# Anisotropic $LMN$ dielectronic resonances from ratios of magnetic-dipole
lines

Yu. Ralchenko<EMAIL_ADDRESS>J.D. Gillaspy  National Institute of
Standards and Technology, Gaithersburg, Maryland 20899-8422

###### Abstract

Signatures of multi-keV $LMN$ dielectronic resonances in highly-charged
$3d^{n}$ ions of tungsten were detected in the intensity ratios of extreme-
ultraviolet magnetic-dipole lines within ground configurations. The
measurements were performed with an electron beam ion trap at beam energies of
about 6 keV. Large-scale collisional-radiative modeling incorporating magnetic
sublevels of autoionizing levels showed the significance of anisotropy effects
due to the monodirectional propagation of the electron beam. The observation
method allows simultaneous resolved registration of dielectronic resonances
from several ions.

###### pacs:

32.30.-r,32.80.Zb,34.50.Fa

##  I Introduction

Dielectronic recombination (DR) Burgess ; GriemBook ; Kunze  is one of the
important processes affecting ionization balance and related parameters (e.g.,
radiative power losses) of plasmas involving highly-charged ions (HCI). DR is
a two-step process involving dielectronic capture (DC), that is, excitation of
the atomic electron with simultaneous capture of the incident electron, and
radiative stabilization of autoionizing states produced during DC. Although a
particular resonant state can only be produced with an electron of specific
energy, a Maxwellian plasma has electrons of all energies and thus the DR rate
is never zero for thermal plasmas. Due to a weak dependence of the DC rates on
the ion charge $z$ and strong $\sim z^{4}$ dependence of the radiative rates,
radiative stabilization is more probable for highly-ionized atoms. That is why
DR is often the most dominant recombination mechanism for HCI plasmas,
especially at low densities. In addition to its effect on the ionization
distribution, DR frequently results in the appearance of strong satellite
lines that may serve as reliable diagnostics of plasma properties.

Dielectronic resonance states have been extensively studied with storage rings
Schi11 ; Muller09 ; Orban09 , electron beam ion traps (EBITs) Knapp89 ;
Knapp91 ; DeWitt92 ; Ba3d ; Naka06 ; Beilmann2010 , tokamaks Urnov07 ;
Bitter03 , and other devices Laser12 . The DR measurements with EBITs are
primarily based on two techniques. One method utilizes extraction of ions at
different beam energies and analysis of their abundances Ali91 ; DeWitt92 ;
Naka06 . It is not unusual, however, that the measured ionization
distributions differ from the originally produced abundances inside the trap,
and thus additional corrections may be required Zhang2010 . Another method is
based on EBIT beam ramping Knapp89 . In this case, a relatively high beam
energy, higher than the ionization potential of a specific ion, is used to
make the ion under study as abundant as possible. This energy is typically
much higher than the energy required to produce outer-shell dielectronic
resonances. Then the beam energy is quickly changed to a lower value (or
series of values) and data is collected before the ionization balance has time
to change substantially. The beam energy is then restored to its initial value
and held there for a relatively long time in order to restore the initial
charge balance. This process is repeated over many duty cycles in order to
integrate the signal acquired during the brief parts of the cycle that the
beam is at the low energies.

Elementary energy balance considerations show that the dielectronic resonances
due to excitation of the outer-shell electrons cannot be produced with the
beam energy larger than the ionization energy of the ion. It is possible,
however, to create resonances due to excitation of inner-shell electrons. For
instance, in Ref. Ba3d  the LMM resonances (i.e., inner L electron excited
into the M shell with the free electron captured into the M shell) were
identified and analyzed in the x-ray spectra from the Ba ions with an open
$3d$ shell, Ba34+ and Ba35+. Note that the ionization potentials of Ba34+ and
Ba35+ are 2142 eV and 2256 eV, respectively ASD , while the beam energies were
about 2400 eV. Good agreement was found between theory and experiment for the
ratios of DR to radiative recombination cross sections.

In this paper we report in situ measurements and analysis of the $LMN$
dielectronic resonances in about 50-times ionized atoms of tungsten with an
open $3d$ shell. Tungsten is currently a leading candidate for the plasma-
facing material in the divertor region of ITER Hawryluk09 , and therefore its
spectroscopic properties are being actively studied in many laboratories
KramidaW . While theoretical papers on DR of tungsten are numerous (see, e.g.,
SafW1 ; LiW ; SafW2 ; ShenW ; BeharW  and references therein), the
experimental data are rather scarce. Over the last decade, dielectronic
resonances in tungsten were measured in I-like W19+ Schi11  and in Si-like to
N-like ions Bie09 . The latter paper reported only the unresolved overlapping
$LMn$ ($n$ = 3 to 10) resonances measured in the x-ray range.

Under typical steady-state EBIT conditions, the balance between the neighbor
ion stages is determined by the corresponding ionization and recombination
processes:

$\frac{N_{z}}{N_{z+1}}=\frac{\tilde{R}_{rr}+\tilde{R}_{cx}+\tilde{R}_{dr}}{\tilde{R}_{ion}},$
(1)

where $N_{z}$ and $N_{z+1}$ are the total populations of the ions with charges
$z$ and $z$+1, and $\tilde{R}$’s are the total rates for radiative
recombination (rr), charge exchange (cx) with the neutrals in the trap,
dielectronic recombination (dr), and ionization (ion). Equation (1) assumes
that multi-electron processes are negligible. For highly-charged high-Z ions
the double charge exchange (CX) may actually not be much weaker than the
single-electron process; however, the lack of data on double CX in
approximately 50-times ionized atoms prevents one from taking it into account.

Since DR is a resonant process, $\tilde{R}_{dr}$ is typically zero for a
nearly monoenergetic EBIT beam. However, when the beam energy $E_{b}$ overlaps
with the resonant energies, $\tilde{R}_{dr}\neq 0$ and the ionization balance
may change drastically. Here we propose to measure multi-keV inner-shell
dielectronic resonances from low-energy spectroscopic footprints of the
changing ionization balance in the EBIT. To demonstrate this, we use the
intensity ratios of forbidden magnetic-dipole (M1) lines from the ground
configurations of neighbor ions. Several dozen M1 lines from all $3d^{n}$ ions
of tungsten were recently identified in the extreme ultraviolet (EUV) range
from 10 nm to 20 nm M1 . It was also found that many of those well-resolved
lines have a rich diagnostic potential for applications in fusion plasmas.
Moreover, we found that the low-lying M1 levels within the ground
configurations are primarily populated by the radiative cascades following
excitation from the lowest ground level. Since in low-density plasmas
(including EBITs) almost all ion population is in its ground state, the M1
line intensities are in effect proportional to the total ion populations.
Accordingly, their ratio is expected to be sensitive to modifications of the
ionization balance. This new method to detect dielectronic resonances with the
M1 line ratios is described below.

##  II Experiment

The EBIT at the National Institute of Standards and Technology (NIST)
Gillaspy_1997  was used to produce and trap highly charged ions of tungsten.
The low charge ions were injected into the trap using a metal vapor vacuum arc
(MEVVA) MEVVA , and then ionized to the required degree. The beam energies in
the present experiment varied between approximately 5500 eV and 6400 eV, and
the beam current was 100 mA.

