# Observation of Time-Reversal Invariant Helical Edge-Modes in Bilayer
Graphene/WSe2 Heterostructure

Priya Tiwari  Department of Physics, Indian Institute of Science, Bangalore
560012, India  Saurabh Kumar Srivastav  Department of Physics, Indian
Institute of Science, Bangalore 560012, India  Sujay Ray  Department of
Physics, Indian Institute of Science, Bangalore 560012, India  Tanmoy Das
Department of Physics, Indian Institute of Science, Bangalore 560012, India
Aveek Bid<EMAIL_ADDRESS>Department of Physics, Indian Institute of
Science, Bangalore 560012, India

###### Abstract

Topological insulators, along with Chern insulators and Quantum Hall insulator
phases, are considered as paradigms for symmetry protected topological phases
of matter. This article reports the experimental realization of the time-
reversal invariant helical edge-modes in bilayer graphene/monolayer WSe2-based
heterostructures – a phase generally considered as a precursor to the field of
generic topological insulators. Our observation of this elusive phase depended
crucially on our ability to create mesoscopic devices comprising both a moiré
superlattice potential and strong spin-orbit coupling; this resulted in
materials whose electronic band structure could be tuned from trivial to
topological by an external displacement field. We find that the topological
phase is characterized by a bulk bandgap and by helical edge-modes with
electrical conductance quantized exactly to $2e^{2}/h$ in zero external
magnetic field. We put the helical edge-modes on firm grounds through
supporting experiments, including the verification of predictions of the
Landauer-B$\mathrm{\ddot{u}}$ttiker model for quantum transport in multi-
terminal mesoscopic devices. Our non-local transport properties measurements
show that the helical edge-modes are dissipationless and equilibrate at the
contact probes. We achieved the tunability of the different topological phases
with electric and magnetic fields, which allowed us to achieve topological
phase transitions between trivial and multiple, distinct topological phases.
We also present results of a theoretical study of a realistic model which, in
addition to replicating our experimental results, explains the origin of the
topological insulating bulk and helical edge-modes. Our experimental and
theoretical results establish a viable route to realizing the time-reversal
invariant $\mathbb{Z}_{2}$ topological phase of matter.

###### keywords:

bilayer graphene, spin-orbit coupling, helical edge-state, topological
insulator, quantized conductance, quantum spin hall, $\mathbb{Z}_{2}$ phase.

##  1 Introduction

Topological phases of matter are characterized by specific global properties
that emerge from symmetry protected, local degrees of freedom 1, 2, 3;
prominent examples being Chern insulators and $\mathbb{Z}_{2}$ invariant
topological insulators. Chern insulators are phases with broken time-reversal
(TR) symmetry which host chiral edge-modes 4 – examples being the Quantum Hall
and the Anomalous quantum Hall phases. The TR invariant $\mathbb{Z}_{2}$
topological phase, on the other hand, has counter-propagating helical edge-
modes. There are two distinct proposals for the realization of a TR invariant
topological phase in two-spatial dimensions. The proposal by Kane and Mele 5,
6 predicted Quantum Spin Hall insulator (QSHI) phase in a carbon honeycomb
lattice with strong spin-orbit coupling (SOC). In this model, the separation
between the helical edge-modes arises from the SOC – the spin-$\uparrow$ and
spin-$\downarrow$ charge-carriers experience opposite forces due to SOC and
form edge-modes of opposite chirality. Alternately, Bernevig and Zhang
considered strained Zinc-Blende semiconductors where the U(1) symmetry is
broken either due to bulk asymmetry or Rashba spin-orbit coupling leading to
the QSHI phase 7, 8. Although the QSHI phase has experimentally been realized
in various systems, including HgTe/HgCdTe, InAs/GaSb 9, 10 quantum wells and
WTe2 11, its confirmation in carbon honeycomb lattice is lacking.

The principal impediments to the presence of QSHI phase in graphene are (i)
lack of a bandgap and, (ii) the small value of its intrinsic SOC ($\sim 2$
$\mathrm{\mu eV}$). In bilayer graphene (BLG), the presence of an external
electric field perpendicular to the plane lifts the $z\rightarrow-z$ inversion
symmetry and creates a bandgap at the $K$ and $K^{\prime}$ points. When such a
gapped BLG is stacked on a material with strong SOC (e.g., monolayer
transition metal dichalcogenides like WSe2), a strong, layer-selective SOC of
spin valley locking nature is inherited by the BLG from the WSe2. Calculations
have shown that this can, under appropriate conditions (which are discussed in
detail later in this article) place the BLG in a topologically non-trivial
phase 12, 13 which hosts metallic helical edge-modes and forms a viable
platform for the observation of QSHI 14, 15, 16, 17, 18.

This article reports the experimental observation of TR invariant helical
edge-modes in high-mobility heterostructures of BLG/monolayer WSe2. The
measured conductance in several multi-probe measurement configurations (both
local and non-local) was found to be precisely quantized over a range of
temperatures (20 mK - 10 K) (see Fig. S4); the quantized value in all cases
being equal to that expected for the helical edge structure of QSHI. The
presence of the helical edge-modes is placed on firm grounds through the
verification of predictions of the Landauer-B$\mathrm{\ddot{u}}$ttiker model
19 for quantum transport in multi-terminal devices with helical edge-modes and
through measurements of non-local transport properties. Through theoretical
studies of a realistic model of our system, we accurately replicate our
experimental results. From our coupled experimental and theoretical study, we
establish that the necessary conditions for the observation of a pair of
helical edge-modes in BLG are: (i) presence of a significant bulk bandgap,
(ii) presence of asymmetric Ising spin-orbit interaction acting predominantly
on one layer of the Bernal stacked BLG, and (iii) the presence of a moderately
strong Rashba spin-orbit interaction that gaps out all low-energy edge-modes
in the BLG except at the $K$ and the $K^{\prime}$ points.

##  2 Results and Discussion

Hall bar devices (B10S5 and B10S7) based on an atomically sharp interface
between a Bernal-stacked BLG and a monolayer of semiconducting crystalline
WSe2 were fabricated using standard dry transfer technique 20, 21. The
BLG/WSe2 stack was encapsulated between hexagonal boron nitride (hBN) crystals
of thicknesses $\sim$51 nm and $\sim$105 nm thicknesses at bottom and top of
the stack (Fig. 1(a)) which acted as gate dielectrics for the back- and top-
gates , respectively (for details of device fabrication see Supplementary
Information (SI)). Fig. 1(b) is the Raman spectra of the BLG/WSe2 stack. The
two Raman peaks centered around 1580 cm-1 (G peak) and 2800 cm-1 (2D peak)
belong to the graphene layer. The spectral decomposition of the 2D peak (see
Fig. S1(C)), as well as the intensity ratio of G and 2D peaks, establish the
graphene to be a bilayer 22. The presence of a peak near 250 cm-1 (and the
absence of one at around 308 cm-1)23 indicate that the WSe2 flake is a
monolayer. Room temperature photoluminescence spectra of the WSe2 flake (Fig.
1(c)) has a peak at $\sim$ 1.65 eV, confirming that it is a monolayer 24. All
the data presented here are from device B10S5 – the data from device B10S7 is
presented in supplementary material (see Fig. S5; SI6).

