Doping a correlated band insulator: A new route to half metallic behaviour
aa r X i v : . [ c ond - m a t . s t r- e l ] J u l Doping a correlated band insulator: A new route to half metallic behaviour
Arti Garg , H. R. Krishnamurthy and Mohit Randeria Theoretical Condensed Matter Physics Division,Saha Institute of Nuclear Physics, 1/AF Bidhannagar, Kolkata 700 064, India Centre for Condensed Matter Theory, Department of Physics, Indian Institute of Science,Bangalore 560 012, and JNCASR, Jakkur, Bangalore 560 064, India Department of Physics, The Ohio State University, Columbus, OH 43210,USA
We demonstrate in a simple model the surprising result that turning on an on-site Coulombinteraction U in a doped band insulator leads to the formation of a half-metallic state. In theundoped system, we show that increasing U leads to a first order transition between a paramagnetic,band insulator and an antiferomagnetic Mott insulator at a finite value U AF . Upon doping, thesystem exhibits half metallic ferrimagnetism over a wide range of doping and interaction strengthson either side of U AF . Our results, based on dynamical mean field theory, suggest a novel route tohalf-metallic behavior and provide motivation for experiments on new materials for spintronics. PACS numbers: 71.10.Fd, 71.30.+h, 71.27.+a, 71.10.Hf, 75.10.Lp
Turning on strong electron correlations in a normalmetallic system is generally believed to result in interest-ing phases like anti-ferromagnetic Mott Insulator, high T c superconductor, pseudogap phase and non fermi-liquidphases. But the effect of e-e interactions in a band in-sulator have not been explored in detail so far. In thispaper we study effects of onsite interaction U on a bandinsulator and present a novel interaction driven route tohalf-metal phase . We show that doping a correlated bandinsulator results in the formation of half-metallic (HM)ferrimagnet. HMs are an interesting class of materialsin which electrons with one spin direction behave as ina metal and electrons with the opposite spin directionbehave as in an insulator, and have applications in spin-tronics as they can generate spin-polarized currents [1].Specifically, in this paper we study a simple tight-binding model with two bands, arising from a staggeredpotential ∆ on the sites of a bipartite lattice, in the pres-ence of an on-site Coulomb repulsion, the Hubbard U .At half filling, when one band is filled and the other isempty, this system is a paramagnetic band insulator (BI).When U is turned on, as we show, an anti-ferromagnetic(AFM) order sets in with a first order phase transitionat some threshold U = U AF . Upon doping, this systembecomes a ferrimagnetic HM over a range of doping and U values.Intuitively the formation of a HM upon doping can beunderstood as follows. Due to the staggered potential,the band gaps for the two spin components in the anti-ferromagnetic insulator (AFMI) phase are different, e.g. Eg ↓ < Eg ↑ . On hole doping, in a rigid band picture, onewould expect that it will be energetically favourable toput all the holes in the down spin band. This will makethe down spin band conducting while the up-spin will re-main insulating, resulting in a HM phase. Similarly for U < U AF , consider the BI in presence of a small stag-gered magnetic field h → U /t hole density x FerrimagneticHalf Metal
Paramagnetic MetalPM Metal U AF U U ∆ =1.0tBIAFMI FIG. 1: Zero temperature phase diagram of the model inEq. (1) obtained within DMFT for the Bethe lattice of infiniteconnectivity. Circles show the data points and the lines areguides to the eye. At half filling, the system is a band insulatorat weak U and becomes an AFMI with a first order phasetransition at U AF . Upon doping, it becomes a ferrimagnetichalf-metal (HM) over a large range of doping and U values. single particle excitation spectrum of the two spin compo-nents is different, being Eg ↑ = Eg + h and Eg ↓ = Eg − h .Following the argument mentioned above, doping this BIwith holes of density x will result in the formation of aHM with a net moment ∼ x which, due to a molecularfield arising from U, will self consistently cause the stag-gered magnetisation to be non zero. The simple rigidband picture used in this argument will hold only forsmall doping and weak coupling regime. The question ofinterest is whether the HM phase exists on finite amountof doping at