Spin Density Wave and Ferromagnetism in a Quasi-One-Dimensional Organic Polymer

Z. Phys. B 104, 27–32 (1997) ZEITSCHRIFT FUR¨ PHYSIK B c Springer-Verlag 1997

Spin density wave and ferromagnetism in a quasi-one-dimensional organic polymer

Shi-Dong Liang1, Qianghua Wang1,2, and Z. D. Wang1

1 Department of Physics, University of Hong Kong, Pokfulam Road, Hong Kong 2 Department of Physics, Nanjing University, Nanjing 210093, People’s Republic of China

Received: 5 February 1997

Abstract. The spin density wave(SDW) – charge density impurities, in which unpaired localized d or f electrons inter- wave(CDW) phase transition and the magnetic properties in act with the itinerant π electrons. Fang et al. [3] investigated a half-filled quasi-one-dimensional organic polymer are in- the stability of the ferromagnetic state and the spin configu- vestigated by the world line Monte Carlo simulations. The ration of the π- electron in a quasi-one-dimensional organic itinerant π electrons moving along the polymer chain are polymer by a mean field approach(the Hatree-Fock approx- coupled radically to localized unpaired d electrons, which imation). They found that not only the interaction between are situated at every other site of the polymer chain. The the π electrons but also the interaction between the π elec- results show that both ferromagnetic and anti-ferromagnetic trons and unpaired d electrons play an important role for radical couplings enhance the SDW phase and the ferromag- the ferromagnetic order, and the spin density wave(SDW) net order of the radical spins, but suppress the CDW phase. states is closely related to the ferromagnetism of this sys- By finite size scaling, we are able to obtain the phase tran- tem. However, the nearest neighbor interaction between the sition line in the parameter space. The ferromagnetic order π electrons has not been taken into account so that the inter- of the radical spins are observed to coexist with the SDW esting CDW-SDW transition as well as the effect of the π π phase. As compared to the system being free of the radical and π d electron interactions on the ferromagnetism cannot− coupling, the phase transition line is shifted upward in the be examined− in their work. On the other hand, the mean field U-V parameter space in favor of larger V , where U is the treatment on the total Hamiltonian introduces some uncertain on-site repulsion and V is the nearest-neighbor interaction approximation due to ignorance of the correlation effect at between the π electrons. All of these findings can be un- the outset. The world-line quantum Monte Carlo simulation derstood qualitatively by a second-order perturbation theory method [4] are, however, almost approximation-free aside starting from the classical state at zero temperature in the from the statistical error. In retrospection, Hirsh [4] used strong coupling limit. We also address the consequences of this technique to study the CDW-SDW transition in the half- the radical coupling for the persistent current if the polymer filled extended Hubbard model and the Su-Schrieffer-Heeger chain is fabricated as a mesoscopic ring. model. When the radical coupling is introduced, the interplay between the π-π electron interaction and π-d electron PACS: 71.20.Rv; 75.30.Fv; 75.30.Hx; 75.40.Mg interaction is by no means trivial, and is in fact important to the ferromagnetism in the organic chain. [3] In this paper, we investigate the CDW-SDW transition and the magnetism of quasi-one-dimensional organic polymer by the world line Monte Carlo technique [6]. We focus on the dependence of I. Introduction the the CDW-SDW phase transition, magnetism and other relevant properties on the interaction between the localized With the rapid development of experimental technology, d electrons and the itinerant π electrons. In Sect. II, we de- low-dimensional systems, such as organic polymers, show scribe the model and Monte Carlo method. The results are attractive potential applications. In particular, the discovery discussed in Sect. III. A summary and some discussion can of organic polymer superconductivity and ferromagnetism be found in the Sect. IV. makes it possible to search superconductors and ferromag- nets based on organic molecules, and brings a challenge to theorists. A quasi-one-dimensional π-conjugated struc- II. Model and Monte Carlo method ture of the organic polymer shows interesting electric and ferromagnetic properties. Ovchinnikov et al. [1] and Cao A typical structure of organic polymer, such as poly-BIPO, et al. [2] discovered the ferromagnetism of poly-BIPO[1,4- consists of a main zigzag chain containing π-conjugated car- bis-(2,2,6,6- tetramethyl-4-oxyl)-4-piperidyl-butadiin]. They bon atoms and a kind of side radical containing an unpaired attributed this kind of ferromagnetism to the transition-metal electron, which may be regarded as a quasi-one-dimensional 28 system. The interaction between electrons dominates the be described in terms of the CDW or SDW phases, which are magnetic properties of the system. Although the electron- related to the ferromagnetism of the system, we will study a phonon interaction can suppress the SDW state of the sys- half-filled chain and assume that the spin-up and spin-down tem, [3] our attention will be focused on the effect of corre- π electron number is equal. The other filling fraction may be lation between the itinerant π electron and the local unpaired addressed elsewhere. In order to investigate the CDW-SDW electron on the CDW-SDW phase transition and the mag- transition, we shall need the CDW order parameter netic properties. Thus, we shall omit the electron-phonon 1 interaction, the effect of which will be argued in the last m = ( 1)i n , (6) c N − h ii section. The simplified system can be described by a Hamil- i tonian (cf. [3, 4]) X and the CDW structure factor

