Thickness and cation control of conductionin nickelate heterostructures

X. Yang and O.K. Andersen

A hundred years after the discovery of super-conductivity, no new superconductor has yetbeen found by design, but merely by chance,or by following empirical rules which the nextdiscovery then proved to be of limited validity.The discoveries of the A15 compounds in theseventies, the cuprates in the eighties, MgB2 in2001, and the iron pnictides in 2008 all testify tothis. Despite 25 years’ intensive research, high-temperature superconductivity in the cuprateshas not been understood and no superconduc-tor with Tc higher than 150 K has been foundsince 1993. This record-holding material istriple layer, pressurized HgBa2Ca2Cu3O8. Evenin cases like MgB2 where the superconductiv-ity mechanism has been understood, this has nothelped to design a better superconductor.

Recently, it has recently become possible togrow oxide heterostructures with great preci-sion, specifically by pulsed laser deposition

(PLD) or molecular beam epitaxy (MBE). Thisopens the possibility to produce materials withdesired physical properties, for instance bymanipulating the nearly localized 3d-electronsin transition metal oxides. This is becominga major activity in our institute. Chaloupkaand Khaliullin [1] speculated that by confin-ing a single layer of LaNiO3 with electronicconfiguration d7 between layers of a lattice-matched, insulating perovskite such as LaAlO3

or LaGaO3, one might be able to force the Ni3d (eg) electron into the planar x2– y2 orbital asin the d9 cuprates, and maybe in this way obtainhigh-temperature superconductivity.

Figure 1 illustrates the principle of this confine-ment: Bulk LaNiO3 is a (pseudo)cubic, para-magnetic metal with electronic configurationNi 3d7, i.e., with one electron in a doubly-degenerate eg band. Its two Wannier orbitals,

Figure 1: Brillouin zone, Ni eg Wannier orbitals (x2– y2 black, 3z2– 1 yellow), eg energy band, and structureof cubic bulk LaNiO3. The Fermi level is at 0 eV.

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x2– y2 and 3z2– 1, are shown respectively insidethe black and yellow frames, and are seen tohave antibonding pdσ tails on the neighboringoxygens. Being formed from orbitals with re-spectively ∣m∣= 2 and 0, the two eg Bloch sumscannot hybridize when k is in a (110) plane,and it is therefore simple to understand the egband structure along the red path in the cubicBrillouin zone (BZ). We see that the x2– y2

band disperses with kx and ky, but not withkz, and that the 3z2– 1 band disperses mainlywith kz. At their common bottom, at (0,0,0),the bands have pure Ni 3d character, i.e., in thek = 0 Bloch sums, the O 2p tails cancel. At theircommon top, at (1,1,1) π

a , the bands have maxi-mal oxygen antibonding character.

Hence, the entire dispersion is due to hoppingvia oxygen. Shown in the second line of the fig-ure are the bands along the blue paths in theBZ. Here, they hybridize as indicated by themixed black and yellow colors. In this cubicbulk structure, the two eg orbitals are degener-ate, e.g., the x2– y2 and 3z2– 1 projected densi-ties of states are identical, and so are the num-bers of x2– y2 and 3z2– 1 electrons obtained byintegrating up to the Fermi level (≡ 0).

However, substitution of every second NiO2

layer by an ‘insulating’ AlO2 layer, as shownon the left-hand side of Fig. 2, forces theBloch waves to vanish near the insulating lay-ers whereby the waves with kz ≲ 1

2πa become

forbidden. This changes the energy of thewaves with 3z2– 1 character, but not of thosewith x2– y2 character. One therefore expects thenearly 2D eg band of such a 1/1 heterostruc-ture to look like the cubic band at some ef-fective kz-value, which depends on the scat-tering properties of the insulator and lies inthe range between 1

2πa and π

a , i.e., like thosein the second line of Fig. 1, with the (000)-(00 1

2) bit excluded. Figure 2 now shows thatthis holds surprisingly well. The band structuresin Fig. 2 result from density-functional (GGA)calculations with structural optimization fortwo 1/1 heterostructures, LaNiO3/LaAlO3 andLaNiO3/YScO3 [2]. Like for the cubic case, theeg band has been colored according to its or-bital character, but all bands are now includedin order to show that the eg band is placed ina 2 – 3 eV gap above the Ni t2g and oxygenp bands, and below the Ni 4s, La or Dy 5d, andAl 3sp or Sc 3d bands.

Figure 2: Structure, DFT energy bands, and Fermi surface of LaNiO3/LaAlO3 heterostructure. Similarlyfor the LaNiO3/YScO3 heterostructure. The bands are plotted along lines with the same horizontal projec-tion as in the second line of Fig. 1 and with kz = π

c ∼ π2a , except Γ = (0,0,0). The red dot-dashed line is at

(kx,ky) = ( 12 , 1

2 ) πa . The Fermi surfaces are plotted in the kz = 1

2πc plane.

