# Atomic Quantum Sensor for Ultralow-Frequency Electric Field Measurements

Yu-Chi Chen  Shao-Cheng Fang  Yi-Hsin Chen<EMAIL_ADDRESS>Department of Physics, National Sun Yat-Sen University, Kaohsiung 80424,
Taiwan

###### Abstract

Rydberg-assisted atomic electrometry with thermal vapors offers a promising
approach for detecting external electric fields. This technique, however, has
proven quite challenging for measuring low frequencies because of the effects
of metal-alkali atoms absorbing on the interior surface of the vacuum
enclosure. We apply high-contrast Rydberg-electromagnetically-induced-
transparency (EIT) spectroscopy to investigate the impact systematically,
including laser power and electric field. Our system enhances frequency
measurements below 10 Hz, demonstrating significant progress in improving the
performance of Rydberg atomic electrometry. Consequently, our study reveals a
standard quantum limit for data capacity and experimentally illustrates it
across bandwidths ranging from 10 Hz to 1 MHz. This study provides valuable
insights for future precision measurements and field sensing applications.

††preprint: APS/123-QED

##  I Introduction

Highly-excited Rydberg atoms exhibit a substantial electric dipole moment,
leading to significant polarizability and robust interactions with light [1,
2, 3]. Moreover, utilizing the dipole-dipole interaction (DDI) between Rydberg
atoms can fine-tune the interactions among quantum bits, offering promising
prospects for advancing quantum computing, including quantum gate operations
[4, 5]. The combination of Rydberg interactions and electromagnetically
induced transparency (EIT) allows for the realization of photon quantum gates
[6, 7, 8]. Furthermore, the interaction strength between Rydberg atoms can be
further enhanced through Förster resonance, achieved by applying external
fields based on the Stark effect [9, 10, 11, 12, 13, 14, 15] with static
fields [9, 10, 11, 12], microwaves [14, 13], or optical fields [15].

Figure 1: (a) Energy levels of $\rm{}^{87}Rb$. The probe beam, with a
wavelength of 780 nm, was frequency locked onto the transition from the ground
state $|5S_{1/2},F=2\rangle$ to the excited state
$|5P_{3/2},F^{\prime}=3\rangle$. The coupling beam, with a wavelength of 480
nm, was swept to the transition from the excited state
$|5P_{3/2},F^{\prime}=3\rangle$ to the Rydberg states $53D_{3/2}$ and
$53D_{5/2}$. The frequency difference between these two states is 78.3 MHz,
used to calibrate the scanned frequency in a spectrum. (b) The schematic of
the optical setup. The probe and coupling beams are counter-propagated, and
their polarization is $\rm{\sigma^{+}}$. A reference cell is used for
frequency calibration. Two stainless steel electrodes were attached on each
side of the science cell, with a distance of 20 mm between them. A heating
block made of aluminum is placed outside the electrodes. DM: dichroic mirror;
$\lambda/4$: quarter wave plate; BS: cubed beam splitter; PD: photodetector.
(c) Rydberg EIT spectrum without external electric field was measured using
probe field transmission in the reference cell.

Rydberg atomic electrometry provides a method for measuring electric-field-
induced frequency shifts created by a Stark effect in high-$n$ Rydberg
resonances. Enhancing the sensitivity of this method is achievable through the
use of the EIT phenomenon, facilitating optical detection of atomic
interactions [16, 17, 18, 19]. However, the presence of alkali-metal atoms on
the inner surface of a vapor cell, due to a few layers of coating, induces
non-zero conductivity in the glass [20]. This leads to an electric-field-
screening effect, a consequence of photodesorption caused by laser irradiation
at the cell’s surface [21, 22]. The phenomenon, originating from the induced
internal electric field by ions, also occurred with cold atoms in high vacuum
chambers [23, 24, 25]. Several strategies are available to counteract the
induced electric field within the cell: (1) The utilization of externally
connectable ring electrodes for applying electric fields within the cell [18].
(2) The insertion of film electrodes on the cell’s interior [22, 26]. (3)
Rapid switching of the electric field [16]. (4) Direct application of UV
light, Argon Laser, or heating to desorb surface metals [27, 28, 29, 30]. (5)
Three-photon excitation with infarred laser fields [31]. The electric field
screening effect allows for detecting high-frequency oscillating electric
field signals ranging from MHz to THz. Notably, the technique can extend the
detection spectrum of atomic electrometry, encompassing superlow-frequency
(SLF) and extremely-low-frequency (ELF) bands, as well as static (dc) regions
[19].

The detecting frequency of the atomic electrometry is limited by the time it
takes for the induced electric field to respond to the external field. In
principle, glass vapor cells are excellent electric insulators; however, the
absorption of alkali-metal atoms leads to nonzero conductivity. We define the
recovery time as the time that free charges redistribute the glass surface to
maintain equal potential on the conductive surface when an external electric
field is applied. The recovery time was 500 $\mu$s in Ref. [21] and it can be
varied from 100 $\mu$s to 700 $\mu$s, depending on the coupling field
intensity in Ref. [16]. A longer recovery time of 200 ms is reported in Ref.
[17]. Our study utilizes Rydberg EIT resonances to characterize the cell’s
screening effect and charge dynamics as we vary parameters like E-field
switching frequency, E-field amplitude, laser power, and vapor temperature.
The recovery time varied from 0.7 ms to 450 ms and a low cutoff frequency
below 10 Hz was successfully achieved. The results demonstrate significant
progress in enhancing the performance of Rydberg atomic electrometry, offering
valuable insights for future precision measurements and field sensing
applications.

