# Spatial Mapping of Torques within a Spin Hall Nano-oscillator

T. M. Spicer  P. S. Keatley  T. H. J. Loughran  Department of Physics and
Astronomy, University of Exeter, EX4 4QL,United Kingdom  M. Dvornik  A. A.
Awad  P. Dürrenfeld  A. Houshang  M. Ranjbar  Department of Physics,
University of Gothenburg, 412 96 Gothenburg, Sweden  J. Åkerman  Department of
Physics, University of Gothenburg, 412 96 Gothenburg, Sweden  Materials
Physics, School of ICT, KTH-Royal Institute of Technology, Electrum 229, 164
40 Kista, Sweden  V. V. Kruglyak  R. J. Hicken  Department of Physics and
Astronomy, University of Exeter, EX4 4QL,United Kingdom

###### Abstract

Time-resolved scanning Kerr microscopy (TRSKM) was used to study precessional
magnetization dynamics induced by a radio frequency (RF) current within a
Al2O3/Py(5 nm)/Pt(6 nm)/Au(150 nm) spin Hall nano-oscillator structure. The Au
layer was formed into two needle-shaped electrical contacts that concentrated
the current in the centre of a Py/Pt mesa of 4 $\mu$m diameter. Due to the
spin Hall effect, current within the Pt layer drives a spin current into the
Py layer, exerting a spin transfer torque (STT). By injecting RF current, and
exploiting the phase-sensitivity of TRSKM and the symmetry of the device
structure, the STT and Oersted field torques have been separated and spatially
mapped. The STT and torque due to the in-plane Oersted field are observed to
exhibit minima at the device centre that is ascribed to spreading of RF
current that is not observed for DC current. Torques associated with the RF
current may destabilise the position of the self-localised bullet mode excited
by the DC current, and inhibit injection locking. The present study
demonstrates the need to characterise both DC and RF current distributions
carefully.

Spin torque oscillators (STOs)Chen  _et al._ (2016) are nanoscale
magnetoresistive devices of great promise for use in microwave assisted
magnetic recording Shiroishi  _et al._ (2009), microwave frequency
telecommunications Chumak  _et al._ (2015), and neuromorphic computing
Grollier  _et al._ (2016). Injection of DC current $I_{DC}$ generates spin
transfer torque (STT) that excites precession of the constituent magnetic
moments. The magnetoresistance (MR) leads to an RF voltage across the device.
Within spin Hall nano-oscillator (SHNO) devices, a charge current is converted
into a pure spin current, via the spin Hall effect (SHE) Kato  _et al._
(2004); Hirsch (1999); Jungwirth  _et al._ (2012); Hoffmann (2013), which
exerts a STT upon the active magnetic layer. Decoupling of spin and charge
currents provides additional freedom in the device design, the magnetic
materials used Ranjbar  _et al._ (2014); Dürrenfeld  _et al._ (2015); Mazraati
_et al._ (2016); Zahedinejad  _et al._ (2018) (including electrical insulators
Hamadeh  _et al._ (2014)) and the precessional modes excited Dvornik  _et al._
(2018). If charge current flows parallel to the plane, without a top contact
obscuring the active region, then optical techniques can probe the
magnetization dynamics Demidov  _et al._ (2012); Awad  _et al._ (2016).
However the spatial current distribution may be highly non-uniform, and
thermal effects may modify both the current distribution and the torques Liu
_et al._ (2013); Demidov  _et al._ (2014); Ulrichs  _et al._ (2014). Knowledge
of the Oersted torque and STT is critical for understanding how auto-
oscillations may be excited, or locked to a reference signal, and until now it
has been impossible to probe their spatial distribution directly.

STT-ferromagnetic resonance (STT-FMR) is widely used to characterise
spintronic devices. Radio frequency (RF) current $I_{RF}$ is injected to
excite the magnetisation, and mixes with the oscillatory MR response to
generate a DC mixing voltage $V_{mix}$, which is recorded as the applied
magnetic field or the frequency of $I_{RF}$ is varied. Analysis of the
resonance field, or frequency, and linewidth allows the torques acting upon
the magnetization to be determined Skinner  _et al._ (2014); Tulapurkar  _et
al._ (2005); Kalarickal  _et al._ (2006); Sankey  _et al._ (2008); Kubota  _et
al._ (2008); Liu  _et al._ (2011); Brataas  _et al._ (2012); Dumas  _et al._
(2014). However, $V_{mix}$ may vanish for certain magnetic field
configurations, and represents a spatial average of magnetization dynamics
that may be highly inhomogeneous.

Recently, time resolved scanning Kerr microscopy (TRSKM) was used to image the
out of plane magnetization induced by $I_{DC}$ and $I_{RF}$ within injection-
locked nano-contact (NC) SHNO devicesSpicer  _et al._ (2018). Each NC-SHNO was
formed on a Py/Pt bilayer disk, with $I_{DC}$ concentrated within a small
central region. While a non-linear bullet mode was observed at the device
centre when $I_{DC}$ exceeded a threshold value, $I_{RF}$ was observed to
stimulate a delocalised and spatially inhomogeneous response. Here TRSKM is
used to determine the spatial variation of the torques generated by the
$I_{RF}$ injected into a NC-SHNO. The spatial variation of both the STT and
Oersted torques is found to diverge strongly from that expected for the DC
current distribution, suggesting that the reactance of the device geometry
strongly modifies the RF current distribution.

Figure 1: (a) SEM image of a NC-SHNO showing electrode separation $d$,
directions of the DC charge current $I_{DC}$, average polarization
$\hat{\sigma}$ of the spin current injected into the Py layer, and field H
orientated at angle $\theta_{H}$ relative to the $x$ axis. COMSOL simulations
for $I_{DC}$ = 4 mA and $d$= 200 nm are shown for (b) current density J within
the Pt layer, (c) in-plane Oersted field within the Py layer that has
magnitude $\sqrt{h_{x}^{2}+h_{z}^{2}}$, and (d), out of plane Oersted field
$h_{y}$ within the Py layer. The arrows and grayscale indicate the direction
and normalized magnitude respectively.

NC-SHNO devices, shown in Figure 1, were fabricated on sapphire substrates by
sputtering and electron-beam lithography Dürrenfeld  _et al._ (2015).
Triangular needle-shaped Au(150 nm) NCs with tip separation of $d$ = 140 - 240
nm were defined on a 4 $\mu$m diameter Py(5 nm)/Pt(6 nm) bi-layer disk. The
$I_{DC}$ from the NCs is concentrated within a small region of the Pt layer
between the tips, and generates a spin current that flows into the Py layer
beneath. The injected spin polarization lies parallel to the +ve $x$ direction
along a horizontal line through the middle of the disk, Emori  _et al._ (2013)
and exerts a STT on the Py magnetization. The charge current also generates an
Oersted field with in and out of plane components. The distributions of
$I_{DC}$ and the Oersted field plotted in Figure 1 were calculated using
COMSOL COM . The STT amplitude has similar spatial distribution to the charge
current within the Pt, while the Oersted field depends upon the current
distribution in both the Pt and Au layers.

