HAL Id: jpa-00247258
https://hal.archives-ouvertes.fr/jpa-00247258
Submitted on 1 Jan 1996
HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.
L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.
Quantum Well States in Metallic Superlattices
C. Sommers, P. Levy
To cite this version:
C. Sommers, P. Levy. Quantum Well States in Metallic Superlattices. Journal de Physique I, EDP Sciences, 1996, 6 (11), pp.1461-1467. �10.1051/jp1:1996152�. �jpa-00247258�
Quantum Weil States in Metallic Superlattices
C. Sommers (~,*) and P-M- Levy (~)
(~) Laboratoire de Physique des Solides, Université de Paris-Sud, Bâtiment 510, 91405 Orsay Cedex, France
(~) Department of Physics, New York University, 4 Washington Place, New York, NY 10003, USA
(Received 5 July1996, accepted 27 August 1996)
PACS.71.20.Be Transitions metals and alloys PACS.73.61.At Metal and metallic alloys
PACS.75.30.Et Exchange and superexchange interactions
Abstract. Spin-polarized bond structures of Fen/Crm superlattices have been computed
for both ferromagnetic and antiferromagnetic configurations of trie iron loyers by using both the
layered KKR and augmented spherical wave formalisms. Along the growth direction trie baud structure and density of states display features which are identified as quantum well-like states.
We discuss the role these confined states play in the interlayer coupling and transport properties
of metallic superlattices.
1. Introduction
The results of spin-polanzed photoemission experiments on epitaxial overlayers of Ag or Cu
on Fe or Co have been interpreted in ternis of quantum well-like states in these metallic
structures [1]. The existence of these confined states has been justified in terms of the mismatch between tue band structures of dissimilar metals. To corroborate the existence of quantum well-like states in layered metallic structures, such as magnetic superlattices, and to determine how the hybridization of the metals at interfaces alters these states we have determined the
spin-polanzed band structures for Fe-Cr superlattices. Specifically we have used the augmented spherical wave (ASW) formalism [2] to compute the complete three-dimensional band structure for Fe3/Cr3, and we have used the layered KKR method [3] to determine the band structure
along the (001) growth direction for Fen/Crm in
= 3, m = 3, 5, 7 and n = 5, m = 7). Band
structures for both ferro and antiferromagnetic configurations of the superlattices were found.
About tue Fermi level we find nearly dispersionless bands which are identified as quantum well-like states. We bave calculated tue partial density of states along certain crystallographic
directions as well the total density of states. Along the growth direction the partial density of
states has gap structure which is a signature of quantum well behavior. Here we present results
obtained for Fe3/Cr31 they are representative of those obtained for other Fen/Crm superlattices (n, m # 3). We discuss the rote these confined states play m interlayer coupling and transport
properties of metallic superlattices.
(*) Author for correspondence (e-mail: [email protected])
@ Les Éditions de Physique 1996
1462 JOURNAL DE PHYSIQUE I N°11
2. Baud Structure
2.1. BULK. In Figure 1 we show the bulk band structures for the majority and minority
spm states of Fe, and for Cr along tue (001) direction jr-H). Tue remarkable feature of these
IRON spin up
1.~ ao.o
le-o 1-a
la-o 1-1
ia,o_
>
g la-o
"
~
~°°i
OE ~ù
~~à
5 2
BO
~ a,o
ao
~ i
r B G N £ r A P DN P F H
CHROMIUM
"~
,~ ai-o
, , ,
i-a is,o
ig o
i-a
p is.op
OE i.o "
3 3
ia_o
~ ' 3 4 ~
w
~
Î. j.
i~ ~~5
OE OE
~
e 1 , ,
~ °UJ
?,o
w i
'
s-o
~ a o
r A B G N Z r A P DN P F H
Fig. 1. Band structure for spin-polarized Fe and Cr, taken from D.A. Papaconstantopolous, Hand-
book of the Baud Structure of Elemental Sohds (Plenum, N. Y., 1986). Note trie resemblance near trie Fermi level of trie Fe down (minority) spin to trie Cr bonds, and trie mismatch between trie Fe up
(majority) spm with trie Cr bands.
IRÙÎ~I S~i~ dOW~
1-S ao o
4
le o
t-a
la-o 1-1
la-o
>
_g la o
"
~
~° ° i
ÎÎ
, e o
Î
z tù ~ù
'~
6 o
_~ A.o
a-o -1
r B G N £ r A P DN P F H
Fig. 1. (Continued)
bands for this direction of growth is the nearly identical character of the Cr and minority
Fe band structures. This implies that minority electrons in Fe see practically no diiference in
potential when they are in Cr layers, so that the superlattice band structure for minority Fe-Cr should be tue saine as their bulk band structures. On tue contrary the majority Fe states have
no equivalent states near tue Fermi level in Cr. On tue basis of this mismatch tue majority electrons are confined to tue Fe layers as they are no available states in Cr; this is tue origin of quantum well-like states.
