# A way to identify whether a DFT gap is from right reasons or error
cancellations: The case of copper chalcogenides

Jiale Shen  Key Lab of advanced optoelectronic quantum architecture and
measurement (MOE), and Advanced Research Institute of Multidisciplinary
Science, Beijing Institute of Technology, Beijing 100081, China  Haitao Liu
Institute of Applied Physics and Computational Mathematics, Beijing 100088,
China  National Key Laboratory of Computational Physics, Beijing 100088, China
Yuanchang Li<EMAIL_ADDRESS>Key Lab of advanced optoelectronic quantum
architecture and measurement (MOE), and Advanced Research Institute of
Multidisciplinary Science, Beijing Institute of Technology, Beijing 100081,
China

###### Abstract

Gap opening remains elusive in copper chalcogenides (Cu${}_{2}X$, $X$ = S, Se
and Te), not least because Hubbard + $U$, hybrid functional and ${GW}$ methods
have also failed. In this work, we elucidate that their failure originates
from a severe underestimation of the 4$s$-3$d$ orbital splitting of the Cu
atom, which leads to a band-order inversion in the presence of an anionic
crystal field. As a result, the Fermi energy is pinned due to symmetry,
yielding an invariant zero gap. Utilizing the hybrid pseudopotentials to
correct the underestimation on the atomic side opens up gaps of experimental
magnitude in Cu${}_{2}X$, suggesting their predominantly electronic nature.
Our work not only clarifies the debate about the Cu${}_{2}X$ gap, but also
provides a way to identify which of the different methods really captures the
physical essence and which is the result of error cancellation.

††preprint: AIP/123-QED

##  I Introduction

Understanding the nature of the gap and accurately calculating its size have
been one of the central themes of modern electronic structure theory,
especially in the study of transition-metal compounds with exotic physics and
promising applicationsRao ; Imada ; Tokura ; Zaanen . Unfortunately, this
remains a formidable challenge for today’s widely used calculation methods.
For example, density-functional theory (DFT) systematically underestimates
gaps and can even incorrectly predict semiconductors as metalsChan ; LiW . To
address this problem, methods such as Hubbard + $U$, hybrid functionals and
${GW}$ have been developed based on DFT, which typically reproduce the
experimental gap under certain conditionsTran ; Anisimov ; Heyd ; Hybertsen .
However, there is a lack of ways to identify which truly capture the physical
essence and which are the result of error cancellation.

Copper chalcogenides (Cu${}_{2}X$, $X$ = S, Se and Te) are a class of low-cost
and environmentally friendly materials for thermoelectric and photovoltaic
applicationsCoughlan ; x-zt-2012 . Their interesting phase shares a common
structure in which the $X$ atoms form an ordered lattice skeleton while the Cu
atoms are randomly distributed on tetrahedral interstitial sites formed by the
$X$ atoms. It has been known for a long time that Cu${}_{2}X$ has an optical
gap in the range 1.08$\sim$1.23 eVs-gap-1965 ; se-gap-1966 ; te-gap-1990 , yet
theoretical study of its electronic structure has been hindered by the
complexity of its crystal structure. Until 2007, Lukashev et al. propose a
highly symmetric antifluorite model that has since been widely adopteds-lda-
prb-2007 ; se-jpcm-2013 ; te-prb-2022 . Using this perfect crystal structure,
their LDA calculation yields a zero gap characterized by the intersection of
the conduction and valence bands at the $\Gamma$ point. This is not
surprising, since it had become generally known by then that DFT
underestimates the gap, especially given the presence of the transition-metal
Cu. What is really surprising is that the Hubbard + $U$, hybrid functionals
and quasiparticle self-consistent ${GW}$ calculations also feature the same
zero gaps-lda-prb-2007 ; se-jpcm-2013 ; x-mbj-jcp-2014 . Remarkably, the self-
consistent ${GW}$ succeeds in solving gap underestimation of analogous
monovalent copper halides and oxidesSchilfgaarde ; Bruneval  that have long
been notorious for DFT and its derivatives. In view of the failure of ${GW}$,
Lukashev et al. explore the possibility of opening the gap through a "so-
called structural mechanism" by considering the deviation of the realistic
structure from the antifluorite models-lda-prb-2007 . However, their
calculations show that this would only yield about half of the experimental
gap. Since then, there has been much debate as to whether the gap of
Cu${}_{2}X$ is opened by an electronic or structural mechanism. In 2012, Xu et
al.Xu  use the HSE to compute much more realistic structures, obtaining gaps
that are almost identical to the experiments. This seems to indicate that the
structural mechanism is indeed at work. However, in 2014, Zhang et al.x-mbj-
jcp-2014  find that the mBJ + $U$ calculations can open up 0.5$\sim$1.2 eV gap
for the perfect antifluorite structure, reigniting the controversy.

Does the gap of Cu${}_{2}X$ arise from an electronic mechanism or from a
structural mechanism due to the over-simplified antifluorite model? Is there
some commonality in gap underestimation for monovalent copper chalcogenides,
halides and oxides? Despite years of attention, these mysteries remain
unknown.

Figure 1:  Band structures of Cu2S by (a) standard PBE, (b)-(e) HYPP@PBE at
different $\alpha$. Red and blue dots represent the contributions from Cu 4$s$
and 3$d$ orbitals, respectively. Three characteristic bands are marked as
$t_{\rm 2g}$, 4$s$ and $e_{\rm g}$ at the $\Gamma$ point. The valence band
maximum is set as energy zero. (f) Imaginary part of the BSE dielectric
function calculated on top of $\alpha$ = 1-HYPP@PBE results. Black and red
dashed lines denote optical and fundamental gaps, respectively.

Perhaps it is both electronic and structural mechanisms that are responsible
for the gap. Under such circumstances, ones must quantitatively compare the
respective sizes of the two. This requires the electronic structure
calculation to give not only the right result (gap value), but also the right
result for the right reason (gap physics). In practice, a straightforward way
of tackling the gap controversy is to treat the "true" gap of the antifluorite
structure obtained under the correct gap physics as the contribution from the
electronic mechanism. The difference between this and the experimental value
is then used to roughly assess the magnitude of the structural contribution.
Nevertheless, DFT-based methods such as aforementioned HSE and mBJ + $U$ both
include tunable parameters. Based solely on the fact that the gap value is
reproduced, it is not possible to determine whether this is a result of the
tunable parameters or whether the gap physics is truly revealed. Therefore,
the key to resolving the Cu${}_{2}X$ gap controversy lies in identifying the
calculation of correctly capturing the gap physics, which is what this work is
addressing.

In this work, we find that the presence or absence of a computational gap in
Cu${}_{2}X$ depends on the competition between the 4$s$-3$d$ splitting of Cu
atoms and the crystal field effect of anions. A severe underestimation of the
former causes 3$d$ and 4$s$ band-order inversion and the Fermi energy pinning
by symmetry, producing a zero gap. This is the reason why many DFT-based
methods fail to open a gap. Using hybrid pseudopotentials (HYPPs) corrects the
atomic splitting to the experimental value, which simultaneously opens a gap
of the experimental magnitude in Cu${}_{2}X$. As such, it not only gets the
right results, but more importantly, it gets the right results for the right
reasons. In contrast, neither Hubbard + $U$ nor hybrid functional calculations
can correct for both atom-side and solid-side underestimations, implying an
error cancellation. These results indicates a primarily electronic nature of
the Cu${}_{2}X$ gap.

