# Dependence of magnetic domain patterns on plasma-induced differential
oxidation of CoPd thin films

Wei-Hsiang Wang  Chak-Ming Liu  Tzu-Hung Chuang  Der-Hsin Wei  Wen-Chin Lin
<EMAIL_ADDRESS>Pei-hsun Jiang<EMAIL_ADDRESS>Department of Physics,
National Taiwan Normal University, Taipei 116, Taiwan  National Synchrotron
Radiation Research Center, Hsinchu 300, Taiwan

###### Abstract

We demonstrate the evolution of the micro-patterned magnetic domains in CoPd
thin films pretreated with e-beam lithography and O2 plasma. During the days-
long oxidation, significantly different behaviors of the patterned magnetic
domains under magnetization reversal are observed via magneto-optic Kerr
effect microscopy on different days. The evolution of the magnetic behaviors
indicate critical changes in the local magnetic anisotropy energies due to the
Co oxides that evolve into different oxide forms, which are characterized by
micro-area X-ray absorption spectroscopy and X-ray photoelectron spectroscopy.
The coercive field of the area pre-exposed to plasma can decrease to a value
10 Oe smaller than that unexposed to plasma, whereas after a longer duration
of oxidation the coercive field can instead become larger in the area pre-
exposed to plasma than that unexposed, leading to an opposite magnetic
pattern. Various forms of oxidation can therefore provide an additional
dimension for magnetic-domain engineering to the current conventional
lithographies.

##  1 Introduction

Metal-oxide heterostructures have been extensively studied in various aspects,
such as interfacial structures and magnetic exchange coupling between
ferromagnetic metals and antiferromagnetic oxides [1, 2], and nanostorage or
catalysis applications of nanosized metals on oxide surfaces [3, 4, 5].
Evolution of metal-oxide growth under different oxidation conditions has also
become an important topic [6, 7, 8, 9, 10], including a special focus on self-
assembly integrated with conventional photo or e-beam lithography that has
been shown to be a powerful tool to achieve complex multiple patterning [11].
Since patterned oxidation can be utilized for sophisticated engineering of
magnetic structures in thin films [12], lithography incorporated with
different types of oxide formation is expected to boost the beneficial
application of magnetic patterning.

The mechanism of oxidation-related magnetic engineering involves several
effects of oxidation on magnetic properties. Oxygen usually exhibits different
affinities with elements of an intermetallic compound, which leads to
selective oxidation. This results in self-assembling of the Co-oxide
nanoclusters and development of a Pd-rich magnetic layer as observed in our
previous study on CoPd alloy thin films [13]. With the change in the
concentrations of the elements, the magnetic easy direction of the sample
gradually turns away from the pristine perpendicular direction to an in-plane
direction. Control of the extent of oxidation by varying the thermal annealing
time under a partial oxygen pressure, on the other hand, has been found to
increase the roughness at the interface of a Co/CoO bilayer structure, leading
to domain-wall pinning and an increase of coercivity [14]. An enhancement of
coercivity is also found possible by implanting oxygen ions into Co films to
promote formation of Co oxides at the Co grain boundaries, which reduces
exchange interactions between Co crystallites, leading to a higher coercivity
[15]. In contrast to metal/metal oxide bilayers formed by surface oxidation,
implantation can result in a different morphology of the interface between the
metal and the oxide, and hence may contribute a radically different
magnetization mechanism to the system studied [15, 16]. More research is
demanded to explore the underlying mechanisms and the full potential of
oxidation methodology for magnetic patterning.

In magnetic engineering, ion irradiation has long been considered a reliable
process to create magnetic domain patterns with the penetration of keV to MeV
ions into the films of magnetic materials [17, 18, 19]. Treatments with ions
of sufficient energy are believed to be a requirement for successful
modification of the magnetic coherence and coupling in a magnetic film in
order to generate artificial magnetic microstructures inside the film.
However, oxidation-induced magnetic patterning may provide a more handy method
to accomplish magnetic engineering with less invasive treatment.

In this study, we demonstrate oxidation-induced magnetic patterning achieved
by surface treatment of CoPd alloy films with low-energy ions generated by a
regular plasma cleaning equipment. Evolution of Co oxides on CoPd alloy films
pretreated with patterned O2 plasma bombardment is investigated by exploring
the behaviors of magnetic domains of the film via magneto-optic Kerr effect
(MOKE) microscopy. As oxidation evolves, intriguing different or even opposite
behaviors of the patterned magnetic domains are observed upon magnetization
reversal, which indicates the existence of various forms of Co oxides during
the oxidation, pointing to the possibility of creating complex magnetic
structures using conventional lithography that provides directed formation of
oxides. Our experiment is the first of its kind to investigate the dependence
of the magnetic domains on oxidation forms for exploring the flexibility and
diversity of patterning of magnetic properties.

##  2 Material and methods

The 8-nm-thick CoPd (50:50) alloy film was deposited on SiO2/Si(001)
substrates through e-beam-heated coevaporation in an ultrahigh vacuum with a
base pressure of $3\times 10^{-9}$ mbar. Two evaporation guns were both
aligned at 45∘ to the normal. This oblique deposition geometry allows uniaxial
magnetic anisotropy to be developed on the surface plane [20]. The evaporation
rates were controlled through the adjustment of the emission currents applied
to the sources. Alloy composition was determined by the deposition rates of Co
and Pd and confirmed through transmission electron microscopy with energy-
dispersive spectroscopy (TEM-EDS) [21]. A layer of polymethyl methacrylate
(PMMA) was then spin-coated on the CoPd film. The PMMA layer was later
patterned through e-beam lithography to create a 10-by-10 array of $50\times
50$ $\upmu$m2 square windows spaced 25 ${\upmu}$m apart. The sample was then
treated with 10.5-watt O2 plasma (PDC-32G, Harrick Plasma) for 90 seconds with
a fairly low flow rate of 0.1 cc/min of oxygen at a typical base pressure of
200 mTorr of the plasma chamber, followed by removal of PMMA. (According to
the Langmuir probe measurement on a Harrick plasma cleaner of a similar model
[22] and the plasma sheath model [23], the ion energy of the plasma is
estimated to be $\sim$50 eV.) After that, the samples were stored in a vacuum
desiccator (internal pressure $\sim 0.15$ bar) to ensure mild and slow
oxidation, which allows us to carefully monitor the different states of
magnetism of the sample. The magnetic properties were inspected after
fabrication using MOKE microscopy as described in the following contents.

The samples are investigated using longitudinal MOKE microscopy with the
external magnetic field ($H$) applied along the magnetic easy axes. All the
measurements are carried out at room temperature. The features to be
demonstrated in this paper are reproducible between the samples. A lower sweep
rate of $H$ with more sampling points per unit field is chosen during the
magnetization reversal, where the magnetic domains exhibit faster and more
prominent changes. For the MOKE measurements presented in Section 3, $H$ is
swept at a rate of $\Delta H=\pm 10$ Oe or $\pm 5$ Oe per 0.5 second for each
data point, except for the magnetization reversal processes (respective $|H|$
ranges are 80–120 Oe for Day 0 (Fig. 1), 50–100 Oe for Day 10 (Fig. 2), and
0–40 Oe for Day 42 (Fig. 3)), during which the sweep rate is decreased to
$\Delta H=\pm 1$ Oe or $\pm 0.5$ Oe per 0.5 second to gently process the
variation of the magnetic domains. The extensively oxidized case on Day 217
(Fig. 6) is an exception, where $H$ is swept at a rate of $\Delta H=\pm 5$ per
0.5 second for the whole measurement. All the MOKE images present the same
region of the sample, which corresponds to the four squares at the top left
corner of the whole array pattern. All the hysteresis loops are measured from
the area inside the single square at the top left corner and from the outside
area next to it on the right, respectively (see the Supplementary Materials).
Note that possible PMMA remnants would not contribute to any field-dependent
MOKE signals, as already confirmed in our previous studies [12].

