The single-ion anisotropy in LaFeAsO 0 1 0 Ren-Gui Zhu 2 College of physics and electronic information, Anhui Normal University, Wuhu, n a 241000, P. R. China J E-mail: [email protected] 4 ] n Abstract. We use Green’s function method to study the Heisenberg model of o LaFeAsO with the striped antiferromagnetic collinear spin structure. In addition to c - the intra-layer spin couplings J1a,J1b,J2 and the inter-layer coupling Jc, we further r consider the contributions of the single-ion anisotropy Js. The analytical expressions p u for the magnetic phase-transition temperature TN and the spin spectrum gap ∆ are s obtained. According to the experimental temperature TN = 138K and the previous . t estimations of the coupling interactions, we make a further discussion about the a m magnitudeandtheeffectsofthesingle-ionanisotropyJs. Wefindthatthemagnitudes - of Js and Jc can compete. The dependences of the transition temperature TN, the d zero-temperature average spin and the spin spectrum gap on the single-ion anisotropy n o are investigated. We find they both increase as Js increases. The spin spectrum gap c at low temperature T → 0 is calculated as a function of Js, the result of which is a [ useful reference for the future experimental researches. 1 v 1 3 4 0 . 1 0 0 1 : v i X r a R G Zhu 2 1. Introduction Itwasrecentlydiscoveredthataniron-basedmaterialLaFeAsOshowshigh-temperature superconductivity when O atoms are partially substituted by F atoms[1]. This discovery has triggered great research interest on the FeAs-based pnictides superconductors and their undoped compounds. It has been theoretically and experimentally confirmed that these pure FeAs-based compounds have a ground state with collinear stripe-like antiferromagnetic(AF) spin order formed by Fe atoms[2, 3, 4, 5, 6, 7, 8, 9, 10, 11]. Thus to establish an effective spin Hamiltonian for them and to elucidate the corresponding antiferromagnetism are helpful in understanding the underlying mechanism to make them superconducting upon doping. For undoped LaFeAsO and other similar parent compounds, a Heisenberg exchange model was suggested to explain their AF structure[2, 4, 12, 13], and was used to explore theirmagneticproperties[14,15,16]. Figure1showstheunitcelloftheorthorhombicAF spin structure of the Fe lattice. This orthorhombic structure exists below a structure transition temperature T , which is 15 ∼ 20K higher than the magnetic transition S temperature T [6, 17]. Usually the nearest neighbor (NN) coupling J (including J N 1 1a and J ), and the next-nearest neighbor (NNN) coupling J in FeAs layers are dominant 1b 2 and must be considered. The NN coupling J between spins on neighboring layers is c regarded to be much smaller than the in-plane couplings[2, 4]. However, J was found c to be essential for the existence of a non-zero magnetic transition temperature T [15]. N A further consideration can include the single-ion anisotropy J . It was estimated to s be even much smaller than J in the model of SrFe As , but a spin spectrum gap was c 2 2 found to be produced by it[14]. For LaFeAsO, so far there is no research report about the magnitude or the effects of the single-ion anisotropy. In this paper, we use Green’s function method[18] to study the Heisenberg model of FeAs-based pure parent compounds. The Hamiltonian of this model in a detailed form is 1 1 H = J S ·S + J S ·S +J S ·S 1a 1i 1j 1a 2i 2j 1b 1i 2j 2 2 Xhiji Xhiji Xhiji +J S ·S +J S ·S −J [(Sz)2 +(Sz)2], (1) 2 1i 2j c 1i 2j s 1i 2i hXhijii Xhiji Xi where the spin coupling J between layers and the single-ion anisotropy J are both c s considered. The subscripts 1 and 2 mean the sublattices 1 and 2 respectively. hiji means NN spin pairs, and hhijii means NNN spin pairs. The self-consistent equations for the average sublattice spin will be derived. An analytical expression for the magnetic transition temperature T will be obtained. For LaFeAsO, according to the recent N estimations of the strengths J ,J ,J and J [15] with the experimental temperature 1a 1b 2 c T = 138K[6, 17], weshall makea furtherestimation ofthesingle-ion anisotropy