Origin of in-plane anisotropic resistivity in the antiferromagnetic phase of Fe 1+x Te
aa r X i v : . [ c ond - m a t . s t r- e l ] J u l Origin of in-plane anisotropic resistivity in the antiferromagnetic phase of Fe x Te Eiji Kaneshita
National Institute of Technology, Sendai College, Sendai 989-3128, Japan ∗ Takami Tohyama
Department of Applied Physics, Tokyo University of Science, Tokyo 125-8585, Japan † (Dated: October 18, 2018)Motivated by a recent experimental report on in-plane anisotropic resistivity in the double-stripedantiferromagnetic phase of FeTe, we theoretically calculate in-plane resistivity by applying a memoryfunction approach to the ordered phase. We find that the resistivity is larger along an antiferromag-netically ordered direction than along a ferromagnetically ordered one, consistent with experimentalobservation. The anisotropic results are mainly contributed from Drude weight, whose behavior isattributed to Fermi surface topology of the ordered phase. PACS numbers: 72.80.-r, 74.70.-b, 75.10.Lp, 75.50.Ee
I. INTRODUCTION
Electronic states of two-dimensional Fe plane withsquare lattice are crucial for the mechanism of supercon-ductivity in iron-based superconductors. At high temper-ature, the electronic state of the plane is isotropic with-out directional difference between two nearest-neighborFe-Fe directions of the square lattice. With decreas-ing temperature, an anisotropic electronic state emergesthrough the breakdown of four-fold symmetry in mag-netic , electric , and electronic properties, result-ing in a nematic state with two-fold symmetry dis-tinguishing two Fe-Fe directions. When antiferromag-netism next to superconductivity in phase diagrams ap-pears, the anisotropy is strongly enhanced, for exam-ple, in resistivity measurements for detwinned samplesof Ba(Fe, T M ) As with T M =Co , Cr, and Mn aswell as (Ba,K)Fe As . Not only BaFe As systems(called 122 system), but also Se- and Cu-substitutedFe x Te systems (called 11 systems) exhibit in-planeanisotropy of resistivity .In the 122 systems, antiferromagnetic (AFM) order oc-curs with a stripe-type spin arrangement characterizedby ordering vector Q = ( π, x and y directions to be nearest-neighbor Fe-Fe directions: AFMarrangement along x , ferromagnetic (FM) along y [seeFig. 1(a)]. The asymmetry gives rise to preference inelectronic transport for the x or y direction. It is veryintuitively supposed that carriers will be scattered morestrongly along the AFM-ordered direction than along theFM-ordered direction. This is tempting us to expectlarger resistivity along the x direction as compared withthe y direction. However, experimental data have clearlyshown that resistivity along the y direction is larger thanthe x direction . This counterintuitive behavior in theAFM phase of the 122 systems is naturally explained ifone takes into account both anisotropic Fermi surfacesand nonmagnetic-impurity scattering .In the 11 system, the ordering vector is close to Q =( π/ , π/ , unlike the 122 systems. The ordering iscalled double stripes, where the AFM spin arrangement occurs along one of the second-neighbor Fe-Fe directionsand the FM arrangement appears perpendicular to theAFM-ordered direction. We call the AFM (FM) direc-tion the a ( b ) direction [see Fig. 1(b)]. A recent exper-iment has shown that resistivity along the a direction(AFM direction) is larger than the b direction (FM di-rection) , which is opposite to the 122 systems and theintuitive view looks accurate. However, it should be ex-amined carefully whether such an intuitive view is reallyaccurate. In order to understand in-plane anisotropy sys-tematically, the procedure applied to the 122 systems inthe previous study would be helpful.In this paper, we theoretically examine in-plane resis-tivity in the AFM phase of the 11 system at zero temper-ature. The