The dual-gated architecture of the device allows independent tuning of both
the charge-carrier number density, $n$, and the displacement field
perpendicular to the device, $D$, using the relation
$n=((C_{tg}V_{tg}+C_{bg}V_{bg})/e)-n_{o}$ and
$D=((C_{tg}V_{tg}-C_{bg}V_{bg})/2\epsilon_{o})-D_{o}$, where, $n_{o}$ is the
residual charge-density due to doping and $D_{o}$ the net internal
displacement field. The values of $n_{o}$ and $D_{o}$ were extracted from the
position of the Dirac point in the $V_{tg}-V_{bg}$ plane. A plot of the four-
probe longitudinal resistance as functions of the top-gate voltage, $V_{tg}$
at the back-gate voltage, $V_{bg}$ equal to $-12$ V is shown in Fig. 1(d).
Arrows show the primary Dirac point (PDP) and the two satellite peaks, called
clone Dirac points (CDP). These two satellite peaks are the result of band
structure reconstruction of BLG due to the moir$\mathrm{\acute{e}}$
superlattice potential caused by a near-perfect alignment of the top hBN layer
with the BLG 25, 26, 27. From the positions of the CDP, the angle between top
hBN and the BLG was estimated to be $\sim 0.9^{\circ}$ (see SI). In Fig. 1(e)
is plotted a contour map of the 2-probe device resistance versus $n$ and $D$.
The asymmetric feature seen near the PDP (outlined by dashed lines) is a
consequence of the band-splitting in the BLG by the induced SOC 28. Depending
on the direction of $D$, this splitting occurs either in the conduction band
(for positive values of $D$) or in the valence band (for negative values of
$D$). Note that the appearance of the moir$\mathrm{\acute{e}}$ potential
induced subgaps in the band structure of the BLG ensures that the $K$ and the
$K^{\prime}$ points are no longer connected in the same band which
significantly suppresses inter-valley scattering in our system 29.

###  2.1 Spin-orbit interaction in BLG

In a system with strong SOC, long-range spin currents can be generated in bulk
by the spin Hall effect (SHE) 30, 31, which produce a non-local signal at
voltage probes remote from the charge current path via the inverse SHE (for a
schematic see Fig. S9, SI). The measured non-local resistance $R_{NL}$ in our
device is peaked sharply near the primary Dirac point (Fig. 2(a)) and is at
least three orders of magnitude larger than that expected from ohmic
contributions alone (see SI). At large $\left|D\right|$, $R_{NL}$ has a split
peak (Fig. 2(b)). Recall that the strength of the Berry curvature is most
significant at the bulk band-edges, at the so-called ‘Berry curvature hot-
spots.’ This causes the effect of the SHE to be most prominent when the
chemical potential of the system lies at one of the band-edges 32. With
increasing $\left|D\right|$, the bandgap in BLG increases causing the Berry
curvature hot-spots in both the valence-band and the conduction-band to move
apart – this can be mapped out in our system by monitoring the location of the
peak of $R_{NL}$ in the $n-D$ plane.

###  2.2 Quantized transport through edge-modes

There are several experimental observations that confirm the appearance of a
strong proximity induced SOC in our BLG system – (i) presence of a substantial
$R_{NL}$ signal at the band-edges, (ii) weak antilocalization correction to
the low-field magnetoresistance (Fig. S6; SI3), and (iii) the asymmetry in the
plot of $R\left(B=0\right)$ (Fig. 1(e)). This leads to the expectation that
the system will host helical edge-modes. (see SI 7). The plots of the measured
4-probe and 2-probe longitudinal resistances for B = 0 T and T = 8 K shown in
Fig. 3 establish that this indeed is the case. Fig. 3(a) shows precise
quantization of the 4-probe longitudinal resistance to $h/\left(2e^{2}\right)$
at the PDP over a range of displacement fields. In Fig. 3(b) is plotted the
measured 2-probe resistance which quantizes to $3h/\left(2e^{2}\right)$. These
are the exact values one would expect for the respective measurement
configurations for a system hosting an odd number of helical edge-modes (see
SI for details). The quantization of the 2-probe and 4-probe longitudinal
resistances at the PDP over a range of values of the displacement field is the
central result of this article and establishes the emergence of helical edge-
modes in our system. Fig. 3(c) and Fig. 3(d) are respectively the plots of the
4-probe, and 2-probe longitudinal resistances plotted versus $n$ and
$\left|D\right|$. The data projected on the $n-R$ plane show that the
quantization in each case is centered around the PDP (with $\Delta n=\pm
2\times 10^{-15}$ m-2).

It might be argued that the observed quantization of the 4-probe longitudinal
resistance can be due to the ballistic nature of the device or a fortuitous
arrangement of scattering centers. To address these issues, the longitudinal
resistance of the device was measured in several multi-terminal
configurations. The data for two such configurations are shown in Fig. 4,
along with the respective measurement configurations (data for additional
configurations are presented in SI4). The gray lines are the values of
longitudinal resistance calculated for the helical edge-modes using the
Landauer-B$\mathrm{\ddot{u}}$ttiker formalism for quantum transport in multi-
terminal devices. (Note that, for chiral edge-modes, the expected values of
the resistance for the configuration in Fig. 4(a) would be zero and for that
in Fig. 4(b) would be $h/e^{2}$). The excellent match between the measured
resistance and the predictions based on helical edge-modes establishes that
these edge-modes are dissipationless and that they equilibrate at the contact
probes. This provides further confirmation that the system is most likely in
the QSHI phase near the PDP. In the SI, we show that the two-probe resistance
measured over two different length scales (1.5 $\mu$m and 5 $\mu$m) quantize
to the expected values. The fact that almost identical quantization results
were obtained over several thermal cycles of the device from room temperature
to 20 mK (comparison of results from two different cool-downs of the device
are presented in Fig. S3 of SI) also establishes the robustness of our
results.

The $R_{NL}$ signal was measured as a function of the magnetic field, $B$
applied perpendicular to the plane of the device. Breaking of TR symmetry by
the magnetic field would lead to a gradual quenching of the QSHI phase and
consequent decay of the non-local signal – the non-local signal can only
persist as long as $\Delta_{z}>E_{Z}$, where $\Delta_{z}$ is the energy-gap
for the QSHI phase and $E_{Z}$ is the Zeeman energy 33, 34. From the
temperature dependence of the quantized conductance, $\Delta_{z}$ was
estimated to be about 7 meV. This yields the maximum magnetic field range,
till which the quantization can persist to be
$B_{max}\sim\Delta_{z}^{2}/(m\mu_{B}v_{F}^{2})=0.1$ T, which matches very well
with our observations (see SI). Here $m$ is the mass of the electron,
$\mu_{B}$ is the Bohr magneton and $v_{F}\sim 1.2\times{10}^{6}$ ms-1 is the
nominal Fermi velocity of the charge carriers in BLG.

To obtain a microscopic understanding of the topological features of our
setup, we use a tight-binding model with both Ising and Rashba SOCs for the
AB-stacked BLG on WSe2 with an applied electric field $\left|D\right|\ \neq
0$. The electric field plays several critical roles in this system: (a) It
lifts the layer degeneracy in the BLG and opens a bandgap at the Fermi level.
(b) Above a certain critical value of the electric field, the top and bottom
layers of the BLG become fully charge polarized. In such a scenario, the
conduction and the valence bands are formed of orbitals localized respectively
in the negatively and positively charged layers of the BLG 12. This leads to a
strong correlation between the bands and the layers of the BLG. (c) This
layer-selective band structure receives SOC differently from the WSe2 layer.
As the WSe2 is proximate to the bottom layer, it induces a much stronger SOC
to the states localized at this layer layer, compared to that in the top
layer. This leads to large ($\sim 2$ meV) spin-splitting of the conduction
(valence) band for positive (negative) values of $D$.

