# Probing material absorption and optical nonlinearity of integrated photonic
materials

Maodong Gao1,∗, Qi-Fan Yang1,∗, Qing-Xin Ji1,∗, Heming Wang1, Lue Wu1, Boqiang
Shen1, Junqiu Liu2, Guanhao Huang2, Lin Chang3, Weiqiang Xie3, Su-Peng Yu4,
Scott B. Papp4,†, John E. Bowers3,†, Tobias J. Kippenberg2,† and Kerry J.
Vahala1,†
1T. J. Watson Laboratory of Applied Physics, California Institute of
Technology, Pasadena, California 91125, USA
2Institute of Physics, Swiss Federal Institute of Technology Lausanne (EPFL),
CH-1015 Lausanne, Switzerland
3ECE Department, University of California Santa Barbara, Santa Barbara, CA
93106, USA
4National Institute of Standards and Technology, Boulder, CO 80305, USA
∗These authors contributed equally to this work.
†Corresponding authors<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
Optical microresonators with high quality ($Q$) factors are essential to a
wide range of integrated photonic devices. Steady efforts have been directed
towards increasing microresonator $Q$ factors across a variety of platforms.
With success in reducing microfabrication process-related optical loss as a
limitation of $Q$, the ultimate attainable $Q$, as determined solely by the
constituent microresonator material absorption, has come into focus. Here, we
report measurements of the material-limited $Q$ factors in several photonic
material platforms. High-$Q$ microresonators are fabricated from thin films of
SiO2, Si3N4, Al0.2Ga0.8As and Ta2O5. By using cavity-enhanced photothermal
spectroscopy, the material-limited $Q$ is determined. The method
simultaneously measures the Kerr nonlinearity in each material and reveals how
material nonlinearity and ultimate $Q$ vary in a complementary fashion across
photonic materials. Besides guiding microresonator design and material
development in four material platforms, the results help establish performance
limits in future photonic integrated systems.

Performance characteristics of microresonator-based devices improve
dramatically with increasing $Q$ factor 1. Nonlinear optical oscillators, for
example, have turn-on threshold powers that scale inverse quadratically with
$Q$ factor 2, 3, 4. The fundamental linewidth of these and conventional lasers
also vary in this way 5, 6, 7. In other areas including cavity quantum
electrodynamics 8, integrated quantum optics 9, 10, 11, 12, cavity
optomechanics 13 and sensing 14, a higher $Q$ factor provides at least a
linear performance boost. In recent years, applications that rely upon these
microresonator-based phenomena, including microwave generation 15, frequency
microcomb systems 16, high-coherence lasers 7, 17, 18 and chip-based optical
gyroscopes 19, 20, 21, have accelerated the development of high-$Q$ photonic-
chip systems 22, 23, 24, 25, 26, 27, 28, 18, 29, 30, 31.

$Q$ factor is determined by material losses, cavity loading (i.e., external
waveguide coupling), and scattering losses (see Fig. 1a). To increase $Q$
factor, there have been considerable efforts focused on new microfabrication
methods and design techniques that reduce scattering loss associated with
interface roughness 32, 22, 33 and coupling non-ideality34, 35. Impressive
progress has resulted in demonstrations of high-$Q$ microresonator systems
with integrated functionality 36, 37, as well as resonators that are
microfabricated entirely within a CMOS foundry 18. With these advancements,
attention has turned towards $Q$ limits imposed by the constituent photonic
material themselves. For example, the presence of water, hydrogen, trace metal
ions 38, 39, 40, 41, 42 and other pathways 43, 44 are known to increase
absorption. In this work, cavity-enhanced photothermal spectroscopy 45, 39,
46, 47, 48, 41 is used to determine the absorption-limited $Q$ factor ($Q_{\rm
abs}$) and optical nonlinearity of state-of-the-art high-$Q$ optical
microresonators fabricated from four different photonic materials on silicon
wafer.

Images of the microresonators characterized in this study are shown in Fig.
1b, where the microresonators are SiO2 4, 49 microdisks and Si3N4 48,
Al0.2Ga0.8As 26, 27 and Ta2O5 50 microrings. Details of the device fabrication
processes are given in the Methods. Typical microresonator transmission
spectra showing optical resonances are presented in Fig. 1b. The transmission
spectra feature Lorentzian lineshapes, but in some cases are distorted by
etalon effects resulting from reflection at the facets of the coupling
waveguide. With such etalon effects accounted for (see Methods), the intrinsic
($Q_{0}$) and external (coupling) ($Q_{e}$) $Q$ factors can be determined. The
measured intrinsic $Q_{0}$ factors are 418 million, 30.5 million, 2.01
million, and 2.69 million, for SiO2, Si3N4, Al0.2Ga0.8As, and Ta2O5 devices,
respectively.

The microresonator intrinsic $Q_{0}$ is determined by scattering and
absorption losses. In order to isolate the absorption loss contribution,
cavity-enhanced photothermal spectroscopy is used. The principle is based on
that the resonant frequencies of dielectric microresonators are shifted by the
Kerr effect and the photothermal effect, both of which result from the
refractive index change that depends on the intracavity optical intensity.
Because these two effects occur on very distinct time-scales (Kerr effect
being ultra-fast and optical absorption occurring at a relatively slow thermal
time scale from milliseconds to microseconds), it is possible to distinguish
their respective contributions to resonant frequency shift and infer their
nonlinear coefficients 46. Two distinct measurements are performed to
determine the absorption-limited $Q_{\rm abs}$. Here, they are referred to as
the “sum measurement” and “ratio measurement”. In the sum measurement,
resonant frequency shift is measured to obtain the sum of Kerr and
photothermal effects. In the ratio measurement, the photothermal frequency
response is measured to distinguish its contribution from the Kerr effect.

Fig. 1: High-$Q$ optical microresonators characterized in this work. a,
Schematic showing optical loss channels for high-$Q$ integrated optical
microresonators. The loss channels include surface (and bulk) scattering loss
and material absorption loss. The intrinsic loss rate is characterized by the
intrinsic $Q$ factor ($Q_{0}$). Bus waveguide coupling also introduces loss
that is characterized by the external (coupling) $Q$ factor ($Q_{e}$). b, Left
column: images of typical microresonators used in this study. Right column:
corresponding low input-power spectral scans (blue points) with fitting (red).
The intrinsic and external $Q$ factors are indicated. M: million.  Fig. 2: The
sum measurement. This experiment measures the sum of Kerr and photothermal
nonlinear coefficients $(g+\alpha)$. a, Experimental setup. ECDL: external-
cavity diode laser; EDFA: erbium-doped fiber amplifier; VOA: voltage-
controlled optical attenuator; PC: polarization controller; PD: photodetector;
MZI: Mach-Zehnder interferometer; AFG: arbitrary function generator; OSC:
oscilloscope. For SiO2 experiment, ECDL is replaced by a narrow-linewidth
fiber laser to achieve a slower frequency tuning speed. As an aside due to
narower tuning range of fiber laser, this experiment is only performed at
1550nm for SiO2. b, Typical transmission spectra of microresonators with
photothermal and Kerr self-phase modulation, where the input power in the bus
waveguide is indicated. Theoretical fittings are plotted in red and discussed
in Methods. The cold transmission spectra measured at low pump power are also
plotted with dashed lines for comparison. WG power: optical power in the bus
waveguide. c, Measured resonant frequency shift versus intracavity power for
microresonators based on different materials. Dashed lines are linear fittings
of the measured data. d, Measured resonant frequency shift versus
microresonator chip temperature for the four materials, with linear fittings.
The fitted shift for Al0.2Ga0.8As, Si3N4, SiO2 and Ta2O5 are -13.1, -2.84,
-1.83 and -0.996, in units of GHz K-1, respectively.  Fig. 3: The ratio
measurement. This experiment measures the ratio of Kerr and photothermal
nonlinear coefficients $g/\alpha$. a, Illustration of the ratio measurement. A
pump laser is stabilized to a resonance and modulated by an intensity
modulator. The intracavity power is thus modulated. As a result of
photothermal effect and Kerr cross-phase modulation, the frequency of a nearby
resonance is also modulated. Another probe laser is stabilized near this
resonance, and its transmission is monitored by a vector network analyzer
(VNA). Inset: the modulation response allows distinguishing the photothermal
and Kerr effects. b, Experimental setup. IM: intensity modulator; CIRC:
optical circulator; LPF: low-pass filter; VNA: vector network analyzer. c,
Typical measured response functions of the probe laser transmission
$\tilde{\mathcal{R}}$ as a function of modulation frequency $\Omega$.
Numerical fittings are outlined as dashed curves. For modulation frequencies
below 1 kHz, the probe response is suppressed by the servo feedback locking
loop. d, Measured wavelength dependence of the ratios between the Kerr
nonlinearity and photothermal effect for three materials.

In the sum measurement, the microresonator is probed by a tunable laser whose
frequency is slowly swept across a resonance from the higher frequency side of
a resonance (i.e., blue-detuned side). The input light polarization is aligned
to the fundamental TE (Si3N4, Al0.2Ga0.8As, and Ta2O5) or TM (SiO2) mode of
the microresonator. The experimental setup is depicted in Fig. 2a. The
frequency scan is calibrated by a radio-frequency calibrated Mach-Zehnder
interferometer (MZI) 51. The probe laser frequency scan is sufficiently slow
(i.e., quasi-static scan, see Supplementary note III for details) to ensure
that scan speed does not impact the observed lineshape through transient
thermal processes within the microresonator. The transmission spectra exhibit
a triangular shape 52 as shown in Fig. 2b. Theoretical fittings of the
transmission spectra are shown in red and discussed in Methods. Also, the cold
resonance spectra (i.e., with very low waveguide power) measured under the
same coupling conditions are plotted for comparison (dashed curve).

