# Optical study of the charge dynamics evolution in the topological insulators
MnBi2Te4 and Mn(Bi0.74Sb0.26)2Te4 under high pressure

M. Köpf  Experimentalphysik II, Institute of Physics, Augsburg University,
86159 Augsburg, Germany  S. H. Lee  2D Crystal Consortium, Materials Research
Institute, Pennsylvania State University, University Park, PA 16802, USA
Department of Physics, Pennsylvania State University, University Park,
Pennsylvania 16802, USA  Z. Q. Mao  2D Crystal Consortium, Materials Research
Institute, Pennsylvania State University, University Park, PA 16802, USA
Department of Physics, Pennsylvania State University, University Park,
Pennsylvania 16802, USA  Department of Materials Science and Engineering,
Pennsylvania State University, University Park, Pennsylvania 16802, USA  C. A.
Kuntscher<EMAIL_ADDRESS>Experimentalphysik II,
Institute of Physics, Augsburg University, 86159 Augsburg, Germany

###### Abstract

The van der Waals material MnBi2Te4 and the related Sb-substituted compounds
Mn(Bi1-xSbx)2Te4 are prominent members of the family of magnetic topological
insulators, in which rare quantum mechanical states can be realized. In this
work, we study the evolution of the charge dynamics in MnBi2Te4 and the Sb-
substituted compound Mn(Bi1-xSbx)2Te4 with $x=0.26$ under hydrostatic
pressure. For MnBi2Te4, the pressure dependence of the screened plasma
frequency, the dc conductivity, and the reflectance at selected frequencies
shows weak anomalies at $\sim$2 and $\sim$4 GPa, which might be related to an
electronic phase transition driven by the enhanced interlayer interaction. We
observe a pressure-induced decrease in the optical gap, consistent with the
decrease in and closing of the energy gap reported in the literature. Both
studied materials show an unusual decrease in the low-energy optical
conductivity under pressure, which we attribute to a decreasing spectral
weight of the Drude terms describing the free charge carrier excitations. Our
results suggest a localization of conduction electrons under pressure,
possibly due to hybridization effects.

##  I Introduction

In recent years, the famous representative MnBi2Te4 (MBT) of topological
insulators with magnetic order was studied extensively with many different
experimental techniques. It is proven by now that MBT is a promising candidate
for hosting rare quantum mechanical effects such as the quantum anomalous Hall
effect (QAHE) or the axion insulator, which makes MBT potentially interesting
for applications in quantum metrology and spintronics Zhang  _et al._ (2019);
Deng  _et al._ (2020); Lei and MacDonald (2021); He (2020); Li  _et al._
(2019a, b). Characteristic for a topological insulator, MBT exhibits a gapped
electronic structure near the Fermi energy $E_{\mathrm{F}}$ in the bulk, while
linear dispersing, gapless surface states are induced by band inversion, which
are protected by spatial and time-reversal symmetry at ambient conditions Hao
_et al._ (2019); Chen  _et al._ (2019a, b); Otrokov  _et al._ (2019a);
Klimovskikh  _et al._ (2020); Swatek  _et al._ (2020); Vidal  _et al._ (2019).
Whether the surface states are gapped or not, has been a matter of
controversy, as contradictory experimental results were reported Lee  _et al._
(2019); Otrokov  _et al._ (2019b). Due to the onset of an $A$-type
antiferromagnetic ordering below $T_{\mathrm{N}}=25\,$K Li  _et al._ (2020a);
Chen  _et al._ (2019b); Ding  _et al._ (2020); Yuan  _et al._ (2020); Rani
_et al._ (2019); Cui  _et al._ (2019), a time-reversally broken system in
coexistence with nontrivial band topology emerges in MBT, which enables the
realization of Weyl fermions close to $E_{\mathrm{F}}$ Lee  _et al._ (2021);
Li  _et al._ (2020a, 2019c, b); Tokura  _et al._ (2020).

MBT crystallizes in the space group $R\overline{3}m$ with the cell parameters
$a=4.33\,$Å and $c=40.91\,$Å Li  _et al._ (2020a) and it possesses a van-der-
Waals type layered structure, consisting of Te-Bi-Te-Mn-Te-Bi-Te septuples
stacked in a rhombohedral ABC order Li  _et al._ (2020a); Hou  _et al._
(2020); Rani  _et al._ (2019). A sketch of the structure is shown in Fig. 1
(a). While the Mn sublattice contributes a magnetic moment to the compound,
the Bi and Te $p_{z}$ bands are responsible for the nontrivial surface states,
which are located at the $\Gamma$-point in momentum space Lee  _et al._
(2019); Lai . Regarding the three-dimensional properties, a bulk energy gap
between 180 and 220 meV is reported Chen  _et al._ (2019a, b), while the
surface Dirac point between the valence and conduction band is located app.
270 meV below the $E_{\mathrm{F}}$ Chen  _et al._ (2019b); Hao  _et al._
(2019); Swatek  _et al._ (2020). A sketch of the described band structure is
shown in Fig. 2 (c), where two electronic bands are crossing $E_{\mathrm{F}}$.
Accordingly, MBT is intrinsically doped, with electrons as the main charge
carriers Lee  _et al._ (2021); Yan  _et al._ (2019).

For further understanding of achieving ideal Weyl semimetal states, ways of
tuning the electronic structure in MBT have been exploited Lee  _et al._
(2021); Li  _et al._ (2021). By gradually exchanging Bi atoms by Sb atoms, the
Sb-substituted compounds Mn(Bi1-xSbx)2Te4 (MBST) are generated, where the
position of the chemical potential as well as the energy gap size are strongly
affected and determined by the Sb content Chen  _et al._ (2019a); Lee  _et
al._ (2021); Ko  _et al._ (2020); Yan  _et al._ (2019); Riberolles  _et al._
(2021). While the structural parameters undergo only minor changes due to Sb
doping, a lowering of the $E_{\mathrm{F}}$ level from the conduction bands
towards the valence bands with increasing Sb content can be observed, where a
nearly insulating state is expected to be reached at a doping level of
$x=0.26$ Chen  _et al._ (2019a); Lee  _et al._ (2021). Accordingly, the
expected electronic band structure of Mn(Bi0.74Sb0.26)2Te4 ($x=0.26$) is
sketched in Fig. 2 (f), where $E_{\mathrm{F}}$ now slightly crosses one of the
lower bands, as will be justified later based on our experimental results. In
addition, the energy gap is reported to be reduced with Sb doping level, where
a gap closing is expected at $x=0.55$, and a reopening of the bands results in
the vanishing of non-trivial bands for Sb-substituted compounds above $x=0.55$
Chen  _et al._ (2019a). Yet, also topological surface states have been
detected in MnSb2Te4 ($x=1$) other studies Wimmer  _et al._ (2021); Liu ;
Eremeev  _et al._ (2021); Mur .