##  II Methodology and models

Three cross PP@functional calculations were performed using the Quantum
Espresso packageQE , i.e., HYPPsYangJ  with PBE functionalPhysRevLett.77.3865
, PBE PPs with PBE + $U$DFT+U  and HSE functionalHeyd , respectively denoted
as HYPP@PBE, PBE@PBE$U$ and PBE@HSE below, for clarity. The HYPPsYangJ  were
derived from the PBE0 functional, but with the amount of exact-exchange set as
an adjustable parameter $\alpha$. Throughout this work, they involved only the
Cu element and were constructed with the OPIUM codeOPIUM  using the same
parameters as beforeWu ; Shen ; Bu ; TanJCP . For other elements, the norm-
conserving Vanderbilt PPsPhysRevB.88.085117  were used with energy cutoffs of
60 Ry. A $12\times 12\times 12$ $k$-grid was used for the Brillouin zone
sampling. Spin-orbit coupling has been neglected. While test calculations show
that it does have an effect on the gap size, especially for Cu2Te containing
the heavy element, it does not affect the nature of the gap, which is the
concern of this work. Optical absorption spectrum was obtained by solving the
Bethe-Salpeter equation (BSE)PhysRevB.62.4927  using the YAMBO
codemarini2009yambo , with three valence and three conduction bands to build
the BSE Hamiltonian.

##  III Results and discussion

Cu2$X$ has similar physics except for the gap size, so Cu2S is illustrated as
an example below. Figure 1(a) shows its standard PBE band structure, which
features a zero gap with the lowest conduction band and the two highest
valence bands intersecting at the $\Gamma$ point. Fat-band analysis shows that
this triple degeneracy originates from the higher-lying $t_{\rm 2g}$ triplet
from the 3$d$-orbital splitting under the S tetrahedral field, while the Cu
4$s$ resides between it and the lower-lying $e_{\rm g}$ doublet. These results
are consistent with previous studiess-lda-prb-2007 ; se-jpcm-2013 ; x-mbj-
jcp-2014 . Despite a zero global gap, it is customary to define the $t_{\rm
2g}$-4$s$ difference at the $\Gamma$ point as a negative gap.

Two anomalies are worth mentioning about Fig. 1(a). On the one hand, it has
been experimentally demonstrated that the oxidation state of Cu in Cu2-xS is
invariably +1 valence regardless of stoichiometryCoughlan . Based on the
unique 3$d^{10}$4$s^{1}$ electronic configuration of Cu, it is natural to
speculate that Cu loses 4$s$ electrons to form a $d^{10}$ semiconductor when
it crystallizes with S as Cu2S. Consequently, 4$s$ and 3$d$ should comprise
the conduction band bottom and the valence band top, respectively, instead of
the zero gap and inverted 3$d$-4$s$ band-order in Fig. 1(a). On the other
hand, one can clearly see that the band-order inversion occurs only very close
to the $\Gamma$ point. A slight deviation will return the expected
4$s$-$t_{\rm 2g}$ order. Such discontinuity in state characteristics is
physically uncommon.

Figure 2:  Schematic of the Cu${}_{2}X$ gap as a result of the competition
between the 4$s$ and 3$d$ energy splitting $\Delta_{4s-3d}$ of Cu atom and the
anionic crystal field effect. Upon crystallization into Cu${}_{2}X$, the
crystal field formed by the four surrounding anions narrows the splitting by
$\Delta_{\rm{cf}}$ = $\Delta_{3d}$ \+ $\frac{2}{5}\Delta T$ in two ways.
Assuming that the charge is uniformly distributed over a sphere, electrostatic
repulsion causes the 3$d$ orbitals to rise $\Delta_{3d}$ as a whole. Consider
further that the charges are distributed tetrahedrally. The 3$d$ orbitals are
split into a higher-lying $t_{\rm 2g}$ triplet and lower-lying $e_{\rm g}$
doublet with an energy difference of $\Delta T$. If $\Delta_{4s-3d}$ surpasses
the crystal field effect, $E_{\rm g}$ is positive. Otherwise, $t_{\rm 2g}$
will lie above the 4$s$, resulting in a negative $E_{\rm g}$.

Figures 1(b)-1(e) show the band structures by different Cu HYPPs of $\alpha$ =
0.25, 0.50, 0.75, and 1, in combination with PBE functional. One can see that
Cu 4$s$ rises with increasing $\alpha$, from below $t_{\rm 2g}$ at $\alpha$ =
0 to coinciding with it at $\alpha$ = 0.5. Continuing to increase $\alpha$, a
global gap appears and gradually increases. Once the gap opens, $t_{\rm 2g}$
and 4$s$ comprise the valence band top and the conduction band bottom,
respectively. At $\alpha$ = 1, the gap reaches 1.20 eV, agreeing almost
perfectly with the experimental 1.21 eVs-gap-1965 . Bader charge analysis
shows that as $\alpha$ increases from 0 to 1, there is a small increase about
10% in charge transfer from Cu to S.

We further calculate the optical absorption spectrum on top of the $\alpha$ =
1-HYPP@PBE band, as plotted in Fig. 1(f). Its first peak appears at 1.06 eV,
which lies within the minimum direct gap. Therefore, its optical gap is set by
an exciton with a binding energy of 0.14 eV.

It is clear that the band-order of $t_{\rm 2g}$ and 4$s$ determines the
presence or absence of a gap in Cu2S. A gap exists only for a positive
4$s$-$t_{\rm 2g}$ order. A negative 4$s$-$t_{\rm 2g}$ order, otherwise, causes
the Fermi energy pinned to $t_{\rm 2g}$, resulting in an invariant zero gap at
the $\Gamma$ point. Because the band-order depends on $\alpha$, an immediate
question is which $\alpha$ is physically reasonable. It determines whether the
experimental result reproduced at $\alpha$ = 1 comes from the right cause or
is merely a coincidence of error cancellation.