##  II Experimental Setup

We carried out the electric field measurement from an Rb vapor cell. The
energy levels of $\rm{}^{87}Rb$ involving Rydberg state and laser excitations
are shown in Fig. 1. The probe beam, with a wavelength of 780 nm, was locked
onto the transition from the ground state $|5S_{1/2},F=2\rangle$ to the
excited state $|5P_{3/2},F^{\prime}=3\rangle$ via the saturation absorption
spectrum, which is not shown in the figure. Meanwhile, the coupling beam, with
a wavelength of 480 nm, was either frequency scanned or locked to the
transition of the excited state $|5P_{3/2},F^{\prime}=3\rangle$ to the Rydberg
state $53D_{5/2}$. The probe and coupling beams were counter-propagated
through a reference cell and a science cell, and their polarizations were both
$\rm{\sigma^{+}}$ to achieve the maximum Rydberg EIT signal [32, 33]. These
beams were collimated to a full width at $e^{-2}$ maximum of 0.81 mm. To
control the sheet resistance on the inner surface of the glass cell, we varied
the coupling laser power from 2.4 mW to 14 mW and changed the probe laser
power from 0.023 mW to 4.1 mW. The optimized probe power was 0.088 mW to
achieve the highest EIT peak height [32]. The Rydberg EIT spectrum in Fig.
1(c) was obtained by measuring the transmission of the probe field in the
absence of electric and magnetic fields in a reference cell. The scanned
detuning was calibrated based on the frequency difference between $53D_{3/2}$
and $53D_{5/2}$ [34]. Using the EIT peak in the reference cell, we can compare
the frequency shift caused by the external electric field.

We constructed an Rb vapor cell with a diameter of 25 mm flat windows. The
thickness of the cell’s windows is 3 mm, while the overall length of the cell
is 19.2 mm (with a path length of 13.2 mm). The vacuum pressure was 2.2$\times
10^{-7}$ torr before sealing the cell. The cell is made of borosilicate glass
and filled with a natural abundance of Rb vapors. Two stainless steel
electrodes were attached to each side of the cell. Two electrodes’ dimensions
are $50\times 50~{}\rm mm^{2}$ to generate a uniform electric field in the
laser propagation direction. We control the electric field by a square
waveform signal generated by a function generator. Laser beams are transmitted
through electrode holes of 4 mm in diameter, with the electric field aligned
parallel to the laser beams. An aluminum heating block is positioned adjacent
to the electrodes. Optical density (OD) is a factor that increases the
sensitivity of the EIT and the electrometry. It can be controlled by vapor
temperature and determined by the probe field absorption signal at the
resonance frequency. The cell temperature is set at 65∘C or 75∘C.

##  III Results

A scanned coupling field with a DC electric field was used to observe the
Stark shift of the Rydberg thermal EIT spectrum. With an electric field of 1
V/cm, our theoretical calculation, using the Alkali Rydberg Calculator (ARC)
PYTHON package [35], predicts the Stark shift of a few hundred MHz. However,
we did not observe significant shift in the Rydberg EIT spectrum. The
phenomenon may be induced by the screening effect. Rb coatings can be excited
by the coupling field, which has a wavelength of 480 nm, so that the internal
E-field induces shielding of the external E-field. In order to gain insights
into the screening effect and charge dynamics within the cell, we then lock
the coupling field to the peak transmission of Rydberg EIT and employ a square
waveform to switch the electric field, as illustrated by the black line in
Fig. 2. Subsequently, we detect the transmission signal from the probe field
at resonant EIT, represented by the blue line. When a sufficiently high-
frequency electric field is applied, specifically at 10 kHz in Fig. 2(a), the
EIT transmission does not have enough time to respond to the electric field’s
rapid switching adequately. Therefore, the transmission experiences a sudden
jump with opposite polarity compared to the electric field. Here we introduce
the transient time, denoted as $\tau$, which characterizes the duration for
the EIT signal $T$ decay to $e^{-1}$ of its amplitude $A$, as expressed by an
exponential function

$T(t)=Ae^{-\frac{t}{\tau}}+y_{0}.$ (1)

The probe field offset transmission, $y_{0}$, indicates that the Stark shift
occurs at steady state. The derived transient time is about few $\rm{\mu}$s.
When the switching frequency of the electric field exceeds the reciprocal of
the transient time, the internal electric field fails to respond effectively.
It constrains the upper boundary of the cutoff frequency within the range of
MHz by this technique.

A vapor cell is generally made from a glass material that is an excellent
electrical insulator. Glass can have non-zero conductivity due to the
optically induced absorption of alkali-metal atoms on its interior surfaces.
If the electric field is applied at a low frequency, e.g., 10 Hz, the EIT
transmission suddenly jumps as shown in Fig. 2(b). Following each transmission
jump, the EIT transmission took several tens of milliseconds to return to its
original level. This indicates that the free charges on the inner surface of
the vapor cell redistribute over a certain timescale, defined as the recovery
time, leading to an equal electric potential on the surface and shielding of
the external electric field. Take the measurement in Fig. 2(b) as an example.
When the electric field is positive, the recovery time measured at 30 ms is
longer than when the electric field is negative, which is 20 ms. Detailed
discussion about the effect of the recovery time can be found in Sec. IV.

Figure 2: Rydberg EIT peak signals varied under square waveform modulations of
the electric field, with frequencies of 10 kHz in (a) and 10 Hz in (b). The
electric field was generated by a function generator with 20 $\rm V_{pp}$,
corresponding to the electric field of 10 V/cm. For fast modulation frequency,
we derive the transient time of $2~{}\mu$s from Eq. (1), which represents the
time that the internal E field is influenced by the external E field. For slow
modulation frequency, we derive the recovery times for the positive and
negative E fields are 30 ms and 20 ms, respectively. The cell temperature was
set to $65^{\circ}$C.  Figure 3: Rydberg EIT spectra with E-field modulation
of 1 kHz (a), 100 Hz (b), and 10 Hz (c). The coupling field frequency was
scanned across the $53D_{5/2}$ Rydberg state. Gray lines represent the E-field
signals. For sufficiently fast E-field modulations, e.g., faster than 100 Hz
(Fig. 3(b)), the EIT signals have the opposite polarity compared to the
E-field signals. For 10-Hz data, the coupling field scanning time was
increased by tenfold to improve clarity. (d) Rydberg EIT spectra with 1 kHz
E-field modulation and 10, 5, 2.4, 0 V/cm E-field strength from top to bottom.

