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Field-induced valence transition in rare-earth system
Journal of Magnetism and Magnetic Materials 208 (2000) 131}136
Field-induced valence transition in rare-earth system A. Chattopadhaya, S.K. Ghatak* Department of Physics and Meteorology, Indian Institute of Technology, Kharagpur 721302, India Received 12 April 1999; received in revised form 4 August 1999
Abstract The magnetic "eld-induced valence transition in rare-earth compound has been examined based on a pseudospin S"1 Ising model proposed earlier for valence transition. The model includes "nite mixing between two pertinent ionic con"gurations (magnetic and non-magnetic) separated by an energy gap and with intersite interaction between rare-earth ions. Using the mean "eld approximation the magnetic behaviour and the critical "eld (H ) for transition are obtained as # a function of energy gap and temperature. The phase boundary de"ned in terms of reduced "eld H /H and reduced # #0 temperature ¹/¹ (¹ being valence transition temperature in absence of "eld) is nearly independent of energy gap. 7 7 These results are in qualitative agreement with experimental observation in Yb- and Eu-compounds. ( 2000 Published by Elsevier Science B.V. All rights reserved. PACS: 75.30.Mb; 75.30.Kz; 75.10.Jm Keywords: Valence #uctuation; Ising model; Metamagnetism; Rare-earth compounds; Magnetic "eld-induced M}I transition
1. Introduction Rare-earth 4f-ion exhibits the valence instability when the two adjacent electronic valence states of ion are energetically close to each other [1}3]. The valence #uctuation phenomena that was extensively observed in Sm-, Eu- and Yb-compounds was induced by pressure and chemical alloying that reduces the energy gap between the ground states of two adjacent valence states. The valence state of these ions changes from lower valence state to higher one abruptly or in continuous way. Between the valence states involved in the transition, the
* Corresponding author. Tel.: #91-3222-55221; fax: #913222-55303. E-mail address: [email protected] (S.K. Ghatak)
ground state of one is magnetic and that of other non-magnetic. For example, Yb2` has non-magnetic ground state whereas ground state of Yb3` is magnetic and reverse is the case for Eu-ion. It is therefore expected that the magnetic "eld can be used to induce the valence transtion. The "eld induced valence transition has been observed in Yb[4}7] and Eu- [8] compounds. The intermetallic compound YbIn Ag Cu with 0(x(0.1 ex1~x x 4 hibits "eld-induced transition from intermediate valence state to trivalent magnetic state. The transition is marked by sharp increase in magnetisation at a critical "eld H obtained from the peak # of dM/dH as a function of "eld H. The transition is also accompanied by change of valence of Yb. The critical "eld H , being maximum at ¹"0, is a de# creasing function of temperature and vanishes at ¹ , valence transition temperature at H"0. Both 7
0304-8853/00/$ - see front matter ( 2000 Published by Elsevier Science B.V. All rights reserved. PII: S 0 3 0 4 - 8 8 5 3 ( 9 9 ) 0 0 5 6 4 - 8
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the critical "elds H and the transition temperature # ¹ are found to be increasing function of x. The 7 phase diagram in H}¹ plane when plotted in terms of reduced "eld H /H and reduced temperature # #0 ¹/¹ , the experimental data on H /H for di!er7 # #0 ent composition of the compound collapse into a single curve. The critical "eld H for Eu in # EuNi (Si Ge ) and Eu(Pd Pt ) Si [8] is 2 1~x x 2 1~x x 2 2 also linear function of ¹ . In this work the e!ects of 7 high magnetic "eld on the valence transition are examined theoretically based on the pseudo-spin model proposed earlier [9]. It is found that the qualitative behaviour of H and the phase diagram # are in tune with the observed situation.
