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10
gap Transitions on Superexchange with Nonequilibrium
Semen I. Polukeev* Vladimir A. Gavrichkov
Kirensky Institute of Physics Federal Research Center KSC SB RAS Akademgorodok, 50/38, Krasnoyarsk, 660036,
Russia
Sergey G. Ovchinnikov
Kirensky Institute of Physics Federal Research Center KSC SB RAS Akademgorodok, 50/38, Krasnoyarsk, 660036 Institute of Engineering Physics and Radio Electronics
Siberian Federal University Svobodny, 79, Krasnoyarsk, 660041
Russia
Received 19.08.2017, received in revised form 28.10.2017, accepted 06.12.2017 We investigated the effects of excited many-electron states in the optical control of the magnetic state in undoped Mott-Hubbard insulator. To derive the spin Hamiltonian in material under optical pumping we have used many-electron approach based on the X-operator representation. Extending the projection operators approach on arbitrary energy spectra of the Mott-Hubbard insulator, we obtained the Hamiltonian of superexchange interaction in analytical form. The Hamiltonian includes the spin-exciton variables which are usually missing in discussion on the magnetic response to optical pumping, and is not additive over contributions from the ground and optically excited states. As a test, a microscopic background for the optical induced superexchange was analyzed in La2CuO4 (further La214).
Keywords: superexchange, optically excited states, Mott-Hubbard insulator. DOI: 10.17516/1997-1397-2018-11-2-159-170.
Introduction
An ultrafast optical manipulation of magnetic order is very actively developing area nowadays. A number of experiments with femtosecond laser pumping of magnetic insulators has revealed unusual magnetic response when the photon energy hv is less then absorption edge Eg [1]. In the absence of the interband excitations these effects results from the intraatomic d-d excitations which form the weak and narrow absorption bands inside the optical gap of the crystal. Such optical spectra are typical for the Mott-Hubbard insulators like La214 [2] and other transition metal oxides [3,4]. These effects are beyond the conventional single electron approach to the
* [email protected]; [email protected] © Siberian Federal University. All rights reserved
УДК 537.9
Effects of Optical Intra-Interaction in La2CuO4 Photoexcited Centers
electronic structure of solids based of different implementation of the density functional theory. Nevertheless, for the materials in the ground state, the exchange interactions has been studied in the ab-initio approach (e.g. in works [5-7]), but the initial many-electron representation still looks more natural and short way to the final result when the material is in an optically excited state.
The intra-atomic optical spectra can be obtained in many-electron approach, where the quasi-particles are just the electron excitations between the many-electron states of relevant symmetry and with different numbers of particles, for La214 and it was demonstrated in the works [8,9]. The undoped Mott-Hubbard insulators, unlike conventional materials have both a singlet and degenerate ground cell states [10].
It is essential that the optically excited states would be introduced to the superexchange theory together with the ground state on equal footing, because the intracell transitions induced by optical pumping occurs much faster than superexchange interaction (Av/W C 1, Av is optical transition width). Therefore, in the process of superexchange the spin and interaction between them can be modified or even break down. A general theory with a fixed spin and orbital degrees of freedom (e.g. the work [11]) is a good starting microscopic point to derive anisotropy effects in the optical absorption at v ~ Eg shows that individual contributions to superexchange are given by virtual charge transfer excitations only. Because, it is necessary to derive the microscopic Hamiltonian here, step by step, where the superexchange constant must be redefined, if possible. To achieve this, there are two acceptable approaches to study the superexchange interaction. First is the calculation with the intermediate states which arise through hopping from ligand to ligand (e.g. between the oxygens) in the perturbation theory of higher order than the fourth [12]. Another approach is the cell perturbation theory taking into account all the excited states. The latter seems more appropriate [12-15], where, however, it is necessary to work with a large number of the virtual charge transfer excitations [16]. Being based on the X-operators representation [17] the LDA+GTB approach [8,18] includes the whole spectrum of localized many-electron states both ground and excited states with different spin and orbital moments. Nevertheless, for the materials in the ground state, the exchange interactions has been studied in the ab-initio approach It's important to understand further that the excited states in our work distinct from the virtual "excited states" occurring in a perturbation theory of superexchange [19]. These excited states may be occupied due to the optical pumping at the frequencies of specific d-d excitations only. As a result, the exchange interaction can be modified. Despite the simplicity of the idea, the magnitude and sign of the pumping effect on the exchange interaction depends on the orbital and spin symmetry of the many-electron excited states.