To ensure that the electric field induces Stark shift and influences the
Rydberg EIT, we simultaneously switched the electric field while sweeping the
coupling field across $|53D_{5/2}\rangle$ Rydberg state. Figures 3(a), 3(b),
and 3(c) are the spectra with the E-field modulation frequencies of 1 kHz, 100
Hz, and 10 Hz. For 10-Hz data (in 3(c)), the coupling field scanning time was
increased tenfold to improve clarity. In the presence of an external E field,
the Rydberg EIT energy shifts from resonance, resulting in a two-level
absorption, subsequently causing a reduction in the EIT signal. In the case of
fast modulation, e.g., faster than 100 Hz, the induced energy shift cannot
return to its original state in time, so the EIT signals have the opposite
polarity compared to the E-field signal. Consequently, the Rydberg EIT
spectrum exhibits a square modulation. When the duty cycle of the EIT signal
is approximately 50%, such as with a 10 Hz E-field modulation, it limits the
lower boundary cutoff frequency attainable using this technique. Figure 3(d)
shows the Rydberg EIT spectra with 1 kHz E-field modulation and varied field
strength. Solid lines are Gaussian fits to EIT resonance envelopes. With the
intensity of 10 V/cm, the linewidth was broadened due to the asymmetrical or
non-uniform generation of charges within the cell, forming of an inhomogeneous
electric field. Besides the square modulation on Rydberg resonance, the Stark
effect is also visible at a detuning of 0 to -30 MHz with long tail EIT
signals above the Gaussian fitting lines. EIT resonance shifts to the same
sign because of quadratic electric field dependence, $\delta\propto E^{2}$.

Figure 4: FFT analysis with Blackman-Harris window to examine the frequency
components of the external field derived from the Rydberg EIT peak signals
under square modulation. The modulation frequencies of the E field were 1 kHz
in (a) and 10 Hz in (b) with square-waveform voltage signals. The span in the
main and inset plots are 1.5 kHz and 1 Hz in (a), and that is 20 Hz in (b).
The AC power with a frequency of 60 Hz is also shown in (a). The inset
spectrum shows that the frequency resolution is 10 mHz. In (b), the 10 Hz
E-field modulation exhibits a distinct signal that remains evident. This
technique enables the low-frequency sensor to achieve high sensitivity.

In extremely low-frequency bands (ELFs), Rydberg atomic electrometry presents
an extremely promising method for measuring external electric fields. We
employ the fast Fourier transform (FFT) analysis with a Blackman-Harris window
to examine the frequency components of the external field derived from the
modulated Rydberg EIT signals. The FFT spectra, depicted in the main and inset
plots of Fig. 4(a), have a span of 1.5 kHz and 1 Hz, respectively. This
technique can determine E-field modulation at 1 kHz with a resolution of up to
10 mHz. The presence of the 60 Hz ambient signal is evident, while the
additional peak observed can be attributed to the square wave modulation.
Subsequently, the E-field modulation was adjusted to 10 Hz. The FFT signals,
which remained distinctly observable, highlight the sensitivity of the low-
frequency sensor. Note that employing sinusoidal waves for the external
electric field at low frequencies prevents the internally induced electric
field from effectively canceling the external disturbances, primarily due to
its slow response. Thus, the utilization of square modulation enables the
electric field sensor to be functional at lower frequencies. However, the
detection of higher frequencies is constrained by the transient time, as
mentioned in Eq. (1). This study shows the transient times were from 1 to 10
$\mu$s, with minor fluctuations attributed to vapor temperature and major
variations correlated with EIT peak signals. These variations establish an
upper limit of 1 MHz for the detected frequency using this technique. Thus,
our study reveals a standard quantum limit for data capacity and
experimentally illustrates it across bandwidths from 10 Hz to 1 MHz.

Figure 5: The recovery times with varied electric field strength in (a) and
varied probe power in (b). The screening rate (reciprocal of the recovery
time) with varied probe power in the inset of (b) and varied coupling power in
(c). Red-open circles and blue-solid circles in (a) are the recovery times
when the electric field is turned on in the positive and negative direction,
respectively. The dashed lines are the fitting line with a growth logistic
function. In (a), the probe and coupling powers were 0.16 mW and 13.8 mW (the
strongest coupling power we applied in this study). In (b), the coupling power
was 14 mW (red circles), 7.7 mW (blue), and 4.0 mW (black); and in (c), the
probe power was 280 $\mu$W (red), 86 $\mu$W (blue), and 27 $\mu$W (black). The
brown data represent the saturation screening rate $\gamma_{0}$ from the inset
of (b). The E-field was set as 10 V/cm in both (b) and (c).

##  IV Discussion

The lowest frequency detection of Rydberg atomic electrometry is limited by
the recovery time of the shielding. The recovery time is the time that free
charges redistribute the glass surface to maintain equal potential on the
conductive surface when an external electric field is applied. We further
investigate the parameters that govern the recovery time. First, we adjust the
E-field strength under the probe and coupling powers of 0.16 and 13.8 mW. The
extracted recovery times are shown in Fig. 5(a), and they can be fitted by a
growth logistic function, $\frac{t_{sat}}{1+e^{-(E-E_{0})/\tau_{E}}}$. Based
on the growth behavior, it appears that the electron concentration increases
as the electric field strength increases, leading to an interplay between
carrier generation, mobility, and interactions in response to external fields.
Mobility refers to how easily electrons can move through the glass surface
when subjected to an electric field. The presence of a strong electric field
can enhance the mobility of charge carriers, allowing them to move more easily
through the material, leading to more efficient charge transport and
accumulation. The application of a strong electric field prolongs the recovery
time due to significant energy level shifts and disturbances to the internal
electric field. The scaling factor $\tau_{E}$ denotes the average electric
field strength, influencing the logistic growth rate or the steepness of the
curve. Smaller values of $\tau_{E}$ imply that a smaller change in the
electric field is required to increase the charge number. The red-open circles
and blue-solid circles in Fig. 5(a) are the recovery times at the positive and
negative electric field transition, respectively. The values of $\tau_{E}$ are
1.8 V/cm and 2.0 V/cm for the red and blue data, respectively. According to
the measurements, the ratio of the recovery times between positive and
negative E fields was approximately 1.5, likely attributed to an inhomogeneous
E field generated by the electrodes. This has been confirmed by exchanging the
direction of E field in space. Thus, we will only discuss the analysis when
the E field is in the positive direction in the following discussion. There
are physical limitations or saturation effects that affect recovery time. For
instance, as the electron concentration increases, interactions between
electrons could lead to recombination processes that limit further growth.