Conventional STT-FMR measurements were made by applying audio frequency
modulation to the $I_{RF}$ injected through the capacitative arm of a bias-
tee, while $V_{mix}$ was measured through the inductive arm using a lock-in
amplifier. The out-of-plane component of the dynamic magnetization was
detected directly by means of TRSKM that has been described elsewhere Keatley
_et al._ (2017).

Both the dynamic magnetization and $V_{mix}$ can be calculated for a
macrospin. The equation of motion for the magnetization driven by an external
field and STT is

$\frac{d\hat{\textbf{m}}}{dt}=-|\gamma|(\hat{\textbf{m}}\times\textbf{H}_{eff})+\alpha\hat{\textbf{m}}\times\frac{d\hat{\textbf{m}}}{dt}\\\
-|\gamma|A\hat{\textbf{m}}\times(\hat{\textbf{m}}\times\hat{\mathbf{\sigma}})+|\gamma|B(\hat{\textbf{m}}\times\hat{\mathbf{\sigma}}),$
(1)

where m̂ is the normalized magnetization vector, $\gamma$ is the gyromagnetic
ratio, $\alpha$ is the Gilbert damping constant, $\hat{\mathbf{\sigma}}$ is
the injected spin polarisation, and $A$ and $B$ are the amplitudes of the “in-
plane” or “anti-damping”, and “out-of-plane” or “field-like” STT respectively.
A large in-plane torque is expected due to the SHE. However, since the Py is
relatively thick and the current is shunted through the Pt, negligible torque
is expected due to the Rashba effect. $\textbf{H}_{eff}$ is the total
effective field acting upon the magnetization, which may be written as
$\textbf{H}_{ext}+H_{d}(\textbf{\^{y}}\cdot\textbf{\^{m}})\textbf{\^{y}}+\textbf{h}$,
where $\textbf{H}_{ext}$ is the static applied field, $H_{d}$ is the out of
plane demagnetizing field, and h is the local Oersted field generated by the
RF current. Other anisotropy fields are expected to be small and have been
neglected.

Equation (1) can be linearized for small amplitude precession, with the out of
plane magnetization written as $m_{y}(\varphi)=Re(ae^{i\varphi})$ where
$\varphi$ is the phase of the RF current and $a$ is the complex amplitude.
$V_{mix}$ and the real and imaginary parts of $a$ have the forms

$V_{mix}=I_{RF}\\\ \Delta Rsin\theta_{H}cos\theta_{H}\\\
\frac{|\gamma^{\prime}|^{2}H_{\perp}(f_{0}^{2}-f^{2})(H_{ext}+H_{d})+|\gamma^{\prime}|f^{2}\Delta(\alpha
H_{\perp}+H_{||})}{(f_{0}^{2}-f^{2})^{2}+f^{2}\Delta^{2}},$ (2)

$Re(a)=\frac{-|\gamma^{\prime}|^{2}H_{ext}H_{||}(f_{0}^{2}-f^{2})+f^{2}\Delta|\gamma^{\prime}|(H_{\perp}-\alpha
H_{||})}{(f_{0}^{2}-f^{2})^{2}+f^{2}\Delta^{2}},$ (3)

$Im(a)=\frac{f|\gamma^{\prime}|(H_{\perp}-\alpha
H_{||})(f_{0}^{2}-f^{2})+f\Delta|\gamma^{\prime}|^{2}H_{ext}H_{||}}{(f_{0}^{2}-f^{2})^{2}+f^{2}\Delta^{2}},$
(4)

where

$H_{\perp}=\left(sin\theta_{H}h_{x}+Bsin\theta_{H}\right),H_{||}=\left(\frac{sin\theta_{H}}{|sin\theta_{H}|}h_{y}-Asin\theta_{H}\right),$
(5)

$\gamma^{\prime}=\gamma/2\pi$, $f$ and $I_{RF}$ are the frequency and
amplitude of the RF current, $\theta_{H}$ is the angle between H and
$\hat{\sigma}$, $H_{\perp}$ and $H_{||}$ represent effective fields, where the
subscripts indicate the direction in which the associated torque acts, and $A$
and $B$ are defined in equation (1). Finally, the linewidth
$\Delta=|\gamma^{\prime}|\alpha(2H_{ext}+H_{d})$, and the FMR frequency
$f_{0}=|\gamma^{\prime}|\sqrt{H_{ext}[H_{ext}+H_{d}]}$. $\Delta R$ = 0.03
$\Omega$ Dürrenfeld  is the change in electrical resistance when the
magnetisation is rotated from orthogonal to parallel to the current.

The above expressions yield $m_{y}$ at different positions within the NC-SHNO
if the observed dynamical magnetization is a response to local torques. This
is a reasonable assumption when spin waves excited due to spatially varying
STT and Oersted torques are similar in frequency. Dispersion due to dipolar
interactions decreases with film thickness. For the 5 nm Py film, the
frequency splitting, of the uniform mode and a spin wave with wavelength equal
to the diameter of the disk, is no more than $20\%$ and lies within the
measured linewidth.

The stroboscopic nature of TRSKM requires that measurements are made at an RF
frequency that is a multiple of the laser repetition rate as $H_{ext}$ and
hence $f_{0}$ are varied. From equations (3) and (4), $Re(a)$ and $Im(a)$ are
seen to contain a minimum in the denominator at the resonance field, and terms
in the numerator that are either slowly varying or antisymmetric (due to the
factor $f_{0}^{2}-f^{2}$) about the resonance field. Hence both expressions
consist of parts that are symmetric and antisymmetric about the resonance
field. The microwave phase $\varphi$ may be chosen in the experiment, so that
$m_{y}$ is a weighted sum of $Re(a)$ and $Im(a)$, and hence a sum of symmetric
and antisymmetric terms. TRSKM measures the polar Kerr rotation
$QM\hat{m}_{y}$ where the constant $Q$ is of order 0.1 mdeg cm3 emu-1. If the
value of $Q$ is known then, by recording the dependence of $m_{y}$ upon
$H_{ext}$ for a number of values of $\varphi$, and fitting $m_{y}$ to
equations (3), (4) and (5), the values $A$ and $B$ can be determined at each
position within the sample.

