Effect of Rashba splitting on RKKY interaction in topological insulator thin films
EE ff ect of Rashba splitting on RKKY interaction in topological insulator thin films Mahroo Shiranzaei, Fariborz Parhizgar, and Hosein Cheraghchi ∗ School of Physics, Damghan University, P.O. Box 36716-41167, Damghan, Iran School of Physics, Institute for Research in Fundamental Sciences (IPM), Tehran 19395-5531, Iran (Dated: November 10, 2018)In this work we have investigated the e ff ect of Rashba splitting on the RKKY interaction in TI thin film bothat finite and zero chemical potential. We find that the spin susceptibility of Rashba materials including TI thinfilm is strongly dependent on the direction of distance vector. Moreover, we find another term in the o ff -diagonalterms of the spin-susceptibility tensor which in contrast to the well-known DM-like term is symmetric. We showhow one can tune the RKKY interaction by using electric field applied perpendicularly to the surface plane andby small chemical doping giving rise to enhance the RKKY term, drastically. We have presented our results fortwo di ff erent situations, namely inter-surface pairing of magnetic impurities as well as intra-surface one. Thebehavior of these two situations is completely di ff erent which we describe it by mapping the density of states ofeach surface on the band dispersion. PACS numbers:
I. INTRODUCTION
Among di ff erent types of magnetic interaction detectedin materials, Ruderman-Kittle-Kasuya-Yosida (RKKY) mechanism, an indirect exchange interaction between twomagnetic adatoms via host itinerant electrons, is one of themain reasons of coupling between magnetic impurities. Thisinteraction is proportional to the spin susceptibility of the hostmaterial and so gives the spin information of the system. Depending on the spin structure of the host material, di ff er-ent types of couplings can occur between magnetic adatomsvia the RKKY interaction. While in spin-degenerate sys-tems, such as graphene, two localized magnetic impuri-ties couple to each other in the form of isotropic collinearHeisenberg-like term, the anisotropic collinear Ising-like termwith di ff erent coe ffi cients in di ff erent spin-directions can beappeared in spin-polarized systems. Moreover in materi-als with Rashba spin-orbit coupling as well as materialswith spin-valley coupling, it has been shown that twist-ing RKKY interaction is possible by the anti-symmetric non-collinear Dzyaloshinskii-Moria-like term.
In general, theRKKY is a long-ranged interaction, (it decays with R − D , D the dimension of the system) which oscillates with respect tothe distance between impurities and electron’s Fermi wave-vector. Fascinating feature of this well-known mechanismis measurable in experimental observations by angle-resolvedphotoemission spectroscopy (ARPES) and scanning tunnel-ing microscopy (STM) in the study of magneto-transport andsingle-atomic magnetometry. Moreover, the RKKY interaction can be in chargeof diverse magnetic phases and ordering in metals andsemiconductors such as ferromagnetic and anti-ferromagnetic as well as spin glass and spi-ral phases.
Recently, quantum anomalous Hall e ff ect(QAHE) have been predicted theoretically and experimen-tally realized in magnetically doped three-dimensional(3D) topological insulators. Since such experiments need theferromagnetic coupling of magnetic adatoms, it brings inten-sive attentions to the mechanism of the coupling among mag-netic impurities in this class of materials. Although the RKKY interaction (and more precisely its zero chemical potential ver-sion known as Van-Vleck mechanism) is thought to be themain mechanism of this coupling, such theory is still underdebate.
3D topological insulators (TI), systems with gapped bulkstates and gapless surface states protected by time reversalsymmetry (TRS), are a novel kind of materials that have beensubject of several researches during past few years.