intermediate and large U values. In this pa-per we show that it is possible to get the HM phase fora finite doping over a range of U values as shown in thephase diagram of Fig. 1. In the HM phase, there occurs afull redistribution of the degrees of freedom as comparedto the half filled case as shown in detail in later sections.The HM phase survives for the widest range of doping inthe intermediate coupling regime where U ∼ H = − t X i ∈ A,j ∈ B,σ [ c † iσ c jσ + h.c ] + ∆ X i ∈ A n i − ∆ X i ∈ B n i + U X i n i ↑ n i ↓ − µ X i n i (1)where t is the nearest neighbor hopping, U the Hubbardrepulsion and ∆ a staggered one-body potential whichdoubles the unit cell. The chemical potential µ is fixedso that the average occupancy is ( h n A i + h n B i ) / n =1 − x . The Hamiltonian (1) is sometimes called the “ionicHubbard model” (IHM) with ∆ the “ionic” potential.In an earlier work, using a dynamical mean field theory(DMFT) approach employing iterated perturbation the-ory (IPT) as the impurity solver, we studied this modelat half filling in the spin-symmetric case and showed howstrong correlations dynamically close the gap in a bandinsulator resulting in an intermediate metallic phase [2].This result was reproduced qualitatively by many othergroups [3]. Here we study this model using the sameDMFT approach in the spin asymmetric case [4]. Oncewe allow for the magnetic order, the system at half-fillingundergoes a first order phase transition from a PM BI toAFMI (see Fig. 3). In the spin asymmetric phase thereare some earlier works at half filling [5] and for the dopedcase [6] but to the best of our knowledge the existence ofa HM phase has not been suggested so far.The DMFT approximation is exact in the limit oflarge dimensionality [7, 8] and has been demonstrated tobe quite successful in understanding the metal-insulatortransition [7, 8] in the usual Hubbard model, which isthe ∆ = 0 limit of eq. (1). We focus in this paper onthe anti-ferromagnetic sector of eq. (1), for which it isconvenient to introduce the matrix Green’s functionˆ G σαβ ( k , iω n ) = (cid:18) ζ Aσ ( k , iω n ) − ǫ k − ǫ k ζ Bσ ( k , iω n ) (cid:19) − (2)where α, β are sub-lattice ( A, B ) indices, σ is the spin in-dex, k belongs to the first Brillouin Zone (BZ) of onesub-lattice , iω n = (2 n + 1) πT and T is the tempera-ture. The kinetic energy is described by the dispersion ǫ k and ζ A ( B ) σ ≡ iω n ∓ ∆ + µ − Σ A ( B ) σ ( iω n ). Within theDMFT approach the self energy is purely local [7]. Thusthe diagonal self-energies Σ ασ ( iω n ) are k -independentand the off-diagonal self-energies vanish (since the latterwould couple the A and B sub-lattices). The DMFT ap-proach includes local quantum fluctuations by mapping[7, 8] the lattice problem onto a single-site or “impu-rity” with local interaction U hybridizing with a self-consistently determined bath as follows. (i) We start with a guess for Σ ασ ( iω n ), n ασ , and compute the local G ασ ( iω n ) = P k G αασ ( k , iω n ) rewritten as G ασ ( iω n ) = ζ ¯ ασ ( iω n ) Z ∞−∞ dǫ ρ ( ǫ ) ζ Aσ ( iω n ) ζ Bσ ( iω n ) − ǫ (3)with α = A ( B ), σ = ↑ , ↓ and ¯ α = B ( A ) with ρ ( ǫ ) isthe bare DOS for the lattice considered (see below). (ii)We next determine the “host Green’s function” [7, 8] G ασ from the Dyson equation G − ασ ( iω n ) = G − ασ ( iω n ) +Σ ασ ( iω n ). (iii) We solve the impurity problem to obtainΣ ασ ( iω n ) = Σ ασ [ G ασ ( iω n )] (iv) We iterate steps (i), (ii)and (iii) till a self-consistent solution is obtained. Weuse as our “impurity solver” in step (iii) a generalizationof the iterated perturbation theory (IPT) [7, 9] schemewhich has the merit of giving semi-analytical results di-rectly in the real frequency domain.For simplicity, here we present the results for the T = 0solution of DMFT equations on a Bethe lattice of con-nectivity z → ∞ . The hopping amplitude is rescaled as t → t/ √ z to get a non-trivial limit and the bare DOSis then given by ρ ( ǫ ) = √ t − ǫ / (2 πt ) which greatlysimplifies the integral in eq. (3). The phase diagram inFig. 1 has been obtained from an analysis of various phys-ical quantities which we describe in detail below. We be-lieve that the results obtained will be qualitatively similarin case of other generic compact DOS. In the discussionbelow, first we describe the detailed results for the half-filled case followed by the results for the doped case.
