P}}$ (full black dots) with error bars according to the standard deviation shown in (b), and giant magnetoresistance ratio, $({\rho }_{\mathrm{AP}}-{\rho }_{\mathrm{P}})/{\rho }_{\mathrm{AP}}$ (white triangles). (d) Electronic Lorenz number $L$. For comparison, we show results from Tsui et al. on an epitaxial [Co(7 ML)/Cu(19 ML)]${}_{215}$ multilayer[12] and from Yang et al. on a [Cu(2.1 nm)/CoFe(1.2 nm)]${}_{40}$ multilayer.[36]

Figure 9

Electronic thermal conductivity, $\kappa$, plotted against the product of temperature, $T$, and electrical conductivity, $\sigma =1/\rho$. Different colors correspond to the respective temperatures during the measurements, as indicated in the legend. At each temperature, the curves are linear due to a constant Lorenz number that does not depend on the magnetic configuration of the Co/Cu multilayer.

Figure 10

Temperature dependence of the thermal conductivity of the fused silica substrate determined from 3$\omega$ measurements.