##  3 Experimental results and discussion

###  3.1 Pinned edge domains (Day 0)

Figure 1: Longitudinal MOKE measurements on Day 0 of the CoPd film pretreated
with e-beam lithography and O2 plasma, with the magnetic field $H$ applied
along the easy axis. (a) The hysteresis loops of the areas inside and outside
a plasma-pretreated $50\times 50$ $\upmu$m2 square, respectively. (b)–(i)
Differential MOKE images of four squares showing the evolution of the magnetic
domains as $H$ is varied. Square edges are marked by white dashed lines in
(f). (j) Illustration of deflected ions during plasma treatment due to the
negatively charged PMMA (not to scale).

Presented in Fig. 1 is the representative sample measured right after the
e-beam lithography and O2 plasma treatment. Fig. 1(a) shows the hysteresis
loops of the areas inside (red curve) and outside (black curve) a square,
respectively, and Figs. 1(b)–(i) show the MOKE images of four squares among
the entire square array on the film simultaneously recorded with Fig. 1(a) as
$H$ is changed gradually from $-90.4$ Oe to $-110.1$ Oe. In Fig. 1(a), no
appreciable difference is found between the curves inside and outside a
square, which is consistent with the observation that the square array pattern
cannot be clearly recognized in Figs. 1(b)–(e), while the total size of the
dark domains with magnetization pointing to the negative direction gradually
increases as $H$ is changed from $-90.4$ to $-100$ Oe. However, for $H=-100$
to $-107.1$ Oe, as shown in Figs. 1(e)–(h), hints of persisting light-gray
domains are observed along the edges of the squares, which are indicated with
yellow arrows. These line segments of pinned, light domains persist until a
more negative $H$ is applied, which eventually results in a completely dark
image shown in Fig. 1(i) at $H=-110.1$ Oe. The magnetic behavior inside a
square does not differ much from that outside, which indicates that the
magnetic properties of the sample do not change significantly right after the
plasma treatment, except that the properties near the edges of the squares
seem to be altered, leading to the pinned magnetic domains near the edges. The
light-gray domain segments are more prominent along the edges parallel to the
easy axis of the film than that along the hard axis.

This phenomenon is reminiscent of the edge domain traps observed in an
experiment of Bauer et al. [24] with voltage-controlled oxidation states in
their device, in which oxidation of the Co layer is most efficient at the edge
of the GdOx gate dielectric, where the triple phases (i.e., O2 gas, O2- ion-
conducting, and electron-conducting phases) meet [25, 26]. In our experiment,
the pinned edge domains are probably also caused by the reactions being most
efficient at the edges, possibly due to trajectories of positive ions being
deflected towards the negatively charged PMMA sidewalls [27, 28], leading to
enhanced bombardment of O2 plasma at the square edges, as illustrated in Fig.
1(j). Denser defects are thereby formed along the edges. It has been shown in
previous studies that small-scale defects in magnetic materials can act as
pinning centers for domain wall movement [29, 30, 31, 15]. This explains the
domain wall pinning observed along the square edges.

A vague contrast between the areas inside and outside the squares can be
observed in Fig. 1(i) when the sample is magnetically saturated. (This slight
contrast can be more clearly seen later in Fig. 2(i) on Day 10.) The remnant
contrast under saturation is often detected in our studies on CoPd or FePd
thin films with their magnetic domain patterns created using lithography and
plasma treatments. Further investigation of these samples using zero-field
polar MOKE microscopy shows clear contrasts of the squares [12], which reveal
the difference in the perpendicular magnetization between and areas inside and
outside the squares. This may explain the existence of the magnetic contrast
between the two regions even under saturation.

###  3.2 Preceding magnetization reversal in plasma-pretreated areas (Day 10)

Figure 2: Longitudinal MOKE measurements of the CoPd film on Day 10. (a) The
hysteresis loops of the areas inside and outside a square, respectively.
(b)–(i) Differential MOKE images of four squares showing the evolution of the
magnetic domains as $H$ is varied.

The sample is re-measured after 10 days of storage in a vacuum desiccator
after the O2 plasma treatment. The hysteresis loops for the areas inside (red)
and outside (black) the top left square are shown in Fig. 2(a). The coercivity
($H_{\mathrm{c}}$) decreases from 105 Oe as shown in Fig. 1(a) to 85 Oe for
outside and to 75 Oe for inside a square, respectively, after mild oxidation
for 10 days. (Statistics of the $H_{\mathrm{c}}$ values over the whole array
pattern and video recordings of the MOKE images are provided in the
Supplementary Materials.) Local magnetic properties are also modified by
oxidation, as revealed in the magnetic patterns observed in the MOKE images in
Figs. 2(b)–(i). During the reversal of the magnetization of the film from the
positive (light gray) to the negative (dark gray) direction as $H$ is changed
from $-70.7$ Oe to $-96$ Oe, the magnetization inside the plasma-pretreated
squares is reversed before that outside the squares, exhibiting dark square
domains under a mediate $H$ around $-85.9$ Oe. The observed magnetic pattern
can be explained by a higher rate of surface oxidation induced by plasma
pretreatment. Ion bombardment is known to stimulate production of point
defects, such as vacancies and interstitials [32], which enhance the mobility
and reactivity of oxygen anions and metal cations in materials. Areas inside
the squares thereby have a higher surface oxidation rate, which leads to a
lower magnetic anisotropy energy (MAE) and hence a smaller $H_{\mathrm{c}}$
[33, 34, 13, 35, 36] inside the squares than that outside, as shown in Fig.
2(a). Therefore, magnetization reversal is easier inside the squares than
outside.

###  3.3 Delayed magnetization reversal in plasma-pretreated areas (Day 42)

Figure 3: Longitudinal MOKE measurements of the CoPd film on Day 42. (a) The
hysteresis loops of the areas inside and outside a square, respectively.
(b)–(i) Differential MOKE images of four squares showing the evolution of the
magnetic domains as $H$ is varied. (j) Atomic force microscopy image of a
corner of a square, with the square edges marked by dashed white lines. A
ridge of the Co oxide clusters can be clearly seen along the edges inside the
square.

The sample is measured again after 42 days of storage in a vacuum desiccator
after the O2 plasma treatment. An intriguing opposite effect of oxidation on
$H_{\mathrm{c}}$ is observed. 42-day oxidation reduces $H_{\mathrm{c}}$
outside the squares more than it does inside the squares, as presented in Fig.
3(a) with $H_{\mathrm{c}}$ inside the top left square (29 Oe) larger than
$H_{\mathrm{c}}$ outside (25 Oe), which is contrary to the result of 10-day
oxidation shown in Fig. 2(a). (Comparison of the dynamic behaviors of the
magnetic domains displayed in Figs. 2 and 3 is more clear in video recordings
provided in Videos S1 and S2 in the Supplementary Materials.) To investigate
the cause of this opposite effect, the MOKE images (Figs. 3(b)–(i)) are
examined closely as follows. The red arrows in Figs. 3(b) and 3(d) indicate
some of the segments of local oxide clusters grown near the edges of the
squares, which can be seen more clearly in an atomic force microscopy image
shown in Fig. 3(j). Also observed in Fig. 3(j) are small nanodots of oxides
away from the edges that have not self-assembled to form larger clusters [13].
When $H$ is changed to more negative values, dark domains are first
established near the edge oxide clusters (see Fig. 3(d) at $-$$23.2$ Oe),
demonstrating that these local oxide clusters behave like domain-wall traps.
As opposed to the Day-10 case where the MAE is uniformly reduced inside the
squares, 42-day oxidation specifically promotes the growth of oxide clusters
near the edges, indicating intensive local oxidation there, which further
reduces the MAE along the edges. Unlike a minor reduction of MAE along the
edges that only pins the magnetic domains on the edges as shown in Fig. 1 when
the sample is measured immediately after the plasma treatment, a local deep
well of MAE shaped in a closed loop can result in an enhancement in
$H_{\mathrm{c}}$ for the entire enclosed area due to the strong trapping
effect of the MAE well on the domain walls [37]. Therefore, the magnetic
domain walls along all the edges of the squares are pinned as the magnitude of
$H$ increases, which leads to a delayed reversal of the magnetization inside
the squares as shown in the MOKE images in Figs. 3(b)–(i). The tendency for
the growth of oxide clusters to start near the edges is likely due to the
enhanced plasma bombardment mentioned in Fig. 1(j). There are other oxide
clusters randomly located inside the squares. These spots also behave like
domain-wall traps, as indicated in Fig. 3(h) with blue arrows.

