Charge pumping and the colored thermal voltage noise in spin valves

Charge pumping and the colored thermal voltage noise in spin valves

Jiang Xiao (萧江兲,1_{Gerrit E. W. Bauer,}1 _{Sadamichi Maekawa,}2,3_{and Arne Brataas}4

1_{Kavli Institute of NanoScience, Delft University of Technology, 2628 CJ Delft, The Netherlands} 2_{Institute for Materials Research, Tohoku University, Sendai 980-8557, Japan}

3_{CREST, Japan Science and Technology Agency, Tokyo 100-0075, Japan}

4_{Department of Physics, Norwegian University of Science and Technology, NO-7491 Trondheim, Norway}

共Received 13 January 2009; revised manuscript received 16 April 2009; published 12 May 2009兲 Spin pumping by a moving magnetization gives rise to an electric voltage over a spin valve. Thermal fluctuations of the magnetization manifest themselves as increased thermal voltage noise with absorption lines at the ferromagnetic resonance frequency and/or zero frequency. The effect depends on the magnetization configuration and can be of the same order of magnitude as the Johnson-Nyquist thermal noise. Measuring colored voltage noise is an alternative to ferromagnetic resonance experiments for nanoscale ferromagnetic circuits.

DOI:10.1103/PhysRevB.79.174415 PACS number共s兲: 85.75.⫺d, 07.50.Hp, 72.25.Pn, 75.70.⫺i

I. INTRODUCTION

A spin valve consists of a thin nonmagnetic metallic 共NM兲 layer sandwiched by two ferromagnetic 共FM兲 layers with variable magnetization direction. One of the FM layers is usually thick and its magnetization is fixed, while the other is thin and its magnetization direction is free to move. Spin valves have a wide range of interesting static and dynamic properties,1–13 _{many of which are related to the} current-induced spin-transfer torque,1,2 _{which can excite} magnetiza-tion dynamics 共and reversal兲. Inversely, magnetization dy-namics generates a current flow or a voltage output. Berger14 first discussed the induced voltage in an FM兩NM兩FM struc-ture by magnetization dynamics. He posited that a voltage of orderប␻/e can be generated when the magnetization of one ferromagnet precesses at frequency ␻. Similar dynamically induced voltages have been studied theoretically15 _and observed16_{in simple FM兩NM junctions and in magnetic} tun-nel junctions 共MTJs兲.17 In spin valves and MTJs, voltage induced by the magnetization dynamics can be understood as a two-step process:共i兲 the moving magnetization of the free layer generates a spin current;共ii兲 the static magnetization of the fixed layer filters the “pumped” spin current and converts it into a charge current or, in an open circuit, a voltage out-put. The electrical voltage induced by moving domain walls can be explained analogously.18–22 _{In the first part of the} present paper, we derive a simple formula for the charge pumping voltage in a spin valve by circuit theory in which magnetization dynamics is taken into account. We find that the magnitude of the voltage is governed by the spin-transfer torque in the same structure. We therefore propose to mea-sure the angular dependence of the spin-transfer torque 共or torkance, i.e., the torque divided by the voltage bias兲 by the angular dependence of the charge pumping voltage.

The charge pumping voltage consists of a dc and an ac component, even when induced by a steady magnetization dynamics such as ferromagnetic resonance共FMR兲. The con-cept can be extended to the case of thermally activated, i.e., fluctuating, magnetization dynamics, which is an extra source of thermal voltage noise that only appears in magnetic structures. Johnson23_{and Nyquist}24 _{showed that in}

nonmag-netic conductors the voltage noise is associated with the ther-mal agitation of charge carriers 共driven by fluctuating elec-tromagnetic modes兲. The power spectrum of this noise is white and proportional to the temperature T and resistance

R : SJN共␻兲=4kBTR up to frequencies of kBT/ប⬃104 GHz at

room temperature.23,24 _{In magnetic structures such as spin} valves, thermal fluctuations of the magnetization direction have to be considered.25_{Some consequences of thermal} fluc-tuation in spin valves, such as noise-facilitated magnetization switching26–28 _{and resistance fluctuations,}29,30 _{have been} studied before. In the so-called thermal ferromagnetic reso-nance, frequencies are studied by means of resistance fluc-tuations without applied magnetic fields.31 _{Foros et al.}30 showed that the time-averaged autocorrelator of the resis-tance fluctuations is significantly affected by the dynamical exchange coupling between the magnetic layers.

In the second part of this paper, we show that a magneti-zation fluctuation-related voltage noise can be of the same order of magnitude as the conventional thermal noise in non-magnetic conductors. This noise is not “white” but displays spectral features related to the FMR. The noise spectrum therefore contains information comparable to that obtained by FMR. For nanoscale ferromagnetic circuits the noise mea-surements might be easier to perform than conventional FMR experiments. Compared to the resistance noise mea-surement, the pumping voltage noise measurement is nonin-trusive because it does not require application of current, which may disturb the system.

This paper is organized as follows. Section IIpresents a general theoretical framework that combines the magneto-electric circuit theory and the Landau-Lifshitz-Gilbert共LLG兲 equation. In Sec. III, we derive a formula for the charge pumping voltage in spin valves. In Sec. IV, by using the magnetic-susceptibility function, we calculate the voltage noise spectrum due to magnetization fluctuations for two dif-ferent magnetic configurations. Section V contains some general remarks on the calculation. In Appendix A we cal-culate the angular dependence of the magnetic susceptibility for a spin valve, and Appendix B presents an alternative calculation of the magnetization-related thermal noise by computing the frequency-dependent impedance of a spin valve.

