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Polarons in organic ferromagnets
Organic Electronics 55 (2018) 133–139
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Polarons in organic ferromagnets
T
Yuanyuan Miao, Shuai Qiu, Guangping Zhang, Junfeng Ren, Chuankui Wang, Guichao Hu
∗
Shandong Province Key Laboratory of Medical Physics and Image Processing Technology, School of Physics and Electronics, Shandong Normal University, Jinan 250100, China
A R T I C L E I N F O
A B S T R A C T
Keywords: Organic spintronics Organic ferromagnet Polaron
With an Anderson-like model including the electron-lattice interaction, the polaron formation in quasi-onedimensional organic ferromagnets with spin radicals is investigated theoretically. In the presence of an additional electron, distortions of spin and charge densities are observed both in the main chain and radicals. A polaron-like lattice distortion confined with fractional charges and spins may be formed in the main chain, where the amplitude is tuned by the electron hopping between the main chain and radicals and the electronelectron interaction. The confined spins and charges are inconsistent in position, which is caused by the asymmetric modification of charges and spins from the low-energy electrons triggered by the polaron lattice distortion. A phase diagram of the polaron formation is given with different electron hopping and electronelectron strengths. The case of a nonzero on-site energy difference between the radicals and the main chain is also discussed. A background charge density wave is found in the main chain due to the charge transfer between the main chain and spin radicals, which hampers the formation of the polaron.
1. Introduction
the hydrogen atoms are replaced alternately by spin radicals containing unpaired electrons. At ground state, the radical spins will form a ferromagnetic order due to the spin coupling between the radical spins and the spins of π electrons in the main chain [10,11]. Recently, the design of organic devices with the utilization of OFs has triggered lots of interests. For example, Yoo et al. [12] has measured the spin dependent electron transport characteristics of V[TCNE]x by connecting the molecule to Au electrodes, which obtained a 2.5% magnetoresistance. Li et al. [13] has studied the magnetoresistance of this material by using magnetic and nonmagnetic electrodes, where even a room temperature magnetoresistance effect was observed when an Alq3 layer was inserted between the magnetic molecule and the electrode. Several theoretical designs of devices were also proposed involving spin filtering and spin diode effects in the frame of quantum transport [14–17]. Prior to the transport study of the OF, the understanding of the charge and spin properties of the carriers is crucial. Due to the spin correlation with radicals, the existence of the above nonlinear excitations as well as their spin-charge properties in OFs is obscure. The picture of soliton has been investigated by Xiong et al. in the organic ferromagnetic polymer m-polydiphenylcarbene (m-PDPC) [18]. The polarons in the poly-BIPO were also discussed recently, where a distinct property of spin-charge disparity of the polaron is obtained [19]. However, in the previous work, an extended Su-Schrieffer-Heeger (SSH) model [20] combined with a Kondo term was used, where the unpaired electron on each radical is looked as a local spin without charge and
Organic spintronics, which manipulates the freedom of electronic spin based on organic materials, has attracted more and more attention in recent years [1–3]. Organic materials are promising in the development of flexible and low-costing electronic devices. Especially, the spinorbit and hyperfine interactions in organic materials are weak, which is advantaged for the spin transport and storage. Another characteristic of organic materials is the strong electron-lattice (e-l) interaction. In the presence of an extra electron, it is known that nonlinear excitations such as polarons, bipolarons or solitons may be formed in polymers [4,5], where the excess charges are surrounded by the lattice distortion and can move along the polymer under an electric field. Polarons are particularly important as a polaron contains one unit charge and spin, which may serve as efficient spin and charge carriers simultaneously in most organic conjugated materials. The formation, injection, and transport of polarons in polymers have been investigated extensively in past decades [6–9]. Organic ferromagnets (OFs) are especially fascinating in organic spintronics, since they combine both the organic and ferromagnetic properties. OFs are usually fabricated artificially by introducing various paramagnetic centers with unpaired electrons, such as transition metal ions or organic spin radicals. The latter is a valid way to obtain pure OFs, such as poly-(1,4-bis(2,2,6,6-tetramethyl-4-piperidyl-1-oxyl)-butadiin (poly-BIPO). poly-BIPO is synthesized from polyacetylene, where
∗
Corresponding author. E-mail address: [email protected] (G. Hu).
https://doi.org/10.1016/j.orgel.2018.01.020 Received 15 November 2017; Received in revised form 2 January 2018; Accepted 21 January 2018 Available online 31 January 2018 1566-1199/ © 2018 Elsevier B.V. All rights reserved.
Organic Electronics 55 (2018) 133–139
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With Hartree-Fork approximation to treat the e-e interaction, the electronic states of the OF can be obtained by solving the eigenequations of the system with an initial lattice configuration:
spin transfer with the main chain. This is incomplete to understand the polarons in real OFs because of the possible spin and charge exchange between the main chain and radicals. In this paper, an improved model, Anderson-like model, is used to study the polarons in the OF, where not only the electron hopping between the main chain and the radicals is included, but also the difference of their on-site energies is considered. The focus is that when an electron is doped into the OF, whether a polaron can be formed and how about the spin and charge relation in the case of the nonlocal spin model for the radicals. The paper is organized as follows. In Section 2, we introduced the model Hamiltonian for the OFs and the details of the theoretical method. The results of the numerical calculations are presented and analyzed in Section 3, where a simple case without on-site energy difference between the main chain and radicals is first discussed including the effects of electron hopping term and electron-electron (e-e) interaction, and then a non-zero on-site energy difference is considered. In Section 4, a brief summary is given.
εμ, σ Zμ, i, σ = −(t0 − αyi ) Zμ, i + 1, σ − (t0 − αyi − 1) Zμ, i − 1, σ + U0 ni, −σ Zμ, i, σ − tmR δi, o Zμ, iR, σ
εμ, σ Zμ, iR, σ = ε1 Zμ, iR, σ − tmR δi, o Zμ, i, σ + U1 niR, −σ Zμ, iR, σ .
n i, σ =
yi = −
+
∑i yi 2 (2)
+ U1 ∑ i
2α K
∑
Zμ, i, σ Zμ, i + 1, σ +
μ, σ , occ .
2α N1 K
N1
∑ ∑ i = 1 μ, σ , occ .
Zμ, i, σ Zμ, i + 1, σ . (7)
3. Results and discussion 3.1. Neutral state at ε = 0 The system considered here contains 80 carbon atoms (N1 = 80) in the main chain and 40 radical sites. We start from a simple case by neglecting the on-site energy difference between the carbon atoms and the radical sites, that is, ε = 0 . The neutral ground state is first calculated in the case of half filling. The e-e interaction u and the electron hopping term t1 between the main chain and radicals are taken as u= 0.6 and t1= 0.3. The lattice configuration, energy band, charge density and spin density are investigated and plotted in Fig. 2. Here a smoothed 1 lattice configuration is used defined as y͠ i = 4 (−1)i (2yi − yi + 1 − yi − 1) . It is found that the lattice is dimerized like most polymers. The energy bands consist of six bands with three spin-up bands and three spindown ones. The two highest and the two lowest cosine-like bands are slight spin-split near the Fermi energy (zero point) with an energy gap of about 2.4 eV. The left two flat bands appear in the gap with a spin splitting of about 0.8 eV. Compared with the normal polymer polyacetylene, the appearance of the two bands in the gap comes from the radicals. It has been proved that the spin splitting of the two flat bands originates from the e-e interaction [18]. In the case of half filling, only the two lowest cosine-like bands and one spin-down flat band in the gap are occupied. Thus, the molecule is ferromagnetic with net up spins S = 1/2 in one unit. The picture is consistent with others' calculations about magnetic property of various OFs using Anderson-like model [18,22–24]. The charge density and spin density at the ground state are given in Fig. 2(c) and (d). It is found that the charge density keeps zero, which is same to our previous work in the frame of local radical spin model [19]. The spin density demonstrates that the net up spins mainly distributes in the radicals with 0.47 spins per radical site. About 0.03 up
i.σ + + δi, o ciR , ↑ciR, ↑ ciR, ↓ciR, ↓.
