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Band contribution to the electronic transport in noncollinear magnetic materials: application to LaMn2 Ge2
ARTICLE IN PRESS
Physica B 354 (2004) 154–157 www.elsevier.com/locate/physb
Band contribution to the electronic transport in noncollinear magnetic materials: application to LaMn2Ge2 S. Di Napolia,, G. Bihlmayerb, S. Blu¨gelb, M. Alouanic, H. Dreysse´c, A.M. Lloisa,d a
Departamento de Fı´sica, Ciudad Universitaria, UBA Pab. I, 1428 Buenos Aires, Argentina b Institut fu¨r Festko¨rperforschung, Forschungszentrum Ju¨lich, D-52425 Ju¨lich, Germany c IPCMS-GEMME, UMR 7504 CNRS-ULP, 23 rue du Loess, F-67034 Strasbourg Cedex, France d Departamento de Fı´sica, CNEA-CAC, Avenida del Libertador 8250,1429 Buenos Aires, Argentina
Abstract The intermetallic ternary compounds of the type RMn2 X2 (R=Ca, La, Ba, Y and X=Si, Ge), crystallizing in the ThCr2 Si2 structure, show a large variety of collinear and noncollinear magnetic ground states (GS) depending on R and X and thus are good candidates for studying the dependence of the band structure contribution to the electronic transport on the different magnetic configurations. In this contribution we focus our analysis on LaMn2 Ge2 : A qualitative understanding of the change in the conductivities with the magnetic structure in this material is provided on the basis of its coherent electronic structure. r 2004 Elsevier B.V. All rights reserved. PACS: 72.15.v; 72.25.Ba Keywords: Noncollinear magnetism; Natural multilayers; Electronic transport properties
In the last decade, characterized by the search for new materials with new magneto-transport properties, much attention has been devoted to the investigation of the electronic and magnetic properties of materials with layered structures. With the observation of giant magnetoresistance (GMR) in La-based manganates [1], research on Corresponding author. Fax: +54 11 4576 3357.
E-mail address: [email protected] (S. Di Napoli).
the magnetoresistant behavior of different materials has been extensively done. In particular, the compounds of the type RMn2 X2 (X=Si, Ge), crystallizing in the ThCr2 Si2 -type structure, are of special interest, as they are the only series of alloys within this structure in which the transition metal carries a magnetic moment (around 3mB ) and shows a large variety of collinear and noncollinear magnetic GS depending on R and X. The naturally layered structure of these compounds provides a
0921-4526/$ - see front matter r 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2004.09.039
ARTICLE IN PRESS S. Di Napoli et al. / Physica B 354 (2004) 154–157
novel route to model artificial multilayer physics. In this contribution we focus our analysis on LaMn2 Ge2 which shows, at low temperatures, a noncollinear magnetic structure with the Mn’s magnetic moments ordered in a conical arrangement with ferromagnetic (FM) coupling along the c-axis and antiferromagnetic (AFM) coupling within the (001) planes [2]. However, the most interesting feature of this compound is that it shows, also at low temperatures, inverse GMR [3]. In a previous work [4] we calculated the band structure contribution to GMR when noncollinearity is explicitly taken into account in the electronic structure calculations. We found that this contribution is always direct, opposite to the experimental observations and stated that in this case the inverse GMR could be attributed to scattering effects induced by