By changing the input pump laser power with a voltage-controlled optical
attenuator (VOA), the quasi-static resonance shift $\delta\omega_{0}$ of the
resonant frequency $\omega_{0}$ versus the intracavity circulating optical
energy density $\rho$ (units of J m-3) is determined (see Supplementary note
III) and summarized in Fig. 2c. The observed linear dependence contains
contributions from the Kerr self-phase modulation and photothermal effects as,

$\frac{\delta\omega_{0}}{\omega_{0}}=-\frac{1}{\omega_{0}}(\alpha+g)\rho,$ (1)

where $\alpha$ and $g$ denote the photothermal coefficient and the Kerr
coefficient given by:

$\displaystyle\alpha$
$\displaystyle=\overline{\kappa_{a}}\frac{\overline{\delta T}}{P_{\rm
abs}}\left(-\frac{\delta\omega_{0}}{\overline{\delta T}}\right)V_{\rm eff},$
(2) $\displaystyle g$
$\displaystyle=\frac{\overline{n_{2}}}{\overline{n_{o}n_{g}}}\omega_{0}c.$

Here, $\kappa_{a}$ is the energy loss rate due to optical absorption, $n_{2}$
is the material Kerr nonlinear refractive index, $n_{o}$ is the material
refractive index, $n_{g}$ is the material chromatic group refractive index,
$c$ is the speed of light in vacuum, $P_{\rm abs}$ is the absorbed optical
power by the microresonator and $T$ is the temperature of the microresonator.
The bar (e.g., $\overline{n_{2}}$) denotes the average value of the underneath
variable weighted by the field distribution of the optical mode. The exact
definition of each average is provided in Supplementary note I.

The energy loss rate $\kappa_{a}$ is related to the material absorption-
limited $Q_{\rm abs}$ factor by

$Q_{\rm abs}=\frac{\omega_{0}}{\kappa_{a}}.$ (3)

To determine $\kappa_{a}$ and hence $Q_{\rm abs}$ from $\alpha$, it is
necessary to determine $V_{\rm eff}$, ${\overline{\delta T}}/{P_{\rm abs}}$
and $\delta\omega_{0}/\delta T$. The effective mode volume $V_{\rm eff}$ is
calculated using the optical mode obtained in finite-element modeling, and
${\overline{\delta T}}/{P_{\rm abs}}$ is further calculated using the finite-
element modeling with a heat source spatially distributed as the optical mode.
The resonance tuning coefficient $\delta\omega_{0}/\delta T$ is directly
measured by varying the temperature of the microresonator chip using a
thermoelectric cooler (TEC), and the results are shown in Fig. 2d. Since the
TEC heats the entire chip, the thermo-elastic effect of the silicon substrate
contributes to the frequency shift and combines with the photothermal effect.
However, this thermo-elastic contribution does not appear in the sum
measurement, where the heating originates only from the optical mode. Thus,
the thermal-elastic contribution of the silicon substrate must be deducted
from the TEC measured results (see Supplementary note I). Other effects that
may lead to frequency shift or linewidth broadening, such as harmonic
generation or multi-photon absorption, are not significant in the samples, as
confirmed by observing the coupling efficiency with respect to power (see
Supplementary note III).

Fig. 4: Absorption $Q_{\rm abs}$, nonlinear coefficients and parametric
oscillation threshold. a, Measured absorption $Q_{\rm abs}$ factors at
different wavelengths in the telecommmunication C-band for the four materials.
Vertical error bars give standard deviations of measurements. b, Comparison of
absorption $Q_{\rm abs}$ factors and normalized nonlinear index
($n_{2}/n_{o}^{2}$) for the four materials. Measured $n_{2}$ values are listed
in Table 1. The $n_{2}$ of SiO2 was not measured here and a reported value of
$2.2\times 10^{-20}$ m2 W-1 is used. Parametric oscillation threshold for a
single material normalized by mode volume ($P_{\rm th}$/$V_{\rm eff}$) is
indicated by the red dashed lines, assuming $\lambda=1550$ nm, intrinsic
$Q_{0}$ equals material absorption $Q$, and $Q_{e}=Q_{0}$ (i.e. critical
coupling condition).  Table 1: Summary of material loss and nonlinearity.
LPCVD: low-pressure chemical vapour deposition; MBE: molecular beam epitaxy;
IBS: ion-beam sputtering. Propagation loss $\sigma_{\rm abs}$ induced by
absorption is calculated as $\sigma_{\rm Mat}=(10/\ln
10)\omega_{0}n_{g}/(Q_{\rm abs}c)$. Error indicates standard deviation. These
numbers should be viewed as state-of-the-art values rather than fundamental
limits. Possible systematic errors of measurement values are discussed in
Supplementary note IIC. The $Q_{\rm abs}$ for Ta2O5 is further discussed in
Supplementary note IIID. Reported $n_{2}$ values are taken from ref. 53
(SiO2), ref. 54 (Si3N4), ref. 55 (Al0.2Ga0.8As) and ref. 50 (Ta2O5).

The measurement associated with Eq. 1 wherein the sum contributions of Kerr
and photothermal effects are measured is supplemented by a measurement that
provides the ratio of these quantities. This second measurement takes
advantage of the very different relaxation time scales of Kerr and
photothermal effects. The experimental concept and setup are depicted in Fig.
3a and 3b. Pump and probe lasers are launched from opposite directions into
the microresonator. The pump laser is stabilized close to one resonance by
directly monitoring the transmission signal, and its power is modulated over a
range of frequencies using a commercial lithium niobate electro-optic
modulator driven by a vector network analyzer (VNA). The probe laser is locked
to another nearby resonance, and is slightly detuned from the center resonant
frequency. With this arrangement, pump power modulations in the first
resonance induce modulations of the output probe power in the second
resonance, as a result of Kerr- and photothermal-induced refractive index
modulations. The probe frequency response measured for three different
microresonators is presented in Fig. 3c. Both pump and probe laser powers are
sufficiently low to minimize the thermal locking effect 52.

The plateau in the frequency response at low frequency gives the combined
quasi-static contributions of photothermal and Kerr effects in the sum
measurement (inset of Fig. 3a), while the high frequency response constitutes
only the Kerr contribution. In addition, the Kerr effect here is the cross-
phase modulation contribution (from the pump to the probe), while, as noted
above, the Kerr self-phase modulation contribution appears in Eq. 1. These two
effects are related by a cross-phase modulation factor $\gamma$ determined by
the mode combinations used (see Methods). For Al0.2Ga0.8As and Ta2O5, pump and
probe modes both belong to the fundamental TE mode. For Si3N4, pump and probe
modes belong to the fundamental TE and TM modes, respectively (see
Supplementary note III). It is noted that this measurement was challenging to
perform in the suspended SiO2 microdisks on account of a very slow thermal
diffusion process (see Supplementary note IIIA). Instead, a published value of
$n_{2}$ for SiO2 ($2.2\times 10^{-20}$ m2 W-1) was used 53. By numerically
fitting the response curves (see Supplementary note IIB and III), the ratio
between Kerr and photothermal effects is extracted over a range of wavelengths
and plotted in Fig. 3d.

Combining results from the above sum and ratio measurements, the photothermal
and Kerr coefficients are obtained individually. The inferred absorption-
limited $Q_{\rm abs}$ values measured over the telecommunication C-band for
each material are summarized in Fig. 4a. It is worth mentioning that the SiO2
microdisk measurement requires a narrow-linewidth, highly-stable fiber laser
on account of the microresonator’s ultra-high $Q$ factor. The use of the fiber
laser limits the measurement range to near 1550 nm. A combined plot of the
measured $n_{2}$ values (normalized by $n_{o}^{2}$) versus the absorption
$Q_{\rm abs}$ is given in Fig. 4b (the $n_{2}$ of SiO2 is taken from the
literature 53). Also, in the cases of critical coupling ($Q_{e}=Q_{0}$) and
absorption-limited intrinsic $Q$ factors ($Q_{0}=Q_{\rm abs}$), the parametric
oscillation threshold per unit mode volume 3, 56, 57 for a single material is
shown by dashed red iso-contours:

${P_{\rm th}\over V_{\mathrm{eff}}}={n_{o}^{2}\,\omega_{o}\over n_{2}Q_{\rm
abs}^{2}c},$ (4)

where $V_{\mathrm{eff}}$ is the effective mode volume. It should be noted that
actual thresholds may be different if the optical field is not tightly
confined in the core of the microresonator heterostructure.

The results described above are further summarized in Table 1, where, for SiO2
and Si3N4, the measured material absorption losses are much lower than the
present microresonator intrinsic losses. Therefore, improvement in
microfabication of SiO2 and Si3N4 to reduce surface roughness, hence to reduce
scattering losses, will benefit photonic integrated circuits using these
materials. For Al0.2Ga0.8As and Ta2O5, the material losses are close to their
respective intrinsic losses, which suggests that both material and scattering
loss contributions should be addressed.

Overall, the absorption $Q_{\rm abs}$ values reported here should be viewed as
state-of-the-art values that are not believed to be at fundamental limits. For
example, silica glass in optical fiber exhibits loss (typically 0.2 dB km-1)
58 that is still over one order of magnitude lower than that reported in Fig.
4b. Likewise, Ta2O5 is the premier material for optical coatings employed, for
example, in the highest performance optical clocks and gravitational-wave
interferometers. However, Ta2O5 exhibits fascinating stoichiometry and
crystallization effects, which require careful mitigation in deposition and
processing. The material-limited $Q$ of Ta2O5 and TiO2:Ta2O5 has been measured
to be 5 million and 25 million, respectively 59. Hence, the nanofabricated
devices and precision-measurement technique reported here highlight the
promise to optimize material-limited performance in the Ta2O5 platform. It is
also noted that in Al0.2Ga0.8As, a compound semiconductors material, surface
defects may generate mid-gap states 43 which cause extra material absorption
loss. This loss mechanism will depend upon process conditions and intrinsic
$Q$ factors as high as 3.52 M for Al0.2Ga0.8As have been reported elsewhere
27. Finally, some of the material parameters used in modeling are impacted by
factors such as the film deposition method. For example, thermal conductivity
of Ta2O5 can depend upon the deposition method as is reflected by a wide range
of values available in the literature (see Supplementary note IIID). Such
effects could also impact other materials used in this study, but we have
nonetheless relied upon bulk values in modeling (see Supplementary note IIC).