2. Model and calculation The electronic con"guration of rare-earth ion can be in mixed state when two successive valence states are nearly degenerate or in terms of oneelectron picture the energy necessary to promote an electron from 4f-level to band states is small. In presence of hybridisation between 4f-level and band states the ion can tunnel between these two nearly degenerate valence states. As in Sm, Eu and Yb, the ground state of one of the con"guration of the ion is magnetic and the other is non-magnetic * for example ground state of 2` state Yb is nonmagnetic whereas that of 3` state carries a magnetic moment. This situation can be described by a pseudo-spin variable S , associated with the iz rare-earth ion at ith site. The state S "0 then iz corresponds to the ground state of non-magnetic con"guration and the states S "$1 represent iz the magnetic one. The Hamiltonian to describe the valence change in terms of pseudo-spin can be written as [9] (1) H"+ [DS2 #uS #HS ]#+ K S2 S2 . iz ix iz ij iz jz ij i The "rst term represents the energy separation between two non-interacting neighbouring con"gurations of the rare-earth ion at ith site. For D'0 the non-magnetic con"guration of the ion has lower energy. This site energy depends on the pressure or chemical alloying. The hybridisation be-
tween the two con"gurations are described by the second term which mixes the states S "0 and iz $1. The mixing energy u scales with hybridisation strength between 4f and 5d}6s band states. The e!ect of magnetic "eld H on the con"gurations is given by the third term. The last term is the interaction between the rare-earth ion at neighbouring sites and its origin can be elastic [10] electronic [11,12] or both [13]. For K (0, the transition ij from non-magnetic to magnetic con"guration is favoured. For K '0 and D(0 the transition ij occurs in reverse direction (e.g. Eu2` (magnetic)PEu3` (non-magnetic) in Eu compounds). The Hamiltonian without u predicts "rst-order transition in some range of D [9] and was considered to simulate the 3He}4He mixture [14]. In terms of spin variable the number of 2` and 3` ions can be expressed as N(2`)"+ [1!SS2 T] and N(3`)"+ SS2 T iz iz i i with the constraint N(2`)#N(3`)"N total number of ions in the system. The magnetisation M"k[N(3`C)!N(3`B)]/N"k + SS T/N, (2) iz i where k represents the magnetic moment of ion in magnetic con"guration and SS T, the average iz value of S . iz We treat the interaction term in Eq. (1) within the mean "eld approximation, and the Hamiltonian H reduces to (3) H"+ [DS2 #uS #HS ], iz ix iz i where D"D!Kn with n"SS2 T being the averiz age number of trivalent ion and K"!+ K . The j ij Hamiltonian equation (3) represents Ising spin in the presence of transverse (u) and longitudinal (H) magnetic "eld, and the eigenvalues (j) of H satisfy a cubic equation j3!2Dj2#(D2!u!H)j#u2D"0.
(4)
The eigenvalues are obtained numerically, and the thermodynamics of the system follows from the Gibbs free energy G"!k¹ ln Z#Kn2/2 where
A. Chattopadhaya, S.K. Ghatak / Journal of Magnetism and Magnetic Materials 208 (2000) 131}136
133
Z is the partition function. For numerical calculation, the parameters D, u, k¹ and H are normalised by intersite interaction strength K. The valence parameter n is obtained from the minimisation of G and the magnetisation M per rare-earth ion is then obtained from Eq. (2).
3. Results and discussions The variation of valence n with magnetic "eld H for D"0.65, 0.7 and 0.75 and with u"0.3 at ¹"0 is displayed in Fig. 1 exihibiting the magnetic "eld-induced valence transition. A sharp change in average con"guration occurs within a narrow region H. The values of n at low and high "elds are determined by relative values of the con"guration mixing (u) and the promotional energy (D). At "nite temperature the transition appears with decrease sharpness. The "eld dependence of the magnetisation M (in units of k) for D"0.7 and u"0.3 is displayed in Fig. 2. The magnetisation at ¹"0 increases slowly at low "eld and a nearly discontinuous change occurs within narrow "eld region where the sharp valence transition appears. At high-"eld region it tends to saturate to a value
Fig. 1. Field variation of valence n at ¹"0 for di!erent values of D and u"0.3.
Fig. 2. Magnetisation (M) as a function of "eld (H) at di!erent temperature ¹ for D"0.7 and u"0.3.
