Thermodynamics of the $\Alpha $-$\Gamma $ Transition in Cerium

$Thermodynamics of the $\Alpha $-$\Gamma $ Transition in Cerium$

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GPa 1.5 = Dtd etme ,2018) 6, September (Dated: ad.In bands. ,3 1, a-egLiu, Hai-Feng α α - γ to method rniinfo h rtpicpecluain ealso We calculation. principle first the from transition α γ f rdi aifcoywy edn ooeetmto of overestimation the to for leading energy way, kinetic satisfactory the a in ered si anycnrbtdb lcrncetoyo lattice or origin? entropy its electronic is by entropy? what contributed en- mainly that transition, fact it the the Is to Given important iii) is role. at tropy no induced pressure), plays negative be entropy by also where example can (for transition temperature zero a the such words, the another not In whether or entropy? is whether by question driven The purely is transition. transition driv- the important quite for the is of force one it ing is pressure, difference entropy ambient that with clear temperature finite at rlgain prxmto GA,tecreainef- correlation the the among (GGA), fects approximation gen- gradient or eral (LDA) cal- approximation phase density (DFT) local structural theory with the density-functional culations pa- of the nature In with the transitions. tool study powerful to free another rameter is calculation principle ain(SA 1,2,2]adLD calculations U + LSDA the and obtain 22] can hand, approxi- 21, density [19, other (LSDA) spin the mation local on corrected and While self-interaction electronic the the of [17–20]. both appearance including entropy the after vibrational for even signal all no at phase is there and the obtained only consequences, the of xeddt h nt eprtr ae Nevertheless, case. temperature finite the the with to associated extended be can that and solutions temperature distinct zero two on at obtain base calculation functionals a density Besides, under hybrid phases picture. is both physical it unified describe still to a but methods these magnetic, for or difficult localized completely either be eprtr.i)Snethe Since ii) temperature. eie h xeietlsuis[,1–6,tefirst the 11–16], [2, studies experimental the Besides ntettlfe nrya ie oueand volume given at energy free total the in ,3 1, γ motn oei the in role important ogWang, Cong hsso e[3,btterslshv o yet not have results the but [23], Ce of phases inbetween tion uigteetoycnrbtdb both by contributed entropy the luding t.Orcluainidct htboth that indicate calculation Our nts. rso eima nt temperature finite at cerium of erms γ dIsiueo Physics, of Institute nd f hs yasmn h 4 the assuming by phase lcrn aentbe ul consid- fully been not have electrons 1 α meaueb Gutzwiller by emperature hn Fang, Zhong and γ ina α plied α f a , - hssprit to persists phases γ to lcrn.Teeoea one as Therefore electrons. transition. α γ hs fcru a be can cerium of phase rniinas happens also transition 2 n iDai Xi and f lcrn to electrons 2 α γ 2 a recent work from density functional theory proposed were performed with the full consideration of the rela- that thermal disorder contributes via entropy to the sta- tivistic effect and a 16×16×16 k mesh for higher energy bilization of the γ phase at high temperature [24], the converged precision. α-γ transition is calculated to occur around 600 K. Our main results in zero temperature are shown in Within the LDA + DMFT method, a combination of Fig. 1(a). The negative curvature region in the to- LDA with dynamical mean field theory (DMFT), early tal energy versus volume curve, which signals the first numerical studies have been carried out to study the order transition with pressure, is present for all inter- phase transitions of Ce at finite temperature [25–31]. action strength, which is slightly different with the re- While since the quantum Monte Carlo methods have sults obtained by G. Kotliar’s group[36]. With interac- been adopted as the impurity solver of DMFT, it is dif- tion strength U =4.0eV , both the experimental volume ficult for LDA + DMFT to study the transition in low (28.0 - 29.0 A˚3 ) and bulk modulus (20.0 - 35.0 GPa) [42– temperature and the full thermodynamic features of the 44] at ambient pressure can be nicely reproduced and we transition in the full temperature range have yet to be will adopt this value for the calculations through out the obtained. paper. The quasi-particle weight and the average occu- In this paper we show that LDA + Gutzwiller method, pation of both the j = 5/2 and 7/2 bands are plotted which incorporate LDA with Gutzwiller variational ap- in Fig. 1(c) and (d). From Fig. 1(d) one can find that proach, can be well applied to study the ground proper- the occupation number of j = 5/2 bands increase dra- ties of the cerium metal and with the generalization to matically from 0.7 to almost one in the volume regime the finite temperature, it can be further applied to study from 29.0 A˚3 (the equilibrium volume of the α phase)to the thermodynamic properties of the α to γ transition in 35.0A˚3 (the equilibrium volume of the γ phase)and at the low temperature. Here we will sketch the most impor- same time the quasi-particle weight drops abruptly (Fig. tant aspects of the method, and leave the details to Refs. 1(c)), indicating that the f -electrons is quite itinerant in [32–34]. The total Hamiltonian to describe the strongly α phase while becomes quite localized in γ phase, which correlated systems can be written as can be better described by the Kondo lattice model.