Thus far, the optical properties of pure MBT and of various Sb-substituted
compounds MBST have been investigated at ambient pressure only Xu  _et al._
(2021); Köpf  _et al._ (2020, 2022a, 2022b). As unusual properties of
pressurized MBT regarding dc electric transport dynamics, electronic
structure, and structual evolution have been reported Pei  _et al._ (2020);
Guo  _et al._ (2021); Chen  _et al._ (2019c); Xu  _et al._ (2022); Shao  _et
al._ (2021); Chong  _et al._ (2023), it is important to also investigate the
frequency-dependent optical response of MBT at high hydrostatic pressures.
Typically, materials show a decreasing resistivity under external pressure, as
bond lengths and interatomic distances are more and more compressed Hazen
(2009). This holds for example for the closely related compounds and
topological insulators Bi2Te3 Matsubayashi  _et al._ (2014) and MnSb2Te4 Yin
_et al._ (2021). In stark contrast, several studies reveal a more complex
behaviour of the dc resistivity for MBT under hydrostatic pressure Pei  _et
al._ (2020); Chen  _et al._ (2019c). Electrical resistivity measurements show
the initial increase in resistivity with increasing pressure up to $\sim$ 12
GPa, followed by a strong resistivity drop for higher pressures. This
unconventional, non-monotonic shift was suggested to result from the competing
nature of gradually localized surface electrons and the bulk electrons
undergoing a delocalization process under pressure Pei  _et al._ (2020). In
another study, the hybridization of Bi-6$p$ and Te-5$p$ electrons with
delocalized Mn-3$d$ electrons, which creates a hybridization gap, or an
increase in electron scattering rate were given as possible mechanisms for the
initial pressure-induced resistivity increase Chen  _et al._ (2019c).
Regarding the crystal structure under pressure, the cell parameters $a$ and
$c$ shrink irregularly with increasing pressure Chen  _et al._ (2019c). This
results in an unusual evolution in the $c/a$ ratio, where an initial drop to a
minimum value at 3 GPa is followed by an increase for higher pressures Pei
_et al._ (2020). A further crucial consequence of external pressure
application is a decrease in the energy gap at the $\Gamma$-point up to $\sim$
15 GPa Pei  _et al._ (2020); Xu  _et al._ (2022), as demonstrated by various
theoretical calculations Pei  _et al._ (2020); Xu  _et al._ (2022); Guo  _et
al._ (2021). In summary, the high-pressure properties of MBT are highly
unusual and many open questions remain.

In this study, we investigate the effect of external pressure on the charge
carrier dynamics in pure MBT and in the Sb-substituted compound
Mn(Bi0.74Sb0.26)2Te4 by optical spectroscopy over a broad frequency range.
According to our optical data, the metallic character of both compounds
weakens under pressure. We also find anomalies in the pressure dependence of
the screened plasma frequency $\omega_{\mathrm{pl}}^{\mathrm{scr}}$, dc
conductivity, and reflectance values at selected frequencies. These pressure-
induced anomalies are in good agreement with the structual parameter
evolutions under pressure reported in the literature Chen  _et al._ (2019c);
Pei  _et al._ (2020). Furthermore, we observe a decrease in the optical gap,
which we relate to the decreasing energy gap under pressure Pei  _et al._
(2020); Xu  _et al._ (2022).

##  II Materials and Methods

Single crystals of MnBi2Te4 and MnBi(0.74Sb0.26)2Te4 were grown by the self-
flux method as reported in Ref. Lee  _et al._ (2021). The 26 % Sb substituted
sample belongs to the batch SL3B1, as described in our previous publication
Köpf  _et al._ (2022b), and accordingly is slightly hole doped. The crystals
have been characterized in detail by temperature-dependent electron transport
and Hall resistivity measurements, as reported in Ref. Lee  _et al._ (2021).
We carried out infrared reflectance measurements by Fourier-transform infrared
(FTIR) spectroscopy with a Bruker 80v spectrometer coupled to a Hyperion
microscope.

Hydrostatic pressure has been applied with EasyLab diamond anvil cells (DAC),
where a small piece of the single crystal was placed into a hole of a CuBe
gasket together with some CsI, which served as pressure transmitting medium
for the realization of quasi-hydrostatic pressure. For the pure compound we
have used a DAC with a 900 $\upmu$m culet size and for the 26 % substituted
sample a DAC with a 800 $\upmu$m culet size has been selected. In order to
determine the pressure inside the DAC, tiny Al2O3 spheres were put next to the
sample. Excited with a green laser, these spheres emit radiation with a
specific wavelength characteristic for the applied pressure. This radiation is
detected with a charge-coupled device (CCD) spectrograph.

The reflectance is calculated according to the equation
$I_{\mathrm{sample}}/I_{\mathrm{reference}}$, where $I_{\mathrm{sample}}$
stands for the intensity reflected by the sample and $I_{\mathrm{reference}}$
for the intensity reflected by the CuBe gasket. The data were measured from
approx. 200 cm-1 to 17000 cm-1 (0.03 eV to 2.11 eV). Over a small range from
app. 1700 cm-1 to 2700 cm-1, the measured spectra had to be interpolated, as
the characteristic multiphonon absorption in diamond is leading to falsified
results Su  _et al._ (2013). Also, extrapolations in the low and high energy
regime are merged to the spectra. These extrapolations are constructed with
the help of literature values from transport measurements and volumetric
information, which is necessary to obtain optical functions, like the optical
conductivity $\sigma_{1}$ and the loss function -Im(1/$\epsilon$). This is
done by applying the Kramers-Kronig relations through programs by David Tanner
Tanner (2015). All data sets are fitted with the Drude-Lorentz model to learn
about pressure-dependent behaviour of various electronic excitations with the
use of the RefFIT software, as described by Kuzmenko et al. Kuzmenko (2005).

Figure 1:  (a) Sketch of the crystal structure of MBST. Pressure-dependent (b)
reflectance and (c) optical conductivity $\sigma_{1}$ of the pure compound as
well as (d) reflectance and (e) optical conductivity $\sigma_{1}$ of the Sb-
substituted compound with $x=0.26$. The corresponding loss functions of both
materials are shown as an inset in (b) and (d), respectively. In (c), the
intercept point of the linear extrapolations (dashed lines) of $\sigma_{1}$
with the frequency axis indicates the onsets of interband transitions,
corresponding to the optical gap size. Figure 2:  Drude-Lorentz fit of
$\sigma_{1}$ of the pure compound at (a) 1.13 GPa and (b) 8.0 GPa as well as
of the 26% substituted compound at (d) 3.4 GPa and (e) 10.2 GPa. (c) and (f)
display sketches of the corresponding band structures close to the Fermi
level, respectively (similar to Ref. Köpf  _et al._ (2022b) and based on
calculations by Chen et al. Chen  _et al._ (2019b)). In (c) and (f), the
vertical arrows display electronic transitions described by the respective
Lorentz oscillators, where L1 describes transitions between two conduction
bands, L1* between valence and conduction band, and L2 corresponds to
transitions across the optical gap. The energy gap $E_{\mathrm{gap}}$ is
indicated by a vertical orange line.