less than its maximum (unity). The deviation from the saturation value of M from unity increases as hybridization is strengthened. With increase in ¹ the variation of M becomes smoother. The system is more magnetic at low "eld but needs higher "eld to saturate compared to that at lower ¹. The maximum value of the slope (dM/dH) is reduced with increase in ¹. The "eld where the magnetisation changes sharply (dM/dH maximum) is termed as critical "eld H . It is evident from # Fig. 2 that H reduces with increase in temperature # and vanishes at a critical temperature ¹ . 7 In order to examine the dependence of H on # D and u the magnetisation has been calculated for di!erent values of D (Fig. 3a) and u (Fig. 3b) at ¹"0. It is clear that larger critical "eld is necessary for the system with higher value of promotional energy. The con"guration mixing alters the "eld variation of the magnetisation and the critical "eld. The plot of the critical "eld (H ) at ¹"0 #0 agianst D (Fig. 4) shows linear relationship between them. This result is in qualitative accord with the experimental observation on EuNi (Si Ge ) 2 1~x x 2 [8]. It is observed that H decreases with x for # 0.74)x)0.95. The Eu-ion is in trivalent nonmagnetic state for x"0 and in divalent magnetic
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A. Chattopadhaya, S.K. Ghatak / Journal of Magnetism and Magnetic Materials 208 (2000) 131}136
Fig. 4. Variation of critical "eld (H ) at ¹"0 with D and ¹ . #0 7
Fig. 3. M versus H at ¹"0: (a) for di!erent D and "xed u"0.3; (b) for di!erent u and given D"0.7.
state in other end (x"1) of solid solution. With increase of x the lattice of the Eu-compound is dilated [8] which in turn increases the energy separation between 4f-level and the conduction band and favours the 4f7-(5d}6s)0 con"guration (Eu2`) as the ground state for x"1. On the other hand, the 4f6-(5d}6s)1 con"guration (Eu3`) lies lowest for x"0 where 4f-level is close to 5d}6s band. Such
variation in energy separation between f-level and the band is represented by decrease of D with x in this pseudo-spin model. The transition temperature ¹ and H also ex7 #0 hibit linear relationship (Fig. 4) and this is in tune with the experimental results in Yb- [5,6] and Eu[8] compounds. The thermal variation of H for # three di!erent values of D and u"0.3 are plotted in Fig. 5. With increase in temperature initial variation of H is small but sharply increases near ¹ . # 7 We note that an uncertainty exists in numerical determination of ¹ related to broad temperature 7 width of dM/dh as ¹ approaches to ¹ . Here ¹ is 7 7 obtained from the extrapolation of H versus # ¹ plot near ¹ . This result is replotted in Fig. 6 in 7 terms of reduced critical "eld h "H /H and # # #0 reduced temperatue t"¹/¹ . The results in Fig. 7 5 merge into a single curve although there is some scattering near t"1 due to error in determination ¹ . It is clear that the reduced "eld is dominantly 7 determined by the reduced temperature. Such functional behaviour h on t has been observed in # Yb-compound [5}7]. The results presented here for chosen set of values of the model parameters. These typical values represent the situation where inter-ionic interaction determines the dominant
A. Chattopadhaya, S.K. Ghatak / Journal of Magnetism and Magnetic Materials 208 (2000) 131}136
135
bridisation energy between f-level and band state is smaller that their separation, D is taken to be larger than u. The Zeeman energy in magnetic state of rare-earth ion is normally smaller compared to energy of electronic in origin, and thus H(D represents the real system. Physical origin of "eldinduced valence transition is related to lowering of the free energy due to the Zeeman energy in the magnetic con"guration of the rare-earth ion. In absence of magnetic "eld the magnetic con"guration (divalent for Eu and trivalent for Yb) ion is energetically separated by D. The Zeeman energy reduces the energy gap between magnetic and nonmagnetic con"gurations by an amount that depends on "eld and the magnetic moment in the magnetic con"guration of the ion. The sharp transition occurs when the Zeeman energy competes favourably with the promotional energy D. Fig. 5. Thermal variation of critical "eld H for u"0.3 and # di!erent D.
Fig. 6. Phase diagram in terms of reduced critical "eld H /H # #0 and reduced temperature ¹/¹ for u"0.3. 7
energy scale. The parameter D that simulates the energy separation between f-level and bottom of the conduction band should be smaller than the interaction energy for transition to occur. As hy-
4. Conclusions The pseudo-spin Ising model considered here points out importance of physical variables, namely: (i) energy gap and (ii) mixing between two pertinent valence states, (iii) intersite interaction between ions and (iv) Zeeman energy of the magnetic state, for the valence transition in rare-earth compounds. The model provides a qualitative description of the "eld-induced valence transition in rare-earth compounds in presence of large magnetic "eld. The maximum critical magnetic "eld H varies almost linearly with energy gap D and #0 transition temperature ¹ . The critical "eld H de7 # creases slowly at low ¹ but drops sharply to zero at ¹ . The phase diagram in reduced unit of critical 7 "eld H /H and temperature ¹/¹ is found to be # #0 # almost independent of the promotional energy. These results are in tune with the experimental observation in Yb- and Eu-based compounds. It is necessary to incorporate the intersite magnetic interaction which become important near H as # more number of ions are in magnetic con"guration. Within the mean "eld approximation this interaction induces an e!ective "eld which is larger than the applied "eld. Therefore, the "eldinduced transition would occur at lower "eld and the quantitative result will be reported in future.