This experimental result is somewhat different from our previous study on a
naturally oxidized CoPd film without plasma pretreatment [13], where no
relation between the magnetic domain walls and the locations of the Co-oxide
clusters was observed in that film via MOKE microscopy measurements. In a
naturally oxidized CoPd film, a uniform Pd-rich layer was formed in the CoPd
film because of diffusion of the Co atoms to the surface to create a Co-oxide
layer with some self-assembled Co-oxide clusters distributed on it. TEM-EDS
elementary mapping confirmed that the concentrations of respective elements of
the naturally oxidized alloy film stayed quite uniform over the whole film
despite the presence of local Co-oxide clusters that were self-assembled on
the oxide surface. This resulted in uniform magnetic properties of the film
with no domain walls observed at the locations of the Co-oxide clusters. This
is different from our present results with plasma pretreatment, where domain
walls are established along the Co-oxide clusters inside the squares (Figs.
3(b)–(i)). This discrimination implies an essential difference in the
properties of the Co-oxide clusters between the CoPd films with plasma
pretreatment and those without it.

#### Surface characterization

Figure 4: $\upmu$-XAS curves of the film measured from inside and outside a
square, respectively. Curves are offset vertically for clarity. The Co L3
peaks at 779.5 and 780.5 eV indicate a mixture of Co and CoO, and a higher
intensity of the 780.5-eV peak inside a square than outside implies a more
extensive oxidation inside. The inset is an XMCD image of the film at 779 eV
from which the $\upmu$-XAS data are extracted.

In order to evaluate the Co oxides at this stage, micro-area X-ray absorption
spectroscopy ($\upmu$-XAS) and X-ray photoelectron spectroscopy (XPS) are used
to inspect the areas inside and outside the squares of a CoPd sample with a
state of oxidation and magnetic property equivalent to that shown in Fig. 3
(Day 42). Fig. 4 is the $\upmu$-XAS results extracted from X-ray magnetic
circular dichroism photoemission electron microscopy (XMCD-PEEM) at beamline
BL05B2 of the Taiwan Light Source in Hsinchu, Taiwan. The top black curve is
obtained from an area outside a square, and the bottom red curve is from
inside a square, with the source areas indicated in the inset XMCD image by
the boxed regions. Both curves show partial oxidation of the metallic Co to
CoO, as evidenced by a broad peak around 780 eV that comprises two Co L3 peaks
at 779.5 and 780.5 eV, which indicate a mixture of Co and CoO [38]. The major
difference between the two curves is that the relative intensity of the
780.5-eV peak inside the square is higher than outside. This indicates more
extensive oxidation of the areas exposed to plasma pretreatment than those
protected from plasma by PMMA masking. The result is consistent with our
optical and atomic force microscopic observations of the cluster structures
grown inside the squares after mild oxidation during the sample storage.
Oxidation thereby is considered a key mechanism that leads to different
magnetic properties inside and outside the squares.

Figure 5: XPS depth profile of the film (a) without and (b) with plasma
pretreatment, respectively. Curves from different depths are offset vertically
for clarity. The spectra from deeper (1–3 nm) locations show a higher relative
intensity of the Co0 peaks with respect to the Co2+ peaks in Fig. 5(a) than in
Fig. 5(b), with the peak ratios shown in (c). This indicates that the
oxidation penetration is deeper in the plasma-pretreated areas than in those
without plasma treatment.

Figs. 5(a) and 5(b), on the other hand, display the depth-dependent XPS
spectra from the areas without and with plasma pretreatment, respectively,
with Co0 (from the Co metal) and Co2+ (from CoO) peaks detected in the
measurements. Each figure has seven curves representing the XPS data at 0 to 3
nm in depth from the film surface, with a sputtering depth resolution of 0.5
nm. It can be seen from the deeper (1–3 nm) curves that the relative intensity
of the Co2+ peaks with respect to the Co0 peaks are higher in Fig. 5(a) than
in Fig. 5(b), with the peak ratios of Co2+ to Co0 shown in Fig. 5(c). This
indicates that the oxidation penetration is deeper in the areas with plasma
pretreatment than in those without it. Another important message from the XPS
data is the energy shifts of the peaks. There exist two kinds of major energy
shifts in the system. First, all the Co0 and Co2+ peaks shift to a higher
energy with plasma pretreatment with $\Delta E$ $\sim 0.5$ eV. Secondly, the
Co2+ peaks at 3 nm are $\sim$1.6 eV higher than those at the surface (0 nm).
The shift in binding energies with plasma pretreatment indicates changes in
the electronic structures of the Co metal and oxide induced by plasma
pretreatment [39], which may depict the structural difference in the oxide
forms between the CoPd alloy films without and with plasma pretreatment. The
higher binding energy of the internal Co valance state than that of the
surface state, on the other hand, may be attributed to the lower coverage of
the Co oxides inside the film than that on the surface [40]. As oxidation
proceeds, Co oxides tend to evolve from CoO into Co3O4 and Co2O3, and the
morphology evolves from scattered oxide clusters into a whole oxide layer. The
Co2+ peak can be mixed with a few Co3+ signals, which have a lower binding
energy than Co2+. The corresponding binding energy of the combined peak
thereby shifts to a lower value as the portion of Co3+ increases when moving
from deep inside the film to the surface.

To interpret the XPS data, an important consideration is whether internal
oxidation [41] or surface oxidation of Co occurs in the samples. Surface
oxidation is known to yield an oxide layer on the surface that provides
protection to some extent [42]. For CoPd films without plasma pretreatment, Co
atoms diffuse to the alloy surface and react with O2 in the air to develop a
complete Co-oxide surface layer, on which self assembly occurs later to build
Co-oxide clusters. For CoPd films with O2 plasma pretreatment, however,
diffusion of oxygen ions into the film during the plasma treatment promotes
internal oxidation from the start before a complete surface oxide can be
formed on the surface to provide protection. This explains the deeper
penetration of the oxidation and the overall change of the electronic
structures observed from the XPS. Our MOKE observations in Fig. 3 are closely
related to the nonuniform distribution of the oxide in the film due to
internal oxidation. It has been known that an antiferromagnet (AFM) can
enhance the $H_{\mathrm{c}}$ of an adjacent ferromagnet (FM) even above the
Néel temperature ($T_{\mathrm{N}}$) of the AFM [43]. This phenomenon is
attributed to an additional anisotropy in the FM induced by the spin
fluctuations in the AFM above its $T_{\mathrm{N}}$. The embedded CoO induced
by the implantation of oxygen ions in our experiment is known to be an AFM
with a $T_{\mathrm{N}}$ of 298 K, surrounded by the CoPd film, which is an FM
[43, 44, 15]. Therefore, the enhancement of $H_{\mathrm{c}}$ inside the
squares pretreated by O2 plasma can also be attributed to the formation of
internal Co oxides in the CoPd film inside the squares, which act as pinning
centers for the domain walls, giving a higher $H_{\mathrm{c}}$ to the
neighboring area around the oxide clusters [44].

###  3.4 Extensive oxidation (Day 217)

Figure 6: (a) The hysteresis loops of the areas inside and outside a square,
respectively, on the CoPd film on Day 217. The inset shows an example of the
differential MOKE image at $-26$ Oe. (b) Coercivity $H_{\mathrm{c}}$ as a
function of time for areas inside and outside a square, respectively.
$H_{\mathrm{c}}$ generally stays lower inside a square than outside, except
that on Day 42 as indicated by an arrow, $H_{\mathrm{c}}$ becomes higher
inside a square.

The sample is measured again after 217 days of storage in a vacuum desiccator
after the O2 plasma treatment. The hysteresis loops inside and outside the top
left square are shown in Fig. 6(a). It can be seen that the $H_{\mathrm{c}}$
of the area outside the squares decreases further to 5 Oe. For the area inside
a square, on the other hand, no magnetization is observed at all. This implies
that the interiors are almost completely oxidized due to internal oxidation or
intergranular penetration [42] that have been actively taking place during
this long time. Deeper penetrations of the internal oxidation in the alloy can
occur near the grain boundaries, which promotes the degradation of the sample.

