Fakhim TiCrMnO2 .pdf



Nom original: Fakhim_TiCrMnO2.pdf
Titre: Exchange mechanism of half-metallic ferromagnetism of TiO2 doped with double impurities A first-principles ASW study
Auteur: A. Fakhim Lamrani; M. Belaiche; A. Benyoussef; A. El Kenz; E.H. Saidi

Ce document au format PDF 1.4 a été généré par Elsevier / , et a été envoyé sur fichier-pdf.fr le 20/12/2012 à 03:01, depuis l'adresse IP 105.153.x.x. La présente page de téléchargement du fichier a été vue 1093 fois.
Taille du document: 427 Ko (5 pages).
Confidentialité: fichier public




Télécharger le fichier (PDF)










Aperçu du document


ARTICLE IN PRESS
Journal of Magnetism and Magnetic Materials 322 (2010) 454–458

Contents lists available at ScienceDirect

Journal of Magnetism and Magnetic Materials
journal homepage: www.elsevier.com/locate/jmmm

Exchange mechanism of half-metallic ferromagnetism of TiO2 doped with
double impurities: A first-principles ASW study
A. Fakhim Lamrani a,b, M. Belaiche b,c,e, A. Benyoussef a,c,e, , A. El Kenz a, E.H. Saidi c,d,e
a

´ des sciences, Rabat, Morocco
Laboratoire de Magne´tisme et de Physique des Hautes Energies (associe´ au CNRST) De´partement de physique, B.P. 1014, Faculte
´tiques, Micro-onde et Ce´ramique, Ecole Normale Supe´rieure, B.P. 9235, Oce´an, Rabat, Morocco
Laboratoire de Magne´tisme, Mate´riaux Magne
c
INANOTECH (Institute of Nanomaterials and Nanotechnology), Rabat, Morocco
d
Laboratoire de Physique des Hautes Energies De´partement de physique, B.P. 1014, Faculte´ des sciences, Rabat, Morocco
e
Hassan II Academy of Sciences and Technologies, Rabat, Morocco
b

a r t i c l e in f o

a b s t r a c t

Article history:
Received 1 September 2008
Received in revised form
14 September 2009
Available online 7 October 2009

The electronic structure and ferromagnetic properties of rutile TiO2 doped with double-impurities
Ti1 2xCrxMnxO2 has been investigated using first-principles calculations within the density-functional
theory (DFT) and the local density approximation (LDA), functional for treating the effects of exchange
and correlation. They were performed using the scalar-relativistic implementation of the augmented
spherical wave (ASW). The advantages of doping TiO2 with double impurities instead of single impurities
are the increase of the total moment of the system and the exhibition of the half-metallic ferromagnetic
nature in Cr- and Mn-doped TiO2 rutile. These behaviors are due to the hybridization of Cr 3d states and
nearest-neighboring O 2p states. The spin–spin interaction between magnetic impurities examined by
the total energy between parallel and antiparallel aligned states indicated that the Cr and Mn impurities
are energetically favorable to be parallel coupled, which mean that the ferromagnetic state is more stable
than the ferrimagnetic one. We proposed a bond magnetic polarons (BMP) model, based on localized
carriers, to explain the mechanism of ferromagnetism in these systems.
& 2009 Elsevier B.V. All rights reserved.

Keywords:
Impurity-doped TiO2 (rutile)
Ab-initio calculation
ASW method
Band structure model
DMS
Magnetic property
Carrier-mediated ferromagnetism

1. Introduction
Titanium dioxide has a variety of interesting physical and
chemical properties. It can be found in many products, ranging
from paint to food to cosmetics, and has therefore been
extensively studied experimentally and using a range of theoretical approaches [1–12]. Recently, it has been observed that when
doping TiO2 with transition-metal (TM) impurities at lowconcentration Ti1 xTMxO2, with x from 0.01 up to 0.14, titanium
dioxide exhibits room-temperature (RT) ferromagnetism [3,4]. In
addition, for iron x 0.07 [11,13] and cobalt x r0.12 [1–3] doping
of thin films, the material remains transparent. The doped
material has potential for use in spintronics and optoelectronics
if the ferromagnetism can be understood and controlled [14].
The origin of the RT ferromagnetism has been studied using a
number of experimental techniques but there is still no clear
consensus about the resultant lattice structure, the sites adopted
by TM ions, the distribution of the ions in the lattice, their
oxidation state or the magnetic moment per ion. Transmission

Corresponding author at: Laboratoire de Magne´tisme et de Physique des
Hautes Energies (associe´ au CNRST) De´partement de physique, B.P. 1014, Faculte´
des sciences, Rabat, Morocco.
E-mail address: benyous@fsr.ac.ma (A. Benyoussef).