To account for the above properties, we use the following model Hamiltonian
for our system:

$\displaystyle H_{BLG}$ $\displaystyle=$ $\displaystyle H_{SLG}^{\rm
T}+H_{SLG}^{\rm B}+t_{\perp}\sum_{i\in T,j\in
B,\alpha}\left(c^{\dagger}_{i\alpha}c_{j\alpha}+c^{\dagger}_{j\alpha}c_{i\alpha}\right)$
(1) $\displaystyle+\frac{U}{2}\sum_{i\in
T,\alpha}c^{\dagger}_{i\alpha}c_{i\alpha}-\frac{U}{2}\sum_{i\in
B,\alpha}c^{\dagger}_{i\alpha}c_{i\alpha}$

where, $H_{SLG}^{T/B}$ is the single-layer Hamiltonian – the superscripts
$\mathrm{T}$ and $\mathrm{B}$ refer to the top and bottom layers,
respectively. $t_{\perp}$ is the inter-layer nearest-neighbor hopping
amplitude. $U$ is the inter-layer onsite potential difference which includes
contribution from: (a) asymmetry between the two layers due to proximity to
the WSe2 layer (minor contribution) and, (b) the applied electric field
(larger contribution). For the single layer Hamiltonian, we include layer-
specific SOC contributions from both Ising SOC and Rashba SOC 14:

$\displaystyle H_{SLG}^{T/B}$ $\displaystyle=$
$\displaystyle-t\sum_{<ij>\alpha}c^{\dagger}_{i\alpha}c_{j\alpha}+i\lambda_{ISO}^{T/B}\sum_{<<ij>>\alpha}\nu_{ij}c^{\dagger}_{i\alpha}\left(s_{\alpha\alpha}\right)_{z}c_{j\alpha}$
(2)
$\displaystyle+it_{R}^{T/B}\sum_{<ij>\alpha\beta}\left(s_{\alpha\beta}\times{\bf
d}_{ij}\right)_{z}c^{\dagger}_{i\alpha}c_{j\beta}$

where, $t$ is nearest neighbor in-plane hopping, $\lambda_{ISO}^{T/B}$ is the
Ising SOC parameter, and $\lambda_{R}^{T/B}$ is the Rashba SOC parameter. The
bandgap near the Fermi levels is primarily controlled by the competition
between $U$ and $\lambda_{\rm R}$, (the Ising SOC is somewhat less effective
here). Therefore, we present a band-gap phase diagram as a function of $U$ and
$\lambda_{\rm R}$ in Fig. 4(d) for a representative values of
$\lambda_{ISO}^{B}=2$ meV , $\lambda_{ISO}^{T}=0.02$ meV12. We find,
consistent with previous calculations 14, that for small $\lambda_{\rm R}$,
the band gap opens in the trivial side (yellow regions) and there is a band
inversion (boundary marked by dark blue regions) to the non-trivial region as
$\lambda_{\rm R}$ increases.

To evaluate the nature of the edge states in the non-trivial topological
phase, we calculate the edge state profile with an open boundary condition
along the zigzag direction and periodic along the other direction (see SI). In
Fig. 4(e) we present the edge state profile extracted from our calculations at
representative values of $\lambda_{\rm R}=40$ meV, and $U=-40$ meV. Indeed we
find helical edge-modes emerging within the bulk bandgap with a Dirac cone – a
representative sketch is shown in Fig. 4(f) for the interpreted helical edge
states. This result is consistent with our experimental observations. Note
that in the absence of Rashba SOC, we do not find any edge-modes. This is in
contrast to the case of symmetric device structure (WSe2/BLG/WSe2) reported
before where there are two pairs of helical edge-modes arising from Quantum
Valley Hall effect giving rise to an edge conductance of $4e^{2}/h$ at
$\lambda_{\rm R}\rightarrow 0$ 14 .

To estimate the strength of the Rashba SOC in our system, we assume that the
Elliot Yafet mechanism is the dominant source of spin scattering in graphene
35 . In that case, the strength of the Rashba SOC is given by
$\lambda_{r}={2E_{F}}\sqrt{\frac{\tau_{P}}{\tau_{asy}}}$, where $E_{F}$ is the
average Fermi energy, $\tau_{asy}$ is time scale corresponding to the spin-
flip processes that breaks the $z\longrightarrow-z$ symmetry and
$\tau_{p}^{-1}$ is the momentum relaxation rate. Using the values
$\tau_{asy}\sim 0.7$ ps and $\tau_{p}$ $\sim 0.3$ ps estimated from our WAL
data (see SI) and reported elsewhere36, $\lambda_{r}$ was found to be $85$
meV. As seen from the phase diagram in Fig. 4(d), this value of $\lambda_{r}$
is enough to create helical edge-modes in the BLG.

The presence of hBN aligned with the top layer of the BLG (with the consequent
moir$\mathrm{\acute{e}}$ superlattice potential) suggests that the observed
$R_{NL}$ may also plausibly arise either from Valley Hall (VH) or Quantum
Valley Hall (QVH) effect. The first scenario can be ruled out by noting that
in VH’s case, transport proceeds through the bulk of the material resulting in
$R_{NL}$ scaling as $\rho^{3}$, where $\rho$ is the local 4-probe longitudinal
resistivity. Fig. 4(c) shows that in our device, $R_{NL}$ scales linearly with
$\rho$, which is what one expects for charge transport confined to the edges
of the device 37, 38. Note also that the bulk bandgap for BLG at
$\left|D\right|=0.2\times{10}^{9}$ V/m is around 10 meV 39, which ensures that
when the Fermi energy is near the center of the bulk bandgap at large
$\left|D\right|$, there is no contribution from the bulk of the BLG to
electrical transport in the system in the temperature range explored
experimentally. In the case of the QVH effect 40, 41, resistance quantization
has been observed primarily near the CDP, not the PDP. On the contrary, the
quantization results presented in this article were all obtained at the PDP.
Measurements of $R_{NL}$ in the presence of a perpendicular magnetic field
shows that for ${B<B}_{max}$, the non-local signal also appears only near the
PDP (see SI). These, and the fact that the 4-probe conductance quantization
observed in our device was $2e^{2}/h$ (and not $4e^{2}/h$ expected in the case
for QVH for BLG) effectively rule out QVH as the origin of our observations.

##  3 Conclusions

In summary, the experimental results presented in this article provide an
unambiguous demonstration of the TR invariant helical edge-modes in bilayer
graphene. We could tune the phase-space with an external displacement field to
induce a topological phase transition between a trivial phase and a possible
QSHI phase. In the topological phase, the linear conductance was quantized to
$2e^{2}/h$, indicating the presence of a helical edge-mode (a pair of counter-
propagating chiral modes). Non-local measurements show that these helical
edge-modes are dissipationless and that they equilibrate at the contact
probes. Our device gives a realization of the much sought after moiré
superlattice with spin-orbit coupling in a mesoscopic system and can form the
platform for the realization of the predicted quantum spin-valley hall
insulator phase 42. Additionally, it provides an experimental realization of a
non-interacting version of a system which, in the limit of strong
interactions, turns into the recently discovered Quantum Spin Liquid 43.

##  4 Methods

###  4.1 Experimental methods

The 2-D flakes used in our device structure were obtained by mechanical
exfoliation from single-crystals purchased from HQ graphene. The size of the
obtained flakes ranged from approximately 100-400 $\mathrm{\mu m^{2}}$. The
exfoliated flakes were stacked on top of each other in the desired sequence
using a home build micro-manipulator under an optical microscope and then
transferred on to SiO2/Si++ wafers. Standard electron beam lithography
followed by etching and metal deposition was used to define the electrical
contacts. The details of device fabrication are described in SI 1. All
electrical measurements were performed in a cryogen-free dilution refrigerator
at a base temperature of 20 mK using standard low-frequency lock-in techniques
with excitation currents of 1 nA.

###  4.2 Theoretical calculations

Theory method: The theoretical calculation is done within a material-specific
tight-binding model with spin-orbit coupling (SOC) and electric field on the
bilayer graphene. The electric field oppositely polarizes the two graphene
layers, and that the proximity effects due to the WSe2 substrate are higher on
the bottom layer than in the top layer. To take into account these effects, we
consider the onsite potential of the two layers to be different, breaking the
sublattice symmetry. The SOC terms include both Ising and Rashba-type terms.
The spin-orbit splitting also becomes anisotropic between the conduction and
valence bands. Their combined effect is shown to garner an inverted bandgap in
the non-trivial topological phase, with a helical edge state as confirmed by
edge state calculations in a lattice model with an open boundary condition.

The authors acknowledge fruitful discussions with Anindya Das, Atindra Nath
Pal, and Hemanta Kumar Kundu. The authors acknowledge device fabrication
facilities in NNFC, CeNSE, IISc. S.K.S. acknowledges PMRF, MHRD for financial
support. A.B acknowledges funding from SERB (HRR/2015/000017), DST
(DST/SJF/PSA-01/2016-17), and IISc.

