Determination of doped charge density in superconducting cuprates from NMR or stripes
DDetermination of doped charge density in superconductingcuprates from NMR or stripes
Manfred BucherPhysics Department, California State University, Fresno,Fresno, California 93740-8031 (Dated: December 3, 2020)
Abstract
Independent investigations of nuclear quadrupole resonance (NQR) and of stripes in high- T c cuprates find a small deviation of doped-hole density h from the doping level of La − x Sr x CuO .The value observed with NQR, x − h ≈ .
02, agrees closely with the density of itinerant holes, ˜ p ,responsible for suppression of 3D-AFM, as obtained from stripe incommensurability. The stripemodel’s assumption that doped holes in La − x Sr x CuO reside at oxygen sites, and that dopedelectrons in Ln − x Ce x CuO ( Ln = P r, N d ) reside at copper sites, is (to a large degree) con-firmed with NQR. The NQR finding of doped-hole probabilities in oxygen and copper orbitals of
HgBa CuO δ and other oxygen-enriched high- T c cuprates, P p (cid:39) P d (cid:39) /
2, as well as of oxygen-doped
Y Ba Cu O y , P p (cid:39) P d (cid:39) /
3, is interpreted with the stripe model in terms of excessoxygen atoms in the
CuO planes and CuO chains. a r X i v : . [ phy s i c s . g e n - ph ] D ec . INTRODUCTION Superconductivity in doped copper oxides of high transition temperature (high T c ) occursin the CuO planes within certain doping ranges. When doped with heterovalent metal, suchas hole-doped La − x Sr x CuO or electron-doped P r − x Ce x CuO , the doped-charge densityin the CuO planes is readily given by stoichiometry, p = x and n = x , respectively. Foroxygen-doped Y Ba Cu O y , on the other hand, p ( y ) cannot be inferred from stoichiome-try because of simultaneous filling of CuO chains. Instead, one needs to resort to indirectmethods. Most commonly used is the the “universal-dome method” which obtains p ( y ) fromexperimental data of T c ( y ) and T c,max . Because of its crucial role in superconductivity, in-dependent methods to determine the doped-charge density are valuable. Two such methodsare discussed and compared here—one using nuclear quadrupole resonance, the other onestripe incommensurability.Nuclear quadrupole resonance (NQR) arises from the interaction of the quadrupole mo-ment of nuclear charge distribution with the electric field gradient (EFG) at the nucleus.For nuclei of nonvanishing quadrupole moments, which holds for nuclear quantum numbers
I > , the method can provide information about the charge distribution in a compound.Particularly, with I ( Cu ) = and I ( O ) = , NQR can give the doped-charge density inthe CuO planes. This was shown in a seminal paper by Haase et al. for La − x Sr x CuO and Y Ba Cu O y a decade and a half ago, followed by the same treatment of oxygen-enrichedcuprates and of electron-doped Ln − x Ce x CuO δ ( Ln = P r, N d ) a decade later.
The doped-hole density p (and doped-electron density n ) of high- T c cuprates can alsobe determined from the incommensurability of charge-order and magnetization stripes, asshown in two recent papers. Shedding new light on the NQR results, here we point outcommonalities with the stripe analysis, but also aspects that have been overlooked in theearlier studies or went unexplained.
II. NQR DETERMINATION OF DOPED-HOLE DENSITY IN La − x Sr x CuO Haase et al. posit the NQR frequency at the oxygen nucleus, ν O , to be proportional tothe hole density n p of the O p orbital, ν O = Q O × n p + C O . (1)2he coefficient Q O is obtained from quantum-mechanical expressions of electric hyperfineinteraction for isolated oxygen ions. Based on electron hopping parameters, n p is determinedfor the undoped CuO plane as n p = 0 .
11. Together with the experimental value of ν O inthe parent crystal, this enables the determination of the material-specific constant C O . TheNQR frequency of the Cu nucleus depends linearly on the hole density n d of the Cu d x − y orbital, but also, by a cross-term, on neighboring O p orbitals, ν Cu = Q Cu × n d − Q OCu × (8 − n p ) + C Cu . (2)The coefficients Q Cu and Q OCu are obtained from quantum-mechanical expressions for isolatedions. With n d = 1 − n p = 0 .