We then adjust the probe and coupling powers and extract the recovery time
under an E field of 10 V/cm, as shown in Figs. 5(b) and 5(c). When the probe
power increases, more photons collide with free electrons, leading to faster
equilibration of the electrons, i.e., shorter recovery time. Here, we define
the screening rate of the electric field as the reciprocal of the recovery
time. In Fig. 5(b), the coupling powers were 14 mW (red circles), 7.7 mW
(blue), and 4.0 mW (black). With the presence of the coupling laser, higher
probe power leads to a higher screening rate, but it is not linear. Screening
rates exponentially increase with probe power and become saturated at a
certain value. Dashed lines indicate the best fit using the function $A\cdot$
Exp[-$P_{p}$/$\tau_{p}$]+$\gamma_{0}$, where $A$ is a constant, and $P_{p}$ is
the probe power. The saturation screening rates, denoted as $\gamma_{0}$, are
89, 42, and 23 (1/s), with corresponding $\tau_{p}$ values of 0.37, 0.54, and
0.60 mW for the strongest to the weakest coupling power. The saturation
screening rate, reflecting the photoelectric current in the photoelectric
effect, is predominantly influenced by the coupling laser power (480 nm).
$\tau_{p}$ serves as a parameter representing the characteristic constant
associated with the photoelectric current’s response to the probe laser power.
A smaller $\tau_{p}$ implies a faster response of the photoelectric current to
change in laser power. Therefore, stronger laser powers result in faster
screening rates, corresponding to shorter recovery times.

Next, we investigate the relationship between the screening rate and coupling
power while keeping the probe power constant. In Fig. 5(c), the screening rate
demonstrates a linear increase with the coupling power, maintaining fixed
probe powers of 280 $\mu$W (red circles), 86 $\mu$W (blue), and 27 $\mu$W
(black). The brown data represent the saturation screening rate $\gamma_{0}$
from the inset of Fig. 5(b). The corresponding slopes are 6.2, 3.2, 2.4, and
1.2 (1/s/mW) from top to bottom. This observation substantiates that the
screening rate of excited Rb atoms is influenced by the 780 nm probe laser and
the 480 nm coupling laser. In principle, the 780 nm photon may not be
sufficient to cause electronic transitions within the solid-state Rb metal
directly. However, the absorption of the 780 nm light may result in localized
effects. As the rubidium metals, encompassing both isotopes, absorb the laser,
they transfer energy to the electrons, increasing their kinetic energy in the
localized region. Consequently, free electrons can readily escape from the
surface when exposed to the 480 nm photons. It is also the phenomenon of
light-induced atom desorption (LIAD), which has been used as a tool to ﬂexible
control the atomic density [27, 28]. A short wavelength light source allows
fast switching of the desired vapor pressure. The recovery time can be varied
from 1 ms to 400 ms by adjusting the laser powers. Therefore, the frequency
detection of Rydberg atomic electrometry can cover the extremely-low-frequency
(ELF) band.

##  V Conclusion and Outlook

Our experiments in a Rb vapor cell illustrate the effectiveness of low-
frequency atomic electrometry through Rydberg EIT measurements. We
successfully identified the screening effect induced by external electric
fields and Rydberg excitation, clarifying its origin in the charge dynamics
resulting from the photoelectric effect of the Rb coating under the influence
of the probe and coupling beams. Utilizing FFT analysis on the E-field
modulated Rydberg EIT signals, we demonstrated the capability to measure
frequency electric fields from 10 Hz to 1 MHz. The frequency detection range
of this Rydberg atomic electrometry is constrained by the recovery and
transition times of the shielding process. Our systematic study revealed that
adjusting the probe and coupling field power, as well as the E-field strength,
can influence the recovery time and screening rate. Remarkably, this atomic
electrometry extends its applicability to the extremely-low-frequency (ELF)
band, presenting a promising technique for sensitively measuring low-frequency
electric fields. This expansion enhances its versatility across various
fields, allowing for the detection of electric fields originating outside the
vacuum environment over a broader range.

###### Acknowledgements.

This work was supported by Grants Nos. 112-2112-M-110-008 and
112-2123-M-006-001 of the National Science and Technology Council, Taiwan. The
authors thank Prof. Tsu-Chiang Yen for the valuable comments on the study.

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# Solid state defect emitters with no electrical activity

Pei Li  School of Integrated Circuit Science and Engineering, Tianjin
University of Technology, Tianjin 300384, China  Beijing Computational Science
Research Center, Beijing 100193, China  HUN-REN Wigner Research Centre for
Physics, P.O. Box 49, H-1525 Budapest, Hungary  Song Li  HUN-REN Wigner
Research Centre for Physics, P.O. Box 49, H-1525 Budapest, Hungary  Péter
Udvarhelyi  HUN-REN Wigner Research Centre for Physics, P.O. Box 49, H-1525
Budapest, Hungary  Department of Atomic Physics, Institute of Physics,
Budapest University of Technology and Economics, Műegyetem rakpart 3., H-1111
Budapest, Hungary  Bing Huang  Beijing Computational Science Research Center,
Beijing 100193, China  Department of Physics, Beijing Normal University,
Beijing, 100875, China  Adam Gali  HUN-REN Wigner Research Centre for Physics,
P.O. Box 49, H-1525 Budapest, Hungary  Department of Atomic Physics, Institute
of Physics, Budapest University of Technology and Economics, Műegyetem rakpart
3., H-1111 Budapest, Hungary

###### Abstract

Point defects may introduce defect levels into the fundamental band gap of the
host semiconductors that alter the electrical properties of the material. As a
consequence, the in-gap defect levels and states automatically lower the
threshold energy of optical excitation associated with the optical gap of the
host semiconductor. It is, therefore, a common assumption that solid state
defect emitters in semiconductors ultimately alter the conductivity of the
host. Here we demonstrate on a particular defect in 4H silicon carbide that a
yet unrecognized class of point defects exists which are optically active but
electrically inactive in the ground state.

