Effective quantum pseudospin-1/2 model for Yb pyrochlore oxides

0.17 MB·

by  Shigeki Onoda

1 2
Effective quantum pseudospin- / model for Yb
1 pyrochlore oxides
1
0
2
Shigeki Onoda

n
a Condensed Matter Theory Laboratory, RIKEN,2-1, Hirosawa, Wako351-0198, Saitama,
J JAPAN

6 E-mail: [email protected]

]
el Abstract. An effective quantum pseudospin-1/2 Hamiltonian for Yb2TM2O7 (TM = Ti
- and Sn) is obtained in terms of the atomic Kramers ground doublet of the LS coupling and
r the crystalline electric field, which is almost described with Jz = ±1/2. It is calculated
t
s microscopically as the sum of Anderson’s superexchange interaction and the magnetic dipole
.
t interaction. It is found that it shows a strong exchange anisotropy.
a
m
-
d
n
o 1. Introduction
c Geometrical frustration and quantum fluctuation provide a promising route to unconventional
[
states in magnetism, including various types of spin liquids characterized by the absence or

1 emergenceofunconventionalordersofchirality ormultipolesinthreespatialdimensions[1,2,3].
v
The pyrochlore lattice structure offers a prototypical stage where the geometrical frustration
0
3 prevents the magnetic dipole long-range order (LRO) [1, 4, 5]. One class of magnetic pyrochlore
2 oxides [6] which have been most intensively studied is a dipolar spin ice, e.g., Dy Ti O
2 2 7
1
and Ho Ti O [6, 7, 8, 9]. Recently, quantum variants of the spin ice have attracted great
. 2 2 7
1 interest [10,11, 12], which includeTb Ti O [13, 14,15], Tb Sn O [15], and Pr Sn O [16, 17],
2 2 7 2 2 7 2 2 7
0
Pr Ir O [18, 19], Yb Ti O [20], and Er Ti O [20].
1 2 2 7 2 2 7 2 2 7
1 In the dipolar spin ice, large 111 Ising magnetic moments ( 10µB) of Dy3+ and Ho3+
h i ∼
: ions, each of which points either inwards (“in”) to or outwards (“out”) from the center of the
v
i tetrahedron, interact with each other mainly through the magnetic dipole interaction [21]. This
X
gives a ferromagnetic nearest-neighbor coupling, forming the “2-in, 2-out” ice rule [22] and
r
a leaving macroscopically degenerate ground states because of geometrical frustration. The spin
dynamics is quite slow and the quantum effects are almost negligible.
Recently, quantum effects have been observed with inelastic neutron-scattering experiments
on the spin-ice related compounds, Tb Ti O [13, 14, 15], Tb Sn O [15], and Pr Sn O [17].
2 2 7 2 2 7 2 2 7
Pr TM O (TM = Zr, Sn, and Ir) shows no magnetic dipole LRO but a (partial) spin freezing
2 2 7
at T 0.1-0.3 K [16, 17, 18, 19, 23]. Pr Ir O shows a metamagnetic transition at low
f 2 2 7
∼
temperatures only when the magnetic field is applied in the [111] direction [19], as in the spin
ice. A chiral spin liquid [3] has been detected through the anomalous Hall effect [24] at zero
magnetic field without magnetic dipole LRO in Pr Ir O [19]. Unlike the dipolar spin ice, the
2 2 7
Curie-Weiss temperature T is antiferromagnetic for the zirconate [23] and iridate [18]. The
CW
stannateshowsasignificantleveloflow-energy short-rangespindynamicsintheenergyrangeup
toafewKelvin[17],whichisabsentintheclassicalspinice. Vitalrolesoftheplanarcomponents
have also been experimentally observed in Yb Ti O and Er Ti O [20]. Obviously, quantum
2 2 7 2 2 7
fluctuations enrich the otherwise classical properties of the spin ice.
