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Crystalline field effects and exchange parameters in metallic beryllides
Journal of Magnetism and Magnetic Materials 0 North-Holland Publishing Company
CRYSTALLINE
7 (1978)
78-81
FIELD EFFECTS AND EXCHANGE PARAMETERS
IN METALLIC BERYLLIDES
*
H.D. DOKTER Kamerlingh Onnes Laboratorium,
Leiden,
The Netherlands
D. DAVIDOV **, J.M. BLOCH, I. FELNER and D. SHALTIEL The Racah Institute of Physics, Jerusalem
We have measured the ESR properties of Er and Dy in the cubic metallic beryllides LaBer 3 and LuBet 3 as a function of temperature. For the low concentration limit (<2%), the data yield information about the crystalline electric field (CEF) splitting and the localized moment-conduction electron (CE) exchange. The higher concentration (570, 7% and 8%) ESR linewidth data reveal anomalous broadening. This effect is attributed to Er spin-spin exchange broadening.
1. Introduction
2. The low concentration single ion effect
The understanding of the interesting magnetic and electric properties of the MBel3 cubic intermetallic compounds (M stands for a rare-earth ion) requires knowledge of the CEF splitting and the localized moment-CE exchange parameters [ 11. EPR has been shown to be an appropriate method to investigate these properties in these materials [2,3]. The present paper reports ESR studies of Er and Dy in the nonmagnetic hosts LaBe, 3 and LuBe I 3 for impurity concentrations between 500 ppm and 8%. The Er and Dy ions in these systems experience a crystalline field, the ground state is the F-, doublet. In the low concentration limit the ESR linewidth versus temperature exhibits the characteristic behaviour expected for a I’, doublet in the presence of Fs low-lying excited states. This behaviour enables us to extract the CEF and the localized moment-CE exchange interaction. The measurements on the higher concentrated samples showed an anomalous thermal broadening. This broadening is attributed to Er spin-spin exchange interaction.
regime (500 ppm G c G 2%)
2.1. LaBel3 : Er and LuBeI
: Er
We have measured the ESR of these compounds in the temperature range 1.4 < T < 16 K. In fig. 1 the ESR linewidth of LaBe 13 : Er 5000 ppm and 500 ppm is shown. As is clearly seen the high temper-
180 ‘;;
160
z
140
z
120
E
100
g
80
Y
60 40 20 0 0
1
2
3
4
5
6
7
8
9 10 11 12 13
TEMPERATURE (OK)
Fig. 1. The ESR linewidth of Er (5000 ppm and 500 ppm) in LaBer3 as a function of temperature. The solid lines represent the theoretical fit of eq. (1) using the parameters (A) b = 6.5 G/K, Al = 10 K, (B) b = 6.5 G/K, A1 = 16 K. The dashed line represents the Korringa linewidth assuming b = 6.5 G/K.
* Supported by the US-Israel Binational Science Foundation and the Stichting voor Fundamenteel Onderzoek der Materie (FOM). ** Also at the Kamerlingh Onnes Laboratorium, Leiden.