II. CIRCUIT THEORY WITH DYNAMICS

Figure 1共a兲shows a spin valve structure under consider-ation. The magnetization in the left FM with direction m0is assumed to be static and m, the one of the right FM, to be free, which can be realized by making the right layer much thinner than the left one. For electron transport, we assume for simplicity that the spin valve is symmetric. Such an as-sumption may be invoked when both FM layers are of the same material and thicker than the spin-flip diffusion length. In that regime, the resistances of the bulk ferromagnet over the spin-flip diffusion length are in series with the interface resistances, whereas the remoter parts of the ferromagnets are magnetically inert series resistances. The regime in which the layers become thinner than the spin-flip diffusion length was treated by Kovalev et al.32_{In this and Sec.III, we} focus on the spactive region in the spin valve, which in-cludes the NM spacer and small part 共of the order of the spin-flip diffusion length兲 of the FM layers as indicated by the dashed box in Fig.1共a兲.

When m depends on time, a spin current is pumped into the metallic spacer layer through the F兩N interface. The magnitude and polarization of the spin-pumping current reads33 Is sp = ប 4␲

冉

grm⫻ dm dt + gi dm dt

冊

, 共1兲

where grand giare the real and imaginary part of the

dimen-sionless transverse spin-mixing conductance.33_{The first term} in Is

sp

corresponds to a loss of angular momentum of the free layer magnetization to the adjacent NM layers, thus provid-ing an extra dampprovid-ing torque.33 _{When the adjacent normal} metal is an ideal spin sink, the spin-pumping current loss can be represented by a Gilbert damping coefficient共see below兲, and each interface contributes to the damping constant by ␣

⬘

=共␥ប/4␲Mtot兲gr, introducing the gyromagnetic ratio ␥

and the total magnetization of the right FM film Mtot. The imaginary part gieffectively modifies the gyromagnetic ratio

for the magnetization under consideration.

In order to use magnetoelectronic circuit theory, the struc-ture has first to be decomposed into nodes 共for bulk兲 and contacts 共for interfaces兲. For each node we may define a charge chemical potential and a 共vector兲 spin chemical po-tential; let␮_L,R,Nand␮_L,R,Nbe the charge and spin chemical potentials in left, right FM leads, and the NM spacer. In the ferromagnet, we may assume that the spin accumulations are aligned with the magnetization, i.e., ␮L=␮L

s

m0 and ␮R=␮R

s

m. The charge current Ic and the spin current IL

through the left interface connecting the left FM and the spacer layer are given by34,35

Ic= eg 2h关2共␮L−␮N兲 + p共␮L s −␮N· m0兲兴, 共2a兲 IL= − eg 2h关2p共␮L−␮N兲 + 共␮L s −␮N· m0兲兴m0 + e hgrm0⫻␮N⫻ m0+ e hgi␮N⫻ m0, 共2b兲

where g = g_↑+ g_↓is the total conductance and p =共g_↑− g_↓兲/g is the polarization of the F兩N interface and the 共longitudinal兲 active regions of the FMs.32 _{Similarly for the right FM lead,}

Ic= − eg 2h关2共␮R−␮N兲 + p共␮R s −␮N· m兲兴, 共3a兲 IR= − eg 2h关2p共␮R−␮N兲 + 共␮R s −␮N· m兲兴m + e hgrm⫻␮N⫻ m + e hgi␮N⫻ m + 2e បIssp, 共3b兲

where the spin current IR at the right interface is modified

due to the additional spin-pumping current Isspemitted by the

moving magnetization m. Additionally, the spin current con-servation in the presence of the spin flips in the NM spacer requires IL+ IR= e h h D␶_sf␮N⬅ e hgsf␮N, 共4兲

where D is the energy density of states at the Fermi energy and␶sfis the spin-flip relaxation time in the NM.

The spin current IR entering the free layer exerts a

spin-transfer torque on m, which is equal to its transverse com-ponent absorbed at the interface,11,12

Nst= ប

2e关IR−共共IR· m兲m兲兴. 共5兲 The LLG equation is therefore modified as

dm dt = −␥m⫻ Heff+␣m⫻ dm dt + ␥ Mtot Nst, 共6兲 where H_effis the total effective magnetic field acting on m, and␣=␣₀+ 2␣

⬘

is the total magnetic damping including both the bulk damping and the spin-pumping enhanced damping from both interfaces.

FIG. 1. 共Color online兲 Spin and charge currents in spin valves. 共a兲 For the steady state case studied in Secs.IIandIII.共b兲 For the thermal magnetization fluctuations studied in Sec.IV

The set of equations in Eqs.共1兲, 共2a兲, 共2b兲, 共3a兲, 共3b兲, and

共4兲–共6兲 describes the charge/spin transport and magnetization

dynamics in the metallic magnetic heterostructures. In many cases, the transport equations in Eqs. 共1兲, 共2a兲, 共2b兲, 共3a兲,

共3b兲, 共4兲, and 共5兲 and the dynamical LLG equation in Eq. 共6兲

can be solved separately by ignoring the spin-pumping con-tribution Issp in IR, in which case the transport only depends

on the instantaneous m but not on m˙ . However, for thin

magnetic layers the spin-pumping modification cannot be ne-glected. It has possibly important consequences, such as a voltage induced by magnetization dynamics, anisotropic magnetic damping and susceptibility tensor, and colored thermal noise, as will become clear from the discussion be-low.

