Percolation via combined electrostatic and chemical doping in complex oxide films
Peter P. Orth, Rafael M. Fernandes, Jeff Walter, C. Leighton, B. I. Shklovskii
PPercolation via combined electrostatic and chemical doping in complex oxide films
Peter P. Orth, Rafael M. Fernandes, Jeff Walter, C. Leighton, and B. I. Shklovskii
1, 3 School of Physics and Astronomy, University of Minnesota, Minneapolis, Minnesota 55455, USA Department of Chemical Engineering and Materials Science,University of Minnesota, Minneapolis, MN 55455, USA Fine Theoretical Physics Institute, University of Minnesota, Minneapolis, MN 55455, USA (Dated: October 11, 2018)Stimulated by experimental advances in electrolyte gating methods, we investigate theoreticallypercolation in thin films of inhomogenous complex oxides, such as La − x Sr x CoO (LSCO), inducedby a combination of bulk chemical and surface electrostatic doping. Using numerical and analyt-ical methods, we identify two mechanisms that describe how bulk dopants reduce the amount ofelectrostatic surface charge required to reach percolation: (i) bulk-assisted surface percolation, and(ii) surface-assisted bulk percolation. We show that the critical surface charge strongly depends onthe film thickness when the film is close to the chemical percolation threshold. In particular, thinfilms can be driven across the percolation transition by modest surface charge densities via surface-assisted bulk percolation. If percolation is associated with the onset of ferromagnetism, as in LSCO,we further demonstrate that the presence of critical magnetic clusters extending from the film sur-face into the bulk results in considerable volume enhancement of the saturation magnetization, withpronounced experimental consequences. These results should significantly guide experimental workseeking to verify gate-induced percolation transitions in such materials. Introduction.–
The rapidly growing field of complex ox-ide heterostructures provides many opportunities for theobservation of new physical phenomena, with promisingapplications in future electronic devices [1–3]. Exam-ples include strain engineering to control structural andelectronic ground states [1–4], realization of novel two-dimensional (2D) electron gases at oxide interfaces [3, 5,6], and the observation of interfacial magnetic [1–3, 7] andsuperconducting states [1–3]. Due to the lower chargecarrier densities in these materials ( n (cid:39) cm − ) com-pared to conventional metals ( n (cid:39) cm − ), surfaceelectrostatic or electrochemical control of these novelproperties via the electric field effect also becomes anexciting possibility [2, 8–10].Stimulated by the above situation, high- κ dielectrics,ferroelectric gating, and electrolyte gating (primarilywith ionic liquids and gels) have been successfully em-ployed to electrostatically induce and control large chargedensities in these materials [2, 8–10]. Particularly promi-nent recent progress has been made with ionic liquidand gel gating, the surface carrier densities achieved rou-tinely exceeding s (cid:39) cm − , corresponding to mod-ulation of significant fractions of an electron (or hole)per unit cell [2, 8–10]. This has, for example, enabledreversible external electrical control of oxide electronicphase transitions from insulating to metallic [11–14], to asuperconducting state [15–17], or from paramagnetic tomagnetically-ordered phases [18, 19]. Nevertheless, at-tainment of sufficient charge density to induce the phasetransitions of interest remains a challenge in many cases,due to the need for s (cid:39) cm − . In such cases oneobvious strategy is to employ a combination of chemi-cal and electrostatic doping, bringing the material closeto some electronic/magnetic phase boundary by chemi-cal substitution, then using surface electrostatic tuning of the carrier density to reversibly traverse the criticalpoint.The work presented here focuses on exactly such com-bined electrostatic surface and bulk chemical doping. Inparticular, we investigate electronic/magnetic percola-tion transitions induced by a combination of chemical V g S - - - - - - - - + + +++ ++ + Ionic liquid/gel V SD Gate D Substrate Sample - + FIG. 1. Schematic setup showing a thin film sample (red) ofthickness ta and area la × la , where a is the lattice constant,with large finite clusters (blue) due to bulk doping. The ionicliquid or gel (light green) on top of the sample induces a num-ber of holes (blue spheres) at the top surface – proportional tothe applied gate voltage V g . Red spheres denote anions in theionic liquid/gel that move towards the surface due to the ap-plied voltage. For bulk doping close to percolation x c − x (cid:28) V SD . The highlighted upperleft cluster shows bulk bridges connecting two surface clus-ters, which is the dominant effect of bulk dopants for x (cid:28) x c (bulk-assisted surface percolation). a r X i v : . [ c ond - m a t . m e s - h a ll ] M a y and electrostatic doping. This is an important situationin complex oxide materials due to the widespread ob-servation of electronic and magnetic inhomogeneity (asin manganites [20], cuprates [21], and cobaltites [22, 23]for example), where many transitions, such as from in-sulator to metal or from short- to long-range magnetism,are percolative in nature. While our analysis and resultsare general, and could apply to percolation transitionsin various materials, in this paper we are motivated byphysics of the perovskite oxide cobaltite, La − x Sr x CoO (LSCO), which is well established to undergo a percola-tion transition from insulator to metal at x c (cid:39) .