The mechanism in which the quantum fluctuations indicated by these observations appear
depend on the rare-earth ions, including the difference in Kramers and non-Kramers ions. A
relatively small localized magnetic moment reduces the magnetic dipole interaction, allowing
for an important role of the superexchange interaction, as in the case of Pr3+ ions [11, 12] and
also expected for Nd3+, Sm3+, and Yb3+ ions because of their small moment amplitudes, 3.3µ ,
B
0.7µ , and 4µ , respectively, for isolated cases. In the case of Tb3+ ions, the presence of low-
B B
energy crystalline electric field levels also plays a role [10]. Remarkably, the large LS coupling
and the crystalline electric field for localized 4f electrons makes the superexchange interaction
highly anisotropic [11, 12]. In fact, the strength of the anisotropy depends on the properties
of local magnetic doublets of rare-earth ions, producing the variety of experimentally observed
low-temperature magnetic properties which are mentioned above. Theoretically, it is expected
that this anisotropic superexchange interaction drives quantum phase transitions among the
spin ice, quadrupolar states having nontrivial chirality correlations, and the quantum spin ice
for non-Kramers magentic doublets.
As for Kramers doublets, on the other hand, understandings of Yb Ti O are rather
2 2 7
controversial. Yb localized 4f magnetic moments are believed to almost lie perpendicular
to the 111 directions, unlike Dy, Ho, and Pr cases. It has a ferromagnetic Curie-Weiss
h i
temperature 0.65 K and a first-order transition has been observed at 0.24 K with the specific
∼
heat measurement [25]. It has been reported that a neutron-scattering experiment on the single
crystal shows a LRO in the low-temperature phase [26], while a combined study using neutron
diffractionandM¨ossbauerandµSRspectroscopies [27]andthepolarizednewtron-scattering [28]
onthepolycrystalshaveindicatedthatthemagneticdipolecorrelationremainsdynamic. Recent
neutron-scatteringexperimentshaveclarifieda[111]rodscatteringintensityintheparamagnetic
phase [29], which has been phenomenologically analyzed by taking into account the anisotropic
superexchange interaction [30].
In this paper, we present a microscopic derivation of an effective quantum pseudospin-1/2
model,whichtakesthesumofthemagneticdipoleinteractionandthesuperexchangeinteraction,
in terms of atomic Kramers doublets of Yb3+ ions for the frustrated magnet Yb TM O (TM
2 2 7
= Ti and Sn) on the pyrochlore lattice.
2. Local Kramers doublets of Yb3+ ions
The ground state of an isolted Yb3+ ion is characterized by the eightfold degenerate manifold
2F7. In the pyrochlore oxide Yb2Ti2O7, the D3d crystalline electric field partially lift the
2
degeneracy, yielding the Kramers ground doublet
7 1 5
σ = ασ J = σ +β J = σ +γσ J = σ , (1)
D z z z
| i − | 2 i | 2 i | −2 i
with σ = , real coefficients α 0.388, β 0.889, and γ 0.242 [31, 32], and the
± ≈ ≈ ≈
eigenstate J = M of the total angular momentum Jz, where the local coordinates (x ,y ,z )
z i i i
| i
(i = 0, ,3) are site-dependent, as explicitly given in Appendix A. In particular, the
···
quantization axis z is always taken to beparallel to the 111 direction that points from the Yb
i
h i
site to the center of one of the two tetrahedrons to which the Yb3+ ion belongs. This ground
doublet is energetically well separated from the first excitated doublet by 620 K [31].
The ground doublet has the following nonvanishing Ising and transverse components of the
total angular momentum J
1
Dhσ|Jˆz|σ′iD = 2Jkσσz,σ′, (2)
(a) h2 cosθ= -1/3 (b)
f h0h 0h2
U f p f
z z' h0h 1h1 h0h 1h1
θ f p f f p f
h1h 0h1 h0h 2h0 h1h 0h1
f p f f p f f p f
hf1 ∆ hf1hp1hf0 h2h 0h0 hf1hp1hf0
0 h f p f
p,m,σ
Figure 1. (a)Locallevel schemeof O2pholes andYb4f holes. (b)Vitualhoppingprocessesof
holes contributing to the superexchange interaction. n (n′) and ℓ in the state hnhℓhn′ represent
f p f
the number of f holes at the Yb site r (r′) and that of p holes at the O site.