##  III Results

###  III.1 Pressure-dependent optical spectra

The room-temperature reflectance spectra of MBT as a function of pressure have
been measured in small steps up to 8.0 GPa. In order to establish a good
sample-diamond interface, which is necessary for reasonable results, the first
spectrum has been taken at 1.1 GPa. The results are shown in Fig. 1 (b). At
the lowest pressure, we find a characteristic metallic spectrum: The
reflectivity level at the lowest measured frequency is quite high, close to 80
%, which is followed by a rather strong decrease corresponding to the plasma
edge. The reflectance shows the plasma minimum at app. 1200 cm-1and a weak but
steady increase up to the highest measured frequency. With increasing
pressure, this profile is maintained. Yet, we observe that the level below
$\sim$700 cm-1 is decreasing with increasing pressure, while above this
wavenumber a reflectance increase is found. The corresponding optical
conductivity spectra [see Fig. 1 (c)] show for all measured pressures a low
level ($\sim$1000 $\Omega^{-1}$cm-1) at low frequencies, which is followed by
a linear-like increase above approx. 2000 cm-1. This quasi-linear increase in
$\sigma_{1}$ has been extrapolated with linear fits, in order to estimate the
size of the optical gap, as will be discussed later. The high-frequency range
is dominated by a pronounced absorption band centered at around 10000 cm-1due
to interband transitions, which is in good agreement with previous
measurements by our group Köpf  _et al._ (2020, 2022a, 2022b) and by Xu et al.
Xu  _et al._ (2021). With increasing pressure this absorption band increases
in strength and shifts slightly to smaller frequencies. The inset of Fig. 1
(b) shows the loss function of MBT as a function of pressure, defined as
-Im(1/${\hat{\epsilon}}$) where $\hat{\epsilon}$ is the complex dielectric
function. A clear plasmon peak in the loss function at the lowest pressure
indicates the metallic character of the material, and its position corresponds
to the value of the screened plasma frequency
$\omega_{\mathrm{pl}}^{\mathrm{scr}}$. With increasing pressure the plasmon
peak shifts from $\sim$800 cm-1at 1.1 GPa to lower frequencies and broadens
considerably, both signalling a weakening of the metallic character of MBT
under pressure, as will be discussed below.

Corresponding optical functions of the 26 % Sb-doped compound are displayed in
Figs. 1 (d) and (e) at various pressures up to 10.2 GPa. The overall pressure
dependence of the optical response is very similar to that of the undoped
material MBT. However, the metallic character of Mn(Bi0.74Sb0.26)2Te4 is much
more reduced compared to MBT: namely, the low-frequency reflectivity level is
low and the plasma edge is less developed with a plasma minimum located at a
lower frequency ($\sim$400 cm-1) see Fig. 1 (d)]. Also the Fabry-Pérot
interferences in the frequency range 500 - 1300 cm-1 signal the much reduced
metallic nature, leading to a partial transparency of the studied crystal. The
Fabry-Pérot interferences are reduced under pressure. At higher frequencies,
the level of the reflectivity is rising very weakly, similar to the pure
compound [see Fig. 1 (b)]. The low-frequency optical conductivity
$\sigma_{1}$, depicted in Fig. 1 (e), has an extremely low level up to
frequency 1000 cm-1, confirming the very low spectral weight of free charge
carrier excitations. As expected, the material is close to insulating. A small
plateau-like feature in $\sigma_{1}$ at approx. 1500 cm-1 is followed by a
steep quasi-linear increase to high values and an absorption band due to
interband transitions. The plasmon peak in the loss function of
Mn(Bi0.74Sb0.26)2Te4 [see inset of Fig. 1 (d)] indicates the pressure-induced
weakening of the (already weak) metallic character like for MBT, namely a
decrease of its height, broadening, and shift to lower frequencies.

Figure 3: (a) Reflectance values of MBT at selected frequencies as a function
of pressure. The dashed, vertical lines highlight the anomalies at $\sim$2 and
$\sim$4 GPa. (b) $\sigma_{\mathrm{dc}}$ values of MBT obtained by the Drude-
Lorentz fits, which are compared to results from Pei et al. Pei  _et al._
(2020) and Chen et al. Chen  _et al._ (2019c). (c) Plasma frequency
$\omega_{\mathrm{pl}}$ of the two Drude terms D1 and D2 and the total plasma
frequency of the combination D1+D2 (see text for definition) for MBT as a
function of pressure. (d) Scattering rate of the Drude terms D1 and D2 for MBT
as a function of pressure. (e) Screened plasma frequency
$\omega_{\mathrm{pl}}^{\mathrm{scr}}$ of the pure and the 26 % doped sample,
as obtained from the plasmon peak position in the loss function. Two anomalies
in the pressure dependence of $\omega_{\mathrm{pl}}^{\mathrm{scr}}$ for MBT at
$\sim$2 and $\sim$4 GPa are indicated by vertical arrows. (f) Interband
transition onset for MBT and $x$=0.26 doped sample as a function of pressure,
determined by the zero-crossing of the linear extrapolations in the
$\sigma_{1}$ spectra, which is an estimate for the size of the optical gap.
The error bars in (a) have been estimated within the accuracy of the
reflection measurement.

###  III.2 Analysis of optical functions and optical parameters as a function
of pressure

For a detailed understanding of the charge carrier dynamics and excitations
under pressure, we have performed fits of the measured optical data with the
Drude-Lorentz model. Fits of the optical conductivity $\sigma_{1}$ of MBT at
the lowest and highest pressure are depicted in Fig. 2 (a) and (b),
respectively. As free charge carrier contributions from two electronic bands
are expected according to the electronic band structure of MBT [Fig. 2 (c)],
and also in order to obtain a reasonable fit of the data, we have implemented
two Drude terms besides six Lorentz terms in the measured range. This fit
model is in accordance with previous analyzes by our group Köpf  _et al._
(2020, 2022b). The “High-Energy” oscillator stands for the sum of all
oscillators lying outside the measured range. From the comparison of the Drude
contributions at lowest and highest measured pressure, we can conclude that
the Drude terms are losing spectral weight under pressure, whereas all the
Lorentz oscillators are gaining in their spectral weight. The electronic band
structure of MBT sketched in Fig. 2 (c) is based on calculations by Chen et
al. Chen  _et al._ (2019b) and has already been discussed in Refs. Köpf  _et
al._ (2020, 2022b). The two energetically lowest interband transitions L1 and
L2 correspond to transitions between the two conduction bands and to the
lowest-energy transitions between the valence and conduction bands, defining
the optical gap, respectively. Especially L1 increases a lot in its oscillator
strength when the external pressure is increased, which lifts the level of
$\sigma_{1}$ around 1000 cm-1.

For the 26 % substituted compound Mn(Bi0.74Sb0.26)2Te4, the behaviour of the
optical conductivity under pressure is partially similar. Comparing the data
shown in Fig. 2 at (d) 3.4 GPa and at (e) 10.2 GPa the Lorentz oscillators in
the high-energy range are gaining in spectral weight. The initial excitation
resulting from interband transitions, also referred to as the optical gap, is
described by the L2 oscillator, which is located just below the quasi-linear
increase. This peak undergoes a slight shift to lower frequencies as well as a
narrowing under pressure. The spectral weight of $\sigma_{1}$ below $\sim$1000
cm-1 has a very low level, with a pressure-induced decrease in the Drude
spectral weights and an increase in the L1* oscillator strength. L1* describes
the electronic transitions between the valence and conduction band as
illustrated in Fig. 2 (f). As compared to MBT, in Mn(Bi0.74Sb0.26)2Te4 the
energy gap $E_{\mathrm{gap}}$ is reduced and the Fermi level is shifted down
in energy and is expected to slightly cut one of the former valence bands Chen
_et al._ (2019a). Also, we obvserve an additional oscillator L1*b located at
around 200 cm-1, which appears for pressures above 8.8 GPa. This weak
excitation seems to be less screened with increasing pressure, as the spectral
weight of the Drude contributions and the overall $\sigma_{1}$ level are
reduced. To conclude, both samples of this compound family show similar
pressure dependences, in particular, a strong reduction of the Drude spectral
weights under pressure.

