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Structural and Thermodynamic Properties of Cerium via First-Principles Plane Wave Method

with a Relativistic Analytic Pseudopotential

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2009 Chinese Phys. Lett. 26 017101

(http://iopscience.iop.org/0256-307X/26/1/017101)

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CHIN. PHYS. LETT. Vol. 26,No. 1 (2009) 017101

Structural and Thermodynamic Properties of Cerium via First-Principles PlaneWave Method with a Relativistic Analytic Pseudopotential *

SUN Li-Li(孙丽丽)1, JI Guang-Fu(姬广富)2, CHEN Xiang-Rong(陈向荣)1,3**, GOU Qing-Quan(芶清泉)11School of Physical Science and Technology, Sichuan University, Chengdu 610064

2Laboratory for Shock Wave and Detonation Physics Research, Institute of Fluid Physics, Chinese Academy ofEngineering Physics, Mianyang 621900

3International Centre for Materials Physics, Chinese Academy of Sciences, Shenyang 110016

(Received 11 August 2008)We investigate the structural and thermodynamic properties of cerium in 𝛼 phase by using the first-principlesplane wave method with a relativistic analytic pseudopotential of Hartwigsen, Goedcker and Hutter (HGH)scheme in the frame of local density approximation (LDA). The obtained lattice constant and bulk modulus areconsistent with the available experimental data. Moreover, dependences of the normalized primitive volume 𝑉/𝑉0

on pressure and the thermodynamic quantities (including the Gruneisen constant 𝛾 and thermal expansion 𝛼) ontemperature and pressure are obtained. The obtained linear thermal expansion parameter 𝛼 (9.857 × 10−6 K−1

at 293.15K and 0 GPa) is slightly larger than the experimental value (6.3 × 10−6 K−1). All the results indicatethat we provide an effective method to deal with the ground properties of the strongly interacting 𝑑- and/or𝑓 -electron systems.

PACS: 71. 15. Mb, 71. 15.Dx, 65. 40.−b, 71. 20. Be

Cerium metal, with its 4𝑓 electrons near theboundary between itinerant and localized behaviours,is an archetypal material in the study of strongly in-teracting 𝑓 -electron systems.[1] It is malleable and ox-idized very readily at room temperature, especiallyin moist air. Generally, cerium has five distinct solidphases, i.e. 𝛼-, 𝛼′-, 𝛽-, 𝛾-, and 𝛿-phases, which isan unusually larger number for an elemental metal.[2]

Cerium also has magnetic[3,4] and superconductingbehaviours.[5] The 𝛾-Ce is magnetic with a magneticmoment and susceptibility consistent with a singly oc-cupied 4𝑓 level.[3,4] For the 𝛼-Ce under pressure above5 GPa, Wittig[5] found its superconductivity at 1.3 K.

Cerium metal exhibits the well-known 𝛾 → 𝛼isostructural phase transition, which is of interest be-cause it involves a change in lattice size without achange in structure. The phase transformation fromthe 𝛾-Ce to the 𝛼-Ce can be induced either by loweringthe temperature below 100 K or by applying a pressureabout 0.8 GPa.[6] The phase transition is characterizedby an unusually large volume change of about 17%and the magnetic character changes.[7] It has been apuzzling subject for a great number of experimentaland theoretical efforts.[8,9] For example, the magneticsusceptibility follows a Curie–Weiss temperature de-pendence for the high-temperature 𝛾-Ce, but it is sup-pressed and follows a Pauli paramagnetic temperaturedependence for the low-temperature 𝛼-Ce.[10]

In a Mott transition scenario,[11] the 4𝑓 electronis treated to be itinerant and binding in the 𝛼 phase,but localized and non-binding in the 𝛾 phase. Thus

the energy for the 𝛾−𝛼 phase transition is provided bythe kinetic energy of the itinerant 4𝑓 electrons. Allenand Martin[12,13] suggested that the phase transition isdue to a Kondo volume collapse. In the Kondo volumecollapse (KVC) picture, the 4𝑓 electron is assumed tobe localized both in the 𝛼 and 𝛾 phases. They saidthat the 4𝑓 cohesive energy contribution is large inboth 𝛼 and 𝛾 phases, while only the Kondo spin fluc-tuation energy and entropy result in the 𝛾 − 𝛼 phasetransition. Recently, Huang and Chen[1] investigatedthe lattice dynamics of 𝛼-Ce and 𝛾-Ce by using anab initio pseudopotentials plane wave method supple-mented with the linear response approach. They con-cluded that with increasing temperature, both vibra-tional and electronic entropy changes increase. Thevibrational entropy change is obviously larger than thecounter part of the electron contribution, especially inthe high temperature region.

