# Thermal laser evaporation of elements from across the periodic table

Thomas J. Smart  Jochen Mannhart  Wolfgang Braun<EMAIL_ADDRESS>Max
Planck Institute for Solid State Research,
Heisenbergstraße 1, 70569, Stuttgart

(August 27, 2024)

###### Abstract

We propose and demonstrate that thermal laser evaporation can be applied to
all solid, non-radioactive elements in the periodic table. By depositing thin
films, we achieve growth rates exceeding 1 Å/s with output laser powers less
than 500 W, using identical beam parameters for many different elements. The
source temperature is found to vary linearly with laser power within the
examined power range. High growth rates are possible using free-standing
sources for most of the elements tested, eliminating the need for crucibles.

thermal laser evaporation, film growth, epitaxy, thin films, deposition

Shortly after the invention of the laser in 1960Maiman_1960 , attempts were
made to synthesize thin films by using a laser to evaporate or ablate
materials.Smith:65 ; Groh  These initial experiments would go on to lay the
groundwork for Pulsed Laser Deposition (PLD):PLDWillmott  a successful vapor
deposition technique that ablates the surface of a source using a pulsed
laser.Schwarz ; pld  The advantages of using lasers for epitaxial growth are
numerous, including near arbitrary power densities, lack of source
contamination and increased efficiency due to the surface of the source being
directly illuminated by the laser as opposed to being indirectly and
inefficiently heated via a crucible.Smith:65 ; Hass:69  The use of lasers also
allows for small working distances, reducing the required source and chamber
sizes when compared to other epitaxy techniques, and correspondingly,
increasing material utilization.

Among the range of established thin film deposition methods,Capper2017
molecular beam epitaxy (MBE)MBE_Henini  may offer the advantage over PLD, e.g.
for the epitaxial growth of complex oxides,Martin_OxideEpitaxy  in that it
exploits adsorption-limited growth modes and the continuous variation of
material composition by varying fluxes from individual sources.

Recently, high-power lasers have become sufficiently affordable to be
practical for the generation of vapors from evaporating individual elemental
sources. The laser heating of such elemental sources forms the basis for
Thermal Laser Evaporation (TLE).braun  TLE aims to use laser beams in order to
evaporate a wide range of possible materials in the form of compact sources. A
co-evaporation from such sources would enable the epitaxial growth of
compounds as films. Substrate heating may be achieved by a heating
lasersubprep , as shown in Fig. 1. By varying the output power of each
evaporation laser, a correspondingly variable composition may be achieved.

Figure 1: Diagram of the proposed TLE chamber for epitaxial growth. Without
breaking the vacuum, both the substrate and the sources are exchangeable via
the use of a load-lock. A gas inlet allows for a gaseous atmosphere to be
present inside the chamber, which features an XHV background pressure. The
pale colored regions show the approximate flux distributions from the sources.

Investigating various extreme cases, we propose that successful evaporation
and deposition can be achieved via TLE for all solid, non-radioactive
elements. Most of these experiments were performed using the same laser, with
the same settings and parameters, varying only the output power. In addition,
we demonstrate how free-standing sources can be utilized for nearly all of
these elements and why the use of these sources is beneficial for epitaxial
growth.

## Experimental Apparatus

The experiments outlined within this paper have been conducted using the
simplified TLE chamber shown in Fig. 2. This chamber had no substrate heater
and only one source laser. The chamber itself was not cooled and was pumped
down to an ultra high vacuum (UHV) with internal pressures between 10-8 and
10-9 hPa. The cylindrical sources were free-standing with their long axes
vertically oriented. These sources rested on three Ta supports with a W-Re
(type C) thermocouple measuring the temperature at the bottom of the source.
The use of this thermocouple provided a large measurement range with an upper
limit at 2320 ∘C. At the same time, it served as a safety control for the
temperature of the bottom of the free-standing sources, in order to avoid
complete melting of the source. For elements that required a crucible, 12 mm
diameter Al_2O_3 or Ta crucibles were used, depending on the properties of the
element in question. The position of the source with respect to the laser beam
could be translated in all three spatial dimensions via a manipulator on which
the source was positioned.

Figure 2: Diagram of the test TLE chamber. This test chamber operates without
substrate heating.

For the dimensions of the elemental sources, we limited ourselves to cylinders
with a maximum diameter of 12.7 mm since materials with these dimensions are
readily available from many suppliers. The source heights along their axes
were 8 mm throughout unless explicitly stated. Deposition rates greater than
0.1 Å/s were deemed to be great enough to confirm successful deposition of the
corresponding element, based on typical growth rates that are suitably large
for many film growth applications.mbegrowthrates

For most of the elements tested, the source was irradiated by a
$\lambda=1030\,$nm fiber-coupled disk laser with a peak power of 2 kW which
was incident at 45∘ to its surface.Trumpf  For seven of these elements (Bi,
Ce, Sc, Cr, Ca, Yb and Mn), we switched to a $\lambda=1070\,$nm fiber
laserTrumpf  with a peak power of 500 W, as it turned out that the required
powers (see also Fig. 4) were consistently below 500 W. In all of these
experiments, the distance between the source and the focal point of the laser
was 62 mm; this produced a laser spot size of approximately 1 mm2 with an
approximately Gaussian beam profile. The laser optics possessed a focal length
of 500 mm, which remained fixed for all the experiments. Elemental films were
deposited on a horizontal 2” Si substrate, centered 60 mm vertically above the
source. Si substrates were used due to their low cost and ease of acquisition.
A quartz crystal microbalance (QCM) was applied to monitor the deposition from
the source on the substrate. The average growth rate was calculated after each
ten-minute deposition test. The growth rate was then checked by cleaving the
Si wafers along their diameter and measuring the thickness of the grown films
via scanning electron microscopy (SEM). This comparison allowed for the
calculation of the tooling factor to calibrate the deposition rate. We
estimate the uncertainty in the recorded growth rate to be less than the
marker size within the relevant figures of this paper. The source was
monitored during deposition with several CCD cameras.