(6)
1
∑ ε1 δi,o ciR+,σciR,σ−tmR ∑ δi,o (ci+,σciR,σ + ciR+,σci,σ ) i, σ
Zμ, iR, σ Zμ, iR, σ .
ρs, i (iR) = 2 (ni (iR), ↑ − ni (iR), ↓ ) in unit of ℏ . During the numerical calculations, the parameters are adopted as those for poly-BIPO [10,11,14,21]: t0 = 2.5 eV, α = 4.5 eV/Å and K = 21.0 eV/Å2. For simplicity, we set U0 = U1 = U , and introduce dimensionless interaction parameters u = U / t0 , t1 = tmR/ t0 , and ε = ε1/ t0 .
The first term describes the hopping of π electrons along the main carbon chain with N1 sites. t0 is the hopping integral of the π electrons along a uniform chain, and α denotes the e-l coupling constant. yi is the lattice distortion with yi ≡ ui + 1 − ui , where ui is the lattice displacement of the carbon atom at the ith site. ci+, σ (ci, σ ) denotes the creation (annihilation) operator of an electron at the ith site with spin σ. The spin index σ assumes the numerical values ↑ ≡ 1 and ↓ ≡ −1. The second term is the elastic energy of the lattice with elastic coefficient K due to the lattice distortion. The last term is the on-site e-e interaction between π electrons described with Hubbard model. U0 is the interaction strength. H1 describes the Hamiltonian of the side radicals and the electron hopping between the radicals and the main chain.
H1 =
∑ μ, occ .
Here a periodic boundary condition is adopted. The electron number is also recalculated with Eq. (6). Eqs. (4)–(7) need to be solved self-consistently until a stable state is achieved. In each iteration, the energy levels are reordered and the electrons occupy the lowest levels. After the iteration, the charge density ρc, i (iR) and spin density ρs, i (iR) along the polymer can be obtained, which are defined as ρc, i (iR) = 1 − (ni (iR), ↑ + ni (iR), ↓ ) in unit of elementary charge e > 0 and
Here H0 is the Hamiltonian of the main chain in the form of an extended SSH model [20], K 2
Zμ, i, σ Zμ, i, σ , niR, σ =
occ. means the occupied states of electrons. The eigenequations are solved with the initial lattice configuration yi and electron density ni (iR), σ . After obtaining the eigen function Zμ, i, σ , the new lattice configuration is calculated with
(1)
U0 ∑i ci+, ↑ci, ↑ ci+, ↓ci, ↓
∑ μ, occ .
Here we considered the quasi-one-dimensional OF poly-BIPO. As shown in Fig. 1, the molecule is composed of a π-conjugated carbon chain and spin radicals attached to the odd sites of the main chain. Each radical has an unpaired electron. An Anderson-like model is used to describe the OF while the e-l interaction in the main chain is included. The Hamiltonian of the whole system is written as
H0 = − ∑i, σ [t0 − αyi ](ci++ 1, σci, σ + ci+, σci + 1, σ ) +
(5)
Here εμ, σ is the eigenenergy and Zμ, i, σ is eigen function in the Wannier space. ni, σ is the average electron number on the ith site with spin σ,
2. Model and method
H = H0 + H1.
(4)
(3)
ε1 is the on-site energy of electrons at the radicals relative to the carbon atoms in the main chain. δi, o means that the radicals only hang on the odd sites of the main chain with δi, o = 1 (δi, o = 0 ) for odd (even) atoms. tmR is the electron hopping integral between the main chain and the side + radicals R. ciR , σ (ciR, σ ) denotes the creation (annihilation) operator of electron at the dangling site iR. U1 describes the on-site Hubbard repulsion strength at the dangling site.
Fig. 1. Simplified structure of quasi-one-dimensional organic ferromagnet.
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Fig. 2. Ground state of the organic ferromagnet with t1 = 0.3 , u = 0.6 , ε = 0 . (a) Smoothed lattice configuration, (b) spin-resolved energy bands, (c) charge density, and (d) spin density. The site number from 80 to 120 indicates the radicals, which corresponds to its adhere site i in the main chain by the relation of i = 2(iR − 80) − 1.
spins distributes in one unit of the main chain with about −0.01 on the site hanging radical and 0.04 on its neighbor site, which generates a spin density wave. The opposite spin density between the radicals and its adherent sites indicates an antiferromagnetic coupling for the electronic spins on the two sites. This picture is consistent with that of the local radical model [19,21], where an antiferromagnetic coupling between the spins on the two sites is assumed.
spin density of the neutral state from the charged one, which stands for the net spin density from the doped electron. The above plots indicate the formation of a polaron in the OF. Unlike the polaron in normal polymers, two distinct characteristics can be observed for the polaron in the OF: (i) The doped electron distributes in both the main chain and radicals. This is induced by the hopping term in the present model, which cannot be obtained in the calculation of a conserved local spin model for the radicals. As a result, a polaron with fractional charges and spins is formed in the main chain. (ii) The distribution of the charge density and spin density of the polaron is not consistent in the main chain, and both of them are not asymmetric about the center of polaron. Such spin-charge disparity was also found in the previous work with a local spin model of the radicals [19]. In the following, we focus on the relation of the polaron charge and spin density in present nonlocal radical spin model. To check the distribution of the doped electron, we first analyze the total quantity of the net charges and spins in the main chain and radicals, respectively. Here
3.2. Charged state at ε = 0 Now we are in the position of the charge state. An additional electron is assumed to occupy the lowest unoccupied molecular orbital (LUMO), which is a spin-down level in the gap according to the band scheme at neutral state. As shown in Fig. 3, a polaron-like lattice distortion is formed in the main chain. A concomitant distortion is also observed in the charge density and spin density shown in Fig. 3(b)–(d). Fig. 3(d) is the net spin density Δρs, i (iR) calculated by subtracting the
Fig. 3. Charged state of the organic ferromagnet. (a) Smoothed lattice configuration, (b) net charge density, (c) spin density and (d) net spin density calculated by subtracting the spin density of the neutral state from that of the charged one. The parameters are the same as those in Fig. 2.
