# 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, )

(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

(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}

(3) |

Here, and . The transition rates per energy are

(4) | ||||

(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)

(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 ,

(7) |

where the spin current and damping
^{2}*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,

(8) | ||||

(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,

(10) | ||||

(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 ,

(12) | ||||

(13) |

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

(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 ,

(15) |

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

(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)

(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}*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,

(18) | ||||

(19) | ||||

(20) | ||||

(21) | ||||

(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,

(23) |

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

(24) | ||||

(25) | ||||

(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,

(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)

(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

(29) |

where , and .

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

(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:

(31) | ||||

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