Kondo effect in f-electron superlattices
aa r X i v : . [ c ond - m a t . s t r- e l ] O c t Kondo effect in f -electron superlattices Robert Peters, ∗ Yasuhiro Tada, and Norio Kawakami Department of Physics, Kyoto University, Kyoto 606-8502, Japan Institute for Solid State Physics, The University of Tokyo, Kashiwa 277-8581, Japan (Dated: July 3, 2018)We demonstrate the importance of the Kondo effect in artificially created f -electron superlattices.We show that the Kondo effect does not only change the density of states of the f -electron layers,but is also the cause of pronounced resonances at the Fermi energy in the density of states of thenon-interacting layers in the superlattice, which are between the f -electron layers. Remarkably, theseresonances strongly depend on the structure of the superlattice; due to interference, the density ofstates at the Fermi energy can be strongly enhanced or even shows no changes at all. Furthermore,we show that by inserting the Kondo lattice layer into a three-dimensional (3D) metal, the gap ofthe Kondo insulating state changes from a full gap to a pseudo gap with quadratically vanishingspectral weight around the Fermi energy. Due to the formation of the Kondo insulating state in the f -electron layer, the superlattice becomes strongly anisotropic below the Kondo temperature. Weprove this by calculating the in-plane and the out-of-plane conductivity of the superlattice. PACS numbers: 71.10.Fd; 71.27.+a; 73.21.Cd; 75.20.Hr; 75.30.Mb
I. INTRODUCTION
Recently, remarkable progress has been made in thecreation of artificial superlattices including f -electronmaterials. These superlattices consist of a periodicarrangement of two-dimensional f -electron layers, whichare inserted into a different material. In the experi-ments, these superlattices show some intriguing magneticand superconducting properties, which are tunable bychanging the superlattice structure. For example, theN´eel temperature in CeIn ( n )/LaIn (4) superlattices decreases to zero when the Cerium layer thickness n isreduced to n = 2, which is accompanied by a linear tem-perature dependence of the resistivity. In other exper-iments, using the heavy fermion CeCoIn and the con-ventional metal YbCoIn , a two-dimensional strong cou-pling superconducting state has been observed. Also inthese experiments, the transition temperature and themagnetic field dependence are strongly affected by thestructure of the superlattice. The ability to create newmaterials with properties that depend on the superlat-tice structure may provide the possibility to constructnew functional devices.In f -electron materials, the competition or coopera-tion of two mechanisms mainly determines the proper-ties of the material, i.e. the Kondo effect and the RKKYinteraction. The quantum criticality arising when botheffects are equally strong, causes non-Fermi liquid be-havior and unconventional superconductivity.
In or-der to understand these intriguing effects in the contextof these superlattices, it is essential to understand howboth mechanisms, which are the cause of most of thephenomena, are influenced by the superlattice structure.Therefore, in this paper we will study the Kondo effectin the paramagnetic state of f -electron superlattices. Wewill clarify how the Kondo effect influences the f -electronlayers and in what way the Kondo effect is observablein the non-interacting layers, which are in between the f -electron layers. Another interesting question which wewill address is the influence of the superlattice structureon the low energy behavior of a Kondo lattice: What hap-pens to the full gap of the isolated Kondo lattice layer,when it is coupled to metallic layers.In order to answer these questions, an essentialpoint is the proper treatment of the superlattice struc-ture. Therefore, we simulate systems which includetwo-dimensional (2D) f -electron layers and 2D non-interacting layers. The number and order of these layerscan be varied, so that a variety of different superlatticescan be studied. We are thus able to analyze these f -electron superlattices and the Kondo effect within thesesystems taking the structure of the superlattice fully intoaccount. We find that the hybridization gap in the f -electron layers changes into a pseudo gap as soon as dif-ferent layers are coupled to each other. However, theKondo effect in the f -electron layers also strongly affectsthe density of states of the non-interacting layers, cor-responding to Yb- or La-layers in the above-mentionedexperiments. We find that resonances, whose shapes de-pend on the superlattice structure and are influenced byinterference from different f -electron layers, appear at theFermi energy. These resonances have strong influence onthe local properties of these superlattices and might beobservable by local probes in experiments.The remainder of this paper is organized as follows: Inthe next section, we explain in detail the model and themethod which we use in our study. This is followed bya section where we study a single f -electron layer withina 3D lattice. This kind of study will give insights intothe physics which can be expected in the superlattice.Finally, we present results for the paramagnetic state in f -electron superlattices, analyzing a wide range of differentsuperlattice structures. A summary will conclude thispaper. II. MODEL AND METHOD
