Spin heat accumulation and its relaxation in spin valves
T. T. Heikkilä,1,*
Moosa Hatami,2and Gerrit E. W. Bauer21Low Temperature Laboratory, Helsinki University of Technology, P.O. Box 5100, Helsinki FIN-02015 TKK, Finland 2Kavli Institute of NanoScience, Delft University of Technology, 2628 CJ Delft, The Netherlands
共Received 19 February 2010; published 12 March 2010兲
We study the concept of spin heat accumulation in excited spin valves, more precisely the effective electron temperature that may become spin dependent, both in linear response and far from equilibrium. A temperature or voltage gradient create nonequilibrium energy distributions of the two spin ensembles in the normal-metal spacer, which approach Fermi-Dirac functions through energy relaxation mediated by electron-electron and electron-phonon coupling. Both mechanisms also exchange energy between the spin subsystems. This interspin energy exchange may strongly affect thermoelectric properties of spin valves, leading, e.g., to violations of the Wiedemann-Franz law.
DOI:10.1103/PhysRevB.81.100408 PACS number共s兲: 75.60.Jk, 72.15.Jf, 75.30.Sg, 85.75.⫺d The electric conductance through ferromagnet兩normal
metal兩ferromagnet spin valves is a function of the magnetic configuration.1It reflects the spin accumulation, i.e., the spin
共index 兲 dependent chemical potential of the
normal-metal island. The latter parametrizes the spin dependence of the energy distribution functions f共E兲, whose description also requires spin-dependent temperatures T.2,3 As shown
below, these should in general be interpreted as effective parameters.
In this Rapid Communication we describe the processes affecting the T and through them the thermoelectric re-sponse in spin valves, which we find to be a sensitive probe for the nonequilibrium state in the nonmagnetic spacer. Whereas the spin accumulation relaxes only by scattering processes that break spin rotation invariance such as spin-orbit interaction and magnetic disorder, the spin heat accu-mulation Ts= T↑− T↓ is sensitive also to electron-phonon 共e-ph兲 and electron-electron 共e-e兲 interactions. Spin-flip scat-tering in Al, Ag, Cu, or carbon is weak and hardly tempera-ture dependent; the typical spin-flip scattering time sfis of
the order 100 ps,4which can be much longer than the dwell
times in magnetoelectronic structures. The interspin energy exchange rate due to inelastic effects is strongly temperature dependent and above cryogenic temperatures typically domi-nates the direct spin-flip scattering in dissipating the spin heat accumulation. The spin heat accumulation in normal-metal spacers should not be confused with the spin 共wave兲 temperature of ferromagnets.5
In a spin valve共Fig.1兲, a nonmagnetic island is coupled
to two ferromagnetic reservoirs with parallel 共P兲 or antipar-allel 共AP兲 magnetic alignments. The chemical potential of the left共right兲 reservoir isL共R兲and the temperature is TL共R兲. The conductances GL/R and Seebeck coefficients SL/R of the contacts between the island and the reservoirs depend on spin 苸兵↑,↓其. Biasing the spin valve with either a voltage ⌬V=共R−L兲/e or a temperature difference ⌬T=TR− TL gives rise to a spin-dependent energy distribution function
f共E兲 of the electrons on the island. As shown below, in
the linear-response regime this can be described exactly by spin-dependent chemical potentials and temperatures, such that f共E兲= f0共E;, T兲, where f0共E;, T兲=兵exp关共E
−兲/共kBT兲兴+1其−1 is the Fermi-Dirac distribution function.
and T are determined by conservation of charge, spin,
and energy 关see Eqs. 共2兲兴. The response matrix of the spin
valve
冉
I Q˙冊
=冉
G GS TGS K冊冉
⌬V −⌬T冊
共1兲relates the charge and heat currents I and Q˙ to the biases ⌬V and ⌬T, respectively. Below, we derive expressions for the heat conductance K and thermopower S, in the presence of interspin energy exchange and for different magnetic con-figurations.
