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Magnetic susceptibility of metallic hydrogen
746
Journal of Magnetism and Magnetic Materials 90 & 91 (1990) 746-748 North-Holland
Magnetic susceptibility of metallic hydrogen K. Ebina and M. Kaburagi College of Liberal Arts, Kobe Unicersity, Nada, Kobe 657, Japan
The effect of the electron-ion interactions on the generalized susceptibility X(q) for metallic hydrogen is studied in the structural expansion method. The leading terms have been calculated to reveal the characteristics. The possibility of appearance of a magnetic phase is discussed by approximately taking account of the exchange and correlat ion effects.
The metallic phase of hydrogen is expected to be obtained in the near future owing to the recent advance in high pressure techniques. Mao and Hemley [1] has recently reported a change in the optical properties at about 2.5 Mbar as an evidence for a change in the electronic structure, which may be thought of as a metallization or at least its precursor effect. It is of interest to investigate the magnetic properties near the metal-insulator transition and monatomic-molecular structural transition (if they are different), because the electron pairs forming a singlet spin state in each molecule will be broken or at least largely modified in the high-density, metallic side. Furthermore, because of the reason stated below we may even expect that a stable magnetically ordered (ferromagnetic (FM), antiferromagnetic (AF) or spin-density wave (SDW» state would appear in the metallic phase, in particular when its structure is an anisotropic one such as filamentary or planar. If we consider the ground state of metallic hydrogen starting from the high-density limit (Ts = 0, where Ts is the usual density parameter), the electronelectron and electron-ion (proton) interactions become more important with decreasing density (increasing Ts )' Then the Pauli susceptibility of uniform free electron gas is to be modified by both interactions. It is wellknown that the susceptibility of an electron gas is enhanced by the exchange effects; the uniform system becomes FM at Ts ::::: 6 in the Hartree-Fock approximation, although the critical Ts becomes considerably larger (toward the low-density side) with the inclusion of the correlation and higher-order effects. It has also been proposed that the electron gas would become an SDW state due to the exchange effect [2]. This tendency of forming magnetically ordered states may further be enhanced by the electron-ion interaction because it makes the electrons to accumulate ncar the nuclear positions, and hence the exchange effect will become stronger. Min et al. [3] have studied the magnetic properties of low-density metallic hydrogen in the bee, fcc and hcp lattices using the linearized muffin-tin orbital method and showed that the metallic hydrogen becomes
an AF state at Ts ::::: 2.5. In such a density, however, the metallic phase becomes already unstable, so the AF state does not appear in actuality. On the other hand, it is highly probable that the metallic phase into which the molecular phase is transformed at a few megabar pressure (Ts ::::: 1.4) is anisotropic. In the anisotropic structures the electron-ion interaction are generally stronger than that for the cubic ones owing to the shorter reciprocal lattice vectors and the AF coupling will be particularly enhanced for the filamentary structures so that a magnetically ordered state may appear in the region where the metallic phase is stable (Ts :5 1.5). the purpose of the present work is to reveal the effects of electron-ion interaction on the trend toward magnetically ordered states of the metallic hydrogen by investigating the Ts dependence of the generalized magnetic susceptibility xts) for several structures. We formulate the problem based on the structural expansion theory [4] combined with the density-functional formalism [5] to investigate systematically the lattice effects. We consider several structures in the rhombohedral family by changing the uniaxial parameter cia. The magnetic susceptibility (generally a matrix in a crystal) is defined by III q
=
LX(q, q + G)hq + G ,
(1)
G
where IIIq and hq are, respectively, the spatial Fourier components of the magnetization and those of the potential from applied magnetic field, and Gs are the reciprocal lattice vectors. For the non-interacting electron system under the potential IVG from the ion lattice, the susceptibility is given in the structural expansion as x'OJ(q, q + G) =
oG,oX'OJ(q)
+L G1
+
IVG2X~OJ(q, q + G)
IVG1IVG- G .[ 2X~O)(q, q + G\, q + G)
0304-8853/90/503,50 ~ 1990 - Elsevier Science Publishers B.V. (North-Holland) and Yamada Science Foundation
(2)
K. Ebina, M. Kaburagi
I
Magnetic susceptibility of metallic hydrogen
where X~OI, X~O), • • . are the non-linear susceptibilities of the free electron gas. The electron-electron interaction modifies eq . (2) essentially in two respects. First. IVG must be replaced by the effective potential, wh ich includes the screening and the vertex effects. Second. in the expression for the inverse susceptibility X(OI-I(q + G I , q + G2 ) (i.e. the inverse of eq. (2» terms due to the exchange-correlation interaction
0.5
I(q, q + G)
=
2(¢ t t (q) -¢ t I (q))8G .o
+ (h.igher-order in 0.0'--
----'-
o
0.0
747
L..-
-.1
2
L..-
o
----J
2
Fig. 1. The '. dependence of the lowest eigenvalues of the inverse susceptibility X(O)-'(q + GI • q + G 2 ) in units of l/Xr(O) for wavevectorse, '" ~(bl - b2 ) and q2 = i(b( + b2 + b3 ) . where bl ' b 2 and b3 are the primitive vectors for the rhombohedral reciprocal lattice. (a) is for the bce (y = 1/2), (b) is for the se (y = 1) and (c) is for the fce (y = 2) lattices respectively, where y = (cla)/(cla) ",.