0304-8853/$ - see front matter & 2009 Elsevier B.V. All rights reserved.
doi:10.1016/j.jmmm.2009.09.075

electron microscopy (TEM) [1,2,4,5,9,10], scanning electron microscopy (SEM) [2,6,7] and atomic force microscopy (AFM) [2] have
been used to demonstrate the solubility of TM atoms in various
forms of TiO2 at a variety of concentrations, for which no sign of
segregation of an impurity phase is evident. This conclusion has
been corroborated by X-ray diffraction (XRD) data, which are
consistent with the incorporation of TM ions into both anatase
and rutile lattices [1–3,4,5,8–10,12].
Great interest has been focused on half-metallic ferromagnetism (HMF) because of their promising application to spintronics.
In HMF one of the two-spin channels is metallic, whereas the
other has an energy gap around the Fermi level.
First half-metallic ferromagnet has been predicted in Heusler
compound NiMnSb in 1983, by Groot et al. [15]. After that, some
half-metallic ferromagnets have been predicted theoretically or
conformed experimentally in ferromagnetic metallic oxides such
as CrO2 [16] and Fe3O4 [17], in binary transition-metal (TM)
pinctides or chalcogenides with zincblende structure such as CrSb
[18], MnBi [19], CrS [20], CrSe, CrTe and VTe [21].
Several works were devoted to investigate the origin of the halfmetallic ferromagnetism and to study its implication in various
physical properties. However, new half-metallic ferromagnets
are looked for, which are more promising in basic properties and
for applications. Actually, some half-metallic ferromagnets have
also been found in diluted magnetic semiconductors (DMS) such as

ARTICLE IN PRESS
A. Fakhim Lamrani et al. / Journal of Magnetism and Magnetic Materials 322 (2010) 454–458

Mn-doped GaAs [22], V- and Cr-doped CdTe [23]. The discovery of
room-temperature ferromagnetism in Co-doped anatase TiO2 by
Matsumoto et al. [1], using a combinatorial molecular beam epitaxy
(MBE) technique, has motivated intensive studies on the structural
and physical properties of this material [24]. Studies on anatase
TiO2-doped by other TM such as Mn, Fe and Ni [25] have also been
made. It is well known that TiO2 has three commonly encountered
polymorphs in nature: anatase, rutile and brookite. Many investigations were devoted to TM-doped rutile TiO2. Geng and Kim [26]
found that Co-doped rutile TiO2 is a ferromagnetic semiconductor by
means of the first-principles ultrasoft pseudopotential calculations
with generalized gradient approximation (GGA). Other TM doping
has been performed: Cr-doped rutile [27], Mn-, Fe-, Ni- and
Cu-doped rutile [28], V-doped rutile [29], Fe-doped rutile [30] and
crystallographically oriented Fe nanocrystals formed in Fe-implanted
TiO2 [31].
To our knowledge, there are no theoretical studies on doped
rutile TiO2 with double impurities of transition-metal Cr and Mn
till now. In this paper, we investigate the electronic structure and
the ferromagnetism of Cr, Mn-codoped rutile TiO2 by spinpolarized calculations with the first-principles augmented spherical wave (ASW) method, and predict that Cr, Mn-codoped rutile
TiO2 is half-metallic. The ferromagnetic stabilization and mechanism are discussed. It may be useful in semiconductor spintronics
and other applications.