AFM state is obtained by a mean-field theoryof a five-orbital Hubbard model. The anisotropy of re-sistivity is obtained by a recently developed multi-orbitalmemory function approach that takes into account non-magnetic impurity scattering. In the approach, resistiv-ity is proportional to scattering rate divided by Drudeweight. Calculated results are consistent with experi-mental data, showing the resistivity in the AFM-ordereddirection larger than that in the FM-ordered direction.In contrast to the 122 systems , the anisotropy of re-sistivity is never reversed, though its magnitude may bechanged as doping related to x , due to a transition ofFermi surface topology. As a result of the contributionfrom Drude weight and scattering rate reflecting the elec-tronic band structure at the Fermi level, the anisotropyremains opposite to the 122 systems. Finding out thatthe anisotropy is attributed to Fermi surface topology ofthe ordered phase, we derive a conclusion that the intu-itive view based on an arrangement of local spins is un-likely to be a good starting point as expected in metallicsystems. II. FORMULATION
We introduce a multiband Hubbard Hamiltonian for d -electron system in two-dimensional square lattices, H d = (a) 122 system abxy (b) 11 system FIG. 1. (Color online) Schematic illustration of spin configu-rations for (a) 122 and (b) 11 systems. Axes x and a ( y and b )show the direction across (along) the stripes. The rectanglesdisplay the unit cell. H + H I , which describes iron pnictides. The noninter-acting Hamiltonian H is given by H = X i,j X µ,ν,σ [ t ( ∆ ij ; µ, ν ) + ε µ δ µ,ν ] c † iµσ c jνσ , (1)where c † iνσ creates an electron of an orbital µ with a spin σ at the i -th Fe site with on-site energy ε µ . The hoppingenergy t ( ∆ ij ; µ, ν ) is for the one from the orbital ν atthe site position r j to µ at r i between the sites distancedby ∆ ij ≡ r i − r j . The interaction Hamiltonian H I canbe written as follows, by assuming that the pair hoppingequals the Hund coupling J : H I = U X i,µ n iµ ↑ n iµ ↓ + ( U − J ) X i,µ = ν n iµ ↑ n iν ↓ − J X i,µ = ν ( c † iµ ↑ c iµ ↓ c † iν ↓ c iν ↑ − c † iµ ↑ c iν ↑ c † iµ ↓ c iν ↓ )+ U − J X i,µ = ν,σ n iµσ n iνσ , (2)where n iµ ↑ = c † iµ ↑ c iµ ↑ and U is the intraorbital Coulombinteraction.In practice, we calculate the electronic state withinthe mean-field approximation by self-consistently solvingthe mean-field equations containing the AFM order pa-rameter. The order parameter is defined by h n l Q µνσ i = N − P k h c † k + l Q µσ c k νσ i with Q being the ordering vector, N being the number of the lattice points, and the Fouriertransform c k µσ = N − P i c iµσ exp (i k · r i ). The order-ing vectors arising from the spin configuration (Fig. 1) are( π,
0) for the 122 system and ( π/ , π/
2) for the 11 sys-tem : Thus, the first Brillouin zone is reduced into1 /N Q , where N Q = 4 for the 11 system. The multiplier l in l Q takes 0 , , . . ., N Q −
1. Finding the solution satis-fying the self-consistent condition P k ,ǫ ψ ∗ µǫσ ( k ) ψ νǫσ ( k + l Q ) = N h n l Q µνσ i , we finally obtain a quasiparticle stateof a band ǫ , γ † k ǫσ = P l P µ ψ ∗ µǫσ ( k + l Q ) c † k + l Q µσ withenergy E k ǫσ . The mean-field Hamiltonian H MF is ex-pressed as H MF = P k ,σ E k ǫσ γ † k ǫσ γ k ǫσ , where k isrestricted within the reduced zone. We refer to the data from an ab initio model basedon the downfolding scheme for the on-site energies andhopping integrals of FeTe. Since Fermi surface topol-ogy in the paramagnetic phase does not fit to a nestingcondition for Q = ( π/ , π/ Q = ( π/ , π/
2) magneticorder. In fact, we get the order with magnetic moment m = 2 . µ B ( µ B is the Bohr magneton) at electron den-sity n = 6 . x = 0 by setting U = 1 . J = 0 .
32 eV, which are larger than the values for the122 systems used before ( U = 1 . J = 0 .