In this Letter, we focus on the structural and ther-modynamic properties of the 𝛼-Ce, which exists be-tween the magnetic 𝛾 phase and the superconduct-ing 𝛼′ phase.[14] We here perform a first-principlesplane wave method with a new relativistic analyticpseudopotential of Hartwigsen, Goedecker and Hut-ter (HGH) scheme[15] in the frame of local densityapproximation (LDA). The HGH potentials allow forthe 𝑓 -electrons to be treated as valence states andso rare-earth materials can be handled. Thermody-namic properties depend on the lattice dynamical be-haviour. A simplified method for the thermodynamicproperties is a Debye model,[16] since the vibrational

*Supported by the National Natural Science Foundation of China under Grant Nos 10776022 and 60436010, and the Fund ofKey Laboratory for Shockwave and Detonation Physics under Grant No 9140C6711010805.

**To whom correspondence should be addressed. Email: xrchen@126.comc○ 2009 Chinese Physical Society and IOP Publishing Ltd

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entropy change is obviously larger than that of theelectron contribution for the 𝛼-Ce.[1] The structuraland thermodynamic properties of the 𝛼-Ce obtainedare in good agreement with the available experimentaldata and other theoretical results.

Before starting work on the ground state and lin-ear response calculations, a set of convergence testshave been carried out in order to choose correctly thegrid of special 𝑘 points and the cut-off for the planewave kinetic energy. We use a plane wave kinetic en-ergy cut-off of 20 Hartree and the Brillouin zone in-tegration is confined to 12× 12× 12 Monkhorst–Packmesh.[17] To investigate the thermodynamic proper-ties of the 𝛼-Ce, we here apply the quasi-harmonicDebye model,[16] in which the phonon effect is consid-ered. Applying this method described above, we havesuccessfully investigated some materials before.[18−20]

First of all, we calculate the equilibrium proper-ties of the 𝛼-Ce for the sake of judging whether thecomputational methodology is reliable and effective.The obtained total energy of the 𝛼-Ce versus vol-ume is plotted in Fig. 1, together with the experimen-tal data[21] and the theoretical results by Huang andChen.[1] The equilibrium cell volume of the 𝛼-Ce is761.80𝑟3

𝐵 , which is only 1.04% smaller than the exper-imental value 769.88𝑟3

𝐵 (corresponding to the primi-tive unit cell volume of about 28 A3 with 𝑟𝐵 being theBohr radius).[21] In Ref. [1], the equilibrium cell vol-ume of the 𝛼-Ce is 793.93𝑟3

𝐵 . Obviously, the presentwork seems to be closer to the experiment.

Fig. 1. Total energies versus volume for the 𝛼-Ce.

In Fig. 2, we show our 𝑃 − 𝑉 curve of the 𝛼-Ce,together with the experimental data[22,23] and thoseobtained by Huang and Chen[1] through the ab ini-tio pseudopotential plane wave method (PPW) andthe full-potential linearized augmented plane wavemethod (FP-LAPW). Similar to the FP-LAPW re-sults of Huang and Chen, the present 𝑃 (𝑉 ) curve alsoshifts somewhat to the left, indicating that the presentequation of state does not describe the 𝛼 phase wellat high pressure. However, we can see that our results

are better consistent with experiment at lower pres-sure (< 2 GPa). Franceschi and Olcese[24] reportedthat the 𝛼-Ce will transform to the 𝛼′-Ce at 5.5 GPa.The shift of the present 𝑃 (𝑉 ) curve for the 𝛼-Ce athigh pressure may be explained by this reason. Obvi-ously, the PPW results of Huang and Chen are some-what bad.[1]

Fig. 2. Theoretical and experimental equations of stateof the 𝛼-Ce. The experimental data are taken fromRefs. [22,23]. Closed down-triangles and open pentagonsrepresent the theoretical result of Huang and Chen bythe ab initio pseudopotentials plane wave method (PPW)and by the full potential linearized augmented plane wavemethod (FP-LAPW).

Fig. 3. Relative volume versus pressure of the 𝛼-Ce at 0,1000 and 2000K, respectively.

The pressure and temperature dependences of therelative volume 𝑉/𝑉0 of the 𝛼-Ce are illustrated inFig. 3. It is seen that, as the pressure 𝑃 increases, therelative volume𝑉/𝑉0 decreases at a given temperature,and the relative volume 𝑉/𝑉0 of higher temperatureis less than that of lower temperature at the samepressure. On the other hand, the volume 𝑉 decreaseswith the elevated pressure 𝑃 , and increases with theelevated temperature 𝑇 . This effect of increasing pres-sure on the 𝛼-Ce is just the same as decreasing tem-perature.

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Table 1. Lattice constant 𝑎 (A), bulk modulus 𝐵0 (GPa) andits pressure derivative 𝐵′0, together with the experimental dataand other theoretical results for the 𝛼-Ce at 0GPa and 0K.

𝑎 𝐵0 𝐵′0Present 4.833 38.63 4.696Expt. 4.85𝑎 35.0𝑏

Others 4.824𝑐, 4.89999𝑑 69.4𝑑,43.0𝑒, 48.4𝑓 , 48.7𝑔

𝑎Ref. [21], 𝑏Ref. [25],𝑐Ref. [26], 𝑑Ref. [1],𝑒Ref. [27],𝑓Ref. [28], 𝑔Ref. [29].