spin fluctuations in the presence of the external magnetic field which affects the relaxation time t [4]. Notwithstanding this, it is interesting to analyze the dependence of the band structure contribution to the electric conductivity tensor on the different magnetic configurations of the system. A qualitative understanding of these dependence should be possible due to its coherent electronic structure. Changes in the electronic structure as a function of magnetic order lead to changes in the Fermi surface, in the Fermi velocities and thereafter in the electric conductivity tensor. The electric conductivity tensor is calculated within the semiclassical Boltzmann approach in the relaxation-time approximation [5]. As we are only considering the band structure contribution to the electric conductivity, the dependence of the relaxation time on k and spin is neglected, as well as the vertex corrections. In a first approximation, spin accumulation and interface disorder effects can be neglected because the systems under study are natural multilayers with perfect interfaces. Taking into account the precedingsimplifications, the electric conductivity tensor is given by sij ¼
e2 t X 8p2 ns
Z
vins ðkÞvjns ðkÞ dðns ðkÞ F Þ d3 k;
155
where vins ðkÞ ¼ ð1=_Þ qns ðkÞ=qki is the semiclassical velocity in the i-direction. When i ¼ z; i.e., i represents the c-axis of the unit-cell, szz corresponds to the current perpendicular to the plane (CPP) electric conductivity, and when i represents the direction perpendicular to the c-axis sii (i ¼ x; y) is called current inplane (CIP) electric conductivity. The integral in Eq. (1) is computed by means of the tetrahedron method [6] using the energies obtained by means of the FLEUR code, an implementation of the FLAPW method [7]. As a general expression for the giant magnetoresistance we use the definition GMRi ¼
sii ðNFÞ 1; sii ðFMÞ
1oGMRo þ 1:
(2)
With NF we indicate a non-ferromagnetic configuration as, for instance, an AFM case or the noncollinear arrangements of the magnetic moments of Mn in LaMn2 Ge2 : Commonly, one observes that sii ðNFÞosii ðFMÞ; the GMR ratio is negative and in this situation one refers to the direct or negative GMR, otherwise an inverse GMR is observed. With the above listed approximations for the relaxation time, t cancels out from Eq. (2). Fig. 1 shows the evolution of the band structure contribution to the CPP and CIP electric conductivities (in arbitrary units) as a function of the
(1)
where s denotes the spin index, n the band index, F the Fermi energy, t the relaxation time, and
Fig. 1. CPP and CIP electric conductivities (in arbitrary units) as a function of the canting angle y for LaMn2 Ge2 : The error bars are related to the change in the electric conductivities values when modifying the Fermi level by 0:3%:
ARTICLE IN PRESS S. Di Napoli et al. / Physica B 354 (2004) 154–157
156
p character
0.4
0.4
0.2
0.2
Energy (eV)
Energy (eV)
s character
0
-0.2
0
-0.2
-0.4
-0.4 Γ
X
M
Γ Z
R
A
Γ
Z
X
M
R
A
Z
R
A
Z
f character
0.4
0.4
0.2
0.2
Energy (eV)
Energy (eV)
d character
Γ Z
0
-0.2
0
-0.2
-0.4
-0.4 Γ
X
M
Γ Z
R
A
Z
Γ
X
M
Γ Z
Fig. 2. Different band character contributions to the band structure for canting angle y=40 at the vicinity of the Fermi surface. The vertical lines show the corresponding weight of the s, p, d and f orbital characters.
Fig. 3. Different band contributions to the Fermi surface for canting angle y=40 ; ordered by increasing band index (see Fig. 2) from (a) to (d).