18 [22–24].In the parent compound LaCoO ( x = 0), the Co (3 d ) ions adopt the S = 0 spin state as T →
0, andthe material is a diamagnetic semiconductor. Substitut-ing Sr for La induces holes, changing the formal va-lence state of a neighboring Co ion to 4+, which is ina S > ) and short- to long-range ferromagneticcorrelations is caused by percolation of nanoscopic ferro-magnetic hole-rich clusters [22–24]. Very thin (few unitcell thick) films of LSCO are the natural target for field-effect gating experiments, as significant modulation ofthe charge carrier density is confined to a narrow layerclose to the surface. The layer width is of the order ofthe electrostatic screening length, which is typically oneor two unit cells [2, 8–10] due to the large carrier densities( n (cid:39) cm − ) in significantly doped LSCO.The theoretical study of percolation phenomena in cor-related systems has a long history [25–33]. However, thecombination of bulk chemical and surface electrostaticdoping defines an interesting and unusual percolationproblem that is so far largely unexplored theoretically.The schematic setup with gate, source and drain elec-trodes is shown in Fig. 1, where the blue parts in theLSCO film denote hole-rich regions and the top surfaceis affected by electrostatic gating. The total (top) surfacecarrier density, s = x + ∆ s (1)arises from doping both by chemical substitution of afraction of lattice sites x and electrostatic gating of afraction of surface lattice sites ∆ s .In this work we identify two different percolation phe-nomena: bulk-assisted surface percolation and surface-assisted bulk percolation, which are schematically de-picted in Fig. 1. In the first case, where the system isinitially far away from the (thickness-dependent) bulkpercolation threshold x c ( t ), percolation on the surface isfacilitated by diluted bulk dopants, which provide bridgesthat connect disjunct finite surface clusters. As a result,the amount of surface charge ∆ s c that must be inducedelectrostatically to reach percolation is insensitive to the film thickness. In the second case, where the bulk chem-ical doping level is close to the percolation threshold, x c ( t ) − x (cid:28)
1, we find that small ∆ s helps to reach bulkpercolation by connecting large finite bulk clusters on thesurface. We show that the surface charge at percolationfollows s c ∝ t ( x c − x ) for films of thickness ta . As a re-sult, ∆ s c grows moderately with ( x c − x ) for thin films,but increases sharply for thicker films. In the particularcase where the percolation transition is associated withferromagnetic order, as in LSCO, the presence of clusters,which extend from the surface into the bulk, greatly en-hance the surface saturation magnetization M s . Numerical modeling of percolation.–
To derive our re-sults, we consider the site percolation problem on thecubic lattice of size la × la × ta along the X , Y and Z axes defined in Fig. 1, where a is the lattice constant and l, t are integers ( t ≤ l ). This geometry describes films ofthickness ta and surface area ( la ) . The percolation prob-lem is solved using the numerical algorithm described inRefs. 34 and 35. Starting from an empty lattice, a frac-tion x of sites are first randomly filled in the whole lat-tice to simulate bulk chemical doping. We verify that thebulk doping percolation threshold on the isotropic cubiclattice ( l = t ) lies at x Dc = 0 .
