Magneto-Electric Effect in a Spin-State Transition System
Makoto Naka Eriko Mizoguchi Joji Nasu Sumio Ishihara

Jittering jets in cooling flow clusters
    Noam Soker1
    December 30, 2023
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Multiferroics are the coexistence phenomena of the ferromagnetic, ferroelectric, and ferroelastic orders. <cit.> 
The cross-correlation effects between the magnetism and dielectrics are termed the magnetoelectric (ME) effect. <cit.> 
These issues have attracted recently much attention not only from the fundamental condensed matter physics, but also the wide potentiality for the technological applications. 
One of the predominant target materials of the multiferroics and the ME effect is the Mott insulating systems, in which the electronic charge degree of freedom is quenched and the spin degree of freedom dominates the low energy physics. 
The well studied examples are the systems showing non-collinear and non-coplanar spin orders owing to the competing exchange interactions. <cit.> 
The spontaneous electric polarizations are induced by the so-called inverse Dzyaloshinsky-Moriya interaction mechanism, and are controlled by an external magnetic field. <cit.> 
Now, the candidates of the multiferroics and ME effect are surveyed extensively in a wide variety of materials. <cit.> 

In some classes of the magnetic ions, multiple spin amplitudes (S) are realized under different external conditions. 
Cooperative changes in the magnetic states induced by the interacting spin-state degree of freedom are termed the “spin-crossover” or “spin-state transition" phenomena, which are often seen in correlated electron materials, <cit.> earth-inner mantels, <cit.> biomolecules, <cit.> and so on. 
These phenomena originate from competition between the crystalline electric-field (CEF) effect and the Hund coupling; the low (high)-spin state with small (large) S is stabilized, when CEF is larger (smaller) than the Hund coupling.   

One of the well-known materials in which the spin-crossover phenomena are realized is the cobalt oxides with the perovskite structure, R_1-xA_xCoO_3 (R: trivalent ion, A: divalent ion). <cit.> 
The nominal valence of the Co ion at x=0 is 3+ and the number of the 3d electrons is six. 
There are three possible spin states, the low-spin (LS) state with S=0 and the configuration of (t_2g)^6(e_g)^0, the intermediate-spin state with S=1 and (t_2g)^5(e_g)^1, and the high-spin (HS) state with S=2 and (t_2g)^4(e_g)^2. 
When the LS (HS) states are realized in all cobalt sites in a crystal, the system is identified as a band (Mott) insulator. 
Thus, the spin-state transition phenomenon is considered as a phase transition between the band and Mott insulators. 
A number of the experimental observations for the spin crossover in the cobaltites have been reported over the past decades. <cit.> 
In recent years, studies of the cobaltites are revived from the viewpoint of the excitonic insulating (EI) state, which is defined as the quantum-mechanical hybridization of the different spin states. <cit.> 

Recently, a multiferroic behavior was discovered in a distorted perovskite cobaltite BiCoO_3. <cit.> 
Oka et al. found a ferroelectric (FE) transition and reduction of the FE transition temperature (T) by applying  pressure. <cit.> 
In the FE phase, the CoO_6 octahedra are largely deformed; the coordination number of the cobalt ion is changed from six in the paraelectric (PE) phase to five in the FE phase. 
From the X-ray structural analyses and X-ray emission spectroscopy, it is suggested that the spin-state change from LS to HS occurs due to this large structural change. 
Thus, this material is expected to be a novel class of ME materials, <cit.> in which the electric polarization couples with the spin-state degree of freedom, and is expected to open a new route to the multiferroics near the boundary between the band and Mott insulators. 

In this Letter, motivated by the experiments in BiCoO_3, we examine the magnetic and dielectric properties in the spin-state transition system. 
We introduce a strong coupling model, derived from the two-orbital Hubbard model, interacting with the FE-type lattice distortion, and investigate the ground-state and finite-T properties using the mean-field approximation. 
We find that the LS-to-HS transition occurs with accompanying the FE transition at which the electronic entropy increases with decreasing temperature. 
The obtained phase diagram well reproduces the experimental pressure-temperature phase diagram. 
In the LS and PE phase close to the spin-state transition boundary, the electric polarization is induced by applying a magnetic field. 
A possible observation of this novel ME effect is discussed. 