canting angle y: This is the semicone angle of the conically arranged magnetic GS of LaMn2 Ge2 ; whereby y=0 corresponds to a FM state and y=58 to the GS. It can be seen from the figure that both CPP and CIP electric conductivities decrease with increasing angle. The error bars
displayed in Fig. 1 are associated to changes in the Fermi level (F ) position of the system which has been allowed to vary by 0.3%; as it could happen if a low density of impurities were present. This change in F leads to changes in the electric conductivity values up to 25%; meaning that the
ARTICLE IN PRESS S. Di Napoli et al. / Physica B 354 (2004) 154–157
157
Fig. 4. Idem Fig. 3 but for canting angle y=60 :
electric conductivity of this system depends highly on the presence of impurities. As it can be inferred from the example shown in Fig. 2 the bands crossing F are mainly of d-character. Since these bands are rather flat, even small changes of the Fermi level can lead to important variations in the topology of the Fermi surface (FS) and thus cause strong variations in the electric conductivity. As examples of these topological modifications we show in Figs. 3 and 4 the band structure contributions to the FS for two different magnetic structures, namely, the ones with canting angles y ¼ 40 and 60 ; respectively. From Fig. 1 we see that the difference in the electric conductivity values between these two magnetic configurations is of 38% in the CPP case and 50% in the CIP case. This drastic change can be rationalized when comparing the FS topologies shown in Figs. 3 and 4, which are very anisotropic and differ substantially for the two magnetic structures. sCIP is basically determined by the in-plane components of the Fermi velocity which mainly stem from cylindrical parts of the FS along the kz -direction. On the other hand, sCPP mostly originates from flat FS contributions in the kx –ky plane with large values of vkz : Comparing the topological evolution of the FS when going from y ¼ 40 to y ¼ 60 the important
variation of the band structure contribution to the electric conductivities of this system can easily be understood. This changing topology should also affect the magnetic susceptibility and thereafter the magnetic structure evolution of this compound. We acknowledge the financial support from UBACyT-X115, from the European Research Training Network RTN1-1999-00145, the French–Argentinian collaboration program ECOS and the German–Argentinian collaboration program DAAD-Antorchas.
References [1] R. Van Helmdt, et al., Phys. Rev. Lett. 71 (1993) 2331. [2] B. Malaman et al., J. Alloys and Comp. 210 (1994) 209. [3] R. Mallik, E.V. Sampathkumaran, P.L. Paulose, Appl. Phys. Lett. 71 (1997) 2385. [4] S. Di Napoli, et al., Phys. Rev. B, accepted. [5] J. Ziman, Electrons and Phonons, Oxford University Press, London, 1960. [6] O. Jepsen, O.K. Andersen, Solid State Commun. 9 (1971) 1763; G. Lehmann, M. Taut, Phys. Stat. Sol. B 54 (1972) 469. [7] Ph. Kurz, et al., Phys. Rev. B 69 (2004) 024415.
Physica B 354 (2004) 154–157 www.elsevier.com/locate/physb
Band contribution to the electronic transport in noncollinear magnetic materials: application to LaMn2Ge2 S. Di Napolia,, G. Bihlmayerb, S. Blu¨gelb, M. Alouanic, H. Dreysse´c, A.M. Lloisa,d a
Departamento de Fı´sica, Ciudad Universitaria, UBA Pab. I, 1428 Buenos Aires, Argentina b Institut fu¨r Festko¨rperforschung, Forschungszentrum Ju¨lich, D-52425 Ju¨lich, Germany c IPCMS-GEMME, UMR 7504 CNRS-ULP, 23 rue du Loess, F-67034 Strasbourg Cedex, France d Departamento de Fı´sica, CNEA-CAC, Avenida del Libertador 8250,1429 Buenos Aires, Argentina
Abstract The intermetallic ternary compounds of the type RMn2 X2 (R=Ca, La, Ba, Y and X=Si, Ge), crystallizing in the ThCr2 Si2 structure, show a large variety of collinear and noncollinear magnetic ground states (GS) depending on R and X and thus are good candidates for studying the dependence of the band structure contribution to the electronic transport on the different magnetic configurations. In this contribution we focus our analysis on LaMn2 Ge2 : A qualitative understanding of the change in the conductivities with the magnetic structure in this material is provided on the basis of its coherent electronic structure. r 2004 Elsevier B.V. All rights reserved. PACS: 72.15.v; 72.25.Ba Keywords: Noncollinear magnetism; Natural multilayers; Electronic transport properties
In the last decade, characterized by the search for new materials with new magneto-transport properties, much attention has been devoted to the investigation of the electronic and magnetic properties of materials with layered structures. With the observation of giant magnetoresistance (GMR) in La-based manganates [1], research on Corresponding author. Fax: +54 11 4576 3357.