H = H + H − H , (1) 240 LDA int DC (b) (a)

500 U = 5.0 eV

U = 4.5 eV

160 , where HLDA is the single particle Hamiltonian obtained U = 4.0 eV 400

0 K by LDA and Hint is the local interaction term for the

300

50 K

4f electrons. HDC is the double counting term rep-

100 K

0 resenting the interaction energy already considered at 150 K 200

200 K Free en erg y (meV ) the LDA level. In the present paper, we compute the Energy (meV) 250 K

-80

100 double counting energy using the scheme described in 300 K

350 K

Ref. [33]. In LDA + Gutzwiller we apply the following -160 0

Gutzwiller trial wave function, |Gi = PG|0i, where PG is 1.0 (c) 1.0 (d) the Gutzwiller projectors containing variational parame- ters to be optimized by the variational principle and the 0.8 0.8

Z n

5/2 non-interacting state |0i is the solution of the eﬀective 5/2 0.6 0.6

Z f

n 7/2 n

Hamiltonian for the quasi-particles Heff ≈ PGHLDAPG. Z 7/2 The ground state properties of cerium metal have been 0.4 0.4 studied using LDA + Gutzwiller by us for the positive 0.2 0.2 pressure case [35] and recently by G. Kotliar’s group for the negative pressure case [36], where they ﬁnd that for 0.0 0.0

20 25 30 35 40 45 25 30 35 40 45

3 O interaction strength U < 5.5eV the ﬁrst order transi- 3

Volume (A ) tion between α and γ phases survives at zero temper- Volume (A ) ature, they also reported a new implementation of the FIG. 1: (Color online) Calculate cold energy for different U Gutzwiller approximation [37–39]. Recently, they im- (upper left panel), and free energy for different temperature plement an LDA + SB method studied the α-γ phase (upper right panel) as a function of atomic volume. Quasipar- transition and phase diagram of cerium at finite temper- ticle renormalization weights of the 7/2 and 5/2 f -electrons ature [40], and obtain results in good agreement with the (lower left panel), 7/2 and 5/2 orbital populations (lower right experiments. panel). In the present paper, we apply the LDA + Gutzwiller method implemented in our pseudo potential plane wave In order to directly compare the first principle results code BSTATE [32, 33, 35, 41] to obtain the ground state with the experimental data, which is always performed energy of cerium metal crystallized in face centered cu- at finite temperature, the calculations of the free energy bic (fcc) structure. The LDA part of the calculations at finite temperature is strongly desired. In the present 3 paper, we generalize the LDA + Gutzwiller method to The free energy curves with different temperature in calculate the free energy by including both the electronic Fig. 1(b) reveals the competition between the α and γ and lattice vibrational entropy. The total Helmholtz free phases, the free energy difference between γ phase and energy can be always written as α phase decreased with increasing temperature, which becomes almost zero at a temperature of 190 K. This F (V,T )= Fvib(V,T )+ Fel(V,T ) (2) results illustrate that the α-γ transition temperature is 190 K at zero pressure, which agrees well with the exper- where Fel(V,T ) and Fvib(V,T ) are the electronic and lat- imental data. [1]. tice vibrational part of the free energy respectively. In The pressure at given volume and temperature can be the present study, we assume that at least in the low tem- estimated as P (V,T )= −∂F (V,T )/∂V , we calculate the perature both the α and γ phases are in the fermi liquid pressure versus volume isotherms of fcc Ce, as shown in region, where the electronic entropy can be calculated by Fig. 2. At given pressure, the first order phase transition counting the thermally excited quasi-particles. Near the can be signaled by the appearance of multiple solutions critical temperature, which is around 500K, the γ phase with different volumes, which has been plotted in the will be no longer in the fermi liquid phase any more, lead- figure by the open symbols indicating the region of ther- ing to possible underestimation of the electronic entropy modynamic instability. Our isotherms of α-Ce (volume in γ phase, which will be discussed in more detail below. little than thermodynamic instability region) are excel- Therefore in the present study, the electronic free en- lent consistent with experimental data [2]. Due to the ergy will be estimated as lack of magnetic entropy in LDA + G method, our results of γ-Ce are slightly underestimate than the experiments, F (V,T )= n (ǫ, V )f(ǫ)ǫdǫ+ el Z qp we will discuss this issue again in this paper later.