In the experiment, we controlled the heating in two ways: firstly, most of the
experiments performed used power as an independent variable, wherein the laser
power was kept constant during each deposition test and the source temperature
was monitored via the thermocouple. For these experiments, the average source
temperature was measured over the time interval after the initial approach
where the thermocouple temperature was relatively stable. Secondly, the
temperature of the bottom of the source could also be PID-controlled. This
control used the thermocouple temperature as input and the laser power as
output which in turn ensured that the temperature of the bottom of the source
remained at a predefined, fixed value for each deposition test.

In order to minimize the effect of surface roughness and contaminants on the
required power to achieve a specific growth rate, an initial deposition was
performed with each source. This melted or sublimated possibly existing small
structures on the surface, yielding a steady-state surface morphology. In the
case of metals, the formation of a smooth surface could be easily observed by
the corresponding increase in reflectivity.

For all of the elements tested, no alteration to the laser parameters was
required for either the disc laser or the fiber laser. Only the output laser
power was changed.

Figure 3: Photographs of Si substrates coated with various elements. The
corresponding scanning electron microscopy images are shown below each wafer,
along with the average thickness of the deposited films. The grid patterns
appearing on the wafer surfaces result from optical reflections. The scale
shown represents 300 nm.

## Results

Thin films with thicknesses greater than 100 Å were grown with growth rates
exceeding 0.1 Å/s for all of the elements tested. Elements from different
parts of the periodic table could be easily deposited in a straightforward
manner as thin films, with particular success for the metals (see also Fig.
3). The films were dense, showing variations in thickness along the substrate
between 5 and 15 $\%$ for most of the elements tested, resulting in highly
reflective films. The results demonstrate the successful deposition of W, C
and Ta using thermal evaporation induced via a CW laser. Like other refractory
metals, these elements have previously required e-beam evaporation to
thermally deposit them due to their low vapor pressures and very high
evaporation temperatures. Carbon ; tungstenebeam ; tantalumebeam

Figure 4: Plot of growth rate as a function of inverse incident power for a
range of elements, accompanied by their associated fits to an Arrhenius-type
dependence. The values of the proportionality constant, $\beta$ (see Eqn. 2)
are indicated for each fit. For each element, the observed melting point is
represented by a colored symbol. Connecting lines act as a guide to the eye.

Different laser power values were required for different elemental sources, as
shown by Fig. 4. This figure shows the relation between growth rate and output
laser power for various elements, accompanied by their associated fits to an
Arrhenius-type dependence. The observed melting points for each material are
shown. These served as a calibration point for the temperature of the source.
For most of the elements tested, deposition rates exceeding 1 Å/s were
possible with laser outputs less than 500 W. By optimizing the source
dimensions, it is likely that this can be achieved for all elements in the
chosen chamber geometry. Assuming a wall-plug efficiency of 35%Trumpf , a
maximum total power requirement of 1.4 kW is obtained for a growth rate of 1
Å/s. This power requirement is similar to the corresponding requirements of
many sputtering or e-beam systems.

Figure 5: Photograph of a free-standing Si source (12.7 mm in diameter and 4
mm thick) during deposition. A liquid phase can clearly be seen on the surface
of the source, indicating a surface temperature of at least 1687 K.vapour
During this deposition test, the average deposition rate was 5.2 Å/s,
resulting from 400 W supplied by the laser. Figure 6: Photograph of the side
of a free-standing Ta source (3 mm in diameter and 10 mm in height) during
deposition. A clear intensity gradient can be seen along the (vertical) axis
of the source, with the temperature difference between the top and the bottom
of the source being $\sim$ 2000 K as determined from the observation of liquid
Ta on the top surface and a thermocouple measurement on the bottom surface.
During this deposition test, the average deposition rate was 0.08 Å/s,
resulting from 280 W supplied by the laser.

We were able to evaporate most of the elements from free-standing cylindrical
sources, with localized laser heating of the source. As a result, small
regions of the source can be melted whilst the rest of the source remains
solid, as shown in Fig. 5. The source then acts as its own crucible, which
offers major advantages. The ductility of metals, combined with the absence of
a mismatch in the thermal expansion between the ’crucible’ and the ’contents’
avoids the issue of crucible failure due to thermally induced strain. The lack
of a crucible also eliminates reactions between the source and the crucible,
since the solid-liquid boundary remains inside the source. This avoids
contaminants generated from the crucible in the deposited film.

Owing to the strong absorption of light with a wavelength around 1000 nm, the
laser heating is not only highly localized via the small beam diameter, but
also along the axis of the source. The resulting absorption corresponds to a
penetration depth on the order of 2 nm for most metals with absorption
coefficients; $\alpha\sim 10^{5}$ cm-1. Due to the combination of heat loss
via conduction and radiation, the high temperature region is found close to
the irradiated surface of the source. Figure 6 shows a side view of a Ta
source during deposition, with a diameter of 3 mm and a height of 10 mm.

In this case, a temperature difference of around 2000 K was observed between
the top surface and the thermocouple touching the bottom of the source. Liquid
Ta was present on the top surface, indicating the temperature exceeding 3290
Kvapour , as denoted in Fig. 6. A temperature of approximately 1200 K was
measured by the thermocouple on the bottom of the source. As a result of the
localized heating from the source laser, the remainder of the source remained
solid and the temperature of the bottom of the source did not exceed the range
of operation for the thermocouple. Depending on the thermal conductivity of
the thermocouple contact to the source, inaccurate thermocouple temperature
readings are expected at high temperatures. Given that vapor pressure is
highly dependent on temperature and proportional to the growth rate braun ,
localized heating of the source also ensured that evaporation from the laser
spot dominated the overall growth rate.

Figure 7: Photograph of a Ti source after approximately 300 nm of Ti was
deposited on the substrate. The surface has melted, generating a dip at the
location of the laser spot. The structural integrity of the source has
remained intact.

