# Pair production in rotating electric fields via quantum kinetic equations:
Resolving helicity states

I. A. Aleksandrov<EMAIL_ADDRESS>Department of Physics, Saint
Petersburg State University, Universitetskaya Naberezhnaya 7/9, Saint
Petersburg 199034, Russia  Ioffe Institute, Politekhnicheskaya street 26,
Saint Petersburg 194021, Russia  A. Kudlis<EMAIL_ADDRESS>Abrikosov
Center for Theoretical Physics, MIPT, Dolgoprudnyi, Moscow Region 141701,
Russia  Russian Quantum Center, Skolkovo, Moscow 121205, Russia

###### Abstract

We investigate the phenomenon of electron-positron pair production from vacuum
in the presence of a strong electric field of circular polarization. By means
of a nonperturbative approach based on the quantum kinetic equations (QKEs),
we numerically calculate helicity-resolved momentum distributions of the
particles produced and analyze the corresponding helicity asymmetry. It is
demonstrated that the external rotating field tends to generate left-handed
and right-handed particles traveling in opposite directions. Generic symmetry
properties of the momentum spectra are examined analytically by means of the
QKEs and also confirmed and illustrated by direct numerical computations. The
helicity signatures revealed in our study are expected to provide a firmer
basis for possible experimental investigations of the fundamental phenomenon
of vacuum pair production in strong fields.

Introduction. Already within the first decade after Dirac proposed his wave
equation dirac_1928  describing the relativistic quantum dynamics of the
electron $e^{-}$, not only was the antiparticle $e^{+}$ (positron) predicted
dirac_1931  and experimentally discovered anderson , but also it became clear
that the vacuum fluctuations of the electron-positron field lead to the
nonlinear effect of light-by-light scattering in the presence of a classical
electric background euler_kockel . Furthermore, the idea that the vacuum state
interacting with a strong electric field can produce electron positron pairs
sauter_1931  was later confirmed within the framework of quantum
electrodynamics (QED) by means of a fully nonperturbative expression for the
one-loop effective action heisenberg_euler ; schwinger_1951 . It was found
that the imaginary part of the effective action, which governs the probability
of the pair-production process, is proportional to $\mathrm{exp}(-\pi
E_{\text{c}}/E_{0})$, where $E_{0}$ is the field strength of a constant
electric background and $E_{\text{c}}=m^{2}c^{3}/|e\hbar|\sim 10^{16}$ V/cm is
the so-called critical (Schwinger) field strength ($m$ and $e$ are the
electron mass and charge, respectively). This result suggests that the
corresponding Sauter-Schwinger mechanism is strongly suppressed unless $E_{0}$
is close to $E_{\text{c}}$. From the theoretical point of view, it is evident
that the effect of pair production in a strong quasistatic field is
nonperturbative with respect to the field amplitude $E_{0}$. Although this
fundamental phenomenon has never been experimentally observed, the continual
development of the laser technologies indicates that forming a setup that
involves sufficiently intense laser fields may become feasible in the near
future (for review, see Ref. fedotov_review ).

There are several theoretical techniques which can be employed for describing
the nonperturbative pair-production effect (see, e.g., Refs.
fradkin_gitman_shvartsman ; BB_prd_1991 ; gavrilov_prd_1996 ; zhuang_1996 ;
zhuang_prd_1998 ; ochs_1998 ; schmidt_1998 ; kluger_prd_1998 ;
pervushin_skokov ; hebenstreit_prd_2010 ; blaschke_prd_2011 ; blinne_gies_2014
; woellert_prd_2015 ; blinne_strobel_2016 ; aleksandrov_prd_2016 ; li_prd_2017
; lv_pra_2018 ; aleksandrov_epjst_2020 ; aleksandrov_kohlfuerst ;
aleksandrov_kudlis_klochai  and references therein). Here we will utilize the
system of quantum kinetic equations (QKEs), which provides an exact treatment
of vacuum pair production in spatially uniform electric backgrounds of
arbitrary polarization. In Ref. aleksandrov_kudlis_klochai  it was shown that
the system involves ten unknown functions and differs from the QKEs obtained
previously pervushin_skokov  due to an inaccuracy in the preceding
derivations. Here we will perform numerical computations based on the correct
form of the QKEs.

Our main focus is placed on the spin effects, which manifest themselves in
setups involving rotating electric fields. Since the number densities of the
particles produced are defined as functions of momentum, the spin projection
onto a given laboratory-frame axis is not a conserved quantum number as the
corresponding spin operator does not commute with the one-particle
Hamiltonian, and it is the total angular momentum that represents a conserved
quantity. Here we propose an experimentally relevant strategy based on
measuring the particle’s helicity, i.e. spin projection onto the propagation
direction. The corresponding helicity-resolved densities can be obtained from
the QKE formalism aleksandrov_kudlis_klochai . Our goal is to describe the
helicity asymmetry of the electrons and positrons produced in the presence of
a rotating external background, which mimics the QED dynamics in a combination
of two counterpropagating laser pulses of circular polarization. The momentum
distributions of left- and right-handed particles represent evident observable
signatures that should facilitate the experimental efforts concerning
practical investigations of the Sauter-Schwinger effect.

Throughout this letter, we assume $\hbar=c=1$ and use $e<0$ as the electric
charge of the electron $e^{-}$.

Theoretical framework. To describe the external classical background, we
employ the temporal gauge $A_{0}=0$, $\mathbf{A}=\mathbf{A}(t)$. The
corresponding electric field $\mathbf{E}(t)=-\dot{\mathbf{A}}(t)$ is assumed
to vanish for $t\leqslant t_{\text{in}}$ and $t\geqslant t_{\text{out}}$. The
electron-positron field is quantized in the presence of the external field,
i.e. within the Furry picture. Since the electromagnetic background is
homogeneous in space, the Heisenberg field operator can be represented by
means of the adiabatic Hamiltonian eigenfunctions in the following form:

$\displaystyle\psi(x)$ $\displaystyle=$ $\displaystyle\sum_{s=\pm
1}\int\\!\frac{d\mathbf{p}}{(2\pi)^{3/2}}\,\mathrm{e}^{{\rm
i}\mathbf{p}\mathbf{x}}\big{[}a_{\mathbf{p},s}(t)u_{\mathbf{p}-e\mathbf{A}(t),s}(t)$
(1) $\displaystyle+$ $\displaystyle
b^{\dagger}_{-\mathbf{p},s}(t)v_{\mathbf{p}-e\mathbf{A}(t),s}(t)\big{]}\,.$

