Small magnetic moment and superconductivity in URu2Si2

Journal of Physics and Chemistry of Solids 63 (2002) 1469±1474

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Small magnetic moment and superconductivity in URu2Si2 Atsushi Tsuruta a,b,*, Akito Kobayashi a, Tamifusa Matsuura a, Yoshihiro Kuroda a,b a

Department of Physics, Nagoya University, Furo-cho, Chikusa-ku, Nagoya 464-8602, Japan b CREST, Japan Science and Technology Corporation (JST), Japan

Abstract One of the most puzzling properties observed in URu2Si2 has been the large jump of speci®c heat around 17 K for small magnetic moments. This speci®c heat jump is attributed to the quadrupolar order rise around 17 K instead of the dipolar order. Neutron experiments showed about 0.03m B at low temperatures whereas no evidences were found in NMR experiments for static magnetic order. These experiments suggest that Bragg peaks observed in the neutron experiments are not really due to static magnetic order but due to quasi-static ones. However, new NMR experiments clearly show the internal ®eld below 17 K, corresponding to the appearance of magnetic moments of 0.2m B. The intensity of the signal is small, suggesting that a small portion of the sample is magnetic while the other portions are non-magnetic. We present a scenario, consistent with these experiments, based on the assumption that the sample is a mixture of dipolar and quadrupolar phases, and also taking account of quasi-static magnetic ¯uctuations. We also discuss the pressure effects in the sample. q 2002 Published by Elsevier Science Ltd. Keywords: A. Electronic materials; A. Superconductors; D. Magnetic properties; D. Electrical properties; D. Superconductivity

The heavy electron state in some uranium compounds is one of the most attractive issues. A puzzling property, the large jump of the speci®c heat at the transition temperature (i.e. 17.5 K), is pointed out in URu2Si2 [1,2] in contrast to the small magnetic moment of 0.04m B below 17.5 K in the ambient pressure from neutron scattering experiments [3±5]. Several authors insisted that the quadrupolar order occurs at the critical temperature and that the large jump of the speci®c heat is attributed to it [6±9]. Then the secondary effects were thought to induce magnetic moment observed in neutron experiments. Until very recently, NMR experiments have never observed the internal ®eld. This implies that the amplitude of the quasi-static magnetic ¯uctuation below the critical temperature is enhanced and observed as Bragg peak within the resolution limit in the neutron experiments. However, the results of a new NMR experiment [10] shows that below around 16 K, Kight shift corresponding to 0.2m B is observed but its signal is weak. As the pressure is applied, the internal ®eld remains unchanged but the * Corresponding author. Address: Department of Physics, Nagoya University, Furo-cho, Chikusa-ku, Nagoya 464-8602, Japan. Tel.: 181-52-789-2912; fax: 181-52-789-2928. E-mail address: [email protected] (A. Tsuruta).

intensity increases. At P ˆ 1 GPa; the intensity reaches 90%. These facts suggest that in URu2Si2, two ordered phases with the dipolar or the quadrupolar moment coexist below the critical temperature, and the small magnetic moment observed in the neutron experiment is nothing but an average magnetic moment in two phases. The ratio of the dipolar moment against the quadrupolar one increases with pressure. Amitsuka et al. [11] have investigated, using neutron experiments, pressure effects on the onset-temperatures and the saturated staggered magnetic moment m0 for pressure P up to 2.8 GPa. They observed that m0 increases almost linearly with P from 0.017 to 0.25m B. At a critical pressure Pc ˆ 1:5 GPa, m0 jumps from 0.25 to 0.4m B. For P . Pc ; the temperature dependence is well described by the 3D Ising model. These facts also suggest that the ratio of dipolar phase increases with pressure. The lattice constant also decreases discontinuously by 0.2% at P ˆ Pc : This implies that a drastic change in the electronic state occurs at P ˆ Pc : We understand this phenomena at ambient pressure as follows: In uranium, the ground state f-electrons is in (5f) 2-states among which the state with quadrupolar moment and the state with dipolar moment have almost the same lowest energies. We set the former energy as 1f and the

0022-3697/02/$ - see front matter q 2002 Published by Elsevier Science Ltd. PII: S 0022-369 7(02)00099-9

