In-gap excitation effect on a superexchange in La{}_{2}CuO{}_{4} by creating nonequilibrium photoexcited centers

# In-gap excitation effect on a superexchange in La2CuO4 by creating nonequilibrium photoexcited centers

## Abstract

We propose a multielectron approach to calculate superexchange interaction in magnetic Mott-Hubbard insulator LaCuO(further La214) that allows to obtain the effect of optical pumping on the superexchange interaction. We use the cell perturbation theory with exact diagonalization of the multiband Hamiltonian inside each CuO unit cell and treating the intercell hopping as perturbation. To incorporate effect of optical pumping we include in this work the excited single-hole local states as well as all two-hole singlets and triplets. By projecting out the interband intercell electron hopping we have obtained the effective Heisenberg-like Hamiltonian with the local spin at site being a superposition of the ground and excited single-hole states. We found that antiferromagnetic contribution to the exchange energy in La214 will increase in accordance to at the resonance light occupation of the excited single hole in-gap state.

###### pacs:
71.35.Cc, 75.78.Jp

## I Introduction

Understanding the energy transfer between charge, orbital, and spin degrees of freedom is the important problem for many fields of solid state physics. Since the first experiments  Beaurepaire et al. (1996); Hohlfeld et al. (1997) optical excitation of electronic spins and ultrafast magnetization dynamics have obtained much attention.  Stanciu et al. (2007); Ostler et al. (2012) A possibility to control the exchange interaction by light is important in many physics areas, from quantum computing Duan et al. (2003); Trotzky et al. (2008); Chen et al. (2011) to strongly correlated materials. Wall et al. (2009); Forst et al. (2011); Li et al. (2013) In many experiments the effect of optical pumping on the exchange interaction in the Mott-Hubbard insulators like manganites, Wall et al. (2009) ferroborates, Kalashnikova et al. (2007, 2008) TmFeO3, ErFeO3 Mikhaylovskiy et al. (2014) etc. has been found. The origin of interatomic exchange interaction in all these oxides is related to the superecxhange mechanism via oxygen. Anderson (1950) There are some simplified model calculations of the super exchange interaction under light irradiation in the three atomic model cation1-oxygen-cation2, Moskvin and Luk’yanov (1979) that in complete theory should be extended to the crystal lattice. The calculation of the superexchange interaction for the crystal lattice can be easily done for some simplified model like the Hubbard model. Bulaevskii et al. (1968); Chao et al. (1977); Hirsch (1987) Within the LDA+DMFT approach the first-priciple calculations of the exchange interaction in correlated materials has been carried out in the work. Katsnelson and Lichtenstein (2000) An idea of generalization of this approach to nonequilibrium optically excited magnetics has been also proposed in work. Secchi et al. (2013) without any practical conclusions. Nevertheless up to now the microscopic calculation of the superexchange interaction in La214 under light irradiation is absent.

It is known that in the Hubbard model the superexchange results from the projecting out the interatomic hopping accompanying the interband excitation from the low Hubbard band (LHB=a) to the upper Hubbard band (UHB=b). Due to the large insulator gap the interband excitation requires too much energy, and only virtual interband excitations from LHB to UHB and back are possible providing the exchange coupling Bulaevskii et al. (1968); Chao et al. (1977); Hirsch (1987) The convenient mathematical tool for projecting out the irrelevant at large UHB is given by the projection operators. Chao et al. (1977)

In this paper we calculate the exchange interaction in La214 under optical pumping within the hybrid LDA+GTB (generalized tight binding) approach. Previously we have carried out similar calculation for La214 in the ground state. Gavrichkov and Ovchinnikov (2008) The LDA+GTB method allows to calculate the electronic structure of strongly correlated oxides like cuprates, Korshunov et al. (2005) manganites, Gavrichkov et al. (2010) boroxide Ovchinnikov (2003); Ovchinnikov and Zabluda (2004) and cobaltates. Orlov et al. (2013) We use the cell perturbation theory with exact diagonalization of the multiband Hamiltonian inside each CuO unit cell with calculated parameters and treating the intercell hopping as perturbation. We restrict ourselves here by the antiferromagnetic undoped cuprate La214, nevertheless all ideas and methods used may be applied to any Mott-Hubbard insulator. To incorporate effect of optical pumping we include in this work the excited single-hole local states as well as all excited two-hole singlets and triplets. It requires a generalization of the projection operators used here in comparison to the papers. Chao et al. (1977); Gavrichkov and Ovchinnikov (2008) Finally we have obtained the modification of exchange interaction induced by the light irradiation.