In this work a Hamiltonian of superexchange interaction was derived in analytical form due to the many-electron approach based on the X-operator representation [17] and technique of projection operators [20] generalized on arbitrary quasiparticle energy spectra of Mott-Hubbard insulator. The Hamiltonian can be reduced to the usual Heisenberg type only under additional assumptions. It has spin-exciton variables missing in the discussion on the optical pumping effects. The superexchange is also not additive over the states of transition element, in contrast the contributions from the virtual "excited states". The superexchange in unexcited La214 is in accordance with the phenomenological Goodenough-Kanamori rules because 180o-superexchange is antiferromagnetic (AFM), and ferromagnetic (FM) contribution from the virtual "excited states" is very small. The induced AFM contribution to the mean energy will increase ~ 4 * 10~3eV/(%) at the optical induced occupation (in %) of the excited states.
1. General theory of superexchange in Mott-Hubbard materials with optically excited states
In this section, we apply the method of projection operators [20] to derive an effective spin Hamiltonian from Hamiltonian of the pd-model with any number of orbitals H = Hd + Hp+
+HPP + Hpd, where
h . = E
f\a
(£X - M) d+fadf + 1 UXn\f nXf +
+ Y1 [-JddXfa d*fa' d+'fa' d^'fa +Y1 vaa' f ny'f y*' v f'
Hp 'y ', {£a — m) p+mapama + 22 Uanamnam +
maa [_ a'm
HPd = {tyaP+ma dfXa + he. + VaxKmKf) ,
mf aXaa'
HPP {taßP+maPßm'a + h-e)
mm' aßa
(1)
and nf
d+fa dfa, n
a
am
P+ma P
amar ama i
f and m run over the positions d\f and pam which are sets of localized atomic orbitals; likewise ey and ea are energy appropriate atomic orbitals, t\a and taß are the hopping matrix elements, Uy, Ua and Jd are the intraatomic Coulomb interactions and Hund exchange, Vay is the Coulomb repulsion between the electrons at the copper and oxygen. In this section a consideration is given to the case with one hole per cell in the undoped materials and arbitrary number of the occupied orbitals. This is relevant for the high-Tc cuprates.
In the many-electron approach it is necessary to introduce a symmetrical cell oxygen states
( Pxk \
pgk by an unitary transformation Pek = AkPak, where the column vectors Pak = I Pyk I and
V Pzk J
( p°lk I
Pek = I Pe2k I at a or n type of bonding. The transformation matrix Ak depends on the
V Pe3k J
cell symmetry and index 0 numbers the irreducible representations of the symmetry group of cell. The transformation matrices can be found in works for cuprates [14,15,21,22], manganites (eg — orbitals) with the orbital ordering [23-25] and cobaltites (t2g — orbitals) [26,27]. In a new symmetric cell representation the dependence of the intra-cluster and inter-cluster interactions, including Coulomb, are renormalized [14] due to strong localization of new symmetrical Wannier cell functions pef. As a result, Coulomb interactions are taken into account in a single cell through a procedure of exact diagonalization, and the interaction between cells is calculated in the framework of cell perturbation theory (see e.g. the diagram approach [28]) The different Coulomb contributions in the cuprates investigated in the work [14,15] in detail. In respect of other oxides the approximation is based on a sharp drop in the renormalization coefficients with increasing distance between cells [22,23,27]. Further, a crystal lattice is divided into unit cells, so that the Hamiltonian is represented by the sum H0 + Iii, where the component H0 is the sum of intracell terms and component IIl takes into account the intercell hoppings and interactions.
The component H0 is exactly diagonalized, and the exact multielectron cell states \p) (\q)) and energies are obtained. Then these states are used to construct the Hubbard operators of the unit cell Rf : Xp'q = \p){q\, where the meaning of the indexes p and q is clear from Fig. 1.