The pressure evolution of several optical parameters is depicted in Fig. 3,
highlighting the most important findings of our study. In Fig. 3 (a) we show
the reflectance values for MBT at selected frequencies below 1000 cm-1 as a
function of pressure. One observes two weak anomalies in the pressure
dependence at around 2 and 4.5 GPa, which signal abrupt changes in the
electronic properties under pressure. The pressure-dependent value of the dc
conductivity $\sigma_{\mathrm{dc}}$ of MBT as extracted from the Drude-Lorentz
fits is depicted in Fig. 3 (b). Obviously, $\sigma_{\mathrm{dc}}$ undergoes a
strong decrease from approx. 2500 $\Omega^{-1}$cm-1 at 1 GPa to 1600
$\Omega^{-1}$cm-1 at 4 GPa followed by a constant behavior. We compare our
experimental data to results from Pei et al. Pei  _et al._ (2020) measured at
room temperature and from Chen et al. Chen  _et al._ (2019c) measured on two
samples at 250 K and 200 K, respectively. Despite the difference in absolute
values, we observe different slopes in the pressure dependence of
$\sigma_{\mathrm{dc}}$ compared to our data. Yet in all cases the dc
conductivity is decreasing with increasing external pressure. A small
pressure-induced increase of the dc conductivity at pressures below 3 GPa was
observed by Pei et al. Pei  _et al._ (2020) and for a sample at 200 K by Chen
et al. Chen  _et al._ (2019c), which is followed by a strong decrease above.
This initial discrepancy is not visible in our data, where
$\sigma_{\mathrm{dc}}$ decreases already from the lowest pressure onwards.

As a measure of the metallic character of a material we can furthermore
consider the plasma frequency, which corresponds to the Drude spectral weight,
and the scattering rate of the Drude terms, as extracted from the Drude-
Lorentz fits. In Fig. 3 (c) the values of the $\omega_{\mathrm{pl}}$ of the
Drude terms D1 ($\omega_{\mathrm{pl,1}}$) and D2 ($\omega_{\mathrm{pl,2}}$)
and the total plasma frequency labelled “D1+D2” and calculated according to
$\omega_{\mathrm{pl}}=\sqrt{\omega_{\mathrm{pl,1}}^{2}+\omega_{\mathrm{pl,2}}^{2}}$
are shown as a function of pressure. The value of $\omega_{\mathrm{pl}}$ (as
well as of $\omega_{\mathrm{pl,1}}$ and $\omega_{\mathrm{pl,2}}$) decreases
under pressure, clearly indicating the weaking of the metallic character. The
scattering rate of the Drude term D1 (see Fig. 3 (d)) shows a small decrease
up to 5 GPa, followed by a stronger drop at higher pressures. The scattering
rate of D2 is approximately pressure independent.

Another characterization of the free charge carrier dynamics is given by the
screened plasma frequency $\omega_{\mathrm{pl}}^{\mathrm{scr}}$, which can be
obtained from the plasmon peak position in the loss function (see insets of
Figs. 1 (b) and (d)). $\omega_{\mathrm{pl}}^{\mathrm{scr}}$ is related to the
plasma frequency $\omega_{pl}$ according to
$\omega_{pl}^{scr}$=$\omega_{pl}$/$\sqrt{\epsilon_{\infty}}$, where
$\epsilon_{\infty}$ is the high-frequency value of $\epsilon_{1}(\omega)$. For
the MBT compound $\omega_{\mathrm{pl}}^{\mathrm{scr}}$ is decreasing under
pressure, with weak anomalies at $\sim$2 and $\sim$4.5 GPa indicated by arrows
in Fig. 3 (e). Like for the pressure-dependent reflectance values, these
anomalies signal abrupt changes in the electronic structure. For
Mn(Bi0.74Sb0.26)2Te4 the value of $\omega_{\mathrm{pl}}^{\mathrm{scr}}$ is
also decreased under pressure. To conclude, the unscreened and screened plasma
frequency are reduced under pressure for both studied materials indicating
that they become less metallic for pressures up to approx. 10 GPa, in good
agreement with published dc transport experiments Pei  _et al._ (2020); Chen
_et al._ (2019c). Such a pressure behavior is very unusual, since the overlap
of atomic orbitals generally increases under pressure, which leads to an
increase in electronic band width and hence improved conductivity.

A further important parameter characterizing the electronic band structure is
the size of the optical gap. In order to extract the optical gap size from our
data, we have extrapolated the quasi-linear increase in $\sigma_{1}$ with a
linear extrapolation, as mentioned above and as sketched in Fig. 1 (c). The
zero-crossing of this extrapolation marks the onset of the interband
transitions and hence is an estimate of the size of the optical gap. The so-
obtained values of the interband transition onset as a function of pressure
are plotted in Fig. 3 (f) for the pure and the 26 % substituted compound. For
MBT we observe an approximately constant value of 2200 cm-1 up to $\sim$6 GPa
followed by a drop to 1600 cm-1 at 8 GPa. In the case of Mn(Bi0.74Sb0.26)2Te4
the interband transition onset decreases from 2700 cm-1 to 2200 cm-1.
Accordingly, in both materials the optical gap is reduced under pressure.

##  IV Discussion

###  IV.1 Pressure-induced anomalies in the optical data

For MBT, we observe weak anomalies at $\sim$2 GPa and $\sim$4 GPa in the
pressure dependence of the reflectance at selected frequencies [Fig. 3 (a)],
the dc conductivity $\sigma_{dc}$ [Fig. 3 (b)], and of the screened plasma
frequency $\omega_{\mathrm{pl}}^{\mathrm{scr}}$ [Fig. 3 (e)], suggesting
abrupt changes in the electronic band structure. These changes could be
interrelated to abrupt pressure-induced changes in the crystal structure
reported in the literature. Chen et al. Chen  _et al._ (2019c) observed a
sudden drop in the pressure dependence of the lattice parameters $a$ and $c$
at 2 GPa, which was interpreted in terms of a lattice softening. According to
experimental and theoretical results in Refs. Pei  _et al._ (2020); Xu  _et
al._ (2022), respectively, the interlayer lattice parameter $c$ is more
sensitive regarding pressure than the intralayer lattice parameter $a$ for low
pressures up to $\sim$2 GPa. However, this behavior changes for higher
pressures, namely above approx. 4 GPa, since the interlayer distance is less
affected in this pressure range. Accordingly, the lattice parameter ratio
$c/a$ initially decreases under pressure and exhibits a minimum between
$\sim$2 and $\sim$4 GPa, followed by an almost linear increase above 4 GPa up
to approx. 14 GPa Pei  _et al._ (2020). The pressure dependence of the $c/a$
ratio can be related to anomalies in the pressure shift of several structural
parameters Pei  _et al._ (2020): For example, various bond distances between
the Mn, Bi and Te atoms, like the Mn-Te and Bi-Te bond lengths, are decreased
under pressure, with an anomalous behavior between 2 and 4 GPa. Such a
pressure dependence is also revealed by the Te-Te bond length, which is a
measure for the interlayer distance, and by the bond angles within the Te-Bi-
Te octahedra. Besides, the intensity ratio of the strongest Raman-active modes
follow this anomalous pressure dependence. It is important to note that no
crystal symmetry change occurs for pressures below 14.6 GPa Pei  _et al._
(2020).