31 [36], and increases for t < l , i.e. , x c ( t ) > x c ( l ) ≡ x Dc [25]. To study the role ofsurface doping, we stop at a bulk doping level x < x c ( t )and subsequently add a fraction ∆ s of sites exclusivelyon the top surface layer to simulate electrostatic gating.The total surface density of sites at the top surface isthen given by Eq. (1). While electrostatically doping thesystem, we continuously monitor whether a percolatingpath exists between the two side surfaces at X = 0 and X = la . We define the critical total density of sites atthe top surface that is required for percolation betweenthe side surfaces as s c . The amount of charge densitythat must be transferred via electrostatic doping is thendenoted ∆ s c .In Fig. 2(a), we show numerical results for ∆ s c as afunction of the starting bulk chemical doping level x ;panel (b) shows s c as a function of x . For pure sur-face doping, x = 0, we find the percolation thresholdof the 2D square lattice, ∆ s c (0) = 0 .
59 [26]. For small x (cid:28) x c ( t ), the behavior of ∆ s c ( x ) depends only weaklyon the film thickness t . In contrast, for x c ( t ) − x (cid:28) s c ( x ) depends strongly on the thickness t ,displaying a sharp enhancement as x decreases for thickfilms but a much more gradual one for thin films. To un-derstand the numerical results, we next employ scalingtheory arguments [25]. Analytical theory.–
To develop an analytical theory,we focus on three limits: (i) x (cid:28) x c ( t ), (ii) x Dc − x (cid:28) x c ( t ) − x (cid:28)
1, which are indicated by yellowrectangles in Fig. 2(a). The first case can be describedas bulk-assisted surface percolation and the other two bysurface-assisted bulk percolation.(i) For x (cid:28) x c ( t ), we have s c (0) − s c ( x ) (cid:28)
1: the sys- (a)(b)
110 2 4 8 16 32
FIG. 2. (a) Surface charge density ∆ s c that must be elec-trostatically induced to reach percolation, as a function ofstarting bulk chemical doping level, x . Different curves cor-respond to different thicknesses t , as indicated, and are ob-tained from extrapolating results for system sizes l × l × t with l = 32 , ,
128 to l − → . × disorder realizations. The curve labelled“3D” is for t = l . The left inset shows that ∆ s c at thebulk percolation threshold x Dc = 0 .
31 obeys Eq. (4) (yel-low line) with c = 0 .
27. The right inset shows the slope of s c − x c = m t ( x c − x ) close to x c ( t ) − x (cid:28) c = 0 .
56. Yellow rectangles mark the three regimes la-belled (i), (ii), and (iii), addressed by our analytical theory.(b) Total surface charge at percolation, s c , as a function of x .The lines are fits of the numerical results according to Eq. (2)with b = 0 .
91 for t = 2 and b = 1 .
12 for t = 4 , , l . Theinset shows the thickness-dependent bulk percolation thresh-old x c ( t ) for purely chemical doping. The yellow line obeysEq. (5) with x Dc = 0 . ν = 0 .
88 [36] and c = 1 . tem is close to the 2D percolation threshold on the sur-face, but far from percolation in the bulk. As a result, thetypical size of bulk clusters is rather small. These smallbulk clusters can still assist percolation at the surface byproviding short bridges across missing links between dis-connected finite large surface clusters. This situation isshown schematically in Fig. 1. Since the smallest possi-ble bulk bridge consists of three sites below the surface, at x (cid:28) s c ( x ) = s c (0) − bx . (2)As shown in Fig. 2(b), this equation, with weakly t -dependent coefficient b , describes the numerical resultswell for s c (0) − s c ( x ) (cid:28)
1; for t = 2 it is even applicableover almost the full range of doping levels up to x c .(ii) In the regime of small x Dc − x (cid:28)
1, the 3D bulkis close to the percolation threshold, but the surface con-centration is far from the surface percolation threshold.Thus, while large critical finite clusters exist in the bulk,with a typical size of ξ ( x ) ∼ a ( x Dc − x ) − ν and correla-tion length exponent ν = 0 .