J . We N s find that the magnitude of J can compete with J , and in some situations even bigger s c than J . The effects of the single-ion anisotropy on the transition temperature T , the c N zero-temperature average spin hS i and the spin spectrum gap are investigated. We z 0 R G Zhu 3 z c b J y c a J1a J 2 x J 1b Figure 1. A unit cell of the orthorhombic Fe spin lattice. a, b, c are the three base vectors. The lattice consists of two sublattices (distinguished by the color gray and black). find they all increase as J increases. In section 2, we shall give our analytical results s derived from the Green’s function method. In section 3, we shall present our numerical results. Finally a conlusion is given in section 4. 2. Green’s function derivation According to the general scheme of Green’s function method to solve an antiferromagnetic spin model with two sublattices, we construct the following Green’s functions: G (ω) = hhS+;S−ii, G (ω) = hhS+;S−ii. (2) 1k,1l 1k 1l 2k,1l 2k 1l The equation of motion is ωhhA;S−ii = h[A,S−]i+hh[A,H];S−ii, (3) 1l 1l 1l where A represents the spin operator S+ or S+. The commutator [A,H] can be derived 1k 2k using Hamiltonian (1) and the basic commutation relations of spin operators:[S+,S−] = i j 2Szδ , [Sz,S±] = ∓S±δ , where S± = Sx ±iSy. i ij i j i ij i i i In order to close the system of equations, the so-called PRA or Tyablikov decoupling[18] is adopted for the terms stemming from the exchange couplings: hhSzS+;S−ii ≈ hSzihhS+;S−ii, i 6= j. (4) i j l i j l While for the terms stemming from the single-ion anisotropy, we adopt the Anderson- Callen(AC) decoupling[19]: hhSzS+ +S+Sz;S−ii ≈ 2hSziΘ(z)hhS+;S−ii, (5) i i i i j i i i j where 1 Θ(z) = 1− [S(S +1)−hSzSzi]. (6) i 2S2 i i The AC decoupling has been demonstrated to be most adequate for the single-ion anisotropy much small compared to the exchange interactions[20, 21]. R G Zhu 4 In order to write the decoupled equations of motion in the k space, we take the following Fourier transformation: 1 G (ω) = G(k,ω)eik·(Rk−Rl), (7) k,l N k X where N is the number of sites in either sublattice, and the summation over k is restricted to the first Brillouin zone of the sublattice. At the same time, the equation δij = N1 keik·(Ri−Rj) is also used. Because of the translation invariant, we have hSz i = hSzi, hSz i = −hSzi and P 1k 2k Θ(z) = Θ(z) = Θ(z) = 1 − 1 [S(S + 1) − hSzSzi]. Finally, we obtain the decoupled 1k 2k 2S2 equations of the two Green’s function in k space: [ω −hSziAk]G11(k,ω)−hSziBkG21(k,ω) = 2hSzi, (8) and [ω +hSziAk]G21(k,ω)+hSziBkG11(k,ω) = 0, (9) where Ak = 2J1acos(kxa)−2J1a +2J1b +4J2 +2Jc +2JsΘ(z), (10) and Bk = 2J1bcos(kyb)+4J2cos(kxa)cos(kyb)+2Jccos(kzc), (11) in which a,b,c are the three lattice constants. Solving equations (8) and (9), we obtain the Green’s function: G (k,ω) = hSzi AkhSzi+ωk − AkhSzi−ωk , (12) 11 ωk " ω −ωk ω +ωk # and the spin spectrum: ωk = hSzi A2k −Bk2. (13) When k → 0, we obtainqa expression for the spectrum gap: ∆ = 2hSzi J Θ(z)[2J +4J +2J +J Θ(z)], (14) s 1b 2 c s q which is similar with the expression given in ref[14] derived from the spin-wave theory, expect for the factor Θ(z). From this expression for the gap, one see that the single-ion anisotropy is essential for the existence of the spectrum gap. Then following the process of solving the average spin, we derive the correlation function hS−S+i using the spectrum theorem: 1 ∞ ImG (k,ω +iǫ) hS−S+i = − dω 11 Nπ eβω −1 k Z−∞ X hSzi Ak βωk = coth −1 , (15) N k  A2k −Bk2 2  X q  in which the equation 1 = P(1) − iπδ(x)(P(···) means taking the principle value) x+iǫ x has been used to obtain the imaginary part of G (ω + iǫ), and β = 1 , k is the 11 kBT B Boltzmann constant, T is the temperature. R G Zhu 5 According to the theory of Callen[22], the average spin for arbitrary S can be calculated using the following equation: (S −Φ)(1+Φ)2S+1 +(S +1+Φ)Φ2S+1 hSzi = , (16) (1+Φ)2S+1 −Φ2S+1 where hS−S+i Φ = 2hSzi 1 Ak βωk = coth −1 . (17) 2N k  A2k −Bk2 2  X q  On the other hand, the correlation functionhSzSzi can be calculated from the equation hSzSzi = S(S +1)−(1+2Φ)hSzi. Using equation (6), we can relate Θ(z) to Φ by hSzi Θ(z) = 1− (1+2Φ). (18) 2S2 Now the equations (16)(17)(18) can be solved self-consistently to obtain the average spin at any given temperature, provided we know the values of the exchange couplings J ,J ,J ,J and the single-ion anisotropy J . 