At first, let us calculate the static共m˙=0兲 magnetoconduc-tance of a spin valve. When a bias voltage V =共␮L−␮R兲/e is

applied, we can calculate the charge current I = IL= IR from

Eqs. 共2a兲, 共2b兲, 共3a兲, 共3b兲, and 共4兲, hence the

magnetocon-ductance G = I/V. By setting␮N= 0 in the spacer and

assum-ing strong spin flips in the ferromagnets 共␮L s

=␮R

s_{= 0兲, we}

find 共with m·m₀= cos␪兲

G共␪兲 =G0g

4

冋

1 − 4p␩共␪兲sin 2␪

2

册

, 共7兲

where G₀= 2e2_{/h is the conductance quantum and} ␩共␪兲 = pg/4 g sin2␪ 2+ 2g˜rcos 2␪ 2+ gsf 共8兲

is the angular-dependent spin current polarization with

g

˜r⬅gr+ 2gi

2_/共2g

r+ gsf兲. The G共␪兲 above agrees with Eq. 共160兲 in Ref. 36.

III. CHARGE PUMPING IN SPIN VALVES

When a voltage difference⌬V is applied over a spin valve which does not excite magnetization dynamics共m˙=0兲, Eqs. 共2a兲, 共2b兲, 共3a兲, 共3b兲, 共4兲, and 共5兲 lead to the spin-transfer

torque1

Nst共␪兲 = ⌬V关␶ip共␪兲m ⫻ m0+␶op共␪兲m0兴 ⫻ m, 共9兲 where ␶ip and ␶op are the so-called 共angular-dependent兲 torkances37_{for the in-plane 共Slonczewski’s兲 component and} out-of-plane共effective-field兲 component

␶ip共␪兲 = e␩共␪兲 2␲ ˜gr and ␶op共␪兲 = e␩共␪兲 2␲ gigsf 2gr+ gsf . 共10兲 When the bias polarity is chosen such that the in-plane torque in Eq. 共9兲 works against the magnetic damping, the

current flow can excite magnetization dynamics, otherwise they are suppressed.

Inversely, magnetization dynamics can induce a current flow by the spin pumping; a moving magnetization 共m兲 pumps a spin current共with zero charge current兲 into adjacent contacts, and the pumped spin current is converted into a charge current IP 共or pumping voltage VP兲 by a static

ferro-magnetic filter 共m0兲.17 In the following, we use the circuit

theory described in Sec.IIto derive a simple expression for the charge pumping voltage 共current兲 induced by FMR in a spin valve. We shall study two different cases:共i兲 when the spin valve is open, no current flow is allowed 共Ic= 0兲, and a

pumping voltage VP is built up;共ii兲 when the spin valve is

closed, i.e., the two ends of the spin valve are short circuited, no voltage difference is allowed at the two ends 共␮L=␮R兲,

and a pumping current IPflows.

共i兲 Open circuit: for an open circuit, the charge current vanishes Ic= 0. By solving Eqs.共1兲, 共2a兲, 共2b兲, 共3a兲, 共3b兲, and

共4兲, we find an electric voltage VP=共␮L−␮R兲/e due to the

spin-pumping current Issp,

VP共␪兲 = R共␪兲关␶ip共␪兲m ⫻ m˙ +␶op共␪兲m˙兴 · m0, 共11兲 with the magnetoresistance R共␪兲=1/G共␪兲. Both the spin-transfer torque in Eq.共9兲 and the charge pumping voltage in

Eq. 共11兲 are governed by the torkances. Equation 共11兲

con-firms the two-step process for the charge pumping: 共1兲 spin current pumped by m˙ ,共2兲 charge current generated by

pro-jecting on m0. Note that Eq.共11兲 entails all multiple scatter-ing in the spacer.

In Eq. 共11兲, the charge pumping voltage is related to the

torkances, which also govern the spin-transfer torque. Cur-rently, the latter can be accessed only indirectly by its effect on the current-induced magnetization dynamics for MTJs.38,39 _{Equation 共11兲 can be employed to measure the} spin-transfer torque or torkances in spin valves via FMR-induced voltages when the magnetoresistance R共␪兲 is ob-tained alongside as done by Urazhdin et al.40

共ii兲 Closed circuit: when the spin valve is short circuited, ␮L=␮R, Eqs.共1兲, 共2a兲, 共2b兲, 共3a兲, 共3b兲, and 共4兲 yield a

pump-ing current

IP共␪兲 = 关␶ip共␪兲m ⫻ m˙ +␶op共␪兲m˙兴 · m0. 共12兲 For comparison, in the presence of an applied current current

I,

Nst共␪兲 = R共␪兲I关␶ip共␪兲m ⫻ m0+␶op共␪兲m0兴 ⫻ m. 共13兲 Usually, there is a passive series resistance in addition to the magnetoresistance R共␪兲. The charge pumping voltage for the open circuit is insensitive to such a passive resistance共for an ideal voltage meter兲.

In transition-metal ferromagnets, giⱕ0.1gr,36 thus from

now on we neglect the imaginary part of the mixing conduc-tance, i.e., gi= 0,␶op= 0, and␶ip= e␩共␪兲gr/2␲.