Here $x=(t,\,\mathbf{x})$ and the bispinors are chosen in the form

$\displaystyle u_{\boldsymbol{p},s}$ $\displaystyle=$ $\displaystyle
C(p^{0})\begin{pmatrix}(p^{0}+m)w_{s}\\\
(\boldsymbol{\sigma}\mathbf{p})w_{s}\end{pmatrix}\,,$ (2) $\displaystyle
v_{\boldsymbol{p},s}$ $\displaystyle=$ $\displaystyle
C(p^{0})\begin{pmatrix}-(\boldsymbol{\sigma}\mathbf{p})w_{s}\\\
(p^{0}+m)w_{s}\end{pmatrix}\,,$ (3)

where $C(p^{0})=[2p^{0}(p^{0}+m)]^{-1/2}$,
$p^{0}=p_{0}=\sqrt{m^{2}+\boldsymbol{p}^{2}}$, and
$w_{-1}=(1,\,0)^{\mathrm{t}}$, $w_{+1}=(0,\,1)^{\mathrm{t}}$ .

Let us denote the Heisenberg state corresponding to the initial vacuum by
$|0,\text{in}\rangle$. It differs from the final vacuum state
$|0,\text{out}\rangle$ due to the vacuum instability fradkin_gitman_shvartsman
. The real-valued QKE functions can be introduced by the following
expresssions for the in-vacuum expectation values aleksandrov_kudlis_klochai :

$\displaystyle\langle
0,\text{in}|a^{\dagger}_{\mathbf{p},s}(t)a_{\mathbf{p}^{\prime},s^{\prime}}(t)|0,\text{in}\rangle$
$\displaystyle=$
$\displaystyle\delta(\mathbf{p}-\mathbf{p}^{\prime})\big{[}f(\mathbf{p},t)\delta_{s^{\prime}s}+\mathbf{f}(\mathbf{p},t)\boldsymbol{\sigma}_{s^{\prime}s}\big{]}\,,$
(4) $\displaystyle\langle
0,\text{in}|b^{\dagger}_{\mathbf{p},s}(t)b_{\mathbf{p}^{\prime},s^{\prime}}(t)|0,\text{in}\rangle$
$\displaystyle=$
$\displaystyle\delta(\mathbf{p}-\mathbf{p}^{\prime})\big{[}f(-\mathbf{p},t)\delta_{ss^{\prime}}-\mathbf{f}(-\mathbf{p},t)\boldsymbol{\sigma}_{ss^{\prime}}\big{]}\,,$
(5) $\displaystyle\langle
0,\text{in}|a^{\dagger}_{\mathbf{p},s}(t)b^{\dagger}_{\mathbf{p}^{\prime},s^{\prime}}(t)|0,\text{in}\rangle$
$\displaystyle=$
$\displaystyle\delta(\mathbf{p}+\mathbf{p}^{\prime})\big{\\{}[\mathbf{u}(\mathbf{p},t)-{\rm
i}\mathbf{v}(\mathbf{p},t)]\boldsymbol{\sigma}_{s^{\prime}s}\big{\\}}\,,$ (6)
$\displaystyle\langle
0,\text{in}|b_{\mathbf{p},s}(t)a_{\mathbf{p}^{\prime},s^{\prime}}(t)|0,\text{in}\rangle$
$\displaystyle=$
$\displaystyle\delta(\mathbf{p}+\mathbf{p}^{\prime})\big{\\{}[\mathbf{u}(-\mathbf{p},t)+{\rm
i}\mathbf{v}(-\mathbf{p},t)]\boldsymbol{\sigma}_{s^{\prime}s}\big{\\}}\,.$ (7)

By utilizing the equation of motion for the Heisenberg field operator, one can
formulate the problem in terms of the QKE system governing the temporal
evolution of the ten unknown components $f$, $\mathbf{f}$, $\mathbf{u}$, and
$\mathbf{v}$ (a detailed derivation is presented in Ref.
aleksandrov_kudlis_klochai ):

$\displaystyle\dot{f}$ $\displaystyle=$
$\displaystyle-2\boldsymbol{\mu}_{2}\mathbf{u}\,,$ (8)
$\displaystyle\dot{\mathbf{f}}$ $\displaystyle=$ $\displaystyle
2(\boldsymbol{\mu}_{1}\times\mathbf{f})-2(\boldsymbol{\mu}_{2}\times\mathbf{v})\,,$
(9) $\displaystyle\dot{\mathbf{u}}$ $\displaystyle=$ $\displaystyle
2(\boldsymbol{\mu}_{1}\times\mathbf{u})+\boldsymbol{\mu}_{2}(2f-1)+2\omega\mathbf{v}\,,$
(10) $\displaystyle\dot{\mathbf{v}}$ $\displaystyle=$ $\displaystyle
2(\boldsymbol{\mu}_{1}\times\mathbf{v})-2(\boldsymbol{\mu}_{2}\times\mathbf{f})-2\omega\mathbf{u}\,.$
(11)

Here

$\displaystyle\boldsymbol{\mu}_{1}(\mathbf{p},t)$ $\displaystyle=$
$\displaystyle\frac{e}{2\omega(\omega+m)}\,[\mathbf{q}\times\mathbf{E}(t)]\,,$
$\displaystyle\boldsymbol{\mu}_{2}(\mathbf{p},t)$ $\displaystyle=$
$\displaystyle\frac{e}{2\omega^{2}(\omega+m)}\,\big{\\{}[\mathbf{q}\mathbf{E}(t)]\mathbf{q}-\omega(\omega+m)\mathbf{E}(t)\big{\\}}\,,$

and $\mathbf{q}\equiv\mathbf{p}-e\mathbf{A}(t)$,
$\omega\equiv\sqrt{m^{2}+\mathbf{q}^{2}}$. All of the QKE functions vanish at
$t=t_{\text{in}}$. While they depend on $\mathbf{p}$, it is $\mathbf{q}$ that
corresponds to the kinetic (gauge-invariant) momentum of the particles, so the
final values of the QKE components will be evaluated as a function of
$\mathbf{q}(t_{\text{out}})=\mathbf{p}-e\mathbf{A}(t_{\text{out}})$. We
underline that the vector potential does not necessarily vanish as
$t\to\infty$.