Point defects in semiconductors play a pivotal role in determining the
electrical and optical properties of the host material. Understanding the
physical fundaments of point defects in semiconductors was a key to arrive at
the concept of opto-electronics devices, photovoltaics and energy storage
devices, and very recently, state-of-the-art quantum information processing
realisations (Polman  _et al._ , 2016; Awschalom  _et al._ , 2013; Zwanenburg
_et al._ , 2013; Awschalom  _et al._ , 2018; Wolfowicz  _et al._ , 2021) which
have been shaped the socio-economical environment at global scale. Point
defects may introduce defect levels (DLs) within the host semiconductor’s
fundamental band gap, thereby influencing its electrical conductivity, i.e.,
electrically active point defects (Queisser and Haller, 1998; Alkauskas  _et
al._ , 2016). Notably, these in-gap DLs and associated states also impact the
optical properties of the material by reducing the optical excitation
threshold energy compared to the perfect semiconductor’s optical gap (Feil
_et al._ , 2023; Redjem  _et al._ , 2020; Li  _et al._ , 2022; Gali, 2023). As
a consequence, a common assumption is that solid-state defect emitters modify
the host material’s electrical conductivity (Tsai  _et al._ , 2022; Ping and
Smart, 2021; Gordon  _et al._ , 2013; Weber  _et al._ , 2010). We show below
that this common assumption is not generally valid.

In Fig. 1, we depict the possible optical transition mechanisms within
semiconductors. Many point defects introduce multiple deep DLs into the
fundamental band gap that could dramatically modify the electrical properties
of the host because these deep levels often participate in carrier trapping
and recombination events. In these defects, optical transition could occur
between the occupied and unoccupied DLs in the gap [see Fig. 1(a)].
Alternatively, the optical transition can occur between localized DLs and the
band edge, either valence band maximum (VBM) or conduction band minimum (CBM)
(e.g. Ref. (Li  _et al._ , 2023)). The respective excited states may be called
pseudo-acceptor and pseudo-donor states as they show a Rydberg-series of the
excited states converging towards the ionization threshold [see Fig. 1(b)]. We
note that the optical excitation threshold energies are lower than the
electrical gap between the occupied and unoccupied states participating in the
optical transition because of the attracting electron-hole interaction in the
excited state. By harnessing this excitonic effect, we suggest a category of
point defects that act as emitters and are electrically inactive at the same
time. A defect may introduce just one occupied DL below VBM without disturbing
the bands close to VBM, so the defect is electrically inactive in the ground
state and its positive charge state is not stable. This defect can be
optically excited where the hole is localized in the resonant DL whereas the
electron occupies a state split from CBM that builds up a pseudo-donor excited
state. The exciton binding energy in the excited state could shift the
excitation energy below the optical band gap of the host semiconductor with
establishing a solid state defect emitter [see Fig. 1(c)]. We label these
defects as EIDE after the expression of electrically inactive defect emitters
in the context.

Figure 1: The possible optical transition mechanisms within indirect band gap
semiconductors. (a) Occupied and unoccupied deep DLs. (b) Deep DL and band
edge. (c) Electrically inactive DL and band edge: establishing EIDE. (d)
Comparison between the values of zero-phonon-line ($E_{\text{ZPL}}$) and band
gap ($E_{\text{g}}$) for the defect category in (c). $E_{\text{ZPL}}$ is the
energy difference between the optical excited (ES) and ground state (GS) of
optical transitions with no-phonon whereas the thinner lines are the optical
transition energies when phonons are participating in the optical transition.

In this study, we demonstrate the principles of EIDE on a tri-carbon
interstitial cluster in 4H silicon carbide (SiC) which produces an ultraviolet
emission below the gap of 4H-SiC. We show by first principles calculations
that the given defect has zero-phonon-line (ZPL) emission with characteristic
local vibration modes in the photoluminescence (PL) spectrum which agrees well
with a previously reported defect emitter, the so-called D${}_{\text{II}}$
center in 4H-SiC (Sridhara  _et al._ , 1997; Steeds  _et al._ , 2006; Sullivan
and Steeds, 2007). The optical excited state is a pseudo-donor type and it
does not show electrical activity. The effect is mediated by the attractive
electron-hole interaction in the optical excited state of the defect enhanced
by the resonant defect states which is strikingly paramount in indirect
semiconductors. This example unveils the EIDE category of point defects in
solids. We discuss potential dopants to engineer such defects in
semiconductors.

##  I Results

4H-SiC is a wide band gap indirect semiconductor which is a platform for high-
power, high temperature electronic devices (Pearton, 2013; Lutz  _et al._ ,
2018; Shur  _et al._ , 2006) as well as quantum information processing (Son
_et al._ , 2020; Csóré and Gali, 2021) that makes it unique among the
technological mature semiconductors. The band gap of 4H-SiC is slightly
reduced at elevated temperatures (see Ref. (Cannuccia and Gali, 2020) and
references therein). The VBM and CBM are located at $\Gamma$-point and
$M$-point, respectively. The excitonic band gap of the material is 3.265 eV at
2 K (Choyke  _et al._ , 1964), where the crystal phonon replica dominates the
PL spectrum upon above-band-gap illumination; nevertheless, the ZPL of the
free exciton can be weakly observed too because the free exciton may gain some
momentum from the defects in 4H-SiC (Ivanov  _et al._ , 1998). The PL spectrum
of the bound exciton of the shallow donors (nitrogen substituting carbon in
the lattice at the so-called quasicubic and hexagonal sites) can be well
observed at 2 K, where the respective ZPL emissions at 3.243 eV and 3.256 eV
are more pronounced as the defect potential breaks the translational symmetry
of the crystal. Nevertheless, the phonon sideband still dominates in the
respective PL spectrum as expected for an indirect semiconductor (Ivanov  _et
al._ , 1998).