The gap $E_{\rm g}$ of Cu2S at the $\Gamma$ point can be understood in terms
of crystal field theory, as illustrated in Fig. 2. An isolated Cu atom has an
energy splitting $\Delta_{4s-3d}$ between 4$s$ and 3$d$ states. Upon
crystallization into Cu2S, the crystal field of S will narrow the splitting by
$\Delta_{\rm{cf}}$ in two waysBethe ; Vleck1 ; Vleck2 : 1) Assuming that the
charge is uniformly distributed on a sphere around the Cu, electrostatic
repulsion leads to an overall increase in the 3$d$ energy by $\Delta_{3d}$;
and 2) Further consideration of the tetrahedral charge distribution, the 3$d$
degeneracy is lifted to the $t_{\rm 2g}$ triplet and $e_{\rm g}$ doublet with
an energy difference of $\Delta T$. According to crystal field theory, $t_{\rm
2g}$ is $\frac{2}{5}$$\Delta T$ higher than that of the original 3$d$Wolfram .
Therefore, we have

$E_{\rm
g}=\Delta_{4s-3d}-\Delta_{\rm{cf}}=\Delta_{4s-3d}-(\Delta_{3d}+\frac{2}{5}\Delta
T).$ (1)

Equation (1) actually provides a criterion for determining whether a
computational method yields the true gap. If a method does capture the gap
physics, it must be able to simultaneously give the correct $E_{\rm g}$,
$\Delta_{4s-3d}$, and $\Delta_{\rm{cf}}$. While $\Delta_{\rm{cf}}$ cannot be
obtained directly from experiment, both $\Delta_{4s-3d}$ and $E_{\rm g}$ can.
If both are correct, it follows immediately from Equation (1) that
$\Delta_{\rm{cf}}$ must also be correct. Otherwise, if only $E_{\rm g}$
matches the experiment but $\Delta_{4s-3d}$ does not, this seemingly correct
$E_{\rm g}$ does not represent the true gap, and is merely a coincidence of
error cancellation between $\Delta_{4s-3d}$ and $\Delta_{\rm{cf}}$.

This criterion makes physical sense. The Hohenberg-Kohn theorem states that
the ground-state electron density of an interacting electron system is
uniquely characterized by the external potentialHK . Changing the PP means
changing the external potential felt by the Cu valence electrons. Here the
$\Delta_{4s-3d}$ serves in a way as a test of the correctness of the external
potential. If a method gets the correct $\Delta_{4s-3d}$ and $E_{\rm g}$ on
both the atom side and the solid side, there is no doubt that the correct gap
is obtained under the correct external potential. So it is true. Otherwise,
only $E_{\rm g}$ being correct and $\Delta_{4s-3d}$ being incorrect implies
that the gap is obtained under an incorrect external potential. Thus, it is an
error-canceling coincidence and the true gap obtained under the correct
external potential will no longer be $E_{\rm g}$.

The calculated $\Delta_{4s-3d}$, $\Delta_{3d}$, $\Delta T$ and $E_{\rm g}$
with different HYPPs are listed in Table I and compared with the experimental
$\Delta_{4s-3d}$ and $E_{\rm g}$. It can be seen that $\Delta_{4s-3d}$ by
standard PBE is only 0.63 eV, which is nearly an order of magnitude smaller
than the experimental 5.04 eVone-electron . It corresponds to the most
negative $E_{\rm g}$ -0.91 eV. Both $\Delta_{4s-3d}$ and $E_{\rm g}$ increase
monotonically with $\alpha$. Interestingly, when $\alpha$ = 1,
$\Delta_{4s-3d}$ = 4.72 and $E_{\rm g}$ = 1.20 eV reproduce the experimental
values (5.04 and 1.21 eV) almost simultaneously. Since $\Delta_{4s-3d}$ is
still slightly underestimated, it makes sense and physical logic that $E_{\rm
g}$ is also slightly underestimated. Now we know that the conventional DFT
underestimation of the Cu2S $E_{\rm g}$ is due to an underestimation of Cu
$\Delta_{4s-3d}$, which incorrectly results in it being smaller than the 3$d$
energy rise induced by the S crystal field. This consequently reverses the
band-order of $t_{\rm 2g}$ and 4$s$, producing a negative $E_{\rm g}$ and
pinning the Fermi energy to the $t_{\rm 2g}$ by symmetry. In light of the
criterion established above, the ability of using HYPPs to synchronously
correct the underestimations of $\Delta_{4s-3d}$ on the atom side and $E_{\rm
g}$ on the solid side indicates that the $\alpha$ = 1-HYPP@PBE indeed captures
the gap physics. Therefore, the produced $E_{\rm g}$ = 1.20 eV is the right
result for the right reason, representing the true gap of the antifluorite
structure.

We also performed PBE@PBE$U$ and PBE@HSE calculations and the results are
summarized in Table I for comparison. Both $\Delta_{4s-3d}$ and $E_{\rm g}$
show a tendency to increase with increasing $U$ or $\alpha$. Specifically, for
the PBE@PBE$U$, $\Delta_{4s-3d}$ = 3.77 eV at $U$ = 7 eV is underestimated by
1.27 eV, corresponding to a negative $E_{\rm g}$ of -0.14 eV. Increasing $U$
to 10 eV, $\Delta_{4s-3d}$ = 5.14 eV is already slightly higher than the
experimental 5.04 eV, but opens only a tiny $E_{\rm g}$ of 0.09 eV. To open
the experimental $E_{\rm g}$ $\sim$1 eV, $U$ would have to reach 45 eV, at
which point $\Delta_{4s-3d}$ is four times larger than the experiment. The
ineffectiveness of applying $U$ can be understood as a consequence of the
rapid increase in Coulomb repulsion with increasing $U$, which leads to a
severe overestimation of the S crystal field effect, thereby canceling out the
vast majority of the $\Delta_{4s-3d}$ increase. The presence of such depletion
causes $E_{\rm g}$ to grow extremely slowly with increasing $U$.

Table 1: The calculated $\Delta_{4s-3d}$, $\Delta_{3d}$, $\Delta T$ and $E_{\rm g}$ from standard PBE and different HYPPs (HYPP@PBE), PBE + $U$ (PBE@PBE$U$) and HSE (PBE@HSE) calculations, along with the known experimental values for comparisons. The value of $\alpha$ denotes the portion of exact exchange used for HYPPs or HSE functionals. All units are in eV. Methods | $\Delta_{4s-3d}$ | $\Delta_{3d}$ | $\Delta T$ | $E_{\rm g}$
---|---|---|---|---
Experiment | 5. | 04one-electron  | — | — | 1. | 21s-gap-1965
PBE | 0. | 63 | 0. | 81 | 1. | 83 | -0. | 91
$\alpha$=0.25-HYPP@PBE | 1. | 59 | 0. | 94 | 2. | 17 | -0. | 22
$\alpha$=0.50-HYPP@PBE | 2. | 13 | 1. | 11 | 2. | 39 | 0. | 06
$\alpha$=0.75-HYPP@PBE | 3. | 27 | 1. | 47 | 2. | 94 | 0. | 62
$\alpha$=1.00-HYPP@PBE | 4. | 72 | 2. | 01 | 3. | 78 | 1. | 20
PBE@PBE$U$=7 | 3. | 77 | 2. | 49 | 3. | 54 | -0. | 14
PBE@PBE$U$=10 | 5. | 14 | 3. | 27 | 4. | 46 | 0. | 09
PBE@PBE$U$=45 | 21. | 67 | 13. | 28 | 18. | 37 | 1. | 04
<EMAIL_ADDRESS>| 2. | 65 | 1. | 24 | 2. | 47 | 0. | 42
<EMAIL_ADDRESS>| 3. | 70 | 1. | 44 | 2. | 85 | 1. | 12
<EMAIL_ADDRESS>| 4. | 77 | 1. | 69 | 3. | 57 | 1. | 65