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Fig. 4. Probability density ψμ, i, σ
2
except the doped one. The results are shown in Fig. 5. Compared with the neutral state, a distortion in the charge density and spin density is generated for the Fermi sea, which occurs both in the main chain and the radicals. Such distortion originates from the disturbance by the polaron lattice distortion as well as the coulombic repulsion from the additional electron. The distortions for the charge and spin densities are different, which brings distinct modification for the polaron spin and charge. This is a source for the spin-charge disparity of the polaron in the main chain. Another source for the spin-charge disparity is the nonuniform charge and spin transfers of the Fermi sea between the main chain and radicals in the presence of the polaron. A total charge transfer of about 0.001 is found from the radicals to the main chain, but the total spin transfer is up to 0.03 up spins. The roles of the charge density modification and the nonuniform charge (spin) transfer from the Fermi sea have not been observed in previous local radical spin model [19]. The electron hopping term t1 and the e-e interaction are key parameters in the nonlocal spin model of the OF. In the following, the effects of the two parameters on the net charge and spin distribution as well as the lattice distortion are discussed. In Fig. 6 (a) and (b), we fixed the e-e interaction with u= 0.6, and changed the strength of the hopping term t1 firstly. It is found that as t1 is increased from 0.1 to 0.5, the net charge (spin) Qm (Sm ) quantity in the main chain increases almost linearly from −0.25 (−0.12) to −0.97 (−0.52), and then keeps the maximum value approximately when t1 is over 0.5. The net charge (QR ) and spin (SR ) quantities in the radicals decrease correspondingly with t1. The results mean that the additional electron tends to locate in the main chain at a larger hopping term t1. The saturation of the change (spin) quantities in the main chain at t1 > 0.5 is reflected from the lattice distortion shown in Fig. 6(b). It is clearly seen that a polaron distortion arises and is strengthened as t1 increases from 0.1 to 0.4. However, the lattice distortion vanishes at t1 = 0.5 and then a uniform lattice appears. This situation holds for a larger t1 which is not shown here. The results indicate that a medium strength of t1 is helpful for the formation of polarons in the OF. The intrinsic physics can be understood as follows. In the limitation of zero t1, the additional electron with whole net charge and spin locates in the state contributed by the radicals. In the case of a nonzero t1, hybridization between the states of the radicals and the main chain happens. For a small t1, the hybridization is weak and the additional electron is not easy to transfer from the radicals to the main chain. So the charge-induced lattice distortion in the main chain is neglectable and the lattice keeps dimerized almost as that of the neutral
of the orbital occupied by the additional spin-down
electron.
the total charge and spin quantities in the main chain (radicals) are calculated as Qm (R) = ∑i (iR) Δρc, i (iR) and Sm (R) = ∑i (iR) Δρs, i (iR) . Δρc, i (iR) (Δρs, i (iR) ) is the difference of the charge (spin) density between the charged state and the neutral one. The results give Qm (R) = −0.618 (−0.382) and Sm (R) = −0.275 (−0.225). This means that for present parameters a large proportion of the additional charges distributes in the main chain bounded with the polaron distortion. However, the spin of the electron distributes almost equally in the main chain and radicals. This result means the spin-charge disparity property of the polaron emerges not only in the contour of spin and charge density curve, but also in the total quantity. To understand the nature of the spin-charge disparity of the polaron in the nonlocal radical spin model, we first plot the probability density of the orbital occupied by the additional electron, which is shown in Fig. 4. A nonlocal distribution close to the distribution of net charge density is observed. One can sum the total probabilities of the orbital in the main chain and radicals, which are 0.619 and 0.381, respectively. So the charge density of the polaron almost follows the distribution probability of the polaron level, while a larger difference appears for the polaron spin density. The intrinsic mechanism can be tracked from the contribution of the Fermi sea, that is, other occupied-state electrons
Fig. 5. Charge density and spin density of the Fermi sea in the presence of the polaron. (a) Charge density, (b) spin density in the main chain, and (c) spin density in the radicals.
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Fig. 6. Dependence of net charge and spin distribution as well as the lattice distortion on hopping term t1 and e-e interaction u. (a) Net charge Qm (R) and spin Sm (R) in the main chain (radicals) with t1. Here u is fixed with u= 0.6. (b) Smoothed lattice distortion y͠ i at several different values of t1. u= 0.6. (c) Net charge Qm (R) and spin
Sm (R) with u. Here t1 is fixed with t1= 0.3. (d) Smoothed lattice distortion y͠ i at several different values of u. t1= 0.3.
state. The charge (spin) transfer between the radicals and the main chain increases with t1, that is, with the enhancement of the hybridization. On the other hand, in the limitation of a large t1(> 0.5), the strong hybridization between the radicals and the main chain will drive the system far away from a quasi-one-dimensional chain, where the dimerization even disappears and of course the polaron cannot form. The role of e-e interaction is investigated and shown in Fig. 6(c) and (d), where the hopping term t1 is fixed at 0.3. It is found that in the involved region u< 2.0, which is reasonable for most polymers, the net charge and spin in the main chain increase with u while those in the radicals decrease correspondingly. The quantity of net charge (spin) Qm (Sm ) in the main chain changes from −0.53 (−0.23) to −0.98 (−0.51) as u is increased from 0.1 to 2.0. The lattice distortion in Fig. 6(d) also indicates an enhanced polaron distortion with u, where an obvious distortion begins to emerge when u> 0.4. Thus a stronger e-e interaction is advantageous to generate a polaron in the OF. This is reasonable since at the neutral state, the total spins in each radical site are up spins. By introducing an extra spin-down electron, the e-e interaction tends to repulse it into the main chain, which is beneficial to the formation of the polaron. The case of very large u is out of our interest, where an irregular lattice distortion is possible because of the strong columbic repulsion in a charged system. To give a comprehensive vision of the parameter dependence of the polaron formation, we drew a phase diagram in Fig. 7 by scanning the values of t1 from 0 to 1.0 and of u from 0 to 2 in a step of 0.1. For simplicity, we judge the existence of the polaron from the smoothed lattice distortion. If a polaron-like distortion exists, and the deviation Δy͠ i of the distortion from the neutral state meets Max( Δy͠ i ) > 10−2 Å, it belongs to the polaron region. In the case of gradual change, it is hard to exactly distinguish the formation of polarons. However, the rough definition is useful to demonstrate the dependence of polaron formation on the two parameters. The gray region in Fig. 7 indicates the existence of the polaron with above criterion, which gives a region between 0.1 ≤ t1 ≤ 0.4 and 0.5 ≤ u ≤ 2.0. Out of this region, the lattice is dimerized or uniform. Furthermore, we note that the net spin of the whole OF is conserved in this region with the value of −1/2, which means that spin flipping process does not occur in the iteration although it is permitted in our algorithm.
Fig. 7. The phase diagram of polaron formation with t1 and u. Other parameters are same as those in Fig. 2. The gray region means the existence of a polaron.
with heterocycles as the spin radicals, the unpaired electron usually locates on the oxygen or nitrogen atom [25]. Consequently, the electronegativity of the radical atom may be larger than the carbon atom. As a result, a negative on-site energy ε of the radicals is reasonable and considered here. With exact diagonalization technique, Wang has demonstrated that the Anderson-like model for the OF may give a ferromagnetic or antiferromagnetic ground state relying on the values of ε and u [23], although the e-l interaction is not included in that work. Here we only focus on the case of the ferromagnetic state, which is guaranteed by an approximate relation of − ε < u according to Wang's work. In the case of a nonzero ε , charge transfer between the main chain and radicals is possible, and may further modifies the ferromagnetic state. We start from a specific case with ε = −0.3. Other parameters are same as those in Fig. 2. The ground state of the OF is first investigated and shown in Fig. 8. Compared with the case without ε , the smoothed lattice configuration, energy bands and spin density are similar as those in Fig. 2, which still indicates a ferromagnetic state for the OF. The two flat bands in the gap are shifted downwards due to the negative on-site energy of radicals. The spin distribution in each unit
3.3. Neutral state at ε ≠ 0 Now we turn to the case of a nonzero ε in the model. In some OFs 137
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Fig. 8. Ground state of the organic ferromagnet with ε = −0.3 . Other parameters are same as those in Fig. 2. (a) Smoothed lattice configuration, (b) energy levels, (c) charge density, (d) spin density.
