A Appendix: Details of derivation of the generating function

# Spin pumping and torque statistics in the quantum noise limit

## Abstract

We analyze the statistics of charge and energy currents and spin torque in a metallic nanomagnet coupled to a large magnetic metal via a tunnel contact. We derive a Keldysh action for the tunnel barrier, describing the stochastic currents in the presence of a magnetization precessing with the rate . In contrast to some earlier approaches, we include the geometric phases that affect the counting statistics. We illustrate the use of the action by deriving spintronic fluctuation relations, the quantum limit of pumped current noise, and consider the fluctuations in two specific cases: the situation with a stable precession of magnetization driven by spin transfer torque, and the torque-induced switching between the minima of a magnetic anisotropy. The quantum corrections are relevant when the precession rate exceeds the temperature , i.e., for .

Spin transfer torque, angular momentum contributed by electrons entering a magnet, can be used to control magnetization dynamics via electrical means, as demonstrated in many experiments. Slonczewski (1996); Tserkovnyak et al. (2005); Ralph and Stiles (2008) Often the effect can be described by considering the ensemble average magnetization dynamics, or taking only thermal noise into account. Brown (1963) The spin transfer torque is in general also a stochastic process, but at bias voltages large enough to drive the magnetization, it is not necessarily Gaussian nor thermal, Foros et al. (2005) especially at cryogenic temperatures. The statistical distribution of electron transfer and the associated torque in magnetic tunnel junctions can be described by counting statistics, Levitov and Lesovik (1993) via a joint probability distribution of charge, energy, and spin transferred into the magnet during time , . The distribution is conditional on the magnetization dynamics during time , which necessitates consideration of back-action effects.

Here we construct a theory describing the probability distribution for electron transfer via a Keldysh action (Eq. (2)) describing a metallic magnet with magnetization , coupled to a fermionic reservoir (another ferromagnetic metal), illustrated in Fig. 1. In the presence of a bias voltage in the reservoir, this coupling may lead to a stochastic spin transfer torque affecting the magnetization dynamics. Unlike some of the earlier discussions of counting and spin torque statistics Utsumi and Taniguchi (2015); Chudnovskiy et al. (2008); Tang and Wang (2014), we follow the approach of Ref. Shnirman et al. (2015) and retain geometric phase factors in the derivation of the generating function. This becomes relevant in the quantum limit where the precession rate is large compared to the temperature .

To study the implications, we suggest two specific settings (Fig. 1b,c), characterized by opposite regimes of the external field and anisotropy field . When , a suitably chosen voltage drives the magnet into a stationary precession with rate around the direction of . Slonczewski (1996); Berger (1996); Kiselev et al. (2003); Rippard et al. (2004) This precession pumps charge Tserkovnyak et al. (2002) and heat into the reservoir, along with the direct charge and heat currents due to the applied voltage. The noise of these currents depends on the intrinsic noise of the pumped current and, at low frequencies, also on the fluctuations of the magnetization, driven by the spin torque noise. The opposite limit is the one relevant for memory applications, as the spin transfer torque can be used to switch between the two stable magnetization directions Koch et al. (2004); Yakata et al. (2009). Our approach allows finding the switching rate at any temperature and voltage, also for .

Besides the average currents and noise, the Keldysh action allows us to calculate the full probability distribution of transmitted charge , energy , or change of the -component of magnetization in a nanomagnet with volume and spin , within a long measurement time . Here is the gyromagnetic ratio. The precise distribution depends on the exact driving conditions and the parameters of the setup. However, symmetries constrain the probability distribution, leading to a spintronic fluctuation relation (here and below, )

 Pt0(δn,δE,δs)=eVδn/TFeδE(T−1F−T−1F′)e−Ωδs/TFP′t0(−δn,−δE,δs), (1)

where corresponds to the case with reversed magnetizations. As in fluctuation relations presented earlier Crooks (1999, 2000); Seifert (2012); Esposito et al. (2009); Tobiska and Nazarov (2005); Utsumi and Taniguchi (2015), this allows for a direct derivation of Onsager symmetries, thermodynamical constraints, and fluctuation-dissipation relations, valid for the coupled charge-spin-energy dynamics (see Appendix).