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latter as 1f 1 D: Then for D . 0; the ground state is with the quadrupolar moment (quadrupolar phase); for D , 0 the ground state is with the dipolar moment (dipolar phase). We ®nd from the NMR experiments that these two phases coexist but are spatially separated. We set the rate of the dipolar phase as x. In the ambient pressure, we assume that the quadrupolar phase dominates and the dipolar phase exists as clusters, i.e. x p 1: First at T . T0 ˆ 17:5 K; we expect a quadrupolar (dipolar) Kondo effect around 100 K in the quadrupolar (dipolar) phase, which causes the large speci®c coef®cient at T . T0 and broad peak of the resistivity around T ˆ 70 K: Next we expect the antiferro-quadrupolar (dipolar) order below T ˆ T0 in the quadrupolar (dipolar) phase. The neutron scattering experiments give about 0.02m B as the dipolar moment through the intensity of Bragg peak. According to the NMR experiment, the internal ®eld corresponds to 0.2m B, suggesting that the ordered dipolar moment 0.2m B is in the dipolar phase (the rate x) and zero is in the quadrupolar phase (the rate 1 2 x), respectively. These dipolar moments give the intensity of Bragg peak corresponding to 0:2xm B. If we assume x ˆ 0:1; we obtain 0.02m B. Since the NMR experiment suggests smaller x, we, however, may not get 0.02m B. To obtain the observed intensity of the Bragg peak, taking into account the resolution limit of the neutron diffraction, we add the contribution of the strong quasi-static dipolar ¯uctuations in the quadrupolar phase following the formation of the quadrupolar order. In this paper, we calculate this contribution with two channel Anderson lattice using 1=N-expansion. We may understand pressure effects observed in NMR and neutron scattering experiments if we take into account pressure dependence of the rate x. Hamiltonian for the two channel Anderson lattice model is given by XX XX Hˆ 1 1k ck1t1 s ckt1 s 1 1 2k ck1t2 s ckt2 s t1 ;s k

1

XX i

t1

t 2 ;s k

1f fi1 t1 fit1

1

XX i

t2

…1f 1

D†fi1 t2 fit2

where kA^ P i Q^ i ll ; Tr‰e2PbHl A^ P i Q^ i Š . =Zl with Zl ; Tr‰e2bHl Š and Hl ; H 1 i li Q^ i : Now we calculate the quasi-static dipolar ¯uctuations. First, we calculate the single particle Green's functions for the slave bosons in the quadrupolar phase; the pseudo fermions and the c-electrons for the diagonal part and the off-diagonal part are shown in Fig. 1. Their explicit form in the limit T ! 0 are given by Bis …inn † ˆ a{inn 2 li 2 1f 1 E0 }21 ;

…3†

Fit1 …ien † ˆ {ien 2 li 2 1f }21 ;

…4†

Fit2 …ien † ˆ {ien 2 li 2 1f 2 D}21 ;

…5†

Gk…D† t1 s …ien † ˆ

…0†21 Gk1Q t1 …ien †

Gk…OD† t1 s …ien † ˆ

Dkt1 s …ien †

;

S…OD† kt1 …ien † ; Dkt1 s …ien †

where t1 (t2 ) denotes the quadrupolar (dipolar) moment with degeneracy N ˆ 2 and s dipole moment with degeneracy M ˆ 2: c kt1 s …ckt2 s † represents an annihilation operator of c-electron with wave vector k, quadrupolar (dipolar) component t1 (t2 ) and dipolar component s: f it1 (fit2 ) and bis represent annihilation operators of the ith …5f†2 localized state with quadrupolar (dipolar) component t1 (t2 ) and the ith …5f†1 localized state with dipolar component s; respectively. N L is the total number of lattice sites. This

…7† …8†

…0†21 …OD†2 Dkt1 s …ien † ˆ Gk…0†21 t1 s …ien †Gk1Qt1 …ien † 2 S kt1 …ien †;

…9†

2

(

…1†

…6†

Gkt2 s …ien † ˆ Gkt…0† …ien †;

G…0† kta …ien †

o X X V 1k n 1 1 p ckt1 s bis fit1 e2ikRi 1 h:c: 1 NL t1 ;s i;k o X X V 2k n 1 1 p ckt2 s bis fit2 e2ikRi 1 h:c: ; 1 NL t2 ;s i;k

model was investigated by Tsuruta et al. [12±14]. We investigate the quadrupolar phase, i.e. D . 0: For guaranteeing the physical equivalence between the present model and the original U ˆ 1 model, the Hamiltonian, Eq. (1), must be treated within the subspace where 1 the local constraint, Q^ i ˆ S s bi1 s bis 1 S a;ta fita fita ˆ 1; holds. In order to calculate the physical quantities within the physical subspace restricted under the local constraint, ^ of a physical quantity we evaluate the expectation value kAl A^ as [15,16] ! kA^ P i Q^ i ll ^ kAl ˆ lim ; …2† {li }!1 kP i Q^ i ll