the formation of polaron in the main chain. To give a polaron picture in such case, a larger e-e interaction u is adopted to form the polaron. Fig. 9 shows a polaron picture with an e-e interaction of u = 1.5. Other parameters are same as Fig. 2 except ε . A localized polaron appears with a narrower width compared with that in Fig. 2. The spin-charge disparity property is conserved, where an extra spin-up peak appears in the net spin density in the polaron region. This spin-up peak originates from the spin transfer of the Fermi sea between the main chain and radicals under a strong e-e interaction. The total charge (spin) quantity of the polaron is calculated, which is about −0.75 (−0.37) in the main chain and −0.25 (−0.12) in the radicals. The dependence of the net charge and spin distribution as well as the lattice distortion on the value of ε is further investigated. As shown in Fig. 10, when ε changes from 0 to −1.0, the net charge quantity Qm in the main chain decreases from −0.91 to −0.04, while QR in the radicals increases from −0.09 to −0.96 correspondingly. This is natural because almost all the charges of the additional electron will locate in the radicals if the magnitude of ε is large enough. The total net spin quantity in the radicals is also increased from −0.05 to −0.5 with ε .
cell changes slightly with 0.46 up spins on the radical site and 0.04 up spins on the atoms of the main chain. The significant change is that a charge density wave arises along the main chain, while a uniform negative charge density appears on the radicals. For each unit cell, −0.03 charges distribute on the radical site, which means electrons are transferred from the main chain to the radicals. 0.01 positive charges are found on the site hanging the radical, and 0.02 on its neighbor site. Thus, there exist 1.2e positive charges in total in the main chain. At ground state, it is found that the transferred charges exist in an extend form without any local distortion in the main chain. The lattice keeps dimerization which is slightly weakened by the background charge density wave.
3.4. Charged state at ε ≠ 0 Then the charged state is investigated by introducing an additional electron into the polymer. We found that the above parameters cannot form a polaron. This is because that in the presence of a negative ε , the electron tends to occupy the radical sites, which is disadvantageous for
Fig. 9. Polaron state of the organic ferromagnet for ε = −0.3 and u = 1.5. Other parameters are same as Fig. 2. (a) Lattice distortion, (b) net charge density, (c) spin density and (d) net spin density.
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formation of the polaron. In the presence of a nonzero on-site energy of the radicals, charge transfer of the Fermi sea between the main chain and radicals occurs, which forms a charge density wave and modifies the lattice distortion of the main chain. A large on-site energy of the radicals is not good for the formation of the polaron in the main chain. This work is helpful for us to understand the distinct spin-charge properties of polarons in OFs. At last, let us comment on the possible experimental probes on the polarons in OFs. In the case of large doped single crystals of OFs, the lattice distortion and spin density could be probed by neutron scattering. The charge density could be accessible with X-ray scattering. In the case of a single-molecule measurement, an X-ray free-electron laser could determine the charge and spin distribution of single molecules in the gas phase. Spin-resolved scanning tunneling microscopy is another direct way to detect the charge and spin density of the molecule deposited on a surface, although the substrate effect will be brought in. Finally, indirect evidence from magnetotransport measurements is expected when the OF is contacted with ferromagnetic electrodes. Acknowledgments Fig. 10. (a) Charge quantity Qm (R) , spin quantity Sm (R) in the main chain (radicals), and (c) lattice distortion with various values of ε . Here t1= 0.3, u= 1.5.
Support from the Natural Science Foundation of Shandong Province (Grant No. ZR2014AM017), the National Natural Science Foundation of China (Grant Nos. 11374195, 11704230, 11674197), and the Taishan Scholar Project of Shandong Province are gratefully acknowledged.
During the increase of ε , whether the polaron is stable can be examined from the lattice distortion shown in Fig. 10(b). A distinct polaron lattice distortion is observed from 0 to −0.4 of ε , where the maximum distortion at the center of the polaron region relative to the dimerization decreases quickly with ε . When ε is beyond −0.6, the lattice distortion at the polaron center becomes quite small. At the same time, the dimerization of the lattice out of the polaron region is reduced, which is due to the more and more charges transfer of Fermi sea between the main chain and radicals. At ε = −1.0, only a bit irregular lattice distortion is left in the previous polaron region, which is not a normal polaron anymore.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12]
4. Summary In summary, we have studied the spin-charge properties as well as its formation condition of polarons in the OFs by doping an extra electron into the magnetic polymer. The results are calculated in two regimes: with or without on-site energy difference between the main chain and radicals. We found that in both cases, a polaron with fractional charges and spins can be formed. The quantities of the spins and charges bound in the lattice distortion depend on the strength of the electron hopping term between the main chain and radicals, and also on the e-e interaction. A spin-charge disparity of the polaron is observed, which is attributed to the asymmetric modification for the charge and spin density by the Fermi sea in the presence of polaron lattice distortion. The dependence of the charge and spin distribution of the polaron on the hopping term and e-e interaction is also calculated, where a medium hopping strength and a large e-e interaction is helpful for the
[13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25]
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W.J.M. Naber, S. Faez, W.G. van der Wiel, J. Phys. D Appl. Phys. 40 (2007) R205. V.A. Dediu, L.E. Hueso, I. Bergenti, C. Taliani, Nat. Mater. 707 (2009) 8. T. Sugawara, M.M. Matsushita, J. Mater. Chem. 19 (2009) 1738. A.J. Heeger, S. Kivelson, J.R. Schrieffer, W.P. Su, Rev. Mod. Phys. 60 (1998) 781. W.P. Su, J.R. Schrieffer, A.J. Heeger, Phys. Rev. B 22 (1980) 2099. X.J. Liu, K. Gao, Y. Li, J.Y. Fu, J.H. Wei, S.J. Xie, Synth. Met. 157 (2007) 380. S.J. Xie, K.H. Ahn, D.L. Smith, A.R. Bishop, A. Saxena, Phys. Rev. B 67 (2003) 125202. E.M. Conweil, S.V. Rakhmanova, Appl. Phys. Lett. 75 (1999) 1518. Åsa Johansson, Sven Stafström, Phys. Rev. Lett. 86 (2001) 3602. Z. Fang, Z.L. Liu, K.L. Yao, Phys. Rev. B 49 (1994) 3916. Z. Fang, Z.L. Liu, K.L. Yao, Z.G. Li, Phys. Rev. B 51 (1995) 1304. J.W. Yoo, R.S. Edelstein, D.M. Lincoln, N.P. Raju, C. Xia, K.I. Pokhodnya, J.S. Miller, A.J. Epstein, Phys. Rev. Lett. 97 (2006) 247205. B. Li, M. Zhou, Y. Lu, C.Y. Kao, J.W. Yoo, V.N. Prigodin, A.J. Epstein, Org. Electron. 13 (2012) 1261. G.C. Hu, Y. Guo, J.H. Wei, S.J. Xie, Phys. Rev. B 75 (2007) 165321. W.Z. Wang, Phys. Rev. B 73 (2006) 235325. G.C. Hu, K.L. He, S.J. Xie, A. Saxena, J. Chem. Phys. 129 (2008) 234708. G.C. Hu, Z. Zhang, Y. Li, J.F. Ren, C.K. Wang, Chin. Phys. B 25 (2016) 057308. B. Xiong, W.Z. Wang, Phys. Rev. B 71 (2005) 174431. G.C. Hu, H. Wang, J.F. Ren, S.J. Xie, C. Timm, Org. Electron. 15 (2014) 118. W.P. Su, J.R. Schrieffer, A.J. Heeger, Phys. Rev. Lett. 42 (1979) 1698. S.J. Xie, J.Q. Zhao, J.H. Wei, S.G. Wang, L.M. Mei, S.H. Han, Europhys. Lett. 50 (2000) 635. W.Z. Wang, Phys. Rev. B 71 (2005) 094407. W.Z. Wang, Phys. Rev. B 73 (2006) 035118. Guang-Shan Tian, Tsung-Han Lin, Phys. Rev. B 53 (1996) 8196. A. Alberola, M. Pilkington, Curr. Org. Synth. 6 (2009) 66.