The EUV spectra from tungsten ions were recorded with a flat-field grazing-
incidence spectrometer that has been described elsewhere Blagojevic_2005 . A
gold-coated concave grating with variable spaced grooves was used in
combination with a liquid-nitrogen-cooled back-illuminated charge-coupled
device (CCD) array having a matrix of 2048 $\times$ 512 pixels. The tungsten
emission was recorded between 10 nm and 20 nm with a resolving power of
approximately 350. As in our previous measurements M1 , this resolution was
sufficient to identify most of the observed lines. Each spectrum consisted of
10 one-minute exposures, and spurious signals from cosmic rays were removed.

One of the measured spectra between 14 nm and 20 nm at the nominal beam energy
$E_{b}$ = 6057 eV is shown in Fig. 1. The strongest lines of the W ions are
the 15.962-nm line in K-like $3d$ ion, the 14.959-nm and 17.080-nm lines in
Ca-like $3d^{2}$ ion, the 17.216-nm and 18.867-nm lines in Sc-like $3d^{3}$,
and the 19.319-nm and 19.445-nm lines in Ti-like $3d^{4}$ ion M1 . Each line
in the measured spectra was fit with a Gaussian function and a constant
background, and the derived intensities (areas) were used to obtain the line
intensity ratios. Since it is the line intensities rather than wavelengths
that are of interest in the present work, even an approximate calibration of
the measured spectra that provides a one-to-one correspondence with the known
wavelengths is sufficient here.

Figure 1:  Measured spectrum (in analog-to-digital units of CCD) at the
nominal beam energy of 6057 eV. The strongest lines of tungsten are identified
by their isoelectronic sequences. A strong impurity line at 15.02 nm due to
emission from oxygen and xenon ions is shown by an arrow.

##  III Modeling

As mentioned above, the $LMn$ dielectronic resonances are produced when the
$L$ electron ($2p_{3/2}$, $2p_{1/2}$, or $2s_{1/2}$ in relativistic notation)
is excited into the $M$ shell and a free electron is captured into an atomic
shell with the principal quantum number $n$. The value of $n$ under EBIT
conditions can be estimated in the limit of a very high ion charge $z$ when
atomic structure becomes quasi-hydrogenic. In units of ($z$+1)2Ry (Ry
$\approx$ 13.6057 eV is the Rydberg constant), the energy difference between
$n$=2 and $n$=3 is 5/36, and the ionization potential for the $n$=3 shell is
IP3=1/9. Numerous EBIT studies show that the beam energy required to make a
specific ion abundant is on the order or higher than its ionization potential.
Therefore, if a free electron has the energy of approximately IP3, then during
dielectronic capture for $L$-$M$ excitation it plunges 1/36(=5/36-1/9) below
the ionization potential, that is, into $n$=6. Certainly, the energy structure
for a 50+-times ionized atom of tungsten cannot be well described by the exact
hydrogenic formulas and the electron energy is normally larger than the
ionization potential but the above estimate of $n\sim 6$ is quite close to
reality, as will be seen below.

One of the widely used parameters representing the contribution of a
particular dielectronic resonance to DR is the dielectronic resonance strength
defined here as

$R_{\alpha}=\frac{g_{\alpha}}{g_{i}}\frac{A^{a}_{\alpha;i}\sum_{f}{A^{rad}_{\alpha;f}}}{\sum_{j}{A^{a}_{\alpha;j}}+\sum_{f}{A^{rad}_{\alpha;f}}},$
(2)

where $g_{\alpha}$ is the statistical weight of the doubly excited state
$\alpha$ in the recombined ion $z$, $g_{i}$ is the statistical weight of the
initial state $i$ in the recombining ion $z+1$, $A^{a}_{\alpha;j}$ is the
autoionization rate from $\alpha$ to state $j$, and $A^{rad}_{\alpha;f}$ is
the radiative rate from $\alpha$ to state $f$. The summation indices $f$ and
$j$ enumerate states in the recombined and recombining ions, respectively.
Note that the summation index $f$ in Eq. (2) extends over both autoionizing
and non-autoionizing states. It is customary to exclude the latter from the
sum in the numerator, however, for highly-charged high-Z ions the radiative
rates become stronger than the autoionization rates. Therefore radiative
transitions between autoionizing states would likely result in decays into
non-autoionizing states and provide additional contribution to the total
dielectronic recombination. Therefore we retain the autoionizing states in the
$f$-summation in the numerator. Then, in the general case, more than one state
in the recombining ion may contribute to dielectronic capture into a state
$\alpha$. However, due to the relatively strong M1 transitions in highly-
charged high-Z ions the populations of the lowest excited states of $3d^{n}$
are much smaller than that of the ground level, and therefore we neglect their
contribution to $R_{\alpha}$ in Eq.(2). Finally, it should be mentioned that
the resonance strength only approximately shows how much a specific doubly
excited state contributes to dielectronic recombination; the accurate answer
can only be obtained with a detailed collisional-radiative (CR) modeling PFR
which is provided below.

Large-scale CR modeling requires the generation of extensive sets of atomic
data, such as energy levels, radiative and autoionization rates, and electron-
impact cross sections. The data used in the present work were calculated with
the relativistic-model-potential Flexible Atomic Code (FAC) FAC . This code is
widely used for highly-charged high-Z ions, and its accuracy has proven to be
sufficient for such atomic systems. The calculated $LMn$ ($n$=3–5) resonance
strengths (Eq. (2))for dielectronic capture from the K-like ion of tungsten
into the Ca-like ion are presented in Fig. 2 as a function of $E_{b}$. The
resonances corresponding to different excitations from the $n$=2 shell are
given in different colors: black for $2p_{3/2}$ (often denoted as $2p_{+}$),
red for $2p_{1/2}$ ($2p_{-}$), and blue for $2s_{1/2}$ ($2s_{+}$). Figure 2
shows that the energy difference between the subshells of $n$=2 is non-
negligible, so that $\Delta E(2p_{3/2}-2p_{1/2})\approx$ 1 keV and $\Delta
E(2p_{3/2}-2s_{1/2})\approx$ 2 keV. This deviation from the quasi-hydrogenic
approximation should certainly modify the $n\approx$ 6 estimate. Then, in
addition to the well-known decrease of resonance strengths with $n$, one can
notice that the lowest resonances do not overlap and thus their contribution
to DR can be pinpointed with the narrow spread of energies in the EBIT beam.
Finally, and most importantly, the range of the beam energies, where both
K-like and Ca-like ions are observed to be most abundant M1  (horizontal arrow
in Fig. 2), overlaps with the $L_{3/2}MN$ resonances at approximately 6 keV.
Hence, one can expect that the measurements of ionization balance between the
K-like and Ca-like ions near 6 keV should indicate presence of such
dielectronic resonances.

Figure 2:  Calculated resonance strengths (in s-1) for the $LMn$ ($n$=3–5)
resonances in Ca-like W. Arrow shows the approximate range of beam energies
where both Ca-like and K-like ions are abundant. The labels show the quantum
numbers for the L-electron and the recombined free electron. For instance,
$2s_{+}3d$ corresponds to the $2s2p^{6}3s^{2}3p^{6}3d^{3}$ configuration.

