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Effect of pressure on the magnetic susceptibility of transition metals
Physica B 159 (1989) 35-38 North-Holland, Amsterdam
EFFECT OF PRESSURE ON THE MAGNETIC SUSCEPTIBILITY OF TRANSITION METALS I.V. S V E C H K A R E V Institute for Low Temperature Physics and Engineering, Ukrainian Academy of Sciences, 310164, Kharkov, USSR
The pressure effect on the magnetic susceptibility of electrons in metals of the sp-, d- and f-type is briefly discussed. The experimentally studied dependences of band spectrum and electron interaction parameters on the atomic volume are presented. The Hubbard model is found to be much better than the local density-functional approximation when describing the exchange-correlation enhancement behaviour of spin paramagnetism in d-metals. A probable cause of the observed paramagnetic moment sensitivity to pressure in the stable f-states of heavy rare-earth metals is considered.
Interest in the steady part of the magnetic susceptibility of weakly magnetic metals arises from its sensitivity both to the single-particle energy band structure of the conduction electrons and to electron-electron interactions. The application of high pressure is one of the methods used to determine the properties of weakly magnetic materials. Pressure effects, unlike the effects of temperature or impurities moreover, do not give rise to additional excitations and scattering, which make theoretical analysis much more complicated. Let us consider the effect of pressure on the band structure and some examples of obtaining useful information from the magnetic susceptibility, which in general is a sum of contributions of different origin: x = x s + Xo,+ X,o + Xi.
(1)
The spin paramagnetism Xs of the conduction electrons, enhanced by the exchange-correlation interaction, has the form Xs = SXp = 1 -XpIXp '
(2)
where Xp = 1~2N(EF) is the Pauli paramagnetism, S is the Stoner enhancement factor, N(EF) is the density of electron states at the Fermi level, and I is the interaction parameter. The orbital contribution Xor consists of several terms, one related
to the quantization of the spectrum, which is the Landau-Peierls diamagnetism XLP, and other complex field-induced interband contributions, for example the Van Vleck paramagnetism XvvConventionally the spin-orbit term Xso is included in one of these contributions. Usually the ionic contribution Xi is not of interest because of its insensitivity to pressure. Rare-earth metal ions, carrying localized magnetic moments of unfilled f-shells are different. They exhibit a Curie-Weiss paramagnetism: Xi = l't'2ff/3 kB ( T -- O)
(3)
but in a metal their free magnetic moment/~eu = I ~ g j ~ J ( J + 1) is augmented by the exchange polarization of conduction electrons, /~e = / ~ ( g J - 1 ) J - I . N(EF),
(4)
which mediates the RKKY interaction and determines the value of the paramagnetic Curie temperature 0. Ions with intermediate valence provide the most interesting spectroscopic possibilities. In metals, sp-type states are described by the nearly free electron model. Pressure changes the widths of the energy bands and their splitting at Brillouin zone boundaries. If the gaps are relatively large and the Fermi level is far from the gaps, as in the alkali metals, spin paramagnetism dominates the electronic susceptibility, and the
0921-4526/89/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
36
I.V. Svechkarev / Magnetic susceptibility of transition metals
pressure effect is governed by a simple scaling factor, din Asp/din V= - 2 / 3 . Information on the spectral behavior in the alkali metals Li, Na, K [1] is obtained from an E P R study of their susceptibilities )t's under pressure. It is valuable because it confirms that the ground state of the electron gas in these metals is a state of uniform charge density, with no spin or charge density waves. Data on xs(P) for alkali metals are indispensible as a test to verify first-principles manybody calculations of the exchange-correlation enhancement of spin susceptibility. Good agreement between these calculations and experiment suggests that the local spin density functional method is adequate to calculate the magnitude and atomic volume dependence of the Stoner factor, at least for this group of metals [1]. Later the local density functional method (LDF) was successfully extended to d-band transition metals. In polyvalent sp-metals the Fermi level is often located near edges of weakly split bands. Such a situation is not realised in pure transition metals but can be found in intermetallic compounds with complex crystal lattices and a large number of atoms in a unit cell. The susceptibility of nearly degenerate states is dominated by an anisotropic orbital contribution which has sharp anomalies at critical points of the spectrum [2]. As a result the value of dx/dP may be as great as 10 -7 emu/g-kbar, while the absolute value of X is small in comparison to that in pure transition metals [3]. The unique dependence of Xor on the chemical potential p. may be used to deduce precise values