Experiment: Ref. [2]

T k n (ǫ, V )[flnf + (1 − f)ln(1 − f)]dǫ (3) 775 K 500 K B Z qp 692 K 481 K

700 K

577 K 414 K

, where nqp(ǫ, V ) is the quasi-particle density of states 377 K

600 K

3 obtained by solving the Gutzwiller eﬀective Hamiltonian 293 K

Heff and f(ǫ) denotes the Fermi-Dirac distribution func- 500 K

440 K tion. The ﬁrst and second terms in the above equation 2 denote the energy and entropy contributions to the elec- 380 K

320 K tronic free energy respectively. 1 The lattice part of the free energy is estimated within 250 K

190 K the mean ﬁeld approximation proposed in the refer- 0

140 K ences [18, 45, 46], where the vibrating motion of the (GPa) Pressure

80 K cerium atoms can be approximately treated as indepen- -1

dent three dimensional oscillators moving under the har- 20 K

monic potential formed by all the surrounding atoms. -2 The strength of such mean ﬁeld potential can be approx- 0 K imately determined by the curvature of the total energy

26 27 28 29 30 31 32 33 34 35

versus volume curve as explained in detail in reference o 3 [18, 45, 46]. Considering the heavy mass of the cerium Vol um e (A ) atoms, we can further treat the atomic motion classically and get the lattice free energy as FIG. 2: (Color online) Pressure versus volume isotherms of fcc Ce. The solid line and open symbols are results of LDA 3 mk T + G calculations, where the open symbols are the region of F (V,T )= −k T ( ln B + lnυ (V,T )) (4) vib B ~2 f thermodynamic instability, i.e., the α-γ transition. The solid 2 2π symbols are previous experimental data [2]. where The calculated phase diagram has been plotted in Fig. g(r, V ) 2 υf (V,T ) = 4π exp(− )r dr (5) 3 together with the comparison to the experimental data Z k T B from several diﬀerent papers. The agreement between 1 g(r, V ) = [Ec(R + r)+ Ec(R − r) − 2Ec(R)] (6) our LDA + Gutzwiller calculation and the experimental 2 data is surprisingly well. where r represents the distance that the lattice ion de- Whether the α to γ transition is mainly driven by en- viates from its equilibrium position, R is the lattice con- tropy is another key question. In this paper, we try to ad- stant, and V = R3/4 in the case of fcc crystal. dress it by comparing the change of three parts of Gibbs 4

T S 5

(a) Experiment

Experiment P V

Ref. [1]

Ref. [2]

Ref. [2], Minima in B

Ref. [3]

Ref. [47]

phase Energy (meV) Energy

300 330 360 390 420 450 480 510

phase Pressure (GPa) Pressure

Temperature (K)

LDA + G

T S -1

(b) LDA + G Phase transition

P V

Minima in B

E T 20

-2

0 100 200 300 400 500 600 700 800

Tem perature (K)

Phonon T S

Electron T S

10 FIG. 3: (Color online) Pressure versus temperature phase

diagram of fcc Ce. Open and close ciecles denote present (meV) Energy 5 LDA + G data. The other symbles are previous experimental data (⋄ [1], △ [2], [2], H [3], ⋆ [47]). 0