Fig. 1. A configuration space of all possible cell states involved in superexchange with one hole per cell. The sectors N0 and N+ correspond to states (6) and (7) respectively (N0 is hole number per cell in undoped material, N_ = N0 — 1, N+ = N0 + 1)
Hi
0 =
Nv
■o f + — m) Xlpla + — 2M)X
la
I
' i
is the sum of intracell terms. A component
H = E Xr Xr'
(2)
(3)
fg rr'
takes into account the intercell hoppings and interactions, where tis defined by the matrix of hopping integrals t^ :
f ^E fXf9 t^Aa (r) IX' a (r') + 7A'a (r) YXa (r')]
W a
where the matrix element is represented by expression
Y fa (r) = {(N+, Ms )v | Cffa |(No, Ms )i) x S (S», Si ± jaj) S (Mv, Mi + a)
(4)
(5)
where cf\a runs over all operators pgfa and d\fa. In the approach one assume that the quasipar-ticles are unit cell excitations which can be represented graphically as single-particle excitations (transitions) between different sectors Nh = ... (N- = N0 — 1), N0, (N+ = N0 + 1),... of the configuration space of the unit cell (N0 is hole number per cell in the undoped material, see Fig. 1) [29]. Each of these excitations forms a r-th quasiparticle band, where the vector band index r = {p, q} in configurational space [30] numerates the initial \p) and final |q) states. The excitations with the number of electrons increasing or decreasing, form the conduction or valence bands, respectively. Note that the the possibility to derive the Hamiltonian (2) and (3) from (1) in terms of arbitrary symmetry of the material has not yet been investigated.
In the case of one hole per cell N0 = 1, the cell states \(No,Ms)t) are a superposition of different hole configurations of the same orbital symmetry:
j(No,Ms)i) = J2ßf (hf) lhf,Ms)■
(6)
In accordance with the spin selection rules, the non-zero matrix elements (5) are possible for the one-hole spin doublet states C^Nx = 2NX in the sector N0 and C1N!x = NS + 3NT of the spin singlets NS = Cfj + NX (low spin partners) and NT = of spin triplets in the sector N+:
\(N+,Ms)v) = EPv (hx,hx')\hx,hw,MS) (7)
XX'
(high spin partners) in the two-hole sector N+ and the NX orbital approach. The superexchange interaction appears at the second order of the cell perturbation theory with respect to hop-pings [13]. 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 t"g with r = {0, la] and r' = {la, v] in expression (3). In the Hub-bard model, there is only one such element, which describes the hoppings between the lower and upper Hubbard bands. To extract them, we extend the projection operator method proposed by Chao et al [20] on the arbitrary energy spectra of Mott-Hubbard material, where the total number of diagonal operators X™ is equal to Nv + Nl + 1 and the sequence indexes l and v (1 < l < Ni, 1 < v < Nv) runs over all electron states in the configuration spaces in Fig. 1. Using a set of operators
po = X + E x1™) (j + E ) , (8)
and
PM
= XfM + XfM - Xmmj2Xjv (9)
with v (p) = 1, 2,...NV we can identify the contribution to the superexchange from the interband transitions. As will be seen below, a approach with the operators (5) and (6) differs from the work [20] just in details. It can be checked that each of operator po and pм is a projection operator p2 = po and p2 = pM. These operators also form a complete and orthogonal system,