88 [26, 36], the largest surfaceclusters remain small.Let us first discuss the case of an infinite isotropic 3Dsystem, before considering finite thickness films. If siteswere randomly added in the bulk, an infinite cluster con-necting X = 0 and X = la , which looks like a network oflinks and nodes with typical separation ξ ( x ), would oc-cur after adding N = N ( x Dc − x ) l sites, with N (cid:39) ξ ( x ) below the surface, the number ofsites ∆ N = N ( x Dc − x ) l ξ ( x ) /a we have added to thislayer is sufficient to induce percolation along the layer.It is now plausible that instead of homogeneously dopingthe sliver of volume ( la ) ξ ( x ), we can reach percolationby adding all these sites to the surface plane only. For a3D system, this yields a critical surface density of s c ( x ) = x Dc + ∆ Nl = x Dc + c ( x Dc − x ) − ν , (3)with a non-universal constant c and x Dc − x (cid:28)
1. Wesee that since ν <
1, connecting bulk clusters on thesurface can be done by very small surface addition ∆ s at x Dc − x (cid:28)
1. Of course, the scaling behavior in Eq. (3)only holds as long as ( x Dc − x ) − ν (cid:28)
1. Since 1 − ν =0 . (cid:28) x close to x c . This explainsthe sharp enhancement of ∆ s c ( x ) observed in the 3Dnumerical results shown in Fig. 2.A finite thickness t of the film introduces anotherlength scale, which cuts off the scaling behavior of Eq. (3)as soon as the correlation length becomes larger than thefilm thickness. We will now show that for bulk dopinglevels such that ξ ( x ) ≥ ta , Eq. (3) is replaced by s c ( x ) = x Dc + c t − /ν , (4)with non-universal constant c . We numerically verifythis scaling behavior at x = x Dc as shown in the (left)inset of Fig. 2(a). To derive Eq. (4), we first notice thatthe bulk percolation threshold x c ( t ) of a film of thickness t is reached when an infinite bulk cluster with correla-tion length ξ [ x c ( t )] ≤ ta appears. From this it followsthat [25]: x c ( t ) = x Dc + c t − /ν , (5)with non-universal constant c = 1 .
21, which is inagreement with our numerical results shown in the in-set of Fig. 2(b). Therefore, to achieve percolation at x (cid:39) x Dc , a film with width t must acquire ∆ N = c (cid:0) x c ( t ) − x Dc (cid:1) tl = c l t − /ν filled sites, where c isa non-universal constant. As above, we assume that wecan reach the percolation threshold by bringing all thesesites into the surface plane by electrostatic gating, yield-ing Eq. (4). Note that Eq. (4) crosses over to Eq. (3) at ξ ( x ) = ta .(iii) We now investigate s c for x c ( t ) − x (cid:28)
1. Inthis regime, it holds that ξ ( x ) > ta since the correla-tion length at x c ( t ) fulfills ξ [ x c ( t )] = ta . In this case, wefind that ∆ N = ( x c ( t ) − x ) l t sites should be added tothe system in order to reach percolation, such that thecritical surface percolation threshold obeys s c ( x ) = x c ( t ) + c t ( x c ( t ) − x ) (6)with non-universal constant c . We demonstrate in the(right) inset of Fig. 2(a) that our numerical results followthis scaling relation of the slope m t = c t with c =0 .