## Ii Effective superexchange hamiltonian

In the GTB approach one can assume that the quasiparticles are unit cell excitations which can be represented graphically as single-particle excitations (transitions) between different sectors of the configuration space of the unit cell ( is hole number per cell in the undoped material, see Fig.1). Ovchinnikov et al. (2012) Each of these transitions forms a -th quasiparticle band, where the vector band index in configurational space  Zaitsev (1975) numerates the initial and final many-electron states in the transition. The transitions, with the number of electrons increasing or decreasing, form the conduction or valence bands, respectively. For undoped La214 due to electroneutrality the proper subspace is with one hole per CuO cluster, it has one hole, . The hole addition requires states . The hole removal results in states that for cuprates is given by a hole vacuum . In the LDA+GTB method the Hamiltonian parameters are calculated ab initioKorshunov et al. (2005) and the GTB cell approach  Gavrichkov and Ovchinnikov (1998); Ovchinnikov et al. (2012) is used to take into account strong electron correlations explicity. A crystal lattice is divided into unit cells, so that the Hamiltonian is represented by the sum , where the component is the sum of intracell terms and component takes into account the intercell hoppings and interactions. The component is exactly diagonalized. The exact multielectron cell states () and energies are determined. Then these states are used to construct the Hubbard operators of the unit cell , where the meaning of the indexes and is clear from Fig.1.

 H0=∑f{ε0X00f+∑lσ(ϵl−μ)Xlσ,lσf+Nν∑ν(Eν−2μ)Xν,νf} (1)

is the sum of intracell terms and component takes into account the intercell hoppings and interactions. Here

 H1 =∑fg∑λλ′σtλλ′fgc+fλσcgλ′σ+h.c. (2) =∑fg∑rr′trr′fg+XrfXr′g,

where is the matrix of hopping integrals, and

 trr′fg = ∑λλ′∑σtλλ′fg (3) ×

where matrix elements:

 γλσ(r) = ⟨(N+,M′S′)μ∣∣cfλσ|(N0,MS)l⟩× (4) × δ(S′,S±|σ|)δ(M′,M+σ),

Consideration is restricted by the case with one hole per cell in the undoped materials and an arbitrary number of the occupied orbitals, i.e. number of electrons . This is relevant for the high- cuprates. In this case of one hole per cell, the cell states are a superposition of different hole configurations of the same orbital (l) symmetry:

 |(N0,MS)l⟩=∑λβl(hλ)|hλ,MS⟩ (5)

Thus, there are one-hole spin doublet states, , where is the number of combinations. Besides, there are of the spin singlets and triplets :

 ∣∣(N+,M′S′)μ⟩=∑λλ′Bμ(hλ,hλ′)∣∣hλ,hλ′,M′S′⟩ (6)

in the two-hole sector (Fig.1) in the -orbital approach. Using the intracell Hamiltonian in the cell function representation the configuration weights and can be obtained by the exact diagonalization procedure for the matrices and in the -eigenvalue problem in different sectors Ovchinnikov et al. (2012) The sum (2) over all the -th excited states with in the sector can not be omitted because of the light pumping. These excited states must be considered along with the - excited states in the nearest sector.