E-mail address: [email protected] (S. Di Napoli).
the magnetoresistant behavior of different materials has been extensively done. In particular, the compounds of the type RMn2 X2 (X=Si, Ge), crystallizing in the ThCr2 Si2 -type structure, are of special interest, as they are the only series of alloys within this structure in which the transition metal carries a magnetic moment (around 3mB ) and shows a large variety of collinear and noncollinear magnetic GS depending on R and X. The naturally layered structure of these compounds provides a
0921-4526/$ - see front matter r 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2004.09.039
ARTICLE IN PRESS S. Di Napoli et al. / Physica B 354 (2004) 154–157
novel route to model artificial multilayer physics. In this contribution we focus our analysis on LaMn2 Ge2 which shows, at low temperatures, a noncollinear magnetic structure with the Mn’s magnetic moments ordered in a conical arrangement with ferromagnetic (FM) coupling along the c-axis and antiferromagnetic (AFM) coupling within the (001) planes [2]. However, the most interesting feature of this compound is that it shows, also at low temperatures, inverse GMR [3]. In a previous work [4] we calculated the band structure contribution to GMR when noncollinearity is explicitly taken into account in the electronic structure calculations. We found that this contribution is always direct, opposite to the experimental observations and stated that in this case the inverse GMR could be attributed to scattering effects induced by spin fluctuations in the presence of the external magnetic field which affects the relaxation time t [4]. Notwithstanding this, it is interesting to analyze the dependence of the band structure contribution to the electric conductivity tensor on the different magnetic configurations of the system. A qualitative understanding of these dependence should be possible due to its coherent electronic structure. Changes in the electronic structure as a function of magnetic order lead to changes in the Fermi surface, in the Fermi velocities and thereafter in the electric conductivity tensor. The electric conductivity tensor is calculated within the semiclassical Boltzmann approach in the relaxation-time approximation [5]. As we are only considering the band structure contribution to the electric conductivity, the dependence of the relaxation time on k and spin is neglected, as well as the vertex corrections. In a first approximation, spin accumulation and interface disorder effects can be neglected because the systems under study are natural multilayers with perfect interfaces. Taking into account the precedingsimplifications, the electric conductivity tensor is given by sij ¼
e2 t X 8p2 ns
Z
vins ðkÞvjns ðkÞ dðns ðkÞ F Þ d3 k;
155
where vins ðkÞ ¼ ð1=_Þ qns ðkÞ=qki is the semiclassical velocity in the i-direction. When i ¼ z; i.e., i represents the c-axis of the unit-cell, szz corresponds to the current perpendicular to the plane (CPP) electric conductivity, and when i represents the direction perpendicular to the c-axis sii (i ¼ x; y) is called current inplane (CIP) electric conductivity. The integral in Eq. (1) is computed by means of the tetrahedron method [6] using the energies obtained by means of the FLEUR code, an implementation of the FLAPW method [7]. As a general expression for the giant magnetoresistance we use the definition GMRi ¼
sii ðNFÞ 1; sii ðFMÞ
1oGMRo þ 1:
(2)
With NF we indicate a non-ferromagnetic configuration as, for instance, an AFM case or the noncollinear arrangements of the magnetic moments of Mn in LaMn2 Ge2 : Commonly, one observes that sii ðNFÞosii ðFMÞ; the GMR ratio is negative and in this situation one refers to the direct or negative GMR, otherwise an inverse GMR is observed. With the above listed approximations for the relaxation time, t cancels out from Eq. (2). Fig. 1 shows the evolution of the band structure contribution to the CPP and CIP electric conductivities (in arbitrary units) as a function of the
(1)
where s denotes the spin index, n the band index, F the Fermi energy, t the relaxation time, and
Fig. 1. CPP and CIP electric conductivities (in arbitrary units) as a function of the canting angle y for LaMn2 Ge2 : The error bars are related to the change in the electric conductivities values when modifying the Fermi level by 0:3%:
ARTICLE IN PRESS S. Di Napoli et al. / Physica B 354 (2004) 154–157
156
p character
0.4
0.4
0.2
0.2
Energy (eV)
Energy (eV)
s character
0
-0.2
0
-0.2
-0.4
-0.4 Γ
X
M
Γ Z
R
A
Γ
Z
X
M
R
A
Z
R
A
Z
f character
0.4
0.4
0.2
0.2
Energy (eV)
Energy (eV)
d character
Γ Z
0
-0.2
0
-0.2
-0.4
-0.4 Γ
X
M
Γ Z
R
A
Z
Γ
X
M
Γ Z
Fig. 2. Different band character contributions to the band structure for canting angle y=40 at the vicinity of the Fermi surface. The vertical lines show the corresponding weight of the s, p, d and f orbital characters.