We simulate a three dimensional system, consisting ofa periodic arrangement of f -electron- and non-interactinglayers, see Fig. 1. The Hamiltonian of the superlatticecan be written as a sum of terms describing 2D non-interacting layers, H NIL , f -electron layers described bya 2D Kondo lattice model, H KLL , and a hopping, H Inter , which connects adjacent layers with each other: H = H NIL + H KLL + H Inter , (1) H NIL = t NIL X σ c † izσ c jzσ ,H KLL = t KLL X σ c † izσ c jzσ + J X i ~S iz · ~s iz ,H Inter = t z X i X
0, between the conduction electrons and thelocalized spins, which is appropriate to describe heavyfermion materials. Without any coupling to the spins inthe Kondo lattice model, the system corresponds to non-interacting electrons in a 3D cubic lattice. In the rest ofthe paper, we will characterize the superlattice structureas (
N, M ), corresponding to KLL N /NIL M ( N : Kondolattice layers (KLL), M : non-interacting layers (NIL)).While we assume that the system is homogeneous withineach plane, i.e. within the same plane all lattice sites areequal, we use open boundary conditions perpendicular tothe layered structure including up to 40 layers in the su-perlattice. Throughout the paper, we focus on physicalproperties of the center-layers, which do not show anysigns of boundary effects.For solving this model, we use a combination of theinhomogeneous dynamical mean field theory (IDMFT)and the numerical renormalization group (NRG). IDMFT originates in the dynamical mean field theory(DMFT). While non-local contributions to the self-energy are neglected, local fluctuations and the layer de-pendence of the self-energy are fully taken into accountwithin IDMFT. The Kondo effect as well as the heavyfermion state are well described, and the phase diagramof the Kondo lattice model within DMFT has been in-vestigated by a number of authors before.
In the original DMFT approach, one is interested in a
Figure 1: (Color online) Schematic picture of a (1,2)-superlattice, consisting of a periodic arrangement of one f -electron layer and two non-interacting layers. homogeneous lattice, so that each site can be describedby the same self-energy. Because in the superlattice thesystem is inhomogeneous, i.e. there are different kinds oflayers with different physical properties, different latticesites cannot be approximated with the same self-energy.Instead, we have to take into account the structure of thesuperlattice and calculate a site dependent self-energy. Inthe case of the f -electron superlattice, Eq. (1), the self-energy for the NILs vanishes. IDMFT works as follows: a) We start with a guessed self-energy, which can alsobe set to zero. b) Using this self-energy, we calculate the site depen-dent local Green’s functions of the superlattice. Becauseour system is translation invariant within each layer,we use Fourier transformation for lattice sites withinthe same layer. The local Green’s function of layer z , G z ( ω + iη ), thus reads G z ( ω + iη ) = 14 π Z dk x dk y (cid:16) ( ω + iη ) I (2) − H ( k x , k y ) − Σ( ω + iη ) (cid:17) − z , where H ( k x , k y ) is a matrix corresponding to the one-particle Hamiltonian of the system, i.e. Eq. (1) with J = 0, in which for the in-plane hopping Fourier transfor-mation has been used, resulting in H NIL = 2 t (cos( k x ) +cos( k y )). Eq. (2) involves a matrix inversion of dimen-sion corresponding to the number of layers in the sys-tem. Interaction effects are taken into account by theself-energy Σ( ω + iη ), which depends on the layer. Fur-thermore, we take η small enough, so that the resultingKondo gap can be described. A typical value in our cal-culations is η = 10 − . c) We now relate the calculated local Green’s functionof each layer with the Green’s function of the correspond-ing quantum impurity model using the same self-energy.The impurity Green’s function can be written as G IMP ( ω + iη ) = 1 ω + iη − ∆( ω + iη ) − Σ( ω + iη ) . (3)∆( ω + iη ) is the hybridization between the impurity siteand the conduction electron bath. Due to the locality ofthe self-energy, we can assume that the calculated lat-tice Green’s function is equal to the impurity Green’sfunction, from which we can determine the hybridizationfunction∆( ω + iη ) = ω + iη − G − z ( ω + iη ) − Σ( ω + iη ) . (4)Within IDMFT, the hybridization function, ∆( ω + iη ),depends on the layer. d) The calculated hybridization functions define foreach layer a quantum impurity model. We solve for eachlayer the corresponding impurity model and calculate itsself-energy. e) We iterate this procedure by calculating new locallattice Green’s functions, see b) . If the self-energies of alllayers between two iterations do not change, the IDMFTsolution is found.Besides the calculation of local Green’s functions, thetime consuming step is the calculation of the self-energyfor each layer. For this purpose we use the NRG. TheNRG is able to solve these quantum impurity models fora wide range of temperatures and is able to resolve ex-ponentially small energy scales like the Kondo resonanceusing the complete Fock space algorithm.