The CR modeling used here differs from the approach implemented in our
previous papers on $3d^{n}$ ions M1 ; M1Hf  where only singly- and doubly-
excited states below the first ionization potential were included. Since
L3/2MN dielectronic recombination is the primary subject of this study, all
autoionizing states $2s^{2}2p^{5}_{3/2}3s^{2}3p^{6}3d^{3}$ and
$2s^{2}2p^{5}_{3/2}3s^{2}3p^{6}3d^{2}4l$ are added to the model. In the
absence of two-electron processes, the ionization balance is determined only
by ionization and recombination between two ion states, and therefore the CR
model included only Ca-like and K-like ions of tungsten. Even within a two-ion
model the total number of fine-structure levels was about 10,500 since in
addition to approximately 1300 autoionizing states in the Ca-like ion one has
to account for all possible autoionization channels.

The electron-impact collisional data were calculated with FAC and fit using
simple formulas with the correct asymptotic behavior, and a database was
created for simulations with the non-Maxwellian code NOMAD NOMAD . Unlike the
previous studies on M1 lines M1 ; M1Hf , we do not group the atomic levels
with high principal quantum number $n\geq$4 into “superterms”; this is done to
provide the most accurate representation of all processes affecting the
observed line emission and populations of autoionizing states.

Generally, CR simulations for EBITs may utilize a number of free parameters
that can be fixed using the experimental data. Thus, the unknown CX rate was
deduced from the spectra measured at non-resonant beam energies ($\approx$6400
eV) and then used in calculations for all other energies. This fit procedure
partially takes into account the higher-order CX processes. The beam electron
energy distribution function (EEDF) was taken to be a Gaussian with full width
at half-maximum of 50 eV which sets the energy resolution. It is not uncommon
in EBIT modeling to use a rectangular EEDF for spectra simulations due to the
smooth energy dependence of collisional cross sections. However, for the
present analysis a Gaussian beam provides a better agreement with the observed
shapes of spectral features. Finally, the effective beam energy that is seen
by ions in the trap is known to be smaller than the nominal value due to space
charge effects. Accordingly, the beam energies in the following discussion and
figures are reduced by 100 eV from their nominal values to match the
calculated positions of resonances.

##  IV Comparisons and discussion

The measured intensity ratio of the 17.080-nm line in Ca-like W54+ and the
15.962-nm line in K-like W55+ is shown in Fig. 3(top). The vertical error bars
correspond to the expanded uncertainty at the level of confidence of 95 % and
are derived from the uncertainties of Gaussian fits of the measured line
profiles using the coverage factor of $k$=2 NISTGuide ; Guide . Simultaneous
with our measurements in W, a small amount of argon was continuously
introduced into the EBIT through the EUV spectrometer and used to monitor
changes in the absolute value of the electron beam energy through the position
of the x-ray radiative recombination (RR) line into the ground state of H-like
argon. These measurements were performed with a wide-band high-purity Ge
solid-state detector. With the exception of a few energies (which may have
involved line blends) the position of the RR line varied linearly with the
nominal electron beam energy, with a standard deviation of 11 eV, consistent
with statistical uncertainties in the fits to the RR line centers. The error
bars in the horizontal direction (beam energy) in the plots of the present
manuscript are taken to be 22 eV (i.e., $k$=2). Using the data at a nominal
beam energy of 5.815 keV, we used the absolute value of the RR line center and
the known value of the ionization energy of He-like Argon to infer
(119$\pm$14) eV (at the level of 1$\sigma$) for the space charge correction to
the beam energy, in good agreement with the value of 100 eV obtained from the
EUV line ratios.

Without dielectronic resonances, the experimental Ca/K line ratio should be
monotonically decreasing with $E_{b}$. This dependence is confirmed by the CR
calculation that excluded autoionizing states (dotted line in Fig. 3(top)).
The measured line ratio, however, clearly shows presence of resonant features,
with the strongest ones near 6020 eV and 6160 eV. The origin of these peaks is
clarified in the bottom panel of Fig. 3 which is the enlarged part of Fig. 2
representing different $4l$ channels. The strongest measured peaks are seen to
correspond to the highest-$l$ channels, $4d$ and $4f$. The contribution from
$4p$ near 5880 eV is visibly smaller, and the weakest $4s$ resonances were
missed on the experimental grid.

Figure 3:  Top: circles – measured intensity ratio of the 17.080 nm line in
W54+ and the 15.962 nm line in W55+; dot-dashed line – simulation with
$m$-averaged autoionizing levels (AL) assuming isotropic collisions; dashed
line – simulation with $m$-averaged AL and effective anisotorpic corrections;
solid line – simulation with $m$-resolved AL; dotted line – simulation without
AL/DR. Both the energy uncertainty of $\pm$22 eV for all data points and the
line ratio uncertainties correspond to $2\sigma$. Bottom: resonance strengths
for 786 autoionizing states $2s^{2}2p^{5}3s^{2}3p^{6}3d^{2}4l$ in W54+.

The non-Maxwellian CR modeling with autoionizing states for the Ca/K line
ratios was performed at several levels of complexity. The dot-dashed line in
Fig. 3(top) presents the 17.080/15.962 line ratio calculated with the model
that included resonant states as J-resolved atomic levels. Here the DC rates
were obtained from the corresponding autoionization rates by integration of DC
cross sections over the beam EEDF. This approach is equivalent to the
assumption of isotropic collisions without any account of the monodirectional
nature of the EBIT beam. Whereas the dot-dashed curve in Fig. 3(top)
reproduces the relative intensities of the resonances reasonably well, it
overestimates their magnitudes. This discrepancy may result from the
anisotropic nature of beam propagation and the ensuing effect on the DC rates.
Due to the conservation of the angular momentum projection, not all magnetic
sublevels of the final recombined state can be populated via dielectronic
capture with monodirectional electrons. Accordingly, the net DC/DR rates are
expected to become smaller and thus reduce the disagreement between modeling
and measurements.

The polarization effects in spectral line emission due to anisotropic
excitation and/or dielectronic capture by monodirectional beam electrons are a
subject of continuing interest (see, e.g., PhysRevA.54.1342 ;
PhysRevA.74.022713 ; Hakel ; GillApJ ; ORourke ; Matula ). To include the
anisotropic effects, one can develop an extensive CR model resolved in
magnetic sublevels for autoionizing states; this model will be described
below. Another approach to this problem can be based on the introduction of
effective corrections that account for non-statistical populations of the
magnetic sublevels. Such corrections can be developed in several ways; here we
use the following multiplicative modification factors for all autoionization
rates and DC cross sections:

$\xi_{\alpha}=\frac{A_{\alpha}^{rad}+A_{\alpha}^{a}}{DC_{\alpha;1}}\sum_{m=-j}^{j}\frac{DC_{\alpha,m;1}}{A_{\alpha}^{rad}+A_{\alpha,m}^{a}},$
(3)

where $A_{\alpha}^{a}$ is the total autoionization rate for the state $\alpha$
with the total angular momentum $j$, $A_{\alpha}^{rad}$ is its total radiative
rate, $DC_{\alpha,1}$ is the isotropic DC cross section from the next ground
state into $\alpha$ (calculated from $A_{\alpha}^{a}$ using the detailed
balance principle), $DC_{\alpha,m;1}^{a}$ is the DC cross section from the
next ground state into the sublevel with the magnetic quantum number $m$, and
$A_{\alpha,m}^{a}$ is the total autonization rate from the same sublevel. This
correction can be justified as follows. The states that contribute most to DR
are those with large radiative transition probabilities $A_{\alpha}^{rad}\gg
A_{\alpha}^{a}$. In this case, the population kinetics of the autoionizing
states becomes similar to the coronal model where a level is populated only by
excitations (dielectronic capture in our case) and depopulated only via
radiative decay. The spectral line intensity for coronal condition does not
depend on the radiative rate and is determined by the excitation rate only.
Reformulating this for the present case, the contribution of a particular
strong dielectronic resonance to dielectronic recombination does not depend on
the radiative stabilization rate but rather on the dielectronic capture rate
only. For such resonances, the correction $\xi_{\alpha}$ becomes simply the
ratio $\sum_{m}{DC_{\alpha,m;1}}/DC_{\alpha}$. This multiplicative factor
applied to both autoionization rates and DC rates effectively modifies only
the latter ($A_{a}$ is too small to compete with $A_{rad}$) and is expected to
bring the total DR rate closer to the magnetic-sublevel-resolved model
results. Note also that correction (3) implies that the population of the
recombining ground state is much larger than the populations of other excited
states which, as mentioned above, is a good aproximation for the present
experiment.

The FAC code allows calculation of DC cross sections into and autoionization
rates from the $m$ states, and such a calculation was performed here. The
result of the “effective” CR calculation with the $\xi_{\alpha}$ factors is
presented by the dashed curve in Fig. 3(top). The agreement with the measured
line ratios is seen to improve, especially for the $4f$ resonances and the
$4p$ resonances near 5900 eV.

The most extensive CR model implemented in the present work splits all 1314
autoionizing levels $2s^{2}2p^{5}_{3/2}3s^{2}3p^{6}3d^{3}$ and
$2s^{2}2p^{5}_{3/2}3s^{2}3p^{6}3d^{2}4l$ into magnetic sublevels, bringing the
total number of such states to 9360, and the total number of all states for a
two-ion CR model to more than 18,000. All possible radiative, autoionization
and dielectronic capture rates connecting the $m$-states were calculated with
FAC and used in NOMAD simulations. The 17.080/15.962 line ratio calculated
with this complete model is presented by the thick solid line in Fig. 3(top).
The “complete” and “effective” models show a comparable level of agreement for
$4p$ and $4f$ resonances, but for the strongest $4d$ resonances near 6020 eV
the complete model is closer to the measured data. These simulations give a
strong indication that anisotropic effects in EBITs are important for analysis
of dielectronic recombination. In particular, this shows that the DR/DC cross
sections measured in EBITs cannot be treated as the isotropic cross sections.

A natural question arises whether monodirectional propagation of the beam
electrons should affect not only dielectronic resonances but also the M1 lines
and their ratios. The important difference between the autoionizing states and
the low excited $3d^{n}$ levels is that the former are populated due to direct
interaction of beam electrons with an ion while the latter are populated due
to radiative cascades from the excited higher states. While excitations from
the ground state are generally polarization-sensitive, the radiative cascades
smear out polarization signatures for the lowest levels. This was explicitly
shown in the recent study of electric-quadrupole and magnetic-octupole lines
in Ni-like W46+ PhysRevA.81.012505 .

A significant advantage of the experimental method presented here consists in
its ability to provide simultaneous resolved measurements of dielectronic
resonances from several ions. Even for the modest resolution spectrometer used
here, the magnetic-dipole lines from different ions are normally well
separated in an EUV spectrum, as exemplified in Fig. 1 (see also Ref. M1 ;
M1Hf ), and therefore line ratios for several ion pairs can be reliably
determined from a single EBIT run. Such analysis was performed here as well,
and Fig. 4 shows the measured ratios for the three pairs of ions, i.e., Ca-
like to K-like (top), Sc-like to Ca-like (middle), and Ti-like to Sc-like
(bottom). The 17.080/15.962 ratio (same as in Fig. 3) and 18.867/17.080 ratio
in the top and middle panels, respectively, are shifted upward by one and two
units, rspectively, for a better visibility. While two other lines, at 17.216
nm in the Sc-like ion and at 19.445 nm in the Ti-like ion, look strong in Fig.
1, we did not use them in this analysis as both are blended by impurity lines
that affect their intensities.

Figure 4:  Measured line ratios for the Ca-like to K-like ions (top), Sc-like
to Ca-like ions (middle), and Ti-like to Sc-like ions (bottom). The
theoretical ratios calculated with the anisotropic collisional-radiative model
are shown by solid (FWHM = 50 eV) and dashed (FWHM = 40 eV) lines in the top
panel. The calculated relative strengths for the L($2s_{1/2}$)MM (blue) and
L($2p_{3/2}$)MN (black) resonances are shown by vertical lines in the middle
and bottom panels. The dotted lines are shown for an eye guide. Both the
energy uncertainty of $\pm$22 eV for all data points and the line ratio
uncertainties correspond to $2\sigma$.

Figure 4(top) shows the experimental 14.959/15.962 Ca/K ratio and the
“complete” anisotropic simulation with the beam FWHM of 50 eV (solid line).
This calculation does not include any free parameters that have not already
been fixed in simulation of the 17.080/15.962 ratio. For the 14.959/15.962
ratio, the agreement between theory and measurements is quite good over the
whole range of energies although for both ratios our simulations slightly
overestimate the DR contribution for the $4f$ resonances at 6150 eV. In
addition to the 50-eV-FWHM calculation, we present the 40-eV-FWHM results
(dashed curve) as well. The difference between the two theoretical simulations
is not very significant, however smaller or larger values of the EBIT beam
width were found to be in worse agreement with the measurements. Obviously,
detailed measurements of DR resonances, especially isolated ones such as the
resonance at $\sim$5925 eV in Fig. 3, can be used to determine the beam width
more accurately.

The available computational facilities did not allow us to perform a full-
scale collisional-radiative modeling for the Sc/Ca or Ti/Sc line ratios,
similar to the results presented for the Ca/K ratios. The total number of
levels in a CR model rapidly increases with the number of electrons $n$ in the
ground configuration $3d^{n}$. Therefore, in the middle and bottom panels of
Fig. 4 we only present the calculated relative dielectronic resonance
strengths which are shown by the blue ($2s_{1/2}$ inner electron excitation,
$LMM$ resonances) and black ($2p_{3/2}$ inner electron excitation, $LMN$
resonances) vertical lines. This comparison shows that for the $3d^{3}$ and
$3d^{4}$ ions of tungsten both positions and strengths of the resonances are
calculated as accurately as for the $3d^{2}$ Ca-like ion (Figs. 2 and 3).