IV. MAGNETIZATION-RELATED VOLTAGE NOISE IN SPIN VALVES

According to the fluctuation-dissipation theorem 共FDT兲, the electrical voltage fluctuations across a nonmagnetic con-ductor are associated with the electron linear momentum dis-sipation by the electrical resistance, which causes Joule heat-ing. In ferromagnets, there are also magnetization fluctuations associated with the angular-momentum dissipa-tion or magnetic damping. In magnetic heterostructures such as spin valves, the two fluctuations 共electric and magnetic兲 are coupled by the dynamical exchange of spin currents.

Electronic noise increases the magnetic fluctuations via the sptransfer effect. Inversely, the magnetization noise in-creases electronic fluctuations via spin/charge pumping.

We discussed in Sec. III the pumping voltage 共current兲 induced by an arbitrary motion of magnetization. Here the formalism is applied to the stochastic magnetization motion at thermal equilibrium; the thermal fluctuations of magneti-zation induce a pumping voltage共current兲, which on filtering by the static layer becomes a noisy voltage signal. In this section, we discuss this magnetization fluctuation-induced voltage noise VM共t兲 in a spin valve at thermal equilibrium,

the power spectrum of which is the Fourier transform of its time-correlation function,

SM共␻兲 = 2

冕

具VM共0兲VM共t兲典e−i␻tdt. 共14兲

As shown by Johnson23 _{and Nyquist,}24 _{the FDT⬘ relates} the noise power spectrum S共␻兲 to the real part of the imped-ance Z共␻兲 which characterizes the dissipation

S共␻兲 = 4kBT Re关Z共␻兲兴. 共15兲

We may calculate the noise spectrum from both sides of the FDT 共1兲 by computing the time correlation 具VM共0兲VM共t兲典

from the response function, then using Eq. 共14兲, and 共2兲 by

computing the frequency-dependent impedance Z共␻兲 of a spin valve, which consists of an electrical and a magnetic contribution, Z共␻兲=RE+ ZM共␻兲, then making use of the

Johnson-Nyquist formula Eq.共15兲. The electric part REgives

rise to a white Johnson-Nyquist noise of SE= 4kBTRE. In this

section, we focus on method 共1兲 and calculate the voltage noise spectrum for two special cases with m_{0 储}xˆ

共perpendicu-lar case兲 and m0 储zˆ共parallel case兲. In Appendix B, we repro-duce the spectrum for m0 储xˆ by using method共2兲.

In bulk ferromagnets, magnetic-moment dissipation is pa-rametrized by the Gilbert damping constant ␣₀, which is associated with thermal fluctuations of the direction of the magnetization vector.25_{The magnetization fluctuations} are caused by a fluctuating torque from the lattice, which is represented by a thermal random magnetic field

h0_{: −M}

totm⫻h0. The autocorrelator of h is25 具␥h_i0共t兲␥h0_j共0兲典 =2␥␣0kBT

M_tot ␦ij␦共t兲 = ⌺0␦ij␦共t兲,

with i , j = x , y 共assuming that the easy axis is along z兲. Simi-larly, the ferromagnet loses energy and angular momentum by spin pumping. The magnetic damping increment␣

⬘

must be accompanied by a fluctuating transverse spin current 共torque兲 Isflfrom the contacts,29which can be represented by

another random magnetic field h

⬘

: Is

fl = −Mtotm⫻h

⬘

with autocorrelator29 具␥hi

⬘共t兲

␥h

⬘共0兲典 =

j 2␥␣

⬘

kBT Mtot ␦ij␦共t兲 = ⌺⬘␦ij␦共t兲.

h

⬘

and h0are statistically independent具hi

⬘

hj

0_典=0.

Including spin pumping from the magnetization fluctua-tions and the fluctuating spin current from the contacts, the total instantaneous spin current through the F兩N interface between the spacer and the free layer is 关see Fig.1共b兲兴

Is共t兲 = Issp+ Isfl= Mtot

␥ 共␣

⬘

m⫻ m˙ −␥m⫻ h⬘兲. 共16兲 Due to the filtering by the static layer magnetization m0, the spin current Is共t兲 is converted into a charge current Ic共t兲

with efficiency ␩共␪兲. If the imaginary part of the mixing conductance is disregarded 共gi= 0 and ␶op= 0兲, an electrical voltage VM共t兲 is given by the same expression as Eq. 共11兲

with m⫻m˙ replaced by m⫻m˙−共␥/␣

⬘

兲m⫻h

⬘

,

VM共t兲 = R共␪兲␩共␪兲

2e

ប关m0· Is共t兲兴 = W共␪兲m0· f共t兲, 共17兲 with W共␪兲=2eR共␪兲␩共␪兲eMtot/␥ប. Assuming that m fluctu-ates around the zˆ axis共m⯝zˆ兲, f reads 共to the leading order in

m and h

⬘兲

fx共t兲 =␥hy

⬘

−␣

⬘

m˙y, 共18a兲 fy共t兲 = −␥hx

⬘

+␣

⬘

m˙x, 共18b兲 fz共t兲 =␥共myhx

⬘

− mxhy

⬘兲 +

␣

⬘

共mxm˙y− mym˙x兲. 共18c兲

The spectrum SM depends on the direction of the

polar-izer. We need to compute, e.g., Fx共t兲⬅具fx共0兲fx共t兲典 when

m0 储xˆ and Fz共t兲⬅具fz共0兲fz共t兲典 when m0 储zˆ. The correlators of f are composed of those between m˙ and/or h

⬘

, which in turn can be expressed by the transverse magnetic susceptibility ␹共␻兲 共in frequency domain兲 as the response to the magnetic field h = h0+ h