78 for the undoped crystals, the constant C Cu can bedetermined. For the doped crystals, Haase et al. make the assumption h = n d + 2 n p − . (3)(In Ref. 2 the doped-hole density h is denoted as δ .) With experimental NQR frequencies ν Cu and ν O of La − x Sr x CuO , along with the calculated coefficients and constants, thequantities n d , n p and h can be determined. They are listed in Table I. The doped-holedensity h is close to the Sr -doping, h ≈ x , which attests to the predictive power of the NQRapproach. A closer look shows that h is systematically slightly lower than x , x − h = ∆ h (cid:39) . . (4)Is this an inaccuracy caused by the approximations involved or the result of some underlyingphysics? We shall return to this question.The hole density in the Cu d x − y orbital arises from the contribution of the parentcrystal and a doping-dependent term, n d = n d + P d h . (5)Here P d is the probability of a doped hole to reside at the Cu ion. The correspondingexpression for oxygen is n p = n p + 12 P p h , (6)where the factor 1/2 accounts for the two oxygen ions in the CuO plane of the unit cell.From the data in Table I, P d (cid:39) .
07 and P p (cid:39) .
93 is obtained. This shows that in La − x Sr x CuO the doped holes in the CuO plane reside almost entirely at the oxygensites—a conclusion that still holds in view of a small uncertainty of n p , as the constant C O in Eq. (1) can be determined only within bounds by the formalism of the NQR approach. ν Cu [MHz] ν O [MHz] n d n p h x − h ˜ p .
00 33 . .
147 0 .
780 0 .
110 0 .
00 0 . .
075 34 . .
18 0 .
784 0 .
137 0 .
058 0 .
017 0 . .
10 34 . .
195 0 .
785 0 .
149 0 .
084 0 .
016 0 . .
15 35 . .
215 0 .
794 0 .
166 0 .
125 0 .
025 0 . .
20 36 . .
245 0 .
797 0 .
190 0 .
190 0 .
023 0 . .
24 37 . .
28 0 .
798 0 .
219 0 .
236 0 .
004 0 . ν Cu and ν O , hole density n d and n p in Cu d x − y and O p c orbitals,respectively, and density h of doped holes in the CuO plane of La − x Sr x CuO , obtained fromNQR data (Ref. 2); difference from nominal Sr doping, x − h ; and density of itinerant holes, ˜ p ,from stripe incommensurability, Ref. 6. III. STRIPES IN La − x Sr x CuO The unit cell of pristine La CuO has a central CuO plane, sandwiched by LaO planes.Consider stepwise ionization, where brackets indicate electron localization at atoms, bothwithin the planes and by transfer from the
LaO planes to the
CuO plane: LaO : La + 3 e − + O → La + [2 e − + O ] + ↓ e − | → La + O − CuO : Cu + 2 e − + 2 O → Cu + [2 e − + O ] + O → O − + Cu + O − LaO : La + 3 e − + O → La + [2 e − + O ] + ↑ e − | → La + O − In the simplest case of doping , Sr substitutes, in some cells, La in both sandwiching planes: SrO : Sr + 2 e − + O → Sr + [2 e − + O ] → Sr + O − CuO : Cu + 2 e − + 2 O → Cu + [2 e − + O ] + O → O − + Cu + ˜O SrO : Sr + 2 e − + O → Sr + [2 e − + O ] → Sr + O − The lack of electron transfer from the sandwiching planes to the
CuO plane leaves someoxygen atoms neutral (marked bold above and below). Compared to the host crystal, theycan be regarded as housing the holes (pairwise). Up to a doping density ˜ p ≤ .
02, such holesare itinerant , enabling ˜O atoms to skirmish long-range antiferromagnetism (3D-AFM). Theremaining lack of electron transfer leaves more oxygen atoms stationary at lattice sites, O .They give rise to static stripes. 4 rO : Sr + 2 e − + O → Sr + [2 e − + O ] → Sr + O − CuO : Cu + 2 e − + 2 O → Cu + [2 e − + O ] + O → O − + Cu + O SrO : Sr + 2 e − + O → Sr + [2 e − + O ] → Sr + O − Relative to the host crystal, both the skirmishing and stationary oxygen atoms, ˜ O and O , appear positive, holding two elementary charges, +2 | e | , each. In this sense, the doping of La CuO with Sr is often called “hole doping” (more accurately, doping the CuO planeswith holes). Both the ˜ O and O atoms have a finite magnetic moment, m ( ˜ O ) = m ( O ) (cid:54) = 0,due to their spin quantum number S = 2 × = 1 according to the spin configuration [ ↑↓ ][ ↑ ] [ ↑ ] of their 2 p subshell (Hund’s rule of maximal multiplicity). The moments of theskirmishing oxygen atoms, m ( ˜ O )—itinerant via anion lattice sites—upset the 3D-AFM thehost [from m ( Cu ) moments] and cause its collapse at hole density ˜ p = 0 . O ions, gives rise to charge-order stripes. Their incom-mensurability, in reciprocal lattice units (r.l.u.), depends on Sr -doping x , q c ( x ) = Ω ± (cid:113) x − ˜ p , x ≤ ˆ x , (7)up to a “watershed” doping ˆ x . The stripe-orientation factor is Ω + = √ x > .