180 210 240 270 300 330 360 390

Temperature (K) free energy (G(P,V,T )= F (V,T )+ P V ) upon the transition, T∆S, P∆V and internal energy ∆E, which are FIG. 4: (Color online) Energy related term across the α-γ plotted in Fig. 4(b) together with the experimental re- transition, upper panel are experimental data ( [13], [48], △ [3]), lower panel are present data (△ electronic entropy, sults in Fig. 4(a) taken from reference ([13],[48], [3]). ▽ entropy of phonon). Entropy term T∆S (solid symbols), Our results show that the biggest contribution to the P∆V (open symbols), and internal energy ∆E (open symbols transition comes from the entropy change which is quite overlaps with plus). consistent with the experimental data [3, 13, 48]. We can further separate the entropy contribution into electronic and lattice parts, which is also illustrated in Fig. 4(b). ment of Science and Technology of China Academy of The electronic entropy change obtained by our LDA + Engineering Physics (Grant No. 2011B0101011). We Gutzwiller calculation is about 5.0 meV /atom, which is acknowledge the helpful discussion with professor G. about 2-3 times smaller than the lattice part. This is Kotliar and Dr. N. Lanat´a. mainly due to the fact that in LDA + Gutzwiller only the quasi-particle entropy has been included but not the magnetic entropy coming from the incoherent motion of the f -electrons. Considering the estimated Kondo temperature for γ phase is about 70 K [12], the magnetic en- [1] D. Koskenmaki and K. A. Gschneidner, Handbook on the Physics and Chemistry of Rare Earths (North-Holland, tropy, may also make sizable contribution to the γ phase Amsterdam, 1978), vol. 1, p. 337. and make the change of electronic entropy to be compat- [2] M. J. Lipp, D. Jackson, H. Cynn, C. Aracne, W. J. Evans, ible to the lattice one upon the transition. and A. K. McMahan, Phys. Rev. Lett. 101, 165703 In conclusion, the thermodynamic features of the (2008). cerium α to γ transition has been obtained by apply- [3] F. Decremps, L. Belhadi, D. L. Farber, K. T. Moore, ing the LDA + Gutzwiller method, from which we got F. Occelli, M. Gauthier, A. Polian, D. Antonangeli, C. M. Aracne-Ruddle, and B. Amadon, Phys. Rev. Lett. 106, the phase diagram and isotherms of cerium, both in 065701 (2011). good agreement with the experiments. Our calculations [4] B. Johansson, Philos. Mag. 30, 469 (1974). also show that the long puzzled transition is persists to [5] B. Johansson, Phys. Rev. B 11, 2740 (1975). the zero temperature with negative pressure, and mainly [6] B. Johansson, I. A. Abrikosov, M. Alde´n, A. V. Ruban, driven by the entropy change, where both the electronic and H. L. Skriver, Phys. Rev. Lett. 74, 2335 (1995). and the lattice part play important roles. [7] J. W. Allen and R. M. Martin, Phys. Rev. Lett. 49, 1106 (1982). This work was supported by the National Science [8] M. Lavagna, C. Lacroix, and M. Cyrot, Phys. Lett. A Foundation of China (Grants No. NSFC 11204015), by 90, 210 (1982). the 973 program of China (No. 2011CBA00108 and [9] J. W. Allen and L. Z. Liu, Phys. Rev. B 46, 5047 (1992). 2013CBP21700), and by the Foundation for Develop- [10] B. Johansson, A. V. Ruban, and I. A. Abrikosov, Phys. 5