Supporting information contains (a) details of device fabrication, (b)
detailed results of magnetoresistance measurements, (c) details of measurement
of quantized conductance in other multi-terminal configurations, and (d)
calculations of edge conductance based on Landauer Büttiker formalism.

##  5 Associated Content

A pre-print version of the article is available at: Priya Tiwari; Saurabh
Kumar Srivastav; Sujay Ray; Tanmoy Das and Aveek Bid. Quantum Spin Hall effect
in bilayer graphene heterostructures. 2020, arXiv:2003.10292 (cond-mat).
ArXiv. https://arxiv.org/abs/2003.10292 (accessed March 23, 2020).

Figure 1:  Device structure and characterization. (a) A schematic of the
device configuration. The co-laminated heterostructure of BLG (shown as
honeycombed structure) and monolayer WSe2 (grey layer) is sandwiched between
two hBN flakes, each of thickness $\sim 20$ nm. (b) Room temperature Raman
spectra of the WSe2 and BLG flakes. The peaks corresponding to monolayer WSe2
and BLG are marked. (c) Room temperature photoluminescence response of WSe2
flake – the peak at 1.65 eV establishes it to be a monolayer. (d) Four-probe
resistance plotted as a function of $V_{tg}$ keeping back gate voltage fixed
at $-12$ V; arrows mark the clone Dirac point (CDP) and primary Dirac point
(PDP). (e) Logarithmic scale color plot of the 2-probe longitudinal resistance
versus the displacement field $\left|D\right|$ and the net charge carrier
concentration, $n$. The dashed lines mark the asymmetry around the primary
Dirac point. The data were acquired at $T=8$ K and $B=0$ T.  Figure 2: (a)
Plot of the non-local resistance, $R_{NL}$ versus $n$ for $D=-0.098$ V/nm.
Left inset: zoom-in of $R_{NL}$ near the PDP. Right inset: schematic of the
measurement configuration. (b) Contour plot of $R_{NL}$ as a function of $n$
and $D$. The yellow dotted lines highlight the splitting observed in the
$R_{NL}$ peak at large $\left|D\right|$.  Figure 3: (a) Quantization of the
4-probe resistance to $h/\left(2e^{2}\right)$ as $V_{tg}$ is varied at fixed
values of $V_{bg}$ such that the Fermi level lies in the bulk bandgap. The
value of $V_{bg}$ for the different plots varies from -12 V (brown line) to
-13.5 V (dark blue line). The inset is a schematic of the 4-probe measurement
configuration - the red and the green lines represent the spin-filtered edge-
modes. (b) Quantization of the 2-probe resistance to $3h/\left(2e^{2}\right)$,
the data were acquired simultaneously with the data presented in (a). The
inset is a schematic of the 2-probe measurement configuration (c) Plot of the
4-probe $R$ versus $n$ and $\left|D\right|$. The projection on the $R-n$ plane
shows that the quantization is around $n=0$, while the projection on the $R-D$
plane shows the quantization appears over a range of values of the
displacement field. (d) Corresponding plot for 2-probe measurements – the
projection on the $R-n$ plane shows that the quantization is again at the PDP.
All the data were acquired at $T=8$ K (quantization obtained at 20 mK and for
other cooldowns are presented in SI).  Figure 4: (a) Quantization of $R$ to
$2h/\left(3e^{2}\right)$ in a 3-probe measurement geometry - the measurement
configuration is sketched in the inset. (b) Quantization of $R$ to
$5h/\left(6e^{2}\right)$ \- the inset shows the measurement configuration. (c)
Plot of $R_{NL}$ versus the local longitudinal resistivity $\rho_{local}$
measured at different $D$. The filled red circles are the measured data
points; the solid blue line is a fit to $R_{NL}\propto\rho_{L}$. The solid
green line represents a plot of $R_{NL}\propto{\rho_{L}}^{3}$. (d) A bandgap
phase diagram as a function of $U$ and $\lambda_{\rm R}$ plotted for
representative values of $\lambda_{ISO}^{B}=2$ meV and
$\lambda_{ISO}^{T}=0.02$ meV. The dark blue region marks the boundary between
the topologically trivial and non-trivial regions of the phase diagram. (e)
Emergence of helical edge-modes within the bulk bandgap for $\lambda_{\rm
R}=40$ meV and $U=-40$ meV. The inset shows a magnified view around the $K$
point of the edge-modes with a Dirac cone. (f) Schematic diagram showing the
helical edge-modes arising from the band structure in panel (e). The blue and
the green arrows represent respectively the counter-clockwise propagating and
clockwise propagating helical edge modes; the red and the orange arrows
indicate the direction of the out-of plane spin-component of these two modes.

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# Femtosecond laser-shockwave induced densification in fused silica

A. Radhakrishnan<EMAIL_ADDRESS>Galatea Laboratory,
IMT/STI, Ecole Polytechnique Fédérale de Lausanne (EPFL), Rue de la Maladière
71b, 2000 Neuchâtel, Switzerland  J. Gateau  Galatea Laboratory, IMT/STI,
Ecole Polytechnique Fédérale de Lausanne (EPFL), Rue de la Maladière 71b, 2000
Neuchâtel, Switzerland  P. Vlugter  Galatea Laboratory, IMT/STI, Ecole
Polytechnique Fédérale de Lausanne (EPFL), Rue de la Maladière 71b, 2000
Neuchâtel, Switzerland  Y. Bellouard  Galatea Laboratory, IMT/STI, Ecole
Polytechnique Fédérale de Lausanne (EPFL), Rue de la Maladière 71b, 2000
Neuchâtel, Switzerland

###### Abstract

Tightly focused femtosecond laser-beam in the non-ablative regime can induce a
shock-wave enough to reach locally pressures in the giga-Pascal range or more.
In a single beam configuration, the location of the highest-pressure zone is
nested within the laser-focus zone, making it difficult to differentiate the
effect of the shock-wave pressure from photo-induced and plasma relaxation
effect. To circumvent this difficulty, we consider two spatially separated
focused beams that individually act as quasi-simultaneous pressure-wave
emitters. The zone where both shock-waves interfere constructively forms a
region of extreme pressure range, physically separated from the regions under
direct laser exposure. Here, we present evidences of pressured-induced
densification in fused silica in between the foci of the two beams, which can
be exclusively attributed to the superposition of the pressure waves emitted
by each focused laser-beam. Specifically, we show how the beams gap and pulses
time-delay affect the structural properties of fused silica using Raman
characterization, beam deflection technique, and selective etching techniques.
The method is generic and can be implemented in a variety of transparent
substrates for high-pressure physics studies and, unlike classical methods,
such as the use of diamond anvils, offers a means to create arbitrary-shaped
laser-induced high-pressure impacted zones by scanning the two beams across
the specimen volume.

††preprint: AIP/123-QED

##  I Introduction

In nature, high-pressure phases of silica are found in meteorite craters
resulting from high-velocity impacts Gillet and Goresy . Studying these phases
in a laboratory setting remains a tedious task, as it requires high-pressure
generation, from tens of giga-pascal (GPa) to tera-pascal (TPa) levels. While
diamond anvil cells (DAC) are commonly used for the laboratory-scale high-
pressure generation, it suffers from intrinsic limitations of volume and
processing time restrictions Jayaraman (1983); Richet and Gillet (1997). Due
to the extreme brevity of the energy deposition and the rapid formation of a
plasma, ultrafast laser interaction with dielectrics creates the conditions
for locally achieving pressure levels in the TPa levels, as reported in
sapphire and fused silica using single spot experiments Juodkazis  _et al._
(2006); A  _et al._ (2011); Gamaly  _et al._ (2002); Juodkazis  _et al._
(2010). There, the high-pressure zone is nested within the laser-affected
zone, making it challenging to differentiate photo-induced from pressure-only
effects as the material is not exclusively subjected to intense pressure
waves, but also to the outcome related to plasma generated at the laser spot.
To circumvent this difficulty and to effectively separate the high-pressure
zone from the regions under direct exposure, we focus two spatially separated
femtosecond. The two act as quasi-simultaneous emitters for strong pressure
waves that interfere one another. In the case of constructive interferences,
this configuration can lead to a higher-pressure zone located outside of the
zone under direct laser exposure. When a femtosecond pulse is absorbed by the
material, it forms shock waves caused by the rapid plasma volume formation,
expansion, and decay Devaux  _et al._ (1993); Zeng  _et al._ (2006); Berthe
_et al._ (1997); Hayasaki  _et al._ (2011); Bergner  _et al._ (2018); Sakakura
_et al._ (2007). These shock-waves propagate radially from the laser-
propagation axis, at an initial velocity faster than the sound wave in the
material (5.9x$10^{3}$ m/s for silica) and decays rapidly into acoustic sound
waves within micron distances Gamaly  _et al._ (2006). Here, we increase the
pressure by beam superposition technique as schematically illustrated in
Figure 1. (left).