1a 1b 2 c s When the temperature T approaches zero, we obtain coth(βωk) → 1. The equation 2 (17) is reduced to 1 Ak Φ| = −1 . (19) T→0 2N k  A2k −Bk2  X q  The zero-temperature average spin hSzi can be obtained by self-consistently solving 0 the equations (16)(18)(19). When the temperature T approaches the magnetic transition temperature T , the N average spin hSzi as well as the spectrum ωk will approach zero. Expanding coth(βωk) 2 in the equation (17), we obtain Γ 1 Φ| ≈ − , (20) T→TN βhSzi 2 where Γ = 1 Ak . Inserting (20) into (16), and expanding the terms in the N k A2−B2 k k denominator and the numerator as the series of hSzi, we finally derive P 12(Γk T )2 T hSzi ≈ B N 1− , (21) v S(2S −1) T u (cid:18) N(cid:19) u t where S(S +1) T = . (22) N 3k Γ B On the other hand, inserting (20) into (18), and using the equation (22), we obtain the reduced expression for Θ(z) near the temperature T : N 2S −1 Θ(z)| ≈ . (23) T→TN 3S R G Zhu 6 0.245 0.24 L kB 0.235 (cid:144) b 1 0.23 J H (cid:144) N 0.225 T 0.22 0.215 0 0.0005 0.001 0.0015 0.002 0.0025 0.003 JS(cid:144)J1 b Figure 2. The transition temperature TN as a function of the single-ion anisotropy Js for J1a = 0.98,J2 = 0.52 and Jc = 0.0004 in the unit of J1b = 50 meV. The black point correspond to the experimental temperature TN =0.238 in the unit of J1b/kB. 3. Numerical results and discussions So far there is no consensus on the magnitudes of the exchange couplings J ,J ,J and 1a 1b 2 J , because of the unclear microscopic origin of the observed AF spin structure. Here c we prefer the estimations in ref[15], which gave J = 50±10 meV, J = 49±10 meV, 1b 1a J = 26 ± 5 meV and J = 0.020 ± 0.015 meV by using the experimental transition 2 c temperature T = 138K of pure LaFeAsO. The main purpose of this paper is to N investigate the magnitude and the effects of the single-ion anisotropy J in LaFeAsO. s Through out our numerical calculation, we take J = 50 meV, J = 49 meV, J = 26 1b 1a 2 meV and the spin S = 1. The result J ∼ 50meV are obtained from the first-principle 1b calculating[4, 23]. In the present systems of units, the Boltzmann constant is taken as k = 0.086 meV/K. B Figure 2 shows the effect of the single-ion anisotropy J on the transition s temperature T . We can see that T increases as J increases. This means that the N N s single-ion anisotropy term is in favor of the AF spin structure. It can be understood from the expression of the single-ion anisotropy term in the Hamiltonian (1). Increasing the magnitude of J will make the spins incline to align along the z axis, and give a lower s total energy, which make the system more stable. To one’s surprise, the magnitude of J s correspondingto theexperimental transition temperatureT isabout0.00143J , which N 1b is much bigger than the magnitude of the exchange coupling J = 0.0004J estimated c 1b in ref[15]. Furthermore, we see from figure 3 that the variation range of hSzi with J 0 s varying from 0 to 0.1 is almost the same as the one produced by J in ref[15]. All these c results imply that the magnitude of J is probably not much small compared with J . s c So the estimation of J maybe need to be adjusted if the single-ion anisotropy term is c considered. As to the estimations of the other exchange couplings J ,J and J , we 1a 1b 2 think there are still reasonable. R G Zhu 7 0.78 0.76 0 > z 0.74 S < 0.72 0.7 0 0.02 0.04 0.06 0.08 0.1 JS(cid:144)J1 b Figure3. Thezero-temperatureaveragespinasafunctionofthesingle-ionanisotropy Js for J1a =0.98, J2 =0.52 and Jc =0.0004 in the unit of J1b =50 meV. 0.0025 0.002 b 1 0.0015 J (cid:144) S J 0.001 0.0005 0 0.0002 0.0004 0.0006 0.0008 0.001 Jc(cid:144)J1 b Figure 4. The competing relation of Jc and Js. The curve is depicted by using the equation (22) for TN =0.238(J1b/kB), J1a =0.98J1b and J2 =0.52J1b. Figure 4 shows the