The system (8)–(11) differs from the QKEs obtained almost two decades ago
pervushin_skokov . In Ref. aleksandrov_kudlis_klochai  it was demonstrated
that the new (correct) form of the QKEs is completely equivalent to the system
that appears within the Dirac-Heisenberg-Wigner formalism BB_prd_1991 ;
zhuang_1996 ; zhuang_prd_1998 ; ochs_1998 ; hebenstreit_prd_2010 ;
blinne_gies_2014 ; blinne_strobel_2016 ; li_prd_2017 ; lv_pra_2018 ;
aleksandrov_kohlfuerst ; aleksandrov_kudlis_klochai ; olugh_prd_2019 ;
kohlfuerst_prd_2019 ; kohlfuerst_arxiv_2022_2 ; yu_prd_2023 ; hu_prd_2023 ;
hu_arxiv_2024 .

In what follows, we will solve the system (8)–(11) numerically and extract the
necessary helicity-resolved densities at $t=t_{\text{out}}$.

Helicity states. Although the QKE functions evolved up to $t=t_{\text{out}}$
can already be used to extract the spin-resolved densities via Eqs. (4) and
(5), the “spin quantum number” $s$ does not provide a physically relevant
quantity as it allows one to merely obtain the populations of the states
corresponding to the basis bispinors (2) and (3). These are not the
eigenfunctions of the spin-projection operator. Moreover, such eigenfunctions
cannot be constructed once one fixes the generalized (canonical) momentum
$\mathbf{p}$ since the spin operator does not commute with the one-particle
Hamiltonian. We propose to unitary transform the basis bispinors, so that they
become the eigenfunctions of the helicity operator
$(\boldsymbol{\Sigma}\mathbf{p})/|\mathbf{p}|$, which does commute with the
Hamiltonian. Accordingly, the new states possess a well-defined quantum number
in addition to $\mathbf{p}$. Note that in order to preserve a relatively
simple form of the QKEs (8)–(11), we will keep the same kinetic functions and
employ them at the very end of the temporal propagation to construct the
helicity-resolved momentum densities. As was shown in Ref.
aleksandrov_kudlis_klochai , in the case of electrons, the latter read

$f^{(e^{-}\text{L/R})}(\mathbf{p},t)=f(\mathbf{p},t)\mp\frac{\mathbf{q}\mathbf{f}(\mathbf{p},t)}{|\mathbf{q}|}\,.$
(12)

Here L (R) corresponds to the left-handed (right-handed) particle, i.e. to
negative (positive) helicity. The positron distributions can be then easily
obtained via

$f^{(e^{+}\text{L/R})}(\mathbf{p},t)=f^{(e^{-}\text{R/L})}(-\mathbf{p},t)\,.$
(13)

The final particle distributions are those calculated at $t=t_{\text{out}}$.
For instance,

$\frac{(2\pi)^{3}}{V}\frac{dN^{(e^{-}\text{L})}_{\mathbf{q}}}{d^{3}\mathbf{q}}=f^{(e^{-}\text{L})}(\mathbf{q}+e\mathbf{A}(t_{\text{out}}),t_{\text{out}})\,,$
(14)

where $V$ is the volume of the system and $\mathbf{q}$ is the particle’s
kinetic momentum. Note that due to the Pauli exclusion principle, these
densities never exceed unity as can also be directly proved by inspecting the
system (8)–(11) aleksandrov_kudlis_klochai .

Numerical results. The rotating external field is chosen in the following
form:

$\mathbf{E}(t)=\frac{E_{0}}{\sqrt{2}}\,F(\Omega t)\big{[}\cos(\Omega
t)\,\mathbf{e}_{x}+\sin(\Omega t)\,\mathbf{e}_{y}\big{]},$ (15)

where $\mathbf{e}_{x}$ ($\mathbf{e}_{y}$) is the Cartesian unit vector along
the $x$ ($y$) direction, $F(\eta)=\mathrm{exp}(-\eta^{2}/\sigma^{2})$ is a
pulse envelope, and $\sigma$ is a dimensionless parameter governing the pulse
duration. We set $t_{\text{out}}=-t_{\text{in}}=T$, where $T$ is sufficiently
large, so the external field is practically zero for $|t|>T$ and the numerical
results are converged. The setup (15) approximates a combination of two
counterpropagating circularly-polarized laser pulses of opposite handedness
(the role of the laser handedness was explored in terms of the number density
$f$ in Ref. kohlfuerst_arxiv_2022_2 ).

Figure 1: Helicity-resolved momentum distributions of the electrons produced
by the electric field (15) as a function of the kinetic momentum projections
$q_{x}$ and $q_{z}$. Left and middle: number densities
$f^{(e^{-}\text{L/R})}$. The right panels correspond to the helicity
asymmetry. The field parameters are (a) $E_{0}=0.2E_{\text{c}}$,
$\Omega=0.5m$, $\sigma=5$; (b) $E_{0}=0.5E_{\text{c}}$, $\Omega=0.2m$,
$\sigma=10$; (c) $E_{0}=0.5E_{\text{c}}$, $\Omega=0.5m$, $\sigma=10$. The
spectra were obtained for $p_{y}=0$ in the gauge $\mathbf{A}(0)=0$.