Previous studyse-jpcm-2013  has found that the PBE@HSE $E_{\rm g}$ for Cu2Se
is sensitive to the range separation parameter $\omega$, which is non-zero
only when $\omega<$ 0.2 Å-1. Here, we fix $\omega$ = 0.106 Å-1 for HSE06Heyd
but change the exact exchange weight $\alpha$. It can be seen from Table I
that the PBE@HSE at typical $\alpha$ = 0.25 gives $\Delta_{4s-3d}$ = 2.65 and
$E_{\rm g}$ = 0.42 eV, respectively, both of which are significantly
underestimated. At $\alpha$ = 0.375, $E_{\rm g}$ = 1.12 eV is close to the
experimental 1.21 eVs-gap-1965 , but the corresponding $\Delta_{4s-3d}$ = 3.70
eV is still underestimated by 1.34 eV. At $\alpha$ = 0.50, $\Delta_{4s-3d}$ =
4.77 eV approaches the experimental 5.04 eVone-electron , but $E_{\rm g}$ =
1.65 eV already exceeds the experimental 1.21 eV by nearly 40%. Even taking
excitons into account (borrowing the binding energy of 0.14 eV from above
$\alpha$ = 1-HYPP@PBE calculation), the relative error is still 25%. These
point out that the PBE@HSE, unlike the $U$ effect, tends to underestimate the
crystal field effect of S, and thus gives the largest $E_{\rm g}$ for the same
$\Delta_{4s-3d}$.

Above results clearly show that neither PBE@PBE$U$ nor PBE@HSE can yield the
correct $\Delta_{4s-3d}$ and $E_{\rm g}$ at the same time. Within typical
parameter ranges ($U$ < 10 eV and $\alpha$ < 0.3), both methods tend to
underestimate $\Delta_{4s-3d}$, which leads to an underestimation of $E_{\rm
g}$. The PBE@PBE$U$ opening $E_{\rm g}$ by 0.09 eV at $U$=10 eV suggests that
the Cu2S gap is largely irrelevant to Mott physics. Even leaving aside the
physical plausibility of the parameters, the gap calculated by the PBE@HSE for
the antifluorite structure under the correct external potential is already
significantly larger than the experimental value (see PBE@HSE-$\alpha$ = 0.50
in Table I). This suggests that the PBE@HSE is not suitable for resolving the
gap controversy, since at this point it implies a clearly implausible negative
structural contribution. Putting these together, neither PBE@PBE$U$ nor
PBE@HSE correctly captures the gap physics of Cu2S, so even if they happen to
get the correct $E_{\rm g}$, it is not true but only the result of an error
cancellation.

In order to fully understand the gap nature, we have also investigated the
following three structural effects. First, the Cu atoms deviate from high-
symmetry positions in the antifluorite structure. Despite the dependence on
the direction and magnitude of the deviation, our test calculations with
$\alpha$ = 1-HYPP@PBE show that this induces a gap correction of no more than
0.1 eV. This finding is consistent with a previous studys-lda-prb-2007 .
Second, we have directly calculated the monoclinic, hexagonal, and cubic Cu2S
with the realistic structures using $\alpha$ = 1-HYPP@PBE and obtained gaps of
1.56, 1.43 and 1.30 eV, respectively. They are at the same level as the HSEXu
and experimental results. Given that the electronic mechanism opens a gap of
1.20 eV, the structural mechanism is indeed of secondary importance. Third, we
have evaluated the effect of copper stoichiometry in terms of cubic Cu1.8Ss-
lda-prb-2007 ; Xu . The standard PBE opens a gap of 0.42 eV while our $\alpha$
= 1-HYPP@PBE opens a gap of 1.68 eV. Note that it has an experimental gap of
1.75 eVReijnen . Compared to GW 1.06 eVs-lda-prb-2007  and HSE 1.20 eVXu , our
result is clearly much better.

Performing $\alpha$ = 1-HYPP@PBE calculations on Cu2Se and Cu2Te open direct
gaps at the $\Gamma$ point of 0.75 and 1.39 eV, respectively, compared to
their experimental values of 1.23 eVse-gap-1966  and 1.08 eVte-gap-1990 . The
former is underestimated by almost 40%, signifying an important complementary
role of Cu structural disorder. Indeed, an extremely strong liquid-like effect
is experimentally found in Cu2Sex-zt-2012 . In contrast, the relatively small
overestimation for Cu2Te comes from our neglect of the spin-orbit couplingx-
mbj-jcp-2014 . Hence, its gap should originate from an electronic mechanism.

##  IV Conclusions

In summary, we compare the electronic structure of Cu2$X$ calculated with
HYPPs, Hubbard + $U$ and hybrid functionals, with a focus on the gap nature.
While each can reproduce experimental gaps under specific parameters, only the
$\alpha$ = 1-HYPP@PBE method complies with the right results for the right
reasons. The other two suffer from significant error cancellations.
Consequently, it is easy to understand that the same $\alpha$ = 1-HYPP also
works well for other monovalent copper halidesWu  and delafossite oxidesShen .
Our results indicate that the time-averaged antifluorite model reflects the
gap physics of Cu2S and Cu2Te well due to their predominantly electronic
nature. However, for Cu2Se, deviations from the antifluorite model and liquid-
like structural disorder must be taken into account.

Traditionally, the PPs are used to reproduce all-electron results with higher
efficiency, so those consistent with exchange correlations are customarily
employed. In practice, there is another side of the coin for PPs as well, as
their specific form also substantially affects the external potential where
the valence electrons are located. This, however, has not yet been emphasized.
The Hohenberg-Kohn theorem restricts the DFT to producing the correct ground-
state properties only at the correct external potential. If calculations are
performed at the wrong external potential, DFT fails in principle. Utilizing
PPs to correct the external potential from the atomic side provides a new
perspective on solving the persistent gap problem that has long plagued the
DFT field.

###### Acknowledgements.

This work was supported by the Ministry of Science and Technology of China
(Grant Nos. 2023YFA1406400 and 2020YFA0308800) and the National Natural
Science Foundation of China (Grant Nos. 12074034 and 11874089).

### AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

### Author Contributions

Jiale Shen: Formal analysis (equal); Investigation (equal); Software (equal);
Validation (equal); Writing-original draft (equal). Haitao Liu: Formal
analysis (equal); Funding acquisition (equal); Methodology (equal); Validation
(equal). Yuanchang Li: Conceptualization (equal); Formal analysis (equal);
Funding acquisition (equal); Methodology (equal); Project administration
(equal); Resources (equal); Supervision (equal); Validation (equal); Writing-
original draft (equal); Writing-review $\&$ editing (equal).