The steady-state potentials and temperatures can be deter-mined from Kirchhoff’s laws for charge and energy for each spin.3For small e⌬V/kB,⌬TⰆT
↑, T↓,
兺
i=L,R Ii,+ Gsf共−−兲/e = 0,兺
i=L,R Q˙i,+ K↑↓共T− T−兲 + Ke-ph共T− Tph兲 = 0. 共2兲Here Ii,= Gi共−i兲/e+GiSi共T− Ti兲 is the charge cur-rent for spin through contact i, Qi,=L0GiT共T− Ti兲 + GiSiT共−i兲/e is the corresponding heat current, Gi and Siare the associated charge conductances and Seebeck coefficients, andL0=2kB2/共3e2兲 is the Lorenz number. Spin decay is described by the 共inter兲spin conductance Gsf = e2F⍀/
sffor an island with volume⍀, density of states at
T-s mL L Tph T mR R T IL,s IR,s R,s Q. L,s Q. Ts spin flip e-e, e-ph m-s ms spin flip
FIG. 1. 共Color online兲 Schematic spin valve biased with a volt-age and/or temperature difference. Spin-flip and inelastic electron-electron and electron-electron-phonon scattering in the normal-metal spacer lead to interspin energy exchange and change the thermoelectric characteristics. I and Q˙ stand for the charge and heat currents flow-ing into the island. Tphis the temperature of the phonon bath.
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the Fermi levelFand spin-flip relaxation timesf. The term Ke-phdescribes the interaction with the phonons at tempera-ture Tph. Interspin energy exchange is governed by the spin
heat conductance K↑↓=L0GsfT + Ke-e↑↓, where the first term originates from the spin-flip scattering and the second is due to e-e interactions. We are allowed to discard the spatial dependence of the distribution functions when the diffusion timeD= L2/D in the island with length L and diffusion con-stant D is shorter than both sfand the spin thermalization
timest=L0e2FT⍀/共Ke-ph+ 2K↑↓兲.
The in general lengthy solutions of Eqs.共2兲 are
consider-ably simplified for left-right symmetric conductances and Seebeck coefficients, parametrized by G0= G↑+ G↓, P =共G↑
− G↓兲/G0, S0=共G↑S↑+ G↓S↓兲/G0, and P
⬘
=共G↑S↑− G↓S↓兲/共G0S0兲 for both junctions. In the antiparallel case the
signs of P and P
⬘
in one of the junctions are inverted. In the parallel configuration the heat conductance becomesKP=L0GPT +
2Ke-phr共1 − P2␥兲
1 − P2␥+ Ke-ph/共L 0G0T兲
共3兲 and in the antiparallel configuration it is
KAP=L0GPT共1 − P2␥兲 +
2Ke-phr 1 + Ke-ph/共L0G0T兲
. 共4兲
The factor r =共Tph− TL兲/共TR− TL兲−1/2 parametrizes the
pho-non temperature on the island: If the phopho-nons are poorly coupled to the substrate, as, for example, in perpendicular spin valves or in suspended structures, Tph=共T↑+ T↓兲/2. For
the P configuration this yields r = 0, whereas for the AP con-figuration we get r = −K↑↓P/关2共Ke-ph+ K↑↓+L0G0T兲兴. In the
opposite limit r =⫾1/2, viz. Tphis fixed to the bath tempera-ture of the left or right reservoir. The coefficient ␥=关1 +共Ke-ph+ 2K↑↓兲/共L0G0T兲兴−1 describes interspin energy
ex-change. Factoring out the temperature dependence of Ke-ph ⬀T4 and K
e-e
↑↓⬀T+1共see the discussion below兲 yields␥=关1
+共T/Tch,ph兲3+共T/Tch,e-e兲+ 2Gsf/G0兴−1, where the
character-istic temperatures are Tch,e-ph=关共L0G0T4兲/Ke-ph兴1/3, Tch,e-e
=关共L0G0T+1兲/共2Ke-e↑↓兲兴1/ for phonon and electron-electron couplings, respectively. The exponent depends on the dimensionality of the sample. We are here mainly inter-ested in three-dimensional 共3D兲 samples 共= 3/2兲 in which all sample dimensions exceed the thermal coherence length
T=
冑
បD/共2kBT兲.In the parallel configuration the thermopower satisfies
SP= S0 and in the antiparallel one6 SAP SP =1 − PP
⬘
+ 2Gsf/G0+␥P共P − P⬘
− 2P⬘
Gsf/G0兲 1 − P2+ 2Gsf/G0 . 共5兲 The temperature dependence of K and S is plotted in Fig.2 for Ke-phⰇKe-e,L0GsfT. For TⰆmin共Tch,e-e, Tch,e-ph兲⬅Tch,
the device operates as a spin heat valve in which the heat current can be controlled by the magnetization configuration. Contrary to the charge conductance, however, the magneto-heat conductance 共KP− KAP兲/KP vanishes for TⰇTch or ␥ →0. Thus the presence of inelastic scattering leads to a
vio-lation of the Wiedemann-Franz law K =L0GT for TⲏTch.