H = t (Ci,σ† Ci+1,σ +H.c.)+U ni, ni, 1 ↑ ↓ iq(Ri Rj ) − S(q)= e − n n , (7) i,σ i N h i ji X X i,j +V n n +1 J δ S S , (1) X i i − i i,R · i i i as well as the SDW order parameter X X where the first-three terms on the right side belong to the 1 i ms = ( 1) ni, ni, , (8) usual extensive Hubbard Hamiltonian, i labels the carbon N − h ↑ − ↓i atoms along the chain, σ = 1 labels spin states (up and i ± X down), t>0 is the hopping matrix element, Ci† and Ci are and the SDW zero-frequency susceptibility the fermion creation and annihilation operators, respectively, β for π electrons with spin σ at the i’th site, U is the coupling 1 χ(q)= dτ (ni, (τ) ni, (τ))(nj, (0) strength between two π electrons at the same carbon atom, N 0 h ↑ − ↓ ↑ i,j Z † X and ni = Ci Ci,, V is a measure of the nearest neighbor iq(Ri Rj ) nj, (0)) e − , (9) Coulomb interaction between the π electrons, and ni = ni, + − ↓ i ↑ ni, . The last term in (1) denotes the ferromagnetic(J>0) where R ( ) is the space coordinate of the i(j)’th site, τ is the ↓ i j or antiferromagnetic(J<0) coupling between the spin Si imaginary time along the Trotter direction of the checker- of π electrons in the chain and the residual spin Si,R.Asin board [4]. The symbol in (3)–(8) represents the statistical [3], we assume that every side radical has a noncompensated average in the canonicalh·i ensemble. spin Si,R and the radical connects with every other carbon If the organic chain is fabricated to form a mesoscopic atom, ring, under an applied magnetic flux, there should be an equilibrium persistent current in the ring [9]. This has been 1, for odd i δ = an interesting topic in recent years [10]. The maximum am- i 0, for even i. plitude of the persistent current is proportional to the charge The radical coupling term can be written as stiffness of the system [7, 8]. It is also interesting in our case to study the influence of the radical coupling on the charge 1 z z + + shiftiness of the system. Physically, the charge stiffness is Si,R Si = Si,RSi + (Si,RSi + Si,R− Si−). (2) · 2 a measure of the modulus of the free energy of the system As usual, by neglecting the spin-flip interaction, [3, 5] we against the change in the twist boundary condition, or in may write other words, the applied magnetic flux. According to [7, 8] the charge stiffness can be simply obtained in the world-line z ni, ni, Si,R Si S [ ↑ − ↓ ]. (3) Monte Carlo simulations by the zero-frequency limit of the · ' i,R 2 Fourier transform of the ‘current-current’ correlation func- 1 Here Sz = becomes an Ising-like spin state on the side tion ( )= (0) ( ) , where the pseudocurrent ( )is i,R ± 2 C τ J J τ J τ radical. defined by [7,h 8] i Interestingly, substituting (3) into (1), we find that the resultant Hamiltonian is invariant under the transformation J(τ)= R [n (τ + ∆τ) n (τ)]. (10) i i,σ − i,σ σ σtogether with J J. Due to this symmetry, i,σ the→− magnetic properties of the→− system is independent of the X sign of the coupling constant J. Therefore in the following it suffices to consider only the ferromagnetic coupling (i.e., III. Results and discussion J 0). ≥ The magnetic properties are reflected by the magnetization In our Monte Carlo simulations, we measure all energies

1 z 1 in units of t and set the imaginary time step ∆τ =0.25 in Mz = Si,Rδi + (ni, ni, ) , (4) N h 2 ↑ − ↓ i the Trotter decomposition [4]. The inverse temperature is set i X to be β = N so that the temperature is sufficiently low to and the static magnetic susceptibility reveal the ground state properties. We apply periodic boundary condition and run at least 105 sweeps to obtain relevant χ = β( M 2 M 2). (5) h i−h i averages after thermalizing the system. In the following fig- Here N is the site number and β =1/kBT is the inverse ures, the lines are presented as a guide for the eye, and the temperature. Since the half-filled chain is an ideal system to statistical errors are within the corresponding symbol size. 29