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We see that the confinement in LaNiO3/LaAlO3

has pushed the bottom of the (3z2– 1)-like bandup to 0.5 eV below the Fermi level, and inLaNiO3/YScO3 has even emptied this upper eg

band.

Hence, the GGA Fermi surface of the alumi-nate has two sheets while that of the scandatehas only one, and the latter is very similar inshape and character (predominantly x2– y2) tothe ones in the cuprates with the highest Tc max

[2,3]. Also Coulomb correlation will tend toreduce the Fermi surface to one sheet [2], butthe effect is smaller than the one seen in Fig. 2which is obtained by change of cations.

Moving apical oxygen closer to Ni by the ap-plication of tensile strain, we have found to beinefficient in emptying the (3z2– 1)-like bandin LaNiO3/LaAlO3 [2]. The reason is, that thebottom of the 3z2– 1 band in the heterostruc-ture has little oxygen character because for thecubic band, the oxygen character and therebythe strain dependence increases from zero at thebottom to a maximum at the top.

Adding more insulating layers helps a bit inemptying the upper band by diminishing thekz-dispersion seen along ΓZ in Fig. 2. Addingmore NiO2 layers, or rather: Failing to growintact single NiO2-layers, should be devastat-ing because interlayer hopping between 3z2– 1orbitals splits the (3z2– 1)-like bands by about0.5 eV and thereby makes the lowest band dipbelow the Fermi level.

What determines the shape of the (x2– y2)-likeFermi-surface sheet is its hybridization withthe axial (m = 0) orbital [3]. As was men-tioned above, this hybridization vanishes forkx = ky and is maximal near (kx,ky) = (1,0) and(0,1) π

a (R), where it pushes the energy of thesaddle-point down. Hence, if we imagine con-tinuing the trend seen in Fig. 2 by moving theenergy of the axial orbital higher and higherabove that of the (x2 – y2)-orbital, the saddle-point would move up towards the center of thatband, whereby the Fermi surface would turn by45∘ around the (1,1) π

a point (A). This shape isthe one found for cuprates such as La2CuO4

with fairly low Tc max, while cuprates with highTc max have Fermi-surface sheets like the onecalculated for LaNiO3/YScO3 and shown in theright-hand side of Fig. 2.

The upper parts of Fig. 3 now show schemati-cally how the energy of the axial orbital is con-trolled in a nickelate heterostructure (left) andin a cuprate (right).

Figure 3: Comparison between nickelate andcuprate axial-orbital levels and conduction bandWannier functions.

In the nickelate, the axial orbital is the fol-lowing linear combination of atomic orbitals:Ni 3d3z2−1 antibonding to the two neighbor-ing apical-oxygen 2pz orbitals, each of whichbond to their neighbor cation axial orbital, i.e.,Al 3s 3pz or Sc 3d3z−1. We wish to push the en-ergy of the axial orbital up above εF so thatthe Fermi surface has merely one sheet. Theeffective O 2pz orbitals should therefore pushthe Ni 3d3z2−1 level up as much as possible,and in order to accomplish this, the cation or-bital should push the O 2pz level down as lit-tle as possible. Hence, a cation should be cho-sen whose covalent interaction with apical oxy-gen is as small as possible. In this respect Sc isfar better than Al, and among the possible insu-lators which are lattice matched with LaNiO3,we have found YScO3 and PrScO3 to be thebest. Habermeier has now been able to pro-duce heterostructures of PrNiO3 and PrScO3,albeit not (yet) with PrNiO3 monolayers. ForPrNiO3/3(PrScO3) we found the bottom of the3z2– 1 band to lie even a bit higher than forLaNiO3/YScO3, and the Wannier orbital calcu-lated for this single conduction band is shownat the bottom left in Fig. 3. On the nickel atomat (0,0,0), this orbital has pure x2– y2 character,which antibonds to x on the oxygens at (± a

2 ,0,0)and to y on those at (0,± a

2 ,0). Each of these

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4 oxygen orbitals bond to 3z2– 1 on the nick-els at (±a,0,0) and (0,±a,0), and the latter an-tibond perpendicular to the plane of the figureto z on the apical oxygens at ±(±a,0,∼ c

4) and±(0,±a,∼ c

4 ). The latter finally bond so weaklywith the Sc 3z2– 1 orbitals at ±(±a,0, c

2 ) and±(0,±a, c

2 ), that this is cannot be seen in the fig-ure.

We now compare with the cuprates. They haveconfiguration d9 with the x2– y2 orbital halffull and 3z2– 1 full, so that there is no pdbond between Cu and apical oxygen, which istherefore far away. In this case, the axial or-bital is Cu 4s antibonding to the two neighbor-ing apical-oxygen 2pz orbitals, each of whichbond to their neighbor cation axial orbital, e.g.,La 3z2– 1, and antibond to Cu 3z2– 1. For thecuprates it has been found, but not understood,that the closer the calculated energy of the ax-ial orbital is above εF, the higher is the mea-sured Tc max [3]. Now, in this case one wants tominimize the pushing up of the Cu 4s level bythe apical-oxygen 2pz orbital, and that happenswhen the distance to apical oxygen is large, asin HgBa2CuO4. The axial level can however notlie lower than the pure Cu 4s level, except inmulti-layer cuprates (HgBa2Ca2Cu3O8), wherethis levels splits.