0 0.2 0.4 0.6 0.8 1 0 2 4 6 m s U/t ∆ =0.1t ∆ =0.5t ∆ =1.0t ∆ =1.5t
0 0.20.4 0.60.8 0 2 4 6 δ n U/t
FIG. 2: Left panel: Staggered magnetization m s plotted asa function of U/t at half-filling. A first order phase transi-tion takes place with the onset of m s at U AF . Right Panel:Staggered occupancy, i.e., the difference in the filling factorof the two sublattices δn plotted as a function of U/t at half-filling. δn is non zero for all values of U/t . For
U < U AF , δn decreases monotonically and a discontinuity occurs in δn at U AF . Half-Filling:
The left panel of Fig. 2 shows our re-sults for the staggered magnetization m s , defined as m s = ( m zB − m zA ) / m zα = n ↑ α − n ↓ α is thesublattice magnetization. For a given value of ∆, thereexists a threshold value U AF at which the staggered mag-netisation turns on with a jump resulting in a first orderphase transition. Due to the presence of the staggeredpotential, the AFM instability does not occur at arbitrar-ily small U , and a finite value of U is required to turnon the magnetisation. Both U AF and the jump in m s at U AF are increasing functions of ∆. Note that sinceat half filling n Aσ + n Bσ = 1, the uniform magnetisation m F = n ↑ − n ↓ = ( m zA + m zB ) / δn = ( n B − n A ) /
2. Due to thestaggered on site potential, δn is always non zero, eventhough the Hubbard U tries to suppress it. δn decreasesmonotonically as a function of U . At U AF , δn also showsa discontinuity. S p ec t r a l G a p U/t E g ↓ E g ↑
0 0.5 1 1.5 0 1 2 3 4 U /t ∆ /tAFMI BI U AF U HM FIG. 3: Left panel: Spin-resolved spectral gaps E g ↑ and E g ↓ plotted as a function of U/t for ∆ = 1 . t at half-filling. For U < U AF , in the BI phase, E g ↑ = E g ↓ and the gaps decreasewith increasing U/t . At U = U AF , there occurs a jump sep-arating the two gaps such that E g ↓ < E g ↑ . For U > U HM ,in the AFMI phase, both the gaps increase with increasing U/t . Right panel: T = 0 phase diagram at half-filling. For U < U AF , the system is a PM BI. At U AF , a first order tran-sition occurs with the onset of an AFM order. A HM AFMpoint is seen at U = U HM > U AF . For all U > U HM , thesystem is an AFMI. Next we discuss the single particle DOS ρ α,σ ( ω ) = − P k Im ˆ G ασ ( k, ω + ) /π where α represents the sublat-tice A, B and σ is the spin. The spectral gap E gσ inthe single particle DOS ρ σ ( ω ) is shown in Fig. 3. For U < U AF , the spectral gap is same for both the spincomponents due to the spin symmetry of the paramag-netic BI phase and E gσ reduces with increase in U/t .This is because the Hubbard U suppresses the effect ofthe staggered potential ∆, which is responsible for a non-zero gap in the BI phase. At U = U AF , there occurs ajump separating the spectral gaps such that E g ↓ < E g ↑ .For U > U AF , E g ↓ keeps decreasing with increase in U/t and vanishes at U HM > U AF , while E g ↑ starts increasingwith increase in U/t and stays non zero at U HM . Thusat half-filling we have an HM AFM point at U = U HM ,details of which will be published elsewhere [10]. For U > U HM , the system is an AFMI in which both thegaps increase with increase in U . The phase diagram on -3-2-1 0 1 2 3 ω U=1.0 ρ ↑ ( ω )[c] [d][a] [b] ρ ↓ ( ω ) U=2.1
Half-metal -3-2-1 0 1 2 3 1 0.5 0 0.5 1 ω U=2.5
Half-metal
U=3.5
FIG. 4: Spin-resolved single particle DOS ρ σ ( ω ) vs ω for x =0 .