po + J2 pм = 1, popM = 0 and pMpv = pM. We separate the diagonal and off-diagonal matrix
m=I
elements in expression
H=(Ho + Hln)+ H°ut. (10)
According to the work [20], one can introduce a Hamiltonian of the exchange-coupled (ij)-th
pair: h = (ho + hin) + h°ut = Йj, where Й = £ Hij, and
ij
ho + h\n = pohpo + Ep^hpv, (11)
ffV
and
(e + (E »)
Kut = P0M Vpm + VpA hpo (12)
are intra- and inter- band contributions in H1 respectively. In the unitary transformation
h = eGhe-G (13)
the matrix G satisfies to equation
po^pj + ( E pm) hpo + ^ ( po^po+E pfhp
Vm/Vm/LV fv
0, (14)
V
and transformed Hamiltonian are given by
( PohPo + EP^PA + 1 GApoh^PM + EPMhPo
\ Mv ) L V M M
(15)
where the contributions from inter-band transitions involving only low-spin partners can be calculated as
Poh
(e pm) = EE tj,l'M
\ M / ll'a M
,l0,l M„/„\ vlao Yl hi V(a)Xi Xj
laO yl'aM
3
(16)
and
(EP^ hpo = EEtMl'a,0lan(v)Xtl aX°la, (17)
M M ll a
where n(a) = ±1 depending on the spin direction. Similar expressions can be obtained and for high-spin partners. The solution of Eq. (14) has the form
,lo,l ' M
G
Am*"(■
M ll a
a) ( XMl'aXola - XlaoXl
3
-l' aM\ 3,
(18)
where All 'M = eo + eM — (el a + el), and the commutator in Eq. (15) can be represented as
Sh = 1 2
E{ Gv,{pohpM + PMhpo) }
1E
M
lo,l'v ij
tlo,l v
E A~n(a) (X^Xfa — XlaoX^ tfk' Mn(s) (xf' sXoks +
ll 'a ll v kk's
+ X"so x" sM
. (19)
Calculating commutator in the above expression (19) we obtain the effective Hamiltonian for the exchange-coupled (j)-th pair:
5h =
(,lo,l' v,ko,k' m\ r {
^AH ¥ X
ll v
■ ,lo,l' v , ko,k'm \ r
ij ij (x'^'k^Xl- ^ + X'^'k^X1'^'k'— X'^'k^X1''
+
I x^''"'x
x /.lo'l'v tko'k >\
LlM xjï'k't) + ]T E -jArj— SklSk l' (xooxmv + XMvX°o) = Shs-ex + Sh„ (20)
ll kk Mv All v
where only a first contribution includes the superexchange interaction SHs-ex = Shs-ex. The
i3
latter can be expanded in powers of variable Xliçs)ls, the mean value of which v+ = X¡°S'ls is an probability of optical intracell excitation:
SHs-ex H s + Hex,
(21)
where the superexchange in unexcited material and contributions from optically excited states are given by:
Hs
{Ns 2(tloo'lom)2 /
E jH
M Al0M
1 s 3Nt (t[oO'loM)2 /
Silo Sjlo — 4 nilo njlo) ^E 3a- ( Silo 3 + 4 nilo 3
3
,
(22)
= —- {(^Z-+KiZ+k + Stk Sil )(Sl0k' Z-' + Slol' Z+k'+
ij ll'kk' H ll'M
+Si'k'SjI') - 4 (Si0kVa + Si0iy+k + Stknu) (Si0k'y-' + Si0i'y+k' +St'k'njI') (23)
and S+ = Xlt'li, 2Szt = Y n H Xla'la, y+ = n10X\01, Z+ = SUoXlot are a spin, electron-exciton
a
operators and spin-exciton at the i-th cell. The commutation relations for the latter operators have the spin character in the mean field approximation Z+ « Siloand the contributions involving the spin-exciton variables result in additional effective field in Eq. (22):
Jj « ^^ ]T U -(v+öioi + v-öiok + Sioiöik) (v+ Slol' + v-Siok' +
Ц ll'kk' ll
loO,loM)2
+Si0,5vk') = 2]T д ' +O (u^ + ... (24)
м
Note that at l = k and l' = k' the contribution in Eq. (23) takes the form
= E E { JAij( Silj' - 4 nil njl') - JF>ij( SilSjl' + 4 nilnjl') f • (25) ij ll' ^ '
It is much like to a conventional superexchange (22), but with other superexchange constants
N 2 I 3N 2 /
J'lij =2 (tj м) / Ди'м and Jp j = j / Дц'м, which is additive over the singlet
and triplet virtual p-th "excited states" respectively.