Through-out this paper, we use a discretization parameter Λ = 2and keep N = 1000 states within each NRG iteration. III. A SINGLE 2D KONDO LATTICE LAYERWITHIN A 3D METALLIC HOST
In this section, we will first analyze a single KLL withina 3D metal. This will give insights into changes occurringdue to the insertion of a KLL into a metallic host.The paramagnetic Kondo lattice model at half fillingexhibits a gap in the density of states (DOS) at the Fermienergy. The system is in the Kondo insulating state. Thefirst question, we want to clarify is, how the DOS of theKondo lattice is modified, if it is inserted into a 3D metal.What happens to the full gap, if it is coupled to a metalliclayer? The electrons of the metallic layer may tunnel intothe KLL, thus, modifying the DOS.We show the DOS of a single KLL within a 3D metalfor different inter-layer hopping amplitudes, t z , in theupper panel of Fig. 2. The insets show magnificationsaround the Fermi energy. First, when increasing theinter-layer hopping amplitude, the gap width decreases.Second, the double-logarithmic inset clearly shows thatthe spectral weight vanishes as a power law for small fre-quencies, if the KLL is coupled to a metallic layer. The -2 -1 0 1 2 ω /W DO S z =0.2tt z =0.6tt z =tt z =1.6t -0.1 0 0.10123 0.0001 0.010.010.11 -1 0 1 ω /W ρ ( ω ) -1 0 1 ω /W -1 0 1 ω /W -1 0 1 ω /W -1 0 1 ω /W Kondolatticelayer cb d ea
1. neighborlayer 2. neighborlayer 3. neighborlayer 10. neighborlayer
Figure 2: (Color online) Results of a single KLL, J = 0 . W ,within a 3D non-interacting host. Upper panel: DOS of theKLL for different hopping amplitudes, t z , between the KLLand the NIL. The insets show magnifications around the Fermienergy. Lower panel: DOS of different layers. a) DOS of theKLL, b) nearest neighbor layer, c) next nearest neighbor layer,d) third-nearest neighbor layer, e) 10 layers away. determined exponent is approximately two and seems tobe independent of the hopping amplitude. The prefac-tor in front of the quadratic term, however, does dependon the hopping amplitude and increases with increasinghopping. An arbitrary small coupling, | t z | >
0, betweendifferent layers is sufficient to transform the full gap of anisolated KLL into a pseudo gap. We note here that evenfor the isolated KLL the edges of the gap are broadeneddue to interaction effects.Another noticeable change occurs in the DOS awayfrom the Fermi energy. Increasing the coupling betweenthe KLL and the NILs, the two-peak structure, whichis a remnant of the van Hove singularity in the two-dimensional square lattice, is flattened and vanishes. Thehopping between different layers creates a 3D structurefor the conduction electrons, and thus destroys these sin-gularities.To get some more insights into the formation of thepseudo gap, we look at a non-interacting analog, consist-ing of two layers, which are coupled via hopping t . Whileone layer is a non-interacting metallic layer, the otherlayer is described by a non-interacting periodic Ander-son model. In the periodic Anderson model each latticesite is hybridized via a hopping V to a localized f -electronlevel. The model can be written as H = X ij,σ t ij c † iσ c jσ + V (cid:16) c † iσ f iσ + f † iσ c iσ (cid:17) , where f † iσ creates an f -electron. Due to this hybridiza-tion, a gap opens around the Fermi energy. In the caseof the non-interacting periodic Anderson model, the spec-tral weight at the edges of the gap jumps from a finitevalue to zero. For the Kondo lattice model, this jumpis smeared out due to two-particle interactions betweenthe conduction electrons and the localized spins. If thelayer described by the periodic Anderson model is cou-pled to the metallic layer via a hopping t , the DOS canbe written as ρ ( ω ) = Z D − D dǫ ω + i − ǫ − V ω + i − t ω + i − ǫ , where we have assumed for simplicity a constant DOSwithin each layer. The result of this integral shows thatthe gap of the f -electron layer is changed by the couplingto a metallic layer from a full gap to a pseudo gap. Theexistence of the gap itself is unchanged. The local hy-bridization within the f -electron layer is sufficient to opena gap. However, inside the original gap, we find that thespectral weight vanishes quadratically. The change be-tween a full gap and the pseudo gap occurs at arbitrarysmall hopping between these two layers and happens inthe