056 whenstripes are parallel to the a or b axis, but Ω − = 1 for x < x when stripes are diagonal. Theoffset value under the radical is ˜ p ≤ .
02. A qualitative change of the incommensurabilityoccurs at a watershed concentration of the dopant, ˆ x , which depends on the species ofdoping and co-doping. It shows up as kinks in the q c ( x ) profile at ˆ x (see Fig. 1), where thesquare-root curve from Eq.(7) levels off to constant plateaus, q c ( x ) = √ (cid:113) ˆ x − ˜ p , x > ˆ x . (8)The charge-order stripes are accompanied by magnetization stripes of incommensurability q m ( x ) = q c ( x ). The square-root dependence of q c ( x ) results from the spreading of the doubleholes by Coulomb repulsion to the farthest available separations. Thus a rising square-root profile of stripe incommensurability signifies an underlying superlattice of lattice-defectcharges (relative to the host crystal). Increasing density of the doped holes, housing pairwiseat lattice-defect O atoms in the CuO planes, raises their Coulomb repulsion energy. Whendoping exceeds a watershed value, x > ˆ x , additional holes overflow to the LaO planes wherethey also reside pairwise in O atoms. This leaves charge-order stripes of constant q c in the CuO planes, Eq. (8). 5 IG. 1. Incommensurability of charge-order stripes, q = q c , and of magnetization stripes, q = 2 q m , in La − z − x Ln z Ae x CuO ( Ln = N d, Eu ; z = 0 , . , .
2) due to doping with Ae = Sr or Ba . Circles show datafrom X-ray diffraction or neutron scattering. The broken solid curve is a graph of Eq. (7). The discontinuityat x = 0 .
056 is caused by a change of stripe orientation from diagonal to parallel, relative to the Cu - O bonds.Doping beyond watershed concentrations, ˆ x , yields constant stripe incommensurabilities, given by Eq. (8)(dashed horizontal lines). IV. COMMONALITIES IN THE STRIPE AND NQR STUDIES OF La − x Sr x CuO The stripe model assumes that the density of doped holes in the
CuO plane equals the Sr -doping, p = x . However, it distinguishes between itinerant holes of density ˜ p and station-ary holes of density x − ˜ p , located pairwise in the O atoms of the oxygen superlattice. Thedistinction between both kind of holes is inferred from the collapse of long-range antifer-romagnetism (3D-AFM) when Sr -doping reaches the N´eel concentration, x = x N = 0 . T N ( x N ) ≡
0, and from stripe incommensurability, Eq. (7).As Fig. 1 shows, a large host of data from neutron scattering, hard X-ray diffraction, andresonant soft X-ray scattering is well described by Eq. (7). For low temperatures and6ow doping ( x < . p in Eq. (7) agrees with the N´eel concentration,˜ p = x N = 0 .
02. With more Sr doping, but still at T ≈
0, it is found that a smaller value,˜ p < x N , suffices to keep 3D-AFM suppressed. Thus the use of ˜ p = 0 .
02 in Eq. (7) becomesinaccurate beyond the low doping range, x > .
09, as it gives too small a value for theincommensurabilty q c,m ( x ). This can be seen in Fig. 1 where in that range most data pointscluster slightly above the drawn q ( x ) curve. Use of a diminished offset value, ˜ p (cid:39) .
015 inthis range (confirmed by recent measurements, as discussed in Ref. 6) shifts that section ofthe curve slightly upward to better agreement with experiment (not shown).Returning to the NQR method, the close agreement of the doped hole density h with Sr -doping of La − x Sr x CuO , h ≈ x , confirms the validity of the approach. Even better is thesystematic slight deviation ∆ h , Eq. (4), to which no significance may have been attributedpreviously. As only stationary holes would contribute, via EFG, to NQR signals, the NQRmethod should detect a doped-hole density x − ˜ p . This strongly suggests the identification∆ h = ˜ p , relating to itinerant holes. Residing pairwise in skirmishing ˜ O atoms, they lead tosuppression of 3D-AFM when x = x N = 0 .
02 and keep 3D-AFM suppressed for x > x N .The values of x − h and ˜ p in Table I scatter somewhat about x N = 0 .
