Recently discovered spin-dependent thermoelectric effects have merged spin, charge, and thermal physics, known as spin caloritronics, of which the spin Seebeck effect is its most puzzling. Here we present a theory of this effect driven by subthermal non-local phonon heat transfer and spectral non-uniform temperature. The theory explains its non-local behaviour from the fact that phonons that store the energy (thermal) and the phonons that transfer it (subthermal) are located in different parts of the spectrum and have different kinetics. This gives rise to a spectral phonon distribution that deviates from local equilibrium along the substrate and is sensitive to boundary conditions. The theory also predicts a non-magnon origin of the effect in ferromagnetic metals in agreement with observations in recent experiments. Equilibration of the heat flow from the substrate to the Pt probe and backwards leads to a vertical spin current produced by the spin-polarized electrons dragged by the thermal phonons.
The key and most intriguing effect in spin caloritronics1,2,3,4,5,6,7,8,9 is the transverse spin Seebeck effect (SSE) in which a thermal gradient in a ferromagnet/substrate structure gives rise to spin currents that vary along the length of the sample and are detected via the inverse spin-Hall voltage10. This effect has been experimentally observed using different ferromagnetic materials: metals1, semiconductors2,3 and insulators4. The magnitude of the SSE is quantified by the transport coefficient , where Vy is the measured ISHE voltage, w is the width of Pt probe, and xT=(T2−T1)/L, where L is the length of the sample, see Fig. 1a. The ISHE voltage is given by , where js is the spin current, s is its polarization, θH is the spin-Hall angle of the probe (in Pt, θH is of the order of one per cent) and ρ is its electric resistivity. The effect is non-local, that is, it depends on the position along the sample rather than the local temperature gradient. In addition, the size of the sample is usually about 1 cm and such a long-range information about position can be transferred only by phonons, propagating along the insulating substrate2. The key role of phonons for the transversal SSE effect was discussed in Adachi et al.5
Here we show that the non-locality of the SSE is a consequence of the non-local energy transfer due to subthermal diffusive phonons that are sensitive to the boundary conditions and give rise to a spectral non-uniform temperature along the sample11,12. In fact, in recent measurements in bilayer F–Pt wire devices, the specific geometry excludes long-ranged propagation of magnons and leaves only phonons as a source of non-locality6. In addition, we demonstrate that while in the insulator the SSE is likely determined by the phonon–magnon mechanism, in the conducting ferromagnet (for example, Ni81Fe19 (ref. 1) and GaMnAs3), the magnon mechanism is not the only one available. The experiments by Jaworski et al.3 were performed on a material with Curie temperature TC=130 K, considerably lower than the Debye temperature θD=350 K, and showed that VISHEM at the vicinity of the Curie point; that is, the SSE signal vanishes with the magnetization M with the same critical behaviour. This latter fact excludes the magnon mechanism for this case: as we have checked, starting from equation 12 of Adachi et al.13, the magnon pumping yields the contribution to the SSE signal, which vanishes as M3/2 (see Supplementary Note 1 for details).