The superexchange interaction appears at the second order of the cell perturbation theory with respect to hoppings.Jefferson et al. (1992) That corresponds to virtual excitations from the occupied singlet and triplet bands through the insulating gap to the conduction band and back. These perturbations are described by the off-diagonal elements with and in expression (2). In the Hubbard model, there is only one such element , which describes the hoppings between the lower and upper Hubbard bands. In order to extract them, we generalize the projection operator method proposed by Chao et al  Chao et al. (1977) to the multiorbital GTB approach. Since the diagonal Hubbard operators are projection operators, the -representation allows us to construct the set of projection operators. The total number of diagonal operators is equal to and the sequence indexes and (, ) runs over all electron states in the configuration spaces in Fig.1. Using a set of generalized operators

 p0=(X00i+∑lXlli)(X00j+∑lXllj), (7)

and

 pμ=Xμμi+Xμμj−Xμμi∑νXννj (8)

with we can identify the contribution to the superexchange from the interband transitions. As will be seen below, a generalized approach with the operators (7) and (8) differs from the work Chao et al. (1977) just in details. It can be checked that each of operator and is a projection operator and . These operators also form a complete and orthogonal system, , and . We highlight the diagonal and off-diagonal matrix elements in expression:

 H =(H0+Hin1)+Hout1, (9)

According to the work we introduce a Hamiltonian of the exchange-coupled -th pair: , where and

 h0+hin1=p0hp0+∑μνpμhpν (10)

and

 hout1=p0h(∑μpμ)+(∑μpμ)hp0 (11)

are intra- and inter- band contributions in respectively. We perform the standard unitary transformation to project out the interband hopping and to derive superexchange interaction

 ~h=eGhe−G, (12)

where the matrix satisfies to the equation

 p0h(∑μpμ) + (∑μpμ)hp0+ (13) + [G,(p0hp0+∑μνpμhpν)]=0,

and transformed Hamiltonian are given by

 ~h ≈ (p0hp0+∑μνpμhpν)+ (14) + 12[G,(p0h∑μpμ+∑μpμhp0)]

where the contributions from inter-band transitions can be calculated as:

 p0h(∑μpμ)=∑ll′μtl0,l′μijXl0iXl′μj (15)
 (∑μpμ)hp0=∑ll′μtμl′,0lijXμl′iX0lj

Note, due to the absence of additivity over -number of the excited state in the projective operator , the solution of Eq.(13) has the form

 G=∑μ∑ll′tl0,l′μijΔll′μ(Xμl′iX0lj−Xl0iXl′μj) (16)

where , and the commutator in (14) can be represented as

 δh=12 ∑μν{[Gν,(p0hpμ+pμhp0)]}= (17) =12∑μν⎧⎪⎨⎪⎩⎡⎢⎣∑ll′tl0,l′νijΔll′ν(Xμl′iX0lj−Xl0iXl′μj),∑kk′tk0,k′μij(X0kiXμk′j+h.c.)⎤⎥⎦⎫⎪⎬⎪⎭.

The right part of the exprexion (14) for effective Hamiltonian can now be derived explicity. Calculating commutator in the above expression (17) hence we obtain the effective Hamiltonian for the exchange-coupled -th pair as

 δ~h=∑ll′kk′∑μν⎛⎜⎝tl0,l′νijtk0,k′μijΔll′ν⎞⎟⎠δμν{Xl↑,k↑iXl′↓,k′↓j+Xl↓,k↓iXl′↑,k′↑j−(Xl↑,k↓iXl′↓,k′↑j+Xl↓,k↑iXl′↑,k′↓j)} + +∑ll′kk′∑μν⎛⎜⎝tl0,l′νijtk0,k′μijΔll′ν⎞⎟⎠δklδk′l′(X00iXμνj+XμνiX00j)=δ~hs−ex+δ~hρ ,

and only a first contribution includes superexchange interaction :

 δHs−ex=∑ij∑ll′kk′∑μ2(tl0,l′μijtk0,k′μij)Δll′μ{(δl1kZ+il+δl1lZ+ik+δlk^Sil)(δl1k′Z+jl′+δl1l′Z+jk′+δl′k′^Sjl′) − −14(δl1ky+il+δl1ly−ik+δlknil)(δl1k′y+jl′+δl1l′y−jk′+δl′k′njl′)} ,

where , , and are a spin–exciton and electron-exciton operators at the -th cell. For simplicity, we assumed that . Note that the contribution in Eq.(LABEL:eq:19) at and

 δHs=∑ij∑ll′∑μ2(tl0,l′μij)2Δll′μ(SilSjl′−14nlnl′),

where , is an analogue of the conventional superexchange with exchange constant .