Pulsing the laser, while still remaining in the thermal evaporation regime,
may further increase thermal gradients, allowing further concentration of the
laser power to the evaporation region.prokhorov1990laser  The experiments were
nevertheless carried out under steady-state conditions, because the thermal
transients at the onset and end of the pulses affect the reproducibility.
Further jitter in the pulse duration adversely affects the stability, adding
additional noise to the steady-state (in)stability of the laser.

Even at high growth rates, the structural integrity of the free-standing
source may be maintained. For the case of a Ti source, 3 mm in diameter (see
Fig. 7), we observed that during deposition, the melt-pool was flat and could
be increased in size until it was near the rim of the source, the remnant of
which can still be seen in Fig. 7. However, upon cooling, a dip formed at the
location of the laser spot, which we attribute to the contraction of Ti due to
rapid cooling at the position of the laser spot, whilst solidifying inwards
from the rim of the melt pool. By reheating and melting the source, the flat
surface was recovered.

Some elements, such as Ge and Pr, required a crucible in order to achieve
growth rates greater than 0.1 Å/s. Whilst (in principle), a free standing
source could be used for these elements too, our deliberate size restriction
to below 12.7 x 8 mm for the current experiments, did not allow a large enough
source for free-standing evaporation of these elements.

## Discussion

It is well-established within the field of film growth that vapor pressure,
$p$, follows the Clausius-Clapeyron equation:

$p\sim e^{-\frac{\Delta H}{RT}},$ (1)

where $\Delta H$ is the enthalpy change between two desired phases and $R$ is
the gas constant. When this equation is plotted on an Arrhenius plot as a
function of inverse temperature, a linear fit is obtained. The growth rate (or
evaporation rate) of an element at a specific temperature is proportional to
the vapor pressure.Langmuir  Since the vapor pressure is strongly dependent on
temperature, the growth rate should be primarily dominated by evaporation from
the laser spot. However, the current experimental setup makes measuring
temperature of the source at the laser spot at any temperature other than the
melting point extremely difficult. Therefore these experiments measured the
growth rate as a function of the output laser power; a variable which is
comparatively easy to measure. Surprisingly, if one plots this growth rate
data on an Arrhenius plot as a function of inverse laser power, as shown in
Fig. 4, the experimental data follows a straight line to a remarkable degree.

This result suggests that the growth rate as a function of inverse power
follows an equation of the form:

$R_{G}\sim e^{-\frac{\beta}{P}},$ (2)

where $R_{G}$ denotes the growth rate, $P$ is the output laser power and
$\beta$ the proportionality constant. Given the prior relation between growth
rate, $R_{G}$, and vapor pressure, $p$, an equivalence can be drawn between
Eqn. 2 and the Clausius-Clapeyron equation. This leads to the interesting
deduction that the relation between incident laser power and peak source
temperature is approximately linear. The proportionality constant from the
$\beta$ values in Fig. 4 can be determined to:

$\beta\sim\frac{\Delta H}{Ra},$ (3)

where $a$ is the constant of proportionality between peak source temperature
and output laser power. We can use the Arrhenius fits obtained in Fig. 4 to
obtain values of $a$ for each source material and geometry. These values,
along with their associated source diameters, are shown in Table I.

Element |  Source Diameter (mm) |  $a$ (W/K)
---|---|---
W |  2 | 0.0127
Zr |  12.7 | 0.005
Ta |  3 | 0.0109
Re |  3 | 0.012
Rh |  5 | 0.004
Mo |  2 | 0.007
Ir |  3 | 0.008
Table 1: Calculated $a$ values from the Arrhenius fits in Fig. 4. The
corresponding source diameters are also shown.

An explicit model of this behaviour would have to account for phase changes,
as these would affect the temperature distribution throughout the source
resulting from convection inside the liquid phase, and the loss of energy due
to vaporisation.

The success we have achieved with our experimental results raises the question
whether thermal laser evaporation can be applied to any solid, non-radioactive
element in the periodic table. To explore this question, we specifically
investigated the growth of those elements with extreme growth parameters. For
example, being able to readily evaporate W, which has the lowest vapor
pressure of any element, and a melting temperature of 3695 Kvapour  indicates
that TLE is not limited by the evaporation temperature. Similarly, successful
evaporation of Cu, Ag and Au, all of which exhibit very high reflectivity
values at 1030 nmnoble  indicates that reflectivity does not present a barrier
to TLE. High and low values of thermal conductivity also pose no issue, as we
were able to successfully evaporate both Ag and S, respectively. Therefore, we
conclude that TLE can be applied to all solid, non-radioactive elements in the
periodic table. Our current experimental state is shown in Fig. 8, with the
deposition of 43 elements being experimentally confirmed.

## Conclusion

We conclude that Thermal Laser Evaporation is a powerful technique to grow any
solid, non-radioactive elements as thin films. We have confirmed this
experimentally for 43 elements by using the same laser beam geometry. Due to
the localized heating induced by the incident source laser, most of these
elements could be deposited using free-standing sources. Some of the
investigated elements required a crucible, as determined by the chamber
geometry. Even for elements like W, C and Ta, deposition rates exceeding 1 Å/s
were achievable with laser powers less than 500 W, with appropriately
optimized source dimensions.

Figure 8: Periodic table showing in dark green the range of elements that in
this work have been successfully deposited at substantial growth rates using
free-standing sources and crucibles. Elements that have been grayed out are
radioactive.

Whilst our focus here has been on film growth, one could easily foresee an
application of this configuration in other areas of physics as well, such as
the generation of atomic vapors for cold ion traps or for spectroscopic
purposes.

## Acknowledgments

The authors thank Hans Boschker, Dong Yeong Kim and Sander Smink for many
valuable discussions and support. In addition, the authors thank Fabian
Felden, Ingo Hagel, Konrad Lazarus, Sabine Seiffert and Wolfgang Winter for
technical assistance.