Animportant branch of these topological insulators is Bismuth-based structures, for instance Bi Se and Bi Te , which madeof Van-der-Waals interacting layers known as Quintuple Lay-ers (QL). For thicknesses above 6QLs, the Bismuth-basedmaterials become topological insulator with gapless surfacestates which have isotropic Dirac-type band dispersion pre-senting with an e ff ective chiral Hamiltonian arising from pureRashba-type spin-orbit coupling. Combination of the pureRashba Hamiltonian with being in the category of Diracmaterials makes TIs a promising candidate for spintronicand electronic applications. Since the bulk band gap of these3D systems are not enough large, in practice, the bulk statesusually play a severe role in experiments and so it is morefavorable to use thin version of these structures in order to re-duce the e ff ect of their bulk states. It has been experimen-tally shown that for 5QLs thickness and less, the states ofdi ff erent surfaces of TI thin film would be hybridized. Al-though these ultra thin films are not 3D topological insulatorswith gapless states, they can share other interesting featuressuch as another topological phase transition from quantumspin Hall insulator to a normal insulator, time reversaltopological superconductivity and band tunability by apply-ing perpendicular electric or in-plane magnetic field. Furthermore, magnetic topological insulators and their thinversion become of much importance since the orderedmagnetic impurities on the surface of TI can create a magneticfield and open a gap in the band dispersion which has beenobserved experimentally. Such intrinsic ferromagnetism canresult in QAHE when the Fermi energy lies within the gap ofthe system. The RKKY interaction in the Rashba materialssuch as TIs have been explored extensively.
Exis-tence of the strong Rashba spin-orbit coupling in these mate- a r X i v : . [ c ond - m a t . m e s - h a ll ] F e b rials makes the RKKY interaction to have a rich physics thatincludes a DM-like term and can result in di ff erent magneticphases such as ferromagnetic, paramagnetic and spin-glass. In addition, such interaction on the surface of TI has been in-vestigated when a superconductor presents in the proximityof TI. Since the magnetic impurities ordered perpendicularto the surface of TI can produce a gap on the surface state,the RKKY interaction together with such gap has been inves-tigated self-consistently. While all these theoretical investi-gations have been done for a thick 3 D TI, the experimentalrealization of QAHE in TI thin films makes it essential to in-vestigate the RKKY interaction in thin version of TIs wheretwo surfaces be hybridized to each other. In this work, we have investigated the spin susceptibility ofTI thin film and so the RKKY interaction both at zero and fi-nite doping. In contrast to the most of previous works on TIs,we found strong spatial anisotropy of the RKKY interactionwith respect to the direction of the connecting line betweenimpurities when one or both impurities have an in-plane spin-component projected on the surface of TI. We tried to ex-plore the e ff ect of parameters such as chemical potential, tun-nelling strength between surfaces and applied biased electricfield on the RKKY interaction. The last one has the benefitthat one can tune the RKKY interaction and as a result themagnetic properties, by using an electric field. We describeour findings by means of contribution of the top and bottomsurface states in the band dispersion. The organization of thepaper is as follow: In section II we introduce the theory ofthe work starting with the model Hamiltonian. In this part,we present contribution of the top and bottom surfaces in theband dispersion separately which is so important to describeour results. Next, we report our method for calculating theRKKY interaction by using the real space Green’s function.To guarantee fluency, we have presented some details of cal-culations and also analytic results for the real space Green’sfunctions in the appendix A. Section III presents our resultswhere we discuss the RKKY interaction between impuritieson the same and di ff erent surfaces. We have summarized andconcluded our results in section IV. II. THEORYA. Model Hamiltonian
The surface states of the TI thin film around the Γ point can be described by the two-dimensional e ff ectiveHamiltonian H ( k ) = − D k σ ⊗ τ + [¯ hv F ( σ × k ) · ˆ z + V σ ] ⊗ τ z + ∆ σ ⊗ τ x , (1)where σ , τ are Pauli matrices in spin and surface space respec-tively, k = ( k x , k y ) represents the wave-vector of surface state’selectrons and v F is their Fermi velocity. The term with coef-ficient D refers to the particle-hole asymmetry in the systemand V shows the potential di ff erence between surfaces whichcan be caused by the e ff ect of substrate or an external electricfield applied perpendicularly to the surfaces. The last term in the above equation shows the tunneling between di ff erent sur-faces and in general it is of the form ∆ − ∆ k where the ∆ term can result in a topological phase transition in the systemwith potentials lower than a critical value V < ¯ hv F √ ∆ / ∆ forspecial thin films in which ∆ ∆ > Restricting ourselvesto the terms upto the first order in k , the energy dispersionwould be obtained as E ( k ) = ± (cid:113) (¯ h v F k + ∓ V ) + ∆ , (2)where the sign ± before the root square is related to the con-duction (C) and valance (V) bands and the sign ∓ before pa-rameter V refers to the di ff erent branches (1 ,
2) in each of(C,V) bands that has been separated as a result of the appliedpotential V known as Rashba splitting in the band dispersion.