78
19
H.D. Dokter j Crystallinefields and exchange in beryllides ature thermal broadening (of the 5000 ppm sample) shows deviation from linearity. This is attributed to the presence of low lying excited CEF states according to the theory of Hirst [4]. A formula describing the EPR thermal broadening of a F7 doublet ground state in the presence of low lying Fs excited states has been derived by Hirst [4] :
200 -
150-
AH=bT+C&
k
Ak exP(&/T)
- 1 ’
where b is the Korringa thermal broadening (see e.g. ref. [5]). Ck are proportional to the squares of the off-diagonal matrix elements, connecting the F7 ground state with the Fik) (k = 1,2,3) excited states (explicit expressions for Ck are given by Davidov et al. [5]). Ak is the energy separation between the F, ground state and the I’r) excited states. Both b and Ck depend on (j’) the average on the Fermi surface of the square of the wave-vector dependent exchange. Formula (1) predicts a linear thermal broadening for T << Ak, but deviation from linearity for T < &. It enables us to derive values for 0 and -0.3 40 K which is consistent with that observed by Bucher et al. for Er in ErBe ia [l]. Using values for I, the density of states at the Fermi surface, from Bucher’s specific heat data, and taking the above values of b, we can extract values for &‘) to be 225 + 30 meV and 220 f 50 meV for LaBera : Er and LuBeia : Er, respectively. 2.2. LaBe13: Dy The ESR linewidth of this compound was measured in the temperature range 1.8 < T G 4.1 K, using the
501 0
I 1
I 2 TEMPERATURE
I 3
I 4
I 5
(OK)
Fig. 2. The ESR linewidth of Dy in LaBel3 (5000 ppm) as a function of temperature. The solid lines represent the theoretical fit of eq. (1) to the experimental results using the parameters (A) b = 16 G/K, Al = 4 K, (B) b = 16 G/K, Al=6K.
naturally abundant Dy isotope (fig. 2). The large residual width (partly attributed to a superposition of the various hyperfine lines, originating from the 161Dy and 163Dy isotopes) and the fast thermal broadening made it difficult to extend the measurements to higher temperatures. If we assume the crystalline field parameters A4 and A, to be close for both Er and Dy in LaBe 13, we are left with an overall splitting of approximately 23 K and a separation <6 K between the F7 ground state and the first excited I’t) level. This is in agreement with Bucher’s results on DyBe 13. From eq. (1) it can be shown that the ratio of the slope of the linewidth at high and at low temperatures amounts 2.25. Combining this value with the “high temperature” slope of fig. 2, we estimated b = 16 G/K. From b we derived d(j’) = 220 f 50 meV, which supports our assumption that the exchange is independent of the 4f occupation number (see also [7]). A fit of the data, using (A) b = 16 G/K, Al = 4 K and (B) b = 16 G/K, A, = 6 K, is shown in fig. 2.
3. The high concentration regime - interaction effect We have extended the measurements on the LaBe 13 : Er system to higher Er concentrations of
H.D. Dokter / Crystalline fields and exchange in beryllides
80 350
T
find for the second moment M2
I
M
=
2
300 -
+
250 -
kJ - lj4 CkJik
C
Z
mfa,b
(Gum
--
exp
,(F
am - Ftd2
‘%n>21
(3)
9
where the summation is over all the levels m excluding the ground state levels a and b. Z is the partition function [Z = C, exp(-E,/T)], and F and G represent the “direct” and “exchange” terms F am
=(a,mlJ~JIa,m)=(alJ,laXmIJ,Im~,
G an, = (a, m IJ * Jlm, a) .
0
1
2
3
4
5
6
7
TEMPERATURE
8
It can be shown that for temperature independent cut-off lorentzian line shape, the Er-Er exchange contribution to the linewidth may be approximated by the square root of the second moment. The total linewidth Mtotal, therefore, neglecting ordering effects, can be written as
9 10 1112 (K)
Fig. 3. The ESR linewidth of Er in LaBer3, as a function of temperature for 5% Er concentration (triangles) and 2% Er concentration (circles). The solid line A represents a theoretical fit with eq. (4), the solid line B is theoretical fit with eq. (1).
5, 7 and 8% (fig. 3 exhibits our results for the 5% sample). For comparison our results for the 2% Er sample are also given in the same figure. Striking differences between the 2 and 5% samples are clearly seen: the last sample exhibits much faster increase of the thermal broadening at higher temperatures. This increase can be explained by off-diagonal transitions between the crystalline field levels, induced by Er spin-spin exchange interaction. Analyzing our data, we start with the hamiltonian X=CXi-(gJ-l)’ i
CJikJi.Jk, i#k
(2)
where the first term represents the “single ion” hamiltonian, including crystalline field and Zeeman interactions. The second term represents the exchange hamiltonian, where Jik gives the exchange interaction between the ith and kth Er ions. Ji and Jk are the angular momenta of the Er ions and gJ is the Er Land6 g-factor. Calculating the line moments, defined in a way similar to that of Pryce and Stevens [8], we
atjtotal = MW_s + AH’ + oI\lM2
(4)
.