Conventional STT-FMR was first performed to obtain $V_{mix}$, as shown in
Figure 2(a). The optical probe was then positioned between the NC tips and the
polar Kerr signal recorded at three values of RF phase as a function of field,
as shown in Figure 2(b). Both optical and electrical resonance curves are a
superposition of components that are either symmetric or antisymmetric about
the resonance field. The $V_{mix}$ data, which does not depend upon $\varphi$,
was fitted to equations (2) and (5), yielding values of $H_{\perp}=-6.1\pm
0.8$ Oe and $H_{||}=29.7\pm 3$ Oe, while values of $|\gamma^{\prime}|=2.94$
MHz/Oe, $\alpha=0.04$ and $H_{D}=8000$ Oe were found to best describe both the
$V_{mix}$ and optical data within the present study Sup . Measurements made at
different $I_{RF}$ values confirmed that the dependence of the extracted
effective fields upon $I_{RF}$ was roughly linear for the range of values
studied here Sup . The large value of $\alpha$ has been attributed to spin
pumping effects Liu  _et al._ (2013). Since the average value of $h_{y}$ is
small due to the NC symmetry, the large value of $H_{||}$ results from the
anti-damping torque. From equation (5), if $\theta_{H}=150^{\circ}$, $h_{x}$
has value of $\sim$12.1 Oe, similar to the 10.2 Oe calculated by COMSOL at the
centre of the disk for a DC current.

Due to the symmetry of the device, one may assume Sup  that the ratio
$H_{\perp}/H_{||}$ determined from optical measurements in Figure 2(b) should
be the same as that determined by fitting $V_{mix}$. Fixing this ratio and
fitting the optical resonance curves yields values for $\varphi$, that have
been used to label each curve. The three phase values were found to be offset
set by the same amount from the values set on the microwave synthesizer Sup ,
justifying the assumed value of $H_{\perp}/H_{||}$. The fitting also yields
$Q$, but this is less reliable because the areas sampled by the electrical and
optical measurements are different, as will be discussed further.

Figure 2: (a) Dependence of mixing voltage $V_{mix}$ upon field $H_{ext}$
applied at $\theta_{H}=150^{\circ}$. The continuous curve is fit to equation
2. (b) Polar Kerr rotation curves recorded with the optical spot between the
tips of the NC for three values of the phase $\varphi$. (c) Polar Kerr (closed
circles) and $V_{mix}$ (open squares) resonance curves recorded with the
optical probe $1\mu$m to the right of centre, for different values of
$\theta_{H}$. (d) Polar Kerr rotation and intensity images for
$\theta_{H}=90^{\circ}$ and $H_{ext}=75$Oe for $\varphi$ values in the range 0
to $180^{\circ}$. $I_{RF}$ had frequency of 2 GHz and amplitude of 2.8, 1.3,
3.2 and 4.0 mA in (a), (b), (c) and (d) respectively. The NC separation $d=$
240 nm in (a) and (b), and 140 nm in (c) and (d)

The dependence of optical and electrical signal strength upon $\theta_{H}$ is
shown in Figure 2(c). Maximum optical signal amplitude is observed for
$\theta_{H}=90^{\circ}$, due to the sin$\theta_{H}$ factor in equation (5). In
contrast $V_{mix}$ vanishes for $\theta_{H}=90^{\circ}$ and $180^{\circ}$, and
so is insensitive to the dynamics when $\theta_{H}=90^{\circ}$. Finally, polar
Kerr images are plotted in Figure 2(d) for different values of $\varphi$. Due
to the symmetry of the current distribution about a vertical line though the
centre of the device, $A$, $h_{x}$ and hence $H_{\perp}$ are expected to be
symmetric about this centre line. On the other hand $h_{y}$ is antisymmetric
so that $H_{||}$ has mixed symmetry. If terms in $\alpha$ are neglected in
equations (3) and (4), then at resonance, when $f=f_{0}$, $Re(a)$ is symmetric
about the centre line, while $Im(a)$ has mixed symmetry. Therefore, the most
symmetric image should occur for $\varphi=0^{\circ}$. The most striking
feature of the images is the minimum at the NC tips, which is unexpected from
the calculation of $I_{DC}$ in Figure 1.

Figure 3: Optically-detected resonance curves acquired at points 1 $\mu$m to
the left and right of the centre of the device, for selected values of RF
phase, with $\theta_{H}$ = 90∘, $f$ = 2 GHz, $I_{RF}$ = 1.3 mA, and $d$ = 240
nm. $\varphi_{RF}$ is the phase value set at the microwave synthesizer, while
$\varphi$ is the phase of $I_{RF}$ at the device. The solid symbols represent
the measured Kerr rotation values, while the continuous curves are fits that
are described within the main text.

| Mean $H_{\perp}$ (Oe) | Mean $H_{||}$ (Oe)
---|---|---
Left | 0.8935 $\pm$ 0.0997 | -5.3400 $\pm$ 0.8643
Right | 0.9141 $\pm$ 0.4807 | -13.4049 $\pm$ 1.7338

Table 1: Mean and standard deviation of the effective field parameters
extracted from the fits shown in Figure 3

.

Optically-detected resonance curves were acquired with the probe spot placed 1
$\mu$m to either side of the centre of the device to demonstrate the spatial
and phase sensitivity of the measurement technique. Figure 3 shows the
resonance curves with fits that assume, as before, that the phase of $I_{RF}$
at the device $\varphi$ is offset by 50∘ from the phase set on the microwave
synthesizer $\varphi_{RF}$. The different fitted curves assume the same set of
experimental and material parameter values apart from $H_{\perp}$ and
$H_{||}$, which were allowed to vary for each curve. The fits generally
describe the data very well, with a few exceptions where either phase wander
of the microwave synthesizer or drift in the position of the focused laser
spot may have led to distortion of the experimental data. The mean and
standard deviation of the extracted effective field values are plotted in
table 1. The extracted values for $H_{\perp}$ are seen to be reasonably
consistent for all values of the phase and on both sides of the disk,
justifying the method used to determine the phase offset. In contrast $H_{||}$
is significantly different between the two sides of the disk, as will be
discussed further below.

To further explore the spatial symmetry of the magnetic response, and hence
the underlying torques, the field dependence of the polar Kerr rotation was
measured at different points on a horizontal line through the middle of the
disk, for values of $\varphi=40^{\circ}$, 85∘ and 130∘. The extracted values
of $H_{\perp}$ and $H_{||}$ are plotted in Figure 4(a). The field values
obtained for the three $\varphi$ values are in good agreement confirming that
the absolute phase has been determined correctly.