1 auVak u2 ˆ ien 2 1ak 2 M ien 2 Ea

S…OD† kt1 …ien †

Lt…i† ˆ 1

p aV1k V1k1Q ˆ ien 2 E1

(

)21 ;

) L…1† t1 1 Lt…2† ; 1 ien 2 E1

X Gk…OD† 1 1 X t1 s …ien † p aV 1k V1k1Q T i; M NL s;k e …i n 2 E1 † en

…10†

…11†

…12†

and Ea ˆ E0 1 D…a 2 1†: We neglect the imaginary part of the self-energy of c-electrons. The diagrams for kQ^ i ll and the vertex in Fig. 1 are shown in Fig. 2. kQ^ i ll ˆ Me2b…li 11f 2E0 † :

…13†

The binding energy of the slave boson, E0 ; and the residue,

A. Tsuruta et al. / Journal of Physics and Chemistry of Solids 63 (2002) 1469±1474

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Fig. 1. Single particle Green's functions for (a) slave bosons, (b) pseudo-fermion, (c1) c-electrons for diagonal part of t1 , (c1 0 ) c-electrons for off-diagonal part of t1 and (c2) c-electrons of t2 .

a, are determined by ReBi21 s …1f 1 li 2 E0 1 id† ˆ 0; a21 ˆ

…14†

d ReBi21 s …n 1 id†un!1f 1li 2E0 : dn

…15†

s; t 1

kn2 l ˆ kn…n† 2 l;

…17†

ntot ˆ kn1 l 1 kn2 l;

…18†

kn…n† a lˆ

1 NL

ta ;s;k

T

X en

( Gk…D† ta s …ien † 1 1

ˆ

X …OD† 1 X t T G …ie †; NL t ;s;k 1 en kt1 s n

…20†

1

The number of quadrupolar moments t1 ; kn1 l; number of dipolar moments t2 ; kn2 l and the total number, ntot ; are, respectively, given by  X  …1† …3† kn1 l ˆ kn…n† 2Lt1 Lt1 1 L…2†2 …16† t1 ; 1 l1

X

…AF† kt…AF† 1c l; and f-electrons, kt1f l; are, respectively, given by D E E 1 X D 1 t…AF† t 1 ck1Qt1 s ckt1 s ˆ 1c NL t1 ;s;k

2

)

auVak u =M : …19† …ien 2 Ea †2

Eqs. (12), (14), (15) and (18) are self-consistent equations …2† for L…1† t1 ; Lt1 ; E0 ; a and chemical potential m: At T , T0 ; antiferro-quadrupolar moments of c-electrons,

D E X D E X t…AF† t1 fit11 fit1 e2iQRi ˆ t1 L…2† ˆ t1 : 1f t1

t1

…21†

In the normal state of the quadrupolar moment t1 ; S kt …OD† …ien † ˆ 0: 1 We calculate the magnetic scattering S…q; v† of the quadrupolar moment t2 under the normal state and the ordered state of the quadrupolar moment t1 : The susceptibility of t1 or t2 of the f-component is given by X xa;ij …t† ˆ kTt t^ ai …t†t^ aj …0†l; t^ ai ˆ ta fi1 …22† ta fita : ta

The diagrams for the susceptibility are shown in Fig. 2(b). In this paper, we set the kinetic energies of the conduction electrons, mixing terms and f-level as e1k ˆ e2k ˆ 2Wcos…k† 2 m; V1k ˆ V1 …1 2 B1 cos…k††; V2k ˆ V2 …1 2 B2 cos…k†† and 1f ˆ 1…0† f 2 m: We use parameters as follows: W ˆ 1; V1 ˆ 0:2; V2 ˆ 0:188; B1 ˆ 0:4; B2 ˆ

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Fig. 2. Diagrams contributing to (a) kQ^ i ll , (b) susceptibility, and vertices (c, d).