The dependence of $H_{\mathrm{c}}$ on the duration of oxidation is summarized
in Fig. 6(b). As $H_{\mathrm{c}}$ decreases with time, one can see that
$H_{\mathrm{c}}$ inside the squares generally stays lower than that outside,
except for the Day-42 case, where $H_{\mathrm{c}}$ inside is higher,
exhibiting an opposite effect of evolving oxidation on $H_{\mathrm{c}}$, which
leads to a reversed outcome of the domain patterns on Day 42. The difference
between $H_{\mathrm{c}}$ inside the square (exposed to plasma) and that
outside (without plasma exposure) is largest on Day 10 with $\Delta
H_{\mathrm{c}}$ = 10 Oe, which has a magnitude comparable with the decrement
in $H_{\mathrm{c}}$ accomplished by 2-keV ion beam erosion [19]. It is quite
intriguing that modification of the surface MAE with the surface plasma
treatment can determine the magnetic domains of the whole continuous magnetic
film. As the duration of oxidation increases from 0 to 10 days and then to 42
days, the oxidation mechanism that is most responsible for the featured
magnetic properties on respective days evolves from edge, to surface, and then
to internal oxidation. This demonstrates the potential of plasma-triggered
oxidation for enhancing the variety of magnetic patterning with variable
oxidation breadths and depths. If further incorporated with precisely
controllable or even reversible oxidation techniques such as a gating
structure that enables voltage-induced oxidation and reduction [24, 45],
oxidation-induced domain patterning can be a powerful technique for future
data-storage and spintronic devices.

##  4 Conclusions

In summary, we demonstrate the oxide-dependent evolution of the magnetic
domain patterns under magnetization reversal of a CoPd thin film that is
pretreated with e-beam lithography and O2 plasma. MOKE measurements reveal
different effects of oxidation on the magnetic micro-domains at different
stages of the Co-oxide evolution. During the days-long oxidation process in a
vacuum storage desiccator, the dynamic behavior of the magnetic micro-patterns
under magnetization reversal evolves into different phases. In the beginning
(Day 0), pinned segments of magnetic domains are observed along the edges of
the plasma-pretreated $50\times 50$ $\upmu$m2 squares. As oxidation continues,
the film then exhibits preceding reversal of the magnetization inside the
squares on Day 10, but later demonstrates an opposite behavior with delayed
magnetization reversal inside the squares on Day 42. On Day 217, the magnetism
inside the squares totally disappears after extensive oxidation. The mechanism
behind these phenomena involves different forms of surface and internal
oxidation of Co in the film, leading to different alterations of the local
magnetic anisotropy energy and coercivity. Various forms of oxidation provide
an additional dimension of magnetic patterning to the existing conventional
lithography. Magnetic-domain engineering via plasma-induced differential
oxidation may be applied to design and fabricate future data-storage and
spintronic devices.

## Acknowledgment

We acknowledge the group of Prof. Hsiang-Chih Chiu for assisting us with the
plasma source. This study is sponsored by the Ministry of Science and
Technology of Taiwan under Grants Nos. MOST 107-2112-M-003-004, MOST
108-2112-M-003-011-MY2, MOST 109-2112-M-003-009, and MOST 110-2112-M-003-019.

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figure table

# Resonant absorption in multilayer quantum-well and quantum-dot solar cells

Maxime Giteau  Research Center for Advanced Science and Technology, The
University of Tokyo, Meguro, Tokyo 153-8904, Japan  Yusuke Oteki  Research
Center for Advanced Science and Technology, The University of Tokyo, Meguro,
Tokyo 153-8904, Japan  Kento Kitahara  Department of Electrical Engineering
and Information Systems, The University of Tokyo, Bunkyo, Tokyo 113-8656,
Japan  Naoya Miyashita  Research Center for Advanced Science and Technology,
The University of Tokyo, Meguro, Tokyo 153-8904, Japan  Ryo Tamaki  Research
Center for Advanced Science and Technology, The University of Tokyo, Meguro,
Tokyo 153-8904, Japan  Yoshitaka Okada  Research Center for Advanced Science
and Technology, The University of Tokyo, Meguro, Tokyo 153-8904, Japan
Department of Electrical Engineering and Information Systems, The University
of Tokyo, Bunkyo, Tokyo 113-8656, Japan

###### Abstract

Epitaxially-grown quantum well and quantum dot solar cells suffer from weak
light absorption, strongly limiting their performance. Light trapping based on
optical resonances is particularly relevant for such devices to increase light
absorption and thereby current generation. Compared to homogeneous media, the
position of the quantum layers within the device is an additional parameter
that can strongly influence resonant absorption. However, this effect has so
far received little attention from the photovoltaic community. In this work,
we develop a theoretical framework to evaluate and optimize resonant light
absorption in a thin slab with multiple quantum layers. Using numerical
simulations, we show that the position of the layers can make the difference
between strong absorption enhancement and completely suppressed absorption,
and that an optimal position leads to an absorption enhancement twice larger
than average. We confirm these results experimentally by measuring the
absorption enhancement from photoluminescence spectra in InAs/GaAs quantum dot
samples. Overall, this work provides an additional degree of freedom to
substantially improve absorption, encouraging the development of quantum wells
and quantum dots-based devices such as intermediate-band solar cells.

###### keywords:

quantum dots, quantum wells, photovoltaics, intermediate-band, resonant
absorption

*Maxime Giteau<EMAIL_ADDRESS>
##  1 Introduction

Quantum layers (QLs), like epitaxially-grown quantum wells (QWs) and quantum
dots (QDs), are broadly used in opto-electronic devices such as lasers,
photodetectors and solar cells [1, 2, 3]. For photovoltaics, they are
investigated to extend the absorption edge of a given material [4], as well as
for high-effiency concepts like intermediate-band [5, 6], multi-exciton
generation [3, 7] and hot-carrier solar cells [8, 9].

Strong light absorption is critical for the performance of solar cells.
However, a single QL absorbs only a small fraction of the incident light. To
increase optical absorption, one must increase the number of QLs [10, 11]
and/or enhance the absorption per layer through light trapping [12, 13, 14,
15, 16, 17, 18, 19, 20, 21, 22, 23, 24]. Depending on the application and
materials required, too many QLs can lead to strain build-up and increased
non-radiative recombinations which can reduce the efficiency of the device
[25]. Light trapping is very attractive as it enables a larger absorption for
a given number of QLs. Resonant light trapping, which can be implemented when
the thickness of the device is comparable with the wavelength of light (the
device acts as a resonant cavity), has been widely applied to homogeneous
absorbers, for example to achieve high broadband absorption in ultra-thin
solar cell absorbers [26, 27, 28]. It has recently received significant
attention for devices with QLs, where different types of resonances have been
considered, including Fabry-Perot (FP) modes [19] and guided modes [20, 21,
22, 23]. Still, light absorption in QL-based solar cells remains stongly
insufficient, especially for enabling intermediate-band solar cells [12, 29],
and additional strategies are required to further increase absorption.

Resonant modes present an interesting feature for inhomogeneous media: the
intensity of the field, and therefore the absorption enhancement, is not
homogeneous in the cavity but oscillates between peaks and nodes depending on
the position [30]. Therefore, as suggested by Behaghel et al., the absorption
of QLs could be further increased by matching the position of the layers with
the intensity peaks of resonant modes [19, 31]. However, while the impact of
the position of multiple QLs on the electric performances of solar cells has
been considered extensively [32, 33, 34], its impact on absorption has been
largely overlooked, the absorber being usually considered as a homogeneous
medium with an effective absorption coefficient [22].

In this work, we introduce a theoretical framework to evaluate the absorption
enhancement in devices with multiple QLs, depending on their position in a
resonant cavity. We derive analytical expressions for resonant absorption
enhancement, and show that it can be twice larger in multilayer devices than
in homogeneous media. We then present numerical simulations of systems with
single and multiple resonant modes, showing the crucial impact of the position
of the QLs on their absorption. We also verify that an optimal QL position
arrangement leads to an absorption twice stronger than a random position, in
average. Finally, we fabricate GaAs devices with multiple InAs QD layers and
confirm, using photoluminescence, that resonances with optimized QD position
show a larger absorption enhancement.