An exciton energy can not exceed the semiconductor gap , because of the divergence of superexchange contributions at . At the exciton cell state decays into an electron-hole pair state. Therefore photocarriers are generated under light pumping with a frequency higher than the absorption edge, and the superexchange on the photoexcited intracell states can be calculated in approach (LABEL:eq:19) only at the light pumping with the frequency in the transparency region of the material. It’s partly colored magnetic nondoped materials.

Let’s obtain the contribution (LABEL:eq:20) to the exchange energy of the system in the framework of mean-field approximation.

 (21)

where and is a probability to detect a cell in the excited state . Thus the light pumping effects in superexchange are frequency selective and linear on the amplitude pumping. In compound La214 the ground cell state is formed by a single hole orbital, the orbital may be excited by the external pumping (Fig.2).The standard mechanism of the superexchange in the ground state is shown in Fig.2b, while the superexchange via optically excited term is shown in Fig.2a. the formation of spin-exciton interaction that is beyond the Heisenberg model is shown in Fig.2c.

## Iii Results for copper oxide La214

We test the approach on the high- parent materal La214. At the LDA parameters of Hamiltonian taken from  Korshunov et al. (2005) 0.15 , =1.78 , 2.00 , and the - band index  Feiner et al. (1996); Gavrichkov et al. (2001) corresponds to first removal electron state.

Using the exact diagonalization procedure with LDA parameters, one obtains the weights , and , at the doublet and singlet, triplet states:

 |(N0,MS)l=1⟩=∣∣2b1⟩=∑λ=dz,pz,aβl=1(hλ)∣∣∣hλ,σ12⟩;|(N0,MS)l=2⟩=∣∣2a1⟩=∑λ=dz,pz,aαl=2(ha)∣∣∣ha,σ12⟩, (22)
 ∣∣(N+,M′S′)μ=1⟩ = |A1⟩=∑λ,λ′=b,dx,a,pz,dzAμ=1(hλ,hλ′)|hλ,hλ′,0⟩, ∣∣(N+,M′S′)μ=2⟩ = ∣∣3B1⟩=∑λ=b,dx∑λ′=a,pz,dzBμ=2(hλ,hλ′)|hλ,hλ′,M1⟩, (23)

where and are the holes in the -symmetrized cell states of oxygen and cooper states of the CuO layer, respectively.

Because of , just two contributions from the doublets and are available in the sum (LABEL:eq:18) over . Due to the symmetry CuO layer at any and therefore . Thus we evaluate the contribution (22) like the next:

 ⟨δHs−ex⟩=−zN2∑μ⎡⎣(tb0,bμ)2Δbμp2b+2⎛⎝(tb0,aμ)2Δbaμ+(ta0,bμ)2Δbaμ⎞⎠papb+(ta0,aμ)2Δbμp2a⎤⎦∼ ∼−zN2⎡⎣0.15(eV)⋅p2b+2(ta0,bA1)2ΔbaA1papb⎤⎦ (24)

Without external irradiation , , and Eq.(24) results in the exchange interaction (the first term in the right side of Eq.(24)) in the ground state obtained earlier in the work. Chao et al. (1977) What are the modifications of the exchange interaction that we can observe in L214 under resonance light pumping? The answer to this question depends on the ratio of the exchange interaction in the ground and excited states. Depletion of the ground state decreases , and a new contribution via excited orbital appears (see Fig.2). Using LDA parameters, and summing over all in the second term in Eq.(24), we finally obtain the result shown in Fig.3. So most likely superexchange contribution (24) will increase at any small population of excited states in La214 by a factor of