In Table 1, we list our lattice constant 𝑎, bulk mod-ulus 𝐵0 and its pressure derivative 𝐵′0, together withthe experimental data and other theoretical results.Clearly, our lattice constant 𝑎 is in good agreementwith the experimental value[21] (with error 0.3%), andour bulk modulus 𝐵0, 38.63 GPa, are also well consis-tent with the experimental value, 35.0 GPa.[25] Othertheoretical results of 𝐵0, for example, 69.4 GPa,[1]

43.0 GPa,[27] 48.4 GPa,[28] and 48.7 GPa,[29] signifi-cantly overestimate the bulk modulus of the 𝛼-Ce.Unfortunately, the experimental values or other the-oretical results for the pressure derivative 𝐵′0 are notyet available for our comparison.

Our results described above suggest that the first-principles plane wave method with the new relativisticanalytic pseudopotential of the HGH scheme is suit-able for investigating the ground state properties ofthe 𝛼-Ce. It may provide an effective method to dealwith the ground properties of the strongly interacting𝑑- and/or 𝑓 -electron systems.

Fig. 4. Bulk modulus 𝐵 of the 𝛼-Ce versus pressure 𝑃 at𝑇 = 0, 800, and 1600K, respectively.

In Fig. 4, we illustrate the bulk modulus 𝐵 as afunction of pressure 𝑃 up to 18 GPa at 𝑇 = 0, 800 and1600 K, respectively. It is seen that, the bulk modu-lus 𝐵 of the 𝛼-Ce at a given temperature increasesswiftly with increasing pressure. At zero pressure andzero temperature, our results agree well with the ex-perimental data. Our results indicate that 𝐵 increaseswith 𝑃 at a given temperature and decreases with 𝑇 ata given pressure. This result also shows the fact thatthe effect of increasing pressure on the 𝛼-Ce is thesame as that of decreasing temperature. Gruneisen

constant 𝛾, which describes the anharmonic effects inthe vibrating lattice, can correctly predict the anhar-monic properties of a material, such as the thermalexpansion coefficient and the temperature dependenceof phonon frequencies and line-widths. In Fig. 5, wepresent the obtained Gruneisen constant 𝛾 at 0 K. Itis noted that the Gruneisen constant 𝛾 decrease withincreasing pressure at zero temperature.

Fig. 5. Gruneisen constant versus pressure for the 𝛼-Ceat zero temperature.

Fig. 6. Thermal expansion versus pressure and tempera-ture for the 𝛼-Ce.

Thermal expansion parameters are important forpredicting the thermodynamic properties of a mate-rial. We thus present the thermal expansion coefficient𝛼 of the 𝛼-Ce under various temperatures (100, 300,and 1200 K) and pressures (0, 3, 9, and 18 GPa). Thepressure and temperature dependences of the thermalexpansion coefficient 𝛼 for the 𝛼-Ce are illustrated inFig. 6. Under a given pressure, the thermal expan-sion coefficient 𝛼 increases dramatically below 400 K,at a given temperature decreases swiftly with increas-ing pressure up to 10 GPa, then trends to a constant.This means that the temperature dependence of ther-mal expansion coefficient 𝛼 of the 𝛼-Ce is very smallat high pressure and high temperature. To furtherinvestigate the thermal expansion 𝛼 of the 𝛼-Ce, wehere calculate the linear thermal expansion parameter

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𝛼. By the following equation

𝛼 =(∆𝑎/∆𝑇 )

𝑎0=

1𝑎0

( 𝜕𝑎

𝜕𝑇

)𝑃

,

where 𝑎0 and 𝑎 represent the lattice constant at 0 Kand some finite temperature 𝑇 respectively, we ob-tain 𝛼 = 9.857 × 10−6 K−1 at 293.15 K and 0 GPa,which is slightly larger than the experimental value6.3 × 10−6 K−1.[30] Our result is appropriate on theorder of magnitude.

In summary, we have investigated the structuraland thermodynamic properties of the 𝛼-Ce by usingthe first-principles plane wave method with the rela-tivistic analytic pseudopotential of Hartwigsen, Goed-cker and Hutter (HGH) scheme in the frame of localdensity approximation (LDA). The obtained latticeconstant and bulk modulus are well consistent withthe available experimental data. Moreover, depen-dences of the normalized primitive volume 𝑉/𝑉0 onpressure and the thermodynamic quantities (includ-ing the Gruneisen parameter 𝛾 and thermal expansion𝛼) on temperature and pressure are successfully ob-tained. The obtained linear thermal expansion param-eter 𝛼 is slightly larger than the experimental value.Our results indicate that we here provide an effec-tive method to deal with the ground properties of thestrongly interacting 𝑑- and/or 𝑓 -electron systems, andthe quasi-harmonic Debye model may be effective atlow temperatures and low pressures.

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