02, possibly due tounderlying approximations. (The x − h value for x = 0 . assumes an average hole density h in the CuO plane, Eq. 3,it cannot discriminate whether the doped holes reside singularly in O − ions or doubly in O atoms. The finding from stripe analysis of double holes in O atoms awaits confirmation (orrefutation) by other experiments. V. NQR AND STRIPES IN n-DOPED Ln − x Ce x CuO , Ln = Pr , Nd The unit cell of pristine Ln CuO ( Ln = P r, N d ) has a central
CuO plane, sandwichedby LnO planes, analogous to La CuO . Consider again crystal formation by stepwise ion-ization, where brackets indicate electron localization at atoms, both within the planes andby transfer from the LnO planes to the
CuO plane: LnO : Ln + 3 e − + O → Ln + [2 e − + O ] + ↓ e − | → Ln + O − CuO : Cu + 2 e − + 2 O → Cu + [2 e − + O ] + O → O − + Cu + O − LnO : Ln + 3 e − + O → Ln + [2 e − + O ] + ↑ e − | → Ln + O −
7n the simplest case of doping , Ce substitutes, in some cells, Ln in both sandwiching planes: CeO : Ce + 4 e − + O → | e − ↓ + Ce + [2 e − + O ] + ↓ e − | → Ce + O − CuO : Cu + 2 e − + 2 O → Cu + [2 e − + O ] + O → Cu + 2 O − CeO : Ce + 4 e − + O → | e − ↑ + Ce + [2 e − + O ] + ↑ e − | → Ce + O − Doping lanthanide-based cuprates with cerium, Ln − x Ce x CuO , partially substitutes Ln by Ce , resulting in electron doping of the CuO plane. As there are no itinerant dopedelectrons in the CuO plane, the doped-electron density equals the Ce -doping, n = x . (9)The doped electrons reside pairwise in lattice-site Cu atoms. Relative to the host crystal,the Cu atoms have a lattice-defect charge of − | e | . Coulomb repulsion spreads the Cu atoms to form a Cu superlattice. Its charge-order incommensurability is given by Eq. (7),with ˜ p = ˜ n = 0 (see Fig. 2).A consequence of smaller ionic radius, r ( P r , N d ) (cid:39) .
26 ˚A, compared to r ( La ) =1 .
30 ˚A, the Ln − x Ce x CuO compounds have the T (cid:48) crystal structure. It differs from the T structure of La − x Sr x CuO by the position of the O − ions in the layers that bracket the FIG. 2. Observed incommensurability q c of charge-order stipes in Ln − x Ce x CuO y , Ln = N d, La (seeRef. 6). The curves are graphs of Eq. (7) without excess oxygen ( δ = 0, dashed line) and with excess oxygenof δ = 0 .
01 (solid line). uO plane. In the T structure those O − ions are at apical positions (above or beneaththe Cu ions) whereas in the T (cid:48) structure they reside above or beneath O − ions of the CuO plane. As a result, the unit cell of the parent crystal Ln CuO is wider and shorter( a (cid:39) .
95 ˚A, c (cid:39) .
19 ˚A) than of La CuO ( a = 3 .
81 ˚A, c = 13 . For reasons of stability, Ln − x Ce x CuO crystals need to be grown with excess oxygen, O δ ,to be eliminated in post-growth annealing. As Eq. (7) is based on Coulomb repulsion of likecharges in the
CuO planes, its success for stripes in electron-doped ‘214’ compounds (of T (cid:48) structure) implies that the excess oxygen must reside interstitially as oxygen ions , O − δ , in orbetween the LnO planes. Any residual excess oxygen results in a hole -doping contribution tothe
CuO planes that correspondingly reduces their electron-doping from Ce . Accordingly,the charge-order incommensurability of Ln − x Ce x CuO δ can be expressed with Eq. (7)using ˜ p = 2 δ . The NQR investigations of Ln − x Ce x CuO yield doped-electron densities very close to the Ce -doping, approximating Eq. (9). It is found that doping with electrons predominantlydecreases n d but only slightly n p . In other words, the doped electrons in Ln − x Ce x CuO reside almost entirely at Cu atoms. The copper signal ν Cu is very sensitive to Ce -doping,having a wide background of broadened satellites such that the signal effectively disappearsat x = 0 .
15. For oxygen, the NQR analysis obtains a hole content n p (cid:39) .
20 in the parent Ln CuO compounds, compared to n p (cid:39) .