2. Method and computational details
2.1. First-principles calculations
The calculations of the present study are performed in the
framework of the density-functional theory using the local
density approximation (LDA), exchange-correlation functional
parametrized by Vosko et al. [32]. The scalar-relativistic augmented spherical wave method [33,34,35] based on the atomic sphere
approximation (ASA) is mainly used. In this method, the wave
function is expanded in atom-centred augmented spherical
waves, which are Hankel functions and numerical solutions of
Schrodinger’s equation, respectively, outside and inside the socalled augmentation spheres. In order to optimize the basis set,
additional augmented spherical waves were placed at carefully
selected interstitial sites, the choice of these sites as well as the
augmentation radii were automatically determined using the
sphere-geometry optimization SGO algorithm [36]. Self-consistency was achieved by a highly efficient algorithm for convergence acceleration [37]. The Brillouin zone (BZ) integrations were
performed, within the irreducible wedge of the Brillouin zone
[35,38], with an increasing number of k-points (4 3 6) in order
to ensure convergence of the results with respect to the space
grid. The geometry was fully relaxed using Hellman–Feynman
force and total energy. The convergence criterion is fixed to
10 8 Ry in the self-consistent procedure and charge difference
DQ= 10 8 between two successive iterations.

been observed for Co concentration up to 12%. The doping
x= 4.167% of Cr and Mn (Ti1 2xCrxMnxO2 ) is based on the
2 3 2 supercell containing 12 primitive unit cells of rutile,
where two Ti atoms are replaced by Cr and Mn atoms.
Ferromagnetic stability is determined by the total energy
difference (DE) of the supercell between spin antiparallel alignment (ferrimagnetic) state and spin parallel alignment (ferromagnetic) state. If DE is negative, the ferrimagnetic state is more
stable; if DE is positive, the ferromagnetic state is more stable. In
this way the structure of Ti22CrMnO48 is obtained. They were
performed using the scalar-relativistic implementation of the
augmented spherical wave method implemented in ASW program
PACKAGE. The radii Rmt of the muffin tins are chosen to be 1.94,
1.94, 1.94 and 1.63 a.u. for Ti, Cr, Mn and O, respectively.

3. Results and discussion
We first calculated the density of states (Figs. 1 and 2) and the
electronic band structure (Fig. 3) of TiO2 without co-doping Cr and
Mn atoms, with an increasing number of k-points (9 9 14). Fig. 3
shows that TiO2 is a directed band gap (G–G) semiconductor. The
direct gap, 2.8 eV, we obtained using the ASW method (LDA-VWN)
(Fig. 1) is comparable to the experimental value 3.00 eV [40]. It is
better than 1.79, 1.85 and 2 eV obtained with LDA, GGA-PBE of
Cambridge serial total energy [41] and LDA of WIEN2K [27],
respectively. This difference mainly arises from the exchangecorrelation functional, which generally underestimates the energy
gap. Fig. 2 shows that the states of O-2p and Ti-3d are mainly
located at the valence band (VB) and the conduction band (CB),
respectively. However, the crystal field split Ti-3d orbital into two
2
2
parts, the t2g (dx2 y2, dxz, dyz) and eg (dxy, d3Z r ) states. The CB is
divided into lower and upper parts. The VB and the upper CB are
composed of O-p and Ti-eg states, whereas the lower CB consists
of the O-p and Ti-2g states, the bottom of the lower CB consisting
of the Ti-dx2 y2, contributes to the metal–metal interaction. The
total (Fig. 4) and partial (Fig. 5) densities of states (DOS) of TiO2
doped with double-impurities Ti1 2xCrxMnxO2, x =0.04167, are
obtained using the tetrahedron method with Bloch corrections.
It is clear from the partial DOS of Cr- and Mn-doped TiO2 rutile
(Fig. 5) that the valence and conduction bands are predominantly
contributed of O-2p and Ti-3d, respectively, so spin polarization
around the Fermi level is mainly composed by the Cr 3d (Fig. 5)

2.2. Computational details
The rutile structure of TiO2 is based on a simple tetragonal
lattice with space group P42/mnm and lattice constants
˚ c= 2.9166 A˚ [39]. The metal atoms are located at the
a =4.4219 A,
Wyckoff positions (2a): (0;0;0), ð12 ; 12 ; 12Þ and the oxygen atoms
occupy the positions (4f): 7(u;u;0); 7 ð12 þ u; 12 u; 12Þ with
u =0.3024. For doped TiO2 with double impurities Cr and Mn, we
mainly take account of the co-doping of 8.34%. The roomtemperature ferromagnetic ordering of Co-doped rutile TiO2 has

455

Fig. 1. Total DOS of rutile TiO2.

ARTICLE IN PRESS
456

A. Fakhim Lamrani et al. / Journal of Magnetism and Magnetic Materials 322 (2010) 454–458

Fig. 2. Partial DOS of rutile TiO2.