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# Unconventional optical selection rules in ZrTe5 under an in-plane magnetic
field

Yi-Xiang Wang<EMAIL_ADDRESS>School of Science, Jiangnan
University, Wuxi 214122, China  School of Physics and Electronics, Hunan
University, Changsha 410082, China  Fuxiang Li<EMAIL_ADDRESS>School of
Physics and Electronics, Hunan University, Changsha 410082, China

###### Abstract

The optical selection rules of an electron system under a magnetic field play
key roles in determining its optical properties, from which the band
structures and underlying symmetries can be derived. In this paper, based on a
three-dimensional strong topological insulator model describing ZrTe5, we
study the Landau levels (LLs) and magneto-optical conductivity under an in-
plane magnetic field. We reveal that in the transverse conductivity
Re$(\sigma_{zz})$, the unconventional optical selection rules $n\rightarrow
n\pm 2$ dominate, with $n$ being the LL index. We attribute the unconventional
selection rules to the peculiar distribution of parity carried by the LLs,
resulting from the chiral symmetry of the sub-Hamiltonians. Moreover, we
predict that, if the strong anisotropic system is tuned to be nearly
isotropic, the LLs would redistribute and the conventional selection rules
$n\rightarrow n\pm 1$ can be recovered.

##  I Introduction

Three-dimensional (3D) zirconium pentatelluride ZrTe5 is a representative of
emerging topological materials and has aroused researchers’ ongoing interests
in recent years Lv . Its nontrivial band topology Weng  together with the
intrinsic dynamics of the Dirac fermion can bring out a lot of unconventional
phenomena, such as the chiral magnetic effect Q.Li , the anomalous Hall effect
T.Liang ; Y.Liu , the 3D quantum Hall effect F.Tang , and the anomalous
thermoelectric effect J.L.Zhang2019 ; W.Zhang . Although there is no consensus
on the ground state of ZrTe5 in experiment, recent measurements of angle-
resolved photoelectron spectroscopy Manzoni , infrared spectroscopy Z.G.Chen ;
B.Xu , magnetoinfrared spectroscopy Y.Jiang2020  as well as magneto-transport
J.Wang  supported its ground state to be a strong topological insulator (TI)
at low temperatures, in which the characteristic linear surface states Manzoni
and the bulk band inversions have been clearly identified Z.G.Chen ; B.Xu ;
Y.Jiang2020 ; J.Wang .

In 3D topological Weyl/Dirac semimetal materials, applying a strong magnetic
field can lead to the formation of Landau levels (LLs) dispersing along the
wave vector parallel to the field. When combining the magnetic field with
scanning tunneling spectroscopy or infrared spectroscopy, they can provide
powerful ways to study the bulk electronic structure and band topology Jeon ;
X.B.Li . Among them, the optical selection rules play key roles in determining
the basic optical properties of the electron system Akrap ; R.Y.Chen ;
E.Martino . In a 3D ZrTe5 crystal, prismatic ZrTe3 chains run along the
crystallographic $a$ axis ($x$ axis) and link along the $c$ axis ($y$ axis)
with the zigzag Te atom chains, which form two-dimensional (2D) layers. Via
weak van-der Waals bondings, the 2D layers stack along the $b$ axis ($z$ axis)
to form a 3D layered orthorhombic structure. In a recent magnetoinfrared
spectroscopy experiment of ZrTe5 Y.Jiang2020 , the magnetic field was applied
not only along the perpendicular $z$ direction, but also in the $x$-$y$ plane.
Since the LLs and magneto-optics under an in-plane magnetic field remain
unexplored in quantum theory, this motivates the present work.

In this paper, based on a strong TI model describing ZrTe5, we study the LLs
and magneto-optical conductivity Re$(\sigma_{\alpha\alpha})$ under an in-plane
magnetic field. By performing numerical calculations in the lattice model, we
find that the zeroth LLs show distinctive behaviors from the $n\geq 1$ LLs,
leading to different resonant peak numbers in Re$(\sigma_{\alpha\alpha})$,
which are consistent with experimental observations Y.Jiang2020 . In the
transverse conductivity Re$(\sigma_{zz})$, we reveal the existence of
unconventional optical selection rules $n\rightarrow n\pm 2$ and attribute
them to the peculiar distribution of parity carried by the LLs. Such a
distribution is shown to be induced by the chiral symmetry of the sub-
Hamiltonians with quadratic terms. Moreover, we predict that, if a strong
anisotropic system is tuned to be nearly isotropic under an external pressure,
the LLs will redistribute and the conventional selection rules $n\rightarrow
n\pm 1$ can be recovered. Our findings will broaden the understanding of
exotic quantum phenomena in topological materials.

##  II Model

We start from the strong TI model of ZrTe5 that was proposed in a recent
magnetoinfrared spectroscopy experiment Y.Jiang2020 . The corresponding low-
energy Hamiltonian can be written as $(\hbar=1)$ Y.Jiang2020 ; H.Zhang ;
C.X.Liu ; Wang2021 ; L.You

$\displaystyle H(\bm{k})$
$\displaystyle=v(k_{x}\sigma_{z}\otimes\tau_{x}+k_{y}I\otimes\tau_{y})+v_{z}k_{z}\sigma_{x}\otimes\tau_{x}$
$\displaystyle+[M-\xi(k_{x}^{2}+k_{y}^{2})-\xi_{z}k_{z}^{2}]I\otimes\tau_{z},$
(1)

where $\sigma$ and $\tau$ are the Pauli matrices acting on the spin and orbit
degrees of freedom, respectively. $v$ and $v_{z}$ are the Fermi velocities,
$\xi$ and $\xi_{z}$ are the band inversion parameters, and $M$ denotes the
Dirac mass. Consider an in-plane magnetic field, $\bm{B}=B\bm{e}_{x}$. To
incorporate it in the system, we use the Peierls substitution,
$\bm{k}\rightarrow\bm{k}-e\bm{A}$, where the magnetic vector potential is
chosen as $\bm{A}=By\bm{e}_{z}$ in the Landau gauge. In the following
calculations, unless specified, we take the model parameters as in Ref.
Y.Jiang2020 : $(v,v_{z})=(6,0.5)\times 10^{5}$ m/s, $(\xi,\xi_{z})=(100,200)$
meV nm2, and $M=7.5$ meV.

