Soft-Theorems for Scalar Particles: The Dilatons Story

Raffaele Marotta

Istituto Nazionale di Fisica Nucleare, Sezione di Napoli, Complesso
Universitario di Monte S. Angelo ed. 6, via Cintia, 80126, Napoli, Italy

We summarize recent results regarding single and double soft theorems of two
different particles named dilatons, the Nambu-Goldstone boson of the
spontaneously broken conformal field theories and the massless scalar particle
of the closed string theories. Similarities and differences between the soft
theorems of these two particles are discussed as well as their connections
with the symmetries of the theories.

##  Introduction

Symmetries are one of the most powerful guiding principle for constructing
consistent mathematical frameworks necessary to describe quantitatively the
fundamental laws of nature. All the theories of the known interactions are
based on such a principle and it turns out to be of great interest to
understand how the symmetries, eventually hidden, are manifest in the
behaviour of the physical observables. It is well known, for example, from the
pioneering works by Low [1, 2, 3, 4, 5] for gauge theories and Weinberg [6, 7,
8] in gravity that gauge symmetries determine the behaviour of the S-matrix in
the infrared region where gauge-bosons or gravitons carry low or soft momentum
[9, 10, 11]. In this soft-regime, scattering amplitudes with a soft graviton
or gauge boson factorize in the product of amplitudes with only hard or finite
energy particles and a soft contribution. This behaviour is universal at
leading order in the low momentum expansion and for tree level amplitudes
extends to subleading order for amplitudes with soft gravitons [11, 12, 13,
14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31]. Loop
amplitudes in four space-time dimensions, instead, are usually ill-defined
because of the presence of infrared divergences. These divergences make even
the leading soft factor not universal for Yang-Mills amplitudes [32] while for
gravity and abelian theories their effect consists in adding logarithmic
corrections to the subleading soft factors [33, 34, 35, 36]. The soft regime
of amplitudes with gravitons also provides predictions on the gyromagnetic
factors of specific massive higher spin states emerging in Kaluza-Klein
theories [37] that they have been confirmed by explicit amplitude calculations
[38].

The deep relation between symmetries and IR-behaviour of the $S$-matrix, have
been recently extended to the asymptotic symmetries of flat space-times
introduced by Bondy, Metzner, van de Burg and Sachs to study the gravitational
radiation at null infinity (BMS-group) [39, 40]. The Ward-identities
associated with the supertranslation generators of the BMS group give the
leading soft behaviour of amplitudes with external gravitons while the
subleading contribution originates from the super-rotation charges [41].

Symmetries determine also the IR behaviour of scattering processes with
massless scalar states [42, 43, 44, 45, 46, 47, 48, 49] although, recently, an
alternative geometric proof of such theorems for scalar particles has been
proposed in Ref. [50]. It is well known, for example, that when an internal
symmetry is spontaneously broken, as in the non-linear $\sigma$-models (NLSM),
the conservation law of the broken current requires the vanishing of
amplitudes with emission at zero momentum of a Goldstone boson(Adler’s zero)
[51, 52, 53, 54].

The situation is different when a space-time symmetry is spontaneously broken
as in the case of the breaking of the conformal group to the Poincaré
symmetry. In this breaking only one Nambu-Goldstone boson, in the following
named CFT-dilaton, appears in the spectrum of the theory [55]. Amplitudes with
a CFT-dilaton and an arbitrary number of massless states are vanishing to
leading order in the soft expansion of the dilaton momentum, but for massive
hard states, differently from the Adler-zero case, these are divergent at zero
dilaton momentum in full analogy with emission of soft-gauge bosons [45]. To
the subleading order instead, amplitudes with one soft dilaton are determined
by the conformal Ward-identity of broken scale invariance [56, 57, 58] while
only recently it has been shown that the subsubleading behaviour of soft
dilaton amplitudes is completely fixed from the Ward-identity associated with
the breaking of the special conformal generators [45, 59]. These relations
hold for any non-anomalous conformal field theory and they have been tested in
${\cal N}=4$ super-Yang Mills perturbatively up to one-loop and non-
perturbatively in the contest of gravity/dual on the Coulomb branch and by
considering instanton effective actions [59, 60].

Symmetries are also manifest in the double soft behaviour of amplitudes
computed in NLSM-models [53, 61, 62, 63, 64, 65]. Amplitudes with two soft
Nambu-Goldstone bosons factorize in amplitudes without the soft particles and
a contribution that capture the algebra of the broken generators. For the
dilaton, the double soft behaviour follows from the Ward-identities associated
with the breaking of the dilatation and special conformal transformation
generators. Furthermore, it turns out to be equal to two consecutive single
soft expansions [59].

There is another particle named dilaton, it is the massless scalar of the
gravitational multiplet of the closed string theories. Amplitudes with this
dilaton, referred to in these notes as gravity dilaton, satisfy single and
double soft-theorems, as well. These have been determined by computing
amplitudes in bosonic, heterotic and superstring theories and expanding them
in the relevant infrared region [11, 42, 44, 46, 48, 49].

The same behaviour has also been derived with a different approach based on
the property of the string theory to provide a unified expression for
amplitudes with external gravitons, dilaton and Kalb-Ramonds states. The
request of gauge invariance fixes, up to subsubleading order in soft
expansion, the soft behaviour of the whole amplitude including the dilaton
contribution [11, 66]. This contribution in the case of a single soft gravity
dilaton is conjectured to be universal being the same in all the cases studied
and shares many features with those of the CFT-dilatons. This similarity has
also led to argue the presence of hidden tree-level conformal symmetry in
ordinary Einstein gravity [67]. However, the resemblance of the two theorems
is less evident in the case of two soft dilatons because for the gravity
dilaton, differently from the CFT-dilaton, the double soft behaviour cannot be
obtained from two consecutive single soft expansions due to the presence of
four-point interactions [68].

The aim of this article, based on the results obtained in the papers of Ref.
[11, 44, 45, 46, 59, 66, 68] is to review the state of art of the single and
double soft-theorems satisfied by dilaton amplitudes. It will be given the
main results achieved in the literature providing few details on their
derivations but highlighting similarities and differences between the IR-
behaviour of the amplitudes with the two dilatons.

The paper is organized as follows. In the Section 2, are summarized the main
features of conformal field theories necessary to introduce the results
considered in the article. In its subsection are discussed the Ward-identities
of the conformal field theory spontaneously broken and their connections with
the infrared properties of amplitudes with CFT-dilatons. In Section 3 are
introduced the main properties of the string amplitudes used in the subsection
to discuss the soft theorems for the gravity-dilatons. In the discussion
section are commented the achieved results.

##  CFT-prelude

A conformal transformation is an invertible map $x\rightarrow x^{\prime}$ that
leave the metric invariant up to a scale factor
$g_{\mu\nu}(x)\rightarrow\Lambda(x)g_{\mu\nu}(x)$ 111More details on conformal
field theories and their breaking can be found in Refs. [69, 70, 55].. The
conformal group is locally isomorphic to $SO(2,d)$ with $d$ the space time
dimensions and it can be seen as an extension with dilatations ${\cal D}$ and
special conformal transformations $\mathcal{K}_{\mu}$ of the Poincarè group.
The generators are:

$\displaystyle\mathcal{P}^{\mu}=i\partial^{\mu}\hskip
76.82234pt\mathcal{J}^{\mu\nu}=-i(x^{\mu}\partial^{\nu}-x^{\nu}\partial^{\mu})-\mathcal{S}^{\mu\nu}$
$\displaystyle\mathcal{D}=i(\Delta+x_{\mu}\partial^{\mu})\,\hskip
31.2982pt\mathcal{K}^{\mu}=i(2x^{\mu}x_{\nu}\partial^{\nu}-x^{2}\partial^{\mu}+2\Delta
x^{\mu})+2x_{\nu}\mathcal{S}^{\mu\nu}$ (2.1)

with $\mathcal{P}^{\mu}$ and $\mathcal{J}^{\mu\nu}$ the generators of the
translations and Lorentz transformations respectively and ${\cal S}^{\mu\nu}$
the spin angular momentum operator. $\Delta$ denotes the scaling dimension of
the fields involved in the transformation. The action of the dilation and
special transformation on a scalar field $\phi(x)$ with conformal dimension
$\Delta$ is:

$\displaystyle\delta\phi(x)=[{\cal
D},\phi(x)]=i\left(\Delta+x^{\mu}\partial_{\mu}\right)\phi(x)$
$\displaystyle\delta_{(\lambda)}\phi(x)=\left[\mathcal{K}_{\lambda},\phi(x)\right]=i\left[(2x_{\lambda}x_{\nu}-\eta_{\lambda\nu}x^{2})\partial^{\nu}+2\,\Delta\,x_{\lambda}\right]\phi(x)\,.$
(2.2)

The Nöther currents associated to the scale and special conformal
transformations

$\displaystyle J^{\mu}_{\cal
D}=x_{\nu}\,T^{\mu\nu}\qquad(J_{\mathcal{K}})^{\mu}_{\rho}=(2x_{\nu}\,x_{\rho}-\eta_{\rho\nu}\,x^{2})T^{\mu\nu}$
(2.3)

are conserved if the improved energy momentum tensor $T^{\mu\nu}$ is conserved
and traceless.

The conformal symmetry is spontaneosly broken when a scalar field acquires a
non zero vacuum expectation value (vev).

The vacuum remains invariant under the Poncaré group but as consequence of the
Goldstone’s theorem, one massless boson, the CFT-dilaton $\xi(x)$, appears in
the spectrum parametrizing the oscillations around the nonconformal vacuum.
The dilaton couples linearly to the energy momentum tensor of the broken
theory and through the equations of motion contributes to its trace:

$\displaystyle T_{\mu}^{\mu}=-v\,\partial^{2}\xi\ ,$ (2.4)

where $v$ is a dimensionful constant related to the vev of the scalar field.
As a consequence, the currents of the broken generators are not conserved
being their divergences related to the trace of the energy momentum tensor
through the identies:

$\displaystyle\partial_{\mu}\,j_{\cal
D}^{\mu}(x)=T_{\mu}^{~{}\mu}(x)=-{v}\,\partial^{2}\,\xi(x)~{}~{};~{}~{}\partial_{\mu}\,j^{\mu}_{\phantom{\mu}(\lambda)}$
$\displaystyle=2\,x_{\lambda}\,T^{\mu}_{\phantom{\mu}\mu}=-2\,{v}\,x_{\lambda}\,\partial^{2}\,\xi(x)$
(2.5)

In the spontaneously broken phase, the theory is not any more conformal but
the memory of the original symmetry is conserved in the infrared behaviour of
the S-matrix through the so called soft-theorems. These follow from the Ward-
identities associated to the broken generators and assert that amplitudes with
one or more soft-dilatons are obtained acting with suitable operators related
to the broken generators on amplitudes without the soft particles.