(3)
must be added. In order to study the instability to magnetic states, we diagonalize the inverse susceptibility X-I(q + G 1• q + G 2 ) to obtain the most frangible mode. To reveal the characteristics of the electron-ion interaction. we first neglect the effect of electron-electron interaction. In fig. 1 we give the inverse susceptibility in units of 1/Xt(O) calculated by eq . (2) for the most frangible mode as a function of 's with the ferro- (q = 0) and some typical antiferromagnetic wave vectors. We show the results for three structures among rhombohedral lattices with y = 1/2 (bee), y = 1 (sc) and y = 2 (fcc) where y is the ratio of cia to that of the sc structure. The overall tendency is not so different among various structures 'and wave vectors. We first consider the FM case. The inverse l/x(O.O) of the bulk susceptibility, which is the one that actually mea sured, increases with increasing 's' but the off-diagonal elements work to reduce drastically the lowest eigen value of X-I for q = O. The obtained eigenvalue is reduced to about 0.2-0.3 times that of the uniform free electron gas at 's"= 1.5. If we estimate the exchange-correlation interaction approximately by that I(q) for uniform electron gas [5], then X-I(O) will further be reduced by about 0.2, By extrapolating the data to larger 'S' an instab ility would occur at = 2-3, which is amazingly consistent with the results of the band calculation [3] in spite of our crudest approximation. For the AF case. we chose ql and q2 (see figure caption). Therefore, the instability to the AF phases will occur in the similar density region. The questions of which inst ability occurs first (with decreasing density) among FM and AF modes is answered only after the more detailed analysis. Ho wever, we note here that the exchange-correlation potential I(q) for an electron gas stays nearly constant for small q and has a hump at around q = 2k F [6]. wh ich may enhance the tendency to become the AF state. Furthermore, the second term in eq. (3) is expected to enhance the exchange instability at a finite wave vector [7]. Calculation including higher-order effects of both electron-ion (eq . (2» and electron-electron interactions (eq. (3» as well as for other structures, particularly filamentary and the planar structures, is now in progress .
's
0.0 ' - - - - - - - - - - - - ' - - - - -_ _-----.J o 2 rs
IVG )
748
K. Ebina, M. Kaburagi / Magnetic susceptibility of metallic hydrogen
References (1) H.K. Mao and R.J. Hemley, Science 244 (1989) 1462.
(2) A.W. Overhauser, Phys. Rev. 167 (1968) 691. (3) B. Min, T. Oguchi, H. Jansen and A. Freeman, Phys. Rev. B 33 (1986) 324. (4) T. Nakamura, H. Nagara and H. Miyagi, Prog. Theor, Phys. 63 (1980) 368, and references therein.
(5) Y. Kawazoe, H. Yasuhara and M. Watabe, J. Phys, C
l(
(1977) 3293. (6) See, e.g. H. Suehiro, Y. Ousaka and H. Yasuhara, J. Phys C 19 (1986) 4247. (7) K. Ebina, H. Nagara and H. Miyagi, Prog. Theor. Phys. 6: (1982) 781.