##  III Main Results

###  III.1 LLs

To solve the LLs, we construct the tight-binding Hamiltonian in the lattice
model [see Sec. I of the Supplemental Material (SM) SM ]. By diagonalizing the
magnetic unit cell, the LL dispersions can be obtained and are shown in Fig.
1. The index $ns\lambda$ are used to label each LL, with $s=\pm 1$ denoting
the conduction/valence band and $\lambda=\pm 1$ the two branches.

Figure 1(a) shows that for the zeroth LLs, the two branches are well separated
in energy, while for the $n\geq 1$ LLs, the two branches are almost
degenerate. At strong magnetic field, all LLs increase with a relation that
can be fitted by the power formula $\varepsilon=(aB+c)^{d}$ as shown in Fig.
1(b). For the lowest LL, the exponent $d=0.80$, whereas for other LLs, the
exponents are close to $\frac{2}{3}$, both of which are distinct from the
characteristic exponent $d=\frac{1}{2}$ of the LLs in a 3D linear Dirac model
Armitage . We note that the lowest $(0+-)$ LL shows anomalous behaviors with
the magnetic field: It first decreases, reducing to zero at a critical
$B^{c}$, and then increases. This anomalous behavior will be analyzed later.

Figure 1: (Color online) The LL dispersions in (a), with magnetic field
$B=10.3$ T. The LL energy $\varepsilon_{ns\lambda}$ at $k_{x}=0$ vs $B$ in
(b). The solid lines are the fittings to the LLs by the power function
$\varepsilon=(aB+c)^{d}$ for $B>10$ T. From $(0+-)$ to $(3++)$ LL, the
exponent $d$ is extracted as $d$=0.80, 0.66, 0.69, 0.68, 0.68, 0.68, 0.68,
0.68.

###  III.2 Magneto-optical Conductivity

The LL structure can be probed by magneto-optical measurements. Within the
linear-response theory, the magneto-optical conductivity
$\sigma_{\alpha\alpha}$ is calculated by using the Kubo formula M.Udagawa ,

$\displaystyle\sigma_{\alpha\alpha}(\omega)=$
$\displaystyle\frac{1}{iV}\sum_{nn^{\prime}}\sum_{ss^{\prime}}\sum_{\lambda\lambda^{\prime}}\frac{f(\varepsilon_{ns\lambda})-f(\varepsilon_{n^{\prime}s^{\prime}\lambda^{\prime}})}{\varepsilon_{ns\lambda}-\varepsilon_{n^{\prime}s^{\prime}\lambda^{\prime}}}$
$\displaystyle\times\frac{|\langle\psi_{ns\lambda}|J_{\alpha}|\psi_{n^{\prime}s^{\prime}\lambda^{\prime}}\rangle|^{2}}{\omega+\varepsilon_{ns\lambda}-\varepsilon_{n^{\prime}s^{\prime}\lambda^{\prime}}+i\tau^{-1}},$
(2)

where $V$ is the volume of the system, $f(x)$ is the Fermi-Dirac distribution
function, $J_{\alpha}=-ie[r_{\alpha},H]$ is the current density operator,
$\alpha=x,z$ is the oscillating direction of the electric field $\bm{E}$, and
$\frac{1}{\tau}$ denotes the linewidth broadening induced by impurity
scattering Wang2021 ; M.Udagawa ; P.Hosur ; Ashby ; Duan2019 . We choose the
Fermi energy at zero energy and set the temperature to be zero, so the index
$s=-1$ and $s^{\prime}=1$. For the linearly polarized light acting on the
system, the optical absorption is proportional to the real part of
$\sigma_{\alpha\alpha}$.

Figure 2: (Color online) The magneto-optical conductivity
Re($\sigma_{\alpha\alpha}$) (in units of $\sigma_{0}=\frac{e^{2}}{2\pi}$) vs
the photon frequency $\omega$ for different $\tau^{-1}$, with $\alpha=x$ in
(a) and $\alpha=z$ in (b). The LL transitions are labeled for each resonant
peak, where a weak peak is indicated by the arrow. The legends are the same in
both figures. We take the same parameters as those in Fig. 1(a).

The numerical results of the magneto-optical conductivity
Re$(\sigma_{\alpha\alpha})$ are plotted in Fig. 2 for different strengths of
linewidth broadening. In the limit of small $\tau^{-1}$ (e.g., 1 meV), one
observes a series of resonant peaks sitting on an increasing background Ashby
, which is due to the dispersive LLs in the 3D system. When $\tau^{-1}$ is
strong, e.g., $\tau^{-1}=20$ meV, the system enters into the diffusive
metallic regime, and the resonant peaks disappear.

The resonant peaks occur at the photon frequency
$\omega=\varepsilon_{n^{\prime}s^{\prime}\lambda^{\prime}}-\varepsilon_{ns\lambda}$
at the van Hove singularity $k_{x}=0$. The corresponding selection rules can
thus be determined for each resonant peak by explicitly matching the energy
difference between the initial and final LLs. In Fig. 2, we carefully label
each LL transition, and observe that the selection rules are $n\rightarrow n$
in the longitudinal conductivity Re$(\sigma_{xx})$, and $n\rightarrow n\pm 2$
in the transverse conductivity Re$(\sigma_{zz})$. The former selection rules
are the same as previous magneto-optic studies of Dirac fermions when
$\bm{E}\parallel\bm{B}$ L.You ; Mukherjee ; Wang2020 ; Duan2020 , whereas the
latter ones are distinctively different from the previous studies of Dirac
fermions when $\bm{E}\perp\bm{B}$, with conventional selection rules given as
$n\rightarrow n\pm 1$ L.You ; Ashby ; Duan2019 ; Mukherjee ; Wang2020 ;
Duan2020 ; R.Y.Chen ; Z.G.Chen ; Y.Jiang2017 ; Wang2017 .

It is interesting to note that, due to the broken twofold degeneracy of the
zeroth LLs, the zeroth LL transitions show two different resonant peaks both
in Re$(\sigma_{xx})$ and Re$(\sigma_{zz})$. In addition, in Re$(\sigma_{zz})$,
there exists a weak peak with the LL transition $0s\lambda\rightarrow
0s^{\prime}\bar{\lambda}$, as indicated by the arrow.