Before we present the results of our numerical calculations, let us discuss a
fundamental symmetry property of the QKE system (8)–(11) which can be
established analytically. One can directly verify that changing the sign of
$p_{z}=q_{z}$, one obtains a solution with opposite signs of $f_{x}$, $f_{y}$,
$u_{z}$, and $v_{z}$ (the other components remain the same). In this case, the
product $\mathbf{q}\mathbf{f}/|\mathbf{q}|$ changes the sign, so the helicity-
resolved distributions “L” and “R” turn to each other upon the reflection
$q_{z}\to-q_{z}$ (or, equivalently, $p_{z}\to-p_{z}$). In other words, the
first (second) term in Eq. (12) is symmetric (antisymmetric) with respect to
$q_{z}\to-q_{z}$. There are several very important consequences of this
symmetry. First, the helicity asymmetry
$f^{(e^{-}\text{L})}-f^{(e^{-}\text{R})}$, which immediately appears once
$\mathbf{q}\mathbf{f}/|\mathbf{q}|\neq 0$, is an asymmetric function of
$q_{z}$. Note that in the case of a linearly polarized external field, the
function $\mathbf{f}$ vanishes identically aleksandrov_kudlis_klochai .
Second, it is sufficient to numerically calculate only one of the two partial
distributions [either $f^{(e^{-}\text{L})}(\mathbf{p},t)$, or
$f^{(e^{-}\text{R})}(\mathbf{p},t)$], and the helicity asymmetry is determined
by the $q_{z}$ asymmetry of the corresponding spectra. Finally, we note that
these features take place once $E_{z}=0$, so they hold within a broad class of
the field configurations (e.g., the field can have arbitrary elliptical
polarization). The total number density summed over helicity is always
symmetric with respect to $q_{z}\to-q_{z}$.

Since the helicity asymmetry vanishes for $p_{z}=q_{z}=0$, in what follows we
will focus on the analysis of $q_{x}q_{z}$ and $q_{y}q_{z}$ momentum
distributions. Even the total number densities $f^{(e^{-})}$, i.e. those
summed over the spin states, being computed within these momentum planes are
not frequently discussed in the literature ($q_{x}q_{y}$ spectra were examined
in numerous studies blinne_gies_2014 ; blinne_strobel_2016 ; li_prd_2017 ;
olugh_prd_2019 ; hu_prd_2023 ; hu_arxiv_2024 ). In Fig. 1 we present the
typical $q_{x}q_{z}$ momentum spectra for three different sets of the field
parameters. The left and middle graphs correspond to negative and positive
helicity, respectively, while in the right column we display the degree
$a^{(e^{-})}$ of the helicity asymmetry defined as the difference
$f^{(e^{-}\text{L})}-f^{(e^{-}\text{R})}$ divided by the maximal value of the
electron number density within the corresponding momentum plane. We observe
that this quantity is indeed an odd function of $q_{z}$ and its sign is
opposite to that of $q_{z}$. This indicates that the right-handed electrons
preferably move along the $z$ axis, whereas the left-handed particles tend to
travel in the opposite direction. Furthermore, due to the property (13), the
asymmetry degree of the positrons produced is also opposite to the sign of the
$z$ momentum projection, so the right-handed positrons are also likely to
propagate in the positive direction of the $z$ axis. The asymmetry degree in
our examples is of the order of $10$–$20\%$, which represents a macroscopic
value from the experimental viewpoint.

Another important feature of the momentum distributions is its symmetry with
respect to the reflection $p_{x}\to-p_{x}$. As was shown in Refs.
blinne_gies_2014 ; dumlu_dunne_prd_2011 , this property appears due to the
time-reversal symmetry. In terms of the QKE components, we find that the
transformation $p_{x}\to-p_{x}$, $t\to-t$ does not alter the QKEs (8)–(11),
provided $f_{x}$, $u_{x}$, $v_{y}$, and $v_{z}$ change the sign. The crucial
point here is that since in this case $q_{x}\to-q_{x}$, the second term in Eq.
(12) remains the same, so the symmetry also holds for the helicity-resolved
distributions. Note that it is essential that the $x$ ($y$) component in Eq.
(15) is symmetric (antisymmetric) under time reversal, and the generalized
momentum $p_{x}$ corresponds to the gauge $\mathbf{A}(0)=0$, so the vector
potential has also symmetry properties. For the field parameters chosen in our
examples, the difference between $q_{x}$ and $p_{x}$ can hardly be identified
in the graphs. One of the important consequences of the $p_{x}$ symmetry is
that it completely prohibits any vortex structures of the helicity-resolved
densities involving the $p_{x}$ axis (cf. Ref. hu_arxiv_2024 , where instead
of the helicity-resolved spectra, more complex distributions were analyzed).
We also note that the total density, i.e. the sum of the left and middle
plots, computed by means of our numerical procedures exhibits the usual
behavior reported, e.g., in Refs. blinne_strobel_2016 ; hu_arxiv_2024 .

Figure 2: Helicity-resolved momentum distributions of the electrons as a
function of $q_{y}$ and $q_{z}$. Left and middle: number densities
$f^{(e^{-}\text{L/R})}$. The right panels correspond to the helicity
asymmetry. The field parameters are (a) $E_{0}=0.5E_{\text{c}}$,
$\Omega=0.5m$, $\sigma=5$; (b) $E_{0}=0.5E_{\text{c}}$, $\Omega=0.2m$,
$\sigma=10$. The spectra were obtained for $p_{x}=0$ in the gauge
$\mathbf{A}(0)=0$.

In Fig. 2 we depict analogous momentum distributions in the $q_{y}q_{z}$
plane. Here we observe again the $q_{z}$ symmetry, while the spectra are not
symmetric with respect to $q_{y}$. Note that in Fig. 2(b) the electron
distributions are considerably shifted in the region of negative $q_{y}$. This
feature appears due to a great net momentum transfer in this direction because
of the large value of the corresponding electric-field area, i.e. large value
of $A_{y}(t_{\text{out}})$. This effect with respect to the $x$ axis (Fig. 1)
is not evident as was pointed out above. Note also that in Fig. 1 we used
$p_{y}=0$, which yields quite large values of the electron densities unlike,
e.g., $q_{y}=0$. Since the external field does not exert a force in the $z$
direction, there is no net acceleration with respect to $q_{z}$ and the
support of the momentum distributions corresponds to $q_{z}\lesssim m$.
Nevertheless, the helicity asymmetry is obvious: the laser pulse tends to push
right-handed electrons toward the $z$ direction, while the left-handed
electrons are mainly accelerated in the opposite direction. In terms of
measuring the scattering angles, this asymmetry should be evidently identified
as the plane $q_{z}=0$ clearly divides the space into two different subspaces.
The generic symmetry properties uncovered above and our numerical examples
demonstrate rather universal patterns. We also verified that if one changes
the handedness of the external rotating field by reversing the sign of the
second term in Eq. (15), then the momentum distributions will yield the
opposite sign of the helicity asymmetry. Finally, we note that in Fig. 2(b)
the momentum distribution has a pronounced interference structure, which is a
well-elaborated effect frequently occurring in the case of many-cycle pulses
blinne_gies_2014 ; blinne_strobel_2016 ; li_prd_2017 ; olugh_prd_2019 ;
hu_prd_2023 ; hu_arxiv_2024 ; dumlu_dunne_prd_2011  [see also Fig. 1(b) and
(c), where $\sigma=10$].