Fig. 3. Different band contributions to the Fermi surface for canting angle y=40 ; ordered by increasing band index (see Fig. 2) from (a) to (d).
canting angle y: This is the semicone angle of the conically arranged magnetic GS of LaMn2 Ge2 ; whereby y=0 corresponds to a FM state and y=58 to the GS. It can be seen from the figure that both CPP and CIP electric conductivities decrease with increasing angle. The error bars
displayed in Fig. 1 are associated to changes in the Fermi level (F ) position of the system which has been allowed to vary by 0.3%; as it could happen if a low density of impurities were present. This change in F leads to changes in the electric conductivity values up to 25%; meaning that the
ARTICLE IN PRESS S. Di Napoli et al. / Physica B 354 (2004) 154–157
157
Fig. 4. Idem Fig. 3 but for canting angle y=60 :
electric conductivity of this system depends highly on the presence of impurities. As it can be inferred from the example shown in Fig. 2 the bands crossing F are mainly of d-character. Since these bands are rather flat, even small changes of the Fermi level can lead to important variations in the topology of the Fermi surface (FS) and thus cause strong variations in the electric conductivity. As examples of these topological modifications we show in Figs. 3 and 4 the band structure contributions to the FS for two different magnetic structures, namely, the ones with canting angles y ¼ 40 and 60 ; respectively. From Fig. 1 we see that the difference in the electric conductivity values between these two magnetic configurations is of 38% in the CPP case and 50% in the CIP case. This drastic change can be rationalized when comparing the FS topologies shown in Figs. 3 and 4, which are very anisotropic and differ substantially for the two magnetic structures. sCIP is basically determined by the in-plane components of the Fermi velocity which mainly stem from cylindrical parts of the FS along the kz -direction. On the other hand, sCPP mostly originates from flat FS contributions in the kx –ky plane with large values of vkz : Comparing the topological evolution of the FS when going from y ¼ 40 to y ¼ 60 the important
variation of the band structure contribution to the electric conductivities of this system can easily be understood. This changing topology should also affect the magnetic susceptibility and thereafter the magnetic structure evolution of this compound. We acknowledge the financial support from UBACyT-X115, from the European Research Training Network RTN1-1999-00145, the French–Argentinian collaboration program ECOS and the German–Argentinian collaboration program DAAD-Antorchas.
References [1] R. Van Helmdt, et al., Phys. Rev. Lett. 71 (1993) 2331. [2] B. Malaman et al., J. Alloys and Comp. 210 (1994) 209. [3] R. Mallik, E.V. Sampathkumaran, P.L. Paulose, Appl. Phys. Lett. 71 (1997) 2385. [4] S. Di Napoli, et al., Phys. Rev. B, accepted. [5] J. Ziman, Electrons and Phonons, Oxford University Press, London, 1960. [6] O. Jepsen, O.K. Andersen, Solid State Commun. 9 (1971) 1763; G. Lehmann, M. Taut, Phys. Stat. Sol. B 54 (1972) 469. [7] Ph. Kurz, et al., Phys. Rev. B 69 (2004) 024415.