136
A. Chattopadhaya, S.K. Ghatak / Journal of Magnetism and Magnetic Materials 208 (2000) 131}136
References [1] J.M. Lawrence, P.S. Riseborough, R.D. Parks, Rep. Prog. Phys. 44 (1981) 1. [2] N.B. Brandt, V.V. Moshchalkov, Adv. Phys. 33 (1984) 373. [3] G.R. Stewart, Rev. Mod. Phys. 56 (1984) 755. [4] H. Aruga Katori, T. Goto, K. Yoshimura, Physica B 201 (1994) 159. [5] K. Yoshimura, T. Nitta, T. Shimizu, M. Mekata, H. Yasuoka, K. Kosuge, J. Magn. Magn. Mater. 90&91 (1990) 466. [6] H. Aruga Katori, T. Goto, K. Yoshimura, J. Magn. Magn. Mater. 140}144 (1995) 1245. [7] I. Felner, I. Nowik, D. Vakim, U. Potizel, J. Moser, G.K. Kalvius, G. Wortmann, G. Schmiester, G. Hilscher,
[8]
[9] [10] [11] [12] [13] [14]
E. Gratz, C. Schnitzer, N. Pillmay, K.G. Prasad, H. deWaard, H. Pinto, Phys. Rev. B 35 (1997) 6956. H. Wada, A. Nakamura, A. Mitsudai, M. Shigai, T. Tanaka, H. Mitamura, T. Goto, J. Phys.: Condens. Matter 9 (1997) 7913. S.K. Ghatak, Physica B 94 (1978) 196. P.W. Anderson, S.T. Chui, Phys. Rev. 89 (1974) 3229. J.M.D. Coey, S.K. Ghatak, M. Avignon, F. Holtzberg, Phys. Rev. B 14 (1976) 3744. C.E.T. Goncalves de Silva, L.M. Falikov, Solid State Commun. 17 (1975) 152. S.K. Ghatak, M. Avignon, K.H. Bennemann, J. Phys. F 6 (1976) 1441. M. Blume, V.J. Emery, R.B. Gri$ths, Phys. Rev. A 4 (1974) 1071.
Field-induced valence transition in rare-earth system A. Chattopadhaya, S.K. Ghatak* Department of Physics and Meteorology, Indian Institute of Technology, Kharagpur 721302, India Received 12 April 1999; received in revised form 4 August 1999
Abstract The magnetic "eld-induced valence transition in rare-earth compound has been examined based on a pseudospin S"1 Ising model proposed earlier for valence transition. The model includes "nite mixing between two pertinent ionic con"gurations (magnetic and non-magnetic) separated by an energy gap and with intersite interaction between rare-earth ions. Using the mean "eld approximation the magnetic behaviour and the critical "eld (H ) for transition are obtained as # a function of energy gap and temperature. The phase boundary de"ned in terms of reduced "eld H /H and reduced # #0 temperature ¹/¹ (¹ being valence transition temperature in absence of "eld) is nearly independent of energy gap. 7 7 These results are in qualitative agreement with experimental observation in Yb- and Eu-compounds. ( 2000 Published by Elsevier Science B.V. All rights reserved. PACS: 75.30.Mb; 75.30.Kz; 75.10.Jm Keywords: Valence #uctuation; Ising model; Metamagnetism; Rare-earth compounds; Magnetic "eld-induced M}I transition
1. Introduction Rare-earth 4f-ion exhibits the valence instability when the two adjacent electronic valence states of ion are energetically close to each other [1}3]. The valence #uctuation phenomena that was extensively observed in Sm-, Eu- and Yb-compounds was induced by pressure and chemical alloying that reduces the energy gap between the ground states of two adjacent valence states. The valence state of these ions changes from lower valence state to higher one abruptly or in continuous way. Between the valence states involved in the transition, the
* Corresponding author. Tel.: #91-3222-55221; fax: #913222-55303. E-mail address: [email protected] (S.K. Ghatak)
ground state of one is magnetic and that of other non-magnetic. For example, Yb2` has non-magnetic ground state whereas ground state of Yb3` is magnetic and reverse is the case for Eu-ion. It is therefore expected that the magnetic "eld can be used to induce the valence transtion. The "eld induced valence transition has been observed in Yb[4}7] and Eu- [8] compounds. The intermetallic compound YbIn Ag Cu with 0(x(0.1 ex1~x x 4 hibits "eld-induced transition from intermediate valence state to trivalent magnetic state. The transition is marked by sharp increase in magnetisation at a critical "eld H obtained from the peak # of dM/dH as a function of "eld H. The transition is also accompanied by change of valence of Yb. The critical "eld H , being maximum at ¹"0, is a de# creasing function of temperature and vanishes at ¹ , valence transition temperature at H"0. Both 7