The magnetothermopower 共SP− SAP兲/SPpersists provided P ⫽ P
⬘
.3The measured heat conductance and thermopower as a function of temperature and magnetic configuration may hence yield unprecedented information on the energy relax-ation in normal metals.We now address the characteristic temperatures Tch,e-ph and Tch,e-e. The former can be obtained directly from the
Debye form for the heat conductance between electrons and acoustic phonons,7K
e-ph=
5
2⌺⍀T4, valid for TⰆTDebye. Here
⌺ is the e-ph coupling constant8and the factor 1/2 takes into
account spin degeneracy. The characteristic temperature for electron-phonon coupling thus reads
Tch,e-ph=
冉
kB 2 15ប⌺⍀冊
1/3冉
G0h e2冊
1/3 . 共6兲For TⲏTDebye, the electron-acoustic phonon scattering and
thereby interspin energy exchange saturates. Optical phonons start to contribute in this temperature regime but are disre-garded here.
The e-e scattering collision integrals with spin-dependent distribution functions contain three terms,
Ie-e,共⑀兲 = I共a兲共⑀兲 + I共b兲,−共⑀兲 + I共c兲,−共⑀兲,
presented by the diagrams in Fig.3.
Processes 共b兲 and 共c兲 induce interspin energy exchange, which can be described in terms of a heat current flowing between two spin ensembles,8
Q˙e-e↑↓=F⍀
冕
d⑀⑀共I共b兲↑↓+ I共c兲↑↓兲. 共7兲 The direct spin current due to e-e interaction vanishes in the absence of spin-orbit scattering, 兰d⑀共I共a兲↑↓+ I共b兲↑↓+ I共c兲↑↓兲=0. In 3D, to lowest order in spin particle and heat accumulation,0 1 2 3 4 5 0 0.5 1 1.5 2 2.5 0 1 2 3 4 5 T /Tcr,e−ph SAP / SP K / (L G0 T )
FIG. 2. 共Color online兲 Temperature dependence of the heat con-ductance K共solid lines, left axis兲 and thermopower S 共dashed line, right axis兲 of a structurally left-right symmetric spin valve with P = 0.9, P⬘= 0.5 and when the electron-phonon relaxation dominates the interspin energy exchange. The lines are plots of Eqs.共3兲–共5兲
and the symbols have been calculated from the full nonequilibrium distribution function 关Eqs. 共10兲 and 共11兲兴. The results have been
calculated for P configuration with r = 0 共circles兲 and r=1/2 共squares兲 and AP configuration with r=0 共stars兲 and r=1/2 共triangles兲.
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共↑−↓兲/共↑+↓兲Ⰶ1 and Ts/共T↑+ T↓兲Ⰶ1, we arrive at Q˙e-e↑↓⬇共K共b兲↑↓+ K共c兲↑↓兲共T↑− T↓兲, where K共b兲↑↓= 105共7/2兲kB 7/2T5/2 32关2ET共1 + F兲兴3/2ប, 共8a兲 K共c兲↑↓= FC 共F + 2兲
冋
2共3/2兲 +35 16共7/2兲册
kB 7/2T5/2 2关2ET共F + 1兲兴3/2ប . 共8b兲Here C = −1 +共F+1兲3/2, F⬎−1 is the spin triplet Fermi-liquid parameter 共F=−1 corresponds to the Stoner instabil-ity兲, ET=បD/⍀2/3is the Thouless energy proportional to the
inverse time it takes to diffuse over a length⍀1/3and共x兲 is the Zeta function. Summing the two contributions from Eqs. 共8兲 yields the characteristic temperature
Tch,e-e= 8
冉
G0h e2冊
2/3E T kB 共F + 1兲 ⫻再
共F + 2兲 48FC2共3/2兲 + 105关6 + F共3 + C兲兴共7/2兲冎
2/3 . 共9兲 In one-dimensional共1D兲 and two-dimensional 共2D兲 struc-tures the spin-flip contribution 共c兲 has an infrared divergence9,10 that needs to be regularized. As a result, the interspin energy exchange due to e-e scattering becomes stronger and the corresponding Tch,e-e lower. This isespe-cially relevant at low temperatures and small structures since
Tmay exceed 100 nm at T⬇1 K. We intend to analyze the resulting interspin energy exchange in reduced dimensions in the future.