We construct a model describing the correlated electrons coupled with the lattice distortion that breaks the inversion symmetry.
The model Hamiltonian is given by  

    H= H_e+ H_l+ H_el ,

where the first and second terms represent the electronic and lattice parts, respectively, and the third term is for the electron lattice coupling. 
The electronic part H_e is derived from the extended two-orbital Hubbard model with CEF. 
This model is decomposed into the on-site interaction term and the inter-site electron hopping term as 
H_e0= H_u+ H_t 
with 

    H_u   =Δ∑_i n_i a +U∑_i ηn_iη↑n_iη↓+U'∑_i n_ian_ib
       +J∑_iσσ'c_iaσ^† c_ibσ'^† c_iaσ'c_ibσ
     +I∑_iη≠η'c_iη↑^† c_iη↓^† c_iη'↓c_iη'↑ ,

and 

    H_t=-∑_ < ij  >∑_ ησ
    t_η 
    (c_iησ^† c_jησ+ H.c.).

We define that c^†_i ησ (c_i ησ) is the creation (annihilation) operator for an electron with the orbital η(=a, b), and the spin σ(=↑, ↓) at the site i. 
We introduce in H_u that Δ, U, U', J, and I represent the energy difference between the orbitals, the intra-orbital Coulomb interaction, inter-orbital Coulomb interaction, the Hund coupling, and the pair hopping interaction, respectively.  
The electron hoppings in H_t are taken into account between the same kinds of orbitals in the nearest neighbor (NN) bonds. 
The average electron number per site is set to be two, and the two-dimensional square lattice is considered for simplicity. 

In the strong coupling limit when the electron-electron interactions and Δ are larger than t_η, the electron configurations at each site in the low energy section are restricted to the spin-singlet state with the a^2b^0 configuration mainly and the spin-triplet state with the a^1b^1 configuration. 
These are referred as the LS and HS states represented by |L⟩ and |H_Γ⟩ with Γ≡ S_z=(+1, 0, -1), respectively. 
In order to describe these local electronic states, we introduce the spin operator S_i with the amplitude S=1, the pseudo-spin operator for the spin-state degree of freedom as   
τ_Γ^x=L⟨H_Γ|+H_Γ⟨L|, 
τ_Γ^y=i(L⟨H_Γ|-H_Γ⟨L|), 
and 
τ_Γ^z=H_Γ⟨H_Γ|-L⟨L|, 
and 
τ^γ=∑_Γτ_Γ^γ for γ=(x, y, z). 

Details are presented in Ref. [nasu]. 
By using the second order perturbational calculations with respect to the hopping integrals, 
the electronic part H_e0 is transformed to the effective model given by 

    H_e   = -h_z∑_iτ_i^z+J_z∑_ijτ_i^zτ_j^z
    +J_s∑_ijS_i·S_j
       -∑_ij∑_Γ=(0, ± 1) ( J_x τ_Γ i^xτ_Γ j^x+J_yτ_Γ i^yτ_Γ j^y  )  ,

where h_z, J_s, J_γ are represented by the parameters introduced in Eqs. (<ref>) and (<ref>). 
The first term originates from the energy difference between the two orbitals, and the second term brings about the attractive interaction between the LS and HS states. 
The last two terms induce the spontaneous hybridization of the LS and HS states, i.e. the EI state. 

As for the lattice part, we consider the local lattice displacement q_i at site i, which breaks the space inversion symmetry, and its canonical momentum p_i. <cit.> 
For simplicity, we assume that the displacements occur along a direction corresponding to the c axis in the distorted perovskite crystal of BiCoO_3.  
The FE-type interaction is introduced between q_is. 
The lattice part of the Hamiltonian is given as 

    H_l=∑_i  [ α/2 p_i^2+V( q_i)  ]
    -1/2∑_i  j v_ijq_i q_j ,

where p_i and q_i are defined to be dimensionless. 
The anharmonic local potential at site i is introduced as V(q_i)=(α/2)q_i^2+(β/4)q_i^4 with α>0 and β>0. 
The FE-type interaction between q's is represented by a positive constant v_0 = ∑_j (≠ i) v_ij in the mean-field approximation without setting of a detailed form of v_ij. 