## Iv Conclusion

In summary, we would like to emphasize that optical pumping results in the occupation of some high energy multielectron states with different overlapping of the excited wavefunctions between neighboring ions vs the ground state orbitals. It is evident that this pumping results in the modification of the exchange interaction. Nevertheless an accurate calculation of a large number of contributions from different multielectron excited states is not a trivial theoretical problem. The gain of the Hubbard operators approach is the ability to control each excited state and its contribution to the ionic spin and orbital moment. Our approach to the exchange interaction via excited states is just a straightforward generalization of the previously developed projection technique for the Hubbard model Chao et al. (1977) and for the ground state of La214 within the realistic multiband model. Gavrichkov and Ovchinnikov (2008) The obtained effective Hamiltonian (LABEL:eq:19) contains not only spin-spin interactions via excited states but also more complicated exchange interactions accompanied with exciton or bi-exciton that are beyond standard Heisenberg model.

For undoped insulating cuprates the theory results in a prediction of the antiferromagnetic coupling strengthening proportional to the concentration of the excited states At the concentration of excited states 1% an increased exchange interaction is estimated by the magnitude K. For simplicity we have assumed stationary pumping with resonance absorbtion. Then the spectral dependence of the modified exchange coupling should coincide with the absorption spectrum. Due to the short time of the local electronic excitations 1 (fs) a dynamics of exchange interaction for the time intervals more then 10 (fs) probably can be also treated in our approach. It is evidently that the spin-exciton effects found here may be important in the dynamical regimes.