Figure 1: Left: Double-beam femtosecond laser bulk exposure: schematic view of
the cross section (top left) and constructive wave interference pattern shown
in the acoustic regime for illustrative purpose (bottom left). The two laser
beams emit shock waves that add up in between the beams location. These
decaying shock waves interfere constructively in between the two foci. Right:
Continuous line-patterns scanning principle. The double-beam exposure can be
applied anywhere through the specimens. The two pulses are temporally
separated by the pulse length to prevent optical interferences, but yet for a
duration order of magnitude shorter than the time scale of shock-wave emission
to consider the two events as instantaneous.

##  II Experimental setup

Figure 2: Optical layout of the double-beam experiment setup. A part of the
incoming laser beam is converted into a frequency-doubled probe beam using a
BBO crystal. The beam gap and the temporal delay are adjusted by a linear
translation stage (LTS 1) and a pump delay line respectively. The front
schematic sketch of the machining platform is shown in the right inset. (BSP:
Beam splitter polarizer cube, HWP: Half wave plate, LTS: Linear translation
stage, BBO: beta barium borate crystal.)

Figure 2 illustrates the optical setup used in these experiments. A
femtosecond laser (s-Pulse from Amplitude) delivering 150 fs pulses with a
center wavelength of 1030 nm at a pulse repetition rate of 100 kHz is focused
inside a fused silica substrate (Coring 7980 OF). The laser beam is split into
two beams of equal energy that are temporally and spatially separated by a
varying time delay ($\Delta$t) and beam gap ($\Delta$t), respectively. The two
beams are focused with the same objective (OptoSigma, NA 0.4). A low-energy
second harmonics probe pulse ($\lambda_{probe}$ = 515 nm), generated in a Beta
Barium Borate (BBO) crystal is used to precisely control the time delay
between the two incoming beams, with a resolution down to tens of femtoseconds
between the pump and probe signals, by visually observing the effect of the
plasma formation on a standard CMOS camera placed above the focusing
objective. The spatial gap between the two machining beams is calibrated by
optimizing the interference fringe pattern of the two beams and further
refined by measuring the distance between laser-affected zones as observed in
the SEM images of the polished cross-section of the written lines. In this set
of experiments, we were able to vary the distance between laser-affected zones
from 480 nm to 2 microns, and the time delay from zero to 66 ps. The machining
was done in the bulk of the material at a distance of 20 microns from the
surface.

##  III results and discussions

###  III.1 Densification measurements

The cross section of the line-patterns produced by the double beams were
analyzed using micro-Raman spectroscopy (Renishaw InVia confocal Raman
spectroscope using a 100X objective, spot size < 1 $\mu$m). Here, we choose
405 nm as the excitation laser wavelength for stimulated Raman emission to
prevent fluorescence excitation of laser-induced defects that could shadow
other Raman peaks. Each measurement point in the scan was obtained with a
laser power of 4.6 mW, and a total exposure time of 20 seconds. Figure 3.b
represents the Raman measurements taken outside and within the laser-modified
zones, polished along their cross-section oriented towards the optical laser
propagation axis. In this illustration, the beams are spatially separated by a
distance of 1.2 microns and temporally, by a time delay of 240 fs. The data
are presented for three characteristic zones, namely, the zone outside the
laser affected zones (labelled ‘Zone O’), the zone located left to the laser
exposed pattern (‘Zone L’), and finally, the zone in between the two laser
affected zones (‘Zone M’), which is the zone where the two shockwaves are
superposed. All Raman spectra are normalized with respect to the $\omega_{4}$
band, which is found to be more invariant to the effect of laser exposure. On
one hand, ‘Zone O’, located one micron away from the laser exposed zone, shows
no visible difference in the Raman spectra compared to a reference (as seen in
Figure 3.a) along all measurement points, while the pressure affected zone
(zone M) was found to have modification in the region of shock superposition,
in particularly among the points M1 and M2 as Figure 3.c. Though we did not
find a large variation in the shift of Raman spectra towards the higher
wavenumbers (as reported in Okuno  _et al._ (1999)), we observe a rise of the
D1 and D2 peak intensity (located around 495 cm-1 and 605 cm-1, respectively)
Galeener (1979); Galeener  _et al._ (1984); Galeener and Lucovsky (1976),
along with a shift of the peaks towards higher wavenumbers as well as the
shrinkage of the main band, which accounts for the reduction in bond angle of
the silica lattice Sen and Thorpe (1977); Sakakura  _et al._ (2011) in Zone M.

Figure 3: (middle) Microscopic image of the point of measurement along the
cross-section of the machined lines; Left & right / Raman spectra of the zones
outside the laser affected zone (denoted by O), and along the pressure
affected zone in the middle of the beams (M) respectively. Figure 4: a. The
Raman parameter $\sigma$ as a function of the maximum pressure reached
$P_{max}$ as adapted from [Sonneville  _et al._ (2012)] and the data from
pressure affected zones was used to estimate the $P_{max}$ after double laser
beam machining; b. The shift of the main band along the pressure affected zone
in the middle of the beams (M); c. The shift of D2 peak and the ratio of the
D2 peak with respect to the D2 reference peaks plotted along the pressure
affected zone (M).

These relative changes, combined in Figure 4, point towards the densification
of the material in between the two laser-affected zones. As this zone is
located outside the laser-exposed zone, we attribute this densification to the
effect of shock wave superposition. By comparing these variations in spectra
to the compressive hydrostatic loading experiments reported elsewhere, we
estimate a pressure development of about 13-14 GPa by estimating the Raman
parameter $\sigma$ and the main band shift as mentioned in Deschamps  _et al._
(2013); Sonneville  _et al._ (2012) (see Fig. 4 a). While this method gives a
first estimate of the pressure, it assumes a hydrostatic loading case, which
differs from our conditions and in fact, underestimates the real pressure. In
our case, the modifications are obtained as a result of dynamic shock waves
superposition. Based on silica shock-wave densification studies Okuno  _et
al._ (1999), we estimate a pressure development of about 25-30 GPa by
correlating the shift and rise of the D2 peak as shown in Figure 4.c. The
latter is of higher pressure since we have the shock superposition just over a
limited period of time, and hence, higher pressure loading conditions. To
explain why points M3-M5 do not show the same behavior as point M1-M2, we note
that there was a shift in one of the beam along the optical propagation axis
direction due to alignment errors, which may have resulted in uneven pressure
distribution. Another possible explanation, as it will be further when
examining Raman spectra inside laser affected zones is the anisotropic
pressure distribution in these zones. These two observations may explain the
lack of evidences for densified zones in between the tails of laser affected
zones.

####  III.1.1 Modification along the laser affected zone

Figure 5: (middle) Microscopic image of the point of measurement along the
cross-section of the machined lines; (Left and right) Raman spectra of the
zones along the laser affected zone (denoted by L). The same Raman spectra
shown for the region in between laser affected zones (denotes M) are shown for
comparison.