The measured line ratios indicate that the beam energy required to produce a
particular $LMN$ resonance decreases with the increase of the ion charge $z$.
As is seen in Fig. 4, the energy of the strongest peak due to $2p_{3/2}4d$
resonances is 6020 eV for Ca-like W54+, about 6070 eV for Sc-like W53+, and
6100 eV for Ti-like W52+. Similar dependences on $z$ are also visible for the
weaker $4f$ peak. Such behavior would seem to contradict the well-known
increase of atomic energies with $z$. The explanation can be derived from a
simple analysis of energy balance. The resonant beam energy is equal to the
difference between the excitation energy $\Delta E_{23}$ from n=$2$ to n=$3$
and the ionization energy IP4 of the $4l$ subshell. Since excitation involves
a deeply screened $2l$ electron, $\Delta E_{23}$ varies with $z$ much less
than IP4 which approximately increases as $z^{2}$. These dependences are
confirmed by our FAC calculations along isoelectronic sequences. As a result,
the resonant beam energies for the $LMN$ resonances in $3d^{n}$ ions of
tungsten weakly decrease with the ion charge.

Among the possible radiative stabilizing channels for the $LMN$ autoionizing
states, the strongest ones are due to electric-dipole transitions into the
$2p_{3/2}$ hole, that is, $2p_{3/2}$-$4l$ (referred to below as A) with
$l$=$s$ or $d$, $2p_{3/2}$-$3d$ (B), and $2p_{3/2}$-$3s$ (C). These
transitions result in emission of photons with the energies of about 11 keV,
8.8 keV, and 7.8 keV, respectively. To record these x-rays, we used the high-
purity Ge solid state detector. The measured spectrum presented in Fig. 5
unambiguously reveals the three radiative channels A, B, and C for the
electron beam energies between 5.7 keV and 6.3 keV. Note that there is a
background of radiative recombination with some RR channels visible as weak
diagonal bands. Unlike the EUV line ratios that are resolved for individual
ions, the stabilizing x-ray photons may simultaneously originate from several
ions that are abundant at a specific beam energy. For instance, as can be seen
from Fig. 4, dielectronic resonances can be produced for both Ca-like and Ti-
like ions at the beam energy of 6140 eV. Since our present CR model is
restricted only to two ions, it is not currently possible to accurately
analyze the observed x-ray emission. Yet some characteristics of the measured
spectrum confirm our identifications. For instance, the B transitions extend
to the beam energies higher than 6.1 keV while there are no A transitions in
that region. This agrees with the preferential population of the $4f$ subshell
at highest energies as $4f$ electrons cannot decay into $2p$ via electric-
dipole transitions.

Figure 5:  Measured x-ray emission (in arbitrary units) between 7.5 keV and 12
keV registered with a Ge solid state detector. A: $2p$-$4l$ transitions, B:
$2p$-$3d$ transitions, C: $2p$-$3s$ transitions.

We mentioned above that the energies of the $2p$-$3l$ inner-shell transitions
(channels B and C) are almost independent of the ion charge. Indeed, the
measured central energies for the B and C features in Fig. 5 do not visibly
change with the beam energy although the ion populations vary significantly.
The changing central energy of the $2p$-$4l$ spectral group A is, however,
clearly discernible as confirmed by the Gaussian fits of this feature. One can
see that the lowest-energy $2p$-$4d$ x-ray photons (from lower-$z$ ions)
appear at higher beam energy and vice versa. This agrees with the above
discussed $z$-dependence of the resonant beam energy and the movement of the
$2p$-$4d$ peak (see Fig. 4).

##  V Conclusions

In this work we proposed a new in situ method of measuring the $LMN$
dielectronic resonances in $3d^{n}$ ions of heavy elements through intensity
ratios of magnetic-dipole lines within ground configurations. Such resonances
were experimentally observed for Ca-, Sc-, and Ti-like ions of tungsten in an
electron beam ion trap with beam energies of about 6 keV. In addition to the
EUV spectra, x-ray photons due to the radiative stabilization phase of
dielectronic recombination were recorded using a high-purity Ge detector. A
detailed non-Maxwellian collisional-radiative model was developed to simulate
the observed line intensity ratios. It was shown that the isotropic CR
simulations overestimate the effect of DR on the measured line ratios. We
proposed effective corrections to account for non-statistical populations of
magnetic sublevels of autoionizing states and also developed a complete
magnetic-sublevel-resolved CR model. Both new models were found to improve
agreement with the measured line ratios thereby pointing out the importance of
anisotropic effects for dielectronic recombination in EBITs. The remaining
difference between the simulations and experiment may stem from imperfection
of atomic data, incompleteness of the CR model, or for instance multi-electron
processes. These issues should be addressed in future works. It is rather safe
to claim, however, that the measured line ratios offer benchmark data for both
atomic structure theories and advanced non-Maxwellian models of plasma
emission.

###### Acknowledgements.

Supported in part by the Office of Fusion Energy Sciences of the U.S.
Department of Energy. We are grateful to J.J. Curry for assitance during the
experimental run and to D. Osin, A. Kramida, Y. Podpaly, J. Reader, and J.N.
Tan for valuable discussions.

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Motivated by the recent synthesis of two-dimensional monolayer AlSb, we
theoretically investigate its ground state and electronic properties using the
first-principles calculations coupled with Bethe-Salpeter equation. An
excitonic instability is revealed as a result of larger exciton binding energy
than the corresponding one-electron energy gap by $\sim$0.1 eV, which is an
indicative of a many-body ground state accompanied by spontaneous exciton
generation. Including the spin-orbit coupling is proven to be a must to
correctly predict the ground state. At room temperature, the two-dimensional
monolayer AlSb is a direct gap semiconductor with phonon-limited electron and
hole mobilities both around 1700 cm2/V$\cdot$s. These results show that
monolayer AlSb may provide a promising platform for realization of the
excitonic insulator and for applications in the next-generation electronic
devices.

# Excitonic Instability and Electronic Property of Two-dimensional AlSb Limit

Shan Dong  Key Lab of advanced optoelectronic quantum architecture and
measurement (MOE), and Advanced Research Institute of Multidisciplinary
Science, Beijing Institute of Technology, Beijing 100081, China  Yuanchang Li
<EMAIL_ADDRESS>Key Lab of advanced optoelectronic quantum architecture and
measurement (MOE), and Advanced Research Institute of Multidisciplinary
Science, Beijing Institute of Technology, Beijing 100081, China

I. Introduction

The successful isolation of graphene in 2004 demonstrated the possibility to
synthesize atomically thin materials from the layered van der Waals solids by
exfoliation. Since then, there is growing theoretical and experimental
interest in the research of other two-dimensional materials such as hexagonal-
BN, transition metal dichalcogenides, phosphorene, metal halides, _etc_. These
crystals exhibit many fascinating electronic, magnetic and optic properties
that are absent in their bulk counterparts, and thus hold a great application
potential ranging from field-effect transistors to spin- and valley-tronics to
photocatalytic water splittingButler ; Bernardi ; WangG . They also provide
new opportunities for some long-standing physical problems, for example, the
excitonic insulator.