Figure 4: (a) Spatial variation of $H_{\perp}$ and $H_{||}$ along the
horizontal line passing through the middle of the disk. Values obtained from
electrical STT-FMR are shown by the dashed line with the error indicated by
the gray shading. The injected $I_{RF}$ had frequency $f$ = 2 GHz and
amplitude of 1.3 mA, while the static field was applied at $\theta_{H}$ = 90∘.
(b) $H_{\perp}$ and the antisymmetric/symmetric parts of $H_{||}$ are plotted
together with the current and Oersted field from Figure 1. (c) Polar Kerr
image acquired for $\varphi=0^{\circ}$, $I_{RF}$ = 4 mA and $H_{ext}$ = 75 Oe
is presented next to the calculated distribution of $h_{x}$. Symmetric and
antisymmetric parts of the polar Kerr data acquired at $\varphi=90^{\circ}$,
are presented next to the vertical component of the current density $J_{z}$
and the out of plane Oersted field $h_{y}$. The calculated images were
convolved with a Gaussian profile of 870 nm half maximum diameter. The NC
separation $d$ was 240 nm in (a) and (b), and 140 nm in (c).

Since negligible out of plane STT is expected, $H_{\perp}$ should be
proportional to $h_{x}$ and spatially symmetric. However, $H_{||}$ should
contain both symmetric and antisymmetric components, (denoted as
$Sym(|H_{||}|)$ and $Asym(H_{||})$) due to the in plane STT and $h_{y}$
respectively. These components can be separated by calculating the mirror
image (reflection about $x=0$) of the $H_{||}$ data, calculating the sum and
difference of the original data with its mirror image, and then dividing both
by a factor of two. Both components are plotted in Figure 4(b), together with
convolutions of the Oersted field and current distribution of Figure 1 with a
Gaussian function of 870 nm half maximum width (denoted Proc($h_{x,y}$) and
Proc($J$)Sup ), to account for the size of the optical spot. In extracting
field quantities from the data, a value of $Q$ = 0.3 mdeg cm3 emu-1 was
assumed so that the Oersted fields towards the edge of the disk were in
agreement with those from Figure 1. However, the experimental $H_{\perp}$ and
$Sym(|H_{||}|)$ curves exhibit a minimum at $x=0$ $\mu$m, in disagreement with
the calculated curves.

As explained above, for $\varphi=0^{\circ}$ and $90^{\circ}$, time resolved
images acquired at resonance reveal the spatial variation of $H_{\perp}$ and
$H_{||}$ respectively. A similar procedure to that applied to the line scan in
Figure 4(b) was used to extract the symmetric and antisymmetric parts of the
image acquired at $\varphi=90^{\circ}$. The resulting images of $H_{\perp}$,
$Sym(|H_{||}|)$ and $Asym(H_{||})$ are plotted next to calculated images of
$h_{x}$, $J$, and $h_{y}$ in Figure 4(c). Each calculated distribution has
been convolved with a 2D Gaussian function of 870 nm half maximum diameter.
The form of the $Asym(H_{||})$ and Proc($h_{y}$) images are in reasonable
agreement. However the Proc($h_{x}$) and Proc($J$) images possess a maximum at
the centre of the disk, whereas the $H_{\perp}$ and $Sym(|H_{||}|)$ images
exhibit a minimum. The convolution with the spot profile accounts for the NCs
partly obscuring the underlying Py/Pt bilayer, so that the minimum corresponds
to a reduction of the in-plane STT and the torque due to the in-plane Oersted
field. The observed minimum initially seems at odds with observations of a
self-localised bullet mode at the centre of this Spicer  _et al._ (2018);
Dürrenfeld  and similar devices Demidov  _et al._ (2012), but may be explained
if the spatial distribution of $I_{RF}$ is different to that of $I_{DC}$ used
to excite the bullet mode.

Figure 5: (a), (b) Scanning electron microscope images of NC-SHNO devices. In
(a) wire bonds have been made to the ground-signal-ground pads of a coplanar
waveguide structure. The signal line tapers to where the NC-SHNO device is
located. The green dashed box indicates the region that is shown at greater
magnification in (b) where a device is located between the end of the tapered
signal line and the point at which the two ground planes meet. (c) Simulated
distribution of $I_{RF}$ within a coplanar waveguide (CPW) structure with
similar lateral dimensions to that of the NC-SHNO device structure at
frequencies of 100 kHz and 6 GHz. The CPW is 1 um thick, has a gold-like
conductivity of 45.6 $\times$106 S/m, and is surrounded by a large air region
with volume of 450 $\times$ 450 $\times$ 400 $\mu$m3. It is excited by a
coaxial lumped port in a ground-signal-ground configuration producing an input
current of 16 mA. A narrow constriction of 4 $\mu$m width is located at the
end of the tapered signal line. The plotted quantity is the magnitude of the
current density in the plane passing through the middle of the gold layer. For
clarity the range of the colour scale has been adjusted in the main images,
with maximum values indicated above the relevant legends. The colour scales of
the 100 kHz plots have been made similar to those of the 6 GHz plots to aid
comparison.

The NC-SHNO devices were located in a constriction of the signal line of a
shorted coplanar waveguide (CPW) as shown in Figure 5 (a) and (b). It is well
established that the distributed reactance of the CPW gives rise to spreading
of current from the centre of the signal line and hence to current crowding at
its edges Chia-Nan Chang  _et al._ (1990); Carlsson and Gevorgian (1997,
1999). The distribution of $I_{RF}$ within the NC-SHNO results from
competition between the confining effect of the NCs and current spreading
within the larger scale device structure that has the form of a short CPW.
Modelling the spatial distribution of $I_{RF}$ within a NC-SHNO using the
COMSOL package is very difficult due to the need to mesh a structure with
feature sizes that vary from the order of 1 nm to 100 $\mu$m. Instead a
simulation was performed for a structure that had lateral dimenions similar to
that of the NC-SHNO device structure, but which was formed from a layer of
gold with uniform thickness of 1 $\mu$m, as shown in Figure 5 (c). The
circular mesa and NCs were replaced by a constriction of the signal line of 4
$\mu$m width. Simulations for frequencies of 100 kHz and 6 GHz are shown.
While the current is uniformly distributed across the width of the
constriction at 100 kHz, spreading towards the edge of the signal line is
observed at 6 GHz, establishing the feasibility of the current spreading
mechanism within the NC-SHNO device. The reduction of the torque due to a
reduction of $I_{RF}$ at the centre may explain why the frequency range for
injection-locking of auto-oscillations observed in NC-SHNOs is reduced
relative to that for nano-constriction SHNOs. Demidov  _et al._ (2014)

In summary, it has been shown that TRSKM can be used to probe the local FMR
driven by a combination of STT and Oersted field torques, and comparison has
been made with a simple theory. By directly probing the local magnetization,
this technique can be applied to magnetic materials or experimental
configurations that exhibit weak MR response. Furthermore the phase and
spatial symmetry of the different torques allows them to be separated and
mapped. It appears that the spatial distributions of $I_{DC}$ and $I_{RF}$ are
significantly different, perhaps due to the reactance of the device that leads
to spreading of $I_{RF}$. The torques due to the RF current exhibit a minimum
at the centre of the device, which suggests that the RF current may act to
destabilise the position of the self-localised bullet mode, and may explain
why the NC-SHNO exhibits a reduced locking range relative to other STOs. The
present study emphasises the need to characterise both the DC and RF current
distributions and the torques that they generate.