In the followings, we focus our attention to the ultraviolet D${}_{\text{II}}$
color center in 4H-SiC recorded near cryogenic temperatures (Sullivan and
Steeds, 2007). The D${}_{\text{II}}$ shows up a sharp ZPL at 3.20 eV which is
only $\approx 60$ meV lower than that of the free exciton in 4H-SiC (Kordina
_et al._ , 1995; Klahold  _et al._ , 2018; Ivanov  _et al._ , 1998). A
characteristic phonon sideband was also observed in the D${}_{\text{II}}$ PL
spectrum with sharp features that were associated with local vibration modes
(LVMs) of the underlying defect (Sullivan and Steeds, 2007). We note that
various reports on the D${}_{\text{II}}$ PL spectrum exhibit different numbers
of sharp features associated with LVMs (Sridhara  _et al._ , 1997; Sullivan
and Steeds, 2007). No single defect observation has yet been carried out for
D${}_{\text{II}}$ center (Bockstedte  _et al._ , 2004; Jiang  _et al._ , 2012;
Mattausch  _et al._ , 2001; Patrick and Choyke, 1973), thus some sharp
features may not belong to the defect and could overlap with the PL spectrum
of other defects. Nevertheless, previous studies attempted to identify this
color center by calculating the LVMs of the defect models and compare those to
the observed features in the phonon sideband Mattausch  _et al._ (2001); Gali
_et al._ (2003); Jiang  _et al._ (2012). The most recent study proposed the
tri-carbon interstitial cluster as the origin of the D${}_{\text{II}}$ center
which was corroborated by molecular dynamics calculations with revealing the
high-temperature stability of the defect in line with the observations (Jiang
_et al._ , 2012). However, the nature of optical transition has not yet been
discussed for any models. Furthermore, it has been recently shown that not all
the calculated LVMs show up in the observed phonon sideband of the PL spectrum
of other carbon clusters in 4H-SiC (Li  _et al._ , 2023), thus identification
of this ultraviolet color center has not yet been established.

Figure 2: (a) The band structure of 4H-SiC. (b) The calculated Kohn-Sham DLs
of the ground state (GS) and excited state (ES) for tri-carbon interstitial
cluster. The occupied and unoccupied DLs are labelled by filled and empty
triangles, respectively. Inset: the configuration of tri-carbon interstitial
cluster. (c) The band structure of 4H-SiC with tri-carbon interstitial
cluster. (d) The local enlarged drawing of (c). The degenerate $e$ states
(DLe) and $a_{1}$ state (DLa) are labelled in green and orange, respectively.

We use the tri-carbon interstitial cluster as a working model for the
D${}_{\text{II}}$ PL center in 4H-SiC. First, the electronic structure and
possible optical transitions of the host semiconductor is analyzed. The band
structure of 4H-SiC is depicted in Fig. 2(a). Our HSE06 calculations yield
3.17 eV electronic band gap for the host 4H-SiC which underestimates the low-
temperature data by about 0.1 eV. The electron-hole pair in a free exciton has
a crystal momentum $\text{k}_{M}$, where $\text{k}_{M}$ is the momentum
corresponding to the $M$-point. Consequently, direct recombination of the
electron at VBM and the hole at CBM is forbidden because of the crystal-
momentum conservation law. To conserve the crystal momentum the free exciton
recombination is only possible with the assistance of another particle (or
quasiparticle). Therefore, the minimum direct optical transition in 4H-SiC
occurs between the band edges at the $M$-point, with a gap of approximately
4.41 eV, and is larger than the indirect band gap.

The tri-carbon interstitial cluster is composed of three carbon interstitials
bridging three adjacent on-axis Si-C bonds at the hexagonal site [see Fig.
2(b)], which is the most stable among all possible configurations (Li  _et
al._ , 2023). The calculated Kohn-Sham DLs for the neutral tri-carbon
interstitial cluster are shown in Fig. 2(b). In the ground state, a double
degenerate $e$ state (VBM$-$0.15 eV) and an $a_{1}$ state (VBM$-$0.20 eV)
appear that show no dispersion unlike the host bands. We find also resonant
states in this energy region when crossed with the host bands that show up
quasilocalization but heavily mixed with the host bands. No further DLs emerge
in the fundamental band gap and the VBM is basically unaffected by the
presence of DLs (see Supplementary Fig. 1), implying that the defect is
electrical inactive [see Figs. 2(c,d)]. Indeed, we find in our density
functional theory (DFT) calculations that the positively charged defect is not
stable (see Supplementary Fig. 2).

It can be recognized that the DLs will be well isolated from the bands around
the $M$-point and it may be expected that the electron from the DLs can be
photoexcited to the CBM that constitutes a pseudo-donor excitonic state. We
first studied the nature of the exciton and the oscillator strength of the
optical transitions by a many-body perturbation method (see Methods) with that
the excitonic effects can be accurately calculated [see Fig. 3(a)]. Here we
focus to the lowest-energy bright transitions that only play the role in the
PL emission of the defect. Bright transitions occur at around 3.2 eV that we
label by $\text{P}_{\text{1}}$ and $\text{P}_{\text{2}}$ in Fig. 3(a). In both
peaks the hole part of the exciton wavefunction is dominantly built up from
the DLs and the electron part is located in the conduction bands at the
$M$-point. The contribution is about 95% and 99% for $\text{P}_{\text{1}}$ and
$\text{P}_{\text{2}}$ peaks, respectively. In contrast to
$\text{P}_{\text{1}}$ and $\text{P}_{\text{2}}$ peaks, the
$\text{P}_{\text{3}}$ and $\text{P}_{\text{4}}$ peaks in Fig. 3(a) are mainly
caused by the excitation from DLs to the conduction bands at the $L$-point,
collectively constituting approximately 87% and 82% of the total intensity. We
here analyze the origin of the $\text{P}_{\text{1}}$ peak in detail whereas
the analysis of the other peaks can be found in Supplementary Note 3. The
calculated many-body perturbation theory band gap between the double
degenerate $e$ state (DLe) and the CBM at the $M$-point is 3.57 eV, thus the
exciton binding energy is about 0.4 eV. The contribution of the electron-hole
pairs to the $\text{P}_{\text{1}}$ exciton is depicted in Fig. 3(b). The
foremost contribution arises from excitation originating from the degenerate
DLe to the CBM, constituting approximately 93.71% of the total intensity. The
excitation from the $a$ state (DLa) to CBM contributes by about 1.40%. Thus,
it can be concluded that the vast majority of the hole wavefunction has a
localized nature whereas the electron wavefunction is from the CBM in the low-
energy bound exciton.