The current method also provides in-situ measurement of $n_{2}$ for integrated
photonic microresonators. We compare the $n_{2}$ values measured here with
other reported values in Table 1. To give a fashion of how the nonlinearity
varies between the four materials, third-order nonlinear susceptibility
$\chi_{(3)}$ is calculated from the measured $n_{2}$ and compared with the
linear susceptibility $\chi_{(1)}$. The Miller’s rule 60, 61
$\chi_{(3)}\propto\chi_{(1)}^{4}$ relating the scaling of these two quantities
is observed (see Supplementary note IV).

In summary, the absorption loss and Kerr nonlinear coefficients of four
leading integrated photonic materials have been measured using cavity-enhanced
photothermal spectroscopy. The material absorption sets a practical limit of
these materials in microcavity applications. The Kerr nonlinear coefficients
have also been characterized, and the results are consistent with a general
trend relating to nonlinearity and optical loss. Overall, the results suggest
specific directions where there can be improvement in these systems as well as
providing a way to predict future device performance.

## References

  * [1] Vahala, K. J.  Optical microcavities.  _Nature_ 424, 839 (2003).
  * [2] Spillane, S., Kippenberg, T. & Vahala, K.  Ultralow-threshold raman laser using a spherical dielectric microcavity.  _Nature_ 415, 621–623 (2002).
  * [3] Kippenberg, T., Spillane, S. & Vahala, K.  Kerr-nonlinearity optical parametric oscillation in an ultrahigh-Q toroid microcavity.  _Phys. Rev. Lett._ 93, 083904 (2004).
  * [4] Lee, H. _et al._ Chemically etched ultrahigh-Q wedge-resonator on a silicon chip.  _Nat. Photon._ 6, 369–373 (2012).
  * [5] Schawlow, A. L. & Townes, C. H.  Infrared and optical masers.  _Phys. Rev._ 112, 1940 (1958).
  * [6] Vahala, K. J.  Back-action limit of linewidth in an optomechanical oscillator.  _Phys. Rev. A_ 78, 023832 (2008).
  * [7] Li, J., Lee, H., Chen, T. & Vahala, K. J.  Characterization of a high coherence, brillouin microcavity laser on silicon.  _Opt. Express_ 20, 20170–20180 (2012).
  * [8] Aoki, T. _et al._ Observation of strong coupling between one atom and a monolithic microresonator.  _Nature_ 443, 671–674 (2006).
  * [9] Lu, X. _et al._ Chip-integrated visible–telecom entangled photon pair source for quantum communication.  _Nat. Phys._ 15, 373–381 (2019).
  * [10] Lu, X. _et al._ Efficient telecom-to-visible spectral translation through ultralow power nonlinear nanophotonics.  _Nat. Photon._ 13, 593–601 (2019).
  * [11] Lukin, D. M. _et al._ 4H-silicon-carbide-on-insulator for integrated quantum and nonlinear photonics.  _Nat. Photon._ 14, 330–334 (2020).
  * [12] Ma, Z. _et al._ Ultrabright quantum photon sources on chip.  _Phys. Rev. Lett._ 125, 263602 (2020).
  * [13] Kippenberg, T. J. & Vahala, K. J.  Cavity optomechanics: back-action at the mesoscale.  _Science_ 321, 1172–1176 (2008).
  * [14] Vollmer, F. & Yang, L.  Label-free detection with high-Q microcavities: a review of biosensing mechanisms for integrated devices.  _Nanophotonics_ 1, 267–291 (2012).
  * [15] Li, J., Lee, H. & Vahala, K. J.  Microwave synthesizer using an on-chip brillouin oscillator.  _Nat. Commun._ 4, 1–7 (2013).
  * [16] Kippenberg, T. J., Gaeta, A. L., Lipson, M. & Gorodetsky, M. L.  Dissipative Kerr solitons in optical microresonators.  _Science_ 361 (2018).
  * [17] Gundavarapu, S. _et al._ Sub-hertz fundamental linewidth photonic integrated Brillouin laser.  _Nat. Photon._ 13, 60–67 (2019).
  * [18] Jin, W. _et al._ Hertz-linewidth semiconductor lasers using cmos-ready ultra-high-q microresonators.  _Nat. Photon._ 14, 346–352 (2021).
  * [19] Li, J., Suh, M. G. & Vahala, K. J.  Microresonator Brillouin gyroscope.  _Optica_ 4, 346–348 (2017).
  * [20] Liang, W. _et al._ Resonant microphotonic gyroscope.  _Optica_ 4, 114–117 (2017).
  * [21] Lai, Y.-H. _et al._ Earth rotation measured by a chip-scale ring laser gyroscope.  _Nat. Photon._ 14, 345–349 (2020).
  * [22] Ji, X. _et al._ Ultra-low-loss on-chip resonators with sub-milliwatt parametric oscillation threshold.  _Optica_ 4, 619–624 (2017).
  * [23] Zhang, M., Wang, C., Cheng, R., Shams-Ansari, A. & Lončar, M.  Monolithic ultra-high-Q lithium niobate microring resonator.  _Optica_ 4, 1536–1537 (2017).
  * [24] Yang, K. Y. _et al._ Bridging ultrahigh-Q devices and photonic circuits.  _Nat. Photon._ 12, 297 (2018).
  * [25] Wilson, D. J. _et al._ Integrated gallium phosphide nonlinear photonics.  _Nat. Photon._ 14, 57–62 (2020).
  * [26] Chang, L. _et al._ Ultra-efficient frequency comb generation in AlGaAs-on-insulator microresonators.  _Nat. Commun._ 11, 1–8 (2020).
  * [27] Xie, W. _et al._ Ultrahigh-Q AlGaAs-on-insulator microresonators for integrated nonlinear photonics.  _Opt. Express_ 28, 32894–32906 (2020).
  * [28] Liu, J. _et al._ Photonic microwave generation in the X- and K-band using integrated soliton microcombs.  _Nat. Photon._ 14, 486–491 (2020).
  * [29] Liu, X. _et al._ Aluminum nitride nanophotonics for beyond-octave soliton microcomb generation and self-referencing.  _Nat. Commun._ 12, 5428 (2021).
  * [30] Gao, R. _et al._ Broadband highly efficient nonlinear optical processes in on-chip integrated lithium niobate microdisk resonators of Q-factor above 108.  _arXiv:2102.00399_ (2021).
  * [31] Biberman, A., Shaw, M. J., Timurdogan, E., Wright, J. B. & Watts, M. R.  Ultralow-loss silicon ring resonators.  _Opt. Lett._ 37, 4236–4238 (2012).
  * [32] Vernooy, D. W., Ilchenko, V. S., Mabuchi, H., Streed, E. W. & Kimble, H. J.  High-Q measurements of fused-silica microspheres in the near infrared.  _Opt. Lett._ 23, 247–249 (1998).
  * [33] Pfeiffer, M. H. P. _et al._ Ultra-smooth silicon nitride waveguides based on the Damascene reflow process: fabrication and loss origins.  _Optica_ 5, 884–892 (2018).
  * [34] Pfeiffer, M. H., Liu, J., Geiselmann, M. & Kippenberg, T. J.  Coupling Ideality of Integrated Planar High- Q Microresonators.  _Phys. Rev. Appl._ 7, 024026 (2017).
  * [35] Spencer, D. T., Bauters, J. F., Heck, M. J. & Bowers, J. E.  Integrated waveguide coupled Si3N4 resonators in the ultrahigh-Q regime.  _Optica_ 1, 153–157 (2014).
  * [36] Liu, J. _et al._ Monolithic piezoelectric control of soliton microcombs.  _Nature_ 583, 385–390 (2020).
  * [37] Xiang, C. _et al._ Laser soliton microcombs heterogeneously integrated on silicon.  _Science_ 373, 99–103 (2021).
  * [38] Gorodetsky, M. L., Savchenkov, A. A. & Ilchenko, V. S.  Ultimate Q of optical microsphere resonators.  _Opt. Lett._ 21, 453–455 (1996).
  * [39] Rokhsari, H., Spillane, S. & Vahala, K.  Loss characterization in microcavities using the thermal bistability effect.  _Appl. Phys. Lett._ 85, 3029–3031 (2004).
  * [40] Liu, J. _et al._ Ultralow-power chip-based soliton microcombs for photonic integration.  _Optica_ 5, 1347–1353 (2018).
  * [41] Puckett, M. W. _et al._ 422 million intrinsic quality factor planar integrated all-waveguide resonator with sub-mhz linewidth.  _Nat. Commun._ 12, 1–8 (2021).
  * [42] Pfeiffer, M. H. _et al._ Ultra-smooth silicon nitride waveguides based on the damascene reflow process: fabrication and loss origins.  _Optica_ 5, 884–892 (2018).
  * [43] Parrain, D. _et al._ Origin of optical losses in gallium arsenide disk whispering gallery resonators.  _Opt. Express_ 23, 19656–19672 (2015).
  * [44] Guha, B. _et al._ Surface-enhanced gallium arsenide photonic resonator with quality factor of 6$\times$ 106.  _Optica_ 4, 218–221 (2017).
  * [45] An, K., Sones, B. A., Fang-Yen, C., Dasari, R. R. & Feld, M. S.  Optical bistability induced by mirror absorption: measurement of absorption coefficients at the sub-ppm level.  _Opt. Lett._ 22, 1433–1435 (1997).
  * [46] Rokhsari, H. & Vahala, K. J.  Observation of kerr nonlinearity in microcavities at room temperature.  _Opt. Lett._ 30, 427–429 (2005).
  * [47] Wang, T. _et al._ Rapid and high precision measurement of opto-thermal relaxation with pump-probe method.  _Sci. Bull._ 63, 287–292 (2018).
  * [48] Liu, J. _et al._ High-yield, wafer-scale fabrication of ultralow-loss, dispersion-engineered silicon nitride photonic circuits.  _Nat. Commun._ 12, 2236 (2021).
  * [49] Wu, L. _et al._ Greater than one billion q factor for on-chip microresonators.  _Opt. Lett._ 45, 5129–5131 (2020).
  * [50] Jung, H. _et al._ Tantala kerr nonlinear integrated photonics.  _Optica_ 8, 811–817 (2021).
  * [51] Li, J., Lee, H., Yang, K. Y. & Vahala, K. J.  Sideband spectroscopy and dispersion measurement in microcavities.  _Opt. Express_ 20, 26337–26344 (2012).
  * [52] Carmon, T., Yang, L. & Vahala, K.  Dynamical thermal behavior and thermal self-stability of microcavities.  _Opt. Express_ 12, 4742–4750 (2004).
  * [53] Agrawal, G. P.  _Nonlinear fiber optics_ (Academic press, 2007).
  * [54] Ikeda, K., Saperstein, R. E., Alic, N. & Fainman, Y.  Thermal and kerr nonlinear properties of plasma-deposited silicon nitride/silicon dioxide waveguides.  _Opt. Express_ 16, 12987–12994 (2008).
  * [55] Pu, M., Ottaviano, L., Semenova, E. & Yvind, K.  Efficient frequency comb generation in algaas-on-insulator.  _Optica_ 3, 823–826 (2016).
  * [56] Li, J., Lee, H., Chen, T. & Vahala, K. J.  Low-pump-power, low-phase-noise, and microwave to millimeter-wave repetition rate operation in microcombs.  _Phys. Rev. Lett._ 109, 233901 (2012).
  * [57] Yi, X., Yang, Q.-F., Yang, K. Y., Suh, M.-G. & Vahala, K.  Soliton frequency comb at microwave rates in a high-Q silica microresonator.  _Optica_ 2, 1078–1085 (2015).
  * [58] Miya, T., Terunuma, Y., Hosaka, T. & Miyashita, T.  Ultimate low-loss single-mode fibre at 1.55 $\mu$m.  _Electron. Lett._ 15, 106–108 (1979).
  * [59] Pinard, L. _et al._ Toward a new generation of low-loss mirrors for the advanced gravitational waves interferometers.  _Opt. Lett._ 36, 1407–1409 (2011).
  * [60] Miller, R. C.  Optical second harmonic generation in piezoelectric crystals.  _Appl. Phys. Lett._ 5, 17–19 (1964).
  * [61] Ettoumi, W., Petit, Y., Kasparian, J. & Wolf, J.-P.  Generalized miller formulæ.  _Opt. Express_ 18, 6613–6620 (2010).