###### Acknowledgements.

We acknowledge financial support from the Engineering and Physical Sciences
Research Council (EPSRC) of the United Kingdom, via the EPSRC Centre for
Doctoral Training in Metamaterials (Grant No. EP/L015331/1), and grants
EP/I038470/1 and EPSRC EP/P008550/1. All data created during this research are
openly available from the University of Exeter’s institutional repository at
https://doi.org/10.24378/exe.1003.

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# Diagnosis of Fast Electron Transport by Coherent Transition Radiation

Yangchun Liu1, Xiaochuan Ning1, Dong Wu2,∗, Tianyi Liang1, Peng Liu1, Shujun
Liu1, Xu Liu2, Zhengmao Sheng1,∗, Wei Hong3, Yuqiu Gu3, Xiantu He1 1Institute
for Fusion Theory and Simulation, School of Physics, Zhejiang University,
Hangzhou, 310058, China  2Key Laboratory for Laser Plasmas and School of
Physics and Astronomy, and Collaborative Innovation Center of IFSA (CICIFSA),
Shanghai Jiao Tong University, Shanghai, 200240, China  3Science and
Technology on Plasma Physics Laboratory, Research Center of Laser Fusion,
China Academy of Engineering Physics, Mianyang, 621900, China
<EMAIL_ADDRESS><EMAIL_ADDRESS>,

###### Abstract

Transport of fast electron in overdense plasmas is of key importance in high
energy density physics. However, it is challenging to diagnose the fast
electron transport in experiments. In this article, we study coherent
transition radiation (CTR) generated by fast electrons on the back surface of
the target by using 2D and 3D first-principle particle-in-cell (PIC)
simulations. In our simulations, aluminium target of $2.7$ g/cc is simulated
in two different situations by using a newly developed high order implicit PIC
code. Comparing realistic simulations containing collision and ionization
effects, artificial simulations without taking collision and ionization
effects into account significantly underestimate the energy loss of electron
beam when transporting in the target, which fail to describe the complete
characteristics of CTR produced by electron beam on the back surface of the
target. Realistic simulations indicate the diameter of CTR increases when the
thickness of the target is increased. This is attributed to synergetic energy
losses of high flux fast electrons due to Ohm heatings and colliding drags,
which appear quite significant even when the thickness of the solid target
only differs by micrometers. Especially, when the diagnosing position is
fixed, we find that the intensity distribution of the CTR is also a function
of time, with the diameter increased with time. As the diameter of CTR is
related to the speed of electrons passing through the back surface of the
target, our finding may be used as a new tool to diagnose the electron energy
spectra near the surface of solid density plasmas.

††: New J. Phys.

  *

Keywords: coherent transition radiation, fast electron transport, laser solid
interaction, particle-in-cell simulation, terahertz radiation

##  1 Introduction

With the invention of chirped pulse amplification technology[1], the laser
intensity increased by six orders of magnitude, and the research work in the
field of high-energy density physics has been effectively promoted. When the
laser intensity is higher than $10^{13}\ \mathrm{W/cm^{2}}$, the laser will
ionize the target material rapidly by multiphoton ionization, tunneling
ionization and so on, forming plasma composed of electrons and ions. When the
laser intensity reaches $10^{18}\ \mathrm{W/cm^{2}}$, it will cause
relativistic oscillation of electrons in the laser field and heat the
temperature of electrons to more than $1\ \mathrm{MeV}$. Such laser can also
be used to accelerate high-energy charged particles, produce ultra-short and
ultra-strong gamma rays and even positive and negative electron pairs. One of
the key subjects in high energy density physics is the transport of fast
electrons. It has many potential applications in various fields, such as laser
accelerators[2, 3], fast ignition[4, 5], positron–electron plasmas [6, 7] and
terahertz source[8, 9]. In experiments, $K\alpha$, extreme ultraviolet (XUV)
emission[10, 11], shadowgraphy [12, 13] and optical emission [14, 15] are
usually used to study fast electron transport. In particular, coherent
transition radiation (CTR) is becoming an effective tool to diagnose the fast
electron population, especially for those electrons micro-bunched in time[15,
16, 17, 18, 19, 14, 20, 21, 22, 23].

Transition radiation is the radiation emitted by charged particles passing
through the interface between two media with different dielectric constants.
The electron beam is periodically produced by different laser heating
mechanisms such as vacuum heating, resonance absorption and
$\textbf{J}\times\textbf{B}$ heating. Therefore, the transition radiation
generated by such electron beam through the interface is easily coherent. It
makes the CTR spectra peak at $n\omega_{0}$[24], where $n$ is a positive
integer and $\omega_{0}$ is laser frequency. We can infer the main laser
heating mechanism of electrons by diagnosing CTR. In addition, CTR is also a
potential scheme for intense terahertz radiation sources[8].

In this article, we study the CTR generated by fast electrons on the back
surface of the target with 2D and 3D particle-in-cell (PIC) simulations.
Realistic simulations containing collision and ionization effects indicate the
diameter of CTR increases when the thickness of the target is increased. This
is attributed to synergetic energy losses of high flux fast electrons due to
Ohm heatings and colliding drags [25, 26] which appear quite significant even
when the thickness of the solid target only differs by micrometers.
Especially, when the diagnosing position is fixed, we find that the intensity
distribution of the CTR is also a function of time, with the diameter
increased with time. As the diameter of CTR is related to the speed of
electrons passing through the back surface of the target, our finding may be
used as a new tool to diagnose the electron energy spectra near the surface of
solid density plasmas. Meanwhile, in order to understand the role played by
collision and ionization physics, artificial simulations without taking into
collision and ionization effects are also performed. When comparing with
realistic simulations, artificial simulations significantly underestimate the
energy loss of electron beam when transporting in the target, which unable to
describe the complete characteristics of CTR produced by electron beam on the
back surface of the target.

##  2 Theoretical Model

Figure 1: (a) Schematic diagram of PIC simulation. (b), (c) is the
polarization characteristics of CTR for different electron beams vertically
and obliquely passing through the interfaces, respectively. Figure 2:
Normalized intensity distribution of CTR when an electron beam vertically
passes through the interface. Different colors correspond to different
electron energies, $\beta=v/c$.

Here, we need to emphasize several properties of CTR. The first characteristic
of CTR is polarization. The polarization characteristic of CTR is called
radial polarization in optics[27]. The polarization component of a certain CTR
ray lies in the plane formed by the ray and the main direction. As shown in
figure 1(a), the polarization direction of ray $OA$ is in plane
$OO^{\prime}A$, where $OO^{\prime}$ is the main direction of radiation. That
is to say, for the electron beam vertically passing through the interface, the
projection of radiation electric field on $XOY$ plane is along the radial
direction, as shown in figure 1(b). For the electron beam obliquely passing
through the interface, CTR is not completely radially polarized, as shown in
figure 1(c). This electric field not only has components parallel to the
radial direction, but also has components perpendicular to the radial
direction, which was observed by Happek et al.[28], the radiation angle
distribution is similar, but the intensity is not completely symmetrical[29].