56. Note that the scaling breaks down for the thinnestsystem, t = 2, which is instead described by Eq. (2) overthe full range of bulk doping levels x (see Fig. 2(b)).The key insight from the combined numerical and ana-lytical results is that bulk chemical doping largely reducesthe amount of electrostatic surface charge ∆ s c requiredto reach percolation (compared to the 2D value) in a re-gion of initial chemical doping levels x Dc < x < x c ( t ).In this regime, the critical surface charge s c scales withthe thickness according to Eq. (6) and therefore growsquickly for thicker films. The underlying physical phe-nomenon is that less surface charge must be transferredby electrostatic gating if percolation is induced by con-necting finite large bulk clusters on the surface ratherthan creating a percolating path that is confined to thesurface alone. The width of this region x c − x Dc ∝ t − /ν rapidly narrows for thicker films. For smaller x the dom-inant effect of the bulk dopants is to act as short bridgesbetween disconnected surface clusters. This reduces thenumber of surface sites that must be filled to reach perco-lation only slightly compared to the 2D case, as describedby Eq. (2). Enhanced surface magnetization.–
If the percolationtransition is associated with ferromagnetic ordering, asfor LSCO, the extension of the percolating cluster fromthe surface into the bulk leads to a dramatic volume en-hancement of the surface saturation magnetization M s in the case of surface-assisted bulk percolation (cases (ii)and (iii)). To capture this phenomenon, in Fig. 3 we showthe size ( i.e. number of sites), of the largest cluster N c (per surface area l ) as a function of electrostatic dop-ing ∆ s . Beyond the percolation threshold ∆ s > ∆ s c ( x ) FIG. 3. Surface density of the largest cluster in the system, N c /l , as a function of electrostatic doping ∆ s in a film ofthickness t = 16. Dots indicate percolation thresholds. Aftercrossing the percolation threshold the largest cluster densityis proportional to the surface saturation magnetization M s .The plot shows the large (volume) enhancement of M s , whichoccurs due to extension of the infinite cluster and its deadends deep into the bulk (see Fig. 1). this cluster percolates and its size is proportional to thesurface saturation magnetization M s ∝ N c /l . Differentcurves correspond to different starting chemical dopinglevels 0 . ≤ x/x c ( t ) ≤ .
976 and the film thickness is t = 16.For small doping levels, we observe regular surface per-colation at ∆ s c = 0 .
59 (the percolation threshold is indi-cated by the dot). The percolated path is almost entirelyconfined to the top surface layer and the magnetizationenhancement is absent: N c /l (cid:46)
1. On the other hand,if the system is initially doped closer to the (bulk) perco-lation threshold x c , the percolating cluster extends sig-nificantly into the bulk and we observe N c /l > x we consider. As the(fractal) dimension of this cluster exceeds d = 2, we findthat N c /l becomes as large as 4 for a film of thickness t = 16 (note that a fully magnetized film corresponds to N c /l = t ). This shows that although bulk doping doesnot assist greatly in reaching percolation, it does ulti-mately generate a much larger saturation magnetizationin the sample, because of the inclusion of preformed clus-ters of spin polarized sites (see also Fig. 1). In addition,we further predict an unusual depth profile of magnetiza-tion M s ( z ) as a function of distance z from the surface,which can be directly experimentally measured, for in-stance using polarized neutron reflectometry. It couldalso be indirectly inferred using perhaps x-ray magneticcircular dichroism (XMCD) or the magneto-optical Kerreffect (MOKE). Conclusions.–
Motivated by existing and ongoing ex-periments on complex oxide thin films, we have studieda new percolation problem, where bulk chemical dopingis combined with electrostatic doping of the surface. Wehave derived new analytical formulae describing universalscaling behavior of the electrostatic percolation thresh-old and explored the full crossover from bulk to surfacepercolation numerically. Experimental predictions thatfollow from our analysis are that: (i) the critical surfacecharge density at percolation s c depends only weakly onthe starting bulk doping level x , except in proximity tothe bulk percolation transition x Dc < x < x c ( t ) . Thecrossover from surface-assisted to bulk-assisted percola-tion occurs more abruptly for thicker films. Given limi-tations of ionic liquid/gel or ferroelectric gating, experi-mental validations of gate-induced percolation may thusrely in most cases on chemically doping close to the per-colation threshold. (ii) Once percolation is reached, thesaturation magnetization M s is largely enhanced due tothe presence of critical clusters extending deep into thebulk. (iii) The existence of ferromagnetic bulk clusterswill also be reflected in the dependence of the magneti-zation M s ( z ) on the distance z from the surface. Ourwork thus shows that “bulk” magnetic properties can becontrolled using “surface” electrostatic gating. We notethat while the percolation threshold x c is a non-universalquantity dependent on microscopic details such as the ge-ometry of the lattice, the scaling behavior of s c ( x ) thatwe derive is universal . Our results thus apply to LSCOand other experimental systems even though the perco-lation threshold in this material is not that of a simplecubic lattice x Dc (cid:39) .
31, but rather x Dc,
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