Methods

Fabrication of optical microresonators. The SiO2 microresonator is fabricated
by thermally growing 8-$\mu$m thick thermal wet oxide on a 4 inch float-zone
silicon wafer, followed by i-line stepper photolithography, buffered oxide
etch, XeF2 silicon isotropic dry etch and thermal annealing 4, 49. The Si3N4
microresonator is fabricated with the photonic Damascene process, including
using deep-ultraviolet stepper lithography, preform etching, low-pressure
chemical vapour deposition, planarization, cladding and annealing 48. The
Al0.2Ga0.8As microresonator is fabricated with an epitaxial Al0.2Ga0.8As layer
bonded onto a silicon wafer with a 3-$\mu$m thermal SiO2 layer, followed by
GaAs substrate removal, deep ultraviolet patterning, inductively coupled
plasma etching, passivation with Al2O3 and SiO2 cladding 26, 27. The Ta2O5
microresonator is fabricated by ion-beam sputtering Ta2O5 deposition followed
by annealing, electron-beam lithography, Ta2O5 etching, ultraviolet
lithography and dicing 50.

Experimental details. In the sum measurement, the scanning speed of the laser
frequency is decreased until the mode’s broadening as induced by the thermo-
optic shift becomes stable (i.e., not influenced by the scan rate). Also, the
waveguide input power is minimized such that it is well below the threshold of
parametric oscillation. The power is calibrated using the photodetector
voltage.

In the ratio measurement, the optical frequencies of the pump and probe lasers
are locked to their respective cavity modes using a servo feedback with 1 kHz
bandwidth. The pump laser is locked near the mode resonant frequency, while
the probe laser is locked to the side of the resonance to increase
transduction of refractive index modulation into transmitted probe power. The
intensity modulator is calibrated in a separate measurement under the same
driving power.

Fitting of spectra in the sum measurement. For Si3N4 and Ta2O5 devices, the
transmission spectrum is the interference of a Lorentzian-lineshaped mode
resonance with a background field contributed by facet reflections of the
waveguide. The transmission function of a cavity resonance is given by

$T_{\rm res}=1-\frac{\kappa_{e}}{\kappa/2+i[\Delta-(\alpha+g)\rho]},$ (5)

where $\Delta$ is the cold-cavity laser-cavity detuning, $\alpha$ and $g$ are
the absorption and Kerr nonlinear coefficients, respectively, and $\rho$ is
the intracavity energy density as defined in the main text. The reflection at
the two waveguide facets forms a low-finesse Fabry–Pérot resonator. Combining
this waveguide reflection with the cavity resonance, the overall amplitude
transmission is given by (see Supplementary note II)

$T\propto\left|\frac{T_{\rm res}}{1-rT_{\rm res}^{2}\exp[i(-\Delta/\omega_{\rm
FP}+\phi)]}\right|^{2},$ (6)

where $r$ is the reflectivity at the waveguide facet, $\omega_{\rm FP}$ is the
free spectral range of the facet-induced Fabry–Pérot cavity (in rad/s units),
and $\phi$ is a constant phase offset.

In the experiment, the above quantities are fitted in three steps. First,
$\omega_{\rm FP}$ and $r$ are obtained by measuring the transmission away from
mode resonances. Next, loss rates $\kappa$ and $\kappa_{e}$ can be determined
by measuring the transmission of the mode at a low probe power. Finally,
launching higher power into the microresonator allows the mode broadening to
be observed and the transmission is fitted with Eq. (6), where $(\alpha+g)$ is
the fitting variable and other parameters are obtained from the previous
steps. For Al0.2Ga0.8As and SiO2 devices which have no Fabry–Pérot background,
$r$ can be set to zero and the first step in the above fitting procedure can
be omitted. The fitting results are presented in Fig. 2b.

Fitting of response in the ratio measurement. The response of the probe mode
resonant frequency $\tilde{\delta}_{\rm b}$ as a result of pump power
modulation $\tilde{P}_{\mathrm{in}}$ can be described by (see Supplementary
note II),

$\frac{\tilde{\delta}_{\rm
b}(\Omega)}{\tilde{P}_{\mathrm{in}}(\Omega)}=-\frac{\alpha\tilde{r}(\Omega)+\gamma
g}{V_{\rm eff}}\frac{2\eta_{\rm p}}{i\Omega+\kappa_{\rm p}/2}.$ (7)

where $\Omega$ is the pump power modulation frequency (in rad/s units),
$\tilde{P}_{\mathrm{in}}$ is the modulation amplitude of the pump power,
$\kappa_{\rm p}$ is the total loss rate of the pump mode, $\eta_{\rm
p}=\kappa_{e,\rm p}/\kappa_{\rm p}$ is the coupling efficiency for pump mode,
$\alpha$ is the absorption coefficient as mentioned in the previous section,
$\tilde{r}$ is the frequency response of modal temperature modulation as a
result of thermal diffusion, and the factor $\gamma$ accounts for cross-phase
modulation of the probe mode by the pump mode.

The frequency response of the transmitted probe mode with respect to its
resonance shift $\tilde{\delta}_{\rm b}(\Omega)$ is derived in Supplementary
note II and has the following form:

$\frac{\tilde{T}_{\rm b}(\Omega)}{\tilde{\delta}_{\rm
b}(\Omega)}=-\frac{2\kappa_{e,\rm b}\Delta_{\rm b}^{(0)}}{\kappa_{\rm
b}^{2}/4+\left(\Delta_{\rm b}^{(0)}\right)^{2}}\frac{\kappa_{\rm
b}-\kappa_{e,\rm b}+i\Omega}{(\kappa_{\rm b}/2+i\Omega)^{2}+\left(\Delta_{\rm
b}^{(0)}\right)^{2}}|a_{\mathrm{in,b}}|^{2},$ (8)

where $\Delta_{\rm b}^{(0)}$ is the steady-state detuning of the probe mode
when no modulation is present, and $\kappa_{\rm b}$ and $\kappa_{e,\rm b}$
refer to the total loss rate and external coupling rate for the probe mode.

The response curve in Fig. 3c is modeled by,

$\tilde{\mathcal{R}}(\Omega)=\frac{\tilde{T}_{\rm b}(\Omega)}{\tilde{P}_{\rm
in}(\Omega)}=\frac{\tilde{T}_{\rm b}(\Omega)}{\tilde{\delta}_{\rm
b}(\Omega)}\frac{\tilde{\delta}_{\rm b}(\Omega)}{\tilde{P}_{\rm in}(\Omega)}.$
(9)

and is fitted according to Eq. (7) and (8). In the fitting, $\kappa$ and
$\kappa_{e}$ have been measured separately, $\tilde{r}$ is determined from
finite element method simulations, and the probe mode $\Delta_{0}$ and ratio
$\alpha/g$ are parameters to be fitted.

Data availability
All data generated or analysed during this study are available within the
paper and its supplementary materials. Further source data will be made
available upon reasonable request.