0304-8853/00/$ - see front matter ( 2000 Published by Elsevier Science B.V. All rights reserved. PII: S 0 3 0 4 - 8 8 5 3 ( 9 9 ) 0 0 5 6 4 - 8
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A. Chattopadhaya, S.K. Ghatak / Journal of Magnetism and Magnetic Materials 208 (2000) 131}136
the critical "elds H and the transition temperature # ¹ are found to be increasing function of x. The 7 phase diagram in H}¹ plane when plotted in terms of reduced "eld H /H and reduced temperature # #0 ¹/¹ , the experimental data on H /H for di!er7 # #0 ent composition of the compound collapse into a single curve. The critical "eld H for Eu in # EuNi (Si Ge ) and Eu(Pd Pt ) Si [8] is 2 1~x x 2 1~x x 2 2 also linear function of ¹ . In this work the e!ects of 7 high magnetic "eld on the valence transition are examined theoretically based on the pseudo-spin model proposed earlier [9]. It is found that the qualitative behaviour of H and the phase diagram # are in tune with the observed situation.
2. Model and calculation The electronic con"guration of rare-earth ion can be in mixed state when two successive valence states are nearly degenerate or in terms of oneelectron picture the energy necessary to promote an electron from 4f-level to band states is small. In presence of hybridisation between 4f-level and band states the ion can tunnel between these two nearly degenerate valence states. As in Sm, Eu and Yb, the ground state of one of the con"guration of the ion is magnetic and the other is non-magnetic * for example ground state of 2` state Yb is nonmagnetic whereas that of 3` state carries a magnetic moment. This situation can be described by a pseudo-spin variable S , associated with the iz rare-earth ion at ith site. The state S "0 then iz corresponds to the ground state of non-magnetic con"guration and the states S "$1 represent iz the magnetic one. The Hamiltonian to describe the valence change in terms of pseudo-spin can be written as [9] (1) H"+ [DS2 #uS #HS ]#+ K S2 S2 . iz ix iz ij iz jz ij i The "rst term represents the energy separation between two non-interacting neighbouring con"gurations of the rare-earth ion at ith site. For D'0 the non-magnetic con"guration of the ion has lower energy. This site energy depends on the pressure or chemical alloying. The hybridisation be-
tween the two con"gurations are described by the second term which mixes the states S "0 and iz $1. The mixing energy u scales with hybridisation strength between 4f and 5d}6s band states. The e!ect of magnetic "eld H on the con"gurations is given by the third term. The last term is the interaction between the rare-earth ion at neighbouring sites and its origin can be elastic [10] electronic [11,12] or both [13]. For K (0, the transition ij from non-magnetic to magnetic con"guration is favoured. For K '0 and D(0 the transition ij occurs in reverse direction (e.g. Eu2` (magnetic)PEu3` (non-magnetic) in Eu compounds). The Hamiltonian without u predicts "rst-order transition in some range of D [9] and was considered to simulate the 3He}4He mixture [14]. In terms of spin variable the number of 2` and 3` ions can be expressed as N(2`)"+ [1!SS2 T] and N(3`)"+ SS2 T iz iz i i with the constraint N(2`)#N(3`)"N total number of ions in the system. The magnetisation M"k[N(3`C)!N(3`B)]/N"k + SS T/N, (2) iz i where k represents the magnetic moment of ion in magnetic con"guration and SS T, the average iz value of S . iz We treat the interaction term in Eq. (1) within the mean "eld approximation, and the Hamiltonian H reduces to (3) H"+ [DS2 #uS #HS ], iz ix iz i where D"D!Kn with n"SS2 T being the averiz age number of trivalent ion and K"!+ K . The j ij Hamiltonian equation (3) represents Ising spin in the presence of transverse (u) and longitudinal (H) magnetic "eld, and the eigenvalues (j) of H satisfy a cubic equation j3!2Dj2#(D2!u!H)j#u2D"0.