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Organic Electronics journal homepage: www.elsevier.com/locate/orgel
Polarons in organic ferromagnets
T
Yuanyuan Miao, Shuai Qiu, Guangping Zhang, Junfeng Ren, Chuankui Wang, Guichao Hu
∗
Shandong Province Key Laboratory of Medical Physics and Image Processing Technology, School of Physics and Electronics, Shandong Normal University, Jinan 250100, China
A R T I C L E I N F O
A B S T R A C T
Keywords: Organic spintronics Organic ferromagnet Polaron
With an Anderson-like model including the electron-lattice interaction, the polaron formation in quasi-onedimensional organic ferromagnets with spin radicals is investigated theoretically. In the presence of an additional electron, distortions of spin and charge densities are observed both in the main chain and radicals. A polaron-like lattice distortion confined with fractional charges and spins may be formed in the main chain, where the amplitude is tuned by the electron hopping between the main chain and radicals and the electronelectron interaction. The confined spins and charges are inconsistent in position, which is caused by the asymmetric modification of charges and spins from the low-energy electrons triggered by the polaron lattice distortion. A phase diagram of the polaron formation is given with different electron hopping and electronelectron strengths. The case of a nonzero on-site energy difference between the radicals and the main chain is also discussed. A background charge density wave is found in the main chain due to the charge transfer between the main chain and spin radicals, which hampers the formation of the polaron.
1. Introduction
the hydrogen atoms are replaced alternately by spin radicals containing unpaired electrons. At ground state, the radical spins will form a ferromagnetic order due to the spin coupling between the radical spins and the spins of π electrons in the main chain [10,11]. Recently, the design of organic devices with the utilization of OFs has triggered lots of interests. For example, Yoo et al. [12] has measured the spin dependent electron transport characteristics of V[TCNE]x by connecting the molecule to Au electrodes, which obtained a 2.5% magnetoresistance. Li et al. [13] has studied the magnetoresistance of this material by using magnetic and nonmagnetic electrodes, where even a room temperature magnetoresistance effect was observed when an Alq3 layer was inserted between the magnetic molecule and the electrode. Several theoretical designs of devices were also proposed involving spin filtering and spin diode effects in the frame of quantum transport [14–17]. Prior to the transport study of the OF, the understanding of the charge and spin properties of the carriers is crucial. Due to the spin correlation with radicals, the existence of the above nonlinear excitations as well as their spin-charge properties in OFs is obscure. The picture of soliton has been investigated by Xiong et al. in the organic ferromagnetic polymer m-polydiphenylcarbene (m-PDPC) [18]. The polarons in the poly-BIPO were also discussed recently, where a distinct property of spin-charge disparity of the polaron is obtained [19]. However, in the previous work, an extended Su-Schrieffer-Heeger (SSH) model [20] combined with a Kondo term was used, where the unpaired electron on each radical is looked as a local spin without charge and
Organic spintronics, which manipulates the freedom of electronic spin based on organic materials, has attracted more and more attention in recent years [1–3]. Organic materials are promising in the development of flexible and low-costing electronic devices. Especially, the spinorbit and hyperfine interactions in organic materials are weak, which is advantaged for the spin transport and storage. Another characteristic of organic materials is the strong electron-lattice (e-l) interaction. In the presence of an extra electron, it is known that nonlinear excitations such as polarons, bipolarons or solitons may be formed in polymers [4,5], where the excess charges are surrounded by the lattice distortion and can move along the polymer under an electric field. Polarons are particularly important as a polaron contains one unit charge and spin, which may serve as efficient spin and charge carriers simultaneously in most organic conjugated materials. The formation, injection, and transport of polarons in polymers have been investigated extensively in past decades [6–9]. Organic ferromagnets (OFs) are especially fascinating in organic spintronics, since they combine both the organic and ferromagnetic properties. OFs are usually fabricated artificially by introducing various paramagnetic centers with unpaired electrons, such as transition metal ions or organic spin radicals. The latter is a valid way to obtain pure OFs, such as poly-(1,4-bis(2,2,6,6-tetramethyl-4-piperidyl-1-oxyl)-butadiin (poly-BIPO). poly-BIPO is synthesized from polyacetylene, where
∗
Corresponding author. E-mail address: [email protected] (G. Hu).
https://doi.org/10.1016/j.orgel.2018.01.020 Received 15 November 2017; Received in revised form 2 January 2018; Accepted 21 January 2018 Available online 31 January 2018 1566-1199/ © 2018 Elsevier B.V. All rights reserved.
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With Hartree-Fork approximation to treat the e-e interaction, the electronic states of the OF can be obtained by solving the eigenequations of the system with an initial lattice configuration:
spin transfer with the main chain. This is incomplete to understand the polarons in real OFs because of the possible spin and charge exchange between the main chain and radicals. In this paper, an improved model, Anderson-like model, is used to study the polarons in the OF, where not only the electron hopping between the main chain and the radicals is included, but also the difference of their on-site energies is considered. The focus is that when an electron is doped into the OF, whether a polaron can be formed and how about the spin and charge relation in the case of the nonlocal spin model for the radicals. The paper is organized as follows. In Section 2, we introduced the model Hamiltonian for the OFs and the details of the theoretical method. The results of the numerical calculations are presented and analyzed in Section 3, where a simple case without on-site energy difference between the main chain and radicals is first discussed including the effects of electron hopping term and electron-electron (e-e) interaction, and then a non-zero on-site energy difference is considered. In Section 4, a brief summary is given.
εμ, σ Zμ, i, σ = −(t0 − αyi ) Zμ, i + 1, σ − (t0 − αyi − 1) Zμ, i − 1, σ + U0 ni, −σ Zμ, i, σ − tmR δi, o Zμ, iR, σ
εμ, σ Zμ, iR, σ = ε1 Zμ, iR, σ − tmR δi, o Zμ, i, σ + U1 niR, −σ Zμ, iR, σ .
n i, σ =
yi = −
+
∑i yi 2 (2)
+ U1 ∑ i
2α K
∑
Zμ, i, σ Zμ, i + 1, σ +
μ, σ , occ .
2α N1 K
N1
∑ ∑ i = 1 μ, σ , occ .
Zμ, i, σ Zμ, i + 1, σ . (7)
3. Results and discussion 3.1. Neutral state at ε = 0 The system considered here contains 80 carbon atoms (N1 = 80) in the main chain and 40 radical sites. We start from a simple case by neglecting the on-site energy difference between the carbon atoms and the radical sites, that is, ε = 0 . The neutral ground state is first calculated in the case of half filling. The e-e interaction u and the electron hopping term t1 between the main chain and radicals are taken as u= 0.6 and t1= 0.3. The lattice configuration, energy band, charge density and spin density are investigated and plotted in Fig. 2. Here a smoothed 1 lattice configuration is used defined as y͠ i = 4 (−1)i (2yi − yi + 1 − yi − 1) . It is found that the lattice is dimerized like most polymers. The energy bands consist of six bands with three spin-up bands and three spindown ones. The two highest and the two lowest cosine-like bands are slight spin-split near the Fermi energy (zero point) with an energy gap of about 2.4 eV. The left two flat bands appear in the gap with a spin splitting of about 0.8 eV. Compared with the normal polymer polyacetylene, the appearance of the two bands in the gap comes from the radicals. It has been proved that the spin splitting of the two flat bands originates from the e-e interaction [18]. In the case of half filling, only the two lowest cosine-like bands and one spin-down flat band in the gap are occupied. Thus, the molecule is ferromagnetic with net up spins S = 1/2 in one unit. The picture is consistent with others' calculations about magnetic property of various OFs using Anderson-like model [18,22–24]. The charge density and spin density at the ground state are given in Fig. 2(c) and (d). It is found that the charge density keeps zero, which is same to our previous work in the frame of local radical spin model [19]. The spin density demonstrates that the net up spins mainly distributes in the radicals with 0.47 spins per radical site. About 0.03 up
i.σ + + δi, o ciR , ↑ciR, ↑ ciR, ↓ciR, ↓.