Code availability
The analysis codes will be made available upon reasonable request.

Acknowledgments The authors thank H. Blauvelt and Z. Yuan for helpful
comments, Z. Liu for discussions on data processing, as well as D. Carlson for
preparing the Ta2O5 SEM image. Q.-X. J. acknowledges the Caltech Student
Faculty Program for financial supports. This work was supported by DARPA under
the DODOS (HR0011-15-C-0055) and the APHi (FA9453-1 9-C-0029) programs.

Author contributions
M.G., Q.-F.Y., Q.-X.J. and J.L. conceived the idea. M.G., Q.-F.Y. and Q.-X.J.
performed the measurement with assistance from H.W., L.W., and B.S.. M.G.,
Q.-F.Y., Q.-X.J. and H.W. devised the theoretical model. L.W. provided the
SiO2 samples. J.L. and G.H. provided the Si3N4 samples. L.C. and W.X. provided
the Al0.2Ga0.8As samples. S.-P.Y. provided the Ta2O5 samples. All the authors
analyzed the data and wrote the manuscript. K.J.V., T.J.K., J.E.B. and S.B.P.
supervised the project.

Competing interests The authors declare no competing financial interests.

Additional information
Supplementary information is available for this paper.
Correspondence and requests for materials should be addressed to K.J.V.


# Damping of the Franz-Keldysh oscillations in the presence of disorder

R. E. Putnam, Jr. and M. E. Raikh  Department of Physics and Astronomy,
University of Utah, Salt Lake City, UT 84112

###### Abstract

Franz-Keldysh oscillations of the optical absorption in the presence of short-
range disorder are studied theoretically. The magnitude of the effect depends
on the relation between the mean-free path in a zero field and the distance
between the turning points in electric field. Damping of the Franz-Keldysh
oscillations by the disorder develops at high absorption frequency. Effect of
damping is amplified by the fact that, that electron and hole are most
sensitive to the disorder near the turning points. This is because, near the
turning points, velocities of electron and hole turn to zero.

##  I Introduction

Application of electric field to an insulator modifies the shape of the
absorption spectrum in two respects: (i) below-threshold part of the spectrum
develops a tail which becomes more voluminous upon increasing the field; (ii)
above-the-threshold part of the spectrum acquires an oscillating component
with a period of oscillations gradually decreasing as the field is increased.
This effect predicted by Franz and KeldyshFranz ; Keldysh  more than 60 years
ago was subsequently observed on a large number of bulk semiconductors, see
e.g. Ref. bulk, .

Figure 1: (Color online) Schematic illustration of the above-the-gap
electroabsorption in an insulator. Initital kinetic energies of electron and
hole are $\hbar\Omega_{e}$ and $\hbar\Omega_{e}$, respectively. Franz-Keldysh
oscillations reflect the interference of incident electron (hole) wave with a
wave reflected at the corresponding turning point. Disorder, illustrated with
dots, leads to the smearing of the interference pattern, and, consequently, to
the smearing of the oscillations. It is crucial that patterns for electron and
hole are smeared by scattering from different impurities.

Later, theoretical and experimental studies were extended to quantum
wellsChemla ; Well  and quantum wires.Wire ; Wire1  Less trivial is that
Franz-Keldysh oscillations were observed in certain polymers,polymer ;
polymer1  and recently, in nanotubes,nanotube ; nanotube1
nanocrystals,nanocrystal  and perovskites.perovskite0 ; perovskite1 ;
perovskite2  Polymers and perovskites are known to be highly disordered
materials with carrier mobilitymobility  $\sim
10~{}\text{cm}^{2}\cdot\text{V}^{-1}\cdot\text{s}^{-1}$. Then the question
arises, how the inherent disorder in these materials affects the Franz-Keldysh
effect. Apparently, the effect of an electric field on the tail part of the
absorption spectrum is weakened by the disorder. This is because the disorder
creates a tail of its own. Effect of disorder on the above-the-threshold
oscillations is of a different physical origin. It is illustrated in Fig. 1.
Upon absorption of a photon, electron and hole have kinetic energies
$\hbar\Omega_{e}$ and $\hbar\Omega_{h}$, respectively. They get, subsequently,
reflected at the points $\pm x_{t}$, shown in the figure, and interfere with
incident waves. The resulting density pattern resembles a standing wave.
Presence of impurities ”scrambles” the standing-wave pattern, leading to the
suppression of the Franz-Keldysh oscillations. This suppression is studied
theoretically in the present paper. Superficially, it is clear that, in order
to distinguish between the weak and strong disorder limits, one has to compare
$x_{t}$ to the mean free path, $l$. For a weak disorder one has $l\gg x_{t}$,
so the oscillations are not affected. However, there is a nontrivial aspect to
this picture, namely, upon approaching the corresponding turning points, $\pm
x_{t}$, kinetic energies of electron and hole turn to zero. Thus, the local
mean free paths decrease dramatically. This contributes to the suppression of
the interference, and thus, to the damping of the oscillations.

##  II Smearing of the Franz-Keldysh oscillations by the disorder

Without the disorder, the oscillating part of the absorption coefficient of
light with frequency, $\omega$, in the presence of electric field, $F$, is
given by

$A(\Omega)\propto\sin\left[c\left(\frac{\hbar\Omega}{E_{0}}\right)^{3/2}\right],$
(1)

where $c$ is a constant, and $\hbar\Omega=\hbar\omega-E_{g}$, where $E_{g}$ is
a band-gap. For equal masses of electron and hole one has
$c=\frac{2^{5/2}}{3}$. Energy, $E_{0}$, in Eq. (1) is defined as

$E_{0}=\left(\frac{\hbar^{2}F^{2}}{m}\right)^{1/3}.$ (2)

It has a meaning of the characteristic electron energy in the field, $F$.

Our main result is that, in the presence of disorder, the spectrum $A(\Omega)$
is multiplied by a damping factor $\exp\big{[}-{{\cal L}(\Omega)}\big{]}$,
where the function ${\cal L}\left(\Omega\right)$ is defined as

${\cal
L}\left(\Omega\right)=\frac{m\gamma}{\hbar^{2}F}\ln\left(\frac{\hbar\Omega}{E_{0}}\right).$
(3)

Here $\gamma$ is the factor in the correlation relation,

$\langle V(x)V(x^{\prime})\rangle=\gamma\delta(x-x^{\prime}),$ (4)

of the disorder potential, $V(x)$. Disorder leads to a finite scattering time.
For electron with energy, $E$, the golden-rule calculation of the scattering
time yields

$\frac{\hbar}{\tau(E)}=\frac{\gamma}{2}\left(\frac{2m}{\hbar^{2}E}\right)^{1/2}.$
(5)

Expression in the square bracket is a 1D density of states. Upon approaching
the turning point, kinetic energy of electron decreases, and, thus, the
scattering time shortens. Decrease of velocity upon approaching the turning
point leads to additional shortening of the mean free path.

$l_{E}=\Big{(}\frac{2E}{m}\Big{)}^{1/2}\tau(E)=\frac{2\hbar^{2}}{m\gamma}E.$
(6)

Note, that oscillating absorption Eq. (1) corresponds to the large-argument
asymptote of the square of the Airy function. It applies for $\hbar\Omega\gg
E_{0}$ when the argument of sine is big.

Physical meaning of the prefactor $\frac{m\gamma}{\hbar^{2}F}$ in Eq. (3) is
the following. Energy scale $E_{0}$ comes from the distance
$x_{F}\sim\frac{E_{0}}{F}$. From Eq. (4) it follows that the r.m.s. random
potential within the distance $x_{F}$ can be estimated as
$V(x_{F})\sim\left(\frac{\gamma}{x_{F}}\right)^{1/2}$. Then the prefactor in
Eq. (3) can be cast in the form $\left(\frac{V(x_{F})}{E_{0}}\right)^{2}$.

##  III General expression for the absorption coefficient

The golden-rule expression for the absorption coefficient of light with
frequency, $\omega$, is the following

$\displaystyle A(\omega)$ $\displaystyle\propto\sum_{\mu,\nu}\Big{|}\int d{\bf
r}\psi_{\mu}^{e}({\bf r})\left(\psi_{\nu}^{h}({\bf
r})\right)^{\ast}\Big{|}^{2}\delta\Big{[}\hbar\Omega_{\mu}^{e}+\hbar\Omega_{\nu}^{h}-\left(\hbar\omega-
E_{g}\right)\Big{]},$ (7)

where $\psi_{\mu}^{e}({\bf r})$, $\psi_{\nu}^{h}({\bf r})$ are the
eigenfunctions of the initial and final states, while $\hbar\Omega_{\mu}^{e}$
and $\hbar\Omega_{\nu}^{h}$ are the corresponding energies. Since the
eigenfunctions carry information about the disorder, it is convenient to
”decouple” the $\delta$-function as follows

$\displaystyle\delta\Big{(}\Omega_{\mu}^{e}+\Omega_{\nu}^{h}-\omega+\frac{E_{g}}{\hbar}\Big{)}=\int
d\Omega_{1}\int d\Omega_{2}$
$\displaystyle\times\delta\left(\omega-\frac{E_{g}}{\hbar}-\Omega_{1}-\Omega_{2}\right)\delta\left(\Omega_{1}-\Omega_{\mu}^{e}\right)\delta\left(\Omega_{2}-\Omega_{\nu}^{h}\right),$
(8)

and rewrite $A(\omega)$ in the form

$\displaystyle A(\omega)=\int d\Omega_{1}\int
d\Omega_{2}\delta\left(\Omega_{1}+\Omega_{2}+\frac{E_{g}}{\hbar}-\omega\right)$
$\displaystyle\times\int d{\bf r}_{1}\int d{\bf r}_{2}\text{Im}G_{e}({\bf
r}_{1},{\bf r_{2}},\Omega_{1})\text{Im}G_{h}({\bf r}_{1},{\bf
r_{2}},\Omega_{2}),$ (9)

where the imaginary parts of the Green functions are defined as

$\displaystyle\text{Im}G_{e}({\bf r}_{1},{\bf
r}_{2},\Omega_{1})=\sum_{\mu}\psi_{\mu}^{e}({\bf
r}_{1})\left(\psi_{\mu}^{e}({\bf
r}_{2})\right)^{\ast}\delta\left(\Omega_{\mu}^{e}-\Omega_{1}\right),$
$\displaystyle\text{Im}G_{h}({\bf r}_{1},{\bf
r}_{2},\Omega_{2})=\sum_{\nu}\left(\psi_{\nu}^{h}({\bf
r}_{1})\right)^{\ast}\psi_{\nu}^{h}({\bf
r}_{2})\delta\left(\Omega_{\nu}^{h}-\Omega_{2}\right).$ (10)

In the presence of disorder, the product $\text{Im}G_{e}\text{Im}G_{h}$ in the
right-hand side should be averaged over different configurations. It is
crucial, that the disorder-induced smearing of the interference patterns for
electrons and holes is dominated by different impurities. This is illustrated
in Fig. 1. As a result, the disorder averaging can be performed independently.