Conclusions. In this study, we numerically implemented a recently derived
system of quantum kinetic equations and investigated the process of
nonperturbative vacuum pair production in a rotating electric background. To
address the spin effects, we focused on the analysis of the particle’s
helicity, which represents a well-defined quantity from the quantum mechanical
perspective. By inspecting the helicity-resolved distributions of the
electrons and positrons, we revealed a significant helicity asymmetry
providing a distinct observable feature. Our findings should further
illuminate the physical intricacies of vacuum pair production and facilitate
experimental research concerning strong-field QED.

Acknowledgments. The investigation was funded by the Ministry of Science and
Higher Education of the Russian Federation (Goszadaniye), project No.
2019-1246. I.A.A. also acknowledges the support from the Foundation for the
advancement of theoretical physics and mathematics “BASIS”.

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Present address: ]School of Engineering, Kwansei Gakuin University, 2-1
Gakuen, Sanda, Hyogo 669-1337, Japan

# Following enhanced Sm spin projection in GdxSm1-xN

J. D. Miller<EMAIL_ADDRESS>J. F. McNulty  B. J. Ruck  MacDiarmid
Institute for Advanced Materials and Nanotechnology,
School of Chemical and Physical Sciences,
Victoria University of Wellington, P.O. Box 600, Wellington 6140, New Zealand
M. Al Khalfioui  S. Vézian  Université Côte d’Azur, CNRS-CRHEA, Valbonne
06560, France  M. Suzuki  [  H. Osawa  N. Kawamura  Japan Synchrotron
Radiation Research Institute (JASRI), 1-1-1 Kouto, Sayo, Hyogo, 679-5198,
Japan  H. J. Trodahl<EMAIL_ADDRESS>School of Chemical and Physical
Sciences, Victoria University of Wellington,
P.O. Box 600, Wellington 6140, New Zealand

###### Abstract

The rare-earth nitrides form a series of structurally simple $intrinsic$
ferromagnetic semiconductors, a rare class of both fundamental interest and
application potential. Within the series there is a wide range of magnetic
properties relating to the spin/orbit contributions to the ferromagnetic
ground states. We report an x-ray magnetic circular dichroism investigation of
the spin/orbit magnetic dipole alignments of Sm and Gd ions in epitaxial
GdxSm1-xN films. The Sm spin-alignment expectation value $\langle
S_{z}\rangle$ is seen to be strengthened by the Gd/Sm exchange interaction,
providing guidance concerning the composition for an angular momentum
compensation point (where the volume-averaged total angular momentum of a film
is zero).

††preprint: APS/123-QED

##  I Introduction

The lanthanides are the series of the periodic table across which the $4f$
shell is sequentially filled, imbuing them with the strongest angular momentum
and magnetic moments among all stable elements. The strong spin-orbit coupling
within the 4$f$ shell ties the orbital moment firmly to the spin so that the
net magnetic moment is of mixed spin and orbital character. Furthermore, the
limited extent of the $4f$ wave function reduces the orbital angular momentum
quenching that is a feature of transition metal compounds; indeed the orbital
moment can dominate the net magnetic moment.

The rare-earth nitrides (RN, with R a lanthanide element) form in the very
simple NaCl structure which provides a fertile ground to investigate the
spin/orbit magnetism that stems from the partially full $4f$ electron levels
[1]. The chemical similarity of the lanthanides and the well-matched lattice
constants in their nitrides invites studies in the mixed-cation environments
offered by multilayers and solid solutions. Multilayer structures of GdN/SmN
and of GdN/NdN have already shown a striking competition between the exchange
and Zeeman interactions across interfaces [2, 3, 4, 5]. The present report
investigates the effects of that competition among randomly-sited Gd3+/Sm3+
ions in the solid solution GdxSm1-xN using x-ray magnetic circular dichroism
(XMCD).

The magnetism of the parent materials, GdN ($x=1$) and SmN ($x=0$) illustrates
the range of magnetism in the RN series. Both are ferromagnetic, with Curie
temperatures (TC) of $\sim 70$ and $\sim 30$ K, respectively, resulting from
an indirect exchange mechanism that links the $4f$ alignments via the R $5d$
and N $2p$ states [6, 1, 7, 8]. The Gd3+ ions in GdN have a half-filled $4f$
shell ([Xe]$4f^{7}$), and in the ground state all seven spins align giving a
spin-number, $S~{}=~{}7/2$. The half-full shell further features zero orbital
angular momentum ($L=0$) so that the material behaves as a spin-only
ferromagnet with $\langle M_{z}\rangle=2\mu_{B}\langle
S_{z}\rangle=7~{}\mu_{B}$ per Gd3+ ion. In contrast, the Sm3+ ions in SmN
possess two fewer electrons giving an electronic configuration of
[Xe]$4f^{5}$. Inter-ion exchange aligns the $4f$ spins, but opposing magnetic
contributions from the spin and orbital angular momenta very nearly cancel to
yield a net magnetisation of $\approx$ 0.035 $\mu_{B}$ per Sm3+ ion in the
ferromagnetic ground state [9, 8]. Crucially, the magnetic moment $\langle
M_{z}\rangle=\mu_{B}(\langle L_{z}\rangle+2\langle S_{z}\rangle)$ is weakly
orbital dominated, with $\langle L_{z}\rangle/2\langle S_{z}\rangle<-1$ [10],
so that the spin contribution to the magnetic moment is opposed to an applied
field. It is this opposition of the spin magnetic moment ($\mu_{S}$) to the
applied field that provides a competition between the Zeeman and exchange
interactions when SmN and GdN are paired in multilayers [10, 3].