(6)
1
∑ ε1 δi,o ciR+,σciR,σ−tmR ∑ δi,o (ci+,σciR,σ + ciR+,σci,σ ) i, σ
Zμ, iR, σ Zμ, iR, σ .
ρs, i (iR) = 2 (ni (iR), ↑ − ni (iR), ↓ ) in unit of ℏ . During the numerical calculations, the parameters are adopted as those for poly-BIPO [10,11,14,21]: t0 = 2.5 eV, α = 4.5 eV/Å and K = 21.0 eV/Å2. For simplicity, we set U0 = U1 = U , and introduce dimensionless interaction parameters u = U / t0 , t1 = tmR/ t0 , and ε = ε1/ t0 .
The first term describes the hopping of π electrons along the main carbon chain with N1 sites. t0 is the hopping integral of the π electrons along a uniform chain, and α denotes the e-l coupling constant. yi is the lattice distortion with yi ≡ ui + 1 − ui , where ui is the lattice displacement of the carbon atom at the ith site. ci+, σ (ci, σ ) denotes the creation (annihilation) operator of an electron at the ith site with spin σ. The spin index σ assumes the numerical values ↑ ≡ 1 and ↓ ≡ −1. The second term is the elastic energy of the lattice with elastic coefficient K due to the lattice distortion. The last term is the on-site e-e interaction between π electrons described with Hubbard model. U0 is the interaction strength. H1 describes the Hamiltonian of the side radicals and the electron hopping between the radicals and the main chain.
H1 =
∑ μ, occ .
Here a periodic boundary condition is adopted. The electron number is also recalculated with Eq. (6). Eqs. (4)–(7) need to be solved self-consistently until a stable state is achieved. In each iteration, the energy levels are reordered and the electrons occupy the lowest levels. After the iteration, the charge density ρc, i (iR) and spin density ρs, i (iR) along the polymer can be obtained, which are defined as ρc, i (iR) = 1 − (ni (iR), ↑ + ni (iR), ↓ ) in unit of elementary charge e > 0 and
Here H0 is the Hamiltonian of the main chain in the form of an extended SSH model [20], K 2
Zμ, i, σ Zμ, i, σ , niR, σ =
occ. means the occupied states of electrons. The eigenequations are solved with the initial lattice configuration yi and electron density ni (iR), σ . After obtaining the eigen function Zμ, i, σ , the new lattice configuration is calculated with
(1)
U0 ∑i ci+, ↑ci, ↑ ci+, ↓ci, ↓
∑ μ, occ .
Here we considered the quasi-one-dimensional OF poly-BIPO. As shown in Fig. 1, the molecule is composed of a π-conjugated carbon chain and spin radicals attached to the odd sites of the main chain. Each radical has an unpaired electron. An Anderson-like model is used to describe the OF while the e-l interaction in the main chain is included. The Hamiltonian of the whole system is written as
H0 = − ∑i, σ [t0 − αyi ](ci++ 1, σci, σ + ci+, σci + 1, σ ) +
(5)
Here εμ, σ is the eigenenergy and Zμ, i, σ is eigen function in the Wannier space. ni, σ is the average electron number on the ith site with spin σ,
2. Model and method
H = H0 + H1.
(4)
(3)
ε1 is the on-site energy of electrons at the radicals relative to the carbon atoms in the main chain. δi, o means that the radicals only hang on the odd sites of the main chain with δi, o = 1 (δi, o = 0 ) for odd (even) atoms. tmR is the electron hopping integral between the main chain and the side + radicals R. ciR , σ (ciR, σ ) denotes the creation (annihilation) operator of electron at the dangling site iR. U1 describes the on-site Hubbard repulsion strength at the dangling site.
Fig. 1. Simplified structure of quasi-one-dimensional organic ferromagnet.
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Fig. 2. Ground state of the organic ferromagnet with t1 = 0.3 , u = 0.6 , ε = 0 . (a) Smoothed lattice configuration, (b) spin-resolved energy bands, (c) charge density, and (d) spin density. The site number from 80 to 120 indicates the radicals, which corresponds to its adhere site i in the main chain by the relation of i = 2(iR − 80) − 1.
spins distributes in one unit of the main chain with about −0.01 on the site hanging radical and 0.04 on its neighbor site, which generates a spin density wave. The opposite spin density between the radicals and its adherent sites indicates an antiferromagnetic coupling for the electronic spins on the two sites. This picture is consistent with that of the local radical model [19,21], where an antiferromagnetic coupling between the spins on the two sites is assumed.
spin density of the neutral state from the charged one, which stands for the net spin density from the doped electron. The above plots indicate the formation of a polaron in the OF. Unlike the polaron in normal polymers, two distinct characteristics can be observed for the polaron in the OF: (i) The doped electron distributes in both the main chain and radicals. This is induced by the hopping term in the present model, which cannot be obtained in the calculation of a conserved local spin model for the radicals. As a result, a polaron with fractional charges and spins is formed in the main chain. (ii) The distribution of the charge density and spin density of the polaron is not consistent in the main chain, and both of them are not asymmetric about the center of polaron. Such spin-charge disparity was also found in the previous work with a local spin model of the radicals [19]. In the following, we focus on the relation of the polaron charge and spin density in present nonlocal radical spin model. To check the distribution of the doped electron, we first analyze the total quantity of the net charges and spins in the main chain and radicals, respectively. Here
3.2. Charged state at ε = 0 Now we are in the position of the charge state. An additional electron is assumed to occupy the lowest unoccupied molecular orbital (LUMO), which is a spin-down level in the gap according to the band scheme at neutral state. As shown in Fig. 3, a polaron-like lattice distortion is formed in the main chain. A concomitant distortion is also observed in the charge density and spin density shown in Fig. 3(b)–(d). Fig. 3(d) is the net spin density Δρs, i (iR) calculated by subtracting the
Fig. 3. Charged state of the organic ferromagnet. (a) Smoothed lattice configuration, (b) net charge density, (c) spin density and (d) net spin density calculated by subtracting the spin density of the neutral state from that of the charged one. The parameters are the same as those in Fig. 2.