The second is characteristic of the angular distribution. Both CTR and
incoherent transition radiation (ITR) follow the same characteristic of
angular distributions, which satisfy the formula described in [24],

$\displaystyle\frac{\mathrm{d}\varepsilon_{\mathrm{single}}}{\mathrm{d}\omega\mathrm{d}\Omega}=\frac{e^{2}}{\pi^{2}c}\frac{\beta^{2}\cos\Theta\left[\sin\theta-\beta\sin\Theta\cos\left(\phi-\Phi\right)\right]^{2}}{\left[\left(1-\beta\sin\theta\sin\Theta\cos\left(\phi-\Phi\right)\right)^{2}-\beta^{2}\cos^{2}\theta\cos^{2}\Theta\right]^{2}},$
(1)

where $(\theta,\phi)$ and $(\Theta,\Phi)$ represent the angle between Z and X
axes and the direction for the radiation and the electron beam, respectively.
As shown in figure 2, when an electron beam vertically passes through the
interface, the angular distribution of the transition radiation presents a
conical distribution. The intensity of transition radiation in the direction
of electron beam exit is zero, and there is a peak at a certain angle
deviating from the direction of electron beam exit. This angle satisfies
$\theta_{\mathrm{max}}=\mathrm{arcsin}(\pm{(\left(\beta-1/\beta\right)^{2}+\left(1-\beta^{2}\right))^{1/2}})$
and decreases with the increase of the electron beam energy[24]. That is to
say, the faster the electron velocity passing through the rear surface of the
target, the smaller the cone angle. In the relativistic limit, this angle is
approximately equal to $1/\gamma$.

The third characteristic of CTR is reflected on the spectrum. Because of
periodic laser heating mechanisms, transition radiation is always coherent.
Resonance absorption and vacuum heating can heat and produce a electron bunch
once in a cycle. This makes the transition radiation coherently superimposed
at the integer multiple of the laser frequency. $\textbf{J}\times\textbf{B}$
heating can heat electrons twice in one laser cycle. This makes the transition
radiation coherently superimposed at even times of laser frequency. Unde the
combined action of various heat mechanisms, the spectrum of transition
radiation will peak at $n\omega_{0}$, and the intensity will decrease
exponentially with the increase of $n$[24]. Where $n$ is a positive integer
and $\omega_{0}$ is laser frequency. Besides, the transition radiation in the
low frequency range is always coherent. This can be seen from a simple model
in [24],

$\frac{\mathrm{d}^{2}\varepsilon_{\mathrm{CTR}}}{\mathrm{d}\omega\mathrm{d}\Omega}=\left(N-1\right)\left|\tilde{n}\left(\omega,\bm{q}\right)\right|^{2}\frac{\mathrm{d}^{2}\varepsilon_{\mathrm{ITR}}}{\mathrm{d}\omega\mathrm{d}\Omega},$
(2)

with
$\tilde{n}\left(\omega,\bm{q}\right)=\tilde{n}_{\ell}\left(\omega\right)\tilde{n}_{\bot}\left(q\right)$,

$\tilde{n}_{\bot}\left(q\right)=\exp\left(-q^{2}a^{2}/2\right)=\exp\left(-\frac{\omega^{2}a^{2}}{2c^{2}}\sin^{2}\theta\right)$

and

$\displaystyle\tilde{n}_{\ell}\left(\omega\right)=$
$\displaystyle\frac{1}{1+\Delta\exp\left(-\omega_{0}^{2}\tau_{0}^{2}/2\right)}\times$
$\displaystyle\left\\{\exp\left(-\frac{\omega^{2}\tau_{0}^{2}}{2}\right)\right.\left.+\frac{\Delta}{2}\exp\left(-\frac{\left(\omega-\omega_{0}\right)^{2}\tau_{0}^{2}}{2}\right)\right\\},$

where
$\mathrm{d}^{2}\varepsilon_{\mathrm{ITR}}/\mathrm{d}\omega\mathrm{d}\Omega$ is
the ITR unrelated to $\omega$, $N$ represents the number of electrons passing
through the interface, $\tilde{n}\left(\omega,\bm{q}\right)$ is Fourier
transform of electron distribution in which $\bm{q}$ is the tangential
radiation wave vector, $a$ is the beam radius, $\Delta$ is the microbunching
amplitude, $\tau_{0}$ is the duration of electron pulse and $\omega_{0}$ is
the microbunching frequency in the beam. According to the above formula,
transition radiation is always coherent for $\omega\leqslant\tau_{0}^{-1}$.
This makes CTR have high intensity in terahertz range[8]. Besides, in the
study of Liu et al. and Batani et al., it is found that the intensity
relationship between CTR and blackbody radiation is related to the thickness
of the target[30, 31]. In a thin target, the optical coherent transition
radiation has the same order of magnitude as the blackbody radiation, and may
even be smaller than the blackbody radiation, but the CTR in terahertz range
is stronger than that in blackbody. However, as the thickness of the target
increases, the intensity of blackbody radiation decreases faster, so the
optical coherent transition radiation dominates in thick targets. Because of
the special polarization characteristics and angular distribution
characteristics of CTR, the CTR for complete diagnosis should be in a ring
shape. As shown by Pakluea’ s plot of CTR intensity distribution according to
theoretical formula[29].

##  3 Simulation Result

###  3.1 Realistic simulations containing collision and ionization effects

Figure 3: Distribution of CTR (Upper right) and distribution of electron
density in different positions of target for a realistic 3D PIC simulation
containing collision and ionization effects (Bottom). Same colorbar is used
for electron density. Figure 4: 2D realistic PIC simulation results including
collision and ionization effects. (a) Spectrum of CTR for $2\ \mu\mathrm{m}$
target. (b) CTR intensity distribution, different colors represent targets
with different thicknesses. (c) Electrons phase space distribution with the
thickness of target of $2\ \mu\mathrm{m}$ at $20\ T_{0}$, where $z=1\
\mu\mathrm{m}$ is the front surface of the target, and $z=3\ \mu\mathrm{m}$ is
the back surface of the target. (d) Energy spectra of electrons. Almost no
electrons reach the rear surface of the target at $5\ T_{0}$. Figure 5:
Distribution of electric field in $1\ \mu\mathrm{m}$ behind target at (a) $10\
T_{0}$, (b) $11\ T_{0}$, (c) $12\ T_{0}$, (d) $13\ T_{0}$ for a 3D realistic
PIC simulation containing collision and ionization effects. Same colorbar is
used.