1
σ Jˆx,y σ′ = J σx,y , (3)
Dh | | iD 2 ⊥ σ,σ′
where J = 7α2 +β2 5γ2 and J = 2√7αγ +4β2. On the other hand, it does not contribute
k ⊥
−
to the atomic quadrupole moment, as is clear from
σ Jˆ±,Jˆz σ′ = σ Jˆ+Jˆ+ σ′ = σ Jˆ−Jˆ− σ′ = 0. (4)
D D D D D D
h |n o| i h | | i h | | i
Let us asign the Pauli matrices σˆr to a psedospin-1 that operates on the subspace for the
2
the atomic doublet at the site r. Then, the localized magnetic dipole moment is given by
1
mˆr = gJµBJˆr = µB g⊥(σˆrxxr +σˆryyr)+gkσˆrzzr , (5)
2 h i
where g = g J 4.18 and g = g J 1.77 are reasonably close to experimental values 4.27
⊥ J ⊥ k J k
≈ ≈
and 1.79, respectively, for Yb Ti O [31], with g = 8/7 being the Land´e factor. This sharply
2 2 7 J
contrasts to the non-Kramers magnetic doublet of Pr3+ ions, where σˆrz only contributes to the
magnetic dipole moment and σˆr± to the magnetic quadrupole moment [11, 12].
3. Magnetic dipole interaction
The magnetic dipole interaction takes the form
Hˆ = µ0 mˆr ·mˆr′ 3(mˆr ·∆r)(∆r·mˆr′) , (6)
D 4π (cid:20) (∆r)3 − (∆r)5 (cid:21)
hrX,r′i
with∆r = r−r′ andthesummation hr,r′i overallthepairsofatomicsites,wherethelocalized
magnetic moment is given by Eq. (5)P.
4. Superexchange interaction
To derive an anisotropic superexchange interaction between Yb magnetic moments, it is
convenient to take the hole picture, where the local ground state is described by a single 4f
hole at each Yb site and no hole at O stites. The local Coulomb repulsion among Yb 4f holes
and the energy level ∆ of the O 2p holes compared with that of a Yb 4f hole provide the largest
energy scales of the problem, as schematically shown in Fig. 1 (a). In fact, the local ground
state is modified by the nonlocal electron transfer. The amplitude is described by two Slater-
Koster parameters V and V , which represent the transfer integrals between p /p and
pfπ pfσ x y
fx(5z2−r2)/fy(5z2−r2) orbitals and between pz and f(5z2−3r2)z orbitals, respectively [33]. Then, we
performthe fourth-orderstrong-coupling perturbation expansion of the f-pelectron transfer, by
fully taking into account the virtual electron transfer processes among O 2p orbitals located at
the center of the tetrahedron and 4f orbitals located at adjacent Yb sites, which are shown in
Fig. 1 (b), in the local coordinate frames. The superexchange Hamiltonian is then obtained as
n.n.
2
Hˆff = ∆2 Vm1Vm′1Vm2Vm′2fˆ†r,m1,σ1fˆr,m2,σ2fˆ†r′,m′1,σ1′fˆr′,m′2,σ2′
hrX,r′imX1,m2mX′,m′ σX1,σ2σX′,σ′
1 2 1 2
1 1
×(∆ + U)(cid:16)Rr†Rr′(cid:17)m1,m′2;σ1,σ2′ (cid:16)Rr†′Rr(cid:17)m′1,m2;σ1′,σ2. (7)
with V±1 = Vpfπ and V0 = Vpfσ as well as the rotation operator Rr of the angular momentum
from the global coordinates to the local ones at the Yb site r, which is given in Appendix A.
Here, fˆ†r,m,σ and fˆr,m,σ represent the creation (annihilation) and annihilation (creation)
operators of 4f electron (hole) with the orbital m and the spin σ at the Yb site r. Rr is
the rotation matrix of the total angular momentum from the glocal coordinates to the local, as
given in Appendix A. Projecting Eq. (7) onto the space of the doublets, we obtain
n.n.