Rev. Lett. 102, 189601 (2009). [31] B. Chakrabarti, M. E. Pezzoli, G. Sordi, K. Haule, and [11] I.-K. Jeong, T. W. Darling, M. J. Graf, T. Proffen, R. H. G. Kotliar, Phys. Rev. B. 89, 125113 (2014). Heffner, Y. Lee, T. Vogt, and J. D. Jorgensen, Phys. Rev. [32] X. Deng, X. Dai, and Z. Fang, Eur. Phys. Lett. 83, 37008 Lett. 92, 105702 (2004). (2008). [12] J.-P. Rueff, J.-P. Itié, M. Taguchi, C. F. Hague, J.-M. [33] X. Y. Deng, L. Wang, X. Dai, and Z. Fang, Phys. Rev. Mariot, R. Delaunay, J.-P. Kappler, and N. Jaouen, Phys. B 79, 075114 (2009). Rev. Lett. 96, 237403 (2006). [34] K. M. Ho, J. Schmalian, and C. Z. Wang, Phys. Rev. B [13] R. I. B. et al., J. Phys. Chem. Solids 15, 234 (1960). 77, 073101 (2008). [14] E. Wuilloud, H. R. Moser, W. D. Schneider, and Y. Baer, [35] M.-F. Tian, X. Deng, Z. Fang, and X. Dai, Phys. Rev. B Phys. Rev. B 28, 7354 (1983). 84, 205124 (2011). [15] D. M. Wieliczka, C. G. Olson, and D. W. Lynch, Phys. [36] N. Lanatá, Y.-X. Yao, C.-Z. Wang, K.-M. Ho, Rev. B 29, 3028 (1984). J. Schmalian, K. Haule, and G. Kotliar, Phys. Rev. Lett. [16] J. W. van der Eb, A. B. Kuzmenko, and D. van der Marel, 111, 196801 (2013). Phys. Rev. Lett. 86, 3407 (2001). [37] N. Lanatá, H. U. R. Strand, Y.-X. Yao, and G. Kotliar, [17] T. Jarlborg, E. G. Moroni, and G. Grimvall, Phys. Rev. Phys. Rev. Lett. 113, 036402 (2014). B 55, 1288 (1997). [38] N. Lanatá, Y.-X. Yao, C.-Z. Wang, K.-M. Ho, and [18] Y. Wang, Phys. Rev. B 61, R11 863 (2000). G. Kotliar, arXiv p. 1405.6934 (2014). [19] M. Lüders, A. Ernst, M. Däne, Z. Szotek, A. Svane, [39] N. Lanatá, Y.-X. Yao, C.-Z. Wang, K.-M. Ho, and D. Ködderitzsch, W. Hergert, B. L. Györffy, and W. M. G. Kotliar, arXiv p. 1407.4862 (2014). Temmerman, Phys. Rev. B 71, 205109 (2005). [40] N. Lanatá, Y.-X. Yao, C.-Z. Wang, K.-M. Ho, and [20] Y. Wang, L. G. Hector, H. Zhang, S. L. Shang, L. Q. G. Kotliar, Phys. Rev. B 90, 161104 (2014). Chen, and Z. K. Liu, Phys. Rev. B 78, 104113 (2008). [41] G. T. Wang, X. Dai, and Z. Fang, Phys. Rev. Lett. 101, [21] Z. Szotek, W. M. Temmerman, and H. Winter, Phys. 066403 (2008). Rev. Lett. 72, 1244 (1994). [42] F. H. Ellinger and W. H. Zachariasen, Phys. Rev. Lett. [22] A. Svane, Phys. Rev. Lett. 72, 1248 (1994). 32, 773 (1974). [23] M. Casadei, X. Ren, P. Rinke, A. Rubio, and M. Scheffler, [43] W. H. Zachariasen and F. H. Ellinger, Acta Cryst. A33, Phys. Rev. Lett. 109, 146402 (2012). 155 (1977). [24] T. Jarlborg, Phys. Rev. B 89, 184426 (2014). [44] J. S. Olsen, L. Gerward, U. Benedict, and J. P. Itié, Phys- [25] M. B. Zölfl, I. A. Nekrasov, T. Pruschke, V. I. Anisimov, ica B 133, 129 (1985). and J. Keller, Phys. Rev. Lett. 87, 276403 (2001). [45] E. Wasserman, L. Stixrude, and R. E. Cohen, Phys. Rev. [26] K. Held, A. K. McMahan, and R. T. Scalettar, Phys. B 53, 8296 (1996). Rev. Lett. 87, 276404 (2001). [46] H.-F. Song and H.-F. Liu, Phys. Rev. B 75, 245126 [27] A. K. McMahan, K. Held, and R. T. Scalettar, Phys. (2007). Rev. B 67, 075108 (2003). [47] K. A. Gschneidner, Valence instabilities and related nar- [28] K. Haule, V. Oudovenko, S. Y. Savrasov, and G. Kotliar, row band phenomena (Plenum, New York, 1977), p. 89. Phys. Rev. Lett. 94, 036401 (2005). [48] A. Schiwek, F. Porsch, and W. B. Hopzapfel, High Press. [29] A. K. McMahan, Phys. Rev. B 72, 115125 (2005). Res. 22, 407 (2002). [30] B. Amadon, S. Biermann, A. Georges, and F. Aryaseti- awan, Phys. Rev. Lett. 96, 066402 (2006).