##  IV Semiclassical calculation of the absorption damping in 1D

Figure 2: (Color online) Evolution of Franz-Keldysh oscillations with applied
electric field is plotted from Eqs. (1) and (3) for three values of electric
field, $F$, measured in the units of $F_{c}$ given by Eq. (21). Energy
$\hbar\Omega$ is measured in the units of
$E_{c}=\left(\frac{m\gamma^{2}}{2\hbar^{2}}\right)^{1/3}$.

Franz-Keldysh oscillations emerge in semiclassical regime, when electron and
hole states can be described by local wave vectors

$k_{e}(x)=\left(\frac{2m}{\hbar^{2}}\right)^{1/2}\left(\hbar\Omega_{1}-Fx\right)^{1/2},$
(11)

$k_{h}(x)=\left(\frac{2m}{\hbar^{2}}\right)^{1/2}\left(\hbar\Omega_{2}+Fx\right)^{1/2}.$
(12)

In accordance with Fig. 1, $k_{e}(x)$ turns to zero at the right turning
point, while $k_{h}(x)$ turns to zero at the left turning point. Electron and
hole Green functions, which enter the light absorption Eq. (III) can be
expressed in terms of $k_{e}(x)$ and $k_{h}(x)$ as follows

$\displaystyle\text{Im}G_{e}^{(0)}(x_{1},x_{2},\Omega_{1})=\frac{\cos\left(\Phi_{e}(x_{1})-\frac{\pi}{4}\right)\cos\left(\Phi_{e}(x_{2})-\frac{\pi}{4}\right)}{\left(k_{e}(x_{1})k_{e}(x_{2})\right)^{1/2}},$
$\displaystyle\text{Im}G_{h}^{(0)}(x_{1},x_{2},\Omega_{2})=\frac{\cos\left(\Psi_{h}(x_{1})+\frac{\pi}{4}\right)\cos\left(\Psi_{h}(x_{2})+\frac{\pi}{4}\right)}{\left(k_{h}(x_{1})k_{h}(x_{2})\right)^{1/2}}.$
(13)

where the semiclassical actions $\Phi_{e}$ and $\Phi_{h}$ are defined as

$\Phi_{e}(x_{1})=\int\limits_{x_{1}}^{\hbar\Omega_{1}/F}dx^{\prime}k_{e}(x^{\prime}),~{}\Psi_{h}(x_{1})=\int\limits_{-\hbar\Omega_{2}/F}^{x_{1}}dx^{\prime}k_{h}(x^{\prime}).$
(14)

Expressions Eq. (IV) are semiclassical and apply away from the turning points:

$\Big{(}\frac{\hbar\Omega_{1}}{F}-x_{1}\Big{)}\gg\Big{(}\frac{\hbar^{2}}{mF}\Big{)}^{1/3}=x_{F},~{}\Big{(}x_{1}+\frac{\hbar\Omega_{2}}{F}\Big{)}\gg
x_{F}.$ (15)

Our goal is to incorporate disorder into the Green functions. For this purpose
we transform the products of the cosines into the sums

$\displaystyle\text{Im}G_{e}^{(0)}(x_{1},x_{2},\Omega_{1})$
$\displaystyle=\frac{\cos\left(\Phi_{e}(x_{1})-\Phi_{e}(x_{2})\right)+\cos\left(\Phi_{e}(x_{1})+\Phi_{e}(x_{2})-\frac{\pi}{2}\right)}{\left(k_{e}(x_{1})k_{e}(x_{2})\right)^{1/2}},$
(16)

$\displaystyle\text{Im}G_{h}^{(0)}(x_{1},x_{2},\Omega_{2})$
$\displaystyle=\frac{\cos\left(\Psi_{h}(x_{1})-\Psi_{h}(x_{2})\right)+\cos\left(\Psi_{h}(x_{1})+\Psi_{h}(x_{2})+\frac{\pi}{2}\right)}{\left(k_{h}(x_{1})k_{h}(x_{2})\right)^{1/2}}.$
(17)

Physically, the first term in Eq. (IV) describes the propagation of electron
from $x_{1}$ to $x_{2}$, while the second term describes the motion of
electron from $x_{1}$ to the turning point, $x=x_{t}$ followed by reflection
from the electrostatic barrier and return to $x_{2}$. It is shown in the
Appendix in great detail that disorder averaging of the first term amounts to
multiplying this term by $\exp(-|x_{2}-x_{1}|/l)$ when the mean free path,
$l$, does not depend on $x$. Since, in electric field, kinetic energy of
electron is a function of position, and $l$ is a function of energy, we
conclude that the mean free path is a function of position. Then the natural
generalization of the damping factor is
$\exp\Big{(}-\int\limits_{x_{1}}^{x_{2}}\frac{dx}{l(x)}\Big{)}$. Substituting
the $x$-dependent kinetic energy into Eq. (6), we get the following
expressions for the mean free paths of electrons and holes

$l_{1}(x)^{-1}=\frac{m\gamma}{2\hbar^{2}(\hbar\Omega_{1}-Fx)},~{}l_{2}(x)^{-1}=\frac{m\gamma}{2\hbar^{2}(\hbar\Omega_{2}+Fx)}.$
(18)

Thus, incorporation of disorder into the Green functions
$\text{Im}G_{e}^{(0)}(x_{1},x_{2})$ and $\text{Im}G_{h}^{(0)}(x_{1},x_{2})$
amounts to ascribing to each cosine in Eqs. (IV and (IV) its corresponding
damping factor.

Turning to the product $\text{Im}G_{e}^{(0)}\text{Im}G_{h}^{(0)}$ we notice
that the piece of this product that captures the Franz-Keldysh oscillations
involves both the reflection of electron from electrostatic barrier at
$x=x_{t}$ and reflection of hole from electrostatic barrier at $x=-x_{t}$.
This process is encoded into the product

$\displaystyle\cos\left(\Phi_{e}(x_{1})+\Phi_{e}(x_{2})-\frac{\pi}{2}\right)\cos\left(\Psi_{h}(x_{1})+\Psi_{h}(x_{2})+\frac{\pi}{2}\right)$
$\displaystyle=\frac{1}{2}\Bigg{[}-\cos\Big{(}\Phi_{e}(x_{1})-\Psi_{h}(x_{1})+\Phi_{e}(x_{2})-\Psi_{h}(x_{2})\Big{)}$
$\displaystyle+\cos\Big{(}\Phi_{e}(x_{1})+\Psi_{h}(x_{1})+\Phi_{e}(x_{2})+\Psi_{h}(x_{2})\Big{)}\Bigg{]}.$
(19)

It is the second cosine that captures two reflections. The damping factor
corresponding to this cosine has the form $\exp\left(-{\cal L}\right)$, where
${\cal L}$ is given by

${\cal
L}=\int\limits_{-\hbar\Omega_{2}/F}^{x_{1}}\frac{dx}{l_{h}(x)}+\int\limits_{x_{1}}^{\hbar\Omega_{1}/F}\frac{dx}{l_{e}(x)}+\int\limits_{-\hbar\Omega_{2}/F}^{x_{2}}\frac{dx}{l_{h}(x)}+\int\limits_{x_{2}}^{\hbar\Omega_{1}/F}\frac{dx}{l_{e}(x)}.$
(20)

The remaining task is to substitute Eqs. (IV) and (20) into Eq. (III) and
perform the integration over $\Omega_{1}$, $\Omega_{2}$, $x_{1}$, $x_{2}$.
Integration is determined by rapidly changing cosine Eq. (IV). Concerning the
damping factor, it is sufficient to substitute
$\Omega_{1}=\Omega_{2}=\frac{E_{g}-\hbar\omega}{2\hbar}$ and set
$x_{1}=x_{2}=0$. Integrals in Eq. (20) diverge logarithmically. A natural
cutoff of the logarithms is $x_{F}$, see Eq. (15). Then Eq. (20) reduces to
our main result Eq. (3).