Interestingly, in the closely related compounds Bi2Te3, Bi2Se3, and Sb2Te3 a
Lifshitz transition Lifshitz (1960), i.e., an electronic topological
transition with changes in the Fermi surface topology without a lattice
symmetry change, has been reported to occur between 3 and 5 GPa Polian  _et
al._ (2011); Gomis  _et al._ (2011); Vil ; Vilaplana  _et al._ (2011). The
layered, polar semiconductor BiTeI, which turns into a topological insulator
under moderate pressure Bahramy  _et al._ (2012), is another example, where an
electronic topological transition has been observed between 2 and 3 GPa Xi
_et al._ (2013): The pressure evolution of the $c$/$a$ lattice parameter ratio
has a minimum in the absence of lattice symmetry change, and the optical
parameters show anomalous behavior, similar to our findings in MBT. Layered
compounds are generally prone to pressure-induced electronic topological
transitions Zhu  _et al._ (2012); Yang  _et al._ (2017); Bassanezi  _et al._
(2018) as also evidenced in BiTeBr Ohmura  _et al._ (2017), 1T-TiTe2 Rajaji
_et al._ , and ZrSiTe Eba ; Kro . A topological phase transition has indeed
been theoretically predicted for MBT under hydrostatic tensile strain Guo  _et
al._ (2021) and suggested to occur in pressurized MBT Pei  _et al._ (2020) and
in the doping series (Mn1-xPbx)Bi2Te4 Qia . The closely related material
MnSb2Te4 undergoes a pressure-induced topological phase transition, when the
interlayer distance is decreased by a critical percentage Zhou  _et al._
(2020). Interlayer interaction therefore seems to be the driving mechanism for
this phase transition. Considering the anomalous behavior of the $c$/$a$ ratio
in MBT under pressure as described above, a similar scenario might hold here
as well. The occurrence of a Lifshitz transition in pressurized MBT due to an
enhanced interlayer interaction thus seems very likely. In the case of
Mn(Bi0.74Sb0.26)2Te4 we cannot draw a conclusion regarding anomalies in the
low-energy optical response, since this compound is less metallic (in fact,
close to insulating) and therefore the pressure-induced changes are much less
developed.

###  IV.2 Optical gap evolution

The energy gap evolution might also be linked to these pressure-induced
structural anomalies. Based on theoretical calculations, Xu et al. Xu  _et
al._ (2022) found a slight increase of the energy gap in MBT up to approx. 2
GPa, followed by an almost linear decrease and a gap closing slightly above 15
GPa. We cannot directly extract the energy gap from our optical data, since it
is smaller than the optical gap, i.e., the onset of interband transitions [see
energy scheme in Fig. 2 (c)]. The size of the optical gap can only serve as an
upper bound for the energy gap size. According to our optical results
presented in Fig. 3 (e), the optical gap is decreasing with increasing
pressure for both studied compounds.

###  IV.3 Charge carrier dynamics under pressure

The weakening of the metallic character of MBT under pressure is puzzling and
possible reasons for it have been discussed in the literature. Pei et al. Pei
_et al._ (2020) ascribed the pressure-induced increase in resistivity to the
competition between the localization of the surface electrons and the
delocalization of the bulk electrons, where the first process is predominant
in the pressure range between 3 and approx. 15 GPa resulting in a resistivity
increase. Such a resistivity increase, i.e., decrease in metallic character is
confirmed by our optical results. However, our optical data do not confirm
this scenario: Since infrared spectroscopy is generally bulk sensitive with
only minor contributions from surface electrons, the observed pressure-induced
decrease in conductivity is a bulk property according to our optical results.

Based on Hall effect measurements, Chen et al. Chen  _et al._ (2019c) related
the pressure-induced increase in resistivity to a decrease in electron
mobility $\mu_{\mathrm{e}}$, despite the observed increase in charge carrier
density $n_{\mathrm{e}}$ under pressure. Two scenarios were discussed as
possible explanations Chen  _et al._ (2019c). In the first scenario, the
pressure-induced suppression of long-range antiferromagnetic order in MBT
causes an enhancement of magnetic fluctuations, which increases the electron
scattering. The electron scattering rate $\gamma$ is related to the mobility
$\mu_{\mathrm{e}}$ according to

$\mu_{\mathrm{e}}=\frac{e}{m^{*}_{\mathrm{e}}\gamma}\quad,$ (1)

where $m^{*}_{\mathrm{e}}$ denotes the effective electron mass Yu and Cardona
(2010); Wooten (1972). Accordingly, an enhanced scattering rate would decrease
$\mu_{\mathrm{e}}$ and hence lead to localization of charge carriers with an
enhancement of the resistivity. Since the carrier scattering determines the
width of the Drude terms in our optical conductivity data, we can test this
scenario. As shown in Fig. 3 (d) the scattering rate of the dominant Drude
term D1 is decreasing with increasing pressure, with a stronger drop above
$\sim$5 GPa. Interestingly, above this pressure a suppression of the
antiferromagnetic ordering occurs Pei  _et al._ (2020); Chen  _et al._
(2019c). For the weaker Drude contribution D2 the scattering rate is pressure
independent. Therefore, our optical results do not support an enhancement of
the scattering rate under pressure, as suggested in Ref. Chen  _et al._
(2019c).

According to Eqn. 1 the mobility of the carriers can also be affected by their
effective mass: A pressure-induced increase in $m^{*}_{\mathrm{e}}$ would
cause a decrease in $\mu_{\mathrm{e}}$ consistent with the Hall effect
experiments. In fact, an increase in $m^{*}_{\mathrm{e}}$ would also explain
the pressure-induced decrease in the plasma frequency $\omega_{pl}$ [see Fig.
3 (c)], since $\omega_{pl}$ depends on $m^{*}_{\mathrm{e}}$ according to Fox
(2001)

$\omega_{\mathrm{pl}}=\sqrt{\frac{n_{\mathrm{e}}e^{2}}{\varepsilon_{0}m^{*}_{\mathrm{e}}}}\quad.$
(2)

Thus, an increase in $m^{*}_{\mathrm{e}}$ under pressure could explain both
the observed decrease in carrier mobility and decrease in plasma frequency,
despite the increase in electron density $n_{\mathrm{e}}$ Chen  _et al._
(2019c). A pressure-induced enhancement of the effective electron mass would
be consistent with the second scenario proposed in Ref. Chen  _et al._
(2019c): In analogy to CaMn2Bi2 Gib ; Lane  _et al._ (2023); Piva  _et al._
(2019), pressure application could induce a partial delocalization of Mn $3d$
electrons and subsequent hybridization with Bi $6p$ and/or Te $5p$ conduction
electrons, causing a localization of the electrons and the opening of a
hybridization gap Chen  _et al._ (2019c). The scenario of a pressure-induced
localization of charges due to hybridization could explain the results from
Hall effect measurements as well as our optical data for both studied
materials.

##  V Conclusion

In this work, we have studied the charge dynamics of the topological
insulators MnBi2Te4 and Mn(Bi0.74Sb0.26)2Te4 at high pressures by determining
the optical response functions. In good agreement with other pressure studies
on MnBi2Te4 Pei  _et al._ (2020); Xu  _et al._ (2022); Chen  _et al._ (2019c);
Guo  _et al._ (2021); Shao  _et al._ (2021), we have detected a pressure-
induced decrease in the metallic strength for both materials based on the low-
energy optical response. This decrease in conductivity is possibly due to an
enhanced effective mass resulting from the creation of a hybridization gap.
Furthermore, several optical parameters show weak anomalies in their pressure
dependence at $\sim$2 and $\sim$4 GPa, which might originate from a
topological electronic transition. In analogy to the closely related material
MnSb2Te4, we suggest the enhancement of interlayer interaction as the driving
mechanism for this phase transition. Besides, we observe a pressure-induced
decrease in the optical gap size for both studied materials, indirectly
confirming the expected reduction of the energy gap under pressure.