Generating function. Consider a magnetic tunnel junction depicted in Fig. 1. The spin transfer torque due to tunnelling, and the corresponding counting statistics can be described by a Keldysh action obtained by integrating out conduction electrons in and . Chudnovskiy et al. (2008); Shnirman et al. (2015) We apply the approach of Ref. Shnirman et al., 2015 to the characteristic function describing the change in particle number and internal energy in the ferromagnetic lead . Esposito et al. (2009); Campisi et al. (2011) In the long-time limit, , this results to the action , where is the Berry phase for total spin . Moreover, the tunneling action is

 ST =i|W|2∫∞−∞dtdt′dϵ2πTrˇP(t)ˇGF′(t−t′)ˇP(t′)†ˇGF(ϵ), (2)

where contains the bias voltage , and the charge and energy counting fields and . The rotation matrix describes the direction of the magnetization in terms of Euler angles and . Keldysh fields are in the basis Kamenev and Levchenko (2010) , where is a Pauli matrix. Below, we fix the gauge Shnirman et al. (2015) so that , . We assume a spin and momentum independent tunneling matrix element . The conduction electrons are described by Keldysh–Green functions , with the exchange field of always parallel to in the rotating frame, .

Consider now the situation depicted in Fig. 1a, where precesses around due to an external magnetic field and/or magnetic anisotropy contributing potential energy . The corresponding action is , with . Separating out the fast motion , the dynamics of , are driven only by the spin transfer torque. We assume this dynamics is slow, and evaluate Eq. (2) under a time scale separation : 1

 ST≃−i∫∞−∞dtdϵ∑σσ′α=±Γσσ′α(ϵ)(eiαησσ′(ϵ)−1). (3)

Here, and . The transition rates per energy are

 Γσσ′α(ϵ) =¯Gσσ′1+σσ′cosθcl(t)2Λα(ϵ,V+Ωσσ′), (4) Λα(ϵ,V) ={fF′(ϵ)[1−fF(ϵ+V)],α=+,fF(ϵ+V)[1−fF′(ϵ)],α=−. (5)

Here, are Fermi distribution functions, and the time-averaged conductance is where , the polarizations are defined as , and is the polarization of the fixed magnet projected onto the precession axis. The densities of states of majority/minority spins are given at the Fermi level. The resulting is independent of , i.e. its dynamics decouples, which constrains (see Appendix).

The result describes Poissonian transport events, each associated with a back-action on due to the spin transfer torque, as described by the dependence on . The rates are proportional to the averaged densities of states and squared spin overlaps , in the frame rotating with the magnetic precession. The transferred energy consists of the voltage bias and the difference of energy shifts on the right and left sides of the junction in the rotating frame Tserkovnyak et al. (2005, 2008). The relation of this additional dependence on to geometric phases is discussed in Ref. Shnirman et al., 2015. It also separates Eq. (3) from the result of Ref. Utsumi and Taniguchi, 2015 for tunneling through a ferromagnetic insulator barrier, where such angular dependencies are not included.

Equation (3) is a main result of this work, as the knowledge of allows access to the statistics of charge, energy and spin transfer in the generic case depicted in Fig. 1a. Below, we describe some applications. First, we can identify the following spintronic fluctuation relation (see Appendix)

 ST(χ,ξ,ϕq)=S′T(−χ+iVTF,−ξ+iTF−iTF′,ϕq+iΩ2TF), (6)

where the prime denotes inverting the magnetizations and the sign of the precession. Identifying the conjugate fields of , and to the number of charges , change of energy and transfer of spin angular momentum , this relation is equivalent with Eq. (1). This relation also implies the Onsager relation relating the pumped current to the torque acting on the angle Brataas et al. (2011). This and further details of the fluctuation relation are discussed in the Appendix.