20:5; 1…0† f ˆ 20:8 and D ˆ 0:001: We set the total number ntot ˆ 2 and the nesting vector Q ˆ p: Then we obtain parameters a ˆ 0:45; E0 ˆ 0:012; m ˆ 20:74 and mp =m ˆ 64: The inset of Fig. 3(a) shows quadrupolar moments of c…AF† electrons, kt…AF† 1c l; and f-electrons, kt1f l: The quadrupolar moment of c-electrons is proportional to the inverse of the mass enhancement, the saturated moment is 1:1 £ 1022: The quadrupolar moment of f-electrons is of the order of 1, the saturated moment is 0.2. Fig. 3(a) shows the temperature dependence of numbers under the normal state of the quadrupolar moment (n…N† ) and the antiferro-quadrupolar state (n…O† ). At T . T0 ; both the numbers are away from 1 because of D and the difference between V1k and V2k ; and the numbers are independent of temperature under the normal state. On the other hand, under the ordered state, the number with t1 should be close to 1 because the nesting vector Q is set to p, and ¯ows to the dipolar state t2 : Fig. 3(b) shows the temperature dependence of the inverse of the susceptibility of quadrupolar moment, …x1Q …0††21 ; the inverse of the susceptibility of dipolar

moment, …x2Q …0††21 : As the quadrupolar moment kt…AF† 1f l increases, the number of t2 closes to one and the nest of Fermi surface of t1 at q ˆ p is enhanced. Thus, the …N† susceptibility under the normal state, x2Q ; is independent of temperature because the number of t2 does not change. On the other hand, the susceptibility under the ordered state, x…O† 2Q ; is enhanced. Using the susceptibility x2q ; the magnetic scattering S…q; v† is given by S…q; v† ˆ

1 Imx2q …2v 1 id† : p e2bv 2 1

…23†

The frequency sum of S…q; v† S…q† ˆ

ZDv 2 Dv

dv S…q; v†;

the wave vector sum of S…q; v† X S…v† ˆ NL21 S…q; v†;

…24†

…25†

q

for some temperatures, are shown in Fig. 4(a) and (b),

A. Tsuruta et al. / Journal of Physics and Chemistry of Solids 63 (2002) 1469±1474

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Fig. 3. (a) Temperature dependence of numbers. The inset shows antiferro-quadrupolar moments. (b) Temperature dependence of susceptibilities.

respectively. We set Dv ˆ 6 £ 1024 : The peak of S…v† around v ˆ 0 is sharper as temperature decreases. The wave vector dependence of S…q† shows the movement of the peak position, which is inconsistent with the experiment. This is the bad effect of one-dimensional dispersion. If we use more realistic band, the movement may not appear. Summation of the wave vector and integration of the frequency of S…q; v† which is observed in neutron scattering is given by X ZDv  Dv† ˆ 1 S…T; dv S…q; v†: N L q 2 Dv

…26†

 Dv† are shown The temperature and Dv dependence of S…T;  Dv† monotonously increases as temperature in Fig. 5. S…T;  ˆ 1:585 £ 1024 ; Dv† ˆ decreases from T ˆ T0 ; and S…T

1:12 £ 1025 : The inset of Fig. 5 shows that S…T; Dv† is almost proportional to Dv: The time scale of neutron scattering is about 10212 s. On the other hand, the time scale of NMR is about 1027 s. Then the Bragg-like peak in the quadrupolar phase can be seen in neutron scattering experiment and not in NMR experiment. We have shown the role of antiferro-magnetic ¯uctuation in the small magnetic moment in URu2Si2. In URu2Si2, the level with quadrupole moment and that with dipole moment of …5f†2 is very close. Thus, the small cluster with antiferro-magnetic moment is observed by NMR experiment. The antiferro-quadrupolar ordering in quadrupolar phase causes the large jump of the speci®c heat coef®cient. The ¯uctuation of the magnetic moment is enlarged by the antiferro-quadrupolar ordering, because of the ¯ow of electrons from quadrupolar phase to dipolar phase. The ¯uctuation

Fig. 4. (a) The wave vector dependence of S…q† for some temperatures. (b) The frequency dependence of S…v† for some temperatures.

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(A.T.) is supported by a Research Fellowship of the Japan Society for the Promotion of Science for Young Scientists. References

 Dv†: Fig. 5. Temperature and Dv (inset) dependence of S…T;

contributes to the Bragg peak observed in neutron scattering experiments. As pressure increases, the ¯uctuation is suppressed and the cluster of the dipolar phase in which antiferro-magnetic moment occurs is enhanced. We will investigate the contribution from both the ¯uctuation in quadrupolar phase and the cluster with antiferro-magnetic moment to the small magnetic moment in URu2Si2 and the origin of superconductivity whose transition temperature Tc ˆ 1:5 K [5]. Acknowledgements The present work was supported by a Grant-in-Aid for Scienti®c Research on Priority Areas from the Ministry of Education, Science, Sports and Culture. One of the authors

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