10 in La CuO .A comparative interpretation of the NQR results from hole-doped and electron-dopedcuprates is complicated by several factors: (i) the EFG at a copper or oxygen nucleusfrom a hole in its own electron shell (here called “self EFG”); (ii) the EFG from differentlycharged nearest neighbors in the CuO plane (abreviated “nn EFG”); (iii) the presence orabsence of O − ions above and beneath Cu ions—called “apical” oxygen ions—and of O − above and beneath O − ions in the CuO plane according to the T or T (cid:48) structure; (iv)hybridization of Cu d x − y and O p σ orbitals leading to covalent bond, (v) deformation ofelectron shells in the crystal, and (vi) higher-order effects.Using the finding from the stripe model that doped holes and electrons are hosted pairwise in O and Cu atoms, respectively, the following diagrams provide a qualitative overview ofpossible cases with combinations of the influences (i), (ii) and (iii). The top diagrams showplanar configurations of the copper sites and the bottom diagrams those of the oxygen sites.The middle panels give the configurations in the parent crystals and the outer panels those9f the doped sites. The largest effect on NQR can be expected from self EFG, noted beneaththe corresponding diagrams. (No self EFG exists at the Cu nucleus due to the spherical4 d s orbitals of the atom.) Noticeable effects can also be expected from planar nn EFGand from the presence or absence of nearest neighbors in the sandwiching layers according tothe compounds’ T or T (cid:48) structure. Here the absence of O − above and beneath Cu + and Cu in P r CuO and P r − x Ce x CuO , respectively, is marked bold in the diagrams ( b, a ).Likewise, the absence of O − ions above and beneath O − and, respectively, O in the CuO plane of La CuO is marked bold in ( g, h ). In all cases the effects of undoped and dopedsites are different. Thus it is not surprising that the corresponding NQR frequencies aredifferent. This renders a comparison difficult.. electron doped parent ( b, f ) parent( c, g ) hole doped .. P r − x Ce x CuO P r CuO La CuO La − x Sr x CuO ..( a ) O − ( b ) O − ( c ) O − ( d ) O − .. O − Cu O − O − Cu + O − O − Cu O − O − Cu O . O − O − O − O − . self EFG self EFG self EFG. + nn EFG.( e ) Cu ( f ) Cu ( g ) Cu ( h ) Cu .. Cu O − Cu Cu O − Cu Cu O − Cu Cu O Cu . Cu Cu Cu Cu . nn EFG self EFG VI. NQR DETERMINATION OF DOPED-HOLE DENSITY IN YBa Cu O + y Using the same procedure as for La − x Sr x CuO , Haase et al. obtain for Y Ba Cu O y the doped-hole densities h listed in Table II. The results are comparable with values from theuniversal-dome method and from stripe incommensurability. This confirms again the valid-ity of the NQR approach to determine h . However, in striking contrast to the heterovalent-metal-doped compounds, La − x Sr x CuO and Ln − x Ce x CuO ( Ln = P r, N d ), where thedoped holes and electrons were found to reside almost entirely at oxygen and, respectively,copper ions, distinctly different orbital occupancies are found with NQR for oxygen-doped10
ABLE II: Doped-hole density h in Y Ba Cu O y , obtained from NQR data for copper d andoxygen p c orbitals, from stripe incommensurability, and from the universal dome method. y h ← NQR h ← Stripes h ← Universal Dome0 .
00 0 . .
31 0 .
00 0 . .
45 0 .
07 0 . .
50 0 .
00 0 .
09 0 . .
60 0 .
10 0 .
12 0 . .
63 0 .
11 0 .
13 0 . .
68 0 .
14 0 .
14 0 . .
75 0 .
16 0 . .
92 0 .
18 0 . .
96 0 .
16 0 . and oxygen-enriched compounds. With Eqs. (5, 6) one can obtain the orbital probabilities P d and P p , according to which a doped hole resides at a copper or oxygen ion in the CuO plane, by the slope, s = n d − n d n p − n p = P d P p , (10)of orbital-occupation data in an n d vs. n p display. Together with P d + P p ≡
1, this gives P d = ss + 1 and P p = 1 s + 1 . (11)In Fig. 5 of Ref. 4 the n d vs. n p values fall along straight lines for compounds families.Their slopes are listed in Table III along with a doped hole’s orbital-occupation probabilities.In oxygen-doped Y Ba Cu O y and Y Ba Cu O , P d ≈ and P p ≈ is found. In oxygen-enriched Hg -, Bi -, and T l -based cuprates, they are essentially equal, P d (cid:39) P p (cid:39) . Howcan these findings be understood and what do they tell us? VII. STRIPES IN YBa Cu O + y The unit cell of
Y Ba Cu O y has two CuO planes, separated by the Y plane andbracketed by BaO layers. At the top (and bottom) of each unit cell of undoped
Y Ba Cu O ABLE III: Slopes s of orbital-occupation data from Ref. 4, Fig. 5, and occupation probabilities P d and P p of doped copper d and oxygen p σ orbitals for heterovalent-metal-doped cuprates (upperpart) and oxygen-doped/enriched cuprates (lower part).Compound s P d P p La − x Sr x CuO .
14 0 .
12 0 . P r − x Ce x CuO .
82 0 .
83 0 . N d − x Ce x CuO .
50 0 .