The simulations is carried out with the PIC code LAPINS developed by one of
the authors. LAPINS is an implicit PIC code [32] which includes collision[33],
ionization[34, 35], radiation[36], strong field QED [37] and nuclear
reaction[38]. Because of the amount of calculation, we decided to use a thin
target for simulation and diagnose the terahertz radiation, so as to obtain
the CTR distribution of 2D and 3D simulations. In our 2D simulations, the
target was modeled as a uniform aluminium (Al) slab with different
thicknesses. The density of the target is $2.7\ \mathrm{g/cc}$ which is the
density of solid aluminum. The initial temperature of the target is set to
room temperature. The laser wavelength is $1\ \mu\mathrm{m}$ and laser radius
is $3\ \mu\mathrm{m}$. The laser is vertically incident on the target and and
the laser is polarized along the y direction. The laser intensity is
$3.425\times 10^{19}\ \mathrm{W/cm^{2}}$ and the duration of the laser pulse
is $10\ T_{0}$, where $T_{0}=\lambda/c$ is the normalized time unit. The 2D
simulations were carried out in a Z-Y Cartesian geometry which is parallel to
the propagation direction of laser. The absorption field and particle boundary
conditions are set in the Y and Z directions. The laser parameters of our 3D
simulation are exactly the same as those of 2D simulations. We selected a
target with a thickness of $2\ \mu\mathrm{m}$ for 3D simulation. Especially,
in order to be close to real situations, collision effect and ionization
effect are both included in all of the simulations displayed in this
subsection. The Monte Carlo collision model in LAPINS includes binary
collisions between electron-electron, electron-ion, and ion-ion[33]. Besides,
the ionization model in LAPINS includes the collision ionization, electron-ion
recombination and ionization potential depression. This model can not only be
applied for plasmas near thermal equilibrium, but also can be used in
relaxation of ionization dynamics[34]. After considering the collision effect
and ionization effect, our PIC simulation is very close to the real situation.

Figure 3(bottom) shows the density distribution of electrons at different
positions in the 3D simulation, where $z=1.0\ \mu\mathrm{m}$ and $z=3.0\
\mu\mathrm{m}$ are the front surface and the back surface of the target,
respectively, and the same colorbar is used between the front surface and the
back surface of the target. We can see that except on the front surface of the
target, the electron distribution in other positions is not looped, and there
is no ring formation in the electron beam transport process. The reason for
the annular distribution of electron density on the front surface is that the
target is deformed by the incident laser. Figure 3(upper right) shows the
annular intensity distribution of CTR at the plane $0.5\ \mu\mathrm{m}$ behind
the target and perpendicular to the laser propagation direction in 3D
simulation. This is obtained by firstly taking Fourier transformation of the
combined electric field strength in X and Y directions at the plane $0.5\
\mu\mathrm{m}$ behind target and then picking up the low frequency limit. This
plot shows that the CTR is indeed annular.

The spectrum of CTR can refer to figure 4(a). It is the spectrum distribution
of CTR for 2D simulation at $2\ \mu\mathrm{m}$ behind target. We diagnose the
electric field in the Y direction along a straight line parallel to the Y axis
and do Fourier transform on it to get the spectrum. As we can see, there is a
peak at low frequency and the optical transition radiation is difficult to
distinguish. Therefore, its intensity distribution and spectrum
characteristics is consistent with the theory. Figure 4(b) shows the intensity
distribution of CTR for target with different thicknesses. As we can see, as
the thickness of the target increases, the intensity of CTR decreases, and the
opening angle of the cone becomes larger and larger. This result can be well
understood if the energy loss of fast electrons in the target is taken into
account, resulting in different speed of the electron beam passing through the
back surface of the target with different thickness. As our 2D simulation is
2D in spatial spaces however 3D in velocity spaces, and the entire collision
and ionization physics are included, therefore, this result can be well
extended to full 3D situations.

To explain more accurately that the radius of the ring increases with the
thickness of the target. In figure 4(c), we have drawn the phase space
distribution of electrons. We can see that the electrons in the target have a
deceleration process. Moreover, there will be a certain deceleration after
passing through the sheath field generated by the front surface. In order to
see the deceleration effect of the target on electrons more clearly, we show
the spectra of electrons in figure 4(d). The orange and purple lines represent
the kinetic energy distribution of electrons in $2\ \mu\mathrm{m}$ target and
$5\ \mu\mathrm{m}$ target at $5\ T_{0}$, respectively. At this time, almost no
electrons reach the rear surface of the target. We can see that the
distribution of electrons produced by targets with different thicknesses is
almost the same. The blue and red lines represent the kinetic energy
distribution of electrons passing through the back surfaces of $2\
\mu\mathrm{m}$ and $5\ \mu\mathrm{m}$ targets at $20\ T_{0}$, respectively.
From this, we can see that the electrons passing through the thick target are
obviously less than those passing through the thin target, because the thicker
the target, the greater the kinetic energy loss of electrons. This reason
makes the thicker the target, the larger the ring of CTR diagnosed. The
research of A. J. Kemp et al. and M. Sherlock et al. shows that the energy
loss of electron beam in the high density target may be due to resistive
heating, the collision effect, and the interaction between wave and plasma[25,
26].

In addition, because of the relatively low frequency of the CTR in our
simulation, we can directly diagnose the electric field to approximately
describe the behavior of the CTR. By diagnosing the distribution of electric
fields in the X and Y directions at different times, we found that the circle
formed by the electric field increased with time, as shown in figure 5. The
(a), (b), (c) and (d) in the figure 5 represents the total electric field
intensity distribution in the X and Y directions at $1\ \mu\mathrm{m}$ behind
the target at the $10\ T_{0}$, $11\ T_{0}$, $12\ T_{0}$ and $13\ T_{0}$ in 3D
PIC simulation, respectively. That is reasonable, because the faster electrons
pass through the back surface of the target preferentially, and the cone angle
of the radiation field formed by them is smaller, while the slower electrons
pass through the back surface of the target latterly, and the cone angle
formed by them is larger, therefore the ring of the radiation field becomes
larger and larger with time. If we can measure the change of the angle of the
CTR with high time resolution, we can infer the energy spectra of the electron
beam near the surface of the target.

###  3.2 Artificial simulations without taking collision and ionization
effects into account

Figure 6: 2D artificial PIC simulation results without taking collision and
ionization effects into account. (a) Spectrum of CTR for $2\ \mu\mathrm{m}$
target. (b) CTR intensity distribution, different colors represent targets
with different thicknesses. (c) Electrons phase space distribution with the
thickness of target of $2\ \mu\mathrm{m}$ at $20\ T_{0}$, where $z=1\
\mu\mathrm{m}$ is the front surface of the target, and $z=3\ \mu\mathrm{m}$ is
the back surface of the target. (d) Energy spectra of electrons. Figure 7:
Distribution of electric field in $1\ \mu\mathrm{m}$ behind target at (a) $13\
T_{0}$, (b) $14\ T_{0}$, (c) $15\ T_{0}$, (d) $16\ T_{0}$ for a 3D artificial
PIC simulation without taking collision and ionization into account. The area
surrounded by the red ring in the figure is the area with high strength. The
red dotted line is to indicate the range of the ring. Same colorbar is used.