###  Soft theorems from Ward-identities

We consider a conformal invariant field theory whose conformal symmetry is
spontaneously broken by the vev acquired by some scalar field. In a such
theory we analyse the matrix element among the divergences of set of broken
currents and a string of scalar fields

$\displaystyle\frac{\partial}{\partial
y_{1}^{\mu_{1}}}\dots\frac{\partial}{\partial y_{m}^{\mu_{m}}}T^{*}\langle
0|j_{1}^{\mu_{1}}(y_{1})\dots
j_{m}^{\mu_{m}}(y_{m})~{}\phi(x_{1})\dots\phi(x_{n})|0\rangle\,,$ (2.6)

$T^{*}$ denotes the usual Time-ordered product with all the derivative placed
outside the time ordering [71]. The derivatives of these matrix elements with
respect the current coordinates determine the Ward-identities of the
spontaneously broken conformal field theory.

In the case of a single current, in momentum space, one gets:

$\displaystyle\int d^{d}x\,\text{e}^{-iq\cdot
x}\Bigl{[}-\partial_{\mu}\,T^{*}\langle
0|j^{\mu}(x)\phi(x_{1})\dots\phi(x_{n})|0\rangle+T^{*}\langle
0|\partial_{\mu}j^{\mu}(x)\phi(x_{1})\dots\phi(x_{n})|0\rangle\Bigr{]}$
$\displaystyle=-\sum_{i=1}^{n}\,\text{e}^{-iq\cdot x_{i}}\,T^{*}\langle
0|\phi(x_{1})\dots\delta\phi(x_{i})\dots\phi(x_{n})|0\rangle\,.$ (2.7)

The matrix element is evaluated up to the order ${\cal O}(q^{0})$ in the
momentum expansion $q$ of the current. The first term on left side of this
equation is neglected after an integration by part under the assumption of
absence of poles at $q=0$. For the broken dilation and special conformal
transformation, Equation (2.7) through the divergences of the currents given
in Equation (2.5) relates the correlation functions with the insertion of a
dilaton field with low momentum to those without the dilaton but depending on
the variations of the fields under the considered transformations. This
identity contains soft-theorems at a very preliminary stage and the LSZ-
reduction translates them in a relation between amplitudes. The LSZ-operator

$\displaystyle\Bigl{[}\text{LSZ}\Bigr{]}\equiv
i^{n}\left(\prod_{j=1}^{n}\lim_{k_{j}^{2}\rightarrow-m_{j}^{2}}\int
d^{D}x_{j}\,\text{e}^{-ik_{j}\cdot
x_{j}}(-\partial_{j}^{2}+m_{j}^{2})\right)\,,$

is applied on both sides of Equation (2.7) with the on-shell limit
$k_{j}^{2}\to-m_{j}^{2}$ performed only to the end of the calculation. The
current whose divergence gives the amplitude with the insertion of the
dilaton, is not truncated by the LZS-reduction formula. The presence in the
divergence of the current of the operator $-\partial^{2}$ amputates the
dilaton propagator determining a relation between two truncated amplitudes.
This in the case of the dilatation current, up to the order ${\cal O}(q^{0})$
in the soft-expansion, gives:

$\displaystyle{v}\,\mathcal{T}_{n+1}(q;k_{1},\ldots,k_{n})=\Bigg{\\{}d-n\,\Delta-\sum_{i=1}^{n-1}k_{i}^{\mu}\frac{\partial}{\partial
k_{i}^{\mu}}-\sum_{i=1}^{n}\frac{m_{i}^{2}}{k_{i}\cdot
q}\left(1+q^{\mu}\frac{\partial}{\partial
k_{i}^{\mu}}\right)\Bigg{\\}}\mathcal{T}_{n}(k_{1},\ldots,\bar{k}_{n})\,.$
(2.8)

with $\bar{k}_{n}=-q+\sum_{i=1}^{n-1}k_{i}$. Equation (2.8) the amplitude with
the insertion of a soft dilaton is obtained acting with the generator of the
dilatation, written in the momentum space, $\hat{\cal
D}=-ik_{\mu}\frac{\partial}{\partial k_{\mu}}$, on the amplitude without the
dilaton. In the case of hard or finite energies massless field, the leading
divergent contribution is vanishing. For massive hard particles, instead, the
presence of the pole in $q$ makes the leading soft-theorem of the dilaton very
similar to that satisfied by the gauge bosons of the fundamental interactions.
More details on such derivation are given in Refs. [45, 59], here it is
worthwhile to comment that in Equation (2.8) we have commuted the generator of
the translations with the delta-function over the momenta and in the leading
divergent term we have used the prescription to perform the on-shell limit
before the soft one.

The same analysis is now performed for broken current generating the special
conformal transformations. The calculation is very similar to the previous one
with the only difference that the operator of the special conformal
transformations commutes with the delta over the momenta and no extra factors,
as the space-time dimension $d$ as in Equation (2.8), is introduced. The
resulting Ward-identity is:

$\displaystyle{v}\,q^{\lambda}\frac{\partial}{\partial
q^{\lambda}}\mathcal{T}_{n+1}(q;k_{1},\ldots,-q-\sum_{j=1}^{n-1}k_{j})$
$\displaystyle=$
$\displaystyle\sum_{i=1}^{n}\left\\{\frac{m_{i}^{2}}{k_{i}\cdot
q}\left(1-\frac{1}{2}q^{\mu}q^{\lambda}\frac{\partial^{2}}{\partial
k^{\mu}\partial
k^{\lambda}}\right)\right.\left.-q^{\lambda}\left[k_{i}^{\mu}\left(\frac{\partial^{2}}{\partial
k_{i}^{\mu}\partial
k_{i}^{\lambda}}-\frac{1}{2}\eta_{\mu\lambda}\frac{\partial^{2}}{\partial
k_{i\nu}\partial k_{i}^{\nu}}\right)+d\frac{\partial}{\partial
k_{i}^{\lambda}}\right]\right\\}$ (2.9)
$\displaystyle\times\mathcal{T}_{n}(k_{1},\ldots,\bar{k}_{n})+\mathcal{O}(q^{2})\,.$

The ${\cal O}(q^{1})$ of the amplitude with one dilaton and $n$-hard states
$\mathcal{T}_{n+1}$ is obtained by decomposition the amplitudes as the sum of
the leading, subleading and subsubleading soft terms:

$\displaystyle\mathcal{T}_{n+1}=$
$\displaystyle\left[\sum_{i=1}^{n}\frac{\mathbf{S}_{i}^{(-1)}(q)}{k_{i}\cdot
q}+\mathbf{S}^{(0)}+q^{\mu}\mathbf{S}^{(1)}_{\mu}\right]\,\mathcal{T}_{n}(k_{1},\ldots,k_{n})+\mathcal{O}(q^{2})\,,$
(2.10)

where the $\mathbf{S}^{(0)}$ and $\mathbf{S}_{\mu}^{(1)}$ are operators
dependent only on the momenta $k_{j}$, while $\mathbf{S}^{(-1)}$ may depend on
$q$ as well and replacing such expression in Equation (2.9). The leading
${\cal O}(q^{-1})$ term cancels on both sides leading to an identity which
determines:

$\displaystyle
v\mathbf{S}_{i}^{(-1)}\Big{|}_{\mathcal{O}(q^{2})}=-m_{i}^{2}\left(\frac{1}{2}q^{\mu}q^{\lambda}\frac{\partial^{2}}{\partial
k^{\mu}\partial k^{\lambda}}\right)\,,$ (2.11) $\displaystyle
v\mathbf{S}_{\lambda}^{(1)}=\sum_{i=1}^{n}\left[\frac{k_{i\lambda}}{2}\frac{\partial^{2}}{\partial
k_{i\nu}\partial k_{i}^{\nu}}-k_{i}^{\mu}\frac{\partial^{2}}{\partial
k_{i}^{\mu}\partial k_{i}^{\lambda}}-d\frac{\partial}{\partial
k_{i}^{\lambda}}\right]$ (2.12)

The Ward-identities of the scale and special conformal transformation
determine through the subsubleading order the amplidude with one soft-dilaton
in terms of the amplitude without the ditaton through the relation:

$\displaystyle{v}\,{\cal{T}}_{n+1}(q;k_{1},\ldots,k_{n})=\Bigg{\\{}-\sum_{i=1}^{n}\frac{m_{i}^{2}}{k_{i}\cdot
q}\left(1+q^{\mu}\frac{\partial}{\partial
k_{i}^{\mu}}+\frac{1}{2}q^{\mu}q^{\nu}\frac{\partial^{2}}{\partial
k_{i}^{\mu}\partial
k_{i}^{\nu}}\right)+d-n\Delta-\sum_{i=1}^{n}k_{i}^{\mu}\frac{\partial}{\partial
k^{\mu}_{i}}$
$\displaystyle\left.-q^{\lambda}\sum_{i=1}^{n}\left[\frac{1}{2}\left(2\,k_{i}^{\mu}\frac{\partial^{2}}{\partial
k_{i}^{\mu}\partial
k_{i}^{\lambda}}-k_{i\,\lambda}\frac{\partial^{2}}{\partial k_{i\nu}\partial
k_{i}^{\nu}}\right)+\Delta\,\frac{\partial}{\partial
k_{i}^{\lambda}}\right]\right\\}\mathcal{T}_{n}(k_{1},\ldots,\bar{k}_{n})+{\cal{O}}(q^{2})\,.$
(2.13)

The terms proportional to the mass of the external states show a pole in the
momentum of the soft particle, in perfect analogy with the the Low and
Weinberg soft behaviours, and are determined by the cubic couplings between a
dilaton and two massive states. It may also be though as a $q$-expansion of
the $n+1$-points amplitude. The regular terms are not vanishing to the ${\cal
O}(q^{0})$ and are fixed by action of broken generators, dilatation ${\cal
D}=-ik_{\mu}\partial_{k_{\mu}}$ and special conformal transformations ${\cal
K}_{\mu}=k_{\mu}\partial_{k}^{2}-2(k\cdot\partial_{k})\partial_{k^{\mu}}$, on
the amplitude without the emission at zero momentum of the dilaton.