of parameters for the corresponding part of the electronic spectrum, in particular, band splitting and its dependence on pressure. In this case the component )(or along the z-axis is included in the expression [4]
(eh] 2 kBT
X~=\ c /
®f
(2~) 3.=~_~
dk
sp(TXgTYgTXgTYg),
(5)
where
g = [i'rrkBT(2n + 1) + / z
- ~]-1,
Y( is the truncated k. p-Hamiltonian for the quasi-degenerate states under consideration, and 7x and '~Y are the matrix elements of the momentum operator in the same representation. The Hamiltonian parameters are evaluated by fitting calculated X(~) to the experiment. A sequence of electronic transitions of 21 order in cadmium can be realized by the combined effect of pressure and isovalent magnesium impurities. The magnitude and the location of the corresponding susceptibility anomalies determine the pressure dependence of both the Coulomb splitting of the bands and the chemical potential, and the value of the electron scattering parameter F in C d - M g alloys [5]. These data are presented in table I; the quantity dWloio/dq is essential for the precise evaluation of the pseudopotential formfactor and demonstrates the importance of the influence of d-states on the spectrum of cadmium, manifested in the non-local nature of the pseudopotential.
Table I Pressure dependence of the pseudopotential form-factor of Cd and scattering in Cdl_xMg x alloys [5]. From susceptibility
Stark-Auluk model
dW~oio dP (mRy/kbar)
0.15 + 0.02
0.16-+ 0.02
dWloio dq (Ry" aB)
0.45 -+ 0.03
0.44
F/x (mRy)
8
Local pseudopotential
0.62
I.V. Svechkarev / Magnetic susceptibility of transition metals
the great difference between the experimental data and the results of first-principles calculations by the L D F method is not clear, especially since the theory yields satisfactory values for the parameter I itself for transition metals. An analysis of the behavior of d In x/d In V in alloys based on V and Pd [10, 11] confirms that the effect of pressure on the spectrum has been treated satisfactorily by this procedure. In the case of Pd we were able to separate the mechanisms for the effect of pressure, i.e. the broadening and the relative shift of bands of sp- and dtypes. We could thus determine values of the parameters characterizing the above mechanisms and electron scattering upon alloying. Values of I and d In I/d In V turn out to be constant in the alloys we have measured [10, 11], and the only characteristic of the spectrum with which the parameters may correlate is the dband width Ad. Such behavior is explained in the Hubbard model by strong correlations of a small number of carriers in a narrow d-band [12], but neither an independent confirmation of the value of intra-atomic correlation energy for Pd (U 0.3 Ry) obtained from this model, nor grounds for application of the model, certainly not to vanadium, exist. The character of exchangecorrelation interactions of electrons in transition metals needs more detailed study.
The spin paramagnetism Xs given in eq. (2) and the orbital paramagnetism X.vv - - 1 / A d (Ad is the d-bandwidth) dominates the magnetic susceptibility of non-magnetic transition metals. Accordingly, the expression for the magnetovolume has the form dlnx =Xvv dlnXvv dlnV X dlnV +
x_ /s -d In- Xp + ( S - 1 ) x L
din V
din/] dlnVd"
(6)
Reliable values of din x/d In V are found for vanadium and palladium by direct measurement of x(P) [6] and from the bulk magnetostriction [7, 8]. Theoretical calculations of all the components in eq. (6) necessary for a comparison with experiment (table II) are also available. To deduce the values of the derivative d l n / / d i n V from the experimental data and to compare them with the results of first-principles calculations, the reliability of which has not yet been verified, it is advisable to use only the spectral characteristics from the table. After making appropriate substitutions in (6), and assuming d In Xvv/d In V = - d In Ad/d In V for V [9] and neglecting the contribution Xvv for Pd, we find the values of the derivative d In I/d In V presented in table II. The reason for
Table II Magnetic susceptibility parts and their volume dependence (see references in [6]). V
Pd
Xvv/Xs
1.25
0
S
2.4
9
d In X din V
1.4"
5.3*
dlnA d dlnV
1.4
1.7
d In N(Er) dlnV
.1.1
1.2
dlnXvv_~ dlnV' dlnxp dlnV d In I d in V
* Experimental results.