22 eV, and m = 0 . µ B ). A tendency toward a large value of U and J for the 11 system is consistent with ab initio low-energymodels based on a constrained random-phase approxi-mation . The obtained m is close to an experimentalvalue of m = 2 . µ B as well as a theoretical value of m = 2 . µ B obtained by an ab initio calculation basedon the local spin density approximation and a value of m = 2 . µ B obtained by a combined density-functionaland dynamical mean-field theory. Fermi surfaces in ourcalculation are qualitatively similar to the ab initio cal-culation in the sense that there are two componentsin the magnetic Brillouin zone at n = 6 [see Fig. 2(a)].Naturally assuming that excess iron of concentration x introduces electrons in the Fe plane, we change n from6.0 to 6.2 and obtain the double-striped AFM order.To investigate the anisotropy of the electronic trans-port, we evaluate the resistivity, Drude weight, and scat-tering rate in the each direction along and across thestripes, i.e., the b and a directions, respectively. Recentlya multiorbital memory function technique that is a multi-orbital version of the memory function theory has beendeveloped , where nonmagnetic impurity is a source ofelastic scattering and a Born approximation is employed.Within the method, resistivity along the α direction isgiven by the ratio of the imaginary part of the memoryfunction M ′′ α to the charge stiffness or Drude weight D α : ρ α = ( ~ /N F ) P k F M ′′ α ( k F )2 D α , (3)where k F represents k points at the Fermi level E F and N F is the number of the points.In the calculation of ρ α of the multi-orbital system,the memory function approach is rather simple and fea-sible, while the application of the Boltzmann equation tothe multiorbital systems is limited and still within a phe-nomenological level This is why we here adopted thememory function approach. D α and M ′′ α are calculated from the current matrix J ǫ,ǫ ′ and the impurity matrix I ǫ,ǫ ′ , which arise fromthe current operator j = − c (cid:0) ∂H∂ A (cid:1) A =0 ( c is the ve-locity of light) and the impurity Hamiltonian H imp = I imp P µ,σ c † ℓµσ c ℓµσ , respectively —we here assume a non-magnetic local potential I imp at a site ℓ and hereafter set r ℓ = 0.The current matrix is defined as j = X ǫ,ǫ ′ ,σ J ǫ,ǫ ′ γ † k ǫσ γ k ǫσ , (4)and the impurity matrix is defined as H imp = 1 N X ǫ,ǫ ′ I ǫ,ǫ ′ ( k , k ′ ) γ † k ǫσ γ k ′ ǫσ . (5)The α component of J ǫ,ǫ ′ is calculated as J ( α ) ǫ,ǫ ′ ( k , σ ) = i N e ~ X l,i,j,µ,ν ( a α · ∆ ij ) t ( ∆ ij ; µ, ν ) × exp [i( k + l Q ) · ∆ ij ] × ψ ∗ µǫσ ( k + l Q ) ψ νǫ ′ σ ( k + l Q ) , (6)where e is the elementary charge and a α is a unit vec-tor pointing to the α direction. The impurity matrix iscalculated as I ( α ) ǫ,ǫ ′ ( k , k ′ , σ ) = I imp X µ,l,l ′ ψ ∗ µǫσ ( k + l Q ) ψ µǫ ′ σ ( k ′ + l ′ Q ) . (7)Drude weight is obtained from D α = 1 N F X ǫ F ,σ X k F | v ( k F ) | (cid:12)(cid:12)(cid:12) J ( α ) ǫ F ,ǫ F ( k F , σ ) (cid:12)(cid:12)(cid:12) (8)where the set ( ǫ F , k F , σ ) is chosen so that E k F ǫ F σ = E F and v ( k F ) is the Fermi velocity at k F . Scattering ratecan be evaluated as M ′′ α ( k F ) = πn c D α N ′ F | v ( k F ) |× X ǫ F ,ǫ ′ F ,σ X k ′ F | v ( k ′ F ) | (cid:12)(cid:12)(cid:12) A ( α ) ǫ F ,ǫ ′ F ( k F , k ′ F , σ ) (cid:12)(cid:12)(cid:12) , (9)where A ( α ) ǫ,ǫ ′ ( k , k ′ , σ ) = I ( α ) ǫ,ǫ ′ ( k , k ′ , σ ) × h J ( α ) ǫ,ǫ ( k , σ ) − J ( α ) ǫ ′ ,ǫ ′ ( k ′ , σ ) i . (10)It can be seen how Fermi velocities affect scatteringrate by taking into account J ( α ) ( k ) ∝ v α ( k ), where thesubscript α means its α component. From Eqs. (9) and(10), M ′′ α ( k F ) is contributed from Fermi velocities as fol-lows: M ′′ α ( k F ) ∝ D α X (cid:12)(cid:12)(cid:12) I ( α ) ǫ,ǫ ′ ( k , k ′ , σ ) (cid:12)(cid:12)(cid:12) | v α ( k F ) − v α ( k ′ F ) | | v ( k F ) | | v ( k ′ F ) | (11)Roughly speaking, M ′′ α can be enhanced for k F with v ( k F ) in the α direction, and diminished for a large v ( k F ). (−π, −π) (π, −π)(−π, π) (c)(a) (−π, −π) (π, −π)(−π, π) (d) (−π, −π) (π, −π)(−π, π) (b) (−π, −π) (π, −π)(−π, π) FIG. 2. (Color online) The Fermi surface and magnitude ofFermi velocity | v F | . Plotted are (a) the Fermi surface and (b) | v F | for n = 6 .
0; panels (c) and (d) show the same for n = 6 . III. RESULTS
We now present the results of the calculation. First,we examine Fermi velocity as a fundamental of the trans-port property together with features of the Fermi surface.Figure 2 shows the distribution of the velocities | v F | [(b)and (d)] on the Fermi surface [(a) and (c)] for differentelectron densities: n = 6 . .
2. In the undopedcase, the Fermi surface has crescent-shaped hole pocketsand circular electron pockets. The former has the largest | v F | on the side facing the b direction. As electrons aredoped, the hole pockets shrink and vanish, while the elec-tron pockets with their radii increased grow into an in-terlocking structure. The n = 6 . ± b rather than ± a . This affords a preference in conductivedirection for b over a . The contribution of velocities tothe transport is closely reflected in that of Drude weightthrough the current operator. The preference for b con-duction is, therefore, interpreted directly as an effect of D α in Eq. (3): The larger the velocity along b , the largerthe value of D b and the smaller the resistivity ρ b along b .We find, as a result, that the Fermi velocity feature tendsto increase the b conduction (or decrease the a resistiv-ity): This is consistent with the experimental results.Since we have perceived the effect of D in Eq. (3) on ρ , we next focus on that of M ′′ , i.e., the scattering rateas an effect of impurities to the transport property. Theintensity of the scattering rate is represented by M ′′ atthe Fermi level [Eq. (9)]. Since the formula of M ′′ α has a (c) (d)(a) (b) (−π, −π) (π, −π)(−π, π)(−π, −π) (π, −π)(−π, π) (−π, −π) (π, −π)(−π, π)(−π, −π) (π, −π)(−π, π)
50 50603 603
FIG. 3. (Color online) Scattering rate arising from the mem-ory functions M ′′ a (a, c) and M ′′ b (b, d) for the cases of n = 6 . n = 6 . factor of inverse D α , it is basically expected to behave inan opposite way: The ratio M ′′ a /M ′′ b tends to increase as D a /D b decreases ( D b /D a increases).Another factor to determine M ′′ α is | v ( k F ) | , which re-lates to its dependence on k F . As mentioned above[Eq. (11)], M ′′ b is enhanced for k F with v ( k F ) in the b direction, and diminished for a large v ( k F ): The resultshown in Fig. 3 is consistent with this basic aspect. Inboth cases n = 6 . .