Fig. 4. Total DOS of Ti1 2xCrxMnxO2 with x= 0.04167.

Fig. 3. Electronic band structure of rutile TiO2.

states and the nearest-neighbouring O 2p states (Fig. 5). From the
calculated total DOS for Ti1 2xCrxMnxO2, x =0.04167 (Fig. 4), we
observe a decrease of energy gap, which is of about 2.05 eV
compared to 2.8 eV for TiO2 without doping. We note that because
Cr (3d54s1) has six valence electrons and Cr4 + (3d2) in
Ti1 2xCrxMnxO2, x =0.04167, has two net electrons, the variation
in the gap is due to the coulomb correlation interaction between
cation Cr4 + and anion O2 and the Jahn–Teller effect. From Fig. 6
one observes a small splitting between t2g (dx2 y2, dxz, dyz) and eg
(dxy, d3z2 r2), which means a low crystal field. Thus the repulsion
2 0
eg Þ and O2 is weak, and then the metal oxide
between Cr4 + ðt2g
distance is lower than in the case of spherical electronic density
and the c parameter decreases.
From Fig. 4, we see that the minority-spin components display
a band gap, while the majority-spin DOS is metallic.
Moreover, in our calculations the total magnetic moment of unit
cell is of 5 mB Cr- and Mn-doped rutile, and partial moments are
2.72, 1.84 and 0.002 mB for Mn, Cr and O, respectively, comparable
to the values obtained by Murugan et al. [42] using VASP Package.
These values show that the exchange coupling between Cr 3d

Fig. 5. Partial DOS of Ti 3d, Cr 3d, Mn 3d and O 2P of Ti1 2xCrxMnxO2 with
x= 0.04167.

(1.84 mB) and O 2p (0.002 mB), coupling p–d, is ferromagnetic (FM).
The Cr–Cr coupling is also ferromagnetic (FM).
Therefore, Fig. 4 shows that Cr- and Mn-doped rutile TiO2 is
half-metal, that is to say, the conduction electrons around the
Fermi level are 100% spin polarized. The calculated total magnetic
moment is also a typical character of half-metallic ferromagnet
due to the hybridization between the Cr 3d states and the nearestneighbouring O 2p states. The Mn impurity band is located at the
bottom of the conduction band (BCB) and the top of the valence
band (TVB) (Fig. 5). In our case (Ti1 2xCrxMnxO2) Cr4 + is both
magnetic doping and electrons donor. The carriers are n-type.
Those latter couple antiferromagnetically with magnetic ion Cr4 +
2 0
ðt2g
eg Þ via phenomenological exchange interaction p–d noted b.
This induces a ferromagnetic coupling between the Cr moments.
Since there are two possible modes (ferromagnetic and antiferromagnetic) of coupling between Cr and Mn, we studied both
configurations (ferromagnetic and antiferromagnetic) for doubleimpurities-doped TiO2 (Fig. 7). We put two magnetic ions in a

ARTICLE IN PRESS
A. Fakhim Lamrani et al. / Journal of Magnetism and Magnetic Materials 322 (2010) 454–458

supercell Ti22CrMnO48 and calculate the total energy difference
between the two cases of the parallel (P) and antiparallel (AP)
spin-ordered states (DE= E(AP) E(P)). Our calculations indicate
that a P-coupled spin is energetically favorable (Table 1).
In Fig. 7, we show the total density of states (DOS) of the
ferrimagnetic Ti22Cr(k)Mn(m)O48 system. The majority-spin states
Mn-d occur close to the bottom of the conduction band and the
top of the valence band of the host system. The minority-spin Cr-d

457

states are metallic. The advantage of doping TiO2 with double
impurities instead of single impurity is to optimize the sp3d
hybridization; this is made easier by selecting double impurities
than single impurity. The net (spin) magnetic moment is exactly
1 mB per unit cell. The magnetic alignment is ferrimagnetic with
roughly a moment of 2.7 mB on Mn and 1.8 mB on Cr and gap of
1.85 eV.
Since the carrier density is low, the Fermi level of diluted
magnetic semiconductor is in the middle of the band gap, and
carriers are localized around the donor ions.
This ferromagnetic behavior may be attributed to the presence
of magnetic polarons due to the localized carriers. It is well known
that the effective radius of polaron depends on the temperature
and concentration of donor ions [43,44]. If the density of polaron
and their radius are large enough to exceed the threshold of
percolation, the magnetic interaction will be ferromagnetic.