37
0.2 0.9 + 0.3*
0.2 0.8 --- 0.1"
38
I.V. Svechkarev / Magnetic susceptibility of transition metals
In heavy rare-earth metals the application of pressure changes not only the Curie t e m p e r a t u r e but also the value of the effective magnetic m o m e n t [13]. It appears in this case however, that the values of the moments themselves practically coincide with the moments of the free ions, i.e., the expected polarization contribution (4) does not appear. M o r e o v e r , the behavior of /Jeff(P) in these metals formally corresponds to the variation of the ionic configuration 4f n, d n / d i n V ~ 2 , i.e., the same increase of ionic valence under pressure as in the case of metallic Ce [14], which has true variable valence. T h e known data on heavy rare-earth metals do not permit us to group them with the metals of variable valence, and the similarity of the properties of the magnetic susceptibility of this class of substances may be related to the large temperature d e p e n d e n c e of 0, possibly due to variation of the lattice anisotropy [15]. Taking into account the corresponding renormalization of parameters according to the Curie-Weiss law of eq. (3), further studies may help to explain the observed behavior of the susceptibility of rareearth metals. Lately there have appeared new classes of metallic systems containing d- and f-elements, having unexpected and puzzling physical properties. Magnetic properties and their pressure dependence will be an attractive probe of the details of their electronic structures and their
inherent electronic interactions. It seems important therefore to continue the effort to understand magnetism in metallic elements.
References [1] L. Wilk, A.H. MacDonald and S.H. Vosko, Can. J. Phys. 57(8) (1979) 1064. [2] I.V. Svechkarev, Izv. AN SSSR, Set. Fiz. 42(8) (1978) 1701. [3] V.N. Manchenko, I.V. Svechkarev and Yu.E Sereda, Fiz. Nizk. Temp. 6(2) (1980) 178. [4] H. Fukuyama, Progr. Theor. Phys. 45(3) (1971) 704. [5] G.E. Grechnev and I.V. Svechkarev, Fiz. Nizk. Temp. 7(9) (1981) 1137. [6] A.S. Panfilov, Yu.Ya. Pushkar and I.V. Svechkarev, Fiz. Nizk. Temp. 14(5) (1988) 532. [7] T.L. Tam, M.O. Steinitz and E. Fawcett, J. Phys. F 2(6) (1972) L129. [8] V. Pluzhnikov and E. Fawcett, J. Phys. F 12(7) (1982) 1467. [9] Y. Ohta and M. Shimizu, J. Phys. F 13(6) (1983) L123. [10] A.S. Panfilov and I.V. Svechkarev, Zh. Eksp. Teor. Fiz. 61(3) (1971) 1087. [11] V.N. Manchenko, A.S. Panfilov and I.V. Svechkarev, Zh. Eksp. Teor. Fiz. 71(6) (1976) 2126. [12] J. Kanamori, Progr. Theor. Phys. 30(3) (1963) 275. [13] H. Fujiwara, H. Fujii, Y. Hidaka, T. Ito, Y. Hashimoto and T. Onamoto, J. Phys. Soc. Jpn. 42(2) (1977) 1194. [14] M.R. MacPherson, G.E. Everett, D. Wohlleben and M.B. Maple, Phys. Rev. Lett. 26(1) (1971) 20. [15] K. Taylor and M. Darby, Physics of rare-earth solids (London, 1972).