##  VI Acknowledgments

The authors acknowledge the fruitful discussions with and the technical
support by Jihaan Ebad-Allah. C. A. K. acknowledges financial support from the
Deutsche Forschungsgemeinschaft (DFG), Germany, through Grant No. KU
1432/15-1. Z.Q.M. and S.H.L. acknowledges the support of the US NSF through
the Penn State 2D Crystal Consortium-Materials Innovation Platform (2DCC-MIP)
under NSF Cooperative Agreement DMR-2039351.

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# A General Scenario of Tunneling Time in Different Energy Regimes

Sheng-Chang Li<EMAIL_ADDRESS>MOE Key Laboratory for Nonequilibrium
Synthesis and Modulation of Condensed Matter, Shaanxi Province Key Laboratory
of Quantum Information and Quantum Optoelectronic Devices, and School of
Physics, Xi’an Jiaotong University, Xi’an 710049, China

###### Abstract

We theoretically study the tunneling time by investigating a wave packet of
Bose-condensed atoms passing through a square barrier. We find that the
tunneling time exhibits different scaling laws in different energy regimes.
For negative incident energy of the wave packet, counterintuitively, the
tunneling time decreases very rapidly with decreasing incident velocity. In
contrast, for positive incident energy smaller than the barrier height, the
tunneling time increases slowly and then reaches a maximum, which is in
agreement with the Larmor clock experiments. The effect of the barrier width
related to the uncertainty principle on the maximum tunneling time is also
addressed. Our work provides a general scenario of tunneling time that can be
used to understand and explain the controversy over tunneling time.

Tunneling is an important quantum mechanical effect, which describes one
microcosmic particle having a finite probability of crossing an energy barrier
when the particle’s energy is less than the barrier. Continuous research on
quantum tunneling stems from both the incomplete understanding of the
tunneling process itself and its extensive applications in many fields. The
tunneling time describing the specific process of tunneling events, which is
closely related to the uncertainty principle, has always been controversial.
The first attempt to calculate tunneling time can be traced back to MacColl’s
work in 1932 [1]. Since then, great efforts have been made to define [2, 3, 4]
and measure [5, 6, 7, 8, 9] tunneling times. Recently, based on the attoclock
[10], the tunneling time defined as the time that the tunneling electron of an
atom spends under a barrier formed by a strong laser field and atomic Coulomb
potential has been detected in strong-field ionization experiments [11, 12,
13, 14]. However, there are controversies over theoretical explanations for
the experimental results [15, 16, 17, 13, 18]. One reason is that in this
case, it is difficult to clearly define the tunneling time due to the time-
dependent laser-Coulomb-formed barrier. Another reason is that such
attosecond-scale tunneling time is deduced from experimental observable (i.e.,
photoelectron momentum distribution) via strong-field theory models and cannot
be measured directly.

On the other hand, ultracold atoms, especially Bose–Einstein condensates
(BECs), as macroscopic quantum matter, provide unprecedented opportunities and
ideal platforms for the experimental detection of tunneling time. In the most
recent study [19], Ramos et al. directly measured the tunneling time of Bose-
condensed 87Rb atoms based on Larmor clock technology for the first time [20].
A Larmor clock uses an auxiliary degree of freedom (i.e., spin precession) of
the tunneling atoms to measure the dwell time inside the barrier. Furthermore,
they observed a slow decrease in the Larmor tunneling time with lower incident
energy [21]. However, these studies limited the incident energy to a range
greater than zero, while the intriguing quantum region where the incident
energy is less than zero remains unsolved.

In this Letter, we study the tunneling time for a wider energy range,
especially for negative energies, which allows us to give a general
description and to obtain a complete picture of the quantum tunneling time. To
directly define the tunneling time without borrowing the auxiliary degrees of
freedom of tunneling atoms, we consider a single-component BEC passing through
a square barrier. We use the time difference between two moments when the
probabilities of finding atoms at two boundaries are maximized to characterize
the tunneling time. We explore the dependence of the tunneling time on the
incident velocity (or energy) of the wave packet and find that the tunneling
time exhibits different scaling laws in different energy regimes.
Particularly, for negative incident energy, we see a counterintuitive
phenomenon in which the tunneling time decreases rapidly with decreasing
incident velocity and find that the tunneling time is maximal at a particular
incident velocity when the incident energy is positive. We further investigate
the effect of barrier width on the maximum tunneling time and find a critical
barrier width that can be used to mark the boundary between classical and
quantum regimes. The physical mechanism behind the results and their
connection to wave–particle duality and uncertainty relation are discussed as
well. Our results are qualitatively consistent with previous experimental
measurements based on the Larmor clock and can be used to understand and
explain the controversy over tunneling time in the field of strong-field
ionization.

A paradigmatic model for discussing the macroscopic tunneling dynamics of
ultracold atoms is the one-dimensional Gross-Pitaevskii (GP) equation. For
convenience, we adopt the dimensionless form, which reads

$\displaystyle i\frac{\partial\psi(x,t)}{\partial
t}=\left[-\frac{\partial^{2}}{2\partial
x^{2}}+V(x)-u|\psi(x,t)|^{2}\right]\psi(x,t),$ (1)

where $V(x)$ denotes a square potential barrier centered on $x=0$ with height
$q$ and width $w$. The left and right boundaries of the barrier are located at
$x_{L,R}=\pm\frac{w}{2}$. The atomic interaction constant $u$ is defined by
$u=4\pi n_{0}l^{2}_{0}$, with $n_{0}$ being the maximum density in the initial
distribution of the atomic condensates. The position $x$, time $t$, and
macroscopic wave function $\psi$ are, respectively in units of $l_{0}=1\mu$m,
$ml_{0}^{2}/\hbar$, and $\sqrt{n_{0}}$, with $m$ being the atom mass and
$\hbar$ being the reduced Planck constant. Initially, we assume that the
matter-wave packet takes the following normalized form:

$\psi(x,t=0)=\frac{1}{\sqrt{2}}\mathrm{sech}(x+x_{0})e^{ivx}.$ (2)

This expression approximates a matter-wave solution comprising one bright
soliton with three parameters: $a_{s}$, $w_{s}$, and $v$, which denote the
amplitude (i.e., ${1}/{\sqrt{2}}$), width (i.e.,
$2\mathrm{sech}^{-1}\frac{1}{2}\simeq 2.634$), and velocity, respectively. The
matter-wave soliton brings a good opportunity to investigate the wave–particle
duality on a macroscopic scale [22] due to its particle-like qualities, such
as localization, nondispersion [23], and center-of-mass trajectories [24, 25,
26]. Soliton tunneling [27, 28, 29, 30] associated with the nonlinear dynamics
of a wave packet colliding with a potential can be used to illustrate the link
between classical and quantum mechanics [22, 31]. Initially, we set
$x_{0}=-15$ (i.e., the initial position of the wave packet) to ensure that the
wave packet is far enough away from the barrier. We take $u=2$ to keep the
shape of the wave packet unchanged as it travels freely. In the calculations,
we numerically solve the time-dependent GP equation via the time-splitting
spectral method for the barrier height $q=2$. By comparing the incident
velocity (or energy) of the wave packet with the barrier height, we can
differentiate whether the motion of particles is in the classical regime or in
the quantum regime.

Figure 1: Schematic of the quantification of tunneling time. We use $t_{in}$
and $t_{out}$ to mark the specific moments when particles enter and exit the
barrier, respectively. The two moments correspond to the maximum probability
of finding particles at the left and right boundaries of the barrier. The
difference between them characterizes the tunneling time.