Figure 3: (a) The optical transition intensity calculated within BSE. (b) The
contributions of excitations originating from VBM, DLs, and resonances state
(R) to CBM and conduction band in the $L$-point (CB-L) for the peak
$\text{P}_{\text{1}}$ in (a). Inset: The diagram of excitation.

Based on our findings about the lowest energy bright excited state in the BSE
spectrum, we employed HSE06 $\Delta$SCF method where an electron was promoted
from the DLe to the CBM, in order to take into account the geometry relaxation
in the optical excited state. This calculation makes it possible to simulate
the ZPL energy and the phonon sideband in the PL spectrum. We assume that the
vertical excitation energy can be well calculated by the many-body
perturbation method and then we apply finite size error corrections of the
supercell method, and then the reorganization energy is subtracted as obtained
by the HSE06 $\Delta$SCF method to arrive at the final ZPL energy. We note
that the $\Delta$SCF procedure will self-consistently change the Kohn-Sham
states and levels accordingly. In the excited state, the hole DL pops up in
the band gap close to VBM by 0.07 eV [see Fig. 2(b)]. The redistribution in
the localization of the defect wavefunctions results in relatively high forces
on the ions and the defect reconstructs in the optical excited state with a
reorganization energy of about 0.13 eV with slightly reducing the symmetry of
the defect. We considered the P1 peak in the BSE spectrum as the vertical
excitation energy for which we applied finite size error corrections (adding
0.10 eV) that account for the pseudo-donor nature of the excited state as
explained in Supplementary Note 4. The final simulated ZPL is at 3.13 eV, in
good agreement with experimental data.

We then computed the phonon sideband of the PL spectrum within Huang-Rhys
theory with using the excited state’s geometry as obtained by HSE06
$\Delta$SCF method (Huang and Rhys, 1950; Alkauskas  _et al._ , 2014; Gali
_et al._ , 2009). The PL spectrum of the tri-carbon interstitial cluster is
shown in Fig. 4 and the respective vibration frequencies are listed in
Supplementary Table I. Because of the symmetry reduction, both symmetry-
breaking $E$ and symmetry conserving $A_{1}$ vibration modes participate to
the phonon sideband of the PL spectrum. The double degenerate ($E$-mode)
highest LVMs at 162 meV are stretching modes from two of the three
approximately vertical C-C bonds. The third LVM is a stretching mode of the
three C-C bonds, which preserves the symmetry of the defect ($A_{1}$-mode).
The fourth LVM is a breathing mode of the triangle formed by the three carbon
interstitials. The respective fifth and sixth LVMs result from the axial
vibration of the carbon atoms right below and above the center of the tri-
carbon interstitial cluster. We conclude that the calculated ZPL energy and
the LVM structure well agree with those of the D${}_{\text{II}}$ color center
in 4H-SiC (see Supplementary Note 5).

Figure 4: The PL spectrum of the tri-carbon interstitial cluster. The $E$ and
$A_{\text{1}}$ modes are marked with green and red vertical lines below the
corresponding LVMs. The LVMs (blue (Sridhara  _et al._ , 1997) and orange
(Sullivan and Steeds, 2007) vertical lines) of $\text{D}_{\text{II}}$ PL
spectrum with frequency higher than that of 4H-SiC bulk phonon spectrum are
also listed for comparison.

##  II Discussion

We demonstrated that the D${}_{\text{II}}$ color center is a point defect
which belongs to the EIDE category. In this particular case, the excited state
has a pseudo-donor nature. This explains the fact that the ZPL energy of the
D${}_{\text{II}}$ center scales with the widths of the band gap of SiC
polytypes, c.f., Refs. (Patrick and Choyke, 1973; Sullivan and Steeds, 2007;
Sridhara  _et al._ , 1997). One can generalize the feasible electronic
structure of EIDE that (i) either an occupied DL should lie below but not too
far from VBM but should be deep enough not to realize a stable positive charge
state (ii) or an empty DL should lie above but not too far from CBM but high
enough not to realize a stable negative charge state or (iii) the combination
of both. The bound exciton state may be easier realized in an indirect than a
direct semiconductor where the attractive electron-hole interaction of the
bound exciton and the geometry relaxation in the excited state could result in
a ZPL emission below the optical gap of the host semiconductor. The wavelength
of these emitters depend on the fundamental band gap the host semiconductors
and the strength of exciton binding energies in the excited of the defect. The
last term depends on the screening effects of the material. Semiconductors
with moderate and low screening can host defects with exciton binding energies
that may exceed hundreds of millielectronvolts which establishes a
considerable playground to realize EIDEs.

Now one may ask how one can engineer EIDEs in various materials. Electrically
inactive defect in semiconductors strongly implies that (i) vacancy or vacancy
complexes can be disregarded as dangling bonds introduce levels in the
fundamental band gap, (ii) the substitutional defect or dopant should be
isovalent with the host atom to avoid donating or accepting electrons to the
crystal, and (iii) interstitial or interstitial cluster defects should
introduce even number of electrons to realize a closed-shell singlet ground
state and they should possess stronger chemical bonds than those constituting
of the host crystal so the occupied and unoccupied defect levels would
presumably lie outside of the fundamental band gap. Certainly, the EIDE defect
models should be checked one-by-one in a given semiconductor. This is not
surprising as this is the case, e.g., for the effective mass donor (n-type
doping) or acceptor (p-type doping) models. Although, the usual recipe for
realizing donor and acceptor states is to substitute the host atom with a
dopant possessing one extra and one less electron, respectively, this recipe
does not always work. As an example, the nitrogen has five valence electrons
that can replace silicon atom with four valence electrons in the silicon
crystal; however, one chemical bond will be broken in the defect and the
silicon dangling bond produces a deep level instead of the shallow effective
mass donor. On the other hand, boron atom with three valence electrons
substituting silicon acts as an effective mass acceptor in silicon crystal
despite the fact that boron and nitrogen atoms sit in the same row of the
periodic table. On the contrary, nitrogen substituting the carbon in SiC
indeed acts as effective mass donor as was anticipated earlier in this paper.
These examples highlight the quest of accurate _ab initio_ predictions of
defect states because simple rules cannot be trustworthy applied for cases
that are thought to be easy such as defect engineering of donors and acceptors
in semiconductors. This study rather establishes the search criteria for
electrically inactive defect emitters in semiconductors as quoted above.
Recent advances in the theoretical spectroscopy of defects in semiconductors
(Gali, 2023) make it possible to systematically search for defects with target
properties that may lead to discoveries of EIDEs, n-type and p-type dopants or
defect qubits in semiconductors.