The double soft behaviour of the amplitude with the emission of two dilatons
at zero momenta is determined starting from the matrix elements with the
insertion of two broken currents and $n$-finite energy states. The two
currents can be either of the same nature, dilatation or special conformal
transformation, or of different kind. Three sets of double-soft Ward-
identities may in principle be computed and they are related, through LZS-
reduction, to the terms in the soft expansion of the amplitudes with the
emission of two dilatons at zero momenta. This calculation has only been
performed for external hard massless states and it is similar but more
involved of the single soft identities. The leading ${\cal O}(q^{0})$ term of
the amplitude is obtained from the Ward-identities with two dilatation
currents while the subleading behaviour is obtained from those with mixed
broken generators. The subsubleading ${\cal O}(q^{2})$ contribution of the
amplitude would be determined by the Ward-identities with two special
conformal transformation currents. These have not been possible to determine
because as discussed in Ref. [59], they would require a formulation of the
single soft-theorem to ${\cal O}(q^{2})$ which is still missing. The universal
behaviour of an amplitude with two dilatons at zero momenta is given in terms
of the amplitude without the soft particles by the expression:

$\displaystyle
v^{2}T_{n+2}(q,k,k_{1},\ldots,\bar{k}_{n})=\Bigg{[}\left(d-\Delta_{\xi}-\sum_{i=1}^{n}\left(\Delta_{i}+k_{i}\cdot\partial_{k_{i}}\right)\right)\left(d-\sum_{i=1}^{n}\left(\Delta_{i}+k_{i}\cdot\partial_{k_{i}}\right)\right)$
$\displaystyle+(q_{\mu}+k_{\mu})\sum_{i=1}^{n}\left(\frac{1}{2}k_{i}^{\mu}\partial_{k_{i}}^{2}-\left(\Delta_{i}+k_{i}\cdot\partial_{k_{i}}\right)\partial_{k_{i}}^{\mu}\right)\left(D-\Delta_{\xi}-\sum_{i=1}^{n}\left(\Delta_{i}+k_{i}\cdot\partial_{k_{i}}\right)\right)\Bigg{]}T_{n}(k_{1},\ldots,\bar{k}_{n})$
$\displaystyle+{\cal O}(q^{2},k^{2},qk)$

It is straightforward to verify that the double soft behaviour coincides with
the one obtained by considering two consecutive single soft expansions. This
observation has led to conjecture that the behaviour of amplitudes with
multiple emission of soft dilatons is universal and it is fixed by the
consecutive soft limits of amplitudes with dilatons at zero momentum emitted
one after the other.

##  String Amplitudes

In string theories, amplitudes are very compact expressions given by complex
integrals of correlation functions of vertex operators evaluated on Riemann
surfaces of a given genus that represent the world-sheet of strings
propagating in the space-time. At genus zero, the sphere is the relevant tree-
level closed string world-sheet and the superstring vertex operators
associated with the massless states in the $(-1,\,-1)$ and $(0,0)$ super-ghost
pictures take the form 222An introduction to superstring amplitudes can be
found in the textbooks of Refs. [72, 73].:

$\displaystyle V^{(-p,\,-p)}=\frac{\kappa_{d}}{2\pi}\int
d\theta~{}\theta^{p}~{}V^{(p)}(z,\,\theta;k)\int
d\bar{\theta}~{}\bar{\theta}^{p}~{}\bar{V}^{(p)}(\bar{z},\,\bar{\theta};k)\,,$
(3.14)

with $\kappa_{d}$ the gravitational coupling constant and $z$ the complex
coordinate parametrizing the insertion on the world-sheet of the vertex
operators

$\displaystyle
V^{(p)}(z,\,\theta;k)=e^{-p\phi(z)}~{}\epsilon_{\mu}~{}DX^{\mu}\,e^{i\sqrt{\frac{\alpha^{\prime}}{2}}k\cdot
X(z,\,\theta)}\,.$ (3.15)

Here, $\theta$ and $\bar{\theta}$ are Grassmannian variables,
$\epsilon_{\mu}\bar{\epsilon}_{\nu}=\epsilon_{\mu\nu}$ is the polarization of
the massless state, $k$ in the momentum of the external state, and the
superfield notation is given by

$\displaystyle X^{\mu}(z,\theta)$ $\displaystyle\equiv
x^{\mu}(z)+\theta\psi^{\mu}(z)\,,\quad
D\equiv\frac{\partial}{\partial\theta}+\theta\frac{\partial}{\partial z}\,.$
(3.16)

This vertex describes collectively all $NS-NS$ massless states, these are the
graviton $g_{\mu\nu}$, the antisymmetric two form $B_{\mu\nu}$ and the scalar
field (gravity dilaton) $\phi$ of the massless gravitational string multiplet.
These fields are identified with the symmetric, anti-symmetric and trace of
the external polarization, through the relation:

$\displaystyle\epsilon_{\mu}\bar{\epsilon}_{\nu}$
$\displaystyle=\left[\frac{\epsilon_{\mu}\bar{\epsilon}_{\nu}+\epsilon_{\nu}\bar{\epsilon}_{\mu}}{2}-\frac{\epsilon_{\mu\nu}^{\perp}}{\sqrt{d-2}}\,\epsilon\cdot\bar{\epsilon}\right]+\left[\frac{\epsilon_{\mu\nu}^{\perp}}{\sqrt{d-2}}\,\epsilon\cdot\bar{\epsilon}\right]+\left[\frac{\epsilon_{\mu}\bar{\epsilon}_{\nu}-\epsilon_{\nu}\bar{\epsilon}_{\mu}}{2}\right]$
(3.17)
$\displaystyle=g_{\mu\nu}(k)+\frac{\epsilon_{\mu\nu}^{\perp}}{\sqrt{d-2}}\phi(k)+B_{\mu\nu}(k)$

where

$\displaystyle\epsilon_{\mu\nu}^{\perp}=\frac{\eta_{\mu\nu}-k_{\mu}\bar{k}_{\nu}-k_{\nu}\bar{k}_{\mu}}{\sqrt{d-2}}\,,\quad
k^{2}=\bar{k}^{2}=0\,,\quad k\cdot\bar{k}=1$ (3.18)

such that $\eta^{\mu\nu}\epsilon_{\mu\nu}^{\perp}=\sqrt{d-2}$, and
$\epsilon_{\mu\nu}^{\perp}{\epsilon^{\perp}}^{\mu\nu}=1$, while $\bar{k}$ is
an unphysical reference momentum. The relevant expectation values for the
world-sheet fields are:

$\displaystyle\langle
X^{\mu}(z_{1},\theta_{1})X^{\nu}(z_{2},\theta_{2})\rangle$
$\displaystyle=-\eta^{\mu\nu}\log(z_{1}-z_{2}-\theta_{1}\theta_{2})\,,$ (3.19)
$\displaystyle\langle e^{-\phi(z_{1})}\,e^{-\phi(z_{2})}\rangle$
$\displaystyle=\frac{1}{z_{1}-z_{2}}\,.$ (3.20)

The superstring amplitude with $n+1$ massless external states is:

$\displaystyle M_{n+1}=$
$\displaystyle\frac{8\pi}{\alpha^{\prime}}\left(\frac{\kappa_{d}}{2\pi}\right)^{n-1}\int\frac{d^{2}z\prod_{i=1}^{n}d^{2}z_{i}d\theta
d\bar{\theta}}{dV_{abc}|z_{1}-z_{2}|^{2}}\bigg{[}\prod_{i=1}^{2}d\theta_{i}\theta_{i}\prod_{i=3}^{n}d\theta_{i}\bigg{]}\bigg{[}\prod_{i=1}^{2}d\bar{\theta}_{i}\bar{\theta}_{i}\prod_{i=3}^{n}d{\bar{\theta}}_{i}\bigg{]}$
$\displaystyle\times\langle 0|\int d\varphi\,\,e^{i\left(\varphi\epsilon
DX(z,\theta)+\sqrt{\frac{\alpha^{\prime}}{2}}qX(z,\theta)\right)}\prod_{i=1}^{n}\left(\int
d\varphi_{i}\,\,e^{i\left(\varphi_{i}\epsilon_{i}D_{i}X(z_{i},\theta_{i})+K_{i}X(z_{i},\theta_{i})\right)}\right)|0\rangle$
$\displaystyle\times\langle 0|\int
d{\bar{\varphi}}\,\,e^{i\left({\bar{\varphi}}{\bar{\epsilon}}{\bar{D}}X({\bar{z}},{\bar{\theta}})+\sqrt{\frac{\alpha^{\prime}}{2}}qX({\bar{z}},{\bar{\theta}})\right)}\prod_{i=1}^{n}\left(\int
d{\bar{\varphi}}_{i}\,\,e^{i\left({\bar{\varphi}}_{i}{\bar{\epsilon}}_{i}{\bar{D}}_{i}X({\bar{z}}_{i},{\bar{\theta}}_{i})+K_{i}X({\bar{z}}_{i},{\bar{\theta}}_{i})\right)}\right)|0\rangle\,$
. (3.21)

Here, $dV_{abc}$ is the volume of the Möbius group, the states with the
indices $1$ and $2$ are in the $(-1,-1)$ picture, while the others are in the
$(0,0)$ picture and the factor
$\prod_{i=1}^{2}\theta_{i}\bar{\theta}_{i}/|z_{1}-z_{2}|^{2}$ comes from the
correlator of the superghosts.

Equation (3.21) is the starting point for computing amplitudes with $n+1$
external massless states. In such an expression, one vertex operator depending
on the complex super-coordinate $Z=(z,\,\theta,\,\bar{\theta})$ and momentum
$q$ is kept separated from all the other external legs. It will be considered
the vertex of the soft particle whose momentum will be expanded up to order
${\cal O}(q)$. Soft theorems will be obtained by integrating over the super-
coordinate $Z$ and expressing the result of the integration by an operator
acting on the amplitude without the soft-leg. This operator, in general,
contains $\alpha^{\prime}$ string corrections, being $\alpha^{\prime}$ the
Regge slope. The leading order in $\alpha^{\prime}$ gives the correct field
theory result. Furthermore, Equation (3.21) provides in a unique expression
the amplitudes for the graviton, the Kalb-Ramond and the gravity dilaton. The
analyses of the IR properties of these amplitudes give, in a single
expression, the soft theorems of these three different particles. In the next
section, we will review those of the gravitational dilaton highlighting the
similarities and differences with those of the conformal dilaton.