## DATA AVAILABILITY

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

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# Channel Estimation for RIS Assisted Wireless Communications: Part I -
Fundamentals, Solutions, and Future Opportunities
(Invited Paper)

Xiuhong Wei, Decai Shen, and Linglong Dai All authors are with the Beijing
National Research Center for Information Science and Technology (BNRist) as
well as the Department of Electronic Engineering, Tsinghua University, Beijing
100084, China (e-mails<EMAIL_ADDRESS><EMAIL_ADDRESS>daill@tsinghua.edu.cn).This work was supported in
part by the National Key Research and Development Program of China (Grant No.
2020YFB1807201) and in part by the National Natural Science Foundation of
China (Grant No. 62031019).

###### Abstract

The reconfigurable intelligent surface (RIS) with low hardware cost and energy
consumption has been recognized as a potential technique for future 6G
communications to enhance coverage and capacity. To achieve this goal,
accurate channel state information (CSI) in RIS assisted wireless
communication system is essential for the joint beamforming at the base
station (BS) and the RIS. However, channel estimation is challenging, since a
large number of passive RIS elements cannot transmit, receive, or process
signals. In the first part of this invited paper, we provide an overview of
the fundamentals, solutions, and future opportunities of channel estimation in
the RIS assisted wireless communication system. It is noted that a new channel
estimation scheme with low pilot overhead will be provided in the second part
of this paper.

###### Index Terms:

Reconfigurable intelligent surface (RIS), wireless communication, channel
estimation.

##  I Introduction

Recently, reconfigurable intelligent surface (RIS) has been proposed to
enhance the coverage and capacity of the wireless communication system with
low hardware cost and energy consumption [1]. In general, RIS consisting of
massive passive low-cost elements can be deployed to establish extra links
between the base station (BS) and users. By reconfiguring these RIS elements
according to the surrounding environment, RIS can provide high beamforming
gain [2]. The reliable beamforming requires accurate channel state information
(CSI). Hence, it is essential to develop accurate channel estimation schemes
for the RIS assisted wireless communication system [3].

Although channel estimation has been widely studied in the conventional
wireless communication system, there are two main obstacles for conventional
schemes to be directly applied in the RIS assisted system [4]. Firstly, all
RIS elements are passive, which cannot transmit, receive, or process any pilot
signals to realize channel estimation. Secondly, since an RIS usually consists
of hundreds of elements, the dimension of channels to be estimated is much
larger than that in conventional systems, which will result in a sharp
increase of the pilot overhead for channel estimation. Therefore, channel
estimation is a key challenge in the RIS assisted system, which will be
investigated in this invited paper composed of two parts.

In the first part, we provide an overview of the fundamentals, solutions, and
future opportunities of channel estimation in the RIS assisted wireless
communication system. Firstly, the fundamentals of channel estimation are
explained in Section II. Then, in Section III, we discuss and compare three
types of overhead-reduced channel estimation solutions that exploit the two-
timescale channel property, the multi-user correlation, and the channel
sparsity, respectively. After that, we point out key challenges and the
corresponding future opportunities about channel estimation in the RIS
assisted system in Section IV. Finally, some conclusions are drawn in Section
V.

Notation: Lower-case and upper-case boldface letters ${\bf{a}}$ and ${\bf{A}}$
denote a vector and a matrix, respectively; ${{{\bf{A}}^{T}}}$ and
${{{\bf{A}}^{H}}}$ denote the transpose and conjugate transpose of matrix
$\bf{A}$, respectively; ${{\left\|\bf{a}\right\|}}$ denotes the
${{l_{2}}}$-norm of vector ${\bf{a}}$; ${\rm{diag}}\left({\bf{x}}\right)$
denotes the diagonal matrix with the vector $\bf{x}$ on its diagonal.

##  II Fundamentals of Channel Estimation in The RIS Assisted System

In this section, we will first illustrate the system model of an RIS assisted
wireless communication system. Then, the channel estimation problem in this
system will be presented. Finally, we will introduce the basic channel
estimation schemes.

Figure 1: An example of RIS assisted wireless communication system.

###  II-A System Model

For the uplink RIS assisted wireless communication system as shown in Fig. 1,
we consider one $M$-antenna BS and one $N$-element RIS to serve $K$ single-
antenna users. Let ${\bf{h}}_{d,k}\in{\mathbb{C}}^{M\times 1}$ denote the
direct channel between the ${k}$th user and the BS,
${\bf{G}}\in{\mathbb{C}}^{M\times N}$ be the channel between the RIS and the
BS, and ${\bf{h}}_{r,k}\in{\mathbb{C}}^{N\times 1}$ be the channel between the
${k}$th user and the RIS. The received signal
${\bf{y}}\in{\mathbb{C}}^{M\times 1}$ at the BS can be expressed by

${{\bf{y}}}=\sum\limits_{k=1}^{{K}}\left({\bf{h}}_{d,k}+{\bf{G}}{{\rm{diag}}\left({\bm{\theta}}\right)}{\bf{h}}_{r,k}\right){s}_{k}+{{\bf{w}}},$
(1)

where $s_{k}$ is the symbol sent by the $k$th user,
$\bm{\theta}=\left[{\theta_{1}},{\theta_{2}},\cdots,{\theta_{N}}\right]^{T}$
is the reflecting vector at the RIS with $\theta_{n}$ representing the
reflecting coefficient for the $n$th RIS element, and
${{\bf{w}}}\in\mathbb{C}^{M\times 1}$ is the received noise at the BS. Note
that $\theta_{n}$ can be further set as ${\theta_{n}=\beta_{n}e^{j\phi_{n}}}$,
with $\beta_{n}\in[0,1]$ and $\phi_{n}\in[0,2\pi]$ representing the amplitude
and the phase for the $n$th RIS element, respectively.

For the RIS assisted system, reliable beamforming requires the accurate CSI
consisting of the direct link and the RIS related reflecting link. We consider
a time division duplex (TDD) RIS assisted system, where the downlink channel
can be obtained based on the estimated uplink channel because of the TDD
channel reciprocity.

###  II-B Channel Estimation Problem

The channel estimation problem for the direct channel ${\bf{h}}_{d,k}$ can be
solved by the conventional schemes in the conventional wireless communication
system. Unfortunately, it is difficult to estimate the RIS related channels
$\bf{G}$ and ${\bf{h}}_{r,k}$ due to passive RIS elements without signal
processing capability.

Let
${\bf{H}}_{k}\triangleq{\bf{G}}{\rm{diag}}\left({\bf{h}}_{r,k}\right)\in{\mathbb{C}}^{M\times
N}$ represent the cascaded channel between the $k$th user and the BS via the
RIS, and the received signal $\bf{y}$ in (1) can be also rewritten as

${{\bf{y}}}=\sum\limits_{k=1}^{{K}}\left({\bf{h}}_{d,k}+{\bf{H}}_{k}{\bm{\theta}}\right){s}_{k}+{{\bf{w}}}.$
(2)

Note that many beamforming algorithms (i.e., how to design the optimal RIS
reflecting vector $\bm{\theta}$ in (2)) aim to optimize the power of the
effective reflecting link, i.e.,
${\parallel{{\bf{G}}{\rm{diag}}\left({\bm{\theta}}\right){\bf{h}}_{r,k}}\parallel}^{2}={\parallel{{\bf{G}}{\rm{diag}}\left({\bf{h}}_{r,k}\right){\bm{\theta}}}\parallel}^{2}={\parallel{{\bf{H}}_{k}{\bm{\theta}}}\parallel}^{2}$.
Therefore, most of existing channel estimation schemes directly estimate the
cascaded channel ${\bf{H}}_{k}$ instead of the individual channels $\bf{G}$
and ${\bf{h}}_{r,k}$.