2V k F1 k F2 ε F E C1 E C2 E V1 E V2 FIG. 1: (Color online) Schematic figure shows the dispersion of TIthin film and illustrates two Fermi wave-vectors, k F , , and the Fermienergy ε F by the dotted line. A schematic figure of these band dispersions has been de-picted in Fig.1. In this figure the horizontal dotted line showsthe chemical potential, which together with the applied po-tential V are tunable parameters of the system. As a re-sult of Rashba splitting, two di ff erent Fermi wave-vectors k F , = ( (cid:113) ε F − ∆ ± V ) / ¯ hv F appear in the systemAlso in this figure, the red solid lines (the blue dashed lines)show the criteria that the band dispersion comes mostly fromthe top (bottom) surface. This can be better understood bylooking at the Green’s function of the system where the localdensity of states (DOS) of the top surface can be studied sepa-rately from the bottom surface and its poles represent the banddispersion. By using G ( k , ε ) = ( ε − H ( k )) − , we have G ( k , ε ) = g t ↑ t ↑ g t ↑ t ↓ g t ↑ b ↑ g t ↑ b ↓ g ∗ t ↑ t ↓ g t ↑ t ↑ g ∗ t ↑ b ↓ g t ↑ b ↑ g t ↑ b ↑ g t ↑ b ↓ g b ↑ b ↑ g b ↑ b ↓ g ∗ t ↑ b ↓ g t ↑ b ↑ g ∗ b ↑ b ↓ g b ↑ b ↑ , (3)where, t ( b ) and ↑ ( ↓ ) refer to the top (bottom) surface and spinup (down) respectively. In addition, similarities between thecomponents have been considered in this matrix. Focusing onjust the diagonal elements of the Green’s function which arerequired for calculation of the DOS, we would have g t ↑ t ↑ ( k , ε ) = A + ( ε − E V ) + A − ( ε − E C ) + B − ( ε − E V ) + B + ( ε − E C ) g b ↑ b ↑ ( k , ε ) = A − ( ε − E V ) + A + ( ε − E C ) + B + ( ε − E V ) + B − ( ε − E C ) , (4)where coe ffi cients A ± and B ± are functions of k , ∆ and V asthe below A ± = (cid:112) ∆ + ( k − V ) ± ( k − V )4 (cid:112) ∆ + ( k − V ) , B ± = (cid:112) ∆ + ( k + V ) ± ( k + V )4 (cid:112) ∆ + ( k + V ) . (5)Using Dos ∝ − π (cid:80) k Im [ G ( k , ε )] and the fact that the imag-inary part of the Green’s function is peaked on the poles ofEq.(4) as δ ( ε − E ( k )), one can interpret the coe ffi cients A ± , B ± as the weight coe ffi cients of the DOS on di ff erent band disper-sions. - 0 . 2 - 0 . 1 0 0 . 1 0 . 200 . 10 . 20 . 30 . 40 . 5 - 0 . 0 5 0 0 . 0 5 0 . 1048 ( b )( a ) A (cid:1) / B (cid:1) k A +
B +
A -
B - V = 0 . 0 5 ¶ A+ / ¶ k k V = 0
FIG. 2: (Color online) (a) The weight coe ffi cients A ± and B ± , as afunction of k for V = .
05 eV and ∆ = .
035 eV. (b) partial derivativeof coe ffi cient A + with respect to k for two di ff erent potentials V = , . ff erent gap sizes ∆ = . ∆ = . Fig.2 (a) shows the behaviour of the weight coe ffi cients A ± , B ± as a function of k . As shown in this figure, the weightof the conduction band E C ( E C ) at large positive (negative) k is dominated by the bottom (top) surface. At k = ∆ and not on theapplied bias V . Figure 2 (b) shows the k-derivative of thecoe ffi cient ∂ A + ∂ k for di ff erent biased potentials V = , . ff erent gap sizes ∆ = . . k with the widths proportional to δ k ∝ ∆ . The biasvoltage V can just change the position of these peaks with noe ff ect on their widths.In addition to the weight coe ffi cients for the band dispersionwhich we will use them in the result section, one can calculatethe density of states (DOS) for the top and bottom surface - 0 . 0 5 0 . 0 0 0 . 0 50 . 00 . 40 . 81 . 21 . 6 - 0 . 0 5 0 . 0 0 0 . 0 5 ( b )( a ) V = 0 . 0 e V
V = 0 . 0 2 e V t o p s u r f a c e
LDOS ( ´ e ( e V ) b o t t o m s u r f a c e e ( e V ) FIG. 3: (Color online) Illustration of the density of states for un-perturbed system for two di ff erent values of voltage V = V = .