Generally, to quantify a tunneling process, one must employ local quantities
to clearly identify the start and end of tunneling events. A classical
particle has a deterministic trajectory, and the time it takes to tunnel is
uniquely determined as long as the position and momentum of the particle at
the beginning and end are determined. A quantum particle is governed by the
principle of uncertainty, and the position and momentum cannot be determined
simultaneously. The motion of a quantum particle has no deterministic
trajectory that can be traced; thus, there is no universal definition of the
beginning and end of tunneling events. In our model (1), the microscopic state
of a large number of condensed atoms is described by a macroscopic quantum
local wave packet (2), and the trajectory of the collective motion of
particles can be characterized by the center of mass. However, considering the
deformation of the quantum wave packet when it collides with the barrier, we
use the maximum probability at the barrier boundary to mark the beginning or
the end of the tunneling event, as illustrated in Fig. 1. To obtain the
tunneling time, we denote the start and end moments of the tunneling process
by $t_{in}$ and $t_{out}$, respectively. These two moments are determined by
the relations

$\displaystyle|\psi(x_{L},t_{in})|^{2}=\max(|\psi(x_{L},t)|^{2}),$ (3)
$\displaystyle|\psi(x_{R},t_{out})|^{2}=\max(|\psi(x_{R},t)|^{2}).$ (4)

The difference between $t_{out}$ and $t_{in}$ gives the tunneling time,
namely, $\Delta t$,

$\Delta t=t_{out}-t_{in},$ (5)

which characterizes the time spent by the atoms inside the barrier.

The main point of this work is to study the tunneling time in different energy
regimes and to determine the distinctive behaviors. Indeed, our system has two
energy scales: the initial energy of the incident wave packet, i.e.,
$E_{0}=\int(\frac{1}{2}|\frac{\partial\psi}{\partial
x}|^{2}-\frac{u}{2}|\psi|^{4})dx$, and the energy corresponding to the barrier
height, i.e., $q$. The initial energy includes two parts: the kinetic energy,
i.e., $E_{k}$, and the potential energy, i.e., $E_{p}$. The kinetic energy
consists of the intrinsic kinetic energy determined by the shape of the
initial wave packet, i.e., $E_{ks}=E_{ks}(a_{s},w_{s})$, and the incident
kinetic energy determined by $v$, i.e., $E_{kv}=E_{kv}(v)=\frac{1}{2}v^{2}$.
For a given wave packet, the initial energy is a function of both the incident
velocity of the wave packet and the strength of the interaction between atoms,
i.e., $E_{0}=E_{0}(v,u)$. For Eq. (2), we have
$E_{0}(v,u)=\frac{1}{6}(1-u+3v^{2})$, which contains $E_{ks}=\frac{1}{6}$,
$E_{kv}=\frac{1}{2}v^{2}$, and $E_{p}=-\frac{u}{6}$. When $u=2$, $E_{0}$ is
simply a function of the incident velocity, i.e.,
$E_{0}(v)=\frac{1}{2}v^{2}-\frac{1}{6}$ or $v(E_{0})=\sqrt{2E_{0}+1/3}$.
Subsequently, we study the dependence of the tunneling time on the initial
energy (or the incident velocity) of the wave packet, and the main results are
shown in Fig. 2. Comparing $E_{0}$ with $q$, we discuss the following three
regimes separately.

(I) If $E_{0}(v)<0$ [see Fig. 2(b)] or $v(E_{0})<v(E_{0}=0)$, we find

$\Delta t\simeq A\log(v)+B,$ (6)

where the constants $A$ and $B$ are determined by the width of the barrier.
For $w=1.0$, we obtain $A\simeq 0.66$ and $B\simeq 0.55$ [see Fig. 2(a)]. In
this region, the initial energy of the wave packet is negative. One can regard
the wave packet as a quasibound state. The De Broglie wavelength is
$\lambda_{D}={2\pi}/{v(E_{0})}$, and its minimum value (i.e.,
${2\pi}/{v(E_{0}=0)}=2\sqrt{3}\pi\simeq 10.883$) is larger than the width of
the wave packet, i.e., $\lambda_{D}>w_{s}$. As a result, the deformation of
the wave packet is very small when interacting with the barrier, and the
tunneling probability is also very small [see Fig. 2(b)]. However, the
tunneling time in this region increases very rapidly with increasing incidence
velocity.

Figure 2: (a) Tunneling times as functions of the incident velocity of the
wave packet. The barrier width $w=1.0$ was used. The numerical data (hollow
circle markers) along with the classical (solid line, $w/v$) and semiclassical
(dashed lines, $w/\sqrt{2|q-E_{0}|}$) results are shown. The fits are
illustrated by the dash-dotted and dotted lines, and their fitting parameters
are reported in the text. The left vertical line represents the position where
the initial energy of the wave packet equals zero [i.e., $E_{0}(v)=0$, red
solid circle in (b)], and the right vertical line denotes the position where
the initial energy matches the barrier height [i.e., $E_{0}(v)=q$, green solid
circle in (b)]. (b) Initial energies (left axis) and tunneling probabilities
(right axis) as functions of the incident velocity.

(II) If $0<E_{0}(v)<q$ or $v(E_{0}=0)<v(E_{0})<v(E_{0}=q)$, we find

$\Delta t\simeq\alpha v+\beta,$ (7)

where the constants $\alpha$ and $\beta$ are determined by the width of the
barrier. For $w=1.0$, we obtain $\alpha\simeq 0.045$ and $\beta\simeq 0.404$
[see Fig. 2(a)]. In this region, the initial energy of the wave packet is
positive but less than the height of the barrier. The De Broglie wavelength is
between ${2\pi}/{v(E_{0}=q)}={2\pi}/\sqrt{2q+1/3}\simeq 3.078$ and
${2\pi}/{v(E_{0}=0)}=2\sqrt{3}\pi$ and is comparable to the wave packet width,
i.e., $\lambda_{D}\sim w_{s}$. Before the wave packet arrives at the left
boundary of the barrier, we find that the deformation of the packet is
remarkable and that the tunneling probability significantly increases [see
Fig. 2(b)]. However, the tunneling time in this region is proportional to the
incidence velocity and changes very slowly. In particular, we find that the
tunneling time is remarkably shorter than that predicted by semiclassical
theory, i.e., $w/\sqrt{2(q-E_{0})}$.

(III) If $E_{0}(v)>q$ [see Fig. 2(b)] or $v(E_{0})>v(E_{0}=q)$, we find

$\frac{w}{v}\leq\Delta t\leq\frac{w}{\sqrt{2(E_{0}-q)}}.$ (8)

In this regime, the tunneling time is more than the classical time $w/v$ but
less than the semiclassical time $w/\sqrt{2(E_{0}-q)}$ [see Fig. 2(a)]. The De
Broglie wavelength is less than ${2\pi}/{v(E_{0}=q)}={2\pi}/\sqrt{2q+1/3}$,
which quickly decreases and is shorter than the width of the wave packet,
i.e., $\lambda_{D}<w_{s}$. It is noted that the tunneling probability in this
region is greater than 50% [see Fig. 2(b)]. When $v^{2}\gg q$ or $E_{0}\gg q$,
the difference between the classical and semiclassical times disappears, and
our numerical results tend to the theoretical predictions [see Fig. 2(a)].

From the perspective of wave–particle duality, the above three regions show
the situations where waves are dominant, waves and particles are comparable,
and particles are dominant, respectively. Thus, we can naturally regard I as
the quantum regime, II as the semiclassical regime, and III as the classical
regime [see Fig. 2]. It is clear that our theoretical calculation of tunneling
time is qualitatively consistent with the recent experimental results of
Larmor tunneling times in the semiclassical and classical regimes [19, 21]. A
maximum tunneling time $\Delta t_{\max}$ at $v\simeq\sqrt{2q}$, where the
total kinetic energy of the wave packet matches the barrier height, is also
found. Surprisingly, our new results in the quantum regime seem
counterintuitive but do support some of the findings in strong-field tunneling
ionization [17]. That is, in the quantum limit $v\rightarrow 0$, the tunneling
of microscopic particles seems instantaneous.