same way in the interacting Kondo lattice model andthe non-interacting periodic Anderson model. The expo-nent of the power law, which determines how the spec-tral weight vanishes around the Fermi energy, seems tobe equal in both models.The coupling between the KLL and the NILs does notonly modify the DOS of the KLL. It has also profoundinfluence on the DOS of the neighboring NILS. In thelower panel of Fig. 2, we show the DOS of different lay-ers in a system, where only one KLL is inserted into a3D metal. The hopping is isotropic, t z = t . The DOS ofthe KLL (panel a) shows the above-described pseudo gaparound the Fermi energy, ω = 0. The DOS of the neigh-boring layer (panel b), which is a non-interacting layer,shows a modification from the usual 3D DOS; there isa peak at the Fermi energy, whose width corresponds tothe gap width of the KLL. The next nearest layer, onthe other hand, shows a dip at the Fermi energy. Thepeak or gap at the Fermi energy alternates with increas-ing distance from the KLL, showing typical 2 k F oscilla-tions. The width of these structures at the Fermi energyis completely determined by the gap width of the KLL, thus, they are supposed to be related to the Kondo effect(Further evidence that the Kondo effect plays a signifi-cant role is shown later). However, the amplitude of thesestructures decreases quickly, roughly as 1 /d κ ( κ ≈ . d :distance between the layers), when going further awayfrom the KLL. These oscillations occur in a similar wayas Friedel oscillations around impurities in metals. Panele) in Fig. 2 shows the DOS of an atom 10 layers awayfrom the KLL, where the DOS resembles the unperturbed3D DOS. IV. f -ELECTRON SUPERLATTICES Next, we will study the Kondo effect in f -electron su-perlattices, i.e. in a periodic arrangement of 2D KLLsand NILs. Figs. 3 and 4 show results for a (1,1)-superlattice, where the system is composed alternativelyof KLLs and NILs. The behavior of the DOS in this su-perlattice is similar to the alternating peak/dip structureat the Fermi energy of the system with a single KLL in3D (Fig. 2). The KLL shows again the formation of apseudo gap, and the DOS of the nearest neighbor NILshows a peak at the Fermi energy. However, if we com-pare the amplitude of the peak at the Fermi energy ofthe NIL to the DOS of the unperturbed 3D metal, theDOS is enhanced nearly by a factor of two. Because eachNIL is sandwiched by two KLLs in this (1,1)-superlatticestructure, the Kondo effect of both KLLs enhances theDOS at the Fermi energy of the NIL. One may refer tothis as “constructive interference”. In Fig. 3, we also in-clude curves for different temperatures, showing how thegap in the KLL and the peak in the NIL are built up withdecreasing temperature. At high temperatures, T ≫ T K ,where T K is the Kondo temperature of the system, onecan clearly see a dip formation in the DOS of the KLLand a bump in the DOS of the NIL. These structuresare formed at temperatures T ≈ J . When lowering thetemperature below the Kondo temperature, the dip be-comes more pronounced and eventually forms the Kondogap, while the peak in the DOS of the NIL increases atthe same time. A closer analysis of the influence ofthe temperature on the DOS at the Fermi energy for the(1,1)-superlattice is presented in Fig. 4. The left panelshows the DOS at the Fermi energy of both layers fordifferent temperatures and different coupling strengths.For all coupling strengths, there is a crossover from thehigh temperature phase, where both layers have nearlyequal weight at the Fermi energy, to the low temperaturephase consisting of Kondo gapped layers and NILs, whichhave enhanced weight at the Fermi energy. The crossovertemperature, which is approximately equal to the widthof the gap of the KLL or the width of the peak of the NILat the Fermi energy, decreases with decreasing couplingstrength. We show these crossover temperatures overthe inverse coupling strength in the right panel of Fig.4, proving the exponential dependence on the couplingstrength J . Because the peak in the DOS of the NIL -1 -0.5 0 0.5 1 ω/ W DO S -0.1 0 0.100.511.522.53 -1 -0.5 0 0.5 1 ω /W DO S -0.1 0 0.1345 KLL NIL d ec r ea s i ng t e m p e r a t u r e d ec r ea s i ng t e m p e r a t u r e Figure 3: (Color online) DOS of a (1,1)-superlattice for dif-ferent temperatures. The insets show magnifications aroundthe Fermi energy.