02 eV for (a) the top and (b) the bottom surfaces. separately. Fig.3 shows the DOS of di ff erent surfaces for fixedtunnelling parameter ∆ = .
035 eV and two di ff erent values ofvoltage V = , .
02 eV. As one can see, Van-Hove singularitiesappear in the DOS due to Rashba splitting ( V (cid:44)
0) when theenergy touches the boundaries of the gap ε = ± ∆ . B. RKKY interaction
By placing two magnetic impurities on the surfaces of TIthin film, the Hamiltonian would be modified to H = H ( k ) + J c (cid:88) i = , S i · ˆ s ( r i ) , (6)where S i shows the spin moment of the localized magneticimpurity, ˆ s ( r i ) = ¯ h / (cid:80) j σ j δ ( r − r j ) denotes the spin of itiner-ant electrons and J c displays the coupling between them. Byapplying the second order perturbation theory, one can trans-form the interaction between magnetic impurities and itiner-ant electrons to an indirect exchange interaction between twomagnetic impurities. Thus, the RKKY interaction would readas H αβ RKKY = J c (cid:88) i , j S α i χ αβ i j ( r , r (cid:48) ) S β j , (7)where χ αβ i j ( r , r (cid:48) ) is the spin susceptibility of the system and canbe evaluated as χ αβ i j ( r , r (cid:48) ) = − π Im (cid:90) ε F −∞ d ε T r [ σ i G αβ ( r , r (cid:48) , ε ) σ j G βα ( r (cid:48) , r , ε )] . (8)Here, α and β denote t / b surface, ( i , j ) = ( x , y , z ) show di ff er-ent directions of magnetic moment’s component, ε F refers tothe Fermi energy and trace is taken over the spin degree offreedom.In order to calculate the spin susceptibility Eq.(8), it isneeded to calculate the unperturbed retarded Green’s functionin real space, G ret ( ε, R ) which reads from the Green’s functionin k-space Eq.(3) by taking Fourier transformation G ret ( ε, R = r − r ) = Ω BZ (cid:90) d k e i (cid:126) k · (cid:126) R G ( k ) . (9)Such Green’s function has a general form of G ret ( ε, ± R ) = G tt ∓ e − i ϕ R G (cid:48) tt ... G tb ∓ e − i ϕ R G (cid:48) tb ± e i ϕ R G (cid:48) tt G tt ... ± e i ϕ R G (cid:48) tb G tb . . . . . . . . . . . . . . . G tb ∓ e − i ϕ R G (cid:48) tb ... G bb ∓ e − i ϕ R G (cid:48) bb ± e i ϕ R G (cid:48) tb G tb ... ± e i ϕ R G (cid:48) bb G bb , (10)where the components of the Green’s functions are given inAppendix A.The impurities can be both located on the same surface(intra-surface) as well as di ff erent surfaces(inter-surface). Al-though in the former case the position of impurities can beassumed to be both on the top or bottom surfaces but us-ing the symmetry of layer inversion together with V → − V ,one can achieve the result of the bottom surface from the topone and so in the following, we discuss two configurationsnamely, impurities to be located on the top surface and impu-rities located on di ff erent surfaces. After some calculations,the RKKY Hamiltonian Eq.(7) can be written as H RKKY = J H S · S + J I ˜ S · ˜ S + J DM · ( ˜ S × ˜ S ) + J xy ( ˜ S x ˜ S y + ˜ S y ˜ S x ) , (11)where the new spinors ˜ S is defined as ˜ S = ( S x cos( ϕ ) , S y sin( ϕ ) , S z ), with ϕ = tan − ( R y / R x ) and alsothe vector J DM = J DM (1 , ,
0) and J xy = J I . The details of theabove terms including some analytic results can be found inappendix A.In conventional two-dimension materials with isotropicband dispersion, the RKKY interaction does not depend onthe direction of R , the distance vector between impurities.However, in systems with Rashba spin-orbit coupling, the spinof itinerant electrons is coupled to the wave-vector k and animpurity with an in-plane magnetic moment would break theisotropy of the system, so the spin-response χ i j ( R ) dependson both magnitude and direction of the vector R . - 1 . 00 . 01 . 0 ( a ) c ( · - ) c t tx x c t ty y c t tz z ( b ) V= 0.0 eV c t tx y c t tx z c t ty z ( c ) c ( · - ) j R ( d ) V= 0.02 eV j R FIG. 4: (Color online) The (a,c) diagonal, (b,d) o ff -diagonal com-ponents of susceptibility tensor, χ αβ i j as a function of polar angle ϕ R are showed for inter surface case. All of them are scaled by( h v F Ω BZ ) . Here we set to ∆ = .