Similarly to multilayer structures, the Gd-Sm exchange interaction in
GdxSm1-xN disrupts the tendency of the Zeeman interaction to align the Sm3+
orbital magnetic moment ($\mu_{L}$) with an applied field. The alignment
(anti-alignment) of the Gd3+ (Sm3+) spin magnetic moments with an applied
magnetic field (Figure 1a) is in direct competition with the ferromagnetic
inter-ion exchange attempting to align the Gd3+ and Sm3+ spins (Figure 1b).
Because the Gd3+ magnetic moment is two orders of magnitude the larger, most
values of $x$ show the alignment of the the Gd3+ spin-moment with the applied
field. Yet, for extremely dilute concentrations of Gd in solution (that is,
$x\approx 0$) it must be that the net Sm3+ magnetic moments align with the
applied field, rather than the Gd3+, due to the same exchange.

The simplest model for the magnetic state of GdxSm1-xN would treat the Sm3+
and Gd3+ ion moments as fixed at their values in SmN and GdN. This model
suggests that under full ferromagnetic order the saturation magnetisation is
linear in $x$ and for some appropriate composition there exists a net zero
magnetisation, a magnetisation compensation point. Within that model the net
magnetisation ($\bar{M}$) ranges from $-0.035~{}\mu_{B}$ to $7~{}\mu_{B}$ per
R ion (with respect to the spin-moment direction in the solid solution) with
the compensation point occurring at $x~{}=~{}0.005$ (Figure 1c). The XMCD
results in this paper indicate that the Sm3+ spin-alignment does $not$ remain
as adopted in SmN, but nearly doubles in Gd-rich films.

Figure 1: Schematic showing the alignment of the magnetic moments of Sm3+ and
Gd3+ in (a) the parent materials where the Sm-Gd exchange is absent and (b)
the solid solution GdxSm1-xN where the Sm-Gd exchange dominates the Zeeman
interaction and aligns the spins. Panel (c) shows the net (low-temperature,
volume averaged) values of the magnetisation ($\bar{M}$) and angular momentum
($\bar{J}$) as the composition ($x$) interpolates between the parent
materials. This schematic refers to to the most simple model and is shown in
Figure 7 adapted in light of our XMCD results.

Furthermore, the same behaviour seen for the net magnetic moment must also
manifest for the net angular momentum, $\bar{J}$, meaning that under full FM
order $\bar{J}$ $must$ also pass through zero as $x$ varies from $0$ to $1$,
giving an angular momentum compensation point (Figure 1c). Here again the
simplest model uses a linear interpolation between the ground state angular
momenta of GdN and SmN, but unlike the magnetisation there is no simple
measurement of the angular momenta of the parent materials. The spin-only
magnetism in GdN clearly indicates that $J=S=7/2$, and Hund’s rules suggest
that $J=L-S=5/2$ in SmN. However, an atom-centred treatment of the magnetism
of SmN based on realistic crystal-field and exchange interactions and
reproducing the orbital-dominant ground state moment of $0.035~{}\mu_{B}$ per
Sm3+ ion yields an eigenstate of neither $L_{z}$ or $S_{z}$ [11]. Rather, it
shows projections $\langle L_{z}\rangle\approx 2\hbar$ and $\langle
S_{z}\rangle\approx-\hbar$, substantially smaller than the maximum possible
values provided by Hund’s rules ($\langle L_{z}\rangle=5\hbar$ and $\langle
S_{z}\rangle~{}=-5\hbar/2$, treating the spin moment direction as positive,
once more). Importantly, the spin-orbit coupling is still strong enough that
the spin and orbital angular momenta remain opposed.

This description of the magnetism of the Sm3+ ion, in which the $4f$ spins are
ferromagnetically aligned in the ground state, but without their maximum
potential magnitudes, accommodates the factor of $\approx$ 2 increase in the
Sm spin polarisation that is seen in XMCD measurements performed on GdN/SmN
superlattices [2, 11]. We thus introduce a parameter $\alpha=\langle
S_{z}\rangle/(5\hbar/2)$, which indicates the expectation value of the spin-
alignment $\langle S_{z}\rangle$ in the $4f$ shell of the Sm3+ ion. Appealing
to the ion-based model of McNulty  _et al._ [11], in homogeneous SmN
$\alpha~{}\approx~{}0.4$, while at the SmN/GdN interface
$\alpha~{}\approx~{}0.6$. One might then anticipate, as seen below, that that
$\alpha$ increases with Gd concentration $x$. The simple model shown in Figure
1 illustrates the dependence of the net magnetic moment and angular momentum
on $x$, but assumes that $\alpha$ is independent of $x$. In the upcoming
discussion we show the equivalent diagram with $\alpha(x)$ as implied by the
XMCD results (Figure 7).

##  II Experimental Details

GdxSm1-xN films were grown in a Riber 32p molecular beam epitaxy (MBE) system
on 100 nm thick (0001) AlN buffered Si, heated during growth to between $450$
and $700^{\circ}$ C depending on the composition. The growth rate was held to
$100$ nm/h. High purity rare-earth metals were evaporated from conventional
effusion cells to achieve the desired composition under nitrogen rich
conditions ($P_{N_{2}}\approx 2.1\times 10^{-5}$ Torr). The composition of the
solid-solutions was measured to within $x\pm 0.04$ using X-ray fluorescence
(XRF) spectroscopy. X-ray diffraction confirmed (111)-oriented FCC NaCl
structure, typical of the rare-earth nitride series.

Field in-plane SQUID magnetometry was performed in a Quantum Design Magnetic
Property Measurements System (MPMS) capable of temperatures as low as 5 Kelvin
and fields up to 7 Tesla. These measurements determine the temperature- and
hysteretic field-dependent net magnetic moment across the para- to ferro-
magnetic transition. In the present system that net magnetic moment is
overwhelmingly dominated by the Gd ions, motivating support of the ionic-
species resolution capability of XMCD to investigate the Sm3+ alignment
separately. That alignment relates specifically to the 4$f$-shell spin and
orbital states, suggesting the use of XMCD at the M4,5 edge, which
interrogates alignment in the 4$f$ shell. However, the surface sensitivity of
XAS at the $\sim 1$ keV M-edge renders it unsuitable for the present study,
and we have used instead XMCD at the L2,3 edges of Gd and Sm. The hard-x-ray
lanthanide L edge measurements probed the full $\approx$ 100 nm thickness of
the films.