T/W ρ F e r m i W /J -4 -3 -2 c r o ss ov e r t e m p e r a t u r e T / W J/W =0.75J/W =0.5J/W =0.4J/W =0.3J/W =0.2 y ~ e xp (- . W / J ) Figure 4: (Color online) DOS at the Fermi energy andcrossover temperature for the (1,1)-superlattice. Left: DOSat the Fermi energy, ρ F ermi for different temperatures andinteraction strengths. The lower (upper) lines correspond tothe KLL (NIL). Right: crossover temperature over inverseinteraction strength, proving the exponential dependence. shows the same temperature dependence as the gap ofthe KLL, we can state that both resonances are inducedby the Kondo effect of the f -electrons. The qualitativebehavior of the system is independent of the couplingstrength J . The coupling strength only determines theKondo temperature below which the resonances at theFermi energy can be observed, as long as a paramagneticstate is formed.If the structure of the superlattice is changed, the res-onances at the Fermi energy are also altered. In Figs.5 and 6, we show the temperature dependent DOS ofthe (1,2)- and (1,3)-superlattices, which consist of twoor three NILs in the unit cell, respectively. For both -1 -0.5 0 0.5 1 ω /W DO S -1 -0.5 0 0.5 1 ω/ W DO S -0.1 0 0.10123 -0.2 -0.1 0 0.1 0.22.52.753 KLL NIL d ec r ea s i ng t e m p e r a t u r e Figure 5: (Color online) DOS of a (1,2)-superlattice for differ-ent temperatures. Due to symmetry, both NILs are identical. -1 -0.5 0 0.5 1 ω /W DO S -1 -0.5 0 0.5 1 ω /W -1 -0.5 0 0.5 1 ω /W -0.1 0 0.1012 -0.1 0 0.1234 -0.1 0 0.1234 D ec r ea s i ng t e m p e r a t u r e D ec r ea s i ng t e m p e r a t u r e KLL NIL (a) NIL (b)
Figure 6: (Color online) DOS of a (1,3)-superlattice for dif-ferent temperatures. configurations, the KLLs form the pseudo gap at theFermi energy at low temperatures. This behavior is notchanged by tuning the structure of the superlattice. In-dependent of the superlattice structure, the KLLs alwaysform Kondo gapped states at half filling and zero tem-perature. The NILs, on the other hand, show differentbehavior depending on the superlattice structure. Forthe (1,2)-superlattice in Fig. 5, both non-interacting lay-ers exhibit identical DOS, because of the symmetry of thesystem. Surprisingly, the DOS at the Fermi energy of theNILs remains unchanged for all temperatures. This is incontrast to the (1,1)-superlattice configuration shown inFig. 3. In the (1,2)-superlattice, we observe only smalltemperature-dependent changes away from the Fermi en-ergy, ω ≈ ± . W in Fig. 5, but there is no peak or dip at T/W ρ F e r m i ρ F e r m i / ρ layers Kondolatticelayer (1,1)(1,2)(1,4)(1,5)(1,3)(1,1) (1,2) (1,3) (1,5) (3,2)
Figure 7: (Color online) Upper panel: DOS at the Fermi en-ergy over temperature for
J/W = 0 .