The ‘Zone L’, located in the left laser affected zone, exhibits clear
modifications in the Raman spectra, which is expected based on previously
reported data Bellouard  _et al._ (2008); Chan  _et al._ (2001) as shown in
Fig. 5. (left), when compared to the pressure affected zone (Fig. 5. right).
Measurement points near the head of the laser affected zone show more
densification in the Raman spectra than the ones measured near the tail of the
LAZ. The tear-shaped geometry of the LAZ and the reduced intensity near the
trailing edge Docchio  _et al._ (1988); Spa  may explain this effect. We
observed a density of 2.35 g/cm3, while comparing the D2 intensity and FWHM of
the main band as mentioned in Wang  _et al._ (2021).

It was also found that the densification inside the laser affected zone (LAZ)
is higher than the densification due to homogeneous modification, a
densification exposure regime found at shorter pulse duration and lower pulse
energy Vlugter and Bellouard (2022). Using the same method proposed by
Deschamps  _et al._ (2013) and used for a the first estimate of the pressure
in between the beams, we estimated a pressure approaching 15 GPa, see Fig
6.This estimate is based on the assumption of an hydrostatic pressure and
therefore, most likely underestimates what the real pressure was, as the
loading case is present for a short duration and. here may liken the one of a
shock wave. However, as we are within the laser-affected zone, and hence in
the zone where the plasma was located, it remains speculative to truly assess
what the pressure conditions as we lack an equation of state for the matter
under such extreme conditions. We would therefore caution that unlike the zone
in between laser affected zones where there is a clear decoupling between
plasma and pressurized zone, this pressure estimates based on Raman data
performed inside the laser-affected zones remain speculative. We also further
noticed that the modification obtained inside the LAZ is independent from the
presence of a shock wave emitted by a neighboring LAZ as identical Raman
spectra were obtained inside LAZ, when the beams were far apart both spatially
and temporally.

Figure 6: The Raman parameter $\sigma$ as a function of the maximum pressure
reached $P_{max}$ as adapted from [Deschamps  _et al._ (2013)]. The data
inside the laser affected zones (Zone L) are superimposed to estimate
$P_{max}$ in our specific case.

###  III.2 Effect of beam gap and time delay in the pressure affected zone

Figure 7: a. Comparison plot between D2/Dref ratio and beam gap ($\Delta$x)
for a fixed time delay (240 fs); b. Contour map between beam gap and time
delay in terms of the extent of the rise in D2 peak.

To further investigate the densification effect, dual-line patterns were drawn
in the fused silica substrates for varying beam gap ($\Delta$x) and time delay
($\Delta$t). The patterns were systematically analyzed using Raman
spectroscopy following the same method than described in the previous
paragraph and we used the ratio D2/Dref as an indirect metric of the
densification. Figure 7 denotes the densification with varying beam gap, but
for a constant time delay of 240 fs. The data corresponds to point M2 in Zone
C, as indicated in Figure 7.a, a point where the maximum densification was
obtained. The maximum densification peaks between 480 nm and 1.2 microns,
which indicates the expected decay of the pressure waves Juodkazis  _et al._
(2006); Zel’dovich and Raizer (2002) after a certain critical radius $r_{c}$
as shown in Figure 1. (left). Beyond $r_{c}$, the pressure superposition is
not sufficient to induce a permanent densification. Figure 7.b. shows a
contour plot for D2/ Dref contrast for varying time delays for various beam
gaps. By taking into account the supersonic velocity approximation from
Hugoniot data of fused silica Koenig  _et al._ (1994); Desjarlais, Knudson,
and Cochrane (2017), we could suppose that above 6.6 ps, the superposition
occurs, near or beyond the boundary of the trailing beam’s laser affected
zone. Further, for lines exposed with ‘infinite time delay’, i.e. lines
written sequentially one after the other, masking each beam sequentially we
did not achieve a much higher densification parameters that the beams written
without masking each other. The map suggests that maximum densification is
obtained when the beam gap is less than one micron, and when the beam delay is
less than 500 fs. This window of parameters can be used for generating a
localized densified zone between two laser-affected zones, thereby having a
varying refractive index regime.

###  III.3  Volume variation measurement in double beam machined fused silica
specimen as evidence of densification

Figure 8: Schematic cross section of a cantilever, used with double-beam laser
exposure conditions. In this illustration, the cantilever bends with a radius
r, and results in an overall deflection $\delta$, due to volume expansion. In
the magnified rectangle A-A, the red ellipses represent the line cross-
section, with individual width $w_{0}$ and separated by a beam gap $\Delta x$.

To complement Raman data and to confirm the presence of densification, we use
a method based on micro-cantilevers Bellouard  _et al._ (2016); VLUGTER and
BELLOUARD (2020) to investigate localized volume variations resulting from the
double-beam femtosecond laser exposure inside the bulk of silica. This method
is highly sensitive and has also been used for investigating coefficient of
thermal expansion changes after laser exposure. The working principle of this
experimental technique, adapted to the double beam exposure, where a series of
twin-lines (red zones in Fig. 8) with a definite spacing ls are written along
the upper part of the cantilever, and towards the anchoring joint. This
results in a bimorph composite with an amplified displacement ($\delta$) in
the transparent glass cantilever. Here, we exposed various cantilevers with
varying time delays and beam gaps.The average stress and strain in the laser
affected zones are extracted from the measured cantilever deflections using
Stoney’s equation Champion and Bellouard (2012). The results are shown in
Figure 9 for four representative exposure conditions (labelled A to D) and
reported in Figure 7.b. The difference in the average stress between case A,
which is the one corresponding to the highest density case suggested by Raman
observations, is in agreement with the formation of a high density amorphous
(HDA) phase Gleason  _et al._ (2015, 2017) in between the laser affected
zones. Indeed, this zone results in a reduced cantilever deflection, due to
the volume compaction found in between laser affected zones, which in turn
leads to a decrease in average stress.

Figure 9: a. Average stress for selected time delays (dt) and beam gaps (A-D).
Exposure conditions for these four cases are shown in Figure 4; b. Strain
along an arc x-x for a beam gap of 1 micron, as shown in the inset of Figure
8.

Further, mathematical modeling was done assuming a uniform stress along the
laser affected zones, to estimate the strain along an arc oriented towards
axis x. The contributions of the different zones to the average strain are
given in equation below, where $\epsilon_{l}$, $\epsilon_{s}$, and
$\epsilon_{sw}$ are strain in the laser affected volume (Zone L), in between
the two set of lines (Zone O) and the zone where the shockwaves superimpose
(Zone M), respectively. The strain in zone O is due to the constant stress
($\sigma_{xx}$) in the LAZ and it is defined as

$\epsilon_{s}=\frac{\sigma_{xx}}{E.\epsilon_{l}}$

is the strain in laser affected zone and is retrieved from the experiments
where no densification due to shock wave is. $V_{l}$, $V_{s}$, and $V_{sw}$
are the respective volume fractions and they are defined as
$V_{l}={2w_{0}}/{l_{s}}$, $V_{s}=1-{2w_{0}}/{l_{s}}$ and $V_{sw}={\Delta
x}/{l_{s}}$. The average stress shall be written as,

$\epsilon_{avg}=\epsilon_{l}V_{l}+\epsilon_{s}V_{s}+\epsilon_{sw}V_{sw}$

. Here, we assume there is no shock-wave contribution when the laser beams are
sufficiently temporarily separated i.e. $\epsilon_{sw}=0$. Solving the average
stress equation for the two extreme cases as shown in Figure 7.b, where there
is limited contribution and maximum contribution from the shockwave, in the
case of a time delay of 66 ps and 300 fs, respectively. Though we may note a
strain of 10$\%$ from the Raman shift of D2 as mentioned in Okuno  _et al._
(1999); It should also be noted that the peak densification obtained at point
M2 as in Figure 4 is diluted for these measurement as we estimate the average
strain from the cantilever deflection. We shall note how the strain varies
along the section and the change in the strain rate in between the laser-
affected zone, clearly emphasizing the effect of a remnant strain due to the
shockwave ($\epsilon_{s}w$), and hence the presence of a shock-induced
densified zone. The strain obtained in our case is an average stress, and
hence it is logical to have a lower value compared to the densification ratio
obtained in shockwave experiments for which the entire element is subjected to
uniform shock loading Okuno  _et al._ (1999).