###  Soft-dilaton behaviour

The analyses of the infrared properties of amplitudes with Gravitational
dilatons have a long history. They were already studied in the old dual models
in connection with the renormalization of the Regge slope [42] and in the
framework of closed string field theories [74]. In this section we review the
recent extension of these old results by studying the soft properties of
amplitudes with scalar particles. The starting point of this analysis is
Equation (3.21) supplemented by the observation that the amplitude after
having integrated over the Grassmann variables associated to the soft vertex
reduces to an expression that is factorized at integrand level:

$\displaystyle M_{N+1}=M_{n}\star S$ (3.22)

where the $\star$ denotes a convolution integral between the $n$-point
amplitude $M_{n}$ and the integral $S$ which collects all the dependence from
the soft-leg. This quantity is the sum of three terms
$S=S_{b}+S_{s}+\bar{S}_{s}$ where $S_{b}$ receives contributions form the
purely bosonic degrees of freedom and it is equal to the similar expression
computed in bosonic string theory while $S_{s}$ and its complex conjugate
$\bar{S}_{s}$ receive contribution from the supersymmetric partners of the
bosonic degrees of freedom. This decomposition turns out to be very useful
because from $S_{b}$ one can deduce the soft function in bosonic string. These
have been computed in Ref. [11], here we only cite their expressions. The
bosonic contribution turns out to be:

$\displaystyle S_{b}=$ $\displaystyle\,\frac{\kappa_{d}}{2\pi}\int
d^{2}z\prod_{l=1}^{n}|z-z_{l}|^{\alpha^{\prime}qk_{l}}\,\prod_{l=1}^{n}{\rm
exp}\left[-\sqrt{\frac{\alpha^{\prime}}{2}}\frac{q\cdot
C_{l}}{z-z_{l}}-\sqrt{\frac{\alpha^{\prime}}{2}}\frac{q\cdot\bar{C}_{l}}{{\bar{z}}-{\bar{z}}_{l}}\right]$
(3.23) $\displaystyle\times\left(\sum_{i=1}^{n}\frac{\epsilon\cdot
C_{i}}{(z-z_{i})^{2}}+\sum_{i=1}^{n}\sqrt{\frac{\alpha^{\prime}}{2}}\frac{\epsilon\cdot
k_{i}}{z-z_{i}}\right)\left(\sum_{j=1}^{n}\frac{\bar{\epsilon}\cdot{\bar{C}}_{j}}{({\bar{z}}-{\bar{z}}_{j})^{2}}+\sum_{j=1}^{n}\sqrt{\frac{\alpha^{\prime}}{2}}\frac{{\bar{\epsilon}}\cdot
k_{j}}{{\bar{z}}-{\bar{z}}_{j}}\right)\,,$ (3.24)

The contribution dues to the superpartner degrees of freedom is:

$\displaystyle\bar{S}_{s}=\frac{\kappa_{d}}{2\pi}\int
d^{2}z\prod_{l=1}^{n}|z-z_{l}|^{\alpha^{\prime}qk_{l}}\,\prod_{l=1}^{n}{\rm
exp}\left[-\sqrt{\frac{\alpha^{\prime}}{2}}\frac{q\cdot
C_{l}}{z-z_{l}}-\sqrt{\frac{\alpha^{\prime}}{2}}\frac{q\cdot\bar{C}_{l}}{{\bar{z}}-{\bar{z}}_{l}}\right]$
(3.25)
$\displaystyle\times\left[\frac{1}{2}\sum_{i=1}^{n}\sqrt{\frac{\alpha^{\prime}}{2}}\frac{q\cdot
A_{i}}{z-z_{i}}\sum_{j=1}^{n}\frac{\epsilon\cdot
A_{j}}{z-z_{j}}\sum_{l=1}^{n}\sqrt{\frac{\alpha^{\prime}}{2}}\frac{q\cdot{\bar{A}}_{l}}{{\bar{z}}-{\bar{z}}_{l}}\sum_{m=1}^{n}\frac{\bar{\epsilon}\cdot{\bar{A}}_{m}}{{\bar{z}}-{\bar{z}}_{m}}\right.$
(3.26) $\displaystyle+\left.\left(\sum_{i=1}^{n}\frac{\epsilon\cdot
C_{i}}{(z-z_{i})^{2}}+\sum_{i=1}^{n}\sqrt{\frac{\alpha^{\prime}}{2}}\frac{\epsilon\cdot
k_{i}}{z-z_{i}}\right)\sum_{j=1}^{n}\sqrt{\frac{\alpha^{\prime}}{2}}\frac{q\cdot{\bar{A}}_{j}}{{\bar{z}}-{\bar{z}}_{j}}\sum_{l=1}^{n}\frac{\bar{\epsilon}\cdot{\bar{A}}_{l}}{{\bar{z}}-{\bar{z}}_{l}}\right]\,,$
(3.27)

The superkinetic quantities introduced in these expressions are given by:

$\displaystyle
A_{i}^{\mu}=\varphi_{i}\epsilon_{i}^{\mu}+\sqrt{\frac{\alpha^{\prime}}{2}}\theta_{i}k_{i}^{\mu}~{}~{};~{}~{}C_{i}^{\mu}=\varphi_{i}\theta_{i}\epsilon_{i}^{\mu}\,,$
(3.28)

with $\varphi$ Grassmannian variables introduced to exponentiate the vertex
operators. All the integrals in $S_{b}$ and $S_{S}$ have been computed in
Refs. [44, 46, 11] up the ${\cal O}(q)$ in the expansion of the momentum of
the graviton chosen to be the soft external leg. The result of the
integrations, when the soft leg is projected on the external symmetric and
traceless polarization, reproduces, according to Equation (3.17), the known
soft theorems for gravitons. The projection on the dilaton state, instead, is
written as the sum of two operators acting on the amplitude depending only on
hard momenta:

$\displaystyle M_{n+1}^{\rm dilaton}=$
$\displaystyle\,\frac{\kappa_{d}}{\sqrt{d-2}}\left[2-\sum_{i=1}^{n}k_{i\mu}\frac{\partial}{\partial
k_{i\mu}}\right.$ (3.30)
$\displaystyle\left.+\frac{1}{2}\sum_{i=1}^{n}\left(q^{\rho}{\hat{K}}_{i\rho}+\frac{q^{\rho}q^{\sigma}}{k_{i}q}\left(\mathcal{S}_{i,\rho\mu}\eta^{\mu\nu}\mathcal{S}_{i\nu\sigma}+D\Pi_{i,\\{\rho,\sigma\\}}\right)\right)\right]M_{n}\,,$

where

$\displaystyle{\hat{K}}_{i\mu}=2\left[\frac{1}{2}k_{i\mu}\frac{\partial^{2}}{\partial
k_{i\nu}\partial k_{i}^{\nu}}-k_{i}^{\rho}\frac{\partial^{2}}{\partial
k_{i}^{\mu}\partial
k_{i}^{\rho}}+i\mathcal{S}_{i,\rho\mu}\frac{\partial}{\partial
k_{i}^{\rho}}\right]~{}~{};~{}~{}\Pi_{i}^{\rho\sigma}=\epsilon_{i}^{\rho}\frac{\partial}{\partial\epsilon_{i\sigma}}+{\bar{\epsilon}}_{i}^{\rho}\frac{\partial}{\partial{\bar{\epsilon}}_{i\sigma}}.$

with $\mathcal{S}_{\rho\mu}$ the spin-operator. Remarkably these operators are
the generators of the dilatations and special conformal transformations acting
in the momentum space. This result has also been confirmed by decomposing the
$n+1$-amplitude in a double copy three-point vertex, with a soft leg
describing collectively a graviton, a dilaton and a Kalb-Ramond, factorized
from an $n$-point amplitude [66]. The expansion of this amplitude in the soft
momentum and the request of the gauge invariance imposed order by order in
soft expansion, reproduces the dilaton infrared behaviour obtained by the
explicit calculation of superstring amplitudes. The same calculation has also
been performed in heterotic and bosonic string theories for massless and
massive external state. Differently from the soft graviton behaviour which is
universal up to subleading order, the tree-level soft behaviour of the dilaton
is universal through the ${\cal O}(q)$ in all string theories. It doesn’t
contain any string correction and therefore is a result that can also be
obtained from amplitudes computed in the framework of the low energy effective
string theories. This result has been extended at $h$-order in the loop
expansion by computing in bosonic string theory amplitudes with one massless
state in interaction with $n$-tachyon states treated as massive particles with
mass $m^{2}=-\frac{1}{\alpha^{\prime}}$. The calculation has been performed in
space-time dimensions bigger than four to avoid infrared logarithmic
divergences, the resulting soft-expansion reads:

$\displaystyle
M_{n;\phi}^{(1)}(k_{i};q)=\frac{\kappa_{d}}{\sqrt{d-2}}\left[-\sum_{i=1}^{n}\frac{m^{2}}{k_{i}q}{\rm
e}^{q\partial_{k_{i}}}+2+h(d-2)-\sum_{i=1}^{n}k_{i}\cdot\frac{\partial}{\partial
k_{i}}+q_{\mu}\sum_{i=1}^{n}{\hat{K}}_{i}^{\mu}\right]M_{n}^{(1)}+{\cal
O}(q^{2})\,.$

Remarkably, the multiloop soft expansion is again free by string corrections
to any order in the loop expansion. Instead, since the multiloop infrared
behaviour has been obtained only in the bosonic string, it is not clear if
universality holds to any loop.