17 and ∆ = 1 . t . [a] For U < U , in the metallic phase, thesystem has spin symmetry with the DOS for both the spincomponents being non zero at ω = 0. For U > U , e.g. at U = 2 . t and 2 . t , ρ ↑ ( ω = 0) = 0 while ρ ↓ ( ω = 0) = 0. Thisis the HM phase shown in [b] and [c]. For U > U , the spinsymmetry is restored with ρ σ ( ω = 0) = 0 and the system is aregular metal as shown in [d]. the basis of above analysis is shown in the right panel ofFig. 3. Note that the spectral gaps are different for theup and down spin components (which is a key feature toget the HM phase) in this model because of the presenceof the staggered potential ∆. In the next section we dis-cuss the formation of the Ferrimagnetic HM in the dopedcase. Doped Case:
The phase diagram (Fig. 1) in thedoped case has a broad Ferrimagnetic HM phase for U < U < U . Below we discuss our results in detailexplaining how we determine U and U from an analysisof the single particle DOS and the magnetic properties. Single particle DOS in the doped case : Fig. 4shows the single particle DOS ρ σ ( ω ) = ρ A,σ ( ω )+ ρ B,σ ( ω )for both the spin components where ω is measured fromthe chemical potential µ . For U < U , the DOS for boththe spin components is same. In this regime the systemis a PM metal since the chemical potential lies inside thelower band for both the spin components (Fig. [4a]). For U > U , magnetic order sets in making the two gapsand the DOS different for the two spin components (Fig.[4b,4c]). The Fermi level lies inside the lower band for thedown-spin component making ρ ↓ ( ω = 0) = 0 while theup-spin DOS ρ ↑ ( ω = 0) = 0 as shown in Fig. 5; hencethe system is a HM. For U > U , the Fermi level liesinside the lower band for both the spin components. Thismakes both the spin components conducting, with equaldensity of up and down spins, and the system becomes a U/t x=0.05
Half Metal ρ ↑ (ω=0)ρ ↓ (ω=0) U/t HM x=0.17 ρ ↑ (ω=0)ρ ↓ (ω=0) FIG. 5: Plot of ρ σ ( ω = 0) vs U/t for x = 0 .
05 and x = 0 . . t . At x = 0 .
05 there exists a broad HM phase inwhich ρ ↑ ( ω = 0) = 0 while ρ ↓ ( ω = 0) = 0. The width of theHM phase shrinks in U space as x increases. paramagnetic metal (Fig. [4d]). Note that there is stilla small band gap (at energies higher then µ ). At evenlarger values of U , this gap will open up again separatingthe lower and the upper Hubbard band with the chemicalpotential being inside the lower Hubbard band. Magnetisation : The curves for U and U in Fig. 1are consistent with the magnetic properties as well whichare shown in Fig. 6. For small U values, magnetic orderis not favoured. As U increases, a first order transitionoccurs at U when both the sublattices acquire non zeromagnetisation m zA and m zB with a jump at U . Sincethe system is doped, these magnetisations are not equaland opposite to each other. This results in a non-zerostaggered magnetisation m s = ( m zB − m zA ) / m F = ( m zA + m zB ) / m F and m s increase with increas-ing x . This is because, in the HM phase, the up-spinband is fully occupied implying n ↑ = 1 / n ↓ = n − / m F , which can alsobe written as n ↑ − n ↓ , goes as 1 − n = x and is indepen-dent of the interaction strength U/t . This is in agreementwith our results in Fig. 6, within the numerical error-bars.The staggered magnetisation m s , however, increases withincreasing U/t . At U > U there occurs another first or-der transition and the system becomes a paramagneticmetal with both m F = m s = 0 for U > U .It is interesting to compare our DMFT phase diagramwith the phase diagram within Hartree-Fock (HF) the-ory, details of which will be published elsewhere [10]. Onecan get a HM phase within a HF theory, but it overesti-mates the tendency to the formation of the HM, and alsopredicts qualitatively wrong results. Within HF theoryfor all U > U , the system is a HM, the reason being thelack of quantum fluctuations in the HF theory which are m s U/t x=0.050.120.17 m F U/t