2, scatterings in the a direc-tion mostly coming from the circular pockets overwhelmthose in b .In total, it results in M ′′ a > M ′′ b as shown in Fig. 4,where it is also demonstrated that the anisotropy of D islarger than that of M ′′ . As a result, M ′′ contributes tothe anisotropy ρ a > ρ b as well as D in a way that it ismuch less than that of D .Hence, we obtain the ratio of ρ a /ρ b consistent with theexperimental result ρ a > ρ b . In addition, this turns outto contribute to the anisotropy of D . D closely reflectsthe structure of Fermi pockets, and so does ρ a /ρ b .The anisotropy ρ a > ρ b is never reversed despite dop-ing. This is different from the 122 systems, where theanisotropy reverses in hole doping . As to the dopingeffect in the 11 system, the fundamental properties areunchanged in terms of anisotropy unless the structure ofFermi pockets is changed. This occurs around n = 6 . ρ a /ρ b decreases until n = 6 .
13 and increases for n > . electron density r a t i o FIG. 4. (Color online) Anisotropy of the resistivity. Plot-ted are the ratio of ρ a /ρ b (squares), D b /D a (triangles), and M ′′ a /M ′′ b (circles). Experimental data indicate that ρ a /ρ b is more thanunity but roughly less than 1.3 . Such a range is roughlylocated around n = 6 .
13 in Fig. 4. This n larger than 6 . x = 0 . , and thus 0 . m electrons will be added to n = 6 assuming Fe m + for the excess Fe. IV. CONCLUSION
We have investigated the origin of the anisotropic re-sistivity based on calculations of memory function andDrude weight. From the calculation, it is revealed thatthe anisotropic property mostly arises from that of Drudeweight, which is closely related to Fermi velocity. Sincethe anisotropy of Drude weight directly represents thatof electronic states at the Fermi level, we simply under-stand that the anisotropic resistivity originates from theanisotropic Fermi surface caused by the magnetic order.We have reached this simple interpretation without intro-ducing any bold, hypothetical assumption. This meansthat the symmetry breaking induced by the magneticorder directly appears in the transport property —notthrough the spin configuration, but through the Fermisurface topology. This is important in advancing thestudy of the 11 system.
ACKNOWLEDGMENTS
This work was supported in part by MEXT as a socialand scientific priority issue (Creation of new functionaldevices and high-performance materials to support next-generation industries) to be tackled by using post-K com-puter and by Grant-in-Aid for Scientific Research fromthe Japan Society for the Promotion of Science (GrantsNo. 26287079 and No. 26400381). ∗ [email protected] † [email protected] S. Kasahara, H. J. Shi, K. Hashimoto, S. Tonegawa, Y.Mizukami, T. Shibauchi, K. Sugimoto, T. Fukuda, T.Terashima, A. H. Nevidomskyy, and Y. Matsuda, Nature(London) 486, 382 (2012). J.-H. Chu, J. G. Analytis, K. De Greve, P. L. McMahon,Z. Islam, Y. Yamamoto, and I. R. Fisher, Science , 824(2010). M. A. Tanatar, E. C. Blomberg, A. Kreyssig, M. G. Kim,N. Ni, A. Thaler, S. L. Bud’ko, P. C. Canfield, A. I. Gold-man, I. I. Mazin, and R. Prozorov, Phys. Rev. B , 184508(2010). S. Ishida, T. Liang, M. Nakajima, K. Kihou, C. H. Lee,A. Iyo, H. Eisaki, T. Kakeshita, T. Kida, M. Hagiwara, Y.Tomioka, T. Ito, and S. Uchida, Phys. Rev. B , 184514(2011). H.