Between the electron and lattice displacement, we introduce the local interaction which stabilizes the HS state associated with the FE order. 
Considering the space inversion symmetry of τ^z, this interaction is represented as 

    H_el=-g ∑_i τ_i^z q_i^2 ,

where g is a positive couping constant. 

The Hamiltonian in Eq. (<ref>) is analyzed by using the mean-field approximation, where the interaction terms in H_e and H_l are decoupled as 
τ_i τ_j →τ_iτ_j+τ_i τ_j-τ_iτ_j, 
S_i  S_j → S_i S_j+ S_i  S_j-  S_i S_j,  and 
q_i q_j →q_i q_j+q_i q_j-q_iq_j. 
We adopt a unit cell including the two sites, and calculate all the mean-field components selfconsistently. 
As for the lattice part, the FE-type order is assumed as q = ⟨ q_i ⟩. 

We introduce the AFM order parameter 

    M_ AF=1/2 ( S^z_A-S^z_B ) ,

and the HS density

    n_ H= ∑_Γ|H_Γ⟩⟨H_Γ| ,

where subscripts A and B represent the two sublattices. 
In all of the numerical calculations except for Fig. <ref>(b) , the parameter values are fixed at t_a=1, t_b=-0.1, U=6J, U'=4J, J=I=2, α=0.1, β=0.2, and v_0=0.4, and the orbital energy difference Δ and the electron-lattice coupling g are changed. 





Before showing the numerical results of the the electron-lattice coupled system, the results at g=0 are briefly mentioned. Details are given in Ref. [nasu]. 
In Fig. <ref>(b), the ground-state phase diagram in the J-Δ plane at g=0 is presented. 
In the large limit of Δ (J), the LS (HS) phase appears as expected. 
An antiferromagnetic (AF) order is realized in the HS phase. 
Between the LS and HS phases, we find the two types of the EI phase, termed the EIM and EIQ phases, as well as the LS/HS ordered phase in which the LS and HS states are aligned alternately. 
The EI phases are identified by the order parameters τ^x and τ^y, and 
the two phases  are distinguished by the magnetic structures: 
the AF order in EIM and the spin nematic order in EIQ. 
The LS/HS ordered phase is characterized by a staggered order of τ^z. 

The ground-state phase diagram in the Δ-g plane is presented in Fig. <ref>(a). 
Hatched areas represent the FE phases. 
At g=0 where the electron and lattice degrees of freedom are independent with each other, the electronic states are changed as LS → EIQ → LS/HS → EIM → HS-AF with decreasing Δ, and the FE-type lattice displacements occur. 
By introducing g which promotes the HS state associated with the FE order, 
both the EIM and LS/HS phases are reduced and disappear around g=0.04. 
The phase is changed as LS → EIQ → HS with decreasing Δ    
in the region of 0.04 < g<0.09. 
With increasing g furthermore, the FE HS phase and the PE LS phase only survive. 

Detailed Δ dependences of the mean fields, i.e. q, M_ AF, and n_H, are shown in Figs. <ref>(c)-<ref>(f) at g=0.02 and 0.1. 
The FE order parameter q is finite except for the LS phase at g=0.1. 
The phase boundary between the HS-AF and EIM phases and that between the LS and EIQ phases are of the second order as shown in Ref. [nasu].  
Figures <ref>(e) and <ref>(f) show that the transition between PE LS and FE HS-AF is of the first order. 