###  III.4  Effect of dual-beam exposure parameters on chemical etching
selectivity

Figure 10: a. Schematic layout for the etching experiment. The sample is
divided into several patterns, where each patch has a specific time delay and
beam gap; b. The bright-field microscope top view image of selected patterns
of the etched sample; c. Contour plot of etching rate with varying time delay
and beam gap.

It is known that femtosecond-laser affects the etching rate Rajesh and
Bellouard (2010); Mouskeftaras and Bellouard (2018), which itself depends on
stress, density and structure of the materials. To understand the effect of
dual-beam exposure on etching rate, a series of lines are drawn in the bulk of
a fused silica substrate as shown Figure 10.a. After exposure, the substrate
is cut using a diamond wire-saw, along the section B-B as in the Figure 10.b.,
and later submerged in 2.5% HF bath for a preferential etching of 4.5 hours.
The effect on etching rate of varying time delays and beam gaps is plotted as
shown in Figure 10.c.. The contour plot suggests a correlation with the
densification experiments, and confirms other experiments reported in Agarwal
et al. Agarwal and Tomozawa (1997). The increased etching rate is attributed
to the densification due to the compressive loading by the shock waves that
resulted in compressive stress similar to a mechanical, hydrostatic loading
Agarwal and Tomozawa (1997).

##  IV Conclusion

Using near simultaneous adjacent, but non-overlapping femtosecond laser beams,
we have demonstrated that one can obtain permanent localized densification in
between laser affected zones as the result of shock-wave superposition, and
this, while preserving the integrity of the laser-exposed zones. Evidences of
a localized densification were obtained using Raman spectroscopy, indirect
remnant strain measurements and indirectly correlated with etching selectivity
enhancement observations. Overall, the exposure method demonstrated here
offers a means for studying the state of matter under high-velocity impact
stress without the need for a direct contact method, like for instance the use
of diamond anvil. Although, the pressure obtained remains moderate (a few tens
of GPa), mainly due to the limit in laser power in our setup, this method
demonstrates a path-way towards non-contact laser-induced high pressure
studies where pressured-zone are separated from laser exposed zones.

### Data Availability Statement

The data that support the findings of this study are available from the
corresponding author upon reasonable request.

###### Acknowledgements.

We acknowledge the Swiss National Science Foundation (FNS) for funding the
Femto-Anvil project (Project number: FNS 200021 169681). We further
acknowledge the financing form the ERC (ERC-2012-StG-307442). We thank Prof.
Céline Hebert and Dr. Farhang Nabiei of the Physics Department, EPFL, and
Ruben Ricca, Dr. Margarita Lesik Galatea lab, EPFL for the fruitful
discussions and helping out with the cantilever experiment. We would also like
to thank Dr. Richard Gaal and Dr. Arnoud Magrez of the Earth and planetary
science department and Crystallographic facility, EPFL, respectively, for the
training with the Raman spectroscope. Finally, the authors would like to thank
Ruben Ricca for aiding in formatting the draft.

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# A Pulsed Muon Source Based on a High-Repetition-Rate Electron Accelerator

Meng Lv<EMAIL_ADDRESS>Jiangtao Wang<EMAIL_ADDRESS>Kim
Siang Khaw<EMAIL_ADDRESS>Tsung-Dao Lee Institute and School of
Physics and Astronomy, Shanghai Jiao Tong University, Shanghai, China

###### Abstract

Muons have established a unique and pivotal role in both fundamental physics
and applied sciences. Given that a typical muon experiment spans roughly ten
muon lifetimes, the optimal muon source should operate at around 50 kHz in
pulsed mode. However, existing muon facilities operate in either the 25-50 Hz
pulsed mode or continuous beam (DC) mode, which results in low-duty cycles for
various muon experiments. As a result, precision muon physics with continuous
muon beam has been limited by statistical uncertainty. In this study, we
investigate the potential of a high-repetition-rate pulsed electron beam at
the Shanghai SHINE facility to serve as a muon source driver. SHINE houses an
8-GeV CW superconducting RF linac, with a 1 MHz bunch rate and 100 pC bunch
charge. Following X-ray production, the electron beam is deflected downstream
of the undulators and absorbed in a beam dump. Using Geant4 Monte Carlo
simulations, we estimated the yield of the muon beam to be approximately
$10^{3}\mu^{\pm}$/bunch. This type of muon beam could be instrumental in a
broad range of muon experiments, including muon lifetime measurement, a search
for muonium to anti-muonium conversion, and the muon spin spectroscopy.

##  I Introduction

Research in fundamental physics and applied science using muons has gained
substantial interests in recent years. A long-standing discrepancy between
theoretical predictions Aoyama:2020ynm  and experimental measurements
Muong-2:2006rrc ; Muong-2:2021ojo  regarding the muon’s magnetic moment
strongly suggests the presence of physics beyond the Standard Model of
particle physics (for a recent review, see Keshavarzi:2021eqa ). Techniques
involving muon spectroscopy, such as muon spin rotation and muon-induced X-ray
emission Hillier:2022nat , have catalyzed advancements in superconductivity,
magnetism, and the elemental analysis of archaeological artifacts.

A common feature of existing and planned facilities is their reliance on high-
power proton accelerators. Most of them operate in a multipurpose mode, where
experiments with muons, neutrons, and pions are conducted simultaneously.
These facilities currently operate either in a pulsed mode (25-50 Hz, e.g.,
J-PARC in Japan Miyake:2014zra , and ISIS in the UK Hillier:2018wkl ) or a
continuous (DC) mode (e.g., PSI in Switzerland Grillenberger:2021kyv , TRIUMF
in Canada Marshall:1991wb ). This is also true for the five new muon
facilities currently under study at CSNS Vassilopoulos:2022ali , HIAF/CiADS
Cai:2019feg , RAON Won:2014gja , and FNAL Gatto:2022olb . Given that a typical
muon experiment spans roughly ten muon lifetimes, current operating modes
result in low-duty cycles for various muon experiments. For instance,
precision muon physics with continuous muon beam has been limited by
statistical uncertainty. Recently, several authors have noted that the optimal
muon source for experiments such as the muon spin rotation (µSR)
Cywinski:2009zz , muon electric dipole moment Adelmann:2010zz ;
Adelmann:2021udj , and muonium to anti-muonium conversion Willmann:2021boq ;
Kuno:2000wn  operates in pulsed mode with a repetition rate of several tens of
kHz. A non-scaling fixed-field alternating gradient (FFAG) proton accelerator
technology with a frequency of a few kHz Kuno:2005zz  has been proposed for
this purpose, but is still under development Seidel:2021jyz . The proton beam
to be delivered to Fermilab’s Mu2e experiment Mu2e:2014fns  has a proton bunch
repetition rate of 0.59 MHz, achieved by resonantly extracting the proton
bunch from a delivery ring. However, it is dedicated to the Mu2e experiment.
Recent work at ORNL’s SNS aimed to extract 50 kHz proton pulses for µSR
applications, employing laser neutralization on a hydrogen ion beam
Liu:2020hcu .

Figure 1: A comparison between pulsed and continuous muon beam.

Apart from the proton beam, an electron beam could also be used to drive a
muon source. Recently, Nagamine et al. proposed using a 300 MeV, 10 µA
electron accelerator for a µSR facility Nagamine:2009zz , predicting a
potential yield of $8\times 10^{3}\mu^{+}$/s taking into account acceptance
effects and transportation losses. A feasibility study for a muon beam using
the Hall-A beam dump at the Continuous Electron Beam Accelerator Facility
(CEBAF) Dudek:2012vr  in Jefferson Lab has also been conducted Fulci:2023hmx .
With a target electron number of about $10^{22}$/year, the anticipated muon
flux is approximately $2\times 10^{15}$/year. The latest advancements in laser
wakefield acceleration (LWFA) technology have compactified electron
accelerators to mere meters rather than kilometers. The development of such
compact muon sources has been extensively studied recently Titov:2009cr ;
Dreesen:2014mt ; Rao:2018njj , in anticipation of the imminent availability of
high-repetition-rate femtosecond multi-PW lasers.