In analogy with the dilaton of the broken conformal field theories, also the
gravitational dilaton obeys a double soft factorization property. This has
been determined by computing in bosonic string theory tree level amplidudes
with $n$-tachyons and two massless states carrying low momenta. The
simultaneos zero momentum limit is performed by rescaling, by a small
parameter $\tau$, the momenta of two massless states and expanding the
amplitude in $\tau$. The projection on the dilatons, up to the subleading
order, gives:

$\displaystyle M_{n+d+d}=\frac{\kappa_{d}^{2}}{d-2}\Bigg{[}$
$\displaystyle\frac{1}{\tau^{2}}\sum_{i,j=1}^{n}\frac{m_{i}^{2}\,m_{j}^{2}}{k_{i}q\,k_{j}l}\left(1+\tau
q\frac{\partial}{\partial k_{i}}+\tau l\frac{\partial}{\partial
k_{j}}\right)-\frac{1}{\tau}\sum_{i=1}^{n}\frac{m_{i}^{4}}{(k_{i}l)(k_{i}q)}\frac{(ql)}{k_{i}(q+l)}$
(3.32)
$\displaystyle-\frac{d-2}{\tau}\sum_{i=1}^{n}\frac{(qk_{i})(lk_{i})}{k_{i}(q+l)\,(ql)}-\frac{1}{\tau}\sum_{i=1}^{n}\frac{2m_{i}^{2}}{k_{i}(q+l)}$
$\displaystyle-\frac{1}{\tau}\sum_{i=1}^{n}\left(\frac{m_{i}^{2}}{k_{i}q}+\frac{m_{i}^{2}}{k_{i}l}\right)\left(2-\sum_{j=1}^{n}k_{j}\frac{\partial}{\partial
k_{j}}\right)\Bigg{]}M_{n}+{\cal O}(\tau^{0})$

This result is independent of the string slope $\alpha^{\prime}$ and it has
also been obtained by starting from Feynman diagrams computed in the low
energy effective string theory. Two classes of exchange Feynman diagrams
contribute to the infrared behaviour of the amplitudes (see Figure 1). The
first class of diagrams involve only three-point vertices with two tachyons
and one dilaton emitted from two different external lines. In the second class
of diagrams, three and four-point vertices are emitted from a single external
line. These are the vertices with two tachyons and one massless state, three
massless states and a four point-vertex with two tachyons and two massless
states. The contribution due to the four-point interaction is collected by the
single pole term in second line in Equation (3.32). Gauge invariance is
obtained by requiring the emission of the massless states from internal lines.
The same expression is also obtained by joining a four-point amplitude, with
two tachyons and two dilatons, to a $n$-point tachyon amplitude. It is
straightforward to see that Equation (3.32) differs from the expression
obtained by considering two consecutive single soft expansions in the term
generated by the four-point interaction.

$k_{i}$$k_{j}$$\varepsilon_{q},\,q$$\varepsilon_{l},\,l$$M_{n}$

$k_{i}$$k_{i}+l+q$$\varepsilon_{l},\,l$$\varepsilon_{q},\,q$$M_{n}$

Figure 1: Diagrams of two massless closed string states with polarization and
momenta $\varepsilon_{q},q$ respectively and $\varepsilon_{l},l$. External
tachyon lines have momenta $k_{i}$, with $i=1,\ldots,n$.

##  Discussions

In this paper, we have summarized the state of the art of the soft-theorems of
two different particles named dilaton, the Nambu-Goldstone boson of the
spontaneously broken conformal theories and the scalar of the massless
gravitational multiplet of the string theories. Symmetries of the theory
determine the single and double soft behaviour for both particles. Ward-
identities associated with the scale and special conformal transformation
control the soft-behaviour of amplitudes with conformal dilatons. Gauge-
invariance of string (or string effective theory) amplitudes fixes the
infrared behaviour of all the massless closed string multiplet. The single
soft theorems for both particles are almost similar. They contain at
subleading order the generators of the scale transformations and kinematically
invariant factors related to the mass and scale dimensions of the amplitudes.
The subsubleading order in both cases is fixed by the generators of the
special conformal transformations. These analogies are less evident in the
double soft behaviour of the two dilatons. Both contain the generators of the
dilations, but for external massless finite energy particles, the amplitudes
with conformal soft dilatons are finite at zero momentum while diverging for
the gravity dilaton. Furthermore, the simultaneously low momentum limit for
the conformal dilatons coincides with the consecutive one. That is not true in
the case of the gravity dilaton for the presence of quartic interaction among
two massive and two dilaton particles. Differences appear also at higher order
in the perturbative expansion. In string theory, the dilaton remains massless
to all loops and satisfies factorization properties similar to the tree-level
one. In conformal theories, the dilaton, in general, become massive at one-
loop because of the breaking of conformal invariance in quantum theories. We
emphasize that it is still missing a symmetry principle that explains the
infrared properties of the gravitational dilaton. These are inferred by direct
amplitude calculations or by the request of gauge invariance of string
amplitudes that describe in a unified expression scattering of gravitons,
Kald-Ramonds and dilatons. Similarities between the soft theorems of the two
dilatons suggest that also the IR properties of the gravity-dilaton might be
determined by the breaking of some “hidden” symmetry of the theory.

Universality of the loop soft theorem for the gravity-dilaton is still an open
issue. The loop factorization properties for this kind of particle have been
derived only in bosonic string theory and for external tachyons treated as
massive scalar fields [75]. They hold also in field theory because the soft
operators are independent on the string slope but a their extension to other
string models and for other kinds of hard states is still lacking.

We conclude these notes by observing that the gravity dilaton is a massless
scalar that appears in all superstring theories and can play an important role
in the so-called string cosmology. They have been proposed, for example,
cosmological models where the primordial evolution of the background is driven
by this massless scalar. On the other side, there are extensions of the
Standard Model and gravity based on scale invariance which could solve the
hierarchy problem. In all these cosmological contests it would be interesting
to explore the role played by soft theorems in establishing relations among
cosmological correlators.

Acknowledgement: I would like to thank Paolo Di Vecchia, Matin Mojaza, Josh
Nole and Mritunjay Verma for having accompanied me on this journey into the
formulation of soft theorems for dilatons. I thank P. Di Vecchia for his
helpful comments on the manuscript.

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# Negative correlation between the linear and the nonlinear conductance in
magnetic tunnel junctions

Shuichi Iwakiri<EMAIL_ADDRESS>Department of Physics,
Osaka University,1-1 Machikaneyamacho, Osaka, Japan  Satoshi Sugimoto
Research Center for Magnetic and Spintronic Materials, National Institute for
Materials Science (NIMS), 1-2-1 Sengen, Tsukuba, Japan  Yasuhiro Niimi
Department of Physics, Osaka University,1-1 Machikaneyamacho, Osaka, Japan
Center for Spintronics Research Network (CSRN), Osaka University, 1-3
Machikaneyamacho, Osaka, Japan  Yusuke Kozuka  Research Center for Magnetic
and Spintronic Materials, National Institute for Materials Science (NIMS),
1-2-1 Sengen, Tsukuba, Japan  Yukiko K. Takahashi  Research Center for
Magnetic and Spintronic Materials, National Institute for Materials Science
(NIMS), 1-2-1 Sengen, Tsukuba, Japan  Shinya Kasai  Research Center for
Magnetic and Spintronic Materials, National Institute for Materials Science
(NIMS), 1-2-1 Sengen, Tsukuba, Japan  JST, PRESTO, 4-1-8 Honcho, Kawaguchi,
Saitama, Japan  Kensuke Kobayashi  Department of Physics, Osaka University,1-1
Machikaneyamacho, Osaka, Japan  Institute for Physics of Intelligence and
Department of Physics, The University of Tokyo, Bunkyo-ku, Tokyo, 113-0033,
Japan  Trans-scale Quantum Science Institute, The University of Tokyo, Bunkyo-
ku, Tokyo, 113-0033 Japan

###### Abstract

The current-voltage ($IV$) characteristics beyond the linear response regime
of magnetic tunnel junction (MTJ) is systematically investigated. We find a
clear negative correlation between the two coefficients to characterize the
linear ($I$$\propto$$V$) and the lowest-order nonlinear ($I$$\propto$$V^{3}$)
currents, which holds regardless of the temperature and the thickness of the
tunnel barrier. This observation cannot simply be explained by the standard
tunneling model such as the Brinkman model, suggesting a mechanism intrinsic
to MTJ. We propose a phenomenological model based on the Julliére model that
attributes the observed negative correlation to the spin-flip tunneling
assisted by a magnon. These results suggest a fundamental law between the
linear and the nonlinear response of MTJ.

††preprint: APS/123-QED

##  I Introduction

Magnetic tunnel junction (MTJ), consisting of ferromagnet-insulator-
ferromagnet heterostructure plays a central role in spintronics. Its
resistance changes with the magnetization configuration from parallel (P) to
anti-parallel (AP) known as the tunneling magnetoresistance (TMR) effect [1,
2, 3]. In particular, the MTJ with MgO barrier exhibits a giant TMR of a few
hundred % [4, 5, 6]. Such giant TMR is due to the spin-dependence of both the
density of states (DOS) in the electrode metal described by the Julliére model
[1, 7] and the decay rates depending on the Bloch states in the barrier [8,
9]. However, it is commonly observed that the TMR is significantly suppressed
by applying the bias voltage, as initially reported by Julliére [1]. From the
physical point of view, this effect stems from the bias voltage dependence of
the conductance (nonlinear conductance) [10, 11], which is the main topic of
this Article.

One of the known mechanisms to account for the nonlinear conductance in tunnel
junctions is the modulation of the tunneling barrier height by the electric
field. This can be addressed through models such as the Simmons model [12] or
the Brinkman model [13], both of which are based on the WKB approximation of
the tunneling. The Brinkman model is an extension of the Simmons model to
include the asymmetry of the barrier height due to the difference of the
electrode materials. It also gives a simple formula of the bias-dependent
conductance at low bias regime. This approach was applied to the nonlinear
conductance in the P configuration of MTJ and the barrier height was estimated
[14, 5].

Other mechanisms to be responsible for the nonlinearity peculiar to MTJ are
the electron’s inelastic tunneling processes due to the interaction with
quasiparticles (e.g. magnons and phonons) [15, 16, 17, 14, 18, 19, 20] or
impurities [20, 21, 22]. Among them, tunneling with emitting/absorbing a
magnon (magnon-assisted tunneling) has been extensively studied. For example,
Zhang $et$ $al.$ proposed an analytical model of the magnon-assisted tunneling
and obtained an agreement with the experiment in MTJ [15]. In addition,
Moodera $et$ $al.$ found a peak/dip structure in the nonlinear conductance
which was also attributed to the magnon [14].

Those prior works have clarified the overall characteristics of the nonlinear
conductance in MTJ. Especially, a lot is known about the nonlinear conductance
at wide voltage range (up to a few hundreds of millivolts or tens of $k_{\rm
B\it}T$ where $k_{\rm B\it}$ is Boltzmann constant and $T$ is temperature) and
its behavior in either P or AP configuration. On the other hand, the
nonlinearity at lower bias regime (comparable to $k_{\rm B\it}T$) is
relatively less understood. For example, little has been known about the
relation between the nonlinear conductance and the linear conductance and/or
the magnetization configuration dependence (e.g., not only the P and AP states
but also the intermediate ones). Such understandings would enable us to
understand the MTJ in more depth.