In order to assess the relevance of our results for realistic samples we consider a disordered island of a spin valve coupled to the reservoirs via tunnel contacts. For example, with F = −0.3 we get Tch,e-e⬇ 0.9 K ⫻ D 0.001 m2/s
冋
0.1 共m兲3 ⍀ G0 0.01 S册
2/3 , Te-ph⬇ 1 K ⫻冋
10 9 W m−3K−5 ⌺ 0.1 共m兲3 ⍀ G0 0.01 S册
1/3 .Making the sample smaller and conductance larger increases both characteristic temperatures, but the increase for Tch,e-ph
is slower. For⍀=0.001 共m兲3 and G
0= 1 S we get Tch,e-ph
= 22 K, whereas Tcr,e-e= 400 K. We may therefore conclude
that in spin valves with metallic contacts and 3D spacers the interspin energy exchange due to e-e interaction can be
ne-glected. The spin thermalization rate with F = −0.3 is 1 st ⬇
冋
1 20 ns冉
T 1 K冊
3/2冉
0.001 m2/s D冊
3/2 + 1 25 ns冉
T 1 K冊
3冉
⌺ 109 W m−3K−5冊
册
⫻ 1047 J−1m−3 F .The first term comes from e-e scattering and the second from e-ph scattering. This rate exceeds the spin-flip scattering rate ⬃10 K at temperatures above Gi共⑀兲⬇Gi0关1+ci共⑀−i兲兴.
Above we assume that the electron energy distribution function is well represented by Fermi-Dirac distributions with spin-dependent chemical potentials and temperatures. This is not true in general since f共⑀兲 has the nonequilibrium form8,11 f共⑀兲 =GLfL+ GRfR+Fe 2⍀I coll关f, f−兴 GL+ GR , 共10兲 where fL/R= f0共⑀;L/R, T兲 are the distribution functions for the reservoirs and Icoll describes all inelastic scattering
events. The charge 共n=0兲 and heat 共n=1兲 currents through contact i then become
Ii兩Q˙i=
兺
冕
d⑀共⑀−i兲nGi
e1+n共⑀兲关f共⑀兲 − fi共⑀兲兴. 共11兲
Thermoelectric effects can be included by adding a weak energy dependence to the conductances, Gi共⑀兲⬇Gi0关1 + ci共⑀−i兲兴, and expanding to linear order in ci,. Identify-ing Si= eL0ciT, we recover Eqs. 共4兲 and 共5兲 in the regime
e⌬V/kB,⌬TⰆTL, TR⬇T even in the absence of collisions 共i.e.,␥= 1兲. For ci= 0 and to linear order in the applied bias, the nonequilibrium distribution 关Eq. 共10兲兴 is identical to the
quasiequilibrium one. Under these conditions, the collision integrals can be calculated by replacing the full distribution functions by the quasiequilibrium ones. Numerical solutions of the kinetic equations 共see Fig. 2兲 indicate that in linear
response collisions and finite cis do not change this conclu-sion.
Beyond linear response spin-dependent temperatures can strictly speaking be invoked only in the presence of strong inelastic scattering such that T↑⬇T↓. Nevertheless we can define effective electron temperatures that satisfy the stan-dard relation with the thermal energy density in the Sommer-feld expansion:12
T=
冑
6kB
冑
冕
−⬁ ⬁关f共⑀兲 − 1 +共⑀−兲兴⑀d⑀. 共12兲
Proceeding with Fermi-Dirac distributions with effective spin-dependent temperatures and chemical potentials,and
Tcan be obtained from Eqs.共2兲 by replacing the expression
for the charge and heat currents through contact i with their nonlinear counterparts, I=Gi, e
再
−i+ ci 2 关L0e 2共T 2− T i 2兲 − 共 −i兲2兴冎
, σ σ σ σ σ σ σ σ ¯ σ ¯ σ ¯ σ ¯ σ + ¯hω + ¯hω + ¯hω − ¯hω − ¯hω − ¯hω ¯hω ¯ hω ¯ hω (a) (b) (c)FIG. 3. Electron-electron scattering vertices.共a兲 Equal-spin scat-tering, which equilibrates the electrons but does not thermalize the spins.共b兲 Spin-conserving scattering and 共c兲 spin exchange scatter-ing, which do thermalize the spins.