Here, .Mr,, is the residual width, and iw’ is the “single ion” Hirst contribution, given by (1). When we , assume AHH,,, to be concentration independent, which is supported by the results of section 2, and by the peculiar structure of the beryllides [ 11, we are left with only one parameter CY to fit the data. This fit is shown in fig. 3. It yields a lower limit for the ion-ion 2 ‘I2 2 0.05 K, which has the same order exchange CCJik) of magnitude as the exchange parameters found in the Van-Vleck beryllides [2]. Details on the high concentration results will be published elsewhere [9]. In conclusion, our experiments on the beryllides enable us to extract the F7-F&‘) splittings and the conduction electron-ion exchange parameters. Furthermore, we found evidence for exchange broadening due to exchange between similar spins. A lower limit for the ion-ion exchange could be given.
References
111 E. Bucher, J.P. Maita, G.W. Hull, R.C. Fulton and A.S. Cooper,
Phys. Rev. Bll
(1975)
440.
121 J.M. Bloch, D. Davidov, 1. Felner and D. Shaltiel, J. Phys. F: Metal Phys. 6 (1976) 1979. and A. Meyer, Solid State Commun.
[31 G. Heinrich 21.
21 (1977)
H.D. Dokter / Crystallinefields and exchange in beryllides [4] L.L. Hirst, Phys. Rev. 181 (1969) 597. [5] D. Davidov, C. Rettori, A. Dixon, K. Baberschke, E.P. Chock and R. Orbach, Phys. Rev. B8 (1973) 3563. [6] K.R. Lea, M.J.M. Leask and W.P. Wolf, J. Phys. Chem. Solids 23 (1962) 1381. [7] J.M. Bloch, D. Davidov, H.D. Dokter, I. Felner and D. Shaltiel, submitted to J. Phys. F.
[8] M.H.L. Pryce and K.W.H. Stevens, Proc. Phys. Sot. (London) A63 (1950) 36. ‘[9] H.D. Dokter, D. Davidov, J.M. Bloch and I. Felner, accepted for publication in Solid State Commun.
81
CRYSTALLINE
7 (1978)
78-81
FIELD EFFECTS AND EXCHANGE PARAMETERS
IN METALLIC BERYLLIDES
*
H.D. DOKTER Kamerlingh Onnes Laboratorium,
Leiden,
The Netherlands
D. DAVIDOV **, J.M. BLOCH, I. FELNER and D. SHALTIEL The Racah Institute of Physics, Jerusalem
We have measured the ESR properties of Er and Dy in the cubic metallic beryllides LaBer 3 and LuBet 3 as a function of temperature. For the low concentration limit (<2%), the data yield information about the crystalline electric field (CEF) splitting and the localized moment-conduction electron (CE) exchange. The higher concentration (570, 7% and 8%) ESR linewidth data reveal anomalous broadening. This effect is attributed to Er spin-spin exchange broadening.
1. Introduction
2. The low concentration single ion effect
The understanding of the interesting magnetic and electric properties of the MBel3 cubic intermetallic compounds (M stands for a rare-earth ion) requires knowledge of the CEF splitting and the localized moment-CE exchange parameters [ 11. EPR has been shown to be an appropriate method to investigate these properties in these materials [2,3]. The present paper reports ESR studies of Er and Dy in the nonmagnetic hosts LaBe, 3 and LuBe I 3 for impurity concentrations between 500 ppm and 8%. The Er and Dy ions in these systems experience a crystalline field, the ground state is the F-, doublet. In the low concentration limit the ESR linewidth versus temperature exhibits the characteristic behaviour expected for a I’, doublet in the presence of Fs low-lying excited states. This behaviour enables us to extract the CEF and the localized moment-CE exchange interaction. The measurements on the higher concentrated samples showed an anomalous thermal broadening. This broadening is attributed to Er spin-spin exchange interaction.