Journal of Magnetism and Magnetic Materials 90 & 91 (1990) 746-748 North-Holland
Magnetic susceptibility of metallic hydrogen K. Ebina and M. Kaburagi College of Liberal Arts, Kobe Unicersity, Nada, Kobe 657, Japan
The effect of the electron-ion interactions on the generalized susceptibility X(q) for metallic hydrogen is studied in the structural expansion method. The leading terms have been calculated to reveal the characteristics. The possibility of appearance of a magnetic phase is discussed by approximately taking account of the exchange and correlat ion effects.
The metallic phase of hydrogen is expected to be obtained in the near future owing to the recent advance in high pressure techniques. Mao and Hemley [1] has recently reported a change in the optical properties at about 2.5 Mbar as an evidence for a change in the electronic structure, which may be thought of as a metallization or at least its precursor effect. It is of interest to investigate the magnetic properties near the metal-insulator transition and monatomic-molecular structural transition (if they are different), because the electron pairs forming a singlet spin state in each molecule will be broken or at least largely modified in the high-density, metallic side. Furthermore, because of the reason stated below we may even expect that a stable magnetically ordered (ferromagnetic (FM), antiferromagnetic (AF) or spin-density wave (SDW» state would appear in the metallic phase, in particular when its structure is an anisotropic one such as filamentary or planar. If we consider the ground state of metallic hydrogen starting from the high-density limit (Ts = 0, where Ts is the usual density parameter), the electronelectron and electron-ion (proton) interactions become more important with decreasing density (increasing Ts )' Then the Pauli susceptibility of uniform free electron gas is to be modified by both interactions. It is wellknown that the susceptibility of an electron gas is enhanced by the exchange effects; the uniform system becomes FM at Ts ::::: 6 in the Hartree-Fock approximation, although the critical Ts becomes considerably larger (toward the low-density side) with the inclusion of the correlation and higher-order effects. It has also been proposed that the electron gas would become an SDW state due to the exchange effect [2]. This tendency of forming magnetically ordered states may further be enhanced by the electron-ion interaction because it makes the electrons to accumulate ncar the nuclear positions, and hence the exchange effect will become stronger. Min et al. [3] have studied the magnetic properties of low-density metallic hydrogen in the bee, fcc and hcp lattices using the linearized muffin-tin orbital method and showed that the metallic hydrogen becomes
an AF state at Ts ::::: 2.5. In such a density, however, the metallic phase becomes already unstable, so the AF state does not appear in actuality. On the other hand, it is highly probable that the metallic phase into which the molecular phase is transformed at a few megabar pressure (Ts ::::: 1.4) is anisotropic. In the anisotropic structures the electron-ion interaction are generally stronger than that for the cubic ones owing to the shorter reciprocal lattice vectors and the AF coupling will be particularly enhanced for the filamentary structures so that a magnetically ordered state may appear in the region where the metallic phase is stable (Ts :5 1.5). the purpose of the present work is to reveal the effects of electron-ion interaction on the trend toward magnetically ordered states of the metallic hydrogen by investigating the Ts dependence of the generalized magnetic susceptibility xts) for several structures. We formulate the problem based on the structural expansion theory [4] combined with the density-functional formalism [5] to investigate systematically the lattice effects. We consider several structures in the rhombohedral family by changing the uniaxial parameter cia. The magnetic susceptibility (generally a matrix in a crystal) is defined by III q
=
LX(q, q + G)hq + G ,
(1)
G
where IIIq and hq are, respectively, the spatial Fourier components of the magnetization and those of the potential from applied magnetic field, and Gs are the reciprocal lattice vectors. For the non-interacting electron system under the potential IVG from the ion lattice, the susceptibility is given in the structural expansion as x'OJ(q, q + G) =
oG,oX'OJ(q)
+L G1
+
IVG2X~OJ(q, q + G)
IVG1IVG- G .[ 2X~O)(q, q + G\, q + G)
0304-8853/90/503,50 ~ 1990 - Elsevier Science Publishers B.V. (North-Holland) and Yamada Science Foundation
(2)
K. Ebina, M. Kaburagi
I
Magnetic susceptibility of metallic hydrogen
where X~OI, X~O), • • . are the non-linear susceptibilities of the free electron gas. The electron-electron interaction modifies eq . (2) essentially in two respects. First. IVG must be replaced by the effective potential, wh ich includes the screening and the vertex effects. Second. in the expression for the inverse susceptibility X(OI-I(q + G I , q + G2 ) (i.e. the inverse of eq. (2» terms due to the exchange-correlation interaction
0.5
I(q, q + G)
=
2(¢ t t (q) -¢ t I (q))8G .o
+ (h.igher-order in 0.0'--
----'-
o
0.0
747
L..-
-.1
2
L..-
o
----J
2
Fig. 1. The '. dependence of the lowest eigenvalues of the inverse susceptibility X(O)-'(q + GI • q + G 2 ) in units of l/Xr(O) for wavevectorse, '" ~(bl - b2 ) and q2 = i(b( + b2 + b3 ) . where bl ' b 2 and b3 are the primitive vectors for the rhombohedral reciprocal lattice. (a) is for the bce (y = 1/2), (b) is for the se (y = 1) and (c) is for the fce (y = 2) lattices respectively, where y = (cla)/(cla) ",.