Hˆex = Jn.n. gkσˆrzσˆrz′ +g⊥ σˆrxσˆrx′ +σˆryσˆry′
− hrX,r′ih (cid:0) (cid:1)
+gq ~σˆr ~nr ~σˆr′ ~nr′ ~σˆr ~n′r,r′ ~σˆr′ ~n′,r,r′
(cid:16)(cid:16) · (cid:17)(cid:16) · (cid:17)−(cid:16) · (cid:17)(cid:16) · (cid:17)(cid:17)
+gK σˆrz ~σˆr′ ~nr,r′ + ~σˆr ~nr,r′ σˆrz′ , (8)
(cid:16) (cid:16) · (cid:17) (cid:16) · (cid:17) (cid:17)i
with J = (2βVpfσ)4 1 + 1 and the dimensionless coupling constants as functions of
n.n. 33(7∆)2 U ∆
(cid:16) (cid:17)
x = V /V ,
pfπ pfσ
9 63
gk = 1 8√6x x2 3√6x3+ x4, (9)
− − 2 − 16
45 9
g⊥ = 1+4√6x+ x2 3√6x3+ x4, (10)
2 − 16
9
gq = 2 1 2√6x+9x2 3√6x3+ x4 , (11)
− (cid:18) − − 4 (cid:19)
45 15 9
gK = 2√2 1+√6x x2+ √6x3 x4 . (12)
(cid:18) − 4 4 − 8 (cid:19)
Here, we have introduced ~σˆr = (σˆrx,σˆry) as well as two unit vectors ~nr,r′ = (cosφr,r′, sinφr,r′)
−
and~n′r,r′ = (sinφr,r′,cosφr,r′). InthechoiceofthelocalcoordinateframesgiveninAppendix A,
the phase takes φr,r′ = 2π/3, 2π/3, and 0, when the pair of Yb sites r and r′ corresponds
−
to that of R + a and R + a with (i,j) = (0,1),(2,3) , (0,2),(1,3) , and (0,3),(1,2) ,
i j
respectively, where a = a(1,1,1), a = a({ 1,1,1), a }={a(1, 1,1),}and a {= a(1,1, 1})
0 −8 1 8 − 2 8 − 3 8 −
with the lattice constant a and an fcc lattice vector R.
Previously, this form of the nearest-neighbor Hamiltonian with and without the JK term
n.n.
is found from the symmetry analysis for the Kramers doublets [12] and microscopically for the
non-Kramersdoublets[11,12],respectively. Thepresentmicroscopiccalculation oftheexchange
couplingconstants fortheKramersdoubletsofYb3+ ionsremarkablyreveals thatthesymmetric
exchange part is ferroic in terms of pseudospins, i.e., “in” or “out”, and antiferromagnetic in
terms of real magnetic moments, unlike the case of Pr moments [11, 12], and that the four
coupling constants are of the same order in the magnitude with their ratios controlled by a
single parameter V /V .
pfπ pfσ
Figure 2. Dimensionless coupling constants as functions of V /V .
pfπ pfσ
5. Summary
We have obtained a highly anisotropic effective pseudospin-1/2 Hamiltonian for interacting Yb
Kramersdoublets of Yb TM O , which includes four adjustable parameters; α and γ describing
2 2 7
crystal-field doublets, J for the magnitude of whole superexchange terms, and V /V
n.n. pfπ pfσ
specifying the ratios of aniotropic exchange couplings. Theoretical analyses of the model in
comparison with experiments will be published elsewhere.
Acknowledgments
The author thanks M. J. P. Gingras for a critical reading of the manuscript. The work was
partly supported by Grants-in-aid for Scientific Research under Grant No. 19052006 from the
Ministry of Education, Culture, Sports, Science, and Technology of Japan and No. 21740275
from Japan Society for the Promotion of Science.
Appendix A. Local coordinate frames
The local coordinate frames are chosen as
1 1 1
x = (1,1, 2), y = ( 1,1,0), z = (1,1,1), (A.1)
0 0 0
−√6 − −√2 − √3
1 1 1
x = (1, 1,2), y = ( 1, 1,0), z = (1, 1, 1), (A.2)
1 1 1
−√6 − −√2 − − √3 − −
1 1 1
x = ( 1,1,2), y = (1,1,0), z = ( 1,1, 1), (A.3)
2 2 2
−√6 − −√2 √3 − −
1 1 1
x = ( 1, 1, 2), y = (1, 1,0), z = ( 1, 1,1), (A.4)
3 3 3
−√6 − − − −√2 − √3 − −
for the Yb sites at R+a (i = 0, ,3). They are invariant under the twofold rotations about
i
···
x, y, and z axes that include the center of the tetrahedron. In particular, all the local z axes
given above point outwards from the center of the tetrahedron. The coordinate frame for the
spins is always attached to that for the orbital space in each case. The rotation matrix of the
total angular momentum j = l+s with the orbital l and the spin s of a single electron takes
the form
Rˆr = exp iϕiˆjz exp iϑiˆjy exp iπˆjz . (A.5)
h− i h− i h− i
with ϕ = π/4, ϑ = arccos 1/√3 , ϕ = 3π/4, ϑ = π + arccos 1/√3 , ϕ = π/4,
0 0 1 1 2
(cid:16) (cid:17) − (cid:16) (cid:17) −
ϑ = π+arccos 1/√3 , ϕ = 3π/4, and ϑ = arccos 1/√3 .