The finite-T phase diagram at g=0.1 is shown in Fig. <ref>(a). 
The temperature dependences of the mean fields at Δ=5.5, 6.0, 6.5 are also shown in Figs. <ref>(b)-<ref>(g). 
In high temperatures around T=0.4, with changing Δ, the spin state is continuously changed, and there is no spin-state transition. 
In the large Δ region around Δ=6.5,  n_ H gradually decreases with decreasing T, and then arrives at n_ H=0, i.e. the LS state. 
On the other hand, in the small Δ region at Δ=5.5, the PE-paramagnetic HS state changes into the FE-paramagnetic HS phase at T ∼ 0.33, and the AF order appears around T=0.19. 
With increasing Δ, the FE-transition temperature decreases monotonically, while the Néel temperature is almost unchanged. 
Around Δ=6.2, the LS/HS ordered phase appears above the HS-AF phase. 
This is attributable to the spin entropy at the HS sites in the LS/HS phase, in which the exchange interactions between the HS sites are blocked by the surrounded LS sites. 
As shown in Figs. <ref>(d) and <ref>(e) at Δ=6, with decreasing T, n_H decreases gradually, and increases discontinuously at the FE transition temperature. 
In the FE phase, n_H increases monotonically with decreasing T. 
This finite-T phase diagram as a function of Δ reproduces qualitatively the experimental phase diagram in BiCoO_3 under pressure, as discussed in more detail later. 






Next, we show the ME effect in the present system. 
The magnetic field is introduced as the Zeeman energy into the Hamiltonian as H_H=- H ∑_i S_i^z with the magnetic field H. 
We examine the magnetic-field effect on a PE LS phase in the ground state marked by the symbol in Fig. <ref>(a).  
The magnetic field dependences of n_ H, M_ AF, q, and ferromagnetic order parameter M_ F (≡ (S_A^z+S_B^z)/2) are presented in Fig. <ref>. 
With increasing H, the LS phase changes into the HS phase around H=0.05 accompanied by the FE distortion. 
That is, the magnetic-field-induced FE transition. 
Further increasing H, the spins are canted gradually and the angle between the sublattice magnetic moments deviates from 180^∘. 
This nonlinear ME effect is owing to the coupling between the spin-state degree of freedom and the FE polarization.





We mention the entropy change across the FE transition. 
As shown schematically in Fig. <ref>(c), the present system is in stark contrast to the conventional spin-crossover system, in which a large spin entropy in the high temperature HS phase is quenched in the low temperature LS phase. 
The total entropy (S_ total) as well as the electronic (S_ ele) and lattice (S_ latt) entropies calculated by the mean-field approximation are shown in Fig. <ref>(a). 
Here, we chose (Δ, g)=(6.5, 0.19) in which the first-order FE transition accompanied by the spin-state change occurs around T=0.23, and the AF transition follows up around T=0.2. 

The detailed T dependence of S_ ele is presented in Fig. <ref>(b). 
The notable feature is that S_ ele increases with decreasing T at the FE transition temperature. 
The electronic part of the entropy is evaluated by the simple analytical formula  
S=-n_ Lln (n_L) - n_ Hln (n_ H/3) with n_ L = 1-n_ H where the LS and paramagnetic HS states are assumed to be mixed in the thermodynamic sense. 
This equation gives
S=ln 4 ∼ 1.39 (≡ S_∞ ) in the high T limit with (n_ H, n_ L)=(0.75, 0.25), 
 S=0.5 ln 2+0.5ln 6 ∼ 1.24 (≡ S_ LH) in the case with the equal weight of the LS and HS states of (n_ H, n_ L)=(0.5, 0.5), 
and S=ln 3 ∼ 1.09(≡ S_ H) in the pure HS state of (n_ H, n_ L)=(1, 0).  
As shown in Fig. <ref>(b), the calculated S_ ele approaches S_∞ in the high T limit, and S_ H in the FE HS phase. 
Between T = 0.23 and 0.3, S_ ele is smaller than S_ LH.

These results imply that with decreasing T the electronic system once approaches the LS state and then changes into the HS state owing to the FE transition. 
This is schematically shown in Fig. <ref>(c), and  is an opposite behavior of a conventional spin-crossover system such as LaCoO_3.