Physical mechanisms of the phonon–electron SSE
The theory of this phonon–electron SSE, which does not involve magnons, has three key physical mechanisms. The first (i) involves the non-local nature of the signal driven by subthermal phonons, which is also relevant for the magnon-phonon mechanism not considered here. In recent measurements of the SSE in insulators14, the temperature difference between thermal magnons and phonons assumed in the current theory13,15 has not been observed, suggesting the necessity of the concept of spectrally non-uniform temperature. This concept originates from the fact that in most dielectrics, and also some semiconductors, the energy transfer is highly non-local11,12 because of the strong dependence of the diffusion coefficient of phonons on frequency: D(ω)ω−4, if the dominant scatterers are point-like16. In the diffusive regime of the experiments the energy relaxation length is given by where the energy relaxation rate is While the thermal phonons, ħω~kBT, are equilibrated, the subthermal low-frequency phonons can deviate from the local equilibrium due to the rapid low-frequency growth of inelastic length lin(ω)=lin(T)(T/ω)5/2, which leads to non-local kinetics. Even the concept of temperature itself is well defined only for phonons of high-enough frequency. For the ‘thermal’ part of the spectrum ħω≥kBT, the distribution function has a Planckian form nT(ω)=()−1 with a local temperature T=T(x). As a result, the phonons that store the energy and phonons that transfer it are located in different parts of the spectrum. As illustrated in Fig. 1b, this spectral separation occurs when lel(T)<<lin(T)<<L, where lel(T)≡lel(ħω=kBT)D(ħω=kBT)T−4, and lin(T)≡lin(ħω=kBT)T−4.5. Then the subthermal phonons whose inelastic length is of the order of (but whose elastic length is much shorter) than L drive the non-local heat propagation along the substrate, giving rise to a steady-state phonon distribution function that deviates from local equilibrium for ħω<<kBT and depends on the position along the substrate. This deviation from local thermal equilibrium in the low frequencies manifest in a T(x) profile that deviates from a linear dependence. To describe this non-local effect, it is essential to formulate the boundary conditions for the equations describing the propagation of the diffusive phonons. (Previously15, the sensitivity to the boundaries entered the theory as a result of the different boundary conditions imposed on phonons and magnons17. This model, however, cannot explain the position dependence of the SSE signal, measured on the sample with a scratched magnet2 or on the bilayer wire device6. Here, instead, we demonstrate that phonons in different parts of the energy spectrum act as the ‘subsystems’ of different sensitivity to the boundaries).
The second (ii) mechanism involves the electron–phonon drag. As the probe is a ‘dead end’, there is a full balance between incoming and outgoing heat fluxes such that net heat flux is zero. However, the incoming and outgoing fluxes have different spectral distributions, because of the inelastic processes in Pt, which average out the spectrum of the incoming flux, and establish a local temperature TPt (x) different from T (x). The spin drag, induced by the phonon flux, is sensitive to the spectral content of the phonon distribution function. Hence, despite zero net heat flux, the spin injection is not zero. In the stationary situation, the drag voltage induced by the phonons is compensated by redistribution of the electron density, so that the total electric current is zero (as well as electrochemical potential gradient). However, in the presence of a spin polarization, there will be a net spin current js=j↑−j↓ polarized along magnetization M: unlike its charge counterpart, spin drag is not blocked by accumulation of the spin density, which is eliminated by the SO interaction in Pt. The magnitude of js depends on the ratio of the thickness of the ferromagnet, dF, and the phonon inelastic scattering length there, . The optimal value of dF for observing the phonon drag SSE is of the order of . For too thin ferromagnet, , the phonons cannot effectively transfer their momentum to electrons to drag them towards the probe. In the opposite limit, , the phonons equilibrate before they reach the region near the probe. An alternative mechanism not considered here is the quantum acoustoelectric pumping18 due to the spectrally non-uniform flux of phonons (O. Tretiakov, K.S.T., J.S., A.M.F., manuscript in preparation).
The final (iii) mechanism involves the conversion of the spin current to an electric signal via the ISHE. This conversion is most optimal if the thickness of the Pt layer is of the same order of magnitude as the spin relaxation length in Pt, which is the case in the discussed experiments19. As shown in detail in the following sections, the resulting theory gives the correct magnitude of the signal, predicts a dependence on magnetization SSSEM, and gives specific temperature and size dependencies that can be tested experimentally.