035 eV, ε F = .
135 eV, R = v F = . × ms and (a,b) V = V = .
02 eV.
The RKKY interaction coe ffi cients J s, introduced inEq.(11) for two considered configurations of impurities, intra-surface case ( tt ) or inter-surface case ( tb ), are defined as fol-low J tt / tbH = − π (cid:90) ε F −∞ d ε ( G tt / tb ( ε, R ) + G (cid:48) tt / tb ( ε, R )) J tt / tbI = π (cid:90) ε F −∞ d ε G (cid:48) tt / tb ( ε, R ) J tt / tbDM = − π (cid:90) ε F −∞ d ε G tt / tb ( ε, R ) G (cid:48) tt / tb ( ε, R ) . (12)The first term in Eq.(11) is similar to the Heisenberg spininteraction which makes no di ff erence between di ff erent spin-directions coupling. However, the second term couples thenew spinors ˜ S instead of S and since ˜ S depends on the angle ϕ , the J I couples spinors of two impurities which have dif-ferent amplitudes in di ff erent directions. This interaction issimilar to the Ising interaction. Both of these terms will resultin collinear alignment of spinors S and S . Moreover, dueto the existence of Rashba spin-orbit coupling in TI thin film,symmetry of spin space is broken and so it is expected thatthe RKKY interaction would also have terms related to theo ff -diagonal components of the spin-susceptibility tensor. The third and forth terms of the above Hamiltonian are of thiskind and contrarily with the first two terms, they can causenon-collinear twisted alignment between spinors of impuri-ties. While the third term is anti-symmetric with respect tothe spinors and resembles the Dzyaloshinskii-Moriya (DM)interaction, the last term is symmetric. ( b )( a )
R2J R ( n m ) J t t H J t t I J t t D M R ( n m ) J t b H J t b I J t b D M
FIG. 5: (Color online) The RKKY interaction terms times R ( R J αβ i for i = H , I , DM ), as a function of the distance in unit of nm, scaledby ( J c ¯ h v F Ω BZ ) . Here we set ∆ = .
035 eV, V = .
02 eV, ε F = .
085 eVand v F = . × ms . Panels (a), (b) refer to intra and inter-surfacecases respectively. III. RESULTS AND DISCUSSIONS
In this section, we present our results for the RKKY interac-tion between two magnetic impurities located on the top sur-face (tt) or on two di ff erent surfaces (tb). As we have shownin the previous section, the SOC in the topological insulatorresults in the angle-dependent of the RKKY interaction whenmagnetic moment of impurities have an in-plan component.We start our result section by presenting this angle depen-dency of the spin susceptibility in Fig.4. In this figure, wehave assumed both impurities to be located on the top surfaceand ε F = .
135 eV, ∆ = .
035 eV and also we have consideredtwo voltages, V = V = .