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Fig. 4. Probability density ψμ, i, σ
2
except the doped one. The results are shown in Fig. 5. Compared with the neutral state, a distortion in the charge density and spin density is generated for the Fermi sea, which occurs both in the main chain and the radicals. Such distortion originates from the disturbance by the polaron lattice distortion as well as the coulombic repulsion from the additional electron. The distortions for the charge and spin densities are different, which brings distinct modification for the polaron spin and charge. This is a source for the spin-charge disparity of the polaron in the main chain. Another source for the spin-charge disparity is the nonuniform charge and spin transfers of the Fermi sea between the main chain and radicals in the presence of the polaron. A total charge transfer of about 0.001 is found from the radicals to the main chain, but the total spin transfer is up to 0.03 up spins. The roles of the charge density modification and the nonuniform charge (spin) transfer from the Fermi sea have not been observed in previous local radical spin model [19]. The electron hopping term t1 and the e-e interaction are key parameters in the nonlocal spin model of the OF. In the following, the effects of the two parameters on the net charge and spin distribution as well as the lattice distortion are discussed. In Fig. 6 (a) and (b), we fixed the e-e interaction with u= 0.6, and changed the strength of the hopping term t1 firstly. It is found that as t1 is increased from 0.1 to 0.5, the net charge (spin) Qm (Sm ) quantity in the main chain increases almost linearly from −0.25 (−0.12) to −0.97 (−0.52), and then keeps the maximum value approximately when t1 is over 0.5. The net charge (QR ) and spin (SR ) quantities in the radicals decrease correspondingly with t1. The results mean that the additional electron tends to locate in the main chain at a larger hopping term t1. The saturation of the change (spin) quantities in the main chain at t1 > 0.5 is reflected from the lattice distortion shown in Fig. 6(b). It is clearly seen that a polaron distortion arises and is strengthened as t1 increases from 0.1 to 0.4. However, the lattice distortion vanishes at t1 = 0.5 and then a uniform lattice appears. This situation holds for a larger t1 which is not shown here. The results indicate that a medium strength of t1 is helpful for the formation of polarons in the OF. The intrinsic physics can be understood as follows. In the limitation of zero t1, the additional electron with whole net charge and spin locates in the state contributed by the radicals. In the case of a nonzero t1, hybridization between the states of the radicals and the main chain happens. For a small t1, the hybridization is weak and the additional electron is not easy to transfer from the radicals to the main chain. So the charge-induced lattice distortion in the main chain is neglectable and the lattice keeps dimerized almost as that of the neutral
of the orbital occupied by the additional spin-down
electron.
the total charge and spin quantities in the main chain (radicals) are calculated as Qm (R) = ∑i (iR) Δρc, i (iR) and Sm (R) = ∑i (iR) Δρs, i (iR) . Δρc, i (iR) (Δρs, i (iR) ) is the difference of the charge (spin) density between the charged state and the neutral one. The results give Qm (R) = −0.618 (−0.382) and Sm (R) = −0.275 (−0.225). This means that for present parameters a large proportion of the additional charges distributes in the main chain bounded with the polaron distortion. However, the spin of the electron distributes almost equally in the main chain and radicals. This result means the spin-charge disparity property of the polaron emerges not only in the contour of spin and charge density curve, but also in the total quantity. To understand the nature of the spin-charge disparity of the polaron in the nonlocal radical spin model, we first plot the probability density of the orbital occupied by the additional electron, which is shown in Fig. 4. A nonlocal distribution close to the distribution of net charge density is observed. One can sum the total probabilities of the orbital in the main chain and radicals, which are 0.619 and 0.381, respectively. So the charge density of the polaron almost follows the distribution probability of the polaron level, while a larger difference appears for the polaron spin density. The intrinsic mechanism can be tracked from the contribution of the Fermi sea, that is, other occupied-state electrons
Fig. 5. Charge density and spin density of the Fermi sea in the presence of the polaron. (a) Charge density, (b) spin density in the main chain, and (c) spin density in the radicals.
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Fig. 6. Dependence of net charge and spin distribution as well as the lattice distortion on hopping term t1 and e-e interaction u. (a) Net charge Qm (R) and spin Sm (R) in the main chain (radicals) with t1. Here u is fixed with u= 0.6. (b) Smoothed lattice distortion y͠ i at several different values of t1. u= 0.6. (c) Net charge Qm (R) and spin
Sm (R) with u. Here t1 is fixed with t1= 0.3. (d) Smoothed lattice distortion y͠ i at several different values of u. t1= 0.3.
state. The charge (spin) transfer between the radicals and the main chain increases with t1, that is, with the enhancement of the hybridization. On the other hand, in the limitation of a large t1(> 0.5), the strong hybridization between the radicals and the main chain will drive the system far away from a quasi-one-dimensional chain, where the dimerization even disappears and of course the polaron cannot form. The role of e-e interaction is investigated and shown in Fig. 6(c) and (d), where the hopping term t1 is fixed at 0.3. It is found that in the involved region u< 2.0, which is reasonable for most polymers, the net charge and spin in the main chain increase with u while those in the radicals decrease correspondingly. The quantity of net charge (spin) Qm (Sm ) in the main chain changes from −0.53 (−0.23) to −0.98 (−0.51) as u is increased from 0.1 to 2.0. The lattice distortion in Fig. 6(d) also indicates an enhanced polaron distortion with u, where an obvious distortion begins to emerge when u> 0.4. Thus a stronger e-e interaction is advantageous to generate a polaron in the OF. This is reasonable since at the neutral state, the total spins in each radical site are up spins. By introducing an extra spin-down electron, the e-e interaction tends to repulse it into the main chain, which is beneficial to the formation of the polaron. The case of very large u is out of our interest, where an irregular lattice distortion is possible because of the strong columbic repulsion in a charged system. To give a comprehensive vision of the parameter dependence of the polaron formation, we drew a phase diagram in Fig. 7 by scanning the values of t1 from 0 to 1.0 and of u from 0 to 2 in a step of 0.1. For simplicity, we judge the existence of the polaron from the smoothed lattice distortion. If a polaron-like distortion exists, and the deviation Δy͠ i of the distortion from the neutral state meets Max( Δy͠ i ) > 10−2 Å, it belongs to the polaron region. In the case of gradual change, it is hard to exactly distinguish the formation of polarons. However, the rough definition is useful to demonstrate the dependence of polaron formation on the two parameters. The gray region in Fig. 7 indicates the existence of the polaron with above criterion, which gives a region between 0.1 ≤ t1 ≤ 0.4 and 0.5 ≤ u ≤ 2.0. Out of this region, the lattice is dimerized or uniform. Furthermore, we note that the net spin of the whole OF is conserved in this region with the value of −1/2, which means that spin flipping process does not occur in the iteration although it is permitted in our algorithm.
Fig. 7. The phase diagram of polaron formation with t1 and u. Other parameters are same as those in Fig. 2. The gray region means the existence of a polaron.
with heterocycles as the spin radicals, the unpaired electron usually locates on the oxygen or nitrogen atom [25]. Consequently, the electronegativity of the radical atom may be larger than the carbon atom. As a result, a negative on-site energy ε of the radicals is reasonable and considered here. With exact diagonalization technique, Wang has demonstrated that the Anderson-like model for the OF may give a ferromagnetic or antiferromagnetic ground state relying on the values of ε and u [23], although the e-l interaction is not included in that work. Here we only focus on the case of the ferromagnetic state, which is guaranteed by an approximate relation of − ε < u according to Wang's work. In the case of a nonzero ε , charge transfer between the main chain and radicals is possible, and may further modifies the ferromagnetic state. We start from a specific case with ε = −0.3. Other parameters are same as those in Fig. 2. The ground state of the OF is first investigated and shown in Fig. 8. Compared with the case without ε , the smoothed lattice configuration, energy bands and spin density are similar as those in Fig. 2, which still indicates a ferromagnetic state for the OF. The two flat bands in the gap are shifted downwards due to the negative on-site energy of radicals. The spin distribution in each unit
3.3. Neutral state at ε ≠ 0 Now we turn to the case of a nonzero ε in the model. In some OFs 137
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Fig. 8. Ground state of the organic ferromagnet with ε = −0.3 . Other parameters are same as those in Fig. 2. (a) Smoothed lattice configuration, (b) energy levels, (c) charge density, (d) spin density.