87 0 . Y Ba Cu O y , Y Ba Cu O .
55 0 .
36 0 . HgBa CuO δ , Bi Ba CaCu O δ , .
96 0 .
49 0 . T l Ba CuO δ , T l Ba CaCu O δ , T l Ba Ca Cu O δ .
96 0 .
49 0 . is a plane of Cu + ions (called the “ Cu (1) plane”). In order to introduce the effects ofdoping we temporarily make a simplifying assumption (here called the “0.5-watershed”),that we’ll later drop when more familiar. By this approximation, oxygen doping y ≤ . Cu (1) plane, O + 2 Cu + → Cu + O − , with the O − ions residingbetween Cu ions along the crystal’s b direction, called CuO chains. The
CuO planesremain unaffected by this doping. Accordingly no charge-order stripes are observed for y ≤ .
5. The filling of Cu O − chains is completed at doping y = 0 .
5, as can be seen bystoichiometry and charge balance of Y Ba Cu O − . . The Cu (1) plane of Y Ba Cu O . alternates with filled and empty CuO chains (called “ortho-II oxygen ordered”).By the 0.5-watershed approximation, doping y > . O atoms either in more CuO chains of the Cu (1) plane or in the CuO ≡ Cu (2) planes, y = χ Cu (1) ( y ) + 2 δ Cu (2) ( y ) . (12)It is the density of embedded oxygen, δ Cu (2) ≡ δ , in each CuO plane that gives rise tocharge-order stripes. Their incommensurability [averaged over stripes along the a and b direction, q c = ( q ac + q bc ) / q c ( y ) = q c ( y onsc ) − γ × [ δ ( y ) − δ ( y onsc )] . (13)Here the superscript ‘ ons ’ indicates the onset of charge-order ( c ) stripes. The onset incom-mensurability is material-specific: q c ( y onsc ) = 0 .
33 for
Y Ba Cu O y . Both the coefficient γ = 0 . δ ( y onsc ) = 0 .
035 hold for the whole family of oxygen-12
IG. 3. Incommensurability of charge-order stripes (solid line) and magnetization stripes (hatched line) in La − z − x Ln z Ae x CuO of the ‘214’ family due to Ae doping with Ae = Sr or Ba (curves on the left, equivalentto Fig. 1) and in oxygen-doped Y Ba Cu O y near the top (and near the bottom, if insufficiently annealed). Figure 3 shows the observed incommensurabilties of charge-orderstripes ( q ac and q bc ), in Y Ba Cu O y (and q am of magnetization stripes if the crystals arenot sufficiently annealed—not discussed here). For comparison, the (re-scaled) result fromFig. 1 is included in Fig. 3 on the left. The doping dependence of charge-order stripesis distinctly different in the alkaline-earth-doped cuprates, La − x Ae x CuO ( Ae = Sr, Ba ),and in the oxygen-doped cuprate,
Y Ba Cu O y , showing a square-root dependence ofthe former, Eq. (7), but almost constant values of the latter, Eq. (13). The square-root dependence results from Coulomb repulsion of the doped holes in the CuO plane(residing pairwise in lattice-site O atoms) to the largest separation, causing formation of an O superlattice. Conversely, the almost constant q c ( y ) values of Y Ba Cu O y result fromthe implementation of neutral O atoms in the CuO planes. (The slight decrease of q c ( y )with increasing O -doping y is caused by secondary effects. ) The distinctly different dopingdependence of charge-order incommensurability disproves the common misconception thatdoped oxygen ionizes in Y Ba Cu O y as O y → O − +2 e + and separates—with O − enteringthe so-called charge-reservoir layer (that is, all planes except CuO ) and the holes, 2 e + ,entering the CuO planes. If this were true, then a rising square-root doping dependence ofcharge-order stripes in Y Ba Cu O y would result, contrary to the observed q c ( y ) ≈ const . It has been proposed that each doped oxygen atom is embedded in the CuO plane atan interstitial site between four O − ions, called “pore” (see Fig. 4). The oxygen atomat the pore site, denoted ˚ O , bonds with two O − neighbors (abbreviated as ¨ O ≡ O − ) toform an ozone molecule ion, ¨ O ˚ O ¨ O . Linked by an intermediate oxygen ion each (markedbold, ¨O ), the ozone molecules line up along the crystal’s a or b direction to form trainsof ¨ O ˚ O ¨ O ¨O motifs that are observed as charge-order stripes in the CuO planes of oxygen-doped Y Ba Cu O y and Y Ba Cu O , as well as of oxygen-enriched, HgBaCuO δ , andthe Bi - and T l -based cuprates. The distinction of oxygen- doping y , and oxygen- enrichment δ , signifies the stoichiometric exactitude of y , in contrast to the uncertainty of δ due todiffusive procedures and thermal treatment. For Y Ba Cu O y , the doping certainty is oflittle help, however, when it comes to the oxygen content of the CuO planes as the dopedoxygen is shared to an uncertain degree with the CuO chains, Eq. (12).The “0.5-watershed” simplification, employed for ease of introduction, holds approxi-mately , but not strictly: To a small degree, O atoms are already embedded in the CuO planes for y < . IG. 4. Doped oxygen in the
CuO plane of Y Ba Cu O y ( y (cid:39) . O − ions (denoted ¨ O for short), the embedded oxygen atom (denoted ˚ O )bonds with two O − neighbors, here in the a -direction, to form an ozone molecule ion, ¨ O ˚ O ¨ O (turquoisecolor). Linked by an intermediate oxygen ion each (marked bold, ¨O ), the ozone molecules line up to formtrains of ¨ O ˚ O ¨ O ¨O motifs. The length of a motif is L (cid:39) . a (red double arrow), slightly larger than 3 a (black double arrow). Its reciprocal gives the incommensurabiliy of charge-order stripes, q c = 1 /L (cid:39) . . < y < .