This study uncovers a class of defects that decouple optical and electrical
effects. These findings propose an unexplored avenue for engineering
semiconductor devices, suggesting the feasibility of creating defect species
that offer independent control over optical and electrical functionalities
within the same platform. The implications of this work extend to the design
and optimization of highly integrated and miniaturized semiconductor devices
that leverage both optical and electrical attributes for enhanced
functionality. Because the modularity of the opto-electronics devices can be
simplified this may lead to reduced cost of production and to contribution to
green technology with lowering the energy consumption of the operation of
these devices.

##  III Conclusion

This study has revealed a phenomenon that expands our understanding of the
interplay between defect-induced optical and electrical effects that was
exemplified on a tri-carbon intersitial defect in 4H-SiC. Unlike the
conventional assumption that solid-state defect emitters in semiconductors
necessarily affect the electrical conductivity of the host, we demonstrate the
existence of defects that introduce optically active but electrically inactive
states. This phenomenon may be observed in rather indirect than direct band
gap semiconductors where all the defect levels lie outside to the fundamental
band gap but reside very close to the band edges that may alter the minimal
optical excitation energy without influencing the electrical conductivity. By
exploiting the attractive interaction between electrons and holes in the
optical excited state, these defects can show significant optical activity
without exerting any discernible impact on the electrical conductivity. This
finding might pave the ways to realize new generation opto-electronics
devices.

##  IV Methods

All the first-principles calculations are performed using density functional
theory (DFT) within the projector augmented wave potential plane-wave method,
as implemented in the Vienna ab initio simulation package (VASP) (Kresse and
Furthmüller, 1996) with the projector augmented wave method (Blöchl, 1994).
The electron wave functions are expanded in plane-wave basis set limited by a
cutoff of 420 eV. The fully relaxed geometries were obtained by minimizing the
quantum mechanical forces between the ions falling below the threshold of 0.01
eV/Å and the self-consistent calculations are converged to $10^{-5}$ eV.

The screened hybrid density functional of Heyd, Scuseria, and Ernzerhof
(HSE06) (Heyd  _et al._ , 2003) is employed to calculate the electronic
structure. In this approach, we could mix part of nonlocal Hartree–Fock
exchange to the generalized gradient approximation of Perdew, Burke, and
Ernzerhof (PBE) (Perdew  _et al._ , 1996) with the default fraction ($\alpha$
= 0.25) and the inverse screening length at 0.2 Å-1. The calculated band gap
is 3.17 eV. We embedded the tri-carbon interstitial cluster in a 576-atom
4H-SiC supercell which is sufficient to minimize the periodic defect-defect
interaction. The single $\Gamma$-point scheme is convergent for the k-point
sampling of the Brillouin zone (BZ). The $M$-point and $L$-point in the BZ are
projected into the $\Gamma$-point in this supercell due to band folding where
the lowest energy conduction bands occur in these k-points. The excited states
were calculated by $\Delta$SCF method (Gali  _et al._ , 2009). We note that
the reorganization energy and the optimized geometry of the optical excited
state can be calculated with $\Delta$SCF method which are both important to
predict the photoluminescence spectrum including the phonon sideband.

For the phonon modes, we calculated the corresponding dynamical matrix
containing the second-order derivatives of the total energy by means of the
PBE (Perdew  _et al._ , 1996) functional where all the atoms in the supercell
were enabled to vibrate. In this case, we apply strict threshold parameters
for the convergence of the electronic structure ($10^{-6}$ eV) and atomic
forces ($10^{-3}$ eV/Å) in the geometry optimization procedure. These
vibration modes are applied together with the HSE06 ground state and excited
state geometries to simulate the PL spectrum of the given defects within
Huang-Rhys theory (Alkauskas  _et al._ , 2014; Gali  _et al._ , 2009). This
strategy worked well for deep defects in diamond (e.g., Refs. Thiering and
Gali, 2017, 2018).

To accurately consider the excitonic effect, many-body perturbation theory
based on GW approximation with Bethe-Salpeter equation (BSE) (Shishkin and
Kresse, 2006; Shishkin  _et al._ , 2007) are used. We here use the supercell
with 576-atom in order to reach close-to-converged calculation for the GW
method which is very computationally demanding in the VASP implementation as
more than 9000 bands were included in the single-shot
$\text{G}_{0}\text{W}_{0}$ calculation. We note that the CBM at the $M$-point
and the conduction band at the $L$-point are projected into the $\Gamma$-point
for this supercell too. The energy cutoff for the response function is set to
be 150 eV. The Tamm-Dancoff approximation was used to solve BSE. The highest
one hundred valence bands and one hundred lowest conduction bands are
considered as the basis for the excited state in the BSE procedure. The
calculations are based on the optimized HSE06 functional which resulted in the
optimized geometry and the electronic structure of the neutral tri-carbon
interstitial cluster in 4H-SiC.

## Author contribution

PL carried out the calculations. All authors contributed to the discussion and
writing the manuscript. AG developed the concept of electrically inactive
defect emitters and led the entire scientific project.

## Competing interests

The authors declare that there are no competing interests.

## Data Availability

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

## Acknowledgement

Support by the National Excellence Program for the project of Quantum-coherent
materials (NKFIH Grant No. KKP129866) as well as by the Ministry of Culture
and Innovation and the National Research, Development and Innovation Office
within the Quantum Information National Laboratory of Hungary (Grant No.
2022-2.1.1-NL-2022-00004) is much appreciated. AG acknowledges the high-
performance computational resources provided by KIFÜ (Governmental Agency for
IT Development) institute of Hungary and the European Commission for the
project QuMicro (Grant No. 101046911). BH acknowledges the NSFC (Grants Nos.
12088101 and 12174404), National Key Research and Development of China (Grant
No. 2022YFA1402400), NSAF (Grant No. U2230402).

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