##  V Concluding remarks

In traditional semiconductor materials the shape of the Franz-Keldysh
oscillations is quite robust.Sipe  Previous theoretical studiesexciton1 ;
Aspnes ; Dow ; Merkulov ; Dow1 ; Aronov  were focused on the effect of
electron-hole attraction on electroabsorption. Numerical calculationsSipe
indicate that the excitonic effect does not alter the shape of the
oscillations. Rather, a smooth background on which Franz-Keldysh oscillations
develop becomes more pronounced due to the electron-hole attraction.Sipe
Effect of disorder on the oscillations is illustrated in Fig. 2, where the
oscillations are plotted from Eqs. (1) and (3) for a given disorder and three
values of electric field. It is seen that, while disorder damps the
oscillations, this damping becomes less effective upon increasing, $F$.
Physically, upon increasing $F$, the travel distance of electron and hole from
the respective turning points to the point of their meeting becomes shorter.
To estimate quantitatively the critical field, $F_{c}$, above which the effect
of disorder becomes negligible, we express $F_{c}$ in terms of the width,
$E_{c}$, of the tail in the density of states, which disorder creates in the
absence of external field. The width, $E_{c}$, can be estimated by equating
the uncertainty, $\frac{\hbar}{\tau(E_{c})}$, given by Eq. (5) to $E_{c}$.
This yields
$E_{c}=\left(\frac{\gamma}{2}\right)^{2/3}\left(\frac{2m}{\hbar^{2}}\right)^{1/3}$.
Now the parameter $\gamma$ can be expressed via the observable $E_{c}$. Then,
equating to one the prefactor, $\frac{m\gamma}{\hbar^{2}F}$, in front of
logarithm in Eq. (3) yields the value

$F_{c}=\frac{\left(mE_{c}^{3}\right)^{1/2}}{\hbar}.$ (21)

For example, setting $m$ equal to the free electron mass, for disordered
system with a tail $E_{c}=0.1~{}\text{eV}$, we conclude that Franz-Keldysh
oscillations are smeared when the applied field is smaller than
$1.5~{}\text{kV}\cdot\text{cm}^{-1}$. Note that, the tail, $E_{c}$, can be
related to the mean free path
$l_{c}=\left(\frac{\hbar^{2}}{mE_{c}}\right)^{1/2}\approx 10~{}\mathring{A}$.
The above values are consistent with those reported for polymers.polymer ;
polymer1  Overall, quantitative comparison of theoretical results with
numerous experimental data is complicated by the fact that experimental papers
do not list the values of electric field, but rather the voltage drop on the
sample. Qualitatively though, experimental data on the Franz-Keldysh
oscillations are in agreement with our prediction: Decrease of the oscillation
period with their number is weak, while the visibility of high-order
oscillations increases with increasing, $F$. To account for this effect, the
authors of Refs. polymer, , polymer1,  adopted a phenomenological approach: in
calculation of absorption spectrum the electron energy, $E$, is replaced by
$E+i\Gamma$, where parameter $\Gamma$ emulates the disorder. It is used as a
fitting parameter. In this regard, within our theory, the role of “coherence
length” in Refs. polymer, , polymer1,  is played by the mean free path due to
impurity scattering and depends strongly on, $F$. We have restricted
consideration to the 1D case. The result in 2D does not differ qualitatively.
This is because the electron travel path towards the turning point lies close
to the travel path of the reflected electron. This is illustrated in Fig. 3.
Quantitatively, the difference between the 2D and 1D cases is that the density
of states in 2D is constant, so that the mean free path grows with energy
slower, as $E^{1/2}$ rather than as $E$.

Figure 3: (Color online) Illustration of the Franz-Keldysh oscillations in 2D.
Semiclassical trajectory of electron from the point $x_{1},y_{1}$ towards the
electrostatic barrier is close to the trajectory from the barrier to the point
$x_{2},y_{2}$. Electron, moving along the semiclassical trajectory is
scattered by the disorder, which “scrambles” the interference pattern
responsible for the oscillations.

##  Appendix A Average Green’s function near a wall

In the presence of a wall at $y=0$, the Green’s function of a free electron in
the half-space $y>0$ has a form

$G^{(0)}\left(E,y,y^{\prime}\right)=\frac{im}{\hbar^{2}k_{E}}\Big{[}e^{-ik_{E}|y-y^{\prime}|}-e^{-ik_{E}(y+y^{\prime})}\Big{]},$
(22)

where $k_{E}=\left(\frac{2mE}{\hbar^{2}}\right)^{1/2}$ is a wave vector. The
function $G^{(0)}$ satisfies the boundary conditions
$G^{(0)}\left(E,0,y^{\prime}\right)=G^{(0)}\left(E,y,0\right)=0$.

Figure 4: (Color online) Illustration of the 2D Green’s function in the
presence of a wall at $y=0$. There are two classical paths between the points
$(x_{1},y_{1})$ and $(x_{2},y_{2})$. One path is along the straight line,
while the other path corresponds to the shortest classical trajectory
reflected from the wall.

With a random disorder potential, $V(y)$, the Green’s function satisfies the
Dyson equation

$G(y_{1},y_{2})=G^{(0)}(y_{1},y_{2})+\int\limits_{0}^{\infty}dyG^{(0)}(y_{1},y)V(y)G(y,y_{2}).$
(23)

This equation is exact. It can be cast into a different form upon substituting
the right-hand side into the integrand

$\displaystyle
G(y_{1},y_{2})=G^{(0)}(y_{1},y_{2})+\int\limits_{0}^{\infty}dyG^{(0)}(y_{1},y)V(y)G^{(0)}(y,y_{2})$
$\displaystyle+\int\limits_{0}^{\infty}dy\int\limits_{0}^{\infty}dy^{\prime}G^{(0)}(y_{1},y)V(y)G^{(0)}(y,y^{\prime})V(y^{\prime})G(y^{\prime},y_{2}).$
(24)

A crucial step which allows to get a closed equation for the average Green’s
function is decoupling of averaging in the integrand of Eq. (A). Assuming that
the potential $V(y)$ is short-ranged

$\langle V(y)V(y^{\prime})\rangle=\gamma\delta(y-y^{\prime}),$ (25)

and performing the averaging, we arrive to the closed RPA equation

$\displaystyle\langle G(y_{1},y_{2})\rangle=G^{(0)}(y_{1},y_{2})$
$\displaystyle+\gamma\int\limits_{0}^{\infty}dyG^{(0)}(y_{1},y)G^{(0)}(y,y)\langle
G(y,y_{2})\rangle.$ (26)

In the absence of the wall, the bare Green’s function is equal to
$G^{(0)}=\frac{im}{\hbar^{2}k_{E}}e^{-ik_{E}|y-y^{\prime}|}$ and the lower
limit in the integral Eq. (A) should be set to $-\infty$. Then the average
Green’s function depends only on the difference $(y_{1}-y_{2})$. This allows
to solve Eq. (A) with the help of the Fourier transform. As a result, the
disorder leads to a factor $\exp\big{(}-\frac{|y_{1}-y_{2}|}{l}\big{)}$ in the
average Green’s function. Here $l$ is the mean free path which can be
expressed via the disorder strength as follows

$l=\frac{\hbar^{4}k_{E}^{2}}{m^{2}\gamma}.$ (27)

The above result applies when the disorder is weak, namely, for $k_{E}l\gg 1$.

In the presence of the wall, the translational symmetry is violated. Taking
the Fourier transform is not permissible. Yet, the form of the disorder-
modified average Green’s function can be established from the following
reasoning. The travel distance between the points $y_{1}$ and $y_{2}$ is
either $|y_{1}-y_{2}|$, when the particle travels directly, or
$(y_{1}+y_{2})$, when the travel involves the reflection from the wall. This
suggests the following form

$\displaystyle\langle G\left(E,y,y^{\prime}\right)\rangle$
$\displaystyle=\frac{im}{\hbar^{2}k_{E}}\Big{[}e^{-ik_{E}|y-y^{\prime}|}e^{-\frac{|y-y^{\prime}|}{l}}-e^{-ik_{E}(y+y^{\prime})}e^{-\frac{(y+y^{\prime})}{l}}\Big{]}.$
(28)

of the average Green’s function.

A nontrivial question is a minimal distance at which Eq. (A) applies. We will
show that the condition of applicability is $y_{1},y_{2}\gg\frac{1}{k_{E}}$.

To solve the equation Eq. (A) approximately, we make a substitution

$\displaystyle\langle G(y,y^{\prime})\rangle$
$\displaystyle=\frac{im}{\hbar^{2}k_{E}}\Big{[}R_{1}(y,y^{\prime})e^{-ik_{E}|y-y^{\prime}|}-R_{2}(y,y^{\prime})e^{-ik_{E}(y+y^{\prime})}\Big{]},$
(29)

and assume that $R_{1}$ and $R_{2}$ are slow functions on the scale
$k_{E}^{-1}$. Substituting the form Eq. (A) into Eq. (A), we cast it into the
form

$\displaystyle\Big{(}R_{1}(y,y^{\prime})-1\Big{)}e^{-ik_{E}|y-y^{\prime}|}-\Big{(}R_{2}(y,y^{\prime})-1\Big{)}e^{-ik_{E}(y+y^{\prime})}$
$\displaystyle=\gamma\left(\frac{im}{\hbar^{2}k_{E}}\right)^{2}\int\limits_{0}^{\infty}dy_{1}\Big{(}e^{-ik_{E}|y-y_{1}|}-e^{-ik_{E}(y+y_{1})}\Big{)}\left(1-e^{-2ik_{E}y_{1}}\right)\Big{(}R_{1}(y_{1},y^{\prime})e^{-ik_{E}|y_{1}-y^{\prime}|}-R_{2}(y_{1},y^{\prime})e^{-ik_{E}(y_{1}+y^{\prime})}\Big{)}.$
(30)