The average dynamics follows the component of the Landau-Lifshitz-Slonczewski equation, Slonczewski (1996) here obtained from stationarity of vs ,

 S˙θ=−sin(θ)τ(θ),τ(θ)=Ωα(θ)+Isz, (7)

where the spin current and damping 2 have been discussed in Ref. Chudnovskiy et al., 2008. The equation describes motion of in an effective potential defined by and the spin torque, illustrated in Fig. 1b. In certain parameter ranges, a fixed point appears — it can be either attractive or repulsive. This can correspond to a stable but fluctuating precession around the angle (Fig. 1b), induced by spin torque, or spin torque-induced switching between two energy minima (Fig. 1c).

Average current and noise. For fast measurements, , we can assume remains fixed, and find the average currents,

 ¯¯¯I =12G0[1+PF′PFzcosθ]V+14G0PFzΩsin2θ, (8) ¯¯¯¯¯¯¯˙QF =12¯¯¯IV+12τ(θ)Ωsin2θ, (9)

where the pumped charge current (second term in Eq. (8)) is that found in Ref. Tserkovnyak et al., 2008. The heat current is a sum of the Joule heat and the magnetic energy lost due to the spin torque, , dissipated equally in and . In contrast to the average values, the energy shifts remain in the noise of the currents,

 SI =∑σσ′¯Gσσ′1+σσ′cosθ2Vσσ′cothVσσ′2T, (10) S˙QF =∑σσ′¯Gσσ′1+σσ′cosθ6(π2T2+V2σσ′)Vσσ′cothVσσ′2T, (11)

where and . In the classical linear regime , the results reduce to a form dictated by the fluctuation-dissipation theorem and Wiedemann-Franz law, , , where is the electrical dc conductance of the magnetic tunnel junction, Huertas-Hernando et al. (2002) and the Lorenz number. The presence of the angle-dependent frequencies is revealed in the quantum noise regime . The noise in the pumped current for is plotted in Fig. 2 — the location of the quantum–classical crossover is pushed up to higher precession frequencies as the tilt angle approaches .

Spin torque induced fluctuating precession. The above results are conditional on a specific value of . For the full probability distribution, the distribution would need to be known.

To find , we assume and take a semiclassical approximation. Defining and , the action reads where is real for real , . The problem can then be analyzed as in Hamiltonian mechanics, , . Kamenev and Levchenko (2010) In a time-sliced discretization of the path integral, the restriction specifying the exact measured value adds a boundary condition that removes one of the integration variables and saddle point equations. This allows for a discontinuity of at , cf. Refs. Pilgram et al. (2003); Heikkilä and Nazarov (2009). The other boundary conditions are , so that relevant paths have integration constant .

Consider now fluctuations close to an attractive fixed point (cf. Fig. 1b). For dynamics driven by an external field, it is located at , and it is attractive if . The phase space picture is shown in Fig. 2b. Expanding around in terms of the torque and torque noise correlator ,

 H ≃[1−s2z][S−1τ(sz)p−S−2D(sz)p2], (12) D(sz) =18∑σσ′α=±(1−σσ′sz)21−s2zΓσσ′α, (13)

where for . The fluctuation contribution comes from following path A from to :

 Extra open brace or missing close brace (14)

where is a normalization constant. This agrees with Ref. Chudnovskiy et al., 2008 in the semiclassical limit , except for the presence of the energy shifts Shnirman et al. (2015) in the spin torque noise correlator , which are relevant in the quantum limit . The variance is plotted in Fig. 3a.