In this Article, the nonlinear electron transport in MTJ is systematically
studied at low bias regime, controlling the magnetization configuration, the
temperature, and the tunneling probability. We find that the two coefficients
to characterize the linear ($I$$\propto$$V$) and the lowest-order nonlinear
($I$$\propto$$V^{3}$) currents are negatively correlated, which cannot be
explained by the above barrier modulation effect described by the Brinkman
model. Instead, we propose a phenomenological model based on the Julliére
model, additionally taking the magnon-assisted tunneling into account. Our
finding would extend the understanding of the nonlinear conductance in MTJ,
which leads to the observed nontrivial negative correlations.

##  II Experimental setup

Figure 1(a) shows the schematic of our sample. The multilayered stack
consisting of Ta (5 nm) / Ru (10 nm) / Ta (5 nm) / Co20Fe60B20 (5 nm) / MgO
($d_{\rm MgO}$ = 1.06, 1.13, 1.27, or 1.33 nm) / Co20Fe60B20 (4 nm) / Ta (5
nm) / Ru (5 nm) is deposited on a thermally oxidized Si substrate by magnetron
sputtering. The number in the parentheses is the thickness.

Figure 1: (a) Schematic of the MTJ nanopillar. The magnetic field is applied
along the long axis of the ellipse. (b) Magnetoresistance of the MTJ ($V=$ 5
mV, $d_{\rm MgO}=$ 1.33 nm) at temperature $T=$ 40, 100, 200, and 260 K. Only
the sweep from positive to negative magnetic field is shown for simplicity.
The schematic diagrams illustrate the magnetization configuration at each
magnetic field.

The film stack is annealed at 400 °C for 30 min. Hereafter, the ferromagnetic
metal, Co20Fe60B20, is denoted as CoFeB. The CoFeB / MgO / CoFeB layers form
an MTJ with in-plane magnetic anisotropy as shown Fig. 1(a). The sample is
patterned into an elliptical pillar of 150 nm×450 nm and the Ta (10 nm) / Au
(100 nm) electrode is sputtered. The thickness of the tunnel barrier $d_{\rm
MgO}$ is set right above the minimal one to obtain the crystallinity at MgO /
CoFeB interface [4, 5]. This enables us to observe nonlinear transport at bias
voltage as low as possible. The $IV$ characteristics are obtained by applying
voltage $V$, measuring current $I$ under in-plane magnetic field $H$ along the
long axis of the pillar.

Figure 1(b) shows the magnetic field dependence of the MTJ resistance. Here,
the data of $d_{\rm MgO}=$ 1.33 nm and $T=$ 40, 100, 200, and 260 K are shown
as a typical example. The bias voltage is $V=$ 5 mV, where the current is
perfectly proportional to the voltage, and one can deduce the linear
resistance ($R=V/I$). As the magnetic field is swept from positive to
negative, the resistance takes a peak at $\sim$ 0 mT and it monotonically
decreases as the field is increased. It finally saturates at around $\pm$20
mT. According to the well-established phenomenological model (Julliére model)
that connects the magnetoresistance to the spin-dependent density of states
(DOS) [1], the resistance minimum $R_{\rm min}$ (maximum $R_{\rm max}$)
corresponds to the magnetization configuration of P (AP) as illustrated in
Fig. 1(b). The magnetoresistance ratio $(R_{\rm max}-R_{\rm min})/R_{\rm min}$
is approximately 300 %, which is a reasonable value for MgO-based MTJ [4, 5].
Unlike the steep switching behavior often realized by designing the MTJ
structure, we intentionally design the present MTJs without inserting an
antiferromagnetic layer which fixes the magnetization direction [4, 5] so that
their resistance gradually change as a function of the magnetic field. This
guarantees that the magnetization configuration is continuously controlled
from P to AP in our samples. Thus we can systematically investigate the
relation between the nonlinear conductance and the magnetization
configuration.

##  III Experimental results

###  III.1 Current-voltage characteristics

The $IV$ characteristics for the samples with four different $d_{\rm MgO}$’s
are measured ($d_{\rm MgO}$ = 1.06, 1.13, 1.27, and 1.33 nm), sweeping $V=\pm$
100 mV, $H=\pm$30 mT, and $T=$40, 100, 200, and 260 K. By numerically
differentiating the $IV$ characteristics, we obtain the differential
conductance as shown in Fig. 2(a).

Figure 2: Characterization of the nonlinear transport obtained for the MTJ
with $d_{\rm MgO}=$ 1.33 nm at 260 K. (a) Bias dependence of the differential
conductance $dI/dV$ at 0, 5, 10, and 20 mT. The black dashed curves are the
result of the parabolic fitting. (b) Magnetic field dependence of the linear
($G_{1}$) and the nonlinear ($G_{3}$) conductance obtained from the fitting.
(c) Correlation between $G_{1}$ and $G_{3}$. The areas of P and AP
configurations are marked as black dotted circles.

We show the data at 260 K and $d_{\rm MgO}$ =1.33 nm as an example. As is
clear from Fig. 2(a), the conductance is symmetric with respect to the sign
reversal of a bias voltage up to $|V|=$ 100 mV.

The nonlinear component is well fitted by the parabolic function $I\simeq
G_{1}V+G_{2}V^{2}+G_{3}V^{3}$. The results of the fitting show that the
magnitude of the $G_{2}V^{2}$ term is only below 1 $\%$ of $G_{1}$ or
$G_{3}V^{3}$ in any condition of magnetic field, temperature, and barrier
thickness, letting us approximately write $I\simeq G_{1}V+G_{3}V^{3}$. The
absence of the asymmetric term $G_{2}$ is reasonable considering that the MTJ
film structure (CoFeB/MgO/CoFeB) is symmetric (the same material is used for
both left/right electrodes), making no difference in transport characteristics
for positive/negative bias. Thus, the differential conductance at low bias
regime can be well approximated by the equation below (black dashed curves in
Fig. 2(a)).

$\frac{dI}{dV}=G_{1}+3G_{3}V^{2}.$ (1)

Henceforth, we refer to $G_{3}$ as nonlinear conductance and investigate how
$G_{1}$ and $G_{3}$ are correlated.

Figure 2(b) shows the magnetic field dependence of $G_{1}$ and $G_{3}$. The
linear conductance $G_{1}$ takes its minimum at $\sim$0 mT and increases as
the field is applied. Note that the data shown in Fig. 1(b) corresponds to
$G_{1}$ via $G_{1}=1/R$. On the contrary, the magnetic field dependence of the
nonlinear conductance $G_{3}$ takes its maximum at $\sim$0 mT and decreases as
the field is increased. Unexpectedly, the behavior of $G_{3}$ is an upside-
down reversal of that of $G_{1}$, suggesting the negative correlation between
them.

Figure 2(c) shows $G_{3}$ as a function of $G_{1}$ when the magnetic field is
varied from -30 mT to 30 mT. The left- (right-) most region where $G_{1}$
takes its minimum (maximum) corresponds to the AP (P) magnetization
configuration (see the circled area). Interestingly, a clear negative linear
correlation between $G_{1}$ and $G_{3}$ is obtained for all the experimental
data. The correlation is well approximated by linear function of
$G_{3}=-kG_{1}+m$, where $k$ ($1/V^{2}$) and $m$ ($1/\Omega/V^{2}$) are
constant coefficients.

###  III.2 Correlation plot for $G_{1}$ and $G_{3}$

Figure 3 shows $G_{3}$ vs $G_{1}$ correlation plots for four different
thickness ($d_{\rm MgO}$ = 1.06, 1.11, 1.21, and 1.33 nm) and temperatures
($T$ = 40, 100, 200, and 260 K).

Figure 3: Summary of the correlation plot for (a) $d_{\rm MgO}$ = 1.33, 1.27,
1.06 and 1.13 nm measured at $T=$ 40, 100, 200, and 260 K. The black dashed
line shows the line with slope -2 ($1/\rm V^{2}$). The area inside the black
dotted line is displayed in (b). (b) The correlation plot zoomed into $d_{\rm
MgO}$ = 1.27 and 1.33 nm. Figure 4: Thickness dependence of (a) the slope $k$
and (b) the intercept $m$ extracted from the correlation plot at four
different temperatures ($T=$ 40, 100, 200, and 260 K). The black dotted line
in (a) shows the line with slope -2 ($1/\rm V^{2}$).

$G_{1}$ and $G_{3}$ are obtained by fitting $\frac{dI}{dV}$ between
$|eV/{k_{\rm B\it}T}|\lesssim 4$, while the result is not sensitive to the
fitting range. As shown in Fig. 3, clear negative linear correlations between
$G_{3}$ and $G_{1}$ are obtained which, again, can be well approximated by
$G_{3}=-kG_{1}+m$ for all the thickness and the temperature.

We perform a linear fitting of the correlation plot and summarize the results
in Fig. 4. It turns out that the slope $k\sim 2$ ($1/V^{2}$) for all the data,
regardless of $T$ and $d_{\rm MgO}$ (see Fig. 4(a)). On the other hand, the
intercept $m$ clearly depends on both $T$ and $d_{\textrm{M}gO}$ and becomes
larger as $T$ or $d_{\textrm{M}gO}$ is decreased (see Fig. 4(b)). Here, as the
slope $k$ has a dimension of $V^{-2}$ which is independent of the geometric
scale such as the junction area, it may be a universal parameter to
characterize the nonlinear behavior of MTJ at low bias regime. These results
prove that the observed correlation holds in wide range of thermal fluctuation
($k_{\rm B\it}T$) and tunneling probability determined by $d_{\rm MgO}$. At
the same time, it implies the relevance between the magnetization
configuration and the nonlinear transport. This simple relation between the
linear and the nonlinear conductance is the central experimental finding in
this Article.

##  IV Discussions

###  IV.1  Barrier height modulation

Now, we discuss the physical origin of the observed nonlinearity and the
negative correlation. First of all, we examine one of the common sources of
nonlinearity in tunnel junction, namely the barrier modulation effect [12,
13].

It is known that the tunneling barrier height can be modulated due to the
application of the bias voltage, which makes the conductance nonlinear. One of
the most accepted models is the Brinkman model [13], an extension of Simmons
model [12] at low bias regime to include the barrier asymmetry. Based on the
WKB approximation, the Brinkman model presents the voltage-dependent
conductance $G(V)$ of a tunnel junction at low bias regime as the polynomial
below.