Finally, relations of the present calculations to the experimental observations are discussed. 
The Δ-T phase diagram shown in Fig. <ref>(a) is compared to the pressure-T phase diagram in BiCoO_3 <cit.>. 
The hydrostatic pressure reduces the lattice constants and increases CEF. 
Therefore, the calculated reduction of the FE transition temperature is consistent qualitatively with the experimental results. 
We predict that, in contrast to the FE transition temperature, the AF Néel temperature is almost unchanged under the pressure, because the HS density n_ H is almost independent of Δ in the FE phase. 
One of the present main message is that the nonlinear ME effect, i.e. the magnetic-field-induced electric polarization, is expected in the PE LS phase close to the FE HS phase associated with the AF order. 
This might be confirmed experimentally in BiCoO_3 under the high pressure around 3 GPa below 420 K (the Néel temperature at ambient pressure). <cit.> 
The present theoretical prediction of the ME effect is also checked in the mixed crystal of BiCo_1-xFe_xO_3 where the fine tuning of the FE transition is possible. <cit.> 

In conclusion, we study the mangetoelectric effects in the spin-state transition system coupled with the ferroelectric lattice distortion. The ferroelectric transition occurs accompanied by the LS-to-HS transition. 
The calculated Δ-T phase diagram is consistent with the experimental pressure-T phase diagram observed  in BiCoO_3. 
It is predicted that the electric polarization is induced by applying the magnetic field in the LS paraelectric phase. 
The present study proposes a concept of the new-class multiferroics in the spin-state transition system with electron correlations. 

We thank M. Azuma for their helpful discussions.
This work was supported by MEXT KAKENHI Grant No. JP26287070, JP15H02100, JP16H00987, JP16K17747, and JP16K17731. 
Some of the numerical calculations were performed using the facilities of the Supercomputer Center, the Institute for Solid State Physics, the University of Tokyo.

9

schmid
H. Schmid, Ferroelectrics 162, 317 (1994). 




curie 
P. Curie, J. Phys. Theor. Appl., 3, 393 (1894).

dzyaloshinskii
I. E. Dzyaloshinskii, Sov. Phys. JETP 10, 628 (1960). 

astrov
D. N. Astrov, Sov. Phys. JETP 13, 729 (1961). 

folen
V. J. Folen, G. T. Rado, and E. W. Stalder, Phys. Rev. Lett. 6, 607 (1961). 

tachiki
M. Date, J. Kanamori, and Y. Tachiki, J. Phys. Soc. Jpn. 16, 2589 (1961).

fiebig
M. Fiebig, J. Phys. D 38, R123 (2005). 




shiratori
K. Shiratori and E. Kita, J. Phys. Soc. Jpn. 48, 1443 (1980).

kimura
T. Kimura, T. Goto, H. Shintani, K. Ishizaka, T. Arima, Y. Tokura, Nature 496, 55 (2003). 

katsura
H. Katsura, N. Nagaosa, and A. V. Balatsky, Phys. Rev. Lett. 95, 057205 (2005).

dagotto
I. A. Sergienko and E. Dagotto, Phys. Rev. B 73, 094434 (2006). 

mostovoy
M. Mostovoy, Phys. Rev. Lett. 96, 067601 (2006). 




n_ikeda
N. Ikeda, H. Ohsumi, K. Ohwada, K. Ishii, T. Inami, K. Kakurai, Y. Murakami, K. Yoshii, S. Mori, Y. Horibe and Hijiri Kito, Nature 436, 1136 (2004).

khomskii
D. I. Khomskii, Nature Mater. 3, 853 (2004).

kimura2
T. Kimura, J. C. Lashley, and A. P. Ramirez, Phys. Rev. B 73, (R)220401 (2006). 

arima
T. Arima, J. Phys. Soc. Jpn. 76, 073702 (2007). 

naka
M. Naka and S. Ishihara, Sci. Rep. 6, 20781 (2016). 




imada 
M. Imada, A. Fujimori, and Y. Tokura, 7010391998.

tokura
Y. Tokura, Y. Okimoto, S. Yamaguchi, H. Taniguchi, T. Kimura, and H. Takagi, 58R16991998.

gretarsson
H. Gretarsson, S. R. Saha, T. Drye, J. Paglione, J. Kim, D. Casa, T. Gog, W. Wu, S. R. Julian, and Y.-J. Kim, Phys. Rev. Lett. 110, 047003 (2013).