2.2. SUPERLATTICE. To corroborate trie picture we bave presented in trie preceding para- grapu, we bave determined tue band structure for tue Fe-Cr superlattices. Even tuougu tue
underlymg positions of tue atours are on b-c-c- lattice sites, tue Brillouin zone of tue super- lattice is a primitive tetragonal Bravais, as suown in Figure 2. For tuis symmetry tue growtu
direction corresponds to r-Z. We bave calculated tue fuit turee-dimensional band structure for Fe3Cr3 using tue ASW formalism; due to limitations on computer memory we dia not consider
larger superlattices. In Figures 3a-c we present tue turee-dimensional band structures for tue
majority and mmority spin bands of tue ferromagnetically aligned superlattice as well as one of tue two equivalent spin directions for tue antiferromagnetically aligned superlattice. For tue ferromagnetic configuration tue minonty spin bands (Fig. 3b) bave tue anticipated dispersion
m tue growtu direction r-Z; m fact if we take tue r-H part of tue b-c-c- Brillouin zone for Cr
(or minority Fe), see Figure 1, and fold it into tue tetragonal zone r-Z we are able to reproduce
the r-Z portion of Figure 3b. Tuerefore, mmority electrons do not distinguisu between Fe and Cr layers. On the contrary the widths of the individual bands m the r-Z direction for the
majority electrons are reduced and show little dispersion in comparison to the minority spm states; see Figure 3a. Tuese reduced band widths are the signature of quantum well behavior
m tuis metallic superlattice. Results similar to tuose suown in Figure 3 were obtained for Fe3Cr5, Fe3Cr7 and Fe5Cr7 in tue r-Z direction using tue layered KKR code [3].
1464 JOURNAL DE PHYSIQUE I N°11
z
R T,
(A
~~r ~
Y
y
Fig. 2. Brillouin zone for primative tetragonal Bravais lattice
It is important to note tuat tue r-Z direction, wuicu is responsible for tue quantum well- like eifects, represents only a small volume of tue complete turee-dimensional zone. In tue remainder of tue zone tue states are less localized as tue bands are wider. To appreciate tue
importance of tue quantum well-like states wuicu are predominantly m tue r-Z direction we bave calcùlated tue partial density of states for tue ferromagnetic configuration along tuis and several otuer principal directions as well as tue total density of states. In tue antiferromagnetic configuration, wuicu we do not show, tuere is similar structure. Amongst tue partial density of
states suown in Figures 4a-d only tue one in tue direction of layer growtu (001) bas a quantum
well-like structure, 1-e-, tuere are band gaps due to tue diiference in tue potentials for Cr and tue majority Fe electrons. Tuis density of states is a reahstic version of an ideal quantum well spectrum wuicu would bave delta function beuavior instead of finite band widtus. However tuese gais
are wasued out in tue total density of states as seen
m Figure 4d. Diiferences occur between tue majority and mmority density of states, e.g., near tue Fermi level tuere are large
gaps in tue density of states (001) direction for tue majonty spin, wuicu are not tuere for tue minority spin electrons.
3. Discussion of Results
In semiconductor superlattices true quantum well states exist turouguout tue Brillouin zone.
From our calculation of tue band structure and density of states of Fecr superlattices we bave
_
a =_,_
) O.B ..' I" '. .".." ',~ l
Î
/
fi0.75 .-" ._"
":
~ ,:. '". ";..' j'
°~ .' .' ""~ "'
~'~ ,~
X A T Z M V A S Z A T
j0.8 1"1
." ";, .;:"
:. 1' ".,
1
~ ./
fi0.75 .. '. '.'
~; ._l,
~ l -,-_'
.. .., ,.'
".;; ,1' ~-
°~ _"... :' "'j. "..._'.,"
,"
'1. "
_..:
~'~
X A r Z M V A S Z A r
.. ..1
_
..~.l."::"' :÷. ".. -' '~
"Y
~ ÷;."
._
7=["~"l" "1" '.. ." ."..."..'