$\frac{G(V)}{G(0)}=1-\left(\frac{A_{0}\Delta\phi}{16\phi^{\frac{3}{2}}}\right)eV+\left(\frac{9}{128}\frac{A_{0}^{2}}{\phi}\right)(eV)^{2}.$
(2)

In this equation, $\phi$ is the effective height (in unit of electron Volt)
and $d$ is the thickness of the barrier (in unit Å). $\Delta\phi$ is the
difference of the barrier height at the interface of the left/right electrode,
which is zero when the same material is used for both electrodes.
$G(0)=3.16\times 10^{10}\frac{\phi}{d}e^{-1.025d\phi^{\frac{1}{2}}}$
$\Omega^{-1}\textrm{m}^{-2}$ is the conductance at zero bias voltage
normalized by the junction area $S$, and
$A_{0}=\frac{4(2m_{\textrm{e}})^{\frac{1}{2}}d}{3\hbar}$ ($m_{\textrm{e}}$ is
the electron mass and $\hbar$ is the reduced Planck constant).

According to Eq. (1) and (2), at first sight, the Brinkman model seems to give
the same voltage dependence of the conductance as observed in the experiment.
Thus, we attribute the linear ($G_{1}^{\textrm{B}}\propto V$) and the
nonlinear ($G_{3}^{\textrm{B}}\propto V^{3}$) conductance by the Brinkman
model as below.

$\displaystyle G_{1}^{\textrm{B}}$ $\displaystyle=3.16\times 10^{10}\frac{\phi
S}{d}e^{-1.025d\phi^{\frac{1}{2}}}.$ (3) $\displaystyle G_{3}^{\textrm{B}}$
$\displaystyle=G_{1}\times\frac{9}{128}\frac{A_{0}^{2}S}{\phi}.$ (4)

Here, the effective barrier height $\phi$ may vary with the magnetization
configuration in our MTJ. This is due to the fact that the tunneling in MgO-
based MTJ is dominated by different Bloch states ($\Delta_{1}$ state in P
configuration and $\Delta_{5}$ state in AP configuration) and their decay rate
is significantly different [8, 9]. Now, we assume that such behaviors of the
Bloch states affect the tunneling probability and modulate the effective
barrier height $\phi$. We compare $G_{1}^{\textrm{B}}$ and
$G_{3}^{\textrm{B}}$ deduced from Eqs. (2), (3), and (4) with the experimental
data to discuss the barrier modulation effect and the role of Bloch states in
the nonlinear conductance.

We start with the calculation of $d$ and $\phi$ dependence of $G_{1}$ and
$G_{3}$ in Eq. (4) as shown in Fig. 5. The barrier thickness $d$ is swept from
0.5 nm to 2 nm, and $\phi$is swept from 0.3 eV to 1 eV, both of which cover
the reasonable value range for CoFeB/MgO/CoFeB. Both $G_{1}$ and $G_{3}$ are
the monotonous decreasing function of $d$ and $\phi$ (see Fig. 5(c)).

Now, we compare these calculations with the experimental data and discuss that
the Brinkman model is not sufficient to explain our experimental observation.
As an example, we use the experimental data of MTJ with MgO thickness 1.33 nm
as shown in Fig. 2. Note that the results shown below hold true for other MgO
thickness (1.06, 1.13, and 1.27 nm) in the experiment.

Figure 5: Calculated results of (a) the linear conductance
$G_{1}^{\textrm{B}}$ and (b) the non-linear conductance $G_{3}^{\textrm{B}}$
by the Brinkman model as functions of $d$ and $\phi$. (c) Correlation plot of
$G_{1}^{\textrm{B}}$ and $G_{3}^{\textrm{B}}$ with changing $\phi$ from 0.3 to
1 eV ($d$ is fixed at 1.3 nm). $G_{1}^{\textrm{B}}$-maximum (minimum) region
is marked as black dotted circle, noting P (AP) configuration and $\Delta_{1}$
($\Delta_{5}$) Bloch state.

The nonlinearity at P configuration where the $\Delta_{1}$ Bloch state is
dominant, the experimental value is $G_{1}\simeq 9\times 10^{-6}$
$\Omega^{-1}V^{-1}$ and $G_{3}\simeq 4\times 10^{-6}$ $\Omega^{-1}V^{-2}$
respectively (see the circled area in Fig. 2(c) annotated “P configuration”).
These values correspond to the ones expected for $\phi=$0.3 eV and $d$=1.3 nm
in the above calculation, which give $G_{1}^{\rm B\it}\simeq 9\times 10^{-6}$
$\Omega^{-1}V^{-1}$ and $G_{3}^{\rm B\it}\simeq 4\times 10^{-6}$
$\Omega^{-1}V^{-2}$ (see the circled area in Fig. 5(c) annotated “P
configuration ($\Delta_{1}$ Bloch state)”). Therefore, it is suggested that
the barrier modulation effect by this $\Delta_{1}$ state is the dominant
source of non-linearity at P configuration. In addition, the estimated
effective barrier height and thickness ($\sim$0.3 eV and $\sim$1.3 nm) agree
with those in the previous report [4].

Now, in the AP configuration where the $\Delta_{5}$ Bloch state is dominant,
the experimental value is $G_{1}\simeq 3\times 10^{-6}$ $\Omega^{-1}V^{-1}$
and $G_{3}\simeq 15\times 10^{-6}$ $\Omega^{-1}V^{-2}$ respectively (see the
circled area in Fig. 2(c) annotated “AP configuration”). Here, the effective
barrier height should be higher than that in P configuration because the
$\Delta_{5}$ state decays more rapidly than $\Delta_{1}$ state [8].
Considering that the $G_{3}$ is the decreasing function of $\phi$ (see Eq.
(4)), however, the barrier modulation effect from this $\Delta_{5}$ state
cannot be larger than that in the $\Delta_{1}$ state (e.g., $G_{3}\simeq
4\times 10^{-6}$ $\Omega^{-1}V^{-2}$ at 20 mT in Fig. 2(c)). Therefore, it is
estimated that the nonlinearity due to the $\Delta_{5}$ state counts at most
$\frac{4\times 10^{-6}}{15\times 10^{-6}}\sim 26\%$ of the total value of
$G_{3}$. In addition, if we directly apply the Brinkman model to the AP
configuration, we obtain $\phi=0.08$ eV and $d=2.4$ nm, which significantly
deviate from the experimental condition and unrealistic. These results suggest
that Bloch states play a role in nonlinearity especially for the P
configuration, while additional mechanism is required to further account for
the nonlinear behavior of MTJ of whole configuration.

In addition, according to Eq.(2), both $G_{1}^{\textrm{B}}$ and
$G_{3}^{\textrm{B}}$ are the monotonous decreasing functions of $d$ and
$\phi$. This means that even when $d$ and $\phi$ changes with experimental
conditions (magnetic field, temperature, and MgO thickness), $G_{1}$ and
$G_{3}$ change in the same way. Therefore, as long as we consider the barrier
modulation mechanism such as the Brinkman model, we would expect positive
correlation between $G_{1}$ and $G_{3}$ (see Fig. 5(c)), which is opposite to
the negative one that we observed in experiment.

The model presented here does not explicitly take the effect of spin-dependent
DOS into account. In principle, we can extend the Brinkman model to include
such an effect. However, it turned out that our conclusion that the
experimental results cannot be explained by the Brinkman model alone remains
the same whether we take the effect of DOS into account or not. To keep the
discussion uncomplicated, therefore, we simply show the Brinkman model without
the effect of DOS here. For this reason, the estimated $\phi$ (the effective
barrier height) should be different from the actual value. We note that the
same conclusion can be obtained based on the Simmons model as well.

###  IV.2  Control experiment

We perform a control experiment with normal metal-insulator-ferromagnet (NIF)
junction of Ta(5 nm)/Ru(10 nm)/Ta(5 nm)/Co20Fe60B20 (5 nm)/MgO(1.0 nm)/Ta(5
nm)/Ru(5 nm) film. From this experiment, we can estimate the effect of
impurity scattering [22] because the film is prepared in the same manner as
MTJ so that it has almost the same quality of tunneling barrier and impurity
concentration. The sample is patterned into an elliptical pillar of 300 nm
$\times$ 450 nm. Note that the area of the ellipse is twice as much as the one
in the MTJ.

Figure 6: (a) differential conductance of the NIF junction. Black dotted line
is the parabolic fit. (b) Magnetic field dependence of $G_{1}$ and $G_{3}$.

We measure the magnetic field dependence of the $IV$ characteristics at room
temperature, which is again well fitted by a parabolic function
$G=G_{1}+2G_{2}V+3G_{3}V^{2}$. Contrary to the case in MTJ, $G_{2}$ is not
negligible in this result. This is consistent with the Brinkman model (see Eq.
2) that expects finite $G_{2}$ when the materials for the left and right
electrodes (CoFeB and Ta) are different. Now we discuss the origin of the
nonlinearity observed in this control experiment. Firstly, the impurity
scattering yields the conductance proportional to $|V|$ [22], which is not the
case in this experiment. Secondly, we consider the barrier modulation effect
according to the Brinkman model. As shown in Fig. 6, neither $G_{1}$ nor
$G_{3}$ depends on the magnetic field in our experimental condition. Such
behavior is totally different from the case with MTJ. Actually, according to
Eq. (2), the estimated barrier height ($\sim$0.3 eV) and barrier thickness
($\sim$1.3 nm) are reasonable compared to the previous works [5, 4]. This
implies the nonlinear conductance in this control experiment originates from
the barrier modulation effect, which does not depend on the magnetic field.

From the above discussion based on the Brinkman model and the control
experiment, we can say that those mechanisms arising from the tunnel junction
structure alone cannot explain the negative correlation and a certain
intrinsic mechanism that stems from the spin-dependent tunneling dynamics
depending on the magnetization dynamics is required.

###  IV.3  Magnon-assisted tunneling

We propose the contribution of the quasiparticles, particularly magnons to the
nonlinear conductance. It is created by the magnetization fluctuation, whose
energy scale is a few tens of meV (a few hundreds of $k_{\textrm{B}}T$) [22].
This implies that the magnon-assisted tunneling is likely to be relevant in
the observed nonlinear conductance. In Ref. [16], it is argued that the
$G\propto V^{2}$ component appears when the electrons interact to magnons that
exist at the junction interface. Besides, it has also been found that the
contribution of such surface magnons is important in the tunnel junctions
based on Co such as Co/$\rm Al_{2}O_{3}$/Co [15] and CoFeB/MgO/CoFeB [22].