lin
J.-F. Lin, V. V Struzhkin, S. D. Jacobsen, M. Y. Hu, P. Chow, J. Kung, H. Liu, H.-K. Mao, and R. J. Hemley, Nature 436, 377 (2005).

antonangeli
D. Antonangeli, J. Siebert, C. M. Aracne, D. L. Farber, A. Bosak, M. Hoesch, M. Krisch, F. J. Ryerson, G. Fiquet, and J. Badro, Science 331, 64 (2011).

hsu
H. Hsu, P. Blaha, M. Cococcioni, and R. M. Wentzcovitch, Phys. Rev. Lett. 106, 118501 (2011).

halder
G. J. Halder, C. J. Kepert, B. Moubaraki, K. S. Murray, and J. D. Cashion, Science 298, 1762 (2002).




mahendiran
R. Mahendiran and A. K. Raychaudhuri Phys. Rev. B 54, 16044 (1996).

sato
K. Sato, A. Matsuo, K. Kindo, Y. Kobayashi, and K. Asai, J. Phys. Soc. Jpn. 78, 093702 (2009).

vanko
G. Vankó, J.-P. Rueff, A. Mattila, Z. Németh, and A. Shukla, 730244242006.

a_ikeda
A. Ikeda, T. Nomura, Y. H. Matsuda, A. Matsuo, K. Kindo, and K. Sato Phys. Rev. B 93, (R)220401 (2016).



kunes1
J. Kuněs and P. Augustinský, 891151342014.

kunes2 
J. Kuněs and P. Augustinský, 902351122014.

kanamori
Y. Kanamori, H. Matsueda, and S. Ishihara, Phys. Rev. Lett. 107, 167403 (2011). 

nasu 
J. Nasu, T. Watanabe, M. Naka, and S. Ishihara, 932051362016. 

tatsuno
T. Tatsuno, E. Mizoguchi, J. Nasu, M. Naka, and S. Ishihara, 850837062016. 













belik
A. A. Belik, S. Iikubo, K. Kodama, N. Igawa, S. Shamoto, S. Niitaka, M. Azuma, Y. Shimakawa, M. Takano, F. Izumi, E. Takayama-Muromachi, Chem. Mater. 18, 798 (2006). 

oka
K. Oka, M. Azuma, W. Chen, H. Yusa, A. A. Belik, E. Takayama-Muromachi,  M. Mizumaki, N. Ishimatsu,  N. Hiraoka, M. Tsujimoto, M. G. Tucker, J. P. Attfield and Y. Shimakawa, J. Am. Chem. Soc., 132, 9438 (2010). 

okimoto
Y. Okimoto, S. Naruse, R. Fukaya, T. Ishikawa, S. Koshihara, M. Azuma, K. Tanaka, and H. Hirori, 
Phys. Rev. Appl. 7, 064016 (2017). 


oguchi
Y. Uratani, T. Shishidou, and T. Oguchi, J. Phys. Soc. Jpn. 78, 084709 (2009). 

chen

X. Y. Chen, R. Y. Tian, J. M. Wu, Y. J. Zhao, H. C. Ding and C. G. Duan, 
J. Phys.: Condens. Matter 23 326005 (2011). 

chen2

X. Y. Chen, L. J. Chen, X. B. Yang, Y. J. Zhao, H. C. Ding, and C. G. Duan, 
J.  Appl. Phys.,  111, 013901 (2012). 

solovyev
I. V. Solovyev, Phys. Rev. B 85, 054420 (2012). 

milosevic

A. S. Milos̆ević, M. V. Lalić, Z. S. Popović, and F. R. Vukajlović, Optical Materials, 
35 , 1765 (2013). 




blink
R. Blink and B. Zeks, Soft Modes in Ferro-electrics and Anti-ferroelectrics, (North Holland, Amsterdam, 1974)




hojo
H. Hojo, K. Onuma, Y. Ikuhara, and M. Azuma, Appl. Phys. Exp. 7, 091501 (2014). 