###### Acknowledgements.
This work was supported by RFFI grants 16-02-00273, No.14-02-00186.

### References

1. E. Beaurepaire, J.-C. Merle, A. Daunois,  and J.-Y. Bigot, Phys. Rev. Lett. 76, 4250 (1996).
2. J. Hohlfeld, E. Matthias, R. Knorren,  and K. H. Bennemann, Phys. Rev. Lett. 78, 4861 (1997).
3. C. D. Stanciu, F. Hansteen, A. V. Kimel, A. Tsukamoto, A. Itoh, A. Kirilyuk,  and T. Rasing, Phys. Rev. Lett. 98, 207401 (2007).
4. T. Ostler, J. Barker, R. Evans, R. Chantrell, U. Atxitia, O. Chubykalo-Fesenko, S. E. Moussaoui, L. L. Guyader, E. Mengotti, L. Heyderman, F. Nolting, A. Tsukamoto, A. Itoh, D. Afanasiev, B. Ivanov, A. Kalashnikova, K. Vahaplar, J. Mentink, A. Kirilyuk, T. Rasing,  and A. Kimel, Nature Communications 3, 666 (2012).
5. L.-M. Duan, E. Demler,  and M. D. Lukin, Phys. Rev. Lett. 91, 090402 (2003).
6. S. Trotzky, P. Cheinet, S. Folling, M. Feld, U. Schnorrberger, A. M. Rey, A. Polkovnikov, E. A. Demler, M. D. Lukin,  and I. Bloch, Science 319, 295 (2008).
7. Y.-A. Chen, S. Nascimbene, M. Aidelsburger, M. Atala, S. Trotzky,  and I. Bloch, Phys. Rev. Lett. 107, 210405 (2011).
8. S. Wall, D. Prabhakaran, A. T. Boothroyd,  and A. Cavalleri, Phys. Rev. Lett. 103, 097402 (2009).
9. M. Forst, R. I. Tobey, S. Wall, H. Bromberger, V. Khanna, A. L. Cavalieri, Y.-D. Chuang, W. S. Lee, R. Moore, W. F. Schlotter, J. J. Turner, O. Krupin, M. Trigo, H. Zheng, J. F. Mitchell, S. S. Dhesi, J. P. Hill,  and A. Cavalleri, Phys. Rev. B 84, 241104(R) (2011).
10. T. Li, A. Patz, L. Mouchliadis, J. Yan, T. A. Lograsso, I. E. Perakis,  and J. Wang, Nature 496, 69 (2013).
11. A. M. Kalashnikova, A. V. Kimel, R. V. Pisarev, V. N. Gridnev, A. Kirilyuk,  and T. Rasing, Phys. Rev. Lett. 99, 167205 (2007).
12. A. M. Kalashnikova, A. V. Kimel, R. V. Pisarev, V. N. Gridnev, P. A. Usachev, A. Kirilyuk,  and T. Rasing, Phys. Rev. B 78, 104301 (2008).
13. R. V. Mikhaylovskiy, E. Hendry, V. V. Kruglyak, R. V. Pisarev, T. Rasing,  and A. V. Kimel, Phys. Rev. B 90, 184405 (2014).
14. P. W. Anderson, Phys. Rev. 79, 350 (1950).
15. A. S. Moskvin and A. S. Luk’yanov, Sov. Phys. Sol. State 21, 1045 (1979).
16. L. N. Bulaevskii, E. L. Nagaev,  and D. I. Khomskii, Journal of Experimental and Theoretical Physics 27, 836 (1968).
17. K. A. Chao, J. Spalek,  and A. M. Oles, Journal of Physics C: Solid State Physics 10, L271 (1977).
18. J. E. Hirsch, Phys. Rev. Lett. 59, 228 (1987).
19. M. I. Katsnelson and A. I. Lichtenstein, Phys. Rev. B 61, 8906 (2000).
20. A. Secchi, S. Brener, A. I. Lichtenstein,  and M. I. Katsnelson, Annals of Physics 333, 221 (2013).
21. V. A. Gavrichkov and S. G. Ovchinnikov, Physics of the Solid State 50, 10811086 (2008).
22. M. M. Korshunov, V. A. Gavrichkov, S. G. Ovchinnikov, I. A. Nekrasov, Z. V. Pchelkina,  and V. I. Anisimov, Phys. Rev. B. 72, 165104 (2005).
23. V. A. Gavrichkov, S. G. Ovchinnikov, Z. V. Pchelkina,  and I. Nekrasov, J. Phys.: Conf. Ser. 200, 012046 (2010).
24. S. G. Ovchinnikov, JETP Lett. 77, 676 (2003).
25. S. G. Ovchinnikov and V. N. Zabluda, JETP 98, 135 (2004).
26. Y. S. Orlov, L. A. Solovyov, V. A. Dudnikov, A. S. Fedorov, A. A. Kuzubov, N. V. Kazak, V. N. Voronov, S. N. Vereshchagin, N. N. Shishkina, N. S. Perov, K. V. Lamonova, R. Y. Babkin, Y. G. Pashkevich, A. G.Anshits,  and S. G. Ovchinnikov, Phys. Rev. B 88, 235105 (2013).
27. S. G. Ovchinnikov, V. A. Gavrichkov, M. M. Korshunov,  and E. I. Shneyder, “Springer series in solid-state sciences,”  (Volume 171, Springer Berlin Heidelberg, Hamburg, 2012) Chap. LDA+GTB method for band structure calculations in the strongly correlated materials. In the Theoretical Methods for strongly Correlated systems, pp. 143–171.
28. R. O. Zaitsev, JETP 41, 100 (1975).
29. V. A. Gavrichkov and S. G. Ovchinnikov, Phys. Solid State 40, 163 (1998).
30. J. H. Jefferson, H. Eskes,  and L. F. Feiner, Phys. Rev. B 45, 7959 (1992).
31. L. F. Feiner, J. H. Jefferson,  and R. Raimondi, Phys. Rev. B 53, 8751 (1996).
32. V. A. Gavrichkov, A. A. Borisov,  and S. G. Ovchinnikov, Phys. Rev. B 64, 235124 (2001).
You are adding the first comment!
How to quickly get a good reply:
• Give credit where it’s due by listing out the positive aspects of a paper before getting into which changes should be made.
• Be specific in your critique, and provide supporting evidence with appropriate references to substantiate general statements.
• Your comment should inspire ideas to flow and help the author improves the paper.

The better we are at sharing our knowledge with each other, the faster we move forward.
The feedback must be of minimum 40 characters and the title a minimum of 5 characters