4. Conclusion
In summary, we have investigated the magnetism and
electronic structure of doped TiO2 with double impurities Cr
and Mn by using the first-principles ASW method.
The results calculated from first principles indicate that the
system has a half-metallic electronic structure and a strong
hybridization between Cr 3d electrons and O 2p electrons, which
gives rise to a stable ferromagnetic ground state, with the
magnetic coupling bound magnetic polaron.

Acknowledgment
Fig. 6. Partial DOS of Cr 3d ðt2g andteg Þ of Ti1 2xCrxMnxO2 with x = 0.04167.

The authors would like to thank V. Eyert for fruitful discussion
concerning the ASW program PACKAGE.
References

Fig. 7. Total DOS of ferrimagnetic state of Ti1 2xCrxMnxO2 with x= 0.04167.

[1] Y. Matsumoto, M. Murakami, T. Shono, T. Hasegawa, T. Fukumura, M.
Kawasaki, P. Ahmet, T. Chikyow, S. Koshihara, H. Koinuma, Science 291
(2001) 854.
[2] Y. Matsumoto, R. Takahashi, M. Murakami, T. Koida, X.-J. Fan, T. Hasegawa, T.
Fukumura, M. Kawasaki, S. Koshihara, H. Koinuma, Jpn. J. Appl. Phys. Part 2 40
(2001) L1204.
[3] W.K. Park, R.J. Ortega-Hertogs, J.S. Moodera, A. Punnoose, M.S. Seehra, J. Appl.
Phys. 91 (2002) 8093.
[4] Z. Wang, J. Tang, L.D. Tung, W. Zhou, L. Spinu, J. Appl. Phys. 93 (2003) 7870.
[5] Z. Wang, W. Wang, J. Tang, L.D. Tung, L. Spinu, W. Zhou, Appl. Phys. Lett. 83
(2003) 518.
[6] N.H. Hong, J. Sakai, W. Prellier, A. Hassini, A. Ruyter, F. Gervais, Phys. Rev. B 70
(2004) 195204.
[7] N.H. Hong, J. Sakai, W. Prellier, J. Magn. Magn. Mater. 281 (2004) 347.
[8] E.C. Kim, S.H. Moon, S.I. Woo, J.H. Cho, Y.G. Joh, D.H. Kim, Solid State Commun.
132 (2004) 477.
[9] Z. Wang, J. Tang, Y. Chen, L. Spinu, W. Zhou, L.D. Tung, J. Appl. Phys. 95 (2004)
7384.
[10] Z. Wang, J. Tang, H. Zhang, V. Golub, L. Spinu, L.D. Tung, J. Appl. Phys. 95
(2004) 7381.
[11] R. Suryanarayanan, V.M. Naik, P. Kharel, P. Talagala, R. Naik, J. Phys.: Condens.
Matter 17 (2005) 755.
[12] K. Inaba, T. Hitosugi, Y. Hirose, Y. Furubayashi, G. Kinoda, Y. Yamamoto, T. Kim,
H. Fujioka, T. Shimada, Jpn. J. Appl. Phys. Part 2 45 (2006) L114.

Table 1
The calculated gap energy, the total variational energy, the total free atoms energies, the cohesive energy and Fermi energy of TiO2, Ti1 2xCrxMnxO2 (ferromagnetic), and
Ti1 2xCrxMnxO2 (ferrimagnetic) for x = 0.04167.

TiO2
Ti1 2xMnxCrxO2 (Ferromagnetic)
Ti1 2xMnxCrxO2 (Ferrimagnetic)

Eg (eV)

Etot (Ryd)

Efae (Ryd)

Ecoh (Ryd)

Ef (Ryd)