5, and low- T c superconductivity in 0 . < y < .
5. Accordingly, some
CuO chain filling with O − ions continues for y > . O atoms. VIII. COMPARISON OF STRIPE AND NQR STUDIES OF HIGH-T c CUPRATES
The relevant properties of heterovalent-metal-doped and oxygen-enriched cuprates aregiven in Table IV. The hole-doped example of La − x Sr x CuO suffices for the compari-son, as electron-doped cases correspond equivalently (see table caption). In order to avoidcomplications from CuO chain filling in oxygen-doped
Y Ba Cu O y , we compare withoxygen-enriched HgBa CuO δ , where all excess oxygen is embedded in the CuO plane.The outstanding distinction is a hole-doping of the CuO plane of La − x Sr x CuO ,with holes residing pairwise at lattice-site O atoms, but of neutral O -atom-embedding in HgBa CuO δ , located at interstial positions (“pores”). Strictly speaking, calling the Property hole-doped oxygen-enrichedExample compound La − x Sr x CuO HgBa CuO δ Doping/enrichment of the crystal with
Sr O causes doping of the
CuO plane with e + ˚ O Net charge accumulation in
CuO plane? yes noFrom stripe analysis: 2 e + + O − → O O → ˚ O Location of doping-affected oxygen at lattice site at interstitial positionDefect charge of doping-affected oxygen Q = 2 | e | ˚ Q = 0Coulomb repulsion of Q’s? yes noStripe incommensurability q c ( x ) ∝ √ x − ˜ p q c ( δ ) ≈ const. NQR: doped orbital probability P d ≈ , P p ≈ P d (cid:39) P p (cid:39) NQR interpretation: doped holes → latt.-site oxy. ?TABLE IV: Comparison of hole-doped and oxygen-enriched cuprates. Electron-doped cases cor-respond equivalently to hole-doping with replacements La − x Sr x CuO → P r − x Ce x CuO ; Sr → Ce ; e + → e − ; 2 e + + O − → O ⇒ e − + Cu → Cu ; Q → − | e | ; P d → , P p → ; dopedelectrons → lattice-site copper. O δ , in the CuO plane “hole doping” is a misnomer—“oxygen-atom doping” would be a more accurate (although awkward) term. Despite their commonlack of two electrons from a closed 2 p shell, lattice-site O atoms and interstitial ˚ O atomshave different properties due to their different position in the crystal. In La − x Sr x CuO ,Coulomb repulsion spreads the effective defect-charges of the lattice-site oxygen, Q = 2 | e | ,to an O superlattice which gives rise to charge-order stripes. Their incommensurabilityhas a rising square-root dependence on Sr doping x . In contrast, the excess ˚ O atoms in HgBa CuO δ , lacking a net defect charge, ˚ Q = 0, are not spread by Coulomb force butaggregate to ¨ O ˚ O ¨ O ¨O trains. They give rise to charge-order stripes with with almost constantincommensurability. The doped orbital probabilities P d,p from NQR confirm that the dopedholes in La − x Sr x CuO reside almost entirely at lattice-site oxygen. Likewise, the dopedelectrons in P r − x Ce x CuO are found by NQR to reside almost entirely at lattice-site copper.This brings us to the doped-orbital probabilities in HgBa CuO δ , P d (cid:39) P p (cid:39) .