By adopting the orthogonal pilot transmission strategy among users, the uplink
channel estimation associated with different users can be independent. Without
loss of generality, the subscript $k$ in ${\bf{h}}_{d,k}$, ${\bf{h}}_{r,k}$,
and ${\bf{H}}_{k}$ can be omitted to represent the corresponding channels
related to any users.

###  II-C Basic Channel Estimation Schemes

If all RIS elements are turned off, i.e., the incident electromagnetic wave
will be perfectly absorbed by the RIS instead of reflected to the
receiver111Note that “turn off” is a widely used expression in the literature
but inaccurate, since an RIS with all elements turned off is also a scatterer
to reflect the incident electromagnetic wave. An implementation method with a
special setting of RIS elements proposed in [5] can realize the perfect “turn
off” for the incident electromagnetic wave., the RIS assisted communication
system can be simplified as the conventional communication system without the
RIS. Hence, the direct channel ${\bf{h}}_{d}$ can be estimated by some
classical solutions such as the least square (LS) algorithm.

As mentioned above, the channel estimation for the RIS related channels
$\bf{G}$ and ${\bf{h}}_{r}$ is challenging. A straightforward solution is to
estimate the cascaded channel $\bf{H}$ in (2) based on the ON/OFF protocol
proposed in [4]. The key idea is to divide the entire cascaded channel
estimation process into $N$ stages, where each stage only estimates one column
vector of ${\bf{H}}\in{\mathbb{C}}^{M\times N}$ associated with one RIS
element. Specifically, the cascaded channel ${\bf{H}}\in{\mathbb{C}}^{M\times
N}$ can be represented by $N$ columns as

${\bf{H}}=\left[\mathbf{h}_{1},\cdots,\mathbf{h}_{n},\cdots,\mathbf{h}_{N}\right],$
(3)

where $\mathbf{h}_{n}\in\mathbb{C}^{M\times 1}$ is the cascaded channel
corresponding to the $n$th RIS element. In the $n$th stage, only the $n$th RIS
element is turned on, while the remained $N-1$ RIS elements are turned off.
Since the direct channel ${\bf{h}}_{d}$ has been estimated in advance, its
impact can be removed from the received pilot signal at the BS. Then,
$\mathbf{h}_{n}$ can be estimated based on the LS algorithm. By following this
similar procedure, $\mathbf{h}_{1}$, $\cdots$, $\mathbf{h}_{N}$ can be
estimated in turn by sequentially turning on the $1$st, $\cdots$, $N$th RIS
element one by one, while the remained $N-1$ RIS elements are turned off.
After $N$ stages, the cascaded channel ${\bf{H}}\in{\mathbb{C}}^{M\times N}$
composed of $N$ columns can be completely estimated. However, since only one
RIS element can reflect the pilot signal to the BS based on the ON/OFF
protocol, the channel estimation accuracy may be degraded.

In order to improve the channel estimation accuracy, [6] further proposed the
discrete Fourier transform (DFT) protocol based channel estimation scheme,
where all RIS elements are always turned on. In this scheme, the entire
cascaded channel estimation process is still divided into $N$ stages. However,
in each stage, the reflecting vector $\bm{\theta}$ at the RIS is specially
designed as one column vector of the DFT matrix. After $N$ stages, based on
the LS algorithm, the cascaded channel $\bf{H}$ can be directly estimated
based on all received pilot signals in $N$ stages at the BS. It is noted that
the overall reflecting matrix for $N$ stages forms a DFT matrix of size
$N\times N$, which has been proved to be the optimal choice to ensure the
channel estimation accuracy [6].

However, the required pilot overhead in [4, 6] is huge. This is mainly caused
by the fact that the number of unknown channel coefficients (e.g., $64\times
256$ with 64 antennas at the BS, 256 elements at the RIS and single antenna at
the user) in the RIS assisted communication system is much larger than that of
unknown channel coefficients (e.g., $64\times 1$ with 64 antennas at the BS
and a single antenna at the user) in the conventional communication system
without the RIS. The huge pilot overhead will significantly decrease the
effective capacity improvement. Thus, it is essential to develop overhead-
reduced channel estimation schemes for the RIS assisted system. In the next
Section III, we will introduce three typical types of overhead-reduced channel
estimation solutions.

##  III Overhead-Reduced Channel Estimation Solutions

In this section, we will introduce three typical types of channel estimation
solutions to reduce the pilot overhead, which exploit the two-timescale
channel property, the multi-user correlation, and the channel sparsity,
respectively.

###  III-A Two-Timescale Based Channel Estimation

The first typical solution to reduce the pilot overhead for channel estimation
is to exploit the two-timescale channel property in the RIS assisted
communication system [7, 8, 9].

Specifically, the two-timescale channel property can be explained as follows.
On the one hand, since the BS and the RIS are usually placed in fixed
positions, the channel $\bf{G}$ between the RIS and the BS usually remains
unchanged for a long period of time, which shows the large timescale property.
On the other hand, due to the mobility of the user, the channel ${\bf{h}}_{r}$
between the user and the RIS and the direct channel ${\bf{h}}_{d}$ between the
user and the BS vary in a much smaller timescale than that of the quasi-static
channel $\bf{G}$, which show the small timescale property.

As shown in Fig. 2, based on this two-timescale channel property, [7] proposed
a two-timescale channel estimation framework, where the two different pilot
transmission strategies are respectively designed for estimating the large
timescale channel $\bf{G}$ and the small timescale channels ${\bf{h}}_{d}$ and
${\bf{h}}_{r}$. Firstly, the high-dimensional channel $\bf{G}$ is estimated
once in a large timescale by using the dual-link pilot transmission strategy
proposed in [7]. Although the pilot overhead required for estimating $\bf{G}$
is large due to its high dimension, such overhead is negligible from a long-
time perspective. Then, based on the widely used uplink pilot transmission
strategy, the low-dimensional channels ${\bf{h}}_{d}$ and ${\bf{h}}_{r}$ can
be estimated before data transmission in a small timescale. Although these
channels have to be estimated more frequently, the required pilot overhead is
small due to their low dimensions. As a result, the average pilot overhead can
be significantly reduced by exploiting the two-timescale channel property.

Figure 2: The two-timescale channel estimation framework [7].