EFFECT OF PRESSURE ON THE MAGNETIC SUSCEPTIBILITY OF TRANSITION METALS I.V. S V E C H K A R E V Institute for Low Temperature Physics and Engineering, Ukrainian Academy of Sciences, 310164, Kharkov, USSR
The pressure effect on the magnetic susceptibility of electrons in metals of the sp-, d- and f-type is briefly discussed. The experimentally studied dependences of band spectrum and electron interaction parameters on the atomic volume are presented. The Hubbard model is found to be much better than the local density-functional approximation when describing the exchange-correlation enhancement behaviour of spin paramagnetism in d-metals. A probable cause of the observed paramagnetic moment sensitivity to pressure in the stable f-states of heavy rare-earth metals is considered.
Interest in the steady part of the magnetic susceptibility of weakly magnetic metals arises from its sensitivity both to the single-particle energy band structure of the conduction electrons and to electron-electron interactions. The application of high pressure is one of the methods used to determine the properties of weakly magnetic materials. Pressure effects, unlike the effects of temperature or impurities moreover, do not give rise to additional excitations and scattering, which make theoretical analysis much more complicated. Let us consider the effect of pressure on the band structure and some examples of obtaining useful information from the magnetic susceptibility, which in general is a sum of contributions of different origin: x = x s + Xo,+ X,o + Xi.
(1)
The spin paramagnetism Xs of the conduction electrons, enhanced by the exchange-correlation interaction, has the form Xs = SXp = 1 -XpIXp '
(2)
where Xp = 1~2N(EF) is the Pauli paramagnetism, S is the Stoner enhancement factor, N(EF) is the density of electron states at the Fermi level, and I is the interaction parameter. The orbital contribution Xor consists of several terms, one related
to the quantization of the spectrum, which is the Landau-Peierls diamagnetism XLP, and other complex field-induced interband contributions, for example the Van Vleck paramagnetism XvvConventionally the spin-orbit term Xso is included in one of these contributions. Usually the ionic contribution Xi is not of interest because of its insensitivity to pressure. Rare-earth metal ions, carrying localized magnetic moments of unfilled f-shells are different. They exhibit a Curie-Weiss paramagnetism: Xi = l't'2ff/3 kB ( T -- O)
(3)
but in a metal their free magnetic moment/~eu = I ~ g j ~ J ( J + 1) is augmented by the exchange polarization of conduction electrons, /~e = / ~ ( g J - 1 ) J - I . N(EF),
(4)
which mediates the RKKY interaction and determines the value of the paramagnetic Curie temperature 0. Ions with intermediate valence provide the most interesting spectroscopic possibilities. In metals, sp-type states are described by the nearly free electron model. Pressure changes the widths of the energy bands and their splitting at Brillouin zone boundaries. If the gaps are relatively large and the Fermi level is far from the gaps, as in the alkali metals, spin paramagnetism dominates the electronic susceptibility, and the
0921-4526/89/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
36
I.V. Svechkarev / Magnetic susceptibility of transition metals
pressure effect is governed by a simple scaling factor, din Asp/din V= - 2 / 3 . Information on the spectral behavior in the alkali metals Li, Na, K [1] is obtained from an E P R study of their susceptibilities )t's under pressure. It is valuable because it confirms that the ground state of the electron gas in these metals is a state of uniform charge density, with no spin or charge density waves. Data on xs(P) for alkali metals are indispensible as a test to verify first-principles manybody calculations of the exchange-correlation enhancement of spin susceptibility. Good agreement between these calculations and experiment suggests that the local spin density functional method is adequate to calculate the magnitude and atomic volume dependence of the Stoner factor, at least for this group of metals [1]. Later the local density functional method (LDF) was successfully extended to d-band transition metals. In polyvalent sp-metals the Fermi level is often located near edges of weakly split bands. Such a situation is not realised in pure transition metals but can be found in intermetallic compounds with complex crystal lattices and a large number of atoms in a unit cell. The susceptibility of nearly degenerate states is dominated by an anisotropic orbital contribution which has sharp anomalies at critical points of the spectrum [2]. As a result the value of dx/dP may be as great as 10 -7 emu/g-kbar, while the absolute value of X is small in comparison to that in pure transition metals [3]. The unique dependence of Xor on the chemical potential p. may be used to deduce precise values