3, and for variety ofsuperlattice configurations. We only show the Kondo latticelayer and the nearest neighbor NIL. Lower panel: Each panelshows the normalized DOS at the Fermi energy for all differ-ent layers in the superlattice. The arrows at the layer indexdenote the KLLs. the Fermi energy. This can be explained by the peak/diposcillations seen in the DOS of a single KLL in 3D, shownin Fig. 2. The nearest neighbor NIL exhibits a peak atthe Fermi energy, while the next nearest layer shows adip. In the (1,2)-superlattice, there is thus a “destruc-tive interference” of both effects. Each NIL is nearestneighbor as well as next nearest neighbor to a KLL. Thisresults in an unaltered DOS of the NIL at the Fermi en-ergy. Therefore, the influence of the Kondo effect on theDOS at the Fermi energy is only visible in the KLLs forthis configuration. The situation is again changed for a(1,3)-superlattice, shown in Fig. 6. This structure favorsa constructive interference. Thus, while the KLL formsthe Kondo gap, the NIL which is nearest to the KLLshows a peak at the Fermi energy. The peak height isreduced compared to the (1,1)-superlattice (see Fig. 3).The second NIL, which is in the middle of three NILs,shows a small dip at the Fermi energy. This agrees withthe peak/dip oscillations found in the results of a singleKLL in 3D.Finally, we compare the DOS at the Fermi energy for avariety of superlattice configurations in Fig. 7. The up-per panel in Fig. 7 shows the temperature dependence ofthe DOS at the Fermi energy for the KLL and the nearestneighboring NIL for fixed interaction strength,
J/W =0 .
3. On the one hand, any KLL forms for any superlatticeconfiguration the Kondo gap at the Fermi energy. On theother hand, we observe that the amplitude of the peakin the neighboring NIL strongly depends on the super-lattice structure, due to interference between the Kondoeffect of different f -electron layers. Because of the above-described constructive/destructive interference, the peakheight changes non-monotonically when inserting addi-tional NILs, and shows even-odd oscillations. Looking DO S -1 -0.5 0 0.5 1 ω /W DO S -1 -0.5 0 0.5 1 ω /W t z /t=0.2t z /t=0.6t z /t=1 (1,1)-superlattice f-electron layer (1,1)-superlattice non-interacting layer(1,2)-superlattice f-electron layer (1,2)-superlattice non-interacting layer Figure 8: (Color online) Spectral functions at T = 0 forthe (1 , , t z . only at an odd number of NILs, which always resultsin a constructive interference for the nearest neighborNIL, the peak height is decreased when inserting newNILs. The maximum peak height can be observed forthe (1,1)-superlattice. On the other hand, for an evennumber of NILs, the peak height is increased when in-serting new layers. For the (1,2)-superlattice, there iscomplete destructive interference and consequently nopeak emerges at the Fermi energy. Furthermore, thecrossover temperature seems to be slightly dependent onthe superlattice structure. This is best visible in the tem-perature dependence of the KLLs in Fig. 7. Note thatthe crossover temperatures of different superlattices arechanged according to the peak heights in the DOS ofthe NIL at the Fermi energy. The lowest crossover tem-perature can be observed for the (1,1)-superlattice, andthe highest crossover temperature can be observed forthe (1,2)-superlattice. The difference between the (1,1)-and the (1,2)-superlattice is about 20%, which should bedetectable in experiments.Up to now, we have focused on periodic configurationsof a single KLL and a certain number of NILs. Thus,there are never two KLLs touching each other. The rea-son for our choice is that the physics, which has been sofar explained, does not change when adding additionalKLLs. There are only small changes in the DOS awayfrom the Fermi energy. Any KLL forms a pseudo gapwithin the superlattice structure, for which the spectralweight around the Fermi energy vanishes quadratically.Only the prefactor in front of the quadratic term doesdepend on the superlattice structure. Furthermore, theresonances observed at the Fermi energy of the NILs, areonly determined by the number of the NILs.In the lower panel of Fig. 7, we show the changesof the DOS at the Fermi energy for all layers in differ-ent superlattices at zero temperature. As in the resultsfor the single KLL embedded in 3D, we observe oscilla-tions of peaks and dips at the Fermi energy of the NILs.Configurations having two NILs are special, as describedabove. We observe that for these superlattices ((1,2)-and (3,2)-superlattice in Fig. 7), the DOS at the Fermienergy is not changed and is independent of the num-ber of KLLs. For the other superlattice