The main difficulty of this scheme is how to estimate $\bf{G}$, since all RIS
elements are passive without signal processing capability. To achieve this
goal, [7] proposed a dual-link pilot transmission strategy as mentioned
before. The key idea is that, the BS firstly transmits pilot signals to the
RIS via the downlink channel ${\bf{G}}^{T}$, and then the RIS reflects pilot
signals back to the BS via the uplink channel $\bf{G}$. There are $(N+1)$ sub-
frames for the dual-link pilot transmission, where each sub-frame consists of
$M$ time slots. In the $m_{1}$th time slot ($m_{1}=1,2,\cdots,M$) of the $t$th
sub-frame ($t=1,2,\cdots,N+1$), the $m_{1}$th antenna of the BS transmits the
pilot signal $s_{m_{1},t}$ to the RIS and other $(M-1)$ antennas of the BS
receive the pilot signal reflected by the RIS. The received pilot signal
$y_{m_{1},m_{2},t}$ at the $m_{2}$th antennas of the BS can be represented as
$(m_{2}=1,2,\cdots,M,m_{1}\neq m_{2})$

$\displaystyle y_{m_{1},m_{2},t}=$
$\displaystyle\left[{\bf{g}}_{m_{2}}^{T}{\rm{diag}}{({\bm{\theta}}_{t})}{\bf{g}}_{m_{1}}+z_{m_{1},m_{2}}\right]s_{m_{1},t}+w_{m_{1},m_{2},t}$
(4) $\displaystyle=$
$\displaystyle\left[{\bf{g}}_{m_{2}}^{T}{\rm{diag}}{({\bf{g}}_{m_{1}})}{\bm{\theta}}_{t}+z_{m_{1},m_{2}}\right]s_{m_{1},t}+w_{m_{1},m_{2},t},$

where ${\bf{g}}_{m_{2}}^{T}\in{\mathbb{C}}^{1\times N}$ is the $m_{2}$th row
vector of the uplink channel $\bf{G}$,
${\bf{g}}_{m_{1}}\in{\mathbb{C}}^{N\times 1}$ is the $m_{1}$th row vector of
the downlink channel ${\bf{G}}^{T}$, ${\bm{\theta}}_{t}$ is the reflecting
vector at the RIS in the $t$th sub-frame, $z_{m_{1},m_{2}}$ is the self-
interference after mitigation from the $m_{1}$th antenna to the $m_{2}$th
antenna of the BS, and $w_{m_{1},m_{2},t}$ is the received noise at the
$m_{2}$th antenna of the BS. After $N+1$ sub-frames, all received pilot
signals $\\{y_{m_{1},m_{2},t}|1\leq m_{1},m_{2}\leq N,m_{1}\neq m_{2},1\leq
t\leq N+1\\}$ can be obtained, which consist of $MN$ unknown variables, i.e.,
$MN$ elements of $\bf{G}$. Then, based on all received pilots, each element of
$\bf{G}$ can be alternately estimated in an iterative manner by utilizing the
coordinate descent algorithm [7].

Some alternative schemes for estimating the channel $\bf{G}$ between the RIS
and the BS were proposed [8, 9]. In [8], two users ($U_{1}$ and $U_{2}$) are
deployed near the RIS to assist its channel estimation. The uplink cascaded
channels ${\bf{H}}_{1}$ for user $U_{1}$, ${\bf{H}}_{2}$ for user $U_{2}$, and
the $U_{1}$-RIS-$U_{2}$ cascaded channel between the user $U_{1}$ and the user
$U_{2}$ via the RIS are estimated based on the pilot symbols transmitted by
the two users, respectively. After that, the entries for BS-RIS channel
$\bf{G}$ can be calculated based on three estimated cascaded channels
mentioned above. By utilizing the long-term channel averaging prior
information [9], the large timescale channel $\bf{G}$ can also be estimated
based on the channel matrix calibration.

After acquiring $\bf{G}$, the low-dimensional channels ${\bf{h}}_{d}$ and
${\bf{h}}_{r}$ can be estimated based on the conventional uplink pilot
transmission strategy. The user transmits the pilot signals to the BS via both
the direct channel ${\bf{h}}_{d}$ and the effective reflecting channel
${\bf{G}}{\bm{\Phi}}{\bf{h}}_{r}$. Based on the received uplink pilot signals
with the known $\bf{G}$ and ${\bm{\Phi}}$, ${\bf{h}}_{d}$ and ${\bf{h}}_{r}$
can be directly estimated at the BS by the conventional channel estimation
algorithms such as the LS algorithm.

Based on the two-timescale channel property, the large timescale channel
$\bf{G}$ and the small timescale channels ${\bf{h}}_{d}$ and ${\bf{h}}_{r}$
can be respectively estimated in different timescales, which can indeed
significantly reduce the average pilot overhead. However, the channel
estimation for $\bf{G}$ is still challenging. In [7], the BS should work in
the full-duplex mode, where different antennas are required to transmit and
receive pilots simultaneously to estimate $\bf{G}$. In [8], the complexity for
user scheduling and the overhead for the $U_{1}$-RIS-$U_{2}$ cascaded channel
feedback are not negligible.

###  III-B Multi-User Correlation Based Channel Estimation

Another solution to reduce the pilot overhead is to directly estimate the
corresponding cascaded channels by utilizing the multi-user correlation. Since
all users communicate with the BS via the same RIS, the cascaded channels
$\\{{{\bf{H}}_{k}}\\}_{k=1}^{K}$ associated with different users have some
correlations. Thus, this multi-user correlation can be exploited to reduce the
pilot overhead required by the cascaded channel estimation [10].

Specifically, the multi-user correlation can be explained as follows. For
convenience, we take the $n$th column $\mathbf{h}_{k,n}\in\mathbb{C}^{M\times
1}$ of the cascaded channel
${\bf{H}}_{k}=\left[\mathbf{h}_{k,1},\mathbf{h}_{k,2}\cdots,\mathbf{h}_{k,N}\right]\in\mathbb{C}^{M\times
N}$ as an example, which can expressed as

$\mathbf{h}_{k,n}=t_{k,n}\mathbf{g}_{n},$ (5)

where $t_{k,n}$ denotes the channel between the $k$th user and the $n$th RIS
element, which is also the $n$th element of ${\bf{h}}_{r}$,
${\bf{g}}_{n}\in\mathbb{C}^{M\times 1}$ denotes the user-independent channel
between the $n$th RIS element and the BS, which is also the $n$th column
vector of ${\bf{G}}$. Since different users enjoy the same channel ${\bf{G}}$
from the RIS to the BS, $\mathbf{h}_{k,n}$ in (5) can be rewritten as

$\mathbf{h}_{k,n}=\lambda_{k,n}\mathbf{h}_{1,n},$ (6)

where

$\lambda_{k,n}=\frac{t_{k,n}}{t_{1,n}}.$ (7)

The key idea of the multi-user correlation based cascaded channel estimation
scheme can be expressed as follows. Firstly, the cascaded channel
${\bf{H}}_{1}=\left[\mathbf{h}_{1,1},\cdots,\mathbf{h}_{1,n},\cdots,\mathbf{h}_{1,N}\right]$
for the first user (which is also called as the typical user) can estimated by
utilizing the DFT protocol based channel estimation scheme discussed in
Subsection II-C. Then, for the $k$th user with $k\geq 2$, the column vector
$\mathbf{h}_{k,n}$ ($n=1,2,\cdots,N$) of ${\bf{H}}_{k}$ can be obtained by
only estimating the unknown scalar $\lambda_{k,n}$ in (7) with only one
unknown coefficient instead of $\mathbf{h}_{k,n}$ with $M$ unknown
coefficients. Hence, there are only $N$ scalars to be estimated in total for
obtaining the cascaded channel ${\bf{H}}_{k}$ of size $M\times N$.