Long measurement times. For , the slow fluctuation of the magnetization contributes low-frequency noise to observables. This contribution is not small in : the typical excursion from the average position is small, , but it lasts for a long time , generating low-frequency noise . The situation is similar to noise induced in tunneling currents by temperature fluctuations on small islands. Laakso et al. (2010)

We now find the result within the semiclassical approximation. The counting fields are switched on in the interval , e.g. . They make the semiclassical path to transition from branch A to B in the time interval following a trajectory of constant . Two such trajectories are shown in Fig. 2a. For simplicity, we consider the limit with full polarization of the free magnet . Then, close to ,

 H ≃eα~χ[1−s2∗][τ(sz)Sp−D(s∗)S2p2]−Γ(sz)(eα~χ−1), (15)

where and . For quadratic , the Hamiltonian equations can be solved exactly (see Appendix). From this approach, we find the current noise:

 SI=Γ(s∗)+4Γ′(s∗)2σ2sτm(1−1−e−t0/τmt0/τm), (16)

where is the slow time scale associated with the spin transfer torque and the variance of the magnetization z-component in Eq. (14). The first term in Eq. (16) is the Poissonian shot noise (10), and the second term originates from magnetization fluctuations. The dependence on the measurement time is shown in Fig. 4. The current noise at frequencies can be used to probe the dynamics and distribution of the magnetization.

Spin torque induced stochastic switching. Magnetic anisotropy field results to an effective magnetic potential with two minima (see Fig. 1c), and the spin torque can induce switching between the two. Here, we take , and . The corresponding semiclassical Hamiltonian picture is shown in Fig. 2c. An unstable fixed point separates the two stable fixed points . The leading exponent of the rate of switching from to is, Utsumi and Taniguchi (2015)

 Γsw ∝e−Δsw=e∫s∗−1dszpsw(sz), H(sz,psw(sz)) =0, (17)

where is shown in Fig. 2c. The switching occurs deterministically () if as becomes unstable. At lower voltages, the switching is stochastic. Numerically computed results are shown in Fig. 3b. At zero temperature, the switching is blocked Utsumi and Taniguchi (2015) at for and otherwise. This occurs because the transition rates vanish for , and because the back-action vanishes for , . 3 The latter constraint is due to the additional angle dependence in the spin torque, which traces back to the geometric phases Shnirman et al. (2015) in the spin dynamics.

Discussion. In conclusion, we have derived a Keldysh action (3), describing the stochastic charge and energy currents affected by a precessing magnetization. We obtain a fluctuation relation for the transferred charge, energy, and magnetization. The noise in the current at low temperatures displays features related to geometric phases, and its low frequency component reflects the magnetization fluctuations. Information about the spin torque noise is also contained in the switching probability of anisotropic magnets. Our predictions are readily accessible in experiments probing spin pumping at low temperatures . Precession frequencies in range have been achieved, Kiselev et al. (2003); Rippard et al. (2004) which translates to .

We thank B. Nikolic and S. van Dijken for discussions. This work was supported by the MIUR-FIRB2013 - Project Coca (Grant No. RBFR1379UX), the Academy of Finland Centre of Excellence program (Project No. 284594) and the European Research Council (Grant No. 240362-Heattronics).

## Appendix A Appendix: Details of derivation of the generating function

We consider a tunneling Hamiltonian model for the ferromagnet/nanomagnet junction,

 H =∑kk′σσ′[c†σk(HF′)σk,σ′k′cσ′k′+d†σk(HF)σk,σ′k′dσ′k′]+HT+Hext, (18) (HF′)σk,σ′k′ =δk,k′[1ϵF′,k+gS⋅σ]σσ′, (19) (HF)σk,σ′k′ =δk,k′ϵF,σσ′k, (20) HT =∑σσ′kk′Wσk,σ′k′c†σkdσ′k′+h.c., (21) Hext =JHext⋅S. (22)

Above, () are conduction electrons in the free (fixed) magnet, is the magnetization in the free (single-domain) magnet, an externally applied field, and and coupling constants. Moreover, describe the noninteracting energy dispersions.