Without the loss of generality we assume that $y^{\prime}>y$. The key step is
dividing the integral in the right-hand-side of Eq. (A) into three domains:
(i) $0<y_{1}<y$, (ii) $y<y_{1}<y^{\prime}$, and (iii) $y_{1}>y^{\prime}$.
Consider first the domain (i). In this domain, we have $|y-y_{1}|=y-y_{1}$,
$|y_{1}-y^{\prime}|=y^{\prime}-y_{1}$. Out of six terms in the integrand only
two terms,
$-e^{-ik_{E}|y-y_{1}|}R_{2}(y_{1},y^{\prime})e^{-ik_{E}(y_{1}+y^{\prime})}=-e^{-ik_{E}(y+y^{\prime})}R_{2}(y_{1},y^{\prime})$,
and
$-e^{-ik_{E}y}R_{1}(y_{1},y^{\prime})e^{-ik_{E}y^{\prime}}=-e^{-ik_{E}(y+y^{\prime})}R_{1}(y_{1},y^{\prime})$
do not oscillate with $y_{1}$, since the functions $R_{1}$ and $R_{2}$ are
slow. All other terms, e.g.
$e^{-ik_{E}|y-y_{1}|}R_{1}(y_{1},y^{\prime})e^{-ik_{E}|y_{1}-y^{\prime}|}=e^{-ik_{E}(y+y^{\prime})}R_{1}(y_{1},y^{\prime})e^{2ik_{E}y_{1}}$,
contain rapidly oscillating exponential factors. These factors suppress the
integral over the domain (i) down to $k_{E}^{-1}$. Similar strategy is applied
to the domains (ii) and (iii). It can be checked that in the domain (iii)
there are no slow terms in the integrand. In the domain (ii) there are two
slow terms in the integrand, namely
$R_{1}(y_{1},y^{\prime})e^{-ik(y^{\prime}-y)}$ and
$R_{1}(y_{1},y^{\prime})e^{ik(y^{\prime}-y)}$. Collecting the terms with slow
integrand and equating coefficients in front of $e^{-ik_{E}|y-y^{\prime}|}$
and $e^{-ik_{E}(y+y^{\prime})}$ yields the system of equations

$R_{1}(y,y^{\prime})=1-\gamma\left(\frac{im}{\hbar^{2}k_{E}}\right)^{2}\int\limits_{y}^{y^{\prime}}dy_{1}R_{1}(y_{1},y^{\prime}).$
(31)

$\displaystyle
R_{2}(y,y^{\prime})=1+\gamma\left(\frac{im}{\hbar^{2}k_{E}}\right)^{2}\int\limits_{0}^{y^{\prime}}dy_{1}R_{1}(y_{1},y^{\prime})$
$\displaystyle+\gamma\left(\frac{im}{\hbar^{2}k_{E}}\right)^{2}\int\limits_{0}^{y}dy_{1}R_{2}(y_{1},y^{\prime}).$
(32)

Exponential decay of $R_{1}$ follows from the fact that the derivative
$\frac{dR_{1}}{dy}$ is proportional to n$R_{1}$. It is easy to check that
$R_{1}=\exp(-\frac{y^{\prime}-y}{l})$ with $l$ defined by Eq. (27) is the
solution of Eq. (31). Equivalently, the solution of Eq. (A) is
$R_{2}=\exp(-\frac{y^{\prime}+y}{l})$. Note that in deriving the system Eqs.
(31), (A) we assumed that $k_{E}y\gg 1$, $k_{E}y^{\prime}\gg 1$, when the
terms with rapidly oscillating integrands can be neglected.

It is straightforward to extend Eq. (A) to the 2D case. The Green function is
dominated by two classical paths. The first path with a length
$\rho_{1}=\left[(x_{1}-x_{2})^{2}+(y_{1}-y_{2})^{2}\right]^{1/2}$ is along the
straight line, while the second path involves the reflection from the wall. As
illustrated in Fig. 4, the minimal length of the path of this type is
$\rho_{2}=\left[(x_{1}-x_{2})^{2}+(y_{1}+y_{2})^{2}\right]^{1/2}$. Within the
numerical factor, the 2D Green’s function has a form

$\displaystyle\langle G(E,x_{1},y_{1},x_{2},y_{2})\rangle$
$\displaystyle=\frac{m}{\left(2\pi\right)^{1/2}\hbar^{5/2}}\Bigg{[}\frac{\exp\left({-ik_{E}\rho_{1}-\frac{\rho_{1}}{l}}\right)}{(k_{E}\rho_{1})^{1/2}}-\frac{\exp\left(-{ik_{E}\rho_{2}-\frac{\rho_{2}}{l}}\right)}{(k_{E}\rho_{2})^{1/2}}\Bigg{]}.$
(33)

##  Appendix B Acknowledgements

The work was supported by the Department of Energy, Office of Basic Energy
Sciences, Grant No. DE- FG02- 06ER46313. We are grateful to Kameron Hansen, a
graduate student’ from Chemistry Department, for picking our interest in
electroabsorption in perovskites.

## References

  * (1) W. Franz, “Einfluss eines elektrischen Feldes auf eine optische Absorptionskante,” Zeitschrift für Naturforschung 13a, 484 (1958).
  * (2) L. V. Keldysh, “Behaviour of Non-Metallic Crystals in Strong Electric Fields,” Sov. Phys. JETP 6, 763 (1958).
  * (3) T. E. Van Eck, L. M. Walpita, W. S. C. Chang, and H. H. Wieder, “Franz-Keldysh electrorefraction and electroabsorption in bulk InP and GaAs,” Appl. Phys. Lett. 48, 451 (1986).
  * (4) D. A. Miller, D. S. Chemla, and S. Schmitt-Rink, “Relation between Electroabsorption in Bulk Semiconductors and in Quantum Wells: The Quantum-Confined Franz-Keldysh Effect,” Phys. Rev. B 33, 6976 (1986).
  * (5) P. J. Hughes, B. L. Weiss, and J. Hosea, “Analysis of Franz$-$Keldysh oscillations in photoreflectance spectra of a AlGaAs-GaAs single-quantum well structure,” Journ. of Appl. Phys. 77, 6472 (1995).
  * (6) T. Y. Zhang and W. Zhao, “Franz-Keldysh effect and dynamical Franz-Keldysh effect of cylindrical quantum wires,” Phys. Rev. B 73, 245337 (2006).
  * (7) D. Li, J. Zhang, Q. Zhang, and Q. Xiong, “Electric$-$Field$-$Dependent Photoconductivity in CdS Nanowires and Nanobelts: Exciton Ionization, Franz-Keldysh, and Stark Effects,” Nano Lett. 12, 2993 (2012).
  * (8) A. Horvath, G. Weiser, C. Lapersonne-Meyer, M. Schott, and S. Spagnoli, “Wannier excitons and Franz-Keldysh effect of polydiacetylene chains diluted in their single crystal monomer matrix,” Phys. Rev. B 53, 13507 (1996).
  * (9) G. Weiser, L. Legrand, T. Barisien, A. Choueiry, M. Schott, and S. Dutremez, “Stark effect and Franz-Keldysh effect of a quantum wire realized by conjugated polymer chains of a diacetylene 3NPh2,” Phys. Rev. B 81, 125209 (2010).
  * (10) V. Perebeinos and P. Avouris, “Exciton Ionization, Franz-Keldysh and Stark Effects in Carbon Nanotubes,” Nano Lett. 7, 609 (2007).
  * (11) M.-H. Ham, B.-S. Kong, W.-J. Kim, H.-T. Jung, and M. S. Strano, “Unusually Large Franz-Keldysh Oscillations at Ultraviolet Wavelengths in Single-Walled Carbon Nanotubes,” Phys. Rev. Lett. 102, 047402 (2009).
  * (12) N. V. Tepliakov, M. Yu. Leonov, A. V. Baranov, A. V. Fedorov, and I. D. Rukhlenko, “Quantum theory of electroabsorption in semiconductor nanocrystals,” Opt. Express 24, A52 (2016).
  * (13) E. Amerling, S. Baniya, E. Lafalce, C. Zhang, Z. V. Vardeny, and L. Whittaker-Brooks, “Electroabsorption Spectroscopy Studies of (C4H9NH3)2PbI4 Organic$-$Inorganic Hybrid Perovskite Multiple Quantum Wells,” J. Phys. Chem. Lett. 8, 4557 (2017).
  * (14) S. Xia, Z. Wang, Y. Ren, Z. Gu, and Y. Wang, “Unusual electric field-induced optical behaviors in cesium lead bromide perovskites,” Appl. Phys. Lett. 115, 201101 (2019).
  * (15) G. Walters, M. Wei, O. Voznyy, R. Quintero-Bermudez, A. Kiani, D.-M. Smilgies, R. Munir, A. Amassian, S. Hoogland, and E. Sargent, “The quantum-confined Stark effect in layered hybrid perovskites mediated by orientational polarizability of confined dipole,” Nat. Commun. 9, 4214 (2018).
  * (16) C. Motta, F. El-Mellouhi, and S. Sanvito, “Charge carrier mobility in hybrid halide perovskites,” Sci. Rep. 5, 12746 (2015).
  * (17) F. Duque-Gomez and J. E. Sipe, “The Franz-Keldysh effect revisited: Electroabsorption interband coupling and excitonic,” Journ. of Phys. Chem. Sol. 76, 138 (2015).
  * (18) H. I. Ralph, “On the theory of Franz-Keldysh effect,” J. Phys. C: Solid State Phys. 1, 378 (1968).
  * (19) D. E. Aspnes, “Electric-Field Effects on Optical Absorption near Thresholds in Solids,” Phys. Rev. 147, 554, (1966).
  * (20) J. D. Dow and D. Redfield, “Electroabsorption in Semiconductors: The Excitonic Absorption Edge,” Phys. Rev. B 1, 3358 (1970).
  * (21) I. A. Merkulov, “Influence of exciton effect on electroabsorption in semiconductors,” Zh. Eksp. Teor. Fiz. 66, 2314 (1974). [Sov. Phys. JETP 39, 1140 (1974)].
  * (22) F. L. Lederrnan and J. D. Dow, “Theory of electroabsorption by anisotropic and layered semiconductors. I. Two-dimensional excitons in a uniform electric field,” Phys. Rev. B 13, 1633 (1976).
  * (23) A. G. Aronov and A. S. Ioselevich, “Effect of electric field on exciton absorption,” Zh. Eksp. Teor. Fiz. 74, 1043 (1978). [Sov. Phys. JETP 47, 548 (1978)].