Fig. 3. The CDW-SDW Phase diagram in parameter space

and N = 32 is plotted in Fig. 1b. The solid line connected open circles is the result of the extended Hubbard model at J = 0, where the transition occurs roughly at V 2.2. This agrees with Hirsh’s work [4]. When the ferromagnetic≈ coupling J = 3 is introduced, we can see from the dashed line Fig. 1. a The CDW ( mc ) SDW ( ms ) order parameter vs nearest- connected open squares that the phase transition point shifts neighbor coupling strength| V| for a 32-site| | chain. b The CDW order pa- to roughly V 2.5. This reveals readily that the radical rameter mc vs V for a 32-site chain | | coupling enhances≈ the SDW phase. In order to determine accurately the CDW-SDW transition point, we perform a finite-size scaling study of the CDW structure factor S(π) and the SDW susceptibility χ(π)asa function of the chain size. The results for U = 2 and U =4 are shown in Fig. 2. From Fig. 2a for U = 2, the system is still in the SDW phase at V =1.5 since χ(π) diverges with increasing size. From Figs. 2b and c where V =1.6, 2.0, respectively, we can see that the CDW structure factor diverges with increasing size. Therefore, the transition point can be estimated to be V 1.55. For U = 4 in Figs. 2d–f, we can see by similar reasoning≈ that the ferromagnetic coupling leads to a transition point V 2.53. For comparison, the SDW-CDW transition for J =≈ 0 in [4], occurs at V 1.15 for U = 2 and V 2.18 for U = 4. Our results imply≈ that the ferromagnetic≈ coupling enhances the SDW phase in ac- cordance with the results of the order parameters in Fig. 1. In particular, the transition point at the same value of U has shifted by ∆V 0.4inV. The underlying mechanism will examined later.≈ After quite extensive calculations of the CDW structure factor S(π) and the SDW susceptibility χ(π) as functions of U, V and chain sizes, we summarize a phase diagram in Fig. 2a–f. The finite size scaling behavior of the CDW structure factor Fig. 3. The lines in Fig. 3 represent the CDW-SDW phase S(π) and the SDW zero-frequency susceptibility χ(π) vs lattice site N for transition lines. The solid line is the mean field result for a,b,c U = 2, and d,e,f U =4 J = 0, i.e., V = U/2; the dotted line is the corresponding result from Monte Carlo simulations by Hirsh [4] and re- produced by our Monte Carlo code. The long dashed line First, we present both the CDW and SDW order param- shows the mean field result at a coupling J = 3, namely, eters for U = 2 in Fig. 1a. The CDW order parameter mc V = U/2+J/8 (see below). The open circles connected by jumps up at V 1.55, while the SDW order parameter ms the short dashed line are the corresponding Monte Carlo re- drops to 0 at the∼ same V . This indicates that the CDW-SDW sult. It can be seen that the CDW-SDW transition line shifts transition occurs at V 1.55, and the transition points ex- upward in V as compared to the mean field result. More- tracted from the CDW∼ and SDW order parameters coincide. over, the transition line near U = 0 develops a slight upward Moreover, these two phases are observed to never coexist. curvature, in contrast to the situation at J =0. Thus, we use only the CDW order parameter in the follow- By the histograms of the CDW order parameter, Hirsh ing. At J = 3 and J = 0, the CDW order parameter for U =4 [4] found that the character of the CDW-SDW transition 30