Subthermal phonon kinetics
On Fig. 1b, we show the spectral phase diagram of frequency regions contributing differently to the kinetics of phonons. There are two characteristic frequencies, ωnl and ωbal, determining the propagation of phonons:
For non-local transport, we require ħωnl=kBT(L/lin(T))−2/5<<kBT. In addition, we will not be interested in phonons in the ballistic part of the spectrum, ω<ωbal. This is legitimate as long as ωbal<<ωnl, and determines a maximum length of the sample, Lmax, given by the point of intersection of the curves lin(ω) and lel(ω), as shown in Fig. 1b. This gives a temperature dependence LmaxT−16/3. For lengths larger than Lmax, the non-local effect is due to the fraction of phonons propagating ballistically and requires a different formalism, which we will not discuss here. The other condition that allows to separate thermal phonons from those which produce non-local effects is lin(T)<<L. This gives a minimum length of the sample LminT−9/2. For length smaller than Lmin even thermal phonons are out of equilibrium and spectral separation does not hold. The large ratio of lin(T)/lel(T) opens the window which we are interested in. Hence, the sample size should be in the range indicated on Fig. 1b. Estimation at T=10 K (when typical phonon energy is 28 K), gives Lmax about few centimetres and Lmin on the scale of milimetres. Recall that the typical size of the sample used for the SSE experiments is 1 cm. With temperature, the width of the region of applicability of the theory behaves as Lmax/LminT−5/6 and we expect it to be relevant up to 50 K. In addition, the temperature is assumed to be much smaller than the Debye temperature, T<<θD, which allows us to ignore Umklapp processes.
With these specific length restrictions, we consider next the theory of propagation of diffusive phonons along the substrate. Owing to the fact that the low-frequency phonons do not primarily interact with themselves but with equilibrated high-frequency phonons, one may use the following kinetic equation that describes propagation of phonons in the insulating substrate, valid for ħω≲kBT:
where δn is the deviation from the local equilibrium
The solution to this second-order differential equation requires two sets of effective boundary condition equations. The first, which establishes T(x) from a given δn(x,ω), is obtained from continuity of the energy density in the system, which in stationary situations reads as .
where ρph(ω)ω2 is the phonon density of states (summed over all branches). The integral for diverges and has to be cutoff at small frequency (the exact value of the cutoff does not enter our results, as the integral for converges). Using equation (2), the energy density continuity equation takes the following form:
This equation should hold for all x. Thus, one has to solve a system of integro-differential equations. For the case of a pulse propagation in an infinite media, the non-local phonon transport has been studied in Levinson11,12 and Wilson and Schaich23. However, we are interested in a stationary solution in the presence of the boundaries which yield the second equation that fully establishes n(ω,x) and T(x).
On the boundary between the substrate and the heater, there is a jump in the phonon distribution function because of the abrupt change in the properties of materials. This leads to a finite thermal boundary resistance (Kapitza resistance), which manifests itself through the jump ΔTK at the contact24. If the scattering in the vicinity of the boundary is mostly elastic, the boundary condition consists of conservation of spectral heat current density across the boundary. It relates the heat flux through the boundary to the jump of the phonon distribution function across it. At the left end of the sample (which is at heat contact with a reservoir at temperature T1) it takes the following form:
The boundary resistance RBd is assumed to be frequency independent. If the heat contacts are in thermal equilibrium, RBd can be related to the thermal boundary conductivity where vs is the averaged sound velocity. Note, that in the absence of the boundary resistance (RBd=0), the locally equilibrated distribution function n(ω,x)=(ω) with T0(x)=T1+(T2−T1)x/L satisfies the kinetic equation, so that δn=0 and the non-local effect vanishes, as the boundaries are effectively at infinity. One may easily see that δnRBd at not too large values of RBd.
With this it is then possible to write down a closed equation for T(x). If the phonon temperature as a function of position x is known, the distribution function can be obtained from equation (2) and reads:
where x<=min (x,x′), x>=max (x,x′). The ‘source’ term nS (ω,x) comes from the boundary condition (equation (6)) and is equal to
Above we have introduced the effective boundary thermal conductance . The second term in equation (7) describes the process of redistribution of phonons along the sample due to diffusion and inelastic scattering. Substituting equation (7) into equation (5), one gets an integral equation for T(x), which can be solved numerically. This procedure is self-consistent: after finding T(x), the distribution function is easily calculated from equation (7). To illustrate the result, we assume the following ratios of characteristic lengths of a thermal phonon
and calculate the correction to the linear temperature behaviour
which is shown on Fig. 2, where we have assumed that RBd=0.1 and T1<T2. Although the deviation from the linear behaviour is small, it ensures the conservation of the energy density of the phonons propagating along the substrate. Ultimately, the non-equilibrium correction δn(ω,x) is responsible for the SSE effect. In Fig. 1b, the frequency dependence of ħωρph (ω)δn (ω,x) is plotted close to the colder end (for x=0.3L). On the hotter end, δn has the opposite sign.