The L edge involves excitation into the $5d$ band, coupled to the $4f$ spin
alignment by the strong $5d$-$4f$ exchange interaction. Thus, we report here
the results of XMCD at the Gd and Sm L2,3 edges to probe separately the
alignment on the two cations. The measurements were performed at temperatures
down to $5$ K and fields up to $6$ T at the BL39XU beamline at the SPring-8
synchrotron in Japan. All XMCD was performed at $10^{\circ}$ from grazing
incidence.

XMCD at the two edges, L2,3, separated by the spin-orbit interaction in the
$2p$ core-hole state, can in principle yield information about the spin and
orbital alignments in the final $5d$ state, though the failure of the XMCD sum
rules for the L2,3 edges prevent their use in the present case [12].
Furthermore, the Sm L2 edge in these materials is masked by magnetic EXAFS
from the Gd L3 edge [13]. We thus focus the present discussion on the
relatively weak L3 edge of Sm and the stronger L2 edge of Gd, interpreted in
terms of the temperature- and composition-dependent spin alignments.

Figure 2: (a) Hysteresis loops of the three GdxSm1-xN films taken at $5$ K.
The inset shows a zoom of the region close to zero, emphasising the different
coercivities of the films. (b) The inverse susceptibility of the three films
with fits included to show the different Curie temperatures extracted from the
measurements.

##  III Results and Discussion

###  III.1 Magnetisation in the para- and ferro-magnetic phases

To aid the interpretation of the XMCD results, we present field-cooled
magnetization and hysteresis loop measurements (Figure 2) on the same three
films for which XMCD was performed. The temperature traces identified for each
sample a single ferromagnetic transition between the 30 and 70 K Curie
temperatures of SmN and GdN [8, 7, 14], as seen in Figure 2b. The most Gd-rich
film ($x=0.92$) has the smallest coercive and saturation fields, comparable
with epitaxial GdN films. With $\approx$ 1/4 of the Gd3+ ions replaced by non-
spherically-symmetric Sm3+ there are increases in both the coercive and
saturation fields. Both trends, which are further enhanced in the most Sm-rich
film, follow from the combination of (i) a weaker Zeeman interaction that
results from a reduced saturation magnetisation, and (ii) a larger crystal
anisotropy experienced by the $J=5/2$ ground state of Sm3+. All films show a
full alignment above $\sim 5$ T, dictating that the spectra below were
collected at the maximum available XMCD field of $6$ T.

###  III.2 XMCD

The near-zero magnetic-moment of Sm in the ferromagnetic state of SmN ensures
that the magnetisation of Figure 2 relates directly to the Gd alignment. We
thus show first in Figure 3 the Gd L2-edge XAS (a) and XMCD (b) in the three
films taken at saturation, 5 K and 6 T. A single feature dominates here, with
the nearly identical magnitudes of the XMCD for different sample compositions
providing a reminder that normalising the XMCD to the XAS signal causes the
XMCD to reflect the _alignment_ of the magnetic moments rather than their
concentration. The few % reduction in the most Sm-rich film then signals that
the Gd3+ ions are less perfectly aligned in that film. This less-than-perfect
alignment is discussed further where we present the temperature-dependence of
the XMCD of the films.

Figure 3: Comparison of XMCD spectra for the three GdxSm1-xN films at the Gd
L2 edge at $5$ K. All measurements were cooled to $5$ K in an applied field of
6 Tesla. Panel (a) shows the average of the x-ray absorption measurements for
left- and right- circularly polarised light, normalised to the continuum edge
jump. Panel (b) shows the XMCD, defined as the difference between the left-
and right- circularly polarised absorption.

Figure 4 shows Sm L3 XAS and XMCD spectra for the three GdxSm1-xN films, with
a spectrum taken from a homogeneous SmN film [10] shown for comparison in
Figure 4c. The spectra in the solid-solution films have the opposite sign to
that seen for pure SmN, a clear signature of the Zeeman-exchange conflict. As
discussed in the Introduction, under solely the Zeeman interaction in SmN the
$4f$ _spin_ magnetic moment aligns in opposition to the applied field. This is
reversed in the GdxSm1-xN films by Gd-Sm exchange that couples the Sm spin
alignment to the strongly Zeeman-aligned Gd3+ spins. Moreover, the magnitude
of the XMCD at the Sm L3 in GdxSm1-xN clearly increases with the value of $x$,
indicating the increasing alignment of $\langle S_{z}\rangle$ in films with
increasing Gd-content.

Figure 4: Comparison of XMCD spectra for the three GdxSm1-xN films at the Sm
L3 edge at $5$ K. All measurements were cooled to $5$ K in an applied field of
$6$ T. Panel (a) shows the average of the x-ray absorption measurements for
left- and right- circularly polarised light, normalised to the continuum edge
jump. Panel (b) shows the XMCD, defined as the difference between the left-
and right- circularly polarised absorptions. Panel (c) shows the XMCD Spectrum
for a SmN film, data taken from Anton  _et al._ [10].

A subtle contrast between the GdxSm1-xN (Figure 4b) and the SmN XMCD (Figure
4c) is an enhanced strength across the range $6.715-6.720$ keV in the present
films, seen most strongly in the most Gd-rich film ($x=0.92$). Significantly,
even in SmN the weak signal at those energies was not predicted by an XMCD-
based calculation [10]. It is, however, exactly the range across which there
are XAS spectral lines from Sm2+ [15, 16, 17], suggesting that their presence
here could relate to a low concentration of divalent Sm. The signal is
exceedingly weak, $\approx 1$ % of the XAS signal at the normalisation point
($\approx 6.735$ keV) so it is no surprise that there appears no clear
signature of this feature in the XAS spectra. In order to limit interference
from the possible divalent signal in our interpretation of the Sm3+ XMCD
signal strength, for our temperature-dependent discussion below we have chosen
to use the XMCD magnitude across the $6.722-6.7256$ keV transition as a
measure of the spin alignment in the Sm3+ final states.

Figure 5: Magnitude of the Sm L3 absorption features as a function of the Gd
fraction, $x$. Panel (a) shows the magnitude for the conventional Sm3+
absorption channel. Panel (b) shows the magnitude of the electric quadrupole
absorption. The star marker represents the data taken from Anton  _et al._
[10].