competing relation of J and J when the transition temperature c s is fixed at the experimental value. The increase of J is accompanied by the decrease c of J , and vice vera. From figure 4, we can see that the ranges of their corresponding s variations are at the same magnitude, which implies they probably have the same status in the viewpoint of theoretical study. As to revealing the actual magnitudes of the two parameters J and J , we think it is not enough to use only the experimental transition c s temperature T . N Figure 5 shows the effect of the single-ion anisotropy J on the spectrum gap at s low temperature. The gap vanishes as J vanishes, and increases as J increases. We s s find the effect of J on the gap is very trivial. The curves for different values of J c c between (0.0001,0.01) are almost the same, while the single-ion anisotropy affects the gap apparently. Considering the gap can be obtained from inelastic neutron-scattering R G Zhu 8 0.6 Jc=0.01 @ 0.5 R@ I@ @ Jc=0.0001 0.4 b 1 J (cid:144) 0.3 D 0.2 0.1 0 0 0.02 0.04 0.06 0.08 0.1 Js(cid:144)J1 b Figure 5. The spin spectrum gaps at low temperature T → 0 as functions of the single-ion anisotropy Js for J1a =0.98J1b, J2 =0.52J1b and Jc =0.0001J1b(the below curve), Jc =0.01J1b(the above curve). experiment[14], we suggest that the magnitude of the single-ion anisotropy be estimated fromthefutureexperimentalresultsofthespectrumgap. Forexample,ifJ = 0.0004J , c 1b we obtain J = 0.00143J from the experimental transition temperature T = 138K. s 1b N Then calculating the spectrum gap with J = 0.00143J , we obtain the magnitude of s 1b the gap ∆ ≈ 3.4 meV, which can be compared with the future experimental result. 4. Conclusion We use Green’s function method to study the Heisenberg model (1) of LaFeAsO with the striped AF spin structure as shown in figure 1. The main purpose of this paper is to investigate the magnitude and the effects of the single-ion anisotropy J . We derive the s self-consistent equations for the average spin, and obtained the analytical expressions for the spin spectrum gap ∆, and the magnetic transition temperature T . We find N that the transition temperature T , the zero-temperature average spin hSzi and the N 0 spin spectrum gap ∆ are all increasing functions of the single-ion anisotropy J . From s our numerical results by using T = 138K and the previous estimations of J ,J ,J N 1a 1b 2 and J in ref[15], we find that the magnitude of J is probably not much small compared c s with J . Because the single-ion anisotropy is essential for the existence of the spin c spectrum gap, we suggest using the experimental result of the spin spectrum gap to fix the magnitude of the single-ion anisotropy J in the future. s References [1] Kamihara Y, Watanabe T, Hirano M and Hosono H 2008 J. Am. Chem. Soc. 130 3296 [2] Yildirim T 2008 Phys. Rev. Lett. 101 057010 [3] Cao C, Hirschfeld P J and Cheng H P 2008 Phys. Rev. B 77 220506(R) [4] Ma F and Lu Z Y 2008 Phys. Rev. B 78 033111 R G Zhu 9 [5] Dong J et al 2008 Europhys. Lett. 83 27006 [6] de la Cruz C et al 2008 Nature 453 899 [7] Chen Y et al 2008 Phys. Rev. B 78 064515 [8] Huang Q et al 2008 Phys. Rev. Lett. 101 257003 [9] Zhao J et al 2008 Phys. Rev. B 78 140504(R) [10] Goldman A I et al 2008 Phys. Rev. B 78 106506(R) [11] McGuire M A et al 2008 Phys. Rev. B 78 094517; arXiv:0804.0796 [12] Si Q and Abrahams E 2008 Phys. Rev. Lett. 101 076401 [13] Fang C et al 2008 Phys. Rev. B 77 224509 [14] Zhao J et al 2008 Phys. Rev. Lett. 101 167203 [15] Liu G B and Liu B G 2009 J. Phys.:Condens. Matter 21 195701 [16] Yao D X and Carlson E W 2008 Phys. Rev. B 72 052507 [17] Klauss H H et al 2008 Phys. Rev. Lett. 101 077005 [18] Tyablikov S V 1959 Ukr. Mat. Zh. 11 289; S. V. Tyablikov 1967 Methods in the Quantum Theory of Magnetism (New York: Plenum Press) [19] Anderson F B and Callen H B 1964 Phys. Rev. 136 A1068 [20] Fro¨brich P, Jensen P J and Kuntz P J 2000 Eur. Phys. J.B 13 477 [21] Henelius P, Fro¨brich P, Kuntz P J et al 2002 Phys. Rev. B 66 094407 [22] Callen H B 1963 Phys. Rev. 130 890 [23] Yin Z P et al 2008 Phys. Rev. Lett. 101 047001