regime (500 ppm G c G 2%)
2.1. LaBel3 : Er and LuBeI
: Er
We have measured the ESR of these compounds in the temperature range 1.4 < T < 16 K. In fig. 1 the ESR linewidth of LaBe 13 : Er 5000 ppm and 500 ppm is shown. As is clearly seen the high temper-
180 ‘;;
160
z
140
z
120
E
100
g
80
Y
60 40 20 0 0
1
2
3
4
5
6
7
8
9 10 11 12 13
TEMPERATURE (OK)
Fig. 1. The ESR linewidth of Er (5000 ppm and 500 ppm) in LaBer3 as a function of temperature. The solid lines represent the theoretical fit of eq. (1) using the parameters (A) b = 6.5 G/K, Al = 10 K, (B) b = 6.5 G/K, A1 = 16 K. The dashed line represents the Korringa linewidth assuming b = 6.5 G/K.
* Supported by the US-Israel Binational Science Foundation and the Stichting voor Fundamenteel Onderzoek der Materie (FOM). ** Also at the Kamerlingh Onnes Laboratorium, Leiden.
78
19
H.D. Dokter j Crystallinefields and exchange in beryllides ature thermal broadening (of the 5000 ppm sample) shows deviation from linearity. This is attributed to the presence of low lying excited CEF states according to the theory of Hirst [4]. A formula describing the EPR thermal broadening of a F7 doublet ground state in the presence of low lying Fs excited states has been derived by Hirst [4] :
200 -
150-
AH=bT+C&
k
Ak exP(&/T)
- 1 ’
where b is the Korringa thermal broadening (see e.g. ref. [5]). Ck are proportional to the squares of the off-diagonal matrix elements, connecting the F7 ground state with the Fik) (k = 1,2,3) excited states (explicit expressions for Ck are given by Davidov et al. [5]). Ak is the energy separation between the F, ground state and the I’r) excited states. Both b and Ck depend on (j’) the average on the Fermi surface of the square of the wave-vector dependent exchange. Formula (1) predicts a linear thermal broadening for T << Ak, but deviation from linearity for T < &. It enables us to derive values for
501 0
I 1
I 2 TEMPERATURE
I 3
I 4
I 5
(OK)
Fig. 2. The ESR linewidth of Dy in LaBel3 (5000 ppm) as a function of temperature. The solid lines represent the theoretical fit of eq. (1) to the experimental results using the parameters (A) b = 16 G/K, Al = 4 K, (B) b = 16 G/K, Al=6K.
naturally abundant Dy isotope (fig. 2). The large residual width (partly attributed to a superposition of the various hyperfine lines, originating from the 161Dy and 163Dy isotopes) and the fast thermal broadening made it difficult to extend the measurements to higher temperatures. If we assume the crystalline field parameters A4 and A, to be close for both Er and Dy in LaBe 13, we are left with an overall splitting of approximately 23 K and a separation <6 K between the F7 ground state and the first excited I’t) level. This is in agreement with Bucher’s results on DyBe 13. From eq. (1) it can be shown that the ratio of the slope of the linewidth at high and at low temperatures amounts 2.25. Combining this value with the “high temperature” slope of fig. 2, we estimated b = 16 G/K. From b we derived d(j’) = 220 f 50 meV, which supports our assumption that the exchange is independent of the 4f occupation number (see also [7]). A fit of the data, using (A) b = 16 G/K, Al = 4 K and (B) b = 16 G/K, A, = 6 K, is shown in fig. 2.