( ~'~ "_
..,
"'~' Î
'~. / '
fj0.75
_", " .. 1' '~. ""..,. /
j ": ..i, [.-. _1' "...." ~ .Î.,Î...1'
0.7
~ ~ ~ z M V A S Z A T
Fig. 3. Band structure for Fe3Cr3 calculated by using the ASW formalism: a) ferromagnetic
majority spin bauds; b) ferromagnetic mmority spin bands and c) antiferromagnetic spin bands. Note the total number of bands in Figures a and b are equal to those in Figure c; we doubled the size of the ferromagnetic cell in order to obtain the same numerical accuracy as for the antiferromagnetic
calculation.
seen tuis beuavior only in tue direction of layer growtu; tuerefore we bave qualified tue use of this phrase as "quantum well-like". The importance of tuese states is reduced wuen considering
the total density of states (summed over ail directions). Therefore to observe quantum well-like
states in metallic superlattices an angle resolved spectroscopy must be used. This is precisely
how the first observation of these states was made by using angle resolved inverse photoelectron
emission experiments iii.
In calculating interlayer coupling between magnetic layers through nonmagnetic spacers,
states in the growth direction are responsible for the dominant contributions [4]. In a manner
of speaking it is the quantum well-like behavior of these states which produce oscillations of the interlayer coupling [Si. For properties of these structures which average over the entire
Brillouin zone, and do not single out the layer growth direction, we do not anticipate quantum well-like behavior from the density of states to show up. For example, solely based on the
density of states we do not expect magnetotransport properties of magnetic superlattices to
display this behavior. However, scattering matrix elements for electrons in confined geometries
bave been shown to produce oscillations in the resistivity of ultra-thin films [6], and in the magnetoresistivity of magnetic multilayers iii.
1466 JOURNAL DE PHYSIQUE I N°11
30 20
io contribution from k-direction 001 majority spin
0 0
10
minority spin ~Q
~
30
30 20
iQ contribution tram k-direction 100 majority spin
0 0
10
~ m>non;ty spin 20
30
30 20
io contribution tram k-direction II1
0 0
10
minority ~Q
c
30
300 200
IQQ total density of states majority spin
0 0
100
d afin°"tY SPin 200
0 0.2 0.4 0.6 0.8 300
Fig. 4. a-c) Partial density of states along selected
axes in the Brillouin zone, and d) total density
of states for Fe3Cr3 from ASW formalism. Note the units for the partial density of states are arbitrary.
The cnteria for the appearance of quantum well-like states in metallic superlattices are:
1) large potential mismatches between layers, 2) thin layers, and 3) transitions (d-band) metals [8]. As recognized as early as 1965 the flat geometry of tue Fermi surface of d-band
metals tends itself to large gaps and tuus quantum well-hke beuavior [9].
Acknowledgments
We wisu to tuank Professor James MacLaren for supplying us witu a copy of bis layered KKR code. Tuis work was clone under a NATO travel grant CRG 890599 and under Office of Naval Researcu grants N00014-91-J-1695 and N00014-96-1-0203. We also wish to tuank tue C.N.R.S.
IDRIS computing center at Orsay, France for Cray computing time.
References
iii Ortega J-E- and Himpsel F-J-, Phys. Reu. Lett. 69 (1992) 844; Smith N-V-, Brookes N-B-, Cuang Y. and Johnson P-D-, Phys. Reu. B 49 (1994) 332.
[2] Williams A.R., Kubler J. and Gellat C.D. Jr., Phys. Reu. B19 (1979) 6094.
[3] MacLaren J-M-, Crampm S., Vvedensky D.D. and Pendry J-P-, Phys. Reu. B 40 (1989)
12164; MacLaren J-M-, Crampin S. and Vvedensky D.D. Phys. Reu. 40 (1989) 12176.
[4] Bruno P. and Cuappert C., Phys. Reu. Lett. 67 (1991) 1602; Bruno P., Phys. Reu. B 52
(1995) 411.
[Si Stiles M.D., Phys. Reu. B 48 (1993) 7238.
[6] Trivedi N. and Asucroft N-W-, Phys. Reu. B 38 (1988) 12298.
iii Vedyayev A. et ai., J. Phys. Condens. Matter 5 (1993) 8289; Europhys. Lett. 25 (1994) 465; Phys. Lett. A185 (1994) iii; Zuang S. and Levy P-M-, Mat. Res. Soc. Sym. Proc.
313 (1993) 53; Levy P-M- et ai., J. Magn. Magn. Mater. 121 (1993) 357.
[8] Levy P.M. and Zuang S., J. Magn. Magn. Mater. lsl (1995) 315.
[9] Freeman A.J., Dimmock J-O- and Watson R-E-, Phys. Reu. Lett. 16 (1966) 94.