In this article, we explore the possibility of utilizing a high repetition
rate electron beam at SHINE Zhao:2018lcl  in Shanghai to power a muon source.
We present a detailed analysis of the anticipated beam intensity for muons and
other secondary particles for two distinct target configurations. Potential
applications of these muon beams in the field of fundamental physics as well
as applied science will also be discussed.

##  II MUON PRODUCTION BY ELECTRON

Muon beams are typically produced as a tertiary beam within a proton
accelerator complex. This production process is driven by a high-intensity
proton beam hitting on a graphite target Bungau:2014rxa ; Berg:2015wna ;
Cook:2016sfz . This results in the production of pions via strong nuclear
interactions. Bending magnets are then used to extract these pions from the
target area. As they travel through a lengthy decay channel, these pions
gradually decay into muons, which are subsequently delivered to muon
experiment zones. A particular category of muon, known as a surface muon, can
be selected by ensuring the beam momentum is approximately 28 MeV/c. This
specific muon beam originates from the pion decaying near the target’s surface
and is typically nearly 100% polarized due to the parity-violating weak decay
of the pion.

Contrarily, the muon production scheme for an electron-driver varies
significantly. Muons can be created as a tertiary beam via photo-nuclear
process. The photo-nuclear process entails the production of a real photon via
the bremsstrahlung process, which is then followed by pion production through
photo-excitation of the nucleus. Muons can also be created via the Bethe-
Heitler process, which uniquely does not proceed through pion production and
decay. The relative sizes of the cross sections are approximately in the ratio
1000:1 for photo-nuclear and pair production processes respectively, for
electrons in the GeV scale Blomqvist:1976mq ; Nagamine:2009zz .

##  III SHANGHAI SHINE FACILITY

Figure 2: Aerial view of the SHINE facility Zhao:2018lcl .

The Shanghai High repetition rate XFEL and Extreme light facility (SHINE)
Zhao:2018lcl , currently under construction at Shanghai’s Zhangjiang High
Technology Park (see Fig. 2), is a fourth-generation light source. This
facility houses a 8-GeV CW superconducting RF linac, which delivers an 8-GeV
bunched electron beam. The beam offers a repetition rate of up to 1 MHz and a
bunch charge of 100 pC, culminating in an average current of 100 µA. SHINE is
equipped with three undulator lines, capable of producing hard X-rays of up to
25 keV. Following X-ray production, each electron beam is diverted and routed
towards a beam dump Xu:2020bd . Interactions between the electron beam and
either the beam dump or a thin target preceding the beam dump could generate
muons and other secondary particles.

##  IV SIMULATION SETUP

In order to simulate the production of muons, positrons, and pions resulting
from electron interactions with the target, we employed the musrSim package
Sedlak:2012 , which is based on the Geant4 toolkit GEANT4:2002zbu . We
modified the physics list to use the FTFP_BERT model for hadronic processes.
The initial phase of our investigation into the electron-beam-driven muon
source focused on optimizing the muon production target. Based on prior study
Rao:2018njj , we selected tungsten as our target material and designed it in a
cylindrical shape. The target was optimized to maximize muon yield at a
distance of 5 cm from the center of the target, as illustrated in Fig. 3.

Figure 3: An event display from the Geant4 simulation for the target
optimization study.

We found that the dimensions yielding the highest surface muon production for
a tungsten target are a thickness of 30 mm and a radius of 6 mm. The energy
and angular distribution of the muons are presented in Fig. 4.

Figure 4: Distribution of muon energy and angle from both production channels
per SHINE electron bunch of 100 pC.

Muons generated from the photo-nuclear process exhibited a lower energy and a
broader angular spread compared to those created from the pair-production
process. This is consistent with our expectations since the former arise from
pion decays. The yield from the photo-nuclear process is approximately
$1.4\times 10^{4}$ per bunch, while the yield from the pair production process
is around $6.9\times 10^{3}$ per bunch. Furthermore, we observed two distinct
peaks in the momentum distribution, as depicted in Fig. 4(top). These peaks
each correspond to the decays of pions and kaons, respectively, and have close
to 100% polarization, suitable for µSR application. The yield of the surface
muon at 4 MeV is approximately 1,400 per bunch, resulting in an intense rate
of $1.4\times 10^{8}$/s for a 100 kHz operation. Surface muons resulting from
kaon decay present a unique prospect for probing extremely dense materials,
owing to their higher penetration power Grinenko:2023 .

Figure 5: Geometry structure and materials of the SHINE beam dump. Adapted
from Xu:2020bd .

In a scenario that minimizes modifications to the SHINE facility, we examined
an alternative target configuration: the SHINE beam dump Xu:2020bd . As shown
in Fig. 5, the SHINE beam dump is cylindrical and composed of aluminum and
copper, with a radius of 16 cm and a total length of 115 cm.

Figure 6: Placement of virtual detection planes for the SHINE beam dump
configuration study.

Given the tiny emittance of the SHINE linac’s electron beam, we approximated
it as a pencil beam to simplify our model. To analyze the muon yield at
various points around the beam dump, we installed five virtual detection
planes: front, left side, right side, back left side, and back right side, as
depicted in Fig. 6.

As anticipated, a substantial rate of muon and pion beams could still be
detected from the beam dump. The energy distribution of the muons and pions is
presented in Fig. 7, and the yield for each particle species is summarized in
Tab 1.

Figure 7: Energy distribution of the muons emitted from the beam dump, per bunch charge of 100 pC at SHINE. Table 1: Intensity of leptons and mesons produced from 8 GeV electrons-on-target. The numbers here are normalized to the bunch charge of 100 pC for SHINE. Location |  $\mu^{+}$/bunch |  $\mu^{-}$/bunch |  $\pi^{+}$/bunch |  $\pi^{-}$/bunch
---|---|---|---|---
Front | $3.1\times 10^{3}$ | $2.9\times 10^{3}$ | $8.9\times 10^{3}$ | $7.6\times 10^{3}$
Side (L) | $5.1\times 10^{3}$ | $3.6\times 10^{3}$ | $2.0\times 10^{5}$ | $1.9\times 10^{5}$
Side (R) | $5.0\times 10^{3}$ | $3.6\times 10^{3}$ | $2.0\times 10^{5}$ | $1.9\times 10^{5}$
Back (L) | $6.6\times 10^{2}$ | $7.5\times 10^{2}$ | $6.8\times 10^{3}$ | $1.0\times 10^{4}$
Back (R) | $6.5\times 10^{2}$ | $7.1\times 10^{2}$ | $6.6\times 10^{3}$ | $1.0\times 10^{4}$

##  V CONCLUSION

In this study, we analyzed the feasibility of leveraging the high-repetition-
rate electron beam at Shanghai’s SHINE facility to generate a muon source.
Through our simulations, we projected a muon yield of approximately
$10^{4}\mu^{\pm}$/bunch using a cylindrical tungsten target with a thickness
of 30 mm and a radius of 6 mm. In the beam dump configuration, we anticipate a
yield of $10^{3}\mu^{\pm}$/bunch or even greater from all sides of the SHINE
beam dump. Such a high-repetition-rate muon beam holds promise for diverse
muon experiments, from muon lifetime measurements Kanda:2022too  to the search
for muonium-to-anti-muonium conversion Bai:2022sxq , as well as the
application in muon spin spectroscopy studies Li:2023gxn . A dedicated study
on the beam extraction will be performed as the next step of our research.

##  VI ACKNOWLEDGEMENTS

We would like to thank Guanghong Wang, Wenzhen Xu, Jianhui Chen, and Dong Wang
from Shanghai Advanced Research Institute for useful discussions regarding the
feasibility of developing a muon beam line using SHINE beam dumps. This work
is supported by the Shanghai Pilot Program for Basic Research (21TQ1400221).

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