The Keldysh action corresponding to is,

 S =S0[S]+∫∞−∞dt(¯cT[i∂t−HF′]c+¯dT[i∂t−HF]d+¯cTWd+¯dTW†c), (23)

where is the standard spin action Abanov and Abanov (2002). We also include source terms in the generating function Kindermann and Pilgram (2004),

 Z[χ,ξ,ζ] =∫D[S,¯c,c,¯d,d]eiS+iSc+iSc2, (24) Sc[χ,ξ] =12∫∞−∞dt¯dTˇγx[˙χ+(HF−μ)˙ξ]d, (25) Sc2[ζ] =∫∞−∞dtζ∂tSclz (26)

so that derivatives vs. and produce cumulants of the charge and energy transfer, and characterizes . The terms added in can be eliminated with a change of variables and conversely for , provided . This results to addition of phase factors in , via . Here and below, we use Larkin-Ovchinnikov rotated Keldysh basis Kamenev and Levchenko (2010): fermion fields have Keldysh structure , where , are related to the fields on the Keldysh branches. Real fields are split similarly, as . is a Pauli matrix in the Keldysh space.

We integrate out the conduction electrons, and expand in small , Shnirman et al. (2015); Chudnovskiy et al. (2008). The resulting tunneling action can be obtained via the same route as in Ref. Shnirman et al., 2015,

 ST[χ,ξ,S]=i∫dtdt′Tr[R(t)GF′(t,t′)R(t′)†We−iγx[HFξ(t′)+χ(t′)]/2GF(t′,t)eiγx[HFξ(t)+χ(t)]/2W†]. (27)

Here, , , and the unitary matrices are defined by and the gauge degree of freedom in is fixed Shnirman et al. (2015) so that and is proportional to time derivatives of classical field components. The conduction electrons may also contribute other terms than , for example change (or generate) the total spin in Abanov and Abanov (2002). However, here we are mainly interested in spin torque and pumping, and therefore concetrate on dynamics implied by and assume any other effects are absorbed to changes in parameters or phenomenological damping terms.

Let us now consider the long-time limit correlation functions of the form , , which characterize a two-measurement protocol Esposito et al. (2009); Campisi et al. (2011). They correspond to choices , and . We can write (see below)

 e−iγxHNξ(t)GF(t,t′)eiγxHNξ(t′)=∫∞−∞dϵ2πe−iϵ(t−t′)e−iϵγxξ(t)GF(ϵ)eiϵγxξ(t′)+a(t,t′), (28)

where the correction term is zero for . As discussed below in more detail, it can be neglected in the long-time limit Kindermann and Pilgram (2004). The tunneling action then reads

 ST ≃i∫∞−∞dt∫∞−∞dϵ2πTr[¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯P(t,ϵ)GF′(t,t′)P(t′,ϵ)†WGF(ϵ)W†], (29)

where , and .

In the case considered in the main text, , and dynamics of and arises from the spin transfer torque. We have

 R(t) =e−i[Ωt+ψ0(t)]σz/2e−iσz[ˇ~ϕ+ˇ~ψ]/2cosˇθ2+e−i[Ωt−ψ0(t)]σz/2e−iσz[ˇ~ϕ−ˇ~ψ]/2(−iσy)sinˇθ2, (30)

where , . Keeping only the non-oscillating parts of Eq. (29) and taking the leading term of the gradient expansion vs. , , we can write the time average:

 ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯eiϵ(t−t′)P(t,ϵ)GF′(t,t′)P(t′,ϵ)†≃∫∞−∞dϵ′∑s=↑/↓σstr( (31) δ(ϵ−ϵ′−V−sΩ−(t))eiη−,s(ϵ,t)ˇγx/2cosˇθ(t)2ˇG1,s(ϵ′)cosˇθ(t)2e−iη−,s(ϵ,t)ˇγx/2 +δ(ϵ−ϵ′−V−sΩ+(t))eiη+,s(ϵ,t)ˇγx/2sinˇθ(t)2ˇG1,−s(ϵ′)sinˇθ(t)2e−iη+,s(ϵ,t)ˇγx/2),

where and . To this order, is independent of . Provided no source fields measuring the statistics of are added, the only part dependent on it is , which implies a constraint and we set . The slow part of the dynamics of the polar angle decouples from the rest of the problem.

The result Eq. (3) in the main text now follows, noting