⬘

+ h

⬙

共h

⬘

and h

⬙

account for the random fields from the left and right interface of the free layer兲,

冋

mx共␻兲 my共␻兲

册

=␹共␻兲

冋

␥hx共␻兲 ␥hy共␻兲

册

. 共19兲

All correlators can be calculated from Eq.共19兲,41 具mi共t兲mj共0兲典 = ⌺ ␣

冕

1 ␻␹ij−共␻兲e−i␻t d␻ 2␲, 共20a兲 具m˙i共t兲mj共0兲典 = − ⌺ ␣

冕

i␹ij −_共␻兲e−i␻td␻ 2␲, 共20b兲 具m˙i共t兲m˙j共0兲典 = ⌺ ␣

冕

␻␹ij −_共␻兲e−i␻td␻ 2␲, 共20c兲 with⌺=⌺₀+ 2⌺⬘共the factor 2 comes from the two pumping interfaces兲 and␹ij −_{共␻兲=关␹} ij共␻兲−␹jiⴱ共␻兲兴/2i, and 具mi共t兲␥hj

⬘共0兲典 = ⌺⬘

冕

␹ij共␻兲e−i␻t d␻ 2␲, 共21a兲 具m˙i共t兲␥h

⬘共0兲典 = − ⌺⬘

j

冕

i␻␹ij共␻兲e−i␻t d␻ 2␲. 共21b兲 By taking all the correlators among m, m˙ , and h

⬘

into ac-count, we confirm that the dc spin/charge current vanishes at thermal equilibrium具Is共t兲典=具Ic共t兲典=0 as required by the

sec-ond law of thermodynamics.

SM x_{共␻兲 = 2W}2_共␲/2兲

冕

−⬁ ⬁ dtei␻t_F x共t兲. 共22兲

Using Eqs.共20兲 and 共21兲, we have

Fx共t兲 = ⌺

⬘

冋

␦共t兲 −␣

⬘

冕

␻␹yy−共␻兲e−i␻t d␻

2␲

册

. 共23兲 From Eqs.共22兲 and 共23兲,

SM

x_{共␻兲 = 2W}2_{共␲/2兲⌺⬘兵1 −␣}

_⬘

_␻Im关␹

yy共␻兲兴其, 共24兲

in terms of the imaginary part of the dynamic susceptibili-ties, i.e., the magnetic dissipation. A measurement of the former therefore determines the latter, serving as an alterna-tive to, e.g., FMR measurements.

When m0 储zˆ, SM z _共

␻兲 follows from Eq. 共22兲 by the

replace-ment Fx共t兲→Fz共t兲 and W共␲/2兲→W共0兲. According to Eq.

共18c兲, this involves four-point correlators, which can be

re-duced to two-point correlators by Wick’s theorem30,42 具abcd典=具ab典具cd典+具ac典具bd典+具ad典具bc典. After some tedious algebra, we reach SM z _{共␻兲 = 2W}2_{共0兲⌺⬘}2

再

1 ␣

⬘

兺

i Re关␹ii共0兲兴 −

冕

d␻

⬘

2␲ ␻− 2␻

⬘

␻

⬘

兺

i,j 共− 1兲␦ij␹ ij −_共␻

_⬘

_兲␹ i j − _共␻ −␻

⬘

兲

冎

, 共25兲 with x¯ = y and y¯ = x. For the antiparallel configuration

共m0 储−zˆ兲, the formula is identical to Eq. 共25兲 except that

W共0兲 is replaced by W共␲兲. Note that␹ for parallel and anti-parallel cases are different, as discussed in Appendix A.

With Eqs. 共24兲 and 共25兲, the calculation of the noise

power spectrum reduces to that of the magnetic susceptibility ␹ for the free layer magnetization. Similar to the Gilbert damping for the free layer magnetization in a spin valve,43_␹ in general depends on the magnetization configuration of the spin valve. We derive the angular dependent␹ in Appendix A.

For simplicity we continue with an isotropic form of the magnetic susceptibility for the free layer magnetization ␹, which includes the effect of the spin-pumping enhanced damping but not the multiple scattering of the spin-pumping current within the spacer

␹共␻兲 = 1

共␻0− i␣␻兲2−␻2

冉

␻0− i␣␻ − i␻

i␻ ␻0− i␣␻

冊

, 共26兲 with␣=␣₀+ 2␣

⬘

and␻₀=␥H_eff. Using this␹, we find

SM x_{共␻兲 = 2W}2_{共␲/2兲⌺⬘} ⫻

再

1 −␣

⬘

␣ 共1 +␣ 2_兲␻4_+␻ 0 2_␻2 关共1 +␣2_兲␻2_−␻ 0 2_兴2_{+ 4␣}2_␻2_␻ 0 2

冎

. 共27兲 A more accurate form of SM

x_{共␻兲 关Eq. 共}

B2兲兴 as calculated in

Appendix B using circuit theory is recovered by the method here by using the angle-dependent susceptibility tensor Eq. 共A11兲 instead of Eq. 共26兲 in Eq. 共24兲.