Figure 3: (Color online) The LLs vs the magnetic field $B$, with
$\varepsilon_{ns}^{a}$ of $H_{aB}^{\prime}$ in (a) and $\varepsilon_{ns}^{b}$
of $H_{bB}^{\prime}$ in (b). The even-(odd-)parity LLs are labeled as solid
(dashed) lines. The arrows indicate the dominated LL transitions in
Re$(\sigma_{zz})$, with the red dashed one related to the weak peak in Fig.
2(b). The matrix element
$\langle\varphi_{n-}^{a(b)}|J_{za(b)}^{\prime}|\varphi_{n^{\prime}+}^{a(b)}\rangle$
between the different initial states $\varphi_{n-}^{a(b)}$ and final states
$\varphi_{n^{\prime}+}^{a(b)}$ in (c), (d). The magnetic field is set as
$B=10.3$ T. For clarity, the neighboring bars are shifted vertically by
$2\times 10^{5}$.

###  III.3 Selection Rules

To understand the behavior of the magneto-optical conductivity and especially
the unconventional selection rules, it would be instructive to consider the
Hamiltonian $H(\bm{k})$ at the van Hove singularity $k_{x}=0$. At this point,
$H(\bm{k})$ can be block diagonalized through a unitary matrix
$U=\frac{1}{\sqrt{2}}(\sigma_{x}+\sigma_{z})\otimes\tau_{z}$:

$\displaystyle H^{\prime}=UH(k_{x}=0)U^{\dagger}=H_{a}^{\prime}\oplus
H_{b}^{\prime},$ (3)

where the sub-Hamiltonians $H_{a}^{\prime}$ and $H_{b}^{\prime}$ are written
as

$\displaystyle H_{a,b}^{\prime}=\mp v_{z}k_{z}\tau_{x}-vk_{y}\tau_{y}+(M-\xi
k_{y}^{2}-\xi_{z}k_{z}^{2})\tau_{z}.$ (4)

Clearly, $H^{\prime}$ owns the chiral symmetry as ${\cal C}H^{\prime}{\cal
C}^{\dagger}=-H$ Asboth , with the operator ${\cal
C}=\sigma_{x}\otimes\tau_{x}$. If the anisotropy is absent, $\xi=\xi_{z}$ and
$v=v_{z}$, $H_{a}^{\prime}$ and $H_{b}^{\prime}$ represent the 2D Chern
insulator models that are expanded to the second-order $k_{i}^{2}$ terms; the
total Hamiltonian $H^{\prime}$ is equivalent to the low-energy effective model
that describes the 2D surface states in the magnetic TI film R.Yu .

We introduce the ladder operators
$a=\frac{1}{\sqrt{2}}(\eta+\frac{\partial}{\partial\eta})$ and
$a^{\dagger}=\frac{1}{\sqrt{2}}(\eta-\frac{\partial}{\partial\eta})$, with the
dimensionless parameter $\eta=l_{B}k_{z}-\frac{y}{l_{B}}$ and the magnetic
length $l_{B}=\frac{1}{\sqrt{eB}}$, so then $H_{a}^{\prime}$ becomes

$\displaystyle H_{aB}^{\prime}=\begin{pmatrix}P-\tilde{Q}&Ta^{\dagger}-Sa\\\
Ta-Sa^{\dagger}&-P+\tilde{Q}\end{pmatrix},$ (5)

with $\tilde{Q}=Qa^{\dagger}a-\frac{R}{2}(a^{2}+a^{\dagger 2})$.
$H_{bB}^{\prime}$ can be obtained from $H_{aB}^{\prime}$ by exchanging $S$ and
$T$. Here the parameters are defined as $P=M-\frac{\xi+\xi_{z}}{2l_{B}^{2}}$,
$(Q,R)=\frac{\xi\pm\xi_{z}}{l_{B}^{2}}$, $(S,T)=\frac{v\pm
v_{z}}{\sqrt{2}l_{B}}$. $H_{a/bB}^{\prime}$ can be diagonalized in the Hilbert
space spanned by $|n\rangle$ that is defined as
$a^{\dagger}a|n\rangle=n|n\rangle$ M.Koshino ; T.Devakul . In the
calculations, one can truncate the Hilbert space at a cutoff $N_{c}$, which is
set as $N_{c}=200$ (see Sec.IV of SM SM ).

Figure 4: (Color online) The LL flowing with $v_{z}$ in (a). The black (red)
lines represent $\varepsilon_{ns}^{a(b)}$, and the solid (dashed) lines denote
the even-(odd-)parity LLs. The inset shows the energy differences,
$\Delta_{1}$ and $\Delta_{2}$. The matrix element
$\langle\varphi_{n-}^{a(b)}|J_{za(b)}^{\prime}|\varphi_{n^{\prime}+}^{a(b)}\rangle$
with $v_{z}$ in (b), (c), where the initial and final state index, $n$ and
$n^{\prime}$, as well as the selection rules are labeled in different regions.
The critical velocity $v_{z}^{c_{1}}$ is the same as that in (a), and
$v_{z}^{c_{2a(b)}}$ marks the point where the matrix element inversion occurs.
We set the magnetic field as $B=10.3$ T and mark the experimental value
$v_{z}=0.5\times 10^{5}$ m/s Y.Jiang2020  with asterisks.