SPIN HEAT ACCUMULATION AND ITS RELAXATION IN… PHYSICAL REVIEW B 81, 100408共R兲 共2010兲
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Q˙i,= Gi,关L0共T2− Ti2兲/2 − 共2−i2兲/共2e2兲兴 + Gici共 −i兲关L0共T2+ Ti2兲/2 − 共2−i2兲/共6e2兲兴. 共13兲 These equations are obtained by a direct integration of Eq. 共11兲 using Fermi-Dirac functions fi共⑀兲 and f共⑀兲. We also
have to replace the linear-response forms of the spin mixing terms in Eqs. 共2兲 by their forms far from equilibrium.
For example, for e-e scattering with F = 0 we use Q˙¯ = 15共7/2兲kB7/2共T7/2− T7/2−兲/关16ប共2ET兲3/2兴.
In the absence of collisions and for weak thermoelectric effects it can be proven by direct integration that the effec-tive temperatures defined by Eq.共12兲 agree with those which
follow from heat conservation. In Fig. 4we present a com-plete numerical solution of the kinetic equations along with the results from the quasiequilibrium heat balance equations from which we conclude that the two approaches for calcu-lating T agree also in the presence of interspin energy ex-change.
Spin heat accumulation cannot be directly measured by two-terminal transport experiments in linear systems. In or-der to prove the presence of a sizable Tsfar from equilibrium it should be probed by spin-selective thermometry, such as a generalization of the tunnel-spectroscopy in Ref.11, by mea-suring the shot noise of the spin valve, or through electron spin resonance.
In conclusion, we have shown that interspin energy ex-change in a spin valve affects the temperature and magnetic configuration dependence of its thermoelectric properties. The different thermalization mechanisms can be quantified by characteristic temperatures关Eqs. 共6兲 and 共9兲兴 above which
interaction effects become important. We introduce the con-cept of spin heat accumulation via the spin-dependent effec-tive electron temperatures T in Fermi-Dirac distribution functions, which can be used to describe transport properties beyond the linear-response regime. We demarcate the regime
in which spin valves can be employed to control heat cur-rents. Other types of operations can be envisaged as well, such as spin-selective cooling of the electrons 共see the left inset of Fig. 4兲.
We thank P. Virtanen for discussions. This work was sup-ported by the Academy of Finland, the Finnish Cultural Foundation, and NanoNed, a nanotechnology program of the Dutch Ministry of Economic Affairs. T.T.H. acknowledges the hospitality of Delft University of Technology, where this work was initiated.
*tero.heikkila@tkk.fi
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−201 −10 0 10 20 1.5 2 2.5 3 eV /kT Tσ / T −1 0 1 1 1.01 1.02 eV /kT Tσ /T −10 0 10 0 0.5 1 E/kT fσ (E ) Spin↓ Spin↑
FIG. 4. 共Color online兲 Spin-dependent effective temperature vs voltage in an asymmetric spin valve with P = 0.9, P⬘= 0.5, and GR
= 0.1GL. The lines are calculated from Eqs.共2兲 and 共13兲 and the
symbols from Eq.共12兲 for numerical solutions of the kinetic
equa-tions. The upper curves are for majority, the lower for minority spins, and different strengths of e-e scattering with F = 0: no scat-tering共solid line and circles兲, weak scattering with ET= 0.05kBT and GL= 100e2/h 共dashed line and squares兲, and strong scattering with
ET= 0.001kBT and GL= 100e2/h 共dash-dotted line and stars兲. Here T
denotes the temperature of the reservoirs. Left inset: behavior at low bias with thermoelectric effects cL= 10cR= 0.02/共kT兲. Right
in-set: distribution function at e⌬V=10ETwith different strengths of e-e scattering.
HEIKKILÄ, HATAMI, AND BAUER PHYSICAL REVIEW B 81, 100408共R兲 共2010兲
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