Applicability of the expessions (24) and (25) is limited to small excitation energy Sllo < Eg, where Sllo = el — elo and Eg = (eMo + eo — 2elo). The usual mechanism of the superexchange (22) in the ground state is shown in Fig. 2b, while the superexchanges (25) via optically excited state is shown in Fig. 2a. At l = l' the last represents a non-symmetrical interaction with one excited cell in the pair. The spin-exciton contribution ) in Eq. (23) beyond Heisenberg model is
shown in Fig. 2c. From Eq. (24) it follows that the optical pumping effects on the superexchange are frequency selective and linear on the amplitude pumping.
Fig. 2. Two circles (dashed line) are a sequence of intracell transitions at the light-induced superexchange Jj (a) and Jj (b) between i and j cells in Eq. (25), (c) illustrates the single
circle (spin-exciton) contribution ~ (taj0'bAttbji'bA) / ДаЬА (for cuprates), which can be reduced to
the spin Hamiltonian using additional assumptions only (see Eq. (24))
2. Optical effects on superexchange in La2CuO4
We test the approach to derive the superexchange in the high-Tc parent materal La214 under optical pumping. The initial Hamiltonian (1) is similar to the one of multiband pd-model [31]. The difference with the low energy three orbital pd-model [32-34] is related to an addition of the z-oriented dz2 orbital and pZap) orbital of the apical oxigen ions. In the LDA+GTB method the Hamiltonian parameters are calculated ab initio [8]. Unlike the work [34], where the method of projection operators has also been used to derive the superexchange interaction, we take into account the intracell excited states in CuO6 cell. Some improvements are also achieved with the help of the cell representation. Indeed, a comparison of the results of the fourth-order with the calculations in higher orders of perturbation theory [12] in atomic representation and the exact diagonalization of finite clusters [12, 35-38] shows that the in-plane superexchange J depends significantly weaker on (pad,\)-hopping tpd), because of the intermediate two-hole states which arise through hopping from oxygen to oxygen tpp). These effects are partly included in the present approach even in the second-order of cell perturbation due to the exact diagonalization procedure for the intracell part of Hamiltonian (1).
Here, it is also useful to obtain the expression for the AFM contribution in Eq. (25) in the mean-field approximation:
{H-ex)
2 EE
ij ii'
jij xliala) (xlsl/sXj
zN
Jj + 2E jlmPiPi0 ++
l = lo
E
l,l'=lo
(ij)pl Pl'
(26)
where plo
1 -
(E Pl)
V=lo J
and pl = (X,
Aflf\
/
X
is a probability to detect a cell in
\(N0,MS)l) excited state. Using the exact diagonalization procedure with LDA parameters: Jbb «0.15 eV, Sllo = Sab=1.78 eV, Eg =2.00 eV taken from the work [8], and the band index r = {26i, Aij associated with the first removal electron state for pair of indexes {/o, no} [14,22], one can obtain the weights al, /3l and BM at the doublet (6) and singlet, triplet (7) states:
|2bi> = \(No,Ms)lo) = E (hx)\hx,ai),
\=dz ,pz ,a
|2ai> = \(No,Ms)l) = Y. al (ha)
\=dz ,pz ,a
(27)
la,* 1
and
|Ai) = \(N+ ,MS' )Mo)
|3B^ = \(N+,M'S' )M)
E AMo (hx,hx')\hx,hx', 0),
X,X' = b,dx ,a,pz,dz
-- E E bM(hx,hx')\hx,hy,Mi),
X=b,dx X' = a,pz ,dz
(28)
where hb and hdx are the holes in the b-symmetrized pgi cell states of oxygen and dx2-y2 cooper states of the CuO2 layer, respectively. Because of Sab < Eg, only two contributions from the doublets |2a^ and |2b^ are available in the sum (26) over / indexes. Due to the symmetry CuO2 layer YX({2ai, Ai}) =0 at any orbital index A, and therefore tbj'aA = taj0'aA = 0. Thus we evaluate the contribution from excited states like the next:
A6M Pb + +% Aba, + Aba, )PaPb + Ab, Pa
^0.15(eV) • p2 + PaPb .