##  2 Theory

###  2.1 Absorption enhancement in multiple quantum layers

We consider an infinite slab of thickness $d$ composed of a host material with
refractive index $n_{host}$ and $M$ identical QLs of thickness $d_{Q}$ with
complex refractive index $\tilde{n_{Q}}=n_{Q}+i\kappa_{Q}$ (Fig.1). We assume
both materials are isotropic, homogeneous and non-magnetic, and the host
material does not absorb light in the wavelength region of interest. The slab
is illuminated from the outside by a monochromatic plane wave of wavelength
$\lambda$. For simplification, we consider normally incident light, although
the discussion can be extended to oblique incidence. The thickness $d$ of the
cavity is comparable to the wavelength of light $\lambda$. The thickness
$d_{Q}$ of the QLs (typically a few nanometers) is much smaller than
$\lambda$, so we can neglect reflection and scattering effects at the
interfaces of the QLs. The spectral power density absorbed in the QLs at a
depth $z$ (in $\mathrm{W.cm^{-3}.nm^{-1}}$) can be directly related to the
local intensity of the electric field $\mathbf{E}$ [35]:

$p_{abs}(\lambda,z)=\frac{1}{2}c\ \varepsilon_{0}\ n_{Q}(\lambda)\
\alpha_{Q}(\lambda)\ |\mathbf{E}(\lambda,z)|^{2},$ (1)

where $c$ is the speed of light in vacuum, $\varepsilon_{0}$ is the dielectric
permittivity of vacuum, and
$\alpha_{Q}(\lambda)=4\pi\kappa_{Q}(\lambda)/\lambda$ is the absorption
coefficient of the QLs.

Each QL $m$ is located between a depth $z_{m}$ and $z_{m}+d_{Q}$. The total
absorption of the QLs is obtained by integrating the absorbed power over the
thickness $d_{Q}$ of each QL, relative to the intensity of the incident field
$\mathbf{E_{inc}}$ (we assume the incident medium is air):

$A(\lambda)=\alpha_{Q}(\lambda)\sum_{m}\int_{z_{m}}^{z_{m}+d_{Q}}{I(\lambda,z)dz},$
(2)

where $I(\lambda,z)$ is the normalized field intensity:

$I(\lambda,z)=n_{Q}(\lambda)\frac{|\mathbf{E}(\lambda,z)|^{2}}{|\mathbf{E_{inc}}(\lambda)|^{2}}.$
(3)

For simplification, we will consider very thin (few nm) QLs, such that the
field can be considered homogeneous within a QL, leading to:

$A(\lambda)=\alpha_{Q}(\lambda)d_{Q}\sum_{m}{F_{m}(\lambda)},$ (4)

with

$F_{m}(\lambda)=I(\lambda,z_{m}).$ (5)

$F_{m}$ corresponds to the absorption enhancement for the $m$-th QL relative
to the single-pass absorption (assuming no front reflection and low
absorption):

$A_{single}(\lambda)=\alpha_{Q}(\lambda)d_{Q}.$ (6)

Finally, we can express the average absorption enhancement $F$ (We call it
"enhancement" since we focus on increasing absorption, although $F$ may be
smaller than 1, as we will see later) in the QLs as:

$F(\lambda)=\frac{1}{M}\sum_{m}{F_{m}(\lambda)}.$ (7)

In the rest of this paper, we consider that host and QLs have the same real
part of the refractive index $n=n_{Q}=n_{host}$, which is a fair approximation
for III-V heterostructures. If we assume a small absorption ($A<0.1$) even in
presence of light trapping, then we can consider that the intensity of the
electric field is not modified by the absorption of the QLs, and the
absorption enhancement $F_{m}$ can be estimated from the intensity
distribution $I$ in absence of QLs, using Eq. 5. For stronger absorption, the
field intensity will be attenuated in the presence of QLs.

Figure 1: General multilayer structure considered in this work: an infinite
slab of thickness $d\approx\lambda$ consisting of a host material real
refractive index $n_{host}$ and $M$ QLs of complex refractive index
$n_{Q}+i\kappa_{Q}$ and thickness $d_{Q}\ll\lambda$ ($M=5$ in the figure). The
slab is illuminated from the top at normal incidence by a monochromatic plane
wave of wavelength $\lambda$.

###  2.2 Maximum enhancement with quantum layers

Applying the temporal coupled-mode theory framework in a homogeneous cavity
[36, 26, 37], we show in Appendix A that, in the limit of low absorption and
assuming a perfect back mirror, the maximum light path enhancement at
resonance wavelength $\lambda_{0}$ is:

$F(\lambda_{0})\approx\frac{8}{\ln(1/R_{f})}.$ (8)

where $R_{f}$ is the front surface reflection. We can generalize this result
to non-homogeneous devices with multiple QLs in the following way. The
amplitude of a wave evolves as a sinewave as it propagates through space.
Therefore, the intensity $I$ of a resonant mode in a cavity oscillates with
depth between 0 and its maximum value $I_{max}$, following a cosine square
function whose average value is $I_{max}/2$. In a homogeneous medium,
absorption occurs as if the field had the same value $I_{max}/2$ everywhere.
As a result, the maximal enhancement achievable with QLs can be twice larger
than for an homogeneous medium if the QLs occupy the intensity peaks of the
resonance (Eq. 5), leading to:

$F_{max}(\lambda_{0})\approx\frac{16}{\ln(1/R_{f})}.$ (9)

##  3 Numerical simulations

Transfer matrix method (TMM) [38] and scattering matrix method [39] are two
common ways to calculate the distribution of the electric field in planar
thin-film stacks. In this work, we use an in-house TMM algorithm that computes
the absorption $A(\lambda)$ as well as the normalized field intensity
$I(\lambda,z)$.

###  3.1 Single resonant mode

We first illustrate the impact of the position of the QLs on a simple and
general example with a single resonance. We consider a 500 nm-thick GaAs[40]
slab on a silver[41] mirror (Fig. 2(a)), forming a FP cavity with air as the
incident medium. We calculate the normalized field intensity $I(\lambda,z)$ in
the wavelength region $[0.9\ \mathrm{\mu m},1.2\ \mathrm{\mu m}]$, beyond the
bandgap of GaAs (Fig. 2(b)). We observe a single FP mode in that region, at a
resonance wavelength $\lambda_{0}=1.040\ \mathrm{\mu m}$. This mode has 4
intensity peaks (including one at the surface) separated by a distance
$\lambda_{0}/2n\approx 150\ \mathrm{nm}$, with an intensity $I_{max}=13.2$ at
each peak.

To validate our model, we calculate the maximum light path enhancement
achievable in this structure, using Eq.9. The front reflection at the GaAs/air
interface is $R_{f}\approx 0.3$, leading to $F_{max}(\lambda_{0})\approx
13.3$, in very good agreement with the value calculated for $I_{max}$. This is
the absorption enhancement we can expect for QLs positioned at the resonance
peaks (assuming a small absorption from the QLs), twice higher than what can
be achieved in homogeneous media.

We now insert three 1 nm-thick QLs in the slab, with a small absorption
coefficient $\alpha_{Q}=10^{4}\ \mathrm{cm^{-1}}$ independent of the
wavelength. We consider two cases (Fig. 2(b)). In the first case, the QLs are
inserted at the intensity peaks of the resonant mode (red dashed lines). In
the second case, the QLs are inserted at the nodes of the resonant mode (blue
dashed lines).

Figure 2:  (a) Structure simulated for a single resonant mode: a 500 nm-thick
GaAs slab with a silver back mirror. (b) Normalized field intensity
$I(\lambda,z)$ in the GaAs slab as a function of the wavelength and vertical
position. We observe a single mode centered around $\lambda_{0}=1.040\
\mathrm{\mu m}$. The contour lines correspond to multiples of 2. Two cases are
considered. In the first, the QLs are inserted at the intensity peak of the
field (red dashed lines). In the second, the QLs are inserted at the nodes of
the field (blue dashes lines). (c) Average absorption enhancement $F$ in the
QLs as a function of the wavelength, in log scale. The single pass absorption
$F=1$ is represented with a dashed black line. When the QLs are at the peak
positions (red line), the absorption enhancement at resonance is
$F(\lambda_{0})\approx 13$. When the QLs are at the nodes (blue line), the
absorption at resonance is reduced by more than 2 orders of magnitude relative
to single-pass absorption.