Excitonic insulator was proposed by theoretical physicists more than 50 years
agoMott ; Knox ; Kohn ; Halperin . It is originated from the excitonic
instability caused by the larger exciton binding energy than the corresponding
one-electron energy gap, alongside with spontaneous exciton condensation. It
has a many-body ground state similar to a superconductor and spontaneous
symmetry-breaking occursKohnRMP . However, to date, the excitonic insulator
has remained a mystery and there is still a lack of compelling evidence in
experimentKogar . Atomically thin materials usually have a significantly
enhanced exciton binding energy owing to their weak electron-hole screening
interaction as compared to their bulk counterpartsWangG ; usPRL . They
naturally provide more opportunities for the occurrence of excitonic
instability and the realization of excitonic insulator stateusEI ; usHEI ;
usgraphone ; usDong ; Varsano ; LiZ . In traditional bulk materials, the
excitonic insulator is often limited to narrow-gap semiconductors or semi-
metals, while in the two-dimensional limit, the existence in moderate or wide
gap semiconductors is also discussed by recent worksusEI ; usHEI ; usgraphone
.

Another outstanding advantage of two-dimensional materials is their suppressed
short-channel effects at the scaling limitYan , which makes them promising in
building the next generation of field-effect transistors. Although graphene
offers the ultra-high carrier mobility, the lack of an energy gap leads to a
low on-off current ratio, limiting its device applicationsLiao . Single-layer
MoS2, which is the mostly studied member of the transition metal
dichalcogenides, possesses a gap $\sim$1.8 eV and shows $n$-type conductivity
with a mobility $\sim$200 cm2/V$\cdot$s Yoon ; Kaasbjerg ; Cai . Monolayer
phosphorene has a direct gap of 1.5 eV and shows high hole mobilityLiu ; Qiao
. The search for new two-dimensional semiconductors with suitable band gap and
high mobility has been a never-ending pursue. Binary III-V semiconductors are
one of the most important class of semiconducting materials with enormous
technological applications. The bulk of these compounds generally crystalize
in zinc-blende or wurtzite structure. Such a non-layered nature thus hinders
to directly exfoliate their two-dimensional counterparts, which, instead, can
be obtained via surface growthButler ; Balushi . The study of two-dimensional
form of III-V semiconductors is still in its infancy. Experimentally, Balushi
et al.Balushi  synthesized a bilayer GaN on the SiC substrate via a migration-
enhanced encapsulated growth technique. Theoretically, Lucking et al.Lucking
predicted that the III-V semiconductors stabilize in a double layer honeycomb
structure at the ultrathin limit. Very recently, Qin et al.Qin  reported the
successful growth of two-dimensional AlSb in the double-layer honeycomb
structure through molecular beam epitaxy on graphene-covered SiC(0001), and a
fundamental gap of 0.93 eV, similar to the Si, is found.

In this paper, we investigate the ground-sate geometric and electronic
structure of monolayer AlSb using the first-principles calculations plus the
Bethe-Salpeter equation (BSE). An excitonic instability is revealed because
the ground-state exciton has a binding energy larger than the corresponding
one-electron gap by $\sim$0.1 eV. This implies that the monolayer AlSb could
be an intrinsic excitonic insulator. Our calculations indicate that spin-orbit
coupling (SOC) must be involved in order to correctly predict the ground state
of monolayer AlSb. Since the many-body ground state tends to become unstable
above the phase transition temperature, we further calculate the intrinsic
carrier mobility of monolayer AlSb within deformation potential theory. Both
electron and hole mobilities are found to be on the level of $\sim$1700
cm2/V$\cdot$s at room temperature, and hence monolayer AlSb may also serve as
a promising candidate for several electronic device applications, such as a
high-energy photon detectorYee .

II. Methodology and models

First-principles calculations were carried out by using the QUANTUM ESPRESSO
packageQE  with the Perdew-Burke-Ernzerhof (PBE) PBE  exchange-correlation
functional. A cut-off energy of 50 Ry was employed for optimized norm-
conserving Vanderbilt pseudopotentialsHamann . An $18\times 18\times 1$
$k$-point grid was used to sample the Brillouin zone. A vacuum space of 20 Å
along the direction normal to the double-layer plane was set to avoid spurious
interactions between adjacent layers. In order to cure the band gap problem of
the Kohn-Sham DFT, single-shot $G_{0}W_{0}$ calculations were performed by the
YAMBO codeyambo . Dielectric functions and low-energy excitonic properties
were calculated by solving the BSE with the Coulomb cutoff technique. Top four
valence bands and bottom two conduction bands are chosen to build the BSE
Hamiltonian. The same $k$ grid, 312 bands, and 12 Ry cutoff were used for the
dielectric function matrix.

Figure 1:  (Color online) Side views (Left panel) and top views (Right panel)
of two kinds of double-layer configuration for two-dimensional AlSb limit. (a)
Proposed by Lucking et al. in Ref. Lucking, . (b) InSe-like structure with
bonding between the same Sb atoms. Black dashed rectangles denote the unit
cell.

III. Results and discussion

The essence that $p$-electrons tend to form covalent bonds often leads to a
buckled geometry at the single-layer honeycomb limit when the compounds do not
contain one of the first row elementsSahin . For III-V semiconductors, this
will be accompanied by a dipole moment along the direction perpendicular to
the basal plane, which is not conductive to lower the system energy. In this
sense, Lucking et al.Lucking  proposed a double-layer honeycomb structure as
illustrated in Fig. 1(a), to compensate the vertical dipole moments through an
“inter-layer” coupling, which consequently becomes more favorable in energy
than the single-layer case. In fact, the double-layer honeycomb structure can
also be presented in another form as depicted in Fig. 1(b). The latter is
often taken by monolayer III-VI semiconductors like InSeLei , which differs
from the one shown in Fig. 1(a) in the bonding between atoms of the same
element. We compare the energetics of two-dimensional AlSb with the above two
configurations. Our calculated results show that the InSe-like AlSb structures
is energetically unfavorable by about 1.32 eV than the one proposed by Lucking
et al.

Figure 2:  (Color online) (a)[(b)] One-electron band structure of monolayer
AlSb without[with] the spin-orbit coupling. Black solid lines are from the PBE
while red dashed lines are from the $G_{0}W_{0}$ on top of the PBE. The Fermi
level is set to energy zero. Band parities are also calculated for the states
near the Fermi level, and in (b) “+” and “-” denote the even and odd parity,
respectively. (c)[(d)] Decomposed charge density for the top valence[bottom
conduction] band at the $\Gamma$ point corresponding to the PBE band shown in
(a), with an isosurface of 0.07 e/Å3.

Next, we explore the ground-state electronic properties of ultrathin AlSb with
the configuration of Fig. 1(a). Shown in Fig. 2(a) is its band structure
without considering the SOC. At the PBE level, it is a semiconductor with a
direct gap very close to zero ($\sim$0.04 eV) at the $\Gamma$ point. This is
in sharp contrast to the bulk AlSb which has an indirect-gap of 1.29 eV from
the PBELucero  and 1.686 eV from the experimentAlibert . The fact that the
calculated gap of bulk AlSb is $\sim$30% smaller than the experimental value
once again proves the persistent problem of PBE’s gap underestimation. A
similar trend is certainly anticipated for the two-dimensional limit, and the
underestimation may be even more severe as a result of the reduced dimension.
To this end, we carried out many-body $G_{0}W_{0}$ calculation and the
corresponding band is plotted in Fig. 2(a) [Red dashed lines] for comparison.
It gives rise to a quasi-particle gap of 1.35 eV, which is much larger than
$\sim$0.04 eV by the PBE. Except for the gap size, the $G_{0}W_{0}$ band
highly resembles the PBE one, including the band dispersion and the direct-gap
characteristic.