2 [ AbaAi J
In unexcited material pb = 1, pa = 0, and Eq. (29) results in the exchange interaction Jbb ~ 0.15eV in the ground state [16]. What are the modifications of the exchange interaction we can expect in L214 under resonance light pumping? The answer to this question depends on the ratio of the exchange interaction at the ground |26i) and excited |2ai) states. Depletion of the ground state pb = 1 — x decreases Jbb contribution, and a new contribution Jba from excited state a1 appears (see Fig. 2a). Summing over all ^-th virtual "excited states" in the second term in Eq. (29), we finally obtain the result that the superexchange AFM contribution in La214 will increase at any small population of excited states by a factor of ~ 4 • 10-3eV(%)-1, i.e. the superexchange interaction is increased by 4 meV at x=0.01.
Conclusions
In summary, we have examined the response of superexchange in magnetic Mott-Hubbard materials with the arbitrary spin under optical pumping. To derive the effective spin Hamiltonian we use the initial pd model Hamiltonian in the Hubbard operators representation (LDA+GTB approach [8]) and method of the projection operators [20]. The effective Hamiltonian (21) contains not only spin-spin interactions involving optically excited states but more complicated interactions of non-Heisenberg type accompanied with exciton or bi-exciton. The Hamiltonian is non-additive over the ground and optically excited states, but it is additive to the virtual "excited states". To test our approach, we have calculated the superexchange interaction and D-M interaction in antiferromagnets La214 under optical pumping.
In cuprates under optical pumping with a frequency of d-d transitions (v < Eg) AFM superexchange enhanced by 40K on 1% of the occupation of excited state. The D-M interaction of also does not undergo radical changes since the ground and the optically excited states of cell are Kramers doublets in La214 (Fig. 2). Under irradiation with a frequency of corresponding to the charge-transfer excitations(v ~ Eg), the optically induced magnetism of electron-hole pair is not appear because the electron and hole are at the many-electron spin singlet states. A spectral dependence of modified superexchange should coincide with the d-d absorption spectra in the transparency window AE ~ 0 ^ 2eV. Let us note here the discrepancy of our results and the general trend given in the work [39], where the excited states contribute to the FM exchange at the low "charge transfer energies" (A;; in our notation).
This work was supported by the Foundation for Assistance to Small Innovative Enterprises (FASIE), Russian Foundation for Basic Research, Government of Krasnoyarsk Territory and Krasnoyarsk Region Science and Technology Support Fund to the research project 17-42-240212. We are also grateful to the RFBR grants: 16-02-00273, No.16-02-00098, NSh-7559.2016.2.
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Эффекты оптических внутрищелевых переходов в суперобменном взамодействии в La2CuO4 с неравновесными фотовозбуждёнными центрами
Семён И.Полукеев Владимир А.Гавричков
Институт физики им. Л. В. Киренского ФИЦ КНЦ СО РАН Академгородок, 50/38, Красноярск, 660036
Россия
Сергей Г.Овчинников
Институт физики им. Л. В. Киренского ФИЦ КНЦ СО РАН Академгородок, 50/38, Красноярск, 660036 Институт инженерной физики и радиоэлектроники Сибирский федеральный университет Свободный, 79, Красноярск, 660041
Россия
Мы иселедовали эффекты возбужденных многоэлектронных состояний в оптическом способе управления магнитным состоянием недопированных Мотт-Хаббардовских материалов. Для вывода спинового гамильтониана в веществе под воздействием оптической накачки был использован многоэлектронный метод, основанный на представлении X-операторов. С помощью обобщенного на произвольный энергетический спектр метода проекционных операторов мы получили гамильтониан суперобменного взаимодействия в аналитической форме. Гамильтониан является не аддитивными по вкладам от основных и оптически возбуждённых состояний, а также содержит спин-экситонные переменные, которые обычно отсутствуют в обсуждении магнитного отклика на оптическую накачку. В качестве конкретного приложения для теории мы рассчитали индуцированный суперобмен для оптически возбужденного купрата La2CuO4(в дальнейшем La214).
Ключевые слова: суперобмен, оптически возбужденные состояния, диэлектрик Мотта-Хаббарда.