To further apply this idea to our analysis, we present a Julliére model [1]
extended to include the spin-flip tunneling assisted by magnons. Fig. 7(a)
illustrates the tunneling processes we consider below. Originally, the
Julliére model describes the linear current as the elastic tunneling ($\it
I_{\rm elastic}$$\propto$$V$), which conserves spin and energy of electron
(see the black dotted arrow annotated “Elastic” in Fig. 7(a)). The current is
proportional to the product of the DOS with the same spin and energy,

$I_{\textrm{elastic}}=G_{1}V=At(D_{L_{\uparrow}}D_{R_{\uparrow}}+D_{L\downarrow}D_{R_{\downarrow}})V.$
(5)

Here, we denote the spin-dependent density of states (DOS) at Fermi level as
$D_{L_{\uparrow}}$, $D_{L_{\downarrow}}$ for the left electrode, and
$D_{R_{\uparrow}}$, $D_{R_{\downarrow}}$ for the right electrode ($\uparrow$
and $\downarrow$ are spin-up and down, respectively). $A$ is a constant
proportional to the junction area and $t$ is the matrix element associated
with the tunneling process.

We define the tilt angle $\theta$ of the magnetization in the right electrode,
seen from the left one. $\theta$ = 0 and $\rm\pi$ correspond to P and AP,
respectively. Note that $D_{R_{\uparrow}}$ and $D_{R_{\downarrow}}$
corresponds to the magnetization direction projected to the $\theta=0$ axis,
and thus is a function of $\theta$.

$\displaystyle D_{L_{\uparrow}}$ $\displaystyle=\frac{1+p}{2}L,$ (6)
$\displaystyle D_{L_{\downarrow}}$ $\displaystyle=\frac{1-p}{2}L,$ (7)
$\displaystyle D_{R_{\uparrow}}$ $\displaystyle=\frac{1+p}{2}R-pR\rm
sin^{2}\frac{\theta}{2},$ (8) $\displaystyle D_{R_{\downarrow}}$
$\displaystyle=\frac{1-p}{2}R+pR\rm sin^{2}\frac{\theta}{2}.$ (9)

We note $D_{R_{\uparrow}}$ and $D_{R_{\downarrow}}$ at $\theta=0$ as
$D^{0}_{R_{\uparrow}}$ and $D^{0}_{R_{\downarrow}}$. The spin-polarization of
left/right electrode is
$p\equiv|(D_{L_{\uparrow}}-D_{L_{\downarrow}})/(D_{L_{\uparrow}}+D_{L_{\downarrow}})|=|(D^{0}_{R_{\uparrow}}-D^{0}_{R_{\downarrow}})/(D^{0}_{R_{\uparrow}}+D^{0}_{R_{\downarrow}})|$.
We also define $L\equiv D_{L_{\uparrow}}+D_{L_{\downarrow}}$ and $R\equiv
D^{0}_{R_{\uparrow}}+D^{0}_{R_{\downarrow}}$. According to the Eq. (6–9), one
obtains the linear conductance as below.

$\displaystyle G_{1}=$ $\displaystyle
AtLR(D_{L\uparrow}D_{R\uparrow}+D_{L\downarrow}D_{R\downarrow})$
$\displaystyle=$ $\displaystyle
AtLR\left[\frac{1-p^{2}}{2}+p^{2}\cos^{2}\frac{\theta}{2}\right].$ (10)

This equation is a straightforward extension of the Julliére model [1] to
include magnetization angle-dependence [7]. The energy dependence of DOS [23]
is neglected because the energy scale of the bias voltage ($eV\sim$ a few tens
of meV) is small enough compared to the Fermi energy.

Next, we further extend the Julliére model to describe the nonlinear
conductance ($G_{3}$). We assume that the nonlinear current corresponds to the
inelastic tunneling current ($I_{\rm inelastic}$$\propto$$V^{3}$), where the
electron changes its energy and spin (see the black arrow in Fig. 7(a)
annotated “Inelastic”). We describe the nonlinear conductance using the
products of DOS with opposite spins,

$I_{\rm
inelastic}=G_{3}V^{3}=BtLR(D_{L_{\uparrow}}D^{*}_{R_{\downarrow}}+D_{L_{\downarrow}}D^{*}_{R_{\uparrow}})\it
V^{\rm 3}.$ (11)

Here, we set an ad-hoc assumption on DOS:

$\displaystyle D^{*}_{R\uparrow}\equiv$ $\displaystyle
D_{R\uparrow}-D^{0}_{R\uparrow}=-pR\sin^{2}\frac{\theta}{2}.$ (12)
$\displaystyle D^{*}_{R\downarrow}\equiv$ $\displaystyle
D_{R\downarrow}-D^{0}_{R\downarrow}=pR\sin^{2}\frac{\theta}{2}.$ (13)

In this assumption, the DOS in the P configurations is subtracted from the
original one, which enables us to subtract the nonlinearity that is already
present at P configuration. This allows us to focus on the continuous changes
of nonlinearity between P and AP configurations. We will examine the relevance
of this assumption later. The inelastic term is simplified to,

$\displaystyle G_{3}=$ $\displaystyle
BtLR\left(D_{L_{\uparrow}}D^{*}_{R_{\downarrow}}+D_{L_{\downarrow}}D^{*}_{R_{\uparrow}}\right)$
$\displaystyle=$ $\displaystyle BtLRp^{2}\sin^{2}\frac{\theta}{2}.$ (14)

In this expression, $G_{3}$ becomes minimum in P configuration ($\theta=0$) in
agreement with the observation shown in Fig. 2.

From Eq. (10) and (14) the above expressions of $G_{1}$ and $G_{3}$, we
obtain,

$G_{3}=-\frac{B}{A}G_{1}+\frac{BtLR}{2}(1+p^{2})=-kG_{1}+m.$ (15)

Here,

$\displaystyle k=$ $\displaystyle B/A.$ (16) $\displaystyle m=$
$\displaystyle\frac{BtLR}{2}(1+p^{2}).$ (17)

According to this formula, only the intercept $m$ depends on the barrier
thickness and the temperature, while the slope $k$ is a constant that is
independent to them. This indicates the linear negative correlation between
$G_{1}$ and $G_{3}$, which qualitatively explains the experimental observation
in Fig. 3.

Figure 7: (a) Schematic energy diagram of the MTJ. The vertical axis
corresponds to the energy. Red and blue areas represent the DOS of the spin up
and down electrons in each electrode respectively, separated by the insulating
barrier (gray rectangle). The energy and spin conserving (“Elastic”, black
dotted arrow) path and the non-conserving path (“Inelastic”, black arrows) are
shown. (b) Temperature dependence of the coefficient $m$. Black dotted curves
are the results of the fitting. (c) MgO thickness dependence of the
coefficient $BtLR$ obtained from the fitting (left axis), plotted together
with the linear conductance in P configuration ($T=260$ K) (right axis). Black
broken line is the fitting with exponential function $\propto$ $e^{-d_{\rm
MgO}/L}$. Here, $L\sim 0.15$ nm is the constant with length dimension obtained
from the fitting.

Now, let us further try to apply the above model to the experimental data.
According to this model, only the intercept $m$ depends on temperature through
polarization $p$. Most simply, $p$ is assumed to obey the Bloch’s law [24, 25]

$p=p_{0}\left[1-\left(\frac{T}{T_{c}}\right)^{\frac{3}{2}}\right].$ (18)

This describes the change of magnetization due to the magnon excitation
($p_{0}$ is the spin polarization at 0 K and $T_{c}$ is the Curie
temperature). The details of the magnon in this system are included in these
macroscopic parameters. Note that such temperature dependence is introduced
through the definition of the nonlinear conductance such as seen in Eq. (14).
Conversely, if we naively defined
$G_{3}=Bt(D_{L_{\uparrow}}D_{R_{\downarrow}}+D_{L_{\downarrow}}D_{R_{\uparrow}})$,
the correlation would become $G_{3}=-\frac{B}{A}G_{1}+\frac{BLRt}{2}$ and no
temperature dependence would appear in the intercept, which fails to explain
the experimental observation. This is the reason why we adopted the ad-hoc
assumption to interpret the behavior of $m$.

Thus, according to Eq. (17) and Eq. (18), the intercept $m$ is obtained as
below.

$m=\frac{BtLR}{2}\left[1+p_{0}^{2}\left(1-\left(\frac{T}{T_{c}}\right)^{\frac{3}{2}}\right)^{2}\right].$
(19)

We show the results of the fitting of $m$ with Eq. 19 in Fig. 7(b). The
parameters ($B,t,L,R,p_{0}$, and $T_{c}$) are set as free parameters. The
estimated Curie temperature $T_{c}\sim$800 K and spin polarization $p_{0}\sim
0.7$ are reasonable for the CoFeB thin film [26, 27, 28]. The agreement
between the data and the fitting is not sensitive to the slight change of
$T_{c}$ and $p_{0}$. Furthermore, as shown in Fig. 7(c), the coefficient
$BtLR$ exponentially decreases with increasing $d_{\rm MgO}$. This behavior is
most likely attributed to the $d_{\rm MgO}$ dependence of the tunneling
probability $t$. In fact, as shown in Fig. 7(c), the linear conductance
$G_{1}$$\propto$$t$ decreases with the same exponential function with $BtLR$.
This agreement ensures that the $t$ obtained from the model actually
corresponds to the transmission probability.

The above model is a simple one but the obtained relation consistency between
$G_{1}$ and $G_{3}$ seems to suggest that this captures the essential
characteristics of the experimental results. The model is nevertheless still
phenomenological that it cannot explain, for example, the microscopic origin
of the constant $k$ and the validity of the ad-hoc assumption. For further
study, the calculation based on the microscopic Hamiltonian of electron
tunneling and the electron-magnon interactions (e.g. non-equilibrium Green
function) would be a powerful approach.

##  V  Conclusion

In conclusion, we have investigated the electron transport right beyond linear
response regime in MTJ at low bias regime. We have found a clear negative
correlation between the linearity and the nonlinearity. This cannot be fully
attributed to the barrier modulation effect (Brinkman model), the rest of
which we attribute to the magnon-assisted tunneling through our
phenomenological model by extending Julliére model. Our findings contribute to
the deeper understanding of the MTJ based on the rigorous understanding of the
linear response regime.

###### Acknowledgements.

This work is partially supported by JSPS KAKENHI Grant Nos. JP17K18892,
JP18J20527, JP19H00656, JP19H05826, JP16H05964, and JP26103002. The authors
acknowledge T. Kato, Y. Ominato, and M. Matsuo for fruitful discussions. The
authors also acknowledge M. Takahagi for technical support.

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