configurations,we see that the amplitude of the resonances decreaseswith inserting new NILs. However, even for the (1,5)-superlattice, where there are five NILs between the KLLs,the enhancement at the Fermi energy is still around 20%,for the (1,3)-superlattice even 40%. This shows that theKondo effect in the f -electron layers has substantial in-fluence on all NILs and should not be dismissed, even ifthere are several layers between the f -electron layers. Inthe experiments, there are 4 or 5 NILs in between the f -electron layers. Our results imply that the Kondo effectpenetrates through the NILs at low temperatures, whichwill lead to a coupling of the separated f -electron layers.An open question, which we want to answer is, how thesuperlattice changes into a 2D system, when the inter-layer hopping is decreased. We show this process exem-plary for the (1 , , t z , each layerbehaves more and more like a 2D square lattice withoutqualitatively altering the behavior at the Fermi energy asdescribed above. The DOS of the f -electron layer exhibitsa hybridization gap at the Fermi energy. The gap widthbecomes slightly increased when decoupling the layers,because the gap width in the 2D Kondo lattice is largerthan in the 3D Kondo lattice. The DOS of the NILs, onthe other hand, is expected to evolve into the DOS of a2D square lattice with van Hove singularity at the Fermienergy. This can be seen in the right panels of Fig. 8. Forthe (1 , t z /t = 1, originates in the Kondoeffect, which strongly depends on the temperature, ashas been shown in Fig. 3. The spectral weight aroundthe Fermi energy also increases for the (1 , J/W = 0 . ~j = e X σ X ~k c † ~kσ ~ v ( ~k ) c ~kσ , (5)where ~ v ( ~k ) = ∂H ( k x ,k y ,k z ) ∂~k , and e is the elementary -1 -0.5 0 0.5 1 ω /W DO S KLLNIL ω /W c ondu c ti v it y [ a . u . ] ω /W ω ω ω ω ω ω σ z σ x Figure 9: (Color online) Real part of the conductivity withinthe layers, σ x , and perpendicular to the superlattice structure, σ z , for J/W = 0 .
5. The lower panel shows the excitations inthe local DOS corresponding to the peaks in the conductivity. charge. For calculating the conductivity, we use periodicboundary conditions in all directions. The conductivityis then given by the Kubo formula via the dynamicalcurrent-current correlation function, where the current istaken in direction µ , and reads, σ µ ( ω ) = 1 iω hh j µ , j µ ii ω + iδ . (6)Due to the locality of the self-energy within DMFT, theresulting two-particle Green’s functions can be writtenas products of one-particle Green’s functions. In Fig. 9, we show the out-of-plane conductivity inthe upper-left panel and the in-plane conductivity in theupper-right panel. The lower panel shows the local DOSof both layers. The anisotropy of the system is clearly vis-ible in the conductivities at low frequencies. The in-planeconductivity, σ x , exhibits a strong peak for ω/W < . σ z , shows a gap at low frequencies. Thus, per-pendicular to the layered structure the f -electron super-lattice is insulating. At the Kondo temperature, the su-perlattice becomes strongly anisotropic, showing metallicbehavior within the planes and insulating behavior per-pendicular to the planes. Analyzing the conductivity athigher frequencies, we observe two distinct peaks. Thesetwo frequencies can be easily identified within the DOSof both layers (lower panel of Fig. 9); one correspond-ing to the excitation energies between the lower bandand the central peak, and the other corresponding to theexcitations between the lower and upper bands. Evenif the superlattice structure is altered, the conductivity -1 0 1 ω /W KLL n=0.93NIL n=0.82 -1 0 1 ω /W KLL n=0.88NIL n=0.68 -1 0 1 ω /W KLL n=0.82NIL n=0.57 -1 0 1 ω /W DO S KLL n=1NIL n=1
Figure 10: (Color online) DOS for the (1,2) superlattice in adoped system
J/W = 0 . remains qualitatively unchanged. The superlattice stillbehaves as an anisotropic metal at low temperatures, i.e.perpendicular to the planes the superlattice is insulat-ing and parallel to the planes metallic. The number andposition of these peaks, however, depend on the superlat-tice structure. At large frequencies, additional peaks arevisible for different superlattice structures, arising fromexcitations between different layers. The temperature-dependent conductivity, especially the influence of thesuperlattice on it, is analyzed in detail in reference. All results, which have been shown until now, havebeen for half filled lattices, for which each lattice site isoccupied in average with one electron and the KLLs formthe Kondo gap at the Fermi energy. If the chemical po-tential is changed, the system is doped away from