The square root of SM

x_{共␻兲 is plotted in Fig.}

2 for the pa-rameters in TableI, where␩is replaced by its ballistic limit

p/2 共left ordinate兲. Assuming that the spin valve resistance R

is dominated by the interface resistances, R⬀1/A, but does not depend on the free layer thickness d. Considering the volume ⍀=Ad and ␣

⬘

⬀1/d, ⌺

⬘

⬀␣

⬘

/⍀⬀1/共Ad2_{兲, the} white-noise background in SM

x_共

␻兲 scales like R2_⍀2_⌺

_⬘

_⬀1/A, and thus does not depend on d. The dip in Fig.2at the FMR frequency is remarkable. Its depth is proportional to ␣

⬘

, hence inversely proportional to the free layer thickness, whereas its width is proportional to␣␻₀. The constant back-ground of the spectrum is

冑

SM

x_{共␻兲⯝50 nV/}

冑

MHz, whereas the dip is about 4 nV/

冑

MHz. For comparison, the root-mean-square of the electrical contribution to the noise is

冑

SE共␻兲=

冑

4kBTR − SM

x_{共0兲⯝87 nV/}

冑

MHz. When m0 储zˆ, Eqs.共25兲 and 共26兲 lead to

SM z_{共␻兲 = 2W}2_共0兲⌺⬘2 ␻0

冋

1 ␣

⬘

−␣ 共1 +␣2_兲␻2_{+ 4␻} 0 2 共1 +␣2_兲2_␻2_{+ 4␣}2_␻ 0 2

册

. 共28兲 The square root of SM

z_{共␻兲 is plotted as the dashed curve in}

Fig.2共right ordinate兲. In contrast to S_Mx共␻兲, the prefactor in

SM

z _{共␻兲⬀R}2_⍀2_⌺

_⬘

2_/␣

_⬘

_⬀1/共A2_{d兲, therefore the noise decreases} TABLE I. Typical spin valve parameters共see text兲.

Quantity Values Ref.

Ms共Co兲 1.42⫻106 A m−1 4 ␥ 共Co兲 1.9⫻1011_{共T s兲}−1 ₄₄ ␣0共Co兲 0.01 45 2␣⬘共Co兩Cu兲 0.01 45 ␻0 10 GHz p 0.35 46and47

R⯝R_P 0.57 ⍀ Deriveda_{from Ref.4}

RSample 1.6 ⍀ 4

T 300 K

⍀=A⫻d 共130⫻130 nm2_{兲⫻2.5 nm 4} a_共R

AP− RP兲/RAP⬇p2 and RAP− RP= 0.073 ⍀4, where P/AP stands

for parallel/antiparallel. 0 10 20 30 40 50 60 0 0.5 1 1.5 20 1 2 3 4 5 6 [S x M(ω )] 1/2 (nV/MHz 1/2 ) [S z M(ω )] 1/2 (nV/MHz 1/2 ) ω/ω₀ T = 300K α = 0.02, α₀= 2α’ = 0.01

FIG. 2. 共Color online兲

冑

S_Mx共␻兲 共solid line with left scale兲 and

with increasing d. This differs from the case m0 储xˆ because the projection on the zˆ axis involves the average deviation of

m from the equilibrium direction, which is inversely

propor-tional to the volume. The divergence at vanishing thickness is caused by the neglect of the finite transparency of very thin magnetic layers for transverse spin currents. Similar to

SM

x_{共␻兲, the depth of the dip at ␻}

= 0 is proportional to ␣

⬘

, hence inversely proportional to the layer thickness and has a width proportional to␣␻0.

V. DISCUSSION The spectrum SM

x_{共␻兲 consists of three contributions,}

which can be seen from the decomposition of

Fx共t兲=具fx共t兲fx共0兲典=Fx sp + Fx fl + Fx ab with Fx sp_{共t兲 =␣}

_⬘

2_具m˙ y共t兲m˙y共0兲典, 共29a兲 Fxfl共t兲 =␥2具h

⬘共t兲h

y y

⬘共0兲典,

共29b兲 Fxab共t兲 = −␣

⬘

␥关具共t兲h

⬘共0兲典 + 具m˙

y y共0兲hy

⬘共t兲典兴.

共29c兲

These three contributions can be interpreted as 共i兲 a spin-pumping current Fx

sp

, which produces a peak at␻=␻0,共ii兲 a random torque 共spin current兲 from the contact Fxfl, whose

spectrum is white, and 共iii兲 the absorption of the random torque from the contact by the magnetization Fx

ab = −2Fx

sp , which gives a dip at␻=␻0 with twice the magnitude of共i兲. The contacts therefore provide a white-noise random torque over the ferromagnetic film, whereas the magnetization ab-sorbs the noise power around␻0. The spectrum SM

z_{共␻兲 also}

consists of three contributions, but the absorption line of the

zˆ component is centered at zero frequency because the

fluc-tuations of the zˆ-component magnetization do not have a characteristic frequency such as the xˆ , yˆ components. In in-homogeneous FM films, the single macrospin mode breaks up into different eigenmodes. The noise power spectrum can then provide a “fingerprint” of the various eigenmode fre-quencies.

The three-point correlators arising in具VM共0兲VM共t兲典 when

m₀ is at arbitrary angles in the xˆ-zˆ plane vanish for normal distributions. The power spectrum is then a linear combina-tion of SM

x

and SM z

, depending on the angle with dips at both the FMR and zero frequencies. The modeling of magnetic anisotropies by an easy axis is appropriate when the free layer magnetization is oriented normal to the plane in axially symmetric pillars. In standard pillars the dominant aniso-tropy is an easy plane, which leads to anisotropic fluctuations of the magnetization. The results remain qualitatively similar but become anisotropic in the xˆ-yˆ plane. For example, when a strong anisotropy constrains the fluctuations of m to the

yˆ-zˆ plane, S_My vanishes. We disregarded the imaginary part of the mixing conductance in our calculation for the noise power spectrum. When it is included, the symmetric dip in the power spectrum in Fig. 2is skewed by gisimilar to for

the spin diode effect discussed by Kupferschmidt et al.48_and Kovalev et al.32

For asymmetric spin valves, a nonmonotonic angular de-pendence of the magnetoresistance and a vanishing torkance

at a noncollinear magnetization configuration has been demonstrated.49–53_{A sign change in torkance leads to a sign} change in the charge pumping voltage. The magnetic contri-bution to the thermal noise vanishes at the zero torkance point.