To understand the role played by collision and ionization physics, artificial
simulations without these effects are performed, as with most of the previous
works. The ionization state of the target in artificial simulations is set to
trivalent.

Figure 6 show the 2D artificial PIC simulation results without taking
collision and ionization into account. Comparing figure 4(a) with figure 6(a),
we can see that the spectrum shapes of CTR are the same for realistic
simulations and artificial simulations. However, the intensity of the
artificial simulations are higher than realistic simulations. Meanwhile, in
the figure 6(b), the opening angles of the CTR remain almost unchanged in the
artificial simulations. This is because electrons transport in the target with
little energy loss when the collision effect is not taken into account. This
weakens the dependence of the energy loss of the electron beam on the
thickness of the target. It can also be drawn from figure 6(c) and (d).
Besides, the intensity of CTR decreases when the thickness of the target is
increased. This is because increase the thickness of the target makes the
electron transport distance longer, and the number of electrons passing
through the back surface of the target is smaller in the same amount of time.
In the figure 6(c), show the electrons phase space distribution for $2\ \mu m$
target at $20\ T_{0}$. We can see that the velocity of electrons is evenly
distributed along the radial direction in the artificial simulation, and there
are also some electrons with large velocity. This also shows that the electron
beam transport in the target with little energy loss. The electrons transport
distance required for thick target is longer. For thick targets, more
electrons are still inside the target, resulting in the CTR intensity of the
thick target in figure 6(b) lower. In the figure 7, we show the spectra of
electrons. The orange and purple lines represent the kinetic energy
distribution of electrons in $2\ \mu\mathrm{m}$ target and $5\ \mu\mathrm{m}$
target at $5\ T_{0}$, respectively. As we can see, for targets with different
thicknesses, the energy spectra of electrons are almost identical, which
reflects the electron beam transport in the target with little energy loss.
Therefore, the opening angles of the CTR in figure 6(b) remain unchanged.

The 3D artificial PIC simulation results without taking collision and
ionization into account are shown in figure 7. Figure 7 show the total
electric field intensity distribution in the X and Y directions at $1\
\mu\mathrm{m}$ behind the target at the $13\ T_{0}$, $14\ T_{0}$, $15\ T_{0}$
and $16\ T_{0}$ for a 3D artificial PIC simulation. The area surrounded by the
red ring in the figure is the area with high strength. The red dotted line is
to indicate the range of the ring. From the results, we can also conclude that
the spatial scale of CTR will increase with time. That’s because the faster
electrons pass through the back surface of the target preferentially. However,
due to the uniform distribution of electrons in the phase space, high-energy
electrons pass through the back surface of the target continuously. This makes
the CTR in small opening angle region with high intensity which has been
marked by a red ring in figure 7 always exist.

Comparing the realistic simulations results, we find that artificial
simulations can also reflect the relationship between the diameter of CTR and
the speed of electrons. However, ignoring the collision effect significantly
underestimate the energy loss of electron beam when transporting in the
target, which fail to describe the complete characteristics of CTR produced by
electron beam on the back surface of the target.

##  4 Discussion and conclusion

Most experiments failed to diagnose the annular CTR. However, in the
experiment of Pakluea[29], they use an electron beam with an energy of MeV to
irradiate the Al target, and diagnose the forward CTR. From their experimental
results, we can see that when the CCD is placed on the focal plane of the
focusing lens, circular CTR is diagnosed, while annular CTR is diagnosed at
the position deviating from the focal plane. We can easily understand this
result from the imaging theory of convex lens. When the image is real and the
CCD is placed at the image distance, the shape and size of CTR at the back
surface of the target can be obtained according to the imaging law. This
result confirms CTR can be measured experimentally. However in most
experiments, CCD is placed at the focus of convex lens[39, 31], circular CTR
is therefore diagnosed. Full PIC simulation of solid density plasma is
extremely difficult. This is because, in order to reduce numerical heating[40,
41, 32, 36, 42] and suppress numerical instability, the grid size for general
explicit PIC codes is of Debye length, which makes PIC simulation of sizeable
solid density plasma almost impossible. In LAPINS code, a fourth-order spatial
difference scheme is combined with an implicit scheme for temporal stepping in
solving electromagnetic fields. This new scheme [32] can completely remove
numerical self-heating and significantly reduce the simulation burden by using
coarse simulation grids when simulating solid-density plasmas. This code
enables us to calculate coupled atomic and plasma processes for intense
laser–solid interaction in a more realistic way than previous codes. Within
the simulations, the ionization charge state and conductivity (or resistivity)
of the target can self-consistently evolve in a precise manner according to
the local plasma and electromagnetic field conditions. Different types of
materials can now be modeled, with account taken of their intrinsic atomic
properties.

However one needs to note, although the simulation burden is significantly
reduced, our 3D PIC simulation still takes more than 170,000 cpu-hours. This
is an unprecedented 3D PIC simulation. Our work figures out the relationship
between CTR and electron transport properties in solid. From our simulation
results, we can see that the annular distribution characteristics of CTR are
closely related to the properties of electron beams. This result provides a
scheme for diagnosing electron transport properties by CTR. It can be used to
study the energy deposition process in inertial confinement fusion and the
improvement of ion beam quality in laser driven ion acceleration.

In conclusion, the characteristics of CTR caused by fast electrons generated
by laser solid target interaction are studied through PIC simulation in two
different situations. Comparing the realistic simulations, artificial
simulations significantly underestimate the energy loss of electron beam when
transporting in the target, which fail to describe the complete
characteristics of CTR produced by electron beam on the back surface of the
target. From the realistic simulation containing collision and ionization
effects, it is found that the diameter of CTR increases when the thickness of
the target is increased. This is due to the more energy loss of thick target
to electron beam. In addition, we find that the radius of the ring formed by
the CTR generated at fixed positions is increasing with time, which can used
to infer the energy spectra of fast electrons. Our finding may be used as a
new tool to diagnose the electron energy spectra near the surface of solid
density plasmas.

This work was supported by the National Natural Science Foundation of China
(Grant No. 12075204, No. 11875235, and No. 61627901), the Strategic Priority
Research Program of Chinese Academy of Sciences (Grant No. XDA250050500) and
Shanghai Municipal Science and Technology Key Project (No. 22JC1401500). Dong
Wu thanks the sponsorship from Yangyang Development Fund.

## reference

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