We simulate both configurations and obtain the mean absorption enhancement in
the QLs from Eq. 7 (Fig. 2(c)). When the QLs are inserted at the peaks of the
mode, the mean absorption enhancement at resonance is $F(\lambda_{0})=12.93$,
almost as large as the maximum field enhancement in absence of QLs (it is
slightly smaller due to the absorption of the QLs). However, when the QLs are
inserted at the nodes of the mode, the absorption at resonance is almost
completely suppressed, less than 1% of the single-pass absorption. Therefore,
the position of the QLs has a tremendous impact on their absorption. In
particular, an optical resonance does not systematically imply an absorption
increase relative to single-pass absorption. We note that the configuration
where the QLs are inserted at the nodes may be useful for applications where
it is beneficial to suppress absorption or emission at a specific wavelength.

###  3.2 Multiple resonant modes

We now consider a 1.8 µm-thick GaAs slab on a silver mirror, for which we
simulate the normalized field intensity (Fig. 3(a,b))). In the spectral region
[0.98 µm, 1.34 µm], we identify 4 resonant modes with many lobes. We focus on
the resonant mode centered around $\lambda=1.192\ \mathrm{\mu m}$, which shows
10 peaks (not counting the one at the surface). Since the peak intensity only
depends on the front surface reflection (Eq.9) and not, for example, on the
absorber thickness, the maximum intensity is the same as in the previous case
for all resonant modes, with $I_{max}\approx 13.3$.

By introducing 10 QLs into the slab, one at each peak, we can fully benefit
from the resonant absorption of this resonant mode, leading to a light path
enhancement over 12 (Fig. 3(c)). For the three non-optimized resonant modes,
the light path enhancement is only slightly over 6, or about half the maximum
value. This is because we can consider the QLs are positioned randomly
relative to the peaks of the non-optimized resonances. As a result, the light
path enhancement $F$ takes the average value of the field enhancement
$I_{max}/2$. Therefore, by optimizing the position of QLs, we can increase the
absorption by a factor 2 compared to a random distribution.

Figure 3:  (a) Structure simulated for several resonant modes: a 1800 nm-thick
GaAs layer with 10 QLs on a silver back mirror. (b) Normalized field intensity
$I(\lambda,z)$ in the slab as a function of the wavelength and position. The
contour lines correspond to multiples of 2. Four resonant modes are observed,
with many lobes indicating high-order FP modes. The QLs are inserted at the
peaks of the resonant mode centered around $\lambda_{0}=1192\mathrm{nm}$. (c)
Light pass enhancement $F(\lambda)$ in the device. At the optimized resonance
peak, the absorption enhancement is over 12, similar to the intensity of the
field, while it is only slightly over 6 (about half) for the non-optimized
resonances.

##  4 Experimental demonstration

In this section, we validate our theoretical framework and confirm
experimentally the numerical observations of Fig. 3(c). To do so, we fabricate
a structure similar to the one in Fig. 3(a), with InAs QD layers as the QLs,
and measure the absorption enhancement for optimized and unoptimized FP modes
using photoluminescence.

###  4.1 Sample fabrication

We grow a 1.8 µm-thick GaAs layer with 10 layers of InAs QDs on a GaAs (001)
double polished substrate by molecular beam epitaxy (MBE). First, a GaAs
buffer layer of about 300 nm is grown at a temperature of
570${\mathrm{{}^{\circ}C}}$, followed by a 200nm-thick AlAs release layer.
Then, the substrate temperature is lowered to 482${\mathrm{{}^{\circ}C}}$ to
grow the 1800 nm-thick GaAs/InAs multilayer. A 1.8 monolayer (ML)-thick InAs
QD layer is grown after every 173.5 nm of GaAs, and the last QD layer is
capped with 65 nm of GaAs. The InAs QD layers are grown at a rate of 0.04 ML/s
under As pressure (${1\times 10^{-4}\,\mathrm{Pa}}$), while the GaAs spacer
layers are grown at a rate of 1.0 ML/s under As pressure (${3\times
10^{-4}\,\mathrm{Pa}}$). From high-resolution X-ray diffraction, we estimate
the spacer thickness to be $172\pm 4\ \mathrm{nm}$, in perfect agreement with
the designed thickness. This as-grown stack constitutes the reference (ref)
sample (Fig. 4(a)).

This sample is then processed to obtain the Fabry-Perot (FP) sample (Fig.
4(b)). A silver mirror (100 nm) is deposited on the front of the epitaxial
stack, and covered with gold (300 nm) for protection, using thermal
evaporation. The sample is then bonded to a silicon wafer with epoxy resin.
The GaAs substrate is etched first in a $\mathrm{NH_{3}OH:H_{2}O_{2}:H_{2}O}$
(1:1:2) solution, and continued in citric acid:$\mathrm{H_{2}O_{2}}$ (1:5),
which is slower but more selective with AlAs. Finally, the AlAs layer is
etched in a buffered HF solution.

Figure 4: Samples fabricated to measure the influence of the position of QLs
on resonant absorption. (a) Reference (as-grown) sample: 1800 nm-thick GaAs
host with 10 regularly spaced InAs QD layers, grown on a GaAs substrate,
separated by an AlAs etch stop layer. (b) FP sample obtained after silver
evaporation, bonding and substrate removal. The optical stack consists of the
reversed 1800 nm thick GaAs/InAs slab on a mirror.

###  4.2 Expressions for the absorption

The absorption of the reference sample is obtained from the Beer-Lambert law
(neglecting the impact of the AlAs layer and the reflection from the back
surface):

$A_{ref}(E)=(1-R_{f}(E))\left[1-\exp(-M\alpha_{Q}(E)d_{Q})\right],$ (10)

where $R_{f}$ is the front surface reflection of the sample and $E=hc/\lambda$
is the photon energy, with $h$ the Planck constant. Since the absorption from
the QDs is small (reflection-transmission measurements indicate that single-
pass absorption is less than 1%), it can be approximated as

$A_{ref}(E)\approx(1-R_{f}(E))M\alpha_{Q}(E)d_{Q},$ (11)

while the absorption of the FP sample writes, according to Eq. 4:

$A_{FP}(E)=\alpha_{Q}(E)d_{Q}\sum_{m=1}^{M}F_{m}(E).$ (12)

###  4.3 Absorption enhancement from photoluminescence

Since absorption is very small, especially in the reference sample,
reflection-transmission measurements do not enable an accurate determination
of the absorption enhancement. Moreover, these measurements include the
absorption from other layers (free-carrier absorption in the GaAs substrate
and parasitic absorption in the mirror), which can be difficult to separate
from that of the QDs. To measure the absorption enhancement in weakly
absorbing media, typical methods include EQE [42] and waveguide absorption
[43], which respectively require electrical contacts and specific fabrication
processes. Photoluminescence (PL), which consists in measuring the spectral
photon flux emitted by a sample under laser illumination, is another approach
to determine the absorption coefficient of weakly absorbing media. It relies
on Kirchhoff’s law to determine absorption from emission, and offers the
advantage of requiring no additional processing. It has been applied
successfully to measure the absorption coefficient near the bandgap in
materials such as silicon [44] or thin-film CIGS [45]. Although the
determination of the absorption coefficient requires the absolute value of the
PL flux [45], we show in the following that the absorption enhancement $F(E)$
can be determined from the ratio of PL spectral intensities alone.