VI. SUMMARY

In conclusion, we find that a pumping voltage arises in a spin valve when the free layer magnetization is in motion. The angular dependence of pumping voltage under FMR condition provides detailed information of the spin transport in spin valves. The pumping voltage induced by the thermal fluctuation of the free layer magnetization gives rise to addi-tional voltage noise, which is associated with the magnetiza-tion dissipamagnetiza-tion. Thus the equilibrium electronic noise in a spin valve consists of two contributions: the Johnson-Nyquist noise associated with the fluctuations of the charge and magnetization-related noise associated with the fluctua-tions of the spins. The magnitude of these two contribufluctua-tions can be comparable. Unlike the white Johnson-Nyquist noise, the latter is found to contain an absorption line at the FMR frequency共and at zero frequency depending on the configu-ration兲 on top of an enhanced white-noise background. The noise spectrum can provide a fingerprint of the magnetic eigenmodes in inhomogeneous structures.

ACKNOWLEDGMENT

This work was supported by EC under Contract No. IST-033749 “DynaMax.” J.X. and G.E.W.B. thank S.M. for his hospitality at Tohoku University. J.X. acknowledges support from H. J. Gao and S. Yi in Beijing and X. F. Jin in Shang-hai.

APPENDIX A: ANGULAR DEPENDENCE OF THE TRANSVERSE MAGNETIC SUSCEPTIBILITY

IN SPIN VALVES

In this appendix we calculate the transverse magnetic sus-ceptibility for the free layer magnetization in a spin valve when no external bias is applied共Ic= 0兲, i.e., the only driving

force is the thermal random field h. For an isolated magnet, the dynamics are described by the Landau-Lifshitz-Gilbert equation,

m˙ = −␥m⫻ 共Heff+ h兲 +␣0m⫻ m˙, 共A1兲 with the thermal magnetic field h and bulk damping param-eter ␣0. When we consider small amplitude precession around the zˆ direction 共assuming Heff 储zˆ兲, the Fourier trans-form of the linearized LLG equation becomes

冋

mx共␻兲 my共␻兲

册

=␹0共␻兲

冋

␥hx共␻兲 ␥hy共␻兲

册

, 共A2兲 with

␹0共␻兲 = 1 共␻0− i␣0␻兲2−␻2

冉

␻0− i␣0␻ − i␻ i␻ ␻0− i␣0␻

冊

. 共A3兲 In spin valves,␹for the free layer magnetization depends on the relative orientation of m₀and m because of the multiple scattering within the spacer, which depends on the orienta-tion of m0, acts as spin-transfer torque on m. We now lin-earize Eq.共6兲, again assuming that m fluctuates around the zˆ

axis with small amplitude共m⯝zˆ兲,

m˙x= −␻0my−␣m˙y+ ␥ Mtot N_stx+␥hy, 共A4a兲 m˙y= +␻0mx+␣m˙x+ ␥ Mtot N_sty−␥hx, 共A4b兲

where ␣=␣0+ 2␣

⬘

and 2␣

⬘

is the enhanced damping from the two interfaces of the free layer.

The circuit theory Eqs. 共1兲, 共2a兲, 共2b兲, 共3a兲, 共3b兲, and 共4兲

are coupled with the LLG equation in Eq.共A4兲 through m˙ in

Eq. 共3兲 and IR in Eq.共A4兲 and they have to be solved

self-consistently. We assume that m0 is static and tilted by an angle ␪ from zˆ, i.e., zˆ · m0= cos␪. Equations 共1兲, 共2a兲, 共2b兲, 共3a兲, 共3b兲, and 共4兲 can be converted to scalar equations by

taking the dot products with m, m₀, and m_⫻= m⫻m₀. Introducing the projections of an arbitrary vector

q :共q0_{, q}m_{, q}⫻_{兲⬅q·共m}

0, m , m⫻兲, setting gsf= 0 for simplicity, Eqs.共1兲, 共2a兲, 共2b兲, 共3a兲, 共3b兲, and 共4兲 become

0 = I =eg 2h共2␮L− p␮N 0_{兲 = −} eg 2h共2␮R− p␮N m_{兲, 共A5a兲} IL0= eg 2h共2p␮L−␮N 0_{兲, 共A5b兲} IL m = eg 2h共2p␮L−␮N 0_兲cos␪ −egr h 共␮N m −␮N 0 cos␪兲, 共A5c兲 IL⫻= − egr h ␮N ⫻_{, 共A5d兲} IR0= − eg 2h共2p␮R− g␮N m_兲cos␪ +egr h 共␮N 0_−␮ N m cos␪兲 +egr 2␲m˙ ⫻_, 共A5e兲 IR m = −eg 2h共2p␮R−␮N m_{兲, 共A5f兲} IR⫻= e hgr␮N ⫻₋egr 2␲m˙ 0_{, 共A5g兲} 0 = IL0− IR0= IL m − IR m = IL⫻− IR⫻, 共A5h兲