The feature that appears at $\approx 6.709$ keV in Figure 4 is also not
predicted in a calculated spectrum, and it lies at an energy that would imply
an empty state deep within the valance band, well below the Fermi energy. It
has been previously identified as an electric quadrupole (EQ) $2p$ to $4f$
transition [10], in which the final $4f$ state appears, as expected within
XAS, at the $4f$ energy reduced in the presence of the $2p$ core hole [12].
Such features have been attributed to electric quadrupole absorption in many
rare-earth insulators [18] and this attribution has been confirmed in
measurements of dysprosium using angular resolved XAS [19]. It is natural to
regard this feature in XMCD spectra as proportional to the spin alignment
within the Sm3+ $4f$ shell, and indeed it does strengthen across the three
films, though not quite commensurate in strength with the $5d$ signals (see
Figure 5). An analogous situation is seen also in SmN/GdN superlattices, and
signals that the spin-alignment of the Sm3+ $5d$ states is enhanced by
neighboring Gd3+ ions. The contraction of the radial $5d$ wavefunction that
stems from $4f-5d$ exchange interaction leads to an enhancement of the
$2p\rightarrow 5d$ transition [12, 20, 21], so it is no surprise to see that
the increased spin alignment of the $4f$ states is reflected with a more
pronounced intensity in the electric dipole absorption channel. It is this
same contraction that prevents the use of the usual XMCD sum rules to separate
the orbital and spin contributions to the magnetic moment [12, 22].

Figure 6: Strengths of the (a) Gd3+ L2-edge and (b) Sm3+ L3-edge features
measured using XMCD as a function of temperature. Error bars are smaller than
the markers where not visible.

The influence of the solid-solution composition on exchange interactions is
also seen in the temperature-dependent XMCD strengths, shown in Figure 6 as
the spin alignments inferred by the Gd L2 and the Sm L3 edges. Looking first
at Gd, there is a factor of $2$ drop of the $80$ K alignment from that at low-
temperature in both of the films with high Gd concentration. Recalling that
these films show a Curie temperature in the range of $60-70$ K, and that the
XMCD was performed at a field of $6$ T, there is no surprise that the
alignment remains strong to this temperature. In contrast, the film with only
$0.21$ Gd shows a much more rapid drop. The Gd concentration for this film is
within uncertainty of the site percolation threshold on the FCC lattice [23];
and this suggests the potential for a reliance in this film on the weaker Gd-
Sm-Gd exchange for full ferromagnetic spin alignment on the Gd ions.The
dominance of the exchange interaction over the Zeeman interaction on the Sm
ions leads to a situation in which the temperature-dependence of the Sm
alignment follows that of Gd ions rather closely in all films.

Returning to the results of Figure 5, a consequence of the varying Sm3+ spin
alignment is its influence on the Gd concentration of the angular momentum
compensation point of Figure 1(c). That compensation occurs at $x\approx 0.2$
for the $\langle S_{z}\rangle\approx\hbar$ found in SmN, but the stronger Sm3+
alignment in these solid solutions shifts the compensation point to higher Gd
concentration. A quantitative interpretation of the shift relies on a measure
of the enhancement of the $4f$ spin-alignment $\langle S_{z}(x)\rangle$, for
which we rely on the increase of the $4f$ quadrupole feature at
$\approx~{}6.709$ keV shown in Figure 5. Those data imply that the magnitude
of the Sm3+ electric quadrupole feature is enhanced by $\sim 1.5$ between
$x=0$ and $x\rightarrow 1$. The strong spin-orbit interaction ensures the Sm3+
net angular momentum (which is negative with respect to the spin direction,
due to the orbital-dominant magnetism) will also increase linearly with $x$,
leading ultimately to the shift of the compensation point by $\approx 0.02$ as
shown in Figure 7.

Figure 7: Model of the net (volume-averaged) angular momentum, $\bar{J}$, as a
function of Gd fraction, $x$. The simple model is a linear interpolation of
total angular momentum between $\langle J_{z}\rangle=-1$ ($\hbar$ per R ion)
for SmN (where $\langle L_{z}\rangle=-2\hbar$ and $\langle
S_{z}\rangle=\hbar$) and $\langle J_{z}\rangle=J=7/2$ for GdN. Taking into
account the variation of $\alpha(x)$ implied by Figure 5 alters the net
angular momentum, shifting the compensation point.

##  IV Conclusions

We have probed the alignment of the moments in GdxSm1-xN by L-edge XMCD
tracing out the spin alignments $\langle S_{z}\rangle$ independently on the
Sm3+ and Gd3+ ions. Special care was taken to follow the XMCD signal
amplitudes and the spin alignments $\langle S_{z}\rangle$ of the two ionic
species as a function of the composition across the GdN/SmN solid solutions.

The XMCD study was supported by thorough magnetisation studies of the films.
It is noted that the magnetisation is overwhelmingly dominated by the Gd3+
spin alignment, since the magnetic moment on the Sm3+ is more than two orders
of magnitude the smaller. Magnetic measurements revealed the temperature-
dependent magnetisation as well and complete hysteresis curves in the
ferromagnetic phase that were used to inform XMCD measurements and limit those
to magnetic fields adequate to ensure saturated magnetisation. There was
excellent agreement between the magnetisation and Gd $\langle S_{z}\rangle$,
but it is only XMCD that is capable of revealing the spin alignment $\langle
S_{z}\rangle$ on the Sm3+ ions, the primary outcome of the study.

The sign of the Sm XMCD, opposite to that in SmN, immediately identifies that
the alignment of the Sm and Gd spins by the Sm-Gd exchange interaction
dominates the Sm3+ Zeeman interaction, which on its own acts to align the
_orbital_ magnetic moment in SmN [10, 2]. The data further show that the
exchange dominance strengthens the Sm3+ spin alignment in the presence of Gd3+
ions; the projection $\langle S_{z}\rangle\approx\hbar$ in SmN, rising to
$\langle S_{z}\rangle\approx 3\hbar/2$ for isolated Sm3+ ions in a
predominantly GdN host. The composition-dependent alignment specifies that the
composition at which an angular-momentum compensation can be expected is at
higher values of $x$ than implied by measurements of pure SmN films.

###### Acknowledgements.

The Synchrotron Radiation experiments were performed at the BL39XU beamline of
SPring-8 with the approval of the Japan Synchrotron Radiation Research
Institute (JASRI), Proposal No: 2020A1250. The MacDiarmid Institute is
supported under the New Zealand Centres of Research Excellence Programme.

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