-H. Kuo, J.-H. Chu, S. C. Riggs, L. Yu, P. L. McMahon,K. De Greve, Y. Yamamoto, J. G. Analytis, and I. R.Fisher, Phys. Rev. B , 054540 (2011). J. J. Ying, X. F. Wang, T. Wu, Z. J. Xiang, R. H. Liu, Y.J. Yan, A. F. Wang, M. Zhang, G. J. Ye, P. Cheng, J. P.Hu, and X. H. Chen, Phys. Rev. Lett. , 067001 (2011). S. Ishida, M. Nakajima, T. Liang, K. Kihou, C. H. Lee, A.Iyo, H. Eisaki, T. Kakeshita, Y. Tomioka, T. Ito, and S.Uchida, Phys. Rev. Lett. , 207001 (2013). A. Dusza, A. Lucarelli, F. Pfuner, J.-H. Chu, I. R. Fisher,and L. Degiorgi, EPL , 37002 (2011). M. Nakajima, T. Liang, S. Ishida, Y. Tomioka, K. Kihou,C. H. Lee, A. Iyo, H. Eisaki, T. Kakeshita, T. Ito, and S.Uchida, Proc. Natl. Acad. Sci. USA , 12238 (2011). M. Nakajima, S. Ishida, Y. Tomioka, K. Kihou, C. H. Lee,A. Iyo, T. Ito, T. Kakeshita, H. Eisaki, and S. Uchida,Phys. Rev. Lett. , 217003 (2012). M. Yi, D. Lu, J.-H. Chu, J. G. Analytis, A. P. Sorini, A. F.Kemper, B. Moritz, S.-K. Mo, R. G. Moore, M. Hashimoto,W.-S. Lee, Z. Hussain, T. P. Devereaux, I. R. Fisher, andZ.-X. Shen, Proc. Natl. Acad. Sci. USA , 6878 (2011). T.-M. Chuang, M. P. Allan, J. Lee, Y. Xie, N. Ni, S. L.Budko, G. S. Boebinger, P. C. Canfield, and J. C. Davis,Science 327, 181 (2010). M. P. Allan, T.-M. Chuang, F. Massee, Y. Xie, N. Ni, S.L. Budko, G. S. Boebinger, Q. Wang, D. S. Dessau, P. C.Canfield, M. S. Golden, and J. C. Davis, Nature Phys. 9,220 (2013). M. Fu, D. A. Torchetti, T. Imai, F. L. Ning, J.-Q. Yan,and A. S. Sefat, Phys. Rev. Lett. , 247001 (2012). T. Kobayashi, K. Tanaka, S. Miyasaka, and S. Tajima, J.Phys. Soc. Jpn. , 094707 (2015). S. Ishida, M. Nakajima, T. Liang, K. Kihou, C.-H. Lee, A.Iyo, H. Eisaki, T. Kakeshita, Y. Tomioka, T. Ito, and S.Uchida, J. Am. Chem. Soc. , 3158 (2013). E. C. Blomberg, M. A. Tanatar, R. M. Fernandes, I. I.Mazin, B. Shen, H.-H. Wen, M. D. Johannes, J. Schmalian,and R. Prozorov, Nature Commun. , 1914 (2013). L. Liu, T. Mikami, M. Takahashi, S. Ishida, T. Kakeshita,K. Okazaki, A. Fujimori, and S. Uchida, Phys. Rev. B ,134502 (2015). K. Sugimoto, P. Prelov˘sek, E. Kaneshita, and T. Tohyama,Phys. Rev. B , 125157 (2014). W. Bao, Y. Qiu, Q. Huang, M. A. Green, P. Zajdel, M. R.Fitzsimmons, M. Zhernenkov, S. Chang, M. Fang,B. Qian,E. K. Vehstedt,J. Yang, H. M. Pham, L. Spinu, and Z. Q.Mao, Phys. Rev. Lett. , 247001 (2009). S. Li, C. de la Cruz, Q. Huang, Y. Chen, J. W. Lynn, J.Hu, Y.-L. Huang, F.-C. Hsu, K.-W. Yeh, M.-K. Wu, andP. Dai, Phys. Rev. B , 054503 (2009). O. J. Lipscombe, G. F. Chen, C. Fang, T. G. Perring, D.L. Abernathy, A. D. Christianson, T. Egami, N. Wang, J.Hu, and P. Dai, Phys. Rev. Lett. , 057004 (2011). A. M. Ole´s, Phys. Rev. B , 327 (1983). T. Miyake, K. Nakamura, R. Arita, and M. Imada, J. Phys.Soc. Jpn. , 044705 (2010). F. Ma, W. Ji, J. Hu, Z.-Y. Lu, and T. Xiang, Phys. Rev.Lett. , 177003 (2009). Z. P. Yin, K. Haule, and G. Kotliar, Nature Mater. ,932 (2011). W. G¨otze and P. W¨olfle, Phys. Rev. B6