where in the third and fifth equations above, we disregard the time dependence of m0共t兲=m共t兲·m0 because m⯝zˆ to

lead-ing order in the deviations. The solutions to Eq.共A5兲 are

IR 0 =␰0 egr 4␲m˙ ⫻_{, I} R m =␰m egr 4␲m˙ ⫻_{, I} R ⫻_{= −}egr 4␲m˙ 0_{, 共A6兲} with ␰0= 1 −␯ 1 −␯2cos2␪ and ␰m= − 共1 −␯兲␯cos␪ 1 −␯2cos2␪ , 共A7兲 and ␯=关2gr− g共1−p2兲兴/关2gr+ g共1−p2兲兴. We now define the

coordinate system in the plane normal to zˆ,

xˆ⬅m0− zˆ cos␪

sin␪ and yˆ⬅ zˆ ⫻ xˆ ⬇

m_⫻

sin␪. 共A8兲 Therefore m˙0_⬇m˙

xsin␪, m˙⫻⬇m˙ysin␪, and

␥ Mtot N_stx⬇ ␥ប 2eMtot IR0− IR m cos␪ sin␪ = 1 2␰␣

⬘

m˙y, 共A9a兲 ␥ M_totNst y_⬇ ␥ប 2eM_tot IR⫻ sin␪= − 1 2␣

⬘

m˙x, 共A9b兲 with ␰=␰0−␰mcos␪, which is related to the

angular-dependent magnetic damping in Ref.43.

Plugging Eq. 共A9兲 into Eq. 共A4兲, after linearization in

terms of small fluctuations about the zˆ axis and Fourier trans-formation, we have

冋

␥hx共␻兲 ␥hy共␻兲

册

=

冉

␻0− i␣y␻ i␻ − i␻ ␻0− i␣x␻

冊

冋

mx共␻兲 my共␻兲

册

, 共A10兲 where␣x=␣− 1 2␰␣

⬘

and␣y=␣− 1

2␣

⬘

reflect the damping an-isotropy. From Eq.共A10兲,

␹共␻兲 =

冉

␻0− i␣y␻ i␻ − i␻ ␻0− i␣x␻

冊

−1 = 1 共1 +␣x␣y兲␻2−␻20+ i共␣x+␣y兲␻0␻ ⫻

冉

␻0− i␣x␻ − i␻ i␻ ␻0− i␣y␻

冊

. 共A11兲

Because ␣y depends on angle ␪ through ␰, ␹共␻兲 also

be-comes angle dependent, i.e., the magnetic-susceptibility function for the free layer magnetization in a spin valve is in general angular dependent, for the same reason as the mag-netic damping in spin valves.43 _{Equation 共A11兲 reduces to} Eq.共26兲 when we identify␣x⯝␣y⯝␣by ignoring the

back-flow correction to the damping.

APPENDIX B: SPIN VALVE IMPEDANCE Z(␻) FOR m0¸ xˆ Here, we calculate the frequency dependence of the im-pedance of a spin valve by applying a small ac current at frequency␻: I共␻兲. We consider the perpendicular case here, i.e., m0 储xˆ or cos␪= 0, so that the circuit theory equations equal Eq. 共A5兲 except that we allow for a nonvanishing

charge current I⫽0. Since we are now interested in the de-terministic response, the thermal random fields h may be ignored. We can then solve Eqs. 共A4兲 and 共A5兲

self-consistently in the frequency domain. We find that the im-pedance of the spin valve Z共␻兲=关␮L共␻兲−␮R共␻兲兴/eI共␻兲

con-sists of two parts, an electric part RE and a magnetic part ZM共␻兲:Z共␻兲=RE+ ZM共␻兲, RE= 4 G0 1 g+␩ 2_R2_G 0g共1 − p2兲 +␩2R2G0gr, 共B1a兲 ZM x =␩2R2G0gr

冋

1 − ␣

⬘

␻共␣y␻+ i␻0兲 共1 +␣x␣y兲␻2−␻20+ i共␣x+␣y兲␻␻0

册

, 共B1b兲 where␩=␩共␲/2兲 and R=R共␲/2兲 are the polarization factor and the dc resistance for the spin valve at ␪=␲/2 or m0 储xˆ.

RE consists of the resistances associated with the electrical

dissipation and half of the magnetic dissipation from the

in-terface with the static magnetization which does not emit a spin-pumping current.

By the Johnson-Nyquist formula Eq.共15兲, the noise

spec-trum SM x_{共␻兲 is given by} SM x = 4kBT Re关ZM x_共␻兲兴 = 2W2共␲/2兲⌺⬘ ⫻

再

1 −␣

⬘

␣y共1 +␣x␣y兲␻ 4_+␣ x␻2␻02 关共1 +␣x␣y兲␻2−␻0兴2+共␣x+␣y兲2␻2␻02

冎

. 共B2兲 This equation is identical to Eq.共27兲 when we that the limit ␣x⯝␣y⯝␣. This difference comes from the approximate

form of ␹ in Eq. 共26兲. If we use Eq. 共A11兲, then the SM x

calculated from Eq. 共24兲 will be exactly the same as Eq.

共B2兲. The frequency-dependent impedance for m0 储zˆ is sec-ond order in mx,y; therefore it is not so straightforward to

calculate, which can also be seen from the nontrivial convo-lution in Eq. 共25兲 calculated from magnetic-susceptibility

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