By exploiting the multi-user correlation, the pilot overhead can be
significantly decreased, since the number of channel coefficients to be
estimated becomes much smaller. However, this scheme proposed in [10] has
assumed that there is no receiving noise at the BS. In the typical scenario of
low SNR for channel estimation, the estimation accuracy will be degraded.

###  III-C Sparsity Based Channel Estimation

The overhead-reduced channel estimation solutions in the previous two
subsections are mainly realized in the spatial domain. By contrast, in this
subsection, we will introduce some overhead-reduced based channel estimation
solutions by exploiting the sparsity of channels in the angular domain [11,
12].

In the conventional wireless communication system, since there are limited
propagation paths, the channel ${\bf{h}}_{d}$ is sparse in the angular domain.
Thus, the channel estimation problem for ${\bf{h}}_{d}$ can be formulated as a
sparse signal recovery problem, which can be solved by compressive sensing
(CS) algorithms with reduced pilot overhead. Similarly, the cascaded channel
${\bf{H}}\in\mathbb{C}^{M\times N}$ in RIS assisted systems also shows the
sparsity when transformed into the angular domain. Specially, by using the
virtual angular-domain representation, the cascaded channel ${\bf{H}}$ can be
decomposed as

${{\bf{H}}}={\bf{U}}_{M}{\tilde{\bf{H}}}{\bf{U}}_{N}^{T},$ (8)

where ${\tilde{\bf{H}}}$ denotes the $M\times N$ angular cascaded channel,
${\bf{U}}_{M}$ and ${\bf{U}}_{N}$ are the ${M\times M}$ and ${N\times N}$
dictionary unitary matrices at the BS and the RIS, respectively. The number of
non-zero elements in ${\tilde{\bf{H}}}$ is determined by the product of the
number of paths between the RIS and the BS and that between the user and the
RIS. Since there are the limited number of scatters around the BS and the RIS,
especially in high-frequency communications, ${\tilde{\bf{H}}}$ is usually
sparse in nature [11], as shown in Fig. 3.

Figure 3: The sparsity of the angular cascaded channel [11].

Based on the sparsity of the angular cascaded channel, the cascaded channel
estimation problem can be also formulated as a sparse signal recovery problem
[11]. Then, some classical CS algorithms, such as orthogonal matching pursuit
(OMP), can be directly used to estimate the angular cascaded channel with
reduced pilot overhead. However, these conventional CS algorithms cannot
achieve the satisfying estimation accuracy, especially in low SNR ranges. In
order to improve the estimation accuracy, a joint sparse matrix recovery based
channel estimation scheme was proposed in [12]. In this scheme, all angular
channels associated with different users can be projected to the same subspace
by considering the fact that different users enjoy the same channel $\bf{G}$
from the RIS to the BS. However, for these sparsity based channel estimation
schemes [11, 12], the required pilot overhead is still high, since the
sparsity of the angular cascaded channel becomes less significant compared
with the angular channel in conventional communications.

Moreover, [13] proposed to divide the entire RIS into several sub-surfaces,
where all RIS elements on the same sub-surface are considered to have the same
channel coefficients. Therefore, the number of the channel coefficients to be
estimated can be significantly decreased. By combining the typical overhead-
reduced channel estimation schemes mentioned above with this idea of dividing
sub-surface, the pilot overhead can be further reduced.

For the sake of simplicity, the above discussions on channel estimation
schemes take the narrow band as an example. The similar idea can be extended
to the wideband orthogonal frequency division multiplexing (OFDM) channel
estimation. Specifically, the channel on each sub-carrier can be estimated
separately as in a narrow band system, such as [13]. It is noted that the
reflecting vector $\bm{\theta}$ at the RIS are the same for all sub-carriers.
Besides, by considering the common sparsity of angular domain channels among
different sub-carriers, [14] proposed a joint overhead-reduced channel
estimation scheme for all sub-carriers, where a denoising convolution neural
network (DnCNN) is used to improve the estimation accuracy.

##  IV Challenges and Future Opportunities for Channel Estimation

In this section, we will point out key challenges for channel estimation in
the RIS assisted wireless communication system, based on which the
corresponding future research opportunities will be discussed.

###  IV-A Ultra-Wideband Channel Estimation

In order to achieve ultra-high-speed wireless transmission, the RIS assisted
ultra-wideband wireless communication will be an important trend. However, the
beam squint effect caused by the ultra-wideband communication brings a serious
challenge for the channel estimation, since the single physical angle will be
transformed to multiple spatial angles. [15] proposed a beam squint pattern
matching based wideband channel estimation in the conventional wireless
communication system. The similar idea may be applied in the wideband channel
estimation for the RIS assisted wireless communication system.

###  IV-B Spatial Non-Stationarity

To further exploit the potential beamforming gains and spatial resolutions of
the RIS, the number of RIS elements may be hundreds of times larger than that
of most scenarios discussed so far, which will result in a large size for the
RIS array. With the significant increase of the array size, the RIS related
channels will present a new characteristic called as the spatial non-
stationarity [16]. This channel characteristic means that the incident
direction and power for the electromagnetic wave at the RIS vary with
different RIS elements. Under this condition, all existing channel estimation
schemes based on the spatial stationarity may not work. The estimation scheme
based on the concept of visibility regions [16] may be used to address this
challenge.

###  IV-C RIS Assisted Cell-Free Network

The cell-free network has been recently proposed to address the inter-cell
interference of the conventional cellular network. In order to further improve
the network capacity with low power consumption, the energy-efficient RISs can
be investigated in the cell-free network. However, with the increase of the
number of RIS, the number of channels to be estimated increases accordingly in
the RIS assisted cell-free network. One possible solution is to exploit the
multi-user correlation mentioned in [10] to reduce the number of channels to
be estimated.

###  IV-D High Pilot Overhead

Since the RIS consists of a large number of elements, the RIS related channels
have many coefficients to be estimated. Although some overhead-reduced channel
estimation solutions have been recently proposed, the required pilot overhead
is still high due to the passivity of RIS elements. By exploiting more channel
characteristics in the RIS, the pilot overhead can be further reduced, which
will be discussed in the second part of this invited paper composed of two
parts.

##  V Conclusions

In the first part of this invited paper, we have investigated the channel
estimation in the RIS assisted wireless communication system. Due to a large
number of passive RIS elements without signal processing capability, the
channel estimation in the RIS assisted system is more challenging than that in
the conventional system. We first explained the fundamentals of channel
estimation. Then, three typical types of overhead-reduced channel estimation
solutions were introduced. Finally, we pointed out several challenges and the
corresponding future research opportunities for channel estimation. Note that
a feasible solution to one of these key challenges, i.e., the high pilot
overhead, will be proposed in the second part of this invited paper.

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