the formation of polaron in the main chain. To give a polaron picture in such case, a larger e-e interaction u is adopted to form the polaron. Fig. 9 shows a polaron picture with an e-e interaction of u = 1.5. Other parameters are same as Fig. 2 except ε . A localized polaron appears with a narrower width compared with that in Fig. 2. The spin-charge disparity property is conserved, where an extra spin-up peak appears in the net spin density in the polaron region. This spin-up peak originates from the spin transfer of the Fermi sea between the main chain and radicals under a strong e-e interaction. The total charge (spin) quantity of the polaron is calculated, which is about −0.75 (−0.37) in the main chain and −0.25 (−0.12) in the radicals. The dependence of the net charge and spin distribution as well as the lattice distortion on the value of ε is further investigated. As shown in Fig. 10, when ε changes from 0 to −1.0, the net charge quantity Qm in the main chain decreases from −0.91 to −0.04, while QR in the radicals increases from −0.09 to −0.96 correspondingly. This is natural because almost all the charges of the additional electron will locate in the radicals if the magnitude of ε is large enough. The total net spin quantity in the radicals is also increased from −0.05 to −0.5 with ε .
cell changes slightly with 0.46 up spins on the radical site and 0.04 up spins on the atoms of the main chain. The significant change is that a charge density wave arises along the main chain, while a uniform negative charge density appears on the radicals. For each unit cell, −0.03 charges distribute on the radical site, which means electrons are transferred from the main chain to the radicals. 0.01 positive charges are found on the site hanging the radical, and 0.02 on its neighbor site. Thus, there exist 1.2e positive charges in total in the main chain. At ground state, it is found that the transferred charges exist in an extend form without any local distortion in the main chain. The lattice keeps dimerization which is slightly weakened by the background charge density wave.
3.4. Charged state at ε ≠ 0 Then the charged state is investigated by introducing an additional electron into the polymer. We found that the above parameters cannot form a polaron. This is because that in the presence of a negative ε , the electron tends to occupy the radical sites, which is disadvantageous for
Fig. 9. Polaron state of the organic ferromagnet for ε = −0.3 and u = 1.5. Other parameters are same as Fig. 2. (a) Lattice distortion, (b) net charge density, (c) spin density and (d) net spin density.
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formation of the polaron. In the presence of a nonzero on-site energy of the radicals, charge transfer of the Fermi sea between the main chain and radicals occurs, which forms a charge density wave and modifies the lattice distortion of the main chain. A large on-site energy of the radicals is not good for the formation of the polaron in the main chain. This work is helpful for us to understand the distinct spin-charge properties of polarons in OFs. At last, let us comment on the possible experimental probes on the polarons in OFs. In the case of large doped single crystals of OFs, the lattice distortion and spin density could be probed by neutron scattering. The charge density could be accessible with X-ray scattering. In the case of a single-molecule measurement, an X-ray free-electron laser could determine the charge and spin distribution of single molecules in the gas phase. Spin-resolved scanning tunneling microscopy is another direct way to detect the charge and spin density of the molecule deposited on a surface, although the substrate effect will be brought in. Finally, indirect evidence from magnetotransport measurements is expected when the OF is contacted with ferromagnetic electrodes. Acknowledgments Fig. 10. (a) Charge quantity Qm (R) , spin quantity Sm (R) in the main chain (radicals), and (c) lattice distortion with various values of ε . Here t1= 0.3, u= 1.5.
Support from the Natural Science Foundation of Shandong Province (Grant No. ZR2014AM017), the National Natural Science Foundation of China (Grant Nos. 11374195, 11704230, 11674197), and the Taishan Scholar Project of Shandong Province are gratefully acknowledged.
During the increase of ε , whether the polaron is stable can be examined from the lattice distortion shown in Fig. 10(b). A distinct polaron lattice distortion is observed from 0 to −0.4 of ε , where the maximum distortion at the center of the polaron region relative to the dimerization decreases quickly with ε . When ε is beyond −0.6, the lattice distortion at the polaron center becomes quite small. At the same time, the dimerization of the lattice out of the polaron region is reduced, which is due to the more and more charges transfer of Fermi sea between the main chain and radicals. At ε = −1.0, only a bit irregular lattice distortion is left in the previous polaron region, which is not a normal polaron anymore.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12]
4. Summary In summary, we have studied the spin-charge properties as well as its formation condition of polarons in the OFs by doping an extra electron into the magnetic polymer. The results are calculated in two regimes: with or without on-site energy difference between the main chain and radicals. We found that in both cases, a polaron with fractional charges and spins can be formed. The quantities of the spins and charges bound in the lattice distortion depend on the strength of the electron hopping term between the main chain and radicals, and also on the e-e interaction. A spin-charge disparity of the polaron is observed, which is attributed to the asymmetric modification for the charge and spin density by the Fermi sea in the presence of polaron lattice distortion. The dependence of the charge and spin distribution of the polaron on the hopping term and e-e interaction is also calculated, where a medium hopping strength and a large e-e interaction is helpful for the
[13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25]
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W.J.M. Naber, S. Faez, W.G. van der Wiel, J. Phys. D Appl. Phys. 40 (2007) R205. V.A. Dediu, L.E. Hueso, I. Bergenti, C. Taliani, Nat. Mater. 707 (2009) 8. T. Sugawara, M.M. Matsushita, J. Mater. Chem. 19 (2009) 1738. A.J. Heeger, S. Kivelson, J.R. Schrieffer, W.P. Su, Rev. Mod. Phys. 60 (1998) 781. W.P. Su, J.R. Schrieffer, A.J. Heeger, Phys. Rev. B 22 (1980) 2099. X.J. Liu, K. Gao, Y. Li, J.Y. Fu, J.H. Wei, S.J. Xie, Synth. Met. 157 (2007) 380. S.J. Xie, K.H. Ahn, D.L. Smith, A.R. Bishop, A. Saxena, Phys. Rev. B 67 (2003) 125202. E.M. Conweil, S.V. Rakhmanova, Appl. Phys. Lett. 75 (1999) 1518. Åsa Johansson, Sven Stafström, Phys. Rev. Lett. 86 (2001) 3602. Z. Fang, Z.L. Liu, K.L. Yao, Phys. Rev. B 49 (1994) 3916. Z. Fang, Z.L. Liu, K.L. Yao, Z.G. Li, Phys. Rev. B 51 (1995) 1304. J.W. Yoo, R.S. Edelstein, D.M. Lincoln, N.P. Raju, C. Xia, K.I. Pokhodnya, J.S. Miller, A.J. Epstein, Phys. Rev. Lett. 97 (2006) 247205. B. Li, M. Zhou, Y. Lu, C.Y. Kao, J.W. Yoo, V.N. Prigodin, A.J. Epstein, Org. Electron. 13 (2012) 1261. G.C. Hu, Y. Guo, J.H. Wei, S.J. Xie, Phys. Rev. B 75 (2007) 165321. W.Z. Wang, Phys. Rev. B 73 (2006) 235325. G.C. Hu, K.L. He, S.J. Xie, A. Saxena, J. Chem. Phys. 129 (2008) 234708. G.C. Hu, Z. Zhang, Y. Li, J.F. Ren, C.K. Wang, Chin. Phys. B 25 (2016) 057308. B. Xiong, W.Z. Wang, Phys. Rev. B 71 (2005) 174431. G.C. Hu, H. Wang, J.F. Ren, S.J. Xie, C. Timm, Org. Electron. 15 (2014) 118. W.P. Su, J.R. Schrieffer, A.J. Heeger, Phys. Rev. Lett. 42 (1979) 1698. S.J. Xie, J.Q. Zhao, J.H. Wei, S.G. Wang, L.M. Mei, S.H. Han, Europhys. Lett. 50 (2000) 635. W.Z. Wang, Phys. Rev. B 71 (2005) 094407. W.Z. Wang, Phys. Rev. B 73 (2006) 035118. Guang-Shan Tian, Tsung-Han Lin, Phys. Rev. B 53 (1996) 8196. A. Alberola, M. Pilkington, Curr. Org. Synth. 6 (2009) 66.