Out-of-plane spin transport
After finding the non-equilibrium distribution function of phonons δn(ω,x), we next concentrate on the heat and spin transport in the vertical direction from the substrate to the probe across the magnet. The temperature of phonons in the Pt probe, TPt(x), is different from the T(x). It is determined by the requirement that the heat flux created by non-equilibrium non-local phonons from the substrate to Pt is compensated by backflow flux of thermal phonons from Pt to the substrate. The resulting temperature difference, δT┴(x)=T(x)−TPt(x), can be found from the heat balance equation:
where δN (ω,x)=nsub−nPt is the difference of the distribution function of phonons entering and leaving the Pt probe, located at x. Here we neglect inelastic scattering of phonons while they pass through the ferromagnetic layer (), and have assumed the sound velocities to be of similar order in the Pt and the substrate. We also assume, that the probe is small enough a<<lel(T), so that the influence of the counterflow on the phonon distribution function in the substrate can be ignored.
It is useful to present δN(ω,x) in the following form:
Then, the temperature difference δT┴(x) can be calculated from the equation:
where h(x) is the dimensionless heat flux supplied to the probe by the non-equilibrium phonons. The function h(x) can be written in a form of where H(x) is a slow function of temperature and boundary resistance RBd. Here ωbal encodes information about scattering of phonons on the disorder and the length of the sample. Function H(x) is plotted on Fig. 3 for the same sample parameters as before.
With this we can finally estimate the scale of the SSE due to conducting electrons, dragged by out-of-equilibrium phonons, in more detail. The guiding idea about the scale of the effect follows from the derivation of the well-known Gurevich formula25,26,27 for the phonon drag. This formula gives for thermoelectric coefficient η=−j/T the following expression: , which is valid when qTl>>1 (here qT=kBT/ħvs is the wavevector of a thermal phonon). For the dirty case qTl<<1, the particle current density dragged to the probe is given by Sergeev and Mitin28:
We write τei, lei for electron-impurity scattering time and length in the ferromagnet. The role of electron-impurity scattering in equation (16) is twofold. It enhances electron–phonon interaction by slowing the motion of electrons (making it diffusive). This is taken into account by the form of W(x). On the other hand, it diminishes the drag effect due to the loss of electron momentum by impurity scattering. The details of the function W(x) depend on the character of the scattering of phonons on defects. We assume that phonons scatter on impurities vibrating with the lattice29 and W(x)=Wvb(x). For temperatures kBT<<ħvs/lei, we may use the asymptotic behaviour Wvb(x<<1)≈x. Recalling that the charge current will be compensated by an unpolarized backflow charge current from the Pt probe, the total spin current is given by the polarized current dragged by the phonons into the Pt probe. Finally, we rewrite the expression for the spin-phonon-drag current injected into the Pt probe as
where Ael(T)=(kBT/εF) (kFlei)2 is a dimensionless constant, determined by electrons, is the level of spin polarization and the dragging factor is
As the spectral densities of the energy and the charge currents are proportional to different powers of the phonon frequency, the electronic drag due to phonons is possible even when the net energy flow is zero. The contribution to J in equation (18) arising due to the temperature difference δT┴(x) between the substrate and the Pt probe (first term in equation (14)) is dominant. In other words, while the non-locality of the effect along the sample is carried by the low-frequency phonons, the dragging force generating the spin current is produced by the thermal phonons. As a result of this intricate joint effort by the phonons in different parts of the spectrum, one gets (restoring units):
Finally, for the magnitude of the SSE, , and recalling that δT↑(x)−Th(x), we obtain:
where is a material-dependent constant. The factor 2.8 takes into consideration that the energy of thermal phonon is 2.8kBT. Assuming that in Pt, ρ=0.9 μΩ·m and , we obtain . Function H(x) is positive at the cold end, meaning the dragging force pushes electrons towards the magnet there, according to equation (19). Note that although the electron–phonon drag is proportional to a high power of temperature, see equation (17), the final result for the SSE coefficient is only weakly temperature dependent, SxyT. It comes out as a result of the strong dispersion of the phonon-scattering time in the substrate. Although function H(x) in equation (20) is also temperature dependent, this dependence comes only from the non-locality of the phonon collision integral in energy, and is relatively weak. Another important property of this function is that its spatial profile varies with temperature rather slowly. This is because the phonons which contribute mostly to the non-local effect have inelastic scattering length of the order of the sample size. Varying the temperature mainly results in the shift of the relevant phonon energy ωnl, so that the corresponding length scale lin (ωnl) remains the same. These observations stress the importance of the strong dispersion of the phonon scattering.