(3)
must be added. In order to study the instability to magnetic states, we diagonalize the inverse susceptibility X-I(q + G 1• q + G 2 ) to obtain the most frangible mode. To reveal the characteristics of the electron-ion interaction. we first neglect the effect of electron-electron interaction. In fig. 1 we give the inverse susceptibility in units of 1/Xt(O) calculated by eq . (2) for the most frangible mode as a function of 's with the ferro- (q = 0) and some typical antiferromagnetic wave vectors. We show the results for three structures among rhombohedral lattices with y = 1/2 (bee), y = 1 (sc) and y = 2 (fcc) where y is the ratio of cia to that of the sc structure. The overall tendency is not so different among various structures 'and wave vectors. We first consider the FM case. The inverse l/x(O.O) of the bulk susceptibility, which is the one that actually mea sured, increases with increasing 's' but the off-diagonal elements work to reduce drastically the lowest eigen value of X-I for q = O. The obtained eigenvalue is reduced to about 0.2-0.3 times that of the uniform free electron gas at 's"= 1.5. If we estimate the exchange-correlation interaction approximately by that I(q) for uniform electron gas [5], then X-I(O) will further be reduced by about 0.2, By extrapolating the data to larger 'S' an instab ility would occur at = 2-3, which is amazingly consistent with the results of the band calculation [3] in spite of our crudest approximation. For the AF case. we chose ql and q2 (see figure caption). Therefore, the instability to the AF phases will occur in the similar density region. The questions of which inst ability occurs first (with decreasing density) among FM and AF modes is answered only after the more detailed analysis. Ho wever, we note here that the exchange-correlation potential I(q) for an electron gas stays nearly constant for small q and has a hump at around q = 2k F [6]. wh ich may enhance the tendency to become the AF state. Furthermore, the second term in eq. (3) is expected to enhance the exchange instability at a finite wave vector [7]. Calculation including higher-order effects of both electron-ion (eq . (2» and electron-electron interactions (eq. (3» as well as for other structures, particularly filamentary and the planar structures, is now in progress .
's
0.0 ' - - - - - - - - - - - - ' - - - - -_ _-----.J o 2 rs
IVG )
748
K. Ebina, M. Kaburagi / Magnetic susceptibility of metallic hydrogen
References (1) H.K. Mao and R.J. Hemley, Science 244 (1989) 1462.
(2) A.W. Overhauser, Phys. Rev. 167 (1968) 691. (3) B. Min, T. Oguchi, H. Jansen and A. Freeman, Phys. Rev. B 33 (1986) 324. (4) T. Nakamura, H. Nagara and H. Miyagi, Prog. Theor, Phys. 63 (1980) 368, and references therein.
(5) Y. Kawazoe, H. Yasuhara and M. Watabe, J. Phys, C
l(
(1977) 3293. (6) See, e.g. H. Suehiro, Y. Ousaka and H. Yasuhara, J. Phys C 19 (1986) 4247. (7) K. Ebina, H. Nagara and H. Miyagi, Prog. Theor. Phys. 6: (1982) 781.