halffilling. However, the main statements about the Kondoeffect, which we have made, remain valid. In Fig. 10, weshow the local DOS of a (1,2)-superlattice for differentchemical potentials. The structures (peaks and dips) inthe DOS of the NILs are shifted. By doping the system,one of the side peaks in the DOS of the NIL is shifted to-wards the Fermi energy. At the same time the gap in theKLLs is shifted away from the Fermi energy. Thus, evenfor the doped system the Kondo effect of the f -electronsuperlattices is of great importance and all our previousstatements about interference of the Kondo effect of dif-ferent KLLs remain valid.Furthermore, we observe that the local occupation ofconduction electrons depends on the layer, see Fig. 10.Thus the system forms a charge density wave correspond-ing to the superlattice structure. This corresponds to thefact that different materials or different microscopic mod-els react differently to a change of the chemical potential.Due to the local singlet formation between conductionelectrons and localized spins in the KLLs, which ener-getically favors a state with one conduction electron perlattice site, we find that the occupation number of the KLLs is closer to one than the occupation of the NILs.However, as the spatial symmetry along the superlatticestructure is broken from the beginning by constructingthe superlattice, there occurs no phase transition, butthe charge density wave corresponds to the superlatticestructure. V. CONCLUSION
We have calculated properties of f -electron superlat-tices. We have elucidated the question, what happens tothe full gap of the isolated Kondo lattice layer, when itis inserted into a metallic host. We have shown that dueto tunneling of electrons from the metallic layer into theKondo lattice, the full gap of the isolated KLL is changedinto a pseudo gap. However, by coupling the KLLs to theNILs, not only the DOS of the KLL is changed. We havedemonstrated that the Kondo effect of the f -electrons alsohas strong influence on the non-interacting layers, whichshow pronounced peaks or dips at the Fermi energy. Re-markably, these peaks and dips in the NILs are stronglyinfluenced by interference between the f -electron layers.While the configurations with an odd number of NILsshow a constructive interference, which enhances the res-onance at the Fermi energy, an even number of NILs re-sults in a destructive interference, canceling all visible ef-fects of the Kondo effect on the DOS at the Fermi energyin the (1,2)-superlattice. The Kondo effect also stronglyaffects the conductivity at low temperatures. Because ofthe formation of the Kondo gap in the KLLs, the super-lattice becomes strongly anisotropic at low temperatures.Decreasing the temperature below the Kondo tempera-ture, the out-of-plane conductivity vanishes, while thein-plane conductivity stays metallic.In this study, we have shown that the Kondo effectplays an important role in the superlattice, not only inthe f -electron layers, but also in the NILs. The possibilityto tune the physical properties such as the behavior of theKondo effect is one of the advantages of a superlattice.So far, we have only analyzed the paramagnetic state.If magnetically ordered states are analyzed, besides theKondo effect the RKKY interaction will also become im-portant. For weak coupling strengths, the RKKY in-teraction will be stronger than the paramagnetic Kondoscreening, which will result in a magnetically orderedstate. However, the heavy fermion materials which havebeen used in the experiments, i.e. CeIn and CeCoIn ,show large regions in the pressure-temperature phase di-agram, where the paramagnetic heavy-fermion state isstable. Our results, which we have shown here, can beexpected to be valid for this paramagnetic heavy fermionphase, where the Kondo screening is stronger than theRKKY interaction. Because the Kondo effect in compe-tition with the RKKY interaction plays an important rolein f -electron materials, we expect that the tunability ofthe superlattice will result in intriguing phase diagramsincluding magnetic phases, which depend on the struc-ture of the superlattice. Including magnetic phases intoour studies is left as a future project.We have focused here on artificially created f -electronsuperlattices. However, there are many naturally occur-ring layered f -electron materials, e.g. CeCoIn . The de-scribed consequences of the Kondo effect on the system,i.e. resonances in the DOS at the Fermi energy, can beexpected to also play an important role in these com-pounds. This is currently under investigation. Acknowledgments
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