2.8
2.05
1.85

4007.459124
49,091.074229
49,091.073504

4003.159815
49,040.270543
49,040.270543

4.299306
50.803685
50.802961

0.746361
0.721346
0.719871

ARTICLE IN PRESS
458

A. Fakhim Lamrani et al. / Journal of Magnetism and Magnetic Materials 322 (2010) 454–458

[13] A.R. Bally, P.E.S.E.N. Korobeinikova, F. Le´vy, F. Bussy, J. Phys. D 31 (1998) 1149.
[14] S. Wolf, D. Awschalom, R. Buhrman, J. Daughton, S. von Molna´r, M.L. Roukes,
A.Y. Chtchelkanova, D.M. Treger, Science 294 (2001) 1488.
[15] R.A. de Groot, et al., Phys. Rev. Lett. 50 (1983) 2024.
[16] S.P. Lewis, P.B. Allen, T. Sasaki, Phys. Rev. B 55 (1997) 1025;
S.M. Watts, et al., Phys. Rev. B 61 (2000) 9621.
[17] F.J. Jedema, A.T. Filip, B.V. Wees, Nature 410 (2001) 345;
S. Soya, et al., Appl. Phys. Lett. 80 (2002) 823.
[18] B.G. Liu, Phys. Rev. B 67 (2000) 172411.
[19] X. Ya-Qiong, L. Bang-Gui, D.G. Pettifor, Physica B 329–333 (2003) 1117.
[20] K.L. Yao, G.Y. Gao, Z.L. Liu, L. Zhu, Solid State Commun. 133 (2005) 301.
[21] W.H. Xie, Y.Q. Xu, B.G. Liu, D.G. Pettrifor, Phys. Rev. Lett. 91 (2003) 037204.
[22] T. Ogawa, et al., J. Magn. Magn. Mater. 196/197 (1999) 428.
[23] K.L. Yao, et al., Physica B 366 (2005) 62.
[24] S.A. Chambers, et al., Appl. Phys. Lett. 79 (2001) 3467;
D.H. Kim, et al., Appl. Phys. Lett. 81 (2002) 2421;
M.S. Park, S.K. Kwon, B.I. Min, Physica B 328 (2003) 120.
[25] M.S. Park, S.K. Kwon, B.I. Min, Phys. Rev. B 65 (2002) 161201.
[26] W.T. Geng, K.S. Kim, Phys. Rev. B 68 (2003) 125203.
[27] G.Y. Gao, et al., J. Magn. Magn. Mater. 313 (2007) 210–213.
[28] Errico, et al., Phys. Rev. B 72 (2005) 184425.

[29] Gao, et al., Phys. Lett. A 359 (2006) 523.
[30] L. Chen, et al., Phys. Rev. B 74 (2006) 235207.
¨
[31] Shengquiang Zhou, K. Potzger, G. Talut, J. von Borany, R. Grotschel,
W. Skorupa, M. Helm, J. Fassbender, J. Appl. Phys. 103 (2008) 083907.
[32] S.H. Vosko, L. Wilk, M. Nusair, Can. J. Phys. 58 (1980) 1200.
[33] A.R. Williams, J. Kubler, C.D. Gelatt Jr, Phys. Rev. B 19 (1979) 6094.
[34] V. Eyert, Int. J. Quantum Chem. 77 (2000) 1007.
[35] V. Eyert, The Augmented Spherical Wave Method: A Comprehensive
Treatment, Lecture Notes in Physics vol. 719, Springer, Berlin, Heidelberg,
2007.
[36] V. Eyert, K.-H. Hock, Phys. Rev. B 57 (1998) 12727.
[37] V. Eyert, J. Comp. Phys. 124 (1996) 271.
[38] P.E. Blochl, O. Jepsen, O.K. Andersen, Phys. Rev. B 49 (1994) 16223.
[39] A.A. Bolzan, C. Fong, B.J. Kennedy, C.J. Howard, Acta Crystallogr. B 53 (1997)
373.
[40] J. Pascual, J. Camassel, H. Mathieu, Phys. Rev. B 18 (1978) 5606.
[41] G.Y. Gao, et al., Physica B 382 (2006) 14–16.
[42] P. Murugan, et al., Meas. Sci. Technol. 16 (2005) 242–247.
[43] A. Kaminski, S.D. Sarma, et al., Phys. Rev. Lett. 88 (2002) 247202.
[44] C. Timm, F. Schafer, F. von Oppen, et al., Phys. Rev. Lett. 89 (2002) 137201.



Documents similaires


fakhim ticrmno2
fakhim snmnwo2
fakhim timoo2
physreva 84 023804
3516iama 2
gyrotrons thz optimisations d etat et possible


Sur le même sujet..