2 3 3
− (cid:16) (cid:17) − (cid:16) (cid:17)
Appendix B. Single-hole f-electron states
The Kramers doublet given in Eq. (1) can be expressed in terms of single f-hole states lz,sz
h
| i
with the orbital lz and the spin sz using
7 σ
Jz = σ = 3σ, ,
h
| 2 i |− −2i
5 1 σ 6 σ
Jz = σ = 3σ, + 2σ, ,
| 2 i √7|− 2ih r7|− −2ih
σ 3 σ 2 σ
Jz = = σ, + 0, . (B.1)
| 2i r7|− 2ih √7| −2ih
References
[1] Anderson PW 1956 Phys. Rev. 102 1008
[2] Anderson PW 1973 Mater. Res. Bull.8 153
[3] WenX G, Wilczek F and Zee A 1989 Phys. Rev. B 39 11413
[4] R.Moessner and J. T. Chalker, Phys. Rev. Lett. 80, 2929 (1998).
[5] Isakov S V,Gregor K, Moessner R and SondhiS L 2004 Phys. Rev. Lett. 93 167204
[6] GardnerJ S, Gingras M J P and Greedan J E, Rev. Mod. Phys. 82 53
[7] Harris M J, Bramwell S T, McMorrow D F, Zeiske T and Godfrey K W 1997 Phys. Rev. Lett. 79 2554
[8] Ramirez A P et al. 1999 Nature (London) 399 333
[9] Bramwell S T and Gingras M J P2001 Science 294 1495
[10] Molavian H R,Gingras M J P and Canals B 2007 Phys. Rev. Lett. 98 157204
[11] OnodaS and TanakaY 2010 Phys. Rev. Lett. 105 047201
[12] OnodaS and TanakaY 2010 Preprint arXiv:1011.4981
[13] GardnerJ S et al. 1999 Phys. Rev. Lett. 82 1012
[14] GardnerJ S, Gaulin B D,Berlinsky A J, Waldron P,Dunsiger S R,Raju N P and Greedan J E 2001 Phys.
Rev. B 64 224416
[15] Mirebeau I, Bonville P and Hennion M 2007 Phys. Rev. B 76 184436
[16] Matsuhira K et al. 2002 J. Phys. Soc. Jpn. 71 1576
[17] Zhou H D, Wiebe C R, Janik J A, Balicas L, Yo Y J, Qiu Y, Copley J R D and Gardner J S 2008 Phys.
Rev. Lett. 101 227204
[18] NakatsujiS et al. 2006 Phys. Rev. Lett. 96 087204
[19] Machida Y,Nakatsuji S, Onoda S,Tayama T and SakakibaraT 2010 Nature (London) 463 210
[20] Cao H, Gukasov A, Mirebeau I, Bonville P, Decorse C and DhalenneG 2009 Phys. Rev. Lett. 103 056402
[21] den Hertog B C and Gingras M J P 2000 Phys. Rev. Lett. 84 3430
[22] Pauling L 1938 The Nature of the Chemical Bonds (Ithaca: Cornel University Press)
[23] Matsuhira K et al. 2009 J. Phys.: Conf. Series 145 012031
[24] Nagaosa N,Sinova J, Onoda S,MacDonald A H and OngN P 2010 Rev. Mod. Phys. 82 1539
[25] Blo¨te W J and Wielinga R F and Huiskamp W J 1969 Physica (Amsterdam) 43 549
[26] YasuiY et al. 2003 J. Phys. Soc. Jpn. 72 3014
[27] Hodges J A et al. 2002 Phys. Rev. Lett. 88 077204
[28] GardnerJ, Ehlers G, Rosov N,Erwin R W and Pertovic C 2004 Phys. Rev. B70 180404(R)
[29] Ross K A, Ruff J P C, Adams C P, Gardner J S, Dabkowska H A, Qiu Y, Copley J R D and Gaulin B D
2009 Phys. Rev. Lett. 103 227202.
[30] Thompson J D, McClarty P A, Ronnow H M, Regnault L P, Sorge A and M J P Gingras 2010 Preprint
arXiv:1010.5476.
[31] Hodges J A, Bonville P, Forget A, Rams M, Kr´olas K and Dhalenne G 2001 J. Phys.: Condens. Matter 13
9301
[32] Similar CEF parameters havebeen obtained in Ref. [20].
[33] SharmaR R, Phys.Rev. B 19, 2813 (1979).