Fig. 4a–d. Histograms of the absolute value of CDW order parameter on the CDW-SDW transition line, a for U =4,bfor U =2,cfor U =0.2 dfor U =0.1 depends on the electron interaction strength in their case Fig. 5. a The magnetization M vs V , b The magnetic susceptibility χ vs (J = 0). For U<3, the CDW-SDW transition is continuous, V but it becomes first order for U>3. It is interesting to find how the character of the CDW-SDW transition would be affected by the radical coupling in our case. For this purpose, U J Along the mean-field phase transition line (V = 2 + 8 ), we present the histograms of the CDW order parameter in J ∆E = ECDW ESDW = > 0. This implies that Fig. 4 for four points on the phase transition line in the U V − V (2V +J) − the system favors the SDW phase as long as the quantum parameter space (see Fig. 3). It can be seen that the double- fluctuation is taken into account, and agrees qualitatively peak structure occurs in two cases in Figs. 4a and b. This with the Monte Carlo results for J = 3 in Fig. 3. Moreover, indicates that there are two metastable states at the transition the smaller V is, the larger ∆E is. This is clearly consistent and the phase transition is clearly of the first order. With with the slight upward curvature in our Monte Carlo results decrease of the on-site interaction of π electrons, the double- in Fig. 3 for J =3. peak structure becomes weak and disappears in Figs. 4c and d. This signals a continuous phase transition. We estimate In order to understand the character of the phase transition, we can employ a ‘droplet’ argument in Hirsh’s line [4]. that Uc 0.2 for J = 3. Evidently, with the increase in J, the critical≈ point moves from U =3toU <3, and would Suppose the system is in the CDW phase near the phase tran- c c sition. The low-lying excitation are ‘droplets’ of the SDW eventually end up at Uc =0. phase, with an energy (n)=V C(J)t n(U 2V+J/4) We can understand the above results as follows. In the − − − strong coupling limit, we can adopt a mean field treatment for a droplet of size n, where V acts as a surface energy, and C(J)t follows from the kinetic energy lowering [4]. When by neglecting the kinetic energy term in (1). The ground state − consists of either electron pairs occupying alternating lattice V is now decreased slightly till 2Vnc is energetically favorable, and that the CDW-SDW transition occurs at V = U/2+− J/8 (see the CDW phase can tunnel to the SDW phase by nucleation. the long dashed line in Fig. 3). The kinetic energy lowering However, if V C(J)t 0, droplets of any size is favorable, in these states can be estimated by a second-order perturba- and the transition− becomes≤ continuous. The critical point is tion theory. The ground state energy in the CDW phase now then given by Vc C(J)t leading to Uc 2C(J)t J/4. ≈ ≈ − reads to the second order in t, In the absence of radical coupling, Hirsh [4] estimated that (0) 1, leading to 2 3. In our case, since a finite 2 2 C Uc U t t radical∼ coupling J lifts the≈ spin-up∼ and spin-down degener- ECDW = N( ). (11) 2 − 3V U J/4 − 3V U + J/4 acy, the quantum fluctuation is suppressed. This is also clear − − − from (11) and (12). As a result, the kinetic energy lowering On the other hand, Since the ground state in the SDW phase decreases in magnitude, or C(J) decreases with increasing is a definite spin configuration state( , , , , ...) for J>0, J. This naturally explains why the critical point U decreases instead of multi-fold degenerated states↑ ↓ for↑ ↓J = 0 (in which c with increasing J. To be more specific, if C(J =3) 0.5, case one would need to map the Hubbard Hamiltonian to we have U 0.25, which is appealing as compared∼ to our the t J model and analyses the ground state energy by the c ∼ − simulation result Uc 0.2. Bethe Ansatz, [4]) we can also use the simplest perturbation ∼ theory to obtain the ground state energy in the SDW phase, At this stage, we proceed to discuss the ferromagnetic properties. The magnetization M and susceptibility χ versus J 2t2 V at fixed U = 2 and J = 3 are shown in Fig. 5, where we E = N(V ) . (12) SDW − 8 − U V + J/4 see that the magnetization drops sharply from 0.25 to 0 as the − 31

ness reaches a maximum at the CDW-SDW transition line and gets reduced as the system is driven away from the transition line. Our results in Figs. 6c is in full agreement with this scenario: As the radical coupling increases, the system goes away from the transition line so that the charge stiffness decreases. In closing this section, we wish to point out that the SDW-CDW transition line in Fig. 3 also serves as a ferromagnetic-paramagnetic phase transition line due to the co- existence of SDW phase and the ferromagnetic phase.