The above results for fixed barrier height and barrier width are
dimensionless. In practical measurements, one should pay more attention to the
dimensional maximum tunneling time. For example, one can use 7Li atomic
condensate to generate the wave packet [32]. For the barrier with height
$q=2\mu$m and width $w=1\mu$m, when the incident velocity of the wave packet
is $v=0.017$mm/s, our theoretical prediction for the maximum tunneling time is
$\Delta t_{\max}=0.053$ms. If one uses 87Rb atoms, when the incident velocity
of the wave packet is $v=1.382$mm/s, our prediction becomes $\Delta
t_{\max}=0.656$ms, which is of the same order as the experimental result
[i.e., $0.61(7)$ms, we have marked this point with an asterisk in Fig. 3.]
based on the Larmor clock [19]. The difference between the two results is
mainly caused by the difference in both the shape of the barrier and the
incident velocity. Actually, both the maximum tunneling time and the
corresponding wave packet incidence velocity depend on the barrier parameters
and the shape of the incident wave packet, i.e., $\Delta t_{\max}=\Delta
t_{\max}(q,w;a_{s},w_{s})$ and $v_{m}(q,w;a_{s},w_{s})$. For our given wave
packet, they are functions of the width and height of the barrier, i.e.,
$\Delta t_{\max}=\Delta t_{\max}(q,w)$ and $v_{m}=v_{m}(q,w)$.

Figure 3: The maximum tunneling time (left axis, black solid squares) and the
corresponding incident velocity of the wave packet (right axis, red solid
circles) as functions of the barrier width. The pink asterisk marks the
experimental result given in Ref. [19]. Two horizontal lines mark the
velocities, which are determined by $E_{0}(v)=q$ ($v_{E}$, olive dash-dotted
line) and $\frac{1}{2}v^{2}=q$ ($v_{q}$, blue dotted line). The vertical gray
dotted line represents the critical barrier width $w_{c}$, which denotes the
boundary between the quantum and classical regions.

To gain insight into the connection between $\Delta t_{\max}$ and $v_{m}$, we
discuss the influence of $w$ and for a fixed barrier height $q$. Figure 3
shows the variation of both $\Delta t_{\max}$ and $v_{m}$ with $w$ for 87Rb
atoms. We mark the width where $v_{m}$ changes most dramatically as $w_{c}$
which approximately equals $0.889\mu$m. When $w<w_{c}$, we find that $\Delta
t_{\max}$ increases slowly with $w$ as $\Delta t_{\max}\simeq(0.15w+0.44)$ms.
When $w>w_{c}$, we see that $\Delta t_{\max}$ increases rapidly with $w$ as
$\Delta t_{\max}\simeq(1.06w-0.44)$ms. It is clear that these two different
kinds of changes in $\Delta t_{\max}$ follow good linear rules. Unlike the
gradual change in the maximum tunneling time (see solid squares in Fig. 3),
the corresponding incident velocity dramatically increases near the critical
barrier width and then reaches a saturation value (see solid circles in Fig.
3). It should be mentioned that the saturation velocity is less than the
critical velocity $v_{E}$ (i.e., $E_{0}(v)=q$, horizontal blue dash-dotted
line) which indicates the boundary between the semiclassical and classical
regions and is approximately equal to the velocity $v_{q}$ (i.e.,
$v=\sqrt{2q}$, horizontal olive dotted line). The critical barrier width
$w_{c}$ corresponds to the position ($\Delta t_{\max}^{\ast}\simeq 0.594$ms,
$v_{m}^{\ast}\simeq 1.068$mm$/$s) where the wave packet kinetic energy changes
most dramatically. In other words, one can use the property that $v_{m}$ is
sensitive to $w$ to identify $w_{c}$.

We can analyze the above results by using the uncertain relation between
energy and time. With the help of $v_{E}=\sqrt{2q+{1}/{3}}\simeq 1.546$mm$/$s,
we have the maximum energy difference $\Delta
E_{\max}=\frac{1}{2}m(v_{E}^{2}-v_{m}^{2})$. When $w<w_{c}$, $\Delta
E_{\max}\cdot\Delta t_{\max}>\Delta E_{\max}^{\ast}\cdot\Delta
t_{\max}^{\ast}\simeq\frac{\hbar}{2}$ with $\Delta
E_{\max}^{\ast}=\frac{1}{2}m[v_{E}^{2}-(v_{m}^{\ast})^{2}]$. In this regime,
the uncertainty relation is satisfied, and thus, the tunneling particles have
pure quantum properties. When $w>w_{c}$, $\Delta E_{\max}\cdot\Delta
t_{\max}<\Delta E_{\max}^{\ast}\cdot\Delta
t_{\max}^{\ast}\simeq\frac{\hbar}{2}$. In this region, the uncertainty
relation is broken down, and therefore, the tunneling particles have obvious
classical properties. We can also analyze the above results by using the
uncertain relation between momentum and space. We use $\Delta p_{\max}$ and
$\Delta x_{\max}$ to record the maximum uncertainty ranges of momentum and
position, respectively, which can be expressed as $\Delta
p_{\max}=m(v_{q}-v_{m})$ and $\Delta x_{\max}=w$ with $v_{q}\simeq
1.486$mm$/$s. Similarly, we find that when $w<w_{c}$, the uncertain relation
$\Delta p_{\max}\cdot\Delta x_{\max}>\Delta p_{\max}^{\ast}\cdot\Delta
x_{\max}^{\ast}\simeq\frac{\hbar}{2}$ holds with $\Delta
p_{\max}^{\ast}=m(v_{q}-v_{m}^{\ast})$ and $\Delta x_{\max}^{\ast}=w_{c}$.
When $w>w_{c}$, the uncertain relation $\Delta p_{\max}\cdot\Delta
x_{\max}>\frac{\hbar}{2}$ is violated. This implies that when discussing
quantum tunneling, both energy requirements (i.e., $E_{0}<q$ or $E_{0}<0$) and
barrier width constraints (i.e., $w<w_{c}=\frac{\hbar}{2m(v_{q}-v_{m})}$)
should be considered, because they together determine the boundary between
quantum and classical regimes.

In summary, we have presented a general and detailed study of the tunneling
time by investigating the collisions between a macroscopic quantum wave packet
and a square barrier. We explore the dependence of the tunneling time on the
incident energy of the wave packet and find three different time scales in
different energy regimes. In particular, the counterintuitive behavior that
the tunneling time increases rapidly with the incidence velocity is shown in
the negative incident energy regime, where the tunneling probability is very
small. We also investigate the effect of the barrier width on the maximum
tunneling time and find a critical barrier width, which together with the
incident energy requirement determines the boundary between the quantum and
classical regimes. We demonstrate the important role of both wave–particle
duality and the uncertainty principle in the study of tunneling time. Our
results are not only qualitatively consistent with the Larmor clock
experiments but also support the claim of instantaneous tunneling in the study
of strong-field ionization. Our study shows a comprehensive physical picture
for understanding and studying the tunneling time and provides a new
perspective to resolve the controversy over the tunneling time in strong-field
ionization.

This work is supported by the National Natural Science Foundation of China
(Grant No. 11974273).

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