FIG. 6: The left panel shows the staggered magnetisation m s vs U/t and the right panel shows the uniform magnetisation m F vs U/t . At U there occurs a first order jump in both m s and m F making them non zero. Note that in the HMphase, m F ∼ x , as explained in the text. At U > U thereoccurs another 1st order transition with m s and m F droppingto zero. ∆ = 1 . t in both the panels. captured within DMFT. ( U - U ) /t x ∆ =0.5t ∆ =1.0t U /t x FIG. 7: The width ( U − U ) /t of the HM phase in U spaceas a function of the hole doping x . Inset shows the phaseboundaries of the HM phase for ∆ = 1 . t and 0 . t . Notethat the HM phase gets wider with increases in the staggeredpotential ∆. We note that, as shown in Fig. 7, the HM phase getswider with increase in the staggered potential ∆. Thiswill be useful from the application point of view. Oneshould look for correlated band insulators with large bareband gap, and an appropriately larger U, to get a robustHM phase.
Conclusions : In conclusion, we have studied a theo-retical model in which e-e interactions in a doped bandinsulator give rise to HM ferrimagnetic phase. Specif-ically, we studied an extension of the Hubbard modelwhich includes a staggered potential that makes the sys-tem a band insulator for U = 0 at half filling. As weturn on the on-site repulsion U , an AFM order sets inwith a first order transition at some threshold value of U = U AF . The AFM phase has different spectral gapsfor the two spin components and on doping away fromhalf filling it becomes a HM ferrimagnet. In the dopedsystem the HM phase survives for U < U AF as well. Thewidth of the HM phase in x space is largest for U ∼ [1] W. E. Pickett and H. Eschrig, J. Phys.: Condens. Mat-ter , 315203 (2007); Xiao Hu, Adv. Materials , 294 (2012), Half-metallic Alloys: Fundamentals and Applica-tions , Lecture Notes in Physics, Vol. 676, I. Galanakis,and P.H. Dederichs, Springer (2005).[2] A. Garg, H. R. Krishnamurthy, and M. Randeria, Phys.Rev. Lett. , 046403 (2006).[3] L.Craco, P. Lombardo, R. Hayn, G. I. Japaridze and E.Muller-Hartmann, Phys. Rev. B , 075121 (2008); N.Paris, K. Bouadim, F. Herbert, G. G. Batrouni, and R.T. Scalettar, Phys. Rev. Lett. , 046403 (2007); A. T.Hoang, J. Phys.: Conden. Matt , 095602 (2010).[4] The true ground state for the IHM for large U indeedhas AFM order and the metallic phase we found is typi-cally overwhelmed by AFMI. However, if we can frustrateAFM order, then the metallic ground state will win.[5] S. S. Kancharla and E. Dagotto, Phys. Rev. Lett. ,016402 (2007); K. Byczuk, M. Sekania, W. Hofstetter,and A. P. Kampf, Phys. Rev. B , 121103 (2009).[6] K. Bouadim, N. Paris, F. Herbert, G. G. Batrouni andR. T. Scalettar, Phys. Rev. B, , 085112 (2007)[7] A. Georges, G. Kotliar, W. Krauth, and M.J. Rozenberg,Rev. of Modern Phys. , 13 (1996)[8] T. Pruschke, M. Jarrell, and J. K. Freericks, Adv. Phys. , 187 (1995).[9] H. Kajueter and G. Kotliar, Phys. Rev. Lett. , 131(1996).[10] A. Garg, H. R. Krishnamurthy, and M. Randeria (un-published).[11] R. Nandkishore, G. Chern, and A. V. Chubukov, Phys.Rev. Lett. , 227204 (2012); Z. Hao, and O. A.Starykh, Phys. Rev. B87