02 eV (panel (c,d)).We have plotted diagonal parts of the spin-susceptibility ten-sor in panels (a,c) where except χ zz , other terms oscillate with2 ϕ R . The o ff -diagonal parts depicted in panels (b,d) oscillatewith ϕ R as expected from Eq.(A3). Besides, by comparing theupper and lower panels it is specified how applying the voltagecan drastically change sign and magnitude of the interactionterms.The behaviour of the RKKY interaction terms are severelya ff ected by distance between two magnetic impurities. In two-dimensional materials, they usually fall o ff with R − and alsooscillate as ∼ sin(2 k F R ), however for materials with severalbands, a more complicated behavior is expected. In Fig.5, wehave plotted J H , J I , J DM , times R and scaled by ( J c ¯ h v F Ω BZ ) , interms of distance R for intra- (panel(a)) and inter- (panel(b))surface case. As one can see in this figure, for the long rangedistances, all interaction terms decay as R − as like as othertwo-dimensional structures. For the intra-surface pairingand in the short distance limit which plays a more prominentrole at higher densities of impurities, the RKKY interactionhas much higher values.In contrast to the intra-surface pairing between impurities,the RKKY interaction multiplied by R behaves in a morestrange way for the inter-surface pairing. First, it starts fromnearly zero values at short distances and then it oscillates in abeating type pattern according to the existence of two di ff er-ent wave-vectors in the system. By looking at the weightcoe ffi cients in Eq.(5), one can see that for the top surface, k F has a more prominent role rather than k F , and that’s why itis seen a roughly monotonic oscillation for the RKKY inter- action in the intra-surface case. However in the inter-surfacecase, both surfaces and as a result both k F s become impor-tant and beating type occurs due to two di ff erent oscillationscharacterized by k F and k F . - 5 . 0- 2 . 50 . 02 . 5 - 0 . 4- 0 . 20 . 00 . 20 . 0 0 0 . 0 3 0 . 0 6 0 . 0 9 0 . 1 2- 2 . 0- 1 . 00 . 01 . 02 . 0 0 . 0 0 0 . 0 3 0 . 0 6 0 . 0 9 0 . 1 2- 2 . 0- 1 . 00 . 01 . 02 . 0 J( ·
10- 4) J t t H J t t I J t t D M J t b H J t b I J t b D M ( d )( c ) ( b ) V= 0.0 eVJ( ·
10- 4) e F ( e V ) ( a ) V= 0.03 eV e F ( e V )
FIG. 6: (Color online) The intra- (a,c) and inter- (b,d) surface RKKYinteraction couplings ( J αβ i for i = H , I , DM ), scaled by ( J c ¯ h v F Ω BZ ) as afunction of the Fermi energy in unit of eV. Here we set ∆ = .
035 eV, R = v F = . × ms , V = V = . Fig.6 shows the e ff ect of the Fermi energy on the RKKYinteraction terms for intra-surface (a,c) and inter-surface (b,d)cases. Here, we chose R =
30 nm, ∆ = .
035 eV and also V = V = .
05 eV in panels (c,d). Asshown in these figures, for the Fermi energy inside the gap, ε F < ∆ , all types of interactions are nearly zero accordingto insulating nature of the material, however they are not ex-actly zero and have small values which refer to the Van-Vleckmechanism. By comparing the insets of panel (a) and (c)in Fig.6, it is observed that all the interaction terms changewith the potential V and none of them can be neglected infavour of another. At the regime of finite doping, the RKKYinteraction would take very larger values than undoped situa-tion and oscillate, however this oscillation doesn’t occur witha constant period which is originated from complicated formof the band structure. As mentioned in the previous section inFig.3, the density of states would have Van-Hove singularitiesat the edge of the band gap for V (cid:44) k F and then after thecritical Fermi energy ε F = √ V + ∆ in which k F becomeszero, they increase. The change in the Fermi wavevector k F is proportional to a change in the electron’s density on the topsurface which justifies this behavior.For the inter-surface case, the dominant parameter for con-trolling the RKKY interaction between two impurities locatedon di ff erent surfaces is not only the DOS on the top and bot-tom surfaces, but also the inter-surface hybridization of thesurface states. In this case, as seen in Fig.6 (b,d), the RKKYinteraction terms decrease with respect to the energy. Thiscan be described by the weight factors explained in Sec.II Awhich say that at higher energies, the surface states are nothybridized any more and they will be purely localized on thetop or the bottom surface which results in weakening of theinter-surface interaction.Furthermore, as one can see in Fig. 6 (b), for V =
0, there isonly the Heisenberg interaction for the inter-surface case andother terms are exactly zero. In this case, the band dispersionsbelonging to di ff erent spin helical states wont split. This prop-erty together with the form of the tunnelling between surfaces, ∆ , which does not couple di ff erent spins, make the RKKY in-teraction to be isotropic collinear.To see the e ff ect of Rashba splitting on the RKKY interac-tion, in Fig.7, the behaviour of all RKKY interaction terms forintra- (panel a) and inter- (panel b) surface pairing are shownwith respect to V . Fixing the chemical potential and chang-ing the biased potential, one can tune the Fermi wave-vectorstogether with DOS and as a result tune the RKKY interac-tion. Tuning the magnetic properties of materials with elec-tric field is so desirable for spintronic technologies. For theintra-surface pairing depicted in Fig. 7 (a), the RKKY in-teraction drops by decrease of k F and then after the criticalbiased voltage V = (cid:113) ε F − ∆ in which k F =
0, it increases.At this critical voltage, the density of electrons on the top sur-face in which mediate the RKKY interaction becomes nearlyzero and that’s why the RKKY strength has its minimum inthe dashed circle. The quenching the RKKY interaction termsin the critical voltage has the same root as its quench in thespecial Fermi energy shown in Fig. 6 (c). This transitionin the RKKY interaction behaviour has been pointed out byan arrow and black dashed circle in the figure. For the inter-surface case both k F s play a role and regardless of oscillationsthe interaction increases. J t t H J t t I J t t D M ( b ) ( a )
J ( ·
10- 4) V ( e V ) V ( e V ) J t b H J t b I J t b D M
FIG. 7: (Color online) The RKKY interaction terms ( J αβ i for i = H , I , DM ), scaled by ( J c ¯ h v F Ω BZ ) , as a function of voltage. Here weset R =
30 nm, ∆ = .