We note, however, that the $G_{0}W_{0}$ gap of 1.35 eV becomes much larger
than the experimental measurement of 0.93 eVQin . One possible reason is the
neglect of SOC given that the compound contains heavy element Sb. Indeed, gap
reduction resulted from the SOC has been reported in the bulk AlSbAli . As can
be seen in Fig. 2(b), the band changes a lot when switching on the SOC. At the
PBE level, it exhibits a metallic behavior and there is a small overlap
between the top valence band and bottom conduction band, yielding a negative
gap of -0.17 eV at the $\Gamma$ point. The $G_{0}W_{0}$ opens up a positive
gap but the size of 0.74 eV is smaller by 0.19 eV than the experimentQin .
Apart from the energy gap, another notable feature caused by the SOC is the
lift of degeneracy at the $\Gamma$ point. This is especially significant for
the top valence band, up to 0.7 eV (0.42 eV) by the $G_{0}W_{0}$ (PBE).

One can understand the SOC induced gap reduction as follows. Plotted in Figs.
2(c) and 2(d) are the decomposed charge densities for the band edge states
from the PBE [see Fig. 2(a)]. One can see that the valence band maximum is
dominantly comprised of the Sb orbitals [see Fig. 2(c)]. The Sb’s $p_{x}$ and
$p_{y}$ orbitals form a $\pi$-like bond and exhibit an in-plane distribution.
On the other hand, the conduction band minimum is dominantly comprised of the
Al orbitals [see Fig. 2(d)]. The Al’s $s$ orbitals form a head-to-head
$\sigma$-like bond and exhibit an out-of-plane distribution. Because the Sb
has a much stronger SOC than that of the Al, the SOC induced splitting
remarkably pushes up the top valence band, resulting in the narrowing of
energy gap.

It is noted that for III-V semiconductors, the energy gap under two-
dimensional limit is generally smaller than their bulk counterparts, for here
observed AlSb or previously reported GaAs and othersusEI ; Lucking .
Nevertheless, the trend is reversed for II-VI and I-VII semiconductorsLucking
, as well as van der Waals layered crystalsQiao ; Zhang ; Gonzalez ; Zhao ,
that is, the energy gap usually increases from the bulk down to the monolayer
as a result of the confinement effect. Such a difference is probably
attributed to their unique double-layer honeycomb structure and deserves
further exploration.

Figure 3:  (Color online) (a)[(b)] Imaginary part ($\varepsilon_{2}$) of the
BSE dielectric function, and (c)[(d)] all exciton states below the quasi-
particle gap without[with] the spin-orbit coupling in the low-energy region.
In (c) and (d), blue and red horizontal lines represent dark and bright
excitons, and $X_{1}$ denotes the ground-state exciton with the lowest
excitation energy. A negative excitation energy means spontaneous generation
of the corresponding exciton. (e)[(f)] Reciprocal and real-space wave
functions of the $X_{1}$ exciton without[with] the spin-orbit coupling. The
density has been normalized by choosing the maximum value to be unity for
reciprocal wave functions. While for the real-space plots, the isosurface
corresponds to an electron density of 0.68 e/Å3. Black dots denote the hole
positions.

The III-VI semiconductors have traditionally been used for luminescence. In
combination with the direct-gap nature of monolayer AlSb, we turn to
investigate its optical properties. In Figs. 3(a) and 3(b), we compare the
optical adsorption spectra without and with the SOC, respectively. Both are
calculated on top of the corresponding $G_{0}W_{0}$ bands. Not considering the
SOC, the first absorption peak appears at 0.82 eV, which is located inside the
quasi-particle gap. Hence, it corresponds to an exciton adsorption, with a
binding energy of 0.53 eV. When the SOC is included, the first adsorption peak
is located at 0.65 eV, which is just 0.09 eV lower than the quasi-particle
gap.

Despite the lack of any spectral signature, the dark exciton states are also
attractive in low-dimensional systems, especially for collective effects such
as a Bose-Einstein condensationusEI ; usHEI ; usgraphone ; usDong ; Beian . In
this regard, we present all low-energy excitons of monolayer AlSb below the
quasi-particle gap in Figs. 3(c) and 3(d), no matter optically active or
inactive. Switching off the SOC, as illustrated in Fig. 3(c), five dark
exciton states emerge below the first bright exciton. The minimum energy
required to generate an exciton (dark) is 0.43 eV, which is 0.39 eV smaller
than the required by photoexcitation. When the SOC is taken into account, many
dark exciton states emerge below the first bright exciton [see Fig. 3(d)]. But
in this case, to our surprise, the lowest exciton state has a negative
excitation energy, namely, -0.10 eV. This means that spontaneous exciton
production will lower the system energy. As a result, an excitonic instability
would occur relative to the one-electron band structure, pointing towards the
formation of a many-body ground state, i.e., the excitonic insulatorKohn .

Two points are noteworthy for here predicted excitonic instability. On the one
hand, it is the role played by the SOC. Above results clearly show the
necessity of including SOC. Otherwise, ones would wrongly predict an one-
electron ground state. In order for a better understanding, we compare the SOC
impact on properties of the ground-state exciton, denoted as the $X_{1}$ in
Figs. 3(c) and 3(d). First, it is doubly degenerate under both cases, in
connection with the transition between valence band maximum and conduction
band minimum. Second, it is not the spin selection rule that forbids the
dipole transition as the $X_{1}$ remains to be dark in the absence of the SOC.
Previous studyusEI  in similar monolayer GaAs indicated that the elimination
comes from the band parity. For this reason, we calculate the parity for
relevant states. It turns out that the frontier states all possess the same
odd parity as shown in Fig. 2(b), so the transitions between them are dipole
forbidden, producing these low-energy dark excitons. Third, we compare the
reciprocal and real-space wave functions of the $X_{1}$ exciton in Figs. 3(e)
and 3(f). No apparent difference is found. Both are concentrated on the
$\Gamma$ point in the Brillouin zone and display an isotropic $s$-wave
character. Localization in the reciprocal space implies delocalization in the
real space. Indeed, they become highly extended, over more than ten unit
cells. Relatively speaking, the exciton wave function is more delocalized when
the SOC is considered. This is consistent with the intuition that the exciton
binding energy of 0.84 eV is slightly less than that of 0.92 eV given by
ignoring the SOC.