(4)
The eigenvalues are obtained numerically, and the thermodynamics of the system follows from the Gibbs free energy G"!k¹ ln Z#Kn2/2 where
A. Chattopadhaya, S.K. Ghatak / Journal of Magnetism and Magnetic Materials 208 (2000) 131}136
133
Z is the partition function. For numerical calculation, the parameters D, u, k¹ and H are normalised by intersite interaction strength K. The valence parameter n is obtained from the minimisation of G and the magnetisation M per rare-earth ion is then obtained from Eq. (2).
3. Results and discussions The variation of valence n with magnetic "eld H for D"0.65, 0.7 and 0.75 and with u"0.3 at ¹"0 is displayed in Fig. 1 exihibiting the magnetic "eld-induced valence transition. A sharp change in average con"guration occurs within a narrow region H. The values of n at low and high "elds are determined by relative values of the con"guration mixing (u) and the promotional energy (D). At "nite temperature the transition appears with decrease sharpness. The "eld dependence of the magnetisation M (in units of k) for D"0.7 and u"0.3 is displayed in Fig. 2. The magnetisation at ¹"0 increases slowly at low "eld and a nearly discontinuous change occurs within narrow "eld region where the sharp valence transition appears. At high-"eld region it tends to saturate to a value
Fig. 1. Field variation of valence n at ¹"0 for di!erent values of D and u"0.3.
Fig. 2. Magnetisation (M) as a function of "eld (H) at di!erent temperature ¹ for D"0.7 and u"0.3.
less than its maximum (unity). The deviation from the saturation value of M from unity increases as hybridization is strengthened. With increase in ¹ the variation of M becomes smoother. The system is more magnetic at low "eld but needs higher "eld to saturate compared to that at lower ¹. The maximum value of the slope (dM/dH) is reduced with increase in ¹. The "eld where the magnetisation changes sharply (dM/dH maximum) is termed as critical "eld H . It is evident from # Fig. 2 that H reduces with increase in temperature # and vanishes at a critical temperature ¹ . 7 In order to examine the dependence of H on # D and u the magnetisation has been calculated for di!erent values of D (Fig. 3a) and u (Fig. 3b) at ¹"0. It is clear that larger critical "eld is necessary for the system with higher value of promotional energy. The con"guration mixing alters the "eld variation of the magnetisation and the critical "eld. The plot of the critical "eld (H ) at ¹"0 #0 agianst D (Fig. 4) shows linear relationship between them. This result is in qualitative accord with the experimental observation on EuNi (Si Ge ) 2 1~x x 2 [8]. It is observed that H decreases with x for # 0.74)x)0.95. The Eu-ion is in trivalent nonmagnetic state for x"0 and in divalent magnetic
134
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Fig. 4. Variation of critical "eld (H ) at ¹"0 with D and ¹ . #0 7
Fig. 3. M versus H at ¹"0: (a) for di!erent D and "xed u"0.3; (b) for di!erent u and given D"0.7.