of parameters for the corresponding part of the electronic spectrum, in particular, band splitting and its dependence on pressure. In this case the component )(or along the z-axis is included in the expression [4]
(eh] 2 kBT
X~=\ c /
®f
(2~) 3.=~_~
dk
sp(TXgTYgTXgTYg),
(5)
where
g = [i'rrkBT(2n + 1) + / z
- ~]-1,
Y( is the truncated k. p-Hamiltonian for the quasi-degenerate states under consideration, and 7x and '~Y are the matrix elements of the momentum operator in the same representation. The Hamiltonian parameters are evaluated by fitting calculated X(~) to the experiment. A sequence of electronic transitions of 21 order in cadmium can be realized by the combined effect of pressure and isovalent magnesium impurities. The magnitude and the location of the corresponding susceptibility anomalies determine the pressure dependence of both the Coulomb splitting of the bands and the chemical potential, and the value of the electron scattering parameter F in C d - M g alloys [5]. These data are presented in table I; the quantity dWloio/dq is essential for the precise evaluation of the pseudopotential formfactor and demonstrates the importance of the influence of d-states on the spectrum of cadmium, manifested in the non-local nature of the pseudopotential.
Table I Pressure dependence of the pseudopotential form-factor of Cd and scattering in Cdl_xMg x alloys [5]. From susceptibility
Stark-Auluk model
dW~oio dP (mRy/kbar)
0.15 + 0.02
0.16-+ 0.02
dWloio dq (Ry" aB)
0.45 -+ 0.03
0.44
F/x (mRy)
8
Local pseudopotential
0.62
I.V. Svechkarev / Magnetic susceptibility of transition metals
the great difference between the experimental data and the results of first-principles calculations by the L D F method is not clear, especially since the theory yields satisfactory values for the parameter I itself for transition metals. An analysis of the behavior of d In x/d In V in alloys based on V and Pd [10, 11] confirms that the effect of pressure on the spectrum has been treated satisfactorily by this procedure. In the case of Pd we were able to separate the mechanisms for the effect of pressure, i.e. the broadening and the relative shift of bands of sp- and dtypes. We could thus determine values of the parameters characterizing the above mechanisms and electron scattering upon alloying. Values of I and d In I/d In V turn out to be constant in the alloys we have measured [10, 11], and the only characteristic of the spectrum with which the parameters may correlate is the dband width Ad. Such behavior is explained in the Hubbard model by strong correlations of a small number of carriers in a narrow d-band [12], but neither an independent confirmation of the value of intra-atomic correlation energy for Pd (U 0.3 Ry) obtained from this model, nor grounds for application of the model, certainly not to vanadium, exist. The character of exchangecorrelation interactions of electrons in transition metals needs more detailed study.
The spin paramagnetism Xs given in eq. (2) and the orbital paramagnetism X.vv - - 1 / A d (Ad is the d-bandwidth) dominates the magnetic susceptibility of non-magnetic transition metals. Accordingly, the expression for the magnetovolume has the form dlnx =Xvv dlnXvv dlnV X dlnV +
x_ /s -d In- Xp + ( S - 1 ) x L
din V
din/] dlnVd"
(6)
Reliable values of din x/d In V are found for vanadium and palladium by direct measurement of x(P) [6] and from the bulk magnetostriction [7, 8]. Theoretical calculations of all the components in eq. (6) necessary for a comparison with experiment (table II) are also available. To deduce the values of the derivative d l n / / d i n V from the experimental data and to compare them with the results of first-principles calculations, the reliability of which has not yet been verified, it is advisable to use only the spectral characteristics from the table. After making appropriate substitutions in (6), and assuming d In Xvv/d In V = - d In Ad/d In V for V [9] and neglecting the contribution Xvv for Pd, we find the values of the derivative d In I/d In V presented in table II. The reason for
Table II Magnetic susceptibility parts and their volume dependence (see references in [6]). V
Pd
Xvv/Xs
1.25
0
S
2.4
9
d In X din V
1.4"
5.3*
dlnA d dlnV
1.4
1.7
d In N(Er) dlnV
.1.1
1.2
dlnXvv_~ dlnV' dlnxp dlnV d In I d in V
* Experimental results.