035 eV, ε F = .
085 eV and v F = . × ms .Panels (a) and (b) refer to intra-surface and inter-surface cases re-spectively. A. Van-Vleck interaction
The RKKY interaction refers to indirect exchange interac-tion via conduction’s electrons which occurs in the metallicphase of systems. However, looking at Eq.(8), one can seethe RKKY interaction is originated from all energies lowerthan the Fermi energy ε F as well, so it would have non-zerovalue even at zero chemical doping ε F =
0. Although, in thisregime the indirect exchange interaction known as Van-Vleckinteraction, is much weaker than the RKKY interaction, it cana ff ect magnetic phases of materials. The zz component ofthe Van-Vleck interaction (related to χ zz ) has been studied inTI thin films to describe the Ferromagnetic phase in QAHEexperiment. Here we investigate all terms of this interactionand its tunability with the biased potential V . J R ( n m ) J t t H J t t I J t t D M
J ( · ( b )( a ) V ( e V ) J t t H J t tI J t tD M FIG. 8: (Color online) The Van-vleck interaction terms ( J αβ i for i = H , I , DM ), scaled by ( J c ¯ h v F Ω BZ ) , as a function of (a) distance, (b)voltage. Here we set ∆ = .
035 eV, ε F = . v F = . × ms and for (a) V = . R =
30 nm.
Figure 8 (a) shows the Van-Vleck interaction with respectto distance R. As shown in this figure, all the interaction termsfalls o ff very rapidly and becomes zero after R ∼ V . Moreover, it is obvious fromthis figure that V make the Van-Vleck interaction stronger. IV. SUMMARY AND CONCLUSION
In summary, we have investigated the e ff ect of Rashba-typeband splitting on the RKKY interaction in topological insu-lator thin films. We explored the RKKY interactions for twodi ff erent situations of magnetic impurities separately, namelyinter-surface pairing and intra-surface pairing where we re-ported completely di ff erent behaviors. We describe this diver-sity by mapping the density of states onto the band dispersionand finding the share of each surface on the band dispersion.We have shown how the RKKY interaction in the Rashba ma-terials have a strong direction-dependency (spatial anisotropy)when at least one of the impurities has a spin component par-allel to the plane. In addition to the conventional RKKY in-teraction terms mentioned in the Rashba materials, namelyHeisenberg-like, Ising-like and DM-like, we found anotherterm of the spin-susceptibility tensor which in contrast to theDM term is a symmetric interaction. We also investigate theRKKY interaction at zero doping where the chemical poten-tial lies within the gap of the TI thin film, (usually known asthe Van-Vleck mechanism). This can shed a light on solvingthe problem of QAHE which has been done experimentallyat zero chemical doping. Furthermore, we show that how theRashba splitting makes a Van-hove singularity in the band dis-persion at the band edges giving rise to large values of theRKKY interaction. So by a small value of the chemical dop-ing, the RKKY interaction can be extremely modified. Acknowledgment
F.P thanks Manuel Pereiro and Alireza Qaiumzadeh foruseful discussions and M.Sh. thanks Saeed Amiri for his sup-portive role in the paper preparation’s progress and also ac-knowledges ”institute for research in fundamental sciences”for their hospitality while the last parts of this paper werepreparing. H.C. thanks the International Center for Theoreti-cal Physics (ICTP) for their hospitality and support during avisit in which part of this work was done.