For such cuprates, the most bonding linear com-bination yields a Fermi surface sheet whoseshape is almost as extreme as the one seen onthe bottom right-hand side of Fig. 2. Increasingthe number of CuO2 layers beyond 3 has notlead to any increase of Tc max, probably becausethe interlayer coherence gets lost, but nick-elate heterostructures point to another route:For those materials, the axial-orbital level ap-proaches εF from below, and the challenge is toshift it to slightly above εF. When this is done,the nickelate conduction band orbital is quitesimilar to that of the cuprate as shown at thebottom of Fig. 3.

In the calculations described so-far, all Ni atomswere forced to be equivalent, i.e., we did not al-low for the charge- and spin-density wave in-stabilities common in bulk nickelates. There-fore, we have now performed LAPW GGA + Ucalculations with larger cells for a numberof nickelate heterostructures, as well as forHgBa2CuO4, searching for the leading insta-bility as we increase the value of the on-site

Coulomb interaction U (J = 0.75 eV). For the90 K cuprate HgBa2CuO4, the leading instabil-ity is antiferromagnetism, as expected, and itoccurs for U ≳ 1 eV. For LaNiO3/YScO3, theresult is the same, thus boosting our hope thatproperly grown scandates will show supercon-ductivity. For LaNiO3/LaAlO3, however, theleading instability, which sets in for U ≥ 4 eVwhen J = 0.75 eV, and earlier when J = 1 eV,is to a magnetic, charge disproportionated(2 d7 −→ d6 + d8) insulator. This is consistentwith recent experiments [4].

Figure 4: Schematic illustration of 2d7 → d6 + d8

charge disproportionation.

So we need to understand why the aluminate,but not the scandate, disproportionates. As il-lustrated in Fig. 4, charge disproportionation isa q = (1,1,0) π

a charge- and spin-density wave,in which the oxygen octahedra breathe alterna-tively in and out around Ni by approximately± 4 pm. Crudely speaking, the contracted siteshave no eg electrons and the expanded siteshave two, d↑3z2−1 and d↑x2−y2 . Such an instabil-ity is believed to be driven by the lattice andHund’s-rule couplings. Obviously, increasingthe crystal-field splitting between the on-siteenergies of the 3z2– 1 and x2– y2 Wannier or-bitals makes charge disproportionation less fa-vorable, and this is consistent with Fig. 2. Theresults of our calculations allow us to be moreexplicit: In the non-magnetic metallic state(Fig. 2), the lower and upper eg bands are sep-arated for all (kx,ky), except at one point, ne-glecting kz, which is along the kx = ky line (AZ)where the bands cannot hybridize. Charge dis-proportionation will now tend to gap each eg

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band around the lines bisecting the q-vectors,and in order that sufficient energy be gained,the result should be an insulator. This meansthat where the two bands come closest, i.e.,along AZ, the two gaps should overlap. Thegap required is thus larger than the separa-tion between the x2– y2 and 3z2– 1 bands atk = ( 1

2 , 12 ) π

a , which is indicated by red stippledlines in Fig. 2. This is the crystal-field splittingrelevant for charge disproportionation, and it isclearly much smaller for the aluminate than forthe scandate. In the insulating, charge dispro-portionated state which our GGA + U calcula-tions find, the direct gap is between the two e↑gbands at ( 1

2 , 12 ) π

a , and is of size ≈ 0.7 eV. In the↓ channel, the gap is larger and is between thet2g and eg bands. Due to oxygen covalency, themagnetic moment is 1.5 rather than 2 µB on theexpanded Ni site, and 0.5 rather than 0 µB on thecontracted Ni site.

We thus predict that in order to produce metal-lic, and maybe even superconducting nickelateheterostructures, it is first of all essential thatthe NiO2 layers be single and, secondly, thatthey be separated by an insulator whose cationcounter to Ni interacts only weakly with apicaloxygen, e.g., to use scandates instead of alumi-nates.

References:[1] Chaloupka, J. and G. Khaliullin. Physical Review

Letters 100, 016404 (2008).

[2] Hansmann, P., X. Yang, A. Toschi, G. Khaliullin,O.K. Andersen and K. Held. Physical Review Letters103, 016401 (2009).

[3] Pavarini, E., I. Dasgupta, T. Saha-Dasgupta,O. Jepsen and O.K. Andersen. Physical ReviewLetters 87, 047003 (2001).

[4] Benckiser, E., M.W. Haverkort, S. Bruck, E. Goering,S. Macke, A. Frano, X. Yang, O.K. Andersen,G. Cristiani, H.-U. Habermeier, A.V. Boris,I. Zegkinoglou, P. Wochner, H.-J. Kim, V. Hinkov andB. Keimer. Nature Materials 10, 189–193 (2011).

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