The LL energies of $H_{aB}^{\prime}$ and $H_{bB}^{\prime}$ are presented as a
function of $B$ in Figs. 3(a) and 3(b), respectively, in which
$\varepsilon_{ns}^{a}$ together with $\varepsilon_{ns}^{b}$ agree well with
the results in Fig. 1(b) by a lattice model calculation. We observe that
particle-hole symmetry is broken in $H_{a(b)B}^{\prime}$, but is still
preserved in the total Hamiltonian $H_{aB}^{\prime}\oplus H_{bB}^{\prime}$.
This can be seen from the relation
$\tau_{x}H_{aB}^{\prime}\tau_{x}=-H_{bB}^{\prime}$. Indeed, this relation
guarantees that the energies and wavefunctions of the two sub-Hamiltonians are
related: $\varepsilon_{ns}^{a}=-\varepsilon_{n\bar{s}}^{b}$ and
$\varphi_{ns}^{a}=\tau_{x}\varphi_{n\bar{s}}^{b}$, with $\bar{s}=-s$.
Actually, in Fig. 1(b), the $(n+-)$ and $(n++)$ LLs with $n>0$ are equivalent
to the $n+$ LLs in $H_{aB}^{\prime}$ and $H_{bB}^{\prime}$, respectively.

With the same unitary matrix $U$, the current density operator matrix at
$k_{x}=0$ is transformed as

$\displaystyle J_{x}^{\prime}=UJ_{x}U^{\dagger}=-ev\sigma_{x}\otimes\tau_{x},$
(6) $\displaystyle J_{z}^{\prime}=UJ_{z}U^{\dagger}=J_{za}^{\prime}\oplus
J_{zb}^{\prime},$ (7)

with

$\displaystyle J_{za,b}^{\prime}=\mp
ev_{z}\tau_{x}-\frac{\sqrt{2}e\xi_{z}}{l_{B}}(a+a^{\dagger})\tau_{z}.$ (8)

As $J_{x}^{\prime}$ in Eq. (6) is anti-diagonal, its nonzero matrix element
must appear between the states of different sub-Hamiltonians. Since
$\langle\varphi_{n+}^{a(b)}|J_{x}^{\prime}|\varphi_{n^{\prime}-}^{b(a)}\rangle=-ev\delta_{nn^{\prime}}$,
the selection rules must be $n\rightarrow n$ and $\lambda^{\prime}=\lambda$.

For $J_{z}^{\prime}$ in Eq. (7), since it is block diagonal, its nonzero
matrix element must appear between the states of the same sub-Hamiltonians.
Now each sub-Hamiltonian $H_{a(b)B}^{\prime}$ owns the parity symmetry with
the operator ${\cal P}_{a}=(-1)^{a^{\dagger}a}\tau_{z}$ commuting with
$H_{a(b)B}^{\prime}$ T.Devakul . Therefore, each LL owns definite even or odd
parity, as labeled in Figs. 3(a) and 3(b) by solid or dashed lines,
respectively. We can see that the neighboring LLs always own opposite
parities. On the other hand, as ${\cal P}_{a}J_{za(b)}^{\prime}{\cal
P}_{a}=-J_{za(b)}^{\prime}$, $J_{za(b)}^{\prime}$ owns odd parity and its
matrix element
$\langle\varphi_{n-}^{a(b)}|J_{za(b)}^{\prime}|\varphi_{n^{\prime}+}^{a(b)}\rangle$
is nonvanishing only when the $n-$ and $n^{\prime}+$ states own opposite
parities (see Sec. V of SM SM ). Thus due to the peculiar distribution of
parity carried by the LLs, the nonvanishing matrix element requires
$n^{\prime}-n=2l$, with $l$ being the integer. This is also demonstrated by
the numerical results in Figs. 3(c) and 3(d). We see that the matrix elements
take large values only when $n^{\prime}=n\pm 2$, and decrease quickly to zero
when $n^{\prime}$ is away from $n$. Therefore one can determine that the
selection rules in Re$(\sigma_{zz})$ are $n\rightarrow n\pm 2$ and
$\lambda^{\prime}=-\lambda$.

###  III.4 Anisotropy

As 3D ZrTe5 is highly anisotropy, we investigate its role in forming the LLs
and selection rules by changing the Fermi velocity $v_{z}$. Experimentally,
tuning $v_{z}$ is quite feasible, because it represents the hopping strength
in the $z$ direction and is expected to increase when the external pressure is
applied along the $z$ direction of the 3D crystal Y.Zhou ; J.L.Zhang2017 . In
Fig. 4(a), the total LLs are plotted with $v_{z}$, with the even-(odd-) parity
LLs denoted by solid (dashed) lines. One observes a transition in the
distribution of parity in the LLs as increasing $v_{z}$, which can be
understood through the chiral symmetry of the sub-Hamiltonian as follows.

In the limit $v_{z}=0$, we have $S=T$ and the sub-Hamiltonian
$H_{a(b)B}^{\prime}$ owns the chiral symmetry
$\tau_{x}H_{a(b)B}^{\prime}\tau_{x}=-H_{a(b)B}^{\prime}$. The chiral symmetry
guarantees that each sub-Hamiltonian has two zeroth LLs but with opposite
parities. When $v_{z}$ increases from zero, the chiral symmetry of the sub-
Hamiltonian is broken, but the two zeroth LLs still persist, until $v_{z}$
reaches a critical value. After $v_{z}$ crosses the critical value, one of the
two zeroth LLs flows away from its partner, and becomes an $n=1$ LL. Thus we
are left with only one single zeroth LL for each sub-Hamiltonian, leading to
the $n+$ and $n-$ LLs carrying the same parities. If the system is nearly
isotropic, $v_{z}\sim v$, we have $S\gg T,\tilde{Q}$ and the sub-Hamiltonian
$H_{a/bB}^{\prime}$ is analogous to that in graphene Z.Jiang ; T.Morimoto ;
M.O.Goerbig  and other Dirac electron systems C.J.Tabert , .

To quantitatively characterize the transition, we define $\Delta_{1}$ and
$\Delta_{2}$ as the energy differences between the neighboring LLs, as
indicated in Fig. 4(a). The inset of Fig. 4(a) presents the relative magnitude
of $\Delta_{1}$ and $\Delta_{2}$, in which $\Delta_{1}=\Delta_{2}$ occurs at
$v_{z}=v_{z}^{c}$. This critical $v_{z}^{c}$ characterizes when the two zeroth
LLs change to be one zeroth LL. The transition also exists in higher $n>0$ LLs
(see Sec. VII of SM SM ). Explicitly, when $v_{z}$ increases, the $ns+$ LL
flows to the $(n+1,s-)$ LL, with a direct consequence being that the index of
the two LLs should be relabeled as $(n+1,s-)$ and $(n+1,s+)$, respectively.

