Theory of the spin Peltier effect
Y. Ohnuma1,2), M. Matsuo1,2), S. Maekawa2)
R.G. for Spin-energy transformation science

In the field of electronics, the thermoelectric effects which are
the mutual conversions of electric and heat currents have beenstudied in terms of the energy conservation. Generation of anelectromotive force from a temperature gradient applied in a
metal or a semiconductor is called the Seebeck effect.
Conversely, creation of a temperature difference from anelectromotive force is called the Peltier effect. Since thethermoelectric conversion enables us to use a waste heat, it hasbeen studied extensively from the viewpoint of the suppressionof the energy consumption.
The spin current, flow of the spin angular momentum ofelectrons, has attracted attention in the field of spintronics
aiming at using and controlling both of the spin and charge
degrees of freedom. In particular, the pure spin current without
the electric current is expected to enable communication with
reduced energy consumption since it is free from the Joule heat.
Recently, the establishment of interconversion between thespin and heat currents has been discussed in the field ofspintronics. When a temperature difference is applied to amagnetic material, the spin current is generated. Thisphenomenon is called the spin Seebeck effect [1]. The inverse
phenomenon in which a temperature difference occurs when thespin current is injected into a magnetic material is called thespin Peltier effect [2, 3]. Clarification of the interconversion
between the spin and heat currents may lead to a new energysaving device. The microscopic theory of the spin Seebeck
effect has been built. In the case of the spin Peltier effect,
however, no microscopic theory has ever been proposed.
In our paper, a microscopic theory of the spin Peltier effecthas been constructed. We assume the bilayer system composedof a paramagnetic metal (PM) and a ferromagnetic insulator (FI),coupled magnetically at the interface (Fig.1).
First, we show the mechanism of the heat current generationfrom the spin current in our system. When the electric current isapplied to PM, the spin current is generated by the spin-orbit
interaction in PM, which is called the spin Hall effect. The spincurrent makes a reservoir of the spin-polarized conduction
electrons at the interface. This is called the spin accumulation.
The spin current is injected into FI by the spin accumulation
through the magnetic coupling at the interface. In FI, the spincurrent is carried by the magnons which is the low energy
magnetic excitation in FI. Since the magnons carry both of thespin and the energy, the heat current is also created. This is our
scenario for describing the mechanism of the spin Peltier effect.
Next, we describe the spin and heat transports at the PM/FIinterface using the non-equilibrium Green's function method.
We define the spin and heat currents as the time derivatives ofthe spin of the conduction electrons in PM and the Hamiltonianin FI, respectively. Performing the perturbative calculation of
the magnetic coupling at the interface, we obtain the expressionof the spin current represented by the product of the spectral
functions of the spin fluctuations in PM and FI, and thedifference of the non-equilibrium distribution functions between
PM and FI. We also derive the expression of the heat current
represented by the product of the spin current and the energy
passing through the PM/FI interface.

Using the obtained formulas, we give a description of the spin
Seebeck and spin Peltier effects. Substituting the difference ofdistribution functions generated by the temperature difference
between PM and FI as well as by the spin accumulation into thespin and heat currents, we obtain the formulas of the spin
Seebeck and spin Peltier effects, respectively. Comparing theseformulas, we confirmed that the spin Seebeck and spin Peltier
effects are summarized using the Onsager's reciprocal relation.
Further, we estimate the temperature change due to the spin
Peltier effect. At the interface, magnons are excited andaccumulated by the spin Peltier effect. The energy change at theinterface is generated by the accumulation of magnons. Then,the temperature change is obtained from the energy change. Ourestimation is consistent with the experimental result [3].
In conclusion, a microscopic theory of the spin Peltier effectin a bilayer system composed of PM and FI was formulatedusing the non-equilibrium Green’s function method. We derived
the spin and heat currents driven by temperature gradient as well
as by spin accumulation at the interface in terms of the spectral
and distribution functions of the spin fluctuations in PM and FI.
These currents have been summarized using Onsager’s
reciprocal relation. In addition, we estimated temperaturechange at the interface due to the spin Peltier effect. Our theory
will provide a microscopic understanding of the interconversion
between spin and heat currents at the PM/FI interface.

Fig.1 Schematic view of the spin Peltier effect. We consider
spin and heat transports in a bilayer structure consisting of aparamagnetic metal (PM) and a ferromagnetic insulator (FI),where the electron spins in PI are coupled with the localized
moments in FI via the magnetic interaction. The spin
accumulation at the interface is found to be a driving force ofspin and heat current.

Present address: 1) Kavli Institute for Theoretical Sciences,
University of Chinese Academy of Sciences. 2) RIKEN Center
for Emergent Matter Science
References[1] K. Uchida et al., Nature (London) 455, 778 (2008).
[2] J. Flipse et al., Phys. Rev. Lett. 113, 027601 (2014).
[3] S. Daimon et al., Nat. Commun. 7, 13754 (2016).