The photon flux spectrum emitted by a semiconductor per unit solid angle can
be described by the generalized Planck law [46]:

$\Phi(E)=A(E)\frac{2E^{2}}{h^{3}c^{2}}\left[\exp\left(\frac{E-\Delta\mu}{k_{B}T}\right)-1\right]^{-1},$
(13)

where $\Delta\mu$ is the quasi-Fermi level splitting (QFLS) between electrons
and holes and $T$ is the lattice temperature. This expression can be extended
to describe the emission of multiple QLs provided these layers are identical
and all share the same $\Delta\mu$. When the incident illumination is not too
strong, we can consider the Boltzmann approximation:

$\Phi(E)\approx
A(E)\frac{2E^{2}}{h^{3}c^{2}}\exp\left(-\frac{E}{k_{B}T}\right)\exp\left(\frac{\Delta\mu}{k_{B}T}\right).$
(14)

The ratio $r$ between the PL spectra from the sample after processing
$\Phi_{FP}$ and before processing $\Phi_{ref}$ thus takes the expression:

$r(E)=\frac{\Phi_{FP}(E)}{\Phi_{ref}(E)}=\frac{A_{FP}(E)}{2A_{ref}(E)}\exp\left(\frac{\Delta\mu_{FP}-\Delta\mu_{ref}}{k_{B}T}\right),$
(15)

where the factor 2 accounts for the emission from the back of the substrate in
the reference case. We assume the illumination intensity is sufficiently low
so that we can neglect band filling and other non-linear effects, and
absorption is obtained from Eqs. 11-12. This is justified because $r$ is
relatively independent of the incident power (see Fig. 6). The PL ratio
becomes:

$r(E)=\frac{F(E)}{2(1-R_{f}(E))}\exp\left(\frac{\mu_{FP}-\mu_{ref}}{k_{B}T}\right).$
(16)

The energy dependence mostly comes from the variations of $F$ (the front
reflection is almost constant in the spectral region considered). Therefore,
the PL ratio $r$ is proportional to the light path enhancement $F$ and is
independent of the exact value of the absorption coefficient.

###  4.4 Experimental validation

Fig.5(a) shows room-temperature PL spectra obtained for the reference and FP
samples (Fig.4), for the same laser excitation ($\lambda=782\ \mathrm{nm}$,
$P=100\ \mathrm{W.cm^{-2}}$ with a spot diameter of 100 µm). The excitation
energy is above the bandgap of GaAs, so the carriers are mainly generated in
the host. Some of these carriers recombine radiatively in the QDs, leading to
the measured PL signal.

Figure 5:  (a) Room-temperature PL spectra measured on reference
($\Phi_{ref}$, blue line) and FP ($\Phi_{FP}$, red line) samples, for a
incident laser wavelength $\lambda=782\ \mathrm{nm}$ and power $P=100\
\mathrm{W.cm^{-2}}$ (log scale). (b) Ratio $r$ of the PL spectra shown in (a)
(red line, left axis) and light path enhancement $F$ simulated for a 1780 nm-
thick GaAs slab with 10 QD layers positioned to maximize the absorption for
the FP resonance at 1.18 µm (dashed blue line, right axis, different scale).
We observe a good agreement between theory and experiment.

The PL of the reference sample $\Phi_{ref}$ shows two peaks, the main one
around $\lambda=1210\ \mathrm{nm}$ corresponding to the fundamental transition
in the QDs. The second peak can be attributed to an excited state or to the
size distribution of the QDs. We calculate the ratio $r$ between both PL
spectra (Fig. 5(b), red line). We observe 4 peaks at 1011 nm, 1088 nm, 1181 nm
and 1294 nm, with a significantly higher intensity for the third peak.

To compute the light path enhancement $F$ (Fig. 5(b), blue dashed line), we
adjust the total stack thickness in the model to 1780 nm. The small difference
(1% compared to the designed structure) can be attributed to the uncertainty
in the growth rate. We observe a very good match in the peak position between
both curves in Fig. 5(b), confirming that the oscillations in the PL ratio
originate from the FP resonances. Furthermore, the relative intensity of the
peaks of $r$ agrees with those of $F$: the peak for which the QD position is
optimized shows an enhancement more than twice that of the other peaks. Also,
$r$ is almost independent of the laser intensity (see Fig. 6), confirming that
this difference in the relative intensity of the FP peaks cannot be attributed
to non-linear effects in the QDs. Overall, we verify experimentally that the
position of multiple QLs can have a significant impact on their absorption,
allowing higher absorption enhancements than with homogeneous media.

We notice two differences in the shape of $r$ compared to $F$: an asymmetry of
the main PL ratio peak, which is broader than expected on the low-wavelength
side, and a ratio larger than 2 between the peaks. These could both originate
from spatial variations of the QD distribution (the PL spectra of the
reference and FP samples were measured in positions separated by about 1 mm on
the original wafer).

Figure 6: PL ratio spectra $r(\lambda)$ for different laser intensities. For
laser powers twice smaller and larger than $100\ \mathrm{W.cm^{-2}}$ (the
power considered in Fig. 5), the changes in the PL ratio are small. In
particular the resonant peak at 1192 nm always shows a much larger ratio than
the other peaks, confirming the impact of the position of the QLs on
absorption.

###  4.5 Difference in quasi-Fermi level splitting between reference and FP
samples

We note that the scale for the PL ratio $r$ and the light path enhancement $F$
in Fig. 5(b) are different, with $F/r\approx 5$. Using Eq. 16 with
$R_{f}\approx 0.30$, we can estimate the difference in QFLS between both
samples, for the same incident power:

$\Delta\mu_{ref}-\Delta\mu_{FP}\approx 33\ \mathrm{meV}.$ (17)

The absorption at the laser wavelength is almost the same for both samples
(63% and 62%, respectively), so the generated carrier density can be
considered identical. Three distinct mechanisms may contribute to explaining
why the QFLS in the FP sample is smaller than in the reference sample (Eq.
17).

A first explanation for the lower QFLS in the FP sample is the difference in
the back interface (Fig. 4). For the reference sample, the AlAs layer acts as
a barrier for the carriers, preventing recombinations. However, for the FP
sample, the silver interface leads to parasitic recombinations, reducing the
carrier density in the sample and therefore the QFLS. This could be prevented
by adding an Al(Ga)As barrier between the epitaxial layer and the mirror in
the FP sample.

A second possible contribution to QFLS reduction is sample quality. Indeed,
the FP sample was slightly degraded after processing (apparition of pinholes),
which can reduce the QFLS due to increased parasitic recombinations.

Even if we could neglect the two first contributions, the QFLS in the FP
sample should be smaller than in the reference sample. Indeed, if we assume
identical generation and non-radiative recombinations in both samples, then,
from detailed balance considerations, radiative recombinations must also be
identical: $\int\Phi_{FP}(E)dE=\int\Phi_{ref}(E)dE$. However, the absorption
differs significantly between both samples. As the absorption is generally
larger in the FP sample, especially close to the emission peak, the QFLS must
be smaller to ensure equal radiative recombinations.

##  5 Conclusion

We have highlighted the crucial impact of the position of multiple QLs within
a host material on their resonant absorption. We introduced a formalism to
calculate the light absorption enhancement factor $F(\lambda)$ for a given
distribution of the QLs, showing that this enhancement can be up to twice
larger than in homogeneous media when the QLs are placed at the peaks of the
field intensity. We then observed numerically how adjusting the position of
the QLs can lead to strongly increasing or suppressing the absorption of a
resonant mode. In particular we showed that an optimal position enables an
absorption enhancement twice larger than a random position. Finally, we
fabricated devices with multiple InAs QD layers. We presented a method to
determine absorption enhancement from PL spectra measured on samples with and
without a resonant cavity, from which we confirmed quantitatively the larger
absorption enhancement for an optimized QL position.

This work is of interest for all application where increasing absorption in
QLs can lead to an improvement in device performance. It is particularly
relevant to increase current generation in high-efficiency PV concepts such as
intermediate-band solar cells. It is important to note that the benefits of
resonant absorption highlighted here can only be obtained in thin devices. The
formalism we developed can be extended to optimize broadband multi-resonant
absorption [26]. Although we focused here on the absorption of interband
transitions within the QLs, this approach could also be used to increase the
absorption of intersubband transitions for intermediate-band solar cells [42].
Finally, it can be adapted to thin superlattice stacks, where several QLs are
grown very close to each other [9].

##  Appendix A Maximum enhancement with quantum layers

The temporal coupled-mode theory framework is helpful to analyze the
absorption spectrum of a resonant mode in a homogeneous cavity [36, 26, 37].
In this section, we consider a frequency description with

$\omega=\frac{2\pi c}{n\lambda},$ (18)

where $c$ is the speed of light in vacuum and $n$ is the refractive index
inside the cavity. Assuming the full-width at half maximum (FWHM) of a
resonance $\Delta\omega$ is small compared to its resonant frequency
$\omega_{0}$, the absorption spectrum follows a Lorentzian function [36, 26,
37]:

$A(\omega)=\frac{4\gamma_{r}\gamma_{nr}}{(\omega-\omega_{0})^{2}+(\gamma_{r}+\gamma_{nr})^{2}},$
(19)

where $\gamma_{r}$ is the radiative decay corresponding to the coupling of the
cavity with the outside environment, and $\gamma_{nr}$ is the non-radiative
decay due to absorption inside the cavity.