Appendix A: Details of Green’s function
Taking the integrals of Eq. (9) according to the Fouriertransformation, one can achieve the Green’s function inreal space. Using two-dimensional polar coordination in k-space, we have exp( i (cid:126) k · (cid:126) R ) = exp( i k R cos( ϕ k − ϕ R )) and so G ret ( ε, ± R ) (Eq. (10)) components would be obtained as thefollowing G tt ( ε, R ) = − πα (cid:88) s = ± a − s ( γ − isV ) K s , G (cid:48) tt ( ε, R ) = − π i α (cid:88) s = ± sa − s (cid:113) − V + is γ ) K s , G tb ( ε, R ) = π i α ∆ γ (cid:88) s = ± s ( V + is γ ) K s , G (cid:48) tb ( ε, R ) = − π i α ∆ γ (cid:88) s = ± s (cid:113) − V + is γ ) K s , G bb ( ε, R ) = − πα (cid:88) s = ± a s ( γ − isV ) K s , G (cid:48) bb ( ε, R ) = − π i α (cid:88) s = ± sa s (cid:113) − V + is γ ) K s (A1a)where, α = / ¯ h v F Ω BZ , γ = √ ∆ − ε and for s = ± , a s = ( εγ + si ) whereas K s / are the zeroth and first order of the modifiedBessel functions of the second kind as below K s = K R (cid:114) − ¯ h v F ( V + si γ ) , K s = K R (cid:114) − ¯ h v F ( V − si γ ) . (A2) Re-writing the spin susceptibility as χ αβ i j = − π Im (cid:82) ε F −∞ d ε F αβ i j ,where F αβ i j = T r [ σ i G αβ ( r , r (cid:48) , ε ) σ j G βα ( r (cid:48) , r , ε )] we can writethe F tt ( tb ) i j s for the intra-surface case (t) and for the inter-surface case (tb) as: F tt ( tb ) xx = G tt ( tb ) − G (cid:48) tt ( tb ) cos(2 ϕ R )) , F tt ( tb ) yy = G tt ( tb ) + G (cid:48) tt ( tb ) cos(2 ϕ R )) , F tt ( tb ) zz = G tt ( tb ) − G (cid:48) tt ( tb ) ) , F tt ( tb ) xy = − G (cid:48) tt ( tb ) sin(2 ϕ R ) , F tt ( tb ) xz = − F tt ( tb ) zx = G tt ( tb ) G (cid:48) tt ( tb ) cos( ϕ R )) , F tt ( b ) yz = − F tt ( b ) zy = G tt ( tb ) G (cid:48) tt ( tb ) sin( ϕ R ) . (A3)which after integration, it gives us the RKKY interactionterms. Introducing new spinors ˜ S = ( S x cos( ϕ ) , S y sin( ϕ ) , S z )the RKKY interaction Eq.(11) can be achieved easily. ∗ Electronic address: [email protected] M. A. Ruderman and C. Kittel, Phys. Rev. , 99 (1954). T. Kasuya, Prog. Theor. Phys. , 45 (1956). K. Yosida, Phys. Rev. , 893 (1957). F. Parhizgar, H. Rostami and R. Asgari, Phys. Rev. B F. Parhizgar, R. Asgari, S. H. Abedinpour and M. Zareyan, Phys.Rev. B M. Sherafati and S. Satpathy, Phys. Rev. B A. M. Black-Scha ff er, Phys. Rev. B , 205416 (2010). F. Parhizgar, M. Sherafati, R. Asgari and S. Satpathy, Phys. Rev.B Karol Szalowski, Phys. Rev. B , 205409 (2011). M. M. Valizadeh, Int. J. Mod. Phys. B, , 1650234 (2016). H. Imamura, P. Bruno and Y. Utsumi, Phys. Rev. B , 121303(R)(2004). J. J. Zhu, D.X. Yao, S.C. Zhang and K. Chang, Phys. Rev. L ,097201 (2011). D. A. Abanin and D. A. Pesin, Phys. Rev. Lett. 106, 136802(2011). M. Zare, F. Parhizgar and R. Asgari, Phys. Rev. B J. Klinovaja and D. Loss, Phys. Rev. B , 045422 (2013). I. E. Dzialoshinskii and et al.,
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