Next, we turn to the selection rules in Re$(\sigma_{zz})$. In Fig. 4(b), for
the matrix element
$\langle\varphi_{n-}^{a}|J_{za}^{\prime}|\varphi_{n^{\prime}+}^{a}\rangle$,
when $v_{z}\ll v$, the initial and final state index are chosen as $n=2$ and
$n^{\prime}=0,2,4$, respectively. Clearly the matrix element with
$n^{\prime}=0,4$ takes a relatively large value, indicating the unconventional
selection rules $n\rightarrow n\pm 2$. When $v_{z}$ increases to be larger
than $v_{z}^{c_{2a}}$, the matrix element with $n^{\prime}=2$ overwhelms that
with $n^{\prime}=0$, thus the selection rules become $n\rightarrow n$ and
$n\rightarrow n+2$. Further increasing $v_{z}$ to cross $v_{z}^{c}$, the LLs
regroup and the initial state index will be increased by one but the final
state index remains unchanged. Consequently, the parity of the $n-$ LL
changes, while the parity of the $n^{\prime}+$ LL remains unchanged, as seen
in Fig. 4(a). Therefore, the conventional selection rules $n\rightarrow n\pm
1$ are recovered. A similar change of the selection rules can also be found
for the matrix element
$\langle\varphi_{n-}^{b}|J_{zb}^{\prime}|\varphi_{n^{\prime}+}^{b}\rangle$ in
Fig. 4(c) and those with a higher index (see Sec. VIII of SM SM ).

###  III.5 Zeroth LLs versus the magnetic field

As analyzed above, zeroth LLs play an important role in forming the
unconventional selection rules in ZrTe5. Here, we make a detailed study of the
anomalous behavior of the zeroth LLs with the magnetic field $B$. In Figs.
3(a) and 3(b), when $B$ increases, the $0-$ ($0+$) LLs in $H_{a(b)B}^{\prime}$
changes its sign, and crosses the zero energy at the same critical magnetic
field $B^{c}=5.24$ T. This can be understood from the effective Dirac mass
term $P$ of $H_{aB}^{\prime}$ in Eq. (5). The “mass” $P$ consists of the
original Dirac mass $M$ and a part proportional to an external magnetic field.
For the case of $M>0$, the effective mass $P$ changes from positive to
negative with increasing magnetic field, leading to the sign inversion of the
lowest LL. However, for the case of $M<0$, there would be no sign inversion of
the lowest LLs. Therefore, the intercept of the lowest LLs with the zero
energy is closely connected to the bulk band inversion, which gives an
important signature of a strong TI phase in ZrTe5. In a recent thermoelectric
effect J.L.Zhang2019  and another magnetoinfrared spectroscopy study Z.G.Chen
, the intercept of the lowest LLs in ZrTe5 with zero energy was also
demonstrated, but under a perpendicular magnetic field. The physical
mechanisms were both attributed to the strong Zeeman splitting, with the
critical magnetic field estimated to be around 13 T J.L.Zhang2019  and 17 T
Z.G.Chen , much higher than the present study. More discussions regarding the
zeroth LLs are presented in Secs. II and III of SM SM .

##  IV Discussions and Conclusions

In this paper, by performing full quantum mechanical calculations, we study
the magneto-optics in 3D ZrTe5 under an in-plane magnetic field and reveal the
unconventional selection rules $n\rightarrow n\pm 2$ in Re$(\sigma_{zz})$. It
should be mentioned that, several studies have been performed for 3D Weyl
semimetals under an in-plane magnetic field, and it was found that the chiral
anomaly can engender the planar Hall effect A.A.Burkov ; S.Nandy ; S.Ghosh .
However, these studies were mainly in the semiclassical framework, and thus
quantum effects are not fully uncovered. We also mention that, in the optical
absorption spectra of MoS2 F.Rose  and the photocurrent of bilayer graphene
L.Ju , similar unconventional selection rules were reported, which were
attributed to a quite different mechanism, the high-order trigonal warping
effect. Moreover, in their studies F.Rose ; L.Ju , the unconventional
selection rules were only connected to very weak peaks and thus are difficult
to be observed in experiments.

For the electron-electron correlations, it is estimated that within the mean-
field framework F.Qin , when the magnetic field is above a critical value, the
correlations can open up a gap with a magnitude of meV around the Fermi level.
However, the parity of the LLs will not be changed by such a gap opening, and
thus the selection rules would not be affected. We also demonstrate that the
selection rules show a certain robustness to the weak in-plane Zeeman
splittings (see Sec. IX of SM SM ).

We make some comparisons with a recent experiment in ZrTe5 Y.Jiang2020  where
the magnetic field was applied along the $a$ axis: (i) The transition energy
extracted from the magnetoinfrared spectra clearly exhibits two branches in
the zeroth LL transitions and one branch in the $n\geq 1$ LL transitions,
which are consistent with our theoretical results; (ii) the magnetic field
implemented in the experiment was in the range $6-17.5$ T with the asymptotic
exponent for the LLs being $\frac{1}{2}$, whereas our calculations indicate
that the asymptotic exponents for the LLs are quite different, and may require
stronger magnetic fields to test the discrepancy. Therefore, further
experimental works in ZrTe5 are expected. We also hope that our results can be
extended to other layered and weak-coupling 3D topological materials, such as
HfTe5 Weng ; P.Wang ; Galeski , Bi2Te3, and Sb2Te3 H.Zhang ; C.X.Liu .

##  V Acknowledgment

This work was supported by the National Key Research and Development Program
of Ministry of Science and Technology (No. 2021YFA1200700) and the Natural
Science Foundation of China (Grants No. 11804122, 11905054, and 12275075), the
China Postdoctoral Science Foundation (Grant No. 2021M690970), and the
Fundamental Research Funds for the Central Universities of China.

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