state in other end (x"1) of solid solution. With increase of x the lattice of the Eu-compound is dilated [8] which in turn increases the energy separation between 4f-level and the conduction band and favours the 4f7-(5d}6s)0 con"guration (Eu2`) as the ground state for x"1. On the other hand, the 4f6-(5d}6s)1 con"guration (Eu3`) lies lowest for x"0 where 4f-level is close to 5d}6s band. Such
variation in energy separation between f-level and the band is represented by decrease of D with x in this pseudo-spin model. The transition temperature ¹ and H also ex7 #0 hibit linear relationship (Fig. 4) and this is in tune with the experimental results in Yb- [5,6] and Eu[8] compounds. The thermal variation of H for # three di!erent values of D and u"0.3 are plotted in Fig. 5. With increase in temperature initial variation of H is small but sharply increases near ¹ . # 7 We note that an uncertainty exists in numerical determination of ¹ related to broad temperature 7 width of dM/dh as ¹ approaches to ¹ . Here ¹ is 7 7 obtained from the extrapolation of H versus # ¹ plot near ¹ . This result is replotted in Fig. 6 in 7 terms of reduced critical "eld h "H /H and # # #0 reduced temperatue t"¹/¹ . The results in Fig. 7 5 merge into a single curve although there is some scattering near t"1 due to error in determination ¹ . It is clear that the reduced "eld is dominantly 7 determined by the reduced temperature. Such functional behaviour h on t has been observed in # Yb-compound [5}7]. The results presented here for chosen set of values of the model parameters. These typical values represent the situation where inter-ionic interaction determines the dominant
A. Chattopadhaya, S.K. Ghatak / Journal of Magnetism and Magnetic Materials 208 (2000) 131}136
135
bridisation energy between f-level and band state is smaller that their separation, D is taken to be larger than u. The Zeeman energy in magnetic state of rare-earth ion is normally smaller compared to energy of electronic in origin, and thus H(D represents the real system. Physical origin of "eldinduced valence transition is related to lowering of the free energy due to the Zeeman energy in the magnetic con"guration of the rare-earth ion. In absence of magnetic "eld the magnetic con"guration (divalent for Eu and trivalent for Yb) ion is energetically separated by D. The Zeeman energy reduces the energy gap between magnetic and nonmagnetic con"gurations by an amount that depends on "eld and the magnetic moment in the magnetic con"guration of the ion. The sharp transition occurs when the Zeeman energy competes favourably with the promotional energy D. Fig. 5. Thermal variation of critical "eld H for u"0.3 and # di!erent D.
Fig. 6. Phase diagram in terms of reduced critical "eld H /H # #0 and reduced temperature ¹/¹ for u"0.3. 7
energy scale. The parameter D that simulates the energy separation between f-level and bottom of the conduction band should be smaller than the interaction energy for transition to occur. As hy-
4. Conclusions The pseudo-spin Ising model considered here points out importance of physical variables, namely: (i) energy gap and (ii) mixing between two pertinent valence states, (iii) intersite interaction between ions and (iv) Zeeman energy of the magnetic state, for the valence transition in rare-earth compounds. The model provides a qualitative description of the "eld-induced valence transition in rare-earth compounds in presence of large magnetic "eld. The maximum critical magnetic "eld H varies almost linearly with energy gap D and #0 transition temperature ¹ . The critical "eld H de7 # creases slowly at low ¹ but drops sharply to zero at ¹ . The phase diagram in reduced unit of critical 7 "eld H /H and temperature ¹/¹ is found to be # #0 # almost independent of the promotional energy. These results are in tune with the experimental observation in Yb- and Eu-based compounds. It is necessary to incorporate the intersite magnetic interaction which become important near H as # more number of ions are in magnetic con"guration. Within the mean "eld approximation this interaction induces an e!ective "eld which is larger than the applied "eld. Therefore, the "eldinduced transition would occur at lower "eld and the quantitative result will be reported in future.
136
A. Chattopadhaya, S.K. Ghatak / Journal of Magnetism and Magnetic Materials 208 (2000) 131}136
References [1] J.M. Lawrence, P.S. Riseborough, R.D. Parks, Rep. Prog. Phys. 44 (1981) 1. [2] N.B. Brandt, V.V. Moshchalkov, Adv. Phys. 33 (1984) 373. [3] G.R. Stewart, Rev. Mod. Phys. 56 (1984) 755. [4] H. Aruga Katori, T. Goto, K. Yoshimura, Physica B 201 (1994) 159. [5] K. Yoshimura, T. Nitta, T. Shimizu, M. Mekata, H. Yasuoka, K. Kosuge, J. Magn. Magn. Mater. 90&91 (1990) 466. [6] H. Aruga Katori, T. Goto, K. Yoshimura, J. Magn. Magn. Mater. 140}144 (1995) 1245. [7] I. Felner, I. Nowik, D. Vakim, U. Potizel, J. Moser, G.K. Kalvius, G. Wortmann, G. Schmiester, G. Hilscher,
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