Taking θH=0.08, εF/kB=103 K, θD=350 K, kFlei=10 and the ratio of characteristic lengths as in equation (11), we find the magnitude of the effect at 10 K to be S~20 μV K−1 × XM.
In this work, we have discussed the main ingredients of the phonon dynamics in the substrate that allows to understand the spatial profile of the SSE signal. As we have shown, to explain the non-local effect, i.e., its dependence on the position of the probe along the substrate, one must consider the spectral non-uniformity of the phonon distribution function, which can be interpreted as spectrally non-uniform temperature. A key aspect of the non-locality is the explicit introduction of the boundaries into the equations describing the propagation of diffusing phonons.
In addition, we have presented a scheme of the non-magnon mechanism in the case when the ferromagnetic element of the device is conducting and obtain the correct magnitude of the effect. (The recent observation30 of the magnetic proximity effect in FM/Pt contact suggests the possibility of another channel to contribute to the SSE voltage: anomalous Nernst effect. However, it is less universal than the electron–phonon drag and in any case must rely on the mechanism, which we propose for generation of the non-local signal: subthermal phonons with energy-dispersive diffusion, which is quite universal.)
Furthermore, the spatial profile of the SSE signal, presented in Fig. 3, is very similar to the one shown as a ‘universal’ profile on the figure 2f of Jaworski et al.2 Although the phonon kinetics at temperatures comparable with θD is strongly modified by Umklapp processes, the measured proportionality between the SSE signal and the magnitude of the magnetization in Jaworski et al.3 clearly indicates that near TC≈130 K, the effect is still dominated by the flux of the spin-polarized electrons, instead of the magnon-mediated spin torque. We believe that the difference between the data presented in figures 2 and 3 of Jaworski et al.3— in particular, the difference in the behaviour near the TC,—supports this picture. Two different samples demonstrate drastically different temperature behaviour. The sample that is thicker and grown on a substrate of a better quality has larger peak value of both Sxy and thermopower αxx and also much faster decay of Sxy at approaching TC. The stronger thermopower observed in the thicker sample demonstrates that in this sample, phonons lose momentum mainly in collisions with electrons, while in the thinner sample, their scattering on the defects is more efficient. However, due to strong sensitivity of the phonon distribution function at the F–Pt boundary to the ratio , in the thicker sample, the SSE decays with temperature much faster than in the thinner one. As we have already discussed, at large , the phonons equilibrate before they reach the probe. Indeed, in the thicker sample (more than three times thicker than the thinner one), the effect was not even resolved near the Curie temperature within the accuracy of the measurement. This suggests the need to study the dependence of the SSE signal on the thickness of the magnetic sample in otherwise identical conditions, that is, keeping the properties of the insulating substrate and semiconductor/substrate boundary the same.
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We are grateful for useful discussion with O. Tretiakov, J. Heremans, R. Myers and G. Jakob. We acknowledge the support by grants ONR-000141110780, NSF-DMR-1006752, NSF-DMR-1105512, NHARP and the Paul and Tina Gardner fund for Weizmann-TAMU collaboration.
The authors declare no competing financial interests.
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Tikhonov, K., Sinova, J. & Finkel’stein, A. Spectral non-uniform temperature and non-local heat transfer in the spin Seebeck effect. Nat Commun 4, 1945 (2013). https://doi.org/10.1038/ncomms2945
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