37
0.2 0.9 + 0.3*
0.2 0.8 --- 0.1"
38
I.V. Svechkarev / Magnetic susceptibility of transition metals
In heavy rare-earth metals the application of pressure changes not only the Curie t e m p e r a t u r e but also the value of the effective magnetic m o m e n t [13]. It appears in this case however, that the values of the moments themselves practically coincide with the moments of the free ions, i.e., the expected polarization contribution (4) does not appear. M o r e o v e r , the behavior of /Jeff(P) in these metals formally corresponds to the variation of the ionic configuration 4f n, d n / d i n V ~ 2 , i.e., the same increase of ionic valence under pressure as in the case of metallic Ce [14], which has true variable valence. T h e known data on heavy rare-earth metals do not permit us to group them with the metals of variable valence, and the similarity of the properties of the magnetic susceptibility of this class of substances may be related to the large temperature d e p e n d e n c e of 0, possibly due to variation of the lattice anisotropy [15]. Taking into account the corresponding renormalization of parameters according to the Curie-Weiss law of eq. (3), further studies may help to explain the observed behavior of the susceptibility of rareearth metals. Lately there have appeared new classes of metallic systems containing d- and f-elements, having unexpected and puzzling physical properties. Magnetic properties and their pressure dependence will be an attractive probe of the details of their electronic structures and their
inherent electronic interactions. It seems important therefore to continue the effort to understand magnetism in metallic elements.
References [1] L. Wilk, A.H. MacDonald and S.H. Vosko, Can. J. Phys. 57(8) (1979) 1064. [2] I.V. Svechkarev, Izv. AN SSSR, Set. Fiz. 42(8) (1978) 1701. [3] V.N. Manchenko, I.V. Svechkarev and Yu.E Sereda, Fiz. Nizk. Temp. 6(2) (1980) 178. [4] H. Fukuyama, Progr. Theor. Phys. 45(3) (1971) 704. [5] G.E. Grechnev and I.V. Svechkarev, Fiz. Nizk. Temp. 7(9) (1981) 1137. [6] A.S. Panfilov, Yu.Ya. Pushkar and I.V. Svechkarev, Fiz. Nizk. Temp. 14(5) (1988) 532. [7] T.L. Tam, M.O. Steinitz and E. Fawcett, J. Phys. F 2(6) (1972) L129. [8] V. Pluzhnikov and E. Fawcett, J. Phys. F 12(7) (1982) 1467. [9] Y. Ohta and M. Shimizu, J. Phys. F 13(6) (1983) L123. [10] A.S. Panfilov and I.V. Svechkarev, Zh. Eksp. Teor. Fiz. 61(3) (1971) 1087. [11] V.N. Manchenko, A.S. Panfilov and I.V. Svechkarev, Zh. Eksp. Teor. Fiz. 71(6) (1976) 2126. [12] J. Kanamori, Progr. Theor. Phys. 30(3) (1963) 275. [13] H. Fujiwara, H. Fujii, Y. Hidaka, T. Ito, Y. Hashimoto and T. Onamoto, J. Phys. Soc. Jpn. 42(2) (1977) 1194. [14] M.R. MacPherson, G.E. Everett, D. Wohlleben and M.B. Maple, Phys. Rev. Lett. 26(1) (1971) 20. [15] K. Taylor and M. Darby, Physics of rare-earth solids (London, 1972).