On the other hand, we have to keep in mind that the $G_{0}W_{0}$ plus SOC
calculations lead to an underestimated one-electron gap for monolayer AlSb,
namely, 0.74 eV vs. 0.93 eV of experiment. Needless to say, this will
profoundly affect the system screening interaction, and may even reverse the
relative size of one-electron energy gap and exciton binding energy, changing
the conclusion of excitonic instability. Although the $G_{0}W_{0}$ calculation
on top of the HSE input succeeded to produce a one-electron gap comparable to
the experimentQin , directly solving the BSE on this level is, however,
unfortunately, too computationally demanding to carry out for us at present.
Herein we circumvent the heavy-cost calculations and chose an alternative way
of using the scissor operator, which is introduced both for the response
function and diagonal part of the BSE kernelSangalli .

Figure 4:  (Color online) Dependence of excitation (Red balls) and binding
(Blue stars) energy for the $X_{1}$ exciton as a function of the quasi-
particle gap that is corrected by a scissor operator. Red and blue dashed
lines are a guide for the eye. Black horizontal dashed line indicates energy
zero.

In Fig. 4, we summarize the dependence of exciton excitation/binding energy as
a function of the quasi-particle gap (after applying a scissor correction).
Above all, the exciton excitation energy is always negative in the whole gap
range studied, from 0.74 eV to 1.14 eV. Note that the gap of 1.14 eV has
exceeded the experimental value of 0.93 eV by 0.21 eV but the exciton
excitation energy remains to be negative, i.e., -0.05 eV. In other words,
although the calculation error on one-electron energy gap inevitably affects
magnitude of the $X_{1}$ exciton excitation energy, it will not change the
conclusion that the $X_{1}$ exciton spontaneously forms and causes the
excitonic instability. In more detail, one can see that the exciton
excitation/binding energy first decreases/increases linearly, reaches its
minimum/maximum near the experimental gap, and then reverses the variation
trend, increasing/decreasing in complexity. Such a non-monotonic behavior
reflects a sophisticated relationship between exciton energy structure and
one-electron energy gap.

Akin to a superconductive state, the excitonic insulator state exists only at
relatively low temperature. Going to high temperature, the many-body state
becomes unstable more and more. Eventually, the exciton is dissociating at
elevated temperature and the system presents the electronic properties
characterized by its one-electron band structure. For example, a semi-
hydrogenated graphene has an exciton binding energy exceeding its one-electron
gap by $\sim$0.1 eV, while a transition temperature around 11 K is derived
through an effective Hamiltonian calculationusgraphone . It is well below the
room temperature albeit high enough for a Bose condensate. Likewise, a similar
performance can be expected for monolayer AlSb. In this regard, two-
dimensional AlSb is more likely to present the one-electron properties instead
of a spontaneous exciton condensate at room temperature. Owing to its
comparable gap to the Si, it is natural to consider monolayer AlSb for
applications in the next generation of electronic and optoelectronic devices.

Table 1: Effective mass ($m^{*}$ in terms of free electron mass m0), deformation-potential constant ($E_{1}$), elastic modulus ($C$), and carrier mobility ($\mu$) of monolayer AlSb at 298 K. All values are calculated on basis of the PBE band structure with the SOC included, as plotted in Fig. 2(b). carrier | $m^{*}$ |  $C$ (N/m) |  $E_{1}$ (eV) |  $\mu$ (cm2/V$\cdot$s)
---|---|---|---|---
electron | 0.07 | 62.95 | -10.38 | 1706
hole | 0.17 | 62.95 | -4.31 | 1695

For this purpose, carrier mobility is a central quantity. Generally speaking,
the carrier mobility needs to be at least on the level $\sim$500
cm2/V$\cdot$s, in order to compete with commercially available silicon-based
devicesPonce . For example, the lower mobility limits monolayer MoS2
transistor for high-performance applicationsYoon . Herein we consider the
acoustic phonon scattering limited mobility within the deformation potential
theory, which can be regarded as an upper limit for the experimentally
achievable measurement. Under this method, the mobility $\mu$ is calculated by
the formulaYoon ; Kaasbjerg ; Cai ; Qiao ; Dai

$\mu=\frac{2e\hbar^{3}C}{3k_{B}T|m^{*}|^{2}E_{1}^{2}},$ (1)

where $T$ is the temperature and $m^{*}$ is the carrier effective mass.
$E_{1}$ is the deformation potential constant and is obtained from the strain
induced shift of conduction/valence-band extrema. $C$ is the elastic modulus,
defined as $C$=$\partial^{2}E/\partial\delta^{2}$/$S^{0}$, where $E$ is the
total energy, $\delta$ is the applied strain, and $S^{0}$ is the area of
adopted supercell. To balance the computational efficiency and accuracy, here
the mobility is estimated based on the PBE band including the SOC.

Figure 5:  (Color online) (a) Change of total energy and (b) shift of band
extrema, as a function of the isotropic in-plane strain. The total energy, as
well as the valence band maximum, for the equilibrium case is set to zero for
reference in the plots. In (b), red and blue lines correspond to the
conduction band minimum and the valence band maximum, respectively. Note that
herein the former is below the latter due to the band gap underestimation by
the PBE [see Fig. 2(b)].

We first derive $m^{*}$ from the band shown in Fig. 2(b). They are 0.07 m0 and
0.17 m0, respectively for electrons and holes (m0 denotes the free electron
mass). We have examined the directional dependence of $m^{*}$ and an almost
completely isotropic behavior is found. It is worth mentioning that the AlSb
has both very light $m^{*}$ for electrons and holes, which is distinguished
from other typical two-dimensional monolayer materials such as the MoS2Yoon ;
Kaasbjerg ; Cai , phosphoreneQiao  and TiS3Dai .

In order to compute $E_{1}$ and $C$, strains up to 1.5% are applied to the
AlSb lattice and then the change of total energy, as well as the energy shift
of band extrema, is measured relative to the strain. The results are shown in
Fig. 5 and the obtained $E_{1}$ and $C$ are summarized in Table I. Inserting
$m^{*}$, $E_{1}$ and $C$ into the formula (1), $\mu$ is computed to be around
$\sim$1700 cm2/V$\cdot$s, either for electrons or holes at 298 K (see Table
I). In spite of its relatively lighter carrier $m^{*}$, the intrinsic mobility
of monolayer AlSb is not high in comparison with other two-dimensional
materials. It is only superior to the MoS2Yoon ; Kaasbjerg ; Cai  and lower
than those of phosphoreneQiao  and TiS3Dai . Anyway, such mobilities, together
with its direct-gap feature and light $m^{*}$, render monolayer AlSb useful in
applications of electronics and optoelectronics.

IV. Conclusions

In summary, we investigate the ground-state properties of newly synthesized
two-dimensional monolayer III-V semiconductor AlSb by using the first-
principles calculations in combination with Bethe-Salpeter equation. We find
two attractive advantages of this material. On the one hand, an excitonic
instability is revealed and thus it serves as a promising candidate for the
long-sought excitonic insulator. On the other hand, monolayer AlSb can be used
for high-performance electronic applications due to its light carrier
effective mass and sufficient carrier mobility. We expect that our work could
attract more attention to the two-dimensional form of traditional III-V
semiconductors.

Acknowledgments

This work was supported by the Ministry of Science and Technology of China
(Grant No. 2020YFA0308800), the National Natural Science Foundation of China
(Grant Nos. 12074034 and 11674071), and the Beijing Institute of Technology
Research Fund Program for Young Scholars.

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