5. TheNQR formalism by Haase et al. assigns doped holes to lattice-site ions, Eqs. (1, 2), butmakes no allowance for doped O atoms at interstitial sites of the CuO plane. The followingdiagram shows a doped oxygen atom at the pore position, ˚ O , and its planar environment.. O − O − O − O − . Cu O − Cu + O − Cu + O − Cu . O − O − ˚ O O − O − . Cu O − Cu + O − Cu + O − Cu . O − O − O − O − The asymmetric positions of its four nearest neighbor O − ions and four next-nearest neigh-bor Cu ions (all marked bold) give rise to EFG’s at their nuclei and thus to NQR contri-butions. However, oppositely to hole-doped La − x Sr x CuO , where the changes of NQR arecaused by a change of hole density n p and n d in O − and Cu orbitals at given EFG’s, Eqs.(6, 5), the changes of NQR in the orbitals of each ˚ O atom’s neighbor ions, O − and Cu ,are caused by changes of the EFG at essentially constant n p (cid:39) n p and n d (cid:39) n d values.There are four such O − and Cu neighbors to each ˚ O atom with comparable distancesand asymmetries. This necessitates a change of interpretation: In HgBa CuO δ it is nota “doped-hole” density in the O − and Cu neighbors of ˚ O , but the changed EFG at theirnuclei, that makes the finding P p (cid:39) P d (cid:39) . O atom, representing its open 2 p shell, and contributes to NQR with its self-EFG.In Y Ba Cu O y , doped oxygen is incorporated (for y > .
5) in both the
CuO planesand the CuO chains, Eq. (12). The NQR contributions from ˚ O atoms and their neighborsin the CuO planes are as already illustrated in the above diagram. An example for con-tributions from the CuO chains, the diagram below gives an ion arrangement in the Cu (1)plane as it may occur for ortho-III oxygen ordered ( y ≈ . a and b direction. The nearest-neighbor configuration of the O − ions and O atoms ismuch more asymmetric than that of the Cu ions. Accordingly, much larger EFG’s andNQR contributions can be expected from oxygen sites than from copper sites. Added to theessentially equal contributions from oxygen and copper sites in the CuO planes, the findingof P p = 0 .
64 and P d = 0 .
36 for
Y Ba Cu O y (Table III) becomes qualitatively plausible.. Cu Cu Cu . O − O . Cu Cu Cu . O − O . Cu Cu Cu . b ↑ O − O . → a Cu Cu Cu IX. CONCLUSION
NQR investigations and stripe analysis of high- T c cuprates complement each other: (i)The small deviation of doped-hole density h from the doping level of La − x Sr x CuO , asobserved with NQR, ∆ h = x − h ≈ .
02, agrees closely with the density of itinerant holes,˜ p , responsible for suppression of 3D-AFM, as obtained from stripe incommensurability. (ii)The residence of doped holes at oxygen sites in La − x Sr x CuO and of doped electrons atcopper sites in Ln − x Ce x CuO ( Ln = P r, N d ), as assumed in the stripe model, is (to a largedegree) confirmed by the NQR studies. (iii) The NQR finding of doped-hole probabilities inoxygen and copper orbitals of oxygen-enriched high- T c cuprates, P p (cid:39) P d (cid:39) , as well as ofoxygen-doped Y Ba Cu O y , P p (cid:39) P d (cid:39) , is interpreted with the stripe model in terms18f excess oxygen atoms in the CuO planes and the CuO chains. M. R. Presland, J. L. Tallon, R. G. Buckley, R. S. Liu, and N. E. Flower, Physica C , 95(1991). J. Haase, O. P. Sushkov, P. Horsch, and G. V. M. Williams, Phys. Rev. B , 094504 (2004). D. Rybicki, M. Jurkutat, S. Reichardt, J. Haase, “Phase diagram of hole-doped cuprates basedon O and Cu NMR quadrupole splittings,” arXiv:1402.4014 M. Jurkutat, D. Rybicki, O. P. Sushkov, G. V. M. Williams, A. Erb and J. Haase, Phys. Rev.B , 140504(R) (2014). M. Jurkutat, J. Haase, and A. Erb, J. Supercond. Nov. Magn. , 2685 (2013). M. Bucher, “Stripes in heterovalent-metal doped cuprates,” arXiv:2002.12116v7 M. Bucher, “Stripes in oxygen-enriched cuprates,” arXiv:2010.06388v2 N. P. Armitage, P. Fournier, and R. L. Greene, Rev. Mod. Phys. , 2421 (2010). Y. Ohta, W. Koshibae, and S. Maekawa, Phys. Soc. Japan , 2198 (1992). E. P. Stoll, P. F. Meier, and T. A. Claxton, Phys. Rev. B , 065432 (2002). C. Bersier, S. Renold, E. P. Stoll, and P. F. Meier, J. Phys.: Condens. Matter , 7481 (2006)., 7481 (2006).