3. The high concentration regime - interaction effect We have extended the measurements on the LaBe 13 : Er system to higher Er concentrations of
H.D. Dokter / Crystalline fields and exchange in beryllides
80 350
T
find for the second moment M2
I
M
=
2
300 -
+
250 -
kJ - lj4 CkJik
C
Z
mfa,b
(Gum
--
exp
,(F
am - Ftd2
‘%n>21
(3)
9
where the summation is over all the levels m excluding the ground state levels a and b. Z is the partition function [Z = C, exp(-E,/T)], and F and G represent the “direct” and “exchange” terms F am
=(a,mlJ~JIa,m)=(alJ,laXmIJ,Im~,
G an, = (a, m IJ * Jlm, a) .
0
1
2
3
4
5
6
7
TEMPERATURE
8
It can be shown that for temperature independent cut-off lorentzian line shape, the Er-Er exchange contribution to the linewidth may be approximated by the square root of the second moment. The total linewidth Mtotal, therefore, neglecting ordering effects, can be written as
9 10 1112 (K)
Fig. 3. The ESR linewidth of Er in LaBer3, as a function of temperature for 5% Er concentration (triangles) and 2% Er concentration (circles). The solid line A represents a theoretical fit with eq. (4), the solid line B is theoretical fit with eq. (1).
5, 7 and 8% (fig. 3 exhibits our results for the 5% sample). For comparison our results for the 2% Er sample are also given in the same figure. Striking differences between the 2 and 5% samples are clearly seen: the last sample exhibits much faster increase of the thermal broadening at higher temperatures. This increase can be explained by off-diagonal transitions between the crystalline field levels, induced by Er spin-spin exchange interaction. Analyzing our data, we start with the hamiltonian X=CXi-(gJ-l)’ i
CJikJi.Jk, i#k
(2)
where the first term represents the “single ion” hamiltonian, including crystalline field and Zeeman interactions. The second term represents the exchange hamiltonian, where Jik gives the exchange interaction between the ith and kth Er ions. Ji and Jk are the angular momenta of the Er ions and gJ is the Er Land6 g-factor. Calculating the line moments, defined in a way similar to that of Pryce and Stevens [8], we
atjtotal = MW_s + AH’ + oI\lM2
(4)
.
Here, .Mr,, is the residual width, and iw’ is the “single ion” Hirst contribution, given by (1). When we , assume AHH,,, to be concentration independent, which is supported by the results of section 2, and by the peculiar structure of the beryllides [ 11, we are left with only one parameter CY to fit the data. This fit is shown in fig. 3. It yields a lower limit for the ion-ion 2 ‘I2 2 0.05 K, which has the same order exchange CCJik) of magnitude as the exchange parameters found in the Van-Vleck beryllides [2]. Details on the high concentration results will be published elsewhere [9]. In conclusion, our experiments on the beryllides enable us to extract the F7-F&‘) splittings and the conduction electron-ion exchange parameters. Furthermore, we found evidence for exchange broadening due to exchange between similar spins. A lower limit for the ion-ion exchange could be given.
References
111 E. Bucher, J.P. Maita, G.W. Hull, R.C. Fulton and A.S. Cooper,
Phys. Rev. Bll
(1975)
440.
121 J.M. Bloch, D. Davidov, 1. Felner and D. Shaltiel, J. Phys. F: Metal Phys. 6 (1976) 1979. and A. Meyer, Solid State Commun.
[31 G. Heinrich 21.
21 (1977)
H.D. Dokter / Crystallinefields and exchange in beryllides [4] L.L. Hirst, Phys. Rev. 181 (1969) 597. [5] D. Davidov, C. Rettori, A. Dixon, K. Baberschke, E.P. Chock and R. Orbach, Phys. Rev. B8 (1973) 3563. [6] K.R. Lea, M.J.M. Leask and W.P. Wolf, J. Phys. Chem. Solids 23 (1962) 1381. [7] J.M. Bloch, D. Davidov, H.D. Dokter, I. Felner and D. Shaltiel, submitted to J. Phys. F.
[8] M.H.L. Pryce and K.W.H. Stevens, Proc. Phys. Sot. (London) A63 (1950) 36. ‘[9] H.D. Dokter, D. Davidov, J.M. Bloch and I. Felner, accepted for publication in Solid State Commun.
81