Assuming the mode is confined within the cavity, the non-radiative decay can
be expressed as a function of the absorption coefficient $\alpha$[26]:

$\gamma_{nr}=\frac{\alpha c}{2n}.$ (20)

In a FP cavity, assuming a perfect back reflector, the radiative decay can be
approximated by [30, 47]:

$\gamma_{r}=\frac{c}{4nd}\ln(1/R_{f}),$ (21)

where $R_{f}$ is the front surface reflection. If absorption is small, we can
assume $\gamma_{nr}\ll\gamma_{r}$, such that:

$A(\omega)\approx\frac{4\gamma_{r}\gamma_{nr}}{(\omega-\omega_{0})^{2}+\gamma_{r}^{2}},$
(22)

which we can also write:

$A(\omega)\approx\frac{4\gamma_{nr}}{\gamma_{r}}\frac{1}{1+\left(\frac{\omega-\omega_{0}}{\gamma_{r}}\right)^{2}}.$
(23)

The absorption at resonance $A(\omega_{0})$, FWHM $\Delta\omega$ and
absorption cross-section $\sigma=\int{A(\omega)d\omega}$ then take the
expressions:

$\displaystyle A(\omega_{0})$
$\displaystyle\approx\frac{4\gamma_{nr}}{\gamma_{r}},$ (24)
$\displaystyle\Delta\omega$ $\displaystyle\approx 2\gamma_{r},$ (25)
$\displaystyle\sigma$ $\displaystyle\approx 4\pi\gamma_{nr}.$ (26)

Combining Eqs.24, 20 and 21, the absorption at resonance can be written:

$A(\omega_{0})\approx\alpha d\frac{8}{\ln(1/R_{f})},$ (27)

and we can define the absorption enhancement at resonance relative to single-
pass absorption:

$F(\omega_{0})\approx\frac{8}{\ln(1/R_{f})}.$ (28)

The higher the front reflection $R_{f}$, the stronger the absorption
enhancement at resonance. However, the absorption bandwidth gets narrower as
the radiative decay $\gamma_{r}$ decreases (Eq. 25). As a result, the
absorption cross-section is independent of $\gamma_{r}$ (Eq.26), meaning that
increasing $R_{f}$ does not improve broadband absorption. We note that in the
general case (not assuming $\gamma_{nr}\ll\gamma_{r}$), the absorption can be
written as a function of $F$, following:

$A(\omega_{0})=\frac{F(\omega_{0})\alpha d}{\left(1+\frac{F(\omega_{0})\alpha
d}{4}\right)^{2}}.$ (29)

### Disclosures

The authors declare having no conflicts of interest.

### Acknowledgments

This work is supported by the New Energy and Industrial Technology Development
Organization (NEDO) and the Ministry of Economy, Trade and Industry (METI),
Japan (Grant No. 20000934-0). This work is also supported by the Japanese
Society for the Promotion of Science (JSPS) KAKENHI (Grant No. 19H02541).

The authors would like to thank Stéphane Collin (C2N) and Jean-François
Guillemoles (CNRS IPVF) for the valuable discussions.

## Data Availability

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

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Maxime Giteau is a postdoctoral researcher at RCAST, The University of Tokyo,
Japan. He received his MSc degrees in engineering and solid state physics,
respectively from Supaero and Paul Sabatier University, France, in 2016, and
his PhD from the University of Tokyo in 2020. His current research interests
include theoretical and practical aspects of light trapping as well as high-
efficiency photovoltaic concepts such as hot-carrier solar cells.

Yusuke Oteki is a doctoral student at RCAST, The University of Tokyo, Japan.
He received his master’s degree in engineering from the University of Tokyo in
2020. His current research interests include light trapping in thin-film solar
cells, in particular its implementation in high-efficiency photovoltaic
concepts such as quantum dot intermediate band solar cells.

Yoshitaka Okada received his bachelor’s degree in electronic and electrical
engineering from the University of London in 1984 and his master’s degree and
PhD from the University of Tokyo in 1987 and 1990, respectively. He is
currently a professor at the RCAST of the University of Tokyo. His research
interests include epitaxial growth of III-V-N dilute nitrides, 2D
dichalcogenides, and low-dimensional quantum nanostructures for applications
to high-efficiency and thin-film solar cells.

Biographies of the other authors are not available.

###### List of Figures

  1. 1 General multilayer structure considered in this work: an infinite slab of thickness $d\approx\lambda$ consisting of a host material real refractive index $n_{host}$ and $M$ QLs of complex refractive index $n_{Q}+i\kappa_{Q}$ and thickness $d_{Q}\ll\lambda$ ($M=5$ in the figure). The slab is illuminated from the top at normal incidence by a monochromatic plane wave of wavelength $\lambda$.
  2. 2  (a) Structure simulated for a single resonant mode: a 500 nm-thick GaAs slab with a silver back mirror. (b) Normalized field intensity $I(\lambda,z)$ in the GaAs slab as a function of the wavelength and vertical position. We observe a single mode centered around $\lambda_{0}=1.040\ \mathrm{\mu m}$. The contour lines correspond to multiples of 2. Two cases are considered. In the first, the QLs are inserted at the intensity peak of the field (red dashed lines). In the second, the QLs are inserted at the nodes of the field (blue dashes lines). (c) Average absorption enhancement $F$ in the QLs as a function of the wavelength, in log scale. The single pass absorption $F=1$ is represented with a dashed black line. When the QLs are at the peak positions (red line), the absorption enhancement at resonance is $F(\lambda_{0})\approx 13$. When the QLs are at the nodes (blue line), the absorption at resonance is reduced by more than 2 orders of magnitude relative to single-pass absorption.
  3. 3  (a) Structure simulated for several resonant modes: a 1800 nm-thick GaAs layer with 10 QLs on a silver back mirror. (b) Normalized field intensity $I(\lambda,z)$ in the slab as a function of the wavelength and position. The contour lines correspond to multiples of 2. Four resonant modes are observed, with many lobes indicating high-order FP modes. The QLs are inserted at the peaks of the resonant mode centered around $\lambda_{0}=1192\mathrm{nm}$. (c) Light pass enhancement $F(\lambda)$ in the device. At the optimized resonance peak, the absorption enhancement is over 12, similar to the intensity of the field, while it is only slightly over 6 (about half) for the non-optimized resonances.
  4. 4 Samples fabricated to measure the influence of the position of QLs on resonant absorption. (a) Reference (as-grown) sample: 1800 nm-thick GaAs host with 10 regularly spaced InAs QD layers, grown on a GaAs substrate, separated by an AlAs etch stop layer. (b) FP sample obtained after silver evaporation, bonding and substrate removal. The optical stack consists of the reversed 1800 nm thick GaAs/InAs slab on a mirror.
  5. 5  (a) Room-temperature PL spectra measured on reference ($\Phi_{ref}$, blue line) and FP ($\Phi_{FP}$, red line) samples, for a incident laser wavelength $\lambda=782\ \mathrm{nm}$ and power $P=100\ \mathrm{W.cm^{-2}}$ (log scale). (b) Ratio $r$ of the PL spectra shown in (a) (red line, left axis) and light path enhancement $F$ simulated for a 1780 nm-thick GaAs slab with 10 QD layers positioned to maximize the absorption for the FP resonance at 1.18 µm (dashed blue line, right axis, different scale). We observe a good agreement between theory and experiment.
  6. 6 PL ratio spectra $r(\lambda)$ for different laser intensities. For laser powers twice smaller and larger than $100\ \mathrm{W.cm^{-2}}$ (the power considered in Fig. 5), the changes in the PL ratio are small. In particular the resonant peak at 1192 nm always shows a much larger ratio than the other peaks, confirming the impact of the position of the QLs on absorption.