IV. Summary

Using the world line quantum Monte Carlo method, we have investigated the CDW-SDW phase transition and magnetic properties in a quasi-one-dimension organic polymer. The magnetic coupling between the itinerant π electron and the localized d electrons enhance the SDW phase, and the ferromagnetic order coexists with the SDW phase. On the other hand, the critical point Uc that separates the first order and the second order transitions decreases with increasing coupling J. A phase diagram has also been obtained, and has been explained qualitatively. In the present paper, we have neglected the electron- phonon interaction, which is well-known to be able to cause a Peierls instability, dimmerizing the polymer chain. This effect could be, of course, included in our simulations. The phonon modes influence the electrons only through the hopping matrix t, (in the context of the SSH Hamiltonian), lead- Fig. 6. a The CDW structure factor S(π) and the SDW zero-frequency χ(π) ing to a site-dependent hopping matrix. In the adiabatic limit susceptibility vs J, b The magnetization M vs J; c The charge shiftiness ~ω t where ω is the Debye frequency of the lattice, of the system vs J. Note that U = 3 and V =1.6 correspond to a transition D D point at J =0 the electron-phonon interaction amounts to a quenched dis- order in t, which is expected to suppress SDW order [3]. Nevertheless, we believe that this effect should not change system goes from SDW phase to CDW phase. Correspond- our qualitative conclusion on the interplay between the π π electron interaction and the π d electron interaction. − ingly, a singular peak appears at the transition point. Clearly, − the ferromagnetic order coexists with the SDW phase. This is understandable since in the CDW state, the d electrons This work was supported by the RGC grant of Hong Kong, under Grant are decoupled from the π electrons effectively. However, in No. HKU 262/95P. the SDW state, the singly occupied electrons can arrange themselves in such a way that the spins of d electrons are polarized. Since the d electrons are situated at every other References site, the most favorable configuration is exactly a sponta- neous aligning of all d electron spins coupled to a SDW 1. Y.V. Korshak, T.V. Medvedeva, A.A. Ovchinnikov, V.N. Spector: state ( , , , , ...). In this way, the SDW phase and the Nature (London) 326, 370 (1987); A.A. Ovchinnikov, V.N. Spector: ferromagnetic↑ ↓ ↑ ↓ ordering of the electrons are mutually sta- Synth. Met. 27, B615 (1988) d 2. Y. Cao, P. Wang, Z. Hu, S.Z. Li, L.Y. Zhang, J.G. Zhao: Synth. Met. bilized. 27, B625 (1988) Finally, we examine the behavior of the system as a 3. Z. Fang, Z.L. Liu, K.L. Yao: Phys. Rev. B 49, 3916 (1994); Z. Fang, function of J. To be specific, we choose U = 3, and V =1.6, Z.L. Liu, K.L. Yao, Z.G. Li, ibid. 1304 (1995); Z. Fang, K.L. Yao, which lies on the CDW side near the transition line in the Z.L. Liu: Z. Phys. B 99, 425 (1996) 4. J.E. Hirsch: Phys. Rev. Lett. 53, 2327 (1984); Phys. Rev. B 31, 6022 case of J = 0. In Figs. 6a–c we plot the J dependence of (1985); J.E. Hirsh: ‘Low-dimensional conductors and superconductors’ S(π) and χ(π), M and the charge stiffness Dc, respectively. p.71 D.Jerme, L.G. Carbon (eds.) New York: Plenum Press 1986 It can be seen from Fig. 6a that the CDW structure factor 5. K. Nasu: Phys. Rev. B 33, 330 (1986) S(π) decreases with increasing coupling J, while the SDW 6. J.E. Hirsch, R.L. Sugar, D.J. Scalapino, R. Blankenbecler: Phys. Rev. susceptibility χ(π) increases drastically. In Fig. 6b, the mag- B 26, 5033 (1982) netization also increases with increasing coupling J, and sat- 7. G. Batrouni, R. Scalettar: Phys. Rev. B 46, 9051 (1992); R. Romer,¨ A. Punnoose: Phys. Rev. B 52, 14809 (1995) urates when J>3. The radical coupling is clearly in favor of 8. Qianghua Wang, Z.D. Wang, Jian-Xin Zhu: Phys Rev. B 54, 8101 the SDW phase. In Fig. 6c, the charge stiffness is suppressed (1996); Qianghua Wang, Jian-Xin Zhu, Z.D. Wang: J. Phys., Condens. with increase in J. In [8], it was found that the charge stiff- Matter L 413 (1996) 32

9. M. Buttiker,¨ Y. Imry, Landauer: Phys. Lett. 96A, 365 (1983); R. Lan- 10. Jian-Xin Zhu, Z.D. Wang: J. Phys. Condens. Matter, 6, L329 (1994); dauer, M. Buttiker:¨ Phys. Rev. Lett. 54, 2049 (1985); D. Mailly, C. J. Phys. A Gen. and Math. 28 L875 (1994); Z.D. Wang, Jian-Xin Zhu: Chapelie, A. Benoit: ibid. 2020 (1993); L.P. Levy, G. Dolan, J. Dun- Phys. Rev. B 52, 5275 (1995); Shi-Dong Liang Z.D. Wang, Jian-Xin smuir, H. Bouchiat: ibid. 2074 (1990); V. Chandraskhar, R.A. Webb, Zhu: Solid State Commun. 98, 909 (1996) N.J. Brady, M.B. Ketchen, W.J. Gallagher, A. Kleinsaser: ibid. 3578 (1991)